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UBC Theses and Dissertations

The structures of clean rhodium surfaces studied by low-energy electron diffraction Watson, Philip Richard 1978

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THE STBUCTUBES OF CLEAN EHODIUJ3 SURFACES ST DDI ED BY LOW-ENEBGY ELECTRON DIFFBACTION by PHILIP BICHABD WATSON E.A. ( Nat. Sex. ) Hons, O x f o r d U n i v e r s i t y , 1974 A THESIS SUBMITTED IN PARTIAL FULFILLHENT OF THE BEQUIBEHENTS FOB THE DEGBEE OF DOCTOB OF PHILOSOPHY i n THE fACUITY OF GRADUATE STUDIES ( Department o f C h e m i s t r y ) He a c c e p t t h i s t h e s i s as c o c f o r m i a g t o the r e q u i r e d s t a n d a r d THE UNIVEBSITY OF BBITISH COLUMBIA O c t o b e r , 1978 @ P h i l i p fiichard Watson, 1978 In presenting th i s thes i s in pa r t i a l fu l f i lment of the requ i rements for an advanced degree at the Univers i ty of Br i t i sh-Columbia, I agree that the L ibrary shal l make it f ree ly ava i l ab le for reference and study. I further agree that permission for extensive copying of th is thesis for scho lar ly purposes may be granted by the Head of my Department or by his representat ives. It is understood that copying or pub l i ca t ion of this thes is for f inanc ia l gain sha l l not be allowed without my written permission. Department of CHEMISTRY The Univers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date Nov. H , 1978 ABSTRACT T h i s t h e s i s d e s c r i b e s t h e d e t e r m i n a t i o n o f the s t r u c t u r e s of the {111), (100) and {110) s u r f a c e s o f rhodium by low-energy e l e c t r o n d i f f r a c t i o n (LEED). I n t h i s work the p h o t o g r a p h i c method of S t a i r e t a l «as r e f i n e d so t h a t LEED p a t t e r n s s t o r e d on p h o t o g r a p h i c f i l m were a n a l y s e d by a c o m p u t e r - c o n t r o l l e d V i d i c o a T-V- camera system t o g i v e i n t e n s i t y v. e n e r g y , o r 1 ( E ) , c u r v e s f o r t h e v a r i o u s d i f f r a c t e d beams. T h i s approach was t e s t e d f o r a Cu(111) s u r f a c e ; I (E) c u r v e s determined i n t h i s new say were compared w i t h i n t e n s i t i e s measured p r e v i o u s l y w i t h a Faraday cup c o l l e c t o r . , S t r u c t u r a l a n a l y s e s were performed by comparing e x p e r i m e n t a l i n t e n s i t i e s w i t h t h o s e c a l c u l a t e d w i t h t h e m u l t i p l e - s c a t t e r i n g programs of Van Hove and l o n g . Checks were made a g a i n s t some p r e v i o u s l y p u b l i s h e d c a l c u l a t i o n s t o ensure t h a t t h e programs were r u n n i n g c o r r e c t l y . F o r c a l c u l a t i o n s on s u r f a c e s o f both rhodium and c opper, an i o n - c o r e s c a t t e r i n g p o t e n t i a l from a band s t r u c t u r e c a l c u l a t i o n - was compared w i t h one c o n s t r u c t e d by s u p e r p o s i n g t h e charge d e n s i t i e s of a cubo-o c t a h e d r a l c l u s t e r o f m e t a l atoms. The degree of agreement between e x p e r i m e n t a l and c a l c u l a t e d d i f f r a c t e d beam i n t e n s i t i e s was d e t e r m i n e d w i t h a r e l i a b i l i t y -i n d e x proposed r e c e n t l y by Z a a a z z i and J o n a . Here, the a n a l y s i s of Z-anazzi and Jona was extended t o e n a b l e assessments t o be made of t h e u n c e r t a i n t i e s i n the s t r u c t u r a l parameters which g i v e the b e s t c o r r e s p o n d e n c e 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 i i 1(E) c u r v e s . The e f f e c t s of independent measurements of i n t e n s i t i e s , and of changes i n t h e n o n - s t r u c t u r a l parameters f o r the c a l c u l a t i o n s upon the r e l i a b i l i t y o f t h e d e r i v e d s t r u c t u r a l r e s u l t s were a l s o examined i n some d e t a i l . A l l t h r e e rhodium s u r f a c e s s e r e found t o be u n r e c o n s t r u c t e d at room tem p e r a t u r e and above, each s u r f a c e b e i n g a si m p l e t e r m i n a t i o n o f the b u l k c r y s t a l l i n e s t a c k i n g seguence. However, t h e topmost i n t e r l a y e r s p a c i n g d i d show some v a r i a t i o n from the b u l k v a l u e . The (110) s u r f a c e showed a c o n t r a c t i o n of 2.7±2.0S r e l a t i v e t o the b u l k i n t e r l a y e r s e p a r a t i o n , w h i l e t h e (111) and (100) s u r f a c e s showed s m a l l {-1±3%) and 0±2.5% c o n t r a c t i o n s r e s p e c t i v e l y . i v TABLE OF COSTENTS A b s t r a c t ................. .............. ..... ........ . . . . . . i i T a ble o f C o n t e n t s ... . . , . . , . i v L i s t o f F i g u r e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v i i L i s t of T a b l e s ............... ..^ ...................... ..xiv Acknowledgements ..».xv Chapter 1; I n t r o d u c t i o n ................................1 1.1 S u r f a c e Technigues .............................. 2 1.2 H i s t o r i c a l Development Of LEED ..................5 1.3 T h e s i s O u t l i n e ..................................8 Chap t e r 2: Low-energy E l e c t r o n D i f f r a c t i o n ( 1 E E . D ) And Auger E l e c t r o n S p e c t r o s c o p y (AES) ....................... 13 2.1 B a s i c C o n s i d e r a t i o n s ............................ 14 2.1(a) Secondary E l e c t r o n D i s t r i b u t i o n ........... 14 . 2.1(b) S u r f a c e S t r u c t u r e C l a s s i f i c a t i o n s .........17 2.1(c) The B e c i p r o c a l Net ........................ 18 2.2 Formation Of The D i f f r a c t i o n P a t t e r n --.-,-..-.--20 2.3 I n t e n s i t i e s Of LEED Beams ----- - ....-.-29 2.3(a) K i n e m a t i c a l Theory ........................30 2.3|b) C h a r a c t e r i s t i c s Of 1(E) c u r v e s ..............37 2.4 Auger E l e c t r o n S p e c t r o s c o p y (AES) .....................39, Chapter 3: 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 ..........-.45 3.1 Parameters P o r T h e o r i e s Of LEED ................-46 3.1(a) The S c a t t e r i n g P o t e n t i a l ..................46 3.1(b) Phase S h i f t s ., ............. -.51 3.1(c) Temperature C o r r e c t i o n s .....................55 3.2 G e n e r a l Schemes Of C a l c u l a t i o n .....-....-.-----.59 3.2(a) L a y e r D o u b l i n g And BFS Methods ..............63 3.21b) Uses Of Symmetry ...................... ,..-..68 3.2(c) Program F l o s -69 Chapter 4: E x p e r i m e n t a l A s p e c t s -75 4- 1 C r y s t a l P r e p a r a t i o n .............................76 4.2 U l t r a High Vacuum . (UHv) A p p a r a t u s ... .......79 4.3 C r y s t a l C l e a n i n g ................................83 4.3(a) Pr o c e d u r e s ................................83 4.3{b) M o n i t o r i n g S u r f a c e C o m p o s i t i o n ............86 4.4 LEED I n t e n s i t y Measurements .....................89 4.4(a) P r e v i o u s Methods 91 4.4{b) V i d i c o n Measurement O f - I JE).-curves ........93 4.4(c) Measurements Of 1(E) c u r v e s For Cu (111) 102 4.4 (d) F u t u r e Developments ----- - . . . . . . 108 Ch a p t e r 5: Leed C r y s t a l l o g r a p h y ....................... 110 5- 1 G e n e r a l C o n s i d e r a t i o n s .-,.,........-..----------111 5-2 B e l i a b i l i t y - i n d i c e s ...--.,,.--,........--..-.---114 5.2(a) The B e l i a b i l i t y - i n d e x Of Z a n a z z i And Jona 5-3 S t r u c t u r a l A n a l y s i s U s i n g The Z J B e l i a b i l i t y F a c t o r : The Cu(111) S u r f a c e As An Example ........... 120 5-4 Other Methods Of O b t a i n i n g S u r f a c e S t r u c t u r a l I n f o r m a t i o n From LEED Data 130 5.5 B i b l i o g r a p h y Of S u r f a c e S t r u c t u r e s Of C l e a n M e t a l s 131 Cha p t e r 6: The (111) S u r f a c e Of Rhodium ......135 v i 6. 1 E x p e r i m e n t a l -..-136 6.2 C a l c u l a t i o n s .................................... 142 6.3 R e s u l t s And D i s c u s s i o n .......................... 144 6.3 (a) N o r m a l . I n c i d e n c e ........ ..... ............. 144 6.3(b) D i r e c t i o n Of I n c i d e n c e ©"=10°,<^=109° ....... 159 6.4 C o n c l u s i o n s .........,.......,.-..,-...----------165 Chapter 7: The Bh (100) S u r f a c e . . . . . - - . . . 168 7.1 E x p e r i m e n t a l ..,....,..,,.---.--.,.-.--,---,----.171 7.2 C a l c u l a t i o n s 178 7-3 R e s u l t s And D i s c u s s i o n .........................-178 7-4 Comparisons l i t h P r e v i o u s S t u d i e s ............... 187 Chapter 8; The Sh(110) S u r f a c e .............. .......188 8.1 E x p e r i m e n t a l .-,.-191 8*2 C a l c u l a t r o n s • .. .:,« . ,, , «. —.—•. ,, — — • — — — — , — , — ... ..«..., 196 8.3 R e s u l t s And D i s c u s s i o n .......................... 197 8.4 Comparisons l i t h P r e v i o u s Work ..202 H €.£ €17 * o * • • <» * « • • * * m -m m • * * • * • * • • * * » • • • • 203 Appendices: ....,....----.---..--,.-,.--,..-------,-,.,-211 v i i LIST OF FIGURES o 2. 1 Schematic diagram o f t h e mean f r e e path L(A) of e l e c t r o n s i n m e t a l l i c s o l i d s as a f u n c t i o n of t h e i r energy ( e V ) . 15 2.2 Schematic energy d i s t r i b u t i o n N{E) o f b a c k s c a t t e r e d s lew e l e c t r o n s as a f u n c t i o n o f t h e i r energy f o r a primary beam energy E0. ........ 15 2.3 Examples o f the 5 d i p e r i o d i c n e t s as r e p r e s e n t e d by models o f s u r f a c e s of f a c e - c e n t r e d c u b i c (FCC) , and b o d y - c e n t r e d c u b i c (ECC) m e t a l s ; s i d e s a,, {sh o r t a x i s ) and a z , i n t e r n a l angleoC a ( = a z , o(= 90 0, e . g„ a) sguare FCC (100) , BCC (100) ; b) p r i m i t i v e r e c t a n g u l a r FCC (110) , BCC (211) ; c) c e n t r e d r e c t a n g u l a r FCC (311) , BCC (110) ; d) h e x a g o n a l FCC{111) ; e) o b l i g u e FCC (321) ; a ( + a i # (<= 9QO, e.g. a, t a.z, d = 90°, e.g. a ( = a z , o< = 60°, e.g. a 2 , d 4 9 0°, e.g. 19 2-4 Angle c o n v e n t i o n s f o r i n c i d e n c e o f an e l e c t r o n beam on a s u r f a c e ................................. 19 2.5 A t w o - d i m e n s i o n a l r e a l n e t , d e s c r i b e d by s, , s z , (dark c i r c l e s ) and i t s a s s o c i a t e d r e c i p r o c a l net s*r s * (open c i r c l e s ) . ............................ 21 2.6 Uni t meshs o f t h e c o r r e s p o n d i n g r e a l and r e c i p r o c a l t w o - d i m e n s i o n a l n e t s a s i n 2.3. ........ 22 2.7 (a) Photograph o f t h e LEED p a t t e r n from a Cu(111). s u r f a c e at {a) normal i n c i d e n c e and 90eV beam energy, (b) ©- = 12°, <f> =6° and 44eV beam e n e r g y , {c) and |d) l a b e l l i n g of t h e d i f f r a c t i o n s p o t s . ....... 24 2.8 The d i r e c t i o n of a d i f f r a c t e d beam i s de t e r m i n e d by the a l l o w e d v a l u e s of k and E. ................. 26 2-9 (a) Schematic d i f f r a c t i o n p a t t e r n from one domain o f Au(100) - (5x1) r e c o n s t r u c t e d s u r f a c e (b) Model s t r u c t u r e of c o i n c i d e n t a l h e x a g o n a l g o l d l a y e r superimposed on the u n d e r l y i n g (100) s u b s t r a t e (c) { a f t e r Palmberg and Rhodin [ 3 3 ] } . 29 2-10 1(E) cur v e s f o r the s p e c u l a r beam from Ni(100) and Cu{100) ,.©-=.3°. The b a r s denote k i n e m a t i c a l Bragg c o n d i t i o n s ( a f t e r Anderson and Kasemo J. 34}) - ...... 31 v i i i 2.11 I n t e n s i t y o f a d i f f r a c t e d LEED beam as a f u n c t i o n of energy i n (a) t h e pure t w o - d i m e n s i o n a l l i m i t (b) the pure t h r e e - d i m e n s i o n a l l i m i t , ( a f t e r S o m o r j a i and F a r r e l l f. 7J) . ........................ 36 2.12 Schematic r e p r e s e n t a t i o n o f t h e L ^ V V auger t r a n s i t i o n : (a) i o n i s a t i o n o f a c o r e l e v e l , (fc) f i l l i n g o f t h e c o r e h o l e , <c) e m i s s i o n of the Auger e l e c t r o n . ................................... 40 2.13 Auger sp e c t r u m o f a h e a v i l y c o n t a m i n a t e d Rh{110) s u r f a c e . Ep=1.5KeV, Ip~10 microamps. ............. 43 3.1 M u f f i n - t i n p o t e n t i a l |a) i n c r o s s - s e c t i o n as c o n t o u r s , (b) a l o n g XX*. Vo i s the c o n s t a n t i n t e r s p h e r e p o t e n t i a l . 48 3.2 I l l u s t r a t i o n of t h e r e l a t i o n s h i p between e n e r g i e s measured w i t h r e s p e c t t o the vacuum l e v e l and those measured w i t h r e s p e c t t o t h e l o w e s t l e v e l o f the c o n d u c t i o n band. .............................. 48 3.3 The c u b o - o c t a h e d r a l M ,3 c l u s t e r used t o model the c r y s t a l p o t e n t i a l o f FCC c r y s t a l s . ................ 52 3.4 An i o n - c o r e immersed i n a p l a n e wave i n d u c i n g s c a t t e r e d s p h e r i c a l waves whose i n t e n s i t i e s a r e f u n c t i o n s o f k, 8 5 and r . A f t e r Pendry £.1.2.J. ----- 54 3-5 Energy dependence of copper phase s h i f t s (1=0-7) ^ 3 a B d <b> C f o r the p o t e n t i a l s : (a) Vc.^ and (b) V®c. ........ 56 3.6 Energy dependence o f rhodium phase s h i f t s (1=0-7) f o r the p o t e n t i a l s : (a) V R M 3 and fb) 7 ^ . ....... 57 3.7 Schematic r e p r e s e n t a t i o n of a s e t of plan e wave i n c i d e n t from t h e l e f t m u l t i p l y s c a t t e r e d by a plane o f i o n c o r e s . 61 3.8 B u i l d i n g up s u b p l a n e s by the l a y e r d o u b l i n g p r o c e s s . I n d i v i d u a l s u b p l a n e s a r e marked A and B; the r e s u l t a n t c o m p o s i t e i s marked C. A f t e r Tong l 15j- . ......................................... 64 3.9 D i a g r a m a t i c r e p r e s e n t a t i o n of 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 (BPS) p r o c e s s . Inward a m p l i t u d e s A^(cj) propagate from vacuum through the 1st l a y e r t o the Nth l a y e r where t h e y a r e t u r n e d around. The e l e c t r o n s a r e then propagated t o t h e 1st l a y e r w i t h outward a m p l i t u d e s B^ (3) . .......... 66 3.10 A f l o w - c h a r t showing t h e p r i n c i p a l s t e p s i n a m u l t i p l e s c a t t e r i n g c a l c u l a t i o n , u s i n g t h e BFS or l a y e r d o u b l i n g methods. ........................... 70 i x 3.11 Comparison o f e x p e r i m e n t a l 1(E) c u r v e s f o r Cu (111) a t normal i n c i d e n c e w i t h c u r v e s c a l c u l a t e d f o r t i e vOU3 a n d vo£l p o t e n t i a l s f o r V o r = -9.5eV and t h r e e d i f f e r e n t v a l u e s o f &d%. .................... 73 4-1 Las e r a l i g n m e n t method t o check the c o i n c i d e n c e of the o p t i c a l f a c e and d e s i r e d c r y s t a l p l a n e . ....... 78 4.2 Diagrammatic r e p r e s e n t a t i o n of the pumping system: I P = 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. .................................. 78 4.3 Schematic of t h e V a r i a n FC12 UHV chamber- ......... 81 4-4 S i m p l i f i e d diagram of an o f f - a x i s e l e c t r o n gun w i t h d e f l e c t i o n e l e c t r o d e and d r i f t tube- ......... 81 4.5 Three methods o f h e a t i n g a c r y s t a l sample: (a) d i r e c t r e s i s t i v e h e a t i n g , (b) u s i n g a V a r i a n c o n d u c t i v e h e a t e r , (c) by e l e c t r o n bombardment. Hatched l i n e s r e p r e s e n t s t a i n l e s s s t e e l and s t i p p l e c e r a m i c i n s u l a t o r s - O t h e r m a t e r i a l s g e n e r a l l y Eh, P t , w" o r f a . , ........................ 85 4.6 Schematic diagram of LEED o p t i c s used a s a r e t a r d i n g f i e l d a n a l y s 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 s e r - ........ 87 4-7 Schematic diagram o f the e l e c t r o n o p t i c s used f o r LEED e x p e r i m e n t s . ................................. 90 4.8 T y p i c a l v a r i a t i o n o f e l e c t r o n gun beam c u r r e n t I p a g a i n s t beam v o l t a g e Vp i n the LEED mode. ......... 90 4.9 Schematic diagram o f the apparat u s used t o a n a l y s e t h e p h o t o g r a p h i c n e g a t i v e s o f LEED p a t t e r n s . ...... S5 4.10 D i g i t i s e r o u t p u t measured f o r d i f f e r e n t r e g i o n s of a Kodak No. 2 s t e p d e n s i t y wedge and p l o t t e d a g a i n s t t h e c o r r e s p o n d i n g c a l i b r a t e d o p t i c a l d e n s i t i e s . ....-... ................................ 97 4-11 D i g i t i s e r o u t p u t measured f o r d i f f e r e n t r e g i o n s of a p h o t o g r a p h i c n e g a t i v e o f t h e s t e p d e n s i t y wedge i n F i g u r e 4. 10, p l o t t e d a g a i n s t t h e o r i g i n a l c a l i b r a t e d o p t i c a l d e n s i t y o f the wedge. The arrows note the p o i n t s on the p l o t which c o r r e s p o n d t o t h e minimum background and the maximum d e n s i t y observed on photographs of t h e LEED p a t t e r n s from Cu (111). ....................... 98 4.12 Flow c h a r t o f t h e computer program which c o n t r o l s t h e s c a n n i n g o f the photographs- .................. 100 4.13 IJE) c u r v e s f o r t h e (11) and (01) beams from X Cu{111) at normal i n c i d e n c e . ...................... 103 4.14 I|E) c u r v e s o f s y m m e t r i c a l l y e q u i v a l e n t beams f o r normal i n c i d e n c e on Cu(111). The i n s e t i n d i c a t e s t h e beam n o t a t i o n and a s p e c i f i c a t i o n o f the a z i m u t h a l a n g l e t h e a s t e r i s k i l l u s t r a t e s t h e p o s i t i o n of the e l e c t r o n gun f o r the non-normal i n c i d e n c e c a s e . ................................... 10 5 4.15 1(E) c u r v e s f o r the s p e c u l a r beam f o r C u (111): (a) fr=12° , £ = 186°; (b) ©- = 12°, </>=70. i h e f i r s t two se r e measured by the method d e s c r i b e d i n t h i s paper, and (c) r e p r e s e n t s measurements made by Woodruff and McDonnell £19] w i t h a Faraday cup c o l l e c t o r . ........................................ 106 5.1 Comparison of some e x p e r i m e n t a l 1(E) c u r v e s f o r Cu(111) wit h c a l c u l a t i o n s f o r the p o t e n t i a l s V<?£ and V a,3 a t 6= 12°, ^ = 6 ° ; V o r=-9.5eV and Ad%=+5, 0 and - 5 %. 121 5.2 P l o t o f r~ r a g a i n s t Ad?i> f o r v a r i o u s v a l u e s o f V0r f o r Cu{111) w i t h t h e V C m >(3 p o t e n t i a l . E r r o r b a r s are s t a n d a r d e r r o r s i n the weighted mean. ......... 124 5.3 P l o t s f o r Cu(111) of • ( r r K f o r 9 i n d i v i d u a l beams versus Ad$ f o r the VCUk\? p o t e n t i a l w i t h V o r =-9-5eV. The dashed l i n e shows the dependence of the energy weighted mean r" r v e r s u s Ad%. ........... 126 5.4 Contour p l o t s f o r Cu (111) o f " r r v e r s u s A cL% and V o r f o r the p o t e n t i a l s (a) and (b) V* c. ..... 128 6.1 Auger s p e c t r a o f Rh (111) s u r f a c e s w i t h a 1.5keV, 10 microamp beam: (a) as mounted, c o n s i d e r a b l e S(152eV) and C(282eV) c o n t a m i n a t i o n ; (b) a f t e r a r g o n - i o n bombardment, reduced S, i n c r e a s e d C; (c) a f t e r a n n e a l i n g , reduced C, i n c r e a s e d S; id) c l e a n e d s u r f a c e . .............................. 137 6.2 Photographs o f t h e (1x1). 1EED p a t t e r n from t h e c l e a n Rh (111) s u r f a c e a t <a) normal i n c i d e n c e (158eV), (b) S- = 1Q0, <f> = 1090 (122eV) i n t b e a n g l e c o n v e n t i o n o f Jona £128J. The l a b e l l i n g scheme i s shown i n (c) and (d) . ............................. 141 6.3 P o s s i b l e r e c o n s t r u c t i o n s of t h e (111) s u r f a c e t h a t p r e s e r v e t h e (1x1) symmetry of the LEED p a t t e r n : (a) n o n - r e c o n s t r u c t e d , CEACBA. ,.C FCC s t a c k i n g ; (.fa) r e c o n s t r u c t e d , CBACBA...A s t a c k i n g ; (c) r e c o n s t r u c t e d , CBACfiA... B, HCP s t a c k i n g . The 4 t h l a y e r C i s i n d i c a t e d by s m a l l dashed x i b a r r e d c i r c l e s , t h e 3 r d l a y e r B by l a r g e blank c i r c l e s , t h e 2nd A by medium d o t t e d c i r c l e s , and the 1st l a y e r i s i n d i c a t e d by s m a l l b a r r e d c i r c l e s . .......................................... 143 6.4 A comparison f o r the <1G) and <01) beams o f 1(E) c u r v e s measured at normal i n c i d e n c e f o r fih(111) w i t h t h o s e c a l c u l a t e d w i t h t h e p o t e n t i a l V ^ j ^ f o r the normal FCC s t a c k i n g seguence and f o r the HCP s t a c k i n g seguence over the t o p t h r e e s u r f a c e l a y e r s . ........................................... 145 6.5 A comparison of e x p e r i m e n t a l I (E) c u r v e s f o r the (10) and (01) beams a t normal i n c i d e n c e f o r Bh(111) w i t h i n t e n s i t y c u r v e s c a l c u l a t e d f o r the p o t e n t i a l s and f o r t h r e e d i f f e r e n t v a l u e s o f A d % assuming t h e normal FCC r e g i s t r y f o r t h e s u r f a c e arrangement. 147 6.6 Contour p l o t s f o r Bh {111) a t normal i n c i d e n c e o f r r v e r s u s V o r and A d i f o r t h e p o t e n t i a l s (a) v££* and |b) - V ^ j s t a r t i n g from 54 eV. ................ 148 6.7 P l o t s f o r Bh(111) of ( r r ) ^ f o r f i v e independent beams a t normal i n c i d e n c e v e r s u s A d % f o r (a) (V o r = -18eV) and (b). V ^ q (V o r=-10eV). The dashed l i n e s show t h e dependence c f r r v e r s u s ^ d % . ....... 152 6.8 Contour p l o t of r~ r v e r s u s V o r and AdSS f o r Bh<111) at normal i n c i d e n c e , u s i n g t h e d a t a o f F i g - , 6.6 o n l y from 66eV f o r t h e p o t e n t i a l s (a) •v^j['1 and (b) ¥ W 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J 154 6-S Contour p l o t o f ~ r r v e r s u s V o r and AdS f o r t h e p o t e n t i a l v^f4 and t h e model o f the Bh{111) s u r f a c e i n which the t o p t h r e e l a y e r s have t h e HCP s t a c k i n g seguence. ................................ 158 6.10 A comparison o f e x p e r i m e n t a l I (E) c u r v e s f o r the C11) and (10) beams a t 10**, <f> = 109° f o r Rh{111) w i t h i n t e n s i t y p r o f i l e s c a l c u l a t e d f o r the p o t e n t i a l s and V^ (3 f o r t h r e e d i f f e r e n t v a l u e s o f &&% assuming t h e normal FCC r e g i s t r y f o r the s u r f a c e . ...................................... 160 €.11 Contour p l o t of F r v e r s u s V 0 r and A d S f or Bh(111) a t ft-=10«, <^> = 109O f o r t h e p o t e n t i a l s |a) v " ^ and (b) V g M J . ........................................ 161 7. 1 Schematic diagram o f the Bh(100) s u r f a c e (a) and the c o r r e s p o n d i n g LEED p a t t e r n (b) i n t h e n o t a t i o n of Jona £128J. The u n i t mesh i s marked i n ( a ) . The complete c i r c l e s a r e f o r atoms i n t h e second l a y e r , and t h e dashed c i r c l e s c o r r e s p o n d t o a topmost l a y e r w i t h the r e g i s t r y b e l o n g i n g t o the x i i b u l k i . e . atoms i n t h e t o p l a y e r a r e above the 4-f o l d s i t e s such as A. Other r e g i s t r i e s c o n s i d e r e d a r e where atoms a r e over the 2 - f o l d s i t e ( l i k e B ) , or d i r e c t l y over atoms i n the l a y e r below (as f o r C ) . .... .. 170 7.2 Auger s p e c t r a o f Bh(100) s u r f a c e s f o r a 1.5KeV, 10 microamp beam: a) s u r f a c e a f t e r p r o l o n g e d h e a t i n g a t 1300K showing s u b s t a n t i a l S i and C i m p u r i t i e s b) a f t e r argon ion-bombardment, showing reduced S i and i n c r e a s e d c a r b o n c) c l e a n s u r f a c e spectrum a f t e r h e a t i n g a t 1000K i n vacuo. 172 7.3 Two-domain (3X1) LEED p a t t e r n from the Bh(1Q0) s u r f a c e a t 100eV, t h o u g h t t o be due t o t h e presence of s i l i c o n . .............................. 174 7.4 LEED p a t t e r n s from the c l e a n Eh (100) s u r f a c e f o r <a) normal i n c i d e n c e (150eV), (to) f o r 6-=9O0 <j> =20» (94eV) and t h e beam l a b e l l i n g scheme (c) and (d) . 175 7.5 I (E) c u r v e s f o r two s e t s o f beams t h a t s h o u l d be e g u i v a l e n t a t normal i n c i d e n c e on the Bh(1Q0) s u r f a c e ; the f o u r t h member o f each s e t i s obscured by t h e sample m a n i p u l a t o r . ........................ 177 7.6 Comparison o f e x p e r i m e n t a l I (E) c u r v e s f o r the (11) and (20) beams a t normal i n c i d e n c e on Bh(100) w i t h c a l c u l a t i o n s , f o r the V ^ k i l p o t e n t i a l , f o r the topmost r e g i s t r i e s d e f i n e d by A,B and C i n E i g . 7. 1. The topmost i n t e r l a y e r s p a c i n g s a r e 1.90, 2.33 and 2.69A f o r the 4 - f o l d , 2 - f o l d and 1-f o l d s i t e s r e s p e c t i v e l y . .......................... 179 7.7 Comparison o f some e x p e r i m e n t a l 1(E) c u r v e s f o r Sh(100) w i t h c a l c u l a t i o n s f o r the and p o t e n t i a l s . V o r=-12eV and Ad%=-5,0 and+5% f o r (a) normal i n c i d e n c e and (b) a t t>-=90, <j>=20 °. ......... 180 7.8 Contour p l o t s f o r Bh(100) of r r v e r s u s V 0 r and A d ^ f o r {a) the Vgfcu and ib) t h e V p o t e n t i a l . E r r o r b a r s a r e t h e s t a n d a r d e r r o r s £j and £v d e f i n e d i n C h a p t e r 5. ............................. 184 8. 1 Diagram o f a) a (110) s u r f a c e and b) the a s s o c i a t e d LEED p a t t e r n . .......................... 190 8.2 Auger s p e c t r a o f the Bh(110) s u r f a c e at a pr i m a r y beam v o l t a g e of 1.5KeV and 10 micrcamp c u r r e n t : a) a f t e r i n i t i a l h e a t t r e a t m e n t s showing S(152eV), P |120eV) and C (272eV) c o n t a m i n a t i o n on the s u r f a c e x i i i b) a f t e r argon ion-bombardment; P and S removed but C i n c r e a s e d c) c l e a n s u r f a c e s p e c t r u m . 192 8.3 LEED p a t t e r n from t h e c l e a n Eh (110) s u r f a c e a t (a) normal i n c i d e n c e (88eV) and (b) ©- = 10°, <b = 135° (90eV). The beam l a b e l l i n g scheme i s shown i n (c) and (d) . .................................. 194 8-4 E x p e r i m e n t a l I(E) c u r v e s f o r the Hh{110) s u r f a c e a t normal i n c i d e n c e f o r t h e 4 - f o l d e q u i v a l e n t {11} and {21J beam s e t s . The 4 t h member of each s e t i s obscured by the sample m a n i p u l a t o r - ............... 195 8.5 Comparisons of two e x p e r i m e n t a l 1(E) c u r v e s f o r the 8h(110) s u r f a c e w i t h c a l c u l a t i o n s , u s i n g the ? £M3 p o t e n t i a l f o r f o u r v a l u e s o f &d?L The v a l u e o f t h e i n d i v i d u a l beam r e l i a b i l i t y - i n d e x ( r ^ ) ^ i s g i v e n i n b r a c k e t s f o r each c a l c u l a t e d c u r v e . ...... 198 8.6 A c o n t o u r p l o t f o r .ah (110) of r> v e r s u s V o r and b&% f o r da t a a t normal i n c i d e n c e and a c a l c u l a t i o n f o r the V O K l^ p o t e n t i a l . ............................ . 200 x i v LIST OF TABLES 1.1 Comparison of some o f the most i m p o r t a n t s u r f a c e t e c h n i g u e s . More d e t a i l s can be found i n r e f e r e n c e £22 J- ................................... 3 3.1 Data used f o r c o n s t r u c t i o n of s u p e r p o s i t i o n p o t e n t i a l s f o r M,^  c l u s t e r s . The c r y s t a l cube s i d e a D a n d muff i n - t i n r a d i u s r„ Tare from £62,6,3 J and the v a l u e s from [ 5 8 , 5 9 j . ................... 52 3.2 R e d u c t i o n , due t o symmetry,, i n the number of beams needed a t normal i n c i d e n c e f o r the t h r e e s i m p l e f a c e s o f rhodium. The maximum g v e c t o r c o r r e s p o n d s t o the beam s e t w i t h the l a r g e s t (hk) v a l u e s needed t o c o v e r the energy range 4Q-250ev. 69 4.1 Sources o f rhodium c r y s t a l s 77 5. 1 Correspondence between v i s u a l match a n d ^ r r ^ f o r a s i n g l e beam ( f i r s t and second row), and between R f o r a s t r u c t u r a l model and i t s r e l i a b i l i t y (second and t h i r d row). A f t e r Z a a a z z i and Jona £23J. . . . . . 118 5.2 Summary of s t r u c t u r a l d e t e r m i n a t i o n o f the Cu(111) s u r f a c e . 123 6.1 Observed and c a l c u l a t e d Auger t r a n s i t i o n s f o r rhodium. .......................................... 139 6.2 C o n d i t i o n s o f be s t agreement between e x p e r i m e n t and 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 t h r e e s e t s of i n t e n s i t i e s measured a t normal i n c i d e n c e onSh ( 1 1 1 ) . . 157 6-3 C o n d i t i o n s o f b e s t agreement between experiment and 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 t h r e e s e t s o f i n t e n s i t i e s measured a t &-=1[0°, <f> =109° on Bh(111) . ..... .. 164 7.1 Summary o f s t r u c t u r a l d e t e r m i n a t i o n s of the Rh(100) s u r f a c e . 183 8.1 C o n d i t i o n s o f b e s t agreement between e x p e r i m e n t and 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 measured f o r two a n g l e s of i n c i d e n c e on 8k<110). ............... 201 ACKNOWLEDGEMENTS When I rev i e w the work t h a t went i n t o t h i s t h e s i s i t g u i c k l y becomes apparent t h a t I owe a l a r g e number of d e b t s t o the many people who c o n t r i b u t e d , i n l a r g e measure or s m a l l , t o i t s e v e n t u a l c o m p l e t i o n . F i r s t of a l l I must thank my c o - s u p e r v i s o r s P r o f e s s o r s K e i t h M i t c h e l l and Dave F r o s t f o r t h e i r guidance and s u p p o r t throughout t h e co u r s e of t h i s work. X would a l s o l i k e t o e x p r e s s my s i n c e r e g r a t i t u d e t o those i n d i v i d u a l s from o t h e r i n s t i t u t i o n s who g e n e r o u s l y s u p p l i e d m a t e r i a l s and computer programs. Dr. D. P. Woodruff ( U n i v e r s i t y of Warwick, U.K.), P r o f e s s o r G. A. S o m o r j a i ( U n i v e r s i t y o f C a l i f o r n i a a t B e r k e l e y ) and Dr. C.W. Tucker ( G e n e r a l E l e c t r i c C o r p o r a t i o n ) l o a n e d c r y s t a l samples. Dr. ,E. Z a n a z z i and P r o f e s s o r F. J o n a (Stonybrook) p r o v i d e d a copy of t h e i r r e l i a b i l i t y - i n d e x program w h i l e Dr. M. A. Van H o v e { U n i v e r s i t y o f C a l i f o r n i a a t B e r k e l e y ) and Dr. S. Y- Tong ( U n i v e r s i t y of Wisconsin) k i n d l y donated a s e t of m u l t i p l e - s c a t t e r i n g programs. Dr. M i c h e l Van Hove was a c o n s t a n t s o u r c e o f a d v i c e and encouragement on t h e use and m o d i f i c a t i o n of thes e programs. At UBC Dr. L o u i s Noodleman a s s i s t e d i n t h e c o n s t r u c t i o n o f s u p e r p o s i t i o n p o t e n t i a l s and Dr. A. Akht a r o f the M e t a l l u r g y Department o f f e r e d a d v i c e on c r y s t a l c u t t i n g and p o l i s h i n g . The ex p e r i m e n t s would have been i m p o s s i b l e t o p e r f o r m w i t h o u t the s k i l l and w i l l i n g c o - o p e r a t i o n o f the s t a f f of the e l e c t r i c a l and mechanical workshops o f the Che m i s t r y Department under t h e x v i a b l e s u p e r v i s i o n o f Joe S a l l o s and B r i n P o w e l l - Many persons t o c k p a r t i n the c o n s t r u c t i o n and r e p a i r of equipment, p a r t i c u l a r l y Don C a t t , B r i a n Greene, Mike H a t t o n and J o e Shim o f t h e E l e c t r o n i c s workshop and B i l l Henderson, C h a r l i e M c C a f f e r t y , E m i l M a t t e r and C e d r i c Neale of the M e c h a n i c a l Borksbop-I thank my f e l l o w w o r k e r s , both p a s t and p r e s e n t , i n the S u r f a c e C h e m i s t r y Group Sunantha Hengrasmee, Tom Moore and Dr. Hick S t r e a t e r f o r t h e i r e n j o y a b l e company and co-o p e r a t i o n . E s p e c i a l mention must be made of my good f r i e n d Dr. Frank Shepherd, who was my c l o s e c o l l a b o r a t o r t h r o u g h o u t the d u r a t i o n o f t h i s work,, and w i t h o u t whose c h e e r f u l e x p e r t i s e , p a r t i c u l a r l y i n mini-computer programming, t h i s p r o j e c t would have been a p r o t r a c t e d and much l e s s e n j o y a b l e e x p e r i e n c e -F i n a l l y , I wish t o thank S h e i l a L i d w i l l f o r her a s s i s t a n c e i n the p r e p a r a t i o n o f t h i s t h e s i s . "rt<£.. 1 CHAPTER 1 INTRODUCTION 2 1,1 S u r f a c e T e c h n i q u e s Many a s p e c t s o f p r e s e n t t e c h n o l o g y depend on p r o c e s s e s t h a t o c c u r upon the s u r f a c e s of s o l i d s . The u l t i m a t e u n d e r s t a n d i n g of many f i e l d s of r e s e a r c h , i n p a r t i c u l a r heterogeneous c a t a l y s i s , would seem to r e g u i r e d e t a i l e d knowledge o f s u r f a c e p r o c e s s e s a t t h e a tomic l e v e l . One approach has been t o s t u d y w e l l - d e f i n e d s i n g l e c r y s t a l s u r f a c e s (of e.g. a c a t a l y t i c metal) r a t h e r t h a n t h e complex and i l l - d e f i n e d s u r f a c e s w i t h s t r u c t u r a l d e f e c t s and u n c e r t a i n c h e m i c a l c o m p o s i t i o n »hich are t y p i c a l o f p r a c t i c a l s i t u a t i o n s . The p r i m a r y g o a l of such fundamental s t u d i e s i s t h u s t o p r o v i d e a c c u r a t e d e s c r i p t i o n s of s u r f a c e s i n g e o m e t r i c a l and e n e r g e t i c terms. There a r e many t e c h n i g u e s a v a i l a b l e f o r the s u r f a c e s c i e n t i s t t o i n v e s t i g a t e the c o m p o s i t i o n , geometry and e l e c t r o n i c s t r u c t u r e of s u r f a c e s . These v a r i o u s s u r f a c e t e c h n i g u e s employ many t y p e s of s u r f a c e probe, d e t e c t an e g u a l l y l a r g e v a r i e t y of p a r t i c l e s o r r a d i a t i o n and have spawned a b e w i l d e r i n g a r r a y o f acronyms. I n T a b l e 1-1 i s p r e s e n t e d a summary of the most i m p o r t a n t methods, t h e i r acronyms, probe and d e t e c t e d p a r t i c l e s and the i n f o r m a t i o n t h a t can be d e r i v e d from each method about th e s u r f a c e . I t can r e a d i l y be seen t h a t each t e c h n i g u e i n g e n e r a l y i e l d s d i f f e r e n t i n f o r m a t i o n and t h a t a complete d e s c r i p t i o n would i n v o l v e c o r r e l a t i n g t h e r e s u l t s from s e v e r a l methods. T h i s has r e s u l t e d i n t h e e v o l u t i o n o f the so-c a l l e d " m u l t i - t e c h n i g u e s t r a t e g y " i n which s e v e r a l d i f f e r e n t t y p e s of e x p e r i m e n t a r e performed i n s i t u on t h e same sample wi t h o u t c h a n g i n g t h e e x p e r i m e n t a l c o n d i t i o n s -3 Low energy e l e c t r o n d i f f r a c t i o n Auger e l e c t r o n s p e c t r o s c o p y U.V. p h o t o e m i s s i o n s p e c t r o s c o p y A n g l e - r e s o l v e d OPS X-ray p h o t o e m i s s i o n s p e c t r o s c o p y I o n - s c a t t e r i n g s p e c t r o s c o p y I o n c h a n n e l i n g Probe  p a r t i c l e LEED e l e c t r o n s 10-300eV AES e l e c t r o n s 1-10KeV UPS photons 10-40eV ABUPS photons io-40eV XPS ISS photons 0. 5-2KeV i o n s 0.5-2KeV i o a s 1-2Me¥ Det e c t e d p a r t i c l e p r i m a r y e l e c t r o n s secondary e l e c t r o n s e l e c t r o n s e l e c t r o n s e l e c t r o n s p r i m a r y i o n s p r i m a r y i o n s S e c o n d a r y - i o n mass SIMS i o n s s p u t t e r e d s p e c t r o m e t r y 1^30KeV i o n s F i e l d - i o n microscopy FIM f i e l d 10* V/cm imaging gas i o n s I n f o r m a t i o n s u r f a c e s t r u c t u r e e l e m e n t a l c o m p o s i t i o n v a l e n c e energy l e v e l s s u r f a c e s t r u c t u r e s ? c o r e l e v e l s e l e me a t a 1 com.p -e l e m e n t a l comp. s u r f a c e s t r u c t u r e s ? e l e m e n t a l comp. s u r f a c e s t r u c t u r e s ? e l e m e n t a l c o m p o s i t i o n d e f e c t s , s u r f a c e m o b i l i t y T a b l e 1.1 . Comparison of some .  o f the most i m p o r t a n t s u r f a c e t e c h n i q u e s . More d e t a i l s can be found i n r e f e r e n c e j. 22}. The most b a s i c i n f o r m a t i o n t h a t we can wish t o know about a s u r f a c e i s ( i ) what i s t h e i d e n t i t y o f the atoms making up t h e s u r f a c e ? ( i i ) where a r e the s e atoms s i t u a t e d ? That i s , we wish t o know t h e e l e m e n t a l c o m p o s i t i o n and g e o m e t r i c a l s t r u c t u r e o f the s u r f a c e . Of the t e c h n i q u e s l i s t e d i n Table 1.1 the most common and c o n v e n i e n t method used t o answer t h e f i r s t • q u e s t i o n i s auger e l e c t r o n s p e c t r o s c o p y ( A E S ) . The r e a s o n s f o r t h i s a r e i t s e x p e r i m e n t a l s i m p l i c i t y compared t o e.g. i o n c h a n n e l i n g , which r e g u i r e s a p a r t i c l e a c c e l e r a t o r and the ease o f i n t e r p r e t a t i o n compared t o e.g. secondary i o n mass s p e c t r o m e t r y (SIHS). Auger e l e c t r o n s p e c t r o s c o p y i s a l s o more s u r f a c e - s e n s i t i v e than X-ray 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 (XPS) owing t o t h e s h a l l o w e r p e n e t r a t i o n depth o f t h e e x c i t i n g e l e c t r o n s compared w i t h t h e X-rays of XPS. Auger s p e c t r o s c o p y i s e a s i l y f i t t e d i n t o the " m u l t i - t e c h n i g u e s t r a t e g y " and has become a r o u t i n e t o o l f o r s u r f a c e c o m p o s i t i o n a n a l y s i s (see Ch- 2) . There i s only, one t e c h n i q u e t h a t has r e a l l y proven i t s e l f t o answer t h e second q u e s t i o n c o n v i n c i n g l y — l o w - e n e r g y e l e c t r o n d i f f r a c t i o n (LEED). F i e l d i o n microscopy(FIM) i s r e s t r i c t e d i n i t s u s e f u l n e s s by the e x p e r i m e n t a l r e q u i r e m e n t s o f v e r y h i g h e l e c t r i c f i e l d s and a sample, o f h i g h c o h e s i v e s t r e n g t h , i n the form of a n e e d l e . The r e l a t i v e l y new methods o f a n g l e - r e s o l v e d u l t r a v i o l e t p h o t o e l e c t r o n spectroscopy(AfiOPS) and i o n c h a n n e l i n g a r e s t i l l i n t h e f i r s t s t a g e s o f development and, w h i l e showing g r e a t p o t e n t i a l f o r t h e d e t e r m i n a t i o n o f s u r f a c e s t r u c t u r e s , do not have as yet t h e base of s u c c e s s f u l r e s u l t s t h a t LEED has p r o v i d e d d u r i n g the l a s t decade. 5 ,1. 2 H i s t o r i c a l Development Of I M P The o r i g i n s of LIED a r e c l o s e l y bound up w i t h the developments of atomic t h e o r y and guantum mechanics- I n h i s t h e s i s , de B r o g l i e £1 ] p o s t u l a t e d i n 1924 the wave n a t u r e o f m a t t t e r , where a f l u x of p a r t i c l e s w i t h v e l o c i t y v and mass m i s c o r r e l a t e d w i t h a wavelength X= h/mv- Hence a beam of 150 eV e l e c t r o n s have a wavelength of about l l and might be e x p e c t e d t o d i f f r a c t from p e r i o d i c c r y s t a l s t r u c t u r e s . S h o r t l y b e f o r e the p u b l i c a t i o n o f de B r o g l i e ' s t h e s i s , e x p e r i m e n t s c o n c e r n i n g e l e c t r o n s b a c k s c a t t e r e d from p o l y c r y s t a l l i n e n i c k e l t a r g e t s showed a n i s o t r o p i c s i n t h e a n g u l a r d i s t r i b u t i o n s £2J b u t t h e s e were i n t e r p r e t e d a s e x p e r i m e n t a l a r t e f a c t s £ 3 ] , The e s s e n t i a l i n s i g h t was, however, a c h i e v e d as t h e r e s u l t of an a c c i d e n t . I n 1925 D a v i s s o n and Germer began a s e r i e s of e x p e r i m e n t s on a p o l y c r y s t a l l i n e n i c k e l sample t h a t was a n n e a l e d at h i g h t e m p e r a t u r e s from time , t o time i n o r d e r ; t o o b t a i n r e p r o d u c i b l e r e s u l t s . D u r i n g one such h e a t i n g c y c l e the g l a s s a p p a r a t u s was damaged and t h e hot n i c k e l sample s e v e r e l y o x i d i s e d . I n o r d e r t o r e s t o r e the i n i t i a l c o n d i t i o n s , t h e c r y s t a l was h e a t e d f o r some t i m e i n hydrogen; t h i s procedure a p p a r e n t l y caused the growth of l a r g e r s i n g l e c r y s t a l g r a i n s which then gave r i s e t o pronounced maxima i n the a n g u l a r d i s t r i b u t i o n o f b a c k s c a t t e r e d slow e l e c t r o n s . S h i l e the f i r s t i n t e r p r e t a t i o n o f t h e s e r e s u l t s was based on a d i r e c t i o n a l " c r y s t a l t r a n s p a r e n c y " t o p a r t i c l e s , more d e t a i l e d i n v e s t i g a t i o n s showed t h a t t h i s was a wave i n t e r f e r e n c e e f f e c t 6 and i n 1927 Da v i s s o n and Germer [ 4 J r e p o r t e d an a n a l y s i s of the d i f f r a c t i o n maxima on the b a s i s o f de B r o g l i e * s e g u a t i o n . T h i s h i s t o r i c e xperiment was performed i n g l a s s a p p a r a t u s which, a f t e r i n i t i a l e v a c u a t i o n t o about 1 0 - * T o r r , was s e a l e d o f f and f u r t h e r pumped below the d e t e c t i o n l i m i t f o r t h a t time of about 1 0 _ 8 T o r r by a c h a r c o a l s o r p t i o n pump and e v a p o r a t e d g e t t e r m e t a l . Even i n t h i s e a r l y work t h e i m p o r t a n c e o f u l t r a -h i g h vacuum {UHV), u s u a l l y d e f i n e d as < 1 0 - 9 T o r r , - t o m i n i m i s e c o n t a m i n a t i o n o f t h e sample s u r f a c e was r e c o g n i s e d . However, the use o f LEED t o s t u d y s u r f a c e s was n o t developed a t t h a t t i m e . Due t o the d i f f i c u l t y of p r o d u c i n g fl'HV c o n d i t i o n s and the l a c k of a s u r f a c e a n a l y s i s t e c h n i g u e , r e s u l t s c o u l d o f t e n r e f l e c t t h e c o n d i t i o n of an o x i d e o r o t h e r s u r f a c e l a y e r . The s m a l l p e n e t r a t i o n o f the sample by low energy e l e c t r o n s { u s u a l l y d e f i n e d as h a v i n g e n e r g i e s between 0 and 500 eV) made t r a n s m i s s i o n e x p e r i m e n t s on even the t h i n n e s t f i l m s d i f f i c u l t whereas i n r e f l e c t i o n o n l y a s m a l l f r a c t i o n of the i n c i d e n t beam i s b a c k s c a t t e r e d , r a i s i n g problems of s e n s i t i v i t y . The more e a s i l y g e n e r a t e d and c o n t r o l l e d h i g h energy e l e c t r o n beams, w i t h t h e i r g r e a t e r p e n e t r a t i n g power, and t h e s i m p l i f i c a t i o n s brought about i n the e f f e c t i v e s c a t t e r i n g power of atoms a t h i g h i n c i d e n t e n e r g i e s , which made t h e t h e o r y more t r a c t a b l e , a l l ensured t h a t a t t h i s s t a g e h i g h energy e l e c t r o n d i f f r a c t i o n was the more i n t e r e s t i n g f i e l d . I n t e r e s t i n LEED r e v i v e d i n t h e e a r l y 1960's. S e n s i t i v i t y was improved by advances i n e x p e r i m e n t a l d e s i g n , and c o n t r o l o f the c o m p o s i t i o n of s u r f a c e s was made p o s s i b l e by ion-bombardment and a n n e a l i n g . Commercial vacuum systems t h a t c o u l d r o u t i n e l y 7 a t t a i n UHV became a v a i l a b l e t o reduce background gases to the p o i n t where d e p o s i t i o n of a l a y e r of f o r e i g n atoms took hours r a t h e r than minutes. I n t h e l a t e 1960*s the t e c h n i q u e s became a v a i l a b l e t o d e t e c t Auger e l e c t r o n s e m i t t e d from s u r f a c e s , and Auger e l e c t r o n , s p e c t r o s c o p y {see C h a p t e r 2) p r o v i d e d a c o n v e n i e n t monitor of s u r f a c e c o m p o s i t i o n . T h e o r e t i c a l e x p l a n a t i o n s of t h e d i f f r a c t i o n phenomena a l s o advanced i n the l a t e 1960*s. A p p l i c a t i o n o f t h e Bragg t r e a t m e n t used i n X-ray d i f f r a c t i o n showed t h a t a complete d e s c r i p t i o n of the d i r e c t i o n s of the d i f f r a c t i o n maxima was p o s s i b l e but not o f t h e i n t e n s i t i e s . The d i r e c t i o n s c o u l d be observed d i r e c t l y i n a t y p i c a l LEED d i s p l a y - t y p e a p p a r a t u s and a l l o w e d t h e s i z e and shape o f t h e s u r f a c e u n i t c e l l , i n d i r e c t i o n s p a r a l l e l t o the s u r f a c e , to be determined. The d e t e r m i n a t i o n of the b a s i s however r e g u i r e d an a n a l y s i s o f the i n t e n s i t i e s of the d i f f r a c t i o n maxima, as i n X-ray d i f f r a c t i o n . That these i n t e n s i t i e s c o u l d not be s i m p l y d e s c r i b e d by t h e Born a p p r o x i m a t i o n adopted f o r X-ray d i f f r a c t i o n had been r e a l i s e d v e ry soon a f t e r the p u b l i c a t i o n o f the r e s u l t s o f D a v i s s c n and Germer. The development of p r a c t i c a l m u l t i p l e - s c a t t e r i n g programs d i d n o t come t o pass u n t i l t h e e a r l y 1970*s, mainly due to the n e c e s s i t y o f l a r g e d i g i t a l computers t o compute the l a r g e numbers of i n t e r a c t i o n s i n v o l v e d . Many d i f f e r e n t methods 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 have been proposed and w i l l be d i s c u s s e d i n Ch., 3. The t h e o r y has advanced t o the s t a g e where the d i f f r a c t i o n p r o c e s s i s e s s e n t i a l l y c o m p l e t e l y u n d e r s t o o d ; t h e d i f f i c u l t y l i e s i n f i n d i n g methods t h a t w i l l p r o v i d e a c c u r a t e i n t e n s i t i e s w i t h o u t massive c o m p u t a t i o n a l 8 e f f o r t f o r systems of c h e m i c a l i n t e r e s t -Many hundreds o f papers have appeared on LEED i n the l a s t decade. Most c o n f i n e t h e m s e l v e s to d e s c r i p t i o n s of t h e d i f f r a c t i o n p a t t e r n s , i n c o m p a r a t i v e l y few c a s e s have i n t e n s i t i e s been measured and compared w i t h a c c u r a t e c a l c u l a t i o n s . Very good agreement between e x p e r i m e n t and t h e o r y has been a c h i e v e d - f o r d i f f r a c t i o n from some c l e a n metal s u r f a c e s ; a p a r t i c u l a r l y good example b e i n g t h e low i n d e x f a c e s of n i c k e l £1.13- s e v e r a l c a s e s unknown s t r u c t u r e s , produced by s i m p l e gaseous a d s o r p t i o n onto m e t a l l i c s u b s t r a t e s , have been determined by i n t e n s i t y a n a l y s e s . A s e l e c t i o n o f i m p o r t a n t papers and r e v i e w s a r e l i s t e d as r e f e r e n c e s £6-10 J ( m a i n l y e x p e r i m e n t a l ) and £ 11-15j ( m a i n l y t h e o r e t i c a l ) -1.3 T h e s i s O u t l i n e  T h i s t h e s i s d e s c r i b e s a d e t a i l e d s t u d y of the s t r u c t u r e s o f t h e c l e a n (111) , {100) and {110) s u r f a c e s of rhodium u s i n g LEED. T h i s c a t a l y t i c a l l y i m p o r t a n t f a c e - c e n t r e d c u b i c m e t a l , i n c o n t r a s t t o most of the o t h e r p l a t i n u m group m e t a l s , has r e c e i v e d s c a n t a t t e n t i o n by s u r f a c e s c i e n t i s t s - S t u d i e s of any s o r t on s i n g l e c r y s t a l s u r f a c e s o f t h i s m e tal a r e r a r e ; a p a r t from e a r l y g u a l i t a t i v e work by T u c k e r £ 16- 17J, who d i d n o t have a s u r f a c e c o m p o s i t i o n m o n i t o r and hence whose r e s u l t s a r e open t o doubt, and r e c e n t c h e m i s o r p t i o n s t u d i e s on t h e Bh{100) and {111) £18,131] , and (110) s u r f a c e s £133 J , the p r e s e n t work r e p r e s e n t s t h e f i r s t d e t a i l e d i n v e s t i g a t i o n o f c l e a n rhodium s u r f a c e s u s i n g modern methods. 9 As s u r f a c e s c i e n c e i s a new f i e l d i n t h i s i n s t i t u t i o n the e x p e r i m e n t a l eguipment had t o be commissioned, or i n some c a s e s b u i l t , and computer programs had t o be adapted f o r use here as w e l l as s e v e r a l s m a l l e r d a t a h a n d l i n g and m a n i p u l a t i o n programs w r i t t e n . The CuJ111) s u r f a c e , which has been s t u d i e d i n some d e t a i l p r e v i o u s l y j. 19-20 J,, was used as a t e s t c a s e f o r both the e x p e r i m e n t a l t e c h n i q u e s and t o check the c o r r e c t n e s s of c a l c u l a t i o n s . Only when i t was c e r t a i n t h a t t h e da t a o f o t h e r workers f o r t h i s s u r f a c e c o u l d be reproduced were s t u d i e s on t h e rhodium s u r f a c e s commenced. Examples from t h i s p r e l i m i n a r y copper work a r e o f t e n used i n t h e t e x t i n an i l l u s t r a t o r y manner. Chapter 2 i s devoted t o a b r i e f r e v i e w of b a s i c i d e a s t h a t u n d e r l i e LEED. S u r f a c e s t r u c t u r e s , t h e r e c i p r o c a l l a t t i c e and a s i m p l e e x p l a n a t i o n o f the f o r m a t i o n of the d i f f r a c t i o n p a t t e r n and i t s i n t e n s i t y a r e d i s c u s s e d i n o u t l i n e . , The p r o d u c t i o n o f Auger e l e c t r o n s and t h e use o f auger e l e c t r o n s p e c t r o s c o p y as a s u r f a c e a n a l y s i s t o o l i s a l s o summarised. Chapter 3 o u t l i n e s the t h e o r e t i c a l b a s i s of the c a l c u l a t i o n s t h a t , when compared w i t h e x p e r i m e n t , y i e l d t he s u r f a c e s t r u c t u r e . The n e c e s s a r y programs are r a p i d l y becoming more a v a i l a b l e and commonplace and hence s t a n d a r d r e s u l t s are guoted w i t h l i t t l e a t tempt t o e x p l a i n t h e c o m p u t a t i o n a l c o m p l e x i t i e s ; i n such c a s e s t h e r e a d e r i s r e f e r r e d t o s t a n d a r d t e x t s . B a t h e r , a t t e n t i o n i s f o c u s s e d on the s t r u c t u r a l and non-s t r u c t u r a l parameters t h a t e n t e r i n t o t h e c a l c u l a t i o n s , t h e i r meaning and i m p o r t a n c e , and the o v e r a l l c o n s t r u c t i o n and f l o w o f th e programs. In p a r t i c u l a r , ; the. i o n - c o r e s c a t t e r i n g p o t e n t i a l 10 t h a t l e a d s to b a c k s c a t t e r i n g of the i n c i d e n t low-energy e l e c t r o n s must be c o r r e c t l y . s p e c i f i e d t o o b t a i n r e l i a b l e t h e o r e t i c a l i n t e n s i t i e s . I n t h i s work two t y p e s o f p o t e n t i a l s were used, b a n d - s t r u c t u r e p o t e n t i a l s and p o t e n t i a l s c o n s t r u c t e d by the method of l i n e a r s u p e r p o s i t i o n o f charge d e n s i t i e s f o r a c l u s t e r of metal atoms. I n C h a p t e r 4 can be found d e t a i l s o f t h e e x p e r i m e n t a l a p p a r a t u s and p r o c e d u r e s . The vacuum system and the p r o d u c t i o n of a c l e a n sample, as judged by AES, a r e d e s c r i b e d . An i n n o v a t i v e new method of measuring t h e e x p e r i m e n t a l i n t e n s i t i e s , employing a c o m p u t e r - c o n t r o l l e d V i d i c o n T - ? . camera system £ 2 1 3 , based on t h e p h o t o g r a p h i c method o f S t a i r e t a l £68J, i s co v e r e d i n some d e t a i l . T h i s method has s i g n i f i c a n t a dvantages o v e r the c o n v e n t i o n a l t e c h n i g u e s of spot-photometry o r use o f a Earaday^-cup c o l l e c t o r , p a r t i c u l a r l y i n terms of speed and c o n v e n i e n c e . The former methods n e c e s s i t a t e d h o u r s of d a t a c o l l e c t i o n t o produce even a s m a l l base of e x p e r i m e n t a l d a t a , d u r i n g which time the s u r f a c e c o u l d become s e r i o u s l y c o n t a m i n a t e d . T h i s new V i d i c o n method reduces d a t a c o l l e c t i o n t i m e s t o a few minutes and p r o v i d e s a hard-copy p h o t o g r a p h i c r e c o r d t h a t can be a n a l y s e d a t l e i s u r e t o y i e l d a v e r y l a r g e d a t a base which unambiguously r e l a t e s t o the same o r i g i n a l s u r f a c e . C hapter 5 d e s c r i b e s p r o g r e s s t o dat e i n LEED s u r f a c e c r y s t a l l o g r a p h y , i n c l u d i n g a b i b l i o g r a p h y o f metal; c l e a n 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 . The p r e f e r r e d s u r f a c e s t r u c t u r e i s found by comparing c a l c u l a t e d r e f l e c t e d i n t e n s i t i e s f o r . a wide v a r i e t y o f p o s s i b l e s t r u c t u r a l models w i t h e x p e r i m e n t a l d a t a . 11 T r a d i t i o n a l l y , c omparisons o f e x p e r i m e n t a l and. t h e o r e t i c a l d a t a have been c a r r i e d out v i s u a l l y o r by v e r y c r u d e n u m e r i c a l matching of major f e a t u r e s . Hence a r e c u r r i n g problem i n LEED c r y s t a l l o g r a p h y has been, t h a t of e s t a b l i s h i n g l i m i t s on t h e a c c u r a c y and r e l i a b i l i t y o f the s u r f a c e s t r u c t u r e s t h u s determined. D e s p i t e a l o n g - s t a n d i n g c o n c e r n over t h i s p r oblem, o n l y v e r y r e c e n t l y has a d e t a i l e d , o b j e c t i v e method been proposed £23J t h a t a t t e m p t s to n u m e r i c a l l y compare a l l the e s s e n t i a l f e a t u r e s o f t h e e x p e r i m e n t a l d a t a w i t h v a r i o u s t h e o r e t i c a l ^ model p r e d i c t i o n s . T h i s r e l i a b i l i t y - i n d e x , o r 8-f a c t o r , method i s d e s c r i b e d t o g e t h e r w i t h some a d d i t i o n s and r e f i n e m e n t s d e v e l o p e d d u r i n g t h e a p p l i c a t i o n o f t h i s i n d e x t o t h e d e t e r m i n a t i o n of t h e s t r u c t u r e s of the £h (111) , (100)< and O10) s u r f a c e s . The a c t u a l d e t e r m i n a t i o n o f t h e s u r f a c e s t r u c t u r e s o f t h e s e t h r e e l o w - i n d e x f a c e s of rhodium i s p r e s e n t e d i n C h a p t e r s 6-8. F i n e d e t a i l s o f the c l e a n i n g p r o c e d u r e s , which can v a r y from sample t o sample depending upon t h e i r o r i g i n s and p r e - t r e a t m e n t , a r e d i s c u s s e d . The s t r u c t u r e d e r i v a t i o n s performed by t h e r e l i a b i l i t y - i n d e x method a r e c l o s e l y examined t o a s s e s s u n c e r t a i n t i e s i n the r e s u l t s from e x p e r i m e n t a l (or c a l c u l a t i o n a l ) a r t e f a c t s and f o r t u i t o u s c i r c u m s t a n c e s . D u p l i c a t e s e t s o f d a t a from d i f f e r e n t e x p e r i m e n t s a r e used t o a s s e s s t h e r e l i a b i l i t y of the e x p e r i m e n t a l d a t a and the r e s u l t s from i n d e p e n d e n t d a t a s e t s , c o l l e c t e d a t v a r i o u s a n g l e s of i n c i d e n c e , compared to t e s t the c o n s i s t e n c y of the method. A l l t h r e e s u r f a c e s r e t a i n the s t r u c t u r e e x p e c t e d f o r a s i m p l e t r u n c a t i o n o f t h e b u l k c r y s t a l . H h i l e t h i s i s normal f o r 12 the c l o s e - p a c k e d {111) s u r f a c e s o f f a c e - c e n t r e d c u b i c m e t a l s , some {110) and (100) f a c e s a r e known to p e r f o r m s u r f a c e s t r u c t u r a l r e c o n s t r u c t i o n s o r r e - a r r a n g e m e n t s of a g e n e r a l l y c o n p l e x and as y e t unknown n a t u r e £24J , The (100) f a c e s o f e.g. p l a t i n u m £25J and i r i d i u m . £26J a r e n o r m a l l y observed i n t h i s c o n d i t i o n . R i g o r o u s e x a m i n a t i o n o f t h e Rh (100) and (110) s u r f a c e s a t room tem p e r a t u r e and above r e v e a l e d no such tendency. A l l t h r e e s u r f a c e s were found t o have the s u r f a c e l a y e r at a d i s t a n c e from the second l a y e r c l o s e t o , or a l i t t l e c o n t r a c t e d f r o m , t h e b u l k i n t e r l a y e r d i s t a n c e . F i n a l l y , a comprehensive s e t o f e x p e r i m e n t a l data f o r t h e t h r e e rhodium s u r f a c e s i s c o l l e c t e d i n t h e a p p e n d i c e s . Examples a r e used i n the main body of the t e x t f o r i l l u s t r a t o r y p u r p o s e s . T h i s t h e s i s i s p r e l i m i n a r y i n t h e sense t h a t a l a r g e p a r t of the work t h a t went i n t o i t was concerned w i t h d e v e l o p i n g t h e s e new, f o r t h i s l a b o r a t o r y , e x p e r i m e n t a l and t h e o r e t i c a l t e c h n i g u e s . Moreover, these new r e s u l t s , w h i l e i n t h e m s e l v e s t h e y form a r e a s o n a b l y c o m p l e t e s t u d y , a r e a l s o p r e l i m i n a r y t o the u n d e r s t a n d i n g of t h e a d s o r p t i o n p r o p e r t i e s and heterogeneous c a t a l y t i c a c t i v i t y of rhodium £26J. J u s t a s we c a n n o t hope t o understand c a t a l y s i s a t t h e a t o m i c l e v e l i f we cannot l o c a t e adsorbed atoms, so we cannot understand c h e m i s o r p t i o n i f we do not have a p r e c i s e d e s c r i p t i o n of t h e c l e a n s u r f a c e . I hope t h i s work p r o v i d e s the l a t t e r i n f o r m a t i o n and w i l l l a y a s o l i d f o u n d a t i o n f o r l a t e r s t u d i e s of rhodium, i t s c h e m i s o r p t i o n p r o p e r t i e s and u l t i m a t e l y i t s c a t a l y t i c a c t i v i t y . 13 CHAPTER 2 LOW-ENERGY ELECTRON DIFFRACTION(LEED) AND A UGER ELECTION SPECTROSCOPY (AES) 14 _2. 1 B a s i c C o n s i d e r a t i o s s 2.1{a) se c o n d a r y e l e c t r o n d i s t r i b u t i o n E l e c t r o n s w i t h e n e r g i e s between about 10 and 1000 eV a r e i d e a l l y s u i t e d t o i n v e s t i g a t e the topmost l a y e r s of s o l i d s because the p r o b a b i l i t i e s f o r i n e l a s t i c s c a t t e r i n g a r e h i g h . A parameter f r e g u e n t l y used i n t h i s c o n t e x t i s t h e e l e c t r o n mean f r e e path l e n g t h , L. T h i s i s t h e mean d i s t a n c e t r a v e l l e d by an e l e c t r o n b e f o r e i t i s s c a t t e r e d i n e l a s t i c a l l y and can be exp r e s s e d by I IE) = I D (E) exp I - l / L CE) j where t h e i n c i d e n t i n t e n s i t y I0{E) f o r energy E i s a t t e n u a t e d t o I <E) on passage through d i s t a n c e 1. The c h a r a c t e r i s t i c dependence o f t h i s p r o p e r t y on the e l e c t r o n energy i s shown i n F i g . , 2.1. A l t h o u g h i t i s somewhat d i f f i c u l t t o o b t a i n e x a c t q u a n t i t a t i v e d a t a , the g e n e r a l f e a t u r e s of t h i s diagram have been w e l l e s t a b l i s h e d - A minimum an t h e mean f r e e p a t h o f o n l y a few A o c c u r s a t e n e r g i e s between 40 and 100 eV. a h i l e t h e c u r v e i n c r e a s e s s t e e p l y a t low e n e r g i e s , a t h i g h e r e n e r g i e s t h e i n c r e a s e i n t h e mean f r e e p a t h i s slow and at 1000 eV i s s t i l l o n l y a few at o m i c l a y e r s -I n t h e LEED experiment a beam o f e l e c t r o n s w i t h a d e f i n i t e p r i m a r y e n e r g y i m p i n g e s on t h e c r y s t a l s u r f a c e and t h o s e 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 a r e c o l l e c t e d . F i g - 2.2 shows a s c h e m a t i c energy d i s t r i b u t i o n , N ( E ) , of the b a c k - s c a t t e r e d e l e c t r o n s as a f u n c t i o n of e n e r g y . T h i s I r — 1 1 1 0 10 100 1000 10,000 00,000 Electron energy (eV) l i s u j e 2*1 Schematic diagram c f the Bean f r e e path L(A) of e l e c t r o n s i t m e t a l l i c s o l i d s as a f u n c t i o n of t h e i r energy O V ) . Ha"!* 2x2 S c h e s a t i c energy d i s t r i b u t i o n M(£) o f b a c k s c a t t e r e d elcv e l e c t r o n s as a f u n c t i o n of t h e i r enerqy f c r a primary bean energy . 16 "secondary e l e c t r o n d i s t r i b u t i o n " can be d i v i d e d i n t o ; t h r e e main r e g i o n s : i ) t h e l a r g e peak a t low e n e r g i e s c o n t a i n s " s e condary e l e c t r o n s " c r e a t e d as a r e s u l t o f i n e l a s t i c c o l l i s i o n s between i n c i d e n t e l e c t r o n s and e l e c t r o n s bound i n the s o l i d . I n each c o l l i s i o n p r o c e s s a r e l a t i v e l y s n a i l amount o f energy i s t r a n s f e r r e d so t h a t a s i n g l e p r i m a r y e l e c t r o n can c r e a t e a c a s c a d e o f s e c o n d a r i e s t h a t c o n t r i b u t e t o t h e l a r g e broad peak a t low e n e r g i e s ; i i ) t he medium energy range i s c h a r a c t e r i s e d by a smooth background upon which a r e superimposed s m a l l f e a t u r e s due m a i n l y t o t h e e m i s s i o n of Auger e l e c t r o n s (see S e c t i o n 2.4) and energy l o s s e s t o c o r e e l e c t r o n s , s i n g l e and c o l l e c t i v e v a l e n c e e l e c t r o n e x c i t a t i o n s J [28J ; i i i ) a s m a l l f r a c t i o n ( t y p i c a l l y a few %) i s b a c k - s c a t t e r e d e l a s t i c a l l y a t energy E 0 t o form t h e " e l a s t i c peak". T h i s peak a c t u a l l y a l s o c o n t a i n s " g u a s i - 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 have undergone phonon s c a t t e r i n g , t h a t i s an i n t e r a c t i o n w i t h the v i b r a t i n g c r y s t a l l a t t i c e . These i n t e r a c t i o n s produce energy changes of t h e o r d e r of 10 meV. The energy r e s o l u t i o n of t y p i c a l LEED i n s t r u m e n t s i s i n s u f f i c i e n t t o o b s e r v e such l o s s e s o r g a i n s and hence, as f a r as LEED i s concerned, t h e whole of the h i g h - e n e r g y peak i s o f t e n l o o s e l y termed the " e l a s t i c peak". I n t h e LEED e x p e r i m e n t 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 17 are f i l t e r e d o u t o f t h e g e n e r a l secondary e l e c t r o n d i s t r i b u t i o n and, u s u a l l y , d i s p l a y e d on a f l u o r e s c e n t s c r e e n - He s h a l l see i n l a t e r s e c t i o n s o f t h i s c h a p t e r how t h e s e 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 behave s p a t i a l l y and t h e i r energy dependence-, 2. 1 (b) s u r f a c e s t r u c t u r e c l a s s i f i c a t i o n s The r e g i o n of an o r d e r e d s u r f a c e probed by LEED has two-d i m e n s i o n a l symmetry p a r a l l e l t o the s u r f a c e b u t no p e r i o d i c i t y normal t o i t - As w i t h t r i p e r i o d i c c r y s t a l s t r u c t u r e s , p e r f e c t d i p e r i o d i c s u r f a c e s t u c t u r e s may be c l a s s i f i e d i n t o c e r t a i n l a t t i c e t y p e s ; f o r many pure s u b s t a n c e s t h e symmetry of t h e s u r f a c e i s g i v e n s i m p l y by t h a t o f t h e p e r f e c t b u l k c r y s t a l i n a plane p a r a l l e l to the s u r f a c e under c o n s i d e r a t i o n . S u r f a c e s a r e u s u a l l y named a f t e r t h e p l a n e s of the b u l k c r y s t a l t o which t h e y a r e p a r a l l e l e.g. (100), J110) e t c - T h e i r s t r u c t u r e s can be d e s c r i b e d i n terms of a u n i t mesh w i t h u n i t v e c t o r s s ( and s t l y i n g i n t h e s u r f a c e p l a n e . A l l the p o i n t s which a r e connected by t r a n s l a t i o n v e c t o r s T = h's f + k » s ^ h«,k« = i n t e g e r s (2.1) form a B r a v a i s n e t . I n t w o - d i m e n s i o n a l c r y s t a l l o g r a p h y the terms " l a t t i c e " and " c e l l " of 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 a re r e p l a c e d by t h e terms " n e t " and "mesh" r e s p e c t i v e l y - The 14 B r a v a i s l a t t i c e s o f 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 a r e reduced t o 5 n e t s f o r d i p e r i o d i c s u r f a c e s t r u c t u r e s . These f i v e t y p e s o f 18 n e t , as r e p r e s e n t e d by models c f some 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 (FCC) and b o d y - c e n t r e d c u b i c {BCC} m e t a l s , a r e shown i n F i g . 2.3. D e t a i l s o f v a r i o u s c o n v e n t i o n s i n s u r f a c e c r y s t a l l o g r a p h y c a n be found i n an a r t i c l e by Hood .£29. J and i n t h e I n t e r n a t i o n a l T a b l e s f o r X—ray C r y s t a l l o g r a p h y £ 30J. F o r c u r purposes i t s u f f i c e s t o observe t h a t i n t h e b u l k , c r y s t a l s t r u c t u r e s can be b u i l t up from i d e n t i c a l 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 . An atom i n one l a y e r i s r e l a t e d t o an i d e n t i c a l atom i n the n e x t by a v e c t o r c; each l a y e r i s r e g a r d e d as s e p a r a t e d by an i n t e r l a y e r d i s t a n c e d. The o r i e n t a t i o n of t h e i n c i d e n t beam o f e l e c t r o n s r e l a t i v e t o t h e s u r f a c e i s d e s c r i b e d by two a n g l e s , &• and 0 ; the p o l a r a n g l e tV i s between t h e beam and the inward-p o i n t i n g s u r f a c e normal and the a z i m u t h a l a n g l e , i s the a n g l e between the plane c o n t a i n i n g the s u r f a c e normal and i n c i d e n t beam, and t h e x - a x i s . F i g . 2.4 c l e a r s up any a m b i g u i t i e s . 2 . 1(c) t h e r e c i p r o c a l net A nother type of B r a v a i s n e t can be c o n s t r u c t e d f o r each of t h o s e shown i n F i g . 2.3 ; t h e s e a r e c a l l e d " r e c i p r o c a l n e t s " . The r e c i p r o c a l n e t c o r r e s p o n d i n g t o a r e a l .: net d e f i n e d by v e c t o r s s- and s ^ i s g. = h s f + ks * (2.2) (a) (b) (c) ^ o o IlS3I§ 2 X 3 Ixamples c f t h e 5 d i p e r i o d i c nets as r e p r e s e n t e d bv a o d e l s of s u r f a c e s of f a c e - c e n t r e d c u b i c (FCC) and t c d y - c e n t r e d c u b i c (ECC) m e t a l s ; s i d e s an olec<: <a j s q u a r e a,« a BCC (100); (t) p r i m i t i v e r e c t a c q u l a r a BCC (2 11); (c) c e n t r e d r e c t a n q u l a r a BCC (110) ; (d) b e x a q c n a l a (e) o b l i q u e a 2 ' ( s h o r t a x i s ) and a^, i n t e r n a l * = 90°, e.q. FCC<100) «* a l # * = 90°, e.q. PCC(110) * a x , * = 90°, e.q, FCC (211) * a x,o<= 600, e.q. ICC(111) a z , 900, e.q. fCC(321) direction of incident beam f i f l S I S 2j.H ADgle c o n v e n t i o n s f o r i n c i d e n c e of an e l e c t r o n beaic cn a s u r f a c e 20 w i t h • s- = 2 IT ( s ^  x z ) / (s ( .s ^ x z ) |2-3) and s ^  = 2 TT ( s ( x z J / j s ^ . s y x z ) where 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 s- and s^, as shown i n f i g - 2-5- The r e c i p r o c a l n e t s c o r r e s p o n d i n g t o t h e r e a l n e t s shown i n F i g - 2-4 a r e g i v e n i n F i g . 2-6-At t h i s s t a g e the r e c i p r o c a l n e t appears as an a b s t r a c t c o n s t r u c t i o n but we s h a l l see i n the f o l l o w i n g s e c t i o n t h a t i t has a very d i r e c t i n t e r p r e t a t i o n i n LEED e x p e r i m e n t s . 2.2 F o r m a t i o n Of The D i f f r a c t i o n P a t t e r n -we s h a l l c o n s i d e r a mono-energetic, c o l l i m a t e d e l e c t r o n beam i n c i d e n t on a p e r f e c t l y c l e a n w e l l - o r d e r e d s u r f a c e . T h i s i s , of c o u r s e , a t h e o r e t i c a l a b s t r a c t i o n a s even t h e most c a r e f u l l y p r e p a r e d s u r f a c e has a degree of roughness t h a t can be seen on e l e c t r o n m i c r o g r a p h s . However, because o f l i m i t s on the c o l l i m a t i o n a c h i e v a b l e i n low energy e l e c t r o n guns, the i n c i d e n t beam i s c o h e r e n t o n l y o v e r r e s t r i c t e d a r e a s {of the o r d e r o f o 1 0 0 A 2 ) , and t h e r e f o r e LEED i s s e n s i t i v e t o a t o m i c o r d e r o n l y e v e r d i s t a n c e s o f t h i s magnitude. A t y p i c a l LEED p a t t e r n from a C u ( 1 1 l ) s u r f a c e i s shown i n Fig., 2.7 £a). , our i n t e r e s t • w i l l c e n t r e on 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 produce most o f t h e s t r u c t u r e i n t h e d i f f r a c t i o n p a t t e r n . T h e i r b e h a v i o u r can be d e s c r i b e d by a S c h r b d i n g e r e g a a t i o n o f the form ( i n a t o m i c u n i t s ) o l A s ^ f e 2x5 A t w o - d i a e n s i o n a l r e a l n e t , d e s c r i b e d by s, , s t , ( d a r k ~ c i r c l e s ) and i t s a s s o c i a t e d r e c i p r o c a l n e t £*, s * (oper c i i c l e s ) . l i f l u r e 2±§. O n i t aeshs of t h e c o r r e s p o n d i n g r e a l and r e c i p r o c a l t w o - d i m e n s i o n a l n e t s as i n F i g u r e 2.3. 23 - V 2 V 2 ^( r ) • V(r )^Cr) = E^{r> (2.4) Far from t h e c r y s t a l t h e e f f e c t of t h e c r y s t a l p o t e n t i a l , V ( r ) , can be n e g l e c t e d and t h e e f f e c t o f t h e i n t e r a c t i o n w i l l be shown by t h e change i n the wave-vector o f t h e plane^wave e i g e n s t a t e s = exp ( ik.r) (2.5) b e f o r e and a f t e r d i f f r a c t i o n . F o r e l a s t i c s c a t t e r i n g by t h e d i p e r i o d i c arrangement o f atoms i n the s u r f a c e r e g i o n , t h e wave-vectors k* of t h e d i f f r a c t e d e l e c t r o n s a r e dete r m i n e d by c o n s e r v a t i o n of energy and momentum p a r a l l e l t o t h e s u r f a c e f 1 2 J E(k«) = E(k) {2.6) b e f o r e (unprimed) and a f t e r (primed) d i f f r a c t i o n = JS|, + <3<hk) (2.7) w i t h t h e r e c i p r o c a l net v e c t o r gjhk) d i s c u s s e d i n S e c t i o n 2. 1 (c) as qjhk) = h s * + k s * (2.8) P h y s i c a l l y t h e n , t he d i f f r a c t e d a a v e f i e l d has, indepe n d e n t of t h e e x a c t s c a t t e r i n g mechanisms i n v o l v e d , t h e form of a s e r i e s of d i s c r e t e beams each w i t h a d i f f e r e n t p a r a l l e l 24 2*1 (a) Photograph o f the LEED p a t t e r n f r c i a Cu(111) s u r f a c e at (a) n o r a a l i n c i d e n c e and 90 eV beam energy (b) 0 = 12°, ^=6° at UHeV bea» energy, (c) and (d) l a b e l l i n g c f the d i f f z a c t i c c s p c t s . component o f momentum (k (| + a.) ; t h i s i s determined by the t r a n s l a t i o n a l symmetry of the r e g i o n t r a v e l l e d by the s c a t t e r e d e l e c t r o n s . The d i r e c t i o n s of the d i f f r a c t e d beams are determined by the wave-vector of the d i f f r a c t e d e l e c t r o n s , k», and hence by k , q and E. T h i s i s i l l u s t r a t e d i n F i q . 2.8. For q i v e n v a l u e s of E and k(|, t h e d i f f r a c t i o n p a t t e r n i s d e t e r m i n e d by jg( h k ) . For c e r t a i n v a l u e s of jg{hk), Jc«^  i s i m a g i n a r y and c o r r e s p o n d s to e v a n e s c e n t , o r s u r f a c e waves, which cannot escape from the s o l i d . Ey c o l l e c t i n g t h e d i f f r a c t e d e l e c t r o n s on a s p h e r i c a l f l u o r e s c e n t s c r e e n w i t h the c r y s t a l a t i t s f o c u s the beams appear as s p o t s , one f o r each v a l u e of £{hk). The s p o t produced by a beam with p a r a l l e l component of momentum 3 (hk) = hs ^  + ks (2.9) i s r e f e r r e d t o as t h e (hk) sp o t or beam, the i n d i c e s o f {2.7-2.9) b e i n g used t o l a b e l t h e d i f f r a c t e d beams. The beam l a b e l l i n g scheme a p p r o p r i a t e to the Cu(111) LEED p a t t e r n of F i g . 2-7 {a) i s shown as F i g . 2.7(b). Thus the {00) beam i s made up of e l e c t r o n s which have i n t e r a c t e d w i t h the s u r f a c e w i t h o u t momentum t r a n s f e r and i s c a l l e d t h e s p e c u l a r l y r e f l e c t e d beam. T h i s s p e c u l a r beam i s e a s i l y r e c o g n i s e d a s i t s d i r e c t i o n r e m a i n s c o n s t a n t as E changes as l o n g as t h e e l e c t r o n s move i n f i e l d - f r e e space and the d i r e c t i o n of the i n c i d e n t beam does not change. As t h e i n c i d e n t energy i s i n c r e a s e d , the p e r p e n d i c u l a r component of momentum i n c r e a s e s , the a n g l e o f d i f f r a c t i o n d e c r e a s e s and the beams 2t f i g u r e 2^8 The d i r e c t i o n of a d i f f r a c t e d team i s d e t e r n i ^ e d oy the a l l o w e d v a l u e s of k' and F. crowd i n towards t h e s p e c u l a r beam. The 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 u s u a l f l u o r e s c e n t s c r e e t c f a LEED a p p a r a t u s p r o v i d e s a d i r e c t p i c t u r e c f the r e c i p r o c a l net o f the s u r f a c e and hence the s i d e s c f the u n i t *esh c f t h a t net a r e e a s i l y d e t e r m i n e d , s * and s*. The u n i t •esh c f the r e a l net i s then o b t a i n e d by i n v e r s i o n of e q u a t i o n s (2.3) : s ( = 2TT( s * x f ) / ( § * . s * x 2 ) (2.10) § =• 2TT| S| X Z ) / ( £ * . S * 1 2 ) 27 A l t h o u g h e q u a t i o n (2-9) g i v e s the most fundamental d e s i g n a t i o n s o f d i f f r a c t e d beams, f o r s u r f a c e s t h a t r e c o n s t r u c t or c o n t a i n adsorbed m o l e c u l e s i t i s o f t e n c o n v e n i e n t t h a t the d i f f r a c t e d beams a r e indexed w i t h r e s p e c t t o the r e c i p r o c a l to j , l a t t i c e v e c t o r s o f the s u b s t r a t e (b ( and b z ) , s i n c e the s u b s t r a t e i s t h e i n i t i a l l y known s t r u c t u r e . T h i s r e l a t i o n i s p p II 12. P P Zl 21. (2. 11) or s = Pb (2. 12) The m a t r i x P f o r a p a r t i c u l a r s t r u c t u r e can o f t e n be o b t a i n e d d i r e c t l y by comparing t h e LEED p a t t e r n of the s u r f a c e w i t h t h a t c a l c u l a t e d f o r the s u b s t r a t e . M a t r i x m a n i p u l a t i o n of e q u a t i o n (2.12) y i e l d s s = P-*b 12.13) which d e f i n e s t h e r e l a t i o n o f t h e mesh v e c t o r s of the a c t u a l s u r f a c e w i t h t h o s e of t h e s u b s t r a t e . The m a t r i x P - 1 p r o v i d e s the most g e n e r a l way o f e x p r e s s i n q t h i s r e l a t i o n [32 1, but when the a n g l e between s- and s ^ i s e q u a l t o t h a t between b^ and b i a more compact d e s i q n a t i c n i s o f t e n used 129J. Then t h e r e l a t i o n between s and b can be s p e c i f i e d by (2. 14) 28 i n terms of the l e n g t h s of the u n i t mesh s i d e s and t h e a n g l e of r o t a t i o n (^) between s and t ( e m i t t e d when & = 0) . T h i s n o t a t i o n i s p r o b a b l y best i l l u s t r a t e d by examples. In the s i m p l e s t c a s e , the s u r f a c e s t r u c t u r e i s g i v e n by the t e r m i n a t i o n of the b u l k s t r u c t u r e a l o n g a g i v e n p l a n e . The s u r f a c e u n i t mesh i n such c a s e s i s b r i e f l y r e f e r r e d t o as (1x1) i n the Wood c o n v e n t i o n £29], i n d i c a t i n g t h a t the s u r f a c e net v e c t o r s a r e i d e n t i c a l t o those o f the u n d e r l y i n g s u b s t r a t e e.g. Eh (100} - {1 x1> . In more g e n e r a l c a s e s t h e s u r f a c e p e r i o d i c i t i e s w i l l d i f f e r from t h o s e of b u l k s u b s t r a t e s . T h i s i s e s p e c i a l l y so f o r the d e p o s i t i o n of f o r e i g n atoms (adsorbates) on the s u b s t r a t e m a t e r i a l . (Such adsorbed c v e r l a y e r s t u c t u r e s w i l l i n q e n e r a l have a p e r i o d i c i t y d i f f e r e n t from t h a t of the s u b s t r a t e ) . Another c a s e more r e l e v a n t t o t h i s t h e s i s c o n c e r n s the r e c o n s t r u c t i o n s of the s u r f a c e r e g i o n s of c h e m i c a l l y c l e a n m a t e r i a l s such as o c c u r f o r many s e m i - c o n d u c t o r s and metals £ 14J . The (100) f a c e of g o l d £33 1 f o r example y i e l d s a d i f f r a c t i o n p a t t e r n shown s c h e m a t i c a l l y i n f i g - 2 . 9 ( a ) . T h i s i s d e s i g n a t e d as a (5x1) LIED p a t t e r n and i s a s s o c i a t e d w i t h a (5x1) u n i t mesh r e l a t i v e t o t h a t o f the b u l k s u b s t r a t e - A model assuming an h e x a g o n a l arrangement o f s u r f a c e g o l d atoms on the u n d e r l y i n g (100) s u b s t r a t e has been proposed t o acc o u n t f o r t h i s p a t t e r n £33] as shown i n F i g . 2.9(b) and ( c ) . The a c t u a l LEED p a t t e r n observed i n t h i s case i s a s u p e r p o s i t i o n of two p a t t e r n s , as i n F i q . 2 . 9 ( a ) , r e l a t e d by a 90° r o t a t i o n . These c o r r e s p o n d to the e x i s t e n c e of two e q u i v a l e n t o r i e n t a t i o n s , or domains, r e l a t e d by a 90° r o t a t i o n o f the s u r f a c e l a y e r on the 29 a) b) c) l i S U I i a) Schematic d i f f r a c t i o n p a t t e r n from one d c n a i n of Au (1C0)-(5X1) r e c o n s t r u c t e d s u r f a c e ; b) Hodel s t r u c t u r e of c o i n c i d e n t a l hexagonal g e l d l a y e r superimposed cn the u n d e r l y i n g (10C) s u b s t r a t e , c ) , ( a f t e r P a l i b e r g and B h c d i n F33 1). u n d e r l y i n g s u b s t r a t e i n t e n s i t i e s Of LEEE Beams The p o s i t i o n s o f the d i f f r a c t i o n s p o t s i n the H I D p a t t e r n y i e l d the s i z e and symmetry o f the s u r f a c e u n i t mesh but no f u r t h e r i n f o r m a t i o n cn the l o c a t i o n s o f the a t c i s i n the u n i t mesh. T h i s i n f o r m a t i o n i s c o n t a i n e d i n t h e i n t e n s i t i e s c f the LEED u e a i s I n t e n s i t i e s a r e a o s t u s u a l l y measured a t c o n s t a n t p o l a r , and a z i m u t h a l a n g l e , ^b, see F i g . 2.4, as a f u n c t i o n o f energy. F l c t s o f i n t e n s i t y a q a i n s t enezqy, 1(E) c u r v e s o r " i n t e n s i t y p r o f i l e s " , a r e the u s u a l form o f data p r e s e n t a t i o n a l t h o u q h " r o t a t i o n d i a g r a m s " , I fc^) f o r co n s t ^- and I, and " r o c x i n q c u r v e s " , I f o r c o n s t a n t <f? and E, remain as n e q l e c t e d 30 a l t e r n a t i v e s . F i q . 2.10 shows t y p i c a l 1(E) c u r v e s f o r the s p e c u l a r beams d i f f r a c t e d from Cu(100) and Ni(100) s u r f a c e s f o r 8 = 30 [ 3 4 ] . Such c u r v e s show c o n s i d e r a b l e s t r u c t u r e e x h i b i t i n g a number of maxima and minima as t h e energy i s v a r i e d . A l s o , as noted i n S e c t i o n 2 . 1 ( a ) , e l e c t r o n r e f l e c t i v i t i e s a r e low f o r e l a s t i c s c a t t e r i n g . We can attempt t o use the k i n e m a t i c a l ( o r e i n q l e -s c a t t e r i n g ) model, v a l i d f o r low s c a t t e r i n g c r o s s - s e c t i o n s , t o understand such I ( E ) c u r v e s . T h i s model has been used w i t h s u c c e s s t o i n t e r p r e t X-ray d i f f r a c t i o n i n t e n s i t i e s . However, low energy e l e c t r o n s c a t t e r i n g c r o s s - s e c t i o n s a r e much l a r q e r , by as much as 1 0 6 , and so we cannot e x p e c t t h i s t h e o r y t o be much more than a rough guide f o r LEED i n t e n s i t i e s . 2.3(a) k i n e m a t i c a l t h e o r y I n the k i n e m a t i c a l t h e o r y o n l y the i n c i d e n t wave i s c o n s i d e r e d t o produce s c a t t e r e d waves w h i l s t double and m u l t i p l e - s c a t t e r i n g a r e i g n o r e d . For an i n c i d e n t p l a n e wave the k i n e m a t i c a l e x p r e s s i o n f o r the t o t a l wave ( c o r r e s p o n d i n g to wave-vector k f) s c a t t e r e d t o the p o i n t of o b s e r v a t i o n (denoted by v e c t o r r) by an assembly o f s c a t t e r e r s i s (2. 15) exp[ i k i r - r - 1 ]/i£-r- | exp (i£.r: ) (2. 16) 31 I l S s I s JLt l f i c u r v e s f o r the s p e c u l a r beaa f r o r Mi(10C) and C u ( 1 0 0 ) , ©=3°. The t a r s denote k i n e a a t i c a l Braqq c o n d i t i o n s ( a f t e r Anderson and Kaseao T3U1). 3 2 where the summation i s oyer the s e t of s p h e r i c a l waves produced a t the s c a t t e r i n g c e n t r e s r - . The f a c t o r f ' i s the atomic r ° s c a t t e r i n g f a c t o r f o r the atom j f o r e l a s t i c s c a t t e r i n g o f e l e c t r o n s of energy E through the ang l e © r (as d e f i n e d by k_=>k^ ) . The f i n a l term denotes the phase s h i f t s f o r the i n c i d e n t waves a t the s c a t t e r i n g atoms. A 0 i s a c o n s t a n t whose v a l u e i s f i x e d by t h e d e n s i t y of atoms i n t h e i n c i d e n t beam. I f the s c a t t e r e d e l e c t r o n s a re d e t e c t e d a t a l o n g d i s t a n c e from the s c a t t e r i n g r e g i o n and i f t h i s r e q i o n i s s m a l l , so t h a t i r - r • | can be approximated by r f o r t h e purpose o f a s s e s s i n g the e f f e c t of d i s t a n c e on s c a t t e r e d a m p l i t u d e , then e g u a t i o n (2. 16) can be r e - e x p r e s s e d as element dT; t h e r e f o r e the nunsber of e l e c t r o n s r e c e i v e d i n u n i t t i n e by a d e t e c t o r of s o l i d a n g l e djv. i s where u i s t h e v e l o c i t y of the e l e c t r o n s and B i s t h e s c a t t e r i n g a m p l i t u d e I n g e n e r a l dT g i v e s the p r o b a b i l i t y t h a t an e l e c t r o n r e p r e s e n t e d by a w a v e - f u n c t i o n 4* i s to be found i n a volume dn = u A* JB* J a j l (2.18) B = X f ^ E ' e s } e x p ( i s . r j ) (2.19) 3 3 f o r the c o l l e c t i o n of atomic s c a t t e r e r s and S = k -k {2.20) i s the s c a t t e r i n g v e c t o r . The number o f e l e c t r o n s c r o s s i n g u n i t a r e a p e r p e n d i c u l a r t o the d i r e c t i o n of k per u n i t time i s I c = u A 2, (2.21) and, on i n t r o d u c i n g the r e l a t i o n dn = I0 ier (2.22) such t h a t d<r- i s the v a l u e of dn when I D i s u n i t y , we o b t a i n the d i f f e r e n t i a l s c a t t e r i n g c r o s s - s e c t i o n d<r/djv = i (j) = fB | 2 (2.23) where I(S) c o r r e s p o n d s t o the f l u x , f o r s c a t t e r i n g v e c t o r S, s c a t t e r e d per u n i t s o l i d a ngle from an i n c i d e n t beam of u n i t f l u x per u n i t a r e a . The e q u a t i o n s (2.23) and (2.19) r e p r e s e n t the g e n e r a l b a s i c e q u a t i o n s f o r k i n e m a t i c a l s c a t t e r i n g of a p l a n e wave t y any assembly of s c a t t e r e r s whose dimensions are s m a l l compared wit h the d i s t a n c e o f o b s e r v a t i o n of the s c a t t e r e d e l e c t r o n frcm the s c a t t e r i n g r e g i o n . One of the i n t e r e s t i n g conseguences o f the e l a s t i c s c a t t e r i n g b e i n g c o n f i n e d by 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 to the 34 v i c i n i t y of the s u r f a c e i s t h a t the f u l l t h r e e - d i m e n s i o n a l p e r i o d i c i t y of t h e c r y s t a l i s not e x p e r i e n c e d by a s c a t t e r e d e l e c t r o n . We can proceed v i a two l i m i t i n g c a s e s , dependinq upon the s t r e n g t h o f the i n e l a s t i c s c a t t e r i n g . The pure two-d i n e n s i o n a l l i m i t c o r r e s p o n d s t o very s t r o n q i n e l a s t i c s c a t t e r i n g , and t h e pure t h r e e - d i m e n s i o n a l t o very weak i n e l a s t i c s c a t t e r i n g . The former i s r a t h e r e a s i e r to f o l l o w ; we c o n s i d e r a two-d i m e n s i o n a l n e t d e f i n e d by mesh v e c t o r s s ( and such t h a t n e t p o i n t s are s i t u a t e d at p o i n t s r r : = m s | + n s ^ (2.24) i n t h i s case t h e a t o m i c s c a t t e r i n g f a c t o r s f ^ c a n be r e p l a c e d by the s c a t t e r i n g f a c t o r of the u n i t mesh, F ( s t r u c t u r e f a c t o r ) , which depends on t h e f ^ and the atomic p o s i t i o n s w i t h i n t h e u n i t mesh. The s c a t t e r e d a m p l i t u d e becomes B = F ( k ^ k 5 ) 2 e x P ( i - S - t f ) (2.25) i and t h e d i f f e r e n t i a l c r o s s - s e c t i o n i s dff?d/i-= I F ( k - ^ k ) J2 5"exp {iS(£.-r ;) 1 = i f a - ^ J J 5 ) l 2 G(S) (2.26) The i n t e r f e r e n c e f u n c t i o n G (J3) has maxima when the s c a t t e r i n g v e c t o r S s a t i f i e s the Laue c o n d i t i o n s 35 S. s ( = 2 i r h and S . s z = 2 IT k (2-27) f o r i n t e g r a l h and k- These c o n d i t i o n s can be r e w r i t t e n as S = <i(hk) (2-28) where the cj{hk) a r e d e f i n e d by e q u a t i o n ( 2 . 8 ) ; t i e Laue" c o n d i t i o n s a re e q u i v a l e n t t o the c o n d i t i o n e x p r e s s e d e a r l i e r i n e q u a t i o n (2.7) as JS], " JS„ = S i n * ) (2.29) That i s , d i f f r a c t e d waves occur o n l y i n c e r t a i n d i r e c t i o n s determined by e q u a t i o n (2.29) t o produce t h e LEED p a t t e r n d e s c r i b e d i n S e c t i o n 2.2. The i n t e r f e r e n c e f u n c t i o n does not t e l l us a n y t h i n q about the energy dependence of the i n t e n s i t y o f a s p o t . The s t r u c t u r e f a c t o r i s however energy-dependent through the energy dependence of the a t o m i c s c a t t e r i n g f a c t o r s f • • The form o f t h i s v a r i a t i o n and hence the i n t e n s i t y v a r i a t i o n of a d i f f r a c t i o n s p o t with energy, an 1(E) c u r v e , i n the t w o - d i m e n s i o n a l k i n e m a t i c a l a p p r o x i m a t i o n i s shown i n F i g . 2.11(a) t o be monotonic. I n the pure t h r e e - d i m e n s i o n a l k i n e m a t i c l i m i t t h e e f f e c t of the s u r f a c e can be i g n o r e d and the e l e c t r o n w i l l move i n p r e d o m i n a n t l y t h e f u l l t h r e e - d i m e n s i o n a l p o t e n t i a l of the b u l k c r y s t a l . Shereas i n the t w o - d i m e n s i o n a l l i m i t t h e p e r p e n d i c u l a r component of t h e wave-vector k , was f r e e t o assume a continuum 36 1(E) 1(E) E(eV) E ( eV ) a) b) l i s u r e 2^.11 I n t e n s i t y of a d i f f r a c t e d LEED t e a s as a f u n c t i o n of energy i n a) the pure t w c - d i n e n s i o n a l l i m i t ar.d t) the pure t h r e e - d i m e n s i o n a l l i m i t ( a f t e r S c E c r j a i a r d F a r r e l l [7"]). c f v a l u e s ( t h a t i s o n l y k i s a qcod quantum nuafcer), i t , the three-din>ensiona 1 l i s i i t i t i s c c n s t r a i r e d t o o t l y c e r t a i n v a l u e s due to the p e r i o d i c i t y i n the p e r p e n d i c u l a r d i r e c t i o n where a, i s a r e c i p r o c a l l a t t i c e v e c t o r p e r p e n d i c u l a r t c the d i f f r a c t e d i n t e n s i t y c f a beam w i l l be z e r o except where e q u a t i o n s (2.29) and (2.30) are s i n u l t a n e c u s l y s a t i s f i e d , ^ r e d u c i n g a v a r i a t i o n of the f o r * shown i n F i q . 2.11(b) . These twc e q u a t i o n s can be c o i t i n e d i r t o (2.30) s u r f a c e . Hence i n the pure t h r e e - d i i e n s i o n a l l i s i t the k» = k • fl (hkl) (2.31) 37 where j (hkl) i s a r e c i p r o c a l l a t t i c e v e c t o r of t h e hulk t h r e e -d i m e n s i o n a l l a t t i c e . 2.3(h) c h a r a c t e r i s t i c s of 1(E) c u r v e s Re-examining F i g . 2.10 we f i n d t h a t r e a l 1(E) c u r v e s show c h a r a c t e r i s t i c s i n t e r m e d i a t e between the two extremes o f the pure t w o - d i m e n s i o n a l and t h e pure t h r e e - d i o e n s i o n a l k i n e m a t i c a l l i n i t s . The c u r v e s a r e not monotonic and do e x h i b i t maxima and Eiinima; however, t h e r e a r e more peaks than a r e p r e d i c t e d by e q u a t i o n ( 2 - 3 1 ) , a t l e a s t a t lower e n e r g i e s . The b a r s on F i g . 2.10 mark the p o s i t i o n s of " p r i m a r y Bragg peaks" where e q u a t i o n (2.31) i s s a t i s f i e d f o r v a r i o u s v a l u e s of 1. In g e n e r a l , maxima f a l l c l o s e to t h e s e v a l u e s but s u b s i d i a r y naxiaia can be seen. Such e x t r a s t r u c t u r e must be a s s o c i a t e d w i t h the n u l t i p l e - s c a t t e r i n q e v e n t s n e g l e c t e d i n the s i m p l e k i n e m a t i c a l t r e a t m e n t . Three energy r e g i o n s a r e o f t e n i d e n t i f i e d i n 1(E) c u r v e s a c c o r d i n g t o t h e magnitude of beam i n t e n s i t i e s , degree of s t r u c t u r e and peak widths [ 3 5 , 1 2 ] , These a r e , r e f e r r i n g to f i g . 2. 10: ( i ) the low-enerqy r e g i o n , t y p i c a l l y below 20eV, where peaks are numerous and narrow w i t h w i d t h s o f 1-2eV and h i g h d i f f r a c t e d i n t e n s i t i e s (sometimes >10% of the i n c i d e n t beam i n t e n s i t y ) ; ( i i ) t h e i n t e r m e d i a t e - e n e r g y range (20-150eV) has wider (up to 10eV) peaks of lower i n t e n s i t y . S t r u c t u r e i s s i m p l e r though t h e r e are s t i l l more maxima than p r e d i c t e d by e q u a t i o n (2.31); ( i i i ) the h i q h - e n e r g y r e q i o n , u s u a l l y above 150eV, has wider (20-30eV) peaks of s t i l l l o w er i n t e n s i t y i n p o s i t i o n s t h a t are b r o a d l y c o n s i s t e n t w i t h e q u a t i o n (2.31) . The k i n e m a t i c a l t h e o r y , i n the form p r e s e n t e d e a r l i e r , does not p r o v i d e i n f o r m a t i o n on peak wid t h s . I n o r d e r t o account f o r peak w i d t h s , r e m o v a l of e l e c t r o n f l u x by i n e l a s t i c s c a t t e r i n q must be i n t r o d u c e d i n t o the model (see Ch. 3) . I t i s p r e c i s e l y the h i q h i n e l a s t i c s c a t t e r i n q c r o s s - s e c t i o n s f o r low-enerqy e l e c t r o n s t h a t make them s u r f a c e s e n s i t i v e . I n g e n e r a l , d i f f r a c t e d team i n t e n s i t i e s d e c r ease w i t h i n c r e a s i n q t e m p e r a t u r e f 36 ] , o f t e n i n an e x p o n e n t i a l manner. C o n s e q u e n t l y , a t t e m p t s have been made, w i t h i n t h e k i n e m a t i c a l t h e o r y , t o r e l a t e such dependencies t o Debye-Waller f a c t o r s i n o r d e r t o e x t r a c t mean-square v i b r a t i o n a l a m p l i t u d e s of s u r f a c e atoms i . 7 ] . However, the d e r i v e d Debye t e m p e r a t u r e s show some v a r i a t i o n (e. q. ±20X) from beam t o beam and w i t h t h e enerqy and d i r e c t i o n of the i n c i d e n t beam. G e n e r a l l y , the k i n e m a t i c a l t h e o r y , even when m o d i f i e d to i n c l u d e i n e l a s t i c s c a t t e r i n q and temperature e f f e c t s , and even double d i f f r a c t i o n e v e n t s £37], i s not a b l e t o r e p r o d u c e e x p e r i m e n t a l I (S) c u r v e s s u c c e s s f u l l y . An e x c e p t i o n t c t h i s r u l e i s the (111) s u r f a c e o f xenon, f o r which 1(E) c u r v e s are e s s e n t i a l l y k i n e m a t i c a l over most of the LEED enerqy ranqe ) 3 8 ] . The much more s u c c e s s f u l m u l t i p l e - s c a t t e r i n q t h e o r i e s are d i s c u s s e d i n C h a p t e r 3. 39 2. .4 Auqjer E l e c t r o n S p e c t r o s c o p y (AES) Seme of the e l e c t r o n s o b s e r v e d as s n a i l f e a t u r e s on a s l o w l y v a r y i n g background i n t h e medium-energy ranqe o f the secondary e l e c t r o n d i s t r i b u t i o n ( S e c t i o n 2. 1) a r e c a l l e d Auger e l e c t r o n s , a f t e r P i e r r e Auqer who f i r s t saw t h e i r t r a c k s i n a H i l s o n c l o u d chamber i n 1925 and c o r r e c t l y e x p l a i n e d t h e i r o r i g i n £39]. These e l e c t r o n s are p a r t i c u l a r l y u s e f u l i n LEED ex p e r i m e n t s a s they a l l o w the atomic c o m p o s i t i o n of the s u r f a c e to be m o n i t o r e d under t h e sane c o n d i t i o n s as which LEED ex p e r i m e n t s a r e c a r r i e d out. S e v e r a l r e v i e w s of such a p p l i c a t i o n s have been p u b l i s h e d e.g. £40-44,10]. The Auger e f f e c t i s a two-stage r a d i a t i o n l e s s t r a n s f e r of energy t o an e l e c t r o n f o l l o w i n g i o n i s a t i o n o f a c o r e e l e c t r o n . F i g . 2.12 shows a s c h e m a t i c r e p r e s e n t a t i o n of the p r o c e s s f o r a s o l i d . I n i t i a l l y , F i g . 2 . 1 2 ( a ) , a c o r e e l e c t r o n i s i o n i s e d by the p r i m a r y e l e c t o n beam, of u s u a l l y 1-10keV. For e l e m e n t s e t h e r than hydrogen, h e l i u m , l i t h i u m atoms o r the L i + i o n i n s o l i d s , an e l e c t r o n from a h i g h e r energy l e v e l i s a b l e t o drop down to t h e i n n e r vacancy, F i g . 2.12(b). The s u r p l u s e n e r g y i s r e l e a s e d e i t h e r as an X-ray photon (X-ray f l u o r e s c e n c e ) , or i s a v a i l a b l e t o a t h i r d e l e c t r o n t h a t i s e l e c t e d as an Auger e l e c t r o n , F i g . 2 . 1 2 ( c ) . I f the o r i g i n a l i o n i s a t i o n i s from a l e v e l w i t h a b i n d i n g energy of l e s s than about 2keV then Auger e m i s s i o n i s more p r o b a b l e than X-ray p r o d u c t i o n [ 4 5 ] . Auger e l e c t r o n s appear a s s m a l l peaks i n t h e medium-energy range of the secondary e l e c t r o n d i s t r i b u t i o n and a r e r e a d i l y d i s t i n g u i s h e d from e n e r g y - l o s s peaks as t h e i r e n e r g i e s are lA-SUIf 2x12 Schematic r e p r e s e n t a t i o n c f t h e VV Auqer t r a n s i t i o n : (a) i o n i s a t i o n of a c o r e l e v e l , <b) f i l l i n q c f the c o r e h o l e , (c) e m i s s i o n of the l u q e r e l e c t r o n . 41 independent o f t h e primary team enerqy, whereas l o s s peaks, due to e.q. plasmon e x c i t a t i o n , s h i f t w i t h chanqes i n t h e i n c i d e n t ream enerqy. Auqer e l e c t r o n s are l a b e l l e d by the n o t a t i o n ABC, where these symbols i n d i c a t e r e s p e c t i v e l y the i n i t i a l h o l e , the f i n a l h o l e , and the l e v e l from which the e m i t t e d e l e c t r o n l e a v e s - For s o l i d s , t h e v a l e n c e band i s denoted by the l e t t e r V, but the i n n e r l e v e l s , b e i n q q u a s i - a t o m i c i n n a t u r e , a r e a s s i q n e d u s i n q the c o n v e n t i o n a l X-ray n o t a t i o n K,L,H... a c c o r d i n q to the p r i n c i p a l quantum number- S u b s c r i p t s a r e used to denote the quantum numoers 1 and - j ; t h u s 1 f o r s s t a t e s (1=0,1=1/2), 2 or 3 f o r p s t a t e s (1=1,1=1/2,3/2) e t c - With t h i s n o t a t i o n the Auqer e l e c t r o n o f F i q . 2.12 would he d e s i g n a t e d as an E ^ V V e l e c t r o n . The k i n e t i c energy of t h a t e l e c t r o n i s a p p r o x i m a t e l y * ^ V V E 3 ( 2 - 3 2 ) The energy l e v e l s can be r e a d i l y e v a l u a t e d u s i n g X-ray j. 46.] or ESCA [ 4 7 ] energy l e v e l e n e r g i e s , but t h i s s i m p l e f o r n u l a i s inadequate i n t h a t i t f a i l s t o t a k e i n t o a c c o u n t the d i f f e r e n t degrees of i o n i s a t i o n and the e x t r a enerqy needed t o remove the second e l e c t r o n from an a l r e a d y i o n i s e d atom, The f o r m u l a has been m o d i f i e d by J e n k i n s and Chung [ 4 8 ] E (Z) = E ( ( Z ) - 1 / 2 [ E i ( Z ) + E ^(Z + 1) ]- 1 / 2 [ E 3 ( Z ) •E ; ?(Z+1) ] (2.33) where Z i s t h e a t o m i c number c f the c h e m i c a l s p e c i e s i n v o l v e d . For atoms t h i s f o r m u l a has a t y p i c a l a c c u r a c y o f about 5eV. 42 The d i s c r e t e l e v e l s i n atcms a r e r e p l a c e d by energy oands i n s o l i d s . The q u a s i - a t o m i c i n n e r l e v e l s i n s o l i d s nay be s h i f t e d r e l a t i v e t o t h e f r e e atoms and the d e n s i t y of s t a t e s i n the v a l e n c e band i s s e n s i t i v e t o c h e m i c a l e n v i r o n m e n t . While the complete assiqnment of Auqer t r a n s i t i o n s c a n be d i f f i c u l t , t h e use of AES f o r q u a l i t a t i v e a n a l y s i s depends o n l y on the a b i l i t y t o a s s i q n peaks to a p a r t i c u l a r e l e ment. T h i s type of assiqnment i s g e n e r a l l y unambiguous and the i d e n t i t y of s u r f a c e atcms can be r e a d i l y found u s i n q the t a b u l a t e d Auqer e n e r g i e s 1.4SJ and r e p r e s e n t a t i v e s p e c t r a [ 5 0 ] now a v a i l a b l e . U n f o r t u n a t e l y , q u a n t i t a t i v e assessments a r e not so e a s i l y aade. The i n t e n s i t y o f Auger e m i s s i o n i s governed not o n l y by the e l e m e n t a l c r o s s - s e c t i o n s but a l s o by the d i s t r i b u t i o n of the element on the s u r f a c e and i n t o t h e c r y s t a l . Q u a n t i f i c a t i o n t h e r e f o r e r e g u i r e s independent c a l i b r a t i o n e x p e r i m e n t s , u s u a l l y employing known g u a n t i t i e s o f d e p o s i t e d m a t e r i a l and assumptions of uniform d i s t r i b u t i o n J. 51-52 J- Auger s p e c t r a c o l l e c t e d w i t h a r e t a r d i n g f i e l d a n a l y s e r based on LEED o p t i c s (see Ch. 4) a l l o w i n p u r i t i e s to be d e t e c t e d t o a p p r o x i m a t e l y 1% o f a monolayer f o r most el e m e n t s . A t y p i c a l example of an Auqer spectrum from the p r e s e n t work i s shown i n F i g . 2. 13. T h i s shows an Auger spectrum of a 8h (110) s u r f a c e t h a t i s h e a v i l y contaminated w i t h s u l p h u r , and to a l e s s e r e x t e n t , c a r b o n and phosphorus. The spectrum i s p r e s e n t e d i n t h e second d e r i v a t i v e form dN(E)/dE t o enhance tfce o t h e r w i s e weak Auger f e a t u r e s and was t a k e n w i t h an i n c i d e n t beam c u r r e n t of about 10 microamps a t 1.5keV. The Auger 4 3 Rh(110) 1 1 1 1 1— 100 200 300 Energy (eV) ZiSSLs. 2i.li Auger spectrum o f a h e a v i l y c o n t a m i n a t e d FM11 s u r f a c e . Ep=1.5KeV, Ip=10 fficrcamps. 44 t r a n s i t i o n energy i s t r a d i t i o n a l l y t aken as the maximum n e q a t i v e e x c u r s i o n of t h e d e r i v a t i v e peak 150]-CHAPTER 3 MULTIPLE-SCATTERING CALCULATIONS lib J . J Parameters For T h e o r i e s Of LEED In t h e l a s t c h a p t e r we saw t h a t i n t e n s i t i e s o f LEED beams depend upon c h a r a c t e r i s t i c s of t h e i n c i d e n t beam, such as enerqy and d i r e c t i o n , and a l s o on the geometry, s c a t t e r i n q power, and dynamics o f the s u r f a c e atoms. Moreover, we a l s o saw t h a t the s i n p l e k i n e m a t i c a l t h e o r y d i d not p r e d i c t 1(E) c u r v e s w i t h any r e a l measure of s u c c e s s and t h a t any t h e o r y t h a t i s t o be used to determine s u r f a c e q e o m e t r i e s , which i s the p r i m a r y o b j e c t o f the e x p e r i m e n t s , must t h e r e f o r e i n c l u d e m u l t i p l e - s c a t t e r i n q of the slow e l e c t r o n s . Such t h e o r i e s a r e c a l l e d " d y n a m i c a l " t h e o r i e s . I n t h i s s e c t i o n , the more i m p o r t a n t f e a t u r e s that-must be i n c l u d e d i n such t h e o r i e s a r e c o n s i d e r e d b r i e f l y ; f u l l d e t a i l s are g i v e n by Pendry A.12J. 3. 1 fa) t h e s c a t t e r i n g p o t e n t i a l I n a t y p i c a l LEED ex p e r i m e n t , o n l y about 1% of the i n c i d e n t e l e c t r o n s are e l a s t i c a l l y b a c k s c a t t e r e d , whereas the remainder ar e removed from the e l a s t i c f l u x by v a r i o u s i n e l a s t i c s c a t t e r i n g p r o c e s s e s , p a r t i c u l a r l y plasmon e x c i t a t i o n 113 1. While such i n e l a s t i c s c a t t e r i n g i n v o l v e s the v a l e n c e e l e c t r o n s of the s o l i d , the e l a s t i c b a c k s c a t t e r i n g a r i s e s from i n t e r a c t i o n s w i t h r e g i o n s of high p o t e n t i a l c l o s e to the n u c l e u s ; such r e g i o n s i n v o l v e the t i g h t l y bound c o r e - s t a t e e l e c t r o n s or " i o n - c o r e s " . The " m u f f i n - t i n " a p p r o x i m a t i o n i s a c o n v e n i e n t model o f the s o l i d t h a t accommodates these two i m p o r t a n t f e a t u r e s o f t h e s c a t t e r i n g p r o c e s s . As i n d i c a t e d i n F i q . 3. 1 the s o l i d i s modelled as n o n - o v e r l a p p i n q r e q i o n s of 47 s p h e r i c a l l y symmetric p o t e n t i a l c e n t r e d on each atom c f the s o l i d w i t h a c o n s t a n t p o t e n t i a l i n the i n t e r s p h e r e r e g i o n . The i n t e r s p h e r e r e q i o n i s m a i n l y o c c u p i e d by t h e v a l e n c e e l e c t r o n s of the s o l i d and t h e r e f o r e i s t h e p r i n c i p a l s i t e f o r i n e l a s t i c s c a t t e r i n g . I n e l a s t i c s c a t t e r i n q of a l l k i n d s can be t r e a t e d i n terms of a l i f e t i m e , T , d e f i n e d as t h e average time an i n c i d e n t e l e c t r o n spends i n the c r y s t a l b e f o r e i t l o s e s energy by i n e l a s t i c s c a t t e r i n g . F o l l o w i n g Pendry [ 1 2 ] , we can d e f i n e t h e t e m p o r a l v a r i a t i o n of w a v e f u n c t i o n a m p l i t u d e f o r an e l e c t r o n w i t h energy E, as e x p ( - i E t ) i n vacuo ( f o r a t o m i c u n i t s h/2ir=1). In t h e r e g i o n of c o n s t a n t p o t e n t i a l VQ, t h i s becomes e x p [ - i (E—V e) t j . I f we a l l o w V Q an i m a g i n a r y ccmponent v o = Kc- i V <3.1) then the i n t e n s i t y of the w a v e f u n c t i o n decays w i t h time as exp (+2V ojt) , as V6^ i s c o n v e n t i o n a l l y taken t o be n e g a t i v e , and V = - V 2 T * 3 - 2 > Hence a t t e n u 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 teams due to i n e l a s t i c s c a t t e r i n q can be s i m u l a t e d by a d d i n g an i m a g i n a r y c capcnent to the p o t e n t i a l , i v ^ , which can be e s t i m a t e d , v i a the time-energy u n c e r t a i n t y p r i n c i p l e , from peak w i d t h s i n 1(E) c u r v e s . The peak width s a t i s f i e s (3.3) tb) Position Ii3U£§ 3.^ .1 M u f f i n - t i n p o t e n t i a l (a) i n c r c s s - s e c t i c n as c c r t c u r s , (b) a l o n q XX'. Vc i s t h e c o n s t a n t i n t e i s p h e r e p o t e n t ia1. Energy Energy ^ Vacuum level ^ Ec+4> E p Fermi energy o Lowest level of conduction band l i f l u i g 3±2 I l l u s t r a t i o n of the r e l a t i o n s h i p between e n e r q i e s •easured w i t h r e s p e c t t c the vacuum l e v e l and those measured w i t h r e s p e c t t o t h e l o w e s t l e v e l c f the c o n d u c t i o n band. 49 V ^ i s t y p i c a l l y about -5eV and weakly energy-dependent f i l l . I n t h i s work, V Q^ was e s t i m a t e d from a primary Braqq type peak i n an 1{E) c u r v e a c c o r d i n g t o e q u a t i o n (3.3) and was a l l o w e d an energy-dependence of the form V = -if<E> (3.4) where JJ(E)=©cE*'3 (3.5) a c c o r d i n g t o t h e e m p i r i c a l r e l a t i o n s h i p e s t a b l i s h e d by Demuth e t a l i 1 1 J . For copper <?C was s e t to 0.89 and to 1.12 f o r rhodium. The r e a l p a r t of t h e i n t e r s p h e r e p o t e n t i a l , V o r i n e q u a t i o n ( 3 . 1 ) , a c c o u n t s f o r the i n c r e a s e i n energy e x p e r i e n c e d by an e l e c t r o n e n t e r i n g t h e c r y s t a l . T h i s i s u s u a l l y c a l l e d the i n n e r p o t e n t i a l . T y p i c a l v a l u e s f o r t h i s q u a n t i t y are -10 t o -20 eV and a r e , i n p r i n c i p l e a t l e a s t , energy-dependent. T h i s dependence appears, from e x p e r i e n c e , t o be s l i g h t , and i s u s u a l l y i g n o r e d . O f t e n ¥ o r i s approximated as t h e sum of the Fermi energy E^ - and t h e work f u n c t i o n ^ , a s shown i n F i g - 3.2. In p r a c t i c e , V t f r i s u s u a l l y e s t i m a t e d a p r i o r i and then e m p i r i c a l l y r e f i n e d by v i s u a l o r n u m e r i c a l comparison of 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 ; c h a n g i n g Vof> amounts, to a good a p p r o x i m a t i o n , to a r i g i d s h i f t i n the energy s c a l e . T h i s t o p i c i s e x p l o r e d i n more d e t a i l i n C h a p t e r 5. I n t h i s work V o r was s e t i n i t i a l l y a t -9.5eV f o r copper [ 65 ] and -12. OeV f o r rhodium £68,69j. The p a r t of t h e p o t e n t i a l t h a t g i v e s the b a c k s c a t t e r i n g i n LEED i s the s p h e r i c a l l y - s y m m e t r i c p o t e n t i a l w i t h i n the E u f f i n -t i n s ; the i o n - c o r e p o t e n t i a l . V a r i o u s methods a r e a v a i l a b l e t o 50 c o n s t r u c t such p o t e n t i a l s . In g e n e r a l , LEED c a l c u l a t i o n s have used e i t h e r f u l l s e l f - c o n s i s t e n t band s t r u c t u r e p o t e n t i a l s o r , more s i m p l y , the p o t e n t i a l g e n e rated by a l i n e a r s u p e r p o s i t i o n of atomic charge d i s t r i b u t i o n s , as o r i g i n a l l y suggested by H a t t b e i s [ 5 3 ] . R e c e n t l y , a s t u d y o f d i f f e r e n t p r e s c r i p t i o n s f o r i o n - c o r e p o t e n t i a l s £54] has suggested t h a t the s i m p l e m u f f i n -t i n l i n e a r s u p e r p o s i t i o n , w i t h the R a p p r o x i m a t i o n of S l a t e r f o r the exchange p o t e n t i a l £55], p r o v i d e s a s u i t a b l e p o t e n t i a l f o r LEED c a l c u l a t i o n s , and t h a t t h e f u l l s e l f - c o n s i s t e n t band s t r u c t u r e c a l c u l a t i o n s do not l e a d t o s i g n i f i c a n t improvements i n t h e degree o f agreement of c a l c u l a t i o n s w i t h e x p e r i m e n t ( i n t h a t i n s t a n c e ) . In the i n i t i a l s t a g e s of t h i s work a band s t r u c t u r e p o t e n t i a l f o r rhodium was not a v a i l a n l e . Hence i n t e r e s t c e n t r e d on g e n e r a t i n g i o n - c o r e p o t e n t i a l s by the s u p e r p o s i t i o n of charge d e n s i t i e s method. I n c o - o p e r a t i o n w i t h Dr. L. Noodleraan, a method was used t h a t employed some o f the i n i t i a l r o u t i n e s of the sea t t e r e d - wave X^k programs developed by Johnson £56.]. The method i s as f o l l o w s : ( i ) u s i n g the t a b l e s of Herman and S k i l l m a n £57], the charge d i s t r i b u t i o n f o r an i s o l a t e d atom was c a l c u l a t e d ; ( i i ) u s i n g the MOLPOT and ENERGY r o u t i n e s o f t h e s c a t t e r e d - w a v e Xet programs, the a t o m i c charge d e n s i t i e s of a c u b o - o c t a h e d r a l M^ c l u s t e r ( c o r r e s p o n d i n g t o an FCC c r y s t a l ) were superimposed. F i g . 3.3. The charge d i s t r i b u t i o n o f t h e c e n t r a l atom was used t o g e n e r a t e an e l e c t r o s t a t i c p o t e n t i a l 51 i n c l u d i n g an exchanqe term u s i n q S l a t e r ' s a p p r o x i m a t i o n [ 5 5 ] . V € y{r) = -6«£ I <3p (r)/3ir]»/3 (3.6) Here p (r) i s the c h a r ge d e n s i t y , w i t h << v a l u e s t a k e n frcm the t a b u l a t i o n s of Schwarz £58,59]. fh e v a l i d i t y of t h i s method was checked by comparinq the s c a t t e r i n g f a c t o r s . S e c t i o n 3.1 ( b ) , and I (E) c u r v e s , S e c t i o n 3.2, produced by a Cu | 3 c l u s t e r s u p e r p o s i t i o n p o t e n t i a l , V- , w i t h those from the band s t r u c t u r e p o t e n t i a l (?c of Burdick-Chodorow J.60,61], . T h i s p o t e n t i a l had p r e v i o u s l y been used i n a s u c c e s s f u l LEED a n a l y s i s c f the Cu(1G0) s u r f a c e £65 ]. A rhodium s u p e r p o s i t i o n p o t e n t i a l , , was a l s o q e n e r a t e d ; th e d a t a used f o r the two p o t e n t i a l s i s q i v e n i n T able 3.1. At a l a t e r staqe a band s t r u c t u r e p o t e n t i a l became a v a i l a b l e f o r rhodium £64 ]. A t o m i c phase s h i f t s c a l c u l a t e d f o r a c l u s t e r of 13 s i l v e r atoms were a l s o shown to c o r r e s p o n d c l o s e l y t o those used by Marcus e t a l £ 104]. 3. 1 (fc) phase s h i f t s LIED i n t e n s i t y c a l c u l a t i o n s can be c o n v e n i e n t l y broken down i n t o two p a r t s ( i ) the d e t e r m i n a t i o n of a s i n g l e i o n - c o r e p o t e n t i a l , and ( i i ) t h e s o l u t i o n of the b e h a v i o u r of an i n c i d e n t e l e c t r o n m u l t i p l y s c a t t e r e d by a l a t t i c e made up of t h e s e s i n g l e i o n - c o r e p o t e n t i a l s . The f i r s t s t e p i n p e r f o r m i n g p a r t ( i i ) i s to e x p r e s s th e e l a s t i c s c a t t e r i n g o f an i n c i d e n t e l e c t r o n by a J i l The c u b o - c c t a b e d r a l «,3 c l u s t e r used t c m c d i l th< i o n - c o r e p o t e n t i a l f o r PCC c r y s t a l s . Cu Eh de 3.615C 3.8C31 A r , 1.2780 1.3449 1 0.70697 0.7C214 J a b i s 3 ^ Data used f o r c c c s t r u c t i c n c f s u p e r p o s i t i o n p o t e n t i a l s f o r B,3 c l u s t e r s . The c r y s t a l cube s i d e a, and • u f f i n - t i n r a d i u s r m are f r c a r 6 2 , 6 3 ] and the v a l u e s f r o i r [58,59 ]. 53 s i n g l e i o n - c o r e p o t e n t i a l . Assuming t h a t the c o r e - e l e c t r o n s a r e not p o l a r i s e d by t h e i n c i d e n t e l e c t r o n , the w a v e f u n c t i o n of an i n c i d e n t 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 o b t a i n 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 £ - 1 / 2 9 2 + ]4>= E4/ <3.7) where .? i s the i o n - c o r e p o t e n t i a l . . For an i n c i d e n t p l a n e wave, s c a t t e r e d waves c o r r e s p o n d t o s o l u t i o n s of e g u a t i o n ( 3 . 7 ) . The i n t e n s i t y s c a t t e r e d t o a p a r t i c u l a r p o i n t depends o n l y on t h e modulus o f t h e wave v e c t o r , k, t h e s c a t t e r i n g a n g l e Qf, and t h e d i s t a n c e r o f t h e p o i n t of o b s e r v a t i o n from the s c a t t e r i n g r e g i o n , as d e p i c t e d s c h e m a t i c a l l y i n F i g . 3.4. At l a r g e r the s o l u t i o n s to e g u a t i o n (3-7) have the form .£12,66,67 J ; + (r ,9 C) oCexp ( i k r c o s B * ) + f u (e x)exp ( i k r / r ) (3.8) where t h e f i r s t term i s t h e i n c i d e n t wave a t the p o i n t o f o b s e r v a t i o n and t h e second i s an o u t g o i n g s c a t t e r e d s p h e r i c a l wave. The a t o m i c s c a t t e r i n g f a c t o r f ^ i s u s u a l l y expanded as £12,66,67J; <*> f t < 9 r ) = iVn (-2l*1)exp(iS j . s i n ^ J P . ^ C c o s e ' ) (3.9) JUo i n terms of t h e Legendre p o l y n o m i a l s Pj^fcosG^) , and the phase s h i f t s Sj^ which c h a r a c t e r i s e t h e i o n - c o r e s c a t t e r i n g f o r each v a l u e of t h e a n g u l a r momentum guantum number 1-5U ( y V VJ p 1L9U££ 3^4 An i o n - c e r e imirersed i n a p l a c e wave i n d u c i n q s c a t t e r e d s p h e r i c a l waves whose i n t e n s i t i e s are f u n c t i o n s c f k, 6* and i . A f t e r Pendry [12"). The s c a t t e r i n g p r o p e r t i e s o f ar i o n - c o r e a r e thus c o n t a i n e d i n the energy-dependent phase s h i f t s . For the p o t e n t i a l s d e s c r i b e d i n S e c t i o n 3.1(a) the r e l e v a n t phase s h i f t s were found by n u m e r i c a l i n t e g r a t i o n of the S c h r o d i n q e r e g u a t i o n i n s i d e the m u f f i n - t i n sphere and - j o i n i n q t h e l a r g e r a s y m p t o t i c forms of the s o l u t i o n s smoothly t o the s o l u t i o n c f the S c h i c d i n g e r e q u a t i o n o u t s i d e the sphere. B h i l s t i n p r i n c i p l e tbe e x p a n s i o n o f e q u a t i o n (3.9) i s o v e r a l l v a l u e s o f 1, i t does c c c v e r q e q u i t e r a p i d l y and f o r c a l c u l a t i o n s up t o 250eV or s o , e phase s h i f t s (1=0-7) ar e u s u a l l y s u f f i c i e n t M 5 T and were used i n a l l the c a l c u l a t i o n s p r e s e n t e d here. P l o t s of the energy-dependence cf the phase s h i f t s f o r the 55 two copper and two rhodium p o t e n t i a l s a r e shown i n F i g . . 3 . 5 and (?<• F i g . 3.6. For copper, t h e band s t r u c t u r e and s u p e r p o s i t i o n V, p o t e n t i a l s produce very s i m i l a r phase s h i f t s which agree w e l l w i t h t h o s e g i v e n p r e v i o u s l y £35J. The rhodium s h i f t s show somewhat g r e a t e r d i f f e r e n c e s . F o r example, the c u r v e s f o r 1=0 and 1=1 show c o r r e s p o n d i n g f e a t u r e s a t lower e n e r g i e s f o r 7 ^ compared w i t h v ^ ) 3 J a l s o t h e r e a r e s i g n i f i c a n t d i f f e r e n c e s i n the s l o p e s o f t h e two s e t s of phase s h i f t s e.g. f o r 1=2 a t 5By (!By=13.6eV) . 3. 1 {c) t e m p e r a t u r e c o r r e c t i o n s I n S e c t i o n 2.3{b) i t was noted t h a t LEED beam i n t e n s i t i e s have a s u b s t a n t i a l t emperature dependence and t h a t s u c h e f f e c t s s u g gest t h a t l a t t i c e motion s h o u l d be i n c l u d e d i n t h e model used to c a l c u l a t e I ( E ) c u r v e s . The l a r g e r a t o m i c d i s p l a c e m e n t s t h a t o c c u r as t h e t e m p e r a t u r e i s r a i s e d i n c r e a s e the i n c o h e r e n t s c a t t e r i n g and t h e r e f o r e d e c r ease t h e i n t e n s i t i e s of s c a t t e r e d beams compared w i t h t h o s e from an i d e a l i s e d s t a t i o n a r y s c a t t e r i n g l a t t i c e . I n many i n s t a n c e s , t h e e x p e r i m e n t a l temperature-dependence of beam i n t e n s i t i e s i s d e s c r i b e d g u i t e w e l l by a k i n e m a t i c a l model [ 7 2 ] . D e t a i l e d t r e a t m e n t s w i t h i n t h e m u l t i p l e - s c a t t e r i n g t h e o r y have been produced £70,71]- a l t h o u g h atoms a r e e f f e c t i v e l y s t a t i o n a r y on the t i m e - s c a l e o f t h e d i f f r a c t i o n p r o c e s s , t h e i r motion i s s u f f i c i e n t l y r a p i d a f o r : the d e t e c t o r t o r e g i s t e r i n t e n s i t i e s t h a t have been averaged o v e r t h e atomic d i s p l a c e m e n t s - The LEED measurement a v e r a g e s ; o v e r many atoms a t • ZiSHIZ 3 X5 Inerqy dependence c f copper phase s h i f t s (1=0-7) f o r the p o t e n t i a l s : (a) V and (b) V? c . 57 l i S H L S Ji§ Enerqy dependence of rhodium phase s h i f t s (1=0-7) f o r th e p o t e n t i a l s : (a) V ana ( t ) 1^"-58 any one moment. Thus the s c a t t e r i n g o f each a t o m i c p o t e n t i a l i s averaged over t h e a t o m i c motions. I f ae assume t h a t the motions are u n c o r r e l a t e d [ 1 2 j then we , o b t a i n a temperature-dependent a t o m i c : s c a t t e r i n g f a c t o r t h a t i s r e l a t e d t o t h a t of t h e r i g i d l a t t i c e but w i t h m o d i f i c a t i o n s i n t h e phase s h i f t s , which become complex. These temperature-dependent phase s h i f t s must i n t e r f e r e i n t h e c o r r e c t f a s h i o n t o produce s t r o n g f o r w a r d , and weak b a c k - s c a t t e r i n g . I n the b a c k - s c a t t e r i n g c a s e t h e r e must be s t r o n g c a n c e l l a t i o n between c o n t r i b u t i o n s from d i f f e r e n t phase s h i f t s . The h i g h e r t h e t e m p e r a t u r e , t h e more t h i s i s s o , hence more s h i f t s a r e needed t o a v o i d t r u n c a t i o n o f the s e r i e s o f e g u a t i o n (3. 9) . fk<e*,T) = exp(r-ig) f ^ f e V 13 . 1 0 ) I n t h e c a s e of i s o t r o p i c v i b r a t i o n s , exp (-H^) = ex p (-M k-jc * J 2 ) ( 3 . 1 1 ) and oC can be a p p r o x i m a t e d , f o r l a r g e T, as a D e b y e - B a i l e r type c f f a c t o r cL = 3 1 i 2 I / 2 m k ( je2 ' ( 3 . 1 2 ) where k g i s Boltzmann's c o n s t a n t , m t h e atomic mass o f an i o n -c o r e and 9 p t h e Debye t e m p e r a t u r e . I n t h i s work, s u r f a c e atomic v i b r a t i o n s were assumed t o be i s o t r o p i c and l a y e r - i n d e p e n d e n t . An e x p e r i m e n t a l v a l u e o f t h e s u r f a c e Debye temperature was used f o r c o p p e r { v i z - 275K [73]) and, f o l l o w i n g van Hove and Tong £74J, a v a l u e o f J0.7^ t i m e s t h e b u l k v a l u e of 480K £75J was used f o r rhodium. 59 3.2 G e n e r a l Schemes Of C a l c u l a t i o n Having computed 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 , w i t h temperature-dependent phase s h i f t s to d e s c r i b e t h e s c a t t e r i n g from v i b r a t i n g atoms, t h e o t h e r , and most d i f f i c u l t , tasJc i s t o s o l v e f o r an i n c i d e n t e l e c t r o n m u l t i p l y s c a t t e r e d by the c r y s t a l l a t t i c e o f s c a t t e r i n g p o t e n t i a l s . Many methods have been deve l o p e d f o r p e r f o r m i n g m u l t i p l e -s c a t t e r i n g , or d y n a m i c a l , c a l c u l a t i o n s ; t h e y d i f f e r c o n s i d e r a b l y i n t h e i r range o f a p p l i c a b i l i t y , speed and computer memory r e q u i r e m e n t s . 111 the u s e f u l methods s t a r t w i t h the c a l c u l a t i o n of the s c a t t e r i n g m a t r i c e s t h a t d e s c r i b e t h e e l e c t r o n f l u x m u l t i p l y s c a t t e r e d , by one p l a n e of the c r y s t a l p a r a l l e l t o the s u r f a c e , from an i n c o m i n g wave l a b e l l e d by the r e c i p r o c a l v e c t o r 3 (or a n g u l a r momentum component 1, depending on the p a r t i c u l a r method) t o a s c a t t e r e d wave l a b e l l e d by a," (or 1»). once t h i s i n t r a l a y e r s c a t t e r i n g m a t r i x i s known, the a m p l i t u d e o f beams d i f f r a c t e d from an assemblage o f n l a y e r s , a p p r o x i m a t i n g an i n f i n i t e c r y s t a l , can be f o u n d . A number of methods of d o i n g t h i s a r e a v a i l a b l e ; they t a k e i n t o account t h e r e m o v a l o f e l e c t r o n f l u x by i n e l a s t i c s c a t t e r i n g p r o c e s s e s and a l l o w f o r i n t e r p l a n a r m u l t i p l e - s c a t t e r i n g e v e n t s . Between any p a i r of l a y e r s i n the c r y s t a l i s p r e s e n t a s e t of p l a n e wa ves [12 J 2. e x p j a j k + a j . r j (3.13) or beams, t h a t have been formed by d i f f r a c t i o n of t h e i n c i d e n t 60 beam. Each o f t h e s e has a d i f f e r e n t p a r a l l e l component of momentum l k d 0 and t r a v e l s e i t h e r f o r w a r d s o r backwards (±). Forward t r a v e l l i n g beams Z u!" e x p ( K ^ -JE) (3-14) 3 I <L where — K* = £<k 0 i i+ja)^ ±(2E-2v^ ^ J % ( i + a J 2 ) A / 2 j (3.15) a r e i n c i d e n t upon the next l a y e r as i n F i g . 3-7- T h i s s e t of beams i s m u l t i p l y - s c a t t e r e d by t h e p l a n e o f i o n - c o r e s as •+ + d e s c r i b e d by t h e l a y e r m a t r i x M ~~, (the ± s i g n s r e f e r r i n g t o | | f o r w a r d o r back s c a t t e r i n g ) t o g i v e ? Z M « 5 0 t e x P < i ^ r -£> (3» 16) w i t h £ 12J <£ & V/ ' = l8nZi/AkoKp-L > X \ ) 11-X 3j» YL, (/(g.)) e 1^' s i n £j/ (3. 16a) In the sum over a n g u l a r momentum components a r e i n c l u d e d s p h e r i c a l harmonics Y L and temperature-dependent phase s h i f t s , Sjj, «hile m u l t i p l e - s c a t t e r i n g i s a c c o u n t e d f o r by t h e i n v e r s e m a t r i x term £12]. + i The e v a l u a t i o n o f M " / i s c o m p l i c a t e d and need not c o n c e r n us h e r e ; however, i t i s worth n o t i n g t h a t i t i s e s s e n t i a l l y composed o f a s t u c t u r e f a c t o r , dependent o n l y on the p o s i t i o n s of the i o n - c o r e s i n the l a y e r , and a s c a t t e r i n g f a c t o r t h a t d e s c r i b e s t h e 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 i n terms of phase s h i f t s . F u l l d e t a i l s have been g i v e n by Fendry £ 12 ]. I t i s i n t h e manner o f a s s e m b l i n g a s t a c k of l a y e r s i n t o a c r y s t a l and computing the r e f l e c t i v i t y of the c r y s t a l t h a t the v a r i o u s c a l c u l a t i o n a l schemes d i f f e r - They can be d i v i d e d i n t o 3 A7 Schematic r e p r e s e n t a t i o n of a s e t of plane wave i n c i d e n t frcni the l e f t m u l t i p l y s c a t t e r e d by a plane c f ioz. c o r e s . a) " e x a c t " aethods ( i ) B l o c h wave nethcds ( i i ) e x a c t T - o a t r i x aethod f 7 6 ] b) p e r t u r b a t i v e a p p r o x i m a t i o n s ( i ) T - a a t r i x e x p a n s i o n s r77 ] ( i i ) l a y e r d o u b l i n q r 1 2 , 7 8 ] ( i i i ) r e n o r a a l i s e d f o r w a r d s c a t t e r i n q (RFS) [ 1 2 , 7 9 ] I n t h i s work the l a s t two aethods were used and w i l l b d i s c u s s e d i n a l i t t l e d e t a i l . The e t h e r s c h e i e s w i l l b 62 d i s c u s s e d i n o n l y the b r i e f e s t o u t l i n e and the i n t e r e s t e d r e a d e r i s r e f e r r e d t o i n d i v i d u a l r e f e r e n c e s and to the r e v i e w o f Tong [ 1 5 ] . . The " e x a c t " methods a r e e x a c t i n t h a t m u l t i p l e - s c a t t e r i n g i s t r e a t e d e x a c t l y both w i t h i n and between l a y e r s . I n the B l o c h wave method the s c a t t e r i n g m a t r i c e s f o r a l a y e r a r e found and the wave a m p l i t u d e s between two l a y e r s c a l c u l a t e d a l l o w i n g f o r i n t e r p l a n a r m u l t i p l e - s c a t t e r i n g . The s c a t t e r e d w a v e f i e l d i s t h e n e x p r e s s e d as a c o m b i n a t i o n of B l o c h waves, or normal modes of the c r y s t a l . A t the s u r f a c e t h e s c a t t e r e d w a v e f u n c t i o n s are matched t o t h o s e o f t h e , i n c i d e n t wave t o g i v e t h e r e f l e c t e d a m p l i t u d e s . I n the most d e v e l o p e d v e r s i o n o f these methods, the l a y e r KKR method [,65], i n t r a l a y e r s c a t t e r i n g i s t r e a t e d by the K o r r i n g a - K o h n - B o s t o k e r (KKR) method o f band t h e o r y . T h i s method i s a c c u r a t e but slow as one has t o s o l v e a (2nX2n) m a t r i x e i g e n v a l u e problem, where n i s t h e number of waves used. C o n s e q u e n t l y , the l a y e r KKR method has o n l y been used t o any g r e a t e x t e n t by t h e group a t IBM [ 6 5 ] because of the v a s t computer time and s t o r a g e r e q u i r e m e n t s i n v o l v e d . The e x a c t T - m a t r i x method [ 76 J e x p r e s s e s t h e l a y e r s c a t t e r i n g m a t r i c e s as s o - c a l l e d T m a t r i c e s i n a n g u l a r fficffientum, r a t h e r than r e c i p r o c a l , space. A l a y e r - d e p e n d e n t T-m a t r i x i s t h e n computed, which i n c l u d e s c o n t r i b u t i o n s to t h e s c a t t e r e d w a v e f i e l d from : one l a y e r due t o waves a l r e a d y s c a t t e r e d from o t h e r l a y e r s . I n o r d e r t o do t h i s , the c r y s t a l must be approximated as an N l a y e r s l a b (N t y p i c a l l y about 5 ) . The t o t a l r e f l e c t i v i t y of t h e N l a y e r s l a b i s then f o u n d . T h i s i n v o l v e s s o l v i n g N e q u a t i o n s each h a v i n g m a t r i c e s o f d i m e n s i o n s 63 m^*> + 1) * x ^ * m x * 1) 2- T h i s method i s a c c u r a t e i n t h e presence o f i n e l a s t i c s c a t t e r i n g i f enough phase s h i f t s and l a y e r s a re used, but i s r a t h e r slow and i s d i f f i c u l t t o extend t o c o m p l i c a t e d s u r f a c e s , The T - m a t r i x e x p a n s i o n method £ 77 j i s an e x t e n s i o n of the l a s t method i n which a l i m i t i s p l a c e d upon t h e o r d e r of m u l t i p l e - s c a t t e r i n g ( i n t e r - and i n t r a p l a n a r ) a l l o w e d . T h i s l i t t l e - u s e d method i s cumbersome above t h i r d o r d e r and can f a i l w i t h s t r o n g s c a t t e r e r s or weak a b s o r p t i o n . 3.2(a) l a y e r d o u b l i n g and SFS methods These two convergent p e r t u r b a t i v e methods, p i o n e e r e d by Pendry, Van Hove and Tong £ 12,78,79 "J, a r e among the most p o p u l a r i n use t o d a y . T h e i r i t e r a t i v e form makes them f l e x i b l e , r e l a t i v e l y easy t o use i n a r o u t i n e manner, f a s t and c o n s e r v a t i v e i n c o r e space r e q u i r e m e n t s . The l a y e r d o u b l i n g method s t a r t s w i t h t h e m u l t i p l e -s c a t t e r i n g m a t r i x f o r a s i n g l e l a y e r ; i n F i g . 3.8 t h e s e a re marked as B { r e f l e c t i o n ) and T ( t r a n s m i s s i o n ) m a t r i c e s f o r two l a y e r s A and B. The i n t e r p l a n a r m u l t i p l e - s c a t t e r i n g i s then s o l v e d e x a c t l y f o r the p a i r t o produce a c o m p o s i t e l a y e r C. The r e s u l t i n g m a t r i c e s R and l c a r e then used t o s o l v e f o r 4 l a y e r s , t h u s d o u b l i n g the number o f l a y e r s i n t h e c r y s t a l . T h i s p r o c e s s o f d o u b l i n g the t h i c k n e s s of the s l a b i s then r e p e a t e d u n t i l the r e f l e c t e d i n t e n s i t i e s converge, t y p i c a l l y i n 8 or 16 l a y e r s . T h i s method i n v o l v e s m a t r i x i n v e r s i o n s and p r o d u c t s of. dimension n, t h e number o f beams use d , t h a t c o n v e r g e even f o r : J i f l a S f i x S E u i l d i n g up sufcplanes by the l a y e r d o u b l i n g p r o c e s s I n d i v i d u a l s u b p l a n e s a r e Barked A and B; the r e s u l t a n t composite i s marked C. A f t e r Tonq M 5 1. 65 s m a l l i n t e r p l a n a r s p a c i n g s . The r e n o r m a l i s e d f o r w a r d s c a t t e r i n g , o r RFS, method once more s t a r t s from t h e s i n g l e l a y e r m u l t i p l e - s c a t t e r i n g m a t r i c e s of e g u a t i o n ( 3 . 1 6 ) . E l e c t r o n s a r e propagated through t h e c r y s t a l w i t h f o r w a r d s c a t t e r i n g b e i n g e v a l u a t e d e x a c t l y ; b a c k s c a t t e r i n g i s t r e a t e d l y p e r t u r b a t i v e l y. R e f e r r i n g t o F i g . 3.9 , f o l l o w i n g Tong £15], we can d e f i n e a column v e c t o r A^(fl) o f l e n g t h n, whose ele m e n t s r e p r e s e n t t h e a m p l i t u d e s i n each of t h e n beams ( l a b e l l e d by j ) p r o p a g a t i n g i n t o t h e c r y s t a l , a t t h e oL l a y e r . The, i n d e x i d e n o t e s the number of t i f f e s e l e c t r o n s a r e p r o p a g a t i n g i n t o t h e c r y s t a l and i s a l s o the i t e r a t i o n o r d e r . O u t s i d e t h e c r y s t a l we have A MS) o 1 \ 0 0 0 (3.17) The c o e f f i c i e n t s A^Ca) a r e e v a l u a t e d midway between p l a n e s and can be found by i t e r a t i o n , assuming a l l l a y e r s a r e i d e n t i c a l , t h u s 66 I i S " I s 1..9 D i a g r a m a t i c r e p r e s e n t a t i o n of the r e n c r a a l i z e d f o r w a r d s c a t t e r i n g (BFS) p r o c e s s . Inward a m p l i t u d e s A'^(g) propagate from vacuum through t h e 1st l a y e r t c the Nth l a y e r where they are t u r n e d around. The e l e c t r o n s a r e t h e n propagated to the 1st l a y e r w i t h outward a m p l i t u d e s E' (gj . 67 A*-(g)- = 2 M ^ ' a o ( 3 > ( 3- T 8 ) and lJ.(g)-.= . . J 8 * * / l f (g) ;- (3- 19) * ft cL-i u T h i s i s c a r r i e d on u n t i l a t t h e N l a y e r t h e e l e c t r o n s a r e damped such t h a t a n e g l i g i b l y s m a l l e l a s t i c f l u x reaches the (N + 1 ) ^ l a y e r , t y p i c a l l y 7-14 l a y e r s . The e l e c t r o n s a r e t h e n t u r n e d around and we f o rm a s e t o f E^tS) f a r a m p l i t u d e s o f e l e c t r o n s g o i n g back o u t . The s e t B 1 ia) a r e a m p l i t u d e s f o r each j-beam a f t e r p a s s i n g i n and outwards t h r o u g h : t h e c r y s t a l once. T h e r e f o r e , r e f e r r i n g a g a i n t o F i g - ,3-9 we have B t (a) = ^ H*~ A * (a) {3.20) F o r B* (a) t h e r e a r e c o n t r i b u t i o n s from b a c k s c a t t e r i n g of A1, (a) from t h e («(#• 1) ^  p l a n e , and t r a n s m i s s i o n o f B* {a) t h u s B* <a) = Z ®W A* m * y M 7,7 B* ((a) {3.21) 1 — / -Each s e t of c o e f f i c i e n t s i s t h u s o b t a i n e d from s e t s p r e v i o u s l y e v a l u a t e d . The e l e c t r o n s a r e e v e n t u a l l y s c a t t e r e d by the f i r s t l a y e r back i n t o t h e c r y s t a l ; i=2. A g a i n t h e c o e f f i c i e n t s A 2 (a) and, a f t e r r e a c h i n g a deepest l a y e r N^SH, the r e f l e c t i o n c o e f f i c i e n t s B 5 i S ) ' a r e e v a l u a t e d . E v e n t u a l l y the r e f l e c t e d a m p l i t u d e s 68 converge and a r e summed to y i e l d t h e r e f l e c t e d i n t e n s i t i e s . The RFS method i s very f a s t , u s u a l l y r e q u i r i n g 3-5 passes t o c o n v e r g e , but c a n f a i l t o converge i f t h e i n e l a s t i c damping i s t oo weak o r t h e i n t e r l a y e r s p a c i n g t o o s m a l l . I n such c a s e s the l a y e r d o u b l i n g method, though s l o w e r , i s t o be p r e f e r r e d . In t h e c a l c u l a t i o n s r e p o r t e d here t h e BPS method was used i n t h e main. O c c a s i o n a l l y , at s m a l l i n t e r l a y e r s p a c i n g s the l a y e r d o u b l i n g method was used; where t h e two methods o v e r l a p p e d , i t was found t h a t the i n t e n s i t i e s produced by both methods were n u m e r i c a l l y , v e r y s i m i l a r . 3.2(b) uses o f symmetry C o n s i d e r a b l e s a v i n g s i n computer s t o r a g e and time can be made by e x p l o i t i n g the symmetry among p l a n e waves p r e s e n t when t h e e l e c t r o n beam i s i n c i d e n t cn the c r y s t a l s u r f a c e a l o n g an a x i s o r a pla n e o f symmetry of the s u r f a c e s t r u c t u r e - Group t h e o r y a l l o w s us t o c o n s i d e r o n l y t h e s y m m e t r i c a l l i n e a r c o m b i n a t i o n s o f s y m m e t r y - r e l a t e d p l a n e waves, a s d e t a i l e d by van Hove £81 j . T h e r e f o r e , i n t h e l i s t o f JJ v e c t o r s t h a t i s i n p u t t o t h e program, as e x p l a i n e d s h o r t l y i n S e c t i o n 3 . 2 ( c ) , o n l y one v e c t o r £ f o r each s e t o f s y m m e t r y - r e l a t e d v e c t o r s i s r e a d , t o g e t h e r w i t h a code-number t h a t i n f o r m s the programs about t h e o m i t t e d s y m m e t r y - r e l a t e d v e c t o r s . T h i s code number i n s t r u c t s t h e programs t o use t h e a p p r o p r i a t e s y m m e t r i c a l w a v e f u n c t i o n s ( r a t h e r than the s i m p l e p l a n e waves l a b e l l e d by CJ) and t o a s s i g n o n l y one row and one column i n d i f f r a c t i o n m a t r i c e s f o r each s y m m e t r i c a l w a v e f u n c t i o n ^ r a t h e r t h a n one f o r 69 S u r f a c e Max. <j Symm. Number o f beams v e c t o r a x i s ncsymm. w i t h symm.-Rh (100) {62} 4 69 13 Bh (111) {4 3} 3 55 13 Rh (110) {42} 2 55 18 T a b l e 3L2 R e d u c t i o n , due t o symmetry, i n t h e number o f beams needed a t normal i n c i d e n c e f o r t h e t h r e e s i m p l e f a c e s of rhodium. The maximum a v e c t o r c o r r e s p o n d s t o t h e beam s e t w i t h t h e l a r g e s t (hk) v a l u e s needed t o c o v e r the energy range 40-250eV. each o f t h e v a r i o u s s y m m e t r y - r e l a t e d p l a n e waves). The s a v i n g s of c o r e s t o r a g e and t i m e can be c o n s i d e r a b l e . T h i s i s i l l u s t r a t e d i n Table 3.2 where the r e d u c t i o n , due t o symmetry, i n the number o f beams, needed a t normal i n c i d e n c e t o c o v e r the energy range up to about 250eV i s shown f o r t h e Rh (100), (111) and (110) s u r f a c e s . 3.2(c) program f l o w The f l o w - c h a r t o f F i g . 3.10 summarises t h e sequence of e v e n t s t h a t o c c u r i n a 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 . The programs s t a r t by r e a d i n g i n a l l the r e l e v a n t - p h y s i c a l and p o t e n t i a l d a t a , and a l s o a l i s t of d i f f r a c t e d beams w i t h t h e i r symmetry code-numbers. 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 a l a y e r , d e s c r i b e d by e g u a t i o n (3. 16) i n v o l v e s sums o v e r r e c i p r o c a l v e c t o r s g_, t h a t i s , o v e r t h e d i f f r a c t e d beams. At any p a r t i c u l a r energy E, o n l y c e r t a i n beams w i t h jg v e c t o r s t h a t permit k*^ t o remain r e a l , S e c t i o n 2 . 2 , c a n l e a v e the c r y s t a l . In a d d i t i o n t h e r e a r e evanescent waves t h a t propagate w i t h o u t emerging because of the p o t e n t i a l s t e p a t t h e s u r f a c e . The program a u t o m a t i c a l l y i n c l u d e s i n t h e s u b s - , o v e r t h e r e c i p r o c a l 70 Bead i n ( i ) geometry ( i i ) v f t r , v e L beans and symmetry ( i v ) t emperature data (v) phase s h i f t s Choose i n i t i a l energy F i n d teams needed at I Compute temperature-dependent phase s h i f t s C a l c u l a t e l a y e r d i f f r a c t i o n m a t r i c e s tr~, J i n d d i f f r a c t e d beam a m p l i t u d e s from s u r f a c e p l u s s u b s t r a t e by FFS F i n d d i f f r a c t i o n m a t r i c e s f o r n s u b s t r a t e l a y e r s by l a y e r d o u b l i n g C a l c u l a t e team i n t e n s i t i e s Add s u r f a c e l a y e r and f i n d d i f f r a c t e d beam a m p l i t u d e s Vary s u r f a c e geometry C a l c u l a t e team i n t e n s i t i e s V Vary s u r f a c e qecmetry Increment E f i s y i f i x l O F l o w c h a r t showing p r i n c i p a l s t e p s i n a m u l t i p l e -s c a t t e r i n g LEED c a l c u l a t i o n , u s i n g t h e RFS o r l a y e r d o u b l i n g programs. 71 v e c t o r s o n l y t h o s e waves t h a t , from one l a y e r t o the n e x t , do n o t decay t o l e s s than 0.00 2 of t h e i r a m p l i t u d e . Having i n c o r p o r a t e d t e m p e r a t u r e e f f e c t s i n t o t h e phase s h i f t s , 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 a r e c a l c u l a t e d . The r e f l e c t e d a m p l i t u d e s , and s u b s e g u e n t l y i n t e n s i t i e s , a r e then found by t h e l a y e r d o u b l i n g c r RFS r o u t i n e s . Each method has the f a c i l i t y t o i n c l u d e a s u r f a c e l a y e r t h a t has a d i f f e r e n t geometry from t h a t o f t h e s u b s t r a t e . I n the c a s e of c l e a n m e t a l s u r f a c e s , the s u r f a c e l a y e r c o u l d be d i s t i n g u i s h e d by a l a t e r a l r e c o n s t r u c t i o n r e l a t i v e t o t h e b u l k s t r u c t u r e or by a s i m p l e change i n the topmost l a y e r s p a c i n g . The two methods d i f f e r s l i g h t l y i n t h a t t h e l a y e r d o u b l i n g , method c a l c u l a t e s 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 f o r t h e s u b s t r a t e once o n l y and then the s e are used r e p e a t e d l y f o r as many s u r f a c e g e o m e t r i e s a s d e s i r e d by a d d i n g t h e s u r f a c e l a y e r as t h e f i n a l s t e p . . U s i n g BFS, t h e whole s u b s t r a t e p l u s s u r f a c e l a y e r system must be r e b u i l t f o r each geometry as the s u r f a c e l a y e r i s i n v o l v e d i n the i n i t i a l s t e p o f the c a l c u l a t i o n o f the i n t e r l a y e r s c a t t e r i n g . The i n t e n s i t i e s 1^(E) of each beam a r e s t o r e d b e f o r e i n c r e m e n t i n g t h e energy and r e p e a t i n g t h e whole p r o c e s s . G e n e r a l l y , the c a l c u l a t i o n s were performed over an energy range of a p p r o x i m a t e l y 40 t o 250eV. These l i m i t s a r e s e t a t the low end by the l a c k o f e x p e r i m e n t a l d a t a , and a t t h e h i g h end by the expense of the c a l c u l a t i o n s and by doubts a s t o whether 8 phase s h i f t s a r e s u f f i c i e n t t o en s u r e convergence o f the atomic s c a t t e r i n g f a c t o r s . &n energy i n c r e m e n t of 2eV was used f o r t h e range 40 t o 100eV, and the i n c r e m e n t was doubled f o r : the range 72 above 100eV. For the l a t t e r range, i n t e n s i t y v a l u e s were; t h e n i n t e r p o l a t e d back onto a 2eV g r i d u s i n g a c u b i c s p l i n e i n t e r p o l a t i o n method £80]. A l l c a l c u l a t e d , i n t e n s i t i e s s e r e s t o r e d on magnetic t a p e and s u b s e g u e n t l y the d a t a c o u l d be t r a n s f e r r e d t o paper tape f o r p l o t t i n g by the Nova 2 minicomputer i n the l a b o r a t o r y . In o r d e r t o check t h a t the programs were r u n n i n g c o r r e c t l y , I|E) c u r v e s were c a l c u l a t e d f o r s u r f a c e s t h a t had p r e v i o u s l y been s t u d i e d t h e o r e t i c a l l y ; c a r e f u l c h e c k s were made f o r d i s c r e p a n c i e s b e f o r e any c a l c u l a t i o n s were performed on the •unknown rhodium s u r f a c e s . These t e s t s were made a g a i n s t ( i ) the c a l c u l a t i o n s o f Laramore £20] f o r Cu (111) and Cu (100) ; ( i i ) the c a l c u l a t i o n s o f Demuth e t a l £11] f o r Ni (1Q0) and N i ( 1 1 0 ) , u s i n g n i c k e l phase s h i f t s s u p p l i e d by M.A. Van Hove In a l l c a s e s comparison w i t h t h e p r e v i o u s c a l c u l a t i o n s showed t h a t the programs used here produced e s s e n t i a l l y i d e n t i c a l 1(E) c u r v e s . T h i s gave us c o n f i d e n c e f o r u s i n g these programs f o r d e t e r m i n i n g the unknown s t r u c t u r e s of rhodiom s u r f a c e s . These type o f c a l c u l a t i o n s were developed t o i n v e s t i g a t e s u r f a c e s t r u c t u r e s , and i t i s g r a t i f y i n g t h a t s i g n i f i c a n t changes can o c c u r i n c a l c u l a t e d 1(E) c u r v e s f o r o n l y minor changes i n t h e assumed s u r f a c e geometry- F o r example, i n F i g . 3.11 we see 1(E) c u r v e s c a l c u l a t e d f o r a Cu (111) s u r f a c e a t normal i n c i d e n c e , u s i n g both t h e Burdick-Chodorow, V> C, and s u p e r p o s i t i o n , V , p o t e n t i a l s d i s c u s s e d e a r l i e r . The 73 I i S l ? I f 3 j . I l Comparison of e x p e r i m e n t a l 1(E) c u r v e s f o r Cu(111) at normal i n c i d e n c e w i t h c u r v e s c a l c u l a t e d f o r the V t a l 3 and v£ p o t e n t i a l s f o r V.r = -9.5eV and t h r e e d i f f e r e n t v a l u e s of A&%. c a l c u l a t i o n s a r e f o r t h e (10) and (11) beams f o r a range o f v a l u e s of the topmost i n t e r l a y e r s p a c i n g d e x p r e s s e d as a percentage change from t h e b u l k v a l u e d^, i n s t e p s of 5%, AdX = £d-d o/d 0] X 10035 0-22) but assuming no o t h e r geometry changes. I n t h i s example, as the s u r f a c e i s c o n t r a c t e d , r e l a t i v e i n t e n s i t i e s o f peaks change s i g n i f i c a n t l y , New s t r u c t u r e i s i n t r o d u c e d , e.g. a new peak grows a t about 200eV f o r the (10) ; beam, and t h e r e l a t i v e i n t e n s i t i e s of the peaks a t about 230 and ,260eV i n t h e (11) beam become r e v e r s e d . CHAPTER H EXPERIMENTAL ASPECTS 76 I n o r d e r to deduce s u r f a c e s t r u c t u r e s v i a a LEED a n a l y s i s s e v e r a l e x p e r i m e n t a l p r e r e q u i s i t e s must be f u l f i l l e d - The f i r s t problem i s t o prepare the s u r f a c e o f i n t e r e s t i n as p r e c i s e a f a s h i o n as p o s s i b l e . T h i s i n v o l v e s a c c u r a t e l y c u t t i n g a s i n g l e c r y s t a l t o expose 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 p o l i s h i n g the f a c e o b t a i n e d t o a h i g h degree of p e r f e c t i o n . Having o b t a i n e d a w e l l - o r i e n t e d s u r f a c e we must be a b l e t o c l e a n and m a i n t a i n i t s s t a t e of c l e a n l i n e s s , a t the l e v e l o f s m a l l f r a c t i o n s of a monolayer of c o n t a m i n a t i o n . D i f f r a c t i o n of low energy e l e c t r o n s o n l y o c c u r s from w e l l - o r d e r e d s u r f a c e s so the c r y s t a l must be t r e a t e d i n such a way as t o p o s s e s s a c l e a n and w e l l - o r d e r e d s u r f a c e b e f o r e t h e LEED e x p e r i m e n t can b e g i n . F i n a l l y , we are f a c e d w i t h t h e problem of p e r f o r m i n g the a c t u a l LEED ex p e r i m e n t which, as we s h a l l s ee, c e n t r e s around problems of data c o l l e c t i o n and a n a l y s i s . 4, ,1 C r y s t a l P r e p a r a t i o n The samples used i n t h i s work /were t y p i c a l l y d i s c s o f about 1mm t h i c k n e s s and 6mm d i a m e t e r . I n some c a s e s , p r e c u t s l i c e s on l o a n from o t h e r l a b o r a t o r i e s were used i n i t i a l l y b ut the e x p e r i m e n t s were r e p e a t e d on new d i s c s c u t h e r e . T a b l e 4. 1 l i s t s the s o u r c e s o f t h e c r y s t a l s used. The s i n g l e c r y s t a l r o d s were c a r e f u l l y o r i e n t e d u s i n g t h e Laue b a c k - r e f l e c t i o n t e c h n i q u e £ 82] such 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 was p e r p e n d i c u l a r t o t h e X-ray beam. V e r t i c a l c u t s p a r a l l e l t o t h e p l a n e were made on a sp a r k e r o s i o n c u t t e r ( " A g i e t r o n " , AGIE, S w i t z e r l a n d ) , The c r y s t a l r o d was 77 Source Treatment Faces used ROC-SIC Cut and; p o l i s h e d a t UBC (111) , (100) , (110) B e r k e l e y As r e c e i v e d (111) GE Cut and p o l i s h e d at UBC (100) T a b l e 4.1 Sources o f rhodium c r y s t a l s 1. BOC-BIC = Research O r g a n i c / I n o r g a n i c C h e m i c a l s Corp. 2. B e r k e l e y = C o u r t e s y of P r o f . ,G. A. Somor j a i , Lawrence B e r k e l e y L a b o r a t o r y , U. o f C a l i f o r n i a , B e r k e l e y , Ca. 3. GE = C o u r t e s y of Dr. C..H-Tucker,General E l e c t r i c Research and Levelopment C e n t r e , Schenectady, N.Y. l e f t i n p l a c e on the goniometer d u r i n g t h i s o p e r a t i o n i n o r d e r t h a t t h e c r y s t a l l o g r a p h i c o r i e n t a t i o n would n o t be compromised. The r e s u l t i n g d i s c was c a r e f u l l y mounted i n a c r y l i c r e s i n i*'Cuickmount", 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 C o r p . , USA) and p o l i s h e d by hand t o a m i r r o r - f i n i s h (0.05 micron a l u m i n a ) . W h i l s t g r e a t c a r e was taken d u r i n g t h e p o l i s h i n g p r o c e s s t o ensure t h a t t h e o r i e n t a t i o n of the d e s i r e d f a c e was not l o s t , i t i s i m p o r t a n t to check a f t e r w a r d s t h a t an e r r o r has n o t o c c u r r e d . A s i m p l e t e c h n i g u e was d e v i s e d t o check .. t h a t the . o p t i c a l p o l i s h e d f a c e was i n f a c t p a r a l l e l t o the d e s i r e d c r y s t a l p l a n e . T h i s i n v o l v e d r e - o r i e n t i n g t h e c r y s t a l s l i c e on the X-ray d i f f r a c t o m e t e r such t h a t the d e s i r e d plane was once a g a i n p e r p e n d i c u l a r to the X-ray beam. The whole goniometer and c r y s t a l ensemble was t h e n removed t o an o p t i c a l bench where the o r i e n t a t i o n o f the o p t i c a l f a c e r e l a t i v e t o t h e c r y s t a l p l a n e c o u l d be checked, a s shown i n F i g . 4. 1, by measurement of the a n g l e of r e f l e c t i o n of the beam from a s m a l l He-Ne l a s e r . T h i s method can e a s i l y d e t e c t m i s o r i e n t a t i o n s o f ±1/2°, 78 He-Ne Laser l i s u r e j ^ J L a s e r a l i g n m e n t method t o check the c o i n c i d e n c e o f the o p t i c a l f a c e and d e s i r e d c r y s t a l plane. GAS LINE 9 9 9 o o A BOI/sI.PJ * EXPTAL. CHAMBER 240 l/s XP. S.P. * S.R n.s.R l i S P I S i L i i Diagrammatic r e p r e s e n t a t i o n of t h e pumping system: I P « Ion pump; ISP = t i t a n i u m s u f c l i m a t i o n pump; SP = s o r p t i o n pump 79 a l t h o u g h they can range up t o ±2°, even w i t h c a r e f u l p o l i s h i n g . In t h e event o f such a ^ D i s o r i e n t a t i o n t h e c r y s t a l was r e p o l i s h e d u n t i l i t was w i t h i n 1/2° of the d e s i r e d c r y s t a l p l a n e . 1.2 U l t r a High Vacuum (t)HV) Apparatus Once h a v i n g p r e p a r e d a w e l l - d e f i n e d c r y s t a l s u r f a c e i t must he kept i n an environment i n which i t w i l l not become con t a m i n a t e d t o any g r e a t e x t e n t b e f o r e t h e e x p e r i m e n t s a r e completed. Simple c o n s i d e r a t i o n of the k i n e t i c t h e o r y of gases shows t h a t a contaminant w i t h u n i t s t i c k i n g p r o b a b i l i t y w i l l form a monolayer on t h e s u r f a c e i n about 1 second a t 1-0-6 T o r r . T h e r e f o r e , i t i s n e c e s s a r y t o work i n t h e OHV p r e s s u r e r e g i o n i . e . i n t h e 1Q-io T o r r p r e s s u r e r a n g e , o r b e t t e r , i n o r d e r t o have s u f f i c i e n t t i m e t o perf o r m an e x p e r i m e n t w i t h o u t the c r y s t a l s u r f a c e becoming s e r i o u s l y . c o n t a m i n a t e d . The c r y s t a l s l i c e i s , t h e r e f o r e , mounted i n s i d e a UHV chamber. I n t h i s work two such systems were used, a V a r i a n 240 and a V a r i a n FC12 chamber. The form e r was o n l y used f o r a s h o r t ; t i m e i n t h e e a r l y s t a g e s s o f u r t h e r d i s c u s s i o n w i l l c e n t r e around the l a t t e r system. A s c h e m a t i c diagram : o f t h i s a p p a r a t u s i s shown i n F i g . 4. 2. The s t a i n l e s s s t e e l chamber can be roughed out t o a p r e s s u r e o f a p p r o x i m a t e l y 1 micron by two 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 c h i l l e d by l i g u i d n i t r o g e n . The main 24 0 1/s i o n pump can t h e n be s t a r t e d and w i l l a t t a i n a base-p r e s s u r e o f <10-*° T o r r f o l l o w i n g an o v e r n i g h t bake-out a t 200°C to remove adsorbed gases from t h e chamber w a l l s . 80 D uring c r y s t a l c l e a n i n g i t i s o f t e n n e c e s s a r y t o admit gases i n t o t i e chamber e.g. argon f o r ion-bombardment or oxygen f o r c h e m i c a l c l e a n i n g - These gases a r e s t o r e d i n g l a s s b u l b s on a gas l i n e c o nnected t o the main chamber through a v a r i a b l e l e a k v a l v e . The gas l i n e can be baked s e p a r a t e l y from the chamber and i s pumped by i t s own s m a l l ion-rpump (20 1 / s ) . Thus, the amount o f e x t r a background i m p u r i t i e s from t h e gas l i n e i n the a d m i t t e d gases can be kept t o a few p a r t s per m i l l i o n - , When the chamber i s f l o o d e d w i t h argon f o r ion-bombardment, t h e main i o n -pump can be t h r o t t l e d o f f by a gate v a l v e and t h e t i t a n i u m s u b l i m a t i o n pump used t o f u r t h e r l o w e r t h e p a r t i a l p r e s s u r e s of a c t i v e gases i n the chamber. F i g - 4-3 shows a s c h e m a t i c r e p r e s e n t a t i o n o f the FC12 chamber. On t h e v a r i o u s p o r t s a r e . f i t t e d : i ) a m a n i p u l a t o r on which i s mounted the sample. The e x t e r n a l c o n t r o l s o f the m a n i p u l a t o r a l l o w the sample t o be t r a n s l a t e d i n a l l t h r e e .. p e r p e n d i c u l a r d i r e c t i o n s , t o g e t h e r w i t h v a r i a t i o n of t h e p o l a r a n g l e o f i n c i d e n c e , & , and the " f l i p a n g l e " i n the v e r t i c a l p l a n e , as i n d i c a t e d i n F i g . 4.3. E l e c t r i c a l f e e d t h r o u g h s t o permit h e a t i n g of the sample and a thermocouple m o n i t o r a r e a l s o p r o v i d e d ; i i ) a "nude" i o n gauge t o monitor the system p r e s s u r e ; i i i ) a hypodermic gas do s e r , a f t e r J o y n e r and S o m o r j a i £83J, con n e c t e d t o the v a r i a b l e l e a k v a l v e . T h i s p e r m i t s r e l a t i v e l y h i g h p r e s s u r e s o f e.g. a c h e m i c a l c l e a n i n g agent, a t ;the c r y s t a l s u r f a c e 81 I o n gauge Gas d o s e r fiS"I§ Hi.3. Schematic of the V a r i a n FC12 UHV chamber. IiS"I§ S i m p l i f i e d diagram of an o f f - a x i s e l e c t r o n que with d e f l e c t i o n e l e c t r o d e and d r i f t t ube. 82 w h i l e k e e p i n g the t o t a l system p r e s s u r e r e l a t i v e l y low; i v ) an ion-bombardment gun f o r c r y s t a l c l e a n i n g ; v) t h e LJSED/Auger e l e c t r o n ; o p t i c s and e l e c t r o n gun. The p r i n c i p l e o f the c o n s t r u c t i o n of the gun i s shown i n F i g . 4.4. The o f f - a x i s i t u n g s t e n f i l a m e n t e m i t s e l e c t r o n s t h a t a r e d e f l e c t e d i n t o a c o l l i m a t i n g l e n s system ; and d r i f t t u b e . The o f f -a x i s geometry p r e v e n t s c o n t a m i n a t i o n of the s u r f a c e by e v a p o r a t i o n o f cathode m a t e r i a l . The e l e c t r o n s l e a v e t h e d r i f t t u b e , which . i s m a i n t a i n e d a t the same p o t e n t i a l as the sample, and t r a v e r s e f i e l d -f r e e space as l o n g as no s t r a y magnetic f i e l d s a r e p r e s e n t . The beam d i a m e t e r i s about 1mm and has t y p i c a l l y an energy spread of about 0.75eV. Secondary and Auger e l e c t r o n s a r e f i l t e r e d by t h e system of g r i d s , and 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 are d i s p l a y e d on a f l u o r e s c e n t s c r e e n ; v i ) a window t o o b s e r ve th e sample and d i f f r a c t i o n p a t t e r n on the f l u o r e s c e n t s c r e e n . The chamber i s s u r r o u n d e d by t h r e e o r t h o g o n a l s e t s o f square H e l m h o i t z c o i l s to r e d uce the r e s i d u a l magnetic f i e l d o f t h e e a r t h , ion-pumps, e t c . t o a l e v e l which w i l l n o t i n t e r f e r e w i t h the motion of slow e l e c t r o n s {about 20 mGauss'for 50eV e l e c t r o n s [84J)> A l t h o u g h t h i s geometry does not s a t i s f y t h e optimum Helmh o i t z c o n d i t i o n the c u b i c arrangement i s adequate 83 and e x p e r i m e n t a l l y more c o n v e n i e n t - S a t i s f a c t o r y n e u t r a l i s a t i o n was a c h i e v e d u s i n g the c r i t e r i o n t h a t t h e s p e c u l a r team i n the LEED p a t t e r n s h o u l d remain s t a t i o n a r y f o r a l l : i n c i d e n t e n e r g i e s . The e f f e c t i v e n e s s of t h i s c a n c e l l a t i o n was a l s o checked by examining 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 g u i v a l e n t beams; f o r example, the {00) beam a t e q u i v a l e n t a n g l e s o f i n c i d e n c e on e i t h e r s i d e o f the s u r f a c e normal or n o n - s p e c u l a r beams of the same o r d e r a t normal i n c i d e n c e . ,fk3 C r y s t a l C l e a n i n q 4. 3 (a) P r o c e d u r e s When the sample i s f i r s t mounted i n t h e UHV chamber i t s s u r f a c e , i n which we a r e i n t e r e s t e d , w i l l be c o n t a m i n a t e d not o n l y by a d s o r p t i o n of a t m o s p h e r i c gases but a l s o by b u l k i m p u r i t i e s p r e s e n t i n t h e o r i g i n a l s i n g l e c r y s t a l rod and by i m p u r i t i e s i n t r o d u c e d onto the s u r f a c e by t h e p o l i s h i n g p r o c e s s . The s u r f a c e must a l s o be a n n e a l e d i n o r d e r t o smooth the i r r e g u l a r i t i e s r e m a i n i n g i n t h e s u r f a c e and t h e r e b y produce a w e l l - o r d e r e d s u r f a c e t h a t e x h i b i t s a sharp LEED p a t t e r n -I n t h i s work a c o m b i n a t i o n o f two t r e a t m e n t s was used t o c l e a n t h e c r y s t a l s u r f a c e s ; of argon ion-bombardment and vacuum a n n e a l s ; t r e a t m e n t s i n o x i d i s i n g o r r e d u c i n g (a) c y c l e s (b) heat 84 atmospheres. > Io-n-fcomhard.men.t-- was a c h i e v e d by f l o o d i n g t h e chamber t o between 10- s and 10~ 6 T o r r o f argon and o p e r a t i n g the ion-bombardment gun, a d e v i c e i n which a heated f i l a m e n t g e n e r a t e s argon i o n s which are a c c e l e r a t e d on t o the s u r f a c e of i n t e r e s t . The impact of these i o n s s p u t t e r s m a t e r i a l from the s u r f a c e t h e r e b y removing i m p u r i t i e s . However, the h o s t atoms are a l s o removed and t h e s u r f a c e may become to o rough t o produce a good LEED p a t t e r n ; i t must t h e n be a n n e a l e d by h e a t i n g . S e v e r a l methods of h e a t i n g the sample were employed. F i g - 4.5 d i s p l a y s these methods:: (a) the c r y s t a l was s p o t - w e l d e d to Bh or P t f o i l and r e s i s t i v e l y h e a t e d . ; T y p i c a l l y a 50A c u r r e n t would produce a s u r f a c e temperature of about 1200 K. The main d i s a d v a n t a g e of t h i s method i s t h a t the s u p p o r t i n g f o i l and m a n i p u l a t o r l e g s h e a t up and can cause s e v e r e o u t g a s s i n g problems; (b) t h e c r y s t a l was p h y s i c a l l y clamped to a c o m m e r c i a l V a r i a n c o n d u c t i v e h e a t e r . T h i s h e a t e r i s l i m i t e d i n the t e m p e r a t u r e i t can produce m a i n l y due to the d i f f i c u l t y of a c h i e v i n g good t h e r m a l c o n t a c t between the c r y s t a l and the main h e a t e r b l o c k ; (c) t h e c r y s t a l was spot-welded to t h i n Kh or P t s t r i p s a t t a c h e d t o a s u p p o r t r i n g and heated by e l e c t r o n bombardment from a f i l a m e n t behind the c r y s t a l which i s f l o a t e d t o t y p i c a l l y +1-5kV. T h i s method l e a d s 85 US [ • k i t H i --V < c/) > o u < > o (/) CL C\J _Q *f\\\ \ \ M h 1 03 l i S S I i i U 5 Three aethcds of h e a t i n q a c r y s t a l sample: (a) d i r e c t r e s i s t i v e h e a t i n g , (fc) n s i n q a V a r i a n c o n d u c t i v e h e a t e r , <c) by e l e c t r o n bombardment. Batcfced l i n e s r e p r e s e n t s t a i r l e s s s t e e l a r c s t i c p l e c e r a m i c i n s u l a t o r s . Other > a t e r i a l s q e c e r a l l y Eh, F t , U or Ta. 86 to v e r y r a p i d h e a t i n g o f j u s t the sample t o h i g h t e m p e r a t u r e s . However, a c o r o l l a r y o f t h i s i s t h a t the c r y s t a l c o o l s down v e r y s l o w l y due t o the poor t h e r m a l c o n d u c t i o n t h a t a l l o w s t h e s e l e c t i v e h e a t i n g . Temperatures were measured by a l u m e l - c h r o m e l or Pt/13XBh-Pt thermocouples a t t a c h e d t o t h e c r y s t a l o r by an o p t i c a l pyrometer ijHartmann and Braun, F r a n k f u r t ) -C h emical c l e a n i n g most o f t e n c o n s i s t e d of h e a t i n g t h e sample i n about 1 0 - 5 T o r r of oxygen^ a p a r t i c u l a r l y u s e f u l procedure f o r removing s u r f a c e c a r b o n , which i s o n l y p o o r l y s p u t t e r e d by argon i o n s . The oxygen r e m a i n i n g on the s u r f a c e c o u l d then u s u a l l y be removed by i o n bombardment and/or h e a t i n g i n hydrogen. D e t a i l s o f t h e a c t u a l h e a t i n g p r o cedures used f o r the i n d i v i d u a l c r y s t a l f a c e s o f rhodium s t u d i e d a r e g i v e n i n the c h a p t e r s d e a l i n g w i t h each s u r f a c e . 4,3|b) M o n i t o r i n g s u r f a c e c o m p o s i t i o n The d i s c u s s i o n so f a r has assumed t h a t we have some method of knowing when the s u r f a c e i s , i n f a c t , c l e a n and when i t i s c o n t a m i n a t e d . F o r t u n a t e l y , such a s u r f a c e c o m p o s i t i o n monitor e x i s t s i n t h e form of Auger e l e c t r o n s p e c t r o s c o p y ( A B S ) , the elements of which were p r e s e n t e d i n C hapter 2. The e l e c t r o n o p t i c s were used as a r e t a r d i n g - f i e l d a n a l y s e r f o r AES , as shown i n F i g . 4.6. The V a r i a n e l e c t r o n gun 87 G u n - ' E l e c t r o n "1 qun c o n t r o l P nrrmr s in cot 3CX)v Neutraliser L o c k - i n Amp . Freq. x 1/ 2 s i n 2o>t X-Y Rotter Scope ll9UI§ J*JL6 Schematic diagram of L I E D o p t i c s used as a r e t a r d i n q f i e l d a n a l y s e r f o r Auqer e l e c t r o n s p e c t r o s c o p y : KCA = m u l t i c h a n n e l a n a l y s e r . 88 o p e r a t i n g i n t h e Auger mode t y p i c a l l y produces a beam of 10-15 microamps f o r a 1-5-2.5 kV p r i m a r y beam v o l t a g e (V^,). The f i r s t g r i d o f t h e o p t i c s i s grounded t o a l l o w the e l e c t r o n s t o move i n f i e l d - f r e e s p a ce, as i s the f o u r t h g r i d t o p r e v e n t f i e l d p e n e t r a t i o n from t h e p o s i t i v e l y b i a s e d f l u o r e s c e n t s c r e e n behind i t , here s i m p l y used as a c o l l e c t o r - In o r d e r t o e x t r a c t t h e Auger e l e c t r o n s , t h e 2nd and 3rd g r i d s a r e modulated by s i n w t about the r e t a r d v o l t a g e V r (V r<Vp), which i s ramped t y p i c a l l y f rom about 30 t o 4Q0eV. The 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 s 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 QPS2Q00).. The f r e q u e n c y doubled m o d u l a t i o n sin2wt was used as a r e f e r e n c e f o r a PAR HR-8 l o c k - i n a m p l i f i e r t u n e d t o d e t e c t the second d e r i v a t i v e o f t h e s i g n a l c o l l e c t e d by t h e s c r e e n . A s i m p l e c i r c u i t based on t h a t of Nathan and Hopkins £85] was used to n e u t r a l i s e the c a p a c i t i v e c o u p l i n g between th e r e t a r d g r i d s and t h e c o l l e c t o r . The o u t p u t of the l o c k - i n a m p l i f i e r was s t o r e d i n the m u l t i c h a n n e l a n a l y s e r and the spectrum swept u n t i l t h e s i g n a l - t o - n o i s e r a t i o was a c c e p t a b l e , a t which p o i n t i t c o u l d be p l o t t e d on an 1-1 p l o t t e r ( H e w l e t t - P a c k a r d 7004B). 89 4-4 LEED I n t e n s i t y Measurements The arrangement o f t h e e l e c t r o n o p t i c s f o r o p e r a t i o n i n the LEED mode i s s k e t c h e d i n F i g , 4.7- The i n n e r g r i d s a r e no* s e t to be a t the p r i m a r y beam v o l t a g e Vp minus ( u s u a l l y ) a s m a l l o f f s e t v o l t a g e { t y p i c a l l y about 5eV) . T h i s arrangement a l l o w s the whole o f t h e " p s e u d o - e l a s t i c " peak { S e c t i o n 2.1) t o be d i s p l a y e d on the f l u o r e s c e n t s c r e e n , now b i a s e d a t +5keV, w i t h t h e optimum c o n t r a s t between t h e d i f f r a c t e d beams and the background. The f i r s t and f o u r t h g r i d s a r e grounded, as b e f o r e f o r t h e measurement of Auger s p e c t r a , t o p r o v i d e f i e l d - f r e e d r i f t space and to p r e v e n t f i e l d p e n e t r a t i o n of t h e s c r e e n v o l t a g e r e s p e c t i v e l y . I n t h i s mode the e l e c t r o n gun i s u s u a l l y r u n i n the range 20 t o 300eV and s u p p l i e s a maximum c u r r e n t o f about 5 microamps. U n f o r t u n a t e l y t h e beam c u r r e n t o u t p u t i s a f u n c t i o n o f the beam energy as d e p i c t e d i n F i g . 4-8. The c u r v e shows a s t e a d y r i s e from v e r y low c u r r e n t a t low beam e n e r g i e s u n t i l a p l a t e a u i s r e a c h e d at about 1 GOe? where the beam c u r r e n t s t a y s c o n s t a n t t o w i t h i n 0.01 microamps- T h i s v a r i a t i o n i s i m p o r t a n t f o r t h e measurement , o f 1(E) c u r v e s s i n c e u n l e s s measured i n t e n s i t i e s a r e n o r m a l i s e d t o u n i t beam c u r r e n t , t h e beam i n t e n s i t i e s w i l l appear a r t i f i c i a l l y reduced at low e n e r g i e s because o f t h e low beam c u r r e n t s -9C Electron Gun control LEED , control l i S ^ I S J4A7 Schematic diagram of t h e e l e c t r o n o p t i c s used f o r L1ID e x p e r i m e n t s . i D ( H A ) 1.0H 0.5H 0.04 0 50 100 l i S - S I f i L i ? T y p i c a l v a r i a t i o n c f e l e c t r o n a g a i n s t beam v o l t a g e Vp i n the 1EED node. Vp(eV) gun bean 150 c u r r e n t I p 91 4.4(a) p r e v i o u s methods LEED i n t e n s i t y measurements i n t h e past have u s u a l l y , been . made by measuring the d i f f r a c t e d beam c u r r e n t e i t h e r d i r e c t l y w i t h a Faraday cup c o l l e c t o r i n s i d e the chamber, o r i n d i r e c t l y by e x t e r n a l measurement, w i t h a c a l i b r a t e d s p o t - p h o t o m e t e r , o f th e b r i g h t n e s s o f t h e s p o t s on the f l u o r e s c e n t - d i s p l a y s c r e e n . number o f comparisons have i n d i c a t e d t h a t b oth methods c a n p r o v i d e r e l i a b l e i n t e n s i t y d a t a £86,19 3. However, b o t h methods possess some d i s t i n c t disadvantages,: ( i ) i n e i t h e r method i t i s d i f f i c u l t t o ensure; t h a t the d e t e c t o r i s measuring the whole of t h e d i f f r a c t i o n s p o t ( i n the a n g u l a r sense) due t o the f i n i t e a p e r t u r e s i z e of t h e d e t e c t o r . T h i s i s e s p e c i a l l y a problem when beams appear t o s h r i n k and grow as they . * t" vary i n i n t e n s i t y ; ( i i ) a l l i e d t o ( i ) i s t h e problem o f p e r f o r m i n g adequate background s u b t r a c t i o n ; u s u a l l y p a r t i c u l a r v a l u e s a r e taken which may, o r may n o t , be r e p r e s e n t a t i v e of the t r u e average background around a d i f f r a c t i o n s p o t ; ( i i i ) b o t h methods a r e time-consuming and become awkward to use f o r complex LEED p a t t e r n s . N o n -specular beams move w i t h i n c i d e n t energy and t h e r e f o r e the d e t e c t o r must be moved t o t r a c k the beams. Hence c o l l e c t i n g a l a r g e amount o f d a t a f o r a l o t o f beams becomes a very clumsy p r o c e s s , r e g u i r i n g f r e g u e n t 92 r e - c a l i b r a t i o n and r e - c l e a n i n g of t h e s u r f a c e to overcome problems from c o n t a m i n a t i o n o r beam-surface i n t e r a c t i o n s I 87 ]. R e c e n t l y S t a i r e t a l £88] i n v e s t i g a t e d t h e p r o c e d u r e o f p h o t o g r a p h i n g t h e d i f f r a c t i o n s p o t s and d e t e r m i n i n g the r e l a t i v e i n t e n s i t i e s by s c a n n i n g t h e n e g a t i v e s w i t h a m e c h a n i c a l l y - d r i v e n m i c r o d e n s i t o m e t e r . T h i s method has t h e advantage of speed of d a t a c o l l e c t i o n , t h u s h e l p i n g t o p r e s e r v e the i n t e g r i t y of t h e s u r f a c e . The l a r g e amount of e x p e r i m e n t a l d a t a p r e s e n t on t h e photographs can be s u b s e g u e n t l y a n a l y s e d a t l e i s u r e . The main d i s a d v a n t a g e o f t h i s approach however, i s i n the a n a l y s i s . The whole o f each frame i s scanned and t h e i n t e n s i t y of each p o i n t on the f i l m , d e f i n e d by the a p e r t u r e o f the m i c r o d e n s i t o m e t e r i s r e c o r d e d on computer magnetic t a p e . The o u t p u t of each frame i s t h e n examined by a l a r g e computer t o deduce t h e l o c a t i o n o f d i f f r a c t i o n s p o t s and t h e i r i n t e g r a t e d i n t e n s i t i e s . P o s s i b l e problems a s s o c i a t e d w i t h l o s i n g s p o t s i n the background, f a l s e i d e n t i f i c a t i o n o f , f o r example, dust p a r t i c l e s as d i f f r a c t i o n f e a t u r e s means t h a t t h e o u t p u t from each frame has t o be checked v i s u a l l y . The a n a l y s i s p r o c e d u r e i s thus r a t h e r slow and i t makes l a r g e demands on computer s t o r a g e and.time. The n e x t s t a g e o f development i s t o combine t h e a t t r a c t i v e f e a t u r e s o f the p h o t o g r a p h i c p r o c e d u r e w i t h a more e f f i c i e n t and f l e x i b l e method o f a n a l y s i s , t h e r e b y s p e e d i n g up the p r o c e s s , c u t t i n g down on s o u r c e s of e r r o r , and r e d u c i n g t h e computer r e g u i r e m e n t s t o t h e p o i n t where t h e a n a l y s i s c o u l d be done on-9 3 l i n e on a mini-computer i n the l a b o r a t o r y . With t h e advent o f c o m p u t e r - c o n t r o l l a b l e V i d i c o n T.V. cameras, such an a n a l y s i s o f photographs of LEED p a t t e r n s became f e a s i b l e and was developed i n t h i s l a b o r a t o r y i n c o l l a b o r a t i o n w i t h Dr.F.R-Shepherd., 4.4(b) V i d i c o n measurement of 1(E) c u r v e s The LEED p a t t e r n s d i s p l a y e d on t h e f l u o r e s c e n t s c r e e n were photographed through the window of the,vacuum chamber u s i n g a Nikon F2 35mm camera w i t h an 85mm f1. 8 l e n s and a K2 e x t e n s i o n r i n g . Photographs were t a k e n g e n e r a l l y i n the 20 t o 250eV range a t 2eV i n t e r v a l s i n t h e i n c i d e n t beam energy u s i n g a f i x e d exposure of 1 sec a t f 4 , t h e i n c i d e n t beam c u r r e n t ( t y p i c a l l y 0.75 microamp) and energy b e i n g r e c o r d e d f o r each photograph. Using a motor d r i v e u n i t and a 250 exposure f i l m back, LEED p a t t e r n s f o r t h i s t y p e of energy range c o u l d be r e c o r d e d e a s i l y w i t h i n f i v e minutes. The s u r f a c e c o n d i t i o n was r o u t i n e l y checked w i t h Auger e l e c t r o n s p e c t r o s c o p y a f t e r measurements t o ensure t h a t no d e t e c t a b l e c o n t a m i n a t i o n had o c c u r r e d d u r i n g d a t a c o l l e c t i o n . A Kodak No. 2 c a l i b r a t e d s t e p d e n s i t y wedge was a l s o photographed on the same l e n g t h o f f i l m . ' S t a n d a r d Kodak T r i - X e m u l s i o n was used and t h e f i l m processed i n a c o n t i n u o u s l e n g t h i n A c u f i n e d e v e l o p e r a t 73°F f o r 7 minutes. The p h o t o g r a p h i c method depends on a knowledge of the d e t a i l e d response c u r v e of the em u l s i o n ( o p t i c a l f i l m d e n s i t y v e r s u s l o g exposure) s o t h a t measurement of the i n t e g r a t e d o p t i c a l d e n s i t y (D) of a d i f f r a c t i o n spot on t h e f i l m p r o v i d e s a 94 d i r e c t measure o f t h e amount o f l i g h t which caused the d a r k e n i n g of t h e f i l m i n t h a t r e g i o n o f t h e n e g a t i v e £89]. I n t u r n , i t i s assumed, as i n t h e s p o t - p h o t o m e t r i c method, t h a t t h e luminance £89] o f the s c r e e n 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 i m p i n g i n g e l e c t r o n f l u x so t h a t D i s r e l a t e d to the c o r r e s p o n d i n g d i f f r a c t e d beam c u r r e n t ( i ) by D = K i (4.1) where K i s a p r o p o r t i o n a l i t y c o n s t a n t . D i v i d i n g e g u a t i o n (4.1) throughout by the i n c i d e n t e l e c t r o n beam c u r r e n t • f i ) g i v e s | D / i 0 ) = K ( i / i c ) = k l ^ = I f ^ (4.2) where t h e d i s t i n c t i o n i s made between the a b s o l u t e d i f f r a c t e d beam i n t e n s i t y I A L j - a n c * * f l e r e l a t i v e d i f f r a c t e d beam i n t e n s i t y I f e l - The l a t t e r g u a n t i t y i s measured i n t h e p r e s e n t work. F i g . 4.9 shows a s c h e m a t i c diagram o f the a p p a r a t u s used t o a n a l y s e t h e p h o t o g r a p h i c n e g a t i v e s o f LEED p a t t e r n s . The V i d i c o n camera and a s s o c i a t e d e l e c t r o n i c s c o m p r i s e p a r t of the Computer Eye System { S p a t i a l Data Systems I n c . , G a l e t a , C a l i f o r n i a ) which was i n t e r f a c e d t o a mini-computer (Data G e n e r a l Nova 2 ) . The f i l m h e l d on t h e l i g h t t a b l e i s scanned c o n t i n u o u s l y by the camera, and the image d i s p l a y e d on the T-V. monitor i n a 512x480 (xy) a r r a y . The i n t e n s i t y ( z - v a l u e ) o f any element of the image may be sampled by t r i g g e r i n g t h e d i g i t i s e r w i t h a p p r o p r i a t e i n s t r u c t i o n s from the computer. A p r o f i l e r shows d i r e c t l y on the monitor t h e v a r i a t i o n of i n t e n s i t y a l o n g 9b vidicon camera film y transport light table scanner digitiser interface nova 2 computer cassette drive teletype joystick scope xy plotter T V monitor profiler f i S H i e 4j_9 S c h e n a t i c diagram of the a p p a r a t u s used t o a n a l y s e the p h o t o g r a p h i c n e g a t i v e s c f LI EE p a t t e r n s . 96 any s e l e c t e d v e r t i c a l l i n e of the image; t h i s may c o r r e s p o n d , f o r example, t o a l i n e through a d i f f r a c t i o n s p o t on the f i l m . The p r o f i l e r a l s o a s s i s t s i n s e t t i n g the range of o p t i c a l d e n s i t y c o v e r e d by the d i g i t i s e r ; t h e l a t t e r c a n r e s o l v e 256 z-v a l u e s o v e r a maximum span o f about 2.5 d e n s i t y u n i t s { c o r r e s p o n d i n g t o a change i n a b s o l u t e i n t e n s i t y by a f a c t o r of about 300). Another u s e f u l f e a t u r e o f t h i s system i s t h e " j o y s t i c k " . T h i s c o n t r o l s the p o s i t i o n o f a f l a s h i n g s p o t on t h e T.V. m o n i t o r , and may be used t o " p o i n t " t o any element of the image whose c o o r d i n a t e s w i t h i n t h e a r r a y a r e t h e n a v a i l a b l e to t h e computer. I n o r d e r to check the performance o f the V i d i c o n camera, a number o f t e s t s were made. The Kodak s t e p d e n s i t y wedge was p l a c e d on t h e l i g h t t a b l e and scanned. F i g . 4.J0 shows t h e d i g i t i s e r o u t p u t p l o t t e d a g a i n s t t h e c a l i b r a t e d v a l u e s c i t h e d e n s i t y wedge, and i t seems t h a t t h e camera response i s l i n e a r t o a good a p p r o x i m a t i o n over a range of a t l e a s t 1.6 d e n s i t y u n i t s . A f t e r a p r e l i m i n a r y warm-up, a number of s cans of t h e d e n s i t y wedge demonstrated t h a t the r e a d i n g s from th e d i g i t i s e r show good s t a b i l i t y o v e r p e r i o d s o f s e v e r a l h o u r s . The presence of f l a r e £89] i n t h e camera o p t i c s c o u l d be observed v i s u a l l y on t h e monitor when v e r y i n t e n s e l i g h t l e v e l s were f o c u s s e d on t h e V i d i c o n d e t e c t o r {e.g. w i t h the a p e r t u r e o f the l e n s f u l l y open). Under normal o p e r a t i n g c o n d i t i o n s a s e r i e s o f d i r e c t c omparisons of r e a d i n g s from the V i d i c o n camera and a c o n v e n t i o n a l f i n e - s p o t m i c r o d e n s i t o m e t e r were made from s c a n s of the d e n s i t y wedge and some t y p i c a l LEED p a t t e r n s ; e x c e l l e n t agreement was found w i t h i n t h e l i n e a r p o r t i o n o f E i g . 4 . 1 0 , i n d i c a t i n g t h a t f l a r e i s n o t a 97 250 0 0.4 0.8 1.2 .1.6 2.0 2A Calibrated optical density 2B l i f l u i e C a q i t i s e r o u tput l e a s u r e d f o r d i f f e r e n t r e q i c n s c f a KcdaJc l i e . 2 s t e p d e n s i t y wedqe and p l o t t e d a q a i n s t the c o r r e s p o n d i n g c a l i b r a t e d o p t i c a l d e n s i t i e s . 98 250 04 0.8 1.2 1.6 2.0 24 2.8 Original optical density l i f l u i f i b . l l D i g i t i s e r o u t p u t measured f o r d i f f e r e n t r e g i c r s of a p h o t o g r a p h i c n e g a t i v e c f t o e s t e p d e n s i t y wedge i n f i g u r e 4.10, p l o t t e d a g a i n s t t h e o r i q i n a l c a l i b r a t e d o p t i c a l d e n s i t y c f the wedge. The ar r o w s note the p o i n t s on the p l o t which c o r r e s p o n d t o t h e m i n i m a background and t h e maximum d e n s i t y observed on r h c t c q r a p h s of the 1EED p a t t e r n s from Cu(111). 99 s i g n i f i c a n t problem i n t h e s e measurements. F i g . 4.11 shows the r e s u l t s o f s c a n n i n g t h e n e g a t i v e image of the d e n s i t y wedge which was photographed a l o n g w i t h the d i f f r a c t i o n p a t t e r n s ; t h i s i n d i c a t e s t h e r e s p o n s e of t h e 35mm f i l m e m u l s i o n t o a s e r i e s of stepped exposures c o n t r o l l e d by the d e n s i t y wedge. In the i n t e r m e d i a t e r e g i o n o f t h e p l o t the response o f t h e f i l m i s a l i n e a r f u n c t i o n of the o r i g i n a l o p t i c a l d e n s i t y , b u t i n the extremes o f h i g h and low d e n s i t i e s t h e l i n e a r r e l a t i o n i s l o s t owing t o r e c i p r o c i t y f a i l u r e £89J. I t i s a l s o noted on F i g - 4.11 t h a t t h e maximum range o f d e n s i t y of d i f f r a c t i o n s p o t s observed on t h e photographs l i e s w e l l w i t h i n t h e l i n e a r p a r t of the response c u r v e , t h e r e b y s i m p l i f y i n g the c o n v e r s i o n o f d i g i t i s e r o u t p u t t o o p t i c a l d e n s i t y . A s i m p l i f i e d f l o w c h a r t of the computer program, w r i t t e n by Dr. F. a. Shepher d, c o n t r o l l i n g t h e s c a n n i n g p r o c e s s i s g i v e n i n F i g . 4-12. Under 'program i n t e r r u p t * , c o n t r o l c h a r a c t e r s i s s u e d from the t e l e t y p e can d i r e c t the program t o perform the v a r i o u s o p e r a t i o n s a s r e g u i r e d . For a g i v e n frame, t h e a n a l y s i s o f a p a r t i c u l a r d i f f r a c t e d beam proceeds by e n t e r i n g t h e energy and c u r r e n t of the i n c i d e n t e l e c t r o n beam through the t e l e t y p e , and the j o y s t i c k i s used t o p o i n t t o t h e spot viewed on the T.V. mo n i t o r . A s i g n a l from the t e l e t y p e i n i t i a t e s a r o u t i n e which scans and d i g i t i s e s an a r e a which i s c e n t r e d on t h e j o y s t i c k c o o r d i n a t e s and i s somewhat l a r g e r than the spot- T h i s i s rep e a t e d and t h e a r r a y of averaged z - v a l u e s i s s t o r e d i n the computer memory; a f t e r a t w o - d i m e n s i o n a l smoothing ope r a t i o n , t h e c o o r d i n a t e s of the maximum are determined and a new a r r a y 1C 0 E ° 3 display spot profile on CRO plot spot profile on XY plotter input beam energy and beam current Store current I (E) value disploy 1(E) curve on CRO plot l ( E ) curve on X Y plotter Store L ( E ) curve on cassette scon and digitise square orroy centered on joystick coords. signal average 4 scons smooth data and find coords of spot maximum scan new orroy around spot moximum signol average 4 scons calculate average background volue integrote z volues obove background level normalise to unit-beam current type integrated spot intensity on teletype Ham* Jlj.12 Flow c h a r t of the computer p r o q r a a which c o n t r o l s the s c a n n i n g o f the photographs. 101 c e n t r e d on t h e s e c o o r d i n a t e s i s scanned and d i g i t i s e d . . I n - o r d e r t o measure t h e background l e v e l , t h i s d i s t r i b u t i o n i s d i s p l a y e d on the T.V. m o n i t o r u s i n g t h e p r o f i l e r , and t h e e x t r e m i t i e s o f the d i f f r a c t i o n s p o t a re f i x e d v i s u a l l y and marked u s i n g the j o y s t i c k . Assuming a Gau s s i a n p r o f i l e , t he s t a n d a r d d e v i a t i o n i s found by d e t e r m i n i n g the f u l l w i d t h of t h e d i s t r i b u t i o n a t h a l f i t s maximum v a l u e above the l i n e j o i n i n g the marked e x t r e m i t i e s . The average of t h e z - v a l u e s of, a l l e l e m e n t s l y i n g on a t h i n a n n u l u s o f mean r a d i u s 2©- i s t a k e n as t h e background l e v e l z ^  ^. E x p e r i e n c e has shown t h a t a preprogrammed v a l u e o f JT can be i n s e r t e d i n t o t h e program i n o r d e r t o automate the background d e t e r m i n a t i o n w i t h o u t d e g r a d i n g the g u a l i t y of t h e a n a l y s i s . The i n t e g r a t i o n procedure i n v o l v e s summing a l l t h e v a l u e s of { z - Z j ^ k ) w i t h i n the c i r c l e o f r a d i u s 26", and d i v i d i n g by the i n c i d e n t beam c u r r e n t to g i v e a measure o f t h e d i f f r a c t e d beam i n t e n s i t y ( I ^ ) * - . T h i s n o r m a l i s a t i o n - i s i m p o r t a n t t o a v o i d d i s t o r t i o n o f i n t e n s i t y v a l u e s a t low e n e r g i e s where t h e v a r i a t i o n of i n c i d e n t beam c u r r e n t w i t h beam energy i s s u b s t a n t i a l , a s was seen i n F i g . 4.8. The a n a l y s i s o f one s p o t t a k e s o n l y . a few seconds. By s t u d y i n g s u c c e s s i v e frames f o r one d i f f r a c t e d beam, a complete 1(E) c u r v e i s g e n e r a t e d and s t o r e d on d i g i t a l c a s s e t t e and may be d i s p l a y e d on * the o s c i l l o s c o p e and p l o t t e d on t h e XY r e c o r d e r . A f t e r t h e i n i t i a l f r a m e, t h e c o o r d i n a t e s o f ; t h e d i f f r a c f i o n s p o t s a r e s t o r e d i n t h e computer s o t h a t when the n e x t frame i s p o s i t i o n e d i n a p p r o x i m a t e l y t h e same p l a c e under t h e camera u s i n g a marker, the a n a l y s i s o f t h i s next frame can s t a r t 102 w i t h o u t r e s p e c i f y i n g t h e p o s i t i o n s of t h e s p o t s v i a the j o y s t i c k . , The movement of the beams f o r a 2eV energy i n c r e m e n t i s s u f f i c i e n t l y s m a l l t h a t t h e i n i t i a l s e a r c h f o r the new d i f f r a c t i o n s p o t u s i n g the c o o r d i n a t e s o f the s p o t on the p r e v i o u s frame i s always s u c c e s s f u l . T h i s r e s u l t s i n s u b s t a n t i a l s a v i n g s i n a n a l y s i s time so t h a t a complete s e t of 1(E) c u r v e s f o r perhaps 10 beams can be produced i n a few h o u r s . 4.4(c) Measurements of I{E) c u r v e s f o r Cu (111) The Cu(111) s u r f a c e was used as a t e s t o f t h e r e l i a b i l i t y of t h e V i d i c o n v e r s i o n of t h e p h o t o g r a p h i c method f o r measuring e x p e r i m e n t a l I{E) c u r v e s . T h i s s u r f a c e was c o n v e n i e n t as some p r e v i o u s e x p e r i m e n t a l £19 3 and t h e o r e t i c a l s t u d i e s f 20 J were a v a i l a b l e . F i g u r e 4. 13 shows 1(E) c u r v e s measured f o r two n o n - s p e c u l a r beams a t normal i n c i d e n c e where the beam energy i s e x p r e s s e d r e l a t i v e t o t h e vacuum l e v e l . A l l t h e f e a t u r e s i n the f i g u r e a r e r e p r o d u c i b l e from one s e t o f measurements t o t h e nex t and r e a n a l y s i n g a p a r t i c u l a r s t r i p of f i l m s u g g e s t s t h a t u n c e r t a i n t i e s i n measuring o p t i c a l d e n s i t i e s {and hence d i f f r a c t e d beam i n t e n s i t i e s ) a r e t y p i c a l l y l e s s t h a n 5%. T h i s s l i g h t v a r i a t i o n must be a s c r i b e d t o a degree of u n c e r t a i n t y i n a p p l y i n g t h e background c o r r e c t i o n . N e v e r t h e l e s s t h e a s s o c i a t e d e r r o r s a re p r o b a b l y l e s s t h a n t h o s e i n v o l v e d , f o r example, w i t h some i n e v i t a b l e v a r i a b i l i t y i n s u r f a c e c o n d i t i o n s . At normal i n c i d e n c e the LEED p a t t e r n from Cu(111) shows t h r e e - f o l d symmetry, see F i g . 2.7, and so the same i n t e n s i t y 1 C 3 USSli 1(E) c u r v e s f o r t h e (11) ana (01) tea»s f r c a C u ( 1 1 1 ) at r e n a l i n c i d e n c e . 10 4 v a r i a t i o n i s e x p e c t e d w i t h i n each group of s y m m e t r i c a l l y - r e l a t e d beams, such as t h e . {10,01 ,-T"l) and (11,10,01) g r o u p s . T h i s i s i l l u s t r a t e d i n F i g . 4.14 where the measured i n t e n s i t i e s c f the (11) and (01) beams a r e c l o s e l y s i m i l a r to t h e (10) and (10) beams r e s p e c t i v e l y . Such b e h a v i o u r i s c o n s i s t e n t w i t h the s c r e e n phosphor r e s p o n d i n g i n a un i f o r m way t o t h e s e d i f f e r e n t , but e g u i v a l e n t , beams and t h i s r e p r e s e n t s an e s s e n t i a l r e g u i r e m e n t f o r both the p h o t o g r a p h i c and sp o t p h o i c f f i e t r i c methods. O f f normal i n c i d e n c e , t h e t h r e e - f e l d symmetry i n the n o n - s p e c u l a r beams i s l o s t and beam i n t e n s i t i e s depend cn b o t h t h e p o l a r and a z i m u t h a l a n g l e s of i n c i d e n c e . I n F i g - 4.15 a r e shown two 1(E) c u r v e s measured from photographs w i t h t h e V i d i c o n camera f o r the s p e c u l a r beam w i t h (a) "©•= 12°, 4>= 186° a a a |b) fe>= 12<>, 0= 6°; (c) r e p r o d u c e s t h e e x p e r i m e n t a l i n t e n s i t y d ata r e p o r t e d by woodruff and McDonnell [ 19j- w i t h a Faraday cup c o l l e c t o r f o r &- = 12°,</> = 7°. a l t h o u g h the a z i m u t h a l a n g l e does d i f f e r by one degree i n t h i s comparison t h e data r e p o r t e d i n r e f e r e n c e 19 showed t h a t the 1(E) c u r v e s f o r the s p e c u l a r beam v a r y s l o w l y w i t h <f> around t h e s e v a l u e s . Some d i f f e r e n c e s i n peak shapes between t h e two a n g l e s i n (a) and {b) a r e a p p a r e n t a l t h o u g h peak e n e r g i e s a r e repr o d u c e d v e r y w e l l . However, t h e c u r v e s (a) and (b) i n F i g . 4.15 s h o u l d be e x a c t l y e g u i v a l e n t because of time r e v e r s a l symmetry £90 1. The s l i g h t d i f f e r e n c e s i n shape observed must t h e r e f o r e be a t t r i b u t e d t o a degree of e x p e r i m e n t a l u n c e r t a i n t y . , To some e x t e n t t h i s c o u l d be due t o t h e beam energy b e i n g i n c r e a s e d i n f i x e d i n c r e m e n t s o f 2eV, even where t h e i n t e n s i t y i s c h a n g i n g 1 0 5 in •+-> E a •+-> ! q -t-» c •T1 01 (10) 1T 10 Cu(111) normal incidence (01) (10) 100 200 100 200 Beam Energy (eV) l i f l H I S i k l i f c u r v e 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 beans f o r t o n a l i n c i d e n c e on C u ( 1 1 1 ) . The i n s e t i n d i c a t e s the beam n o t a t i o n and a s p e c i f i c a t i o n of the a z i a u t h a l anqle if\ the a s t e r i s k i l l u s t r a t e s t he p o s i t i c r of the e l e c t r o n gun f c r the r c r - n c r a a l i n c i d e n c e case. 1 0 6 Cu (111) specular beam I 1 - i 1 - i 1 1 0 100 200 300 Beam Energy (eV) fiS5I§ ^U15 1(E) c u r v e s f o r t h e s p e c u l a r t e a i f o r Co (111): (a) *-=12° , ^ = 1860; (b) ©-=12<># «*=7°. The f i r s t twc were •easured by t h e »ethod d e s c r i b e d i n t h i s paper, and (c) r e p r e s e n t s • e a s u r e i e n t s Bade by Woodruff and McDonnell ( 1 9 1 «itb a Faradav cup c o l l e c t o r . 107 r e l a t i v e l y r a p i d l y w i t h energy, but o t h e r p o s s i b l e f a c t o r s i n c l u d e u n c e r t a i n t i e s i n the a n g l e s o f i n c i d e n c e ( t o the o r d e r o f one degree i n each s e t of measurements) and i n s u r f a c e c o n d i t i o n s . The p o i n t of s p e c i a l s i g n i f i c a n c e i s t h a t a l l t h r e e c u r v e s e x h i b i t the same f e a t u r e s a t the same beam e n e r g i e s t o w i t h i n our 2eV u n c e r t a i n t y . T h i s encourages us t o b e l i e v e t h a t the method d e s c r i b e d here i s a b l e t o approach t h e r e l i a b i l i t y of t h e Faraday cup method f o r measuring r e l a t i v e beam i n t e n s i t i e s . I n c o n t r a s t . S t a i r e t a l compared i n t e n s i t y d ata from the p h o t o g r a p h i c and s p o t p h o t o m e t r i c methods on t h e same s u r f a c e , b u t - s i n c e t h e s e methods have a number of a s s u m p t i o n s i n common, the comparison made h e r e , a l t h o u g h i t . i s o f l i m i t e d e x t e n t , may perhaps be seen as b e i n g t h e more s t r i n g e n t . .. I n t h i s method of measurement t h e r e i s no p h y s i c a l a p e r t u r e to d e f i n e t h e a n g u l a r s i z e o f a s p o t ; t h e s i z e i s f i x e d by the background s u b t r a c t i o n p r o c e d u r e which i s s u f f i c i e n t l y f l e x i b l e t h a t t h e e f f e c t i v e a p e r t u r e can be made l a r g e enough t o i n c l u d e the whole of the s p o t above background t o a good a p p r o x i m a t i o n and, i f n e c c e s s a r y , can be r e d e f i n e d d u r i n g t h e a n a l y s i s . Background s u b t r a c t i o n i s h a n d l e d i n an a c c u r a t e manner u s i n g an averaged v a l u e o f many elements i n an a n n u l u s s u r r o u n d i n g the s p o t . T h i s seems a p r e f e r a b l e ; p rocedure to the more common p r a c t i c e o f a v e r a g i n g o n l y a r e s t r i c t e d number of a r b i t r a r y background v a l u e s . The s h o r t d a t a c o l l e c t i o n t i m e s l i m i t s c o m p l i c a t i o n s due t o c o n t a m i n a t i o n o r beam-surface i n t e r a c t i o n s . The c o n v e n i e n c e of a "hard-copy" r e c o r d , which may be s u b s e g u e n t l y r e a n a l y s e d a t l e i s u r e , r e p r e s e n t s a n o t h e r 108 a t t r a c t i v e f e a t u r e of t h e p h o t o g r a p h i c method. 4. 4 (d) F u t u r e developments F u r t h e r improvements i n . the c o l l e c t i o n o f i n t e n s i t y d a t a a r e needed. As Jona £ 138J has emphasised, the i d e a l s o l u t i o n w i l l not o n l y be f a s t , but be o n - l i n e i . e . i n t e n s i t y d a t a w i l l be a v a i l a b l e w h i l e the e x p e r i m e n t i s i n p r o g r e s s . The p r e s e n t V i d i c o n system i n use h e r e has shown under t e s t t h a t i t i s not s e n s i t i v e enough to r e c o r d beam i n t e n s i t i e s d i r e c t from th e f l u o r e s c e n t s c r e e n . However, . i t i s c e r t a i n t h a t the n e x t g e n e r a t i o n of such cameras w i l l have i n c r e a s e d s e n s i t i v i t y . I n d eed, s h o r t l y a f t e r the f i r s t s e t of measurements, Heilmann e t a l £91] r e p o r t e d the s u c c e s s f u l use of a d i f f e r e n t make o f V i d i c o n camera to make such d i r e c t measurements; t h e s e a u t h o r s r e q u i r e d about 5 minutes t o c o l l e c t i n t e n s i t y data f o r 10 beams to 200eV. Many p r e s e n t V i d i c o n cameras would be adequate f o r d i r e c t measurements i f t h e b r i g h t n e s s of t h e d i f f r a c t i o n s p o t s seen by the camera c o u l d be i n c r e a s e d . I n p r i n c i p l e t h i s can be a c h i e v e d e i t h e r e x t e r n a l l y w i t h the use of an i m a g e - i n t e n s i f i e r , or i n t e r n a l l y w i t h a c h a n n e l p l a t e e l e c t r o n m u l t i p l i e r . The l a t t e r method has been used t o measure p h o t o e m i s s i o n a n g u l a r d i s t r i b u t i o n s £92], and f o r the p r o d u c t i o n of v i s i b l e LEED p a t t e r n s w i t h very weak (nanoamp) beam c u r r e n t s £137]. An a t t r a c t i v e a s p e c t o f p o s s i b l e f u t u r e r e d u c t i o n s i n i n c i d e n t beam c u r r e n t s i s the a s s o c i a t e d p o s s i b i l i t y of reduced beam-surface i n t e r a c t i o n s . 10 9 One o f t h e problems of any image a n a l y s i n g d e v i c e i s the amount of time s p e n t i n d i g i t i s i n g unwanted background p o r t i o n s o f the LEED p a t t e r n . At normal ; i n c i d e n t e n e r g i e s t h e a r e a of the s c r e e n o c c u p i e d by t h e d i f f r a c t e d beams i s o n l y a s m a l l f r a c t i o n of t h e t o t a l . We have a v o i d e d t h i s problem by r e c o r d i n g the d a t a i n a hard-copy form and t h e n o n l y a n a l y s i n g the f e a t u r e s o f i n t e r e s t i n a s e m i - i n t e r a c t i v e manner. The p r o v i s i o n o f a f a s t o n - l i n e ^ r e c o r d i n g s y s t e m , at a r e a s o n a b l e c o s t , t h a t a u t o m a t i c a l l y r e c o r d s o n l y s i g n i f i c a n t f e a t u r e s , and y e t can d i f f e r e n t i a t e t r u e d i f f r a c t i o n s p o t s from s p u r i o u s e v e n t s , i s s t i l l a major task., 110 CHAPTER 5 LEED CEYSTALLOGBAPHY 1.11 5.1 G e n e r a l C o n s i d e r a t i o n s We have seen t h a t the i n t e n s i t i e s of e x p e r i m e n t a l LEED beams c o n t a i n i n f o r m a t i o n ; on the s t r u c t u r e o f t h e s u r f a c e i n b o t h t h e p a r a l l e l and p e r p e n d i c u l a r d i r e c t i o n s . . F o r t u n a t e l y , 1(E) c u r v e s c a l c u l a t e d f o r t r i a l s t r u c t u r e s a r e g u i t e s e n s i t i v e t o changes i n t h e s u r f a c e geometry as we saw i n S e c t i o n 3 . 2 ( c ) . Changes i n n o n - s t r u c t u r a l parameters such as the i n e l a s t i c damping, v i b r a t i o n a l p r o p e r t i e s e t c . produce u n i f o r m s m a l l changes .in... i n t e n s i t y c r p o s i t i o n f o r a l l the peaks i n an 1(E) c u r v e 1 1 2 J . The u s u a l p r o c e d u r e f o l l o w e d t o o b t a i n s t r u c t u r a l i n f o r m a t i o n w i t h LEED has been a t r i a l and e r r o r p r o c e s s c o n s i s t i n g o f t h r e e s t e p s : i ( i ) e x p e r i m e n t a l d a t a I s c o l l e c t e d f o r a number o f d i f f r a c t e d beams a t s e v e r a l a n g l e s of i n c i d e n c e ; ( i i ) a t h e o r e t i c a l s e t of da t a i s produced f o r the same beams by p o s t u l a t i n g a s t r u c t u r a l model and c h o o s i n g a number o f n o n - s t r u c t u r a l parameters; ( i i i ) s t r u c t u r a l (and p r e f e r a b l y , but by no means o f t e n , n o n - s t r u c t u r a l parameters) are v a r i e d i n the c a l c u l a t i o n s u n t i l an agreement i s found between 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 d a t a . agreement between t h e e x p e r i m e n t a l d a t a and the c a l c u l a t i o n s f o r a p a r t i c u l a r s t r u c t u r a l model i s u s u a l l y a c c e p t e d as the pr o o f f o r the c o r r e c t n e s s o f the p o s t u l a t e d 112 model, U n d e r l y i n g , t h i s i s the assumption t h a t s u c h agreement, e s p e c i a l l y i f i t i n v o l v e s a , l a r g e d a t a b a s e , c a n n o t be a c c i d e n t a l . S m a l l d i s c r e p a n c i e s between t h 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 s e t s o f d a t a a r e u s u a l l y a t t r i b u t e d t o such f a c t o r s as e x p e r i m e n t a l e r r o r s i n s e t t i n g t h e i n c i d e n c e a n g l e , poor p r e p a r a t i o n o f the c r y s t a l s u r f a c e o r s l i g h t l y i n c o r r e c t . v a l u e s of the p o o r l y known n o n - s t r u c t u r a l parameters t h a t e n t e r i n t o the c a l c u l a t i o n s , such as t h e s u r f a c e Debye t e m p e r a t u r e . T h e r e f o r e d e t e r m i n i n g the g u a l i t y of f i t between the c a l c u l a t i o n s and experiment i s c r u c i a l . I n g e n e r a l , t h i s e v a l u a t i o n has been performed v i s u a l l y . I h i l e the eye i s a b l e to make f i n e d i s t i n c t i o n s between c l o s e l y s i m i l a r d a t a , i t i s i m p o s s i b l e t o o b j e c t i v e l y e v a l u a t e t h e l a r g e d a t a base needed i f one i s t o have c o n f i d e n c e i n t h e f i n a l r e s u l t . F o r example, l e t us suppose t h a t we have e x p e r i m e n t a l data f o r 10 d i f f r a c t e d beams and have c a l c u l a t e d t h e c o r r e s p o n d i n g I ( E ) c u r v e s f o r 7 d i f f e r e n t s t r u c t u r a l models, as would be t h e c a s e f o r a t y p i c a l range o f e x p a n s i o n s and c o n t r a c t i o n s of the s u r f a c e l a y e r r e l a t i v e t o the b u l k . V i s u a l e x a m i n a t i o n would t h e n i n v o l v e 70 d i f f e r e n t comparisons t o be s y n t h e s i s e d i n t o one f i n a l r e s u l t . I f a n o n - s t r u c t u r a l parameter such as t h e i n n e r p o t e n t i a l i s then a l s o a l l o w e d t o v a r y , t h e t o t a l number of comparisons t o be made, r a p i d l y becomes unmanageable i n a v i s u a l a n a l y s i s , even when o b v i o u s m i s f i t s a re d i s c o u n t e d . The d i f f i c u l t i e s a r e f u r t h e r i n c r e a s e d i f , a s i s o f t e n t h e c a s e , a change i n one parameter improves t h e c o r r e s p o n d e n c e w i t h e x p e r i m e n t f o r some c u r v e s but worsens those o f o t h e r s . The conseguence o f t h e s e d i f f i c u l t i e s i s an unknown u n c e r t a i n t y i n the v a l u e s of the 113 " c o r r e c t " parameters. T h e r e f o r e , t h e r e i s a r e a l need i n LEED c r y s t a l l o g r a p h y f o r a n u m e r i c a l f a c t o r which i d e a l l y can both s e l e c t t h e b e s t s t r u c t u r e from t h e e x p e r i m e n t a l and c a l c u l a t e d i n t e n s i t i e s and can g i v e some measure o f t h e r e l i a b i l i t y of t h a t r e s u l t . Such a r e l i a b i l i t y - f a c t o r or i n d e x i s n e c e s s a r i l y complex because a whole s e t of f e a t u r e s has t o =be e v a l u a t e d i n the c o m p a r i s o n be ween c a l c u l a t i o n s and o b s e r v a t i o n s . These f e a t u r e s i n c l u d e : ( i ) the g e n e r a l shape o f t h e I|E) c u r v e s , r e g a r d l e s s o f t h e a b s o l u t e i n t e n s i t i e s ; <ii) the number and e n e r g i e s o f maxima,minima, s h o u l d e r s e t c . ; ( i i i ) the presence o f p o r t i o n s of c u r v e s w i t h marked p e c u l i a r i t i e s e.g. narrow peaks, deep t r o u g h s . i S e v e r a l a t t e m p t s have been made t o c o n s t r u c t r e l i a b i l i t y -i n d i c e s t h a t t a k e r e g a r d o f seme or a l l of the above p o i n t s and they a r e o u t l i n e d i n the n e x t s e c t i o n . The most complete i n d e x i s t h a t o f Z a n a z z i and Jona | 2 3 J which i s d i s c u s s e d i n d e t a i l -114 5- 2 R e l i a M l i t y - i a d i c e s S e v e r a l a u t h o r s have attempted t o c o n s t r u c t s i m p l e r e l i a b i l i t y - i n d i c e s t o complement v i s u a l e v a l u a t i o n s . One o f the most s t r a i g h t f o r w a r d i n v o l v e s c a l c u l a t i n g the mean v a l u e o f the d i f f e r e n c e i n c o r r e s p o n d i n g peak p o s f i o o s i n t h 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 c u r v e s where the E ^ a r e t h e e n e r g i e s a t which t h e i peak o c c u r s i n the c a l c u l a t e d and o b s e r v e d c u r v e s and N t h e number of peaks i n t h e beam under s t u d y £93,94]. The most i m p o r t a n t d e f e c t o f t h e &E c r i t e r i o n i s t h a t i t c o m p l e t e l y d i s r e g a r d s the peak i n t e n s i t i e s and weighs e x c l u s i v e l y t h e i r p o s i t i o n s . Other d i s a d v a n t a g e s a r e p o s s i b l e a m b i g u i t i e s i n t h e c h o i c e o f peaks to be matched and degrees of s u b j e c t i v i t y i n p o s i t i o n i n g p o o r l y -r e s o l v e d f e a t u r e s and d e c i d i n g whether a minor peak s h o u l d , o r s h o u l d not be c o n s i d e r e d . R e c e n t l y Van Hove e t a l £9.5], co m b i n i n g t h e e x p e r i e n c e o f e a r l i e r a t t e m p t s , have proposed t h e use c f f i v e s e p a r a t e r e l i a b i l i t y - i n d i c e s , each o f which t e n d t o p i c k out d i f f e r e n t f e a t u r e s of I (E) c u r v e s . The f i r s t two a r e s i m i l a r t o t h o s e employed i n s t r u c t u r e d e t e r m i n a t i o n s by X-ray c r y s t a l l o g r a p h y «S. 1) £ 9 6 , 9 7 J 115 R1 .= Z. §c,I- | - I - i J / 2 I- l r (5.2) £tL f t -and H2 = ^ ^ l 1 ^ * ! ' X . o b * 2 ' ^ ^ l , * ^ 2 * 5 - 3 ) where c • i s a s c a l i n g f a c t o r t h a t p l a c e s t h e obser v e d and c a l c u l a t e d i n t e n s i t i e s on t h e same s c a l e f o r an energy range E((. to E ^ . The f a c t o r s fi1 and R2 tend t o emphasise the match i n p o s i t i o n s , h e i g h t s and w i d t h s o f peaks and v a l l e y s w h i l s t i g n o r i n g s h o u l d e r s and bumps w i t h i n a peak. a t h i r d , R3, e v a l u a t e s the f r a c t i o n of the energy range f o r which t h e s l o p e s of 1(E) c u r v e s have d i f f e r e n t s i g n . T h i s can measure, b e t t e r t h a n - R l o r 82, d i f f e r e n c e s i n minor s t r u c t u r e . The f a c t o r s R4 = 2. , - 1»- [• 1 (5.4) Eli tiC and R5 = 2 1 ^.i'* V ^ 1 Z'l . l r 3* * 5 * 5 > where the primes i n d i c a t e d e r i v a t i v e s , match i n g r e a t e r d e t a i l t h e s l o p e s o f 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 1(E) c u r v e s . I n p r i n c i p l e , when experiment and t h e o r y match c l o s e l y , a l l f i v e R-f a c t o r s s h o u l d s i m u l t a n e o u s l y a t t a i n t h e i r minimum v a l u e s . However, i f t h e proposed s u r f a c e s t r u c t u r e i s wrong, then the v a r i o u s f a c t o r s c o u l d be e x p e c t e d t o show a s c a t t e r i n g o f minima at v a r i o u s s p a c i n g s . The most complete r e l i a b i l i t y - i n d e x proposed so f a r t h a t a t t e m p t s t o i n c l u d e a l l t h e f e a t u r e s mentioned above i s t h a t of Z a n a z z i and Jona (ZJ) £23], T h i s i n d e x has been used i n t h i s 116 work t o determine t h e s t r u c t u r e s of rhodium s u r f a c e s . 5.2 (a) t h e r e l i a b i l i t y - i n d e x of Z a n a z z i and Jona (ZJ) The Z J i n d e x compares d i r e c t l y t h e shapes o f t h e two c u r v e s under s c r u t i n y by comparing t h e i r d e r i v a t i v e s , and a t the same time t a k e s i n t o a c c o u n t t h e f e a t u r e s d e s c r i b e d i n S e c t i o n 5 .1 . Thus t h e r e l i a b i l i t y - i n d e x f o r a s i n g l e beam {..23J i s r r • = w(E)|cI». , - I ! . |dE / I I- i dE (5.6) The n o t a t i o n f o l l o w s t h a t above; t h e s c a l i n g c o n s t a n t c^ a g a i n a l l o w s f o r an a r b i t r a r y i n t e n s i t y s c a l e i n t h e e x p e r i m e n t a l c u r v e s : r c; = | I c ^ d E / c I . c J d E (5.7) The weight f u n c t i o n w(E) w ( E ) = -|c-I« ,- I " , I / JI«- | I + € (5.8) emphasises the extrema.of the o b s e r v e d c u r v e t h r o u g h t h e i n v e r s e dependence on i - < } j ) / # a s s e l l as t h o s e prominent f e a t u r e s w i t h h i g h c u r v a t u r e i n b o t h s e t s of c u r v e s through t h e use o f second d e r i v a t i v e s . The d i f f e r e n c e f u n c t i o n i n the numerator p r e v e n t s w{E) from v a n i s h i n g a t i n f l e c t i o n p o i n t s , u n l e s s both c u r v e s have an i n f l e c t i o n p o i n t a t t h e same energy. A p o s s i b l e d i s a d v a n t a g e o f e q u a t i o n (5.8) i s t h a t w{E) v a n i s h e s i f the two 117 c u r v e s have s t r a i g h t p o r t i o n s i n t h e same energy range, even i f of o p p o s i t e s l o p e . The f a c t o r £, p r e v e n t s d i v e r g e n c e of the i n t e g r a l f o r s m a l l H » . J and i s chosen £23] t o he LyobJ £- = H I , \ (5.9) T h i s c h o i c e e n s u r e s t h a t w (E) i s independent of t h e s c a l e o f bot h t h e 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 . As e g u a t i o n (5.6) s t a n d s , t h e i n d e x r ^ has no t h e o r e t i c a l upper l i m i t . For a g i v e n p a i r of c u r v e s i t has a s p e c i f i c v a l u e ; the s m a l l e r t h i s v a l u e , the b e t t e r t h e match between t h e two c u r v e s . Thus i t i s c o n v e n i e n t t o i n t r o d u c e the reduced r -f a c t o r £23] (r . ) = r- /0.027 (5. 10) where 0.027 i s an average v a l u e o f r ^ f o r random p a i r s o f c u r v e s £23]. Eandom p a i r s o f c u r v e s t h u s g i v e v a l u e s of ( r )• of the o r d e r of u n i t y . Z J a p p l i e d t h i s r e l i a b i l i t y - i n d e x t o many examples of s i n g l e beams and made a d i r e c t a s s o c i a t i o n between v a l u e s o f £ r r ) t and degrees of f i t a s s e s s e d v i s u a l l y by a number o f i n d i v i d u a l s . T h i s e n a b l e d ZJ t o produce a corres p o n d e n c e between v i s u a l e v a l u a t i o n s and (r^.)^ v a l u e s which i s repro d u c e d i i n T a b l e 5.1. The d e t e r m i n a t i o n of a s u r f a c e s t r u c t u r e i n v o l v e s more; than one beam, e i t h e r at t h e same o r d i f f e r e n t a n g l e s of i n c i d e n c e . The concept of a s i n g l e beam reduced r - f a c t o r was t h e r e f o r e 11.8 V i s u a l match f o r Good Mediocre Bad a s i n g l e beam S i n g l e beam i n d e x (r,),; 0.20 0.3 5 0.50 or many-beam i n d e x fi R e l i a b i l i t y o f Very p r o b a b l e P r o b a b l e D o u b t f u l s t r u c t u r e T a b l e 5.1 Correspondence between v i s u a l , m a t c h and (t,)^ f o r a s i n g l e beam ( f i r s t and second r o w ) , and between B f o r a s t r u c t u r a l model and i t s r e l i a b i l i t y (second and t h i r d row). A f t e r Z a n a z z i and Jona £ 23J. extended by 2J t o a s e t of beams where A E • = E . -E 21 It The q u a n t i t y r r s a t i s f i e s many of the r e q u i r e m e n t s f o r a r e l i a b i l i t y - i n d e x i n LEED c r y s t a l l o g r a p h y . However, LEED s t r u c t u r a l a n a l y s e s have been performed on d a t a bases o f v a r y i n g degrees of co m p l e t e n e s s . S t r u c t u r a l models c o u l d be proposed on the b a s i s of o n l y a few beams a t one angle of i n c i d e n c e , o r , p r e f e r a b l y from a much l a r g e r range o f e x p e r i m e n t a l d a t a . To accommodate these c o n s i d e r a t i o n s , ZJ proposed an o v e r a l l 8-f a c t o r 8 = C (P/n) +g] r" r (5. 12) where p and g a r e c o n s t a n t s , and n-•. i s t h e number of beams t e s t e d . T h i s form was used so t h a t the b r a c k e t e d term s h o u l d decrease w i t h i n c r e a s i n g n but should.; never become s m a l l e r than 119 some a s y m p t o t i c v a l u e q. T h i s a v o i d s the s i t u a t i o n i n which a "wrong" model f o r which a v e r y l a r g e number of beams were t e s t e d c o u l d produce a s m a l l v a l u e o f B. The n u m e r i c a l v a l u e s o f p and q a r e d e t e r m i n e d by e s t a b l i s h i n g a r e l a t i o n s h i p between the n u m e r i c a l v a l u e o f fi and t h e r e l i a b i l i t y o f t h e c o r r e s p o n d i n g s t r u c t u r a l model. For s i m p l i c i t y Z J found i t c o n v e n i e n t t o m a i n t a i n t h e assignment t h a t was e s t a b l i s h e d f o r s i n g l e beams i n the f i r s t two rows of T a b l e 5-1-, Thus, by u s i n g the d e n o m i n a t i o n s o f " v e r y p r o b a b l e " e t c . f o r a g i v e n s t r u c t u r e , they a r r i v e d a t the B v a l u e a s s i g m e n t s shown i n t h e second and t h i r d rows. T h i s assignment then s u g g e s t s p o s s i b l e v a l u e s f o r p and g i n e g u a t i o n ( 5 . 1 2 ) . The a s y m p t o t i c value g must be such t h a t , where th e ( r r ) j v a l u e s are o f t h e same o r d e r as those o b t a i n e d from comparing random c u r v e s , t h e c o r r e s p o n d i n g s t r u c t u r e B - f a c t o r must remain worse t h a n " d o u b t f u l " r e g a r d l e s s o f t h e number of beams used i n the a n a l y s i s . The v a l u e of g was t h e r e f o r e s e t t o 2/3. A l s o , t h e v a l u e o f the b r a c k e t e d f u n c t i o n i n e g u a t i o n (5.12) s h o u l d i n c r e a s e r a p i d l y f o r n<3, so t h a t s t r u c t u r a l models w i l l not be judged as very p r o b a b l e on the b a s i s of l e s s t h a n 3 beams. The o v e r a l l R - f a c t o r proposed by ZJ i s B = I (3/2n) • (2/3) J \ r ^ (5-13) The c h o i c e o f p=3/2 has t h e advantage t h a t t h e n u m e r i c a l v a l u e of fi does not change much beyond n=10. B remains a l m o s t c o n s t a n t f o r s t r u c t u r e s t h a t are determined w i t h 10,15 or 20 120 teams. , The d e t a i l s o f the n u m e r i c a l e v a l u a t i o n o f e q u a t i o n <5.6J a r e g i v e n i n t h e o r i g i n a l paper o f ZJ .£23 . To a v o i d i n s t a b i l i t y problems w i t h t h e i n t e g r a l s i n v o l v e d , a l l 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 d a t a a r e put onto a f i n e l y - s p a c e d common ^ l i n e a r energy g r i d . H o i s e and s c a t t e r i n the e x p e r i m e n t a l data must be removed p r i o r t o t h e c a l c u l a t i o n o f t h e r e l i a b i l i t y - i n d e x t o e l i m i n a t e a b r u p t changes i n the d e r i v a t i v e s . The programmed i n d e x was used as s u p p l i e d by t h e a u t h o r s w i t h o n l y minor m o d i f i c a t i o n s t o accommodate i n p u t -o u t p u t r e g u i r e m e n t s . 5.3 S t r u c t u r a l A n a l y s i s U s i n g The ZJ fie1ia h i l i t y F a e % o r : T he Cu (111) S u r f a c e As An Example R e s u l t s from a Cu{111) s u r f a c e w i l l be used t o i l l u s t r a t e t h e use of t h e ZJ r e l i a b i l i t y - i n d e x and t h e r e f i n e m e n t s and a d d i t i o n s made t o i t d u r i n g the c o u r s e o f t h i s work. R e f e r r i n g t o F i g . 3.11 and F i g - 5.1, we can see 1{E) c u r v e s f o r a t o t a l o f f o u r beams a t two a n g l e s o f i n c i d e n c e , t o g e t h e r »ith the c o r r e s p o n d i n g c u r v e s c a l c u l a t e d f o r an u n r e c o n s t r u c t e d s u r f a c e u s i n g t h e V, and V, p o t e n t i a l s . The c a l c u l a t i o n s were performed f o r an i n i t i a l i n n e r p o t e n t i a l o f -9.5eV and f o r a range of t o p l a y e r s p a c i n g s from Ad%= -10S t o +-1055 o f the bulk s p a c i n g i n s t e p s o f 2.51 (Q.Q52A). The: c u r v e s shown i n the f i g u r e s a r e o n l y f o r t h r e e v a l u e s o f &&% t o a v o i d o v e r c r o w d i n g . The 1(E) c u r v e s c a l c u l a t e d f o r Cu {111) from the two p o t e n t i a l s a r e almost i d e n t i c a l , c o n s i s t e n t w i t h t h e s i m i l a r i t i e s noted f o r 1 2 1 Energy <eV) IiSi>I£ 5 .1 Comparison of some e x p e r i m e n t a l 1(E) c u r v e s f o r Cu(111) mith c a l c u l a t i o r s f o r the p o t e n t i a l s 1*- and V,., at «=12°, ^ = 6 0 ; ? = - 9 . 5 e V and Ad* = *5, 0 and 5 X . * 122 the phase s h i f t s i n S e c t i o n 3.1(b) -In a s u r f a c e s t r u c t u r e a n a l y s i s , t h e f i n a l B a t c h i n g o f 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 i n v o l v e s n o t - o n l y a v a r i a t i o n of A d % but a l s o a v a r i a t i o n of t h e i n n e r p o t e n t i a l . T h i s l a t t e r parameter i s f i x e d a p r i o r i f o r the c a l c u l a t i o n s but i s a p o o r l y known g u a n t i t y . The i n n e r p o t e n t i a l i s c l e a r l y an i m p o r t a n t g u a n t i t y s i n c e , i n a s t r u c t u r a l a n a l y s i s , i t i s n e c e s s a r y t o d i s t i n g u i s h i t s e f f e c t s from those a s s o c i a t e d w i t h Ad$. To a good a p p r o x i m a t i o n , a change i n t h e i n n e r p o t e n t i a l r e s u l t s i n a t r a n s l a t i o n o f t h e c a l c u l a t e d c u r v e a l o n g the energy a x i s . A v i s u a l e s t i m a t i o n o f the " b e s t - f i t " v a l u e of &&% t h e r e f o r e a l s o i n v o l v e s a s i m u l t a n e o u s d e t e r m i n a t i o n o f t h e " b e s t - f i t " v a l u e of t h e i n n e r p o t e n t i a l - T h i s d e l i c a t e procedure was attempted f o r a l l 16 beams measured from the Cu(111). s u r f a c e and these r e s u l t s a r e shown, f o r each p o t e n t i a l , as the l o w e r rows of T a b l e 5.2. T h i s t y p e o f comparison i s very d i f f i c u l t t o make as a v i s u a l e s t i m a t i o n o f t h e degree of f i t may v a r y from very good f o r some beams e.g. t h e (00) beam o f f i g . 5.2 a t &&%--5%t t o g u i t e poor f o r o t h e r s , e.g.. t h e (10) beam of E i g . , 3 . 1 1 a t /\,&%=+5%. B a l a n c i n g good f i t s a g a i n s t bad f i t s f o r so many p a i r s of e x p e r i m e n t a l and c a l c u l a t e d c u r v e s i s d i f f i c u l t and t h e u n c e r t a i n t y i n V o p and b&% a r e i n e v i t a b l y a p p r e c i a b l e . Moreover t h i s a n a l y s i s shows the two p o t e n t i a l s t o be i n d i s t i n g u i s h a b l e . The o v e r a l l v i s u a l e s t i m a t e s u g g e s t s b e s t - f i t v a l u e s o f V o f =-9±2eV and &d$=-2.5±2. 5%. The e g u i v a l e n t o p e r a t i o n u s i n g the ZJ r e l i a b i l i t y - i n d e x i s as f o l l o w s : ( i ) an 1(E) c u r v e i s c a l c u l a t e d w i t h a r e a s o n a b l e 123 E x t e n t of C o n d i t i o n s of Comparison b e s t a g r e e m e n t Pot. n EfeV) A n a l y s i s A d J -,8± v i s u a l -2. 5±2-5 -9 ±2 V, 16 2382 R - f a c t o r -4. 1±0. 6 -9- 0.6 0.132 Cc 16 2342 v i s u a l -2-5±2.5 -9 ±2 V, 16 2342 R-f a c t o r -4.2±0.6 -9.0 + 0.6 0. 136 CU. T a b l e 5.2 Summary of s t r u c t u r a l d e t e r m i n a t i o n s o f t h e Cu(111) s u r f a c e . v a l u e of Var; ( i i ) t h e v a l u e s of ( r r ) j a r e computed f o r a number c f p a i r s formed by the observed c u r v e on one hand and t h e c a l c u l a t e d c u r v e t r a n s l a t e d by v a r i o u s amounts A Vor on the o t h e r . The minimum v a l u e of ( r r ) j i n d i c a t e s t h e b e s t match, and the c o r r e s p o n d i n g "best 1* i n n e r p o t e n t i a l f o r the c a l c u l a t e d c u r v e i s V6(. *&V o r. , For a number o f beams, i s c a l c u l a t e d from t h e s i n g l e - b e a m d a t a and the. minimum i n ~iTr f o u n d - The p r o c e s s can then be r e p e a t e d f o r o t h e r models e.g- a change i n Ad35. In t h e i r o r i g i n a l a n a l y s i s ZJ p l o t t e d t h e r e s u l t s of such a procedure as ~E~r ( t h e energy- weighted mean o f t h e i n d i v i d u a l beam r - f a c t o r s ( ^ ) L ) # a s i n e g u a t i o n (5. 11), a g a i n s t V f t r or &d% and t h e b e s t c o m b i n a t i o n of minima »as chosen by i n s p e c t i o n . An example of t h i s method of p r e s e n t a t i o n i s shown f o r the Cuj1.11) d a t a i n F i g . 5.2. T h i s a n a l y s i s uses t h e V p o t e n t i a l , a t o t a l o f 16 beams and e x p e r i m e n t a l . 1(E) c u r v e s o v e r a t o t a l energy range o f 2342eV. The minimum v a l u e o f "ry- i s e s t i m a t e d by t h i s method t o be f o r AdI=-4±2& and V o f =-9±2eV. We can n o t e two p o i n t s from such p l o t s . F i r s t , the e s t i m a t e s of t h e p o s i t i o n of F r a l o n g t h e &&% and V o r axes a r e s u b j e c t t o e r r o r owing t o t h e l a r g e s p a c i n g between p o i n t s a l o n g 12a ' — i r 1 r~ -10 -5. 0 +5 +10 f i a u i f 5^2 P l o t of r r a q a i n s t Ad% f o r v a r i o u s v a l u e s c f Vor f o r Co (111) w i t h t h e ¥ ^ 3 p o t e n t i a l . E r r o r fcars are s t a n d a r d e r r o r s i n the weighted l e a n . 125 the &d% a x i s (2.5%). > and the V o r a x i s j 2 e V ) . A f i n e r g r i d , p a r t i c u l a r l y o f Q&% v a l u e s , would however be r a t h e r c o s t l y i n terms of computer time-Second, each p o i n t on one o f t h e c u r v e s i n F i g . 5-2 i s a weighted mean over the 16 beams a v a i l a b l e f o r : a n a l y s i s - The s t a n d a r d e r r o r on the weighted mean i s £98] Cv = £ Ji^rk _ Y r > 2 i n " 1> ^ * E C 3 1 / z (5. 13) where n i s t h e number of beams and t h e o t h e r symbols are as b e f o r e . Some t y p i c a l v a l u e s of Er a r e i n d i c a t e d by b a r s on F i g . 5-2 aud can be s u b s t a n t i a l . T h i s opens up the g u e s t i o n o f why t h e c u r v e s shown i n F i g . 5.2 a r e so smooth, g i v e n t h a t the e r r o r s can be so l a r g e . A r a t i o n a l i s a t i o n o f t h i s apparent anomaly i s suggested by F i g - 5-3 where v a l u e s of t r ^ ) ^ , f o r i n d i v i d u a l beams from Cu (111), a r e p l o t t e d as a f u n c t i o n of AdX f o r Vof =-9.5eV. Of the 16 beams a v a i l a b l e o n l y 9 a r e shown f o r c l a r i t y , the o t h e r s have a combined w e i g h t i n g o f l e s s t h a n 20%. A l s o shown i n F i g . 5.3 , as t h e dashed c u r v e , i s a p l o t of whose ainimum v a l u e of 0.173 c o r r e s p o n d s t o Ad$=-4-15L The degree o f agreement i n the i n d i v i d u a l c u r v e s v a r i e s f r o m (tr)^ =0.057 t o ( r f )c-=0-240 w i t h a s t a n d a r d e r r o r £ r =0.047. The mean {dashed) c u r v e at { r r ) ± £ r { f o r 68% p r o b a b i l i t y ) i n d i c a t e s a c o n t r a c t i o n w i t h t h e topmost i n t e r l a y e r s p a c i n g c o r r e s p o n d i n g t o -4±3%. However, i t i s apparent t h a t t h e minima o f a l l the i n d i v i d u a l c u r v e s i n F i g . 5.3 l i e w e l l w i t h i n these l i m i t s , and t h i s s u g g e s t s t h a t t h e e r r o r i n t h e v a l u e of J^d% t h a t 0.40 H 0.30H ( r r ) , 0.20H 0.10-Cu(lll) V Cu13 A d % f i g u r e 5^3 P l o t s f o r Cu(111) of ( r r ) . t f o r 9 i n d i v i d u a l beaas v e r s u s Ad* f o r t h e V C k i j p o t e n t i a l w i t h ? # r«-9.5eV. The dashed l i n e shows t h e dependence of the enerqy weiqhted aean T, versus Ad?. r 127 c o r r e s p o n d s t o a b e s t f i t f o r a p a r t i c u l a r v a l u e o f V s h o u l d r . rr or perhaps be e v a l u a t e d from t h e s e t of i n t e r l a y e r s e p a r a t i o n s ( 6 d J ^ ) c o r r e s p o n d i n g t o t h e minimum i n each o f t h e i n d i v i d u a l c u r v e s - A c c o r d i n g l y we can d e f i n e A i ^ A = Z AB-Ad^- / 2 > E L (5.14) c L £± = C ^ A E - C A d ^ -Ad~ r t )2 / ( i i - 1 ) X A E c ] ' / 2 (5.15) where the Ad V. are det e r m i n e d , a f t e r e x t r a p o i n t s have been f i t t e d t o the c u r v e s by i n t e r p o l a t i o n ; A d ^ ±2 £J ( f o r 95$ p r o b a b i l i t y ) c o r r e s p o n d s t o -4.1±1.2S. , T h i s range i s i n d i c a t e d i n F i g . . 5 . 3 and i t p o i n t s t o a much s m a l l e r , and a p p a r e n t l y more r e p r e s e n t a t i v e , e r r o r i n t h e geometry t h a n t h a t s u g g e s t e d above. I t i s a l s o c l e a r from F i g . 5.3, p r o v i d e d the i n d i v i d u a l c u r v e s f o r lTf)i show s i m i l a r v a r i a t i o n s w i t h ^ d f , t h a t the mean c u r v e T r w i l l be a smooth f u n c t i o n of odS even though the (r ) i n d i v i d u a l l y e x h i b i t v a r y i n g degrees o f f i t . S i n c e t h e T ^ are f u n c t i o n s of a t l e a s t two i m p o r t a n t v a r i a b l e s , namely &&% and V f t /r, a more i n f o r m a t i v e method o f p r e s e n t i n g the d a t a i s i n t h e form of a c o n t o u r p l o t o r diagram; t h i s i s d e p i c t e d i n E i g . ,5.4 f o r both t h e p o t e n t i a l s . These p l o t s were c o n s t r u c t e d fcy t a k i n g the o r i g i n a l g r i d of ry, v a l u e s and i n t e r p o l a t i n g s e p a r a t e l y a l o n g l i n e s of c o n s t a n t Vor and c o n s t a n t &&% t o o b t a i n an expanded g r i d of about 900 p o i n t s ; t h e e r r o r i n v o l v e d i n not u s i n g a t r u e t w o - d i m e n s i o n a l i n t e r p o l a t i o n i s s m a l l compared w i t h o t h e r e r r o r s a l r e a d y p r e s e n t i n t h e d a t a . The p o s i t i o n o f the minimum i n r t h a t f i S S I S S±!L Contour p l o t s f o r Cu (111) o f r f v e r s u s A d * acd f o r the p o t e n t i a l s (a) and (b) v£ c. 129 d e t e r m i n e s the best, f i t v a l u e s of ? . and A d S i s found by i n s p e c t i o n . A l s o shown i n F i g . 5.4 a r e t h e e r r o r b a r s c o r r e s p o n d i n g t o ±£j and ± £\j (the s t a n d a r d e r r o r i n the minimum f o r t h e i n n e r p o t e n t i a l , d e f i n e d a n a l o g o u s l y t o Q ) which p l a c e s 68% c o n f i d e n c e l i m i t s on the l o c a t i o n o f the minimum o f xr; t h e s e a r e e v a l u a t e d at t h e n e a r e s t g r i d p o i n t and p r o v i d e e s t i m a t e s of the e r r o r s i n t h e l o c a t i o n o f the a c t u a l minimum. The p l o t s o f F i g . 5.4 i l l u s t r a t e t h e . v a l u e of the r e l i a b i l i t y - i n d e x approach i n a n a l y s i n g LEED d a t a ; w i t h such d i s p l a y s the v a l u e s o f y; parameters g i v i n g b e s t ^ f i t w i t h experiment can be s e l e c t e d i m m e d i a t e l y . Each of t h e s e c o n t o u r p l o t s p r o v i d e s an immediate assessment of 784 c u r v e s o f t h e t y p e shown i n F i g . 5.1 { t h a t i s 1(E) c u r v e s f o r 16 beams a t 7 d i f f e r e n t v a l u e s o f Ad$ and 7 d i f f e r e n t v a l u e s of ? 0 / * ) -The v a l u e s of odS and Vor c o r r e s p o n d i n g t o minimum : ~xf a r e i d e n t i c a l t o w i t h i n one s t a n d a r d e r r o r f o r t h e two p o t e n t i a l s . T a b l e 5.2.. Both p o t e n t i a l s i n d i c a t e a c o n t r a c t i o n i n the topmost i n t e r l a y e r s p a c i n g of -4.1±0.6% and =-9.4±0.6eV. The low v a l u e s o f t h e o v e r a l l Z J r e l i a b i l i t y f a c t o r , R, (0.132 and 0.136 f o r V^i2 a n a " v<^ r e s p e c t i v e l y ) and t h e s m a l l e r r o r i n the b e s t - f i t v a l u e s of Ad% and V o r suggest t h a t t h i s r e p r e s e n t s a r a t h e r w e l l - d e f i n e d s t r u c t u r e . P r e v i o u s s t u d i e s of the (111) s u r f a c e s o f FCC metals w i t h LEED have i n d i c a t e d no l a r g e e x p a n s i o n s o r c o n t r a c t i o n s o f t h e s u r f a c e l a y e r , see S e c t i o n 5.5. F o r al u m i n i u m , one s t u d y £11] has suggested an e x p a n s i o n of 5% w h i l e o t h e r s t u d i e s £11,120] i n d i c a t e t h e topmost s p a c i n g i s e g u a l t o t h e b u l k w i t h i n 5%. F o r both n i c k e l [ 11,120] and p l a t i n u m £122], LEED a n a l y s e s 130 i n d i c a t e t h i s s p a c i n g i s w i t h i n 2.5% of the b u l k v a l u e ; s t u d i e s on Pt{111) w i t h i o n - c h a n n e l i n g , £ 123 ] now suggest an e x p a n s i o n o f about 2%. The one p r e v i o u s s t r u c t u r a l d e t e r m i n a t i o n of Cu(111) : by JLaramore |_ 20 J c o n c l u d e d t h a t the s u r f a c e i s not d i l a t e d b u t t h a t a s m a l l c o n t r a c t i o n between 0 and 5% i s p o s s i b l e . The c o n t r a c t i o n determined here f o r Cu{111), namely -4.1±0.6^, i s c o n s i s t e n t w i t h t h i s and i s t h e l a r g e s t yet r e p o r t e d f o r a c l o s e - p a c k e d s u r f a c e , a l t h o u g h l a r g e r c o n t r a c t i o n s a r e known f o r more l o o s e l y packed s u r f a c e s , e.g. 10 2 f o r A l {110) [101,110,111 1 and 7% f o r Ag (110) £ 101, 106, 107 J . 5.4 Other Bethods Qf 0 b t a i n i n g S u r f a c e S t r u c t u r a l I n f o r m a t i o n From LEED Data w h i l e most 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 by LEED have been performed by c o m p a r i s o n 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 f o r v a r i o u s s t r u c t u r a l ; models, t h e r e have been a t t e m p t s t o f i n d d a t a i n v e r s i o n p r o c e d u r e s t h a t w i l l produce a s u r f a c e s t r u c t u r e d i r e c t l y from e x p e r i m e n t a l d a t a - T h i s procedure would a v o i d the i n e v i t a b l y l e n g t h y and e x p e n s i v e t r i a l and e r r o r p r o c e s s a s s o c i a t e d w i t h c o n v e n t i o n a l m u l t i p l e -s c a t t e r i n g c a l c u l a t i o n s . Two approaches have been proposed so f a r ; b o t h , a r e based on the k i n e m a t i c a l t h e o r y , a l t h o u g h n e i t h e r can y e t be c o n s i d e r e d to be w e l l - e s t a b l i s h e d - One method a t t e m p t s t o m a n i p u l a t e t h e e x p e r i m e n t a l d a t a s o as t o enhance t h e s i n g l e - s c a t t e r i n g s t r u c t u r e and c o n c o m i t a n t l y d i m i n i s h t h e m u l t i p l e - s c a t t e r i n g .131 f e a t u r e s . , I f s u c c e s s f u l , t h i s would a l l o w a n a l y s i s with, the c o m p a r a t i v e l y s i m p l e k i n e m a t i c a l t h e o r y ; t h i s i s t h e p h i l o s o p h y of the d a t a - a v e r a g i n g (DA) approach £1,02,103], The a l t e r n a t i v e approach uses the F o u r i e r t r a n s f o r m (FT) method £ 99,100 ], i n ana l o g y w i t h t h e approach i n X-ray s t r u c t u r e a n a l y s i s of g e n e r a t i n g a P a t e r s o n map of t h e s c a t t e r i n g c e n t r e s . Because of d i f f i c u l t i e s a s s o c i a t e d w i t h each method, and a r e l u c t a n c e on the p a r t of LIED p r a c t i t i o n e r s t o d i s c a r d d a t a of m u l t i p l e -s c a t t e r i n g o r i g i n , t hese methods have s u f f e r e d some c r i t i c i s m . However, t h e r e i s a c o n t i n u i n g hope t h a t they can become u s e f u l , a t l e a s t i n the sense o f r e s t r i c t i n g the volume of "parameter space" t h a t must be s e a r c h e d i n a t y p i c a l f u l l m u l t i p l e -s c a t t e r i n g a n a l y s i s -5.5 B i b l i o g r a p h y Of S u r f a c e S t r u c t u r e s Qf- C l e a n fletals I n t h i s s e c t i o n i s c o m p i l e d a b i b l i o g r a p h y 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 s of c l e a n m e t a l s by LEED. I t i s not i n t e n d e d t o be an e x h a u s t i v e l i s t o f every paper p u b l i s h e d about c l e a n metal s u r f a c e s . Many e a r l y papers c o n t a i n e d e x p e r i m e n t a l d a t a or c a l c u l a t i o n s now r e c o g n i s e d t o be o f d u b i o u s g u a l i t y o r of such a l i m i t e d scope as t o be o f l i t t l e use. I n such c a s e s t h e more r e c e n t and complete s t u d i e s a r e quoted, a l t h o u g h some e a r l y s o r k / o f h i g h g u a l i t y i s g i v e n . A few s t r u c t u r e s a r e now b e i n g i n v e s t i g a t e d by t e c h n i g u e s s uch as medium-energy e l e c t r o n d i f f r a c t i o n (HEED) , i o n - c h a n n e l i n g (IC) and medium-energy i o n s c a t t e r i n g (IS) and have been i n c l u d e d f o r c o m p l e t e n e s s . The m e t a l s are l i s t e d a l p h a b e t i c a l l y and by f a c e . The 132 c r y s t a l h a b i t i s a l s o l i s t e d as, l n v e s t . i g a t i . p i i s of phase changes ar e now b e i n g c a r r i e d oat. The methods used t o d e t e r m i n e the s u r f a c e s t u c t u r e i s l i s t e d as v i s u a l {V), r e l i a b i l i t y - i n d e x (HI) , F o u r i e r t r a n s f o r m (FT) o r d a t a a v e r a g i n g (DA). The s u r f a c e s p a c i n g i s c h a r a c t e r i s e d by A d % and i s s u p p l i e d w i t h the e s t i m a t e d e r r o r s . I n some c a s e s doubts are r a i s e d by s e v e r e l y l i m i t e d ranges of e i t h e r the c a l c u l a t e d o r , e x p e r i m e n t a l data-The l i s t i s c u r r e n t as o f Aug..,.1978.., I n a l l these c a s e s t h e m e t a l s u r f a c e s do not r e c o n s t r u c t i n a l a t e r a l sense but do show v a r y i n g degrees of c o n t r a c t i o n o r e x p a n s i o n o f the s u r f a c e l a y e r . I n g e n e r a l , t h e c l o s e - p a c k e d (111) f a c e s o f FCC m e t a l s can be seen t c show very s m a l l d i f f e r e n c e s of the s u r f a c e l a y e r s p a c i n g from t h a t o f the b u l k , w i t h p o s s i b l y a s l i g h t : p r e f e r e n c e f o r c o n t r a c t i o n s . The s i t u a t i o n seems t o be s i m i l a r f o r the (100) s u r f a c e s of FCC m e t a l s w h i l e (100) s u r f a c e s o f BCC metals appear i n g e n e r a l t o show c o n t r a c t i o n s o f t h e o r d e r o f 5 t o 10%. The i n t e r e s t i n g example of Co, which has a phase change from HCP t o FCC a t about 600°C behaves u n e x c e p t i o n a l l y i n e i t h e r phase. The HCP m e t a l s have been l i t t l e s t u d i e d but t h e b a s a l p l a n e s of T i and Zn a r e unremarkable. The o n l y h i g h e r i n d e x f a c e s t u d i e d so f a r , t h a t of Cu{311) shows a c o n t r a c t i o n o f s i m i l a r magnitude t o the FCC (110) f a c e s . S e v e r a l o f the n o t a b l e o m i s s i o n s from t h i s l i s t e. g. Pt (100), Au{100) show by t h e i r complex LEED p a t t e r n s t h a t they a r e l a t e r a l l y r e c o n s t r u c t e d ; t h e i r s u r f a c e s t r u c t u r e s have not y e t been s o l v e d by LEED i n t e n s i t y a n a l y s e s u s i n g 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 . M e t a l Face C r y s t a l Method A d j . h a b i t 1 Ref. {100) FCC V 0±? 104 1111) FCC V 0±? 105 (110) FCC FT . -7±5 101 R l -10±1 . 106 MEED -5±5 107 A l (100) FCC 7 0±1 ? 108 y Q+? 109 V 0±5 110 (111); • FCC • V 0±5 110 V + 5±5 111 V 0±5 112 FT -3±5 99 (110) FCC FT . -4±5 101 . V -10±5 110 V -10±5 111 V -10±1 108 v -12*3 112 Co (0001) HCP V 0±5 113 (111) FCC V 0±5 113 Cu (100) FCC . FT :. 0±1 99 DA 0±1.. 114 V 0±5 20 (111) FCC V -2.5±2-5 20 RI -4- 1±0.6 t h i s a o r k (311) FCC HI -5.0±1.5 115 Fe (100) BCC V -1. 5±2.5 116 I r (111) FCC V -2.5+5 117 Ho (100) BCC V .  . -11 ±5 118 Na (110) BCC V 0±5 119, B i (100) FCC FT = 0±1 99 V +5±5 11 V 0±5 120 DA . 0±4 121 (111) FCC V -1±1 11 0±5 120 (110) FCC v -5±1 11 FT -5±5 101 . I S -4±? 135 M e t a l Face C r y s t a l h a b i t .Method A H Ref. Rh (100) (111) (110) FCC FCC FCC BI 81 BI 0±2.5 -1+3 -2.7±2.0 t h i s t h i s t h i s work work work Pt (111) FCC V IC I S •2.5±2.5 + 2±1 . •:i..„5±.1 : 122 123 136 I i (0001) HCP V -2±1 124 w (100) (110) BCC BCC V V -11±5 Q±5 125 126 Zn (0001) HCP . v -2±2 127 135 CH&PTER 6 THE (111) SURFACE OF SBODI0& 136 P r e v i o u s LEED s t u d i e s on t h i s s i n g l e c r y s t a l f a c e a r e l i m i t e d to t h a t of Grant and Haas £130], which c o n c e n t r a t e d mainly cn Auger s p e c t r o s c o p y , and a r e c e n t c h e m i s o r p t i o n s t u d y by C a s t n e r e t a l £18]. Both t h e s e a u t h o r s r e p o r t e d a (1X1) LEED p a t t e r n f o r the c l e a n s u r f a c e but d i d not perfor m any i n t e n s i t y measurements- An as yet u n p u b l i s h e d LEED s t u d y of t h i s s u r f a c e which c e n t r e s e s p e c i a l l y on the v i b r a t i o n a l ; p r o p e r t i e s of t h e s u r f a c e atoms, but which does r e p o r t some i n t e n s i t y d a t a , has been r e c e n t l y undertaken £131]. 6, 1 E x p e r i m e n t a l E x p e r i m e n t a l data f o r t h e fin (111.) s u r f a c e s e r e t a k e n from two c r y s t a l s (see T a b l e 4.1)2 one was a p r e - c u t s l i c e l o a n e d by P r o f - G. A. S o m o r j a i ( B e r k e l e y ) , and t h e second was c u t and p o l i s h e d h e r e . D e s p i t e t h e d i f f e r e n t p r e h i s t o r i e s of the s e c r y s t a l s , t hey behaved i d e n t i c a l l y w i t h i n e x p e r i m e n t a l e r r o r . Auger e l e c t r o n s p e c t r o s c o p y of both the (111) s u r f a c e s as mounted i n t h e vacuum chamber r e v e a l e d a p p r e c i a b l e amounts of carbo n (272eV) and sul p h u r ( 1 5 2 e V ) c o n t a m i n a n t s as shown i n t h e Auger spectrum of F i g - 6.1(a). Argon i o n bombardment removed S from t h e s u r f a c e b u t the G Auger s i g n a l showed a r e l a t i v e i n c r e a s e . F i g - 6.1 (b)- T h i s appears t o be a common s i t u a t i o n caused by t h e low s p u t t e r i n g c r o s s - s e c t i o n of ca r b o n . S p u t t e r i n g t e n d s t o l e a v e i s l a n d s o f C o r a s u r f a c e l a y e r e n r i c h e d i n carbon- A n n e a l i n g a t about 13QQK reduces the l e v e l of carbon c o n t a m i n a t i o n on t h e s u r f a c e by b a c k - d i f f u s i o n i n t o t h e b u l k c r y s t a l , b u t w i l l i n d u c e the re-ap p e a r a n c e o f s u l p h u r 137 Rhdn) i 1 1 1 1 100 200 300 eV l i ^ u i S 6^1 Auger s p e c t r a of Eh (111) s u r f a c e s witb a 1.5keV, 10 nicroamp bean: (a) as aounted, c o n s i d e r a b l e S(152eV) and C |282eV) c c r t a a i n a t i o n ; (b) a f t e r a r g o n - i o n b c a t a r d a e n t , reduced S, i n c r e a s e d C; (c) a f t e r a n n e a l i n g , reduced C, i n c r e a s e d S; <d) c l e a n e d s u r f a c e . 138 as i s shown i n F i g - 6.1(c) ., The l e v e l o f S and C c o n t a m i n a t i o n subseguent t o bombardment can be r e g u l a t e d by c a r e f u l t e mperature c o n t r o l - A c l e a n s u r f a c e showing v e r y low C and S Auger s i g n a l s c o u l d be produced by many c y c l e s of hot i o n -bombardment (about 1 microamp a t 2kV f o r TO minutes w i t h t h e c r y s t a l a t 1000K) f o l l o w e d by a 10 minute a n n e a l , a t 1300K-A f t e r s e v e r a l such c y c l e s t h e s u l p h u r c o n t e n t of the s u r f a c e r e g i o n of the c r y s t a l becomes d e p l e t e d , and c l e a n i n g i s made e a s i e r ; o f t e n o n l y a s h o r t a n n e a l i s then r e g u i r e d t o remove s u r f a c e c a r b o n . The spectrum of the c l e a n e d s u r f a c e i s shown i n F i g . 6 . 1 ( d ) . The rhodium peak e n e r g i e s and r e l a t i v e i n t e n s i t i e s a r e l i s t e d i n columns 1 and 5 of T a b l e 6.1. The r e s u l t s of the o t h e r e x p e r i m e n t a l ; s t u d i e s of t h i s s u r f a c e a r e l i s t e d f o r comparison i n columns 2 t o 4, /Some v a r i a t i o n i n r e p o r t e d peak e n e r g i e s i s o b v i o u s ; i n p a r t t h i s must be a t t r i b u t e d t o d i f f i c u l t i e s i n e s t a b l i s h i n g a s t r i c t l y l i n e a r energy s c a l e and t o c o n t a c t p o t e n t i a l d i f f e r e n c e s . A l s o e s t i m a t i o n of the p o s i t i o n o f weak peaks i s p r o b a b l y s u b j e c t to s e v e r a l eV e r r o r , e s p e c i a l l y i f t h e s i g n a l t o n o i s e r a t i o of the spectrum i s n o t v e r y good, which may account f o r t h e r e l a t i v e l y l a r g e spread i n t h e guoted e n e r g i e s of the weak f e a t u r e s below 220e7. The c a l c u l a t e d e n e r g i e s and a s s i g n m e n t s of P a c k e r and U i l s o n £49j are a l s o l i s t e d i n the f i n a l two columns. These c a l c u l a t i o n s a r e f o r f r e e atoms w i t h an i o n i s a t i o n c o r r e c t i o n but n e v e r t h e l e s s a r e i n r e a s o n a b l y good agreement w i t h the e x p e r i m e n t a l d a t a . No s i g n i f i c a n t amount of boron (Auger peak a t 180eV) c o u l d 1 3 9 O B S E R V E D • R E L . I i l a ; J : A S S I G N M E N T a) b) c) a>. a) e) 1 3 9 1 4 5 5 1 4 1 1 7 0 1 7 4 1 7 5 1 7 0 5 1 7 5 2 0 3 2 0 8 2 1 0 2 0 0 5 2 0 8 M a N 2 2 5 2 2 6 2 2 7 2 2 2 1 5 2 2 5 2 5 7 2 6 0 2 5 9 2 5 6 3 0 2 5 8 3 0 2 3 0 6 3 0 3 3 0 2 1 0 0 3 0 3 t h i s work, b) r e f e r e n c e 1 3 0 , c) r e f e r e n c e 1 3 1 , d) r e f e r e n c e 1 8 , e) r e f e r e n c e 4 9 T a b l e 6 , 1 - O b s e r v e d and c a l c u l a t e d Auger t r a n s i t i o n s f o r rhodium. 1 4 0 be d e t e c t e d i n the c o n t a m i n a t e d or c l e a n s u r f a c e s , T h i s i s i n c o n t r a s t t o another s t u d y I 18 J i n which a b u l k boron c o n t a m i n a n t d i f f u s e d t o the s u r f a c e d u r i n g c l e a n i n g p r o c e d u r e s and proved v e r y d i f f i c u l t t o remove. One c f the c r y s t a l s used i n the work r e p o r t e d here was the same c r y s t a l as used i n t h i s e a r l i e r study- The d i f f e r e n c e i n b e h a v i o u r i n r e l a t i o n t o boron seems to be the r e s u l t o f t h e s e workers h a v i n g removed n e a r l y a l l the b u l k boron c o n t a m i n a t i o n t h r o u g h many c l e a n i n g c y c l e s ( p r i v a t e c o mmunication). I n the second c a s e , the c r y s t a l was p u r c h a s e d from a d i f f e r e n t s o u r c e from t h a t o f t h e f i r s t c r y s t a l ; i n t h i s c a s e we have t o c o n c l u d e t h a t n e i t h e r t h e m a n u f a c t u r i n g p r o c e s s , nor t h e p o l i s h i n g method employed, i n t r o d u c e d any s i g n i f i c a n t amounts cf boron i n t o t h e c r y s t a l . A f t e r s e v e r a l c y c l e s of bombardment and a n n e a l i n g , a sharp (1X1) LEED p a t t e r n was o b s e r v e d , i n agreement w i t h e a r l i e r work J. 18,130,131 J , w i t h a low background and n e g l i g i b l e Auger s i g n a l s f o r s p e c i e s o t h e r t h a n rhodium. Sample photographs of the LEED p a t t e r n and the beam l a b e l l i n g scheme a r e shown i n F i g - 6.2. The i n t e n s i t i e s o f d i f f r a c t e d beams were measured as d e s c r i b e d i n C h apter 4 up to about 250eV a t normal i n c i d e n c e and f o r 0- = 1OO,<£ =109° i n t h e angle c o n v e n t i o n of Jona £ 128 J . Each s e t of e x p e r i m e n t a l i n t e n s i t y measurements was r e p e a t e d t h r e e t i m e s on s e p a r a t e o c c a s i o n s t o a s s e s s t h e e f f e c t o f s m a l l e x p e r i m e n t a l v a r i a t i o n s on t h e s t r u c t u r a l a n a l y s i s . 141 a ) b) 21 t JO .11 .12 •01 #01 (180- ffi\ 01 yi <* ' » 0 0 *01 #02 '11 '10 »11 '21 c) .11 .12 • »10 »11 11 d ) f i g u r e &j2 Photographs of the (1i1) LEED p a t t e r n f r o i the c l e a n Rh (111) s u r f a c e at (a) n o r a a l i n c i d e n c e (158eV) , (b) 9- = 10°, y*=109° (122eV) i n the angle c o n v e n t i o n o f Jona T128]. Ihe l a b e l l i n g s c h e i e i s shove i n (c) and (d). 142 6.2 C a l c u l a t i o n s The (1x1) LEED p a t t e r n o b t a i n e d i n d i c a t e s t h a t the s u r f a c e does not r e c o n s t r u c t i n such a way a s t o change t h e s u r f a c e symmetry from t h a t o f the b u l k s t r u c t u r e . However i t i s p o s s i b l e f o r t h e s u r f a c e t o r e c o n s t r u c t i n such a way as t o p r e s e r v e t h e (1x1) p a t t e r n , t h a t i s by a l t e r i n g the r e g i s t r y of the s u r f a c e l a y e r w i t h r e s p e c t t o those underneath. Such p o s s i b i l i t i e s a r e shown i n F i g - ., 6.3- I f the s u r f a c e : does not r e c o n s t r u c t t h e n the f a c e - c e n t r e d c u b i c s t a c k i n g seguence i s c o n t i n u e d to t h e s u r f a c e , as shown i n fa) namely CBACEA..-C. Two p o s s i b l e r e c o n s t r u c t i o n s a r e t o a CBACBA.-.A sequence, (b) , or to a CBACBA-,,B s t a c k i n g , <c), the l a t t e r s t u c t u r e p o s s e s s i n g t h e h e x a g o n a l c l o s e packed seguence f o r the t o p t h r e e l a y e r s . T h i s l a s t s u r f a c e r e c o n s t r u c t i o n seems p h y s i c a l l y the more p l a u s i b l e and was the o n l y , r e c o n s t r u c t i o n , model t h a t was i n v e s t i g a t e d h e r e w i t h d i r e c t c a l c u l a t i o n s . C a l c u l a t i o n s were performed as d e t a i l e d i n C h a p t e r 3 f o r b o t h the v ^ and V^ | 3 p o t e n t i a l s i n the energy range 30-260eV; a t o t a l of 55 beams were a v a i l a b l e t o d e t e r m i n e 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 depending upon the energy and a n g l e of i n c i d e n c e . 143 c) ZiasiS. £x3 P o s s i b l e r e c o n s t r u c t i o n s o f t h e (111) s u r f a c e t h a t p r e s e r v e the (1x1) s y i a e t r y of the LEED p a t t e r n : <a) n o n - r e c o n s t r u c t e d , CBACBA...C FCC s t a c k i c q ; <b) r e c o n s t r u c t e d , CBACEA...A s t a c k i n g ; <c) r e c o n s t r u c t e d , CBACBA...B, HCP s t a c k i n g . The 4th l a y e r C i s i n d i c a t e d by s a a l l dashed b a r r e d c i r c l e s , the 3r d l a y e r B by l a r q e blank c i r c l e s , t h e 2nd A by aed i u a d o t t e d c i r c l e s , and the 1st l a y e r i s i n d i c a t e d by s i a l l b a r r e d c i r c l e s . 6, 3 R e s u l t s And D i s c u s s i o n 6.3(a) Normal i n c i d e n c e Three; i n d e p e n d e n t s e t s o f i n t e n s i t i e s s e r e measured f o r normal i n c i d e n c e . Although t h e s e independent s e t s showed a good g e n e r a l agreement w i t h each o t h e r , t h e r e were s m a l l d i f f e r e n c e s i n d e t a i l a s , f o r example, i n r e l a t i v e peak i n t e n s i t i e s and i n s t r u c t u r e such as s h o u l d e r s ( a l l t h e e x p e r i m e n t a l d a t a i s shown i n Appendices A1-A3). W i t h i n each s e t of data the measured i n t e n s i t i e s o f t h e d i f f r a c t e d beams show c l o s e l y t h e symmetry ex p e c t e d i f t h e a c t u a l s u r f a c e arrangement m a i n t a i n s the t h r e e -f o l d r o t a t i o n a x e s which a r e p e r p e n d i c u l a r t o (111) p l a n e s i n the bulk ( a l t e r n a t i v e l y : t h e symmetry i n the LEED p a t t e r n ; c o u l d be a s s o c i a t e d w i t h e g u a l p o p u l a t i o n s o f a p p r o p r i a t e domains). To minimise any a r t i f a c t s i n the comparison w i t h c a l c u l a t e d i n t e n s i t i e s , measured 1(E) c u r v e s f o r s e t s o f beams which a r e t h e o r e t i c a l l y e g u i v a l e n t , on t h e above b a s i s , were averaged and d i g i t a l l y smoothed {by two o p e r a t i o n s o f a t h r e e p o i n t smoothing f i l t e r ) p r i o r t o comparing w i t h t h e c a l c u l a t i o n s . F i g u r e 6.4 compares e x p e r i m e n t a l I ( E ) c u r v e s f o r t h e {10) and (01) beams a t normal i n c i d e n c e w i t h I {E) c u r v e s c a l c u l a t e d f o r two s t a c k i n g seguences; a c o n t i n u a t i o n - o f the b u l k s t a c k i n g seguence. F i g . 6.3 { a ) , and the h e x a g o n a l r e c o n s t r u c t i o n model* F i g . 6 . 3 ( c ) , w i t h t h e topmost l a y e r s p a c i n g e g u a l t o the b u l k o v a l u e o f 2.195A. A d e t a i l e d matching of peaks and t r o u g h s between t h e e x p e r i m e n t a l and c a l c u l a t e d c u r v e s c o u l d o n l y be 145 Rh Cm) V ^ r Ad % = 0.0 50 100 J50 200 250 50 100 150 200 250 Energy (eV) Energy (eV) Z i S U I S * comparison f o r the (10) and (01) cea»s c£ 1 ( E ) c u r v e s measured at n o r t a l i n c i d e n c e f o r Rh(111) with those c a l c u l a t e d w i t h t h e p o t e n t i a l f o r the n o r a a l FCC s t a c k i n g sequence and f o r the HCF s t a c k i D q sequence ever the top thrfre s u r f a c e l a y e r s . 146 a c h i e v e d f o r the normal f a c e - c e n t r e d c u b i c p a c k i n g sequence; the o t h e r d i f f r a c t e d beams a r e s i m i l a r l y i n c o n s i s t e n t w i t h an HCP arrangement f o r the t o p t h r e e l a y e r s . To d e t e r m i n e t h e s u r f a c e r e l a x a t i o n , c a l c u l a t i o n s were performed f o r d i f f e r e n t v a l u e s o f the. t o p i n t e r l a y e r s p a c i n g , e x p r e s s e d as i n the Cu (111) case as percentage changes frem the b u l k v a l u e T h i s q u a n t i t y was varied--..from +10% { d i l a t i o n ) t o -108 ( c o n t r a c t i o n ) i n s t e p s of 2.5% (0.0551). F i g u r e 6.5 compares 1(E) c u r v e s f o r t h e (10) and (01) beams w i t h those c a l c u l a t e d f o r &d%= -5,0 and *5% f o r . t h e two i o n c o r e p o t e n t i a l s . I t i s d i f f i c u l t even w i t h t h i s l i m i t e d data s e t t o assess v i s u a l l y which v a l u e o f t h e s u r f a c e r e l a x a t i o n b e s t d e s c r i b e s t h e e x p e r i m e n t a l d a t a , but F i g - 6.6 a t t e m p t s a more complete assessment by showing c o n t o u r p l o t s o f t h e mean reduced r e l i a b i l i t y i n d e x r r as • f u n c t i o n s of and V o r , from one o f th e s e t s o f independent d a t a , as d e s c r i b e d i n C h a p t e r 5. With the a v e r a g i n g r e f e r r e d t o above, t h e e x p e r i m e n t a l d a t a used i n th e s e c o n t o u r p l o t s f o r normal . i n c i d e n c e - i n v o l v e s the (10)> (0 1 ) , ( 1 1 ) , (20) and (02) beams. The minimum o f r~ r i n F i g . 6-6 (a) o c c u r s f o r AdS = -3-7± 1. 1% and V o r = -19. 1±0.9eV, where the u n c e r t a i n t i e s i n 6d% and V o r (± £j and ± £ v r e s p e c t i v e l y ) a r e the s t a n d a r d e r r o r s i n t h e energy-weighted means deduced from t h e spread i n v a l u e s g i v e n by th e minima o f t h e r e l i a b i l i t y i n d e x ( r ^ ) ^ f o r t h e v a r i o u s i n d i v i d u a l beams a s d e t a i l e d i n Chapter 5, To t h e e x t e n t t h a t t h e s e e r r o r s r e a l l y do r e p r e s e n t a gauge of t h e . u n c e r t a i n t y i n each r e s u l t then t h e r e i s a 68% p r o b a b i l i t y t h a t the a c t u a l minimum l i e s w i t h i n t h e a r e a d e f i n e d by the e r r o r b a r s i n 147 Rh(iii) e= o e (01) beam >0% W5% 100 150 200 250 Energy (eV) 50 Rh (in)G = 0° (10) beam Exp WJM R h l 3 >-5% v R h ' Rh13 ,WJM >+5% Rh!3 100 150 200 250 Energy (eV) J i f l u i e * comparison o f e x p e r i m e n t a l 1(E) c o r v e s f o r the (10) • r d (01) beams a t normal i n c i d e n c e f o r Bh_(111) w i t h i n t e n s i t y c u r v e s c a l c u l a t e d f o r the p o t e n t i a l s v" ftk and V f o r t h r e e d i f f e r e n t v a l u e s o f Ad* assuminq t h e normal FCC r e q i s t r y f c r the s u r f a c e arrangement. 146 fiSi?J§ 6^ .6 Contour p l o t s f o r Fh (111) a t n o r a a l i n c i c e n c e of r f v e r s u s V^ . and AdX f o r the p o t e n t i a l s <a) and (b) F^,, s t a r t i n g f r o a 54 eV. 149 150 F i g - 6 . 6 ( a ) , F i g u r e 6.6(b) shows the c o r r e s p o n d i n g c o n t o u r p l o t which compares t h e e x p e r i m e n t a l c u r v e s wi t h those c a l c u l a t e d from the s u p e r p o s i t i o n p o t e n t i a l I t i i e minimum of IT now o c c u r s f o r Ad%= +3.5±1..2X and VQr =- 9-7±1.2eV. a l t h o u g h both s e t s of c a l c u l a t e d c u r v e s a r e b a s i c a l l y s i m i l a r (see F i g . 6.5) the minima i n t h e c o n t o u r p l o t s are a t r a t h e r d i f f e r e n t v a l u e s of the parameters Ad$ and v„„ . *• or He can see t h a t t h e o v e r a l l l e v e l of agreement, as judged by the v a l u e o f r f., i s very good w i t h i n the Z a n a z z i - J c n a (ZJ) framework. The v a l u e s of t h e r e f i n e d v a l u e s o f V d i f f e r by over 9eV f o r t h e two p o t e n t i a l s - . P a r t o f t h i s d i f f e r e n c e can be t r a c e d back to the d i f f e r e n c e i n p o t e n t i a l a t the edge of the m u f f i n ^ t i n s p h e r e s . T h i s amounts t o about 6eV and r e s u l t s i n an energy s h i f t o f the two s e t s of phase s h i f t s r e l a t i v e t o one a n o t h e r a s was d i s c u s s e d i n Chapter 3 (see F i g - 3 - 6 ) . C o r r e s p o n d i n g l y , t h i s w i l l r e s u l t i n a r e l a t i v e s h i f t i n c a l c u l a t e d 1(E) c u r v e s a s , f o r example, i n F i g - 6.5 where th e l a r g e peaks i n t h e c a l c u l a t e d c u r v e s a t +5% f o r the (01) beam at normal i n c i d e n c e are d i s p l a c e d r e l a t i v e t o one a n o t h e r . The e x t r a s h i f t of about 3eV, deduced from the r e l i a b i l i t y - i n d e x a n a l y s i s , may r e f l e c t f u r t h e r d i f f e r e n c e s i n the two p o t e n t i a l s . The d i f f e r e n c e i n t h e v a l u e s o f Ad& g i v e n by t h e two p o t e n t i a l s appears more s e r i o u s . The two minimum v a l u e s of ~r~ g i v e n i n F i g . 6.6 seem s u f f i c i e n t t o p r e c l u d e c h o o s i n g e i t h e r p o t e n t i a l as g i v i n g t h e " b e s t f i t " . We can probe i n t o the o r i g i n s o f t h e d i f f e r e n t v a l u e s of from V and V by n o t i n g t h a t the c o n t o u r p l o t s i n F i g . 6.6 e x h i b i t e l o n g a t e d v a l l e y s r u n n i n g d i a g o n a l l y a c r o s s t h e 151 diagrams; a s i m i l a r e f f e c t can fee seen i n the a n a l o g o u s p l o t s f o r C u ( l i t ) i n the d i s c u s s i o n o f Chapter 5 (see F i g . 5 - 4 ) - , As a r e s u l t of t h e s e v a l l e y s i n the c o n t o u r p l o t s , r e l a t i v e l y good agreement between experiment and c a l c u l a t i o n s - , (as judged by Tr ) can be a c h i e v e d f o r a number o f c o m b i n a t i o n s o f V 6 r and kdSI. For F i g - 6 . 6 ( b ) , a l l p a i r s o f v a l u e s which f a l l on the dashed l i n e from V o r=-16eV, Ad£=-6g t o Vor =-6eV., Ad«= + 6% have r r <0.25, and t h e r e f o r e g i v e "good" agreement a c c o r d i n g t h e c r i t e r i o n o f ZJ- S i n c e t h e degree of f i t worsens r a p i d l y i n d i r e c t i o n s p e r p e n d i c u l a r t o the v a l l e y , i t seems p r o b a b l e t h a t any minor d e f i c i e n c i e s i n t h e c a l c u l a t e d or measured 1(E) c u r v e s would mainly move t h e l o c a t i o n o f ; t h e minimum of ~Fr a l o n g the v a l l e y . To see the e x t e n t o f t h i s f o r s m a l l changes i n e x p e r i m e n t a l i n t e n s i t i e s , the c o r r e s p o n d i n g r e l i a b i l i t y - i n d e x c a l c u l a t i o n s w i t h t h e o t h e r two independent s e t s o f e x p e r i m e n t a l d a t a were urn made. The r e s u l t s o f the a d d i t i o n a l c o n t o u r p l o t s w i t h . g i v e v a l u e s o f A a ? e q u a l to -5.6±1.3S and -4.8±0.8$, and t h e r e f o r e an o v e r a l l mean v a l u e o f -4.7%. The f u r t h e r p l o t s w i t h V ^ - j g i v e Ad%=*3. 8±1.3% and +3.5±1.2%, and mean value of +3-6$. I t seems c l e a r t hat> w h i l e t h e s e mean v a l u e s of A d % a r e o u t s i d e t h e 96% c o n f i d e n c e l i m i t s (±2€j) , t h i s d i s c r e p a n c y i n geometry from the two p o t e n t i a l s i s not p r i m a r i l y a s s o c i a t e d w i t h s m a l l u n c e r t a i n t i e s i n the e x p e r i m e n t a l d a t a . F i g u r e 6.7 shows p l o t s of ( r r )^  f o r ; the i n d i v i d u a l beams a g a i n s t &&% f o r i a (a) , and 7 ^ ( 3 i n - (by ; t h e d o t t e d l i n e s show t h e c o r r e s p o n d i n g v a r i a t i o n s of r p . In a c c o r d w i t h t h e above r e s u l t s , t h e minima o f c u r v e s f o r the i n d i v i d u a l beams i n F i g . 6.7(b) a r e g e n e r a l l y s h i f t e d t o more p o s i t i v e v a l u e s o f Ad$ f i S " I f 4x2 P l o t s f o r Bh (111) of ( r r ) ; f o r f i v e independent beams at nc«al~incidence v e r s u s AdX f o r (a) l V , r = 18eV) and (b) VjlVO (v»r=**10eV) . T i e dashed l i n e s shew the dependence of r r v e r s u s AdX. 153 compared w i t h t h e c o r r e s p o n d i n g v a l u e s i n F i g . 6.7 (a) . HoSever, f o r the (10) and £01) beams t h e r e i s a much s t e e p e r r i s e a t n e g a t i v e v a l u e s o f o d l i n F i g . 6-7 (b); i n p a r t i c u l a r the degree of f i t f o r the £01) beam appears t o d e t e r i o r a t e much more r a p i d l y i n t h i s r e g i o n f o r ( F i g - 6.7 (b)) than f o r ( F i g - 6 . 7 ( a ) ) . The o r i g i n o f t h i s b e h a v i o u r i n t h e r e l i a b i l i t y - i n d e x a n a l y s i s may not be i m m e d i a t e l y o b v i o u s from a v i s u a l e v a l u a t i o n o f the I (E) .curves i n F i g . 6.5, a l t h o u g h t h i s d i f f e r e n c e i s a s c r i b e d here as b e i n g a s s o c i a t e d w i t h the l a r g e peak i n the c a l c u l a t e d I ( E ) c u r v e s f o r t h e (10) beam a t around 54eV. T h i s peak i s a p p r o x i m a t e l y ; t w i c e as l a r g e a t A d S = - 5 $ f o r VfcM3 a s ^ G r * a n a * s t b e dominant f e a t u r e i n both 1(E) c u r v e s . The presence o f such a s t r o n g f e a t u r e r i g h t a t t h e b e g i n n i n g of I (E) c u r v e s b e i n g compared w i t h the r e l i a b i l i t y -i n d e x t u r n s o u t t o be u n d e s i r a b l e . T h i s i s because the v a l u e s of t h e s c a l i n g c o n s t a n t c ^ can change s u b s t a n t i a l l y o v er t h e range o f V Q r s h i f t s employed; t h i s problem a r i s e s from the a r e a s under the 1(E) c u r v e s c h a n g i n g a p p r e c i a b l y a s the energy range i s t r u n c a t e d a t d i f f e r e n t p o i n t s w i t h i n t h i s peak when the v a l u e o f the i n n e r p o t e n t i a l i s v a r i e d . To a v o i d t h i s problem, i t was found n e c e s s a r y t o s t a r t the comparison of the 1(E) c u r v e s a t a h i g h e r e n e r g y . F i g u r e 6.8 shows c o n t o u r p l o t s f o r t h e same s e t of normal i n c i d e n c e d a t a as i n F i g . 6,6 when t h e comparison i s s t a r t e d a t 66eV i n s t e a d of 54eV as was done e a r l i e r . By comparing these two f i g u r e s se can see t h a t t h e b e s t f i t of t h e i n n e r p o t e n t i a l i s lowered by about 1eV f o r the band s t r u c t u r e p o t e n t i a l by t h i s s m a l l energy range t r u n c a t i o n , but l i t t l e e f f e c t on V , i s d i s c e r n a b l e f o r the 154 f i g u r e .ii.8 Contour p l o t o f r r v e r s u s and A d * f o r Bh(111) a t r c r j a l i n c i d e n c e , u s i n g the d a t a c f F i g . 6.6 o n l y frcm 66eV f o r t h e p o t e n t i a l s (a) v"^ and (b) V R M i . 155 1 5 6 s u p e r p o s i t i o n p o t e n t i a l . The o p p o s i t e appears t o be t r u e f o r A d % however; i n the 7 ^ c a s e , the minimum v a l u e o f tr o c c u r s a t a s l i g h t l y s m a l l e r c o n t r a c t i o n a f t e r t r u n c a t i o n , whereas f o r Vfchl3 an a p p r e c i a b l e change of almost 2% i s s e e n . The r e s u l t s from t h e s h o r t e n e d energy range comparison a r e c o l l e c t e d i n T a b l e 6.2. S t a r t i n g a t 54eV r a t h e r than 66eV adds o n l y 1/2% t o the t o t a l energy ra n g e , and y e t t h i s e x t e n s i o n has a s t r o n g e f f e c t on r e s u l t s from the r e l i a b i l i t y i n d e x a n a l y s i s f o r t h e p o t e n t i a l V ^ , j . Thus w h i l e t h e v a l u e of ~z~r i s i n c r e a s e d between o n l y 0.01 and 0.02, t h e v a l u e o f &&% s h i f t s t o more p o s i t i v e v a l u e s by over 3% on average when t h e comparison i s s t a r t e d from 54eV. F o r , where t h e peak c l o s e t o 54eV i n the c a l c u l a t e d 1(E) c u r v e i s somewhat s m a l l e r than f o r v^/^ » t h e change i n the v a l u e of d$ f o r t h e two energy ranges i s o n l y about 0-4$. The same problem w i t h n o r m a l i s a t i o n c o u l d a r i s e when a l a r g e peak i n an 1(E) c u r v e o c c u r s c l o s e to the end of t h e energy range o f t h e c o m p a r i s o n . To c o n s i d e r f u r t h e r t h e t o p l a y e r r e g i s t r y , i n F i g u r e 6-9 i s p r e s e n t e d a c o n t o u r p l o t of r<- showing the c o mparison c f one s e t o f the e x p e r i m e n t a l d a t a f o r normal i n c i d e n c e w i t h c a l c u l a t i o n s f o r V ^ f o r the s u r f a c e arrangement w i t h the l a t e r a l l y s h i f t e d t op l a y e r (CBA.,B). These c o n t o u r s c o n t r a s t w i t h t h e s t e e p v a l l e y and l o c a l i s e d minimum found i n each of the p l o t s i n F i g u r e 6.6 f o r the b u l k s t a c k i n g sequence (CBA-i.C); the c o n t o u r s f o r t h e HCP s u r f a c e arrangement have a r a t h e r f l a t topography and much h i g h e r v a l u e s of r ^ . A l t h o u g h o t h e r r e g i s t r i e s have n o t been i n v e s t i g a t e d , t h e s e r e s u l t s p o i n t s t r o n g l y t o a c o n t i n u a t i o n of the b u l k s t a c k i n g seguence a t the 157 c 41 H 01 01 u CO CO 4J 03 01 J 3 O CO e o c o u o U) +1 -a > v/1 d ao I 94 d cn u-i o d •H C J \ I oo d cn I o O > 11 o • ao ~* i m d CN i > d •H 00 • ao • o ft 00 o d d > 01 O d o CN O > 4 ) O •••I m »4 CO d d i > 4 1 O •H s O d I »4 o +i C I d + 01 3 0 c CO w -a 01 u > > > 3 0 CO 01 01 01 u a. oi S 3 0 o >» c o C N CM o M U CO •CM > 01 o CM > 01 o to e <0 0> -n -a 01 0 CO Q . • B O U O z u-i c 01 o cu x 3 06 > x r 3 2 3 OS cn cn rn -C OS OS 06 c 01 e CU - H O U ' 01 • a. o x z u en 01 3 »H CO > C CO 01 X CM co 01 a CO > e CO 01 X Table 6j_2 C o n d i t i o n s o f best aqreement between experiment and ffuItiple-scatterinq c a l c u l a t i o n s f o r t h r e e s e t s of i n t e n s i t i e s measured at normal i n c i d e n c e on E h ( 1 1 1 ) . 158 Rh(11i)V" J Me = O e CBA--B A d % ll.9u.If £i5 Contour p l o t of r r v e r s u s V^r a n d A d S f o r the p o t e n t i a l ' v 1 ^ and the model of the Rh(111) s u r f a c e i n which the top t h r e e l a y e r s have the HCP s t a c x i n q sequence. 159 Bh(111) s u r f a c e , as suggested above w i t h r e f e r e n c e t o F i g u r e 6.4., 6.3(b) D i r e c t i o n o f i n c i d e n c e O=l0o,^>=1 QS° Depending on the d a t a s e t i n v o l v e d up t o 16 i n d i v i d u a l beams were measured at t h i s a n g l e of i n c i d e n c e f o r s the Rh(1.11) s u r f a c e ; t h e s e d a t a a r e c o l l e c t e d i n Appendices A4-:A6.: - A p l o t of e x p e r i m e n t a l 1(E) c u r v e s f o r the (11)-- and (10) beams w i t h c u r v e s c a l c u l a t e d f o r the two p o t e n t i a l s i s p r e s e n t e d i n F i g . 6.10, f o r the normal b u l k s t a c k i n g sequence. F o l l o w i n g on the p r e v i o u s e x p e r i e n c e , the, c o m p a r i s o n s of i n d i v i d u a l c a l c u l a t e d and measured 1(E) c u r v e s were c l o s e l y checked t o a v o i d m i s l e a d i n g c o n c l u s i o n s a s s o c i a t e d w i t h -dominant peaks i n 1(E) c u r v e s a t the b e g i n n i n g or end of t h e c o n s i d e r e d energy range. For d i f f r a c t i o n s p o t s c l o s e to the edge of the c u r v e d LEED s c r e e n , our measured i n t e n s i t i e s a r e a r t i f i c i a l l y d i m i n i s h e d because b o t h t h e s o l i d a n g l e subtended a t the p o i n t of o b s e r v a t i o n and the g r i d t r a n s m i s s i o n £129] a r e l o w e r compared w i t h those f o r s p o t s near t h e c e n t r e o f t h e s c r e e n . To a s s e s s t h e i n f l u e n c e o f such a d d i t i o n a l f a c t o r s , c o m p a r i s o n s of e x p e r i m e n t a l and t h e o r e t i c a l 1(E) c u r v e s were made w i t h d i f f e r e n t energy r a n g e s (and d i f f e r e n t numbers of beams) f o r the i n d e p e n d e n t l y measured s e t s of d a t a . Sample c o n t o u r p l o t s a r e p r e s e n t e d i n F i g . 6.11 and the r e s u l t s from t h e c o n t o u r p l o t s o f f o r t h e s e d a t a a t . o f f - n o r m a l : i n c i d e n c e a r e g i v e n i n Table 6.3 as w e l l as s p e c i f i c a t i o n s of t h e e x t e n t o f t h e c o m p a r i s o n s made between 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 each s e t of t h e 160 50 I00 150 200 250 ~I00 150 200 250 Energy (eV) Energy (eV) E i S S I f 6 A J 0 * c c i p a r i s o n o f e x p e r i m e n t a l I ( E ) c u r v e s f c r the (IT) and (10) l e a n s at *=10°, + = 109° f o r Bh(111) w i t h i n t e n s i t y p r o f i l e s c a l c u l a t e d f o r the p o t e n t i a l s V^™ and ¥<^u f ° r t h r e e d i f f e r e n t v a l u e s o f AdX assuminq t h e normal F C C r e q i s t r y f o r t h e s u r f a c e . 161 163 e x p e r i m e n t a l d a t a . A l l t h r e e i n d e p e n d e n t s e t s o f d a t a measured at0"?10°, ^=109° g i v e c o n s i s t e n t v a l u e s of Gd% and lgr . , F u r t h e r m o r e , f o r t h i s d i r e c t i o n o f i n c i d e n c e , both p o t e n t i a l s i n d i c a t e a s m a l l c o n t r a c t i o n (about 1%) f o r the topmost l a y e r s p a c i n g , i n c o n t r a s t w i t h t h e da t a a t normal i n c i d e n c e where a d i s c r e p a n c y was noted (Table 6.2). I t i s a l s o i n t e r e s t i n g t h a t the d i f f e r e n c e between t h e mean v a l u e s of V . i n T a b l e 6-3 f o r V*,,, and i s v e r y c l o s e to the 6 eV expected f r o m t h e d i f f e r e n t p o t e n t i a l s a t the m u f f i n - t i n ; r a d i u s . Ey c o n t r a s t t h e mean v a l u e s o f V o f i n T a b l e 6-2 a r e more n e g a t i v e (-18.8eV) and more p o s i t i v e (-10.6eV) than t h e v a l u e s i n Ta b l e .6-3 f o r ,7^^ and V r e s p e c t i v e l y - The c o r r e s p o n d i n g v a l u e s o f Ad& i n T a b l e 6.2 a r e more n e g a t i v e (-4.3%) and more p o s i t i v e (+0-3%) th a n the v a l u e s i n T a b l e 6.3; i n r e l a t i o n t o t h e r e s u l t s f o r o f f - n o r m a l i n c i d e n c e , a t normal i n c i d e n c e the minima i n r"r have been d i s p l a c e d s l i g h t l y i n o p p o s i t e d i r e c t i o n s a l o n g t h e v a l l e y s i n the c o n t o u r p l o t s o f F i g u r e 6-6- Even though t h e mean v a l u e s of &d% and Vor f o r t h i s o f f - n o r m a l a n g l e of i n c i d e n c e c o r r e s p o n d c l o s e l y f o r the two p o t e n t i a l s , t h e r e l i a b i l i t y i n d e x r ^ does not i n d i c a t e t h e o v e r a l l agreement i s any b e t t e r t h a n t h a t a c h i e v e d a t normal i n c i d e n c e . , A l t h o u g h t h e minimum values...of;,,!^ i n T a b l e 6.3 i n d i c a t e t h a t t h e g e n e r a l l e v e l of agreement between c a l c u l a t e d and measured 1(E) c u r v e s d i m i n i s h e s as t h e comparisons a r e made ov e r i n c r e a s e d energy r a n g e s (and f o r more beams), the c o r r e s p o n d i n g v a l u e s of O d l , and V 0 f a r e c l o s e l y s i m i l a r f o r each s e t of d a t a . T h i s s u g g e s t s t h a t t h e n e g l e c t o f v a r i a t i o n s i n the g r i d I u U -4 M S o C M C M o o CM O C M CM > V e a 0> 01 u 60 «,. M S • W O « +' « 01 o o I o •I > at > Ml o o > o > as o © > 41 > • • O *M o o -* I I o •I GO > 01 M S o 0> > 01 CM co S3 c < o »4 O O •w CM CO co o o *4 CO o I o I co co • o o d *4 MS O •H -9 O I * 4 • O • O I »4 MS o •I o *4 MS I 01 c <B > > > U ed 01 01 01 u CM >» <o lA CO CM a. •M MS o 01 s CM c 0 u > > > 01 01 01 C M M S >»• u i —1 «»• - M M S « H C M > 01 CM o CM > 0) >»• CM o CM 01 e <8 0) •a "o 01 o ra a • E o o z u U - l O e c 0 o i -w 10 c - H 90 3 •H O z Ifi 01 a 3 oe a u > e 0) e i* O u 01 • a 0 X z *4 z 2 OS CM at 0) 9 -M CO > e « C M co 0) 3 <0 > c 3 X e o e CO II as Z > e o a e CM 3 ad > l a t l e 6 j j C o n d i t i o n s o f t e s t aqreenent between experi»erit • u l t i t l e - E c a t t e r i n g c a l c u l a t i c r s f o r t h r e e s e t s of i n t e r s i • easured a t *-=10<>, «< = 1090 on Eh(111). 165 t r a n s m i s s i o n and t h e angl e subtended a t the p o i n t of o b s e r v a t i o n a r e not s t r o n g l y a f f e c t i n g c o n c l u s i o n s about s u r f a c e geometry, a l t h o u g h t h e r e may be r e d u c t i o n s i n t h e o v e r a l l agreement. A n o t h e r source o f e x p e r i m e n t a l u n c e r t a i n t y i s i n s e t t i n g t h e a n g l e of i n c i d e n c e . To a s s e s s t h i s f o r t h e data o f e x p e r i m e n t 3, i n f o r m a t i o n i s g i v e n i n T a b l e 6.3 comparing t h e l e v e l o f agreement w i t h experiment f o r a d d i t i o n a l c a l c u l a t i o n s made w i t h £-=80, ^=109° and 9-= 12°, <^  = 1G9°. The f i r s t of t h e s e a d d i t i o n a l a n g l e s o f i n c i d e n c e does not improve the l e v e l of agreement a c h i e v e d w i t h © " = 1 0 0 , ^  =109°, as m o n i t o r e d b y ~ r r , but t h e second i n d i c a t e s a much p o o r e r agreement. T h i s s u g g e s t s t h a t the v a l u e 10° i s c o r r e c t t o w i t h i n 1 ° , and t h i s e x p e r i m e n t a l u n c e r t a i n t y does n ot s e r i o u s l y a f f e c t t h e , d e t e r m i n a t i o n o f the topmost i n t e r l a y e r s p a c i n g . 6.4 C o n c l u s i o n s By a v e r a g i n g t h e r e s u l t s o f t h e r e l i a b i l i t y - i n d e x a n a l y s i s f o r the t h r e e i n d e p e n d e n t s e t s of measurements a t each a n g l e of i n c i d e n c e we f i n d { I a b l e s 6.2 and 6.3) t h e f o l l o w i n g v a l u e s f o r t h e s u r f a c e s t r u c t u r a l parameter and t h e i n n e r p o t e n t i a l ( i ) from Ad%=-4.,3±0. 751 and 7 o r =-18.8±0.5eV a t normal i n c i d e n c e and 6dS=-0.8±Q.8% and V =-17.5±0.5eV a t or 4>=10o, <p =109o 166 ( i i ) from Adg= + 0.3±0-9S and Vof =- 1 0.6±0.7eV a t normal i n c i d e n c e and £>d%=-0. 5±C.7% and V 0 f N =- 11.0±0.7eV a t 9-=10o, 0=109° As a f i n a l s t e p , r e s u l t s from each p o t e n t i a l need a v e r a g i n g over t h e two a n g l e s of i n c i d e n c e . T h i s c o u l d be done by a d i r e c t s u p e r p o s i t i o n of the i n d i v i d u a l c o n t o u r p l o t s , a f t e r w e i g h t i n g them a c c o r d i n g t o t h e a p p r o p r i a t e energy range. T h i s would be e g u i v a l e n t to the approach o f Z J . However, a t t h i s s t a g e the o v e r a l l r e s u l t s f o r Rh (111) can be gauged j u s t as o b j e c t i v e l y by t a k i n g an energy-weighted mean o f parameter v a l u e s g i v e n f o r each a n g l e of i n c i d e n c e . With t h e a p p r o p r i a t e r o u n d i n g o f s i g n i f i c a n t f i g u r e s t h i s y i e l d s a f i n a l r e s u l t ; ( i ) from V ^  &&%=-2±2% and ? o r=-18±1e? ( i i ) from v . . , , o.d%= 0±2$ and V =-11±1eV (CKI3 of The f i n a l u n c e r t a i n t i e s guoted here appear t o encompass t h e e f f e c t s of s m a l l e r r o r s o f an e x p e r i m e n t a l n a t u r e , i n c l u d i n g minor d i f f e r e n c e s i n 1(E) c u r v e s from independent measurements and e r r o r s i n s e t t i n g the a n g l e o f i n c i d e n c e . The r e s u l t s from t h e V ^  p o t e n t i a l c l e a r l y i n d i c a t e t h e n e c e s s i t y f o r i n c l u d i n g more th a n one a n g l e of i n c i d e n c e i n 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 ; whether two a n g l e s o f i n c i d e n c e n e c e s s a r i l y p r o v i d e s s u f f i c i e n t d a ta i s s t i l l perhaps d e b a t a b l e . The r e l i a b i l i t y - i n d e x approach i n t h i s case does n ot a l l o w a statement t o the e f f e c t t h a t one p o t e n t i a l i s r i g h t and the o t h e r i s wrong. B i t h i n the p r e s e n t c a l c u l a t i o n a l scheme 167 both p o t e n t i a l s d e s c r i b e the e x p e r i m e n t a l d a t a e q u a l l y w e l l , a l t h o u g h some c l a r i f i c a t i o n of t h e i r r e l a t i v e m e r i t s may become apparent i f f u r t h e r r e f i n e m e n t s a r e made e.g. i n c l u d i n g t h e energy dependence o f V o r and b e t t e r d e s c r i p t i o n s of t h e atomic v i b r a t i o n s i n the c a l c u l a t i o n s . T h i s work does show t h a t s p e c i a l c a r e i s needed when t h e r e a r e dominant f e a t u r e s i n 1(E) c u r v e s a t the b e g i n n i n g (or end) o f t h e energy range c o n s i d e r e d . The z e r o o r s m a l l c o n t r a c t i o n found here f o r : the Eh(111) s u r f a c e i s t y p i c a l of the v a l u e s found f o r o t h e r (111}. f a c e s o f f a c e - c e n t r e d c u b i c m e t a l s . As was l i s t e d i n S e c t i o n 5.5 the (111) s u r f a c e s o f Ag,Al>Co,Ir and N i a l l show s m a l l t o z e r o c o n t r a c t i o n s w h i l e the work p r e s e n t e d e a r l i e r on Cu (111) i n S e c t i o n 5. 3 i n d i c a t e s a somewhat l a r g e r c o n t r a c t i o n of about 5%. As y e t P t remains the o n l y FCC metal thought t o show a d i l a t i o n of i t s (111) s u r f a c e , t h o u g h , t h e m o d i f i c a t i o n i s o f s m a l l p r o p o r t i o n s . 168 CHAPTEB 7 THE BHfTOO) SO BE ACE 169 The (1.00.) s u r f a c e s of t h e t r a n s i t i o n m e t a l s have been i n t e n s e l y s t u d i e d f o r t h e i r c h e m i s o r p t i o n p r o p e r t i e s £ 14 j and f o r the f a c t t h a t the most s t a b l e (100) s u r f a c e s o f F t , I r and Au a r e r e c o n s t r u c t e d £-25,26,33 J a t room t e m p e r a t u r e . As d i s c u s s e d i n S e c t i o n 2.2, t h e s e r e c o n s t r u c t i o n s of FCC m e t a l s a r e thought t o e s s e n t i a l l y i n v o l v e a h e x a g o n a l s u r f a c e l a y e r superimposed on t h e b u l k (100) s u b s t r a t e . A s i d e from the p r e s e n t work, the o n l y o t h e r LEED s t u d i e s o f Eh (100) a r e those of Tucker £16,171 and o f C a s t n e r e t a l J. 18 J , both concerned mainly w i t h c h e m i s o r p t i o n p r o p e r t i e s , and n e i t h e r of which r e p o r t e d any 1(E) d a t a . T u c k e r * s r e s u l t s s u f f e r e d from the l a c k o f a s u r f a c e c o m p o s i t i o n m o n i t o r , b u t both s e t s of a u t h o r s r e p o r t e d a (1x1) p a t t e r n f o r the c l e a n s u r f a c e . A s t a b l e r e c o n s t r u c t e d Eh (100) s u r f a c e t h e r e f o r e appears u n l i k e l y ; i f such a r e c o n s t r u c t i o n s h o u l d o c c u r , i t would have t o p r e s e r v e the symmetry of the s u r f a c e . Two p o s s i b l e r e c o n s t r u c t i o n s t h a t f u l f i l l t h i s c r i t e r i o n by simple, r e g i s t r y s h i f t s of the s u r f a c e l a y e r a r e shown i n F i g . 7^.1. The normal b u l k s t a c k i n g seguence r e g u i r e s t h e s u r f a c e atoms t o occupy the A s i t e s . Two o t h e r p o s s i b l e r e g i s t r i e s a r e w i t h the t o p - l a y e r atoms on the on-top C s i t e s or on the b r i d g e s i t e s B. l i S P I s Jj.1 S c h e n a t i c d i a g r a n of the Rh(100) s u r f a c e (a) and the c o r r e s p o n d i n g LIED p a t t e r n (b) i n the D o t a t i o n of Jona r 1281. The u n i t aesh i s Barked i n ( a ) . The complete c i r c l e s a re f o r a t c a s i n the second l a y e r , and the dashed c i r c l e s c o rrespond t c a t o p a o s t l a y e r w i t h t h e r e g i s t r y b e l c n g i n g t c the bulk i . e . a t c a s i n the top l a y e r are above the 4 - f o l d s i t e s such as A. Cther r e g i s t r i e s c o n s i d e r e d are where atoms a r e ever the 2-f c l d s i t e ( l i k e B ) , or d i r e c t l y over atons i n the l a y e r below la s f o r C ) . 171 7-iJ E x p e r i m e n t a l The f i r s t e x p e r i m e n t s were c a r r i e d out on the same s i n g l e c r y s t a l as used by Tucker £ 16,17J- The (100) s l i c e was f o u n d , however, t o be m i s o r i e n t e d by some 4°, so the s l i c e was c a r e f u l l y r e p o l i s h e d t o the c o r r e c t (100) o r i e n t a t i o n -Tucker r e p o r t e d s h a r p LEED p a t t e r n s on h e a t i n g t o 1600K-In the p r e s e n t work w i t h t h i s c r y s t a l . Auger e l e c t r o n s p e c t r o s c o p y showed t h a t p r o l o n g e d h e a t i n g a t 1300K, t h e t e m p e r a t u r e l i m i t o f t h e h e a t e r used i n t h i s c a s e , caused a p p r e c i a b l e amounts o f C and S i t o accumulate on t h e s u r f a c e as shown i n F i g - 7 . 2 ( a ) - The S i _ c o u l d be removed by argon i o n bombardment (5G0 eV, about 5 microamps/cm 2) f o r 10 minutes. F i g - 7 . 2 ( b ) . , A n n e a l i n g a t 1000K i n vacuum, or s l i g h t l y l o wer t e m p e r a t u r e s i n oxygen, reduced the C Auger s i g n a l t o below the d e t e c t a b l e l i m i t . F i g . 7 . 2 ( b ) . However, depending upon the p r e c i s e time and temperature o f a n n e a l i n g . S i was sometimes found t o r e a p p e a r on t h e s u r f a c e . I t was found t h a t a sharp (1x1),LEED p a t t e r n , w i t h low background i n t e n s i t y and m i n i m a l c o n t a m i n a t i o n 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 , c o u l d be o b t a i n e d from s e v e r a l c y c l e s of argon i o n bombardment and heat t r e a t m e n t i n oxygen f o l l o w e d by a f i n a l a n n e a l i n g f o r a few minutes a t 900K i n vacuum., H e a t i n g a t or below 600K i n hydrogen ( 1 x 1 0 - 7 T o r r ) was found t o be u s e f u l f o r removing any r e s i d u a l oxygen d u r i n g t h e f i n a l s t a g e s o f c l e a n i n g . On s e v e r a l o c c a s i o n s d u r i n g the c l e a n i n g and o r d e r i n g p r o c e d u r e s , f a i n t f r a c t i o n a l - o r d e r d i f f r a c t i o n s p o t s were found 172 RhflOO) I a) A A I r r 1 If. b) /V^ /^ VV/ r V f I I I I I 100 200 300 eV I i 5 i J I g 2*2 Auger s p e c t r a of Eh (100) s u r f a c e s f o r a .5KeV, 10 l i c r c a m p bean: a) s u r f a c e a f t e r prolonged h e a t i n g a t 1300K showing s u b s t a n t i a l S i and C i m p u r i t i e s t) a f t e r a r g c n ion-fccmbardment, showing reduced S i ana i n c r e a s e d carbon c) c l e a n s u r f a c e spectrum a f t e r h e a t i n g a t 1000K i n vacuo-173 f o r l i m i t e d energy r a n g e s ; t h i s LEED p a t t e r n c o r r e s p o n d e d t o a two-domain (3x1) s u r f a c e s t r u c t u r e (see F i g - 7-3)- However, Auger e l e c t r o n s p e c t r o s c o p y i n d i c a t e d t h a t t h i s p a t t e r n was n o t s i m p l y due t o a r e c o n s t r u c t e d top m etal l a y e r but r a t h e r was a s s o c i a t e d w i t h t h e presence o f S i i m p u r i t y { Auger peak a t 92eV) which had s e g r e g a t e d t o and o r d e r e d on the s u r f a c e . I t would be i n t e r e s t i n g t o know the a c t u a l s u r f a c e s t r u c t u r e i n v o l v e d ; a c o i n c i d e n c e s i t e s u p e r p o s i t i o n , perhaps i n v o l v i n g a rhodium s i l i c i d e l a y e r , may seem more p l a u s i b l e than a 1/3 monolayer coverage of a t o m i c S i . L a t e r e x p e r i m e n t s u s i n g s l i c e s c u t from a c r y s t a l from a d i f f e r e n t s o u r c e , see T a b l e 4.2, f a i l e d t o d u p l i c a t e t h i s (3x1) p a t t e r n . Auger e l e c t r o n s p e c t r o s c o p y d i d n o t r e v e a l any S i i m p u r i t y ; hence t h e occurence of t h i s ( 3 x i ) , p a t t e r n does i n f a c t seem t o be a s s o c i a t e d w i t h the p r e s ence o f S i on t h e s u r f a c e . When t h i s second c r y s t a l was c l e a n e d by the method d e s c r i b e d e a r l i e r , a s u r f a c e e x h i b i t i n g a sharp (1x1) LEED p a t t e r n and a " c l e a n " Auger spectrum s i m i l a r t o t h o s e of t h e c l e a n Eh (111) s u r f ace. F i g . 6 . 3 ( d ) , was o b t a i n e d - F o r b oth c r y s t a l s , i t was n o t i c e a b l e t h a t t h e low energy f e a t u r e s a t 170 eV and 139 eV were r e l a t i v e l y i n t e n s e f o r : t h e (100) f a c e compared w i t h t h o s e from t h e (111) f a c e s , see F i g - 6. 1 (d) and 7.2 (c) . A t y p i c a l LEED p a t t e r n and the beam l a b e l l i n g scheme i s d e p i c t e d i n F i g . 7.4. 1(E) c u r v e s were measured a t normal i n c i d e n c e and f o r 6^=9°, ^ =20°, i n t h e a n g l e c o n v e n t i o n o f Jona I 1283- The i n t e g r a t e d beam i n t e n s i t i e s , n o r m a l i z e d to u n i t i n c i d e n t c u r r e n t and c o r r e c t e d f o r background i n t e n s i t y , were 174 U f l B I S 2a.3 Two-doiain (3X1) l i E D p a t t e r n from the Eh (100) s u r f a c e ~ a t 100eV, th o u g h t t o be due t c the presence o f s i l i c o n 175 F i g u r e 7^4 LEEC p a t t e r n s from the c l e a n Eh (100) s u r f a c e f c r (a) normal i n c i d e n c e (150eV) , (b) f c r ©^9°, £=20<> (94eV) and the team l a b e l l i n g scheme (c) and (<3) . 176 s t o r e d on d i g i t a l c a s s e t t e p r i o r t o t r a n s f e r t o an IBB 370/168 computer f o r t h e r e l i a b i l i t y - i n d e x c a l c u l a t i o n s . The e x p e r i m e n t a l d a t a a r e c o l l e c t e d i n Appendices k7~&9. E x p e r i m e n t a l 1(E) c u r v e s f o r beams t h a t . > s h o u l d be e g u i v a l e n t by symmetry a t normal i n c i d e n c e s e r e a v e r a g e d , a l t h o u g h the d i f f e r e n c e s were s m a l l . The I (E) c u r v e s measured f o r t h e (11) s e t o f beams, which s h o u l d be e g u i v a l e n t a t normal i n c i d e n c e , a r e shown i n the upper h a l f o f F i g - 7.5; the same peak p o s i t i o n s and o v e r a l l i n t e n s i t y v a r i a t i o n i s observed f o r the (11), (11) and (11) beams. S i m i l a r l y , the ( 2 0 ) , (02) and (02) beams have a l m o s t i d e n t i c a l p r o f i l e s . S m a l l v a r i a t i o n s i n the r e l a t i v e peak i n t e n s i t i e s f o r beams i n t h e (11} s e t were always p r e s e n t i n independent e x p e r i m e n t s a t normal i n c i d e n c e , w h i l s t the agreement between i n d i v i d u a l beams i n t h e {20}, (22} and o t h e r beam s e t s was i n g e n e r a l s l i g h t l y b e t t e r - Such v a r i a t i o n s have t o be a t t r i b u t e d t o e x p e r i m e n t a l e r r o r s ( i n v o l v i n g such f a c t o r s as uneven response of the s c r e e n , i m p e r f e c t i o n s i n t h e c r y s t a l s u r f a c e , and u n c e r t a i n t i e s i n s e t t i n g t h e a n g l e o f i n c i d e n c e ) , •an:a-•vthey^.Ma•it^:A.tie• :le.v!el!;.:Of agreement t o be e x p e c t e d between c a l c u l a t i o n and e x p e r i m e n t s 1(E) c u r v e s f o r a l l beams were smoothed by two o p e r a t i o n s of a t h r e e - p o i n t smoothing ' . f i l t e r . ..©:rior- t o t h e : r e l i a b i l i t y - i n d e x c a l c u l a t i o n s . To a v o i d s p u r i o u s r e s u l t s i n t h e s e c a l c u l a t i o n s , background and s c a t t e r i n the e x p e r i m e n t a l data were e l i m i n a t e d . 1 7 7 fJ9y£§ 2*_5 1(E) c u r v e s f o r two s e t s c f teams t h a t 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 on the Bh (100) s u r f a c e ; the f o u r t h member of each s e t i s o t s c u r e d by the sample m a n i p u l a t o r . 17.8 ls.2. C a l c u l a t i o n s The c a l c u l a t i o n s were performed as d e t a i l e d i n C h a p t e r s 3 and 5, f o r the two a l t e r n a t i v e r e g i s t r i e s shown i n F i g - 7-1 as w e l l as f o r a t r u n c a t e d h u l k c r y s t a l ; the topmost l a y e r s p a c i n g s o ranged from a -10 to +101 change from t h e bulk v a l u e o f 1.9022A, •o i n s t e p s o f 2.5% or 0.04751. Both t h e band s t r u c t u r e p o t e n t i a l and the s u p e r p o s i t i o n p o t e n t i a l V^tj were used w i t h a t o t a l c f 69 beams b e i n g a v a i l a b l e t o determine 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 f o r an energy range of 40-30Oey. 7.3 R e s u l t s and D i s c u s s i o n F i g - 7-6 compares e x p e r i m e n t a l 1(E) c u r v e s a t normal i n c i d e n c e f o r the (11) and (20) beams w i t h t h o s e c a l c u l a t e d f o r t h e d i f f e r e n t r e g i s t r i e s u s i n g the 7^^|3 p o t e n t i a l ; I n t h e s e c a l c u l a t i o n s the t o p i n t e r l a y e r s p a c i n g s a r e f i x e d by the h a r d -sphere model f o r atomic r a d i i determined by the b u l k s t r u c t u r e of rhodium £62,63]- I t i s c o n c l u d e d from F i g - 7.6 t h a t t h e t h e o r e t i c a l c u r v e s f o r r e g i s t r i e s B and C do n o t agree as w e l l as t h o s e f o r t h e b u l k s t a c k i n g seguence ( r e g i s t r y &} , a l t h o u g h the l e v e l of agreement f o r the l a t t e r i s by no means i d e a l -C o n s i d e r a t i o n o f o t h e r d i f f r a c t e d beams and o t h e r ;topmost i n t e r l a y e r s p a c i n g s f u r t h e r p r e c l u d e t h e r e g i s t r i e s E and C; i t seems t h a t t h e s u r f a c e s t r u c t u r e of Rh (100) i s i n t e r p r e t e d b e s t by an u n r e c o n s t r u c t e d p a c k i n g seguence f o r the t o p l a y e r -F i g - 7.7 shows 1(E) c u r v e s measured f o r s i x beams d i f f r a c t e d from t h e 8h (100) s u r f a c e , t o g e t h e r w i t h c u r v e s c a l c u l a t e d f o r an u n r e c o n s t r u c t e d s u r f a c e u s i n g both p o t e n t i a l s , 179 J i j g u r e 7^6 Comparison o f e x p e r i m e n t a l 1(E) c u r v e s f o r the (11) and (20) beans at normal i n c i d e n c e on fib (100) v i t b c a l c u l a t i o n s , f c r the ¥^(j p o t e n t i a l , f o r the topmost r e g i s t r i e s d e f i n e d by A,E and C i n F i g . 7.1. a The topmost i n t e r l a y e r s p a c i n q s a re 1. 90, 2. 33 and 2.69A f o r t h e i»-fold, 2 - f o l d and 1 - f o l d s i t e s r e s p e c t i v e l y . 1 8 C fi5i?£§ 2x2 Comparison of some e x p e r i m e n t a l 1(E) cu r v e s f o r Bh(100) w i t h c a l c u l a t i o n s f o r t h e vJf H and Mfk0 p o t e n t i a l s V o r =-12eV and Ad%=-5,Q and+5* f o r (a) n o r i a l i n c i d e n c e and (b) at £- = 9°, i =200. Energy (eV) 182 f o r Adg=-5I, 0 and - 5 % . F o r each beam and s p a c i n g , the two s e t s of c a l c u l a t e d c u r v e s a r e c l o s e l y s i m i l a r a p a r t from a s l i g h t s h i f t t o h i g h e r energy f o r r e l a t i v e t o 7 ^ . Oth e r w i s e the d i f f e r e n c e s noted e a r l i e r i n t h e b e h a v i o u r o f the phase s h i f t s do not appear t o have much v i s u a l e f f e c t on the c a l c u l a t e d 1(E) c u r v e s . The observed s h i f t r e l a t e s t o t h e d i f f e r e n c e i n p o t e n t i a l a t the r a d i u s o f the m u f f i n - t i n s p h e r e s of about 6eV, a s mentioned i n Chapter 6, and assuming t h i s p r o v i d e s t h e main d i f f e r e n c e i n t h e p o t e n t i a l s , the two s e t s o f c a l c u l a t e d 1(E) c u r v e s would show peaks a t the same e n e r g i e s i f the r e a l p a r t s of t h e i n n e r p o t e n t i a l ( i . e . .. V r ) : a r e t a k e n as -12eV f o r and - 18eV f o r V ^ . To a good a p p r o x i m a t i o n , making V o r more n e g a t i v e causes a r i g i d s h i f t i n t h e c a l c u l a t e d 1(E) c u r v e s to lower energy. With t h e s e v a l u e s o f Vflr , the agreement w i t h e x p e r i m e n t may be e s t i m a t e d by v i s u a l c o m parison t o vary from v e r y good f o r some beams (e.g. (11) beam a t normal i n c i d e n c e , Ad$=-5%) t o q u i t e poor f o r o t h e r s (e.g. (11) beam a t © - = 9 0 , (/>=20o, o d S 5--5S). A v i s u a l e s t i m a t i o n o f t h e o v e r a l l degree o f f i t cannot d i s t i n g u i s h between t h e two p o t e n t i a l s , but i t s u g g e s t s t h a t &% has a v a l u e between 0 and +51-., T h i s e s t i m a t e i s i n c l u d e d i n Tab l e 7. 1. The r e s u l t s of a r e l i a b i l i t y - i n d e x a n a l y s i s f o r the t o t a l of 16 beams a n a l y s e d a t the two a n g l e s of i n c i d e n c e a r e t a b u l a t e d i n T a b l e 7.1 and shown as c o n t o u r p l o t s i n F i g - 7.8 f o r both p o t e n t i a l s . Once a g a i n we n o t i c e a pronounced " v a l l e y of b e s t f i t " as n o t e d i n C h a p t e r 6 f o r the (111) f a c e . The p r e c a u t i o n s noted t h e r e c o n c e r n i n g s t r o n g f e a t u r e s a t the 183 E x t e n t o f C o n d i t i o n s of Comparison b e s t agreement Po t . 9 p n J l e V l A n a l y s i s Adj§ J 6 r R-V 402 16 1548 8 - f a c t o r +2.5±0.9 -11.5±0.7 0-155 w v i s u a l t-2-5±2.5 -12 ±2 WW V , 402 16 1684 E-f a c t o r -1-8± 1-0 -19-6±0-8 0.167 v i s u a l +2-5±2.5 -17 ±2 V f t k ( 3 250 16 1786 R-f a c t o r *2.7±1-0 -11.0±0.9 0.188 Table 7.1 Summary of s t r u c t u r a l d e t e r m i n a t i o n o f t i e Rh(1Q0) s u r f a c e -b e g i n n i n g or end o f 1(E) c u r v e s were ta k e n i n t o a c c o u n t . D e s p i t e the s i m i l a r degree o f v i s u a l agreement between the e x p e r i m e n t a l d a t a and both s e t s of c a l c u l a t e d c u r v e s , t h e c o n t o u r p l o t s e x h i b i t minima a t somewhat d i f f e r e n t v a l u e s o f A d % and V 4 r (Table 7. 1). From t h e d i s c u s s i o n above, a 6eV s h i f t i n V o r i s e x p e c t e d because of d i f f e r e n c e s i n p o t e n t i a l a t the edge of the m u f f i n - t i n s p h e r e ; the observed d i f f e r e n c e i s 8eV, t h e e x t r a 2eV presumably r e f l e c t i n g f u r t h e r d i f f e r e n c e s i n t h e p o t e n t i a l s . The v a l u e s o f the s u r f a c e r e l a x a t i o n p r e d i c t e d by the two p o t e n t i a l s a re o u t s i d e t h e 68% c o n f i d e n c e l i m i t s ( 1 s t a n d a r d e r r o r ) , a l t h o u g h t h e y a r e w i t h i n 95% c o n f i d e n c e l i m i t s i 2 s t a n d a r d e r r o r s ) - , . The minimum v a l u e of R i s s l i g h t l y l o w e r f o r V ^ K D (0- 155) than f o r V ^  (0. 167) -As an e x p e r i m e n t a l v a l u e of t h e s u r f a c e Debye t e m p e r a t u r e |°/) was not a v a i l a b l e f o r rhodium, a f u r t h e r c a l c u l a t i o n u s i n g the p o t e n t i a l V ^ r j was performed w i t h ®^=2 50K- The b e s t f i t v a l u e s o f A d % and Vor have been changed o n l y s l i g h t l y ( Table 7,1) a l t h o u g h R has i n c r e a s e d t o 0-188 from 0- 155 f o r c9-0=402K. A l l v a l u e s o f a guoted i n T a b l e 7.1 a r e below 0.20, 184 f i s u i e l i . 8 Contour p l o t s f o r Bh(100) of r r v e r s u s V* r and AdX f o r (a) t h e V ^ j and (h) the p o t e n t i a l . E r r o r t a r s are the s t a n d a r d e r r o r s ana d e f i n e d i n Chapter 5. 185 186 and axe t h e r e f o r e w i t h i n the, range f o r which Z a n a z z i and Jona proposed a s t r u c t u r e i s " v e r y p r o b a b l e " . The a n a l y s i s h e r e puts such c o n s i d e r a t i o n s on a more g u a n t i t a t i v e b a s i s . F u r t h e r s t u d i e s w i t h c o n t o u r p l o t s f o r r e g i s t r i e s t h a t a r e i n c o r r e c t , on the b a s i s of v i s u a l a n a l y s i s , g i v e h i g h v a l u e s of S a t l o c a l minima as w e l l as l a r g e u n c e r t a i n t i e s as measured by c j and c"v . Comparison of t h e d e t e r m i n a t i o n s of &&% from th e two p o t e n t i a l s , w i t h b oth the v i s u a l and r e l i a b i l i t y - i n d e x a n a l y s e s , as w e l l as t h e s l i g h t l y h i g h e r B v a l u e f o r t h e V ^ p o t e n t i a l , make i t t e m p t i n g t o l a b e l the '^13 p o t e n t i a l as t h e " b e t t e r " p o t e n t i a l and t h e r e f o r e t o quote a more d e f i n i t e &% v a l u e . U n f o r t u n a t e l y a t p r e s e n t i t i s not p o s s i b l e t o a s s e s s the f u l l s i g n i f i c a n c e of t h e d i f f e r e n t n v a l u e s o f ft f o r the two p o t e n t i a l s . C a r e f u l e x a m i n a t i o n of the d a t a d i d not produce any a n o m a l i e s of the s o r t noted f o r t h e (111) s u r f a c e where a r e l a t i v e l y minor f e a t u r e i n an I (£) c u r v e s i g n i f i c a n t l y i n t e r f e r e d w i t h the comparison between ex p e r i m e n t and t h e o r y . N e v e r t h e l e s s t h a t e x p e r i e n c e s u g g e s t s c a u t i o n i n a t t a c h i n g u n c e r t a i n t i e s a t t h i s s t a g e of LEED c r y s t a l l o g r a p h y ; we t h e r e f o r e b e l i e v e t h a t the v a l u e o f AdS f o r Eh (100) must be guoted as 0±2.5%. 187 7. 4 Comparisons With P r e v i o u s .Studies The b i b l i o g r a p h y o f S e c t i o n 5-5 shows t h a t none of the (100) f a c e s of f a c e - c e n t r e d c u b i c m e t a l s s t u d i e d up t i l l now show any s u b s t a n t i a l c o n t r a c t i o n or d i l a t i o n . Thus f a r , no e t h e r r e l i a b i l i t y - i n d e x c a l c u l a t i o n s have been performed f o r (100) s u r f a c e s a l t h o u g h F o u r i e r t r a n s f o r m £99J and d a t a a v e r a g i n g £114] a n a l y s e s on data from Cu(100) c l a i m low e r r o r margins w i t h r e s u l t s of a s u r f a c e c o n t r a c t i o n o f 0±1%. As t h e r e s u l t s from t h e two p o t e n t i a l s used h e r e appear e q u a l l y p r o b a b l e , we can see t h a t the r e s u l t of &d%=0±2.5% i s c o n s i s t e n t w i t h t h e c o n c l u s i o n s r e a c h e d f o r e t h e r FCC (10 0) s u r f a c e s -CHAPTER 8 TEE RH O 10) SURFACE 189 The {110) s u r f a c e s o f s e v e r a l FGC me t a l s have been s t u d i e d by 1EED; the b i b l i o g r a p h y o f S e c t i o n 5.5 i n d i c a t e s t h a t t h e s e s u r f a c e s o f t e n e x h i b i t c o n t r a c t i o n s f o r the s u r f a c e l a y e r of up t o 10% from the b u l k v a l u e . S t u d i e s o f t h e Rh{110) s u r f a c e appear t o be c o n f i n e d t o t h e e a r l y oxygen a d s o r p t i o n e x p e r i m e n t s of Tucker J 1 6 J , a s h o r t i n f r a r e d s t u d y o f CO a d s o r p t i o n on (110) o r i e n t e d Rh f i l m s 1132 J and, most r e c e n t l y , a d e t a i l e d s t u d y of CO a d s o r p t i o n on the s i n g l e c r y s t a l s u r f a c e by Marbrow and Lambert £133 3- I n t h i s l a t t e r work the a u t h o r s f o u n d t h a t t h e c l e a n Eh (110) s u r f a c e showed a s i m p l e (1x1) LEED p a t t e r n and hence was u n l i k e l y to be l a t e r a l l y r e c o n s t r u c t e d . A diagram o f an i d e a l (110) s u r f a c e i s shown i n F i g . 8.1, t o g e t h e r w i t h the a s s o c i a t e d LEED p a t t e r n . T h i s s u r f a c e p o s s e s s e s o n l y a t w o - f o l d symmetry a x i s w i t h 2 m i r r o r p l a n e s and i s r e l a t i v e l y open compared w i t h t h e (111) o r (100) s u r f a c e s . The b u l k i n t e r p l a n a r s p a c i n g i s c o r r e s p o n d i n g l y s h o r t , and t h i s may a c c o u n t i n p a r t f o r t h e se e m i n g l y l a r g e c o n t r a c t i o n s r e p o r t e d f o r t h e Aq{110) and Al(110) s u r f a c e s ; t h u s t h e a b s o l u t e c o n t r a c t i o n s may be s i m i l a r t o those found on the l e s s open s u r f a c e s but t h e s h o r t e r bulk i n t e r l a y e r d i s t a n c e s l e a d t o l a r g e r percentage changes. A r e c u r r i n g g u e s t i o n i n the LEED s t u d i e s o f t h e (110) s u r f a c e s o f A l and Ag i n v o l v e s t h e r o l e of s u r f a c e roughness. For A l £ 101, 108, 110- 112J and Ag C 101,106- 107] o n l y modest agreement between 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 (E) c u r v e s has been o b t a i n e d . The d i s c r e p a n c i e s have been a t t r i b u t e d t o s u r f a c e roughness, as e v i d e n c e d by poor c o n t r a s t i n t h e LEED p a t t e r n s i n some c a s e s , but a s i m p l e attempt t o i n c l u d e roughness 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 d i d n o t 190 191 s u b s t a n t i a l l y improve the agreement w i t h experiment £108 ]. Host r e c e n t l y a UPS/XPS stu d y £134} has shown t h a t the a n g u l a r dependence o f p h o t o e m i s s i o n from the c l e a n Ag(110}-(1x1) s u r f a c e i s markedly a l t e r e d by the n a t u r e o f the c r y s t a l p o l i s h i n g . However, both c h e m i c a l l y and n o n - c h e r a i c a l l y p o l i s h e d specimens gave a p p a r e n t l y i d e n t i c a l LEED p a t t e r n s ; u n f o r t u n a t e l y no 1(E) c u r v e s were measured. The a u t h o r s s u g g e s t t h a t c h e m i c a l c l e a n i n g does i n f a c t produce a "smoother" s u r f a c e and t h a t i n c o m p l e t e r e m o v a l o f the d i s o r d e r e d m a t e r i a l i n d u c e d by m e c h a n i c a l p o l i s h i n g may. be r e s p o n s i b l e f o r some of the d i f f i c u l t i e s e x p e r i e n c e d i n o b t a i n i n g good agreement between e x p e r i m e n t a l and t h e o r e t i c a l LEED d a t a . 8.J - E x p e r i m e n t a l E x periments were performed on a s i n g l e c r y s t a l s l i c e c u t from a r o d purchased from Research O r g a n i c / I n o r g a n i c C h e m i c a l s C o r p o r a t i o n . The i n i t i a l h e at t r e a t m e n t s produced the Auger spectrum o f F i g - 8 . 2 ( a ) . Peaks due t o s m a l l amounts of C (272eV) and S(152eV) can be seen i n a d d i t i o n t o the main Sh peaks. A s m a l l peak a t about 116eV i s a s s i g n e d t o phosphorus; t h i s element i s u s u a l l y found a t . 120eV but no o t h e r common co n t a m i n a n t s have an Auger peak i n t h i s r e g i o n . Rhodium does not have a c a l c u l a t e d t r a n s i t i o n near t h i s energy £49] and the peak i s not a p parent i n t h e c l e a n s u r f a c e spectrum, F i g . 8 . 2 ( c ) . Once a g a i n no t r a c e o f a boron peak a t 180eV c o u l d be d e t e c t e d . T h i s i s i n marked c o n t r a s t to t h e r e s u l t s of Marbrow and Lambert £133 J who o b t a i n e d B s i g n a l s almost as s t r o n g as the 192 Rh(110) ~ i 1 1 1 1 100 2 0 0 300 eV Ii^Jl£g JL2 Auqer s p e c t r a o f the Bh(110) s u r f a c e a t a primary bea« v o l t a g e of 1.5KeV and 10 a i c r o a n p c u r r e n t : a) a f t e r i n i t i a l heat t r e a t a e n t s showing S ( 1 5 2 e V ) , P(120eV) and C{272eV) c o n t a a i n a t i o n on the s u r f a c e t ) a f t e r a r g c n i c n - b c a b a r d a e n t ; P and S reaoved t u t C i n c r e a s e d c) c l e a n s u r f a c e spectrum. 193 main Rh peak at 302eV upon h e a t i n g t o 1300K. I t can o n l y be c o n c l u d e d a g a i n t h a t these c o n t r a s t i n g o b s e r v a t i o n s must o r i g i n a t e i n d i f f e r e n c e s i n manufacture o r p o l i s h i n g p r o c e d u r e s . The S and P c o n t a m i n a n t s c o u l d be removed from t h e s u r f a c e ty a r g o n - i o n bombardment (1keV a t 5 microamps f o r 20 minutes) but a g a i n o n l y a t the expense o f i n c r e a s i n g the s u r f a c e c o n c e n t r a t i o n of C a s shown i n the Auger spectrum of F i g - 8.2(b). However, t h i s s u r f a c e c a r b o n d i f f u s e d back i n t o t h e b u l k on h e a t i n g a t 1000K. A f t e r s e v e r a l ; c y c l e s of i o n -bombardment and a n n e a l i n g an a p p a r e n t l y c l e a n s u r f a c e was o b t a i n e d h a v i n g t h e Auger spectrum of F i g . 8.2 (c) w i t h a s h a r p (1x1) LEED p a t t e r n . The Auger spectrum of t h e c l e a n s u r f a c e i s r o u g h l y i n t e r m e d i a t e i n appearance . between t h a t o f Eh (100) ( F i g . 7.2) and t h a t of R h ( 1 1 1 ) , F i g . 6-1- The major peaks remain v e r y s i m i l a r i n a l l t h r e e c a s e s . The minor peaks at 139 and 170ev appear t o i n c r e a s e i n r e l a t i v e i n t e n s i t y as (111)< (110) < (100) -A t y p i c a l LEED p a t t e r n and the l a b e l l i n g scheme i s shown i n F i g . 8.3 f o r Rh{110). I n t e n s i t y data were r e c o r d e d f o r t h e range 50 t o 250eV f o r two d i r e c t i o n s o f i n c i d e n c e d e f i n e d by 6>=0°, and 6>= 1Q<>,<^ . =1350 i n the a n g l e c o n v e n t i o n Of Jona I 128]-1(E) c u r v e s t h a t s h o u l d be e g u i v a l e n t by symmetry a t normal i n c i d e n c e were averaged p r i o r t o smoothing- E g u i v a l e n t members of the {11} and {21} beam s e t s a r e d e p i c t e d i n F i g - 8.4- The s i m i l a r i t y of each member of an e q u i v a l e n t s e t t o t h e o t h e r members o f t h a t s e t i s not as s a t i s f a c t o r y a s t h a t t y p i c a l l y found f o r the (111) and (100) s u r f a c e s - T h i s may perhaps be a s s o c i a t e d w i t h some degree of s u r f a c e roughness of t h i s (110) 194 c) d) Ii51i£f 5x3 LEED p a t t e r n from the c l e a n Bh<110) s u r f a c e a t (a) t c r i a l i n c i d e n c e (88eV) and (b) P-=10«, ^ = 135° <90eV). The fceai l a b e l l i n g scheme i s shewn i n (c) and (d). 1 9 5 RhCllO) I 1 1——I —I 1 1 1 100 200 eV 1 0 0 200 eV f i g u r e 8.4 E x p e r i a e n t a l 1(E) c u r v e s f o r t b e Rh(110) s u r f a c e a t u o r i a l i n c i d e n c e f o r the 4 - f c l d e q u i v a l e n t |11> and 121} beam s e t s . The 4th nember o f each s e t i s obscured by the sample t a r i c u l a t o r . 196 s u r f a c e . Two complete s e t s c f independent e x p e r i m e n t s were performed f o r each a n g l e o f i n c i d e n c e , and the i n t e n s i t y d a t a i s g a t h e r e d t o g e t h e r i n A p p e n d i c e s A10-A13, where the good agreement between the two s e t s can be seen. 8.2 C a l c u l a t i o n s Due to t h e low symmetry o f the (110) s u r f a c e m u l t i p l e -s c a t t e r i n g c a l c u l a t i o n s a r e r e l a t i v e l y slow and c o s t l y . Hence i t i s d e s i r a b l e i n t h i s case to l i m i t c o m p u t a t i o n a l e f f o r t by t a x i n g note o f s e v e r a l l e s s o n s l e a r n t d u r i n g the s t u d i e s of the (111) and (100) s u r f a c e s . F i r s t l y , d i f f e r e n t t o p - l a y e r r e g i s t r i e s t h a t p r e s e r v e t h e symmetry of the s u r f a c e , and hence a (1x1) LEED p a t t e r n , do i n f a c t g i v e c l e a r l y d i s t i n g u i s h a b l e c a l c u l a t e d 1(E) c u r v e s . I t seems r e a s o n a b l e t h e r e f o r e t h a t i f the 1(E) c u r v e s f o r a l a t e r a l l y u n r e c o n s t r u c t e d s u r f a c e a r e a good match t o t h e e x p e r i m e n t a l d a t a then i t i s v e r y u n l i k e l y t h a t a n o t h e r r e g i s t r y w i l l produce t h e o r e t i c a l 1(E) c u r v e s w i t h a b e t t e r f i t . F o r Bh( 110) t h e s u r f a c e was i n i t i a l l y assumed t o have t h e r e g i s t r y o f t h e a p p r o p r i a t e b u l k l a y e r and c a l c u l a t i o n s w i t h a l t e r n a t i v e r e g i s t r y were d e f e r r e d u n t i l t h e f i t of the b u l k s t a c k i n g c a l c u l a t i o n s w i t h experiment had been a s s e s s e d . S e c o n d l y , the e x p e r i e n c e o f the (111) and (100) s u r f a c e s r e v e a l s t h a t n e i t h e r the V ^ p s u p e r p o s i t i o n p o t e n t i a l nor the band s t r u c t u r e p o t e n t i a l c o u l d be judged t o be a more s u i t a b l e p o t e n t i a l f o r LEED; both gave e s s e n t i a l l y i d e n t i c a l degrees o f agreement o f t h e o r y w i t h e x p e r i m e n t . As d i s c u s s e d p r e v i o u s l y , t h e y d i d d i f f e r however i n t h e i n n e r p o t e n t i a l 197 needed and i n t h e v a l u e of s u r f a c e c o n t r a c t i o n . s u g g e s t e d , . T h i r d l y i t appears t h a t r e s u l t s d e r i v e d from d i f f e r e n t d i r e c t i o n s of i n c i d e n c e can vary a t l e a s t a s much a s r e s u l t s f o r the same a n g l e s but w i t h a d i f f e r e n t p o t e n t i a l . T a k i n g t h e s e p o i n t s t o g e t h e r , d i f f r a c t e d beam i n t e n s i t i e s were c a l c u l a t e d u s i n g both the r e n o r m a l i s e d f o r w a r d s c a t t e r i n g and l a y e r d o u b l i n g f o r m a l i s m s f o r , o n l y the s u p e r p o s i t i o n p o t e n t i a l V^, 3 a s the use o f t h e second p o t e n t i a l seems u n l i k e l y t o be very h e l p f u l a t t h i s s t a g e , , The same non-s t r u c t u r a l parameters used f o r the (111) and (100) s u r f a c e s were t a k e n , and c a l c u l a t i o n s were made assuming a t r u n c a t e d bulk s t r u c t u r e f o r which t h e s u r f a c e s p a c i n g was a l l o w e d t o vary from o a 15% c o n t r a c t i o n t o a 5% e x p a n s i o n from the b u l k v a l u e (1..345.A) i n s t e p s o f 2.5%, 8-3 B e s u i t s And D i s c u s s i o n S e t 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 I (E) c u r v e s a r e compared i n F i g . .8.5 f o r the (11) beam a t normal i n c i d e n c e and f o r t h e (00) beam a t ©"=1QQ, 4> =135°. A v i s u a l comparison i n d i c a t e s a b e t t e r f i t f o r the (11) beam; v a l u e s of t h e r e l i a b i l i t y i n d i c e s f o r i n d i v i d u a l beams, ( r ^ . ) c , a r e a l s o g i v e n i n F i g . 8.5. These v a l u e s seem c o n s i s t e n t w i t h v i s u a l e v a l u a t i o n s , e s p e c i a l l y f o r t h e (11) beam. The (00) team a t o f f - n o r m a l i n c i d e n c e i s judged by t h e ( r ^ ) ^ v a l u e s t o show a r e l a t i v e l y poor f i t , presumably because s t r u c t u r e i n t h e c a l c u l a t e d 1(E) c u r v e s between 50 and 100eV i s n o t reproduced i n the e x p e r i m e n t a l data., 198 RhOlO) Energy (eV) Energy (eV) f i g u r e 8 A5 C c a r a r i s o n s o f two e x p e r i a e n t a l 1 ( f ) c u r v e s f o r the Eh ( 1 1 0 ) ~ s u r f a c e w i t h c a l c u l a t i o n s , u s i n g the V^,3 p o t e n t i a l f o r f o u r v a l u e s o f The v a l u e of the i n d i v i d u a l beam r e l i a b i l i t y - i n d e x ( r r ) ^ i s g i v e n i n b r a c k e t s f c r each c a l c u l a t e d c u r v e . 199 F i g u r e 8.6 shows a c o n t o u r p l o t of the energy-weighted mean r e l i a b i l i t y - i n d e x r f , as a f u n c t i o n o f V Q r a n d A d S f o r a s i n g l e s e t of d a t a a t normal i n c i d e n c e . As noted p r e v i o u s l y f o r t h e (111) and (100) s u r f a c e s and f o r Cu (111) , t h e r e i s a v a l l e y of "good f i t " r u n n i n g d i a g o n a l l y a c r o s s the p l o t and a pronounced minimum. The v a l u e of r r a t the minimum ( v i z . 0.10) i n d i c a t e s a c l o s e c o r r e s p o n d e n c e between t h e 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 f o r t h i s s e t of da t a a t normal i n c i d e n c e ; i n t h i s c ase t h e c o n d i t i o n s f o r t h e minimum a r e V =-11.*2±0.6eV and £dI--2. 5±1-.23L . The r e s u l t s of t h e r e l i a b i l i t y - i n d e x a n a l y s e s f o r the two independent s e t s o f e x p e r i m e n t a l d a t a a t the two a n g l e s o f i n c i d e n c e employed a r e c o l l e c t e d i n T a b l e 8-1. The agreement both between d i f f e r e n t a n g l e s of i n c i d e n c e and d i f f e r e n t e x p e r i m e n t s i s a t l e a s t a s good as t h a t f o u n d f o r t h e o t h e r f a c e s o f rhodium. As was found f o r Rh (111) , the o f f - n o r m a l i n c i d e n c e d a t a appear t o show a g r e a t e r p e r c e n t a g e c o n t r a c t i o n than do t h e n o r m a l j i n c i d e n c e d a t a . , For both a n g l e s t h e second e x p e r i m e n t shows a s m a l l e r c o n t r a c t i o n than t h e f i r s t ; a g a i n i t appears t h a t e x p e r i m e n t a l e r r o r must l i m i t t h e a c c u r a c y o f a r e s u l t a t a p a r t i c u l a r a n g l e t o 1 or 2% of the b u l k s p a c i n g . The ener g y - w e i g h t e d mean o f the r e s u l t s i n T a b l e 8.1 y i e l d s a c o n t r a c t i o n o f the topmost i n t e r l a y e r s p a c i n g of 2.1±2.Q%, compared w i t h t h e b u l k s p a c i n g , and V =-10.7±1-OeV- T h i s v a l u e c f the i n n e r p o t e n t i a l compares r a t h e r w e l l w i t h t h o s e found w i t h t h e same i o n - c o r e p o t e n t i a l f o r t h e e t h e r rhodium f a c e s ( v i z . -11.5±0.7 and -11. 0.±O. 6ev) . 200 f i s a i i 8^6 A c o n t o u r r l c t f o r Eh (110) of r r v e r s u s V« and A&% f o r d ata a t n o n a l i n c i d e n c e and a c a l c u l a t i o n f o r the p o t e n t i a l . 201 u u c i i -ai 01 u 60 CO 01 01 «o u o u o 00 c o C K O T 3 o O N m m f t O C M C M • • • • o o o O > > > > > > 01 0) V cu 01 CU o o o o o • • • • • • o o o <-l f t f t +1 +1 +1 +1 +1 +1 C M C O o in • • • • • • i -H o o O N o i - t 1 i H 1 r - l • H 1 1 f t i • 4 » 4 C M C M C M O i - l O • • • • • • f t f t C M C M C M +1 +1 +1 +1 +1 m O 00 in • • • • • C N f t •* C M cn 0) oc c CO U T5 OJ u 00 co u o. 01 e c o til u > 01 co > 01 00 C M 01 oo 00 > HI as oo en 6 CO 01 £> "0 0> U - l t - i O CO a • E o o z u CO 1 - " 3 CO CO z > > 01 i - t C M f t « N X w e o> o o •H C u oi C J " O c E CO CO 01 01 e e e e m m r-> P I a £ ? ? V V O 5 ct> a> co ce l a f c l e J A 1 C o n d i t i o n s of best aqreeaent between e x p e r i a e n t and l u l t i F l e - s c a t t e r i n q c a l c u l a t i o n s Measured f o r two a n q l e s of i n c i d e n c e on Rh (110). 8.4 Comparisons H i t h P r e v i o u s Work On a p e r c e n t a g e b a s i s , t h e Rh(110) s u r f a c e appears to be s l i g h t l y more c o n t r a c t e d t h a n t h e (11.1} o r (100) s u r f a c e s . The c o n t r a c t i o n of about 3.% f o r Rh(110) i s t o be compared w i t h the c o n t r a c t i o n s o f 555 f o r N i (110) £ 1,1,1 35 J , 5 t o 1015 f o r Ag (110) £ 106,107 J and between 10 and 1555 f o r t h e (110) s u r f a c e o f Al£ 108, 11Q-112 "] r e p o r t e d p r e v i o u s l y 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 i m i l a r or s l i g h t l y l o w e r c o n t r a c t i o n s have been i n d i c a t e d f o r t h e l a t t e r s u r f a c e s by the con volution-:-transform method £ 1 0 1 j . However, w i t h t h e e x c e p t i o n of the l a t t e r approach and t h e l a t e s t work on Ag( 110)£ 1 0 6 j , a l l the o t h e r comparisons 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 n t e n s i t i e s have been made v i s u a l l y and a r e t h e r e f o r e l i a b l e t o unknown degrees of e r r o r a s s o c i a t e d w i t h s u b j e c t i v e e v a l u a t i o n of the d a t a . The g u e s t i o n o f s u r f a c e roughness r a i s e d o v e r Al(110)£108] and Ag (110) £ 106,134 j cannot be c o n c l u s i v e l y r e f u t e d f o r Rh (110). 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On S o l i d S u r f a c e s , V i e n n a , 1977, pi2515 M. D. Chinn and S. C. F a i n , J . Vac. S c i . T e c h n o l . 14 (1977) 314 F. J o n a , S u r f a c e S c i . 68 (1977) 204 APPENDICES 212 The f o l l o w i n g a p p e n d i c e s c o n t a i n a l l the e x p e r i m e n t a l data from rhodium s u r f a c e s used d u r i n g t h i s work. I n a l l c a s e s except one, the d a t a i s as c o l l e c t e d and has n o t been smoothed or m a n i p u l a t e d i n any way save f o r a c o r r e c t i o n t o u n i t i n c i d e n t beam c u r r e n t . The e x c e p t i o n (A7) shows d a t a t h a t has been averaged and smoothed; u n f o r t u n a t e l y the f a i l u r e of a tape c a s s e t t e has l e d to i r r e c o v e r a b l e l o s s of t h e o r i g i n a l d a t a i n t h i s i n s t a n c e . The c o n t e n t s of the a p p e n d i c e s a r e l i s t e d below. UJ?JNDIX 5 PR FACE ANGLE DATA SET A1 (111) ^ = 00,d>= 00 1 A2 2 A3 3 A l 6-= 100,gS= 1090 1 A5 2 A6 3 A7 (100) 00,(0= 00 1 A8 2 A9 ©-= 9 0 , ^ = 20O 1 A10 (110) 9-= 0o^i= 00 1 A11 2 A12 ©- = 100,^= 1350 1 A13 2 U 3 Rh(H1) 0=0 Expt.1 x4 A1 Alt (02) 1 1 r 200 300 eV A1 2/5" -•. x4 RhdH) 9=0° Expt.2 (10) x4 x4 x4 (11) x4 (01) x4 100 200 300 eV 200 300 A 2 ZI4 (02) (20) 200 300 eV A 2 2R-x4 Rh(111) e=o Expt3 (10) (01) •• x4 ~ X 4 x4 X 4 x4 (11) i r T — — r i r 100 200 300 eV 200 300 A3 (02) (20) 200 300 eV A3 211 RK111) e=ioV=io9 Expt.1 x4 (00) (11) (10) (11) - i 1 1 r 100 200 (TO) (2D (12) (01) ' " ( 2 2 ) i r eV 100 200 A4 22o Rh(111) 0=10,0=109 Expt. 2 x2 (00) <H) (1T) < ™ (10) (22) (10) ~ (12) (01) (21> • ". . -r 1 1 1 f T 1 TTI ' e V 100 200 100 200 A5 (02) (23) Rh(111) 6=10,jzM09 Expt.3 (00) (ii) (10) (01) (01) (TD (11) do) (02) •. (12) — i 1 1 1 — — i 1 1 i r 100 200 eV 100 200 A6 (21) (12) (22) (21) 1 1 1 — 200 eV (23) (13) / T 1 — H r~ 100 200 eV A6 RhfiOO) e=o° Expt.1 x4 —i 1 1 1 r i 1 1 100 200 300 100 200 A7 22 b 200 — I -3 0 0 eV 200 300 A7 Rhdoo) e--oc Expt.2 x4 x4 (11) (20) X4 100 200 300 100 i r -200 eV A 8 229-(22) (31) 200 300 eV 200 300 A8 (20) 21* RhClOO) 0=9,0=20 (00) (20) (1T) •' (02) (11) • (22) (11) . (22) . (13) (02) (3D — , 1 r 1 1 f 100 200 100 200 eV A Q Rh(HO) 0=0° Expt.1 (10) (20) (01) (21) 250 e 150 (11) ~r~ 50 150 250 eV A10 Rh(110) e=o° Expt.2 2 3 0 (10) (01) (11) (20) (21) 100 200 eV 100 200 A11 Z3I Rh(110) 0-10° 0=135° Expt. 1 (00) (11) (01) (TT) (02) (11) (20) (21) "lOO ' 200 eV 100 200 A12 232 Rh(1lO) e^00^3b° Expt.2 ( i f ) (02) (1T) (11) 100 200 eV 100 200 A13 PUBLICATIONS "Use of a Vidicon camera for the measurement of LEED beam intensities by the photographic method," D.C. Frost, K.A.R. Mitchell, F.R. Shepherd and P.R. Watson, Journal of Vacuum Science and Technology, 13 (1976), 1196-98. "Structure determination of the (100) surface of rhodium by LEED," K.A.R. Mitchell, F.R. Shepherd, P.R. Watson and D.C. Frost, Surface Science, 64 (1977), 737-750. "Surface Structures of Rhodium Studied by LEED," D.C. Frost, K.A.R. Mitchell, F.R. Shepherd and P.R. Watson, Pro-ceedings of the 7th International Vacuum Congress and 3rd International Conference on Solid Surfaces, Vienna, 1977, pA-2725 (Poster). "LEED intensities from Cu(311) and Ni(311)," D.C. Frost, K.A.R. Mitchell, W.T. Moore, R.W. Streater and P.R. Watson, Ibid, p2403-2406. "Applications of the rel iabi l i ty factor proposed for LEED by Zanazzi and Jona to structure determinations of the Rh(100) and Cu(lll) Surfaces," P.R. Watson, F.R. Shepherd, D.C. Frost and K.A.R. Mitchell, Surface Science, 72 (1978), 562-576. "The Structure of the (311) surface of copper as determined by multiple scattering calculations of LEED intensi-ties," R.W. Streater, W.T. Moore, P.R. Watson, D.C. Frost and K.A.R. Mitchell, Surface Science, 72 (1978), 744-748. "An investigation with LEED of the structure of the (111) surface of rhodium," F.R. Shepherd, P.R. Watson, D.C. Frost and K.A.R. Mitchell, Accepted for publication by Journal of Physics. "The structure of the (110) surface of rhodium,"D.C. Frost, I S. Hengrasmee,K.A.R. Mitchell,F.R. Shepherd and P.R. Watson,Surface Science,76(1978)1585-1589. 

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