UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Studies in LEED crystallography Hengrasmee, Sunantha 1980

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1980_A1 H36.pdf [ 8.88MB ]
Metadata
JSON: 831-1.0060930.json
JSON-LD: 831-1.0060930-ld.json
RDF/XML (Pretty): 831-1.0060930-rdf.xml
RDF/JSON: 831-1.0060930-rdf.json
Turtle: 831-1.0060930-turtle.txt
N-Triples: 831-1.0060930-rdf-ntriples.txt
Original Record: 831-1.0060930-source.json
Full Text
831-1.0060930-fulltext.txt
Citation
831-1.0060930.ris

Full Text

STUDIES IN LEED CRYSTALLOGRAPHY by SUNANTHA  HENGRASMEE  B.Sc.(Hons), The U n i v e r s i t y M.Sc.  , The U n i v e r s i t y  o f Otago, 1971 o f Otago, 1972  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  in THE FACULTY OF GRADUATE STUDIES (Department  o f Chemistry)  We a c c e p t t h i s t h e s i s as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA J u l y , 1980  SUNANTHA HENGRASMEE, 1980  In presenting this thesis  in partial fulfilment of the requirements f o r  an advanced degree at the University of B r i t i s h Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of t h i s  thesis  for scholarly purposes may be granted by the Head of my Department or by his representatives.  It  is understood that c o p y i n g or p u b l i c a t i o n  of this thesis for financial gain shall not be allowed w i t h o u t my written permission.  Department of  ^trnii-^-^y  The University of B r i t i s h Columbia 2075 W e s b r o o k P l a c e V a n c o u v e r , Canada V6T 1W5  Date  Abstract T h i s t h e s i s i s i n v o l v e d w i t h the use (LEED) f o r d e t e r m i n i n g the g e o m e t r i c a l surfaces  of single c r y s t a l s .  of low-energy e l e c t r o n  structures of well-characterized  S p e c i f i c a p p l i c a t i o n s are t o s u r f a c e s  both c l e a n and when c o n t a i n i n g  adsorbed  o f rhodium,  species.  A p r e l i m i n a r y problem concerned d i s c r e p a n c i e s d e t a i l s o f the g e o m e t r i c a l  diffraction  reported  previously  (100)  (111)  s t r u c t u r e s f o r the c l e a n  and  i n the  surfaces  when u s i n g rhodium p o t e n t i a l s from e i t h e r a band s t r u c t u r e c a l c u l a t i o n or the  linear superposition  c o r r e c t i o n has  now  o f charge d e n s i t y procedure f o r a metal c l u s t e r .  been made i n the c a l c u l a t i o n o f phase s h i f t s  s t r u c t u r e p o t e n t i a l , and  r e i n v e s t i g a t i o n s o f the  (100),  (110)  o f rhodium w i t h t h i s p o t e n t i a l r e s o l v e the d i s c r e p a n c i e s . support the s u g g e s t i o n ,  from  as shown p r e v i o u s l y  t h a t the s u p e r p o s i t i o n p o t e n t i a l p r o v i d e s  in this  A  f o r the band  and  (111)  surface  These r e s u l t s  laboratory for  now  Cu(lll),  a good a p p r o x i m a t i o n to a band s t r u c -  t u r e p o t e n t i a l f o r the purpose o f LEED c r y s t a l l o g r a p h y . In the s t r u c t u r a l d e t e r m i n a t i o n s made h e r e , the degree o f  correspondence  between i n t e n s i t y v e r s u s energy curves f o r d i f f e r e n t beams from experiment and  from m u l t i p l e - s c a t t e r i n g c a l c u l a t i o n s were a s s e s s e d w i t h the  index r ^ proposed by use  of this  Zanazzi  and Jona.  aspect  index f o r d e t e r m i n i n g the n o n - s t r u c t u r a l  the m u l t i p l e - s c a t t e r i n g c a l c u l a t i o n s . v a r i a t i o n s o f the imaginary p a r t o f the m u f f i n - t i n spheres and conclusions  A new  from  are  the s u r f a c e  constant  latter for Rh(lll)  p o t e n t i a l (V ^) between  compared w i t h v i s u a l analyses o f the  involved  parameters r e q u i r e d  i n the  Debye temperature  t h i s work g e n e r a l l y supports the use crystallography.  Included  considered  reliability-  (B^  s u r  £)•  the in are the  Structural  wherever p o s s i b l e ,  Zanazzi-Jona index i n LEED  and  The  e x p e r i m e n t a l p a r t of t h i s study i n v o l v e d  o f rhodium.  A series of d i f f r a c t i o n patterns  s o r p t i o n of 0^  and  H^S.  ( 3 x l ) - 0 , Rh(100)-p(2x2)-S and  surface  structures  Rh(110)-c(2x2)-S.  layer-doubling  methods) and  each case S atoms adsorb on the  single-bond  centre  atoms at a d i s t a n c e  v a l u e 2.29  and  (110)  were observed f o r the  a n a l y z e d by m u l t i p l e - s c a t t e r i n g c a l c u l a t i o n s  n e i g h b o u r i n g Rh  (100)  surfaces  chemi-  I n t e n s i t y v e r s u s energy curves were measured f o r  a v a i l a b l e d i f f r a c t e d beams f o r the  s c a t t e r i n g and  the  o  A), and  The  d e s i g n a t e d Rh(100)l a t t e r two  systems were  ( u s i n g the r e n o r m a l i z e d  surface  structures  o  A  (very  forward  determined.  s i t e s ; on Rh(100) S bonds to  o f 2.30  the  c l o s e t o the  four  Pauling  o  on Rh(110) each S atom i s 2.12  In  A from the  Rh  o  atom d i r e c t l y below i n the Rh  second l a y e r and  2.45  A from the  atoms i n the top m e t a l l i c l a y e r . An  i n v e s t i g a t i o n was  a l s o made f o r the use  i n LEED c r y s t a l l o g r a p h y  the q u a s i d y n a m i c a l method r e c e n t l y proposed by Van includes  i n d i v i d u a l l a y e r s , and  s a v i n g s i n computing time and the  Hove and  i n t e r l a y e r m u l t i p l e - s c a t t e r i n g p r o p e r l y , but  scattering within  for  four neighbouring  c l e a n and  has  core s t o r a g e .  sulphur-adsorbed  (100)  and  Tong.  neglects  considerable  T h i s method was  investigated  (110)  and  surfaces,  The  dynamical method appears to have some promise f o r making i n i t i a l trial  T h i s scheme  multiple-  the p o t e n t i a l f o r  compared w i t h the more-complete m u l t i p l e - s c a t t e r i n g methods.  o f the most s i g n i f i c a n t  of  results quasiselections  s t r u c t u r e s p r i o r to the m o r e - d e t a i l e d t e s t i n g  With f u l l m u l t i p l e - s c a t t e r i n g c a l c u l a t i o n s .  -iv-  T a b l e o f Contents  Page  Abstract  i i  T a b l e o f Contents List  :  iv  o f Tables  vii  L i s t of Figures  ix  Acknowledgement  xvii  Chapter 1:  Introduction  1  1.1  Modern S u r f a c e  1.2  I n t r o d u c t i o n to Low  1.3  Surface  1.4  Auger E l e c t r o n S p e c t r o s c o p y  17  1.5  Aims o f T h e s i s  20  C a l c u l a t i o n o f LEED I n t e n s i t i e s  22  2.1  C h a r a c t e r i s t i c s o f 1(E)  23  2.2  P h y s i c a l Parameters r e q u i r e d i n LEED Theory  24  2.3  T - M a t r i x Method  32  2.4  B l o c h Wave Method  2.5  P e r t u r b a t i o n Methods  39  (a) Layer Doubling Method  40  (b) Renormalized Forward S c a t t e r i n g Method  42  2.6  F u r t h e r M u l t i p l e S c a t t e r i n g Methods  45  2.7  G e n e r a l Aspects o f Computations  47  Chapter 2:  Science  2  Energy E l e c t r o n D i f f r a c t i o n  Crystallography  13  curves  (a) S t r u c t u r a l Parameters and (b) Program Flow  4  :  Use  o f Symmetry  34  47 51  -V-  T a b l e o f Contents 2.8  Page  Evaluation o f Results  •  (a) I n t r o d u c t i o n  53  (b) Zanazzi and Jona's P r o p o s a l s  Chapter 3: 3.1  3.2  53  •  54  (c) F u r t h e r Developments  56  P r e l i m i n a r y Work  60  General  Experimental  Procedures  61  (a) LEED Apparatus  61  (b) C r y s t a l P r e p a r a t i o n  65  (c) D e t e c t i o n o f S u r f a c e I m p u r i t i e s  68  (d) I n t e n s i t y Measurements  71  S t r u c t u r a l Determinations  o f Low Index S u r f a c e s o f Rhodium  75  (a) P r e v i o u s LEED I n t e n s i t y C a l c u l a t i o n s f o r Rhodium Surfaces  75  (b) F u r t h e r S t u d i e s 3.3  77  S t u d i e s w i t h the R e l i a b i l i t y  index o f Zanazzi and Jona  (a) I n t r o d u c t i o n  82  (b) R e l a t i o n s between R e l i a b i l i t y  Index and t h e Imaginary  Potential (c) R e l i a b i l i t y  82 Index and t h e V a r i a t i o n o f S u r f a c e Debye  Temperature 3.4  82  89  S t u d i e s o f A d s o r p t i o n o f some Gaseous M o l e c u l e s on Rhodium S u r f a c e s  95  (a) B i b l i o g r a p h y o f O v e r l a y e r S t r u c t u r e s on Rhodium Surfaces (b) A d s o r p t i o n o f 0  95 ?  on Rh(100)  97  -vi-  T a b l e o f Contents Chapter 4:  LEED A n a l y s i s o f Rh(100)-p(2x2)-S S u r f a c e  Page Structure  101  4.1  Introduction  4.2  Adsorption  4.3  Computational Scheme  107  4.4  Results  108  4.5  Discussion  117  Chapter 5:  102  o f . ^ S on Rh(100)  —  102  LEED A n a l y s i s o f t h e Rh(110)-c(2x2)-S S u r f a c e  Structure  125  5.1  Introduction  5.2  Experimental  5.3  Calculations  131  5.4  Results  134  5.5  Discussion  141  Chapter 6:  Studies with  126 .  126  the Quasidynamical Method  145  6.1  Introduction  146  6.2  Calculations  148  6.3  Results  150  and D i s c u s s i o n  (a) Rh(110) and Rh(110)-c(2x2)-S  -— -  (b) Rh(100) and Rh(100)-p(2x2)-S  6.4  Concluding  Remarks  150 161  :  168  References  171  Appendices  179  -vii-  List 2.1  of Tables  Numbers o f s y m m e t r i c a l l y - i n e q u i v a l e n t beams a c t u a l l y used i n c a l c u l a t i o n of various surface structures. o v e r l a y e r s t r u c t u r e s are d e s i g n a t e d and  3.1  Page  The models f o r the  as i n f i g u r e  1.8  2.8.  50  Observed and  c a l c u l a t e d Auger t r a n s i t i o n e n e r g i e s f o r  rhodium. 3.2  70  S t r u c t u r a l d e t e r m i n a t i o n o f low  index s u r f a c e s o f rhodium.  (Watson e t a l . ) 3.3  76  S t r u c t u r a l d e t e r m i n a t i o n o f low  index s u r f a c e s o f rhodium.  (This work.) 3.4  76  C o n d i t i o n s f o r b e s t agreement between e x p e r i m e n t a l at  normal i n c i d e n c e f o r R h ( l l l ) and r  potential r  curves  MJWT  1(E)  curves  calculated with  ] a c c o r d i n g t o the r e l i a b i l i t y  the  i n d i c e s r ^ and  f o r d i f f e r e n t v a l u e s o f a.  86  m 3.5  S u r f a c e s t r u c t u r e s r e p o r t e d f o r a d s o r p t i o n o f s m a l l gaseous molecules  4.1  on  low  index s u r f a c e s o f rhodium.  C o n d i t i o n s f o r minima o f r Rh(100)-p(2x2)-S.  4.2  Effective radii  r  96  f o r d i f f e r e n t models o f 116  o f chemisorbed s u l p h u r atoms on v a r i o u s  metal  surfaces. 4.3  122  Comparisons o f M-X bond d i s t a n c e s f o r chalcogen on  (100)  atoms  adsorbed  s u r f a c e s o f fee metals w i t h P a u l i n g ' s s i n g l e bond  l e n g t h s [133]. 6.1  123  Comparisons o f c o n d i t i o n s f o r minimum r ^ f o r v a r i o u s s u r f a c e s t r u c t u r e s o b t a i n e d from e v a l u a t i n g e x p e r i m e n t a l with corresponding and  curves  from quasi-dynamical  1(E)  curves  from m u l t i p l e - s c a t t e r i n g c a l c u l a t i o n s  calculations.  151  -viii-  List 6.2  of Tables  Page  A d e m o n s t r a t i o n o f t h e correspondence between peak p o s i t i o n s 1(E)  curves c a l c u l a t e d w i t h the q u a s i d y n a m i c a l method f o r t h e  f o u r models o f R h ( 1 1 0 ) - c ( 2 x 2 ) - S a t the s p e c i f i e d S-Rh s p a c i n g and t h o s e g i v e n by experiment f u l l multiple-scattering beam, the denominator in  the r e l e v a n t  calculations.  In the e n t r i e s  f o r each  s p e c i f i e s the number o f s i g n i f i c a n t peaks o r from t h e  full  c a l c u l a t i o n s , and the numerator g i v e s t h e  number o f t h o s e peaks t h a t quasidynamical  interlayer  and by t h e c o r r e s p o n d i n g  1 ( E ) c u r v e from experiment  multiple-scattering  6.3  in  a r e matched t o w i t h i n  7 eV by  the 15S  calculations.  A d e m o n s t r a t i o n o f the correspondence  between peak p o s i t i o n s  in  1 ( E ) curves c a l c u l a t e d w i t h t h e q u a s i d y n a m i c a l method f o r t h e f o u r models o f R h ( 1 0 0 ) - p ( 2 x 2 ) - S at the s p e c i f i e d S-Rh s p a c i n g and t h o s e g i v e n by full  multiple-scattering  beam, t h e denominator in  the r e l e v a n t  experiment  calculations.  and by t h e c o r r e s p o n d i n g In the e n t r i e s  f o r each  s p e c i f i e s t h e number o f s i g n i f i c a n t peaks  1(E) curve from experiment  multiple-scattering  interlayer  c a l c u l a t i o n s , and  number o f t h o s e peaks t h a t  or from t h e  the numerator g i v e s t h e  a r e matched t o w i t h i n  quasidynamical c a l c u l a t i o n s .  full  7 eV by  the 16'  -ix-  L i s t of Figures 1.1  Schematic diagram o f the mean f r e e path l e n g t h L (A) i n a m e t a l l i c s o l i d as a f u n c t i o n o f energy  1.2  Page of electrons  (eV).  5  Schematic energy d i s t r i b u t i o n N(E) o f b a c k - s c a t t e r e d e l e c t r o n s f o r a primary beam o f energy E .  5  q  1.3  (a) Schematic diagram o f the LEED experiment. (b) The p r i n c i p l e o f the f o r m a t i o n o f a d i f f r a c t i o n p a t t e r n i n LEED experiment.  1.4  8  Conventions f o r the i n c i d e n t a n g l e o f an e l e c t r o n beam on a s u r f a c e ; 6 i s a p o l a r a n g l e r e l a t i v e t o a s u r f a c e normal  and  4> an a z i m u t h a l a n g l e r e l a t i v e to a major c r y s t a l l o g r a p h i c  axis  i n the surface plane. 1.5  11  1(E) curves f o r the s p e c u l a r beam from Ni(100) at 6=3°.  The  bars i n d i c a t e e n e r g i e s where p r i m a r y Bragg c o n d i t i o n s are satisfied 1.6  ( a f t e r Andersson [ 4 8 ] ) .  12  A schematic comparison o f o v e r l a y e r and s u b s t r a t e r e g i o n s , both o f which a r e d i p e r i o d i c i n the x and y d i r e c t i o n s .  1.7  Schematic d i f f r a c t i o n p a t t e r n s o f c l e a n and  12  overlayer  structures. 1.8  15  Four p o s s i b l e s t r u c t u r a l models f o r Rh(110)-c(2x2)-S which a r e c o n s i s t e n t w i t h the observed d i f f r a c t i o n p a t t e r n . s u l p h u r atoms are r e p r e s e n t e d by the f i l l e d  1.9  The p r o d u c t i o n o f an  The  adsorbed  circles.  16  ^TV Auger e l e c t r o n i n aluminum.  X-ray energy l e v e l s are i n d i c a t e d r e l a t i v e t o the Fermi level. 1.10  18  Auger spectrum o f a h e a v i l y contaminated Rh(110) s u r f a c e , E  Q  = 1.5  keV,  I  q  = 10 microamps  19  -X-  L i s t of Figures 2.1  Muffin-tin  potential  (a) i n c r o s s - s e c t i o n as (b) a l o n g V 2.2  q  contours,  xx'.  i s the constant intersphere p o t e n t i a l .  25  I l l u s t r a t i o n o f the r e l a t i o n s h i p between e n e r g i e s measured w i t h r e s p e c t t o the vacuum l e v e l and the lowest  2.3  Page  those measured w i t h r e s p e c t ,to  l e v e l o f the c o n d u c t i o n band.  M u f f i n - t i n model o f an adsorbate  25  covered s u r f a c e  (after  Marcus et a l . [ 5 9 ] ) . 2.4  Schematic  28  r e p r e s e n t a t i o n o f a s e t o f p l a n e wave i n c i d e n t  from  the l e f t and m u l t i p l y s c a t t e r e d by a p l a n e o f i o n - c o r e s . 2.5  Schematic the a  diagram o f t r a n s m i s s i o n and r e f l e c t i o n m a t r i c e s a t  subplane.  The broken l i n e s a r e the c e n t r a l  between t h e subplanes. 2.6  form the t w o - l a y e r four-layer slab.  Planes A and  35 illustrate  B are f i r s t  the  stacked to  s l a b C; the p r o c e s s i s c o n t i n u e d t o form a ( A f t e r Tong [ 6 5 ] ) .  41  (a) I l l u s t r a t i o n o f the r e n o r m a l i z e d forward Vertical  lines  ;  S t a c k i n g o f p l a n e s t o form a c r y s t a l s l a b and l a y e r - d o u b l i n g method.  2.7  35  lines represent layers.  s c a t t e r i n g method.  Each t r i p l e t  o f arrows  r e p r e s e n t s the complete s e t o f p l a n e waves t h a t t r a v e l l a y e r to  from  layer.  (b) P r o p a g a t i o n  steps o f the i n w a r d - t r a v e l l i n g waves.  (c) P r o p a g a t i o n steps o f the o u t w a r d - t r a v e l l i n g waves. ( A f t e r Van 2.8  Schematic  Hove and Tong [ 8 l ] . )  43  diagram o f t h r e e s i m p l e models f o r Rh(100)-p(2x2)-S.  In r e c i p r o c a l space, s e t s o f s y m m e t r i c a l l y e q u i v a l e n t beams are i n d i c a t e d by a common symbol.  48  -xi-  L i s t o f Figures 2.9  Flowchart  showing p r i n c i p a l  Page  s t e p s i n a m u l t i p l e - s c a t t e r i n g LEED  c a l c u l a t i o n , u s i n g t h e RFS o r l a y e r d o u b l i n g programs. 2.10  P l o t s f o r C u ( l l l ) o f ( r ) . f o r 9 i n d i v i d u a l beams versus Ad% with V  = -9.5 eV.  or b i l i t y index  2.11  52  r  i  The dashed l i n e shows t h e reduced  ( r ) f o r the t o t a l  relia-  16 beams.  ( A f t e r Watson et a l . [ 4 3 ] ) .  57  Contour p l o t f o r C u ( l l l ) o f r v e r s u s Ad% and V . * r or ( A f t e r Watson e t a l . [ 4 3 ] ) .  59  K  3.1  (a) Schematic  o f the V a r i a n FC12 UHV chamber.  (b) D i a g r a m a t i c  r e p r e s e n t a t i o n o f t h e pumping system:  IP = Ion Pump; TSP= T i t a n i u m S u b l i m a t i o n Pump; SP = S o r p t i o n Pump. 3.2  (a) Schematic  62  diagram o f t h e e l e c t r o n o p t i c s used  f o r LEED  experiments. (b) Diagram showing sample mounted on a tantalum s u p p o r t i n g r i n g . (c) E l e c t r o n bombardment sample h e a t e r .  Hatched l i n e s r e p r e s e n t  s t a i n l e s s s t e e l p a r t s while the s t i p p l e p a t t e r n i n d i c a t e s the ceramic 3.3  insulator.  63  Auger s p e c t r a o f c l e a n Rh(110) s u r f a c e as a f u n c t i o n o f c r y s t a l temperature  i n d i c a t i n g carbon  c o n c e n t r a t e d around t h e s u r f a c e  r e g i o n a t 200°C and d i f f u s e d back i n t o t h e b u l k a t 300°C. 3.4  Schematic  diagram o f LEED o p t i c s used as a r e t a r d i n g  a n a l y z e r f o r Auger e l e c t r o n  field  spectroscopy:  MCA = m u l t i c h a n n e l a n a l y z e r . 3.5  Schematic  67  diagram o f t h e apparatus  69 used t o a n a l y s e t h e photo-  g r a p h i c n e g a t i v e s o f LEED p a t t e r n s .  74  -xii-  L i s t of Figures 3.6  Energy dependence o f rhodium phase s h i f t s potential  '3.7  r  Page (£=0-7) f o r t h e  [V^ ] .  78  W  (a) Schematic  diagrams o f t h e (100),  o f rhodium.  (110) and (111) s u r f a c e s  The d o t t e d c i r c l e s r e p r e s e n t rhodium atoms  i n t h e second  layer,  (b) The c o r r e s p o n d i n g  LEED p a t t e r n s i n d i c a t i n g the beam  n o t a t i o n as used i n t e x t . 3.8  The e x p e r i m e n t a l  80  1(E) curve f o r the (01) beam a t normal  i n c i d e n c e from t h e R h ( l l l ) s u r f a c e compared w i t h  five  [V^j^j  c o r r e s p o n d i n g curves c a l c u l a t e d w i t h t h e p o t e n t i a l and Ad% = -2.5% f o r the parameter a v a r y i n g from  1.17  to 2.34. 3.9  84  Contour p l o t o f r  v e r s u s 6_ _ and V f o r normal i n c i d e n c e r D,surf or d a t a from R h ( l l l ) where t h e c a l c u l a t i o n s use t h e p o t e n t i a l r  [Vo^ 3 W  3.10  ot=1.76 and 6 . ,,=480 K. Rh D,bulk Contour p l o t o f r versus 6^ _ and Ad% f o r normal i n c i d e n c e r D,surf data from R h ( l l l ) where t h e c a l c u l a t i o n s use t h e p o t e n t i a l  90  [V™]  91  w  i  t  h  n  r  L  3.11  Rh  w i t h a=1.76 and 6  . ,.=480 K.  D,bulk  J  The experimental  n  1(E) curve f o r t h e (01) beam a t normal  i n c i d e n c e from t h e R h ( l l l ) s u r f a c e compared w i t h  five  c o r r e s p o n d i n g curves c a l c u l a t e d w i t h t h e p o t e n t i a l Ad% = -2.5%, and a = 1.76 f o r t h e parameter 6^ 3.12  s  u  r  f  from 200 t o 600 K. Photographs o f some p(2><2) and (3x1) LEED p a t t e r n s  J, varying 93 observed  at normal i n c i d e n c e from t h e a d s o r p t i o n o f oxygen on a Rh(100) surface. (a) Rh(100)-p(2x2)-0 a t 70 eV; (b) R h ( 1 0 0 ) - ( 3 x l ) - 0 ,  s i n g l e domain a t 174 eV;  (c) R h ( 1 0 0 ) - ( 3 x l ) - 0 ,  2 e q u a l l y p o p u l a t e d domains a t 100 eV;  (d) R h ( 1 0 0 ) - ( 3 x l ) - 0 ,  2 e q u a l l y p o p u l a t e d domains at 152 eV.  99  -xiii-  L i s t of Figures 4.1  Page  Photographs o f LEED p a t t e r n s observed at normal  incidence  from a d s o r p t i o n o f S on Rh(100) s u r f a c e . R h ( 1 0 0 ) - c ( 2 x 2 ) - S at 80  eV;  (b) R h ( 1 0 0 ) - p ( 2 x 2 ) - S at 72  eV;  (a)  4.2  (c)  R h ( 1 0 0 ) - p ( 2 x 2 ) - S a t 114  Cd)  R h ( 1 0 0 ) - p ( 2 x 2 ) - S at 168 eV.  eV; 103  Auger s p e c t r a o f Rh(100) s u r f a c e s w i t h 1.5 keV and 10 m i c r o amp  beam at d i f f e r e n t  stages d u r i n g the p r e p a r a t i o n o f  Rh(100)-p(2x2)-S. 4.3  104  Beam n o t a t i o n f o r t h e LEED p a t t e r n o f  Rh(100)-p(2x2)-S  structure. 4.4  Comparison different  4.5  106 f o r t h e (-^j) and  (01) beams o f 1(E) curves from  two  experiments measured at normal i n c i d e n c e .  Comparison  o f e x p e r i m e n t a l 1(E) curves f o r v a r i o u s  and f r a c t i o n a l - o r d e r d i f f r a c t e d beams from  109 integral-  Rh(100)-p(2x2)-S  w i t h t h e c a l c u l a t e d curves f o r S adsorbed on-the 4F, 2F and IF  s i t e s at the topmost  Rh-S  i n t e r l a y e r spacing indicated f o r  each c u r v e .  111 1  4.6  Comparison  o f e x p e r i m e n t a l 1(E) curves f o r t h e (0-^) and  beams from t h e R h ( 1 0 0 ) - p ( 2 x 2 ) - S for  11 (— -)  s u r f a c e w i t h those c a l c u l a t e d  S adsorbed on t h e 4F s i t e f o r a range o f topmost  Rh-S  i n t e r l a y e r spacings. 4.7  Contour p l o t s o f f  r  model.  115 f o r Rh(100)-p(2x2)-S versus V  i n t e r l a y e r s p a c i n g f o r (a) 4F model,  and  (b) 2F model, and  Rh-S (c) IF  E r r o r b a r s i n d i c a t e s t a n d a r d e r r o r s as d e f i n e d i n  chapter 2.  118  -xiv-  L i s t o f Figures 5.1  Page  Auger s p e c t r a f o r a R h ( l l O ) s u r f a c e when c l e a n e d and when containing a c ( 2 x 2 ) overlayer o f sulphur.  5.2  Photographs o f LEED p a t t e r n s observed from  128  a t normal i n c i d e n c e  a d s o r p t i o n o f S on R h ( l l O ) s u r f a c e .  (a) R h ( l l O ) a t 1 4 4 eV; (b) R h ( 1 1 0 ) - c ( 2 x 2 ) - S Rh(110)-c(2x2)-S  a t 1 0 2 eV;  (d) R h ( 1 1 0 ) - c ( 2 x 2 ) - S  a t 1 5 0 eV.  (c)  5.3  a t 7 8 eV;  129  Beam n o t a t i o n f o r t h e LEED p a t t e r n from  the R h ( l l O ) - c ( 2 x 2 ) - S 130  surface structure. 5.4  Experimental  1 ( E ) curves  f o r two s e t s o f beams which a r e expected  t o be e q u i v a l e n t f o r t h e R h ( l l O ) - c ( 2 x 2 ) - S s t r u c t u r e . 5.5  Comparison o f some e x p e r i m e n t a l  1 ( E ) curves  132  from R h ( l l O ) - c ( 2 x 2 ) - S  w i t h those c a l c u l a t e d f o r t h e f o u r s t r u c t u r a l models over a range of  topmost i n t e r l a y e r s p a c i n g s : 31 and (c) (—j) beam.  5.6  Comparison o f e x p e r i m e n t a l  (a) ( 0 1 ) beam, (b) ( 1 0 ) beam,  1 ( E ) curves  135 f o r some i n t e g r a l - and  f r a c t i o n a l - o r d e r beams from R h ( l l O ) - c ( 2 x 2 ) - S w i t h t h o s e c a l culated  5.7  f o r t h e 4 F model w i t h s u l p h u r e i t h e r  o  0 . 7 5 or 0 . 8 5 A  above t h e topmost rhodium l a y e r .  139  Contour p l o t s o f r f o r R h ( 1 1 0 ) - c ( 2 x 2 ) - S v e r s u s V and Rh-S r or i n t e r l a y e r s p a c i n g f o r f o u r d i f f e r e n t s t r u c t u r a l models.  140  r  5.8  Schematic s p e c i f i c a t i o n o f i n t e r a t o m i c d i s t a n c e s i n t h e v i c i n i t y o f an o v e r l a y e r s u l p h u r atom i n t h e s u r f a c e s t r u c t u r e Rh(110)-c(2x2)-S.  5.9  143  D i s t a n c e s i n Angstrom.  I n t e r a t o m i c distances, f o r t h e s p e c i f i c a t i o n o f h a r d radii  sphere  i n t h e neighbourhood o f an oxygen atom i n t h e  F e ( 1 0 0 ) - ( l x l ) - 0 structure. ( A f t e r Legg e t a l . [ 1 5 3 ] ) .  o  D i s t a n c e s i n Angstrom. 143  -XV-  L i s t of Figures 6.1  Page  Comparison o f e x p e r i m e n t a l 1(E) curves f o r normal i n c i d e n c e on R h ( l l O ) w i t h those c a l c u l a t e d w i t h t h e q u a s i d y n a m i c a l method and t h e f u l l m u l t i p l e - s c a t t e r i n g method when t h e topmost i n t e r l a y e r s p a c i n g equals t h e b u l k v a l u e  (0%) and when i t i s  c o n t r a c t e d by 10%. 6.2  Contour  152  plots of r  f o r Rh(110)-c(2x2)-S  versus V  and t h e  Rh-S i n t e r l a y e r s p a c i n g f o r f o u r d i f f e r e n t s t r u c t u r a l models c a l c u l a t e d w i t h t h e q u a s i d y n a m c i a l method.  154 33  6.3  Comparison o f 1(E) curves measured f o r t h e (01) and (-^j) d i f f r a c t e d beams f o r normal i n c i d e n c e on Rh(110)-c(2x2)-S  with  those  c a l c u l a t e d by t h e q u a s i d y n a m i c a l method and by t h e f u l l m u l t i p l e - s c a t t e r i n g method f o r t h e f o u r s t r u c t u r a l models d e s c r ibed 6.4  i n text.  157  Comparisons o f some experimental  1(E) curves f o r f r a c t i o n a l -  o r d e r beams f o r normal i n c i d e n c e on Rh(110)-c(2x2)-S  and  Rh (100) -p(2><2) -S w i t h those c a l c u l a t e d f o r t h e c e n t r e a d s o r p t i o n s i t e s w i t h t h e q u a s i d y n a m i c a l method and w i t h t h e f u l l s c a t t e r i n g method.  The topmost Rh-S i n t e r l a y e r s p a c i n g s i n t h e o  quasidynamical  o  c a l c u l a t i o n s a r e 1.15 A and 1.3 A f o r  Rh(110)-c(2x2)-S  and Rh(100)-p(2x2)-S r e s p e c t i v e l y ; t h e  corresponding values f o r the m u l t i p l e - s c a t t e r i n g o  6.5  multiple-  calculations  o  are 0.75 A and 1.3 A. Comparisons o f some experimental  159  1(E) curves f o r normal i n c i d e n c e  on Rh(100) w i t h t h o s e c a l c u l a t e d w i t h t h e q u a s i d y n a m i c a l method and w i t h t h e f u l l 6.6  Contour  m u l t i p l e - s c a t t e r i n g method.  plots of r r  r  162  f o r Rh(100)-p(2x2)-S v e r s u s V and t h e Rh-S or  i n t e r l a y e r s p a c i n g f o r t h e 4F and 2F s t r u c t u r a l models by t h e q u a s i d y n a m i c a l method: and f r a c t i o n a l - o r d e r beams; o r d e r beams o n l y .  (a) comparisons (b) comparisons  calculated  with a l l i n t e g r a l -  with f r a c t i o n a l 164  -xvi-  List  of Figures  Page 13  6.7  Comparisons o f 1(E) curves measured  f o r t h e (01) and (-^j)  d i f f r a c t e d beams f o r normal i n c i d e n c e on Rh(100)-p(2x2)-S w i t h those c a l c u l a t e d by t h e q u a s i d y n a m i c a l method and by the f u l l m u l t i p l e - s c a t t e r i n g method f o r t h r e e p o s s i b l e s t r u c t u r a l models.  166  -xvii-  Acknowledgement I t has been a rewarding e x p e r i e n c e t o work under P r o f e s s o r s K.A.R. M i t c h e l l and D.C.  F r o s t d u r i n g the course o f t h i s work.  have p r o v i d e d i n v a l u a b l e s u p p o r t , and f o r t h i s I am v e r y g r a t e f u l t o Dr. C.W.  Tucker  T h e i r guidance I g i v e them my  and  sincere  York, Stony  programs and to Dr. M.A. Dr. S.Y.  Brook) f o r p r o v i d i n g t h e i r  (State  reliability-index  Van Hove ( U n i v e r s i t y o f C a l i f o r n i a a t B e r k e l e y )  Tong ( U n i v e r s i t y o f Wisconsin)  and quasidynamical  thanks.  (General E l e c t r i c C o r p o r a t i o n ) f o r  p r o v i d i n g a Rh(100) c r y s t a l , Dr. E. Z a n a z z i and P r o f e s s o r F. Jona U n i v e r s i t y o f New  encouragement  f o r copies o f t h e i r  and  multiple-scattering  computer programs.  I would l i k e t o acknowledge the c o n t r i b u t i o n s o f every member o f the s u r f a c e s c i e n c e group. T.W.  In the p a s t , Dr. R.W.  S t r e a t e r and a t p r e s e n t ,  Moore and Dr. S.J. White f o r e x p e r i m e n t a l a s s i s t a n c e , s t i m u l a t i n g  d i s c u s s i o n and  i n p a r t i c u l a r f o r t h e i r comments d u r i n g t h e p r e p a r a t i o n o f t h i s  thesis.  Among t h e s e , I owed a s p e c i a l thank t o Dr. F.R.  Dr. P.R.  Watson who  had a s s i s t e d and  Shepherd  c o l l a b o r a t e d i n t h i s work  and  throughout  the d u r a t i o n o f time they were h e r e . I am who  i n d e b t e d t o many members o f the mechanical  and e l e c t r i c a l workshops  have c o n t r i b u t e d so much i n m a i n t a i n i n g the working  c o n d i t i o n s o f the  instruments. I am v e r y  g r a t e f u l to B i l l  Ng f o r support  and u s e f u l s u g g e s t i o n s  e s p e c i a l l y f o r h i s p r o f e s s i o n a l job i n t y p i n g t h i s  thesis.  F i n a l l y , but foremost, a deep sense o f g r a t i t u d e and toward my husband, D h i t i Hengrasmee, who and s p i r i t u a l l y supported me dedicate this  thesis.  throughout  and  love i s d i r e c t e d  has been concerned w i t h my the course o f my  study.  progress  To him,  I  CHAPTER 1 Introduction  -2-  1.1  Modern S u r f a c e Studies  o f the p r o p e r t i e s o f s o l i d surfaces  o v e r t h e past various  Science have assumed g r e a t i n t e r e s t  decade, i n p a r t because such s u r f a c e s  t e c h n o l o g i c a l processes  have dominant r o l e s i n  (e.g. f r i c t i o n and wear, e l e c t r o n i c d e v i c e s  and heterogeneous c a t a l y s i s ) [ l , 2 ] .  T r a d i t i o n a l research  emphasized t h e  p r o p e r t i e s o f r e a l s u r f a c e s , u s u a l l y o f p o l y c r y s t a l l i n e m a t e r i a l s , which c o u l d n o t be w e l l - c h a r a c t e r i z e d surface  s c i e n c e has i n t r o d u c e d  characterized surfaces  at the atomic l e v e l  [3],  However, modern  the " c l e a n s u r f a c e " approach where c a r e f u l l y  are studied with the o b j e c t i v e o f developing  prin-  c i p l e s which can l e a d t o b e t t e r understandings o f t h e a t o m i s t i c a s p e c t s o f surface processes, the  i n c l u d i n g those o f t e c h n o l o g i c a l i n t e r e s t [ 4 , 5 ] ,  In  c l e a n s u r f a c e approach, s i n g l e c r y s t a l s are used and t h e p r o p e r t i e s o f  surfaces  corresponding t o w e l l - d e f i n e d  under c o n d i t i o n s impurities.  such t h a t t h e s u r f a c e  This requires  c r y s t a l l o g r a p h i c planes are studied i s not contaminated by unwanted  experiments t o be c a r r i e d out under u l t r a - h i g h  _9 vacuum  (<10  torr).  The n e c e s s i t y f o r t h i s p r o v i s i o n f o l l o w s  k i n e t i c t h e o r y which p r e d i c t s t h a t , f o r an ambient p r e s s u r e a surface  from t h e  o f 10 ^ t o r r ,  can be covered by an adsorbed monolayer i n 1 second, assuming  t h a t a l l c o l l i d i n g molecules s t i c k t o t h e s u r f a c e . With t h e a v a i l a b i l i t y o f u l t r a - h i g h vacuum f a c i l i t i e s , many e x p e r i mental t e c h n i q u e s have been developed r e c e n t l y f o r t h e c h a r a c t e r i z a t i o n of s o l i d surfaces with regard  t o chemical c o m p o s i t i o n , g e o m e t r i c a l and  e l e c t r o n i c s t r u c t u r e as w e l l as chemical bonding, v i b r a t i o n a l s t r u c t u r e and energy exchange w i t h impinging m o l e c u l e s .  Among t h e t e c h n i q u e s a v a i l a b l e ,  -3-  Auger e l e c t r o n s p e c t r o s c o p y analyses  (AES)  i s commonly used f o r q u a l i t a t i v e  o f s u r f a c e s , whereas u l t r a v i o l e t p h o t o e m i s s i o n  [6] i s p o p u l a r f o r i n d i c a t i n g e l e c t r o n i c s t r u c t u r e and diffraction  and  spectroscopy low  (RHEED) [7] and  (UPS)  energy e l e c t r o n  (LEED) g i v e s i n f o r m a t i o n on g e o m e t r i c a l s t r u c t u r e .  h i g h energy e l e c t r o n d i f f r a c t i o n  chemical  Reflection  the s c a t t e r i n g o f  molecular  i o n beams [8,9] a l s o have h i g h p o t e n t i a l s f o r s u r f a c e s t u d i e s .  Research  on w e l l - d e f i n e d s u r f a c e s w i t h a v a r i e t y o f techniques  has  established that  s u r f a c e p r o p e r t i e s depend not o n l y on the p a r t i c u l a r m a t e r i a l i n v o l v e d , but a l s o on the s p e c i f i c c r y s t a l l o g r a p h i c p l a n e exposed [io].  For  example,  c h e m i s o r p t i o n and m o l e c u l a r beam s c a t t e r i n g s t u d i e s have shown t h a t p r o b a b i l i t i e s and r e a c t i o n r a t e s can be v e r y d i f f e r e n t on stepped of  platinum  compared w i t h  At the p r e s e n t time  low-index s u r f a c e s o f the same metal  in  1927  surfaces  [ll].  LEED appears as the most d i r e c t t e c h n i q u e  determination of surface geometrical s t r u c t u r e . when the experiment was  first  T h i s p o t e n t i a l was  performed by Davisson  experimental  difficulties,  and  f o r the recognized  and Germer [ 1 2 ] .  However the development o f t h i s t e c h n i q u e t o i t s f u l l p o t e n t i a l was by many t h e o r e t i c a l and  sticking  i t was  inhibited  only during  the 1970's t h a t these problems were overcome s u f f i c i e n t l y f o r some s u r f a c e s t r u c t u r e s t o be determined. antageous, i f not  e s s e n t i a l , t o u t i l i z e o t h e r techniques  c h a r a c t e r i z i n g the s u r f a c e . - i s AES. in  1925  In c u r r e n t LEED s t u d i e s i t i s c o n s i d e r e d advsimultaneously f o r  The most commonly-used complementary  technique  H i s t o r i c a l l y , e l e c t r o n s produced by the Auger p r o c e s s were d i s c o v e r e d [ 1 3 ] , and a l t h o u g h t h e i r p o t e n t i a l  by Lander [14] i n 1953,  i t was  not u n t i l  i n s u r f a c e a n a l y s i s was  recognized  the l a t e 1960's t h a t they c o u l d  be  -4-  detected o f AES  r o u t i n e l y i n s u r f a c e experiments [15-17].  Indeed the  development  as a method f o r q u a l i t a t i v e s u r f a c e a n a l y s i s encouraged the  ment o f r e p r o d u c i b l e LEED experiments, and adequate t h e o r i e s f o r LEED.  The  develop-  i n t u r n the development  e x i s t e n c e o f the  of  l a t t e r represented  n e c e s s a r y requirement f o r the development o f LEED c r y s t a l l o g r a p h y the d e t e r m i n a t i o n  1.2  o f surface geometrical  I n t r o d u c t i o n to Low  s t r u c t u r e by  (i.e.,  LEED).  Energy E l e c t r o n D i f f r a c t i o n  A LEED experiment i n v o l v e s d i r e c t i n g a beam o f low-energy (typical  a  energy <500eV) w i t h  known angles  s u r f a c e o f a c r y s t a l l i n e s o l i d and  electrons  o f i n c i d e n c e onto a w e l l - d e f i n e d  observing  the i n t e n s i t y d i s t r i b u t i o n  e l e c t r o n s which are e l a s t i c a l l y b a c k - s c a t t e r e d  from the s u r f a c e .  The  of  de o  B r o g l i e hypothesis according  r e l a t e s e l e c t r o n energy  (E i n eV)  t o wavelength  (X i n A)  to  * =JM3 ; e l e c t r o n s i n the comparable w i t h  (1.1,  low-energy range t h e r e f o r e have wavelengths which i n t e r l a y e r spacings  i n the s o l i d .  Low-energy e l e c t r o n s  a r e p a r t i c u l a r l y " s u r f a c e s e n s i t i v e " because they e x p e r i e n c e elastic scatterings i n solids.  strong i n -  A h e l p f u l parameter f o r d i s c u s s i n g  s c a t t e r i n g i s the e l e c t r o n mean f r e e p a t h l e n g t h  are  (L) which can be  inelastic expressed  i n terms o f I  =  where the i n c i d e n t i n t e n s i t y I on passage through a d i s t a n c e £ .  I  Q  exp  [J/L]  ,  (1.2)  at a p a r t i c u l a r energy i s a t t e n u a t e d The  general  to I  form o f the dependence o f  mean f r e e p a t h l e n g t h on e l e c t r o n energy i s shown i n f i g u r e 1.1.  the  Electrons  -5-  1,000-1  ioo-4  °<  100,000  F i g u r e 1.1:  Schematic diagram o f the mean f r e e path l e n g t h L (A) o f e l e c t r o n s i n a m e t a l l i c s o l i d as a f u n c t i o n o f energy ( e V ) .  true  elastic  secondary  peak  Ul  Z  ENERGY  Figure  1,2:  Schematic energy d i s t r i b u t i o n N(E) o f b a c k - s c a t t e r e d f o r a primary beam o f energy E . Q  electrons  -6-  i n t h e low-energy range a r e a s s o c i a t e d w i t h v a l u e s  o f L o f j u s t a few  o  Angstroms, and t h e r e f o r e they a r e i d e a l l y s u i t e d f o r i n v e s t i g a t i o n o f the top  few l a y e r s o f a s o l i d .  lengths  Further  information  has been reviewed by Brundle [ 1 8 ] ,  on e l e c t r o n mean f r e e path  Ibach [19] and Powell [ 2 0 ] .  A monoenergetic beam o f low-energy e l e c t r o n s surface  t y p i c a l l y gives  an energy d i s t r i b u t i o n f o r the b a c k - s c a t t e r e d  e l e c t r o n s o f the type shown i n f i g u r e 1.2.  The narrow " e l a s t i c peak" on the  r i g h t hand s i d e i n v o l v e s the e l e c t r o n s which LEED experiment.  i n c i d e n t upon a s o l i d  a r e s t u d i e d i n the c o n v e n t i o n a l  T h i s peak i n c l u d e s t h e g e n u i n e l y  elastically-scattered  e l e c t r o n s , as w e l l as those e l e c t r o n s which have undergone phonon s c a t t e r i n g with small be  energy changes  ( ^0.1 eV ) .  This  r e f e r r e d t o as q u a s i e l a s t i c e l e c t r o n s .  electrons contribute  to the " e l a s t i c peak".  i n e l a s t i c scattering, associated excitations  l a t t e r group o f e l e c t r o n s can  T y p i c a l l y only  1-5% o f the i n c i d e n t  Most e l e c t r o n s  experience  e s p e c i a l l y w i t h s i n g l e - e l e c t r o n and plasmon  [21,22], and those e x c i t a t i o n s c o n t r i b u t e t o t h e  s h o r t mean f r e e path l e n g t h e l e c t r o n s , which t y p i c a l l y  strong  i n d i c a t e d i n f i g u r e 1.1. corresponds t o a c u r r e n t  comparatively  The e m i s s i o n o f Auger -12  o f ~10  A on a back-  _7 ground o f -10  A, appears as s m a l l peaks superimposed on a  background i n t h e i n t e r m e d i a t e electrons  range o f f i g u r e 1.2.  slowly-varying  Peaks due t o Auger  can be d i s t i n g u i s h e d from l o s s peaks due t o plasmon or s i n g l e - e l e c t r o n  e x c i t a t i o n s because t h e former occur at e n e r g i e s - p r i m a r y e l e c t r o n energy.  which a r e independent o f t h e  The l a r g e peak a t low energy i n f i g u r e 1.2 i n v o l v e s  the s o - c a l l e d " t r u e secondary" e l e c t r o n s which a r e a s s o c i a t e d w i t h a s e r i e s o f i n e l a s t i c s c a t t e r i n g s i n a cascade-type p r o c e s s [ 2 3 ] .  -7-  The incident  p r i n c i p l e o f t h e LEED experiment i s i l l u s t r a t e d  The  e l e c t r o n s a r e s c a t t e r e d by t h e s u r f a c e r e g i o n and t h e e l a s t i c a l l y  back-scattered The  i n f i g u r e 1.3a.  e l e c t r o n s are separated  from others by energy s e l e c t i n g g r i d s .  e l a s t i c a l l y s c a t t e r e d waves i n t e r f e r e c o n s t r u c t i v e l y t o g i v e  beams along  diffracted  c e r t a i n d i r e c t i o n s , and each beam shows as a b r i g h t spot when  t h e s e e l e c t r o n s a r e a c c e l e r a t e d onto a f l u o r e s c e n t s c r e e n . o f t h e s e spots  i s r e f e r r e d t o as t h e LEED p a t t e r n .  a s t i c s c a t t e r i n g , the e l a s t i c a l l y - s c a t t e r e d  The d i s t r i b u t i o n  Because o f s t r o n g  e l e c t r o n s do not n o r m a l l y  inelexperi-  ence a r e g u l a r p e r i o d i c i t y normal t o t h e c r y s t a l s u r f a c e and c o n s e q u e n t l y the r e g i o n probed by t h e LEED e l e c t r o n s i s d i p e r i o d i c ( i . e . , c h a r a c t e r i z e d by two u n i t t r a n s l a t i o n v e c t o r s d i f f r a c t i o n pattern  a^ and a ^ ) .  i t can be  The  corresponding  ( f i g u r e 1.3b) i n v o l v e s the a s s o c i a t e d t r a n s l a t i o n a l v e c t o r s  i n r e c i p r o c a l space, namely a^f and £i* d e f i n e d by  a* ~1  =  2TT  a *z ^2 ~  ,  a* ~z  a,, a xz  =  2TT  axz ~1 ^  (1.3)  a . a/z 0  where £ i s the u n i t v e c t o r p e r p e n d i c u l a r  t o a^ and a. .  Pendry [24] has given a d e t a i l e d a n a l y s i s showing how a LEED p a t t e r n i s a direct  consequence o f t h e s u r f a c e t r a n s l a t i o n a l symmetry.  Assuming the  i n c i d e n t e l e c t r o n s can be d e s c r i b e d by a p l a n e wave V o where B i s an a p p r o p r i a t e . v e c t o r and k ~o  +  =  B exp[ik . r ] ~o — +  r  normalization  ,  constant,  (1.4) r i s a general p o s i t i o n  i s t h e i n c i d e n t wave v e c t o r which r e l a t e s t o e l e c t r o n energy  through ~ 2m  |k | '~o +  1  2  , '  (1.5)  -8-  Figure  1.3:  a) Schematic diagram o f t h e LEED experiment. b) The p r i n c i p l e  o f the formation  LEED experiment.  of a d i f f r a c t i o n pattern i n  -9-  then wave v e c t o r s k  f o r the d i f f r a c t e d  e l e c t r o n s a r e determined by  conserva-  t i o n o f energy E(k")  =  E(k^)  and by t h e c o n s e r v a t i o n o f momentum p a r a l l e l k~  =  (1.6)  to the surface  k ,, + g(hk) +  ,  (1.7)  ,  (1.8)  where g(hk) h and k b e i n g i n t e g e r s .  =  ha* + ka*  As i l l u s t r a t e d , i n f i g u r e 1.3b, t h e d i r e c t i o n o f +  each d i f f r a c t e d beam (wave v e c t o r k ,,.-.) i s determined by E, k ~g(hk) ~o For g i v e n v a l u e s c i a t e d with (hk).  o f lc* and E, each spot i n a d i f f r a c t i o n p a t t e r n i s asso-  a p a r t i c u l a r g, and hence may be i d e n t i f i e d w i t h t h e i n d i c e s  F o r a g i v e n energy, o n l y a l i m i t e d number o f beams can reach t h e  s c r e e n ; i f |g| i s s u f f i c i e n t l y ~ an evanescent The  l a r g e k~ becomes complex and corresponds ~g  to  ( o r s u r f a c e ) wave which cannot escape from t h e s o l i d .  (00) beam i s made up o f e l e c t r o n s which have i n t e r a c t e d w i t h the  s u r f a c e without it  and g. ~  momentum t r a n s f e r p a r a l l e l  t o the s u r f a c e  i s f r e q u e n t l y c a l l e d the " s p e c u l a r beam".  beam remains constant  Qc | =J< ||)» and +  (  The d i r e c t i o n o f t h e s p e c u l a r  as E changes, as long as t h e e l e c t r o n s move i n a  f i e l d - f r e e space o u t s i d e t h e c r y s t a l and t h e d i r e c t i o n o f t h e i n c i d e n t beam is fixed.  With i n c r e a s i n g energy, more d i f f r a c t e d beams a r e observed, t h e  non-specular  beams move towards t h e (00) beam, t h e symmetry o f t h e LEED  p a t t e r n remains unchanged, but the beam i n t e n s i t i e s v a r y  continuously.  In p r a c t i c e , i n c i d e n t e l e c t r o n beams i n LEED a r e coherent restricted  distances  (-10  2 ° A) [ 2 5 ] , and t h i s  only  over  l i m i t s t h e range over which  -10-  s u r f a c e order  can be r e c o g n i z e d  i n the d i f f r a c t i o n experiment.  disorder i s i n e v i t a b l y present  at s u r f a c e s , and  by b r o a d e n i n g the d i f f r a c t e d beams, by s p l i t t i n g s , and by LEED p a t t e r n s  t h i s can a f f e c t  spot  domain s t r u c t u r e i n which two  presence o f domain s t r u c t u r e , p r o v i d e d  or more e q u i -  the i n d i v i d u a l domains. systems, and  For a g i v e n s u r f a c e , the the  and  the  from  adsorption  1(E)  ( s p e c i f i e d by angles  with 0, cf>;  presented  curves f o r each d i f f r a c t e d beam) w i t h A typical  D a v i s s o n and  example o f  1(E)  Germer [ 1 2 ] , at the time o f  LEED experiment, r e a l i z e d beam i n t e n s i t i e s c o n t a i n i n f o r m a t i o n  s u r f a c e bond d i s t a n c e s , but geometries c o u l d be  n e a r l y 50 years  elapsed before  e x t r a c t e d from measured i n t e n s i t i e s .  u t i l i z e d at the p r e s e n t J(E)  important f o r  Most o f t e n i n t e n s i t y data are  o t h e r parameters b e i n g h e l d c o n s t a n t .  first  o f the p a t t e r n s  i n t e n s i t i e s o f the d i f f r a c t e d beams v a r y  ( i . e . , as  curves i s g i v e n i n f i g u r e 1.5.  are  later.  the temperature.  as a f u n c t i o n o f energy  the  the i n c i d e n t e l e c t r o n beam,  superpositions  e l e c t r o n energy E, the d i r e c t i o n o f i n c i d e n c e  see f i g u r e 1.4)  all  direct  T h i s can be p a r t i c u l a r l y  examples are g i v e n  In  t h a t the dimensions o f the domains  g r e a t e r than the coherence width a s s o c i a t e d w i t h represent  spot  Frequently  v a l e n t o r i e n t a t i o n s o f the s t r u c t u r e are p o s s i b l e on the s u r f a c e .  observed LEED p a t t e r n s  patterns  i n t r o d u c i n g s t r e a k s , r i n g s and  i n c r e a s i n g the background i n t e n s i t y [ 2 6 ] .  are a f f e c t e d by  Some  on  d e t a i l e d surface The  b a s i c method  time i n v o l v e s the t r i a l - a n d - e r r o r approach i n which  curves are c a l c u l a t e d f o r d i f f e r e n t p o s s i b l e s u r f a c e geometries, and  search  i s made f o r t h a t geometry which allows  perimental  1(E)  curves f o r the v a r i o u s d i f f r a c t e d beams.  t h i s t h e s i s i s involved with crystallography.  the b e s t match up w i t h The  the  main content  the a p p l i c a t i o n o f t h i s approach to LEED  a exof  -11-  -z  direction of incident  Figure  1.4:  Conventions surface;  f o r the i n c i d e n t  8 i s a polar  angle o f an e l e c t r o n beam on a  angle r e l a t i v e t o a s u r f a c e  <(> an azimuthal angle r e l a t i v e t o a major a x i s i n the s u r f a c e  bear  plane.  normal and  crystallographic  -12-  1 2  ELECTRON ENERGY (eV) Figure  1.5:  1(E) curve f o r the s p e c u l a r beam from Ni(100) a t 9=3°. The b a r s i n d i c a t e e n e r g i e s where p r i m a r y Bragg are s a t i s f i e d  F i g u r e 1.6:  ( a f t e r Andersson  A schematic comparison o f which  conditions  [48]).  o f o v e r l a y e r and s u b s t r a t e r e g i o n s , both  are d i p e r i o d i c i n the x and y d i r e c t i o n s .  -13-  1.3  Surface The  Crystallography  d e f i n i t i o n o f s u r f a c e i s v e r y much a f u n c t i o n o f t h e p a r t i c u l a r  probe used t o study  it.  F o r LEED from a c r y s t a l l i n e s o l i d ,  i t i s convenient  t o r e f e r t o t h e " s u r f a c e r e g i o n " as t h e r e g i o n probed by t h e LEED e l e c t r o n s ( i . e . , over t h e range o f mean f r e e path scattered electrons).  Figure  length corresponding  to the e l a s t i c a l l y -  1.6 a l s o i n d i c a t e s t h e " s u b s t r a t e " whose  s t r u c t u r e i s g e n e r a l l y known from X-ray c r y s t a l l o g r a p h y and i s t h a t f o r which the b u l k t r i p e r i o d i c i t y  i s established.  The o b j e c t i v e o f s u r f a c e  crystallo-  graphy i s then t o determine t h e p o s i t i o n o f a l l atoms beyond t h e s u b s t r a t e surface  ( i . e . , t h e topmost s u b s t r a t e p l a n e ) .  The s u r f a c e r e g i o n may  an o v e r l a y e r whose d i p e r i o d i c t r a n s l a t i o n a l symmetry i s d i f f e r e n t of a substrate plane.  The a p p r o p r i a t e p e r i o d i c t r a n s l a t i o n a l  involve  from t h a t  symmetry f o r  LEED i s t h a t f o r t h e o v e r a l l s u r f a c e r e g i o n , and i s d e s c r i b e d by t h e u n i t t r a n s l a t i o n a l v e c t o r s a- and a_.  These v e c t o r s may r e s u l t  from t h e combination  o f t h e d i p e r i o d i c symmetries o f t h e s u b s t r a t e and the o v e r l a y e r . The  vectors  and a  d e f i n e a u n i t mesh which i s analogous t o t h e u n i t  c e l l of t r i p e r i o d i c crystallography. t  =  The v e c t o r ma, + na_  (1.9)  t r a n s l a t e s from one p o i n t i n a s u r f a c e r e g i o n t o another w i t h an i d e n t i c a l environment, and a two-dimensional net can be generated from a l l i n t e g r a l v a l u e s o f m and n; t h i s  i s t h e d i p e r i o d i c analogue o f the t r i p e r i o d i c  used i n X-ray c r y s t a l l o g r a p h y . and  F i v e types  they a r e analogous t o t h e 14 B r a v a i s  lattice  o f d i p e r i o d i c nets a r e p o s s i b l e  lattices  in triperiodic  crystallo-  graphy.  There a r e 17 p o s s i b l e space groups i n d i p e r i o d i c c r y s t a l l o g r a p h y ,  and they  a r e d e t a i l e d i n t h e I n t e r n a t i o n a l Tables  f o r X-ray C r y s t a l l o g r a p h y [ 2 7 ] .  -14-  Adsorption dicities  and  e x t r a spots  on c l e a n s u r f a c e s t y p i c a l l y g i v e s i n c r e a s e d  t h e r e f o r e e x t r a LEED s p o t s , as shown i n f i g u r e 1.7.  clean surface s t r u c t u r e .  t i o n f o r s u r f a c e s t r u c t u r e s and substrate.  discussed  i s p r i m i t i v e (one  (110)  are d e s i g n a t e d latter  n o t a t i o n has  E s t r u p and McRae [30]) .  designated  surface  been  introduced  With Wood's n o t a t i o n ,  f r e q u e n t l y added to i n d i c a t e whether the s u r f a c e mesh  atom per u n i t mesh) or c e n t r e d  surfaces  o f rhodium  as Rh(100)-p(2x2)S and  i s always used f o r  (with an  the s t r u c t u r e s  the on  obtained  Rh(110)-c(2x2)S r e s p e c t i v e l y ; the as Rh(110)-(/3x/3/2)54-S although  simplicity.  A d i f f r a c t i o n p a t t e r n u s u a l l y allows  a s p e c i f i c a t i o n o f the  p e r i o d i c i t y , but never o f the a c t u a l s u r f a c e s t r u c t u r e . a n a l y s i s o f beam i n t e n s i t i e s .  e x t r a atom at  For the examples o f S adsorbed  ( f i g u r e 1.7)  c o u l d a l t e r n a t i v e l y be d e s i g n a t e d  the f i r s t  the  ( f o r more complex s u r f a c e s , where such  c e n t r e o f the u n i t mesh), r e s p e c t i v e l y . and  o f the  nota-  G i s the angle o f r o t a t i o n between  a p p l i c a b l e , a matrix  f u r t h e r by  a  d i f f r a c t e d beams which i s based on  s u b s t r a t e u n i t meshes  the symbols p or c a r e  (100)  s t r u c t u r e as  i t i s convenient t o use  (Jb^.b^) are the u n i t d i p e r i o d i c v e c t o r s  an angle o f r o t a t i o n i s not and  Generally  s u b s t r a t e r e s p e c t i v e l y , and  the s u r f a c e and  [29]  beams from the a d s o r p t i o n  For example i n Wood's nomenclature [ 2 8 ] , a s u r f a c e i s  where ( a ^ a ^ ) and r e g i o n and  Such  are f r e q u e n t l y c a l l e d " f r a c t i o n a l o r d e r " spots when the same  n o t a t i o n i s used f o r corresponding f o r the  surface perio-  For S adsorbed on the  t h e r e are f o u r p a r t i c u l a r l y important  The  (110)  surface  latter  requires  s u r f a c e o f rhodium,  l o c a t i o n s f o r the S atoms.  These  are  -15-  real  reciprocal  space  space  ft  %  Rh (100)  oTi  ffi  ir  oo;  10  20  2  9 ft 9 I  2  R h ( 1 0 0 ) - p(2 x 2 ) S  o 0 0 o 0 ft-®"}  OCDOO OCXBOO COBOO CXX)QO  012  112  Rh(110)  00  20  2  O Rh(1 1 0 ) - c ( 2 x 2 ) S On  o  2  Ti  O 00  Figure  1.7:  Schematic d i f f r a c t i o n p a t t e r n s o f c l e a n and o v e r l a y e r s t r u c t u r e s .  -16-  On-top(IF) model  Centre ( 4 F ) model  Short-bridge (2 SB)  Long-bridge ( 2 LB)  model  model  Four p o s s i b l e s t r u c t u r a l models f o r Rh(llO)-c(2*2)S which are consistent with the observed d i f f r a c t i o n p a t t e r n . Th adsorbed sulphur atoms are represented by the f i l l e d  circ  -17-  shown i n f i g u r e 1.8, pattern.  The  and a l l are c o n s i s t e n t w i t h t h e c(2x2) d i f f r a c t i o n  a d s o r p t i o n s i t e s a r e d e s i g n a t e d as c e n t r e or f o u r - f o l d ( 4 F )  s i t e s , on-top or o n e - f o l d (IF) s i t e s , s h o r t - b r i d g e (2SB) bridge  (2LB)  sites.  To determine  s i t e s or l o n g -  the a c t u a l a d s o r p t i o n s i t e i t i s n e c e s s a r y  t o c a l c u l a t e the 1(E) curves o f the d i f f r a c t e d beams f o r the v a r i o u s models and  compare them w i t h the experimental  1(E) curves t o assess which model  g i v e s the b e s t agreement.  1.4  Auger E l e c t r o n  Spectroscopy  The Auger p r o c e s s i s d e p i c t e d i n f i g u r e 1.9.  I t i s i n i t i a t e d by  ionisation  o f a core e l e c t r o n e i t h e r by e l e c t r o n impact  action.  e l e c t r o n from a h i g h e r energy  An  i n n e r vacancy, (e.g. X-ray energy  or by photon  l e v e l then drops  and t h i s p r o c e s s r e l e a s e s energy  the inter-  down t o f i l l  the  e i t h e r by photon p r o d u c t i o n  f l u o r e s c e n c e ) or by e j e c t i o n o f an Auger e l e c t r o n whose k i n e t i c  depends d i r e c t l y on the energy  l e v e l s i n v o l v e d i n the p r o c e s s  G e n e r a l l y Auger e m i s s i o n i s the more p r o b a b l e p r o c e s s t i o n i n v o l v e s an e l e c t r o n whose b i n d i n g energy  [23,3l].  i f the i n i t i a l  i s l e s s than -2keV.  ionisaThe  key  p o i n t f o r s u r f a c e a n a l y s i s i s t h a t the k i n e t i c e n e r g i e s o f Auger e l e c t r o n s are c h a r a c t e r i s t i c o f the p a r t i c u l a r element from which the e l e c t r o n s o r i g i n a t e ; chemical s h i f t  e f f e c t s are observed, but t h e s e e f f e c t s a r e s m a l l  compared w i t h the d i f f e r e n c e s between d i f f e r e n t  elements [32,33].  a n a l y s i s i n p r a c t i c e i n v o l v e s comparing the e n e r g i e s o f observed peaks w i t h t h e l i s t e d v a l u e s hydrogen and h e l i u m , a surface region.  Most elements,  Auger  with the exception of  can be d e t e c t e d u n i q u e l y even i f s e v e r a l a r e p r e s e n t i n  A typical  shown i n f i g u r e 1.10;  [34-36],  Qualitative  this  example o f an Auger spectrum  from- t h i s work i s  i s f o r a Rh(110) s u r f a c e contaminated  with  sulphur,  -18-  Figure  1.9:  The p r o d u c t i o n o f an L energy  2  VV Auger e l e c t r o n  i n aluminum.  l e v e l s are i n d i c a t e d r e l a t i v e t o the Fermi  level.  X-ray  -19-  T 100  r  200  l  r  300  ENERGY/eV Figure 1.10:  Auger spectrum o f a h e a v i l y contaminated R h ( l l O ) s u r f a c e , E  Q  = 1.5 keV,  I  = 1 0 microamps.  r 4  0  0  -20-  carbon and phosphorus.  The  spectrum i s p r e s e n t e d  ( dN(E)/dE ) t o enhance the weak Auger f e a t u r e s . as a r e t a r d i n g f i e l d a n a l y z e r can  be d e t e c t e d  i n the d e r i v a t i v e form Using  standard  [16,17], amounts o f around 1-5%  f o r most elements; h i g h e r  a c y l i n d r i c a l mirror analyzer  [37].  The  LEED o p t i c s  o f a monolayer  s e n s i t i v i t i e s are p o s s i b l e w i t h f l u x o f Auger e l e c t r o n s produced  depends e s p e c i a l l y on the i o n i z a t i o n c r o s s - s e c t i o n o f i n d i v i d u a l and  this  g e n e r a l l y v a r i e s with  In t h i s t h e s i s , AES although  energy.  i s used o n l y f o r q u a l i t i a t i v e chemical  analysis,  t h e r e a r e c o n t i n u i n g attempts to develop t h i s t e c h n i q u e  t a t i v e a n a l y s i s [38,39]. g i v e important  for quanti-  With s u i t a b l e c a l i b r a t i o n s , t h i s t e c h n i q u e  i n f o r m a t i o n on s u r f a c e k i n e t i c s  v a l u e f o r a s s e s s i n g aspects  1.5  elements,  [40].  can  I t a l s o has p o t e n t i a l  o f s u r f a c e band s t r u c t u r e s [ 4 1 ] .  Aims o f T h e s i s The  overall  objective of this  i n knowledge a s s o c i a t e d w i t h  t h e s i s i s to c o n t r i b u t e to an  LEED c r y s t a l l o g r a p h y , both by  unknown s u r f a c e s t r u c t u r e s and by  a s s e s s i n g p o s s i b l e new  or  increase  determining  some  modified  procedures. The [42] but tigated.  c a t a l y t i c aspects  o f rhodium have been w e l l known f o r a long time  the c r y s t a l l o g r a p h y o f i t s s u r f a c e s has  not been t h o r o u g h l y  inves-  In e a r l i e r work, Watson et a l . [43,44] r e p o r t e d d i s c r e p a n c i e s i n  the g e o m e t r i c a l  s t r u c t u r e s o f the  c i a t e d w i t h the use  (100)  and  (111)  o f atomic p o t e n t i a l s from two  s u r f a c e s o f rhodium, different  were expected to g i v e e s s e n t i a l l y e q u i v a l e n t r e s u l t s . are r e s o l v e d i n t h i s  thesis.  sources  asso-  which  These d i s c r e p a n c i e s  -21-  An  important r e c e n t  emphasis i n LEED c r y s t a l l o g r a p h y  development o f s u i t a b l e r e l i a b i l i t y between experimental and appears t o be  that  f a c t o r s f o r making r o u t i n e  c a l c u l a t e d 1(E)  introduced  by  Zanazzi  curves. and  assessing  i t s value  The  the  comparisons  most complete  Jona [45].  f a c t o r i s s t u d i e d here b o t h i n a c t u a l s u r f a c e by  involves  This  R-factor  reliability  structure determinations  f o r f i x i n g some n o n - g e o m e t r i c a l parameters  and  required  i n the m u l t i p l e s c a t t e r i n g c a l c u l a t i o n s o f LEED i n t e n s i t i e s . In the sulphur  experimental p a r t s o f t h i s t h e s i s , the a d s o r p t i o n s  on the  (100)  and  (110)  surfaces  o f oxygen  o f rhodium have been s t u d i e d  d i f f r a c t e d beam i n t e n s i t i e s measured f o r v a r i o u s  structures.  and  Complete LEED  c r y s t a l l o g r a p h i c analyses  w i t h f u l l m u l t i p l e - s c a t t e r i n g c a l c u l a t i o n s have  been made f o r the s u r f a c e  s t r u c t u r e s designated  Rh(100)-p(2x2)S.  schemes f o r c a l c u l a t i n g LEED i n t e n s i t i e s  l a r g e computational times arid computer core storages  scheme c a l l e d the q u a s i d y n a m i c a l method has  by Tong and Van  Hove [ 4 6 ] ; i t i s f a s t e r and  than the complete methods. for  and  chemical bonding.  A problem w i t h the p r e s e n t  A simpler  Rh(110)-c(2x2)S  These s t r u c t u r e s have proved u s e f u l f o r g a i n i n g some  insights into surface  concerns the  and  Initial  studies  systems o f weak s c a t t e r e r s [46,47], and  reported  here, p a r t i c u l a r l y  surfaces  o f rhodium.  required.  r e c e n t l y been proposed  r e q u i r e s much l e s s core  storage  i n d i c a t e t h a t i t c o u l d be further investigations  f o r s t r u c t u r e s i n v o l v i n g sulphur  useful  are  adsorbed  on  -22-  CHAPTER 2 C a l c u l a t i o n o f LEED  Intensities  -23-  2.1  C h a r a c t e r i s t i c s o f 1(E) curves A typical  1(E) curve has a l r e a d y been i l l u s t r a t e d  i s s p e c i f i c a l l y f o r the s p e c u l a r beam d i f f r a c t e d  i n f i g u r e 1.5; t h i s  from a Ni(100)  surface.  Such a curve shows c o n s i d e r a b l e s t r u c t u r e , t h a t i s the i n t e n s i t y e x h i b i t s a number o f maxima and minima as t h e energy section  1.2, the e l a s t i c r e f l e c t i v i t y  the t o t a l  incident electrons.  i s varied.  A l s o as noted i n  corresponds t o o n l y a few p e r c e n t o f  E a r l y attempts  t o e x p l a i n 1(E) curves i n LEED  based on the k i n e m a t i c a l t h e o r y (which i s a p p l i c a b l e when s c a t t e r i n g s e c t i o n s a r e v e r y low e.g. X-ray d i f f r a c t i o n a s u r f a c e whose s t r u c t u r e corresponds  [49]) were u n s u c c e s s f u l . F o r  to that o f the bulk, the kinematical  t h e o r y p r e d i c t s peaks i n 1(E) curves f o r t h e t r i p e r i o d i c d i f f r a c t i o n  where g(hkil) i s a v e c t o r o f .the r e c i p r o c a l LEED, e q u a t i o n  lattice.  condition  F o r the (hk) beam i n  (2.1a) becomes e q u i v a l e n t t o k" ~g  =  Peaks i n 1(E) curves which s a t i s f y  k ~o +  1(E) curves o f the type i n f i g u r e  + g(00il) ~  .  (2.1b) a r e termed  and may be d e s i g n a t e d by the index I.  1)  cross-  (2.1b) "primary Bragg  peaks"  F u r t h e r r e l e v a n t o b s e r v a t i o n s from  1.5 a r e as f o l l o w s :  Peaks i n e x p e r i m e n t a l 1(E) curves which a r e c l o s e t o s a t i s f y i n g t h e  primary Bragg than expected.  condition  ( e q u a t i o n 2.1b) a r e g e n e r a l l y found a t lower e n e r g i e s  T h i s suggests an i n n e r p o t e n t i a l  - a consequence o f t h e reduced p o t e n t i a l  c o r r e c t i o n i s n e c e s s a r y as  e x p e r i e n c e d by an e l e c t r o n i n s i d e the  c r y s t a l [50]. 2)  O f t e n more peaks a r e observed i n e x p e r i m e n t a l 1(E) curves than  expected  -24-  from e q u a t i o n 2.1.  T h i s suggests m u l t i p l e - s c a t t e r i n g i s s i g n i f i c a n t ;  i s c o n s i s t e n t w i t h the c r o s s - s e c t i o n s f o r s c a t t e r i n g o f low-energy  this  electrons  b e i n g o f the o r d e r o f u n i t mesh areas [51] (and hence s e v e r a l o r d e r s o f magnitude g r e a t e r than those f o r the s c a t t e r i n g o f X - r a y s ) . 3)  Peaks i n 1(E) curves g e n e r a l l y show i n c r e a s i n g widths w i t h  energy [ 5 2 ] . to f i n i t e  Peak widths  increasing  are r e l a t e d t o u n c e r t a i n t i e s i n energy and hence  l i f e - t i m e s v i a the u n c e r t a i n t y p r i n c i p l e [ 2 4 ] ; the average  life-  time can be i n t e r p r e t e d as t h e time f o r the e l e c t r o n t o t r a v e r s e the mean f r e e path l e n g t h (L) i n t r o d u c e d i n s e c t i o n 4)  1.2.  The d i f f r a c t e d beam i n t e n s i t i e s d e c r e a s e w i t h i n c r e a s i n g temperature  often  i n an e x p o n e n t i a l manner [53,54]. Such o b s e r v a t i o n s suggest t h a t the LEED p r o c e s s i s a dynamical i n which the n o n - g e o m e t r i c a l parameters description.  The  process  p l a y an important r o l e i n i t s  f i x i n g o f t h e s e parameters,  together with m u l t i p l e - s c a t t e r i n g  o f e l e c t r o n s through ordered s u r f a c e r e g i o n s , r e p r e s e n t c o m p l i c a t i o n s f o r an a n a l y s i s o f the d i f f r a c t i o n p r o c e s s . 2.2  P h y s i c a l Parameters  r e q u i r e d i n LEED Theory  I t has a l r e a d y been i n d i c a t e d t h a t the i n c i d e n t e l e c t r o n s i n LEED e x p e r i e n c e s t r o n g e l a s t i c and i n e l a s t i c s c a t t e r i n g s ; c l e a r l y the c r y s t a l p o t e n t i a l must be chosen  c a r e f u l l y to accommodate t h e s e two  f e a t u r e s i n LEED i n t e n s i t y c a l c u l a t i o n . a convenient model f o r t h i s purpose. the p o t e n t i a l  important  The " m u f f i n - t i n " p o t e n t i a l p r o v i d e s  In t h i s approximation  i s taken as s p h e r i c a l l y symmetric  (figure  2.1),  i n the v i c i n i t y o f atoms and  gure 2.1:  Muffin-tin  potential  a)  i n c r o s s - s e c t i o n as c o n t o u r s ,  b)  a l o n g xx' ( V  i s t h e c o n s t a n t i n t e r s p h e r e p o t e n t i a l ).  Energy T 01  Energy Vocuum level —  F  E Fermi energy p  o Lowest level of conduction bond gure 2.2:  I l l u s t r a t i o n o f t h e r e l a t i o n s h i p between e n e r g i e s measured wi r e s p e c t t o t h e vacuum the lowest  level  l e v e l and t h o s e measured w i t h r e s p e c t t  o f t h e c o n d u c t i o n band.  -26-  c o n s t a n t elsewhere. equated  The r e a l p a r t o f t h e c o n s t a n t p o t e n t i a l  t o t h e e m p i r i c a l i n n e r p o t e n t i a l noted above;  t o t h e sum o f t h e Fermi 2.2.  V  o r  muffin-tin  energy  i s n e g a t i v e and i t can be regarded zero below t h e vacuum l e v e l ;  range from -10 t o -20 eV.  up t h e i n c i d e n t  I i s roughly  equal  i n figure  as g i v i n g t h e p o s i t i o n o f t h e  i t i s associated with the p o t e n t i a l  electrons to s o l i d s . Typical values o f  The e f f e c t  o f t h i s p o t e n t i a l w e l l i s t o speed  e l e c t r o n s i n s i d e the c r y s t a l .  endent on energy  ) i s often  and t h e work f u n c t i o n as i l l u s t r a t e d  w e l l that confines the conduction V  1^  (V  Although  is strictly  [ 5 5 ] , because o f exchange and c o r r e l a t i o n e f f e c t s ,  dep-  this  dependence i s o f t e n s u f f i c i e n t l y weak t h a t i t can be i g n o r e d f o r t h e purpose of c a l c u l a t i n g give a r i g i d  1(E) curves  shift  [ 5 6 ] . To a good a p p r o x i m a t i o n ,  i n calculated  1(E) c u r v e s ; t h i s  changes i n V  enables v a l u e s o f V^^ used  i n c a l c u l a t i o n t o be r e f i n e d by t r a n s l a t i n g the c a l c u l a t e d 1(E) curves the energy  scale u n t i l  o p t i m a l matching with the c o r r e s p o n d i n g  along  experimental  1(E) curves i s o b t a i n e d [ 5 7 ] . Inelastic scattering i s conveniently incorporated into schemes by g i v i n g an imaginary  calculation  c o n t r i b u t i o n t o the intersphere p o t e n t i a l ,  t h a t i s e x p r e s s i n g t h e constant p a r t o f t h e p o t e n t i a l as V  =  o  V  or  + iV . oi  .  (2.2)  For an e l e c t r o n wave f u n c t i o n w i t h time dependence Vfr.t)  Y f r )e  =  l E t  ;  (2.3)  2V • t the i n t e n s i t y decays w i t h time as e  0  provided V  1  i s negative.  Pendry  [24] e s t a b l i s h e d t h e r e l a t i o n AE  = w  .I 01  2IV 1  1  (2.4)  -27-  where AE^ i s t h e peak width at h a l f maximum h e i g h t i n an 1(E) curve and 2 the  analysis  estimating  uses atomic u n i t s  2  ( n =ro  =e e  =1).  Equation  v a l u e s o f V ^ from e x p e r i m e n t a l i n t e n s i t i e s ; t y p i c a l l y V  around -5 eV w i t h a f a i r l y weak energy dependence [58]. proposed t h e use o f t h e f u n c t i o n a l form V . = -aE oi In p r a c t i c e , e s p e c i a l l y f o r an o v e r l a y e r , the  (2.4) i s h e l p f u l f o r  1 / 3  a schematic r e p r e s e n t a t i o n  .  (2.5)  i s indicated  I d e a l l y t h e p o t e n t i a l used i n LEED c a l c u l a t i o n s calculations  close to  o f the substrate region  of the c r y s t a l p o t e n t i a l  s e l f - c o n s i s t e n t band s t r u c t u r e  Demuth et a l . [56]  the c r y s t a l p o t e n t i a l  topmost atoms can be d i f f e r e n t from t h a t  [60],  potentials  f i n i t e clusters by  an e l e c t r o n  local density  from the s u p e r p o s i t i o n  [22,6l].  i s constructed  However s u i t a b l e  <K,L\) i s most o f t e n  ex  The  potentials involves  experienced  r e p r e s e n t e d by S l a t e r ' s  approximation [62]  V (£)Kr) where p ( r )  from  o f atomic charge d e n s i t i e s i n  In e i t h e r case, t h e exchange p o t e n t i a l  o f wave f u n c t i o n  [59];  i n f i g u r e 2.3,  o f t h i s type a r e not always a v a i l a b l e , and a p l a u s i b l e a l t e r n a t i v e constructing  is  i s the l o c a l  charge  =  -6(ig&?)  1 / 3  *fr)  (2.6)  density.  s c a t t e r i n g o f an e l e c t r o n p l a n e wave by a s p h e r i c a l l y  symmetric  i o n - c o r e p o t e n t i a l y i e l d s a s p h e r i c a l wave, and t h e t o t a l w a v e f i e l d a t large  |'rj has t h e form [63,64] 4< ( r ) = s ~  *  e ^'^ + f(9) 1  ,ii*iur  (2.7)  IMAGINARY POTENTIAL VACUUM LEVEL  ADSORBATE LAYER SPACING SUBSTRATE LAYER SPACINGS  i t  REAL POTENTIAL  SUBSTRATE NO REFLECTION MATCHING ADSORBED LAYER TRANSITION REGION A t a n M B z * C) k  2 3:  M u f f i n T i n model o f an a d s o r b a t e covered s u r f a c e  ( a f t e r Marcus et a l . [ 5 9 ] ) .  -29-  The  s c a t t e r i n g amplitude f ( 8 )  f(8) where 6^  J~T  =  i s commonly expanded  C2£+l)expCi6 )sin<5 P (cos9)  L  £  1  i s the phase s h i f t which c h a r a c t e r i z e s  a n g u l a r momentum SL, and  sphere and  at the boundary o f the  found by  j o i n i n g the  f(G)  sphere to t h o s e s o l u t i o n s  as  (2.8)  i s about  The  factor.  max  solution solving  inside smoothly the  f o r LEED i t i s  o n l y a l i m i t e d number o f I f o r e n e r g i e s up  to  and  ) needed i n e x p r e s s i o n s such r  thermal motion o f i o n - c o r e s i s g e n e r a l l y  i s o t r o p i c Debye-Waller-type c o n t r i b u t i o n Jepsen et a l . [57]  showed t h a t  w i t h some m o d i f i c a t i o n s  i n t o the  treated  atomic  ( f(6)  t o the phase s h i f t s .  for  ) o f the r i g i d  Specifically  by  scattering  the atomic s c a t t e r i n g f a c t o r  such a v i b r a t i n g l a t t i c e can be r e l a t e d to t h a t but  for  7.  e f f e c t o f the  adding an  o b t a i n e d by  In p r a c t i c e  (i.e. I  ion-cores  Schro'dinger e q u a t i o n  T y p i c a l l y i n LEED c a l c u l a t i o n s  eV, the maximum v a l u e of I '  (2.8)  For a p a r t i c u l a r atomic  a s y m p t o t i c form o f the  converges f a i r l y r a p i d l y so t h a t  v a l u e s are needed. around 200  s c a t t e r i n g by  s o l v i n g the  SchrOdinger e q u a t i o n f o r the o u t s i d e r e g i o n . found t h a t  ,  Jl  i s a Legendre p o l y n o m i a l .  p o t e n t i a l , phase s h i f t s are the m u f f i n - t i n  as  f o r the  lattice p^  atom, f(0,T)  =  f(6)exp[-M  (k'-kj ]  ,  2  P  (2.9)  I  where a wave c h a r a c t e r i z e d  by  Ic i s s c a t t e r e d  M  and  u  p.  "  ^< p>T U  into k ,  '  i s the v i b r a t i o n a l amplitude i n the d i r e c t i o n o f the momentum t r a n s f e r P  -30-  (k -k).  In the h i g h temperature l i m i t (0^)  temperature  /  v  P  3h T  2  i s the  i s r e l a t e d to the  Debye  T M k e p I D  '  2  T  atomic mass and  ,, ... (2.10)  2  =  ~ p  D  by  <*OT where M  (T>0 ), u  k  i s the  n  B  Boltzmann c o n s t a n t ,  Computational procedures f o r LEED i n t e n s i t i e s developed r a t h e r i n p a r t because o f the However, d u r i n g the  complexity associated  w i t h the  multiple-scattering.  1970's a number o f schemes have been d e r i v e d ,  reviews have been g i v e n by  Duke [ 2 2 ] ,  Tong [65]  and  slowly,  and  Stoner et a l . [ 6 6 ] .  e a r l i e s t c a l c u l a t i o n s n e g l e c t e d i n e l a s t i c s c a t t e r i n g [ 6 7 ] ; Duke and [68] were among the  first  t o emphasize the n e c e s s i t y  scattering  i n computational schemes.  calculated  and  e x p e r i m e n t a l 1(E)  Jepsen et a l . [57]  on  These c a l c u l a t i o n s  assumed:  i)  the  first  curves was  (100)  S u r f a c e geometries that  The  surface  helpful  for including  substantial  produced i n 1972  Tucker inelastic  agreement between i n the work o f  o f aluminium, s i l v e r and  correspond t o u n d i s t o r t e d  copper.  truncations  o f the  structures. ii)  Electron-ion  band s t r u c t u r e iii)  core i n t e r a c t i o n s  A b s o r p t i o n e f f e c t s can be  Lattice vibrations  indicated v)  The  be  r e p r e s e n t e d by p o t e n t i a l s  incorporated  can be  w i t h an  imaginary  potential  [62].  treated  by  a Debye-Waller type f a c t o r  above.  inner  from  calculations.  from u n i f o r m e l e c t r o n - g a s t h e o r y iv)  can  potential correction  a l i g n i n g t h e o r e t i c a l and  (V  )  c a n  e x p e r i m e n t a l 1(E)  he  chosen e m p i r i c a l l y  curves.  The  by  as  bulk  -31-  T h i s work o f Jepsen e t a l . e s t a b l i s h e d t h a t t h e dominant a s p e c t s o f the  e l a s t i c LEED p r o c e s s were e s s e n t i a l l y u n d e r s t o o d , even though n u m e r i c a l  agreement was not o b t a i n e d f o r a b s o l u t e  intensities.  The l a t t e r appears t o  r e l a t e e s p e c i a l l y t o incomplete o r d e r f o r the s u r f a c e s , b u t i n any event t h i s d i s c r e p a n c y d i d not i n h i b i t t h e development o f LEED  crystallography,  s i n c e i t was found t h a t the p o s i t i o n s o f s t r u c t u r e i n 1(E) curves c o u l d be c a l c u l a t e d to w i t h i n Calculations  experimental  error.  o f LEED i n t e n s i t i e s g e n e r a l l y  involve t r e a t i n g the scatter-  i n g o f a p l a n e wave by a s u r f a c e r e g i o n o f p e r f e c t d i p e r i o d i c symmetry. t o t a l wave f i e l d  outside  o f t h e c r y s t a l has t h e form ik  nr)  •r  = 4>(r) + Z c e ~& ~ g  where <K£) i s t h e i n c i d e n t p l a n e wave. reflectivities  ,  ~ The o b j e c t i v e i s t o c a l c u l a t e beam  'k g  k  c |  2  g  B r i e f d e s c r i p t i o n s o f some o f  important procedures now a v a i l a b l e f o r c a l c u l a t i n g beam i n the f o l l o w i n g  sections.  (2.12)  o  which r e l a t e t o the measured i n t e n s i t i e s .  are g i v e n  (2.11)  ,  R  the  The  reflectivities  -32-  2.3  T - M a t r i x Method The  T-matrix method was f o r m u l a t e d by Beeby [70] and has s i n c e been  d e t a i l e d f u r t h e r by Tong [ 6 5 ] . function  T h i s method s t a r t s by w r i t i n g t h e wave  f o r an e l e c t r o n i n s i d e t h e s o l i d as *C£)  =  •(£)  +  /G(r-r')  V(r')  where t h e Green s f u n c t i o n GQr-r, ) d e s c r i b e s from r, t o £ .  *(r') dr'  ,  (2.13)  t h e p r o p a g a t i o n o f an e l e c t r o n  T h i s e q u a t i o n can be s o l v e d by d e f i n i n g a t o t a l  scattering  m a t r i x (T) f o r t h e s o l i d V(r')nr')  =  / T(r'r)*Q:)dr  .  (2.14)  With t h e m u f f i n - t i n approximation f o r t h e p o t e n t i a l , s u b s t i t u t i o n o f (2.14) into  (2.13) y i e l d s  TCCa^)  EVr -R,r R)  =  2  R  E  +  r  ~  ^-^v^  {t .(r -R'.£ -R') ~ R  R*R  ~ ' ^  )  2  3  v  d  ^ 4 r  —  ( 2  -  1 5 )  where t  KtXrE.IrJP  •  V  E (  r -R) 2  6 x i l 2  2  2  i s t h e t - m a t r i x f o r t h e s i n g l e i o n c o r e a t R.  r  In (2.15), t h e f i r s t  covers a l l s i n g l e i o n c o r e s c a t t e r i n g , t h e second term r e p r e s e n t s s c a t t e r i n g events, etc. atomic and i n t r a - a t o m i c  (2.16)  */v,(r -R)Gtr -r)Vr-R,r R)dr  E q u a t i o n 2.15 t h e r e f o r e sums a l l p o s s i b l e  term  a l l double inter-  s c a t t e r i n g events i n v o l v e d w i t h t h e e l e c t r o n going  from r , t o r» i n s i d e t h e s o l i d ~1 ~2 For t h e a c t u a l e v a l u a t i o n  o f the c  g  i n (2.11), and hence.the beam  -33-  reflectivities,  the c r y s t a l  i s d i v i d e d i n t o subplanes p a r a l l e l t o t h e  s u r f a c e such t h a t each subplane has t h e same B r a v a i s s t r u c t u r e and c o n t a i n s the  same k i n d o f atoms.  c  =  y  ^ E  The f i n a l r e s u l t i s  Y. (k~)Y*,(k ;) * ~° I>  i ( k - k ~ ) - d a .., ° & ^ T (k )  +  +  L  L  where L stands f o r t h e a n g u l a r momentum quantum numbers I,m, a s s o c i a t e d s p h e r i c a l harmonic, and d  (2.17)  L  i s the  t h e second summation i s over a l l subplanes  i s t h e v e c t o r from t h e o v e r a l l o r i g i n a t t h e i n t e r f a c e t o t h e o r i g i n  chosen f o r subplane a.  In (2.17) T^  (k ) ^  t h e element  s  Q  o f the t o t a l  s c a t t e r i n g m a t r i x f o r subplane a i n t h e a n g u l a r momentum r e p r e s e n t a t i o n i  L L 1  T  ex  (k ) i s t h e LL o  Qfa  2  1 2  o f the planar s c a t t e r i n g matrix ( f ) f o r the »a  element  r  subplane a  T  (  k  )  =  t (k ) [ l - G ( k . ) t  (k ) ]  S p  _  1  ,  (2.19)  and t (k ) i s t h e d i a g o n a l t - m a t r i x f o r a s i n g l e i o n core i n subplane a. «a o The non-zero  elements  o f t h i s m a t r i x r e l a t e t o t h e phase s h i f t  4- \\' q2  t (k ) a o• M  v  =  by  1  [  2m  6  2ik  3  •  ( - °) 2  2  o  A l s o needed i n (2.18) and (2.19) a r e t h e i n t r a p l a n a r s t r u c t u r a l p r o p a g a t o r s G P and t h e i n t e r p l a n a r p r o p a g a t o r s G ^. S  a  are  dependent  the  i o n core s i t e s .  These a r e complex m a t r i c e s which  on t h e i n e l a s t i c s c a t t e r i n g and t h e geometries a s s o c i a t e d w i t h  -34-  S u c c e s s f u l c a l c u l a t i o n s have been made w i t h t h i s method f o r c l e a n metal s u r f a c e s .  In p r i n c i p l e i t i s exact  and can work f o r any type o f  s u r f a c e s t r u c t u r e ; i n p r a c t i c e , however, t h e s o l v i n g o f t h e s e t o f equations '(2.18) t o g i v e t h e m a t r i x  T  i s v e r y time consuming and r e q u i r e s a l a r g e  amount o f computer core s t r o a g e t o be i n c l u d e d .  i f an a p p r e c i a b l e number o f subplanes have  T h i s method i s o n l y p r a c t i c a l  s c a t t e r i n g , a f e a t u r e t h a t Beeby n e g l e c t e d  i n the presence o f i n e l a s t i c  i n the i n i t i a l  formulation.  The  e x t e n s i o n t o i n c l u d e thermal motion o f t h e i o n cores was made by Tong and Rhodin i n 1971 f o r t h e (100) s u r f a c e o f aluminum [ 7 1 ] ,  2.4  Bloch Wave Method T h i s method was i n t r o d u c e d by McRae [67,73] and developed by Pendry [ 7 4 ] ,  Kambe [75,76] and Jepsen et a l . [57,77]. shed i n Pendry s book [ 2 4 ] .  In t h i s approach, the m u f f i n - t i n a p p r o x i m a t i o n  i s a g a i n used and an i n f i n i t e c r y s t a l the r e g i o n o f constant  A d e t a i l e d account has been p u b l i -  i sbuilt  up o f p a r a l l e l  p o t e n t i a l between s u c c e s s i v e  can be expanded i n terms o f p l a n e waves.  layers. For  l a y e r s , each Bloch wave  The s c a t t e r i n g s i t u a t i o n a t a s i n g l e  l a y e r i s d e p i c t e d i n f i g u r e 2.4, where a s e t o f i n c i d e n t p l a n e waves Y.(r) i ~  =  Z b exp(ik -r) _ g ~g ~ +  +  (2.21)  i s d i r e c t e d onto t h e c r y s t a l , and s c a t t e r e d waves Y (r) = $ ~  Y ~ i  gg  M*f b e x p f i k V r ) g g g ~ J ~ ~ ~ ~  propagate both i n t h e outward d i r e c t i o n (k*t).  The m a t r i c e s  +  (k ) and i n t h e inward f  i n v o l v e d i n t h i s f o r m u l a t i o n a r e expressed  (2.22)  direction i n terms o f  -35-  ft  /  £b;exp(ik -r) +  g  F i g u r e 2A\  8  EE  ~  M^b;.xp(l|4:r)  Schematic r e p r e s e n t a t i o n o f a s e t o f p l a n e wave i n c i d e n t from the  left  p  and m u l t i p l y s c a t t e r e d by a p l a n e o f i o n c o r e s  p  +  1  t h a  layer  -+  Figure  2.5:  Schematic diagram of transmission and reflection'matrices at the a  t h  subplane.  The broken lines are the central lines  between the subplanes.  -36-  t h e l i n e a r momentum (K-space) r e p r e s e n t a t i o n ; t h i s c o n t r a s t s w i t h t h e a n g u l a r momentum  (L-space) r e p r e s e n t a t i o n i n t h e T-matrix method. d i f f r a c t i o n m a t r i x M ++ where both i n c i d e n t  of the layer  and d i f f r a c t e d beams  A  move i n t o t h e c r y s t a l .  ++ The n o t a t i o n M , c o v e r s - a l l g'g  M*t i s an element g g  f o u r combinations o f  directions. It i s c l e a r  f o r the s i t u a t i o n  beams become coupled t o g e t h e r . layers  i n figure  The c o e f f i c i e n t s  a and a+1 can be expressed b i »a+l +  b" ara  2.4 t h a t a l l t h e d i f f r a c t e d  ==  =  f o r p l a n e waves between  i n a compact m a t r i x n o t a t i o n T b «a wa  + R b , «a a+1  (2.23a)  " i a "a " a+1  +  R" b aa «a  (2.23b)  T  +  +  b  +  where, f o r example, t h e components o f t h e column v e c t o r b* a r e t h e v a r i o u s values o f b layers,  +  between l a y e r s  a and a+1.  For a c r y s t a l  composed o f i d e n t i c a l c,  which a r e s e p a r a t e d by a constant displacement  the transmission  and r e f l e c t i o n m a t r i c e s can be expressed as T | +  =  P i +  I I  T"  &&  +  =  J  & fL  + M"T  ) P"  i  (2.24a)  +  M  l  (2.24b)  l  =  P . M T P" g | | g  (2.24c)  R~* g g  =  P", M"t P g g g g  (2.24d)  +  +  +  r e p r e s e n t s inward p r o p a g a t i o n  o f an i n t e r l a y e r  ) P  +  M&  P", ( I ,  &  + M t  J  R*7  2 8  where P  ( I i  g  distance while P  p r o p a g a t i o n w i t h wave v e c t o r k  w i t h wave v e c t o r k  +  through one h a l f  r e p r e s e n t s t h e c o r r e s p o n d i n g outward  -37-  P~ g The I i  i n equation  =  ik  e  +  (2.25)  s  (2.24) are elements o f a u n i t m a t r i x .  Schematic r e p r e -  S I s e n t a t i o n s o f the r e f l e c t i o n and t r a n s m i s s i o n m a t r i c e s  a r e shown i n f i g u r e  C o r r e s p o n d i n g c o e f f i c i e n t s between s u c c e s s i v e l a y e r s must  2.5.  s a t i s f y the  Bloch c o n d i t i o n s +  a+1  a  e  ik-c , + ~ b a  (2.26a)  e  -ik-c , ~ " b  (2.26b)  (2.24) i n t o  l"b  1  +  a+1  (2.26) y i e l d s t h e e i g e n v a l u e  = X  equation  (2.27) b" , «a+l  " i »a+l b  where  T L  R  + +  (2.28)  =  -(T"")" R"V 1  T""-(T"") R"V" _ 1  +  and X  =  Pendry [24] has d i s c u s s e d t h e  ik- c exp~~ *  .  (2.29)  e v a l u a t i o n o f the l a y e r d i f f r a c t i o n  ±± matrices M  i n terms o f the s c a t t e r i n g p r o p e r t i e s o f the i n d i v i d u a l  ion-cores  For a l a y e r which i n v o l v e s a s i n g l e atom per u n i t mesh, the elements s a t i s f y ++ M"T  g g  rS.' ftgO[l-X]-J;.Y .fk*)exp(i6 ,Dsin6 ,Y  L  Ak k* LL' o~gi  L  1  1  (2.30)  -38-  where  ++  d e s c r i b e s m u l t i p l e s c a t t e r i n g w i t h i n the l a y e r .  Given M~  for a  p a r t i c u l a r system, the t r a n s m i s s i o n and r e f l e c t i o n m a t r i c e s i n equations (2.24) can be s e t up and hence (2.27) can be s o l v e d by s t a n d a r d methods t o g i v e e i g e n v e c t o r s , which f i x the Bloch waves, and the c o r r e s p o n d i n g v a l u e s which f i x p o s s i b l e wave v e c t o r s a l o n g w i t h the requirement v a t i o n o f momentum p a r a l l e l Only h a l f o f the 2n  possible solutions  (where n  i s the number o f v e c t o r s  ) are p h y s i c a l l y a c c e p t a b l e  to waves which e i t h e r propagate  ( i . e . correspond  or decay e x p o n e n t i a l l y i n the  To complete the c a l c u l a t i o n o f d i f f r a c t e d beam r e f l e c t i v i t i e s to match each wave f u n c t i o n , and both s i d e s o f the solid-vacuum  its first  interface.  are i n v o l v e d i n extending  top l a y e r s are d i f f e r e n t  o f conser-  t o the s u r f a c e .  g i n c l u d e d i n the c a l c u l a t i o n  dures  eigen-  z-direction). i t i s necessary  d e r i v a t i v e w i t h r e s p e c t t o z, at Corresponding  wave matching p r o c e -  t h i s scheme t o s i t u a t i o n s where one  from the r e s t . (e.g. f o r an adsorbed  or more  layer).  This  b a s i c approach i n v o l v e s l e s s computer• core s t o r a g e than the T-matr-ix method, but the s o l u t i o n o f e q u a t i o n  (2.27) becomes time consuming when n  g  i s large.  -39-  2.5  P e r t u r b a t i o n Methods The  T-matrix and  the Bloch wave methods are exact  i n the sense t h a t  they i n c l u d e a l l m u l t i p l e s c a t t e r i n g events i n the c r y s t a l . have proved v a l u a b l e  f o r c a l c u l a t i n g LEED i n t e n s i t i e s o f c l e a n  a l t h o u g h they r e q u i r e long computational times and considerations  These methods surfaces,  l a r g e core s t o r a g e .  Such  l i m i t the use o f t h e s e exact m u l t i p l e s c a t t e r i n g methods to  the more complex s u r f a c e s t r u c t u r e s o f i n t e r e s t i n LEED c r y s t a l l o g r a p h y , t h e r e f o r e encourage the development o f approximate schemes based on b a t i o n expansions. that with  P a r t o f the m o t i v a t i o n  for this  i n e l a s t i c s c a t t e r i n g the c o m p a r a t i v e l y  must l i m i t the order  comes from the  perturrealization  s h o r t mean f r e e path  o f m u l t i p l e s c a t t e r i n g t h a t can be  important.  c a l c u l a t i o n to t h i r d o r d e r , and t e r i n g metals l i k e aluminum.  Tong et a l . [72] made the  length  This  suggests t h a t i t should be p o s s i b l e to reduce computational times by l a t i n g i n terms o f p e r t u r b a t i o n t h e o r y .  and  formu-  T-matrix  showed t h a t i t can work w e l l f o r weak s c a t -  However the appoach o f u t i l i z i n g  perturbation  t h e o r y w i t h i n the T-matrix method seems l e s s h e l p f u l f o r s t r o n g e r s c a t t e r e r s ; b a s i c a l l y t h i s appoach becomes too Pendry has  have the s i g n i f i c a n t p r o p e r t y  from each a d d i t i o n a l o r d e r has (this  described  order.  These new  methods are the  on  t h a t the c o n t r i b u t i o n  the same b a s i c form as those from the  i s u n l i k e the s i t u a t i o n f o r the t h i r d order  noted above f o r A l ( 1 0 0 ) ) . renormalized  unwieldy at above t h i r d  developed convenient i t e r a t i v e schemes which are based  the Bloch wave method and  orders  clumsy and  previous  c a l c u l a t i o n [72]  layer doubling  and  forward s c a t t e r i n g methods; m u l t i p l e s c a t t e r i n g c a l c u l a t i o n s  in this thesis utilized  t h e s e methods e x t e n s i v e l y .  -40-  2.5  (a)  Layer D o u b l i n g Method  T h i s method [24,78] r e q u i r e s t h a t i n e l a s t i c s c a t t e r i n g i s s u f f i c i e n t l y s t r o n g so t h a t a s e m i - i n f i n i t e c r y s t a l f i n i t e thickness. at  Two  can be approximated  layers are considered f i r s t ,  by a s l a b o f  then f o u r l a y e r s , and  each l e v e l o f i t e r a t i o n the number o f l a y e r s i s doubled.  T h i s method  s t a r t s w i t h a c a l c u l a t i o n o f the r e f l e c t i o n and t r a n s m i s s i o n m a t r i c e s as in  equations  s t a c k o f two  (2.24), and then generates the c o r r e s p o n d i n g m a t r i c e s f o r a layers. T!  +  +  Ic  =  -  T !  R~  +  +  -  §  -  IA  A  ( I - C O "  + T~~R~ AA * B  by C.  _ 1  T *A  (2.31b)  + + y  (2.3id)  a r e denoted by A , B and t h e r e s u l t i n g  composite  The d o u b l i n g p r o c e s s i s shown s c h e m a t i c a l l y i n  s t a c k t o 2, 4, 8, 16... l a y e r s . t i o n amplitudes have converged;  (2.31)  a r e used t o extend t h e c r y s t a l  This process i s continued u n t i l the r e f l e c t y p i c a l l y t h i s r e q u i r e s 8 or 16 atomic  layers.  r e f l e c t i v i t i e s have been o b t a i n e d f o r the s u b s t r a t e , s u r f a c e  l a y e r s can be s y s t e m a t i c a l l y added s t i l l f e a t u r e o f t h i s method or  +  J  1  f i g u r e 2.6; t h e same s e t o f equations  Once convergent  ri-R "R" ) i SA * B +  l  (2.31a)  +  a-CsI")" I""  where t h e i n d i v i d u a l subplanes l a y e r i s denoted  +  T+  1  u s i n g equations  (2.31).  A convenient  i s t h a t a s u r f a c e l a y e r can be s h i f t e d e i t h e r  v e r t i c a l l y , without h a v i n g t o recompute t h e b u l k  reflectivities.  laterally  -41-  Figure 2.6:  Stacking  of planes to form a c r y s t a l slab and i l l u s t r a t e th.  layer-doubling  method.  Planes A and B are f i r s t stacked to  form the two-layer slab C; the process i s continued to form four-layer slab.  (After Tong [65].)  -42-  T h i s method i s c o n s i d e r a b l y f a s t e r than the f u l l yet  i t can p r o v i d e good n u m e r i c a l a c c u r a c y .  s i o n s o f two m a t r i c e s o f dimension calculation).  (b)  and  Each i t e r a t i o n i n v o l v e s i n v e r -  (the number o f beams i n c l u d e d i n the  A l i m i t a t i o n i s t h a t t h i s method i s not s u i t a b l e f o r v e r y s m a l l  i n t e r l a y e r spacings  2.5  n  Bloch wave method  (c<0.5A) when n^ i s r e q u i r e d t o be e x c e s s i v e l y l a r g e [ 7 9 ] .  Renormalized  Forward S c a t t e r i n g method  The r e n o r m a l i z e d forward  scattering  (RFS) method was  i n t r o d u c e d by  Pendry [24] and d i s c u s s e d f u r t h e r by Tong [ 6 5 ] ; i t s c h a r a c t e r i s t i c f e a t u r e s are t h a t the i n t r a l a y e r s c a t t e r i n g s are c a l c u l a t e d e x a c t l y , w h i l e t h e l a y e r s c a t t e r i n g s are i t e r a t e d  f o r the v a r i o u s p o s s i b l e paths  The p r i n c i p l e o f t h i s method i s i l l u s t r a t e d The  crystal  i n the  schematically i n figure  i s a g a i n r e p r e s e n t e d by a f i n i t e number o f l a y e r s ;  a c t u a l number used  (n) i s such t h a t the t o t a l  elastically  intercrystal. 2.7.  the  s c a t t e r e d amplitude  til  t h a t would r e a c h the (e.g.  (n+1)  l a y e r i s l e s s than a predetermined  0.003) o f i t s i n c i d e n t amplitude.  C l e a r l y the s t r o n g e r the  s c a t t e r i n g , the s m a l l e r i s the number o f l a y e r s t h a t are needed. Tong [ 6 5 ] , A th and  (a+1)  •  (g) d e s i g n a t e s the amplitude a ~  fraction inelastic Following  at the l o c a l o r i g i n between t h e a  l a y e r s f o r the e l e c t r o n wave c h a r a c t e r i z e d by g p r o p a g a t i n g  t ll  into  the c r y s t a l ; the index i i s the o r d e r o f i t e r a t i o n which i d e n t i f i e s the number of  times the e l e c t r o n has propagated  - path.  For the f i r s t  i t e r a t i o n we  i n t o the c r y s t a l a l o n g t h i s  have  V&> =. CtsOVi^ ' z  g  particular  -  (2  32)  -43-  incident  beam _  It  (a  (a-1)  +  l)  A (S a  (go.)  .++  (28)  Figure 2.7:  a) I l l u s t r a t i o n of the renormalized forward scattering method. V e r t i c a l lines represent layers. Each t r i p l e t o f arrows represents the complete set of plane waves that t r a v e l from layer to layer. b) Propagation steps of the inward-travelling waves. c) Propagation steps of the outward-travelling waves. (After Van Hove and Tong [ 8 l ] . )  -44-  but no waves propagating i n the inward d i r e c t i o n are included after the n*^ layer.  Waves propagating i n the outward d i r e c t i o n are represented by  in an analogous notation.  Except at the deepest layer, the outward  l i n g waves consist of two components (figure 2.7c):  B (g) 1  travel-  the r e f l e c t e d portions  of the inward t r a v e l l i n g waves, and the transmitted portion of the outwardt r a v e l l i n g waves.  In general, the amplitudes of the outward-directed waves  satisfy  I  J  (2.33)  ( a = n-1, n-2,  ... 0 ),  where n i s the deepest subplane reached i n the appropriate i t e r a t i o n .  The  corresponding expression for the inward-directed waves i s A (g) X  a ~  =  Z  R "(gg')B " (g') +  ' a  1  ~2  a  1  ~  '  g  + E  T (gg')A ++  i  a  ~~  .(g ) a-1 2  1  1  J  ;  '  I  (2.34)  ( a = 1, 2, 3, ... n ).  Equations (2.33) and (2.34) are solved i t e r a t i v e l y i n the RFS method u n t i l the r e f l e c t i v i t y has converged.  This approach i s computationally  convenient since no eigenvalue equations or matrix inversions are involved. 2 The computation times scale as n , where n i s the number o f beams included; this i s more favorable than the layer doubling method for which 3 time scales as n .  g  computation  The RFS method has proved to be an excellent method for  c a l c u l a t i n g LEED i n t e n s i t i e s for many systems provided the electron damping is sufficient.  Otherwise i t s only l i m i t a t i o n i s a f a i l u r e to converge when o  . any two layers are closer than about 1 A. doubling method may be applicable.  In the l a t t e r event the layer  -45-  2.6  F u r t h e r M u l t i p l e S c a t t e r i n g Methods The  venient  RFS  and  layer doubling  methods have proved to be r e l i a b l e and  f o r LEED c r y s t a l l o g r a p h i c a n a l y s e s  o f many c l e a n and  simple  con-  overlayer  surface structures.  A l i m i t a t i o n i n a l l procedures which u t i l i z e the  K-  space r e p r e s e n t a t i o n  ( i n c l u d i n g the f u l l  the  number o f p l a n e waves r e q u i r e d decreasing  Bloch-wave method ) i s t h a t  i n the c a l c u l a t i o n s i n c r e a s e s r a p i d l y w i t h  i n t e r l a y e r separations  [79].  Once m a t r i c e s  o f dimension o f  the  2 order and  o f 10  are i n v o l v e d the K-space methods become i n c r e a s i n g l y unwieldy  numerically  unreliable; effective  l i m i t s are s e t w i t h  i n t e r l a y e r spacing  o  o f around 0.5  o  A f o r both the  s e t s the lower l i m i t  layer doubling  f o r the RFS  and  the two  the  are r e q u i r e d , t h e r e are  two  ( i ) to s t a y w i t h  the K-space r e p r e s e n t a t i o n but  l a y e r s as a composite l a y e r (with consequent i n c r e a s e i n  dimensions and with  ( ~  requirements f o r computing time and  L-space r e p r e s e n t a t i o n  of matrices  s t o r a g e ) , or  (as i n the T-matrix method).  matrix ( i i ) to work  The  dimensions  c a l c u l a t i o n s , hence t h i s approach s t a r t s to have  advantages over the K-space r e p r e s e n t a t i o n when n  i s large.  T-matrix method more e f f i c i e n t ,  [80]  Zimmer and  Holland  method i n the L-space r e p r e s e n t a t i o n .  accounts f o r forward s c a t t e r i n g events,  but  an  Holland  a  reverse-  equivalent  T h i s approach a g a i n  approximates the  r e v e r s e s c a t t e r i n g p r o c e d u r e o f Zimmer and  To make the  introduced  s c a t t e r i n g i t e r a t i v e procedure which e s s e n t i a l l y r e p r e s e n t s  The  treat  i n v o l v e d i n L-space c a l c u l a t i o n s are independent o f the number  o f beams r e q u i r e d f o r the  the RFS  lA  method ).  For models where c l o s e i n t e r l a y e r s p a c i n g p o s s i b l e approaches:  the Bloch wave methods  of  fully  back-scattering.  requires matrices  of  -46-  dimensions 200  2 (I +1) . max  Typically I +1^8 f o r e l e c t r o n energies max 1  V  l e s s than  J  eV, thus t h i s i t e r a t i v e method appears advantageous over t h e RFS method  i f t h e number o f beams r e q u i r e d exceeds about 64.  However t h i s  i t e r a t i o n approach r e q u i r e s t h e e v a l u a t i o n and s t o r a g e matrices  (3°^ f o r an n - l a y e r  recalculated  L-space  of n(n-l)  square  c r y s t a l , and moreover these m a t r i c e s  f o r every change made t o t h e s u r f a c e  layer.  have t o be  This represents  a  l e s s s a t i s f a c t o r y f e a t u r e o f the method. Recently  Van Hove and Tong [79] d e s c r i b e d  a combined-space method which  u t i l i z e s both t h e L-space and K-space r e p r e s e n t a t i o n t o a c h i e v e antages o f each.  optimal  adv-  S p e c i f i c a l l y the c a l c u l a t i o n i s made i n the L-space  representation  f o r those l a y e r s which a r e c l o s e l y spaced, w h i l e  representation  i s used f o r t h e r e s t o f the c a l c u l a t i o n where t h e i n t e r l a y e r  spacings  are larger.  Discussions  o f approaches and t h e a s s o c i a t e d  programs f o r t h e v a r i o u s methods now a v a i l a b l e f o r s u r f a c e with  LEED have been d e s c r i b e d  t h e K-space  computer  crystallography  i n a r e c e n t book by Van Hove and Tong [ 8 l ] .  i  -47-  2.7 2.7  General Aspects o f Computations (a) The  S t r u c t u r a l Parameters and  Use  o f Symmetry  b a s i c approach to s u r f a c e  c r y s t a l l o g r a p h y w i t h LEED i n v o l v e s pos-  t u l a t i n g a s e r i e s of t r i a l  s t r u c t u r e s and  model which g i v e s the best  agreement between c a l c u l a t e d and  1(E)  curves.  The  models p o s t u l a t e d  must be  i n d i c a t e d by the observed LEED p a t t e r n . known from X-ray c r y s t a l l o g r a p h y , but s u r f a c e need not crystal.  f o r that p a r t i c u l a r experimental  c o n s i s t e n t w i t h the  The  substrate  symmetries  structure i s generally  atoms i n the upper l a y e r s o f a  occupy e x a c t l y the p o s i t i o n s they would i n the  clean  infinite  Many c l e a n metals have s u r f a c e s whose t r a n s l a t i o n a l symmetries  are found by structure  searching  LEED t o be  (the s u r f a c e  i d e n t i c a l w i t h those o f the c o r r e s p o n d i n g i s s a i d to be u n r e c o n s t r u c t e d  are m a i n t a i n e d , although t h e r e may contrast  LEED p a t t e r n s  general,  f o r both c l e a n s u r f a c e s  l a y e r spacing  be  must be v a r i e d i n the  v a r i a t i o n s should  a l s o be  are p o s s i b l e a p p r o p r i a t e l a t i o n s i n order  changes i n the v e r t i c a l  adsorption  spacing); [86].  systems, the topmost  For models where domain  In  interlateral  calcu-  expectation, t h a t the i n c i d e n t beam i n For example, f o r  sulphur  s i t e s o f the Rh(100) s u r f a c e , as i n f i g u r e 2;8,  i n t e n s i t i e s o f the  by  structures  beam i n t e n s i t i e s need t o be averaged i n the  the experiment samples a l l the domain t y p e s .  a v e r a g i n g of the  registries  LEED i n t e n s i t y c a l c u l a t i o n s and  considered.  t o accommodate the  adsorbed on the b r i d g e  i f the normal  show d i r e c t l y t h a t many are r e c o n s t r u c t e d and  substrate  (10)  and  an  (01) beams i s n e c e s s a r y f o r  the c a l c u l a t i o n s to become c o n s i s t e n t w i t h the f o u r f o l d symmetry observed i n the e x p e r i m e n t a l LEED p a t t e r n .  •48-  Rh(l00)-P(2x2)-S real reciprocal  11  - $ — ©  ® -  m  0  <s> ©  i  &  © 11  2 mirror  planes + 1 C  A  axis  1F  2 mirror p l a n e s  2.8:  only  Schematic diagram o f three simple models f o r Rh(100)-p(2*2>S In r e c i p r o c a l space, sets o f symmetrically equivalent beams ; i n d i c a t e d by a common symbol.  -49-  Th e computational  effort  can be reduced when t h e d i r e c t i o n . o f i n c i d e n c e  c o i n c i d e s w i t h a symmetry a x i s or a symmetry p l a n e [ 8 1 ] ; t h i s depends on the i n e v i t a b l e e q u i v a l e n c e s i n the d i f f r a c t e d beams as a r e s u l t e t r y elements i n t h e model.  The s i m p l i f i c a t i o n s  o f t h e symm-  i n the m u l t i p l e - s c a t t e r i n g  c a l c u l a t i o n s r e p r e s e n t a s t a n d a r d a p p l i c a t i o n o f t h e group t h e o r y . symmetry reduces  Utilizing  the dimensions o f t h e m a t r i c e s r e q u i r e d w i t h i n the K-space  r e p r e s e n t a t i o n [ 8 2 ] , s p e c i f i c a l l y o n l y one g v e c t o r i s needed f o r each s e t o f symmetry-related  beams.  For the p a r t i c u l a r  examples o f t h e model types shown  i n f i g u r e 2.8 f o r Rh(100)-p(2x2>S, i t i s r e a d i l y seen t h a t , w i t h normal i n c i d e n c e , the 4F and IF models p r e s e r v e two m i r r o r p l a n e s o f symmetry p e r p e n d i c u l a r t o each o t h e r as w e l l as a model c o n t a i n s o n l y two m i r r o r p l a n e s .  r o t a t i o n a x i s , whereas t h e 2F A consequence o f the  a x i s i s the  e q u i v a l e n c e o f t h e f o l l o w i n g 8 f r a c t i o n a l o r d e r beams  ( l j ) = ( l | )  = ( l | ) E ( i j ) E ( i l ) E ( i l ) E c f l )  f o r both t h e 4F and IF models.  =( i l )  The s i t u a t i o n f o r t h e 2F model i s t h a t t h e s e  f r a c t i o n a l o r d e r beams s e p a r a t e i n t o two s e t s o f 4 e q u i v a l e n t beams, ( 1 | ) = ( 1  y ) =( 1 y ) =( 1 \  ) f  ( \  1 ) = ( \ "1 ) = ( y l ) = ( J 1 )  S i m i l a r l y , the 4F and IF models have the e q u i v a l e n c e s ( 0 1 ) = ( 0 1 ) = ( 1 0 ) = ( 1 0 ) whereas t h e c o r r e s p o n d i n g s i t u a t i o n f o r the 2F model i n v o l v e s  (oi )  M  oi )  M  l  o ) = ( I o ).  The c a l c u l a t i o n s f o r t h e 2F model t h e r e f o r e r e q u i r e more beams, and c o r r e s p o n d i n g l y l a r g e r m a t r i c e s , than t h e 4F and IF models.  Table 2 . 1 :  Numbers o f s y m m e t r i c a l l y - i n e q u i v a l e n t beams a c t u a l l y used i n c a l c u l a t i o n o f v a r i o u s s u r f a c e structures.  Surface s t r u c t u r e  The models f o r t h e o v e r l a y e r s t r u c t u r e s a r e d e s i g n a t e d  S u r f a c e model  Number o f s y m m e t r i c a l l y  Equivalent  i n e q u i v a l e n t beams used  t o t a l number  Type o f symmetry  in Rh(100)  unreconstructed  as i n f i g u r e 1 . 7 and 2 . 8 .  calculation  o f beams 53  2 perpendicular mirror planes +  Rh(llO)  unreconstructed  23  71  10  37  same as R h ( 1 0 0 )  35  221  2 perpendicular mirror  52  177  2 perpendicular mirror  Rh(lll)  unreconstructed  planes  3 m i r r o r planes a t 6 0 ° to  each o t h e r +  along  z-axis  Rh(100)-p(2x2)-S  4F.1F 2F  planes  Rh(110)-c(2x2)-S  4F,1F  same as R h ( 1 1 0 )  49  175  2SB.2LB  same as R h ( 1 1 0 )  49  175  -51-  C a l c u l a t i o n s r e p o r t e d here w i t h the RFS l i z e symmetry as i n the d i s c u s s i o n and and Tong [ 8 1 ] .  and  l a y e r d o u b l i n g methods u t i -  computer programs g i v e n by Van  In t h e s e r o u t i n e s symmetry i s accommodated by  l i s t i n g the g  v e c t o r s i n the i n p u t data t o g e t h e r w i t h a p p r o p r i a t e code numbers to the symmetry type o f each beam.  The  code number enables  use the a p p r o p r i a t e symmetrized wave f u n c t i o n s and d i f f r a c t i o n matrices.  L i s t e d i n T a b l e 2.1  Hove  identify  the program to  to s e t up the  a r e the numbers o f  simplified  symmetrically  i n e q u i v a l e n t beams needed f o r c a l c u l a t i o n s on the v a r i o u s s u r f a c e s s t u d i e d in this  2.7  (t>) The  occur  thesis.  Program Flow flow-chart  i n f i g u r e 2.9  summarises the sequence of events  in a multiple-scattering calculation.  The  programs s t a r t by  that reading  i n a l l the r e l e v a n t s t r u c t u r a l and n o n - s t r u c t u r a l parameters as w e l l as a list  o f d i f f r a c t e d beams w i t h t h e i r symmetry code numbers.  the dimensions o f the m a t r i c e s  a r e s e t equal t o the number o f  beams ( i . e . those beams w i t h r e a l k ±± The  layer d i f f r a c t i o n matrices M  ) p l u s the f i r s t  the l a y e r d o u b l i n g or the RFS with  s t r u c t u r e s which d i f f e r  The  few  propagating  evanescent beams.  ~ are c a l c u l a t e d ; d i f f e r e n t  a v a i l a b l e depending on whether the l a y e r corresponds o r t o a composite l a y e r - t y p e .  At each energy,  subroutines  to a simple  B r a v a i s net  s t a c k i n g o f l a y e r s i s performed by  methods.  are  either  Each method can i n c l u d e o v e r l a y e r s  from the a p p r o p r i a t e l a y e r o f the s u b s t r a t e ; a  s p e c i a l case o f t h i s i n v o l v e s a v a r i a t i o n o f the topmost l a y e r s p a c i n g f o r example f o r c l e a n metal s u r f a c e s .  G e n e r a l l y the c a l c u l a t i o n s a r e made f o r  the energy range 40-200 eV i n increments  o f 2 eV up to 80 e V . a n d i n  increments  •52Read i n ( i ) geometry (iv)  V .  (ii)V  QT>  ( i i i ) beams and symmetrv  01  temperature data  Choose i n i t i a l  (v) phase  shifts  energy  F i n d beams needed a t E  Compute temperature-dependent phase  shifts  C a l c u l a t e layer d i f f r a c t i o n matrices M  Find d i f f r a c t e d  Find d i f f r a c t i o n  beam  for a substrate  amplitudes from s u r f a c e p l u s  by  s u b s t r a t e by RFS  Add C a l c u l a t e beam  ,  intensities  layer  matrices layers  doubling  s u r f a c e l a y e r and f i n d  diffracted  beam amplitudes  C a l c u l a t e beam i n t e n s i t i e s Vary s u r f a c e geometry  )  k-  1  Vary  s u r f a c e geometry  Increment E  ZZl F i g u r e 2.9:  Flowchart  showing p r i n c i p a l  steps  i n a multiple-scattering  LEED c a l c u l a t i o n , u s i n g t h e RFS or l a y e r d o u b l i n g  programs.  -53-  of  4 eV above 80 eV; t h e r e f l e c t e d i n t e n s i t i e s  i n t h e h i g h energy  then i n t e r p o l a t e d t o g i v e v a l u e s i n 2 eV i n t e r v a l s . sities  The c a l c u l a t e d  a r e s t o r e d on magnetic tape and can be p l o t t e d f o r v i s u a l  with the experimental  range a r e  1(E) c u r v e s ; a l t e r n a t i v e l y t h e c a l c u l a t e d  inten-  comparison intensities  can be compared w i t h experimental v a l u e s by means o f a r e l i a b i l i t y  index as  d i s c u s s e d i n t h e next s e c t i o n s .  2.8 E v a l u a t i o n o f R e s u l t s 2.8 (a) I n t r o d u c t i o n In  LEED c r y s t a l l o g r a p h y i t i s n e c e s s a r y t o f i n d t h e s t r u c t u r a l model  which g i v e s t h e b e s t correspondence 1(E) c u r v e s .  between the c a l c u l a t e d and experimental  T h i s opens the need t o be a b l e t o e v a l u a t e t h e s i m i l a r i t y , o r  o t h e r w i s e , between two s e t s o f curves on v a r y i n g s t r u c t u r a l , and some nons t r u c t u r a l , parameters. comparisons  Such a s e a r c h has most o f t e n been done by v i s u a l  (e.g. by matching up the p o s i t i o n s and r e l a t i v e i n t e n s i t i e s o f  peaks, d i p s and o t h e r s t r u c t u r a l disadvantage  o f b e i n g unwieldy  meters a r e l a r g e . for  f e a t u r e s ) , b u t t h i s approach s u f f e r s t h e  when t h e numbers o f beams and v a r i a t i o n  para-  As a consequence t h e r e has been c o n s i d e r a b l e encouragement  t h e development o f n u m e r i c a l  i n d i c e s f o r guiding these  comparisons.  Among t h e s i m p l e s t p o s s i b i l i t i e s i s  AE  =  |E i=l  which o n l y compares peak p o s i t i o n s . at which t h e  I E?  a l  - E? '|  (2.35)  b s  In (2.35), t h e E^ r e p r e s e n t  peak o c c u r s i n t h e c a l c u l a t e d and observed  energies  c u r v e s , and N  -54-  i s the t o t a l number o f peaks compared [83,84].  C l e a r l y the b e t t e r the c o r r e s -  pondence i n peak p o s i t i o n s between the e x p e r i m e n t a l and t h e lower the v a l u e o f AE. it  In p r a c t i c e t h i s  calculated  1(E)  curves,  c r i t e r i o n seems incomplete  because  i g n o r e s the a c t u a l i n t e n s i t y v a l u e s , i t g i v e s an equal w e i g h t i n g t o each  peak, and  i t i s ambiguous when a peak p r e s e n t i n one  appears  as an i n c o m p l e t e l y developed  curve.  Van  curve i s e i t h e r absent  f e a t u r e (e.g. a s h o u l d e r ) i n the o t h e r  Hove et a l . [85] proposed  an e x t e n s i o n i n v o l v i n g f i v e  i n d i c e s , where each g i v e s a d i f f e r e n t  emphasis i n the comparison.  the most complete index so f a r i s t h a t proposed T h i s index attempts visual  or  simple However,  by Z a n a z z i and Jona [ 4 5 ] .  t o compare n u m e r i c a l l y a l l the f e a t u r e s i n c l u d e d i n a  comparison.  i 2.8  (b)  Z a n a z z i and Jona s P r o p o s a l s  The r e l i a b i l i t y  index proposed  shapes v i a t h e i r d e r i v a t i v e s .  ri . = Jf  21  w(E) |i ci , lc a!l 1  c  b  li  by  Z a n a z z i and Jona [45] compares  For the i  th  beam the r e l i a b i l i t y  | d /l f, / J  - i! i,obs  E  p  1  t  u  obs  index i s :  I.dE  (2.36)  l i  where i n t e n s i t i e s are compared between e n e r g i e s E ^ indicate f i r s t  E2i  curve  and ^2\>  d e r i v a t i v e s f o r the c a l c u l a t e d and observed  a n c  * ^ t  P ^  i e  r  m e s  1(E) c u r v e s .  The  weight f u n c t i o n w  ^ = c c.i:;^ -  i':  >obs  )/(ii:  emphasizes the extrema o f the experimental  )0bs  i  • ii;  >obs  i  max  )  .  curve and o t h e r p o r t i o n s w i t h  h i g h c u r v a t u r e ; the double primes i n (2.37) i n d i c a t e second  derivatives.  & w  -55-  Th e s c a l i n g constant  c. l  =  f J  I. . dE /(* i,obs / J  2 1  E  I. , dE i.eal  2 1  E  li  (2.38)  l i  a l l o w s f o r an a r b i t r a r y s c a l e o f i n t e n s i t y i n the e x p e r i m e n t a l c u r v e s ; comparisons o f r e l a t i v e i n t e n s i t i e s are s u f f i c i e n t  f o r LEED c r y s t a l l o g r a p h i c  s t u d i e s a t the p r e s e n t time. One  total r e l i a b i l i t y  index g i v e n by Z a n a z z i and Jona f o r a s e t o f  d i f f r a c t e d beams i s r  where AE. l  r  =  E (r ) . A E . / E AE. . r l l . I l l 6  r  of  curves.  (2.39)  '  = E_. - E.. . , ( r ) . i s the reduced s i n g l e beam index 2i l i ' r l (r )  and p was  ,  equated  to 0.27,  i  =  r./p  ,  (2.40)  a mean v a l u e o f r ^ found by matching  In (2.39), an average  i s taken over the s i n g l e beam  random p a i r s indices,  where they a r e weighted  a c c o r d i n g t o the energy range over which the compar-  i s o n between experiment  and c a l c u l a t i o n i s made.  A v a r i a t i o n o f (2.39), a l s o proposed by Z a n a z z i and Jona, i s  R  =  ••§)  r  r  ,  (2.41)  where n i s the number o f d i f f e r e n t beams t r e a t e d i n the comparisons. advantage of  o f (2.41), over  the o v e r a l l r e l i a b i l i t y  The  (2.39), i s t h a t i t m i t i g a t e s a g a i n s t a low v a l u e index r e s u l t i n g from a comparison  involving  just  a s m a l l number o f beams; i t i s g e n e r a l l y b e l i e v e d t h a t a r e l i a b l e LEED c r y s t a l l o g r a p h i c a n a l y s i s r e q u i r e s comparisons  i n v o l v i n g I ( E ) •curves f o r 10  -56-  d i f f e r e n t d i f f r a c t e d beams.  R i n (2.41) was  s e t up w i t h the o b j e c t i v e o f  being c o n s i s t e n t with the f o l l o w i n g p o s s i b i l i t i e s comparisons o f experimental  and  calculated  from  1(E) curves f o r a p a r t i c u l a r  proposed  model:  suggests  the model i s " p o s s i b l e " , and R>0.35 suggests  2.8  F u r t h e r Developments  (c)  R<0.20 suggests  f o r values obtained  t h e model i s "very p r o b a b l e " , 0.20<R<0.35 the model i s u n l i k e l y .  As p a r t o f an i n v e s t i g a t i o n o f the p r o p o s a l o f Z a n a z z i and Jona, Watson et a l . [43] p l o t t e d  Cr )^ as a f u n c t i o n o f topmost s p a c i n g f o r the  s u r f a c e o f copper.  T h i s i s shown i n f i g u r e 2.10  s p a c i n g expressed  as the percentage  where Ad%  (111)  g i v e s t h e topmost  change from the b u l k s p a c i n g , ( i . e . d-d  Ad%  x  =  100  d o where d  Q  i s the b u l k i n t e r l a y e r s p a c i n g and d i s the topmost  spacing).  The  curves shown i n f i g u r e 2.10  are s p e c i f i c a l l y f o r V  o f the '16 beams a v a i l a b l e o n l y 9 a r e shown f o r c l a r i t y . a b i l i t y index f i g u r e , and e  interlayer = -9.5  The reduced  eV;  reli-  (r ) f o r the 16 beams i s p l o t t e d as the dashed l i n e i n the same  the a s s o c i a t e d e r r o r {[ Z AE.((r ).-f ) r  2  r  ] / [ (n-1) Z AE.  ] } *  ,  (2.42)  r  c o r r e s p o n d i n g t o the minimum o f r ^ i s i n d i c a t e d by the arrows.  In  (2.42),  n i s the number o f beams c o n s i d e r e d . Watson et a l . [43] concluded e r r o r i n Ad%  that  f o r the data a v a i l a b l e .  i n d i c a t e s an u n r e a l i s t i c a l l y l a r g e In f a c t the top l a y e r s p a c i n g s  indicated  •57-  CuOlO  Vara  0.40  0.30H  (0, 0.20H  0.KH  Ad%  Figure 2.10:  Plots f o r C u ( l l l ) of ( r ) . f o r 9 i n d i v i d u a l beams versus Ad% r  with V  or  = -9.5 eV.  The dashed l i n e shows the reduced r e l i -  a b i l i t y index (f ) f o r the t o t a l 16 beams. (After Watson et a l . [43].)  -58-  by the minima f o r a l l the i n d i v i d u a l curves for  the minimum i n the dashed l i n e , and  are r a t h e r c l o s e t o . t h e  t h i s suggests  spacing  t h a t the u n c e r t a i n t y  i n s p a c i n g a s s o c i a t e d w i t h the minimum v a l u e o f r ^ c o u l d be g i v e n by standard  e r r o r found  from t h e d i s t r i b u t i o n o f t o p l a y e r spacings  i n d i c a t e d by the minimum f o r each i n d i v i d u a l  e, d  =  ( [ Z AE. J  .  I  I  ( d . mm  . mm  =  1  - d . mm  ) ]  (°^ ) n  curve,  / [ (n-1)  2  the  J  E AE. ] 1 ! ;  ,  J  m  (2.43)  I  where d  F i g u r e 2.10  shows d  6  of  ( E AE.d . . I mm l  ) / ( E AE.  1  . ±2e ,; t h i s mm d  . I  corresponds r  I  )  .  t o -4.1±1.2%.  from LEED c r y s t a l l o g r a p h y .  C e r t a i n l y numerical  are r e q u i r e d f o r t h i s purpose; i t i s v e r y hard to see how be h e l p f u l l y  e v a l u a t e d s o l e l y from v i s u a l  advantage o f n u m e r i c a l  2.11  The i n t r o d u c t i o n  by Watson et a l . makes a s t a r t on the problem o f e s t i m a t i n g u n c e r t a i n t i e s  in results  form.  (2.44)  Again  t h i s was  r e l i a b i l i t y indices u n c e r t a i n t i e s could  e v a l u a t i o n s o f 1(E) c u r v e s .  i n d i c e s i s t h a t t h e y can be e a s i l y p l o t t e d i n contour i n t r o d u c e d by Watson et a l . , and the example i n f i g u r e  shows a contour p l o t o f r r  versus V r  and  Ad%  for C u ( l l l ) .  A c c o r d i n g to  or  the p r o p o s a l o f Zanazzi and Jona, the o v e r a l l minimum i n r ^ i n f i g u r e corresponds  to the v a l u e s o f V  Q r  the complete s e t o f e x p e r i m e n t a l  Ad% which g i v e the b e s t agreement between  and  calculated  w i t h the d i s t r i b u t i o n o f v a l u e s o f V  1(E) c u r v e s .  ± 6 ^ (the standard f o r t h e minima o f  or to e ^ ) ; these i n d i c a t e 68%  minimum o f r _ .  2.11  and  - shown f o r the minimum r e p r e s e n t ±e^ and  analogously  Another  E r r o r bars  error associated (r ). and d e f i n e d r l  confidence l i m i t s associated with  the  -59-  F i g u r e 2.11:  Contour p l o t  for C u ( l l l ) of r  ( A f t e r Watson et a l . [43].)  r  v e r s u s Ad% and  V . o r  -60-  CHAPTER 3 P r e l i m i n a r y Work  -61-  3.1 3.1  General (a) As  Experimental  Procedures  LEED Apparatus i n a l l work on w e l l d e f i n e d c r y s t a l  s u r f a c e s , LEED experiments must  -9 be  c a r r i e d out  the g e n e r a l  at low  pressure  (i.e.^  10  f e a t u r e s o f the c o n v e n t i o n a l  torr).  This section describes  type o f LEED apparatus which  been used i n the m a j o r i t y o f LEED experiments made so f a r .  The  has  discussion  w i l l be b r i e f , but a l o t more i n f o r m a t i o n can be o b t a i n e d from the r e f e r e n c e s provided. A review o f the v a r i o u s m o d i f i c a t i o n s o f LEED instruments i s available  [87],  A schematic diagram o f the LEED apparatus used i n t h i s work i s shown i n f i g u r e 3.1.  T h i s i n v o l v e s a V a r i a n FC12  magnetic s t a i n l e s s s t e e l  and  chamber which i s c o n s t r u c t e d  i s connected to a s e r i e s o f pumping  below the main chamber i n d i c a t e d i n f i g u r e 3.1. i s done w i t h h i g h  s u r f a c e area m o l e c u l a r  which are c o o l e d by the system to -10 started.  -3  liquid nitrogen.  sieves  The  initial  s o r p t i o n pumping  These pumps can reduce the p r e s s u r e  the whole system f o r -12  £ s  -1  ) can  to out-gas  components o f the system t h a t a r e heated d u r i n g an experiment.  periods.  Gases f o r a d s o r p t i o n  bombarding i n the c l e a n i n g p r o c e s s through a leak v a l v e from a gas pumped by  i t s own  can be  introduced  i n l e t manifold.  s m a l l i o n pump (20 I s *) and  be  thoroughly A  titaniu  s u b l i m a t i o n pump i s a v a i l a b l e f o r e x t r a pumping d u r i n g both o u t - g a s s i n g „- the a c t u a l experimental  of  hours at 200°C (to remove  adsorbed gases from the chamber w a l l s ) , i t i s n e c e s s a r y all  facilities  (zeolites) i n containers  t o r r when the main s p u t t e r i o n pump (200  A f t e r baking  o f non-  and  s t u d i e s or f o r i o n -  i n t o the whole chamber  T h i s p a r t o f the system i s i t can be baked s e p a r a t e l y  -62-  manipulator I o n gun  ca)  GAS LINE S.P.  (b)  eovsipJ  S.P  EXPTAL. tHAMBER  T.S.P  200 l/s IP  F i g u r e 3.1.:  (a) Schematic o f t h e V a r i a n FC12 UHV chamber. (b) D i a g r a m a t i c r e p r e s e n t a t i o n o f t h e pumping system: IP = Ion Pump; TSP= T i t a n i u m S u b l i m a t i o n Pump; SP = S o r p t i o n Pump.  -63-  (a)  Crystal  (c)  ( b) F i g u r e 3.2:  (a) Schematic diagram o f the e l e c t r o n o p t i c s used f o r LEED experiments. (b) Diagram showing sample mounted on a tantalum (c) E l e c t r o n bombardment  sample h e a t e r .  supporting  Hatched.lines  ring  represent  s t a i n l e s s s t e e l parts while the s t i p p l e pattern i n d i c a t e s the ceramic  insulator.  -64-  from the main chamber. i n the  The  o b j e c t i v e here i s t o l i m i t the amount o f  admitted gases t o v e r y  low  proportions  i n the main chamber.  o f pumping methods, measurement o f p r e s s u r e  and  associated  given  and  Tom  i n reviews by Hobson [ 8 8 ] , Lange [89] The  sample m a n i p u l a t o r  enables the  c r y s t a l to be  impurities Details  techniques  are  [90].  ( V a r i a n 981-2528) h o l d s the  c r y s t a l sample  t r a n s l a t e d as w e l l as r o t a t e d both about the  and axis  1" o f the chamber  (to enable the sample which i s o f f - s e t by  to d i f f e r e n t f a c i l i t i e s ) and the beam from the respect  to the  ment h e a t i n g  about an  e l e c t r o n gun  crystal).  The  e l e c t r o n gun  has  facilities  tunsten  The  typical  i n t h i s work (energy range 30-230 eV)  has  a current  beam diameter at the sample o f ^ 0.75  mm.  a n a l y s i s at a t y p i c a l  and  [91]  and  energy o f 1 keV  t e c h n o l o g y o f low  and  Kohl  The  electron optics  voltage  The  accelerated  i n c i d e n t beam used f o r LEED o f about  same gun  current  are  therm-  was  o f 10 yA.  1 yA,  and  a  used f o r Auger Reviews o f  e l e c t r o n guns i n c l u d e s t h o s e by  the  Rosebury  [92].  s p h e r i c a l phosphor s c r e e n  ( V a r i a n 981-0127) ( f i g u r e 3.2a) and  four concentric  the sample i s p o s i t i o n e d at the screen  i s measured w i t h a  sample.  cathode; these e l e c t r o n s  c o l l i m a t e d through anode p l a t e s .  design  f o r e l e c t r o n bombard-  the temperature o f the c r y s t a l w i t h the  (9) w i t h  ( V a r i a n 981-2125) produces an e l e c t r o n beam by  i o n i c e m i s s i o n from a hot and  directed  to make d i f f e r e n t angles o f i n c i d e n c e  Pt/13%Rh-Pt thermocouple j u n c t i o n i n c o n t a c t The  to be  a x i s i n the h o r i z o n t a l p l a n e (to enable  sample h o l d e r  (figure 3.2(c));  2^  f o r LEED.  In the u s u a l  common c e n t r e  c o n s i s t s o f a hemi-  g r i d s each o f -80% of curvature  mode o f o p e r a t i o n ,  c l o s e s t to the sample are grounded t o ensure t h a t  transparency;  o f the g r i d s  the specimen and  the  and grid  e l e c t r o n s t r a v e l through  an  -65-  e l e c t r o s t a t i c a l l y f i e l d - f r e e space between the sample and f i n a l anode o f the e l e c t r o n gun g r i d s are connected t o g e t h e r  i s a l s o grounded).  and  e l e c t r o n s which have l o s t energy on only the  the  The  o b j e c t i v e i s to stop  spot.  The whole d i f f r a c t i o n p a t t e r n on  Another a c c e s s o r y  three  onto the phosphor  the s c r e e n  shows up  can be observed  screen,  as a b r i g h t directly  needed f o r the  LEED experiment i s the  c r y s t a l by  sputtering  i o n bombardment.  The  3.1  fh) The  o r t h o g o n a l s e t s o f square Helmholtz c o i l s to reduce  Preparation  experiments r e p o r t e d  (99.99% p u r i t y ) , the other was  f a c e p l a n e by  process,  the  II  erosion  i n t h i s thesis involve surfaces  sources o f s i n g l e c r y s t a l ; one  the p r e p a r a t i o n  of  s t u d i e d i s minimized.  Crystal  cut from two  gun  chamber  the r e s i d u a l magnetic f i e l d t o a l e v e l where i t s e f f e c t on the motion electrons being  :  photographed.  (Varian 981-2043) f o r c l e a n i n g the i s surrounded by  The  e l a s t i c a l l y scattered electrons, a f t e r penetra-  from an ordered c r y s t a l s u r f a c e  through the g l a s s window and  third  those  e l e c t r o n s to pass through.  t i n g t h i s g r i d , are a c c e l e r a t e d through about 5 keV where each beam d i f f r a c t e d  the  i n t e r a c t i n g w i t h the sample, w h i l e  e l a s t i c a l l y scattered  fourth g r i d i s earthed.  second and  are h e l d at a p o t e n t i a l which i s c l o s e to  t h a t on the cathode i n the e l e c t r o n gun;  permitting  The  Cthe  the o p t i c s  ( Agietron  Laue  provided  by  was  purchased  commercially  another l a b o r a t o r y  the s i n g l e c r y s t a l  o f rhodium  [94].  To  i s o r i e n t e d t o the r e q u i r e d  X-ray b a c k - r e f l e c t i o n t e c h n i q u e [95] and  cut by  [93] start surspark  II  , AGIE, S w i t z e r l a n d ) .  o r i e n t a t i o n from the d e s i r e d c r y s t a l  To  c o r r e c t f o r small deviations  f a c e , the  c r y s t a l s l i c e i s mounted i n  of  -66-  it II a c r y l i c r e s i n ( Quickmount F u l t o n M e t a l l u r g i c a l Produce Corp., USA) and p o l ii i s h e d w i t h 5, 3 and 1 micron diamond p a s t e on a p o l i s h i n g wheel  ( Universal  II  Polisher  , Micrometallurgical  process,  i t i s n e c e s s a r y t o check again  the r e q u i r e d  Limited, T h o r n h i l l , Ontario.).  c r y s t a l l o g r a p h i c plane.  After  that the f i n i s h e d surface  This  this  s t i l l has  i s done by p l a c i n g t h e c r y s t a l  on t h e Lau^ X-ray d i f f r a c t o m e t e r so t h a t t h e d e s i r e d p l a n e i s p e r p e n d i c u l a r to t h e X-ray beam; t h e whole goniometer and c r y s t a l assembly i s then f e r r e d t o an o p t i c a l bench where a Ne-He l a s e r beam i s d i r e c t e d onto t h e s u r f a c e and t h e angle o f r e f l e c t i o n i s d e t e c t e d . t e s t o f whether t h e p h y s i c a l s u r f a c e  supporting  the -1x10  plane.  o f the desired  At t h i s stage t h e back o f t h e sample i s spot welded onto a which i n t u r n i s mounted onto t h e  The sample and m a n i p u l a t o r i s then p l a c e d  i n t h e vacuum chamber,  l a t t e r i s c l o s e d and t h e chamber i s pumped down t o a base p r e s s u r e o f ^  t o r r a f t e r t h e standard  out-gassing  AES  indicates that sulphur,  phosphorus and carbon a r e t h e i m p u r i t i e s  g e n e r a l l y present  some o t h e r impurity  processes.  i n the rhodium c r y s t a l s used i n our experiments; no sub-  s t a n t i a l amounts o f boron  -  a  coincides with the required c r y s t a l 1°  tantalum r i n g ( f i g u r e 3 . 2 ( b ) ) ,  manipulator.  perpendicularly  This provides  G e n e r a l l y we aim t o have t h e s u r f a c e o r i e n t e d t o w i t h i n — c r y s t a l plane.  trans-  research  (Auger peak a t 180 eV) has been d e t e c t e d  groups [96,97] have r e p o r t e d  i n t h e i r rhodium samples.  formed by c y c l e s o f heat treatment  appreciable  although  amounts o f t h i s  The c l e a n i n g p r o c e s s e s a r e g e n e r a l l y p e r (700-1000°C f o r 10-60 min.) t o d r i v e most  b u l k i m p u r i t i e s t o t h e s u r f a c e , and argon i o n bombardment t o s p u t t e r o f f the impurities at the surface.  A l l i m p u r i t i e s except c a r b o n , can be removed from  -67-  F i g u r e 3.3:  al  -68-  rhodium s u r f a c e s by argon i o n bombardment microamps and  ~1 keV  f o r 10-30  •carbon Auger s i g n a l to  (282 eV)  min.).  (typically  s p u t t e r i n g c r o s s - s e c t i o n o f carbon.  a f t e r a n n e a l i n g at 700°C f o r a few minutes, AES  i n t o the b u l k ) and  0.1-1  Immediately a f t e r s p u t t e r i n g , the  always showed a r e l a t i v e i n c r e a s e ; t h i s  be a s s o c i a t e d w i t h the low  carbon c o n t a m i n a t i o n  10 ^ t o r r o f Ar at  on the s u r f a c e i s reduced  appears  However,  i n d i c a t e s t h a t the  level  of  (presumably by back d i f f u s i o n  LEED i n d i c a t e s t h a t the s u r f a c e has become ordered  again.  In p r e l i m i n a r y s t u d i e s , Auger s p e c t r a o f the c l e a n R h ( l l O ) s u r f a c e were s t u d i e d as a f u n c t i o n o f c r y s t a l temperature  ( f i g u r e 3.3),  and  i t was  t h a t below 300°C carbon d i f f u s e s to the s u r f a c e whereas above t h i s temperature carbon a p p a r e n t l y d i f f u s e s back i n t o the b u l k .  found  critical  Further  general  d i s c u s s i o n s on the p r e p a r a t i o n o f c l e a n s u r f a c e s are g i v e n i n reviews Farnsworth [ 9 8 ] , Bauer [99] and Jona  3.1  (c)  Detection of Surface  [100].  Impurities  S u r f a c e i m p u r i t i e s were d e t e c t e d spectroscopy  i n t h i s work by means o f Auger e l e c t r o n  u s i n g the LEED o p t i c s as a r e t a r d i n g f i e l d  Auger e l e c t r o n s o f c h a r a c t e r i s t i c e n e r g i e s are p r e s e n t imposed on the h i g h r e g i o n s o f N(E)  (but r e l a t i v e l y  vs E curve  electronic differentiation anode o f t h e gun,  [116],  AV=Vsinu>t  and  super-  intermediate  t h e s e peaks can be enhanced by the  final  f o u r t h g r i d s a r e grounded as f o r  f o r d e t e c t i n g Auger e l e c t r o n s the r e t a r d i n g  m i d d l e g r i d s has  a s m a l l modulating  ( t y p i c a l v a l u e s o f V used i n t h e s e experiments a r e <10  t h i s modulating  [16,17].  as s m a l l peaks  With r e f e r e n c e t o f i g u r e 3.4,  the sample, t h e f i r s t and  t h e normal LEED experiment, but  analyzer  c o n s t a n t ) background o f the  ( f i g u r e 1.2),  p o t e n t i a l a p p l i e d on the two  by  voltage eV).  v o l t a g e , t h e t o t a l c u r r e n t c o l l e c t e d on t h e s c r e e n  With  (held at  -69-  Gun  JElectron 1 Qun  control  300v  V «sin ut r Lock-in  Neat rail ser  Amp. sin 2wt  sin Qt Freq. x/ 1  2  Scope  X-Y Plotter F i g u r e 3.4:  Schematic diagram o f LEED o p t i c s used as a r e t a r d i n g a n a l y z e r f o r Auger e l e c t r o n MCA = m u l t i c h a n n e l a n a l y z e r .  spectroscopy:  field  -70-  T a b l e 3.1:  Observed and c a l c u l a t e d Auger t r a n s i t i o n e n e r g i e s f o r rhodium.  Relative  Observed (a)  (b)  (c)  (d)  144  Ce)  Intensity  145  10  %  Ca)  Calculation  Assignment  Cf)  (f)  145.0  M  M 4  1 1 N  176  174  165  170  175  7  174.0  M N N T l 2,3  207  208  200  200  210  10  208.0  M N N 5 2,3 2,3  223  226  222  222  227  27  221.5  M N N 5 1 4,5  255  260  256  256  259  55  255.5  M N N 5 2,3 4,5  302  306  302  302  303  100  303.0  M N N 5 4,5 4,5  (a)  t h i s work  Cb)  Grant and Haas [102]  (c)  Palmberg  Cd)  Castner e t a l .[96]  (e)  Chan e t a l . [118]  (f)  Coghlan and C l a u s i n g [103]  et a l .. [36]  m  1  N  -71-  a p o s i t i v e p o t e n t i a l o f about 300 the  components o f the  (frequencies  U J and  current  eV)  i s modulated.  c o r r e s p o n d i n g to the  2OJ r e s p e c t i v e l y )  the  1.10  [l6,17].  [3l].  i s p l o t t e d i n dN(E)/dE form.  spectra  a function  secondary e l e c t r o n d i s t r i b u t i o n N(E)  d e r i v a t i v e dN(E)/dE r e s p e c t i v e l y figure  first  measured by  The  o f the  at the  A p l o t of  the  retarding  energy  and  its first  t y p i c a l Auger spectrum shown i n T h e o r e t i c a l l y , the  sensitivity  of  t h i s method i s a p p r o x i m a t e l y 1% o f a monolayer  Higher s e n s i t i v i t i e s t o i m p u r i t i e s  available  amplifier  second harmonics  ( f i g u r e 1.2)  are p o s s i b l e when Auger  are measured w i t h a c y l i n d r i c a l m i r r o r a n a l y z e r [101]. not  and  are r e a d i l y i d e n t i f i e d .  •amplitude o f these harmonic components as E produces the  Using a l o c k - i n  time the  spectra  Such an a n a l y z e r  was  experimental work r e p o r t e d i n t h i s t h e s i s  was  done. Measured peak e n e r g i e s and o f rhodium are  r e l a t i v e peak h e i g h t s f o r the Auger spectrum  summarized i n T a b l e 3.1.  o t h e r p u b l i s h e d measurements must be and  to the  i n e v i t a b l y increased  Coghlan and  are a l s o  variations  listed  Clausing  [103]  i n peak e n e r g i e s from  a t t r i b u t e d to e r r o r s  lack o f an a p p r o p r i a t e c o n t a c t p o t e n t i a l  t a i n t i e s are by  The  for low-intensity  f o r f r e e atoms w i t h an  i n the  correction; peaks.  energy also  uncer-  Energies  ionization  scale  calculated  correction  i n the t a b l e ; t h e s e v a l u e s are h e l p f u l f o r g u i d i n g  the  assignment t o p a r t i c u l a r Auger t r a n s i t i o n s .  3.1  (d)  LEED I n t e n s i t y Measurements  D i f f r a c t e d beam i n t e n s i t i e s i n LEED have most o f t e n been measured d i r e c t l y as d i f f r a c t e d beam c u r r e n t s i n s i d e the  chamber [104]  w i t h a moveable Faraday cup  either  collector  or i n d i r e c t l y as the b r i g h t n e s s o f spots on  the  -72-  phosphor s c r e e n w i t h an e x t e r n a l spot photometer [105]. l a t t e r approach i s the p h o t o g r a p h i c and developed  technique  A v a r i a n t o f the  i n t r o d u c e d by S t a i r et a l . [106]  f u r t h e r by F r o s t et a l . [107], who  employed a c o m p u t e r - c o n t r o l l e d  V i d i c o n camera t o a n a l y z e the p h o t o g r a p h i c  f i l m and thereby produce e x p e r i -  mental 1(E) c u r v e s .  has been used  T h i s l a t t e r procedure  i n the p r e s e n t work.  B a s i c a l l y photographs o f the LEED s c r e e n are taken at a s e r i e s o f e l e c t r o n e n e r g i e s and measurements a r e made o f the i n t e g r a t e d o p t i c a l d e n s i t i e s f o r the d i f f r a c t e d density  spots on the f i l m n e g a t i v e s .  Assuming the measured  optical  (D) f o r a spot i s p r o p o r t i o n a l t o the amount o f l i g h t which  caused  the d a r k e n i n g , and hence t o the a s s o c i a t e d e l e c t r o n f l u x which h i t s  the s c r e e n ,  then D d i v i d e d by the i n c i d e n t e l e c t r o n c u r r e n t i s p r o p o r t i o n a l t o t h e r a c t e d beam i n t e n s i t y .  diff-  Such measurements i n e v i t a b l y g i v e r e l a t i v e beam  intensities. The through an 85 mm  LEED p a t t e r n s d i s p l a y e d on the phosphor s c r e e n were photographed the window o f the vacuum chamber u s i n g a Nikon F2 35 mm f l . 8 l e n s and a K2 e x t e n s i o n r i n g .  camera w i t h  Photographs were taken g e n e r a l l y  f o r the range o f i n c i d e n t beam e n e r g i e s 30-250 eV i n 2 eV i n t e r v a l s u s i n g fixed  exposures  o f 1 s at f 4 , the i n c i d e n t c u r r e n t and  f o r each photograph.  Using a m o t o r - d r i v e n  unit  energy b e i n g  recorded  t o wind the f i l m and a  250  exposure f i l m back, LEED p a t t e r n s c o u l d be photographed over t h i s  energy  range i n l e s s than 5 minutes.  the s u r f a c e  " p u r i t y was  A f t e r t a k i n g a set o f photographs,  r o u t i n e l y checked w i t h AES  occurred during data  collection.  t o a s s e s s whether any  contamination  -73-  Standard  Kodak T r i - X emulsion  i n a continuous photographic  length i n  Acufine  f i l m was  used and  developer  the f i l m was  processed  a t 73 F f o r 7 minutes.  The  n e g a t i v e s were a n a l y s e d w i t h the system i n d i c a t e d i n f i g u r e  3.5.  The v i d i c o n camera and a s s o c i a t e d e l e c t r o n i c s comprise p a r t o f the Computer Eye System  ( S p a t i a l Data System Inc.) which was  (Data General  Nova 2).  The  i n t e r f a c e d to a mini-computer  f i l m h e l d on the l i g h t t a b l e i s scanned  nuously by the camera and the image i s d i s p l a y e d on the TV monitor 480  (xy) a r r a y .  The  intensity  (z v a l u e ) o f any  contiin a  512x  element o f the image may  be  sampled by t r i g g e r i n g the d i g i t i z e r w i t h a p p r o p r i a t e i n s t r u c t i o n s from  the  computer.  of  The p r o f i l e r d i s p l a y s d i r e c t l y on the monitor  i n t e n s i t y a l o n g any the p o s i t i o n  selected v e r t i c a l  the v a r i a t i o n  l i n e o f the image.  The  joystick controls  ( c o o r d i n a t e s ) o f the f l a s h i n g spot on the TV monitor,  used to s t a r t the a n a l y s i s [107] by p o i n t i n g at the spot to be Assuming a Gaussian intensity  ( ^ j .) ^ z  ac  s  <  estimated by a v e r a g i n g the  the i n t e n s i t y d i s t r i b u t i o n ) . ( ~ \ z  z  i a c  z-value o f a l l elements  The  i n t e g r a t i o n procedure  i n v o l v e s summing a l l  ] ) w i t h i n t h e c i r c l e o f r a d i u s 2 a and (  t h i s value i s  A f t e r the i n t e g r a t i o n , the c o o r d i n a t e s o f the  maximum are determined next  frame.  and  s t o r e d as the new  starting  intensity  c o o r d i n a t e s f o r the position  frame, the computer can a u t o m a t i c a l l y f o l l o w each  spot as i t moves toward the c e n t r e o f the s c r e e n w i t h The whole a n a l y s i s o f each spot takes may  diffracted  S i n c e the a r e a scanned f o r each spot always i n c l u d e s the  o f t h a t spot on the next  lying  at h a l f maximum o f  d i v i d e d by the i n c i d e n t beam c u r r e n t to g i v e a measure o f the beam i n t e n s i t y .  analysed.  d i s t r i b u t i o n f o r the spot i n t e n s i t y , the background  i n an annulus o f mean r a d i u s 2 a (where 2 a i s the width  the v a l u e s o f  and i s  l e s s than  i n c r e a s i n g energy.  30 seconds and  the 1(E)  be d i s p l a y e d on an o s c i l l o s c o p e and p l o t t e d on an xy r e c o r d e r .  curves  -74-  TV  digitiser interfoce  film V ' transport  novo 2 computer cossette drive teletype  light table  F i g u r e 3.5:  Schematic  monitor  sconner  vidicon . T comero  diagram  o f the apparatus  joystick  profiler  scope xy plotter  used  p h o t o g r a p h i c n e g a t i v e s o f LEED p a t t e r n s .  to analyse the  -75-  3.2 3.2  S t r u c t u r a l Determinations (a)  o f Low  Index S u r f a c e s o f Rhodium  P r e v i o u s LEED I n t e n s i t y C a l c u l a t i o n s f o r Rhodium S u r f a c e s  Watson et a l . [43,44,108,109] a n a l y s e d measured 1(E) curves from index s u r f a c e s o f rhodium w i t h m u l t i p l e - s c a t t e r i n g c a l c u l a t i o n s and s t r u c t u r a l c o n c l u s i o n s are summarized i n T a b l e 3.2. two  types o f atomic  i)  The  low  their  These c a l c u l a t i o n s  potential:  s e l f - c o n s i s t e n t band s t r u c t u r e p o t e n t i a l p r o v i d e d by M o r u z z i , Janak  and W i l l i a m s [110]; t h i s p o t e n t i a l was  designated  V^^.  i i ) The s u p e r p o s i t i o n p o t e n t i a l c a l c u l a t e d f o r the c e n t r a l atom i n a c l u s t e r by a l i n e a r s u p e r p o s i t i o n o f atomic was  designated  ween the two c t u r e s and  This p o t e n t i a l  Watson et a l . o b t a i n e d d i s c r e p a n c i e s b e t -  atomic p o t e n t i a l s which r e s u l t e d  in different  geometrical  i n n e r p o t e n t i a l v a l u e s r e p o r t e d f o r the same s u r f a c e .  a band s t r u c t u r e p o t e n t i a l i s l i k e l y suggested  charge d e n s i t i e s .  Rh^  ^£^13'  With r e f e r e n c e t o T a b l e 3.2,  to  used  et a l . [43] supported  Generally  t o be p r e f e r r e d [60] although i t has  [ 6 l ] t h a t the s u p e r p o s i t i o n p o t e n t i a l  the band s t r u c t u r e p o t e n t i a l  stru-  can produce v e r y s i m i l a r  been results  f o r the purpose o f LEED c r y s t a l l o g r a p h y .  Watson  t h i s s u g g e s t i o n i n a d e t e r m i n a t i o n o f the g e o m e t r i c a l  s t r u c t u r e o f the C u ( l l l ) s u r f a c e . However f o r rhodium, upon e v a l u a t i n g the l e v e l o f agreement between e x p e r i m e n t a l and ability  calculated  1(E) c u r v e s , both w i t h v i s u a l a n a l y s e s and  i n d i c e s , Watson et a l . were unable  p o t e n t i a l s as b e i n g p r e f e r r e d .  t o s e l e c t one  T h i s thereby  left  significant  i n the d e t a i l s o f the s t r u c t u r e s o f the Rh(100) and the o b j e c t i v e s o f my  i n i t i a l r e s e a r c h was  surfaces i n order to e l u c i d a t e t h i s  to perform  problem.  of these  (111)  reli-  atomic  uncertainties  surfaces.  One  f u r t h e r s t u d i e s on  of these  -76-  Table 3.2:  Structural determination of low index surfaces of rhodium. ( Watson et a l . ) ^JW Rh  v Rhl3 V ie or v (eV)  V  v  W  V ie or v (eV)  Rh(100)  -1.8+1.0  Rh(lll)  -4.210.5  Surface  Ad%±c  d  Ad%le  f  d  r r  r  (%)  -19.610.8  0.17  2.510.9  -11.510.7  0.16  -18.610.5  0.16  -0.710.8  -11.310.7  0.12  Rh(llO)  —  —  --  -2.511.2  -11.210.6  0.10  Rh(llO)  —  --  --  -1.011.2  -10.510.8  0.09  Table 3.3: ''  Structural determination of low index surface of rhodium. ( This work ) ^iJW _Rh .  (%)  V le or v (eV)  Rh(100)  1.010.9  Rh(lll)  Surface  Ad%le  d  V Ad%le  r  d  r  (%)  -12.810.4  0.09  0.511.2  -1.610.8  -11.210.6  0.08  Rh(110)  -3.311.5  -10.910.8  0.12  Rh(110)  -0.510.7  - 9.610.9  0.09  Rhl3 V ie or v (eV) -14.010.6  r r 0.09 —  —  —  --  --  --  -773.2  (b)  Further Studies  In t h i s work, m u l t i p l e s c a t t e r i n g c a l c u l a t i o n s were r e p e a t e d f o r normal i n c i d e n c e on the  (100),  (110) and  (111)  s u r f a c e s o f rhodium, and  a t e d LEED i n t e n s i t i e s were compared w i t h the e x p e r i m e n t a l o u s l y produced a new  by Watson et a l . f o r the  set of experimental  and used  (110)  and  data f o r normal i n c i d e n c e on Rh(100) was experimental  o f phase s h i f t s investigated.  from the two  d i f f e r e n t atomic p o t e n t i a l s was  In doing t h i s an error.was  i o u s l y f o r the p o t e n t i a l  previ-  Although obtained,  1(E)  s i g n i f i c a n t d e v i a t i o n s from the p r e v i o u s data  P r i o r to making the m u l t i p l e s c a t t e r i n g c a l c u l a t i o n s , the  calcul-  curves  (111) s u r f a c e s .  i n the comparison i n t h i s work, these new  d i d not show any  1(E)  the  curves  [ill]. calculation  completely r e -  d e t e c t e d i n the v a l u e used  prev-  at the m u f f i n - t i n r a d i u s ^ , and t h i s r e s u l t e d  i n an  MJW  incorrect  s e t o f phase s h i f t s a s s o c i a t e d w i t h the p o t e n t i a l  .  making the c o r r e c t i o n f o r the band s t r u c t u r e p o t e n t i a l , a new s h i f t s was These new  generated  for different  multiple-scattering calculations (110)  and  (111)  f o r the  The n o n - s t r u c t u r a l parameters were kept  i n the p r e v i o u s work.  atomic v i b r a t i o n s were assumed t o be *The p o s s i b i l i t y o f a n u m e r i c a l  S p e c i f i c a l l y , the surface  i s o t r o p i c and  e r r o r was  (Univ. o f Washington).  Watson and W.T.  from the band s t r u c t u r e p o t e n t i a l ,  s u r f a c e s assuming r e g u l a r p a c k i n g arrangements as i n -  unchanged from t h o s e used  P.R.  to avoid confusion with  f o r normal i n c i d e n c e were r e p e a t e d  - d i c a t e d p r e v o u s l y [43,44,108,109].  by J . J . Rehr  of  o f Watson et a l .  With the c o r r e c t e d phase s h i f t s  (100),  3.6).  from the band s t r u c t u r e p o t e n t i a l  M o r u z z i , Janak and W i l l i a m s are d e s i g n a t e d as [ V ^ ^ ] MJW V_, Rn  s e t o f phase  Jl t o a maximum v a l u e o f 7 ( f i g u r e  phase s h i f t s v a l u e s generated  the erroneous  After  first  layer-independent,  suggested  The a c t u a l e r r o r was  the  t o K.A.R. M i t c h e l l l a t e r d e t e c t e d by  Moore w h i l e c a l c u l a t i n g some phase s h i f t s f o r z i r c o n i u :  -78-  E n e r g y ( Ry )  F i g u r e 3.6:  Energy dependence o f rhodium phase s h i f t s •  ,  potential  rw JW-| M  LV , R  J.  (£=0-7) f o r the  -79-  s u r f a c e Debye temperature v a l u e o f 480  b e i n g taken as 406  K [112,115]).  The  K ( i . e . /0.7  times, the b u l k  imaginary p a r t o f the i n n e r p o t e n t i a l  ( ^) v  0  1/3 was  equated  t o -1.17E  , guidance b e i n g p r o v i d e d by the widths  Bragg-type peaks i n e x p e r i m e n t a l and the energy  1(E) c u r v e s a c c o r d i n g t o e q u a t i o n  dependence proposed  spacings below the second  i n equation  topmost i n t e r l a y e r s p a c i n g s second  (110)  i n increments  (10), (01), (11), (02) and  was  used  (100)  f o r the  and  (110)  a 10% c o n t r a c t i o n f o r the  (100)  expansion.  and  the The  range o f 30-250 eV were done  (12) beams f o r a l l t h r e e s u r f a c e s (beam  i n f i g u r e 3.7). (111)  o f 2.5%  c o n t r a c t i o n t o a 2.5%  m u l t i p l e - s c a t t e r i n g c a l c u l a t i o n s over the energy  method f o r the  The  s u r f a c e c a l c u l a t i o n s were made w i t h  topmost s p a c i n g v a r y i n g from a 12.5%  notations are i l l u s t r a t e d  interlayer  2.1960 A f o r R h ( l l l ) ) .  rhodium l a y e r s ) were allowed to v a r y from  s u r f a c e s , w h i l e f o r the  f o r the  A l l the  ( i . e . the p e r p e n d i c u l a r d i s t a n c e between the  from the b u l k v a l u e t o a 5% expansion (111)  (2.5).  (2.4)  rhodium l a y e r were f i x e d at the b u l k v a l u e s ( i . e .  1.9022 A f o r Rh(100), 1.3452 A f o r Rh(110) and  f i r s t and  o f primary  The  calculations u t i l i z e d  the  RFS  s u r f a c e s , whereas the l a y e r - d o u b l i n g method  s u r f a c e t o a v o i d any p o s s i b i l i t y t h a t the  i n t e n s i t i e s might not converge f o r the s m a l l e r i n t e r l a y e r  reflected  spacings.  I n t e n s i t i e s c a l c u l a t e d w i t h the c o r r e c t e d phase s h i f t s from t h e band structure potential mental 1(E) c u r v e s . - o c c u r r e d when Ad%  f o r the  (100)  s u r f a c e were compared w i t h the new  V i s u a l a n a l y s i s suggested  i s between 0 and  2.5%  t h a t the b e s t  (here Ad%  experi-  correspondence  i n d i c a t e s the topmost  i n t e r l a y e r s p a c i n g (d) expressed as the percentage change from the b u l k value d ( i . e . Ad% = [ ( d - d )/d ] x l 0 0 ) . The a n a l y s i s w i t h the r e l i a b i l i t y o " - ^ o o v  •80-  9,  real space  reciprocal space  22  RhdOO) 21  -9,  20  02  12  22  01  111  21  Rh(110)  O C X )  1 1 1  666 066  • 22  1  00  •  10 •  g  v  20  X  (111)  surfaces  12  Rh(H1)  02  11 10  00 10  01  s  11  20"  21 22  F i g u r e 3.7:  (a) Schematic diagrams o f the (100), o f rhodium. i n the second  (110) and  The d o t t e d c i r c l e s r e p r e s e n t rhodium  atoms  layer,  (b) The c o r r e s p o n d i n g  LEED p a t t e r n s i n d i c a t i n g the beam  n o t a t i o n as used i n t e x t .  -81-  index proposed r  r  was  by  Z a n a z z i and Jona [45] i n d i c a t e d t h a t the minimum v a l u e f o r  0.085 and o c c u r r e d when Ad% = 1.0±0.9% and V  To assess the correspondence another and  the curves c a l c u l a t e d  r ^ was V  or  comparison w i t h r ^ was  between the two  or  = -12.8±0.4  eV.  potentials Vp^g  and  made between the same e x p e r i m e n t a l  1(E)  from  [108].  [V j^], R  curves  T h i s time the minimum v a l u e o f  a g a i n 0.085, a l t h o u g h f o r the c o n d i t i o n s Ad% = 0.5±1.2% and  = -14.0±0.6 eV.  These two  r e s u l t s , which are summarized i n T a b l e  a r e i n c o n t r a s t to the p r e v i o u s r e p o r t o f Watson et a l . (Table 3.2). summarized i n T a b l e 3.3  are the c o n d i t i o n s f o r minimum r ^ from  3.3, Also  comparisons  MJWT  of i n t e n s i t i e s calculated using [ V f o r the  (110)  i n the energy  and  (111)  ] w i t h one s e t o f experimental  R n  data  s u r f a c e s ; each s e t o f e x p e r i m e n t a l d a t a covers 5 beams  range 30-200 eV.  Corresponding  r e s u l t s from the p o t e n t i a l  VRhl3 o b t a i n e d p r e v i o u s l y by Watson et a l . are i n T a b l e 3.2. Comparisons o f our new r e s u l t s o b t a i n e d from the c o r r e c t e d phase s h i f t s r  from the band s t r u c t u r e p o t e n t i a l rhodium  (Table 3.3)  potential Vj^jg V  MJWn  LV  Rh  J f o r t h r e e low-index s u r f a c e s o f  w i t h those o b t a i n e d p r e v i o u s l y from the s u p e r p o s i t i o n  (Table 3.2), a l l o w s the c o n c l u s i o n t h a t the v a l u e s o f Ad%  g i v e n by the two  p o t e n t i a l s a r e equal t o w i t h i n the i n d i c a t e d  f o r each s e t o f experimental measurements.  T h i s suggests  and  uncertainties  t h a t the two  rhodium  p o t e n t i a l s a r e e q u i v a l e n t f o r t h e purpose o f LEED c r y s t a l l o g r a p h y , and  provides  support  cluster  f o r the s u g g e s t i o n [ 6 l ] t h a t the s u p e r p o s i t i o n p o t e n t i a l s from  c a l c u l a t i o n s can be u s e f u l when s e l f - c o n s i s t e n t band s t r u c t u r e p o t e n t i a l s unavailable.  T h i s s i t u a t i o n f o r the rhodium s u r f a c e s i s now  t h a t found p r e v i o u s l y f o r C u ( l l l ) [ 4 3 ] .  are  c o n s i s t e n t with  -82-  3.3 3.3  S t u d i e s w i t h the R e l i a b i l i t y (a)  Index o f Zanazzi  and  Jona  Introduction  The  b a s i c approach f o r s u r f a c e  s t r u c t u r a l and  non-structural  t i o n s i n order  t o f i n d the b e s t  mental 1(E)  c r y s t a l l o g r a p h y w i t h LEED i n v o l v e s  varying  parameters i n the m u l t i p l e - s c a t t e r i n g c a l c u l a correspondence between c a l c u l a t e d and  curves f o r a l l d i f f r a c t e d beams [113].  o f the m u l t i p l e - s c a t t e r i n g c a l c u l a t i o n s i n h i b i t s  At p r e s e n t  a full  experi-  the high  cost  v a r i a t i o n o f non-  s t r u c t u r a l parameters t o maximize the agreement i n t h e s e comparisons, and far  only V  q i >  has  been s u b j e c t e d  t o much v a r i a t i o n [ 5 7 ] .  been because o f a common f e e l i n g t h a t the o t h e r not  have a s t r o n g  parameters  (e.g. V ^ , Q  involves  8^)  u l a r l y w i t h the r e l i a b i l i t y The  [56].  This philosophy  of  Hence one  3.3  (b) The  loglcal the  life  o f P.R.  S.J. White  R e l a t i o n s between R e l i a b i l i t y i m a g i n a r y p a r t o f the d e s c r i p t i o n o f the  keeping t h e s e  and  Jona  a  fairly  o b j e c t i v e o f t h i s work i s t o a s s e s s where  a l r e a d y been p u b l i s h e d Watson and  non-structural  Zanazzi  parameters i s a b l e t o  r e s u l t s s i m i l a r to a v i s u a l a n a l y s i s and where i t does not.  vations  procedure  parameters appears to p r o v i d e  o f r ^ f o r the v a r i a t i o n o f n o n - s t r u c t u r a l  t h i s s e c t i o n has  do  i s tested here, p a r t i e -  index r ^ suggested f o r LEED by  v a r i a t i o n of non-structural  s t r i n g e n t t e s t of r ^ . the use  f i n d i n g a p l a u s i b l e choice  has  parameters  Thus, the u s u a l  a t the s t a r t o f the c a l c u l a t i o n and  parameters f i x e d from then on  [45].  non-structural  e f f e c t on determined g e o m e t r i e s .  i n LEED c r y s t a l l o g r a p h y  In p a r t t h i s  so  The  content  give of  a l o n g w i t h some supplementary obser[150].  Index and  the  Imaginary P o t e n t i a l  i n n e r p o t e n t i a l (V ^) p r o v i d e s  a phenomeno-  i n e l a s t i c s c a t t e r i n g o f e l e c t r o n s by  times o f e l e c t r o n s o f w e l l d e f i n e d  a  solid;  energy i n the s o l i d p l a c e  a  -83-  restriction  ( v i a the u n c e r t a i n t y p r i n c i p l e ) on peak widths  according t o equation  (2.4).  Increase i n V  . corresponds 01  i n 1(E)  to a reduction i n  t h e p r o p o r t i o n o f the e l a s t i c a l l y s c a t t e r e d e l e c t r o n s and t o a o f peaks i n the 1(E) c u r v e s . The i n i t i a l s e l e c t i o n o f a v a l u e o f V  curves  f o r rhodium was  broadening  made  (utilizing  oi equations  (2.4) and  i n experimental  (2.5)) from the measured widths  of  kinematical  1(E) c u r v e s ; on t h i s b a s i s a p l a u s i b l e expansion  peaks  for V  is  Q i  1/3 -aE  w i t h a equal t o about 1.17.  changes i n a would modify  A p o i n t o f i n t e r e s t h e r e i s t o see whether  c o n c l u s i o n s on the geometries  o f rhodium s u r f a c e s ,  and whether the r e l i a b i l i t y - i n d e x a n a l y s i s would i n d i c a t e t h a t ct=1.17 i s the most a p p r o p r i a t e v a l u e o f a.  In o r d e r t o examine t h i s ,  further multiple-  s c a t t e r i n g c a l c u l a t i o n s were made f o r normal i n c i d e n c e on the o f rhodium a t a s e r i e s o f v a l u e s o f a, s p e c i f i c a l l y and  2.34,  1.17,  (111)  1.47,  surface  1.76,  w i t h a l l o t h e r n o n - s t r u c t u r a l parameters f i x e d a t the v a l u e s  p r e v i o u s l y i n s e c t i o n 3.2  (b).  The  (111)  s u r f a c e was  convenient  2.05 used  for this  study, s i n c e the c a l c u l a t i o n s r e q u i r e d a c o m p a r a t i v e l y s m a l l number o f beams and the RFS method was a p p l i c a b l e . 1(E) curves f o r the  (01) beam f o r the f i v e v a l u e s o f a w i t h a 2.5%  a c t i o n o f the topmost l a y e r a r e shown i n f i g u r e 3.8 mental d a t a .  contr-  t o g e t h e r w i t h the e x p e r i -  The main f e a t u r e s o f each i n d i v i d u a l curve a r e m a i n t a i n e d , a l -  though i n c r e a s e i n a g i v e s a g e n e r a l l o w e r i n g o f i n t e n s i t i e s and, f i c a n t l y , a broadening beams suggested  o f t h e peaks.  Visual  signi-  evaluations of a l l d i f f r a c t e d  the b e s t agreement between e x p e r i m e n t a l  curves o c c u r r e d when a i s i n t h e range 1.47  most  t o 1.76.  and  calculated  1(E)  These comparisons were  -84(01)  BEAM  ENERGY/eV F i g u r e 3.8:  The e x p e r i m e n t a l  I ( E ) curve f o r the (01) beam a t normal  i n c i d e n c e from the R h ( l l l )  s u r f a c e compared w i t h  five  c o r r e s p o n d i n g curves c a l c u l a t e d w i t h t h e p o t e n t i a l and Ad% = -2.5% f o r the parameter ct v a r y i n g from t o 2.34.  C^^l  1.17  -85-  a l s o made w i t h t h e n u m e r i c a l r e l i a b i l i t y r  index, and c o n d i t i o n s f o r minimum  f o r each v a l u e o f a a r e summarized i n T a b l e 3.4.  These r e s u l t s  indicate  r t h a t v a r i a t i o n o f a has o n l y a minor e f f e c t on t h e determined l a y e r s p a c i n g , and t h i s  supports t h e common assumption  i s not e s s e n t i a l i n LEED c r y s t a l l o g r a p h y .  inter-  that v a r i a t i o n o f  I t i s s a t i s f y i n g also that the  i n s e n s i t i v i t y o f geometrical s t r u c t u r e to V l e s s i t must be noted  topmost  i s r e c o g n i z e d by r ^ .  Neverthe-  t h a t even though c l o s e l y s i m i l a r g e o m e t r i c a l s t r u c t u r e s  a r e i n d i c a t e d by t h e d i f f e r e n t v a l u e s o f a , t h e v a l u e s o f r ^ a t t h e d i f f e r e n t minima a r e not e q u i v a l e n t . 1.76,  and t h i s suggest  Both v i s u a l than  The lowest r ^ v a l u e corresponds  that the i n i t i a l  to a close to  c h o i c e o f 1.17 may n o t be o p t i m a l .  and r - i n d e x e v a l u a t i o n s a r e c o n s i s t e n t i n i n d i c a t i n g a i s l a r g e r  1.17 and t h i s  supports t h e use o f t h e index r .  On t h e other hand,  v a l u e s o f a l a r g e r than 1.17 seem l e s s c o n s i s t e n t w i t h d e t e r m i n i n g V  from  e q u a t i o n 2.4. The v a l u e s o f r ^ r e p o r t e d i n T a b l e 3.4 a r e u n u s u a l l y low, e s p e c i a l l y those f o r t h e h i g h e r v a l u e s o f a. with t h e o r i g i n a l the tendency  The t r e n d s found d i d not seem c o n s i s t e n t  c o n c l u s i o n s o f Z a n a z z i and Jona, and we wondered whether  f o r low v a l u e s o f r ^ t o be found  f o r h i g h a c o u l d be an a r t e f a c t  a s s o c i a t e d w i t h t h e v a l u e o f p b e i n g f i x e d a t 0.027 i n t h e c a l c u l a t i o n o f (r ) . i n e q u a t i o n r i  (2.40).  p was o b t a i n e d by a v e r a g i n g  A c c o r d i n g t o Z a n a z z i and Jona [ 4 5 ] , t h i s v a l u e o f (  r  ) ^ f ° matching random p a i r s o f experimental r  r  II  and  calculated  1(E) c u r v e s .  One u n c e r t a i n t y was whether  complexity o f  II  structure  was f u l l y b u i l t  one would expect  i n t o t h e scheme o f Z a n a z z i and Jona.  t h a t an e x p e r i m e n t a l  In g e n e r a l  1(E) curve t h a t c o n t a i n s a l o t o f  -86-  T a b l e 3.4:  C o n d i t i o n s f o r b e s t agreement between e x p e r i m e n t a l 1(E) curves a t normal i n c i d e n c e f o r R h ( l l l ) and curves with the p o t e n t i a l indices r  Ad%  [vJJ ] W n  calculated  according to the r e l i a b i l i t y  and r f o r d i f f e r e n t v a l u e s o f a. r m  (%)  V  o r  (eV)  r  r  r  m  1.17  -1.6±0.8  -11.210.6  0.080  0.985  1.47  •2.510.5  -11.810.7  0.042  0.510  1.76  -2.310.6  -11.710.6  0.035  0.430  2.05  -2.310.5  -11.610.7  0.037  0.440  2.34  -2.010.6  -11.010.8  0.041  0.490  -87-  s t r u c t u r e would be more d i f f i c u l t l e s s s t r u c t u r e , and  to match t o c a l c u l a t e d curves than one  t h e r e f o r e i n s e t t i n g up a many-beam r e l i a b i l i t y - i n d e x  perhaps the former s h o u l d have a r e l a t i v e l y  g r e a t e r w e i g h t i n g than the  approach to t h i s i s t o a l l o w the v a l u e o f p to v a r y f o r each  One  curve.  In o r d e r to make an i n i t i a l  (2.40) w i t h a new  r  structure i t involves.  Equation  by comparing an experimental I. , = I.' i,cal i,cal  (2.40), we r  m  .  ' obs  then s e t up a new  m  1(E) curve a c c o r d i n g t o the amount  (3.1) i s o b t a i n e d from e q u a t i o n line  (2.36)  corresponding  U s i n g r, «... i n s t e a d o f p i n e q u a t i o n (st.line,expt) &  o v e r a l l reduced  reliability  index d e s i g n a t e d  as  f o r t h e v a r i a t i o n o f a v a l u e s are a l s o summarized i n  However, i t t u r n e d out f o r t h e case c o n s i d e r e d here t h a t m i n i -  m i z i n g r ^ gave i d e n t i c a l v a l u e s o f Ad% and r^;  replaced p i n  max  1  1(E) curve w i t h a s t r a i g h t  = i " , = 0. i,cal  The v a l u e s o f r  T a b l e 3.4.  we  (st.line,expt)  T h i s q u a n t i t y v a r i e s w i t h each e x p e r i m e n t a l  to  experimental  quantity  E^  of  latter  assessment o f whether such e f f e c t s c o u l d  be r e l e v a n t to the t r e n d s o f r ^ w i t h a shown i n T a b l e 3.4, equation  with  n u m e r i c a l v a l u e s o f the two  t o t h o s e found by  minimizing  i n d i c e s are d i f f e r e n t , but t o a good a p p r o x i -  mation c o r r e s p o n d i n g v a l u e s o f r ^ can be o b t a i n e d by d i v i d i n g v a l u e s o f r ^ by  12.1.  T h i s o b s e r v a t i o n does not support the p o s s i b i l i t y t h a t the  ^values o f r ^ found  f o r high a  v a l u e o f p used i n e q u a t i o n  (hence h i g h V ^ )  (2.40).  was  low  a s s o c i a t e d w i t h the  constant  -88-  F u r t h e r i n v e s t i g a t i o n suggested  t h a t t h e h i g h v a l u e o f a needed f o r  b e t t e r matching between c a l c u l a t e d and experimental be a s s o c i a t e d w i t h the way t h a t the e x p e r i m e n t a l i n the analysis.  The i n i t i a l  1(E) curves appears t o  i n t e n s i t i e s were handled  v a l u e o f a=1.17 was o b t a i n e d by c o n s i d e r i n g  i n d i v i d u a l l y measured 1(E) c u r v e s , whereas the experimental a c t u a l l y used  i n the comparisons w i t h t h e c a l c u l a t e d  1(E) curves were ave-  raged over a p p r o p r i a t e s e t s o f beams which a r e expected approximation  are, symmetrically e q u i v a l e n t .  1(E) curves  t o be, and t o a good  However, minor e r r o r s i n t h e II  experiment  [107,108] can l e a d t o c o r r e s p o n d i n g peak p o s i t i o n s i n  equivalent  II  sets  o f beams b e i n g s h i f t e d s l i g h t l y  (e.g. by 1 o r 2 eV) from t h e mean  v a l u e s and t h i s i n e v i t a b l y l e a d s t o some b r o a d e n i n g averaged  1(E) c u r v e s .  Upon i n v e s t i g a t i n g the averaged  a c h o i c e o f a as suggested by equations is  1.65.  o f structure i n the experimental  1(E) c u r v e s ,  (2.4) and (2.5) f o r the R h ( l l l ) s u r f a c e  T h i s v a l u e i s i n r e a s o n a b l e agreement w i t h the c o n c l u s i o n s noted  above from the v i s u a l e v a l u a t i o n and the r - f a c t o r  analysis.  These s t u d i e s i n d i c a t e the f o l l o w i n g c o n c l u s i o n s : 1)  Determined s u r f a c e g e o m e t r i c a l s t r u c t u r e i s i n s e n s i t i v e t o changes i n V  values.  T h i s s u p p o r t s t h e u s u a l approach o f keeping V  Q i  f i x e d i n the m u l t i p l e  s c a t t e r i n g c a l c u l a t i o n s , and o f choosing s u i t a b l e v a l u e s o f V  from  equation  (2.4). 2)  The index r ^ proposed  by Z a n a z z i and Jona i s c o n s i s t e n t w i t h a v i s u a l  analysis f o r i d e n t i f y i n g values of  which o p t i m i z e agreement between ex-  p e r i m e n t a l and c a l c u l a t e d 1(E) c u r v e s . - 3)  F u r t h e r improvements a r e needed i n the e x p e r i m e n t a l measurements f o r  e n s u r i n g t h a t 1(E) curves from s y m m e t r i c a l l y - r e l a t e d beams r e a l l y a r e equivalent.  -89-  3.3 (c) R e l i a b i l i t y - I n d e x and t h e V a r i a t i o n o f S u r f a c e Debye Temperature The  e f f e c t s o f atomic v i b r a t i o n s a r e i n c o r p o r a t e d  into multiple  scat-  t e r i n g c a l c u l a t i o n s by means o f temperature-dependent atomic s c a t t e r i n g f a c t o r s i n v o l v i n g the Debye temperature (2.9)- (2.10).  (8^) as i n d i c a t e d i n e q u a t i o n s  S t r i c t l y t h e atomic v i b r a t i o n s  are expected t o be l a y e r  dependent and t o decrease i n t o the b u l k [ 1 1 4 ] . have used a s i n g l e e f f e c t i v e Debye temperature probed by t h e a n a l y s e d e l e c t r o n s .  However, most LEED (6^ f f ) f ° e  In p r i n c i p l e , a b e t t e r ,  ra  ^  layers  although  II  simple, p o s s i b i l i t y and  i s t o g i v e t h e topmost l a y e r a  ( 8 ^ ^uUp  [ 5 6 ] . In the p r e v i o u s m u l t i p l e  made so f a r i n t h i s t h e s i s , e , D,bulk n  D  surface  value  eff w  a  s  e  s  t  i  m  a  t  e  d  a  s  y /  °~^ 9  n  b  u  l  k  scattering  (8^  grounds: moreover f o r a s s e s s i n g ' b  6  s  u  r  f)  by the  calculations  f o r rhodium was taken as 480 K [115] and  [112].  J  Although t h i s t y p e o f c h o i c e  seems p l a u s i b l e , i t i s n e v e r t h e l e s s made on i n t u i t i v e , r a t h e r  than  rigorous,  f u r t h e r the choice o f 9 -~ i t would seem D.surf n  h e l p f u l t o determine t h e e f f e c t o f i t s v a r i a t i o n on the s t r u c t u r a l as  still  II  t o assume the second and a l l deeper l a y e r s can be c h a r a c t e r i z e d  bulk value  9  studies  considered f o r v a r i a t i o n s o f V . i n section  3.3 ( b ) . F o r t h i s  conclusions, investi-  01  gation, multiple by v a r y i n g 1  6  s c a t t e r i n g c a l c u l a t i o n s f o r the R h ( l l l ) s u r f a c e were made  6„ . over the range o f 200-600 K i n 100 K s t e p s , D.surf  non-structural  a l l other  parameters b e i n g f i x e d a t t h e v a l u e s g i v e n i n s e c t i o n 3.2(b)  (except a was r e s t r i c t e d t o 1.76). Figures against " zontal  3.9 and 3.10 show two d i f f e r e n t s e t s o f contours o f r ^ p l o t t e d  8^ D,surf  -For both t h e contours a r e r e a s o n a b l y symmetrical about a h o r i J  J  l i n e , and minimum v a l u e s o f r ^ a r e c l o s e l y i n d i c a t e d t o correspond t o t h e  -90-  -Figure 3.9: •  Contour p l o t  of f  r  versus  9  and V f o r normal U,surr or n  o  r  data from R h ( l l l ) where the c a l c u l a t i o n s [ V ^ ] w i t h a=1.76 and 6 M  W  D b u l k  =480  K.  incidence  use t h e p o t e n t i a l  -91-  200  300  400  300  ®D,SURF  F i g u r e 3.10:  Contour p l o t o f r  versus 6  D  600  < K )  g  u  r  f  and Ad% f o r normal  incidence  d a t a from R h ( l l l ) where t h e c a l c u l a t i o n s use the p o t e n t i a l  -92-  values V  or  = -11.5  eV and Ad%  = -2%.  r e p o r t e d p r e v i o u s l y i n T a b l e s 3.3  These v a l u e s are comparable w i t h  those  r  and  3.4  f o r a f i x e d v a l u e o f 6^  An  unexpected f e a t u r e o f these p l o t s , however, i s t h a t they p o i n t to v a l u e s 6„ _ i n the p h y s i c a l l y unreasonable D,surf  ranee o f b e i n g g r e a t e r than 0„ , ,, D.bulk  r  (i.e.  480  K f o r rhodium  b  multiple-scattering calculations i n t e n s i t i e s but without  calculated  1(E)  calculated  f o r the  b  [115]).  W i t h i n the c o n v e n t i o n a l procedure  overall  of  curves.  f o r i n c l u d i n g atomic v i b r a t i o n s i n  [24,65],  the main e f f e c t o f 6^  a p p r e c i a b l y a f f e c t i n g s t r u c t u r e i n the  T h i s can be seen i n f i g u r e 3.11  (01) beam o f R h ( l l l ) w i t h Ad%  v a l u e s o f 6„ „ between 200 D.surf  i s to modify  and  600  K.  where 1(E)  = -2.5%  curves  are p l o t t e d f o r  The most n o t i c e a b l e t r e n d i s t h a t  the lower v a l u e s o f 9^ _ e s p e c i a l l y g i v e r e l a t i v e l y lower c a l c u l a t e d i n t e n D,surf r / & i s i t i e s at the h i g h e r e n e r g i e s . T h i s c o n t r a s t s w i t h the t r e n d observed i n the experimental  1(E) curves where r e l a t i v e l y h i g h e r i n t e n s i t i e s are found  the h i g h e r e n e r g i e s .  T h i s suggests  0^ _ p i c k e d out by the use o f r D.surf r r  J  compared, a l t h o u g h than 0^ k u i k '  0  u  r  t h a t the tendency t o h i g h v a l u e s  f  e  e  l  i g n  f o r 0^  s  u  r  f  t  0  be  s  i n g r i d transparency  and  i n the s o l i d angle p r e s e n t e d  curves  greater  i - t h a t the o r i g i n o f t h i s d i s c r e p a n c y may  a s s o c i a t e d w i t h g e n e r a l problems i n the data c o l l e c t i n g p r o c e s s e s . both  of  i s r e f l e c t i n g a r e a l t r e n d i n the  i t i s p h y s i c a l l y unreasonable  be Changes  t o the camera as  the spots move toward the c e n t r e o f t h e s c r e e n f o r i n c r e a s i n g e n e r g i e s • caused  apparent  variations  in relative intensities.  c o r r e c t i o n s f o r t h e s e f a c t o r s and  energies.  can  Legg et a l . [117] made  demonstrated a consequent  r e l a t i v e beam i n t e n s i t i e s at t h e lower  at  lowering i n  In f u t u r e LEED i n t e n s i t y  -93Rh(l1l)  ENERGY / eV " Figure 3.11: The experimental 1(E) curve for the (01) beam at normal incidence from the R h ( l l l ) surface compared with f i v e r  M  J  corresponding curves calculated with the potential L V  Rn  Ad% = -2.5%, and a = 1.76 f o r the parameter e from 200 to 600 K.  Q  s  u  r  f  W  i J»  varying  -ad-  measurements we a r e p l a n n i n g t o i n c o r p o r a t e c o r r e c t i o n s f o r these a l s o i t i s p o s s i b l e t h a t some refinement be needed a t h i g h e n e r g i e s when spots  effects;  i n t h e background c o r r e c t i o n  could  crowd t o g e t h e r i n t h e LEED s c r e e n .  At p r e s e n t we f e e l t h a t t h e source o f d i s c r e p a n c y i n d i c a t e d by t h e untenable l a r g e v a l u e o f 8^ - i s a s s o c i a t e d w i t h aspects o f t h e experimental D,surf &  r  surements, and although  t h i s has not y e t been unambiguously confirmed, two  c o n c l u s i o n s do seem s e c u r e . index  The f i r s t  i s t h a t t h e Zanazzi-Jona  reliability  appears a b l e t o g i v e a f a i r assessment o f t h e r e l a t i v e  intensities  o f 1(E) curves when 9_ _ i s varied i n the calculations D,surf J  appreciable s e n s i t i v i t y  in r  r  to r e l a t i v e intensities  o f 1(E) curves has not been r e c o g n i z e d p r e v i o u s l y ) . tural  mea-  r  (although an  i n successive sections Secondly,  surface struc-  c o n c l u s i o n s seem u n a f f e c t e d by v a r i a t i o n o f 6„ i n the c a l c u l a t i o n s . D.surf Although  t h e r e would be advantages i n r e f i n i n g t h e treatment  o f atomic  v i b r a t i o n s i n LEED i n t e n s i t y c a l c u l a t i o n s [ 6 5 ] , t h e evidence p r e s e n t e d does suggest affect  that modifying values o f 8  D  s  u  r  i s not going t o s i g n i f i c a n t l y  f  c o n c l u s i o n s about s u r f a c e geometry.  here  T h i s s u g g e s t i o n i s supported by  an independent a n a l y s i s o f t h e R h ( l l l ) s u r f a c e w i t h m u l t i p l e - s c a t t e r i n g c a l c u l a t i o n s by Chan et a l . [ 1 1 8 ] .  Using 6  D  b  u  l  k  as 300 K and 6  D  s  u  r  Chan et a l . o b t a i n e d Ad% f o r t h e topmost rhodium l a y e r as 0±5%.  as 250 K,  f  Although  these e r r o r l i m i t s seem r a t h e r l a r g e , n e v e r t h e l e s s t h i s c o n c l u s i o n i s cons i s t e n t w i t h our d e t e r m i n a t i o n o f t h e R h ( l l l ) s u r f a c e - we f e e l ,  f o r the present  (Table 3.2).  Generally  stage o f development o f LEED c r y s t a l l o g r a p h y , the  m u l t i p l e - s c a t t e r i n g c a l c u l a t i o n s might j u s t as w e l l c o n t i n u e t o use 8^ o b t a i n e d from t h e experimental  measurements [ 2 5 ] or a l t e r n a t i v e l y 6^  f o r t h e topmost l a y e r s p e c i f i e d as a c e r t a i n f r a c t i o n o f 8^ ^ u i ^  s  g  u  C^ ]. 2  r  ^ f  -95-  3.4 3.4  S t u d i e s o f A d s o r p t i o n o f some Gaseous M o l e c u l e s (a)  on Rhodium s u r f a c e s  B i b l i o g r a p h y o f O v e r l a y e r S t r u c t u r e s on Rhodium S u r f a c e s  The p r o p e r t i e s o f w e l l - d e f i n e d s u r f a c e s o f rhodium have been l e s s e x t e n s i v e l y i n v e s t i g a t e d than those o f many o t h e r t r a n s i t i o n m e t a l s , though rhodium shows a h i g h degree o f c a t a l y t i c a c t i v i t y T a b l e 3.5  even  f o r many r e a c t i o n s [42]  summarizes s t u d i e s where g e n e r a l c h e m i s o r p t i v e p r o p e r t i e s o f rhodium  have been i n v e s t i g a t e d w i t h LEED.  Auger e l e c t r o n s p e c t r o s c o p y was  l a b l e f o r m o n i t o r i n g s u r f a c e p u r i t y i n the i n i t i a l a l t h o u g h t h i s t e c h n i q u e was T a b l e 3.5.  s t u d i e s by Tucker  avai-  [119-122]  a v a i l a b l e f o r a l l other studies reported i n  Much o f the work on rhodium t h a t has  years has been concerned  not  emerged o v e r the p a s t s e v e r a l  mainly w i t h e i t h e r LEED p a t t e r n s o r a d s o r p t i o n  kinetics. An important was  t o determine  on rhodium.  The  o b j e c t i v e f o r p a r t o f the r e s e a r c h r e p o r t e d i n t h i s  thesis  some d e t a i l e d s u r f a c e s t r u c t u r e s w i t h LEED f o r a d s o r p t i o n initial  aim was  t o i n v e s t i g a t e some c o m p a r a t i v e l y  s t r u c t u r e s i n v o l v i n g 0 o r S adsorbed  on  low-index  s u r f a c e s and  simple  to compare  w i t h s i m i l a r systems a l r e a d y i n v e s t i g a t e d w i t h LEED c r y s t a l l o g r a p h y , f o r example a d s o r p t i o n on n i c k e l . p t i o n o f l ^ S on the chapters 4 and  (100) and  5 respectively.  The (110)  LEED a n a l y s e s r e s u l t i n g from the  adsor-  s u r f a c e s o f rhodium a r e d e s c r i b e d i n  The next s e c t i o n reviews  a d s o r p t i o n of 0^ on Rh(100), a system t h a t was i n v e s t i g a t e d v i a a n a l y s e s o f LEED i n t e n s i t i e s .  o b s e r v a t i o n s f o r the  o r i g i n a l l y planned  t o be  T a b l e 3.5:  S u r f a c e s t r u c t u r e s r e p o r t e d f o r a d s o r p t i o n o f s m a l l gaseous molecules  on  low  index  s u r f a c e s o f rhodium.  surface  surface  ref.  structure  ref.  disorder  [c]  c(2x2)-0  [a,f]  c(2x4)-0  [c]  (3xl)-0  [b,f]  c(2x8)-0  [c]  c(2x2)-CO  CO -  hexagonal  [a] overlayer  [a]  1—1  (1x3)-0  1—I  (lx2)-0  1—1  (2x3)-0  1—1  (2x2)-0  (2x1)-CO  [d]  c(2x2)-C  [a,e,g]  1 1  [b]  (2x2)-0  ref.  1  c(2x8)-0  structure  1—1  [a,b,f]  1—1  p(2x2)-0  surface  1  °2  structure  Rh(lll)  Rh(llO)  Rh(100)  Adsorbate  [d]  (/3x/3)R30°-CO  [a] [a,e]  (2x2)-CO  (4x1)-CO CO.  c(2x2)-CO  (/3x/3)R30°-CO  [a]  Z  NO  HS 2  (2x2)-CO c(2x2)-N0  [a]  p(2x2)-S  [f]  c(2x2)-S  [f]  [ a ] - C a s t n e r et a l . [ 9 6 ] ; [ b ] - Tucker [ e ] - Grant  and Haas [ 1 0 2 ] ; [ f ] -  c(2x2)-S  [119]; [ c ] - Tucker  T h i s work [123,124];  [f]  [a] [a,e]  c(4x2)-NO  [a]  c(2x2)-NO  [a]  —  —  —  [ l 2 0 , 1 2 l ] ; [ d ] - Marbrow and  [ g ] - Weinberg et a l . [127]..  Lambert [ 9 7 ] ;  -97-  3.4  (b) A d s o r p t i o n The  o f 0^ on Rh(100)  sample used i n t h i s study was c u t from the s i n g l e c r y s t a l  by Tucker [ 9 4 ] ,  provided  and i t was p r e v i o u s l y used by Watson e t a l . f o r a LEED  a n a l y s i s o f the c l e a n Rh(100) s u r f a c e [ 1 0 8 ] .  P r i o r to s t a r t i n g the adsorption 1°  work, t h e s u r f a c e was r e p o l i s h e d and checked t o ensure t h a t i t was w i t h i n — o f the  (100) p l a n e .  A f t e r mounting and i n s t a l l i n g  a c c o r d i n g t o the procedures d e s c r i b e d i n s e c t i o n 3 . 1 , and  sample was cleaned  annealed u n t i l t h e LEED p a t t e r n e x h i b i t e d a sharp  (99.99%,Matheson) was i n t r o d u c e d torr.  ( l x l ) pattern with  i n t o the vacuum chamber at a p r e s s u r e o f  A f t e r 5 minutes a sharp  (3x1)  LEED p a t t e r n c o r r e s p o n d i n g  d i f f e r e n t domains was o b s e r v e d , and an Auger spectrum taken a f t e r the o f t h i s p a t t e r n f a i l e d t o d e t e c t the p r e s e n c e o f any i m p u r i t i e s . peaks o f oxygen a t around 510 eV c o u l d n o t be d e t e c t e d .  i t appears t o be a s s o c i a t e d w i t h t h e low i o n i z a t i o n  i n i t i a t i n g t h e Auger process  f o r adsorbed oxygen.  t o two formation  The Auger  T h i s e f f e c t has been  observed p r e v i o u s l y f o r oxygen a d s o r p t i o n on some t r a n s i t i o n metals and  low back-  The sample was heated t o 3 0 0 ° C b e f o r e h i g h p u r i t y 0^  ground i n t e n s i t i e s .  10  i n the vacuum chamber, t h e  [125,126]  cross-section for  A sharp  c h a r a c t e r i s t i c o f the c l e a n Rh(100) s u r f a c e can be r e s t o r e d  ( l l ) pattern x  (presumably by  d e s o r p t i o n o f the oxygen [96,127]) upon h e a t i n g a t 1 0 0 0 ° C f o r 10 minutes. A f t e r r e t u r n i n g t o the base p r e s s u r e  the p r o c e s s  c o u l d be r e p e a t e d  with  a new dose o f oxygen a p p l i e d under t h e same c o n d i t i o n s as i n d i c a t e d above. Sharp  (3x1) p a t t e r n s  c o u l d always be o b t a i n e d ,  v a r i a t i o n s were found i n the domain s t r u c t u r e . populated  domains  f  t o two u n e q u a l l y  populated  although  on d i f f e r e n t  occasions  These ranged from two e q u a l l y domains and even t o t h e appearance  -98-  (figure 3 . 1 2 ) .  o f a s i n g l e domain  From time t o time f a i n t h a l f - o r d e r  diffr-  spots were observed superimposed on t h e ( 3 x 1 ) p a t t e r n , but t h e p a t t e r n  acted  never developed i n t o a complete exposed t o  f o r longer periods  removed by h e a t i n g termperature,  at 700°C  o f time.  Furthermore these  the c l e a n R h ( 1 0 0 )  i f t h e c r y s t a l was l e f t 3 0 minutes.  c o u l d be  ( 3 x 1 ) pattern.  ( f i g u r e 3 . 1 2 ) c o u l d be observed when  LEED p a t t e r n  s u r f a c e was exposed t o 0  apparent, but i n c o m p l e t e l y  spots  f o r a few seconds; then a f t e r c o o l i n g t o room  the LEED p a t t e r n showed o n l y t h e sharp  A well-defined p(2x2)  further  ( 2 x 2 ) p a t t e r n even though t h e c r y s t a l was  9  a t 1 0 ^ t o r r f o r 5 minutes.  developed, c ( 2 x 2 )  p a t t e r n c o u l d a l s o be d e t e c t e d  atmosphere o f 0 ^ at 1 0 ^ t o r r f o r a  i n t h e constant  T h i s was observed as an i n c r e a s e i n i n t e n s i t i e s  t i o n a l - o r d e r spots o f type  (^j),  while  An  t h e other  of frac-  f r a c t i o n a l - o r d e r spots  showed  r e l a t i v e decreases i n i n t e n s i t i e s . These r e s u l t s  f o r the adsorption  o f oxygen on R h ( 1 0 0 )  e a r l i e r work done by Tucker [ 1 1 9 ] and C a s t n e r (2x2), not  ( 3 x 1 ) and ( 2 x 8 ) p a t t e r n s  observe t h e c ( 2 x 2 )  which transformed (3x1)  t o the c ( 2 x 2 )  p a t t e r n was d e t e c t e d  pressure. with  pattern.  My o b s e r v a t i o n  a c(2x2)  Castner  between t h e s e  et a l . reported a p ( 2 x 2 )  pattern  i n t h a t work over a wide range o f temperature and of the p ( 2 x 2 )  p a t t e r n seems b r o a d l y  two s t u d i e s .  oxygen exposure.  different  reported  p a t t e r n a t h i g h e r oxygen exposures, b u t no  Also  LEED p a t t e r n s , f o r t h e t r a n s f o r m a t i o n  p a t t e r n with  Tucker  with  f o r i n c r e a s i n g oxygen exposures, but he d i d  those observed i n these o t h e r  ' through f a i n t  et a l . [ 9 6 ] .  agree p a r t l y  I had some  of a p(2x2)  One p o s s i b i l i t y  studies could involve other  i n agreement evidence, pattern  into  f o r the discrepancies  gases  (e.g. CO) b e i n g  -99-  I'igure  3.12:  P h o t o g r a p h s o f some p ( 2 x 2 ) at normal Rh(lOO)  incidence  a n d (3><1) LEED p a t t e r n s  from t h e a d s o r p t i o n  observed  o f o x y g e n on a  surface.  (a)  Rh[100)-p(2*2)-0  a t 70 eV,  (b)  Rh(100)-(3*l)-0,  s i n g l e d o m a i n a t 174  (c)  Rh(100)-(3xl)-0,  2 equally populated  d o m a i n s a t 100 eV.  (d)  Rh(100)-(3*l)-0,  2 equally populated  d o m a i n s a t 152 eV.  eV,  -100d i s p l a c e d from the w a l l s o f t h e vacuum chamber on a d m i t t i n g oxygen t o t h e system.  U n f o r t u n a t e l y , t h e mass s p e c t r o m e t e r d i d not f u n c t i o n p r o p e r l y d u r i n g  t h e s e experiments and so we had no independent chamber.  However, no e v i d e n c e was  assessment  o f t h e gases i n t h e  found f o r t h e b u i l d up o f i m p u r i t i e s  t h e s u r f a c e on a d d i n g oxygen t o t h e system, a l t h o u g h i t was  again unfortunate  t h a t t h e r e t a r d i n g f i e l d a n a l y z e r as used at the time o f t h i s work was s e n s i t i v e enough t o d e t e c t t h e oxygen.  on  N e v e r t h e l e s s c a r e was  not  taken during  t h e heat t r e a t m e n t s t o o p e r a t e under c o n d i t i o n s where carbon does not apprec i a b l y m i g r a t e from t h e b u l k ; t h e Auger s p e c t r a c o n f i r m e d t h a t carbon i m p u r i t i e s remained a t low l e v e l s d u r i n g t h e s e e x p e r i m e n t s . Two  complete s e t s o f photographs  f o r t h e ( 3 x 1 ) p a t t e r n s were t a k e n on  d i f f e r e n t o c c a s i o n s o v e r t h e energy range  30-200  eV f o r normal  incidence.  The f i l m s were a n a l y s e d t o y i e l d the 1(E) c u r v e s shown i n Appendices A l and A 2 ; t h e f i r s t i s f o r two e q u a l l y p o p u l a t e d domains and the second i s f o r a s i n g l e domain t y p e o n l y .  These 1(E) c u r v e s have not y e t been a n a l y s e d w i t h  m u l t i p l e - s c a t t e r i n g c a l c u l a t i o n s e s p e c i a l l y because we have no c l u e s a t p r e s e n t t o the p o s s i b l e s t r u c t u r e , and some g e o m e t r i c a l models t h a t s h o u l d be a r e complex.  tested  An a t t a c k on t h e problem o f the s t r u c t u r e o f t h e ( 3 x 1 ) s u r f a c e  would be a i d e d by t h e a v a i l a b i l i t y o f more d e t a i l e d e x p e r i m e n t a l d a t a , f o r example on s u r f a c e coverage  (from AES w i t h a c y l i n d r i c a l m i r r o r a n a l y z e r ) and  on p o s s i b l e oxygen bonding s i t e s from h i g h - r e s o l u t i o n e l e c t r o n energy . spectroscopy  [128].  loss  -101-  CHAPTER 4 LEED A n a l y s i s o f Rh(100)-p(2x2)-S S u r f a c e  Structure  X -102-  4.1  Introduction Knowledge o f t h e s t r u c t u r e s adopted by atomic and m o l e c u l a r  species  adsorbed on s u r f a c e s o f rhodium i s o f importance f o r an understanding o f the c a t a l y t i c p r o p e r t i e s o f t h i s m e t a l . with  LEED f o r the (2x2)  surface [123]; for  s t r u c t u r e formed by a d s o r b i n g  t h i s appears t o r e p r e s e n t  a d s o r p t i o n on rhodium.  structural  information i s a v a i l a b l e f o r sulphur  H^S on t h e c l e a n (100)  such s t r u c t u r a l a n a l y s i s initial  adsorption  study  s i n c e some  ( v i a H^S) on o t h e r  One immediate o b j e c t i v e i s t o g a i n  bonding a t these  surfaces.  o f H S on Rh(100)  Adsorption A clean  r e p o r t s an a n a l y s i s  p r o v i d i n g points o f reference f o r assess-  t h e s t r u c t u r e o f Rh(100)-p(2x2)-S.  i n f o r m a t i o n about t h e chemical  4.2  the f i r s t  H^S was chosen f o r t h i s  t r a n s i t i o n metal s u r f a c e s , thereby ing  T h i s chapter  2  (100)  s u r f a c e o f rhodium w i t h  a sharp  ( l x l ) LEED p a t t e r n  (obtained  by t h e procedures d e s c r i b e d i n s e c t i o n 3.1) was exposed t o h i g h p u r i t y H^S -8 (Matheson) a t 10"  t o r r f o r 1 min.  A f t e r pumping excess gas,  annealed a t 300°C f o r 1 min.  and a sharp p(2x2)  good c o n t r a s t  Auger s p e c t r a  ( f i g u r e 4.1).  t h e s u r f a c e was  LEED p a t t e r n o b t a i n e d  ( f i g u r e 4.2)  with  taken a f t e r t h e f o r -  mation o f t h i s p a t t e r n i n d i c a t e d S as the main f o r e i g n component w i t h peak h e i g h t  ratios  152eV(S)/302eV(Rh)=2/3.  Small  Auger  t r a c e s o f C c o u l d a l s o be  d e t e c t e d , b u t i t s p r o p o r t i o n s were minimized by t h e low temperature  annealing.  We b e l i e v e t h a t H S d i s s o c i a t e d on t h e Rh(100) s u r f a c e , i n p a r t because 2  we a l s o o b t a i n e d sulphur  t h i s p(2x2)-S LEED p a t t e r n by h e a t i n g the metal such t h a t  i m p u r i t y segregated  t o t h e s u r f a c e from t h e b u l k .  Exactly  similar  -d-  - c!• i  gure 4.1:  Photographs o f  LHHD  p a t t e r n s observed at normal i n c i d e n c e from  a d s o r p t i o n o f S on Rh(100) s u r f a c e . eV,  (a)  Rh(100)-c(2*2)-S at 80  (b)  Rh(100)-p(2x2)-S at 72 eV,  (c)  Rh(100)-p(2*2)-S at 114  (d)  Rh(100)-p(2*2)-S at 168 eV.  eV,  -104-  Rh 100  Figure  4.2:  200  Energy (eV )  Auger s p e c t r a o f Rh(100) s u r f a c e s w i t h beam at d i f f e r e n t Rh(100)-p(2*2)-S.  stages  300  1.5 keV and 10 microamp  d u r i n g the p r e p a r a t i o n o f  -105-  observations  have been r e p o r t e d by G a u t h i e r  et a l . [129]  and  Demuth et a l .  [130,131] i n t h e i r p r e p a r a t i o n s  o f Ni(100)-p(2x2)-S and  and  i n t h e i r s t u d i e s o f the Rh(100) s u r f a c e .  a l s o by C a s t n e r et a l . [96]  1(E)  curves measured from the Rh(100)-p(2x2)-S s u r f a c e o b t a i n e d  t i o n o f the bulk adsorption. H^S  Ni(100)-c(2x2)-S,  sulphur  impurity  This provided  agreed c l o s e l y w i t h  some t e n t a t i v e evidence t h a t the a d s o r p t i o n  d i s s o c i a t i n g on a metal s u r f a c e was  provided  H^S  and  thermal d e s o r p t i o n  spectroscopy  d i s s o c i a t e d upon a d s o r p t i o n  [132].  2  c o u l d be removed o n l y by  This required heating  sphere o f H^s  (lxlO  - 7  o f uv  Fischer s  photo-  T h i s work i n d i c a t e d t h a t  extensive  atoms are h e l d s t r o n g l y to Ar  +  bombardment.  i n g the Rh(100) s u r f a c e , a c(2x2) p a t t e r n c o u l d a l s o be t o H S.  of  over the e n t i r e range o f coverage.  In Rh (100)-p (2x2)-S, the adsorbed s u l p h u r s u r f a c e and  H^S  D i r e c t evidence  by Keleman and  study on the Ru(100) s u r f a c e w i t h the a d d i t i o n a l t e c h n i q u e s emission  by the migra-  those p r e p a r e d by  on t h i s rhodium s u r f a c e i n v o l v e s d i s s o c i a t i v e a d s o r p t i o n .  for  The  t o r r ) , and  on  A f t e r clean-  formed on  the c r y s t a l at 400°C f o r 4 min.  exposure  i n the atmos-  c o o l i n g the LEED p a t t e r n o f the  e x h i b i t e d a c(2x2)-S o v e r l a y e r p a t t e r n s u r f a c e gave a r a t i o o f peak h e i g h t s  ( f i g u r e 4.1).  the  surface  Auger s p e c t r a f o r t h i s  152eV(S)/302eV(Rh)=4/3 which suggests  t h a t the S coverage f o r t h i s s t r u c t u r e i s a p p r o x i m a t e l y twice  that of  the  Rh(100)-p(2x2)-S s t r u c t u r e . 1(E)  curves were measured f o r Rh(100)-p(2x2)-S f o r normal i n c i d e n c e  - t h e beams (01),  (11),  (02),  (12),  (0±) , ( l | ) , (^|),  beam n o t a t i o n shown i n f i g u r e 4.3. the LEED s c r e e n  collected.  (y|),  i n s e c t i o n 3.1.  Two  with  using  the  These measurements i n v o l v e d p h o t o g r a p h i n g  at 2 eV i n t e r v a l s over the energy range 40-200 eV,  z i n g the p h o t o g r a p h i c n e g a t i v e s described  ( o | ) , and  for  the c o m p u t e r - c o n t r o l l e d  and  analy-  V i d i c o n camera as  independent s e t s o f experimental  data were  -106-  9x  F i g u r e 4.3:  Beam n o t a t i o n f o r the LEED p a t t e r n o f Rh(100)-p(2x2)-S s t r u c t u r e .  -107-  4.3  Computational 1(E)  Scheme  curves were c a l c u l a t e d w i t h the l a y e r - d o u b l i n g method [ 2 4 ] , u s i n g  a c o n v e n t i o n a l m u f f i n - t i n - t y p e p o t e n t i a l , f o r some s u r f a c e models i n which o n l y s u l p h u r was  present  i n an o v e r l a y e r .  The  s c a t t e r i n g s by the  p o t e n t i a l s were d e s c r i b e d by e i g h t p h a s e - s h i f t s . was  used f o r the atomic  atomic  A band s t r u c t u r e p o t e n t i a l  r e g i o n s i n the s u b s t r a t e [ 1 1 0 ] .  For the atomic  reg-  ions i n the s u l p h u r o v e r l a y e r , the s u p e r p o s i t i o n p o t e n t i a l o b t a i n e d by Demuth et  a l . [131] was  used.  Hove and Tong [146]  T h i s s u p e r p o s i t i o n p o t e n t i a l was  for  Van  i n an a n a l y s i s o f s u r f a c e s t r u c t u r e s formed by S on  The r e a l p a r t o f the i n n e r p o t e n t i a l (although t h i s was  a l s o used by  refined  (V  ) was  initially  s e t at -12.0  or l a t e r i n the comparison w i t h e x p e r i m e n t a l  both the o v e r l a y e r and the s u b s t r a t e , w h i l e the imaginary  part  Ni(100).  eV data)  (V ^)  was  1/3 equated t o -1.51E for  rhodium  eV.  The  e f f e c t i v e Debye temperatures  (as d i s c u s s e d i n s e c t i o n 3.4)  and  236  were taken as 406  K  K f o r sulphur f o l l o w i n g  Demuth et a l . [ l 3 l ] . The g e o m e t r i c a l models c o n s i d e r e d f o r Rh(100)-p(2x2)-S were s i m p l i f i e d  by  o  f i x i n g a l l i n t e r l a y e r spacings  i n t h e metal  a t the b u l k v a l u e  (1.9022 A ) ;  this  f o l l o w s our p r e v i o u s c o n c l u s i o n f o r c l e a n Rh(100) t h a t t h i s s u r f a c e i s not r e c o n s t r u c t e d and ( s e c t i o n 3.2). to  i t s topmost s p a c i n g i s w i t h i n 2.5%  o f the b u l k  value  Three types o f s t r u c t u r a l model were t e s t e d , a l l c o r r e s p o n d i n g  a q u a r t e r monolayer o f S atoms.  These models a r e shown i n f i g u r e 2.8  they a r e d e s i g n a t e d a c c o r d i n g t o the number o f n e a r e s t - n e i g h b o u r (as a l r e a d y d e s c r i b e d i n s e c t i o n 2.7) spheres, with r a d i i  as 4F,  IF and  g i v e n by P a u l i n g [ 1 3 3 ] , was  2F.  metal  and  atoms  The p a c k i n g o f hard  used t o guide the p o s s i b l e  -108-  v a l u e s o f topmost i n t e r l a y e r s p a c i n g f o r each model type; t h i s a n a l y s i s s p e c i es  fically  c o n s i d e r e d spacings  between 2.1 and 2.7 A f o r t h e IF model, between  o  o  1.4 and 2.2 A f o r t h e 2F model and between 1.0 and 1.6 A f o r t h e 4F model. Symmetry c o u l d be used i n t h e c a l c u l a t i o n s a t normal i n c i d e n c e and the number of beams used i n t h e c a l c u l a t i o n s a r e summarized i n T a b l e model i t i s n e c e s s a r y  F o r t h e 2F  t o average a p p r o p r i a t e c a l c u l a t e d beam i n t e n s i t i e s  according to the p o s s i b l e symmetrically-equivalent 4.4  2.1.  domains.  Results 1(E) curves  measured f o r normal i n c i d e n c e f o r t h e (01) and (~)  beams a r e shown i n f i g u r e 4.4 f o r two independent experiments.  sets of  Beams w i t h i n  each s e t s h o u l d be s y m m e t r i c a l l y - e q u i v a l e n t , both w i t h r e g a r d t o peak p o s i t ions and o t h e r s t r u c t u r a l suggest  features.  t h a t t h e experimental  t h e i r general r e l i a b i l i t y . a t t r i b u t e d t o experimental of  The correspondences seen i n t h e f i g u r e  data a r e c l o s e l y r e p r o d u c i b l e , and t h i s  The s m a l l v a r i a t i o n s which do occur must be e r r o r s ( i n v o l v i n g such f a c t o r s as uneven response  the screen, imperfections o f the c r y s t a l  s u r f a c e , and some u n c e r t a i n t y i n  s e t t i n g t h e angle o f i n c i d e n c e ) ; such e r r o r s , a l t h o u g h limit  supports  s m a l l , do i n e v i t a b l y  t h e l e v e l o f agreement p o s s i b l e between c a l c u l a t i o n and experiment.  To  minimize any a r t e f a c t s i n t h e comparisons w i t h t h e c a l c u l a t e d i n t e n s i t i e s , measured 1(E) curves  f o r s e t s o f beams which a r e t h e o r e t i c a l l y  were averaged and d i g i t a l l y  equivalent  smoothed (by two o p e r a t i o n s o f t h e t h r e e - p o i n t  smoothing f i l t e r ) p r i o r t o comparing w i t h t h e c a l c u l a t i o n s .  -109-  01  so F i g u r e 4.4:  100  iTo  energy(eV)  Comparison different  01  1  50  200  f o r the (~)  and  1  100  1  energy  150  200  (eV)  (01) beams o f 1(E) curves from  experiments measured at normal  incidence.  two  -110-  Some comparisons p(2x2)-S  o f e x p e r i m e n t a l and c a l c u l a t e d 1(E) curves f o r Rh(100)-  a r e shown i n f i g u r e 4.5.  V i s u a l comparisons  o f a l l data  available  p o i n t s t o t h e c o n c l u s i o n t h a t t h e c e n t r e (4F) model g i v e s a b e t t e r correspondence (IF) models.  t o t h e experimental  overall  1(E) curves than t h e b r i d g e (2F) and on-top  F o r t h e i n t e g r a l - o r d e r beams a l o n e , r e a s o n a b l e match-ups b e t -  ween e x p e r i m e n t a l and c a l c u l a t e d  1(E) curves a r e found f o r t h e (01) and (02) o  beams w i t h a l l t h e t h r e e models  o  ( i . e . 4F a t 1.3 A, 2F a t 1.9 A and IF a t 2.3 A ) ,  but t h e 4F model a l s o g i v e s a good correspondence the 4F and IF models f a i l  o  i n t h i s regard.  f o r t h e (11) beam whereas  As expected, t h e f r a c t i o n a l - o r d e r  beams a r e g e n e r a l l y more s e n s i t i v e t o t h e l o c a t i o n s o f t h e o v e r l a y e r atoms, and t h e o v e r a l l c o n c l u s i o n from a v i s u a l a n a l y s i s o f a l l d a t a f o r t h e f r a c t i o n a l order beams i s t h a t t h e 4F model g i v e s t h e b e s t account o f t h e e x p e r i o  mental  1(E) curves w i t h the Rh-S i n t e r l a y e r s p a c i n g c l o s e t o 1.3 A.  the agreement i s not complete, accounted  However,  r e l a t i v e peak i n t e n s i t i e s a r e not p r o p e r l y  f o r and i n a few i n s t a n c e s t h e 4F model f a i l s  i n t h e e x p e r i m e n t a l 1(E) c u r v e s .  t o reproduce f e a t u r e s  In p a r t i c u l a r , t h e c a l c u l a t e d  1(E) curve  f o r t h e (0^-) beam f o r t h e 4F model w i t h t h e Rh-S i n t e r l a y e r s p a c i n g equal t o 1.3 A does not reproduce  t h e peak p r e s e n t i n t h e experimental curve at 110 eV;  3 a l s o f o r t h e (0-) beam t h e 4F model shows an e x t r a s m a l l peak a t 130 eV which c o u l d not be d e t e c t e d i n t h e e x p e r i m e n t a l  curve. 3  For some f r a c t i o n a l - o r d e r beams  (especially  13  (0^) and C^))»  calculated  1(E) curves from the b r i d g e (2F) model g i v e r e a s o n a b l e agreement w i t h t h e o  experimental  1(E) curves f o r t h e topmost s p a c i n g o f 1.9 A, b u t t h i s a d s o r p t i o n  s i t e i s l e s s f a v o r a b l e than t h e 4F s i t e f o r t h e (o|) and ( ™ ) beams. top ( I F ) model g i v e s poor v i s u a l agreement between c a l c u l a t i o n and  The on-  experiment  -111-  F i g u r e 4.5:  Comparison o f experimental  1(E) curves f o r v a r i o u s i n t e g r a l -  and f r a c t i o n a l - o r d e r d i f f r a c t e d beams from w i t h the c a l c u l a t e d  curves f o r S adsorbed  IF s i t e s at the topmost Rh-S each  curve.  Rh(100)-p(2x2)-S on the 4F, 2F  and  i n t e r l a y e r spacing i n d i c a t e d f o r  Electron energy (eV)  T — " — i — i — i — 1 — i — i  I—i—i—i—i—i—i—i—i  40  80  120 160 200  1  1  I — | — | — i — i — i — i — i — i  I — i — i — i — i — i — i — i — i  40  80  120  160 200  1  1  ( — i — i — i — i — i — i — i — I  I — i — i — i — i — i — i — i — i  40  80  120  Electron energy (eV)  160 200  1  I — i — i — r — i  i  I—i—i—i—i—i  40  80  120  i  i  r  i  i  i  160 200  -114-  13 f o r most beams, a l t h o u g h some agreement i s p r e s e n t f o r the (— —) beam f o r the o  topmost s p a c i n g o f 2.7 mental 1(E) curves  A.  Illustrated  f o r the  (o|)  and  i n f i g u r e 4.6  (-—)  are comparisons o f e x p e r i -  beams w i t h those c a l c u l a t e d  from  the 4F model f o r v a r i o u s v a l u e s o f the topmost i n t e r l a y e r s p a c i n g r a n g i n g from  1.0  A t o 1.6  A.  b e s t correspondence 1.2  and  1.3  The  Although  the l e v e l o f agreement i s not complete,  seems t o occur w i t h the S-Rh  the  i n t e r l a y e r s p a c i n g between  A.  correspondence  between the experimental  and  calculated  1(E)  curves  f o r the Rh(100)-p(2x2)-S s u r f a c e were a l s o a s s e s s e d by e v a l u a t i n g the ability  index  ( r ^ ) proposed  by Z a n a z z i and Jona [ 4 5 ] .  g i v e contour p l o t s o f r ^ as a f u n c t i o n o f the o f the t h r e e models when compared w i t h one  Rh-S  i n T a b l e 4.1.  f o r each  set of experimental data.  Compari-  s i m i l a r r e s u l t s , as sum-  The a n a l y s i s w i t h r ^ unambiguously showed t h a t the 4F  model g i v e s the b e s t correspondence.between the e x p e r i m e n t a l 1(E) c u r v e s .  4.7(a)-4.7(c)  s p a c i n g and V  son w i t h the o t h e r s e t o f e x p e r i m e n t a l d a t a produced marized  Figures  reli-  For t h i s model, r ^ i s minimized  and  calculated.  ( f i g u r e 4.7(a)) w i t h the  Rh-S  o  i n t e r l a y e r s p a c i n g equal t o 1.30±0.03 A and V ^ equal -13.6±0.9 eV, when the u n c e r t a i n t i e s are g i v e n as i e . and d b  u n c e r t a i n t i e s correspond Watson et a l . [ 4 3 ] .  ±e  v  as i n d i c a t e d i n s e c t i o n 2.8.  t o 68% p r o b a b i l i t i e s a c c o r d i n g t o t h e a n a l y s i s  The minimum v a l u e o f r  r  r e p r e s e n t s a moderate l e v e l o f agreement and - at l e a s t The b r i d g e Rh-S  probably  The  f o r the 4F model i s 0.26; suggests  of this /  t h a t the s t r u c t u r e i s  c o r r e c t a c c o r d i n g t o a c r i t e r i o n o f Z a n a z z i and Jona  L45J.  (2F) model a l s o g i v e s a l o c a l i z e d minimum, s p e c i f i c a l l y at the o  i n t e r l a y e r s p a c i n g o f 1.94±0.08 A and V ^  equal t o -11.611.4 eV.  The  -115-  I  40  1  1  80  1  1  120  1  1  1  160  —'  200  1  I  40  1  1  80  1  1  —I  1  120  1  160  1  200  Electron energy (eV) Figure 4.6:  Comparison of experimental  I(E) curves f o r the (0^) and (~,~)  beams from the Rh(100)-p(2x2)-S surface with those c a l c u l a t e d for S adsorbed on the 4F s i t e f o r a range o f topmost Rh-S i n t e r l a y e r spacings.  -116-  T a b l e 4.1:  C o n d i t i o n s f o r minima o f r  r  for different  models o f  Rh(100)-p(2x2)-S.  s u r f a c e model  centre  site  expt. no.  (A)  V , (eV) or  856  1.3010.03  •13.6+0.9  0.26  932  1.3110.03  •13.810.8  0.25  856  1.9410.08  •11.6+1.4  0.30  932  1.9410.08  -13.511.2  0.28  AE (eV)  S-Rh  nT  (4F)  bridge  site  (2F)  on-top  site  856  no l o c a l i z e d  minimum  932  no l o c a l i z e d  minimum  (IF)  t o t a l range o f energy compared.  -117-  c o r r e s p o n d i n g minimum v a l u e o f  (0.30) i s h i g h e r than t h a t o f the 4F model  (0.26), although these r ^ v a l u e s are c l o s e r than expected the v i s u a l a n a l y s i s .  on the b a s i s o f  F u r t h e r s u g g e s t i v e support f o r the 4F model, from  r e l i a b i l i t y i n d e x a n a l y s i s , i s i n d i c a t e d by the l a r g e r u n c e r t a i n t i e s ated w i t h the b r i d g e model.  The  contour p l o t o f r ^ i n f i g u r e 4.7  not i n d i c a t e a l o c a l i z e d minimum f o r the on-top r^  are c o m p a r a t i v e l y h i g h over the complete  spacing considered.  However i t was  associ-  (c) does  (IF) model, a l s o v a l u e s o f  ranges  observed  of V  and Rh-S  interlayer  i n separate c a l c u l a t i o n s  the contour p l o t s o f r ^ f o r the i n t e g r a l - o r d e r beams alone and  that  f o r the  t i o n a l - o r d e r beams alone d i d show l o c a l minima c o r r e s p o n d i n g to Rh-S o  the  frac-  inter-  o  l a y e r s p a c i n g s o f 2.3 A and 2.7 A r e s p e c t i v e l y ; t h i s i n d i c a t e s the r e a s o n the c a l c u l a t e d 2.7 4.5  1(E)  curves shown i n f i g u r e 4.5  are f o r the s p a c i n g s 2.3  why  and  X. Discussion The  evidence p r e s e n t e d above i n d i c a t e s t h a t the s u r f a c e s t r u c t u r e  Rh(100)-p(2x2)-S has the s u l p h u r atoms adsorbed  on the f o u r - f o l d  (4F)  sites  o  of  the Rh(100) s u r f a c e a t about  1.30  A above the topmost rhodium  layer. o  T h i s corresponds dence t h a t t h i s  t o a n e a r e s t neighbour  S-Rh  d i s t a n c e equal t o 2.30  i s a r e a s o n a b l e bond d i s t a n c e i s suggested by the  A.  Evi-  average  o  v a l u e s found by X-ray (2.37 X)  c r y s t a l l o g r a p h y i n Rhj^S  [135]; a l s o Rh-S  g e n e r a l l y range  from 2.23  (2.33 A)  d i s t a n c e s i n unhindered t o 2.38  A [136-138J.  and  in  Rh^S^  c o o r d i n a t i o n complexes  O f t e n s t r u c t u r e s from  II  c r y s t a l l o g r a p h y are d i s c u s s e d i n terms o f  ( _ 134J  LEED  II  effective radii  (r  © I T  ) f o r the  -118-  (a) Rh(100)-P(2x2)S 4F  S-Rh D I S T A N C E Figure  4.7:  (A)  Contour p l o t s o f r ^ f o r Rh(100)-p(2x2)-S v e r s u s i n t e r l a y e r s p a c i n g f o r (a) 4F model, model.  and Rh-S  (b) 2F model, and (c) IF  E r r o r b a r s i n d i c a t e s t a n d a r d e r r o r s as d e f i n e d i n  c h a p t e r 2.  -119-  RhdOO)-p(2x2)S 2F o  S-Rh DISTANCE (A )  -120-  -121-  adsorbed  species [139].  By c o n s i d e r i n g Rh as b e i n g unchanged by a d s o r p t i o n o  so t h a t i t r e t a i n s t h e m e t a l l i c r a d i u s o f 1.34  A, an e f f e c t i v e r a d i u s o f S  i s o b t a i n e d by s u b t r a c t i n g the rhodium m e t a l l i c r a d i u s from the Rh-S  nearesto  neighbour  d i s t a n c e , and  T h i s v a l u e can be with it  t h i s g i v e s the v a l u e o f r £^ f o r S equal t o 0.96  compared w i t h o t h e r  values f o r S  (Table 4.2)  LEED c r y s t a l l o g r a p h y f o r a d s o r p t i o n on m e t a l l i c s u r f a c e s .  i s c l e a r that r  A.  deduced  From T a b l e  o f S o b t a i n e d i n t h i s work i s s i m i l a r to v a l u e s  4.2  obtained  eff from some o t h e r s t u d i e s , although o f S to be constant d i f f e r e n t metal coordination Although  i t i s p r o b a b l y not r e a s o n a b l e  i n different.bonding situations  atoms, d i f f e r e n t  s u b s t r a t e dimension  t o expect  r  £^  ( i n v o l v i n g f o r example, and  especially  different  sites). hard sphere r a d i i  (e.g. r  ) have o f t e n been used f o r i n t e r -  p r e t a t i o n s o f s u r f a c e bond d i s t a n c e s , i t would c l e a r l y be p r e f e r a b l e to r e l a t e such d i s c u s s i o n s more c l o s e l y t o the concepts M-X  s u r f a c e bond lengths correspond  o f c o v a l e n t bonding.  i n a good approximation  That  some  to single-bond  v a l u e s i s e s t a b l i s h e d i n T a b l e 4.3 where some comparisons are g i v e n f o r d i s t a n c e s f o r the h e a v i e r chalcogens  on  n a l i z a t i o n s o f such  t h e i r e x t e n s i o n s to o t h e r s u r f a c e systems,  c o r r e l a t i o n s and  have been g i v e n by M i t c h e l l metals  (100)  surfaces of fee metals.  M-X Ratio-  [143,144] based on h y b r i d i z a t i o n schemes f o r  g i v e n by Altmann, Coulson  and Hume-Rothery [145]  v a l e n c i e s and the bond l e n g t h - bond o r d e r r e l a t i o n  and  on  relative  g i v e n by P a u l i n g  The p o i n t o f immediate i n t e r e s t , however, i s t h a t the Rh-S  [133],  bond l e n g t h  found  o  i n the LEED a n a l y s i s o f Rh(100)-p(2x2)-S i s w i t h i n 0.01 v a l u e , thereby  A o f the  i n d i c a t i n g a g e n e r a l c o n s i s t e n c y w i t h s u r f a c e bond  single-bond lengths  T a b l e 4.2:  Effective radii  o f chemisorbed s u l p h u r atoms on v a r i o u s metal  Overlayer System  surface  structure  Bonding site  r  M-S bond distance  surfaces.  o (A)  r  of  r  9 sulphur. (A) e  f  f  References  S/Ni(100)  c(2x2)  4F  2.18  0.94  131  S/Ni(100)  P(2x2)  4F  2.18  0.94  131  S/Ni(110)  p(2x2)  4F  2.17, 2.35  0.93  140  S/Ni(lll)  P(2x2)  3F  2.02  0.78  140  3F  2.28  0.92  147  0.77  123  S/Ir(lll)  (/3x/3)R30°  +  S/Rh(110)  c(2x2)  4F  2.12, 2.45  S/Rh(100)  p(2x2>  4F  2.30  0.96  124  S/Fe(100)  c(2x2)  4F  2.30  1.06  142  +  +  E a c h S atom i s c l o s e r t o a metal atom i n t h e second l a y e r than the atoms i n the f i r s t  layer.  Table 4 . 3 :  Comparisons o f M-X  bond d i s t a n c e s f o r chalcogen atoms adsorbed on  f e e m e t a l s w i t h P a u l i n g s s i n g l e bond lengths  overlayer  bonding  M-X  ( 1 0 0 ) surfaces o f  [133].  distance o  M-X  single  references 0  surface structure  site  c(2x2)  4F  2.18  2.19  131  p(2x2)  4F  2.18  2.19  131  c(2x2)  4F  2.28  2.32  131,140  P(2x2)  4F  2.32  2.32  131,140  c(2x2)  4F  2.59  2.52  131,140,149  p(2x2)  4F  2.52  2.52  131,140  Te/Cu(100)  p(2x2)  4F  2.48  2.54  148  S/Rh(100)  p(2x2)  4F  2.30  2.29  123  S/Ni(100)  Se/Ni(100)  Te/Ni(100)  by LEED (A)  ;  bond l e n g t h (A)  .  -124-  reported This our  from o t h e r  example o f S, Se and Te a d s o r p t i o n  c o r r e l a t i o n had not been r e c o g n i z e d LEED a n a l y s i s f o r Rh(100)-p(2x2)-S Generally  i t is felt  on f e e (100) s u r f a c e .  at t h e time we i n i t i a l l y [123].  t h a t the s u r f a c e s t r u c t u r e r e p o r t e d  Rh(100)-p(2x2)-S gives bond dimensions which are b r o a d l y c r y s t a l l o g r a p h i c data f o r S-Rh bond lengths t i o n o f S atoms on other  surfaces.  published  here f o r  c o n s i s t e n t w i t h X-ray  and w i t h LEED r e s u l t s f o r adsorp-  The l e v e l o f agreement reached between  the c a l c u l a t e d and experimental 1(E) curves i s not complete, and t h e o r i g i n s o f the d e f i c i e n c i e s are p r e s e n t l y unknown. considered complicated present  The number o f model  structures  i n t h e c a l c u l a t i o n f o r t h i s work i s l i m i t e d ; i n p r i n c i p l e more models a r e p o s s i b l e , but s i n c e no c o n f l i c t  seems to be  w i t h the p r i n c i p l e s o f s u r f a c e s t r u c t u r a l c h e m i s t r y , as they are  p r e s e n t l y e v o l v i n g , we do not f e e l t h a t f u r t h e r m u l t i p l e - s c a t t e r i n g c a l c u l a t i o n s on more complex s u r f a c e models a r e r e q u i r e d at t h i s time.  An i n e v i t a b l e  problem w i t h the t r i a l - a n d - e r r o r approach i n LEED c r y s t a l l o g r a p h y  i s that,  however good t h e correspondence may be between experimental and c a l c u l a t e d 1(E) curves f o r a given  s t r u c t u r e , there  the p o s s i b i l i t y t h a t some other b e t t e r agreement.  i s no a b s o l u t e  (untested) s t r u c t u r e s  way o f r u l i n g out  could  Although t h e o r i g i n o f some d i s c r e p a n c i e s  g i v e even between t h e  experimental and c a l c u l a t e d i n t e n s i t i e s found here a r e not y e t c l e a r , we b e l i e v e the r e s u l t s i n d i c a t e t h a t t h e s t r u c t u r e most l i k e l y i n v o l v e s S atoms "adsorbed at 1.3 A above t h e f o u r - f o l d s i t e s o f the Rh(100) s u r f a c e .  -125-  CHAPTER 5 LEED A n a l y s i s o f the R h ( 1 1 0 j - c ( 2 x 2 ) - S S u r f a c e  Structure  -126-  5.1  Introduction Having determined the s u r f a c e geometry f o r s u l p h u r  adsorbed on the  (100)  s u r f a c e o f rhodium, we were i n t e r e s t e d i n comparing w i t h t h e s i t u a t i o n f o r S adsorbed on the more open  (110)  surface.  a n a l y s i s o f t h i s a d d i t i o n a l s t r u c t u r e was two  A second reason  suggested by e a r l i e r r e p o r t s t h a t  d i f f e r e n t a d s o r p t i o n s i t e s are i n d i c a t e d by  atomic a d s o r p t i o n on  (110)  LEED c r y s t a l l o g r a p h y f o r  surfaces of face-centered  atoms are r e p o r t e d to adsorb on the s h o r t - b r i d g e and  ( i m p u r i t y - s t a b i l i z e d ) unreconstructed  adsorb on the c e n t r e  f o r making a LEED  Ir(110)  cubic metals.  Oxygen  s i t e s o f both Ni(110) [151], whereas s u l p h u r  ( f o u r - f o l d ) s i t e s o f Ni(110) [140].  5.2  atoms  I t i s hoped t h a t  an i n v e s t i g a t i o n o f the a d s o r p t i o n o f S on the Rh(110) s u r f a c e may t h e r i n s i g h t s i n t o s u r f a c e chemical  [59]  give  fur-  bonding.  Experimental The  first  rhodium, and t h e s i s and  p a r t of t h i s , study  involved obtaining a clean  (110)  surface of  t h i s f o l l o w e d c l o s e l y the procedures d e s c r i b e d e a r l i e r  i n o t h e r work r e p o r t e d from our  performed on a s i n g l e c r y s t a l s l i c e from Research O r g a n i c / I n o r g a n i c vacuum chamber, the i n i t i a l phosphorus, s u l p h u r  and  . t h e s u r f a c e by a r g o n - i o n  l a b o r a t o r y [109].  in this  T h i s study  cut from a r o d o f p u r i t y 99.99% purchased  Chemical Corp.  A f t e r pumping down i n the  Auger spectrum i n d i c a t e d some contamination  carbon.  was  The  bombardment  S and (1 keV  from  P i m p u r i t i e s c o u l d be removed from at 5 microamps f o r 20  minutes),  but, as p r e v i o u s l y , a r e l a t i v e i n c r e a s e i n the s u r f a c e c o n c e n t r a t i o n o f C indicated. at  300°C.  However, t h i s i m p u r i t y a p p a r e n t l y  was  d i f f u s e d i n t o the b u l k on h e a t i n j  A f t e r s e v e r a l c y c l e s o f ion-bombardment and  annealing,  the  surface  -127-  showed both an e s s e n t i a l l y - c l e a n Auger spectrum ( l x l ) LEED p a t t e r n . f o r t h e cleaned  (100)  ( f i g u r e 5 . 1 ( a ) ) and a sharp  T h i s r e s u l t i n g Auger spectrum i s s i m i l a r t o t h a t (figure 4.2).  s u r f a c e o f rhodium  A f t e r o b t a i n i n g the w e l l - d e f i n e d Rh(110) s u r f a c e , h i g h p u r i t y l ^ S  LEED p a t t e r n c h a r a c t e r i s t i c o f t h e c l e a n  (Mathe-son) was allowed  s u r f a c e by t h e f o l l o w i n g p r o c e d u r e s .  obtained  t o adsorb on t h e  F i r s t t h e sample was heated a t 3 0 0 ° C  f o r 1 minute and U^S was l e t i n t o the vacuum chamber a t t h e p r e s s u r e o f _7 5x10  t o r r f o r 1 minute.  A f t e r pumping out t h e excess gas,  LEED showed a  d i f f u s e r i n g p a t t e r n i n d i c a t i n g t h a t H^S (or S) had adsorbed w i t h o r d e r i n g on t h e s u r f a c e . and  allowed  t o c o o l down.  only  partial  The sample was then heated a t 3 0 0 ° C f o r 3 minutes At t h i s point  LEED i n d i c a t e d t h a t t h e r i n g  pattern  had been r e p l a c e d by t r a c e s o f (^j) s p o t s , c h a r a c t e r i s t i c o f a c(2x2) p a t t e r n , but  t h e spot  i n t e n s i t i e s were weak.  With f u r t h u r h e a t i n g  minutes, LEED showed a s t a b l e and sharp c(2x2) p a t t e r n c o u l d be removed o n l y by argon i o n bombardment. no  other detectable  at 700°C f o r 2  ( f i g u r e 5.2)  The Auger spectrum i n d i c a t e d  i m p u r i t i e s and a r a t i o o f t h e Auger peak  S(152):Rh(302) a p p r o x i m a t e l y equal  t o 3:4  which  heights  (figure 5.1(b)).  For t h e purposes o f beam i n t e n s i t y measurements, two s e t s o f photographs were t a k e n : one a t normal i n c i d e n c e over t h e energy range 22 t o 220 eV and the o t h e r  f o r off-normal  22 t o 160 eV.  incidence  (specifically  The p h o t o g r a p h i c n e g a t i v e s  c o n t r o l l e d V i d i c o n camera as d e s c r i b e d  6= 10°,  were analyzed  i n section 3.4.  cf>=135°[l00]) with  from  t h e computer-  F o r normal  incidence,  1(E) curves were measured f o r 9 i n t e g r a l - o r d e r beams and f o r 5 f r a c t i o n a l order beams u s i n g t h e beam n o t a t i o n i n d i c a t e d i n f i g u r e 5 . 3 .  These a r e  -128-  Energy (eV) F i g u r e 5.1:  Auger s p e c t r a f o r a R h ( l l O )  s u r f a c e when cleaned  c o n t a i n i n g a c(2*2) o v e r l a y e r o f s u l p h u r .  and when  -129-  -  r i gure 5.2:  Photographs  o f LEED p a t t e r n s  observed  from a d s o r p t i o n  o f S on R h ( l l O )  (a)  Rh (110) a t  144 e V ,  (b)  Rh(110)-c(2x2)-S  at  78 e V ,  (c)  Rh(110)-c(2x2)-S  at  102 e V ,  (d)  Rh(110)-c(2*2)-S  at  150 e V .  fo-  at normal  surface.  ^  incidence  -130-  9,  22  1 2 3 3  22  21  11 3 1 2 2  20  10  1 1  F i g u r e 5.3:  01  11  21  Beam n o t a t i o n f o r t h e LEED p a t t e r n from t h e Rh(110)-c(2><2)-S surface structure.  -131-  The  1(E)  (01)  (02)  (03)  (10)  (11)  11 ^22^  13  ^22^  15 ^"22^  31 ^22^  33 ^22^  (12)  (13)  (20)  "  curves f o r the i n t e g r a l - o r d e r beams were found t o be  s i m i l a r to those o f the c l e a n  (110)  changes i n the p o s i t i o n s o f the Rh  The  experimental  similarities  1(E)  i n v o l v e any  atoms from those i n the c l e a n  surface.  curves f o r normal i n c i d e n c e are shown i n f i g u r e  f o r the beams which should be  larger deviations  equivalent  are not  as  (110)  surface.  The  5.4,  close  This  probably  from normal i n c i d e n c e , a l t h o u g h t h e r e may  e x t r a degrees o f roughness f o r the i n t e n s i t y data  pro-  appreciable  as those g e n e r a l l y found from the Rh(100)-p(2*2)-S s t r u c t u r e . indicates  rather  s u r f a c e ; t h i s suggested t h a t the  d u c t i o n o f the Rh(110)-c(2x2)-S s t r u c t u r e d i d not  Typical  (21)  complete s e t s  be of  f o r both d i r e c t i o n s o f i n c i d e n c e are c o l l e c t e d i n Appendices  A5-A6.  5.3  Calculations The  s i m p l e s t models f o r the c(2x2) t r a n s l a t i o n a l symmetry  w i t h atoms adsorbed on an u n r e c o n s t r u c t e d  (110)  surface of a  associated  face-centred  c u b i c metal have a l r e a d y been shown i n f i g u r e 1.8.  These models are  nated a c c o r d i n g  c e n t r e or f o u r - f o l d  t o the s i t e s o f a d s o r p t i o n namely: (IF) model, s h o r t - b r i d g e  bridge  curves f o r the v a r i o u s r e q u i r e d d i f f r a c t e d beams  model.  1(E)  were c a l c u l a t e d u s i n g the The  model and  a  (4F)  model, on-top or o n e - f o l d (2LB)  (2SB)  desig-  long-  l a y e r - d o u b l i n g method f o r a l l o f t h e s e models.  computing times were reduced by  e x p l o i t i n g the symmetry at normal  dence, and by adding the a d s o r b a t e l a y e r s e p a r a t e l y t o b o t h the bottom  inciand  -132-  F i g u r e 5.4:  E x p e r i m e n t a l 1(E) curves f o r two t o be e q u i v a l e n t  s e t s o f beams which are expected  f o r the Rh(110)-c(2x2)-S  structure.  -133-  the t o p o f t h e s u b s t r a t e  s t a c k , a f t e r t h e r e f l e c t i o n and t r a n s m i s s i o n  have converged f o r t h e s u b s t r a t e a l o n e to g i v e d i f f r a c t e d beam i n t e n s i t i e s  (this t y p i c a l l y requires  matrices  8 to 16 layers),  from the 4 F and I F models from a s i n g l e  set o f m u l t i p l e - s c a t t e r i n g c a l c u l a t i o n s ( s i m i l a r l y t h e 2 L B and 2 S B models c o u l d be t r e a t e d t o g e t h e r ) .  A l l f o u r s t r u c t u r a l models c o n s i d e r e d  perpendicular  4 9 symmetrically  mirror planes;  have two  i n e q u i v a l e n t beams were  included  i n t h e c a l c u l a t i o n t o ensure convergence. The  same n o n - s t r u c t u r a l  parameters were used i n the m u l t i p l e - s c a t t e r i n g  c a l c u l a t i o n s on R h ( 1 1 0 ) - c ( 2 x 2 ) - S as f o r t h e a n a l y s i s o f t h e R h ( 1 0 0 ) - p ( 2 x 2 ) - S structure.  S p e c i f i c a l l y the Rh p o t e n t i a l was c h a r a c t e r i z e d by phase s h i f t s from a band s t r u c t u r e c a l c u l a t i o n [ 1 1 0 ] ;  (to 1=1) d e r i v e d the  constant  -12.0  the r e a l part o f  potential  (V ) between t h e atomic spheres was s e t i n i t i a l l y a t or eV; a s u p e r p o s i t i o n p o t e n t i a l [ 1 3 1 ] was used f o r S ; t h e s u r f a c e Debye  temperatures were taken as 4 0 6 and 2 3 6 K f o r Rh and S r e s p e c t i v e l y , w h i l e t h e imaginary p a r t  (V ^) o f t h e constant  p o t e n t i a l between a l l spheres was equ-  1/3  ated  to - 1 . 5 1 E  eV.  The s t r u c t u r a l parameters f o r t h e R h ( 1 1 0 ) - c ( 2 * 2 ) - S  s u r f a c e were s i m p l i f i e d by f i x i n g a l l i n t e r l a y e r s p a c i n g s f o r R h ( 1 1 0 ) a t the b u l k v a l u e clean Rh(110) spacing  (1.345 surface  i s contracted  A); this  follows  our p r e v i o u s  i s not r e c o n s t r u c t e d by o n l y  observations  that the  and t h a t t h e topmost i n t e r l a y e r  3% from t h e b u l k v a l u e  [109,150].  The Rh-S  o  s p a c i n g s were v a r i e d o v e r t h e f o l l o w i n g r a n g e s :  0.65 -  1 . 2 5 A f o r the 4 F  model, 2 . 0 - 2 . 6 A f o r t h e I F model, 1 . 1 - 1.7 A f o r t h e 2 L B model and 1.6  - 2 . 2 A f o r the 2 S B model.  -134-  P r e l i m i n a r y attempts sities  were made t o c a l c u l a t e the d i f f r a c t e d beam i n t e n -  f o r the c o n d i t i o n s measured i n the experiment  dence (6 = 10°, <J>=135°).  Symmetry c o u l d not now  f o r off-normal  inci-  be e x p l o i t e d and hence the  t o t a l number o f beams needed i n the c a l c u l a t i o n i s g r e a t l y i n c r e a s e d over t h a t f o r normal i n c i d e n c e . and we  Around 175 beams would be r e q u i r e d at 200  found t h a t the consequent  computational requirements were too expen-  s i v e f o r us t o proceed w i t h these c a l c u l a t i o n s .  The  e x p e r i m e n t a l data f o r  o f f - n o r m a l i n c i d e n c e has however been c o l l e c t e d i n the  5.4  eV,  appendix.  Results Some comparisons  i n f i g u r e s 5.5  and  o f e x p e r i m e n t a l and c a l c u l a t e d  5.6.  F i g u r e s 5.5(a)-5.5(c)  1(E) curves are g i v e n  compare e x p e r i m e n t a l  1(E)  31 curves f o r the IF, 2SB and  (10), (01) and  (--)  beams w i t h those c a l c u l a t e d f o r the  2LB models f o r v a r i o u s Rh-S  i n t e r l a y e r spacings.  sons show poor agreement f o r the s h o r t - b r i d g e (2SB) models, and w h i l e the on-top  (IF) model produced  f o r the  little  (10) beam, t h e r e was  comparisons  over the complete  best correspondence  X  a reasonable  compari(2LB)  correspondence Visual  range o f data unambiguously i n d i c a t e d t h a t  between the e x p e r i m e n t a l and  ( f i g u r e 5.6).  long b r i d g e  agreement f o r o t h e r beams.  p r o v i d e d by the 4F model w i t h the Rh-S t o 0.85  and  Visual  4F,  calculated  1(E) curves i s  i n t e r l a y e r s p a c i n g i n .the range  D i s c r e p a n c i e s are apparent,  the  0.75  e s p e c i a l l y f o r some  r e l a t i v e peak h e i g h t s , although at the p r e s e n t stage o f development o f LEED c r y s t a l l o g r a p h y the g e n e r a l correspondence II  as  it  good .  can  (we b e l i e v e ) be  classified  -135-  F i g u r e 5.5:  Comparison o f some e x p e r i m e n t a l 1(E) curves from w i t h those c a l c u l a t e d  f o r t h e f o u r s t r u c t u r a l models over a  range o f topmost i n t e r l a y e r s p a c i n g s :  31  Rh(110)-c(2x2)-S  beam, and (c) ( r r ) beam.  (a) (01) beam, (b) (10)  2LB  40  80  (01) beam  120  160  200  lb)  Electron energy (eV)  I—I—I—I—I—I—I—I—I—•  40  80  120 160  200  Electron energy  (eV)  -139-  V A  •A  -i—i—i—i—i—i—i—i  -i—i—i—I—I—i—i—r  —I—1—1—1—1—1—1—1  (20) beam  (II) beam  (10) beam  -  A O . 85 A  ft i i'  • *  i  11 a  40 «>  120  • i i i i i i i -I—I—I—I—I—I—'—I—  i  (12) beam  "c =>  i  i  200  40  i T  i  i  120  i  —  L  200  T—I—I—l—l—l—"—'—  (01) beam  >» k»  o  w  J5  w O  >10 z UJ  120  200 40  40 I l i I—r—i—i I  -i—i—i—i—i—i—i— ~ r  (i  ( I L) beam  —1—1—» 1 1  i  i—i—i—  i—i  l—r—i—i—  (Jf)beam  V| l)beam  |)beam  i  200  A K. / <  A  k 200  i  40  1  120  1  200  i  40  J  i  i1iii 120  1  i i  i I i i i. 120 200  ELECTRON ENERGY (eV) F i g u r e 5.6:  Comparison o f e x p e r i m e n t a l  1(E) curves f o r some i n t e g r a l - and  f r a c t i o n a l - o r d e r beams from Rh(110)-c(2x2)-S w i t h t h o s e f o r t h e 4F model w i t h s u l p h u r e i t h e r 0.75 topmost rhodium  layer.  calculated  o r 0.85 X above t h e  -140-  4F  model  I F model  R h - S spacing (A) F i g u r e 5.7:  Contour p l o t s interlayer  of r  r  f o r Rh(110)-c(2x2)-S v e r s u s V  spacing f o r four d i f f e r e n t  structural  Q r  and  models.  -141-  The  comparisons  between e x p e r i m e n t a l and c a l c u l a t e d 1(E) curves were  a l s o a s s e s s e d by e v a l u a t i n g the r e l i a b i l i t y Jona  [45].  F i g u r e 5.7  s p a c i n g and V  index proposed by Z a n a z z i and  gives contour p l o t s o f r ^ as a f u n c t i o n o f  f o r each o f the f o u r models c o n s i d e r e d here.  i s c l e a r evidence t h a t the c e n t r e (4F) model g i v e s the b e s t between the e x p e r i m e n t a l and c a l c u l a t e d i n t e n s i t i e s . r^ V  = -12.210.8 eV and a Rh-S  correspondence 5.5  correspondence  i t corresponds  i n t e r l a y e r s p a c i n g o f 0.7710.04 A. r  J  other models, r ^ was  Again there  The minimum v a l u e o f  (0.165) r e p r e s e n t s a good l e v e l o f agreement [ 4 5 ] , and  or  Rh-S  to  For the  t>  always s u f f i c i e n t l y l a r g e  (>0.35) to i n d i c a t e a poor  between the experimental and c a l c u l a t e d 1(E) c u r v e s .  Discussion The evidence j u s t p r e s e n t e d i n d i c a t e s t h a t the Rh(110)-c(2x2)-S  t u r e has the s u l p h u r atoms adsorbed  struc-  on the c e n t r e (4F) s i t e s o f the Rh(110)  o  s u r f a c e at about  0.77  A above the topmost rhodium l a y e r .  The  s c a t t e r i n g c a l c u l a t i o n s made here assumed t h a t a l l metal-metal correspond t o the normal b u l k v a l u e s .  t h i s index t o a s s e s s the l e v e l o f correspondence 1(E) curves f o r the beams ( 1 0 ) , ( 0 1 ) , (11) and  was  minimized  at the v a l u e o f 0.22  s p a c i n g o f rhodium b e i n g expanded by j u s t  index r ^ ; we  between the  used  experimental  (12) f o r the o v e r l a y e r s t r u c -  t u r e and those c a l c u l a t e d f o r the c l e a n s u r f a c e . r  distances  T e n t a t i v e evidence i n support i s  p r o v i d e d by an a d d i t i o n a l a n a l y s i s w i t h the r e l i a b i l i t y  ...found r  multiple-  For these c o n d i t i o n s , we  w i t h the topmost  interlayer  1% over the b u l k v a l u e .  -142-  F i g u r e 5.8 sulphur  i n d i c a t e s i n t e r a t o m i c d i s t a n c e s i n the v i c i n i t y o f adsorbed  atoms i n the Rh(110)-c(2x2)-S s t r u c t u r e assuming t h e r e i s no r e l a x a -  t i o n f o r the rhodium s t r u c t u r e .  I t i s apparent t h a t the f o u r - f o l d h o l e i n  the R h ( l l O ) s u r f a c e i s s u f f i c i e n t l y q u i t e deeply;  l a r g e t h a t the s u l p h u r  atom can  i n f a c t s u l p h u r becomes c o n s i d e r a b l y c l o s e r to the rhodium  atom d i r e c t l y below i n the second metal l a y e r than to the f o u r rhodium atoms i n the f i r s t Rhj Similar observations  layer.  -S = 2.12  The  A and  respective distances Rh-S  = 2.45  are  A.  f o r which the  corresponding  are N i j j - S = 2.17  and  A and  Ni -S = 2.35  f o r 0 adsorbed on the Fe(100) s u r f a c e F e - 0 = 2.02  A and  n  Fe^S  A,  ( f i g u r e 5.9) = 2.08  f o r which  i n t e r a c t i o n o f S to the second l a y e r Fe atom [142].  i s too l a r g e to s i n k deeply The  centre  0 appears too s m a l l to adsorb on  case,  the  i n t e r a c t w i t h metal o r b i t a l s d i r e c t e d at t h i s s i t e i n  terms o f h y b r i d i z a t i o n model o f Altmann, Coulson and "Bonding p o s s i b i l i t i e s  In t h i s  S  S chemisorbed on fee(110) s u r f a c e s can p l a u s i b l y  size effects.  (4F) s i t e and  involve signi-  i n t o the f o u r - f o l d h o l e o f the Fe(100) s u r f a c e .  d i f f e r e n c e s between 0 and  be a s s o c i a t e d w i t h  [153]  A.  By c o n t r a s t , a d s o r p t i o n o f S on t h e Fe(100) s u r f a c e does not ficant  neighbouring  have a l s o been made from LEED c r y s t a l l o g r a p h i c a n a l y s e s  f o r S adsorbed on the N i ( l l O ) s u r f a c e [140], distances  penetrate  Hume-Rothery  [145].  f o r 0 seem b e t t e r on the s h o r t - b r i d g e s i t e s  [143].  -143-  F i g u r e 5.9:  I n t e r a t o m i c d i s t a n c e s f o r the s p e c i f i c a t i o n radii  o f hard  i n the neighbourhood o f an oxygen atom i n the o  Fe(100)-(lxl)-O  structure.  ( A f t e r Legg et a l . [ 1 5 3 ] ) .  Distances  i n Angstroms.  sphere  -144-  Th e most s i g n i f i c a n t comparison f o r the new  results  f o r S on R h ( l l O ) i s  w i t h the s t r u c t u r e formed by a d s o r p t i o n o f the same s p e c i e s on Mitchell  [143] has  o f f e r e d a t e n t a t i v e a n a l y s i s o f these s t r u c t u r e s ,  i n d i c a t e d a tendency  it  l a y e r and  atoms i n the topmost l a y e r .  • d i s t a n c e s found  from  and  f o r S t o form a s i n g l e c o v a l e n t bond t o the metal  d i r e c t l y below i n the second metal  Ni(llO).  f o u r 3/4  o r d e r bonds to the  atom  neighbouring  An i n t e r e s t i n g p o i n t i s t h a t w h i l e  the  LEED f o r S on N i ( l l O ) are b r o a d l y c o n s i s t e n t w i t h  this,  i s p h y s i c a l i m p o s s i b l e f o r the c o r r e s p o n d i n g d i s t a n c e s to be s i m u l t a n e o u s l y  s a t i s f i e d f o r S on R h ( l l O ) , and t h i s  i s a d i r e c t consequence o f t h e l o n g e r  Rh-Rh d i s t a n c e compared w i t h the N i - N i d i s t a n c e .  M i t c h e l l concluded  that t h i s  r e s u l t s i n S b e i n g h e l d a t t h a t h e i g h t above the R h ( l l O ) s u r f a c e where the combined s t r e n g t h s o f the f i v e bonds are o p t i m i z e d , and t h i s r e q u i r e s some t  i  l  l  squeezing  o  o f the R h j j - S d i s t a n c e from the s i n g l e bond v a l u e  (2.29  A) i n  order t o get r e a s o n a b l e i n t e r a c t i o n s t o the f o u r Rh atoms i n the f i r s t An important  aspect o f t h i s d i s c u s s i o n i s t h a t i t r e p r e s e n t s a s t a r t on  i z i n g c o v a l e n t bonding  concepts  emphasized hard sphere r a d i i . is  0.77  A; t h i s  can be  f o r chemisorption. The  compared w i t h o t h e r v a l u e s r e p o r t e d from  lography v a r y i n g from 0.78  A t o 1.04  o  A as noted  util-  Most a n a l y s e s so f a r have  e f f e c t i v e r a d i u s i n d i c a t e d f o r S on  o  layer.  i n section  4.5.  Rh(llO)  LEED c r y s t a l -  -145-  CHAPTER 6 S t u d i e s w i t h the Quasidynamical Method  -146-  6.1  Introduction R e l i a b l e s u r f a c e s t r u c t u r e s so f a r r e p o r t e d by  LEED c r y s t a l l o g r a p h y  have come from s t u d i e s which used the t r i a l - a n d - e r r o r approach wherein experimental  I(E) curves are compared w i t h  those c a l c u l a t e d f o r a range o f  p o s s i b l e s u r f a c e models and  a s e l e c t i o n i s made o f the g e o m e t r i c a l  t h a t g i v e s the b e s t  correspondence.  overall  Generally  model  the c a l c u l a t i o n s have  used m u l t i p l e - s c a t t e r i n g methods which are e i t h e r f o r m a l l y exact  (e.g.  the  T-matrix or Bloch-wave methods) or i n v o l v e good i t e r a t i v e approximations the f u l l m u l t i p l e - s c a t t e r i n g methods (e.g. This provides  the o n l y g e n e r a l l y - a c c e p t e d  at the present  time.  Aside  mental measurements, and  methods).  approach t o LEED c r y s t a l l o g r a p h y  from l i m i t a t i o n s s t i l l  l i m i t a t i o n s introduced  the model assumed f o r the p o t e n t i a l and the p r e s e n t  l a y e r - d o u b l i n g or RFS  present  i n the  experi-  i n t o the c a l c u l a t i o n s through  l a t t i c e v i b r a t i o n s , the a c c u r a c y  approach t o LEED c r y s t a l l o g r a p h y i s l i m i t e d e s p e c i a l l y by  a t i o n time and  core s t o r a g e .  s t r y concerns the  to  of  comput-  A s e r i o u s problem f o r s u r f a c e s t r u c t u r a l chemi-  l i m i t a t i o n s s e t on t h i s approach f o r complex s u r f a c e s t r u c -  t u r e s , f o r which m u l t i p l e - s c a t t e r i n g c a l c u l a t i o n s i n e v i t a b l y become p r o h i b i tively  expensive.  which m a i n t a i n  T h i s opens the need to search  r e l i a b i l i t y while reducing  In p r i n c i p l e the s i m p l e s t method [49]  f o r new  c a l c u l a t i o n schemes  the computational burden.  LEED c a l c u l a t i o n i n v o l v e s the  i n which s c a t t e r i n g by  ion-cores  o n l y s i n g l e s c a t t e r i n g events are i n c l u d e d .  kinematical  i s assumed to be weak so The  that  discussion in section  . e s t a b l i s h e s t h a t t h i s method i s inadequate f o r d e c r i b i n g the a c t u a l observed i n the s c a t t e r i n g o f low-energy e l e c t r o n s by  2.1  features  a s o l i d surface.  Attempts  -147-  have been made t o make the k i n e m a t i c a l t h e o r y u s a b l e f o r LEED by p r o c e s s i n g experimental  d a t a such t h a t the m u l t i p l e - s c a t t e r i n g c o n t r i b u t i o n s a r e aver-  aged out and the r e s i d u a l i n t e n s i t i e s can then be a n a l y z e d w i t h t h e k i n e m a t i c II  theory.  These data p r o c e s s i n g procedures  include the  c o n s t a n t momentum  II  t r a n s f e r averaging  it  method i n t r o d u c e d by L a g a l l y et a l . [ 2 5 ] , t h e II  it  averaging  energy  method i n t r o d u c e d by Tucker and Duke [154], and t h e F o u r i e r II  transform cannot  method [157].  Although  a t t r a c t i v e i n p r i n c i p l e , these methods  y e t be c o n s i d e r e d w e l l - e s t a b l i s h e d f o r d e t e r m i n i n g unknown s u r f a c e  structures involving adsorption. A new approximate m u l t i p l e - s c a t t e r i n g scheme f o r c a l c u l a t i n g sities  i s the quasidynamical  i s i n c l u d e d w i t h i n an atomic  method [ 4 6 ] ,  LEED i n t e n -  In t h i s method, o n l y s i n g l e  scattering  l a y e r , w h i l e the i n t e r l a y e r s c a t t e r i n g i s c a l c u -  l a t e d p r o p e r l y , f o r example by t h e RFS method.  The o r i g i n a l authors  proposed  t h a t t h i s approach s h o u l d be most r e l i a b l e f o r s u r f a c e systems i n v o l v i n g  light  atoms i n r e l a t i v e l y open s t r u c t u r e s , where t h e n e g l e c t o f i n t r a l a y e r m u l t i p l e s c a t t e r i n g i s expected  t o be l e s s s e r i o u s .  I n i t i a l a n a l y s e s f o r t h e unrecon-  s t r u c t e d model o f G a A s ( l l O ) and f o r r e c o n s t r u c t e d S i ( 1 0 0 ) gave p r o m i s i n g ment w i t h f u l l m u l t i p l e - s c a t t e r i n g c a l c u l a t i o n s d a t a [46,47] r e s p e c t i v e l y . can g i v e r e a s o n a b l e accounts  [155] and w i t h  agree-  experimental  Such t e s t s i n d i c a t e d t h a t t h e q u a s i d y n a m i c a l  method  o f t h e p o s i t i o n s o f t h e main peaks i n e x p e r i -  mental 1(E) c u r v e s , as w e l l as much o f the secondary  s t r u c t u r e , although t h e  a b s o l u t e i n t e n s i t i e s and t h e r e l a t i v e i n t e n s i t i e s o f n e i g h b o u r i n g peaks are . o f t e n not p r e d i c t e d r e l i a b l y . The purpose o f t h e p r e s e n t study i s t o i n v e s t i g a t e f u r t h e r t h e q u a s i dynamical  method by comparing w i t h experimental  and c a l c u l a t e d 1(E) curves  -148-  a l r e a d y r e p o r t e d i n t h i s t h e s i s , e s p e c i a l l y f o r t h e a d s o r p t i o n systems R h ( 1 0 0 ) - p ( 2 x 2 ) - S and Rh ( 1 1 0 ) - c ( 2 x 2 ) - S .  A p a r t i c u l a r o b j e c t i v e i s to assess  whether t h i s method can i d e n t i f y c e r t a i n s u r f a c e models as g i v i n g i e n t l y poor correspondences  suffic-  w i t h t h e e x p e r i m e n t a l 1(E) c u r v e s t h a t  these  models need not be c o n s i d e r e d i n t h e r e f i n e m e n t s t a g e s o f LEED c r y s t a l l o graphic analyses.  Analyses  f o r the corresponding  clean surfaces  o f rhodium  a r e made, and t h e y p r o v i d e c o n v e n i e n t r e f e r e n c e p o i n t s f o r t h e a d s o r p t i o n systems .6.2  Calculations A fundamental p a r t o f c a l c u l a t i o n s o f LEED i n t e n s i t i e s i n v o l v e s  a t i o n of the layer d i f f r a c t i o n matrices M  ( e q u a t i o n 2.30)  evalu-  f o r each atomic  p l a n e ; then the p l a n e s a r e s t a c k e d i n o r d e r t o determine t h e s c a t t e r i n g from a c r y s t a l slab evaluation of M  (of e i t h e r f i n i t e or s e m i - i n f i n i t e e x t e n t ) .  Generally the  i s t h e most t i m e consuming p a r t o f t h i s whole  s p e c i f i c a l l y because i t i n v o l v e s  calculating Q - X l  - 1  process,  which describes  m u l t i p l e - s c a t t e r i n g events w i t h i n an a t o m i c l a y e r ( e q u a t i o n 2 . 3 0 ) .  all The  q u a s i d y n a m i c a l scheme makes use o f a commonly-found o b s e r v a t i o n , t h a t i n t e r l a y e r m u l t i p l e s c a t t e r i n g i s much s t r o n g e r  than i n t r a l a y e r m u l t i p l e - s c a t t e r i n g  [ 4 6 ] , by e q u a t i n g t h e p l a n a r s c a t t e r i n g m a t r i x  to zero.  gives s u b s t a n t i a l r e d u c t i o n s i n computation times.  This  assumption  The i m p o r t a n t q u e s t i o n now  concerns whether t h e g a i n i n c o m p u t a t i o n a l convenience i s o f f s e t , or n o t , by t o o g r e a t a l o s s o f a c c u r a c y i n t h e c a l c u l a t e d 1(E)  curves.  The p r e s e n t t e s t s w i t h t h e q u a s i d y n a m i c a l method use t h e same t y p e s s u r f a c e models as t h o s e c o n s i d e r e d i n t h e p r e v i o u s s t u d i e s w i t h t h e  full  of  -149-  multiple-scattering calculations [ 1 2 3 , 1 2 4 , 1 5 0 ] .  Thus o n l y t h e r e g u l a r f a c e -  c e n t r e d c u b i c r e g i s t r i e s were c o n s i d e r e d h e r e f o r t h e c l e a n s u r f a c e s , but relaxations  o f t h e topmost i n t e r l a y e r  f o r the S overlayer a r e designated  spacings  were a l l o w e d .  D i f f e r e n t models  as i n f i g u r e s 1 . 8 and 2 . 8 ; a l l Rh-Rh  distances are f i x e d a t the a p p r o p r i a t e bulk values.  Unless  otherwise i n d i c a t e d  here t h e same n o n - s t r u c t u r a l parameters were used i n t h e q u a s i d y n a m i c a l l a t i o n s as i n t h e c o r r e s p o n d i n g previously  (chapters  calcu-  multiple-scattering calculations described  3 - 5 ) . The o n l y m o d i f i c a t i o n s made i n t h i s r e g a r d were  t o t h e c o n s t a n t p o t e n t i a l s between t h e s p h e r i c a l l y - s y m m e t r i c atomic p o t e n t i a l s . The  imaginary  part  (V ^) o f t h i s p o t e n t i a l was f i x e d a t - 6 . 8 eV f o r t h e I F ,  2 S B and 2 L B models o f R h ( 1 1 0 ) - c ( 2 x 2 ) - S whereas t h e energy dependent  form  1/3  V  . = -1.76E  eV was used f o r a l l o t h e r s u r f a c e s c o n s i d e r e d ,  01  except  clean  1/3  R h ( l l O ) f o r which V potential  (  v o r  was r e p r e s e n t e d by - 2 . 0 5 E  ) was f i x e d  eV.  The r e a l p a r t o f t h i s  at - 1 2 eV f o r a l l c a l c u l a t i o n s , although  was e f f e c t i v e l y r e f i n e d d u r i n g comparisons w i t h  experimental  this  value  1 ( E ) curves f o r  each system. Quasidynamical c a l c u l a t i o n s were made f o r normal i n c i d e n c e over t h e energy range 4 0 t o 2 0 8 eV f o r c l e a n R h ( 1 0 0 ) f o r a l l o t h e r systems c o n s i d e r e d h e r e . atomic p l a n e s , these allowed  and over  t h e range 5 0 t o 1 7 8 eV  The RFS method was used f o r s t a c k i n g  c a l c u l a t i o n s were made w i t h 9 1 beams and e l e c t r o n s were  t o t r a v e l through upto 1 2 l a y e r s i n t h e c r y s t a l .  R h ( 1 1 0 ) - c ( 2 x 2 ) - S , i t was n e c e s s a r y  t o combine t h e s u l p h u r  F o r t h e 4 F model o f l a y e r and t h e top-  most rhodium l a y e r as a composite l a y e r because o f t h e i r c l o s e s p a c i n g .  -150-  6.3  R e s u l t s and D i s c u s s i o n  6.3  (a)  R h ( l l O ) and  Experimental  Rh(110)-c(2*2)-S  1(E) curves f o r normal i n c i d e n c e on the c l e a n Rh(110) s u r f a c e  a r e compared w i t h those from q u a s i d y n a m i c a l in  f i g u r e 6.1.  G e n e r a l correspondences  c a l c u l a t i o n s f o r Ad% = 0 and  i n peak p o s i t i o n s are apparent f o r  every p a i r o f c u r v e s , a l t h o u g h r e l a t i v e i n t e n s i t i e s are o f t e n not Comparisons between q u a s i d y n a m i c a l  (QD)  -10%  and m u l t i p l e - s c a t t e r i n g  satisfactory.  (MS)  calculated  1(E) curves are a l s o shown i n the same f i g u r e ; a g a i n major peak p o s i t i o n s match, although the r e l a t i v e i n t e n s i t i e s have changed i n the quasidynamical experimental and c a l c u l a t e d 1(E) curves were a l s o a s s e s s e d w i t h the index r ^ [ 4 5 ] , and T a b l e 6.1  case.  reliability  l i s t s the c o n d i t i o n s f o r b e s t correspondence ( i . e .  minimum r ) between e x p e r i m e n t a l and c a l c u l a t e d s c a t t e r i n g and q u a s i d y n a m i c a l  1(E) curves  (from both m u t i p l e -  c a l c u l a t i o n s ) f o r the v a r i o u s s u r f a c e s c o n s i d e r e d .  The p r e v i o u s m u l t i p l e - s c a t t e r i n g c a l c u l a t i o n s on c l e a n R h f l l O ) i n d i c a t e d the b e s t correspondence 3.3%  The  c o r r e s p o n d i n g a n a l y s i s w i t h the  quasidynamical  method p o i n t s t o a c o n t r a c t i o n o f 10.8%, however d e t a i l e d s t u d i e s o f the 1(E) curves suggested t h a t the index  p a r t i c u l a r purpose.  may  indi-  be l e s s h e l p f u l f o r t h i s •  T h i s c o n c l u s i o n depends on r ^ b e i n g q u i t e s e n s i t i v e t o  r e l a t i v e i n t e n s i t i e s over s u c c e s s i v e p o r t i o n s o f i n d i v i d u a l and the f a c t  that  i s w i t h the topmost i n t e r l a y e r s p a c i n g c o n t r a c t e d by  from the b u l k v a l u e .  vidual  The  (as seen from f i g u r e 6.1  1(E) curves  and noted above f o r G a A s ( l l O ) and  [150], Si(100)  [46]) t h a t the quasidynamical method i s o f t e n u n r e l i a b l e f o r peak magnitudes w i t h i n each 1(E)  curve.  T a b l e 6.1:  Comparisons o f c o n d i t i o n s f o r minimum from e v a l u a t i n g scattering  Surface structure  f o r various surface structures  e x p e r i m e n t a l 1(E) curves w i t h c o r r e s p o n d i n g curves from  calculations  and from quasidynamical  Multiple-scattering Ad%  d _ (A) R h  s  o r  r  r  multiple-  calculations.  Quasidynamical  calculations  V (eV)  obtained  A  d  %  d  Rh-S*  calculations V  or  C e V )  Rh(llO)  -3.3  -11.9  0.12  •10.8  -16.0  0.23  Rh(100)  1.0  •12.8  0.09  3.2  •18.0  0.17  -12.2  0.17  0.83  -24.4  0.23  1.02  •18.0  0.26  0.72  •16.4  0.30  1.32  -21.0  0.28  Rh(110)-c(2x2)-S  0.77  (4F model)  Rh(100)-p(2x2)-S (4F model)  1.30  -13.6  0.26  -152-  -10% ( 0 1 ) beam EXPT  -i—i—i—i—'—r—r  A  ( 1 0 ) beam  A •  \  „  - i — i — i  1—i  /\  iJ  f  _  r — i — i  EXPT  QD MS 1  n  1  1 1  1 r—I  -i  1  1—I  1  ,  ( 0 2 ) beam  x  1 1 1  (02) beam  /  80  120 160 200 240  Electron energy (eV) F i g u r e 6.1:  I  K.J V..--'~-- ^ P T  i  40  1  l  Comparison o f e x p e r i m e n t a l  40  i—i—r  80  120 160 200 240  Electron energy (eV) 1(E) curves f o r normal i n c i d e n c e on  R h ( l l O ) w i t h those c a l c u l a t e d w i t h t h e q u a s i d y n a m i c a l method and t h e f u l l m u l t i p l e - s c a t t e r i n g method when t h e topmost  inter-  l a y e r s p a c i n g equals t h e b u l k v a l u e (0%) and when i t i s c o n t r a c t e d by 10%.  -153-  1(E) curves f o r d i f f e r e n t models o f t h e Rh(110)-c(2x2)-S s u r f a c e c a l c u l a t e d by t h e quasidynamical  (QD) method were compared, by d i r e c t o b s e r v a t i o n ,  w i t h t h e e x p e r i m e n t a l 1(E) curves and a l s o w i t h t h e c o r r e s p o n d i n g curves c a l c u l a t e d with the m u l t i p l e - s c a t t e r i n g  (MS) method f o r t h e 4F model w i t h t h e o  topmost Rh-S i n t e r l a y e r s p a c i n g ( d _ g ) equal t o 0.75 A.  Overall  R h  difficult  i t was  t o p i n - p o i n t t h e s t r u c t u r a l model from t h e quasidynamical  calculation  which g i v e s t h e b e s t agreement w i t h t h e e x p e r i m e n t a l c u r v e s ; i n p a r t t h i s was because o f t h e e f f e c t s o f e r r o r s i n r e l a t i v e i n t e n s i t i e s f o r s u c c e s s i v e portions o f the c a l c u l a t e d  1(E) c u r v e s .  Also there are systematic s h i f t s  p o s i t i o n s f o r t h e q u a s i d y n a m i c a l l y - c a l c u l a t e d 1(E) c u r v e s .  i n peak  However i t d i d  seem p o s s i b l e t o c o n c l u d e , from t h e v i s u a l a n a l y s i s , t h a t t h e b e s t match w i t h the 1(E) curves from t h e f u l l  m u l t i p l e - s c a t t e r i n g c a l c u l a t i o n s occurred f o r o  the 4F model i n t h e quasidynamical  c a l c u l a t i o n s w i t h d ^ _ - 1.15 A. R  C o n c l u s i o n s on c o n d i t i o n s f o r correspondence c a l c u l a t e d and e x p e r i m e n t a l of  Z a n a z z i and Jona.  0T  between q u a s i d y n a m i c a l l y -  1(E) curves were a i d e d w i t h t h e r e l i a b i l i t y  Two-dimensional contour p l o t s o f r r  V >  g  Comparisons o f con-  t h a t t h e 4F model g i v e s t h e lowest r ^ v a l u e  l o c a l minima a r e found minima i n T  versus d , and . Rh-S n  r  f o r each s t r u c t u r a l model, a r e shown i n f i g u r e 6.2.  t o u r p l o t s suggest  index  (0.23).  No  f o r t h e 2SB model whereas, f o r t h e 2LB and IF models,  occur with r a t h e r high values  models a r e l e s s p r o b a b l e .  (>0.42) which suggests t h a t  these  The contour p l o t s o f r ^ f o r t h e 4F and 2LB models  show t h e common f e a t u r e o f e x h i b i t i n g more than one l o c a l minimum  ( f i g u r e 6.2). o  For t h e 4F model, t h e f i r s t and V _ nr  minimum  = -24.4 eV, t h e second  (with r ^ = 0.23) occurs f o r d ^ R  g  = 0.83 A  (with a s l i g h t l y h i g h e r v a l u e o f r , v i z . 0.28)  -154-  F i g u r e 6.2:  Contour p l o t s  of r  r  f o r Rh(110)-c(2x2)-S versus V  Q r  and the Rh-S  i n t e r l a y e r s p a c i n g f o r f o u r d i f f e r e n t s t r u c t u r a l models w i t h the q u a s i d y n a m i c a l  method.  calculated  -155-  o  occurs w i t h d_, „ = 1.02 Rh-S at  d,,, „ = 0.72 Rh-S  A and V  A and V  or  or  = -16.4  = -18.0 eV  eV and  the t h i r d  (Table 6.1).  _  (r  r  = 0.30)  occurs  T h i s s i t u a t i o n i s t o be  "compared w i t h a s i n g l e minimum f o r the c o r r e s p o n d i n g c o n t o u r p l o t o f r ^ f o r the same system when the c a l c u l a t i o n s u t i l i z e the f u l l  multiple-scattering  o  procedures « r  r  = 0.17 In  to  i n t h i s case d^  ( f i g u r e 5.7); (Table  ^ = 0.77  A, V  = -12.2  eV  and  6.1).  p r i n c i p l e the e x i s t e n c e o f more than one  l o c a l minimum c o u l d r e l a t e  m u l t i p l e - c o i n c i d e n c e s i n a d s o r b a t e - s u b s t r a t e s p a c i n g s as d i s c u s s e d by  Andersson and Pendry [ l 5 6 j .  However, a g a i n s t t h i s p o s s i b i l i t y a r e the  fol-  lowing o b s e r v a t i o n s : i)  no  such e f f e c t was  scattering ii)  calculation  d e t e c t e d i n the p r e v i o u s a n a l y s i s w i t h the m u l t i p l e ( f i g u r e 5.7),  and  v i s u a l a n a l y s i s o f the i n d i v i d u a l  1(E) curves c a l c u l a t e d w i t h the q u a s i o  dynamical  method f o r the s p a c i n g s 0.75,  0.85  s a t i s f a c t o r y than t h o s e c a l c u l a t e d f o r 1.15 Two of  e f f e c t s seem t o be i n v o l v e d h e r e . the q u a s i d y n a m i c a l  method, and  The  o  and  1.05  A a r e on b a l a n c e  less  A. first  the second  concerns  appears  the incomplete  nature  t o be a s s o c i a t e d w i t h  the r e l i a b i l i t y - i n d e x a n a l y s i s b e i n g l e s s r e l i a b l e f o r a s s e s s i n g i n t e r l a y e r s p a c i n g s when the r e l a t i v e i n t e n s i t i e s o f s u c c e s s i v e p o r t i o n s o f i n d i v i d u a l 1(E) curves a r e not c a l c u l a t e d c o r r e c t l y , even though a r e a s o n a b l e match i n p o s i t i o n s o f s t r u c t u r e may calculated  1(E)  curves.  still  be r e c o g n i z e d between the experimental  and  -156-  For quasidynamical c a l c u l a t i o n s associated with values o f V  or  f o r t h e 4F model, minima i n r ^ a r e  i n t h e range  -16.4 t o -24.4 eV.  These v a l u e s  b  a r e s u b s t a n t i a l l y changed from t h e v a l u e o f -12.2 eV r e p o r t e d from t h e multiple-scattering calculations. the i n d i v i d u a l  T h i s shows up i n t h e v i s u a l a n a l y s i s o f  1(E) c u r v e s ; f e a t u r e s from t h e q u a s i d y n a m i c a l  occur on average a t about  6 eV lower i n energy  than do t h e c o r r e s p o n d i n g  f e a t u r e s from t h e m u l t i p l e - s c a t t e r i n g c a l c u l a t i o n s . shift  calculations  T h i s need f o r a s y s t e m a t i c  i n t h e 1(E) curves must be a s s o c i a t e d w i t h t h e n e g l e c t o f i n t r a l a y e r  m u l t i p l e - s c a t t e r i n g i n the quasidynamical have a l s o been observed  calculations.  f o r the quasidynamical  S i m i l a r changes i n  calculations of Rh(llO)  (Table 6.1) and o f S i (100) [ 4 7 ] . 33 F i g u r e 6.3 compares e x p e r i m e n t a l  1(E) curves f o r t h e (01) and (--) d i f f -  r a c t e d beams w i t h t h e c o r r e s p o n d i n g q u a s i d y n a m i c a l l y - c a l c u l a t e d 1(E) curves f o r p a r t i c u l a r geometries  o f t h e f o u r d i f f e r e n t s t r u c t u r a l models.  A l s o shown  are t h e c o r r e s p o n d i n g 1(E) curves c a l c u l a t e d by t h e m u l t i p l e - s c a t t e r i n g method f o r d ^ g = 0.75 A.  F o r these two r e p r e s e n t a t i v e beams, q u a s i d y n a m i c a l  calcu-  l a t i o n f o r t h e 2SB and 2LB models do not show any agreement w i t h t h e e x p e r i mental  1(E) c u r v e s .  Significant  l e v e l s o f agreement a r e apparent  f o r both  beams f o r t h e 4F model, whereas f o r t h e IF model t h e q u a s i d y n a m i c a l 33  calculation  g i v e s some r e a s o n a b l e agreement f o r t h e (yr) beam b u t l i t t l e agreement f o r the  (01) beam.  These comparisons  emphasize t h e matching  when a l l a v a i l a b l e d a t a from t h e q u a s i d y n a m i c a l the 4F model appears b o t h experiment  o f peak p o s i t i o n s ;  c a l c u l a t i o n s are considered  t o g i v e t h e b e s t correspondence  w i t h 1(E) curves  and from t h e r e f e r e n c e m u l t i p l e - s c a t t e r i n g  from  calculations.  -157-  fth (IIO)-c(2*2)-S (  3|)b«am  EXPT  ........ EXPT t  :  j ; • *•  •;  *\  A  '' •  A  OD 2LB.I5 A  \.-v :  ly  ; \  \ \  /  \  \  * \!\  i •  f:  \  ''•  i\ /  OD . 2SB ,1 6A  OD . ,2LB,I.5A  \  ^  {'•  I  .*> »  /v \ v  ; /  oD \2SB,I.6A  "A  •  \.. •-.  1  \\ •  '.  /  i \ •.  \  '•  *\ \  /  :  1 \  \  IF , 2.2A  QD IF, 2.2A]  / •"'  *  OD 4F,I 15 A \  •"•  MS 4F 0.75 A  120  •ncrgy ( « V )  t o o  40  • n t r g y («V )  33 F i g u r e 6.3:  Comparison o f 1(E) curves measured f o r the (01) and (-^j) d i f f r a c t e d beams f o r normal i n c i d e n c e on Rh(110)-c(2x2)-S w i t h  those  c a l c u l a t e d by the q u a s i d y n a m i c a l method and by the f u l l  multiple-  s c a t t e r i n g method f o r the f o u r s t r u c t u r a l models d e s c r i b e d i n text.  -158-  Table 6.2:  A demonstration o f the correspondence between peak p o s i t i o n s i n 1(E) curves c a l c u l a t e d w i t h the q u a s i d y n a m i c a l method f o r the f o u r models o f Rh(110)-c(2*2)-S  at the s p e c i f i e d S-Rh  s p a c i n g and those g i v e n by experiment f u l l multiple-scattering beam, the denominator i n the r e l e v a n t  calculations.  In the e n t r i e s  f o r each  s p e c i f i e s the number o f s i g n i f i c a n t peaks or from the  c a l c u l a t i o n s , and the numerator  number o f those peaks that quasidynamical  and by the c o r r e s p o n d i n g  1(E) curve from experiment  multiple-scattering  interlayer  are matched to w i t h i n  full  g i v e s the  7 eV by the  calculations.  S-Rh=1.6A 2SB F u l l MS Expt  2 LB Expt  S-Rh=1.5A F u l l MS  Beam  S-Rh=1.15A 4F F u l l MS Expt  S-Rh=2.2A IF F u l l MS Expt  (01)  2/4  3/5  2/4  3/5  2/4  1/5  1/4  1/5  (02)  2/3  3/4  2/3  3/4  1/3  3/4  1/3  2/4  (03)  0/1  1/3  0/1  1/3  0/1  1/3  1/1  1/3  (10)  3/5  4/5  3/5  5/5  1/5  2/5  3/5  4/5  (11)  3/4  2/2  2/4  2/2  2/4  2/2  2/4  1/2  (12)  3/4  4/4  1/4  2/4  2/4  2/4  1/4  3/4  (13) •  0/2  0/1  0/2  1/1  0/2  0/1  0/2  0/1  (20)  1/2  2/4  0/2  2/4  1/2  3/4  0/2  1/4  (21)  1/1  2/2  0/1  1/2  0/1  1/2  1/1  1/2  (hh)  2/2  2/2  1/2  2/2  1/2  1/2  2/2  2/2  (h 3/2)  2/4  1/4  3/4  2/4  3/4  2/4  1/4  1/4  (h  5/2)  2/2  4/4  0/2  2/4  2/2  3/4  2/2  3/4  h)  2/2  2/3  1/2  1/3  2/2  1/3  1/2  1/3  4/4  0/2  1/4  1/2  2/4  1/2  2/4  34/47  15/38  28/47  18/38  24/47  17/38  23/47  (3/2  (3/2 3/2)1/2  Total  24/38  -159-  F i g u r e 6.4:  Comparisons o f some e x p e r i m e n t a l  1(E) curves f o r f r a c t i o n a l -  o r d e r beams f o r normal i n c i d e n c e on R h ( 1 1 0 ) - c ( 2 x 2 ) - S and Rh(100)-p(2x2)-S  with those c a l c u l a t e d  f o r the centre  a d s o r p t i o n s i t e s w i t h t h e q u a s i d y n a m i c a l method and w i t h t h e full  multiple-scattering  method.  The topmost Rh-S  intero  layer spacings i n the quasidynamical and  1.3 A forRh(110)-c(2*2)-S  calculations  are 1 . 1 5 A  and Rh ( 1 0 0 ) - p ( 2 x 2 ) - S  the c o r r e s p o n d i n g v a l u e s f o r the m u l t i p l e - s c a t t e r i n g o  o  are 0 . 7 5 A and 1 . 3 A.  respectively; calculations  -160-  Rh(100)-p(2x2)-S  Rh(HO)-c(2x2)-S  (O^)beam VA-V'^EXPT  EXPT  A (^|)beom  (l^)beam vEXPT  (^)beam  £|)beom  - EXPT  V'  (fl)beom  (Ol)beom \ /v EXPT v  40  80 120 160 200  Electron energy (eV)  40  " i — i — i — i  80  i  120 160 200  Electron energy (eV)  -161-  Evidence i s p r o v i d e d i n T a b l e 6.2, where the d e t a i l s i n matching p o s i t i o n s f o r each beam and f o r each model a r e summarized. peak p o s i t i o n s o f up to 7 eV was modate v a r i a t i o n s o f V  o f peak  A spread i n  a l l o w e d i n t h i s matching  i n o r d e r to accom-  f o r the d i f f e r e n t s u r f a c e models. or  From t h e comparisons  i n d i c a t e d i n T a b l e 6.2,  i t appears  f o r the 4F model  t h a t the l e v e l o f agreement, between the q u a s i d y n a m i c a l c a l c u l a t i o n s e i t h e r experiment  o r f u l l m u l t i p l e - s c a t t e r i n g c a l c u l a t i o n s , i s b e t t e r f o r the  f r a c t i o n a l - o r d e r beams than f o r the i n t e g r a l - o r d e r beams. t i o n was  and  A similar  observa-  a l s o r e p o r t e d by Tong and Maldonado f o r the S i ( 1 0 0 ) s u r f a c e [ 4 7 ] ,  F i g u r e 6.4  (a) d e t a i l s some s p e c i f i c 1(E) curves f o r the  fractional-order  beams c a l c u l a t e d w i t h the quasidynamical method f o r R h ( 1 1 0 ) - c ( 2 x 2 ) - S , compares w i t h those from experiment 6.3  (b)  and from m u l t i p l e - s c a t t e r i n g  and  calculations.  Rh(100) and Rh (100)-p(2x2)-S  The p r e v i o u s a n a l y s i s o f LEED i n t e n s i t i e s from Rh(100), based on  multiple-  s c a t t e r i n g c a l c u l a t i o n s and the use o f the r e l i a b i l i t y - i n d e x r , i n d i c a t e d t h a t the topmost i n t e r l a y e r s p a c i n g i s v e r y c l o s e to the b u l k v a l u e , t h e r e b e i n g a s u r f a c e l a y e r c o n t r a c t i o n o f about  1% [ 4 3 , 1 5 0 ] .  A similar  analysis  made here w i t h beam i n t e n s i t i e s c a l c u l a t e d w i t h the quasidynamical method a l s o suggests a s m a l l c o n t r a c t i o n , t h i s time by 3%  (Table 6.1).  f o r c l e a n u n r e c o n s t r u c t e d Rh(100) a p p r e c i a b l e correspondence  F i g u r e 6.5  between peaks i n  1(E) curves c a l c u l a t e d w i t h the q u a s i d y n a m i c a l method and those from experiment mental  or m u l t i p l e - s c a t t e r i n g c a l c u l a t i o n s .  either  w i t h the e x p e r i -  1(E) c u r v e s , the q u a s i d y n a m i c a l l y - c a l c u l a t e d 1(E) curves needed  to loweii energy by a p p r o x i m a t e l y 6 eV. mized  In matching  indicates  at V  = -18.0 or  eV.  shifting  This i s consistent with r ^ being mini-  energy (eV ) Comparisons o f some e x p e r i m e n t a l  e nerg y (eV ) 1(E) curves f o r normal  on Rh(100) w i t h those c a l c u l a t e d w i t h the quasidynamical and w i t h the f u l l m u l t i p l e - s c a t t e r i n g  method.  incidence method  -163-  For Rh(100)-p(2x2)-S, p r e v i o u s a n a l y s i s w i t h the m u l t i p l e - s c a t t e r i n g calculations d  Rh  S  ^ §i  =  curves  ( s e c t i o n 4.4) v e s  (Table 6.1).  a l s o produced  p o i n t e d t o the c o n c l u s i o n t h a t t h e 4F model w i t h  tbe b e s t correspondence  w i t h the e x p e r i m e n t a l  In t h i s e a r l i e r a n a l y s i s we  values  noted t h a t the 2F model  a minimum r ^ which i s comparable w i t h t h a t from t h e 4F model.  S i m i l a r a n a l y s e s here w i t h the q u a s i d y n a m i c a l ponding  1(E)  f e a t u r e s ; b o t h 4F and ( f i g u r e 6.6a)  calculations highlight  corres-  2F models g i v e l o c a l minima w i t h comparable r ^  a l t h o u g h no minimum i s found  f o r the IF model.  With  o  quasidynamical n  calculations, r  r  i s minimized  at d„. „ = 1.32 Rh-S  A and V  or  = -21.0 o  f o r the 4F model, whereas f o r the 2F model the c o r r e s p o n d i n g v a l u e s a r e 1.70 and  -12.2  eV  respectively.  To a s s e s s t h i s curves and  f u r t h e r we made a v i s u a l  evaluated r  e v a l u a t i o n o f the i n d i v i d u a l  j u s t f o r the f r a c t i o n a l - o r d e r beams.  r are expected  A  The  I(E)  l a t t e r beams  J  to be e s p e c i a l l y a s s o c i a t e d w i t h the adsorbate l a y e r and T a b l e  notes f o r Rh(110)-c(2x2)-S  t h a t the q u a s i d y n a m i c a l method appears  6.2  to work  b e t t e r f o r the f r a c t i o n a l - o r d e r beams than f o r the i n t e g r a l - o r d e r beams. F i g u r e 6.6(b) shows contour p l o t s o f r ^ f o r Rh(100)-p(2x2)-S, from q u a s i dynamical  c a l c u l a t i o n s , where o n l y the f r a c t i o n a l o r d e r beams are i n c l u d e d i n  the comparison  with.experiment.  Both the 4F and  2F models g i v e d e f i n i t e  minima i n the contour p l o t s , although the minimum v a l u e o f r ^ f o r the 4F model (with d . „ = 1.34 A, V = -21.0 eV) i s now c l e a r l y b e t t e r than t h a t from Rh-S or v  n  the 2F model  (with d , Rh-S n  = 1.91  model from the q u a s i d y n a m i c a l  A and V  or  = -27.2  eV).  Support r  f o r the 4F  r  c a l c u l a t i o n i s p r o v i d e d by the o b s e r v a t i o n t h a t  the v a l u e s o f dr,, „ and V which g i v e minimum r Rh-S or r b  from the  fractional-order  eV  -164-  F i g u r e 6.6:  Contour Rh-S  plots  o f r ^ f o r Rh (100) -p (2x2)-S v e r s u s  and the  i n t e r l a y e r s p a c i n g f o r the 4F and 2F s t r u c t u r a l models  c a l c u l a t e d by t h e q u a s i d y n a m i c a l method:  (a)  w i t h a l l i n t e g r a l - and f r a c t i o n a l - o r d e r beams; sons w i t h f r a c t i o n a l - o r d e r beams o n l y .  comparisons (b) compari-  -165Rhl100)-P(2x2J-S (•)  intagralffractional  ; Ouaatdynamical  calculation  C°) fractional only  -166-  Rh HOO)-p(2x2)-S  I J L i beam  (o 1 ) beam  (ii)  A' • EjXPT  EXPT  t  : A •  *;• »  • ; :  3  QD  /•:  \  / \  >»  • •  "V./  2F , 1.8 A  '••  \  i  QD  QD IF , 2.2A  w CO  QD 2 F, 1.8 A  IF , 2.2 A  QD •  4F , 1.3 A  MS  4FJ3A  4 F , 1.3 A  40  120  4FJ.3A  200  2ffo  energy (eV )  energy ( eV )  13 F i g u r e 6.7:  Comparisons o f 1(E) curves measured  f o r t h e (01) and  (^j) d i f f r a c t e d  beams f o r normal i n c i d e n c e on Rh(100)-p(2*2)-S wi-th those by t h e q u a s i d y n a m i c a l method and by t h e f u l l method f o r t h r e e p o s s i b l e  s t r u c t u r a l models.  calculated  multiple-scattering  -167-  Tab1e 6 . 3 :  A d e m o n s t r a t i o n o f t h e correspondence between peak p o s i t i o n s i n 1 ( E ) curves c a l c u l a t e d w i t h t h e q u a s i d y n a m i c a l method f o r t h e f o u r models o f R h ( 1 0 0 ) - p ( 2 x 2 ) - S  a t t h e s p e c i f i e d S-Rh i n t e r l a y e r  s p a c i n g and t h o s e g i v e n by experiment full  multiple-scattering  beam, t h e denominator  calculations.  In t h e e n t r i e s  multiple-scattering  a r e matched t o w i t h i n  gives the  7 eV by the  calculations.  O  O  IF S--Rh=2.2A F u l l MS Expt  O  2F Expt  S-Rh=1.8A F u l l MS  Beam  AF Expt  S-Rh=l.3A F u l l MS  (01)  2/2  1/2  2/2  1/2  1/2  2/2  (ID  2/2  1/3  2/2  2/3  2/2  2/3  (02)  1/1  1/1  0/1  0/1  1/1  1/1  (12)  1/1  1/1  1/1  1/1  1/1  1/1  (hh)  2/3  2/4  1/3  1/4  1/3  1/4  (h 3/2)  2/3  3/4  2/3  3/4  1/3  1/4  (0  h)  4/5  4/5  2/5  1/5  2/5  1/5  (l  h)  2/4  2/4  2/A  1/4  3/4  2/4  (0 3/2)  3/3  A/5  2/3  0/5  2/3  3/5  (3/2 3/2)  -  2/2  -  1/2  0/2 '  19/24  21/31  1A/2A  14/31  peaks  o r from t h e f u l l  c a l c u l a t i o n s , and t h e numerator  number o f t h o s e peaks t h a t  Total  f o r each  s p e c i f i e s t h e number o f s i g n i f i c a n t  1 ( E ) c u r v e from experiment  i n the relevant  quasidynamical  and by t h e c o r r e s p o n d i n g  14/24  14/31  -168-  beams a l o n e a r e v e r y s i m i l a r t o those from the combination o r d e r and r  i n t e g r a l - o r d e r beams.  are very d i f f e r e n t  situations  t h e n , we b e l i e v e t h a t the q u a s i d y n a m i c a l model g i v e s the b e s t correspondence r  F i g u r e 6.7  n L  r  Rh-S  fractional-  By c o n t r a s t , the c o n d i t i o n s f o r minimum  i n t h e s e two  Rh(100)-p(2x2)-S w i t h d  of  = 1.32  from the 2F model.  Overall,  c a l c u l a t i o n i n d i c a t e s t h a t the 4F  w i t h experimental A and V  or  = -21.0  1(E) curves f o r eV.  compares q u a s i d y n a m i c a l l y - c a l c u l a t e d 1(E) curves f o r the  13 (01) and  (^j) beams o f Rh(100)-p(2x2)-S w i t h those from experiment  from m u l t i p l e - s c a t t e r i n g c a l c u l a t i o n s . are apparent  f o r a l l models w i t h the 13  b e s t match f o r the  (^j) beam.  i n peak p o s i t i o n s  (11) beam, but the 4F model shows the  D e t a i l s o f comparisons  curves are summarized i n T a b l e 6.3. matching  Correspondences  and  of individual  1(E)  A g a i n t h i s t a b l e shows t h a t the b e s t  f o r the f r a c t i o n a l - o r d e r beams i s p r o v i d e d by the 4F model.  a c t u a l 1(E) curves a r e i l l u s t r a t e d 6.4 C o n c l u d i n g Remarks  in figure  The r e s u l t s p r e s e n t e d i n T a b l e 6.2  and  (Some  6.4(b)).  6.3  i n d i c a t e t h a t a d s o r p t i o n o c c u r s i n the 4F s i t e s  f o r the q u a s i d y n a m i c a l method f o r both  Rh(110)-c(2x2)-S  and Rh(100)-p(2x2)-S; c o m f o r t i n g l y t h e s e are j u s t the a d s o r p t i o n s i t e s c a t e d by the f u l l m u l t i p l e - s c a t t e r i n g c a l c u l a t i o n s . the quasidynamical  indi-  For Rh(100)-p(2x2)-S  c a l c u l a t i o n , i n c o n j u n c t i o n w i t h the Zanazzi-Jona  reli-  o  ability-index r  , i n d i c a t e s a topmost i n t e r l a y e r s p a c i n g o f 1.32  A, i n v e r y  o  c l o s e agreement w i t h t h a t  (1.30 A) from the m u l t i p l e - s c a t t e r i n g  calculation  (Table 6.1); however the s i g n i f i c a n c e o f t h i s c l o s e correspondence  must be  -169-  tempered by the a p p r e c i a b l e d i s c r e p a n c i e s found f o r both c l e a n R h ( l l O ) and Rh(110)-c(2*2)-S.  In g e n e r a l , t h e index r ^ seems l e s s r e l i a b l e f o r a s s e s s i n g  i n t e r l a y e r s p a c i n g s and V  Q r  from the quasdynamica'l  s i n c e t h i s method can be erroneous  calculations,  f o r c a l c u l a t i n g r e l a t i v e peak  especially intensities  over s u c c e s s i v e p o r t i o n s o f 1(E) c u r v e s . Comparisons i n f i g u r e 6.4 curves a r e e i t h e r absent  show t h a t some peaks i n e x p e r i m e n t a l  i n the q u a s i d y n a m i c a l l y - c a l c u l a t e d curves o r a r e  r e p r e s e n t e d o n l y by s h o u l d e r s . of  In p a r t the l a t t e r may  the r e l a t i v e l y l a r g e v a l u e s o f V  r e p r e s e n t a consequence  t h a t are needed i n our  c a l c u l a t i o n s to avoid occasional d i f f i c u l t i e s  in  quasidynamical  convergence.  Although the q u a s i d y n a m i c a l method c l e a r l y i s not e x a c t , we l e s s encouraged  1(E)  by our o b s e r v a t i o n s f o r Rh(110)-c(2x2)-S  and  are neverthe-  Rh(100)-p(2x2)-S  t h a t i t i s a b l e t o s e l e c t the c o r r e c t a d s o r p t i o n s i t e s as p r o v i d i n g the most l i k e l y models f o r t h e s e s u r f a c e s . for  Moreover the c a l c u l a t i o n s here were made  a metal which i s a r e l a t i v e l y s t r o n g s c a t t e r e r and t h e r e f o r e does not  correspond t o the s i t u a t i o n s f o r which the q u a s i d y n a m i c a l method was ially of  judged t o be most h e l p f u l .  These o b s e r v a t i o n s support the  possibility  u s i n g the q u a s i d y n a m i c a l method f o r making p r e l i m i n a r y assessments  t r i a l models t h a t need more d e t a i l e d a n a l y s e s w i t h f u l l  applicable.  of scat-  g e o m e t r i c a l types f o r which t h i s c o n c l u s i o n may  I f such ranges  o f those  multiple-scattering  methods, a l t h o u g h f u r t h e r t e s t s a r e needed t o d e l i n e a t e the ranges t e r i n g s t r e n g t h s and  init-  be  can be o b t a i n e d , then t h i s would c l e a r l y p r o v i d e  a most s i g n i f i c a n t r o l e f o r t h e ' q u a s i d y n a m i c a l method i n LEED c r y s t a l l o g r a p h y . In  any  event t h i s method s h o u l d have v a l u e i n making p r e l i m i n a r y assessments  -170-  of adsorption  systems which i n v o l v e w e a k l y - s c a t t e r i n g  adsorbates a t low  coverage, p a r t i c u l a r l y where t h e number o f f r a c t i o n a l - o r d e r beams i s l a r g e and t h e c o n v e n t i o n a l m u l t i p l e - s c a t t e r i n g procedures r a p i d l y become i n t r a c t a b l e .  -171-  REFERENCES 1.  G.A. S o m o r j a i , " P r i n c i p l e s o f S u r f a c e Chemistry", P r e n t i c e H a l l , Englewood C l i f f , New J e r s e y (1972).  2.  Abdus Salam, ed. " S u r f a c e S c i e n c e " , L e c t u r e s P r e s e n t e d at an I n t e r n a t i o n a l Course a t T r i e s t e o r g a n i z e d by the I n t e r n a t i o n a l Centre f o r T h e o r e t i c a l P h y s i c s , T r i e s t e , I n t e r n a t i o n a l Atomic Energy Agency, Vienna  (1975).  3.  J.M. B l a k e l y , " I n t r o d u c t i o n t o the P r o p e r t i e s o f C r y s t a l S u r f a c e s " Pergamon, New York (1973).  4.  S. Andersson, S u r f a c e S c i . , 18, 325  5.  R. Vanselow and S.Y. Tong, "Chemistry and P h y s i c s o f S o l i d S u r f a c e " CRC P r e s s , Inc. C l e v e l a n d , Ohio (1977).  6.  E.W. J.R.  7.  S. Ino, Japanese J . A p p l . Phys.  8.  H.H.  (1969).  Plummer and T. G u s t a f s s o n , S c i e n c e 198, 165 (1977); S c h r i e f f e r and P. Soven, P h y s i c s Today 28(4), 24 (1975). 16, 891  (1977).  Brongersma and J.B. Theeten, S u r f a c e S c i . 54, 519  J . F . Van 79, 219  der Veen, R.G.  Smeenk, R. M.  (1976);  Tromp and F. S a r i s , S u r f a c e S c i .  (1979) .  9.  M.J.  C a r d i l l o and G.E.  Becker, Phys. Rev.  10.  H.P.  B o n z e l , S u r f a c e S c i . 68, 236  11.  K. Baron, D.W.  12.  C.J. D a v i s s o n and L.H.  13.  P. Auger, J . Phys. Radium 6, 205  (1925).  14.  J . J . Lander, Phys. Rev. A l , 1382  (1953).  15.  L.N.  Tharp and E . J . S c h e i b n e r , J . A p p l . Phys.  16.  R.E.  Weber and W.T.  17.  P.W.  Palmberg  18.  C.R.  B r u n d l e , J . Vac. S c i . Techn.  19.  H. Ibach i n " E l e c t r o n S p e c t r o s c o p y f o r S u r f a c e A n a l y s i s " , T o p i c s i n C u r r e n t P h y s i c s V o l . 4, ed. H. Ibach, S p r i n g e r - V e r l a g (1974).  B l a k e l y and G.A.  (1979).  (1977).  Somorjai; Surface S c i . H ,  Germer, Phys. Rev.  P e r i a , J . A p p l . Phys.  and T.N.  L e t t . 42, 508  3 J , 705  45  (1974).  (1927).  38, 3320 (1967).  38, 4355  (1967).  Rhodin, J . A p p l . Phys. J59, 2425 (1968). 11, 212  (1974).  -172-  20.  C . J . P o w e l l , S u r f a c e S c i . J 4 , 29 (1974).  21.  H. Raether, S u r f a c e S c i . £ , 233 (1967).  "22.  C.B. Duke, Adv. Chem. Phys.  27, 1 (1974).  23.  T.A. C a r l s o n , " P h o t o e l e c t r o n and Auger S p e c t r o s c o p y " , Plenum, New York (1975).  24.  J.B. Pendry, "Low Energy E l e c t r o n D i f f r a c t i o n " , Academic P r e s s , New York (1974).  25.  M.B. Webb and M.E. L a g a l l y , S o l i d S t a t e P h y s i c s , 28, 301 (1973).  26.  G.E. Rhead, S u r f a c e S c i . 68, 20 (1977).  27.  N.F.M. Henry and K. L o n s d a l e , eds. " I n t e r n a t i o n a l T a b l e s f o r X-ray C r y s t a l l o g r a p h y " , V o l . 1, The Kynoch P r e s s , Birmingham (1952).  28.  E.A. Wood, J . A p p l . Phys.  29.  R.L. Park and H.H. Madden, S u r f a c e S c i . 11, 188 (1968).  30.  P.J. E s t r u p and E.G. McRae, S u r f a c e S c i . 25, 1 (1971).  31.  C C . Chang, S u r f a c e S c i . 25, 53 (1971).  32.  T.W. Haas and J.T. Grant, Phys. Rev. L e t t . 30A, 272 (1969)-, J . Vac. S c i . T e c h n o l . 2, 43 (1970).  33.  F . J . S z a l k o w s k i and G.A. S o m o r j a i , J . Chem. Phys.  34.  K. Siegbahn et a l . , "ESCA: Atomic, M o l e c u l a r , and S o l i d S t a t e S t r u c t u r e s S t u d i e d by t h e Means o f E l e c t r o n S p e c t r o s c o p y " , Almquist and W i k s e l l s , Uppsala (1967).  35.  Y. S t r a u s s e r and J . J . Uebbing, " V a r i a n Chart o f Auger E l e c t r o n E n e r g i e s " , V a r i a n Corp., P a l o A l t o (USA) (1970).  36.  P.W. Palmberg, G.E. R i a c h , R.E. Weber, and N.C. MacDonald, "Handbook o f Auger E l e c t r o n S p e c t r o s c o p y " , Phys. E l e c . Ind. Inc., E d i n a , Minnesota (1972).  37.  P.W.  .38. 39.  Palmberg,  35, 1306 (1964).  61, 2065 (1974).  G.K. Bohn and J.C. T r a c y , A p p l . Phys.  C C . Chang, S u r f a c e S c i . 48, 9 (1975). M.P. Seah, S u r f a c e S c i . 32, 703 (1972).  Lett.  15, 254 (1969).  -173-  40.  H.P. B o n z e l , S u r f . S c i . 27, 387 (1971).  41.  W.M. M u l a r i e and W.T. P e r i a , S u r f a c e S c i . 26,  -42.  43.  125 (1971).  A.E. Rae and M. Bebbington, "An Annotated B i b l i o g r a p h y o f Ruthenium, Rhodium and I r i d i u m as C a t a l y s t s " , I n t . N i c k e l Co. Inc., New York (1959). P.R. Watson, F.R. Shepherd,  D.C. F r o s t , and K.A.R. M i t c h e l l , S u r f a c e S c i .  72, 562 (1978). 44.  F.R. Shepherd, P.R. Watson, D.C. F r o s t , and K.A.R. M i t c h e l l , J . Phys. C 11, 4591 (1978).  45.  E. Z a n a z z i and F. Jona, S u r f a c e S c i . 62, 61 (1977).  46.  S.Y. Tong, M.A. Van Hove, and B.J. M r s t i k , Proc. 7 I n t e r n . Vacuum Congr. and t h i r d I n t e r n . Conf. on S o l i d S u r f a c e s , V i e n n a , p. 2407 (1977).  47.  S.Y. Tong and A.L. Maldonado, S u r f a c e S c i . 78., 459 (1978).  48.  S. Andersson and B. Kasemo, S u r f a c e S c i . 25, ? 3 (1971).  49.  R.W. James, "The O p t i c a l P r i n c i p l e s o f D i f f r a c t i o n o f X-ray", C o r n e l l U n i v e r s i t y P r e s s , I t h a c a (1965).  50.  T.B. Rymer, " E l e c t r o n D i f f r a c t i o n " , Methuen  51.  N.F. Mott and H.S.W. Massey, "The Theory O x f o r d U n i v e r s i t y Press (1965).  52.  R.M. S t e r n and F. B a l i b a r , Phys. Rev. L e t t . 25, 1338 (1970).  53.  R.L. Dennis and M.B. Webb, J . Vac. S c i . T e c h n o l . IQ, 192 (1973).  54.  D. Tabor, J.M. W i l s o n , and T . J . Bastow, S u r f a c e S c i . 20, 471 (1971).  55.  L. Hedin and S. L u n d q v i s t , S o l i d S t a t e Phys.  56.  J . E , Demuth, P.M. Marcus and D.W. Jepsen, Phys. Rev. B 11, 1460 (1975).  57.  D.W. J e p s e n , P.M. Marcus and F. Jona, Phys. Rev. B5, 3933 (1972).  58.  D.P. J e p s e n , P.M. Marcus and F. Jona, Phys. Rev. B8, 5523 (1973).  59.  P.M. Marcus, J . E . Demuth, and D.W. J e p s e n , S u r f a c e S c i . 53, 501 (1975).  60.  T.N. Rhodin and S.Y. Tong, P h y s i c s Today, 28(10), 23 (1975).  til  2  (1970).  o f Atomic  Collisions",  23, 1 (1969).  -174-  61.  S.Y.  62.  J.C. S l a t e r , Phys. Rev.  63.  L . J . S c h i f f , "Quantum Mechanics",  64.  R.G. New  Tong, J.B. Pendry and L.L. Kesmodel, S u r f a c e S c i . 54, 21 81, 385  (1951). McGraw-Hill, New  York  (1968).  Newton, " S c a t t e r i n g Theory o f Waves and P a r t i c l e s " , York  (1976).  McGraw-Hill,  (1966).  65.  S.Y.  Tong, Prog, i n S u r f . S c i . 7, 1 (1975).  66.  N. S t o n e r , M.A. Van Hove, and S.Y. Tong, i n " C h a r a c t e r i z a t i o n o f M e t a l and Polymer S u r f a c e s " , ed. L.H. Lee, Academic P r e s s , New York (1976).  67.  E.G.  McRae, J . Chem. Phys.  68.  C.B.  Duke and C.W.  69.  B.I. L u n d q v i s t , Phys. S t a t e . S o l . 32, 273  70.  J . L . Beeby, J . Phys. C l , 82  71.  S.Y.  Tong and T.N.  72.  S.Y.  Tong, T.N.  73.  E.G.  McRae, S u r f a c e S c i . i l ,  74.  J.B. Pendry, J . Phys. C4, 2501;  75.  K. Kambe, Z. N a t u r f o r s c h , 22a,  332  76.  K. Kambe, Z. N a t u r f o r s c h , 23a,  1280  77.  D.W.  Jepsen, P.M.  78.  M.A.  Van Hove and S.Y.  79.  S.Y.  Tong and M.A.  80.  R.S.  Zimmer and B.W.  81.  M.A.  Van Hove and S.Y.  Springer-Verlag 82.  M.A.  45., 3258 (1966).  Tucker, S u r f a c e S c i . 15, 231  (1969).  (1969).  (1968).  Rhodin, Phys. Rev.  Rhodin, and R.H.  Lett.  26, 711  T a i t , Phys. Rev.  479  (1971).  B8, 421; 430  (1973).  (1968). 2514  (1971). (1967). (1968).  Marcus and F. Jona, Phys. Rev.  Lett.  26, 1365  Tong, J . Vac. S c i . T e c h n o l . 12, 230  Van Hove, Phys. Rev.  B16,  H o l l a n d , J . Phys. C8,  1459  (1975).  (1977).  2395 (1975).  Tong, " S u r f a c e C r y s t a l l o g r a p h y by LEED",  (1979).  Van Hove and J.B. Pendry, J . Phys. C8,  1362  (1975).  (1971).  -175-  83.  J . E . Demuth, D.W.  84.  M.A.  Van Hove and S.Y.  85.  M.A.  Van Hove, S.Y.  Tong and E. E l c o n i n , S u r f a c e S c i . 64, 85  86.  D.G.  Fedak and N.A.  Gjostein, Surface S c i .  87.  A. Dulong, i n "LEED-Surface S t r u c t u r e o f S o l i d s " ed. M. L a z n i c k a , Union o f Czechoslovak Mathematicians and P h y s i c i s t s , Prague (1972).  88  J.P. Hobson, Adv.  89.  W.J.  90.  T. Tom,  91.  Jepsen and P.M.  Tong, Phys. Rev.  Colloid  L e t t . 35, 1092  77  I n t e r f a c e Sci._4_, 79  Lange, P h y s i c s Today 25, 40 P h y s i c s Today 25, 32  F. Rosebury,  Marcus, S o l i d S t a t e Comm. 13, 1311  (1973)  (1975). (1977).  (1967).  (1974).  (1972).  (1972).  "Handbook o f E l e c t r o n Tube and Vacuum Techniques", Addison-  Wesley, Messachusettes  (1965).  92.  W.H. K o h l , "Handbook o f M a t e r i a l and Technique R e i n h o l d , New York (1967).  93.  Research O r g a n i c / I n o r g a n i c Chemicals Corp.  94.  C o u r t e s y o f Dr. C.W.  T u c k e r , General E l e c t r i c  Centre, Schenectady,  New  f o r Vacuum D e v i c e s " ,  USA. Research and Development  York.  95,  N.F.M. Henry, H. L i p s o n and W.A. Wooster, "The I n t e r p r e t a t i o n o f X-ray D i f f r a c t i o n Photographs", M a c M i l l a n , London (1960).  96.  D.G.  C a s t n e r , B.A.  97.  R.A.  Marbrow and R.M.  98.  H.E.  Farnsworth, i n "The  Dekker, New 99.  York  Sexton and G.A.  S o m o r j a i , S u r f a c e S c i 7J, 519  Lambert.Surface  S o l i d - G a s I n t e r f a c e s " ed. E.A.  100.  F. Jona, J . Phys. Chem. 11, 4271  101.  P.W.  102.  J.T. Grant and T.W.  103.  W.A.  Coghlan  Flood, Marcel  G.K.  (1969). (1978).  Bohn, and J . C . T r a c y , A p p l . Phys. Haas, S u r f a c e S c i . 21, 76  and R.E.  (1971);  L e t t . 15, 524  (1964).  (1970).  C l a u s i n g , "A C a t a l o g o f C a l c u l a t e d Auger T r a n s i t i o n s  f o r the Elements", USAEC Report 0RNL-TM-3576, Oak Laboratory  (1977).  (1967).  E. Bauer, Tech. M e t a l Res. 2, 502  Palmberg,  S c i . 67, 489  (1978).  Atomic Data j i ,  317  (1973).  Ridge N a t i o n a l  -176-  104.  J . E . Demuth and T.N. Rhodin, S u r f a c e S c i . 42, 261 (1974).  105.  L. McDonnell  106.  P.C. S t a i r , T . J . Kaminska, L.L. Kesmodel and G.A. S o m o r j a i ,  and D.P. Woodruff,  Phys. Rev. B l l ,  S u r f a c e S c i . 46, 505 (1974).  623 (1975).  107.  D.C. F r o s t , K.A.R. M i t c h e l l , F.R. Shepherd J . Vacuum S c i . T e c h n o l . J_2, 1196 (1976).  108.  K.A.R. M i t c h e l l , F.R. Shepherd,  and P.R. Watson,  P.R. Watson and D.C. F r o s t ,  S u r f a c e S c i . 64, 737 (1977). 109.  D.C. F r o s t , S. Hengrasmee, K.A.R. M i t c h e l l , F.R. Shepherd S u r f a c e S c i . Z£, L585  and P.R.Watson,  (1978).  110.  V.L. M o r u z z i , J . F . Janak and A.R. W i l l i a m s , " C a l c u l a t i o n s o f E l e c t r o n i c p r o p e r t i e s o f m e t a l s " , Plenum P r e s s , New York (1978).  111.  P.R. Watson, Ph.D. T h e s i s , U n i v e r s i t y o f B r i t i s h Columbia  112.  M.A. Van Hove and S.Y. Tong, S u r f a c e S c i . 54, 91 (1976).  113.  J.A. S t r o z i e r , D.W. Jepsen and F. J o n a , i n " S u r f a c e P h y s i c s o f M a t e r i a l s " v o l . 1 ed. J.M. B l a k e l y , Academic P r e s s , New York (1975).  114.  M.G. L a g a l l y , i n " S u r f a c e P h y s i c s o f M a t e r i a l s " v o l . I I . ed. J.M. B l a k e l y , Academic P r e s s , New York  (1978).  (1975).  115.  K.A. G s c h n e i d e r , S o l i d S t a t e Phys.  16, 275 (1964).  116.  L.A. H a r r i s , J . A p p l . Phys.  117.  K.O. Legg, M. P r u t t o n and C. K i n n i b u r g h , J . Phys. Chem. 7, 4236 (1974).  118.  C M . Chan, P.A. T h i e l , J.T. Yates and W.H. Weinberg, S u r f a c e S c i . 76, 296 (1978)  119.  C.W. Tucker, J r . , J . A p p l . Phys.  37, 3013 (1966).  120.  C.W. Tucker, J r . , J . A p p l . Phys.  3 J , 4147 (1966).  121.  C.W. Tucker, J r . , J . A p p l . Phys.  38, 2696 (1967).  122.  C.W. Tucker, J r . , A c t a Met. 15, 1465 (1967).  123.  S. Hengrasmee, P.R. Watson, D.C. F r o s t and K.A.R. M i t c h e l l , S u r f a c e S c i . , 87, L249 (1979).  38, 1419 (1968).  -177-  124.  S. Hengrasmee, P.R. Watson, D.C. F r o s t and K.A.R. M i t c h e l l , S u r f a c e S c i . 92, 71 (1980).  125.  L. McDonnell, Ph.D. T h e s i s , U n i v e r s i t y o f Warwick, 1974.  126.  M. Salmeron  127.  P.A. T h i e l , J . T . Y a t e s , J r . , and W.H. Weinberg, S u r f a c e S c i . 82, 22 (1979).  128.  H. F r o i t z h e i m , i n " E l e c t r o n S p e c t r o s c o p y f o r s u r f a c e a n a l y s i s " ed. H. Ibach, T o p i c s i n C u r r e n t P h y s i c s , v o l . 4, S p r i n g e r - V e r l a g , B e r l i n H e i d e l b e r g , New York (1977).  129.  Y. G a u t h i e r , D. Aberdam and R.R. Baudoing,  130.  J . E . Demuth, D.W.  Jepsen, and P.M. Marcus, S u r f a c e S c i . 45, 733 (1974).  131.  J . E . Demuth, D.W.  Jepsen, and P.M. Marcus, Phys. Rev. L e t t . £ 1 , 540 (1973).  132.  S.R. Keleman and T.E. F i s c h e r , S u r f a c e S c i . 8_Z, 53 (1979).  133.  L. P a u l i n g , "The Nature o f The Chemical Bond" I t h a c a , New York (1960).  134.  S. G e l l e r , A c t a C r y s . 15, 1198 (1962).  135.  E. Parthe\ D. Mohnke and F. H u l l i g e r , A c t a C r y s t . 23, 832 (1967).  136.  P. Colamanno and P. O r i o l i , J . Chem. Soc. D a l t o n T r a n s . 845 (1976).  137.  R.J. Butcher and E. S i n n , J . Am. Chem. Soc. 98, 2440 (1976).  138.  R.H. M o r r i s , Ph.D. T h e s i s , U n i v e r s i t y o f B r i t i s h Columbia  139.  F. Jona, S u r f a c e S c i . 68, 204 (1977).  140.  J . E . Demuth, D.W.  141.  C M . Chan and W.H. Weinberg, J . Chem. Phys.  142.  K.O. Legg, F. J o n a , D.W.  143.  K.A.R. M i t c h e l l , S u r f a c e S c i . 9_2, 79 (1980).  "144. 145.  and G.A. S o m o r j a i , S u r f a c e S c i . 91, 373 (1980).  S u r f a c e S c i , 78, 339 (1978).  Cornell U n i v e r s i t y Press,  Jepsen and P.M. Marcus, Phys. Rev. L e t t .  (1978).  32, 1182 (1974).  21, 5988 (1979).  Jepsen and P.M. Marcus, S u r f a c e S c i . 6J>, 25 (1977)  K.A.R. M i t c h e l l , S u r f a c e S c i . , ( i n p r e s s ) . S.L. Altmann,  C A . Coulson and W. Hume-Rothery, Proc. Roy. Soc. (London)  A240, 145 (1957).  -178-  146.  M.A.  Van Hove and S.Y.  147.  CM.  Chan and W.H.  148.  A. Salwen and J . Rundgren, S u r f a c e S c i . 53, 523  149.  J . E . Demuth, D.W.  150.  S. Hengrasmee, K.A.R. M i t c h e l l , P.R. Watson and S.J. Canadian J o u r n a l o f P h y s i c s . 58 ( 2 ) , 200 (1980).  151.  CM.  Chan, K.L.  Tong, J . Vacuum S c i . T e c h n o l . 12, 230  Weinberg, J . Chem. Phys.  Jepsen and P.M.  Luke, M.A.  S u r f a c e S c i . 78, 386  Van Hove, W.H.  153.  K.O.  Legg, F. J o n a , D.W.  5271  (1977).  154.  C.W.  Tucker and C.B.  155.  F. Jona, H.D.  _  157.  White,  Weinberg and S;P.  Withrow,  Smeenk and F.W.  Jepsen and P.M.  Marcus, Phys. Rev.  Duke, S u r f a c e S c i . 23, 411  S h i h , D.W.  Jepsen and P.M.  Saris,  B  16,  (1970); 29, 237  and J.B. Pendry,  S o l i d S t a t e Comm. 16., 563  Adams and U. Landman, Phys. Rev.  (1972).  Marcus, J . Phys. Chem. 12,  (1979).  A. Andersson D.L.  (1975).  (1978).  J . F . Van der Veen, R.M. Tromp, R.C S u r f a c e S c i . 82, 468 (1979).  1 5 6  71, 3988 (1979).  Marcus, J . Phys. Chem. _6, L307 (1973).  152.  L455  (1975).  B 15, 3775 (1977).  (1975).  -179-  Appendices  The  f o l l o w i n g appendices  rhodium s u r f a c e s c o l l e c t e d  contain a l l the experimental data  d u r i n g t h i s work.  from  In a l l c a s e s , t h e d a t a  i s as c o l l e c t e d and has not been smoothed.  Appendix  Surface  Al  Rh(100)-(3xl)-0  Angle 6=0, <j>=0  2 equal domains  A  2  Rh(100)-(3xl)-0 single  A3  9=0, <f>=0  domain  Rh(100)-p(2x2)-S  6=0, ty=0  Expt. 1 A4  Rh(100)-p(2x2)-S  6=0, <$>=0  Expt. 2 A5  Rh(110)-c(2x2)-S  6=0, c|,=0  Expt. 1 A6  Rh(110)-c(2x2)-S Expt. 2  6=10, c}>=135  -181-  -182-  -184-  n ergy (ev )  -186-  

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0060930/manifest

Comment

Related Items