Introduction to Photoelectron Angular Distributions: Theory and Applications (Springer Tracts in Modern Physics, 286) 3031080262, 9783031080265

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Springer Tracts in Modern Physics 286

V. T. Davis

Introduction to Photoelectron Angular Distributions Theory and Applications

Springer Tracts in Modern Physics Volume 286

Series Editors Mishkatul Bhattacharya, Rochester Institute of Technology, Rochester, NY, USA Yan Chen, Department of Physics, Fudan University, Shanghai, China Atsushi Fujimori, Department of Physics, University of Tokyo, Tokyo, Japan Mathias Getzlaff, Institute of Applied Physics, University of Düsseldorf, Düsseldorf, Nordrhein-Westfalen, Germany Thomas Mannel, Emmy Noether Campus, Universität Siegen, Siegen, NordrheinWestfalen, Germany Eduardo Mucciolo, Department of Physics, University of Central Florida, Orlando, FL, USA William C. Stwalley, Department of Physics, University of Connecticut, Storrs, USA Jianke Yang, Department of Mathematics and Statistics, University of Vermont, Burlington, VT, USA

Springer Tracts in Modern Physics provides comprehensive and critical reviews of topics of current interest in physics. The following fields are emphasized: – Particle and Nuclear Physics – Condensed Matter Physics – Light Matter Interaction – Atomic and Molecular Physics Suitable reviews of other fields can also be accepted. The Editors encourage prospective authors to correspond with them in advance of submitting a manuscript. For reviews of topics belonging to the above mentioned fields, they should address the responsible Editor as listed in “Contact the Editors”.

V. T. Davis

Introduction to Photoelectron Angular Distributions Theory and Applications

V. T. Davis University of Nevada, Reno Reno, NV, USA

ISSN 0081-3869 ISSN 1615-0430 (electronic) Springer Tracts in Modern Physics ISBN 978-3-031-08026-5 ISBN 978-3-031-08027-2 (eBook) https://doi.org/10.1007/978-3-031-08027-2 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022, Corrected Publication 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, 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. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

For my parents

Acknowledgments

The author gratefully acknowledges the help of Dr. Kiattichart Chartkunchand, Professor Till Jahnke, and Professor Joshua Williams, all of whom contributed directly to the preparation of this manuscript by providing the results of their research. Professor Till Jahnke and Professor Joshua Williams further assisted by graciously inviting me to participate as a member of their research teams. Special thanks go to Professor Jeff Thompson, who first suggested this project, and to Professor Aaron Covington who provided support along the way. In addition, Professor Jeff Thompson and Professor Aaron Covington have allowed me to participate as a member of their research teams for over 20 years now. I would also like to thank Nemul Khan, Dinesh Vinayagam, Dr. Ute Heuser, and Dr. Sam Harrison of the editorial team for helping to bring this book to fruition. Finally, most of the credit for any success that I may have goes to my wife, Anne, who has been at my side for 40 years and counting.

vii

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

2

Angular Momentum in Quantum Mechanics . . . . . . . . . . . . . . . . . 2.1 Commutation Relations of Angular Momentum Operators . . . . . 2.2 Construction of Eigenstates and the Spectrum of Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Matrix Elements of Angular Momentum Operators . . . . . . . . . . 2.4 Orbital Angular Momentum and the Spherical Harmonics . . . . . 2.5 The Addition Theorem for Spherical Harmonics . . . . . . . . . . . . 2.6 Rotations in Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . 2.7 Matrix Elements of the Rotation Operators . . . . . . . . . . . . . . . . 2.8 The Coupling of Two Angular Momenta . . . . . . . . . . . . . . . . . 2.9 The Clebsch-Gordan Coefficients . . . . . . . . . . . . . . . . . . . . . . . 2.10 The Clebsch-Gordan Series . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.11 The Coupling of Three Angular Momenta . . . . . . . . . . . . . . . . 2.12 Spherical Tensor Operators and the Wigner-Eckart Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3 5 7 9 18 21 25 28 30 39 43 46

3

Classical Model of Photoelectron Angular Distributions . . . . . . . . .

61

4

Quantum Treatment of Photoelectron Angular Distributions (Dipole Approximation) . . . . . . . . . . . . . . . . . . . . . . .

77

Higher-Order Multipole Terms in Photoelectron Angular Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

97

6

Relativistic Theory of Photoelectron Angular Distributions . . . . . .

117

7

Angular Momentum Transfer Theory . . . . . . . . . . . . . . . . . . . . . .

153

5

ix

x

Contents

8

Molecular Photoelectron Angular Distributions . . . . . . . . . . . . . . .

189

9

Measuring Photoelectron Angular Distributions in the Laboratory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

203

Applications of Photoelectron Angular Distribution Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

227

Correction to: Introduction to Photoelectron Angular Distributions . . .

C1

Appendixes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

241

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

327

10

Chapter 1

Introduction

The interaction of electromagnetic radiation with matter provides the primary physical process by which physicists and chemists study atoms and molecules. In some cases, the absorption of a photon (or photons) by atoms or molecules leads to bound-free transitions in which one or more electrons (“photoelectrons”) are emitted in a photodissociation (pd) reaction. In photoelectron spectroscopy experiments, these photoelectrons are collected and analyzed. Analysis of collected photoelectrons provides information on fundamental atomic and molecular properties such as electronic structure and photoionization (or photodetachment) cross-sections. As it turns out, photoelectrons are not emitted randomly but instead in particular directions determined by the nature and dynamics of the material under study, and these directions are calculable by employing the appropriate physical theory. Photoelectron angular distribution (PAD) measurements are important because they can shed light on possible atomic and molecular electronic configurations and system dynamics that are not obtainable from total cross-sections. The measurement of differential cross-sections in particular can provide information not only on the magnitude of quantum transition amplitudes but also on their relative phases, as well as information on the nature of the interaction between light and matter. For example, recent measurements of molecular-frame photoelectron angular distributions (MFPADs) have been used to extract photoelectron emission delays in the attosecond range, delays which can provide ultrasensitive, time-dependent maps of molecular potentials. Also, photoelectron angular distribution measurements are particularly useful in the study of negative ions. The masking of the nuclear-electron interaction due to increased screening in negative ions elevates short-range electron-electron correlation forces to a dominant position within these ions, forces without which (for

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 V. T. Davis, Introduction to Photoelectron Angular Distributions, Springer Tracts in Modern Physics 286, https://doi.org/10.1007/978-3-031-08027-2_1

1

2

1 Introduction

example) atomic anions could not exist. PAD measurements, being particularly sensitive to these types of many-body forces, are an essential tool for understanding these systems. Before engaging in photoelectron angular distribution measurements, experimentalists should understand the basic analytical models involved because they reveal the conservation laws, dynamical processes, and geometrical effects that ultimately determine the shape of the measured angular distributions. It is often found that, after laborious computations, relatively simple expressions remain. This is no accident, as these simple expressions reveal deeper properties of the underlying physical processes involved in atomic or molecular photoionization (or photodetachment) under rotational and inversion symmetry [1]. The various physical influences on photoelectron angular distributions are revealed in analytical models primarily through the use of angular momentum coupling algebra and the mathematics of spherical tensor operators, as will be amply demonstrated. Key derivations are presented in (sometimes gruesome) detail in hopes of increasing the understanding of ultimate results. This manuscript breaks no new ground but instead attempts only to arrange the relevant information in such a way as to make these important theories more accessible (at an introductory level) to those engaged in experimental photoelectron spectroscopy studies in the laboratory. The information contained herein may also be useful to those interested in deepening their knowledge of the interaction of atoms and molecules with light, so this book may serve as a useful supplement in a standard course on atomic and molecular physics. In addition to the basic theory of PADs, common laboratory techniques used to measure PADs and areas of current research are briefly described.

Chapter 2

Angular Momentum in Quantum Mechanics

Since angular momentum plays a prominent role in the dynamics of photon-atom/ molecule interactions, a brief review of the formalism of quantum angular momentum is warranted. To be clear, angular momentum in quantum mechanics is a vast topic about which entire tomes can be (and have been) written. Here we concentrate on those aspects of quantum angular momentum that are most directly applicable to an examination of the theory of photoelectron angular distributions. We develop all our results from first principles starting with the definition of angular momentum operators in quantum mechanics.

2.1

Commutation Relations of Angular Momentum Operators

In standard quantum theory, angular momentum operators are vector operators whose Cartesian components are the Hermitian operators Jx, Jy, Jz, all of which obey the commutation relations:


X  J l , J j ¼ iħ εljk J k ; l, j, k ¼ x, y, z

ð2:1Þ

k

The operator for the square of the total angular momentum is

Primary references for this chapter: [2–9] The original version of this chapter was revised. The correction to this chapter is available at https://doi.org/10.1007/978-3-031-08027-2_11 © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022, Corrected Publication 2023 V. T. Davis, Introduction to Photoelectron Angular Distributions, Springer Tracts in Modern Physics 286, https://doi.org/10.1007/978-3-031-08027-2_2

3

4

2

Angular Momentum in Quantum Mechanics

J 2 ¼ J 2x þ J 2y þ J 2z

ð2:2Þ

This operator commutes with any of the component operators Ji. The proof is simple:
 P

  PP
J i , J j J j þ J j J i , J j ¼ iħ εijm J m J j þ J j J m ; Ji, J2 ¼ m

j

j

i, j, m ¼ x, y, z

ð2:3Þ

where (2.1) and the commutator relation: ½A, BC  ¼ B½A, C  þ ½A, BC

ð2:4Þ

were used. The Levi-Civita tensor εijm is defined in Problem 3.8. In (2.3), i, j, m must be different from one another, else εijm ¼ 0. If they are all different, then the sums over m and j will generate an even number of terms which will subtract in pairs due to the antisymmetric nature of εijm. Therefore,


 Ji, J2 ¼ 0

ð2:5Þ

which completes the proof. It will prove convenient to introduce the so-called ladder operators: J  ¼ J x  iJ y

ð2:6Þ

Note that these two operators are not Hermitian, although they are Hermitian conjugates of each other. We can now construct some additional commutation relations, as shown in the following example: Example 2.1 Evaluate the commutators [Jz, J] and [J+, J]

 ½J z , J   ¼ J z , J x  iJ y ¼ ½J z , J x   i J z , J y ¼ iħJ y  ðiÞ2 ħð1ÞJ x ) ½J z , J   ¼ ħJ 

ð2:7Þ


 ½J þ , J   ¼ J x þ iJ y ,
J x  iJy

 ¼ ½J
x , J x  þ  iJ
y , J x þ  J x , iJ y þ iJ y , iJ y ¼ i J y , J x  i J x , J y ) ½J þ , J   ¼ 2ħJ z

ð2:8Þ █

2.2

Construction of Eigenstates and the Spectrum of Eigenvalues

2.2

5

Construction of Eigenstates and the Spectrum of Eigenvalues

The ladder operators are useful in finding alternate expressions for J2:   
 J  J  ¼ J x  iJ y J x  iJ y ¼ J 2x  iJ y J x  iJ x J y þ J 2y ¼ J 2x  i J y , J x þ J 2y ¼ J 2x þ J 2y  ħJ z

ð2:9Þ

) J  J  þ J 2z  ħJ z ¼ J 2x þ J 2y þ J 2z ) J 2 ¼ J  J  þ J 2z  ħJ z

The fact that the square of the angular momentum commutes with each of the Cartesian angular momentum components [c.f. (2.5)] means that we can choose a basis in which we simultaneously diagonalize the matrix representations of J2 and any one component Ji. By convention, we choose the component Jz. Thus, we can construct the states |j, mi which are simultaneously eigenstates of J2 and Jz. Let J 2 j j, mi ¼ ħ2 λj j, mi

ð2:10aÞ

J z j j, mi ¼ ħmj j, mi

ð2:10bÞ

where j, m label the simultaneous eigenstates of J2 and Jz, and the factors of ħ2 and ħ are chosen by appealing to dimensional analysis. Because the jj, mi are the eigenfunctions of Hermitian operators, they form a complete set. We also assume that we can make the jj, mi orthonormal. Note the inequality 



 λħ2 ¼ j, m J 2 jj, m ¼ j, m J 2x j j, m þ hj, mjJ 2y j j, mi þ j, m J 2z j j, mi  2 ¼ kJ x j j, mik2 þ J y j j, mi þ ħ2 m2  0

ð2:11Þ

The first two terms on the RHS of (2.11) are positive or zero. Hence, λ  m2

ð2:12Þ

Consider the state (Jjj, mi) acted on by the operator Jz, J z J  j j, mi ¼ ð½J z , J   þ J  J z Þj j, mi ¼ ðħJ  þ J  ħmÞj j, mi ¼ ħðm  1ÞJ  j j, mi

ð2:13Þ

where (2.7) and (2.10b) were used. It is clear that the state (Jjj, mi) is an eigenfunction of Jz with eigenvalues ħ(m  1). Also, since (2.5) implies that J2 commutes with the ladder operators, we have

6

2

Angular Momentum in Quantum Mechanics

J 2 J  j j, mi ¼ J  J 2 j j, mi ¼ ħ2 λJ  j j, mi

ð2:14Þ

where (2.10a) was used. Equation (2.14) tells us that the state (Jjj, mi) is an eigenfunction of J2 with eigenvalues ħ2λ. The fact that the (Jjj, mi) are simultaneous eigenfunctions of Jz and J2, combined with the structure of (2.10b) and (2.13) allow us to infer that J  j j, mi ¼ C  j j, m  1i

ð2:15Þ

where C are constants that depend on the quantum numbers j and m and where we have assumed that the eigenfunctions jj, m  1i are normalized. Clearly, the ladder operators J act on the states jj, mi to raise or lower the value of m by one while leaving the value of j unchanged. Thus, (2.12) and (2.15) tell us that, for a given value of j, we must have a finite sequence of values of m separated by integers and bounded by a minimum value mmin and a maximum value mmax. We must be able to reach jj, mmaxi by applying J+ to jj, mmini a finite number of (integer) times or, conversely, we must be able to reach jj, mmini by applying J to jj, mmaxi a finite number of times. Physically, we expect mmax ¼  mmin by the symmetry between z and -z, but we can also show this explicitly. To do this, first note that we can expect J þ j j, mmax i ¼ J  j j, mmin i ¼ 0

ð2:16Þ

Then, using (2.9), 

J ¼ JJ þ 2

J 2z

+

 λ ¼ mmax ðmmax þ 1Þ  ħJ z j, m max ) λ ¼ mmin ðmmax þ 1Þ min

) m2max þ mmax  m2min þ mmin ¼ 0

) mmax

 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ¼ 1  1 þ 4m2min  4mmin 2 ( mmin  1 1 ¼ ½1  ð2mmin  1Þ ¼ 2 mmin

ð2:17Þ

The upper result leads to a contradiction, leaving us to conclude mmax ¼ mmin

ð2:18Þ

Because successive values of m differ by integer units, the quantity mmax  mmin is a positive definite integer which we set equal to 2j, where j is a positive integer or half-integer. So, from (2.18), we now have

2.3

Matrix Elements of Angular Momentum Operators

mmax  mmin ¼ 2j mmax þ mmin ¼ 0

)

7

mmax ¼ j mmin ¼ j

ð2:19Þ

and if we let mmax ¼ j, then (for a given value of j) the m-values run in a finite sequence as m max ,  mmax þ 1, ⋯, mmax  1, mmax , j,  j þ 1, ⋯, j  1, j

ð2:20Þ

Substitution of mmax ¼ j into (2.17) also leads to the result λ ¼ jðj þ 1Þ

ð2:21Þ

1 3 5 J 2 j j, mi ¼ ħ2 jðj þ 1Þj j, mi; j ¼ , 1, , 2, , ⋯ 2 2 2

ð2:22aÞ

J z j j, mi ¼ ħmj j, mi; m ¼ j,  j þ 1, ⋯, j  1, j

ð2:22bÞ

To summarize,

Finally, we see that there are 2j þ 1 values of m (m is the projection of the angular momentum j onto the z-axis) for a given value of j.

2.3

Matrix Elements of Angular Momentum Operators

To find the matrix elements of the angular momentum operators, first recall that the ladder operators are Hermitian conjugates and again use (2.9) 

 h j, m J 2 ¼ J  J þ þ J 2z þ ħJ z jj, m ) ħ2 jðj þ 1Þ ¼ jC þ j2 þ ħ2 m2 þ ħ2 m ) jC þ j2 ¼ ħ2 ½jðj þ 1Þ  mðm þ 1Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ) C þ ¼ eiφþ ħ jðj þ 1Þ  mðm þ 1Þ ¼ ħ jðj þ 1Þ  mðm þ 1Þ

ð2:23Þ

where we used (2.15) and (2.22a and 2.22b) and adopted a phase convention of eiφþ ¼ 1. We also used the fact that the jj, mi are normalized. In a similar fashion, we would also find jC  j2 ¼ ħ2 ½jðj þ 1Þ  mðm  1Þ

ð2:24Þ

Now consider ( 2

jC  j ¼ h j, mjJ  J  j j, mi ) But at the same time,

Cþ ðj, mÞ ¼ eiφþ ħ½ jðj þ 1Þ  mðm þ 1Þ1=2 C ðj, mÞ ¼ eiφ ħ½ jðj þ 1Þ  mðm  1Þ1=2

ð2:25Þ

8

2

Angular Momentum in Quantum Mechanics

h j, mjJ þ J  jj, mi ¼ ½C ðj, mÞ ½C þ ðj, m  1Þ i h ih ¼ eiφ ħ½jðj þ 1Þ  mðm  1Þ1=2 eiφþ ħ½jðj þ 1Þ  ðm  1Þm1=2 ¼ eiðφþ þφ Þ ħ2 ½jðj þ 1Þ  mðm  1Þ ð2:26aÞ Or, alternatively, ðh j, mjJ þ ÞðJ  j j, miÞ ¼ ½C ðj, mÞ  ½C  ðj, mÞ ¼ jC  j2 ¼ ħ2 ½jðj þ 1Þ  mðm  1Þ ð2:26bÞ We have already chosen the phase convention φ+ ¼ 0. Equating (2.26a) and (2.26b) forces us to conclude that, with the chosen phase convention, we must have φ ¼ 0 as well. Thus, C ¼ ħ½jðj þ 1Þ  mðm  1Þ1=2

ð2:27Þ

J  j j, mi ¼ ħ½jðj þ 1Þ  mðm  1Þ1=2 j j, m  1i

ð2:28Þ

And so,

The matrix elements for our three operators Jx, Jy, Jz (for a given value of j in the basis in which Jz is diagonal) are summarized below: h j, m0 jJ z j j, mi ¼ ħmδm0 m

ð2:29aÞ

1 h j; m0 jJ x j j; mi ¼ h j; m0 j ðJ þ þ J  Þj j; mi 2 i pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ħhpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jðj þ 1Þ  mðm þ 1Þδm0 ,mþ1 þ jðj þ 1Þ  mðm  1Þδm0 ,m1 ¼ 2 ð2:29bÞ i h j; m0 jJ y j j; mi ¼ h j; m0 j ðJ   J þ Þj j; mi 2 i pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi iħ h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  jðj þ 1Þ  mðm þ 1Þδm0 ,mþ1 þ jðj þ 1Þ  mðm  1Þδm0 ,m1 ¼ 2 ð2:29cÞ The Kronecker delta δm0 m is defined in Problem 3.8. Example 2.2 Using (2.29), construct the matrix representations for the operators Jx, Jy, Jz for angular momentum j ¼ 1. For angular momentum j ¼ 1, (2.20) tells us that the possible values for the z-projection of the angular momentum range from 1 to 1 in integer steps.

2.4

Orbital Angular Momentum and the Spherical Harmonics

9

Therefore, m0, m ¼  1, 0, 1. The matrix representations for the angular momentum operators for angular momentum j ¼ 1 are thus represented by 3x3 matrices. From (2.29b) and (2.29c), we see that the Kronecker deltas allow for nonzero elements only on the off-diagonals in the matrix representation of Jx and Jy. Filling in the numbers, 0

0 ħ B pffiffiffi Jx ¼ @ 2 2 0

pffiffiffi 2 0 pffiffiffi 2

1 0 pffiffiffi C 2A

0

pffiffiffi 0 i 2 ħ B pffiffiffi Jy ¼ @ i 2 0 2 pffiffiffi 0 i 2

0

1 0 pffiffiffi C i 2 A 0

As for the matrix representation of Jz, (2.29a) allows for nonzero elements only along the diagonal (as expected) 0

1 B J z ¼ ħ@ 0

0 0

0

0

1 0 C 0 A 1 █

2.4

Orbital Angular Momentum and the Spherical Harmonics !

The orbital angular momentum operator L in quantum mechanics is defined by the relation !

!

!

L¼ rp

ð2:30Þ

!

where r is the single-particle position operator in three-dimensional coordinate ! ! ! ! space and p is the single-particle momentum operator. L , r , and p are vector operators whose (orthogonal) components are Hermitian operators. The components of the position and momentum operators obey the fundamental commutation relations:
 ð2:31aÞ r i , pj ¼ iħδij ; i, j ¼ x, y, z

 ð2:31bÞ r i , r j ¼ pi , pj ¼ 0; i, j ¼ x, y, z In the position representation, the momentum operator is given by !

!

p ¼ iħ∇

ð2:32Þ

10

2

Angular Momentum in Quantum Mechanics

Using (2.30) and (2.31a and 2.31b), it is easy to show that the components of the orbital angular momentum operator obey the following commutation relations:


X  Ll , r j ¼ iħ εljk r k ; l, j, k ¼ x, y, z

ð2:33aÞ

k

X
 Ll , pj ¼ iħ εljk pk ; l, j, k ¼ x, y, z

ð2:33bÞ

k




X  Ll , Lj ¼ iħ εljk Lk ; l, j, k ¼ x, y, z

ð2:33cÞ

k

Example 2.3 Using (2.30) and (2.31a and 2.31b), evaluate the commutators [Lx, Ly], [Lx, y], and [Lz, px].


       Lx , Ly ¼ ypz  zpy zpx  xpz  zpx  xpz ypz  zpy      
 ¼ xpy zpz  pz z  ypx zpz  pz z ¼ xpy  ypx z, pz   ¼ iħ xpy  ypx ¼ iħLz    
 ½Lx , y ¼ ypz  zpy y  y ypz  zpy ¼ z y, py  ½y, ypz ¼ iħz     ½Lz , px  ¼ xpy  ypx px  px xpy  ypx ¼ y½px , px   ½px , xpy ¼ iħpy where (2.4) was also used. █ Because the components of the orbital angular momentum operator obey (2.33c), ! we are justified in identifying L as an angular momentum operator [c.f. (2.1)]. As such, we can immediately write the following results: L2 jl, mi ¼ ħ2 lðl þ 1Þjl, mi

ð2:34aÞ

Lz jl, mi ¼ ħmjl, mi

ð2:34bÞ

L jl, mi ¼ ħ½lðl þ 1Þ  mðm  1Þ1=2 jl, m  1i

ð2:34cÞ

where the jl, mi are the simultaneous eigenkets of the operators L2 and Lz with the indicated eigenvalues. Many physical systems are conveniently described in spherical coordinates. In spherical coordinates, the unit vector b n designates a direction in space that is specified by the spherical angles (θ, ϕ). If we are going to be operating in threedimensional coordinate space using spherical coordinates, we will need to define the direction eigenkets jb ni. We can then project the jl, mi into the coordinate representation as follows: njl, mi Y lm ðθ, ϕÞ hb

ð2:35Þ

2.4

Orbital Angular Momentum and the Spherical Harmonics

11

For now, all we will say about Ylm(θ, ϕ) is that it is the amplitude for the state characterized by l,m to be oriented in the direction b n specified by the angles θ and ϕ. To construct the form of the angular momentum operators in three-dimensional coordinate space, we start with the transformation equations between Cartesian coordinates and spherical coordinates: x ¼ r cos ϕ sin θ

ð2:36aÞ

y ¼ r sin ϕ sin θ

ð2:36bÞ

z ¼ r cos θ

ð2:36cÞ

and the inverse transformations pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x 2 þ y2 þ z 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi tan θ ¼ x2 þ y2 =z

ð2:37bÞ

tan ϕ ¼ y=x

ð2:37cÞ

r¼

ð2:37aÞ

The Cartesian components of the spherical unit vectors are br ¼ bx cos ϕ sin θ þ by sin ϕ sin θ þ bz cos θ

ð2:38aÞ

b θ ¼ bx cos ϕ cos θ þ by sin ϕ cos θ  bz sin θ

ð2:38bÞ

b ¼ bx sin ϕ þ by cos ϕ ϕ

ð2:38cÞ

Using the above transformation equations, the form of the gradient operator in spherical coordinates is found to be !

∇ ¼ br

∂ b1 ∂ ∂ b 1 þθ þϕ r ∂θ r sin θ ∂ϕ ∂r

ð2:39Þ

We now have enough information to find the orbital angular momentum operator and the eigenvalue equations in spherical coordinates:   ħ! ! ħ ∂ 1 ∂ b b br  θ þ br  ϕ L ¼ r ∇¼ i i sin θ ∂ϕ ∂θ   ħ b ∂ b 1 ∂ ¼ ϕ θ i sin θ ∂ϕ ∂θ

!

where (2.32) was also used.

ð2:40Þ

12

2

Angular Momentum in Quantum Mechanics

Example 2.4 Verify (2.39). From (2.38a, 2.38b and 2.38c) ∂br ¼0 ∂r ∂br ¼ bx cos ϕ cos θ þ by sin ϕ cos θ  bz sin θ ¼ b θ ∂θ ∂br b sin θ ¼ bx sin ϕ sin θ þ by cos ϕ sin θ ¼ ϕ ∂ϕ Also, since br ¼ br ðθ, ϕÞ, !

d r ¼ dðrbr Þ ¼ br dr þ rdbr ¼ br dr þ r

  ∂br ∂br ∂br dr þ dθ þ dϕ ∂r ∂θ ∂ϕ

b ¼ drbr þ rdθb θ þ r sin θdϕϕ   ! Suppose u ¼ u r ¼ uðr, θ, ϕÞ. Then, du ¼

∂u ∂u ∂u dr þ dθ þ dϕ ∂r ∂θ ∂ϕ

By definition, we also have !

!

du ¼ ∇u d r Therefore,

!  !  !  ∂u ∂u ∂u dr þ dθ þ dϕ ¼ ∇u dr þ ∇u rdθ þ ∇u r sin θdϕ ∂r ∂θ ∂ϕ r θ ϕ giving us the gradient in spherical polar coordinates !  !  !  ! 1 ∂u b 1 ∂u b ∇u ¼ br ∂u þ b θ ∇u þ ϕ θ þϕ ∇u ¼ br ∇u þ b r ∂θ r sin θ ∂ϕ ∂r r θ ϕ Thus, verifying (2.39). █ Combining (2.38a, 2.38b and 2.38c) and (2.40), and taking (2.30) into account gives us the forms of the component operators: 

Lx ¼ ypz  zpy





∂ cos ϕ ∂ ¼ iħ sin ϕ þ tan θ ∂ϕ ∂θ

 ð2:41aÞ

2.4

Orbital Angular Momentum and the Spherical Harmonics



Ly ¼ zpx  xpz





sin ϕ ∂ ∂ ¼ iħ  cos ϕ tan θ ∂ϕ ∂θ

13



  ∂ Lz ¼ xpy  ypx ¼ iħ ∂ϕ

ð2:41bÞ ð2:41cÞ

We can also construct the form of the orbital angular momentum ladder operators, and the operator for the square of the orbital angular momentum:    ∂ ∂ ∂ ∂ iϕ ¼ ħe i  cot θ  þ i cot θ L ¼ iħe ∂ϕ ∂ϕ ∂θ ∂θ    2  ∂ 1 ∂ ∂ 1 2 2 2 2 2 L ¼ Lx þ Ly þ Lz ¼ ħ sin θ þ sin θ ∂θ ∂θ sin 2 θ ∂ϕ2 iϕ



ð2:42aÞ ð2:42bÞ

In the position representation, the orbital angular momentum operator acts on the states jl, mi as follows: !

!

!

njL jl, mi ¼ iħ r  ∇Y lm ðθ, ϕÞ hb

ð2:43Þ

To see this more clearly, let the following operator 

    δϕ  δϕ xpy  ypx 1i Lz ¼ 1  i ħ ħ

ð2:44Þ

!

act on a position eigenket j r i ¼ jx, y, zi . Keeping in mind that the momentum operator is the generator of translations in quantum mechanics, we get          δϕ δϕ  Lz jx, y, zi ¼ 1  i xpy  ypx jx, y, zi 1i ħ ħ ¼ jx  yδϕ, y þ xδϕ, zi

ð2:45Þ

By the way, this result is exactly what we would expect if the operator of (2.44) rotated the position eigenket by an infinitesimal angle δϕ about the z-axis. Furthermore, we see that Lz is the generator of this rotation (more on this later). Now consider the ket jl, mi projected into the coordinate representation in spherical coordinates: hx, y, zjl, mi ! hr, θ, ϕjl, mi After an infinitesimal rotation about the z-axis, we have

ð2:46Þ

14

2

Angular Momentum in Quantum Mechanics

    δϕ Lz jl, mi ¼ hx þ yδϕ, y  xδϕ, zjl, mi hx, y, zj 1  i ħ ! hr, θ, ϕ  δϕjl, mi

ð2:47Þ

Expanding to first-order in δϕ, hr, θ, ϕ  δϕjl, mi  hr, θ, ϕjl, mi  δϕ

∂ hr, θ, ϕjl, mi ∂ϕ

ð2:48Þ

and substituting into (2.47) gives,     ∂ δϕ Lz jl, mi ¼ hr, θ, ϕjl, mi  δϕ hr, θ, ϕjl, mi hr, θ, ϕj 1  i ħ ∂ϕ

ð2:49Þ

Comparing both sides of (2.49) allows us to conclude hr, θ, ϕjLz jl, mi ¼ iħ

∂ hr, θ, ϕjl, mi ∂ϕ

ð2:50Þ

We could perform the same analysis on the other components of the orbital angular momentum operator, thereby validating (2.43). We now strip the radial ! nj ¼ hθ, ϕj and then dependance out of (2.50) by first letting h r j ¼ hr, θ, ϕj ! hb multiplying both sides of (2.34b) from the left by the bra hb nj, njl, mi ) iħ njLz jl, mi ¼ ħmhb hb ) i

∂ njl, mi ¼ ħmhb njl, mi hb ∂ϕ

∂ Y lm ðθ, ϕÞ ¼ mY lm ðθ, ϕÞ ∂ϕ

ð2:51Þ

This equation implies that the ϕ-dependance of Ylm(θ, ϕ) goes as eimϕ. The requirement that Ylm(θ, ϕ) be single-valued means that m must be an integer. It then follows from (2.20) that l must also be an integer. From (2.34a) and (2.42b) we could construct, in a similar way, 

1 ∂ sin θ ∂θ



  2 ∂ 1 ∂ þ þ lðl þ 1Þ Y lm ðθ, ϕÞ ¼ 0 sin θ ∂θ sin 2 θ ∂ϕ2

ð2:52Þ

And from (2.34c) and (2.42a),   ∂ ∂ Y lm ðθ, ϕÞ ¼ ½lðl þ 1Þ  mðm  1Þ1=2 Y l,m1 ðθ, ϕÞ ieiϕ i  cot θ ∂θ ∂ϕ ð2:53Þ The solutions to (2.51), (2.52) and (2.53) are the well-known spherical harmonics:

2.4

Orbital Angular Momentum and the Spherical Harmonics

15

8 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l > > ð2l þ 1Þðl þ mÞ! imϕ 1 ð 1 Þ dlm < e sin 2l θ; m  0 m sin θ d ð cos θÞlm 4π ðl  mÞ! Y lm ðθ, ϕÞ ¼ 2l l! > > : ð1Þm Y  ðθ, ϕÞ; m < 0 l,m

ð2:54Þ where l ¼ 0, 1, 2, . . . and l m l. Equations (2.54) and (2.55) reflect standard phase conventions and are consistent with the phase conventions of (2.29b) and (2.29c). Below are listed some properties of the spherical harmonics, which may be deduced from (2.54): Y lm ðθ, ϕÞ ¼ ð1Þm Y l,m ðθ, ϕÞ ðcomplex conjugationÞ

ð2:55Þ

Y lm ðπ  θ, ϕ þ π Þ ¼ ð1Þl Y lm ðθ, ϕÞ ðspace inversionÞ

ð2:56Þ

Y lm ðθ, ϕÞ ¼ ð1Þl Y lm ðθ, ϕÞ ¼ Y lm ðθ, 2π  ϕÞ ¼ Y lm ðθ, ϕÞ rffiffiffiffiffiffiffiffiffiffiffiffi 2l þ 1 l for l þ m ¼ even Y lm ðπ, ϕÞ ¼ ð1Þ δ 4π m0 rffiffiffiffiffiffiffiffiffiffiffiffi 1=2 lm 2l þ 1 ½ðl  mÞ!ðl þ mÞ! 2 for l þ m ¼ even Y lm ðπ=2, 0Þ ¼ ð1Þ 4π ðl  mÞ!!ðl þ mÞ!! rffiffiffiffiffiffiffiffiffiffiffiffi 2l þ 1 P ð cos θÞ Y l0 ðθ, ϕÞ ¼ 4π l

ð2:57Þ ð2:58Þ ð2:59Þ ð2:60Þ

where the Pl(cosθ) are the Legendre polynomials (see App. A). Continuing, rffiffiffiffiffiffiffiffiffiffiffiffi 2l þ 1 δ Y lm ð0, ϕÞ ¼ 4π m0

ð2:61Þ

From the orthogonality relation, hl0 , m0 jl, mi ¼ δll0 δmm0

ð2:62Þ

and the completeness relation for the position eigenkets, Z dΩjb nihb nj ¼ 1 we get the orthonormality condition for the spherical harmonics:

ð2:63Þ

16

2

Z2π

Z1 dϕ

0

Angular Momentum in Quantum Mechanics

d ð cos θÞY l0 m0 ðθ, ϕÞY lm ðθ, ϕÞ ¼ δll0 δmm0

ð2:64Þ

1

Since the spherical harmonics are the eigenfunctions of a Hermitian operator, they form a complete set. We can see this directly from the properties of the position eigenkets jθ, ϕi: hθ, ϕjθ0 , ϕ0 i ¼ δðΩ  Ω0 Þ )

XX hθ, ϕjl, mihl, mjθ0 , ϕ0 i ¼ δðΩ  Ω0 Þ l

)

XX l

m

Y lm ðθ, ϕÞY lm ðθ0 , ϕ0 Þ ¼ δðΩ  Ω0 Þ ¼

m

δðθ  θ0 Þδðϕ  ϕ0 Þ sin θ

ð2:65Þ

where we also used the completeness relation XX l

jl, mihl, mj ¼ 1

ð2:66Þ

m

The following is a list of the first few spherical harmonics in their spherical and Cartesian forms: 1 Y 00 ðθ, ϕÞ ¼ pffiffiffiffiffi 4π rffiffiffiffiffi rffiffiffiffiffi 3 3 z cos θ ¼ Y 10 ðθ, ϕÞ ¼ 4π 4π r rffiffiffiffiffi rffiffiffiffiffi 3 iϕ 3 ðx  iyÞ e sin θ ¼  Y 1,1 ðθ, ϕÞ ¼  8π 8π r rffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffi  5  5 2z2  x2  y2 3 cos 2 θ  1 ¼ Y 20 ðθ, ϕÞ ¼ 16π 16π r2 rffiffiffiffiffi rffiffiffiffiffi 15 iϕ 15 ðx  iyÞz Y 2,1 ðθ, ϕÞ ¼  e cos θ sin θ ¼  8π 8π r2 rffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffi 2 15 2iϕ 15 ðx  iyÞ sin 2 θ ¼ Y 2,2 ðθ, ϕÞ ¼ e 2 32π 32π r rffiffiffiffiffi  rffiffiffiffiffi 5 2 3 2  7 5 3 7 2z  2r cos 3 θ  cos θ ¼ z Y 30 ðθ, ϕÞ ¼ 4π 2 2 4π r3

ð2:67aÞ ð2:67bÞ ð2:67cÞ ð2:67dÞ ð2:67eÞ ð2:67fÞ ð2:67gÞ

2.4

Orbital Angular Momentum and the Spherical Harmonics

rffiffiffiffiffiffiffiffi   7 iϕ 15 3 Y 3,1 ðθ, ϕÞ ¼  sin θ cos 2 θ  e 2 48π 2 rffiffiffiffiffiffiffiffi 15 2 3 2  7 2 z  2 r ðx  iyÞ ¼ 48π r3 rffiffiffiffiffiffiffiffiffiffi  7 2iϕ  e 15 cos θ sin 2 θ Y 3,2 ðθ, ϕÞ ¼ 480π rffiffiffiffiffiffiffiffiffiffi 7 15zðx2  y2  2ixyÞ ¼ 480π r3 rffiffiffiffiffiffiffiffiffiffiffiffiffi 7 3iϕ e 15 sin 3 θ Y 3,3 ðθ, ϕÞ ¼  2880π rffiffiffiffiffiffiffiffiffiffiffiffiffi 7 15½x3  3xy2  ið3x2 y  y3 Þ ¼ 2880π r3

17

ð2:67hÞ

ð2:67iÞ

ð2:67jÞ

A recurrence relation for the spherical harmonics is given in (2.68). This relation will be required later. For a proof of this relation, see App. A (see also Problem 4.9).1 

cos θY lm ¼

1=2 ð l  m þ 1Þ ð l þ m þ 1Þ Y lþ1,m ð2l þ 1Þð2l þ 3Þ  1=2 ðl  mÞðl þ mÞ þ Y l1,m ð2l  1Þð2l þ 1Þ

ð2:68Þ

Example 2.5 Equation (2.51) implies that we can write the eigenfunctions of the orbital angular momentum operators as Y lm ðθ, ϕÞ ¼ N l eimϕ Pm l ðθ Þ. Use (2.42a and 2.42b), (2.53), and the appropriate normalization integrals to find the general form of Yll(θ, ϕ). From (2.42a and 2.42b) and (2.53), we operate on Yll(θ, ϕ) with the raising operator, knowing that the result will vanish: 

   ∂ ∂ ∂ ∂ ilϕ l Y ll ðθ, ϕÞ ¼ 0 ) icot θ e Pl ðθÞ ¼ 0 ħe icot θ þ þ ∂ϕ ∂θ ∂ϕ ∂θ   ∂Pl ðθÞ=∂θ ∂ ∂ ∂ Pll ðθÞ ¼ 0 ) l l ) l cot θ þ ¼ l cot θ ) ln Pll ðθÞ ¼ l ln ð sin θÞ ∂θ ∂θ ∂θ Pl ðθÞ iϕ

) Pll ðθÞ ¼ ð sin θÞl We use (2.64) to find the normalization constant Nl

1

For an exhaustive list of formulas involving the spherical harmonics, see also [10].

18

2

Z2π

Z1 dϕ

0

Angular Momentum in Quantum Mechanics

dð cos θÞY ll ðθ, ϕÞY ll ðθ, ϕÞ ¼ 1 ) 2π jN l j2

1

Z1 dð cos θÞð sin θÞ2l 1

Z1 ¼ 2π jN l j2

 2l dx 1  x2 ¼ 1

1

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2l þ 1Þ! ð1Þl ð2l þ 1Þ! 1 ) jN l j ¼ l ) Nl ¼ l 4π 4π 2 l! 2 l! where we have used the indicated phase convention and the result Z1 1

 2l 22lþ1 ðl!Þ2 dx 1  x2 ¼ ð2l þ 1Þ!

This gives us the general form for Yll(θ, ϕ), ð1Þl Y ll ðθ, ϕÞ ¼ l 2 l!

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2l þ 1Þ! ilϕ e ð sin θÞl 4π █

2.5

The Addition Theorem for Spherical Harmonics !

!0

Suppose we have two vectors x and x whose orientations with respect to a fixed set of axes are described by the spherical angles as shown in Fig. 2.1. Fig. 2.1 The angle γ is the ! angle between the vectors x 0 ! and x

z

y

x

2.5

The Addition Theorem for Spherical Harmonics

19

Recall the completeness relation for the spherical harmonics from (2.65) and note that the term δ(Ω  Ω0) in that equation depends only on the angle γ, which is the same as the angle between the two vectors shown in Fig. 2.1. If we imagine the ! vector x as the polar axis of a new set of coordinate axes, then it is natural to want to expand δ(Ω  Ω0) in the Legendre polynomials Pl(cosγ): δ ð Ω  Ω0 Þ ¼

X bl Pl ð cos γ Þ

ð2:69Þ

l

where 2l þ 1 bl ¼ 2

Z1

δðΩ  Ω0 ÞPl ð cos γ Þd ð cos γ Þ

1

2l þ 1 ¼ 2ð2π Þ

Z2π

Z1

δðΩ  Ω0 ÞPl ð cos γ Þdð cos γ Þ

dψ 0

ð2:70Þ

1 !

and where we have introduced an integration over azimuthal angle ψ about “axis x ” (γ would be the polar angle about this axis), and we used the orthogonality integral for the Legendre polynomials: Z1 dxPl ðxÞPl0 ðxÞ ¼ 1

2 δ0 2l þ 1 ll

ð2:71Þ

We further recognize that dΩ ¼ dð cos γ Þdψ

ð2:72Þ

We have (in our double integral) a solid-angle integration over a unit sphere. The integration was constructed for some arbitrary γ, so we are free to choose any γ. So, for purposes of the integration, we choose γ ¼ 0 ) cos γ ¼ 1: 2l þ 1 ) bl ¼ P ð 1Þ 2ð2π Þ l

Z unit sphere

where we used the boundary condition

δðΩ  Ω0 ÞdΩ ¼

2l þ 1 4π

ð2:73Þ

20

2

Angular Momentum in Quantum Mechanics

P l ð 1Þ ¼ 1

ð2:74Þ

Since we are integrating over the entire unit sphere, it does not matter which set of angles we use. Hence, Z unit sphere

δðΩ  Ω0 ÞdΩ ¼

Z unit sphere

δðΩ  Ω0 ÞdΩ0 ¼

Z unit sphere

δðΩ0  ΩÞdΩ0 ¼ 1 ð2:75Þ

Combining (2.65), (2.69), and (2.73) X2l þ 1 X  0 Pl ð cos γ Þ Y lm ðΩ ÞY lm ðΩÞ ¼ ) 4π lm l

ð2:76Þ

We are not quite finished until we satisfy ourselves that we can equate terms in l on both sides of (2.76). First, it is always possible to define a new set of axes via some rotation. All the Ylm(Ω0) measured in the new coordinate system can then be expressed as linear combinations of the Ylm(Ω) from the old coordinate system, all having the same l: Y lm ðΩ0 Þ ¼

l X m0 ¼l

Dlm0 m Y lm0 ðΩÞ

ð2:77Þ

where Dlmm0 describes the rotation (see the section on rotations below). Remember that Ω and Ω0 refer to the same direction in space but are expressed in different coordinate systems. Furthermore, from (2.60), note that Pl(cosγ) can be similarly expressed: rffiffiffiffiffiffiffiffiffiffiffiffi l X 4π Dl0m0 Y lm0 ðΩÞ Pl ð cos γ Þ ¼ Y l0 ðΩ0 Þ ¼ 2l þ 1 m0 ¼l

ð2:78Þ

Thus, we are satisfied that we can equate terms in l on both sides of (2.76): Pl ð cos γ Þ ¼

l 4π X  Y ðΩ0 ÞY lm ðΩÞ 2l þ 1 m¼l lm

ð2:79Þ

This is the addition theorem for spherical harmonics. This theorem is valid for any γ as indicated by the symmetry in the expression between the prime and the unprimed coordinates. Either function on the RHS could be the complex conjugate, since the sum is over positive and negative values of m, and spherical harmonics are related to their complex conjugates by (2.55).

2.6

Rotations in Quantum Mechanics

21

Example 2.6 As an example of the usefulness of addition theorem for spherical harmonics, let us apply it under the condition γ ¼ 0:

)

l 2l þ 1 X ¼ jY lm ðθ, ϕÞj2 4π m¼l

ð2:80Þ

This result is called the sum rule for spherical harmonics. █ Example 2.7 As another example, we apply the addition theorem for the case l ¼ 1. There are three terms in the summation:    0  1 sin θeiϕ þ cos θ0 cos θ P1 ð cos γ Þ ¼ cos γ ¼ sin θ0 eiϕ 2    0  1  sin θeiϕ þ  sin θ0 eiϕ 2 h 0 i 0 1 ) cos γ ¼ cos θ0 cos θ þ sin θ sin θ0 eiðϕ ϕÞ þ eiðϕϕ Þ 2 ) cos γ ¼ cos θ0 cos θ þ sin θ sin θ0 cos ðϕ  ϕ0 Þ

ð2:81Þ

thereby verifying a trigonometric result that can be obtained using elementary methods (but only after considerable effort). █

2.6

Rotations in Quantum Mechanics

In the study of classical mechanics, one often encounters rotations. In classical mechanics, rotations are intimately connected with angular momentum, and it is no different in quantum mechanics. Just as the components of a vector change when the vector is rotated, so do state kets of rotated systems differ from their unrotated progenitors. We will confine ourselves to the formalism of the active transformation in which the state itself is rotated as opposed to passive transformation in which the basis kets (i.e., the axes) are rotated. We define a rotation operator R in the appropriate ket space as follows: jα0 i ¼ R jαi , hα0 j ¼ hαjR {

ð2:82Þ

where jα0i/hα0j and jαi/hαj represent the rotated and unrotated kets/bras, respectively. The first observation we make is that probabilities should be unaffected by rotations, hα0 jα0 i ¼ hαjαi ) hαjR { R jαi ¼ hαjαi ) R { R ¼ 1 The conclusion is that the rotation operator is unitary.

ð2:83Þ

22

2

Angular Momentum in Quantum Mechanics

Our next observation is that the expectation of an observable A should not change under rotation: hα0 jA0 jα0 i ¼ hαjAjαi ) hαjR { A0 R jαi ¼ hαjAjαi ) A ¼ R { A0 R ) A0 ¼ R AR {

ð2:84Þ

Equation (2.84) tells us how operators transform under   rotations. We have already seen [c.f. (2.44)] that the operator R ^z ðδϕÞ ¼ 1  i δϕ ħ Lz rotated an orbital angular momentum state by an infinitesimal angle δϕ about the z-axis. Furthermore, ! we saw that if the operator p is a generator of translations, then the orbital angular ! ! ! momentum operator L ¼ r  p is the generator of rotations in three-dimensional coordinate space. There are, however, other types of angular momentum—such as ! ! spin angular momentum—that have nothing to do with r and p . In quantum mechanics, we therefore define the rotation operator R ^n ðδαÞ so that it is the operator responsible for a rotation about an axis oriented in the b ndirection by an infinitesimal angle δα and is given by ! n J b R ^n ðδαÞ 1  i δα ħ !

ð2:85Þ

!

Since J is a Hermitian operator, R ^n ðδαÞ will be unitary, as required. We also expect that the rotation operator should approach the identity operator in the limit δα ! 0. The operator of (2.85) meets this requirement. Example 2.8 Equation (2.84) shows us how operators transform under rotations. Find the condition under which an operator is invariant under a rotation. First recall that the operator responsible for a rotation about an axis oriented in the b ndirection by an infinitesimal angle δα is given by ! n i J b R ^n ðδαÞ ¼ 1  i δα ¼ 1  J n δα ħ ħ !

Apply to (2.84):     i i i A0 ¼ R ðδαÞAR { ðδαÞ ¼ 1  J n δα A 1 þ J n δα ¼ A  δαðJ n A  AJ n Þ þ ⋯ ħ ħ ħ i  A  δα½J n ,A ħ So, the operator A is invariant under a rotation about an axis oriented in the b ndirection if [Jn, A] ¼ 0.

2.6

Rotations in Quantum Mechanics

23

Comment: We know from quantum theory that the operator A is a constant of the motion under the Hamiltonian H if [A, H] ¼ 0. If [Jn, H] ¼ 0, then Jn is a constant of the motion, and we see that angular momentum conservation corresponds to invariance under rotation. █ A finite rotation (about a specified axis u) through an angle α can be built out of a succession of n equal infinitesimal rotations of angle δα such that δα ¼ α/n. The operator that describes this finite rotation is  n   i α iα R u ðαÞ ¼ ½R u ðδαÞn ¼ n!1 lim 1  J u ¼ exp  J u ħn ħ !

) R u ðαÞ ¼ eiα J bu=ħ

ð2:86Þ

The form of the rotation operator of (2.86) and the fact that angular momentum operators obey the commutation rules of (2.1) together guarantee that rotations about different axes will not commute. This is as it should be, because physical successive finite rotations about axes in different directions generally do not commute. In three-dimensional coordinate space, one could uniquely characterize a finite rotation of angle α about the b n axis by specifying the three components of the vector αb n . A more convenient characterization is achieved by making use of the Euler-angle description of rotations. The Euler angles can be used to specify the orientation of a rotated body with respect to a set of (unrotated) axes that remain fixed. The two sets of coordinates have a common originn. In Fig. o 2.2, the “old” b b b (unrotated) axes are symbolized by the (red) unit vectors X , Y , Z , and the “new” (rotated) axes are symbolized by the (green) unit vectors fbx, by,bzg . One transitions from the old coordinates to the new coordinates via a series of three sequential rotations as follows:

Zˆ

Fig. 2.2 The Euler angles and thenrelativeoplacements b Y, b Z b -axes and of the X,

 xˆ

the fbx, by,bzg-axes. The angle γ in this application is not necessarily the same as the angle γ in Fig. 2.1 and (2.79) and (2.81)

zˆ

yˆ

Xˆ

Yˆ

24

2

Angular Momentum in Quantum Mechanics

b through an angle α. The Y b axis is thus • Rotate counterclockwise about axis Z brought into alignment with the “line of nodes,” denoted by a dotted line in the figure. The angle α is called the precession angle. • Rotate about the line of nodes counterclockwise through an angle β. This action b b brings the Z-axis to its final orientation. The Z-axis now becomes the bz-axis. The angle β is called the nutation angle. • Rotate about the bz-axis counterclockwise through an angle γ. Now the axes are all in their final resting places. The angle γ is called the body angle. The rotation operator that describes these three successive rotations is      iγ iβ iα R ðα, β, γ Þ ¼ exp  J z exp  J N exp  J Z ð2:87Þ ħ ħ ħ 

The operator JN (the angular momentum along the line of nodes) is actually the operator JY transformed by the rotation operator exp(iαJZ/ħ). Therefore, according to (2.84),     iα iα J N ¼ exp  J Z J Y exp JZ ħ ħ

ð2:88Þ

The same transformation applies to all powers of J:   P iβn  1  iβ exp  J N ¼ ħ ðJ Þn n! N ħ n  i P iβn  1 h  iα  n iα ¼ ħ JZ exp  J Z J Y exp ħ n! ħ n       iβ iα iα ¼ exp  J Z exp  J Y exp J ħ ħ Z ħ

ð2:89Þ

Similarly,         iβ iβ iγ iγ JN exp  J z ¼ exp  J N exp  J Z exp ħ ħ ħ ħ

ð2:90Þ

So, substituting (2.90) into (2.87) gives           iβ iγ iβ iα iβ R ðα,β, γ Þ ¼ exp  J N exp  J Z exp J N exp  J N exp  J Z ħ ħ ħ ħ ħ

      iα iγ iβ ¼ exp  J N exp  J Z exp  J Z ħ ħ ħ

ð2:91Þ

2.7

Matrix Elements of the Rotation Operators

25

Example 2.9 Put R ðα, β, γ Þ into a form that consists entirely of rotations in the same frame. We use (2.89) to substitute for the first term on the RHS of (2.91)          iα iβ iγ iα iα R ðα,β,γ Þ ¼ exp  J Z exp  J Y exp J Z exp  J Z exp  J Z ħ ħ ħ ħ ħ       iβ iα iγ ¼ exp  J Z exp  J Y exp  J Z ħ ħ ħ 

ð2:92Þ where we used the fact that JZ commutes with itself to cancel some terms. Equation (2.92) has the advantage in that it consists entirely of rotations in the same frame. It also shows that the Euler rotations may be carried out in the fixed frame if the order of the rotations is reversed. █

2.7

Matrix Elements of the Rotation Operators

The matrix elements of the rotation operator that describes a rotation through an angle α about an axis whose direction is specified by the unit vector b u (in the basis that diagonalizes J2 and Jz) are defined by !

Djm0 m ðαÞ h j, m0 jR u ðαÞj j, mi ¼ h j, m0 jeiα J ^u=ħ j j, mi

ð2:93Þ

We do not need to consider matrix elements between states of different j because the rotation operator commutes with J2, and so those elements will vanish trivially. This result reflects the fact that the angular momentum of any system should depend neither on the direction in which we are looking nor the orientation of any set of axes. In this circumstance, the completeness relation for the angular momentum eigenstates reduces to X XX j j, m0 ih j, m0 j ¼ 1 ! j j, m0 ih j, m0 j ¼ 1 j

m0

ð2:94Þ

m0

Making use of (2.94), we perform a rotation on the state jj, mi:

R u ðαÞj j, mi ¼

X m0

! 0

0

j j, m ih j, m j R u ðαÞj j, mi ¼

X j Dm0 m ðαÞj j, m0 i

ð2:95Þ

m0

So, although the rotated states R u ðαÞj j, mi are eigenfunctions of J2, they do not (generally) form a basis that diagonalizes Jz. In fact, (2.95) tells us that the rotated

26

2 Angular Momentum in Quantum Mechanics

state is a linear superposition of eigenstates jj, m0i, each with a different value of m0. The coefficients of the expansion in (2.95) are the Djm0 m (also known as the Wigner D-functions), which are the elements of a (2j þ 1)  (2j þ 1) unitary matrix. In the case of a rotation characterized by the Euler angles, we have iβ

0

iγ

Djm0 m ðα, ϕ, γ Þ ¼ h j, m0 j e ħ J Z e ħ J Y e ħ J Z j j, mi ¼ eiαm djm0 m ðβÞeiγm iα

ð2:96Þ

where we have defined iβ

d jm0 m ðβÞ h j, m0 j e ħ J Y j j, mi

ð2:97Þ

j D

ð2:98Þ

Note that m0 m ðα, β, γ Þ

2 j ¼ d 0 ðβÞ 2 mm

A closed-form expression for the d jm0 m ðβÞ is given by the Wigner formula [6]: djm0 m ðβÞ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðj þ mÞ!ðj  mÞ!ðj þ m0 Þ!ðj  m0 Þ! ¼ ð1Þ n!ðj  m0  nÞ!ðj þ m  nÞ!ðm þ m0  nÞ! n  2jþmm0 2n  m0 mþ2n β β  cos  sin 2 2 X

n

ð2:99Þ

where the summation is over values of n for which the arguments of the factorials in the denominator are not negative. Eq. (2.99) can be used to find a number of useful relations for the d jm0 m ðβÞ. For example, (2.99) is invariant under the replacements m ⇄  m0. As a result, we have djm0 m ðβÞ ¼ d jm,m0 ðβÞ

ð2:100Þ

As another example, we replace β by β in (2.99): d jm0 m ðβÞ ¼ djmm0 ðβÞ

ð2:101Þ

Combining, 0

j m m j djm0 m ðβÞ ¼ dj d m0 ,m ðβÞ m0 m ðβ Þ ¼ d mm0 ðβÞ ¼ ð1Þ 0

¼ ð1Þm m djmm0 ðβÞ

ð2:102Þ

For the case β ¼ 0, only the n ¼ 0 term in the summation of (2.99) contributes, and then only if m0 ¼ m,

2.7

Matrix Elements of the Rotation Operators

27

djm0 m ð0Þ ¼ δm0 m

ð2:103Þ

Similarly, 0

djm0 m ðπ Þ ¼ ð1Þjm δm0 ,m

ð2:104Þ

We are now in the position of being able to relate the spherical harmonics to the rotation matrices. Let us apply the rotation operator of (2.92) to the position eigenket ni. Since we are in three-dimensional jbzi to achieve the general position eigenket jb coordinate space, we are obviously talking about orbital angular momentum with integer values of l. The symbolism is adjusted accordingly. With (2.92) in mind, we imagine first rotating about the Y-axis through an angle θ, and then about the Z-axis through an angle ϕ: ni ¼ R ðα ¼ ϕ, β ¼ θ, γ ¼ 0Þjbzi jb

ð2:105Þ

Applying the completeness relation for the orbital angular momentum eigenfunctions, XX l

jl, mihl, mj ¼ 1

ð2:106Þ

m

to (2.105) gives ni ¼ R ðϕ, θ, γ ¼ 0Þ jb

XX

jl, mihl, mjbzi ¼

m

l

XX R ðϕ, θ, 0Þjl, mihl, mjbzi l

m

X Dlm0 m ðϕ, θ, 0Þhl, mjbzi ) hl , m jb ni ¼ 0

0

ð2:107Þ

m

where Dlm0 m ðϕ, θ, γ Þ are the matrix elements defined in (2.96) for orbital angular momentum and for the angles indicated. From (2.35), (2.54), and (2.61), we have rffiffiffiffiffiffiffiffiffiffiffiffi 2l þ 1 δ hl, mjbzi ¼ Y lm ðθ ¼ 0, ϕ ¼ undeterminedÞ ¼ 4π m0

ð2:108Þ

Combining (2.107) and (2.108), )

Y lm0 ðθ, ϕÞ

rffiffiffiffiffiffiffiffiffiffiffiffi 2l þ 1 l ¼ D 0 ðϕ, θ, 0Þ ) Dlm0 ðα ¼ ϕ, β ¼ θ, 0Þ 4π m 0 rffiffiffiffiffiffiffiffiffiffiffiffi 4π  ¼ Y lm ðθ, ϕÞ 2l þ 1

ð2:109aÞ

θ¼β,ϕ¼α

Taking the complex conjugate and using (2.55) and example 2.10, we find

28

2

Dl0m ðα

Angular Momentum in Quantum Mechanics

rffiffiffiffiffiffiffiffiffiffiffiffi 4π ¼ ϕ, β ¼ θ, 0Þ ¼ Y lm ðθ, ϕÞ 2l þ 1

ð2:109bÞ θ¼β,ϕ¼α

For the case m ¼ 0, Dl00 ðβ ¼ θÞ ¼ d l00 ðβÞ ¼ Pl ð cos θÞ

ð2:110Þ

where we also used (2.96).2 Example 2.10 Find an expression P j Dkm ðα, β, γ ÞDjk0 m ðα, β, γ Þ ¼ δkk0 .

Dj m0 m ðα, β, γ Þ: Also

for

show

that

m

0 Dj m0 m ðα, β, γ Þ ¼ hj, m jR ðα, β, γ Þjj, mi



¼ hj, mjR { ðα, β, γ Þjj, m0 i ¼ hj, mjR 1 ðα, β, γ Þjj, m0 i 0

¼ Djmm0 ðα,  β, γ Þ ¼ ð1Þm m Djm0 ,m ðα, β, γ Þ where we also used the fact that the rotation operator is unitary. Continuing, X m

j Dj nm ðα, β, γ ÞDmn0 ðα, β, γ Þ ¼

X

h j, n R 1 ðα, β, γ Þj j, m h j, mjR ðα, β, γ Þj j, n0 i m

¼ h j, njR 1 ðα, β, γ ÞR ðα, β, γ Þj j, n0 i ¼ hj, nj j, n0 i ¼ δnn0 █

2.8

The Coupling of Two Angular Momenta

When examining a physical system, often we find that we must add two or more independent angular momenta. For example, to find the total orbital angular momentum of an atom, we must add the orbital angular momenta of all the electrons in the atom. To find the total angular momentum of an individual atomic electron, we add the electron’s spin angular momentum to its orbital angular momentum. In the latter case, ! the total angular momentum J is the vector sum of the two individual angular momentum vectors: !

!

!

J ¼LþS

2

For an exhaustive list of properties of the Wigner D-functions, see also [10].

ð2:111Þ

2.8

The Coupling of Two Angular Momenta

29 !

!

where L is the orbital angular momentum operator we first saw in (2.30) and S is the single-particle spin operator. This example leads us to first consider a more general ! ! case of adding two angular momenta from different subspaces, J 1 and J 2. The total angular momentum for such a system is !

!

!

J ¼ J1 þ J2

ð2:112Þ

J k ¼ J 1k þ J 2k ; k ¼ x, y, z

ð2:113Þ

Equation (2.112) implies

!

!

It is important to remember that J 1 and J 2 obey the angular-momentum commutation relations within their respective subspaces:


X  J 1l , J 1j ¼ iħ εljk J 1k ; l, j, k ¼ x, y, z

ð2:114aÞ

k




X  J 2l , J 2j ¼ iħ εljk J 2k ; l, j, k ¼ x, y, z

ð2:114bÞ

k

but that angular momenta from different subspaces commute


 J 1l , J 2j ¼ 0; l, j ¼ x, y, z

ð2:115Þ

A direct consequence of (2.114a and 2.114b) and (2.115) is that the Cartesian ! components of the total angular momentum J obey the angular-momentum commutation relations:


X  J l , J j ¼ iħ εljk J k ; l, j, k ¼ x, y, z

ð2:116Þ

k !

Thus, J is, by definition, an angular momentum operator in the sense described in Sect. 2.1. We can therefore (for example) define the ladder operators J+ and J for the composite system and the operator J2 for the square of the total angular ! momentum of the composite system. J has the standard set of eigenvectors, eigenvalues, and matrix elements and is the generator of rotations for the entire system. We can also derive the following commutation relations:
2 2
2 2

 J , J 1 ¼ J , J 2 ¼ J z , J 21 ¼ J z , J 22 ¼ 0

ð2:117Þ

We therefore have two complete sets of mutually commuting observables that we can use to describe the composite system. We can either use the set of

30

2 Angular Momentum in Quantum Mechanics

observables J 21 , J 1z , J 22 , J 2z with their simultaneous eigenkets { |j1, m1i |j2, m2i , jj1, j2; m1, m2i } as a basis set, or we can use the set of observables J 2 , J z , J 21 , J 22 with their simultaneous set of eigenkets {jj1, j2; j, mi , jj, mi } as a basis set. The first representation is called the uncoupled representation or the uncoupled basis. The second is called the coupled representation or the coupled basis. Note that, for the coupled representation, the symbol for the eigenket jj, mi is shorthand for jj1, j2; j, mi, since the quantities j1, j2 are understood (or given). In the same vein, we could also shorten the symbol for the eigenket in the uncoupled basis: jj1, j2; m1, m2i ! jm1, m2i, and this is sometimes done in the literature.

2.9

The Clebsch-Gordan Coefficients

The two representations described above are equivalent descriptions that span a composite space of dimensionality N, where N¼

j¼j 1 þj2 X

ð2j þ 1Þ ¼ ð2j1 þ 1Þð2j2 þ 1Þ

ð2:118Þ

j ¼ j1 j2 j1  j2

We connect the two descriptions using the Clebsch-Gordan (C-G) coefficients. The C-G coefficients hj1, j2; m1, m2| j, mi are the elements of a unitary transformation which connect the uncoupled angular momentum basis {jj1, j2; m1, m2i} with the coupled basis{jj, mi }. They arise in the expansion of the coupled basis on the uncoupled basis as follows: " jj, mi ¼

j1 P

j2 P

m1 ¼j1 m2 ¼j2

¼

PP m1 m2

# jj1 , j2 ; m1 , m2 ihj1 , j2 ; m1 , m2 j jj, mi

ð2:119Þ

hj1 , j2 ; m1 , m2 jj, mijj1 , j2 ; m1 , m2 i

The C-G coefficients are sometimes also known as the Wigner coefficients or as the vector-coupling coefficients. Because Jz ¼ J1z þ J2z, we can say, hj1 , j2 ; m1 , m2 jðJ z  J 1z  J 2z Þ j j, mi ¼ 0 ) ðm  m1  m2 Þ hj1 , j2 ; m1 , m2 jj, mi ¼ 0 ) hj1 , j2 ; m1 , m2 jj, mi ¼ 0 unless m ¼ m1 þ m2

ð2:120Þ

Therefore, the double sum in (2.119) is actually a single sum, the sum over m2 being superfluous: j j, mi ¼

X m1

hj1 , j2 ; m1 , m  m1 jj, mijj1 , j2 ; m1 , m  m1 i

ð2:121Þ

2.9

The Clebsch-Gordan Coefficients

31

In addition, the C-G coefficients must also satisfy the triangle selection rule(s). That is, the C-G coefficients are all identically zero unless the following conditions are simultaneously satisfied: ð2:122aÞ m ¼ m1 þ m2 jj1  j2 j j j1 þ j2

ð2:122bÞ

It is clear that the (so-called) magnetic quantum numbers m add algebraically, while the angular momenta add as vectors. The general tenants of angular momentum theory derived above require that, for the ket jj, mi to exist, we must also have m ¼ j, j  1, j  2, ⋯,  j

ð2:123aÞ

m1 ¼ j1 , j1  1, j1  2, ⋯,  j1

ð2:123bÞ

m2 ¼ j2 , j2  1, j2  2, ⋯,  j2

ð2:123cÞ

If the conditions in (2.122) and (2.123a, 2.123b and 2.123c) are not met, then the C-G coefficients are not defined. The C-G coefficients are further defined by the initial condition: hj1 , j2 ; m1 , m2 jj1 þ j2 , j1 þ j2 i ¼ 1

ð2:124Þ

and by the phase convention that the C-G coefficients are all real. Finally, the C-G coefficients are sometimes represented by the abbreviated notations hm1, m2j j, mi, j,m C j,m j1 ,m1 ; j2 ,m2 , or C m1 ,m2 . We will use some of these alternate notations later when it proves convenient. If the jj, mi are normalized, then h j0 , m0 jj, mi ¼ δj0 j δm0 m P 0 0 ) hj0 , m0 jj, mi ¼ h j , m jj1 , j2 ; m01 , m02 i m01 , m02

h j1 , m01 jhj2 , m02 j

P

m1, m2

hj1 , j2 ; m1 , m2 j j, mijj1 , m1 ijj1 , m1 i

¼ δj0 j δm0 m δm0 m

¼

P

1 1

δm0 m

2 2

zfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflffl{ h j0 , m0 jj1 , j2 ; m01 , m02 ihj1 , j2 ; m1 , m2 j j, mi hj1 , m01 jj1 , m1 ihj2 , m02 jj2 , m2 i

m1 , m01 , m2 , m02

¼ δj0 j δm0 m P 0 0 ) hj , m jj1 , j2 ; m1 , m2 ihj1 , j2 ; m1 , m2 jj, mji ¼ δj0 j δm0 m m1 , m2

ð2:125Þ where we have used the fact that the C-G coefficients are real. Note again that, strictly speaking, we need only one index in the summation, since m1 and m2 are constrained by (2.122a).

32

2

Angular Momentum in Quantum Mechanics

Example 2.11 Find the normalization condition for the C-G coefficients. If we set j ¼ j0 and m0 ¼ m ¼ m1 þ m2 in (2.125), we get, X jhj1 , j2 ; m1 , m2 jj, mij2 ¼ 1

ð2:126Þ

m1 , m2 ¼mm1

which is the normalization condition for C-G coefficients. █ Example 2.12 Invert (2.119) and use the result to find a second orthogonality condition for the C-G coefficients. Using (2.122a and 2.122b), (2.123a, 2.123b and 2.123c), and (2.124), jj, mi ¼ ¼

j P

j1P þj2

j0 ¼jj1 j2 j m0 ¼j

P j0 , m0

) jj, mi ¼



P

m1 , m2

j

j P

j1P þj2

j0 ¼jj1 j2 j m0 ¼j

hj0 , m0 jj, mijj0 , m0 i

 hj , m jj1 , j2 ; m1 , m2 ihj1 , j2 ; m1 , m2 jj, mi jj0 , m0 i 0

XX 0

δj0 j δm0 m j j0 , m0 i ¼ 0

hj1 , j2 ; m1 , m2 jj0 , m0 ihj1 , j2 ; m1 , m2 jj, mijj0 , m0 i

ð2:127Þ

, m0 m1 , m2

But, by definition, we have X j j, mi ¼ hj1 , j2 ; m1 , m2 jj, mi jj1 , m1 i jj2 , m2 i

ð2:128Þ

m1 , m2

Equating (2.127) and (2.128), X

0 hj1 , j2 ; m1 , m2 jj, mi@

X

1 hj1 , j2 ; m1 , m2 jj , m i jj , m i  jj1 , m1 i jj2 , m2 iA ¼ 0

j0 , m0

m1 , m2

) jj1 , j2 ; m1 , m2 i ¼

0

0

0

0

X hj1 , j2 ; m1 , m2 jj, mij j, mi

ð2:129Þ

j, m

Equation (2.129) is the inverse of (2.119). We can now deduce a second orthogonality condition from (2.129) as follows: 



XX j1 , m01 jj1 , m1 j2 , m02 jj2 , m2 ¼ hj1 , j2 ; m1 , m2 jj, mi |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} j, m j0 , m0 δm

)

0 1 m1

X

δm

0 2 m2



 j0 , m0 jj1 , j2 ; m01 , m02 hj0 , m0 jj, mi |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} δj0 j δm0 m



hj1 , j2 ; m1 , m2 jj, mi j1 , j2 ; m01 , m02 jj, m ¼ δm1 m01 δm2 m02

ð2:130Þ

j, m

Of course, we already knew this because the C-G coefficients form a real unitary matrix, and a real unitary matrix is an orthogonal matrix. █

2.9

The Clebsch-Gordan Coefficients

33

Since the jj1, j2; m1, m2i form a standard angular momentum basis, we have pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi J 1 j j1 , j2 ; m1 , m2 i ¼ ħ j1 ðj1 þ 1Þ  m1 ðm1  1Þj j1 , j2 ; m1  1, m2 i pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi J 2 j j1 , j2 ; m1 , m2 i ¼ ħ j2 ðj2 þ 1Þ  m2 ðm2  1Þj j1 , j2 ; m1 , m2  1i

ð2:131aÞ ð2:131bÞ

And, by construction, the kets jj, mi satisfy J  j j, mi ¼ ħ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jðj þ 1Þ  mðm  1Þj j, m  1i

ð2:132Þ

for J  ¼ J 1 þ J 2

ð2:133Þ

So, for m >  j we have, applying (2.131a, 2.131b, 2.132 and 2.133) to (2.119): pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jð j þ 1Þ  mðm  1Þj j, m  1i ¼ # " pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j1 ð j1 þ 1Þ  m01 ðm01  1Þj j1 , j2 ; m01  1, m02 i P h j1 , j2 ; m01 , m02 j j, mi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m01 , m02 þ j2 ð j2 þ 1Þ  m02 ðm02  1Þj j1 , j2 ; m01 , m02  1i ð2:134Þ Multiplying by the bra h j1, j2; m1, m2| gives pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jð j þ 1Þ  mðm  1Þh j1 , j2 ; m1 , m2 j j, m  1i pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ j1 ð j1 þ 1Þ  m1 ðm1  1Þh j1 , j2 ; m1  1, m2 j j, mi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ j2 ð j2 þ 1Þ  m2 ðm2  1Þh j1 , j2 ; m1 , m2  1j j, mi

ð2:135Þ

A more convenient form of (2.135) is pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð j  mÞð j  m þ 1Þh j1 , j2 ; m1 , m2 j j, m  1i pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ð j1  m1 þ 1Þð j1  m1 Þh j1 , j2 ; m1  1, m2 jj, mi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ ð j2  m2 þ 1Þð j2  m2 Þhj1 , j2 ; m1 , m2  1jj, mi

ð2:136Þ

Similarly, we could calculate h j1, j2; m1, m2|J|j, m  1i to get the recursion relation pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð j  mÞð j  m þ 1Þh j1 , j2 ; m1 , m2 j j, mi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ð j1  m1 þ 1Þð j1  m1 Þh j1 , j2 ; m1  1, m2 jj, m  1i pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ ð j2  m2 þ 1Þð j2  m2 Þhj1 , j2 ; m1 , m2  1jj, m  1i

ð2:137Þ

34

2

Angular Momentum in Quantum Mechanics

Equations (2.124), (2.126), and (2.136), along with the given phase convention are all that are needed to uniquely determine all the C-G coefficients. That notwithstanding, the following closed-form expression for the C-G coefficients was developed by Racah [2]: 1  ð2j þ 1Þð j1 þ j2  jÞ!ð j1  j2 þ jÞ!ð j þ j2  j1 Þ! 2 ð j1 þ j2 þ j þ 1Þ! pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X ð j1 þ m1 Þ!ð j1  m1 Þ!ð j2 þ m2 Þ!ð j2  m2 Þ!ð j  mÞ!ð j þ mÞ!  ð1Þz ð þ j  j  zÞ!ð j1  m1  zÞ!ð j2 þ m2  zÞ!ð j  j2 þ m1 þ zÞ!ð j  j1  m2 þ zÞ! z! j 1 2 z hj1 , j2 ;m1 , m2 jj3 , m3 i ¼ δm1 þm2 , m

ð2:138Þ To put all the angular momentum indices on a completely equal footing [and thereby completely highlight the symmetric nature of (2.138)], let j ! j3; m ! m3, and let s j1 þ j2 þ j3, which gives [3] 1  ð2j3 þ 1Þðs  2j3 Þ!ðs  2j2 Þ!ðs  2j1 Þ! 2 ðs þ 1Þ! pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X ð j1 þ m1 Þ!ð j1  m1 Þ!ð j2 þ m2 Þ!ð j2  m2 Þ!ð j3 þ m3 Þ!ð j3  m3 Þ!  ð1Þz z! j ð þ j  j3  zÞ!ð j1  m1  zÞ!ð j2 þ m2  zÞ!ð j3  j2 þ m1 þ zÞ!ð j3  j1  m2 þ zÞ! 1 2 z hj1 ,j2 ;m1 ,m2 jj3 ,m3 i ¼ δm1 þm2 , m3

ð2:139Þ Many symmetry relations for the C-G coefficients can be readily obtained from (2.138) or (2.139) [10]. Example 2.13 Use (2.139) to find an expression for hj2, j3;  m2, m3| j1, m1i in terms of hj1, j2; m1, m2| j3, m3i. Making the substitution n ¼ ( j2 þ m2)  z and summing over n in (2.139), hj1 ,j2 ;m1 ,m2 jj3 ,m3 i ¼ δm1 þm2 ,m3 ð1Þj2 þm2



X n



ð2j3 þ 1Þ ð2j1 þ 1Þ

1  2

ð2j1 þ 1Þðs  2j3 Þ!ðs  2j2 Þ!ðs  2j1 Þ! ðs þ 1Þ!

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð j þ m1 Þ!ð j1  m1 Þ!ð j2 þ m2 Þ!ð j2  m2 Þ!ð j3 þ m3 Þ!ð j3  m3 Þ! ! 3 ð1Þn 2 1 j  j m  m þ n 1 2 6 ð j2 þ m2  nÞ!ð j1  j3  m2 þ nÞ! 1 2 |fflfflfflfflfflffl{zfflfflfflfflfflffl} !n! 7 7 6 m3 7 6 ! 7 6 7 6 5 4  j3 þm1 þ m2  n !ð j  j þ j  nÞ! |fflfflfflfflfflffl{zfflfflfflfflfflffl} 3 1 2 þm3

1 2

2.9

The Clebsch-Gordan Coefficients

35

1 1  ð2j3 þ 1Þ 2 ð2j1 þ 1Þðs  2j3 Þ!ðs  2j2 Þ!ðs  2j1 Þ! 2 ¼ δm1 þm2 ,m3 ð1Þ  ðs þ 1Þ! ð2j1 þ 1Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X ð j1 þ m1 Þ!ð j1  m1 Þ!ð j2 þ m2 Þ!ð j2  m2 Þ!ð j3 þ m3 Þ!ð j3  m3 Þ! " # ð1Þn n!ð j3  j1 þ j2  nÞ!ð j2 þ m2  nÞ!ð j3 þ m3  nÞ! n j2 þm2



ð j1  j3  m2 þ nÞ!ð j1  j2  m3 þ nÞ!  12 j2 þm2 ð2j3 þ 1Þ ) hj1 , j2 ; m1 , m2 jj3 , m3 i ¼ ð1Þ hj2 , j3 ;  m2 , m3 jj1 , m1 i ð2j1 þ 1Þ ð2:140Þ █

Similarly, by making the substitution n ¼ ( j1 þ j2  j3)  z and summing over n, one can easily verify that hj1 , j2 ; m1 , m2 jj3 , m3 i ¼ ð1Þj1 þj2 j3 hj1 , j2 ;  m1 ,  m2 jj3 ,  m3 i ¼ ð1Þj1 þj2 j3 hj2 , j1 ; m2 , m1 jj3 , m3 i

ð2:141Þ

And by making the substitution n ¼ j1  m1  z and summing over n, one could also show sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2j3 þ 1Þ hj1 , j2 ; m1 , m2 jj3 , m3 i ¼ ð1Þj1 m1 hj , j ; m ,  m3 jj2 ,  m2 i ð2:142Þ ð2j2 þ 1Þ 1 3 1 and so forth. We must be cautious when interpreting (2.138) and (2.139). In these equations, the summation index z ranges over all integral values for which the factorial arguments in the denominator of the last term on the RHS are positive; the factorial of a negative number being undefined in this context. This is demonstrated below explicitly for (2.139) as follows [3]: h j1 ,j2 ;m1 ,m2 jj3 ,m3 i ¼ 2 312 ðs  2j3 Þ!ðs  2j2 Þ!ðs 2j1 Þ! 6 ð2j3 þ 1Þ 7 ðs þ 1Þ! δm1 þm2 ,m3 4 5 ð j1 þ m1 Þ!ð j1 m1 Þ!ð j2 þ m2 Þ!ð j2  m2 Þ!ð j3 þm3 Þ!ð j3  m3 Þ! zup P ð1Þz  z! ð j þ j  j  z Þ! ð j  m  z Þ! ð j þ ð j3  j2 þ m1 þzÞ! 1 zlow 1 2 3 1 2 m2  zÞ! |fflfflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflfflffl ffl} |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} 0 0 0 0 gives an upper limit forz gives an upper limit forz gives an upper limit for z

gives a lower limit for z

ð j3  j1  m2 þzÞ! |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} 0 gives a lower limit for z

) zup ¼ j1 þ j2  j3 ) zup ¼ j1  m1 ) zup ¼ j2 þm2 ) zlow ¼ j2  j3  m1 ) zlow ¼ j1  j3 þ m2 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} zup ¼minimum of these

zlow ¼maximum of these

ð2:143Þ

:

36

2

Angular Momentum in Quantum Mechanics

For those who are curious, the Racah formula for the C-G coefficients is derived in App. B. For some parts of this document, we use not the C-G coefficients, but the Wigner 3-j symbols, which are defined as 

j1

j2

j3

m1

m2

m3



¼ ð1Þj1 j2 m3 ð2j3 þ 1Þ2 hj1 , j2 ; m1 , m2 jj3 ,  m3 i 1

hj1 , j2 ; m1 , m2 jj3 , m3 i ¼ ð1Þ

j1 j2 þm3

ð2j3 þ 1Þ

1 2



j1

j2

j3

m1

m2

m3

ð2:144aÞ

 ð2:144bÞ

We use the 3-j symbols because they tend to have better symmetry properties upon exchange of angular momentum quantum numbers. For example, the 3-j symbols have the advantage that an even permutation of the columns is cyclic, that is, leaves the numerical value of the 3-j symbol unchanged (the presence of the phase factors in (2.144a and 2.144b) ensures this). The proof is simple (showing one cyclic permutation as an example): h j1 , j2 ; m1 , m2 jj3 , m3 i  12 j2 þm2 ð2j3 þ 1Þ ¼ ð1Þ h j2 , j3 ;  m2 , m3 jj1 , m1 i ð2j1 þ 1Þ!  1 ð2j3 þ 1Þ 2 ð1Þj1 þj2 j3 h j2 , j3 ; m2 ,  m3 jj1 ,  m1 i ¼ ð1Þj2 þm2 ð2j1 þ 1Þ! ! j1 j2 j3 1 j1 j2 þm3 2 ) ð1Þ ð2j3 þ 1Þ m1 m2 m3 !  12 j2 j3 j1 1 j2 þm2 ð2j3 þ 1Þ j1 þj2 j3 j2 j3 m1 ð1Þ ð1Þ ð2j1 þ 1Þ2 ¼ ð1Þ ð2j1 þ 1Þ! m2 m3 m1 ! ! j1 j2 j3 j2 j3 j1 4j2 2j3 m3 þm2 m1 ) ¼ ð1Þ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} m1 m2 m3 m2 m3 m1 ¼

j2

j3

j1

!

¼ð1Þ4j2 2j3 2m1 ¼1 ðfor integer or halfintegerÞ

m2 m3 m1 ð2:145aÞ

where we used (2.122a and 2.122b), (2.140), (2.141), and (2.144a and 2.144b). From (2.141) and (2.144a), we see that an odd permutation is equivalent to a multiplication by ð1Þj1 þj2 þj3 :

2.9

The Clebsch-Gordan Coefficients



j2 m2

j1 m1

j3 m3

37

 ¼ ð1Þ

j1 þj2 þj3



j1 m1

j2 m2

j3 m3

 ð2:145bÞ

Again, from (2.141) and (2.144a), we have also 

j1 m1

j2 m2

j3 m3



¼ ð1Þj1 þj2 þj3



j1 m1

j2 m2

j3 m3

 ð2:145cÞ



 j 1 j2 j3 must have Equation (2.145c) tells us that any 3-j symbol of the form 0 0 0 j1 þ j2 þ j3¼even integer; otherwise, the 3-j symbol is zero. The restriction that the summation index z in (2.143) ranges only over all integral values for which the factorial arguments in the denominator of the last term on the RHS are positive has consequences for the 3-j symbols.  For example, from this restriction, it follows that, for the 3-j symbol j1 j2 j3 , we must have (for the 3-j symbol to be real): m1 m2 m3 j1 þ j2 þ j3 ¼ integer, m1 þ m2 þ m3 ¼ integer, and j1  j2  m3 ¼ integer

ð2:145dÞ

We must also have. j1 þ j2  j3 , j2 þ j3  j1 , and j3 þ j1  j2

ð2:145eÞ

Since there is a sign change in the relation between the C-G coefficients and the 3-j symbols [c.  f. (2.144)], the relation between the projection quantum numbers for  j1 j2 j3 is the 3-j symbol m1 m2 m3 m1 þ m2 þ m3 ¼ 0

ð2:145fÞ

Equations (2.145d, 2.145e and 2.145f) represent the triangle rules for the 3-j symbols. We will see that the triangle rules are often essential for evaluating sums involving 3-j symbols. The utility of the Racah formula for evaluating 3-j symbols can be demonstrated by a few simple examples.   j j 0 Example 2.14 Calculate m m 0

38

2

j

j

0

m

m X

0

!

 ¼

Angular Momentum in Quantum Mechanics

ð2jÞ!ðj  mÞ!ðj þ mÞ!ðj  mÞ!ðj þ mÞ! ð2j þ 1Þ!

12

ð1Þz z z!ð2j  zÞ!ðj  m  zÞ!ðj  m  zÞ!ðj þ m þ zÞ!ðj þ m þ zÞ! X 1 ð1Þz ¼ ð2j þ 1Þ2 ½ðj  mÞ!ðj þ mÞ! 2 2 z z!ð2j  zÞ!½ðj  m  zÞ! ½ðj þ m þ zÞ!   1 1 ð1Þjm ¼ ð2j þ 1Þ2 ð1Þjm ¼ ð2j þ 1Þ2 ½ðj  mÞ!ðj þ mÞ! ðj  mÞ!ðj þ mÞ! 

ð2:146Þ where using the guidance from (2.143), we see that the summation over z collapsed to the range j  m z j  m (a single value for z). █ Example 2.15 0 B Calculate @

1 2

j

j

m

m 

1 2

1 1 2C A 1 2

1 1 1  j B 2 C ¼ ð2j  1Þ!ð j  mÞ!ð j þ mÞ!ð j þ mÞ!ð j  m  1Þ! 2 A @ 1 1 ð2j þ 1Þ! m m  2 2

0

1 j 2

X 

ð1Þz z z!ð2j  1  zÞ!ð j  m  zÞ!ð j  m  1  zÞ!ðj þ 1 þ m þ zÞ!ðj þ m þ 1 þ zÞ!  1   ð1Þjm1 ðj  mÞ 2 ¼ ðj þ mÞ!ðj  m  1Þ! 2jð2j þ 1Þ ðj  m  1Þ!ðj þ mÞ! 1 0 1 1  1 j j B 2 2 C ¼ ðj  mÞ 2 ð1Þjm1 )@ A 1 1 2jð2j þ 1Þ m m  2 2 ð2:147Þ

where using the guidance from (2.143), we see that the summation over z collapsed to the range j  m  1 z j  m  1 (a single value for z). █ The Racah formula (and the symmetry relations derived from the Racah formula) can be used to compute the values of many of the C-G coefficients and 3-j symbols. Many tables of these algebraic expressions exist (see, e.g., [3, 5, 10]).

2.10

The Clebsch-Gordan Series

39

We finish this section by noting that to successfully navigate the many different available sources that discuss angular momentum in atomic and molecular applications, one must be able to move effortlessly between using C-G coefficients and 3-j symbols.

2.10

The Clebsch-Gordan Series

Recall from (2.129): j j1 , j2 ; m1 , m2 i jj1 , m1 ijj2 , m2 i ¼

X hj1 , j2 ; m1 , m2 jj, mij j, mi

ð2:129Þ

j, m

Apply a rotation R to both sides: XX m001 m002





XX j j Dm1 00 m1 Dm2 00 m2 j1 , m001 j2 , m002 ¼ hj1 , j2 ; m1 , m2 jj, miDjm0 m j j, m0 i 1

2

j, m m0

ð2:148Þ   where we used (2.95). Now multiply both sides from the left by j1 , m01 j2 , m02 , j

j

) Dm1 0 m1 Dm2 0 m2 ¼ 1

2

X

 hj1 , j2 ; m1 , m2 jj, mi j1 , j2 ; m01 , m02 j j, m0 iDjm0 m

ð2:149Þ

j

Notice that the summations over m and m0 have disappeared from the RHS of (2.149). This is because (2.122a) tells us that, for a given m1 and m2, m is determined, and for a given m01 and m02, m0 is determined. Equation (2.149) is called the ClebschGordan series. We can use the C-G series to solve useful integrals. Consider, for example, Z

j 

Z j

dΩDm1 0 m1 Dm2 0 m2 ¼ 1

2

0

1 2 dΩð1Þm1 m1 Dm 0 ,m Dm0 m 1 2

j

j

1

2

ð2:150Þ

where we have used (2.102) which also applies to the Djm0 m . Let us apply (2.149) to an integral over two rotation matrices Djm0 m ðα, β, γ Þ for which dΩ ¼ dα sin βdβdγ Z

j 

j

dΩDm1 0 m1 Dm2 0 m2 1 2 X 0 ¼ ð1Þm1 m1 h j1 , j2 ;  m1 , m2 jj3 ,  m1 þ m2 ihj1 , j2 ;  m01 , m02 jj3 , m01 þ m02 i j3

Z2π 

dαe

iαðm01 þm02 Þ

Z2π dγe

iγ ðm1 þm2 Þ

Zπ j

3 dβ sin βd m 0 þm0 ,m þm 1 2 1

0

0

0

2

40

2 Angular Momentum in Quantum Mechanics

where we also used (2.96). Examining the integrals, Z2π dαe

iαðm01 þm02 Þ

Z2π dγe

iγ ðm1 þm2 Þ

Zπ j

3 dβ sin βdm 0 þm0 ,m þm 1 2 1

0

0

2

0

Zπ j

¼ ð2π Þδm01 m02 ð2π Þδm1 m2

3 dβ sin βdm 0 þm0 ,m þm 1 2 1

2

0

Zπ ¼ ð2π Þ

j

2

3 dβ sin βd00

0

Zπ ¼ ð2π Þ2

dβ sin βPj3 ð cos βÞ 0



Zπ ¼ ð2π Þ

2

2

dβ sin βPj3 ð cos βÞP0 ð cos βÞ ¼ ð2π Þ

 2 δ ¼ 8π 2 δj3 0 2j3 þ 1 j3 0

0

where we used (2.110) and (2.71) and noted that P0(cosβ) ¼ 1. Putting everything together, Z

 0 j  j dΩDm1 0 m1 Dm2 0 m2 ¼ 8π 2 ð1Þm1 m1 hj1 , j2 ;  m1 , m1 j0, 0i j1 , j2 ; m01 , m01 j0, 0i 1

2

ð2:151Þ Using (2.144b) and (2.146) and realizing that (2.122b) implies that we must have j1 ¼ j2:   j1 j2 0 j1 j2 hj1 , j2 ;  m1 , m1 j0, 0i ¼ ð1Þ m1 m1 0 ð2:152aÞ ¼ ð2j1 þ 1Þ2 ð1Þj1 þm1 δj1 j2 ! j1 j2 0  0 1 ¼ ð1Þj1 þm1 ð2j1 þ 1Þ2 δj1 j2 j1 , j2 ; m01 , m01 j0, 0i ¼ ð1Þj1 j2 0 0 m1 m1 0 1

ð2:152bÞ giving us finally Z

j 

j

dΩDm1 0 m1 ðα, β, γ ÞDm2 0 m2 ðα, β, γ Þ ¼ 1

2

¼

0 8π 2 ð1Þ2ðj1 þm1 Þ δj1 j2 δm01 m02 δm1 m2 2j1 þ 1

8π 2 δ δ 0 0δ 2j1 þ 1 j1 j2 m1 m2 m1 m2

ð2:153Þ

2.10

The Clebsch-Gordan Series

41

0 where we see that ð1Þ2ðj1 þm1 Þ ¼ 1 for j1 , m01 integer or half-integer. Equation j (2.153) serves as a normalization condition for the Dm2 0 m2 ðα, β, γ Þ. 2

Example 2.16 Use the C-G series to solve the integral over the spherical harmonic triple product: Z2π

Z1 dϕ

d ð cos θÞY lm ðθ, ϕÞY l1 m1 ðθ, ϕÞY l2 m2 ðθ, ϕÞ

1

0

In (2.149), let j1 ! l1, j2 ! l2, m1 ! 0, m2 ! 0, j ! l3, ) Dlm1 0 0 Dlm2 0 0 ¼ 1

X

2

 hl1 , l2 ; 0, 0jl3 , 0i l1 , l2 ; m01 , m02 jl3 , m0 iDlm3 0 0

ð2:154Þ

l3

where we used the fact that, since m1 ¼ m2 ¼ 0, (2.122a) tells us that we must also have m ¼ 0. Now let m01,2 ! m1,2 , m0 ! m3, and invoke (2.109a and 2.109b) )

Y l1 m1 ðθ, ϕÞY l2 m2 ðθ, ϕÞ

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2l1 þ 1Þð2l2 þ 1ÞX ¼ ð2l3 þ 1Þ1=2 hl1 , l2 ; 0, 0jl3 , 0i 4π l 3

 hl1 , l2 ; m1 , m2 jl3 , m3 iY l3 m3 ðθ, ϕÞ ) ð1Þm1 þm2 m3 Y l1 m1 ðθ, ϕÞY l2 m2 ðθ, ϕÞ ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2l1 þ 1Þð2l2 þ 1ÞX ð2l3 þ 1Þ1=2 hl1 ,l2 ; 0,0jl3 , 0ihl1 , l2 ;m1 , m2 jl3 ,m3 iY l3 m3 ðθ,ϕÞ 4π l 3

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2l1 þ 1Þð2l2 þ 1ÞX ð2l3 þ 1Þ1=2 hl1 , l2 ; 0, 0jl3 , 0i ) Y l1 m1 ðθ, ϕÞY l2 m2 ðθ, ϕÞ ¼ 4π l 3

 hl1 , l2 ; m1 , m2 jl3 , m3 iY l3 m3 ðθ, ϕÞ ð2:155Þ where we observed that ð1Þm1 þm2 m3 ¼ 1 since m1 þ m2 ¼ m3. Now multiply both sides by Y lm ðθ, ϕÞ and integrate Z1

Z2π dϕ 0

d ð cos θÞY lm ðθ, ϕÞY l1 m1 ðθ, ϕÞY l2 m2 ðθ, ϕÞ

1

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2l1 þ 1Þð2l2 þ 1Þ hl1 , l2 ; 0, 0jl, 0ihl1 , l2 ; m1 , m2 jl, mi ¼ 4π ð2l þ 1Þ

ð2:156Þ

42

2

Angular Momentum in Quantum Mechanics

Notice that the spherical harmonic orthonormality condition (2.64) collapsed the summation over l3 on the RHS of (2.155). █ Let us manipulate (2.156) to put the result in terms of 3-j symbols: Z1

Z2π

dð cos θÞð1Þm Y l,m ðθ, ϕÞY l1 m1 ðθ, ϕÞY l2 m2 ðθ, ϕÞ

dϕ 1

0

¼ ð1Þ

2ðl1 l2 Þþm

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2l1 þ 1Þð2l2 þ 1Þð2l þ 1Þ l1 4π 0

l2

l

0

0

!

!

l1

l2

l

m1

m2

m ð2:157Þ

Relabel m !  m and note that ð1Þ2ðl1 l2 Þ ¼ 1 for l1, l2 integer or half-integer: Z1

Z2π

dð cos θÞY l,m ðθ, ϕÞY l1 m1 ðθ, ϕÞY l2 m2 ðθ, ϕÞ

dϕ 0

1

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2l1 þ 1Þð2l2 þ 1Þð2l þ 1Þ l1 ¼ 4π 0

l2

l

0

0



l1

l2

l

m1

m2

m

 ð2:158Þ

We now use the C-G series and (2.153) to evaluate an integral over a triple product of rotation matrices (all with the same angular arguments): Z

j 

j

j

dΩDm3 0 m3 ðRÞDm2 0 m2 ðRÞDm1 0 m1 ðRÞ ¼ 3 2 1 2P 3 hj1 , j2 ;m1 , m2 j j, m1 þ m2 ih j1 ,j2 ; m01 ,m02 j j, m01 þ m02 iðRÞ 6 j 7 6 Z 7 4 5 j   dΩDm3 0 m3 ðRÞDjm0 þm0 ,m1 þm2 ðRÞ 3

¼

P j

Z )

1

2

hj1 ,j2 ; m1 ,m2 j j, m1 þ m2 ihj1 , j2 ; m01 , m02 j j, m01 þ m02 i

j 

j

j

dΩDm3 0 m3 ðRÞDm2 0 m2 ðRÞDm1 0 m1 ðRÞ ¼ 3

2

1

8π 2 δ δ 0 0 0δ 2j3 þ 1 j3 j m1 þm2 ,m3 m1 þm2 ,m3



8π 2 hj1 ,j2 ;m1 ,m2 jj3 ,m3 i j1 ,j2 ;m01 ,m02 j3 ,m03 2j3 þ 1 ð2:159aÞ

Or in terms of 3-j symbols, 

Z j j j dΩDm1 0 m1 ðRÞDm2 0 m2 ðRÞDm3 0 m3 ðRÞ 1 2 3

¼ 8π

2

j1

j2

j3

m01

m02

m03



j1

j2

m1

m2

j3



m3 ð2:159bÞ

Note that (2.156) is a special case of (2.159a).

2.11

The Coupling of Three Angular Momenta

2.11

43

The Coupling of Three Angular Momenta

In the previous sections, we found that the states {jj, mi } completely characterized the resultant system when two angular momenta were added together. However, when coupling together three angular momenta, the states {jj, mi } are no longer unique since there is more than one way to add three angular momenta together to get the same total angular momentum. That is, there are several different sets of mutually commuting angular momentum operators that can be combined to describe the same physical system. For example, one may first add j1 and j2 and add the result to j3. Or, one may first add j2 to j3 and add the result to j1. In the first case, (2.121) gives jj12 , m12 i ¼

X m1

¼

X m12

j ,m

12 Cj12 jj , m1 ijj2 , m12  m1 i ) jðj12 , j3 Þj, mi 1 ,m1 ; j2 ,m12 m1 1

Cj,m j12 ,m12 ; j3 ,mm12 jj3 , m  m12 i

X m1

j ,m

Cj121 ,m112; j2 ,m12 m1 jj1 , m1 i ð2:160Þ

 jj2 , m12  m1 i

Notice that the resultant states for this coupling are symbolized by j( j12, j3)j, mi (here we are using alternate symbols for the C-G coefficients for ease of tracking). Similarly, for the second case, jj23 , m23 i ¼

X j ,m 23 C j23 jj , m2 ijj3 , m23  m2 i ) jðj1 , j23 Þj, mi 2 ,m2 ; j3 ,m23 m2 2 m2

X j,m X j ,m C j1 ,mm23 ; j23 ,m23 jj1 , m  m23 i Cj232 ,m223; j3 ,m23 m2 jj2 , m2 i ¼ m23

m2

ð2:161Þ

 jj3 , m23  m2 i

where the resultant states for this coupling are symbolized by j( j1, j23)j, mi. These two representations of jj, mi are physically equivalent and so must be connected by a unitary transformation: jðj12 , j3 Þj, mi ¼

X

hðj1 , j23 Þj, mjðj12 , j3 Þj, mi jðj1 , j23 Þj, mi

ð2:162Þ

j23

Substituting (2.160) and (2.161) into (2.162) gives P

j ,m

m1 , m12

¼

C j121 ,m112; j2 ,m12 m1 C j,m j12 ,m12 ; j3 ,mm12 j j1 , m1 ijj2 , m12  m1 ij j3 , m  m12 i P P m2 , m23 j23

j ,m

23 23 hðj1 , j23 Þj, mjðj12 , j3 Þj, miC j,m j1 ,mm23 ; j23 ,m23 C j2 ,m2 ; j3 ,m23 m2

jj1 , m  m23 ijj2 , m2 ijj3 , m23  m2 i

ð2:163Þ

44

2

Angular Momentum in Quantum Mechanics

The objects introduced in (2.162), h( j1, j23)j, mj ( j12, j3)j, mi, are known as the recoupling coefficients. Multiplying both sides by hj1, μ1jhj2, μ2jhj3, μ3j, X

j ,m

m1 , m12

¼

C j121 ,m112; j2 ,m12 m1 Cj,m j12 ,m12 ; j3 ,mm12 δm1 ,μ1 δm12 m1 ,μ2 δmm12 ,μ3

XX hðj1 , j23 Þj, mjðj12 , j3 Þj, miCj,m j1 ,mm23 ; j23 ,m23 m2 , m23 j |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Rj23 ,j12

23

j ,m Cj232 ,m223; j3 ,m23 m2 δmm23 ,μ1 δm2 ,μ2 δm23 m2 ,μ3 j ,m ) C j121 ,μ1 ;12j2 ,μ2 C j,m j12 ,m12 ¼μ1 þμ2 ; j3 ,μ3 δμ1 þμ2 þμ3 ,m ¼ δμ1 þμ2 þμ3 ,m X j ,m23  Rj23 ,j12 Cj23 C j,m j1 ,μ1 ; j23 ,m23 ¼μ2 þμ3 2 ,μ2 ; j3 ,μ3 ¼mμ1 μ2 j23 j,μ þμ þμ

j ,m

2 3 ) C j121 ,μ1 ;12j2 ,μ2 C j12 1,μ1 þμ ¼ 2 ; j3 ,μ3

ð2:164Þ

X j ,m23 j,μ þμ þμ Rj23 ,j12 Cj23 C j1 ,μ1 1 ; j232 ,μ23þμ3 2 ,μ2 ; j3 ,μ3 j23

where we have labeled the recoupling coefficients as the unitary transformation Rj23 ,j12 (just as a reminder that the transformation is a unitary one). Now multiply by C j,m j2 ,μ2 ; j3 ,μ and sum over μ2 with μ2 þ μ3 fixed 3

X j ,m X j ,m X j,μ þμ2 þμ3 j,μ þμ þμ 23 C j121 ,μ1 ;12j2 ,μ2 C j12 1,μ1 þμ Cj,m Cj23 C j,m Rj23 ,j12 C j1 ,μ1 1 ; j232 ,μ23þμ3 j2 ,μ2 ; j3 ,μ ¼ j2 ,μ2 ; j3 ,μ 2 ; j3 ,μ3 2 ,μ2 ; j3 ,μ3 3

μ2

3

μ

2 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

j23

δj23 ,j δm,m23

ð2:165Þ where (as indicated) we have used (2.125) with μ2 þ μ3 fixed (which determines μ3). The δj23 ,j collapses the sum over j23 to a single value j; and after relabeling j ! j23 on both sides of (2.165), and letting m ! m23, X j ,m j,μ þμ2 þμ3 j ,m23 j,μ þμ þμ C j121 ,μ1 ;12j2 ,μ2 C j12 1,μ1 þμ Cj23 ¼ Rj23 ,j12 Cj1 ,μ1 1 ; j232 ,μ23þμ3 2 ; j3 ,μ3 2 ,μ2 ; j3 ,μ 3

μ2

ð2:166Þ

Multiply both sides of (2.166) by CJ,M j1 ,μ1 ; j23 ,μ2 þμ3 and sum over μ1 (keeping μ1 þ μ2 þ μ3 fixed): X μ1 , μ2

j,μ þμ þμ

j ,m

j ,m

23 12 2 3 C j12 C j12 1,μ1 þμ Cj23 CJ,M j1 ,μ1 ; j23 ,μ2 þμ3 1 ,μ1 ; j2 ,μ2 2 ; j3 ,μ3 2 ,μ2 ; j3 ,μ 3

X j,μ þμ þμ ¼ Rj23 ,j12 C j1 ,μ1 1 ; j232 ,μ23þμ3 C J,M j1 ,μ1 ; j23 ,μ2 þμ3 μ1

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} δj,J δM,μ1 þμ2 þμ3

ð2:167Þ

2.11

The Coupling of Three Angular Momenta

45

where (as indicated) we have used (2.125) with μ1 þ μ2 þ μ3 fixed (which determines μ2 þ μ3). So now we have X μ 1 , μ2

j,μ þμ þμ

j ,m

j ,m

23 2 3 Cj121 ,μ1 ;12j2 ,μ2 Cj12 1,μ1 þμ C j23 C j,m j1 ,μ1 ; j23 ,μ2 þμ3 ¼ Rj23 ,j12 2 ; j3 ,μ3 2 ,μ2 ; j3 ,μ 3

ð2:168Þ

Since both coupling schemes result in the same projection m of the total angular momentum, the recoupling coefficients must be independent of m, which allows us to write hðj1 , j23 Þj, mjðj12 , j3 Þj, mi ! hðj1 , j23 Þjjðj12 , j3 Þji

ð2:169Þ

Rewriting in terms of the original variables gives hðj1 , j23 Þjjðj12 , j3 Þji ¼

X m1 , m12

j ,mm

j ,m

j,m 1 12 23 Cj12 C j,m j12 ,m12 ; j3 ,mm12 C j2 ,m12 m1 ; j3 ,mm12 C j1 ,m1 ; j23 ,mm1 1 ,m1 ; j2 ,m12 m1

ð2:170Þ In (2.170), the sum is still over all six m-values, but only two summation indices are shown, since, by the preceding analysis, they serve to fix all the other m-values (i.e., only two of the six summation indices are independent). Equation (2.170) can be used to find all the recoupling coefficients. However, again for reasons of symmetry, we prefer to use the Wigner 6-j symbols, which are defined by the formula

j2 j

j1 j3

j12 j23

ð1Þj1 þj2 þj3 þj ½ð2j12 þ 1Þ ð2j23 þ 1Þ2 hðj1 , j23 Þjjðj12 , j3 Þji 1

ð2:171Þ The 6-j symbols can also be computed via the Racah formula (

a

b

e

d c X " 

f

z0

) ΔðabeÞΔðacf ÞΔðbdf ÞΔðcdeÞ 0

ð1Þz ðz0 þ 1Þ! # ðz0  a  b  eÞ!ðz0  c  d  eÞ!ðz0  a  c  f Þ!ðz0  b  d  f Þ! xða þ b þ c þ d  z0 Þ!ða þ d þ e þ f  z0 Þ!ðb þ c þ e þ f  z0 Þ! ð2:172Þ

46

2 Angular Momentum in Quantum Mechanics

where (for example) ΔðabeÞ

  ða þ b  eÞ!ða  b þ eÞ!ða þ b þ eÞ! ða þ b þ e þ 1Þ!

ð2:173Þ

In addition, we must remember that the Δ(abe) vanish unless the triangle condition of (2.122b) is met for a, b, and e. The same is true for all the like terms in (2.172). For those who want more visibility on this topic, the Racah formula for the 6-j symbols is developed in App. C.

2.12

Spherical Tensor Operators and the Wigner-Eckart Theorem

Earlier in this chapter, we described how operators transformed under rotations [c.f. (2.84) and example 2.8]. In this section, we explore this subject in more detail. Operators transform in various ways when subject to a rotation and can be classified according to their resultant behavior. Such a classification scheme allows us to create a convenient operator basis for the development of photoelectron angular distribution theory (among other things). As it turns out, Cartesian tensor operators are inconvenient for the purpose of describing spherically or axially symmetric systems because they can be decomposed (“reduced”) into several different operators, each of which may transform differently under rotation (see example 2.19). Spherical tensor operators, on the other hand, are irreducible, making them excellent vehicles for the study of photoelectron angular distributions as well as a host of other topics in atomic and molecular physics. Operating with the language of spherical tensor operators also allows us to derive the most powerful theorem in our study of quantum angular momentum, the Wigner-Eckart theorem. Example 2.17 A scalar operator A is one that is invariant under a rotation so that A0 ¼ A. Therefore, according to the result from example 2.8, a scalar operator obeys the following commutation relation: ½J n , A ¼ 0

█

A tensor operator is a natural generalization of a vector operator. A spherical tensor operator of rank k (k ¼ integer) is actually a set of 2 k þ 1 functions, T ðqkÞ ; q ¼ k,  k þ 1, ⋯, k  1, k, that transform under the rotation R as follows:

R T ðqkÞ R 1

X ðk Þ Dkq0 q T q0 q0

ð2:174Þ

2.12

Spherical Tensor Operators and the Wigner-Eckart Theorem

47

An entirely equivalent definition of a spherical tensor operator is that they obey the following commutation relations with the (various manifestations of the) angular ! momentum operator J : h

h

i J z , T ðqkÞ ¼ ħqT ðqkÞ

ð2:175aÞ

i pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðkÞ J  , T ðqkÞ ¼ ħ kðk þ 1Þ  qðq  1Þ T q1

ð2:175bÞ

A spherical tensor operator of rank k ¼ 1 corresponds to a vector operator. The operators T ðq1Þ are related to the Cartesian components of the vector operator !

A ¼ Axbex þ Aybey þ Azbez as follows:  1  ð1Þ T 1 ¼  pffiffiffi Ax  iAy ; 2

ð1Þ

T 0 ¼ Az

ð2:176Þ

According to (2.175a and 2.175b), spherical tensor operators of rank 1 (vector operators) also have the following commutator properties with the angular momentum operators:

h

h i J z , T ðq1Þ ¼ ħqT ðq1Þ ðq ¼ 1, 0, 1Þ

ð2:177aÞ

i pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1Þ J  , T ðq1Þ ¼ ħ 1ð1 þ 1Þ  qðq  1Þ T q1

ð2:177bÞ

Example 2.18 Show that (2.174) and (2.175a and 2.175b) are equivalent definitions of a spherical tensor operator. From (2.29a) we have hk, q0 jJ z jk, qi ¼ ħqδq0 q )

k X q0 ¼k

ðkÞ

T q0 hk, q0 jJ z jk, qi ¼ ħqT qðkÞ

From (2.29b) and (2.29c), we can construct hk, q0 jJ  jk, qi ¼ ħ k X q0 ¼k

ðkÞ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kðk þ 1Þ  qðq  1Þδq0 ,qþ1 )

T q0 hk, q0 jJ  jk, qi ¼ ħ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðkÞ kðk þ 1Þ  qðq  1ÞT q1

48

2 Angular Momentum in Quantum Mechanics

Combine these results with (2.177): k P q0 ¼k

and

h i ðkÞ T q0 hk, q0 jJ z jk, qi ¼ J z , T qðkÞ k P

q0 ¼k

h i ðkÞ T q0 hk, q0 jJ  jk, qi ¼ J  , T qðkÞ

!

If J is decomposed in a spherical basis [c.f. (2.185) below], then we can merge the last two results into one form: h! i h! i k ! P ðkÞ J , T qðkÞ ¼ T q0 hk, q0 j J jk, qi ) J ^ n, T qðkÞ q0 ¼k

¼

k P

!

ðkÞ

q0 ¼k

T q0 hk, q0 j J ^njk, qi

From example 2.8, we can say h i i R ðδαÞT ðqkÞ R { ðδαÞ ¼ T ðqkÞ  δα J n , T ðqkÞ ħ Combining, k X ! i ðkÞ R ðδαÞT qðkÞ R { ðδαÞ ¼ T qðkÞ  δα T 0 hk, q0 jJ ^ njk, qi ħ q0 ¼k q

¼ ¼

  i ! ðkÞ T q0 hk, q0 j 1  δαJ ^ n jk, qi ħ q0 ¼k k P

k P q0 ¼k

ðkÞ

!

T q0 hk, q0 jeiδα J ^n=ħ jk, qi

This result also holds in the limit (for finite rotations):   iα ðkÞ T q0 hk, q0 jexp  J n jk, qi ħ q0 ¼k P k ðkÞ ¼ Dq0 q ðαÞT q0

R ðαÞT qðkÞ R 1 ðαÞ ¼

k P

q0

█

2.12

Spherical Tensor Operators and the Wigner-Eckart Theorem

49

We now have all we need to derive the Wigner-Eckart (W-E) theorem. From (2.95) and (2.174), under a rotation, we have h i P ðkÞ P 0 R T qðkÞ jα0 j0 m0 i ¼ R T qðkÞ R 1 R jα0 j0 m0 i ¼ Dkq0 q ðRÞT q0 Djμ0 m0 ðRÞjα0 j0 μ0 i ¼

P q0 , μ0

q0

h i 0 ðkÞ Dkq0 q ðRÞDjμ0 m0 ðRÞ T q0 jα0 j0 μ0 i

μ0

ð2:178Þ where the label “α” stands for all other quantum numbers needed to uniquely identify the state. h i Multiply the quantity T qðkÞ jα0 j0 m0 i from the left by the bra hαjm| and use (2.178) h i h i hαjmj T ðqkÞ jα0 j0 m0 i ¼ hαjmjR 1 R T ðqkÞ jα0 j0 m0 i h i P P k ðk Þ 0 0 0 j0 ¼ Dj ð R Þhαjμj D ð R ÞD ð R Þ T j μ i ) hαjmjT ðqkÞ jα0 j0 m0 i jα 0 0 0 0 μm qq μm q μ

¼

X μ

q 0 , μ0

hαjμj

X μq0 , μ0

0

ðk Þ

j k 0 0 0 Dj μm ðRÞDq0 q ðRÞDμ0 m0 ðRÞhαjμjT q0 jα j μ i

ð2:179Þ

Integrate both sides of (2.179) over the Euler angles dΩ ¼ dϕ sin θ dθdχ, hαjmjT ðqkÞ jα0 j0 m0 i

Z dΩ ¼

X

ðk Þ

μ, q0 , μ0

) 8π 2 hαjmjT ðqkÞ jα0 j0 m0 i ¼ ) hαjmjT ðqkÞ jα0 j0 m0 i ¼

hαjμjT q0 jα0 j0 μ0 i X

μ, q0 , μ0

ðk Þ

Z

0

dΩDjμm ðRÞ Djμ0 m0 ðRÞDkq0 q ðRÞ

hαjμjT q0 jα0 j0 μ0 i

X Cj,m k,q; j0 ,m0 ð2j þ 1Þ μ, q0 , μ0

8π 2 C j,m 0 Cj,μ0 0 ð2j þ 1Þ k,q; j ,m0 k ,q0 ; j ,μ0

ðk Þ

hαjμjT q0 jα0 j0 μ0 iC j,μ k 0 ,q0 ; j0 ,μ0

ð2:180Þ

where we used (2.159a). Now define the reduced matrix elements:   0 1 X ðk Þ hα jT ðkÞ α0 j0 i ð1Þkj þj ð2j þ 1Þ2 hα jμjT q0 jα0 j0 μ0 iC j,μ k,q0 ; j0 ,μ0 μ, q0 , μ0

which gives

ð2:181Þ

50

2

Cj,m k,q; j0 ,m0

0

hαjmjT ðqkÞ jα0 j0 m0 i ¼ ð1Þkþj j

ð2j þ 1Þ

¼ ð1Þ

kþj0 j

¼ ð1Þ

kþj0 j

1 2

  hα jT ðkÞ α0 j0 i

ð1Þ

kj0 þm

ð1Þ

kj0 þm

j

¼ ð1Þjþm

Angular Momentum in Quantum Mechanics

k

0

j

m

m j

k

j0

m !

q

m0

m q m  j k jm m

j 0

q

0

) hαjmjT ðqkÞ jα0 j0 m0 i ¼ ð1Þ

j0

k

q

! !

  hα jT ðkÞ α0 j0 i   hα jT ðkÞ α0 j0 i

  hα jT ðkÞ α0 j0 i 

j0 0

m

  hα jT ðkÞ α0 j0 i

ð2:182Þ

where we used (2.144b) and (2.145b and 2.145c). We also used the fact that for any angular momentum quantum number j and its projection m, the quantity j  m¼integer, and for any integer k, (1)k ¼ (1)k. The theorem (2.182) is called the Wigner-Eckart (W-E) theorem, and it has profound physical implications. In a nutshell, the W-E theorem states that the matrix elements of a spherical tensor operator between the states jj, mi and jj0, m0i can be written as the product of a C-G coefficient (or equivalently, a 3-j symbol), which contains all the information on the orientation (geometry) of the system (i.e., depends only on m0, m, and q), and a ðkÞ “reduced” matrix element, hj0 jjT^ jjji which contains all the information on the dynamics of the system (i.e., depends only on j, j0, and k). Equation (2.182) is the usual form of the W-E theorem and reflects standard conventions [3, 5, 10]. We can also write the W-E theorem as follows:   0 0 0 ,m 0 0  ðk Þ  αji hα0 j0 m0 jT ðqkÞ jαjmi / ð1Þkþjj C jk,q; j,m hα j T     0 0 kþj0 j kþjj0 j0 ,m0 ,m 0 0  ðk Þ  0 0  ðk Þ  ð1Þ C j,m; k,q hα j T ¼ ð1Þ αji ¼ C jj,m; αji k,q hα j T ð2:183Þ The question now arises as to how we can create spherical tensor operators. One way to do so is by forming the spherical tensor operator T qðkÞ from the product of two other spherical tensor operators T ðqk11 Þ and T ðqk22 Þ as follows: T ðqkÞ ¼

XX hk 1 , k 2 ; q1 , q2 jk, qiT ðqk11 Þ T ðqk22 Þ q1

ð2:184Þ

q2

To prove that T ðqkÞ is, in fact, a spherical tensor operator, we need to show that the indicated product transforms according to (2.174):

2.12

Spherical Tensor Operators and the Wigner-Eckart Theorem

1 R T ðkÞ ¼ q R

¼

PP q1 q2

51

hk 1 ,k 2 ; q1 , q2 jk,qiR T qðk11 Þ R 1 R T qðk22 Þ R 1

P PPP ðk Þ ðk Þ hk1 ,k2 ; q1 ,q2 jk,qiDkq10 q Dkq20 q T q0 1 T q0 2 1 1

q1 q2 q0 q0 1 2

2 2

1

2

PPPXX 00 ðk Þ ðk Þ ¼ hk 1 ,k2 ; q1 ,q2 jk00 ,q00 ihk 1 , k2 ;q1 ,q2 jk, qihk1 ,k 2 ; q01 ,q02 jk 00 ,q0 iDkq0 q00 T q0 1 T q0 2 1 2 q01 q02 k00 q1 q2 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ¼ ¼

PP q01 q02

δkk00 δqq00

ðq0 Þ ðq0 Þ hk1 ,k 2 ; q01 ,q02 jkq0 iT k1 1 T k2 2 Dkq0 q

PPP q0

q01

q02

!

ðk Þ ðk Þ



PP P q0 q01 q02

hk1 ,k 2 ; q01 , q02 jk,q0 iT q0 1 T q0 2 Dkq0 q ¼ 1

2

P q0

ðk Þ ðk Þ

hk 1 , k2 ;q01 ,q02 jk, q0 iT q0 1 T q0 2 Dkq0 þq0 ,q 1

2

1

2

ðkÞ

Dkq0 q T q0

where we used the Clebsch-Gordan series (2.149) andPthe Porthogonality PPP relation of ! , which does the C-G coefficients (2.125). We also made the move, q01 q02

q0 q01 q02

not change the value of the equation since q0 ¼ q01 þ q02 . We see that the RHS of (2.184) transforms correctly, thereby validating the fact that we can construct spherical tensor operators of higher or lower rank by multiplying two tensor operators together. It is important to realize that the components Li of the orbital angular momentum vector operators can themselves be written in the form of spherical tensor operators of rank one. If written in a spherical basis, the orbital angular momentum operators would appear as ð1Þ L1

rffiffiffi rffiffiffi 1 1 ð1Þ ð1Þ ¼ L ; L ¼ Lz ; L1 ¼ L 2 þ 0 2 

ð2:185Þ

We would then find that the orbital angular momentum operators obey the following commutation relations:

h

h i Lz , LðqkÞ ¼ qLðqkÞ

ð2:186aÞ

i 1 ðk Þ L , LðqkÞ ¼ ½k ðk þ 1Þ  qðq  1Þ2 Lq1

ð2:186bÞ

making them (by definition) spherical tensor operators of rank one. Example 2.19 As an exercise in operating with angular momentum operators in their spherical tensor form, we will now use (2.184) to decompose the !

!

operator L2z in terms of the rank-1 vector operators U and V as follows [3, 4, 6]. First consider

52

2

ð2Þ

T0 ¼

PP q1 q2

Angular Momentum in Quantum Mechanics

h1, 1; q1 , q2 j2, 0iU 1ðq1 Þ V 1ðq2 Þ

¼ h1, 1; 0, 0j2, 0iU 0 V 0 þ h1, 1;  1, 1j2, 0iU 1 V þ1 þ h1, 1; 1,  1j2, 0iU þ1 V 1 ð2:187Þ There are only three non-zero terms in the expansion. All other terms in the double summation are zero because the C-G coefficients in those terms violate the triangle condition 0 ¼ q ¼ q1 þ q2. Using standard tables of C-G coefficients [easier than trying to apply (2.143)], ð2Þ T0

rffiffiffi rffiffiffi rffiffiffi 2 1 1 ¼ U V þ U V þ U V 3 0 0 6 1 þ1 6 þ1 1 rffiffiffi r ffiffi ffi rffiffiffi      1  1   2 1 1 U  iU y V x þ iV y  UV  U þ iU y V x  iV y ¼ 3 z z 2 6 x 2 6 x rffiffiffi  1 2U z V z  U x V x  U y V y ¼ 6 ð2:188Þ

where we also used (2.176). ! ! ! Now let U ¼ V ¼ L , ð2Þ T0

!

ð2Þ L0

rffiffiffi rffiffiffi   1 2 1 3Lz  L2 ¼ 2Lz Lz  Lx Lx  Ly Ly ¼ 6 6 pffiffiffi ð2Þ 6L0 þ L2 2 ) Lz ¼ 3

ð2:189Þ

ð2Þ

L0 is a spherical tensor angular momentum operator of rank 2, and L2 is a scalar invariant. We say that we have “reduced” the Cartesian tensor L2z to the sum of a second-rank spherical tensor and a rank-zero (scalar) spherical tensor. █ P Example 2.20 Evaluate the sum: m2 jY lm j2 . m

First, consider the following [4, 6]: X X 2 m2 dlm0 m ðθÞ ¼ m2 hl,m0 jeiθLy jl,mihl, mjeiθLy jl,m0 i m

m

¼

X

hl,m0 jeiθLy L2z jl,mihl, mjeiθLy jl, m0 i

m

¼ hl, m0 jeiθLy L2z eiθLy jl, m0 i ¼ hl, m0 jR ðα ¼ 0,β ¼ θ,γ ¼ 0ÞL2z R 1 ðα ¼ 0, β ¼ θ, γ ¼ 0Þjl, m0 i ð2:190Þ

2.12

Spherical Tensor Operators and the Wigner-Eckart Theorem

53

Substituting (2.189) into (2.190) gives X





2 m2 dlm0 m ðθÞ

m

! pffiffiffi ð2Þ 6L0 þ L2 ¼ hl, m j R R 1 jl, m0 i 3 0

ð2:191Þ

We apply the definition in (2.174) to the RHS interior term of (2.191): ! pffiffiffi ð2Þ 6L0 þ L2 hl, m jR R 1 jl, m0 i 3 pffiffiffi X 2 6 1 ð2Þ ¼ D20 hl, m0 jLq0 l, m0 ji þ hl, m0 jR L2 R 1 jl, m0 i 3 q0 ¼2 q 0 3 pffiffiffi 6 2 1 ð2Þ ¼ D00 hl, m0 jL0 jl, m0 i þ hl, m0 jL2 jl, m0 i 3 3 pffiffiffi     6 1 1 0 2 2 0 ¼ P ð cos θÞ pffiffiffi hl, m j 3Lz  L jl, m i þ hl, m0 jL2 jl, m0 i 3 2 3 6 0

ð2:192Þ

In (2.192) only the term q0 ¼ 0 contributed in the summation, since ð2Þ hl, m0 jLq0 6¼0 jl, m0 i ¼ 0 , (which can be easily proved using the W-E theorem), and since L2 was unchanged by the rotation because it is a scalar invariant. We also used (2.110). Substituting the results from (2.192) into (2.191), X 2 m2 dlm0 m ðθÞ

¼

m

¼ ¼

pffiffiffi    6 1  P2 ð cos θÞhl, m0 j pffiffiffi 3L2z  L2 jl, m0 i 3 6

1 þ hl, m0 L2 jl, m0 h 3 i 3m0 2  lðl þ 1Þ 3

ð2:193Þ

1 P2 ð cos θÞ þ lðl þ 1Þ 3

1 2 m0 P2 ð cos θÞ þ l ðl þ 1Þ sin 2 θ 2

For m0 ¼ 0, we have X 2 1 m2 dl0m ðθÞ ¼ l ðl þ 1Þ sin 2 θ 2 m

ð2:194Þ

Recalling (2.98) and (2.109) gives us finally [6] X m

m2 jY lm j2 ¼

l ðl þ 1Þ ð2l þ 1Þ sin 2 θ 8π

ð2:195Þ

█

54

2

Angular Momentum in Quantum Mechanics

The W-E theorem is essential for quantitatively evaluating operators that represent the interaction of atoms and molecules with photons. To use this theorem, however, we must be able to evaluate the reduced matrix elements that result from the use of the W-E theorem. Fortunately, there are ways to find reduced matrix elements that commonly appear. We start with the matrix elements for the normalized spherical harmonic tensor operators of (4.11). According to the W-E theorem, hl, mjCðqkÞ jl0 , m0 i

 ¼ ð1Þ

l

k

l0

m

q

m0

lm



hlkC ðkÞ kl0 i

ð2:196Þ

But from (2.157), rffiffiffiffiffiffiffiffiffiffiffiffiffi Z2π Z1 4π dϕ dð cos θÞY lm ðθ,ϕÞY kq ðθ,ϕÞY l0 m0 ðθ,ϕÞ 2k þ 1 0 1 ð2:197Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l k l0 ! l k l0 ! 0 ¼ ð1Þ2ðkl Þþm ð2l þ 1Þð2l0 þ 1Þ 0 0 0 m q m0     qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l k l0 l k l0 m 0 ð2l þ 1Þð2l þ 1Þ ¼ ð  1Þ m q m0 0 0 0

hl,mjCðqkÞ jl0 , m0 i ¼

0

where we have used the fact that ð1Þ2ðkl Þ ¼ 1. Comparing (2.196) and (2.197), qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l hlkC kl i ¼ ð1Þ ð2l þ 1Þð2l0 þ 1Þ 0 ðk Þ

0

l

k

l0

0

0

 ð2:198Þ

We can combine (2.145a) and (2.146) with (2.198) to derive the following results: hlkC ð0Þ kl0 i ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2l þ 1Þδll0

ð2:199Þ

and hlkCðkÞ k0i ¼ ð1Þl h0kC ðkÞ kli ¼ δlk

ð2:200Þ

Since this is fun, we can keep going for a while longer. Example 2.21 Consider the matrix elements of the unit tensor operator. By the W-E theorem, hαjmjI jα0 j0 m0 i ¼ ð1Þjm



j

0

j0

m

0

m0



hα jk1kα0 j0 i

ð2:201Þ

2.12

Spherical Tensor Operators and the Wigner-Eckart Theorem

55

Equation (2.145e) implies that, inside the 3-j symbol, we must have j ¼ j0, 0 0



0

) hαjmjI jα j m i ¼ ð1Þ

jm

j m

0 0

j m0



hα jk1kα0 j0 i

ð2:202Þ

Using (2.145a) and (2.146), 0

j

0

B hαjmjIjα0 j0 m0 i ¼ ð1Þjm @ m |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}

C 0 0 0 |{z} m0 A hα jk1kα j i ¼m

¼δαα0 δjj0 δmm0

j

m

m

0

¼ ð1Þ ð2j þ 1Þ |fflfflfflfflfflffl{zfflfflfflfflfflffl}

1=2

¼ ð1Þ

2ðjmÞ

0

!

j

jm

1

j

¼1

) hα jk1kα0 j0 i ¼ δαα0 δjj0

h α j k 1k α 0 j 0 i hα jk1kα0 j0 i

pffiffiffiffiffiffiffiffiffiffiffiffi 2j þ 1

ð2:203Þ █

Example 2.22 Consider hαjjjJ z jα0 j0 j0 i ¼ jδαα0 δjj0

ð2:204Þ

But by the W-E theorem [and anticipating the delta functions in (2.204)], 0 0 0

hαjjjJ z jα j j i ¼

ð1Þ hαjjjJ 0 jα0 j0 j0 i

 jj

¼ ð1Þ

j

1

j0

j

0

j0



hα jkJ ð1Þ kα0 j0 i

ð2:205Þ

|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}   j 1 j

¼

j

0

j

We use the relation [3] (which can be proved using (2.138), see Problem 4.2) 

j

1

j

j

0

j



rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j ¼ ðj þ 1Þð2j þ 1Þ

ð2:206Þ

Thus, rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j hα jkJ ð1Þ kα0 j0 i hαjjjJ z jα j j i ¼ ðj þ 1Þð2j þ 1Þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j hα jkJ ð1Þ kα0 j0 i ) jδαα0 δjj0 ¼ ðj þ 1Þð2j þ 1Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ) hα jkJ ð1Þ kα0 j0 i ¼ jðj þ 1Þð2j þ 1Þδαα0 δjj0 0 0 0

ð2:207Þ █

56

2

Angular Momentum in Quantum Mechanics

Example 2.23 Extending our thoughts from the previous example, let us consider hαjmjJ z jα0 j0 m0 i ¼ mδαα0 δjj0 δmm0

ð2:208Þ

But, by the W-E theorem, ð1Þ

hαjmjJ z jα0 j0 m0 i ¼ hαjmjJ 0 jα0 j0 m0 i   j 1 j0 ¼ ð1Þjm hα jkJ ð1Þ kα0 j0 i m 0 m0 |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}   j 1 j ¼

m

0

ð2:209Þ

m

Combining (2.207, 2.208 and 2.209), 



j 1 j  α jkJ ð1Þ kα0 j0  m 0 m  j 1 j pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ð1Þjm jðj þ 1Þð2j þ 1Þδαα0 δjj0 m 0 m   j 1 j m ) ¼ ð1Þjm pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m 0 m jðj þ 1Þð2j þ 1Þ

mδαα0 δjj0

¼ ð1Þjm

ð2:210Þ

So, the W-E theorem is also useful for finding the value of 3-j symbols. █ We can use similar techniques to find the reduced matrix elements of the Pauli spin matrices of (6.7) and (6.8). First, we consider the spin operator of (6.9). We could write its components in a spherical basis much as we did for the orbital angular momentum operators in (2.185). The spin operators, so written, would then obey the relations of (2.186). We then argue as we did above [3] D   E pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sSð1Þ s0 ¼ sðs þ 1Þð2s þ 1Þδss0

ð2:211Þ

For spin 1/2, rffiffiffi D   E D pffiffiffi  E 2 1 1  ð1Þ 1 Sð1Þ 1 ¼ 2 3 ¼ 6 ¼ σ 2 ħ 2 2 ħ 2 2

in units of ħ ¼ 1:

ð2:212Þ

2

Angular Momentum in Quantum Mechanics

57

Problems 2.1. Angular momentum j ¼ 1. (a) Using the expressions for Jx and Jy from example 2.2, find, by matrix multiplication, the matrices JxJy and JyJx and thus find the commutator [Jx, Jy]. Ensure your result satisfies (2.1). (b) Find the matrix representations for the operators J ¼ Jx  iJy, J 3z , J 3 . Which of these operators, if any, are Hermitian? (c) Find the eigenvalues and (normalized) eigenvectors for Jx in the given basis. (d) Suppose you have a particle of spin 1. What are the probabilities that, having measured the angular momentum of the particle in the x-direction to be ħ, a subsequent measurement in the z-direction yields the values ħ, 0, or ħ? (e) When a beam of spin ¼ 1 particles is analyzed in the z-direction, it is found that the probability of measuring Jz ¼ ħ is 50% and the probability of measuring Jz ¼ 0 is 50%. Furthermore, when the beam is analyzed in the x-direction, it is found that hJ x i ¼ pħffiffi2 g. In terms of g, what is the value, or possible range of values, of hJyi? HINT: Expand the state in a linear superposition of eigenvectors of the Jz operator in the most general form possible consistent with the given information. 2.2. Starting with (2.36a, 2.36b and 2.36c) and (2.37a, 2.37b and 2.37c), verify (2.41a, 2.41b, and 2.41c) and (2.42a and 2.42b). 2.3. In example 2.5, we found the general form for Yll(θ, ϕ) to be. ð1Þl Y ll ðθ, ϕÞ ¼ l 2 l!

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2l þ 1Þ! ilϕ e ð sin θÞl 4π

(a) Operate on this form with the lowering operator of (2.34c), (2.42a), and (2.53) to verify (2.54). (b) Use this form to construct Y22(θ, ϕ). Then use (2.53) and (2.55) to form Y21(θ, ϕ), Y20(θ, ϕ), Y2,  1(θ, ϕ), and Y2,  2(θ, ϕ), thus verifying (2.67d, 2.67e and 2.67f). Use your results to verify (2.80) for the family Y2m(θ, ϕ). 2.4. Under a rotation of coordinates through an angle dα about an axis along b u, the unit vectors beðlÞ along the rectangular axes transform as be0

ðlÞ

¼ beðlÞ þ dαb u  beðlÞ. !

Use this information to show that any component of a vector observable V has a !

commutator with the components of the angular momentum operator J given by,


X  V i , J j ¼ iħ εijk V k : k

58

2

Angular Momentum in Quantum Mechanics

2.5. Verify (2.141) and (2.142). 2.6. Prove the following recursion relation for the C-G coefficients: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðj þ mÞ ðj  mÞ ðj  j1 þ j2 Þ ðj þ j1  j2 Þ ðj þ j1 þ j2 þ 1Þ ðj1  j þ j2 þ 1Þ j1,m C j1 ,m1 ; j2 ,m2 4j2 ð2j  1Þ ð2j þ 1Þ   j ðj þ 1Þ  j2 ðj2 þ 1Þ þ jðj þ 1Þ j,m ¼ m1  m 1 1 C j1 ,m1 ;j2 ,m2 2jðj þ 1Þ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi i uh u ðj þ 1Þ2  m2 ðj  j1 þ j2 þ 1Þ ðj þ j1  j2 þ 1Þ ðj þ j1 þ j2 þ 2Þ ðj1  j þ j2 Þ t Cjjþ1,m  1 ,m1 ; j2 ,m2 4ðj þ 1Þ2 ð2j þ 1Þ ð2j þ 3Þ !

!

!

2.7. Suppose we have the total angular momentum operator J ¼ J 1 þ J 2 . Prove the following result for the matrix elements of the operator J1z in the basis ! j j, mi of the total angular momentum operator J . h j, mjJ 1z j j, mi ¼

½jðj þ 1Þ þ j1 ðj1 þ 1Þ  j2 ðj2 þ 1Þ ħm 2jðj þ 1Þ

2.8. Find the 3-j version of (2.125). 2.9. Prove the following: X  l1

l2

l

 Y l1 m1 ðθ, ϕÞY l2 m2 ðθ, ϕÞ

m1 m2 m sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2l1 þ 1Þð2l2 þ 1Þ l1 ¼ 4π ð2l þ 1Þ 0

m1 , m2

HINT: Form the expansion Y l1 m1 Y l2 m2 ¼

l2 0 P

 l Y lm ðθ, ϕÞ 0

alm Y lm , and solve for the expan-

l, m

sion coefficients. 2.10. Find the reduced matrix elements for the solid spherical harmonic operators. Note: The definition of the solid harmonic operator can be found in Problem 4.3. !

2.11. A particle of spin S moves in a central potential in a state with orbital angular !

!

momentum L and total angular momentum J . A uniform magnetic field

!

!

B ¼ Bbz is applied to the system. Calculate the matrix elements h j, mjS

!

B j j, m0 i.

2

Angular Momentum in Quantum Mechanics

59

2.12. Prove the commutation relation: ii Xh h J i , J i , T ðqkÞ ¼ ħ2 k ðk þ 1ÞT ðqkÞ i

2.13. A particle moves in a central potential in a state of orbital angular momentum ! L . For purposes of calculating dipole interactions, one may need to know the matrix elements:   1 hl, mj ni nj  δij jl, m0 i 3 The W-E theorem guarantees that within the jl, mi subspace, any two traceless symmetric rank-2 tensor operators are proportional. Therefore, one can write     1 2 hl, mj ni nj  δij jl, m0 i ¼ chl, mj Li Lj þ Lj Li  L2 δij jl, m0 i 3 3 Find the coefficient c. Note: In this problem, the components ni, nj and Li, Lj are the spherical !

components of the vector operators br and L , respectively. The unit vector br is defined in Problem 3.8. Note: The Kronecker delta δij is defined in Problem 3.8. 2.14. The quantum mechanical momentum operator in the position representation is given in (2.32). (a) show that, p2 ¼ ħ2

1 ∂ 2 ∂ ħ2 2 r þ L . r 2 ∂r ∂r r 2

(b) Find the reduced matrix elements for the momentum operator. 2.15. Prove the projection theorem, !

!

h j, mj J A j j, mi h j, m jAq j j, mi ¼ h j, m0 jJ q j j, mi ħ2 jðj þ 1Þ 0

!

where Aq is the qth spherical component of the vector operator A and Jq is the !

qth spherical component of J .

Chapter 3

Classical Model of Photoelectron Angular Distributions

To begin our discussion of the classical theory of atom/molecule-photon interactions, we start with Maxwell’s equations (here presented in Gaussian units) [11]: ! !
!  ∇  B r,t ¼ 0 ð3:1aÞ
 ! !
 ! ! ! 1 ∂B r , t ¼0 ð3:1bÞ ∇  E r,t þ c ∂t
 ! !
!  ! ð3:1cÞ ∇  E r , t ¼ 4πρ r , t
 ! !

 ! ! ! 4π ! ! 1 ∂E r , t ¼ ð3:1dÞ ∇  B r,t  J r,t c c ∂t

 ! ! ! where ρ r , t and J r , t are the charge and current densities, respectively, that

are the sources of the fields. The fields can be written in terms of scalar and vector potentials as follows:
 ! !
 ! B r,t ¼ ∇  A r,t

! !

!
!  !
!  1 ∂A r , t  ∇Φ r , t E r,t ¼  c ∂t




! !



ð3:2aÞ

ð3:2bÞ

A major portion of this chapter follows closely the development presented in chapter 3, application 6 of [3]. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 V. T. Davis, Introduction to Photoelectron Angular Distributions, Springer Tracts in Modern Physics 286, https://doi.org/10.1007/978-3-031-08027-2_3

61

62

3

Classical Model of Photoelectron Angular Distributions

We can manipulate the mathematical forms of the potentials in certain ways as long as the fields remain unchanged by these manipulations. These manipulations are called gauge transformations. One such gauge transformation ! !
!  ∇  A r,t ¼ 0

ð3:3Þ

is called the Coulomb, or transverse, gauge transformation, and is convenient under certain circumstances [11, 12]. With this condition, the source-dependent Maxwell equations can be written as

 ! ! ∇2 Φ r , t ¼ 4π ρ r , t

ð3:4Þ



 ! 2! !
 ∂ A r , t ∂Φ r,t ! 4π ! ! 1 1 ¼ ∇  J r,t ∇ A r,t  2 2 c c c ∂t ∂t
2! !



ð3:5Þ

Example 3.1 Verify (3.4) and (3.5). Start with (3.1c): !

!

∇  E ¼ 4πρ ) 


 !2 !2 1 ∂ ! ! ∇  A  ∇ Φ ¼ 4πρ ) ∇ Φ ¼ 4πρ c ∂t |fflfflfflfflffl{zfflfflfflfflffl} ¼0

where we also used (3.2b). Now from (3.1d), ! !
! ! ! 1 ∂ 1 ∂A ! 1 ∂E 4π !  ¼ J )∇ ∇A   ∇Φ ∇B c ∂t c ∂t c c ∂t !

!

!
!

!

! ¼

4π ! J c

2!

1 ∂ A 1 ! ∂Φ 4π ! )∇ ∇A ∇ Aþ 2 2 þ ∇ ¼ J c c c ∂t ∂t |fflfflfflfflffl{zfflfflfflfflffl} 2!

¼0

2!

1 ∂ A 1 ! ∂Φ 4π ! )∇ A 2 2 ¼ ∇  J c c c ∂t ∂t 2!

where we used (3.2a) and (3.2b), and the vector identity,
! ! !
! ! ! ! ∇  ∇  A ¼ ∇ ∇  A  ∇2 A █

3

Classical Model of Photoelectron Angular Distributions

63

Under the Coulomb gauge and in regions far from any sources, (ρ ¼ J ¼ 0), (3.2a and 3.2b) give
 ! !
 ! B r,t ¼ ∇  A r,t

! !

ð3:6aÞ

!
! 
 ∂A r,t ! ! 1 E r,t ¼  c ∂t

ð3:6bÞ

where  we also noted that Φ ¼ 0
from (3.4). Thus, finding
under these conditions, !
!  ! ! ! ! A r , t is equivalent to finding E r , t and B r , t . So, how do we find A r , t ?

!
!

Under the Coulomb gauge and in regions far from any sources, (ρ ¼ J ¼ 0), the vector potential solves the wave equation [c.f. (3.5)]:
 2! ! ∂ A r,t 1 ! ¼0 ∇ A r,t  2 2 c ∂ t ! 2 !




ð3:7Þ

The solution to this equation is !
!



!

A r , t ¼ A0e

! !

i k  r ωt



!

þ c:c:; k2 ¼ ω2 =c2 ; A 0 ¼ A0bε

ð3:8Þ !

!

The source-free Maxwell equations can also be manipulated to show that E and B solve similar wave equations: 2!

1 ∂ E ∇ E¼ 2 c ∂t 2 2!

2!

1 ∂ B ∇ B¼ 2 c ∂t 2 2!

ð3:9aÞ

ð3:9bÞ

with solutions ! !  ! E ¼ E 0 ei k  r ωt þ c:c:

ð3:10aÞ

! !  ! B ¼ B 0 ei k  r ωt þ c:c:

ð3:10bÞ

!

!

64

3

Classical Model of Photoelectron Angular Distributions

Example 3.2 Verify (3.9a) and (3.9b). We start with the source-free Maxwell equations:
! ! ! )∇ ∇E ¼∇ !

!

1 ∂B  c ∂t

!

!
!



!

2!

!

)∇ ∇E ∇ E ¼∇ |fflfflfflfflffl{zfflfflfflfflffl}

!

1 ∂B  c ∂t

!

¼0

2! 2!
 ! 1∂ E 1∂ E 1∂ ! ! ¼ ∇  B ¼  2 2 ) ∇2 E ¼ 2 2 c ∂t |fflfflfflfflfflffl{zfflfflfflfflfflffl} c ∂t c ∂t ! 1∂ E c ∂t

Similarly,
! ! ! ∇ ∇B ¼∇ !

!

1 ∂E c ∂t

!


 !
! ! ! 1 ∂ ! ! ) ∇ ∇  B  ∇2 B ¼ ∇E c ∂t |fflfflfflfflfflffl{zfflfflfflfflfflffl} |fflfflfflfflffl{zfflfflfflfflffl} ¼0

!

1c∂∂tB

2!

1 ∂ B )∇ B¼ 2 c ∂t 2 2!

█

!

!

Our choice of gauge implies k  A 0 ¼ 0, so that the fields described by (3.2a and 3.2b) [or equivalently, by (3.6a and 3.6b)] can be written as  !   ! i k  r ωt þ c:c: E ¼ ikA 0 e

ð3:11aÞ

 !   ! i k  r ωt þ c:c: B ¼ i k  A0 e

ð3:11bÞ

!

!

!

!

!

By convention, the monochromatic plane electromagnetic wave is said to be polarized in the direction the electric field vector points. For a monochromatic plane ! electromagnetic wave propagating in an arbitrary direction k ¼ kb k ¼ ðω=cÞb k¼ b ð2π=λÞk with polarization direction bε, we write ! ! 
 E r , t ¼ E 0bεei k  r ωt þ c:c:

ð3:12aÞ


 ! kE B r,t ¼ b

ð3:12bÞ

! !

! !

k ¼ 0. Notice that our choice of gauge implies that bε  b

3

Classical Model of Photoelectron Angular Distributions

65

A useful approximation is realized when we consider interactions of atomic (or molecular) electrons

with photons of wavelengths in the visible range 0 0 ! λ ¼ 4000 A 7000 A . The size of r in (3.8) and (3.12) is on the order of the size (or molecule), that is, on the order of a0, the Bohr radius of the atom

0 a0  0:5 A . Under these circumstances, we have !

!

kr ¼




 2π b ! k  r > ð l þ 1 Þ
m l
m     > 2 2 > > k  ℜ l
1 þ k  ℜ lþ1 > Y l
1,m b Y lþ1,m b > > > > ð 2l þ 1 Þ ð 2l
1 Þ ð 2l þ 3 Þ ð 2l þ 1 Þ > > > > > > = < 1 1



P 2 2 2 2 2 2 ð l þ 1Þ
m  l
m þ > m> > > > > ð2l þ 1Þð2l
1Þ ð2l þ 3Þð2l þ 1Þ > > > > h i



 > > > > > > :  Y lþ1,m b k Y l
1,m b k e
iðδlþ1
δl
1 Þ þ Y l
1,m b k Y lþ1,m b k eiðδlþ1
δl
1 Þ ℜ lþ1 ℜ l
1 ;

ð4:26Þ We now call upon (2.68), which we reproduce here in a slightly different form,

84

4

Quantum Treatment of Photoelectron Angular Distributions (Dipole. . .



cos θY lm

1

12 ðl þ 1Þ2
m2 2 l2
m 2 ¼ Y þ Y ð2l þ 3Þð2l þ 1Þ lþ1,m ð2l þ 1Þð2l
1Þ l
1,m

ð4:27Þ

First, we square both sides of (4.27) and then sum over m,

X

cos 2 θjY lm j2 8 9



2 2 2 2 > > ð l þ 1 Þ
m l
m 2 2 > > > > jY lþ1,m j þ jY l
1,m j > > < = X ð2l þ 3Þð2l þ 1Þ ð2l þ 1Þð2l
1Þ ¼

1

12   > > > m > ðl þ 1 Þ2
m 2 2 l 2
m2 > > > Y lþ1,m Y l
1,m þ Y lþ1,m Y l
1,m > :þ ; ð2l þ 3Þð2l þ 1Þ ð2l þ 1Þð2l
1Þ " # X X 1 ðl þ 1Þ2 jY lþ1,m j2
m2 jY lþ1,m j2 ¼ ð2l þ 3Þð2l þ 1Þ m m " # X X 1 2 2 2 2 m jY l
1,m j þ l jY l
1,m j
ð2l þ 1Þð2l
1Þ m m 1

12 X ðl þ 1Þ2
m2 2   l 2
m2 þ Y lþ1,m Y l
1,m þ Y lþ1,m Y l
1,m ð2l þ 1Þð2l
1Þ m ð2l þ 3Þð2l þ 1Þ m

ð4:28Þ Rearranging, 1

12 X ðl þ 1Þ2
m2 2   l2
m2 Y lþ1,m Y l
1,m þ Y lþ1,m Y l
1,m ð2l þ 1Þð2l
1Þ m ð2l þ 3Þð2l þ 1Þ # " X X X 1 2 2 2 2 2 2 ¼ cos θjY lm j
ð l þ 1Þ jY lþ1,m j
m jY lþ1,m j ð2l þ 3Þð2l þ 1Þ m m m " # X X 1 2 2 2 2
l m jY l
1,m j jY l
1,m j
ð2l þ 1Þð2l
1Þ m m

ð4:29Þ

If you look closely, you will notice that (4.29) is symmetric in the quantity: 1

12 X ðl þ 1Þ2
m2 2   l2
m 2 Y lþ1,m Y l
1,m þ Y lþ1,m Y l
1,m ð2l þ 1Þð2l
1Þ m ð2l þ 3Þð2l þ 1Þ

which can be seen by taking the complex conjugate of (4.27) before squaring and summing over m. This means that [13]

4

Quantum Treatment of Photoelectron Angular Distributions (Dipole. . .

85

2 X ðl þ 1Þ2
m2 2   l2
m2 Y lþ1,m Y l
1,m ð 2l þ 3 Þ ð 2l þ 1 Þ ð 2l þ 1 Þ ð 2l
1 Þ m 1

12 X ðl þ 1Þ2
m2 2  l2
m2 ¼ Y lþ1,m Y l
1,m ð2l þ 1Þð2l
1Þ m ð2l þ 3Þð2l þ 1Þ " #9 8 X X > > P 1 2 2 2 2 > 2 > > ð l þ 1Þ jY lþ1,m j
m jY lþ1,m j > cos 2 θ jY lm j
> > > > ð2l þ 3Þð2l þ 1Þ = < m m m 1 ¼ " # 2> > X X > > > > 1 > > > >
l2 jY l
1,m j2
m2 jY l
1,m j2 ; : ð2l þ 1Þð2l
1Þ m m 1

1

ð4:30Þ Applying (2.80) and (2.195) to the RHS of (4.30) gives 1

12 X ð l þ 1Þ 2
m 2 2   l2
m 2 Y lþ1,m Y l
1,m ð 2l þ 3 Þ ð 2l þ 1 Þ ð 2l þ 1 Þ ð 2l
1 Þ m 1

12 X ðl þ 1Þ2
m2 2  l2
m2 ¼ Y lþ1,m Y l
1,m ð2l þ 1Þð2l
1Þ m ð2l þ 3Þð2l þ 1Þ 9 8 1 > > ð4:31Þ 2 ð2l þ 1Þ > > cos θs
> > > > 4π ð2l þ 3Þð2l þ 1Þ > > > > > >

> > = < ðl þ 1Þðl þ 2Þ ð2l þ 3Þ 1 2 ð2l þ 3Þ 2
sin θs  ðl þ 1Þ ¼ 4π 8π 2> > > > > > >

> > > > > > > ð 2l
1 Þ ð l
1 Þ ð l Þ ð 2l
1 Þ 1 2 2 > ; :
l
sin θs > 4π 8π ð2l þ 1Þð2l
1Þ

After some additional algebra, and noting that cos2θ ¼ 1
sin2θ, we get 1

12 X ð l þ 1Þ 2
m 2 2   l2
m 2 Y lþ1,m Y l
1,m ð2l þ 1Þð2l
1Þ m ð2l þ 3Þð2l þ 1Þ 1

12 X ð l þ 1Þ 2
m 2 2  l2
m2 ¼ Y lþ1,m Y l
1,m ð2l þ 1Þð2l
1Þ m ð2l þ 3Þð2l þ 1Þ  2 lðl þ 1Þ 2
3 sin θs ¼ 8π ð2l þ 1Þ

We now apply the results of (4.32) to (4.26):

ð4:32Þ

86

4

Quantum Treatment of Photoelectron Angular Distributions (Dipole. . .




2 dσ pd 16π 2 X l2
m2   k  ℜ 2l
1 ¼ Y l
1,m b dΩ ð2l þ 1Þ m ð2l þ 1Þð2l
1Þ

 
2 ðl þ 1Þ2
m2   k  ℜ 2lþ1 þ Y lþ1,m b ð2l þ 3Þð2l þ 1Þ 
2  16π 2 lðl þ 1Þ 2
3 sin θs þ e
iðδlþ1
δl
1 Þ þ eiðδlþ1
δl
1 Þ ℜ lþ1 ℜ l
1 8π ð2l þ 1Þ ð2l þ 1Þ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 2 cos ðδlþ1
δl
1 Þ

ð4:33Þ Rearranging again, dσ pd 16π 2 ¼ dΩ ð2l þ 1Þ 9 8


2 P 
2

ℜ 2l
1 > >    2 P 2 > > b b > > l Y l
1,m k 
m Y l
1,m k  > > > > ð 2l þ 1 Þ ð 2l
1 Þ m m > > > > >



> = <     2

 2 2 P P ℜ lþ1     2 2 b b  þ ðl þ 1Þ Y lþ1,m k 
m Y lþ1,m k  > > ð2l þ 3Þð2l þ 1Þ m m > > > >  > > > > 2 > > θ l ð l þ 1 Þ 2
3 sin > > s > > ; :þ 2 cos ðδlþ1
δl
1 Þℜ lþ1 ℜ l
1 8π ð2l þ 1Þ ð4:34Þ Equation (4.34) is in a form against which we can again apply (2.80) and (2.195): dσ pd 16π 2 ¼ dΩ ð2l þ 1Þ 8



9 ℜ 2l
1 ðl
1ÞðlÞ ð2l
1Þ > > 2 2 ð2l
1Þ > > l sin
θ > s > > > 4π 8π > ð 2l þ 1 Þ ð 2l
1 Þ > > > > > > > > >

> > 2 > > ℜ > > lþ1 > > þ > > = < ð2l þ 3Þð2l þ 1Þ 

> > > > ðl þ 1Þðl þ 2Þ ð2l þ 3Þ 2 ð2l þ 3Þ 2 > > > >
sin  ð l þ 1 Þ θ s > > > > 4π 8π > > > > > > > >  > > 2 > > sin θ l ð l þ 1 Þ 2
3 > > s > > ; :þ 2 cos ðδlþ1
δl
1 Þℜ lþ1 ℜ l
1 8π ð2l þ 1Þ

ð4:35Þ

After some more algebra, and again noting that sin2θ ¼ 1
cos2θ, and after collecting like terms, we get

4

Quantum Treatment of Photoelectron Angular Distributions (Dipole. . .

I ðθ s Þ ¼

87

dσ pd dΩ

 9 8 lðl þ 1Þ ℜ 2l
1 þ ℜ 2lþ1 þ 2ℜ l
1 ℜ lþ1 cos ðδlþ1
δlþ1 Þ > > > > = < " # 2π 2 2 þ ð l þ 1 Þ ð l þ 2 Þℜ l ð l
1 Þℜ ¼ l
1 lþ1 > ð2l þ 1Þ2 > > > cos 2 θs ; :þ
6lðl þ 1Þℜ l
1 ℜ lþ1 cos ðδlþ1
δl
1 Þ ð4:36Þ

Multiply the term in the last bracket on the RHS by identifications:

2 3 3

2

and make the following

  α  lðl þ 1Þ ℜ 2l
1 þ ℜ 2lþ1 þ 2ℜ l
1 ℜ lþ1 cos ðδlþ1
δlþ1 Þ   χ  lðl
1Þℜ 2l
1 þ ðl þ 1Þðl þ 2Þℜ 2lþ1
6lðl þ 1Þℜ l
1 ℜ lþ1 cos ðδlþ1
δl
1 Þ   ξ  ð2l þ 1Þ lℜ 2l
1 þ ðl þ 1Þℜ 2lþ1 ð4:37Þ which gives [13] h i dσ pd 2 3 2π 2 α þ χ cos θ ¼ s 3 2 dΩ ð2l þ 1Þ2



2π 2χ ξ 3 1 2χ ξ 1 2 ¼ α þ θ þ
cos þ χ s 3 ξ2 2 3 ξ 3 ð2l þ 1Þ2 2 3
 1 7 2π 2 6 3α 1χ χ 3 þ cos 2 θs
7 ξ6 þ ¼ 2 3 42 ξ 2 5 2ξ ξ 2 ð2l þ 1Þ |fflfflfflfflffl{zfflfflfflfflffl}

I ðθ s Þ ¼

ð4:38Þ

¼1

σ pd ¼ ½1 þ βP2 ð cos θs Þ 4π where σ pd ¼ " β¼

χ ¼ ξ

h i 16π 2 l0 ℜ 2l0
1 þ ðl0 þ 1Þℜ 2l0 þ1 3ð2l0 þ 1Þ

l0 ðl0
1Þℜ2l0
1 þ ðl0 þ 1Þðl0 þ 2Þℜ2l0 þ1
6l0 ðl0 þ 1Þℜl0
1 ℜl0 þ1 cosðδl0 þ1
δl0
1 Þ i h ð2l0 þ 1Þ l0 ℜ2l0
1 þ ðl0 þ 1Þℜ2l0 þ1 P2 ð cos θs Þ ¼

1 3 cos 2 θs
1 2

ð4:39aÞ # ð4:39bÞ ð4:39cÞ

Z1 ℜ l0 1 ¼

r 3 Rnl0 ðr Þ Gk,l0 1 ðr Þ dr 0

ð4:39dÞ

88

4

Quantum Treatment of Photoelectron Angular Distributions (Dipole. . .

Here we have relabeled slightly (for the sake of clarity in what is to follow), so that the label l0 refers to the orbital angular momentum of the electron prior to photodetachment. Equations (4.38) and (4.39) are the main results of what is known as the CooperZare (C-Z) theory of photoelectron angular distributions. Eq. (4.38) has the same form as (3.29), and, as before, for I(θs) to be positive, we must have
1 β 2. The key feature of the C-Z theory is that the anisotropy of the photoelectron angular distribution is determined entirely by the asymmetry parameter β. For example, for s-electron photodissociation, β(l0 ¼ 0) ¼ 2, which implies a cos2 distribution [c.f. (3.36)]. A cos2 distribution is consistent with what we might expect from the dipole selection rules (c.f. Problem 4.4), which, for an s-electron pd. event, predict p-waves emitted in the same direction (and anti-direction) as the photon polarization vector. Equation (4.39a), contains the sum of the two radial dipole integrals ℜ 2l0
1 and ℜ 2l0 þ1 : We conclude that the total pd. cross-section is the sum of contributions from the transitions l0 ! l0
1 and l0 ! l0 þ 1. Since there are no other terms on the RHS of (4.39a), we also conclude that these transitions do not interfere in the total pd. cross-section. Example 4.4 Find the expression for the asymmetry parameter for p-electron photodissociation. For p-electron detachment, we have l0 ¼ 1: βðl0 ¼ 1Þ ¼

2ℜ 22
4ℜ 0 ℜ 2 cos ðδ2
δ0 Þ  2 ℜ 0 þ 2ℜ 22

ð4:40Þ █

The dependence of β on the kinetic energy of the photoelectrons is realized through the radial dipole integrals and can be made explicit by rearranging (4.39b) in terms of their ratios [(4.39b) also indicates that β depends on phase differences between continuum partial waves]. In anion photodetachment, for example, these ratios are often found to vary linearly with the photoelectron kinetic energy, a consequence of the Wigner threshold law, demonstrated as follows. First, we recall from the Wigner threshold law that near threshold [17, 18], 2l0 þ1

σ pd / k 2l0 þ1 / E c 2

ð4:41Þ

where k is the wavenumber and Ec is the (center-of-mass) kinetic energy of the continuum photoelectron, respectively. It is also found that (see Problem 7.3) [17, 19] σ pd / kjℜ l0 j2 So, it must be that [20]

ð4:42Þ

4

Quantum Treatment of Photoelectron Angular Distributions (Dipole. . .

jℜ l0 j / k l0 ) jℜ l0 1 j / kl0 1

89

ð4:43Þ

If the final state wavelength is large compared to the size of the initial state, and if we neglect any interactions between the photoelectron and the residual neutral, we can then say [20, 21] ℜ l0 þ1 =ℜ l0
1 / k 2 ) ℜ l0 þ1 =ℜ l0
1 ¼ AE c

ð4:44Þ

where A gives the relative size of the of the two radial matrix elements. Example 4.5 Find an expression for the asymmetry parameter for the photodetachment of an atomic anion near threshold [17]. Divide the numerator and denominator of (4.39b) by the factor ℜ 2l0
1 and apply (4.44) )β¼

l0 ðl0
1Þ þ ðl0 þ 1Þðl0 þ 2ÞA2 E2c
6l0 ðl0 þ 1ÞAE c cos ðδl0 þ1
δl0
1 Þ   ð2l0 þ 1Þ l0 þ ðl0 þ 1ÞA2 E2c ð4:45Þ

Although the Wigner threshold law is strictly valid only near threshold, partial wave cross-section ratios have been found to be valid even at energies several electron volts above the threshold regime [18, 20, 22]. █ Now, for p-electron detachment (l0 ¼ 1) near threshold, from (4.44), ℜ 2 =ℜ 0 ¼ A20 E c

ð4:46Þ

where A20 gives the relative size of the two radial dipole integrals ℜ2 & ℜ0. Combining (4.40) or (4.45) with (4.46), we have (after a little algebra) the following expression for p-electron detachment near threshold [20]: β¼

2ðA20 E c Þ2
4A20 E c c 1 þ 2A220 E2c

ð4:47Þ

where c ¼ cos (δ2
δ0). Eq. (4.47) is known as the Hanstorp model. A typical plot of the Hanstorp model for p-electron detachment can be found in Fig. 4.1. The plot in Fig. 4.1 shows the spectral dependence of the asymmetry parameter for p-electron photodetachment. According to the dipole selection rules (see Problem 4.4), the photodetachment of a p-electron gives rise to s- and d-waves. In accordance with the Wigner threshold law, in a central potential, the centrifugal barrier suppresses the d-wave near threshold, leaving an isotropic

90

4

Quantum Treatment of Photoelectron Angular Distributions (Dipole. . .

Fig. 4.1 Plot of the Hanstorp model for p-electron photodetachment. The plot shown is for the value A20 ¼ 1. Note the location of the minimum at the ratio ℜ2/ℜ0 ¼ 0.5 [23]. Reproduced from [23] with permission from the author

s-wave distribution with the concomitant asymmetry parameter β ¼ 0. Away from threshold, the two waves interfere destructively, driving the asymmetry parameter into negative territory and the plot reaches a minimum when ℜ2/ℜ0 ¼ 0.5. For full destructive interference (δ2
δ0 ¼ 0 ) c ¼ 1), the asymmetry parameter will reach its minimum possible value β ¼
1. At higher photoelectron kinetic energies, the dwave becomes increasingly dominant, and β becomes positive, eventually approaching the asymptotic limit β ¼ 1. Note that for the photoionization of a pelectron from a neutral or positive atom, β will not display the same behavior, and the plot of Fig. 4.1 will not apply [20, 24, 25]. For p-electron photodetachment near threshold, we have [17, 20–22] jℜ 2 j2

> > > > 3 Y 1,1 þ Y 1,1 > > > > > = < m m 0 0 1 1  ! ! > 1 2 3

> > pffiffiffi 1 2 3 > > > > >  > > þ 7 Y þ Y 3,1 : 3,1 ; m m 0 0 1 1 

ð5:39Þ Tabulated values of 3-j symbols allow us to evaluate the following 3-j symbols [3, 7, 10]:



1

2

m

m

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2  mÞ ð2 þ mÞ ¼ ð1Þ 30 rffiffiffiffiffi  2 1 1 ¼ 10 0 1 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   ð2  mÞ ð2 þ mÞ 3 mþ1  2 ¼ ð1Þ 3  5m 7543 0 r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

 1 2 3 1 ¼4 7532 0 1 1

1 m

2 1 m 0

1



m

ð5:40aÞ ð5:40bÞ ð5:40cÞ ð5:40dÞ

106

5 Higher-Order Multipole Terms in Photoelectron Angular Distributions

Substituting, pffiffiffiffiffi 4π 6π X l0 l00 iðδl0 δl00 Þ i e ℜ l0 Ql00 ð2l0 þ 1Þð2l0 þ 1Þð2l00 þ 1Þ hzi hxzi ¼ 3ð2l0 þ 1Þ 0 00 ll

 

 l0 2 l00 X 1 l0 l0 l0 1 l0  ð1Þm 0 0 0 0 0 0 m m m 0

 2 l00 l0  m m 0 9 8  

 > > ð1Þmþ1 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > ð2  mÞð2 þ mÞ Y 1,1 þ Y 1,1 = < 10 r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi    > > 7ð2  mÞð2 þ mÞ

2 > ; : þð1Þmþ1 Y 3,1 þ Y 3,1 > ð3  5m2 Þ 2 753 ð5:41Þ 

The triangle conditions of the 3-j symbols limit the sum over m to the terms m ¼ 0, 1: pffiffiffiffiffi 4π 6π X l0 l00 iðδl0 δl00 Þ i e|fflfflfflfflffl{zfflfflfflfflffl} ℜ l0 Ql00 ð2l0 þ 1Þð2l0 þ 1Þð2l00 þ 1Þ 3ð2l0 þ 1Þ 0 00 ll 2cos ðδl00 δl0 Þ ! ! 0 l0 1 l l0 2 l00  0 0 0 0 0 0 9 82 ! ! 3   1 l0 l0 l0 2 l00 > > 2 > > > > > > 7 6 > > > > 10 7 6 > > 0 0 0 0 0 0 > > 7 6 > > > > 7 6 > > ! ! > > p ffiffi ffi 0 00 7 6

 > > 1 l l l 2 l > >6 0 0 7

 > > 3  > > 7 6 > > Y þ þ Y 1,1 > > 1,1 7 6 > > 10 > > 7 6 1 1 0 1 1 0 > > > > 7 6 > > > > ! ! 7 6 > > 0 00 p ffiffi ffi

 > > 7 6 1 l l 2 l l > > 0 0 > > 3 5 4 > > > > þ > > > > 10 = < 1 1 0 1 1 0 ð1Þ 2 3

pffiffiffiffiffiffiffiffi 1 l l0 ! l 2 l00 ! > > > > 0 0 > > 2 7  2 > > > > 7 6 > > > > 7  5 7 6 > > > > 0 0 0 0 0 0 7 6 > > > > 7 6 > > ! ! > > r ffiffiffiffiffiffiffi ffi 0 00 7 6 > >   > > 1 l l 2 l l 0 0 7 6

 > > 4 7  3 > >  7 6 > > Y þ þ Y þ > > 3,1 3,1 7 6 753 > > 2 > > 7 6 > > 1 1 0 1 1 0 > > 7 > > 6 > > ! ! 7 6 > > r ffiffiffiffiffiffiffi ffi 0 00 > > 7 > 6   > 1 l l l 2 l 0 0 > > 4 7  3 > 5 4 > > > þ > > ; : 753 2 1 1 0 1 1 0

hzi hxzi ¼

ð5:42Þ

5

Higher-Order Multipole Terms in Photoelectron Angular Distributions

107

Recalling that the symmetry properties of the 3-j symbols appearing in (5.42) 00 demand that l0 + l0 + 1¼even and l0 + l + 2¼even, we can say,

1

l0

l0

1

1

0



l0

2

l00

1

1

0



¼

1

l0

1 1

l0 0



l0

2

l00

1

1

0

 ð5:43Þ

Combining (5.42) and (5.43), pffiffiffiffiffi 4π 6π X l0 l00 iðδl0 δl00 Þ i e ℜ l0 Ql00 ð2l0 þ 1Þð2l0 þ 1Þð2l00 þ 1Þ 3ð2l0 þ 1Þ 0 00 ll ! ! 1 l0 l0 2 l00

hzi hxzi ¼ 

l0 0

0 0 0 0 0 9 82 0 10 1 3 > > 1 l0 l0 l0 2 l00   > > > > 1 @ > > A@ A > 6 7 > > > > > 6 5 7 > > > > 6 7 0 0 0 0 0 0 > >

 > > 6 7  > > > Y þ Y > 6 7 1,1 1,1 0 1 0 1 > > > > 6 7 0 00 p ffiffi ffi

 > > l 2 l l 1 l > >6 0 0 7 > > 3 > > 4 5 @ A@ A > > þ > > > > 5 > > = < 1 1 0 1 1 0 ð1Þ 0 1 0 1 3 > > 2 pffiffiffiffiffiffiffiffiffi 1 l0 l0 > > l0 2 l00 > > > > 2 72 @ > > A @ A > 6 7 > > > > 6 7 > 75 > > > > 6 7 0 0 0 0 0 0 > >

 > > 6 7  > > > þ6 7 Y 3,1 þ Y 3,1 > 0 1 0 1 > > > > 6 7 0 00 r ffiffiffiffiffiffiffiffi ffi > > l0 2 l > > 6  8  7  3 1 l0 l 7 > > > > 4þ 5 @ A @ A > > > > > > 753 2 ; : 1 1 0 1 1 0

ð5:44Þ Now we make explicit all the angular factors: rffiffiffiffiffi 3 sin θ cos ϕ ¼ Y 1,1 þ 2π rffiffiffiffiffi  

 1 21  sin θ 5 cos 2 θ  1 cos ϕ Y 3,1 þ Y 3,1 ¼ 4 π

Y 1,1



Combining (5.44) and (5.45) and rearranging,

ð5:45aÞ ð5:45bÞ

108

5

Higher-Order Multipole Terms in Photoelectron Angular Distributions

X 0 00 4π il l cos ðδl00  δl0 Þℜ l0 Ql00 ð2l0 þ 1Þð2l0 þ 1Þ 3ð2l0 þ 1Þ 0 00 ll ! ! l0 2 l00 l0 1 l 0 00  ð2l þ 1Þ 0 0 0 0 0 0 9 8" ! !# 1 l0 l0 l0 2 l00 > > > > > > ½ sin θ cos ϕ > > > > > > > > 1 1 0 1 1 0 > > > > > > 0 1 0 1 > > 2 3 > > 0 00 > > l l 2 l 1 l > > 0 0 p ffiffi ffi = <  pffiffiffi @ A@ A 7 6 3  2 3 6 7 > > 6 7 0 0 0 0 0 0 > > > > 6 7 > > 2 > > ½ θ cos ϕ sin θ cos þ  6 7 > > 0 1 0 1 > > 6 7 0 00 > > > > 6 1 l0 l l0 2 l 7 > > > > 4 5 > > @ A @ A > > 2 > > ; : 1 1 0 1 1 0

hzi hxzi ¼

ð5:46Þ Again, we invoke the triangle rules and symmetries inherent in the 3-j symbols to 00 00 see that the sums over l0and l are limited to the terms l0 ¼ l0 1 and l ¼ l0, l0 2. Tabulated values of 3-j symbols allow us to calculate the following factors [3, 7, 10]: 00

For l0 ¼ l0  1, l ¼ l0  2, 1 l0 l0 2 0 00 l l p ffiffi ffi ð2l0 þ 1Þð2l þ 1Þð2l þ 1Þ i 3 0 0 0 il ðl þ 1Þðl0  1Þ ¼ 0 0 ð2l0  1Þð2l0 þ 1Þ 0

!

00

l0 2 l00

!

1 l0 l0

!

1 1 0

0 0 0

l0 2 l00

!

1 1 0

ð5:47aÞ pffiffiffi

2 2 l0 2 l00 1 l0 l0 0 00 2  3il20 ðl0  1Þ il l pffiffiffi ð2l0 þ 1Þð2l0 þ 1Þð2l00 þ 1Þ ¼ ð2l0  1Þð2l0 þ 1Þ 0 0 0 0 0 0 3 ð5:47bÞ 00

For l0 ¼ l0  1, l ¼ l0, i

2 l0 l00 p ffiffiffi 3

0

00

ð2l0 þ 1Þð2l þ 1Þð2l þ 1Þ

1 l0 l0 0 0 0

!

l0 2 l00 0 0 0

!

1 l0 l0 1 1 0

!

l0 2 l00

!

1 1 0

il0 ðl0 þ 1Þ ¼ ð2l0 þ 3Þð2l0  1Þ ð5:47cÞ

5

Higher-Order Multipole Terms in Photoelectron Angular Distributions

109

2 2 l0 2 l00 1 l0 l0 0 00 2 2il20 ðl0 þ 1Þ il l pffiffiffi ð2l0 þ 1Þð2l0 þ 1Þð2l00 þ 1Þ ¼ pffiffiffi 0 0 0 0 0 0 3 3ð2l0 þ 3Þð2l0  1Þ ð5:47dÞ 00

For l0 ¼ l0  1, l ¼ l0 + 2, 1 l0 l0 2 0 00 l l p ffiffi ffi ð2l0 þ 1Þð2l þ 1Þð2l þ 1Þ i 3 0 0 0 il ðl þ 1Þðl0 þ 2Þ ¼ 0 0 ð2l0 þ 3Þð2l0 þ 1Þ 0

!

00

l0 2 l00

!

1 l0 l0

!

1 1 0

0 0 0

l0 2 l00

!

1 1 0

ð5:47eÞ pffiffiffi

2 2 l0 2 l00 1 l0 l0 0 00 2  3il0 ðl0 þ 1Þðl0 þ 2Þ il l pffiffiffi ð2l0 þ 1Þð2l0 þ 1Þð2l00 þ 1Þ ¼ ð2l0 þ 3Þð2l0 þ 1Þ 0 0 0 0 0 0 3 ð5:47fÞ 00

For l0 ¼ l0 + 1, l ¼ l0 + 2, i

2 l0 l00 p ffiffiffi 3

0

00

ð2l0 þ 1Þð2l þ 1Þð2l þ 1Þ

1 l0 l0

!

0 0 0

l0 2 l00

!

1 l0 l0

!

1 1 0

0 0 0

l0 2 l00

!

1 1 0

il ðl þ 1Þðl0 þ 2Þ ¼ 0 0 ð2l0 þ 3Þð2l0 þ 1Þ ð5:47gÞ

2 2 pffiffiffi l0 2 l00 1 l0 l0 0 00 2 3il0 ðl0 þ 1Þðl0 þ 2Þ il l pffiffiffi ð2l0 þ 1Þð2l0 þ 1Þð2l00 þ 1Þ ¼ ð2l0 þ 3Þð2l0 þ 1Þ 0 0 0 0 0 0 3 ð5:47hÞ 00

For l0 ¼ l0 + 1, l ¼ l0, i

2 l0 l00 p ffiffiffi 3

0

00

ð2l0 þ 1Þð2l þ 1Þð2l þ 1Þ

1 l0 l0 0 0 0

!

l0 2 l00 0 0 0

!

1 l0 l0 1 1 0

!

l0 2 l00

!

1 1 0

il0 ðl0 þ 1Þ ¼ ð2l0 þ 3Þð2l0  1Þ ð5:47iÞ

110

5

Higher-Order Multipole Terms in Photoelectron Angular Distributions

pffiffiffi

2 2 l0 2 l00 1 l0 l0 0 00 2  3il0 ðl0 þ 2Þ2 il l pffiffiffi ð2l0 þ 1Þð2l0 þ 1Þð2l00 þ 1Þ ¼ ð2l0 þ 3Þð2l0  1Þ 0 0 0 0 0 0 3 ð5:47jÞ 00

For l0 ¼ l0 + 1, l ¼ l0  2, 1 l0 l0 2 il l pffiffiffi ð2l0 þ 1Þð2l0 þ 1Þð2l00 þ 1Þ 3 0 0 0 il0 ðl0 þ 1Þðl0  1Þ ¼ ð2l0  1Þð2l0 þ 1Þ 0

00

!

l0 2 l00

!

0 0 0

1 l0 l0

!

1 1 0

l0 2 l00

!

1 1 0

ð5:47kÞ

1 l0 0 00 2 il l pffiffiffi ð2l0 þ 1Þð2l0 þ 1Þð2l00 þ 1Þ 0 0 3 pffiffiffi 3il0 ðl0 þ 1Þðl0  1Þ ¼ ð2l0  1Þð2l0 þ 1Þ

l0 0

2

l0 0

2 l00 0 0

2

ð5:47lÞ

We can now combine (5.18), (5.46), and all the iterations of (5.47). The result is usually presented in the form [40, 41]:    dσ pd σ pd

¼ 1 þ βP2 ð cos θÞ þ δ þ γ cos 2 θ sin θ cos ϕ dΩ 4π

ð5:48Þ

where σ pd is given by (4.39a) and β by (4.39b). The factors δ and γ are given by [49] γh

8π 2 ħ2 αω2 lℜ 2l0 1 þ ðl0 þ 1Þℜ 2l0 þ1

i

X Al0 l00 ℜ l0 Ql00 cos ðδl00  δl0 Þ

ð5:49aÞ

l0 l00

X 8π 2 ħ2 αω2 i Bl0 l00 ℜ l0 Ql00 cos ðδl00  δl0 Þ δh lℜ 2l0 1 þ ðl0 þ 1Þℜ 2l0 þ1 l0 l00

ð5:49bÞ

Al0 l00 and Bl0 l00 are found by combining (the imaginary part of) the appropriate iterations of (5.47) with the appropriate factors inside of the curly brackets on the RHS of (5.46).

5

Higher-Order Multipole Terms in Photoelectron Angular Distributions

111

Example 5.3 Find Bl0 1,l0 2 and Al0 1,l0 2 . For l0 ¼ l0  1, l00 ¼ l0  2, Bl0 l00 ¼ ð5:47aÞ ¼ ¼

l0 ðl0 þ 1Þðl0  1Þ , while Al0 l00 ð2l0  1Þð2l0 þ 1Þ

pffiffiffi l ðl þ 2Þðl0  1Þ : 3  ð5:47bÞ  2  ð5:47aÞ ¼ 0 0 ð2l0  1Þð2l0 þ 1Þ

█

The complete list of the factors Al0 l00 and Bl0 l00 is found in [43]. Notice that the interference between the electric dipole and electric quadrupole terms has introduced odd powers of cosθ into the differential pd. cross-section as parity considerations demand [1, 51]. A few remarks are in order here. Eq. (5.48) shows us that the non-dipolar terms are greatest in the direction (and anti-direction) of photon propagation, ϕ ¼ 0, π. Also, the parameter δ multiplies the factor sinθ cos ϕ, which is the direction of the photon momentum vector along the x-axis (direction of photon propagation) [52]. It appears that the inclusion of the first higher-order multipole term in the expression for the photoelectron angular distribution reveals an asymmetry that depends on the direction of photon propagation [53]. Eq. (5.48) also shows that the term with δ is rotationally symmetric about the direction of photon travel but is forward-backward asymmetric with respect to the reversal of the photon propagation direction as is the term with γ. The term with γ also has reflection symmetry in the xy and xz planes [54]. Example 5.4 Explain under what conditions experimentalists will be able to isolate higher-order multipole corrections to photoelectron angular distributions. First, experimentalists attempting to measure total cross-sections should note that the concept of the “magic angle” θ ¼ 54.70 holds only in the ϕ ¼ π/2 plane. Experimentalists should also note that measurements carried out in a plane perpendicular to the photon direction, θ ¼ 0, ϕ ¼ π/2, will not reveal any of the interference terms. Or, putting it another way, higher-order multipole corrections to photoelectron angular distributions can only be measured in a plane that is not the polarization plane (for linearly polarized light). On the other hand, it is possible to isolate the higher-order terms if measurements are made at the magic angle, θ ¼ 54.70 (and in the direction ϕ ¼ 0). In this case, (5.48) gives [54, 55] " # rffiffiffiffiffi dσ pd σ pd 2 ¼ 1þ ð3δ þ γ Þ dΩ 4π 27

ð5:50Þ

although only the combined quantity (3δ + γ) can be extracted from such a █ measurement.

112

5

Higher-Order Multipole Terms in Photoelectron Angular Distributions

Example 5.5 Find expressions for the higher-order asymmetry parameters for photodissociation from s- and p-subshells. Using (5.46), all relevant iterations of (5.47), and (5.49a and 5.49b), and the dipole and quadrupole selection rules, we can determine the expressions for δ and γ for photodissociation from s- and p-subshells [49, 56, 57]: δs ¼ 0 γs / δp /

Q2 cos ðδ2  δ1 Þ ℜ1

ð5:51aÞ ð5:51bÞ

fℜ 0 ½Q1 cos ðδ1  δ0 Þþ Q3 cos ðδ3  δ0 Þþ ℜ 2 ½Q1 cos ðδ1  δ2 Þþ Q3 cos ðδ3  δ2 Þg

2  ℜ 0 þ2ℜ 22 ð5:51cÞ

γp /

f5ℜ 0 Q3 cos ðδ3  δ0 Þ þ 2ℜ 2 ½2Q3 cos ðδ3  δ2 Þ  3Q1 cos ðδ1  δ2 Þg

2  ℜ 0 þ 2ℜ 22 ð5:51dÞ

Equation (5.51a) can be derived most directly by considering (5.33) for the case m ¼ 0.

 pffiffiffi D100 D210  D210 ¼ 6 cos 2 θ sin θ cos ϕ

ð5:52Þ

where we also used (2.109) and (2.67e). This result shows that the m ¼ 0 component of the sum over m in (5.41) contributes only to γ. Nonzero values of δ arise from the m ¼ 1 terms in the sum [49]. By measuring the angular distributions of electrons photodissociated from s-subshells at the magic angle θ ¼ 54.70 and in the direction ϕ ¼ 0, one could isolate the non-dipole parameter γ s, which would then be determined by (5.50) under the condition δs ¼ 0. █ It should be remembered that the higher-order multipole correction terms in (5.48) generally will not manifest themselves for photon energies below 100 eV (although important exceptions exist for which the electric dipole-electric quadrupole interference term is comparable or larger than the dipole term [58–63]). Conversely, at photon energies greater than ~10 keV, (5.48) is no longer strictly valid since the magnetic dipole term (and additional higher-order multipole terms) will become important [48]. On a final note, expressions other than (5.48) have been derived that include higher-order multipole terms. These expressions differ in the details, but not in the physics [50, 64, 65].

5

Higher-Order Multipole Terms in Photoelectron Angular Distributions

113

Problems 5.1. Explain why a spin 1/2 particle cannot have a permanent quadrupole moment. 5.2. Fill in the following table: 00

l0 l0  1 l0  1 l0  1 l0 + 1 l0 + 1 l0 + 1

l l0  2 l l0 + 2 l0  2 l l0 + 2

Al0 l00

Bl0 l00

5.3. Prove. 1 l  X r<   Pl ð cos αÞ lþ1 ! ! 0  ¼ r> r  r  l¼0

1

! !0

where r>/r< is the greater/lesser of r = r . Use your analysis to derive (2.71).

Finally, show that

l  1 X l X r< 1  0  1 0  ¼ 4π ð Ω ÞY ð Ω Þ Y lm lm lþ1 ! !  2l þ 1 r > r  r  l¼0 m¼l  0 ! can be 5.4. The potential due to a continuous volume charge density ρ r described as.

 0 Z ρ !   r ! Φ r ¼ k ! !0  d3 r 0 r  r 

For regions outside of the distribution

    0  !  ! r  >  r  , we can construct a

multipole expansion of the potential in spherical coordinates using spherical harmonics. We might imagine that such an expansion would take the form

114

5

Higher-Order Multipole Terms in Photoelectron Angular Distributions

 0 Z ρ ! 1 X l   r X Y ð ΩÞ 4π ! Φ r ¼ k ! !0  d3 r 0 ¼ k qlm lmlþ1 2l þ 1 r r  r  l¼0 m¼l where the qlm are the multipole moments. Find an expression for the qlm. 5.5. From Problem 5.4, we can infer that the electric quadrupole moment is proportional to the term r2 C ðq2Þ . Use this fact to find the electric quadrupole selection rules for single-particle transitions between states of definite angular momentum. Ignore spin. 5.6. In (5.13), we found that the magnetic dipole operator was proportional to the !

term hf j iω 2c L y jii. Use this term to find the magnetic dipole selection rules for single-particle transitions between states of definite angular momentum. Ignore spin. 5.7. Average displacement of the electron from the nucleus in hydrogenic atoms. The radial dipole and quadrupole integrals introduced in (4.20) and (5.31), respectively, usually cannot be evaluated analytically. One set of radial integrals that can be evaluated analytically and are of interest [e.g., see Problem 6.12(c)] is the set of integrals that give the expectation values of various powers of the average displacement of the electron from the nucleus in hydrogenic atoms. When expressed in spherical coordinates, the Schrodinger equation HΨ ¼ EΨ is separable into a radial equation and an angular equation with Ψ ¼ Y lm ðθ, ϕÞREl ðr Þ where REl(r) is the radial portion of the wave function. REl(r) solves the radial Schrodinger eq. [15]:

   lðl þ 1Þħ2 d2 2 d 2μ 2μE REl ðr Þ þ 2 REl ðr Þ ¼ 0 þ R ð r Þ  V ð r Þ þ El 2μr 2 dr 2 r dr ħ2 ħ

where μ is the reduced mass of the system. If we let unl(r)  rREl(r), the radial equation becomes   lðl þ 1Þħ2 d2 2μ unl ðr Þ ¼ 0 unl ðr Þ þ 2 E  V ðr Þ  2μr 2 dr 2 ħ For hydrogenic atoms, the potential is

5

Higher-Order Multipole Terms in Photoelectron Angular Distributions

V ðr Þ ¼ 

115

Ze2 r

where Z is the number of protons in the nucleus. This potential results in the energy eigenvalues [15]: 2

ðZαÞ e2 Z 2 1 jEn j ¼ μc2 2 ¼ 2 2 n 2n a0 e ħ where α ¼ ħc and a0 ¼ μe 2. 2

2

(a) Let, 2μjE n j ħ2



2 1=2  1=2 μ Ze2 μc Z λ 2 ¼ Zα ¼ κa 2 E 2 E j j j j 0 ħ n n

1=2 8μjE n j 2μcZα 2Z ρ  2κr ¼ r¼ r r¼ ħn na0 ħ2 κ2 

Using these definitions, show that the radial equation can be written as [15]

 d2 u lðl þ 1Þ λ 1  uþ  u¼0 ρ 4 dρ2 ρ2 Note that, lim u ρlþ1 and lim u ! 0 ρ!1

ρ!0

(b) Use this form of the radial equation to prove Kramer’s relation [9]: 

 i a2    sh a0 sþ1 s 2 s1 2 0  ð 2s þ 1 Þ r s2  s þ r r ð 2l þ 1 Þ h i 2 2 Z 4 n Z

¼ 0; s þ 2l þ 3 > 0 where Z hr inl ¼ s

1

drr sþ2 ½Rnl ðrÞ2

0

Hint: Multiply the radial equation of part (a) by [9]:

116

5

Higher-Order Multipole Terms in Photoelectron Angular Distributions

h

ρsþ1 u0 

  i sþ1 s ρu 2

and integrate over ρ. (c) Use Kramer’s relation to compute various radial integrals hrsinl of interest (i.e., for all integer values s ¼  6 ! 4). 5.8. The Cartesian components of a 2nd-rank tensor can be formed from the Cartesian components of two vectors via, T ij ¼ U i V j Follow the procedure outlined in example 2.19 to compute the spherical ð1Þ of this 2nd-rank tensor in terms of the product U ð1Þ components T ð2Þ q q1 V q2 . Also, express the T ð2Þ q in terms of the Tij.

Chapter 6

Relativistic Theory of Photoelectron Angular Distributions

In general, relativistic interactions have little effect on photoelectron angular distributions for atoms of low or moderate atomic weight. For most (open-shell) atoms, the asymmetry parameter β is energy-dependent due to non-relativistic (term-dependent) electron-core interactions, which permit several additional final-state channels to open (see Chap. 7). But for heavier alkali and for closed-shell atoms, the energy dependence of β has been found to be the result of relativistic (mainly spin-orbit) effects [51, 66– 70]. In addition, experimental evidence has revealed different values of β for different values of j ¼ l  1/2, indicating that relativistic jj-coupling may influence photoelectron angular distributions in some systems [68, 71]. Discrepancies between the angular distributions predicted in non-relativistic and relativistic theories also tend to grow with increasing Z [51, 72]. What follows is a simplified, relativistic, independentparticle treatment in which core-continuum interactions are neglected. In relativistic quantum theories, the “non-radial” part of the total wavefunction (for a particle in a central potential) is a combination of spherical harmonics and two-component spinors. These combinations are called spin-angular or spin-orbital functions. Single-particle spin-orbital functions are classified by their total angular momentum j and their projection onto the z-axis mj in accordance with (2.119) and (2.144b) as follows [66, 73, 74], j rÞ  h^rjλjmj i ¼ χm κ ð^

X ms

ð1Þ

λ12mj

0 pffiffiffiffiffiffiffiffiffiffiffiffi 2j þ 1@

λ mj  ms

1 2 ms

! j mj

Y λ,mj ms ð^r Þχ ms

ð6:1Þ

The original version of this chapter was revised. The correction to this chapter is available at https://doi.org/10.1007/978-3-031-08027-2_11 © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022, Corrected Publication 2023 V. T. Davis, Introduction to Photoelectron Angular Distributions, Springer Tracts in Modern Physics 286, https://doi.org/10.1007/978-3-031-08027-2_6

117

118

6

Relativistic Theory of Photoelectron Angular Distributions

where χ ms is the two-component Pauli spinor; ms being the projection of the spin angular momentum onto the z-axis, χ

1=2

¼

  1 0

χ

1=2

¼

  0

ð6:2Þ

1

m

Note that the χ κ j ðbr Þ are simultaneous eigenfunctions of L2, S2, J2 and Jz [72]. The spin-orbit functions are normalized as follows [73], Z

 mj { m0j χ κ ðbr Þ χ κ0 ðbr ÞdΩ ¼ δκκ0 δmj m0j

ð6:3Þ !

The label κ stands for the eigenvalue of the operator K ¼ L  e σ þ ħ and takes the values   1 1 κ ¼ l  1 ¼  j þ for j ¼ l þ 2 2   1 1 κ ¼l¼ jþ for j ¼ l  2 2

ð6:4aÞ ð6:4bÞ

which we write as [66, 74],   1 κ ¼ jþ a 2

ð6:5Þ

with a ¼ þ1 for l ¼ j 

1 2

ð6:6aÞ

a ¼ 1 for l ¼ j þ

1 2

ð6:6bÞ

e σ  bxσ x þ byσ y þ bzσ z

ð6:7Þ

and

And, of course, σ x, σ y, σ z are the Pauli spin matrices,  σx ¼

 0 1 ; 1 0

 σy ¼

 0 i ; i 0

 σz ¼

1 0

0 1

 ð6:8Þ

The spin operators are related to the Pauli spin matrices as follows, ħ Si ¼ σ i ; i ¼ x, y, z 2

ð6:9Þ

6

Relativistic Theory of Photoelectron Angular Distributions

119

Example 6.1 Verify the relations in (6.4a and 6.4b) h i !   

2 ! ! 2 1 2 mj 2 2 mj j L e σ þ ħ χm κ ¼ ħ L  S þ ħ χκ ¼ ħ 2 J  L  S þ ħ χκ h   i 1 1 j þ 1 þ 1 ħχ m ¼ ħ jðj þ 1Þ  lðl þ 1Þ  κ 2 2 h i 1 j j ¼ ħ jðj þ 1Þ  lðl þ 1Þ þ ħχ m ¼ κħχ m κ 4 κ

j Kχ m κ ¼

where we used (2.111), (2.22a), (2.34a), and the fact that s ¼ 1/2 for electrons. So, we have, h i 1 κ ¼  jðj þ 1Þ  lðl þ 1Þ þ 4 Now, if j ¼ l þ 12, then, κ¼

h   i 1 3 1 lþ lþ  lðl þ 1Þ þ ¼ ðl þ 1Þ 2 2 4

and if j ¼ l  12, then, κ¼

h

1 2

l

  i 1 1 lþ  l ð l þ 1Þ þ ¼ l 2 4

Thus, the relations in (6.4) are verified. █ Example 6.2 The conventional forms of the Pauli spin matrices are given in (6.8). What physical arbitrariness has been exploited in reducing the results from Problem 6.2(a) to these forms? In Problem 6.2, we found the most general forms for the Hermitian matrices σ x and σ y that satisfy the given angular momentum commutation rules to be,  σx ¼

0

iC

iC 

0



 and σ y ¼

0

C

C

0



where CC  ¼ 1:

Because CC ¼ 1 is our only constraint, we can write C ¼  i.  ) σ x ¼ eiφ

0

1

1

0



 and σ y ¼ eiφ

0

i

i

0



where we have explicitly expressed any lingering phase factors that may remain. These phase factors are global, and therefore cannot be measured, which allows us to arbitrarily set them to unity. █

120

6

Relativistic Theory of Photoelectron Angular Distributions

  ! For a central potential V r ¼ V ðr Þ , the relativistic single-particle bound orbitals are written as [6, 66, 69, 74], " # mj   1 Pnκ ðr Þχ κ ðbr Þ ! ψ nκmj r ¼ j r iQnκ ðr Þχ m rÞ κ ðb

ð6:10Þ

where Pnκ(r) and Qnκ(r) solve the coupled radial Dirac eqs. [71–74],

∂Pnκ ðr Þ 1 κ E  V ðr Þ þ mc2 Qnκ ðr Þ ¼  Pnκ ðr Þ þ cħ r ∂r

ð6:11aÞ

∂Qnκ ðr Þ κ 1 ¼ Qnκ ðr Þ  E  V ðr Þ  mc2 Pnκ ðr Þ r cħ ∂r

ð6:11bÞ

and E is the total (relativistic) energy of the particle. These radial wave functions are usually normalized as follows [73, 75], Z

 

! ψ {nκmj r

Z1    2  ! 3 Pnκ ðr Þ þ Q2nκ ðr Þ dr ¼ 1 ψ nκmj r d r ¼ 1 )

ð6:12Þ

0

Since j ¼ jκj  12, we can instead use the label κ, and write the 4-component wave function for the initial bound state/orbital (in Dirac notation) as [66, 74], " !

ψ nκmj ð r Þ ¼

# hrjnκmj , ς ¼ 1  hrjnκmj , ς ¼ þ1

ð6:13Þ

where,   hr jnκmj , ς ¼ hr jnκςihbr jλjmj δλ,jþ12aς

ð6:14Þ



1 Pnκ ðr Þ, ς ¼ 1 hr jnκςi ¼ r iQnκ ðr Þ, ς ¼ þ1

ð6:15Þ

and

  ! The spin-angular part of ψ nκmj r , h^rjλjmj i, is given in (6.1).

6

Relativistic Theory of Photoelectron Angular Distributions

121

Example 6.3 Show that the operator K anti-commutes with the operator σ r ¼ br  e σ [73]. That is, show that, fK, σ r g ¼ 0 1 ! ! fK, σ r g ¼ ½K r  σ~ þ r  σ~K r ! 1 ! ! ! ¼ ½ðL  σ~ þ ħÞð r  σ~Þ þ ð r  σ~ÞðL  σ~ þ ħÞ r ! 1 ! ! ! ! ¼ ½ðL  σ~Þð r  σ~Þ þ ð r  σ~ÞðL  σ~Þ þ 2ħð r  σ~Þ r ! ! ! 1 ! ! ! σ  ðL  r Þ þ |ffl{zffl} L  r ¼ ½|ffl{zffl} L  r þ i~ r ¼0

¼0

! 1 ! ! þ ½i~ σ  ð r  L Þ þ 2ħð r  σ~Þ r ! ! 1 ! ! ! ¼ fi~ σ  ½ðL  r Þ þ ð r  L Þ þ 2ħð r  σ~Þg r

Examining the z-component of this equation, fK, σ r gz

¼ ¼ ¼

1 ½iσ ðL y  Ly x þ xLy  yLx Þ þ 2ħσ z z r z x 1 ½iσ ð½L , y  ½Ly , xÞ þ 2ħσ z z r z x 1 1 ½iσ ðiħz þ iħzÞ þ 2ħσ z z ¼ ð2ħσ z z þ 2ħσ z zÞ ¼ 0 r r z

where we used the results of Problem 6.2(b)(x) below. Similar results for the other components prove the desired relationship. One final remark. As a consequence of this relationship, we observe, j j j j Kσ r χ m r Þ ¼ σ r Kχ m r Þ ¼ σ r ðħκ Þχ m r Þ ¼ ħκσ r χ m rÞ κ ðb κ ðb κ ðb κ ðb

For this result to be consistent with example 6.1, we must have, j j r Þ ¼ χ m rÞ σr χm κ ðb κ ðb

█

The final state consists of a continuum photoelectron, which can be expanded in a series of partial waves either of the form in (F.53) or in the equivalent form [66, 73],

122

6

ψ pκmj ðpr Þ ¼ 4π

X κmj

Relativistic Theory of Photoelectron Angular Distributions

" #   rjpκm , ς ¼ 1 j   rjpκmj , ς ¼ þ1

j ! exp ðiδκ Þχ m p κ

ð6:16Þ

!

Here p is the momentum of the photoelectron. The function in the brackets in (6.16) must have the correct asymptotic behavior [66, 73], Pκ ðpr Þ ! jl ðpr Þ cos δκ r

ð6:17Þ

The phase shift δκ is given by [73], tan δκ ¼

pca jl ðpr Þ½Qκ ðpr Þ=Pκ ðpr Þ  Eþmc 2 jl ðpr Þ pca ηl ðpr Þ½Qκ ðpr Þ=Pκ ðpr Þ  Eþmc2 ηl ðpr Þ

ð6:18Þ

where [73] l ¼ l þ 1 ¼ κ for κ < 0

ð6:19aÞ

l ¼ l  1 ¼ κ  1 for κ > 0

ð6:19bÞ

In the relativistic case, the matrix elements arising from multipole transitions in an external field have the form [66, 74], !

α  A jnκmj i D ¼ hpκ 0 m0j je

ð6:20Þ

where e α ¼ bxαx þ byαy þ bzαz are the (4x4) Dirac matrices,  αi ¼

0

σi

σi

0

 ; i ¼ x, y, z

ð6:21Þ

!

and A the electromagnetic vector potential. The ket jnκmji represents the initial (bound) state, and jpκ0 m0j i the continuum photoelectron. The vector potential can be written as [cf. (3.8)], !!

A r

!  ! / beq exp i k  r

ð6:22Þ

!

where beq is the polarization vector direction, k is the propagation vector of the !

photon which causes the pd. event, and ω ¼ cj k j is the angular frequency of the photon.

6

Relativistic Theory of Photoelectron Angular Distributions

123

!

If we take the z-axis to be along k and considering (for now) circularly polarized light, we can write the polarization vector as [74],  1  beq ¼ pffiffiffi bex þ ibey ¼ qξq ; q ¼ 1 2

ð6:23Þ

This choice means we are working in the spherical vector basis {ξ0, ξ1, ζ 1}. ! With the z-axis along k , we can expand the vector potential as a sum of the normalized spherical harmonics using (F.47), (2.60), and (4.11) [66, 73, 74], !  X ! ðiÞl ð2l þ 1Þjl ðkr ÞCl0 ðbr Þ exp i k  r ¼

ð6:24Þ

l !

α operates in spin The operator A operates in coordinate space and the operator e space. With that in mind, and using (6.22, 6.23 and 6.24), we construct the tensor product [3, 66], ! ! XX pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 l L  ðLÞ   1lþq e e ð1Þ α C l q ðiÞl ð2l þ 1Þjl ðkr Þ 2L þ 1 α  A ¼ αq  A  ¼ |fflfflfflfflffl ffl {zfflfflfflfflffl ffl } q1 0 q l q1 !

¼ð1Þl

ð6:25Þ where we also used (2.184) and (2.144b). Note that αq is the component of e α in the  ðLÞ q-direction and is a rank-one tensor. Defining X ðq1l,LÞ  e α Cl q , and seeing that the triangle relations inherent in the 3-j symbol in (6.25) require q1 ¼ q (thus collapsing the sum over q1), gives us,  X pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 l l l e ð1Þ ðiÞ 2L þ 1ð2l þ 1Þ αA ¼ q 0 l !

L q



jl ðkr ÞX ðq1l,LÞ

ð6:26Þ

Example 6.4 Using what we have done so far, find expressions for the electric 2L-pole operators and the magnetic 2L-pole operators. !

We can write the qth component of the interaction e α  A as a linear combination of tensors of rank L 1as follows [74],  ! X X pffiffiffiffiffiffiffiffiffiffiffiffiffiffi e ð1ÞL ð1Þl ðiÞl 2L þ 1ð2l þ 1Þ αA ¼ q

L





l

1

l

L

q

0

q

 jl ðkr ÞX ðq1l,LÞ

ð6:27Þ

124

6

Relativistic Theory of Photoelectron Angular Distributions

The triangle rule of the 3-j symbol in (6.27) tells us that the allowed values of l are l ¼ L, L  1. Thus, the tensors in the summation of (6.27) fall into two classes of opposite parity. Those for which the parity is (1)l ¼ (1)L  1 are the electric 2L-pole operators, and those for which the parity is (1)l ¼ (1)L are the magnetic 2L-pole operators. We now evaluate the sum over l in (6.27), by finding the needed 3-j symbols. Realizing that (L  1) + 1 + L is even, and recalling that q ¼  1 allows us to compute the following [7, 10], 1L

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð L þ 1Þ 2ð2L þ 1Þð2L  1Þ

1L

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L 2ð2L þ 1Þð2L þ 3Þ

!

1

L1

L

q

0

q

¼ ð1Þ

ð6:28Þ

Similarly, 



Lþ1 L 0 q

1 q

¼ ð1Þ

ð6:29Þ

Thus, we find the expression for the electric 2L-pole operators to be [66, 74],  !L   e αA 

L1

q ðeÞ

¼ ði Þ

"rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2L  1ÞðL þ 1Þ jL1 ðkr ÞX ðq1L1,LÞ 2 # rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2L þ 3ÞL  jLþ1 ðkr ÞX ðq1Lþ1,LÞ 2

ð6:30Þ

We also find [7, 10], 



1

L

L

q

0

q

¼ ð1Þ

1L

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2ð2L þ 1Þ

ð6:31Þ

so that the magnetic 2L-pole operators are [66, 74],  !L  e α  A 

q ðmÞ

1 ¼ pffiffiffi ð2L þ 1ÞðiÞL jL ðkr ÞX ðq1L,LÞ 2

ð6:32Þ

█

!

Since we are writing the interaction e α  A as a linear combination of tensors of rank L 1, we re-write (6.20) as follows [66, 74], D¼

X L

 !L h pκ00 m00j j e α  A jnκmj i q

ð6:33Þ

6

Relativistic Theory of Photoelectron Angular Distributions

125

As in earlier chapters, when computing the differential cross-section, we sum over all possible final states, square the interaction matrix element, and then average over all possible initial states [3, 66], dσ pd dΩ

¼

X 2 m00  m0 1 16π 2 X X χ κ00j ðpÞχ κ0 j ðpÞ jDj ¼ ð2j þ 1Þ mj ð2j þ 1Þ mj 00 0 00 0 00 LL mj mj j j  !L00 ! L α  A jnκjmj ihnκjmj jðe α  A Þq jpκ0 j0 m0 j i hpκ00 j00 m00 j j e

ð6:34Þ

q

The explicit dependence on j has been reinstated in the matrix elements to remind us that we need to sum over those states as well. We begin the process of working out the terms in the expression of the differential cross section in detail. First, we take care of the possible terms in ς by writing [74], 

 !L  0 0 0  !L  0 0 0  X  nκjmj , ς e α  A pκ j m j , ς α  A pκ j m j ¼ nκjmj  e q

q

ς

ð6:35Þ

If we write a typical term of the operators in (6.30) and (6.32) as alL ðr ÞX ðq1l,LÞ, we find that [73, 74],  !L  nκjmj j e α  A jpκ 0 j0 m0 j ¼ hnκjmj jalL ðr ÞX ðq1l,LÞ jpκ0 j0 m0 j i q P ¼ hnκjmj , ςjalL ðr ÞX ðq1l,LÞ jpκ0 j0 m0 j , ςi ς     P ¼ hnκj, ςjalL ðr Þjpκ0 j0 , ςi λjmj X ðq1l,LÞ λ0 j0 m0 j δλ,jþ12aς δλ0 ,j0 12a0 ς



ð6:36Þ

ς

    We operate on λjmj X ðq1l,LÞ λ0 j0 m0 j with the Wigner-Eckhart theorem, 

   λjmj X ð1l,LÞ λ0 j0 m0 j ¼ ð1Þjmj q

j

L

j0

mj

q

m0j

! hλjkX ð1l,LÞ kλ0 j0 i

ð6:37Þ

As stated earlier, this product of two tensors consists of one operator which operates in spin space, and the other which operates in coordinate space, and we can form the tensor product via (D.15) [3, 74], 8 >λ pffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffi< 1  ð1l,LÞ 0 0 hλjX kλ j i ¼ 2j þ 1 2j0 þ 1 2L þ 1 > :2 j  ðlÞ 0 ð 1 Þ  hλC kλ ih1=2ke σ k1=2i where we have rewritten the 6-j symbol as follows,

λ0 1 2 j0

9 l> = 1 > ; L ð6:38Þ

126

6

8 > < λ 1=2 > : j

Relativistic Theory of Photoelectron Angular Distributions

9 8 l> = >

; > : l L j0 λ0

1=2 1=2 1

9 j> = j0 > ; L

ð6:39Þ

and realized that the summation over ς in (6.36) has reduced the Dirac matrix e α to the Pauli spin matrix e σ. We now appeal to the identities (2.198) and (2.212) [3, 16], pffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffi hλkC kλ i ¼ ð1Þ 2λ þ 1 2λ0 þ 1 ðlÞ

λ

0



λ l 0 0

λ0 0

 ð6:40Þ

and h1=2ke σ ð1Þ k1=2i ¼

pffiffiffi 6

ð6:41Þ

Combining (6.36) through (6.41), pffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffi hnκjmj jalL ðrÞX ð1l,LÞ jpκ 0 j0 m0 j i ¼ ð1Þjmj þλ 2j þ 1 2j0 þ 1 2L þ 1 2λ þ 1 q ! ! pffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffi λ l λ0 j L j0 0  2λ þ 1 6  mj q m0j 0 0 0 9 8 λ λ0 l > > > > =X < 1 1 1 hnκj, ςjalL ðrÞjpκ0 j0 ,  ςiδλ,jþ12aς δλ0 ,j0 12a0 ς  2 2 > > > > ς ; : j j0 L ð6:42Þ We can simplify (6.42) by invoking the following identity based on (C.73) (for l ¼ L + 1) [10, 76], λ

Lþ1

0

0 λþjþ32

¼ ð1Þ

8 λ > ! > λ0 < 1 >2 0 > : j

λ0 1 2 j0

9 L þ1> > = 1 > > ; L

½ðλ  jÞð2j þ 1Þ þ ðλ0  j0 Þð2j0 þ 1Þ þ L þ 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 6ðL þ 1Þð2L þ 1Þð2L þ 3Þð2λ þ 1Þð2λ0 þ 1Þ

j

L

j0

1=2

0

1=2

!

ð6:43Þ Triangle rules inherent in the 9-j symbol in (6.43) tell us that λ ¼ j  1/2 and λ0 ¼ j0  1/2. So, we can write,

6

Relativistic Theory of Photoelectron Angular Distributions

127

½ðλ  jÞð2j þ 1Þ þ ðλ0  j0 Þð2j0 þ 1Þ þ L þ 1 2 3 κ κ0 zfflfflfflfflfflfflfflffl ffl}|fflfflfflfflfflfflfflfflffl{ zfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflffl{     6 7 ¼ 4ς 1 ð2j þ 1Þ ς 1 ð2j0 þ 1Þ þL þ 15 2 2

ð6:44Þ

Giving us (for l ¼ L + 1), λ 0

8 λ λ0 !> > < Lþ1 λ 1 1 2 2 > 0 0 > : j j0 0

¼ ð1Þ

λþjþ32

9 L þ 1> > = 1 > > ; L

½ςðκ 0  κÞ þ L þ 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 6ðL þ 1Þð2L þ 1Þð2L þ 3Þð2λ þ 1Þð2λ0 þ 1Þ

!

j

L

j0

1=2

0

1=2 ð6:45Þ

We also have the identity (for l ¼ L  1) [10, 76],

¼ ð1Þ ¼ ð1Þ

λþjþ32

λþjþ32

λ

L1

0

0

8 λ !> > λ > : j 0

λ0 1 2 j0

9 L 1> > = 1 > > ; L

½ðλ  jÞð2j þ 1Þ þ ðλ0  j0 Þð2j0 þ 1Þ  L pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 6Lð2L þ 1Þð2L  1Þð2λ þ 1Þð2λ0 þ 1Þ ½ςðκ 0  κÞ  L pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 6Lð2L þ 1Þð2L  1Þð2λ þ 1Þð2λ0 þ 1Þ

!

j

L

j0

1=2

0

1=2

j

L

j0

1=2

0

1=2

!

ð6:46Þ And finally (for l ¼ L ) [10, 76],

λ

L

0

0

8 !> λ 0 > λ 2 0 > : j

0

λ 1 2 j0

ςðκ 0 þκ Þ

9 zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{   L> 1 > =  ½ð2j þ 1Þ þ ð2j0 þ 1Þ λþλ0 þL 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ¼ð1Þ > 6L ð L þ 1Þð2L þ 1Þð2λ þ 1Þð2λ0 þ 1Þ > ; L ! j L j0  1=2 0 1=2 ð6:47Þ

128

6

Relativistic Theory of Photoelectron Angular Distributions

Combining (6.30) and (6.42, 6.43, 6.44, 6.45 and 6.46), we have, for the electric multipoles [66], pffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffi  !L  0 0 0  1 α  A j pκ j m j ¼ qð1Þmj þ2 ðiÞL1 2j þ 1 2j0 þ 1 nκjmj  e q ð eÞ !   j L j0 j L j0  mj q m0j 1=2 0 1=2 9 8 2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 > > ðL þ 1Þ > > 0 0 0 > > ½ςðκ  κ Þ  Lhnκj, ςjjL1 ðωr Þjpκ j ,  ςi =

> L > > ς 4 > > ½ςðκ0  κÞ þ L þ 1hnκj, ςjLþ1 ðωr Þjpκ0 j0 ,  ς  ; : 2 ðL þ 1Þ 

ð6:48Þ Combining (6.32), (6.42), and (6.47), we find, for the magnetic term [66], 

 !L  nκjmj  e αA  q

pffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffi  00 0 pκ j m j ¼ ð1Þmj þ12 ðiÞL 2j þ 1 2j0 þ 1 ! ðmÞ   j L j0 j L j0  mj q m0j 1=2 0 1=2 ð2L þ 1Þ X  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½ ς ðκ 0 þ κ Þ 2LðL þ 1Þ ς hnκj, ςjjL ðωr Þjpκ 0 j0 ,  ςiδλ,jþ12aς δλ0 ,j0 12a0 ς

ð6:49Þ

where we have noted (realizing that mj + 1/2 is an integer), ð1Þjmj þλ ð1Þλþjþ2 ¼ ð1Þ2ðjþλÞmj þ1þ2 ¼ ð1Þ2ðjþj2Þmj þ1þ2 3

1

1

1

¼ ð1Þ4j ð1Þ1þ1 ð1Þmj þ2 ¼ ð1Þmj þ2 ¼ ð1Þmj 2 ¼ ð1Þmj þ2 1

1

m0 

1

1

ð6:50Þ

m00

We now evaluate the term χ κ0 j ðpÞχ κ00j ðpÞ by first recalling, 0

χ ms  χ ms ¼ δms ,ms0

ð6:51Þ

which results in [cf. equation (6.1)], 0 00 0 00 pffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffi m0  m00 χ κ0 j ðpÞχ κ00j ðpÞ ¼ ð1Þν þv mj mj 2j0 þ 1 2j00 þ 1 ! ! 1 1 X ν0 ν00 j0 j00 2 2  ms m0j  ms ms m0j m00j  ms ms m00j

 Y ν0 ,m0 ms ð^r ÞY ν00 ,m00j ms ð^r Þ j

ð6:52Þ

6

Relativistic Theory of Photoelectron Angular Distributions

129

We invoke the following identity based on (C.74) [3, 10], ! 1 j00 2 m0j  ms m0j m00j  ms ms m00j 9 8 0 < j j00 ^ = P 0 00 0 0 00 ¼ ð1Þν þj þj þmj ms mj ð2 ^ þ1Þ : v00 v0 1 ; ^ 10 0 2 00 ν00 ν0 ^ j j A@  m00j  ms ðm0j  ms Þ m^ m0j m00j 1 2 ms

ν0

j0

!

ν00

ð6:53Þ ^

1 A

m^

leaving us with, 0 00 0 00 pffiffiffiffiffiffiffiffiffiffiffiffiffi m0  m00 χ κ0 j ðpÞχ κ00j ðpÞ ¼ ð1Þν þv mj mj 2j0 þ 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi X 0 0 00 0 00 Y ν0 ,m0 ms ð^rÞY ν00 ,m00j ms ð^r Þð1Þν þj þj þmj ms mj ð2 ^ þ1Þ  2j00 þ 1

ms ,^

j

9 8 ! ! < j0 j00 ^ = ν0 ν00 ^ j0 j00 ^  : v00 v0 1 ; m00j  ms ðm0j  ms Þ m^ m0j m00j m^ 2 ð6:54Þ

The two 3-j symbols in (6.54) are incompatible with one another unless m^ ¼ 0 (this also means that m00j ¼ m0j). Also, there are no other terms in m00j anywhere else in the expression for the pd. differential cross-section (6.34) except for those in (6.54) above. So, we can “pull forward” that summation to write, m0 

m00

χ κ0 j ðpÞχ κ00j ðpÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiP P 2j0 þ 1 2j00 þ 1 ð2 ^ þ1Þ ^

Y ν0 ,m0 ms j |fflfflfflffl{zfflfflfflffl}

ms , m00j ð1Þ

0

 ð1Þ2ν þv 0

j0

B @ m0 j

00

2m0j þj0 þj00 ms

j00 m00j

¼m0j

^

1

8 0

L ð2L þ 1Þ 0 0 0 0 0 0 > > p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð i Þ j ½ ð κ þ κ Þ  nκj, ς ð ωr Þjpκ j , ς ς h j i > > L > > > > 2L ð L þ 1 Þ > > > > > > > > r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi = < L1 ðL þ 1Þ 0 0 0 0 0 0 ς  ð i Þ ½ ð κ  κ Þ  L  h nκj, ς j j ð ωr Þjpκ j , ς i L1 > > 2L > > > > > > r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > >   > > L > L1 0 0 0 0 0 0 > > > j þ ð i Þ ½ ð κ  κ Þ þ L þ 1  nκj, ς ð ωr Þjpκ j , ς ς h ; : Lþ1 2 ð L þ 1Þ

ð6:65Þ We perform the sums over mj and m0j by first noting the following, 0 @

j0

j00

^

1

j00

mj q m0j mj q 0 0 1 j j00 ^ L 00 00 A ¼ ð1Þ3ðjþj Þ ð1ÞL þ^ @ q m0j m0j 0

m0j

m0j

m0j

0

A

L

j0

!

L00

j

j

!

j0

j

m0j

mj

!

j

j00

L00

mj

m0j

q

!

ð6:66Þ and by considering the identity [again based on (C.73)] [3, 10],

6

Relativistic Theory of Photoelectron Angular Distributions

X

0 ð1Þ

0

00

00

j þj þL þ^ @

mj , m0j

¼ ð1ÞLþjþq

j0

j00

m0j

m0j

L00

L

^

q

q

0

^ 0 !(

1

133

L

j0

j

q

m0j

L

j0

j

mj )

j00

L00

^

A

!

!

j

j00

L00

mj

m0j

q

ð6:67Þ giving us [66], X 1 dσ pd ð1Þjþ2þ^þq ð2j0 þ 1Þð2j00 þ 1Þð2 ^ þ1Þ ¼ 4π dΩ L L00 j0 j00 ^ ! !   j L j0 j00 ^ j0 L L00 ^  q q 0 1=2 !( 1=2 00 0 ) 1=2 0 1=2 00 00 L L ^ L j j P^ ð cos θÞ  j00 j0 j " 1=2 0 1=2

 P  0 0 0 0 00 00 00 00 00 00 00  00  hnκj, ς jal¼L,L1; L ðr Þjpκ j , ς ihpκ j , ς jal ¼L ,L 1; L ðr Þjnκj, ς i ς0 , ς00

ð6:68Þ where we have also made liberal use of the symmetry properties of the 3-j symbols to arrive at our final phase factor. It is clear that this expression is more complicated than the non-relativistic expressions seen earlier, since it allows for additional interference between different multipole terms (except when ^ ¼ 0) [66]. !  ! We can realize the electric dipole approximation by setting exp i k  r  1, which translates to setting L ¼ 1 and l ¼ 0 in (6.42) [c.f. (6.24) and (6.25)] [66]. Under these circumstances, the top term on the RHS of (6.65) is the magnetic dipole term, and the bottom term on the RHS of (6.65) is the electric quadrupole 00 term. It is the middle term that represents the electric dipole. Setting L, L ¼ 1; l ¼ 0 in (6.68) and using the middle term on the RHS of (6.65), ! ! 1 1 ^ X j0 j00 ^ dσ pd 0 00 ¼4π ð2j þ 1Þð2j þ 1Þð2 ^ þ1Þ dΩ 1=2 1=2 0 q q 0 j0 j00 ^ ( ) ! ! 1 1 ^ j 1 j0 j00 1 j P^ ð cos θÞ  j00 j0 j 1=2 0 1=2 1=2 0 1=2 " # X 0 0 00 00 0 0 0 0 00 00 00 00 ½ς ðκ  κ Þ  1½ς ðκ  κ Þ  1jhnκj,ς jpκ j ,  ς ihpκ j , ς jnκj,  ς ij  ς0 , ς00

ð6:69Þ where we have also noted that j0(kr) ! 1 [c.f. (F.26)] in the long-wavelength limit (dipole approximation).

134

6

Relativistic Theory of Photoelectron Angular Distributions

Triangle rules and symmetry properties of the 3-j and 6-j symbols in (6.69) limit the sum over ^ to the terms ^ ¼ 0, 2, giving us, X 1 dσ pd ¼ 4π ð1Þjþ2 ð2j0 þ 1Þð2j00 þ 1Þ dΩ j0 j00 8 ! 9 ! ! 1 1 0 j00 0 j 1 j0 j0 > > > > > > > > > > > > > > q q 0 1=2 0 1=2 1=2 1=2 0 > > > > > > ! ( ) > > 00 > > 1 1 0 j 1 j > > > > > >  > > > > = < 00 0 j j j 1=2 0 1=2  ! ! ! 0 > > 1 1 2 j 1 j j0 j00 2 > > > > > > þ5 > > > > > > 1=2 1=2 0 > > q q 0 1=2 0 1=2 > > > > > ! ( ) > > > 00 > > 1 1 2 1 j j > > > > > > P2 ð cos θÞ  > > ; : 00 0 1=2 0 1=2 j j j " # P 0  ½ς ðκ  κ 0 Þ  1½ς00 ðκ  κ00 Þ  1jhnκj,ς0 jpκ0 j0 ,  ς0 ihpκ00 j00 ,ς00 jnκj,  ς00 ij ς0 , ς00

ð6:70Þ For linearly-polarized photons, we set q ¼ 0 in the above expression, X 1 dσ pd ¼ 4π ð1Þjþ2 ð2j0 þ 1Þð2j00 þ 1Þ dΩ j0 j00 1 9 8 ! !0 j 1 j0 j0 j00 0 > > 1 1 0 > > > > @ A > > > > > > > > 0 0 0 1=2 1=2 0 > > 1=2 0 1=2 > > |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} > > > > p ffi q ffiffiffiffiffiffiffiffi ffi > > > > 1 0 j þ1=2 > >  3 1 > > ð 1 Þ δ 0 00 0 þ1Þ j j > > 2j ð > > > > > > > > 0 00 1 ( > > ) > > > > j 1 j > 1 1 0 > > > > > @ A > >  > > > > > > 00 0 > > j j j 1=2 0 1=2 > > > > |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} = < q ffiffiffiffiffiffiffiffiffiffi 0 þ1 jþj 1  ð1Þ δ 0 00 3ð2j0 þ1Þ j j > > > > > > > > 0 1 0 1 > > ! > > 0 00 0 > > j j 2 j 1 j > > 1 1 2 > > > > @ A @ A > > þ 5 > > > > > > > > 0 0 0 1=2 1=2 0 1=2 0 1=2 > > > > |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} > > > > p ffiffi ffi > > 2 > > > > 15 > > > > > > 9 8 0 1 > > > 00 > > > 1 1 2 1 j j = < > > > > > > @ A > > P ð cosθ Þ  2 > > ; : ; : 00 0 j j j 1=2 0 1=2 hP i  ½ς0 ðκ  κ0 Þ  1½ς00 ðκ  κ00 Þ  1jhnκj, ς0 jpκ 0 j0 ,  ς0 ihpκ 00 j00 , ς00 jnκj,  ς00 ij ς0 , ς00

6

Relativistic Theory of Photoelectron Angular Distributions

135

X 0 dσ pd ð2j0 þ 1Þð2j00 þ 1Þ ¼ 4π ð1Þ2ðjþj Þþ1 4π dΩ j0 j00 0 1 0 1 9 8 j00 1 j 1 j0 j > > > > 1 > > @ A@ A > > > > 0 > > 3 2j ð þ 1 Þ > > > > 1=2 0 1=2 1=2 0 1=2 > > > > > > > > 0 1 0 1 0 1 > > 0 00 0 00 ffiffiffiffiffi r > > j 2 j 1 j 1 j j j = < 2@ A@ A A@  þ5 15 > > > 1=2 1=2 0 1=2 0 1=2 > 1=2 0 1=2 > > > > > > > > 8 9 > > > > > > 1 1 2 < = > > > > > > > > P ð cos θ Þ  2 > > ; : : 00 0 ; j j j " # X 0 0 00 00 0 0 0 0 00 00 00 00  ½ς ðκ  κ Þ  1½ς ðκ  κ Þ  1jhnκj,ς jpκ j ,  ς ihpκ j ,ς jnκj,  ς ij

)

ς0 , ς00

ð6:71Þ 00

Again the 3-j symbols come to our rescue, limiting the sums over j0 and j to the 00 values j0, j ¼ j, j  1. We can now evaluate all nine possible combinations of terms using tabulated values of 3-j and 6-j symbols, which allow us to calculate the following factors [3, 7, 10], For j0 ¼ j + 1, j00 ¼ j  1, ð1Þ2ð2jþ1Þþ1 ð2j þ 3Þð2j  1Þ 0 10 1 9 8 j 1 jþ1 j1 1 j > > > > 1 > > @ A@ A > > > > > > 3 ð 2j þ 3 Þ > > > > 1=2 0 1=2 1=2 0 1=2 > > > > > > > > 0 1 0 1 0 1 > > rffiffiffiffiffi > > jþ1 j1 2 j 1 jþ1 j1 1 j = < 2@ A @ A @ A  5 15 > > > 1=2 1=2 0 1=2 0 1=2 1=2 0 1=2 > > > > > > > > > 9 > > 8 > > > > 1 1 2 = < > > > > > > > > P ð cos θ Þ  2 > > ; : : ; j1 jþ1 j # " P 0 0 0 00 00 0 0 0 00 00 00  ½ς ðκ  κ Þ  1½ς ðκ  κ Þ  1jhnκj, ς jpκ j ,  ς ihpκ j  1, ς jnκj, ς ij ς0 , ς00

#   ð2j þ 3Þð2j  1Þ ð2j  1Þ   pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ ¼ P2 ð cos θÞ ℜ jþ1 ℜ j1  16j ð j þ 1 Þ 12 jðj þ 1Þ "

ð6:72aÞ

136

6

Relativistic Theory of Photoelectron Angular Distributions

00

For j0 ¼ j  1, j ¼ j + 1, ð1Þ2ð2j1Þþ1 ð2j  1Þð2j þ 3Þ 1 0 10 9 8 jþ1 1 j j 1 j1 > > > > 1 > > A @ A@ > > > > > > 3 ð 2j þ 3 Þ > > > > 1=2 0 1=2 1=2 0 1=2 > > > > > > > > 0 1 0 1 0 1 > > ffiffiffiffiffi r > > j1 jþ1 2 j 1 j1 jþ1 1 j = < 2@ A @ A @ A  5 15 > > > 1=2 1=2 0 1=2 0 1=2 1=2 0 1=2 > > > > > > > > > 8 9 > > > > > > 1 1 2 < = > > > > > > > > P ð cos θ Þ  2 > > ; : : ; jþ1 j1 j " # X 0 0 00 00 0 0 0 0 00 00 00  ½ς ðκ  κ Þ  1½ς ðκ  κ Þ  1jhnκj, ς jpκ j ,  ς ihpκ j þ 1, ς jnκj, ς ij ς0 , ς00

"

#   ð2j  1Þ ð2j þ 3Þð2j  1Þ   pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ ¼ P2 ð cos θÞ ℜ j1 ℜ jþ1  16jðj þ 1Þ 12 jðj þ 1Þ ð6:72bÞ

00

For j0 ¼ j + 1, j ¼ j, ð1Þ2ð2jþ3Þþ1 ð2j þ 3Þð2j þ 1Þ 0 10 1 9 8 jþ1 1 j j 1 j > > > > 1 > > A @ A@ > > > > > > 3 ð 2j þ 3 Þ > > > > 1=2 0 1=2 1=2 0 1=2 > > > > > > 0 1 0 1 0 1 > > > > rffiffiffiffiffi j þ 1 = < j 2 j 1 jþ1 j 1 j 2 @ A@ A@ A  5 > > 15 > > > 1=2 1=2 0 1=2 0 1=2 1=2 0 1=2 > > > > > > > > > ( ) > > > > 1 1 2 > > > > > > > > P2 ð cos θÞ ; : j jþ1 j " # X 0 0 00 00 0 0 0 0 00 00 00  ½ς ðκ  κ Þ  1½ς ðκ  κ Þ  1jhnκj, ς jpκ j ,  ς ihpκ j, ς jnκj,  ς ij ς0 , ς00

"

#   ð2j þ 1Þ ð2j þ 3Þ   pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ P ð cos θ Þ ℜ ℜ ¼  2 jþ1 j 12 2jðj þ 1Þ 16jðj þ 1Þ2 ð6:72cÞ

6

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137

00

For j0 ¼ j, j ¼ j + 1, ð1Þ2ð2jÞþ1 ð2j þ 1Þð2j þ 3Þ 1 0 10 9 8 jþ1 1 j j 1 j > > > > 1 > > A @ A@ > > > > > > 3 ð 2j þ 3 Þ > > > > 1=2 0 1=2 1=2 0 1=2 > > > > > > > > 0 1 0 1 0 1 > > ffiffiffiffiffi r > > j jþ1 2 j 1 j jþ1 1 j = < 2@ A @ A @ A  5 15 > > > 1=2 1=2 0 1=2 0 1=2 1=2 0 1=2 > > > > > > > > > 8 9 > > > > > > 2 1 1 < = > > > > > > > > P ð cos θ Þ  2 > > ; : : ; jþ1 j j " # X 0 0 00 00 0 0 0 0 00 00 00  ½ς ðκ  κ Þ  1½ς ðκ  κ Þ  1jhnκj, ς jpκ j ,  ς ihpκ j þ 1, ς jnκj,  ς ij ς0 , ς00

"

#   ð2j þ 1Þ ð2j þ 3Þ    pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ ¼ P ð cos θ Þ ℜ ℜ  2 j jþ1  12 2jðj þ 1Þ 16jðj þ 1Þ2 ð6:72dÞ 00

For j0, j ¼ j, ð1Þ2ð2jÞþ1 ð2j þ 1Þ2 9 8 0 12 > > j 1 j > > > > 1 > > @ A > > > > > > > > 3ð2j þ 1Þ = < 1=2 0 1=2  9 10 12 8 > > rffiffiffiffiffi0 j > > j 1 j j 2

> > > 2 > > @ A @ A > > P 5 ð cos θ Þ > > 2 > > 15 ; : ; : 1=2 1=2 0 1=2 0 1=2 j j j # " X  ½ς0 ðκ  κ 0 Þ  1½ς00 ðκ  κ 00 Þ  1jhnκj, ς0 jpκ0 j0 ,  ς0 ihpκ 00 j, ς00 jnκj,  ς00 ij ς0 , ς00

"

# ð2j  1Þð2j þ 3Þ 1  P2 ð cos θÞ ℜ 2j ¼ 12jðj þ 1Þ 48j2 ðj þ 1Þ2 ð6:72eÞ

138

6

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00

For j0, j ¼ j + 1, ð1Þ2ð2jþ3Þþ1 ð2j þ 3Þ2 1 0 10 9 8 jþ1 1 j j 1 jþ1 > > > > 1 > > A @ A@ > > > > > > 3 ð 2j þ 3 Þ > > > > 1=2 0 1=2 1=2 0 1=2 > > > > > > 0 1 0 1 > > ! > > = < rffiffiffiffiffi j þ 1 j þ 1 2 j 1 jþ1 jþ1 1 j 2 @ A@ A  5 > > 15 1=2 1=2 0 > > > 1=2 0 1=2 1=2 0 1=2 > > > > > > > > > ( ) > > > > 1 1 2 > > > > > > > > P2 ð cos θÞ ; : jþ1 jþ1 j " # X 0 0 00 00 0 0 0 0 00 00 00  ½ς ðκ  κ Þ  1½ς ðκ  κ Þ  1jhnκj, ς jpκ j ,  ς ihpκ j þ 1, ς jnκj,  ς ij ς0 , ς00

"

# ð2j þ 3Þ ð2j þ 5Þð2j þ 3Þ þ P2 ð cos θÞ ℜ 2jþ1 ¼ 12ðj þ 1Þ 48ðj þ 1Þ2 ð6:72fÞ 00

For j0 ¼ j, j ¼ j  1, ð1Þ2ð2jÞþ1 ð2j þ 1Þð2j  1Þ 0 10 1 9 8 j 1 j j1 1 j > > > > 1 > > @ A@ A > > > > > > 3 ð 2j þ 1 Þ > > > > 1=2 0 1=2 1=2 0 1=2 > > > > > > > rffiffiffiffiffi0 10 10 1> > > > > j j1 2 j 1 j j1 1 j = < 2@ A @ A @ A  5 15 > > > 1=2 0 1=2 > 1=2 1=2 0 1=2 0 1=2 > > > > > > > > 9 > > 8 > > > > 1 1 2 = < > > > > > > > > P ð cos θ Þ  2 > > ; : : ; j1 j j " # X 0 0 00 00 0 0 0 0 00 00 00  ½ς ðκ  κ Þ  1½ς ðκ  κ Þ  1jhnκj, ς jpκ j ,  ς ihpκ j  1, ς jnκj,  ς ij ς0 , ς00

"

#   ð2j  1Þ ð2j þ 1Þ   pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 2 P2 ð cos θÞ ℜ j ℜ j1  ¼ 12j ð2j þ 1Þðj þ 1Þ 16j ðj þ 1Þ ð6:72gÞ

6

Relativistic Theory of Photoelectron Angular Distributions

139

00

For j0 ¼ j  1, j ¼ j, ð1Þ2ð2j1Þþ1 ð2j  1Þð2j þ 1Þ 1 0 10 9 8 j 1 j j 1 j1 > > > > 1 > > A @ A@ > > > > > > 3 ð 2j þ 1 Þ > > > > 1=2 0 1=2 1=2 0 1=2 > > > > > > 0 1 0 1 0 1 > > > > = < rffiffiffiffiffi j  1 j 1 j j 2 j 1 j1 2 A@ A @ A@  5 > > 15 > > > 1=2 0 1=2 > 1=2 1=2 0 1=2 0 1=2 > > > > > > > > ( ) > > > > 1 1 2 > > > > > > > > P2 ð cos θÞ ; : j j1 j # " X 0 0 00 00 0 0 0 0 00 00 00  ½ς ðκ  κ Þ  1½ς ðκ  κ Þ  1jhnκj, ς jpκ j ,  ς ihpκ j, ς jnκj,  ς ij ς0 , ς00

"

#   ð2j  1Þ ð2j þ 1Þ   pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 2 P2 ð cos θÞ ℜ j1 ℜ j  ¼ 12j ð2j þ 1Þðj þ 1Þ 16j ðj þ 1Þ ð6:72hÞ 00

For j0, j ¼ j  1, ð1Þ2ð2j1Þþ1 ð2j  1Þ2 0 10 1 9 8 j 1 j1 j1 1 j > > > > 1 > > @ A@ A > > > > > > 3 ð 2j  1 Þ > > > > 1=2 0 1=2 1=2 0 1=2 > > > > > > > rffiffiffiffiffi0 10 10 1> > > > > j1 j1 2 j 1 j1 j1 1 j = < 2@ A @ A @ A  5 15 > > > 1=2 0 1=2 > 1=2 1=2 0 1=2 0 1=2 > > > > > > > > 9 8 > > > > > > 1 1 2 = < > > > > > > > > P ð cos θ Þ  2 > > ; : : ; j1 j1 j " # X 0 0 00 00 0 0 0 0 00 00 00  ½ς ðκ  κ Þ  1½ς ðκ  κ Þ  1jhnκj, ς jpκ j ,  ς ihpκ j  1, ς jnκj,  ς ij ς0 , ς00

¼



ð2j  1Þ ð2j  1Þð2j  3Þ þ P ð cos θ Þ ℜ 2j1 2 12j 48j2 ð6:72iÞ

We use the symbol ℜ j0 for the radial portion of the matrix element, written below in terms of the large and small radial components of the radial Dirac equation [66, 73],

140

6

ℜj0 ¼

P ς0

Relativistic Theory of Photoelectron Angular Distributions



½ς0 ðκ  κ0 Þ  1jhnκj, ς0 jpκ0 j0 ,  ς0 ijexp iδj0



¼ ifðκ  κ 0  1ÞhQj0 jPj i þ ðκ  κ0 þ 1ÞhPj0 j Qj igexp iδj0

ð6:73Þ

Substituting (6.72) and (6.73) into (6.71) results in [67], dσ pd σ pd ½1 þ βP2 ð cos θÞ ¼ dΩ 4π

ð6:74Þ

where [66, 67, 77], σ pd ¼ 16π 2



ð2j þ 3Þ 2 2j  1 2 1 ℜ 2j þ ℜ jþ1 ℜ j1 þ 12j 12jðj þ 1Þ 12ðj þ 1Þ

ð6:75aÞ

and the relativistic asymmetry parameter is (in the dipole approximation) [66, 67, 77], β¼ 9 8 ð2j  3Þð2j  1Þ 2 ð2j  1Þð2j þ 3Þ 2 ð2j þ 5Þð2j þ 3Þ 2 > > > > ℜ  ℜ þ ℜ > > j1 j jþ1 = < 48j2 48j2 ðj þ 1Þ2 48ðj þ 1Þ2     > > ð2j  1Þ  ð2j þ 3Þ  ð2j þ 3Þð2j  1Þ   >    > > ℜ ℜ ℜ ℜ þ ; :þ 2 ℜ j ℜ j1  þ    > j j1 jþ1 jþ1 8jðj þ 1Þ 8j ðj þ 1Þ 8jðj þ 1Þ2



ð2j  1Þ 2 ð2j þ 3Þ 2 1  ℜ j1 þ ℜ 2j þ ℜ 12j 12jðj þ 1Þ 12ðj þ 1Þ jþ1

1

ð6:75bÞ Other expressions for the relativistic version of β (using different normalization schemes) have also been proposed [69, 77]. It now remains only to show that the above expression for the relativistic asymmetry parameter reduces to the familiar non-relativistic expression in the appropriate limit. That is the subject of the following example. Example 6.6 Show that (6.75b) reduces to the appropriate expression in the non-relativistic limit. First, we shift the energy by a constant amount: E ¼ W þ mc2. Equations (6.11a) and (6.11b) now appear as,

∂Pnκ ðr Þ κ 1 ¼  Pnκ ðr Þ þ W  V ðr Þ þ 2mc2 Qnκ ðr Þ r cħ ∂r

ð6.11a-modifiedÞ

∂Qnκ ðr Þ κ 1 ¼ Qnκ ðr Þ  ðW  V ðr ÞÞPnκ ðr Þ r cħ ∂r

ð6.11b-modifiedÞ

In the limit c ! 1, we have from (6.11a-modified),

6

Relativistic Theory of Photoelectron Angular Distributions

Q

  1 κ P0 þ P 2c r

141

ð6:76Þ

Taking the derivative of (6.11a-modified) after realizing in the non-relativistic limit that ðW  V Þ ∕ c > 2 > > ℜ  ℜ > > l0 1 l þ1 2 > > > > 48ðl0 þ 1=2Þ2 ðl0 þ 3=2Þ2 0 > > 48ðl0 þ 1=2Þ > > > > > > = < ð2l þ 4Þð2l þ 6Þ   ð2l0 Þ 0 0 2    ℜ þ ℜ  ℜ β¼ l0 þ1 l0 1 l0 þ1 2 2 > > 48ðl0 þ 3=2Þ 8ðl0 þ 1=2Þ ðl0 þ 3=2Þ > > > > > > > > > >   > > ð2l0 þ 4Þ ð2l0 þ 4Þð2l0 Þ 2  > >   > þ ℜ  ℜ ℜ l0 1 l0 þ1 > ; : l0 þ1 2 8ðl0 þ 1=2Þðl0 þ 3=2Þ 8ðl0 þ 1=2Þðl0 þ 3=2Þ  1 ð2l0 þ 4Þ 2l0 1  ℜ 2l0 1 þ ℜ 2l0 þ1 þ ℜ2 12ðl0 þ 1=2Þ 12ðl0 þ 3=2Þ l0 þ1 12ðl0 þ 1=2Þðl0 þ 3=2Þ h 9 8 > 4l0 þ ðl0 þ 3Þð2l0 þ 1Þ2 > > > ð þ 2 Þ l > > 0 2 > > > > = < l0 ðl0  1Þℜ l0 1 þ 2 1 ð2l0 þ 3Þ 2 ð þ 1 Þℜ þ6 2l ¼ 0 l0 þ1 2> > 3ð2l0 þ 1Þ > > > >   > > 6l > > 0  ; : ½1 þ ðl0 þ 2Þð2l0 þ 1Þℜ l0 þ1 ℜ l0 1  ð2l0 þ 3Þ  h i1 1 2 2  þ ðl þ 1Þℜ l0 þ1 l ℜ 3ð2l0 þ 1Þ 0 l0 1   l0 ðl0  1Þℜ 2l0 1 þ ðl0 þ 1Þðl0 þ 2Þℜ 2l0 þ1  6l0 ðl0 þ 1Þℜ l0 þ1 ℜ l0 1  h i ¼ ð2l0 þ 1Þ l0 ℜ 2l0 1 þ ðl0 þ 1Þℜ 2l0 þ1 ð6:83Þ This result is identical to the form in (4.39b). █ Similarly, we find that (6.75a) reduces to (4.39a) if we account for the appropriate definition (normalization) for the non-relativistic cross-section (see Problem 6.9) [66, 67]. So, our relativistic analysis agrees with our non-relativistic one in the appropriate limit. One thing example 6.6 shows us is that a change in coupling schemes cannot, in and of itself, lead to a change in the asymmetry parameter. But it does show that the value of β will deviate from the non-relativistic value when the matrix elements (and relative phase shifts) are different in the relativistic theory [66]. For example, in the case of s-electron detachment, we find by substituting j ¼ 1/2 into (6.75b),

β¼

    2ℜ 2p3=2 þ 4ℜ p3=2 ℜ p1=2  ℜ 2p1=2 þ 2ℜ 2p3=2

ð6:84Þ

It is only when the two radial integrals in (6.84) are equal (and the relative phase shifts are identical) that we recover the familiar (non-relativistic) result β(l0 ¼ 0) ¼ 2. Furthermore, if the two radial matrix elements ℜ p3=2 and ℜ p1=2 were to go to zero at the same photon energy, then β would have no meaning (since there would be a zero cross-section). Fortunately, it is well-known that the two matrix elements go to zero

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143

at slightly different energies, which is why a non-zero minimum is observed at the “Cooper minimum” in certain photodetachment and photoionization spectra [66]. If ℜ p1=2 alone is zero, then (6.84) tells us that β ¼ 1. If ℜ p3=2 alone is zero, then (6.84) tells us that β ¼ 0. So, if rapid oscillations in β are observed near the (non-zero) minimum of certain photoelectron spectra, it may be an indication of (possibly large) deviations from the predictions of the non-relativistic theory for that species. These effects may also be present in systems having angular momentum of some value other than j ¼ 1/2; certainly (6.75b) seems to predict that, in many cases, β will be different for the two components of any l value. For example, if two photoelectrons from two states split by spin-orbit coupling have two different kinetic energies when detached by photons of the same energy (because their pd. potentials are slightly different, for example), then it may be necessary to account for spin-orbit effects even in a non-relativistic approximation, especially near pd. cross-section minima [67, 78–86]. This effect can be further magnified because the two electrons have different radial wave functions in the relativistic theory, and hence, different dipole matrix elements with the continuum orbitals. So, while it may be obvious that relativistic descriptions are necessary for high-Z elements and for high photon energies, it may also be necessary to consider relativistic effects for lighter species at low photon energies, even within the dipole approximation [51, 66–69].

144

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Problems 6.1. Spherical Bessel functions. In App. F, we developed a series representation of the spherical Bessel functions by first identifying an integral form for these functions in the complex plane. In this problem, we will use standard series solution methods to find an expression for the spherical Bessel functions. The standard form of Bessel’s equation is,   ν2 d2 y 1 dy y¼0 þ þ 1  x2 dx2 x dx The parameter v is a given number, so we can consider v 0 without loss of generality. When v ¼ p, p¼integer, we refer to the above equation as Bessel’s equation of order p. (a) Use the Method of Frobenius (MOF) to construct a series solution for the above equation about the regular singular point x ¼ 0. (b) Denote the solutions to Bessel’s equation as Jν(x). For ν6¼integer, Jν(x) and Jν(x) are linearly independent. The general solution to Bessel’s equation is then, yðxÞ ¼ C 1 J ν ðxÞ þ C 2 J ν ðxÞ For v ¼ p, p¼integer, Jp(x) and Jp(x) are no longer linearly independent. To prove this, show that Jp(x) ¼ (1)pJp(x). Note: To get the second linearly independent solution of Bessel’s equation of integer order, we define, N v ð xÞ ¼

J v ðxÞ cos νπ  J ν ðxÞ sin πν

Nv(x) is called the Neumann function or the Bessel function of the second kind. Thus, for the case p¼integer, the general solution of Bessel’s equation of integer order is, yð xÞ ¼ C 1 J p ð xÞ þ C 2 N p ð xÞ Note that Jν(x) and Nν(x) are linearly independent even if ν6¼integer. Also note that Nν(x) is not finite at x ¼ 0, but is still needed for solutions in regions “away” from origin. The differential equation for the spherical Bessel functions arises from expressing the Helmholtz equation in spherical coordinates. The differential equation for the spherical Bessel functions is,

6

Relativistic Theory of Photoelectron Angular Distributions

145

  r 2 R00 þ 2rR0 þ k2 r 2  lðl þ 1Þ R ¼ 0; l ¼ integer; prime denotes d=dr: (c) Let R(r) ¼ r1/2S(r) and show that the above differential equation can be written as Bessel’s equation of order l þ 12 , and that the solutions are therefore,   Rðr Þ ¼ r 1=2 c1 J lþ1=2 ðkr Þ þ c2 N lþ1=2 ðkr Þ (d) The standard representations of the spherical Bessel functions of the first and second kind are, jl ðxÞ 

rffiffiffiffiffi rffiffiffiffiffi π π J lþ1=2 ðxÞ and nl ðxÞ  N ðxÞ: 2x 2x lþ1=2

Using your results from parts (a) and (c), develop a series representation for jl(x). Show that it matches (F.26). 6.2. The Pauli spin matrices. There are several ways to deduce the forms of the spin matrices of (6.8). One way is based on physical arguments (see, for example, Feynman, Leighton, and Sands, The Feynman Lectures of Physics: The New Millennium Edition, Vol III: Quantum Mechanics, Basic Books, New York, 2010). Another way to construct the matrices is by assuming they obey the commutation rules of (2.1). (a) Consider the set of 2x2 spin matrices which satisfy the commutation relations of (2.1), 

X  Sl , Sj ¼ iħ εljk Sk ; l, j, k ¼ x⇄1, y⇄2, z⇄3 k

Since the eigenvalues of these matrices are ħ/2, it is conventional to introduce the Pauli matrices σ k such that Sk  ħ2 σ k , giving us, 

X  σ l , σ j ¼ 2i εljk σ k ; l, j, k ¼ x⇄1, y⇄2, z⇄3 k

 Given σ 3 ¼

1

0



, find the general Hermitian matrices σ 1 and σ 2 that 0 1 (along with σ 3) satisfy the commutation rules above.

(b) Using your results from part (a), establish the following relations: P (i) σ i σ j ¼ i εijk σ k þ δij I 2 where I2 is the unit matrix for the 2x2 space. k

(ii) σ iσ j ¼  σ jσ i (iii) σ 1σ 1σ 3 ¼ iI2

146

6

Relativistic Theory of Photoelectron Angular Distributions

Trσ i ¼ 0 Detσ i ¼ 1 σ 2i ¼ I 2 Tr(σ iσ j) ¼ 2δij {σ i, σ j} ¼ 2δijI2 where {σ i, σ j}  σ iσ j + σ jσ i e σe σ ¼ 2ie σ  ! !   ! ! ! ! ! ! (x) e σA e σ AB þ σB ¼i e σ  A  B þ A  B I 2 ¼ ie ! ! ! ! A  B I 2 where A and B are not spin operators.

(iv) (v) (vi) (vii) (viii) (ix)

!

(xi) For an arbitrary vector α , show that, eiα eσ ¼ cos jαj þ i !

!

σ α e sin jαj jαj

6.3. The Dirac matrices are defined in (6.21). Prove the following properties of the Dirac matrices,   (a) αi , αj ¼ 2δij I 4   n o 0 I2 β ¼ 0; e β (b) αi , e 0 I 2 2 2 β ¼ I4 (c) ðαi Þ ¼ e   σi 0 σ 4  bxσ 4x þ byσ 4y þ bzσ 4z ; σ 4i ¼ (d) e αe α ¼ 2ie σ 4 where e 0 σi  ! !      ! ! ! ! ! ! ! ! αB ¼i e σ 4  A  B þ A  B I 4 ¼ ie (e) e αA e σ4  A  B þ A  B I 4   0 I 2 (f) e γ5e σ 4 ¼ e α and eγ 5 e γ5  . α ¼ e σ 4 where e I 2 0 6.4. Prove the following,  !  ! ! ∂ (a) ∇ ¼ br br  ∇  br  br  ∇ ¼ br ∂r  ħi

!

r jr j

!

L

(b) From part (a), we can say,

  ! ! ! ! ∂ e α  ∇ ¼ iħe α  br ∂r  j1rj e α  p ¼ iħe α r  L

Thus, show that, !

e α  p ¼ iħαr

  ∂ i βK  ħ þ αr e ∂r jr j !

α  br and the operator e βK ¼ L  e σ 4 þ ħ. where αr ¼ e

6

Relativistic Theory of Photoelectron Angular Distributions

147

6.5. The Dirac Hamiltonian for a free particle is, !

α p þe βmc2 H 0 ¼ ce The eigenvalue equation H0ψ ¼ Eψ can therefore be written as,   ! E  ce α p e βmc2 Ψ ¼ 0 !

∂ If we identify E ¼ iħ ∂t and p ¼ iħ∇ , then the time-dependent Dirac equation is,



   ! ∂ ! 2 e iħ þ ciħe α  ∇  βmc Ψ r , t ¼ 0 ∂t

! !

    ! ! Solve this equation to find Ψ r , t :Assume the form Ψ r , t ¼ e uei k  r ωt where the elements of e u are just numbers. Remember that, since e α and e β are 44 matrices, e u must be in the form of a 41 column matrix. 6.6. The Dirac equation for a spherically symmetric potential is, h i ! ce α p þe βmc2 þ V ðr Þ ψ nκmj ðr Þ ¼ Eψ nκmj ðr Þ Substitute the following form into the Dirac equation,   ! ψ nκmj r ¼

"

m

F nκ ðr Þχ κ j ðbr Þ m iGnκ ðr Þχ κj ðbr Þ

#

and using results from the previous problem, show that the functions of (6.10) solve the resulting radial eq. (6.11). 6.7. Interaction of an electron with the electromagnetic field in the Dirac formalism. (a) An expression for the operator of the time-rate-of-change of an observable O can be found from the Heisenberg equation of motion [6],

∂O ∂O i ¼ þ ½H, O ∂t op ∂t ħ

where H is the Hamiltonian. The Dirac Hamiltonian for a free particle is,

148

6

Relativistic Theory of Photoelectron Angular Distributions !

H 0 ¼ ce α p þe βmc2 Use the Heisenberg equation of motion to find the form of the “velocity” operator for the Dirac Hamiltonian of a free particle. (b) The Hamiltonian for the interaction of an electron with an electromagnetic ! field Hint is realized by replacing the canonical momentum operator p with !

!

the kinetic momentum operator p  ec A in the Dirac Hamiltonian for a free particle, so that,   e! e! ! H ¼ ce α p  A þe α A βmc2 ¼ H 0 þ H int ; H int ¼ ce c c Show that this interaction Hamiltonian reduces to the correct form given in chapter four [in (4.4a), for example] in the non-relativistic limit. h !i h !i and ∂A for the (c) Use the Heisenberg equation of motion to find ∂∂tp ∂t op

op

Dirac Hamiltonian,      e! ! ! ! H ¼ ce α p  A r þV r þe βmc2 c (d) Use the above results along with Newton’s second law to find the force on an electron. 6.8. The Dirac equation in the presence of an electromagnetic field. From Problem 6.5 above, the Dirac equation for a free particle can be written as,   ! βmc2 Ψ ¼ 0 E  ce α p e In the presence of an electromagnetic field, we make the replacements, ! e! ! ! ! ! E ! E  eΦ and p ! p  A ) c p ! c p  eA : c

giving us, h  i ! ! ðE  eΦÞ  e α  c p  eA  e βmc2 Ψ ¼ 0 i h  ! ! Multiply from the left by ðE  eΦÞ þ e α  c p  eA þ e βmc2 . Manipulate the result to show that the Dirac equation in the presence of an electromagnetic field can be written as,

6

Relativistic Theory of Photoelectron Angular Distributions



149

    ! !2 ! ! 2 4 ðE  eΦÞ  c p  eA  m c þ ħec e σ 4  B  eiħc e αE Ψ¼0 2

Give a physical interpretation of the factors on the LHS of this equation. Hint: To interpret the last two terms on the LHS, take the non-relativistic limit. 6.9. Show that (6.75a) reduces to the appropriate form in the non-relativistic limit. 6.10. Relativistic photon-electron scattering. The energy of a photon can be expressed as E ¼ hν, where h is Planck’s constant and ν is the frequency of the photon. The momentum of a photon is expressed as hν/c. Show that, if the photon scatters from a free electron (of mass me), the scattered photon has an energy,

1 E E ¼E 1þ ð1  cos θÞ m e c2 0

Show also that the electron acquires a (relativistic) kinetic energy, " # E2 1  cos θ K¼ me c2 1 þ mEc2 ð1  cos θÞ e Potentially useful formula: The total relativistic energy of a particle of mass m, speed v, and relativistic !rel

3-momentum p E TOT ¼

is,

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   !rel 2

ðmc2 Þ2 þ c p



1=2 ¼ mc2 γ v ; γ v ¼ 1  v2 =c2

Use your results to show that a free electron can neither absorb nor emit a photon.

150

6

Relativistic Theory of Photoelectron Angular Distributions

6.11. Matrix elements of tensor products [3, 7, 16]. Equations (6.38) and (D.15) give the matrix elements of a product of tensor operators from two different angular momentum spaces. In this problem, you will find the matrix elements of the products of different types of tensor operators. (a) Find the matrix elements of an operator that is a scalar product of two tensor operators, each of which act on separate parts of a coupled system. ðk Þ

ðk Þ

Hint: The scalar product T 1  T 2 is related to the tensor contraction h i ðk Þ ðk Þ ð0Þ T1  T2 by, 0

h ið0Þ ðk Þ ðk Þ ðk Þ ðk Þ T 1  T 2 ¼ ð1Þk ð2k þ 1Þ1=2 T 1  T 2 0

(b) Find the reduced matrix elements of a tensor operator that acts only on space one of a two-space coupled system. (c) Find the reduced matrix elements of a tensor operator that acts only on space two of a two-space coupled system. (d) Find the reduced matrix elements of an operator that is the product of two tensor operators that act in a space that is not decomposable into two subspaces. 6.12. Tensor products and the hyperfine interaction in hydrogen [8, 11, 15]. The hyperfine interaction in hydrogen results from the interaction of the !

!

magnetic moment of the proton mI (due to its intrinsic spin I ) with the magnetic field created by the electron. We can use the results of Problem 3.8 to model the magnetic field produced by the electron’s magnetic dipole !

!

moment ms (due to its intrinsic spin S ). In other words, because the dipole term in the magnetic multipole expansion is the dominant term, the magnetic interaction due to the spin of the two particles can be approximated as a magnetic dipole-dipole interaction. To this interaction we must add the effect !

of the magnetic field due to the electron’s (nonrelativistic) orbital motion L . The Hamiltonian for the hyperfine interaction can then be modeled after the classical formula for the energy of interaction of the proton’s magnetic !

moment with the magnetic field B produced by the electron evaluated at the location of the proton, !

!

U ¼ mI  B

(a) Show that the Hamiltonian for the hyperfine interaction can be written as,

6

Relativistic Theory of Photoelectron Angular Distributions

H hf ¼ 

151

 h    i   μ0 8π ! ! ! e ! ! 1 ! ! ! ! b b L  m þ 3 m  r m  r  m  m  m δ r þ m I s I s I 3 s I 4π me r 3 r3

where br is a unit vector that points from the proton to the electron. Note that the ! charge e is negative and that ms points in the opposite direction of the electron’s spin. The results of Problem 3.8(d) might be helpful. Rewrite this Hamiltonian in terms of the angular momentum operators involved. Take the gyromagnetic ratio of the electron to be ge ¼ 2, and let the gyromagnetic ratio of the proton be gp. Otherwise, treat the proton as a point particle (i.e., ignore the magnetic field that may exist inside the proton due to its internal structure). Also, ignore any relativistic effects. (b) By first noting that br ¼ C ðq1Þ (see note in Problem 4.3), show that, h! !  ið1Þ pffiffiffiffiffih ið1Þ S  3 S  br br ¼ 10 Sðq11Þ C ðq22Þ q

q

where C ðqkÞ is defined in (4.11). (c) Use your results from parts (a) and (b), and the results from Problem 6.11 to find the hyperfine splitting for hydrogen states of l 6¼ 0. That is, find the matrix elements,    D    E   n, ½lsj , I F, mF H hf n, ½l0 sj , I F 0 , m0F where F is the eigenvalue of the total angular momentum operator, !

!

!

!

!

!

F¼LþSþ I ¼J þI

There is an elementary method for evaluating these matrix elements, but in this problem you will show how tensor methods deliver the same results. You will need tabulated values of some 3-j, 6-j, and 9-j symbols found in [3, 10, 89]. You will also need the quantity h1/r3i found in Problem 5.7. Show that the matrix elements are diagonal in j, l, F, mF. 6.13. Tensor products and the Zeeman effect in hydrogen [6, 8, 15]. The Hamiltonian HZ for the Zeeman effect in hydrogen describes the !

interaction of the atom with an externally-imposed constant magnetic field B,   ! ! ! ! H Z ¼  mL þ mS þ mI  B !

!

!

where mL , mS , mI are the magnetic dipole moments due to the electron orbital !

!

!

motion L , the electron spin S , and the proton spin I , respectively. Note that

152

6

Relativistic Theory of Photoelectron Angular Distributions

we are neglecting higher-order terms in the magnetic multipole expansion and other small effects. (a) Make the case that we can neglect the interaction of the external magnetic field with the spin of the proton and write the remaining Zeeman Hamiltonian in terms of the resultant angular momentum operators. (b) Use your results from Problem 6.11 to find the matrix elements, D

    E   ½lsj , I F, mF H Z  ½l0 sj0 , I F 0 , m0F

where F is the eigenvalue of the total angular momentum operator, !

!

!

!

!

!

F¼LþSþ I ¼J þI

Chapter 7

Angular Momentum Transfer Theory

A more general way of examining photoelectron angular distributions is via angular momentum transfer theory. Angular momentum transfer theory analyzes the pd process by focusing on the net angular momentum deposited by the photon into the target. The angular momentum transfer is equal to the difference between the angular momentum input to the target and the angular momentum output from the target, which (in this theory) is equal to the photoelectron’s final-state orbital angular momentum. Angular momentum transfer theory can account for non-isotropic effects (among other phenomena) and so should be considered a more complete theory (in that respect) than some of the treatments presented earlier. Consider a standard, one-photon photodetachment reaction in which a photon collides with an unpolarized target and detaches an electron (we use a photodetachment reaction as an example, but we could just as easily have used a photoionization process):
 A
ðLA
, SA
, jA
, π A
Þ þ γ jγ , mγ , π γ ! AðLA , SA , jA , π A Þ   þ e
l, s, je , π e ¼ ð
1Þl

ð7:1Þ

where spin S, orbital angular momentum L, total angular momentum j, and parity π are labeled with respect to the initial anionic state A
, and residual neutral core A, and l, s, je label the photoelectron’s post-reaction orbital, spin, and total angular momentum, respectively. The total angular momentum, projection quantum number, and parity of the photon γ are similarly labeled. We are working here only with states which are well-characterized by these quantum numbers, that is, by states that are defined in the LS-coupling approximation. Note that the parity of the photoelectron

The original version of this chapter was revised. The correction to this chapter is available at https://doi.org/10.1007/978-3-031-08027-2_11 © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022, Corrected Publication 2023 V. T. Davis, Introduction to Photoelectron Angular Distributions, Springer Tracts in Modern Physics 286, https://doi.org/10.1007/978-3-031-08027-2_7

153

154

7 Angular Momentum Transfer Theory

π e is a function of its post-pd-reaction orbital angular momentum l. The photodetachment interaction can be characterized by the total angular momentum ! J which is conserved in the reaction !

!

!

!

!

J ¼ j A
þ j γ ¼ j A þ j e

ð7:2Þ

or by the angular momentum transferred during the pd reaction to the residual neutral core !

!

!

jt  jγ
l ¼

!  ! ! j A þ s
j A


ð7:3Þ

Here we assume that the target A
is unpolarized (i.e., its orientation has not been preselected), and the angular momentum/orientation of the residual neutral core (and the photoelectron spin-which can be regarded as still coupled to the residual core) remain unobserved. One advantage of angular momentum transfer theory is that it allows for averaging over magnetic quantum numbers of reactants whose orientation is ! ! not observed. The transferred angular momentum j t can either be used to eliminate J entirely or used in tandem with it. In any case, one can always introduce one or the other via the appropriate recoupling transformation, as will be shown below. Given the conditions stated above, the differential cross-section for photodetachment dσ pd/dΩ is proportional to the scattering matrix for the reaction, averaged over all possible initial target orientations, and summed over all possible (undetected) final states of the residual neutral core [87, 88]:     D E2 X X dσ pd 1   ðk Þ  /  Y ðθ, ϕÞ lm, jA mA Sq jA
mA
 dΩ  ð2jA
þ 1Þ mA
, mA  lm lm

ð7:4Þ

Equation (7.4) is structured so that S(k, q) represents the 2q + 1 spherical tensor operators responsible for the pd process. The initial state is characterized by the symbol jii ¼ j jA
mA
i , keeping in mind that the properties of the photon are contained in S(k, q). The final state is given by |fi ¼ |lm, jAmAi, with the caveat that the spin of the photoelectron, since it remains unobserved, is considered to be coupled to the residual neutral core. The quantity in the bracket represents the probability amplitude for photoelectron emission into the direction (θ, ϕ) while in the state l, m and we notice that it has therefore been expanded in a series of spherical harmonics Ylm (θ, ϕ). As stated above, the differential cross-section for photodetachment is then proportional to the square of the probability amplitude, averaged over all initial possible target orientations, and summed over all possible undetected final states. In short, the essence of angular momentum transfer theory lies in performing the indicated sums over the magnetic quantum numbers and expanding the results into a series of spherical harmonics, which serve as convenient functions with which to represent the final angular distribution [87, 88]. We begin our operation by recoupling to the basis of the total angular momentum ! J [c.f. (2.119)] and executing the indicated square modulus:

7

Angular Momentum Transfer Theory

155

2 X X X dσ pd 1  /  Y lm ðθ, ϕÞ hl, jA ; m, mA jJ, M ihJMjSðqkÞ jjA
mA
i dΩ ð2jA
þ 1Þ mA
, mA lm JM X XXXX 1 ¼ Y Y 0 0 hl, jA ; m, mA jJ, M ihl0 , jA ; m0 , mA jJ 0 , M 0 i ð2jA
þ 1Þ mA
, mA lm 0 0 JM 0 0 lm l m lm

JM

hJMjSðqkÞ jjA
mA
ihJ 0 M 0 jSðqkÞ jjA
mA
i ð7:5Þ Operating on the matrix elements via the Wigner-Eckart theorem [3], hJMjSðqkÞ jjA
mA
ihJ 0 M 0 jSðqkÞ jjA
mA
i ! ! J k jA
J 0 k jA
J
M J 0
M 0 ¼ ð
1Þ ð
1Þ hJkSðkÞ kjA
ihJ 0 kSðkÞ kjA
i 0


M q mA
M q mA ð7:6Þ and then recasting all C-G coefficients as 3-j symbols [c.f. (2.144a)], gives us [3, 89] Xpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi     dσ pd 1 ð2J þ 1Þ ð2J 0 þ 1ÞhJ SðkÞ jA
ihJ 0 SðkÞ jA
i / dΩ ð2jA
þ 1Þ 0 JJ 3 2 0 0 Y lm Y l0 m0 ð
1Þlþl
2jA þJþJ 7 6 ! 0 ! 7 6 J l jA l jA J0 7 6 7 6 7 6 0 0 7 6 m mA
M m mA
M 7 6 7 6 ! ! 7 PP P P 6 J k jA
k jA
J0 7 6  7 6 lm l0 m0 mA
, mA MM 0 6  7 7 6 0 7 6
M q mA

M q mA
7 6 |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} 7 6 ! !7 6 0 j A
k J j A
k J 7 6 0 5 4 ð
1ÞjA
þkþJ ð
1ÞjA
þkþJ
mA



q M


mA



q M 0

ð7:7Þ !

!

where we also used (2.145c) as indicated. We recouple from the J basis to the j t basis as follows, using (C.74) [3, 87–89]

156

7





Angular Momentum Transfer Theory



 jA
jA J k J mA
M
mA

q M 2
ð2j þ 1Þð
1ÞlþjA
JþjA
þkþjt
mþmA X6 (t ) ! 6 k j jt jA ¼ 4 x l jA J jt jA
k jt
q m mt mA

l m

l0 m0

jA

J0



jA


k

J0

3 !7 7 5

jA


jt


mA



mt

ð7:8aÞ




mA

q M 0 mA
M 0 2
 0 0 0 0
2j0t þ 1 ð
1Þl þjA
J þjA
þkþjt
m þmA X6 ( ) ! 6 jA k l0 j0t l0 jA J 0 ¼ 4 0 x jt mA jA
k j0t
q m0 m0t

3 !7 7 5

jA


j0t


mA



m0t

ð7:8bÞ

Combining (7.7) and (7.8a and 7.8b), Xpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dσ pd 1 ð2J þ 1Þ ð2J 0 þ 1ÞhJkSðkÞ kjA
ihJ 0 kSðkÞ kjA
i / dΩ ð2jA
þ 1Þ 0 JJ XX X X 0 0  Y lm Y l0 m0 ð
1Þlþl
2jA þJþJ 2

lm l0 m0 mA
, mA MM 0

ð
1Þ

jA
þkþJ

3


 0 ð
1ÞjA
þkþJ ð2jt þ 1Þ 2j0t þ 1

6 7 6 7 ¼1 for all cases 6 7 zfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflffl{ 6 7 0 0 0 0 6 ð
1Þlþl
J
J þ2jA þ2kþjt þjt
m
m ð
1Þ2ðjA
þmA
Þ 7 6 7 6 ( 7 )( 0 ! 0 6 0) 0 !7 l j k l j J jA J jt 7 l k l 6 A t 6 7 7 0 0 0 X6 6
q m mt jA
k jt jA
k jt
q m mt 7 6  ! ! 777 6 0 jt j0t 6 jA jA jA
jt jA
jt 6 7 6 7 6 7 6 7 0 mA
mA

mt mA
mA
mt 6 7 6 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ! |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 77 6 ! 6 jA jA
jt jA jA
j0t 7 6 7 0 þj j þj þj j þj

4 ð
1Þ A A t 5 ð
1Þ A A t
mA

mA


mt


mA

mA


m0t

ð7:9Þ Using the orthogonality properties of the 3-j symbols (see Problem 2.8) [3, 10, 89]:

7

Angular Momentum Transfer Theory

X

jA

jA


jt

mA
, mA


mA

mA


mt

!

157

jA

jA


j0t


mA

mA


m0t

! ¼ ð2jt þ 1Þ
1 δjt ,j0t δmt ,m0t ð7:10Þ

The recoupling in (7.8) results in the fact that there are no longer any terms in M and M0 in (7.9). The sums over those terms will therefore serve only to bring in the multiplicative factors (2J + 1)(2J0 + 1), leaving us with X 3 3 dσ pd 1 ð2J þ 1Þ2 ð2J 0 þ 1Þ2 hJkSðkÞ kjA
ihJ 0 kSðkÞ kjA
i / dΩ ð2jA
þ 1Þ 0 JJ XX 0   Y lm Y l0 m0 ð
1Þ2lþ2l lm l0 m0

2

0

3

0

ð
1ÞjA
þkþJ ð
1ÞjA
þkþJ ð2jt þ 1Þð
1Þ2jA
þ2jA þ2kþ4jt
m
m X6 ( )( 0 ) ! 6 l jA J k l jt jA J 0 l k l0  4  jt
q m mt jA k jt jA
k jt
q m0

!7 7 5

jt mt

ð7:11Þ From the two 3-j symbols, we observe that {mt ¼ q
m; mt ¼ q
m0} ) m ¼ m0 which allows us to eliminate the sum over m0: X 3 3 dσ pd 1 ð2J þ 1Þ2 ð2J 0 þ 1Þ2 hJkSkjA
ihJ 0 kSðkÞ kjA
i / dΩ ð2jA
þ 1Þ 0 JJ X 0   Y lm Y l0 m ð
1Þ2ðlþl Þ |fflfflfflfflfflffl{zfflfflfflfflfflffl} 0 ll m

2

¼1; l, l0 ¼integer

¼1

3

¼1

zfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflffl{ zfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflffl{ 6 jA
þkþJ jA
þkþJ 0 6 ð
1 Þ ð
1 Þ ð 2j þ 1 Þ ð
1Þ2ðjA
þjA þjt Þ ð
1Þ2ðkþjt
mÞ X6 t ! )( 0 )  6 ( l jA J k l jt 6 jA J 0 k l0 l jt 4 x
q m q
m jA
k jt jA
k jt
q m

jt

7 7 7 !7 7 5

q
m ð7:12Þ

We make use of the following identity, which is easily derived using (2.155) and (2.60) [see also Problem 4.6(a)] [3, 87–89]: Y lm Y l0 m ¼ Y lm ð
1Þm Y l0
m ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X ð2l þ 1Þð2l0 þ 1Þ m ¼ ð
1Þ hl, l0 ; 0, 0jL, 0ihl, l0 ; m,
mjL, 0iPL ð cos θÞ 4π L ð7:13Þ

158

7

Angular Momentum Transfer Theory

to get X 3 3 dσ pd  1 ð2J þ 1Þ2 ð2J 0 þ 1Þ2 hJkSðkÞ kjA
ihJ 0 kSðkÞ kjA
i / dΩ ð2jA
þ 1Þ 0 JJ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi XX ð2l þ 1Þð2l0 þ 1Þ 0  ð
1Þm hl, l0 ; m,
mjL, 0i PL ð cos θÞ hl,l ; 0, 0jL, 0i 4π |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} 0 ! ll m L l l0 L pffiffiffiffiffiffiffiffiffi 0 ¼ð
1Þl
l

2

ð
1ÞjA

k
J ð
1ÞjA

k
J

3

0

2Lþ1

m
m 0

7 6 zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{0 7 6 ð
1ÞjA
þkþJ ð
1ÞjA
þkþJ 7 6 7 6 9 9 8 8 7 6 < l jA J =< l0 jA J 0 = 7 6 7 6 7 6  ð2jt þ 1Þ : j k j ;: j
k j ; 7 6 7 6 A t A t X6 7 ! 7 6 0 1  0 7 6 k l j k l j t 7 t jt 6 7 6 @ A 7 6 6
q m q
m
q m q
m 7 7 6 6 |fflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflffl1 ffl} 7 7 6 0 7 6 l0 k jt 7 6 4 @ A5
m q m
q ð7:14Þ

Now to eliminate the sum over the last remaining magnetic quantum number by noting that what should be a triple sum in (7.15) below [c.f. (C.73)] in our case reduces to the sum over one term, m (see problem 7.1) [3, 87–89],

X k m ð
1Þ
q m

l m 0

jt q
m

¼ ð
1Þ2kþl
jt þl þLþq





l0
m

k

l

jt

l0

k

L

k q

jt m
q





l m

l0
m

k

k

L


q

q

0



L 0

|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}



k l

Putting it all together,

k

j

0

L jt

ð
1Þ2kþL

k q

k L
q 0



ð7:15Þ

7

Angular Momentum Transfer Theory

159

( XXX l 3 dσ pd 1 jA

k
J ðkÞ ð
1Þ ð2J þ 1Þ2 hJkS kjA
i / dΩ ð2jA
þ 1Þ 0 j J jA
ll t ( 0 ) l jA J 0 3 P 0   ð
1ÞjA

k
J ð2J 0 þ 1Þ2 hJ 0 kSðkÞ kjA
i jA
k jt J0 ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 P ð2l þ 1Þð2l þ 1Þ  ð
1Þ
jt þq hl, 0; l0 , 0jL, 0iPL ð cos θÞð2jt þ 1Þ 4π L ( )( ) k k L k k L pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ð2kþLÞ 2l ð2L þ 1Þ  ð
1Þ ð
1Þ |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} |fflfflffl{zfflfflffl} l l0 jt q
q 0 ¼1 ¼1:l integer |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

jA

J

k

jt

)

ð
1Þk
k hk, k; q,
qjL, 0i

ð7:16Þ The differential cross-section is usually presented in the following form [87, 88]: XX dσ pd K { hJkSðjt ÞkjA
ihjA
kS ðjt ÞkJ 0 iΘðjt ; kq; ll0 ; θÞ ¼ dΩ ð2jA
þ 1Þ j 0 t

ð7:17Þ

ll

where the scattering amplitudes/matrices are hJkSðjt ÞkjA
i 

X

( ð
1Þ

jA

k
J

3 2

ðk Þ

ð2J þ 1Þ hJkS kjA
i

J

l

jA

J

jA


k

jt

X 3 0 hjA
kS ðjt ÞkJ i  ð
1ÞjA

k
J ð2J 0 þ 1Þ2 hJ 0 kSðkÞ kjA
i {

0

J0

(

) ð7:18aÞ

l0

jA

J0

jA


k

jt

)

ð7:18bÞ and Θ( jt; kq; ll0; θ) is called the geometrical function pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ð2l þ 1Þð2l0 þ 1Þ 4π L ( ) k k L ð2jt þ 1Þhl, l0 ; 0, 0jL, 0ihk, k; q,
qjL, 0iPL ð cos θÞ l l0 jt X ð
1Þ
jt þq Θðjt ; kq; ll ; θÞ  0

ð7:18cÞ

The constant of proportionality K is chosen in terms of the wavelength of the photon λ (divided by 2π) to match boundary conditions between incoming and outgoing states [87, 88]

160

7

Angular Momentum Transfer Theory

K ¼ 3πD2 ; D  λ=2π

ð7:19Þ

Notice from (7.17) that the differential cross-section has taken the form of an incoherent sum over the angular momentum transfer quantum number [3, 87, 88]: dσ pd X dσ ðjt , θÞ ¼ dΩ dΩ j

ð7:20Þ

t

To recap, we have now a general expression for the photodetachment process of ! (7.1) in terms of separate components characterized by alternate magnitudes of j t . ! The allowed values of j t must be consistent with the conservation of the total ! angular momentum J and the conservation of parity [87, 88, 90, 91]1: π A
π γ ¼ π A π e

ð7:21Þ

For most purposes, the most interesting and applicable event is the low-energy pd event (Eγ  100 eV), for which the electric dipole approximation is valid and for which the incident photon carries an angular momentum jγ ¼ 1 and parity π γ ¼
1 to the target atom. For this case (7.2) and (7.3) read [90] !

!

!

!

!

J ¼ j A
þ 1 ¼ j A þ j e

ð7:22Þ

and !

!

!

jt  1
l ¼

!  ! ! j A þ s
j A


ð7:23Þ

In addition, because the electric dipole interaction is spin-independent (under the LS-coupling approximation), the angular momentum imparted by the photon affects ! ! only the orbital angular momentum of the system. As a result, the total spin S ¼ S A
! and orbital angular momentum L are conserved separately [90]: !

!

!

!

!

!

!

!

S A
¼ S A þ s ; L ¼ L A
þ 1 ¼ L A þ l

ð7:24Þ

and (7.3) reduces to [90] !

!

!

!

!

j t  1
l ¼ L A
L A


1

ð7:25Þ

The fact that photoelectron angular distributions we have seen up to this point have no linear term in cosθ is due to parity conservation [1].

7

Angular Momentum Transfer Theory

161

Equation (7.21) now takes on a particularly meaningful form for our discussion [90]: π A
π γ ¼ π A π e ) π A
ð
1Þð
1Þ ¼ π A ð
1Þl ð
1Þ ) π A
¼ π A ð
1Þlþ1 |{z} |{z}
1

ð
1Þl

) π A
π A ¼ ðπ A Þ2 ð
1Þlþ1 |ffl{zffl} ¼1

) π A
π A ¼ ð
1Þlþ1

ð7:26Þ

In the dipole approximation, and under LS-coupling, allowed values of jt are determined by (7.25) and (7.26) [90]. For a photodetachment caused by a linearly polarized photon, we have k ¼ 1; q ¼ 0. The geometrical function now reads pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ð2l þ 1Þð2l0 þ 1Þ ð2jt þ 1Þ Θðjt ; 10; ll ; θÞ ¼ ð
1Þ 4π ( ) X 1 1 L  hl, l0 ; 0, 0jL, 0ih1, 1; 0, 0jL, 0iPL ð cos θÞ l l0 jt L 0

jt

ð7:27Þ

Triangular conditions and symmetry properties of the 6-j symbol limit the sum to just two terms, L ¼ 0, 2, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ð2l þ 1Þð2l0 þ 1Þ ð2jt þ 1Þ Θðjt ; 10; ll ; θÞ ¼ ð
1Þ 4π rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3 1 ¼ð
1Þjt þlþ1 δ0 1 3ð2l þ 1Þ ll p ffiffi ffi ¼
1 6 zfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflffl{ ¼ð
1Þl ð2lþ1Þ
2 ( ) 7 3 ¼1 6 zfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflffl 7 ffl{zfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflffl{zfflfflfflfflffl}|fflfflfflfflffl{ 6 7 1 1 0 6 h1, 0; 1, 0j0, 0ihl, 0; l0 , 0j0, 0iP0 ðcos θÞ 7 6 7 6 7 l l0 jt 6 7 6 rffiffiffi 7 6 7 2 6 7 ¼ 6 7 3 ( ) 6 zfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflffl{ 7 1 1 2 4 5 0 þ h1, 0; 1, 0j2, 0i hl, 0; l , 0j2, 0iP2 ðcos θÞ 0 l l jt 0

jt

ð7:28Þ Parity-unfavored transitions are defined as transitions for which jt + l + ( jγ ¼ 1)¼ odd, or jt ¼ l, l0. For this case, (7.26) implies [87, 88, 90–93] π A
π A ¼ ð
1Þjt þ1 and the geometric function becomes

ð7:29Þ

162

7

Angular Momentum Transfer Theory

2

6 ð2jt þ 1Þ2 6 6 Θðjt ; 10; jt jt ; θÞ ¼ ð
1Þ 6 4π 6 4 jt

( )  1 1 0 1
pffiffiffi hjt , jt ; 0, 0j0, 0i 3 jt jt jt ( rffiffiffi 1 1 2 þ hj , j ; 0, 0j2, 0iP2 ð cos θÞ 3 t t j j t

t

3 7 7 7 )7 2 7 5 jt ð7:30Þ

Tabulated values of C-G coefficients and 6-j symbols [see also (C.75) and (C.76)] allow us to compute the following [3, 7, 10]: hjt , jt ; 0, 0j0, 0i ¼ ð
1Þjt ð2jt þ 1Þ
2 1

ð7:31aÞ

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð
jt Þðjt þ 1Þ 5 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hjt , jt ; 0, 0j2, 0i ¼ ð
1Þ ð2jt þ 1Þ ð2jt þ 3Þðjt þ 1Þðjt Þð2jt
1Þ jt





1 jt

1 jt

2 jt

1

1

0

jt

jt

jt

1 1 ¼ ð
1Þ pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 ð2jt þ 1Þ

ð7:31bÞ

ð7:31cÞ

2½6
8jt ðjt þ 1Þ ¼ ð
1Þ2jt þ1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð7:31dÞ 5!ð2jt þ 3Þð2jt þ 2Þð2jt þ 1Þð2jt Þð2jt
1Þ

Example 7.1 Verify (7.31a). Equation (7.31a) can be verified by combining (2.146) and (2.144a): j

j

0

m


m

0

! ¼ hj, j; m,
mjj3 , 0i ¼ ð2j þ 1Þ
2 ð
1Þj
m 1

) hjt , jt ; 0, 0j0, 0i ¼ ð
1Þjt ð2jt þ 1Þ
2 1

█

7

Angular Momentum Transfer Theory

163

Combining (7.30) and (7.31a, 7.31b, 7.31c and 7.31d), Θðjt ; 10; jt jt ; θÞ ¼ ð
1Þjt  2

3 1 1 1 1
pffiffiffi ð
1Þjt þ1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6 7 3 ð2jt þ 1Þ 3 ð2jt þ 1Þ 6 7 6 rffiffiffi 7 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 26 ð2jt þ 1Þ 6 ð
jt Þðjt þ 1Þ 2 5 7 3jt þ1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7  ð
1Þ 6þ 4π 3 6 ð2jt þ 1Þ ð2jt þ 3Þðjt þ 1Þðjt Þð2jt
1Þ 7 6 7 6 7 4 5 2½6
8jt ðjt þ 1Þ P2 ð cos θÞ  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5!ð2jt þ 3Þð2jt þ 2Þð2jt þ 1Þð2jt Þð2jt
1Þ ð7:32Þ which, after some algebra, reduces to [88] Θðjt ; 10; jt jt ; θÞ ¼

ð2jt þ 1Þ ½1
P2 ð cos θÞ 12π

ð7:33Þ

Parity-favored transitions are defined as transitions for which jt + l + ( jγ ¼ 1)¼ even, and which consist of two functions diagonal in l, l0; l ¼ l0 ¼ jt  1 and one interference term l ¼ jt + 1; l0 ¼ jt
1. For this case, (7.26) implies [87, 88, 90–93] π A
π A ¼ ð
1Þjt

ð7:34Þ

Tabulated values of C-G coefficients and 6-j symbols allow us to compute the following factors [3, 7, 10]: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5ðjt þ 1Þðjt þ 2Þ hjt þ 1, jt þ 1; 0, 0j2, 0i ¼ ð
1Þjt ð2jt þ 1Þð2jt þ 3Þð2jt þ 5Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5ðjt
1Þðjt Þ jt hjt
1, jt
1; 0, 0j2, 0i ¼ ð
1Þ ð2jt
3Þð2jt
1Þð2jt þ 1Þ

hjt þ 1, jt
1; 0, 0j2, 0i ¼ ð
1Þ

1

1

2

jt þ 1

jt þ 1

jt

jt
1

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 15ðjt þ 1Þðjt Þ 2ð2jt
1Þð2jt þ 1Þð2jt þ 3Þ

ð7:35aÞ

ð7:35bÞ

ð7:35cÞ

2ðjt þ 2Þð2jt þ 5Þ ffi ¼ ð
1Þ2jt þ2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5!ð2jt þ 5Þð2jt þ 4Þð2jt þ 3Þð2jt þ 2Þð2jt þ 1Þ ð7:35dÞ

164



7

1 jt
1

1 jt
1

2 jt

Angular Momentum Transfer Theory

2ðjt
1Þð2jt
3Þ ¼ ð
1Þ2jt pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5!ð2jt þ 1Þð2jt Þð2jt
1Þð2jt
2Þð2jt
3Þ ð7:35eÞ



1 jt þ 1

1 jt
1

2 jt

¼ ½5ð2jt þ 1Þ
1=2

ð7:35fÞ

Combining (7.28) and the appropriate iterations of (7.35) (and after grinding through some algebra), we get [87, 88]   ð2jt þ 1Þ ðjt þ 2Þ 1þ P ð cos θÞ Θðjt ; 10; jt þ 1 jt þ 1; θÞ ¼ 12π ð2jt þ 1Þ 2

ð7:36aÞ

  ð2jt þ 1Þ ð j
1Þ 1þ t P2 ð cos θÞ 12π ð2jt þ 1Þ

ð7:36bÞ

h i
1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðjt Þðjt þ 1ÞP2 ð cos θÞ 4π

ð7:36cÞ

Θðjt ; 10; jt
1 jt
1; θÞ ¼

Θðjt ; 10; jt þ 1 jt
1; θÞ ¼

the last term being the interference term. Combining (7.17), (7.18a, 7.18b and 7.18c), (7.19), (7.33), and (7.36a, 7.36b and jt ,P jt 1 , we 7.36c), and accounting for all the possible values of l, l0 in the summation ll0

get, for a photodetachment caused by a linearly polarized photon:   9 8  2 ð2jt þ 1Þ ðjt þ 2Þ > >   > > S ð j Þ ð cos θ Þ P 1 þ > > þ t 2 > > 12π ð þ 1 Þ 2j > > t > > > > > >   > > > >  2 ð2jt þ 1Þ ð j
1 Þ > > t > >   = < P 1 þ ð j Þ ð cos θ Þ þ S 2
2 X t dσ pd 3πD 12π ð þ 1 Þ 2j t ¼ dΩ ð2jA
þ 1Þ j > > > >   3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   > t > > > > > j ð j ÞS ð j Þ þ S ð j ÞS ð j Þ ð Þ ð j þ 1 Þ P ð cos θ Þ
S þ t
t
t þ t 2 t t > > > > 12π > > > > > > > >   > > ð 2j þ 1 Þ 2 t ; : þ  S ðj Þ  ½1 þ ð
1ÞP2 ð cos θÞ 0 t 12π 9 8      > > Sþ ðjt Þ2 þ S
ðjt Þ2 ð2jt þ 1Þ þ P2 ð cos θÞ > > > > > > > > > > 2 3 > >     2 2 > > = <     2 X ð j Þ ð j þ 2 Þ þ S ð j Þ ð j
1 Þ S þ
t t t t 3πD 4 5 ¼  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
 > 12π ð2jA
þ 1Þ j >   > > t >
3 ðjt Þðjt þ 1Þ Sþ ðjt ÞS
ðjt Þ þ S
ðjt ÞSþ ðjt Þ > > > > > > > > > > >   2 ; :   þ S0 ðjt Þ ð2jt þ 1Þ½1 þ ð
1ÞP2 ð cos θÞ

7

Angular Momentum Transfer Theory

165

dσ pd 3πD2 ¼ dΩ 12π ð2jA
þ 1Þ i 9 8 h     Sþ ðjt Þ2 þ S
ðjt Þ2 ð2jt þ 1Þ > > > > > > > > > > > > 2 3     > > 2 3 2 2 > >     > > S ð j Þ ð j þ 2 Þ þ S ð j Þ ð j
1 Þ
> > þ
t t t t > > > > 4 5 > > 6 7 > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
 = 7>   X< 6 6 7 Þðjt þ 1Þ Sþ ðjt ÞS
ðjt Þ þ S
ðjt ÞSþ ðjt Þ  61 þ 3 ðjt 7   P ð cos θ Þ 2     > 6 7> jt > Sþ ðjt Þ2 þ S
ðjt Þ2 ð2jt þ 1Þ > > 6 7> > > > 4 5> > > > > > > > > > > > > > > > > > > 2 ; :   þ S0 ðjt Þ ð2jt þ 1Þ½1 þ ð
1ÞP2 ð cos θÞ )

ð7:37Þ Now we do a little rearranging 8 2 9 2  2 i πD ð2jt þ 1Þ h > > > >    Sþ ðjt Þ þ S
ðjt Þ > > > > > > ð2jA
þ 1Þ > > > > > > 2 3 > >     2 3 > > 2 2 > > Sþ ðjt Þ ðjt þ 2Þ þ S
ðjt Þ ðjt
1Þ
> > > > > > 4 5 > > 6 7 > > ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p > > 
6 7   < = X 6 7 dσ pd 1 3 ðjt Þðjt þ 1Þ Sþ ðjt ÞS
ðjt Þ þ S
ðjt ÞSþ ðjt Þ 61 þ 7 ¼   P ð cos θ Þ  2     7> dΩ 4π j > Sþ ðjt Þ2 þ S
ðjt Þ2 ð2jt þ 1Þ > 6 6 7> t > > > 4 5> > > > > > > > > > > > > > > > > > > > > 2 > >   > > ð 2j þ 1 Þ πD 2 t >þ > S0 ðjt Þ ½1 þ ð
1ÞP2 ð cos θÞ : ; ð2jA
þ 1Þ n h i h io 1X σ ðjt Þfav 1 þ βðjt Þfav P2 ð cos θÞ þ σ ðjt Þunf 1 þ βðjt Þunf P2 ð cos θÞ ¼ 4π j t

ð7:38Þ where [91–93]. σ ðjt Þfav ¼

2  2 i πD2 ð2jt þ 1Þ h Sþ ðjt Þ þ S
ðjt Þ ð2jA
þ 1Þ

ð7:39aÞ

2 πD2 ð2jt þ 1Þ  S ðj Þ ð2jA
þ 1Þ 0 t

ð7:39bÞ

σ ðjt Þunf ¼

166

7

Angular Momentum Transfer Theory

h    pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
i Sþ ðjt Þ2 ðjt þ 2Þ þ S
ðjt Þ2 ðjt
1Þ
3 ðjt Þðjt þ 1Þ Sþ ðjt ÞS ðjt Þ þ S
ðjt ÞS ðjt Þ
þ  βðjt Þfav ¼    Sþ ðjt Þ2 þ S
ðjt Þ2 ð2jt þ 1Þ

ð7:39cÞ βðjt Þunf ¼
1

ð7:39dÞ

    and S ðjt Þ and S0 ðjt Þ represent photodetachment scattering amplitudes for photoelectron escape angular momenta l ¼ jt  1 and l ¼ jt, respectively. Let us extend our manipulation of (7.38) a little further: n h i h io dσ pd 1X σ ðjt Þfav 1 þ βðjt Þfav P2 ð cos θÞ þ σ ðjt Þunf 1 þ βðjt Þunf P2 ð cos θÞ ¼ dΩ 4π j t

¼

i nh 1X σ ðjt Þfav þ σ ðjt Þfav βðjt Þfav P2 ð cos θÞ 4π j t

h

þ σ ðjt Þunf þ σ ðjt Þunf βðjt Þunf P2 ð cos θÞ

io

P 8 σ¼ σ ðjt Þ > >

> > 3 cos þ Z cos θ þ Z θ  1 Z 10 T 20 T > > 00 > > 2 > > > > 2 3 > >   > > 1 Re Im > > > > pffiffiffi Z 11 cos ϕT  Z 11 sin ϕT > > > > p ffiffiffiffiffi 6 7 = < 2 2 6 7 dσ 16π 4π 6 rffiffiffi 7 sin θT ¼ 4 5  dΩT 3 > > 3 Re > > > > þ Z 21 cos ϕT  Z Im > > 21 sin ϕT cos θT > > 2 > > > > > > rffiffiffi > > > >   > > 3 Re 2 > Im > ; :þ Z 22 cos 2ϕT  Z 22 sin 2ϕT sin θT 8 m

p Im where Z Re KM =Z KM refers to the real/imaginary part of Z KM . How many parameters are needed to completely specify the differential cross-section if linearly polarized light is used? (e) For a target molecule of cylindrical symmetry, we have the restriction m ¼ mγ + m0 and m0 ¼ m0γ þ m0 . Show that, if linearly polarized light is used, the differential cross-section takes the form

200

8

Molecular Photoelectron Angular Distributions

σ dσ ¼ ½1 þ βT P2 ð cos θT Þ dΩT 4π Find σ and βT. Compare your results to the forms in (3.27) and (3.28). 8.5. Molecular PADs and angular momentum transfer theory [111, 118, 119] If we express the differential cross-section of (8.7) (in the dipole approximation) in terms of its angular momentum transfer components and integrate over all molecular orientations, then, following the procedures outlined in Chap. 7, we will get X dσ ðj Þ dσ t ¼ b b dke dke jt where X dσ ðj Þ t

jt

db ke

/

X X ðÞj  ðÞj 1 X ll0 iðδl δl0 Þ ðiÞ e Θðjt ; 10; ll0 ; θÞ Dlmm0t Dlmmγt δmmγ ,m0 m0γ γ 2jt þ 1 0 0 0 mm mγ mγ

ll

and

ðÞΓ ðÞj Dlmmγt  ð1Þmγ 1, l; mγ , m jt , m  mγ Dlmmγ 0 (a) Find the normalization integral Z2π Zπ 0

sin θdθdϕΘðjt ; 10; ll0 ; θÞ

0

(b) Using your result from part (a) and arguing by analogy from the results of Chap. 7 and this chapter, find σ( jt), σ, β, β( jt). 8.6. Molecular rotations and PADs [114] b that points in the direction of motion of a Suppose we have a unit vector D photoelectron that is about to be ejected from a rotating molecule. As the b ð0Þ describe molecule rotates, this vector changes its orientation in space. Let D b this unit vector in its initial orientation at time t ¼ 0 and Dðt Þ its orientation at some later time t. According to (8.6), the rotation R ¼ fα0 , β0 , γ 0 g carries the (fixed) lab frame OXYZ onto the molecular frame Ox0y0z0(M0) at time t ¼ 0. A further rotation dΩ(t) carries this frame onto the molecular frame Oxyz(Mt) as it is at some later time t.

8

Molecular Photoelectron Angular Distributions

201

(a) The initial pd probabilityhdensity iis given in (3.19). We want to average the b ðt Þ with respect to this probability density Legendre polynomial Pl bε D over the initial orientations of the molecule: D h iE Z h i b ðt Þ ¼ Pl bε D b ðt Þ Ppd ðα0 , β0 , γ 0 Þdα0 sin β0 dβ0 dγ 0 Pl bε D Transform the appropriate quantities into the molecular frame to show that   Z 4π b ðt Þi ¼ 1 hPl ½bε D dα0 sin β0 dβ0 dγ 0 8π 2 2l þ 1 h i 0 1 b M 0 ðt Þ Y lm ðbεÞDlm2 m ðR ÞY lm2 D l B C X B ( )C  B C 2 X

 8π @ A  m, m2 ¼l b  1þ D2 ð R ÞY b μ ð 0 Þ Y ð ε Þ 2m eM 1 2m3 0 5 m , m ¼2 m3 m1 1

3

(b) Evaluate the integral(s) to show that 2 X

b ðt Þi ¼ 8π hPl ½^ε D 25

m2 ¼2

h i

 b M 0 ðt Þ Y 2m b Y 2m2 D μeM 0 ð0Þ 2

(c) After a time t, the molecule has rotated according to the following transformation via (2.95): h i X h i b M 0 ðt Þ ¼ b M 0 ð 0Þ Y 2m2 D D2m1 m2 ½δΩðt ÞY 2m1 D m1

Apply this transformation and then average over the angular momenta and reorientation angles to show that 2 h i E D h iE X

D 2 b M 0 ð0Þ Y 2m b b ðt Þ ¼ 8π Pl bε D Y 2m1 D μ ð 0 Þ D ½ δΩ ð t Þ  eM m1 m2 0 2 5 m , m ¼2 1

2

(d) Apply the reasoning in example 3.3 to argue that this average is the asymmetry parameter β for this molecule. In the event that the pd event is very much faster than the rotational motion of the molecule, show that the asymmetry parameter reduces to β ¼ 2P2 ð cos θÞ where θ is the angle between the dipole moment of the molecule and the momentum vector of the photoelectron.

202

8

Molecular Photoelectron Angular Distributions

8.7. Chiral molecules [144] (a) Modify (8.26) to include circular polarization. Average the result over all molecular orientations to show that  XXX 0 dσ ¼ 16π 2 8π 2 ð1Þmþmγ mp ðiÞll eiðδl δl0 Þ b dke ll0 m mγ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2l0 þ 1Þð2l þ 1Þ ðÞΓ0  ðÞΓ0 Dl0 mmγ Dlmmγ  4π ! ! 0 1 1 Ke l l0 Ppffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l l K e  2K e þ 1 Ke 0 0 0 mp mp m m 0 !  γ 1 1 Ke  PK e ^ke mγ mγ 0

Ke

!

0

(b) Use your result from part (a) to prove (8.27). ðÞΓ

HINT: For chiral molecules, the matrix elements Dlmmγ 0 are not equivalent for positive and negative values of the azimuthal quantum numbers m, mγ . Note: You should also end up proving the Yang theorem of reference [1] while completing this problem. The Yang Theorem states that, “If only incoming waves of orbital angular momentum L contribute appreciably to the interaction, the angular distribution of the outgoing particles in the center-ofmass system will be a function of even polynomials of cos θ with maximum power not higher than 2L.”

Chapter 9

Measuring Photoelectron Angular Distributions in the Laboratory

For the pd process to be possible, the energy of an incident photon of frequency ν and energy Eγ ¼ hν must exceed the energy (BE) binding the electron to the atom or molecule. This condition may be expressed by the Einstein energy-balance equation [145]: E ¼ hν  BE  0

ð9:1Þ

where E is the kinetic energy of the ejected electron. A photon of sufficient energy Eγ can access different pd channels, possibly leaving the residual core in an excited state. The single-photon pd process, in which a parent P in an initial state i absorbs a photon γ, resulting in a residual core C in a final state f and a photoelectron e, is described by the general bound-free equation: PðiÞ þ γ ! C ðf Þ þ e ðif γ Þ

ð9:2Þ

where, in the rest frame of the parent/core, the kinetic energy of the photoelectron for the dissociation channel i ! f is [24, 25]
 E ðif γ Þ ¼ E γ  E f  E n ¼ E γ  Efi |fflfflfflfflfflffl{zfflfflfflfflfflffl}

ð9:3Þ

E fi

where En is the energy of the parent in the initial state and Ef is the energy of the residual core in the final state. Measuring photodissociation cross-section and asymmetry parameters is only possible if one has a reliable source of known-frequency light. Whether it be light from a laser or light from a synchrotron, or some other source of photons, all photospectroscopic measurements rely on these sources, and various techniques are used to make the relevant measurements. Most techniques involve a beam of ions intersecting a collimated photon beam. In such cases, it may be necessary to make © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 V. T. Davis, Introduction to Photoelectron Angular Distributions, Springer Tracts in Modern Physics 286, https://doi.org/10.1007/978-3-031-08027-2_9

203

204

9

Measuring Photoelectron Angular Distributions in the Laboratory

Laser Polarization Vector qs qcm Laser Beam

ql ®

vc

qc

h

Ion Beam ®

vl

®

vi

Fig. 9.1 Nonrelativistic kinematic transformation diagram for a typical crossed-beams photodissociation experiment. A fast-moving beam of ions is crossed at right angles with a laser beam. Photoelectrons created at the intersection of the two beams are collected and their kinetic energies measured in a kinetic energy (KE) analyzer. The figure shows the relationship between the velocity ! ! ! vectors of a collected photoelectron in the lab frame v l , and the CM frame v c. v i is the velocity of ! ! ! ! ! an ion in the beam for a given ion beam energy Ei, and v l is the vector sum v l ¼ v c þ v i. v l is the velocity of a photoelectron as seen by a stationary observer in the lab frame. The angle η ¼ θcθl ! can always be determined after the fact, since v i is a function of the experimental parameters (which ! are controlled by the experimentalist), and v c is ultimately determined by the Einstein energy balance equation. Also shown is the polarization vector for a linearly polarized laser photon. In this crossed-beams setup, the polarization vector is perpendicular to the laser propagation direction and ! ! ! is thus coplanar with v l , v c , and v i . θs is the angle between the polarization vector of the linearly ! polarized photon and v l , and θcm is the angle between the polarization vector of the linearly ! polarized photon and v c .

kinematic corrections and to account for the effects of the Doppler shift. Photoelectron angular distribution measurements are made in the laboratory, and we must have a way to convert these lab-frame measurements into the parent/center-of-mass (CM)1 frame to accurately describe the physics. To illustrate this point, we will highlight three common techniques used to measure PADs in the laboratory. One experimental method that uses a crossed-beams geometry to measure PADs is the laser photodetachment electron spectroscopy (LPES) technique. In the LPES technique, photoelectrons are collected, and their kinetic energies measured with a kinetic energy (KE) analyzer (see, e.g., [46] and problem 9.4). Using the LPES technique, angular distributions are measured one angle at a time, and the kinematic transformation from the lab frame to the CM frame is accomplished as follows (see Fig. 9.1).

1

We assume the parent/core is infinitely heavy, i.e., we neglect the recoil of the core during the pd process.

9

Measuring Photoelectron Angular Distributions in the Laboratory

205

Let θcm be the angle measured from the photon polarization vector to the photoelectron velocity vector in the CM frame. Now, consider the following expression (we are working with linearly polarized photons in the dipole approximation): h  i  π I cm ðθcm ¼ 0Þ þ I cm θcm ¼  I cm ðθcm ¼ 0Þ sin 2 θcm 2 

σ pd σ pd σ pd β  ¼ ð1 þ β Þ þ 1 ð1 þ βÞ sin 2 θcm 4π 4π 2 4π  

 σ pd σ pd 3β 3β
2 2 ¼ 1þβ sin θcm ¼ 1þβ 1  cos θcm 4π 2 4π 2 

  i σ σ pd h 1 3 3 cos 2 θcm  1 pd 2 1 þ β  þ cos θcm ¼ 1þβ ¼ 2 4π 2 2 4π σ pd ¼ ½1 þ βP2 ð cos θcm Þ ¼ I cm ðθcm Þ 4π h  i  π ) I cm ðθcm Þ ¼ I cm ðθcm ¼ 0Þ þ I cm θcm ¼  I cm ðθcm ¼ 0Þ sin 2 θcm 2

ð9:4Þ

where we have also used (3.29). From Fig. 9.1: η ¼ θcm  θs ¼ θc  θl

ð9:5Þ

where η is the angle pictured in Fig. 9.1 and given by η ¼ θc  θl ¼ sin 1



vi sin θl vc

 ð9:6Þ

Equation (9.5) tells us that when θs ¼ 0, then θcm ¼ η, which in turn means I lab ðθs ¼ 0Þ ¼ I cm ðθcm ¼ ηÞ

ð9:7Þ

h   i π I lab ð0Þ ¼ I cm ð0Þ þ I cm  I cm ð0Þ sin 2 η 2

ð9:8Þ

Combine (9.4) with (9.7):

Again, from (9.5), θs ¼ π2  η ) θcm ¼ π2, which implies  I lab Next, we combine (9.3) and (9.6):

   π π  η ¼ I cm 2 2

ð9:9Þ

206

9

Measuring Photoelectron Angular Distributions in the Laboratory

" 
# 12 2 E  E γ fi η ¼ sin 1 vi sin θl me

ð9:10Þ

Before we can use (9.10), we must first find a kinematic transformation relating the photoelectron kinetic energy in the ion frame E(ifγ)  Ec, with its corresponding kinetic energy as measured in the lab frame El. From Fig. 9.1,   ! ! ! ! ! v l ¼ v c þ v i ) v c ¼ v l  v i ) v2c  ¼ v2l þ v2i  2jvl j jvi j cos θl   1 1 1 1 ) me v2c ¼ me v2l þ me v2i  2vl vi me cos θl 2 2 ffl{zfflfflffl} |fflffl 2 ffl{zfflfflffl} 2|fflfflffl{zfflfflffl} |fflffl   E E !

c

l

ε

me mi

Ei

 2 12 1 ) E c ¼ E l þ ε  2 v2l v2i me cos θl 2 pffiffiffiffiffiffiffi ) E c ¼ E l þ ε  2 εE l cos θl

ð9:11Þ

We can use (9.11) to derive the inverse transformation
pffiffiffiffiffi2
pffiffiffi pffiffiffiffiffi El þ 2 ε cos θl E l þ ðε  Ec Þ ¼ 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi 1  pffiffiffi 2 ε cos θl  4ε cos 2 θl  4ðε  E c Þ ) El ¼ 2 0 12 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ffiffi ffi B C ) El ¼ @ ε cos θl  ε cos 2 θl  ε þ E c A |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}  ¼

ð9:12Þ

ε sin 2 θl

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 pffiffiffi ε cos θl  E c  ε sin 2 θl

As an aside, note that ε  me v2i =2 ¼ ðme =mi ÞE i is the kinetic energy of an electron moving with the same speed vi as an ion in the beam. Nonrelativistic kinematic shifts can also cause an apparent shift in angular coordinates. For example, the solid angle of acceptance of photoelectrons into the kinetic energy (KE) analyzer changes depending upon one’s reference frame. In spherical coordinates, the ratio of the solid angle in the lab frame Ωl, to the solid angle in the CM frame Ωc, is sin θl dθl dΩl ¼ dΩc sin θc dθc

ð9:13Þ

9

Measuring Photoelectron Angular Distributions in the Laboratory

207

where we have noted that dϕl ¼ dϕc due to axial symmetry. We use (9.11) and (9.12) to find pffiffiffiffiffiffiffi dEc ffi dE c ¼ 2 εE l sin θl dθl ) sin θl dθl ¼ pffiffiffiffiffiffi 2 εEl

ð9:14aÞ

pffiffiffiffiffiffiffi dE l dEl ¼ 2 εEc sin θc dθc ) sin θc dθc ¼  pffiffiffiffiffiffiffi 2 εE c

ð9:14bÞ

Substituting these results into (9.13) gives rffiffiffiffiffi    dΩl  dEc E c  ¼ dΩc  dE l El

ð9:15Þ

But from (9.11), we see dEc ¼1 dEl

rffiffiffiffiffi ε cos θl El

ð9:16Þ

which, when substituted into (9.15), gives   rffiffiffiffiffi   rffiffiffiffiffi  dΩl    ¼ E c 1  ε cos θl dΩc  El El

ð9:17Þ

This equation is used when transforming differential cross-sections from one frame to the other:    dσ  dΩl dσ  ¼   dΩ c dΩ l dΩc

ð9:18Þ

As a reminder, we note that (9.6) gives the relative angle of emission of photoelectrons in the two frames. As stated earlier, most photo-spectroscopic experiments involve ions moving within a laser field. These ions will experience a shift in the photon frequency ν due to the relativistic Doppler effect [11, 24, 25]: 1  βi cos θ ν0 ¼ ν qffiffiffiffiffiffiffiffiffiffiffiffiffi 1  β2i

ð9:19Þ

where ν0 is the laser frequency as seen in the ion frame, and ν is the laser frequency in the lab frame. Here θ is the angle between the laser beam and the ion beam. The numerator in (9.19) is the nonrelativistic (first order) Doppler effect and is zero for θ ¼ π/2. Therefore, to first-order, the geometry of the experiment depicted in Fig. 9.1

208

9 Measuring Photoelectron Angular Distributions in the Laboratory

will not suffer any Doppler shift. The denominator of (9.19) is a consequence of relativistic time dilation and gives rise to second-order effects regardless of the crossing angle. The shift associated with this term (for θ ¼ π/2) is approximated as follows: ν Δν2ndorder ¼ ν0  ν ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffi  ν 1  β2i 2 3

β2 ν 1 6 7 ¼ ν4qffiffiffiffiffiffiffiffiffiffiffiffiffi  15  ν 1 þ i  1 ¼ β2i 2 2 2 1  βi

ð9:20Þ

and is second-order in the velocity parameter βi ¼ vi/c (as expected from the name) and is negligible for nonrelativistic ion beams. To maximize the photoelectron signal, some experiments invoke a collinear/ merged-beams geometry (see, e.g., [146, 147]). One major advantage of a collinear-beam geometry setup is that, if one could measure the relative shift in the absorbed laser frequency when the laser and ion beams are parallel and antiparallel, the uncertainty due to the Doppler effect can be eliminated to all orders of βi by taking the geometric mean as follows [24, 25, 146]:

ν0p,a

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 10 1 u0 u q ffiffiffiffiffiffiffiffi ffi u 1  βi B 1 þ βi CB 1  βi C pffiffiffiffiffi ¼ ν qffiffiffiffiffiffiffiffiffiffiffiffiffi ) ν0p ν0a ¼ u t@ν qffiffiffiffiffiffiffiffiffiffiffiffiffiA@ν qffiffiffiffiffiffiffiffiffiffiffiffiffiA ¼ νν ¼ ν 1  β2i 1  β2i 1  β2i ð9:21Þ

Example 9.1 Estimate the error associated with ion beam divergence Δθ. For nonrelativistic beams crossed at right angles, the error associated with ion beam divergence Δθ is derived (to first order) as follows [20, 24, 25]: Δνeff ¼ ν½1  βi cos ðθ þ ΔθÞ  ν ð1  βi cos θÞ ¼ νβi ½ cos θ  cos ðθ þ ΔθÞ In the case that the divergence is from the right angle π/2, we have cos θ ¼ 0; cos



 π π π þ Δθ ¼ cos cos Δθ  sin sin Δθ ¼  sin Δθ  Δθ 2 2 2

for a small divergence. So, we have

9

Measuring Photoelectron Angular Distributions in the Laboratory

Δνeff  νβi Δθ ¼

209

    vi ν 1 vi Δθ; ¼ νΔθ ¼ c λL c λL

Note: For a typical crossed-beams apparatus, we might have vi  5  105 m\s, Δθ  0.02 rad, and λL ¼ 1064 nm, which gives Δνeff  5.9μeV. For collinear-beam geometries, the divergence is from θ ¼ 0, which gives [146]   Δθ2 Δνeff ¼ νβi ½ cos θ  cos ðθ þ ΔθÞ ¼ νβi ½1  cos ð0 þ ΔθÞ  νβi 1  1 þ 2  2 Δθ ) Δνeff  νβi 2 █

Once a pd experiment has been completed, and a photoelectron binding energy (BE) computed, all the quantities in (8.10) are known, allowing us to determine the angle η. Therefore, the quantity Icm(π/2) in (9.9) can be computed from a measurement in the lab frame and normalized to one. When this is done, (9.8) becomes I lab ð0Þ ¼ I cm ð0Þ þ ½1  I cm ð0Þ sin 2 η ) I cm ð0Þ ¼

I lab ð0Þ  sin 2 η 1  sin 2 η

ð9:22Þ

The asymmetry parameter is now easily calculated by determining Icm(θcm ¼ 0) and Icm(θcm ¼ π/2) and by using (3.29) [146]   I ð0Þ σ pd 2 2 ½ 1 þ β P ð cos 0 Þ  I ð 90 Þ ð 1 þ β Þ I cm ðθcm ¼ 0Þ 2 4π
   ¼ ) β ¼ ¼ σ pd I ð0Þ I cm θcm ¼ π2 1  β2 þ2 4π ½1 þ β P2 ð cos 90Þ

ð9:23Þ

I ð90Þ

If Icm(θcm ¼ π/2) is normalized to unity, then (9.23) becomes β¼

2I cm ð0Þ  2 I cm ð0Þ þ 2

ð9:24Þ

Another crossed-beams (or merged-beams) method now commonly used to measure PADs is the velocity-map imaging (VMI) method. In a VMI spectrometer, a series of three electrostatic plates is used to collect and focus photoelectrons produced in a pd process. The photoelectrons are then projected onto a positionsensitive detector (see Fig. 9.2) [148]. A major advantage of the VMI method over other methods is the near 100% collection efficiency of photoelectrons achieved over the entire 4π solid angle, resulting in the ability to collect a full angular distribution

210

9

Measuring Photoelectron Angular Distributions in the Laboratory

IMAGE ACQUISITION PROGRAM 5 kV CCD CAMERA

POWER SUPPLY

1 kV POWER SUPPLY PHOSPHOR SCREEN MCPs F F L I I E G L H D T | F T R U E B E E

ION BEAM

1 kV POWER SUPPLY

e-

PULLER ELECTRODE

e-

ACCELERATION ELECTRODE

VACCEL

VPULL

INTERACTION REGION PUSHER ELECTRODE

VPUSH

µ-METAL SHIELDING

Fig. 9.2 Schematic view of a VMI spectrometer. The trajectories of photoelectrons created in the interaction region via a photodetachment or photoionization process are projected onto a positionsensitive detector by a three-element electrostatic lens and a series of aperture electrodes. In this design, the detector consists of microchannel plates (MCPs), a phosphor screen, and a chargecoupled device (CCD) camera. The VMI spectrometer maps photoelectrons with the same ejection direction and speed onto the same annular region of the position-sensitive detector, regardless of where the photoelectron was initially created in the interaction region. All elements are kept at the appropriate potentials to avoid distortions of the photoelectron trajectories, to focus the image on the detector, and to ensure a near-unity photoelectron collection efficiency. The design of the VMI spectrometer thus preserves the PAD as a two-dimensional projection which is later reconverted back into the original three-dimensional pattern using the appropriate analysis procedures (e.g., see Problem 9.6) [23, 149] Reproduced from [23] with permission from the author

pattern in one scan. Because the VMI technique ensures that photoelectrons with the same velocity strike the detector at the same radius, one can also collect photoelectron (relative) kinetic energy spectra while simultaneously collecting the PAD (see Fig. 9.3).

9

Measuring Photoelectron Angular Distributions in the Laboratory

211

Fig. 9.3 VMI image analysis process (see also Problem 9.6). A photoelectron (relative) kinetic energy spectrum can be extracted from the PAD (middle right figure), and the asymmetry parameter β is calculated via (9.25) and (9.26) (bottom right figure). The VMI images in this figure are from the photodetachment of copper anions using 532 nm linearly polarized laser photons [23]. Reproduced from [23] with permission from the author

The projection method involved in velocity-map imaging requires that the PAD possess azimuthally symmetry, which means the differential cross-section must be in the form of (3.29).2 If a full angular distribution pattern is collected in one scan via 2

To ensure that the VMI is oriented properly to capture the axially symmetric photoelectron distribution, the polarization vector of the (linearly) polarized laser photon must be oriented ! horizontally, collinear with the ion velocity vector v i . In that case, θcm ¼ θc and θs ¼ θl (see Fig. 9.1).

212

9

Measuring Photoelectron Angular Distributions in the Laboratory

VMI techniques, then one may determine β by producing a linear plot of photoelectron intensity Icm verses P2(cosθ). The asymmetry parameter is then determined by measuring the slope of the resulting line (see Fig. 9.3). However, one still needs to convert the intensity pattern, which is measured in the lab frame, into the true pattern, which is a pattern in the CM frame. Once corrections for photoelectron energies are made, additional corrections are necessary to transform angles and solid angles from the lab frame to the CM frame (see Fig. 9.4).

Fig. 9.4 Polar plot of a photoelectron angular distribution for the photodetachment of Ge. The single-photon photodetachment at a photon wavelength of 532 nm proceeded via the following transitions: Ge4p3(4S3/2) + hν(532 nm) ! Ge4p2(3P0, 1, 2) + e(s, d ). The radial coordinate in each plot represents (collected) photoelectron relative intensity for one of these transitions. The polarization vector of the linearly polarized laser photons are in the horizontal direction with respect to every diagram. The first diagram (upper left) represents a PAD to which a kinematic correction has not yet been applied. The second diagram (upper right) is the same data as shown in the first diagram, against which the solid angle kinematic correction has now been applied. The third diagram (lower left) is the same as the second diagram, except now plotted against the CM angle. The final diagram (lower right) shows the photoelectron angular distribution fully corrected for both kinematic effects and systematic errors. The small circles represent experimental data. The solid (red) curves indicate the fit to (9.26). Note that the measured asymmetry parameter will be erroneous unless the full kinematic correction is applied [149]

9

Measuring Photoelectron Angular Distributions in the Laboratory

213

Starting with (3.29) and using the nonrelativistic kinematic transformation from above, we get

I cm

dσ pd σ pd ¼ ¼ ½1 þ β P2 ð cos ðθcm  ηÞÞ dΩc 4π

"sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi# rffiffiffiffiffi Ec ε 1 cos θl El El

ð9:25Þ

where η is determined by (9.10) and the sign in the θcm  η term depends on the quadrant in which the kinematic transformation is made (see Fig. 9.5). Note that the term in the brackets [last term in (9.25)] represents the solid angle kinematic correction. The form for Icm in (9.25) allows for a straightforward process to fit experimental data with standard curve-fitting routines via the equation I ðαÞ ¼ a½1 þ β P2 ð cos ðα  cÞÞ

ð9:26Þ

where a, c, and β serve as fitting parameters [41]. Although a kinematic transformation is absolutely necessary to obtain accurate results, one problem that exists with using (9.26) to transform VMI images and PADs from the lab frame to the CM frame is that the photoelectron kinetic energy Ec (and El) must be known in advance (or measured during the experiment) for the transformation equation to be useful. Although LPES methods are well-suited to measure absolute photoelectron kinetic energies, other methods (such as VMI techniques) are not. One way to overcome this limitation is to use the displacement of the photoelectron image in the VMI imaging apparatus (for example) to extract the photoelectron kinetic energy (see Fig. 9.5). In Fig. 9.5, we choose as our origin OL the center of the photoelectron image as it is viewed in the lab frame. Angular measurements with respect to the z-axis will then correspond to θL. Radial measurements from the lab-frame origin, however, correspond to rC. To see this, we first realize that the origin in the lab frame is shifted from the CM origin by an amount Δz (which can be measured by the experimentalist). Since rL is defined with respect to the CM origin, it also becomes shifted by Δz when moving to the lab origin. Then by construction, the portion of rL that intersects the photoelectron distribution is equivalent to rC, while the difference rL  rC is unobserved. Thus, in choosing the center of the photoelectron distribution as our origin, we can measure radii, and hence energies, in the CM frame, while angular measurements are made in the lab frame [consequently, angular measurement must still be frame-corrected via (9.25)]. Our analysis of laboratory techniques for measuring PADs would not be complete without mentioning a recently developed type of reaction microscope called COLTRIMS. Cold Target Recoil Ion Momentum Spectroscopy (COLTRIMS)3 is an imaging technique in which a beam of collimated cold molecules is bombarded with a beam of high-energy monochromatic photons. A collision between a molecule and a

The name “COLTRIMS reaction microscope” is sometimes abbreviated as C-REMI in the literature.

3

214

9

Measuring Photoelectron Angular Distributions in the Laboratory

Fig. 9.5 Kinematic effect on a PADs image collected by a VMI spectrometer. The labels in the diagram are explained in the text [149]

photon may cause the molecule to undergo a “Coulomb explosion,” leaving in its wake a number of positive ions (“recoils”) and electrons, each of which now has an initial momentum vector pointing in some direction in space. These reaction products are then collected in coincidence for the purpose of reconstructing the orientation of the molecule (at the time of the explosion) with respect to a lab-fixed reference frame. This allows for a PAD from fixed-in-space molecules to be constructed. This type of PAD, called a molecular-frame photoelectron angular distribution, or MFPAD (a photoelectron angular distribution measured in the frame of the molecule) offers a rich source of information about the nature of molecular potentials and photonmolecule interactions. One advantage of the COLTRIMS technique is that, by judicious use of electric and magnetic fields, the COLTRIMS apparatus can collect all particles in the entire 4π solid angle (within limits of detector construction) [150–152]. Another advantage of the COLTRIMS reaction microscope is that it is easily transportable and can be integrated into a synchrotron facility quickly and with little effort (see Figs. 9.6, 9.7 and 9.8). As a result, COLTRIMS devices are ubiquitous in many locations. Synchrotrons serve as bright sources of collimated, high-energy photons that are needed to penetrate deep into the cores of atoms and molecules. When this light is used to study inner-shell pd processes (e.g., with a COLTRIMS), the core electrons that are ejected can be used to illuminate the molecule from within, allowing one to measure photoemission delays and create detailed maps of the way in which molecular potentials evolve during the pd process (see Chap. 10) [135, 153–158]. COLTRIMS has also been used to study circular dichroism in small molecules [159]. For a basic description of the COLTRIMS device, see App. G. Finally, it should be mentioned that, although most of the analysis in this chapter was based on linearly polarized incident photons, other polarizations can be used. However, these alternate polarizations do not generally offer any additional dynamical information on the pd process since the differential cross-sections expressed in (3.29) and (4.39) can always be expressed in terms of σ pd and β, regardless of the photon polarization (see examples below) [90].4

4

Excepting the aforementioned interaction between circularly polarized light and chiral molecules

Fig. 9.6 Two views of a COLTRIMS attached to beamline U49/2-PGM-1 of the BESSY II synchrotron facility, Helmholtz-Zentrum Berlin, Germany. In the top picture, the COLTRIMS is located to the right of the center of the picture and can be identified by the stacked, circular Helmholtz coils surrounding the apparatus. The bottom picture shows a closer view of the COLTRIMS. The COLTRIMS shown here was designed and constructed by members of the Experimental Atomic and Molecular Physics Group at the Goethe-Universitat, Frankfurt Am Main

216

9

Measuring Photoelectron Angular Distributions in the Laboratory

Fig. 9.7 Two views of a COLTRIMS device attached to beamline 9.0.1 of the Advanced Light Source (ALS) synchrotron facility, Berkeley National Laboratory, USA. The COLTRIMS itself can be identified by the stacked, circular, copper-sheathed Helmholtz coils surrounding the apparatus. The long “pipe” is the beamline through which collimated light is delivered to the device. The COLTRIMS shown here was designed and constructed by members of the Atomic, Molecular, Optical and Chemical Physics Group at the University of Nevada, Reno

9

Measuring Photoelectron Angular Distributions in the Laboratory

217

Fig. 9.8 A COLTRIMS attached to beamline SEXTANTS of the SOLEIL synchrotron facility, Saint-Aubin, France. The COLTRIMS itself is located in the left of the picture and can be identified by the stacked, circular Helmholtz coils surrounding the apparatus. The long “pipe” extending the length of the picture is the beamline through which collimated light is delivered to the device. The COLTRIMS shown here was designed and constructed by members of the Experimental Atomic and Molecular Physics Group at the Goethe-Universitat, Frankfurt Am Main

Example 9.2 Find an expression for the differential cross-section when unpolarized/circularly polarized light is used. Unpolarized/circularly polarized light is equivalent to two incoherent/coherent linearly polarized beams of equal intensity pointing along mutually orthogonal (x and y) axes. In that case, the differential cross-section would be written as [90]

dσ pd dΩ




 1 σ pd 1 σ pd ½1 þ β P2 ð cos θx Þ þ 1 þ β P2 cos θy 2 4π 2 4π h i σ pd 1 ¼ 1  β P2 ð cos θz Þ 4π 2 ¼

unpol=circ pol

ð9:27Þ

where the z-axis is taken as the beam propagation direction, and we have used the definition of P2(cosθ) found in (4.39c) and the relation [9, 90]
 cos 2 ðθx Þ þ cos 2 θy þ cos 2 ðθz Þ ¼ 1

ð9:28Þ

For partially polarized light, the differential cross-section has the same form as in (9.27), except that each component is scaled by a weighting factor describing the fraction of polarized light along each orthogonal axis [90, 160]:

218

9



dσ pd dΩ

¼ par pol

Measuring Photoelectron Angular Distributions in the Laboratory


 I y σ pd I x σ pd ½1 þ β P2 ð cos θx Þ þ 1 þ β P2 cos θy I 0 4π I 0 4π

ð9:29Þ

where I0 ¼ Ix þ Iy

ð9:30Þ

We manipulate portions of the RHS of (9.29) as follows:
 Iy Ix ½1 þ β P2 ð cos θx Þ þ 1 þ β P2 cos θy I0 I0

 Iy
Ix Iy Ix ¼ þ þ β P2 ð cos θx Þ þ P2 cos θy I0 I0 I0 I0

3 Ix 1 Ix 3 Iy 1 Iy 2 2 cos θx  þ cos θy  ¼1þβ 2 I0 2 I0 2 I0 2 I0

Iy β I ¼ 1  1  3 x cos 2 θx  3 cos 2 θy 2 I0 I0    

2I y β 3 2I x 3  cos 2 θy ¼ 1  1  cos 2 θx 2 2 I0 I0 2    

Iy I Iy β 3 3 I ¼ 1  1  cos 2 θx 1  þ x  cos 2 θy 1  x þ 2 2 I0 I0 I0 I0 2 Iy β I ¼ 1  ð3  1Þ  3 cos 2 θx  3 cos 2 θy  3 x cos 2 θx þ 3 cos 2 θx I0 I0 4

Iy I þ3 x cos 2 θy  3 cos 2 θy I0 I0


 Ix  Iy Ix  Iy β ¼ 1  3 1  cos 2 θx  cos 2 θy  1  3 cos 2 θx þ3 cos 2 θy I0 I0 4


  I I β x y cos 2 θx  cos 2 θy ¼ 1  3 cos 2 θz  1  3 4 I0 h i β 3
¼1 P2 ð cos θz Þ  p cos 2 θx  cos 2 θy 2 2 ð9:31Þ where we have used (9.28) and (9.30), and the degree of polarization is given by [90] p¼

Ix  Iy Ix þ Iy

Now we can rewrite (9.29) in its final form [90, 146]:

ð9:32Þ

9

Measuring Photoelectron Angular Distributions in the Laboratory



dσ pd dΩ

¼ par pol

  h i σ pd β 3
1  P2 ð cos θz Þ  p cos 2 θx  cos 2 θy 4π 2 2

219

ð9:33Þ

Equation (9.33) also applies to elliptically polarized light as long as the orthogonal axes are understood to be the major and minor axes of the ellipse which characterizes █ the incident laser light. Example 9.3 Repeat the above example and this time include higher-order multipole terms. To include the higher-order multipole terms, manipulations similar to the ones above can be performed on (5.48). For unpolarized/circularly polarized light, the result is [49, 52] h

i

dσ pd dΩ unpol=circ pol

¼

    σ pd β γ þ δ þ sin θ cos ϕ 1þ 4π 4 2

3β γ 2 2  sin θ cos ϕ sin 3 θ cos 3 ϕ 4 2

ð9:34Þ

Notice that (9.34) is a function only of sinθ cos ϕ, the cosine of the angle between the photoelectron momentum and the photon propagation vectors. For partially polarized light, using (9.32), we get [49, 52]



dσ pd dΩ par

pol

8

9

ðp  1Þ > β 3 3 > 2 2 > > 1 þ  pβ þ pβ cos θ þ δ þ γpcos θ  γ > > > > > > 2 4 4 2 > > = < σ pd  sin θ cos ϕ ¼ 4π > >



> > > > > > ð p  1 Þ 3β > > 2 3 2 3 > > sin þ θ cos ϕ þ γ θ cos ϕ ð p  1 Þ sin ; : 2 4

9 8
 β > > > > 1 þ ð1 þ 3pÞ 3cos 2 θ  1 > > > > 8 > > > > > >

> > > > > >
 ð p  1 Þ > > 2 2 = < þ δ þ γ cos θ þ γ θ  1 5cos σ pd 8 ¼ 4π >

>

> > > > > ðp  1Þ > 3β 2 > > > >  sin θ cos ϕ þ ðp  1Þ sin θ cos 2ϕ þ γ > > > > 8 8 > > > > > > ; : 3  sin θ cos 3ϕ ð9:35Þ █

220

9

Measuring Photoelectron Angular Distributions in the Laboratory

Problems 9.1. In most experiments in which atomic or molecular ion beams are used, a method must be found to select the desired atomic or molecular species by mass and/or charge (such a device is known as a momentum analyzer). One selection technique centers around a 90 bending electromagnet. The magnet is so designed as to produce a radial magnetic field in its interior. Show that, to select a particle of mass mi and charge qi that has been accelerated by a potential Vi, a magnet of radius Rm (the radius along which the selected ion must travel) must be set to a magnetic field BRm , whose magnitude is given by BRm

sffiffiffiffiffiffiffiffiffiffiffiffi 2mi V i ¼ qi R2m

9.2. Another type of momentum analyzer is a Wien filter. In a Wien filter, the magnetic field is supplied by an electromagnet, or by a pair of permanent magnets [161]. The electric field is supplied by a pair of parallel conducting plates separated by a distance d and maintained at a potential difference V. In ! ! this type of selector, the magnetic and electrostatic fields (B &E , respectively) are arranged perpendicularly, as shown in the diagram below:

l a/2 a E

B

d D

Schematic diagram of a Wien filter If one were to scan the mass spectrum of an ion beam entering the filter by varying the potential V, only ions with a well-defined mass-to-charge ratio will be transmitted. (a) Show that, for ions of mass mi and charge qi that have been accelerated from rest by a potential Vi, the mass of the transmitted ions is given by mi ¼

2qi V i B2 d2 V2

(b) Consider a filter that is set to allow an ion of mass m1 and charge qi to pass through undeflected. An ion with the same charge, but of slightly higher

9

Measuring Photoelectron Angular Distributions in the Laboratory

221

mass m2 (m1  m2), will experience a (slight) centripetal deflection. Find the angle ϕ (in terms of the relevant given parameters) between the trajectories of m1 and m2 for a Wien filter of linear extent a (neglecting end effects). Show that the mass resolution of the Wien filter is Δmi 4DV i d ¼ mi alV where D is the length of the post-filter aperture/slit, and l is the distance from the center of the filter to the post-filter aperture. 9.3. Electrostatic beam optics [19, 161] In experiments designed to measure PADs, ions must be delivered from the ion source to the region in which they will interact with incoming photons. To focus the ion beam as it travels from the source to the interaction region, a series of electrostatic focusing lenses are usually employed. An Einzel lens consists of hollow conducting tubes in which axially symmetric electrostatic potentials Φ are realized to focus charged particle beams without changing the energy of the beam particles. We will show this as follows. In the paraxial approximation, we can expand this potential in a Taylor series expansion [19]: Φðr, zÞ ¼ Φð0, zÞ þ b2 ðzÞr 2 þ b4 ðzÞr 4 þ ⋯

ð1Þ

where the z-axis is parallel to the axis of the cylindrical tube and is the axis of symmetry of the field, r is the radial coordinate, and ϕ is the azimuthal angle, and where, due to axial symmetry, Φ(r) ¼ Φ(r), only the even terms in the expansion are nonzero. Note also that Φ(0, z) is the potential along the z-axis. The paraxial approximation tells us that bnrn 2 can be neglected. Physically, the paraxial approximation assumes that the ion beam does not stray very far from the symmetry axis (the z-axis) of the lens. (a) Show that, in the paraxial approximation and for axially symmetric potentials, the potential at a point close to the symmetry axis can be calculated 00 from solely its values Φ(r ¼ 0) and Φ (r ¼ 0) [19]. Also show that ions which stray from the axis are subject to a restoring force which drives them back to the z-axis and thus focuses the beam.  2 
 r. Note: The paraxial approximation also implies that ∂∂rΦ2 r 2 > > > þ exp  x  < exp x  = 2 2

ðl þ mÞ! 1 m mþ1 m! ð l  m Þ!2 > x > m¼0 :

π ð l  m þ 1Þ cos x  l X 2 ðl þ mÞ! ¼ m mþ1 m! ð l  m Þ!2 x m¼0

jl ðxÞ ¼

2

> > ;

ðF:20Þ This particular series expression has advantages in that it makes clear not only the relation of the spherical Bessel functions to the trigonometric functions but also the asymptotic behavior of the spherical Bessel functions. Specifically, as x gets very large, (F.20) shows the dominant term in the sum will be the m ¼ 0 term, so that

302

Appendixes

lim jl ðxÞ !

h i Þ cos x  πðlþ1 2

x!1

x

¼

  sin x  lπ2 x

ðF:21Þ

On the other hand, this particular form obscures the fact that the leading term in the series for small x is of order xl. Let us see if we can find another series representation that makes this fact clear. We revisit (F.1) and make the change of variable ξ  xz, l 1 ð2Þ l! ) jl ðxÞ ¼ 2π xlþ2

I Cj



eiξ lþ1 dξ ξ2  1 2 x

ðF:22Þ

We pause here and note that the contour Cj is a positively oriented, closed contour that contains the singularities at ξ ¼  x. Rewriting, 1 jl ðxÞ ¼ ð2Þl l!xl 2π

 l1 eiξ x2 1 2 dξ 2lþ2 ξ Cj ξ

I

ðF:23Þ

For any region in the complex ξ plane outside the circle |ξ| ¼ x, the quantity in the brackets is uniformly convergent and can be expanded in a power series:  1

x2 ξ2

l1

x2 1 x4 þ ð l þ 1Þ ð l þ 2Þ 4 þ ⋯ 2 2 ξ ξ 2 m 1 X ðl þ mÞ! x ¼ m!l! ξ2 m¼0 ¼ 1 þ ð l þ 1Þ

ðF:24Þ

If we deform the contour Cj so that all parts of it lie outside the disk(s) |ξ| ¼ x, then we can evaluate the integral using the residue theorem. First, we interchange the order of summation and integration (which we can do because of the uniform convergence) and then evaluate j l ð xÞ

"I # I iξ eiξ e 1 l l ¼ ð2Þ l!x dξ þ ðl þ 1Þx2 dξ þ ⋯ 2lþ2 2lþ4 2π Cj ξ Cj ξ   2lþ1 2lþ3 l l ðiÞ 2 ðiÞ ¼ ið2Þ l!x þ ðl þ 1Þx þ⋯ ð2l þ 1Þ! ð2l þ 3Þ!   2l l! xlþ2 xl  þ⋯ ¼ ð2l þ 1Þ! 2ð2l þ 3Þ

ðF:25Þ

Appendixes

303

where (F.11) was used to evaluate the residues of all the poles at ξ ¼ 0. Equation (F.25) makes it abundantly clear that xl is the leading term in this expansion. We finish by writing equation (F.25) in summation form, j l ð x Þ ¼ 2l x l

1 X

ð1Þm

m¼0

ðl þ mÞ!x2m m!ð2l þ 2m þ 1Þ!

pffiffiffi X 1 ðx=2Þ2mþl π ð1Þm ¼ 2 m¼0 m!Γðl þ m þ 3=2Þ

ðF:26Þ

In a similar fashion, we can construct l 1 ð2Þ l! nl ðxÞ ¼ lþ1 2πi x

I

eixz dz; C n ¼ C 1  C 2 ðz  1Þlþ1 C n ðz þ 1Þ lþ1

ðF:27Þ

This construction allows us to realize the series representation of the spherical Neumann functions nl(x) right away. All we have to do is change the sign of the term eix in (F.20) to eix and then divide by i to get nl ð xÞ ¼

1 X m¼0

  π sin x  lmþ1 ðl þ mÞ! 2 2m m!ðl  mÞ! xmþ1

ðF:28Þ

Expansion of Plane Waves in Terms of Spherical Functions In this section, we derive the formula for the expansion of a plane wave in terms of spherical waves; a formulation that first appears in (4.14). We start by assuming eikrx ¼

1 X

al jl ðkr ÞPl ðxÞ

ðF:29Þ

l¼0

Our task will be to find the coefficients al. Multiplying both sides of (F.29) by Pn(x) and integrating Z1 e 1

ikrx

Pn ðxÞdx ¼

Z1 X 1 1

al jl ðkr ÞPl ðxÞPn ðxÞdx

l¼0

Using the orthogonality of the Legendre polynomials (2.71),

ðF:30Þ

304

Appendixes

Z1 eikrx Pn ðxÞdx ¼

1 X l¼0

1

al jl ðkr Þ

2 2 δ ¼ a j ðkr Þ 2l þ 1 nl 2n þ 1 n n

ðF:31Þ

and differentiating both sides n times with respect to kr, we get dn d ðkr Þn

Z1 eikrx Pn ðxÞdx ¼ 1

dn 2 j ðkr Þ an 2n þ 1 d ðkr Þn n

Z1 ) ðiÞ

n

xn eikrx Pn ðxÞdx ¼ 1

2 dn an j ðkr Þ 2n þ 1 dðkr Þn n

ðF:32Þ

To evaluate the derivative on the RHS of (F.32), we appeal to (F.26): 2mþn pffiffiffi n X 1 ð1Þm π d kr dn j ð kr Þ ¼ 2 d ðkr Þn m¼0 m!Γðn þ m þ 3=2Þ 2 dðkr Þn n 2m pffiffiffi X 1 π ð1Þm ð1=2Þn ð2m þ nÞð2m þ m  1Þ⋯ð2m þ 1Þ kr ¼ 2 m¼0 2 m!Γðn þ m þ 3=2Þ 2m pffiffiffi X 1 m n ð1Þ ð1=2Þ ð2m þ nÞ! kr π ¼ 2 m¼0 m!Γðn þ m þ 3=2Þð2mÞ! 2 ðF:33Þ So now we have Z1 ðiÞn

xn eikrx Pn ðxÞdx ¼ 1

2m pffiffiffi X 1 π ð1Þm ð1=2Þn ð2m þ nÞ! kr 2 an 2 m¼0 m!Γðn þ m þ 3=2Þð2mÞ! 2 2n þ 1 ðF:34Þ

Taking the limit of both sides as kr ! 0, Z1 lim ðiÞ

n

n ikrx

x e

kr!0

1

Z1 ) ð i Þn

2m pffiffiffi X 1 ð1Þm ð1=2Þn ð2m þ nÞ! kr π 2 Pn ðxÞdx ¼ lim an 2 m¼0 m!Γðn þ m þ 3=2Þð2mÞ! 2 kr!0 2n þ 1

2m pffiffiffi X 1 π ð1Þm ð1=2Þn ð2m þ nÞ! kr 2 an 2 m¼0 m!Γðn þ m þ 3=2Þð2mÞ! 2 kr!0 2n þ 1

xn Pn ðxÞdx ¼ lim 1

ðF:35Þ

Appendixes

305

To evaluate the LHS, we use (A.32) Z1 xn Pn ðxÞdx ¼ 1

2nþ1 ðn!Þ2 ð2n þ 1Þ!

ðA:32Þ

To evaluate the RHS, we note that, in the limit, only the first term of the summation is non-negligible: 2m pffiffiffi X pffiffiffi 1 ð1Þm ð1=2Þn ð2m þ nÞ! kr π an π ð1=2Þn n! 2 ) lim ! an 2 m¼0 m!Γðn þ m þ 3=2Þð2mÞ! 2 2n þ 1 Γðn þ 3=2Þ kr!0 2n þ 1 ðF:36Þ Making use of the following identity [172],  pffiffiffi π ð2n þ 1Þ! 3 Γ nþ ¼ 2 22nþ1 n!

ðF:37Þ

which, after mating the RHS and the LHS back together gives ði Þn

an ðn!Þ2 2nþ1 2nþ1 ðn!Þ2 ¼ ) an ¼ ðiÞn ð2n þ 1Þ ð2n þ 1Þ! ð2n þ 1Þð2n þ 1Þ!

ðF:38Þ

The derivation of (F.38) was based, in part, on (A.31), which we never proved. For that reason, the previous derivation may be unsatisfactory to some. In that case, we offer the following alternative derivation. Here we assume !!

ei k  r ¼

1 X l X

alm jl ðkr ÞY lm ðθ, ϕÞ

ðF:39Þ

l¼0 m¼l !

Let the z-axis lie along the direction of k : !!

) ei k  r ¼ eikr cos θ ¼

1 X l X

alm jl ðkr ÞY lm ðθ, ϕÞ

ðF:40Þ

l¼0 m¼l

The quantity on the LHS has no ϕ-dependance; therefore, we must have m ¼ 0 on the RHS: !!

) ei k  r ¼ eikr cos θ ¼

1 X l¼0

Using (2.60),

al jl ðkr ÞY l0 ðθ, ϕÞ

ðF:41Þ

306

Appendixes !!

) ei k  r ¼

1 X l¼0

al jl ðkr Þ



1=2 2l þ 1 Pl ð cos θÞ 4π

ðF:42Þ

Again, we use the orthogonality of the Legendre polynomials (2.71) to establish 1 al jl ðkrÞ ¼ ½4π ð2l þ 1Þ1=2 2

Z

1

1

dzPl ðzÞeikrz

As before, we take the limit of both sides as kr ! 0. To evaluate the LHS, we again note that, in the limit, only the first term of the summation for jl(kr) in (F.26) is non-negligible. Using (F.37), we get l Z1 pffiffiffi π 2l l! 22lþ1 kr 1 1=2 pffiffiffi ¼ ½4π ð2l þ 1Þ dzPl ðzÞeikrz al 2 ð2l þ 1Þ! π 2 2 1

2l l! 1 ðkr Þl ¼ ½4π ð2l þ 1Þ1=2 ) al 2 ð2l þ 1Þ!

ðF:43Þ

Z1 dzPl ðzÞeikrz 1

The oscillatory nature of the exponential term on the RHS of (F.43) makes it hard to evaluate (even in the limit as kr ! 0), but we can proceed by equating the correct power of kr on the RHS with the corresponding power of kr on the LHS: 1 2l l! ðkr Þl ¼ ½4π ð2l þ 1Þ1=2 ðikr Þl ) al 2 ð2l þ 1Þ! 2l ðl!Þ2 1 ) al ¼ ½4π ð2l þ 1Þ1=2 ðiÞl ð2l þ 1Þ! 2

Z1 dzPl ðzÞ 1

Z1

zl l! ðF:44Þ

dzPl ðzÞzl 1

We can evaluate the integral by remembering that Pl(z) is an lth-degree polynomial in z. The coefficient of the leading power, zl, can be obtained from (A.3) l 2lð2l  1Þð2l  2Þ⋯ðl þ 1Þ l   1 dl  2 z þ O zl1 z 1 ¼ l l l 2 l! dz 2 l! l l! 2 P ðzÞ þ terms involving Plþ1 ðzÞ and higher ) zl ¼ 2lð2l  1Þð2l  2Þ⋯ðl þ 1Þ l ðF:45Þ

Pl ðzÞ ¼

Substituting this into the integral of (F.44) and again using (2.71) gives us (to leading order)

Appendixes

307

2l ðl!Þ2 1 2l l! 2 ¼ ½4π ð2l þ 1Þ1=2 ðiÞl 2lð2l  1Þð2l  2Þ⋯ðl þ 1Þ 2l þ 1 ð2l þ 1Þ! 2 l! 1 ¼ ½4π ð2l þ 1Þ1=2 ðiÞl ) al ð2lÞ! 2lð2l  1Þð2l  2Þ⋯ðl þ 1Þ l! 1 ¼ ½4π ð2l þ 1Þ1=2 ðiÞl ) al ð2lÞð2l  1Þð2l  2Þ⋯ðl þ 1Þl! 2lð2l  1Þð2l  2Þ⋯ðl þ 1Þ

al

) al ¼ ½4π ð2l þ 1Þ1=2 ðiÞl ðF:46Þ Finally resulting in, e

!!

ik r

¼

1 X

1=2

½4π ð2l þ 1Þ

l¼0

!!

) ei k  r ¼

1 X



1=2 2l þ 1 ðiÞ jl ðkrÞ Pl ðcosθÞ 4π l

ð2l þ 1ÞðiÞl jl ðkr ÞPl ð cos θÞ

ðF:47Þ

l¼0

as before. If we want to account for the spin of the photoelectron (e.g., in relativistic applications), then our continuum photoelectron plane wave would have the form !!

ψ pκmj ðpr Þ ¼ ei p  r =ħ χ ms

ðF:48Þ

We now expand in the usual way [73]: !!

ψ pκmj ðpr Þ ¼ ei p  r =ħ χ ms ¼

X κ, mj

j aκmj jl ðkr Þχ m rÞ κ ðb

ðF:49Þ

where k ¼ p/ħ is the wave number of the lth partial wave of the continuum photoelectron. To find the coefficients aκmj , we start by multiplying both sides of m0 {

(F.49) by χ κ0 j ðbr Þ and integrating over the solid angle. Using (6.3) we get Z aκmj jl ðkr Þ ¼

!!

χ κmj { ðbr Þei k  r χ ms dbr

ðF:50Þ

Expand the exponential in terms of spherical harmonics using (F.47), (6.1), and (2.79):

308

Appendixes

Z aκmj jl ðkrÞ ¼ 4π

χ κmj { ð^r Þ

2

1 X l0 , m0

  0 ðiÞl jl0 ðkrÞY l0 m0 ^k Y l0 m0 ð^r Þχ ms d^r 0

3

1

1 2

l 1 pffiffiffiffiffiffiffiffiffiffiffiffiB 6P ð1Þl2mj 2j þ 1@ Z 6 6 m0 6 s mj  m0s m0s ¼ 4π 6 6 6 1   0 P 4  ðiÞl jl0 ðkrÞY l0 m0 ^k Y l0 m0 ð^r Þd^r

j C 0 7 AY l,mj m0s ð^r Þχ ms χ ms 7 |fflffl{zfflffl} 7 mj δm0 ms 7 7 s 7 7 5

0 P l12mj pffiffiffiffiffiffiffiffiffiffiffiffi@ ¼ 4π ð1Þ 2j þ 1

1

l0 , m0

m0s



1 P l0 , m0

l m0s

1 2 m0s

mj    ðiÞl jl0 ðkrÞY l0 m0 ^k δll0 δmj m0s ,m0 δm0s ms

¼ 4π ð1Þ

j

A

mj

0

l12mj

0 pffiffiffiffiffiffiffiffiffiffiffiffi 2j þ 1@

l mj  ms

1   j A l ðiÞ Y l,mj ms ^k jl ðkrÞ mj

1 2 ms

ðF:51Þ where we also used (6.2) and (2.64). Comparing the LHS of (F.51) with the RHS of (F.51), l

) aκmj ¼ 4πðiÞ ð1Þ

l12mj

pffiffiffiffiffiffiffiffiffiffiffiffi 2j þ 1

l mj  ms

1 2 ms

! j mj

  Y l,mj ms ^k

ðF:52Þ

Giving us, finally [73], !! P pffiffiffiffiffiffiffiffiffiffiffiffi 1 ψ pκmj ðprÞ ¼ ei p  r =ħ χ ms ¼ 4π ðiÞl ð1Þl2mj 2j þ 1 κ, mj 0 1 1   l j mj ^ AY 2 @ rÞ l,mj ms k jl ðkrÞχ κ ð^ mj  ms ms mj

This form is equivalent to the one given in (6.16).

ðF:53Þ

Appendixes

309

Appendix G: Basic Theory of the Design of the COLTRIMS Reaction Microscope10 Recoil Ion Detection This appendix details the basic kinematic ideas behind the design of the COLTRIMS device. A schematic of the COLTRIMS (or C-REMI) spectrometer is shown in the figure below. Photons are introduced into the spectrometer along the x-axis in the positive x-direction. A well-collimated jet of cold molecules is introduced along the y-axis in the positive y-direction. The intersection of the photon beam and the gas jet is called the interaction region (IR). For now, we assume that the IR is small enough that it can be approximated by a point in space which we place at the origin of a set of lab-fixed axes (see Fig. G.1). Magnetic and electrostatic fields are used to accelerate positive ion (recoil) fragments and electrons onto their respective position-sensitive detectors via the Lorentz force: h! i ! ! F¼q Eþ v B

!

®

y (xi,yi)

Recoil Ion Detector

½p0½ ®

ðG:1Þ

®

½p^0½

½pz0½

Drift Region

q0

z

x

Zr

Electron Detector

rD Interaction Region

Ze

Zd

Fig. G.1 Schematic of COLTRIMS device. The solid (blue) curve shows two typical ion trajectories (in the left portion of the diagram). The impact coordinates (x, y) of one of the trajectories onto the recoil ion detector is also shown. The initial momentum vector triangle (for one of the trajectories) is shown near the origin. The initial momentum vector triangle is reproduced again at the top of the diagram for clarity. A typical electron trajectory is shown as a (green) dashed curve. The recoil ion detector is a fixed distance Zr along the z-axis from the IR. Both detectors are flat and lie in planes parallel to the x-y plane. The distances Ze and Zd indicated in the figure are also constants (and are, !

!

along with the E and B fields, controlled by the experimenter). Adapted from [152] and republished with permission of Elsevier; permission conveyed through Copyright Clearance Center, Inc

10

Primary references for App. G: [150–152, 177].

310

Appendixes

where electric and magnetic fields fill the entire volume of the spectrometer according to !



E¼

E 0bz; Z r  z  Z e



!

B ¼ B0bz; Z r  z  ðZ e þ Z d Þ

0 ; z Z e

¼ const

E0 , B0 ðG:2Þ

By measuring the time of flight (TOF) of the ion, and the x- and y-coordinates of the ion’s impact on the recoil ion detector, it is possible to reconstruct the components of the ion’s initial momentum vector (and by extension, the orientation of the molecule in space at the time of its break-up) with respect to the lab-fixed axes. That is, it is possible to find and separately express each of the three individual Cartesian components of the recoil ion’s initial momentum vector as a function of the time of flight of the ion, the x- and y-coordinates of its impact on the recoil ion detector (as appropriate), and other constants of the problem. The initial momentum vector of the recoil ion can be written in component form as !

!

!

!

p 0 ¼ p x0 þ p y0 þ p z0

where p⊥0 ¼

ðG:3Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p2x0 þ p2y0 is the magnitude of the transverse component of the initial

ion momentum. If the time of flight ti of the ion is an integer multiple of the cyclotron period, T ¼ 2π/ω, where ω ¼ qB0/m is the cyclotron frequency for a particle of charge q and mass m, then the impact coordinates of the ion onto the detector will be (0, 0, Zr). To start, we assume that, right before the collision with the photon (and the resulting Coulomb explosion), the molecule is at rest with respect to the lab frame. That is, we ignore the mechanical momentum imparted to the molecule by the photon (because it is negligible) and the constant initial velocity in the positive ydirection that all the particles will have by their being part of a collimated, fastmoving jet (we do this because it is easy to account for this initial velocity in postexperiment analysis, and so it is not essential to the main concept at this point). We may also assume that the trajectory of the ion does not cross into the drift region of the spectrometer. All motions are nonrelativistic. A couple of quick notes: • The high-energy photons needed for these types of experiments are generally produced by a synchrotron. The synchrotron facility provides an electronic timing marker when photons are produced to assist the experimenter in measuring times of flight. • The drift region and the fields in the drift region are designed in such a way as to confine and capture fast-moving electrons (to be discussed below). We start with the Lorentz force equation for the force on a particle of charge q and ! mass m, moving with velocity v ,

Appendixes !

F

311

h! i ! ! ¼q Eþ v B  3 2  bx by bz       7 ! 6 ) m a ¼ q4E 0bz þ  vx vy vz 5 ¼ q bxB0 vy  byB0 vx þ E 0bz    0 0 B0  qB0 qB qE ) v_ x ¼ v ; v_ ¼  0 vx ; v_ z ¼ 0 m y y m m

ðG:4Þ

Examining the z-component: dvz dt

qE ¼ 0)m m

Zvz

qE dv0z ¼ 0 m

vzo

)

dz qE 0 t þ vz0 ) ¼ m dt

Zz z0

Zt

0 1 qE dt 0 ) ðvz  vz0 Þ ¼ 0 @t  t 0 A |{z} m

t0

dz0 ¼

qE 0 m

Zt

t 0 dt 0 þ vz0

Zt

0

¼0

dt 0

0

qE qE ) z  z0 ¼ 0 t 2 þ vz0 t ) mz ¼ 0 t 2 þ pz0 t |{z} 2m 2 ¼0

qE 1 mi Z r  0 t 2i ) pz0 ¼ 2 ti ðG:5Þ where ti is the time of flight of the ion, mi is the mass of the ion, and we have noted that the z-coordinate of all ions striking the detector will be Zr. Note also that we have started the clock at the instant of the Coulomb explosion of the molecule. Although the z-equation is decoupled, the x- and y-equations are not: v_ x ¼

qB0 qB v ; v_ ¼  0 vx m y y m

ðG:6Þ

These two coupled equations can be easily solved:

2 qB0 qB0 qB0 vx ¼ ω2 vx etc: v ) €vx ¼ v_ ¼  v_ x ¼ m y m y m ) vx ¼ A cos ðωt þ φÞ and vy ¼ A sin ðωt þ φÞ

ðG:7Þ

where ω ¼ qB0/m is the cyclotron frequency for a particle of charge q and mass m, and A and φ are constant to be determined from the initial conditions.

312

Appendixes

Continuing, vx

¼ A½ cos ωt cos2φ  sin ωt sin φ 3 Zx Zt Zt ) dx0 ¼ A4 cos φ cos ωt 0 dt 0  sin φ sin ωt 0 dt 0 5 0

)x¼

0

ðG:8Þ

0

A ½ cos φ sin ωt þ sin φð cos ωt  1Þ ω

Similarly, vy

¼ A½ sin ωt cos2φ þ cos ωt sin φ 3 Zt Zt Zy sin ωt 0 dt 0 þ sin φ cos ωt 0 dt 0 5 ) dy0 ¼ A4 cos φ 0

0

)y¼

ðG:9Þ

0

A ½ cos φð1  cos ωt Þ þ sin φ sin ωt  ω

To find A and φ, we use (G.7): v2x þ v2y ¼ A2 ) A ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffi   ! v2x þ v2y   v ⊥ 

ðG:10Þ

Since A is a constant, we must have !  !  A ¼  v ⊥  ¼  v ⊥0 

ðG:11Þ

which makes sense, because the magnetic field does no work and hence cannot change the transverse momentum of the particle. Now evaluate (G.7) at t0 ¼ 0: vx0 ¼ v⊥0 cos φ ¼ v⊥0 cos ðφÞ and vy0 ¼ v⊥0 sin φ ¼ v⊥0 sin ðφÞ

ðG:12Þ

But, vx0 ¼ v⊥0 cos ϕ0 and vy0 ¼ v⊥0 sin ϕ0

ðG:13Þ

ϕ0 ¼ φ

ðG:14Þ

Thus,

So, the phase angle is the negative of the initial azimuthal angle ϕ0 at which v⊥0 was directed in the x-y plane.

Appendixes

313

Equations (G.4) can now be written as vx ¼ v⊥0 cos ðωt  ϕ0 Þ

ðG:15aÞ

vy ¼ v⊥0 sin ðωt  ϕ0 Þ

ðG:15bÞ

Equations (G.8) and (G.9) can be written as v sin ðϕÞ A A ð cos ωt  1Þ sin φð cos ωt  1Þ þ cos φ sin ωt ¼ ⊥0 ω ω ω v cos ðϕÞ sin ωt þ ⊥0 ω py0 p ¼ ð cos ωt  1Þ þ x0 sin ωt mω mω ðG:16Þ

x ¼

and y

v cos ðϕÞ A A ð1  cos ωt Þ cos φð1  cos ωt Þ  sin φ sin ωt ¼  ⊥0 ω ω ω v sin ðϕÞ sin ωt  ⊥0 ω py0 px0 ¼ ð1  cos ωt Þ þ sin ωt mω mω ðG:17Þ ¼

Now combine (G.16) and (G.17): px0

  1 sin ðωt=2Þ sin ðωt=2Þ px0 ¼ þ 2 sin ðωt=2   Þ sin ðωt=2Þ   cos ðωt=2Þ cos ðωt=2Þ sin ðωt=2Þ sin ðωt=2Þ 1  þ px0 þ ¼ p 2 8 y0 sin ðωt=2Þ sin ðωt=2Þ sin ðωt=2Þ sin ðωt=2Þ9   cos ðωt=2Þ cos ðωt Þ þ sin ðωt=2Þ sin ðωt Þ cos ðωt=2Þ > > > > > > p  < y0 sin ðωt=2Þ sin ðωt=2Þ = 1   ¼ 2> cos ðωt=2Þ sin ðωt Þ  sin ðωt=2Þ cos ðωt Þ sin ðωt=2Þ > > > > > þ þp ; : x0 sin ð ωt=2 Þ sin ð ωt=2 Þ 8 9 py0 cos ðωt=2Þð cos ðωt Þ  1Þ px0 cos ðωt=2Þ sin ðωt Þ > > > > > > þ = <  mω mω sin ðωt=2Þ sin ðωt=2Þ mω   ¼ 2 > > > þ px0 sin ðωt=2Þð1  cos ðωt ÞÞ  py0 sin ðωt=2Þ sin ðωt Þ > > > ; : mω mω sin ðωt=2Þ sin ðωt=2Þ  h i mω ωt i ) px0 ¼ i xi cot  yi 2 2 ðG:18Þ

where xi and yi are the x- and y-coordinates of the ion’s impact on the recoil ion detector, respectively.

314

Appendixes

Similarly, py0

¼ ¼

¼

¼

)

  1 sin ðωt=2Þ sin ðωt=2Þ þ p 2 sin ðωt=2Þ sin ðωt=2Þ y0      sin ðωt=2Þ sin ðωt=2Þ cos ðωt=2Þ cos ðωt=2Þ 1 þ px0 þ  py0 2 sin ðωt=2Þ sin ðωt=2Þ sin ðωt=2Þ sin ðωt=2Þ  9 8  cos ðωt=2Þ sin ðωt Þ  sin ðωt=2Þ cos ðωt Þ sin ðωt=2Þ > > > > p þ > > = < y0 sin ð ωt=2 Þ sin ð ωt=2 Þ 1   2> > sin ðωt=2Þ sin ðωt Þ  þ cos ðωt=2Þ cos ðωt Þ cos ðωt=2Þ > > > > ; : þpx0  sin ðωt=2Þ sin ðωt=2Þ   9 8 px0 cos ðωt=2Þð1  cos ðωt ÞÞ py0 cos ðωt=2Þ sin ðωt Þ > > > > þ  > > = < mω mω sin ðωt=2Þ sin ðωt=2Þ mω   2 > py0 sin ðωt=2Þð cos ðωt Þ  1Þ px0 sin ðωt=2Þ sin ðωt Þ > > > > > ; : þ mω mω sin ðωt=2Þ sin ðωt=2Þ  h i mω ωt i py0 ¼ i yi cot þ xi 2 2 ðG:19Þ

Thus, we have seen that the x- and y- motions are coupled, whereas the motion in the z-direction is characterized by a constant acceleration. We continue by manipulating (G.16) and (G.17): r 2 ¼ x2 þ y2  2 n A ¼ ½ cos φ sin ωt þ sin φð cos ωt  1Þ2 þ ½ cos φð cos ωt  1Þ ω  sin φ sin ωt2 g i  2  2 h A A ð cos ωt  1Þ2 þ sin 2 ωt ¼ ½2ð1  cos ωt Þ ¼ ω ω i  2 h  2πt A 2 1  cos ¼ T ω rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2πt v ) rðt Þ ¼ ⊥0 2 1  cos ω T

ðG:20Þ

It is clear that whenever t ¼ nT for n¼integer, r(t) is zero.11 Thus, if ti ¼ nT, the x and y impact coordinates are both zero. The experimenter will have to disregard these events, because they are indeterminate (they do not uniquely define an initial momentum vector). Equation (G.20) also implies we must design the spectrometer so that the detector radius rD > 2v⊥0/ω ensures that all the charged particles strike the detector [c.f. (G.23) below].

11

Note that this result could also be obtained by evaluating equations (G.16) and (G.17) at t ¼ nT.

Appendixes

315

Geometric Analysis Notice that, from (G.20), the projection of the ion trajectory onto the x-y plane is a circle, which can be written as v pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v rðt Þ ¼ ⊥0 2ð1  cos ωt Þ ¼ ⊥0 ω ω

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi     ωt  ωt v  2  2 sin 2 ¼ 2 ⊥0  sin  ðG:21Þ ω 2 2

A projection of a typical ion trajectory onto the x-y plane is shown in Fig. G.2. The view in Fig. G.2 is from behind the ion detector looking back (in the negative zdirection) toward the IR. From this point of view, the ion is traveling in a clockwise direction around the circle. From the figure (remembering that ϕ is measured clockwise from the x-axis), jϕ0 j ¼

ωt  ϕ0 2

ðG:22aÞ

ϕ þ ðϕ0 Þ ¼ 2π ) ϕ0 ¼ ϕ  2π cos ϕ0 ¼

ðG:22bÞ

x r

sin ϕ0 ¼ 

ðG:22cÞ

y r

ðG:22dÞ

We also have v⊥ ¼ Rω ) R ¼

Fig. G.2 Projection of a typical ion trajectory onto the x-y plane. Adapted from [151] and reproduced with permission from IOP publishing; permission conveyed through Copyright Clearance Center, Inc

v⊥ v⊥0 ¼ ω ω

ðG:23Þ

y ®

½p^0½

f⬘ r R

f0 x

wt 2

316

Appendixes

Now we have enough information to proceed px0

py0

h  i ωt ¼ p⊥0 cos ϕ0 ¼ p⊥0 cos  ϕ0 h  2   i x ωt y ωt ¼ p⊥0 cos cos 1 cos þ sin sin 1 sin 2 i h r  2  i h  r ωt ωt x ωt y ωt mv ¼ mi v⊥0 cos  y sin  sin ¼ i ⊥0 x cos 2 2 r 2 h r  2  r i ωt ωt mv ¼ v  i ⊥0ωt  x cos  y sin 2 2 2 ⊥0  sin  ω h2 i  mω ωt i ) px0 ¼ i xi cot  yi 2 2 ðG:24Þ h  i ωt ¼ p⊥0 sin ϕ0 ¼ p⊥0 sin  ϕ0 2 h     i x ωt y ωt ¼ p⊥0 cos cos 1 sin  sin sin 1 cos 2 i h r  2  i h  r ωt ωt x ωt y ωt mv ¼ mi v⊥0 sin þ y cos þ cos ¼ i ⊥0 x sin 2 2 r 2 h r  2  r i ωt ωt mv ¼ v  i ⊥0ωt  x sin þ y cos 2 2 2 ⊥0  sin  ω h2 i  mω ωt i ) py0 ¼ i yi cot þ xi 2 2 ðG:25Þ

as before. Because the cyclotron period scales linearly with the mass of the ion, as a practical matter the ion will not be much affected by the magnetic field before it hits the ion detector. Therefore, we may neglect the influence of the magnetic field on the recoil ions and rewrite the last two equations of (G.4) as v_ y ¼ 0 and v_ x ¼ 0

ðG:26Þ

If we now include the constant initial velocity in the positive y-direction that all the particles will have by their being part of a collimated, fast-moving jet, vjet, we can solve the first of equations (G.26) as Zvy v_ y ¼ 0 )

dv0y

Zy ¼ 0 ) vy ¼ vyinitial ¼ vy0 þ vjet )

vyinital

0

  ) yðt Þ ¼ vy0 þ vjet t which means

  dy ¼ vy0 þ vjet 0

Zt

dt 0

0

ðG:27Þ

Appendixes

317



mi y ¼ mi vy0 þ mi vjet



  mi yi  vjet t ) py0 ¼ ti

ðG:28Þ

We also have xðt Þ ¼ vx0 t

ðG:29Þ

mi xi ti

ðG:30Þ

which leads to px0 ¼

Note that (G.27) and (G.29) imply that the ion trajectory lies in the plane given by xð t Þ ¼

vx0 y vy0 þ vjet

ðG:31Þ

Because we now have cylindrical symmetry, we are free to reorient the x-axis so that vx0 ¼ 0. In that case, we can rewrite (G.27) as yðt Þ ¼ v0 sin θ0 t þ vjet t

ðG:32Þ

We combine (G.32) with (G.5), eliminating the common TOF to show that the ion trajectory is a parabola (as suggested in Fig. G.1): z ð yÞ ¼

qE0 v0 cos θ0 2 yþ  2 y v0 sin θ0 þ vjet 2m v0 sin θ0 þ vjet

ðG:33Þ

Set z( y) ¼ Zr in (G.5) to find the TOF of the ion:

ti ¼

v0 cos θ0 þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðv0 cos θ0 Þ2 þ 2qEm0i Z r qE 0 =mi

ðG:34Þ

where we have carefully chosen the positive root of the radical to ensure the TOF is positive. The good news is that all the factors under the radical are inherently positive, so there is no chance of having an imaginary TOF. The y-coordinate of the ion’s impact on the recoil ion detector can now be found by evaluating (G.32) at the time indicated in (G.34):

318

Appendixes

y

¼ ¼

v0 sin θ0 t i þ vjet t i rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 2 2qE 0 Z r 2 6v0 cos θ0 þ ðv0 cos θ0 Þ þ mi 7 7 v0 sin θ0 6 4 5 qE 0 =mi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 2qE0 Z r 2 6v0 cos θ0 þ ðv0 cos θ0 Þ þ mi 7 7 þ vjet 6 4 5 qE 0 =mi 2

 v2 sin 2θ0 v0  þ sin θ0  vjet cos θ0 ¼  0 2qE0 =mi qE0 =mi

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2qE 0 Z r ðv0 cos θ0 Þ2 þ mi ðG:35Þ

Electron Detection We can apply the same analysis used to find the components of the ion initial momentum vector to find the components of the initial momentum vectors of electrons produced in the Coulomb explosion. For example, from the vantage point of a position behind the electron detector, looking back toward the IR (in the positive z-direction), the electron describes a clockwise circle (see Fig. G.3). From the figure, ωt 2 x cos ϕ ¼ r y sin ϕ ¼ r

ϕ0 ¼ ϕ 

ðG:36aÞ ðG:36bÞ ðG:36cÞ

The rest of analysis is the same as that which led to (G.24) and (G.25), giving us,

Fig. G.3 Projection of a typical electron trajectory onto the x-y plane. Adapted from [151] and reproduced with permission from IOP publishing; permission conveyed through Copyright Clearance Center, Inc.

wt/2

R y

®

½p^0½

x

f0

f

r

Appendixes

319

px0 ¼

h  i me ω ωt e þ ye xe cot 2 2

ðG:37Þ

py0 ¼

h  i me ω ωt e  xe ye cot 2 2

ðG:38Þ

where te is the time of flight of the electron, me is the mass of the electron, and xe and ye are the x- and y-coordinates of the electron’s impact on the electron detector, respectively. We cannot neglect the influence of the magnetic field on the electrons (in fact, the magnetic field is there confine the electrons), but because the magnetic field does to !  !  no work (i.e.,  v ⊥  ¼  v ⊥0 ), we can calculate the TOF for the electrons as we did for the ions above. First, we use (G.5) to find the time it takes the electron to travel from the IR to the coordinate z ¼  Ze,

tZe ¼

v0 cos θ0 þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðv0 cos θ0 Þ2 þ 2qEm0e Z e qE 0 =me

ðG:39Þ

Here q refers to the electron charge, and me is the electron mass. From above, ðvz  vz0 Þ ¼

qE0 qE t ) vz ¼ vz0 þ 0 t m m

ðG:40Þ

Therefore, the z-component of the speed of the electron when it reaches the end of the acceleration region will be vz jtZ e ¼ vz0 þ

qE0 t me Z e

¼ v0 cos θ0 þ

2

qE 0 4 me

v0 cos θ0 þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 ðv0 cos θ0 Þ2 þ 2qEm0e Z e 5 qE0 =me

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2qE 0 Z e ¼ ðv0 cos θ0 Þ2 þ me

ðG:41Þ

The time td required for the electron to travel from z ¼  Ze to z ¼  (Ze + Zd) is Z d =vz jtZ e . The TOF of the electron is t Z e þ t d : te ¼ tZ e þ td ¼

v0 cos θ0 þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðv0 cos θ0 Þ2 þ 2qEm0e Z e qE 0 =me

Zd ffi þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðv0 cos θ0 Þ2 þ 2qEm0e Z e

ðG:42Þ

320

Appendixes

Including a drift region in the spectrometer is a way to account for the fact that the IR is not truly a point but as a practical matter has a finite size for which one must account. An electron that originates at a point some small distance h (along the zaxis) from the origin (so that z ¼ h + Ze) will have a TOF given by

t e ðzÞ ¼

v0 cos θ0 þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0z ðv0 cos θ0 Þ2 þ 2qE me qE 0 =me

Zd þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0z ðv0 cos θ0 Þ2 þ 2qE me

ðG:43Þ

Because the distance h is small, we can expand te(z) in a Taylor series about the point Ze:  d½t ðzÞ t e ðzÞ ¼ t e ðZ e Þ þ h e  þ⋯ ðG:44Þ dh z¼Z e We wish to ensure that the TOF deviation ΔTOF  te(z)  te(Ze) is zero: ΔTOF  t e ðzÞ  t e ðZ e Þ ¼

 1 X hn dn ½t e ðzÞ ¼0 n! dzn z¼Z e n¼1

ðG:45Þ

where we have noted that dh ¼ dz. From (G.45), it is clear that, to achieve our aim, all the indicated derivatives must simultaneously be zero. As it turns out, this is not possible (given the spectrometer design shown above), so we settle for an approximate solution by requiring only the first derivative be zero:  d½t e ðzÞ ¼0 dz z¼Z e 2 3 Z d qE0 =me qE0 =me 1 h i3=2 5 ) 4qE =me qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2qE 0 z 0z 2 0 ðv0 cos θ0 Þ2 þ 2qE ð v cos θ Þ þ 0 0 me me

ΔTOF 

¼0 z¼Z e

ðG:46Þ We make the further approximation that the z-component of the electron’s initial velocity, vz0, is small compared to the velocity the electron gains while in the acceleration region: v0 cos θ0