UBC Theses and Dissertations

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UBC Theses and Dissertations

Studies in LEED crystallography Moore, William Thomas 1981

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STUDIES IN LEED CRYSTALLOGRAPHY by WILLIAM THOMAS MOORE Sc., The U n i v e r s i t y of B r i t i s h C o lumbia, 1975 THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES ( Department of C h e m i s t r y ) We ac c e p t t h i s t h e s i s as c o n f o r m i n g t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA J u l y , 1981 (c) W i l l i a m Thomas Moore, 1981 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the requirements f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head o f my department o r by h i s o r her r e p r e s e n t a t i v e s . I t i s understood t h a t c o p y i n g or p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department of 0 h iS'Vr The U n i v e r s i t y o f B r i t i s h Columbia 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 Date i i A b s t r a c t The work i n t h i s t h e s i s i n v o l v e d assessments and a p p l i c a t i o n s of p r o c e d u r e s used f o r d e t e r m i n i n g 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 s w i t h low energy e l e c t r o n d i f f r a c t i o n (LEED), a s u b j e c t t h a t i s now o f t e n r e f e r r e d t o as LEED c r y s t a l l o g r a p h y . S p e c i f i c s t u d i e s were made f o r the (311) s u r f a c e s of copper and n i c k e l , and f o r the (0001) s u r f a c e of z i r c o n i u m . These .surfaces were c u t from h i g h p u r i t y s i n g l e c r y s t a l samples, s t u d i e d under c o n d i t i o n s of u l t r a - h i g h vacuum, and c h a r a c t e r i z e d ' by LEED and Auger e l e c t r o n s p e c t r o s c o p y . The LEED c r y s t a l l o g r a p h i c s t u d i e s of the (311) s u r f a c e s of f a c e - c e n t e r e d c u b i c m e t a l s r e p r e s e n t t h e f i r s t i n v e s t i g a t i o n s of stepped s u r f a c e s which employed 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 g e n e r a l , i n t h i s t h e s i s , 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 used the r e n o r m a l i z e d f o r w a r d s c a t t e r i n g and the l a y e r d o u b l i n g methods, and the e x p e r i m e n t a l i n t e n s i t y v e r s u s energy c u r v e s f o r the d i f f r a c t e d e l e c t r o n beams were measured w i t h our combined v i d i c o n - p h o t o g r a p h i c method. . In these s t u d i e s , many of the p r o c e d u r e s c u r r e n t l y employed i n LEED c r y s t a l l o g r a p h y were i n v e s t i g a t e d , p a r t i c u l a r l y i n r e g a r d t o t h e i r e f f e c t s on s u r f a c e geometry d e t e r m i n a t i o n s . These p r o c e d u r e s i n c l u d e d smoothing raw e x p e r i m e n t a l d a t a , a v e r a g i n g beams t h a t are e x p e c t e d t o be e q u i v a l e n t from symmetry c o n s i d e r a t i o n s , c o r r e c t i n g measured i n t e n s i t i e s f o r e f f e c t s of n on-uniform g r i d t r a n s p a r e n c y , and s e t t i n g a p p r o p r i a t e v a l u e s f o r the s u p p r e s s o r g r i d v o l t a g e . In a d d i t i o n , c o n s i d e r a t i o n was g i v e n t o i n v e s t i g a t i n g ways of d e a l i n g w i t h n u m e r i c a l i n s t a b i l i t i e s , which sometimes oc c u r i n the p e r t u r b a t i v e methods used 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 , and assessments were made of the r e l i a b i l i t y index r e c e n t l y proposed by Pendry. The l a t t e r g i v e s n u m e r i c a l v a l u e s f o r the 'goodness of f i t ' between 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 y c u r v e s , and was compared e s p e c i a l l y w i t h the index proposed p r e v i o u s l y by Z a n a z z i and Jona. Some c o n s i d e r a t i o n was a l s o g i v e n t o the q u e s t i o n of a s s e s s i n g u n c e r t a i n t i e s i n s t r u c t u r a l a n a l y s e s based on the use of r e l i a b i l i t y i n d i c e s . The topmost i n t e r l a y e r s p a c i n g i n the (311) s u r f a c e s of copper and n i c k e l a r e i n d i c a t e d t o be c o n t r a c t e d by about 5% and 14.5% r e s p e c t i v e l y from the b u l k s p a c i n g s . P o s s i b l e reasons f o r t h i s d i f f e r e n c e a r e d i s c u s s e d i n the t h e s i s . By c o n t r a s t , the (0001) s u r f a c e of z i r c o n i u m has a topmost s p a c i n g which i s v e r y c l o s e t o the b u l k v a l u e . E x p e r i m e n t a l i n t e n s i t y v e r s u s energy c u r v e s have a l s o been measured f o r t h r e e s u r f a c e s i n v o l v i n g oxygen a d s o r p t i o n on Z r ( 0 0 0 l ) ; one of thes e s u r f a c e s i s d e s i g n a t e d Z r ( 0 0 0 1 ) - ( 2 x 2 ) - 0 w h i l e the o t h e r two are both d e s i g n a t e d Z r ( 0 0 0 1 ) - ( 1 x 1 ) - 0 . For the l a t t e r , one i s b e l i e v e d t o i n v o l v e a monolayer coverage of oxygen atoms w h i l e the o t h e r p r o b a b l y i n v o l v e s oxygen i n c o r p o r a t i o n i n t o the s u r f a c e r e g i o n . The LEED i n t e n s i t y d a t a f o r thes e oxygen s t r u c t u r e s , once they have been a n a l y z 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 , s h o u l d p r o v i d e a b a s i s f o r d e t a i l i n g s t r u c t u r a l a s p e c t s of the i n i t i a l s t a g e s of the o x i d a t i o n of Z r ( 0 0 0 l ) . i v T a b le Of C o n t e n t s page A b s t r a c t i i Ta b l e of C o n t e n t s i v L i s t of T a b l e s v i i L i s t of F i g u r e s v i i i Acknowledgements x i v Chapter 1: SURFACE SCIENCE AND LOW ENERGY ELECTRON DIFFRACTION 1.1 I n t r o d u c t i o n 2 1.2 Some S u r f a c e Nomenclature 7 1.3 Low Energy E l e c t r o n D i f f r a c t i o n 10 1.3.1 E l e c t r o n S c a t t e r i n g i n S o l i d s 10 1.3.2 The LEED Experiment 13 1.3.3 The LEED D i f f r a c t i o n P a t t e r n 15 1.3.4 LEED C r y s t a l l o g r a p h y 18 1.4 I n s t r u m e n t a l Response, Domains and D i s o r d e r 19 1 .5 T h i s Work 24 Chapter 2: LEED INTENSITY CALCULATIONS 2.1 I n t r o d u c t i o n 27. 2.2 P h y s i c a l Parameters f o r LEED C a l c u l a t i o n s 28 2.2.1 The Ion Core P o t e n t i a l 29 2.2.2 The Constant P o t e n t i a l Vo 32 i ) Vor 32 i i ) V o i 33 2.2.3 Atomic M o t i o n s 34 2.2.4 The Solid-Vacuum I n t e r f a c e 36 2.3 M u l t i p l e S c a t t e r i n g Methods 37 2.4 P e r t u r b a t i v e Methods 40 V 2.4.1 R e n o r m a l i z e d Forward S c a t t e r i n g 41 2.4.2 L a y e r D o u b l i n g 43 2.4.3 Comparison of Methods 45 2.5 Program Flow and the use of Symmetry 46 Chapter 3: EXPERIMENTAL METHODS 3.1 Sample P r e p a r a t i o n 51 3.2 The UHV Chamber 52 3.3 Auger E l e c t r o n S p e c t r o s c o p y 55 3.4 C l e a n i n g the Sample 60 3.5 LEED I n t e n s i t y A n a l y s i s 61 3.6 The R - f a c t o r Programs 64 3.6.1 The R e l i a b i l i t y Index of Z a n a z z i and Jona .... 66 3.6.2 The Pendry R-f a c t o r 67 3.6.3 R - f a c t o r s and E r r o r E s t i m a t i o n 68 Chapter 4: STUDIES OF THE (311) SURFACES OF COPPER AND NICKEL 4.1 I n t r o d u c t i o n 75 4.2 Copper (311) 78 4.2.1 E x p e r i m e n t a l 78 4.2.2 C a l c u l a t i o n s and R - f a c t o r R e s u l t s 81 4.3 N i c k e l ( 3 1 1 ) 88 4.3.1 E x p e r i m e n t a l 88 4.3.2 C a l c u l a t i o n s and R - f a c t o r R e s u l t s 91 4.4 The R - f a c t o r and the N i c k e l E x p e r i m e n t a l Data 102 4.4.1 The E f f e c t s of G r i d T r a n s p a r e n c y 102 4.4.2 Smoothing 104 4.4.3 A v e r a g i n g s y m m e t r i c a l beams ' ..107 4.4.4 Comparison of two s e t s of e x p e r i m e n t a l d a t a ..109 v i 4.4.5 V a r i a b l e Inner P o t e n t i a l 114 4.5 C a l i b r a t i o n 114 4.6 Sup p r e s s o r G r i d V o l t a g e 118 4.7 D i r t y N i c k e l ( 3 H ) 120 4.8 C o n c l u s i o n 1 23 Chapter 5: STUDIES ON THE (0001) SURFACE OF ZIRCONIUM 5.1 The Clea n e d (0001) S u r f a c e Of Z i r c o n i u m ...128 5.1.1 E x p e r i m e n t a l 128 5.1.2 C a l c u l a t i o n s 132 5.1.3 R - f a c t o r R e s u l t s 133 5.1.4 Comparison With Other Work 143 5.2 Oxygen A d s o r p t i o n on the (0001) S u r f a c e of Z i r c o n i u m 144 5.2.1 E x p e r i m e n t a l 145 5.2.2 R e s u l t s 152 5.2.3 F u t u r e Work • 156 REFERENCES: 158 APPENDIX: 164 v i i L i s t Of T a b l e s page 4.1 Pendry R - f a c t o r , Ad%, and u n c e r t a i n t y v a l u e s where Rp i s m i n i m i z e d , f o r t h e comparison of c a l c u l a t e d 1(E) c u r v e s f o r N i ( 3 1 1 ) w i t h e x p e r i m e n t a l d a t a t h a t has been smoothed 0,1,2,3 t i m e s . These v a l u e s were o b t a i n e d f o r Vor=-l0eV 108 4.2 S i n g l e beam r - f a c t o r v a l u e s f o r t h e comparison of p a i r s of n o m i n a l l y symmetric beams w i t h i n a s i n g l e experiment f o r normal i n c i d e n c e from N i (311) 111 4.3 S i n g l e beam r - f a c t o r v a l u e s f o r the comparison of c o r r e s p o n d i n g beams measured a t normal i n c i d e n c e from N i ( 3 1 1 ) from two independent e x p e r i m e n t s . S u p e r s c r i p t s on beam l a b e l s i n d i c a t e the s o u r c e of the 1(E) c u r v e ( i . e . e i t h e r e x periment 1 or 2) 111 5.1 S i n g l e beam Z a n a z z i and Jona r - f a c t o r s f o r the comparison of 1(E) c u r v e s measured from t h e (0001) s u r f a c e of z i r c o n i u m w i t h c a l c u l a t e d 1(E) c u r v e s f o r the u n r e c o n s t r u c t e d (hep) and r e c o n s t r u c t e d (hep+fee t o p l a y e r ) s u r f a c e models 139 5-.2 S i n g l e beam Pendry r - f a c t o r s f o r the comparison of c o r r e s p o n d i n g beams from two independent measurements of beams from the Z r ( 0 0 0 1 ) - ( 2 x 2 ) - 0 s t r u c t u r e . W i t h i n an e x p e r i m e n t a l s e t , s y m m e t r i c a l l y e q u i v a l e n t beams were ave r a g e d and smoothed once ( e q u a t i o n 4.3) 155 v i i i L i s t Of F i g u r e s page 1.1 Energy l e v e l diagram of aluminum i l l u s t r a t i n g the p r o d u c t i o n of Auger e l e c t r o n s . The energy l e v e l s were o b t a i n e d from x r a y s t u d i e s and a r e g i v e n r e l a t i v e t o t h e Fermi l e v e l . 6 1.2a Schematic i l l u s t r a t i o n of the ' s u r f a c e r e g i o n ' probed by a LEED experiment showing o v e r l a y e r and s u b s t r a t e , b oth of which a r e d i p e r i o d i c i n the x,y d i r e c t i o n 8 1.2b D i p e r i o d i c s u r f a c e n e t s and the c o r r e s p o n d i n g n e t s i n r e c i p r o c a l space 8 1.3a Energy d i s t r i b u t i o n of e l e c t r o n s b a c k s c a t t e r e d from a s u r f a c e f o r an i n c i d e n t beam energy Eo 12 1.3b Mean f r e e p a th l e n g t h L(A) of 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 of e l e c t r o n energy (eV) 12 1.4a Schematic diagram of the LEED experiment 14 1.4b Schematic diagram showing how the c o n s e r v a t i o n c o n d i t i o n s determine a d i f f r a c t e d beam's d i r e c t i o n . The (0,0) beam r e s u l t s from s p e c u l a r r e f l e c t i o n 14 1.5 Leed I (E.) c u r v e s f o r normal i n c i d e n c e d a t a from a c l e a n C U ( 3 1 1 ) s u r f a c e compared w i t h 1(E) c u r v e s c a l c u l a t e d f o r d i f f e r e n t topmost i n t e r l a y e r s p a c i n g s 16 1.6 S t r u c t u r e of the hcp(OOOl) s u r f a c e showing the two p o s s i b l e domains t h a t r e s u l t from the t r u n c a t i o n of the b u l k s t r u c t u r e . L a r g e r c i r c l e s r e p r e s e n t atoms i n the topmost atomic l a y e r , f u l l and open c i r c l e s atoms i n l a y e r s A and B r e s p e c t i v e l y . The two domains are r e l a t e d t o each o t h e r by a 180° r o t a t i o n . Axes s e l e c t i o n and u n i t c e l l v e c t o r s a f t e r Van Hove [21] 21 1.7 Three 120° r e l a t e d domains of p2x1 o v e r l a y e r s t r u c t u r e s on a hcp(OOOl) s u r f a c e . X's r e p r e s e n t atoms i n the topmost l a y e r of the hcp(OOOl) s u b s t r a t e and dark c i r c l e s r e p r e s e n t the atoms i n the o v e r l a y e r 22 2.1 Model f o r p o t e n t i a l of an a b s o r b a t e c o v e r e d s u r f a c e r e g i o n ( a f t e r Marcus [56]) 31 2.2a Schematic r e p r e s e n t a t i o n of a s e t of p l a n e waves (kg) 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 an atomic l a y e r i n t o o u t g o i n g waves (kg') 39 2.2b Schematic diagram i l l u s t r a t i n g the t r a n s m i s s i o n and r e f l e c t i o n of waves at the n t h atomic l a y e r . Dashed l i n e s r e p r e s e n t the m i d p o i n t s between the n t h p l a n e and the n-1 and n+1 atomic l a y e r s 39 i x 2.3 Schematic i l l u s t r a t i o n of the comp u t a t i o n p r o c e d u r e of Re n o r m a l i z e d Forward S c a t t e r i n g (RFS). The t r i p l e t s of arrows r e p r e s e n t the e n t i r e s e t of p r o p a g a t i n g p l a n e waves. a) RFS procedure f o r t h r e e i t e r a t i o n s b) P r o p a g a t i o n of inward moving waves c) P r o p a g a t i o n of outward moving waves ( a f t e r Van Hove and Tong. [21]) 42 2.4 I l l u s t r a t i o n of the l a y e r d o u b l i n g p r o c e d u r e of s t a c k i n g l a y e r s . L a y e r s A and B a r e s t a c k e d t o g e t h e r and 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 a r e c a l c u l a t e d . The double l a y e r C i s then s t a c k e d t o g e t h e r t o form a 4 - l a y e r s l a b and the d i f f r a c t i o n m a t r i c e s a r e a g a i n c a l c u l a t e d . ( a f t e r T o n g [ 5 l ] ) 44 2.5 A map of r e c i p r o c a l space f o r n i c k e l ( 3 1 l ) w i t h c i r c l e s A,B,C,which e n c l o s e ; A-the number of beams used i n the c a l c u l a t i o n B-the number of beams t h a t emerge from the c r y s t a l C-the number of beams t h a t appear on the LEED sc r e e n f o r an i n c i d e n t energy of 200eV 47 2.6 F l o w c h a r t of the LEED p r o g r a m s [ 2 l ] used t o c a l c u l a t e d i f f r a c t e d beam i n t e n s i t i e s as a f u n c t i o n of energy u s i n g the l a y e r d o u b l i n g or RFS methods. 49 3.1 (a) Schematic diagram of the FC12 UHV chamber (b) Diagrammatic r e p r e s e n t a t i o n of the pumping system: IP = i o n 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 53 3.2 Schematic diagram of LEED o p t i c e s used as a r e t a r d i n g f i e l d a n a l y z e r f o r Auger e l e c t r o n s p e c t r o s c o p y . MCA = m u l t i c h a n n e l a n a l y z e r ( F a b r i t e k 1062) 57 3.3 Schematic diagram of the e x p e r i m e n t a l s e t up on the V a r i a n FC12 vacuum chamber u s i n g the 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 g l a n c i n g i n c i d e n c e e l e c t r o n gun t o tak e Auger s p e c t r a 59 3.4 Model r - f a c t o r v e r s u s geometry p l o t s f o r two s i t u a t i o n s i n v o l v i n g two d i f f e r e n t beams. Top i l l u s t r a t i o n s u g g ests a p o s s i b l e f a i l u r e of U{1} t o r e p r e s e n t the d i f f e r e n c e i n 'best f i t ' g e o m e t r i e s ( g m i n ( 1 ) , g m i n ( 2 ) ) found f o r the i n d i v i d u a l beams. Bottom diagram shows the l a r g e U{1} v a l u e t h a t can be found because of t h e d i f f e r e n c e s i n the a b s o l u t e r <' l- , v a l u e s even when both beams show t h e i r best agreement w i t h the same s t r u c t u r a l model. ... 70 4.1 I l l u s t r a t i o n of the c e n t e r e d - r e c t a n g u l a r net t h a t d e s c r i b e s a f c c ( 3 1 1 ) s u r f a c e 76 4.2 R e c i p r o c a l net v e c t o r s and beam l a b e l s c o r r e s p o n d i n g t o the s u r f a c e net shown i n 4.1 77 X 4.3a Auger spectrum of a c o n t a m i n a t e d G U ( 3 1 1 ) s u r f a c e taken u s i n g the LEED o p t i c s as an energy a n a l y z e r (Ep=1500eV, Ip=20juA) 80 4.3b Auger spectrum of the c l e a n e d Cu(311) s u r f a c e . ......... 80 4.4 (0,-2) beam measured a t normal i n c i d e n c e from C U ( 3 1 1 ) compared w i t h c a l c u l a t e d 1(E) c u r v e s f o r A<a% v a l u e s from -15 t o +5. . 83 4.5 (1,-3) and (-1,-3) beams measured at normal i n c i d e n c e from C U ( 3 1 1 ) compared w i t h c a l c u l a t e d 1(E) c u r v e s f o r Ad% v a l u e s from -15 t o +5. 84 4.6 (1,5) beam measured a t normal i n c i d e n c e - ' from Cu(311) compared w i t h c a l c u l a t e d 1(E) c u r v e s f o n A d % v a l u e s from -15 t o +5 85 4.7 (1,1) and (-1,1) beams measured a t normal i n c i d e n c e from C U ( 3 1 1 ) compared w i t h c a l c u l a t e d 1(E) c u r v e s f o r A d % v a l u e s from -15 t o +5 86 4.8 A c o n t o u r p l o t of the multi-beam R z j v a l u e s f o r the c o p p e r ( 3 1 l ) normal i n c i d e n c e d a t a as a f u n c t i o n of Vor and A d% 87 4.9a Auger spectrum of a c o n t a m i n a t e d N i ( 3 1 1 ) s u r f a c e t aken u s i n g the CMA on the FC12 vacuum system w i t h Ep=2500eV, Ip=100pA , 90 4.9b Auger spectrum of a c l e a n e d N i ( 3 1 1 ) s u r f a c e 90 4.10 a) Sum of emergent beam i n t e n s i t i e s vs.Ad% a t 200eV. b ) , c ) R e f l e c t e d i n t e n s i t y v s . ^ d % a t 200eV f o r (0,0) and (1,3) beams. The c a l c u l a t i o n s w i t h 42 or 51 beams y i e l d the same i n t e n s i t i e s (on t h i s s c a l e ) e xcept a t Ad%=-18 where the c a l c u l a t i o n w i t h 51 beams f a i l s t o converge. : d) RA ( e q u a t i o n 4.1) p l o t t e d as a f u n c t i o n of the number of beams used i n the c a l c u l a t i o n of the b u l k d i f f r a c t i o n m a t r i c e s a t 200eV ' 93 4.11 (0,4) beam measured a t normal i n c i d e n c e from N i ( 3 1 1 ) compared w i t h 1(E) c u r v e s c a l c u l a t e d f o r A d% from -24 (0.80A) t o 0 ( 1 .06A) . 95 4.12 (0,2) beam measured a t normal i n c i d e n c e from N i ( 3 1 1 ) compared w i t h 1(E) c u r v e s c a l c u l a t e d f o r A d % from -24 (0.80A) t o 0 ( 1 .06A) 96 4.13 (0,-2) beam measured a t normal i n c i d e n c e from N i ( 3 1 l ) compared w i t h 1(E) c u r v e s c a l c u l a t e d f o r A d % from -24 '•.:•< (0.80A) t o 0 ( 1 .06A) 97 4.14 (0,-4) beam measured a t normal i n c i d e n c e from N i ( 3 1 1 ) compared w i t h 1(E) c u r v e s c a l c u l a t e d f o r A d % from -24 x i (0.80A) t o 0 { 1 .06A) . 98 4.15 (1,-1) beam measured a t normal i n c i d e n c e from Ni.(31 1 compared w i t h 1(E) c u r v e s c a l c u l a t e d f o r A.d% from -24 (0.80A) t o 0 ( 1 .06A) 99 4.16 Contour p l o t of v a l u e s of multi-beam index R z j f o r 1(E) c u r v e s measured a t normal i n c i d e n c e from a N i ( 3 l l ) s u r f a c e as a f u n c t i o n of i n n e r p o t e n t i a l (Vor) and per c e n t a g e change i n the topmost i n t e r l a y e r s p a c i n g (Ad%) ..100 4.17 Contour p l o t of v a l u e s of multi-beam index Rp f o r 1(E) c u r v e s measured a t normal i n c i d e n c e from a N i ( 3 1 1 ) s u r f a c e as a f u n c t i o n of i n n e r p o t e n t i a l (Vor) and per c e n t a g e change i n the topmost i n t e r l a y e r s p a c i n g (Ad%) 101 4.18 E x p e r i m e n t a l arrangement f o r LEED i n t e n s i t y measurements showing the a n g l e s and d i s t a n c e s r e l e v a n t t o e q u a t i o n 4.2 which c o r r e c t s f o r the d i f f e r e n c e s i n g r i d t r a n s m i s s i o n as a f u n c t i o n of beam p o s i t i o n on the LEED sc r e e n ( a l l d i m e n s i o n s i n mm) 103 4.19 The (0,-4) beam measured a t normal i n c i d e n c e from N i ( 3 1 1 ) b e f o r e and a f t e r c o r r e c t i n g f o r the d i f f e r i n g t r a n s m i s s i o n of the LEED g r i d s as a f u n c t i o n of energy. L a r g e s t changes occur a t the low e s t e n e r g i e s . .... 105 4.20 M u l t i - b e a m index Rp p l o t t e d as a f u n c t i o n of Ad%. The Rp v a l u e s a r e from the comparison of 1(E) c u r v e s measured at normal i n c i d e n c e from N i ( 3 1 1 ) t h a t have been smoothed up t o 3 ti m e s w i t h c a l c u l a t e d 1(E) c u r v e s f o r changes i n the topmost i n t e r l a y e r s p a c i n g s i n the range from -24% t o 0% 106 4.21 (1,-1) and (-1,-1) beams and t h e i r average from d a t a measured at normal i n c i d e n c e f o r the N i ( 3 1 1 ) s u r f a c e . ..110 4.22 (1,-3) and (-1,-3) beams measured f o r normal i n c i d e n c e on N i ( 3 1 1 ) . Upper p a i r of 1(E) c u r v e s a re f o r s y m m e t r i c a l l y r e l a t e d beams from one e x p e r i m e n t , and lower p a i r of c u r v e s a r e from an independent e x p e r i m e n t . 112 4.23 (1,1) and (-1,1) beams measured f o r normal i n c i d e n c e on N i ( 3 1 l ) . Upper p a i r of 1(E) c u r v e s a re f o r s y m m e t r i c a l l y r e l a t e d beams from one e x p e r i m e n t , and lower p a i r of c u r v e s a r e from an independent experiment 113 4.24 D i g i t i z e d i n t e n s i t i e s of d i f f r a c t e d beams measured a t l75eV f o r normal i n c i d e n c e from a N i ( 3 1 1 ) • s u r f a c e p l o t t e d as a f u n c t i o n of d i f f r a c t e d beam c u r r e n t . A V i d i c o n camera a p e r t u r e s e t a t f2.8 B V i d i c o n camera a p e r t u r e s e t a t f2.0 x i i C V i d i c o n camera a p e r t u r e s e t a t f2.8 and camera moved away from the f i l m so t h a t the e n t i r e d i f f r a c t e d beam p r o f i l e i s i n c l u d e d i n the i n t e g r a t i o n . 116 4.25 D i g i t i z e d d i f f r a c t e d beam i n t e n s i t y r a t i o s f o r two p a i r s of d i f f r a c t e d beams measured a t 175eV f o r normal i n c i d e n c e from a N i ( 3 1 1 ) s u r f a c e p l o t t e d as a f u n c t i o n of the Sup p r e s s o r g r i d v o l t a g e . The g r e a t e s t changes oc c u r f o r s u p p r e s s o r g r i d v o l t a g e s of l e s s than 3 v o l t s 119 4.26 Contour p l o t of v a l u e s of multi-beam index R z j f o r 1(E) c u r v e s measured a t normal i n c i d e n c e from a c o n t a m i n a t e d Ni ( 3 1 1 ) s u r f a c e as a f u n c t i o n of i n n e r p o t e n t i a l (Vor) and p e r c e n t a g e change i n the topmost i n t e r l a y e r s p a c i n g (Ad%) 122 5.1a Auger spectrum of a c o n t a m i n a t e d Z r ( O O O l ) s u r f a c e measured u s i n g the CMA on the FC12 vacuum chamber (Ep=2500eV, Ip=l00juA) 130 5.1b Auger spectrum of a c l e a n e d Z r ( 0 0 0 l ) s u r f a c e 130 5.2 Diagram showing how the a z i m u t h a l a n g l e i s s p e c i f i e d f o r the o f f - n o r m a l i n c i d e n c e experiment on Z r ( 0 0 0 l ) . The a z i m u t h i s d e f i n e d a c c o r d i n g t o the f o r m a l i s m of Z a n a z z i et a l [ l l 7 ] f o r the s e l e c t i o n of s u b s t r a t e axes i l l u s t r a t e d 131 5.3 (1 , 0 ) and (1,1) beams measured a t normal i n c i d e n c e from the (0001) s u r f a c e of z i r c o n i u m compared w i t h the c a l c u l a t e d 1 (E) c u r v e s which gave the lo w e s t s i n g l e beam r - f a c t o r s f o r the hep and hep+fee t op l a y e r s u r f a c e models 134 5.4 ( 0 , 0 ) beam measured f o r o f f - n o r m a l i n c i d e n c e (0=9° 0=13) from the c l e a n e d Z r ( 0 0 0 l ) s u r f a c e , and compared w i t h c a l c u l a t e d 1 (E) c u r v e s f o r A d % v a l u e s r a n g i n g from -7.5 t o 7.5 135 5.5 ( 0 , 1 ) beam measured f o r o f f - n o r m a l i n c i d e n c e (9=9°, 0=13) from the c l e a n e d Z r ( 0 0 0 l ) s u r f a c e , and compared w i t h c a l c u l a t e d .I(E) c u r v e s f o r A d % v a l u e s r a n g i n g from -7.5 t o 7.5 136 5.6 (-2,1) beam measured f o r o f f - n o r m a l i n c i d e n c e (9=9° 0=13°) from the c l e a n e d Z r ( 0 0 0 l ) s u r f a c e , and compared w i t h c a l c u l a t e d 1 (E) c u r v e s f o r Ad% v a l u e s r a n g i n g from -7.5 t o 7.5 137 5.7 (-1,1) beam measured f o r o f f - n o r m a l i n c i d e n c e (9=9°, 0=13°) from the c l e a n e d Z r ( 0 0 0 l ) s u r f a c e , and compared w i t h c a l c u l a t e d 1 ( E ) c u r v e s f o r A d % v a l u e s r a n g i n g from -7.5 t o 7.5 138 5.8 Contour p l o t of the multi-beam R z j v a l u e s as a f u n c t i o n x i i i of Vor and A d % f o r the comparison o f 1(E) c u r v e s measured a t normal i n c i d e n c e from the Zr(OOOl) s u r f a c e w i t h c a l c u l a t e d 1(E) c u r v e s f o r an u n r e c o n s t r u c t e d (hep) s u r f ace w i t h A d% v a l u e s r a n g i n g from -7.5 t o 7.5 140 5.9 Contour p l o t of t h e multi-beam R z j v a l u e s as a f u n c t i o n of Vor and A d % f o r t h e comparison of 1(E) c u r v e s measured a t normal i n c i d e n c e from the Z r ( 0 0 0 l ) s u r f a c e w i t h c a l c u l a t e d 1(E) c u r v e s f o r a r e c o n s t r u c t e d (hep+fee t o p l a y e r ) s u r f a c e w i t h ^ d % v a l u e s r a n g i n g from -7.5 t o 7.5 141 5.10 Contour p l o t s of the m u l t i - b e a m R z j v a l u e s as a f u n c t i o n of Vor and A d % f o r the comparison of 1(E) c u r v e s measured f o r o f f - n o r m a l i n c i d e n c e (9=9? 0=13°) from t h e Z r ( 0 0 0 l ) s u r f a c e w i t h : a) c a l c u l a t e d 1(E) c u r v e s f o r an u n r e c o n s t r u c t e d (hep) s u r f a c e w i t h A d % v a l u e s r a n g i n g from -7.5 t o 7.5; b) c a l c u l a t e d 1(E) c u r v e s f o r a r e c o n s t r u c t e d (hep+fee t o p l a y e r ) s u r f a c e w i t h A d % v a l u e s r a n g i n g from -7.5 t o 7.5 142 5.11 Auger peak h e i g h t r a t i o 0(510eV)/Zr(174eV) p l o t t e d as a f u n c t i o n of oxygen exposure f o r two independent e x p e r i m e n t s ( l a r g e and s m a l l c i r c l e s ) . The Auger peak h e i g h t r a t i o s have been d i v i d e d by 2.8 t o y i e l d a p p r o x i m a t i o n s t o the f r a c t i o n a l monolayer c o v e r a g e . ...147 5.12 Auger peak h e i g h t r a t i o s f o r C(272eV)/Zr(174eV) and 0(510eV)/Zr(174eV) p l o t t e d as a f u n c t i o n of exposure f o r the a d s o r p t i o n of CO on a (0001) s u r f a c e of z i r c o n i u m . .149 5.13 Auger peak h e i g h t r a t i o s f o r C(272eV)/Zr(174eV) and 0(510eV)/Zr(174eV) p l o t t e d as a f u n c t i o n of CO exposure on a Z r ( 0 0 0 l ) s u r f a c e t h a t c o n t a i n e d a d s orbed oxygen and which showed a (2x2) d i f f r a c t i o n p a t t e r n 151 5.14 1(E) c u r v e s f o r (1,0) beams a t normal i n c i d e n c e from two independent measurements f o r the Z r ( 0 0 0 l ) - ( 2 x 2 ) - 0 s t r u c t u r e . The c u r v e s i l l u s t r a t e d a r e formed by a v e r a g i n g s y m m e t r i c a l l y e q u i v a l e n t 1(E) c u r v e s w i t h i n each s e t , and each average has been smoothed once ( e q u a t i o n 4.3) 153 5.15 1(E) c u r v e s f o r (0,3/2) beams a t normal i n c i d e n c e from two independent measurements f o r the Z r ( 0 0 0 1 ) - ( 2 x 2 ) - 0 s t r u c t u r e . The c u r v e s i l l u s t r a t e d a r e formed by a v e r a g i n g s y m m e t r i c a l l y e q u i v a l e n t 1(E) c u r v e s w i t h i n each s e t , and each average has been smoothed once ( e q u a t i o n 4.3) 154 x i v Acknowledgements I would f i r s t l i k e t o thank my s u p e r v i s o r s , 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 , f o r t h e i r s u p port of my r e s e a r c h . I am e s p e c i a l l y g r a t e f u l t o Dr. M i t c h e l l f o r h i s c a r e f u l r e a d i n g of t h i s t h e s i s and f o r h i s a d v i c e and comments c o n c e r n i n g i t s p r e p a r a t i o n . Many of the computer programs used i n c a l c u l a t i n g the LEED i n t e n s i t i e s and i n comparing e x p e r i m e n t a l and c a l c u l a t e d d a t a were o b t a i n e d from o u t s i d e s o u r c e s . My s i n c e r e thanks a r e owed t o Dr. M.A. Van Hove ( U n i v . C a l i f o r n i a , B e r k e l e y ) and t o Dr. S.Y. Tong ( U n i v . W i s c o n s i n , Milwaukee) f o r c o p i e s of t h e i r m u l t i p l e s c a t t e r i n g programs, and a l s o t o P r o f e s s o r F. Jona (S.U.N.Y, Stony Brook) and t o P r o f e s s o r M. P r u t t o n ( U n i v . York, U.K.) f o r c o p i e s of the r e l i a b i l i t y index programs used i n t h i s work. I t has been a p l e a s a n t and s t i m u l a t i n g e x p e r i e n c e t o work i n the s u r f a c e s c i e n c e l a b o r a t o r y . I owe a s p e c i a l g r a t i t u d e t o Frank Shepherd, R i c k S t r e a t e r , and P h i l Watson f o r t h e i r p a t i e n c e i n g u i d i n g me as a new grad u a t e s t u d e n t , and l a t e r , t o Stephen White, who p r o v i d e d v a l u a b l e a s s i s t a n c e i n many e x p e r i m e n t a l u n d e r t a k i n g s . I have a l s o a p p r e c i a t e d my i n t e r a c t i o n s w i t h Sunantha Hengrasmee and Kachung H u i . For t h e i r p a t i e n c e i n a d a p t i n g t o the trauma of b u i l d i n g u l t r a - h i g h vacuum equipment and f o r p u t t i n g up w i t h c o n s t a n t q u e s t i o n s and a f i n i c k y c l i e n t , I would l i k e t o acknowledge the a s s i s t a n c e of the m e c h a n i c a l shop employees and e s p e c i a l l y C h a r l i e M c C a f f e r t y , B r i n -Powell, Ed Gomm and B i l l Henderson. I am a l s o i n d e b t e d t o B r i a n Greene, Mike H a t t o n , Don C a t t , and Joe X V S a l l o s of the e l e c t r o n i c s shop f o r t h e i r h e l p f u l a s s i s t a n c e . F i n a l l y , and most i m p o r t a n t l y , I w i s h t o thank my f a m i l y f o r t h e i r encouragement, and e s p e c i a l l y t o thank my w i f e , L e s l i e , f o r her u n d e r s t a n d i n g , s u p p o r t and good humor over the p a s t year of extended p e r s o n a l ' d e a d l i n e s ' . CHAPTER 1 SURFACE SCIENCE AND LOW ENERGY ELECTRON DIFFRACTION 2 1.1 INTRODUCTION Many im p o r t a n t i n d u s t r i a l p r o c e s s e s depend upon the p r o p e r t i e s of s o l i d s u r f a c e s , f o r example the f i e l d s of heterogeneous c a t a l y s i s , f r i c t i o n and wear, c o r r o s i o n , and semiconductor t e c h n o l o g y . Fundamental t o a thorough u n d e r s t a n d i n g of these a r e a s i s knowledge of the s u r f a c e p r o c e s s e s which occur a t the atomic l e v e l , i n f o r m a t i o n t h a t has been o n l y s l o w l y becoming a v a i l a b l e . H i s t o r i c a l l y , c h e m i c a l r e s e a r c h on s u r f a c e s emphasized the study of m a t e r i a l s w i t h h i g h s u r f a c e / v o l u m e r a t i o s and measurements were made on s u r f a c e s w i t h u n c e r t a i n d i s t r i b u t i o n s of exposed c r y s t a l l o g r a p h i c p l a n e s and i m p u r i t y and d e f e c t p o p u l a t i o n s . A l t h o u g h u s e f u l e m p i r i c a l r e l a t i o n s h i p s were d e r i v e d from t h e s e t r a d i t i o n a l s t u d i e s , they d i d not p r o v i d e a d e t a i l e d u n d e r s t a n d i n g of the e v e n t s t h a t were o c c u r r i n g on the atomic s c a l e . I n the 1960's, w i t h the development of the means f o r r o u t i n e l y p r o d u c i n g u l t r a - h i g h vacuum (UHV), an a l t e r n a t i v e ' c l e a n - s u r f a c e ' approach became p o s s i b l e . T h i s approach s t u d i e d s u r f a c e s t h a t c o u l d be w e l l - c h a r a c t e r i z e d ( i . e . h i g h p u r i t y s i n g l e c r y s t a l s c u t t o expose a s i n g l e c r y s t a l l o g r a p h i c p l a n e ) and an emphasis was p l a c e d on i n v e s t i g a t i o n s t h a t would p r o v i d e knowledge of the b a s i c p r o c e s s e s ( a d s o r p t i o n , d e s o r p t i o n , d i f f u s i o n , d i s s o c i a t i o n , e t c . ) which t a k e p l a c e at such s o l i d s u r f a c e s . The n e c e s s i t y of u l t r a - h i g h vacuum f o r s t u d i e s of w e l l d e f i n e d s u r f a c e s f o l l o w s from the k i n e t i c t h e o r y of gases which p r e d i c t s t h a t a monolayer of m o l e c u l e s can be adsorbed i n about one second a t a p r e s s u r e of 10- 6 t o r r . The f o r m a t i o n of monolayer coverage assumes t h a t each m o l e c u l e - s u r f a c e c o l l i s i o n 3 r e s u l t s i n a d s o r p t i o n ( i . e . a s t i c k i n g p r o b a b i l i t y of u n i t y ) . UHV i s g e n e r a l l y d e f i n e d as p r e s s u r e s l e s s than 1x10- 9 t o r r (1 torr=133 p a s c a l ) , and i t i s e v i d e n t t h a t such p r e s s u r e s a r e e s s e n t i a l t o produce a c l e a n s u r f a c e and t o c a r r y out c o n t r o l l e d e x p e r i m e n t s on i t . At the p r e s e n t time a range of t e c h n i q u e s a r e a v a i l a b l e f o r s u r f a c e c h a r a c t e r i z a t i o n w i t h i n t h e ' c l e a n s u r f a c e approach'. These t e c h n i q u e s g e n e r a l l y i n v o l v e s t u d y i n g the i n t e r a c t i o n of the sample w i t h photons, e l e c t r o n s , or i o n s t h r o u g h the measurement of the a n g u l a r and/or energy d i s t r i b u t i o n s of the e m i t t e d p a r t i c l e s . Some of t h e s e e x p e r i m e n t a l methods are low energy e l e c t r o n d i f f r a c t i o n [ 1 , 2 ] (LEED), Auger e l e c t r o n s p e c t r o s c o p y [ 3 , 4 ] (AES), e l e c t r o n energy l o s s s p e c t r o s c o p y [ 5 ] , p h o t o e l e c t r o n s p e c t r o s c o p y [ 6 ] , f i e l d i o n m i c r o s c o p y [ 7 ] , m o l e c u l a r beam s c a t t e r i n g [ 8 ] , and i o n n e u t r a l i z a t i o n s p e c t r o s c o p y [ 9 ] . V a r i o u s g e n e r a l r e v i e w s a r e a v a i l a b l e d e s c r i b i n g the numerous t e c h n i q u e s used i n modern s u r f a c e s t u d i e s under UHV c o n d i t i o n s [ 1 0 - 1 4 ] . The work i n t h i s t h e s i s i n v o l v e s the use of two of t h e s e t e c h n i q u e s , LEED and AES, i n the d e t e r m i n a t i o n of s u r f a c e g e o m e t r i e s . The f i r s t LEED experiment was performed by D a v i s s o n and G e r m e r [ l 5 ] i n 1927. They observed t h a t the a n g u l a r d i s t r i b u t i o n of e l a s t i c a l l y b a c k s c a t t e r e d e l e c t r o n s from an a n n e a l e d n i c k e l sample was h i g h l y a n i s o t r o p i c , and they c o r r e c t l y i n t e r p r e t e d the b a c k s c a t t e r e d e l e c t r o n d i s t r i b u t i o n as r e s u l t i n g from d i f f r a c t i o n by the p e r i o d i c a r r a y of s u r f a c e atoms. I n i t i a l l y t he importance of t h i s experiment r e s u l t e d from i t s c o n f i r m a t i o n of the w a v e - l i k e n a t u r e of e l e c t r o n s a c c o r d i n g t o De B r o g l i e ' s h y p o t h e s i s , a l t h o u g h D a v i s s o n and Germer d i d r e c o g n i z e the p o t e n t i a l of t h i s t e c h n i q u e f o r d e t e r m i n i n g s u r f a c e s t r u c t u r e s . E q u a t i o n 1.1 shows the r e l a t i o n s h i p between the wavelength o (X i n A) and energy (E i n eV) f o r n o n - r e l a t i v i s t i c e l e c t r o n s . X = V l50-4/E (1.1) A t y p i c a l LEED experiment i n v o l v e s e l e c t r o n s w i t h e n e r g i e s around lOOeV and t h e r e f o r e w a velengths of the o r d e r of an angstrom. Such e l e c t r o n s a r e w e l l s u i t e d as probes of the i n t e r a t o m i c and i n t e r l a y e r s e p a r a t i o n s i n s o l i d s . A d i s t i n c t i o n i s made between LEED and o t h e r e l e c t r o n d i f f r a c t i o n e x p e r i m e n t s such as r e f l e c t i o n h i g h energy e l e c t r o n d i f f r a c t i o n RHEED[16], and i n e l a s t i c low energy e l e c t r o n d i f f r a c t i o n I L E E D [ 1 7 ] . The LEED experiment i n v o l v e s d i r e c t i n g an i n c i d e n t e l e c t r o n beam of w e l l - d e f i n e d energy and d i r e c t i o n a t a w e l l - c h a r a c t e r i z e d s u r f a c e and s t u d y i n g e l e c t r o n s which a r e 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 RHEED e x p e r i m e n t , which i n p r a c t i c e e s p e c i a l l y probes the s u r f a c e t o p o g r a p h y , d i r e c t s a beam of h i g h energy (10-lOOkeV) e l e c t r o n s a t g r a z i n g a n g l e s of i n c i d e n c e on t o a s u r f a c e and o b s e r v e s 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 . ILEED uses i n c i d e n t e l e c t r o n s w i t h e n e r g i e s s i m i l a r t o tho s e i n LEED and i t i n v o l v e s the st u d y of t h e i n e l a s t i c a l l y b a c k s c a t t e r e d e l e c t r o n s where b o t h energy l o s s and d i f f r a c t i o n p r o c e s s e s have o c c u r r e d . ILEED c o u l d become a u s e f u l t e c h n i q u e f o r s t u d y i n g a s p e c t s of the s u r f a c e v a l e n c e e l e c t r o n i c s t r u c t u r e ( e . g . s u r f a c e plasmon d i s p e r s i o n r e l a t i o n s ) . Even though the f i r s t LEED experiment was performed over 50 5 y e a r s ago, d i f f i c u l t i e s i n the c o n t r o l and c h a r a c t e r i z a t i o n of s u r f a c e s h i n d e r e d i t s a p p l i c a t i o n t o s u r f a c e s t u d i e s . In the 1960's, when UHV and Auger e l e c t r o n s p e c t r o s c o p y f a c i l i t i e s became r e a d i l y a v a i l a b l e , t h e r e o c c u r r e d a renewed i n t e r e s t i n LEED. To date LEED has emerged as the p r i n c i p a l method f o r s u r f a c e s t r u c t u r e d e t e r m i n a t i o n . Other s u r f a c e t e c h n i q u e s t h a t can g i v e t h i s type of i n f o r m a t i o n i n c l u d e a n g l e - r e s o l v e d p h o t o e m i s s i o n [ 1 8 ] , i o n s c a t t e r i n g [ 1 9 ] , and extended X-ray a b s o r p t i o n f i n e s t r u c t u r e [ 2 0 ] e x p e r i m e n t s . These a l t e r n a t i v e t e c h n i q u e s have not y e t been a b l e t o produce enough r e s u l t s t o a s s e s s t h e i r f u l l u s e f u l n e s s f o r s u r f a c e s t r u c t u r e d e t e r m i n a t i o n s . A r e c e n t c o m p i l a t i o n of LEED s t r u c t u r e d e t e r m i n a t i o n s f o r both c l e a n and a b s o r b a t e c o v e r e d s u r f a c e s i s g i v e n by Van Hove and T o n g [ 2 l ] . Auger e l e c t r o n s were f i r s t o b s e r v e d i n a c l o u d chamber i n 1925 by P i e r r e A u g e r [ 2 2 ] , and f o l l o w i n g a s u g g e s t i o n of L a n d e r [ 2 3 ] i n 1953, Auger e l e c t r o n s a r e now used t o g i v e i n f o r m a t i o n about the e l e m e n t a l c o m p o s i t i o n of s u r f a c e s . Auger e l e c t r o n s a r e e m i t t e d d u r i n g the r e l a x a t i o n of an i n i t i a l c o r e h o l e • s t a t e which i s u s u a l l y produced by bombarding the sample w i t h e l e c t r o n s . F i g u r e 1.1 shows s c h e m a t i c a l l y the e m i s s i o n of Auger e l e c t r o n s from aluminum. In the l a t e 1960's Auger e l e c t r o n s p e c t r o s c o p y (AES) became an e s s e n t i a l p a r t of ' c l e a n s u r f a c e ' s t u d i e s w i t h the development of c o n v e n i e n t ways f o r t a k i n g Auger s p e c t r a [ 2 4 , 2 5 ] . More d e t a i l s of the t h e o r e t i c a l and e x p e r i m e n t a l a s p e c t s of Auger e l e c t r o n s p e c t r o s c o p y are g i v e n i n Chapter 3 of t h i s t h e s i s . 6 e m i t t e d Auger e l e c t r o n i n i t i a l core h o l e T 3 118 1560 2,3 L | K Fig.1.1 Energy l e v e l diagram of aluminum i l l u s t r a t i n g the p r o d u c t i o n of L 2,3W Auger e l e c t r o n s . The energy l e v e l s were obtained from xray s t u d i e s and are given r e l a t i v e to the Fermi l e v e l . 7 1.2. SOME SURFACE NOMENCLATURE The term ' s u r f a c e ' i s somewhat ambiguous i n s o f a r as i t s meaning depends on the type of e x p e r i m e n t a l probe b e i n g used. In LEED, the t o p 2-5 atomic l a y e r s are t y p i c a l l y probed; t h i s t h i c k n e s s i s d e t e r m i n e d by the mean f r e e p a t h l e n g t h of the e l e c t r o n s ( S e c t i o n 1.3.1). The s e n s i t i v i t i e s of o t h e r s u r f a c e methods such as f i e l d i o n m i c r o s c o p y or i o n n e u t r a l i z a t i o n s p e c t r o s c o p y are not governed by the e l e c t r o n mean f r e e p a t h l e n g t h , and i n consequence t h e s e t e c h n i q u e s probe d i f f e r e n t a s p e c t s of a s u r f a c e . F i g u r e 1.2a s u g g e s t s t h a t LEED e l e c t r o n s o f t e n probe r e g i o n s where the t r i p e r i o d i c i t y of the s o l i d i s e s t a b l i s h e d . The s u r f a c e r e g i o n s t u d i e d i s c o n v e n i e n t l y d i v i d e d i n t o a ' s u b s t r a t e ' which p o s s e s s e s the t r i p e r i o d i c i t y of the b u l k and d i p e r i o d i c ' o v e r l a y e r s ' . The d i p e r i o d i c i t y of the ' s u r f a c e r e g i o n ' s t u d i e d i n v o l v e s a s u p e r p o s i t i o n of the d i p e r i o d i c o v e r l a y e r w i t h an e f f e c t i v e l y d i p e r i o d i c s u b s t r a t e . Assuming the s u p e r p o s i t i o n of the s u b s t r a t e and o v e r l a y e r p e r i o d i c i t i e s i s c o h e r e n t , the d i p e r i o d i c i t y of the s u r f a c e r e g i o n can be r e p r e s e n t e d by a t w o - d i m e n s i o n a l net w i t h u n i t v e c t o r s SJ and S2, such t h a t a l l e q u i v a l e n t p o i n t s i n the s u r f a c e r e g i o n can be g e n e r a t e d by the t r a n s l a t i o n v e c t o r s T. T = nS1 + mS2 where n,m a r e i n t e g e r s (1.2) There a r e 5 d i p e r i o d i c s u r f a c e n e t s analogous t o the 14 B r a v a i s l a t t i c e s used i n t r i p e r i o d i c c r y s t a l l o g r a p h y . These a r e shown i n F i g u r e 1.2b w i t h the c o r r e s p o n d i n g r e c i p r o c a l n e t s . The 8 F i g . 1 . 2 a Schematic i l l u s t r a t i o n of t h e ' s u r f a c e r e g i o n ' probed by a LEED experiment showing o v e r l a y e r and s u b s t r a t e , both of which a r e d i p e r i o d i c i n the x,y d i r e c t i o n . REAL f • * * SQUARE RECIPROCAL <> • 4 <> RECTANGULAR r CENTERED HEXAGONAL \&0 O B L I Q U E • f F i g . 1 . 2 b D i p e r i o d i c s u r f a c e n e t s and the c o r r e s p o n d i n g n e t s i n r e c i p r o c a l space. 9 r e c i p r o c a l net v e c t o r g(h,k) i s d e f i n e d i n e q u a t i o n 1.3 where S1* and S2*, the r e c i p r o c a l u n i t v e c t o r s , a r e r e l a t e d t o the r e a l net v e c t o r s S1,S2 by e q u a t i o n 1.4. g(h,k) = h S I * + kS2* where h,k a r e i n t e g e r s (1.3) S1* = 2f<(S2xZ)/(S1-S2xZ) S2* = 2<rf(S1xZ)/(S2-S1xZ) (1.4) r""—' r^r r+J r+~r /*—' Here Z i s a 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 the p l a n e d e f i n e d by SJ and S2. As i n o t h e r d i f f r a c t i o n phenomena, the r e c i p r o c a l net c o n s t r u c t i o n p r o v i d e s a c o n v e n i e n t means of d e s i g n a t i n g d i f f r a c t e d beams and i s u s e f u l i n the d e s c r i p t i o n of the s c a t t e r i n g of e l e c t r o n s from the s u r f a c e . The d i f f r a c t i o n p a t t e r n s observed from incommensurate s u r f a c e s t r u c t u r e s , where the s u p e r p o s i t i o n s of o v e r l a y e r and s u b s t r a t e a r e i n c o h e r e n t , have been d i s c u s s e d by E s t r u p and M c R a e f l ] . C r y s t a l l o g r a p h i c a l l y w e l l - d e f i n e d s u r f a c e s a r e n a t u r a l l y i d e n t i f i e d w i t h the M i l l e r i n d i c e s ( h k l ) of the p l a n e t h a t i s p a r a l l e l t o the s u r f a c e (e.g. the ( 1 0 0 ) , ( 1 1 0 ) , and (111) s u r f a c e s of f a c e - c e n t e r e d c u b i c s o l i d s ) . Low-index s u r f a c e s of m e t a l s o f t e n have the s t r u c t u r e e x p e c t e d from the t r u n c a t i o n of the b u l k except f o r an e x p a n s i o n or c o n t r a c t i o n of the topmost i n t e r l a y e r s p a c i n g . In c o n t r a s t , s u r f a c e s where c o v a l e n t bonds ar e i n v o l v e d commonly ' r e c o n s t r u c t ' (e.g. S i ( l 1 l ) ) . R e c o n s t r u c t i o n r e f e r s t o the rearrangement of the atoms i n the t o p few l a y e r s . 'Adsorbate' s t r u c t u r e s (e.g. Z r ( 0 0 0 1 ) - ( 2 X 2 ) - 0 ) ) c o n s i s t of a s u b s t r a t e and one or more adsorbed o v e r l a y e r s . 10 Adsor b a t e and r e c o n s t r u c t e d s u r f a c e s t r u c t u r e s a r e u s u a l l y named w i t h r e f e r e n c e t o the s u b s t r a t e , net v e c t o r s . The most s i m p l e s u r f a c e s t r u c t u r e s i n v o l v e v e c t o r s which are p a r a l l e l t o those of the s u b s t r a t e and have l e n g t h s which are i n t e g r a l m u l t i p l e s of the s u b s t r a t e v e c t o r l e n g t h s ; such a s t r u c t u r e i s commonly i d e n t i f i e d as a nxm s t r u c t u r e , where n,m are the r a t i o s of the l e n g t h s of the s u r f a c e t o s u b s t r a t e v e c t o r s . F u r t h e r d i s c u s s i o n s of s u r f a c e c r y s t a l l o g r a p h y have been g i v e n by Wood[26], and f o r more g e n e r a l s i t u a t i o n s , Park and Madden[27] have i n t r o d u c e d a m a t r i x n o t a t i o n . 1.3 LOW ENERGY ELECTRON DIFFRACTION 1.3.1 E l e c t r o n S c a t t e r i n g i n S o l i d s When a monoenergetic beam of e l e c t r o n s impinges on a s o l i d s u r f a c e the energy d i s t r i b u t i o n of the b a c k s c a t t e r e d e l e c t r o n s i s l i k e t h a t shown i n F i g u r e 1.3a. The energy d i s t r i b u t i o n of the e m i t t e d e l e c t r o n s i s c o n v e n i e n t l y d i v i d e d i n t o t h r e e r e g i o n s ; a) the e l a s t i c peak b) the i n t e r m e d i a t e energy range c) the low energy secondary peak The e l a s t i c peak i s c o m p r i s e d of the e l e c t r o n s of i n t e r e s t i n a LEED ex p e r i m e n t . T h i s peak, which c o r r e s p o n d s t o a p p r o x i m a t e l y 1-5% of the i n c i d e n t c u r r e n t a t l 0 0 e V [ 2 8 ] , i n v o l v e s e l e c t r o n s t h a t have been t r u l y e l a s t i c a l l y s c a t t e r e d as w e l l as t h o s e e l e c t r o n s t h a t have s u f f e r e d s m a l l energy l o s s e s due t o phonon s c a t t e r i n g ( the q u a s i - e l a s t i c e l e c t r o n s ) ; the l a t t e r a r e not r e s o l v e d from 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 11 i n a t y p i c a l LEED e x p e r i m e n t . The reason t h a t the e l e c t r o n s s t u d i e d i n LEED a r e s e n s i t i v e t o t h e s u r f a c e r e g i o n of the s o l i d i s i l l u s t r a t e d by F i g u r e 1.3b which shows the mean f r e e p a t h l e n g t h L, as a f u n c t i o n of e l e c t r o n energy. The mean f r e e p a t h l e n g t h i s d e f i n e d by I (I ) = I C e x p ( - V L ) (1.5) where I c i s the i n c i d e n t i n t e n s i t y a t a p a r t i c u l a r energy a t t e n u a t e d t o I (£) a f t e r p r o p a g a t i n g a d i s t a n c e S i n c e the mean f r e e p a t h l e n g t h f o r e l e c t r o n s w i t h t y p i c a l LEED e n e r g i e s i n the 30-300eV range c o r r e s p o n d s t o j u s t a few atomic l a y e r s , then i t f o l l o w s t h a t the e l e c t r o n s t h a t c o mprise the e l a s t i c peak must have been t u r n e d around by the t o p few l a y e r s . The c o m p a r a t i v e l y s h o r t mean pa t h l e n g t h i n s o l i d s , f o r the e l e c t r o n e n e r g i e s used i n LEED, r e f l e c t s the e x i s t e n c e of s t r o n g i n e l a s t i c s c a t t e r i n g . Above .1 OeV the major l o s s mechanism i s the e x c i t a t i o n of plasmons[29] which can be e n v i s a g e d as the q u a n t i z e d o s c i l l a t i o n s of the ' e l e c t r o n gas' i n the s o l i d . S p e c i f i c peaks c o r r e s p o n d i n g t o plasmon l o s s e s a r e i n d i c a t e d i n F i g u r e 1.3a a t e n e r g i e s lO-20eV below the e l a s t i c peak. The i n t e r m e d i a t e energy range of the b a c k s c a t t e r e d e l e c t r o n d i s t r i b u t i o n i n F i g u r e 1.3 shows peaks c o r r e s p o n d i n g t o Auger e l e c t r o n s and c h a r a c t e r i s t i c energy l o s s e s r e s u l t i n g from i o n i z a t i o n s and e x c i t a t i o n s of the v a l e n c e and c o r e e l e c t r o n s . The peaks due t o Auger e l e c t r o n s can be d i s t i n q u i s h e d from o t h e r l o s s peaks s i n c e t h e i r energy i s independent of the i n c i d e n t beam energy. The l a r g e low energy secondary peak r e s u l t s from 12 N(E) secondary peak e l a s t i c peak Auger and c h a r a c t e r i s t i c l o s s peaks plasmon l o s s e s ENERGY Eo Fig.1.3a) Energy d i s t r i b u t i o n of e l e c t r o n s b a c k s c a t t e r e d from a s u r f a c e f o r an i n c i d e n t beam energy Eo. t.t.oJ 1*0,000 E N E R G Y / » V Fig.1.3b) Mean f r e e path l e n g t h L(A) of 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 of e l e c t r o n energy (eV). 13 cascade p r o c e s s e s [ 4 ] where the i n c i d e n t e l e c t r o n e x c i t e s a number of s e c o n d a r i e s t h rough i n t e r a c t i o n s w i t h e l e c t r o n s i n the s o l i d . The secondary e l e c t r o n s can i n t e r a c t w i t h o t h e r e l e c t r o n s and produce t e r t i a r i e s e t c . ; a l l of which c o n t r i b u t e t o the i n t e n s i t y of the 'secondary' peak. 1.3.2 The LEED Experiment The e l e c t r o n energy a n a l y s e r of a t y p i c a l LEED experiment i s shown i n F i g u r e 1.4a. The e l e c t r o n gun d i r e c t s a beam of n e a r l y monoenergetic e l e c t r o n s towards t h e sample. The f u l l w i d t h a t h a l f maximum (FWHM) of the energy d i s t r i b u t i o n of the i n c i d e n t e l e c t r o n s i s d e t e r m i n e d p r i m a r i l y by the temperature of the e m i t t i n g f i l a m e n t ; i t i s about 0.5eV f o r a t u n g s t e n e m i t t e r a t 2000°C. The e l e c t r o n s t h a t a r e 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 sample pass through the f o u r h e m i s p h e r i c a l g r i d s ( the LEED o p t i c s ) and are a c c e l e r a t e d towards a phosphor c o a t e d s c r e e n which glows where the e l e c t r o n s s t r i k e i t . The i n n e r g r i d (G1) i s grounded and keeps the r e g i o n s u r r o u n d i n g the sample f i e l d f r e e . E l e c t r o n s w i t h e n e r g i e s g r e a t e r than the r e t a r d i n g v o l t a g e on the double g r i d (G2,G3) can pass t h r o u g h t h e s e g r i d s and then through another grounded g r i d (G4) t o be d i s p l a y e d on the phosphor s c r e e n . The use of a double g r i d f o r the r e t a r d i n g g r i d s i s t o reduce the e f f e c t s of f i e l d i n h o m o g e n i t i e s and thus improve the i n s t r u m e n t ' s energy r e s o l u t i o n ; t h i s i s p a r t i c u l a r l y i m p o r t a n t when the LEED o p t i c s a re used f o r t a k i n g Auger e l e c t r o n s p e c t r a . G4 i s grounded t o p r e v e n t the h i g h v o l t a g e on the s c r e e n (about 5kV) from p e n e t r a t i n g the r e g i o n of the double g r i d s . 14 VIEW PORT | FLUORESCENT SCREEN GRIDS F i g . 1 . 4 a ) Schematic d iagram of the LEED expe r imen t . Fig.1.4b) Schematic diagram showing how the c o n s e r v a t i o n c o n d i t i o n s determine a d i f f r a c t e d beam's d i r e c t i o n . The (0,0) beam r e s u l t s from s p e c t u l a r r e f l e c t i o n . 15 In the c o n v e n t i o n a l e x p e r i m e n t a l arrangement, the f l u o r e s c e n t s c r e e n can be viewed t h r o u g h a window from o u t s i d e the vacuum chamber and thus an e l e c t r o n d i f f r a c t i o n p a t t e r n can be o b s e r v e d d i r e c t l y . When the i n c i d e n t e l e c t r o n s a r e e l a s t i c a l l y s c a t t e r e d from a w e l l - o r d e r e d s u r f a c e , the LEED p a t t e r n i s seen t o be a r e g u l a r a r r a y of s p o t s of d i f f e r i n g i n t e n s i t i e s . The o b j e c t i v e of q u a n t i t a t i v e LEED e x p e r i m e n t s i s t o measure, a t a known d i r e c t i o n of i n c i d e n c e , the i n t e n s i t i e s of t h e s e d i f f r a c t e d beams over a range of i n c i d e n t beam energy and t h e r e b y produce i n t e n s i t y v e r s u s energy s p e c t r a or 1(E) c u r v e s . An a l t e r n a t i v e , but l e s s common, approach i s t o v a r y the a n g l e of i n c i d e n c e a t a f i x e d beam energy and produce i n t e n s i t y v e r s u s a n g l e p r o f i l e s . Examples of some 1(E) c u r v e s a r e shown i n F i g u r e 1.5 f o r d i f f r a c t e d beam i n t e n s i t i e s measured at normal i n c i d e n c e from the Cu(311) s u r f a c e compared w i t h c a l c u l a t e d 1(E) c u r v e s f o r d i f f e r e n t topmost i n t e r l a y e r s p a c i n g s . 1.3.3 The LEED D i f f r a c t i o n P a t t e r n The i n c i d e n t and d i f f r a c t e d e l e c t r o n s i n a LEED experiment t r a v e l i n f r e e space and, t o a good a p p r o x i m a t i o n , can be r e p r e s e n t e d by p l a n e waves of t y p e ; where k i s the wavevector w i t h a p p r o p r i a t e d i r e c t i o n and w i t h magnitude e q u a l t o 2 t r / x . The wavevector and energy a r e r e l a t e d by; E = t i 2 | k 2 | /2m (1.6) The d i f f r a c t i o n of the e l e c t r o n s i n t o d i s c r e t e beams by a 16 EXPERIMENT r. EXPERIMENT 50 100 150 200 50 100 150 200 Energy (eV) F i g . 1 . 5 Leed 1(E) c u r v e s f o r normal i n c i d e n c e d a t a from a c l e a n Cu(311) s u r f a c e compared w i t h 1(E) c u r v e s c a l c u l a t e d f o r d i f f e r e n t topmost i n t e r l a y e r s p a c i n g s . 17 w e l l o r d e r e d s u r f a c e i s a d i r e c t consequence of the d i p e r i o d i c t r a n s l a t i o n a l symmetry which n e c e s s i t a t e s the c o n s e r v a t i o n of momentum c o n d i t i o n i l l u s t r a t e d by e q u a t i o n 1.7. 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 t h a t a r e a n a l y z e d i n a LEED experiment a l s o s a t i s f y energy c o n s e r v a t i o n as g i v e n by e q u a t i o n 1.8. kg-„=ko +„+ g(h,k) (1.7) s>+s r^r r^t |kg-| 2 = |ko'| 2 (1.8) Here g(h,k) i s the r e c i p r o c a l net v e c t o r as d e f i n e d i n e q u a t i o n 1.2, and kp + 1 ( and kg- ( 1 are the p a r a l l e l w avevectors of the i n c i d e n t and d i f f r a c t e d e l e c t r o n waves r e s p e c t i v e l y . F i g u r e 1.4b shows how the d i r e c t i o n of a d i f f r a c t e d beam (wavevector k g " ) , depends on the v a l u e of g(h,k) and E. The c o n s e r v a t i o n of p a r a l l e l momentum a r i s e s from the c o n d i t i o n t h a t the phases of waves s c a t t e r e d a t e q u i v a l e n t s i t e s i n the s u r f a c e r e g i o n must d i f f e r by 2nri (n i n t e g e r ) f o r c o n s t r u c t i v e i n t e r f e r e n c e . The l i m i t e d p e n e t r a t i o n of 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 p r e c l u d e s s u b s t a n t i a l c o n s t r u c t i v e i n t e r f e r e n c e a s s o c i a t e d w i t h the p e r i o d i c i t y normal t o the s u r f a c e . N e v e r t h e l e s s , i n t e n s i t i e s of the d i f f r a c t e d beams are sometimes l a r g e near the e n e r g i e s t h a t s a t i s f y ; k g - - k o + = g ( h k i ) i^, r-^> ^ where g(hk£) i s a t r i p e r i o d i c r e c i p r o c a l l a t t i c e v e c t o r . The peaks i n the 1(E) c u r v e s which c o r r e s p o n d t o t h i s c o n d i t i o n a r e c a l l e d Bragg peaks. The low energy e l e c t r o n s s t u d i e d by LEED i n t e r a c t w i t h the 18 s o l i d s u r f a c e much more s t r o n g l y than do X - r a y s . E l a s t i c c r o s s - s e c t i o n s f o r TOOeV e l e c t r o n s a re of the o r d e r of 10A 2[30] and a r e t h e r e f o r e about 10 7 times g r e a t e r than t y p i c a l c r o s s - s e c t i o n s f o r X-ray d i f f r a c t i o n [ 3 1 ] . A s s o c i a t e d w i t h t h i s s t r o n g i n t e r a c t i o n , e x t r a s t r u c t u r e o c c u r s i n the 1(E) c u r v e s because many e l e c t r o n s undergo more than one s c a t t e r i n g e v e nt. These ' m u l t i p l e s c a t t e r i n g ' peaks are o f t e n more i n t e n s e than the Bragg peaks. C l e a r l y , t h e s i n g l e - s c a t t e r i n g ( k i n e m a t i c a l ) t h e o r i e s of X-ray d i f f r a c t i o n a r e inadequate f o r LEED and ' m u l t i p l e s c a t t e r i n g ' t h e o r i e s a r e r e q u i r e d (Chapter 2 ) . 1.3.4 LEED C r y s t a l l o g r a p h y The p r e s e n t l y employed method of s u r f a c e s t r u c t u r e d e t e r m i n a t i o n w i t h LEED i s t o propose a t r i a l s t r u c t u r e and then c a l c u l a t e 1(E) c u r v e s f o r t h e e l e c t r o n beams d i f f r a c t e d from t h i s s t r u c t u r e . The proposed s t r u c t u r e i s then a l t e r e d u n t i l the c a l c u l a t e d and e x p e r i m e n t a l 1(E) c u r v e s show the c l o s e s t c o r r e s p o n d e n c e . The comparison of the 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) c u r v e s has o f t e n been done v i s u a l l y by matching the e n e r g i e s and shapes of a l l the peaks i n the 1(E) c u r v e s . F i g u r e 1.5 shows the comparison of two e x p e r i m e n t a l 1(E) c u r v e s w i t h c a l c u l a t e d 1(E) c u r v e s f o r v a r i o u s topmost i n t e r l a y e r s p a c i n g s . N o r m a l l y , 10 or more 1(E) c u r v e s a re measured, and o f t e n f o r more than one a n g l e of i n c i d e n c e . The l a r g e amount of d a t a i n v o l v e d when t h e r e a re many p o s s i b l e s t r u c t u r e s , and many e x p e r i m e n t a l 1(E) c u r v e s , has made the v i s u a l assessment approach e x t r e m e l y cumbersome. R e c e n t l y , n u m e r i c a l ' r e l i a b i l i t y i n d i c e s ' have been proposed which p u r p o r t t o q u a n t i f y l e v e l s of 19 c o r r e s p o n d e n c e . The work i s t h i s t h e s i s uses the r e l i a b i l i t y i n d e x approach t o LEED c r y s t a l l o g r a p h y and examines many of the p r o c e d u r e s t h a t a r e f o l l o w e d i n s t r u c t u r a l d e t e r m i n a t i o n s . A l t e r n a t i v e methods t o d e r i v e s t r u c t u r a l i n f o r m a t i o n from the e x p e r i m e n t a l LEED da t a a r e b e i n g i n v e s t i g a t e d . These methods i n c l u d e the c o n s t a n t momentum t r a n s f e r a v e r a g i n g approach!32] (CMTA), and F o u r i e r t r a n s f o r m t e c h n i q u e s ! 3 3 ] . The CMTA method av e r a g e s l a r g e s e t s of e x p e r i m e n t a l i n t e n s i t y d a t a a t c o n s t a n t momentum t r a n s f e r t o the s u r f a c e , w i t h the hope of r e d u c i n g 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 so t h a t comparisons can be made w i t h k i n e m a t i c c a l c u l a t i o n s f o r d i f f e r e n t proposed s u r f a c e models. The F o u r i e r t r a n s f o r m t e c h n i q u e s attempt t o p r o v i d e ' d i r e c t ' approaches t o s t r u c t u r a l i n f o r m a t i o n , analogous w i t h the P a t t e r s o n f u n c t i o n method of X-ray c r y s t a l l o g r a p h y . Both of t h e s e approaches a v o i d complex 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 , but so f a r they have not y e t p r o v i d e d g e n e r a l l y a p p l i c a b l e p r o c e d u r e s f o r d e t e r m i n i n g s u r f a c e s t r u c t u r e s . 1.4 INSTRUMENTAL RESPONSE, DOMAINS AND DISORDER The LEED e l e c t r o n gun u s u a l l y produces a beam t h a t i s about 1 mm i n d i a m e t e r and so i n t e r s e c t s an a r e a on the t o p s u r f a c e l a y e r t h a t c o n t a i n s about 1 0 1 3 atoms. However, the i n c i d e n t beam p o s s e s s e s a n g u l a r and energy d e v i a t i o n s a s s o c i a t e d w i t h the f i n i t e s i z e and h i g h t e m p e r a t u r e of the s o u r c e , and these f a c t o r s d e f i n e an i n s t r u m e n t a l response f u n c t i o n which y i e l d s a c h a r a c t e r i s t i c d i m e n s i o n , the t r a n s f e r w i d t h W, over which the i n s t r u m e n t i s s e n s i t i v e as a probe of t h e s u r f a c e p e r i o d i c i t y [ 3 4 ] . The LEED i n s t r u m e n t can o n l y r e s o l v e a n g u l a r 20 d i s t r i b u t i o n s of e l e c t r o n s which r e s u l t from d i f f r a c t i o n by f e a t u r e s on the s u r f a c e w i t h p e r i o d i c i t i e s s m a l l e r than W, or a l t e r n a t i v e l y , the i n s t r u m e n t o b s c u r e s i n f o r m a t i o n about s u r f a c e s t r u c t u r e s w i t h p e r i o d i c i t i e s g r e a t e r than the t r a n s f e r w i d t h . Park e t a l [ 3 4 ] found t h a t t y p i c a l t r a n s f e r w i d t h s ranged from 20 o t o 100A. The r e s t r i c t i o n of the a r e a s t h a t c o n t r i b u t e t o LEED i n t e n s i t i e s i s s i g n i f i c a n t s i n c e i t ens u r e s t h a t the observed d i f f r a c t i o n p a t t e r n i s the sum of a v e r y l a r g e number of i n d i v i d u a l d i f f r a c t i o n p a t t e r n s ; each from a r e g i o n on the o r d e r of 10*A 2. An i n t e r e s t i n g o b s e r v a t i o n f o r (0001) s u r f a c e s of hep m e t a l s i s p r o v i d e d by t h e i r LEED p a t t e r n s which show 6 - f o l d symmetry r a t h e r than the 3 - f o l d symmetry e x p e c t e d f o r the p e r f e c t t r u n c a t i o n of the b u l k . The c o n v e n t i o n a l e x p l a n a t i o n can be i n d i c a t e d w i t h r e f e r e n c e t o F i g u r e 1.6. T h i s shows the presence of two d i f f e r e n t 'domains' or i s l a n d s w i t h s u r f a c e net v e c t o r s r e l a t e d by a 180° r o t a t i o n . These domains r e s u l t from the p r e s e n c e of monoatomic s t e p s on the s u r f a c e . For domains t h a t a r e l a r g e r than the ar e a a s s o c i a t e d w i t h the i n s t r u m e n t a l t r a n s f e r w i d t h and a r e s m a l l e r than the e l e c t r o n beam w i d t h , the observ e d d i f f r a c t i o n p a t t e r n s i n v o l v e a summation over p a t t e r n s from the d i f f e r e n t domains. When the p o p u l a t i o n s of the two s e t s of domains a r e a p p r o x i m a t e l y e q u a l , 6 - f o l d symmetry r e s u l t s f o r the 'domain-averaged' LEED p a t t e r n . For o v e r l a y e r systems, LEED p a t t e r n s u s u a l l y show the r o t a t i o n a l symmetry of the s u b s t r a t e , even though each domain may have lower symmetry. T h i s can i n t r o d u c e a m b i g u i t i e s i n i n t e r p r e t a t i o n . For example, the same LEED p a t t e r n s a r e produced from (2x2) o v e r l a y e r s oh a h c p ( 0 0 0 l ) 21 Fig.1.6 S t r u c t u r e of the hcp(OOOl) surface showing the two p o s s i b l e domains that r e s u l t from the t r u n c a t i o n of the bulk s t r u c t u r e . Larger c i r c l e s represent atoms i n the topmost atomic l a y e r , f u l l and open c i r c l e s atoms i n l a y e r s A and B r e s p e c t i v e l y . The two domains are r e l a t e d to each other by a 180° r o t a t i o n . Axes s e l e c t i o n and u n i t c e l l v e c t o r s a f t e r Van Hove [21]. 22 Fig.1.7 Three 120° r e l a t e d domains of (2x1) o v e r l a y e r s t r u c t u r e s on a hcp(OOOl) s u r f a c e . X's rep r e s e n t atoms i n the topmost l a y e r of the hcp(OOOl) s u b s t r a t e and dark c i r c l e s r epresent the atoms i n the o v e r l a y e r . 23 s u r f a c e , as from t h r e e (2x1) domains r e l a t e d by 120° r o t a t i o n s as i n d i c a t e d i n F i g u r e 1.7. An a n a l y s i s of beam i n t e n s i t i e s i s n e c e s s a r y i n o r d e r t o d i s t i n g u i s h between the s e p o s s i b i l i t i e s . The d i s c u s s i o n of the f o r m a t i o n of LEED p a t t e r n s assumed d i f f r a c t i o n from p e r f e c t l y o r d e r e d s u r f a c e s . R e a l s u r f a c e s i n e v i t a b l y have i m p e r f e c t i o n s ; the presence of s t e p s , k i n k s , and d i s l o c a t i o n s i s w e l l - k n o w n [ 3 5 ] . I t i s i m p o r t a n t t o know what e f f e c t s such f e a t u r e s can have on LEED p a t t e r n s and beam i n t e n s i t i e s . The LEED p a t t e r n i s not always s e n s i t i v e t o the presence of s u r f a c e d i s o r d e r . S t u d i e s of S i ( l 1 1 ) s u r f a c e s [ 3 6 ] have shown t h a t up t o 20% of the S i atoms on the s u r f a c e can be randomly d i s t r i b u t e d w i t h o u t c a u s i n g any o b s e r v a b l e changes i n the LEED p a t t e r n . The presence of r e g u l a r s t e p s , f a c e t i n g , and domains w i t h d i m e n s i o n s l e s s than the t r a n s f e r w i d t h , l e a d s t o the s p l i t t i n g or b r o a d e n i n g of the LEED d i f f r a c t i o n s p o t s . I n f o r m a t i o n about the d i s t r i b u t i o n of domain s i z e s can be o b t a i n e d by s t u d i e s of the a n g u l a r p r o f i l e s of the d i f f r a c t e d beams[37]. G e n e r a l l y , the main e f f e c t of the d i s o r d e r e d a r e a s i s t o reduce the a b s o l u t e beam i n t e n s i t i e s and t o produce a smoothly v a r y i n g background. However, t h e r e i s e v i d e n c e t h a t d i s o r d e r may a l s o a l t e r r e l a t i v e peak i n t e n s i t i e s ! 3 8 ] and t h i s p o s s i b i l i t y s h o u l d be c o n s i d e r e d when t a k i n g i n t e n s i t y measurements. D i s o r d e r r e s u l t i n g from the t h e r m a l v i b r a t i o n s of the atoms i s n o r m a l l y a c c o u n t e d f o r when c a l c u l a t i n g the d i f f r a c t e d beam i n t e n s i t i e s (Chapter 2 ) . 24 1.5 THIS WORK The work d e s c r i b e d i n t h i s t h e s i s r e p r e s e n t s a c o n t r i b u t i o n t o the development of the s u b j e c t of LEED c r y s t a l l o g r a p h y . S p e c i f i c s t u d i e s have been made f o r c l e a n (311) s u r f a c e s of the f a c e - c e n t e r e d c u b i c ( f e e ) m e t a l s copper and n i c k e l , and f o r (0001) s u r f a c e s of z i r c o n i u m b oth when c l e a n and when c o n t a i n i n g adsorbed oxygen. The (311) s u r f a c e s were s t u d i e d as c o m p a r a t i v e l y s i m p l e examples of s t e p p e d s u r f a c e s , and t h i s work was the f i r s t LEED study of a s tepped s u r f a c e t h a t employed 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 t o determine the s u r f a c e geometry. Stepped s u r f a c e s a r e of i n t e r e s t t o c h e m i s t s as they have been shown t o p o s s e s s d i f f e r e n t r e a c t i v i t i e s ( e . g . f o r c a t a l y t i c r e a c t i o n s ) from the f l a t low index s u r f a c e s of the same m e t a l [ 3 9 , 4 0 ] . These (311) s u r f a c e s were amenable t o d e t a i l e d LEED a n a l y s e s because they can be r e p r e s e n t e d by a p r i m i t i v e l a t t i c e which i s s u f f i c i e n t l y s m a l l f o 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 t o be p r a c t i c a l . As p a r t of the study of N i ( 3 1 1 ) , a comparison was made of two r e l i a b i l i t y i n d i c e s t h a t have been proposed f o r use i n LEED c r y s t a l l o g r a p h y [ 4 1 , 4 2 ] . S t u d i e s on Z r ( 0 0 0 l ) were i n i t i a t e d f o r s e v e r a l r e a s o n s . No q u a n t i t a t i v e LEED work has been done on t h i s m e t a l , a l t h o u g h T i ( 0 0 0 l ) shows some i n t e r e s t i n g f e a t u r e s i n i t s 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 [ 4 3 , 4 4 ] . I t was hoped t h a t some comparisons c o u l d be made w i t h T i ( 0 0 0 1 ) by s t u d y i n g the a d s o r p t i o n of oxygen on Z r ( 0 0 0 l ) . Z i r c o n i u m i s a l s o a s p e c i a l c h a l l e n g e because i t i s h i g h l y r e a c t i v e and because i t undergoes a phase t r a n s i t i o n t o the body c e n t e r e d c u b i c form a t around 865°C[45]; b o t h t h e s e f a c t o r s c o m p l i c a t e the p r o c e d u r e s f o r c l e a n i n g s u r f a c e s of t h i s 25 m e t a l . B r i e f l y the o u t l i n e of t h i s t h e s i s i s as f o l l o w s . Chapter 2 o u t l i n e s m u l t i p l e s c a t t e r i n g t h e o r i e s of low energy e l e c t r o n d i f f r a c t i o n and d e s c r i b e s the s a l i e n t f e a t u r e s of the computer programs used here f o r c a l c u l a t i n g d i f f r a c t e d beam i n t e n s i t i e s . Chapter 3 i n c l u d e s an e x a m i n a t i o n of the e x p e r i m e n t a l a s p e c t s of LEED/Auger s t u d i e s , as w e l l as d e s c r i b i n g the r e l i a b i l i t y index r o u t i n e s used i n the comparisons of the 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) c u r v e s . Chapter 4 d e t a i l s the work on the (311) s u r f a c e s of copper and n i c k e l and i t s i m p l i c a t i o n s t o broader a s p e c t s of LEED c r y s t a l l o g r a p h y , e s p e c i a l l y w i t h r e g a r d t o the use of r e l i a b i l i t y i n d i c e s . F i n a l l y , Chapter 5 d e s c r i b e s s t u d i e s on the (0001) s u r f a c e of z i r c o n i u m , both when c l e a n and when c o n t a i n i n g adsorbed oxygen. 26 CHAPTER 2 LEED INTENSITY CALCULATIONS 27 T h i s c h a p t e r w i l l b r i e f l y d e s c r i b e the t h e o r e t i c a l approach t o the problem of e l e c t r o n s . s c a t t e r e d by a s o l i d s u r f a c e . The emphasis i s on the p h y s i c a l n a t u r e of the s c a t t e r i n g p r o c e s s and on the u t i l i z a t i o n of the computer programs t h a t were a v a i l a b l e f o r our LEED c r y s t a l l o g r a p h i c s t u d i e s . 2.1 INTRODUCTION The f i r s t a t t e m p t s at d e v e l o p i n g r e a l i s t i c 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 took p l a c e i n the l a t e 1960's[46,47]. S i g n i f i c a n t advances were the i n c l u s i o n of an i m a g i n a r y s e l f - e n e r g y t o account f o r a b s o r p t i o n due t o i n e l a s t i c s c a t t e r i n g [ 4 8 ] and improved models f o r the s c a t t e r i n g p o t e n t i a l [ 4 9 ] . V a r i o u s schemes of c a l c u l a t i o n have been d e v e l o p e d t o c a l c u l a t e LEED 1(E) c u r v e s and d e t a i l e d s t a t e m e n t s have been g i v e n i n the r e v i e w s by Duke[50], T o n g [ 5 l ] , and P e n d r y [ 5 2 ] , The c o m p u t a t i o n s made i n t h i s work are based on programs s u p p l i e d by M. A. Van Hove; these a r e d e s c r i b e d i n the monograph by Van Hove and T o n g [ 2 1 j . W i t h i n a c r y s t a l , two d i f f e r e n t r e p r e s e n t a t i o n s of the s c a t t e r e d w a v e f i e l d are n o r m a l l y used, and they a r e suggested by two a s p e c t s of the o v e r a l l s c a t t e r i n g problem. In the K-space, or beam, r e p r e s e n t a t i o n the w a v e f i e l d i s r e p r e s e n t e d by p l a n e waves, which a r e the n a t u r a l f u n c t i o n s f o r e l e c t r o n s moving i n r e g i o n s of c o n s t a n t p o t e n t i a l . The L-space, or s p h e r i c a l wave, r e p r e s e n t a t i o n i s n a t u r a l f o r c o n s i d e r i n g the w a v e f i e l d i n the r e g i o n of an atom, where a s p h e r i c a l p o t e n t i a l p r o v i d e s a u s e f u l a p p r o x i m a t i o n . In g e n e r a l p l a n e waves can be d e s c r i b e d i n terms of a s e r i e s e x p a n s ion of s p h e r i c a l waves[53], and v i c e v e r s a , 28 but f o r p a r t i c u l a r a s p e c t s of the o v e r a l l s c a t t e r i n g problem one of t h e s e r e p r e s e n t a t i o n s may be more c o n v e n i e n t [ 5 4 , 5 5 ] . Most methods 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 break down the whole s c a t t e r i n g problem i n t o s i m p l e r u n i t s , which s t a r t by c a l c u l a t i n g s c a t t e r i n g a m p l i t u d e s from i n d i v i d u a l atoms, which a r e assumed t o be s p h e r i c a l l y symmetric. Next the s c a t t e r i n g w i t h i n whole l a y e r s of atoms p a r a l l e l t o the s u r f a c e i s c a l c u l a t e d , and then the d i f f e r e n t l a y e r s a r e s t a c k e d and 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 . F i n a l l y , the w a v e f i e l d o u t s i d e the c r y s t a l i s matched t o the w a v e f i e l d i n s i d e the c r y s t a l a t the s o l i d - v a c u u m i n t e r f a c e [ 4 7 ] . The w a v e f i e l d o u t s i d e the c r y s t a l t a k e s the form Y ( r ) = e x p ( i k o + - r)+]>C g e x p ( i k g " . r ) (2.1) y The o b j e c t i v e of a LEED c a l c u l a t i o n i s t o compute the beam r e f l e c t i v i t i e s , Rg= k£-ko* C o g | 2 (2.2) which r e l a t e t o the r e l a t i v e i n t e n s i t i e s of the measured 1(E) c u r v e s . 2.2 PHYSICAL PARAMETERS FOR LEED CALCULATIONS As w e l l as a s t r u c t u r a l model, v a r i o u s n o n - g e o m e t r i c a l parameters a r e r e q u i r e d f o r the c o m putation of 1(E) c u r v e s ; the l a t t e r s p e c i f i c a l l y i n v o l v e the atomic s c a t t e r i n g f a c t o r s , s c a t t e r i n g by the e l e c t r o n gas of the s o l i d , and the e f f e c t s of the atomic v i b r a t i o n a l m o t i o n s . A d i s c u s s i o n of t h e s e parameters i s i n c l u d e d i n t h i s s e c t i o n . 29 2.2.1 The Ion Core P o t e n t i a l A h e l p f u l model f o r the c r y s t a l p o t e n t i a l i s t h a t c o n v e n t i o n a l l y used i n band s t r u c t u r e c a l c u l a t i o n s . T h i s model i s c a l l e d the m u f f i n t i n a p p r o x i m a t i o n , and i s i l l u s t r a t e d i n F i g u r e 2 . 1 [ 5 6 ] . The l a r g e s t p o s s i b l e n o n - o v e r l a p p i n g spheres a r e drawn about each atom c e n t e r . A s p h e r i c a l l y symmetric p o t e n t i a l i s used i n s i d e each sphere and the p o t e n t i a l between the spheres i s t a ken t o be c o n s t a n t . In LEED, e l e c t r o n s a r e e s p e c i a l l y b a c k s c a t t e r e d by the i o n c o r e p o t e n t i a l s (where the e f f e c t i v e n u c l e a r p o t e n t i a l i s s t r o n g ) . The w a v e f u n c t i o n of an e l e c t r o n i n s i d e a m u f f i n t i n sphere i s c a l c u l a t e d by s o l v i n g the S c h r o d i n g e r e q u a t i o n (-h 2v 2/2m+ V ) Y =E¥ , (2.3) where V i s the sum of the c o u l o m b i c p a r t of the i o n c o r e p o t e n t i a l and a c o n t r i b u t i o n f o r exchange. In LEED c a l c u l a t i o n s , the a tomic p o t e n t i a l s used a r e o f t e n 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 , examples of which are g i v e n by M o r u z z i et a l [ 5 7 ] , but p o t e n t i a l s c a l c u l a t e d f o r c l u s t e r models are a l s o u s e d [ 5 8 ] . The l a t t e r c o u l d be i m p o r t a n t f o r a d s o r p t i o n systems where the a p p r o p r i a t e band s t r u c t u r e p o t e n t i a l may not be a v a i l a b l e . The exchange term i n the p o t e n t i a l i s o f t e n r e l a t e d t o the charge d e n s i t y p ( r ) by S l a t e r ' s e x p r e s s i o n Vexc = - 6 A [ 3 p ( r ) / 8 r l ] 1 / 3 , (2.4) w i t h v a l u e s of A t a b u l a t e d by Schwarz[59,60]. I t has been suggested t h a t (2.4) i n c l u d e s some c o r r e c t i o n f o r c o r r e l a t i o n . 30 For 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 f o r the C U ( 1 1 1 ) s u r f a c e , Tear e t a l [ 6 l ] v a r i e d A t o g i v e the best f i t between the c a l c u l a t e d and e x p e r i m e n t a l 1(E) c u r v e s . The a s y m p t o t i c form of the s o l u t i o n t o e q u a t i o n ( 2 . 3 ) , t h a t i s a t a l o n g d i s t a n c e from the atom where V=0, i s y ~ A [ e x p ( i k - r ) + f (9 S,E)r~ 1 e x p ( i k - r ) ] (2.5) where f (9 s, E) , the s t r u c t u r e f a c t o r , can be expanded i n terms of s p h e r i c a l waves f (0s,E) = 1 / 2 i k ^ ( 2 i + 1 )sin (S t)exp ( iS x)P e ( c o s ( 9 ) (2.6) In ( 2 . 6 ) , the Sj> are c a l l e d phase s h i f t s , the P^(cose) are Legendre p o l y n o m i a l s , and 9 s i s the a n g l e t h r o u g h which the e l e c t r o n i s s c a t t e r e d . The phase s h i f t s can be e v a l u a t e d by m atching the d e r i v a t i v e s of the s o l u t i o n of ( 2 . 3 ) , f o r the a p p r o p r i a t e i o n c o r e p o t e n t i a l , w i t h the a s y m p t o t i c form i n e q u a t i o n (2.5) a t the m u f f i n t i n r a d i u s . The phase s h i f t s a r e dependent on e l e c t r o n energy, as w e l l as the p a r t i c u l a r atom, and i n p r i n c i p l e t h e e x p a n s i o n f o r f(9 S,E) i n v o l v e s an i n f i n i t e number of A. v a l u e s . However, i n p r a c t i c e convergence i s s u f f i c i e n t l y r a p i d t h a t o n l y a c o m p a r a t i v e l y s m a l l number of phase s h i f t s a r e r e q u i r e d . IMAGINARY POTENTIAL VACUUM LEVEL REAL POTENTIAL ADSORBATE LAYER SPACING SUBSTRATE LAYERS PACINGS SUBSTRATE 'NO REFLECTION MATCHING ADSORBED LAYER TRANSITION REGION Atanh(Bz+ CI 32 2.2.2 The Constant P o t e n t i a l Vo The c o n s t a n t p o t e n t i a l between the m u f f i n t i n spheres changes the energy of the e l e c t r o n s i n s i d e the c r y s t a l . The energy i n s i d e the c r y s t a l can be r e l a t e d t o the e l e c t r o n energy i n vacuum by E ( i n s i d e ) = E ( v a c u u m ) - V o (2.7) where Vo i s , i n g e n e r a l , c o n s t r u c t e d o-f a r e a l p a r t (Vor) and an i m a g i n a r y p a r t ( V o i ) Vo=Vor + i V o i (2.8) i ) Vor The r e a l p a r t of the c o n s t a n t p o t e n t i a l (Vor) accommodates the i n c r e a s e i n energy as a r e s u l t of the a t t r a c t i v e p o t e n t i a l 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 . Vor u s u a l l y has a v a l u e between -5 and -15 eV. The v a l u e of |Vor| i s a p p r o x i m a t e l y e q u a l t o 0+Ef where 0 i s the work f u n c t i o n and Ef i s the Fermi energy. In p r i n c i p l e exchange and c o r r e l a t i o n e f f e c t s g i v e Vor an energy dependence[62], but n e v e r t h e l e s s i t i s o f t e n an adequate a p p r o x i m a t i o n t o t r e a t Vor as c o n s t a n t . I n c r e a s i n g |Vor| s h i f t s the peaks i n the c a l c u l a t e d 1(E) c u r v e s t o lower v a l u e s of E(vacuum). In the p r e s e n t work, a p l a u s i b l e v a l u e of Vor i s used f o r 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 t h e n , d u r i n g the comparison 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 d a t a , the c a l c u l a t e d 1(E) c u r v e s a r e g i v e n a c o n s t a n t s h i f t t o p r o v i d e t h e b e s t agreement w i t h the e x p e r i m e n t a l 1(E) c u r v e s . T h i s p r o c e d u r e e f f e c t i v e l y r e f i n e s the v a l u e of Vor, which i s t h e r e f o r e an e m p i r i c a l parameter. In t h i s t h e s i s , Vor i s o f t e n c a l l e d the i n n e r p o t e n t i a l . 33 i i ) V o i The importance of i n e l a s t i c s c a t t e r i n g t o LEED was e s t a b l i s h e d i n S e c t i o n 1.3.1; t h i s i s i n c l u d e d p h e n o m e n o l o g i c a l l y i n the c a l c u l a t i o n s by an i m a g i n a r y c o n t r i b u t i o n ( V o i ) t o the c o n s t a n t p o t e n t i a l between the m u f f i n t i n s p h e r e s . The e f f e c t of V o i i s t o i n t r o d u c e an i m a g i n a r y component i n t o the wavevectors f o r e l e c t r o n s i n s i d e the c r y s t a l , and i n t u r n t h i s r e s u l t s i n a b s o r p t i o n . For the g e n e r a l time dependence T ( r , t ) = ( p ( r ) e x p ( - i E t / h ) , the i n t e n s i t y \~Y\ 2 i s damped, a f t e r p r o p a g a t i n g f o r time t , by the f a c t o r exp(2Voit/ft) when V o i i s a n e g a t i v e q u a n t i t y . T h i s s e t s a minimum w i d t h of 2| V o i | on peaks i n the c a l c u l a t e d 1(E) c u r v e s . The dependence of peak w i d t h on V o i i s a consequence of the H e i s e n b e r g u n c e r t a i n t y p r i n c i p l e and the f i n i t e l i f e t i m e of the d e c a y i n g wave[2]. For LEED e n e r g i e s i n the range 30 t o 300eV, v a l u e s of V o i from -2 t o -8eV a r e t y p i c a l , w i t h V o i g e n e r a l l y b e i n g a s l o w l y v a r y i n g f u n c t i o n of energy. A s u i t a b l e v a l u e f o r V o i i n the c a l c u l a t i o n s i s n o r m a l l y o b t a i n e d by r e f e r e n c e t o peak w i d t h s o b s e r v e d i n the e x p e r i m e n t a l 1(E) c u r v e s . Emphasis i s on peaks which appear e s s e n t i a l l y k i n e m a t i c a l ; c a r e i s needed because m u l t i p l e s c a t t e r i n g r e s u l t s i n b r o a d e n i n g due t o the presence of o v e r l a p p i n g peaks. Jepsen et a l [ 6 3 ] proposed the f u n c t i o n a l form Voi=-BE 1/3 (2.9) f o r s i l v e r . T h i s energy dependence i s assumed i n the p r e s e n t 34 work, and has been f a i r l y w i d e l y used f o r m e t a l s . The parameter B i s d e t e r m i n e d from an e x a m i n a t i o n of the narrowest peaks i n the e x p e r i m e n t a l 1(E) c u r v e s . 2.2.3 Atomic M o t i o n s The atomic motions a s s o c i a t e d w i t h a v i b r a t i n g l a t t i c e cause many e l e c t r o n s t o be s c a t t e r e d i n c o h e r e n t l y . The i n t e n s i t i e s of the d i f f r a c t e d e l e c t r o n beams are thus reduced and the background i n t e n s i t y i n c r e a s e s compared w i t h the c a s e of a r i g i d i d e a l s o l i d . S i n c e the time an i n c i d e n t e l e c t r o n i n t e r a c t s w i t h an atom i s much s h o r t e r than the p e r i o d of an atomic v i b r a t i o n , the LEED experiment measures the average atomic p o s i t i o n s by sampling over the 'snap s h o t ' p o s i t i o n s of many atoms. The i n t e n s i t i e s a r e f u r t h e r averaged s i n c e the p e r i o d of d a t a a c q u i s i t i o n i s r e l a t i v e l y l o n g (on the o r d e r of a s e c o n d ) . For k i n e m a t i c c a l c u l a t i o n s , a d i f f r a c t e d beam i n t e n s i t y I from a v i b r a t i n g l a t t i c e i s reduced from t h a t ( I o ) f o r a r i g i d l a t t i c e by I=Ioexp[-2M] (2.10a) M = 3 | Kg" -Ko* | 2T/2mk 69 I ) 2 (2.10b) and exp[-M] i s o f t e n r e f e r r e d t o as the Debye W a l l e r f a c t o r . In (2.10b), k B i s the Boltzmann c o n s t a n t , m i s the atomic mass, and 9 P i s t h e Debye te m p e r a t u r e . The dependence on kg~-ko + r e s u l t s i n a g r e a t e r r e d u c t i o n i n i n t e n s i t y f o r h i g h e r e l e c t r o n e n e r g i e s . 9j> has the u n i t s of temperature and i s i n v e r s e l y r e l a t e d t o the atomic v i b r a t i o n a m p l i t u d e s . In the h i g h 35 t e m p e r a t u r e l i m i t (T>>9 D) r 9 p 2=9T/mk B<(*r) 2>, (2.11) where < ( ^ r ) 2 > i s the mean square v i b r a t i o n a m p l i t u d e . With m u l t i p l e s c a t t e r i n g , o n l y the a m p l i t u d e s s t r i c t l y f o l l o w a Debye-Waller type dependence, a l t h o u g h i n p r a c t i c e , e x p r e s s i o n s such as (2.10a) o f t e n p r o v i d e a,good a p p r o x i m a t i o n . Averaged Debye t e m p e r a t u r e s f o r the s u r f a c e r e g i o n have o f t e n been determined by measuring LEED i n t e n s i t i e s as a f u n c t i o n of t e m p e r a t u r e [ 6 4 , 6 5 ] . S u r f a c e v a l u e s a r e g e n e r a l l y lower than b u l k Debye t e m p e r a t u r e s from X-ray d i f f r a c t i o n s t u d i e s ; t h i s r e f l e c t s the enhanced v i b r a t i o n a l a m p l i t u d e s of atoms i n the topmost atomic l a y e r s . For i s o t r o p i c s c a t t e r e r s , and n e g l e c t i n g c o r r e l a t i o n s between the motions of n e i g h b o u r i n g atoms, the e f f e c t s of t h e r m a l v i b r a t i o n s can be i n c l u d e d i n 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 m o d i f y i n g the phase s h i f t s a p p r o p r i a t e f o r the r i g i d l a t t i c e a c c o r d i n g t o T h i s has the e f f e c t of making the temperature dependent phase s h i f t s complex; a l s o an i n c r e a s e d number of phase s h i f t s a re r e q u i r e d f o r convergence a t any g i v e n energy compared w i t h the r i g i d l a t t i c e . For e n e r g i e s up t o 200eV, e i g h t phase s h i f t s (2=0 t o 7) are u s u a l l y adequate f o 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 of 1(E) c u r v e s . f (ef,T) = f (G)exp[-M] (2.12) 36 2.2.4 The Solid-Vacuum I n t e r f a c e Both Vor and V o i te n d t o z e r o i n vacuum a t a l o n g d i s t a n c e from the s o l i d . However, the d e t a i l e d s c a t t e r i n g of i n c i d e n t e l e c t r o n s by the c r y s t a l must depend on the s p e c i f i c v a r i a t i o n s of Vor and V o i through the s o l i d - v a c u u m i n t e r f a c e . A p a r t i c u l a r s i t u a t i o n i s i l l u s t r a t e d i n F i g u r e 2.1. O f t e n i t t u r n s out t o be adequate t o i g n o r e t h i s e f f e c t and t o s e t both Vor and V o i e q u a l t o z e r o a t a s t e p a t the outermost e x t e n t of the t o p atomic l a y e r . T h i s a p p r o x i m a t i o n i s o n l y , inadequate f o r s h a l l o w a n g l e s of i n c i d e n c e or f o r low e n e r g i e s (<40eV) where narrow e x t r a s t r u c t u r e can be observed i n e x p e r i m e n t a l 1(E) c u r v e s [ 6 6 ] . T h i s s t r u c t u r e i s a t t r i b u t e d t o a resonance phenomenon, which i n v o l v e s evanescent e l e c t r o n waves b e i n g t r a p p e d between the s u r f a c e p o t e n t i a l b a r r i e r and the s u b s t r a t e p r i o r t o s c a t t e r i n g i n t o d i f f r a c t e d beams t h a t can emerge from the c r y s t a l [ 67']. Such f e a t u r e s a r e u s u a l l y seen i n 1(E) c u r v e s a t e n e r g i e s j u s t below t h a t f o r emergence of another d i f f r a c t e d beam. R e f r a c t i o n of e l e c t r o n s a t the s o l i d - v a c u u m i n t e r f a c e i s accommodated 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 by Vor. At normal i n c i d e n c e r e f r a c t i o n o n l y o c c u r s when the d i f f r a c t e d beams emerge from the c r y s t a l , and so the s c a t t e r i n g p a t h s w i t h i n the c r y s t a l a re not a f f e c t e d . For o f f - n o r m a l i n c i d e n c e , an i n c i d e n t beam r e f r a c t s upon e n t e r i n g a c r y s t a l . The i n c l u s i o n of r e f r a c t i o n 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 i s t h e r e f o r e more i m p o r t a n t f o r i n c i d e n t a n g l e s w e l l away from the normal. The c a l c u l a t i o n s made i n t h i s work • i n c l u d e the e f f e c t s of r e f r a c t i o n , but n e g l e c t s c a t t e r i n g by the s u r f a c e b a r r i e r a t the 37 s o l i d - v a c u u m i n t e r f a c e . I t would be p o s s i b l e t o i n c l u d e such e f f e c t s by i n c l u d i n g an a d d i t i o n a l ' o v e r l a y e r ' w i t h 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 p p r o p r i a t e f o r a model p o t e n t i a l of the s u r f a c e b a r r i e r . ( S e c t i o n 2.3) 2.3 MULTIPLE SCATTERING METHODS The f i r s t s t e p i n the c a l c u l a t i o n of beam r e f l e c t i v i t i e s i s the c o m p u t a t i o n of the s c a t t e r i n g a m p l i t u d e s of the i n d i v i d u a l atoms. T h i s r e q u i r e s a s u i t a b l e i o n c o r e p o t e n t i a l and a v a l u e f o r the Debye t e m p e r a t u r e , i n o r d e r t o determine the temperature dependent phase s h i f t s [ 2 ] . The next s t e p i s t h e c a l c u l a t i o n of the l a y e r d i f f r a c t i o n m a t r i c e s M. The p a r t i c u l a r element Mg'g g i v e s the a m p l i t u d e of the p l a n e wave w i t h wavevector kg' which i s s c a t t e r e d from a l a y e r f o r an i n c i d e n t p l a n e wave of u n i t a m p l i t u d e and wavevector kg. S u p e r s c r i p t s i g n s a r e a l s o added t o i n d i c a t e the p l a n e wave d i r e c t i o n s ; s p e c i f i c a l l y , a + s i g n f o r p r o p a g a t i o n i n t o the c r y s t a l and a - s i g n f o r p r o p a g a t i o n out of the c r y s t a l . F i g u r e 2.2a s c h e m a t i c a l l y i l l u s t r a t e s t h i s s i t u a t i o n . The g e n e r a l e x p r e s s i o n [ 2 ] f o r Mg'g i s ± t ^ - -Mg'g = 8 r r 2 i , Z, Y,(kg' ) [ 1-X] - \ Y ,(kg)exp( i§n)sin(Sp ) (2.12) Ako kg^_ LL' ~ ~ LU where Y L ( k g ) are s p h e r i c a l harmonics of the a n g l e between kg and the s u r f a c e normal, the L n o t a t i o n i m p l i e s a sum over a l l JL and the a s s o c i a t e d m v a l u e s , and the m a t r i x X d e s c r i b e s the 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 [ 2 ] . The e x a c t B l o c h wave method ( e x a c t w i t h i n the m u f f i n t i n 38 a p p r o x i m a t i o n ) expands the i n t e r l a y e r w a v e f i e l d i n terms of p l a n e waves. I t i s c o n v e n i e n t t o c o n s i d e r l o c a l i n t e r l a y e r o r i g i n s a t the m i d p o i n t s between n e i g h b o u r i n g atomic p l a n e s . For the p l a n e s n and n+1, the w a v e f i e l d a t the l o c a l o r i g i n can be e x p r e s s e d 'V(r) = ^ , C + n <,exp(ikg +- r) + C" exp( ikg- - r) (2.13) The m a t r i x elements which d e s c r i b e t r a n s m i s s i o n T* + , T"" and r e f l e c t i o n R +", R ~ * w i t h p r o p a g a t i o n between i n t e r l a y e r o r i g i n s a r e T g ' g + + = P g ' + ( l g ' g + M g ' g + + ) P g + (2.14a) T g ' g " = P g ' - ( l g ' g + Mg'g-")Pg- (2.14b) Rg'g +- = Pg'*Mg'g +"Pg- (2.14c) Rg'g- + = Pg''Mg'g- +Pg + (2.14d) where, f o r example, Pg +=exp[ikg +»d/2) r e p r e s e n t s p r o p a g a t i o n t h r o u g h one h a l f of an i n t e r l a y e r s p a c i n g d. In equatons 2.14, Ig'g i s an element of the u n i t m a t r i x and r e p r e s e n t s the u n s c a t t e r e d p l a n e wave. A schematic r e p r e s e n t a t i o n of 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 s shown i n F i g u r e 2.2b. U s i n g the a p p r o p r i a t e 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 the r e l a t i o n between waves p r o p a g a t i n g between l a y e r s n and n+1 can be e x p r e s s e d as (2.15c) (2.15b) c;+1= T ; - C + „ + R V C m-l c « = T V c ; i l + R - ; C ; 39 E b + e x p C i k ^ r ) Rl / / / / F i g . 2 . 2 a Schematic r e p r e s e n t a t i o n of a set of plane waves (kg) 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 an atomic l a y e r i n t o outgoing waves (k g ' ) . n-1 n+1 p+ p • O -M ~ \ / M ~ + ^ M + ~ p i A-T " R + « + _ Fig.2.2b Schematic diagram i l l u s t r a t i n g the t r a n s m i s s i o n and r e f l e c t i o n of waves at the nth atomic l a y e r . Dashed l i n e s r epresent the midpoints between the nth plane and the n-1 and n+1 atomic l a y e r s . 40 where, f o r example, the elements of column m a t r i x . C + n + l are the v a l u e s of C + q between l a y e r s n and n+1. The v a l u e s of CoQ i n n + l e q u a t i o n (2.2) a re found by matching the w a v e f u n c t i o n s a t the so l i d - v a c u u m i n t e r f a c e . D e t a i l s of t h i s p r o c e d u r e a r e g i v e n by Jepsen e t a l [ 6 2 ] f o r the c a l c u l a t i o n of LEED i n t e n s i t i e s from the A1(001) s u r f a c e . T h i s c a l c u l a t i o n employed the B l o c h wave method and t r e a t e d i n t r a l a y e r s c a t t e r i n g u s i n g the ' l a y e r KKR' method from band s t r u c t u r e c a l c u l a t i o n s . Another e x a c t method, the T- m a t r i x method, works i n a n g u l a r momentum space. The r e f l e c t i v i t y of an N - l a y e r s l a b i s found by s o l v i n g N e q u a t i o n s , each w i t h m a t r i c e s of dimen s i o n s (imax+1) 2 x (i m a x + 1 ) 2 , where J>.max i s the number of phase s h i f t s needed ( t y p i c a l l y 8 f o r e n e r g i e s l e s s than ,200eV) . For i n f o r m a t i o n on the T-matrix method, the i n t e r e s t e d reader i s r e f e r r e d t o the rev i e w by T o n g [ 5 l ] . 2.4 PERTURBATIVE METHODS Both the B l o c h wave and T- m a t r i x methods have f o r m i d a b l e computing time and s t o r a g e space (about 0.4-1 Mb) r e q u i r e m e n t s [ 5 1 ] , and, i n the development of LEED f o r p r a c t i c a l s u r f a c e s t r u c t u r e d e t e r m i n a t i o n s , t h e r e a r e c o n s e q u e n t l y c o n s i d e r a b l e p r e s s u r e s f o r d e v e l o p i n g f a s t e r methods w i t h l e s s s e v e r e s t o r a g e r e q u i r e m e n t s . As d e t a i l e d i n S e c t i o n 1.3.1, i n e l a s t i c l o s s e s r e s u l t i n a s h o r t mean f r e e p a t h l e n g t h and t h i s may l i m i t the o r d e r of m u l t i p l e s c a t t e r i n g s t h a t w i l l be i m p o r t a n t f o r d e t e r m i n i n g beam r e f l e c t i v i t i e s . The K-space methods d e s c r i b e d below, r e n o r m a l i z e d f o r w a r d s c a t t e r i n g (RFS) and l a y e r d o u b l i n g , 41 u t i l i z e t h i s by a p p r o x i m a t i n g the f i n a l r e f l e c t i v i t y by the sum over terms w i t h p r o g r e s s i v e l y h i g h e r o r d e r of m u l t i p l e s c a t t e r i n g . 2.4.1 R e n o r m a l i z e d Forward S c a t t e r i n g The r e n o r m a l i z e d f o r w a r d s c a t t e r i n g (RFS) method was de v e l o p e d by P e n d r y [ 5 5 ] . The RFS proce d u r e i s an i t e r a t i v e method which e v a l u a t e s the f o r w a r d s c a t t e r i n g t o a l l o r d e r s . As s c h e m a t i c a l l y i l l u s t r a t e d i n F i g u r e 2.3, the computation proceeds l a y e r by l a y e r u n t i l the sum of the a m p l i t u d e s of the f o r w a r d s c a t t e r e d beams i s l e s s than some p r e d e t e r m i n e d f r a c t i o n of the a m p l i t u d e i n c i d e n t on the f i r s t l a y e r (e.g. 0.2%). I n e l a s t i c l o s s e s l i m i t the p e n e t r a t i o n and hence the number of atomic l a y e r s r e q u i r e d t o r e p r e s e n t the c r y s t a l i n t h i s c o n t e x t . The a m p l i t u d e s of waves p r o p a g a t i n g from the n t h t o the n+1th l a y e r can be r e p r e s e n t e d by a column v e c t o r a ^ + ( ( L > where i i s the i t e r a t i o n o r d e r . Then g i v e s the a m p l i t u d e s i n c i d e n t on the n+1th l a y e r . P r o c e e d i n g from the deepest l a y e r towards the topmost l a y e r , the b a c k s c a t t e r e d a m p l i t u d e s f o r each l a y e r a r e summed w i t h t h e t r a n s m i t t e d f l u x t o c a l c u l a t e the a m p l i t u d e s of beams on the way out of the c r y s t a l . 1 0 0 0 r e p r e s e n t s the i n c i d e n t beam and (2.16a) (2.16b) 42 F i g . 2 . 3 Schematic i l l u s t r a t i o n of the c o m p u t a t i o n p r o c e d u r e of R e n o r m a l i z e d Forward. S c a t t e r i n g (RFS). The t r i p l e t s of arrows r e p r e s e n t the e n t i r e s e t of p r o p a g a t i n g p l a n e waves. a) RFS p r o c e d u r e f o r t h r e e i t e r a t i o n s b) P r o p a g a t i o n of inward moving waves c) P r o p a g a t i o n of outward moving waves ( a f t e r Van Hove and Tong) 43 • b- n ( 1>= R - - a V V ' +T--b- n + l' 1> (2.17) The c a l c u l a t i o n proceeds l a y e r by l a y e r u n t i l the topmost l a y e r i s reached and y i e l d s b - 0 ( 1 > , the f i r s t a p p r o x i m a t i o n t o the r e f l e c t e d a m p l i t u d e s . The p r o c e s s i s i t e r a t e d t o g i v e a + n + i < 2 > v a l u e s by p r o c e e d i n g back i n t o . t h e b u l k u s i n g the b a c k s c a t t e r e d a m p l i t u d e s R" B*b; The p r o c e d u r e i s r e p e a t e d u n t i l the r e f l e c t e d a m p l i t u d e s converge; u s u a l l y a f t e r 3-5 i t e r a t i o n s . 2.4.2 L a y e r D o u b l i n g The l a y e r d o u b l i n g method, s c h e m a t i c a l l y i l l u s t r a t e d i n F i g u r e 2.4, b e g i n s w i t h the c o m p u t a t i o n of M. The c a l c u l a t i o n then proceeds by e v a l u a t i n g 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 f o r a t w o - l a y e r s l a b c, composed of atomic l a y e r s a and b. T c + + = T b + + ( I - R a * - R b " + ) " 1 T a + + (2.18a) T e " = Ta--(I-Rb" +Ra + - ) - 1 T b " - (2.18b) ft ?c fz ft & Rc-* = Ra' + + T a ' " R b + " ( I - R a + " R b " + ) - v T a + + (2.18c) R c + - = Rb +" + T b + + R a ' + ( I - R b " + R a + " ) " 1 T b " - ( 2 . l 8 d ) % ca. fat ft <K ^-The two l a y e r s l a b s a r e s t a c k e d t o g e t h e r and the d i f f r a c t i o n m a t r i c e s a r e c a l c u l a t e d , a g a i n u s i n g e q u a t i o n s ( 2 . l 8 a - d ) . T h i s p r o c e d u r e i s r e p e a t e d f o r 8,16.. l a y e r s l a b s u n t i l the c a l c u l a t e d r e f l e c t i v i t i e s c o nverge. The l a y e r d o u b l i n g method i n v o l v e s the i n v e r s i o n and m u l t i p l i c a t i o n of m a t r i c e s of o r d e r nj,, where n b i s the number of beams r e q u i r e d i n the c a l c u l a t i o n (see b e l o w ) . 44 Fig.2.4 I l l u s t r a t i o n of the l a y e r d oubling procedure of s t a c k i n g l a y e r s . Layers A and B are stacked together and 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 are c a l c u l a t e d . The double l a y e r C i s then stacked together to form a 4 - l a y e r s l a b and the d i f f r a c t i o n m a t r i c e s are again c a l c u l a t e d . ( a f t e r Tong{56}) 45 2.4.3 Comparison of Methods Computing t i m e s f o r the l a y e r d o u b l i n g and RFS methods s c a l e as n b 3 l o g ( L ) and n b 2 L r e s p e c t i v e l y , where L i s the number of l a y e r s . However, the RFS method can f a i l t o converge f o r s m a l l i n t e r l a y e r s p a c i n g s ( a p p r o x i m a t e l y 1A or l e s s ) , and i n such c a s e s the l a y e r d o u b l i n g method i s a u s e f u l a l t e r n a t i v e . An a t t r a c t i v e f e a t u r e of the l a y e r d o u b l i n g method i s t h a t once the l a y e r d i f f r a c t i o n m a t r i x M has been computed f o r the s u b s t r a t e i t can be s t o r e d and the LEED i n t e n s i t i e s f o r d i f f e r e n t o v e r l a y e r s i t e s can be c a l c u l a t e d , whereas u s i n g the RFS p r ocedure each o v e r l a y e r model must be c a l c u l a t e d s t a r t i n g from the o n e - l a y e r v a l u e s of Mg'g. Some l i m i t a t i o n s on m u l t i p l e s c a t t e r i n g c o m p u t a t i o n s a r e s e t by p o s s i b l e n u m e r i c a l e r r o r s . W i t h i n the K-space methods, n u m e r i c a l i n s t a b i l i t i e s can be e s p e c i a l l y troublesome when the number of beams r e q u i r e d i n a c a l c u l a t i o n i s l a r g e [ 6 8 ] , and t h i s i s always the case f o r s m a l l i n t e r l a y e r s p a c i n g s . The r e l a t i o n s h i p between number of beams r e q u i r e d f o r the K-space c a l c u l a t i o n s and the i n t e r l a y e r s p a c i n g d has been e s t i m a t e d by Van Hove and T o n g [ 2 l ] as n b=A/4it [2(E-Vo) + { l o g ( t ) / d } 2 ] (2.19) where A i s the a r e a of the u n i t c e l l and t i s the f r a c t i o n of the o r i g i n a l a m p l i t u d e of a d i f f r a c t e d beam a f t e r p r o p a g a t i n g t h rough the i n t e r l a y e r s p a c i n g d. In the computing programs used here i n the c a l c u l a t i o n s f o r the (311) s u r f a c e s of copper and n i c k e l , t was s e t t o 0.002. 46 In c a s e s where n b i s r e q u i r e d t o be of the o r d e r of 100 or more the computing r e q u i r e m e n t s of a l l t h e K-space methods ( i n c l u d i n g the l a y e r d o u b l i n g method) become u n f a v o r a b l e compared w i t h L-space methods. T h i s has encouraged the use of 'combined space' m e t h o d s [ 2 l ] where the L-space r e p r e s e n t a t i o n i s used f o r the s c a t t e r i n g by p l a n e s of atoms w i t h s m a l l i n t e r l a y e r s p a c i n g s (composite l a y e r s ) and then the K-space r e p r e s e n t a t i o n i s used t o c a l c u l a t e the i n t e r l a y e r s c a t t e r i n g between the composite l a y e r s and the o t h e r l a y e r s . A p e r t u r b a t i v e approach d e v e l o p e d w i t h i n the L-space f o r m a l i s m , the r e v e r s e s c a t t e r i n g p e r t u r b a t i o n method[69], i s u s e f u l when i n t e r l a y e r s p a c i n g s a r e s m a l l . To i l l u s t r a t e the number of beams r e q u i r e d i n t h i s work f o r 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 f o r N i ( 3 1 1 ) , F i g u r e 2.5 shows a map of r e c i p r o c a l h a l f - s p a c e w i t h c i r c l e s A,B, and C which e n c l o s e the numbers of beams used i n the c a l c u l a t i o n , those t h a t emerge from the c r y s t a l , and those t h a t appear on the LEED s c r e e n f o r an i n c i d e n t energy of 200eV. I t can be seen t h a t a much l a r g e r number of beams a r e r e q u i r e d f o r 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 than a r e a v a i l a b l e f o r d i r e c t measurement. 2.5 PROGRAM FLOW AND THE USE OF SYMMETRY C o m p u t a t i o n a l r e q u i r e m e n t s can be g r e a t l y reduced when the d i r e c t i o n of 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 symmetry p l a n e [ 7 0 ] . In the K-space r e p r e s e n t a t i o n , the w a v e f u n c t i o n s f o r s y m m e t r i c a l l y r e l a t e d beams a r e r e p l a c e d by a symmetrized l i n e a r c o m b i n a t i o n of p l a n e waves. T h i s r e s u l t s i n a 47 MIRROR PLANE I Fig.2.5 A map of r e c i p r o c a l h a l f - s p a c e f o r n i c k e l ( 3 1 1 ) with c i r c l e s A,B,C,which e n c l o s e : A-the number of beams used i n the c a l c u l a t i o n B-the number of beams that emerge from the c r y s t a l C-the number of beams t h a t appear on the LEED screen f o r an i n c i d e n t energy of 200eV. 48 r e d u c t i o n i n the number of w a v e f u n c t i o n s r e q u i r e d f o r the d i f f r a c t i o n m a t r i c e s , and s u b s t a n t i a l r e d u c t i o n s i n computation t i m e s can o c c u r (e.g. by a f a c t o r of 10 or more). A f l o w c h a r t i n d i c a t i n g the g e n e r a l p r o c e d u r e s f o l l o w e d by the LEED programs used here i s shown i n F i g u r e 2.6. The beam s e t and symmetry l a b e l s , v a l u e s f o r the p h y s i c a l p a r a m e t e r s , and the geometry of the proposed s t r u c t u r a l model are i n p u t t o the program and beam i n t e n s i t i e s a r e c a l c u l a t e d f o r one energy. For the purpose of economy, the energy i n c r e m e n t s a r e chosen as the w i d e s t p o s s i b l e s p a c i n g t h a t does not r e s u l t i n a s i g n i f i c a n t l o s s of i n f o r m a t i o n about peak and t r o u g h p o s i t i o n s i n the 1(E) c u r v e s . For the v a l u e s of V o i used i n t h i s work ( v i z . Voi=-4 or -5eV a t lOOev i n c i d e n t energy and v a r i e d w i t h energy as i n e q u a t i o n 2.9), the m u l t i p l e s c a t t e r i n g i n t e n s i t i e s were c a l c u l a t e d a t 2eV s p a c i n g s f o r e n e r g i e s l e s s than lOOev and 4eV s p a c i n g s above. The c a l c u l a t e d 1(E) c u r v e s are then i n t e r p o l a t e d t o the same energy g r i d as the e x p e r i m e n t a l d a t a (2eV) i n p r e p a r a t i o n f o r the r - f a c t o r a n a l y s i s (Chapter 3 ) . 49 Read in (i) geometry ( i i ) V O T , V . 01 ( i i i ) beams and symmetrv (iv) temperature data (v) phase shifts Choose i n i t i a l energy Find beams needed at E c I Compute temperatuTe-ojependent phase shift? Calculate layer diffraction natrices M^, Find diffracted bearr, amplitudes from surface plus substrate by R F S Calculate beam intensities J Vary surface geometry Find diffraction matrices for a substrate layers bv laver doubling Add surface layer and f i n d diffracted beam a m p l i t u d e s Calculate beam intensities *- Vary surface geometry i { i n c r e m e n t E f-I Fig.2.6 Flowchart of the LEED programs[21] used to c a l c u l a t e d i f f r a c t e d beam i n t e n s i t i e s as a f u n c t i o n of energy using the l a y e r doubling or RFS methods. CHAPTER 3 EXPERIMENTAL METHODS 5.1 LEED c r y s t a l l o g r a p h i c s t u d i e s r e q u i r e t h e o r e t i c a l and e x p e r i m e n t a l 1(E) c u r v e s f o r c omparison. The t h e o r e t i c a l 1(E) c u r v e s f o r d i f f e r e n t proposed s u r f a c e models a r e g e n e r a t e d v i a the LEED computer programs d e s c r i b e d i n the p r e v i o u s c h a p t e r . The e x p e r i m e n t a l c u r v e s are o b t a i n e d from s u r f a c e s t h a t have been s p e c i a l l y p r e p a r e d and c h a r a c t e r i z e d . T h i s c h a p t e r r e v i e w s the g e n e r a l t e c h n i q u e s of sample p r e p a r a t i o n and c h a r a c t e r i z a t i o n , and the c o l l e c t i o n of 1(E) d a t a . A d e s c r i p t i o n i s t hen g i v e n of the r - f a c t o r programs t h a t a r e used t o f i n d the b e s t f i t between the two s e t s of 1(E) c u r v e s . 3.1 SAMPLE PREPARATION Sample p r e p a r a t i o n , b e f o r e e n t e r i n g the UHV chamber, s t a r t s by c u t t i n g a h i g h p u r i t y s i n g l e c r y s t a l r o d t o expose the d e s i r e d s u r f a c e . T h i s i n v o l v e s mounting the c r y s t a l on an aluminum b l o c k w i t h a c o n d u c t i v e m i x t u r e of aluminum powder and p l a s t i c cement, and then f i x i n g t o a goniometer which a l l o w s the o r i e n t a t i o n t o be a d j u s t e d . The c r y s t a l i s o r i e n t e d t o the d e s i r e d c r y s t a l l o g r a p h i c p l a n e u s i n g back r e f l e c t i o n Laue X-ray d i f f r a c t o g r a p h y [ 71 ]. S l i ' c e s about 1mm t h i c k a r e c u t from the r o d u s i n g spark e r o s i o n ( ' A g i e t r o n ' , A g i e , S w i t z e r l a n d ) and the c r y s t a l d i s c s are then mounted i n an 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 P r o d u c t s ) . The c r y s t a l i s p o l i s h e d u s i n g a p l a n e t a r y l a p p i n g system (DU 172, Canadian T h i n F i l m s L t d . ) t h a t uses a p o l i s h i n g j i g[7 2] w i t h a l i g n m e n t m i c r o m e t e r s t h a t a l l o w the sample o r i e n t a t i o n t o be a d j u s t e d w i t h o u t removing i t from the j i g . The whole assembly can be mounted on the X-ray d i f f T a c t o m e t e r and t h u s the c r y s t a l 52 o r i e n t a t i o n can be r e a d j u s t e d t o compensate f o r e r r o r s i n a l i g n m e n t i n t r o d u c e d d u r i n g the c u t t i n g and p o l i s h i n g s t a g e s . C r y s t a l s a r e m e c h a n i c a l l y p o l i s h e d w i t h p r o g r e s s i v e l y f i n e r g rades of diamond p a s t e , f i n i s h i n g w i t h e i t h e r 0.05 micron a l u m i n a or 1 micron diamond p a s t e . A f t e r p o l i s h i n g , the o p t i c a l f a c e i s a l i g n e d by back r e f l e c t i o n of a Ne/He l a s e r f i x e d on an o p t i c a l bench t o l e s s than 1/2° from normal. The assembly i s then moved t o t h e X-ray d i f f r a c t o m e t e r , and a Laue p i c t u r e i s taken t o ensure t h a t the d e s i r e d c r y s t a l l o g r a p h i c p l a n e and the o p t i c a l f a c e a r e a l i g n e d w i t h i n ±1/2°. An o r i e n t e d s u r f a c e i s o f t e n a c i d e t c h e d t o remove f o r e i g n m a t e r i a l i n c o r p o r a t e d i n t o the s u r f a c e d u r i n g the p o l i s h i n g p r o c e s s . F i n a l l y the sample i s u l t r a s o n i c a l l y r i n s e d i n d e i o n i z e d w a t e r , acetone and methanol. 3.2 THE UHV CHAMBER U l t r a h i g h vacuum (UHV), r e q u i r e d t o c o n t r o l the c o n d i t i o n s of a s u r f a c e d u r i n g an e x p e r i m e n t , i n v o l v e s a number of t e c h n i q u e s t h a t a re i l l u s t r a t e d by F i g u r e 3.1. Two vacuum chambers ( V a r i a n 240 and FC12 models) were used i n t h i s work. Both chambers a r e c o n s t r u c t e d of s t a i n l e s s s t e e l and have s e a l s made by com p r e s s i n g OFHC copper g a s k e t s between k n i f e - e d g e ( C o n f l a t ) f l a n g e s . The i n i t i a l s t age of pumpdown from a t m o s p h e r i c p r e s s u r e i n v o l v e s the use of a m e c h a n i c a l pump or s o r p t i o n pumps c o n t a i n i n g m o l e c u l a r s i e v e . S o r p t i o n pumping i s p r e f e r a b l e because of o i l - f r e e o p e r a t i o n , and p r e s s u r e s of about 1 m i c r o n ( 1 0 - 3 t o r r ) a r e a c h i e v e d a f t e r c h i l l i n g the pumps w i t h l i q u i d n i t r o g e n . T h i s i n i t i a l pumping reduces p r e s s u r e s i n t o the range where i o n pumps and t i t a n i u m s u b l i m a t i o n pumps can be 53 G A S LINE 122 ( b ) S.P. I S P LEAK V A L V E o EXPT'L CHAMBER <T.SP 200 l/s IP Pig.3.1 (a) Schematic diagram of the FC12 UHV chamber (b) Diagrammatic'representation .of the pumping system: IP = ion pump TSP = t i t a n i u m sublimation pump SP = s o r p t i o n pump 54 s t a r t e d ; the l a t t e r y i e l d p r e s s u r e s around 1 x 1 0 - 1 0 t o r r a f t e r b a k i n g the chamber f o r about 12 hours a t 200-250°C and d e g a s s i n g a l l the f i l a m e n t s . Ion pumps depend on a d i s c h a r g e t o produce i o n s , and the s e i o n s a r e t r a p p e d by magnetic f i e l d s and d i r e c t e d on t o r e a c t i v e ' g e t t e r ' p l a t e s ( o f t e n t i t a n i u m ) which remove the gases e i t h e r by the f o r m a t i o n of s t a b l e s o l i d compounds (e.g. as o x i d e s , h y d r i d e s or n i t r i d e s ) , or by b u r i a l f o r the noble gases. A t i t a n i u m s u b l i m a t i o n pump s i m p l y e v a p o r a t e s f i l m s of t i t a n i u m which have v e r y h i g h pumping r a t e s f o r r e a c t i v e gases. More d e t a i l e d i n f o r m a t i o n about UHV equipment and t e c h n i q u e s i s g i v e n by H o l l a n d e t a l [ 7 3 ] ; a s p e c t s of the h a n d l i n g and s e l e c t i o n of m a t e r i a l s f o r UHV a r e d i s c u s s e d by R o s e b u r y [ 7 4 ] . The f o l l o w i n g f a c i l i t i e s were a v a i l a b l e i n both UHV chambers: a s e t of LEED o p t i c s ( V a r i a n 981-0127), a g l a s s v i e w p o r t t o observe the f l u o r e s c e n t LEED s c r e e n , an i o n gun f o r c l e a n i n g by i o n bombardment, a nude i o n gauge t o m o n i t o r the p r e s s u r e , a sample h o l d e r w i t h h e a t i n g s t a g e , a m a n i p u l a t o r which a l l o w e d x,y,z l i n e a r t r a n s l a t i o n and two degrees of r o t a t i o n a l freedom, a quadrupole mass s p e c t r o m e t e r (EAI 150A) t o a n a l y z e the system gas c o m p o s i t i o n , and a v a r i a b l e l e a k v a l v e and c a p i l l a r y t o admit gases f o r c l e a n i n g or a d s o r p t i o n e x p e r i m e n t s . The FC12 system was a l s o equipped w i t h a s i n g l e - p a s s 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 ( V a r i a n 981-2607) and a g l a n c i n g i n c i d e n c e e l e c t r o n gun f o r Auger e l e c t r o n s p e c t r o s c o p y . The m a n i p u l a t o r used on the V a r i a n 240 chamber has the c r y s t a l sample on the r o t a t i o n a x i s ( V a r i a n 981-0523). The FC12 chamber i s b u i l t f o r m u l t i - t e c h n i q u e a n a l y s i s and t h e r e f o r e uses a m a n i p u l a t o r ( V a r i a n 981-2530) w i t h the sample 2 1/2" o f f - a x i s . 55 On both chambers, the v a r i a b l e l e a k v a l v e i s conn e c t e d t o a gas m a n i f o l d which has 3 gas b o t t l e s a t t a c h e d . Each m a n i f o l d i s pumped by a 20 i o n pump. Magnetic f i e l d s i n the l a b o r a t o r y a r e a n n u l l e d as f a r as p o s s i b l e by t h r e e o r t h o g o n a l s e t s of square 'Helmholtz c o i l s ' [ 7 5 ] u s i n g a H a l l probe t o s e t z e r o f i e l d a t the sample p o s i t i o n . For a l l e x p e r i m e n t a l d a t a r e p o r t e d i n t h i s t h e s i s , the r e s i d u a l f i e l d was reduced s u f f i c i e n t l y t h a t movement of the s p e c u l a r beam was not d e t e c t e d a t e n e r g i e s as low as 10eV. The r e d u c t i o n of the magnetic f i e l d s seemed adequate f o r LEED s t u d i e s s i n c e s y m m e t r i c a l l y r e l a t e d beams showed the ex p e c t e d e q u i v a l e n c e even a t low e n e r g i e s . 3.3 AUGER ELECTRON SPECTROSCOPY Auger e l e c t r o n s p e c t r o s c o p y i s an e s s e n t i a l t e c h n i q u e f o r c h a r a c t e r i z i n g s u r f a c e s i n modern s u r f a c e s c i e n c e . The b a s i c p r o c e s s f o r the e m i s s i o n of Auger e l e c t r o n s i s i n d i c a t e d i n F i g u r e 1.1. The competing p r o c e s s e s a s s o c i a t e d w i t h r e l a x a t i o n of an i n i t i a l c o r e h o l e a r e photon e m i s s i o n or e j e c t i o n of an Auger e l e c t r o n , but f o r c o r e l e v e l b i n d i n g e n e r g i e s l e s s than 2keV the Auger p r o c e s s i s the dominant o n e [ 7 6 ] . The i n i t i a l c o r e h o l e s a r e u s u a l l y formed by e l e c t r o n bombardment, but X-ray or io n beams can a l s o be used. Auger t r a n s i t i o n s a r e c l a s s i f i e d by the t h r e e energy l e v e l s of t h e e l e c t r o n s i n v o l v e d i n the r e l a x a t i o n . A WXY t r a n s i t i o n r e f e r s t o an o r i g i n a l c o r e h o l e i n l e v e l W b e i n g f i l l e d by an e l e c t r o n from l e v e l X, w h i l e the Auger e l e c t r o n i s e m i t t e d from l e v e l Y t o • l e a v e a doubly p o s i t i v e l y charged i o n . In s o l i d s the X and Y l e v e l s may be i n 56 the v a l e n c e band i n which case the t r a n s i t i o n i s d e s i g n a t e d WW. The use of AES f o r q u a n t i t a t i v e a n a l y s i s , band s t r u c t u r e d e t e r m i n a t i o n , and f o r c h e m i c a l s h i f t i n f o r m a t i o n has been re v i e w e d by Chang[3]. F o r the s t u d i e s i n t h i s t h e s i s , AES i s used e s p e c i a l l y 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 ; the assignment of o b s e r v e d Auger e n e r g i e s i s made by r e f e r e n c e t o a handbook[77] which c a t a l o g u e s e x p e r i m e n t a l Auger s p e c t r a f o r the v a r i o u s e l e m e n t s . When t h e e x c i t a t i o n s o u r c e i s an e l e c t r o n beam, Auger e l e c t r o n s produce v e r y s m a l l peaks on a s l o w l y v a r y i n g background. The r a t i o of the Auger s i g n a l t o background i n the i n t e r m e d i a t e r e g i o n of F i g u r e 1.3a i s a p p r o x i m a t e l y 1/10 [ 3 ] , and t h e f e a t u r e s i n the spectrum a r e n o r m a l l y enhanced by e l e c t r o n i c d i f f e r e n t i a t i o n [ 7 8 ] . T h i s can be done u s i n g the c o n v e n t i o n a l LEED o p t i c 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 , as shown i n F i g u r e 3.2. T a k i n g an Auger spectrum i n v o l v e s ramping the r e t a r d i n g v o l t a g e on g r i d s G2 and G3 ( F i g u r e 1.4a) and c o l l e c t i n g t he e l e c t r o n s r e a c h i n g the f l u o r e s c e n t s c r e e n . D e r i v a t i v e s p e c t r a a r e o b t a i n e d by a d d i n g a s m a l l m o d u l a t i o n (1kHz) v o l t a g e t o the ramp a p p l i e d t o the r e t a r d i n g g r i d s and m o n i t o r i n g t h e 2kHz component of t h e output w i t h a l o c k - i n a m p l i f i e r (PAR HR8). The s c r e e n t o g r i d c a p a c i t a n c e , which has a l r e a d y been s u b s t a n t i a l l y reduced by the use of the grou n d i n g g r i d between the r e t a r d i n g g r i d s and the f l u o r e s c e n t s c r e e n c o l l e c t o r , g e n e r a t e s a l a r g e 1kHz s i g n a l a t the c o l l e c t o r . T h i s s i g n a l must be reduced t o a v o i d s a t u r a t i o n of the l o c k - i n a m p l i f i e r . The n e u t r a l i z a t i o n c a p a c i t o r between the m o d u l a t i o n t r a n s f o r m e r and the c o l l e c t o r f e e d s back an out of phase s i g n a l 5? SAMPLE Vr« sin D t • f t , Sin wt RAMP GEN. D GUN CONTROL 300V TUNED PREAMP. LOCK-IN AMP. FREQ. HALVER Sin 2u\ H.C.A. O X-T PLOTTER SCOPE Fig.3.2 Schematic diagram of LEED o p t i c s used as a r e t a r d i n g f i e l d analyzer f o r Auger e l e c t r o n spectroscopy: MCA=multichannel analyzer ( F a b r i t e k 1062) 58 which f u r t h e r reduces the c a p a c i t i v e l y c o u p l e d g r i d - c o l l e c t o r s i g n a l . The v o l t a g e ramp i s c o n t r o l l e d by a m u l t i - c h a n n e l a n a l y z e r ( F a b r i t e k 1062) l i n k e d t o a programmable power s u p p l y (Kepco OPS2000). The output from the l o c k - i n a m p l i f i e r " i s s t o r e d i n the F a b r i t e k and s i g n a l averaged u n t i l the S/N r a t i o i s a c c e p t a b l e . The tuned p r e a m p l i f i e r i s based on the d e s i g n by Nathan and H o p k i n s [ 7 9 ] , The c i r c u i t i s b i a s e d a t +300V t o f a c i l i t a t e t h e c o l l e c t i o n of the e l e c t r o n s . M o d u l a t i o n v o l t a g e s from 3 t o 10V (peak t o peak) are used w i t h t y p i c a l p r i m a r y beam c u r r e n t s (Ip) of 20uA f o r p r i m a r y beam e n e r g i e s (Ep) of 2000eV. An Auger spectrum f o r the energy range 50-550eV can be o b t a i n e d i n 3-5 m i n u t e s ; i t i s f i n a l l y p l o t t e d on an X-Y r e c o r d e r ( H e w l e t t - P a c k a r d 7004b). The e x i s t e n c e of 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 (CMA) on the FC12 system a l l o w s a much more s e n s i t i v e a n a l y s i s . T h i s a r i s e s because the r e s t r i c t e d energy range of c o l l e c t e d e l e c t r o n s o b t a i n e d w i t h a CMA reduces the shot n o i s e t h a t l i m i t s the s e n s i t i v i t y of r e t a r d i n g f i e l d a n a l y z e r s (LEED o p t i c s ) [ l 3 ] . The e x p e r i m e n t a l arrangement f o r u s i n g the CMA i s shown i n F i g u r e 3.3. The m o d u l a t i o n on the o u t e r c y l i n d e r i s about 3V(peak t o peak) a t 5-lOkHz and a p r e a m p l i f i e r i s used t o i s o l a t e the h i g h v o l t a g e (2-2.5kV) on the c o l l e c t o r of the c h a n n e l e l e c t r o n m u l t i p l i e r [ 8 0 ] . The f r o n t end of the c h a n n e l t r o n i s u s u a l l y grounded, but i s h e l d a t +300V when e l e c t r o n s of l e s s than 50eV ar e t o be d e t e c t e d . The g l a n c i n g i n c i d e n c e gun i n c r e a s e s the s u r f a c e / b u l k s e n s i t i v i t y [ 8 1 ] . The i n c i d e n t beam t y p i c a l l y has an energy of 2-3keV, a c u r r e n t of 100-200 J U A , and a c r o s s - s e c t i o n a l a r e a . <1mm2. The CMA i s a d i s p e r s i v e a n a l y z e r , and as a 59 F i g . 3 . 3 Schematic diagram of the experimental set up on the V a r i a n FC12 vacuum chamber u s i n g the 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 g l a n c i n g i n c i d e n c e e l e c t r o n gun to take Auger s p e c t r a . 60 consequence d e r i v a t i v e s p e c t r a a r e o b t a i n e d by s e t t i n g the l o c k - i n a m p l i f i e r a t the m o d u l a t i o n fre q u e n c y a p p l i e d t o the o u t e r c y l i n d e r [ 1 3 ] . S p e c t r a s u p e r i o r t o those o b t a i n e d w i t h the r e t a r d i n g f i e l d LEED o p t i c s can be taken i n l e s s than 30 seconds w i t h the CMA; moreover they can be d i s p l a y e d d i r e c t l y on an o s c i l l o s c o p e . The l a t t e r i s c o n v e n i e n t f o r q u i c k m o n i t o r i n g of the c o m p o s i t i o n of d i f f e r e n t a r e a s of a c r y s t a l . 3 .4 CLEANING THE SAMPLE The g e n e r a l method f o r c l e a n i n g c r y s t a l specimens under UHV i n v o l v e s Ar+ bombardment, f o l l o w e d by a n n e a l i n g t o r e l i e v e the bombardment damage. I t i s g e n e r a l l y p r e f e r a b l e t o s t a r t by bombarding the sample at room t e m p e r a t u r e , and then o n l y h e a t i n g the sample a f t e r Auger s p e c t r a show the s u r f a c e i s e s s e n t i a l l y c l e a n . T h i s a v o i d s c o n t a m i n a t i n g the bu l k w i t h i m p u r i t i e s i n t r o d u c e d d u r i n g sample p r e p a r a t i o n , and so h i n d e r s the f o r m a t i o n of compounds ( e . g . m e t a l l i c c a r b i d e s ) which can be d i f f i c u l t t o remove. D u r i n g Ar+ bombardment, the t i t a n i u m s u b l i m a t i o n pump i s used .to pump i m p u r i t y gases ( t h i s pump has n e g l i g i b l e pumping speed f o r a r g o n ) . A f t e r a few hours of i o n bombarding, the argon i s pumped away by the i o n pumps, and a new s u p p l y of argon i n t r o d u c e d v i a the v a r i a b l e l e a k v a l v e . When Ar+ bombarding, the system i s t y p i c a l l y i n the 10- 5 t o 10-" t o r r range and p a r t i a l p r e s s u r e s of i m p u r i t i e s a r e u s u a l l y <1x10- 8 t o r r . . Argon i o n s w i t h e n e r g i e s i n the range 300 t o 3000 eV are used f o r bombarding, a l t h o u g h i o n s w i t h e n e r g i e s a t the lower end of t h i s range a r e p r e f e r a b l e when the sample becomes c l e a n e r i n o r d e r t o reduce the bombardment damage. Ion c u r r e n t s a re 61 measured u s i n g a K e i t h l e y e l e c t r o m e t e r (DCVTVM 200B) and c u r r e n t s around 1-10;JA a r e t y p i c a l l y used. For a n n e a l i n g , and h e a t i n g of samples t o d r i v e out i m p u r i t i e s from the b u l k , samples a r e mounted on r e s i s t i v e h e a t e r s ( V a r i a n 981-2058) w i t h t a n t a l u m c l i p s . The sample t e m p e r a t u r e i s m o n i t o r e d e i t h e r w i t h a 0.005" a l u m e l - c h r o m e l thermocouple spot-welded t o the c r y s t a l , or w i t h an o p t i c a l pyrometer (Hartmann and Braun, F r a n k f u r t ) . The p r o c e d u r e s t h a t r e s u l t i n c l e a n , w e l l - o r d e r e d s u r f a c e s have t o be d i s c o v e r e d by t r i a l and e r r o r . T h i s i n v o l v e s v a r y i n g the Ar+ bombardment and h e a t i n g c o n d i t i o n s , and u s i n g AES and LEED t o m o n i t o r the s u r f a c e c o m p o s i t i o n and s u r f a c e o r d e r r e s p e c t i v e l y . F u r t h e r d e t a i l s f o r the s p e c i f i c c r y s t a l s s t u d i e d here a r e d i s c u s s e d i n the e x p e r i m e n t a l s e c t i o n s of C h a p t e r s 4 and 5. 3.5 LEED INTENSITY ANALYSIS An i m p o r t a n t parameter f o r LEED i n t e n s i t y measurements i s the d i r e c t i o n of i n c i d e n c e . T h i s i s c o n t r o l l e d here by V a r i a n m a n i p u l a t o r s which have r e p r o d u c i b l e motions t o b e t t e r than 1°. A LEED analogue of the X-ray Laue t e c h n i q u e i s used t o determine normal i n c i d e n c e . P o l a r o i d f i l m i s exposed by the l i g h t from the LEED s c r e e n as the i n c i d e n t e l e c t r o n energy i s v a r i e d from 40 t o 250 eV. The sample i s then t r a n s l a t e d ( w i t h r o t a t i o n a l motions l o c k e d ) so t h a t a f u r t h e r exposure of the l i g h t from the f i l a m e n t of the e l e c t r o n - gun can be made. The r e s u l t i n g p i c t u r e i n d i c a t e s t r a c k s from the movement of the d i f f r a c t e d beams as the energy i n c r e a s e s . Normal i n c i d e n c e i s i n d i c a t e d when the beam paths i n t e r s e c t a t the c e n t e r of the e l e c t r o n gun. T h i s 62 c o n d i t i o n can be d e t e r m i n e d t o w i t h i n 1/2° ( l i m i t a t i o n s b e i n g s e t by the f i n i t e a p e r t u r e of the e l e c t r o n gun and the u n c e r t a i n t i e s i n a l i g n i n g the camera). T h i s p r o c e d u r e i s a l s o u s e f u l as a method of i n d i c a t i n g s t r a y f i e l d s which cause c u r v a t u r e i n the paths of the d i f f r a c t e d - beams. A common method f o r d e t e r m i n i n g normal i n c i d e n c e i n v o l v e s measuring 1(E) c u r v e s f o r s y m m e t r i c a l l y e q u i v a l e n t beams; t h i s approach i s not a p p l i c a b l e w i t h the p h o t o g r a p h i c method, where the 1(E) c u r v e s a r e o n l y a v a i l a b l e a f t e r d e v e l o p i n g and a n a l y z i n g the f i l m . A d e s c r i p t i o n of the LEED o p t i c s was g i v e n i n Chapter 1; here c o n s i d e r a t i o n i s g i v e n t o the measurement of d i f f r a c t e d beam i n t e n s i t i e s * One s t a n d a r d approach i n v o l v e s an i n t e r n a l Faraday cup[82] which can i n t e r c e p t i n d i v i d u a l d i f f r a c t e d beams, so p r o v i d i n g a d i r e c t measurement of beam c u r r e n t s . W i t h knowledge of i n c i d e n t beam c u r r e n t s , they can be c o n v e r t e d t o a b s o l u t e beam i n t e n s i t i e s ( d i f f r a c t e d beam c u r r e n t 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 ) , which a r e r e l a t e d t o the c a l c u l a t e d r e f l e c t i v i t i e s d e f i n e d i n e q u a t i o n ( 2 . 2 ) . The Faraday cup i s e s p e c i a l l y c o n v e n i e n t f o r measurements of the s p e c u l a r beam, s i n c e the beam d i r e c t i o n does not change w i t h energy ( p r o v i d e d the e l e c t r o n s move i n f i e l d - f r e e s p a c e ) . However, i t i s l e s s c o n v e n i e n t f o r the n o n - s p e c u l a r beams whose d i r e c t i o n s change w i t h energy. Another s t a n d a r d method of LEED d a t a c o l l e c t i o n i n v o l v e s the use of an e x t e r n a l spot photometer[83] t o measure the l i g h t from the f l u o r e s c e n t s c r e e n , but t h i s method a l s o s u f f e r s from an i n a b i l i t y t o measure a l l beams q u i c k l y . S i n c e 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 a u t o m a t i c a l l y g i v e i n t e n s i t i e s f o r a l l d i f f r a c t e d beams, i t i s advantageous t o 63 employ d a t a c o l l e c t i o n t e c h n i q u e s which can s i m u l t a n e o u s l y measure a l l the d i f f r a c t e d beam i n t e n s i t i e s . A l s o speed of measurement i s i m p o r t a n t i n o r d e r t o a v o i d c h a n g i n g s u r f a c e c o n d i t i o n s w h i l e the measurements are b e i n g made. Newer developments i n LEED da t a a c q u i s i t i o n methods i n c l u d e the p h o t o g r a p h i c method of S t a i r e t a l [ 8 4 ] , and most r e c e n t l y the use of low l i g h t l e v e l TV cameras[85] and the use of m u l t i c h a n n e l p l a t e s . With the l a t t e r , s u f f i c i e n t a m p l i f i c a t i o n of d i f f r a c t e d beam c u r r e n t s can be o b t a i n e d so t h a t LEED p a t t e n s can be d i s p l a y e d w i t h i n c i d e n t e l e c t r o n beam c u r r e n t s on the o r d e r of a nanoampere[86]. The method f o r i n t e n s i t y measurements used here i n v o l v e s a combined v i d i c o n - p h o t o g r a p h i c p r o c e d u r e [ 8 7 ] . Photographs of the LEED s c r e e n are taken t h r o u g h the v i e w p o r t w i t h a Nikon F2 camera, u s i n g Kodak T r i - X 35mm b l a c k / w h i t e f i l m , w i t h the l e n s (85mm-f1.8 w i t h K2 e x t e n s i o n r i n g ) n o r m a l l y s e t f o r 1 second e x p o s u r e s and an a p e r t u r e of f1.8 or f 2 . 8 . A s e t of photographs from 30 t o 300eV i n 2eV i n t e r v a l s can be t a k e n i n about 4 m i n u t e s . As the energy i s v a r i e d the i n c i d e n t beam c u r r e n t i s r e c o r d e d f o r n o r m a l i z a t i o n of the i n t e n s i t y d a t a . A f t e r t a k i n g a s e t of photographs an Auger spectrum i s always r e c o r d e d t o a s s e s s the degree of s u r f a c e c l e a n l i n e s s . F i l m i s d e v e l o p e d i n a tank u s i n g A c u f i n e d e v e l o p e r f o r 5-10 minutes at 75-80°C w i t h g e n t l e a g i t a t i o n . A f t e r f i x i n g (Kodak R a p i d F i x e r - 5 m i n u t e s ) a permanent r e c o r d of the d i f f r a c t i o n p a t t e r n s as a f u n c t i o n of energy i s a v a i l a b l e f o r d e t e r m i n i n g 1(E) c u r v e s . The a n a l y s i s f o l l o w s the p r o c e d u r e s d e s c r i b e d by F r o s t e t a l [ 8 7 ] . P h o t o g r a p h i c n e g a t i v e s a r e scanned 64 by a v i d i c o n camera; each r e g i o n around a d i f f r a c t i o n spot i s d i g i t i z e d ( v i d i c o n and d i g i t i z e r , S p a t i a l Data Systems, G a l e n a , C a l i f o r n i a ) and the d e n s i t y v a l u e s a r e i n t e g r a t e d s u b j e c t t o a background s u b t r a c t i o n t o y i e l d one i n t e n s i t y v a l u e per d i f f r a c t e d spot per energy. I t i s assumed here t h a t the d e n s i t y of the f i l m i s d i r e c t l y p r o p o r t i o n a l t o the luminance of the s c r e e n , which i n t u r n i s assumed t o be p r o p o r t i o n a l t o the i m p i n g i n g e l e c t r o n c u r r e n t . F u r t h e r , the l i n e a r i t y of the i n t e g r a t e d d i g i t i z e d output w i t h f i l m d e n s i t y i s assumed so t h a t the v i d i c o n o utput i s a measure of the r e l a t i v e i n t e n s i t y of a d i f f r a c t e d beam. The r e l a t i v e i n t e n s i t i e s a r e s t o r e d on c a s s e t t e tape and can be t r a n s f e r r e d v i a paper tape t o the U n i v e r s i t y Computing C e n t r e (IBM 370/168 l a t e r r e p l a c e d by Amdahl 470 V8) f o r comparison w i t h the c a l c u l a t e d 1(E) c u r v e s . 3.6 THE R-FACTOR PROGRAMS The p r o c e d u r e of s e l e c t i n g the best model f o r the s u r f a c e geometry was o r i g i n a l l y done by v i s u a l comparison of the 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) c u r v e s . However, w i t h 10 or more e x p e r i m e n t a l beams and o f t e n a t l e a s t 20 g e o m e t r i e s t o be c o n s i d e r e d , the v i s u a l t method i s i n e v i t a b l y u n w i e l d y . T h i s i s p a r t i c u l a r l y t r u e s i n c e the i n n e r p o t e n t i a l , Vor, used i n the c a l c u l a t i o n s , a l s o needs some r e f i n e m e n t . Such c o n s i d e r a t i o n s p o i n t t o the advantages of n u m e r i c a l i n d i c e s which s i g n i f y the l e v e l s of agreement between two s e t s of 1(E) c u r v e s ; such i n d i c e s may a l s o i n d i c a t e measures of the ' r e l i a b i l i t y ' of a d e t e r m i n a t i o n . S e v e r a l r e l i a b i l i t y i n d i c e s , or r - f a c t o r s , have been p r o p o s e d f 8 8 ] . Many'compare the e n e r g i e s and/or shapes of 6 5 peaks and t r o u g h s i n the c a l c u l a t e d and e x p e r i m e n t a l 1(E) c u r v e s , w i t h o u t r e g a r d t o the a b s o l u t e v a l u e s of the i n t e n s i t i e s . Two r - f a c t o r s are used and d i s c u s s e d h e r e ; namely those proposed by Z a n a z z i and J o n a [ 4 l ] , and by P e n d r y [ 4 2 ] , For l a t e r r e f e r e n c e i t i s u s e f u l t o d i s t i n g u i s h here between an r - f a c t o r ( r ( L ) ) f o r an i n d i v i d u a l ( t h e i t h ) beam and an o v e r a l l r - f a c t o r , R, b u i l t up from a s e t of r ( L ) . One u s e f u l parameter f o r d e s c r i b i n g the geometry of a c l e a n s u r f a c e i s the topmost i n t e r l a y e r s p a c i n g (d) which may v a r y s i g n i f i c a n t l y from t h a t of the b u l k v a l u e ( d Q ) , even though the second and s u c c e s s i v e s p a c i n g s e s s e n t i a l l y e g u a l d e . The changes i n the topmost s p a c i n g are o f t e n e x p r e s s e d as p e r c e n t a g e changes, namely as * d % = l 0 0 ( d - d o ) / d o (3.7) For the a n a l y s i s i n v o l v e d i n a s s e s s i n g u n c e r t a i n t i e s i n the d e t e r m i n e d s u r f a c e geometry, g m i n ( t ) i s used t o i n d i c a t e the v a l u e of A d % where r ( L ) i s m i n i m i z e d , w h i l e Gmin i n d i c a t e s the topmost s p a c i n g c o r r e s p o n d i n g t o the minimum i n R. An r - f a c t o r comparison n o r m a l l y a l l o w s f o r v a r i a t i o n i n Vor by s h i f t i n g the energy s c a l e f o r one s e t of 1(E) c u r v e s r e l a t i v e t o the o t h e r . A c o n v e n i e n t way of p r e s e n t i n g t h i s comparison i s w i t h c o n t o u r s of R v a l u e s p l o t t e d as f u n c t i o n s of Vor and ^ d % (e.g. F i g u r e s 4.8 and 4.16). 66 3.6.1 The R e l i a b i l i t y Index of Z a n a z z i and Jona The Z a n a z z i and Jona index f o r an i n d i v i d u a l beam t a k e s the shapes of the 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) c u r v e s i n t o account by comparing f i r s t and second d e r i v a t i v e s . The s p e c i f i c e x p r e s s i o n used f o r the reduced r ( L ) v a l u e i s r ( 1 '=Sw(E) | CI ' t h - I 'exp|dE/ Jj 0.027IexpdE C = $IexpdE/ j >IthdE w ( E ) = | C I " t h - I " e x p | / ( | I ' e x p | + X) Here Iexp and I t h a r e the e x p e r i m e n t a l and t h e o r e t i c a l i n t e n s i t i e s , a s i n g l e prime r e p r e s e n t s the f i r s t d e r i v a t i v e w i t h r e s p e c t t o energy, a double prime r e p r e s e n t s the second d e r i v a t i v e . The I'exp v a l u e i n the e x p r e s s i o n f o r w(E) i n c r e a s e s the w e i g h t s , i n the compar i s o n , of maxima and minima i n the e x p e r i m e n t a l c u r v e s ; X i s chosen as the maximum a b s o l u t e v a l u e of I'exp, namely X=|I'exp|max (3.5) to p r e v e n t the denominator i n (3.4) from e q u a l l i n g z e r o . The c o e f f i c i e n t 0.027 i n (3.2) i s a n o r m a l i z a t i o n f a c t o r d e t e r m i n e d by matching random s e t s of d a t a [ 4 l ] . To a p p l y t o many-beam a n a l y s e s , a multi-beam ind e x ( r r i n n o t a t i o n of Z a n a z z i and Jona) i s c a l c u l a t e d by w e i g h t i n g the c o n t r i b u t i o n s of the i n d i v i d u a l beams by t h e i r energy range (3.2) (3.3) (3.4) Rz j = 2; r < "L > E < ' ' -/ % E ( 1 > ( 3 . 6 ) .67 where E ( L ) i s the energy range of a v a i l a b l e 1 (E) d a t a f o r the i t h beam. Z a n a z z i and Jona i n t r o d u c e d another q u a n t i t y R which was dependent on the number of beams used i n the a n a l y s i s . However, p r o v i d e d a r e a s o n a b l e number of beams ( e . g . >8) i s used, R seems not t o have any advantages over the use of R z j . C l e a r l y , the b e t t e r the l e v e l of corre s p o n d e n c e between a s e t of 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) c u r v e s , the lower the v a l u e of R z j . Z a n a z z i and Jona proposed t h a t v a l u e s of Rzj<0.2 i n d i c a t e t h a t the s t r u c t u r a l model used i n the c a l c u l a t i o n s i s p r o b a b l y c o r r e c t , whereas v a l u e s of Rzj>0.5 i n d i c a t e poor agreement and t h a t the proposed model i s p r o b a b l y i n c o r r e c t . 3.6.2 The Pendry R - f a c t o r Pendry has c o n s t r u c t e d an r - f a c t o r t h a t uses o n l y f i r s t d e r i v a t i v e s and t h e r e f o r e i n v o l v e s l e s s c o m putation than the Z a n a z z i and Jona r - f a c t o r . Pendry's multi-beam index i s Rp=3 ^ ( Y i t h - Y i e x p ) 2 d E / Z ^ ( Y 2 i t h + Y 2 i e x p ) d E (3.8) Y = L " 1 / ( L - 2 + V o i 2 ) (3.9) L = I ' / I * (3.10) where, f o r example, Y i t h i s the l o g a r i t h m i c f u n c t i o n f o r the i t h c a l c u l a t e d beam. The v a l u e of V o i used f o r both Y i t h and Y i e x p i n (3.9) i s the same ' v a l u e t h a t i s 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 of the 1(E) c u r v e s . The Pendry r - f a c t o r i s most s e n s i t i v e t o the p o s i t i o n s of peaks and t r o u g h s i n the 1(E) c u r v e s and, because of the l o g a r i t h m i c d e r i v a t i v e s i n v o l v e d , i t g i v e s a l l the maxima and minima a p p r o x i m a t e l y e q u a l 68 w e i g h t s i n the comparisons. Rp v a l u e s of z e r o i n d i c a t e t h a t a l l c o r r e s p o n d i n g 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) c u r v e s a r e i d e n t i c a l ; v a l u e s of Rp c l o s e t o 1.0 i m p l i e s t h e r e i s no c o r r e l a t i o n between the c a l c u l a t e d and e x p e r i m e n t a l c u r v e s . 3.6.3 R - f a c t o r s and E r r o r E s t i m a t i o n An i m p o r t a n t o b j e c t i v e of the r e l i a b i l i t y index approach i s t o p r o v i d e an e s t i m a t e of the u n c e r t a i n t y i n the v a l u e of an i n t e r l a y e r s p a c i n g found f o r a minimum R v a l u e . We have examined t h r e e approaches f o r e s t i m a t i n g t h e s e u n c e r t a i n t i e s ; a l l b e i n g based on s t a n d a r d e r r o r s [ 8 9 ] . In the d i s c u s s i o n s t h a t f o l l o w s , Vor i s assumed t o be f i x e d a t the v a l u e which m i n i m i z e s R. The f i r s t method f o r e s t i m a t i n g u n c e r t a i n t i e s i n v o l v e s c a l c u l a t i n g the s t a n d a r d e r r o r from the r ( L ) v a l u e s a t minimum R, and then r e l a t i n g t h i s t o an u n c e r t a i n t y i n geometry by u s i n g the s p e c i f i c dependence of R on ^ d % . For a comparison i n v o l v i n g N e x p e r i m e n t a l 1(E) c u r v e s g i v e s the s t a n d a r d e r r o r i n R. In ( 3 . 1 1 ) , W ( L 1 i s the weight of an i n d i v i d u a l beam ( d e t a i l e d f u r t h e r b e l o w ) . F i g u r e 3.4b shows how U{1}G, the c o r r e s p o n d i n g u n c e r t a i n t y i n geometry, can be d e t e r m i n e d from U{1}R. D i f f i c u l t i e s w i t h U{1}R f o r e s t i m a t i n g u n c e r t a i n t i e s can a r i s e because of an i m p l i c a t i o n t h a t the s e t of r ( L ) v a l u e s f i t a normal d i s t r i b u t i o n ( i . e . i m p l i e s t h a t ' d e v i a t i o n s i n r ( L > from t h e i r mean ar e the. r e s u l t of random e r r o r s ) . In p r i n c i p l e , i t i s U{1 }R (3.11) 69 p o s s i b l e t h a t a l l r ( ' L > v a l u e s c o u l d go through minima a t the same A d % but show l a r g e d i f f e r e n c e s i n the a b s o l u t e v a l u e of r ( L > [ 9 0 ] . F u r t h e r m o r e , the multi-beam R z j v s . A d % c u r v e s a r e i n p r a c t i c e r a t h e r smoother than would be e x p e c t e d from the c o m p a r a t i v e l y l a r g e u n c e r t a i n t i e s i n d i c a t e d by U{1}R, w h i l e the g m i n ( L ) v a l u e s f o r the i n d i v i d u a l beams can a l l be grouped w e l l w i t h i n the range of *d% c o r r e s p o n d i n g t o U{1}R ( i e . U{1}G). T h i s i s i l l u s t r a t e d i n F i g u r e 3.4b f o r a model s i t u a t i o n i n v o l v i n g the comparison of two p a i r s of 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) c u r v e s (and hence two s i n g l e beam i n d i c e s r ( 1 ) and r ( 2 > ) , where both comparisons show best agreement at the same v a l u e of A d % , but the a b s o l u t e v a l u e s of r ( 1 ) and r ( 2 ) a r e q u i t e d i f f e r e n t . For such a s i t u a t i o n the d i f f e r e n c e s i n r ( L ' v a l u e s may not be due t o random e r r o r s , but r a t h e r t o s y s t e m a t i c e x p e r i m e n t a l u n c e r t a i n t i e s (e.g. a s s o c i a t e d w i t h s u r f a c e roughness or e r r o r s i n a n g l e s of i n c i d e n c e ) which a f f e c t some beams more than o t h e r s . For example, beams f u r t h e s t from the s u r f a c e normal a r e l i k e l y t o be more a f f e c t e d by p r o t u b e r a n c e s i n the s u r f a c e . Another d i f f i c u l t y w i t h t h e use of U{1] o c c u r s i n a s i t u a t i o n where t h e r e i s c o n s i d e r a b l e s c a t t e r i n the i n d i v i d u a l beam minima, but i t i s s t i l l p o s s i b l e t o o b t a i n s m a l l e r r o r s i n R. F i g u r e 3.4a shows a model s i t u a t i o n where two beams i n d i c a t e v e r y d i f f e r e n t topmost i n t e r l a y e r s p a c i n g s ( g m i n ( 1 ) , g m i n ( 2 > ) , and y e t U{1] would be n e g l i g i b l e s i n c e the r ( L > v a l u e s are e q u a l a t the minimum i n R. A second method f o r e s t i m a t i n g u n c e r t a i n t i e s c a l c u l a t e s the s t a n d a r d e r r o r of the weighted mean of the g m i n ( L ) v a l u e s [ 9 0 ] 70 gmin 1 Cm i n o *-I_J a. Lu or GEOMETRY Fig.3.4 Model r - f a c t o r versus geometry p l o t s f o r two s i t u a t i o n s i n v o l v i n g two d i f f e r e n t beams. Top i l l u s t r a t i o n suggests a p o s s i b l e f a i l u r e of U{1} to re p r e s e n t the d i f f e r e n c e i n 'best f i t ' geometries ( g m i n ( 1 ) , gmin (*>) found f o r the i n d i v i d u a l beams. Bottom diagram shows the l a r g e U{1} value that can be found because of the d i f f e r e n c e s i n the absolute & r ( 1 ) v a l u e s even when both beams show t h e i r best agreement with the same s t r u c t u r a l model. U{2}G = / l w ( L ' (gmin< L'-G'min) 2/(N-1 ) 71 (3.12) where G'min i s the weigh t e d mean of the i n d i v i d u a l gmin. ( L > v a l u e s G'min=Z*W( l ) g m i n ( L ,/^W ( L> L i-U{2}G has been used f o r a s s e s s i n g u n c e r t a i n t i e s i n much of t h i s work, a l t h o u g h i t has a p o t e n t i a l weakness i n s o f a r as i t does not take i n t o account the 'shapes' of the i n d i v i d u a l r ( L ) v s . geometry c u r v e s . O f t e n some of t h e s e c u r v e s do not show a s t r o n g s t r u c t u r a l dependence, and t h e r e f o r e do not s i g n i f i c a n t l y i n f l u e n c e the o v e r a l l s t r u c t u r a l d e t e r m i n a t i o n . I t i s suggested here t h a t such f l a t c u r v e s s h o u l d not c o n t r i b u t e s u b s t a n t i a l l y t o e r r o r e s t i m a t e s , a l t h o u g h t h i s i s not the case f o r U{2}G.. A t h i r d approach, which does take c u r v e shape i n t o a c c o u n t , i s U{3}R = } ( ^ r ( l ) ) V ( N - l ) (3.13) where the A r ( t ) v a l u e s a r e the d i f f e r e n c e s between r < L ) v a l u e s at Gmin and r < L > m i n a t g m i n ( u ) . A r < 1) and £ r ( 2 > a r e shown i n F i g u r e 3.4a. E r r o r e s t i m a t e s u s i n g U{3}R g i v e the g r e a t e s t w e i g h t i n g t o the s h a r p e s t r ( L > v s . geometry c u r v e s . For the s i t u a t i o n i n F i g u r e 3.4a, beam 1 would be g i v e n a l a r g e r w e i g h t i n g i n the c a l c u l a t i o n of U{3}R than beam 2 s i n c e A r < 1 ) i s l a r g e r than A r ( 2 > . These c a l c u l a t i o n s of u n c e r t a i n t i e s r e q u i r e a w e i g h t i n g f a c t o r W(L' f o r each beam. In the multibeam r e l i a b i l i t y index of 72 Z a n a z z i and Jona[40] t h i s i s taken as E ( 1 '/ % E ( L > (3.14) and the same w e i g h t i n g f a c t o r s a r e used i n e q u a t i o n s ( 3 . 1 1 ) - ( 3 . 1 3 ) f o r a n a l y s e s i n v o l v i n g the Z a n a z z i and Jona i n d e x . However, the c a l c u l a t i o n of the multibeam index of Rp i n v o l v e s a w e i g h t i n g f a c t o r f o r each i n d i v i d u a l beam depending on t h e v a l u e of Y 2 i exp+Y 2 i t h . To p r o v i d e an o v e r a l l c o n s i s t e n c y , c a l c u l a t i o n s of u n c e r t a i n t i e s based on Rp v a l u e s use the weight of an i n d i v i d u a l beam g i v e n by The t h r e e methods noted f o r e s t i m a t i n g u n c e r t a i n t i e s t r e a t each e x p e r i m e n t a l 1(E) c u r v e as an independent measurement; t h i s i s i n d i c a t e d by d i v i s i o n by (N-1) where N i s the number of n o n - e q u i v a l e n t beams. However, the. energy ranges over which i n t e n s i t i e s f o r i n d i v i d u a l beams can be measured a r e o f t e n q u i t e d i f f e r e n t s i n c e the h i g h e r index beams appear on the f l u o r e s c e n t s c r e e n o n l y a t h i g h e r e n e r g i e s . T h i s opens the q u e s t i o n of ,whether the u n i t of i n t e n s i t y i n f o r m a t i o n , here i d e n t i f i e d w i t h 'a whole 1(E) curve of an i n d i v i d u a l beam, s h o u l d be d e f i n e d i n some a l t e r n a t i v e way, f o r example by a f i x e d energy range, or by a c e r t a i n amount of s t r u c t u r e (e.g. number of peaks, t r o u g h s , s h o u l d e r s ) i n the 1(E) d a t a . However, such p o s s i b i l i t i e s have not y e t been a n a l y z e d . I t must be assumed t h a t t h e s e a n a l y s e s a r e a p p l i c a b l e t o s i t u a t i o n s where the b e s t - o b t a i n a b l e agreement between W ( L ) ( Y 2 i lexp+Y 2 i t h ) d E (3.15) 73 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) c u r v e s has been reached t h r o u g h the v a r i a t i o n of both g e o m e t r i c a l and n o n - g e o m e t r i c a l parameters (e.g. phase s h i f t s , V o r , V o i , and Debye tem p e r a t u r e ) 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 . The q u e s t i o n b e i n g c o n s i d e r e d i n t h i s s u b s e c t i o n concerns the u n c e r t a i n t y i n geometry a s s o c i a t e d w i t h the r e s i d u a l d i s c r e p a n c y , , as shown by the non-zero v a l u e s of Rp or R z j . The v a l u e of the methods i n d i c a t e d here f o r e s t i m a t i n g u n c e r t a i n t i e s can o n l y be a s s e s s e d t h r o u g h t h e i r a p p l i c a t i o n t o d a t a f o r many d i f f e r e n t s u r f a c e s . CHAPTER 4 STUDIES OF THE ( 3 1 1 ) SURFACES OF COPPER AND NICKEL 75 4.1 INTRODUCTION The c l e a n , p r i n c i p a l l o w - i n d e x s u r f a c e s of copper and n i c k e l have been s t u d i e d p r e v i o u s l y by the methods of LEED c r y s t a l l o g r a p h y [ 9 1 - 9 4 ] . The work p r e s e n t e d here r e p r e s e n t s the f i r s t s t u d i e s of stepped s u r f a c e s which employ d y n a m i c a l c a l c u l a t i o n s t o - d e t e r m i n e s u r f a c e g e o m e t r i e s . The s u r f a c e net v e c t o r s of a fee (311) s u r f a c e are shown i n F i g u r e 4.1; a diagram of the c o r r e s p o n d i n g r e c i p r o c a l net v e c t o r s i s i n F i g u r e 4.2. A c c o r d i n g t o the c o n v e n t i o n s of the I n t e r n a t i o n a l T a b l e s of X-ray C r y s t a l l o g r a p h y [ 9 5 ] , the (311) s u r f a c e b e l o n g s t o the cm t w o - d i m e n s i o n a l space group and can be r e p r e s e n t e d by a c e n t e r e d - r e c t a n g u l a r n e t . The i n t e n s i t i e s of the (h,k) beams ar e z e r o when h+k i s odd. The (311) s u r f a c e of a fee m e t a l i s a stepped s u r f a c e c o n s i s t i n g of (111) t e r r a c e s and (100) s t e p s of monoatomic h e i g h t and would be d e s i g n a t e d {2(111)x(100)} f o l l o w i n g the n o t a t i o n of Lang, Joyner and S o m o r j a i [ 9 6 ] . I t can a l s o be r e p r e s e n t e d by a p r i m i t i v e mesh of c o m p a r a t i v e l y s m a l l a r e a , which makes i t a s u i t a b l e c a n d i d a t e f o 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 . The study of the (311) s u r f a c e of copper i s p r e s e n t e d f i r s t . The d e t e r m i n a t i o n of the N i ( 3 1 1 ) s u r f a c e geometry f o l l o w s , and t h i s Chapter a l s o i n c l u d e s an e x a m i n a t i o n of the s e n s i t i v i t y of the r - f a c t o r a n a l y s i s t o some of the e x p e r i m e n t a l and d a t a h a n d l i n g methods commonly employed i n LEED c r y s t a l l o g r a p h y . . 76 Fig.4.1 I l l u s t r a t i o n of the c e n t e r e d - r e c t a n g u l a r net that d e s c r i b e s a fcc(311) s u r f a c e . 11-• ( 06 ) • (2 6 ) •(15) • (1 5) • ( 0 4 ) • ( 2 4 ) • (13) • (1 3) • ( 0 2 ) • (2 2 ) • (T 1) • (1 1) 0 | ( 0 0 ) # ( 0 2) • (1 T) a* • d T) • ( 0 2 ) • (2 2 ) • (1 3) •o 3) f.cc. (311) reciprocal net F i g . 4 . 2 R e c i p r o c a l net v e c t o r s and beam l a b e l s c o r r e s p o n d i n g t o t h e s u r f a c e net shown i n F i g . 4 . 1 78 4.2 COPPER (311) 4.2.1 E x p e r i m e n t a l Data f o r the C U ( 3 1 1 ) s u r f a c e were o b t a i n e d i n the V a r i a n 240 chamber d e s c r i b e d i n Chapter 3. The copper specimens were c u t by spark e r o s i o n from a 99.999% p u r i t y c r y s t a l r o d . The c r y s t a l s l i c e was p r e p a r e d f o l l o w i n g the p r o c e d u r e g i v e n i n S e c t i o n 3.1 except t h a t a s a t u r a t e d s o l u t i o n of CuCl i n HCI was used i n the l a s t p o l i s h i n g s t a g e w i t h the one micron diamond p a s t e [ 9 7 ] . The sample was mounted on a s t a n d a r d V a r i a n r e s i s t i v e h e a t e r and a c h r o m e l - a l u m e l thermocouple was a t t a c h e d . A f t e r bakeout, l a r g e amounts of c a r b o n and s u l p h u r as w e l l as some phosphorus, and e i t h e r boron or c h l o r i n e , were seen i n the Auger s p e c t r a taken u s i n g the LEED o p t i c s . The sample was c l e a n e d by c y c l e s of Ar+ bombardment (1 -2juA,300-400eV) and a n n e a l i n g a t 400-700°C. With t h i s method, a l l c o n t a m i n a n t s except carbon c o u l d be removed t o below the d e t e c t i o n l i m i t of the r e t a r d i n g f i e l d a n a l y z e r . Most of the r e s i d u a l carbon c o u l d be removed by h e a t i n g the sample f o r 15-20 minutes i n 1x10- 6 t o r r of oxygen, f o l l o w e d by a f u r t h e r Ar+ bombard and a f i n a l a n n e a l f o r 10 minutes a t 400-500°C. S u r f a c e s c o u l d be o b t a i n e d w i t h carbon Auger peak h e i g h t s reduced t o l e s s than 1/100 of t h a t f o r the 63eV peak of copper. The d e t e c t a b l e l i m i t f o r Auger s i g n a l s was about 1/200 of the Cu(63eV) peak h e i g h t . U s i n g the r e l a t i v e s e n s i t i v i t i e s of the Auger s i g n a l s f o r carbon and copper from the s p e c t r a g i v e n by P a l m b e r g [ 7 7 ] , the r e s i d u a l carbon c o n t a m i n a t i o n appears t o c o r r e s p o n d t o l e s s than 5% of a monolayer. From the v a l u e s of S(150eV)/Cu(63eV) peak r a t i o s g i v e n by A r g i l e and Rhead[98], the minimum d e t e c t a b l e s u l p h u r 79 ' c o n t a m i n a t i o n would be about 1-2% of a monolayer. An Auger spectrum of a c l e a n e d s u r f a c e i s shown i n F i g u r e 4.3b w i t h the spectrum of a c o n t a m i n a t e d s u r f a c e shown f o r comparison i n F i g u r e 4.3a. No e v i d e n c e of f a c e t i n g or r e c o n s t r u c t i o n of the s u r f a c e was i n d i c a t e d from o b s e r v a t i o n s of the LEED d i f f r a c t i o n p a t t e r n s . Adsorbed o v e r l a y e r s of both oxygen and s u l p h u r d i d not produce any w e l l - o r d e r e d LEED p a t t e r n s , but o f t e n showed s t r e a k s t h a t c o r r e s p o n d t o p a r t i a l o r d e r i n g a l o n g the s t e p s on the (311) s u r f a c e . Photographs of the normal i n c i d e n c e d i f f r a c t i o n p a t t e r n d i s p l a y e d on the LEED f l u o r e s c e n t s c r e e n were taken a t 2eV i n t e r v a l s from 50 t o 230eV. A second independent s e t of da t a was taken t o check f o r the r e p r o d u c i b i l i t y of the e x p e r i m e n t a l 1(E) c u r v e s . The photographs were a n a l y z e d u s i n g the v i d i c o n c a m e r a - d i g i t i z e r system and d a t a f o r 17 beams, i n c l u d i n g 4 e q u i v a l e n t p a i r s , were output on paper tape f o r subsequent comparison w i t h the t h e o r e t i c a l c a l c u l a t i o n s . 80 copper sulphur carbon 11 "l^ r-/ 50 r- 1-ELECTRON ENERGY (eV) 550 F i g . 4 . 3 a ) Auger spectrum of a c o n t a m i n a t e d Cu(311) s u r f a c e t aken u s i n g the LEED o p t i c s as an energy a n a l y z e r (Ep=1500eV, Ip=20;uA). F i g . 4 . 3 b ) Auger spectrum of the c l e a n e d Cu(311) s u r f a c e . 81 4.2.2 C a l c u l a t i o n s and R - f a c t o r R e s u l t s Due t o the r e l a t i v e l y c l o s e b u l k i n t e r l a y e r s p a c i n g , (1.09A), t h e l a y e r d o u b l i n g method was chosen f o r the c a l c u l a t i o n s . E i g h t phase s h i f t s were used, and they were o b t a i n e d from the p o t e n t i a l of B u r d i c k and Chodorowf99]. The v i b r a t i o n a l a m p l i t u d e s f o r both the t o p l a y e r and the b u l k l a y e r s - were s e t e q u a l u s i n g the s u r f a c e Debye temperature (0j, = 27O°K) measured f o r the Cu( 111) s u r f ace [ 1 00 ]. The maximum number of beams made a v a i l a b l e f o r the c a l c u l a t i o n of the d i f f r a c t i o n m a t r i c e s was 67 and t h i s r e s u l t e d i n CPU times of about l O O s f o r each energy p o i n t on the IBM 370/168 computer. An i n i t i a l v a l u e of -9.5eV (b e s t v a l u e found f o r C U ( 1 1 1 ) by Watson et a l [ 9 0 ] ) was chosen f o r Vor 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 ; t h i s v a l u e was then r e f i n e d d u r i n g the r - f a c t o r comparison w i t h the e x p e r i m e n t a l I ( E ) c u r v e s by r i g i d l y s h i f t i n g the energy s c a l e . A f t e r examining the narrowest peaks i n the e x p e r i m e n t a l d a t a , V o i was s e t t o -4.5eV at lOOeV energy and was g i v e n the 1/3 power energy dependence i n e q u a t i o n ( 2 . 9 ) . The t o p l a y e r s p a c i n g was the o n l y g e o m e t r i c a l parameter which was a l l o w e d t o v a r y ; c a l c u l a t i o n s were made f o r n i n e ^ d % v a l u e s from +5.0% t o -15.0%. F i g u r e s 4.4-4.7 compare some of the e x p e r i m e n t a l 1(E) c u r v e s w i t h t h o s e from the c a l c u l a t i o n s f o r d i f f e r e n t topmost l a y e r s p a c i n g s . The r e m a i n i n g e x p e r i m e n t a l 1(E) c u r v e s a r e p r e s e n t e d i n Appendix 1. The r - f a c t o r a n a l y s i s of Z a n a z z i and Jona was used i n the comparison between the c a l c u l a t e d and e x p e r i m e n t a l 1(E) c u r v e s and a minimum v a l u e of Rzj=0.088 was found w i t h the c o r r e s p o n d i n g A d % e q u a l t o -5.0±1.5%. A c o n t o u r p l o t of the 82 multi-beam R z j v a l u e s f o r v a r i o u s Vor and topmost i n t e r l a y e r s p a c i n g s i s shown i n F i g u r e 4.8. The e x p e r i m e n t a l 1(E) c u r v e s used i n the comparison were smoothed once w i t h a t h r e e - p o i n t smoothing r o u t i n e u s i n g 1:2:1 w e i g h t s . The e r r o r l i m i t s on the f i n a l d e t e r m i n a t i o n a r e g i v e n by U{2}G (eqn.3.12). 83 Energy (eV) F i g . 4 . 4 (0,-2) beam measured at normal i n c i d e n c e from Cu(311) compared with c a l c u l a t e d 1(E) curves f o r v a r i o u s topmost i n t e r l a y e r s p a c i n g s . 84 Energy (eV) Fig.4.5 (1,-3) and (-1,-3) beams measured at normal i n c i d e n c e from Cu(311) compared with c a l c u l a t e d 1(E) curves f o r v a r i o u s topmost i n t e r l a y e r s p a c i n g s . 85 - 1 I I J I 5 0 100 150 2 0 0 250 Energy (eV) Fig.4.6 (1,5) beam measured at normal i n c i d e n c e from Cu(311) compared with c a l c u l a t e d 1(E) curves f o r v a r i o u s topmost i n t e r l a y e r s p a c i n g s . 86 -I 1 I I > 5 0 100 150 200 250 Energy (eV) Fig.4.7 (1,1) and (-1,1) beams measured at normal i n c i d e n c e from Cu(311) compared with c a l c u l a t e d 1(E) curves f o r v a r i o u s topmost i n t e r l a y e r s p a c i n g s . 87 Fig. 4.8 A contour p l o t of the multi-beam R z j v a l u e s f o r the copper(311) normal i n c i d e n c e data as a f u n c t i o n of Vor e n d &d%. 88 4.3 NICKEL(311) The study of the s t r u c t u r e of the (311) s u r f a c e of n i c k e l was more e x t e n s i v e than t h a t f o r the c o r r e s p o n d i n g s u r f a c e of copper i n s o f a r as i t i n c l u d e d an i n v e s t i g a t i o n of many of the p r o c e d u r e s t h a t a r e used i n LEED c r y s t a l l o g r a p h y . T h i s work r e p r e s e n t s a c o n t i n u a t i o n of s t u d i e s a l r e a d y done by t h i s group i n a s s e s s i n g the s e n s i t i v i t y of the r - f a c t o r a n a l y s i s t o the v a l u e s of t h e n o n - s t r u c t u r a l parameters i n v o l v e d i n . the t h e o r e t i c a l c a l c u l a t i o n s [ 1 0 1 ] . These s t u d i e s were done t o f i n d g u i d e l i n e s f o r p r o c e d u r e s t h a t w i l l a l l o w more d e f i n i t i v e r e s u l t s t o be o b t a i n e d from the comparison of the c a l c u l a t e d and e x p e r i m e n t a l 1(E) c u r v e s . 4.3.1 E x p e r i m e n t a l The (311) f a c e s used i n th e s e e x p e r i m e n t s were c u t by spark e r o s i o n from a 99.999% p u r i t y , 1/4" d i a m e t e r s i n g l e c r y s t a l n i c k e l r o d o r i e n t e d u s i n g Laue photographs t o w i t h i n 1/2° of the (311) d i r e c t i o n . Each sample was then m e c h a n i c a l l y p o l i s h e d (0.05 micron a l u m i n a ) , mounted on a V a r i a n h e a t e r assembly, and a 0.005" c h r o m e l - a l u m e l thermocouple was spot-welded t o the c r y s t a l edge. The specimen was i n t r o d u c e d i n t o the V a r i a n FC12 UHV chamber where the s u r f a c e was c l e a n e d and o r d e r e d by c y c l e s of Ar+ bombardment ( 500-2000eV, 1 - 1 O J J A ) and h e a t i n g a t 700°C. Su l p h u r and carbon were the p r i n c i p a l c o n t a m i n a n t s and s u l p h u r p r o v e d v e r y troublesome s i n c e i t d i f f u s e d t o the s u r f a c e q u i c k l y a t t e m p e r a t u r e s above 600°C. A f t e r many hours of a l t e r n a t i n g Ar+ bombardment and a n n e a l i n g , the s u l p h u r c o n t a m i n a t i o n c o u l d be reduced t o m i n i m a l l e v e l s p r o v i d e d the a n n e a l a f t e r argon 89 bombarding d i d not exceed 1/2 hour a t 600°C. Photographs of the LEED p a t t e r n a t normal i n c i d e n c e were taken a t 2eV i n t e r v a l s over the energy range 50 t o 230eV; t h i s was r e p e a t e d for" an i n d e p e n d e n t l y c l e a n e d and a l i g n e d sample, and y e t a g a i n f o r a s l i g h t l y , c o n t a m i n a t e d s u r f a c e which had been a l l o w e d t o 'soak' f o r about 3 hours a t 2 x l O - 1 0 t o r r . F i g u r e s 4.9a,b show Auger s p e c t r a of the c o n t a m i n a t e d and c l e a n e d s u r f a c e s . U s i n g the r e l a t i v e Auger peak h e i g h t s S ( 1 5 0 e V ) / N i ( 8 6 5 e V ) , f o r s u l p h u r adsorbed on n i c k e l s u r f a c e s g i v e n by D e m u t h [ i 0 2 ] , the s u l p h u r i m p u r i t y l e v e l f o r our c l e a n e d N i ( 3 1 1 ) s u r f a c e s c o r r e s p o n d s t o l e s s than 2% of a monolayer. The LEED p a t t e r n s i n d i c a t e d t h a t c o n t a m i n a n t s on the N i (31 1 ) s u r f a c e o f t e n o r d e r e d ; ( 1 x 2 ) , ( 2 x 5 ) , ( 2 x 1 ) , and (2x3) p a t t e r n s were observed f o r s u l p h u r d i f f u s e d from the b u l k . Oxygen and carbon i m p u r i t i e s produced s t r e a k s i n the S2* d i r e c t i o n , w i t h i n d i c a t i o n s of a p a r t i a l l y o r d e r e d (1x2) s t r u c t u r e . No measurements of 1(E) c u r v e s were made f o r these o v e r l a y e r s t r u c t u r e s . 90 n i c k e l F i g . 4 . 9 a Auger spectrum of a c o n t a m i n a t e d N i ( 3 1 1 ) s u r f a c e taken u s i n g the CMA on the FC12 vacuum system w i t h Ep=2500eV, Ip=100;uA. Fig.4.9b) Auger spectrum of a c l e a n e d N i ( 3 1 1 ) s u r f a c e . 91 4.3.2 C a l c u l a t i o n s and R - f a c t o r r e s u l t s The l a y e r d o u b l i n g method used f o r 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 C U ( 3 1 1 ) was a l s o used f o r N i ( 3 1 1 ) i n c o n j u n c t i o n w i t h the f o l l o w i n g v a l u e s f o r the n o n - s t r u c t u r a l p a r a m e t e r s ; Vor = - l 0 . 0 e V , Voi=-0.9E 1/ 3eV (E i n e V ) , 9=380°K. The v a l u e f o r V o i was o b t a i n e d w i t h e q u a t i o n 2.9 a f t e r measuring the w i d t h s of the narrowest peaks i n the e x p e r i m e n t a l 1(E) c u r v e s . The v a l u e used f o r the Debye temperature was o b t a i n e d by a v e r a g i n g the v i b r a t i o n a l a m p l i t u d e s d e t e r m i n e d by C l a r k e t a l [ l 0 3 ] f o r the f i r s t f i v e l a y e r s , w i t h w e i g h t i n g 0.38, 0.26, 0.17, 0.12, 0.07 suggested by the a t t e n u a t i o n of lOOeV e l e c t r o n s p r o p a g a t i n g between the l a y e r s w i t h Voi=-4ev. These n o n - s t r u c t u r a l parameters are r e a s o n a b l y s i m i l a r t o those used by Demuth e t a l [ 9 4 ] f o r s t u d i e s of the low index s u r f a c e s of n i c k e 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 f o r N i ( 3 1 1 ) used e i g h t energy dependent phase s h i f t s d e r i v e d from Wakoh's band s t r u c t u r e p o t e n t i a l [ 1 0 4 ] , and were run f o r the energy range 50 t o 200eV, f o r n i n e topmost i n t e r l a y e r s p a c i n g s (-24^ A d % £0). The range of c o n t r a c t i o n s c o n s i d e r e d was d e t e r m i n e d a f t e r a p r e l i m i n a r y c a l c u l a t i o n had i n d i c a t e d a c o n t r a c t i o n of 12% or g r e a t e r . Up t o 51 n o n - e q u i v a l e n t beams were i n c l u d e d i n the c a l c u l a t i o n s of the bu l k d i f f r a c t i o n m a t r i c e s . The 1(E) c u r v e s so c a l c u l a t e d showed u n p h y s i c a l s p i k e s or 'c r a s h e s ' a t some p a r t i c u l a r e n e r g i e s and g e o m e t r i e s . Such n u m e r i c a l i n s t a b i l i t i e s have been obser v e d by p t h e r s [ l 0 5 ] , and they pr o v e d p a r t i c u l a r l y troublesome a t h i g h e r e n e r g i e s . In o r d e r t o f a c i l i t a t e the development of a u t o m a t i c assessments of e x p e r i m e n t a l and c a l c u l a t e d . I ( E ) c u r v e s , i t i s im p o r t a n t t h a t 92 such i r r e g u l a r f e a t u r e s can be r e a d i l y i d e n t i f i e d . O f t e n t h e i r e x i s t e n c e i s o b v i o u s , however, s i n c e t h i s i s not always the c a s e , methods a r e needed f o r , d e t e c t i n g and c o r r e c t i n g any i n t e n s i t i e s t h a t have been so a f f e c t e d . In t h i s study i t was found t h a t a p l o t of the sum of the r e f l e c t e d i n t e n s i t i e s v s . A d % f o r p a r t i c u l a r e n e r g i e s p r o v i d e d a c o n v e n i e n t and e f f e c t i v e way of d e t e c t i n g the i n c o r r e c t l y c a l c u l a t e d d a t a p o i n t s (which appear d i s p l a c e d from an o t h e r w i s e smooth c u r v e ) , even when t h e r e were no o b v i o u s i n d i c a t i o n s i n the 1(E) c u r v e s t h a t an e r r o r had o c c u r r e d . An example of t h i s p r o c e d u r e i s d e t a i l e d i n F i g u r e s 4 . l 0 a,b,c. When e r r o r s were found, the i n t e n s i t i e s were r e c a l c u l a t e d f o r d i f f e r e n t numbers of beams. The 'crashed' d i f f r a c t e d i n t e n s i t i e s were r e p l a c e d by i n t e n s i t i e s c a l c u l a t e d f o r the l a r g e s t number of beams t h a t c o u l d be used b e f o r e any n u m e r i c a l e r r o r s became a p p a r e n t . The s u c c e s s of t h i s approach i s i l l u s t r a t e d by F i g u r e s 4 . l 0 a-d which show, p l o t s of i n t e n s i t y v s . geometry f o r c a l c u l a t i o n s u s i n g d i f f e r e n t number of d i f f r a c t e d beams. For c a l c u l a t i o n s a t e n e r g i e s t h a t d i d not show any n u m e r i c a l e r r o r s the i n t e n s i t i e s o b t a i n e d u s i n g 42 beams were u s u a l l y changed by l e s s than 5% from those o b t a i n e d from c a l c u l a t i o n s i n v o l v i n g 51 beams. F i g u r e 4 . l 0 d shows a p l o t , f o r d i f f e r e n t numbers of beams, of the sum of r e f l e c t e d a m p l i t u d e s (RA) of the d i f f r a c t e d beams. RA=^, |R: ( r e a l ) | + | R"L( i m a g i n a r y ) | (4 .1) L where R~L i s the r e f l e c t e d a m p l i t u d e f o r the i t h beam i n the -z d i r e c t i o n . RA v a l u e s were o b t a i n e d from the s u b r o u t i n e which 93 0 . 0 1 5 i-u fa. wi 0 . 0 1 2 - 2 1 • . S I C ) 4 2 , 5 1 , - 2 B I-•21 K U K B E R O F B E A K S / ' OS E D • ) —r— 3 0 0 1 0 0 j N U M B E R O F B E A M S U S E D S 0 ta *3 ec / \ 2B \ 3 5 \ \ \ \ \. * v. \ ( 0 , 0 ) B E A K r x \ x \ v *»\ •> \ e) £> 51 t 2B 35 .» s / / • ( 1 , 3 ) B E A M \ ^ - 2 4 -24 Fig.4 .10 a) Sum of emergent beam i n t e n s i t i e s vs.^d% at 200eV. b),c) R e f l e c t e d i n t e n s i t y v s . A d % at 200eV for (0,0) and (1,3) beams. The c a l c u l a t i o n s with 42 or 51 beams y i e l d the same i n t e n s i t i e s (on t h i s scale) except at ^ d%=-18 where the c a l c u l a t i o n with 51 beams f a i l s to converge.) d) RA (equation 4.1) p l o t t e d as a f u n c t i o n of the number of beams used i n the c a l c u l a t i o n of the bulk d i f f r a c t i o n matrices at 200eV. 94 c a l c u l a t e s the d i f f r a c t i o n m a t r i c e s f o r the b u l k l a y e r s . The assessments of the 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) c u r v e s u s i n g the r e l i a b i l i t y index of Z a n a z z i and Jona y i e l d e d o p t i m a l agreement when Ad%=-14.7±2.6 (Rzj=0.12), w h i l e t h a t of Pendry gave Ad%=-14.4±2.2,(Rp=0.32). The u n c e r t a i n t i e s quoted are the v a l u e s of U{2}G (eqn 3.12). Examples of 1(E) c u r v e s f o r N i ( 3 1 1 ) a r e shown i n F i g u r e s 4.11-4.15. Here each e x p e r i m e n t a l c u r v e i s p l o t t e d above the s e t of c u r v e s c a l c u l a t e d f o r A d % v a l u e s r a n g i n g from -24 t o 0. Contour p l o t s of R z j and Rp v e r s u s A d % and Vor are shown i n F i g u r e s 4.16 and 4.17 r e s p e c t i v e l y . Appendix 1 c o n t a i n s f u r t h e r 1(E) c u r v e s measured f o r normal i n c i d e n c e on N i ( 3 1 1 ) . 95 i i r 4 0 6 0 8 0 '^^^PY ^EvP 2 2 0 .4.11 (0,4) beam measured at normal i n c i d e n c e from Ni(311) compared with 1(E) curves c a l c u l a t e d f o r A d % from -24 ( 0 . 8 0 A ) to 0 ( 1 . 0 6 A ) . 96 Fiq.4.12 (0,2) beam measured at normal incidence from Ni 311.) 9 compared with 1(E) curves c a l c u l a t e d f o r * d % from -24 (0.80A) t o 0 (1.06A). 97 F i a 4 13 (0,-2) beam measured at normal incidence from Ni(311) 9 compared w i t h 1(E) curves c a l c u l a t e d f o r * d % from -24 (0.80A) to 0 ( 1 . 0 6 A ) . 9 8 (0,-4) beam measured a t normal i n c i d e n c e from Ni(311) compared with I(E) curves c a l c u l a t e d f o r * d % from 24 rn.ftol) t o 0 ( 1 . 0 6A ) . F i g 4.14 ( O o l J ' t o 6 (1.06 99 Fig.4.15 (1,-1) beam measured at normal incidence from Ni(311) compared w i t h 1(E) curves c a l c u l a t e d f o r * d % from -24 (O.BOA) t o 0 (1.06A). 100 2*J RFRCTOR FOR NJ(311) -6 -8 uj »— o CL. a: -10 -12 -14 -24 -IB -12 -6 * CHANGE IN TOP SPACING Fig.4.16 Contour p l o t of v a l u e s of multi-beam index Rzj f o r 1(E) curves measured a t normal i n c i d e n c e from a N:(311) s u r f a c e as a f u n c t i o n of inner p o t e n t i a l (Vor) ai.d percentage change i n the topmost i n t e r l a y e r s p a c i n g (Ad%). 101 PENDRT RFfiCTOR FOR NK311) -6 -8 L U •— O -10 -32 -34 -24 -38 -32 -6 % CHflNGE JN TOP SPACING Fig.4 .17 Contour p l o t of v a l u e s of multi-beam index Rp f o r 1(E) curves measured at normal i n c i d e n c e from a Ni(311) s u r f a c e as a f u n c t i o n of inner p o t e n t i a l (Vor) and percentage change i n the topmost i n t e r l a y e r spacing ( A d % ) . 102 4.4 THE R-FACTOR AND THE NICKEL EXPERIMENTAL DATA S i n c e the d e t e r m i n a t i o n of s u r f a c e geometry i n v o l v e s the comparison of 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) c u r v e s , i t i s im p o r t a n t t o know whether the r e s u l t s d e r i v e d from such comparisons a r e s e n s i t i v e t o changes i n the mode of tr e a t m e n t of the e x p e r i m e n t a l d a t a . An i n v e s t i g a t i o n was made here of the e f f e c t s on Rp and R z j of smoothing the e x p e r i m e n t a l d a t a , of a v e r a g i n g e x p e r i m e n t a l 1(E) c u r v e s t h a 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 , and of c o r r e c t i n g the measured i n t e n s i t i e s f o r d i f f e r e n c e s i n g r i d t r a n s p a r e n c y . R e l i a b i l i t y index comparisons were a l s o made between e x p e r i m e n t a l beams which s h o u l d be e q u i v a l e n t a t normal i n c i d e n c e , and between c o r r e s p o n d i n g beams i n i n d e p e n d e n t l y measured s e t s of d a t a . 4.4.1 E f f e c t s of G r i d T r a n s p a r e n c y When a LEED p a t t e r n i s observ e d through the window of the vacuum chamber, the h e m i s p h e r i c a l g r i d s have d i f f e r e n t t r a n p a r e n c i e s f o r l i g h t coming from d i f f e r e n t p a r t s of the s c r e e n . F o r the e x p e r i m e n t a l s e t up shown i n F i g u r e 4.18, Legg et a l [ l 0 6 ] have c o r r e c t e d f o r the v a r i a t i o n i n g r i d t r a n s p a r e n c y w i t h a t r e a t m e n t which assumes t h a t the l i g h t i s e m i t t e d from the phosphor a c c o r d i n g t o the c o s i n e law (e x p e c t e d f o r a rough s u r f a c e ) . From s i m p l e geometry f o r a f o u r - g r i d system, an a c t u a l i n t e n s i t y I i s changed t o an apparent i n t e n s i t y I ' by I'=T(w)I, where T(w) has the form T(w)=t«(1-d/s)'tsin(w) (4.2) 103 k—d—>l F i g . 4 . 1 8 E x p e r i m e n t a l arrangement f o r LEED i n t e n s i t y measurements showing the a n g l e s and d i s t a n c e s r e l e v a n t t o e q u a t i o n 4.2 which c o r r e c t s f o r the d i f f e r e n c e s i n g r i d t r a n s m i s s i o n as a f u n c t i o n of beam p o s i t i o n on the LEED s c r e e n ( a l l d i m e n s i o n s i n mm). 104 where t = 1 - d / [ s s i n ( w ) ] , d i s the diameter of the g r i d w i r e s and s t h e i r s e p a r a t i o n . F i g u r e 4 . 1 9 shows an example of the e f f e c t of t h i s c o r r e c t i o n on a p a r t i c u l a r e x p e r i m e n t a l 1(E) c u r v e . The e f f e c t of the t r a n s p a r e n c y c o r r e c t i o n i s t o i n c r e a s e the beam i n t e n s i t i e s f o r lower e n e r g i e s . A q u e s t i o n of importance t o the p r e s e n t study c o n c e r n s whether s u r f a c e s t r u c t u r a l d e t e r m i n a t i o n s a r e a f f e c t e d by t h i s c o r r e c t i o n . The e f f e c t s of t h i s p r o c e d u r e f o r d a t a from N i ( 3 1 1 ) f o r the r e l i a b i l i t y i n d i c e s R z j and Rp were found t o be r a t h e r s m a l l . The R v a l u e s were reduced s l i g h t l y w i t h the c o r r e c t i o n ( s p e c i f i c a l l y Rpmin=0.33 u n c o r r e c t e d , Rpmin=0.32 c o r r e c t e d , Rzjmin=0.14 u n c o r r e c t e d , Rzjmin=0.12 c o r r e c t e d ) , but changes i n Gmin t u r n e d out t o be l e s s than 1% (0.01 A ) . 4.4.2 Smoothing Some smoothing of e x p e r i m e n t a l 1(E) c u r v e s i s g e n e r a l l y n e c e s s a r y t o remove n o i s e and t h e r e b y a v o i d e x t r a n e o u s s t r u c t u r e which can a f f e c t the v a l u e s of the d e r i v a t i v e s r e q u i r e d f o r the r - f a c t o r c a l c u l a t i o n s . F i g u r e 4.20 shows a p l o t of Rp v s . A d % f o r the comparison of c a l c u l a t e d 1(E) c u r v e s w i t h e x p e r i m e n t a l c u r v e s t h a t have undergone up t o t h r e e smoothing o p e r a t i o n s . A s i n g l e o p e r a t i o n of smoothing was done a c c o r d i n g t o I ( E ) = [ l r ( E - e ) + 2 I r ( E ) + I r ( E + e ) 1 / 4 (4.3) where the I r i n d i c a t e raw or i n p u t i n t e n s i t i e s , and e i s the s p a c i n g of the energy g r i d f o r the e x p e r i m e n t a l i n t e n s i t i e s . 105 CORRECTED FOR EFFECTS OF GRID TRANSMISSION to to K < *-< z w 2 l ! UNCORRECTED \ U > I I ' l l / / -M--V 40 80 (0,-4) \ i •\+-i i i .A.... t i "i / / AJ ---J 1 20 1 60 ENERGY (eV) 200 240 F i g . 4 . 1 9 The (0,-4) beam measured a t normal i n c i d e n c e from N i ( 3 1 1 ) b e f o r e and a f t e r c o r r e c t i n g f o r the d i f f e r i n g t r a n s m i s s i o n of the LEED g r i d s as a f u n c t i o n of energy. L a r g e s t changes o c c u r a t the l o w e s t e n e r g i e s . 1 0 6 4.20 Multi-beam index Rp p l o t t e d as a f u n c t i o n o f ^ d % . The Rp v a l u e s are from the comparison of 1(E) curves measured at normal i n c i d e n c e from N i ( 3 1 1 ^ t h a t have been smoothed up t o 3 times, with c a l c u l a t e d 1(E) curves f o r changes i n the topmost i n t e r l a y e r spacings i n the range from -24% t o 0%. 107 F i g u r e 4.20 i n d i c a t e s t h a t a l i m i t e d amount of smoothing g i v e s a r e d u c t i o n of Rp v a l u e s , a l t h o u g h a f t e r e x c e s s i v e smoothing v a l u e s of Rp i n c r e a s e . Rpmin, Gmin and a s s o c i a t e d u n c e r t a i n t i e s are l i s t e d i n Ta b l e 4.1 f o r the comparison of 1(E) c u r v e s c a l c u l a t e d f o r C l d % v a l u e s i n the range -24 t o 0 w i t h e x p e r i m e n t a l d a t a t h a t has been smoothed .0,1,2,3 t i m e s . The u n c e r t a i n t y i n the s t r u c t u r a l d e t e r m i n a t i o n i s l a r g e s t f o r unsmoothed d a t a and t h e r e a r e o n l y s m a l l d i f f e r e n c e s i n the v a l u e s of Gmin o b t a i n e d f o r d a t a t h a t has undergone d i f f e r e n t amounts of smoothing. 4.4.3 A v e r a g i n g S y m m e t r i c a l Beams At normal i n c i d e n c e the m i r r o r p l a n e p r e s e n t i n the (311) s u r f a c e r e s u l t s i n the e q u i v a l e n c e of the (h,k) and (-h,k) d i f f r a c t e d beams. The e x p e r i m e n t a l d a t a a v a i l a b l e f o r N i ( 3 1 1 ) c o n s i s t e d of 14 i n d i v i d u a l beams, i n c l u d i n g 4 p a i r s of s y m m e t r i c a l l y e q u i v a l e n t beams. R - f a c t o r comparisons were made f o r the d a t a i n two w a y s : ( i ) f o r a l l 14 beams, and ( i i ) f o r 10 beams where p a i r s of 1(E) c u r v e s which are e x p e c t e d t o be e q u i v a l e n t a r e averaged. In t h i s a n a l y s i s , e v e r y beam was t r e a t e d as a s e p a r a t e measurement f o r the multi-beam Rp v a l u e s and the u n c e r t a i n t i e s . The Pendry r - f a c t o r d e t e r m i n a t i o n s g i v e f o r ( i ) Ad%=-14.4+2.2 a t Rpmin=0.3l5, and f o r ( i i ) Ad%=-13.8±2.0 at Rpmin = 0.323... The change i n Gmin and the i n c r e a s e d Rpmin v a l u e are a s s o c i a t e d w i t h the e q u i v a l e n t beams b e i n g g i v e n a s m a l l e r w e i g h t i n g i n ( i i ) compared w i t h ( i ) . There i s a r e d u c t i o n i n U{l,2,3} v a l u e s f o r the averaged s e t and the r ( L ) v a l u e s f o r an averaged beam were always s l i g h t l y s m a l l e r than the average r ( L ) J 0 8 T a b l e 4.1 Pendry R - f a c t o r , * d % , and u n c e r t a i n t y v a l u e s where Rp i s m i n i m i z e d , f o r the comparison of c a l c u l a t e d 1(E) c u r v e s f o r N i ( 3 1 1 ) w i t h e x p e r i m e n t a l d a t a t h a t has been smoothed 0,1,2,3 t i m e s . These v a l u e s were o b t a i n e d f o r Vor=-10eV. Rpmin Gmin U{1}G U{2}G U{3}G RAW DATA 0.378 -14.1 ±2.6 ±2.1 ±2.9 SMOOTHED ONE TIME 0.332 -14.1 ±2.4 ±1.9 ±2.8 SMOOTHED TWO TIMES 0.320 -14.4 ±2.5 ±1.9 ±2.6 SMOOTHED THREE TIMES 0.329 -14.4 ±2.8 ±2.0 ±2.5 109 v a l u e of the two e q u i v a l e n t beams. I t may be t h a t a v e r a g i n g s i m u l a t e s a c o r r e c t i o n f o r d e v i a t i o n s i n t h e a n g l e of i n c i d e n c e . For d e v i a t i o n s from normal i n c i d e n c e , one of the s e t of e q u i v a l e n t beams pr o p a g a t e s so t h a t i n t e r l a y e r s p a c i n g s a r e s l i g h t l y s m a l l e r than those e x p e r i e n c e d a t normal i n c i d e n c e and the s y m m e t r i c a l beam e x p e r i e n c e s s l i g h t l y l o n g e r i n t e r l a y e r s p a c i n g s . G e n e r a l l y , t h i s would r e s u l t i n the peaks i n the 1(E) c u r v e s s h i f t i n g t o h i g h e r and lower e n e r g i e s r e s p e c t i v e l y , and thus i t i s not u n r e a s o n a b l e t h a t the averaged d a t a can have peaks a t p o s i t i o n s which a re c l o s e r t o e x p e c t a t i o n f o r normal i n c i d e n c e . F i g u r e 4.21 shows the e x p e r i m e n t a l 1(E) c u r v e s f o r a p a i r of beams which a r e ex p e c t e d t o be e q u i v a l e n t i n the i d e a l s i t u a t i o n , as w e l l as t h e i r a v e r a g e . 4.4.4 Comparison of two s e t s of e x p e r i m e n t a l d a t a S i n g l e beam r - f a c t o r comparisons were made f o r two independent s e t s of e x p e r i m e n t a l 1(E) c u r v e s , and v a l u e s from b o t h Z a n a z z i and Jona's and Pendry's r e l i a b i l i t y i ndex c a l c u l a t i o n s a r e l i s t e d i n Ta b l e 4.3. Ta b l e 4.2 i n c l u d e s d a t a on the r - f a c t o r comparison of s y m m e t r i c a l l y r e l a t e d beams from the same ex p e r i m e n t . Comparisons of 1(E) c u r v e s between c o r r e s p o n d i n g beams i n the two e x p e r i m e n t s , or f o r e q u i v a l e n t beams w i t h i n a s i n g l e e x p e r i m e n t , always y i e l d e d lower s i n g l e beam r - f a c t o r s than t h o s e d e t e r m i n e d f o r one e x p e r i m e n t a l 1(E) cu r v e compared w i t h t h e c a l c u l a t e d 1(E) c u r v e w h i c h g i v e s b e s t agreement. F i g u r e s 4.22 and 4.23 show some examples of the e x p e r i m e n t a l 1(E) c u r v e s i n v o l v e d i n the comparisons i n Ta b l e 4.2 110 Fig.4.21. (1,-1) and (-1,-1) beams and t h e i r average from data measured at normal incidence f o r the Ni(311) surface. 1 1 1 T a b l e 4.2 S i n g l e beam r - f a c t o r v a l u e s f o r the comparison of p a i r s of n o m i n a l l y symmetric beams measured a t normal i n c i d e n c e from N i (311), \ BEAMS ZANAZZI AND JONA PENDRY (1,1) (-1,1) 0.028 0.041 (1,-1) (-1,-1) 0.042 0.125 (1,-3) (-1,-3) 0.030 0.180 (1,3) (-1,3) 0.039 0.064 T a b l e 4.3 S i n g l e beam r - f a c t o r v a l u e s f o r the comparison of c o r r e s p o n d i n g beams measured a t normal i n c i d e n c e from N i ( 3 1 1 ) from two independent e x p e r i m e n t s . S u p e r s c r i p t s on beam l a b e l s i n d i c a t e the s o u r c e of the 1(E) c u r v e ( i . e . e i t h e r e x periment 1 or 2 ) . BEAMS ZANAZZI AND JONA PENDRY ( 0 , - 4 ) 1 (0,-4) 2. 0.056 0.045 ( 0 , - 2 ) 1 ( 0 , - 2 ) 2 0.028 0.041 ( 0 , 2 ) * ( 0 , 2 ) 2 0.034 0.040 (0,4) 1 (0,4) 2 0.045 0.091 ( 1 , - D 1 ( 1 , - 1 ) 2 0.104 0.119 ( 1 , 1 ) * ( 1 , 1 ) 2 0.050 0.104 ( 1 , - 3 ) * ( l , - 3 ) 2 0.048 0.061 ( 1 , 3 ) ' ( 1 , 3 ) 2 0.056 0.109-( 0 , - 3 ) 1 ( 0 , - 3 ) 2 0.140 0.098 1 1 2 80 120 160 ENERGY (eV) 200 240 F i g . 4 . 2 2 . (1,-3) and (-1,-3) beams measured f o r normal i n c i d e n c e on N i ( 3 1 1 ) . Upper" p a i r of 1(E) c u r v e s a r e f o r s y m m e t r i c a l l y r e l a t e d beams from one- e x p e r i m e n t , and lower p a i r of c u r v e s a r e from an independent e x p e r i m e n t . 113 ENERGY (eV) F i g . 4 . 2 3 . (1,1) and (-1,1) beams measured f o r normal i n c i d e n c e on N i ( 3 1 1 ) . Upper p a i r of 1(E) c u r v e s a r e f o r s y m m e t r i c a l l y r e l a t e d beams from one e x p e r i m e n t , and lower p a i r of c u r v e s a r e from an independent-e x p e r i m e n t . \ 1 1 4 4.4.5 V a r i a b l e Inner P o t e n t i 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 f o r N i ( 3 1 1 ) used a c o n s t a n t v a l u e of Vor e q u a l t o -lOeV. From LEED i n v e s t i g a t i o n s of the n i c k e l (100),(110) and (1 1 1 ) s u r f a c e s , Demuth et a l [ 9 4 ] p r o v i d e d e v i d e n c e t h a t i t c o u l d be advantageous t o make Vor energy dependent, p a r t i c u l a r l y so i t s v a l u e becomes l e s s n e g a t i v e w i t h i n c r e a s i n g energy. The v a l u e s of Vor t a b u l a t e d by Demuth et a l were used here t o a l t e r the energy s c a l e of the c a l c u l a t e d i n t e n s i t i e s , and new i n t e n s i t y v a l u e s were then d e t e r m i n e d by i n t e r p o l a t i o n t o the o r i g i n a l energy g r i d . An r - f a c t o r comparison of the e x p e r i m e n t a l d a t a w i t h t h e s e 1(E) c u r v e s r e s u l t e d i n a poorer f i t (Rp=0.36) and a s h i f t i n Gmin towards a l e s s c o n t r a c t e d topmost l a y e r s p a c i n g ( Ad%=-10). The d i r e c t i o n of the s h i f t i n Gmin i s c o n s i s t e n t w i t h the form of the Vor dependence, s i n c e , as noted by Demuth et a l , the o b s e r v e d Vor dependence and t o p l a y e r c o n t r a c t i o n s s h i f t peaks i n the 1(E) c u r v e s i n the same d i r e c t i o n . An energy-independent Vor i s f a v o r e d by the p r e s e n t a n a l y s i s ; i t i s p o s s i b l e t h a t the Vor dependence suggested by Demuth et a l a r i s e s i n p a r t from s l i g h t u n d e r e s t i m a t e s of the topmost l a y e r c o n t r a c t i o n s f o r the n i c k e l ( 1 0 0 ) , (111), and (110) s u r f a c e s . 4.5 CALIBRATION The c a l i b r a t i o n of the v i d i c o n - d i g i t i z e r system has been p r e v i o u s l y checked f o r l i n e a r i t y [ 8 7 ] by a n a l y z i n g a s e t of photographs of a s t e p d e n s i t y wedge. I t was shown t h a t t h e d i g i t i z e r output v s . o p t i c a l d e n s i t y (O.D.) i s a p p r o x i m a t e l y l i n e a r over a range of about 1.6 O.D. u n i t s . In the p r e s e n t 1 15 work, t e s t i n g of the p h o s p h o r - f i l m - v i d i c o n system was a c c o m p l i s h e d from a s e r i e s of photographs taken a t a f i x e d i n c i d e n t beam v o l t a g e ( l 7 5 e V ) f o r d i f f e r e n t i n c i d e n t beam c u r r e n t s . The cur v e A i n F i g u r e 4.24 i s the r e s u l t of a c a l i b r a t i o n i n v o l v i n g the d i g i t i z e r o u tput v s . i n c i d e n t beam c u r r e n t . T h i s c u r v e was c o n s t r u c t e d u s i n g s e v e r a l of the d i f f r a c t e d beams a v a i l a b l e a t 175eV so t h a t the range of d i f f r a c t e d beam c u r r e n t s (from weakest beam a t lo w e s t i n c i d e n t c u r r e n t t o the most i n t e n s e beam a t the h i g h e s t i n c i d e n t c u r r e n t ) i s not dependent on the a c c u r a t e measurement of s m a l l v a l u e s ( l e s s than 0.2juA) of i n c i d e n t beam c u r r e n t . The r a t i o of the d i g i t i z e d i n t e n s i t i e s was c o n s i d e r e d t o be an a c c u r a t e measure of the r a t i o of the r e l a t i v e beam c u r r e n t s i n the ' l i n e a r range' of the d a t a ( i . e . d i g i t i z e d i n t e n s i t i e s i n the range 25 t o 275). The c a l i b r a t i o n c u r v e A shown i n F i g u r e 4.24 c o r r e s p o n d s t o the f-s.top ( f 2 . 8 ) of the v i d i c o n camera which gave the l a r g e s t l i n e a r response f o r the range of f i l m d e n s i t i e s e n c o u n t e r e d . T h i s a p e r t u r e s e t t i n g was a l s o used f o r the measurement of the LEED 1(E) c u r v e s f o r N i ( 3 1 1 ) . A s a t u r a t i o n e f f e c t i s e v i d e n t a t h i g h e r i n t e n s i t i e s ; the out p u t a c t u a l l y d e c r e a s e s w i t h i n c r e a s i n g i n c i d e n t beam c u r r e n t s i n the h i g h range of the l a t t e r . T h i s e f f e c t i s i n p a r t due t o the bro a d e n i n g of the d i f f r a c t e d ' beam p r o f i l e on the p h o t o g r a p h i c f i l m . I t can be p a r t i a l l y compensated, as shown i n cur v e C of F i g u r e 4.24, by making the a r r a y s i z e t h a t samples the spot i n t e n s i t y on the d i g i t i z e r s u f f i c i e n t l y l a r g e t o i n c l u d e the e n t i r e c r o s s - s e c t i o n of the beam. The use of s m a l l e r a p e r t u r e s ( f 4 . 0 ) on the v i d i c o n camera 1 16 MEflSURED INTENSITY VS. BEAM CURRENT 0 60 3 20 3 80 240 300 BEAM CURRENT F i g . 4 . 2 4 . D i g i t i z e d i n t e n s i t i e s of d i f f r a c t e d beams measured a t 175eV f o r normal i n c i d e n c e from a N i ( 3 1 1 ) s u r f a c e p l o t t e d as a f u n c t i o n of d i f f r a c t e d beam c u r r e n t . A V i d i c o n camera a p e r t u r e s e t a t f2.8 B V i d i c o n camera a p e r t u r e s e t a t f2.0 C V i d i c o n camera a p e r t u r e s e t a t f2.8 and camera moved away from the f i l m so t h a t t h e e n t i r e d i f f r a c t e d beam p r o f i l e i s i n c l u d e d i n the i n t e g r a t i o n . 117 causes an even more severe s a t u r a t i o n of the c u r v e a t h i g h e r beam c u r r e n t s . T h i s i s because the b l a c k l e v e l (maximum d i g i t i z e d v a l u e ) o c c u r s a t s m a l l e r O.D. v a l u e s on the f i l m , and thus the u s a b l e i n t e n s i t y range i s reduced. The e f f e c t of a l a r g e r a p e r t u r e ( f 2 . 0 ) f o r the v i d i c o n camera i s shown i n F i g u r e 4.24, c u r v e C. There i s e v i d e n t l y l e s s s a t u r a t i o n a t the same f i l m d e n s i t y as i n c u r v e A, but t h e r e i s a s i g n i f i c a n t n o n - l i n e a r i t y a t lower v a l u e s of beam c u r r e n t . The l a r g e s t d i f f r a c t e d beam c u r r e n t s encountered i n the study of the N i ( 3 1 1 ) s u r f a c e c o r r e s p o n d r o u g h l y t o the s t a r t of s a t u r a t i o n i n the c a l i b r a t i o n c u r v e A of Fig.4.24 (about 300 u n i t s of d i g i t i z e d i n t e n s i t y ) . T h i s means t h a t the i n t e n s i t i e s of the s t r o n g e s t peaks i n the measured 1(E) c u r v e s a r e a c t u a l l y s m a l l e r than they s h o u l d be. The l i n e a r range of d i g i t i z e d f i l m i n t e n s i t y v e r s u s i n c i d e n t c u r r e n t appears t o be adequate f o r r - f a c t o r comparisons t h a t are not t o o s e n s i t i v e t o peak shape. M o d i f y i n g the 1(E) c u r v e s measured from the N i ( 3 1 1 ) s u r f a c e u s i n g the r e s u l t s of the measured i n t e n s i t y v s . d i f f r a c t e d beam c u r r e n t ( c u r v e A i n F i g . 4 . 2 4 ) d i d not s i g n i f i c a n t l y a l t e r the v a l u e s of Rp o b t a i n e d f o r the comparison w i t h the c a l c u l a t e d 1(E) c u r v e s ( i . e . changes i n Rpmin and Gmin were much l e s s than •1 %) . I t i s apparent from t h i s study t h a t more thought w i l l have t o be g i v e n t o the s e l e c t i o n of v a r i o u s camera s e t t i n g s ( v i d i c o n and Nikon) and p h o t o g r a p h i c p r o c e s s i n g methods i n o r d e r t o ensure t h a t the measurements f a l l w i t h i n the r e s t r i c t e d l i n e a r range of d i g i t i z e d i n t e n s i t i e s . In t h i s r e g a r d , a r e a l - t i m e d a t a a n a l y s i s system would be advantageous as i t would a l l o w d i r e c t 118 c a l i b r a t i o n of the LEED experiment f o r the n o n - l i n e a r response of the phosphor and TV camera. 4.6 SUPPRESSOR GRID VOLTAGE The r e t a r d i n g or s u p p r e s s o r g r i d s (G2,G3 i n F i g u r e 1.4a) a r e used t o f i l t e r out the i n e l a s t i c a l l y b a c k s c a t t e r e d e l e c t r o n s from the e l a s t i c e l e c t r o n s which a r e of i n t e r e s t i n the LEED ex p e r i m e n t . The v o l t a g e on t h e s e g r i d s , the s u p p r e s s o r g r i d v o l t a g e , d e t e r m i n e s the e n e r g i e s of the e l e c t r o n s t h a t w i l l be t r a n s m i t t e d t h rough t o the s c r e e n . Suppressor g r i d v o l t a g e s are chosen so t h a t o n l y 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 those e l e c t r o n s t h a t have undergone s m a l l energy l o s s e s (e.g. a few eV) can r e a c h the c o l l e c t o r . F i g u r e 4.25 shows a p l o t of the i n t e n s i t y r a t i o s f o r two p a i r s of d i f f r a c t e d beams v s . s u p p r e s s o r g r i d v o l t a g e . I d e a l l y the c u r v e s would be h o r i z o n t a l l i n e s . However, the i n t e n s i t i e s a r e dependent on the s u p p r e s s o r v o l t a g e due t o e f f e c t s such as v a r i a t i o n s i n t h e work f u n c t i o n on d i f f e r e n t p a r t s of the r e t a r d i n g g r i d s (as a r e s u l t of uneven a c c u m u l a t i o n of d e p o s i t e d m a t e r i a l d u r i n g h e a t i n g and bombarding o p e r a t i o n s ) . I f the energy window se t by the s u p p r e s s o r v o l t a g e i s too r e s t r i c t i v e , then the t r a n s m i s s i o n ' c u t o f f w i l l o c cur on the h i g h l y - s l o p e d e l a s t i c peak. For these v a l u e s of s u p p r e s s o r v o l t a g e , s m a l l work f u n c t i o n changes on d i f f e r e n t p a r t s of the g r i d s can l e a d t o a p p r e c i a b l e changes i n t r a n s m i t t e d c u r r e n t . To l e s s e n the e f f e c t s of such v a r i a t i o n s , the d a t a r e p o r t e d i n t h i s t h e s i s was c o l l e c t e d w i t h g r i d s u p p r e s s o r v o l t a g e s 3-6eV more p o s i t i v e than the i n c i d e n t energy. T h i s a l l o w s the 119 1 . B 5 tn a ar ZD tn CE LU o r 1 .BO 1 . 7 5 1.70 I— 1.65 h— 1 .60 0 . 0 2 . 5 5 . 0 7 . 5 ia.o SUPPRESSOR GRID VOLTAGE F i g . 4 . 2 5 . D i g i t i z e d d i f f r a c t e d beam i n t e n s i t y r a t i o s f o r two p a i r s of d i f f r a c t e d beams measured at 175eV for normal incidence from a Ni(311) surface p l o t t e d as a f u n c t i o n of the suppressor g r i d v o l t a g e . The greatest changes occur f o r suppressor g r i d v oltages of l e s s than 3 v o l t s . 120 c o l l e c t i o n of the e n t i r e e l a s t i c peak ( r e s o l u t i o n of a n a l y z e r i s about 1%, i . e . 2eV f o r 200eV e l e c t r o n s ) and a l s o those e l e c t r o n s which c o n t r i b u t e t o the f l a t r e g i o n i n the energy d i s t r i b u t i o n between the e l a s t i c peak and the energy l o s s peaks due t o plasmons i n F i g u r e 1.3a. The s e l e c t i o n of t h i s range of s u p p r e s s o r g r i d v o l t a g e s ( i . e . 3-6eV) i s c o n s i s t e n t w i t h t h e r e s u l t s i n F i g u r e 4 .25 . The p l o t s a r e r e a s o n a b l y f l a t above the lower l i m i t of 3eV, and the 6eV upper l i m i t s h o u l d r e s t r i c t the e n e r g i e s of the c o l l e c t e d e l e c t r o n s so t h a t no e l e c t r o n s t h a t have undergone energy l o s s e s due t o plasmons w i l l be c o l l e c t e d . To a l l e v i a t e work f u n c t i o n v a r i a t i o n s f o r f u t u r e work, we i n t e n d t o c o v e r the LEED o p t i c s w i t h a p r o t e c t i v e s h u t t e r when the sample i s b e i n g h e a t e d or Ar+ bombarded t o p r e v e n t m a t e r i a l from b e i n g d e p o s i t e d on t o the g r i d s . 4.7 DIRTY NI (311) A s e t of 1(E) c u r v e s was measured from a N i (3 1 1 ) s u r f a c e which was exposed f o r about t h r e e hours t o the r e s i d u a l gases c o m p r i s i n g the background a t 1x1 0 - 1 0 t o r r . The photographs of the LEED p a t t e r n showed f a i n t s t r e a k s and o v e r l a y e r s p o t s a t some p a r t i c u l a r e n e r g i e s . An Auger spectrum showed the presence of some carbon and s u l p h u r ; u s i n g the r e l a t i v e s e n s i t i v i t i e s g i v e n by Demuth e t a l [ l 0 2 ] , c o v e r a g e s of about 0.20 of a monolayer C, and about 0.25 of a monolayer S were i n d i c a t e d . A r e l i a b i l i t y index comparison between the e x p e r i m e n t a l 1(E) c u r v e s f o r t h i s c o n t a m i n a t e d s u r f a c e and the I ( E ) c u r v e s c a l c u l a t e d f o r the c l e a n N i ( 3 l l ) s u r f a c e gave be s t agreement of Rp=0.49 at ^,d%=-8.7. The l e v e l of agreement was not as good as 121 f o r the c l e a n N i ( 3 1 1 ) d a t a (Rp=0.32), but t h i s s t u d y i l l u s t r a t e s the p o s s i b i l i t y t h a t a s i g n i f i c a n t r e l a x a t i o n of the top l a y e r s p a c i n g may occur i n t h e p r e s e n c e of adsorbed i m p u r i t i e s ( A d % ( c l e a n e d s u r f a c e ) = - 1 4 . 5 ) ) . A l t e r n a t i v e l y , i t i s p o s s i b l e t h a t t h i s r e l a x a t i o n . i s not r e a l , but r a t h e r a s s o c i a t e d w i t h the d i f f e r e n t phase s h i f t s i n t r o d u c e d by the c o n t a m i n a t i n g o v e r l a y e r . F i g u r e 4.26 shows a c o n t o u r p l o t of Rp as a f u n c t i o n of Vor and A d % . 122 -24 -16 -8 0 %CHANGE IN TOP LAYER SPACING Fig.4.26 Contour p l o t of v a l u e s of multi-beam index Rzj f o r 1(E) curves measured at normal i n c i d e n c e from a contaminated Ni(311) s u r f a c e as a f u n c t i o n of inner p o t e n t i a l (Vor) and percentage change i n the topmost i n t e r l a y e r s p a c i n g (^d%). 123 4.8 CONCLUSION V a l u e s of A d % found i n t h e s e s t u d i e s ( v i z . -5% f o r Cu, -14.5% f o r N i ) can be compared w i t h A d % v a l u e s r e p o r t e d f o r f c c ( l l O ) s u r f a c e s . T h i s comparison i s r e a s o n a b l e s i n c e f c c ( 1 l 0 ) s u r f a c e s a re more l i k e the stepped (311) s u r f a c e s than the f l a t fee (100) and (111) s u r f a c e s . A d % v a l u e s r e p o r t e d f o r f c c ( l 1 0 ) s u r f a c e s a r e l i s t e d below: Ni= -5% [94] Al= -9 t o -15% [107,108] Ag = -6 t o -10% [109,110] Rh= -3% [111] A 14.5% c o n t r a c t i o n f o r the topmost i n t e r l a y e r s p a c i n g of Ni (31 1 ) r e p r e s e n t s a N i - N i bond l e n g t h c o n t r a c t i o n of o n l y 0.075A ( 3 % ) . The c o n t r a c t i o n found f o r N i ( 3 l l ) i s r a t h e r l a r g e r than the 5% c o n t r a c t i o n found f o r the topmost i n t e r l a y e r s p a c i n g of N i ( l l O ) , where the bond l e n g t h c o n t r a c t i o n i s o n l y 1.5%[94]. The p o s s i b i l i t y t h a t the l a r g e p e r c e n t a g e c o n t r a c t i o n i n d i c a t e d f o r N i ( 3 1 l ) i s a s s o c i a t e d w i t h s u r f a c e roughness, as suggested p r e v i o u s l y f o r A l (1 1 0) [ 1 07 ] and Ag (1 1 0) [ 1 1 0 ], seems u n l i k e l y - . The sharp LEED p a t t e r n , the o v e r a l l agreement between the t h e o r e t i c a l and e x p e r i m e n t a l s e t s of 1(E) c u r v e s , as w e l l as the agreement between independent e x p e r i m e n t a l s e t s of d a t a t e n d t o argue a g a i n s t such a problem f o r N i ( 3 l l ) . A l s o , t h e r e was no e v i d e n c e of spot s p l i t t i n g or s t r e a k i n g i n the observed LEED p a t t e r n s t h a t c o u l d have i n d i c a t e d a p p r e c i a b l e amounts of s u r f a c e roughness e x i s t e d . The o v e r a l l c o n t r a c t i o n of 5% found f o r the C U ( 3 1 1 ) topmost i n t e r l a y e r s p a c i n g i s s i g n i f i c a n t l y s m a l l e r than the 14.5% v a l u e 124 found f o r n i c k e l . T h i s d i f f e r e n c e may be r e a l , a l t h o u g h u n e x p l a i n e d , or i t c o u l d be a s s o c i a t e d w i t h the pr e s e n c e of d i s o r d e r e d oxygen t h a t was not d e t e c t e d i n the Auger s p e c t r a . I t has been n o t e d p r e v i o u s l y t h a t t h e r e a re d i f f i c u l t i e s i n d e t e c t i n g the Auger s i g n a l of oxygen a t 5l0eV when oxygen i s adsorbed on copper s u r f a c e s [ 9 3 , 9 8 ] . The presence of the oxygen c o u l d cause an a c t u a l r e l a x a t i o n of the t o p l a y e r of copper atoms towards the b u l k l a y e r s p a c i n g , or an apparent r e l a x a t i o n due t o the d i f f e r e n t phase s h i f t s i n t r o d u c e d by the o v e r l a y e r . T h i s p o s s i b i l i t y i s s u p p o r t e d by the c o n c l u s i o n of S e c t i o n 4.7 where a s i g n i f i c a n t l y s m a l l e r c o n t r a c t i o n was found f o r the t o p l a y e r s p a c i n g of a c o n t a m i n a t e d N i ( 3 1 l ) s u r f a c e compared w i t h the c l e a n s u r f a c e . I t i s c o m f o r t i n g t h a t the e f f e c t s of smoothing, a v e r a g i n g and c o r r e c t i n g e x p e r i m e n t a l i n t e n s i t i e s f o r v a r i a t i o n s i n g r i d t r a n s p a r e n c y produced o n l y v e r y s m a l l changes i n the f i n a l s t r u c t u r e d e t e r m i n a t i o n . The l a r g e r change observed w i t h the i n c l u s i o n of a v a r i a b l e i n n e r p o t e n t i a l i s of more c o n c e r n , but i t a l s o may be a u s e f u l means of d e t e r m i n i n g the dependence of Vor on e l e c t r o n energy. The i n v e s t i g a t i o n of o t h e r e x p e r i m e n t a l p a r a m e t e r s , such as s u p p r e s s o r g r i d v o l t a g e and the c a l i b r a t i o n of the LEED i n t e n s i t y measurements h e l p s e s t a b l i s h ' g u i d e l i n e s f o r f u t u r e work and may a l s o y i e l d i n f o r m a t i o n t h a t w i l l be im p o r t a n t when u p d a t i n g our p r e s e n t LEED system t o p e r m i t r e a l - t i m e a n a l y s i s . An i n i t i a l e v a l u a t i o n of the r e l i a b i l i t y index of P e n d r y [ 4 2 ] , compared w i t h Z a n a z z i and Jona's i n d e x , i s p r e s e n t e d h e r e , based on the l i m i t e d use of the s e r - f a c t o r s f o r the 1 25 comparison of 1(E) c u r v e s i n t h i s t h e s i s . G e n e r a l l y , the main advantage of the Pendry r - f a c t o r over t h a t of Z a n a z z i and Jona i s the s i g n i f i c a n t r e d u c t i o n i n computing c o s t s . For c o m p u t a t i o n s on the Amdahl 470 V6 (a l a r g e main-frame computer), the c o s t of c o m p u t a t i ons f o r the Pendry r - f a c t o r s were about 8 t o 10 times l e s s than the c o r r e s p o n d i n g c a l c u l a t i o n s f o r the Z a n a z z i and Jona programs. The Pendry r - f a c t o r s were a p p r e c i a b l y more s e n s i t i v e t o a v a r i a t i o n i n V o r , and were somewhat more s e n s i t i v e t o changes i n A d % (as evidenced, by the s m a l l e r u n c e r t a i n t i e s ) . The multibeam r - f a c t o r s from both procedures g e n e r a l l y y i e l d e d the same s t r u c t u r a l c o n c l u s i o n s . However, t h e r e was a s i g n i f i c a n t d i f f e r e n c e between the v a l u e s of Vor and Ad%, found i n the two r - f a c t o r a n a l y s e s , t h a t gave the best agreement f o r an i n d i v i d u a l e x p e r i m e n t a l 1(E) c u r v e . A l t h o u g h t h e r e a r e no apparent advantages t o u s i n g the more e x p e n s i v e Z a n a z z i and Jona r - f a c t o r program, i t i s p r o b a b l y u s e f u l t o c o n t i n u e w i t h the p r a c t i c e of comparing the r e s u l t s of both r - f a c t o r s u n t i l a wide range of s u r f a c e s have been i n v e s t i g a t e d . CHAPTER 5 STUDIES ON THE ( 0 0 0 1 ) SURFACE OF ZIRCONIUM 127 A l t h o u g h z i r c o n i u m has i m p o r t a n t t e c h n o l o g i c a l uses (e.g.. n u c l e a r r e a c t o r m a t e r i a l , g e t t e r f o r e l e c t r o n i c t u b e s ) , z i r c o n i u m s u r f a c e s have not so f a r been w i d e l y s t u d i e d by the ' c l e a n s u r f a c e ' methods i n u l t r a - h i g h vacuum. In p a r t , t h i s may be a t t r i b u t e d t o the d i f f i c u l t i e s of o b t a i n i n g w e l l - c h a r a c t e r i z e d s u r f a c e s , due e s p e c i a l l y t o t h e i r h i g h r e a c t i v i t i e s and t o the b u l k phase change (hep t o bec a t around 865°C). The (0001) s u r f a c e of an hep m e t a l ( o f t e n r e f e r r e d t o as the b a s a l p l a n e ) has two p o s s i b l e t e r m i n a t i o n s ; w i t h the hexagonal l a y e r s s t a c k e d i n o r d e r A,B,A,B..., e i t h e r A or B may be the topmost l a y e r . These t e r m i n a t i o n s are i l l u s t r a t e d i n F i g u r e 1.6. The LEED experiment averages over the two t e r m i n a t i o n s (see S e c t i o n 1.5) and t h u s the d i f f r a c t e d beams d i s p l a y 6 - f o l d symmetry a t normal i n c i d e n c e . In the p r e s e n t work, 1(E) c u r v e s have been measured f o r both the c l e a n Z r ( 0 0 0 l ) s u r f a c e and f o r Z r ( 0 0 0 l ) s u r f a c e s c o n t a i n i n g adsorbed oxygen. A complete 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 has been made f o r the c l e a n s u r f a c e . 128 5.1 THE CLEANED (0001) SURFACE OF ZIRCONIUM 5.1.1 E x p e r i m e n t a l . Specimens were c u t by spark e r o s i o n from 1/4", 99.9% p u r i t y s i n g l e c r y s t a l rods ( p r e p a r e d by A. A k h t a r , Department of M e t a l l u r g y , U n i v e r s i t y of B r i t i s h C o l u m b i a ) . A f t e r c u t t i n g , the samples were m e c h a n i c a l l y p o l i s h e d (1JU diamond paste.) and then p o l i s h e d c h e m i c a l l y i n a c i d s o l u t i o n (45% HN03, 50%water, 5%HF by v o l u m e ) [ 1 1 2 ] . T h i s p r o c e d u r e r e s u l t e d i n a s h i n y s u r f a c e w i t h an ' o r a n g e - p e e l ' appearance when viewed w i t h a 10X m a g n i f i e r . The c r y s t a l s l i c e s were mounted on a V a r i a n r e s i s t i v e h e a t e r and a c h r o m e l - a l u m e l thermocouple was spotwelded t o the sample edge. F i g u r e 5.1a shows an Auger spectrum measured w i t h the 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 from a Z r ( 0 0 0 l ) s u r f a c e on e n t e r i n g the FC12 chamber. Large q u a n t i t i e s of carbon and oxygen were i n d i c a t e d , and c l e a r l y the sample was i n i t i a l l y s t r o n g l y c o n t a m i n a t e d . Argon i o n bombardment ( l - 2 k e V , 5uA) was then c a r r i e d out a t room temperature u n t i l carbon was the o n l y d e t e c t a b l e c o n t a m i n a n t ; but i t proved i m p o s s i b l e t o reduce the carbon c o n t a m i n a t i o n below d e t e c t a b l e l i m i t s . The t o t a l bombardment time a t room tem p e r a t u r e was g r e a t e r than 50 h o u r s . The l o w e s t l e v e l s of c o n t a m i n a t i o n c o r r e s p o n d e d t o Auger peak h e i g h t r a t i o s C(272eV)/Zr(174eV) of around 0.05 t o 0.10. Such r a t i o s a r e b e l i e v e d t o c o r r e s p o n d t o coverages of l e s s than 3% of a monolayer; t h i s . e s t i m a t e i s based on the r e s u l t s of e x p e r i m e n t s t h a t measured the Auger peak h e i g h t s as a f u n c t i o n of CO e x p o s u r e ( S e c t i o n 5.2.1). W e l l - o r d e r e d s u r f a c e s , c l e a n e d t o the l e v e l of minimum carbon c o n t a m i n a t i o n , were produced by c y c l e s of Ar+ bombardment(400-600eV) and a n n e a l i n g a t 600°C. An 129 Auger spectrum of such a c l e a n e d s u r f a c e i s shown i n F i g u r e 5.1b. A n n e a l i n g a t t e m p e r a t u r e s above 600°C r e s u l t e d i n the a c c u m u l a t i o n of s u l p h u r on the s u r f a c e . The s u l p h u r coverage was d i f f i c u l t t o q u a n t i f y s i n c e the S d 5 0 e V ) and Z r d 4 7 e V ) peaks c o u l d not be r e s o l v e d even when u s i n g m o d u l a t i o n v o l t a g e s l e s s than 1Vpp. To m o n i t o r the p r e s ence of s u l p h u r i t was n e c e s s a r y t o measure the r e l a t i v e peak h e i g h t s of the z i r c o n i u m Auger peaks a t 92,116,124,147, and 174eV. S i n c e s u r f a c e o r d e r a l s o a f f e c t e d the r e l a t i v e peak h e i g h t s , an i n c r e a s e i n s u l p h u r c o n t a m i n a t i o n was o n l y unambiguously i n d i c a t e d when the Auger peak at around 150eV i n c r e a s e d r e l a t i v e t o a l l the o t h e r peaks a s s o c i a t e d w i t h z i r c o n i u m . The LEED p a t t e r n c o u l d a l s o be changed by the p r e s ence of s u l p h u r . Low c o v e r a g e s produced an i n c r e a s e d background and a f a i n t 3x3 p a t t e r n . H i g h e r s u l p h u r c o v e r a g e s r e s u l t e d i n a v e r y f a i n t and c o m p l i c a t e d LEED p a t t e r n t h a t p o s s i b l y i n d i c a t e d the f o r m a t i o n of z i r c o n i u m s u l p h i d e . For the c l e a n e d Zr(OOOl) s u r f a c e , two independent s e t s of 1(E) c u r v e s were measured f o r normal i n c i d e n c e , and a f u r t h e r s e t was measured f o r o f f - n o r m a l i n c i d e n c e (0=9°, 0=13°) f o l l o w i n g the p r o c e d u r e s of S e c t i o n 3.5. F i g u r e 5.2 shows how the a z i m u t h a l a n g l e 0 i s s p e c i f i e d f o l l o w i n g the c o n v e n t i o n of Z a n a z z i e t a l [ 1 1 3 ] . 130 NITROGEN OXYGEN ZIRCONIUM CARBON 100 ENERGY (eVj 50 0 g.5.1a Auger spectrum of a contaminated Zr(0001) s u r f a c e measured u s i n g the C M A on the FC12 vacuum chamber (Ep=2500eV, Ip«100juA). 50 CARBON ZIRCONIUM 1 1 ELECTRON ENERGY (eV) 5 5 0 3.5.1b Auger spectrum of a c l e a n e d Zr(0001) s u r f a c e . 131 Fig.5 . 2 Diagram showing how the azimuthal angle i s s p e c i f i e d f o r the o f f - n o r m a l i n c i d e n c e experiment on Zr(0001). The azimuth i s d e f i n e d f o l l o w i n g the c o n v e n t i o n of Zanazzi et a l [ l 1 3 ] f o r the s e l e c t i o n of s u b s t r a t e axes i l l u s t r a t e d . 132 5.1.2 C a l c u l a t i o n s 1(E) c u r v e s were c a l c u l a t e d u s i n g the r e n o r m a l i z e d f o r w a r d s c a t t e r i n g (RFS) method w i t h the f o l l o w i n g n o n - s t r u c t u r a l p a r a m e t e r s : Vor=-lOeV, Voi=-1 .08E 1/ 3eV (where E i s vacuum energy i n e V ) , 6D=270°K. E i g h t phase s h i f t s were o b t a i n e d from the band s t r u c t u r e p o t e n t i a l of M o r u z z i e t a l [ 5 7 ] . The v a l u e of G p was an average v a l u e from a c o m p i l a t i o n by S c h n e i d n e r [ 1 1 4 ] ; V o i was e s t i m a t e d from an e x a m i n a t i o n of the narrowest peaks i n the e x p e r i m e n t a l 1(E) c u r v e s , and was f i t t e d t o the energy dependence e x p r e s s e d i n e q u a t i o n 2.9. The e f f e c t of domains due t o the two t e r m i n a t i o n s was c o n s i d e r e d . At normal i n c i d e n c e t h i s i n v o l v e d a v e r a g i n g the 1(E) c u r v e s f o r beams r e l a t e d by a 180° r o t a t i o n . For the o f f - n o r m a l i n c i d e n c e c a l c u l a t i o n s , 1(E) c u r v e s were c a l c u l a t e d f o r the two p o s s i b l e t e r m i n a t i o n s and then a veraged. A p o s s i b l e s u r f a c e r e c o n s t r u c t i o n , hep t o hep+fee t o p l a y e r (ABABA... t o CBABA...), was a l s o i n v e s t i g a t e d . RFS c a l c u l a t i o n s f o r t h i s r e c o n s t r u c t e d s u r f a c e , a g a i n a l l o w i n g f o r the two t e r m i n a t i o n s , were a l s o made f o r normal i n c i d e n c e and f o r G=9°, 0=13°. The two s e t s of c a l c u l a t i o n s were made f o r A d % v a l u e s i n the range -7.5 t o +7.5, and f o r e n e r g i e s a t 2eV i n t e r v a l s i n the range 50 t o lOOeV, and a t 4eV i n t e r v a l s from 100 t o 230eV. The maximum number of beams a v a i l a b l e f o r the c a l c u l a t i o n was 61. F i g u r e 5.3 compares some 1(E) c u r v e s measured at normal i n c i d e n c e w i t h c a l c u l a t e d c u r v e s f o r the hep and hep+fee t o p l a y e r s u r f a c e models t h a t showed c l o s e s t agreement. F i g u r e s 5.4-5.7 show 1(E) c u r v e s measured a t 6=9°, 0=13° compared w i t h c u r v e s c a l c u l a t e d f o r the hep model w i t h A d % v a l u e s r a n g i n g from -7.5 t o 7.5. V i s u a l assessments of 133 the c u r v e s i n d i c a t e d t h a t the c a l c u l a t e d 1(E) c u r v e s f o r the two s u r f a c e models are v e r y s i m i l a r , and t h a t t h e comparison w i t h experiment seemed perhaps t o f a v o u r the u n r e c o n s t r u c t e d model. 5.1.3. R - f a c t o r R e s u l t s T a b l e 5.1 l i s t s the s i n g l e beam r - f a c t o r v a l u e s f o r the comparison of e x p e r i m e n t a l 1(E) c u r v e s w i t h both the u n r e c o n s t r u c t e d and the hcp+fcc t o p l a y e r s u r f a c e models. The r ( L ) v a l u e s i n Table 5.1 a r e the minimum v a l u e s found f o r the ranges of Vor and A d % v a l u e s i n v e s t i g a t e d . The r e l i a b i l i t y index approach was unambiguous i n a l l c a s e s w i t h the b e s t agreement ( s m a l l e s t r ( L > ) always found f o r the u n r e c o n s t r u c t e d s u r f a c e model. The minimum multi-beam R z j v a l u e s f o r the u n r e c o n s t r u c t e d s u r f a c e model were 0.086 at Ad%=0.2±1.0 f o r the normal i n c i d e n c e d a t a and 0.132 a t Ad%=-2.4±0.8 f o r the experiment w i t h 9=9°, 0=13°. The c o r r e s p o n d i n g v a l u e s f o r the hcp+fcc top l a y e r s u r f a c e model were 0.150 a t Ad%=3.0±1.7 and 0.254 a t Ad%=-2.6±1.5, r e s p e c t i v e l y . The quoted u n c e r t a i n t i e s i n Ad% a r e U{2} v a l u e s ( e q u a t i o n 3.12). The magnitudes of both the u n c e r t a i n t y and R z j v a l u e s a l s o i n d i c a t e d t h a t the u n r e c o n s t r u c t e d s u r f a c e model was f a v o r e d over the hcp+fcc t o p l a y e r model. A c o n t o u r p l o t of the multibeam R z j v a l u e s as a f u n c t i o n of Vor and A d % f o r the normal i n c i d e n c e d a t a i s shown i n F i g u r e 5.8 f o r the u n r e c o n s t r u c t e d s u r f a c e model; the R z j c o n t o u r s f o r comparison w i t h the hcp+fcc top l a y e r model a r e p l o t t e d i n F i g u r e 5.9. Contour p l o t s f o r the o f f - n o r m a l i n c i d e n c e d a t a f o r both s u r f a c e models are shown i n F i g u r e s 5.10a and 5.10b. 134 1 I I I I I 1 1 1 T I *0 60 120 160 200 240 BERM ENERGY (EV) F i g . 5 . 3 (1,0) and (1,1) beams measured at normal i n c i d e n c e from the (0001) s u r f a c e of z i r c o n i u m compared with the c a l c u l a t e d 1(E) curves which gave the lowest s i n g l e beam r - f a c t o r s f o r the hep and hep+fee top l a y e r s u r f a c e models. 135 CO CD CL" LO L U EXPT RNG. 2.75 J 2.70 J 2.63 J 2.57 J 2.50 2.44 2.3B 40 80 120 160 BEAM ENERGY (EV) 200 Fig.5.4 (0,0) beam measured f o r o f f - n o r m a l i n c i d e n c e (6=9°, 0=13°) from the c l e a n e d Z r ( 0 0 0 l ) s u r f a c e , and compared with c a l c u l a t e d 1(E) curves f o r A.d.% v a l u e s ranging from -7.5 to 7.5. 136 BEAM ENERGY (EV) Fig.5.5 (0,1) beam measured f o r o f f - n o r m a l i n c i d e n c e (9=9°, 0=13°) from the c l e a n e d Z r ( 0 0 0 D s u r f a c e , and compared w i t h c a l c u l a t e d 1(E) curves f o r A d % v a l u e s ranging from -7.5 to 7.5. 137 Z R t O D O l ) ^ e = 9 i ^ t 1 3 B E A M E N E R G Y ( E V ) F i g . 5 . 6 (-2,1) beam measured f o r o f f - n o r m a l i n c i d e n c e (©=9 , 0=13°) from the c l e a n e d Zr (OOOl) s u r f a c e , and compared wit h c a l c u l a t e d 1(E) curves f o r Ad% va l u e s ranging from -7.5 to 7.5. BEAM ENERGY IEV) Fig.5.7 (-1,1) beam measured f o r of f - n o r m a l i n c i d e n c e (9=9°, 0=13°) from the c l e a n e d Zr(0001) s u r f a c e , and compared w i t h c a l c u l a t e d 1(E) curves f o r &d% v a l u e s ranging frcm -7.5 t o 7.5. 139 T a b l e 5.1 S i n g l e beam Z a n a z z i and Jona r - f a c t o r s f o r the comparison of 1(E) curves measured from the (0001) s u r f a c e of z i r c o n i u m with c a l c u l a t e d 1(E) curves f o r the u n r e c o n s t r u c t e d (hep) and r e c o n s t r u c t e d (hep+fee top l a y e r ) s u r f a c e models. BEAK LABEL HCP MODEL HCP+FCC TOP LATER THETA=0° (1,0) 0.045 0.066 (1,1) 0.055 0.102 (2,0) 0. 123 0.146 (2,1) 0.062 0.084 THETA=9° (0,0) 0.021 0.085 (1,-1) 0.038 0.080 (1,0) 0.249 0.373 (0,1 ) 0. 149 0. 1 6 8 (-1,0) 0.051 0.059 (-1,1) 0.080 0. 160 (-2,0) 0.207 0.270 (-2,-1) 0.096 0.151 140 Zr(0001) NORMAL INCIDENCE HCP STACKING %CHANGE IN TOP LAYER SPACING Fig.5 . 8 Contour p l o t of the multi-beam Rzj v a l u e s as a f u n c t i o n of Vor and A d % f o r the comparison of 1(E) curves measured a t normal i n c i d e n c e from the Zr(000l) s u r f a c e . with c a l c u l a t e d 1(E) cu r v e s f o r an u n r e c o n s t r u c t e d (hep) s u r f a c e with Ad% v a l u e s ranging from -7.5 to 7.5. 141 HCP+FCC TOP LAYER Zr(0001) NORMAL INCIDENCE -7.5% -5.0% -2.5% 0.0% 2.5% 5.0* 7.5% %CHANGE IN TOP LAYER SPACING F i g . 5 . 9 Contour p l o t of the multi-beam Rzj v a l u e s as a f u n c t i o n of Vor and A d l f o r the comparison of 1(E) curves measured at normal i n c i d e n c e from the Zr(OOOl) s u r f a c e with c a l c u l a t e d 1(E) curves f o r a r e c o n s t r u c t e d (hcp+fcc top l a y e r ) s u r f a c e with Ad% v a l u e s ranging from -7.5 to 7.5. 142 H C P H C P + F C C Fig.5.10 Contour p l o t s of the Rzj v a l u e s as a f u n c t i o n of Vor •nd A d % f o r the comparison pf 1(E) c u r v e s measured f o r o f f - n o r m a l i n c i d e n c e (9«9°, 0=13) from the Z r ( 0 0 0 1 ) s u r f a c e with c a l c u l a t e d 1(E) c u r v e s f o r A d % v a l u e s r a n g i n g from -7.5 t o 7.5: a) an u n r e c o n s t r u c t e d (hep) s u r f a c e , b) a r e c o n s t r u c t e d (hep+fee top l a y e r ) s u r f a c e . 143 5.1.4 Comparison With Other Work The comparison of the normal and o f f - n o r m a l i n c i d e n c e e x p e r i m e n t a l data w i t h 1(E) c u r v e s c a l c u l a t e d f o r the two s u r f a c e models su g g e s t s t h a t the (0001) s u r f a c e i s u n r e c o n s t r u c t e d and has a topmost i n t e r l a y e r s p a c i n g t h a t i s c l o s e t o the bulk v a l u e . An o v e r a l l c o n t r a c t i o n of -1±2% seems r e a s o n a b l e t o c o v e r the s l i g h t d i s c r e p a n c y between the normal and o f f - n o r m a l i n c i d e n c e e x p e r i m e n t s . S i m i l a r c o n c l u s i o n s have been reached by LEED f o r the (0001) s u r f a c e s of o t h e r hexagonal c l o s e packed m e t a l s . C o r r e s p o n d i n g v a l u e s of A d % r e p o r t e d i n the l i t e r a t u r e a r e : Be 0% [115] Cd 0% [116] Co 0% [117] T i -2% [43] Zn -2% [118] The most complete s t u d i e s a r e those f o r t i t a n i u m and c o b a l t , and i t i s c o m f o r t i n g t h a t those s t r u c t u r a l r e s u l t s a r e i n v e r y c l o s e agreement w i t h t h a t found here f o r z i r c o n i u m . The o t h e r s t u d i e s a r e l e s s c o m p l e t e . For example, the study f o r z i n c r e l i e d on a v e r a g i n g methods t o determine the s t r u c t u r e , the d a t a f o r the B e ( 0 0 0 l ) s u r f a c e were taken w i t h o u t any means of c h e m i c a l a n a l y s e s , and the study f o r cadmium i n v o l v e d a f o u r - l a y e r d e p o s i t i o n on T i ( 0 0 0 l ) . N e v e r t h e l e s s , t h e r e does seem t o be a g e n e r a l c o n c l u s i o n t h a t (0001) s u r f a c e s of hep m e t a l s a r e u n r e c o n s t r u c t e d w i t h a topmost i n t e r l a y e r s p a c i n g v e r y c l o s e t o the b u l k v a l u e . 144 5.2 OXYGEN ADSORPTION ON THE (0001) SURFACE OF ZIRCONIUM I n i t i a l exposure of a c l e a n e d Zr(0001) s u r f a c e t o oxygen a t room tem p e r a t u r e y i e l d e d a f a i n t (2x2) LEED p a t t e r n w i t h d i f f u s e d i f f r a c t i o n s p o t s , a f t e r an exposure of about 1 L; however, the s p o t s became b r i g h t e r and sh a r p e r when the sample was a n n e a l e d a t a p p r o x i m a t e l y 250°C. A n n e a l i n g a t t e m p e r a t u r e s above 400°C f o r a few m i n u t e s r e s u l t e d i n a d e p l e t i o n of oxygen, presumably due t o d i f f u s i o n i n t o the b u l k . W i t h i n c r e a s i n g oxygen e x p o s u r e , the (2x2) p a t t e r n became f a i n t e r and e v e n t u a l l y a (1x1) p a t t e r n appeared w i t h 1(E) c u r v e s d i f f e r e n t from th o s e of the c l e a n e d s u r f a c e . F o r even g r e a t e r oxygen e x p o s u r e s , the LEED p a t t e r n remained (1x1) but became more d i f f u s e w i t h an i n c r e a s e d background. Because the oxygen a d s o r p t i o n s t r u c t u r e s n e c e s s a r i l y c o n t a i n e d some carbon which c o u l d not be c o m p l e t e l y removed i n the c l e a n i n g p r o c e s s , i t seemed a p p r o p r i a t e t o expose a Z r ( 0 0 0 l ) s u r f a c e t o carbon monoxide i n o r d e r t o have a means f o r c a l i b r a t i n g the r e l a t i v e Auger peak h e i g h t s of the C(272eV) and O ( 5 1 0 E V ) peaks. CO a d s o r p t i o n y i e l d e d i n i t i a l l y a b r i g h t (2x2) p a t t e r n , which c o n v e r t e d a t h i g h e r c o v e r a g e s t o a (1x1) p a t t e r n . T h i s l a t t e r p a t t e r n was s h a r p and b r i g h t and, u n l i k e the oxygen s t r u c t u r e , i t d i d not r a p i d l y degrade w i t h i n c r e a s i n g CO exp o s u r e . G e n e r a l l y , the LEED p a t t e r n s of the CO s t r u c t u r e s were s h a r p e r and b r i g h t e r than the p a t t e r n s of the oxygen s t r u c t u r e s . These o b s e r v a t i o n s a r e c o n s i s t e n t w i t h t h o s e of S h i h e t a l [ l l 9 -121] f o r s t u d i e s of oxygen and carbon monoxide a d s o r p t i o n on the (0001) s u r f a c e of t i t a n i u m . A d i f f e r e n c e between t i t a n i u m and z i r c o n i u m , which may be 145 s i g n i f i c a n t t o the q u e s t i o n of oxygen a d s o r p t i o n on the (0001) s u r f a c e , i s t h a t z i r c o n i u m forms o n l y one b u l k o x i d e ( Z r 0 2 ) [ l 2 2 ] , w h i l e both T i O a n d T i 0 2 a r e s t a b l e compounds of t i t a n i u m . I t i s t h e r e f o r e p l a u s i b l e t h a t the s t r u c t u r e formed f o r the h i g h e s t oxygen exposures on z i r c o n i u m w i l l be Zr02, whereas f o r t i t a n i u m , the o x i d e formed upon exposure t o oxygen a t room temperature i s r e p o r t e d t o be T i O [ l l 9 ] . A q u e s t i o n of i n t e r e s t i n the study of oxygen a d s o r p t i o n on the Z r ( 0 0 0 l ) s u r f a c e i s whether the adsorbed oxygen i s an o v e r l a y e r or an u n d e r l a y e r ( i . e . between the two topmost l a y e r s of z i r c o n i u m ) . S h i h et a l [ 4 4 ] p r o v i d e d e v i d e n c e t h a t an o r d e r e d u n d e r l a y e r of n i t r o g e n was formed on the T i ( 0 0 0 l ) s u r f a c e a t room t e m p e r a t u r e . T h i s u n d e r l a y e r s t r u c t u r e precedes the f o r m a t i o n of any o v e r l a y e r s . The p r o b a b i l i t y of u n d e r l a y e r f o r m a t i o n was a l s o mentioned i n a LEED i n v e s t i g a t i o n of the s t r u c t u r e of CO adsorbed on T i ( 0 0 0 1 ) [ 1 2 1 ] . Because of the marked s i m i l a r i t y between t i t a n i u m and z i r c o n i u m , i t seems p o s s i b l e t h a t u n d e r l a y e r s may p l a y a p a r t i n the a d s o r p t i o n of oxygen on the (0001) s u r f a c e of z i r c o n i u m . 5.2.1 E x p e r i m e n t a l The i n t r o d u c t i o n of oxygen i n t o the FC12 vacuum chamber always r e s u l t e d i n the p r o d u c t i o n of CO and C02, as w e l l as s m a l l e r q u a n t i t i e s . of CH4 and H20, from d i s p l a c e m e n t p r o c e s s e s and from r e a c t i o n s of the oxygen w i t h the w a l l s , hot f i l a m e n t s and the i o n pumps. The gas c o m p o s i t i o n was m o n i t o r e d w i t h the EAI150 q u a d r u p o l e mass s p e c t r o m e t e r . Matheson UHP grade (99.95% p u r i t y ) and r e s e a r c h grade (99.99% p u r i t y ) oxygen was used i n 146 t h e s e s t u d i e s ; no d e t e c t a b l e changes were observed i n the mass s p e c t r a or i n the s u r f a c e c o n t a m i n a t i o n l e v e l s , as i n d i c a t e d by Auger e l e c t r o n s p e c t r o s c o p y , f o r the d i f f e r e n t gas p u r i t i e s . The c l e a n e s t oxygen-covered s u r f a c e s were o b t a i n e d by l e a k i n g oxygen i n t o the chamber w i t h b o t h . t h e t i t a n i u m s u b l i m i n a t i o n pump and the main, i o n pump o p e r a t i n g . The CO p a r t i a l p r e s s u r e was f u r t h e r reduced u s i n g the f o l l o w i n g two p r o c e d u r e s : i ) ' c o n d i t i o n i n g ' the i n l e t system and chamber by f l o w i n g oxygen th r o u g h a t 10" 6 t o r r , and then s u b s e q u e n t l y c l e a n i n g the sample and r e p e a t i n g the a d s o r p t i o n e x p e r i m e n t ; i i ) p o s i t i o n i n g the sample d i r e c t l y i n f r o n t of the c a p i l l a r y gas i n l e t and a d j u s t i n g the l e a k v a l v e f o r the minimum d e t e c t a b l e l e a k . For i i ) the l e a k r a t e d i d not i n c r e a s e the system p r e s s u r e of about 2 x l O " 1 0 t o r r ( i . e . no o b s e r v a b l e i n c r e a s e of the i o n i z a t i o n gauge r e a d i n g ) , but i t c o u l d be d e t e c t e d w i t h the CMA by m o n i t o r i n g an i n c r e a s e i n the oxygen Auger peak a t 5l0eV, r e l a t i v e t o the carbon peak a t 272eV and the z i r c o n i u m peaks a t 92,116,124,147, and 174eV. F i g u r e 5.11 shows p l o t s of t h e Auger peak h e i g h t r a t i o 0(510eV)/Zr(174eV) as a f u n c t i o n of oxygen exposure. There were no apparent breaks f o r the range of coverages t h a t c o r r e s p o n d t o the (2x2) LEED p a t t e r n , a l t h o u g h the s l o p e d e c r e a s e s a t h i g h e r oxygen c o v e r a g e s ; n e v e r t h e l e s s the oxygen t o z i r c o n i u m Auger r a t i o c o n t i n u e s t o i n c r e a s e s l o w l y w i t h f u r t h e r e x p o s u r e s . T h i s r a t i o i s e x p e c t e d t o l e v e l o f f when the o x i d e l a y e r becomes t h i c k e r than the escape depth of the e m i t t e d Auger e l e c t r o n s . To q u a n t i f y the coverage of carbon observed d u r i n g oxygen 147 3 . 5 Oxyqen(5lOeV) Zirconium!174eV) 3 . 0 ^ 2.5-2.0\ g 1 . 5 J X 8 12 16 20 200 i £ ' .04 K U < 0.6-0 . 6 S 0.4-0.2. 1.0 J 12 16 20 24 — i — 28 OXYGEN EXPOSURE (ARB. UNITS) Fig.5.11 Auger peak h e i g h t r a t i o 0(510eV)/Zr(174eV) p l o t t e d as a f u n c t i o n of oxygen exposure f o r two independent experiments ( l a r g e and sm a l l c i r c l e s ) . The Auger peak h e i g h t r a t i o s have been d i v i d e d by 2.8 to y i e l d a pproximations t o the f r a c t i o n a l monolayer coverage. 148 a d s o r p t i o n , carbon monoxide was i n t r o d u c e d i n t o the vacuum system. F i g u r e 5.12 shows a p l o t of the Auger peak h e i g h t r a t i o s C ( 2 7 2 e V ) / Z r ( l 7 4 e V ) and 0(51OeV)/Zr(174eV) as a f u n c t i o n of CO exposure. The C(272eV)/0(5lOeV) Auger peak h e i g h t r a t i o was found t o be a p p r o x i m a t e l y 1.5 f o r a l l CO c o v e r a g e s . U n l i k e the oxygen exposure e x p e r i m e n t , the a d s o r p t i o n of CO has a sharp break a t s a t u r a t i o n coverage (which presumably c o r r e s p o n d s t o h a l f monolayer coverage f o r C and h a l f monolayer coverage f o r 0 ) . The s a t u r a t i o n r a t i o s a r e C(272eV)/Zr(174eV)=2.1 and O ( 5 l 0 e V ) / Z r ( l 7 4 e V ) = 1 . 4 . The CO (2x2) LEED p a t t e r n was observed f o r t o t a l C+0 coverages r a n g i n g from 0.2 t o 0.6 of a monolayer. These r a t i o s a r e c o n s i s t e n t w i t h the c o n c l u s i o n s of S h i h e t a l f 1 2 1 ] , t h a t the (2x2) CO s t r u c t u r e on Ti(OOOl) c o r r e s p o n d s t o 1/4 monolayer coverage each f o r C and 0. U s i n g the i n f o r m a t i o n about r e l a t i v e Auger peak h e i g h t s from the CO a d s o r p t i o n e x p e r i m e n t s , the c o v e r a g e s f o r the oxygen on Zr(OOOl) s t r u c t u r e s were e s t i m a t e d . The oxygen (2x2) p a t t e r n was o b s e r v e d f o r oxygen co v e r a g e s between 0.25 and 0.6 of a monolayer, w h i l e a (1x1) p a t t e r n o c c u r r e d f o r oxygen coverages from 0.6 t o g r e a t e r than monolayer c o v e r a g e . The (2x2) oxygen p a t t e r n i s thought t o c o r r e s p o n d t o a (2x1) s t r u c t u r e of 1/2 monolayer c o v e r a g e . The presence of s y m m e t r i c a l l y e q u i v a l e n t domains would g e n e r a t e the o b s e r v e d (2x2) LEED p a t t e r n ( F i g u r e 1.7). For coverages g r e a t e r than a monolayer, the l i n e a r e x t r a p o l a t i o n of the oxygen t o z i r c o n i u m Auger peak h e i g h t r a t i o i s e x p e c t e d t o g i v e a l e s s r e l i a b l e i n d i c a t i o n of coverage i f some of the oxygen i s i n c o r p o r a t e d between the l a y e r s of z i r c o n i u m atoms. 149 AUGER PEAK HEIGHT RATIO ver s u s EXPOSURE IN CM O ID CM 00 o < X •—( w x carbon(272eV) zirconium(174eV) s. e g / oxyqen(510eV) &< ^ zirconium(174eV) £ • A A ©1 | ' *S 4 , » ! • If I I i I il -| 1 1 \— r-4 6 12 16 32. EXPOSURE ( a r b i t r a r y u n i t s ) Fig.5.12 Auger peak h e i g h t r a t i o s f o r C(272eV)/Zr(174eV) and O(510eV)/Zr(174eV) p l o t t e d as a f u n c t i o n of exposure f o r the a d s o r p t i o n of CO on a (0001) s u r f a c e of z i r c o n i u m . 150 A t h i r d a d s o r p t i o n experiment i s i l l - u s t r a t e d by the Auger peak r a t i o v e r s u s exposure p r o f i l e s i n F i g u r e 5.13. A f t e r a d s o r b i n g oxygen u n t i l a (2x2) LEED p a t t e r n was o b s e r v e d , the s u r f a c e was exposed t o carbon monoxide. The f i n a l Auger peak h e i g h t r a t i o s i n d i c a t e d c o v e r a g e s of 0.38 of a monolayer of carbon and 0.65 of a monolayer of oxygen; a g a i n d e m o n s t r a t i n g a s a t u r a t i o n coverage of one monolayer. For measurements of 1(E) c u r v e s , two s e t s of LEED photographs were taken f o r independent p r e p a r a t i o n s of the oxygen (2x2) p a t t e r n . F u r t h e r s e t s of the (1x1) oxygen p a t t e r n s were t a k e n f o r oxygen coverages t h a t were thought t o be near monolayer coverage and f o r much h i g h e r oxygen c o v e r a g e s . The carbon c o n t a m i n a t i o n of the oxygen (2x2) s t r u c t u r e was e s t i m a t e d t o be about 5% of a monolayer, and about 10% of a monolayer f o r the (1x1) oxygen s t r u c t u r e s . 151 < X o •—I W X tc < DC ca o D 2.0 1 . 6 o»yqen(510ey) 8irconium(174eV) - • carbon(272ey) xirconium(174eV) l - 2 4 . ' » / / o.e 0 . 4 ESTIMATED COVERAGE INITIAL FINAL CARBON 0.10 0.38 OXYGEN 0.43 0.65 0 . 0 — i — 12 32 —r— 1 0 O CO EXPOSURE (ARBITRARY UNITS) Fig.5.13 Auger peak h e i g h t r a t i o s f o r C(272eV)/Zr(174eV) and O(510eV)/Zr(174eV) p l o t t e d as a f u n c t i o n of CO exposure on a Z r ( 0 0 0 l ) s u r f a c e t h a t c o n t a i n e d adsorbed oxygen and which showed a (2x2) d i f f r a c t i o n p a t t e r n . 1 52 5.2.2 R e s u l t s Some examples from the two s e t s of 1(E) c u r v e s measured f o r the (2x2) oxygen p a t t e r n are shown i n F i g u r e s 5.14 and 5.15. An r - f a c t o r a n a l y s i s of c o r r e s p o n d i n g beams from the two d a t a s e t s y i e l d e d t h e s i n g l e beam r ( L ) v a l u e s i n Ta b l e 5.2. G e n e r a l l y , the corresp o n d e n c e i s q u i t e good, e s p e c i a l l y f o r the i n t e g r a l o r d e r beams. Each 1(E) c u r v e i n the comparison i s the r e s u l t of a v e r a g i n g the s y m m e t r i c a l l y e q u i v a l e n t beams w i t h i n an e x p e r i m e n t a l s e t and the average was then smoothed once w i t h a t h r e e p o i n t t r i a n g u l a r smooth ( e q u a t i o n 4.3). The 1(E) c u r v e s from the s a t u r a t i o n (1x1) oxygen s t r u c t u r e a r e g i v e n i n Appendix 1. These c u r v e s a r e d i f f e r e n t from the 1(E) c u r v e s measured f o r the c l e a n s u r f a c e and are a l s o d i f f e r e n t from c u r v e s measured f o r the lower coverage (1x1) p a t t e r n . The l a t t e r d a t a a r e i n c l u d e d i n Appendix 1. 153 Fig.5.14 1(E) curves f o r (1,0) beams at normal i n c i d e n c e from two independent measurements f o r the Z r ( 0 0 0 l ) - ( 2 x 2 ) - 0 s t r u c t u r e . The curves i l l u s t r a t e d a re formed by a v e r a g i n g s y m m e t r i c a l l y e q u i v a l e n t 1(E) curves w i t h i n each s e t , and each average has been smoothed once (equation 4.3). 1 54 .5.15 1(E) curves f o r (0,3/2) beams at normal i n c i d e n c e from two independent measurements f o r the Z r ( 0 0 0 1 ) - ( 2 x 2 ) - 0 s t r u c t u r e . The curves i l l u s t r a t e d a r e formed by a v e r a g i n g s y m m e t r i c a l l y e q u i v a l e n t 1 ( E ) curves w i t h i n each s e t , and each average has been smoothed once (equation 4 . 3 ) . 1 5 5 T a b l e 5.2 S i n g l e beam Pendry r - f a c t o r s f o r the comparison of corr e s p o n d i n g beams from two independent measurements of beams from the Zr(0001)-(2x2)-0 s t r u c t u r e . W i thin an e x p e r i m e n t a l s e t , s y m m e t r i c a l l y e q u i v a l e n t beams were averaged and smoothed once (equation 4.3). BEAN LABEL R-FACTOR (0,1/2) 0.081 (1/2,1/2) 0.158 (1,0) 0.043 (1,1) 0.066 (0,3/2) 0.126 (1,3/2) 0.128 156 5.2.3 F u t u r e Work Se t s of 1(E) c u r v e s f o r a l a r g e number of p o s s i b l e s i t e s , each w i t h a range of i n t e r p l a n a r s p a c i n g s , would have t o be c a l c u l a t e d w i t h the m u l t i p l e s c a t t e r i n g methods i n o r d e r t o e s t a b l i s h the b e s t models f o r the (2x2) and (1x1) oxygen a d s o r b a t e s t r u c t u r e s . Such s t u d i e s are t o be u n d e r t a k e n , but . i t i s b e i n g hoped t h a t these i n v e s t i g a t i o n s can be g u i d e d w i t h h i g h r e s o l u t i o n e l e c t r o n energy l o s s s p e c t r o s c o p y (HREELS). T h i s t e c h n i q u e g i v e s v i b r a t i o n a l f r e q u e n c i e s , which i n p r i n c i p l e can be i n t e r p r e t e d i n terms of a d s o r p t i o n s i t e ( i n c l u d i n g symmetry and t y p e ) . A HREELS s p e c t r o m e t e r , based on the d e s i g n of C a r e t t e [ 1 2 3 ] , has been c o n s t r u c t e d i n our l a b o r a t o r y , and i t i s c u r r e n t l y b e i n g e v a l u a t e d . I n i t i a l s t u d i e s have i n d i c a t e d the s p e c t r o m e t e r d e s i g n i s w o r k a b l e , and the r e c e n t a d d i t i o n of magnetic s h i e l d i n g s h o u l d a l l o w us t o a c h i e v e s u f f i c i e n t s e n s i t i v i t y w i t h the r e q u i r e d r e s o l u t i o n ( l e s s than 20meV). Another planned development i s the m o d i f i c a t i o n of the LEED a n a l y s i s system u s i n g a more s e n s i t i v e v i d i c o n camera. T h i s m o d i f i c a t i o n would enable us t o r e c o r d 1(E) c u r v e s w h i l e the a d s o r p t i o n experiment i s i n p r o g r e s s and a p a r t i c u l a r s u r f a c e s t r u c t u r e c o u l d be i n v e s t i g a t e d w i t h o p t i m i z a t i o n of the e x p e r i m e n t a l c o n d i t i o n s . T h i s development s h o u l d be p a r t i c u l a r l y h e l p f u l f o r d i s t i n q u i s h i n g between the two p o s s i b l e (1x1) s t r u c t u r e s . C u r r e n t l y , i t i s not p o s s i b l e t o o b t a i n LEED 1(E) c u r v e s u n t i l a f t e r the p h o t o g r a p h i c f i l m has been dev e l o p e d and the spot i n t e n s i t i e s have been d i g i t i z e d . F u r t h e r m o d i f i c a t i o n s t o the FC12 vacuum chamber have been proposed t o a l l o w us t o c a r r y out s t u d i e s on a d s o r p t i o n systems 1 57 more e f f e c t i v e l y . 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I b a c h , S p r i n g e r V e r l a g , B e r l i n (1977) 1 64 APPENDIX 1 6 5 *0 80 120 160 200 240 ENERGY (eV) 166 COPPER(311): EXPERIMENT INTENSITY (ARB. UNITS) INTENSITY (ARB. UNITS) 169 170 ENERGY (eV) 171 ENERGY (eV) ENERGY (eV) 172 N I C K E L O l l ) : CONTAMINATED SURFACE *0 80 120 160 200 40 80 120. " 160 200 ENERGY (eV) ENERGY (eV) I N T E N S I T Y (ARB. U N I T S ) 174 2 r ( 0 0 0 l ) - ( 2 x 2 ) - 0 EQUIVALENT BEAMS AVERAGED 1ST AND 2ND EXPERIMENTS ENERGY (eV) 175 • 1 » . I 40 80 120 160 200 ENERGY (eV) 176 40 80 120 160 200 240 ENERGY (eV) 

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