Structural Studies of Surfaces (Springer Tracts in Modern Physics) 3540109641, 9783540109648

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Springer Tracts in Modern Physics 91

Editor: G. HShler Associate Editor: E.A. Niekisch Editorial Board: S.FI0gge H.Haken J.Hamilton H. Lehmann W.Paul

Springer Tracts in Modern Physics 68* Solid-Stat~ Physics With contributions by D. B&uerle, J. Behringer, D. Schmid 69* Astrophysics

With contributions by G. Borner, J. Stewart, M. Walker

70* Quantum Statistical Theories of Spontaneous Emission and their Relation to Other Approaches By G. S. Agarwal 71

Nuclear Physics With contributions by J. S. Levinger, P. Singer, H. 0berall

72 Van der Waals Attraction:

Theory of Van der Waals Attraction By D. Langbein

73 Excitons at High Density Edited by H. Haken, S. Nikitine. With contributions by V. S. Bagaev, J. Biellmann, A. Bivas, J. Goll, M. Grosmann, J. B. Grun, H. Haken, E. Hanamura, R. Levy, H. Mahr, S. Nikitine, B. V. Novikov, E. I. Rashba, T. M. Rice, A. A. Rogachev, A. Schenzle, K. L. Shaklee 74 Solid-State Physics With contributions by G. Bauer, G. Borstel, H. J. Falge, A. Otto 75 Light Scattering by Phonon.Polaritons

By R. Claus, L. Merten, J. Brandm011er

76 Irreversible Properties of Type II Superconductors

By H. UIImaier

77 Surface Physics With contributions by K. M011er, P. Wi6mann 78 Solid.State Physics With contributions by R. Dornhaus, G. Nimtz, W. Richter 79 Elementary Particle Physics With contributions by E. Paul, H. Rollnick, P. Stichel 80* Neutron Physics With conlributions by L. Koester, A. Steyerl 81 Point Defects in Metals I: Introduction to the Theory 2nd Printing By G. Leibfried, N. Breuer 82 Electronic Structure of Noble Metals, and Polariton.Mediated Light Scattering With contributions by B. Bendow, B. Lengeler 83 Electroproduction at Low Energy and Hadron Form Factors By E. Amaldi, S. P. Fubini, G. Furlan 84 Collective Ion Acceleration With contributions by C. L. Olson, U. Schumacher 85 Solid Surface Physics With contributions by J. HOlzl, F. K. Schulte, H. Wagner 86 Electron-Positron Interactions

By B. H. Wiik, G. Wolf

87 Point Defects in Metals I1: Dynamical Properties and Diffusion Controlled Reactions With contributions by P. H. Dederichs, K. Schroeder, R. Ze~ler 88 Excitation of Plasmons and Interband Transitions by Electrons

By H. Raether

89 Giant Resonance Phenomena in Intermediate.Energy Nuclear Reactions By F. Cannata, H. 0berall 90* Jets of Hadrons

By W. Hofmann

91 Structural Studies of Surfaces With contributions by K. Heinz, K. M011er,T. Engel, and K. H. Rieder 92 Single-Particle Rotations in Molecular Crystals

By W. Press

93

Coherent Inelastic Neutron Scattering in Lattice Dynamics

94

Exciton Dynamics in Molecular Crystals and Aggregates V. M. Kenkre and P. Reineker

By B. Dorner

With contributions by

* denotes a volume which contains a Classified Index starting from Volume 36.

Structural Studies of Surfaces Contributions by K. Heinz K. M011er T. Engel

K.-H. Rieder

With 120 Figures

Springer-Verlag Berlin Heidelberg New York 1982

Professor Dr. Klaus Heinz Professor Dr. Klaus M011er Institut fQr Angewandte Physik, Universit&t Erlangen, Erwin-RommeI-StraBe 1, D-8520 Erlangen, Fed. Rep. of Germany Dr. Thomas Engel Department of Chemistry, University of Washington, Seattle, WA 98195, USA Dr. Karl-Heinz Rieder IBM Zurich Research Laboratory, CH-8803 RQschlikon, Switzerland

Manuscripts for publication should be addressed to:

Gerhard HOhler Institut for Theoretische Kernphysik der Universit&t Karlsruhe Postfach

6380, D-7500 Karlsruhe

1, F e d . R e p . o f G e r m a n y

Proofs and all correspondence concerning papers in the process of publication should be addressed to:

Ernst A. Niekisch

Haubourdinstrasse6, D-5170 J01ich,Fed. Rep. of Germany

ISBN 3-540-10964-1 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-10964-1 Springer-Verlag New York Heidelberg Berlin Library of Congress Cataloging in Publication Data. Main entry under title; Structural studies of surfaces. (Springer tracts in modern physics; 91). Includes bibliographical references and index. Contents: LEED intensities: experimental progress and new possibilities of surface structure determination / by K. Heinz and K. M011erStructural studies of surfaces with atomic and molecular beam diffraction/'E Engel and K.-H. Rieder. 1. Surfaces (Physics) -Addresses, essays, lectures. 2. Diffraction- Addresses, essays, lectures. I. Heinz, Klaus. I1. Series QC1.$797. voL 91 [QC173.4S94] 539s[530.4] 81-9327AACR2 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, reuse of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under w 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to ,,Verwertungsgesellschaft Wort", Munich. 9by Springer-Verlag Berlin Heidelberg 1982 Printed in Germany The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations end therefore free for general use. Offset printing and bookbinding: BrJhlsche Universit~.tsdruckerei, Giessen 2153/3130 -- 54321 0

Contents

LEED Intensities - Experimental Progress and New Possibilities of Surface Structure Determination By K, Heinz and K. M~ller. With 29 Figures I.

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

I

2. S t r u c t u r a l Determination by Comparison of Experimental and Calculated Intensities

............................................................

3

2.1

Comparison o f T h e o r e t i c a l and Experimental Spectra . . . . . . . . . . . . . . . .

4

2.2

Comparison of Experimental Spectra from D i f f e r e n t Measurements . . . .

7

2.3

Sources o f Experimental E r r o r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.4

C l a s s i c a l Methods f o r I n t e n s i t y Measurements . . . . . . . . . . . . . . . . . . . . . .

3. New Experimental Methods f o r I n t e n s i t y Data C o l l e c t i o n . . . . . . . . . . . . . . . . .

9 13 18

3.1

Photographic Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

3.2

TV Computer Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

3.2.1

Data A c q u i s i t i o n Rate Lower than TV Rate . . . . . . . . . . . . . . . . . . .

21

3.2.2

Data A c q u i s i t i o n Rate Equal to TV Rate . . . . . . . . . . . . . . . . . . . . .

27

4. Examples f o r R e l i a b l e I n t e n s i t y Data Obtained by the New Methods . . . . . . . 4.1

33

I n f l u e n c e o f Sample Misalignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

4.2

I n f l u e n c e of Background S u b t r a c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

4.3

I n f l u e n c e of Residual Gas Adsorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

4.4

I n f l u e n c e o f Adsorbate Decomposition and Desorption . . . . . . . . . . . . . . .

38

5. New P o s s i b i l i t i e s

Using Modern ! n t e n s i t y Measurement Methods . . . . . . . . . . .

5.1

Integral Intensities

5.2

Spot P r o f i l e s of Rapidly Varying Surface Systems . . . . . . . . . . . . . . . . . .

of Rapidly Varying Surface Systems . . . . . . . . . . .

39 39 42

5.3

Extension of I n t e n s i t y Measurements to Varying Temperature . . . . . . . .

43

5.4

Extension o f I n t e n s i t y Measurements to the Medium-Energy Range . . . .

45

6. Summary and Outlook

...................................................

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

48 49

VI

Structural Studies of Surfaces with Atomic and Molecular Beam Diffraction By T. Engel and K.--E. Rieder. With 91 Figures I.

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

2. The P a r t i c l e - S u r f a c e I n t e r a c t i o n P o t e n t i a l

55

...............................

57

2.1

Physical Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

2.2

Short Survey of T h e o r e t i c a l E f f o r t s

58

2.3

Determination of the Surface P o t e n t i a l from Bound-State Energy Data

3. Quantum Theory of P a r t i c l e D i f f r a c t i o n

.................................

..................................

65 .

71

3.1

The Corrugated Hard-Wall Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

3.2

Diffraction

72

Condition and Ewald Construction . . . . . . . . . . . . . . . . . . . . . . . .

3.3

Calculation of Diffraction

3.4

C a l c u l a t i o n of I n t e n s i t i e s - R a y l e i g h

Intensities-General

Method . . . . . . . . . . . . . .

Hypothesis . . . . . . . . . . . . . . . . . . . . .

3.4.1

The GR Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.4.2

The Eikonal Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75 78 79 79

3.5

Calculation of Intensities-lterative

Series . . . . . . . . . . . . . . . . . . . . . . . .

3.6

A Few I l l u s t r a t i v e

3.7

The I n v e r s i o n Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

90

3.8

E f f e c t s Due to the Softness of the Repulsive P o t e n t i a l

92

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83 86

..............

4. I n e l a s t i c S c a t t e r i n g of Atoms from Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93

4.1

The Dependence of the S c a t t e r i n g on the Time Scale o f the I n t e r a c t i o n

93

4.2

The Debye-Waller Factor in the Time-Dependent I n t e r a c t i o n Regime . . . .

95

4.3

The Size E f f e c t in the Debye-Waller Factor f o r Atom S c a t t e r i n g . . . . . .

97

4.4

Experimental I n v e s t i g a t i o n s of the Debye-Waller Factor f o r AtomSurface S c a t t e r i n g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5. I n f l u e n c e o f the A t t r a c t i v e Part of the P o t e n t i a l on D i f f r a c t i o n sities

98 Inten-

................................................................... Intensities

103

5.1

M o d i f i c a t i o n s f o r the C a l c u l a t i o n of D i f f r a c t i o n

5.2

Bound Surface States and Resonant T r a n s i t i o n s . . . . . . . . . . . . . . . . . . . . . . .

........

5.3

Theory of Atom S c a t t e r i n g from a Corrugated Hard Wall w i t h an

103 105

A t t r a c t i v e Well . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

111

I n e l a s t i c E f f e c t s in Resonant S c a t t e r i n g . . . . . . . . . . . . . . . . . . . . . . . . . . . .

114

6. Experimental Aspects o f Gas-Surface S c a t t e r i n g . . . . . . . . . . . . . . . . . . . . . . . . . . .

118

5,4

6.1

Requirements on an Apparatus to Perform Gas-Surface S c a t t e r i n g Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

118

Vll 6.2

Beam Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

!19

6.2.1

E f f u s i v e Beam Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

119

6.2.2

Nozzle-Beam Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

119

6.3

Beam Energy V a r i a t i o n f o r E f f u s i v e and Nozzle-Beam Sources . . . . . . . . . .

123

6.4

The Design o f Nozzle-Beam Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

124

6,5

Molecular-Beam D e t e c t o r s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

126

6.6

Detector Rotation ...................................................

128

6.7

Sample M a n i p u l a t o r s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

129

6.8

Beam-Modulation Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

131

6.9

Experimental Systems f o r D i f f r a c t i v e

133

S c a t t e r i n g from Surfaces . . . . . . .

6,10 The I n f l u e n c e o f the T r a n s f e r Width o f the Apparatus and o f Surface P e r f e c t i o n on Measured I n t e n s i t i e s 7. S t r u c t u r a l

Investigations

..................................

on Surfaces o f l o n i c C r y s t a l s . . . . . . . . . . . . . . . . . .

137 140

7.1

Diffraction

Studies on L i F ( I O 0 )

.....................................

140

7,2

Diffraction

Studies on NiO(IOO) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

146

7.3

Diffraction

from Other I o n i c M a t e r i a l s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

147

8. S t r u c t u r a l

Investigations

on Semiconductor Surfaces . . . . . . . . . . . . . . . . . . . . . .

8.1

Helium-Diffraction

Studies on S i ( 1 1 1 ) and Si(100)

8.2

Helium D i f f r a c t i o n

from GaAs(110) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8,3

Diffraction

from G r a p h i t e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

154

8.4

Diffraction

from Layer Compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

155

8.5

Helium-Diffraction

9. S t r u c t u r a l

...................

148

Studies from Other Surfaces . . . . . . . . . . . . . . . . . . . . . .

Investigations

on Metal Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

152

156 157

9.1

Introduction

9.2

Helium and Hydrogen D i f f r a c t i o n

9.3

Helium D i f f r a c t i o n

from f c c ( 1 1 0 ) and bcc(112) Planes . . . . . . . . . . . . . . . .

158

9.4

Helium D i f f r a c t i o n

from Stepped Metal Surfaces . . . . . . . . . . . . . . . . . . . . . .

160

10. S t r u c t u r a l

........................................................

148

from Close-Packed Metal Surfaces . . . .

Studies on Adsorbate-Covered Surfaces . . . . . . . . . . . . . . . . . . . . . . . . .

10.1 I n t r o d u c t i o n

........................................................

10.2 Hydrogen A d s o r p t i o n on Ni(110)

......................................

157 157

162 162 163

10.3 Oxygen A d s o r p t i o n on Ni(110) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

171

10.4 Oxygen A d s o r p t i o n on Cu(110) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

173

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

174

LEED Intensities - Experimental Progress and New Possibilities of Surface Structure Determination K. Heinz and K. M011er

1. Introduction The structure of a crystal surface is one of i t s important properties. Other quantities,

such as the surface density of states, the electronic work function, and

chemical bonding or chemical reactions in the presence of adsorbates, are correlated to structure parameters. I t is therefore one of the basic aims of surface science to investigate the surface structure, i . e . , the geometric arrangement of atoms in the f i r s t

few layers of a c r y s t a l .

Among the techniques sensitive to surface structure is d i f f r a c t i o n of low-energy particles with De Broglie wavelenghts of the order of the l a t t i c e constant resulting from ionic scattering (e.g., / 1 . 1 - 3 / ) and atomic or molecular scattering (e.g., / 1 . 4 , 5 / ) . Methods more sensitive to local order, such as angular-resolved photoemission spectroscopy (e.g., / 1 . 6 , 7 / ) or angular resolved Auger-electron spectroscopy (e.g., / 1 . 8 - 1 3 / ) , are also in progress. Low-energy electron d i f f r a c t i o n (LEED), however, is the oldest and most commonly used method for surface structure determination (e.g., /1.14-30/). A more or less coherent beam of electrons in the energy range of about 20-600 eV, i . e . , with wavelengths between 3 ~ and 0.5 ~, impinges on the c r y s t a l l i n e surface. The e l a s t i c a l l y backscattered electrons, separated from an i n e l a s t i c background by retarding f i e l d grids, display the d i f f r a c tion pattern of the surface on a luminescent screen. Both electron-beam generation by an electron gun and detection of the d i f f r a c t i o n pattern can be e a s i l y performed by commercially available UHV equipment. In spite of the s i m p l i c i t y of the experimental arrangement, only since about 1974 have the number of surface structure determinations by LEED been notably increasing as shown in Fig.1.1. The reason for this comparatively late s t a r t is the complexity of low-energy electron scattering. The geometry of the d i f f r a c t i o n pattern is determined by the translational symmetry and the p e r i o d i c i t y of the surface layers only, and so shape and size of the surface unit cell can be deduced by simple arguments. However, essential details of the unit c e l l , i . e . , the geometric arrange-

N/a 60-

50

40

30

20

10

70

71

72

73

74

75

76

77

78 79 year

Fig.1.1. Development of the number of publ i c a t i o n s appearing per year on surfacestructure determination by LEED during the l a s t decade as taken from the Surface and Vacuum Physics Index

ment of i t s atoms, can be determined only by independent model calculations of the i n t e n s i t i e s of d i f f r a c t e d beams and subsequent comparison with the experiment. This holds also for the mutual orientation between adjacent surface layers. Unfortunatel y , the interpretation of d i f f r a c t i o n i n t e n s i t i e s is highly complicated by multiple d i f f r a c t i o n , and kinematic or f i r s t Born approximation is not v a l i d as in X-ray d i f f r a c t i o n . In most cases eve~ the extension to higher Born approximations does not lead to convergence, and i t was only in the early 1970's that suitable numerical methods with tolerable computer e f f o r t were developed, becoming improved in the following years (for comprehensive reviews see / I . 3 1 , 3 2 / ) . So the recent increase in structural determinations as shown in Fig.1.1 results from computational progress. The peak occurring at about ~977 is believed to be at least p a r t i a ] l y due to repeated structure analyses and therefore should be reduced to some extent to give a more reasonable increase in a c t i v i t y . Up to now, more than a hundred structural determinations have been performed f o r clean and adsorbate-covered surfaces ( f o r reviews see /1.33,34/). Today computer programs are available which can be applied also by those experimentalists who do not wish to consider theoretical d e t a i l s . However, only comparatively simple systems can be treated successfully and they s t i l l

necessitate considerably large amounts of computer time, allowing only a

few structural or nonstructural parameters to be varied. Nevertheless, a great deal of surface models with an unit cell area below about 25 ~2 and containing not more than 4 atoms can be calculated y i e l d i n g an accuracy of atomic positions of about 0.1 ~. Even so, this or determinations of even higher

accuracy can only be obtained i f s u f f i c i e n t l y accurate measurements are available f o r comparison. In contrast to the s i t u a t i o n a few years ago, theory has reached a stage where the improvement of experimental r e l i a b i l i t y

becomes important. I t is

the purpose of t h i s paper to report the most recent developments in data c o l l e c t i o n and data handling in LEED. We s t a r t the next chapter with a description of the current standard of theoryexperiment f i t .

I t w i l l be demonstrated in more d e t a i l why more r e l i a b l e measure-

ments are necessary. In Chap.3 several new methods f o r LEED i n t e n s i t y measurements are reported which promise more accuracy by fast and convenient measurements. The f o l l o w i n g chapter discusses results obtained with the new methods as compared to the usual measurements of i n t e n s i t y spectra, and the l a s t chapter concentrates on new p o s s i b i l i t i e s which are feasible by the new methods.

2. Structural Determination by Companson of Expenmental and Calculated Intensities Experimental data f o r the determination of surface structures by LEED come from the i n t e n s i t i e s of d i f f e r e n t d i f f r a c t i o n spots recorded as a function of energy or angle of incidence of the primary electrons impinging on the sample. So sets of energy-int e n s i t y spectra I(E) or r o t a t i o n diagrams I(@,~) are usually collected. In some cases also so-called i s o - i n t e n s i t y maps are presented / 2 . 1 , 2 / which contain intens i t y contour l i n e s in a plane whose axes are defined by the energy and the polar angle of incidence. The spectra have to be matched by model calculations because no d i r e c t method f o r model determination e x i s t s . There is no evident reason why one of the methods, energy or angle v a r i a t i o n , should be favored. I(E) measurements can be generated more e a s i l y because only a voltage has to be varied, while I ( 8 , ~ ) curves at constant energy have the advantage that energy dependences of the scattering potential and of the optical potential do not enter into the c a l c u l a t i o n . In both cases, however, the problem s t i l l

remains of how to properly compare calcu-

lated and experimental spectra with detailed features in order to develop the surface model which best approximates r e a l i t y . F i r s t , a measurement of the agreement between two spectra must be defined since comparison by simple inspection is too inaccurate. We w i l l stress t h i s point in the f o l l o w i n g section and demonstrate the necessity for precise measurements. In Sect.2.2 measurements of d i f f e r e n t laboratories on the same surface structure w i l l be compared. I t w i l l be shown that the agreement between d i f f e r e n t experimental spectra

of the same surface can be much worse than the best f i t

of experimental and calcu-

lated spectra reached by variation of surface-structure parameters. As the great majority of existing calculated or experimental spectra are I(E) curves, we w i l l concentrate on that class of data.

2.1

Comparison of Theoretical and Experimental Spectra

The simplest description of the difference of two sets of data by a least mean square f i t

does not consider the peak structure of i n t e n s i t y spectra. More suitable

c r i t e r i a , e.g., the average deviation of peak positions, have been t r i e d . Though this measure was successful, at least in some cases / 2 . 3 - 5 / , i t has the disadvantage of neglecting the r e l a t i v e i n t e n s i t i e s of d i f f e r e n t peaks as well as t h e i r widths. Other more-or-less simple c r i t e r i a have been t r i e d , also with limited success / 2 , 5 . 7 / . However, a proposal of ZANAZZI and JONA / 2 . 8 / is more sophisticated. The so-called r e l i a b i l i t y

or r - f a c t o r is defined in order to emphasize the struc-

ture of the spectra rather than t h e i r peak heights. Therefore, the i n t e n s i t y spectra to be compared, II(E) and 12(E), are normalized with respect to t h e i r average values, and , and only the differences of the d e r i v a t i v e s , I I = dll/dE and I~ = dl2/dE, are considered:

r

=

iI

lr

A~/EW(E) dE.

The integral is taken over the energy range AE which defines the range of overlap between the spectra, and W(E) is a factor which additionaly increases the weight of p a r t i c u l a r l y narrow maxima using the second deriatives IV and 12 ,

It -

II

W(E) = I ~ +

i > D and z ~(R) V(z) = ~ for z ~ ~(B)

(3.1)

with z denoting the direction of the surface normal. The f u l l crystallographic information obtainable from a d i f f r a c t i o n experiment with atomic beams is contained in the two-dimensional corrugation function ~(R) = ~(x,y) with periods ~I and ~2" As outlined in Chap.2, ~(R) is a replica of the spatial modulation of the valence electron density of the scattering surface. Due to the fact that in r e a l i t y the repulsive part of the potential does not have i n f i n i t e slope, ~(R) is in principle a function of the energy Ei of the incoming p a r t i c l e s . However, in the range of energies applicable in actual d i f f r a c t i o n experiments (Ei ~ 20-100 meV), both the

72 maximum amplitude Cm of ~(R) as well as its shape w i l l be affected only s l i g h t l y (Sects.2.2,3.9). Due to its two-dimensional p e r i o d i c i t y ,

the "hard-wall corrugation function"

~(R) can be conveniently written as a Fourier series r

: Z ~G eiw G -

'

(3.2)

where the reciprocal l a t t i c e vectors G = J~1 + ~ 2 lated to the u n i t - c e l l

( j ' ~ = O, • I, • 2 . . . ) are re-

vectors ~I and ~2 of the direct l a t t i c e through

apbq = 2~6pq

(3.3)

with ~pq denoting the Kronecker symbol. Equation (3.3) implies that ~I and b2 are normal to ~2 and ~I' respectively. With ~ being the angle between ~I and ~2' the lengths of bl and ~2 are determined by 2~ [~I I = alsinS

2~ Ib21 = a2sin6

(3.4)

The relation between direct and reciprocal l a t t i c e s is i l l u s t r a t e d

y

(o)

in Fig.3.1.

y'

(b)

Fig.3.1a,b. Graphical representation of the relation between a surface l a t t i c e and the corresponding reciprocal l a t t i c e (b) according to (3.3,4)

3.2

(a)

Diffraction Condition and Ewald Construction

Consider a beam of particles with energy Ei impinging on a surface at an angle s as measured from the surface normal. The particle energy E. is related to the beam 1

73 wavelength ~. according to the de Broglie r e l a t i o n l x.

h

-

(3.5) l

with m denoting the mass of the p a r t i c l e s , and h being Planck's constant. The wavevector k. is determined by the wavelength and the d i r e c t i o n of the beam -I

2TF

ki _ i k i l :~-T. 1

(3.6)

We separate k. into components p a r a l l e l and perpendicular to the surface: -I

(3.7)

~i z (~' kiz) = ( k i s i n O i ' - k i c ~ The well-known Bragg condition f o r d i f f r a c t i o n

to occur is

K + _G = KG In the d i f f r a c t i o n

(3.8) process, the t o t a l energy of the p a r t i c l e s remains unchanged,

k. 2 = ~ G2 -1

(3.9)

In w r i t i n g (3.7), we have again separated the wavevectors f o r the d i f f r a c t e d beams into p a r a l l e l and perpendicular components: ~G = (~G' kGz) = (kisinSG' kic~

(3.10)

Equation (3.9) r e s t r i c t s the number of reciprocal l a t t i c e vectors f o r which d i f f r a c tion can occur to the f i n i t e

set {F} f o r which

kFz 2 = ki 2_ (K+ F)2 > 0

(3.11)

In the case of a one-dimensional corrugation function ~(x) with period a f o r which the G vectors are given by G = j ( 2 ~ / a )

(j = O, • I , • 2), (3.8) y i e l d s

X. sinO. = sine. + j --Zl j 1 a

(3.12)

The conditions (3.7,8) can be represented graphically with the Ewald construction shown for the example of a one-dimensional surface in Fig.3.2.

74 G_(j) 8

7

6

5

4

5

2

0

2

z

qo, /k-~-k-~:9~. -';'G-(~

Fig.3.2. The Ewald construction for diffraction from surfaces

Ewald

Fig~3.3. Specification of the d i f fraction angles 0G and @G in accordance with possible motions of a goniometer (see Fig.6.11)

For two-dimensional corrugation functions ~(x,y), two angles specifying the scattering direction have to be determined. According to the motions usually possible with a goniometer (Fig.6.11), we choose these angles in the following manner (Fig. 3,3). ~G describes the angle by which the detector has to be rotated in the scattering plane, which is spanned by the wave vector -ik" and the surface normal', ~G is measured from the surface normal. @Gdenotes the angle through which the detector has to be moved out of plane. For an arbitrary two-dimensional lattice with unitcell vectors ~I and a2 comprising the angle B, 0G and ~G can be calculated from the following formulae (G = J~1 + ~ 2 ): sin~j~

:

[-

~

-

~ cosy7

(siny + cosy cot~) + a2 s i n a i >~i

(3.13a)

75

I [sinSi + ~i aT j (cosy - siny cot~) + hi sinejs = cos~js

s siny a2

sin6]

(3.13b)

In these equations, y denotes the angle between K and ~I (Fig.3.1a).

3.3

Calculation of Diffraction

A method to calculate d i f f r a c t i o n

Intensities-General

Method

i n t e n s i t i e s on a quantum-mechanical basis for gen-

eral hard corrugated surfaces ((R) and general scattering geometries has been developed in the last few years as a result of the e f f o r t of several groups / 3 . 3 - 9 / . The starting point is the Lippman-Schwinger equation, which can be solved for the hard corrugated potential

(3.1), yielding for the wave function of the incoming and

scattered particles ~(r) = exp[i(KR+kizZ) ] +

unit cell

dR'f(R ') ~ 1-~--expEi(K+G)(R-R'~ exp[ikGzlZ-~(R)l] G kGz (3.14)

with f(R) denoting the density of sources. I t should be mentioned that the numerical value of k. is negative (3.7), and iz koz = - kiz, with koz denoting the z-component of the wave vector of the specular beam G = O. The f a r - f i e l d solution, obtained with z § ~, ~([) : exp[i(KR+kizZ)] + ~ AG exp[i(K+G)R] exp[ikGzZ] G -

(3.15)

contains the (complex) scattering amplitudes AG which determine the scattered intensities PG through IkGzl PG = - ~ IAw 2

(3.16)

As all the scattering from the r i g i d hard wall is e l a s t i c , the diffracted intensities have to satisfy the u n i t a r i t y condition PF = I F -

(3.17)

Comparison of (3.14) and (3.15) yields for the scattering amplitudes AG = I f dR'f(R') expE-i(K+G)R:] exp[-ikGz((R') ] kGz _ _ _

(3.18)

76 To be able to calculate AG from (3.18), knowledge of the source function f(R) is required. With the boundary condition ~[_R,z

: ~(R)]

~= 0

(3.19)

,

which simply expresses the fact that the particles cannot penetrate the surface, we obtain from (3.14), _ exp[ikiz~(R) ]_

= Z 11_ exp[iw G kGz

I dR'f(R') exp[-i(K_+G)R'] exp[ikGzlC(R)-c(R')l] unit cell (3.20)

This equation has to be solved to obtain the source function f(R). GARCIA and CABRERA /3.7,8/ have proposed a numerical procedure to deduce f(R) from (3.20) called the RR' method, which we shall sketch in the following. We f i r s t multiply both sides of (3.20) by exp(iKR) and exchange summation and integration on the right-hand side: - exp{i[KR+kiz~(R) ] } = f dR'f(R') ~ exp[i(~+w G kGz

)] exp[ikGzl~(R)-~(R')I] "

(3.21)

Now, we substitute the integration by a summation and for simplicity r e s t r i c t ourselves to a one-dimensional corrugation ~(x). Taking 2N equidistant points x n=na/2N within the length a of the elementary c e l l , we obtain a set of 2N linear equations which has to be solved for the 2N unknowns fn' e f(Xn')a/2N' 2N Bn = n '~=I fn'Mnn'

'

(3.22)

whereby the Bn and Mnn, are given by Bn : exp{i [Kna/2N+kiz~(Xn) ] }

(3.23)

Mnn, = ~ kGz exp[i (K+G)(n-n' )a/2N] exp [ikGz I ~(Xn)-~(x n , )I]

(3.24)

and

For calculation of the M ,, the summation has to be performed over a s u f f i c i e n t set nn of G vectors so that Mnn, is "close enough" to the value for G § ~. This can be checked numerically /3.8/. For n = n ' , (3.22) cannot be used, as the summation over the G's diverges. GARCIA and CABRERA/3.7,8/ have derived an analytic expression using the linear approximation ]~(x n) - ~(Xn,) I ~ rc'(Xn)Ilx n - Xn, I , which should read

77 2N exp{i [(K+G)+l~'(Xn) IkGz]a/4N}-1 Mnn, = ~ k G z a i[(K+G)+l~'(Xn)IkGz] (3.25) 1 - e x p { - i [ (K+G) _ _ - I '(Xn)IkGz]a/4N} §

i[(K+w - I ~ ' (Xn)IkGz] With (3.22) solved for the f n ' the scattering amplitudes AG are obtained by using (3.16) in discretized form, 2N AG = kGz I n! I fn exp[-i(K+G)an/2N] exp[-ikGz~(Xn) ]

(3.26)

Model calculations based on this numerical procedure for one-dimensional corrugations of d i f f e r e n t shape and amplitude seem to confirm that the method is applicable to any kind of corrugation with no r e s t r i c t i o n or scattering geometry (ki,@i,~).

concerning the maximum corrugation amplitude

Even corrugations with discontinuities

in ~(x)

and/or d~(x)/dx can be treated /3.10/. Although such corrugations are highly unl i k e l y to play a role in atomic-beam scattering, the same formalism also applies to other problems, for example, the scattering of acoustic waves from solid walls. Examples for the spatial variation of the real and imaginary parts of the source function f ( x ) for d i f f e r e n t l y shaped corrugations ~(x) with increasing amplitude are given in / 3 . 8 / .

I t is worthwhile mentioning that for small corrugation ampli-

tudes, the real part of f(x) is nearly constant with a value of ~kiz, and the imaginary part is very small. The solution of (3.22) requires handling of very large matrices, and even for the one-dimensional model corrugations considered in /3.7-10/ the number 2N had to be taken between 100 and 200. As a consequence, although the method is in principle applicable to two-dimensional corrugations ~(x,y), up to now no such calculation has been reported. The r e l i a b i l i t y

of the calculated i n t e n s i t i e s is usually judged

using the following two c r i t e r i a ,

a) The u n i t a r i t y relation (3.17) has to be veri-

fied within a fraction of a percent, b) The calculation must yield the correct threshold behaviour of the i n t e n s i t i e s :

the intensity PF of a beam F disappearing

at the horizon (GF = 90 ~ , kFz = O) must decrease to zero with a vertical tangent as k i , 6i , or ~ vary so that eF approaches 90 ~ /3.8,11/. observed in the model calculations of Figs.3.7,8.)

(This behaviour can be

78 3.4

Calculation of I n t e n s i t i e s - R a y l e i g h Hypothesis

An appreciable s i m p l i f i c a t i o n of the calculational procedure to obtain the scattering amplitudes AG is achieved by an assumption put forward almost a century ago in Lord RAYLEIGH's f T r s t theoretical investigations of the r e f l e c t i o n of acoustic waves from hard walls / 3 . 1 / . Rayleigh assumed that the f a r - f i e l d solution (3.15) is s t r i c t l y v a l i d a l l the way to the surface. Imposing the boundary condition (3.19) on (3.15), we obtain exp{i [KR+kiz~(R) ] } + ~ AG exp{i [_KR+_GR+kGz~(R)] } = 0 G -

(3.27)

AG exp[ikGz~(R)] exp[iGR] =- exp[ikizC(R) ] G -

(3.28)

or

These equations must be f u l f i l l e d

for every point R at the surface (the two-dimen-

sional p e r i o d i c i t y permits r e s t r i c t i o n to a single surface unit cell in actual calculations) and allows d i r e c t determination of the AG'S without the necessity of knowing the source function f(R) e x p l i c i t l y . Of course, there is a price to pay for this simplifying Rayleigh assumption, which consists in the limited convergence range of (3.28). The l i m i t s of convergence were investigated a n a l y t i c a l l y in /3.1215/. For a one-dimensional sinusoidal corrugation with p e r i o d i c i t y a, I 27 ~(x) = ~ ~mCOST X

,

(3.29a)

the Rayleigh method converges up to Cm = 0.143 a

(3.29b)

For a two-dimensional quadratic corrugation with l a t t i c e constant a, 1 ( 2"rr C(x,y) = ~ Cm cos T x

2"rr ) + cos T y

,

(3.30a)

the l i m i t of convergence is given by ~m = 0.188 a

(3.30b)

Both (3.29) and (3.30) are written in such a way that ~m denotes the maximum corrugation amplitude. VAN DEN BERG and FOKKEMA/3.15/ have investigated f i n i t e Fourier

79 series of t r i a n g u l a r and rectangular one-dimensional corrugation p r o f i l e s and have shown that the convergence l i m i t s become smaller as the number of terms increases, or in other words, as the smallest radius of curvature present at the surface approaches zero. Consequently, the Rayleigh approach breaks down for corrugations exh i b i t i n g discontinuities in ~(R) and/or d~(R)/dR.

3.4.1

The GR Method

This very powerful procedure to compute the AG'S from (3.28) was developed by GARClA et a l . /3.16-19/. Rearranging (3.28) by multiplying each side by exp[-ikiz~(R) ] , we obtain AG MGR G - --

I

(3.31)

MGR = exp{i[(kGz-kiz)~(R)+w

(3.32)

=

-

with

=_

Equation (3.32) must be s a t i s f i e d for every point R in the unit c e l l . I f one now chooses a f i n i t e set of n vectors R uniformly distributed over the surface unit -n

c e l l and relates them to the same number of uniformly distributed reciprocal l a t t i c e vectors G, one can regard (3.31) as a set of n l i n e a r equations which can be solved for the AG'S. The computational a p p l i c a b i l i t y of this method was proven by GARCIA /3.19/ in-many one- and two-dimensional model calculations, and especially in the f i r s t

surface crystallographic investigation using He d i f f r a c t i o n /3.18/,

which was based on the experimental data of BOATO et a l . /3.20/, y i e l d i n g the corrugation function for the (100) surface of LiF (Chap.7). Garcia also showed that r e l i a b l e numerical results could be obtained for maximum corrugation amplitudes s l i g h t l y above the analytical convergence l i m i t s given above. Depending on the form of the corrugation, i t s maximum amplitude, and the scattering conditions (ki,@i), the dimension n of the matrix MGR may be very large (in his LiF study, Garcia used n between 100 and 200) and the calculations may therefore be rather time-consuming.

3.4.2

The Eikonal Approximation

Starting again from (3.28), we m u l t i p l y both i t s sides by exp[-i(_G'R+kG,z~(R)] , then integrate over the unit cell and obtain

80 0

(3.33)

Z MGG, AG = AG, with

I

MGG, = ~ f

exp[i(G-_G')R+i(kGz-kG,z)~(R)]dR

(3.34)

exp{-i[GR+(kGz-kiz)~(_R)]}dR

(3.35)

and 0 I AG = - ~ f

,

where~denotes the unit-cell area. Restricting the possible G's to a f i n i t e set of n vectors, (3.33) yields a set of n linear equations which can be solved for the AG'S. However, as values of n of the same magnitude have to be used in the case of tee GR method /3.21/, actual calculations using this formulation are even more timeconsuming, because in addition to handling the large matrix MGG,, the integrals (3.34,35) also have to be evaluated with good accuracy. Nevertheless, from (3.33) one can readily derive an important approximation /3.2/, which is very convenient to use for calculations of scattering amplitudes. As one can see from (3.34), the diagonal elements of the matrix MGG, are a l l unity. Under certain conditions, the out-of-diagonal elements w i l l be small and can be neglected so that (3.33) becomes 0 AG = AG

(3.36)

This is the so-called eikonal approximation, from which the diffraction probabilities can readily be calculated by simply evaluating the integral (3.35). The conditions under which (3.35) can be used with satisfactory accuracy are the following. The corrugation function ~(R) has to be smooth, and i t s maximumamplitude should be small compared to the l a t t i c e constant (~m ~ O.la). Furthermore, the angle of incidence must be small, so that all intense diffraction beams appear far from grazing emergence. The l a t t e r condition is a consequence of the fact that neglect of the non-diagonal terms in (3.33) corresponds to a neglect of the contribution of evanescent waves (kGz2 < 0), which belong to w vectors outside the Ewald sphere. As a consequence, the eikonal approximation f a i l s to describe correctly the threshold behaviour of the intensities of the beams near horizon. The classical analogue (hi § O) for the condition that evanescent waves do not play a role is that mult i p l e collisions of the particles with the hard wall do not occur. The unimportance of evanescent waves has a remarkable nontrivial consequence for surface structural investigations using atomic-beam diffraction: the diffraction intensities of ~(R) and -~(-R) are the same as long as coupling to evanescent waves is negligible /3.22/. For surfaces with two-dimensional inversion symmetry,

81 ~(R) = ~(-R), which form the overwhelming number of real surfaces, this means that from an analysis of d i f f r a c t i o n i n t e n s i t i e s under conditions where the eikonal approximation works w e l l , one cannot decide whether +~(R) or -~(R) describes the surface p r o f i l e . To prove t h i s statement, we consider (3,31), which for +~(R) reads in f u l l form Z AG exp{i[(kGz-kiz)~(R)+GR]} = - I G -

(3.37)

With no contribution of evanescent waves, a l l the kGz are r e a l . An analogous equation holds for -~(-R); for this case we denote the scattering amplitudes by AG, (3.38)

AG exp{i[-(kGz-kiz)~(-R)+GR ] } = - I G -

We can take the complex conjugate of (3.37) to obtain the equivalent equation for AG. Furthermore, as this equation has to be f u l f i l l e d

at every point of the surface,

we can perform the transformation R § -R without changing i t s v a l i d i t y . Combining both steps, we obtain AG exp{i[-(kGz-kiz)~(-R)+G~} = - I G -

(3.39)

Comparison of (3.38,39) shows that AG = AG. Therefore, for both +~(R) and -~(-R), the d i f f r a c t i o n i n t e n s i t i e s PG ~ IAGi2 = IAG12 = AGAG * are the same. This proof does not hold for evanescent waves. Indeed, w r i t i n g kGz = i• z in any of the terms of (3.37,38), which correspond to evanescent waves, one e a s i l y v e r i f i e s that complex conjugation plus inversion y i e l d s for (3.37) AG exp[-(ikiz+•

+iGR]

whereas the analogous term in (3.38) y i e l d s AG exp[-(ikiz-XGz )~(~)+iw Therefore, opposite corrugations w i l l be impossible to distinguish in practice i f the contribution of evanescent waves is small or, in other words, i f multiple scattering is n e g l i g i b l e . We i l l u s t r a t e this in Fig.3.4, using the example of classical e l a s t i c scattering of p a r t i c l e s from a simple one-dimensional hard-wall model corrugation with inversion symmetry ~(R) = ~(-R). For the case of classical scattering, the t r a j e c t o r i e s of the p a r t i c l e s can be traced. In Fig.3.4a, the maximum corrugation is small, and with the p a r t i c l e beam impinging at normal incidence, three dis-

82

(o)

,,,,i,,,,,,,......

.......................... 5

........................... ,,,,

.! ..

. !1

(b) i

!

/

!~ ............

/

i

/

/. c.','; /

//

.|

~

(c) "

~ l l l l l r t l l l l l l l l l

,lllllll

Fig.3.4a-c. Examples using classical p a r t i c l e Scattering to demonstrate the influence of multiple scattering. In the case (a) of a shallow corrugation and normal incidence (gi = O) where no multiple scattering occurs, i t is impossible to distinguish between +~(x) and - ~ ( x ) , as both p r o f i l e s give rise to the same i n t e n s i t y d i s t r i b u tion of scattered p a r t i c l e s ; the quantummechanical analogue of t h i s " s i n g l e - h i t " situation corresponds to the range of val i d i t y of the eikonal approximation. In cases (b) and (c), for grazing incidence and large corrugation amplitudes, +~(x) and -~(x) y i e l d d i f f e r e n t i n t e n s i t y d i s t r i b u t i o n s . In quantum-mechanical d i f f r a c t i o n , for such cases e i t h e r the general method (Sect.3.3) or the GR method (Sect.3.4.1) or the i t e r a t i v e series (Sect.3.5) have to be used for i n t e n s i t y calculations

t i n c t scattering directions I , 2a, and 2b are observed for both +~(x) and -~(x). In Fig.3.4b, the maximum corrugation amplitude is enhanced, and only for the case of +~(x) do we observe three scattering directions as in Fig.3.4c. For -~(x), however, due to double scattering in the trough, two further scattering directions 3a and 3b appear. A s i m i l a r e f f e c t happens for the shallow corrugation of Fig.3.4a i f we approach grazing angles of incidence (Fig.3.4c). In t h i s case, the corrugation +~(x) gives rise to beams scattered in three d i f f e r e n t directions, whereas -~(x) gives only two scattering directions. For quantum-mechanical d i f f r a c t i o n , the general conclusions are the same: for s u f f i c i e n t l y small corrugations, no differences in the

83 Bragg peak i n t e n s i t i e s w i l l occur for near-normal incidence for +~(R) and -~(R). This constitutes the region of v a l i d i t y of the eikonal approximation. For small corrugation amplitudes looked at by using grazing angles of incidence, as well as for large corrugations for any 0i , differences in the Bragg i n t e n s i t i e s between +~(R) and -~(R) w i l l occur. In the l a t t e r cases, for the calculation of intensities e i t h e r the general method (Sect.3.3), the GR method (Sect.3.4.1) or the i t e r ative series method of Sect.3.5 must be used. I t should be emphasized, however, that in practice in many cases for grazing incidence and small corrugations as well as for large corrugations for any ei , the a t t r a c t i v e part of the potential is no longer negligible and a r e l i a b l e intensity determination is very d i f f i c u l t cause of resonant scattering involving bound states of the potential To check the r e l i a b i l i t y

be-

(Chap.5).

of the results obtained by using the eikonal approxi-

mation, the u n i t a r i t y condition (3.17) has to be f u l f i l l e d

within at least a few

percent. GARIBALDI et al. / 3 . 2 / have developed kinematic factors which help to satisfy u n i t a r i t y .

We c i t e here an expression for the scattering amplitudes, which

seems to be generally accepted to give the best results for intensity calculations within the eikonal approximation: k . ( k i - k G) AG, = -I - 0~ _ kGz(-kiz+kGz ) A -

,

(3.40a)

-

or e x p l i c i t l y expressed as a function of the angles eG and ~G' (3.13): -

1+c~176176176

0 =

AG' =

-

_

coseGcOS~G(COSOi+cosSGcOS~G) AG

-

(3.40b)

=

We close this section with the remark that the frequently quoted Kirchhoff approximation is obtained by replacing kGz by - kiz in (3.35).

3.5

Calculation of I n t e n s i t i e s - l t e r a t i v e

Series

In this section, we outline a method developed by LOPEZ, YNDURAIN, and GARCIA /3.23/ which allows d i f f r a c t i o n conditions (~i,Oi,u

i n t e n s i t i e s to be calculated correctly for any scattering

The limits of convergence of this method have been shown for

several numerically studied examples to be beyond that of the Rayleigh approach, and for small corrugations the method has the further advantage of fast convergence. Here, we follow a derivation provided by SCHLUP /3.24/ which starts from the hardwall equation (3.28) in the Rayleigh l i m i t

84 (3.41)

AG exp[ikGzm~(R) ] exp[iGR] : - exp[ikizm~(R) ] G -

Expanding the AG'S into a series using the strength of the corrugation m as an expansion parameter, oo

AG = Z n A~n) -

,

(3.42)

n=O

and writing also exp[ikGzm~(R) ] and exp[ikizm~(R) ] in their Taylor series representation, (3.41) becomes


denotes thermal averaging. Note that for a given r e l a t i v e o r i e n t a t i o n of the incoming beam of the surface, W w i l l be d i f f e r e n t f o r each d i f f r a c t i o n peak. The assumptions underlying (4.1,2) are that the i n t e r a c t i o n in the scattering event is both weak and short in duration. This can be j u s t i f i e d for x-ray scattering and also f o r neutron scattering because of the short-range potential which leads to short-duration i n t e r a c t i o n despite the thermal neutron v e l o c i t i e s . However, the a p p l i c a t i o n of the standard Debye-Waller treatment to atom scattering from surfaces is not straightforward, since the atom-surface i n t e r a c t i o n is strong and long-ranged and the incoming v e l o c i t i e s are thermal. Therefore, the correction f a c t o r analogous to that of (4.1) for atom scattering w i l l depend on the mass and v e l o c i t y of the i n coming atom, the time scale of the scattering i n t e r a c t i o n and the v i b r a t i o n a l spectrum of the surface.

9ast atoms (conventional theory) increasing/ W D~6 / slow atoms B hard crystal (~ime-dependent interaction)

A

~

increasing

\

C slow atoms sos crystal (Born-Oppenheimer)

increasing

uJD Fig.4.1. A graphical description of the d i f f e r e n t regimes of quantum scattering of atoms from surfaces / 4 . 4 /

95 This is i l l u s t r a t e d in Fig.4.1, in which three l i m i t s to the scattering i n t e r action are shown. For f a s t atoms, a r e s u l t i d e n t i c a l to that f o r x-rays and neutrons can be derived / 4 . 4 / .

In t h i s regime, which corresponds to vertex A in Fig.

4.1, the scattering atom is fast enough that the surface motion during the i n t e r action can be neglected. The incoming atom then sees the disorder present at a given time. For increasing i n t e r a c t i o n time ~c' two d i f f e r e n t l i m i t s are observed depending on whether the crystal has a hard or soft phonon structure. A measure of the i n t e r a c t i o n time Tc is given by ( ~ v i ) - I , where ~ is a reciprocal range parameter f o r the p o t e n t i a l , and v i is the incoming-atom v e l o c i t y ; the hardness of the crystal surface increases with i t s Debye frequency ~D" For slow heavy atoms i n t e r acting with a soft c r y s t a l , the s o l i d has time to adjust to the presence of the i n coming atom, and the s i t u a t i o n is s i m i l a r to the Born-Oppenheimer approximation with the surface atoms and the scattering p a r t i c l e s taking the roles of the electrons and n u c l e i , respectively. This l i m i t is shown as vertex C in Fig.4.1. For slow atoms i n t e r a c t i n g with a hard c r y s t a l , the r e c o i l of the surface atom can be neglected, and the predominant e f f e c t of the increased i n t e r a c t i o n time is to average-out some of the surface disorder. This leads to an enhancement of the d i f f r a c tion i n t e n s i t y when compared to fast-atom scattering and is shown as vertex B of Fig.4.1. This l i m i t , which LEVI and SUHL / 4 . 4 / c a l l the time-dependent i n t e r a c t i o n regime, is the relevant l i m i t f o r He, H2, or Ne scattering from most surfaces. These authors have dealt with t h i s case using a semiclassical theory assuming a Morse potential for the atom-surface i n t e r a c t i o n . The most important consequence of these c a l c u l a t i o n s , which w i l l be discussed below, is that W in (4.2) cannot be separated into separate momentum change and v i b r a t i o n a l amplitude functions as can be done for the fast-atom case. Due to the long-range a t t r a c t i v e force (Chap.2) the part i c l e w i l l be accelerated along i t s t r a j e c t o r y , and in order to calculate the DebyeWaller f a c t o r , the time-dependent atom-surface force and the time-dependent surfaceatom displacement must be calculated along the t r a j e c t o r y of the incoming atom. These two q u a n t i t i e s cannot be decoupled as for the fast-atom case.

4.2

The Debye-Waller Factor in the Time-Dependent I n t e r a c t i o n Regime

For a s o l i d with a Debye spectrum of v i b r a t i o n a l frequencies, LEVl and SUHL / 4 . 4 / obtain the r e s u l t 12mE. kT IZ W = ~--~--D2 C

(4.3)

96 where C is a function of p and K, whereby p = ( I / 2 ) ~mDTc and K = Eiz/D. In (4.3) M and m are the masses of a surface and gas atom, respectively. In these expressions Eiz = ( I / 2 ) mViz2, where Viz is the perpendicular component of the incoming-atom v e l o c i t y at i n f i n i t y ,

and D is the well depth of the Morse p o t e n t i a l . C is shown as

a function of p for several values of K in Fig.4.2. I t is seen that the presence of an a t t r a c t i v e potential w i l l lead to a decrease of the d i f f r a c t i o n i n t e n s i t i e s ( C > I ) f o r small p values which correspond to short i n t e r a c t i o n times. This consequence of the a t t r a c t i v e potential was i n i t i a l l y v e r i f i e d by HOINKES e t a ] .

reported by BEEBY / 4 . 5 / and experimentally

/ 4 . 6 / . BEEBY proposed that the e f f e c t of the long-range

a t t r a c t i v e potential on the d i f f r a c t i o n i n t e n s i t i e s would be averaged out, since i t is due to an i n t e r a c t i o n with many surface atoms over a s u f f i c i e n t l y long time. The remaining e f f e c t of the a t t r a c t i v e potential in Beeby's treatment is to accelerate the incoming p a r t i c l e such that E. is increased by D. This model always predicts iz a reduction in the d i f f r a c t e d i n t e n s i t i e s for D > O.

K=I

.~ ,,12o 1

I

P Fi~.4.2: The dependence of the correction factor to the Debye-Waller exponent, C, as a function of p, for several values of K / 4 . 4 /

However, the inclusion of thermal motion in the a t t r a c t i v e potential in Levi and Suhl's c a l c u l a t i o n shows that d i f f r a c t i o n i n t e n s i t i e s can also be enhanced (C < I) by the presence of an a t t r a c t i v e potential f o r large p corresponding to long i n t e r action times as is seen in Fig.4.2. In practice, t h i s requires both a c o l l i s i o n time longer than the Debye period and an energy which is high compared to the well depth. In these c a l c u l a t i o n s , the correction for the perpendicular energy of the scattering p a r t i c l e depends on p and K, but is always less than Eiz+D.

97

lira K m - 37 PolO l

/ ,

l /// // 10

20

p~

Fig.4.3. Constant C lines drawn in the (po,K) plane. The dashed l i n e connects the K values y i e l d i n g a minimum value of C for each PO" The area in the lower l e f t for which C > I is Beeby's region for which the d i f f r a c t i o n i n t e n s i t i e s are reduced /4.4/ The e f f e c t i v e correction factor C of Levi and Suhl is shown in Fig.4.3 as a function of PO and K, where PO = p KI/2" This r e d e f i n i t i o n has the advantage that PO is independent of the energy of the incoming atom. The shaded area shows Beeby's region f o r which C > I . I f PO and K are known, and the use of a Morse i n t e r a c t i o n potential and a Debye spectrum f o r the s o l i d are j u s t i f i e d ,

C can be determined from Fig.4.3.

Typical values f o r several gas-surface systems are shown in Table 4.1. Beam v e l o c i t i e s c h a r a c t e r i s t i c of nozzle sources have been used, and ~ has been set equal to 2 ~ - I . The l a s t column of Table 4.1 shows values of C which have been obtained from Fig.4.3. With the exception of argon scattering from graphite, C is less than u n i t y , showing that in general, the Beeby correction overestimates the influence of the a t t r a c t i v e potential on the Debye-Waller factor. C is smallest for heavy atoms so that neon d i f f r a c t i o n w i l l be enhanced when compared with helium d i f f r a c t i o n .

4.3

The Size Effect in the Debye-Waller Factor for Atom Scattering

The treatment discussed in Sect.4.2 ~n p a r t i c u l a r , see (4.3)] has assumed that the scattering atom interacts with a single atom of the s o l i d . However, the atom-surface potential is long-ranged so that t h i s assumption is c e r t a i n l y not v a l i d . Physi c a l l y , t h i s means that short wavelength o s c i l l a t i o n s in the surface w i l l not be f e l t by the incoming atom. This leads to a reduction in W and to an enhancement of

98

Table 4.1. PO, K, and p values for a number of gas-surface scattering systems. The v e l o c i t y d i s t r i b u t i o n assumed is c h a r a c t e r i s t i c of a nozzle beam, and the p o t e n t i a l range parameter is approximately 2 ~ - I . Also included in the l a s t column are the C values taken from Fig.4.3 / 4 . 4 / PO

K

p

C

He/LiF

0.12

9

4

0.6

9

0.4

12

0.5

Ne/LiF

0.19

4

Ar/LiF

0.16

1.6

He/Graphite

7.5

4.5

Ne/Graphite

11

Ar/Graphite

9.5

2 0.8

3.5

0.9

8

0.7

10.5

1.2

the d i f f r a c t i o n i n t e n s i t i e s . A correction f o r t h i s e f f e c t was o r i g i n a l l y proposed by HOINKES et a l . / 4 . 6 / who used a model in which the motion of the surface atoms was uncorrelated. ARMANDet a l . / 4 . 7 , 8 / have refined t h i s idea to include the correlated motion of surface atoms on the (100) plane of face-centred-cubic (fcc) crystals.

In t h i s model, only harmonic forces between nearest neighbours are included

and the force constant is adjusted to give the correct value of the bulk Debye temperature. Assuming that the incoming atom interacts with the four atoms of the surface u n i t c e l l of Cu(100), the normal component of can be calculated for t h i s configuration / 4 . 9 / .

I t is found that the correlated mean-square displacement is

only 40% of that for an i n d i v i d u a l surface atom. Of course, the number of surface atoms with which the incoming atom interacts depends on the range of the i n t e r a c tion so that at the present state of knowledge of atom-surface p o t e n t i a l s , i t is not easy to correct for t h i s e f f e c t unambiguously. However, since the mean-square correlated displacement w i l l be smaller, the greater the number of atoms considered, the enhancement of the d i f f r a c t i o n i n t e n s i t i e s is greater, the longer the range of the i n t e r a c t i o n p o t e n t i a l .

4.4

Experimental Investigations of the Debye-Waller Factor for Atom-Surface Scattering

Experimentally, W can be determined by measuring the dependence of a d i f f r a c t e d beam i n t e n s i t y on e i t h e r the surface temperature Ts or on the normal component of the i n coming-atom energy Eiz. For the l a t t e r case, e i t h e r the t o t a l energy of the atom or the angle of incidence can be varied, since Eiz = Eicos28 i . The determination of W by varying the angle of incidence is only possible for very small corrugations f o r

99 H1 ~

-In%

LiF[001) e ~ = e i

D : 0 rneV

,/ +/

/, /, 9176

9,#. x. ..", 9 9

o/

/x o~ @x~X x

I-

-~"

'

2~::)OK'

l TSF 9 : 725 K

2

x TSF : &75K oTsF : 255K

~

i

i

i

D : 71meV

""

I ~t~ x~

f

4~)0K TSF' 6(~0K '>8~K^D '

10~K i

"{COS2~i+ ZK IB }

O~

' [

'

~

I

'

400K 8(~K I~O0K',5100

)

Fig.4.4. Logarithmic p l o t of the specular i n t e n s i t y f o r H scattering from LiF(IO0) versus the product of surface temperature and momentum t r a n s f e r perpendicular to the surface. The l e f t side shows a f i t f o r an a t t r a c t i v e potential D = 17.8 meV. The r i g h t side shows plots for D = 0 and 71 meV [Ref. / 4 . 6 / , the corrected version of Fig.5] which only the specular peak is intense, since otherwise the angular v a r i a t i o n can be dominated by s t r u c t u r a l effects. Figure 4.4 shows the results of a measurement of the specular-beam i n t e n s i t y f o r H d i f f r a c t i o n from LiF(IO0) in which the angle of incidence has been varied f o r three surface temperatures and a Beeby correction has been made / 4 . 6 / . The a t t r a c t i v e well depth is known to be 17.8 meV from resonant scattering measurements /4.10/. For t h i s value of D, i t is seen that a l l points l i e near the s t r a i g h t l i n e f o r the Beeby correction, but that substantial deviations are observed for D = 0 and 71 meV. Using a model of uncorrelated surface-atom displacements, the surface Debye temperature is found to be 415•

K. This is rather small

when compared with the bulk value of 732 K and may be due to the assumptions of uncorrelated motion and that C = I . Detailed studies of the Debye-Waller f a c t o r f o r He and Ne scattering from Cu(100) have been carried out by LAPUJOULADE et a l . / 4 . 1 1 / . Figure 4.5 shows an angular scan f o r Ne scattering at two substrate temperatures. I t is seen that the i n e l a s t i c a l l y scattered i n t e n s i t y has a broad lobular d i s t r i b u t i o n whose center is s h i f t e d from the specular d i r e c t i o n towards the surface normal. Whereas the e l a s t i c scattering can be c l e a r l y seen in Fig.4.5 f o r Cu(IO0), t h i s is not the case for a l l metal surfaces. Figure 4.6 shows angular d i s t r i b u t i o n s taken in the authors' laboratory f o r Ne scattering from the (I•

reconstructed surface of Au(110) for which He d i f f r a c -

t i o n traces are shown in Figs.6.7 and 9.4. A double loire which is f a r removed from the specular angle is seen for in-plane scattering with the beam i n c i d e n t in the [001] azimuth, and a single lobe centered near the specular angle is seen f o r the

100 Z/io 0.0050

0.0025

0 0.0"/5

15

85

T-?0K 0.(T-~

Fi9.4.5. Specular-beam i n t e n s i t y as a function of the scattering angle f o r Ne scattered from Cu(100) f o r surface temperatures of 70 K and 473 K. E. = 63 meV, e. = 75 ~ /4.11/ 1

0.02~

055

65

75

85

1

90

(a)

9:

~

I

I

I

I

I

I

f

f

I

r

I

-~0

0

t0

20

50

40

50

60

70

80

90

8f (o)

Fig.4,6a,b. In-plane scattered i n t e n s i t y as a function of the s c a t t e r i n g angle for Ne scattering from the (Ix2) reconstructed Au(110) surface in (a) the [ I T ~ and (b) the DO0] azimuths. The surface and nozzle temperatures are 300 K

beam i n c i d e n t in the [1i0] azimuth. No e l a s t i c a l l y scattered i n t e n s i t y was observed for surface temperatures as low as 100 K and for a wide range of beam energies and angles of incidence. This shows that the Debye-Waller factor is appreciably smaller f o r Au(110) than f o r Cu(100). Figures 4.7,8 show experimental results of LAPUJOULADEet a l . /4.11/ for the dependence of the specular-beam i n t e n s i t y on the surface temperature f o r Ne and He scattering from Cu(100). Also shown in these figures are theoretical f i t s data using the fast-atom l i m i t f o r W,

to the

101 1 (Ak)2

W=

,

(4.4)

where only the v i b r a t i o n s perpendicular to the surface have been taken i n t o account. The mean-square c o r r e l a t e d displacement has been c a l c u l a t e d as o u t l i n e d in Sect. 4.3 and m u l t i p l i e d by a c o r r e c t i o n f a c t o r X which was used as a f i t

parameter. The

a t t r a c t i v e well depth was introduced using the Beeby c o r r e c t i o n . The b e s t - f i t

para-

meters g i v i n g the s o l i d l i n e s in F i g s . 4 . 7 , 8 are c = 9.8 meV f o r He and 124 meV f o r

10~I LOG(I/Io) 71.6~

10-2

x~,X~.O+"

10 I~

10 o

I

100

,

I

I

I

3o0

I

I

~oo

I

~

I , I , I I~ ~ , 6o0

~o

~

I 9o0

, I

.

t, lOOO ~oo

T, K

t , ~zoo

=

~.~.7

LOG(IIIo}

\ 104

10.2

1~

\~

\

~

\

11% ~ \\

,

\ ~ 6~0 ~.o L , I ~AI ,~ I ,Ol~, 1~ 2~ 300 400 500

I , =, i, 6~ 700 ~0

I,

9m

I,

I,

1000 11~

~K

= , =

12m

Fi9.4.8

Fig.4.7~ Specular-beam i n t e n s i t y versus the surface temperature T f o r He s c a t t e r i n g from Cu(100). The parameters are the angles of incidence and Ei = 63 meV. The s o l i d l i n e s are a t h e o r e t i c a l f i t to the data / 4 . 1 1 / F i 9 . 4 . 8 . Specular-beam i n t e n s i t y versus the surface temperature T f o r Ne s c a t t e r i n g from Cu(100). The parameters are the angles of incidence and E. = 63 meV. The s o l i d and dashed l i n e s are t h e o r e t i c a l f i t s to the data / 4 . 1 1 / 1

102 Ne and ~ = 0.855 f o r He and 0.775 f o r Ne. The authors discuss several i n t e r p r e t a tions f o r the deviation from ~ = I , which would be expected i f the incoming atom interacted with four Cu atoms coupled by harmonic forces, and the force constant was determined as outlined in Sect.4.3. One p o s s i b i l i t y is that the force constant between the surface atoms is higher than assumed. However, the deviation of I from u n i t y may also be due to the range of the p o t e n t i a l , and ~ > I indicates that the incoming atom interacts with more than four Cu atoms. The well depth deduced for He is w i t h i n the range of what is expected f o r metal surfaces, but that f o r Ne is much higher than would be expected. However, the assumption of D = 60 meV gives a poorquality fit

as shown by the dashed l i n e in Fig.4.8.

These measurements have been reinterpreted by BUHEIM et a l . / 4 . 1 2 / . They point out that besides the f i n i t e - t i m e and f i n i t e - s i z e effects outlined in Sects.4.2,3, there are f u r t h e r contributions to the Debye-Waller exponent r e s u l t i n g from inelast i c processes with momentum transfer p a r a l l e l to the surface. This can lead to an increase of the exponent by factors of the order of two to three, thus compensating the f i n i t e - s i z e effects to a large extent. B~HEIM et a l . consider i t best to neglect both corrections and to compare the f i t

parameters of the potential with additional

information on i n e l a s t i c processes such as energy accommodation c o e f f i c i e n t s using a theory /4.13/ consistent with the approximations f o r the Debye-Waller factor. Their f i t

to the neon data of Fig.4.8 using a Morse potential with D = 33 meV is

shown in Fig.4.9. Note the curvature in the p l o t at low temperatures, which is due to the inclusion of the zero-point motion of the s o l i d . The value of D obtained with t h i s treatment is w i t h i n the range expected for the Ne-metal surface potential /4.14/ and the good agreement obtained with experiments shows that the t h e o r e t i c a l formulation of the Debye-Waller f a c t o r for atom scattering has advanced to the stage where q u a n t i t a t i v e agreement can be achieved. lOo

-LOG (I/I o)

10-1

10.2

I0 "3

E

~, 9 dx, 45.0', 80.0~, 75.0', I 'Aj '91 n~' 200 400

]

600

I

Ts

800

Fi9.4.9. Specular-beam i n t e n s i t y versus the surface temperature T for Ne scattering from Cu(100). The data points and the dashed l i n e are from /4.11/. The s o l i d l i n e s are a f i t with D = 33 meV taken from /4.12/

103 In closing this section, we shall b r i e f l y mention a number of theoretical and experimental studies relevant to the Debye-Waller factor in atom-surface scattering. ASADA /4.15/ has published a detailed study of He and H2 scattering from Ag(111). This paper also includes an extensive l i s t of references to these investigations. MOLLER-HARTMANNet a l . /4.16/ have suggested that i n e l a s t i c scattering through d i r e c t excitation of electrons near the Fermi level may be an important mechanism in metals. MASONand WILLIAMS /4.17/ have reported results for He scattering from Cu(110) in which the specular i n t e n s i t y shows an anomalous dependence on Ak. This r e s u l t has been discussed elsewhere /4.11/. Recent results by CANTINI et a l . /4.18/ show that resonant scattering can also couple with i n e l a s t i c scattering.

5, Influence of the Attractive Part of the Potential on Diffraction Intensities

5.1

Modifications for the Calculation of D i f f r a c t i o n I n t e n s i t i e s

In Chap.3, we presented an extensive discussion of how d i f f r a c t i o n i n t e n s i t i e s can be calculated for a given corrugation and a given scattering geometry, neglecting the influence of the a t t r a c t i v e part of the p o t e n t i a l . The existence of the a t t r a c t i v e well leads, however, to the important phenomenon of resonant scattering or sel e c t i v e adsorption, which can influence the Bragg i n t e n s i t i e s considerably. We w i l l discuss t h i s e f f e c t in the following sections. In t h i s section, we consider simple modifications in the i n t e n s i t y calculations which must be performed in situations where the depth of the a t t r a c t i v e part D cannot be neglected in comparison to the incoming energy Ei , but where selective adsorption does not play a role. For this purpose, i t is s u f f i c i e n t to characterize the a t t r a c t i v e well solely by i t s depth D /3.2,5.1/. Due to t h e i r a t t r a c t i v e interaction with the surface, the p a r t i c l e s are accelerated in the region of the a t t r a c t i v e w e l l , and t h e i r e f f e c t i v e energy perpendicular to the surface E~]z is increased against Eiz according to 2m 2m E! ~ k~ 2 = k. 2+2~ D e (Eiz+D) ~-~ ~2 I Z IZ IZ

(5 I)

This has the consequence that the p a r t i c l e beam is refracted toward the surface normal in the region of the a t t r a c t i v e w e l l , so that the beam h i t s the repulsive wall under a smaller e f f e c t i v e angle of incidence 0i ' given by

104 k.

sin8 i, =_~.,sinO i

(5.2)

,

1

the total effective energy of the particles then being T12 12 1~2 Ei ' _=~-~ k = ~-~ (kiz+K)2+D_ _ -= E.+DI

3o

~o

Xo

oo

(5.3)

io

zo

I

/ \'\ ~q

I '&lip

Fi9.5.1. Ewald construction for d i f f r a c t i o n of particles with wave vector ~ and angle of incidence 8i . Due to the a t t r a c t i v e potential of depth D, the particles are accelerated towards the surface according to (5.1). The incoming beam is therefore refracted towards the surface normal. The Ewald construction for the e f f e c t i v e k~ and 8~ is also shown. The dotted lines represent a geometrical proof that the effective angles of emergence e~ correspond to the ones actually observed, 8~, as the particles lose the extra energy D when leaving the surface region. [Compare (5.4); A = (2mD/~2) I/2]

The situation is shown in Fig.5.1. The solid lines refer to the Ewald construction for the i n i t i a l

wave vector ~ i ' and to the Ewald construction for -ik[" For the l a t t e r ,

the z-component of the diffracted beams G is given in analogy to (5.1) by k~z2 = ki_ ( K + G ) 2 2rod ~2

(5.4)

The diffracted particles w i l l lose t h e i r extra energy D when leaving the region of the attractive well and are therefore refracted from the surface normal, so that they f i n a l l y occur at d i f f r a c t i o n angles corresponding to ~i" A geometrical proof that the effective d i f f r a c t i o n angles @~ and the real angles 0G are in accordance with (5.4) is given by the dotted construction shown in Fig.5.1. A (2mD/~2)I/2]. =

[Note that

105 Therefore, to a f i r s t

approximation the a t t r a c t i v e well can be taken into account

by replacing k. and 8. by k~ and 8! in the corresponding expressions of Chap.2 used 1

1

1

1

for i n t e n s i t y c a l c u l a t i o n s . This is c e r t a i n l y a v a l i d procedure, as long as the Ewald spheres f o r k. and k~ cover the same range of F vectors as is the case in 1 1 Fig.5.1. I f the Ewald sphere for -ikl comprises more F vectors than that f o r ~ i ' i n t e n s i t y w i l l be calculated f o r these extra beams although in r e a l i t y they cannot be observed (0~ < 90 ~ , 8F > 90~

Hence, t h i s simple procedure w i l l give r e l i a b l e

r e s u l t s only as long as the i n t e n s i t i e s of the extra beams come out very small. This w i l l always be the case for small corrugations, i f the beam comes in near normal incidence. Otherwise, the f u l l mathematical apparatus outlined in Sect.5,3, which takes into account resonant s c a t t e r i n g , has to be applied.

5.2

Bound Surface States and Resonant Transitions

Already in t h e i r early H e - d i f f r a c t i o n experiments, STERN and coworkers observed c h a r a c t e r i s t i c structures (usually minima) in the d i f f r a c t i o n peaks as a function of angle of incidence 0. / 5 . 2 / or as a function of azimuthal angle u (Fig.3.1) of i the sample r e l a t i v e to the incoming beam / 5 . 3 / . In Fig.5.2, we show a recent example for the l a t t e r case obtained by FINZEL et a l . / 2 . 2 / using the d i f f r a c t i o n of atomic deuterium from a NaF(IO0) surface. LENNARD-JONESand DEVONSHIRE / 5 . 4 / att r i b u t e d these features c o r r e c t l y to resonant t r a n s i t i o n s of the incoming p a r t i c l e s i n t o bound states of the a t t r a c t i v e potential and referred to the phenomenon as " s e l e c t i v e adsorption". The basic underlying idea is simply related to the Bragg condition for conservation of momentum p a r a l l e l to the surface K+_G = KG

(3.8)

in connection with the condition f o r conservation of the t o t a l p a r t i c l e energy k2 2 (K+G) +k 2 i = kG = - Gz

(3 9)

The normal component of the energy of the p a r t i c l e s I~2

~n2 k2 EGz -= 2-m Gz = 2-m

- (K+ - -G) 2]

E i - E(_KG)

(5.5)

is always > 0 for p a r t i c l e s scattered into Bragg peaks _F [compare (3,11)]. The e x i s t ence of the a t t r a c t i v e part of the potential has the consequence that the p a r t i c l e s can gain k i n e t i c energy at the expense of potential energy i f conditions (3.8,9) per-

106

F

I

1

'

I

D1--~ I~F (001) Ei = 53.2meV

I

TSF= 2/,0K 1,1) N,I) (-1,4)(I,1) II

! !,1)(-I,-1) H,-I)(1,1)

I

1

!

I

~

I

]E. =-13.3meV IEn =-S.erneV i En = -l.6meV

(-1r2)(0,-1)

(0,-1)(0,1)

l i

(0,-1!,(-1z~

(0,4)(0,1)

(0,1)(-12)

i

(0,-1l~ '(0,1) (0,-1) (0,1)

e~

(-1,2~(0,1I

t

"I IIf',i [,"---",,il./>"-~.-~.I/'.i !,~, I,'..--"-".--*~i ."-_

..~

1-1rl )

'~

.~.,(~.4)(11)' 9

(~,! l(o,~)(~,-m (o,4) m~l (oln (o.i) ..... .-,,~ i -.....,.,--'~ ,S i ,,~ ~r-,, I

~,z).m.1:i

........~, .I

~SOc

4.

i ..!.,.....,I

1,4'

.o70-

\t./".......... (0,-II( 1,1) ,, ,

(11)

',,

(0,-I)(I,1) (0o-II I

.,

:

(0,I)~'II(I,-I)(0,I) : ~

.' 'X.,---....................... "- ........... :'".._, t../!

I'~

:

it66~

90-

-L,5o -L.Oo

-200

-10 ~

100 ozimutho( ong(e 'y

F i g . 5 . 2 . Resonant structures observed w i t h atomic deuterium scattered from NaF(O01). The dependence of the specular i n t e n s i t y on the azimuthal o r i e n t a t i o n y of the sample r e l a t i v e to the incoming beam is shown f o r several angles of incidence O< and ~ a f i x e d i n c i d e n t energy Ei . T r a n s i t i o n s to bound states w i t h the correspondlng rec i p r o c a l l a t t i c e vectors and binding energies are i n d i c a t e d / 2 . 2 /

107 mit a t r a n s i t i o n into one of the allowed energy levels ~ of the a t t r a c t i v e potent i a l via a vector G outside of the Ewald sphere. In such a case, the p a r t i c l e s move parallel to the surface in the direction ~G with a total energy~2k#/2m+Ic I, but are bound perpendicular to the surface with the binding energy c

= EGz < 0

,

(5.6)

where EGz is determined by (5.5). Whenever such a t r a n s i t i o n to a bound state is possible, the i n t e n s i t y of the observed Bragg peaks F w i l l be modified, and the location of the resonant structures as a function of hi , ei , and u can be used for determination of the bound-state energies according to (5.5). With the i n t e n s i t y data f o r the (00) beam shown in Fig.5.2, FINZEL et a l . / 2 . 2 / were able to determine the three deepest energy levels of the system DI/NaF(IO0). The corresponding G vectors by which the resonant transitions are obtained are indicated in the figure. Another energy level ~3 could be traced by studying the angular dependence of the (10) beam. All the bound-state energies l i s t e d in Table 2.1 were obtained from simil a r experiments. Rearranging (5.5) to the form (~i +G)2 = (Ei + IEGzl) ~2m

(5.7)

,

one can see that the ~G vectors leading to a p a r t i c u l a r t r a n s i t i o n ~ = EGz l i e on a c i r c l e centred at G with the radius [2m(Ei + IEGzl)~2] I/2. Figure 5.3 depicts this s i t u a t i o n graphically, and Fig.5.4 shows an experimental r e s u l t obtained by MEYERS and FRANKL / 5 . 5 / for He d i f f r a c t i o n from NaF(IO0), proving that (5.8) is f u l f i l l e d reasonably well. This implies that the bound p a r t i c l e s may be treated as nearly free in two dimensions p a r a l l e l to the surface, i . e . ~2 2 E(~G) = ~ KG

(5.8)

In Fig.5.3, we have plotted a p a r t i c u l a r vector K which leads to f u l f i l l m e n t of -

(5.8) for both the G vectors (oT) and (12). In Fig.5.Sa, we present the experimen-

tal results of HOINKES et alo / 2 . 3 / for exactly the same s i t u a t i o n using DI d i f fraction from NaF(IO0). The arrows indicate the crossover of the resonance minima for ~0 =-13"3 meV as expected from (5.9) f o r the reciprocal l a t t i c e vectors (OT) and (T2). In contrast to this expectation, two well-separated minima are always observed in the experiment. In accordance with the theoretical investigations of CHOW and THOMPSON/ 5 . 6 / , this e f f e c t could be explained by energy s p l i t t i n g of mixed degenerate bound states. The two resonant bound states Iv(J~)> = 10(0T)> and I0(~2)>

108

% i o 3

r2n

'i =

DO1", ,

%#

9 "\

,

/

!

/

I

+1

+

x~L

,Eo

~ EIOI~

o

' "1

+;,, i

2

\

%, ,i

+

%;',, ?/]A+ ''~

+

t.

E~',k

"~r ~

+

e

,%

('t~) Fig.5.3. Graphical representation of (5.8) showing that the ~Gvect~ leading to a particular resonant transition (~,G) l i e on a c i r c l e with radius [ ~ I ( ~ + ~ ,)2~/~211/2 centred at G. For the case lndlcated, resonant transitions to both G = (07) and G = (T2) are possible. In such situations, band-splitting effects can be observed

9 9

r.9

,+

Fig.5.4. x- and y-components of the wave vector K for the (01) selective adsorption transitions (v = 0-3) f o r He/NaF(IO0), proving experimentally (5.8) 15.5/

belonging to the same total energy EO(O~) = EO(~ ) are mixed near the resonance angle because of the influence of the periodic term V(o~)_(7~ ) = v11 of the potential,

(2.1). For the mixed states, perturbation theory yields the energy eigen-

val ues Ea,b = g

LE0(0+)+

(07)-E0(12)2 +

with H12 = b11Dexp(o2~/mw) [compare (2.13)] and w = 2(D-IEoI)~. Using (5.9), the only unknown parameter b11 could be calculated from the experimentally observed s p l i t t i n g . The results are summarized in Fig.5.5b. The circles indicate the experimentally observed locations of the minima, their darkness being roughly proportional to their intensity. The f u l l heavy line and the dashed line correspond to the f u l l and open arrows of Fig.5.5a, respectively. These lines cross in accordance with (5.8). The f u l l l i g h t lines correspond to a calculation with (5.9) using the b e s t - f i t value for b11 = 0.02. The corresponding contribution to the potential is shown in Fig.5.2. The observed energy s p l i t t i n g corresponds to a two-dimensional band-structure effect and is analogous to the energy s p l i t t i n g near zone boundaries of electrons

109 q O - - NoF fOOl) / TSF = 2~OK E~O) (-1,-2)I_E(o0)(O _11 Oi : 75" ,*~, , \ ~ **,'*% ,''' Ei =88.3 meV

30"

et

30.

9

--.. ~ I 99

. . . .

~30

.

"%,,~,,,"

oma~

~, %9

oo 9 t '

~oJ

"-..,

~e

! ~"

E{ :81.5 meV 9

f

-300

i

i

-28 ~

9

78.6

9

g

% ; ""

i

~ ( ~ \",,\

86.0

Ei :78.6 meV

6g.3 --

QI 0

E i : 5g.3 meV

%,,"

a

88.3

74.2

9

Z,O

E~O}(o _1)

Ei = 86.0 meV

~

35'

,,_E (0) (-1, 2)

81.5

AI

,,9 ~,oo

~3o

,""

E i meV

i

-26 ~

65.0

r

I

-2'4~

i

-2'2~

-20 ~

b

\

-

-27.0

-2d.s

-2&o

-2;.s -2;.2 y

Y

Fig.5.5. (a) Experimentally observed s p l i t t i n g of bound-state resonance minima in the specular i n t e n s i t y of atomic deuterium scattered from NaF(IO0) as a function of azimuthal orientation u f o r d i f f e r e n t incoming energies Ei . (Compare Fig.5.4.) The contributing bound channels are indicated by arrows. (b) Calculated dependence of resonances on Ei and ~ for independent bound channels (dashed and f u l l heavy l i n e s ) and for mixed bound channels ( f u l l l i g h t l i n e s ) . The c i r c l e s indicate the experiment a l l y observed locations of the resonances / 2 . 3 /

moving in periodic potentials. I t shows that (5.8) is only approximately true; the particles moving in bound states parallel to the surface feel the two-dimensional p e r i o d i c i t y of the surface potential and t h e i r wave functions are not plane waves as for free p a r t i c l e s , but Bloch waves, whose energy spectrum contains gaps. Therefore, (5.8) f a i l s to hold near intersections of the c i r c l e s described by (5.7), (Fig.5.2). The most important consequence of this effect l i e s in the fact that the dominant higher-order terms in the potential series can be determined to a high accuracy. Similar band s p l i t t i n g has been observed for He/NaF(IO0) / 5 . 7 / , He/LiF(IO0) / 5 . 8 / and He/graphite(1000) /2.8,9,37/. I t should be noted that in the i n t e n s i t y variation of the specular beam due to resonant scattering, mostly minima, but sometimes also maxima and Fano-type a n t i resonances are observed. On the basis of a purely e l a s t i c theory, WOLFE and WEARE / 5 . 9 / have worked out three rules which predict under which conditions which structures should appear. As these rules can be helpful in analyzing experimental data and may give a rough idea of which Fourier components of the potential are important, i t is worth c i t i n g them here: a) Specular minima w i l l be observed when the channel in resonance couples d i r e c t l y and strongly to the specular channel (more strongly

110 than through i n d i r e c t coupling) and to at least one other open channel, b) Mixed maxima-minima (maxima predominant) in the specular i n t e n s i t y w i l l be observed when the only open channel to which the resonant state couples strongly and d i r e c t l y is the specular, c) Specular maxima w i l l be observed for resonant channels that couple only i n d i r e c t l y to the specular through strong Fourier components. Examples for these rules are given in the original paper of WOLFE and WEARE /5.9/ as well as in /5.10/. I t must, however, be emphasized that the "Weare-Wolfe" rules are derived on the basis of a purely e l a s t i c theory, and therefore, deviations can occur in cases where i n e l a s t i c effects play an important role. CHOW and THOMPSON/5.6/ have also considered t h e o r e t i c a l l y the e f f e c t of strong coupling of bound channels to diffracted channels. An example for such a situation is shown in Fig.5.6 which refers to H-diffraction from KCI(IO0) covered with H20 as

/

15 z_

~z

w 10 z

60- H,-KQ(001)

x:13~,A

u 5e-

~

2(1,O)

~ V

.~

/

#

3(1,0)

~. H / H ~ - - KCI 1001)

~'/

TsF= 150 K

5

\

(O,l)-beom

10"

30" 50 ?O" 90" Angle of incidence i~i

,510,~, , ,5=56,. ,610~ , . ,6~5& , , ANGLE OF INCIDENCE ~1i

Fi9.5.6~ (a) Specular i n t e n s i t y as a function of angle of incidence ei for atomic hydrogen (E i = 46 meV) scattered from KCI(O01). Bound-state minima are indicated. (b) Diffracted i n t e n s i t i e s for (71) and (01) beams versus 0i . At resonance the (01) beam shows pronounced maxima due to strong coupling of the resonant channel (10) to the open channels (0 • via the strong potential term v11 /2.3/

investigated by HOINKES et a l . / 2 . 3 / . In this case, i n t e n s i t y minima are found for the specular as well as for the (17) beam at rather large angles of incidence ei . These minima correspond to i n t e n s i t y losses via the channel (10) into the bound states ~2 =-15"9 meV and ~3 =-10"3 meV / 2 . 4 / . The (01) beam, however, shows int e n s i t y maxima at the same values ei . As is easily observed from geometrical considerations, the symmetric open d i f f r a c t i o n channels (0 •

are those coupled strong-

l y via v11 (2.13) to the bound channel (10). Therefore, according to the theoretical results of CHOWand THOMPSON/ 5 . 6 / , the beams (0 •

can show i n t e n s i t y maxima at

111 the resonance angles, as is a c t u a l l y found. The adjacent (-I •

beams which are

not d i r e c t l y coupled to the bound channel (v12 is vanishing small) indeed do show i n t e n s i t y minima. Expressed in a p i c t o r i a l way, t h i s r e s u l t means that atoms which were scattered into the bound state ev by a d i f f r a c t i o n corresponding to the (01) undergo a second d i f f r a c t i o n of type (-I •

and reappear in the (0 •

beam / I . 9 / .

Analogous observations were reported by FRANKL et a l . /5.11/ for He scattering from LiF(IO0).

5=3

Theory of Atom Scattering from a Corrugated Hard Wall with an A t t r a c t i v e Well

Numerous theoretical attempts can be found in the l i t e r a t u r e aiming at solving the f u l l problem of p a r t i c l e scattering from s o l i d surfaces, whereby effects due to the a t t r a c t i v e part of the potential as well as the i n t e r a c t i o n with l a t t i c e v i b r a t i o n s were considered / 3 . 2 ; 5 . 1 , 4 , 6 , 1 2 - 1 9 / . The r e l a t i o n between the d i f f e r e n t methods proposed has been discussed in an a r t i c l e by WOLFE et a l . /5.20/.

Instead of surveying

a l l these d i f f e r e n t approaches and t h e i r r e l a t i v e merits, we f i n d i t more useful for the present purpose to r e s t r i c t ourselves to a detailed description of a rather recent formulation of the problem developed by CELLI et a l . / 5 . 2 1 / , which has the advantages of being mathematically closely related to the formalism outlined in Chap.3, of being p h y s i c a l l y transparent, and of allowing a convenient numerical parameterization. Thus, i t permits q u a n t i t a t i v e calculations f o r real systems, provided the hard-wall corrugation function ~(R) and the bound-state energies c

are known. In-

deed, the method has already been applied with considerable success in describing resonant effects in atom d i f f r a c t i o n from i o n i c crystals / 5 . 2 2 , 2 3 / , graphite /5.24-27/, and the adsorbate system Ni(110)+H(Ix2) / 5 . 2 8 / . CELLI et a l . /5.21/ s t a r t from a simple model potential including a short-range repulsion Vr due to a hard wall described by ~(R) and a long-range a t t r a c t i o n Va, as shown in Fig.5.7. There are two planes z = z 0 and z = z 0 - ~

(~ > 0), so that

= ~ for z ~ z 0 - 5 Vr[Z - ~(R)]

= 0 for z > z 0 -

(5.10)

and Va(Z) = -D for z < z 0

(5.11)

Thus, the a t t r a c t i v e well has a f l a t bottom of depth D and a width > ~. The choice of such a potential has the consequence that the wave functions for V and V can r a be found separately and that they can then be matched at z = z O. In the re-

first

112

i-D Ei

f

Zo-a

f

-

Z

Fig.5.7. Model potential for hard corrugated wall with a t t r a c t i v e wall used by CELLI et a l . /5.21/

zo

gion z O- 6 < z < z O, the p a r t i c l e wave function has the form +

~(r) = G~ B-G exp i FK+G)R+k~z] + BG exp i[(K+G)R-kG~Z ] -1 UL --" with k ' given by (5.4) Gz

(5.12)

The f a r - f i e l d solution has the usual form [compare (3.15~ (5.13)

# ( r + ~ ) = exp i[(KR+kizZ)] + ~ AG exp i[(K+G)R+kGz ~ G The scattering amplitudes AG are the quantities desired. For the region of the potential Vr - D, the incoming waves are of the form +

exp i[(K+w

] and t h e i r amplitudes BG are related to the amplitudes BG of the

d i f f r a c t e d waves by B+G = ~ S(G,G')BG, G' " With the corrugation function ((R) known, the amplitudes S(w

(5.14) can be calculated

by using the methods outlined in Chap.3 taking into account the modifications of Sect.5.1. The scattering problem for Va is e a s i l y solved, as Va does not e x h i b i t l a t e r a l periodic modulation, and therefore, p a r a l l e l momentum is conserved. This means that with the wave exp i[(K+G)R+kG~Z] incident from the l e f t on Va, only exp i[(K+G)R-kG~Z~ and exp i[(K+G)R+kGzZ] can be corresponding outgoing waves. Therefore, for-G ~ O, +

AG = TG BG

(5.15)

and +

BG = RG BG

(5.16)

113 with RG and TG the r e f l e c t i o n and transmission coefficients for incidence from the l e f t , which a~e related through kGzlTGI2 + kGzlRGI 2 : kGz' For G = O, one must also include the i n i t i a l

(5.17) incoming wave exp i(KR-kozZ) and

therefore, A0 = R~ +T O B~

(5.18)

= T&+R o

(5.19)

with R~ and T~ being the coefficients for incidence from the r i g h t , where ,2 TO = koz/koz in accordance with (5.17). Combining (5.15,16,18,19) one obtains an infinite

set of linear equations of the form

+

+

BG : Z S(G,G')R~ BG, +S(G,O)T~ G

-

(5.20)

that, according toCELLI et al. /5.21/, can be r e s t r i c t e d in most cases to the set of vectors {N}_ lying within the two Ewald spheres with radii k i and kll (compare Fig.5.1). Thus, (5.20) becomes a matrix equation of f i n i t e (and usually small) size + involving the unknowns BN. The RN can be specified /5.21,26,27/ through a phase s h i f t ~N that characterizes r e f l e c t i o n of the particles from the a t t r a c t i v e potential.

~N can be obtained easily from an interpolation of the experimental bound-

state energies ~

i f they obey approximately the relation

where the parameters D, ~, and A f i x the depth, range, and steepness (or asymmetry) of the p o t e n t i a l , respectively /5.26/. The r e f l e c t i o n amplitudes desired can be obtained by writing (5.21) as a function of v, v(~) = X

I- -

- g

(5.22)

through ~N = 2TP~(~V) and

(5.23)

114 RN : exp[i6 N ]

(5.24)

Thus, i t suffices to characterize the a t t r a c t i v e part of the potential by the phase s h i f t s 6N, and therefore, the assumption of the potential described by (5.10,11) constitutes only a mathematical t r i c k and does not r e s t r i c t the a p p l i c a b i l i t y of the method to the peculiar shape of the potential shown in Fig.5.7. +

The set of equations (5.20) can now be solved for the BG, and the total scattering i n t e n s i t i e s for the beams F within the Ewald sphere of-radius k. can be calcu1 lated according to k

i

kFz 2 Fz + 12 PF = ~ IAFI =-=-- I~ S(F,N)B N RN/T~+S(F,O) -

koz

-

k

Oz '

N

(5.25)

- -

A detailed analysis of the conditions leading to minima, maxima, or Fano-type a n t i resonance structures in the angular or energy dependence of the i n t e n s i t i e s can be found in the original paper /5.21/. As the theory does not take into account ine l a s t i c e f f e c t s , the sum of the calculated i n t e n s i t i e s PF obeys the u n i t a r i t y cond i t i o n (3.17). The theoretical approach outlined above has been applied by GARCIA et a l . /5.22/ to the case of He/LiF(IO0). Figure 5.8 shows, in the upper part, the r e s u l t of the e l a s t i c calculation and, in the lower part, experimental data of FRANKL et a l . /5.11/. The calculation was based on the b e s t - f i t corrugation function for LiF obtained by GARCIA /3.19/ (Chap.7) and the bound-state energies obtained by MEYERS and FRANKL / 5 . 5 / . A comparison shows that a l l the experimentally observed features are reproduced quite w e l l , although a l l the calculated structures are sharper than the experimental ones. The theoretical i n t e n s i t y had to be m u l t i p l i e d by an overall factor of 0.43 to obtain agreement with the experimental i n t e n s i t y . This is due to i n e l a s t i c losses in the experiment and corresponds to a Debye-Waller correction (Chap.4).

5,:4 I n e l a s t i c Effects in Resonant Scattering In t h e i r work on the He/LiF /5.22/, GARCIA et a l . conjectured that the differences in the width and sometimes also the shapes of the resonant structures between the experimental and theoretical results might be due to i n e l a s t i c effects. Their conjecture was corroborated by theoretical results concerned with He scattering from graphite(O001) /5.24/, as i n e l a s t i c effects should play a larger role in graphite than in LiF because of i t s smaller Debye temperature. Figure 5.9a shows the experi-

115

(0,0)

]

,

o

20

IO

30

40

Fi9.5.8. Elastic theory (upper part) and experiment (lower p a r t ) f o r the azimuthal dependence of the specular i n t e n s i t y 0i = 70 ~ for the He/LiF(IO0) at k i = 5.76 ~ - I . The theoretical i n t e n s i t i e s are m u l t i p l i e d by a factor 0.43. Energy levels and resonant channels are i d e n t i fied in the experimental plot /5,22/

45

o,4

T~3

II:~}-(121 ,l (01}-11~)

8 0.3

(10)

u

~

| I I

I

t

o~

0

0

~ 0.2 llll-12ii

u '0'

0

12oi, , .../r~

.,"i~7~

V'

C18 0.5 Ct~ 02 cl

~" 0.3 0.2

O"

20"

Z,O~ Potar angle

60"

80 ~

Fig.5.9a-d. Specular scattering of 22 meV helium atoms from the (0001) surface of graphite. (a) Experimental results of BOATO et a l . / 2 . 9 / , (b) e l a s t i c theory, (c) with Debye-Waller attenuation according to (5.27), (d) Debye-Waller attenuation plus convolution with the energy dispersion of the experimental beam. Energy levels and resonant channels are labelled in (a). The asterisk indicates a channel strongl y coupled only to the specular beam that changes from a maximum in e l a s t i c theory to a minimum when Debye-Waller attenuation is included /5.26/

mental r e s u l t of BOATO et a l . /2.37/ for the angular dependence of the specular beam. In Fig.5.9b the r e s u l t obtained with the e l a s t i c theory /5.24/ is shown. The calculations were based on the corrugation function obtained by BOATO et a l . /2.36/ (compare Chap.8) and the bound-state energies cited in the same publication. Comparison

116

of Figs.5.9a,b reveals large deviations of the purely e l a s t i c theory from experiment concerning both the i n t e n s i t y and the d e t a i l s of the resonance structures. Note in p a r t i c u l a r that the observed widths of the resonance features are a factor of three larger than those of the e l a s t i c c a l c u l a t i o n . The structure marked by an asterisk in Fig.5.9a is observed as a minimum, whereas the e l a s t i c calculations give a maximum. An analysis of the kinematic conditions (5.5) shows that in t h i s case the bound state is coupled strongly only to the specular and to no other d i f f r a c t i o n channel. This corresponds to a s i t u a t i o n where the Weare-Wolfe rule 2 is applicable (Sect.5.2). As the i n t e n s i t i e s obtained from e l a s t i c theory did not adequately reproduce the resonant structures observed f o r He/graphite, HUTCHISON et al. /5.25/ proposed i n corporating the Debye-Waller factor i n t o the theory outlined in Sect.5.3 by making the f o l l o w i n g replacement f o r a l l

hard-wall amplitudes,

S(G,G') § exp[-W(G,G')] S(G,G~)

,

(5.26)

where the exponent in f u l l analogy to (4.2), W(w

= ~I 2 EkGz, + k~,~2

,

(5.27)

contains the perpendicular momentum t r a n s f e r with the well correction of Sect.5.1 included and the mean-square displacement of the surface atoms perpendicular to 2 the surface . As outlined in Chap.4, for the scattering from a hard surface where the c o l l i s i o n process is of short duration compared to a t y p i c a l phonon f r e quency, the Debye-Waller correction of the form (5.27) holds for both completely correlated and uncorrelated motions of the surface atoms / 4 . 4 / . The r e s u l t of t h i s rather straightforward introduction of the Debye-Waller factor for the case He/graphite is shown in Fig.5.8c /5.26,27/. Now, the overall i n t e n s i t i e s as well as the shapes and widths of the bound-state resonances are in remarkable agreement with experiment. The agreement becomes even more convincing when the f i n i t e - e n e r g y spread of the experimental beam is also taken into account (Fig. 5.8d). Note that the resonant t r a n s i t i o n marked by an asterisk in Fig.5.8a, which produces a maximum in purely e l a s t i c theory, is changed in the i n e l a s t i c theory to a minimum in agreement with observation (for a detailed discussion of when and why t h i s can happen, see /5.21,25-27/). As mentioned at the end of Sect.5.3, in the purely e l a s t i c theory, u n i t a r i t y is conserved. This is no longer the case when the Debye-Waller correction is taken i n to account according to (5.26). An experimental and theoretical i n v e s t i g a t i o n of t h i s s i t u a t i o n has been performed recently by GREINER et a l . /5.23/. Figure 5.10 shows the r e s u l t of a D-scattering experiment from LiF(O01), which corresponds to

117 '1 ....

I ....

1 ....

I';''1

....

I''';]-l-r-rr~

A

~

90-

o

80-

Ilo:o,ll

(1;0~=}) (2;0,1) (2;0, 11 [1;0,1}

.= r

~60. D --LiF

50,

i

i

= 79 =

Ei = 5 6 m e V - . t 3 m W I

J

.~ 53-

-g 0

.10 o

-5 ~

0 9

5o

10 o

15 o

20 ~

azimuthal angle y

Fig.5.10. Selective adsorption resonances of DI/LiF(IO0) appearing in the specular and the ( i i ) beams when azimuthal orientation y is varied. Only part of the intens i t y disappearing in the (00) reappears in the (11) /5.23/

strong coupling of the (TI) beam to the resonant channel (01) through the strongest periodic term in the potential v10. Therefore, at y = 6 ~, a minimum appears in the specular, whereas a maximum is observed for the (71) beam. However, only part of the i n t e n s i t y disappearing in the (00) reappears in the (71). According to the discussion at the end of Sect.5.2 (Fig.5.6), t h i s can be e a s i l y understood as the p a r t i c l e s in the bound state moving for an appreciable time parallel to the surface and therefore being able to suffer several i n e l a s t i c events before being released from the surface. Calculations on the basis of the i n e l a s t i c theory showed that this intens i t y loss is reproduced at least semiquantitatively /5.23/. In a further recent study, HUTCHISON and CELLI /5.27/ have shown that although band s p l i t t i n g is described by the theory outlined in Sect.5.3, a periodic modulation of the well depth D has to be included to reproduce the observed s p l i t t i n g for the He/graphite system /2,8,9,37/ properly. F i n a l l y , we mention the work of SOLER et a l . /5.28/, who have shown for the case of He scattering from the adsorbate system Ni(IIO)+H(I• resonance can be used to decide whether +r

that the shape of the

or -~(R) describes the real surface

p r o f i l e . In this case, the i n t e n s i t y analysis on the basis of the eikonal approximation has l e f t this question open (compare Sect.3.4.2). For small corrugations, the scattering amplitudes for +~(R) and -~(R) are nearly complex conjugate to each other, so that they lead to almost the same i n t e n s i t i e s in cases of nonresonant scattering (3.16). On the other hand, complex conjugation of the amplitudes has a strong influence on resonant structures as is e a s i l y seen from (5.25). Therefore, a decision which sign is the correct one is possible by analyzing resonant l i n e shapes.

118

6. ~4~edment~l Aspects of Gas-Sur~ce Scattedng

6.1

Requirements on an Apparatus to Perform Gas-Surface Scattering Experiments

The requirements placed on an apparatus to perform d i f f r a c t i o n experiments on s i n g l e crystal surfaces are numerous. Since i t is described to carry out studies on react i v e surfaces such as metal and semiconductors, the scattering chamber should be capable of a base pressure of better than I•

-10 Torr. The surface structure and

chemical composition should be known, which necessitates including several of the standard surface-analysis techniques such as LEED, AES, UPS, XPS, EELS, SIMS, and ion scattering. Most materials can be cleaned only by a combination of heating cycles, ion bombardment, and chemical reaction, so that a sputtering gun and gas-inlet facilities

should be present in the apparatus. Since the angle between the beam

and the surface normal should be v a r i a b l e , the sample manipulator should allow angles of r o t a t i o n about two perpendicular axes, one of which is p a r a l l e l to the surface normal. This assumes that the detector can be rotated out of the plane defined by the beam d i r e c t i o n and the surface normal. I f the detector can be rotated only in t h i s plane, the manipulator should allow r o t a t i o n s about three mutually perpendicul a r axes, one of which is p a r a l l e l to the surface normal. Apart from the need that the substrate be cleaned, characterized, and i t s orient a t i o n r e l a t i v e to the beam changed, additional requirements are placed on the beam source and the detector f o r the scattered p a r t i c l e s . The beam should have an intens i t y at the surface of approximately 1014-1015 p a r t i c l e s cm-2 s -I and should be as monoenergetic as possible for d i f f r a c t i o n studies. The angular divergence of the beam and i t s diameter at the surface should be small to increase the angular resolution a t t a i n a b l e . The detector should also have a small aperture and a s e l e c t i v e l y high s e n s i t i v i t y f o r the beam gas. As UHV techniques / 6 . 1 , 2 / and surface-analysis methods / 6 . 3 , 4 / have been discussed elsewhere, we shall r e s t r i c t our review of experimental aspects to beam sources, beam detectors, and sample manipulators suitable f o r surface studies. In the f i n a l section we shall discuss the t r a n s f e r widths of molecular beam systems and the e f f e c t of c r y s t a l l i n e imperfections on the measured i n t e n s i t i e s . In discussing beam sources, we have drawn on e a r l i e r reviews / 6 . 5 - 8 / for much of the f o l l o w i n g material.

119 6.2

Beam Sources

6.2.1

Effusive Beam Sources

The simplest sources which can be constructed are of the e f f u s i v e type. From elementary k i n e t i c theory / 6 . 9 / , the number of p a r t i c l e s which emerge from a t h i n o r i f i c e of area A at an angle e to the normal of the o r i f i c e plane into a s o l i d angle d~ S

is given by d l ( e , r ) = ~d~ n~AsCOSs

,

(6.1)

where n is the density of the gas, and ~ is the mean molecular v e l o c i t y . In terms of source pressure PO in Torr, temperature T, and molecular weight m, (6.1) becomes l ( e , v ) = 1.1•

PoAs cosS(mT) - I / 2 p a r t i c l e s sr -I s -I

(6.2)

Both (6.1,2) are v a l i d only f o r source pressures such that the mean free path behind the o r i f i c e ~ is greater than or equal to the smallest dimension of the o r i f i c e d. Because of the low density, 1(O,v) w i l l be to f i r s t

order independent of

the gas used and is 5xi016 p a r t i c l e s sr -I s -I at 300 K. At a distance of I meter from the source, t h i s corresponds to a beam i n t e n s i t y of 5•

p a r t i c l e s cm-2 s - I

Two disadvantages of e f f u s i v e sources for d i f f r a c t i o n studies are the Maxwellian v e l o c i t y d i s t r i b u t i o n of the beam and the low beam i n t e n s i t y integrated over a l l v e l o c i t i e s given by (6.2). Since a monoenergetic source is necessary f o r d i f f r a c t i v e i n v e s t i g a t i o n s (otherwise, in general only f i r s t - o r d e r d i f f r a c t i o n can be resolved), i t is necessary to combine the e f f u s i v e source with a v e l o c i t y s e l e c t o r , which adds to the system complexity. With s l o t t e d disc selectors / 6 . 1 0 , 1 1 / , a suff i c i e n t l y monoenergetic beam can be generated, but only with an appreciable loss in i n t e n s i t y since the v e l o c i t y d i s t r i b u t i o n of the effusive source is broad. The poor d i r e c t i v i t y

of the single o r i f i c e effusive source can be improved by using

long c a p i l l a r y arrays /6.12,13/. However t h i s increase in d i r e c t i v i t y

is most pro-

nounced f o r low source pressures which y i e l d low i n t e n s i t y beams. As the source pressure is increased, the d i r e c t i v i t y

becomes comparable to that of a simple o r i -

f i c e source.

6.2.2

Nozzle-Beam Sources

H i s t o r i c a l l y , JOHNSON /6.14/ showed using Hg beams that the i n t e n s i t y increased even as the Knudsen number Kn (Kn = h/d) was decreased far below u n i t y . Their i n i t i a l

ex-

120

periments showed that the detector signal increased l i n e a r l y with source pressure up to values for K of 10-3 . This corresponds to continuum flow in the o r i f i c e ren gion which is followed by a t r a n s i t i o n to molecular flow as the beam density decreases f a r from the o r i f i c e . KANTROWITZand GREY /6.15/ f i r s t

proposed the use of

continuum sources to achieve high beam i n t e n s i t i e s , and the f i r s t

experimental re-

a l i z a t i o n s were by KISTIAKOWSKY and SLICHTER /6.16/ and BECKER and BIER / 6 . 1 7 / .

In

t h e i r designs, a converging-diverging nozzle was combined with a sharp-edged conical skimmer whose function is to avoid a detached shock wave in f r o n t of the skimmer. The sharp edge and shape of the skimmer also minimize interference effects which can occur from molecules scattered from i t s inside and outside surfaces. Later work has shown that the nozzle form is not important f o r most purposes, and simple free jets have been used by most workers in recent studies. The skimmer q u a l i t y is an important factor in designing h i g h - i n t e n s i t y highly monoenergetic beams, and design c r i t e r i a /6.18/ and interference effects /6.19/ have been discussed elsewhere. A free j e t source can be simply fabricated by welding or p r e s s - f i t t i n g commercially available electron-microscope apertures into a suitable holder, and skimmers have also become commercially available /6.20/ in recent years. The gas dynamics of free j e t sources has been studied by a number of investigators /6.21-26/. In the o r i g i n a l study of KANTROWITZand GREY / 6 . 1 5 / , the v e l o c i t y d i s t r i b u t i o n at the skimmer entrance is given by m 13/2 I (u,v,w)dudvdw = /~2--~s / exp - I ~ m

[(V-Us)2 +u 2+w2]}dudvdw

,

(6.3)

where v is the p a r t i c l e v e l o c i t y p a r a l l e l to the beam a x i s , u and w are the perpend i c u l a r components, and Ts is the gas temperature at the skimmer entrance. This d i f fers from the v e l o c i t y d i s t r i b u t i o n from an effusive source in that v s is nonzero and in that T is the gas temperature at the skimmer rather than the o r i f i c e tems perature as for the effusive source. Since Ts is much lower than the skimmer temperature, the v e l o c i t y d i s t r i b u t i o n is narrow and is centred near the flow v e l o c i t y v s . This can be seen in Fig.6.1, in which Ms is the Mach number of the beam. Ms is defined as the r a t i o of us to c s, where c s is the local sound v e l o c i t y given by (YTs/m)I/2, and y is the specific-heat r a t i o Cp/Cv. Assuming an i s e n t r o p i c expansion, the maximum in the v e l o c i t y d i s t r i b u t i o n f o r a nozzle beam is greater than that f o r an e f f u s i v e source by the factor [2y/3(y-1)] I / 2 , which is 1.29 f o r a monoatomic gas such as He. In recent years, i t has become more common to characterize the v e l o c i t y d i s t r i b u tion of a nozzle beam by the speed r a t i o S given by

121

M,O I

"~f

~\

M=I0

..i

~o

O

I

RELATIVE VELOCITY ,

2

v / (~)~'lt

Fi9.6.1 ~ Theoretical v e l o c i t y d i s t r i b u t i o n s of e f f u s i v e and nozzlebeam sources /6.27/

where T is the temperature describing the mean k i n e t i c energy of the molecular motion in the gas which moves with the flow v e l o c i t y v s . For large speed r a t i o s , Av/v s and T/T 0 (where Av is the f u l l width at h a l f maximum (FWHM) of the v e l o c i t y d i s t r i bution about the average value v s, and TO is the temperature of the s t a t i o n a r y gas p r i o r to the expansion) are given by /6.26/ Av/v s ~ 1.65/S

(6.5)

and T/T 0 ~ 2.5/S 2

(6.6)

Speed r a t i o s as high as 350 have been reported f o r He beams / 6 . 2 8 / , whereas much lower l i m i t i n g values are obtained for gases such as Ne due to beam condensation at high values of PO" The speed r a t i o s which can be attained depend on the stagnation pressure PO and the nozzle diameter d as can be seen in Fig.6.2 which shows calculated values for He beams. The assumption of an isentropic expansion even at large distances from the nozzle leads to speed ratios independent of POd . However, t h i s assumption does not c o r r e c t l y take into account that no f u r t h e r cooling takes place a f t e r the trans i t i o n from continuum to molecular flow has taken place. As can be seen from the results in Fig.6.2, for increasing values of POd the distance at which t h i s t r a n s i t i o n takes place is f u r t h e r from the nozzle, allowing the expansion to reduce the temperature T more, leading to an increase in S. Since the t o t a l flow through the nozzle is proportional to POd2, large values of S can best be achieved for a given throughput using small nozzle diameters. However, nozzle clogging by impurities sets a practical lower l i m i t of about 3xi0 -2 mm for d.

122 '

''

....

I

'

''

....

i

'

'

~ ....

i

'

'

'i

....

1000 HELIUM

/

7 .~ ~

u~

~

600 tort cm

,9

o

100

expansinn

~-~

- "-

60 torr cm-

m

"~:~

10

- .......

6 tort cm

i/"

101

......

r

-

q.m.

-

|02 Reduced

distQnce

103 from

the

nozzle

10t" (x/d)

Fig.6.2. Calculated p a r a l l e l speed ratios based on classical (- - -) and quantum mechanical ( ) c a l c u l a t i o n s . Also shown is the calculated curve for an isent r o p i c expansion ( .) /6.26/

105

104

~ 1o2

3

'~ 10 /

ff

/,

e~ ""~15B

10-4

I0 -3

10 .2

10 -~

Total flow in Tort" IiteP/s

1

Fi9.6.3. Theoretical performance of several beam sources. The v e l o c i t y integrated beam i n t e n s i t y is shown as a function of the t o t a l nozzle throughput. ( I ) refers to a molecular effusion source, (2) to a multichannel array with the same t o t a l area as ( I ) , (3) to the t r a n s i t i o n region from molecular to hydrodynamic f l o w , and (4) to a hydrodynamic flow source. The nozzle area is the same as ( I ) , M = 10,

y = 1.4

/6.8/

The advantages of nozzle sources when compared with effusive sources for d i f f r a c t i v e gas-surface investigations l i e in the high i n t e n s i t y coupled with the high degree of monoenergicity which can be achieved without v e l o c i t y selection. A comparison of the v e l o c i t y integrated i n t e n s i t y obtainable with e f f u s i v e , multichannel eff u s i v e , and nozzle sources is shown in Fig.6.3. One standard beam is defined as the

123 maximum obtainable e f f u s i v e i n t e n s i t y which is 5x10

16

p a r t i c l e s sr

-I

s

-I

. I t is seen

that the t o t a l available i n t e n s i t y is much higher f o r a nozzle source, but at the expense of a much increased throughput. The advantage of a nozzle source for d i f f r a c t i o n experiments becomes even clearer when i t is considered that the e f f u s i v e source i n t e n s i t y in Fig.6.3 must be reduced considerably, since v e l o c i t y selection is necessary before impingement on the surface.

6.3

Beam Energy Variation for Effusive and Nozzle-Beam Sources

For e f f u s i v e sources the energy can be varied by heating and cooling the source. The most probable v e l o c i t y is given by (3kTo/m)I/2, and the v e l o c i t y d i s t r i b u t i o n , by (6.3) with v s = 0 and Ts = T0. The lower l i m i t is set by the vapor pressure of the gas being used, whereas the upper l i m i t is set by the mechanical properties of the oven. I f r e f r a c t o r y metals such as tungsten can be used, temperatures in excess of 2000 K can be reached. The energy of nozzle sources can be varied in the same way, although the most probable v e l o c i t y is s h i f t e d to higher values by the factor [2y/3(y-1)] I / 2 , and the nozzle cannot be cooled to very low temperatures, since condensation can occur in the expansion. The speed r a t i o w i l l decrease as TO increases, but a simple r e l a t i o n s h i p v a l i d f o r a l l gases cannot be derived. Additional methods are also available to extend the energy range f o r nozzle sources to higher and lower values. One of these techniques is seeding / 2 . 2 9 / , in which a seed gas is added in small q u a n t i t i e s (less than 10%) to a c a r r i e r gas. The seedgas p a r t i c l e s are swept through the nozzle as i f they were carrier-gas p a r t i c l e s and, depending On the r e l a t i v e molecular weights of the seed and c a r r i e r gases, w i l l be accelerated or decelerated. Very high energies can be achieved with arc sources / 6 . 3 0 / , but t h i s energy range is not useful f o r d i f f r a c t i o n experiments. For He beams, the energy range available by heating and cooling the nozzle is s u f f i c i e n t for d i f f r a c t i o n experiments on surfaces. The de Broglie wavelength corresponding to the most probable v e l o c i t y of a nozzle beam of mass m and source pressure TO is given by = 19.58 (mTo)-I/2

(6.7)

Nozzle sources with TO as low as 3 K have been reported / 6 . 3 1 / , and temperatures of 2000 K should be accessible. This corresponds to a range from 0.2 to 5.6 ~ which is more than s u f f i c i e n t for s u r f a c e - d i f f r a c t i o n studies.

124

I TO PUMPS

SCATTERING CHAMBER

1

Fig.6.4. Schematic drawing of a nozzle-beam system consisting of a nozzle-skimmer chamber, several d i f f e r e n t i a l pumping stages, and a scattering chamber

6.4

The Design of Nozzle-Beam Systems

A schematic drawing of a nozzle-beam system is shown in Fig.6.4. I t consists of a nozzle-skimmer chamber and a number of d i f f e r e n t i a l l y pumped chambers intermediate between the nozzle-skimmer and UHV scattering chambers. The intermediate chambers are necessary to reduce the pressure from the high value in the nozzle-skimmer chamber to a value compatible with UHV investigations and to insure that the f l u x into the scattering chamber is almost e n t i r e l y due to the direct beam. The number of d i f f e r e n t i a l chambers needed depends on the pressure in the nozzle-skimmer region, the beam diameter, and the pumping speeds in the d i f f e r e n t i a l stages. A simple design c r i t e r i o n is that the effusion rate from the l a s t d i f f e r e n t i a l stage calculated using (6.1) should be much smaller than the beam f l u x . Conventional nozzle-beam systems for surface studies have been pumped in a l l stages with baffled o i l - d i f f u s i o n pumps, which l i m i t the nozzle-skimmer region pressure to at most 10-3 Torr. At these pressures, c o l l i s i o n s between the beam and background particles can substantially reduce the speed r a t i o and beam i n t e n s i t y . At a throughput of I Torr 1 s - I , which for helium beams corresponds approximately to POd values of 10 and a speed r a t i o of 10, pumps with speeds of 103 1 s -I above the baff l e are required. Such pumps require flange openings of 200-300 mm diameter, r e s u l t ing in rather bulky and expensive systems. Since more than 99% of the nozzle throughput in a typical system is evacuated in the nozzle-skimmer chamber, the subsequent pumps can be substantially smaller i f several d i f f e r e n t i a l l y pumped stages are included. However, because S rises with

POd to a power of roughly 0.4 /6.27/, the

generation of highly monoenergetic beams requires very large pumping speeds in the nozzle-skimmer chamber. An a l t e r n a t i v e design for a nozzle-beam system has been proposed and realized by CAMPARGUE /6.28,32,33/. Rather than eliminating background-beam c o l l i s i o n s by de-

125 creasing the nozzle throughput, in t h i s design the j e t density is increased to the point that a barrel shock is formed about the free j e t which shields the beam from c o l l i s i o n s with the background gas. Due to the high-pressure tolerance in the nozzleskimmer region of 0.25 Torr, POd values of up to 500 Torr cm have been used with res u l t i n g speed r a t i o s of 350 for helium / 6 . 2 8 / .

In t h i s pressure range, high through-

puts can be pumped through r e l a t i v e l y small flange diameters with Roots or o i l - e j e c tor pumps, which are much less expensive than comparable throughput o i l - d i f f u s i o n pumps operating in the 10-3-10 -4 Torr range. Another recent advance in nozzle-beam systems is the development of sources which can be pulsed on for periods as short as 10 Ns / 6 . 3 4 / .

I f the time between successive

pulses is much longer than the pulse width, a high instantaneous beam i n t e n s i t y can be achieved with a low average throughput. Such a design requires much smaller pumps than are needed f o r a continuous source. However, such sources are l i m i t e d to a p p l i cations such as i n e l a s t i c scattering studies in which a low duty cycle does not necess a r i l y lead to a corresponding increase in the measuring time. In closing t h i s section, a few examples of nozzle constructions suitable f o r gas surface-scattering studies w i l l be presented. The construction used in the authors' laboratory f o r He, H2 and Ne beams is shown in Fig.6.5. The source is a I mm O.D. platinum tube, in whose side a hole of ~ 8•

-2 mm has been spark eroded. The tube

is clamped between massive copper supports, which can be cooled with water or l i q u i d nitrogen. The temperature of the tube can be regulated to better than I K between 80 K and 1500 K by d i r e c t current heating, whereby the thermoelement output is used to drive the temperature regulator. The speed r a t i o can be s u b s t a n t i a l l y improved by using smaller o r i f i c e sizes. Other designs have been published which allow e i t h e r heating /6.35/ or cooling /6.36/ to extremely high or very low temperatures. NOZZLE

Fig.6.5. Schematic drawing of the variabletemperature nozzle source used in the authors' laboratory. ( I ) platinum tube in which the nozzle is spark eroded, (2) and (3) copper supports, (4) and (5) copper tubing f o r l i q u i d - n i t r o g e n cooling. The heating current also passes through the tubing. (6) thermoelement, (7) t e f l o n i n s u l a t o r

126 The only p a r t i c l e s other than He, H2, and Ne for which d i f f r a c t i o n from surfaces has been observed are atomic hydrogen and deuterium. Although thermal sources can be constructed e a s i l y / 6 . 3 7 / , most operate in the effusive mode, which necessitates t h e i r use together with a v e l o c i t y selector f o r d i f f r a c t i o n studies. However, supersonic thermal H sources have also been reported /6.38/ with a lower degree of dissociation than would be obtained at the same temperature f o r lower pressures, but with a more desirable v e l o c i t y d i s t r i b u t i o n f o r d i f f r a c t i o n experiments. Microwave discharge sources have been successfully used to generate supersonic beams of atomic species, and one example of such a source /6.39/ is shown in Fig.6.6. Source pressures of up to 200 Torr have been used in t h i s design.

J\-, |

,

25 mm

IG G

o

t

rA [.~ L.rd

Fig.6.6. Schematic drawing of a microwave discharge source. (A) g a s - i n l e t tube, (B) quartz discharge tube, (C) microwave c a v i t y , (D) tuning electrode, (E) support disc, (F) gear housing, (G) water-cooling tube, (H) nozzle, ( I ) skimmer, (J) drive shaft, (K) shaft guides, (L) axis of coupling stab, (M) O-ring seal, (N) level gears, (0) t e f l o n washer, (P) framework /6.39/

6.5

Molecular-Beam Detectors

Molecular-beam detectors should have a high selective s e n s i t i v i t y for the beam gas to increase the s i g n a l - t o - n o i s e r a t i o and to reduce the measurement time necessary f o r a d i f f r a c t i o n trace to a minimum. This is p a r t i c u l a r l y important f o r h i g h l y reactive surfaces f o r which the measurement time is l i m i t e d to about 30 minutes to avoid surface contamination even at reactive gas p a r t i a l pressures of I x i 0 -I0 Torr.

I27 Most investigators have used i o n i z a t i o n detectors combined with mass spectrometers to achieve the selective s e n s i t i v i t y required. The detectors can be e i t h e r of the flow-through type /6.40-42/, in which the density is measured, or of the stagnation type / 6 . 4 3 / , in which the f l u x is measured. I f the apparatus is to be capable of t i m e - o f - f l i g h t measurements in addition to d i f f r a c t i o n experiments, onl y the flow-through detector can be used. For d i f f r a c t i o n experiments, a choice between these two types should be based on the s e n s i t i v i t y , time constant, and ease of construction of the detector. For flow-through i o n i z a t i o n detectors which are not d i f f e r e n t i a l l y

pumped, a s e n s i t i v i t y of 4•

-4 1O, where I 0 is the d i r e c t beam

i n t e n s i t y , has been obtained in the authors' laboratory with a commercially a v a i l able mass spectrometer. (Phase-sensitive detection at 75 Hz, 1:1 s i g n a l - t o - n o i s e r a t i o , I s time constant.) Stagnation detectors in s i m i l a r l y designed systems have been able to detect 10-3 10 /6.44/. The time constant

9of the stagnation detector

is given by V/Sp, where V is the enclosed volume about the detector, and Sp is the conductance of the aperture. Although T in many applications is as high as 30 seconds /6.43/ f o r small detector volumes,

9 can be made s u f f i c i e n t l y low that stagnation

detectors can be combined with chopped beams and phase-sensitive detection to y i e l d better s i g n a l - t o - n o i s e r a t i o s than conventional flow-through detectors /6.45/. The s e n s i t i v i t y l i m i t s cited above are also dependent on the background pressure at the detector and can be s u b s t a n t i a l l y increased e i t h e r by d i f f e r e n t i a l l y

pumping the

detector or by incorporating extremely high-speed pumps in the scattering chamber. S e n s i t i v i t i e s of I x i 0 -5 10 /6.46/ and better than 10-4 10 /4.17/ have been obtained with a single d i f f e r e n t i a l pumping stage around the detector and with a cryogenicall y pumped system and stagnation detector, respectively. For well-ordered surfaces with not too many intense d i f f r a c t i o n beams, detectors which are not d i f f e r e n t i a l l y pumped are s u f f i c i e n t f o r d i f f r a c t i v e studies. Figure 6.7 shows a H e - d i f f r a c t i o n trace from the (I•

reconstructed Au(110) surface, which was obtained with a time

constant of 0.3 s in less than 180 s. Such a rapid rate of data accumulation is often necessary as in the above case in which 50 such spectra at d i f f e r e n t angles of incidence were measured in a 2-hour period to analyze resonant scattering from the Au-He potential w e l l . Surface contamination would have f a l s i f i e d the r e s u l t s i f a s u b s t a n t i a l l y longer measurement time had been used. Another type of detector, which has been used p a r t i c u l a r l y in combination with cryogenically pumped systems, is the bolometer. This detector is s e n s i t i v e to the energy of the impinging p a r t i c l e s rather than to the f l u x or density. Investigations using bolometers in the d i f f r a c t i o n experiments have achieved a s e n s i t i v i t y of better than 10-4 10 /2.11/. Recently, a superconducting bolometer, whose sensit i v e element is a t i n or indium f i l m , has been described /6.47/ which has a response time of I ~s. The minimum detectable signal ( s i g n a l - t o - n o i s e 1:1, I s time constant)

128 He Diffraction from Au (1t0) (Ix2) Ts = lOOK

x : i.o9~, 04-

8i = 25~ lOOt] ozimuth

O5

I

oo O~

~167176 /

Scattering angle e

ot

P

Fig.6.7~ D i f f r a c t i o n scan from the Au(110) (I• surface i l l u s t r a t i n g the s i g n a l - t o noise r a t i o for a rapid scan with a time constant of 0.3 s

is 1.2xi08 Ar atoms s -I in a detector area of 4xi0 -2 cm2, The fast response time also allows such bolometers to be used for t i m e - o f - f l i g h t measurements. Atomic hydrogen can best be detected by monitoring the change in the surface c o n d u c t i v i t y of a ZnO crystal upon H adsorption /6.48/. The detector is highly sel e c t i v e and responds only to atomic hydrogen and oxygen. I t is l i n e a r with hydrogen coverage over a wide range and can be p e r i o d i c a l l y regenerated by desorbing the adsorbed atoms by a mild heating. The minimum detectable signal is about 1011 atoms -2 -I

cm

s

Other detector modifications which have been described include a cold-cathode ion source, which enables a quadrupole mass spectrometer to be used in a cryogenic environment /6.49/ and a f i e l d i o n i z a t i o n detector for helium atoms /6.50/.

6.6

Detector Rotation

In d i f f r a c t i o n experiments, the i n t e n s i t y of as many possible d i f f r a c t i o n features should be measured f o r f i x e d angles of incidence of the molecular beam. This requires r o t a t i o n of the detector about the scattering centre. One approach has been to rotate the detector w i t h i n an all-metal sealed vacuum system using rotary-motion feedthroughs to transmit the motion through the chamber walls. The design used in the authors' laboratory is of t h i s type and is shown schematic a l l y in Fig.6.8. I t allows f o r two independent r o t a t i o n s , one of which is in the scattering plane and the second perpendicular to t h i s plane. The bearing for the in-plane r o t a t i o n consists of an upper plate and a lower ring with a V-shaped c i r cular groove. Three equally-spaced carbide b a l l s of 15 mm diameter separate the

129

Fi9.6.8. Schematic drawing of the detector r o t a t i o n mechanism used in the authors' laboratory. ( I ) bearing for in-plane rot a t i o n , (2) bearing for out-of-plane rot a t i o n , (3) quadrupole mass-spectrometer detector, (4) and (5) coupling gears, (6) and (7) drive shafts f o r in-plane and out-of-plane rotations

r i n g s , and allow low f r i c t i o n r o t a t i o n without a bearing cage /6.51/. The bearing f o r the out-of-plane r o t a t i o n is s i m i l a r in design but contains a perforated ring segment to keep the b a l l s equally spaced. The angular motion requires f l e x i b l e detector cables which in our design l i m i t the motion to 200 ~ in plane as well as 15~ above and 40 ~ below the scattering plane. Although this design has the important advantage that out-of-plane scans can be recorded, i t cannot e a s i l y be combined with a differentially-pumped detector. A second approach is to f i x the detector on a large flange and to rotate the ent i r e flange as has been done by AUERBACHet a l . / 6 . 5 2 / . Their design is shown schema t i c a l l y in Fig.6.9. The seals are made with spring-loaded MoS2 f i l l e d

t e f l o n O-rings.

Due to the d i f f e r e n t i a l pumping between adjacent O-rings, the s t a t i c leak rate is less than 10-13 Torr 1 s - I . The pressure rise while r o t a t i n g the flange is less than I x i 0 -iO Torr. This design allows f o r convenient d i f f e r e n t i a l pumping of the detect o r , but must be combined with a three-axis manipulator in order to measure out-ofplane scattering.

6.7

Sample Manipulators

Examples of two- and three-axis manipulators suitable for use in UHV w i l l be presented in t h i s section. In order to carry out H e - d i f f r a c t i o n studies, i t would be desir-

130 Rofotion Axis ",- R

18cm-~,=

Atm__ To .

TO

~

~

~

Spring' Looded

Vocuum

I LL

....

J

5 cm

Fig.6.9. Schematic drawing of a r o t a t i n g flange f o r use in UHV showing the springloaded t e f l o n seals and the d i f f e r e n t i a l pumping stages /6.52/

able to modify these or other s i m i l a r designs to be able to reach as low temperatures as possible. In this way, the uncertainty in correction f o r i n e l a s t i c effects (Chap.4) can be minimized. The two-axis manipulator shown in Fig.6.10 /6.53/ features an e l e c t r i c a l readout f o r the azimuthal angle based on a wire-wound potentiometer in vacuum. The mechanism can be added onto commercial manipulators which have a f l i p actuator. The polar angle can be varied by •

~ , and the azimuthal angle v a r i a t i o n is •

~ with an e l e c t r i c a l

resolution of 0.5 ~ The construction is completely nonmagnetic, and the eight adjustment screws allow a 5 mm l i n e a r displacement of the sample as well as a 5 ~ t i l t

to

align the crystal surface p a r a l l e l to the bearing plane. The three-axis manipulator shown in Fig.6.11 /6.54/ allows independent angular v a r i a t i o n s of 240~

~ and 40 ~ about the A, B, and C axes, r e s p e c t i v e l y , with an

angular r e s o l u t i o n of 0.02 ~. The r o t a t i o n about the A axis is carried out with the central shaft of the manipulator, A r o t a t i o n of one of the outer two of the three coaxial drive shafts causes a keyed lead nut to t r ans la t e along the threaded lower end. The movement is converted back to a r o t a t i o n a l movement by means of a rack and pinion. The sample is mounted in a gimbal assembly which allows i t to rotate about i t s normal axis regardless of the t i l t

setting. The high angular resolution is

achieved by spring loading the keyed nuts to eliminate backlash and by using large gearing-down r a t i o s of 360:1 for the B and C r o t a t i o n and 100:1 f o r the A r o t a t i o n .

131

SIDE VIEW

FRONT VIEW

I

s

T R

SE~:==~I~S

Fig.6.10. Drawing of the manipulator attachment for azimuthal rotation with elect r i c a l readout. (T) target, (R) W rod, (S) alumina support, (AS) adjustment screws, (CuL) copper current leads, (BB) ball bearings with sapphire b a l l s , (TC) thermocouple, (F) holder, (C) transmission cord, (MS) manipulator shaft, (FA) f l i p actuator, (L) lever, (TA) travel adjustment, (P) pulleys of the reversed block and tackle, (Sp) return spring, (LP) linear potentiometer /6.53/

6.8

Beam-Modulation Devices

In d i f f r a c t i o n experiments which u t i l i z e fast detectors, beam chopping combined with the use of phase-sensitive techniques can lead to a significant improvement in the signal-to-noise r a t i o . In contrast to reactive scattering-surface techniques for which frequency variation is an important asset /6.55/, a fixed frequency chosen to l i e in a low-noise regime is s u f f i c i e n t in d i f f r a c t i o n experiments. In gas phasescattering experiments, modulation has often been carried out with a motor mounted in the beam system, but other designs have been used which avoid the contamination problems associated with this solution. Tuning-fork choppers /6.56,57/, piezoelect r i c devices /6.58/, and magnetically-coupled feedthroughs /1.5/ have been incorporated into scattering system designs. The beam-modulation device used in the authors' laboratory is shown in Fig.6.12 /6.59/. I t consists of a rotary feedthrough flanged onto the system onto which a chopper wheel is mounted from the inside. I t is compatible with UHV environments and can be rotated between 1,500 and 15,000 rpm. I t is based on an asynchronous

132

Fi9.6.11~ Schematic drawing of a UHV three-axis goniometer showing the r o t a t i o n axes A, B, and C which i n t e r a c t at the surface /6.54/

motor; in t h i s application the rotor is supported by bearings inside the vacuum system and the s t a t o r , which is more prone to outgassing, is mounted outside the vacuum envelope. Perhaps the most elegant modulation device is that based on a magnetic bearing which was developed at the Kernforschungsanlage JUlich /6.60/. Since there is no f r i c t i o n in operation, the r o t a t i o n frequency is l i m i t e d only by the mechanical properties of the rotor and chopper wheel. This and the rotary feedthrough discussed above are the most v e r s a t i l e devices, since they can be used for reactive scattering and t i m e - o f - f l i g h t investigations in addition to d i f f r a c t i o n studies.

133

Millim~ers

Fig.6.12. Schematic drawing of the beam-modulation device used in the authors' laboratory. ( I ) motor s t a t o r , (2) motor rotor, (3) shaft, (4) support cylinder, (5) 2 3/4" flange, (6) bearings, (7) and (8) stater supports, (9) vacuum envelope, (10) spring washers, (11) and (12) chopper-disc clamps, (13) chopper disc, (14) fastening screw /6.59/

6.9

Experimental Systems for D i f f r a c t i v e Scattering from Surfaces

The requirements for the individual parts of a gas-surface scattering apparatus have been described in Sects.6.1-7. In this section we shall describe four d i f ferent approaches to combine these parts to construct an apparatus for d i f f r a c t i v e scattering. Figure 6.13 shows the apparatus used by the Genoa group /3.20/ to study d i f f r a c tion from a number of surfaces. The crystal surface S is mounted on a three-axis goniometer and can be rotated in the scattering plane. The goniometer is in thermal contact with the l i q u i d helium in cryostat I I which allows substrate temperatures of 10 K to be reached. The detector is a liquid-helium cooled bolometer which can be rotated in the scattering plane. Rotating cryostats I and I I also act as high-speed pumps for gases other than He and H2 and generate a UHV environment in the v i c i n i t y of the surface. The helium generated by the beam is pumped away by o i l - d i f f u s i o n pumps mounted on the nozzle-skimmer and scattering chambers. The sensitivity

of the apparatus is approximately 5•

-4 10.

A second apparatus which uses both o i l - d i f f u s i o n and cryopumping techniques is shown in Fig.6.14 /6.43/. I t consists of nozzle-skimmer (A) and collimator (B) chambers pumped by o i l - d i f f u s i o n pumps and the UHV scattering chamber (C) pumped by liquid-helium cooled copper surfaces. Their surfaces were coated with a porous Ca-Ag a l l o y which can be used to pump a l i m i t e d amount of helium. The capacity is s u f f i c i e n t l y high that no other pumps are needed in the scattering chamber. The crystal surface and the stagnation-detector o r i f i c e are located in the region l a -

134

SHIELD rAT

I

rAT

II

All

Fi9.6.13. Schematic drawing of a cryogenic system with a bolometer detector for gas-surface scattering studies /3.20/

E

J

i

N

Fig.6.14. Schematic drawing of a cryogenic system with a stagnation detector for gas-surface scattering. (A) nozzle skimmer chamber, (B) different i a l pumping chamber, (C) scattering chamber, (L) detector. The substrate and detector o r i f i c e are in the region labeled J /6.43/

135 beled J. The i o n i z a t i o n gauge detector L is mounted outside the low-temperature zone and is coupled to the scattering region with small diameter tubing and f l e x i b l e s t a i n less-steel bellows. The time constant is 30 s which l i m i t s the data accumulation rate. The s e n s i t i v i t y of the apparatus was not stated, but based on a l a t e r p u b l i c a t i o n / 6 . 6 1 / , we estimate i t to be between 10-3 and 10-4 1O, An apparatus which uses commercially-available UHV components is shown in Fig.6.15 /6.62/.

I t has the novel feature that the beam source is rotated w i t h i n the scat-

t e r i n g chamber, while the detector remains f i x e d . A l l other pumps are e i t h e r of the l i q u i d - n i t r o g e n baffled o i l - d i f f u s i o n or titanium-sublimation type. Typical operating pressures during a He-beam scattering experiment are 10-3 Pa in the nozzle-skimmer chamber and 10-8 Pa in the detector chamber. Although the pumping speed in the nozzle-skimmer chamber is conductance l i m i t e d to 500 1 s - I , a POd value of 10 can be attained for He. The s e n s i t i v i t y of the apparatus was not stated, but based on a l a t e r publication / 6 . 6 3 / , we estimate i t to be I•

LN2

CRYO6ENIC--~ PUMP ~

/I I

// / I{ {

\\ \ ~ .

~

[~---~.I"--

~ l

tl

SKIMMER~V_=."'~

"

....... < Z//~ > \ NOZZLE~ V ~ ~

-4 10.

AUGER ELECTRON

SPECTROMETER ~ ION ~

~

1 ~\ ~~

/ z ~

. r

/ - - MASSSPECTROMETER

"--:

III

ALIGNMENT

~ Ill TELESCOPE '-DETECTION ~--DETECTION

%- SCATTERING CHAMBER

WINDOW

Fig.6.15. Schematic drawing of a molecular-beam surface-scattering apparatus in which the nozzle-skimmer assembly is rotated and the detector is fixed /6.62/

Figure 6.16 shows an apparatus which incorporates LEED and AES for surface anal y s i s , and which u t i l i z e s a differentially-pumped i o n i z a t i o n detector /6.64/. Chamber I contains the nozzle and skimmer whose arrangement can be seen in more d e t a i l at the lower l e f t .

This chamber is pumped with a water-baffled 500 1 s -I o i l - d i f -

fusion pump. Chamber 2 contains the motor-driven chopper and is pumped with a l i q u i d nitrogen baffled 1250 1 s -I o i l - d i f f u s i o n pump. Chamber 3 is the UHV scattering chamber and is pumped with three 15 cm l i q u i d - n i t r o g e n baffled o i l - d i f f u s i o n pumps. The

136

Fig.6.16~ Schematic drawing of a molecular-beam surface-scattering apparatus. The three chambers are labeled I , 2, and 3. At the lower l e f t is an enlargement of the nozzle (NZ) and skimmer assembly (SK) contained in chamber I. Chamber 2 contains a chopper (CH) mounted on a micrometer head. Between chambers I and 2 is a butterfTy valve, and between chambers 2 and 3, a gate valve (GV) with three collimation apertures. Inside chamber 3 is an Auger spectrometer (AES) and LEED apparatus on bellows, a rotatable d i f f e r e n t i a l l y pumped quadrupole mass spectrometer (MS), a precision manipulator (PM), and an ion gun (IG) /6.64/

quadrupole-mass-spectrometer detector is housed in the large cylinder v i s i b l e in the scattering chamber. I t rotates together with the housing, and the d i f f e r e n t i a l pumping is achieved by including a very small gap between the rotating cylinder and the stationary pump flange opening. The s e n s i t i v i t y of the apparatus is approximately ID -5 10 for long time constants /6.46/ Each of these form designs has advantages for d i f f e r e n t types of investigation. I f a cryopumped design is used, a fast bolometer detector /6.47/ would be a better choice than a stagnation detector, since modulated-beam and t i m e - o f - f l i g h t measure-

137 ments are possible. The Genoa design is well suited f o r high-angular-resolution measurements f o r which a high s e n s i t i v i t y is required. Furthermore, cryogenic systems are i d e a l l y suited for low-temperature

studies such as rare-gas adsorption

on solids. The metal UHV systems are better suited for experiments on reactive surfaces f o r which LEED and AES are necessary. However, i t is more d i f f i c u l t

to work

with l i q u i d - h e l i u m cooled detectors and substrates in such systems because of the thermal r a d i a t i o n load.

6.10

The Influence of the Transfer Width of the Apparatus and of Surface Perfection on Measured I n t e n s i t i e s

Both LEED and atom d i f f r a c t i o n are techniques in which the d i f f r a c t i o n pattern arises through the interference between the plane-wave components of the wave packet which represents each incoming p a r t i c l e . ty is 1012-1013 cm-2 s - I ,

(For both methods, the incoming p a r t i c l e i n t e n s i -

i . e . so low that i n t e r a c t i o n s between p a r t i c l e s can be neg-

lected.) As the detectors in both techniques are not s u f f i c i e n t l y sensitive to detect a single event, the experimental i n t e n s i t i e s are enhanced by the superposition of many i n d i v i d u a l d i f f r a c t i o n events in a given time i n t e r v a l . Each wave packet gives rise to a d i f f r a c t i o n pattern and the i n t e n s i t i e s of a l l these events are added to produce the f i n a l measured i n t e n s i t y . The question which arises in i n t e r p r e t i n g such d i f f r a c t i o n data is to what extent the instrument i t s e l f and to what extent the cryst a l l i n e perfection of the surface l i m i t the widths of the d i f f r a c t e d wave packet(s) characterizing an i n d i v i d u a l d i f f r a c t i o n event. Since the observed d i f f r a c t i o n spots represent the superposed i n t e n s i t i e s of many such events, the widths in question are closely related to those of the measured spots as long as the l a t t e r do not overlap appreciably. The problem of the instrumental l i m i t a t i o n in LEED has been discussed by a number of authors /6.65-67/. PARK et a l . /6.65/ has introduced the concept of a t r a n s f e r width which represents the minimum l a t e r a l dimension over which the surface must be perfect to give d i f f r a c t i o n spots whose widths are l i m i t e d by instrumental resolution alone. I t depends on the energy spread in the electron beam and on geometrical parameters of the instrument. For most LEED instruments, the t r a n s f e r width is of the order of 100 ~ / 6 . 6 7 / . COMSA/6.68/ has derived an e x p l i c i t formula for the t r a n s f e r width in atom d i f f r a c t i o n experiments which again depends on the energy spread in the incoming beam and the geometrical parameters of the source and detect o r . The geometry of a t y p i c a l experiment is shown in Fig.6.17. The angular spread at the detector due to the source and detector geometry is given approximately by /6.68/

138

' k

\

i

Y~deteclor F!9.6.17. Schematic diagram of an apparatus for atom d i f f r a c t i o n studies. 5s, 6a, and 6d are the source, c o l l i mation aperture, and detector aperture diameters, respectively /6.68/

%

~

+

~

+

0d 0a/ + t d/

'

(6.8)

and the contribution to the transfer width which arises from this geometrical f a c t o r , w0, is given by

we - x / ( I A o eflcose f)

(6.9)

The energy-spread contribution to the transfer width arises from the fact that i n t e n s i t y maxima in the d i f f r a c t i o n pattern from a p a r t i c l e of a given energy can overlap with i n t e n s i t y minima from a p a r t i c l e of d i f f e r e n t energy. This contribution to the transfer width can be written as WE ~ X/{isinOi_sinSfl[(AE)2/E ~ I/2 }

(6.10)

where (AE)2 is the mean-square energy spread in the beam. Combining the two v e c t o r i a l l y , Comsa obtains w ~ X/[(A 0 Of)2cos2Of+(sinOi-sinOf)2 (AE)2/E2] I/2

(6.11)

For 0i = ef, corresponding to specular r e f l e c t i o n , the energy dependence disappears and the transfer width when extracting information from the specular beam w i l l be greatest as grazing angles are approached. Large transfer widths can be obtained by minimizing the angular divergence in the beam and the beam diameter at the surface, as well as by using a small detector aperture and as large a surface to detector distance as possible. However, a l l of these steps w i l l lead to a signal reduction at the detector; therefore, when designing a atom-diffraction system, there w i l l be a tradeoff between transfer width and signal

139 i n t e n s i t y . At present, i t appears that using beam diameters and detector apertures of roughly 0.2 mm diameter transfer widths roughly equal to those in LEED can be obtained with He nozzle beams. The above factors describe the l i m i t a t i o n s which the instrument imposes on the d i f f r a c t i o n experiment. An additional angular broadening of the observed beams w i l l be observed i f the crystal is not well-ordered over dimensions which are small in comparison with the t r a n s f e r width. This problem has been considered by LAPUJOULADE et a l . / 4 . 1 1 / . They derived a formula f o r the case in which scattering occurs coherently over a l a t e r a l length ~c and the i n t e n s i t i e s from a l l such areas on the surface add incoherently. For in-plane scattering they obtain the f o l l o w i n g expression f o r the angular d i s t r i b u t i o n in the i n t e n s i t y I:

where the in-plane tangential momentum change AK is given by (6.13)

AK = I k l ( s i n O i - s i n e f )

This leads to a broadening in addition to the instrumental effects discussed above, and the t o t a l mean-square c o n t r i b u t i o n s from both effects can be w r i t t e n in the form 9

. + Ao 7 z AO~xp = AO~ Ins~r

(6

'

14)

where the subscripts indicate the experimentally determined, instrument l i m i t e d , and f i n i t e - s i z e l i m i t e d contributions to the angular widths

respectively. Since Ae~ . inscr can be determined by measuring the angular d i s t r i b u t i o n of the i n t e n s i t y in the d i '

rect beam, Ae~z_ can be determined. LAPUJOULADEet a l . /4.11/ have carried out such an analysis for He and Ne scattering from Cu(100) and, assuming that the specular peak broadening is due to purely e l a s t i c s c a t t e r i n g , obtain a c h a r a c t e r i s t i c length of 78 ~. The assumption that a l l the i n t e n s i t y in a measured scattered beam envelope is purely e l a s t i c is not always j u s t i f i e d ,

although i t should be v a l i d for cases such

as He s c a t t e r i n g from metal surfaces at low temperatures where the specular i n t e n s i ty can be as high as 70% of the incoming i n t e n s i t y . However, in some cases a pronounced broadening of a d i f f r a c t e d beam is observed, as f o r H scattered from KCI(IO0) f o r which r e s u l t s are shown in Fig.6.18 / 6 . 6 9 / . This d i s t r i b u t i o n can be separated into three parts: e l a s t i c , q u a s i - e l a s t i c , and i n e l a s t i c plus incoherent e l a s t i c . The form of the e l a s t i c c o n t r i b u t i o n is given by the d i r e c t beam p r o f i l e ; WILSCH et a l , assumed that the q u a s i - e l a s t i c c o n t r i b u t i o n has a Gaussian d i s t r i b u t i o n and that the

140 >~- 1.0 z uJ

!

I

I

I

I

I

I

~1~# H~KCI ~ 8o =60~

5"

0 0

_

F- 1.0

0

,~176 ~ 58 ~

A

D

o

UO0"~ 9 oo

/ 60 ~

62 ~

SCATTERING ANGLE e Fig.6.19. Separation of the specular beam shown in Fig.6.18 into cosine background (A), coherent e l a s t i c (o) and incoherent (o) i n t e n s i t i e s /6.69/

i n e l a s t i c plus incoherent e l a s t i c background has a cosine d i s t r i b u t i o n . With these assumptions, the i n d i v i d u a l contributions to the specular peak shown in Fig.6.18 are presented in Fig.6.19. Angular i n t e g r a t i o n shows that the coherent e l a s t i c , q u a s i - e l a s t i c and cosine contributions represent 38.8%, 26.7%, and 34.5% of the t o t a l i n t e n s i t y , respectively. These results show that careful examination of the peak p r o f i l e s is necessary to establish whether the raw data must be deconvoluted to obtain the e l a s t i c part before carrying out a s t r u c t u r a l c a l c u l a t i o n .

7. Structural Investigations on Surfaces of Ionic Crystals

7.1

D i f f r a c t i o n Studies on LiF(IO0)

The very f i r s t

atom-diffraction studies were performed in 1929 with effusion sources

by ESTERMANNand STERN / 1 . 1 / on the system He/LiF(IO0) in order to establish the de Broglie r e l a t i o n (3.1). Since then t h i s system has been investigated in great d e t a i l , and the f i r s t

corrugation function from which s t r u c t u r a l implications were deduced

has been reported f o r t h i s system. More than f o r t y years a f t e r ESTERMANNand STERN, BOATO and coworkers / 7 . 1 , 3 . 2 0 / published the f i r s t

d i f f r a c t i o n patterns obtained with He nozzle beams of good mono-

chromaticity, and examples of t h e i r results with ~i = 0.57 ~ are shown in Fig.7.1 for two d i f f e r e n t in-plane scans corresponding to d i f f e r e n t azimuthal orientations

141 H e - Li F ( O 0 1 )

(-1,0)

(110) (-2,0) .001

, o

0

a)

ULI3L, .o2

-30 o

>,. )-

(1,0)

0o

30 ~

,

60 ~

90 ~

Z uu

I,-

z

(1,1)

e.,, tlJ u,,i i-

(100)

o r

b) (-2,-2)

~ (-1,-11

,002

- 30 ~

0o

300

SCATTERING

60o

90 ~

ANGLE Gf (0,1) and (1,0)

He - Li F (001) .00,

bound state resonances

a)

0 "''""...

~- ~)2

..."" 9

(IM/Io) x 10 /

..

"'.... ,

i

Fig.7.1a~b. H e - d i f f r a c t i o n spectra from LiF(IO0) at 80 ~ K f o r 0i = 30 ~ w i t h ~i = 0.57 ~. I n c i d e n t beam along (a) [110] azimuth, (b) [100] azimuth / 3 . 1 9 /

.''""

. .., ..., .......

;~i :' :"

." .... ""i . . . . . . . . . ~ , , , "

,."i"

I

I

I

a (11~)

b)

:-ll

0z DE

Oo

........ ' " " ( . I M ~ IO)~10

f

i

~:

I

30o

"'r"

I

/ ..i'"

60 o

"k,.,:"

J

~

f

=NCmENT ANGLE el

Fig.7A2a,b, I n t e n s i t y of the specular beam as a f u n c t i o n of 0i f o r He/LiF (Xi = 0.57 ~). I n c i d e n t beam along (a) [110] azimuth, (b) [100] azimuth. Bound-state resonances play a ro]e f o r 0. > 50 ~ / 3 . 1 9 / 1

of the sample. Figure 7.2 shows the i n t e n s i t y of the specular beam f o r these two o r i e n t a t i o n s as a f u n c t i o n of the angle of incidence. These curves e x h i b i t a smooth variation until

0. ~ 50 ~ where s e l e c t i v e adsorption features s t a r t to cause rapid 1

142 angular intensity variations. These data constituted the basis for the f i r s t thorough structural surface analysis based on He d i f f r a c t i o n ,

which was performed by GARCIA

/3.17,18/. Using the GR method (Sect,3.4.1), he established the b e s t - f i t corrugation function by f i t t i n g the angular variation of the intensities of several beams for two azimuthal orientations as shown in Fig.7.3. The b e s t - f i t corrugation has the analytical form ~(x,y) =21 ~I0 [cos--2~X+cosa 2 _ ~ ] + 12 ~11 [cos2~ (x+Y)a + cos2~ (x-Y)a ]

1oI

i

..,.~-,,~.=" ({,o)

l

(7.1)

(~'i)

:J//' 1o~ ~'~ /

t0 0

'~

V 0

20

40

/

80

2,2)

J

i

~

i

I0i (3.

t

S~ t

i

'~ / ,'//" X / / / 1 / I < , , , >I I /1 i'l I

I0e~

i0 (

o

60

~162

~ I

20 40 60 INCIDENT ANGLE 81(~

2'o

'q 40

~3 '

Incident Angle el(O)

Bb

O

0

_,,02) with L being the rotational quantum number, kB Boltzmann's constant, and Tr (~ 85 K) the characteristic rotation temperature of H2. The peaks observed in Fig.7.5 correspond to transitions 0 + 2, 2 § O, and 3 § I . The t r a n s i tion I § 3 is not possible in this experiment, since the beam energy is smaller than the rotational-energy change Erot(1§

= 10 kBTr.

An analysis of the corrugation function was not performed for H2/LiF, because of the appreciable r o t a t i o n a l l y i n e l a s t i c contributions and because of the enhanced ine l a s t i c effects due to phonon excitations (compared to He d i f f r a c t i o n ) . The l a t t e r can be observed from the smaller f r a c t i o n of e l a s t i c a l l y scattered p a r t i c l e s and by the large t a i l s around the d i f f r a c t i o n peaks. The reason for this enhanced i n e l a s t i c scattering may be due to the fact that although the mass of H2 is half that of He, i t s e f f e c t i v e incident energy El = E.+D with D ~ 40 meV is about doubled. A rough 1

1

estimate of the corrugation amplitude could be performed by using the rainbow angles (3.52) and yields ~I0 ~ 0.2 ~, which is not unreasonable in view of the He and Ne results.

7.2

D i f f r a c t i o n Studies on NiO(100)

Another substance with highly ionic bonding character, whose (100) surface has been investigated in detail with p a r t i c l e beam d i f f r a c t i o n , is NiO. Both scattering of He and H2 were reported by CANTINI et a l . /2.10/. The data for He d i f f r a c t i o n were caref u l l y analyzed l a t e r by using Patterson series /2.11/, and the best agreement between experiment and calculation was obtained with the Fourier coefficients ~I0 = 0.139 ~, ~11 = 0.009 ~, ~20 = 0.007 ~, and ~21 = 0.010 ~, where (3.2) for a quadratic l a t t i c e with inversion symmetry was used. A contour map of this corrugation function is shown in Fig.7.6. The maximum corrugation amplitude is ~m = 0.28 ~, which is much smaller than the difference of the bulk ionic r a d i i of 02- and Ni2+: r(O 2 - ) - r ( N i 2§ = 1.4-0.72 = 0.68 ~ / 7 . 2 / . LEED calculations

/7.13,14/ f i t t i n g experimental inten-

!47

1/2ao

1/4

1/4 J

l

,/"

./"

1/2ao

/.//

/./

/

,/

,/

J

! !

I

// !

/

/

//Jf~ - 0.10 ! ! !

i

/

/

i

Fi9.7.6. Contour map of the corrugation function for NiO(100) as obtained by CANTINI et a l . /3.27/. Contour levels are given in Angstrom units. The maximum corrugation amplitude is 0.28

s i t y versus voltage curves /7.15/ indicate that the location of the surface ion cores does not d i f f e r from the corresponding bulk locations by more than •

~. Also,

theoretical model calculations for surface relaxations of oxide ions predict only s l i g h t changes of the core locations /7.16/. The small corrugation amplitude obtained with He d i f f r a c t i o n suggests therefore that at the surface of NiO(100) an appreciable charge r e d i s t r i b u t i o n must take place, the location of the cores being very nearly the same as in the bulk. D i f f r a c t i o n of H2 from NiO(100) again e x h i b i t s additional peaks due to r o t a t i o n al transitions and also shows appreciably stronger i n e l a s t i c contributions due to phonon scattering than He d i f f r a c t i o n . In analogy to H2/LiF, the l a t t e r e f f e c t is probably again due to the larger potential depth for H2/NiO (D ~ 60 meV) than for He/NiO (D ~ 20 meV). From the positions of the rainbow maxima, CANTINI et a l . e s t i mated the parameter ~I0 ~ 0.25 ~. I t is remarkable, but not at a l l understood, that the NiO(100) surface is f l a t t e r than LiF(IO0) when seen by He, while i t is more corrugated and probably looks more complex when seen by H2 molecules /2.10/.

7.3

D i f f r a c t i o n from Other lonic Materials

Rather early He-diffraction studies with nozzle beams from NaCI(IO0) and LiF(IO0) were reported by BLEDSOE and FISHER /7.17/. Since at low sample temperatures, contaminations of the surfaces (especially of NaCI) due to the poor background pressure of 10-7 Torr were obtained, the experiments were performed with the samples at room temperature. Whereas the d i f f r a c t i o n patterns of LiF show rather small ine l a s t i c contributions, the scattering from NaCI is dominated by i n e l a s t i c events due to the lower Debye temperature. Nevertheless, GARCIA et a l . /7.18/ attempted to estimate the corrugation amplitude and found ~I0 = 0.34 2. This amounts to a

148 maximum corrugation amplitude Cm ~ 0.7 R which is reasonably close to the d i f f e r ence of the bulk i o n i c r a d i i of CI- and Na+: r ( C l - ) - r ( N a +) = 1 . 8 1 - 0 . 9 8 = 0.83 /7.2/. HI d i f f r a c t i o n from KCI(IO0) was investigated by FRANK et al. / 2 . 4 / . For sample temperatures ~ 170 K, the small residual pressure of water gave rise to an ordered water overlayer. A q u a l i t a t i v e i n t e n s i t y analysis yielded ~I0 = 0.76 R which is very large and l i e s outside the v a l i d i t y range of the eikonal approximation applied. However, in view of the one water molecule adsorbed per u n i t c e l l of KCI(IO0), t h i s res u l t does not seem unreasonable and a refined analysis using a large amount of d i f f r a c t i o n data seems very worthwhile. We close t h i s section by mentioning the experiments of ROWE and EHRLICH /7.19,20/ who scattered He, H2, HD, and D2 from MgO(100). These authors used an effusion source and were mainly interested in the r o t a t i o n a l t r a n s i t i o n s during scattering of H2, HD, and D2. For He d i f f r a c t i o n , only beams of f i r s t

order were well resolved and no i n -

formation on the corrugation function could be obtained. This substance would be a favourable candidate for more detailed studies, as r e l i a b l e LEED analyses e x i s t /7.21-23/.

8. Structural Investigations on Semiconductor Surfaces

8.1

Helium-Diffraction Studies on Si(111) and Si(100)

Semiconductor surfaces show a more extensive reconstruction than metal surfaces due to the d i r e c t i o n a l nature of the dangling bonds l e f t when the surface is exposed. Furthermore, the reconstruction w i l l extend deeper into the s o l i d , and f o r Si(100) is believed to extend into the f i f t h

atomic layer / 8 . 1 / .

In view of the technolo-

gical relevance of these surfaces and the large changes in the e l e c t r o n i c propert i e s which accompany small s t r u c t u r a l modifications / 8 . 2 / , s t r u c t u r a l determinations on semiconductor surfaces is an important research area. However, to date, not much progress has been made in t h i s f i e l d using LEED due to the complexity of the reconstructions which occur. One example in which the structure has been solved with LEED is GaAs(110) / 8 . 3 , 4 / , but neither the Si(111) nor the Si(100) surface phases are s t r u c t u r a l l y understood. Atom d i f f r a c t i o n could be of use here, since the scattering mechanism is simpler than in LEED and m u l t i p l e - s c a t t e r i n g effects w i l l not be as pronounced. However, the reconstructed semiconductor surfaces w i l l have a more open structure than metal surfaces, and therefore the incoming atom

149 (a)

)

Fig.8.1

9

m - - .

r-

~

~

F

i

g

,

8

,

2

Fig.8.1. (a) Model proposed for the (7• surface of Si(111). (b) The hexagonal arrangement of the surface atoms about a corner atom of the unit cell is shown. Raised and lowered atoms are represented by open and shaded circles, respective-

ly /8.5/

Fig.8.2. Various periodicities for the Si(100) surface based on the t i l t e d dimer model of CHADI /8.2/. The shaded atom is raised with respect to its dimer partner. The alternation of the t i l t will yield p(2• or c(2• periodicities depending on the phasing of adjacent rows /8.6/

w i l l , in general, see more than the topmost layer of the surface. This will lead to a corrugation function which is strongly structured. Before discussing recent atom-diffraction studies on silicon surfaces, we show models which have been proposed for the Si(111) (7•

and the reconstructed Si(100)

surfaces in Figs.8.1,2. Figure 8.1 shows the buckled ring-like structure proposed by CHADI /8.5/ for Si(111) (7•

in which the open and shaded circles represent

raised and lowered atoms, respectively. The model is consistent with the six-fold rotational symmetry seen with LEED at low energies at which diffraction from the

150 topmost layer should dominate. Models for the Si(100) surface based on a t i l t e d dimer model proposed by CHADI /8.2/ are shown in Fig.8.2. Depending on the surface preparation and annealing techniques, both (2xi) and c(2•

periodicities

have been reported / 8 . 6 / . The dominant feature of the reconstruction is the pairing of Si rows to form t i l t e d dimers, which leads to doubling of the p e r i o d i c i t y . The phases of the dimer t i l t s

between adjacent paired rows allow (2•

p(2•

c(2x4), and c(2x2) structures to be formed. He-diffraction studies on Si(100) by CARDILLO and BECKER /8.6/ show that several of the structures in Fig.8.2 are present together on the surface. Diffraction traces for various cuts through the reciprocal l a t t i c e are shown in Fig.8.3. Dominant peaks are seen for the integral and h a l f - i n t e g r a l beams as expected from two perpendicular domains of a (2•

structure. However, additional d i f f r a c t i o n inten-

• r

8t=70 =

XH==0,57A

Fig.8.3. He-diffraction traces from Si(lO0) for four d i f f e r e n t azimuthal angles labeled A through D. The i n t e n s i t i e s normalized to the incoming i n t e n s i t y are plotted against the parallel momentum change AK, and the abscissa is rotated by the azimuthal angle @. This allows a v e r t i c a l correspondence between the r e c i procal l a t t i c e and the d i f f r a c t i o n trace, ei = 70 ~ ~i = 0.57 ~ /8.6/

151 s i t y is seen in the regions of the reciprocal l a t t i c e shown shaded in Fig.8.3. This d i f f u s e region was also observed with LEED at low energies. The streaked pattern can be interpreted as small domains of (2x2) and (2•

structures which are longer, par-

a l l e l to the row p a i r i n g d i r e c t i o n than perpendicular to i t .

I f the phases of ad-

jacent domains are l a r g e l y uncorrelated, streaking such as that seen in Fig.8.3 w i l l be observed. The lack of long-range order is consistent with the observation that inplane d i f f r a c t i o n traces showed only I-2% of the incoming beam i n t e n s i t y . A d i f f r a c t i o n trace for the Si(7x7) surface obtained by CARDILLO and BECKER / 8 . 7 / is shown in Fig.8.4. The i n d i v i d u a l peaks can be c l e a r l y resolved f o r ki = I R, and a marked o s c i l l a t i o n in the beam i n t e n s i t i e s is observed. Again, the in-plane scatt e r i n g is only I-2% of the incoming beam i n t e n s i t y , but considerable out-of-plane i n t e n s i t y is seen using a larger detector aperture. This suggests that 10-20% of the incoming i n t e n s i t y appears in the d i f f r a c t i o n peaks. For both these surfaces, which are expected to show a large corrugation, numerous d i f f r a c t i o n peaks are observed whose i n t e n s i t i e s vary r a p i d l y with the wavelength and angle of incidence. However, at present no model consistent with the i n t e n s i t i e s observed has been reported. The added complexity of incomplete long-range order together with a strong two-dimensional corrugation make these surfaces less accessible to a s t r a i g h t f o r ward i n t e n s i t y analysis than weakly corrugated surfaces. However, with advances in our understanding of the gas-surface i n t e r a c t i o n potential and in the c a l c u l a t i o n of corrugation functions from measured d i f f r a c t i o n i n t e n s i t i e s , i t is to be expected that these surfaces can be analyzed.

(00)

.

o)

(oi)

~-(o

t(o:i

II

X

%

e

oo)

[oi]* O

9. . . . . . . . .

H

6

30"

2 o

0

10

.~* . .

/

.

20

30

t"

I

40 50 8r (DEGREES)

T

/

HI'

II

I

70

80

60

90

Fig.8.4. H e - d i f f r a c t i o n trace on the (7x7) structure of 5i(111) f o r (a) ~i = 0.57 ~n _ and (b) ~i = 1.0 ~. The expected angular positions of the seventh-order beams are " dicated by arrows. The inset shows the reciprocal l a t t i c e . 0. = 70 ~ , q~ = 0 ~ 1

152 8.2

Helium D i f f r a c t i o n from GaAs(110)

The complexity of the scattering data f o r s i l i c o n surfaces suggests that atom d i f f r a c t i o n should f i r s t

be carried out on a semiconductor surface of known s t r u c t u r e .

This has been done recently by CARDILLO et a l . / 8 . 8 / who have investigated He d i f f r a c t i o n from GaAs(110). The structure of t h i s surface, which has been established with dynamical LEED calculations / 8 . 3 , 4 / is shown in Fig.8.5. The side view i l l u s -

GoAs (I I0~_

",l,y --H I

I d

s,oE ~ ~ ~ _ ~ L ViEW. - ~ - }

~

I

"C15~=-)~-~-~-r~-

o 144~

=-~

_),....