Spectrophysics Principles and Applications [illustrated] 3540651179

Citation preview

A. Thorne U. Litzen S. Johansson

QQ:

Springer

FF, FALVEY& ny ©) AG wepiey :

\s ~C)

“RN

O

Digitized by the Internet Archive in 2022 with funding from Kahle/Austin Foundation

https://archive.org/details/spectrophysicspr0000thor

>

a

Spectrophysics

Springer Berlin

Heidelberg New York Barcelona Hong Kong London Milan Paris Singapore

Tokyo

A. Thorne

U.Litzén

S. Johansson

Spectrophysics Principles and Applications

With 172 Figures and 13 Tables

&:

Springer

Dr. Anne Thorne

Dr. Ulf Litzén

Imperial College

Dr. Sveneric Johansson

The Blackett Laboratory Prince Consort Road

Lund University Physics Department

London SW7 2BZ United Kingdom

Box 118 S-22100 Lund

E-mail: [email protected]

Sweden E-mail: [email protected] sveneric. [email protected]

ISBN 3-540-65117-9 Springer-Verlag Berlin Heidelberg New York Library of Congress Cataloging-in-Publication Data Thorne, Anne P.

Spectrophysics : principles and applications / A. Thorne, U. Litzén, S. Johansson. p. cm. Includes bibliographical references and index. ISBN 3-540-65117-9 1, Spectrum analysis. I. Litzén, Ulf. II. Johansson, S. (Sveneric), 1942- . II. Title.

QC451.T49 1999 535.8 4-dc21 99-10573 CIP

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from

Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1999

Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Data conversion by Steingraeber Satztechnik GmbH, Heidelberg Cover design: design & production GmbH, Heidelberg

Computer to film: Saladruck, Berlin SPIN: 10685446

56/3144/di - 5 43 210 — Printed on acid-free paper

Preface

Spectrophysics covers those applications of spectroscopy that investigate the interactions of radiating atoms and molecules with their environment, with particular reference to the fields of astrophysics, plasma physics and atmospheric physics. The book has three parts. The first part describes the structure of atoms and simple molecules and the way in which it is related to their absorption and emission spectra, including the spectra of complex atoms, which are not usually covered in introductory texts but which are astrophysically important. The second part deals with spectral intensities, again a subject largely neglected in introductory texts, and with radiation transfer, equilibrium conditions, and the effects of temperature, density, pressure, collisions with other particles, and so on. Much of this material is usually found only in specialized texts on astrophysics or plasma physics. Finally, the experimental methods of optical spectroscopy, from the mid-infrared to the far ultraviolet regions, are described and compared, with a final chapter addressing such practical matters as wavelength and intensity calibration and signal-to-noise ratios. The text is based on the first author’s book Spectrophysics, the second edition of which was published in 1988 and is now out of print. The material has been rearranged and mostly rewritten; in particular, there are substantial changes and additions to the first part, and the chapters on experimental techniques have been updated. In addition to the specific references cited in the text, a list of literature for further reading is given at the end of each chapter. We have included the books we consider to be most useful, and, although some of them are now out of print, they are likely to be available in most libraries. The text is directed at first degree students and at postgraduates starting research in astrophysics, plasma or atmospheric physics, or spectrochemical analysis. It should also be useful as a quick reference to established researchers in these fields. We would like to thank the many colleagues at Imperial College, the University of Lund, and elsewhere who have given help and advice. We also thank our families for their forbearance. London and Lund

February 1999

Anne P. Thorne

Ulf Litzén Sveneric Johansson

7

_ >=

:

eu

i

=

@D

-

. -

«enh

dite

hon)

tiie

Wi

OY ee

4

areal! ithe

im meg’

pe

sft

fied)

4

bev

«

Pio

emeetg

hed;

iee

wl

ot

tiv] ‘vgeee alt

Ts

ni \

WGe

Petia

Riv.

«743

heen

Peer

J-nmpte

Pitan

104)

fae

ea ah

we

Hepesle @

eer att

dats

jaat-

Oita

ofr

:

Pevie lindtins W 4

Af

wee

Lae

Op

OST

serie

tne

i

GA ag

7

Bie,

Otel

mar

© 10 togte Meena a

ra ws

oo

o;

a)

a paved O. clam

0) Vilalleee

fe lows Sirsionny

ay

Aut

ey 1

oe ati2ihn

ten,

taut

ee ee ee ee vive

(L-eege lt

ot mortiihs eet

%

am co beinaniioes amanat

ee. ee

omnes tS >

we sso

are Owls 2

8A wh YOON

eae

a

:

é

Wie bh: tos ws tn nal @ aoe vetrw) « dar gil. kujé filtro, eee ‘i ahytenss oor

wee oe -

cog M

BRP

war

Gra

”‘ota

ihheep ine inegeenanns anal

Mul

rb

Me ci

hl

a

Ve

it Inde sidlary eat itnabye be

ee seni

¢ wid

jr

a 2 a)

oe able @aljuay lecahegy ie ‘Raber

| wy:

¢

lop beticeh

Wiguslesgy as Gant Saute

“pi

abo marta

pet ifinhn ott tewtcpalgens

wm creghgriem fue feet

eA)

at Fi

?

reuen / igo. eedewe grat Oy meily gd? be semaines Witragr & tei PelGus ae]? ee Dapld 1 om Pag

_

:

:

few wheal a terre

insrlepe Git

Dee

ee

ol Vd

The

ayet

Wer

eee tall at aad

ee

Xe

iC

ea

ee

)

ae yg mal ow

eeien-44

a

ohne

ee Vr ~- -F codtengen datas end Be 9 dies

vetementn! tefl ef wallimad cas 3 4

ob

4

‘

:

ee mon

a}

«

a

®

ry

(a

hora

|

=

aa ET

=

Table of Contents

Tot rOduction See. aera tres ee ot cha ee ts SMO RRO air? 151 SCope Ole HIS OOhe hte wen ar oko oc ta teh ate NTS A A oe 12 SURE PlecteoimaeneuespeceLuin ikaw cas. 6 os ceed Tatas yt Roe 3° “Spectrallbines\and nergy Uevelsr i. sy. < seo css tay ore aoe Ao Unies in Optical Spectroscopy. - asamp ee eco aa oe fae awe ops Purrthier elteaeliade Paget, ci pete lc caret ch cough a ye 6 052 chert cal

Part

2.

I. Atomic

and Molecular

1 3 4 5 8 9

Structure

Basie Atomica heoryec ee te ee eer ost eee DAs One: hlectromntouisueiet. eee he ee os ee ee eo ne 2.1.1 Schrédinger’s Equation for One-Electron Atoms ...... 2.1.2 Quantum Numbers and Wave Functions............. Priss al hey Pr bapiitweLensityese wane wk pee ss os ces 2.424>-Wleciron Spimand fine Structures. :.- +... eas oe OD > wo MichitonwAtonieigerst... ete tt Se ne oa Cee os 2.2.1 Schrédinger’s Equation for Two-Electron Atoms...... 2.2.2 The Pauli Principle and Antisymmetric Wave Functions........,...+... 44. 2 OSA The Exchange INCeraculOl wag COUPE Sumit oc «us ons oe oes D3" ey oc hg ea asa yuo pubes DS AD Ciga hoya hie SuerC 2.4 Radiative Transitions and Selection Rules ................. 1. one aie O77 ie srimne- Dependent Perturbations 5.626 92.4.2. The Hlectromagnetic Imteraction sa oye. woos nue eects «2 um A Se The Hlectmc Dipole Approximation oso «0 «re «nate 2.4.4 Selection Rules for Electric Dipole Transitions ....... 9.4.5 Selection Rules and Multiplets in LS Coupling ....... 2.4.6 Forbidden Lines and Higher-Order Radiation ........ bat Sen ap wOe nen: Acai enn arSeana tns ie Co ciercrr [hha Nope eyerelel

13 14 14 1y 19 22 25 25 26 29 31 32 33 36 42 44 45 45 46 48 AQ 49 be 52

Table of Contents

Vill

Atomic Structure and Atomic Spectra.................m. 3.1 -One-Hlectromvsyetemis ac aseertee aie Alem ene narra Be "The AlkaltaMe tals ieee em eit aie areata 3.1.2) “Spectral Series 43 oo tec na ete a > came eee 3.43 Other One-Electron Syetems > 2.2.52. 0s.seeeee es ore 3:20 "Two-Hlectrom: Sv Stenis ae, acento measta teeta ern ieaers aoe 3.2.1 Systems with an s* Ground Configuration ........... 3.2.2 Systems with a p* Ground Configuration ............ 3253: oLhe Rate GaspoyStenis tte sores wines ieee 3.07 “QOmp lee A COMIS sve Wee ee See ieee coe ee eee tae a 3.3.1 p-Shell Atoms with Multiple Parent Terms .......... Sios2), he “Uransition blemient eu: saite ee ie fee ee ees 33.3) lanthanides andcACimiles 2 ae as ee eae ere 3.4 Interpretation and Understanding of the Observations....... 3.5 Inner-Shell Excitation and Autoionization ................. 3.6 Isoelectronic Sequences and Highly Charged Ions ........... 3.7 Atomic Structure and jhe Periodig lables 4... ewe6 oo Stil lrendseA lonsthe Perodicw kablas=-aes eee ee 3.0.2 Regularities: Within Periods: 2 gansta) as qe ae See 3.03 4Ditterent Coupling=ly pes .4. ass i eee ee 3.0: JNielégn, Eieot ster ste ck beac Se otis jcth Sines Sewn os Se Ol. Hy Penne, SURICT UTC. oh ea le see aoe eee, ae gore [SOtOPexotructure OpticallyeMhicksinaite Aes) I ts 10.6" Intensity"en CominuomeRadiation i: Beh. tasks....: HurcherbReadin® See ei sisicmcscdn SR PURI at Aare ata

Part III. Experimental

XI

251 251 202 200 253 200 254

Methods

11. Introduction to Experimental Methods ................... TD. Dispersion and*Resolvinges Power wees fh Anta eet. ce 1152) Throughputrendalilummations stern as ae 64 Meee oot 11.3 WavelengthandiIntensity Measurements j)1n.nets 2 fa). . o Further Bead mies ares ee eee ot. ee en fe ee ecore

200 258 262 265 266

12. Dispersive

267

Spectrometers:

POPUPS UGC

ete

Prisms

nee ee ores

and Gratings...........

Sarek a otra

NS Bf

Me aenian. 268

T2-2> Prism’ Instruments < * saad teat Pye Erevan beorrasnd > fegh cea ccr ate 270 122 ypesolbrism Instrument) jielf0 ate qititemie 5 ace 270 1927 Deviation aud Dispersion... .. 12.3.1 Basic Properties of Diffraction Gratings. !.:..5:..... 200 278 PD eee. as 13'3°9, Condition tor Maxima... 12.3.3 Intensity Distribution from Ideal Diffraction Grating .. 279 ¢ Jase Jt. il 283 12.3.4 Dispersion and Resolving.Power wisgnus aeee. tere. SAL. 286 12.3:5) Goncaya Gratingelihoahs heme 12.4 Mountings for Diffraction Grating Spectrometers ........... 290 ok! 290 12.4.1. Mountings for. Plane Gratingsiitivs« wettest. i... 4.0.60.eal 292 12:4.2 Mountings for Concave’ Gratings''.. Be 295 thre. 52% 0). Spee 124:3) Ieisee Delectsicg 295 ..........0...... Gratings of Cheracteristics Production-and 12.5 SORE 298 12.6 Gratings#er Special Purposes. ai! 0. on «5 whale aie onan 298 iat Pe bare torGhle Kehelia: Gratingewinw TE en 300 2a 20). SR 12:6:2) Gratingsin the Infrared ch 301 Miakiite Ulirayiolentyilons Bar inthe: 12-6.3.Gretings 303 Oe Re s ese e ee encuns Furthess-Reading

XII

Table of Contents

13. Interferometric Spectrometers .......2....205.02.0.00.030 13.1 Basic Concepts of Interferometric Spectroscopy............. 13:2 Resolutioniand Throughpieete pees Geeeeee Poker cee 13.3 Fabry-Perot Interferometers: Intensity, Distribution and Resolution -ehegeee: 6-662... .13.3.1. The Airy) Distributiow and lisiProperties .7.L5.1..-.«. 13.3:2. Resolving: Powerit aah ons ook Se eee 134 Usevot Fabry=Perot Interierometers «2.2 -e es peeegeseo-seese 13.4.1. Limitations an. RA a eh wseeterw eemeppie teh eee 13:4:2 Methods of Use ses>. Ft Ae eeeee 34:3 Particularen pplicstions save ve se -ncre eens © are rae 13.5 Michelson Interferometers and Fourier. Transform epectroseopy (PTS ile a ctiogees. 2.2 13.5.1 The Interferogram and the Spectrum ............... 13.5.2. Instrumentibunction and Resolutionm®. poses 9a: -oe lS-o.0 oampling and.” lass sn. - «cen ae eee eee eee 13.6 Practical Aspects of Fourier Transform Spectroscopy........ 13:61 “Rie Scanning turerferometers aes setae. ese eee 13-6:25 Computing and Control er. oe en eee ee 136.3 Lamitations-and=Advantages 4222 en oe eee eee. eee 13.7 Spatially Heterodyned, Non-scanning Interferometers tor: FS Se eek ee ae ae ST Rs ris = eek Further ‘Reading =. tex .0 2 en oe ee ee ee ene ea 4:

305 305 307 309 310 Bs 315 315 317 319

320 320 324 an 330 330 332 5 334 337

Laser Speéctroscopyaew 24 +0eok a ee ee 6 i ae 14.1 Introduction o.Lasersaseey ae le ees Ge Sete A oa ee 14.2 ZLy pes. 0b Lasemeiic.4.cmewe te aon ROR EA, Oe eeeee. 14.2.1 Pimed= Freqneneyslyasers ris eek, t oma. 6 2E 94... 2 es 14.9.2 enable Lasers: S20 Seale UAB. Satin cca la2 oebrequency Doubling and Mixing? ame). Bante. ccs I4/3 Optical: Pumping and aturaion?, ate.aeeeea. bane, ee 14.4 Spectral and Temporal Laser Bandwidths ................. 14.5 Laser Absorption and Excitation Experiments.............. 14.6, Doppler-Free*Speckrostooy 4 Ghee 4 am nee «1 age et oe 14.7 Trapping’and Coolingof Atommandwlonsmigie! 2.8 98... 2. 14.8 Comments on Laser versus’ ‘Conventional apectroseopy ae. samen dae Melmee as ee Further Reading’ dicen 2c eee A ee ee ee

339 339 343 343 344 346 347 349 aol 354 357 358 358

io: LightsSources:and Detectorst, eau oecaeeret Oe. ue 15,1 TMdissiomeS DeRosa aanette nee nen een ne en VO cLSL| EAS eis FAR Gettin, urs. 0.8 hae deh a ee ee LOSE 2 ATCOIR wi crc erate ene ts eae ent ne pe eee 152143 ©Spearhisthy vaedeaee cn ee, ark eee ame Lo, L4 8Glowe D ischiae Ses eerie ate. 1 noone in on er

359 360 360 360 361 362

Table of Contents

Tea. ouslnductively, Coupled Plasmas.cc5..c.yomnn cua Sake 15:1.6 {Beant Vol Sources L260... 6..0 Re: ee. ae 15:1 falesermProduced Plasiiae stage A 2 eb ee: Se aie. eared Iaabas 15akSieShotio'TubesOJ Se Me 2 Ae nei Plasmobevices? 1511.9. Fusion, a. ae URE 4. lonvirap.....2saa0eite Beam 15.1c10 Electron, «Aca eeeite! Seen 15.2: Absorption Spectroscopy... 15.2.1 Continuum Sources for the Visible Region ........... 15.2.2 Continuum Sources for the Infrared eee and: Ultraviolet: Regiond oy wana snes cei Eee 8 i. Detectors om Remarks 153¥General eee ee ee iebetectorsitor thenuirarcd™ a on. 0...) Ultraviolet. and Visible the for 15:5 Detectors Se 2 oe AOS es. Detectors: iG6aMulticharinel De Ss Se £5. Crim Arrays Detectors ie 520 WU. ee an. +e Mimulsionsws hotographic 15.6:2°P es A AE Bi a oar ce Further Reading

XIII

363 363 364 364 365 366 366 367 368 370 313 OVO 3th Ot 378 380

16. Experimental Determination of Transition Probabilities and Radiative Lifetimes ....... 381 oe, «eer = 383 1641" Bamission Measurements. 4... oa cacs aches Gee te 16.2 Absorption and Dispersion Measurements ...........+-++-- 385 all 386 16.2.1 IntepratedsAbsorption caso. 2 ms ohare 2h hace eae as 386 a ep ocean ea 1622, ctiivalemtty WV yea cabo > oeisankce oa 387 16.3334 Hooks Pechimiguieic’: vio.crespsituc 16 Se [atetime. Messurements ie.) 3 cate bic cus dare tee aco eae 391 see font deems vets one. pice eta ie te 391 16.3: Delay MehnOdsce santa ollie ca 394 16:3.2 beam Measurements. ances, vi leiscs uel ieee 396 abbr 16.9.5 Banie, Uileet cos Gio ate tear ake 164. Combinationavol Methods sn ose occu oc cia ays ne Bate > A a 398 aaa hes Bea pies 400 Puarthet: Reade ioc Ackgcontee Hetst apes ae ta te

Lie Uncertainties in Experimental Measurements...........-. 403 lo Pees 404 PRY Signal-te-Noise Ratios yy 024 Vo hd Me ae eae aa 404 8 . Fee Oe IN ees NGIse SauréenoL Peel ...... 405 (FTS) Spectrometry Transform Fourier in SNR 17.1.2 406 ............. Spectrometry FT and Grating in SNRs 17.1.3 407 ees ee e eee teen 20sec .......... Measurements 17.2 Wavelength 407 .. Calibration Wavelength and Precision Measurement 17.2.1 408 Standards.............-. Wavelength to Background 17.2.2 410 essere 5s Standards............... 17.2.3 Frequency-Based All ees eee eee ee dee 00.60 ...... Measurements 17:3 Intensity 411 eee eee eee see 7.0... ..... Measurements 17.3.1 Absorption A412 Ee PANE 50. 0s. Measurements, 1723.9 Britesion’ 413 oo Mealy. oie veh. en see. [73 SAP aciometPemstanderds encom age Sete enn 8 se hm die we ce clesah 414 PiEvner Reade

XIV

Table of Contents

Appendixes waz.ck ood ere eee ae ea ie ee ae Adl: Perturbations heotrysaaheal oe ee ee eee A.1.1 Time-Independent First-Order Perturbation......... A.1.2 Time-Independent Second-Order Perturbation....... A.Ic3...Timeée-Dependent, Perturbationst. nisl. .2.05..-... A.2,.Physical:Constantsieoetueaen teen ieee one nL Le Ai2sl) General.Constants:.).... Anwar tee A225 Specitoseopie Constantine seen. 8 fae... 5. AL236 .Energya Conversion yi eats. eee eee ee

415 415 415 A417 417 419 419 420 420

References ai) wax oninseaiot a cept

421

bien Eee

eee

oe

1. Introduction

At the end of the nineteenth century physical science had evolved into a state that appeared to be close to full understanding: almost all observations and measurements from widely different fields could be described and explained by means of theories that were derived from a few fundamental principles or laws, and the theories were tested and verified through new experiments. But a small number of intriguing phenomena that could not be understood within the framework of classical physics remained to be explained. Two of these phenomena stemmed from spectroscopic observations, viz. the continuous spectrum emitted by a heated body (the black body radiation), and the spectral lines observed in flames, electric discharges and astronomical objects. The first step into modern physics was taken by Planck in 1900, when he introduced quantization of energy in order to develop an equation for the spectral distribution of black body radiation. In 1913 Bohr was able to derive a theoretical description of the spectrum of hydrogen, based on Rutherford’s model for the atom with additional assumptions about quantization of radiation and angular momentum, thereby creating a unified theory of atomic spectra and atomic structure. However, line spectra had been used as a tool in other branches of science long before any attempts were made to explain them. Fraunhofer’s discovery at the beginning of the nineteenth century of spectral lines in flames and in sunlight was made during his search for monochromatic light that could be used for measuring refractive indices of optical glasses at different wavelengths. In fact his primary interest in the spectral lines seems to have remained at their use in optical measurements. Some fifty years later Bunsen and Kirchhoff discovered that the spectral lines were characteristic of the different chemical substances, and thus laid the foundations for spectrochemical analysis.

The work of Bunsen and Kirchhoff also led to the identification of a large

number

of the Fraunhofer

lines in the sun and in stellar spectra,

and con-

sequently to knowledge about the chemical composition of celestial objects. The classification of stars according to spectral type was to become one of

the cornerstones of stellar astronomy. Optical spectroscopy has continued to be closely connected with the development of atomic and molecular theory, and also to be used as a tool

2

1. Introduction

within basic physics research and in other branches of science. During the early years of quantum mechanics the atomic and molecular structure derived from observed spectra as well as the spectra themselves provided test cases for the new theoretical methods. The fruitful exchange between theory and experimental spectroscopy has continued — for example, in the development and application of new theoretical methods for the analysis of the complex spectra of lanthanide and actinide elements and in the theoretical treatment of molecular structure. When the Lamb shift was discovered and quantum electrodynamics developed, high precision spectroscopy was immediately applied for quantitative tests of the theory — work that is still going on. Today powerful computers and computing methods allow the details of energy level structure and transition probabilities to be calculated with such accuracy that, in the case of the simple few-electron systems, new spectroscopic measurements are needed for validating the different theoretical approaches, while, for the more complex systems, the calculations form an essential guide to the interpretation of the experimental data. Spectrochemical analysis has also developed steadily, and in different forms is now used routinely in chemistry, medicine, biology, geology and other fields of research as well as in innumerable industrial applications. Spectroscopic methods are increasingly applied to investigating the detailed composition and chemistry of the earth’s atmosphere and to monitoring, both locally and globally, the changes caused by pollution and other human activities. Astronomical spectroscopy has evolved constantly ever since Fraunhofer’s observation of the solar line spectrum, and it has been applied to new classes of objects. Observations from high altitude rockets and satellites have opened the ultraviolet region for astronomical research, and the need for spectroscopic data to interpret the observations has led to a boom also for laboratory spectroscopy. Recently new demands for data appeared when observations with the Hubble Space Telescope yielded ultraviolet spectra with such high resolution and wavelength accuracy that old laboratory atomic data proved to be inadequate for positive line identifications and derivations of elemental abundances. Further demands can be expected when new spaceborne instruments with high resolution spectrometers for the infrared and extreme ultraviolet regions are launched in the near future. Similarly, the use of spectrometers on satellites and high-flying aircraft for observations of the earth’s atmosphere has highlighted deficiencies in the molecular data base, both for naturally occurring species and for pollutants. It is not only wavelengths and transition probabilities that are used for deriving information about astrophysical plasmas from the spectroscopic observations — the line profiles provide information, such as temperature, density, turbulence and stellar rotation, about the physical environment of the emitting or absorbing atoms. Line profile measurements are also used as diagnostic tools in the study of high temperature plasmas such as thermonuclear fusion devices. These applications in plasma physics have further increased

1.1 Scope of This Book

3

the demand for spectroscopic data and the development of new spectroscopic techniques. Optical spectroscopy on atoms and molecules thus plays such an important role both in basic research and in practical applications that its principles and fundamental techniques deserve a comprehensive name: Spectrophysics.

1.1 Scope of This Book This book is concerned with the use of atomic and molecular spectroscopy, both in the basic study of atomic and molecular structure and radiative transitions, and in applications in science and technology. It can be divided into three parts:

I Atomic and Molecular Structure (Chaps. 2-6). II Emission and Absorption of Radiation (Chaps. 7-10). III Experimental Methods (Chaps. 11-17). The purpose of Part I is to provide a basic knowledge about spectra and structure, needed by those who use optical spectroscopy as a tool. It is also intended to be comprehensive enough to give an overview of different types of atomic and molecular systems, useful as an introduction for specialists in atomic and molecular physics. In particular it gives introductory descriptions of the structure and spectra of complex atoms and highly charged ions, derived from observations — subjects which are usually omitted in atomic physics texts. Chapter 2 surveys the foundations of the theoretical treatment of atomic structure and radiative transitions, presuming a knowledge of basic quantum mechanics. Chapters 3 and 4 give more comprehensive descriptions of observed atomic structure, including complex atoms and highly charged ions, and review some methods used in the analysis of atomic spectra. Chapters 5 and 6 survey the basics of molecular structure and spectra, mainly for diatomic molecules but including some introductory material on atmospherically important triatomic molecules. Part II describes the radiative transitions between the states discussed in Part I. Chapter 7 contains the basic definitions and treats the processes of line emission and absorption. The factors affecting the widths and shapes of spectral lines are treated in Chap. 8. Chapter 9 discusses radiation transfer, population distributions and conditions for equilibrium, and Chap. 10 describes how these considerations affect the radiation emitted or absorbed

in a plasma environment. Part III describes the most important spectroscopic techniques used in research and in the applications mentioned in this introduction. After the basic definitions in Chap. 11, dispersive and interferometric spectroscopy are discussed in Chaps. 12 and 13. For the sake of completeness Chap. 14 contains

4

1. Introduction

a brief survey of laser spectroscopy, a subject that is treated in much more detail in numerous specialized textbooks. The components at the entrance and the exit of a spectrometer, i.e. the light source and the detector, are discussed in Chap. 15. Chapter 16 describes how transition probabilities are measured. The factor limiting the accuracy of all measurements, the signalto-noise ratio, as well as the attainable accuracy, are discussed in the last chapter of the book.

1.2 The Electromagnetic

Spectrum

The spectrum of electromagnetic waves extends from the longest radio waves, whose wavelengths are measured in kilometres, to X-rays in the wavelength

region down to 1071! m. At still shorter wavelengths the y rays also belong to the electromagnetic spectrum, but are associated with nuclear rather than atomic or molecular transitions. Figure 1.1 is a diagram of the spectrum on which are marked the conventional spectral regions. The divisions are necessarily rather arbitrary, except for the closely defined visible region. The wavelength A, the wavenumber o, the frequency v, and the energy FE are marked in units to be discussed in Sect. 1.5. Different experimental techniques have to be used in the different spectral regions — in fact the partitions are to some extent connected to these differ-

Wavelength

Energy tom! 33 MHz

10° cm'

33 GHz

10° cm"

10° cm"

33 THz 1.2 eV

Radio wave Medium

Microwave

Infra-red

Short

Far IR

Atomic and molecular

phenomena

TICES ry | em materials

Ultra-violet

Near IR

VUV

X-ray

XUV

Atomic transistions Molecular rotation

lonic transitions

Molecular vibration-rotation Inner shell transitions Mol. electron transitions

cre we

et

eed

ir

Fig. 1.1. The electromagnetic spectrum.

MomiCaks

1.3 Spectral Lines and Energy Levels

5

ences. If we take ‘visible’ to reach a little way into both the infrared and the ultraviolet, we have a well-defined wavelength region from 1 1m to 200 nm, which is by far the easiest in which to work. Air is transparent throughout this region, and so is quartz and — down to 300 nm ~ glass. A wide range of detectors and light sources is available. Going below 200 nm, first air (or oxygen, to be precise) and then — at 175 nm — quartz start to absorb. To overcome the air absorption the light path has to be evacuated — hence the name vacuum ultraviolet for this region. Lenses have to be replaced by mirrors, and as the reflectivity decreases at still shorter wavelengths, the number of mirrors must be reduced. Concave gratings, imaging and dispersing at the same time, are used, and at the shortest wavelengths (< 30 nm) the reflectivity of mirrors and gratings is improved by operating at grazing incidence. Below the limit where no transparent materials are available, the experimental complexity is drastically increased by the difficulties of running light sources and absorption cells without windows. However, the high energy of the photons makes the detection relatively easy, and background problems are small compared with those of the infrared. Going the other way from the visible into the infrared, water vapour and carbon dioxide in the air absorb in certain wavelength bands. Mirrors and gratings are very efficient, and different transparent materials are available. Photomultipliers, photodiodes and photographic detectors have to be replaced by semiconductors or non-selective energy detectors. The principal difficulty in this region is the low intensity of infrared sources, together with background and noise problems that become increasingly severe as the photon energy approaches the energy kT’ of the background radiation. Detailed descriptions of sources, detectors, and spectroscopic instruments for different regions are found in the last part of this book.

1.3 Spectral Lines and Energy Levels The fact that a spectral line is named after its shape, i.e. that it really looks like a line when it is observed in a spectroscope, deserves to be pointed out today, when most spectroscopic techniques display the lines not as images of a line-shaped slit, but as curves showing the intensity distribution as a function of wavelength or frequency. Spectral lines are commonly classified by wavelength for historic and experimental reasons, based on the fact that conventional spectroscopic instruments, dispersive or interferometric, deflect or select the radiation according to some relation between wavelength and difference of optical path lengths. However the important quantity from the point of view of atomic or molecular structure is the energy of the emitted or absorbed radiation because this is equal to the energy difference between two stationary atomic or molecular states. This energy AF is directly proportional to the frequency v through

6

1. Introduction

AE = hv, where h is Planck’s constant. Frequency is related to the wavelength in vacuum by Pel

Ava ;

(1.1)

where c is the velocity of light in vacuum. Frequency, being proportional to energy, is often used as a measure of energy difference, but in optical spectroscopy it is more usually replaced by the wavenumber a, defined by

HE

SY

ples

As far back as the late 1880’s Rydberg derived his famous formula for spectral series, showing that the wavenumbers of a series could be written as a difference between two ‘terms’. This fact was later restated by Ritz. The Ritz combination principle comprehensively describes the connection between our spectral observations and our knowledge of atomic and molecular energy levels and can be expressed as follows: “Every spectral line we observe represents the energy difference between two energy levels of an atomic or molecular system. By combining all our knowledge of the spectral lines we can derive the relative positions of energy levels in the system.” The last sentence describes the branch of atomic and molecular physics known as term analysis. Wavelengths derived from experimentally established energy levels in this way are known as Ritz wavelengths. The energy level structure can be displayed in a diagram, as shown in Fig. 1.2 for hydrogen, the simplest of all atomic systems. In a conventional Energy (cm’')

Continuum states

0

ee

ONIZQuomilne

100 000

Excited states

-50 000 50 000

-100 000

0

Ground state

Fig. 1.2. Energy level diagram of the hydrogen atom (with fine structure omitted). The energy unit is cm~! (Sect. 1.5).

1.3 Spectral Lines and Energy Levels

7

energy level diagram the levels are plotted vertically upwards in order of increasing energy. Since we measure only energy differences, we may place the zero of the energy scale wherever we want. In the theoretical treatment of atomic structure the energy of the system is defined as zero when the electron is at an infinite distance and at rest relative to the nucleus. The bound states will then have negative energies. We use the symbol F for this binding energy. At E = 0 the atom is ionized, and the continuum of states above this ionization

limit indicate that the free electron can have any kinetic energy relative to the nucleus. The lowest level is called the ground state of the atom, while the rest of the system consists of excited levels. The numerical value of the energy of a particular level on this scale, i.e. the distance from the ionization limit, is conventionally called the term value, as this is the value appearing in the Rydberg and all other series formulae. The symbol T is used for the term value. Experimentally we establish the energy levels through the measured energies of the spectral lines. The strongest lines in a spectrum are emitted or absorbed in transitions involving the ground state or other low levels, while the lines involving higher levels get progressively weaker as the ionization limit is approached. It is only in exceptional cases that the ionization limit itself can be directly observed in a spectrum, and even then its position cannot be established with the same accuracy as the spectral lines. The energy level structure of the system is thus derived relative to the ground state and not relative to the ionization limit, and the obvious choice is to use an energy scale having the ground state at zero. The energy of an excited state then represents the excitation energy which is the energy needed to lift the atom from the ground state to the excited state. We shall use this energy scale whenever we discuss observed energy levels in this book. If we designate the excitation energy by W, then E = W — J, where I is the ionization energy. Both energy scales are shown in Fig. 1.2, where transitions between bound states representing emission and absorption of electromagnetic radiation are also indicated. If we represent the energy of the upper and lower state of a transition by FE, and FE), the wavenumber (OF =| Fj 3

Ey

| 5

of a transition is (1.3)

Spectroscopic designations of levels, based on a theoretical description of atomic or molecular structure, will be discussed at length later in this book. Let us for the moment label the upper and lower levels A, and Aj. A transition between these two levels is always written as A; — Ay by atomic spectroscopists, regardless of whether the process is emission or absorption. Confusingly, the opposite convention is used in molecular spectroscopy, and often also in plasma physics, even for atomic spectra. If it is considered necessary to indicate whether the transition involves emission or absorption, the dash may be replaced by an arrow, thus A, Au. This possibility is not often used because there is seldom any ambiguity.

8

1. Introduction

Ionization of an atom would be represented in Fig. 1.2 by a transition from one of the bound states to any state in the continuum. Conversely, recombination or capture of an electron by an ion would be represented by a transition from the continuum to one of the bound states. Such bound-free transitions will be discussed briefly in Chap. 10.

1.4 Units in Optical Spectroscopy In the SI system, wavelengths are measured in metres and the various sub-

units such as wm and nm, as shown in Fig. 1.1. The Angstrom (A) and X (XU) units are also recognized and still used in the literature. Since 1 A is defined

as 10-!° m, the nanometre and the Angstrom are related by 1 nm = 10 A. The XU was originally defined independently as 1002.06 XU = 1 A. It is now

defined as exactly 10-3 A or 107! m. The unit of wavenumber in the SI system should logically be m~+, but with very few exceptions cm~! is the only unit used in practice. This unit has been designated by the name Kayser (K), but that name is rarely seen except as the multiple unit kK, for highly charged ions, or the sub-unit mK, for line widths and hyperfine and isotope structure intervals. The frequency is seldom used to characterize a spectral line, except in the microwave and radio-frequency regions. However, in laser spectroscopic work line widths and distances between lines are usually described in frequency units. A convenient relation to remember is 1 mK ~ 30 MHz. Finally, very high photon energies, corresponding to very short wavelengths, are often expressed in the unit electron volt (eV). 1 eV is equal to 1.602 x 10~!° J and corresponds to 8066 cm~!. These three energy units are marked below the wavelength scale in Fig. 1.1. With the commonly used units, (1.2) takes the forms a =10' x i DPE

g¢=10°X

1/Ayac

Eetteiachoee

Wirral nm),

(@ in em", Nin A).

(1.4)

The use of the wavelength measured in vacuum in these formulae must be stressed. If the wavelength is measured in air we must write (1.1) and (1.3) as

V

Gata /ecain =

c/(nXair)

;

C= V/E= 1) (nAgiv) » The correction imposed by n, the refractive index of air, is about 3 parts in 10*, an amount that is far from negligible even at modest spectroscopic accuracy. For high accuracy the internationally adopted formula for n must

be used [1] :

1.4 Units in Optical Spectroscopy

n = 1+ 8342.13 x 10-8 +

2 406 030 15 997 130 x 108 — o? bs88.0 52108

02

9

(1:5)

where o is the wavenumber in cm~'. Even this formula is strictly accurate only for dry air containing 0.03 % by volume of carbon dioxide. The effects of water vapour and carbon dioxide are significant in the near infrared. Differentiating (1.4) we obtain the following relations between wavenumber and wavelength intervals:

\50| = 10" x 6A/X? (5c in cm7!,

in nm),

\5o| = 108 x 5A/d?_ (60 in em=, d in A).

(1.6)

The direct relation between the measured wavenumbers and distances between energy levels expressed in (1.3) implies that it is convenient to use cm! as energy unit also for the energy levels. This is common practice in all experimental work in atomic and molecular spectroscopy, and cm! will be used with few exceptions in this book. In certain cases, e.g. for work in astronomy and plasma physics, eV is often used. Ionization energy is sometimes, especially in older literature, replaced by ionization potential, measured in V. In theoretical work one can find the unit Ry, rydberg, where 1 Ry = 109 737.3 cm~!, as well as the unit hartree, where 1 hartree = 2 Ry.

Further Reading References in the text are found in a list at the end of the book. At the end of each chapter we present a list of books and general review articles covering the contents of the chapter in more detail or in a wider context. Historic reviews of the early development of spectroscopy can be found in the introductory chapter of older books on atomic theory, many of which are, however, out of print. We mention here two of them: — Condon, E.U., Shortley, G.H., Theory of Atomic Spectra (Cambridge University Press, London, 1967), (reprint of 1935 book). — Condon, E.U., Odabasi, U., Atomic Structure (Cambridge University Press,

Cambridge, 1980). A general description of spectroscopic concepts is found in the first chapter of the book by Cowan:

— Cowan, R.D., The Theory of Atomic Structure and Spectra (University of California Press, Berkeley, CA, 1981).

iti = -

s

aineres

See!

4

tulle parnein wihiedow

"

an

. fi)

~y wa

‘

ow

Gili

de gervele When

om

‘arirgh

oe Fhe pale mecPome teesevenleny caoailie oe

"

ome

;

ons

if

uw

‘

Ob; ‘ait

suber

Wa

A

ne

SEP OOGR i 16) wet

:

: o ia ‘om aaa Ptah

rie

of ” mua,

0!

ne

etl ci sings jeryti ek

dt ester:

7 lin ruts

ab

in

—

engin

a

Mey ya

Felt

alahrus

CA

ietus

ofav

rm pea)

>

9

ji ieeimp

B

rae aahime:

16.1) a el

ior

heat org Hl

A

(doo

kewgere,

ey

len

y] 10d

re

Pct

el Miycnt ~

be

>

MiiaMaia Pty mul oa

ee

opind

«

-

hey

Sars ls

aloe

se

te ie

Same

tate :¢se

te

qhintn

catalina)

wilt Ob

iy

tamaly

ems,

utes

WS

ee

_—

aio

i=

“ie

we Vie

en pa

nalboot

wmdiwt

oe ul.

ot)

ie

Gam

Phi eh al er

ts

pot Heme he, fake Maliornp toe Gell wp adengh( wg “wileomleagheeth ie

Ane

Hulhie@ Se

i

>

os

7

VS

:

~

OP,

;

7

J

go

.

wes aitin, +eal wae ere ;

es

‘e?,? “i ois ieeane é

. AW

vw me

+ oitolames

ery

|

ald l

par leg

vin

Ful:

=)

ieee

Lewy’ ara deee wht hye wht Iyatl ce

;

a

e ing

Qu

wt

tlre

CPi Pty Ihe ts

be TF SARE 7

oH fea)

nee?

ie

@@ Sh ape i Setvpalo a) hy erehegy gil? & wae ties’) Viuew ot? Si weivut Heetdlt en! Mle le Siigeks reoiniiwiini o& ni aA jetl

woah

i

wwall

a6.

iid 1g

atypia

>

'

CY

FLU) vet?

rye

nies)

Partyhe 914

—

chery

email

SD

edule

TA

alee) —

ent hea!

g OYkeen alt) 0°) 5k cp funy (oni "al tp

Wrens

to!

iy

ny

ti)

© a

Te enya

si

wiiyiee

a)

.

ror’é

vlaneth 1,

ko

geal

@fY

‘ oo,

in

@

iAqiet

»

i? , 1 ait

7

.OF) gael

A) asl!)

Lenares

A

et ‘see? wii Wy

bat

fume

oak)

_

=

=the dil’) it

© Gears

OC aie

6a hte 5 a

_

Part I

Atomic

and Molecular

Structure

etmsjouie

a

talwoalolty

bas siarosA

2. Basic Atomic Theory

A quantum-mechanical treatment of atoms and molecules, including relativistic effects, provides full understanding and generally a reasonably good qualitative description of their energy level structure over a wide range of energies. For a one-electron system, for which Schrédinger’s equation can be solved exactly, the agreement is quantitatively very close, but it is necessary to include relativistic and quantum-electrodynamical effects to account for the details of the fine structure observed in very accurate measurements. Two-electron systems constitute a three-body problem for which no exact solution exists either in classical or in quantum mechanics; nevertheless, similar agreement with observations can be reached with appropriate approximations and computational techniques. Progress is being made in the same way for atoms and ions with three and four electrons. For more complex systems the theoretical treatment becomes more difficult, but it still provides the basis for understanding the structure, even though the differences between calculated and observed energy levels are still orders of magnitude larger than the experimental uncertainties. In this chapter we discuss elementary atomic theory, assuming that the reader is familiar with basic quantum mechanics and its application to simple systems, including one-electron atoms. The fundamental technique and results for one-electron systems are reviewed in the first section, followed by overviews of the application of quantum mechanics to two- and many-electron systems. The last section of the chapter deals with radiative transitions. The treatment is necessarily rather superficial, and for a more comprehensive and stringent approach the reader is referred to specialized texts on atomic structure, some of which are listed at the end of this chapter. Simpler models are sometimes useful for intuitive predictions and interpretations of observations, and in certain cases application of such a model leads to results that are in agreement with a more elaborate theoretical treatment. Accordingly, we occasionally use the concept of orbits from Bohr’s atomic model, and we shall also use the vector model for visualizing angular momenta.

14

2. Basic Atomic Theory

2.1 One-Electron

Atoms

We assume that the discussion in this section is well known to the reader, and

its main purpose is to review the methods and results that will be referred to later in the chapter. A one-electron system consists of a nucleus with charge +Ze and one electron, —e. With Z equal to the atomic number the model includes not only hydrogen but also the hydrogen-like ions, He*?, Lit?, etc. The total energy of such a system includes the kinetic energy of the motions both of the electron and of the nucleus around the common centre of gravity. The treatment is simplified by choosing the nucleus as the origin, and studying the motion of the electron relative to the nucleus. It should be well known from elementary classical mechanics that the motion relative to the centre of gravity is accounted for by replacing the mass m, of the electron by the reduced mass pp = m-M/(me+M), where M is the mass of the nucleus.

2.1.1

Schrodinger’s Equation for One-Electron

Atoms

In the first sections of this chapter we consider stationary states, described by the time independent Schrodinger equation:

HW (r) = EV(r) ,

(2:1)

where r is the position variable. EF represents the energy eigenvalues of the Hamilton energy operator H. Because of their physical interpretation as prob-

ability amplitudes (Sect. 2.1.3) the eigenfunctions WY must satisfy certain mathematical conditions: they must be single valued, continuous and finite everywhere (i.e. possible to normalize). In classical mechanics the Hamiltonian #7 is 2

H=f

4y,

2

where V is the potential energy and p is the linear momentum of the electron. Using the quantum-mechanical substitution p—>-ihV

,

and inserting the Coulomb potential energy with the nucleus at the origin Ves

V=-

4nTegr

;

ed

we obtain ieee ( QU

Va

Ze" a)

V(r) = BV(r) . ©)

in)

=)

The spherical symmetry of the potential leads to the choice of a system of spherical polar coordinates r,@ and @ (Fig. 2.1). Transforming the Laplacian

2.1

Fig.

One-Electron

2.1.

Spherical

Atoms

5

polar coordi-

nates.

rsinéd@e

operator V? to spherical polar coordinates puts (2.3) into a separable form: as V(r) depends only on r, a solution to the equation can be written as

(2.4)

W(r,6,¢) = R(r)Y(8, 4) , and inserting this into (2.3), we get, after rearrangement, two equations: re de Raga Rl +e =

2»pr?

E- VO)

(2.5)

and

Q2 laaas (sino) + = 9 so: Y(6,¢) =

—aY (9, ¢) ,

(2.6)

where a is the separation constant. By setting Y(@,¢) = O(0)®(@), we can separate (2.6) into two further equations, one containing only functions of 0,

Q(@), and the other only functions of ¢, ®(¢), where d

2

(2.7)

sab as(sino) fsay @(0) = a0(8) , 2

—

ss —m?P(¢)

;

(2.8)

The separation constant in this case is designated by m?. The solutions ® to the last equation contain the function expim@, which is single valued [i.e.

®($) = &(¢+2r) only if m = 0, +1, +2,.... The condition that the function

should be single valued thus gives us a quantum number m. Correspondingly the solution to the equation containing the function O(@) introduces a new quantum number I. The requirement that the solution should be finite over the range 0 < 6 < m imposes the conditions tatoo

rite ed

l(b

bee Od, 2

16

2. Basic Atomic Theory

Equation (2.6) contains only the angular parts of the functions, and it is independent of the potential energy term. The equation and its solutions Yim (9, 6) = O(0)G(¢), the so-called spherical harmonics, are well known from classical mechanics. In quantum mechanics these functions also appear as eigenfunctions of the operators representing the square of the angular momentum I? and its component along the z axis in spherical coordinates. In fact the bracket on the left-hand side of (2.6) is the 1? operator (apart from

a factor —h”). Equation (2.6) thus tells us that ah? = 1(1+1)h? are eigenvalues of the I? operator, i.e. the magnitude of the angular momentum of our one-electron atom is quantized with the values ,//(/ + 1)h. Similarly we find that its component in any specified direction, usually taken as the z axis, can have only the values mh.

We can now insert a = /(/+1) in the radial equation (2.5), and search for a physically acceptable solution. For energies E < 0, we obtain a new quantum number n, and radial eigenfunctions R(r) depending on the quantum numbers n and I. The energy eigenvalues are 4 2 2

——

;

we

(4me9)?h*

5) a = Sa gs

1

n

.

(2.9)

The gross structure of the hydrogen energy level system is shown in Fig. 1.2. Ry is the Rydberg constant for a system in which the reduced mass uw of the electron corresponds to a nucleus with the atomic weight M. This is exactly the same expression as the formula for the energy levels derived from the Bohr atomic model.

Ry

can be calculated from the physical constants

of (2.9). It can also be very accurately determined by means of spectroscopic measurements of wavelengths of one-electron systems and application of Rydberg’s formula. In fact Rjz is one of the most accurately measured constants of physics, and (2.9) is used together with a set of other relations and other measurements for the determination of the fundamental physical constants (Appendix A.2). If jz is replaced by the electronic mass Me, (2.9) describes the energy eigenvalues of a fictitious system where the centre of gravity coincides with the nucleus, i.e. where the nucleus is infinitely heavy. It is the corresponding Rydberg constant, R, that is usually quoted as the fundamental constant. The relation between Ray and RR. is Lh

M

(2.10) Ri = Reo Ie Roo M+me ’ where M is the mass of the nucleus. The conversion factor can be related to

the atomic weight by writing me and M in atomic mass units. From (2.10) we can then derive the approximate expression

60.20 (2.110) M’ which is useful for practical purposes, e.g. in connection with series formulae to be discussed in the next chapter. M is the atomic weight in atomic mass

Ru = Ro

units and for R,, we can use 109737.32 cm-!.

2.1

2.1.2

Quantum

Numbers

and Wave

One-Electron Atoms

I1Y7/

Functions

To summarize the previous section: solution of the Schrodinger equation gives us a set of quantum numbers, required by the physical constraints on the wave functions. Each possible state of a one-electron atom or ion is described by the three quantum numbers n, / and m, defining the eigenfunction of that state.

The principal quantum number n is connected to the solution of the radial part of the equation, and it is the only quantum number that appears in (2.9) for the energy eigenvalues. n can have any positive integer value from 1 to NOW AA WHO'R 0 (> cil URPors eet The J quantum number appears in the solution of the angular part of the equation, and it is also connected to the radial function. It can take any

positive integer value from 0 to n — 1,1 = 0,1,...,(n— 1), ie. n different values. For historic reasons, states with 1 =0, 1, 2, 3, 4, 5, ... are called s, p, dot @,h. ... states: The m quantum number appears only in the angular function. It can take any integer value between —/ and +1, m =0,+1,+2,...,+1, ie. 21+1 different values. As the energy depends only on n, there are n—1

$7 (2 +1) =n I=0

1+2(n—1

a

&

Ma

1

=i

different states having the same energy. The energy levels E, are said to be

n?-fold degenerate. We mentioned above dinger equation appears the angular momentum this angular momentum system is mh. To stress

that the angular part of the solution to the Schroalso as an eigenfunction of the I? operator, and that I has the magnitude \//(J + 1) h. The component of in the z direction of the spherical polar coordinate its connection with 1, we use the designation m, for

this quantum number. It should be noted that the maximum tum

on

the z axis, 1h, is always

smaller

projection of the angular momenthan

the length of the angular

momentum vector itself, ,/I(1 + 1) h. Figure 2.2 shows the vector model representation: the angular momentum vector can have 2] + 1 different angular directions relative to the z axis, but it can never be parallel to the z axis. This behaviour of the l vector can be interpreted as a precession around the z axis so that 1. is known, while /, and 1, are undetermined,

as required by

the Heisenberg uncertainty principle. Using the quantum numbers we can write the wave functions (2.4) as Wrlea

ls 0, ~) a

nee

tan (0, ~) ;

(2.12)

where R,, .(r) represents the solutions to the radial equation (2.5). The explicit

expressions for the radial hydrogenic wave functions for infinite nuclear mass are shown in Table 2.1 for n = 1 and 2.

18

2. Basic Atomic Theory

Fig. 2.2. Space quantization of angular momentum for | = 2. The length of the vector is ,/2(2 + 1) = 2.45. The unit of length is h.

The expressions have been abbreviated by using Anegh” ago =

Mee€

5

(2.13)

ao is the radius of the first circular orbit in the Bohr atomic model for infinite nuclear mass, used as the unit of length in the system of atomic units. The numerical value of ag is 5.29 x 107!! m. The radial functions Ryri(r) are shown in Fig. 2.3 for n = 1, 2, 3 and 4. The spherical harmonics Y,, are given in Table? ifort Os leand.2. An important property of the wave functions that we will use later is the parity. When the parity operator P acts on an arbitrary function f(r) it causes an inversion of the position coordinate r through the origin,

Uceal NS de e oat il

(2.14)

IfP-w(r) = ~(r), ie. o(—r) = w(r), then the function wv is said to be even. In the opposite case, y(—r) = —~(r), the function is odd. The parity operator in polar coordinates causes the changes

r > r, 0 > 7 —6,¢>

2+.

The

radial part of the wave function in (2.12) is therefore unaffected, whereas the angular part can be shown to change according to Yim, (8, ) > (—1)'Yim(0, 6). Table 2.1. Radial nuclear mass.

hydrogenic

ls

Rio

—

(2)? Je

2s

Roo

=

G2

2a9

Py

functions

27/20

\toe

3

wave

Zt)

2r/2a0

2a0

Zr.)e —Zr/2a9 day | =f= ee) Ne 93 (eer

Ry,

for

n =

1 and 2 for infinite

2.1

0

2

=

BieOn)

0

6

8

ee 20

r

O

2S ae

©

A

446

8

lO

One-Electron

Atoms

19

8 d0-12 84°16 36s

Ws

20

25

/s; have to be evaluated only for electrons outside closed subshells because, as noted in Sect. 2.3.1, the total orbital and spin angular momenta of a full subshell are both zero. Each of our new wave functions is, again, the product of an angular and a radial function. The angular part has exactly the same form as in the hydrogenic wave functions, so it is exactly known, and the corresponding part of the perturbation integral can always be evaluated exactly. The radial part on the other hand depends on the potential used to obtain the original product functions by, for example, the Hartree-Fock method. In general, the energy contribution from the electrostatic 1/rj; perturbation can be written

as the sum

BMS) SP preg.

a™,

(2.44)

where f,F* and gprG® represent the direct. and the exchange parts respectively. fj, and g, are derived from the angular part of the wave functions, while F* and G*, known as Slater integrals, are integrals over the radial parts of the wave functions. The number of terms in the sum and the suband superscripts k for which the coefficients f, and-g, are non-zero depend on the configuration. As an example we will look at the predicted structure of a pd configuration, e.g. 1s?2s?2p°3s?3p3d in a neutral silicon atom. Because the angular

38

2. Basic Atomic Theory

Table 2.4. Term structure of a pd configuration. ib

(OSS

(eSSienm

way

Sp h aps Sb 0e te: Ta Lees Daa OMEN Leeviclbones

3p 1p 2D 1D 3p

Fo + 2F> = 6G1423Ga Fo + 2Fo + 6G1 + 3G3 Fo = 715s +384) 21s Foitlo = SGaGteiGs Fock Fo Go63Ge

1

Ip

Fo + 7F>+

0

1

Gi

+ 63G3

“ The Fy and G, integrals of the table are related to the F* and G* of (2.44) in the following way: fF) = FP? 135: Gi = G* /15; G3 = G? /490. This kind of notation,

which avoids repeating common

factors of the angular coefficients, is often used.

momenta of the filled subshells add to zero we have to consider only the 3p and 3d electrons having n = 3, 1 = 1 and n = 3, 1 = 2. As the total orbital angular momentum L is the vector sum of the angular momenta of the individual electrons, the possible values of L are obtained as the set of ly tlg > LD > |h — yl, ie. in this example L = 3,2 or 1. For the spin angular momentum we have only two possibilities: the spins can be parallel,

with S = 1/2 + 1/2 = 1, or antiparallel, with S$ = 1/2 — 1/2 = 0. The six possible combinations of L and S are shown in Table 2.4. Each LS combination is called a term, and the spectroscopic notation for each term, of the form 25+! is shown in the table. Capital letters S, P, D, ... are used for LE =0,1,2,...in analogy to the designations of individual electrons, and the superscript 2.5 + 1 is called the multiplicity of the term, for reasons that will soon become obvious. Terms with multiplicity 1,2,3, etc. are called singlets, doublets, triplets, etc. The parity of the configuration to which the term belongs is sometimes indicated in the notation by adding a superscript ° to terms of odd parity, while terms of even parity have either a superscript © or no superscript at all. All terms in Table 2.4 are odd because they arise from a pd configuration, so the $F term, for example, would be written as 3°. A 3F term from an even configuration, such as d?, could be written °F°, However, the parity notation is often omitted, as the parity is evident from the configuration.

The last column of Table 2.4 shows the energy contribution Fes(LS) ac-

cording to (2.44) for each term. As the angular coefficients are independent of n, these formulae are valid for all pd configurations, but the magnitudes of F* and G* depend on the values of n and the nuclear charge Z. For electrons having the same n and 1, the so-called equivalent electrons, the Pauli principle restricts the number of possible terms. Consider as an example two p electrons having different n. By the method used above they are found to produce the terms 1$,38,1P.°P2D and 3D. If the electrons have

the same n, states having the same value of n,l,m, and m, must be excluded.

2.3 Many-Electron Atoms

39

Table 2.5. Allowed LS terms for equivalent s, p and d electrons. N is the number of terms in each configuration.

ihe

Terms

s

he)

1

s?

's

1

pps

@2P

es,’D

Sp

p°

=P 419

46

3

“por 4p 4B Pere Ger Perl PAD, FG

1 5 8 1G eto

pap

deduemecD) LD, G geo, a di Gf oD D-r, fae ee oe Pe DE. G, G, ! Aa pHey rer COC) Seep i

N

1

1) con

The allowed terms are found by combining all allowed sets of m; and ms to produce the set of possible values of My, and Ms, which in turn determine the possible combinations of L and S. For example, My, = 2 combined with Mg = 1 is forbidden for equivalent p electrons because it requires both electrons to have m; = +1 andm, = +1/2, and therefore the term 3D cannot exist. In this

way it can be deduced that a p? configuration has the terms 'S, 3P-and *D:

In fact this is exactly the same procedure as that used in the previous section to find that the ground configuration of helium, 1s”, has only a singlet state, while the excited configuration 1s2s consists of one singlet and one triplet. We can now label the ground state of helium as a 'S term, while the 1s2s

terms are 'S and °S. When a configuration consists of more than two electrons outside the closed shells, the 1 and the s quantum numbers are added step by step to give the resulting L and S. We shall discuss this procedure briefly in the next chapter. For equivalent electrons the situation is simplified by the fact that the configuration /* can be shown to have the same terms as J~-* where

w = 2(21+1) is the number of electrons in a closed subshell. Table 2.5 shows

the allowed terms for the s*, p* and d* configurations. It should be noticed that some d* configurations contain more than one term having the same L and §. One more quantum number is then needed to distinguish between these terms. This quantum number, called the seniority number, is derived from a theoretical treatment of complex atomic systems. A simple empirical rule, known as Hund’s rule, determines the order in which the terms of a configuration of equivalent’ electrons are arranged. Hund’s rule states that the term of lowest energy is the one that has the highest multiplicity, and, if there are several of these, the lowest is the one

40

2. Basic Atomic Theory

with the highest value of L. The rule is in accordance with more detailed theoretical predictions of the structure, but it must be remembered that it applies only to equivalent electrons. The Spin—Orbit Interaction. We still have not considered the last part of the Hamiltonian (2.36), representing the interaction between the magnetic moment of the electrons and the magnetic field caused by the orbital motion. The perturbation is

Fe

N

rae

(eae

(2.45)

cI

It is thus a sum of one-electron spin-orbit energies, each proportional to l;-s; with a proportionality factor € that depends on the potential V; in which each electron moves. In the one-electron case discussed in Sect. 2.1.4 the spin-orbit interaction was related to the total angular momentum 7 of the electron. In the same way, the interaction here is related to the total angular momentum J of all the electrons, defined by

A) J lpsavsy According to the rules for the addition of angular momenta the possible values of the corresponding quantum number J are obtained as

aS

ea

RE EN)

(2.46)

and the total angular momentum of the electrons takes the values

Bile a

ta a:

For evaluation of the spin-orbit perturbation, the functions W(yLSM,Ms),

where y represents the configuration nil1,ngl2,..., are coupled to new functions W(yLSJMj7), where hM, is the component of J along the z axis. As an example consider a °P term, where L = 1 and S = 1. The possible values of J are 2, 1 and 0. As the magnitude of the spin-orbit energy depends on J, the °P term is split into three fine structure levels, designated °P», °P; and 3Pp9. These should be read as “triplet P two”, etc. Note that the number of levels is equal to the multiplicity of the term, 25 + 1 = 3. In fact, (2.46) shows that 2.5 + 1 values for J are possible if L > S. If L < § , the number of levels is 20 + 1, but 25 + 1 is still used to denote the multiplicity of the term.

The energy does not depend on the quantum number M,, and for each energy level the states having M; = J, J—1,... , —d all have the same energy, i.e. a level has the degeneracy

C= 2d ree

(2.47)

g is known as the statistical weight of the level. According to first-order perturbation theory, the magnitude of the spin— orbit contribution to the energy is proportional to the scalar product D-S.

2.3 Many-Electron Atoms

41

The expression for this energy is therefore similar to that for the one-electron

case, (2.23):

Eyo(L8J) = A(LS) 5pide boosh Wate ay ence

my

(2.48)

Using a ?P term as an example again, we find that (2.48) predicts that the spin-orbit interaction shifts the J = 2, 1 and 0 levels by A, —A and —2A, respectively. The splitting factor A(LS) is a linear combination of the individual splitting factors ¢,; of the electrons with coefficients depending on L and S. The ¢,, factors or spin-orbit integrals can be evaluated through integration of the radial wave functions. (Equation 3.48 is valid only for the hydrogenic case.) A general relation for the intervals between two levels of a term can be

derived from (2.48): AB

J

fle)

— fobst

1

A(LS) 5 [J(J+1)-J'(J'+1)]

.

If we consider the two adjacent levels J and J’ = J — 1, we find that

AE(J,

J —1) = A(LS) - J,

(2.49)

i.e. the interval is proportional to the larger of the two J values. This relation

is known as Landé’s interval rule. A *D term (S = 3/2, L = 2), for example, has J = 7/2,5/2,3/2 and 1/2. According to the Landé rule the intervals are in the ratio 7:5:3. The energy level structure of a pd configuration in LS coupling is shown in Fig. 2.10, with the different energy contributions indicated below. It must be emphasized that the discussion in this section and the structure shown in the figure are based on LS coupling, which is valid when the non-central part of the electrostatic interaction is very much larger than the spin-orbit interaction. In the resulting energy level structure, the terms are widely spaced compared to the small fine-structure splittings within one term. The corresponding treatment for the opposite case, known as jj coupling, where the spin-orbit interaction is the larger of the two, leads to different energy contributions and to different level designations to describe the radically different level structure. Other cases are also possible; for example, in a two-electron configuration the spin-orbit interaction of one electron may be much larger than the non-central electrostatic interaction between the electrons, while the spin-orbit interaction of the other electron is much smaller. We will show the structure in such a case in the next chapter (Sect. 3.7.3). Whatever the coupling scheme, the total number of levels and their J values are uniquely determined by the electron configuration because the total angular momentum J is a constant of motion of an isolated atom (i.e. an atom which is not acted upon by any external forces). The possible values of the corresponding quantum number J are independent of the approximations made in the coupling schemes, which are only different ways of adding the various interactions, resulting in different relative positions for the levels.

42

2. Basic Atomic Theory

term 7 pete i a yi is

configuration

level --

3

==

1

Sp

2c

es

3

a:

3 *

Fig. 2.10. The structure of a pd configuration in L,S coupling, cf Table 2.4. The different contributions to the energy are shown below the diagram. The spin-orbit splittings are greatly exaggerated.

7

/

Ci

!

\

!i /

ee eee

/ /

i

1

—

f A

St

0

' \

De \

ly

1

\\

\

\

\

\

\

i"

D

M

i

mh

\

Ha

\

1

ee

\

Sy,

:

a

PI

4

eee

‘

ott

%

direct

eB

{

oo central

3

ae

non-central exchange

D spin-orbit

LS coupling is found to be a very good approximation for the lightest elements, and it is satisfactory also for medium-heavy elements. For the heaviest elements jj designations are better suited to describe the observed structure. In the general case of so-called intermediate coupling, both interactions are treated simultaneously in the theoretical calculations, but LS designations are still used to label the terms. We will now take a brief look at the cases for which LS coupling may still be a reasonable approximation, although not as good as it is for the lightest elements. 2.3.3 Deviation from Pure LS Coupling

The evaluation of the spin-orbit contribution to the energy in the previous section was carried out by first-order perturbation theory. In that treatment (Appendix A.1.1) the energy is given by the diagonal matrix element

Ey = GLE (Heol yo) .

(2.50)

where ¥ is the configuration. A second-order perturbation (Appendix A.1.2) from non-diagonal matrix elements of the form

pe — WALST |Hool YL'S' J)? sO

E

=

E'

’

involves

contributions

(2.51)

2.3 Many-Electron Atoms

43

splitting Energy

-6

4

1

0

1

1

4

3 2 Spin-orbit energy

L

4

1

5

Fig. 2.11. The deviation from LS coupling at increasing spin-orbit energy. The broken lines show the predicted trend when the second-order perturbation is neglected. The same energy units are used on both axes.

where FH is the unperturbed energy of the term LS and E’ is the unperturbed energy of another term L/S’. The LS coupling approximation is based on the assumption that the spin-orbit interaction is much smaller than the noncentral electrostatic electron—electron interaction. It is the latter interaction that is responsible for the separations between terms, and the LS approximation implies that the numerator is much smaller than the denominator

of (2.51) so that the second-order contribution is negligible. If, on the other

hand, the spin-orbit interaction cannot be regarded as small compared to the distance between the terms, |H — E’|, the second-order perturbation shifts

the levels LSJ (with energy E) and L’S’J (with energy EF") in opposite di-

rections — away from one another. Note that this perturbation only affects states with the same J and the same parity. As an example Fig. 2.11 shows the predicted structure of an sp configuration as the spin-orbit interaction integral ¢, increases from 0 to 5 units, while the electrostatic Slater integral G; remains constant at 5 units. In pure LS coupling the !P, level should be unaffected by the varying spin-orbit interaction, while the levels with J = 0, 1 and 2 of the triplet should change Dy Gi

44

2. Basic Atomic ‘Theory

—0.5¢, and +0.5¢, respectively. [See eq. (2.48) for an sp configuration, with A = 0.5¢,.] The J = 0 and 2 levels are seen to follow this linear trend, while the second-order contribution shifts the ?P; and 'P; levels quadratically in opposite directions. A second-order perturbation like (2.51) that has the effect of shifting the levels also affects the wave functions. The wave function for a pure LS state has to be replaced by a sum in which small fractions of other states L/S’ have been added to the original function. The coefficients of the states that are “mixed in” are, to first order,

(LSJ |Heol yL'S'J) E-—E' In our example in Fig. 2.11, the wave function for the upper J = 1 state consists mainly of a 'P, function, but with an increasingly large fraction of a °P, function added when we move to the right in the diagram. Correspondingly,

an equal fraction of 'P; is added to the ?P, function. 2.3.4

Configuration Interaction

The non-central part of the electrostatic interaction may also have secondorder contributions that are not negligible. In this case the energy shift is E@ = (yLS |Hes |LS)?

ce

E-—E

(2.52)

The perturbation acts between states having the same L and S as well as the same J, but from different configurations of the same parity. In the same way as the spin-orbit mixing, this configuration mixing or configuration interaction causes shifts of energy levels, and it also mixes the wave functions. The configuration interaction can be large, even if the matrix element is small, provided the interacting configurations are close. In fact different configurations often overlap in complex many-electron atoms, causing considerable level shifts. For example, the nd and (n + 1)s electrons in the transition metals have similar energies, and the configurations including them have the same parity; as a result there is often strong mixing between levels arising from the nd*, nd*-!(n+1)s and nd*~?(n+1)s? configurations. If the matrix element in the numerator of (2.52) is large, as a result of a large overlap of the wave functions, the interaction may be significant even when the states of the two configurations are widely spaced. Any apparent configuration interaction between states of different L and S can generally be attributed to departures from LS coupling, so that the LS states are not in fact pure.

2.4

2.4 Radiative

Radiative Transitions and Selection Rules

Transitions

45

and Selection Rules

Radiative transitions between the stationary states discussed in the previous parts of this chapter can be treated as interactions between the atom and an electromagnetic field. There are three possible types of transition — absorption, stimulated emission and spontaneous emission — and the relations between them are derived in Chap. 7. For absorption and stimulated emission one can use a semiclassical treatment, in which the field is described by classical electromagnetic theory whereas the atomic states are described by quantum mechanics. It is the perturbation on the Hamiltonian due to the field that causes the transition between two stationary states. Spontaneous emission cannot be dealt with in this way because, according to the classical description, no radiation field is present before the transition takes place. A full treatment of spontaneous emission requires quantum electrodynamics (QED). Fortunately, the Einstein relations between the three types of transition that are derived in Chap. 7 provide us with everything we need to know about spontaneous emission. 2.4.1

Time-Dependent

Perturbations

A transition between stationary states is a time-dependent phenomenon, so we must use the time-dependent Schrodinger equation

ow

HoW° = 1ih— 0 at .

(DDS )

The solutions of this equation can be written as

WO

(2.54)

r (arctica shadeeet

where the time-independent functions 7, satisfy the time-independent equation Aon

=

EnWn

(2.55)

.

Let us now suppose that a small time-dependent perturbation H’(t) is switched on at time t = 0, changing the Hamiltonian into H = Ho + H'(t). The time-dependent Schrodinger equation is now

AW

= =ih ow ey

(256 )

and we write the general solution as W(r,t). This solution can be expanded as a sum of the solutions to the unperturbed problem (Appendix A.1.3):

(2.57)

vr.) = ee k

P

where the coefficients c;,(t) are time-dependent. We can assume that we know the functions 7, and the eigenvalues FE), — these are the stationary states that

46

2. Basic Atomic Theory

we have dealt with in the previous sections of this chapter. Solving (2.56) then consists of finding the coefficients cy. With the c, determined in such a way

that W is normalized, lc. (t)|? can be interpreted as the probability of finding the system in the stationary state kh at time f. To evaluate a transition probability between an intial state (designated by the index 7) and a final state (designated by f), we need to determine the value of the particular coefficient cy at time t, given that all coefficients except c; are zero at time t = 0. Time-dependent perturbation theory (Appendix A.1.3) leads to the equation

ine =(hy|| yi)ele , .

de

(2.58)

iw

where wy; is the angular frequency of the transition between states f and 7 given by |E, — E;| = hus; = hwz;. The solution of this equation determines

Icrl’, the probability of finding the system in state f. To solve (2.58) we must know the explicit form and time dependence of the perturbation H’(t). 2.4.2 The Electromagnetic

Interaction

According to classical electrodynamics, an electromagnetic field in empty space can be described by means of a vector potential A. For a one-electron atom the Hamiltonian is modified by the interaction between the electron and the field to take the form

H= a1 Pt eA) 2 +V(r). We will not pursue the detailed treatment of this operator, which can be found in textbooks on atomic theory. The final result, if we consider processes involving only one photon, is that the time-dependent perturbation of the Hamiltonian can be written as

ieeem Ag

eae m Ate

(2.59)

The most general description of the time-dependent electromagnetic field is a superposition of plane waves of all frequencies w. This can be introduced by writing the vector potential as Ate

Ne jeltoe—en) me e

i(wt-k-r)

Here k is the wave vector having a magnitude of 27/ and a direction perpendicular to the wave fronts. We start with a single frequency w in our equations, and later integrate over all frequencies. We are now ready to rewrite (2.58), the differential equation for the timedependent coefficients: —

=

C

(vy | AG

F; pe

*

|i ) ellwfitw)t

m

e

f

+ (vy | 7 {10‘pea

:

i) ellus-w)t

(2.60)

2.4 Radiative Transitions and Selection Rules

AT

With the boundary condition c¢(0) = 0, the solution is

ett) = (bp |S Ao ve** |s) (Rwy + w) + (¥s| 3, etc. The resonance line is the 3p *P—4s 2S doublet. In these systems the s? subshell is not as stable as the filled rare gas shell formed by the inner electrons in the alkalis, and the energy needed for exciting one of the s electrons is relatively small. The lowest configuration where one of the s electrons is excited in Al I is 3s3p?, which has three electrons outside the filled shells. Addition of the angular momenta in order to derive the LS terms from this configuration is most easily done by starting with the equivalent p electrons. According to Table 2.5, p? gives the terms 'S, 3P and 'D. The additional s electron contributes only spin angular momentum, producing the

terms *S, ?P, 4P and 7D. The *P has in fact a very low energy, as can be seen in Fig. 3.2. The doublet terms of the 3s3p? configuration are higher, all of them except 7D appearing above the ionization limit. Due to configuration interaction (Sect. 2.3.4) terms of 3s3p? may affect the 3s?nl series having the ? Note the unfortunate use of the letter ¢ in spectroscopic literature for two different quantities: ¢ is the net charge of the atomic core, i.e. the charge seen by an outer electron, whereas ¢,; is the spin-orbit energy.

3.1 One-Electron Systems

25

|

—

61

4s

cm’') (10° Energy

A. 3s°3p ns

np

nd

nf

ng

Fig. 3.2. The Al I energy level system. The fine structure is too small to be visible on this scale.

same parity and terms with the same L and S. In this case 3s3p? 7D perturbs the 3s?nd 7D series so severely that it cannot be described by a simple Ritz formula, and the mixing between the perturber and the series is so complete that it is impossible to determine which of the observed 7D terms is the perturber. For this reason it is not marked in Fig. 3.2. The metals Cu, Ag and Au also form one-electron systems, having ground

configurations 3d!° 4s for Cu I, 4d!° 5s for Ag I, and 5d!° 6s for Au I. Excitation of the 4s electron gives rise to a system of 3d!°nl configurations in Cu I. As with Al, the situation is complicated by excitation of an electron from

the closed shell, in this case the 3d shell. The nd and (n+ 1)s electrons have approximately the same binding energies, and so another system of configurations with slightly higher energy is formed when one of the nd electrons is

excited instead of the (n + 1)s electron. Such configurations, e.g. 3d?4snl in Cu I, are known as inner-shell excited. The whole system is sometimes called the displaced system.

62

3. Atomic Structure and Atomic Spectra

3.2 Two-Electron 3.2.1

Systems

Systems with an s* Ground

Configuration

The simplest two-electron system is He I, which we discussed in detail in

the previous chapter. The ground term is 1s? ‘So. Excitation of one electron

gives snl with the terms 'Z and °L (L =1). This system forms series that converge towards the parent term Is 2S, which is the ground term of the hydrogen-like system He II. Similar energy level systems appear in the alkaline earths Be, Mg, Ca, Sr and Ba, and isoelectronic ions (Sect. 3.6), which have two s electrons outside the stable p® shell. There is, however, a difference between the first three elements, He, Be and Mg, and the second three, Ca, Sr and Ba, which arises from the similarity in energy of the nd and (n+ 1)s electrons that was noted for Cu I. As 3d is the lowest d shell, this complication arises first for Ca I, the energy level system of which is depicted in Fig. 3.3. The complete ground configuration is 1s?2s?2p°3s?3p°4s”, with the 3d subshell empty. The ground term is 'So, and excitation of one of the s electrons gives rise to the

4snl system of 'L and 3Z terms (L = 1), which has the parent 4s *S of Ca II.

dr Dag», 6g 60 [

50

7

E

mi5d _ af 2

| 4s °S eS }

==

4o |

—

bs

2 panel, SD

eS oe

IO®

one

ne 6S

(e.Ss

Ata —

ope

aS

Oat —

st

—

adi te

tequed S

4d

a

Ppeerie

|

3d LI 4p

si

2|

|

oO coe

4p

10 | 27

0 |

4s

Gkwisy

So

Jajey

|

latel — fahi—Jnke) kent

ya}o)

Jako

tabi

Fig. 3.3. Survey diagram of the Ca I energy level system. Each configuration contains singlet and triplet terms, except the ground configuration 4s”. The boxes indicate the energy range of each configuration, but the individual terms or levels are not shown. The two systems 4snl and 3dnl have the parents 4s 7S and 3d 7D, respectively. Note that the configuration 3d4s, shown as a dotted box, is identical to 4s3d. The configuration 4p” to the far right is the lowest configuration of a third system 4pnl.

3.2 Two-Electron Systems

63

However, 3d and 4s are sufficiently close in energy that excitation of both of the 4s electrons, one of them to 3d and the other to higher n and 1, gives a new level system 3dnl, which is just slightly higher than the 4sn/ system, as seen in Fig.3.3. As in the case of Cu I, this doubly excited system is sometimes called the displaced system. In Ca I it has the parent 3d 7D, and it is a two-electron system where the terms are singlets and triplets. There are, however, more possible values for L than in the 4snl system, as the terms are formed by adding the | to the 3d 7D parent. A 3dnp configuration has

the LS terms !P, °P,'D, °D, 'F and 3F. Although Ca I can be considered as consisting of two simple term systems, the occurrence of both systems below the ionization limit makes the spectrum somewhat complex. There is another group of elements, Zn, Cd and Hg, having an s* 'S ground term. The ground configuration of these elements can be written as

(n—1)d!°ns?, e.g. 3d!°4s? in the case of Zn I. The normal system consists of nsn/l 'L and ?L terms. A displaced system is formed by excitation of both the ns electrons, with the lowest configuration np?.

3.2.2 Systems with a p? Ground Configuration The elements C, Si, Ge, Sn and Pb have the ground configuration ns?np?. The two equivalent p electrons form the terms ?P, 'D and 1S, of which 3P has the lowest energy (Hund’s rule). The normal, excited two-electron system npn'l of singlets and triplets has the parent term np 2P. Some terms of the displaced system, where one of the s electrons is excited, are low, e.g. 2s2p? 5S in CI, which is the lowest excited term above the ground configuration. Atoms and ions with a d? (or f?) ground configuration might also be considered as two-electron systems, but in practice they have at least one completely overlapping displaced system, giving a situation similar to Ca I but much more complicated. We therefore consider them as belonging to the complex systems, which are discussed below. 3.2.3 The Rare Gas Systems

It is obvious that He I is a two-electron atom, but in fact the other rare gases Ne, Ar, Kr, Xe and Rn can also be treated as two-electron systems. They all have the ground configuration np®, and excitation of one electron gives a system np°n’'l with the parent configuration np? and hence one possible parent term, 7P, cf. Table 2.5. The LS terms of np?n’'l are found by adding the orbital and spin angular momenta of the excited electron n‘l to L = 1 and S = 1/2 of the parent term, giving a system of singlets and triplets. The

fine structure splitting of the ?P parent is large because of the large spin— orbit energy of the np° configuration, and as a result the observed structure of the np®n'l configurations is far from that predicted by the LS coupling approximation.

64

3. Atomic Structure and Atomic Spectra

Excitation of a rare gas from the ground state to the lowest excited configuration, np® > np°n/s, means breaking the stable p® shell, which involves a large change of energy. The excited levels are therefore far above the ground term, and the resonance lines of the rare gases appear in the far UV region.

3.3 Complex Atoms All the atoms we have considered until now have a simple level structure built on only one parent term, giving doublet terms in the one-electron cases and singlet and triplet terms in the two-electron cases. The terms are formed by adding the angular momenta of the outer electron to those of the parent. The only complication is the displaced system that appears when one electron from a filled inner shell or subshell, or more than one electron in a partly filled shell, is excited. The complexity is increased in systems like Ca I where the displaced system has a low excitation energy, giving many terms below the ionization limit. The next stage of complexity appears when the parent configuration contains several terms, and it is still more complex when such a system overlaps other parent systems. The structure of a multiple parent system is best described by starting from the parent terms and adding to them the spin and orbital angular momenta of the outer electron.

3.3.1

p-Shell Atoms

with Multiple Parent

The simplest multiple parent case arises when

Terms the parent configuration

is

np’, with the three terms °P, 'D and !S. The relevant atoms are N I, P I, As I, Sb I and Bi I (and the isoelectronic ions), which all have the ground configuration np?. We take as an example N I (Fig. 3.4), having the ground

configuration 2p? with the terms 4S (lowest), 7D and 2P. Besides 2p°, three different sets of configurations are formed, built on each

of the parent terms 2p? 3P, 'D and 'S; the first two of these sets are shown in Fig. 3.4. In cases like this the parent term must be included notation, as several identical LS terms (with different energies) in a configuration. This is done by writing the parent term in after the parent configuration. If one of the 2p electrons in N I form the 2p?3d configuration, we obtain three subconfigurations: —

in the term may appear parentheses is excited to

2p4("P jad

— 2p?('D)3d

— 2p*(*8)3d. _ By adding / = 2 and s = 1/2 of the 3d electron to L = 1 and S = 1 of the 3P parent term, we get L = 3,2,1 and § = 3/2,1/2, so the possible terms are quartets and doublets of F, D and P: 2p?(°P)3d ?P,?2D,?F,4P,4D and

3.3 Complex Atoms

65

£2)BDwnk olen, 120 } 2p? Sp [ =

=,

—4d —4f

~4P —3 —4, Osp

100 Te

—4s

80

|

4P =3d

= Af

= = 65

Oi3s

2s2p'

|

oO

Xo)

=

60}

ra) ay

Ww

40 |

20 |

| 2p°

OL

2p°(°P) ns

np

ond

nf 2p°('D)ns ajo)

fu

:

fakol

Tani

4

4

Fig. 3.4. The N I energy level system. The configurations are represented by boxes, as in Fig. 3.3. Only configurations based on the parent terms 3P and 'D are shown. 2s2p* is the lowest configuration of the system 2s2p°nl.

4. In the same way, building on the other parent terms, we get 2p?(1D)3d 25 2P2D,2F,2G and 2p?(!S)3d 7D. The 2p73d configuration thus contains 12 LS terms, among them two ?P terms, two 7F terms and three 2D terms. The energy level system becomes very complex when each configuration may comprise such a large number of terms, and each term further split into several levels by the spin-orbit interaction. The number of possible transitions in such a system increases very rapidly, and a series structure like that observed in the simple spectra is no longer obvious. In practice it is difficult to establish experimentally all the levels of the excited configurations, even for low values of n. In particular, all the levels in N I built on the very high 1S parent lie above the ionization limit defined by the lowest parent level. As Table 2.5 shows, the p* configuration has the same terms as Di, So the atoms and ions with the ground configuration np” and the excited system np‘n’l have energy level systems and spectra with the same degree of complexity as those with an np® ground configuration, except that the p° ground configuration itself (like a single p electron) has only one term, ze: The halogen elements F, Cl, Br and I all have this type of energy system. Elements with the ground configuration np* and the excited system np?n’'l have the ground configuration terms 3P 1D and 1S, (as for np?) and the parent terms 4S, 7D and 2P. The complexity is similar to the cases discussed above. The elements O, S, Se and Te are in this category.

66

3. Atomic Structure and Atomic Spectra

All the p* systems we have electrons, two s electrons in the to the normal systems, in which system nsnp*t+! may introduce is shown in Fig. 3.4. 3.3.2

The

Transition

treated here have, besides the equivalent p ground configuration, i.e. ns?np*. In addition one of the p electrons is excited, the displaced terms below the ionization limit; one of these

Elements

The transition elements are those sequences of elements in the periodic table where the binding energies of nd and (n + 1)s electrons are approximately equal; they comprise the so-called iron (n = 3), palladium (n = 4) and platinum (n = 5) groups. Compared to the elements with ground and parent configurations formed by equivalent p electrons, the transition elements with d* configurations generally have a larger number of parent terms (Table 2.5). The situation is further complicated by the overlap of systems. We have already seen an ex-

ample of this in Ca I (Fig.3.3), caused by the similarity of the 3d and 4s binding energies. In the iron group elements there are three low overlapping configurations, 3d", 3d*—!4s and 3d*~?4s?. All these configurations have the same parity, and configuration interaction may therefore occur. Moreover, the first ion of each element has a similar set of three low configurations 3d*~}, 3d*-24s and 3d*~4s?, all of which act as parent configurations of the atom, each of them giving a number of parent terms. As an example, we look at the relatively simple case of singly ionized zirconium, Zr II. It has the three low configurations 4d°, 4d?5s and 4d5s?. The

next ion, Zr III, has the three low configurations 4d?, 4d5s and 5s”, giving

eight parent terms: (4d) 3F, 3P, 1G, 'D, 1S; (4d5s) 3D, 1D; (5s?) 1S. Excitation of the most loosely bound electron of Zr II to nl gives a large number of

terms, which we can group in eight subconfigurations: 4d?(?F)nl, 4d?(°P)nl, 4d? (*G)nl, 4d?('D)nl, 4d?(1S)nl, 4d5s(?D)nl, 4d5s(1D)nl, 5s?(1S)nl. The resulting terms of Zr II are derived by adding the spin and angular momenta of the nl electron to the parent term, as was done in the simpler cases above

(Sect. 3.3.1). A simplified term diagram of Zr I is shown in Fig.3.5. This is a type of diagram suitable for displaying the structure of complex systems. As in Figs. 3.3 and 3.4, the configurations are represented by boxes, and individual terms or levels are not shown. The different parent configurations and terms are displayed at the top of the figure, where the location of the lowest parent defines the ionization limit. All observed subconfigurations based on a certain parent term are joined by vertical lines, with even and odd configurations in the same vertical line. Obviously, the parent structure is closely reproduced by the subconfiguration structure. The lines of the 5s—5p transition give — together with 4d—5p — the strongest lines in the spectrum. As the distance between the 5s and the 5p subconfigurations is approximately the same for all 4d? parents, the corresponding lines will appear in the same

3.3 Complex Atoms

140

Zr Ill

1 ap

120 + } — —

a

= (s)

{

PT

ae —

—

—

—

oo)

—- ~

+

~~

spa

C

— Gf Sy

67

Ses

Gels ID) IP rt ie vale

|

ae

100

2,

=J

pe 80 +

4fSs

> 2

wi

60 r

era

ewe

40

5d

eames

6sCa-

4.0. a

Ot

sol| all

sp]. a 20 +

A upeiabs

| 5s(_]

C

+

Zr Il

De

mate

Fig. 3.5. The Zr II term system. The terms of the lowest Zr HI configurations form the parent terms of Zr II. Configurations with the same n and | but based on different parents are joined by dotted lines.

narrow wavelength region, giving a very high line density in that region of the spectrum. It should be noted that the 4d° configuration is not split up according to parent terms. Due to the restrictions imposed by the Pauli principle, the terms of a configuration of k equivalent electrons cannot be derived by adding | and s to the terms of the configuration of k — 1 electrons (Sect. 2.3.2 and Table 2.5). No definite parent can therefore be ascribed to a term of such a configuration. In the theoretical treatment of equivalent electrons a concept called fractional parentage is introduced to describe the relations between

the terms of configurations 1’ and 1*~. Table 2.5 shows that certain configurations of equivalent electrons contain more than one term with the same L and S. For the reasons just mentioned they cannot be assigned to different parent terms, but they are instead distinguished by means of a new quantum number, the seniority number, appearing in the theory of complex spectra. 3.3.3

Lanthanides

and Actinides

The elements with Z = 57-71, known as the lanthanides, and the group beginning with Z = 89, the actinides, have the most complex spectra of all. There is a high density of lines over wide wavelength regions as a result of the high density of energy levels, formed by overlapping complex configurations. In these elements the 4f and 5f subshells respectively are partially filled,

68

3. Atomic Structure and Atomic Spectra

giving large numbers of parent terms. As with the transition elements, several electrons have approximately the same binding energy — 4f, 5d, 6s and 6p for the lanthanides — giving several overlapping systems of energy levels. As an example of the complexity, the configuration 4f"5d?6p, observed in Gd I,

has 24662 possible LS terms, with 78 822 levels [2]. Only a small fraction of these levels has been experimentally established from the observed spectrum. The lanthanide and actinide spectra are actually even more complex than might be expected from the high density of levels, because deviations from LS coupling and interactions between overlapping configurations lead to a breakdown of the LS selection rules.

3.4 Interpretation and Understanding of the Observations Observations and theoretical predictions agree within the experimental error limits for hydrogen and hydrogen-like ions, and for helium and helium-like ions the agreement is also extremely good. For systems with more electrons, the approximations involved in the theoretical treatment cause larger deviations, and in the many-electron systems the theoretical predictions can generally not be used for unambiguous identification of spectral lines. In spite of this, it must be stressed that the theoretical treatment, outlined in its most basic form in the previous chapter, provides a complete interpretation and understanding of all observations of atomic energy level systems and spectra. But it must also be stressed that whenever individual spectral lines are to be used in spectroscopic applications, in astrophysics and plasma physics for example, experimental wavelengths or transition energies are needed for line identifications or simulations. Experimental values for line strengths are also generally necessary, particularly in complex systems with severe mixing of states where the theoretical predictions may be wrong by orders of magnitude.

3.5 Inner-Shell

Excitation

and Autoionization

The ionization energy of an atom or ion is the smallest amount of energy needed to transfer it from its ground state to the ground state of the next higher ion with the emission of a free electron. In the simple case of an alkali atom the ground state is a 7S term, and the ground state of the alkali ion is the parent 'S term formed by the filled inner shells. As all the series in the alkali atom

converge

towards

this term, the parent is identical with

the ionization limit in this case. In atoms having a parent term with a fine structure splitting, it is the lowest level of the term that defines the ionization

3.5 Inner-Shell Excitation and Autoionization

69

limit, and in atoms with multiple parent terms and overlapping level systems, the limit is the lowest level of the lowest parent term (Fig. 3.5).° We have seen several examples where energy levels of a displaced system appear above the ionization limit of the normal system. It may seem strange that a bound state can exist when the energy transferred to the atom exceeds the energy needed to release an electron, but it is not so difficult to understand if the displaced system consists of doubly excited states, as in Ca I, with the energy shared by two electrons, neither of which receives enough energy to be ejected. The existence of high, bound, inner-shell excited states can also be understood by taking into account the difference in the screening of the nuclear charge produced by excitation of an inner electron. Compare for example an excited state nl in the normal Na I term system 2p°nl with an inner-shell excited 1s?2s?2p°3snl state. The outer electron in the normal term is rather loosely bound because of the efficient screening of the nucleus by the spherically symmetric filled inner shells, whereas in the doubly excited system the screening is less complete and the outer electron is more tightly bound. The inner-shell excited configurations can thus form bound states above the ionization limit of the normal system. Let us now consider what happens when an atom is ionized to form a system consisting of an ion and a free electron. Again we take the simple case of the sodium atom, with the outer electron excited from its ground state to 1s22s22p°np ?P° by absorption of a photon. (The superscript ° indicates the odd parity of the state.) With high photon energies, high np 2P° states can be reached, and if the photon energy is greater than the ionization energy the electron is ejected. The ion is left in its ground state 2p® 1S. The conservation of energy requires that the ejected electron carries the excess energy. In the same way, conservation of parity and angular momentum requires that the system consisting of the ion and the ejected electron has the same parity and the same angular momentum J (and for LS coupling the same L and S) as the series of bound states. The continuum of energy states above the series 1s22s?2p°np ?P° can therefore be described as 1s?2s?2p®«p 7P°, where « indicates that the free electron may have any positive energy. A similar continuum of states exists above the limit for each of the series 2p°Ks 7S, 2p®«p 2P°, 2p°«d 2D, 2p°«f 2F°, etc., as shown in Fig. 3.6. Like any other state, the continuum states are described by wave functions. The bound states above the first ionization limit that were discussed above are embedded in this continuum of free states. For Na I the lowest system of inner-shell excited states is 2p°3snl. The configurations 2p°3sns have odd parity and give rise to the terms 2P° and *4P°. The overlap between the discrete, bound 2p°3sns 7P° terms and the 2p°«p 7P° continuum of free states, having the same parity and angular momenta, leads to a mixing of 3 To avoid confusion with the energy needed to produce an ion in a higher state, i.e. a higher parent, the ionization limit as defined here is sometimes called the first ionization limit.

70

3. Atomic Structure and Atomic Spectra

=

250 |

ee

.

_

4p

—-2p°3s3p_ |

2p°3s° °P°

200 | "e

| KS °S

Kp Pol

xd ?D|

Kf °F

oO

© 150}

|

oO

ex > ®O (es,

ww

100 |

|

20 op SA «Bie

AB

0

2p°ns 2c

np *P

nd°D

nf °F

Fig. 3.6. Discrete and continuum states of Na I. The lowest observed inner-shell excited states are seen to the right. The dotted lines indicate the interaction between the continuum and the inner-shell excited configurations leading to autoionization.

the states. This means that there is a high probability for ionization of the bound state by the process

2p°3sns 7P° > 2p°(1So)Kp 2P° = 2p® 1Sp +7

,

as indicated in Fig.3.6. This process is called autoionization. It can take place only when there is an interaction between the bound state and the continuum. The 2p°3sns 4P states are not autoionized if LS coupling is a good approximation because there is no quartet continuum. A high probability for autoionization means that the lifetime of the autoionizing state is short, generally so short that spectral lines involving this state have an appreciable natural linewidth (Sect.8.2). In many cases an autoionizing state cannot be observed in an emission spectrum because the probability that it emits a photon before it is autoionized is negligible. In an absorption spectrum the bound states above the first ionization limit give rise to discrete absorption lines superimposed on the continuous absorption into the continuum. The absorption lines involving the autoionizing states have line profiles that can be very asymmetric. This behaviour is theoretically well

3.6 Isoelectronic Sequences and Highly Charged Ions

if

understood as depending on the relative phases of the wave functions of the discrete and the continuum states. The inverse of autoionization is often an important process in astrophysical and other plasmas. In this process a free electron collides with an ion in The impact electron loses its ground state and excites one of its electrons. inner-shell excited state or excited doubly a form energy, and is captured to excited and one is capis one — involved are of the atom. As two electrons The doubly excited recombination. dielectronic tured — the process is called process) or by capture the (reversing autoionization state can decay either by state through ground the to finally and state excited a transition to a singly photon emission.

3.6 Isoelectronic Sequences and Highly Charged Ions A useful aid to interpreting or predicting energy level systems is to follow their behaviour along an isoelectronic sequence. The effective nuclear charge increases monotonically along such a sequence, whereas the electron configurations do not change, so the energy levels also vary monotonically, and interpolation or extrapolation can often provide useful information. The energy levels of hydrogen-like ions, ignoring fine structure, are exactly described by the relation

RZ? EE, = =

p)

in which the increase in Z describes the evolution of the level structure along the isoelectronic sequence H I, He IH, Li III, etc. In systems with more than

one electron the electron—nuclear energy still increases with Z 2 The electron—

electron energy depends on 1/r;; = 1/ |ri—1j |which increases only linearly

with Z. The increasing importance of the nuclear attraction indicates that the structure should become increasingly hydrogen-like so that the energy tends to depend more on the quantum number n and less on l. Section 3.1.2 identified two different ways of describing terms of manyelectron systems, of which the quantum defect method (3.6) was found to be more useful for series of terms within one system. The alternative method, involving partial screening of the nuclear charge, is described by (3.4) and (3.5): 2 2

pe fie

n?

ea

ar

n?

where s and p depend on n and |. This description is more useful for isoelectronic sequences because the parameters s and p are found to be almost independent of Z and ¢, at least after the first few members of a sequence. This is true not only for one-electron systems, but also for complex configurations.

ibe,

3. Atomic Structure and Atomic Spectra

60

a,

eee 8

>

fo) Sy

aay

2500 |

;

a

f

cae

[J 3d

|

2000

So ws =

Ge aes,

imine ms

2

4p

40

a= 4f

2,

1500;

4s

Cae

ae

CoO

4. ==

5

Lise

2

3p

Ww

1000 + oy

L 3p3d

10

(J 3s3d

500 |

Mg 0

g

|

np

Of nd

nf 3pns

|

i

Cr XIll

1] asap

== 8s) 3sns_

C1] 3p

— 38"

np

3sns

np

A

:

ey

nd

nf 3pnsnp

nd

. nf3dns

np

;

1

nd

nf

Fig. 3.7. Two term systems in the Mg-like isoelectronic sequence. Dotted boxes show the predicted position of configurations in which no levels have been established experimentally.

To see how (3.9) can be applied, consider first the energy difference AW between two configurations of the same system that differ only in one electron nl: AW

—

VG ate

“3

R(Z

i

=

Wrsis

82)?

a

n5

es

+ Pa) 2 Ns

=

R(Z

nglo

=

—

AYO

81)?

ni RC Pats + Pa) Ne .

(3.10)

My

If the electron has the same n = n, = no in the two configurations, this relation is reduced to AW

wi 2R(s1 -4 S52) 5 [Z nr

(s1+82)/2]

=

Pa

Pde + (pat) /2 Gs Bip

Obviously the distance between two configurations having the same n increases with Z, while the distance according to (3.10) increases with Z? when the configurations differ in n. This means that as Z increases the order of the configurations depends relatively more on n than on I. This effect is illustrated by the magnesium-like isoelectronic sequence depicted in Fig. 3.7. The ground configuration is 3s? and the normal energy level system is 3snl. In Mg I the configurations are not ordered according to n: 4s is lower than 3d, 5s is lower than 4d and 4f, etc. Only one term of the configuration 3p” is observed below the ionization limit in Mg I, and the rest of the 3pnl configurations will appear far above the limit. In Cr XIII, belonging to the same isoelectronic

3.6 Isoelectronic Sequences and Highly Charged Ions Phir

T

SS

S=—=

SS

3

3p3d “F P

i

Se ees Ss

==

T

ee SS

T

+

T

73

=

4

Se eae

60 +

j

50

+

=

Sele) De

Her:

ee

-

°

3 Dre

a,

=:

.

es eeeer

oyeS on) EE etrie,re| =eas

€

3p:

Oo oO

=, 40}

_

2 0

Ende

PSS

&

|

ee

0

= 30

4

: SS opp biwaeeanegs

Souk

20 +

°P SSS 10

Ca IX

TiX!

CrXill

pW

, netted

ie FeXV

Be tf NiXVIl

.

Zn XIX

Fig. 3.8. Some 3/31’ configurations in the Mg-like isoelectronic sequence. The 3p” 1D has been omitted, as it would be difficult to distinguish from 3P on this scale. The scale is also too small to show the 3s3d °D fine structure.

sequence (12 electrons), the configurations are grouped together according to n, with an | dependence within each n group. All the possible configurations 3131' and 314l' are observed or predicted to appear below the first ionization limit. The ordering by n with increasing Z can also be illustrated by the transition elements. In the neutral atoms of the iron group elements the lowest configuration is either 3d'-24s? or 3d*—!4s, while 3d” is the ground configuration in all the doubly and more highly ionized elements.

According to (3.11) the distance between configurations with the same principal quantum numbers is a linear function of ¢ so that AW/¢ should be roughly constant. This is indeed observed, as shown by the graphs repre-

senting some 3s3/ and 3p3l configurations in the Mg-like sequence shown in Fig. 3.8. Not only are the graphs for different configurations approximately parallel and horizontal as predicted, but also the graphs for different terms within one configuration are approximately parallel. This illustrates the fact that the electrostatic electron—electron interaction, described by the Slater integrals (2.44), also increases approximately linearly with ¢. It can also be seen from the figure that the fine structure splitting of each of the 3P terms increases along the sequence, showing a faster increase with ¢; this is expected

74

3. Atomic Structure and Atomic Spectra

from the Z4 dependence of the spin-orbit interaction given in the hydrogenic formula (3.8), which can be adapted for estimating the one-electron spin— orbit energy in non-hydrogenic cases if Z is replaced by (Z — s).

The 3p? !D term, not shown in Fig. 3.8, is found between the fine structure levels of 3p? °P. This is too low according to the L.S approximation, which predicts that it should appear between the 'S and the °P terms. The 3s3d 'D term on the other hand is higher than expected. These deviations are due to configuration interaction, causing the two 'D terms to be shifted in opposite directions. In Mg I, on the other hand, 3p? !D is much higher than

3s3d 'D, and it is therefore not seen in Fig. 3.7. With increasing Z, 3p? 1D should move down through the 3snl system, and already in Al II it is lower than 3s3d 'D. The 3p3d and the 3d? configurations also move down to the positions shown for Cr XIII, and in the same way the higher 3pn/ and 3dnl configurations are displaced downwards. In the ions close to the positions where these so-called plunging configurations cross 3snl, n > 4, the mixings between LS terms of the same kind are large, and due to the shifts of the energy levels the isoelectronic diagrams show pronounced discontinuities in the crossing regions. The highly ionized systems have a number of other characteristics. Because the distances between configurations increase with Z (An = 0) or Z? (An # 0), the spectrum is shifted towards the UV and XUV regions as Z increases. The Z* variation of the spin-orbit interaction implies increasing deviation from LS coupling conditions. Spin-forbidden transitions appear with increasing intensities, and the corresponding mixing of LS states causes slow changes of relative intensities in the spectrum. More drastic changes appear close to the crossings with plunging configurations. Another important change along a sequence is the fact that the probabilities of magnetic dipole and electric quadrupole transitions, the so-called forbidden transitions, increase with high powers of Z, and at very high charge states, they may be of the same strength as the electric dipole transitions.

3.7 Atomic

Structure

and the Periodic

Table

In the previous section we dealt with the regular evolution of atomic structure along an isoelectronic sequence, where the nuclear charge varies and the number of electrons is constant. Here we take a more general look at the atomic structure in different regions of the periodic table, and compare atoms having the same charge state, mostly neutral atoms.

3.7.1

Trends Along the Periodic Table

The building-up principle of the periodic table, based on the Pauli principle and the central field approximation, explains the periodicity of atomic

3.7 Atomic

Structure and the Periodic Table

®

(eV) lonization energy

Fig. 3.9. Ionization energies of neutral atoms.

properties. These periodically changing properties include the energy level structure and the spectra, so we can study general trends of the structure by comparing elements having the same position in different periods (i.e. in the same vertical column of the table), known as homologous elements. Examples include the rare gases, the alkali metals and the alkaline earths. The plot of ionization energies of the neutral elements in Fig. 3.9 shows the well-known fact that the rare gases with filled s*p® shells have high ionization energies, whereas those of the alkali metals with a single s electron in the outer shell are low. It also reveals that the ionization energies of the rare gases and — to a smaller extent — the alkali metals decrease with increasing Z. This effect can be attributed to a higher electron density close to the nucleus as a result of the higher nuclear charge. The screening by the inner electrons is therefore more efficient, and so the penetrating s and p electrons in the outer

shell are less tightly bound, leading to lower ionization energies (JF). The same effect is evident when comparing the homologous elements immediately to the left of the rare gas peaks, where the ground configuration is formed

by s and p electrons: F (IE = 17.4 eV), Cl (13.0), Br (11.8), I (10.5). The d electrons in the outer shell of the transition elements on the other hand are less penetrating and therefore less affected by the increased screening close

to the nucleus, and the ionization energies of homologous 3d (Z = 21-28), 4d (Z = 39-46) and 5d (Z = 71-78) elements are quite similar. The decreased ionization energies with increasing Z imply a smaller energy range for the system, so that transitions between levels involve smaller energy changes and the spectrum is shifted towards longer wavelengths. For example, the wavelengths of the rare gas resonance lines increase from 58.4 nm in He to 179 nm in Rn. In the alkali metals the corresponding change of the resonance doublet is smaller, from 670.776 and 670.791

nm for the two

lines in Li to 852.1 and 894.3 nm in Cs. The drastic change of the doublet splitting in the alkali resonance lines from Li to Cs illustrates the overall increase of the spm-—orbit energy through the periodic table. As another example, the 2s?2p ?P doublet splitting in B I is 15 cm~!, while the splitting in the corresponding 6s6p ?P term in TI I is 7793 cm~!. This increased spin-orbit energy is caused by the larger rate of

76

3. Atomic Structure and Atomic Spectra

change of the potential close to a high Z nucleus: the spin-orbit integral Gn

contains the factor (1/7r)(dV/dr). As the electron-electron interaction stays

approximately constant, a gradual deviation from LS coupling towards jj coupling takes place, as will be discussed later in this section. 3.7.2 Regularities Within

Periods

Fig. 3.9 shows that the ionization energy increases rapidly with Z in that part of each period where the outer shell of p electrons is being filled, from B to Ne, Al to Ar, Ga to Kr, etc. This is caused by the penetration of a p electron into the core of the inner electrons where the screening of the nuclear charge is not complete. When the nuclear charge is increased by one unit and another p electron is added, the screening by the core is essentially the same. The mutual screening by the p electrons is not sufficient to compensate the increase in effective charge in the penetrating part of the orbits, and the ionization energy increases. The same increased binding is seen when the s subshells are filled in going from Li to Be, Na to Mg, etc. The changes in the periods where a d shell is built up are much smaller, as the d electrons are less penetrating, and the mutual screening by the d electrons compensate the increase in nuclear charge to a greater extent. Corresponding behaviour is seen in the 4f (Z = 57-70) and 5f (Z > 89) periods. The spin-orbit energy of equivalent electrons changes rapidly with the number of electrons within a period. The fine structure splittings of DS terms — the A factors of (2.48) — are combinations of the spin-orbit energies Cn) of the individual electrons. In the first half of a subshell the A factors are positive so that levels with the highest J have the highest energy; this is known as normal splitting. In the second half of a subshell the A factors are mostly negative, so that the terms are inverted. First-order theory predicts that the fine structure splitting should be zero when a subshell is half-filled. Observations confirm the predicted order of the fine structure levels, and the splittings are indeed found to be very small for half-filled subshells. Like the spin-orbit energy, the non-central electrostatic electron—-electron interaction increases smoothly along each period. This behaviour allows accurate predictions of the energy level structure within a period by interpolation or extrapolation of both the spin-orbit ¢, and the Slater F* and G* integrals.

3.7.3 Different

Coupling Types

Up to now, we have described energy level structure in terms of the LS coupling scheme, for which the non-central electrostatic interaction is much larger than the spin-orbit interaction. As explained, this is a good approximation at the beginning of the periodic table. Formally the LS coupling scheme for a two-electron configuration can be written as [(dile)L, (sis2) S| J °

(3.12)

3.7 Atomic Structure and the Periodic Table Cl

Si|

Ge |

Sn |

Hats

Pb |

== (3/212), — (3/2,1/2)

cm’) (10° W

©

| [

i]

(1/2,1/2),

0

25/2) 2p3s

3p4s

4p5s

5p6s

yal

6p7s

Fig. 3.10. The change from LS to 77 coupling in the homologous spectra C I to Pb I. The level positions are shown relative to the J = 0 level for each element. LS and jj designations are shown for the C I and Pb I levels. The splitting of the upper group in Pb I is larger than expected because of a perturbation.

This notation signifies a vectorial addition of the orbital angular momentum vectors to the resulting vector D, the spin angular momenta to S, and finally addition of L and S to J. The levels appear in groups — terms — described by the quantum numbers L and S, with small distances between the levels with different J. Using the Slater and spin-orbit integrals we can express the conditions for LS coupling as F k G¥ >> G1, Co. The general form for the LS

notation is 2°+!L,. The opposite extreme case is j7j7 coupling, where the spin-orbit interaction for each electron is larger than the non-central electrostatic interaction between the electrons. It can be written as

)41, (lo82) Ja] J [181

(3.13)

The result is still the same number of levels with the same values of J, but

they are now assembled in groups described by j; and jg. The condition for jj coupling is ¢y,¢2 >> F*,G*. The notation is (j1, j2) J. The spin-orbit interaction increases through the periodic table, but it is only in the heaviest elements that it reaches sufficiently large values for clear cases of 77 coupling to appear. As examples of LS and jj structures Fig. 3.10 shows the observed levels of the configurations np(n+1)s in the homologous spectra C I to Pb I. The LS and jj coupling schemes presume that the spin-orbit interaction of both electrons is either much smaller or much larger than the non-central

78

3. Atomic Structure and Atomic Spectra

63.90

i

df= 125

3 —

63.88

63.86

.

(10° cm’) Energy 63.84

ho

63.82 |

60.36 | 60.34

oO

——

60.32

6

60.30

a

Jp=2.5

ae)

60.28

K= 1/2

3/2

5/2

7/2

9/2

11/2

Fig. 3.11. The configuration 4d°(?D)4f in Pd I. The value of J is shown to the left of each level. The distance between the two groups of levels designated by the J. values 2.5 and 1.5 is approximately equal to the spin-orbit splitting 3539 cm~* of the parent term, and the K values are obtained by the vectorial addition of lz = 3. The pair splittings in this example are affected by a small perturbation.

electrostatic interaction. However, if the spin-orbit interactions of the electrons are widely different in magnitude, one being larger and the other smaller than the electrostatic interaction, we can describe a different coupling scheme

by [((1151) 91, l2)K, so] .

(3.14)

The conditions for this coupling are ¢; > F*, G* > Gy. The notation is

(ji [A] yThis scheme represents the vectorial addition of the orbital and spin angular momenta of one electron with large spin-orbit interaction to give a total angular momentum Jj, for this electron. By virtue of the electrostatic interaction between the electrons, 7; is added to the orbital angular momen-

tum ly of the other electron to give the intermediate angular momentum K. Finally, the weak spin-orbit interaction of the second electron is taken care of by adding K and sy to make J. The quantum number Kk can take the values

jitle, jitl,—1,...|j1 —lo|. As s2 = 1/2, J can take only the values K +1/2 and kK — 1/2. The energy levels thus appear in pairs with small splittings. For this reason the coupling scheme has previously been called pair coupling. In analogy with LS and jj, where the schemes are named according to the intermediate quantum numbers, the pair coupling is now called 7K coupling. In a system with more than two electrons 1, s; and j; are replaced by L, S

3.8 Nuclear Effects

79

and J of the parent term. The J of the parent is often called J, where c stands for core, and the coupling scheme represents the addition of an electron to this core. As the spin-orbit interaction decreases rapidly with increasing 1, 7K coupling generally gives a good description of the structure for l2 > 3 if the spin— orbit interaction of the parent is large. For atoms with a p* or d* parent, this coupling is often appropriate towards the end of a period for configurations with smaller lz because of the increase of the spin-orbit interaction of the parent along the period, as described in Sect. 3.7.2. As an example of 7K

coupling the configuration of 4d°(?D)4f in Pd I is shown in Fig. 3.11.

3.8 Nuclear

Effects

The only property of the nucleus that has been considered so far is its charge Z. However, there are other properties of the nucleus that affect the electronic energy levels and hence the spectrum. The effects are generally small, but in certain cases they are not negligible. Apart from the need to account for them in interpreting the spectrum, they form an important link between atomic and nuclear physics. The phenomena to be discussed are of two kinds:

— hyperfine structure (hfs) caused by interactions involving the nuclear spin, giving rise to splittings of energy levels of an atom or ion.

— isotope shift (IS) caused by differences in the nuclear mass and/or volume between isotopes (nuclei of the same element with different numbers of neutrons), giving shifts of energy levels of one isotope relative to those of another isotope of the same atomic species.

3.8.1

Hyperfine Structure

Magnetic

Dipole Interaction.

The intrinsic spins of the individual pro-

tons and neutrons in the nucleus may give rise to a resultant angular momen-

tum J, called the nuclear spin. This spin has the magnitude |I| = \/J(J + 1)h, where the quantum number J has a small integer or half-integer value according to whether the total number of protons and neutrons (the mass number) is even or odd. Associated with this spin is a magnetic moment fy, and the relation between them can be expressed in a way analogous to the relation for electron spin eq. (2.19):

ip

(3.15)

where m, is the proton mass. The g; factor arises from the addition of the magnetic moments of the protons and the neutrons, and it can have small

80

3. Atomic Structure and Atomic Spectra

positive or negative values. gr factors between -4.3 and +5.3 have been observed. It is the interaction of this magnetic moment with the magnetic field at the nucleus due to the electrons that gives a small additional contribution to the atomic energy levels. The magnetic field is produced by the combined orbital motion and spins of the electrons, so it is proportional to and in the same direction as the total angular momentum J. The quantized orientation of the nuclear magnetic moment in this field By yields the potential energy contribution to the Hamiltonian Hee

=

—pyr-

By

=C-1l-

J ..

(3.16)

py is smaller than pg, the magnetic moment of the electron, by the factor Me/Mp, or 1/1836, while Bz is of the same order of magnitude as the field

B_, responsible for the spin-orbit interaction eq. (2.22). The ratio of hyperfine to fine structure splittings is thus of the order 10~%. As a result, hyperfine splittings are only a few hundredths of a cm~! in the light elements, rising

to a few cm! in the heavy elements. The hyperfine structure interaction depends on the relative orientation of J and I, in the same way as the spin-orbit interaction depends on LZ and S. Just as J was defined by the vector addition of Z and S, so the total angular momentum F is defined by

(ae

(3.17)

The calculation of the perturbation energy given by (3.16) is analogous to the calculation of the spin-orbit energy, with the result

IN ake a ee)

Tee

ei soe

where A is called the hyperfine structure constant

(3.18) and F is the quantum

number related to the angular momentum in the usual way, ie. |F| = VF (F +1)h. The component of F' in the z direction is hMp. The quantum numbers can take the values

eee te) tlds ert ete Malt oR AlgeeieBi: A relation corresponding to Landé’s interval rule eq. (2.49) can obviously be

derived from (3.18). Although the same letter A is conventionally used for both fine and hyperfine structure splitting, it must be remembered that the hyperfine coupling constant is about three orders of magnitude smaller. Its actual value depends on gy; and on the electronic wave functions that determine the relation between By and J. It has its largest value for an unpaired s electron. Transitions between hyperfine structure levels can be treated in the same way as in Sect. 2.4. A new selection rule is obtained:

AP =(0ekl,

F=0=4F

=0 notallowed.

(3.19)

3.8 Nuclear Effects

81

Fig. 3.12. Hyperfine structure in Nb II.

Left

aaron 30 874

30 876

wed! WWW Wyre 33 572

33 574 (cm")

The statistical weight of a hyperfine level is 2F'+1, and the relative intensities of the hyperfine components are governed by sum rules corresponding to those of Sect. 2.4.5. The fact that the hyperfine interaction is so much smaller than other interactions means that the sum rules and the Landé interval rule generally provide a very good description of the hfs structure and the relative intensities. Figure 3.12 shows two “lines” in Nb II split into hfs components. The lower configuration, in both cases 4d°5s, is responsible for the hyperfine structure, while the splitting is negligible in the upper configuration, 4d%5p. The lower levels for the two lines are °P» (left) and °P;. The nuclear spin is

1 = 9/2, giving F = 13/2, 11/2, 9/2,7/2 and5/2 and f= 11/2, 9/2.and ¥/2 for the two levels. The intensities of the components are closely proportional to 2F' + 1, and the intervals follow the Landé rule.

Electric Quadrupole Interaction. Small but significant deviations from the Landé interval rule for hfs levels are observed in some spectra. In some cases they can be explained by mixing of states, in the same way as the deviations from LS' coupling discussed in Sect. 2.3.3, but in other cases they are attributable to the existence of a nuclear electric quadrupole moment Q in addition to the nuclear magnetic moment fy. The value of @ is a measure of the deviation from a spherical charge distribution in the nucleus, and the sign of Q depends on whether the nucleus is elongated along the I direction

(Q > 0) or flattened (Q < 0). The quadrupole interaction contributes to the hfs when @ and the gradient of the non-spherical electric field produced at the nucleus by the electrons are both non-zero.

This requires both J > 1 and J > 1. The electric

quadrupole shift can be expressed as

ABesB iE

J),

where the function f differs from the corresponding expression in (3.18). The quadrupole interaction therefore gives rise to a departure from the interval

82

3. Atomic Structure and Atomic Spectra

rule. The Q derived from observations provides information about the shape of the nucleus if the electron wave functions are known well enough. In free atoms departures from spherical symmetry are not large, so the quadrupole interaction constant B dipole constant A, and the electric as small corrections to the magnetic quadrupole contributions are much of molecular and crystalline electric 3.8.2

is generally smaller than the magnetic quadrupole effects usually show up only dipole effects. In molecules and solids the larger because of the strong asymmetry fields.

Isotope Structure

In the lighter elements it is the differences in mass that give rise to differential energy level shifts between isotopes, and hence to line shifts. This effect decreases rapidly with increasing mass through the periodic table. In the heavier elements the shifts are due to changes in the nuclear size when additional neutrons are added, the so-called volume shift. In the middle of the periodic table the two effects may be of comparable size.

Mass Shift. In Sect. 2.1 we introduced the reduced mass wp = meM/(me + M), making a small correction to the energy levels in one-electron systems to take account of the finite mass of the nucleus. In a many-electron system we must consider this correction in more detail. The kinetic energy term T in the Hamiltonian including the motion of the electrons and the nucleus is

1

peo

pe re ye

2M

2

(82)

where the first term is the kinetic energy of the nucleus moving relative to the centre of gravity, and the second term sums the kinetic energy of the electrons. In our previous treatment of many-electron systems we neglected the first term, assuming the nucleus to be “infinitely heavy”. This is justified when calculating absolute energy levels because the error introduced is small compared to errors from other approximations. However, the differences in the first term are important if energy levels in different istotopes are to be compared, and the full equation (3.20) must then be used. Conservation of linear momentum requires

N 1

=~

SUF: )

i=1 leading to le —

Cy 2M

a POS i=

- 2Me

il ah 2M

U

ere

S>p? eneta)

ay M oP

=

a

pj + Ses

(32 JF)

Using the definition of the reduced mass 4 = m_.M/(me + M), we can write this as

3.8 Nuclear Effects

(322)

Dj -

Le abe ae

83

My

In one-electron atoms the second term vanishes, and the kinetic energy is reduced to the form used in Sect. 2.1, with the energies expressed in terms of the Rydberg constant Rj, incorporating the reduced mass jz. Using eq. (2.10), we can derive the difference 6H in energy Py, for a level in an atom with nuclear mass M relative to the level E,, in an atom with an infinitely heavy nucleus:

oS

anh

Sar

ite n

ey

en

Be (

M

wm)

st

2

n

=e

m

M+me

;

The difference between the levels of two isotopes with masses called the normal mass shift, is then

AE

ee) M, and Mo,

5E(M;) — 6E(Mp) Loe

Me (a7 +m.

Me = Mot |

—Eoo

Me(Mz — Mi) > (Sn: M M2 Be)

5E in (3.23) is positive because B. < 0. All energy levels are therefore higher for mass M than for infinite mass. Similarly, (3.24) shows that the level of isotope M, is higher than that of M2 if M; < Mg. As it is rather easy to get confused with positive shifts of negative energy levels, it is probably easier to remember that a smaller nuclear mass gives a smaller Rydberg constant and hence a more compressed energy level system. Since the energy shift is proportional to the energy, it follows that the shift in the wavenumber of a transition between two energy levels is proportional to the wavenumber of that transition: AVo — (f+

Me(M2

=

M,)

M, M2

The line from the heavier isotope has the larger wavenumber and hence the shorter wavelength. For a given mass difference M2 —M, the isotope splitting obviously decreases approximately with 1/M ? and is largest for hydrogen and

deuterium, where AE/E = Aa/c is about 3 x 10~*. This amounts to 0.179 nm for Hy. In all atoms

and ions with more than one electron the second term of

(3.22) gives an additional mass-dependent shift, the so-called specific mass

shift. This shift depends in a complicated way on the correlation of the electron motions, and it is very difficult to calculate reliably. Generally the normal and the specific mass shifts are of comparable size, but the specific shift may have the opposite sign. When the shift is in the same direction as the normal shift, it is said to be positive. Because of the rapid decrease with increasing M, both effects can usually be ignored in the last third of the periodic table,

84

3. Atomic Structure and Atomic Spectra

Fig. 3.13. Ni II.

Isotope

shift

in

60

14521.0

14521.5

14522.0

14522.5

(cm)

but in the middle third the specific shift may be large enough to introduce significant uncertainties in the evaluation of the remaining shift, the volume shift. Isotope components of a Ni II line are shown in Fig. 3.13. The shift is dominated by the specific mass effect, which has the opposite sign to the normal effect so that the order of the isotopes is reversed. The intensity ratios of the components are equal to the abundance ratios of the isotopes: mass 58

(687%), 60 (26%), 61 (1%), 62 (4%), 64 (1%). The odd mass isotope, M = 61, is not seen because of its position between the more abundant isotopes. Moreover, it may be split up in hfs components.

60 and 62

Volume Shift. The volume or field shift is caused by the difference in nuclear volume for different isotopes. Adding neutrons alters the size and sometimes the shape of the nuclear charge distribution. The radial probability density curves in Fig. 2.4 show that s electrons have a non-negligible probability of penetrating into the nucleus. Assuming that the charge is distributed throughout the volume of the nucleus, the penetrating electron experiences a smaller positive charge inside the nucleus, and is less tightly bound the larger the nuclear volume. As this effect is usually small except for s electrons, significant shifts of spectral lines are to be expected only for transitions where the number of such electrons changes. The magnitude of the volume effect increases with the size of the nucleus. It is difficult to calculate because it depends on both the electronic and the nuclear charge distributions; however,

measurements of the relative spacing between different pairs of isotopes for the same transition, which have the same electronic factor, have contributed important data to nuclear models.

3.9 The Influence of External Fields

3.9 The Influence of External

85

Fields

The quantum numbers mj, ms, m;, Mr, Ms, Mj, have appeared in several places in this and the previous chapter. Each is related to the component of an angular momentum vector in the z direction. For example,

Jz =Myh.

(3.25)

The 2J + 1 states with the same J and different Mj 7 have the same energy and are therefore (2.J + 1)-fold degenerate. This is to be expected in isotropic space, where the direction of the z axis of the coordinate system is completely arbitrary. However, if one direction in space is singled out by means of a static electric or magnetic field, the states of different space quantization are no longer degenerate because of the interaction of the electric charge or the magnetic moment of the atom with the field. We will first treat the interaction with the magnetic field, the Zeeman effect, in a semiclassical way, and then present an outline of the theory for the interaction with the electric field, the Stark effect. 3.9.1

The

Zeeman

Effect

We discuss the Zeeman effect in a many-electron atom, assuming LS to be a good approximation. To evaluate the interaction between the field B and the atom, we need the total magnetic moment pj of the analogy with eq. (2.21) for a one-electron atom, the total magnetic associated with the electron spins can be written as

ls = shies ;

coupling magnetic atom. By moment

(3.26)

where g, is the gyromagnetic ratio of the electron, equal to 2 apart from the small QED correction, and jug is the Bohr magneton. Similarly, the magnetic moment associated with the orbital part of the angular momentum is

ES ae!

HB

pare

a

(3.27)

where gz both experimentally and theoretically is unity. As a result of the inequality between gz and gs, the two vectors J = LD+S and w= pr + ws do not point along the same line, as illustrated in Fig. 3.14. According to the vector model for the atom, the vectors L and S rapidly precess around the resultant J, causing fp and prs to precess around the same line. The components of zp and yg perpendicular to J then average to zero. Only the components in the J direction contribute to the resultant time-averaged magnetic moment yy. Alternatively, one can think of js as precessing round the direction of —J so that only the component py along this direction is effective. Therefore

ad Leth ein Tog Doel Lbier ee iene aa UB eines Taha

eT +28)

(3.28)

86

3. Atomic Structure and Atomic Spectra

Fig. 3.14. The vector model for an atom in a weak magnetic field.

where we have used (3.26) and (3.27) and inserted the values for gp and gs. The scalar products of (3.28) are evaluated through the relations

SoS)

De

es Aes

2d

(3.29)

ST

(3.30)

SiS) tel ts Se

Inserting these two scalar products in (3.28) we find HB

=

3J°

2

2

— E* + §*)

2

‘ S51

.

47 1s parallel to —J, so in analogy with (3.26) and (3.27) we can write it

[Bip =

=95

(232)

J;

which defines g7, the so-called Landé g factor. Equation (3.31) can be written as

Wy

LB

My —

|J|

J

(33? — L? +87) J.

.

———

2h |J|?

(3.33)

Comparing (3.32) and (3.33) gives an expression for the Landé g factor:

Bd i

=D?

54 oH a J(J+1)-L(L+1)+S(S54+1)

2F(J +1)

ree

The interaction between the magnetic moment and the magnetic causes {17 to precess slowly round B. The interaction energy is

ANT

ey gy ze L

(3.34) field

(3.35)

3.9 The Influence of External Fields

Fig. 3.15.

B

Zeeman

87

splitting of a

J = 3/2 level.

M; +3/2

My, tooo

+1/2

et/ 2) JO) 2 eee ee

>

ee GjlteB

a

|

23/25

eer

an

-3/2

As B defines the z axis, J-B = J,B. By (3.25) this is equal to MjhB, where M,=J,J—1,...—J. Therefore, AEs

=

gripupBMy

:

(3.36)

The magnetic field splits each level into 2J + 1 sublevels of equal separation gjtpB. This removes the last degeneracy in the wave function: each observed energy level corresponds to one and only one set of quantum numbers. As an example the splitting of a J = 3/2 level is depicted in Fig. 3.15. The selection rules for dipole radiation depend on the polarization of the radiation and on the direction of the radiation relative to the magnetic field. Viewed perpendicularly to the field, the o and 7 components of the radiation are defined as having the electric field vector polarized perpendicular and parallel, respectively, to B. The selection rules are

AM;

=

+1,

o. components;

AM;

=

0.,

7 components.

Gmyre)

Viewed along the B direction, the o components are circularly polarized and components do not appear at all. The predicted Zeeman patterns for the two different types of transition are displayed in Fig. 3.16. For the special case of a transition between singlet terms, for which S = 0, J = L and gy = 1, the separation of the magnetic sublevels is the same in both the upper and lower terms of the transition. The Zeeman pattern then consists of a single 7 component, AM, = 0, at the zero-field position, and a pair of g components, AM, = +1, at distances tupB. This pattern can be explained purely classically and was originally known as a ‘normal Zeeman triplet’. It was the observation of the so-called ‘anomalous Zeeman splitting’ that led to the first postulates of electron spin. The proliferation of

88

2

Pap0

3. Atomic Structure and Atomic Spectra +3/2

|

ee = SST = See

ee

ee

+1/2

ee

-1/2 -3/2

Qu= 4/3

2

S US

So

eee

+1/2

| ee il ae

gu= 2

-1/2

: root+ Le (Poms |eo | ae + ; 5 3 oO

oO

a

TT

oO

r

oO

zero field

+2 +1

:Ds

0

-1 Qy= 7/6

(tem

=

=2)

=H

+1

°P,

0

Qu= 3/2

=f

ae Le bell bedser a

er

on Yel

a ‘aejn

=

Gor or

Fig. 3.16. Examples of Zeeman splitting of spectral lines. The height of the component lines on the \ axis shows the relative intensities predicted by theory (not treated in this book). a+ and o~ refer to the helicity of the circular polarization of the radiation, i.e. the direction of rotation of the electric field vector.

components is the result of the different values of g; in the upper and lower levels. If LS coupling is not a good approximation, g cannot be evaluated from (3.34), but it may still be determined from measurements of Zeeman patterns via (3.36). Comparison between the measured gz value and that determined from (3.34) is in fact a good diagnostic of the type of coupling. As gj is not very different from unity, the order of magnitude of the splitting is determined by zgB, which is 0.046 cm>! for a field of 0.1 T

(1 kG). This is comparable with the Doppler linewidth for a light element in

a conventional light source (Sect 8.3).

3.9 The Influence of External Fields

89

In this treatment it has been assumed that the interaction with B is small compared to the spin-orbit interaction. This was implicit in the assumption that L and S precess rapidly around J, which precesses slowly around B. If the external magnetic field interaction is strong compared to the spin— orbit interaction, then Z and S are “decoupled” and precess directly around B. This is known as the Paschen—Back effect. It is of little importance as far as fine structure levels of low configurations are concerned, because the fields required are so large. The analogous effect in hyperfine structure, where the intrinsic coupling is so much weaker, is important. It forms the basis for a number of magnetic resonance and level-crossing techniques involving transitions in the radio-frequency region of the spectrum, which are outside the scope of this book. The Paschen—Back effect can also be important for high levels, close to the ionization limit, where the spin-orbit interaction is small. 3.9.2

The

Stark Effect

The Stark effect differs from the Zeeman effect in that atoms have no permanent electric dipole moment to interact with an electric field. The effect cannot be treated in a simple semiclassical fashion. We shall outline here the perturbation treatment, with the main aim of establishing a base for the description of pressure broadening in Chap. 8. We start by disregarding hydrogen, which must be treated separately because of the | degeneracy.

Quadratic Stark Effect. The electric field may be either externally applied, or it may be a microscopic field produced by electrons or ions in a

plasma or a crystal. One can think of the electric field

& as distorting the

electron distribution in the atom so as to induce a dipole a€, where a is the polarizability. The interaction between this induced dipole and the field is then a€2, which is the reason for calling it the quadratic Stark effect. a may reasonably be expected to depend on the angular distribution of the electron charge, and thus on M;. The distributions represented by +M, and —M), have the same symmetry with respect to the z axis, which is defined

by the direction of &€, so the Stark interaction depends only on |My], and each sublevel except M7 = 0 remains doubly degenerate. In the perturbative treatment the perturbation operator is given by

Hiei f= Chee:

(3.38)

The first-order contribution from this operator is zero because (4; |z| qi) = 0, as can be deduced both from the absence of a permanent dipole moment and from the discussion in connection with the selection rule Al = +1 in Sect. 9.4.4. We therefore go on to the second-order perturbation [Appendix A.1.2,

(A.11) and A.12)], yielding the perturbed wave functions

v= vit > cede , k#i

(3.39)

90

3. Atomic Structure and Atomic Spectra

where eS

(Wk lez| Wi ) ¢

SSS

bE,— Ex

(3.40)

ws

The resulting energy perturbation is given by (he lez| Wi le AE, = Dae (bp |ezE| 4) =i nda eh,

The matrix

element

2

is seen to be identical with the z component

(3.41) of the

electric dipole line strength eq. (2.67) for a transition between states 7 and k. Hence, the rules that determine which states k are to be mixed with state 7

are identical with the selection rules for electric dipole transitions: Al = +1, AJ = 0,+1 and AM; = 0 (the AM; = +1 contributions vanish because the

electric field is along the z axis). From the denominator in (3.41) it can be seen that state 7 is pushed away from the perturbing state k. The electric field € causes the two states to repel each other by an amount proportional to €? and inversely proportional to their separation. Evaluation of the matrix elements shows that the repulsion depends on |M | and has a magnitude that increases with decreasing /,. Figure 3.17 shows the Stark splitting of the alkali resonance lines as an example of the quadratic Stark effect. The ground state is not much perturbed because the first excited configuration is a rather long way above it. Since the density of levels increases as the excitation increases, all levels tend to be shifted downwards, the shift increasing with the excitation. As a result most transitions show a ‘red shift’. The |4/;| dependence leads to an asymmetric splitting of the shifted line. Fields in the range 10°-10" V/m are usually required to produce observable Stark splitting — say a few tenths of acm~'. The splitting increases with the excitation because of the greater density of states. At values of the principal quantum number n so high that Stark splitting becomes comparable with the energy differences between states of different 1, the latter become effectively degenerate, the second-order perturbation treatment breaks down, and the Stark effect becomes hydrogenic, as discussed below. Another consequence of the | mixing induced by the field is the breaking of the electric dipole radiation selection rule Al = +1. For example, the admixture of p waves in the s-wave function allows the ‘forbidden’ transitions s-d and s-s to appear in the alkali spectra in the presence of an electric field. In effect, the mixing of wave functions of opposite parity means that 1 is no longer a ‘good’ quantum number.

Linear Stark Effect.

In hydrogen the Stark effect is much larger than in

other atoms, it is symmetric, and it is a linear function of the electric field strength. This is a consequence of the | degeneracy in hydrogen, i.e. the fact that all states of the same n have the same ener gy. The fine structure from relativistic effects is small enough to be ignored in this context. The

3.9 The Influence of External Fields

2

5

M

ak

a ede eae

J

+3/2 Iss

:

Sif

2 |

ae

M

Pie

ea gee Bee

gee

2

ie

|

J

ND

eee sa yy

Lex

re

TO:

“~~ zero field

91

TiO

er

tielt

Fig. 3.17. Stark splitting and shift of the alkali resonance lines.

perturbation theory for degenerate states must be used to avoid such obvious problems as a singularity in (3.40) when the denominator goes to zero. The treatment for degenerate states involves using linear combinations of all wave functions having the same eigenvalue as the zero-order functions so that states of different parity are mixed. This mixing introduces an asymmetry into the electron distribution, which is equivalent to a permanent dipole moment. Consequently the first-order perturbation energy evaluated with these new functions no longer vanishes. The matrix elements are of the form (Wik |Z| Vix’), Where 7 shows that the functions have the same eigenvalue, while k and k’ indicate that they have different parities. The perturbation

energy is then proportional to |&|. The result of a full treatment for hydrogen is that a state of given n splits symmetrically into n — 1 sublevels, whose separation is proportional to |& and to n. The line pattern is therefore also symmetric around the field-free

line. Figure 3.18 shows the effect for the Balmer a line, n = 2 6 n = 3. The splitting is a few cm~! in a field of order 10° V/m, which is very much larger than the quadratic effect shown by other atoms. The Stark splitting of hydrogen is of great importance in plasma spectroscopy because hydrogen is a common constituent of most plasmas and its Stark pattern can be calculated with reliability. Theoretical and experimental line shapes from the fields due to the charged particles in the plasma can be accurately compared, and as a result electron densities can be deduced from measured hydrogen line profiles.

92

3. Atomic Structure and Atomic Spectra

m) 0

|

+1

n=o

of the Balmer

Hz, line, showing symmetric splitting.

OreZ

HT See

Fig. 3.18. Linear Stark effect

:

z a

nona amen WS

0

| ZETO field

nee eee i

Further Reading A classical description of atomic structure has been published by Edlén: — Edlén, B., Atomic Spectra, in Handbuch der Physik, Vol.

XX VII, ed. by S.

Fligge (Springer, Berlin, Heidelberg, 1964). A survey of more recent work on highly charged ions has also been presented by Edlén:

— Edlén, B., Energy Structure of Highly Ionized Atoms, in Progress in Atomic Spectroscopy, Part D, ed. by H.J. Beyer and H. Kleinpoppen (Plenum, New York, 1987). Cowan’s book on atomic theory contains two chapters with general descriptions of energy level structure in simple and complex configurations, as well as a chapter called “External Fields and Nuclear Effects”: — Cowan, R.D., The Theory of Atomic Structure and Atomic Spectra (Uni-

versity of California Press, Berkeley, 1981). An introduction to atomic theory and a survey of atomic structure including hyperfine and isotope stucture is given in a book by Kuhn, now out of print: — Kuhn, H.G., Atomic Spectra (Longman, London, 1969).

4. The Analysis of Atomic

Spectra

As stated already in the Ritz combination principle (Sect. 1.4), a spectral line is emitted or absorbed in a transition between two energy levels. What we actually observe is therefore the energy differences between levels. The analysis of an atomic spectrum, known as term analysis, aims at identifying the upper and lower levels of the observed spectral lines and deriving the energies of these levels by means of the measured energy differences. In a simple one-electron system, such as an alkali atom or an alkali-like ion, this is a rather easy problem to solve, provided the observed spectrum is of reasonable quality as regards resolution and wavenumber accuracy. On the other hand, the analysis of a complex spectrum with thousands of lines, emitted in transitions between hundreds of levels, is a formidable task, requiring suitable tools and a certain amount of experience. A pertinent description of the difficulties in this kind of work has been given by Cowan in the first chapter

of his book [2]: The problem is rather like that of trying to put the pieces of a complicated jigsaw puzzle together when the pieces never fit exactly, some pieces fit spuriously, some critical pieces are missing, and there are pieces present that belong to one or more entirely different puzzles. Carefully planned and performed observations are crucial in order to reduce these difficulties, as discussed in the first section of this chapter. To guide the analysis, some knowledge of the expected energy level structure is needed. This knowledge may be based on purely theoretical predictions, but often empirical or semiempirical relations are very useful — sometimes even necessary. Such predictions are treated in Sect. 4.2. Section 4.3 describes the analysis procedure in some detail. We discuss in Sect. 4.4 methods for determination of the ionization energy from the observed energy levels. Finally, we give some information about sources for the most recent information about atomic energy level systems and tables of spectral lines.

4.1 Observations Most of the knowledge about atomic energy level systems is obtained through the analysis of emission spectra, and the discussion in this section will there-

94

4. The Analysis of Atomic Spectra

fore refer primarily to emission spectroscopy. Absorption spectroscopy is a useful tool for establishing very high energy levels both below and above the first ionization limit, in the latter case including autoionizing levels. Laser spectroscopic ultra-high resolution methods are used for studies of hyperfine or isotope structure in individual lines, as described in Chap. 14. The use of broad-band absorption spectroscopy and laser spectroscopy is generally limited to neutral (and to some extent singly ionized) atoms by purely practical difficulties. Emission spectroscopy, on the other hand, provides large numbers of transitions involving all parts of the energy level systems up to high n and l, for all charge states from neutral atoms to highly charged ions. The first step in a spectroscopic investigation is to find a suitable light source that emits spectral lines from as many levels as possible. The photon flux in each line should be high, as the signal-to-noise ratio sets the limit

for the wavenumber accuracy (Chap. 17). In order to reduce the number of “pieces present that belong to one or more entirely different puzzles” the source should contain as few impurity elements as possible. Even with only one element, several charge states may be present. It is then necessary to be able to distinguish between lines from the different charge states. This can be done in various ways, usually by comparing spectra recorded with different light source conditions. Spectroscopic light sources are further discussed in Chap. 15. The spectroscopic instrument must provide high spectral resolution. In simple spectra this is needed for resolving the small fine structure intervals in the light elements and — at least partially — hyperfine and isotope structures of heavier elements. In complex spectra it is also needed to reveal spurious close coincidences of lines from entirely different transitions. High spectral resolution, high dispersion, careful calibration and a high signal-to-noise ratio together give a high wavenumber accuracy. According to the quotation above, “the pieces never fit exactly, some pieces fit spuriously”. The higher the wavenumber accuracy, the better the “pieces” (the lines) will fit. The tolerances can then be narrowed, reducing the probability that lines may “fit spuriously”. Moreover, high absolute wavenumber accuracy is becoming increasingly important for applications other than term analysis, for example, for unambiguous line identifications in stellar spectra observed with astronomical spectrometers of high dispersion. Finally, if energy levels are to be established over a wide energy range, spectral lines must be observed over a wide wavenumber range. Simultaneous recording of large spectral regions is therefore a great advantage. Spectroscopic instruments are discussed in some detail in Chaps. 12 and 13.

4.2 Predictions As stated in Chaps. 2 and 3, the theory of atomic structure and atomic spectra can explain in principle all observed features. In general, theoretical pre-

4.2

fs i

Predictions

95

Fig. 4.1. The ns and np series of Mg II. The two

3s

fine structure levels of the

1.37

np series are shown.

1.36

4s

6s

5s

0.86

0.85

0

10

20

30

40

= T (cm')

dictions are not accurate enough for direct identification of spectral lines, although they are needed for the analysis of a complex spectrum. In fact the basic work on simple and moderately complex systems was performed by empirical methods, before useful theoretical predictions were available. These methods, further developed in an interplay with the theory, are still important and can give the first clue to the analysis of an atomic spectrum. We describe a few of the large number of such methods; they are usually referred to as semiempirical because they are based on theoretically derived relations. We then give a brief review of how theoretical methods can be used.

4.2.1

Semiempirical

Relations

Term Series. The first regularities discovered in atomic spectra were the series in one-electron systems, described in Sect. 3.1.2. As soon as the beginning of a series has been established it can be described by a series formula, which allows accurate predictions of new levels by extrapolation to higher n values. The Ritz formula, which is a relation between the quantum defect and the term value, was discussed

in Sect. 3.1.2, and an example was shown

in

Table 3.1. The quantum defect as a function of the term value for two series in Mg II is shown in Fig. 4.1. Configurations with / > 3 in a one-electron system can (with some exceptions) be described in one series formula, and extrapolation to higher values of both n and J is thus possible. The formula is based on a theoretical description of the polarization of the atomic core in the field of the outer electron

96

4. The Analysis of Atomic Spectra

| n=7

0.00

6

5

‘ 0.01

4

1 0.02

rs 0.03

0.04

|

= (1)

Fig. 4.2. Convergence diagram showing the 3snp ?P levels of Mg I plotted versus Gane:

[5]. This simple polarization formula can be used for easy and rapid predictions of levels with high n and / with an accuracy that is typically 0.01-0.001 cm! in neutral or singly ionized atoms. The formula has been shown to be applicable also in more complex cases than one-electron systems. The spin-orbit fine structure splitting can also be extrapolated in a series. According to (3.8), the splitting varies as n~° in a hydrogen-like system. In other systems the splitting is found to be proportional to (n*)~°, where n*

is defined in (3.5): ie

jae :

(4.1)

T;, is the term value of the level — that is, its distance from the ionization limit. The evolution of the fine structure intervals in a series can be displayed in a so-called convergence diagram, in which the relative level positions are

plotted as functions of (n*)~?. Figure 4.2 shows the 3snp 3P levels of Mg I

as an example. Such a diagram can be used for predicting fine structure splittings, but it is also useful for showing up any irregularities caused by configuration interaction.

Promotion Energies and Binding Energies. In complex systems the series structure is not obvious, and in many cases only levels belonging to a few — perhaps very complicated — configurations are observed. Approximate positions of configurations can, however, still be predicted by means of promotion and binding energies.

4.2

Predictions

97

The promotion energy is the energy needed to promote an electron from one n,/ orbit to another, n’,/’. We take an example from the Ca I term system shown in Fig.3.3. The difference between the 4s? ground configuration and the excited configuration 4s4p is the promotion of one electron from 4s to 4p, but this is also the difference between the 3d4s and the 3d4p configurations. The energy differences in the two cases are found to be approximately equal, showing that the 4s + 4p promotion energy is approximately independent of the other electrons present. This independence of the one-electron promotion energy is found to hold generally, and it can therefore be used for predictions. The relation is particularly useful when different elements are compared. Another example where the approximation is good can be seen in Fig. 3.7, showing Cr XIII. Here the 3p—3d promotion energy is similar in the 3snl and 3pnl systems, and so also are the 3d—4/ promotion energies in all three of the systems 3snl, 3pnl and 3dnl. As a result the spectral lines of the 3d—-4f transition in all three systems appear within a narrow spectral range. In Sect. 3.3.2 we pointed out that the parent structure in a system with multiple parent terms is closely reproduced by the subconfiguration structure, as can be seen in Figs. 3.4 and 3.5. This means that the distance between a configuration and the corresponding parent term, which is the binding energy as calculated from that parent, is approximately the same for all parents. If a subconfiguration based on one parent is known, the approximate position of the corresponding subconfiguration from another parent can be predicted. Isoelectronic and Isoionic Relations. The regular evolution of the structure along an isoelectronic sequence, discussed in Sect. 3.6 and displayed in Fig. 3.8, can be used for predicting unknown levels, but with some reservations. Isoelectronic comparisons are generally not useful for the lowest charge states, where the relative positions of configurations change as the energies depend increasingly on n rather than on /, see for example Fig. 3.7. When the order between two configurations changes, the energy shifts caused by configuration interaction change direction, and the regular behaviour of the sequence is therefore upset. Similar disturbances may occur at higher charge states when plunging configurations move down through the term system, as discussed in Sect. 3.6. In regions of the periodic table where the binding energies are similar, that is, when the d or f shells are being filled (Sect. 3.7.2), so-called isoionic comparisons between the same charge state of different elements may be used for predictions. As an example, the regular behaviour of the term value of the lowest term of the configurations 3d”5s and 3d*~'4s5s in the singly ionized

iron group elements is shown in Fig. 4.3 [7]. Obviously the positions of the missing terms can be predicted within a few hundred em! in such a diagram. 4.2.2

Ab initio Theoretical

Predictions

Theoretical methods for the calculation of atomic structure are constantly being developed and improved, and today highly accurate calculations are made

98

she

4. The Analysis of Atomic Spectra

;

Fig.

4.3.

Term

of the

% =

3d*~'4s5s terms of the singly ionized iron group

;

elements. The term value is in this case the distance from the lowest term

56|

of the parent

8

lowest

value

pe

tion 3d* and

al |

3d*’5s

and

configura3d*~14s

re-

spectively.

52|

oo

3d‘5s

48;

7

46

SC

neil

Ve

Cherian

Penn

COme

Nin

Cle

nea

for atoms and ions with up to 4-5 electrons. Calculations of this kind are performed by specialists in the field, using powerful computers and large computer codes, often specially adapted for a particular problem. These methods are beyond the scope of this book; some literature references are given at the end of this chapter. There are less sophisticated computational methods and computer codes that can be used for routine work over the whole range of atoms and ions in the periodic table. These methods generally involve two stages, of which the first is an ab initio calculation, made without any assumptions based on prior knowledge of the predicted structure. In the second stage the accuracy of the predictions is improved by using adjustable parameters to fit the theoretical energy levels to the experimental levels. The most commonly used program package for such ab initio and parametric calculations has been designed by R.D. Cowan. It was originally written to run on a large computer system, but versions are now available that have been adapted for systems ranging from supercomputers to PCs. The input for the ab initio part consists of the atomic number and the charge state of the atom or ion and a list of the configurations to be included in the calculation. Only the electrons outside the last, filled rare gas shell have to be specified. The first part of the chain of programs uses the Hartree-Fock method to calculate the radial wave functions of all the electrons. The second part uses the wave functions to calculate the average energies of the configurations and the Slater and spin-orbit integrals. In the third program the

4.2

Predictions

99

energy eigenvalues are derived by means of the matrix method:! coefficients fe and gx (2.44) for the energy integrals are derived, and matrix elements are formed by adding the contributions from the different integrals (2.44)

and (2.48). The second-order perturbations, (2.51) and (2.52), are automatically included in this method, appearing as non-diagonal matrix elements. One such matrix is created for each J. The problem is solved numerically by transforming the matrix to a diagonal form, where the diagonal elements are the energy eigenvalues. If both odd and even configurations are included in the calculation, the electric dipole moments for all allowed transitions are derived in the second step. The third step will then include calculation of wavenumbers, wavelengths and transition probabilities of all electric dipole transitions. A comparison between observations and the calculated structure shows generally good qualitative agreement. However, a detailed scrutiny reveals serious deficiencies that reduce the usefulness of the primary results for predicting energy levels. An understanding of the reasons for the discrepancies indicates how some simple adjustments can improve the results. In general the calculated distances between the LS terms within a configuration are too large. This can be understood as an effect of configuration interaction from an infinite number of higher, distant configurations. All terms are pushed down by this interaction, the higher terms in a configuration being more affected than the lower because they are closer to the perturbers. (This effect is analogous to the ‘red shift’ in the Stark effect described in Sect. 3.9.2.) As the initial calculation makes the distances between the electrostatic terms too large, the discrepancy can be reduced by decreasing the electrostatic integrals by 5-20%. This task can be automatically performed as an option in the second program of the Cowan code. The calculation can then no longer be considered as purely ab initio, but should rather be described as semiempirical. Discrepancies are also observed that are due to errors in the calculated distances between configurations belonging to different systems, such as nd*n'l and nd‘*-))(n + 1)sn’l. In such cases it is possible to correct the error by adjusting the values of the average energy of the configurations. An approximate value for the correction may be derived by means of the promotion energy or the binding energy discussed in the previous section. 4.2.3

Parametric

Predictions

As soon as a few levels are known in a configuration, the predictions can be improved by means of the parametric method. In the Cowan chain of programs this is performed in the fourth and last program. The configuration average energies, the electrostatic Slater integrals (including configuration 1 The so-called Slater-Condon

matrix method

The reader is referred to Cowan’s book [2].

is beyond the scope of this book.

100

4. The Analysis of Atomic Spectra

interaction), and the spin-orbit integrals can all be treated as parameters that are adjusted to give the best agreement between the observed and the calculated energy levels. The adjusted parameters are then used in a new calculation, which is expected to provide improved predictions for the unknown levels. The fitting procedure has to be performed with much care, and it requires a good deal of experience for a successful result. The number of free parameters must be restricted to a value well below the number of observed levels. The fitted parameters should be in reasonable agreement with the ab initio values, with deviations for the electrostatic parameters within the limits given by the scaling factors discussed above. The spin-orbit parameters should show very small deviations from the ab initio values. The parametric calculation is an iterative process, in which the predictions made with fitted parameters lead to new identifications, and these in turn are used to generate improved parameters, and so on. In moderately complex cases with neutral or singly ionized atoms the fitting produces an rms deviation between the observed and calculated energy levels of typically 5-50 cm7?.

4.3 Term Analysis in Practice We are now ready to describe step by step how a term analysis of a complex spectrum can be performed, and we do so by referring to the flow chart in Fig. 4.4. Most of the steps are performed by means of computer programs, indicated by shaded frames in the flow chart. The observations, as discussed in the first section of this chapter, should have produced a line list of wavenumbers and intensities. Intensities on a calibrated scale are not necessary — even rather rough estimates may be sufficient. All known impurity and carrier gas lines should be identified. If possible the unknown lines should be labelled according to the most probable charge state. The spectral region covered by the list depends on the purpose of the analysis. For a complete analysis of the term system of a neutral atom it should ideally cover the region 170 — 5000 nm. For higher charge states the region is shifted to higher energies. The first step is to identify all lines from transitions between previously known levels. This is done by means of a simple computer code using the known energy levels as the input. The code calculates all possible energy differences within the spectral range of the observations, with the restriction that the two fundamental selection rules for electric dipole transitions must be obeyed: change of parity and AJ = 0,+1 (not 0 — 0). The differences are sorted and written to a list. This list is compared with the primary list of observed lines, and, when a coincidence within a given tolerance is found, the identification is written to the list of observations. The tolerance is determined by the uncertainties both in the observations and in the energies

4.3 Term Analysis in Practice

101

All obs. lines

{

Tables

-—|

Identify lines |»— "Impurity" lines TS Primary line list y

|Predict lines

| Known levels

| Fit levels

jf Identify lines

=

'

|

+——Identified lines

|Unidenttt. lines

!

’

! Improved levels#—] Search levels |

Predict levels

ij Analysis

ae

i

J —

New levels

=—

Fit parameters

Fig. 4.4. Flow chart showing the main steps of term analysis. The shaded frames indicate computer programs, while the single frames contain input or output lists of the programs.

of the previously known levels. The line list is then split into two lists, one of which contains the identified lines and the other the remaining, unidentified lines. The energies of the known levels can now be improved by a least squares fit that minimizes the differences between the observed wavenumbers and those derived from the energy levels. The input to the fitting program consists of the identified lines and the known levels. The next step is to search for new levels and to identify new lines. The input for the search routine consists of the unidentified lines and the subset of the improved, previously known levels that has the opposite parity to the levels for which we are searching. This is where the predictions discussed in the previous section are used — we must know what to look for and have an idea about where to look for it. The basic principle of the level search is illustrated in the left hand part of Fig. 4.5. Suppose the unknown level is predicted to appear in the region between FE) and £2 and that it is expected to have strong transitions to the levels at A and B. The search is performed by looking for two lines separated

102

4. The Analysis of Atomic Spectra

es

Fig. 4.5. The basic principles of a levels search. The left hand side of the figure shows a level in the interval between F£; and Eo, established by means of two transitions to the known levels A and B. To the right a search is performed on a system of four known levels and four lines. Three to determine

of the lines are

oom

seen

/

marked with a solid line.

the

new

level

I

H af j!

=

i!

jt Hil

B— '!

A

by the wavenumber difference B — A in the wavenumber region between E, — B and E2 — A. A more general case is shown to the right in the figure. In principle the wavenumbers of all the unidentified lines are added to and subtracted from all the known energy levels having the opposite parity to the level that is being searched for. All the sums or differences with values between FE and EF» are stored and sorted. If several of these values match to within a given tolerance, we may have found a new level. In a complex system with many levels and in a spectral region with a high line density, the probability of spurious coincidences is high unless the accuracy both of the observed lines and of the previously known levels is good enough to specify a narrow matching tolerance. An example of a printout of the result of a computerized level search is shown in Fig. 4.6; one real level and several spurious coincidences appear in this example. The next step can be considered as the heart of the matter — the analysis. A proposed new level must be confirmed by careful checking of the lines by which it is established. The lines must of course obey the selection rules for parity and J, and their relative intensities should be in reasonable agreement with theoretical predictions or empirical rules, taking into account the appropriate coupling scheme. The position of the new level should be compared with the theoretically or empirically predicted position, and an explanation should be sought for any deviation that is larger than expected. The procedure is now continued in an iterative manner as shown in the lower half of Fig. 4.4: the whole level system is recalculated to provide improved level values for the next search, more

new

levels are found, and so

on. When a number of new levels are found, they are used for improving the theoretical predictions by the parametric method.

4.4 Determination of the Ionization Energy KNOWN LABEL

LEVEL J

ENERGY

WAVENUMBER

LINE INTENSITY

LEVEL

6d (oxeh.

164 4.0 SHethy alait)

SMS eNS SPSS 63792.446

-

SKS (OMy) 3484.130

5d yeh 5d

3F4 shgsl 364

4.0 Aly 4.0

55025 a5 63792.446 54811.482

soies: +

5283 .096 3484.011 5496 .956

38 10 101

60308.431 60308.435 60308 .438

5d Cues

3'P2> 2-0 DSieorG

54820.803 54947.871

+ +

5488.544 5361.476

419 26

60309 .347 60309 .347

xeh Sa SSZes Bd 5d Saupe Sd

hei 2510) 3 F380 rane 0 303) 3-0 3F4 4,0 D2 e250 3 P2 42.30

63896-01589 = 65): SS01223 oe 85,0415 7p 54947.871 + 5o0ZSe aon 54998.665 + + 54820.803

Eisele yaZee 52985505 PSRs) ABEL) bseZ2e999 52855309 §312.206 5490.068

Pig| 127 22 124 224 243 502

60310.867 60310.870 60310.870 60310.870 60310.870 60310.871 60310.871

53746.763 63 Sis 3585.246

33 40 14

60310.908 60310.912 603210.912

bse goDag ..0 sS2 220 Sa 6a 9 3D1l92 -10

6564.145 Sa 57 422 62896.158

+ Beet =

Be) §)

NEW

103

60308.316 60308.316

7p J=3

Fig. 4.6. Printout from a computerized level search in Pd I. A 7p J = 3 level is found, while the rest of the groups are formed by spurious coincidences. Simplified level designations like those shown in the figure are generally used in the practical term analysis work, and subscripts and superscripts are not used in the Printouts formats for the computer codes. For example, 6d 1G4 in the figure means 6d AEM

When no more levels can be found and the majority of the lines in the original list are identified, the analysis is finished. If important levels are still missing, new experimental material may be needed. When the analysis is considered to be complete, a new parametric calculation including all the levels gives a final set of fitted parameters. These parameters are used for a calculation providing level designations in different coupling schemes. ‘Transition probabilities may also be calculated by using the fitted parameters.

4.4 Determination of the Ionization Energy The energy levels derived from observed spectra form a self-consistent set relative to (usually) the ground level, but they are not directly related to the ionization limit. The observed structure can, however, be used for deriving the ionization energy, which is needed in many applications in astrophysics and plasma physics. The series of energy levels in a simple, one-electron system converge towards a single ionization limit. In complex atoms with multiple parent levels or parent terms, there are multiple series systems converging towards the different parents. If a sufficient number of members of a series is known, the limit can in principle be determined through extrapolation. Provided that the positions of the series members are known relative to the ground state, the

104

4. The Analysis of Atomic Spectra

ionization energy can be derived. If the series converges towards the lowest parent level, the series limit gives directly the ionization energy. If it converges towards a higher level or a higher parent term, the ionization energy is obtained by subtracting the corresponding energy interval from the value of the series limit. This interval is usually known from the analysis of the parent ion.

The accuracy of the extrapolation is increased if a series can be followed up to high values of the principal quantum number n. For neutral atoms this is best done in absorption, preferably by laser spectroscopy, but limits can be determined for all stages of ionization with good accuracy from a smaller number of series members established by means of term analysis of an emission spectrum. All limit determinations are based on a series formula such as the Ritz formula (3.5), or the polarization formula mentioned in Sect. 4.2.1. The quantum defect in the Ritz formula can be expressed as a series, (3.7), which we rewrite here as

Ri,”

== 1,

aye Wiim i

Se

Wr

ag B(Wiim a W,) =F (Wim

= W,)? =a

ae

(4.2)

where Wjim is the position of the ionization limit. By choosing an approximate starting value, Wim can be determined either by iteration or by a standard fitting procedure. For high series members the quadratic and higher terms are not significant and the series is linear, as illustrated in Fig. 4.1. An approximate value for the limit can thus be found graphically by plotting n—n* versus Wim — W,, for high series members, varying Wi; until a straight line

is obtained. If a series is perturbed, it may be difficult to determine the limit in this way. Perturbations may arise from configuration interaction with a term from a displaced system (Sect.3.1.3) or from an accidental coincidence with a term of a series converging to a different parent (Sect.3.3). Even when the perturbation is small, the series may be useless for a limit determination by means of a simple series formula, although it may be possible to use quantum defect theory, QDT [6], to obtain the limit. Another kind of perturbation of a series may occur in atoms with more than one electron outside the closed shells, where several series of LS terms

converge to the same parent. The fine structure splitting in the LS terms is determined by a combination of the spin-orbit energies of the different electrons (Sect. 2.3.2), and as the n value of only one electron is changing along the series, the spin-orbit splitting may decrease more slowly than (n*)~%. The 3snp series in Mg I (Fig. 4.2) and the 3pns series in Si I are examples of completely different variation of the fine structure. In the 3snp series the splitting is determined by the np electron, as the spin-orbit energy of an s electron is zero, and the splitting therefore varies as (n*)~3, as seen in

4.5 Databases: Spectral Lines and Energy Levels

105

Fig. 4.2. In the 3pns series on the other hand, the 3p electron determines the splitting, which should stay constant along the series. The electrostatic interaction that determines the distance between the 3pns 'P and ?P terms does, however, decrease with (n*)~*. The distance between the J = 1 levels of the 'P and °P terms thus decreases faster than the spin-orbit interaction, the resulting perturbation shifts the levels, and the series cannot be described by the simple Ritz formula. The shifts of two interacting levels are opposite in direction but equal in magnitude (2.51), and a value for the series limit can still be derived by applying the Ritz formula to the series of average values of the interacting levels. The series of levels with a unique J, in the present example J = 0 and 2, are uperturbed, and can be used directly for the limit determination. An experimental complication in emission spectroscopy is the Stark effect mixing described in Sect. 3.9.2, which may occur in plasma light sources. The levels with high n and / that are best suited for the limit determination are also those that are most sensitive to perturbations of this type.

4.5 Databases:

Spectral Lines and Energy Levels

Tables of spectral lines and atomic energy levels have been compiled for as long as spectroscopic wavelength measurements and term analyses have been performed. Extended and improved measurements and term analyses are continually produced and published, and it is difficult to know where to find the most reliable spectroscopic data. Some of the old, very comprehensive compilations of wavelengths and energy levels that have been extremely valuable in the past are still in use. However, much of the information in these tables is out of date. We give here some of the more recent compilations, which should be used in preference to the older data whenever possible. Detailed references are provided at the end of this chapter. A very useful, critical compilation of wavelengths of the stronger lines of neutral atoms and ions up to four stages of ionization for all elements was

published in 1980 by Reader et al. [8] at the National Institute of Standards

and Technology, NIST (former National Bureau of Standards, NBS), Washington, DC. The tables contain the wavelengths of 47000 lines, arranged by element and in a finding list. They do not identify the upper and lower levels of each transition, but a comprehensive bibliography providing references to the measurements and analyses of the spectra is included. The tables have also been published in the 60th and later editions of the CRC Handbook of

Chemistry and Physics [9]. A compilation of wavelengths of all observed lines below 200 nm of all

atoms and ions of the elements hydrogen through krypton was published in 1987 by Kelly [10]. These tables include upper and lower levels of the lines, as well as references to the original work.

106

4. The Analysis of Atomic Spectra

The classical work on energy levels is the “Atomic Energy Levels” (AEL),

Vol. I-III, by Moore [11], published 1949-1958. For many years these tables were indeed the best source for energy levels, as regards both energies and designations. Today almost all the data have been superseded by the results of new analyses with improved accuracy, revised designations, increased numbers of levels and extended range of ionization stages. Nevertheless, the AEL tables are still useful for a first rapid glimpse of the structure of an atom or ion. The AEL tables did not include the rare earth elements and the actinides. Tables of these elements that are still up to date were published later by

Martin et al. [12], and by Blaise and Wyart[13].

A compilation of the levels

of the iron group elements was published in 1985 by Sugar and Corliss [14]. Much new work on these elements has been performed more recently, leading to significant extensions and revisions. This new information is generally published in different scientific journals, but not yet included in any printed compilations. For many other elements of special interest for astrophysics and plasma physics, compilations of energy levels and wavelengths have been published by NIST. Rather than continue to print tables of forever-changing and increasing material, the obvious course is to exploit the techniques available today to make them available in an electronic database that is constantly updated. The drawback is that such a database, universally available on the Internet, can be created by anybody, by copying from other data bases. Several such data bases of wavelengths and energy levels already exist, and it is difficult for the user of spectroscopic data to find out which base contains the most recent and the most reliable data. As in the case of printed tables, critically evaluated data bases are created and maintained at NIST, and their use is recommended for experimental data. For spectra not yet included by NIST, information has to be sought in the scientific journals, for example by means of the INSPEC data base on publications, or asked for directly at the laboratories at universities and research institutes performing term analyses. Tables and data bases on transition probabilities and oscillator strengths are discussed in Chap. 16, in connection with the experimental techniques for measuring these quantities.

Further Reading A comprehensive survey of empirical and semiempirical methods in atomic spectroscopy can be found in

— Edlén, B., Atomic Spectra, in Handbuch der Physik, Vol. XXVII ed. by S. Fligge (Springer, Berlin, Heidelberg, 1964). Theoretical ab initio and parametric calculations are described, e.g., in

Further Reading

107

— Froese Fischer, C.F., Brage, T., and Jonsson, P., Computational Atomic Structure (Institute of Physics Publishing, Bristol, 1997). — Cowan, R.D., The Theory of Atomic Structure and Atomic Spectra (University of California Press, Berkeley, CA, 1981).

The following compilations of wavelengths and energy levels are mentioned in the text (The Natl. Stand. Ref. Data Series and J. Phys. Chem. Ref. Data series are the NIST compilations referrred to in the text): — Reader, J., Corliss, C.H., Wiese, G.L., and Martin, G.A., Wavelengths and Transition Probabilities for Atoms and Atomic Ions, Part I. Wavelengths. Part II. Transition Probabilities, Natl. Stand. Ref. Data Ser., Natl. Bur. Stand. (U.S.) 68 (1980). — Reader, J. and Corliss, C.H. (Eds.) Line Spectra of the Elelements, in CRC Handbook of Chemistry and Physics (CRC Press, Boca Raton, FL, 1979

and later editions). — Kelly, R.L., Atomic and Ionic Spectrum Lines below 2000 Angstroms: Hydrogen through Krypton, J. Phys. Chem. Ref. Data 16, Suppl. 1 (1987). — Moore, C.E., Atomic Energy Levels, Natl. Stand. Ref. Data Ser., Natl. Bur.

Stand. (U.S.) 35, Vol. I-III (1971). (Reprint from 1949, 1952 and 1958). — Martin, W.C., Zalubas, R., and Hagan, L., Atomic Energy Levels — The Rare Earth Elements, Natl. Stand. Ref. Data Ser., Natl. Bur. Stand. (U.S.) 60 (1978). — Blaise, J. and Wyart, J.-F., Energy Levels and Atomic Spectra of Actinides. Tables Internationales de Constantes, Paris 1992.

— Sugar, J. and Corliss, C., Atomic energy levels of the iron period elements: potassium through nickel, J. Phys. Chem. Ref. Data 14, Suppl. 2 (1985). The NIST Atomic Spectroscopic Database, which is a critical compilation, can be found in the World Wide Web at

— http://physics.nist.gov — select the Physical Reference Data link. The Kurucz database contains Ritz wavelengths, i.e. wavelengths derived from experimentally established energy levels for both observed and predicted lines. It is found at the addresses

— http://cfa-www.harvard.edu/amp/data/amdata.html — http://www.pmp.uni-hannover.de/projekte/kurucz/sekur.html The Cowan computer code is available at different Web sites, e.g. a PC version at

— http://plasma-gate.weizmann.ac.il This site also contains other computer codes for calculation of atomic parameters and links to various atomic data bases.

ind ato el ie

;

Ae

» ban 2

wos f

«

mee

4

ap

mee

Wel

sie

Cops

-_

ee) eruy

®

eigw\s\

‘7

gut he

hae heal

ys

Oe

7

é

yO

7

a

|

AeW:

ea

#

ar

.

ct

Tiel

Tul

é

¢

sir

‘,

'

ig

wars

ai®

7

7

we

4

»

o

(48

fila.

ini)?

FG

asi

oad

meats) Ciba?aan etfs

cee

(Cl

p

®

mune

L@pPeitiy waligangee.,

hae

on

ld

r

fi

naeal

®

f

~~,

a

at

eee

’

'

.

Ved

‘ robin 7 Io

ee e


; wwe A

to Ghee

vy om of

:

=—* ‘eG

7

»

ie

4. “)

=e

Miodedat. y aw

ee :

Te

oat tye

o

ai)

ashe

aly a

&

—"

i>?

iT

bibs {

ad

Sey

ol

: a

}

vids

4 ~¢

a7)

ie

¢

‘

-

~ey

weprnity

aes

7

=

*

a

«|| zat

a

ymin Alow

Sale

= |

rector

roirageep

wantiedy

=

waa

6 Hgdag

:

i

iT

Teeel

© ate

Wietous’ i)

Si

Ney?

« ty!

ou Nevle

he

OO ye 098

‘?

eo & aden, _

7

5.

A Summary

of Molecular Structure

As two atoms approach one another the valence electrons on each begin to experience the attractive potential of the other nucleus as well as the repulsive force of the electrons of the other atom. There is a rearrangement of the electron distribution, which may result in a net attractive force between the atoms. Such a bound system containing more than one nucleus is called a molecule. In this chapter we show how the attractive force is created and how the electronic states of the individual atoms are related to the resulting molecular states. We will also discuss the dependence of the total energy of a molecule on the nuclear motion — vibration and rotation — as well as the electronic energy. Except for the last section, the treatment is confined to diatomic molecules, i.e. systems containing two nuclei. If the two nuclei are identical, the molecule is homonuclear; if they are different, then it is heteronuclear.

5.1 The Born—Oppenheimer

Approximation

Two isolated atoms have each three degrees of freedom, specified by motion along the three coordinate axes. When the two atoms are bound together in a molecule, three of these degrees of freedom describe the motion of the centre

of mass of the whole system. The other three are represented by relative motion: one by vibration along the internuclear axis and the other two by rotation about two axes perpendicular to the internuclear axis. (Rotation about the internuclear axis does not change the position of either nucleus. ) As we are dealing with a system of bound particles, the vibrational and rotational energies are quantized. Besides the electronic energy levels, the energy system of a molecule also comprises vibrational and rotational levels, and the wave functions contain vibrational and rotational as well as electronic quantum numbers. The Hamiltonian of a diatomic molecule, including all the electrostatic interactions, consists of several parts: H

=

Th

+7. -+ Vn. # Vee + Ven-

(5.1)

Here Ty and JT, represent the kinetic energy of the nuclei and the electrons respectively, Vyn is the potential energy of the electrostatic repulsion be-

110

5. A Summary of Molecular Structure

tween the two nuclei, Vee the electron—electron electron—nucleus attraction. Let the two nuclei and Mp, take the centre of mass as the origin, of the nuclei and the electrons Ra, Rp and r;.

repulsion energy, and Ven the A and B have the masses Ma

and call the position vectors We can then write the terms

of 5.1 in the following way: hi

=

Ty

ine eek 2m

-

:

3

UV

yes i.

Zr Zpe

haa

Vee

h

Sw

2Ma VRa

(og V;

‘

ATrE9 |Ra

=a

Vine

—

Tet

;

oe Aneg |ri — 1 5| Gre

=

"

Rp|

ess

;

awa ST,

eter

Zpe?

Pe ot Oe

ee

lr;

—

Rp

evs (

)

The Schrodinger equation with this Hamiltonian is impossible to solve, and simplifications are needed. The so-called Born—Oppenheimer approximation is based on the fact that the nuclei, because of their much greater mass, move much more slowly than the electrons. We can therefore solve the Schrodinger equation for different, fixed values of Ra and Rp, and then study the variation of the eigenvalues as a function of R = Ra — Rp. For each fixed position of A and B, Ty is equal to zero and Vyn is just a number.

The remaining electronic Hamiltonian consists of He

—

de + Vee + Von

;

(5.8)

The approximation is equivalent to a separation of the molecular wave function W into an electronic and a nuclear part:

WY =

(ri, R)xn(Ra, Rp) ,

(5.4)

where w. depends parametrically on R. The solution of the electronic equation Hae

= Eee

(5.5)

gives an energy /.(R) that forms part of the potential to be inserted into the remaining nuclear part of the equation to yield the total energy. A further simplification can be made by writing yy as a product of two functions, ~y,(2) representing changes in the relative positions of the nuclei (vibration along the internuclear axis in the case of a diatomic molecule) and w(@, p) representing rotation of the molecule. It is then possible to write

Y=

Petr

Exe

Bb

,

Bebe

(5.6)

5.2 Electronic Energy of Diatomic Molecules

10

The empirical justification for (5.6) is the condition E, >> Ey > E,. It will be seen that the vibrational quanta are usually a few thousand cm~!, or a few tenths of an eV, while rotational quanta are some two orders of magnitude smaller. Since the electronic energies should be of the same order of magnitude as the excitation energies of atoms, i.e. a few to several eV, the separation of the wave functions is usually justified. However, the Born—Oppenheimer approximation breaks down when different excited electron configurations are close to one another. The electronic wave functions y. are not those of the isolated atoms. The different approaches to calculating them are described in the next section. If

known with sufficient accuracy, they can be used to calculate E.(R), which combines

with the nuclear

repulsion

Vyn(R)

to form the potential curve

governing the nuclear vibrational motion (Sect. 5.3). The rotational energy, which is determined to a first approximation by the moments of inertia of the molecule about the relevant axes of rotation, depends only on the equilibrium internuclear separation, and not on the shape of the interatomic potential. Rotation is discussed in Sect. 5.4.

5.2 Electronic Energy of Diatomic Molecules Assuming the validity of the Born—Oppenheimer approximation, we need to solve Schrodinger’s equation for the electrons in the field of the two nuclei. The central field method is obviously inappropriate for two centres of force, but one can still use the atomic one-electron wave functions as starting points. There are two general approaches to the problem. The molecular orbital (MO) approach treats each electron outside closed shells as belonging to both nuclei, usually by writing its wave function as a linear combination of the two atomic orbitals (LCAO). There are necessarily two such combinations satisfying the requirement that the electron density be invariant against exchange of electrons; as shown below for the hydrogen molecular ion, one of these increases the electron density in the region between the nuclei, thus attracting the nuclei together, while the other concentrates the electron density outside the internuclear region and has the reverse effect. These combinations are known as bonding and antibonding orbitals respectively. The second approach is known as the valence bond or Heitler-London method. This starts with the separated atoms, for which the wave function is just the product of the individual wave functions, but incorporates an interchange of the two electrons that allows either electron to be near either nucleus, in rather the same way as the two-electron wave functions discussed in Sect. 2.2. Both these approaches need further modification. The simple LCAO treatment fails at large internuclear distance, and some fraction of antibonding orbital has to be added to the bonding orbital to represent the approach to two separate atoms. Similarly, the simple valence bond treatment does

Ht

5. A Summary of Molecular Structure

not adequately describe the sharing of the electrons at small internuclear distance, and this is remedied by adding some fraction of an ionic bond in which both electrons are localized around each nucleus. With these modifications, the two approaches can be made to yield essentially similar wave functions and binding energies. The MO treatment is perhaps conceptually simpler,

and we shall illustrate it by describing the H} molecular ion. 5.2.1

The Hydrogen

Molecular Ion, HS

The Hj ion consists of two protons and one electron and so is the simplest possible molecule. The electronic Hamilton operator of (5.5) for fixed values of Ra and Rp is

he

e?

e?

2m

Areg|r —Ra|

4reo|r —Rp|

Hy = —-=—V>1

as

e?

42¢0|Ra — Rp

(5.7)

Set r — Ra = ra, f — Rg = rp and |Ra — Rp| = A. The relations between the position vectors can be found from Fig. 5.1. We use the LCAO method to find approximate solutions to this equation. When the distance R between the two protons is large, the electron is localized at either A or B. In the ground state of the three-particle system the electron must be in a ls state, described by the one-electron atomic orbitals @),(T4)

or ¢1s(rg). At smaller distances, when a molecule is formed, the electron is shared between the protons, and the molecular orbital should be symmetric with respect to A and B. We can construct two normalized linear combinations symmetric about the midpoint between A and B:

g(r,R)= a (die(ra) + dre(ra)]

(5.8)

ay ee me /2 Tee Pare IR

(5.9)

and

Obviously wz is even and wy is odd under reflection in the midpoint. The commonly used subscripts g and u stand for “gerade” and “ungerade” (German for even and odd). Note that w, and y, are functions of r, the position

2" aN

H* A =

r a

Ra

O -

-

R

ey Fs B _

Fig. 5.1. Coordinates for H}.

5.2 Electronic Energy of Diatomic Molecules

Wel”

e

2

113

dul?

eo

e

Fig. 5.2. Wave functions and charge densities along the molecular axis for the 1s state in the molecular hydrogen ion.

of the electron, but they also depend on R, the distance between the protons

through the last term in the Hamiltonian (5.7). The wave functions W, and yy and the electronic charge densities wel?

and |w,|" along the molecular axis are shown in Fig. 5.2 (compare with the 1s wave function in Fig. 2.3). Two-dimensional pictures of the charge densities

can be seen in Fig. 5.3 (cf. Fig. 2.6). The charge density is zero at the midpoint of the ~, function, whereas the ~, function shows a greater probability of finding the electron between the protons. The next step is to apply the variational method to find the equilibrium internuclear distance R. This is done by evaluating the total energy EF, and FE, for trial functions y, and y, with different values of R and finding the

ates Meet)

aM a AN

9, .

Ky x) o oe

a ,% Cy, e,“e,ex) %Ses e sees’ | “ane soeanene casenee tetMe me afiS ox ‘iene Ssxy“ote i aeeXsee9,i o,a S, oiee

Ne

o,° e,

i oN%.

Fig. 5.3. Two-dimensional diagrams of the charge density of the 1s bonding (left) and antibonding states in H}. Note the difference in charge density in the region between the protons.

114

5. A Summary of Molecular Structure

(eV)

. 0

1

2

3

4

5

6

if

8

R 9

(au)

Fig. 5.4. The electronic potential energy curves for the bonding and antibonding

states in H}. The distance between the protons is given in atomic units, au (the Bohr radius). 1 au © 0.05 nm = 0.5 A.

minimum value. Figure 5.4 shows the result. The total energy is found to have a minimum at a certain nuclear distance Ro for the w, orbital. The system is stable at this distance, and the particles are bound together to a molecule. ~, is said to be a bonding orbital. Y, on the other hand gives no minimum at a finite distance R and is an antibonding orbital. We can understand the bonding or antibonding properties of the orbitals by studying Figs. 5.2 and 5.3. When there is an enhanced electronic charge density between the protons, the protons are attracted toward this negative charge. This attractive force increases with decreasing distance, and the potential energy decreases. At the same time the repulsion between the positive protons increases, and at a sufficiently small distance the repulsion dominates over the attraction and the potential energy rises again, giving a minimum for the W, orbital. For the ~, orbital on the other hand, there is a reduction of negative charge in the centre, and the repulsion between the protons is enhanced by the negative charge on the opposite sides of the protons. The distance Ro in Fig. 5.4 is the equilibrium distance between the protons. D is the dissociation energy, which is the energy needed to separate the system into a state where the distance between the protons is infinite, and the electron is localized at one of them. The Schrédinger equation with the Hamilton operator (5.7) can be solved numerically to any degree of accuracy, yielding the values Ro = 0.106 nm and D = 2.79 eV = 22500 cm™!. Like an atom, a molecule may be excited to electronic states with higher energy. The excited states may likewise be described by means of atomic

5.2 Electronic Energy of Diatomic Molecules

115

orbitals; for example, in Hj there is a bonding @95(ra) + ¢2s(rpB) and an antibonding @¢25(ra) — ¢2s(rB). However, an important difference between atoms and molecules must be briefly discussed before we proceed. 5.2.2

Symmetry

Properties of Molecular Orbitals

To construct molecular orbitals from atomic 2p orbitals, we must distinguish between the cases with m; = 0 and m; = +1. In contrast to an atom, the diatomic molecule has an internally defined direction — the molecular axis. This axis is taken as the z axis in our coordinate system, and the angular distribution of charge for the one-electron orbitals is shown in Figs. 2.5 and 2.6 with respect to this axis. The fact that m,; is an important quantum number for molecules can be understood from basic physical principles. When the force on the electron is central the angular momentum is conserved, i.e. it is a constant of motion, and so the quantum number J for the total angular momentum is fundamental in atomic physics. When an axial force is added, in the direction of the z axis, the angular momentum vector precesses around the z axis, as has already been seen in the Zeeman effect (Sect. 3.9.1). In this case the projection of the angular momentum is constant, and the quantum number describing this projection, m; in the one-electron case, is the good quantum number. The sign of mj; corresponds to the direction of the precession, and for symmetry reasons this cannot make any difference in a diatomic molecule.

A new quantum number A’ = |m,| is therefore defined. States with A > 0 are doubly degenerate because the direction of the precession cannot affect the energy. By analogy with the atomic designations, letters are used for the different values of A: 0,7, 6,y,..., for A =0,1,2,3,.... A molecular orbital is designated by the value of A and the atomic orbital used to construct it. The HJ orbitals from 1s should then be written as og 1s and o*ls, where the antibonding orbital is distinguished by the asterisk. A 2p atomic orbital can give six molecular orbitals g,2p, o}2p, T,2p and m*2p, where the 7 orbitals are doubly degenerate. With this nomenclature established, we can proceed to a discussion of molecules more complex than

Ht.

5.2.3

General

Structure

of Diatomic

Molecules

The filling of molecular orbitals is governed by the Pauli exclusion principle in the same way as that of atomic orbitals. Each orbital can contain two electrons, provided their spins are opposed. When the lowest energy bonding orbitals are filled, any additional valence electrons must go into antibonding orbitals. For example, the two 1s electrons from two’ hydrogen atoms both go into the bonding orbital gg1s with spins opposed, and a stable H2 molecule is formed. The two additional 1s electrons in the combination He+He have to

116

5. A Summary of Molecular Structure

go into the antibonding o*1s, and the net bonding energy is effectively zero. This is the reason why He is a monatomic gas. Three of the six possible 2p orbitals are bonding. The oxygen atom has four 2p valence electrons, so in the Og molecule six of the eight valence electrons go into the three bonding orbitals and two into antibonding orbitals; the net score is two filled bonding orbitals, constituting a “double bond”. For a homonuclear molecule the equilibrium electron distribution is necessarily symmetric with respect to the two nuclei. The centroids of the electron and the nuclear charge distributions must coincide at the midpoint between the nuclei, so the molecule has no permanent electric dipole moment. For a heteronuclear molecule the energy balance is generally favoured by a concentration of electron density near one nucleus. In NaCl, for example, the 3s electron contributed by the sodium atom and the unpaired 3p electron from the chlorine atom both go into a molecular orbital whose probability density is much greater near the chlorine atom than near the sodium atom. As a result the centroid of the electronic charge distribution is shifted towards the chlorine atom, and the molecule does possess a permanent dipole moment. To some extent an NaCl molecule may be regarded as if it were formed by two well-defined ions, Nat and Cl—, approximating to the valence-bond approach. However, the electron transfer is by no means complete: there is an infinite gradation possible between pure covalent (shared electrons) and pure ionic (localized electrons) bonding, and the electric dipole moment of any molecule is a measure of its degree of ionic bonding. As shown in the next chapter, the appearance of pure vibrational and rotational spectra depends on the existence of this dipole moment. If one or both of the atoms is in an excited state, further sets of bonding and antibonding orbitals can be constructed, giving potential curves of the same general shape as that of Fig. 5.4, but with the zero of energy (R = oo) shifted up by the atomic excitation energy. A few of the low states of Op» are shown in Fig.5.5. It can be seen that several stable molecular states may result from each pair of atomic states: this is the result of the different ways in which their angular momenta may be coupled. A brief description of this coupling is necessary to understand the labelling of these states and the selection rules governing transitions between them. 5.2.4 The Labelling of Electronic

States

The situation for a molecule is similar to that for an atom, in which a given electron configuration gives rise to several different energy levels characterized by resultant angular momentum vectors and their corresponding quantum numbers. In a molecule the quantum numbers A of the molecular orbitals are added to form a resultant quantum number A. States with A = 0,1,2,... are designated by X’, 17, A,.... The individual electron spins form a resultant S$ as in atoms, but for a rotationless molecule it is the axial component

designated by X’, that is coupled to A, forming a resultant Q:

of S,

5.2 Electronic Energy of Diatomic Molecules

117

10°(cm") r

60

40;

20;

Ot 0.0

;

if 0.1

: 0.2

Hh

: 0.3

; (nm)

Fig. 5.5. The lowest electronic states of Oo.

Q=|A+5|. ») takes 2+1 different values. A state with S = 0 has only one value of 3} (X’ = 0) and hence only one value of (2; such states are singlets and are written as 1D’, 'J7,..., for A = 0, 1,.... States with S = 1 are triplets; for

A=1and

SX’ = —1,0,+1 there are three values of 2 (0,1,2), and the states

are written as ?J/p, °JJ, and ?/7y. The designations are thus analogous to the

atomic LS notation. There are two complications for molecular states, both concerned with the symmetry of the electron orbital. The first affects homonuclear molecules only. As discussed in connection with the H} molecule, the electron density in a homonuclear molecule is always symmetric with respect to the midpoint between the two nuclei, but the wave function itself can be either symmetric — gerade — or antisymmetric — ungerade. The symmetry is shown by a subscript g or u. (Homonuclear in this context means nuclei with the same charge — for instance, '®O!8O as well as !®Og.) In the Hz molecule, for example, the

excited binding molecular states formed by the atoms H(1s) and H(2p) are Ty

ee

ny.

Secondly, one has to take note of the symmetry with respect to reflection in any plane through both nuclei. Again, the electron density |eb|* is necessarily symmetric with respect to such a plane, in heteronuclear as well as homonuclear molecules, but y can go to either +W or —y. As »’ states are non-degenerate, only one of these options is possible for any given state, and

118

5. A Summary of Molecular Structure

they are distinguished by labelling them 2’* or XY’. States with A 4 0 are doubly degenerate. By taking suitable linear combinations of the two basis functions for, say, a IT state, it is possible to construct IJ* and IT~ states, but both states have the same energy, just as the states +My and —M, of an atom have the same energy in an electric field (Sect. 3.9.2) — until the molecule starts rotating. The degeneracy in a rotating molecule is removed by the coupling between orbital and rotational angular momentum, resulting in a small splitting known as A doubling. In the ground state of many — especially homonuclear — molecules the electrons are paired off to make both A and S' zero, giving 1X’ states. Such molecules have no permanent magnetic dipole moment in their ground states. An important exception is O2, for which the ground state is ° jz , with spin equal to 1. Molecules and molecular radicals with an odd number of valence

electrons (e.g. NO, OH) form doublets (S = 1/2). One more aspect of the labelling of the electronic states of diatomic molecules should be mentioned: X is used to designate the ground state, while A, B, C, ... are used for excited states of the same multiplicity. States with multiplicity different to that of the ground state are labelled a, b,c, ... . The complete designations of the lowest states of O2 are shown in Fig. 5.5.

5.3 Vibrational Energy of Diatomic Molecules The energy curves of Figs. 5.4 and 5.5 showing the energy of the molecule in different electronic states as a function of the distance R between the nuclei are the potential energy curves governing the nuclear motion. The general shape of a bonding potential curve is shown in Fig. 5.6, with the zero point of the energy scale at the minimum of the curve. Classically, if the nuclei have zero kinetic energy they must be at rest relative to each other at separation Ro, while if they have kinetic energy Ey they can oscillate between R, and Ry. If the energy exceeds D, the molecule dissociates into two free atoms. In Sect. 5.2.1 it was shown how the basic principles of the LCAO method could be used for calculating the ln curve shown in Fig. 5.4. This and other methods are used in quantum chemistry for accurate calculations of complex molecular structures. In practice the potential curves are often approximated by empirical expressions, the simplest of them known as the Morse potential:

V = D.(l= eg)":

(5.10)

Here V is the energy referred to the minimum of the curve, D. is the depth of the well, «(= R— Ro) is the displacement of the nuclei from their equilibrium separation and (3 is a constant. For small displacements from the equilibrium position (small 2), this potential can be expanded as

= D.6?aga hee MP Sp stall eal e

(5.11)

5.3 Vibrational Energy of Diatomic Molecules

119

Fig. 5.6. Vibrational energy molecule. R,

of a diatomic

and Ro are the classical turn-

r E |

;

ing points for vibrational energy Ey.

In fact, any potential can be expanded in a similar way near its minimum, so we shall not lose much generality by using the Morse potential in this section. The leading term

We Dig?x?

(5:12)

represents a parabola. This corresponds to the classical potential for simple harmonic motion under a restoring force proportional to the displacement, F, = —kz, for which the potential energy is given by V = — [ F, dz = ka?/2. The classical vibrational frequency is

Y=

1 k —\/—.

oon

Vim

Inserting the molecular parameters

k = 2D.6? and the reduced mass pp =

Ma Mp/(Ma + Mg), we find Vo

Lb

S)2D;

aria

B.

(5.13)

The classical vibrational energy is related to the amplitude of the vibrations and may have any arbitrary value. Quantum mechanically we have to solve the Schrodinger equation for the variable x and the potential of (5.10). For the simple harmonic approximation with the potential (5.12) the equation is

Rea (-Sae

x 4

Dep

G

. ) Wy

=

Be

(5.14)

120

5. A Summary of Molecular Structure

Fig. 5.7. Potential energy curves and energy levels for a parabolic potential (dotted)

and the Morse potential (solid lines). Do is the dissociation energy. The zero-point energy and the spacing between the levels are greatly exaggerated for clarity — in a real case a larger number of levels appear in the region where the parabola and the Morse curve coincide.

This equation can be solved analytically, and the solutions may be found in any standard book on quantum mechanics. Physically acceptable solutions are possible only for values of FE’ given by Ey = AV (c+ 5).

(5,15)

where vp is the classical frequency given by (5.13), and the vibrational quantum number v takes the integer values 0, 1, 2,...The energy levels are evidently equally spaced. The parabolic potential curve and the corresponding energy levels are shown as broken lines in Fig. 5.7. For practical purposes the vibrational energy is usually expressed in wavenumbers and designated by G(v):

E,

Glo)\= eX

|

1

1077 = we (cao 5) ome),

where we, the vibrational constant,

yO] 2s

Oy

(5.16)

is in units of cm~!. we depends on the

molecule and on the particular electronic level considered, but it is usually in the range 300-3000 cm~!. An important difference between the quantum and the classical oscillator is the existence in the former of the so-called zero-point energy w./2 associated with the lowest level, v = 0. The dissociation energy of the molecule, Do, is measured from the lowest vibrational energy level and is therefore less than D., the depth of the well, by w./2; this is typically around 0.1 eV, a

5.4 Rotational Energy of Diatomic Molecules

ai

not insignificant difference in a total of 2-3 eV. For comparison, kT’ at room temperature is 0.025 eV, or 200 cm™!. When the Morse potential (5.10) is inserted in the Schrédinger equation, instead of just the first term in the expansion, the allowed energy levels are found to be 2 G@y=u4 («- 5)— LeWe (c+ 5) cms (5.17) where the second term corresponds to the deviation from the parabolic approximation and thus from the simple harmonic motion. ze, the so-called anharmonicity constant, is small and positive (~ 0.01). The vibrational levels therefore crowd together as v increases, but they remain finite in number. They do not converge towards a dissociation limit like atomic energy levels converging towards the ionization limit. The Morse potential curve and the corresponding energy levels are shown in Fig. 5.7 as solid lines. For large values of v, terms of yet higher order in v may be required to describe the energy correctly, since the Morse potential itself is not necessarily a good approximation for large displacements. Values of we and x¢ for different electronic states of molecules and molecular ions can be found in different tables and databases, e.g. the books by Huber and Herzberg and by Herzberg given in Further Reading.

5.4 Rotational Energy of Diatomic Molecules We start by disregarding the vibration and treating the molecule as a rigid dumbbell, consisting of masses M, and Mg joined by a rigid bar of length Ro, rotating about an axis through its centre of mass, Fig. 5.8. The moment of inertia I about the axis of rotation is given by J = R2, where yu is the reduced mass My Mp/(Ma + Mp). Classically, for an angular velocity w the system has the angular momentum L = Jw and energy Iw?/2 = L?/27. Quantum

im, 30B

J Loree bs

12B

+—

Nae

( Tie,

() maw R Tipe ae ee

Fig. 5.8.

6B 2B OB

383

2 — +—

1. 0

molecule

Rotation with

of a diatomic

angular

velocity

w

about an axis through the centre of mass. The rotational energy levels are shown to the right. B is the rotational constant and J is the rotational quantum number.

122

5. A Summary of Molecular Structure

mechanically, one must solve the Schrédinger equation for the nuclear coordinates to obtain the energy eigenvalues and the corresponding wave functions. The potential V in this equation is a function of R only, so for fixed R = Ro the angular equation is identical with that for the hydrogen atom, (2.6). The solutions are the spherical harmonics Y (9, y), and the allowed values of angular momentum are ,/J(J +1) h, with J =0,1,2,.... The energy eigenvalues are given by di (etaliie Ey ieee) =J +1) 2

5.18 (5.18)

which is identical with the classical relation between angular momentum and energy. For practical purposes the energy is expressed in wavenumbers and usually designated by the symbol F’:

Ey os ine Ele or é

8 ie a Bere i

where the rotational constant B is in cm!

ne h B= ~— x 10* =——, The ©

AncuR2

(5.19) and is given by

10”.

5.20

een

For two approximately equal nuclei of mass M, yp ~ M/2. We have previously seen that Ro ~ 2 x 10719 m. For M ~ 40 we have B ~ 1 cm“!?. If one nucleus is much lighter than the other, Ma < Mg say, then uw ~ Ma. A diatomic hydride such as HCl, NaH, etc., therefore has a particularly large B value, in the range 10-20 cm~!. For a given molecule the value of B varies from one electronic state to another because of the variation of Ro (cf.

Fig. 5.5). The rotational energy levels given by (5.19) are 0, 2B, 6B, 12B,..., as shown in Fig. 5.8. The spacing between levels increases by 2B at each step. The spacing is always much less than the room temperature energy of about 200 cm~!, in contrast with the spacing of vibrational levels. We must now consider corrections to the simple rigid rotator. The first correction arises from the centrifugal distortion, which tends to stretch the molecule at high J values. The corrected value of R is found by equating

centrifugal force to the restoring force D.B?(R— Ro)? of (5.12); it leads toa correction term in the rotational energy proportional to J?(J +1)?:

FD)

BIG)

= DI? (ie iF emt

(5.21)

The centrifugal distortion constant D (not to be confused with the dissociation energy!) depends on the attractive force between the two rotating masses, and it can therefore be expressed as a function of the vibrational constant We: 4B p47

moor We

5.5 Remarks on Polyatomic Molecules

123

With B~ 1 cm7! and we ~ 1000 cm, the correction factor for J ~ 30 is a few tenths of a percent. The second correction depends on the fact that the molecule is vibrating at the same time as it is rotating so that R is not in fact constant. As the vibrational frequency is orders of magnitude larger than the rotational frequency (as shown by the different orders of magnitude of the vibrational and rotational energies), the vibration can be taken into consideration by

using an average value for R. The fixed value 1/R? in (5.20) is replaced by the expectation value of 1/R? for the particular vibrational state:

(jg

h? x 1072

[4 ce fou wy( av (R)dR.

(5.22)

Evaluation of the integral shows that B can be represented by the expansion 1

i!

ree = Bo — ae(u + 5) + elu + a 42 )te cms

(5.23)

As might be expected from Fig. 5-7, the correction depends mainly on the increasing average distance between the nuclei at increasing vibrational energy. This gives a larger moment of inertia and smaller energy (5.20), so that a is positive. Higher terms in (5.23) can usually be neglected. The dependence of B on the vibrational quantum number v is recognized by writing it as By. Bo, representing the rotational constant at Ro, is never observed because of the zero-point energy. The rotational energy levels should now be written:

F,(J) = ByJ(J+1)-D, (J +1)’,

(5.24)

where B, and D, depend on the electronic as well as the vibrational state. So far in this section the angular momentum has been assumed to come

entirely from nuclear rotation, but if the electronic angular momentum is not zero, the total angular momentum is the resultant of all three contributions — rotational, spin and orbital. There are several different ways of coupling these angular momenta, according to the relative strengths of the interactions. We will return to this problem in the next chapter, when we study transitions between different electronic states.

5.5 Remarks on Polyatomic Molecules The structure and spectra of polyatomic molecules are beyond the scope of this book. Symmetry considerations are crucial, and group theory is therefore an indispensable tool. Nevertheless, it is possible to understand some of the characteristics of both structure and spectra from what has been said about diatomic molecules, and in this section we discuss some simple ideas of structure. For a full treatment, a book on molecular structure such as one of those listed at the end of this chapter, should be consulted.

124

5. A Summary of Molecular Structure

Starting with the electronic structure, the first question is whether the molecular orbitals should embrace the whole molecule or whether they should be localized, each involving in the simplest case just a pair of atoms. It turns out that in many simple molecules the localized picture is the most useful. It is possible to compile tables of bond energies in which, for example, the C-H and O-H bonds have almost the same energy regardless of the molecule of which they are a part. The concept of localized molecular orbitals leads to predicitions about the shape of simple molecules. The general principle is to maximize the binding energy by maximizing the overlap of the atomic wavefunctions that make up the molecular orbital. In H2O, for example, each H atom has one 1s electron to contribute to the bond, whereas the O atom has effectively four p electrons outside closed shells. The p electrons can be thought of as having three orthogonal spatial distributions along the x, y and z axes (cf. Fig. 2.5). Let us take, say, the p, electrons to be paired (having opposite spins); then the maximum overlap with the H atom wavefunctions is achieved by siting the H atoms in the xy plane on the x and y axes respectively. The bond angle is therefore expected to be close to 90° (Fig. 5.9). It is actually slightly larger (104°), partly because of the Coulomb repulsion between the two H nuclei. In a similar way, the ammonia molecule, NHs3, has the shape of a right pyramid with the three H atoms located along the x,y and z axes bonding respectively with the pz, p, and p, orbitals of the three unpaired p electrons in N. This simple picture is, unsurprisingly, not always valid, and in particular it does not explain the carbon compounds. The ground state of the C atom is 1s?2s?2p”, suggesting that it should form two bonds at roughly right angles in the same way as O. However, C is normally tetravalent: the 2s and 2p electrons are close in energy, and promotion of one of the 2s electrons to 2p makes available four electrons for bonding. The problem that the 2s orbital is spherically symmetric while the 2p orbitals are directional is solved by mixing them, a process known as hybridization. For example, in the case of methane, CHy, linear combinations of the four orbitals can be constructed to form four new orbitals that are identical in shape and are directed at the diagonal corners of a cube that has the C atom at its centre. Each of these bonds with the 1s orbital of a H atom. This is known as sp® hybridization. It is not necessary for all of the p electrons to be involved in the hybridization: sp? and sp hybrids also occur. It is the second of these that accounts for the linear form of the COz molecule. If we take the molecular axis to be the x axis, we can use the 2s and 2p, orbitals of the C atom to form two hybrid orbitals directed along the + and — x axes. These bond with the 2pz orbitals of the two O atoms. Of the remaining eight electrons, C has a 2p, and a 2p,

and each of the O atoms has one (inactive) pair and one free p electron. If these free electrons are taken to be p, for one O atom and p, for the other, they can be combined with the corresponding p electrons of C to form the

5.5 Remarks on Polyatomic Molecules

O O “

Var

H

C S

O Z

125

Fig. 5.9. A bent (H20O) and a linear (CO2) triatomic molecule.

H

two additional C-O bonds. Symmetry requires that the roles of the py and p- electrons be interchangeable, so that the O=C=O molecule has full axial symmetry, Fig. 5.9. Unfortunately the structure of many common triatomic molecules cannot be predicted so easily. As will be seen in Sect.6.5, the form of the spectrum of the molecule — the frequencies of the vibrational modes and whether they are infrared or Raman active, in particular — can rule out certain types of structure and suggest others. The results are sometimes not what one might intuitively expect. For example, nitrous oxide, N2O, which has the same number of electrons as COsz, is likewise a linear molecule, but it is not symmetric,

having the form N-N-O rather than N-O-N. Nitrogen dioxide, NOz, has one more electron and is a bent molecule. Ozone, O3, has yet one more electron, and one might expect it to be either linear or in the form of an equilateral triangle, whereas in fact it is an isosceles triangle. Going to vibration and rotation of polyatomic molecules, we must consider the degrees of freedom available for the different modes of motion. A molecule consisting of n atoms has 3n degrees of freedom, one for each of the translational coordinates of each of the atoms. In the molecule three degrees of freedom are used to specify the translation of the centre of mass, leaving 3n — 3 for rotation and vibration. Let us take a look at triatomic molecules with 3x3 —3 = 6 internal degrees of freedom, and begin with the linear CO molecule, see Fig. 5.10. Rotation is possible around two axes, perpendicular to each other and to the molecular axis. Rotation around the molecular axis is not included, as it does not change the nuclear coordinates. Four degrees of freedom thus remain for vibration, one symmetric stretch, one antisymmetric stretch and two (degenerate) bending modes, Fig. 5.10. A bent, triatomic molecule like H2O, Fig. 5.11, can rotate around three

perpendicular axes. This leaves us with three possible modes of vibration: symmetric stretch, antisymmetric stretch and bend. It is possible to choose coordinates in such a way that each classical equation of vibrational motion involves one coordinate only. Quantummechanically the Schrédinger equation can then be separated into the same number of independent equations. Each equation describes a “normal mode” of vibration, which to a first approximation is simple harmonic with a characteristic frequency v,. The energy associated with each of these modes is separately quantized: 1 Ey, = hy, (0ae 5):

126

e

5. A Summary of Molecular Structure

eos

D) Oa

ey UNC ee py

d

pe ae

=

ae

O—-O— 0 ae)

as

Fig. 5.10. The six rotational and vibrational modes of a linear AX2 molecule. (a) Rotation about two perpendicular axes. (b) Symmetric stretch. (c) Antisymmetric stretch. (d) Bend. In a second bending mode the motion is perpendicular to the plane shown here.

aS

Sos

'

=

Seat r an

Fig. 5.11. The six vibrational and rotational modes of a bent AX» molecule. (a) Rotation about three perpendicular axes. (b) Symmetric stretch. (c) Antisymmetric stretch. (d) Bend.

The actual motion of each nucleus is a complicated superposition of all normal modes, but in simple molecules it is often possible to regard each normal mode as either a stretching or a bending vibration. In more complex molecules the normal modes may describe vibrations either of the whole “molecular skeleton” or of particular groups of atoms such as OH, NH and CH3. Vibrations of the second kind are characteristic of the group and almost independent of the particular molecule to which it is attached. For the purpose of considering the rotational structure, it is necessary to classify the molecules according to their moments of inertia about three perpendicular axes, Iq, Ip and I, of which J, is conventionally taken to be the smallest and J, the largest. In a linear molecule, dg = Sd” ee eich is also the case for a diatomic molecule. The class known as symmetric top again has two equal moments of inertia, but the third is not zero; an example is ammonia, NH3, in which J, = I, but I, 4 0, where the a axis is the line

Further Reading

Le

joining the N atom to the centre of the triangular base of the pyramid formed by the three H atoms. A spherical top has all three moments of inertia equal; we have also seen an example of this structure in methane, CHy, where all the C-H bonds make equal angles with one another. Finally, the asymmetric top class has all three moments of inertia unequal; the majority of molecules belong to this class, including H20O. The task of calculating the full term structure is clearly far more difficult even for a triatomic molecule than for a diatomic, and the information on symmetry and basic shape that can be deduced from the spectra is absolutely essential. As an example of the complexity, the H2O molecule consists of three nuclei and ten electrons: just to write down the full Hamiltonian for this molecule requires a full sheet of A4 paper.

Further Reading The classical texts on molecular structure and molecular spectra are the three volumes by G. Herzberg, published by Van Nostrand, New York, and now out of print:

— Spectra of Diatomic Molecules (1959). — Infrared and Raman Spectra of Polyatomic Molecules (1947).

— Electronic Spectra and Electronic Structure of Polyatomic Molecules (1966). A fourth volume in the same series lists the spectroscopic constants for diatomic molecules: — Huber, K.P. and Herzberg, G., Constants of Diatomic Molecules (Van Nos-

trand, New York, 1979). A modern treatment of molecular spectroscopy with an introduction to group theory is found in the following books:

— Bernath, P.F., Spectra of Atoms and Molecules (Oxford University Press, New York, 1995). — Steinfeld, J.I., Molecules and Radiation, 2nd ed. (MIT Press, Cambridge, MA, 1985). — Hollas, J.M., Modern Spectroscopy, 2nd ed. (Wiley, Chichester, 1992). A rather clear description of molecular bonding is given by — Coulson, C.A., Valence, 3rd ed. (Oxford University Press, London, 1979).

An introductory given, e.g., by

quantum-mechanical

treatment

of molecular

structure is

— Bransden, B.H. and Joachain, C.J., Physics of Atoms and Molecules (Long-

man, London, 1983).

ociedslaetinaswi Seed

es nolsi k ED

thahed alteUi n) iilianGnn! anger ama lh in eolarm plow =e

wily yo

=a

es

eavet© +i

|

4

+

ere

¢ -

oye

Cha

fr Wi

naisire es

sete

2 ae eraaAe

GRRE neye 47s hot. Gino s. ei @: dN, UT fivd Comiaethy a

Gruiphels jo OT hal a7 ima

teasinti

;

Saree

ee

ee Pe

tL tow. Agi set sapere age

named

nee i th pertheoiwe ce hth ereegand Me

te

owed OA tee be Ma @:

Z

|

avlboss i

or

piel

Gane

tata

a wie?

Cubrise Quay =yrt Ge 4

48

Kew"

ov =

olin

la

bowhalia ny Nindegel?

=e me

aun dia alaMd, caine ~ WEE: edly) r

kaj tebe

ap

qiwreradli le Wiese wuaich. *

@ eee)

S&S a Geeley

= Peng 0.08 wren gull ow (TPUIGD. ae es A Seer & et, @aatecag foes

9 2

=)

Pee

eae

¢

eed

i ofee

ng

4

Saxe

2m, ge

pe

p>

:

wndsa wiitto malt spiced es oneasen =

bax

—_

Fer

po

ae)

lo

7

7i5

-

is

se

7A

Ss, .

:

.

:

me

3

-

lect

50

Vine

| Vem,

WOES

:

.

Lg

ate

(0g

« 7

ifs 7

: 1

2

voamas

ve

IViyeed

a ‘.

’

eva ore

© Any

| aieae

Top.

p

:

u

) Ai Aicges

‘Wh,

oledia@nmeeet .% el Midve «s/Dagn

Chee ip ihe iia

%

one

fee ee ae i. qulnies i ae aa 2

ey

ay

ny hacer

om

7 -

—_ y

’

a.

7. Emission and Absorption of Radiation

The rules governing radiative transitions in atomic and molecular spectra were discussed in Chaps. 2 and 6, respectively. The purpose of this chapter and the next one is to show how the theoretical expressions are related to the usual descriptions of ‘line strengths’ and to actual measurements of emission and absorption for different conditions in the emitting or absorbing gas. Whereas Part I (Chaps. 2-6) was mainly concerned with the ‘x axis’— fre-

quency, wavelength, wavenumber — Part II (Chaps. 7-10) brings in the y axis — radiometric flux, intensity, absorption — and it is useful to start with some definitions.

7.1 Definitions of Light Flux The quantitative description of optical radiation has been plagued by changes of terminology, by a wide variety of units, and by the ambiguity of words such as intensity which have both a general and a particular meaning. In this section we aim to assemble the minimum necessary set of definitions and to show the relations between them, using the currently accepted terminology [15,16].

If radiant power W (watts) falls on an area A (Fig. 7.1a), the flux density at the surface is given by

I=W/A-

(7.1) |

ee

solid angle dQ

power W

\

\

|0

LA

dW

eo

area A

area A

I=W/A

a)

power

Ly

=adWKA cosé dQ)

b)

Fig. 7.1. The definition of (a) flux density (irradiance) J and (b) radiance L.

162

7. Emission and Absorption of Radiation

This quantity is called irradiance. It is measured in units of W m~? and may apply to a radiating surface, to the power crossing an imaginary surface in space, or to the power falling on a real surface. We denote it here by J rather than the ‘preferred’ FE to avoid confusion with energy or electric field. I tells us nothing about the angular spread of the incident power. Consider now the power dW incident at an angle @ to the normal to the surface and contained within a solid angle dQ (Fig. 7.1b). The radiance L is defined as the power per unit projected area per unit solid angle: 8,

eS

6

dw

(yen

“Acos6 dQ ”

This quantity was formerly called brightness, and its units are W m~? sr —] Again, it may apply to a source or to a real or imaginary detecting surface. I is obtained from L by integrating over the relevant solid angle:

I at ts =a

q 4A

2

hte cosy dl -

(7.3)

If the radiation is isotropic, which means that Lg has a constant independent of 6, then using d{2 = sin6 dé dé, Qn

a2

es L| d@ cos @sin@ dé = an ¢=0 6=0

value L

i

f sin@d(sin@) =a7L. 0

(7.4)

(The spherical polar coordinate system is that of Fig. 2.1 with the normal as

the z axis.) The familiar word intensity should properly be restricted to the power per unit solid angle radiated by a source, which makes it a property of the source rather than of the radiating surface. For a small radiating surface of area A we can regard the intensity as the product L.A (W sr~!), but this is strictly true only for a point source, and intensity is not a useful quantity for discussing the interaction between radiation and matter. The adoption of two such similar names as radiance and irradiance for two rather easily confused quantities appears to the occasional user to be a curious and unnecessary complication. In what follows we shall use radiance but avoid irradiance wherever possible. In fact, radiance is the more useful of these quantities in the context of optical systems because it has the same value throughout the system so long as power is not lost from the beam by absorption or scattering. Constant radiance is a consequence of constant throughput, defined as the product of the cross-sectional area A of the light beam and the solid angle delimiting it. Figure 7.2 shows this for a simple lens system, though it can be proven quite generally. The linear magnification

here is u'/v, so A’/A = (v'/v)?, whereas Q'/Q = (v/v')?. Therefore throughput = AQ = A’).

(7.5)

From (7.2), with A and A’ normal to the optic axis, we have W = LAQ, and hence, if W is conserved,

7.1 Definitions of Light Flux

163

Fig. 7.2. Conservation of throughput: AQ = A‘Q’.

L=L.

(7.6)

It should be noted in passing that a further string of names and units attaches to these three basic quantities in measurements of visible light flux. For example, radiance becomes luminance and is measured in lumens m~? sr~! or candela m~?. There are 680 lm per watt at 555 nm, the peak sensitivity of the human eye. These units are not used in this book. If one considers radiation from a thermodynamic rather than a radiometric starting point, the basic quantity is energy density, p, measured in J m~°. The energy density associated with an electric field is given by p = D- €/2, where € and D are the electric field and displacement, respectively. The magnetic field of the electromagnetic wave contributes an equal amount of energy, so for an unpolarized electromagnetic wave in vacuum

p=enee Im”.

(7.7)

where & is the amplitude of the electric field vector and € is the permittivity of vacuum. If the light is plane polarized, a factor of 1/2 is required. To relate p to the radiometric quantities defined above, consider first a collimated beam of cross-sectional area A. All the energy in a layer of thickness ct (where c is the speed of light) will cross A in time t — i.e. Actp joules. The power, or flux W, crossing A is thus Acp watts, and the flux density (irradiance) is

(collimated radiation) J = pc.

(7.8)

T is related to the radiance in the direction of the beam by (7.3), L = dI/dQ = pc/AQ. The angular spread d2 of a collimated beam is never zero: even in a ‘perfect’ collimator there has to be a diffraction spread, so although L may be large in the direction of the beam, it is never infinite. At the other extreme, if the radiation is isotropic, the radiance in all directions is by definition the

same, and (7.8) becomes (isotropic radiation) L = pc/4r .

(7.9)

Finally, the subject matter of this book is usually concerned with the spectral distribution of all these quantities; this is specified by inserting the word ‘spectral’ into the name — spectral radiance, spectral energy density, etc. Another potential confusion arises at this point because a power distribution can be expressed either as power per unit wavelength interval or as power per

164

7. Emission and Absorption of Radiation

unit frequency interval, and both of these are used in practice. From v = c/A we have

dv} = = dal .

(7.10)

Thus, one unit wavelength interval (d\ = 1) is c/A? unit frequency intervals, and there is c/\? times as much power per unit wavelength interval as there is per unit frequency interval.

7.2 Blackbody Radiation For a solid or for a gas in thermodynamic equilibrium, the absolute temperature T’ can be defined thermodynamically. The spectral energy density of radiation in the gas or in a cavity in the solid is given by the Planck blackbody distribution, which can be expressed either as energy density per unit wavelength interval or as energy density per unit frequency interval: pla, T)

8mrhe 1 “48 ghepaat

=

8rhv3

p(v,T) =

a.

2 7 J me

wer

1

"nm

5

Bes:

Sa

ioe

;

p

(7.11)

where k is Boltzmann’s constant. The radiation in such a cavity is necessarily isotropic, so (7.9) can be used to give the radiance of a blackbody source, again on either a wavelength or a frequency scale:

PPA

[Dy

=

2he?

a3)

2hv C2

1

Gay

1 ehv/kT

it

W m

pee

=

’sr-'nm~?

W m?sr-1Hz~? -

(7.12)

It has to be assumed that the radiation escapes from a hole small enough not to disturb the thermodynamic equilibrium. Figure 7.3 shows the Planck distribution for two different temperatures. It does not matter whether we plot p or L as ordinate, one being simply proportional to the other, but it does matter whether the abscissa is \ or

v because, as seen at the end of the previous section, the ratio p(A)/p(v) is wavelength-dependent. At short wavelengths the p(v) curve (Fig. 7.3b) falls off more rapidly than the p(A) curve (Fig.7.3a). Similarly, for any given temperature, the maxima of the two curves occur at different wavelengths. The broken line in Fig. 7.3a shows the locus of the maximum for different values of T’. The shift to shorter wavelengths as T increases is expressed by Wien’s displacement law:

Ned = COnst: = 2288498 107? mae

(7.13)

7.2 Blackbody Radiation

200

400

600

800

165

1000 1200 1400 nm

a)

b)

Fig. 7.3. Spectral distribution of blackbody radiation (a) as a function of A and (b) as a fuction of v. The broken lines show the positions of the maxima at different temperatures. The curves representing the same temperature in the two figures have different shapes and are not mirror images of each other.

Am here refers to the maximum of the p(A) distribution. One can bring out the difference in the two distributions rather clearly by noting that A, is given to a very good approximation by hc/\y,kT = 5, whereas the maximum

in the frequency distribution p(v) is given by hyy,/kT = 3, which amounts to increasing A,, by a factor of 5/3 for a given T. Two approximations to the Planck function are often useful. The Rayleigh— Jeans formula is valid when hv < kT: 2kTv? Ler

c and the Wien approximation when hy > kT:

Ln

2hv*

Ts Se c

(7.14)

(7.45)

In the visible and ultraviolet regions Wien’s approximation is valid for most laboratory sources. For A < 600 nm the correction is within 5% for T < 7600

K, and the criterion for a 1% error is

Xam) < 3 x 10°/7F (K). A 5% accuracy for the Rayleigh—Jeans expression requires \ >14 jum even for T as high as 10 000 K, so its applicability really is limited to the far infrared. No equilibrium source can have greater radiance at any wavelength than a black body at the same temperature, a fact that may be expressed by writing

LOT) = Lo, 7),

(7.16)

where the emissivity € < 1. € is in general a function of wavelength, but, if it is almost constant, the source is known as a grey body.

166

7. Emission and Absorption of Radiation

The terms ‘brightness temperature’ and ‘colour temperature’ are sometimes used to specify the characteristics of near-blackbody radiation from a source of true temperature 7’. The brightness temperature Tj, at any wavelength is the temperature of a black body radiating the same energy at that wavelength, i.e.

DORT) = LCT

ret c(AN ie CAE

€ < 1 implies T, < T, and evidently, T, varies with wavelength unless € is constant. The colour temperature 7. of a source is the temperature of a black body having the same relative radiance as the source at two different wavelengths A; and Ag, so that LOG,

2) =

Leer

TAOS)

eb,

Tf)

Piet LE Oo, Pires LPOG)

by using (7.16). For a grey body (€; = €2) the colour temperature is identical to the true temperature. Lamp filaments are usually nearly grey, and this makes them satisfactory intensity standards for any application in which relative rather than absolute intensities are required.

7.3 The Emission and Absorption of Line Radiation At the opposite extreme to blackbody radiation are the line spectra generated by transitions of atoms or molecules between discrete energy levels. There is naturally a continuous gradation between these two extremes as the optical density of the radiating gas increases. This will be discussed in the next chapter; here we shall consider only isolated atoms, taking ‘atoms’ to include molecules with resolved rotational structure. The power radiated by an atom at frequency v;2 as a result of a radiative transition between two discrete states (2) and (1) is determined by the probability of finding the atom in the initial state (2) and the inherent probability of the particular transition 2 — 1. The first of these probabilities is determined, in the particular case of thermodynamic equilibrium, by the Boltzmann distribution of population; the second is designated by the Einstein probability coefficients. The latter are of course related to the transition rates discussed in terms of time-dependent perturbation theory in Sect. 2.4, and they can be calculated if the wave functions are sufficiently well known. They are also related to the other ways of describing line strengths: oscillator strengths or f values (much used in astrophysics) and absorption coefficients. The equations linking these alternative descriptions are derived in this chapter and summarized in Tables 7.1 to 7.4. We start with the Boltzmann distribution because although it is valid only in thermodynamic equilibrium it serves as a link between the emission and absorption line strengths.

7.5 The Einstein Probability Coefficients

7.4 Boltzmann

167

Distribution of Population

It can be shown by statistical mechanics that for a system in thermodynamic equilibrium at temperature 7 the ratio of the numbers of atoms occupying the two energy states Ey and F is given by Boltzmann’s equation:

M2 _ 92 .—(B2—B1)/kT mY

CiLt)

fu

where n is the number density in atoms per m®. The statistical weight g of a level is equal to its degeneracy. The degeneracy of a level with total angular momentum quantum number J is 2 +1 in the absence of an external

field (2.47), but in a magnetic field each J level is split into 2.7 + 1 sublevels, each with a different value of Mj, and unit statistical weight. Similarly, each rotational level of the simple vibrator—rotator diatomic molecule has a statistical weight of 2J + 1. Vibrational levels are non-degenerate. The conditions in which Boltzmann’s relation may be expected to hold are discussed in Chap. 9. Its relevance to line spectra is at first sight anomalous: a source of line spectra cannot be in complete thermodynamic equilibrium, because if the radiating gas is enclosed in a blackbody cavity, its spectrum is described by the Planck distribution and is necessarily continuous. It is shown in Chap.9 that the Boltzmann relation holds in a condition known as local thermodynamic equilibrium, commonly abbreviated to LTE, in which atomic states are populated and depopulated predominantly by collisions rather than by radiation. This condition can actually be achieved in real sources, both laboratory and astrophysical. However, in this chapter we shall use the Boltzmann distribution only to derive the relation between the Einstein probability coefficients.

7.5 The Einstein Probability Coefficients It is shown in Sect. 2.4 that the probability of a transition between two states induced by electromagnetic wave radiation can be calculated from timedependent perturbation theory if the wave functions of the two states are known. But this treatment says nothing about spontaneous emission, which we know to occur in the absence of resonant electromagnetic radiation and which we might expect to be related in some way to absorption. In fact, the relation between the probabilities for absorption and emission was first derived by Einstein from thermodynamical considerations before anything was known about wave functions. The derivation is based on a ‘two-level atom’

in a blackbody enclosure. Figure 7.4 shows the energy levels 2, and E2 of an atom (or molecule) with populations n; and ng, respectively. There are three possible radiative processes connecting the two levels, two obvious and one less so. First, an

168

7. Emission and Absorption of Radiation

ES

Ng

P42)

Wer E,

ny

Fig. 7.4. The Einstein probability coefficients. n1 and nz are the population densities of the levels of energy FE; and F2, and 42 = (E2 — F;)/h.

atom in level 2 may undergo spontaneously a transition to level 1 with emission of energy hi;2; the probability of this occurring in unit time is defined by the coefficient A2;. The number of such transitions per second is therefore A21n2. Secondly, in the presence of radiation of density p = p(v12) of the appropriate frequency, an atom in level 1 may jump to level 2 with absorption of energy hvj2; the probability of this process is defined by Bi2p s—!, and the number of transitions per second is B,2n,p. Finally, an atom in state 2 in the presence of radiation p(v12) may undergo stimulated emission to level 1 with emission of energy hvj2; the probability of this is defined by Bo; p, and it gives rise to Bz;n2p additional downward transitions per second. This

last process is the less obvious one. Its existence is a necessary consequence of the Planck distribution, and quantum-mechanically, it arises from exactly the same time-dependent perturbation as absorption. It is often helpful to regard it as negative absorption because the emitted radiation has exactly the same direction and phase as the stimulating radiation. To obtain the relations between the coefficients, assume an assembly of atoms to be enclosed in a blackbody cavity in thermodynamic equilibrium at temperature 7’. The population ratio n2/n, and the radiation density p(Viz) are given by the Boltzmann and Planck relations, respectively [(7.17) and (7.11)]. In equilibrium the rate at which atoms enter level 2 must be equal to the rate at which they leave it. The thermodynamical principle of detailed balance (or microscopic reversibility) states that this equality must hold separately for collisional and radiative processes and for all pairs of levels individually. In particular, the rate of radiative transitions downward 2-1 must equal the rate upward 1 — 2:

Aging + Boi p(12)n2 = Byop(vy2)n1 .

(7.18)

Rearranging this, 4 p12)

=

Agno TiS. oe Bygny — Boyne

By Boltzmann’s equation (7.17),

Ag) By2(n4/nz2) = Boy

:

7.5 The Einstein Probability Coefficients

169

ny/n2 = (gi/g2)e”?/*7 , where (£2 — FE) has been set equal to hv;2. Therefore, p(V12) =

Agi

By2(g1/g2) eh12/kT — Bo ©

This is identical with the Planck expression, (7.11), for all values of T if and only if MNBi2

=

goBr ,

Aon = (8mhv2,/c?)Bai.

(7.19)

The Einstein coefficients are intrinsic atomic properties, depend on the environment of the atoms. Hence the relations although derived for thermodynamic equilibrium, must still atoms are moved out of the blackbody enclosure — i.e. they

so they cannot between them, hold when the are universally

true.

The role of stimulated emission is made a little clearer by rewriting (7.18) to express it as negative absorption:

n2 Ao = 11 p[Biz — (n2/n1) Bai] = nipBia[1 — (g1/g2)(m2/m1)] .

(7-20)

In thermodynamic equilibrium this fractional decrease in absorption, gin2/g2n1, is equal by Boltzmann’s relation to exp(—hv12/kT), and the decrease is negligibly small for hvj2/kT > 1. This inequality is just the condition for the validity of Wien’s approximation (7.15), so the stimulated emission can be held responsible for the —1 term in the Planck distribution; this is also clear from a comparison of the two expressions above for p(v12). Stimulated emission can be important even at short wavelengths if no is greater than its thermal equilibrium value. Evidently, stimulated emission exceeds absorption if nog; > n1g2. This condition, known as population inversion, is necessary for laser action. Spontaneous emission is a loss mechanism for lasers; since the ratio of stimulated to spontaneous emission increases with 3, it is not surprising that masers were achieved before lasers. The relations between the Einstein coefficients tell us nothing about their absolute values. The next section relates them to the radiative transition rates derived from time-dependent perturbation theory in Sect. 2.4. Note that there is an important pitfall to look out for in using the B coefficients: they are sometimes defined in terms of isotropic spectral radiance rather than spectral

L = pc/4m for isotropic radiation. Since the

radiation density. From (7.9),

two definitions must give the same actual transition probability,

B(L)L(v) = Blp)py) ; leading to

An B(L)

=

~, Ble) and

2hv®? Ao

=

C2

Bio

:

‘pee

170

7. Emission and Absorption of Radiation

Actually the pitfall is even more extensive because there is yet another variant: if p is defined in terms of unit wavelength interval, the relation

p(A) = c/d? p(v) (Sect. 7.1) gives

BOVa=\7

fe Bia).

722)

The ambiguity of definition is confusing and regrettable. One must always make sure which convention is being used by any particular author.

7.6 Absorption Coefficient In practice absorption is more often described in terms of an absorption coefficient than a B coefficient, although the two are evidently related. The

absorption coefficient k(v) is defined as the fractional decrease in flux density at frequency v per unit path length through the absorbing medium:

—dI(v) =I(v)k(v)da ,

(Ces)

where [(v) is the flux density incident on the layer of thickness 62. The units of k are reciprocal length, or m~'. For a homogeneous layer of thickness | this equation can be integrated to give

I(v) = Ipn(v)e7*

(7.24)

The product k(v)l defines the optical depth 7(v) at frequency v. If the absorbing medium is not homogeneous, l

T(v) =| eB obes

(7.25)

The absorption from an isolated spectral line is spread over a finite frequency range in the ways discussed in the next chapter. Figure 7.5 shows (a) a notional absorption line and (b) the absorption coefficient derived from it by means of (7.24):

k(v) = (1/1) Inflo/fi(v)] ,

(7.26)

where Jo is taken to have a constant value I(vg) across the line. The absorption coefficient can be easily related to the Einstein B coefficient by considering the total power absorbed from the incident beam. Integrating

(7.23) over the line gives

—AI = I(vp)6x | k(v)dv

Wm”.

line

From the definition of By), a slab of gas of unit area and thickness 6x containing n,dx2 atoms in level 1 absorbs n16xp(Vp)Bi2 photons per second, where vy is the central frequency of the line. The power absorption is therefore

—AI =n 6z29(¥9)Bighvyo

Wm-?.

7.6 Absorption Coefficient

all

i

Fig. 7.5. An absorption line (a) and the absorption coefficient (b).

I

The absorption line shows the effect of increasing optical depth on line shape. a, b and c refer, respectively, to the optically thin, intermediate and optically thick regimes.

a) Up

al

k, | ky

0.5k,

b)

lie

ny

—tVo

=V

For collimated radiation the flux density J is related to the energy density p by I = pc (7.8), and equating the two expressions for AJ gives

/ k(v)dv = ni Bighvo/e

(7.27)

m7 Hz.

line

Thus Bj» is proportional to the total area under the absorption coefficient curve in Fig. 7.5b, independently of the line shape. Absorption is sometimes expressed in terms of the absorption coefficient per absorbing atom, a(v). This has the dimensions of area and is often referred to as the (radiative) atomic cross section. Thus

a(v) =k(v)/n,

and / a(v) dv = Byhvo/c line

m? Hz.

(7.28)

ie

7. Emission and Absorption of Radiation

It is important to remember that the relationship between k(v) and I(v) is logarithmic: unless the absorption is weak, the area under the k(v) curve in Fig.7.5b does not scale with the area defined by the absorption line in Fig. 7.5a. The latter, normalized to the incident flux density, is known as the equivalent width and is given by

equivalent width

i!

=

i, O

=

[lo — I(v)] dv = / (1 — I(v)/Io] dv /line

line

/ [1 — e PON) ay line

= i [k(v)l — k?(v)I?/2+---]dv.

(7.29)

line

The requirement for the two areas to match is therefore k(v)l < values of v. We shall return to this subject in the next chapter.

1 for all

7.7 Einstein Coefficients and Line Strengths The object of this section is to relate the Einstein coefficients to the radiative transition probability found in Chap. 2 by the time-dependent perturbation method. As this method is concerned only with transitions induced by an electromagnetic field, it can be used directly to obtain the B coefficients but indirectly only, by way of (7.19), for the A coefficient governing spontaneous transition probability. Direct evaluation of A requires quantum electrodynamics (quantization of the electromagnetic field as well as of the atomic energy levels) and is beyond the scope of this book; it gives exactly the same result for A so long as the radiation field is weak enough for a perturbation treatment to be valid. Equation (2.66) gives the transition probability per unit time for an electric dipole transition as

where the electric dipole line strength is defined by (2.67): 2 2 2

Siz = |(2Jer|1)[" = |(2 lea]1)|° + |(2 Jey]1)" + [(2 ez] 1)/? .

The modulus terms here are shorthand for the electric dipole moment uated between states 1 and 2:

(2 er|1)|=

eval-

[ver dr

From the definition of Byy in terms of frequency, using p(v) = 27 p(w), we get

i

Py

p(vi2)

=

S12

=

6eqh?

Qn?

Sada

3€oh?

(7 30)

7.7 Einstein Coefficients and Line Strengths

173

Table 7.1. Conversion factors for transition probabilities. Read: Agi = (8mhv? /c3) Bai, etc. Sea is electric dipole line strength.

Aoi

Bo

Bie

i

Sed

1

8rhv? oe

gi 8rhv? OD

gi 2me?v? g2 e€omc3

1 167°v? g2 3éeohc?

3

1

gu

G2

1

ye

Bei » ogeeCc

8rhv

Bowe f

8

_

Gus

3

g2

gi 8rhv3

91

g2 Eomc?

g2 4degmhv

gi 2me2v2

gy _~—s e?

whee weGy 3eohc?

‘

1673v3

g2 4eqomhv

Qn?

2

degmhv

1

e?

2

ee

g2 3éoh? 1 8x?mv

gi

3eqh?

oe

2

g2 3eoh

ee degmhv

3eoh?

de

2

CN died _ AA

3e7h

Qn?

3e7h 1

ce 8r2mv

Notes

1. The Bs are defined in terms of energy density per unit frequency interval. (a) For energy density per unit wavelength interval: Ao

=

(8rhv*/c*),

Bo

=

(8mhc/°)

Bai,

etc.

(b) For isotropic radiance per unit frequency interval: Aoi = (2ahv?/c?), Bor = (2thc/*) Bor, ete. (c) For isotropic radiance per unit wavelength interval: Agi =

(2hv* /c?), Boi

(2hc? /d°) Boi, etc.

2. The integrated absorption coefficient is related to the f value by fi. _h(v) dv = (e? /4eqmc) nif, where n; is the number density of absorbing atoms. 3. The formulae can be converted to CGS units by replacing ¢9 with 1/47.

No account has been taken of degeneracy so far, and, as the subscripts 1 and 2 can be interchanged without altering the value of the integrals, (7.30) serves for both Byzg and Bo. The wave functions 7, and wy» refer to single sets of quantum numbers, and degeneracy must now be considered. If levels 1 and 2 have respectively gi and gz sublevels, the total transition probability to level 2 from any one of the level 1 sublevels is found by summing the matrix elements over all of the go sublevels, specified by the quantum number m2. The symmetry of

the individual matrix elements ((2 |ex|1) = (1 |ea| 2), etc.) makes it logical to define the line strength in a symmetric form by extending the sum over

174

7. Emission and Absorption of Radiation

Table 7.2. Numerical conversion factors for transition probabilities. A is in s~', d is in nm, S is in (C m)?, and f is dimensionless. Aoi

F

1/499 =

Sed

6.670 x 10'3g, gor?

2.819 x 107% gor?

7

4.226 x 10°9

10 ““q, 7

iS

Sea.

f

gl

gir

93.5489610 “*garX®

2:366 5410, “aah

1

Other relations:

1. Bo: = 6.01 x 10*A° Aoi, where B is ins~' (J

m~* Hz~1)7?.

De ie. k(v) dv = 2.65 x 10-°nif, where k is in m~!, v is in Hz and n; is in m~°.

all values of mj, but, as all the g; sublevels of level 1 must have the same upward transition probability in order that their populations remain equal, this is tantamount to multiplying the total transition probability 1 + 2 by

gi. As this g factor has already been included in the Boltzmann expression for the population of level 1, it is necessary to divide the double sum by gj; to avoid summing twice over these sublevels. Bj. is then given by

Be

aS

= ——_ ae

gi

3€0h?

my,

e

Oe

mg

with the line strength defined by

$=Si2=S = S> [|(2lec 1)? + |(2leyl1)l? +|(2lez4)P] . (7.31) m4

,TMNa

The final expressions for the Einstein coefficients are then Ag

=

167?v?

3eohc’

S$

ete

go

By = 5, ef

on?

§

3e€9h?

g2

2n* Bio

==

|

(7 32)

SS 5) aes

3eoh

gi

These relations will be found in the collection of Table 7.1, with the corre-

sponding numerical relations in Table 7.2. In using them, or comparing them with expressions elsewhere in the literature, one has to beware of tripping over factors of 27 in no fewer than three places — over and above the ATEG

7.8 Oscillator Strength

175

factor relating to the CGS system. The questions to ask are: (i) Does the expression use w or v? (ii) Does it use h or h? (iii) (the most insidious of all) Is the spectral radiation density that is hidden in the B definition expressed as p(w) or p(v) — i.e. is it energy per unit angular frequency or energy per invA S has the dimensions of electric dipole squared, (er)?; indeed the expression for A in the above equation is exactly what would be expected classically for the radiation of an oscillating electric dipole of moment VS. As r must have a maximum value of the order of the Bohr radius ag, the order of mag-

nitude of S is given by S ~ (eao)?.

7.8 Oscillator Strength The other quantity used to specify transition probability is the oscillator strength, or f value. This originated from the classical explanation of absorption and dispersion as forced, damped oscillations of atomic oscillators driven by an electromagnetic wave. The f value was introduced to represent the fraction of the atomic oscillator effective for each resonant frequency. Its convenience as a simple dimensionless number that is approximately unity for a strong resonance transition has kept it in wide use, especially by astrophysicists. In this section we shall use the classical model of absorption and dispersion to relate the oscillator strength to the other quantities, B, A and S, governing radiative transitions. For simplicity we consider oscillations in the 2 direction only, produced by a plane-polarized electromagnetic wave travelling in the z direction. The solution of Maxwell’s equations for the electric vector E in a medium of permittivity € is

E(z) = &(0) expliw(t — z/v)] , where the complex phase velocity v is given by 1 Vv =

c

c

=

Eo

\/ (€/€0)

(7.33)

oa

£9 and jig are the permittivity and permeability of vacuum, complex refractive index, which can be written as

n' = /(e/eo) =n—ik-. Therefore

giving

E(z) = €o(0) exp[—(wk/c)z] exp[iw(t — nz/c)] .

and n’ is the

(7.34)

176

7. Emission and Absorption of Radiation

Thus « and n determine the absorption and the phase velocity respectively. K can be related to the more familiar absorption coefficient k by evaluating the attenuation of radiative flux passing through the layer z. As I is proportional to |E ie we can compare the expression

E(2)|? = £5 (0) exp[—2(wee/c)z] with (7.24) to give

Rak C Shee

Cs)

The next step is to relate n and « to the properties of the atomic oscillators. The equation of motion for a bound electron driven by a force —e&(t) at angular frequency w is

i ’ Cie #+ yh + wer = ——Epe™ m

,

where wo is the resonant frequency of the oscillator and y covers all forms of damping. The equation has the well-known solution r=xre with fn

A

—{e/m)£

ea (wg —

re w?) + iwy

The connection to permittivity is made via susceptibility. The instantaneous value of the dipole formed by one electron displaced a distance x is —er = —ex pe .

If there are no oscillators per unit volume, the susceptibility y, defined as polarization per unit field, is

__ —noero e* as

a

noe? /m ~

(we —w®) + iwy |

The standard relation between permittivity and susceptibility, leads to E.

E09

noe

=]+4+

¢ = e9 + x,

1

5 ’ —. Eom (w6 — w?) + iwy

By (7.34) this expression is the square of the complex refractive index n/ — 1.€., 2

(n’)? = (nin)? =14 20 Eom

:

(we — w?) + iwy |

Expanding the square on the left and equating real and imaginary parts gives noe 2 Wi 2)— WwW a ) a ae et (wo

Eom (we — w?)? + wy? ’ is

=

noe? wy Eom (we — w?)? + w2y2 ©

(7.36)

7.8 Oscillator Strength

177

This pair of equations can be put into a simpler form if we are dealing with gases or vapours, for which n — 1 at STP! is of order 10~°-107*. The approximation n ~ | in the second equation gives « directly:

k=

noe?

Wy

(7.37)

;

2eqm (we — w?) + wy?

Putting n = 1+ in the first equation and neglecting the two second order terms 67 and Kk? gives

noe? we — Ww? ~ Yeqm (we — w?)? + 24? ? and therefore

m= 1+

noe? we — w 2eqm (we — w?)? + w?7? ©

(7.38)

The transition from classical oscillators to real atoms is made by (i) allowing for more than one resonant frequency and damping coefficient (wo, Y > wiz,%;), and (ii) replacing no with n,fi;, where n; is the number density of atoms in the initial state of the transition at w;; characterized by the oscillator strength f;;. The f value is thus the effective number of electrons per atom for that particular transition. In principle, all transitions contribute to the values of n and « at any one frequency, so it is also necessary to sum over all initial and final states. Incorporating these changes and rewriting the equations in terms of v instead of w, we get a

=

:2

2

:

Ni Fish Jig v2. i) =

y2 )

5

8rteqm (Yj, — ¥*)? + (vj /2T) *

_

1 Fig fig Vis Vig VagVi [2H

e

=

8n%eqm 2 (ug; — v7)? + (vrij /20)?

7.39

aah

For practical purposes these equations can usually be simplified. In spec-

tral regions outside the wings of any absorption line, where |v — %;| > ij, « is negligible, and the refractive index and dispersion are found from 2

nv) =1=—— >> s 8m-Egm

ap

Mis

;

(7.40)

Alternatively, in the spectral region near one particular absorption line at frequency V12, the term involving this transition dominates both sums. With the approximations

Vig —V? = 242(Vi2 — v) and vyi2 = M1212 | together with (7.35) relating « to k, the two equations (7.39) become 1 Standard temperature and pressure: 0°C and 1.01 x 10° Pa (1 atm).

178

7. Emission and Absorption of Radiation

k |

Fig.

7.6.

Absorption

coefficient

k

(top) and refractive index n (bottom) as a function of frequency for a classical oscillator with resonance frequency vo and damping constant Va

—

SS

Ws

n-1

n(v) —1 i

etnafia

aes

l6m2egmry2 (142 — V)? + (F12/47)? ” 2 e*ny

ee

Ji:

2/Am

v12/

Amegme (112 — Vv)? + (M12/47)? ©

(7.41) i (7.42)

These functions are displayed in Fig. 7.6.

Equations (7.41) and (7.42) imply that it is possible to obtain the f value of a transition from the dispersion near the line as well as from the absorption. This is indeed the basis of the ‘hook method’ described in Sect. 16.2.3. Equation (7.42) can also be used in conjunction with (7.27) to derive the relation between the f value and the Einstein B coefficient of a transition. Integrating over the line gives

ja

| k(v)dv 0

=

e?ny

fi:

_e mhz 4m egme

Wyg—v| a

arctan —————

7.9 The f Sum Rule

Ee

(3 — --

Amegme \ 2 Since 2

Atr142

arctan ———

a

179

} .

is much larger than the linewidth 7/47, the second term is also

1/2. Therefore, le k(v)dv (v) V

=

e2 teammate?

(7 43)

Equating this with the integral in (7.27) gives fiz =

degmh = 142B}p2 .

(7.44)

Although this relation was obtained by integrating over the particular line shape of (7.42), derived from classical dispersion theory, the integral [ kdv, representing the total energy absorbed from the incident light beam, is necessarily independent of the line shape. Equation (7.44) therefore completes the interlinkage of the intensity parameters. Table 7.1 summarizes the relations between the five quantities A2,, Boy, Bio, fig and S for electric dipole radiation. Table 7.2 gives the numerical

conversion factors, and [ k(v)dv is also included in both tables. Note that f has been taken to represent the absorption line strength, which is its normal meaning. One does occasionally come across the emission f value, f21; this is related to fig by

for = —(91/92) fir -

(7.45)

The reason for the minus sign becomes apparent in the next section. It is common practice to give oscillator strengths in the form of “the gf values” so that they can be applied either to emission or to absorption. The relations between f and S$ for molecular bands and lines are considerably more complex. It is not apparent at first sight what, if anything, is meant by the oscillator strength of a band, and the summing of line strengths to give a band strength is often confused by the different normalization conventions in use. Molecular transitions are therefore treated in the last section of this chapter, which can be omitted without loss of continuity.

7.9 The f Sum Rule From the interpretation of f as the number of electrons per atom that is optically effective, one would expect some connection between f and the number of valence electrons (one for hydrogen and the alkalis, two for the alkaline

earths, etc.). Each electron can take part in several different transitions, so the total oscillator strength should be split in some way between several lines, with the f value of a strong line close to unity.

180

7. Emission and Absorption of Radiation

WYIIIWJW,-—~**'s. 77. Transitions involved in f sum rule.

The Thomas—Kuhn-—Reiche sum rule states that the sum of all transitions from a given state should equal the number of ‘optical’ or valence electrons,

Dhar uw

fut f fade=z.

(7.46)

I

The subscripts 7u and il refer to transitions upwards and downwards, respectively, from the initial state i, while ie refers to transitions to the continuum (Fig. 7.7). The downward transitions represent stimulated emission. This is the reason for the minus sign in (7.45): if the downward transitions are expressed in terms of absorption oscillator strengths f);, they have to be subtracted from the sum. In terms of absorption oscillator strengths only,

er. > S (gi /9) fui +f uri

ficede =z.

(7.47)

J’ transition consists of g lines, and the accepted normalization gives for the sum rule

0 Syn = g(2I" +1),

(7.51)

where the sum is taken over all g sublevels of a given J” and over all upper levels with which they can combine. This electronic degeneracy g is the nub of the confusion, and a little more needs to be said about it. The electronic degeneracy consists of an electron spin factor 2S + 1 and a A doubling factor 2 for all except 2’ states (Sect. 5.2.4). The degeneracy of

a 3) state is thus (25 + 1) and that of a I state is 2(25 + 1). Should the normalization of the Hénl-London factors then be different for 1) + IT and II + » transitions? The answer by the earlier recommendations of 1967 was ‘yes’, but it is now ‘no’. The current recommendation [17, 18] is to normalize the electronic part of the transition moment so as to give g in (7.51) the same value for upward and downward transitions. Formally, this is expressed as

g = (25 + 1)(2 — d0,a'4.4”) 5 but it is considerably easier to remember the rule

g = g

(2S +1) for XY — » transitions, 2(2S + 1) for all other transitions. A

Paty

Vi tt

We now return to the radial part of the transition moment, R?”

(r, R).

As the vibrational wave functions do not depend on the coordinates r of the

electron(s), we can define

(7.53)

/peeryer dr = and write Ru

4! | verre

aim

Unfortunately R. does in general depend on R, so the integral cannot be exactly separated into electronic and vibrational parts. Nevertheless, Re can

184

7. Emission and Absorption of Radiation

usually be assumed to vary sufficiently slowly with R over the range relevant to a particular vibrational transition that it can be taken outside the integral sign to give

Re Sk. /wry dR. J vw

(7.54)

dR is known as the overlap integral, and its square is the Franck—

Condon factor qy/,. If the interatomic potential can be expressed in a standard form such as the Morse potential (5.10), the vibrational wave functions and hence the Franck—Condon factors can be computed, giving relative band strengths. Vibrational levels have unit statistical weight, and the Franck— Condon factors are normalized to give SO dure

=

ye aye

=1.

(7.55)

Thus, insofar as the approximation that R. is independent of R is justified, the rotational line strength can indeed be expressed as the product of three factors, the last two of which can be calculated by fairly straightforward means: 2

Surya = |Rel” duu Saray

(7.56)

The calculation of R. to give absolute line or band strengths demands electronic wave functions, which are much harder to calculate or to estimate reliably.

The recommended definition of the band strength is [17, 18] Fall

Ola!

2

are! Re ‘

>)

~g

|Rel? du'v!

-

(7.57)

The relation between line and band strengths found by combining this with

(7.56) is 1 Syn

yin

=

See

:

The quantities derived from probabilities and f values, so we between these quantities and the From Table 7.1, fi2 and Ag; are ; fiz

(7.58)

experimental measurements are transition now need to make sure that the relations line and band strengths are unambiguous. related to S' by

aes, =

C1—S

a

Cy—S,

4

’

p? Ag

92

(7.59)

where C and C2 comprise all the constants and gi, gz are the degeneracies of the lower and upper levels, respectively. For a single line from a single sublevel of the level J”, the degeneracy gz is 2J" +1, so, using (7.56),

7.11

f Values in Diatomic Molecules 2

fog

yj?

=

Cl aval” a7 ra wt

tI"

=

Civ |R.|

Sy dv'v" J" +1 1 :

185

(7.60)

Similarly, the transition probability for a single line from a single sublevel of

the level J’ is 3 Ay

yg

=

Ons

7Sutera”

=

Cov” |Rel? du'v” we

:

(7.61)

These relations present no problem for lines, but a band oscillator strength is a more awkward quantity because of the factor v in (7.59). If the frequency spread over the band is not too large, v can be set equal to some effective frequency Vp, and the band oscillator strength f,y, can be defined by (7.59)

in terms of the band strength S$,” in (7.57):

Foro = Cr VoTFSoren = C10 Rel? uw

(7.62)

where g” is the electronic degeneracy of the lower state, 25 + 1 for Y’ and

2(25+1) for IT... . Actually the only case for which g and g” do not cancel is a

} > IT absorption transition. Finally, we can combine this equation with

(7.60) to relate the line and band oscillator strengths:

fur

V

1g" = ai Dy

g

Sy yu Grae

2d

Or

ee

cS

g Syrgu STE TE

LPT

hos

(7.63)

The variation of v across the band — usually not more than 1-2% — has been ignored in making the last approximation. This equation is important in that it allows the entire band absorption to be deduced from the integrated absorption of a single line, on condition only that vibration—rotation interaction is negligible. In an analogous way, A,» is defined as the probability of a transition from the vibrational level v’ of the upper electronic state to uv” of the lower one, irrespective of the particular rotational transition. Applying (7.59) and

(7.57) again,

ne =e,DS

= OweLa a

re

(7.64)

Again, g and g’ef except in the case of a Y’ > II transition. As with oscillator strengths, vp represents a weighted mean over the band, but as it is here raised to the third power, A is more sensitive than f to the weighting. It must be remembered that the weighting is determined by the distribution of population over the rotational levels so that the band transition probability, and to a lesser extent the band oscillator strength, are actually temperaturedependent. The frequency spread presents a much worse obstacle to any attempt to define an electronic oscillator strength for a band system, which may extend over a hundred nm or so. The quantity f,) is still met in the literature, despite efforts to discourage its use. It is defined as the sum of the band oscillator strengths for a given progression:

186

7. Emission and Absorption of Radiation

C fel = Sh

irg a

a So u(v',v") Suu

vu!

2

vu!

The frequency v(v',v’’) is usually given some sort of effective mean value V for the whole band system and brought outside the sum. Since }>,, Sy" = g|Re|? So Qiu, We then have, using the normalization of the Franck— Condon factors,

fei = OPT [Ese

(7.65)

The value of depends on the relative weighting of the bands, which in turn depends on the relative vibrational level populations. f.; therefore depends rather strongly on the temperature. Moreover, the assumption that R, is independent of R and hence of the particular vibrational transition is not necessarily valid over the whole band system. All in all, f,) is really not a very meaningful quantity.

Further Reading — Corney, A., Atomic and Laser Spectroscopy (Oxford University Press, Lon-

don, 1977). — Kuhn, H.G., Atomic Spectra (Longman, London, 1969). — Loudon, R., The Quantum Theory of Light (Oxford University Press, Lon-

don, 1983).

8. The Width and Shape of Spectral Lines

The last chapter was concerned with the total amount of energy absorbed or emitted in a radiative transition, which is independent of the frequency spread, or line profile. However, accurate measurements of wavelengths and intensities, as well as sensible choices of experimental parameters, require some knowledge of line profiles, and these profiles are of interest in themselves for the information they can give on source conditions and for the study of interatomic interactions and collision processes. They are indispensable in the analysis of astronomical spectra, for example for determining stellar abundances of the elements. The same is true for investigations of laboratory plasmas, produced, for instance, in combustion processes, thermonuclear fusion experiments or by focusing high power laser beams on different targets. Experimentally observed line profiles are frequently broadened by the instrument with which they are observed, as described in the third part of this book. In this chapter we consider only the true line shape, as it would be if observed by an ideal spectrometer of infinite resolving power.

8.1 Line Profiles in Absorption and Emission Four different processes may contribute to the finite width of a spectral line: natural broadening, Doppler broadening, interactions with neighbouring particles and power broadening. In the optical region the last of these occurs only in very intense laser fields, and it is beyond the scope of this book. The third process could take us all the way from low-pressure gases to the solid state, but the discussion here is limited to free atoms and molecules and comes under the description of pressure broadening. Even with this limitation the most that can be done in one chapter is to present a semiquantitative type of treatment. Line shapes may be studied experimentally either in absorption or emission. If an absorption line is scanned with an ideal spectrometer, the output

signal J(v) and the absorption coefficient k(v) derived from it as described in Sect. 7.6 will look something like Figs. 7.5a,b, respectively. Figure 7.5b can also be taken to represent an emission line scanned with an ideal spectrometer, the ordinate in this case being j(v) instead of k(v). The emissivity j(v)

188

8. The Width and Shape of Spectral Lines

\

/ Ip

a)

Vo

V

b)

Yo

VY

Fig. 8.1. Effect of increasing optical depth on line profile (a) in emission and (b) in absorption.

is defined as the power emitted per unit solid angle per Hz by unit volume of the gas. Equivalently, it is the spectral radiance per unit thickness.

Just as the integral of k(v) over the line is related to the Bj2 coefficient and the population of the lower state by (7.27), which we repeat here for reference,

i: k(v) dv =m Byghrvo/c,

(8.1)

line

so the integral of j(v) is related to Aj; and the population of the upper state. As there are nz excited atoms in unit volume, the number of photons emitted in unit time over the whole line is n2A21, and the power per unit solid angle is (1/47)n2Ao1hv, giving i

j(v) dv

=

n2 Aoi hvo/(47)

é

(8.2)

It is important to remember that the true shape of the absorption line (as defined by the absorption coefficient) is not the same as that of the dip in the output signal unless the absorbing layer is optically thin, as explained in Sect. 7.6. The same is true in emission: the observed and true line shapes are the same only if the emitting layer is optically thin. Figure 8.1 shows output signals in both emission and absorption for three different optical thicknesses, and only the smallest of these represents respectively k(v) and j(v). The interpretation of ‘optically thin’ and the effect of increasing optical depth on the observed signal (the radiation transfer problem) will be discussed in the next chapter. It is assumed in this chapter that we are always dealing with true line shapes. The importance of any line-broadening process is usually measured by its full width at half the maximum (FWHM). The line shape depends on the particular broadening process, and the FWHM does not necessarily give useful information about the wings of the line. The term ‘half-width’ in the literature, incidentally, must be treated with some caution: while it usually

8.2 Natural Broadening

189

has the meaning of full width at half maximum, as used here, it is occasionally taken to be half of this.

8.2 Natural Broadening Natural broadening appears in both the classical and the quantum-mechanical theories of radiation. In both cases it is a consequence of the finite lifetime of the excited state due to spontaneous emission. The bandwidth theorem of classical radiation Aw At ~ 1 implies the same frequency spread, Av ~ 1/(27At), as does the Heisenberg uncertainty principle in the form AE At ~ h with AE = hw. There is, however, a conceptual difference between the two pictures in that classical radiation theory attributes the frequency spread to the decay of the emitted wave train while quantum mechanics attributes it to the finite width of the discrete energy levels involved in the transition. The spectral line shape can be derived from either picture by a Fourier transform from the time domain to the frequency domain. In both cases the observed signal decreases exponentially with time. For the two-level atom of Sect.7.5 in the absence of external radiation the population of level 2 decreases according to —dn2/dt = A21n2 by definition of A. Thus nz decays exponentially:

no(t) = n2(0) e421! = n2(0) e/?

(8.3)

with a mean lifetime 72 equal to 1/A2, and so too does the radiated flux L(t), which is proportional to n2. For the classical case we go back to the oscillating bound electrons of Sect. 7.8 and switch off the external field: the amplitude of the oscillations then decreases exponentially with a decay constant of 7/2, and the decrease in energy is given by E(t) = E(0)e~7. The flux radiated by an oscillating dipole is proportional to its energy, so again we have an exponential decay:

Kip LO)e™ .

(8.4)

By a well-known Fourier transform theorem, multiplying in the time domain by an exponential e~7! is equivalent to convolving in the frequency domain

with the Lorentzian function 1/[v?— (7/47)°]. The line profile of a spectral line at frequency Vp is therefore

1 9/40 tv) = — 5 2 mt (v — %)? + (7/47) This function has been normalized to make the ae In terms of the peak intensity Jp it becomes

ms

(7/4)?

TO) = ToGo)? + (y/4n

(8.5) an area equal to unity.

ae

190

8. The Width and Shape of Spectral Lines

Vo

V

Fig. 8.2. The Lorentzian, or dispersion, distribution, with FWHM

doar.

évy, = 7/27 =

i

Figure 8.2 shows the Lorentzian function. The frequency at half maximum is

given by |44/2 — % |= y/4a. The FWHM Oy ="y)

20"

is therefore (8.7)

Equations (8.5) to (8.7) are valid whatever the interpretation of y. Classically, if the damping is entirely due to radiative loss, Peg? _ 2me*Vp

Ae tcems 3egmc3 Yc depends only on the frequency, and in the visible region the corresponding lifetime 1/7, works out to be about 16 ns. As one might expect, the classical value is of the right order for a strong resonance line having an oscillator strength of unity, but it does not describe the wide range of observed lifetimes. From (8.3) the quantum-mechanical value of y for the two-level atom is given by

y= An =1/72.

(8.8)

To go from the mythical two-level atom to a real atom with many levels requires two modifications. First, there are in general several allowed transitions from a given upper level FE; to different lower levels Ey. The total transition probability from H; is the sum of these independent probabilities >>, Aje, and the lifetime of level j is therefore Ty

~ Sao

(8.9)

Second, unless the lower level i of the transition j — 7 is the ground or a metastable state, (7; > oo), the lower level is also broadened, as indicated in

Fig. 8.3. The line profile for the transition between these broadened levels is obtained by convoluting the two Lorentzian functions. This convolution gives another Lorentzian whose width is the sum of the separate widths:

8.3 Doppler Broadening

6v>= AE,/h

191

Fig. 8.3. Natural broadening of spectral lines. The energy levels are smeared out as indicated by the probability distributions on the right of the diagram. The FWHM of the line is given by 6142 = OV, + OV.

1

=|é6v, = AE,/h

OV; = OV; + Oyj; .

(8.10)

[This result is easy to understand if one starts with the product of the two decaying exponentials since exp(—7,;t) exp(—yit) = exp[—(7; + %4)t].] Combining (8.8) and (8.10) gives the generalized expression for the FWHM due to natural broadening: 1

a

1

LU igen “2NT%

with 7; given by (8.9) and 7; by a similar equation.

Lifetimes of non-metastable states are typically in the range 10~°-107° s, so natural line widths are of order 0.1-100 MHz, or 107°-10~? cm~!. This is significantly smaller than the range of Doppler widths normally encountered. Nevertheless, natural broadening can be important in a number of contexts: in Doppler-free laser spectroscopy; in autoionizing transitions, where lifetimes may be as short as 10~!% s (Sect. 3.5) and in the far ultraviolet, where the v® dependence of the A value leads to high transition probabilities. Even when the Doppler width is much greater than the natural width, the wings of the Lorentzian distribution can still make a significant contribution to the radiation emitted or absorbed, as will be shown in Sect. 9.2.

8.3 Doppler Broadening Doppler broadening is a result of the well-known ‘Doppler effect’, which is the apparent shift in wavelength of the signal from a source moving towards or away from the observer — a decrease in wavelength for motion towards and an increase for motion away. The latter is associated with the cosmological ‘red shift’ of the radiation from distant galaxies, and Doppler shifts in both directions are used to identify the individual stars in binary systems.

192

8. The Width and Shape of Spectral Lines

In a laboratory source the shifts arise from the thermal motion of the emitting atoms or molecules. The observer sees a spread of shifts corresponding to the spread of velocities in the line of sight, and this is tantamount to broadening the line. In laser terminology this is known as ‘inhomogeneous’ broadening in contrast with the ‘homogeneous broadening’ exhibited by any subset of atoms with the same line-of-sight velocity. The Doppler broadening line shape follows the velocity distribution of the atoms and can be derived easily for a gas in thermodynamic equilibrium. For an emitter approaching the observer with a velocity v, the Doppler shift is given by AX

Vay

do

Vo

=

“a,

:

8.12

Cc

(

)

In equilibrium at temperature T the fraction of atoms with velocity between v, and vz, + dv, follows the Maxwell distribution dN(v

iL

we lta exp(—v2/a”)duz ,

(8.13)

where a is the ‘most probable speed’, given by

ET \ eo ORE

a= | —

4

IMA

M

In this expression m, is the actual atomic or molecular mass, while MW is the mass number; k and R are Boltzmann’s constant and the universal gas constant, respectively. a can also be interpreted as \/2 times the standard deviation of the v, distribution.

Substituting v, from (8.12) into (8.13), we have for the fraction of atoms emitting in the frequency interval v to v + dv

dN(v) N

=

C JTVoa

cAv\* Cx

rc

q

—- | Ps

V ,

Yar

where Av = vy — v9. The intensity at v is proportional to dN(v), so the line profile is given by the Gaussian function

Ae z

C is / TU

c(v — Vv) a

nee

Vom

:

(8.14)

where g(v) is normalized for unit area. It is usually more convenient to write this function in terms of the FWHM. The half maximum points V1/2 are defined by me) 2 Sa ou Ay) / ) hat >,

giving for the FWHM

oa dVp = 2 |4/2 — 4 |= 2VIn 2

(8.15)

8.3 Doppler Broadening

193

Equation (8.14) then becomes 2

g(v)

pind tad es sm: (“ = “)|

5 wee

FORTIN \7 = 7.16 x 1077,/T/M . (A*)

(8.16)

OVD

OVD

with

Vo

(8.17)

€

In this dimensionless form

dup_ Bo _ don Vue

aa

AO

The expression for the line shape in terms of its peak intensity Jo is

fSi,e

(8.18)

where

igo

fayip

laa, OVD

The Doppler line shape is shown in Fig.8.4. It is much more rectangular than the Lorentzian. The product of peak height and FWHM, from (8.16), is

2,/In 2/7, which is 0.94, very nearly equal to the area (unity). By contrast, the same product for the Lorentzian is only 2/7, or 0.64, reflecting the large contribution made by the wings of this function. This is an important factor in the consideration of optical depth. The range of Doppler widths can easily be found by putting specimen

figures into (8.17). For atoms and diatomic molecules and for most laboratory sources M and T are, respectively, in the ranges 1-200 and 300-10 000, so the extreme values for the right-hand side of the equation are roughly 7 x 107° to 7 x 1077. In the visible region the corresponding widths are 0.04—0.0004 nm,

0.51

Vo

Vv

Fig. 8.4. The Doppler, or Gaussian, distribution, with FWHM

10°-7,/T/M.

dp = vo x 7.16 x

194

8. The Width and Shape of Spectral Lines

or 1.0-0.01 cm~!. These widths, measured on a wavenumber scale, go down by a factor of 10 in the infrared and up by a factor of 10 in the far ultraviolet. The instrumental resolving powers required to match the Doppler widths are

found from (8.17) simply by taking reciprocals, and with the above figures they range from 15000 to 1.5 x 10°. In the past much effort has gone into reducing Doppler widths in order to obtain accurate peak wavelength measurements or to resolve close components or to study other broadening processes. For example, in low-current hollow cathode sources cooled with liquid nitrogen, or even liquid helium, the gas kinetic temperature can be well below 100 K, but the reduction in width is still only a factor of about 3. An order-of-magnitude gain can be made by using a beam source, in which a series of collimated slits selects only those atoms having a small velocity in the line of sight, but the number density of atoms is then usually very low. The real breakthrough came when the development of tunable lasers made possible various methods of Doppler-free spectroscopy; these are described briefly in Chap. 14. This discussion of Doppler broadening has assumed thermal kinetic motion, but a similar effect is produced by macroscopic motion in the form of turbulence or mass motions of the emitting gas, an effect that is found in many astrophysical and some laboratory plasmas. In such cases the profile is not necessarily Gaussian.

8.4 Pressure Broadening It is a well-known experimental fact that spectral lines are broadened, sometimes asymmetrically, and often also shifted by increasing gas pressure and by the presence of ions and electrons. Sometimes additional lines from ‘forbidden’ transitions also appear. These perturbations — widening, shifting and mixing of energy levels — arise from interactions between the emitting or absorbing atom or molecule and the other particles in the gas, and unfortunately a proper treatment of them is a very long and complex matter. The best that can be done here is to present an outline, focusing on those aspects that are of most experimental importance. More information can be found in the section Further Reading at the end of the chapter. To set the scene, one can start with the simplest possible model (first used by Lorentz) in which the wave trains from the radiating atoms are cut off abruptly by collisions after an average time to. From the usual probability arguments, the number of undisturbed atoms after a time ¢ is related to the mean lifetime by

N(t) = N(0)e7*/* , This expression is exactly the same as that describing spontaneous decay, and consequently the line profile is Lorentzian with a FWHM given by

8.4 Pressure Broadening

ene

etbme newer tiacinatMe 0, Ly

Fig. 9.5. Approach to saturation of radiance of gas column as optical depth is increased. LB(T) is the blackbody curve for temperature 7’.

9.4 Thermodynamic Equilibrium and Collisional Transition Rates

Daly

Li(v) > S"(v)k'(v)l = j(v)L, which is simply the optically thin relation. The saturation of emission of radiation in a gas at constant temperature is not to be confused with the self-absorption and self-reversal that occur if the edges are cooler than the middle. Whereas in the homogeneous case each layer contributes to the emission as well as to the absorption, in the case of a temperature gradient there are fewer excited atoms at the edges, and therefore more absorption than emission. The line peak is first flattened and then reversed, forming a dip in the middle of the line. When emission in the cooler layers is almost negligible, we get back to the case of a pure absorption line, seen against the broad emission line from the hotter background gas. Line shape and intensity can thus be used as a temperature diagnostic, to be further discussed in Sect. 10.11.

9.4 Thermodynamic and

Collisional

Equilibrium

Transition

Rates

It is shown in texts on statistical mechanics that in thermodynamic equilibrium the three types of energy distribution that have already been used in this book are all valid with the same temperature parameter 7’. These three are the Planck blackbody function for radiation energy (7.11) the Boltzmann distribution of population among excited states (7.17) and the Maxwell distribution of particle velocities, given by (8.13) for one component of velocity. In thermodynamic equilibrium the Maxwell distribution applies to all types of particle — molecules, atoms, ions and electrons. A fourth distribution func-

tion, the Saha equation relating the number densities of neutral atoms, ions and electrons, will be discussed in Sect. 9.6.1. If radiation and particles are all bottled up together in an enclosure maintained at constant temperature, there is no doubt about the existence of thermodynamic equilibrium, but the system is of no spectroscopic interest because the radiation is, by definition, blackbody radiation, independent of the type or number density of particles in the enclosure. Fortunately, there can exist a state known as local thermodynamic equilibrium in which populations and velocities are still described by the equilibrium relations even though the radiation density is below the blackbody level. In order to characterize this state it is necessary to consider collisional excitation and de-excitation. Figure 9.6 represents the two-level atom of Fig. 7.4 with the collisional transition rates Cy. and Co; shown as well as the radiative coefficients. These transitions are primarily due to collisions with fast electrons, as mentioned in Sect. 8.4.7. The principle of detailed balance, or microscopic reversibility, states that in thermodynamic equilibrium every process must be balanced by its exact inverse. This principle was invoked in equating the radiative transition rates in Sect. 7.5. Applying it now to the collisional rates gives

9. Radiative Transfer and Population Distributions

218

p(V42)

£. [=

WW

collisions

radiation

ny

Fig. 9.6. Collisional and radiative processes for a two-level atom.

m4 C2 = n2C21 ,

(9.14)

and this relation must hold separately for every type of collision. 9.4.1

Collisional

Transition

Rates

Collisional excitation and de-excitation of atoms by electrons are represented by the process

Ate

+AE

A*+e

,

where AB is the kinetic energy transferred to or from the electron. The crosssections for these interactions depend both on the transition moment of the atom for the pair of states involved and on the velocity v of the electron. For excitation there must be a threshold velocity v, defined by the conversion of all the electron kinetic energy into excitation energy: ik YR ane = hy— FE, .

(9.15)

A ‘hard sphere’ classical argument suggests that for v >> v;, the cross-section should decrease as 1/v?, and it is shown below that the quantum-mechanical Born approximation gives almost the same v dependence. For neutral atoms the order of magnitude of most excitation cross-sections is around 7a. For transitions in the optical range of 2 or 3 eV and for an electron temperature of a few thousand degrees (for which kT’ is less than 1 eV), it is only the electrons in the high energy tail of the Maxwell distribution that satisfy (9.15) and are effective in collisional excitation. To relate the collisional transition rates to the cross-sections, consider an electron of velocity v incident on a cylinder of unit area and length v, which it traverses in unit time. If the number density of target atoms is n, the cylinder contains nv atoms, and for a collisional cross-section a(v) the total target area presented is nva(v). The collision probability per unit time per electron is therefore nvo. To get the total number of collisions per second

9.4 Thermodynamic Equilibrium and Collisional Transition Rates

Zug

atv)

itv)

a)

v;

Vv

iv)ve

b)

Ve

;

v

Fig. 9.7. (a) Collisional excitation cross-section o(v) as a function of v, together with the Maxwell velocity distribution function f(v). v¢ is the threshold velocity. (b) The product f(v)vo(v). The area under the curve is the total collisional probability per atom for unit electron density.

per unit volume this must be multiplied by the number density of electrons with velocity v, dne(v), and integrated over all v. From the definition of the collisional rate C,

Ce where

0

‘iva(v) dne(v) = ne

f(v) is the electron

0

crva(v)f(v) du = ne(va(v)) ,

velocity distribution,

shown

(9.16)

in Fig.9.7 for a

Maxwell distribution, and (va(v)) is the average value of vo weighted with this velocity distribution. It is represented by the area under the curve in Fig. 9.7b. The two collisional transition rates are then given by

Cia = ne(vor2(v)) and C21 = ne(vae1(v)) .

(9.17)

The calculation of collision cross-sections is beyond the scope of this book, but is treated in a number of standard textbooks (for example [21, 22, 24, 29]. Here we give the result of the widely used Born approximation. A free electron travelling in the z direction is represented by the wave function w = ae'**, where hk is equal to the momentum mv. The Born approximation is essentially a first-order perturbation treatment in which this unperturbed wave function is used to evaluate the interaction with the transition moment of the atom. The resulting cross-section for an electric dipole transition is

O12

=

me?

id

mv?

=

S eeA

2mv?

Amekht k2 in(= = =) can ei n( i =) i

9.18

ety

220

9. Radiative Transfer and Population Distributions

where S$, is the z component of the electric dipole line strength of the transition. Cross-sections for transitions that are optically ‘forbidden’ — i.e. S, = 0 — depend on the quadrupole term (2 |z| 1); they are smaller, although by not such a large factor as the radiative electric quadrupole transitions, and fall off faster with energy above the threshold. The Born approximation is strictly valid only for electron energies well above threshold, but it often works surprisingly well at lower energies if one accepts that it overestimates the cross-sections near the threshold. Extensive formulae for more accurate calculations and tables of numerical

results are given in [24, 25]. 9.4.2 Local Thermodynamic

Equilibrium

In any case of spectroscopic interest a radiating mass of gas is not optically thick at all frequencies, so its radiation energy is not represented by a blackbody distribution. In the state known as local thermodynamic equilibrium, or LTE, it is possible to find a common temperature, which may vary from place to place, that fits the Boltzmann population distribution and the Maxwell distribution for the velocities of the electrons. The criterion for LTE is that collisional processes must be more important than radiative, so that the shortfall of radiative energy does not matter. More precisely, an excited state must have a higher probability of de-excitation by collision than by spontaneous radiation. This is tantamount to setting a minimum value for the electron density. The first step in showing this is to find the relation between the upward and downward collisional cross-sections. Consider the two-level atom of Fig. 9.6 in an enclosure in complete thermodynamic equilibrium, so that Boltzmann’s relation is valid. From (9.14)

and (9.17) we have

Cig _= mg _. (voie(v)) —= —e 92 (mm) /eT Co

ny

(vo21(v))

;

(9.19)

gi

The cross-sections depend only on atomic parameters and on v, so this relation must still hold when the system is taken out of the enclosure provided that the electron velocities still have a Maxwellian distribution at the temperature 7’ that appears on the right-hand side of the equation. The fact that the relation between the two cross-sections is temperature-dependent means that it is not so general as that connecting the A and B coefficients. Now consider this two-level atom in any sort of equilibrium, meaning simply that the populations n; and ng do not change with time. Equating the total number of transitions per unit time upwards with those downwards, we have

Mm pBi2 + ny nNeCi2 = nepBoi + n2A21 + N2NeCd1 ,

(9.20)

where p is the radiation density at the frequency V9. Obviously, if ne is very small this equation reduces to (7.18), the relation between the radiative coefficients. At the other extreme, if p is very small,

9.4 Thermodynamic

Equilibrium and Collisional Transition Rates

2ail

N{NeCj2 = N2(NeC21 + Agr) , giving n

a

ny

Mle:

a,

(9.21)

NMeC21 + Agi

The collisional equilibrium is perturbed by the spontaneous radiation. If the conditions are such that

Ne > Aoi /Cr1

(9.22)

then, using (9.19), io a Che =e 92 .—(E2—E1)/kT i NY C21 f

(9.23)

Thus, provided the electron density is high enough to satisfy (9.22), the population density is described by the Boltzmann equation with a temperature parameter 7 identical with that for the electron velocity distribution. This is the condition known as local thermodynamic equilibrium (LTE). T may vary from one place to another in the gas (hence the name ‘local’), and it does not necessarily follow that the velocities of the atoms and ions can be described by a Maxwell distribution at the same temperature. The frequency and temperature dependence of the inequality represented

by (9.22) can be estimated. From (9.18) the cross-sections are proportional to electric dipole line strength S$ and inversely proportional to v7, giving C x S/v, whereas Ay; is proportional to v?S (Table 7.1). Therefore Aoi /C21 Oo vv? x V/T(AE)®

;

where T is the electron temperature in K and AF is the energy difference between the level in question and any neighbouring level to which it can make

a transition. A full treatment [26] leads to the condition

me 16 al0 VITALY m~,

(9.24)

where AF is in eV. This criterion is most difficult to satisfy for low-lying states where AE

may be 2 or 3 eV. At 10 000 K ne must be of the order of 10!© cm~3. (In practice, electron densities are almost invariably expressed in cm~* rather than in m~°.) In a low pressure discharge n. is well below this value, and even in a free-burning arc at atmospheric pressure running at a few amperes, Ne is too small for LTE to exist in the lower states. It is for this reason that stabilized arcs running at high currents have been developed for use in experiments requiring knowledge of populations. The temperature dependence is actually greater than would appear from (9.24) because at high temperature the relevant spectra are likely to be from ions rather than neutral atoms, and AE scales with z?, where z is the degree of ionization. The critical value of n, therefore increases as 2°. Nevertheless, for any ne and any z it is possible at high enough excitation to reach a limit

Dae

9. Radiative Transfer and Population Distributions

where the states are crowded closely enough together for (9.24) to hold; this is known as the ‘thermal limit’ for the particular values of ne,T and z, and the plasma is said to be in partial LTE. 9.4.3

Coronal Equilibrium

The importance of LTE is that relative populations can be found without any knowledge of cross-sections and transition probabilities. In other forms of equilibrium at least some of these quantities have to be known. The most important of these is called coronal equilibrium because it is applicable to the Sun’s corona where temperature is high (~ 10° K) and electron density low (~ 10" cm~?). The radiation density is also low, and it is assumed that upward transitions are due to electron collisions while downward transitions occur by spontaneous radiation. The two-level atom is not a very realistic model, but it will serve to show the considerations involved. For Az; > neC21,

(9.21) becomes uP)

Tle

Fore

Aa,

(9.25)

so that the electron density and two rate coefficients must be known to determine the population ratio. Comparing this equation with (9.23), it can be seen that neC2; has been replaced with the larger quantity A21, so the ratio n2/n, must be below the Boltzmann value. This is demonstrated by the plot of n2/n, against ne in Fig. 9.8. The ratio increases linearly at first, following (9.25), but at large Ne it approaches the LTE value. For forbidden lines, having small A values, the LTE condition is reached at lower electron densities (as noted above, collision cross-sections are less sensitive than radiative transition probabilities to selection rules). The appearance of forbidden lines is, indeed, one of the features of low density plasmas in coronal equilibrium. Although excitation

No/n

Boltzmann value ——

allowed line

forbidden line

Ne

Fig. 9.8. Population ratio n2/n for a two-level atom as a function of ne when the two levels are connected by an allowed transition (solid curve) or a forbidden transition (broken curve).

9.5 Energy Distributions and Partition Functions

223

rates are low, the atom or ion, once excited, has very little chance of decaying other than by spontaneous radiation. The arguments used to show that LTE is more easily established among higher excited states show also that coronal equilibrium is likely to persist for the lower states when the electron density is high enough to invalidate it for the upper states. It is quite possible to have the populations of the higher states following LTE while those of the lower states follow coronal equilibrium.

9.5 Energy Distributions and Partition Functions 9.5.1

The

Boltzmann

Distribution

The two-level atom represents a special case of the application of the Boltzmann distribution of population. The general case, specifying the fraction of the total number of atoms or molecules in any given state, requires consideration of all possible energy levels. It is necessary to evaluate a quantity called the partition function or state sum which can be thought of as a normalization factor. If all excitation energies are measured relative to the ground state, the Boltzmann relation gives for the population of any state 7:

sip

haslg ion) Ct

where no and go refer to the ground state. The sum of all n; must equal the total number density n of the species, so

=

ya, = (no/go) Mage j

where Q(T)

hie

= (n0/90) Q(T) ,

is defined by

ou

a

(9.26)

Therefore no/go = n/Q(T), and the population of the jth level is given by

Gee eee Bik Ty

n

Q(T)”

(9.27)

The partition function Q(T) is sometimes designated as U(T’) or Z(T). The number of terms that has to be included to evaluate @ depends both on the temperature and on the structure of the atom. For example, the resonance lines of most simple atoms are in the visible or near ultraviolet region, corresponding to E; ~ 25 000 cm™!. Since k = 0.7 cm! per degree, T has to be of the order of 30000 K for E,; ~ kT. At moderate temperatures the populations of the higher levels are very small, and Q converges rapidly. However,

224

9. Radiative Transfer and Population Distributions

in complex atoms there may be a number of low-lying states that are appreciably populated even at fairly low temperatures. At very high temperatures the sum to infinity may not converge at all; in such cases the series has to be terminated, a point to which we return in Sect. 10.5. In molecules one can often ignore the excited electronic states, but the rotational and (usually) the vibrational levels of the ground electronic state have to be included in the state sum. Luckily this does not have to be done level by level, because the sums can be evaluated analytically in terms of the vibrational and rotational constants. From eqs. (5.19) and (5.16) we have

F(J) =BI(d--+81) em- wand G@)s ws

(o+5) em.

The statistical weight of a rotational level with total angular momentum quantum number J is 2J + 1, and it is usually permissible to regard the rotational states as sufficiently close together that the sum may be replaced by an integral. The rotational partition function is then given by co

OF

=

S403 a DhenbePd

Gre

a [es is Herre

0

J=0

kt o—heBI(I4+1)/kT PT:

-.

Tones a

ng ehcB™

(9.28)

Vibrational levels all have unit statistical weight, giving for the vibrational partition function

Qvib= Yep (= me) ae! es

:

(9.29)

For the temperature range in which molecules remain undissociated, the population of levels other than v = 0 is usually small enough for the approximations

ik exp (—Shawkr

Quip. = .

9.5.2

eee

The

:

—vhcwe xpa| $=

PET

Maxwell

‘ (9.30)

Distribution

The Maxwell distribution of velocities can also be expressed in terms of a partition function. Equation (8.13) gave the fraction of particles having a velocity v, in terms of ‘the most probable speed’, \/2kT'/m. To extend this

to three dimensions, the independent probabilities dn(v,),dn(v,) and dn(v-) can be multiplied to give

9.6 Dissociation and Ionization Equilibrium

dn(ve, Vy, Uz)

n

( m

i

None

m(v2 0, +?)

(oNe

QkT

225

du, dv,dvz. (9.31)

In terms of the total speed v this can be rewritten as

f(v)dv= wee = Ges

exp (-52) Aru? dv ,

(9.32)

where E(v) is the kinetic energy mv?/2. Comparing this with (9.27), it can be seen that both have the same exponential (—E/kT), that there is a normalizing factor corresponding to Q(T) and that v? dv must represent some form of statistical weighting. To obtain the translational partition function and statistical weight explicitly, we rewrite the equation in terms of the momenta, mvz = Pz, etc.:

An(pes Py, Pz) _

n

1

(QnmkT)3/2

aka

P\

E(v)

KP

)dp,dpydpz .

The uncertainty principle does not allow the momentum and the position of a particle to be specified simultaneously with a precision greater than that given by ApAz = h. In three dimensions, therefore, Ap, Ap, Ap, ArAyAz = h°. In a six-dimensional momentum-space carved into cells of volume h°, a particle cannot be specified more precisely than by assigning it to one such cell, but it is quite possible for the particle to have the same energy in several cells, just as a bound electron may have several (degenerate) states of the same energy. The statistical weight of a free particle is the number of momentum-space cells in which it may be found:

pe

Ae

n osApion

(9.33)

where V is the actual volume. With this interpretation of g, (9.33) becomes identical with (9.27), if the translational partition function Q;(Z) is defined by

(2nmkT)3/? Q(T)

=

=

(9.34)

9.6 Dissociation and Ionization Equilibrium The partition functions determine the equilibrium number densities in any form of dissociation represented by

Ka Y ee Le Statistical mechanics gives for the numbers (that is, absolute numbers, not number densities) of the three species in equilibrium: 0 (0

sh

Nz

meth as

Oe

(9.35)

226

9. Radiative Transfer and Population Distributions

where each Q° is the total partition function of the species — that-is, the product of the internal partition function Q(T) and the translational partition

function Q(T). For dissociation of the molecule Z into the atoms X and Y, this equation gives

-

=

Ny

2 V

= [2n(Mx My /Mz)kT}*”” rsC2

oo DI

where the first term on the right comes from the three translational partition

functions, (9.34), and the Qs are now the internal partition functions, the zero of energy being in each case the ground term of the atom or molecule. D is the dissociation energy, and the factor exp(—D/kT) appears because Qz

has to be multiplied by exp(D/kT) to give the state sum of the molecule the same zero of energy as that of the separated atoms. This equation can be written in terms of the number densities and the reduced mass pz given by

_ MxMy — MxMy ME magia to give

= nxny _ (2npkT)*? 8/2 QxQy gc Dikr NZ

he

(9.36)

Qz

In many atoms and ions the first excited state is several eV above the ground state, and only the first term in the partition function is significant, at any rate for temperatures up to about 10000 K. Thus Q(T) ~ go is often a permissible approximation. Moreover, for most simple atoms and ions go is close to unity, so QxQy is at most a few units. Conversely, Qz is very large, because even at moderate temperatures a large number of rotational states, and perhaps a few vibrational states, are occupied; for T = 1000 K anda typical value of B (~ 1 cm~!) in (9.28), Qrot is about 700. Consequently, an appreciable number of molecules may stay stuck together even when kT is larger than D. Ionization equilibrium, corresponding to the process At+e

/? gig Qz4i1(f) i Ra h Wage eal.)

( ee

Fa

=)

.

(9.39)

.

228

9. Radiative Transfer and Population Distributions

To get an idea of the degree of ionization in real situations, we shall put some numbers into Saha’s equation. Inserting numerical values for the

constants, (9.37) becomes NeNi ies =4.

—— Ne

4.83 x 10

Vi 2173/2 ay Tote a

exp

LLB F lOc 4 TD ne

(9.40)

where the number densities are in m~* and J is the ionization potential in volts. Since the internal partition functions of both atoms and ions are likely to be only a few units, as remarked in the previous section, the ratio Qi/Qa is usually of order unity and is not very strongly temperature-dependent. There is an appreciable degree of ionization even when y >> kT’. For an ionization energy of 10eV (which is mid-range for the periodic table) and T = 10000 K, the exponential factor is only about 10~°, but the right-hand side of the

equation works out to be 4.8 x 10%. For an electron density of 10'7 cm~%, or 1072 m~3, such as might be found in a high-current arc, we have nj ~ Na, or 30% ionization. It is worth contrasting this figure with the population of excited states in the neutral atom: an excited level 10 eV above the ground state at this temperature has a population only about 10~° of that of the ground state. As a particular example, we calculate the degree of ionization of calcium

(I = 6.1 V) at this same temperature and electron density. For Ca and Ca™ the ground states are 'Sp and 7S, /2, respectively, giving Qi/Qa = 2. The ratio ni/na then comes to about 100. If only 1% of the calcium atoms are neutral, what about the next stage of ionization? Cat has J = 11.9 V, and as

Cat* has a 'So ground state, the ratio of the partition functions is 1/2. The Cat*/Ca* ratio is found to be only about 2%; about 97% of the calcium exists as Ca’. In stellar atmospheres, where the temperature is likely to be a few thousand K, the degree of ionization is higher than in laboratory plasmas of comparable temperature because of the much lower electron density. For example, in the Sun’s lower chromosphere T ~ 7000 K and ne ~ 10!° cem73. Astrophysicists usually replace the electron density by the electron pressure, P. = nekT’, when using Saha’s equation, and in this example the appropriate electron pressure is 10~? N m-*. Substituting the calcium values in Saha’s equation, one finds that only about one Ca atom in 10” is neutral in these conditions.

9.6.2

Conditions

for Ionization Equilibrium

Like excitation, ionization and the inverse process of recombination can proceed by both radiative and collisional processes:

A+hy

—>

A+e”

At+2e~+AE,

At+e"+AE,

(9.41)

(9.42)

Further Reading

229

where AF is the energy balance in the form of electron kinetic energy. The continuous radiation absorbed or emitted in the first of these processes will be described in Sect. 10.6. Here we are concerned only with the conditions for the validity of Saha’s equation. The arguments are similar to those given for the validity of Boltzmann’s relation. If the radiation density is low, ionization is predominantly due to electron collisions, and Saha’s equation is valid if the three-body recombina-

tion rate, (9.42), exceeds the radiative recombination rate, (9.41). There is also an astrophysically important category of plasmas, stellar atmospheres, in which photoionization exceeds collisional ionization, and Saha’s equation is valid on the basis of the balance of radiative transitions only. However, in coronal conditions, we find, as for excitation, collisional ionization balancing radiative recombination, and we are back to having to know the individual cross-sections. Both the upward collisional rate A +e7 — At +2e7 and the downward radiative rate At + e~ — A+ hy are proportional to electron density, which suggests that the ratio nj/n, should be a function of T’ only,

independent of ne in contrast to (9.37). However, the two-level model is inadequate to describe the equilibrium because ionization is predominantly from the more populated lower levels, whereas recombination can take place into any of the bound states. A full discussion is given in Mihalas’ book.

Further Reading — Mihalas, D., Stellar Atmospheres (Freeman, San Fransisco, CA, 1978). — Cowley, C.R., The Theory of Stellar Spectra (Gordon & Breach, New York, 1970). — Unséld, A., Physik der Sternatmosphdren (Springer, Berlin, Heidelberg, 1968). — Aller, L.H., Atoms, Stars and Nebulae (Cambridge University Press, Cambridge, 1991). — Mitchell, A.C.G. and Zemansky, M.S., Resonance Radiation and Excited Atoms (Cambridge University Press, Cambridge, 1934, reprinted 1971).

10. Elementary Plasma Spectroscopy

The preceding chapters have shown that the widths, shapes and intensities of spectral lines depend on the temperature, pressure and electron density of the environment of the atom or molecule, as well as on its intrinsic properties. If the broadening and other physical processes are properly understood and the necessary atomic parameters are known, the spectral lines can give information about the physical conditions in the emitting or absorbing gas or plasma. Under laboratory conditions spectroscopic techniques have the advantage over some other methods — those involving probes, for example — that they do not interfere in any way with the plasma, and in the early days of plasma physics spectroscopy was an important diagnostic tool. Other optical diagnostic techniques involving lasers have now supplemented and to a considerable extent superseded the purely spectroscopic ones, but there remains an important role for spectroscopy in determining the physical processes going on in the plasma. Moreover, the understanding of laboratory plasmas is an important prerequisite to the use of spectroscopic methods on astrophysical plasmas for which no alternative methods are available. In this chapter a brief account of plasma parameters (Debye radius and plasma frequency) is followed by discussions of the depression of ionization potential and the constraints on the attainment of thermodynamic equilibrium in a transient plasma. Continuous emission and absorption are then treated, and the remainder of the chapter is concerned with a range of applications.

10.1 Composition of Plasmas The word ‘plasma’ is sometimes used loosely both in this book and elsewhere to describe a mass of hot gas regardless of its degree of ionization. The proper definition of a plasma is quite unambiguous, as shown in Sect. 10.2, but the word is a convenient shorthand to describe any vapour or gas that is partly ionized. For the moment, then, we consider a plasma to be an assembly of atoms, molecules, ions and electrons that may be treated statistically in the same way as a solid, liquid or unionized gas. The plasma as a whole is electrically neutral, the total ionic charge being equal to the number of free electrons. It

Day

10. Elementary Plasma Spectroscopy

can easily be shown that the charge neutrality must apply to quite-a small volume: if n; and n, are the number densities of ions (assumed singly charged) and electrons, respectively, a sphere of radius r contains a net charge Q of ar (nj —n,.)e, and the corresponding rise in potential V is given by

Q P) A4mTeEor

_ e(nj—ne)r? 3&0

With a typical value of ne, 107? m~%, and a 1% difference between n; and Ne, the rise in potential at the surface of a sphere of radius 1 mm

works out

to be nearly a million volts. Such a large potential difference cannot possibly be maintained in a gas, where the particles can move freely to neutralize the excess charge. It is therefore quite safe to take the average particle densities as given by n; = ne until one gets down to distances so small that the out-ofbalance potential energy is comparable with the thermal energy. The order of magnitude of such a distance is given by the Debye screening radius pp. Note

on Units

In the SI system the unit of number density is m~°, but in the references cited in this chapter and in nearly all current work on plasma spectroscopy number densities are given in cm~*. We have used m7? in the equations in this chapter for consistency, but we have deliberately used cm~° descriptively from time to time because the advantage of familiarity with the actual numerical values seems to outweigh the disadvantage of having to multiply occasionally by 10° when inserting a number density into an equation. Plasma physicists frequently measure temperature in electron volts. Since LeV=1.6x10719 J, and kT = 1.4x10-8T J, the temperature corresponding to 1 eV is 11400 K. For semiquantitative purposes, 1 eV= 10000 K is a sufficiently good approximation.

10.2 Debye Radius The distinguishing characteristic of a plasma is the importance of collective effects. Every charged particle interacts with several other particles at the same time because of the long range of electrostatic forces, and the motions of the particles are therefore correlated. The Debye shielding radius pp is a criterion of the importance of collective effects: it measures the distance to which the electric field of an individual ion or electron extends before it is effectively shielded by the oppositely charged particles. Individual interactions between particles are important only over distances less than pp, while over larger distances the collective effects dominate. The criterion for the existence of a plasma, or the predominance of collective effects, therefore amounts to requiring pp to be much smaller than the dimensions of the assembly of particles.

10.2 Debye Radius

233

The expression normally used for the Debye radius is that originally derived by Debye for an electrolyte. Suppose a particle of charge gq is injected into an electrically neutral plasma, for which the mean potential is zero and nN; = Ne. In the neighbourhood of g the potential has some value V, and there is a local charge imbalance giving a net charge density p = e(n;i — ne). If the particle density is sufficiently high for V to vary smoothly, unaffected by individual ions or electrons, Poisson’s equation must be satisfied:

fy atoll E0

E0

An electron at potential V has the energy —eV. In equilibrium the number of electrons with this energy is given by Boltzmann’s equation (7.17) in the form Ne = me

since Ne = Ne, when V = 0. Similarly for the ions,

n; = THe

=Heen

«

The differential equation determining V becomes V2V

=

_

Me (g=eV/kT ae a

lice :

EO

Assuming that eV < kT, which will be justified below, VV

a

ene 2eV

2e7?Ne

do gill tect ee

For the notional point charge disturbance, V has spherical symmetry, so that

iro OT

OV Or

VV=s>—

Gea

For the boundary conditions V + 0 as r > co and V > q/(4me0r) as r > 0, this equation has the solution (easily verified by substitution) q ree

=

—r

/pd

.

where al Pp =

eEgkL 20271. :

pp determines the ‘decay distance’ of the potential distribution produced by the charge q, and this is defined as the Debye shielding radius, i.e.

pp (3r-) =

EegkL

| ———

1/2 :

(10.1) IOS

An expression of the same order of magnitude is obtained if one considers the penetration of an external field from the boundary of the plasma, in which case pp may be regarded as a skin depth.

234

10. Elementary Plasma Spectroscopy i

10°

1

10

2

10°

4

temperature

(ev)

,

2

>

5

ip)

5 is) 2

= =®

(oe

oO

eae

a &

10

2a ro

‘3

3D

&

OD

8

*

10

8

cS

© (2) Oo

&fe)

10 6

10°

|

1

interstellar

4

10

10

4

10

6

| 0° temperature

(K)

Fig. 10.1. Temperature and electron density for different types of plasma. The corresponding plasma frequencies are marked upon the right-hand side of the diagram, and the diagonals are lines of constant Debye radius pp. The region marked X corresponds to plasmas generated by lasers, vacuum sparks and fusion devices, and to stellar interiors.

In the cases of transient plasmas or very rapid disturbances, the ions may not be able to move fast enough to conform to the instantaneous equilibrium charge distribution. Then only the electrons can be effective in the shielding, and p?, must be increased by a factor of two. Putting the numerical constants into (10.1) gives

1/2 pp = 50 (=

m,

(10.2)

where T is in K and n, is in m~®. For an arc having T = 10* K and n, = 1022

m~* (i.e. 10'° cm~) pp comes to about 5 x 1078 m, which is obviously

10.3

Plasma Oscillations

235

very much smaller than the dimensions of the arc column. High-temperature

pinches (JT ~ 10° K) of relatively low density (ne ~ 10'4 cm~3) have pp ~ 10~° m, which is still orders of magnitude smaller than the pinch dimensions. In the solar corona, with T ~ 10° K and n. ~ 10% cm~°, pp is of the order of several cm, but, of course, the dimensions are very many orders of magnitude greater. Figure 10.1 shows values of pp for various types of plasma. The derivation of pp was subject to the restrictions that V be a smoothly varying function and that eV < kT’. In fact, these two conditions amount to the same, at least so long as the perturbing potential is attributable to an isolated ion or electron. The first requires that the Debye sphere contains a large number of particles:

Eliminating 72 between (10.1) and (10.2) gives the inequality: 3e?

i

>

——..

IPTG,

The mean electrostatic energy eV of the electrons surrounding a charge e is of order e?/(4e9pp), leading directly to the second condition

a

2 ATEQ PD

3

(10.4)

Plasmas in which the Debye sphere contains only about one particle, so that (10.3) is not satisfied, are known as non-Debye plasmas. They exist in stellar interiors and are of considerable theoretical interest. Their formation requires a combination of high ionization with relatively low temperature which may possibly be achievable in the laboratory in laser-compressed plasmas or dense pinch discharges. The spectroscopy of dense plasmas has been

reviewed in [27].

10.3 Plasma

Oscillations

Just as the Debye radius represents a typical length for collective action, so the plasma frequency — or rather, its reciprocal — is a typical time. The plasma frequency is the resonance frequency for collective oscillations of the electrons about their equilibrium positions. It can be derived as follows. Consider all the electrons in a slab of plasma of unit cross-section and length | to be displaced simultaneously a distance « from their equilibrium positions, where « < |. The amount of displaced charge is enex. The slab now has surface charges +o at one side and —o at the other, with o = enex

(Fig. 10.2). The electric field produced within the slab by this charge is

a/€o. Each electron therefore experiences a force eg€o, OF e*nex/E0, and its

equation of motion, if one neglects collisions and thermal energy, is

236

=

10. Elementary Plasma Spectroscopy

Fe

>

—~>x

=

Fig. 10.2. Plasma oscillations — see text.

2 We . ecita (geo ——

es

£0

This is the familiar equation of undamped solution is

simple harmonic

motion.

The

L=asinwpe ,

where the angular plasma frequency wy is given by

ole

:

(10.5)

Egm

The plasma frequency in Hz is

Wp >=

21

Ne ag (=) == ATeEgm

/ Me,Ze

(10.6)

where the numerical value is for ne in m3. For the laboratory plasmas dis-

cussed in the previous section, with n. in the range 109-1074 m~% (or 10!4— 10'8 cm), one finds Vy» ~ 10''-1013 Hz, which puts the plasma frequency in the far infrared, between 3 mm and 30 um in wavelength. Values of Up for the various plasmas in Fig. 10.1 are shown on the right-hand side of the diagram. It should be emphasized that plasma oscillation is not just a resonance oscillation of one electron bound to a nucleus, of the type dealt with in classical absorption and dispersion theory: it is a collective motion of free electrons, and the restoring force exists only because all of the electrons are displaced together. This collective behaviour tends to disappear if the electron motions are randomized by collisions, and a necessary condition for its existence is that the time between collisions must exceed the oscillation period: the collision frequency v, must be smaller than vp. The fact that the plasma frequency frequently corresponds to that of radiation in the infrared region suggests that the propagation of radiation through a plasma should undergo some change when the two frequencies coincide. This change is actually a cutoff, as shown below. A preliminary qualitative look is worthwhile for the insight it gives into plasma processes.

10.3

Plasma Oscillations

DEW

If we eliminate the electron density between (10.1) and (10.4), we have, for the relation between Debye length and plasma frequency, kT PDYp

=

(

iy? )

f

(10.7)

The mean electron velocity is given by (1/2)mv? = (3/2)kT, so the right-hand side of (10.6) is approximately plasma oscillation the electron what it needs to do in order to field such as an electromagnetic

equal to v. Thus, in the time taken for one travels about one Debye length. This is just cancel out the disturbance from an external wave. When w > wp, the disturbance is too fast for the electrons, but when w < wp, the electrons can move fast enough to annul the wave and therefore prevent it from propagating. For a quantitative treatment we need an expression for the refractive index of a plasma. Equation (7.36) for the refractive index of a medium arising from the oscillations of bound electrons can be adapted for free electrons by setting the resonance frequency wo equal to zero on the grounds that there is no restoring force for individual free electrons in a plasma. If there is no absorption the damping factor 7¥ is also zero, and (7.36) becomes (with « = 0) g nes SeEgmw 2 =1--. Ww 2 Wace

2

Ww

(10.8)

As w decreases from infinity to wp, Ne goes from 1 to 0. The phase velocity of the electromagnetic wave is given in the usual way by v = c/n, so it is greater than c when w > wp, approaches infinity as w + wp, and becomes

imaginary for w < wy. In terms of the wave vector k(= 27/A), we have v=

Ww c = = eo on rt awe yer

(10.9)

giving

wi — oe ee

(10.10)

c

It can be seen that as w decreases, k tends to zero at the plasma frequency and then becomes imaginary. The propagation of electromagnetic energy is determined by the group velocity u, obtained by differentiating (10.10) dw Se

y

dk

w?

-(

gi

=)

10.11

(

)

u is, of course, always less than c, and it approaches zero at the plasma frequency, where energy propagation ceases. The behaviour of v,u and k as a function of w is illustrated in Fig. 10.3. At some distance from the cutoff, when w >> wp, and 7¢) is fairly close to

unity, (10.8) can be expanded to give

238

10. Elementary Plasma Spectroscopy

Fig. 10.3. Propagation of electromagnetic radiation for plasma frequency wp. The diagram shows phase velocity v, group velocity u, and wave vector k as a function of

velocity

See hae

2

2

EG =

Qw2

Tae eee Yegmw?

2

ele a 8r2egmc2

10.12 )

(

The comparable expression for bound electrons is eq. (7.39), with the damping term negligible so long as we are well away from an actual absorption line: 2

€

TN

ot oryTWILL

Ni fig

p Merc calCe

ij

2°

J

Most strong absorption lines fall in the visible and ultraviolet, so in the infrared it is reasonable to assume that v < y;;, in which case (7.39) can be expanded as

€

2

apt)

Ni fi

ij

+... te. iad aeThies Naa oa8m2Egm py Ui, ( reas ) =

e

8m2egme2

oy :

a)

ni fig Ai;

1+

Mi

2

Sao

F

(10.13)

This expression varies very slowly with A in contrast with the dependence of the electron refractive index in (10.12), and it is therefore possible to separate the two contributions to the total refractive index by measuring the latter at two different wavelengths. Such a pair of measurements gives vp and hence Tas The propagation of electromagnetic waves through a real plasma is considerably more complicated than this simple treatment indicates. Thermal motions and collisions tend to smear out the cutoff frequency, and electrons in highly excited states occupy a position intermediate between free and bound. Moreover, we have not considered at all the effects of external magnetic fields, which are important in many astrophysical and laboratory plasmas. These

10.4 Thermodynamic Equilibrium in Transient Plasmas

239

produce a different type of collective motion, with the electrons spiralling round the field lines with the cyclotron frequency w. = eB/m and the ions

with the much lower frequency (w.); = eB/M.

10.4 Thermodynamic

Equilibrium in Transient Plasmas

The general conditions for the existence of thermodynamic equilibrium — or, more particularly, local thermodynamic equilibrium (LTE) — were discussed in Sect. 9.4, and Sect. 9.6 applied them to the case of ionization equilibrium. This discussion implicitly assumed a steady state. In a transient plasma it is also necessary to consider the time required to reach this steady state. Many high temperature plasmas last for only a microsecond or so. In the initial stages the particles are accelerated by strong fields, and it is not until their velocities have been randomized by collisions that it is valid to define a temperature. The time required varies inversely with n,. A full discussion

may be found in Griem’s book and in references such as [26, 28]. To give an example, full LTE in a plasma at 10 000 K would take about 1 us for ne ~ 10'° em~® if collisional processes only were involved, but the time may be significantly shortened by radiative excitation. The time required to establish partial LTE among the more highly excited states would be about 10 ns in this example. Although LTE demands a well-defined electron temperature, it does not necessarily follow that the gas kinetic temperature determining the translational velocities of the ions is equal to the electron temperature. Elastic collisions are very inefficient in transferring energy from electrons to heavy particles because of the large mass difference. As only about 2m/M of energy is transferred at each collision, about 1000 collisions are required for approximate equipartition of energy. Nevertheless, in the example given above, the time for kinetic equilibrium between electrons and hydrogen ions is only about 10 ns, and the transfer from H-ions to other ions and neutrals is an efficient

one. The same considerations apply to the establishment of ionization equilibrium. For example, in a laboratory plasma of relatively low density and high

temperature such as a pinch discharge (ne ~ 10'° cm~3, T’' ~ 10° K), (9.38) gives nz41/nz ~ 5 x 10° exp(—I/100), where J is the ionization potential of the z times ionized atom in volts. An appreciable density of the stage z + 1 should be attained for a value of I of about 2000 volts (making exp(—J/100) about 10~°). This corresponds to some 10 to 12 stages of ionization for most species. Such high ionization is not in fact reached in most pinch discharges because the electron density is too low to establish LTE during the lifetime of the discharge. The much denser plasma focus devices can attain ionization stages up to 18 or so.

240

10. Elementary Plasma Spectroscopy

10.5 Depression of Ionization Potential and the Inglis—Teller Limit An atom in an environment of charged particles can no longer be treated as isolated; the energy levels of the valence electrons are affected by the surrounding fields, to an ever greater extent as their excitation increases, and the distinction between a bound and a free electron becomes somewhat blurred. There are in fact. two different effects to be considered, one related to screening and the other to Stark broadening. The distance (r) of a valence electron from its parent nucleus increases as n*, and when (r) is approximately equal to pp, the limit of influence of the nucleus, the electron is no longer effectively bound. Thus ‘ionization’ takes place when the potential energy of the electron is approximately —ze?/(4me9pp) instead of zero; ze is the nuclear charge seen by the valence electron, so z = 1 for a neutral atom, 2 for a singly charged ion, etc. The effective reduction in ionization potential Ay is given

by

ere

X=

ze 2

e-n

: ney (5)

bie

Sei!

dia

Me \ 1/2

es (=)

V

In a typical high current arc, with T ~ 10000 K and ne © 101” cm~3, Ay is only about 0.1 eV, which is too small to measure accurately. The experimental uncertainties make it difficult to decide conclusively in favour of any one of the various more sophisticated expressions for Ay. The depression of the ionization potential removes a difficulty about the divergence of partition functions at high temperatures. In any atom there are an infinite number of bound states crowding up towards the ionization limit, and unless the exponential function exp(—E;/kT) is vanishingly small for these states — i.e. x > kT — the sum diverges; but if the series is terminated at a value of n, say n*, such that (r;«) = pp the sum is necessarily finite. n* is related to Ay by

(1/n*)? = Ay, where Ay is in rydbergs. This ‘depressed series limit’ must be distinguished from the superficially similar Inglis—Teller limit. The latter is defined by the n value at which the individual energy levels are sufficiently Stark-broadened to merge with one another. ‘The dependence of this limit on n. can be estimated by treating all atoms as hydrogen-like, which is a good approximation at high n values. The spread of the Stark pattern (Fig. 3.18) is then proportional to n(n—1)é, as shown at the end of Sect. 3.9.2. Averaging over the fields € from different neighbours converts the Stark splitting to Stark broadening, proportional at

large n to (E)n*. However, the separation of states of high n, d(1/n?)/dn is proportional to 1/n°. The value of n, say n», at which the broadening catches up with the separation, defines the Inglis—Teller limit. In the quasistatic ap-

proximation discussed in Sect.8.4.4, (€) can be taken as the electric field

10.6 Continuous Emission and Absorption

241

from a particle at the mean interparticle separation R — i.e. (€) « 1/R?. Since R x ne au 2 the broadening should be proportional to no! °n2 and we 5

—1/4

expect to find n?, one’.

Mm is in general lower than n*. This is understandable, in that n,, limits the radius of the atom to the mean interparticle distance R, while n* limits it to pp, which has already been assumed to be greater than R. The functional dependence of n* on ne is also different from that of n,,: from the relations

n* «x (Ay)?

and Ay « ne!” it follows that n* « no !/*.

10.6 Continuous Emission and Absorption The discussion of emission and absorption of radiation up to now has been almost entirely concerned with line radiation, involving transitions between two bound states. When there is an appreciable degree of ionization, radiative transitions between bound and free states become important, and so too does radiation emitted or absorbed by free electrons in the neighbourhood of ions. As the free states are not quantized, all such transitions involve continuous rather than line radiation. The bound-free transitions correspond to the photoionization and radiative recombination processes mentioned in Sect. 9.6.2. Transitions to or from the bound state j are represented by

Aj thy

Att+e

+e,

(10.14)

where

Wie — Lo G. In this equation y is the ionization energy and ¢(= smv?) is the kinetic energy of the electron (Fig. 10.4). The radiation, in both emission and absorption, is

a continuum extending in the short-wavelength direction from the line series

Fig.

10.4.

Bound-free

free transitions — see text.

and

free—

242

10. Blementary Plasma Spectroscopy

limit corresponding to the particular lower state H;. One such continuum extends from each bound state, so the whole bound-free (bf) continuum is characterized by steps, or edges, as shown in Fig. 10.6 below, corresponding to the different series limits; as v is increased, a new chunk of continuum is added whenever hy reaches a new bound level.

Free-free (ff) transitions, also illustrated in Fig. 10.4, are associated with loss or gain of energy by an electron in the field of an ion. This type of radiation is expected classically from a charged particle constrained to follow a curved path because the particle is necessarily accelerated in the process. It is also known as ‘Bremsstrahlung’ (literally, braking radiation). By energy conservation hy = Ae, and there are no characteristic edges. At long wavelengths the free-free emission approaches the blackbody function, as discussed in Sect. 10.8. Bound-free and free-free transitions are important in both laboratory and astrophysical plasmas. The principal features are discussed in this and the next two sections. Reference [28] or books on stellar atmospheres such as those listed in the section Further Reading should be consulted for a fuller treatment.

For bound-free radiation there must exist a general relation between the cross-sections for photoionization and radiative recombination analogous to that between the Einstein coefficients for line radiation. The relation can be obtained by assuming the system to be in LTE, so that the source function (the ratio of emissivity to absorption coefficient) is equal to the blackbody

function as in eqs. (9.8) and (9.11): V ae =13(y,T)

or

aV ase

(10.15)

k’(v) here is the absorption coefficient corrected for stimulated emission (9.9) and LP is the Planck function; in the many cases of interest where hy > kT, the second relation using Wien’s approximation is valid. It is convenient to work with cross-sections rather than absorption and emission coefficients, so we define the continuous absorption cross-section in the same way as for a

line (7.28): a(v) = k(v)/n, .

(10.16)

a(v) can be corrected for stimulated emission in the same way as k(v) for line radiation — that is, in LTE

a!(v) = a(v)(1— eT hY/ FP) |

(10.17)

The cross-section for the inverse process of radiative recombination from the continuum to a bound level 7 depends on the electron velocity v as well as

the particular bound level and is designated by a;(v) (Fig. 10.5). The rate of capture of electrons of velocity v to the state j is then dne(v)va;(v) per ion. For each recapture the energy radiated per unit solid angle is hv/4a. The emissivity (per unit volume) for a number density nj; of ions is given by

10.6 Continuous Emission and Absorption

243

Fig. 10.5. Photoionization and radiative recombination. n; is the total number density of ions and dne is the num-

ber density of electrons with velocities in the range v to dv.

é hv j(v)dv = Jy rive (v)dne(v) : where dv is the frequency spread corresponding to the velocity spread dv. The two are related by putting € = smv? in (10.14) and differentiating:

hdv = mvdv . With this substitution and Maxwell’s relation for dne, we have for LTE :

is

Hee

as

m tomer

3/2 ) nnevv"a;(v)e/ FF :

(10.18)

We can now evaluate j/k and set it equal to the Wien function (the same

result is obtained by setting j/k’ equal to the Planck function):

ee k(v) nj;

(

nr er

Ug

—

O26

y= Ee re

exp |——_>—_] :

sll

p

—hv

exp | —~ ] f

od

The exponentials conveniently vanish by virtue of the relation of (10.14), leaving

upeal greats

Q;

(10.20)

mActu-Gio

This is known as the Milne or Einstein—Milne relation. Although it was derived for a system in LTE, T does not appear in it, and we can use the usual arguments,

that the cross-sections

are intrinsic

atomic

constants,

to

take the relation as valid in any conditions. One of its practical uses is that capture cross-sections, and hence emissivities, can be found from the more easily measured photoionization cross-sections.

244

10. Elementary Plasma Spectroscopy

10.7 The Continuous Absorption Cross-Section The photoionization cross-section may be calculated in the same way as the cross-section for a bound—bound transition if the relevant atomic and ionic wave functions are known. In practice such calculations are difficult and somewhat unreliable except for hydrogen and the H-like ions. The most widely used expression for a(v) is based on a semiclassical calculation that extends the cross-sections for bound—bound H-like transitions into the realm of imaginary quantum numbers beyond the series limit. For absorption from the level of principal quantum number n the result is 3 x

Oa

arate (=)

Gn

Ys 2,

(10.21)

where apo is the first Bohr radius, a is the fine-structure constant,

yy is the

ionization energy of neutral hydrogen, and z is the effective nuclear charge (1 for a neutral atom, 2 for a single ion, etc.). G>'(v,z) is the Gaunt correction factor, introduced to take care of the discrepancy between the semiclassical and quantum-mechanical calculations. It is a function of n,v and z that is close to one for H-like ions. As all atoms and ions become increasingly H-like towards the more highly excited states, (10.21) is actually a reasonably good approximation for the higher states of any species. However, the detailed calculations that have been carried out for a number of atoms and ions (mainly those of astrophysical importance) show departures from H-like behaviour in the lower states, even for the alkali-like spectra, in which the continuous absorption passes through a minimum beyond the series limit instead of

decreasing steadily with 1/v°. Insofar as (10.21) can be taken as an acceptable estimate for the crosssection for atoms in state n, the total bound-free

absorption coefficient at

frequency v is found by multiplying by n, (the population of level n) and summing over all bound levels n that lie within a distance hy of the ionization limit. For the first step we have to assume Boltzmann’s relation to hold in order to get anywhere at all, remembering that only partial LTE is required so long as hy is much smaller than the ionization energy so that the lower states are not involved. For the H-like case the statistical weight of level n is 2n”, and its energy can be put in the form 2?yy(1 — 1/n?), giving Mn =

Na

——-_2n’

Cn

9

e3

8

=

a XH

leer

(1 — 1/n7)

where ng and Q, are the total number density and partition function of the absorbing atoms. (For simplicity we refer to the initial state of the absorbing

particle as ‘the atom’ and the final state as ‘the ion’, but the process can be understood to include photoionization from an ion of charge z to one of charge z+ 1.) The cutoff for the sum over bound levels can be expressed in terms of a quantum number n, determined by

10.7 The Continuous Absorption Cross-Section

245

n=1

= S

40

ES

000 k

aw

20 000 K

10 000 K

log v ao

Fig. 10.6. Bound-free absorption coefficient as a function of frequency for a H-like atom.

2 Wee

2 XH

Avie

(10.22)

The final expression is then Rf

a

64

=

DeveH

2 VA

ai a TON ee 2

2

1

XH

x ps as e€xp (3)

Gr

OT

pee Re Coe

4

(10.23)

A plot of log k(v) against log v has the sawtooth form shown in Fig. 10.6. Whenever v gets large enough to reach a new bound level, the absorption rises sharply. It then falls off as 1/v? until the next bound level is reached. The physical reason for the strong temperature dependence illustrated in the figure is that at low temperatures only the lower levels are sufficiently populated to contribute significantly to photoionization; k is thus very small except at the short wavelengths (high photon energies) that can reach the continuum from these lower levels. At all temperatures the edges get less pronounced as n increases because the population increments become smaller. For this reason the sum in (10.21) can be replaced by an integral for the higher levels, n > n’, where n’ is conventionally taken as 5. If we set the Gaunt factor

equal to unity and write (z*y"/kT) = , this integral becomes ~ ;

L @/n? reN /

—7

rive &/n? “~< 1 [ ae dn = “3

=n &/n? 49) Ml c d(1/n )

n=n

=e! il

12

= 1),

‘

(10.24)

246

10. Elementary Plasma Spectroscopy

From this we get a convenient long-wavelength approximation for Are). Uf Fi, is above n’, the condition for which is ZX Se or

hv
pm,

A>

the sum term disappears, leaving only the integral with n_ as its lower limit:

fone eee 64 ot 0" 2 75tN H exp 8Q,(T) 35 k*(v) = —=N, ;

KH 7 2 LT

(e hu/kT _ 4 ) :

[The last exponential is obtained by using (10.22) for n¢.] If the Bohr radius, the fine-structure constant, and also y7, where it appears outside the exponential, are expressed in terms of e,m,h and c, this becomes 327?

ila

e

\°

naz?kT

3V3 (=)

.

chv5Q,(T) eal

zxuk) is lane aa

Free—free transitions, which have so far not been considered, become important at high temperature and long wavelength. A semiclassical expression for the free-free cross-section of a H-like ion can be obtained by the same device of imaginary quantum numbers as was used for the bound-free crosssection. The result for an electron with velocity between v and v + dv is Ar

a,(v)

e?

dv = —=

ol)

‘

|—

3V3 (=)

oe

roa

} ———G'

m?c?vhv3

(v,z,T) dv,

(

10.26

)

(

)

where G® is another Gaunt correction factor, again close to unity so long as kT is below the ionization energy z7yq. To turn the cross-section into an absorption coefficient, a,(v) must first be multiplied by the number density of ion-electron pairs with electron velocity v, nidne(v), and then integrated over all uv for a fixed v. On the assumption of a Maxwellian electron velocity distribution,

167?

aa

k®(v) = nine—=

ke"

[=—

3/3

S75

\4me0/

—= Ge,

ch(2rm)3/2(kT)4/2v3

Still assuming LTE, nine can be obtained from Saha’s equation, with Q; equal to unity because a H-like ion is just a bare nucleus. Finally,

f

kf (v) = w)

16n2

/ e2

\° 22?nakT

Be

(—)

Sn

cn Qa(t)e

zy P

jul

eli

10.27

(

)

k>f and k™ have the same 1/v* dependence. At long wavelengths, where the edge structure can be ignored, k@ in (10.27) is actually identical with k! in (10.25) except for the last term and the Gaunt factor. If the latter is set equal to unity, the combined absorption coefficient assumes the relatively simple form peor

(yy)

=

kPf(v)

167?

3/8

+ b® (v)

e2

(—)

\°

227*nakT

ch4Qa(t)v3 exp(

2? yVH — Av

kr

).

ee

10.8

The Continuous

Emission

247

It must be remembered that this expression is valid only if there is at least partial LTE and that it is a long wavelength approximation, ignoring the edge structure in the visible and ultraviolet.

10.8 The

Continuous

Emission

The emission counterpart to k>!(v) in (10.23) and (10.25) is j>f(v), the spectral radiance per unit volume at frequency v for free-bound radiative recombination, integrated over all electron velocities and summed over all bound levels within a range hv below the ionization limit. The easiest way to derive

it is to combine (10.15) for the source function with (10.23) for k>f(v) and then use Saha’s equation to express the result in terms of the number densities of the initial state particles, ions and electrons, rather than that of neutral atoms. This route gives a misleading impression of dependence on LTE, as can be seen by using the alternative derivation. This starts from the semiclas-

sical calculation of @,,(v) given in (10.21) and uses the Milne relation (10.20), which is independent of LTE, to find the radiative capture cross-section to

state n, n(v). Equation (10.18) converts o,,(v) to j(v) for state n, assuming a Maxwellian velocity distribution for the electrons. It remains to sum over

all states n as far as the cutoff n. given by (10.22), Te

eS 2° xn/hv

;

to reach the same end point as the first route: 5 (v) fe, 1287 4

e€ee

8/3

\Anen)

Ga sy n?

Mz A

|

|

nythe en RY/RT

coh 2amkT3/4

peat

xn)

Gotmee”

n2kT

(10.29) (

N>Ne

The only equilibrium assumption made is that the electron velocities obey Maxwell’s relation.

Evidently (10.29) has the same edge structure as (10.23), but the emission falls off exponentially between the edges instead of as 1/v3. This can be seen on the right of Fig. 10.7. At wavelengths long enough to ignore the edge

structure we have as the counterpart to (10.25):

Leh ee oa

:

3/3

gh

A

\4ne0/

me3(2amkT)1/2

nite (1 — ehe/*T) * ©

:

(10.30)

where, as before, G has been set equal to unity. The free-free emission can be derived in a similar fashion from either

(10.26) or (10.27). Again, ignoring the Gaunt correction, one obtains f(y)

:

=

167

3/3

e2

3

2?

\4re9 / mc3(2rmkT)/2

rien

* *

eu

/kT

(10.31)

10. Elementary Plasma Spectroscopy

248

b

@ = 5S

S Ss

ay

s

Vy!

Vv

Fig. 10.7. Continuous emission coefficient as a function of frequency. v,” is the frequency below which edge structure is insignificant.

on condition only that the electron velocities obey Maxwell’s equation. As with the absorption, we can combine the two types of emission to find the total radiance at long wavelengths:

eom'(v) = iv) + 5%)

4)

167 e? : Zz angels (aie ie: a3 (—) me (2amkT)/2 1"

10.32 sae!

This expression has the interesting property of being independent of frequency — that is, the total continuous emission per unit frequency interval is constant for v < Vp, the frequency at which the edge structure becomes significant. This relation does not depend on the existence of LTE, the only requirement being that the electrons follow a Maxwellian distribution. At very small values of v, as hv/kT — 0, j°°"" is increasingly dominated by the free-free term. It cannot continue constant to indefinitely long wavelengths because, as shown in Fig. 10.7, it must eventually hit the blackbody curve, represented in this region by the Rayleigh Jeans approximation of eq. (7.4) — in other words, the plasma becomes optically thick. Thereafter the radiance follows the blackbody curve, falling off as v2. The reason for the increasing optical depth is clear from (10.28): for hv/kT — 0, the absorption coefficient increases as 1/v* and must eventually become very large, whatever the particle density. As a matter of fact, the optically thick limit may not be reached before the plasma frequency 1, (10.6), in which case the emission falls abruptly to zero at v = yp. Energy losses through Bremsstrahlung are a significant cause of cooling in laboratory plasmas. The z? factor in (10.31) shows that it is important to eliminate as far as possible impurities of high atomic number from a hydrogen or helium plasma if the object is to get it hot.

10.9

10.9 Other

Other Continua

249

Continua

The equations for continuous emission and absorption given in the last two sections are strictly applicable only to H-like ions. The general behaviour for many-electron ions is similar, but significant deviations are to be expected at shorter wavelengths when low-lying states are involved in the free-bound transitions. An additional complication may arise from autoionization, which causes anomalous bumps, or resonances, in the photoionization cross-section (Sect. 3.5). The inverse process of dielectronic recombination (capture of an electron into a bound state above the ionization limit) may enhance the electron capture cross-section sufficiently to have a significant effect on the calculation of rate processes in plasmas. The negative hydrogen ion, a proton with two electrons, is an important constituent of many stellar atmospheres (see the section Further Reading at the end of the chapter), and other negative ions play significant roles in some laboratory plasmas. Negative ions have only one stable state, and their emission and absorption is therefore all continuous. Photoionization leaves the neutral H-atom in one of its bound states, the extra energy being carried away by the electron: H

+hvy—-—H+e

+e

with

Ahyv=yxy

+Fy,te.

x” is the detachment energy, or electron affinity. For H~, y~ is 0.75 eV, which gives 1.65 um for the photoionization threshold. As the wavelength is decreased below this value, the absorption cross-section rises to a peak at about 850 nm and then drops again. Photodetachment cross-sections are of the same order as photoionization cross-sections, and the importance of negative ions therefore depends on their relative number densities. Saha’s equation is applicable in the form

MaNe _ (2nmkT)3/2 2Qa(T) —x-/kr ne he Cali) Not surprisingly, the formation of negative ions is favoured by large ne and small T. Detailed information can be found in [29, 30]. Another possible source of Bremsstrahlung is the cyclotron radiation from electrons moving in magnetic rather than ionic fields. The best-known example is the electron synchrotron described as a spectroscopic source in Sect. 15.2.2. In pinch discharges cyclotron radiation in the magnetic field containing the plasma may make a significant contribution to the free—free radiation. Its importance increases as T’, in contrast with the 1/7’ dependence of thermal Bremsstrahlung. Cyclotron radiation is important in the many astrophysical plasmas where there are strong magnetic fields, and it is held responsible for the strong continuous emission from certain supernovae.

250

10. Elementary Plasma Spectroscopy

10.10 Applications of Plasma Spectroscopy Plasma spectroscopy developed initially as a method of determining number densities and temperature parameters in plasmas. With the development of other diagnostic techniques — notably laser scattering and laser interferometry — much of the interest in laboratory plasma spectroscopy has switched into three different directions. First, a plasma may be regarded simply as a light source for the spectra of highly ionized species or for the determination of oscillator strengths and collision cross-sections, with T and n. determined by other methods. Secondly, in a plasma of known composition and temperature, the interactions of the particles with each other, with the plasma fields and with the radiation can be investigated through the broadening and shifts of spectral lines, the appearance of forbidden lines and satellites, the relaxation rates of the plasma after excitation or following a perturbation by a laser pulse, and so on. These aspects of plasma spectroscopy may be fol-

lowed up in [26, 28, 31, 32], which themselves give many further references. Thirdly, in very dense laser-generated plasmas the main non-spectroscopic diagnostics (refractive index measurements, Thompson scattering, etc.) tend to be unusable, and plasma spectroscopy again becomes a diagnostic as well as a means of studying radiation transport and plasma dynamics [27]. In astrophysics, spectroscopic methods remain the only ones available. The study of astrophysical plasmas has benefitted greatly from the ‘calibration’ of spectroscopic techniques in laboratory plasmas, as well as from the greatly improved and increased data on oscillator strengths and collisional cross-sections stemming from laboratory work. Measurements of temperature, elemental abundances and transition probabilities are inevitably bound together because line intensities, measured in emission or absorption, depend on all three quantities. The experimental determination of transition probabilities is discussed in Chap. 16, where it will be seen that on the whole the most accurate techniques are those that involve relative measurements and do not depend on the existence of LTE and a well-defined temperature. However, the inverse process of using line intensity ratios to establish the existence of local temperatures still finds applications, particularly in astrophysics, and this chapter will conclude with a brief review of spectroscopic temperature

measurements.

10.11 Spectroscopic Measurement It is important

to remember

of Temperature

that a spectroscopic measurement

of tempera-

ture is actually a measurement of a parameter determining some equilibrium distribution. Unless the equilibrium exists, the measurement

will give limited

information, and in any case different distributions may be characterized by different temperature parameters. Of the methods described here, the first measures the velocity distributions of atoms or ions and the last measures

10.11

Spectroscopic Measurement of Temperature

251

that of electrons. The others measure population (Boltzmann) or ionization (Saha) temperatures and therefore depend on the existence of LTE over the excited states involved. 10.11.1

Gas Kinetic Temperature

It is shown in Sect. 8.3 that the FWHM of a Doppler broadened line is determined entirely by the atomic or molecular weight of the atom or ion and the temperature parameter describing its velocity distribution. By (8.17)

Sip OAp) -2/f2RTIn2\*? weil scien =e a ) = 716 X10 4 When using this relation to determine T, it is essential to make sure that instrumental width and pressure broadening have been allowed for properly. As pressure effects usually show up first in the line wings, the FWHM may not be greatly affected, and it may be possible to avoid the effects altogether by using a transition between inner shells, sufficiently shielded by outer electrons to have a pure Doppler profile. In certain cases Doppler broadening may be caused by bulk motions of the gas - macroscopic turbulence or ions in an electric field, for example. Non-random velocities of this kind do not define a temperature.

10.11.2

Population or Excitation Temperature

The temperature parameter in the Boltzmann distribution is usually found by measuring relative intensities of emission lines of known transition probability. By eq. (8.2) the integrated intensity in the line 2—r1 is To

Xx ih q(v) OW — line

nz Aoihv42/(47)

;

For a Boltzmann population distribution this becomes Io, = const. QAsvieen

F

(10.33)

where the constant includes all the geometric and instrumental factors common to all lines in the measurement. A plot (Boltzmann plot) of log(JA/gA) against E for several spectral lines should therefore be a straight line of slope

be 1/kT. A number of precautions need to be taken. First, the source must

optically thin in the direction of observation for all lines used. Second, the integration over the line profile must extend far enough to include the far wings, which may mean several times the FWHM if the line profile has a significant Lorentzian component. Third, the base line has to be carefully drawn into exclude scattered light or underlying continuum. Fourth, the method is usually this and range, energy wide reasonably a sensitive unless E covers implies a wide wavelength range, which makes the calibration of the spectrometer more difficult. Finally, A values may be lacking or inaccurate.

202

10. Elementary Plasma Spectroscopy

Vibrational and rotational levels of molecules give useful Boltzmann plots for temperatures below about 5000 K. They have the advantage that their relative transition probabilities can usually be calculated rather accurately. The relative rotational line strengths are the Honl—London factors Sy jv discussed in Sect. 7.11. In terms of these factors and (7.61) relating transition probability to line strength, we have I=

const. v?

Syn

Pipl

iA

Colstr i

oem

ee

5

(10.34)

Since By = BJ'(J’ +1), a plot of log(IA*)/S jy against J’(J’ + 1) should give a straight line with a slope of B/kT. A modification of this method is based on band maxima: as J increases the intensity in any branch of a band first increases because of the Hénl-London factor, which is approximately proportional to 2.J’+1, and then decreases because of the exponential factor. For two lines of equal intensity on either side of the maximum, follows from the last equation that

log(S,v4) = log (Shui) =

Hue

ae 1) —

J, and Jp, it

Jy (Je a 1)] ’

from which T can be found by matching rather than measuring intensities. Since the lines have the same intensity and approximately the same frequency, the method is applicable even when the plasma is not truly optically thin, but it is not easy to check its accuracy. The rotational lines of certain molecules whose spectra are nearly always observed as impurities in arcs and flames, notably OH and CN, are frequently used for thermometric purposes. High resolution is required to resolve the rotational lines, especially if the molecule is not a hydride. 10.11.3

Ionization Temperature

A much wider effective range of upper levels can be obtained by using different stages of ionization, together with Saha’s equation to relate the number densities. The relative populations of excited levels of different stages of ionization are given by the combined Saha—Boltzmann equation (9.39). In an optically thin plasma the intensity of a line J;,, from the kth level of the ion relative to that of a line from the jth level of the atom I,; is given by

Hy PUAN y 2amk?)P Lae siAAjer h3 ne

oe e~(X+En—Hs)/kT Gan

Note that one has to know ne as well as the A values and that there is an additional, although relatively slowly varying, term involving T’. The degree of ionization is more useful as an estimate of temperature than as a measurement with any degree of accuracy.

10.11

10.11.4

Spectroscopic Measurement of ‘Temperature

253

Reversal Temperature

This is actually a variant of the population temperature method, applicable only to laboratory plasmas at relatively low temperatures (e.g. flames or lowcurrent discharges) because it requires a continuous background source of variable temperature. The principle is quite simple. A spectral line from, say, a flame seen against a cold background appears in emission. If the temperature of the background source is turned up, the line must eventually appear in absorption against a bright background. At the reversal point, when the line vanishes against the background, the energy absorbed from the background is exactly equal to that put back by the gas. The brightness temperature of the background source is then equal to the Boltzmann temperature of the gas for the two states concerned in the transition. The advantages of this method, when it can be used, is that no transition probabilities are needed and the plasma need not be optically thin.

10.11.5

Optically Thick Limit

As shown in Sect. 9.3, the peak height of any emission line increases with increasing optical depth until it hits the blackbody ceiling as k(v)l — oo. In principle, therefore, the temperature of an optically thick plasma in LTE can be found from an absolute intensity measurement. Whether or not the line is optically thick can usually be checked by attempting to increase the intensity by reflecting the radiation back through the plasma or by viewing it end-on instead of side-on, but it is usually necessary to use sufficient spectral resolution to pick out the centre of the line. The method is inherently rather inaccurate because for T > 10000 K, the peak of the blackbody distribution is in the ultraviolet where absolute intensity measurements are difficult, and in the visible region where the measurements are easier, the intensity varies rather slowly with T’. An alternative is to measure the absolute intensity of the continuous emission at a wavelength sufficiently long that the plasma is optically thick (Sect. 10.8 and Fig. 10.7). This is feasible only if the optically thick limit is reached at a frequency above the plasma frequency, which in turn requires a critical electron density for any given temperature. 10.11.6

Intensity of Continuous

Radiation

Equations (10.30) and (10.31) for the bound-free and free-free emission of an optically thin hydrogen-like plasma show that the frequency dependence is governed by the term exp(—hv/kT). In the regions between the absorption edges a plot of logj, versus v should give a straight line of slope h/kT’, where T pertains to the Maxwellian velocity distribution of the electrons and not to any population ratios.

254

10. Elementary Plasma Spectroscopy

Further Reading — Mihalas, D., Stellar Atmospheres (Freeman, San Fransisco, CA, 1978). — Unsold, A., Physik der Sternatmosphdren (Springer, Berlin, Heidelberg, 1968). — Griem, H.G., Principles of Plasma Spectroscopy (Cambridge University Press, Cambridge, 1997).

art Lit

Experimental Methods

7

a

24 wo is given by dAgit = (dA/dl) w, and

(12.12) for the dispersion gives

iy

dA w _ dcosé!’ w ae rere ees a

(12.20)

Again, this expression can be simplified for the Littrow-type mounting,

which 6 = 6’ and mA = 2dsin@, to give

in

12. Dispersive Spectrometers: Prisms and Gratings

286

Relit =

A

_ 2ftané

dAslit ”

(12.21)

Ww

Comparison with (12.17) shows that, not surprisingly, the slit-limited resolving power is degraded by the factor wo/w from the diffraction limit: (1222)

LIL) Pe t Fe literal Wiley)

Throughput is inversely related to resolution by way of the slit width w. The relation is often quantified by taking their product RE to define a figure of merit that can be used to compare different types of spectrometer. From

(11.12), with A = wh, we have E = wh(1/4)(D/f)?, which can be written as

E = (w/f)(h/f)(aD*/4) . Using (12.20) for w/f gives

pales

mr hrD? dcos6’ f 4

°

We now replace the notional circular area of the grating, (7D?)/4, with the projected area of the rectangular grating, Wcos6’ x H, and also set the angular height of the slit, h/f, equal to (3, to give

jy tS —r WH

(12.23)

In the Littrow-type mounting this takes the extremely simple form

Rank =2sm0 BW

hia BW

A

The limits on the grating area are obviously the culties of ruling a large grating. In practice @ is slit-image curvature and alignment problems, to An interesting sideline on the figure of merit

(12.24) expense and technical diffialso limited, in this case by a value of 1/100 or less. is obtained by putting W =

Nd in (12.23) or (12.24); RE is then seen to be proportional to the total length NH of the grooves, or the overall distance travelled by the ruling diamond. 12.3.5

Concave

Gratings

It was first shown by Rowland in 1882 that if a grating is ruled on a concave mirror instead of on a plane surface it can act as its own collimator and camera lens. The spectrometer is reduced to the three essential elements of slit, grating and detector, which is particularly advantageous in the far ultraviolet where lenses do not exist and mirror reflectivities are very low. Rowland showed that if the slit and grating both lie on a circle whose diameter is equal to the radius of curvature of the grating, and if the grating is tangent to this circle, then for small apertures (the equivalent of paraxial rays in a

12.3 Grating Instruments

287

P

Fig. 12.15. Angles of incidence @ and diffraction 0’ for concave grating. C is the centre of curvature of the grating, A and B are adjacent rulings (with AB much exaggerated), S is the slit and P its image.

lens system) the spectral lines are in focus on the circle. The angular positions of the lines are given, as for the plane grating, by (12.7):

d(sin6@ + sin 6’) = mA , where d is the distance between rulings as measured along a chord across the erating arc. The Rowland condition can be derived quite easily for rays in a plane perpendicular to the grating if the width of the grating is small enough for it to be considered coincident with the circle instead of tangent to it. Figure 12.15 shows two adjacent rulings A and B of a grating whose centre of curvature is at C. Rays from the slit S are incident at angle 0, diffracted at angle 0’, and intersect at P. To a first approximation the path difference between the two incident rays SA and SB is dsin@, and that between the two diffracted rays AP and BP is dsin@’. Equation (12.7) therefore defines the condition for a

Fig. 12.16. Ray diagram for concave grating. The grating GQ’ of width W and radius of curvature R subtends angles a, 3,7 at C, S, P, respectively. v and v’ are the object (slit) and image distances, respectively.

288

12. Dispersive Spectrometers: Prisms and Gratings

principal maximum at P so far as this little bit of the grating is concerned. We now need to show that all rays from S that satisfy (12.7) intersect at the same point P, wherever they hit the grating. Figure 12.16 shows the grating GQ’ of width W and radius of curvature R, with S at a distance v and P at a distance v’. The angles of incidence and diffraction are @ and 6’ at G. If the small aperture condition W < v,v’ is fulfilled, the corresponding angles

at G’ can be written as

9+ d6 and 0’ + dé’, where differentiation of (12.7)

requires

cos @d6 + cos 6’ dé’ =0. The angles a, 3 and y subtended by GG’ at C, S and P respectively are given

by a=W/R,

B=Weosd/v,

y=WeoosO'/v .

It can be seen from the geometry of the figure that

@=a=04d0+6

and? +a=6 +d? +4.

Therefore,

dé = a— 3 and dl’ =a—-y. The condition that rays from S to P satisfy (12.7) over the whole width of the grating is therefore 1

cos 6 (5_ =

0

)+ cos

1

@’

(5- ae )= ()

i225)

In mountings based on the Rowland circle this is satisfied by making both expressions in parentheses identically zero — i.e. v = Rcos@ and v’ = Reos 6’. As shown in Fig. 12.17, S and P then lie on a circle of diameter R, which is the Rowland circle. This is the simplest and most aberration-free way of mounting the concave grating, but (12.25) can be satisfied by other geometric arrangements. Actual mountings are discussed in Sect. 12.4.2. Since plane and concave gratings obey the same basic grating equation, the expressions already derived for dispersion and resolving power still hold. The only modification is that the focal lengths of the lenses or mirrors are replaced by the radius of curvature of the grating, so that the reciprocal dispersion becomes

dX

Tee

1 di . dcos6’

orm

The ‘standard’ values for R are 1, 3 and 6 or 6.6 m, the last two being very large instruments. In fact, gratings of up to 10 m radius are used, but the angular aperture, and hence the light throughput, of these large instruments is very small; manufacturing a good large grating is even harder if the grating is concave than if it is plane. Astigmatism is an important defect of concave gratings that is not shared by plane gratings. The imaging properties of a concave grating are those of

12.3 Grating Instruments

Fig. 12.17.

289

Geometry of Rowland circle. The letters have the same meanings as

in Figs. 12.15 and 12.16. S’ is the Sirks focus (see text). The dotted line shows the construction for finding the Sirks focus when 0’ ~ 0.

a concave mirror used off-axis. If the slit is accurately parallel to the grating rulings, each point of the slit is imaged at the optimum horizontal focus as a vertical line. For a short slit and moderate astigmatism there need be no loss of definition, but the flux density is reduced. Moreover, the point-to-point correspondence between the slit and its image is lost, so that it is not in general possible to resolve spatially an image of the source, a diaphragm or a set of interference fringes on the slit. It is sometimes possible to get around this limitation by placing the source at the Sirks focus, indicated by S’ on Fig. 12.17. A point object at this position is imaged on the Rowland circle as a sharp horizontal line. The distance from the slit to the Sirks focus can be calculated [33]. For spectral lines close to the normal, for which 6’ ~ 0, the Sirks focus can be found as the position where the tangent to the Rowland circle at the centre of curvature of the grating, C in Fig. 12.17, intersects the line GSS’. Alternatively, it is actually possible to eliminate astigmatism altogether by using an auxiliary concave mirror, as is done in the Wadsworth mounting described in the next section. The measure of astigmatism is the length z of the line image from a point on the slit. z is proportional to the vertical aperture H/R, where H is the length of the rulings, and it has a minimum value of 2H sin? @ for 6’ = 6; this works out to be about H/2 for the second order of a 1200 line mm! grating in the visible region. The flux density in the central part of the image would be little reduced if it were possible to work with a slit length approximately equal to z, but this is usually too long to be practicable. Even if a slit some centimetres long could be illuminated, the useful length is limited by the curvature of the image, as discussed in Sect. 12.4.3.

290

12. Dispersive Spectrometers: Prisms and Gratings

12.4 Mountings

for Diffraction Grating Spectrometers

Many different types of mounting have been designed for diffraction gratings, some for special problems. This section describes the types in most general use; further details are given in the references at the end of the chapter. The distinction between multichannel (photographic or array) and single channel photoelectric detection is not as important as might at first appear, because the focal plane P of the camera lens defines either the location of the plate or the region over which the detector plus exit slit must be scanned. The alternative of scanning by rotating the grating and keeping the detector fixed is generally not sufficiently accurate for high resolution, although it is satisfactory for a monochromator.

12.4.1

Mountings

for Plane Gratings

The simplest mounting for plane gratings is the Littrow, or autocollimating, type, for which 6 = 6’. In the most basic version, the half-prism of the prism Littrow mounting illustrated in Fig. 12.4 is replaced by a grating, and to change the spectral region it is necessary to rotate the grating, refocus the lens and adjust the tilt of the plateholder P. More commonly, chromatic aberration is eliminated by replacing the lens with a spherical mirror, acting as both collimator and camera (Fig. 12.18). However, use of the mirror in this off-axis configuration introduces another aberration, coma, that gives asymmetric spectral lines and thus degrades the resolution. In high resolution instruments the spherical mirror is replaced with an off-axis paraboloid, but this is much more difficult to adjust and align properly as well as adding to the cost of the instrument. The comatic distortion of the wavefront produced by the first reflection at a spherical Littrow mirror is doubled by the second reflection as the beam returns towards the slit. The coma can be reduced by modifying the Littrow

Fig. 12.18. Littrow mounting of plane grating. The slit S and the plane mirror R are both above (or below) the plane of the rest of the diagram. G is the grating and M is the mirror, spherical or off-axis paraboloidal, acting both as collimator and camera. The detector or the exit slit is placed at P.

12.4 Mountings for Diffraction Grating Spectrometers

291

s, Fig. 12.19. Ebert mounting with plane grating G and spherical mirror M. This diagram shows the in-plane arrangement with the entrance and exit slits S; and S2 side by side. In a spectrograph S2 is replaced by a photographic plate or an array detector.

instrument to an Ebert mounting, Fig. 12.19. It can be shown that in this case the wavefront distortion at the second reflection is opposite to that produced at the first reflection, and the coma is thus reduced. The spectral region is changed simply by rotating the grating. The mounting shown in the figure is the in-plane arrangement, with the entrance and exit slits side by side, known as the Ebert—Fastie mounting. The effects of spectral line curvature (Sect. 12.4.3) can be overcome in this arrangement by using curved slits. An alternative is the out-of-plane, or up-and-over, mounting, in which the entrance slit S; is below the plane of symmetry and the exit slit S2 is above it. The variation along the slit of the angle of incidence at the grating causes in this case not a curvature of the spectral lines, but a tilt that varies with the wavelength. This mounting has therefore mainly been used for photographic work; it has the advantage of a low level of scattered light. A variant of the Ebert-type mounting in which the single mirror is replaced by two adjacent mirrors is known as the Czerny—Turner mounting. The coma correction is not complete in the true Ebert type because the widths of the diffracted and incident beams are not the same (Fig. 12.19): if the width of the diffracted beam is the smaller of the two, the ‘correcting’ coma at the second reflection is too small. The Czerny—Turner arrangement

s,

s

|

M, (ay

Sa

|

B

M>

Ce aa Fig. 12.20. Czerny—Turner mounting with coma correction, The angles a and at the collimator M; and the camera M2 are different, a < (3.

292

12. Dispersive Spectrometers: Prisms and Gratings

can be designed to increase the compensation by increasing the tilt of the camera mirror, as shown in Fig. 12.20, or by decreasing its focal length, and coma can be completely eliminated — at least for a certain range of grating angles. Moreover the astigmatism is almost negligible. 12.4.2

Mountings

for Concave

Gratings

Most common mountings of the concave grating for photographic or array detectors use the Rowland circle in some form, while other arrangements fulfilling the conditions of (12.25) are used for monochromators and scanning instruments where the wavelength change is achieved by rotating the grating. Among the circle instruments Rowland’s original mounting has the simplest geometry, but is now really only of historical interest. The closest modern equivalent is the Paschen—Runge mounting shown in Fig. 12.21: the slit and grating are both fixed on the Rowland circle. Most common today is the ‘normal incidence’ variant, where the angle of incidence is small, ~ 15°, and the spectrum is observed around the grating normal. This variant gives an

almost linear wavelength scale (12.12), and small coma and astigmatism. The astigmatism can be reduced further by placing the source at the Sirks focus

(Fig. 12.17). The volume of the instrument is comparatively small, and it is therefore suitable for vacuum work. The most widely used and most compact mounting for the concave grating is that devised by Eagle. It is a modified Littrow arrangement, using the grating in autocollimation. The axis of the spectrometer forms a chord of the Rowland circle with the grating and focal plane at its ends, as shown in Fig. 12.22. In instruments intended for the visible and near infrared the slit is usually mounted ‘off-plane’, slightly above or below the axis GP and offset to the side, with a small quartz prism to reflect in the incident light. The alternative ‘in-plane’ mounting has the slit displaced laterally instead of vertically; this is the form usually adopted for monochromators and farultraviolet instruments. In both cases the slit is fixed, and the spectral region is changed by rotating the grating and at the same time moving it along

seein

os

Fig. 12.21. Normal incidence variant of the Paschen—Runge mounting of concave grating. The slit S, the grating G and the plate holder P are fixed at the Rowland circle. The instrument can be enclosed in a vacuum tank, indicated by the dotted line.

12.4 Mountings for Diffraction Grating Spectrometers

293

Fig. 12.22. Eagle mounting of concave grating, showing the in-plane arrangement with the slit S and the plate holder P in the same plane. The figure shows how the grating G and the plate holder have to be rotated and displaced when the wavelength range is changed.

the spectrometer axis to maintain the correct geometry. A related rotation of the plate holder or scanning mechanism P is required. Various mechanical links for the necessary adjustment have been devised, but for high resolution photoelectric work the detector is scanned along the focal plane at a fixed grating position. Again 6 and 6’ are small, thus minimizing the astigmatism. The Wadsworth mounting (Fig. 12.23) gets away from the Rowland circle by using an auxiliary mirror as a collimator. The principal advantage is that the spectrum near the normal to the grating is stigmatic, and coma can be kept small provided that the slit is close to the grating. The distance from the grating to the detector plane is approximately R/2 instead of R, as can

be seen by putting v = 00 and cos@’ ~ 1 in (12.25):

Fig. 12.23. Wadsworth mounting of concave grating, using an auxiliary collimating mirror M. The exit slit or the plate holder is positioned at the grating normal. The wavelength range is changed by rotating the grating G. At the same time the distance between the grating and the focus on the curve P, a parabola, is changed.

294

12. Dispersive Spectrometers: Prisms and Gratings

Fig. 12.24. Scanning concave grating monochromator with combined rotation and translation of the grating G. A good image of the entrance slit S; is obtained at the exit slit S2 when 6; = 6p.

cos0/R+1/R—1/v' =0, leading to

v' = R/(1+cos6) ~ R/2. The linear dispersion is therefore only half as large as in the other types of mounting. The spectral region is changed by rotating the grating, but at the same time v’ changes and the focal plane is displaced. The focal curve is parabolic instead of spherical, which makes for difficulties if the spectrum is to be scanned by rotating the grating. The priority for a monochromator is usually ease of scanning rather than high resolution, and the complicated linkages required in some of the concave grating mountings so far described are undesirable. A number of mountings achieve simplicity at the expense of some loss of resolution, and of these probably the best known is the Seya. The exit and entrance slits of this instrument are both fixed, and the spectrum is scanned by rotating the grating about an axis through its surface. It can be shown that the optimum angle between

the exit and entrance beams is 70°, and with a 1 m grating of 1200 lines/mm the focus is within 0.1 mm of the exit slit over the whole visible and near ultraviolet regions. In another arrangement, used in a number of commercially available instruments, the resolution is improved by translating the grating while it is rotated, in such a way that the Rowland circle is equally distant from, but on opposite sides of the entrance and the exit slit, see Fig. 12.24.

12.5 Production and Characteristics of Gratings

12.4.3

295

Image Defects

Some of the grating spectrometer image defects have already been touched on, in particular coma and astigmatism. Any effect that increases the width of a spectral line is undesirable because it degrades the resolution of the instrument. However, the most serious adverse effects for a spectrometer arise from image defects or combinations of defects that result in asymmetric spectral lines. A symmetrically blurred line can still be assigned a centre, albeit with some loss of accuracy, whereas it is difficult to know even how to define the centre of an asymmetric line. Spectral line curvature is an inescapable feature of grating spectrometers, just as it is with prism spectrometers, because the path difference imposed by the grating is a little greater for the out-of-plane rays from the ends of the slit than for the rays from the centre of the slit. The ends of the slit image therefore curve towards the grating normal. This defect limits the usable length of slit even for plane gratings. For concave gratings the limit is more stringent because the combination of astigmatism with curvature results in asymmetric broadening. Focusing errors arise when the beam incident on the grating is imperfectly collimated. In zero order small errors in collimation can be compensated by small offsets in the camera lens or mirror so that the slit image is still sharp. For the diffracted orders, however, curvature of the incident wavefront varies the path difference across the area of the grating, giving an asymmetric blurred image similar to that produced by a lens used off-axis and suffering from coma.

12.5 Production and Characteristics of Gratings The early gratings were ruled on speculum metal with ruling engines based on the original Rowland engine of the 1880s. The gratings produced in the early part of the present century by such experts in the required combination of art and science as Rowland,

Michelson,

Wood

and Strong were not

greatly improved on for some 40 years, except that the speculum metal was replaced in the 1930s by glass blanks coated with a thin evaporated layer of aluminium. Two important advances of the period 1950-1970 are both developments of original ideas of Michelson’s: interferometric control of ruling engines and ‘holographic’ gratings formed from an optical interference pattern. Almost equally important in practice was the development of grating

replication techniques [34, 35}. For ruled gratings the two criteria to be met by the grooves are equal spacing and specified profile, the latter being both consistent and smooth. Periodic errors in the spacing give rise to ‘ghosts’, which are satellites of each spectral line at a distance from it inversely proportional to the period

296

12. Dispersive Spectrometers: Prisms and Gratings

Fig. 12.25. Geometry of blazed grating. N and N’ are the normals to the grating and the step, respectively, and the blaze wavelength corresponds to specular

reflection from the step (see text).

of the error — i.e. an error with a period of only a few grooves produces distant ghosts. Very slow variations produce wings to the lines that are largely responsible for the difference between theoretical and actual resolution. Random errors in the spacing produce a general background intensity between the spectral lines, known as ‘grass’. The groove shape controls the distribution of intensity among the diffracted orders, and any roughness or random errors in it contribute to the grass. Once a good master grating has been ruled, replicas of it can be made with epoxy-resin castings that are aluminized after separation from the master. The process can be repeated, replicating the original replicas, for several generations. The quality of the replicas may actually be better than that of the master, because the burrs at the edges of the rulings are transferred to the bottoms of the grooves, but the main purpose of replication is to spread the high cost of ruling a good grating. Modern gratings are usually blazed, which means that the groove is shaped to concentrate a large fraction of the incident intensity into diffraction at a particular angle. The principle is illustrated in Fig. 12.25: the preferred direction is that corresponding to specular reflection from the ‘step’ of each groove. If the step is inclined at an angle a to the surface of the grating, and N and N’ are the normals to the grating and the step respectively, then

6-a=—-'+a. (0’ is negative because it is on the opposite side of N to 0.) Therefore,

a=(6+6')/2. Combining this with (12.7) gives (mA)plaze = 2d sin a cos(O — a) . In the Littrow configuration @ = 6’ = a, so the steps are perpendicular to the axis of the spectrometer and the wavelength of maximum efficiency is given by

2dsin@ Aplane =

2dsina

mn

(12.27)

12.5 Production and Characteristics of Gratings

297

Nplaze for m = 1 is normally defined as the blaze wavelength. Although it is

fixed once and for all by the grating ruling, the efficiency is maintained over an appreciable range on each side of the optimum, falling to half maximum at about (2/3)Abiaze on one side and (3/2)Apiaze on the other. Since the blaze efficiency depends only on the angle of use, it is equally good for the second order of Apiaze/2, etc. It is possible to achieve a peak efficiency of about 80%

in the blaze direction. It should be noticed that (12.27) is exactly valid only for a Littrow mounting. When a grating is used in another mounting, the peak efficiency may appear at a significantly different wavelength. This is the case at grazing incidence (Sect. 12.6.3), where @ is close to 90°, and Apiaze Must be corrected

by the factor cos(@ — a) 3A. [This is not the same as the limit for diffraction to take place at all, which can be

seen from (12.7) to be d > \/2.] There is a marked difference in efficiency for the two directions of polarization along and perpendicular to the grooves, the latter deviating most from simple theory. This is understandable in a microscopic picture, because the oscillations of the conduction electrons in the direction across the grooves are affected by finite groove width. In fact, abrupt changes in efficiency with wavelength were first observed by Wood in 1902 and are called after him ‘Wood’s anomalies’. It is only relatively recently that the problem of relating efficiency to groove shape, size and material has been successfully treated theoretically in terms of collective oscillations of the

conduction electrons, known as surface plasmons [34]. The most serious problems of ruling a large grating with fine pitch are the wear on the ruling diamond and the control of environmental conditions over a period of several days. Both of these are avoided with interference (‘holographic’) gratings. The notion of generating a grating by optical interference was first put forward by Michelson, but its practical implementation had to await the advent of the laser. Such gratings have now been in use for some 30 years and are still being improved. The name ‘holographic’ arose because the optical arrangement for generating the fringes is similar to that for producing holograms, but ‘interference’ is a more accurate description. The gratings are produced by exposing a glass blank coated with photoresist to the interference pattern produced by two plane wavefronts crossing at an angle, as shown in Fig. 12.26. The interfering wavefronts are derived from the same laser source via a beam splitter. Exposure to light changes the solubility of the photoresist, and subsequent development leaves a surface following the sinusoidal variation of intensity of the interference fringes. From Fig. 12.26 it can be seen that the fringe spacing x from two beams intersecting at an angle 2¢ is given by « = A/(2sin@). The spacing of the grating grooves can be adjusted by inclining the blank at an angle w to the plane perpendicular to the fringes, giving

298

12. Dispersive Spectrometers: Prisms and Gratings

%. Ve A

|

A

wavefronts

Xa cose A= § sin2¢=2x sing

Fig. 12.26. Production of interference gratings. The two plane wavefronts crossing at an angle 2¢ produce interference maxima spaced by « = A/(2sin@), as shown by the vertical lines in the diagram.

a

~ cosy

mN

2sin¢dcosy ”

Fine spacing requires ~ = 0 and the largest possible value of @, about 60° in practice, giving a minimum value of about 0.6A for d. For the argon ion laser line at 458 nm this amounts to 3500 lines/mm, but various tricks can

be used to obtain up to 6000 lines/mm. Interference gratings have several distinct advantages: the pitch is easily variable and can be finer than that of any ruled grating; diamond wear is avoided; and ghosts and grass should both be absent, though it is necessary to be very careful about stray light and spurious interference patterns during the production of the gratings. Their principal disadvantage used to be the generally lower efficiency, which was a result of the basically sinusoidal groove profile. Considerable effort has been put into generating a sawtooth blaze profile, either during exposure or by subsequent ion etching, and in many circumstances the efficiency can now match that of a blazed ruled grating. Further details of both types of grating can be found in [34].

12.6 Gratings for Special Purposes 12.6.1

Echelle Gratings

It was noted in relation to (12.14), R = Wm/d, that high resolution can be obtained from a coarse ruling in a high order. With m get, in round numbers, a resolving power of a million from and (1/d) = 100 lines/mm. This relatively large value of d control of the groove angle so that a very high efficiency can high orders.

= 100 one can W = 100 mm allows accurate be obtained at

12.6 Gratings for Special Purposes

299

Fig. 12.27. Geometry of echelle grating. The blaze angle corresponds to specular reflection from the narrow side of the step, giving mA = 2dsin@ = 2dcosa = 2t.

Figure 12.27 represents Fig. 12.25 redrawn to bring out the characteristics of echelles. Since they are always used in or close to the Littrow configuration with the light incident on the narrow side of the step, the groove angle a is related to the angle of incidence by a = 90° — 6, and this condition also defines the angle made by the plane of the grating with the spectrometer

axis. The blaze condition (12.27) becomes for this geometry

mN = 2d sin

f= '2d cosa’:

(12.28)

Since m is large, this equation can be satisfied by a whole set of wavelengths with neighbouring values of m. An echelle is therefore specified by its blaze

angle rather than blaze wavelength and (12.28) simply tells us the order number for any particular wavelength. In practice blaze angles are in the range 50-70°. From (12.28) and Fig. 12.27 it can be seen that the grating equation can be written as

mA = 2t,

(12.29)

where t is the long side of the step. The echelle is beginning to look like an interferometer with a plate separation of t. As a consequence of the high-order number, the order overlap problem is

very severe. If the mth order of \ coincides with the (m—1)th order of \+ A), the wavelength band A) is known as the free spectral range. For large m is found by differentiating the grating equation and setting Am = 1:

it

|AA| = A/m. For a typical value of m, in the range 50-100, the free spectral range is only a few nm. Expressed in wavenumbers, it has a constant value for a given grating:

Ao

Ale

~

bag

1

2

(12.30)

mA 2deosa’ to an interferometer is. brought out by using similarity The (12.28). from (12.29) to give

Ag = limNies Ly

2t.,

(12.3.1)

300

12. Dispersive Spectrometers: Prisms and Gratings echelle order

ic

xe) Y

m

oO

2o

m+1

vp)

8

m+2

ae —

Oo

—

X echelle

Fig. 12.28. Echelle spectrum. The horizontal echelle dispersion is crossed by vertical dispersion from a prism or a small grating to avoid order overlap.

which is identical with the free spectral range of a Fabry—Perot interferometer

of plate separation t (Sect. 13.3.1). The overlap problem is dealt with by introducing cross-dispersion in the form of a small prism or low-resolution grating orientated with its dispersion perpendicular to that of the echelle. Successive orders are then displaced vertically, and the spectrum takes the form of a square array crossed by sloping bands, as indicated in Fig. 12.28. This is obviously a very compact way of stacking up a great deal of spectral information for detection with a twodimensional array detector. Echelle instruments are widely used nowadays at laboratories and astronomical observatories — both on earth and in space — instead of conventional photographic spectrographs when a wide spectral range is to be covered in one exposure. 12.6.2

Gratings in the Infrared

It is in the infrared region that Fourier transform spectroscopy (FTS) offers the most pronounced gains in signal-to-noise ratios over grating spectroscopy as a result of the multiplex and throughput advantages discussed in Chaps. 13 and 17. The advent of cheap and fast computers has effectively eliminated the computational drawbacks of FTS, and it is now rare to find a grating spectrometer designed to operate at wavelengths greater than a few pm.

As already noted, the maximum wavelength for which a given grating can be used is limited by A < 2d. Gratings for the infrared therefore require coarser rulings. [t is usually desirable to work in the first order because of the difficulty of filtering out overlapping orders. Reflectivities in the infrared are so high that extra reflections cost almost nothing, and infrared spectrometers generally use plane gratings in an Ebert-Fastie or Czerny—Turner mounting, taking advantage of the stigmatic imaging and the possibility of using long curved slits to maximize detected energy. Resolving power is usually slitrather than grating-limited because of the low radiance of the sources available. The elimination of unwanted orders is a particular problem in the infrared. A grating blazed for first order of, say, 2 um is also blazed for second

12.6 Gratings for Special Purposes

301

order of 1 um and third order of 670 nm, and the shorter wavelengths may be nearer the peak of the distribution of the source. Ideal filters, cutting off all wavelengths shorter than half the wavelength of interest, can rarely be found in practice. Filters relying on selective absorption, selective reflection (‘rest-

strahlen’) and selective scattering (Christiansen filters) have all been used for different spectral regions; details may be found in the book by Chantry (see Further Reading). Two other practical points affect the design of infrared spectrometers. Atmospheric air shows strong absorption bands from water vapour and carbon dioxide, starting between 1 and 2 um, and it is necessary either to evacuate the spectrometer or to operate it in a dry gas. Secondly, as seen in Chap. 15, it is usually necessary to provide for cooling the detector.

12.6.3

Gratings in the Far Ultraviolet

Spectroscopic techniques at short wavelengths are governed by the necessity of working in a vacuum from 200 nm to the soft X-ray region, by the lack of transmitting materials below 110 nm (the quartz limit is ~ 170 nm) and by the decreasing reflectivities of mirrors and gratings. Although interferometers and echelle gratings can be used for a little way into the vacuum ultraviolet (VUV), the concave grating reigns supreme through most of the region, for it requires only one reflection, that from the grating itself. Below about 250 nm dielectric coatings, giving close to 100% reflectivity in the visible and near ultraviolet, have to be replaced by aluminium, or aluminium with a magnesium fluoride overcoating, and the reflectivity drops with wavelength from about 90% to perhaps 70% near 100 nm, below which other metals perform better. Gold, platinum and osmium have all been used below 100 nm, but reflectivities rarely exceed about 20%. Until recently high-quality ruled gratings of 1200 or 2400 lines/mm were used for the highest resolution. The theoretical resolution is seldom attained because ruling errors and surface defects all become relatively more important at short wavelengths and detract from performance. Interference gratings should offer significant advantages in this respect, and they are being increasingly used at short wavelengths. Further, it is actually possible to control the focal properties of the grating and to correct some of the aberrations, notably astigmatism, by appropriate shaping of the wavefronts of the interfering beams used to produce the grating. Below about 30 nm reflectivities are so low that it becomes necessary to use gratings at grazing incidence. The refractive index of any material is less than unity at very short wavelengths, so radiation incident at a sufficiently small angle to the surface is totally externally reflected. In terms of the grazing angle @ (the complement of the angle of incidence); the condition for total reflection is @ < @c, with ¢, given by

12. Dispersive Spectrometers: Prisms and Gratings

302

1/2

be =A (a) An%eqmc? where e and of electrons round, this wavelength have

m are the charge and mass of the electron and ne per cubic metre in the surface material. Looked at expression gives, for a given angle of incidence, for which there is total reflection. Putting in the

Amin (nm) = 3.3 x 10!nz1/2¢.

(1232) is the number the other way the minimum constants we

(12.33)

For ¢@ = 1°, this gives Amin & 0.66 nm for aluminium or glass and about 0.25 nm for a heavy metal. To get down to 0.1 nm requires a grazing angle of about 20 arc minutes. In practice these limits are always affected adversely

by surface imperfections, and A (nm) > @ (degrees) is a good rule of thumb. One effect of working at grazing incidence is that the angular dispersion is increased by the factor 1/(sin¢) © 1/(cos 6’) > 1 (eq. 12.12). For ¢ = 1°, this factor of about 50 means that a grating ruled with 1200 lines/mm is equivalent to a 60,000 lines/mm grating at normal incidence and can still be used in low orders (Fig. 12.29). The smaller effective width of the grating reduces its theoretical resolving power; but the resolution of a grazing incidence spectrometer is in practice slit-limited anyway because it is impossible to make slits narrow enough to reach the diffraction limit. Great care has to be taken over surface polish and smoothness in order to reduce surface scattering at these short wavelengths. This can be better done with laminar than with blazed grooves. These have a shallow square profile, as shown in Fig. 12.29, and actually come rather close to the original bar-and-space model discarded earlier in the chapter. The grazing incidence mounting is shown in Fig. 12.30. The slit, grating and plateholder all lie on the Rowland circle, but as only a small segment of the circle is used the instrument is actually quite small, even for a grating of 2m radius. The mounting and its adjustments have to be carefully designed and machined with great accuracy because such fine angles are involved, and the astigmatism from a standard concave grating is enormous. This defect, with its accompanying light loss, can be largely overcome if the grating is formed on a toroidal rather than a spherical surface — that is, a surface for

SS

|

es oo

Fig. 12.29. Grating at grazing incidence. The effective spacing of the rulings is dsiny. The order numbers are shown on the right.

Further Reading

303

Fig. 12.30. Grazing incidence mounting of concave grating. The broken arc is part of the Rowland circle. The grazing angle ¢ is the complement to the angle of incidence @.

which the curvature along the length of the grooves is much greater than that perpendicular to them. Further details are given in the book by Samson and

Ederer and in [34, 36].

Further Reading — Bousquet, P., Spectroscopy and its Instrumentation (Hilger, London, 1971). — Harrison, G.R., Lord, R.C., and Loofbourow, J.R., Practical Spectroscopy

(Prentice-Hall, Englewood Cliffs, NJ, 1948). — Hutley, M.C., Diffraction Gratings (Academic Press, London, 1982) — Sawyer, R.A., Experimental Spectroscopy (Dover, New York, 1961). For the infrared

— Chantry, 1984).

G.W., Long-Wave

For the vacuum

Optics, 2 vols. (Academic Press, New York,

ultraviolet

— Samson, J.A.R. and Ederer, D.L. (Eds.) Vacuum

2 vols. (Academic Press, New York, 1998).

Ultraviolet Spectroscopy,

—_—

7

waa

jeer of? Ting eee

CO Th.ae Pt),

OE

deatathe al ine ‘pies See,

pl Se

4el Oe ERR Ore

eewte ate htt aby

ot AMok we] Wi PASSO?

iy a

dee

(902 (ante Lovet? hiantusthndinadh ventas Pe eerToe ee, oy ere he _ ? Me piencgeeRi i cig ives

=

Bi

:

= al port ant) Grebo) fatty >>tan... nvr ate Gta LM oll stp Degreei> Georg: expt a

7

et

fts

oy

‘

2@

Ai .

S40

i

eee

i te

.

>

1

a

Matt.

-

,

ins SD

A

;

‘

ind

b

ia

iam Sim

ne

veel). b>

ee

a

*

\

.

i

1

me oO

os = :

>am

-

a)

Ae

or

eee

Tis 2k _

fea) * hed

A

8

.-

ca

ee..

_&

;

f

|

i

easel) -menqadest/® (he eeu

@Gaieytog!

es a

@

heer?

ng

=f

7

aay

ra

oa wdedieneae

( 0)

ba

OHy

Th

Weer aa it ig

be

ale

= Pie

Vew

vw

wyage!

7

ev ®

fi ality >

Spies

q

aaa

Asset

jay ba aiagn 4

Lae

Vio!

omit

oe

a on >

a A

a>

od imite

way

aa

coy

i @

ni

»

Ve

13. Interferometric Spectrometers

Different forms of interferometer are now used for a large number of very different purposes — high precision metrology, measurements of refractive index and density, optical testing, surface studies and microscopy, and vibrational monitoring, amongst others. Only two types are of practical importance for spectroscopy: the Fabry-Perot and the Michelson. The Fabry—Perot is a multiple-beam interferometer, capable of extremely high resolution from the near infrared to the quartz ultraviolet. The Michelson is a two-beam interferometer whose present importance comes from its use as a scanning instrument in Fourier transform spectroscopy. Many other types of interferometer in current use are variations of the Michelson, and one of these, the Mach-Zehnder, is relevant to the subject matter of this book in that it is used

as a refractometer for the measurement of oscillator strengths (Sect. 16.2.3) and electron densities in plasmas.

13.1 Basic Concepts of Interferometric

Spectroscopy

Both the Fabry-Perot and the Michelson are division-of-amplitude interferometers, which means that the interfering beams are produced by splitting the incident beam at a partially reflecting surface. In the Fabry—Perot a series of such beams is produced by multiple reflections between two parallel plates with partially reflecting coatings. The Michelson uses a single partially reflecting plate to produce two beams, which are reflected back by plane mirrors and recombined at the beamsplitter. Figure 13.1 shows the optical arrangement for (a) a Michelson and (b) a Fabry-Perot interferometer. In each case light from an extended source behind an entrance aperture A is collimated by the lens L; before impinging on the Fabry-Perot plates F or the beamsplitter B of the Michelson. The lens Ly images the aperture A at A’ in the plane P. (The lenses are usually replaced by concave mirrors in the Michelson interferometer to avoid chromatic problems.) Usually the two plane mirrors M; and Mz of the Michelson are adjusted so that the image M‘ of M, in the beamsplitter is parallel to Mg, as shown in Figs 13.1a and 13.2a, with a distance t between them. It can be seen from Fig. 13.2a that for rays incident at an angle @ the optical path difference x« between reflections from the two surfaces is given by

306

13. Interferometric Spectrometers

My

b)

Ly

G

Lo

P

Fig. 13.1. Optical arrangements for (a) Michelson and (b) Fabry—Perot interferometer. In both diagrams A is the entrance aperture and A’ is the image formed by the collimating and camera lenses L; and Lz in the plane P. In (a) B is the beamsplitter and M;, Mp2 are plane mirrors; Mj is the image of M; in B. In (b) F is the Fabry—Perot interferometer.

x = s(1+ cos 26) , where

Si sec.

2

Therefore

x = 2tsec6 cos” 6 = 2tcosé .

13.0)

Similarly, the optical path difference for each double passage of the Fabry— Perot gap in Fig. 13.2b is 2tcos@. In both types of interferometer the wavefronts interfere constructively when « is an integral number of wavelengths, mA, and destructively when «x is an odd number of half-wavelengths. The interference fringes are localized in the plane of intersection of the wavefronts, which, since they are parallel in this configuration, is at infinity. The lens L focuses these fringes in the plane P. For a given wavelength the fringes are loci of constant 6 and are known as fringes of equal inclination. It follows from the axial symmetry of the arrangement that the maxima are circles in the plane P of radius f@ where @ satisfies

13.2 Resolution and Throughput

Fig. 13.2.

307

Optical path differences for two parallel plates separated by distance

t. In (a) the plates are path difference between text). In (b) the optical is also 2tcos@, and the

2t cos

2OCCOS 0

]

the mirrors Mz and the image Mj of Mi, the two reflections is 2tcos@ for angle of path difference between successive rays in rays are focused at a distance f@ from the

and the optical incidence 6 (see

the Fabry—Perot optic axis.

ma ,

ae

digepey,

for successive values of m. Several points about these circular fringes should be noted. First, the order number m decreases outwards from the centre. Secondly, the circles move outwards as t increases (for a given m, larger t requires larger @); but, thirdly, increasing t crowds the circles more closely together because a given range of 6 accommodates more values of m. The number of rings that can be seen at any particular value of t depends on the maximum value of @, which, with the geometry of Fig. 13.1, is determined by the size of the source or of the aperture A and the focal length of the lens L; . This lens is not essential for the formation of the fringes; its function is to increase the illumination,

and in the case of a scanning interferometer it also serves to image the input aperture A in the plane of the fringes.

13.2 Resolution and Throughput There are two ways of using an interferometer as a spectrometer. First, for fixed t (13.2) gives \ as a function of #, and the wavelengths can be measured from the interference fringes photographed in the plane P. Secondly, for fixed

6 (0 ~ 0) (13.2) gives A as a function of t, and wavelengths are found by recording the fringes photoelectrically as t is scanned. The scanning mode is now almost always used for the Fabry-Perot and always for the Michelson. In this section we shall consider only this mode, for which the aperture A

and/or another aperture at A’ (Fig. 13.1) ensure that the detector sees only the centre of the ring pattern. It was shown in Sect. 11.1 that the resolving power of a spectrometer is

of order L/X (11.5), where L is the extreme path difference between inter-

308

13. Interferometric Spectrometers

fering beams. The conventional definition of the resolving power in Fourier transform spectroscopy (eq. 13.33) leads to the expression for the Michelson

RL N Ate) Ae

(1873)

where tm is the maximum effective separation of the mirrors. For the Fabry—

Perot, ‘extreme’ should refer to the last of the multiple reflections, but the intensity just drops steadily with section we define an effective number on the reflectivity of the coatings and

R=

= ON hen

path difference between the first and of course there is actually no ‘last’ — the number of reflections. In the next Ny of interfering beams that depends is normally in the range 20-50. Then

(13.4)

Thus, in both types of interferometer the resolving power can be increased, in principle without limit, simply by increasing tm. The usual price — low throughput — must be paid for high resolution. Just as high resolution in a grating spectrometer demands a narrow slit, so in an interferometer it demands a small input aperture. The reason for this can be readily understood from the description of the ring pattern in the previous section. For large t the rings become small. If the aperture is large enough to encompass a complete order — all the way from one maximum to the next of any given wavelength — there is no net change of intensity as t is changed and thus no way of distinguishing wavelengths. To find a quantitative limit on the limiting size of the aperture we go back to the dependence of path difference on angle of incidence given in (13.1):

x = 2tcos@ ~ 2t(1 — 67/2) ,

(13.5)

where @ is always small enough to justify the expansion, and use the criterion that xz should not vary by more than half a wavelength over the field of view at the maximum value of t (i.e. a phase change of 7 from centre to edge):

D5 a a ae Oa Using (13.3) for the Michelson,

eh

b= 21

R,

which can be written in terms of the maximum radius a of the aperture and the focal length of the collimator as Omax

~

. a

‘

= :

fe

(13.6)

Equation (13.6) holds also for the Fabry-Perot interferometer. For R = 2 x 10°, equivalent to a high resolution grating, @max is 3 mrad, so for a focal length of 1 m the radius of the entrance aperture is 3 mm. This is a very large input aperture when compared with a slit of 10mm x10 um. " The factor 2 difference between (13.3) and (13.4) arises from the arbitrary definition of resolving power applied to two different instrument functions.

13.3 Fabry—Perot Interferometers: Intensity Distribution and Resolution

309

The throughput advantage of interferometric over dispersive spectrometry is a direct consequence of this large aperture. Equation (11.12) defined the throughput F as the product of the area A of the entrance aperture and the

solid angle 7D?/(4f*) subtended by the grating or interferometer plate at the aperture. With A = 7a”, the equation can be rearranged to give

B = x(a/f)? (nD?/4) , and using (13.6) we get

RE = 2n(nD?/4) .

(aleera)

If this is compared with the result for a grating spectrometer with slit-limited resolving power (12.24) it can be seen that for the same resolving power the throughput advantage of the interferometer is given by

High, Topi where

S; and S, are the areas of the interferometer

(13.8) plate and the grating

respectively. In general, Fabry—Perot plates and Michelson beam splitters are likely to be somewhat smaller than diffraction gratings, but with 3 < 1/100 for the reasons given in Chap. 12 the throughput advantage is still about two orders of magnitude. At this stage it is more useful to discuss the two types of interferometer separately. We take first the Fabry-Perot, its intensity distribution and the ways in which it is used, and then the Michelson interferometer and the technique of Fourier transform spectroscopy.

13.3 Fabry—Perot Interferometers: Intensity Distribution and Resolution A Fabry-Perot interferometer normally consists of two glass or fused silica plates, usually 2-10 cm in diameter, with their facing surfaces flat and parallel to better than 1/50 of a wavelength. (Modified versions of the interferometer for the infrared and far ultraviolet are mentioned briefly in the next section.) The facing surfaces are coated with metal or dielectric films to give a reflectivity which is usually in the range 85-95%. The back surfaces have a wedge angle sufficient to throw reflections from them clear of the main fringe pattern. The plates are sometimes held at a constant distance by a fixed spacer — this arrangement is known as an etalon — but they may also be mounted on piezoelectric crystals or other devices suitable for making small changes in separation. From (13.1) and Fig. 13.2 the path difference for each double passage of the etalon at angle of incidence 6 is 2tcos@. Allowing for a medium of refractive index n between the plates, the optical path difference is actually

310

13. Interferometric Spectrometers

x = 2ntcosé.

(13.9)

Strictly, @ in this equation should be the angle in the medium rather than the angle of incidence, but so long as the medium is a gas the difference is about one part in 10° and is unimportant. For the same reason n can usually be set equal to unity; the reason for introducing it is that one method of changing the path difference between the plates is to vary the gas pressure, as will be seen in Sect. 13.4. 13.3.1

The Airy Distribution and Its Properties

Although we already know that x = m4 defines the fringe maxima, it is necessary to find the intensity distribution in the fringes — i.e. the instrument function, which is quite different from that of a grating — to quantify the properties of the Fabry—Perot as a spectrometer. Figure 13.3 shows the amplitudes of the interfering beams in terms of the incident amplitude Apo; s and r are the amplitude coefficients of transmission and reflection of each of the films (assumed to be the same for both plates). The first transmitted beam has amplitude Ags’, and each double passage introduces a factor r? to the amplitude and a phase shift 6 given by

OG =] 2hr A= 2roe =

4Anotcosé.

(13.10)

When all these beams are superposed resultant complex amplitude is

Ae Ags

Te ay ee

ee

in the focal plane of the lens, the

Ags?

1 — rei

Fig. 13.3. The amplitude of successive transmissions and reflections in a Fabry— Perot interferometer. s and r are the amplitude coefficients of transmission and reflection of the plate coatings. The first backward-going reflection, Agr’ has a phase shift of 7 with respect to the other beams.

13.3 Fabry—Perot Interferometers: Intensity Distribution and Resolution

311

To get from complex amplitude to intensity, A is multiplied by its complex conjugate, as is done for the grating in Sect. 12.3.3: 2.4

1 — r2(el) + e- 19) + r4 We can now introduce the intensity coefficients, transmissivity T= s? and reflectivity R =r’, and the incident intensity Jj = A?, and use the relation

Cr (ees

eee a = =

Reco (1— R)* + 2R(1 — cos6) (l= R)? 4 4Rsin? (6/2).

Therefore,

‘cana

I=I1p)

| ——

(=z)

1

|) ———————>—_—_

1+ [4R/(1 — R)?]sin?(6/2)

.

Tax hl

(

)

Equation (13.11) is known as the Airy distribution. The maximum values of I occur, as expected, for 6/2 = mm — ie.

Chi NOG

MAG

(13512)

which is identical with (13.2). The peak intensity is rT \2 Imax =

I

‘ (,Z 2)

(13.13) NBs

.

ares

In the ideal case of no absorption, 7 + R = 1, and the peak intensity is equal to the incident intensity. The minima occur at values of 6 half-way between the maxima, when 6/2 = (m+ 4), and their intensity is

pT

\?

13.14

| Baia Sle \ (, + z

Seute

The contrast C is defined by the ratio of maxima to minima:

Y= si

Fast neg (sli Ri | oe Imin

lja

13.15

“i

\

For R ~ 90%, C is a few hundred. The Airy distribution can conveniently be written in terms of Imax as

i

—

1s

>.@

1+ fsin*(6/2)

=

_

IES

x

1+ fsin* rox

:

(13.16)

where f is a function of R only:

4R faa @—F)

|

Figure 13.4 shows the distribution plotted as a function of x for two different reflectivities, 70% and 90%. The sharpness of the fringes obviously depends critically on R.

uly

13. Interferometric Spectrometers

mx

(m+1/2)\

(m+1)r

Fig. 13.4. Airy distribution as a function different reflectivities, R = 0.7 and R= 0.9.

x

of optical path difference x for two

By rescaling the abscissae we can equally well regard Fig. 13.4 as a plot of intensity versus wavenumber at fixed zx. In this guise it represents the output of the instrument from a monochromatic input — in other words the instrument function. Figure 13.5 presents the distribution for R = 90% appropriately relabelled. The single monochromatic line has been reproduced at intervals Ao with a FWHM (full-width half maximum) of 60;/2. The repeat distance Ao represents one order of interference and is the free spectral range, while the FWHM determines the resolution. Expressions for Ao and 60,/2 can be found from the Airy distribution. The free spectral range comes from differentiating (13.12) for constant x and setting Am = 1:

Do] Am/ eal

= —-

0.5Ac

ea V2.

SF

(13.17)

oO oO

Fig. 13.5. Airy distribution as a function of wavenumber o. Aq is the free spectral

range. The distribution shown is for R = 0.9, and the FWHM

4604/2 is (1/28)Ao.

13.3 Fabry—Perot Interferometers: Intensity Distribution and Resolution

313

Fig. 13.6. Rayleigh’s criterion applied to the Airy distribution. The peak intensity Jp is the sum of the maximum

from one line, Imax, and

the intensity J, in the wing of the other at a distance do from its centre.

This equation is identical to (12.31) for the free spectral range of an echelle grating. However, t, and hence also m, is typically from one to three orders of magnitude higher in the Fabry-Perot, and the problem of overlapping orders is correspondingly more severe. For t = 1 mm, Ao = 5 cm~!, which is only about 0.1 nm in the visible region, and it is quite usual to work at considerably higher values of t than this. Cross-dispersion is nearly always essential to avoid a huge ambiguity problem: this is discussed further in Sect. 13.4. The FWHM 60; /2 of the Airy distribution is found by stepping off by an amount (1/2)éo/2 on either side of one of the maxima in Fig. 13.5, such that I in (13.16) is equal to (1/2)Imax- This occurs for

1+ f sin? (1501 /22/2) = 2 => sin(604/22/2) = L/h The argument of the sine is expressed as a fraction of an order by setting

x = 1/Ao (13.17). The quantity (7/2)(d01/2/Ac) is necessarily small if the fringes are sharp, so we can use the small angle approximation to give

2 —

6014/2 =

walt

Ao.

(i348)

The ratio of FWHM to free spectral range measures the fine-ness, or ‘finesse’, of the fringes, and the coefficient of finesse is defined by jie

Ao 6014/2

13.3.2

_avf _ aVR =

2

il —R

f

13.19 (

)

Resolving Power

It is reasonable to expect the FWHM to be approximately equal to the resolution limit of the etalon, and we now show that indeed the two are effectively

314

13. Interferometric Spectrometers

the same. Rayleigh’s criterion for resolution is directly applicable only to a diffraction-limited instrument function, but it can be adapted for other line shapes, as stated at the end of Sect. 11.1, by taking a 20% dip between the peaks of two equal lines as the resolution criterion. At first sight the FWHM does not meet this criterion for the resolution limit, as it would appear that the superposed profiles of two equal lines separated by 601/2 would show no dip at all at the midpoint where each has intensity Tnax/2; but the Airy distribution does not fall to zero in the wings of the lines and, as shown in Fig. 13.6, each of the line peaks is raised by the intensity J,, in the wing of the other to a value Ip = Imax + Jw. The value of I, can be found from the

Airy function by going out a distance 60/2 on either side of a maximum.

Equations (13.16) and (13.17) give eee: w

Peciiat f sin? 1(604/2/Ac) ©

The sine can again be replaced by the angle, and using (13.18) we get Yer

ad

a

paced

( Ao

ww mr?

)-

bo

pee VE)

(Ao

ei

)

_—

—

if

Therefore, ip. ==) WwW

ihes i ae

ee

leading to 6 if =

5 fmax

:

The intensity in the dip is thus 5/6, or 83%, of the peak intensity, which is quite close enough to the value of 81% required by the strict application of Rayleigh’s criterion. For all practical purposes the resolution limit do is identical with the FWHM 4017/2 and is given by (13.19): or =

Aa /N¢ 5

The resolving power is then a a R=

ce =

Agi

=

mN¢

ci

(13.20)

Comparing the last expression with that for the diffraction-limited resolving power of a grating, (11.9) or (12.13), it is clear that the finesse Nr can be interpreted as the effective number of interfering beams. Table 13.1 shows the dependence of Ne on R. A large value of Nr is desirable in order to obtain high resolving power without the restriction of free spectral range imposed

by a large value of m, but there are a number of practical considerations (described in Sect. 13.4.1) limiting it to about 50. It is obvious from putting in a few figures that the maximum resolving powers attainable from gratings can easily be exceeded with Fabry Perots.

13.4 Use of Fabry—Perot Interferometers

315

Table 13.1. Finesse Nr of Fabry—Perot for different values of reflectivity R.

R(%) Ne

70 Some

75 80 O mee Oo

85 OG

90 20.

95 ome)

For N¢ = 25 aresolving power of a million in the visible region requires a plate separation of only 10mm, and it is not difficult to increase this by a factor of up to 10. The Fabry-Perot is, in fact, equivalent to a grating of width Net, with the additional advantage that the ‘width’ can be easily varied according to the needs of a particular experiment simply by using different spacers. As an interferometer also has the substantial advantage over a grating in light throughput discussed in Sect. 13.2, there must be good reasons why it is so much less widely used. These are considered in the next section. We conclude this section by attacking a popular misconception about the Fabry-Perot. It is easy to imagine that the transmitted intensity through two plates, each with reflectivity say 90%, should be reduced to 1% . This is indeed true of the average level, but it is not true of the peaks. Provided there are no absorption losses, the peak intensity is 100% of the incident intensity,

as can be seen from (13.13) with T = 1 — R. What the interferometer does is to redistribute the energy, piling it up in the narrow bright fringes at the expense of the regions between peaks. A related point concerns the interference pattern formed by the rays reflected back towards the source. These rays are shown in Fig. 13.3; the first of them, marked Apr’ in the figure, has a phase shift of 7 relative to the others (if absorption in the plate coatings is negligible), because it is the only one reflected from the substrate side of the coating. It can be shown by summing the complex amplitudes in the same way as for the transmitted rays that the intensity distribution in the reflected beam is exactly complementary to that

described by (13.11) — that is, it consists of narrow dark fringes on a bright background. Indeed, this pattern can be deduced without any algebra at all from the principle of conservation of energy.

13.4 Use of Fabry—Perot Interferometers 13.4.1

Limitations

The ways in which Fabry—Perot interferometers are used are largely dictated by their limitations, which is the reason for taking this topic first. The principal limitation is the small free spectral range, Ao = 1/2t. For a very sparse line spectrum it may be possible to isolate the relevant spectral ranges with interference filters, but it is nearly always necessary to use an auxiliary prism or grating spectrograph. To reduce the demands on this cross-dispersing instrument, one would like to use the largest possible Nr with a correspondingly

316

13. Interferometric Spectrometers

small value of t to attain a given resolution. N¢ is normally limited to 20—50, depending on the wavelength region, for two possible reasons. The first concerns loss of intensity from absorption in the films. Writing T = 1— R— A, where A is the absorption, in the expression for Imax, (13.13), gives A

Fane = Ip (1- 3)

2

(13.21)

The peak intensity is close to Jy only if A is small compared with (1 — R), a condition that inevitably becomes harder to meet as R is increased. Multilayer dielectric films are, in fact, so efficient in the visible and near infrared that absorption is not usually a serious limitation, but very high reflectivities can be obtained only over a rather narrow spectral band near the wavelength for which the film thicknesses have been optimized, so that a compromise between efficiency and band width is usually necessary. In the ultraviolet region the materials of high refractive index in the multilayer coatings tend to start absorbing. Reflectivities of about 90% can be attained down to about 250 nm, but at shorter wavelengths aluminium films have to be used, and 80-85% is a more likely figure at 200 nm. Fabry-Perots of even lower finesse

have been used down to about 140 nm [37], but for most purposes they are not useful much below 200 nm. The second limitation on finesse is set by the imperfections of the actual interferometer plates. The effect of any departure from complete flatness is that the rays emerging from the interferometer at a given angle have varying phase differences, depending on the region of the plate from which they are reflected. A bump of A/50 changes the local path difference by \/25 and hence changes the phase by 1/25 of an order. If the fringe maximum can wander about through 1/25 of an order, the finesse cannot be greater than 25. Although interferometer plates have been worked to 4/100, or even better over a small area, A/50 is a more usual figure — and by \ we mean visible light, usually 633 nm, by which this very skilled work is monitored. The plates are relatively ‘rougher’ in the ultraviolet. Little is gained by using a reflection finesse appreciably greater than the plate finesse, and a detailed treatment shows that the two should be approximately matched (see the books by Bousquet and Hernandez). The effective finesse is then 0.6 of either, usually 25-30 in the visible and near ultraviolet and somewhat higher in the infrared. A disadvantage of quite a different kind arises from the non-zero minimum in the Airy intensity distribution. This tends to mask weak lines with strong neighbours. For the same reason the Fabry—Perot is not ideal for absorption spectroscopy: the absorption lines are to some extent filled in by the minima from the surrounding continuum. Finally, Fabry—Perot interferometers are much more limited in spectral range than gratings because they require transmitting and very flat substrates for the partially reflecting coatings. We have seen that their effective

short-wavelength limit is close to 200 nm. The infrared is less of a problem,

13.4 Use of Fabry—Perot Interferometers

cop

Fig. 13.7. Fabry-Perot ring pattern crossed with spectrograph slit. The full and broken circles are maxima for two different wavelengths, A; and Ao.

even beyond the quartz cutoff at a few um, because the flatness tolerances become easier to meet as the wavelength increases. Fabry—Perots with stretched polymeric films (‘melinex’) have been used above the limits of the alkali halide crystals, and indeed metallic mesh ‘plates’ have been taken right into the submillimetre range. However, the main competitor with grating spectrometers in the infrared is Fourier transform rather than Fabry-Perot interferometry. 13.4.2

Methods

of Use

The traditional way of using the Fabry-Perot was to photograph the ring pattern and measure the ring diameters with a travelling microscope to obtain wavelengths. Nowadays Fabry—Perots are normally used in the photoelectric scanning mode. Replacement of the photographic plate by an array detector with digital signal processing appears to be a useful method only for certain astronomical applications (see below). Either way, an auxiliary spectrograph or monochromator is nearly always necessary because of the small free spectral range. Prism instruments usually provide adequate resolution. For photographic use the spectrograph must be stigmatic. The interferometer can be mounted either outside the spectrograph, the fringes being focused on the slit with a lens of high quality, or inside it in the parallel beam between the collimator and the prism (Littrow instruments are not suitable for internal mounting). The slit image defines a thin rectangular section through the ring system, and the interference fringes are imaged as short arcs in the focal plane of the spectrograph, as shown in Fig. 13.7. To avoid to the ambiguity the slit must be narrow enough to exclude contributions spectrum complete The order. different pattern from neighbouring lines in a in forms a two-dimensional array a little like that of thé echelle, except that low the and slit the of length the this case the high resolution runs along scale is resolution dispersion is perpendicular to the slit. The wavenumber

318

13. Interferometric Spectrometers

quadratic in fringe diameter D, as can be seen from (13.2): within a given order m, Ay — Ae

(2t/m)(cos 6; — cos 62)

x (A/2)(83 — 87) = (A/8f?)(D3 — Dr) , where f is the focal length of the fringe-forming lens. The scaling factor \/ f? can be found by measuring diameters for successive orders of one wavelength. Wavelength measurements made in this way can be extremely accurate, but intensity measurements suffer from the usual photographic difficulties of plate calibration and small dynamic range. The photoelectric scanning mode requires a circular aperture to isolate the centre of the ring pattern, as described in Sect. 13.2. Ideally, to exploit fully the potentially high throughput, the slit width of the auxiliary spectrometer should match the diameter of the aperture. As the latter decreases only as the inverse square root of the resolving power (13.6), whereas the free spectral range decreases as 1/R for a given finesse (13.20), a combination of high resolution and a crowded spectrum may require for the match an impracticably large linear dispersion. The actual scan length can be very short. In principle, all the available information can be obtained from a single order, a change in t of only 4/2. Usually the scan length is a few um so as to cover several orders. It is necessary to keep the plates parallel to about 4/50 during the scan. This can be achieved by physically moving one plate with a piezoelectric, magnetic or spring-strip device, but it can also be done by enclosing an etalon with a fixed spacer in an airtight box and changing the pressure. Equation (13.9) in the form x = 2nt shows that a scan of one order, Ax = X, requires

gamesie attaSN Tee | alse For a finesse of 30 and a resolving power of a million, An = 3 x 1075, which amounts to a change in pressure of 0.1 atm. The wavelength scale, given by A\ = Azx/m, is effectively linear in x and hence in t or n because for such large values of m (m ~ 3 x 10* in the above example) it varies by only about a part in 104 over the scan.

The problem of limited spectral range is sometimes tackled by using two, or even more, etalons in series. If one spacing is an exact multiple (5, say) of the other, the maxima of the two etalons coincide except that the longer etalon has five maxima for every one of the shorter. The latter acts as a filter, transmitting every fifth order of the longer, high-resolution etalon. The fine tuning necessary to obtain the exact ratio can be made by adjusting the pressure in one of the etalons. The advantage of the double etalon is that the intensity in the transmitted peaks is scarcely reduced at all, provided the absorption in the coatings is small, whereas an auxiliary spectrometer is likely to give at best 50% efficiency even if it does not cut the throughput. The filter etalon may solve a problem sometimes met in studying Zeeman or hyperfine

13.4 Use of Fabry—Perot Interferometers

319

structure, where the high resolving power required of the main etalon to separate the components may spread the whole pattern over a wavelength range exceeding its free spectral range. The disadvantage is that the nonzero minima of the filter etalon do not entirely suppress the intermediate orders of the main etalon, and these have to be allowed for in interpreting the spectrum. 13.4.3

Particular Applications

The very high resolution of the Fabry-Perot and the wavelength accuracy that could be achieved led to its extensive use, before the advent of laser spectroscopy, for measurements of hyperfine structure and isotope shifts and for establishing wavelength standards. The first important example of the latter was the historic determination of the absolute wavelength of the red line of cadmium in terms of the International Metre in Paris in 1905 by Fabry, Perot and Benoit. When the international standard of length became the orange krypton line, the same type of measurement was reinterpreted as a measurement of the metre in terms of wavelength. A large number of secondary and tertiary wavelength standards were subsequently established by Fabry—Perot techniques. Now that the fundamental standards are frequencyderived — see Chap. 17 — the role of the Fabry—Perot in this field has switched towards calibrating laser wavelength scans. Tunable laser spectroscopy has also taken over to a large extent, although not completely, measurements of hyperfine structure and isotope shifts in individual spectral lines. As will be seen in Chap. 14, not only can the linewidth of the scanning laser be made very narrow, but also the various techniques of Doppler-free spectroscopy greatly reduce the effective linewidth of the spectral features scanned. Astrophysics is the one field in which Fabry—Perot interferometers have no competition from lasers. Scanning Fabry—Perots behind telescopes are used to observe line profiles or small line shifts from stellar, galactic, interstellar and other sources, usually for a single spectral line (frequently the hydrogen Balmer a line) that can be isolated with an interference filter. These shifts arise from bulk motions or rotations of the object, and astronomers measure them — and the resolving power of the spectrometer — in km/s; a resolving power of 10°, for example, corresponds to 3 km/s. The availability of array detectors has made it possible to use Fabry—Perots in three-dimensional spectroscopy. A two-dimensional spatial image of the nebula or galaxy or star cluster is formed in the plane of the interference fringes, and the third, spectral, dimension is obtained by scanning the Fabry-Perot.

320

13. Interferometric Spectrometers

13.5 Michelson Interferometers

and Fourier Transform Spectroscopy (FTS) The Michelson interferometer set up as in Figs. 13.la and 13.2a gives fringes of equal inclination in the same way as the Fabry—Perot. Regarded simply as a Fabry-Perot interferometer with a finesse of 2, it obviously has no merit. Its value as a spectrometer is due to the fact that the output signal as a function of the path difference « between the two beams — the interferogram —is the Fourier transform of the spectrum; the spectrum can be recovered unambiguously by taking the inverse Fourier transform of the interferogram. 13.5.1

The Interferogram and the Spectrum

Figure 13.8 is part of Fig. 13.1a redrawn to show two important features of the Michelson

interferometer

as used for FTS.

First, one mirror

is moved

parallel to itself to change the path difference x between the two interfering beams. Second, a compensating plate matched to the beamsplitter is inserted in the beam that undergoes the first reflection from the beamsplitter in order to equate the optical paths through the beamsplitter material. Without this compensating plate one beam would have three passages through the substrate and the other beam only one, and the additional optical path cannot be compensated at all wavelengths by simply increasing the air (or vacuum) path in the other beam because of the dispersion of the substrate material. The complex amplitude from the superposition of two monochromatic. coherent beams of amplitudes a; and a2 and wavenumber o with an optical path difference x between them is given by id , A=a,+age" where

0=I27ox

.

Multiplying by the complex conjugate, we get

Fig. 13.8. Compensating plate used to match optical paths through beamsplitter material.

13.5 Michelson Interferometers and Fourier Transform Spectroscopy (FTS)

———@_—q“-

1a

0

321

x

Fig. 13.9. The interferogram for a monochromatic source of wavenumber a: (ge) =

I,(1 + cos 2102).

I= AA*

a? + a2 + ayag(e? + ant) = at +a} +2a,a2 C086.

(13.22)

If both beams have the same amplitude a, then J = 2a?(1 + cos), and the signal falling on the detector for a given wavenumber o is given by

I,(x) =I,(1+cos2nox) ,

(13.23)

where I, is the mean value. The interferometer is used in the scanning mode, with the centre of the ring pattern isolated by a circular aperture so that @ ~ 0 in (13.1), and @ is changed by scanning one or both of the mirrors. it The signal recorded as a function of x is known as the interferogram; 13.9. Fig. in d illustrate varies sinusoidally between the limits 21, and 0, as the We get from monochromatic radiation to a real light source by summing between coherence no contributions from all spectral elements, as there is range them. If the source has spectral radiance L(c), the average flux in the varying slowly some to and do reaching the detector is proportional to L(a)do ion function of o which describes the spectral dependence of the transmiss modified this notation, and reflection coefficients. To conform with the usual per cm7?. source function is designated B(c); its units are W (or photons /s)

Equation (13.23) then becomes | B(a)(1 + cos 2mox) da 0

Io(x) |

ih B(o) do + f B(o) cos 2na0a do 0 0

(13.24) B(o) cos 2roxdo . 0 ion, The first term in (13.24) is the constant level in the absence of modulat =

7+

m. We define and the second term contains all the information on the spectru

Io(x) — I. I(x) as the modulated part of the interferogram only — i.e. I(x) =

Then

oa

13. Interferometric Spectrometers

!

Fig. 13.10. Schematic interferogram for a polychromatic

source Bic): I(x) = le B(a) cos 2na0z do.

l(a

i B(o) cos2noa deo ,

I +

(13.25)

where the lower limit of integration of this term can be taken as —oo instead

of zero because B(o) =0 for negative o. Figure 13.10 illustrates the type of interferogram obtained from a multiline source.

I(x) is the cosine transform of B(c), and B(c) can be recovered from it by the inverse cosine transform. The Fourier transform pair is

Naepes ‘PeB(o) cos(27a0x) do (W) , 12,8)

Bia)

/ I(x) cos(2rox) dx (W per cm).

(13.26)

OO)

Looked at from a slightly different point of view, ometer imposes a characteristic modulation on each enables the signal to be subsequently unscrambled. moved at a speed v/2, the path difference changes

(13.25) becomes WKN

fe B(f) cos(2rft) df ,

the Michelson interferoptical frequency which If one of the mirrors is at a rate v; as x = vt,

(13:27)

—co

where

J Sv0 =vvic. The modulation frequency f is in the audio-frequency range; for example, a wavelength of 1 um with v = 1 mm/s gives f = 1 kHz. The detector receives signals simultaneously from every spectral element in the source, each encoded with a modulation frequency proportional to its optical frequency. As the proportionality constant is v/c, the interferometer can be thought of as a device for reducing optical frequencies from 10!4 Hz or so to a region in which they can be followed by a photoelectric detector. This form of spectroscopy is given the name ‘multiplex’, by contrast with the dispersive techniques in which the different spectral elements are spread out spatially and recorded either by one detector sequentially (scanning mode) or by many detectors

13.5 Michelson Interferometers and Fourier Transform Spectroscopy (FTS)

!

Te |

| i | |Mi in | |

P | \ | ATTY

323

Fig. 13.11. Asymmetric interferogram from unmatched optical paths. MA

NAsA

0

simultaneously (multichannel mode, applicable both to array detectors and photographic emulsions). So far we have implicitly assumed the interferogram to be symmetric about x = 0, so that the spectrum can be accurately recovered by the cosine transform. In practice some degree of asymmetry is almost inevitable; the most probable reasons for this are that the sampling grid (Sect. 13.5.3 below) does not coincide exactly with x = 0, that the beamsplitter and compensating plate are not exactly matched, and that the signal-processing electronics has some frequency-dependent delay. Figure 13.11 shows such a ‘chirped’ interferogram with a wavelength-dependent path difference. This additional path difference, «(a) say, can be expressed as a phase angle ¢(a) = 27a€, so that

(13.25) becomes co

Fey= | B(c) cos[2r70a + ¢(c)] do .

(13.28)

—co

To recover all the spectral information requires a full (complex) Fourier transform, as can be understood from symmetry considerations. Any asymmetric function can be represented as the sum of a completely symmetric (even) and a completely antisymmetric (odd) function; the cosine transform recovers the information from the even component of the interferogram and is zero for the odd component, while the sine transform recovers the odd component and is zero for the even one. The recovered spectrum is the complex quantity S(c) given by Co

Slay = |

I(x) exp(—27iox) dx = Bia)?

(13.29)

=29

Although it is possible just to take the spectrum as being represented by

the modulus of S(c), a better procedure for several reasons (connected with

reducing the effects of noise) is to ‘phase-correct’: a low resolution phase spectrum (a) is evaluated from a short, symmetrically truncated section

of the interferogram (Sect. 13.5.2) and used to multiply the high resolution spectrum by e'? so as to rotate the complex quantity S(o) into the real plane. These two ways of proceeding are represented by

324

13. Interferometric Spectrometers

B(a) = {Re[S(c)]? + Im[S()]?}"7,, 1? , Bic) = S(c)e

(13.30)

where

tan ¢= (Im[S(c)])/(Re[S(o)]) . The ideal mathematical transform, (13.26) or (13.29), is never realized in practice. There are two fundamental departures from these equations. First, the interferogram is recorded up to a finite path difference « = L, not to infinity. According to Sect. 11.1, this should limit the resolving power to about

L/X, or oL. We need to look at the instrument function before we can define R more exactly. Secondly, the Fourier transform is computed digitally, using discrete sampling points at intervals Ar in the interferogram to give B(c) at discrete spectral intervals. The relations in the above equations should therefore be Fourier sums rather than integrals, with the mathematical consequence (explained in Sect. 13.5.3 below) that the computed spectrum is replicated at wavenumber intervals determined by 1/Az. 13.5.2

Instrument

Function

and Resolution

We can find the instrument function by following the fortunes of a monochromatic input (i.e. a delta function), B(a) = b(ao), in a symmetric interfero-

gram, I(x) = b(00) cos(2mox), recorded from x = —L to x = +L. Ignoring for the moment the discrete sampling, the spectrum recovered from the transform is

L so) = vf cos(2709x) cos(2r0x) dz . =i The integrand can be expanded as 1

cos(27o9x) cos(27ox) = 5 {cos [27 (09 + a)| + cos [27 (09 —o)]} , and as it is an even function of 2, we get

sin[27(o9 D(a) =D |

+0)a] | sin[27(o09 — o)a] *

21(o9 + 0)

21(o9 — 0c)

(so

0

Of these two terms, the first is negligibly small for all positive values of 7. We shall have to take account of it when we come to consider aliasing, but for the moment we ignore it and take the second term to represent the instrumental profile of the line at a9. By multiplying top and bottom by L this can be written as

S(o) = bL

sin|27(o9 — 7) L] 2m(o9 — a)L

The instrument function is therefore the normalized sinc function

13.5 Michelson Interferometers and Fourier Transform Spectroscopy (FTS)

325

Fig. 13.12. The intensity distribu-

tion S(o) = sinc2(o — o0)L for a monochromatic line at oo. The first

zeros are at t1/(2L).

90

1/2L

Lsinc(2oL) = L

sin(27o0L)

(1d252)

2nroL This function is plotted in Fig. 13.12. At first sight the negative lobes are disconcerting, but it must be remembered that the instrument function is a mathematical distortion of the original delta function input, not a real signal obtained by scanning a detector across the line. The integrated area (with the normalizing factor L) is equal to unity, the height of the original delta function.

The first zeros of sinc(2cL) are at +1/(2L), and this distance is normally taken to define the resolution limit of the spectrometer:

resolution dg = resolving power R =

1/(2L), 2L/2.

(13.33)

This seemingly arbitrary choice will be justified when we come to discuss sampling in Sect. 13.5.3. It does not conform to the Rayleigh criterion, for two equal sinc functions separated by the distance to the first zero do not show any dip at all in the composite peak. On the other hand, if the separation is

doubled to 1/L the intensity falls to zero at the midpoint between the two

peaks. The FWHM of the function, as measured from the zero line, is 1.260, or 0.6/L, and in fact two equal lines separated by this amount come very close to meeting the Rayleigh criterion. As is evident from the derivation

of (13.31), d0 depends only on the extreme path difference L, not on the total length of the interferogram. The function is unchanged, except for a multiplicative factor 1/2, if we take the integration limits from 0 to L instead of —L to +L. The side lobes of the sinc function, the first of which reach 20% of the peak,

are more pronounced than those of the sinc? diffraction function of CLEL0);

but as they alternate in sign their integrated area actually contributes less to the background. This ‘ringing’ is a mathematical consequence of the abrupt truncation at {EL and can actually be modified if necessary. Indeed, a feature of FTS is the ability to vary the instrument function mathematically, as we shall now show. An important property of Fourier transforms is that the transform of the product of two functions is the convolution of their individual transforms.

326

13. Interferometric Spectrometers

Let A(x) represent the function that truncates the interferogram, so that the finite interferogram is described by A(x) x I(a). Then the recovered spectrum is given by

S(a) = FT[A(a) x I(x)] = A(o) * B(o) ,

(13.34)

where the tilde indicates the Fourier transform (FT) and * means convolution. The instrument function A(c) is the Fourier transform of the truncation function A(x). There is nothing peculiar to Fourier transform spectroscopy about the right-hand side of this equation: the output from a grating spectrometer or a Fabry—Perot can similarly be expressed as the convolution of the true spectrum with the appropriate instrument function, the sinc? or the Airy distribution, respectively. It is helpful for what follows to show that (13.31) can be derived directly from the above convolution relation. The truncation function used here was the top-hat, or boxcar, function 1, defined by

Al@) =) for |e

La

Ale) 20 for kel Sak.

One needs to know two Fourier transform pairs: (i) the FT of a top-hat is a sinc, 1.e. Rim

sincroln,

and (ii) the Fourier transform of cos 270 is a delta function pair at op and —og. Then, (13.34) becomes

S(a) = sinc(20L) x [d(0 + o9)] . The relations in the two Fourier domains are illustrated in Fig. 13.13. The mirror image of the spectrum at @ = —go corresponds to the first term in (13.31). Obviously this mirror image is not peculiar to the mythical monochromatic spectrum: the Fourier transform process always gives both the real spectrum and its mirror image at negative frequencies. The instrument function can be changed by varying the truncation function A(a); this can be done mathematically, simply by multiplying each interferogram point by an appropriate factor before transforming. The best known example, although not the best function to use in practice, is the triangular function A; falling linearly from 1 at « = 0 to 0 at x = +L. This is the convolution of two equal top-hats with limits +L/2, Mlz/2, so its FTis the product of the two corresponding sinc functions: A(o) = since”oL, which is identical to the diffraction-limited instrument function of a grating spectrometer. ‘The distance to the first zero is now 1/L, twice the value for the original sinc function, and the FWHM is also increased, from 1.2 x (1/2L) to ~ 1.8 x (1/2L), but the side lobes are reduced to about 4% of the maximum. For this reason the tapering of the truncation function to reduce the ringing is known as apodization ~ literally ‘cutting off the feet’. Several different apodization functions are in common use and can be chosen to suit particular problems. For a given value of L, the more gentle the cutoff the more the

13.5 Michelson Interferometers and Fourier Transform Spectroscopy (FTS) x-domain

By

o-domain

!

B

AANA =

elas

x

*

Fig. 13.13. Multiplication of the infinite cosine interferogram by the top-hat truncation function A(x) in the x domain, corresponding to convolution of the delta functions at +oo with the sinc function A(c) in the o domain. The two functions in each horizontal row are Fourier transforms of one another.

ringing is reduced, though always at the expense of increased FWHM. If L is large enough for the true spectral linewidth to exceed 1/2L, then the interferogram is effectively self-apodized — i.e. the modulation falls to zero at the ends of the interferogram, and the ringing disappears. In terms of (13.34),

the convolution approximates to B(a) when A(c) is appreciably narrower than B(o). The line is then said to be fully resolved. Again, this situation is not peculiar to FTS, but it is not so frequently encountered with gratings because of their lower potential resolving power.

13.5.3

Sampling and Aliasing

We come now to the replication of the spectrum arising from the discrete sampling. This is a general feature of Fourier series. B(o) in (13.26) should actually be

Bo) =

j=N >a I(jAz) cos[2ra(jAz)] j=—N

Ar is the ‘fundamental

.

(13.35)

frequency’ in this Fourier series, and the function

B(c) that the series represents is replicated at intervals Ao =

1/Az in

the wavenumber domain. This follows mathematically from the fact that cos 2ajaAzx repeats itself when o increases by 1 /Az, but it can also be understood from the convolution theorem, (13.34). Sampling the interferogram at intervals Ar means multiplying it by a so-called Dirac comb, or shah

328

13. Interferometric Spectrometers x-domain

o-domain

!

B \

“

=O ll On

x

*

j

S

AX

a

yay

|

fie 1/Ax

I We ate |

_KAKAIMAKA |

X

oe

|

Ses

o

Zoe

Fig. 13.14. Sampling (i.e. multiplication) of the interferogram by the Dirac comb of spacing Az in the x domain, corresponding to convolution of the spectrum with the Dirac comb of spacing Ag = 1/Az (i.e. replication at intervals Ac). As in Fig. 13.13 the two functions in each row are Fourier transforms of one another.

function, as shown in Fig. 13.14. The Fourier transform of this Dirac comb is another Dirac comb with spacing Ao = 1/Az (see the Fourier transform literature cited). This second comb has to be convolved with the spectrum, producing the replication shown in Fig. 13.14. We can now define a minimum ‘safe’ sampling interval. If the original spectrum extends from o = 0 to some maximum value, o,,, then the recovered spectrum has a total width of 20,, because of the mirror image. It can be

seen that the replications will begin to overlap unless the replication interval 1/Az is at least equal to 20,,. Thus the minimum sampling interval is given by

AN ings

L

20m

= Amin 2

(13.36)

In other words, it is necessary to sample the interferogram at least every halfwavelength of its shortest wavelength in order to recover the true spectrum. This criterion is well-known in signal processing; the minimum sampling rate, twice the highest frequency in the band, is known as the Nyquist frequency. Figure 13.15a shows a spectrum recovered from an interferogram sampled at the Nyquist interval Az = 1/20,;. If one is working in the visible or ultraviolet regions, much of the wavenumber range from 0 to oy is likely to be completely empty (as in the illustrations in Fig. 13.15) because of the long-wavelength cutoff of the detector, the source, or any optical filters used. For example, in the ultraviolet below 300 nm it is common to use ‘solar-blind’ detectors that are insensitive above

13.5 Michelson Interferometers and Fourier Transform Spectroscopy (FTS)

329

a) -Om

0

Om

1/Ax

2/Ax

og

Fig. 13.15. Representation of aliasing. In (a) the interferogram is correctly sampled to give a free spectral range of om. In (b), where the spectrum is band-limited to the region between a; and a2, the interferogram can be undersampled to give a free spectral range Ao > o2 — 01. (c) represents the recovered spectrum, without ambiguity provided it is known to exist in the fold between 2Ao and 3Ao.

this wavelength. The spectrum is then effectively band-limited to the region 30000-50000 cm~! (the upper end being the oxygen absorption cutoff), and if the interferogram is correctly sampled, more than half of the recovered spectrum is empty. Band-limiting by means of optical filters may in fact be used in any spectral region. In these circumstances it is possible to undersample without any ambiguity in the recovered spectrum. The sampling interval defines a free spectral range according to

free spectral range

= Ao = 1/(2Az) ,

(13.37)

and, provided the band limits 7, and a2 satisfy la, —02|< Ao,

the recovered spectrum is contained within one replication. This is illustrated in Fig. 13.15b,c. The high frequency band shown shaded is aliased by undersampling into the empty lower frequency regions from 0 to Ao and Aco to 2Ac (as well as into the empty higher frequency regions above 3Aqa), as shown in (b). Provided the true spectrum is known to exist only between 2Ac and 3Ac there is no ambiguity. It is often convenient to think of the spectrum as

folded backwards and forwards at intervals of Ao as shown in (c); each fold

represents an alias, and the spectrum in this case is in the third alias. Prior knowledge of the band limits is required to determine the correct alias. Two conditions must be met if undersampling is not to degrade the spectrum. First, some combination of source, detector and filter characteristics must ensure that the signal really is restricted to the free spectral range Ao, and second, a suitable electrical or digital bandpass filter is needed to prevent noise from other aliases from being folded back into the region of interest, as

explained in Sect. 13.6.1.

330

13. Interferometric Spectrometers

The discussion of sampling in this section can be used to justify the apparently arbitrary definition of resolution limit do in (13.33). As the x and ao domains are linearly connected by Fourier transforms, it must be possible to go back and forth between them without loss of information. Suppose we start with the spectrum in Fig. 13.15a and ask how often we should sample it in order to reconstruct the original interferogram. Even if the latter was recorded only from 0 to +L, the reconstruction by Fourier transforming the spectrum will include the mirror image from 0 to —L, a total length of 2L. If the spectrum is sampled at intervals of do, the recovered interferogram is

replicated at intervals of 1/é0, so overlap of the replications is just avoided if 1/60 = 2L. The set of samples at intervals of do = 1/2L therefore carries all the useful information in the spectrum. Sampling more coarsely degrades the information, and sampling more finely adds nothing to it. With these definitions the number N of independent samples is the same in both domains:

wal peg Axa 60

(13.38)

with

il do

=

57 and

1 An

=

5A5

Moreover, if the interferogram is sampled at the Nyquist frequency so that the spectrum is in the first alias, N = o,/d0 = R. There is an obvious analogy here with a grating, which samples the wavefront at N points to

give a (diffraction-limited) resolving power of N. The relation still holds for undersampling because the maximum wavenumber in the mth alias is mAc, giving

ReamAclio=imN,

(13.39)

just as for a grating of N rulings used in the mth order.

13.6 Practical Aspects of Fourier Transform Spectroscopy Some important characteristics of PTS, in particular the accuracy and the signal-to-noise ratios attainable, will be discussed in Chap. 17, where comparisons are made with grating and Fabry—Perot spectroscopy. In this section we consider the requirements first for the interferometer and then for the control and sampling systems and the computation of the spectrum.

13.6.1

The Scanning Interferometer

The optical tolerances for a two-beam interferometer are normally specified as \/4 for the recombining wavefronts. As several surfaces are involved, this

13.6 Practical Aspects of Fourier Transform Spectroscopy

331

Fig. 13.16. Retroreflectors used instead of plane mirrors to make Michelson interferometer tilt-invariant.

amounts to about 4/8 for each surface. This sounds like an easy requirement to meet after the \/50 of the Fabry—Perot, but it has to be maintained over much longer scans of the moving mirror: a resolving power of one million,

for example, requires L = 1 m at a wavelength of 2 um and L = 10 cm at 200 nm. Although the necessary scan length decreases with wavelength, the tolerances for both optical and mechanical precision get increasingly difficult to meet, and this is certainly one of the principal reasons why FTS, though a well-established technique in the infrared, has only rather recently been extended to the visible and ultraviolet. One way of easing the demands on the scanning system is to use retrore-

flectors (cube corners or catseyes) instead of plane mirrors (Fig. 13.16). These

are tilt-invariant, and they have the additional advantage that the complementary interferogram returning towards the source can be laterally displaced from the input beam and picked up by a second detector for the improvement The arrangement shown in the figure, with the of the signal-to-noise ratio. two halves of the beamsplitter coated on opposite sides, also dispenses with the need for a separate compensating plate. Errors in the recombining wavefronts, whether due to the optical surfaces or to misalignment during scanning, decrease the depth of modulation so that the cosine term in the basic equations (13.22)—(13.26) is multiplied by some factor less than unity while the mean value is unaffected. This is undesirable because the modulated part of the signal contains all the useful information while the constant part contributes only noise. The modulation depth is also reduced if the assumption made after (13.22) that the two interfering beams have the same amplitude is not valid. As normally arranged, each beam has one reflection and one transmission at the partially reflecting coating of the beamsplitter, so equal amplitudes are achieved regardless of the actual value of the reflectivity. For maximum efficiency, and however, the reflection and transmission coefficients should be equal, returning output this condition also maximizes the modulation in the second the towards the source, for which one beam has undergone two reflections and the of half other two transmissions. If there is no absorption in the coatings,

Boe

13. Interferometric Spectrometers

incident energy then goes into each beam. Absorption is scarcely a problem in the infrared and visible regions but becomes increasingly serious into the ultraviolet, not only for the semireflecting beamsplitter coatings but also for the collimating and focusing and any beam-steering mirrors before and after the beamsplitter. Instruments designed for the infrared can afford to be far more lavish with such mirrors than those operating in the ultraviolet. We have seen that although the spectral resolution depends only on the maximum path difference L on one side of zero, the calculation of the phase (a7) requires a symmetrically truncated part of the interferogram (since otherwise the instrument function for the phase spectrum itself contains a phase factor). It is thus always necessary to record a section of the interferogram for x —Om| are aliased in the same way as in a scanning FT spectrometer that is undersampled. It is possible to get around the ambiguity problem by rotating one of the gratings through a small angle @ about an axis in its plane, so that the grooves of the two gratings are no longer parallel. This produces an unheterodyned low resolution interferogram along the y axis, in which o, + do are no longer degenerate, thereby doubling the free spectral range. By an extension of this idea, the gratings can be used in higher orders to produce a two-dimensional interferogram that can be transformed to yield a spectrum in an echelle-like format, with the overlapping orders of the high resolution spectrum along the x axis separated by low resolution cross-dispersion along

the y axis [41]. The advantages or otherwise of SHS over a straightforward dispersion arrangement using the same gratings (and hence the same resolving power) and the same detector are mainly based on the larger throughput and hence better signal-to-noise ratio of the interferometer. As against conventional scanning F'TS, SHS is much less flexible, its resolving power is limited by grating rulings, and its intrinsic wavelength accuracy is lower. On the other hand, a non-scanning instrument can be much more robust and less sensitive to fluctuations of the light source. Moreover, it lends itself more readily to an all-reflection configuration, with a grating acting as beamsplitter, that can be

Further Reading

ool

used at short wavelengths beyond the transmission limits of optical materials

suitable for beamsplitters [42].

Further Reading — Bousquet, P., Spectroscopy and its Instrumentation (Hilger, London, 1971). — Steel, W.H., Interferometry, 2nd ed. (Cambridge University Press, Cambridge, 1983). — Hernandez, G., Fabry-Perot Interferometers (Cambridge University Press, Cambridge, 1986). — Chamberlain, J.C., The Principles of Interferometric Spectroscopy (Wiley, New York, 1979). — Bracewell, R.N., The Fourier Transform and its Applications (McGraw Hill, New York, 1965). For the infrared

— Chantry, G.W., Long-Wave Optics, 2 vols. (Academic Press, New York, 1984). — Griffiths, P.R. and deHaseth, J.A., Fourier Transform Infrared Spectrome-

try (Wiley, New York, 1986). For the vacuum

ultraviolet

— Samson, J.A.R. and Ederer, D.L. (Eds.), Vacuum Vol. II (Academic Press, New York, 1998).

Ultraviolet Spectroscopy,

a.

al

ind

@

‘4(tur th SU) owe


+:sapphire. The energy level scheme is similar to that shown for the ruby laser in Fig. 14.la, with the important difference that both the levels 1 and 2 of the Ti+ ion are coupled to the lattice and so form wide bands. Level 2 is fed, as before, by rapid radiationless decay from the pumped band above it, and lasing can take place over a wide band from 660 to 1180 nm. The output power can be significantly higher than that of a dye laser, allowing efficient frequency mixing (see next section) to extend the tunable range to both longer and shorter wavelengths. For the infrared, in addition to the semiconductor laser diodes described above, which have a very limited tuning range, there are colour centre or F-centre lasers, which are crystals of the heavier alkali halides doped with Li or Na (FA and FB centres, respectively). They are pumped and tuned in the same way as dye lasers, each type of crystal having an absorption band at shorter wavelength than its emission band. They cover the range from the far red to about 4 um, and their tuning ranges and output powers are comparable with those of dye lasers. Tunable lasers for the ultraviolet exist in the form of excimers. The word is a portmanteau of ‘excited dimer’, the first such lasers being the inert gas dimers Xe, Krg and Arg. The word is also applied, semantically incorrectly, to inert-gas-halogen combinations such as KrF and XeF. The common characteristic of all these ‘dimers’ is a stable excited state and an unstable ground state. The dimers are formed in a highly excited state by an electron beam of high current density in gas at a pressure of a few atmospheres, and they then cascade down to the lowest excited electronic state. For Xe, for example, this is Xe (5p°6s) + Xe(5p°). There is always population inversion of this state 11 GHz

& 0.03 cm~?.

346

14. Laser Spectroscopy

with respect to the ground state because emission to the latter is followed immediately by dissociation. Emission takes place over a band of some tens of nm on the long-wavelength side of the atomic resonance line. The required spectral band is selected by a tuning element in the optical cavity. Excimer lasers operate in pulses of a few ns with a peak power that is usually of the order of 1 MW, depending on bandwidth, excimer type, etc. They are used both directly as tunable ultraviolet lasers, and also in frequency mixing and frequency doubling and tripling arrangements to reach wavelengths in the extreme ultraviolet. The inert gas monohalide lasers, which give very large powers at wavelengths around 200-300 nm, also serve as pumps for dyes with absorption bands in the ultraviolet. 14.2.3

Frequency Doubling and Mixing

Frequency mixing exploits the non-linear terms in the expression for the polarizability of a medium. The electric dipole moment p induced by an electromagnetic wave, E = €, cosuwt, is to first order proportional to €,, and the polarizability and refractive index of the medium are usually calculated on the basis of this linear relationship. However, if €, is large, higher-order terms must be included, and the expression for p becomes D=Vl

ree

PVE

+...

(14.3)

where a, @ and y are tensors unless the medium is isotropic. It can be seen

that the second term in (14.3), GE? cos? wt, when rewritten as GE2(cos 2wt + 1), contains the second harmonic of the original wave. In the same way 3w appears in the third term. If all the dipoles in the medium are oscillating coherently, the emerging wave contains these harmonics along with the original frequency. For 3 to be non-zero the medium must have some anisotropy, a condition that can be met in crystals but not in gases. Moreover, for the coherence condition to be satisfied the refractive index of the medium has to be the same for the waves at w and 2w (or 3w), a condition known as phase-matching. Frequency doubling and tripling is often used to enable ruby or neodymium lasers to pump dyes, many of which have absorption bands corresponding to half the ruby or one-third the neodymium wavelengths. It is also used to extend the range of tunable dye lasers into the ultraviolet. Starting with an excimer laser and using suitable mixtures of metal vapour and inert gas for the tripling and phase-matching, it is possible to achieve tunable laser radiation down to about 100 nm. In frequency mixing the same process is applied to radiation from two lasers at different frequencies w; and wa. The second-order term in (14.3) contains the sum-and-difference frequencies w; + Ww. If w; represents a fixed frequency, the output can be tuned by varying we, provided the phase-matching conditions for all three waves are met. Similarly, the third-order term contains not only the third harmonics, but also the combinations

2W, + We and

14.3 Optical Pumping and Saturation

347

w +2w>. In this way frequency mixing can be used to generate tunable radiation in both the ultraviolet and the infrared. For example, a Ti:sapphire laser pumped by the second harmonic of a Nd:YAG can be doubled or tripled and mixed with the fundamental of the pump laser to give tuning ranges right down to 200 nm. Stimulated Raman scattering represents another frequency extension method that should be mentioned, although it is not strictly frequency mixing. An intense laser pump beam can excite Raman scattering at sufficient intensity for coherent amplification, either at longer wavelength (Stokes scattering) or at shorter wavelength (anti-Stokes), the shifts corresponding to the energy differences between the initial and final vibrational or electronic energy levels in the scattering medium.

14.3 Optical Pumping

and Saturation

Optical pumping is the name given to the process of increasing the population of an atomic or molecular level above its equilibrium value either by absorption of radiation or by a combination of absorption and re-emission. We have already met it in this chapter as one of the ways of achieving population inversion in a laser; more generally, it is a way of preparing a collection of atoms for further experiments or observations. Optical pumping experiments were carried out long before lasers existed, but they are much easier with tunable lasers because of the very high photon flux density within the absorption bandwidth of the transition that is being pumped. As an example of the use of optical pumping to prepare a true two-level atom, consider the process illustrated in Fig. 14.4, which represents an atom with two states having J = 1 and 2, respectively, in a magnetic field. There are three magnetic sublevels in the lower state and five in the upper (Sect. 3.9.1, Fig. 3.16). If the atoms are irradiated with right-circularly polarized light of the correct frequency, they are excited to one of the upper sublevels corresponding to AM, = +1. For re-emission AM; = —1,0,+1 are equally Ss

J=2 now40o-n

J=1

Y =

Fig. 14.4. Example of optical pumping. Absorption of right-circularly polarized light corresponds to AM; = +1, and ideally all atoms are eventually cycled between the two levels joined by the double arrows.

348

14. Laser Spectroscopy

probable, but the next absorption again increases Mj 7 by 1. All the atoms are eventually pumped into the M7 = 1 sublevel, and while the radiation is still on they continue to cycle between this level and the M7 = 2 level of the upper state. A similar process can be applied to hyperfine magnetic sublevels. With high enough flux density at the resonance frequency it is possible to ‘saturate’ a transition. From Fig. 7.4 and eq. (7.18), spontaneous emission can be neglected compared with stimulated emission if Bj;p >> Az. The equilibrium condition then becomes Bz\ng = By2n,. Equation (7.19) for the relations between the A and B coefficients shows that the populations of the two levels are in the ratio of their statistical weights m2

Mm

= © for the condition >

91

Ao

8rhv

Boy

c

aia

Soe

(14.4)

In terms of [(112), the spectral flux density of a collimated beam (in W m~?

Hz~'), the saturation condition is

I> a

3

= seh

(14.5)

which can be rather more vividly pictured as 87 photons per second per Hz over an area of one square wavelength. Once saturation is attained, the gas becomes transparent to radiation of that particular wavelength because absorption and stimulated emission are equally likely. Actually (14.4) is a necessary but not a sufficient condition for saturation; the laser has to supply sufficient power to replenish the losses from spontaneous emission, and in a gas of high column density this may be a more stringent criterion. If one assumes that all spontaneously emitted photons escape from the gas (i.e., radiation trapping is neglected), the power loss from a column of gas with

area S (m*), length LZ (m) and number density n (m7) is nSLAos1 photons per second. The laser spectral flux density required to keep pace with this is nL Ao, hv

ia

Av

Wm?

Hz"! ,

where Av is the effective linewidth for absorption. If the laser bandwidth is smaller than the Doppler width of the absorbing atoms, Saturation occurs only for a subset of atoms within the Doppler spread. In laser spectroscopy, natural (or natural plus pressure) and Doppler broadening are often referred to as homogeneous and inhomogeneous broadening, respectively. The reason for the distinction is that Doppler (inhomogeneous) broadening is the result of observing atoms with a range of line-of-sight velocities. If a particular subset of atoms all having essentially the same velocity can be selected in some way, the linewidth is determined by pressure effects

or natural lifetime, and the line is said to be homogeneously broadened. The significance of the distinction for saturation is that narrow-band laser radiation at a frequency Vv; travelling in the x direction is absorbed only by the particular subset of atoms with an 2 component of velocity v; such that their rest resonance frequency Vo is Doppler-shifted to match 1:

14.4 Spectral and Temporal Laser Bandwidths

349

Fig. 14.5. Hole burning. Laser radiation at is absorbed by atoms with a vz component of v1, creating a ‘hole’ in the ground-state population at this value of v4.

ns het aS c

(14.6)

Vo

It is thus possible to saturate a narrow band within the Doppler profile, the width of which is determined by whichever is the greater of the laser bandwidth and the homogeneous linewidth of the absorbing atoms. This process is known as ‘hole-burning’ and is illustrated in Fig. 14.5. Saturation leads on to the important techniques of Doppler-free spectroscopy discussed in Sect. 14.6. It is also the mechanism for producing and stabilizing narrow tunable laser bandwidths, and we shall consider this first.

14.4 Spectral and Temporal Laser Bandwidths It was seen in Chap.8 that Doppler widths in the visible region are typically of order 1 GHz (0.03 cm~'), whereas homogeneous broadening (natural linewidths) may be two or three orders of magnitude smaller. Evidently ‘subDoppler’ laser bandwidths are needed to perform Doppler-free spectroscopy. There is a fundamental relation between spectral and temporal resolution set by the bandwidth theorem, Av At ~ 1: a pulse of at least 10 ns is required for a linewidth of 100 MHz, for example. Instability and shot-to-shot reproducibility also affect the linewidths of pulsed lasers, and the narrowest laser linewidths are obtained with CW lasers. A good CW dye laser can have an inherent linewidth, determined by its resonance cavity, of about 10 MHz, and since this ‘width’ is really due to small fluctuations, thermal and otherwise, in the length of the cavity it can be improved — to about 1 MHz — by locking the frequency to a stabilized reference cavity. One can think of this as a Fabry-Perot etalon of large separation, through which some fraction of the laser output is passed. If the

laser frequency wanders, the transmission of this cavity changes, and a correction signal is generated to the laser cavity. The reference cavity itself may be locked to a second, fixed-frequency, stabilized laser. In normal use the dye laser has to be scanned through the spectral region under investigation. This can be done either by changing the optical path in the reference cavity

350

14. Laser Spectroscopy N

a)

b)

Fig. 14.6. Laser stabilization by saturation. For detuning by Av, ‘holes’ are burned at +v1, as shown in (a). When Av = 0, as in (b), the ‘holes’ coalesce.

or by using this cavity as a Fabry—Perot to record successive maxima. The continuous tuning range of most CW lasers is only about 30 GHz; there is then a jump to a new tuning band. The saturation phenomenon described in the previous section can be used to stabilize a reference laser. A gas cell with an absorption line at a frequency Yo somewhere within the laser gain curve is placed in the laser cavity. If the laser is operating at a frequency vp — dv, the light is absorbed by atoms travelling towards the incident radiation, which see it as blue shifted into resonance, and a hole is burned in the population distribution at the velocity v1 given by (14.6); but the cavity contains a wave of equal amplitude travelling in the opposite direction, and this is absorbed by atoms of equal and opposite speed, so another hole is burned at the velocity —v,, as shown in Fig. 14.6a. When the laser is tuned (by adjusting the cavity length) to exactly vo the two holes coalesce (Fig. 14.6b), and the gain of the cavity increases, as shown in Fig. 14.7. The pip on the Doppler distribution has the homogeneous width of the absorbing transition and is often called the ‘inverted Lamb dip’. The true Lamb dip refers to an analogous saturation effect in the lasing medium itself: the laser gain has a sharp dip at v9 because fewer atoms can contribute to the stimulated emission when both left- and right-travelling waves are using the same group of atoms. Any atom or molecule with sharp absorption lines in the appropriate spectral region can be used for locking on the inverted Lamb dip in this way; methane (CH4) is the most useful in the near infrared and iodine (Iz) in the visible region. The long-term stability and reproducibility of such lasers can be as good as 5 and 50 kHz, respectively — well below the actual molecular linewidths. Temporal resolution, obtained from very short laser pulses, offers the possibility of measuring lifetimes of excited states and following collision rates and fast chemical and biological reactions. Pulses of a few ns can be obtained by inserting in the laser cavity a Pockels cell, which is essentially a variable polarizer. This is switched off until pumping of the laser transition is

14.5 Laser Absorption and Excitation Experiments

351

Fig. 14.7. Frequency locking. The laser gain curve is reduced by the loss from the absorber, but the hole in the absorption

(Fig. 14.6b) gives a reference pip on the net gain curve.

complete; when it is switched on the gain increases so rapidly that the level population reaches equilibrium in a few ns. This so-called Q switching is used to produce very high powers as well as short pulses. Much shorter pulses can be generated by mode-locking. A cavity of length L without additional tuning can support a number of modes, as shown in Fig. 14.2, the wavelengths of which are defined by integral values of m in the relation mA = 2L. If the phase relation between the corresponding frequencies is constant, which is the condition for mode-locking, then there is a constant beat frequency at the mode separation, c/2L Hz — i.e. all the modes come into phase at time intervals of 2L/c. They stay in phase only for a time At inversely proportional to the frequency band over which they are spread,

which is the full width of the gain curve in Fig. 14.2. If this is 10’? Hz, which

is 1 nm in the visible region, At is 1 ps. There are different techniques for mode-locking, which will not be discussed here. Pulses of a few femtoseconds

have actually been obtained [43]; this is almost unbelievably short when one remembers that it corresponds to only a few cycles of the optical frequency. This ultrashort pulse work does not have direct spectroscopic applications, and we shall not pursue it here.

14.5 Laser Absorption and Excitation Experiments Atomic and molecular absorption lines can be recorded without the need for a spectrometer by scanning a tunable laser across the line profile. Apart from Doppler-free spectroscopy (Sect. 14.6), laser spectroscopy offers several possibilities that can be realized either with great difficulty or not at all by conventional spectroscopy. Perhaps the most important is that the high photon flux can populate the upper level of a transition with great efficiency. Two different types of experiment follow from this. First, the fluorescence from this upper level is an exceedingly sensitive diagnostic of the absorption resonance, following the usual rule that a small positive signal is more readily detectable than a small change in a large signal. Indeed, laser-induced fluorescence is usually the most sensitive of all analytical techniques for detecting trace quantities of elements. It will be seen in Sect. 14.7 that fluorescence from single atoms or ions confined in traps can be detected. Second, efficient

3O2

14. Laser Spectroscopy

laser 1

laser 2

discharge or laser 1

i a) Fig. 14.8. (a) Stepwise excitation and (b) two-photon excitation. Level k in both cases is a real level. The two-photon probability is enhanced when the virtual level n (shown broken) is close to a real level k.

stepwise excitation becomes possible. Figure 14.8a shows a schematic example. The intermediate level, level k in the figure, may be populated either by a discharge or by a tunable laser, and the final level is reached with a second tunable laser. By the ordinary dipole selection rules, this final level has the same parity as the initial level and so cannot be reached by a single transition from the latter. With sufficient laser intensity, two-photon transitions become possible, as shown in Fig. 14.8b. The intermediate level is in this case a virtual level. labelled n in the figure. The transition probability is found from second-order perturbation theory to be proportional to the square of the laser intensity

and to the product of the transition moments i + k and k + f [see (2.67)] summed over all real states k:

aryl x iJer|&) (kler|)P (Wri =

;

(14.7)

Wik)?

where w,,; is the laser frequency. The process is somewhat similar to Rayleigh and Raman scattering, which involve absorption to a virtual level and emis-

sion to either the initial level (Rayleigh) or another real level (Raman). It is evident from (14.7) that the probability of the two-photon transition is much enhanced if the virtual level is close to a real level. P; does not, of course, actually become infinite when w,,;= w;, because there is a damping term that has not been included in the equation but becomes significant near resonance. ‘T'wo-photon transitions are important in several respects. They connect the ground state with states of the same parity so that, for exam-

ple, s-s and s—d transitions can take place, and transitions in the ultraviolet can be probed with visible region tunable lasers. Two-photon Doppler-free spectroscopy (see next section) is a particularly powerful technique. There is no reason for stopping at two photons; multiphoton excitation. and ionization have proved entirely feasible. The three-photon transition probability depends on the cube of the laser intensity and on terms analogous

14.5 Laser Absorption and Excitation Experiments

353

Yk, ivoo_— Yfflb03: Rydberg

states

isotope

(Shite |

a)

b)

Fig. 14.9. Selective ionization. A narrow-band laser is tuned to the resonance frequency of a particular isotope. The excited atoms can then be ionized by a broadband laser, as in (a), or further excited to a high n state, as in (b), from

which field ionization is possible (see text).

to (14.7), but with the product of three matrix elements in the numerator

and two frequency differences in the denominator. Again the cross-section is much enhanced whenever one of the virtual levels coincides with a real level. Efficient selective population of excited levels, whether by one or by more photons, has made possible a wide variety of experiments, of which the following must be regarded as examples, not a complete list. The absorption spectrum from the populated excited level to higher levels or to the continuum can be probed, either by a second laser or by conventional methods. The lifetime of the level can be measured, as further discussed in Chap. 16. If the state is reasonably long-lived its structure can be probed by various magnetic resonance and level-crossing techniques that are beyond the scope of the present book [44]; such techniques were originally confined to atoms and molecules in their ground states. Selective excitation is also important as a method of isotope shift measurement and isotope separation. If the laser bandwidth is smaller than the isotope shift (Sect. 3.82) in the transition concerned, one isotope at a time can be excited, or at least preferentially excited. If a second photon from the same laser, or from a second broadband laser, photoionizes the atom, as shown in Fig. 14.9a, the ion can be extracted with an electric field by virtue of its positive charge. Actually, it is not necessary for the second laser to take the atom all the way to its ionization limit: if one of the closely packed high-lying states is populated, as shown in Fig. 14.9b, the ionization process can be completed by applying an electric field, a process known as field ionization. The strength of field required depends on the

energy gap to be bridged. The selective excitation /field ionization method is

extremely sensitive, as in principle every atom capable of absorbing a photon of the particular frequency to which the laser is tuned can be ionized and of detected. It is actually possible to detect one atom against a background [45]. frequency about 1012 atoms or isotopes of slightly different resonance

354

14. Laser Spectroscopy

The properties of the states of high n shown in Fig. 14.9b — the so-called Rydberg states — are themselves of great interest. As the mean distance of an electron from the nucleus increases with n?, a highly excited electron sees the nucleus and core electrons almost as a point charge of +e, and the atom becomes increasingly hydrogen-like. A number of strange effects result both from its macroscopic ‘size’ (for n = 100, the ‘diameter’ of the atom is 1 um) and from the weakness of the electron-nuclear binding energy, which decreases as 1/n? and is therefore only about 1 meV for n = 100. Relatively weak external fields appear strong to Rydberg atoms, allowing the study of quadratic Zeeman and higher-order field effects, the mixing of states of different n value, and so on. The large atomic radius gives rise to a large electric dipole moment, with interesting consequences for interactions with

other atoms and for collisional cross-sections [46]. Another feature of Rydberg atoms of high / value, excited by multiphoton absorption, is that they cannot decay directly to the ground state because of the Al selection rule, and as transitions to nearby high n states are in the microwave frequency range, where spontaneous transition probabilities are very small, they actually have rather long lifetimes. By introducing such atoms into a microwave cavity it is possible to conduct very fundamental studies of the interactions between a few atoms and a few photons [47].

14.6 Doppler-Free Spectroscopy In Doppler-free spectroscopy the smearing out of the homogeneous line profile by the velocity spread of the absorbing atoms is avoided by selecting a subset of atoms with a particular line-of-sight velocity — normally zero velocity. There are various ways of achieving this, but the principle can be explained in the context of the original ‘hole-burning’ method. As shown in Fig. 14.5, a sufficiently powerful narrow-band laser tuned to a frequency Av from the centre vp of an absorption line can burn a hole through the absorption profile by saturating the transition — i.e. equilibrating the upper and lower state populations — for all atoms with the particular value of v;, given by (14.6). A probe laser tuned to the same frequency finds the gas transparent if it is going in the same (+2) direction, but if its direction is reversed the atoms it probes are those with velocity —v,, and the gas no longer appears transparent. Only if both lasers are tuned to vp does the probe see the same set of atoms as the bleaching laser — those with zero Vz. In practice, a single laser with a beamsplitter and appropriate mirror arrangement forms both bleaching and probe beams, as shown in Fig. 14.10. The laser is scanned through the line profile, and the probe detector registers a sharp peak at line centre. This phenomenon is essentially the same as the inverted Lamb dip locking described in Sect. 14.4. The technique is made much more sensitive by modulating the bleaching laser so that the line centre is detected by the appearance of modulation

14.6 Doppler-Free Spectroscopy

355

Fig. 14.10. Doppler-free spectroscopy by saturation. The laser beam L is divided at the beamsplitter S into an intense bleaching beam B and a weak probe beam P, which pass in opposite directions through the gas G. Tuning to the line centre is registered by an increase in the probe beam signal at the detector D.

in the probe detector. This reduces the effects of background and scattered light to the extent that it is possible to detect, for example, Doppler-free transitions in a discharge. The hole in the Doppler distribution is constantly refilled by velocity-changing collisions in the gas, so the Doppler-free peak actually rises from a platform of the full Doppler width (Fig. 14.11). Various other modifications of the basic absorption saturation technique have been devised to improve the sensitivity and reduce noise so as to detect weaker signals. If the bleaching laser is circularly polarized, the atoms are pumped to particular Mj, sublevels in the manner illustrated in Fig. 14.4, and the Doppler-free peak can be detected by a change in polarization of the (plane-polarized) probe laser, a small change in polarization being more easily detected than a small change in intensity. Further clever tricks can be played by modulating both beams at different frequencies or by modulating the polarization. The ultimate bandwidth limit is the natural linewidth, which is of order

10 MHz for a strong line and a few kHz for ‘forbidden’ lines (Sect. 8.2). This has been approached, or even reached, in some experiments. The difficulties of working to such limits are increased by the necessity of using low gas pressures and reasonably low laser intensities to avoid pressure and power broadening, respectively. Probably the most fundamental measurement made by saturation absorption spectroscopy has been the absolute wavelengths of the fine structure components of the Balmer lines of hydrogen and deuterium

Yo

ue

Fig. 14.11. Doppler-free signal for an arrangement such as that of Fig. 14.10 with modulation of the B beam. The Dopplerfree spike rises from a base of the full Doppler width.

356

14. Laser Spectroscopy

(Sect. 2.1.4): these have yielded a value of the Rydberg constant accurate to about three parts in 10°, an improvement of a couple of orders of magnitude on Doppler-limited measurements. The technique of two-photon Doppler-free spectroscopy is essentially a complementary one, applicable to transitions forbidden to single-photon spectroscopy as discussed in the last section. In this case the laser beam is split into two oppositely propagating beams of equal intensity, and the Dopplerfree transition occurs for atoms capable of absorbing one photon from each beam, which means zero velocity in the beam direction. There is again a Doppler-broadened plateau, in this case corresponding to absorption of two photons from one beam. The most fundamental application of the two-photon technique is also to the fine structure of the hydrogen atom — in this case to the two photon 1s?S—2s 7S transition. At densities sufficiently low that collisional transfers from 2s 7S to the almost degenerate 2p 7P level are rare, the former is metastable, and the intrinsic natural linewidth is only a few Hz. By the Balmer formula, the frequency of the 1s—2s transition is (1 — 1/4)Ry =

(3/4) Ry, corresponding to two photons at 243 nm. Now the frequency of the Balmer ( line, n = 2 —n = 4, is given by v = (1/4 — 1/16) Ry = (3/16) Ry, corresponding to 486 nm. A laser tuned near HG can therefore serve as the fundamental for frequency doubling to produce the 243 nm radiation for the two-photon transition, and the difference from an exact multiple of four between the two resonances measures the Lamb shift of the 1s state of hydrogen, ~ 8150 MHz, a most important quantity for comparison with quantum electrodynamical theory (Sect. 2.1.4). The final limiting accuracy on this measurement seems likely to be a few tens of kHz [48]. The high powers required for two-photon spectroscopy are much more easily achieved with pulsed lasers than with CW lasers. The limiting resolution is often the frequency spread 1/7 associated with the finite pulse length T. In the case of CW lasers a similar limit is imposed by the time of flight of an atom through the laser beam. The possibility of using optical interference effects to get around this limitation has been demonstrated for both spatial and temporal separation of the two absorptions. In the first case, applied to an atomic beam, an atom experiences two coherent laser fields separated by the time of flight 7’ between them; in the second case T is the time delay between two coherent laser pulses (an incident pulse and its reflection from a mirror). In both cases the resonance manifests itself as a two-beam interference pattern in the frequency domain: a set of interference fringes of separation 1/T limited by an envelope of width 1/7. The effect can be thought of as a temporal equivalent to a Young’s slits setup, with 7 replacing the slit widths and 7’ their separation. An analogous device was used by Ramsey to improve resolution in molecular beam magnetic resonance experiments, and for this reason the name ‘optical Ramsey fringes’ is often used.

14.7 Trapping and Cooling of Atoms and Ions

357

14.7 Trapping and Cooling of Atoms and Ions Methods of confining atoms or ions in a small region of space ~ i.e. ‘trapping’ them — for long enough to conduct sophisticated experiments on them have developed considerably over recent years. Ion traps use some combination of electric, or electric and magnetic, fields to create a potential minimum of several volts over a mm or so, such that a finite number of ions, from one to

several thousands, can be trapped there for minutes or even hours [49]. The trapping of neutral atoms is a more difficult problem but can be accomplished by exploiting the strong field gradients in an intense laser beam focused to a narrow waist. This gradient interacts with the electric dipole moment of the atom induced by the laser when it is tuned slightly off resonance to give a net force on the atom, driving it in the direction of increasing field. For full

explanations of such traps see [50]. We consider here only the laser-cooling of the trapped particles. The mechanism for laser cooling is the slowing down of an atom or ion confined in a trap by transfer of momentum from the photons in the laser beam. Radiation from a laser tuned to the long-wavelength side of a resonance line is absorbed by atoms travelling towards the laser because they see the radiation as blue-shifted into resonance. Their momentum in this direction is decreased with each absorption by hiy,/c, where vy is the laser frequency. Subsequent re-emission is in a random direction and does not affect the average momentum. The average kinetic energy loss per absorption is h(vp — vy), and about 10* absorptions are needed to get from thermal velocities to a few degrees K. For a strong transition the lifetime of the excited state is of order 10ns, and the required recycling can be achieved in a few ms. There are some obvious practical difficulties to be overcome. As the atoms cool, the laser needs to be tuned closer to resonance, which can be done either by sweeping the laser frequency vy, or by Zeeman-shifting the resonance frequency vo with an external magnetic field. Secondly, the laser reduces momentum in one direction only, and the random recoil momentum from spontaneous emission is tantamount to heating. In principle six lasers are required to cool free atoms efficiently. The cooling of ions whose motion is constrained by electric and magnetic fields — i.e. trapped ions — requires only one laser. Temperatures of a few mK, corresponding to ion velocities of

order 1 cms! can be realized [51]. Experiments on cooled atoms or ions differ from the Doppler-free experiments described in the previous section in that all atoms or ions participate, instead of just a subset with a particular velocity. Moreover, the secondorder transverse Doppler shift is also eliminated. In the case of neutral atoms some cooling is an essential prerequisite to trapping because the traps are too shallow to contain fast atoms. Experiments on trapped particles have covered several aspects of atomic physics. The principal spectroscopic applications have been to hyperfine structure (yielding an improved value for the ratio of electron to proton

358

14. Laser Spectroscopy

mass), and to Lamb shifts and isotope shifts. The sensitivity can be made so high that fluorescence from single ions can be detected. This makes possible measurements on minute quantities of rare isotopes, the shifts of which yield

important data on nuclear shapes and distortion factors [49].

14.8 Comments on Laser versus ‘Conventional’ Spectroscopy The advantages of sub-Doppler resolution, efficient selective excitation and ultrahigh detection sensitivity offered by laser spectroscopy clearly cannot be approached by grating or interferometric spectroscopy. However, the two different methods should be seen as complementary rather than competitive. Laser spectroscopy is not a good broad-band technique because of the difficulties of tuning accurately over wide ranges and maintaining stability over long scans. Its strength is in the extraction of maximum information from a narrow spectral region. The equipment, particularly for frequency doubling and mixing and for sophisticated Doppler-free spectroscopy, is complex and expensive, even by the standards of high resolution grating or Fourier transform spectroscopy. The applicability is limited in practice to neutral and singly charged ions because of the difficulty of creating tunable radiation at very short wavelengths. It needs to be remembered also that laser spectroscopy is an ‘active’ technique, confined to probing atomic and molecular spectra in laboratory conditions. Astrophysicists and atmospheric physicists observing solar or stellar radiation are unable to influence their sources and must use conventional ‘passive’ techniques.

Further Reading — Svanberg, S., Atomic delberg, 1996).

and Molecular Spectroscopy (Springer, Berlin, Hei-

~ Siegman, A. E., Lasers (University Science Books, Mill Valley, 1986). — Svelto, O., Principles of Lasers, 4th ed. (Plenum, New York, 1998). — Demtroder, W., Laser Spectroscopy (Springer, Berlin, Heidelberg, 1998).

15. Light Sources and Detectors

Light sources for spectroscopy, other than lasers, may be divided into broadband (or continuum) sources suitable for intensity standards or as backgrounds for absorption spectroscopy, and narrow-band (or line) sources for emission spectroscopy. Intensity, or radiometric, standards will be discussed in Chap. 17, and we shall consider here sources for absorption and emission spectroscopy.

The simplest background source for absorption spectroscopy is an incandescent solid, but solids necessarily have an upper temperature limit of about 3650 K, the melting point of tungsten. The peak wavelength for such a source is in the red, and its power falls rapidly in the ultraviolet. Sources of continuum for the far infrared and the ultraviolet are described in Sect. 15.2.1. For the most part they are not in thermodynamic equilibrium at a well-defined temperature, and their spectral energy distribution may bear little relation to the Planck function. The same is true of the pulsed sources used to observe absorption in very hot gases or plasmas. Emission sources must obviously be chosen to match the temperature, or degree of excitation, required to generate the spectrum, but there are a number of other characteristics that may be important, such as the width of the spectral lines emitted, the extent to which thermal equilibrium is attained, the spectral region over which the source is useful, whether it be pulsed or continuously running, and so on. It was seen in Chap.8 that the two most important line-broadening processes are Doppler, due to the thermal motion of the emitting atoms, and pressure broadening. In low-pressure sources the first of these dominates, whereas at sufficiently high densities, particularly electron densities, pressure broadening may become the more important. In sources such as low-pressure glow discharges, which are not in local therthe modynamic equilibrium, the gas kinetic temperature, which determines temperDoppler width, may be close to room temperature while the electron In ature, which determines the excitation, may be several thousand degrees. (a similar an arc at atmospheric pressure the two temperatures are usually be few thousand degrees), and Doppler and pressure broadening may both

quite large. Sources suitable for the measurement of transition probabilities, which um, have to satisfy particular requirements as to thermodynamic equilibri

360

15. Light Sources and Detectors

are further discussed in Chap. 16. The next few sections describe traditional spectroscopic emission sources and some recently developed devices, including sources suitable for the infrared and the far ultraviolet. Since many of the new types of source do not appear in standard textbooks, a fairly extensive list of references is given at the end of the chapter.

15.1 Emission

Spectroscopy

Various criteria could be used for classification of the various types of light source for emission spectroscopy — for example, according to use (spectrochemical analysis, research on atomic or molecular structure, determination of transition probabilities), physical conditions (plasma, beam of atoms or ions), excitation mechanism (electric discharge, thermal collisions), charge state (neutrals, moderately or highly charged ions). Instead of using such a classification scheme we list a number of widely used sources and describe their characteristic properties. 15.1.1

Flames

Most flames have temperatures of the order of 2000 K, although certain stoichiometric flames! can reach about 4000 K. The low excitation energy (~ 0.25 eV) tends to produce a rather simple atomic spectrum, with the resonance lines dominating; indeed, the temperature of the flame is sometimes measured by seeding one of the alkali metals into it and using the reversal method described in Sect.10.11.4. The principal uses of flames are for the study of the molecules and radicals formed during the combustion process, and for spectrochemical analysis by atomic absorption. 15.1.2

Arcs

The traditional type of free-burning arc runs typically at a few amperes and a few tens of volts, producing temperatures of the order of 5000 K: this is high enough to excite neutral spectra of atoms and molecules and a number of ionic lines. The arc is struck between metal or carbon electrodes, usually in air at atmospheric pressure, and has a column about 10mm long. Where practical the arc pole pieces can be made of the metal whose spectrum is required. Other elements may be introduced as salts in an arc run between carbon electrodes, or as trace impurities in copper electrodes. Arcs may also be run in gases other than air, and under reduced pressure. The lines emitted from an arc are often wide, asymmetric

and shifted, due to collisions close

to the electrodes where the plasma density and the interionic field strength stoichiometric: constituents of the fuel mixed in proportions giving a complete combustion.

15.1

Emission Spectroscopy

361

may be high. Traditionally the arc was one of the most important sources for systematic investigations of atomic spectra — the spectrum of a neutral atom was known as the ‘are spectrum’, while the ‘spark spectrum’ contained lines from various stages of ionization. Today arcs of this kind are used in some spectrochemical applications. Various special arcs have been developed to reach higher temperatures. High-current arcs (up to 500 A) with water-cooled electrodes can produce temperatures in the centre of the column as high as 30 000 K. The arc column is restricted to a few mm diameter, either by a helical swirl of gas or water (vortex arc) or by a series of water-cooled diaphragms with axial holes (wallstabilized arc). It is possible to introduce powdered solids as well as vapours into these arcs. Wall-stabilized arcs are particularly important in the measurement of transition probabilities, when high stability and a close approach

to LTE are essential (Sect. 16.1). One particular type of wall-stabilized arc, the argon mini-arc, has been developed not as a line emission source but as a continuum source suitable as a radiometric standard for the ultraviolet

(Sect. 17.3.3). Run at currents of up to 40 A at a pressure of 1-2 atmm., with an

axial temperature of 11000-13000 K, it gives an almost line-free continuum between about 350 and 90 nm, arising from the recombination of argon ions and electrons. 15.1.3

Sparks

Sparks require high voltage to break down an insulating gap. For sparks used in spectrochemical analysis of metals a transformer may be connected directly across the spark gap, but normally it charges a capacitor, and the stored energy is discharged across the gap every half-cycle. Several degrees of ionization can be reached, both in the atoms of the metal forming the spark electrodes and in the gas through which it passes. The traditional spark source used for producing spectra of more highly charged ions consists of two electrodes in a vacuum separated by a gap of a few mm. The electrodes are connected to a capacitor (0.1-0.5 pF) that is charged by a high voltage power supply until breakdown occurs in the spark gap, typically at 50-100 kV. The current in the spark may reach 50-100 kA. The capacitor is recharged, and the procedure is repeated with a frequency limited by the charging current from the power supply. Stages of ionization from 10 to 20 are produced by this kind of spark. The degree of ionization can be controlled by varying the circuit inductance and hence the peak current. The highest charge states are reached by reducing the inductance in the spark circuit as far as possible. A low inductance capacitor with large capacitance is used at a lower voltage than in the traditional spark, and various triggering mechanisms are used to induce breakdown in the spark gap. Extremely high charge states from the electrode metals have been produced with such triggered low inductance sparks, for example hydrogen-like molyb-

denum, Mot?*!.

362

15. Light Sources and Detectors

Rather less violent ionization is produced by a sliding spark, in which the voltage breakdown takes place across the surface of an insulator. The spectrum contains lines of electrode and insulator materials and of any lowpressure gas introduced into the gap. Stages of ionization from 2 to 5 are produced in such sparks. 15.1.4

Glow

Discharges

Gases and vapours are often conveniently excited in the positive column of a glow discharge, or in an electrodeless discharge, at pressures of a few torr. Molecular, atomic and ionic spectra may all be excited in this way. Excitation is by electron bombardment throughout the length of the positive column, which can be made to extend for a metre or so if necessary. A discharge of this type has to be run off a high voltage source (1 kV upwards) with a ballast resistance to stabilize it and to limit the current to something of the order of an ampere.

Another form of glow discharge, the hollow cathode [52], uses the cathode glow instead of the positive column. The cathode is in the form of a hollow cylinder with a bore of a few mm in diameter. At a suitable pressure, of the order of a few mbar, the cathode glow jumps into and completely fills the hole, producing an intense and fairly compact light source. Material from the walls of the hole is sputtered into the discharge and there excited, so that metals and salts as well as gases can be studied. A choice of carrier gas from among the inert gases, or mixtures of them, allows the sputtering and the excitation to be varied [54], so that the spectra of neutral atoms and singly and — in some cases — doubly charged ions can be observed. The voltage between anode and cathode varies from below one hundred to several hundred volts, depending on the cathode material, the carrier gas and the pressure. At one extreme the hollow cathode can be run up to 10 amperes or more [53] to bring up weak spectral lines; at the other it can be run at a few mA with liquid nitrogen cooling to reduce the Doppler widths of the lines, and for high resolution work it has the further advantage that the electric field in the cathode is small so that Stark broadening and shift are also small. The hollow cathode can also be run as a condensed discharge. A capacitor of about 10 nF, charged to say 500 V, is repeatedly discharged through the hollow cathode by means of a mechanical or electronic switch. The peak current may reach several hundred amperes. In this way the excitation of ionic spectra is enhanced, particularly to high levels, but higher charge states

than in the continuous discharge mode are usually not observed [55]. The Grimm discharge lamp [56] is somewhat similar to the hollow cathode in that it relies on sputtering from the cathode to introduce the required atoms into the discharge. It was developed primarily for metallurgical analysis and uses plane cathodes that can be easily machined from the metal under study. The cylindrical anode is mounted only a mm or so from the cathode,

15.1

Emission Spectroscopy

363

and the spectra are excited in the negative glow immediately adjacent to the cathode, a region where the electric field is no longer small. The Penning discharge lamp [57] is another variant of a glow discharge which generates the emission spectrum of material sputtered from the cathode, but in this case an axial magnetic field is applied. The carrier gas pressure can be two orders of magnitude lower than in a standard glow discharge because the electrons, spiralling round the magnetic field lines, have a much longer path through the gas. The lamp was developed as a discrete line radiometric source, but it has also proved useful in exciting higher states of ionization than can be achieved with a d.c. hollow cathode lamp. In contrast to lamps that rely on sputtering, electrodeless discharges are particularly useful for producing spectra of materials that are only available in small amounts, for example, monoisotopic samples of elements naturally containing several isotopes, or unstable, artificially produced isotopes. The pure element or a salt, usually a halide, is enclosed in a small, sealed quartz container together with a carrier gas. The lamp is excited in a microwave cavity, free of contamination from electrode material. Another type of electrodeless lamp, mainly used for gases, is excited in the oscillating field inside the capacitor or the inductance of a radio-frequency resonance circuit. The degree of ionization is low in these lamps; with the exception of some specially designed excitation devices they produce only the spectra of the neutral atoms.

15.1.5

Inductively Coupled Plasmas

The ICP [58] has effectively replaced the arc for spectrochemical analysis. A stream of argon at approximately atmospheric pressure forms a ‘torch’ excited by a radio-frequency loop, into which can be injected either gases or solids in solution in the form of fine droplets produced by a nebulizer. The source is close to LTE at about 6000 K, appropriate for the spectra of neutral atoms and the first stage of ionization. Its merit for spectrochemistry is that it can be made accurately quantitative: the emission of a given species of atom depends linearly on its concentration in solution over a wide range. 15.1.6

Beam

Foil Source

Beam foil spectroscopy (BFS) uses a source of an entirely different nature. A beam of fast ions is fired through a thin foil (thickness typically 10 pg em~2 or about 50 nm), usually made of carbon. The ions may be further ionized through collisions with the atoms in the foil and the beam emerges as a mixture of ions of various degrees of ionization, but with its beam character

unimpaired and its velocity essentially unaltered. The ions are excited to a wide range of excited states through the collisions in the foil, and a spectrum is emitted in the region downstream of the foil as the ions decay to lower

364

15. Light Sources and Detectors

states. Compared to the excitation processes in plasma sources, the beam foil mechanism produces higher populations of states with high n and / and doubly excited states. A wide range of accelerators has been used for BFS experiments. Ion energies ranging from 0.5 MeV to several hundred MeV have been used, and spectra emitted from 40- to 50 times ionized atoms have been recorded in several experiments. The upper limit of the charge states is obviously set by the accelerator system, and transitions in helium-like and hydrogen-like

uranium, U9°+ and U®!+ have been observed at the BEVALAC

accelerator

[59]. The particle density in the beam is low. This eliminates perturbations from collisions and interionic fields, but it also means that the emitted spectrum is weak. High spectrometer throughput and thus low resolution (Sect. 11.2 and (12.23)) is needed to obtain a good signal-to-noise ratio in a reasonable recording time, and line blending can therefore be a serious problem in complex spectra. The beam foil source has a unique property that we will discuss further in Chap. 16: the spatial resolution of the spectrum along the beam corresponds to a temporal resolution related to it by the known beam velocity. BFS can therefore be used for measuring lifetimes of excited states. 15.1.7

Laser-Produced

Plasmas

A pulse from a high-energy laser, such as a Q switched ruby or neodymium laser, focused on a solid target causes emission of ions and electrons from the surface. After the initial emission the electromagnetic energy in the pulse is absorbed by the free electrons, increasing the temperature of the plasma and increasing the ionization through electron-ion collisions. At a pulse energy of a few Joules the temperature can be in the range 100 eV to 1 keV, with an

electron density of the order of 107° cm~%. Stages of ionization up to about 20 can then be attained. Spectra from more than 50 times ionized atoms have been recorded at large multiple-beam laser systems with beam energies

of several hundred Joules [60]. The most efficient production with laser pulse lengths of 0.1—10 with a high power laser having weaker prepulse can be used for by the short, high power pulse. 15.1.8

Shock

of spectra of highly charged ions is obtained ns. In order to obtain a strong line spectrum a pulse length in the femtosecond range a producing the plasma, which is then heated

Tubes

Shock tubes are aerodynamic devices that have been used as spectroscopic sources. There are several different ways of producing a shock wave in a gas; for example, a pressure-driven shock is generated by bursting a diaphragm

15.1

Emission Spectroscopy

365

between a region of high-pressure light gas and one of low-pressure heavier gas. As the wave travels down the tube through the low-pressure gas it leaves behind it an almost uniform region of hot gas at a few thousand degrees and at a pressure of a few atmospheres, lasting, at any given observation point, a few ms. The temperature and pressure range is comparable with that of an arc, but the shock tube has the advantage of a relatively large homogeneous region in which both temperature and pressure can be measured and independently controlled. Atomic species that do not form suitable volatile compounds can be introduced as powdered solids. The transit time is sufficiently long for easy time resolution, and the spectra of the constituent atoms or molecules can be studied either in emission or, using a flash as background, in absorption. The heating effect may be increased by reflecting the shock wave from the end of the tube. 15.1.9

Fusion

Plasma

Devices

The high-temperature plasmas produced for thermonuclear fusion research have proved to be excellent sources for spectra of highly charged ions. The first type of fusion device to be used for spectroscopy was the pinch discharge. It works on the principle that the magnetic field associated with a pulse of high current, produced by discharging a high voltage capacitor, can be used to squeeze a column of ionized gas rapidly in towards the axis. As the collapse is completed, the ordered radial velocities are converted by collisions to random or thermal velocities, giving a narrow column of very hot gas lasting, typically, a few microseconds before it expands and cools. The high degree of ionization attained in the pinch and its short duration make it comparable with a spark source, but with the advantages of much larger dimensions and greater reproducibility. Small pinch devices with an energy of a few kJ stored in the capacitor have been used for studies of gases at a few stages of ionization, e.g. N IV, N V, O V and O VI. Larger pinches have been used for producing spectra of higher charge states, e.g., Ar X-Ar XV. The recombining plasma that follows the pinch also constitutes a useful source at a temperature of order 10000 K and duration about 1 ms. The fact that temperatures and electron densities can be measured independently is a great asset for studies of line broadening and the appearance of forbidden

lines. The toroidal plasma devices — primarily the Tokamaks — used in current research on controlled thermonuclear fusion can be considered as spectroscopic light sources with unique properties. In certain cases a Tokamak has been used intentionally as a light source for spectroscopic experiments, but usually the observed spectra are emitted either by impurities present in the plasma or by ions injected into the plasma for diagnostic purposes. In the large Tokamaks like JET (Joint European Torus) in Culham, U.K., the ion

366

15. Light Sources and Detectors

and electron temperatures are high — 10 keV or more — and spectra from very high charge states have been recorded, e.g. Mo XXXIX and Mo XL. The transition probability for forbidden transitions increases with a high power of the net charge ¢ along an isoelectronic sequence (Sect. 3.6). In Tokamak plasmas the electron density may be low enough (order of Loe Seine) to make the probability for collisional de-excitation of metastable levels at high charge states smaller than the probability for magnetic dipole (M1) radiation, and transitions between levels within the ground configurations of highly charged ions can therefore be observed [61, 62]. 15.1.10

Electron Beam

Ion Trap

The EBIT, Electron Beam Ion Trap, is a device designed for high precision spectroscopic and other measurements on highly charged ions. An electron beam is accelerated to an energy of 30-200 keV and compressed by an axial magnetic field created by superconducting coils to a diameter less than 100 um. The element to be studied is injected into the beam, where it is ionized through electron—atom collisions and trapped, radially by the potential created by the charge of the electron beam and axially by a field from electrodes at the ends of the trap. Besides ionizing and trapping, the electrons also excite the ions, and a spectrum is emitted. The ions in an EBIT are virtually at rest, in contrast to the ions moving at high velocity in BFS experiments or high temperature plasmas. This enables accurate spectroscopic measurements to be made, and it also creates new possibilities for lifetime measurements (Chap. 16). The charge of the trapped ions depends on the energy of the ionizing electron beam and can be varied within wide limits. The EBIT at the US National Institute of Standards and Technology (NIST) can accelerate the electrons up to 30 keV. As one example of its use, the charge states of barium ions have been studied up to Ba*®+. The SuperEBIT at Lawrence Livermore National Laboratory has a maximum voltage of 200 kV, and spectra from hydrogen-like and helium-like uranium have been recorded (see Further Reading).

15.2 Absorption Spectroscopy Absorption spectroscopy as it is understood by a physicist or astrophysicist requires a background source of radiation that is continuous over the relevant spectral region. In this it differs from the technique of atomic absorption spectroscopy (AAS) used by analytical chemists, which uses a set of line sources as background as explained below. The other requirement is an absorption cell. Unless the spectrum under investigation is that of one of the permanent gases, the cell is normally an oven, running usually in an inert or reducing atmosphere, but may be a reaction vessel. The temperature attainable is limited by the melting or softening point of the furnace material,

15.2

Absorption Spectroscopy

367

which means in practice some 3000 K. Various flash-heating techniques have been developed for studying the absorption of short-lived molecular radicals (flash photolysis) and of very refractory solids (flash pyrolysis), and it is possible to use many of the emission sources described in the previous section in absorption, provided there is available a background source of effectively greater brightness temperature — in practice some type of flash tube. The ‘cell’ windows tend to present a problem in absorption spectroscopy: they are apt to get fogged with material sputtered or condensed on them, or to be chemically attacked by certain vapours, and in any case 110nm is the lower-wavelength limit of transparency. Depending on the experiment, a few mbar of buffer gas or a differential pumping system may be used as protection. For some metals over an appropriate temperature range a heat pipe is a very effective absorption cell: the metal vapour condensing at the cooler ends of the tube is returned to the hot centre by the capillary action of a suitable mesh, forming a sort of diffusion pump which pumps the buffer gas to the ends of the hot region. The metal vapour pressure is then determined entirely by the pressure of the buffer gas; winding up the power input to the furnace simply increases the length of the vapour column.

15.2.1

Continuum

Sources for the Visible Region

The simplest form of background continuum in the visible and near infrared is the tungsten filament lamp, which is close to being a ‘grey body’ in the temperature range around 3000 K. Higher temperatures, and therefore more blue and ultraviolet radiation, are produced by high-pressure gas arcs. The commercial version of the xenon arc, running at 50-80 atm. pressure, has a colour temperature of about 6000 K and gives a usable continuum between about 750 and 190 nm. If a higher brightness temperature is required, say to obtain an absorption spectrum from a hot plasma, a pulsed source is probably necessary. Pulsed xenon lamps are available commercially, as are rather large low-pressure flash tubes of the type described in Sect. 15.2.2, reaching brightness temperatures up to 50000 K in a flash of a few ps. Background sources for the infrared and ultraviolet regions are discussed in Sect. 15.2.2. Absorption spectra feature largely in astrophysics, starting of course with the Fraunhofer lines in the visible and ultraviolet spectrum of the sun. Observations from space of the atmospheres of hotter stars yield absorption spectra extending well into the vacuum ultraviolet. The absorption spectra of interstellar gas clouds are studied when they intersect the line of sight to a more distant bright star that acts as the background continuum. The specialized technique of absorption spectroscopy applied to analytical chemistry, AAS, is directed at maximum sensitivity in order to detect trace quantities of particular elements. The spectral lines of choice are therefore usually the resonance lines, and the sensitivity is greatly increased by using as background a line source of the element in question rather than a broad-band continuum. Low-current (~ 10 mA) hollow cathode lamps for all common

368

15. Light Sources and Detectors

elements are manufactured commercially for this purpose. The lines from these lamps are narrower than the absorption lines generated in a flame or a furnace, and it is the widths of these emission lines that determine the effective resolution, with the spectrometer acting simply as a monochromator to exclude other emission lines. Sensitivity is further increased by chopping the hollow cathode light and observing the signal with a detector amplifier circuit tuned to the chopping frequency so as to eliminate emission from the absorption furnace. 15.2.2

Continuum

Sources

for the Infrared

and Ultraviolet Regions

The intensity of the tungsten halogen filament lamp peaks at about 900 nm, and the quartz envelope ceases to transmit at about 3 um. At longer wavelengths it is preferable to use a filament made of a mixture of rare-earth oxides operating at about 2000 K, known as the Nernst glower. At wavelengths above 10 um this is beaten by the globar, a silicon carbide strip source, and by its more modern equivalent, a nichrome strip in a silica envelope; these are good greybodies from 2 to 40 um. The most widely used source in the far infrared is the mercury arc lamp at a pressure of a couple of atmospheres. In the region 3-60 um, where the quartz envelope is opaque, this radiates as a blackbody, but at longer wavelengths the radiation is that of a plasma continuum in LTE at a few thousand degrees. The useful short-wavelength limit of the tungsten lamp is about 400 nm. Continuous emission over the region from the quartz ultraviolet down to the resonance line of helium can be obtained in one way or another from discharges in hydrogen or the inert gases, although below 110nm these must be run in windowless tubes. The high-pressure xenon arc referred to in Sect. 15.2.1 is overtaken in radiance at the end of the quartz ultraviolet (220 nm) by the argon wall-stabilized arc (Sect. 15.1.2) and by the deuterium lamp. The radiation from the latter consists mainly of a molecular recombination continuum analogous to that described in Sect. 6.3.1 for hydrogen. The commercial lamps run at 300 mA (30 W) and give a fairly uniform continuum from 350 nm to about 165 nm, where it breaks up into molecular band structure. The laboratory counterpart of the deuterium lamp, using hydrogen in a water-cooled discharge tube running at an ampere or so, radiates more power than the commercial deuterium lamp, but has the same short-wavelength limit of 165 nm. Good continua extending some tens of nm to the long-wavelength side of their respective resonance lines can be obtained from all of the inert gases by running either a condensed or a microwave discharge at a pressure of a few hundred mbar. The origin of these continua is the bound—free emission described in Sect. 6.3.1. Between them, Xe, Ar, Ne and He cover the region from 180 to 60 nm.

15.2

Absorption Spectroscopy

369

The whole ultraviolet region is covered by the flash-tube source that was mentioned in Sect. 15.2.1 as a general purpose source of high-brightness temperature for the visible. In the original version devised by Lyman the flash was produced by discharging a capacitor through a capillary tube containing low-pressure gas. This had the disadvantage that the capillary was eroded rather rapidly and the continuous spectrum was overlaid by emission and absorption lines from the material scoured off the walls; the Garton modification reduces these problems by using a wider bore tube in a low-inductance circuit so that the necessary current density is achieved in a very short flash (~ lus). A discharge of a few uF at 10kV produces a continuum usable down to about 30 nm, but increasingly overlaid with lines towards the short-wavelength end. Shortwards of about 50 nm the best flash continuum is provided by the BVR source, called after its developers Balloffet, Vodar and Romand. This is a condensed spark in vacuum with a uranium pin acting as anode. The continuous radiation is emitted from a very small blob of hot gas just off the tip of the anode. Under typical operating conditions the flash from a 0.1 HF capacitor charged to 30 kV lasts about 1 ps and produces a continuum down to 1 nm or less. Laser-produced plasmas have already been mentioned as emission sources. With suitable target material they can also act as reasonably line-free continua. The rare earths, probably because of the sheer complexity of their partly filled f- and d-shell configurations, have quasicontinuous ionic spectra all through the region from 200 to about 4 nm. Typically, a joule of radiation from a ruby or neodymium laser is focused on the target to a spot less than 1 mm in diameter. The ultraviolet radiation is emitted as a pulse lasting about as long as the laser pulse (5-30 ns) and is fairly reproducible from shot

to shot. A source of an entirely different type is the Bremsstrahlung radiation emitted by the electrons in a synchrotron accelerator or storage ring. The electrons are accelerated towards the centre of the circular orbit, and at low energies

they emit radiation in all directions perpendicular to the orbital radius, in accordance with classical electromagnetic theory. At high energies, when relativistic effects become important, the radiation is concentrated into a narrow

cone in the direction of the instantaneous motion of the electron. The greater the energy, the smaller is the angle of this cone. The radiation is virtually confined to the orbital plane, and it is plane-polarized with the electric vector in this plane. The power radiated at any wavelength can be calculated from the radius of the orbit and the energy of the electrons, so that this source is in the same category as the blackbody as a true absolute intensity standard, albeit an even less accessible one to most experimenters. Figure 15.1 shows the general form of the spectral power distribution, which has something of the appearance of a set of blackbody curves, but with a different functional dependence on energy and wavelength. The important parameter is the ‘critical wavelength’, A., which is given by Ac = 0.56R/E?% nm, where R is the

370

15. Light Sources and Detectors

= &

g o =

a o

2

i

e

—

350 MeV 325 MeV 300 MeV

250 MeV

0

Oe

OMn OME OME OMG OO OME OMeni)

Fig. 15.1. Spectral distribution of synchrotron radiation for different electron energies.

radius of the orbit in metres and F is the energy in GeV. The peak of the distribution occurs at 0.4\. and therefore varies as E~° for a given radius. The peak power varies as E” and the total power as E+, again for a fixed R. For A > A, the power is almost independent of £. Over the past 25 years or so, synchrotron radiation has acquired so many important applications, in chemistry and biology as well as in several branches of physics besides spectroscopy, that a number of so-called storage rings have been built specifically to use the radiation rather than the fast electrons themselves; the injected electrons circulate in an ultrahigh vacuum for several hours, during which the power drops slowly as electrons are lost. This is an intrinsically ‘quieter’ source than a synchrotron, in which there is inevitably some pulse-to-pulse variation. Further refinements involve the insertion of additional magnets as ‘undulators’ or ‘wigglers’ to modulate the effective radius, and hence the acceleration, periodically round the circuit, thereby tuning the wavelength at which the radiation has its maximum brightness.

15.3 General

Remarks

on Detectors

Detectors may respond either to incident power or to incident photon flux. The first type — thermopiles and bolometers for example — are non-selective in wavelength response, and the second type — photodiodes and photoconductors for example — are selective. Although different photon detectors can

15.3

General Remarks

on Detectors

371

be used from the far ultraviolet to about 15 um, a natural division between types occurs at about 1 um, corresponding to a photon energy of 1.2 eV. Below this wavelength photons have sufficient energy to eject an electron from a photocathode or to sensitize the silver halide grains of a photographic emulsion. Above it, they can modify the characteristics of a semiconductor; photon detection by photoconductors or semiconducting photodiodes is possible up to about 15 um. At longer wavelengths certain photoconductors are still usable, but thermal detectors are more common. These, being relatively slow and insensitive, are not normally used at the shorter wavelengths except for absolute measurements. The useful information obtained from any spectroscopic system depends on the signal-to-noise ratio in the final output. Noise may originate from fluctuations of the light source itself; from photon noise associated with the random arrival of photons from a steady source; from photon noise associated with the background radiation incident on the detector; from the detector itself; or from the amplifying system that follows it. For the present purpose we are concerned with the last three categories. Background radiation makes an important contribution to noise in the infrared because the spectrometer and its surroundings (including the spectroscopist) all radiate most effectively around 10 um; the background can usually be ignored for A < lum. Detector noise comes from fluctuations in the dark current, which is the signal in the absence of incident radiation, and it is essentially thermal in origin. For the photoemissive detectors, working at cutoff energies much greater than kT (remember that hy at 1 um is 1.2 eV whereas kT at room temperature is 0.025 eV) the detector noise also is usually negligible, although for very low light levels there can be an advantage in cooling the detector. Likewise, amplifier noise may limit the performance of photodiodes at low light levels, but this limitation is avoided by photomultipliers, in which the amplification takes place in the vacuum tube itself with little or no degradation of the initial signal-to-noise ratio. In practice, performance in the visible and ultraviolet is usually limited by the photon noise in the signal. In the infrared region the detector, amplifier and background noise may all be important. Dark current in the semiconducting devices is reduced by cooling, as is the detector noise in thermal detectors, which arises from fluctuations in the ‘dark’ temperature of the detector. To reduce background noise and make a.c. amplification possible it is general practice to chop the light signal at some frequency f and use a phase-sensitive amplifier of restricted bandwidth Af. The choice of f is limited by the requirement that 1/f must be large compared with the response time of the detector, and it may range from 10 Hz for slow thermal detectors to a few kHz for photoconductors. The noise power associated with random fluctuations is approximately independent of frequency (‘white noise’), so Af sets the noise power passed by the ; amplifier.

372

15. Light Sources and Detectors

It is useful to define here a few terms commonly used in the literature on detectors. The responsivity of a detector is defined as the ratio of the rms

(root mean square) output voltage Vo to the rms power input W:

R=Vo/W

(VW).

(15.1)

The noise equivalent power (NEP) is defined as the input power that would be required to give an output voltage just equal to the rms noise voltage — in other words, the input power P, required for a signal-to-noise ratio of unity. From (15.1), we have P, = V,/R, where V, is the rms noise in the output voltage. The detectivity D is simply the inverse of the NEP, but the quantity much more widely used than D is the specific detectivity D*, which is D normalized to a detector area of 1 cm? and a bandwidth of 1 Hz according to the relation

AN Py

ema Fg

We he

(15.2)

The logic of this apparently strange definition is that for many sources of noise the NEP is proportional to the square root of the detector area A, and for white noise it is also proportional to the square root of the bandwidth Af. Thus D* serves as a figure of merit for most types of detector at any particular wavelength and temperature. The square root dependence can be justified as follows. Dark current and current due to background radiation are both proportional to A. According to Poisson statistics the rms fluctuations of these currents, Va, is proportional to VA. Similarly, V,, is proportional to 1/\/t, where t is the integration time; and as t is limited by the bandpass ac-

cording tot ~ 1/Af

it follows that V, is proportional to \/Af. Consequently,

PaxVn, x VW AAT .

Having dealt at some length with noise, we now consider the signal levels recorded by a detector in association with a spectrometer. The dispersion of a grating is almost constant with wavelength, so a thermal detector with a grating records watts per nm. However, the dispersion of a Fourier transform spectrometer is effectively constant with frequency or wavenumber, and a thermal detector therefore records watts per cm~!. The distinction is important because, as noted in Sect. 7.2, the spectral radiance of a thermal

source decreases towards short wavelengths faster (by a factor of 1/A?) for the frequency scale than for the wavelength scale. Furthermore, the number of photons per watt is proportional to A, so for a photon detector there is a further factor of 1/\ against short wavelengths. The position of the effective maximum of the blackbody distribution is a good illustration of this effect. Equation (7.15) gave Wien’s displacement law for the watts per unit wavelength distribution as A,,7 = 2880, with \ in um and T in K. For the dis-

tribution in watts per unit wavenumber the constant is 5100, and for photons per unit wavenumber it is 9030. The peak wavelength for a 5000 K blackbody

15.4

Detectors for the Infrared

373

(the Sun, say, or an arc discharge running at a current of about 10 A) shifts from 580 nm to 1 um to 1.8 um for these three types of response. The remainder of this chapter gives some more detail on detectors for the infrared, meaning A > 1m, and detectors for the visible and ultraviolet, \ < 1um. There is also a section on multichannel detectors; array detectors are superseding photographic emulsions for this purpose. The technology of these detectors is advancing so rapidly that we shall give only an overview here.

15.4 Detectors

for the Infrared

These fall into two classes: photon detectors, which have a long-wavelength cutoff, and thermal detectors, which respond to radiative power irrespective of wavelength. The photon detectors above 1 um are all semiconductors of one kind or another. They may be formally divided into photoconductors, in which the conductance of the material is increased by the creation of a free electron (or hole) when a photon is absorbed, and p-n junction photodiodes, in which the current-voltage characteristics are changed by the hole-electron pair produced by absorption of a photon near the boundary layer. In fact, the second class may be used either in the photovoltaic mode as voltage generators or in the photoconductive mode as current generators, and we shall not worry about the distinction here. A wide variety of both types is available commercially with different spectral ranges, specific detectivities and time constants. A specialized text [63] or the commercial catalogues should be consulted for details. Figure 15.2 gives a few typical curves of D* as a function of wavelength. D* has the same general spectral dependence for all semiconductor detectors, rising to a peak near the cutoff and then dropping sharply. This can be understood as follows. All detectors integrate background noise at all wavelengths up to the cutoff, so the noise power for any given detector is set by its cutoff. The number of photons per watt increases linearly with A, so at longer wavelengths less power is required to generate a given input signal. The approximately linear rise in D* of a given detector is a consequence of this effect: the input power required to generate enough photons to equal the fixed noise power is at its lowest when the wavelength is as large as possible — i.e. just before cutoff. The drop in the peak D* for detectors with longer cutoff wavelengths is due to the increase in the integrated background noise power. Once the cutoff reaches about 10 jum, the 300 K peak, the integrated background increases relatively slowly and the D* peaks change little. The curve labelled ‘ideal photoconductor’ is the locus of these peaks for a 300 K background and a 60° field of view. Most photodetectors (lead sulphide, lead selenium and indium antimonide, for example) have cutoff wavelengths in the range up to 5 pm, but

374

15. Light Sources and Detectors

=

=

\

T

10

ideal

£

\photoconductor

©

\300 K \

13

\

\ \ \

i 293 K 10

\

12

\ \ \ \ K

10"

PbS (Kaye

InSb 77 K 1

_ideal thermal

10 0|

0.1

4

Gpoieler

‘ae

VAS K opeae

|

300K

1

10

100

um

Fig. 15.2. Specific detectivity D* as a function of wavelength for some typical detectors. The curve labelled ‘ideal photoconductor’ is the limiting value for the peaks for a 300 K background-limited detectivity.

the various mercury cadmium-telluride alloys work up to 15 um. Peak val-

ues of D* are mostly in the range 10!°-10!! cm Hz!/2 W7-!, but absolute values must be treated with caution because they depend on temperature, acceptance angle, composition and (to a small extent) chopping frequency as well as cutoff wavelength. Extrinsic photoconductors, consisting of germanium doped with small amounts of impurity, have cutoff wavelengths as high as 130 um, but will work only with liquid-helium cooling. Indeed, InSb detectors can be used right up to the millimetre-wave region because the mobility of the free electrons is increased by the absorption of photon energy — i.e. the effective temperature of the electrons is raised above that of the crystal. For this reason they are known as ‘hot electron’ detectors. The response times of all these detectors depend on temperature and method of use as well as on type and area. They range from a few ns to about 100 wus. Thermal detectors are most widely used in the far infrared. The flat detectivity of the ideal thermal detector shown in Fig. 15.2 simply reflects its power as opposed to photon — response, which integrates the entire background spectrum regardless of measurement wavelength. There are several different ways of measuring temperature changes. Thermopiles measure thermoelectric emf, bolometers measure resistance, pyroelectric detectors measure the change in polarization of a ferroelectric crystal, and the Golay cell measures

15.5

Detectors for the Visible and Ultraviolet

375

the change in pressure of a gas cell. Of these, thermopiles are the most useful for absolute measurements at shorter wavelengths, but it is actually rather difficult to make them totally ‘black’ in the far infrared. Bolometers, measuring change in resistance from absorbed power, need to be cooled with liquid helium for high sensitivity. Either carbon or germanium may be used for the resistor, the latter having higher detectivity but a slower response time of several ms. Bolometers can be used all the way from the near infrared to the millimetre-wave region and can be made to approach the ‘ideal’ background limit for D*. Pyroelectric detectors measure changes with temperature of the bulk dipole moment of a ferroelectric crystal, essentially by using the crystal as a capacitance in a suitable circuit. The emphasis is on the word ‘changes’: the incident radiation must be chopped for this type of detector, and the response depends on the chopping frequency. The detectivity is smaller by an order of magnitude than that of the carbon bolometer and the time constant is somewhat greater, but these detectors have the great advantage for many purposes of working at room

15.5 Detectors

temperature.

for the Visible and Ultraviolet

In the region below 1 um photoemission from a solid surface becomes possible. Vacuum photodiodes use this type of detection in its simplest form, measuring the current due to the electrons ejected from a suitable photocathode exposed to radiation. Current is proportional to photon flux over 6-8 orders of magnitude, and the useful spectral range is about 600-200 nm. However, silicon p-n junction photodiodes are usually preferable to vacuum photodiodes because of their smaller size and greater quantum efficiency (up to 80%). Their spectral range extends from about 1 j1m into the ultraviolet, with specially constructed types on very thin substrates sensitive down to about 180 nm. Time constants can be in the ns range, and a typical detectivity curve is included in Fig. 15.2. At low light levels photodiodes tend to be limited by amplifier noise, and photomultipliers come into their own. The photomultiplier starts with a photocathode, just like the vacuum photodiode, but this is followed by an assembly of dynodes, each at a potential of about +100 V with respect to the last, and each emitting several secondary electrons for every one that hits it. The total gain of the multiplier depends on the number of stages and the applied voltage, but is typically 10° for 10 stages with a total applied voltage of 1 kV. The noise of a photomultiplier comes almost entirely from the photocathode dark current; the signal-to-noise ratio is normally degraded very little by the amplification. Dark currents are typically 0.1-1 nA, compared with a maximum output of some tens of wA, and can be reduced further by cooling. The detectivity is some two orders of magnitude above that of the silicon diode in Fig. 15.2. Standard photomultipliers have a time

376

15. Light Sources and Detectors

response of about 2 ns, and special tubes can be made about 10 times faster. Properly used, with the gain adjusted to avoid saturation, a photomultiplier can give a linear response over about nine orders of magnitude. At extremely low light levels (below about 10° photons s~') photomultipliers are used in photon-counting mode — one pulse per detected photon. The spectral response of photomultipliers is determined by the photocathode material and, where relevant, the window. Quantum efficiency? peaks at about 25% in the visible or near UV region, falling, even for the most favourable materials, to 1% by 1 um and falling also below 10% in the far ultraviolet. For ultraviolet applications, useful discrimination against visible radiation is afforded by ‘solar-blind’ photocathodes which cut off at 300 or even 180 nm. Photomultipliers intended for wavelengths below 180 nm have MegF»2 windows with a transmission limit close to 100 nm. For still shorter wavelengths the choice is between a phosphor-coated window and an open (windowless) tube. A channel multiplier is a special kind of windowless photomultiplier tube in which the multiplication is achieved down a single curved channel coated with a thin resistive layer, instead of along a discrete dynode chain. Photoionization becomes a possible detection mechanism below about 130 nm, where the photon energy of greater than 9 eV is sufficient to ionize gases. For continuous (d.c.) detection an ionization chamber is used, working on the plateau or saturation region of the curve of ion current versus voltage, because the ion current is then virtually independent of the chamber voltage and is proportional to the incident photon flux. The quantum efficiency, in terms of ion pairs per photon, may easily be 100% — indeed, if the photon energy is high enough for double ionization it may actually be greater than this. The pulsed version of the detector is essentially a Geiger counter: the original photoelectron from the incident photon is accelerated to make several collisions with gas molecules so that an avalanche takes place, using gas amplification. Both types are difficult to use in the wavelength region 104-30 nm because of the lack of transmitting windows. Below 30 nm cellulose and thin metal films start to transmit, and the detector can be used from there right on to the X-ray region. The filler gas at longer wavelengths is usually nitric oxide or a similar molecule, but the inert gases are preferable as soon as the photon energy reaches their respective ionization potentials. With a clever choice of window material and filler gas it is possible to arrange the long- and short-wavelength cutoffs to leave a fairly narrow band of sensitivity. Photoionization detectors have also been used to measure absolute intensities and to calibrate sources as intensity standards in the vacuum ultraviolet. For 100% quantum efficiency the photon flux can be calculated from the output current of an ion chamber. The inert gases fulfil this requirement successively from 102 nm, the ionization limit of xenon, to 25 nm, where the ejected electrons have sufficient energy to cause secondary ionization even ? Quantum efficiency is the ratio of photoelectrons to incident photons.

15.6

Multichannel

Detectors

Set

in helium so that the 1:1 correspondence breaks down. Photon counters can take over at this point, since they record separate pulses from each absorbed photon, irrespective of the number of secondary electrons.

15.6 Multichannel

Detectors

Photographic emulsions as multichannel detectors are being steadily superseded by array detectors, which have the advantages of linear response, large dynamic range, time resolution and direct interface to analogue or digital electronics, although their limited size restricts the wavelength interval that can be recorded in an ordinary grating spectrograph. We give here an overview of array detectors and follow it with a brief description of the properties of photographic emulsions, partly because they are still used to a limited extent with some high resolution grating instruments, but also because so much of the experimental work reported in the spectroscopic literature depended on them. 15.6.1

Array Detectors

Early array detectors had the disadvantages of a limited number of elements, or pixels, large pixel size (limiting the spectral resolution), and high cost, but these drawbacks can now be said to have been overcome. The pixel size is now much the same as the grain size of a photographic emulsion. The rate of development of these devices is such that details of their design and performance are likely to be soon out of date. We shall simply summarize the more important features of the devices currently available, making no attempt to discuss read-out and data-handling systems. Three classes of array detectors are currently suitable for spectrometers. First, there are photodiode arrays, which are linear arrays of up to 1024 silicon p-n junctions similar to those described in Sect. 15.5. They have the same spectral range, 1 wm to 200 nm, and quantum efficiency (up to 60% in the visible), with dynamic range of 104—10°. Linear arrays suitable for grating spectrometers may have 1024 sensors, typically 25 jm (or even 12.5 um) by 2.5 mm. With a digital system the exposure, or readout, time can be set to suit the experimental requirements. Useful exposure times are, however, limited by the increase with exposure of dark current, or rather dark-current noise. Cooling the detector to a few tens of degrees below room temperature decreases this noise dramatically, and, for integration times of the order of 10s or so, the noise limit is then set by the read-out noise, which is independent of integration time and temperature. The second class of array detector is the CCD, or charge coupled device

[64]. This consists of a two-dimensional array of photodiodes, essentially the

same as those used in imaging devices such as video cameras. Strips of electrodes deposited on the surface between the pixels carry voltages that create

378

15. Light Sources and Detectors

potential wells at each pixel where charge is accumulated while light is falling on the pixel. ‘Readout’ is accomplished by manipulating the voltages to move the charge across the detector. CCDs for spectroscopic applications are constructed for high dynamic range, low dark current, low readout noise and wide spectral range. The latter can be extended into the ultraviolet by using back-illumination or phosphor-coated windows. Pixel sizes are in the range 5-30 um, with typically 1024 pixels along the dispersion direction and 256 in the orthogonal direction. A smaller pixel size means a higher resolution, but it also gives a smaller storage capacity and thus a lower dynamic range. The stored charges are read off sequentially at anything up to 1 MHz. In the vertical binning mode the signal is integrated along the length of the slit, but it is possible to increase the time resolution by illuminating only a strip along the bottom of the array and using the rest of it as a temporary storage area. The readout noise is some two orders of magnitude lower than that of a photodiode array, so the sensitivity is correspondingly higher for weak signals. On the other hand, the saturation level, corresponding to the filling of the potential wells, is also some two orders of magnitude lower, so that at high light levels the signal-to-noise ratio is better for the photodiode array. CCDs can also be connected to image intensifiers by fibre-optic couplers for fast triggering and gating applications. There is another type of multichannel detector that is related to the photomultiplier rather than the photodiode. The channel multipliers mentioned in Sect. 15.5 can be miniaturized to form a microchannel array or microchannel plate, MCP, with individual channel (pixel) sizes down to 25 um x 25 um. As in a conventional photomultiplier tube, the spectral range is set by the photocathode and window materials anywhere from the visible to the soft Xray region. Quantum efficiencies, at around 5-15%, and multiplication factors of about 10° are as described in Sect. 15.5. 15.6.2

Photographic

Emulsions

All the detectors considered up to this point give an output signal proportional to the incident radiant flux over a wide dynamic range. The response of a photographic emulsion to radiation is quite different. It is usually represented in the form shown in Fig. 15.3, in which density d is plotted against the logarithm of flux density (irradiance) J for constant exposure time tf. The density d is defined as logio(1/T), where T is the transmissivity of the blackened emulsion — i.e. the ratio of transmitted to incident intensity. The properties of the emulsion are characterized by three parameters, threshold, contrast and saturation, as shown on Fig. 15.3. Schematic response curves for slow (insensitive) and fast (sensitive) emulsions are illustrated. A fast emulsion means a low threshold, but as the contrast is usually low the film is not necessarily fast for long exposures. As a photographic emulsion is essentially an integrating device, it might be thought that d should depend on the product Jt, in which case Fig. 15.3

15.6

Multichannel

Detectors

379

saturation

contrast y = tan @

a) log /

threshold

medium

b)

—— y/4a, this equation reduces to n

é 16n2e9mvp Vo

nf

e? —Y

16 m2Egmc?

nfr% A— Ao

(16.1)

The refractive index can be tracked as a function of wavelength by crossing a two-beam interferometer with a spectrograph. This is normally done by inserting a cell containing the absorbing gas in one arm of a Jamin-type two-beam interferometer (a Mach-Zehnder interferometer is generally used

388

S

16.

Transition Probabilities and Radiative Lifetimes

B,

Mz

C My

\

mal aad

SG

OB: zt

E

Fig. 16.1. Mach—Zehnder interferometer used for hook measurements. Bi, Bz are beamsplitters and M;, M2 are mirrors. C is the gas cell and C’ is the compensating plate. The source S and the interference fringes are focused on the spectrograph SG.

in practice, as shown in Fig. 16.1) illuminated by a continuous source. With suitable small tilts of the mirrors this interferometer can be arranged to give horizontal interference fringes that are focused on the slit of a stigmatic spectrograph — i.e. the path difference between the two beams is proportional to the distance y up the slit. Taking into account the additional path difference introduced by the absorption cell, length 1, the fringe maxima are specified

by by+(n-—1)l=pda, where p is a small integer and b is a geometrical constant. In the absence of absorption lines, n changes very slowly with wavelength, so the fringes extend almost horizontally across the spectrum, as in Fig. 16.2a. However, near an absorption line, the optical path difference changes rapidly, and the fringes are curved in the hyperbolic form shown in Fig. 16.2b, which reproduces the n — 1 curve of Fig. 7.6. In principle the f value can be found by tracing the path of a given fringe. The hook method is simply a more convenient way of using the fringe geometry. If a ‘compensating plate’ of thickness lI’ and refractive index n’ is inserted in the lower beam, the fringe equation becomes

iW tit otal ala

A

a)

suchen. ee a ae ree ——

b)

A

Fig. 16.2. Interference fringes (a) far from and (b) close to an absorption line at Ao. The figures show \ increasing towards the left, so as to conform with the plot of refractive index against \ in Fig. 7.6.

16.2 Absorption and Dispersion Measurements

389

Ul. =

d

a)

b)

Fig. 16.3. High-order interference fringes (a) far from and (b) close to an absorption line at \o. Hooks are formed at the two wavelengths A; at separation Ay.

(16.2)

by + (n—1)l—(n'-1)l =p),

where p is now a large negative integer. The variation of both n and n’ with \ can be ignored in any small region clear of absorption lines. The slope of the fringes in such a region, found by differentiating (16.2) for constant p, is now large:

(5¢) da Shy Paks

Figure 16.3a shows these high-order fringes; the direction of the slope depends on the sign of b, taken as negative in this figure. Near an absorption line the rate of path increase in the upper beam from the rapidly rising refractive index n must eventually overtake this constant rate, reversing the slope of the fringes. The turning points on either side of the line form the hooks, shown in Fig. 16.3b. The hook wavelength A, is defined by dy/dX\ = 0 at constant p. From (16.2), with dn’/dA ~ 0, together with (16.1), we get dn wort

dA

e?

nfr%

1622e9mc? (An — Ao)?

The two solutions to this equation determine the distance of each hook from the line:

my do st (=I i

2

1612e9mc?

nfr3

1/2

|

Pp

The distance Aj, between the hooks is given by Ap = 2|An — Aol. Since p is negative, we set it equal to —A and obtain for the hook separation

gehen e? ranfl nh An2egmc? K

(16.3)

The ‘hook constant’ K, which is just the negative of the order number, can

be found from the undistorted fringes, either by measuring the slope or by counting fringes at constant height across a wavelength interval AX:

pAr = —AAp or

K = —p=AAp/Ar.

390

16.

Transition Probabilities and Radiative Lifetimes

If several close lines contribute to the refractive index, as in a multiplet,

(16.1) has to be summed over all relevant lines:

e? ~ 16n2e9mc?

nr

a

ei ene Ae Np :

and the hook condition becomes 2 3

Z

1672e9mc?

RL

NN

gene l

(16.4)

In general, there are 2r such equations, corresponding to two hooks for each of r lines, and the r unknowns

n,/f, are overdetermined

from the solution

of these 2r simultaneous equations. As the pair of hooks near any one line depends mainly on the nf value of that line, with the other lines contributing corrections, the solutions can usually be obtained quite easily by successive approximations. The same method can be applied to the rotational lines in a molecular band, provided that the rotational structure is sufficiently well spaced for hooks to be measured

between the lines; if this is not the case,

the corresponding roots of (16.4) are imaginary. An alternative approach for close rotational structure if the band has a well-defined head is to measure the single hook near the band head; if the relative rotational line strengths (the Honl—London factors of Sect.7.11) are known, this one measurement determines the band oscillator strength. The hook method has two great advantages over the absorption methods, and, in this form, two disadvantages. The first disadvantage is that it is essentially a photographic technique, whereas spectroscopy is increasingly geared to photoelectric detection methods. The second disadvantage is the sensitivity: the magnitude of nfl required for a measurable hook is at least a factor of 10 greater than that required for respectable absorption. The advantages are first that the accuracy does not depend on resolution, optical thinness or any assumptions about line shape and, secondly, that wavelength differences can nearly always be measured more accurately than intensity differences, and the problems of scattered light, absorption from trace impurities, and the far wings of the line are all irrelevant in the hook method. A number of variants of the basic method have been devised to improve both accuracy and sensitivity and to get away from the photographic plate. The ‘hook vernier’ makes use of information from the y shift of the fringes between the two hook positions on either side of the line. The phase shift method measures the increase in fringe separation at a constant value of y in going from the undistorted region towards the line; this method can actually be used with the compensating plate in the same beam as the absorbing gas, in which case the fringe spacing decreases towards the line, but no hooks appear. The phase shift variant is the one most easily adapted to photoelectric rather than photographic measurement, and it can also be used with a narrow-band scanning laser instead of a spectrometer. The additional complexity of these and other variants over the simple hook measurements is

16.3

Lifetime Measurements

391

justified by their greater sensitivity in the case of weak lines. Further details can be found in the review by Huber and Sandemann. The combination of hook and absorption measurements offers a neat way of determining f values over a wide dynamic range. Strong lines are ‘hooked’ in the usual interferometer, and the equivalent widths of weak lines are measured simply by blanking off the lower interferometer beam.

16.3 Lifetime

Measurements

There are three rather different ways of determining lifetimes of excited states: delay-time measurements, beam measurements and Hanle effect. Accounts of these methods may be found in the review by Imhof and Read. None of them is universally applicable. Some work only for states that can be excited directly from the ground state; others are no good if the lifetime is too short (strong line) or the intensity too low (weak line). Amongst them, they cover a very wide range of transitions, both atomic and molecular. Lifetime measurements get away from the uncertainties of population densities and assumptions about LTE, but they do not lead directly to oscillator strengths, unless there is only one transition of any significance from the excited state. The combination of lifetime with relative emission measurements to obtain f values from branching fractions is described in Sect. 16.4. 16.3.1

Delay Methods

If a large number of atoms is excited to the required level with a very short pulse, of either electrons or radiation, the intensity of a line starting from this level should be a decaying exponential, I(t) ox A21Nn2(t)

=

Aoin2(0)e~/”

,

the time constant of which gives T2. The trouble with this rather simple idea is the noise associated with the low photon flux attainable. Fluctuations due to the random arrival of photons follow Poisson statistics, for which the noise is equal to the square root of the signal. A signal-to-noise ratio of 100 requires at least 10* photons per measurement interval. If the lifetime is itself of order 10 ns, a measurement interval can be only a few ns. Taking into account detection efficiency and restrictions on gas density (see below), this method is not usually feasible with conventional methods of excitation, but it certainly can be used with tunable laser excitation. Lifetime measurements with pulsed tunable lasers have the advantage that of only one level is excited by the laser pulse, and the recorded signal consists the a single exponential decay. Stepwise excitation makes it possible to study (Fig. decay of levels that cannot be reached directly from the ground state 14.8), but the method is limited to the wavelength range where tunable laser

radiation is avialable.

392

me

16. Transition Probabilities and Radiative Lifetimes

BA

decay

ae pulsed laser

target

a

a

I

el

| ol filter detector excitation

|

transient

digitizer

Fig. 16.4. Lifetime measurement cence.

by direct observation of laser-induced fluores-

The basic principle, known as time-resolved laser-induced fluorescence (LIF), has been adapted for different types of problems (see Further Reading, review by Lawler et al.). In the most straight-forward method the decay signal detected by a fast photomultiplier is sampled at constant time intervals by a transient digitizer and stored in a computer. The signal-to-noise ratio can be improved by adding the data from a large number of laser pulses. Due to the selective excitation all the radiation recorded by the detector comes from the decay of one level, and in principle no monochromator is needed. In practice a low resolution monochromator or a filter is used to eliminate background light and scattered light from the laser. Generally the exciting wavelength is excluded from the detection, which is performed at longer wavelengths, as shown in Fig. 16.4. With sufficiently short exciting laser pulses and fast electronics, lifetimes down to and even below 1 ns can be measured. However, there is a potential difficulty with the short pulses needed for very short lifetimes: as stated by the bandwidth theorem, such pulses have large spectral

width — at least 10’* Hz or 30 cm—! for a 1 ps pulse. This means that more

than one level may be excited in a complex system where the density of levels is high, and a high resolution monochromator is then needed to separate the different decay signals.

CF

\- >

pulsed

target

-

i

laser

|

|

start

time-to-

Schematic

3

& 2

stop

pulse-height converter Fig. 16.5. dence.

z

;

filter or /, monochromator

arrangement

ew

5

multichannel

analyzer

for lifetime measurement

channel oe) by delayed coinci-

16.3

Lifetime Measurements

393

One of the difficulties with the direct recording of the whole decay curve is caused by the large number of photons at the start of the pulse, which may cause deviation from the linearity of the photomultiplier. Such problems are avoided with the delayed coincidence method, which is based on the detection of single photons. The method was developed for electron or radiative excitation and is now frequently used with pulsed lasers. The principle of the method is shown schematically in Fig. 16.5. The short exciting pulse also triggers the ‘start’ of a time-to-pulse-height converter. This is basically a capacitor charged by a steady current, so that the final charge is proportional to the time the current is allowed to flow. The ‘stop’ signal is provided from the first photon received by the photomultiplier detecting the fluorescence. The pulse height is fed into the appropriate channel of a multichannel pulse height analyzer, which displays the result of repetitive pulsing as a histogram of number of counts per channel against channel number, or time delay. The probability that more than one photon reaches the photomultiplier after each exciting pulse must be kept small: the first photon stops the converter, so the later pulses would never be recorded. The experiment can be continued until the signal-to-noise ratio on the histogram is acceptable, and the time constant is then extracted. A variant of this method uses modulated instead of pulsed excitation. The emitted radiation is then also modulated, but with a phase shift @ relative to the excitation which depends on the lifetime of the excited state according to tan@ = 27,72, where Ym is the modulating frequency. The modulation method has been used with both electron and radiative excitation, but the latter is much more common, particularly since lasers have become avail-

able. The modulation can be done either electro-optically, with Kerr cells, or acousto-optically, with ultrasonic waves. Apart from difficulties associated with scattered light and sometimes insufficient spectral resolution, the delay methods have three principal potential sources of systematic error. The first, cascading, is relevant only to electron or broad-band radiative excitation. If higher levels are excited and subsequently decay to the level under investigation, the lifetime of the latter is apparently prolonged by the repopulation. This difficulty can evidently be avoided by selective population of one level only, a great bonus provided by tunable lasers. The second problem affects all levels emitting to the ground or a metastable state and is known as imprisonment of resonance radiation. If the gas is not optically thin, some of the photons are absorbed and re-emitted one or more times before they eventually get out, and the effect is, again, to prolong the apparent lifetime of the state. Finally, longer-lived states may be collisionally depopulated, or ‘quenched’, which of course shortens the apparent lifetime. Both of these last problems can be dealt with in principle by going to suf ficiently low pressure, but one may then run into difficulties with low light in intensities. Systematic errors of this type have often been underestimated the experimental data.

394

16.

Transition Probabilities and Radiative Lifetimes

The delay methods as described here can be used for lifetimes from a few microseconds to less than a nanosecond. The decay rates of metastable levels by forbidden transitions with lifetimes in the range from microseconds to seconds have also been measured, but special precautions are required. Besides the need for a very low particle density, and thus a very good vacuum, the ions must also be prevented from disappearing from the field of view of the detection system before they decay. This can be done by means of different kinds of ion traps, for example a radio-frequency trap (Sect. 14.7) or an EBIT

(Sect. 15.1.10). 16.3.2

Beam

Measurements

The beam foil light source is described in Sect.15.1.6. A beam of ions of various degrees of ionization emerges from the thin foil in different states of excitation. The excited states decay as the ions travel downstream from the foil, and the rate of decrease of the intensity of any particular line as a function of distance from the foil gives directly the lifetime of the relevant excited state. Historically this method is the successor to the experiments of Wien in the 1920s on canal rays, in which the lifetime was measured from the decay of emitted radiation as the excited ions in a discharge tube travelled beyond the cathode. The beam foil method has intrinsically much greater accuracy because of the much higher ion velocity. In practice the spectrometer is kept fixed and the intensity is measured as the foil is moved upstream. It is necessary to monitor the constancy of the beam while this is going on, either by measuring the total charge collected at the end or by using a second photomultiplier at a constant distance from the foil. The principle of the method is shown in Fig. 16.6. The main difficulties of the method are cascading from higher excited states, low light intensity and large Doppler broadening. The first of these x, —>——_———_———_|

beam

2 .

observed decay

i

foil

D

Ae)

monochromator

foil position

Fig. 16.6. Schematic arrangement for lifetime measurement with the beam-foil method. The effects of cascades and the finite acceptance angle of the spectrometer are shown.

16.3

Lifetime Measurements

395

was discussed in connection with delay methods, and it is also illustrated in Fig. 16.6. If the lifetimes of the repopulating cascades are much shorter than the lifetime to be measured, only the first part of the decay curve is affected, and this part can be omitted in the data analysis. The lifetime of a repopulating transition having a much longer lifetime can be determined from the long tail of the decay curve, and the contribution from this decay can easily be subtracted from the primary decay curve. The real problem appears when the lifetimes are of the same order of magnitude, but even in this case it can be solved with a technique known as ANDC, “Arbitrarily Normalized Decay Curves”, where both the primary decay and the separately recorded decays of the most important cascades are included in the analysis; see e.g. [66]. The cascading problem can also be overcome by using selective laser excitation as described below. The second difficulty, low light intensity, is a consequence of the small population of a particular level of a particular stage of ionization in a beam that is in any case of low density (~ 10° ions/cm*). Photon-counting methods are used, and fast spectrometers with low resolution are usually necessary. The latter may lead to problems with line blending in complex spectra. The low density in the beam can, however, also be considered as a great advantage of the beam-foil method: there are no difficulties with imprisonment of radiation or collisional quenching. The reason for the Doppler problem is the high velocity of the beam, typically of order 10’ ms~'. The Doppler shift is minimized by observing the radiation perpendicularly to the beam direction, but it cannot be eliminated in this way for all rays because of the finite acceptance angle of the monochromator (Fig. 16.6). The effect can, however, be completely compensated because the blueshifted “upstream” photons and the redshifted “downstream” photons hit different parts of the grating. By translating the grating slightly from its normal position, the angles of incidence of the “red” and “blue” rays is changed. A new focus can then be found where this difference exactly compensates the Doppler width, and a sharp image is formed. This refocusing technique is now common practice in beam-foil experiments [67].

The beam foil method has been used over the range 10~1?-10~7 s. The

short lifetimes present problems of spatial resolution because the decay distance is itself so short, whereas longer lifetimes would require observation of the decay over impractically long distances along the beam. The latter problem can be solved by injecting the excited beam into a storage ring where the decay can be obseved as the ions circulate and repeatedly pass the detection area. In cases where the photon emission rate is too low for direct optical detection, the decay of the excited state has been measured by observing the decaying yield from a resonant electron-ion reaction involving the excited state. Precision measurements in the millisecond range have been reported

(68}.

396

16. Transition Probabilities and Radiative Lifetimes

B source

polarizer

y

gas

Meee

x

Fig. 16.7. Schematic arrangement for the Hanle method. See text for explanation.

A powerful alternative to foil excitation has been furnished by tunable lasers. Very accurate lifetime measurements have been made by crossing a laser beam tuned to a particular excitation frequency with a fast-ion beam. Cascading and spectral overlap problems are both eliminated in this technique, which is, however, limited to wavelength regions where tunable laser radiation is available. It has therefore mainly been used for neutral and singly ionized atoms. Much of the importance of the beam-foil method is that it is applicable to the higher stages of ionization, which are usually beyond the reach of all the other methods. Data obtained by following up iso-electronic sequences are important both theoretically and practically and have been used to extrapolate to yet higher degrees of ionization. 16.3.3

Hanle

Effect

This method is again a resurrection from the 1920s, when the effect, investigated by Hanle, was known as magnetic depolarization of resonance radiation. It is now also known as zero-field level-crossing. For a very brief qualitative description of the effect it is simplest to use a semiclassical model, but when there is hyperfine structure or close fine structure in the excited level, a proper quantum-mechanical treatment is required. A typical experimental layout is shown in Fig. 16.7. Light from the source of resonance radiation travelling in the y direction is plane-polarized in the x2 direction before entering the resonance vessel. The emitted radiation is due to electrons oscillating along the x axis and has the same polarization. A detector on the x axis therefore records no signal. If now a magnetic field B is applied in the z direction, the resonance radiation is partly depolarized, and the intensity rises at strong fields to half that in the absence of the polarizer. The depolarization can be ascribed to the precession of the polarization direction about the z axis with angular frequency w given in terms of the Landé g factor by

w=gsupB/h.

16.3

a)

Lifetime Measurements

397

b)

Fig. 16.8.

Classical interpretation of Hanle effect. (a) represents small damping

(w > 1/r) and (b) represents large damping (w < 1/7). The length of each spoke is proportional to the intensity of radiation for that particular direction of polarization.

If w is sufficiently rapid, the components of the oscillation along the x and y axes are equalized, and the y oscillations, representing half the total intensity, radiate along the x axis. The lifetime enters into the story because for small w some of the atoms decay before the precessional cycle is completed, so that the depolarization is only partial. The signal thus depends on the relative magnitudes of the precessional and decay times, as shown schematically in Fig. 16.8: in terms of the damping constant y = 1/7, Fig. 16.8a represents w > y, and Fig. 16.8b shows w < y. The semiclassical treatment gives for

the signal (see, for example, [69]) 4? J const... | l= s-———— ns ( i) | This expression has the inverted Lorentzian signal has half its maximum value when

¥2 1 —_——.= —

~+4u2

2

:

ie

form shown

in Fig. 16.9. The

leadingtow=7>~==—.

2”

Or

The lifetime can therefore be determined from the half-width of the plot of signal versus field if g7 is known from some other experiment: =

h/(2up By/2) .

The quantum-mechanical interpretation of the Hanle effect starts from the basis that in zero field all the magnetic sublevels of the upper state are degenerate and are populated coherently by the absorbed radiation. When this is polarized in the x direction, the emitted radiation from the sublevels interferes destructively in the x direction. To lift the degeneracy, it is necessary to separate the sublevels by an amount AF greater than their natural

width h/r, leading to the condition AB

=

g7jtpe = ae

398

16. Transition Probabilities and Radiative Lifetimes

/

Fig. 16.9. Signal as a function of magnetic field in the Hanle effect. The curve is an inverted Lorentzian, and

gs7 can be found from its FWHM

(see

text).

AB

which is fulfilled beyond the half-width point. Further degeneracies, with consequent changes of polarization, occur whenever the field is such that the energies of two hyperfine components overlap. These other degeneracies may have to be taken into account at the fields of order 10~°-10~? T required for the ‘zero-field’ measurements. This technique has no cascading or collisional quenching problems, and Doppler and power broadening are irrelevant. Radiation trapping is still a potential source of error: absorbing atoms in the same magnetic field are in a sense coherent with the emitting atoms, and the polarization information is passed on to them by the photon. However, with due care the method can be made to give accuracies of a few percent for lifetimes in the micro- to nanosecond range, and with selective laser excitation it has been applied to many molecular as well as atomic transitions.

16.4 Combinations

of Methods

Given that excited-state lifetimes can often be measured with considerable accuracy independently of the number density, there is a strong incentive for using them to put relative measurements made by emission or absorption methods on an absolute scale. A lifetime does not in itself give a transition probability except in the few cases where a resonance line is the only transition of any strength from the level in question. If there is more than one allowed transition from the level k, as shown in Fig. 16.10, we have

Z

!states / !

Fig. 16.10. Branching fractions. The lifetime of state k is given by tT, = 1/ Se Api, Summed over all states i with which k can combine.

16.4

Combinations of Methods

399

Fig. 16.11. Combination of emission and absorption measurements between the four levels k, 1, i and j to obtain relative oscillator strengths independent of population.

Gee

oy cua

and we need relative intensity measurements for all transitions from k to obtain an absolute A value for any one of them. Thus, the first important combination is that of lifetimes with branching fractions. The measured intensity J; of an optically thin line k — 2 is given

by iC

Ap hin, ,

where C; contains not only all geometric factors but also the spectrometer and detector responses at the frequency 1;. The transition probability of any one line, k > 7, can be written as

Ary

_

45 /Civ; iy

a Aki

Ci)

giving Ag;

a

i

Lift

pes

Tea Coles). The experimental difficulties of this seemingly simple method are mainly associated with the C factors. It is not a trivial matter to calibrate detectors and spectrometers accurately over a spectral range that may extend from the

infrared well into the ultraviolet. It also requires sufficient knowledge of the spectrum to ensure that all significant transitions from level k are included in the sum. The increasing availability of rather good relative oscillator strengths for the same set of lines from emission, absorption and dispersion methods has led to the exploitation of a technique first used by Ladenburg in the 1930s and now known by various names: ‘Ladenburg’ and ‘bow-tie’ among them. It is best explained with reference to Fig. 16.11. Suppose we have two upper levels k and | that can combine with two lower levels 7 and 7 by the four transitions shown. From two pairs of absorption (or dispersion) measurements we can find, say, equivalent widths W:

Wa/Wir = ful fin

fir and Wyi/Wje = fyi/

Similarly, from two pairs of emission measurements,

400

16. Transition Probabilities and Radiative Lifetimes

lng

/

WegAnj

Veg lok

oye

e

Tgi

Vi Ani

3 fF Si Vig

a

Li,

ii

Mig fit

Tepe

ce

Ths

St

fi;eee

where J’ = I/C represents the intensity corrected for instrument response, and g values have been omitted for simplicity. In the first place the consistency of the data can be checked by comparing the emission value of fi fjx/ fix fj with the same quantity obtained from the equivalent widths. Second, if one of the f values is taken as reference we have four equations to determine three unknowns, so a least-squares best fit can be used to put the four f values on the same scale, which becomes absolute if the reference value can be made absolute. The method can be extended to a larger system of transitions so long as each additional level is connected to the network by at least one absorption (or dispersion) and one emission measurement. The larger the number of interlinked levels, the more are the equations overdetermined, and the easier it becomes to identify a bad measurement. Ideally the method is combined with an absolute lifetime measurement to put all the f values on an absolute scale. If k + i and k — j are the only transitions from level k in Fig. 16.11, a measurement of 7; combined with the branching ratio of these two lines serves to put all the f values in the network on an absolute scale.

Further Reading Compilations and data bases — Wiese, W.L., Smith, M.W., and Glennon, B.M., Atomic transition probabilities — hydrogen through neon, NSRDS-NBS 4 (US Governement Printing Office, Washington, D.C., 1966). — Wiese, W.L., Smith, M.W., and Miles, B.M., Atomic transition probabilities — sodium through calcium, NSRDS-NBS Office, Washington, D.C., 1969).

22 (US Governement Printing

— Martin, G.A., Fuhr, J.R., and Wiese, W.L., Atomic transition probabilities ~ scandium through manganese. J. Phys. Chem. Ref. Data 17, Suppl. 3 (1988). ~ Fuhr, J.R., Martin, G.A., and Wiese, W.L., Atomic transition probabilities ~ iron through nickel, J. Phys. Chem. Ref. Data 18, Suppl. 4 (1988). — Wiese, W.L., Fuhr, J.R., and Deters, T.M., Atomic transition probabilities of carbon, nitrogen and oxygen, J. Phys. Chem. Ref. Data, Monograph 7 (1995). Internet adresses

— Fuhr, J.R. and Felrice, H.R., Atomic transition probability bibliographic database, http://physics.nist.gov/PhysRefData/fvalbib /reffrm0.html

Further Reading

— International Astronomical Union, home page of Commission

401

14, Atomic

and molecular data, http://cfa-www.harvard.edu/amp/iaul4/iaualls.html General

description of classical methods

— Mitchell, A.C.G.

and Zemansky,

M.S., Resonance

Radiation

and Excited

Atoms (Cambridge University Press, Cambridge, 1934, reprinted 1971). Emission

and absorption measurements

— Huber, M.C.E. and Sandemann, R.J., The strengths, Rep. Progr. Phys. 49, 397 (1986).

measurement

of oscillator

— Kock, M., Atomic oscillator strengths from emission measurements: achieve-

ments and limitations, Physica Scripta T 65, 43 (1996). Lifetime

measurements

— Imhof, R.E. and Read, F.H., Measurement of lifetimes of atoms, molecules

and ions, Rep. Progr. Phys. 40, 1 (1977). — Svanberg, S., Atomic and Molecular Spectroscopy, 2nd ed. (Springer, Berlin, Heidelberg, 1997). — Lawler, J.E., Bergeson, S.D., and Wamsley, R.C., Advanced experimental techniques for measuring oscillator strengths of vacuum ultraviolet lines. Physica Scripta T 47, 29 (1993). — Trabert, E., Radiative-lifetime measurements on highly charged ions, in Accelerator-Based Atomic Physics. Techniques and Applications, ed. by S.M. Shafroth and J.C. Austin (AIP Press, New York, 1998).

br

varmines Renae

ee va

>

a

ee

Aero ct

Tee

cane ont

=p

sciininmpantiensel

digest (te

ee

eee

vote

o> 429 eo! ot

te

Palen? Mitoteieee

ry

EG

:

os

3 (dale oh& G1) 1

rie!

met

awit

aw

anki Fe eae manasa

pty ape | eerie

ge

tear ita rtenrgmati bin

ers

1iBake

ed

St

poigize

—?

ot

oy.

¢

ban eens

is Upse hem, yi regret |

Dee et

eee

ce

ee ee etn iow tn {79

femteuse hinger hs “118 oslo fom 02 oekl as pees me Lepratlontivtedincapue \o ce H Wy exp(—iEgt/h) + S- ceH by exp(—iEgt/h) k

k

= it (=a

ve exp(—iFE,t/h) + X Ch,

Vk e(-s8t/n)

(A.13) and (A.14) show that the first and the last terms of this equation

cancel, leaving us with the equation

SS CrH'Wy,exp(—iE,t/h) = ih Sy aby exp(—iF,t/h). k k

(A.18)

Before the perturbation was switched on at t = 0. the system existed in a stationary state described by (A.15). We denote this state by 7 for initial. This means that the coefficient c;(0) = 1 and cx(0) = 0 for all k # a. Tf

the perturbation H’(t) is small we can assume that the coefficients do not

change very much with time, implying that c; stays much larger than all the other cj, i.e. ¢;(t) can be approximated by c;(0) = 1 and the other cz can be

neglected on the left-hand side of (A.18):

: t dex A", exp(—iE;t/h) = ih a a be exp(—ik,t/h).

(A.19)

k

We can now multiply from the left by w; and integrate over all space, and obtain because of the orthonormality of the wy functions:

(thy |H’ |Ui) exp(—-iE\t/h) = ih = : exp(—iB;t/h) a

:

or, using the relation £; — EF; = hv54 = hwy, rer

ih? = (iby |H! | ds) expliwyit).

(A.20)

This is the differential equation for c 3, which we need to solve in order to

derive |c; |?, ie. the probability to find the system in state ei

A.2 Physical Constants

419

A.2 Physical Constants Internationally adopted values for physical constants can be found in a database at NIST (National Institute of Standards and Technology) available at the Internet address

http://physics.nist.gov/cuu/Constants/index.html An introduction describing the historical development of the system of constants and an overview of the present measurements is found at

http://physics.nist.gov /cuu/Constants/introduction.html The constants given below are rounded off to a smaller number of digits than the NIST values. A.2.1

General

Constants

Quantity

Symbol

Value

elementary charge

e

L602i8310

electron mass

Me

9.10939 x 10-3! kg

charge/mass ratio of electron

e/me

175882501044 Cheer

proton mass

ite

1.672623

proton/electron mass ratio

Mp/Me

1836.15

Planck’s constant

h

G62607 C10

h bar (h/2z)

h

L05457-< 106

Boltzmann’s constant

ke

Cc

x10-*" ke

Js

188066.% 107 SK = 0.69504 cm~! K7!

speed of light

c

2.997925 % 10° ma

Avogadro constant

Na

6.02214 x 1078 mole~

Loschmidt’s number

Ny,

D 6eh

permittivity of vacuum

E0

Sead Oro On

permeability of vacuum

Lo

An x 10-7

oe Lu" ia

NA~?

aie Nk rae orn my

420

Appendix

A.2.2 Spectroscopic Constants

Constant 2

Rydberg constant

(< ) 0

21? Touahenl

e2 wie

for hydrogen

Bohr radius

2 4 ae (= Nite Neve

fine-structure constant

odes

1

e?

Cc

TEQ

\?

0

5eh

Bohr magneton

Value

Hie,

1.09737315 x 10° cm~+

Reale

13.6057 eV

Ry

1.09677587 x 10°

ao

5 29L

Thee 1 : 0.529177 A

a

1.297353 x 10

1/a

137.0360

LB

9.274015 x 10 ~24 Sih 7-1

2X2

a

Rydberg constant

Symbol

Me

nuclear magneton

5

5

cm

one

Ae

0.4668644 cm~! T-!

eh Mp

LN

my Boe = 0.05079 * 10“ JT

A.2.3 Energy Conversion Conversion factors and automatic conversion of energy values are available at the Internet address

http://physics.nist.gov /cuu/Constants/energy.html 1 J = 6.242 x 1018 eV

= 5.034 x 102? em7!

1 eV = 1.60218 x 10- J = 8065.54 cm=!

1 cm fer 1.986

k = 1.3807 x 10°78

10-29"

= 1940

10 Sev

JK~1 = 0.8617 x 10-4 eVK-—! = 0.69504 em—! K—!

he = 1.98645 x 10-7" Jem"

4

= 1.93084

10-8 eV m

References

. Edlén, B., Metrologia 2, 71 (1966). . Cowan, R.D., The Theory of Atomic Structure and Spectra (University of California Press, Berkeley, 1981). . Risberg, P., Ark. Fys. 10, 583 (1956). Martin, W.C., J. Opt. Soc. Am. 70, 784 (1980). Edlén, B., Atomic Spectra, in Handbuch der Physik, Vol. XXVII, Fliigge (Springer, Berlin, Heidelberg 1964). Lu, K.T. and Fano, U., Phys. Rev. A 2, 81 (1970).

ed.by S.

Johansson, S., Litzén, U. Sinzelle, J., and Wyart, J.-F., Physica Scripta 21, 40

(1980). Reader,

.

. . .

J., Corliss,

C.H., Wiese,

G.L., and

Martin,

G.A., Wavelengths

and

Transition Probabilities for Atoms and Atomic Ions, Part I. Wavelengths. Part II. Transition Probabilities, Natl. Stand. Ref. Data Ser., Natl. Bur. Stand. (U.S.) 68 (1980). Reader, J. and Corliss, C.H. (Eds.), Line Spectra of the Elelements, in CRC Handbook of Chemistry and Physics (CRC Press, Boca Raton, FL, 1979 and later editions). Kelly, R.L., Atomic and ionic spectrum lines below 2000 Angstroms: Hydrogen through Krypton, J. Phys. Chem. Ref. Data 16, Suppl. 1 (1987). Moore, C.E., Atomic Energy Levels, Natl. Stand. Ref. Data Ser., Natl. Bur. Stand. (U.S.) 35, Vol. I-III (1971), (Reprint from 1949, 1952 and 1958). Martin, W.C., Zalubas, R., and Hagan, L., Atomic Energy Levels — The Rare

Earth Elements. Natl. Stand. Ref. Data Ser., Natl. Bur. Stand. (U.S.) 60 (1978).

Blaise, J. and Wyart, J.-F., Energy Levels and Atomic Spectra of Actinides, Tables Internationales de Constantes,

Paris 1992. Sugar, J. and Corliss, C., Atomic energy levels of the iron period elements: potassium through nickel, J. Phys. Chem. Ref. Data 14, Suppl. 2 (1985). Bass, M. (Ed.) Handbook of Optics, 2nd ed. (McGraw Hill, New York, 1995),

(Vol. II, Chap. 24). Grun, F. and Becheren, R.J. Optical Radiation Measurements, Vol. I (Academic Press, New York, 1979). . Schadee, A., J. Quant. Spect. Rad. Transf. 19, 451 (1978). . Whiting, E.E., Schadee, A., Tatum, J.B., Hougen, J.T. and Nicholls, R.W., J.

Molec. Spectros. 80, 249 (1980). . Herzberg, G., Spectra of Diatomic Molecules (Van Nostrand, New York, 1950). _ Kovacs, I. Rotational Structure in the Spectra of Diatomic Molecules (Hilger,

London, 1969). _ Bransden, B.H. and Joachim, C.J., Physics of Atoms and Molecules Longman, London, 1983). . Hasted, J.B., Physics of Atomic Collisions (Butterworths, London, 1964).

422

References

23. Massey, H.S.W. and Burhop, E.H.S., Electronic and Ionic Impact Phenomena

(Oxford University Press, London, 1952). 24. Sobelman, I.I., Vainshtein, L.A., and Yukov, E.A., Excitation of Atoms and Broadening of Spectral Lines (Springer, Berlin, Heidelberg, 1981). 25. Bell, K.L., Gilbody, H.B., Hughes, J.G., Kingston, A.E., and Smith, F.J., J. Phys. Chem. Ref. Data, 12, 891 (1983). 26. Huddlestone, R.H. and Leonard, S.L. (Eds.) Plasma Diagnostic Techniques

(Academic Press, New York, 1968). Maile More, R., in Atomic and Molecular Physics of Controlled Thermonuclear Fusion, ed. by Joachain, C.J. and Post, D.E. (Plenum Press, New York, 1983). 28. Lochte-Holtgreven, W. (Ed.) Plasma Diagnostics (North-Holland, Amsterdam, 1968). 29. Massey, H.S.W. Negative Ions, 3rd ed. (Cambridge University Press, Cam-

bridge, 1976). . Hotop, H. and Lineberger, W.C., J. Phys. Chem. Ref. Data, 14, 731 (1985). . Key, M.H. and Hutcheon, R.J., Adv. Atomic & Molec. Physics, 16, 201 (1980). . Barnett, C.F. and Harrison, M.F.A. (Eds.) Applied Atomic Collision Physics,

. . -

Vol. 2 (Academic Press, New York, 1984). Beutler, H.G., J. Opt. Soc. Am. 35, 311 (1945). Hutley, M.C., Diffraction Gratings (Academic Press, New York, 1982). Loewen, E.G., J. Phys. E 3, 953 (1970). Hunter, W.R., in Spectrometric Techniques IV ed. by Vanasse, G.A. (Academic

Press, New York, 1985).

Bideau-Méhu, A., Guern, Y., Abjean, R., and Johannin-Gilles, A., J. Phys. E 13, 1159 (1980). . Brigham, E.O., The Fast Fourier Transform (Prentice-Hall, New Jersey, 1974).

. Thorne, A., Physica Scripta T 65, 31 (1996).

. Dohi, T. and Suzuki, T., Appl. Optics 10, 1137 (1971). . Harlander, J., Reynolds, R.J., and Roesler, F.L., Astrophys. J. 396, 730 (1992). . Chakrabati, S., Cotton, D.M., Vickers, J.S., and Bush, B.C.. Appl. Optics 33, 2596, (1994). Nisoli, M., De Silvestri, S., Svelto, O., Szipécs, R., Ferencz, K.. Spielmann, Ch., Sartania, S., and Krausz, F., Optics Letters 22, 522 (1997). . Neumann, R., Trager, F., and zu Putlitz, G., Laser-Microwave Spectroscopy, in Progress in Atomic Spectroscopy, ed. by H.J. Beyer and H. Kleinpoppen

(Plenum Press, New York, 1987) p. 1.

5.

Hurst, G.S. and Letokhov, V.S., Physics Today 47, 38 (1994).

}. Gallagher, T.F., Rep. Progr. Phys. 51, 143 (1988). . Gallagher, T.F., Rydberg Atoms in Strong Microwave Fields, in Atoms in Intense Laser Fields, ed. by M. Gavrila (Academic Press, Boston, 1992) p. 67. . Zimmerman,

(1990).

C., Kallenbach, R., and Hansch, T.W.., Phys. Rev. Lett. 65, 571

. Horvarth, G.Z.K., Thompson, R.C., and Knight, P.L., Contemporary Physics 38, 25 (1997). Bradley, C.C. and Hulet, R.G., Laser Cooling and Trapping of Neutral Atoms, in Experimental Methods in the Physical Sciences, Vol. 29B, ed. by F.B. Dunning and R.G. Hulet (Academic Press, San Diego, 1996) p. 67. - Cohen-Tannoudji, C.N. and Phillips, W.D., Physics Today 43, 33 (1990). . Caroli, 5. (Ed.) Improved Hollow Cathode Lamps for Spectroscopy (Horwood, Chichester, 1985). Danzmann, K., Guenther, M., Fischer, J., Kock, M., and Kihne, M.., Appl. Opt. 27, 4947 (1988). Johansson, S. and Litzén, U., Physica Scripta 10; 121 (1974),

References

423

. Johansson, S. and Litzén, U., J. Phys. B: Atom. Molec. Phys. 11, L703 (1978). . Brockaert, J.A.C., Appl. Spectrosc. 49, 12A (1995). . Heise, C., Hollandt, J., Kling, R., Kock, M., and Kiihne, M., Appl. Opt. 33,

511 (1994).

. Montaser, A. and Golightly, D.W. (Eds.) Inductively Coupled Plasmas in Analytical Atomic Spectrometry (VCH, New York, 1987). Munger, C.T. and Gould, H., Phys. Rev. Lett. 57, 2927 (1987). . Seely, J.F., Ekberg, J.O., Brown, C.M., Feldman, U., Behring, W.E., Reader, J. and Richardson, M.C., Phys. Rev. Lett. 57, 2924 (1986). . Hinnov, E., Suckewer, S., Cohen, S., and Sato, K., Phys. Rev. A, 25, 2293,

(1982).

. Finkenthal, H., Bell, R.E., and Moos, H.W., J. Appl. Phys. 56, 2012 (1984).

. Wolfe, W.L. and Zissis, G.J. (Eds.) The Infrared Handbook, Environmental Research Institute of Michigan, Ann Arbor (1985). . Sweedler, J.V., Ratzlaff, K.L., and Denton, M.B. (Eds.) Charge-Transfer Devices in Spectroscopy (VCH, New York, 1994). . Blackwell, D.E., Petford, A.D., Shallis, M.J., and Simmons, G.J., Mon. Not. R. Astr. Soc. 199, 43 (1982). . Engstrém, L., Nucl. Instr. Meth. 202, 369 (1982). . Stoner, Jr., J.O. and Leavitt, J.A., Appl. Phys. Lett. 18, 477 (1971). . Schmidt, H.T., et al., Phys. Rev. Lett. 72, 1616 (1994). . Corney, A., Atomic and Laser Spectroscopy (Oxford University Press, London,

1977).

. Chamberlain, J., The Principles of Interferometric Spectroscopy (Wiley, Chichester, 1979) p. 291. . Chamberlain, J., The Principles of Interferometric Spectroscopy (Wiley, Chichester, 1979) p. 221. . Griffiths, P.R. and de Haseth, J.A., Fourier Transform Infrared Spectrometry (Wiley, Chichester, 1986) p. 33. . Cafias, A.A.D., J. Quant. Spectr. Rad. Trans. 56, 803 (1996). . Norlén, G., Phys. Scr. 8, 249 (1973). . Whaling, W., Anderson, W.H.C., Carle, M.T., Brault, J.W., and Zarem, H.A., J. Quant. Spectr. Rad. Trans. 53, 1 (1995). _ Learner, R.C.M. and Thorne, A.P., J. Opt. Soc. Am. B 5, 2045 (1988). . Nave, G., Learner, R.C.M., Thorne, A.P., and Harris, C.J., J. Opt. Soc. Am. B 8, 2028, (1991). . Nave, G., Learner, R.C.M., Murray, J., Thorne, A.P., and Brault, J.W., J. Phys. II France 2, 913 (1992). Nave, G., Johansson, S., and Thorne, A.P., J. Opt. Soc. Am. B, 14, 1035 (1997). Giachetti, A., Stanley, R.W., and Zalubas, R., J. Opt. Soc. Am. 69, 474 (1970). . Palmer, B.A. and Engleman, R., Atlas of the thorium spectrum, Rep. LA-9615 (Los Alamos National Laboratory, Los Alamos, NM, 1983). . Maki, A.G. and Wells, J.S., J. Res. NIST 97, 409 (1992).

7

ae

~~

ft

ie

har:

a) Vimo : " 7 a due? 15? coteill

:

i

— 7

a:

f

Se

Pas

Py

: Pal ws :

=

ao

Q

yg)

: Thee Sh a TULA

AT

YW ae

or ry

lt OP, publ

pe C35 oD _ >

tet

wet ee) 0

;

(GN Tate at rig Viet,

.

endl ed NCOP i © Bidet4 vim

4 BF

POI)

fea-o!

i~out

¥

|

‘bot

Si

b

— A -

ay

)

A)

et en

:

ve)

4

ervit

a.

pS

thin

has

ou

iy

ie iliett yece)

ce

OL

&f

yy

gue

2

hs te

valli aa49

bin.

wet tow

Peas

fe @

Yeah be

ol

j~

Le

ets

HOS oD

- i

ge

Fees

1

9oer.s

aia

Seis

Se wlio

; Doe

OR ets ted

sé

iheVvL

Hurd

wd

bal

‘i

AeA

4

,

MALO

Wiwleetua

hs.

Glo

(S@bwar,

po

yA

La

Ch

yet

ro

ee

senaah

pied

asl

-ee)

fate

y he

il fi

pee!

1S clea

“ele

TAP hl

tute)

Mey

Lb

yi)

th

Mad STUY

TES 8]-

oo

= ay.’

7

+?

Aoureed)

UN)

al Repl),

,

eat ch aha

$F

Medline

.

naeI>

ad? hrgrmempannng hed

rd

)

(el)

oO

rik

Cie

aes

IRR

oak)

rl

eee

ted: OS eee

ee Ge

Sede

y Oo

‘! tereks’).

Adu

bh

OL,

ee)

be 8

ainda h ieee 0)

a

dj

vA

ee

se

oor

4. 26% (pe Wisin fase ti " ee Ly#)

pec?

iv ala

‘ck

.

CPOT (Fe

7

6amtll,

ice 7 (IRE;

Frain

:

(0) sete ‘hLO:

oe

eyqewi- tte

il

’

PAE

|

|

Pies

D1;

‘s

OE

,

Rint

i>

UG

sinks, Chdske dad At

a0 (5 MCT cei¥t =

GA

ott

Boeing

jonni) 4 eas

ii-=

aA

i

ek elas

Ge wll?” ee

ie hd

bes

6

Cred

aahieete&

:

i

>.

and

ppt

4A

ate

wd

wht fein AL Slo 1.46 raked er

wal

Wl

i

hie awe? a | ores

“1s @

104908 S' 8S GA ood" iid | Gein S Allele.

GB

+

itees

¥

7 Ce

:

i

4) leeks ER

Mee GN

a

ay

a

é

Se &

eer ee

SIC Doae

6

ae

eo

1 egih ¢

wee goniOremelide, Dicer TB @, Ihaababid-ak Sad /s

ait ieee

.ceisivvel

ein?

‘ennial) anmelh eal}

CPA od A h.geat AF EE ahem Tt

)

up

2

re

Le

I

1

if

»

haIS a

edie

At

eh de Pane

i

Gi rhea

;

tit

. ij

e Walk’

\

1

ya

Y at

F

oiey 4yed

OR pian

ee

Tae

Met =

7

aa

Index

Absorption — by atmospheric gases 5, 156-158 — coefficient 170-172 — continuous 241-247 —— cross sections 244-247 —— molecular 147-148 — integrated 170,171, 173,179, 386

— probability coefficient see Einstein coefficients — spectroscopy 94, 366-370 —— sources for 367-370 — total see Equivalent width Abundance — astrophysical 187, 214, 250, 387 — isotopic 84 — of atmospheric gases 157 Adiabatic approximation 195 — breakdown of 201 Airy distribution see Intensity distribution from Fabry—Perot Aliasing 327-330 Alternation of intensities 145-146 Angstrém unit 8 Angular momentum — nuclear 79 — orbital 16 — rotational 122 — spin 22 — total 24, 40, 80, 136 Anharmonicity constant 121 Anti-Stokes lines 150-152 Antibonding orbitals 111,114 Antisymmetric function see Wave function Apodization 326 Arcs 360 — argon mini413 — argon mini-arc 361 — mercury 368 — wall-stabilized 361, 368, 385

Array detector 377-378 Astigmatism 276, 288 Autoionization 70-71 Balmer series 55 Band — origin 134,141 — strength 184 Bandwidth theorem 189, 259 Beam foil spectroscopy 363, 394-396 Beamsplitter 305, 320, 331, 334 Bibliography — oscillator strengths 381 — wavelengths 105 Binary approximation 195, 205 Binding energy 7,58, 97 Blackbody 413 — radiation 164-166 Blaze see Grating Bohr — magneton 23 — radius 18 Bolometer 375 Boltzmann — distribution 167, 223-224 — equation 167 —plot 251 Boltzmann—Saha equation 227, 252 Bonding orbitals 114 Born approximation 218, 219 Born—Oppenheimer approximation 110 Bound-free transition 241-246, see also Photoionization; Radiative recombination Branching fraction

382, 399

Bremsstrahlung 242, 248, 249, 369 Brightness 162 Brightness temperature see Temperature

426

Index

Broadening, homogeneous and inhomogeneous 192, 348 Camera

lens or mirror

Cyclotron frequency 239 Cyclotron radiation 249 Czerny—Turner mounting

291

267

Cascading 393 Cavity — blackbody 164 —laser 341 Central field approximation 33 Centrifugal constant 122 Classical oscillator 119,175 Classical path approximation 195 cm~' unit 8 Collimating lens or mirror 267 Collisional broadening see Pressure broadening Collisional cross-sections 218-220 Coma 276, 290 Compensating plate 320 Compilation — energy levels 105 — oscillator strengths 381 — wavelengths 105 Condensing lens 263 Configuration interaction 44 Configuration, electron 35 Constant deviation mounting 271 Continuum states 69 Contrast — of Fabry—Perot fringes 311 — of photographic emulsion 378 Convergence diagram 96 Conversion factors — for energy units 420 — for intensity parameters INf(eiy Leal 181, 182 Convolution 207 Convolution theorem 326 Cooley—Tukey algorithm 332 Core electrons 58 Coronal equilibrium see Equilibrium Coulour temperature see ‘Temperature

Covalent bonding 116 Cross-dispersion ~ for echelle 300 — for Fabry-Perot 313,315 Curvature of field 276 Curvature of lines ~— from Ebert—Fastie mounting — from grating 295 — from prism 276 Curve of growth 211-214, 387

291

Damping — constant 176,178,190, 205, 213, 397 = ratio, 22 Dark current 371

Data base — atmospheric molecules 157, 158 — energy levels 106, 107 — wavelengths 106, 107 Database — oscillator strengths 381, 400 De-mixing 384 Debye — radius 232-235 — sphere 235 Degeneracy 17, 23, 40,118, 183, see also Statistical weight Delayed coincidence 393 Detachment energy see Electron affinity Detailed balance 168, 217 Detectivity 372 — specific 372 Detectors 370-380 — for the infrared 373-375 — for the ultraviolet 375-377 — multichannel 377-380

— photographic Deuterium lamp Deviation,

378-380 368

minimum

271, 274

Dielectronic recombination 71,249 Diffraction — limited resolution 268 — Fraunhofer 260 Dirac = combye2 7 — notation 416 — relativistic quantum mechanics 22 Dispersion — angular 258 —— of grating 283

= Olprishi meine — linear 258 — prism materials 273 = reciprocally 258 —— of grating instrument —— of prism instrument Displaced system 61 Dissociation —energy 114

284 273

Index

see Equilibrium Doppler 191-194 — broadening — profile 192 — shift 191 — width 193 Doppler-free spectroscopy 354-356 Doubly excited state 69

— equilibrium

Eagle mounting 292 Ebert mounting 291 Ebert—Fastie mounting 291 Echelle see Grating Effective quantum number 59 Eigenfunction 14 Eigenvalue 14 Binstein coefficients 167-170, 172-175 Einstein—Milne relation 243 Blectric dipole line strength 48,172 Electric dipole moment of molecule 130-132 Blectric dipole radiation 48 Electric quadrupole moment, nuclear 81 52, 74, Electric quadrupole radiation 180 363 Electrodeless discharge Electron affinity 249 Electron pressure 228 Electron spin 22 see ‘TemperaElectron temperature ture

—-NI 65 — Nal 57 —ZrIl 67 Equilibrium 222=223 — coronal — dissociation 225 — ionization 226-229 — local thermodynamic, LTE — partial LTE 222 Pall — thermodynamic Equivalent — electrons 38 — width 172,210, 386 Etalon 309 Etendue 262 Exchange degeneracy 26 Excitation 218 — cross-section — energy 7,58 Excited levels 7

Fine structure — constant a

INIA xayil = Oa leo

Fourier — transform

helium

— hydrogen

32

6

220-222

fsum rule 180 fvalue see Oscillator strength Fabry-Perot interferometer 305-319 — photographic 317 — scanning 318 see Multiplex Felgett advantage advantage 332 FFT (fast Fourier transform) Field — ionization 353 — shift see Volume shift

Electron volt unit 8 Electronic energy of diatomic molecules 111-118 Electronic spectra of diatomic molecules 136-145 Electrostatic integral — direct 30 — exchange 30 Emission f value 179 241-243, Emission, continuous 247-249 — molecular 147 — of hydrogen 148 — of inert gases 148 Emissivity 165, 187 163 Energy density of radiation Energy level diagram

—

427

60 — half-filled shell 76 — hydrogen 25 — inverted 76 — isoelectronic 73 — normal 76 — two-electron 28 Finesse 313 Flames 360 Flash 367 — photolysis and pyrolysis — tubes 369 Flatness of interferometer plates Flux density 161 Forbidden

lines

— Stark induced

316

52, 74, 222, 366, 394

91, 206

322 —~— spectroscopy 320-337 — spatially heterodyned 335

428

Index

Fractional parentage 67 Franck—Condon — factor 145 — principle 143-145 Fraunhofer — Aline 157 — Dlines 56 — diffraction 260 Free spectral range — echelle grating 299 — Fabry-Perot 312 — Fourier transform spectroscopy Free-free transition 242, 246, 247 Frequency — doubling and mixing 346-347 — standard 410 FWHM — collisional 200 — definition 188 — Doppler (Gaussian) 192 — Fabry-Perot fringes 312 — natural 191 — of diffraction distribution 268 — of Fourier transform instrument function 325 g, statistical weight 40 gr, nuclear g factor 79 gs, Landé g factor 86 Js, Syromagnetic ratio of electron y-ray region 4 Gaunt factor 244-246 Gaussian profile 192,211 Gerade and ungerade states 112 gf value see Oscillator strength Ghost

Gyromagnetic ratio

Hamilton operator — central field approximation — diatomic molecule 109 —many-electron 32 — one-electron 14 — two-electron 25 Hanle effect 396-398 Hartree 36 ——(unaiiy

329

- blazed — concave - echelle

296 286-288 298-300

grazing incidence 301 holographic 297 - interference 297 replica 296 - ruled 295 Grazing incidence mounting Grey body 166 Ground state 7

49,

— energy levels

23

34

Hartree-Fock calculation 36 Heitler-—London method 111 HITRAN = 157, 158 Hole-burning 349 Hollow cathode 362 Holtsmark approximation 201 Honl-London factors 182 Hook method 387-391 Hund’s — coupling cases 136-137 — rule 39 Hydrogen — fine structure — line series 55 — molecular ion

— negative ion — Stark effect Hydrogen-like Hydrogenic

16 24 112-115

249 91-92 55

— term values 59 — wave functions 17 Hyperfine structure (9=82

Image defects

= IES) Bes! — grating 295 Globar 368 Glow discharge 362-363 Grain of photographic emulsion Grating

23,85

379

— grating instruments 295 — prism instruments 276-277 Impact — approximation 195, 198-201 — parameter 198 Induced emission see Stimulated emission

Inductively coupled plasma (ICP) Infrared region 4 Inglis—Teller limit 240 Inner-shell excitation 61, 69-71 Instrument function — definition 258 302

— Fabry—Perot

312 SANs) pul ~ prism and grating Intensity 162

260

363

Index

Intensity distribution — from Fabry—Perot 310-313 — from FTS instrument 324 — from grating 279 — from slit 268 see Radiometric Intensity standards standards Intercombination line 51 Interferogram 320-322 Interferometer — Fabry—Perot 305 — Mach-Zehnder 387 — Michelson 305 Inverted term 76 Ion trap 357 Ionic bonding 116 Ionization — energy 7,68, 75 —— depression of 240 — limit 7,103 — potential 9 see EquilibIonization equilibrium rium Tons

— highly charged 71-74 — negative 249 Irradiance 162 55, T1=74, 97 Isoelectronic sequence Isoionic comparison 97 Isotope separation 353 Isotope structure 82-84 Isotropic radiation 162 Jaquinot advantage advantage jj coupling 77 jK coupling 78-79

see Throughput

Kayser unit 8 Kirchhoff’s law 215 Kuhn model for pressure broadening 202 Lamb dip 350 Lamb shift 24, 356 A doubling 118 Landé — g factor 86 — interval rule Laser

— — — —

41,81

cooling 357 mode structure 342 spectral resolution 349 stabilization of 350

429

— temporal resolution 350 Laser-induced fluorescence 392 Laser-produced plasma 364, 369 Lasers — diode 344 dye 344 — excimer 345 — F-centre 345 — fixed frequency 344 — fixed-frequency 343 — helium—neon 340

— ion (argon and krypton)

343

— neodymium 343 — nitrogen 343 —ruby 340 — Ti-sapphire 345 — tunable 344-346 Lennard—Jones potential see Potential Lens aberrations 276 Level search 101 Lifetime — measurement 391-398 — of excited state 189,190, 391, 398 Line strength 48, 172-175 — conversion factors for 173,174, 181, 182 Linear combination of atomic orbitals (LCAO) 111, 112 Littrow mounting — grating 290 — prism 270 Lorentzian profile 190,211 LS coupling 36, 76 — deviation from 42 LS notation 38 LTE see Equilibrium Luminance 163 Lyman =a od — series

55

Magnetic dipole radiation Magnetic moment — classical 23 — electronic 23 — nuclear 79 — total atomic 85 Magnetic sublevels 87 Maser 339 Mass shift — normal

83

— specific 83 Matrix element

416

52, 74, 180

430

Index

— diagonal 29 — non-diagonal 42 Matrix method 99 Maxwell velocity distribution 192, 219, 224-225 Metastable state 52 Metre unit 409 Michelson interferometer 305-309, 320, 330-332 Microwave region 4 Millikayser, mk 8 Milne relation 243 Missing rotational lines 145-146 Molecular orbital 111 Moment of inertia of molecules 121, 153 Monochromator 267 Morse potential see Potential Multiphoton excitation 352 Multiplet 51 Multiplex advantage 333, 406 Multiplex spectroscopy 322 Multiplicity 38 Natural broadening Nernst glower 368 Noise 404 — detector 371,405 — instrumental 405 — photon 405 — shot 405 — source

— white

189-191

404

333

Noise equivalent power (NEP) 372 Normal incidence mounting 292 Normal mode of vibration 125 Normalization of wave function 20 Nuclear spin 79 Nyquist see Sampling theorem Optical depth 170, 209-211 Optical path difference 259, 275, 305, 320

Optical pumping 347 Optically thick 171, 248 Optically thin

188, 209

Orbital 34 Order number — echelle 299 Fabry—Perot 307 - grating 279 Oscillator strength 175-179 — absolute 383

— conversion factors for 173, 174, 181, 182 — in diatomic molecules 181-186 —— band 185 —— electronic 186 —— rotational line 185 — relative 382 Overlap integral 132 Overlapping orders 299, 313 P branch 135 Pair coupling 78 Parent term

60, 64

Parity 18, 35,38, 49 Partition function 223-224

— divergence of

240

— translational 225 Paschen—Back effect

89 Paschen—Runge mounting Pauli exclusion principle

Penetration

292 27

34,58

Perturbation theory Phase shift method Photoconductors Photodiodes 373

415-418 ~393 373

Photographic emulsions

378

Photoionization 229, 241-243 — detectors 376 Photomultipliers 375 Photon noise see Noise

Physical constants 419 ™ component 87 Pinch discharge 365 Pixel ps7 Planck distribution Plasma

164

— frequency 235-237 — oscillations 235-239 Plasma, definition of 232 Plate factor 258 Plunging configuration 74, 97 Pockels cell 350 Polarizability — in scattering 150 — in Stark effect 89 — non-linear 346 Polarization formula 96, 104 Population inversion 169, 340, 341 Potential — Coulomb 14 — Lennard—Jones 198 — Morse 118 — van der Waals 197

Index

17, 85 Precession Predictions — parametric 99 — semiempirical 95 — theoretical 97 Predissociation 149-150 Pressure broadening 194-206 Prism materials 273 Probability density 19 Progression of molecular bands Promotion energy 97 Pyroelectric detectors 375

141

Radiance 162 Radiative atomic cross section 171 Radiative recombination 228, 242 Radiative transfer 214-216 Radiometric standards 413-414 Radiowave region 4 Raman spectroscopy 150-153 Ramsey fringes, optical 356 Rayleigh criterion 261 Rayleigh scattering 150 Rayleigh—Jeans formula 165 Reciprocity law 379 Reduced

Q branch 139 Q switching 351 QDT see Quantum defect theory QED © see Quantum electrodynamics Quantum — defect 59 — defect theory 58 — efficiency 376 — electrodynamics 24 — number — F 80 —f[ 79 = J) 740 =17 24 — J (rotational) 122 — K (jK coupling) 78 — K (molecular) 137 —L 37 —l 17 — A 116 = A adald — Mr 80 — M; 40 =m, 24 — Mr, 37 —

Mi

AG

— Ms 37 = Ms_—22 —N 137 —n 17 — (2 116 — 9 37 = 22 = DF = 116 = 120

mass

14, 82

Reflectivity 309, 315, 316 — of Fabry—Perot plates — of grating 301 — of laser mirrors 341 — of Michelson beamsplitter 331 Refractive index 175-177, 237 — complex 175 —ofair 8 — of prism materials 273 Relativistic effects 23,24 see Aliasing Replication of spectrum Resolution limit 259 — Fabry-Perot 314 — FTS 325 — grating 284 Resolving power — definition 258 — diffraction limited 260, 263 —echelle 298 — Fabry-Perot 314 = FITS 325 — grating 284 — prism 274 Resonance — broadening 197 — line 56,75 Response curve, photographic 378 Responsivity 372 Retroreflektor 331 Ritz — combination principle 6, 93 — series formula 59, 104 — wavelengths 6, 107, 409

Quasimolecule 197 Quasistatic approximation

201-204

R branch 135 Radial probability density

21

431

Rotational — constant 122 — energy of diatomic molecules 122-123 — spectra of diatomic molecules 132-133 Rowland circle 288

432

Index

Russell-Saunders coupling see LS coupling Rydberg — constant 16 — constant, measurement of 356 — formula 6,57 — states 354 =unit 9 Saha’s equation 226-228 — for negative ions 249 Sampling 327-330 — errors 333 — interval 328 — theorem 328 Satellite 51,204 — branch 145 Saturation 216,348, 355 Schrodinger equation — time dependent 45 — time independent 14 Schumann—Runge band 140, 146, 157 Schwarzchild factor 379 Screening 34,58, 71 Selection rules — for LS coupling 49 — for electric dipole radiation 49 — for electronic transitions in molecules 138-140 — for hyperfine structure 80 — for magnetic dipole and electric quadrupole radiation 52 ~ for Raman scattering 152 — for rotational transitions — for vibrational transitions

— in magnetic field

87

Selective excitation

353,392

130,132 131, 134

Self-absorption see Radiative transfer Self-consistent field 36 Self-reversal 217 Seniority 39,67 Series - alkali metals 57-60 ~ in hydrogen 55 = limit 104 — limit, depressed 240 - perturbed 61,104 Series formula 95, see also Ritz formula; Polarization formula Seya mounting 294 Shah function see Dirac comb Shell 35 Shell structure 35

Shift of spectral lines

90,191,195,

196, 198, 200 Shock tubes 364 ao component 87 Signal-to-noise ratio (SNR) Sirks focus 289 Slater integral 37 Slit width 268-270 Solar corona 52, 222, 235 Source function 215 Spark sources 361-362 Spectral region 4 Spectrograph 267 Spectrometer 267 Spectroscope 267 Spectroscopic constants Spherical harmonics

404-407

420 16, 20, 122

Spherical polar coordinates 14 Spin see Angular momentum Spin function 27 Spin-orbit energy 23,33, 40 Spin-orbit integral 41

Splitting factor ¢,,

60

Spontaneous emission Spurious coincidence Stark — broadening 196 — effect —— linear 91 —— quadratic 89

168, 340 102

State sum see Partition function Stationary state 14 Statistical weight 40, 81, 146, 167, 181 — of free particle 225 Stepwise excitation 352,391 Stimulated emission 168,339 Stokes lines 150 Se are Subconfiguration 64, 97 Subshell 35 Sum rule for LS multiplets 51 Suppressed orders 318 Synchrotron radiation 369 ‘Temperature — blackbody

— — — = — — =

164 brightness 166 colour 166 électron 221,239 (excltation 992k gas kinetic 239, 251 ionization 252 reversal), 253

Index

— spectroscopic

measurement

of

250-253 Term 6,38 Term — analysis 6,93 — value

7,58

Thermodynamic equilibrium — see Equilibrium Thermopile 374 Throughput 162, 262 Throughput advantage 263, 309, 333 Transition probability 46, 48 — accuracy of 382 — bibliography of see Bibilography — calculation of 381 — conversion factors for see Conversion factors 19S 32 — diatomic molecules — measurement of —— by dispersion 387-391 —— in absorption 385-387 —— inemission 383-385 — two-photon transition 151 Two-photon excitation 352

Ultraviolet region 4 Undersampling 329 Unntses—9 Vector model

17,85, 136

Velocity of light 410 Vibration—rotation spectra of diatomic molecules 134-136 Vibrational — constant

120

120 — frequency, classical 119 Visibility of interference fringes Visible region 4 Voigt profile 207-208 Volume shift 84

433

—energy

332

Wadsworth mounting — grating 293 = jor

27

Wave function — angular 16 — antisymmetric 27,37 — hydrogenic 17 — molecular 110 — radial 16 — symmetric

27

Wavelength —inair 8 —in vacuum

— standards Wavenumber

6,8

408-411 6

Weisskopf radius 199-201 Wien’s — approximation 165 — displacement law 164 Wood’s anomalies 297 X unit 8 X-ray region

4

Zeeman — effect 85-89 — splitting see Magnetic sublevels Zero-point energy 120

Computer to Film: Saladruck, Berlin Binding: Buchbinderei Liideritz & Bauer, Berlin

Hay Ls

A. Thorne:U,Litzen-5. Johansson

Spectrophysies

‘Thettext is suireees at fi Ist: degreeSedans and ateases starting = research iinastrophysics, plasma oratmospheric physics, or spectro- _ chemical analysis. It should gn beuseful as. a quick reference to etablshed pet elds. in thesefi researchers |

ISBN 3-540-65117-9

19

7835406500978