UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Studies in LEED crystallography Hengrasmee, Sunantha 1980

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

Notice for Google Chrome users:
If you are having trouble viewing or searching the PDF with Google Chrome, please download it here instead.

Item Metadata

Download

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

Full Text

STUDIES IN LEED CRYSTALLOGRAPHY by SUNANTHA HENGRASMEE B.Sc.(Hons), The Un i v e r s i t y of Otago, 1971 M.Sc. , The Un i v e r s i t y of Otago, 1972 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES (Department of Chemistry) We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA J u l y , 1980 SUNANTHA HENGRASMEE, 1980 In presenting this thesis in partial fulfilment of the requirements f o r an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of t h i s t he s i s for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or p u b l i c a t i o n of this thesis for financial gain shall not be allowed without my written permission. Department of ^trnii-^-^y The University of British Columbia 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 Date Abstract This thesis i s involved with the use of low-energy electron d i f f r a c t i o n (LEED) for determining the geometrical structures of well-characterized surfaces of s i n g l e c r y s t a l s . S p e c i f i c applications are to surfaces of rhodium, both clean and when containing adsorbed species. A preliminary problem concerned discrepancies reported previously i n the d e t a i l s of the geometrical structures for the clean (100) and (111) surfaces when using rhodium po t e n t i a l s from either a band structure c a l c u l a t i o n or from the l i n e a r superposition of charge density procedure for a metal c l u s t e r . A correction has now been made i n the c a l c u l a t i o n of phase s h i f t s for the band structure p o t e n t i a l , and r e i n v e s t i g a t i o n s of the (100), (110) and (111) surface of rhodium with t h i s p o t e n t i a l resolve the discrepancies. These r e s u l t s now support the suggestion, as shown previously i n t h i s laboratory for C u ( l l l ) , that the superposition p o t e n t i a l provides a good approximation to a band struc-ture p o t e n t i a l for the purpose of LEED crystallography. In the s t r u c t u r a l determinations made here, the degree of correspondence between i n t e n s i t y versus energy curves for d i f f e r e n t beams from experiment and from m u l t i p l e - s c a t t e r i n g c a l c u l a t i o n s were assessed with the r e l i a b i l i t y -index r ^ proposed by Zanazzi and Jona. A new aspect considered involved the use of t h i s index for determining the non-structural parameters required i n the m u l t i p l e - s c a t t e r i n g c a l c u l a t i o n s . Included i n the l a t t e r f or R h ( l l l ) are v a r i a t i o n s of the imaginary part of the constant p o t e n t i a l (V ^) between the muffin-tin spheres and the surface Debye temperature (B^ s u r £ ) • S t r u c t u r a l conclusions from are compared with v i s u a l analyses wherever p o s s i b l e , and t h i s work generally supports the use of the Zanazzi-Jona index i n LEED crystallography. The experimental part of t h i s study involved the (100) and (110) surfaces of rhodium. A se r i e s of d i f f r a c t i o n patterns were observed for the chemi-sorption of 0^ and H^S. Intensity versus energy curves were measured f o r the a v a i l a b l e d i f f r a c t e d beams for the surface structures designated Rh(100)-(3xl)-0, Rh(100)-p(2x2)-S and Rh(110)-c(2x2)-S. The l a t t e r two systems were analyzed by m u l t i p l e - s c a t t e r i n g c a l c u l a t i o n s (using the renormalized forward s c a t t e r i n g and layer-doubling methods) and surface structures determined. In each case S atoms adsorb on the centre s i t e s ; on Rh(100) S bonds to four o neighbouring Rh atoms at a distance of 2.30 A (very close to the Pauling o o single-bond value 2.29 A), and on Rh(110) each S atom i s 2.12 A from the Rh o atom d i r e c t l y below i n the second layer and 2.45 A from the four neighbouring Rh atoms i n the top m e t a l l i c layer. An i n v e s t i g a t i o n was also made for the use i n LEED crystallography of the quasidynamical method recently proposed by Van Hove and Tong. This scheme includes i n t e r l a y e r m u l t i p l e - s c a t t e r i n g properly , but neglects multiple-s c a t t e r i n g within i n d i v i d u a l layers, and has the p o t e n t i a l for considerable savings i n computing time and core storage. This method was investigated for the clean and sulphur-adsorbed (100) and (110) surfaces, and r e s u l t s compared with the more-complete m u l t i p l e - s c a t t e r i n g methods. The quasi-dynamical method appears to have some promise for making i n i t i a l selections of the most s i g n i f i c a n t t r i a l structures p r i o r to the more-detailed t e s t i n g With f u l l m u l t i p l e - s c a t t e r i n g c a l c u l a t i o n s . - i v -Table of Contents Page Abstract i i Table of Contents : i v L i s t of Tables v i i L i s t of Figures i x Acknowledgement x v i i Chapter 1: Introduction 1 1.1 Modern Surface Science 2 1.2 Introduction to Low Energy Electron D i f f r a c t i o n 4 1.3 Surface Crystallography 13 1.4 Auger Electron Spectroscopy 17 1.5 Aims of Thesis 20 Chapter 2: C a l c u l a t i o n of LEED I n t e n s i t i e s 22 2.1 C h a r a c t e r i s t i c s of 1(E) curves 23 2.2 Physical Parameters required i n LEED Theory 24 2.3 T-Matrix Method 32 2.4 Bloch Wave Method : 34 2.5 Perturbation Methods 39 (a) Layer Doubling Method 40 (b) Renormalized Forward Scattering Method 42 2.6 Further M u l t i p l e S c a t t e r i n g Methods 45 2.7 General Aspects of Computations 47 (a) S t r u c t u r a l Parameters and Use of Symmetry 47 (b) Program Flow 51 - V -Table of Contents Page 2.8 Evaluation o f Results • 53 (a) Introduction 53 (b) Zanazzi and Jona's Proposals • 54 (c) Further Developments 56 Chapter 3: Preliminary Work 60 3.1 General Experimental Procedures 61 (a) LEED Apparatus 61 (b) C r y s t a l Preparation 65 (c) Detection of Surface Impurities 68 (d) Intensity Measurements 71 3.2 S t r u c t u r a l Determinations of Low Index Surfaces of Rhodium 75 (a) Previous LEED Intensity Calculations f o r Rhodium Surfaces 75 (b) Further Studies 77 3.3 Studies with the R e l i a b i l i t y index of Zanazzi and Jona 82 (a) Introduction 82 (b) Relations between R e l i a b i l i t y Index and the Imaginary Poten t i a l 82 (c) R e l i a b i l i t y Index and the V a r i a t i o n of Surface Debye Temperature 89 3.4 Studies o f Adsorption o f some Gaseous Molecules on Rhodium Surfaces 95 (a) Bibliography of Overlayer Structures on Rhodium Surfaces 95 (b) Adsorption of 0? on Rh(100) 97 - v i -Table of Contents Page Chapter 4: LEED Analysis of Rh(100)-p(2x2)-S Surface Structure 101 4.1 Introduction 102 4.2 Adsorption o f . ^ S on Rh(100) — 102 4.3 Computational Scheme 107 4.4 Results 108 4.5 Discussion 117 Chapter 5: LEED Analysis of the Rh(110)-c(2x2)-S Surface Structure 125 5.1 Introduction 126 5.2 Experimental . 126 5.3 Calculations 131 5.4 Results 134 5.5 Discussion 141 Chapter 6: Studies with the Quasidynamical Method 145 6.1 Introduction 146 6.2 Calculations 148 6.3 Results and Discussion 150 (a) Rh(110) and Rh(110)-c(2x2)-S - — - 150 (b) Rh(100) and Rh(100)-p(2x2)-S 161 6.4 Concluding Remarks : 168 References 171 Appendices 179 - v i i -L i s t of Tables Page 2.1 Numbers of symmetrically-inequivalent beams a c t u a l l y used i n c a l c u l a t i o n of various surface structures. The models for the overlayer structures are designated as i n figure 1.8 and 2.8. 50 3.1 Observed and calculated Auger t r a n s i t i o n energies f o r rhodium. 70 3.2 S t r u c t u r a l determination of low index surfaces of rhodium. (Watson et a l . ) 76 3.3 S t r u c t u r a l determination of low index surfaces of rhodium. (This work.) 76 3.4 Conditions f o r best agreement between experimental 1(E) curves at normal incidence for R h ( l l l ) and curves calculated with the r MJWT p o t e n t i a l ] according to the r e l i a b i l i t y indices r ^ and r for d i f f e r e n t values of a. 86 m 3.5 Surface structures reported for adsorption of small gaseous molecules on low index surfaces of rhodium. 96 4.1 Conditions f o r minima of r f o r d i f f e r e n t models of r Rh(100)-p(2x2)-S. 116 4.2 E f f e c t i v e r a d i i of chemisorbed sulphur atoms on various metal surfaces. 122 4.3 Comparisons of M-X bond distances for chalcogen atoms adsorbed on (100) surfaces of fee metals with Pauling's s i n g l e bond lengths [133]. 123 6.1 Comparisons of conditions for minimum r ^ for various surface structures obtained from evaluating experimental 1(E) curves with corresponding curves from m u l t i p l e - s c a t t e r i n g c a l c u l a t i o n s and from quasi-dynamical c a l c u l a t i o n s . 151 - v i i i -L i s t of Tables Page 6 . 2 A demonstration of the correspondence between peak posit i o n s i n 1(E) curves calculated with the quasidynamical method for the four models of Rh ( 1 1 0)-c ( 2 x 2)-S at the s p e c i f i e d S-Rh i n t e r l a y e r spacing and those given by experiment and by the corresponding f u l l m u l t i p l e - s c a t t e r i n g c a l c u l a t i o n s . In the entries f or each beam, the denominator s p e c i f i e s the number of s i g n i f i c a n t peaks i n the relevant 1(E) curve from experiment or from the f u l l m u l t i p l e - s c a t t e r i n g c a l c u l a t i o n s , and the numerator gives the number of those peaks that are matched to within 7 eV by the quasidynamical c a l c u l a t i o n s . 15S 6 . 3 A demonstration of the correspondence between peak posit i o n s i n 1(E) curves calculated with the quasidynamical method for the four models of Rh ( 1 0 0 )-p ( 2 x 2 )-S at the s p e c i f i e d S-Rh i n t e r l a y e r spacing and those given by experiment and by the corresponding f u l l m u l t i p l e - s c a t t e r i n g c a l c u l a t i o n s . In the entries f or each beam, the denominator s p e c i f i e s the number of s i g n i f i c a n t peaks i n the relevant 1(E) curve from experiment or from the f u l l m u l t i p l e - s c a t t e r i n g c a l c u l a t i o n s , and the numerator gives the number of those peaks that are matched to within 7 eV by the quasidynamical c a l c u l a t i o n s . 1 6 ' - i x -L i s t of Figures Page 1.1 Schematic diagram of the mean free path length L (A) of electrons i n a m e t a l l i c s o l i d as a function of energy (eV). 5 1.2 Schematic energy d i s t r i b u t i o n N(E) of back-scattered electrons for a primary beam of energy E q . 5 1.3 (a) Schematic diagram of the LEED experiment. (b) The p r i n c i p l e of the formation of a d i f f r a c t i o n pattern i n LEED experiment. 8 1.4 Conventions f o r the incident angle of an electron beam on a surface; 6 i s a polar angle r e l a t i v e to a surface normal and 4> an azimuthal angle r e l a t i v e to a major c r y s t a l l o g r a p h i c axis i n the surface plane. 11 1.5 1(E) curves for the specular beam from Ni(100) at 6=3°. The bars i n d i c a t e energies where primary Bragg conditions are s a t i s f i e d (after Andersson [48]). 12 1.6 A schematic comparison of overlayer and substrate regions, both of which are d i p e r i o d i c i n the x and y d i r e c t i o n s . 12 1.7 Schematic d i f f r a c t i o n patterns of clean and overlayer structures. 15 1.8 Four possible s t r u c t u r a l models for Rh(110)-c(2x2)-S which are consistent with the observed d i f f r a c t i o n pattern. The adsorbed sulphur atoms are represented by the f i l l e d c i r c l e s . 16 1.9 The production of an ^TV Auger electron i n aluminum. X-ray energy levels are indicated r e l a t i v e to the Fermi l e v e l . 18 1.10 Auger spectrum of a heavily contaminated Rh(110) surface, E Q = 1.5 keV, I q = 10 microamps 19 -X-L i s t of Figures Page 2.1 M u f f i n - t i n p o t e n t i a l (a) i n cross-section as contours, (b) along xx'. V q i s the constant intersphere p o t e n t i a l . 25 2.2 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 energies measured with respect to the vacuum l e v e l and those measured with respect ,to the lowest l e v e l of the conduction band. 25 2.3 M u f f i n - t i n model of an adsorbate covered surface (a f t e r Marcus et a l . [59]). 28 2.4 Schematic representation of a set of plane wave incident from the l e f t and multiply scattered by a plane of ion-cores. 35 2.5 Schematic diagram of transmission and r e f l e c t i o n matrices at the a subplane. The broken l i n e s are the c e n t r a l l i n e s between the subplanes. ; 35 2.6 Stacking of planes to form a c r y s t a l slab and i l l u s t r a t e the layer-doubling method. Planes A and B are f i r s t stacked to form the two-layer slab C; the process i s continued to form a four-layer slab. (After Tong [65]). 41 2.7 (a) I l l u s t r a t i o n of the renormalized forward s c a t t e r i n g method. V e r t i c a l l i n e s represent layers. Each t r i p l e t of arrows represents the complete set of plane waves that t r a v e l from layer to layer. (b) Propagation steps of the inward-travelling waves. (c) Propagation steps of the outward-travelling waves. (After Van Hove and Tong [ 8 l ] . ) 43 2.8 Schematic diagram of three simple models for Rh(100)-p(2x2)-S. In r e c i p r o c a l space, sets of symmetrically equivalent beams are indicated by a common symbol. 48 - x i -L i s t of Figures Page 2.9 Flowchart showing p r i n c i p a l steps i n a mu 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 , using the RFS or layer doubling programs. 52 2.10 Plots for C u ( l l l ) of (r ). for 9 i n d i v i d u a l beams versus Ad% r i with V = -9.5 eV. The dashed l i n e shows the reduced r e l i a -or b i l i t y index (r ) for the t o t a l 16 beams. (After Watson et a l . [43]). 57 2.11 Contour p l o t f or C u ( l l l ) of r versus Ad% and V . * K r or (After Watson et a l . [43]). 59 3.1 (a) Schematic of the Varian FC12 UHV chamber. (b) Diagramatic representation of the pumping system: IP = Ion Pump; TSP= Titanium Sublimation Pump; SP = Sorption Pump. 62 3.2 (a) Schematic diagram of the electron optics used f o r LEED experiments. (b) Diagram showing sample mounted on a tantalum supporting r i n g . (c) Electron bombardment sample heater. Hatched l i n e s represent s t a i n l e s s s t e e l parts while the s t i p p l e pattern indicates the ceramic i n s u l a t o r . 63 3.3 Auger spectra of clean Rh(110) surface as a function of c r y s t a l temperature i n d i c a t i n g carbon concentrated around the surface region at 200°C and d i f f u s e d back into the bulk at 300°C. 67 3.4 Schematic diagram of LEED optics used as a retarding f i e l d analyzer for Auger electron spectroscopy: MCA = multichannel analyzer. 69 3.5 Schematic diagram of the apparatus used to analyse the photo-graphic negatives of LEED patterns. 74 - x i i -L i s t of Figures Page 3.6 Energy dependence of rhodium phase s h i f t s (£=0-7) for the po t e n t i a l [ V ^ W ] . 78 r'3.7 (a) Schematic diagrams of the (100), (110) and (111) surfaces of rhodium. The dotted c i r c l e s represent rhodium atoms i n the second layer, (b) The corresponding LEED patterns i n d i c a t i n g the beam notation as used i n text. 80 3.8 The experimental 1(E) curve f o r the (01) beam at normal incidence from the R h ( l l l ) surface compared with f i v e corresponding curves calculated with the p o t e n t i a l [V^j^j and Ad% = -2.5% for the parameter a varying from 1.17 to 2.34. 84 3.9 Contour p l o t of r versus 6_ _ and V f o r normal incidence r r D,surf or data from R h ( l l l ) where the ca l c u l a t i o n s use the p o t e n t i a l [Vo^ W3 w i t h ot=1.76 and 6 n . ,,=480 K. 90 Rh D,bulk 3.10 Contour p l o t of r versus 6^ _ and Ad% for normal incidence r r D,surf data from R h ( l l l ) where the calcu l a t i o n s use the p o t e n t i a l [ V ™ ] with a=1.76 and 6 n . ,.=480 K. 91 L Rh J D,bulk 3.11 The experimental 1(E) curve f o r the (01) beam at normal incidence from the R h ( l l l ) surface compared with f i v e corresponding curves calculated with the p o t e n t i a l J , Ad% = -2.5%, and a = 1.76 for the parameter 6^ s u r f varying from 200 to 600 K. 93 3.12 Photographs of some p(2><2) and (3x1) LEED patterns observed at normal incidence from the adsorption of oxygen on a Rh(100) surface. (a) Rh(100)-p(2x2)-0 at 70 eV; (b) Rh(100)-(3xl)-0, s i n g l e domain at 174 eV; (c) Rh(100)-(3xl)-0, 2 equally populated domains at 100 eV; (d) Rh(100)-(3xl)-0, 2 equally populated domains at 152 eV. 99 - x i i i -L i s t of Figures Page 4.1 Photographs of LEED patterns observed at normal incidence from adsorption of S on Rh(100) surface. (a) Rh(100)-c (2x2)-S at 80 eV; (b) Rh(100)-p (2x2)-S at 72 eV; (c) Rh(100)-p (2x2)-S at 114 eV; Cd) Rh(100)-p (2x2)-S at 168 eV. 103 4.2 Auger spectra of Rh(100) surfaces with 1.5 keV and 10 micro-amp beam at d i f f e r e n t stages during the preparation of Rh(100)-p (2x2)-S. 104 4.3 Beam notation for the LEED pattern of Rh(100)-p (2x2)-S structure. 106 4.4 Comparison for the (-^ j) and (01) beams of 1(E) curves from two d i f f e r e n t experiments measured at normal incidence. 109 4.5 Comparison of experimental 1(E) curves for various i n t e g r a l -and f r a c t i o n a l - o r d e r d i f f r a c t e d beams from Rh(100)-p (2x2)-S with the calculated curves for S adsorbed on-the 4F, 2F and IF s i t e s at the topmost Rh-S i n t e r l a y e r spacing indicated f o r each curve. 111 1 11 4.6 Comparison of experimental 1(E) curves f o r the (0-^) and (— -) beams from the Rh(100)-p (2x2)-S surface with those calculated for S adsorbed on the 4F s i t e f o r a range of topmost Rh-S i n t e r l a y e r spacings. 115 4.7 Contour pl o t s of f f o r Rh(100)-p (2x2)-S versus V and Rh-S i n t e r l a y e r spacing for (a) 4F model, (b) 2F model, and (c) IF r model. Error bars indi c a t e standard errors as defined i n chapter 2. 118 -xiv-L i s t of Figures Page 5 . 1 Auger spectra for a Rh(llO) surface when cleaned and when containing a c ( 2 x 2 ) overlayer of sulphur. 1 2 8 5 . 2 Photographs of LEED patterns observed at normal incidence from adsorption of S on Rh(llO) surface. (a) Rh(llO) at 144 eV; (b) R h ( 1 1 0 ) - c ( 2 x 2 ) - S at 7 8 eV; (c) R h ( 1 1 0 ) - c ( 2 x 2 ) - S at 102 eV; (d) Rh ( 1 1 0)-c ( 2 x 2)-S at 1 5 0 eV. 1 2 9 5 . 3 Beam notation f o r the LEED pattern from the Rh(l l O ) - c ( 2 x 2 ) - S surface structure. 1 3 0 5 . 4 Experimental 1(E) curves for two sets of beams which are expected to be equivalent f o r the R h ( l l O ) - c ( 2 x 2 ) - S structure. 1 3 2 5 . 5 Comparison of some experimental 1(E) curves from R h ( l l O ) - c ( 2 x 2 ) - S with those calculated f o r the four s t r u c t u r a l models over a range of topmost i n t e r l a y e r spacings: (a) ( 0 1 ) beam, (b) ( 1 0 ) beam, 31 and (c) (—j) beam. 1 3 5 5 . 6 Comparison of experimental 1(E) curves f o r some i n t e g r a l - and fr a c t i o n a l - o r d e r beams from R h ( l l O ) - c ( 2 x 2 ) - S with those c a l -o culated f o r the 4 F model with sulphur either 0 . 7 5 or 0 . 8 5 A above the topmost rhodium layer. 139 5 . 7 Contour plo t s of r for R h ( 1 1 0 ) - c ( 2 x 2 ) - S versus V and Rh-S r r or i n t e r l a y e r spacing for four d i f f e r e n t s t r u c t u r a l models. 1 4 0 5 . 8 Schematic s p e c i f i c a t i o n of interatomic distances i n the v i c i -n i t y of an overlayer sulphur atom i n the surface structure Rh ( 1 1 0)-c ( 2 x 2)-S. Distances i n Angstrom. 1 4 3 5 . 9 Interatomic distances, for the s p e c i f i c a t i o n of hard sphere r a d i i i n the neighbourhood of an oxygen atom i n the o F e ( 1 0 0 ) - ( l x l ) - 0 structure. Distances i n Angstrom. (After Legg et a l . [ 1 5 3 ] ) . 1 4 3 - X V -L i s t of Figures Page 6.1 Comparison of experimental 1(E) curves f o r normal incidence on Rh(llO) with those calculated with the quasidynamical method and the f u l l m u l t i p l e - s c a t t e r i n g method when the topmost i n t e r l a y e r spacing equals the bulk value (0%) and when i t i s contracted by 10%. 152 6.2 Contour plots of r for Rh(110)-c(2x2)-S versus V and the Rh-S i n t e r l a y e r spacing for four d i f f e r e n t s t r u c t u r a l models calculated with the quasidynamcial method. 154 33 6.3 Comparison of 1(E) curves measured f or the (01) and (-^j) d i f f r a -cted beams for normal incidence on Rh(110)-c(2x2)-S with those calculated by the quasidynamical method and by the f u l l m u l t i p l e - s c a t t e r i n g method f or the four s t r u c t u r a l models descr-ibed i n text. 157 6.4 Comparisons of some experimental 1(E) curves f o r f r a c t i o n a l -order beams f o r normal incidence on Rh(110)-c(2x2)-S and Rh (100) -p(2><2) -S with those calculated f o r the centre adsorption s i t e s with the quasidynamical method and with the f u l l multiple-s c a t t e r i n g method. The topmost Rh-S i n t e r l a y e r spacings i n the o o quasidynamical c a l c u l a t i o n s are 1.15 A and 1.3 A for Rh(110)-c(2x2)-S and Rh(100)-p(2x2)-S r e s p e c t i v e l y ; the corresponding values f o r the m u l t i p l e - s c a t t e r i n g c a l c u l a t i o n s o o are 0.75 A and 1.3 A. 159 6.5 Comparisons of some experimental 1(E) curves f o r normal incidence on Rh(100) with those calculated with the quasidynamical method and with the f u l l m u l t i p l e - s c a t t e r i n g method. 162 6.6 Contour pl o t s of r for Rh(100)-p(2x2)-S versus V and the Rh-S r r or i n t e r l a y e r spacing f o r the 4F and 2F s t r u c t u r a l models calculated by the quasidynamical method: (a) comparisons with a l l i n t e g r a l -and f r a c t i o n a l - o r d e r beams; (b) comparisons with f r a c t i o n a l -order beams only. 164 - x v i -L i s t of Figures Page 13 6.7 Comparisons of 1(E) curves measured for the (01) and (-^j) d i f f r a c t e d beams for normal incidence on Rh(100)-p(2x2)-S with those calculated by the quasidynamical method and by the f u l l m u l t i p l e - s c a t t e r i n g method for three po s s i b l e s t r u c t u r a l models. 166 - x v i i -Acknowledgement It has been a rewarding experience to work under Professors K.A.R. M i t c h e l l and D.C. Frost during the course of t h i s work. Their guidance and encouragement have provided invaluable support, and f o r t h i s I give them my sincere thanks. I am very g r a t e f u l to Dr. C.W. Tucker (General E l e c t r i c Corporation) for providing a Rh(100) c r y s t a l , Dr. E. Zanazzi and Professor F. Jona (State University of New York, Stony Brook) for providing t h e i r r e l i a b i l i t y - i n d e x programs and to Dr. M.A. Van Hove (University of C a l i f o r n i a at Berkeley) and Dr. S.Y. Tong (University of Wisconsin) for copies of t h e i r m u l t i p l e - s c a t t e r i n g and quasidynamical computer programs. I would l i k e to acknowledge the contributions of every member of the surface science group. In the past, Dr. R.W. Streater and at present, T.W. Moore and Dr. S.J. White for experimental assistance, stimulating discussion and i n p a r t i c u l a r f o r t h e i r comments during the preparation of t h i s t h e s i s . Among these, I owed a s p e c i a l thank to Dr. F.R. Shepherd and Dr. P.R. Watson who had ass i s t e d and collaborated i n t h i s work throughout the duration of time they were here. I am indebted to many members of the mechanical and e l e c t r i c a l workshops who have contributed so much i n maintaining the working conditions of the instruments. I am very g r a t e f u l to B i l l Ng for support and useful suggestions and e s p e c i a l l y for h i s professional job i n typing t h i s t h e s i s . F i n a l l y , but foremost, a deep sense of gratitude and love i s dir e c t e d toward my husband, D h i t i Hengrasmee, who has been concerned with my progress and s p i r i t u a l l y supported me throughout the course o f my study. To him, I dedicate t h i s t h e s i s . CHAPTER 1 Introduction -2-1.1 Modern Surface Science Studies of the properties of s o l i d surfaces have assumed great i n t e r e s t over the past decade, i n part because such surfaces have dominant roles i n various technological processes (e.g. f r i c t i o n and wear, e l e c t r o n i c devices and heterogeneous c a t a l y s i s ) [ l , 2 ] . T r a d i t i o n a l research emphasized the properties of r e a l surfaces, usually of p o l y c r y s t a l l i n e materials, which could not be well-characterized at the atomic l e v e l [3], However, modern surface science has introduced the "clean surface" approach where c a r e f u l l y characterized surfaces are studied with the objective of developing p r i n -c i p l e s which can lead to better understandings of the atomistic aspects of surface processes, including those of technological i n t e r e s t [4,5], In the clean surface approach, s i n g l e c r y s t a l s are used and the properties of surfaces corresponding to well-defined c r y s t a l l o g r a p h i c planes are studied under conditions such that the surface i s not contaminated by unwanted impurities. This requires experiments to be c a r r i e d out under u l t r a - h i g h _9 vacuum (<10 t o r r ) . The necessity f o r t h i s p r o v i s i o n follows from the k i n e t i c theory which predicts that, for an ambient pressure of 10 ^ t o r r , a surface can be covered by an adsorbed monolayer i n 1 second, assuming that a l l c o l l i d i n g molecules s t i c k to the surface. With the a v a i l a b i l i t y of ul t r a - h i g h vacuum f a c i l i t i e s , many experi-mental techniques have been developed r e c e n t l y f o r the ch a r a c t e r i z a t i o n of s o l i d surfaces with regard to chemical composition, geometrical and el e c t r o n i c structure as well as chemical bonding, v i b r a t i o n a l structure and energy exchange with impinging molecules. Among the techniques a v a i l a b l e , -3-Auger electron spectroscopy (AES) i s commonly used for q u a l i t a t i v e chemical analyses of surfaces, whereas u l t r a v i o l e t photoemission spectroscopy (UPS) [6] i s popular for i n d i c a t i n g e l e c t r o n i c structure and low energy electron d i f f r a c t i o n (LEED) gives information on geometrical structure. R e f l e c t i o n high energy electron d i f f r a c t i o n (RHEED) [7] and the s c a t t e r i n g of molecular and ion beams [8,9] also have high p o t e n t i a l s f o r surface studies. Research on well-defined surfaces with a v a r i e t y of techniques has established that surface properties depend not only on the p a r t i c u l a r material involved, but also on the s p e c i f i c c r y s t a l l o g r a p h i c plane exposed [io]. For example, chemisorption and molecular beam sca t t e r i n g studies have shown that s t i c k i n g p r o b a b i l i t i e s and r e a c t i o n rates can be very d i f f e r e n t on stepped surfaces of platinum compared with low-index surfaces of the same metal [ l l ] . At the present time LEED appears as the most d i r e c t technique for the determination of surface geometrical structure. This p o t e n t i a l was recognized in 1927 when the experiment was f i r s t performed by Davisson and Germer [12]. However the development of t h i s technique to i t s f u l l p o t e n t i a l was i n h i b i t e d by many t h e o r e t i c a l and experimental d i f f i c u l t i e s , and i t was only during the 1970's that these problems were overcome s u f f i c i e n t l y f o r some surface structures to be determined. In current LEED studies i t i s considered adv-antageous, i f not e s s e n t i a l , to u t i l i z e other techniques simultaneously for characterizing the surface. The most commonly-used complementary technique - i s AES. H i s t o r i c a l l y , electrons produced by the Auger process were discovered in 1925 [13], and although t h e i r p o t e n t i a l i n surface analysis was recognized by Lander [14] i n 1953, i t was not u n t i l the l a t e 1960's that they could be -4-detected r o u t i n e l y i n surface experiments [15-17]. Indeed the development of AES as a method f o r q u a l i t a t i v e surface analysis encouraged the develop-ment of reproducible LEED experiments, and i n turn the development of adequate theories f o r LEED. The existence of the l a t t e r represented a necessary requirement for the development of LEED crystallography ( i . e . , the determination o f surface geometrical structure by LEED). 1.2 Introduction to Low Energy Electron D i f f r a c t i o n A LEED experiment involves d i r e c t i n g a beam of low-energy electrons ( t y p i c a l energy <500eV) with known angles of incidence onto a well-defined surface of a c r y s t a l l i n e s o l i d and observing the i n t e n s i t y d i s t r i b u t i o n of electrons which are e l a s t i c a l l y back-scattered from the surface. The de o Broglie hypothesis r e l a t e s electron energy (E i n eV) to wavelength (X i n A) according to * =JM3 ; (1.1, electrons i n the low-energy range therefore have wavelengths which are comparable with i n t e r l a y e r spacings i n the s o l i d . Low-energy electrons are p a r t i c u l a r l y "surface s e n s i t i v e " because they experience strong i n -e l a s t i c scatterings i n s o l i d s . A h e l p f u l parameter f o r discussing i n e l a s t i c s c a t t e r i n g i s the electron mean free path length (L) which can be expressed i n terms o f I = I Q exp [ J / L ] , (1.2) where the incident i n t e n s i t y I at a p a r t i c u l a r energy i s attenuated to I on passage through a distance £. The general form of the dependence of the mean free path length on electron energy i s shown i n f i g u r e 1.1. Electrons -5-1,000-1 ioo-4 °< Figure 1.1: 100,000 Schematic diagram of the mean free path length L (A) of electrons i n a m e t a l l i c s o l i d as a function of energy (eV). Ul Z true secondary elastic peak E N E R G Y Figure 1,2: Schematic energy d i s t r i b u t i o n N(E) of back-scattered electrons for a primary beam of energy E Q . -6-i n the low-energy range are associated with values of L of just a few o Angstroms, and therefore they are i d e a l l y suited f o r i n v e s t i g a t i o n of the top few layers of a s o l i d . Further information on electron mean free path lengths has been reviewed by Brundle [18], Ibach [19] and Powell [20]. A monoenergetic beam of low-energy electrons incident upon a s o l i d surface t y p i c a l l y gives an energy d i s t r i b u t i o n for the back-scattered electrons of the type shown in figure 1.2. The narrow " e l a s t i c peak" on the ri g h t hand side involves the electrons which are studied i n the conventional LEED experiment. This peak includes the genuinely e l a s t i c a l l y - s c a t t e r e d electrons, as well as those electrons which have undergone phonon s c a t t e r i n g with small energy changes ( ^ 0.1 eV ). This l a t t e r group of electrons can be r e f e r r e d to as q u a s i e l a s t i c electrons. T y p i c a l l y only 1-5% of the incident electrons contribute to the " e l a s t i c peak". Most electrons experience strong i n e l a s t i c s c a t t e r i n g , associated e s p e c i a l l y with s i n g l e - e l e c t r o n and plasmon excitations [21,22], and those excitations contribute to the comparatively short mean free path length indicated i n figure 1.1. The emission of Auger -12 electrons, which t y p i c a l l y corresponds to a current of ~10 A on a back-_7 ground of -10 A, appears as small peaks superimposed on a slowly-varying background i n the intermediate range of figu r e 1.2. Peaks due to Auger electrons can be distinguished from loss peaks due to plasmon or s i n g l e - e l e c t r o n excitations because the former occur at energies which are independent of the - primary electron energy. The large peak at low energy i n figure 1.2 involves the so-called "true secondary" electrons which are associated with a serie s of i n e l a s t i c scatterings i n a cascade-type process [23]. -7-The p r i n c i p l e of the LEED experiment i s i l l u s t r a t e d i n fig u r e 1.3a. The incident electrons are scattered by the surface region and the e l a s t i c a l l y back-scattered electrons are separated from others by energy s e l e c t i n g g r i d s . The e l a s t i c a l l y scattered waves i n t e r f e r e c o n s t r u c t i v e l y to give d i f f r a c t e d beams along c e r t a i n d i r e c t i o n s , and each beam shows as a bright spot when these electrons are accelerated onto a fluorescent screen. The d i s t r i b u t i o n of these spots i s r e f e r r e d to as the LEED pattern. Because of strong i n e l -a s t i c s c a t t e r i n g , the e l a s t i c a l l y - s c a t t e r e d electrons do not normally experi-ence a regular p e r i o d i c i t y normal to the c r y s t a l surface and consequently the region probed by the LEED electrons i s d i p e r i o d i c ( i . e . , i t can be characterized by two unit t r a n s l a t i o n vectors a^ and a^). The corresponding d i f f r a c t i o n pattern (figure 1.3b) involves the associated t r a n s l a t i o n a l vectors in r e c i p r o c a l space, namely a^ f and £i* defined by a *z a x z ^2 ~ ~1 ^ a* = 2TT , a* = 2TT (1.3) ~1 ~z a,, a xz a 0. a / z where £ i s the unit vector perpendicular to a^ and a. . Pendry [24] has given a d e t a i l e d analysis showing how a LEED pattern i s a d i r e c t consequence of the surface t r a n s l a t i o n a l symmetry. Assuming the incident electrons can be described by a plane wave V = B exp[ik +. r ] , (1.4) o r ~o — where B i s an appropriate normalization constant, r i s a general p o s i t i o n . vector and k + i s the incident wave vector which r e l a t e s to electron energy ~o through ~ |k +| 2 , (1.5) 2m '~o1 ' -8-Figure 1.3: a) Schematic diagram of the LEED experiment. b) The p r i n c i p l e of the formation of a d i f f r a c t i o n pattern i n LEED experiment. -9-then wave vectors k f o r the d i f f r a c t e d electrons are determined by conserva-t i o n of energy E(k") = E(k^) (1.6) and by the conservation of momentum p a r a l l e l to the surface k~ = k+,, + g(hk) , (1.7) where g(hk) = ha* + ka* , (1.8) h and k being integers. As i l l u s t r a t e d , i n f i g u r e 1.3b, the d i r e c t i o n of + each d i f f r a c t e d beam (wave vector k ,,.-.) i s determined by E, k and g. ~g(hk) ~o ~ For given values of lc* and E, each spot i n a d i f f r a c t i o n pattern i s asso-ciated with a p a r t i c u l a r g, and hence may be i d e n t i f i e d with the indices (hk). For a given energy, only a li m i t e d number of beams can reach the screen; i f |g| i s s u f f i c i e n t l y large k~ becomes complex and corresponds to ~ ~g an evanescent (or surface) wave which cannot escape from the s o l i d . The (00) beam i s made up of electrons which have interacted with the surface without momentum transfe r p a r a l l e l to the surface Qc |(=J<+||)» and i t i s frequently c a l l e d the "specular beam". The d i r e c t i o n of the specular beam remains constant as E changes, as long as the electrons move i n a f i e l d - f r e e space outside the c r y s t a l and the d i r e c t i o n of the incident beam i s f i x e d . With increasing energy, more d i f f r a c t e d beams are observed, the non-specular beams move towards the (00) beam, the symmetry of the LEED pattern remains unchanged, but the beam i n t e n s i t i e s vary continuously. In p r a c t i c e , incident electron beams i n LEED are coherent only over 2 ° r e s t r i c t e d distances (-10 A) [25], and t h i s l i m i t s the range over which -10-surface order can be recognized i n the d i f f r a c t i o n experiment. Some disorder i s i n e v i t a b l y present at surfaces, and t h i s can a f f e c t spot patterns by broadening the d i f f r a c t e d beams, by introducing streaks, rings and spot s p l i t t i n g s , and by increasing the background i n t e n s i t y [26]. Frequently LEED patterns are affected by domain structure i n which two or more equi-valent orientations of the structure are possible on the surface. In the presence of domain structure, provided that the dimensions of the domains are greater than the coherence width associated with the incident electron beam, observed LEED patterns represent d i r e c t superpositions of the patterns from the i n d i v i d u a l domains. This can be p a r t i c u l a r l y important for adsorption systems, and examples are given l a t e r . For a given surface, the i n t e n s i t i e s of the d i f f r a c t e d beams vary with the electron energy E, the d i r e c t i o n of incidence ( s p e c i f i e d by angles 0, cf>; see f i g u r e 1.4) and the temperature. Most often i n t e n s i t y data are presented as a function of energy ( i . e . , as 1(E) curves for each d i f f r a c t e d beam) with a l l other parameters being held constant. A t y p i c a l example of 1(E) curves i s given i n f i g u r e 1.5. Davisson and Germer [12], at the time of the f i r s t LEED experiment, r e a l i z e d beam i n t e n s i t i e s contain information on surface bond distances, but nearly 50 years elapsed before d e t a i l e d surface geometries could be extracted from measured i n t e n s i t i e s . The basic method u t i l i z e d at the present time involves the t r i a l - a n d - e r r o r approach i n which J(E) curves are calculated for d i f f e r e n t possible surface geometries, and a search i s made for that geometry which allows the best match up with the ex-perimental 1(E) curves for the various d i f f r a c t e d beams. The main content of t h i s thesis i s involved with the a p p l i c a t i o n of t h i s approach to LEED crystallography. -11--z direction of incident bear Figure 1.4: Conventions for the incident angle of an electron beam on a surface; 8 i s a polar angle r e l a t i v e to a surface normal and <(> an azimuthal angle r e l a t i v e to a major c r y s t a l l o g r a p h i c axis i n the surface plane. -12-1 2 ELECTRON ENERGY (eV) Figure 1.5: 1(E) curve for the specular beam from Ni(100) at 9=3°. The bars i n d i c a t e energies where primary Bragg conditions are s a t i s f i e d (after Andersson [48]). Figure 1.6: A schematic comparison of overlayer and substrate regions, both of which are d i p e r i o d i c in the x and y d i r e c t i o n s . -13-1.3 Surface Crystallography The d e f i n i t i o n of surface i s very much a function of the p a r t i c u l a r probe used to study i t . For LEED from a c r y s t a l l i n e s o l i d , i t i s convenient to r e f e r to the "surface region" as the region probed by the LEED electrons ( i . e . , over the range o f mean free path length corresponding to the e l a s t i c a l l y -scattered e l e c t r o n s ) . Figure 1.6 also indicates the "substrate" whose structure i s generally known from X-ray crystallography and i s that f o r which the bulk t r i p e r i o d i c i t y i s established. The objective of surface c r y s t a l l o -graphy i s then to determine the p o s i t i o n of a l l atoms beyond the substrate surface ( i . e . , the topmost substrate plane). The surface region may involve an overlayer whose d i p e r i o d i c t r a n s l a t i o n a l symmetry i s d i f f e r e n t from that of a substrate plane. The appropriate p e r i o d i c t r a n s l a t i o n a l symmetry for LEED i s that f o r the o v e r a l l surface region, and i s described by the unit t r a n s l a t i o n a l vectors a- and a_. These vectors may r e s u l t from the combination of the d i p e r i o d i c symmetries of the substrate and the overlayer. The vectors and a define a unit mesh which i s analogous to the unit c e l l of t r i p e r i o d i c crystallography. The vector t = ma, + na_ (1.9) translates from one point i n a surface region to another with an i d e n t i c a l environment, and a two-dimensional net can be generated from a l l i n t e g r a l values o f m and n; t h i s i s the d i p e r i o d i c analogue of the t r i p e r i o d i c l a t t i c e used i n X-ray crystallography. Five types of d i p e r i o d i c nets are possible and they are analogous to the 14 Bravais l a t t i c e s i n t r i p e r i o d i c c r y s t a l l o -graphy. There are 17 possible space groups i n d i p e r i o d i c crystallography, and they are d e t a i l e d i n the International Tables f o r X-ray Crystallography [27]. -14-Adsorption on clean surfaces t y p i c a l l y gives increased surface perio-d i c i t i e s and therefore extra LEED spots, as shown i n figure 1.7. Such extra spots are frequently c a l l e d " f r a c t i o n a l order" spots when the same notation i s used for corresponding beams from the adsorption structure as for the clean surface structure. Generally i t i s convenient to use a nota-t i o n f o r surface structures and d i f f r a c t e d beams which i s based on the substrate. For example i n Wood's nomenclature [28], a surface i s designated where (a^a^) and (Jb^.b^) are the unit d i p e r i o d i c vectors of the surface region and substrate r e s p e c t i v e l y , and G i s the angle of r o t a t i o n between the surface and substrate unit meshes (for more complex surfaces, where such an angle of r o t a t i o n i s not applicable, a matrix notation has been introduced [29] and discussed further by Estrup and McRae [30]) . With Wood's notation, the symbols p or c a r e frequently added to indicate whether the surface mesh i s p r i m i t i v e (one atom per unit mesh) or centred (with an extra atom at the centre of the unit mesh), r e s p e c t i v e l y . For the examples of S adsorbed on (100) and (110) surfaces of rhodium (figure 1.7) the structures obtained are designated as Rh(100)-p(2x2)S and Rh(110)-c(2x2)S r e s p e c t i v e l y ; the l a t t e r could a l t e r n a t i v e l y be designated as Rh(110)-(/3x/3/2)54-S although the f i r s t i s always used for s i m p l i c i t y . A d i f f r a c t i o n pattern usually allows a s p e c i f i c a t i o n of the surface p e r i o d i c i t y , but never of the actual surface structure. The l a t t e r requires analysis of beam i n t e n s i t i e s . For S adsorbed on the (110) surface of rhodium, there are four p a r t i c u l a r l y important locations for the S atoms. These are -15-rea l space reciprocal space ft % Rh (100) oTi ffi ir2 oo; 10 20 9 ft 9 I2 Rh(100)- p(2 x 2)S ft-®"} o 0 0 o 0 OCDOO OCXBOO COBOO CXX)QO 012 Rh(110) 112 00 20 Rh(1 10)-c(2x2)S On 00 O O 2 2 o Ti Figure 1.7: Schematic d i f f r a c t i o n patterns of clean and overlayer structures. -16-On-top(IF) model Centre ( 4 F ) model Short-bridge (2 SB) Long-bridge ( 2 LB) model model Four possible structural models for Rh(llO)-c(2*2)S which are consistent with the observed d i f f r a c t i o n pattern. Th adsorbed sulphur atoms are represented by the f i l l e d c i r c -17-shown i n fi g u r e 1.8, and a l l are consistent with the c(2x2) d i f f r a c t i o n pattern. The adsorption s i t e s are designated as centre or four-fold(4F) s i t e s , on-top or one-fold (IF) s i t e s , short-bridge (2SB) s i t e s or long-bridge (2LB) s i t e s . To determine the actual adsorption s i t e i t i s necessary to c a l c u l a t e the 1(E) curves of the d i f f r a c t e d beams f o r the various models and compare them with the experimental 1(E) curves to assess which model gives the best agreement. 1.4 Auger Electron Spectroscopy The Auger process i s depicted i n f i g u r e 1.9. It i s i n i t i a t e d by the i o n i s a t i o n of a core electron either by electron impact or by photon i n t e r -action. An electron from a higher energy l e v e l then drops down to f i l l the inner vacancy, and t h i s process releases energy either by photon production (e.g. X-ray fluorescence) or by e j e c t i o n of an Auger electron whose k i n e t i c energy depends d i r e c t l y on the energy le v e l s involved i n the process [23,3l]. Generally Auger emission i s the more probable process i f the i n i t i a l i o n i s a -t i o n involves an electron whose binding energy i s less than -2keV. The key point for surface analysis i s that the k i n e t i c energies of Auger electrons are c h a r a c t e r i s t i c of the p a r t i c u l a r element from which the electrons o r i g i n a t e ; chemical s h i f t e f f e c t s are observed, but these e f f e c t s are small compared with the differences between d i f f e r e n t elements [32,33]. Q u a l i t a t i v e analysis i n p r a c t i c e involves comparing the energies of observed Auger peaks with the l i s t e d values [34-36], Most elements, with the exception of hydrogen and helium, can be detected uniquely even i f several are present i n a surface region. A t y p i c a l example of an Auger spectrum from- t h i s work i s shown i n f i g u r e 1.10; t h i s i s for a Rh(110) surface contaminated with sulphur, -18-Figure 1.9: The production of an L 2 VV Auger electron i n aluminum. X-ray energy levels are indicated r e l a t i v e to the Fermi l e v e l . - 1 9 -T r l r r 100 200 300 4 0 0 ENERGY/eV Figure 1.10: Auger spectrum of a heavily contaminated Rh(llO) surface, E Q = 1.5 keV, I = 10 microamps. -20-carbon and phosphorus. The spectrum i s presented in the d e r i v a t i v e form ( dN(E)/dE ) to enhance the weak Auger features. Using standard LEED optics as a retarding f i e l d analyzer [16,17], amounts of around 1-5% of a monolayer can be detected for most elements; higher s e n s i t i v i t i e s are possible with a c y l i n d r i c a l mirror analyzer [37]. The f l u x of Auger electrons produced depends e s p e c i a l l y on the i o n i z a t i o n cross-section of i n d i v i d u a l elements, and t h i s generally varies with energy. In t h i s t h e s i s , AES i s used only for q u a l i t i a t i v e chemical a n a l y s i s , although there are continuing attempts to develop t h i s technique for quanti-t a t i v e analysis [38,39]. With s u i t a b l e c a l i b r a t i o n s , t h i s technique can give important information on surface k i n e t i c s [40]. It also has p o t e n t i a l value for assessing aspects of surface band structures [41]. 1.5 Aims of Thesis The o v e r a l l objective of t h i s thesis i s to contribute to an increase i n knowledge associated with LEED crystallography, both by determining some unknown surface structures and by assessing possible new or modified procedures. The c a t a l y t i c aspects of rhodium have been well known for a long time [42] but the crystallography of i t s surfaces has not been thoroughly inves-tigated. In e a r l i e r work, Watson et a l . [43,44] reported discrepancies i n the geometrical structures of the (100) and (111) surfaces of rhodium, asso-ciated with the use of atomic p o t e n t i a l s from two d i f f e r e n t sources which were expected to give e s s e n t i a l l y equivalent r e s u l t s . These discrepancies are resolved i n t h i s t h e s i s . -21-An important recent emphasis i n LEED crystallography involves the development of s u i t a b l e r e l i a b i l i t y factors for making routine comparisons between experimental and calculated 1(E) curves. The most complete R-factor appears to be that introduced by Zanazzi and Jona [45]. This r e l i a b i l i t y f a c tor i s studied here both i n actual surface structure determinations and by assessing i t s value f o r f i x i n g some non-geometrical parameters required i n the multiple s c a t t e r i n g c a l c u l a t i o n s of LEED i n t e n s i t i e s . In the experimental parts of t h i s t h e s i s , the adsorptions of oxygen and sulphur on the (100) and (110) surfaces of rhodium have been studied and d i f f r a c t e d beam i n t e n s i t i e s measured for various structures. Complete LEED cr y s t a l l o g r a p h i c analyses with f u l l m u l t i p l e - s c a t t e r i n g c a l c u l a t i o n s have been made for the surface structures designated Rh(110)-c(2x2)S and Rh(100)-p(2x2)S. These structures have proved useful for gaining some insights into surface chemical bonding. A problem with the present schemes for c a l c u l a t i n g LEED i n t e n s i t i e s concerns the large computational times arid computer core storages required. A simpler scheme c a l l e d the quasidynamical method has recently been proposed by Tong and Van Hove [46]; i t i s f a s t e r and requires much less core storage than the complete methods. I n i t i a l studies i n d i c a t e that i t could be useful for systems of weak scatterers [46,47], and further investigations are reported here, p a r t i c u l a r l y for structures involving sulphur adsorbed on surfaces of rhodium. -22-CHAPTER 2 C a l c u l a t i o n of LEED I n t e n s i t i e s -23-2.1 C h a r a c t e r i s t i c s of 1(E) curves A t y p i c a l 1(E) curve has already been i l l u s t r a t e d i n figure 1.5; t h i s i s s p e c i f i c a l l y f o r the specular beam d i f f r a c t e d from a Ni(100) surface. Such a curve shows considerable structure, that i s the i n t e n s i t y exhibits a number of maxima and minima as the energy i s varied. Also as noted i n section 1.2, the e l a s t i c r e f l e c t i v i t y corresponds to only a few percent of the t o t a l incident electrons. Early attempts to explain 1(E) curves i n LEED based on the kinematical theory (which i s applicable when scattering cross-sections are very low e.g. X-ray d i f f r a c t i o n [49]) were unsuccessful. For a surface whose structure corresponds to that of the bulk, the kinematical theory predicts peaks i n 1(E) curves f o r the t r i p e r i o d i c d i f f r a c t i o n condition where g(hkil) i s a vector of .the r e c i p r o c a l l a t t i c e . For the (hk) beam i n LEED, equation (2.1a) becomes equivalent to k" = k + + g(00il) . (2.1b) ~g ~o ~ Peaks i n 1(E) curves which s a t i s f y (2.1b) are termed "primary Bragg peaks" and may be designated by the index I. Further relevant observations from 1(E) curves of the type i n figure 1.5 are as follows: 1) Peaks i n experimental 1(E) curves which are close to s a t i s f y i n g the primary Bragg condition (equation 2.1b) are generally found at lower energies than expected. This suggests an inner p o t e n t i a l correction i s necessary as - a consequence of the reduced p o t e n t i a l experienced by an electron inside the c r y s t a l [50]. 2) Often more peaks are observed i n experimental 1(E) curves than expected -24-from equation 2.1. This suggests m u l t i p l e - s c a t t e r i n g i s s i g n i f i c a n t ; t h i s i s consistent with the cross-sections f or s c a t t e r i n g of low-energy electrons being of the order of unit mesh areas [51] (and hence several orders of magnitude greater than those for the s c a t t e r i n g of X-rays). 3) Peaks i n 1(E) curves generally show increasing widths with increasing energy [52]. Peak widths are r e l a t e d to uncertainties i n energy and hence to f i n i t e l i f e - t i m e s v i a the uncertainty p r i n c i p l e [24]; the average l i f e -time can be interpreted as the time for the electron to traverse the mean free path length (L) introduced i n section 1.2. 4) The d i f f r a c t e d beam i n t e n s i t i e s decrease with increasing temperature often i n an exponential manner [53,54]. Such observations suggest that the LEED process i s a dynamical process i n which the non-geometrical parameters play an important r o l e i n i t s d e s c r i p t i o n . The f i x i n g of these parameters, together with m u l t i p l e - s c a t t e r i n g of electrons through ordered surface regions, represent complications f o r an analysis of the d i f f r a c t i o n process. 2.2 Physical Parameters required i n LEED Theory It has already been indicated that the incident electrons i n LEED experience strong e l a s t i c and i n e l a s t i c s c a t t e r i n g s ; c l e a r l y the c r y s t a l p o t e n t i a l must be chosen c a r e f u l l y to accommodate these two important features i n LEED i n t e n s i t y c a l c u l a t i o n . The " m u f f i n - t i n " p o t e n t i a l provides a convenient model for t h i s purpose. In t h i s approximation (figure 2.1), the p o t e n t i a l i s taken as s p h e r i c a l l y symmetric i n the v i c i n i t y of atoms and gure 2.1: M u f f i n - t i n p o t e n t i a l a) i n cross-section as contours, b) along xx' ( V i s the constant intersphere p o t e n t i a l ). Energy Energy T Vocuum level 01 — F E p Fermi energy o Lowest level of conduction bond gure 2.2: 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 energies measured wi respect to the vacuum l e v e l and those measured with respect t the lowest l e v e l of the conduction band. -26-constant elsewhere. The r e a l part of the constant p o t e n t i a l (V ) i s often equated to the empirical inner p o t e n t i a l noted above; 1 ^ I i s roughly equal to the sum of the Fermi energy and the work function as i l l u s t r a t e d i n f i g u r e 2.2. V o r i s negative and i t can be regarded as giving the p o s i t i o n of the muffin-tin zero below the vacuum l e v e l ; i t i s associated with the p o t e n t i a l well that confines the conduction electrons to s o l i d s . Typical values o f V range from -10 to -20 eV. The e f f e c t of t h i s p o t e n t i a l well i s to speed up the incident electrons i n s i d e the c r y s t a l . Although i s s t r i c t l y dep-endent on energy [55], because of exchange and c o r r e l a t i o n e f f e c t s , t h i s dependence i s often s u f f i c i e n t l y weak that i t can be ignored f o r the purpose of c a l c u l a t i n g 1(E) curves [56]. To a good approximation, changes i n V give a r i g i d s h i f t i n calculated 1(E) curves; t h i s enables values of V^^ used i n c a l c u l a t i o n to be r e f i n e d by t r a n s l a t i n g the calculated 1(E) curves along the energy scale u n t i l optimal matching with the corresponding experimental 1(E) curves i s obtained [57]. I n e l a s t i c s c a t t e r i n g i s conveniently incorporated i n t o c a l c u l a t i o n schemes by gi v i n g an imaginary contribution to the intersphere p o t e n t i a l , that i s expressing the constant part of the p o t e n t i a l as V = V + iV . . (2.2) o or o i For an electron wave function with time dependence V f r . t ) = Y f r ) e l E t ; (2.3) 2V • t the i n t e n s i t y decays with time as e 0 1 provided V i s negative. Pendry [24] established the r e l a t i o n AE = 2IV .I (2.4) w 1 011 -27-where AE^ i s the peak width at h a l f maximum height i n an 1(E) curve and 2 2 the analysis uses atomic units ( n =ro e = e =1). Equation (2.4) i s h e l p f u l for estimating values of V ^ from experimental i n t e n s i t i e s ; t y p i c a l l y V i s around -5 eV with a f a i r l y weak energy dependence [58]. Demuth et a l . [56] proposed the use of the functional form V . = - a E 1 / 3 . (2.5) o i In p r a c t i c e , e s p e c i a l l y f o r an overlayer, the c r y s t a l p o t e n t i a l close to the topmost atoms can be d i f f e r e n t from that of the substrate region [59]; a schematic representation of the c r y s t a l p o t e n t i a l i s indicated i n f i g u r e 2.3, Ideally the p o t e n t i a l used i n LEED ca l c u l a t i o n s i s constructed from s e l f - c o n s i s t e n t band structure c a l c u l a t i o n s [60], However s u i t a b l e p o t e n t i a l s of t h i s type are not always a v a i l a b l e , and a p l a u s i b l e a l t e r n a t i v e involves constructing p o t e n t i a l s from the superposition of atomic charge densities i n f i n i t e c l u s t e r s [22,6l]. In either case, the exchange p o t e n t i a l experienced by an electron of wave function <K,L\) i s most often represented by Slater's l o c a l density approximation [62] V e x(£)Kr) = - 6 ( i g & ? ) 1 / 3 * f r ) (2.6) where p(r) i s the l o c a l charge density. The s c a t t e r i n g of an electron plane wave by a s p h e r i c a l l y symmetric ion-core p o t e n t i a l y i e l d s a sphe r i c a l wave, and the t o t a l wavefield at large |'rj has the form [63,64] * ,ii*iur 4< (r) = e 1^'^ + f(9) s ~ (2.7) ADSORBATE LAYER SPACING SUBSTRATE LAYER SPACINGS IMAGINARY POTENTIAL VACUUM LEVEL REAL POTENTIAL SUBSTRATE kNO REFLECTION MATCHING ADSORBED LAYER TRANSITION REGION AtanMBz* C) i t 2 3: Muffin T i n model of an adsorbate covered surface (after Marcus et a l . [59]). -29-The s c a t t e r i n g amplitude f(8) i s commonly expanded as f(8) = J~T L C2£+l)expCi6 1)sin<5 £P J l(cos9) , (2.8) where 6^ i s the phase s h i f t which characterizes s c a t t e r i n g by ion-cores for angular momentum SL, and i s a Legendre polynomial. For a p a r t i c u l a r atomic p o t e n t i a l , phase s h i f t s are found by solving the Schro'dinger equation in s i d e the m u f f i n - t i n sphere and j o i n i n g the asymptotic form of the s o l u t i o n smoothly at the boundary of the sphere to those solutions obtained by solving the SchrOdinger equation for the outside region. In p r a c t i c e for LEED i t i s found that f(G) converges f a i r l y r a p i d l y so that only a l i m i t e d number of I values are needed. T y p i c a l l y i n LEED ca l c u l a t i o n s for energies up to and around 200 eV, the maximum value of I ( i . e . I ) needed i n expressions such ' max r as (2.8) i s about 7. The e f f e c t of the thermal motion of ion-cores i s generally treated by adding an i s o t r o p i c Debye-Waller-type contribution into the atomic sc a t t e r i n g f a c t o r . Jepsen et a l . [57] showed that the atomic s c a t t e r i n g f a c t o r f o r such a v i b r a t i n g l a t t i c e can be r e l a t e d to that ( f(6) ) of the r i g i d l a t t i c e but with some modifications to the phase s h i f t s . S p e c i f i c a l l y for the p ^ atom, f(0,T) = f(6)exp[-M ( k ' - k j 2 ] , (2.9) P I where a wave characterized by Ic i s scattered into k , Mp. " ^< Up>T ' and u i s the v i b r a t i o n a l amplitude i n the d i r e c t i o n of the momentum tra n s f e r P -30-(k -k). In the high temperature l i m i t (T>0 D), u i s r e l a t e d to the Debye temperature (0^) by / v 2 3h 2T ,, ... <*OT = T ' (2.10) P T M k e 2 ~ p I D where M i s the atomic mass and k n i s the Boltzmann constant, p B Computational procedures for LEED i n t e n s i t i e s developed rather slowly, i n part because of the complexity associated with the m u l t i p l e - s c a t t e r i n g . However, during the 1970's a number of schemes have been derived, and h e l p f u l reviews have been given by Duke [22], Tong [65] and Stoner et a l . [66]. The e a r l i e s t c a l c u l a t i o n s neglected i n e l a s t i c s c a t t e r i n g [67]; Duke and Tucker [68] were among the f i r s t to emphasize the necessity for including i n e l a s t i c s c a t t e r i n g i n computational schemes. The f i r s t substantial agreement between calculated and experimental 1(E) curves was produced i n 1972 i n the work of Jepsen et a l . [57] on the (100) surface of aluminium, s i l v e r and copper. These ca l c u l a t i o n s assumed: i ) Surface geometries that correspond to undistorted truncations of the bulk structures. i i ) Electron-ion core in t e r a c t i o n s can be represented by p o t e n t i a l s from band structure c a l c u l a t i o n s . i i i ) Absorption e f f e c t s can be incorporated with an imaginary p o t e n t i a l from uniform electron-gas theory [62]. i v ) L a t t i c e v i b r a t i o n s can be treated by a Debye-Waller type factor as indicated above. v) The inner p o t e n t i a l correction (V ) c a n he chosen emp i r i c a l l y by a l i g n i n g t h e o r e t i c a l and experimental 1(E) curves. -31-This work o f Jepsen et a l . established that the dominant aspects of the e l a s t i c LEED process were e s s e n t i a l l y understood, even though numerical agreement was not obtained for absolute i n t e n s i t i e s . The l a t t e r appears to r e l a t e e s p e c i a l l y to incomplete order for the surfaces, but in any event t h i s discrepancy did not i n h i b i t the development of LEED crystallography, since i t was found that the positions o f structure i n 1(E) curves could be calculated to within experimental erro r . Calculations of LEED i n t e n s i t i e s generally involve t r e a t i n g the scatter-ing of a plane wave by a surface region of perfect d i p e r i o d i c symmetry. The t o t a l wave f i e l d outside o f the c r y s t a l has the form ik • r n r ) = 4>(r) + Z c e ~& ~ , (2.11) g ~ where <K£) i s the incident plane wave. The objective i s to cal c u l a t e beam r e f l e c t i v i t i e s , ' k R g k o c | 2 (2.12) g which r e l a t e to the measured i n t e n s i t i e s . B r i e f descriptions of some of the important procedures now ava i l a b l e for c a l c u l a t i n g beam r e f l e c t i v i t i e s are given i n the following sections. -32-2.3 T-Matrix Method The T-matrix method was formulated by Beeby [70] and has since been d e t a i l e d further by Tong [65]. This method s t a r t s by w r i t i n g the wave function for an electron i n s i d e the s o l i d as *C£) = •(£) + / G ( r - r ' ) V(r') *(r') dr' , (2.13) where the Green s function GQr-r, ) describes the propagation of an electron from r, to £. This equation can be solved by defining a t o t a l s c a t t e r i n g matrix (T) f o r the s o l i d V ( r ' ) n r ' ) = / T(r'r)*Q:)dr . (2.14) With the muffi n - t i n approximation for the p o t e n t i a l , s u b s t i t u t i o n of (2.14) int o (2.13) y i e l d s TCCa^) = EVr 2-R,r rR) + E {t R.(r 2-R'.£ 3-R') R ~ R*R v ~ -^^ v^  ~ ' ^ ) d ^ 4 r — ( 2- 1 5 ) where t KtXrE.IrJP • V E ( r 2 - R ) 6 x i l 2 * / v , ( r 2 - R ) G t r 2 - r ) V r - R , r r R ) d r (2.16) i s the t-matrix f o r the s i n g l e ion core at R. In (2.15), the f i r s t term covers a l l s i n g l e ion core s c a t t e r i n g , the second term represents a l l double sca t t e r i n g events, etc. Equation 2.15 therefore sums a l l p o s s i b l e i n t e r -atomic and intra-atomic s c a t t e r i n g events involved with the electron going from r , to r» in s i d e the s o l i d ~1 ~2 For the actual evaluation of the c g i n (2.11), and hence.the beam -33-r e f l e c t i v i t i e s , the c r y s t a l i s divided into subplanes p a r a l l e l to the surface such that each subplane has the same Bravais structure and contains the same kind of atoms. The f i n a l r e s u l t i s ^ Y. (k~)Y*,(k +;) i ( k + - k ~ ) - d a .., c = y E L * ~° I> ° & ^ T L L (k ) (2.17) where L stands f o r the angular momentum quantum numbers I,m, i s the associated sp h e r i c a l harmonic, the second summation i s over a l l subplanes and d i s the vector from the o v e r a l l o r i g i n at the in t e r f a c e to the o r i g i n chosen f o r subplane a. In (2.17) T^ (k Q) ^ s the element of the t o t a l s c a t t e r i n g matrix for subplane a in the angular momentum representation i L 1 L 2 Qfa 1 2 T (k ) i s the LL element of the planar s c a t t e r i n g matrix ( f ) for the ex o r »a subplane a T ( k ) = t (k ) [ l - G S p ( k . ) t (k ) ] _ 1 , (2.19) and t (k ) i s the diagonal t-matrix f o r a si n g l e ion core i n subplane a. «a o The non-zero elements of t h i s matrix r e l a t e to the phase s h i f t 6 by t M ( k ) = 4- [ q2\\'1 3 • (2-2°) a v o • 2m 2ik o Also needed i n (2.18) and (2.19) are the intraplanar s t r u c t u r a l propagators G SP and the interplanar propagators Ga^. These are complex matrices which are dependent on the i n e l a s t i c s c a t t e r i n g and the geometries associated with the ion core s i t e s . -34-Successful c a l c u l a t i o n s have been made with t h i s method f or clean metal surfaces. In p r i n c i p l e i t i s exact and can work for any type of surface structure; i n p r a c t i c e , however, the solving of the set of equations '(2.18) to give the matrix T i s very time consuming and requires a large amount of computer core stroage i f an appreciable number of subplanes have to be included. This method i s only p r a c t i c a l i n the presence of i n e l a s t i c s c a t t e r i n g , a feature that Beeby neglected i n the i n i t i a l formulation. The extension to include thermal motion of the ion cores was made by Tong and Rhodin i n 1971 for the (100) surface of aluminum [71], 2.4 Bloch Wave Method This method was introduced by McRae [67,73] and developed by Pendry [74], Kambe [75,76] and Jepsen et a l . [57,77]. A d e t a i l e d account has been p u b l i -shed i n Pendry s book [24]. In t h i s approach, the muffin-tin approximation i s again used and an i n f i n i t e c r y s t a l i s b u i l t up of p a r a l l e l layers. For the region of constant p o t e n t i a l between successive layers, each Bloch wave can be expanded i n terms of plane waves. The s c a t t e r i n g s i t u a t i o n at a si n g l e layer i s depicted i n figu r e 2.4, where a set of incident plane waves Y.(r) = Z b + e x p ( i k + - r ) (2.21) i ~ _ g ~g ~ is d i r e c t e d onto the c r y s t a l , and scattered waves Y (r) = Y M*f b + e x p f i k V r ) (2.22) $ ~ ~ i g g g ~ J ~ gg ~ ~ ~ propagate both i n the outward d i r e c t i o n (k f) and i n the inward d i r e c t i o n ( k * t ) . The matrices involved i n t h i s formulation are expressed i n terms of -35-£ b ; e x p ( i k + - r ) g 8 ~ ft / E E M^b;.xp(l|4:r) Figure 2A\ Schematic representation of a set of plane wave incident from the l e f t and multiply scattered by a plane of ion cores p + 1 p -+ at h layer Figure 2.5: Schematic diagram of transmission and reflection'matrices at the a t h subplane. The broken lines are the central lines between the subplanes. -36-th e l i n e a r momentum (K-space) representation; t h i s contrasts with the angular g g momentum (L-space) representation i n the T-matrix method. M*t i s an element ++ A move into the c r y s t a l . The notation M , c o v e r s - a l l four combinations of of the layer d i f f r a c t i o n matrix M where both incident and d i f f r a c t e d beams ++ g'g d i r e c t i o n s . It i s clear f o r the s i t u a t i o n i n figu r e 2.4 that a l l the d i f f r a c t e d beams become coupled together. The c o e f f i c i e n t s f o r plane waves between layers a and a+1 can be expressed i n a compact matrix notation b + i = »a+l = T b «a wa + R+ b , «a a+1 (2.23a) b" ara = T " " b " i aa a+1 + R"+b + aa «a (2.23b) where, for example, the components of the column vector b* are the various values of b + between layers a and a+1. For a c r y s t a l composed of i d e n t i c a l l a y e r s , which are separated by a constant displacement c, the transmission and r e f l e c t i o n matrices can be expressed as T + | = P + i ( I i + M +t ) P + (2.24a) I I g J J M & M T " = P", ( I , + M"T ) P" (2.24b) & & & & fL i l l R*7 = P +. M +T P" (2.24c) 2 8 g | | g R~* = P", M"t P + (2.24d) g g g g g g where P + represents inward propagation with wave vector k + through one h a l f of an i n t e r l a y e r distance while P represents the corresponding outward propagation with wave vector k -37-+ i k P~ = e s g (2.25) The I i i n equation (2.24) are elements of a unit matrix. Schematic repre-S I sentations of the r e f l e c t i o n and transmission matrices are shown i n f i g u r e 2.5. Corresponding c o e f f i c i e n t s between successive layers must s a t i s f y the Bloch conditions + a+1 i k - c , + e ~ b a (2.26a) a - i k - c , e ~ " b a+1 (2.26b) (2.24) into (2.26) y i e l d s the eigenvalue equation l"b + 1 = X (2.27) b " i »a+l b" , «a+l where L = T + + R - ( T " " ) " 1 R " V + T""-(T"") _ 1R " V " (2.28) and ik- c X = exp~~ * . (2.29) Pendry [24] has discussed the evaluation of the layer d i f f r a c t i o n ±± matrices M i n terms of the s c a t t e r i n g properties of the i n d i v i d u a l ion-cores For a layer which involves a s i n g l e atom per unit mesh, the elements s a t i s f y ++ M"T g g r S . ' Y L f t g O [ l - X ] - J ; . Y L . f k * ) e x p ( i 6 1 , D s i n 6 1 , -Ak k* LL' o~gi (2.30) -38 -+ + where describes multiple s c a t t e r i n g within the layer. Given M~ for a p a r t i c u l a r system, the transmission and r e f l e c t i o n matrices i n equations (2.24) can be set up and hence (2.27) can be solved by standard methods to give eigenvectors, which f i x the Bloch waves, and the corresponding eigen-values which f i x possible wave vectors along with the requirement of conser-vation of momentum p a r a l l e l to the surface. Only h a l f of the 2n possible solutions (where n i s the number of vectors g included i n the c a l c u l a t i o n ) are p h y s i c a l l y acceptable ( i . e . correspond to waves which either propagate or decay exponentially i n the z - d i r e c t i o n ) . To complete the c a l c u l a t i o n of d i f f r a c t e d beam r e f l e c t i v i t i e s i t i s necessary to match each wave function, and i t s f i r s t d e r i v a t i v e with respect to z, at both sides of the solid-vacuum i n t e r f a c e . Corresponding wave matching proce-dures are involved in extending t h i s scheme to s i t u a t i o n s where one or more top layers are d i f f e r e n t from the r e s t . (e.g. for an adsorbed l a y e r ) . This basic approach involves less computer• core storage than the T-matr-ix method, but the s o l u t i o n of equation (2.27) becomes time consuming when n g i s large. -39-2.5 Perturbation Methods The T-matrix and the Bloch wave methods are exact i n the sense that they include a l l multiple s c a t t e r i n g events i n the c r y s t a l . These methods have proved valuable for c a l c u l a t i n g LEED i n t e n s i t i e s of clean surfaces, although they require long computational times and large core storage. Such considerations l i m i t the use of these exact multiple s c a t t e r i n g methods to the more complex surface structures of i n t e r e s t i n LEED crystallography, and therefore encourage the development of approximate schemes based on pertur-bation expansions. Part of the motivation for t h i s comes from the r e a l i z a t i o n that with i n e l a s t i c s c a t t e r i n g the comparatively short mean free path length must l i m i t the order of multiple s c a t t e r i n g that can be important. This suggests that i t should be possible to reduce computational times by formu-l a t i n g i n terms of perturbation theory. Tong et a l . [72] made the T-matrix c a l c u l a t i o n to t h i r d order, and showed that i t can work well for weak scat-t e r i n g metals l i k e aluminum. However the appoach of u t i l i z i n g perturbation theory within the T-matrix method seems less h e l p f u l for stronger s c a t t e r e r s ; b a s i c a l l y t h i s appoach becomes too clumsy and unwieldy at above t h i r d order. Pendry has developed convenient i t e r a t i v e schemes which are based on the Bloch wave method and have the s i g n i f i c a n t property that the contribution from each a d d i t i o n a l order has the same basic form as those from the previous orders (this i s unlike the s i t u a t i o n for the t h i r d order c a l c u l a t i o n [72] noted above for Al(100)). These new methods are the layer doubling and renormalized forward s c a t t e r i n g methods; multiple s c a t t e r i n g c a l c u l a t i o n s described i n t h i s t hesis u t i l i z e d these methods extensively. -40-2.5 (a) Layer Doubling Method This method [24,78] requires that i n e l a s t i c s c a t t e r i n g i s s u f f i c i e n t l y strong so that a s e m i - i n f i n i t e c r y s t a l can be approximated by a slab of f i n i t e thickness. Two layers are considered f i r s t , then four layers, and at each l e v e l of i t e r a t i o n the number of layers i s doubled. This method starts with a c a l c u l a t i o n of the r e f l e c t i o n and transmission matrices as in equations (2.24), and then generates the corresponding matrices f o r a stack of two layers. T ! + = T ! + ( I - C O " 1 T+ + (2.31a) + - R~+ + T~~R~+ r i - R + " R " + ) _ 1 T + + (2.31b) - § A AA * B l i SA * B J *A y Ic - IA a-CsI")"1 I"" (2.3id) where the i n d i v i d u a l subplanes are denoted by A , B and the r e s u l t i n g composite layer i s denoted by C. The doubling process i s shown schematically i n fi g u r e 2.6; the same set of equations (2.31) are used to extend the c r y s t a l stack to 2, 4, 8, 16... layers. This process i s continued u n t i l the r e f l e c -t i o n amplitudes have converged; t y p i c a l l y t h i s requires 8 or 16 atomic lay e r s . Once convergent r e f l e c t i v i t i e s have been obtained f or the substrate, surface layers can be systematically added s t i l l using equations (2.31). A convenient feature of t h i s method i s that a surface layer can be s h i f t e d either l a t e r a l l y or v e r t i c a l l y , without having to recompute the bulk r e f l e c t i v i t i e s . -41-Figure 2.6: Stacking of planes to form a crystal slab and i l l u s t r a t e th. layer-doubling method. Planes A and B are f i r s t stacked to form the two-layer slab C; the process is continued to form four-layer slab. (After Tong [65].) -42-This method i s considerably f a s t e r than the f u l l Bloch wave method and yet i t can provide good numerical accuracy. Each i t e r a t i o n involves inver-sions of two matrices of dimension n (the number of beams included i n the c a l c u l a t i o n ) . A l i m i t a t i o n i s that t h i s method i s not s u i t a b l e f or very small i n t e r l a y e r spacings (c<0.5A) when n^ i s required to be excessively large [79]. 2.5 (b) Renormalized Forward Scattering method The renormalized forward s c a t t e r i n g (RFS) method was introduced by Pendry [24] and discussed further by Tong [65]; i t s c h a r a c t e r i s t i c features are that the i n t r a l a y e r scatterings are c a l c u l a t e d exactly, while the i n t e r -layer scatterings are i t e r a t e d f or the various possible paths i n the c r y s t a l . The p r i n c i p l e of t h i s method i s i l l u s t r a t e d schematically i n f i g u r e 2.7. The c r y s t a l i s again represented by a f i n i t e number of l a y e r s ; the actual number used (n) i s such that the t o t a l e l a s t i c a l l y scattered amplitude t i l that would reach the (n+1) layer i s less than a predetermined f r a c t i o n (e.g. 0.003) of i t s incident amplitude. C l e a r l y the stronger the i n e l a s t i c s c a t t e r i n g , the smaller i s the number of layers that are needed. Following • t l l Tong [65], A (g) designates the amplitude at the l o c a l o r i g i n between the a a ~ th and (a+1) layers f or the electron wave characterized by g propagating into the c r y s t a l ; the index i i s the order of i t e r a t i o n which i d e n t i f i e s the number of times the electron has propagated into the c r y s t a l along t h i s p a r t i c u l a r - path. For the f i r s t i t e r a t i o n we have V&> = z. CtsOVi^ ' (2-32) g -43-incident beam _ It (a-1) (go.) . + + (28) A a ( S ( a + l) Figure 2.7: a) Illustration of the renormalized forward scattering method. Vertical lines represent layers. Each t r i p l e t of arrows represents the complete set of plane waves that travel from layer to layer. b) Propagation steps of the inward-travelling waves. c) Propagation steps of the outward-travelling waves. (After Van Hove and Tong [8l].) -44-but no waves propagating in the inward direction are included after the n*^ layer. Waves propagating in the outward direction are represented by B 1(g) in an analogous notation. Except at the deepest layer, the outward travel-ling waves consist of two components (figure 2.7c): the reflected portions of the inward travelling waves, and the transmitted portion of the outward-travelling waves. In general, the amplitudes of the outward-directed waves satisfy I J (2.33) ( a = n-1, n-2, ... 0 ), where n is the deepest subplane reached in the appropriate iteration. The corresponding expression for the inward-directed waves is A X(g) = Z R +"(gg')B 1" 1(g') + E T + +(gg')A 1 .(g 1) ; a ~ ' a ~2 a ~ ' i a ~~ a-1 2 J ' g I (2.34) ( a = 1, 2, 3, ... n ). Equations (2.33) and (2.34) are solved iteratively in the RFS method until the r e f l e c t i v i t y has converged. This approach is computationally convenient since no eigenvalue equations or matrix inversions are involved. 2 The computation times scale as n , where n is the number of beams included; this is more favorable than the layer doubling method for which computation 3 time scales as n . The RFS method has proved to be an excellent method for g calculating LEED intensities for many systems provided the electron damping is sufficient. Otherwise i t s only limitation is a failure to converge when o . any two layers are closer than about 1 A. In the latter event the layer doubling method may be applicable. -45-2.6 Further M u l t i p l e Scattering Methods The RFS and layer doubling methods have proved to be r e l i a b l e and con-venient for LEED c r y s t a l l o g r a p h i c analyses of many clean and simple overlayer surface structures. A l i m i t a t i o n i n a l l procedures which u t i l i z e the K-space representation ( including the f u l l Bloch-wave method ) i s that the number of plane waves required i n the c a l c u l a t i o n s increases r a p i d l y with decreasing i n t e r l a y e r separations [79]. Once matrices of dimension of the 2 order of 10 are involved the K-space methods become incr e a s i n g l y unwieldy and numerically u n r e l i a b l e ; e f f e c t i v e l i m i t s are set with i n t e r l a y e r spacing o o of around 0.5 A for both the layer doubling and the Bloch wave methods ( ~ l A sets the lower l i m i t for the RFS method ). For models where close i n t e r l a y e r spacing are required, there are two possible approaches: (i) to stay with the K-space representation but t r e a t the two layers as a composite layer (with consequent increase i n matrix dimensions and requirements for computing time and storage), or ( i i ) to work with the L-space representation (as i n the T-matrix method). The dimensions of matrices involved i n L-space c a l c u l a t i o n s are independent of the number of beams required for the c a l c u l a t i o n s , hence t h i s approach s t a r t s to have advantages over the K-space representation when n i s large. To make the T-matrix method more e f f i c i e n t , Zimmer and Holland [80] introduced a reverse-s c a t t e r i n g i t e r a t i v e procedure which e s s e n t i a l l y represents an equivalent of the RFS method i n the L-space representation. This approach again f u l l y accounts for forward s c a t t e r i n g events, but approximates the back-scattering. The reverse s c a t t e r i n g procedure of Zimmer and Holland requires matrices of -46-2 dimensions (I +1) . T y p i c a l l y I +1^8 for electron energies less than max 1 V J max 200 eV, thus t h i s i t e r a t i v e method appears advantageous over the RFS method i f the number of beams required exceeds about 64. However t h i s L-space i t e r a t i o n approach requires the evaluation and storage of n(n-l) square matrices (3°^ for an n-layer c r y s t a l , and moreover these matrices have to be recalculated for every change made to the surface layer. This represents a less s a t i s f a c t o r y feature o f the method. Recently Van Hove and Tong [79] described a combined-space method which u t i l i z e s both the L-space and K-space representation to achieve optimal adv-antages of each. S p e c i f i c a l l y the c a l c u l a t i o n i s made i n the L-space representation for those layers which are c l o s e l y spaced, while the K-space representation i s used for the re s t of the c a l c u l a t i o n where the i n t e r l a y e r spacings are larger. Discussions of approaches and the associated computer programs for the various methods now a v a i l a b l e f o r surface crystallography with LEED have been described i n a recent book by Van Hove and Tong [ 8 l ] . i -47-2.7 General Aspects of Computations 2.7 (a) S t r u c t u r a l Parameters and Use of Symmetry The basic approach to surface crystallography with LEED involves pos-t u l a t i n g a s e r i e s of t r i a l structures and searching for that p a r t i c u l a r model which gives the best agreement between calculated and experimental 1(E) curves. The models postulated must be consistent with the symmetries indicated by the observed LEED pattern. The substrate structure i s generally known from X-ray crystallography, but atoms i n the upper layers of a clean surface need not occupy exactly the positions they would i n the i n f i n i t e c r y s t a l . Many clean metals have surfaces whose t r a n s l a t i o n a l symmetries are found by LEED to be i d e n t i c a l with those of the corresponding substrate structure (the surface i s s a i d to be unreconstructed i f the normal r e g i s t r i e s are maintained, although there may be changes i n the v e r t i c a l spacing); by contrast LEED patterns show d i r e c t l y that many are reconstructed [86]. In general, for both clean surfaces and adsorption systems, the topmost i n t e r -layer spacing must be varied i n the LEED i n t e n s i t y c a l c u l a t i o n s and l a t e r a l v a r i a t i o n s should also be considered. For models where domain structures are possible appropriate beam i n t e n s i t i e s need to be averaged i n the calcu-lations i n order to accommodate the expectation, that the incident beam i n the experiment samples a l l the domain types. For example, for sulphur adsorbed on the bridge s i t e s of the Rh(100) surface, as i n f i g u r e 2;8, an averaging of the i n t e n s i t i e s of the (10) and (01) beams i s necessary for the c a l c u l a t i o n s to become consistent with the four f o l d symmetry observed i n the experimental LEED pattern. •48-Rh(l00)-P(2x2)-S real 11 reciprocal - $ — © ® -m 0 & <s> © i © 11 2 mirror planes + 1 C A axis 1F 2 mirror planes only 2.8: Schematic diagram of three simple models for Rh(100)-p(2*2>S In reciprocal space, sets of symmetrically equivalent beams ; indicated by a common symbol. -49-Th e computational e f f o r t can be reduced when the d i r e c t i o n . o f incidence coincides with a symmetry axis or a symmetry plane [81]; t h i s depends on the i n e v i t a b l e equivalences i n the d i f f r a c t e d beams as a r e s u l t of the symm-etry elements i n the model. The s i m p l i f i c a t i o n s 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 represent a standard a p p l i c a t i o n of the group theory. U t i l i z i n g symmetry reduces the dimensions of the matrices required within the K-space representation [82], s p e c i f i c a l l y only one g vector i s needed f o r each set of symmetry-related beams. For the p a r t i c u l a r examples of the model types shown in f i g u r e 2.8 for Rh(100)-p(2x2>S, i t i s r e a d i l y seen that, with normal incidence, the 4F and IF models preserve two mirror planes of symmetry perpendicular to each other as well as a r o t a t i o n a x i s , whereas the 2F model contains only two mirror planes. A consequence of the axis i s the equivalence of the following 8 f r a c t i o n a l order beams ( l j ) = ( l | ) = ( l | ) E ( i j ) E ( i l ) E ( i l ) E c f l ) = ( i l ) for both the 4F and IF models. The s i t u a t i o n f o r the 2F model i s that these f r a c t i o n a l order beams separate into two sets of 4 equivalent beams, ( 1 | ) = ( 1 y ) = ( 1 y ) = ( 1 \ ) f ( \ 1 ) = ( \ "1 ) = ( y l ) = ( J 1 ) S i m i l a r l y , the 4F and IF models have the equivalences ( 0 1 ) = ( 0 1 ) = ( 1 0 ) = ( 1 0 ) whereas the corresponding s i t u a t i o n f o r the 2F model involves ( o i ) M o i ) M l o ) = ( I o ). The calc u l a t i o n s f o r the 2F model therefore require more beams, and corres-pondingly larger matrices, than the 4F and IF models. Table 2 . 1 : Numbers of symmetrically-inequivalent beams act u a l l y used i n c a l c u l a t i o n of various surface structures. The models f or the overlayer structures are designated as in figu r e 1 . 7 and 2 . 8 . Surface structure Surface model Type of symmetry Number of symmetrically inequivalent beams used i n c a l c u l a t i o n Equivalent t o t a l number of beams Rh (100) unreconstructed 2 perpendicular mirror planes + 5 3 Rh(llO) unreconstructed 2 perpendicular mirror planes 2 3 71 R h ( l l l ) unreconstructed 3 mirror planes at 6 0 ° to each other + along z-axis 10 37 Rh ( 1 0 0)-p ( 2 x 2)-S 4 F . 1 F 2 F same as Rh (100) 2 perpendicular mirror planes 35 52 2 2 1 1 7 7 Rh ( 1 1 0)-c ( 2 x 2)-S 4 F , 1 F 2 S B . 2 L B same as Rh (110) same as Rh ( 110 ) 49 4 9 1 7 5 1 7 5 -51-Calculations reported here with the RFS and layer doubling methods u t i -l i z e symmetry as i n the discussion and computer programs given by Van Hove and Tong [81]. In these routines symmetry i s accommodated by l i s t i n g the g vectors i n the input data together with appropriate code numbers to i d e n t i f y the symmetry type of each beam. The code number enables the program to use the appropriate symmetrized wave functions and to set up the s i m p l i f i e d d i f f r a c t i o n matrices. L i s t e d i n Table 2.1 are the numbers of symmetrically inequivalent beams needed f o r c a l c u l a t i o n s on the various surfaces studied i n t h i s t h e s i s . 2.7 (t>) Program Flow The flow-chart i n f i g u r e 2.9 summarises the sequence of events that occur 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 reading i n a l l the relevant s t r u c t u r a l and non-structural parameters as well as a l i s t of d i f f r a c t e d beams with t h e i r symmetry code numbers. At each energy, the dimensions of the matrices are set equal to the number of propagating beams ( i . e . those beams with r e a l k ) plus the f i r s t few evanescent beams. ±± ~ The layer d i f f r a c t i o n matrices M are c a l c u l a t e d ; d i f f e r e n t subroutines are a v a i l a b l e depending on whether the layer corresponds to a simple Bravais net or to a composite layer-type. The stacking of layers i s performed by either the layer doubling or the RFS methods. Each method can include overlayers with structures which d i f f e r from the appropriate layer of the substrate; a s p e c i a l case of t h i s involves a v a r i a t i o n of the topmost layer spacing for example for clean metal surfaces. Generally the c a l c u l a t i o n s are made f o r the energy range 40-200 eV i n increments of 2 eV up to 80 eV.and i n increments •52-Read in (i) geometry ( i i ) VQT> V . 0 1 ( i i i ) beams and symmetrv (iv) temperature data (v) phase s h i f t s Choose i n i t i a l energy Find beams needed at E Compute temperature-dependent phase s h i f t s Calculate layer d i f f r a c t i o n matrices M , Find d i f f r a c t e d beam amplitudes from surface plus substrate by RFS Calculate beam i n t e n s i t i e s Vary surface geometry k-1 Find d i f f r a c t i o n matrices f o r a substrate layers by layer doubling Add surface layer and f i n d d i f f r a c t e d beam amplitudes Calculate beam i n t e n s i t i e s ) Vary surface geometry Increment E ZZl Figure 2.9: Flowchart showing p r i n c i p a l steps 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 , using the RFS or layer doubling programs. -53-of 4 eV above 80 eV; the r e f l e c t e d i n t e n s i t i e s i n the high energy range are then interpolated to give values i n 2 eV i n t e r v a l s . The calculated inten-s i t i e s are stored on magnetic tape and can be plo t t e d f o r v i s u a l comparison with the experimental 1(E) curves; a l t e r n a t i v e l y the calculated i n t e n s i t i e s can be compared with experimental values by means of a r e l i a b i l i t y index as discussed i n the next sections. 2.8 Evaluation of Results 2.8 (a) Introduction In LEED crystallography i t i s necessary to f i n d the s t r u c t u r a l model which gives the best correspondence between the calculated and experimental 1(E) curves. This opens the need to be able to evaluate the s i m i l a r i t y , or otherwise, between two sets of curves on varying s t r u c t u r a l , and some non-s t r u c t u r a l , parameters. Such a search has most often been done by v i s u a l comparisons (e.g. by matching up the positi o n s and r e l a t i v e i n t e n s i t i e s of peaks, dips and other s t r u c t u r a l features), but t h i s approach suffers the disadvantage of being unwieldy when the numbers of beams and v a r i a t i o n para-meters are large. As a consequence there has been considerable encouragement for the development of numerical indices for guiding these comparisons. Among the simplest p o s s i b i l i t i e s i s AE = | E I E ? a l - E ? b s ' | (2.35) i = l which only compares peak p o s i t i o n s . In (2.35), the E^ represent energies at which the peak occurs i n the calculated and observed curves, and N -54-i s the t o t a l number of peaks compared [83,84]. C l e a r l y the better the corres-pondence i n peak posit i o n s between the experimental and calculated 1(E) curves, the lower the value of AE. In p r a c t i c e t h i s c r i t e r i o n seems incomplete because i t ignores the actual i n t e n s i t y values, i t gives an equal weighting to each peak, and i t i s ambiguous when a peak present i n one curve i s either absent or appears as an incompletely developed feature (e.g. a shoulder) i n the other curve. Van Hove et a l . [85] proposed an extension involving f i v e simple i n d i c e s , where each gives a d i f f e r e n t emphasis i n the comparison. However, the most complete index so f a r i s that proposed by Zanazzi and Jona [45]. This index attempts to compare numerically a l l the features included i n a v i s u a l comparison. i 2.8 (b) Zanazzi and Jona s Proposals The r e l i a b i l i t y index proposed by Zanazzi and Jona [45] compares curve th shapes v i a t h e i r d e r i v a t i v e s . For the i beam the r e l i a b i l i t y index i s : u dE (2.36) obs r. = f 2 1 w(E) | cl! - i! | d E / f E 2 i I. i J c 1 i i , c a l i,obs / J p l , b l i 1 t l i where i n t e n s i t i e s are compared between energies E ^ and ^2\> a n c * t^ i e P r ^ m e s i n d i c a t e f i r s t d e r i vatives f o r the calculated and observed 1(E) curves. The weight function w^ = c c . i : ; ^ - i': > o b s) / ( i i : ) 0 b s i • i i ; > o b s i m a x ) . & w emphasizes the extrema of the experimental curve and other portions with high curvature; the double primes i n (2.37) i n d i c a t e second d e r i v a t i v e s . -55-Th e s c a l i n g constant c. = f 2 1 I. . dE /(* 2 1 I. , dE (2.38) l J E i,obs / J E i . e a l l i l i allows for an a r b i t r a r y scale of i n t e n s i t y i n the experimental curves; com-parisons o f r e l a t i v e i n t e n s i t i e s are s u f f i c i e n t f o r LEED c r y s t a l l o g r a p h i c studies at the present time. One t o t a l r e l i a b i l i t y index given by Zanazzi and Jona for a set of d i f f r a c t e d beams i s r = E (r ) .AE . / E AE. , (2.39) r . r l l . I ' l l where AE. = E_. - E.. . , (r ). i s the reduced s i n g l e beam index l 2i l i ' r l 6 ( r r ) i = r./p , (2.40) and p was equated to 0.27, a mean value of r ^ found by matching random pai r s of curves. In (2.39), an average i s taken over the s i n g l e beam in d i c e s , where they are weighted according to the energy range over which the compar-ison between experiment and c a l c u l a t i o n i s made. A v a r i a t i o n of (2.39), also proposed by Zanazzi and Jona, i s R = • • § ) r r , (2.41) where n i s the number of d i f f e r e n t beams treated i n the comparisons. The advantage of (2.41), over (2.39), i s that i t mitigates against a low value of the o v e r a l l r e l i a b i l i t y index r e s u l t i n g from a comparison inv o l v i n g just a small number of beams; i t i s generally believed that a r e l i a b l e LEED cr y s t a l l o g r a p h i c analysis requires comparisons involving I(E) •curves f o r 10 -56-d i f f e r e n t d i f f r a c t e d beams. R i n (2.41) was set up with the objective of being consistent with the following p o s s i b i l i t i e s f o r values obtained from comparisons of experimental and calculated 1(E) curves for a p a r t i c u l a r proposed model: R<0.20 suggests the model i s "very probable", 0.20<R<0.35 suggests the model i s "possible", and R>0.35 suggests the model i s u n l i k e l y . 2.8 (c) Further Developments As part of an i n v e s t i g a t i o n o f the proposal of Zanazzi and Jona, Watson et a l . [43] p l o t t e d Cr )^ as a function of topmost spacing f o r the (111) surface of copper. This i s shown i n fi g u r e 2.10 where Ad% gives the topmost spacing expressed as the percentage change from the bulk spacing, ( i . e . d-d Ad% = x 100 d o where d Q i s the bulk i n t e r l a y e r spacing and d i s the topmost i n t e r l a y e r spacing). The curves shown i n f i g u r e 2.10 are s p e c i f i c a l l y f o r V = -9.5 eV; of the '16 beams a v a i l a b l e only 9 are shown f o r c l a r i t y . The reduced r e l i -a b i l i t y index (r ) for the 16 beams i s p l o t t e d as the dashed l i n e i n the same fi g u r e , and the associated error { [ Z A E . ( ( r r ) . - f r ) 2 ] / [ (n-1) Z AE. ] } * , (2.42) e r corresponding to the minimum of r ^ i s indicated by the arrows. In (2.42), n i s the number of beams considered. Watson et a l . [43] concluded that indicates an u n r e a l i s t i c a l l y large error i n Ad% for the data a v a i l a b l e . In fact the top layer spacings indicated •57-0 . 4 0 0.30H (0, 0.20H 0.KH CuOlO Vara Ad% Figure 2.10: Plots for Cu(lll) of ( r r ) . for 9 individual beams versus Ad% with V = -9.5 eV. The dashed line shows the reduced r e l i -or a b i l i t y index (f ) for the total 16 beams. (After Watson et a l . [43].) -58-by the minima for a l l the i n d i v i d u a l curves are rather close to.the spacing for the minimum i n the dashed l i n e , and t h i s suggests that the uncertainty i n spacing associated with the minimum value of r ^ could be given by the standard error found from the d i s t r i b u t i o n of toplayer spacings ( ° ^ n ) indicated by the minimum for each i n d i v i d u a l curve, e, = ( [ Z AE. ( d 1 . - d . ) ] 2 / [ (n-1) E AE. ] 1 , (2.43) d J . I mm mm J m ! J ; I I where d . = ( E AE.d 1. ) / ( E AE. ) . (2.44) mm . I mm . I l I Figure 2.10 shows d . ±2e ,; t h i s corresponds to -4.1±1.2%. The introduction 6 mm d r of by Watson et a l . makes a s t a r t on the problem of estimating uncertainties i n r e s u l t s from LEED crystallography. C e r t a i n l y numerical r e l i a b i l i t y indices are required for t h i s purpose; i t i s very hard to see how uncertainties could be h e l p f u l l y evaluated s o l e l y from v i s u a l evaluations of 1(E) curves. Another advantage of numerical indices i s that they can be e a s i l y p l o t t e d i n contour form. Again t h i s was introduced by Watson et a l . , and the example i n f i g u r e 2.11 shows a contour p l o t of r versus V and Ad% for C u ( l l l ) . According to r r or the proposal of Zanazzi and Jona, the o v e r a l l minimum i n r ^ i n f i g u r e 2.11 corresponds to the values of V Q r and Ad% which give the best agreement between the complete set of experimental and calculated 1(E) curves. Error bars - shown for the minimum represent ±e^ and ± 6 ^ (the standard error associated with the d i s t r i b u t i o n of values of V f o r the minima of (r ). and defined or r l analogously to e^); these i n d i c a t e 68% confidence l i m i t s associated with the minimum of r _ . -59-Figure 2.11: Contour p l o t f o r C u ( l l l ) of r r versus Ad% and V o r . (After Watson et a l . [43].) -60-CHAPTER 3 Preliminary Work -61-3.1 General Experimental Procedures 3.1 (a) LEED Apparatus As i n a l l work on well defined c r y s t a l surfaces, LEED experiments must -9 be c a r r i e d out at low pressure ( i . e . ^  10 t o r r ) . This section describes the general features of the conventional type of LEED apparatus which has been used i n the majority of LEED experiments made so f a r . The discussion w i l l be b r i e f , but a l o t more information can be obtained from the references provided. A review of the various modifications of LEED instruments i s a v a i l a b l e [87], A schematic diagram of the LEED apparatus used i n t h i s work i s shown in fig u r e 3.1. This involves a Varian FC12 chamber which i s constructed of non-magnetic s t a i n l e s s s t e e l and i s connected to a series of pumping f a c i l i t i e s below the main chamber indicated i n f i g u r e 3.1. The i n i t i a l sorption pumping i s done with high surface area molecular sieves ( z e o l i t e s ) i n containers which are cooled by l i q u i d nitrogen. These pumps can reduce the pressure of -3 -1 the system to -10 t o r r when the main sputter ion pump (200 £ s ) can be started. After baking the whole system for -12 hours at 200°C (to remove adsorbed gases from the chamber w a l l s ) , i t i s necessary to out-gas thoroughly a l l components o f the system that are heated during an experiment. A t i t a n i u sublimation pump i s a v a i l a b l e for extra pumping during both out-gassing and „- the actual experimental periods. Gases for adsorption studies or for ion-bombarding i n the cleaning process can be introduced into the whole chamber through a leak valve from a gas i n l e t manifold. This part of the system i s pumped by i t s own small ion pump (20 I s *) and i t can be baked separately -62-ca) manipulator I o n g u n (b) GAS LINE e o v s i p J EXPTAL. tHAMBER 200 l/s I P S.P. T.S.P S.P F igure 3.1.: (a) Schematic o f the Va r i an FC12 UHV chamber. (b) D iagramat ic r e p r e s e n t a t i o n o f the pumping system: IP = Ion Pump; TSP= T i tan ium Sub l imat ion Pump; SP = So rp t i on Pump. -63-( a ) ( b) Figure 3.2: Crystal ( c ) (a) Schematic diagram of the electron optics used for LEED experiments. (b) Diagram showing sample mounted on a tantalum supporting r i n g (c) Electron bombardment sample heater. Hatched.lines represent s t a i n l e s s s t e e l parts while the s t i p p l e pattern indicates the ceramic i n s u l a t o r . -64-from the main chamber. The objective here i s to l i m i t the amount of impurities i n the admitted gases to very low proportions i n the main chamber. Details of pumping methods, measurement of pressure and associated techniques are given i n reviews by Hobson [88], Lange [89] and Tom [90]. The sample manipulator (Varian 981-2528) holds the c r y s t a l sample and enables the c r y s t a l to be translated as well as rotated both about the axis 1" of the chamber (to enable the sample which i s o f f - s e t by 2^ to be directed to d i f f e r e n t f a c i l i t i e s ) and about an axis i n the h o r i z o n t a l plane (to enable the beam from the electron gun to make d i f f e r e n t angles of incidence (9) with respect to the c r y s t a l ) . The sample holder has f a c i l i t i e s f o r electron bombard-ment heating (figure 3.2(c)); the temperature of the c r y s t a l i s measured with a Pt/13%Rh-Pt thermocouple junction i n contact with the sample. The electron gun (Varian 981-2125) produces an electron beam by therm-i o n i c emission from a hot tunsten cathode; these electrons are accelerated and collimated through anode p l a t e s . The t y p i c a l incident beam used for LEED i n t h i s work (energy range 30-230 eV) has a current of about 1 yA, and a beam diameter at the sample of ^ 0.75 mm. The same gun was used for Auger analysis at a t y p i c a l energy of 1 keV and current of 10 yA. Reviews of the design and technology of low voltage electron guns includes those by Rosebury [91] and Kohl [92]. The electron optics (Varian 981-0127) (figure 3.2a) consists of a hemi-sph e r i c a l phosphor screen and four concentric grids each of -80% transparency; the sample i s positioned at the common centre of curvature of the grids and screen for LEED. In the usual mode of operation, the specimen and the gri d closest to the sample are grounded to ensure that electrons t r a v e l through an -65-e l e c t r o s t a t i c a l l y f i e l d - f r e e space between the sample and the optics Cthe f i n a l anode of the electron gun i s also grounded). The second and the t h i r d grids are connected together and are held at a p o t e n t i a l which i s close to that on the cathode i n the electron gun; the objective i s to stop those electrons which have l o s t energy on i n t e r a c t i n g with the sample, while permitting only the e l a s t i c a l l y scattered electrons to pass through. The : fourth g r i d i s earthed. The e l a s t i c a l l y scattered electrons, a f t e r penetra-t i n g t h i s g r i d , are accelerated through about 5 keV onto the phosphor screen, where each beam d i f f r a c t e d from an ordered c r y s t a l surface shows up as a bright spot. The whole d i f f r a c t i o n pattern on the screen can be observed d i r e c t l y through the glass window and photographed. Another accessory needed for the LEED experiment i s the sputtering gun (Varian 981-2043) for cleaning the c r y s t a l by ion bombardment. The chamber i s surrounded by three orthogonal sets of square Helmholtz c o i l s to reduce the r e s i d u a l magnetic f i e l d to a l e v e l where i t s e f f e c t on the motion of electrons being studied i s minimized. 3.1 fh) Crystal Preparation The experiments reported i n t h i s thesis involve surfaces of rhodium cut from two sources of s i n g l e c r y s t a l ; one was purchased commercially [93] (99.99% p u r i t y ) , the other was provided by another laboratory [94]. To s t a r t the preparation process, the s i n g l e c r y s t a l i s oriented to the required sur-face plane by the Laue X-ray b a c k - r e f l e c t i o n technique [95] and cut by spark I I I I erosion ( Agietron , AGIE, Switzerland). To correct for small deviations of o r i e n t a t i o n from the desired c r y s t a l face, the c r y s t a l s l i c e i s mounted i n -66-it I I a c r y l i c r e s i n ( Quickmount Fulton M e t a l l u r g i c a l Produce Corp., USA) and po l -ii ished with 5, 3 and 1 micron diamond paste on a p o l i s h i n g wheel ( Universal I I Polisher , Micrometallurgical Limited, T h o r n h i l l , Ontario.). A f t e r t h i s process, i t i s necessary to check again that the f i n i s h e d surface s t i l l has the required c r y s t a l l o g r a p h i c plane. This i s done by plac i n g the c r y s t a l on the Lau^ X-ray diffractometer so that the desired plane i s perpendicular to the X-ray beam; the whole goniometer and c r y s t a l assembly i s then trans-ferred to an o p t i c a l bench where a Ne-He laser beam i s direc t e d perpendicularly onto the surface and the angle of r e f l e c t i o n i s detected. This provides a te s t of whether the phys i c a l surface coincides with the required c r y s t a l plane. 1° Generally we aim to have the surface oriented to within — of the desired c r y s t a l plane. At t h i s stage the back of the sample i s spot welded onto a supporting tantalum r i n g (figure 3.2(b)), which i n turn i s mounted onto the manipulator. The sample and manipulator i s then placed i n the vacuum chamber, the l a t t e r i s closed and the chamber i s pumped down to a base pressure of -1x10 ^ t o r r a f t e r the standard out-gassing processes. AES indicates that sulphur, phosphorus and carbon are the impurities generally present i n the rhodium c r y s t a l s used i n our experiments; no sub-s t a n t i a l amounts of boron (Auger peak at 180 eV) has been detected although some other research groups [96,97] have reported appreciable amounts of t h i s impurity i n t h e i r rhodium samples. The cleaning processes are generally per-- formed by cycles of heat treatment (700-1000°C for 10-60 min.) to drive most bulk impurities to the surface, and argon ion bombardment to sputter o f f the impurities at the surface. A l l impurities except carbon, can be removed from -67-Figure 3.3: a l -68-rhodium surfaces by argon ion bombardment ( t y p i c a l l y 10 ^ t o r r of Ar at 0.1-1 microamps and ~1 keV for 10-30 min.). Immediately a f t e r sputtering, the •carbon Auger s i g n a l (282 eV) always showed a r e l a t i v e increase; t h i s appears to be associated with the low sputtering cross-section of carbon. However, a f t e r annealing at 700°C for a few minutes, AES indicates that the l e v e l of carbon contamination on the surface i s reduced (presumably by back d i f f u s i o n into the bulk) and LEED indicates that the surface has become ordered again. In preliminary studies, Auger spectra o f the clean Rh(llO) surface were studied as a function of c r y s t a l temperature (figure 3.3), and i t was found that below 300°C carbon d i f f u s e s to the surface whereas above t h i s c r i t i c a l temperature carbon apparently d i f f u s e s back into the bulk. Further general discussions on the preparation of clean surfaces are given i n reviews by Farnsworth [98], Bauer [99] and Jona [100]. 3.1 (c) Detection of Surface Impurities Surface impurities were detected i n t h i s work by means of Auger electron spectroscopy using the LEED optics as a retarding f i e l d analyzer [16,17]. Auger electrons of c h a r a c t e r i s t i c energies are present as small peaks super-imposed on the high (but r e l a t i v e l y constant) background of the intermediate regions of N(E) vs E curve (figure 1.2), and these peaks can be enhanced by e l e c t r o n i c d i f f e r e n t i a t i o n [116], With reference to f i g u r e 3.4, the f i n a l anode of the gun, the sample, the f i r s t and fourth grids are grounded as for the normal LEED experiment, but for detecting Auger electrons the retarding p o t e n t i a l applied on the two middle grids has a small modulating voltage AV=Vsinu>t ( t y p i c a l values of V used i n these experiments are <10 eV). With t h i s modulating voltage, the t o t a l current c o l l e c t e d on the screen (held at -69-JElectron Gun 1 Q u n control V «sin ut r 3 0 0 v Neat rail ser Lock- in Amp. sin Qt Freq. x1/2 sin 2wt X-Y Plotter Scope Figure 3.4: Schematic diagram of LEED optics used as a retarding f i e l d analyzer for Auger electron spectroscopy: MCA = multichannel analyzer. -70-Table 3.1: Observed and calculated Auger t r a n s i t i o n energies f o r rhodium. Observed Relative C a l c u l a t i o n Assignment (a) (b) (c) (d) Ce) Intensity % Ca) Cf) (f) 144 145 10 145.0 M4M 1 N 1 176 174 165 170 175 7 174.0 M N N T l 2,3 207 208 200 200 210 10 208.0 M N N 5 2,3 2,3 223 226 222 222 227 27 221.5 M N N 5 1 4,5 255 260 256 256 259 55 255.5 M N N m51 2,3N4,5 302 306 302 302 303 100 303.0 M N N 5 4,5 4,5 (a) t h i s work Cb) Grant and Haas [102] (c) Palmberg et a l . . [36] Cd) Castner et a l . [96] (e) Chan et a l . [118] (f) Coghlan and Clausing [103] -71-a p o s i t i v e p o t e n t i a l of about 300 eV) i s modulated. Using a l o c k - i n amplifier the components of the current corresponding to the f i r s t and second harmonics (frequencies U J and 2OJ respectively) are r e a d i l y i d e n t i f i e d . A p l o t of the •amplitude of these harmonic components as a function of the retarding energy E produces the secondary electron d i s t r i b u t i o n N(E) (figure 1.2) and i t s f i r s t d e r i v a t i v e dN(E)/dE r e s p e c t i v e l y [ 3 l ] . The t y p i c a l Auger spectrum shown i n figure 1.10 i s p l o t t e d i n dN(E)/dE form. T h e o r e t i c a l l y , the s e n s i t i v i t y of the spectra measured by t h i s method i s approximately 1% of a monolayer [l6,17]. Higher s e n s i t i v i t i e s to impurities are possible when Auger spectra are measured with a c y l i n d r i c a l mirror analyzer [101]. Such an analyzer was not a v a i l a b l e at the time the experimental work reported i n t h i s thesis was done. Measured peak energies and r e l a t i v e peak heights for the Auger spectrum of rhodium are summarized i n Table 3.1. The v a r i a t i o n s i n peak energies from other published measurements must be a t t r i b u t e d to errors i n the energy scale and to the lack of an appropriate contact p o t e n t i a l correction; also uncer-t a i n t i e s are i n e v i t a b l y increased for low-intensity peaks. Energies calculated by Coghlan and Clausing [103] for free atoms with an i o n i z a t i o n correction are also l i s t e d i n the t a b l e ; these values are h e l p f u l f o r guiding the assignment to p a r t i c u l a r Auger t r a n s i t i o n s . 3.1 (d) LEED Intensity Measurements D i f f r a c t e d beam i n t e n s i t i e s i n LEED have most often been measured either d i r e c t l y as d i f f r a c t e d beam currents with a moveable Faraday cup c o l l e c t o r inside the chamber [104] or i n d i r e c t l y as the brightness of spots on the -72-phosphor screen with an external spot photometer [105]. A variant of the l a t t e r approach i s the photographic technique introduced by S t a i r et a l . [106] and developed further by Frost et a l . [107], who employed a computer-controlled Vidicon camera to analyze the photographic f i l m and thereby produce experi-mental 1(E) curves. This l a t t e r procedure has been used i n the present work. B a s i c a l l y photographs of the LEED screen are taken at a series of electron energies and measurements are made of the integrated o p t i c a l d e n s i t i e s f o r the d i f f r a c t e d spots on the f i l m negatives. Assuming the measured o p t i c a l density (D) for a spot i s proportional to the amount of l i g h t which caused the darkening, and hence to the associated electron f l u x which h i t s the screen, then D divided by the incident electron current i s proportional to the d i f f -racted beam i n t e n s i t y . Such measurements i n e v i t a b l y give r e l a t i v e beam i n t e n s i t i e s . The LEED patterns displayed on the phosphor screen were photographed through the window of the vacuum chamber using a Nikon F2 35 mm camera with an 85 mm f l . 8 lens and a K2 extension r i n g . Photographs were taken generally for the range of incident beam energies 30-250 eV i n 2 eV i n t e r v a l s using fixed exposures of 1 s at f4, the incident current and energy being recorded for each photograph. Using a motor-driven unit to wind the f i l m and a 250 exposure f i l m back, LEED patterns could be photographed over t h i s energy range i n less than 5 minutes. A f t e r taking a set of photographs, the surface " p u r i t y was r o u t i n e l y checked with AES to assess whether any contamination occurred during data c o l l e c t i o n . -73-Standard Kodak Tri-X emulsion f i l m was used and the f i l m was processed i n a continuous length i n Acufine developer at 73 F for 7 minutes. The photographic negatives were analysed with the system indicated i n fi g u r e 3.5. The v i d i c o n camera and associated e l e c t r o n i c s comprise part of the Computer Eye System (Spatial Data System Inc.) which was interfaced to a mini-computer (Data General Nova 2). The f i l m held on the l i g h t t able i s scanned c o n t i -nuously by the camera and the image i s displayed on the TV monitor i n a 512x 480 (xy) array. The i n t e n s i t y (z value) o f any element of the image may be sampled by t r i g g e r i n g the d i g i t i z e r with appropriate i n s t r u c t i o n s from the computer. The p r o f i l e r displays d i r e c t l y on the monitor the v a r i a t i o n of i n t e n s i t y along any selected v e r t i c a l l i n e of the image. The j o y s t i c k controls the p o s i t i o n (coordinates) of the f l a s h i n g spot on the TV monitor, and i s used to s t a r t the analysis [107] by pointing at the spot to be analysed. Assuming a Gaussian d i s t r i b u t i o n for the spot i n t e n s i t y , the background i n t e n s i t y ( z^ a cj <.) ^ s estimated by averaging the z-value of a l l elements l y i n g i n an annulus of mean radius 2a (where 2 a i s the width at h a l f maximum of the i n t e n s i t y d i s t r i b u t i o n ) . The i n t e g r a t i o n procedure involves summing a l l the values o f ( z~ z\ i a c] () within the c i r c l e of radius 2 a and t h i s value i s divided by the incident beam current to give a measure of the d i f f r a c t e d beam i n t e n s i t y . After the i n t e g r a t i o n , the coordinates of the i n t e n s i t y maximum are determined and stored as the new s t a r t i n g coordinates for the next frame. Since the area scanned for each spot always includes the p o s i t i o n of that spot on the next frame, the computer can automatically follow each spot as i t moves toward the centre of the screen with increasing energy. The whole analysis of each spot takes less than 30 seconds and the 1(E) curves may be displayed on an o s c i l l o s c o p e and p l o t t e d on an xy recorder. -74-vidicon . T comero film V ' transport light table sconner digitiser interfoce novo 2 computer cossette drive teletype joystick scope xy plotter TV monitor profiler Figure 3.5: Schematic diagram of the apparatus used to analyse the photographic negatives of LEED patterns. -75-3.2 Structural Determinations of Low Index Surfaces of Rhodium 3.2 (a) Previous LEED Intensity Calculations for Rhodium Surfaces Watson et a l . [43,44,108,109] analysed measured 1(E) curves from low index surfaces of rhodium with m u l t i p l e - s c a t t e r i n g c a l c u l a t i o n s and t h e i r s t r u c t u r a l conclusions are summarized i n Table 3.2. These c a l c u l a t i o n s used two types of atomic p o t e n t i a l : i ) The s e l f - c o n s i s t e n t band structure p o t e n t i a l provided by Moruzzi, Janak and Williams [110]; t h i s p o t e n t i a l was designated V^^. i i ) The superposition p o t e n t i a l c a lculated f or the ce n t r a l atom i n a R h ^ c l u s t e r by a l i n e a r superposition of atomic charge d e n s i t i e s . This p o t e n t i a l was designated ^£^13' With reference to Table 3.2, Watson et a l . obtained discrepancies bet-ween the two atomic p o t e n t i a l s which res u l t e d i n d i f f e r e n t geometrical s t r u -ctures and inner p o t e n t i a l values reported for the same surface. Generally a band structure p o t e n t i a l i s l i k e l y to be preferred [60] although i t has been suggested [ 6 l ] that the superposition p o t e n t i a l can produce very s i m i l a r r e s u l t s to the band structure p o t e n t i a l for the purpose of LEED crystallography. Watson et a l . [43] supported t h i s suggestion i n a determination of the geometrical structure of the C u ( l l l ) surface. However f o r rhodium, upon evaluating the l e v e l o f agreement between experimental and calculated 1(E) curves, both with v i s u a l analyses and r e l i -a b i l i t y i n d i c e s , Watson et a l . were unable to select one of these atomic po t e n t i a l s as being preferred. This thereby l e f t s i g n i f i c a n t u ncertainties i n the d e t a i l s of the structures of the Rh(100) and (111) surfaces. One of the objectives of my i n i t i a l research was to perform further studies on these surfaces i n order to elucidate t h i s problem. -76-Table 3.2: Structural determination of low index surfaces of rhodium. ( Watson et a l . ) ^JW VRh v vRhl3 Surface Ad%±c d W V ie or v (eV) f r Ad%le d (%) V ie or v (eV) r r Rh(100) -1.8+1.0 -19.610.8 0.17 2.510.9 -11.510.7 0.16 Rh(lll) -4.210.5 -18.610.5 0.16 -0.710.8 -11.310.7 0.12 Rh(llO) — — -- -2.511.2 -11.210.6 0.10 Rh(llO) — -- -- -1.011.2 -10.510.8 0.09 Table 3.3: Structural determination of low index surface of rhodium. '' ( This work ) ^iJW _Rh . V Rhl3 Surface Ad%le d (%) V le or v (eV) r r Ad%le d (%) V ie or v (eV) r r Rh(100) 1.010.9 -12.810.4 0.09 0.511.2 -14.010.6 0.09 Rh(lll) -1.610.8 -11.210.6 0.08 — — Rh(110) -3.311.5 -10.910.8 0.12 — Rh(110) -0.510.7 - 9.610.9 0.09 -- -- --- 7 7 -3.2 (b) Further Studies In t h i s work, multiple s c a t t e r i n g c a l c u l a t i o n s were repeated for normal incidence on the (100), (110) and (111) surfaces of rhodium, and the c a l c u l -ated LEED i n t e n s i t i e s were compared with the experimental 1(E) curves p r e v i -ously produced by Watson et a l . for the (110) and (111) surfaces. Although a new set of experimental data for normal incidence on Rh(100) was obtained, and used i n the comparison i n t h i s work, these new experimental 1(E) curves did not show any s i g n i f i c a n t deviations from the previous data [ i l l ] . P r i o r to making the multiple s c a t t e r i n g c a l c u l a t i o n s , the c a l c u l a t i o n of phase s h i f t s from the two d i f f e r e n t atomic p o t e n t i a l s was completely re-investigated. In doing t h i s an error.was detected i n the value used prev-iously for the p o t e n t i a l at the muffin-tin radius^, and t h i s r e s u l t e d in an M J W incorrect set of phase s h i f t s associated with the p o t e n t i a l . A f t e r making the c o r r e c t i o n for the band structure p o t e n t i a l , a new set of phase s h i f t s was generated for d i f f e r e n t Jl to a maximum value of 7 (figure 3 . 6 ) . These new phase s h i f t s values generated from the band structure p o t e n t i a l of Moruzzi, Janak and Williams are designated as [ V ^ ^ ] to avoid confusion with MJW the erroneous V_, of Watson et a l . Rn With the corrected phase s h i f t s from the band structure p o t e n t i 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 for normal incidence were repeated for the (100), (110) and (111) surfaces assuming regular packing arrangements as i n -- dicated prevously [43,44,108,109]. The non-structural parameters were kept unchanged from those used i n the previous work. S p e c i f i c a l l y , the surface atomic v i b r a t i o n s were assumed to be i s o t r o p i c and layer-independent, the *The p o s s i b i l i t y of a numerical error was f i r s t suggested to K.A.R. M i t c h e l l by J . J . Rehr (Univ. of Washington). The actual error was l a t e r detected by P.R. Watson and W.T. Moore while c a l c u l a t i n g some phase s h i f t s f o r zirconiu: -78-Energy ( Ry ) Figure 3.6: Energy dependence of rhodium phase s h i f t s (£=0-7) for the • , r w M J W - | p o t e n t i a l LVR, J. -79-surface Debye temperature being taken as 406 K ( i . e . /0.7 times, the bulk value of 480 K [112,115]). The imaginary part o f the inner p o t e n t i a l ( v 0^) 1/3 was equated to -1.17E , guidance being provided by the widths of primary Bragg-type peaks i n experimental 1(E) curves according to equation (2.4) and the energy dependence proposed i n equation (2.5). A l l the i n t e r l a y e r spacings below the second rhodium layer were fixed at the bulk values ( i . e . 1.9022 A f o r Rh(100), 1.3452 A f o r Rh(110) and 2.1960 A f o r R h ( l l l ) ) . The topmost i n t e r l a y e r spacings ( i . e . the perpendicular distance between the f i r s t and second rhodium layers) were allowed to vary from a 10% contraction from the bulk value to a 5% expansion i n increments o f 2.5% for the (100) and (111) surfaces, while for the (110) surface c a l c u l a t i o n s were made with the topmost spacing varying from a 12.5% contraction to a 2.5% expansion. 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 over the energy range of 30-250 eV were done for the (10), (01), (11), (02) and (12) beams for a l l three surfaces (beam notations are i l l u s t r a t e d i n figure 3.7). The c a l c u l a t i o n s u t i l i z e d the RFS method for the (100) and (111) surfaces, whereas the layer-doubling method was used for the (110) surface to avoid any p o s s i b i l i t y that the r e f l e c t e d i n t e n s i t i e s might not converge f o r the smaller i n t e r l a y e r spacings. I n t e n s i t i e s c a lculated with the corrected phase s h i f t s from the band structure p o t e n t i a l for the (100) surface were compared with the new experi-mental 1(E) curves. V i s u a l analysis suggested that the best correspondence - occurred when Ad% i s between 0 and 2.5% (here Ad% indicates the topmost i n t e r l a y e r spacing (d) expressed as the percentage change from the bulk value d ( i . e . Ad% = [(d-d )/d ]xl00). The analysis with the r e l i a b i l i t y o v " - ^ o o real space RhdOO) Rh(110) O C X ) 666 066 Rh(H1) 10 01 •80-9, 02 01 •00 22 • 11 00 11 reciprocal space 12 10s 12 111 1 1 1 1 10 22 21 20 - 9 , 02 22 21 • g v 20 X 20" 21 22 Figure 3.7: (a) Schematic diagrams of the (100), (110) and (111) surfaces of rhodium. The dotted c i r c l e s represent rhodium atoms i n the second layer, (b) The corresponding LEED patterns i n d i c a t i n g the beam notation as used i n text. -81-index proposed by Zanazzi and Jona [45] indicated that the minimum value for r was 0.085 and occurred when Ad% = 1.0±0.9% and V = -12.8±0.4 eV. r or To assess the correspondence between the two p o t e n t i a l s V p ^ g and [ V R j ^ ] , another comparison with r ^ was made between the same experimental 1(E) curves and the curves calculated from [108]. This time the minimum value of r ^ was again 0.085, although for the conditions Ad% = 0.5±1.2% and V = -14.0±0.6 eV. These two r e s u l t s , which are summarized i n Table 3.3, or are in contrast to the previous report of Watson et a l . (Table 3.2). Also summarized i n Table 3.3 are the conditions for minimum r ^ from comparisons MJWT of i n t e n s i t i e s c a l c ulated using [ V R n ] with one set of experimental data for the (110) and (111) surfaces; each set of experimental data covers 5 beams in the energy range 30-200 eV. Corresponding r e s u l t s from the p o t e n t i a l VRhl3 obtained previously by Watson et a l . are in Table 3.2. Comparisons of our new r e s u l t s obtained from the corrected phase s h i f t s r MJWn from the band structure p o t e n t i a l LV R h J for three low-index surfaces of rhodium (Table 3.3) with those obtained previously from the superposition p o t e n t i a l V j ^ j g (Table 3.2), allows the conclusion that the values of Ad% and V given by the two potentials are equal to within the indicated uncertainties for each set of experimental measurements. This suggests that the two rhodium pot e n t i a l s are equivalent f o r the purpose of LEED crystallography, and provides support for the suggestion [ 6 l ] that the superposition p o t e n t i a l s from c l u s t e r c a l c u l a t i o n s can be useful when s e l f - c o n s i s t e n t band structure p o t e n t i a l s are unavailable. This s i t u a t i o n f or the rhodium surfaces i s now consistent with that found previously for C u ( l l l ) [43]. -82-3.3 Studies with the R e l i a b i l i t y Index of Zanazzi and Jona 3.3 (a) Introduction The basic approach f o r surface crystallography with LEED involves varying s t r u c t u r a l and non-structural parameters i n the m u l t i p l e - s c a t t e r i n g c a l c u l a -tions i n order to f i n d the best correspondence between calculated and experi-mental 1(E) curves for a l l d i f f r a c t e d beams [113]. At present the high cost of the m u l t i p l e - s c a t t e r i n g c a l c u l a t i o n s i n h i b i t s a f u l l v a r i a t i o n of non-s t r u c t u r a l parameters to maximize the agreement i n these comparisons, and so far only V q i > has been subjected to much v a r i a t i o n [57]. In part t h i s has been because of a common f e e l i n g that the other non-structural parameters do not have a strong e f f e c t on determined geometries. Thus, the usual procedure i n LEED crystallography involves f i n d i n g a p l a u s i b l e choice of non-structural parameters (e.g. V Q ^ , 8^) at the s t a r t of the c a l c u l a t i o n and keeping these parameters f i x e d from then on [56]. This philosophy i s tested here, p a r t i e -u l a r l y with the r e l i a b i l i t y index r ^ suggested f o r LEED by Zanazzi and Jona [45]. The v a r i a t i o n of non-structural parameters appears to provide a f a i r l y stringent t e s t of r ^ . Hence one objective of t h i s work i s to assess where the use of r ^ for the v a r i a t i o n of non-structural parameters i s able to give r e s u l t s s i m i l a r to a v i s u a l analysis and where i t does not. The content of t h i s section has already been published along with some supplementary obser-vations of P.R. Watson and S.J. White [150]. 3.3 (b) Relations between R e l i a b i l i t y Index and the Imaginary Potential The imaginary part of the inner p o t e n t i a l (V ^) provides a phenomeno-l o g l c a l d e s c r i p t i o n of the i n e l a s t i c s c a t t e r i n g of electrons by a s o l i d ; the l i f e times of electrons of well defined energy i n the s o l i d place a -83-r e s t r i c t i o n (via the uncertainty p r i n c i p l e ) on peak widths i n 1(E) curves according to equation (2.4). Increase i n V . corresponds to a reduction i n 01 the proportion of the e l a s t i c a l l y scattered electrons and to a broadening of peaks i n the 1(E) curves. The i n i t i a l s e l e c t i o n o f a value of V f o r rhodium was made ( u t i l i z i n g o i equations (2.4) and (2.5)) from the measured widths of kinematical peaks i n experimental 1(E) curves; on t h i s basis a p l a u s i b l e expansion for V Q i i s 1/3 -aE with a equal to about 1.17. A point of i n t e r e s t here i s to see whether changes i n a would modify conclusions on the geometries of rhodium surfaces, and whether the r e l i a b i l i t y - i n d e x analysis would i n d i c a t e that ct=1.17 i s the most appropriate value of a. In order to examine t h i s , further multiple-s c a t t e r i n g c a l c u l a t i o n s were made for normal incidence on the (111) surface o f rhodium at a ser i e s of values of a, s p e c i f i c a l l y 1.17, 1.47, 1.76, 2 .05 and 2.34, with a l l other non-structural parameters f i x e d at the values used previously i n section 3.2 (b). The (111) surface was convenient for t h i s study, since the c a l c u l a t i o n s required a comparatively small number of beams and the RFS method was applicable. 1(E) curves f o r the (01) beam for the f i v e values of a with a 2.5% contr-action o f the topmost layer are shown i n f i g u r e 3.8 together with the experi-mental data. The main features o f each i n d i v i d u a l curve are maintained, a l -though increase i n a gives a general lowering of i n t e n s i t i e s and, most s i g n i -f i c a n t l y , a broadening o f the peaks. V i s u a l evaluations of a l l d i f f r a c t e d beams suggested the best agreement between experimental and calculated 1(E) curves occurred when a i s i n the range 1.47 to 1.76. These comparisons were -84-(01) B E A M E N E R G Y / e V Figure 3.8: The experimental I(E) curve for the (01) beam at normal incidence from the R h ( l l l ) surface compared with f i v e corresponding curves c a l c u l a t e d with the p o t e n t i a l C ^ ^ l and Ad% = -2.5% for the parameter ct varying from 1.17 to 2.34. -85-also made with the numerical r e l i a b i l i t y index, and conditions f o r minimum r f o r each value of a are summarized i n Table 3.4. These r e s u l t s i n d i c a t e r that v a r i a t i o n of a has only a minor e f f e c t on the determined topmost i n t e r -layer spacing, and t h i s supports the common assumption that v a r i a t i o n of is not es s e n t i a l i n LEED crystallography. It i s s a t i s f y i n g also that the i n s e n s i t i v i t y of geometrical structure to V i s recognized by r ^ . Neverthe-less i t must be noted that even though c l o s e l y s i m i l a r geometrical structures are indicated by the d i f f e r e n t values of a, the values of r ^ at the d i f f e r e n t minima are not equivalent. The lowest r ^ value corresponds to a close to 1.76, and t h i s suggest that the i n i t i a l choice of 1.17 may not be optimal. Both v i s u a l and r-index evaluations are consistent i n i n d i c a t i n g a i s larger than 1.17 and t h i s supports the use of the index r . On the other hand, values of a larger than 1.17 seem less consistent with determining V from equation 2.4. The values of r ^ reported i n Table 3.4 are unusually low, e s p e c i a l l y those for the higher values of a. The trends found did not seem consistent with the o r i g i n a l conclusions of Zanazzi and Jona, and we wondered whether the tendency for low values of r ^ to be found for high a could be an art e f a c t associated with the value of p being fixed at 0.027 i n the c a l c u l a t i o n of (r ). i n equation (2.40). According to Zanazzi and Jona [45], t h i s value of r i p was obtained by averaging ( r r ) ^ f ° r matching random pairs of experimental II and calculated 1(E) curves. One uncertainty was whether complexity of I I structure was f u l l y b u i l t into the scheme of Zanazzi and Jona. In general one would expect that an experimental 1(E) curve that contains a l o t of -86-Table 3.4: Conditions f o r best agreement between experimental 1(E) curves at normal incidence for R h ( l l l ) and curves calculated with the p o t e n t i a l [vJJnW] according to the r e l i a b i l i t y indices r and r for d i f f e r e n t values of a. r m 1.17 1.47 1.76 2.05 2.34 Ad% (%) V o r (eV) r r r m -1.6±0.8 -11.210.6 0.080 0.985 •2.510.5 -11.810.7 0.042 0.510 -2.310.6 -11.710.6 0.035 0.430 -2.310.5 -11.610.7 0.037 0.440 -2.010.6 -11.010.8 0.041 0.490 -87-structure would be more d i f f i c u l t to match to calculated curves than one with less structure, and therefore i n s e t t i n g up a many-beam r e l i a b i l i t y - i n d e x perhaps the former should have a r e l a t i v e l y greater weighting than the l a t t e r One approach to t h i s i s to allow the value of p to vary f o r each experimental curve. In order to make an i n i t i a l assessment of whether such e f f e c t s could be relevant to the trends of r ^ with a shown i n Table 3.4, we replaced p i n equation (2.40) with a new quantity r ( s t . l i n e , e x p t ) E^ ' obs 1 max This quantity varies with each experimental 1(E) curve according to the amount of structure i t involves. Equation (3.1) i s obtained from equation (2.36) by comparing an experimental 1(E) curve with a s t r a i g h t l i n e corresponding to I. , = I.' = i " , = 0. Using r, «... instead of p i n equation i , c a l i , c a l i , c a l & (st.line,expt) (2.40), we then set up a new o v e r a l l reduced r e l i a b i l i t y index designated as r . The values of r f o r the v a r i a t i o n of a values are also summarized i n m m Table 3.4. However, i t turned out for the case considered here that mini-mizing r ^ gave i d e n t i c a l values of Ad% and to those found by minimizing r ^ ; numerical values of the two indices are d i f f e r e n t , but to a good approxi-mation corresponding values of r ^ can be obtained by d i v i d i n g values of r ^ by 12.1. This observation does not support the p o s s i b i l i t y that the low ^values of r ^ found for high a (hence high V ^ ) was associated with the constant value of p used i n equation (2.40). -88-Further i n v e s t i g a t i o n suggested that the high value of a needed f or better matching between calculated and experimental 1(E) curves appears to be associated with the way that the experimental i n t e n s i t i e s were handled i n the ana l y s i s . The i n i t i a l value of a=1.17 was obtained by considering i n d i v i d u a l l y measured 1(E) curves, whereas the experimental 1(E) curves a c t u a l l y used i n the comparisons with the calculated 1(E) curves were ave-raged over appropriate sets of beams which are expected to be, and to a good approximation are, symmetrically equivalent. However, minor errors i n the I I experiment [107,108] can lead to corresponding peak posit i o n s i n equivalent I I sets of beams being s h i f t e d s l i g h t l y (e.g. by 1 or 2 eV) from the mean values and t h i s i n e v i t a b l y leads to some broadening of structure i n the averaged 1(E) curves. Upon i n v e s t i g a t i n g the averaged experimental 1(E) curves, a choice of a as suggested by equations (2.4) and (2.5) for the R h ( l l l ) surface i s 1.65. This value i s i n reasonable agreement with the conclusions noted above from the v i s u a l evaluation and the r - f a c t o r a n a l y s i s . These studies i n d i c a t e the following conclusions: 1) Determined surface geometrical structure i s i n s e n s i t i v e to changes i n V values. This supports the usual approach of keeping V Q i f i x e d i n the multiple sc a t t e r i n g c a l c u l a t i o n s , and of choosing s u i t a b l e values of V from equation (2.4). 2) The index r ^ proposed by Zanazzi and Jona i s consistent with a v i s u a l analysis for i d e n t i f y i n g values of which optimize agreement between ex-perimental and calculated 1(E) curves. - 3) Further improvements are needed i n the experimental measurements for ensuring that 1(E) curves from symmetrically-related beams r e a l l y are equivalent. -89-3.3 (c) R e l i a b i l i t y - I n d e x and the V a r i a t i o n of Surface Debye Temperature The e f f e c t s of atomic vi b r a t i o n s are incorporated i n t o multiple scat-t e r i n g c a l c u l a t i o n s by means of temperature-dependent atomic s c a t t e r i n g factors i n v o l v i n g the Debye temperature (8^) as indicated i n equations (2.9)- (2.10). S t r i c t l y the atomic vi b r a t i o n s are expected to be layer dependent and to decrease into the bulk [114]. However, most LEED studies have used a sing l e e f f e c t i v e Debye temperature (6^ e f f ) f ° r a ^ layers probed by the analysed electrons. In p r i n c i p l e , a better, although s t i l l II II simple, p o s s i b i l i t y i s to give the topmost layer a surface value (8^ s u r f ) and to assume the second and a l l deeper layers can be characterized by the bulk value (8^ ^uUp [56]. In the previous multiple s c a t t e r i n g c a l c u l a t i o n s made so far i n t h i s t h e s i s , e n , for rhodium was taken as 480 K [115] and D,bulk J 9D e f f w a s e s t i m a t e d a s y / ° ~ ^ 9 n b u l k [112]. Although t h i s type of choice seems p l a u s i b l e , i t i s nevertheless made on i n t u i t i v e , rather than rigorous, grounds: moreover f o r assessing further the choice of 9 n -~ i t would seem b ' 6 D.surf h e l p f u l to determine the e f f e c t of i t s v a r i a t i o n on the s t r u c t u r a l conclusions, as considered for v a r i a t i o n s of V . i n section 3.3 (b). For t h i s i n v e s t i -01 gation, multiple s c a t t e r i n g c a l c u l a t i o n s f o r the R h ( l l l ) surface were made by varying 6„ . over the range of 200-600 K i n 100 K steps, a l l other 1 6 D.surf non-structural parameters being f i x e d at the values given i n section 3.2(b) (except a was r e s t r i c t e d to 1.76). Figures 3.9 and 3.10 show two d i f f e r e n t sets o f contours of r ^ pl o t t e d against 8^ -For both the contours are reasonably symmetrical about a h o r i -" D,surf J J zontal l i n e , and minimum values of r ^ are c l o s e l y indicated to correspond to the -90--Figure 3.9: Contour plot of f versus 9 n o r and V for normal incidence • r U,surr or data from R h ( l l l ) where the ca l c u l a t i o n s use the po t e n t i a l [ V M ^ W ] with a=1.76 and 6 D b u l k = 4 8 0 K. -91-200 300 400 300 600 ® D , S U R F < K ) Figure 3.10: Contour p l o t of r versus 6 D g u r f and Ad% for normal incidence data from R h ( l l l ) where the c a l c u l a t i o n s use the p o t e n t i a l -92-values V = -11.5 eV and Ad% = -2%. These values are comparable with those or r reported previously i n Tables 3.3 and 3.4 for a f i x e d value o f 6^ An unexpected feature of these p l o t s , however, i s that they point to values of 6„ _ i n the p h y s i c a l l y unreasonable ranee of being greater than 0„ , ,, D,surf r b b D.bulk ( i . e . 480 K for rhodium [115]). Within the conventional procedure for including atomic v i b r a t i o n s i n m u l t i p l e - s c a t t e r i n g c a l c u l a t i o n s [24,65], the main e f f e c t of 6^ i s to modify o v e r a l l i n t e n s i t i e s but without appreciably a f f e c t i n g s tructure i n the calculated 1(E) curves. This can be seen i n f i g u r e 3.11 where 1(E) curves calculated for the (01) beam of R h ( l l l ) with Ad% = -2.5% are p l o t t e d f or values of 6„ „ between 200 and 600 K. The most noticeable trend i s that D.surf the lower values of 9^ _ e s p e c i a l l y give r e l a t i v e l y lower calculated inten-D,surf r / & i s i t i e s at the higher energies. This contrasts with the trend observed i n the experimental 1(E) curves where r e l a t i v e l y higher i n t e n s i t i e s are found at the higher energies. This suggests that the tendency to high values of 0^ _ picked out by the use of r i s r e f l e c t i n g a r e a l trend i n the curves D.surf r J r compared, although i t i s p h y s i c a l l y unreasonable for 0^ s u r f t 0 be greater than 0^ kuik' 0 u r f e e l i n g i - s that the o r i g i n of t h i s discrepancy may be associated with general problems i n the data c o l l e c t i n g processes. Changes both i n g r i d transparency and i n the s o l i d angle presented to the camera as the spots move toward the centre of the screen for increasing energies can • caused apparent v a r i a t i o n s i n r e l a t i v e i n t e n s i t i e s . Legg et a l . [117] made corrections for these factors and demonstrated a consequent lowering in r e l a t i v e beam i n t e n s i t i e s at the lower energies. In future LEED i n t e n s i t y -93-Rh(l1l) ENERGY / eV " Figure 3.11: The experimental 1(E) curve for the (01) beam at normal incidence from the Rh(lll) surface compared with five r M J W i corresponding curves calculated with the potential LV R n J» Ad% = -2.5%, and a = 1.76 for the parameter e Q s u r f varying from 200 to 600 K. - a d -measurements we are planning to incorporate corrections for these e f f e c t s ; also i t i s possible that some refinement i n the background correction could be needed at high energies when spots crowd together i n the LEED screen. At present we f e e l that the source of discrepancy indicated by the untenable large value of 8^ - i s associated with aspects of the experimental mea-& D,surf r r surements, and although t h i s has not yet been unambiguously confirmed, two conclusions do seem secure. The f i r s t i s that the Zanazzi-Jona r e l i a b i l i t y index appears able to give a f a i r assessment of the r e l a t i v e i n t e n s i t i e s of 1(E) curves when 9_ _ i s var i e d i n the cal c u l a t i o n s (although an J D,surf appreciable s e n s i t i v i t y i n r r to r e l a t i v e i n t e n s i t i e s i n successive sections of 1(E) curves has not been recognized p r e v i o u s l y ) . Secondly, surface struc-t u r a l conclusions seem unaffected by v a r i a t i o n of 6„ i n the c a l c u l a t i o n s . D.surf Although there would be advantages i n r e f i n i n g the treatment of atomic vibrations i n LEED i n t e n s i t y c a l c u l a t i o n s [65], the evidence presented here does suggest that modifying values of 8 D s u r f i s not going to s i g n i f i c a n t l y a f f e c t conclusions about surface geometry. This suggestion i s supported by an independent analysis of the R h ( l l l ) surface with m u l t i p l e - s c a t t e r i n g c a l -culations by Chan et a l . [118]. Using 6 D b u l k as 300 K and 6 D s u r f as 250 K, Chan et a l . obtained Ad% for the topmost rhodium layer as 0±5%. Although these error l i m i t s seem rather large, nevertheless t h i s conclusion i s con-s i s t e n t with our determination of the R h ( l l l ) surface (Table 3.2). Generally - we f e e l , for the present stage of development of LEED crystallography, the mul t i p l e - s c a t t e r i n g c a l c u l a t i o n s might just as well continue to use 8^ g ^ obtained from the experimental measurements [25] or a l t e r n a t i v e l y 6^ s u r f f o r the topmost layer s p e c i f i e d as a ce r t a i n f r a c t i o n of 8^ ^ u i ^ C ^ 2 ] . -95-3.4 Studies of Adsorption of some Gaseous Molecules on Rhodium surfaces 3.4 (a) Bibliography of Overlayer Structures on Rhodium Surfaces The properties of well-defined surfaces of rhodium have been less extensively investigated than those of many other t r a n s i t i o n metals, even though rhodium shows a high degree of c a t a l y t i c a c t i v i t y for many reactions [42] Table 3.5 summarizes studies where general chemisorptive properties of rhodium have been investigated with LEED. Auger electron spectroscopy was not avai-l a b l e f or monitoring surface p u r i t y i n the i n i t i a l studies by Tucker [119-122] although t h i s technique was a v a i l a b l e f o r a l l other studies reported i n Table 3.5. Much o f the work on rhodium that has emerged over the past several years has been concerned mainly with either LEED patterns or adsorption k i n e t i c s . An important objective for part of the research reported i n t h i s thesis was to determine some d e t a i l e d surface structures with LEED for adsorption on rhodium. The i n i t i a l aim was to investigate some comparatively simple structures i n v o l v i n g 0 or S adsorbed on low-index surfaces and to compare with s i m i l a r systems already investigated with LEED crystallography, for example adsorption on n i c k e l . The LEED analyses r e s u l t i n g from the adsor-p t i o n o f l ^ S on the (100) and (110) surfaces of rhodium are described i n chapters 4 and 5 r e s p e c t i v e l y . The next section reviews observations f o r the adsorption of 0^ on Rh(100), a system that was o r i g i n a l l y planned to be investigated v i a analyses of LEED i n t e n s i t i e s . Table 3.5: Surface structures reported for adsorption of small gaseous molecules on low index surfaces of rhodium. Adsorbate Rh(100) surface structure r e f . Rh(llO) surface structure r e f . R h ( l l l ) surface structure r e f . ° 2 p ( 2 x 2)-0 [a,b,f] disorder [c] ( 2 x 2 )-0 [a,e,g] c ( 2 x 2)-0 [ a , f ] c ( 2 x 4)-0 [c] ( 3 x l ) - 0 [b,f] c ( 2 x 8)-0 [c] c ( 2 x 8)-0 [b] ( 2 x 2 )-0 ( 2 x 3 )-0 ( l x 2 ) - 0 ( 1 x 3 )-0 1—I 1—1 1—1 1—1 1 1 1 1 1—1 1—1 CO c (2x2)-CO [a] (2x1)-CO [d] (/3x/3)R30°-CO [a] - hexagonal overlayer [a] c ( 2 x 2)-C [d] (2x2)-CO [a,e] (4x1)-CO CO. c (2x2)-CO [a] (/3x/3)R30°-CO [a] Z (2x2)-CO [a,e] NO c ( 2x2)-N0 [a] c(4x2)-NO c(2x2)-NO [a] [a] H 2S p ( 2 x 2)-S c ( 2 x 2)-S [ f ] [ f ] c ( 2 x 2)-S [ f ] — — — [ a ] - Castner et a l . [96]; [ b ] - Tucker [119]; [ c ] - Tucker [l20,12l]; [d]- Marbrow and Lambert [97]; [ e ] - Grant and Haas [102]; [ f ] - This work [123,124]; [ g ] - Weinberg et a l . [127].. -97-3.4 (b) Adsorption of 0^ on Rh(100) The sample used i n t h i s study was cut from the s i n g l e c r y s t a l provided by Tucker [94] , and i t was previously used by Watson et a l . f o r a LEED analysis of the clean Rh(100) surface [108]. P r i o r to s t a r t i n g the adsorption 1° work, the surface was repolished and checked to ensure that i t was within — of the (100) plane. A f t e r mounting and i n s t a l l i n g i n the vacuum chamber, the sample was cleaned according to the procedures described i n section 3 .1 , and annealed u n t i l the LEED pattern exhibited a sharp ( l x l ) pattern with low back-ground i n t e n s i t i e s . The sample was heated to 300°C before high p u r i t y 0^ (99.99%,Matheson) was introduced into the vacuum chamber at a pressure of 10 t o r r . After 5 minutes a sharp (3x1) LEED pattern corresponding to two di f f e r e n t domains was observed, and an Auger spectrum taken a f t e r the formation of t h i s pattern f a i l e d to detect the presence of any impurities. The Auger peaks of oxygen at around 510 eV could not be detected. This e f f e c t has been observed previously for oxygen adsorption on some t r a n s i t i o n metals [125,126] and i t appears to be associated with the low i o n i z a t i o n cross-section for i n i t i a t i n g the Auger process for adsorbed oxygen. A sharp ( l x l ) pattern c h a r a c t e r i s t i c of the clean Rh(100) surface can be restored (presumably by desorption of the oxygen [96,127]) upon heating at 1000°C f o r 10 minutes. After returning to the base pressure the process could be repeated with a new dose of oxygen applied under the same conditions as indicated above. Sharp (3x1) patterns could always be obtained, although on d i f f e r e n t occasions v a r i a t i o n s were found i n the domain structure. These ranged from two equally populated domains f to two unequally populated domains and even to the appearance - 9 8 -of a s i n g l e domain (figure 3 . 1 2 ) . From time to time f a i n t half-order d i f f r -acted spots were observed superimposed on the ( 3 x 1 ) pattern, but the pattern never developed into a complete ( 2 x 2 ) pattern even though the c r y s t a l was exposed to for longer periods of time. Furthermore these spots could be removed by heating at 700°C for a few seconds; then a f t e r cooling to room termperature, the LEED pattern showed only the sharp ( 3 x 1 ) pattern. A well-defined p ( 2 x 2 ) LEED pattern (figure 3 . 1 2 ) could be observed when the clean Rh ( 1 0 0 ) surface was exposed to 0 9 at 10 ^  t o r r f o r 5 minutes. An apparent, but incompletely developed, c ( 2 x 2 ) pattern could also be detected i f the c r y s t a l was l e f t i n the constant atmosphere of 0 ^ at 10 ^ t o r r f o r a further 30 minutes. This was observed as an increase i n i n t e n s i t i e s of f r a c -tional-order spots of type (^j), while the other f r a c t i o n a l - o r d e r spots showed r e l a t i v e decreases i n i n t e n s i t i e s . These r e s u l t s f o r the adsorption of oxygen on Rh ( 1 0 0 ) agree p a r t l y with e a r l i e r work done by Tucker [ 1 1 9 ] and Castner et a l . [ 9 6 ] . Tucker reported ( 2 x 2 ) , ( 3 x 1 ) and ( 2 x 8 ) patterns f o r increasing oxygen exposures, but he did not observe the c ( 2 x 2 ) pattern. Castner et a l . reported a p ( 2 x 2 ) pattern which transformed to the c ( 2 x 2 ) pattern at higher oxygen exposures, but no ( 3 x 1 ) pattern was detected i n that work over a wide range of temperature and pressure. My observation of the p ( 2 x 2 ) pattern seems broadly i n agreement with those observed i n these other two studies. Also I had some evidence, ' through f a i n t LEED patterns, for the transformation of a p ( 2 x 2 ) pattern into a c ( 2 x 2 ) pattern with oxygen exposure. One p o s s i b i l i t y f o r the discrepancies between these d i f f e r e n t studies could involve other gases (e.g. CO) being -99-I'igure 3 . 1 2 : Photographs o f some p ( 2 x 2 ) and (3><1) LEED p a t t e r n s o b s e r v e d at normal i n c i d e n c e from t h e a d s o r p t i o n o f oxygen on a Rh(lOO) s u r f a c e . (a) Rh[ 1 0 0 )-p ( 2 * 2 ) - 0 at 70 eV, (b) R h ( 1 0 0 ) - ( 3 * l ) - 0 , s i n g l e domain a t 174 eV, (c) R h ( 1 0 0 ) - ( 3 x l ) - 0 , 2 e q u a l l y p o p u l a t e d domains at 100 eV. (d) R h ( 1 0 0 ) - ( 3 * l ) - 0 , 2 e q u a l l y p o p u l a t e d domains at 152 eV. -100-d i s p l a c e d from the w a l l s o f the vacuum chamber on a d m i t t i n g oxygen to the system. U n f o r t u n a t e l y , the mass spectrometer d i d not f u n c t i o n p r o p e r l y during these experiments and so we had no independent assessment of the gases i n the chamber. However, no evidence was found f o r the b u i l d up of i m p u r i t i e s on the surface on adding oxygen to the system, although i t was again unfortunate that the r e t a r d i n g f i e l d analyzer as used at the time of t h i s work was not s e n s i t i v e enough to detect the oxygen. Nevertheless care was taken during the heat treatments t o operate under c o n d i t i o n s where carbon does not appre-c i a b l y migrate from the b u l k ; the Auger sp e c t r a confirmed t h a t carbon impuri-t i e s remained at low l e v e l s during these experiments. Two complete sets o f photographs f o r the ( 3 x 1 ) p a t t e r n s were taken on d i f f e r e n t occasions over the energy range 3 0 - 2 0 0 eV f o r normal incidence. The f i l m s were analysed to y i e l d the 1(E) curves shown i n Appendices A l and A 2 ; the f i r s t i s f o r two e q u a l l y populated domains and the second i s f o r a s i n g l e domain type only. These 1(E) curves have not yet been analysed w i t h m u l t i p l e - s c a t t e r i n g c a l c u l a t i o n s e s p e c i a l l y because we have no clues at present to the p o s s i b l e s t r u c t u r e , and some geometrical models t h a t should be t e s t e d are complex. An attack on the problem of the s t r u c t u r e o f the ( 3 x 1 ) surface would be aided by the a v a i l a b i l i t y of more d e t a i l e d experimental data, f o r example on surface coverage (from AES w i t h a c y l i n d r i c a l m i r r o r analyzer) and on p o s s i b l e oxygen bonding s i t e s from h i g h - r e s o l u t i o n e l e c t r o n energy loss . spectroscopy [ 1 2 8 ] . -101-CHAPTER 4 LEED A n a l y s i s o f Rh(100)-p(2x2)-S Surface S t r u c t u r e X -102-4.1 Introduction Knowledge of the structures adopted by atomic and molecular species adsorbed on surfaces of rhodium i s of importance for an understanding o f the c a t a l y t i c properties of t h i s metal. This chapter reports an analysis with LEED for the (2x2) structure formed by adsorbing H^S on the clean (100) surface [123]; t h i s appears to represent the f i r s t such s t r u c t u r a l analysis for adsorption on rhodium. H^S was chosen for t h i s i n i t i a l study since some st r u c t u r a l information i s ava i l a b l e for sulphur adsorption (via H^S) on other t r a n s i t i o n metal surfaces, thereby providing points of reference for assess-ing the structure of Rh(100)-p(2x2)-S. One immediate objective i s to gain information about the chemical bonding at these surfaces. 4.2 Adsorption of H 2 S on Rh(100) A clean (100) surface of rhodium with a sharp ( l x l ) LEED pattern (obtained by the procedures described i n section 3.1) was exposed to high p u r i t y H^S - 8 (Matheson) at 10" t o r r for 1 min. A f t e r pumping excess gas, the surface was annealed at 300°C for 1 min. and a sharp p(2x2) LEED pattern obtained with good contrast (figure 4.1). Auger spectra (figure 4 .2) taken a f t e r the f o r -mation of t h i s pattern indicated S as the main foreign component with Auger peak height r a t i o s 152eV(S)/302eV(Rh)=2/3. Small traces of C could also be detected, but i t s proportions were minimized by the low temperature annealing. We believe that H 2S dissoc i a t e d on the Rh(100) surface, i n part because we also obtained t h i s p(2x2)-S LEED pattern by heating the metal such that sulphur impurity segregated to the surface from the bulk. Exactly s i m i l a r - c- -d-!• i gure 4.1: Photographs of LHHD patterns observed at normal incidence from adsorption of S on Rh(100) surface. (a) Rh(100)-c(2*2)-S at 80 eV, (b) Rh(100)-p(2x2)-S at 72 eV, ( c ) Rh(100)-p(2*2)-S at 114 eV, (d) Rh(100)-p(2*2)-S at 168 eV. -104-Rh 100 200 300 Energy (eV ) Figure 4.2: Auger spectra of Rh(100) surfaces with 1.5 keV and 10 microamp beam at d i f f e r e n t stages during the preparation of Rh(100)-p(2*2)-S. -105-observations have been reported by Gauthier et a l . [129] and Demuth et a l . [130,131] in t h e i r preparations of Ni(100)-p(2x2)-S and Ni(100)-c(2x2)-S, and also by Castner et a l . [96] i n t h e i r studies of the Rh(100) surface. The 1(E) curves measured from the Rh(100)-p(2x2)-S surface obtained by the migra-t i o n of the bulk sulphur impurity agreed c l o s e l y with those prepared by H^S adsorption. This provided some tentative evidence that the adsorption of H^S on t h i s rhodium surface involves d i s s o c i a t i v e adsorption. Direct evidence for d i s s o c i a t i n g on a metal surface was provided by Keleman and Fischer s study on the Ru(100) surface with the a d d i t i o n a l techniques o f uv photo-emission and thermal desorption spectroscopy [132]. This work indi c a t e d that H^S d i s s o c i a t e d upon adsorption over the e n t i r e range of coverage. In Rh (100)-p (2x2)-S, the adsorbed sulphur atoms are held strongly to the surface and could be removed only by extensive A r + bombardment. A f t e r clean-ing the Rh(100) surface, a c(2x2) pattern could also be formed on exposure to H 2S. This required heating the c r y s t a l at 400°C for 4 min. i n the atmos-sphere of H^ s ( l x l O - 7 t o r r ) , and on cooling the LEED pattern of the surface exhibited a c(2x2)-S overlayer pattern (figure 4.1). Auger spectra f o r t h i s surface gave a r a t i o of peak heights 152eV(S)/302eV(Rh)=4/3 which suggests that the S coverage for t h i s structure i s approximately twice that of the Rh(100)-p(2x2)-S structure. 1(E) curves were measured for Rh(100)-p(2x2)-S for normal incidence f o r - the beams (01), (11), (02), (12), (0±) , ( l | ) , (^ |), ( o | ) , and (y|), using the beam notation shown i n figure 4.3. These measurements involved photographing the LEED screen at 2 eV i n t e r v a l s over the energy range 40-200 eV, and analy-zing the photographic negatives with the computer-controlled Vidicon camera as described i n section 3.1. Two independent sets of experimental data were c o l l e c t e d . -106-9x Figure 4.3: Beam notation for the LEED pattern of Rh(100)-p(2x2)-S structure. -107-4.3 Computational Scheme 1(E) curves were calculated with the layer-doubling method [24], using a conventional muffin-tin-type p o t e n t i a l , f or some surface models i n which only sulphur was present i n an overlayer. The scatterings by the atomic po t e n t i a l s were described by eight phase-shifts. A band structure p o t e n t i a l was used for the atomic regions i n the substrate [110]. For the atomic reg-ions i n the sulphur overlayer, the superposition p o t e n t i a l obtained by Demuth et a l . [131] was used. This superposition p o t e n t i a l was also used by Van Hove and Tong [146] i n an analysis of surface structures formed by S on Ni(100). The r e a l part of the inner p o t e n t i a l (V ) was i n i t i a l l y set at -12.0 eV or (although t h i s was r e f i n e d l a t e r i n the comparison with experimental data) for both the overlayer and the substrate, while the imaginary part (V ^) was 1/3 equated to -1.51E eV. The e f f e c t i v e Debye temperatures were taken as 406 K for rhodium (as discussed i n section 3.4) and 236 K for sulphur following Demuth et a l . [ l 3 l ] . The geometrical models considered for Rh(100)-p(2x2)-S were s i m p l i f i e d by o f i x i n g a l l i n t e r l a y e r spacings i n the metal at the bulk value (1.9022 A); t h i s follows our previous conclusion for clean Rh(100) that t h i s surface i s not reconstructed and i t s topmost spacing i s within 2.5% of the bulk value (section 3.2). Three types of s t r u c t u r a l model were tested, a l l corresponding to a quarter monolayer of S atoms. These models are shown i n fi g u r e 2.8 and they are designated according to the number of nearest-neighbour metal atoms (as already described i n section 2.7) as 4F, IF and 2F. The packing of hard spheres, with r a d i i given by Pauling [133], was used to guide the possible -108-values of topmost i n t e r l a y e r spacing for each model type; t h i s analysis speci-es f i c a l l y considered spacings between 2.1 and 2.7 A for the IF model, between o o 1.4 and 2.2 A for the 2F model and between 1.0 and 1.6 A f o r the 4F model. Symmetry could be used i n the ca l c u l a t i o n s at normal incidence and the number of beams used i n the c a l c u l a t i o n s are summarized i n Table 2.1. For the 2F model i t i s necessary to average appropriate calculated beam i n t e n s i t i e s according to the possible symmetrically-equivalent domains. 4.4 Results 1(E) curves measured for normal incidence f o r the (01) and (~) sets of beams are shown i n figu r e 4.4 for two independent experiments. Beams within each set should be symmetrically-equivalent, both with regard to peak p o s i t -ions and other s t r u c t u r a l features. The correspondences seen i n the figu r e suggest that the experimental data are c l o s e l y reproducible, and t h i s supports t h e i r general r e l i a b i l i t y . The small v a r i a t i o n s which do occur must be att r i b u t e d to experimental errors (involving such factors as uneven response of the screen, imperfections of the c r y s t a l surface, and some uncertainty i n se t t i n g the angle of incidence); such e r r o r s , although small, do i n e v i t a b l y l i m i t the l e v e l of agreement possible between c a l c u l a t i o n and experiment. To minimize any artefacts i n the comparisons with the calculated i n t e n s i t i e s , measured 1(E) curves f o r sets of beams which are t h e o r e t i c a l l y equivalent were averaged and d i g i t a l l y smoothed (by two operations of the three-point smoothing f i l t e r ) p r i o r to comparing with the c a l c u l a t i o n s . -109-01 01 so 100 iTo energy(eV) 200 1 1 1 50 100 150 energy (eV ) 200 Figure 4.4: Comparison for the (~) and (01) beams of 1(E) curves from two d i f f e r e n t experiments measured at normal incidence. -110-Some comparisons of experimental and cal c u l a t e d 1(E) curves f o r Rh(100)-p(2x2)-S are shown i n figure 4.5. V i s u a l comparisons of a l l data a v a i l a b l e points to the conclusion that the centre (4F) model gives a better o v e r a l l correspondence to the experimental 1(E) curves than the bridge (2F) and on-top (IF) models. For the integral-order beams alone, reasonable match-ups bet-ween experimental and calculated 1(E) curves are found f o r the (01) and (02) o o o beams with a l l the three models ( i . e . 4F at 1.3 A, 2F at 1.9 A and IF at 2.3 A), but the 4F model also gives a good correspondence for the (11) beam whereas the 4F and IF models f a i l i n t h i s regard. As expected, the f r a c t i o n a l - o r d e r beams are generally more s e n s i t i v e to the locations of the overlayer atoms, and the o v e r a l l conclusion from a v i s u a l analysis of a l l data f o r the f r a c t -i o n a l order beams i s that the 4F model gives the best account o f the experi-o mental 1(E) curves with the Rh-S i n t e r l a y e r spacing close to 1.3 A. However, the agreement i s not complete, r e l a t i v e peak i n t e n s i t i e s are not properly accounted for and i n a few instances the 4F model f a i l s to reproduce features i n the experimental 1(E) curves. In p a r t i c u l a r , the calculated 1(E) curve for the (0^-) beam for the 4F model with the Rh-S i n t e r l a y e r spacing equal to 1.3 A does not reproduce the peak present i n the experimental curve at 110 eV; 3 also for the (0-) beam the 4F model shows an extra small peak at 130 eV which could not be detected i n the experimental curve. 3 13 For some f r a c t i o n a l - o r d e r beams (e s p e c i a l l y (0^) and C^))» calculated 1(E) curves from the bridge (2F) model give reasonable agreement with the o experimental 1(E) curves for the topmost spacing of 1.9 A, but t h i s adsorption s i t e i s less favorable than the 4F s i t e for the (o|) and ( ™ ) beams. The on-top (IF) model gives poor v i s u a l agreement between c a l c u l a t i o n and experiment -111-Figure 4.5: Comparison of experimental 1(E) curves for various i n t e g r a l -and f r a c t i o n a l - o r d e r d i f f r a c t e d beams from Rh(100)-p(2x2)-S with the calculated curves for S adsorbed on the 4F, 2F and IF s i t e s at the topmost Rh-S i n t e r l a y e r spacing indicated for each curve. Electron energy (eV) T — " — i — i — i — 1 — i — i 1 I — | — | — i — i — i — i — i — i 1 ( — i — i — i — i — i — i — i — I 1 I — i — i — r — i i i i r I — i — i — i — i — i — i — i — i 1 I — i — i — i — i — i — i — i — i 1 I — i — i — i — i — i — i — i — i I — i — i — i — i — i i i i 40 80 120 160 200 40 80 120 160 200 40 80 120 160 200 40 80 120 160 200 Electron energy (eV) -114-13 for most beams, although some agreement i s present for the (— —) beam f o r the o topmost spacing of 2.7 A. I l l u s t r a t e d i n figure 4.6 are comparisons of experi-mental 1(E) curves for the (o|) and (-—) beams with those calculated from the 4F model for various values of the topmost i n t e r l a y e r spacing ranging from 1.0 A to 1.6 A. Although the l e v e l of agreement i s not complete, the best correspondence seems to occur with the S-Rh i n t e r l a y e r spacing between 1.2 and 1.3 A. The correspondence between the experimental and calculated 1(E) curves for the Rh(100)-p(2x2)-S surface were also assessed by evaluating the r e l i -a b i l i t y index (r^) proposed by Zanazzi and Jona [45]. Figures 4.7(a)-4.7(c) give contour pl o t s of r ^ as a function of the Rh-S spacing and V f o r each of the three models when compared with one set of experimental data. Compari-son with the other set of experimental data produced s i m i l a r r e s u l t s , as sum-marized i n Table 4.1. The analysis with r ^ unambiguously showed that the 4F model gives the best correspondence.between the experimental and calculated. 1(E) curves. For t h i s model, r ^ i s minimized (figure 4.7(a)) with the Rh-S o i n t e r l a y e r spacing equal to 1.30±0.03 A and V ^ equal -13.6±0.9 eV, when the uncertainties are given as i e . and ±e as indicated i n section 2.8. The b d v uncertainties correspond to 68% p r o b a b i l i t i e s according to the analysis of Watson et a l . [43]. The minimum value of r for the 4F model i s 0.26; t h i s r / represents a moderate l e v e l of agreement and suggests that the structure i s - at least probably correct according to a c r i t e r i o n of Zanazzi and Jona L 4 5 J . The bridge (2F) model also gives a l o c a l i z e d minimum, s p e c i f i c a l l y at the o Rh-S i n t e r l a y e r spacing of 1.94±0.08 A and V ^ equal to -11.611.4 eV. The -115-I 1 1 1 1 1 1 1 — ' 1 I 1 1 1 1 1—I 1 1 40 80 120 160 200 40 80 120 160 200 Electron energy (eV) Figure 4.6: Comparison of experimental I(E) curves for the (0^) and (~,~) beams from the Rh(100)-p(2x2)-S surface with those calculated for S adsorbed on the 4F s i t e for a range of topmost Rh-S interlayer spacings. -116-Table 4.1: Conditions f o r minima of r r for d i f f e r e n t models of Rh(100)-p(2x2)-S. surface model expt. no. AE (eV) S-Rh (A) V n T, (eV) or centre s i t e (4F) 856 932 1.3010.03 1.3110.03 •13.6+0.9 •13.810.8 0.26 0.25 bridge s i t e (2F) 856 932 1.9410.08 1.9410.08 •11.6+1.4 -13.511.2 0.30 0.28 on-top s i t e (IF) 856 932 no l o c a l i z e d minimum no l o c a l i z e d minimum t o t a l range of energy compared. -117-corresponding minimum value of (0.30) i s higher than that of the 4F model (0.26), although these r ^ values are closer than expected on the basis of the v i s u a l a n a l y s i s . Further suggestive support for the 4F model, from the r e l i a b i l i t y index a n a l y s i s , i s indicated by the larger uncertainties a s s o c i -ated with the bridge model. The contour p l o t of r ^ i n figure 4.7 (c) does not ind i c a t e a l o c a l i z e d minimum for the on-top (IF) model, also values of r ^ are comparatively high over the complete ranges of V and Rh-S i n t e r l a y e r spacing considered. However i t was observed i n separate calculations that the contour plots of r ^ for the integral-order beams alone and for the f r a c -t i onal-order beams alone did show l o c a l minima corresponding to Rh-S i n t e r -o o layer spacings of 2.3 A and 2.7 A r e s p e c t i v e l y ; t h i s i n d i c a t e s the reason why the calculated 1(E) curves shown in figure 4.5 are for the spacings 2.3 and 2.7 X. 4.5 Discussion The evidence presented above indicates that the surface structure Rh(100)-p(2x2)-S has the sulphur atoms adsorbed on the f o u r - f o l d (4F) s i t e s o of the Rh(100) surface at about 1.30 A above the topmost rhodium layer. o This corresponds to a nearest neighbour S-Rh distance equal to 2.30 A. E v i -dence that t h i s i s a reasonable bond distance i s suggested by the average o values found by X-ray crystallography i n Rhj^S (2.33 A) (_ 134J and i n Rh^S^ (2.37 X) [135]; also Rh-S distances i n unhindered coordination complexes generally range from 2.23 to 2.38 A [136-138J. Often structures from LEED I I I I crystallography are discussed i n terms of e f f e c t i v e r a d i i (r ) for the © I T -118-(a) Rh(100)-P(2x2)S 4F S-Rh DISTANCE (A) Figure 4.7: Contour plo t s of r ^ f o r Rh(100)-p(2x2)-S versus and Rh-S i n t e r l a y e r spacing f o r (a) 4F model, (b) 2F model, and (c) IF model. Error bars i n d i c a t e standard errors as defined i n chapter 2. -119-RhdOO)-p(2x2)S 2F o S-Rh DISTANCE (A ) -120--121-adsorbed species [139]. By considering Rh as being unchanged by adsorption o so that i t retains the m e t a l l i c radius of 1.34 A, an e f f e c t i v e radius of S i s obtained by subtracting the rhodium m e t a l l i c radius from the Rh-S nearest-o neighbour distance, and t h i s gives the value of r £^ for S equal to 0.96 A. This value can be compared with other values f o r S (Table 4.2) deduced with LEED crystallography for adsorption on m e t a l l i c surfaces. From Table 4.2 i t i s clear that r of S obtained i n t h i s work i s s i m i l a r to values obtained e f f from some other studies, although i t i s probably not reasonable to expect r £^ of S to be constant i n different.bonding s i t u a t i o n s (involving for example, d i f f e r e n t metal atoms, d i f f e r e n t substrate dimension and e s p e c i a l l y d i f f e r e n t coordination s i t e s ) . Although hard sphere r a d i i (e.g. r ) have often been used for i n t e r -pretations of surface bond distances, i t would c l e a r l y be preferable to r e l a t e such discussions more c l o s e l y to the concepts of covalent bonding. That some M-X surface bond lengths correspond i n a good approximation to single-bond values i s established in Table 4.3 where some comparisons are given for M-X distances for the heavier chalcogens on (100) surfaces of fee metals. Ratio-n a l i z a t i o n s of such correlations and t h e i r extensions to other surface systems, have been given by M i t c h e l l [143,144] based on h y b r i d i z a t i o n schemes f o r metals given by Altmann, Coulson and Hume-Rothery [145] and on r e l a t i v e valencies and the bond length - bond order r e l a t i o n given by Pauling [133], The point of immediate i n t e r e s t , however, i s that the Rh-S bond length found o i n the LEED analysis of Rh(100)-p(2x2)-S i s within 0.01 A of the single-bond value, thereby i n d i c a t i n g a general consistency with surface bond lengths Table 4.2: E f f e c t i v e r a d i i of chemisorbed sulphur atoms on various metal surfaces. System Overlayer surface structure Bonding s i t e M-S bond o distance (A) r r r of e f f 9 sulphur. (A) References S/Ni(100) c (2x2) 4F 2.18 0.94 131 S/Ni(100) P(2x2) 4F 2.18 0.94 131 S/Ni(110) p (2x2) 4F 2.17, 2.35+ 0.93 140 S / N i ( l l l ) P (2x2) 3F 2.02 0.78 140 S / I r ( l l l ) (/3x/3)R30° 3F 2.28 0.92 147 S/Rh(110) c (2x2) 4F 2.12, 2.45+ 0.77 123 S/Rh(100) p(2x2> 4F 2.30 0.96 124 S/Fe(100) c (2x2) 4F 2.30 1.06 142 +Each S atom i s closer to a metal atom in the second layer than the atoms i n the f i r s t layer. Table 4 . 3 : Comparisons of M-X bond distances for chalcogen atoms adsorbed on ( 1 0 0 ) surfaces of fee metals with Pauling s s i n g l e bond lengths [ 1 3 3 ] . overlayer bonding M-X distance ; M-X s i n g l e references o 0 surface structure s i t e by LEED (A) bond length (A) S/Ni ( 1 0 0 ) c ( 2 x 2 ) 4 F 2 . 1 8 2 . 1 9 131 . p ( 2 x 2 ) 4F 2 . 1 8 2 . 1 9 131 Se/Ni ( 1 0 0 ) c ( 2 x 2 ) 4F 2 . 2 8 2 . 3 2 1 3 1 , 1 4 0 P ( 2 x 2 ) 4F 2 . 3 2 2 . 3 2 1 3 1 , 1 4 0 Te/Ni ( 1 0 0 ) c ( 2 x 2 ) 4F 2 . 5 9 2 . 5 2 1 3 1 , 1 4 0 , 1 4 9 p ( 2 x 2 ) 4F 2 . 5 2 2 . 5 2 1 3 1 , 1 4 0 Te/Cu (100) p ( 2 x 2 ) 4 F 2 . 4 8 2 . 5 4 1 4 8 S/Rh (100) p ( 2 x 2 ) 4 F 2 . 3 0 2 . 2 9 1 2 3 -124-reported from other example of S, Se and Te adsorption on fee (100) surface. This c o r r e l a t i o n had not been recognized at the time we i n i t i a l l y published our LEED analysis for Rh(100)-p(2x2)-S [123]. Generally i t i s f e l t that the surface structure reported here f o r Rh(100)-p(2x2)-S gives bond dimensions which are broadly consistent with X-ray c r y s t a l l o g r a p h i c data for S-Rh bond lengths and with LEED r e s u l t s f o r adsorp-t i o n of S atoms on other surfaces. The l e v e l of agreement reached between the calculated and experimental 1(E) curves i s not complete, and the o r i g i n s of the d e f i c i e n c i e s are presently unknown. The number of model structures considered i n the c a l c u l a t i o n for t h i s work i s l i m i t e d ; i n p r i n c i p l e more complicated models are po s s i b l e , but since no c o n f l i c t seems to be present with the p r i n c i p l e s of surface s t r u c t u r a l chemistry, as they are presently evolving, we do not f e e l that further m u l t i p l e - s c a t t e r i n g c a l c u l -ations on more complex surface models are required at t h i s time. An i n e v i t a b l e problem with the t r i a l - a n d - e r r o r approach i n LEED crystallography i s that, however good the correspondence may be between experimental and calculated 1(E) curves f or a given s t r u c t u r e , there i s no absolute way of r u l i n g out the p o s s i b i l i t y that some other (untested) structures could give even better agreement. Although the o r i g i n of some discrepancies between the experimental and ca l c u l a t e d i n t e n s i t i e s found here are not yet c l e a r , we beli e v e the r e s u l t s i n d i c a t e that the structure most l i k e l y involves S atoms "adsorbed at 1.3 A above the f o u r - f o l d s i t e s of the Rh(100) surface. - 1 2 5 -CHAPTER 5 LEED Analysis of the Rh ( 1 1 0 j-c ( 2 x 2)-S Surface Structure -126-5.1 Introduction Having determined the surface geometry for sulphur adsorbed on the (100) surface of rhodium, we were interested i n comparing with the s i t u a t i o n f or S adsorbed on the more open (110) surface. A second reason f o r making a LEED analysis of t h i s additional structure was suggested by e a r l i e r reports that two d i f f e r e n t adsorption s i t e s are indicated by LEED crystallography for atomic adsorption on (110) surfaces of face-centered cubic metals. Oxygen atoms are reported to adsorb on the short-bridge s i t e s of both Ni(110) [59] and (impurity-stabilized) unreconstructed Ir(110) [151], whereas sulphur atoms adsorb on the centre (four-fold) s i t e s of Ni(110) [140]. It i s hoped that an i n v e s t i g a t i o n of the adsorption of S on the Rh(110) surface may give fur -ther i n s i g h t s into surface chemical bonding. 5.2 Experimental The f i r s t part of this, study involved obtaining a clean (110) surface of rhodium, and t h i s followed c l o s e l y the procedures described e a r l i e r i n t h i s thesis and i n other work reported from our laboratory [109]. This study was performed on a single c r y s t a l s l i c e cut from a rod of p u r i t y 99.99% purchased from Research Organic/Inorganic Chemical Corp. Af t e r pumping down i n the vacuum chamber, the i n i t i a l Auger spectrum indicated some contamination from phosphorus, sulphur and carbon. The S and P impurities could be removed from . the surface by argon-ion bombardment (1 keV at 5 microamps for 20 minutes), but, as previously, a r e l a t i v e increase i n the surface concentration of C was indicated. However, t h i s impurity apparently d i f f u s e d i n t o the bulk on heatinj at 300°C. A f t e r several cycles of ion-bombardment and annealing, the surface -127-showed both an e s s e n t i a l l y - c l e a n Auger spectrum (figure 5 . 1(a)) and a sharp ( l x l ) LEED pattern. This r e s u l t i n g Auger spectrum i s s i m i l a r to that obtained for the cleaned (100) surface of rhodium (figure 4 . 2 ) . A f t e r obtaining the well-defined LEED pattern c h a r a c t e r i s t i c of the clean Rh(110) surface, high p u r i t y l ^ S (Mathe-son) was allowed to adsorb on the surface by the following procedures. F i r s t the sample was heated at 3 0 0 ° C for 1 minute and U^S was l e t i n t o the vacuum chamber at the pressure of _7 5x10 t o r r f or 1 minute. After pumping out the excess gas, LEED showed a d i f f u s e r i n g pattern i n d i c a t i n g that H^S (or S) had adsorbed with only p a r t i a l ordering on the surface. The sample was then heated at 300°C for 3 minutes and allowed to cool down. At th i s point LEED indicated that the r i n g pattern had been replaced by traces of (^j) spots, c h a r a c t e r i s t i c of a c(2x2) pattern, but the spot i n t e n s i t i e s were weak. With furthur heating at 700°C for 2 minutes, LEED showed a stable and sharp c(2x2) pattern (figure 5.2) which could be removed only by argon ion bombardment. The Auger spectrum indicated no other detectable impurities and a r a t i o of the Auger peak heights S(152):Rh(302) approximately equal to 3:4 (figure 5 . 1(b)). For the purposes of beam i n t e n s i t y measurements, two sets of photographs were taken: one at normal incidence over the energy range 22 to 220 eV and the other for off-normal incidence ( s p e c i f i c a l l y 6 = 1 0 ° , c f > = 1 3 5 ° [ l 0 0 ] ) from 22 to 160 eV. The photographic negatives were analyzed with the computer-con t r o l l e d Vidicon camera as described i n section 3 .4. For normal incidence, 1(E) curves were measured for 9 integral-order beams and for 5 f r a c t i o n a l -order beams using the beam notation indicated i n f i g u r e 5 .3 . These are -128-Energy (eV) Figure 5.1: Auger spectra f o r a Rh(llO) surface when cleaned and when containing a c(2*2) overlayer of sulphur. - 1 2 9 -- fo-r i gure 5.2: Photographs o f LEED p a t t e r n s o b s e r v e d a t normal i n c i d e n c e f rom a d s o r p t i o n o f S on R h ( l l O ) s u r f a c e . (a) Rh (110) a t 144 eV, ( b ) Rh(110)-c(2x2)-S a t 78 eV, (c) Rh (110 ) - c ( 2 x2 ) - S a t 102 eV, ^ (d) Rh (110 ) - c (2 *2 ) - S a t 150 eV. -130-9, 1 2 11 10 3 3 2 2 3 1 2 2 22 21 20 1 1 01 11 21 Figure 5.3: Beam notation for the LEED pattern from the Rh(110)-c(2><2)-S surface structure. -131-(01) (02) (03) (10) (11) (12) (13) (20) (21) 11 13 15 31 33 ^22^ ^22^ "^22^  ^22^ ^22^ " The 1(E) curves for the integral-order beams were found to be rather s i m i l a r to those of the clean (110) surface; t h i s suggested that the pro-duction of the Rh(110)-c(2x2)-S structure did not involve any appreciable changes i n the p o s i t i o n s of the Rh atoms from those i n the clean surface. Typical experimental 1(E) curves f o r normal incidence are shown i n f i g u r e 5.4, The s i m i l a r i t i e s for the beams which should be equivalent are not as close as those generally found from the Rh(100)-p(2*2)-S structure. This probably indicates larger deviations from normal incidence, although there may be extra degrees of roughness f o r the (110) surface. The complete sets of i n t e n s i t y data for both di r e c t i o n s of incidence are c o l l e c t e d i n Appendices A5-A6. 5.3 Calculations The simplest models for the c(2x2) t r a n s l a t i o n a l symmetry associated with atoms adsorbed on an unreconstructed (110) surface of a face-centred cubic metal have already been shown i n figure 1.8. These models are desig-nated according to the s i t e s of adsorption namely: centre or f o u r - f o l d (4F) model, on-top or one-fold (IF) model, short-bridge (2SB) model and a long-bridge (2LB) model. 1(E) curves for the various required d i f f r a c t e d beams were calculated using the layer-doubling method for a l l of these models. The computing times were reduced by e x p l o i t i n g the symmetry at normal i n c i -dence, and by adding the adsorbate layer separately to both the bottom and -132-Figure 5.4: Experimental 1(E) curves for two sets of beams which are expected to be equivalent f or the Rh(110)-c(2x2)-S structure. - 1 3 3 -the top of the substrate stack, a f t e r the r e f l e c t i o n and transmission matrices have converged f o r the substrate alone (this t y p i c a l l y requires 8 to 16 l a y e r s ) , to give d i f f r a c t e d beam i n t e n s i t i e s from the 4 F and I F models from a s i n g l e set of m u l t i p l e - s c a t t e r i n g c a l c u l a t i o n s ( s i m i l a r l y the 2 L B and 2 S B models could be treated together). A l l four s t r u c t u r a l models considered have two perpendicular mirror planes; 49 symmetrically inequivalent beams were included i n the c a l c u l a t i o n to ensure convergence. The same non-structural parameters were used i n the m u l t i p l e - s c a t t e r i n g c a l c u l a t i o n s on R h ( 1 1 0 ) - c ( 2 x 2 ) - S as for the analysis of the R h ( 1 0 0 ) - p ( 2 x 2 ) - S structure. S p e c i f i c a l l y the Rh p o t e n t i a l was characterized by phase s h i f t s (to 1=1) derived from a band structure c a l c u l a t i o n [ 1 1 0 ] ; the r e a l part of the constant p o t e n t i a l (V ) between the atomic spheres was set i n i t i a l l y at or - 1 2 . 0 eV; a superposition p o t e n t i a l [ 1 3 1 ] was used f o r S ; the surface Debye temperatures were taken as 4 0 6 and 2 3 6 K for Rh and S r e s p e c t i v e l y , while the imaginary part (V ^) of the constant p o t e n t i a l between a l l spheres was equ-1 / 3 ated to - 1 . 5 1 E eV. The s t r u c t u r a l parameters f o r the R h ( 1 1 0 ) - c ( 2 * 2 ) - S surface were s i m p l i f i e d by f i x i n g a l l i n t e r l a y e r spacings f o r Rh ( 1 1 0 ) at the bulk value ( 1 . 3 4 5 A); t h i s follows our previous observations that the clean Rh ( 1 1 0 ) surface i s not reconstructed and that the topmost i n t e r l a y e r spacing i s contracted by only 3% from the bulk value [ 1 0 9 , 1 5 0 ] . The Rh-S o spacings were varied over the following ranges: 0 . 6 5 - 1 . 2 5 A f o r the 4 F model, 2 . 0 - 2 . 6 A for the I F model, 1 .1 - 1.7 A for the 2 L B model and 1 . 6 - 2 . 2 A for the 2SB model. -134-Preliminary attempts were made to c a l c u l a t e the d i f f r a c t e d beam inten-s i t i e s f o r the conditions measured i n the experiment for off-normal i n c i -dence (6 = 10°, <J>=135°). Symmetry could not now be exploited and hence the t o t a l number of beams needed i n the c a l c u l a t i o n i s greatly increased over that f or normal incidence. Around 175 beams would be required at 200 eV, and we found that the consequent computational requirements were too expen-sive f or us to proceed with these c a l c u l a t i o n s . The experimental data for off-normal incidence has however been c o l l e c t e d i n the appendix. 5.4 Results Some comparisons of experimental and calculated 1(E) curves are given i n figures 5.5 and 5.6. Figures 5.5(a)-5.5(c) compare experimental 1(E) 31 curves for the (10), (01) and (--) beams with those calculated for the 4F, IF, 2SB and 2LB models for various Rh-S i n t e r l a y e r spacings. V i s u a l compari-sons show poor agreement for the short-bridge (2SB) and long bridge (2LB) models, and while the on-top (IF) model produced a reasonable correspondence for the (10) beam, there was l i t t l e agreement for other beams. V i s u a l comparisons over the complete range of data unambiguously indicated that the best correspondence between the experimental and calculated 1(E) curves i s provided by the 4F model with the Rh-S i n t e r l a y e r spacing i n .the range 0.75 to 0.85 X (figure 5.6). Discrepancies are apparent, e s p e c i a l l y for some r e l a t i v e peak heights, although at the present stage of development of LEED crystallography the general correspondence can (we believe) be c l a s s i f i e d II it as good . -135-Figure 5.5: Comparison of some experimental 1(E) curves from Rh(110)-c(2x2)-S with those calculated for the four s t r u c t u r a l models over a range of topmost i n t e r l a y e r spacings: (a) (01) beam, (b) (10) 31 beam, and (c) (rr) beam. 2LB (01) beam 4 0 8 0 120 160 2 0 0 l b ) Electron energy (eV) I—I—I—I—I—I—I—I—I—• 4 0 8 0 120 160 2 0 0 Electron energy (eV) -139-—I—1—1—1—1—1—1—1 V A (10) beam ft •A AO. 85 A i i' • * i 11 a • i i i i i i i i -i—i—i—I—I—i—i—r (II) beam i i i T i i i — L 40 120 200 40 120 200 «> "c => >» k» o w J5 w O >-10 z UJ -I—I—I—I—I—I—'—I— (12) beam T—I—I—l—l—l—"—'— (01) beam 40 I l i I—r—i—i I ( I L) beam - i — i — i — i — i — i — i — r ~ ( i |)beam i 1 1 - i — i — i — i — i — i — i — i -(20) beam 200 40 —1—1—» 1 1 i i—i—i— V| l)beam A A K. / < k J i i i 1 i i i 1 120 200 i—i i l—r—i—i— (Jf)beam i i i I i i i. 120 200 200 40 120 200 40 120 ELECTRON ENERGY (eV) Figure 5.6: Comparison of experimental 1(E) curves f o r some i n t e g r a l - and fra c t i o n a l - o r d e r beams from Rh(110)-c(2x2)-S with those calculated for the 4F model with sulphur either 0.75 or 0.85 X above the topmost rhodium layer. -140-4 F model I F model Rh-S spacing (A) Figure 5.7: Contour plots of r r for Rh(110)-c(2x2)-S versus V Q r and i n t e r l a y e r spacing for four d i f f e r e n t s t r u c t u r a l models. -141-The comparisons between experimental and c a l c u l a t e d 1(E) curves were also assessed by evaluating the r e l i a b i l i t y index proposed by Zanazzi and Jona [45]. Figure 5.7 gives contour pl o t s of r ^ as a function of Rh-S spacing and V for each of the four models considered here. Again there i s c l e a r evidence that the centre (4F) model gives the best correspondence between the experimental and calculated i n t e n s i t i e s . The minimum value of r ^ (0.165) represents a good l e v e l of agreement [ 4 5 ] , and i t corresponds to V = -12.210.8 eV and a Rh-S i n t e r l a y e r spacing of 0.7710.04 A. For the or J r t> other models, r ^ was always s u f f i c i e n t l y large (>0.35) to i n d i c a t e a poor correspondence between the experimental and calculated 1(E) curves. 5.5 Discussion The evidence j u s t presented indicates that the Rh(110)-c(2x2)-S struc-ture has the sulphur atoms adsorbed on the centre (4F) s i t e s of the Rh(110) o surface at about 0.77 A above the topmost rhodium layer. The multiple-s c a t t e r i n g c a l c u l a t i o n s made here assumed that a l l metal-metal distances correspond to the normal bulk values. Tentative evidence i n support i s provided by an a d d i t i o n a l analysis with the r e l i a b i l i t y index r ^ ; we used t h i s index to assess the l e v e l of correspondence between the experimental 1(E) curves for the beams (10), (01), (11) and (12) for the overlayer struc-ture and those calculated for the clean surface. For these conditions, we ...found r r was minimized at the value of 0.22 with the topmost i n t e r l a y e r spacing of rhodium being expanded by j u s t 1% over the bulk value. -142-Figure 5.8 indicates interatomic distances i n the v i c i n i t y of adsorbed sulphur atoms i n the Rh(110)-c(2x2)-S structure assuming there i s no relaxa-t i o n for the rhodium structure. It i s apparent that the f o u r - f o l d hole i n the Rh(llO) surface i s s u f f i c i e n t l y large that the sulphur atom can penetrate quite deeply; i n f a c t sulphur becomes considerably closer to the rhodium atom d i r e c t l y below i n the second metal layer than to the four neighbouring rhodium atoms i n the f i r s t layer. The respective distances are Rhj -S = 2.12 A and Rh-S = 2.45 A . Similar observations have also been made from LEED c r y s t a l l o g r a p h i c analyses for S adsorbed on the N i ( l l O ) surface [140], for which the corresponding distances are N i j j - S = 2.17 A and Ni -S = 2.35 A, and for 0 adsorbed on the Fe(100) surface (figure 5.9) for which [153] F e n - 0 = 2.02 A and F e ^ S = 2.08 A. By contrast, adsorption o f S on the Fe(100) surface does not involve s i g n i -f i c a n t i n t e r a c t i o n of S to the second layer Fe atom [142]. In t h i s case, S i s too large to sink deeply into the f o u r - f o l d hole of the Fe(100) surface. The differences between 0 and S chemisorbed on fee(110) surfaces can p l a u s i b l y be associated with s i z e e f f e c t s . 0 appears too small to adsorb on the centre (4F) s i t e and i n t e r a c t with metal o r b i t a l s directed at t h i s s i t e i n terms of h y b r i d i z a t i o n model of Altmann, Coulson and Hume-Rothery [145]. "Bonding p o s s i b i l i t i e s f o r 0 seem better on the short-bridge s i t e s [143]. -143-Figure 5.9: Interatomic distances for the s p e c i f i c a t i o n of hard sphere r a d i i i n the neighbourhood of an oxygen atom i n the o F e ( 1 0 0 ) - ( l x l ) - O structure. Distances i n Angstroms. (After Legg et a l . [153]). -144-Th e most s i g n i f i c a n t comparison for the new r e s u l t s for S on Rh(llO) i s with the structure formed by adsorption of the same species on N i ( l l O ) . M i t c h e l l [143] has offered a t e n t a t i v e analysis of these structures, and indicated a tendency for S to form a s i n g l e covalent bond to the metal atom d i r e c t l y below i n the second layer and four 3/4 order bonds to the neighbouring metal atoms i n the topmost layer. An i n t e r e s t i n g point i s that while the • distances found from LEED for S on N i ( l l O ) are broadly consistent with t h i s , i t i s physical impossible for the corresponding distances to be simultaneously s a t i s f i e d f o r S on Rh(llO), and t h i s i s a d i r e c t consequence of the longer Rh-Rh distance compared with the Ni-Ni distance. M i t c h e l l concluded that t h i s r e s u l t s i n S being held at that height above the Rh(llO) surface where the combined strengths of the f i v e bonds are optimized, and t h i s requires some t i l l o squeezing of the Rhjj-S distance from the s i n g l e bond value (2.29 A) i n order to get reasonable i n t e r a c t i o n s to the four Rh atoms i n the f i r s t layer. An important aspect of t h i s discussion i s that i t represents a s t a r t on u t i l -i z i n g covalent bonding concepts f o r chemisorption. Most analyses so f a r have emphasized hard sphere r a d i i . The e f f e c t i v e radius indicated f o r S on Rh(llO) i s 0.77 A; t h i s can be compared with other values reported from LEED c r y s t a l -o o lography varying from 0.78 A to 1.04 A as noted i n section 4.5. -145-CHAPTER 6 Studies with the Quasidynamical Method -146-6.1 Introduction Rel i a b l e surface structures so far reported by LEED crystallography have come from studies which used the t r i a l - a n d - e r r o r approach wherein experimental I(E) curves are compared with those calculated for a range of possible surface models and a s e l e c t i o n i s made of the geometrical model that gives the best o v e r a l l correspondence. Generally the c a l c u l a t i o n s have used m u l t i p l e - s c a t t e r i n g methods which are e i t h e r formally exact (e.g. the T-matrix or Bloch-wave methods) or involve good i t e r a t i v e approximations to the f u l l m u l t i p l e - s c a t t e r i n g methods (e.g. layer-doubling or RFS methods). This provides the only generally-accepted approach to LEED crystallography at the present time. Aside from l i m i t a t i o n s s t i l l present i n the experi-mental measurements, and l i m i t a t i o n s introduced i n t o the c a l c u l a t i o n s through the model assumed for the p o t e n t i a l and l a t t i c e v i b r a t i o n s , the accuracy of the present approach to LEED crystallography i s l i m i t e d e s p e c i a l l y by comput-ation time and core storage. A serious problem for surface s t r u c t u r a l chemi-s t r y concerns the l i m i t a t i o n s set on t h i s approach for complex surface struc-tures, for which m u l t i p l e - s c a t t e r i n g c a l c u l a t i o n s i n e v i t a b l y become p r o h i b i -t i v e l y expensive. This opens the need to search for new c a l c u l a t i o n schemes which maintain r e l i a b i l i t y while reducing the computational burden. In p r i n c i p l e the simplest LEED c a l c u l a t i o n involves the kinematical method [49] i n which s c a t t e r i n g by ion-cores i s assumed to be weak so that only s i n g l e s c a t t e r i n g events are included. The discussion i n section 2.1 . establishes that t h i s method i s inadequate for decribing the actual features observed i n the s c a t t e r i n g of low-energy electrons by a s o l i d surface. Attempts -147-have been made to make the kinematical theory usable f o r LEED by processing experimental data such that the mu l t i p l e - s c a t t e r i n g contributions are aver-aged out and the res i d u a l i n t e n s i t i e s can then be analyzed with the kinematic II theory. These data processing procedures include the constant momentum II it transfer averaging method introduced by Lagally et a l . [25], the energy i t II averaging method introduced by Tucker and Duke [154], and the Fourier II transform method [157]. Although a t t r a c t i v e i n p r i n c i p l e , these methods cannot yet be considered well-established for determining unknown surface structures i n v o l v i n g adsorption. A new approximate m u l t i p l e - s c a t t e r i n g scheme for c a l c u l a t i n g LEED inten-s i t i e s i s the quasidynamical method [46], In t h i s method, only s i n g l e s c a t t e r i n g i s included within an atomic layer, while the i n t e r l a y e r s c a t t e r i n g i s calcu-lated properly, for example by the RFS method. The o r i g i n a l authors proposed that t h i s approach should be most r e l i a b l e f o r surface systems i n v o l v i n g l i g h t atoms i n r e l a t i v e l y open structures, where the neglect of i n t r a l a y e r multiple-s c a t t e r i n g i s expected to be less serious. I n i t i a l analyses f o r the unrecon-structed model of GaAs(llO) and for reconstructed Si(100) gave promising agree-ment with f u l l m u l t i p l e - s c a t t e r i n g c a l c u l a t i o n s [155] and with experimental data [46,47] r e s p e c t i v e l y . Such tests indicated that the quasidynamical method can give reasonable accounts of the positions of the main peaks i n experi-mental 1(E) curves, as well as much of the secondary structure, although the absolute i n t e n s i t i e s and the r e l a t i v e i n t e n s i t i e s of neighbouring peaks are . often not predicted r e l i a b l y . The purpose of the present study i s to inves t i g a t e further the quasi-dynamical method by comparing with experimental and calculated 1(E) curves -148-a l ready repor ted i n t h i s t h e s i s , e s p e c i a l l y f o r the ad so rp t i on systems Rh(100)-p(2x2)-S and Rh (110)-c(2x2)-S. A p a r t i c u l a r o b j e c t i v e i s to assess whether t h i s method can i d e n t i f y c e r t a i n su r face models as g i v i n g s u f f i c -i e n t l y poor correspondences w i t h the exper imenta l 1(E) curves t ha t these models need not be cons idered i n the ref inement stages of LEED c r y s t a l l o -graph ic ana ly ses . Analyses f o r the corresponding c lean su r faces of rhodium are made, and they p rov ide convenient r e f e rence po i n t s f o r the ad so rp t i on systems .-6.2 C a l c u l a t i o n s A fundamental pa r t o f c a l c u l a t i o n s of LEED i n t e n s i t i e s i n vo l v e s e v a l u -a t i o n o f the l a ye r d i f f r a c t i o n matr i ces M (equation 2.30) f o r each atomic p l ane ; then the planes are s tacked i n order to determine the s c a t t e r i n g from a c r y s t a l s l ab (of e i t h e r f i n i t e o r s e m i - i n f i n i t e e x t e n t ) . Gene ra l l y the e va l ua t i on o f M i s the most t ime consuming pa r t o f t h i s whole p roces s , s p e c i f i c a l l y because i t i n vo l ve s c a l c u l a t i n g Q - X l - 1 which desc r ibes a l l m u l t i p l e - s c a t t e r i n g events w i t h i n an atomic l a y e r (equat ion 2.30). The quas idynamical scheme makes use o f a commonly-found ob se r v a t i on , t ha t i n t e r -l a y e r m u l t i p l e s c a t t e r i n g i s much s t ronger than i n t r a l a y e r m u l t i p l e - s c a t t e r i n g [ 4 6 ] , by equat ing the p l ana r s c a t t e r i n g ma t r i x to zero . This assumption g ives s u b s t a n t i a l r educ t i on s i n computation t imes . The important ques t i on now concerns whether the ga in i n computat ional convenience i s o f f s e t , or n o t , by too great a lo s s o f accuracy i n the c a l c u l a t e d 1(E) curves . The present t e s t s w i t h the quas idynamical method use the same types o f s u r f a ce models as those cons idered i n the p rev ious s tud ie s w i t h the f u l l -149-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 [ 1 2 3 , 1 2 4 , 1 5 0 ] . Thus only the regular face-centred cubic r e g i s t r i e s were considered here f o r the clean surfaces, but relaxations of the topmost i n t e r l a y e r spacings were allowed. D i f f e r e n t models for the S overlayer are designated as i n figures 1 . 8 and 2 . 8 ; a l l Rh-Rh distances are fi x e d at the appropriate bulk values. Unless otherwise indicated here the same non-structural parameters were used i n the quasidynamical calcu-la t i o n s as i n the corresponding 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 described previously (chapters 3 - 5 ) . The only modifications made i n t h i s regard were to the constant p o t e n t i a l s between the spherically-symmetric atomic p o t e n t i a l s . The imaginary part (V ^ ) of t h i s p o t e n t i a l was f i x e d at - 6 . 8 eV for the IF, 2SB and 2 L B models of Rh ( 1 1 0 )-c ( 2 x 2 )-S whereas the energy dependent form 1 / 3 V . = - 1 . 7 6 E eV was used f o r a l l other surfaces considered, except clean 01 1 / 3 Rh(llO) f o r which V was represented by - 2 . 0 5 E eV. The r e a l part of t h i s p o t e n t i a l ( v o r ) was fixed at - 1 2 eV for a l l c a l c u l a t i o n s , although t h i s value was e f f e c t i v e l y r e f i n e d during comparisons with experimental 1 ( E ) curves for each system. Quasidynamical c a l c u l a t i o n s were made for normal incidence over the energy range 4 0 to 2 0 8 eV for clean Rh ( 1 0 0 ) and over the range 50 to 178 eV for a l l other systems considered here. The RFS method was used f o r stacking atomic planes, these c a l c u l a t i o n s were made with 9 1 beams and electrons were allowed to t r a v e l through upto 12 layers i n the c r y s t a l . For the 4F model of Rh ( 1 1 0 )-c ( 2 x 2 )-S, i t was necessary to combine the sulphur layer and the top-most rhodium layer as a composite layer because o f t h e i r close spacing. -150-6.3 Results and Discussion 6.3 (a) Rh(llO) and Rh(110)-c(2*2)-S Experimental 1(E) curves for normal incidence on the clean Rh(110) surface are compared with those from quasidynamical c a l c u l a t i o n s f or Ad% = 0 and -10% i n f i g u r e 6.1. General correspondences i n peak posit i o n s are apparent f o r every p a i r of curves, although r e l a t i v e i n t e n s i t i e s are often not s a t i s f a c t o r y . Comparisons between quasidynamical (QD) and m u l t i p l e - s c a t t e r i n g (MS) calculated 1(E) curves are also shown i n the same fi g u r e ; again major peak posit i o n s match, although the r e l a t i v e i n t e n s i t i e s have changed i n the quasidynamical case. The experimental and c a l c u l a t e d 1(E) curves were also assessed with the r e l i a b i l i t y index r ^ [45], and Table 6.1 l i s t s the conditions for best correspondence ( i . e . minimum r ) between experimental and calculated 1(E) curves (from both mutiple-s c a t t e r i n g and quasidynamical c a l c u l a t i o n s ) f or the various surfaces considered. The previous m u l t i p l e - s c a t t e r i n g c a l c u l a t i o n s on clean RhfllO) indicated that the best correspondence i s with the topmost i n t e r l a y e r spacing contracted by 3.3% from the bulk value. The corresponding analysis with the quasidynamical method points to a contraction of 10.8%, however d e t a i l e d studies of the i n d i -v i d u al 1(E) curves suggested that the index may be less h e l p f u l f o r t h i s • p a r t i c u l a r purpose. This conclusion depends on r ^ being quite s e n s i t i v e to r e l a t i v e i n t e n s i t i e s over successive portions of i n d i v i d u a l 1(E) curves [150], and the fact (as seen from fig u r e 6.1 and noted above for GaAs(llO) and Si(100) [46]) that the quasidynamical method i s often u n r e l i a b l e f o r peak magnitudes within each 1(E) curve. Table 6.1: Comparisons of conditions for minimum for various surface structures obtained from evaluating experimental 1(E) curves with corresponding curves from multiple-s c a t t e r i n g c a l c u l a t i o n s and from quasidynamical c a l c u l a t i o n s . Surface structure M u l t i p l e - s c a t t e r i n g calculations Ad% d R h _ s ( A ) V o r(eV) r r Quasidynamical calculations A d % d R h - S * V o r C e V ) Rh(llO) -3.3 -11.9 0.12 •10.8 -16.0 0.23 Rh(100) 1.0 •12.8 0.09 3.2 •18.0 0.17 Rh(110)-c(2x2)-S (4F model) 0.77 -12.2 0.17 0.83 -24.4 1.02 0.23 •18.0 0.26 0.72 •16.4 0.30 Rh(100)-p(2x2)-S (4F model) 1.30 -13.6 0.26 1.32 -21.0 0.28 -152-( 0 1 ) beam E X P T A A • \ ( 10 ) beam „ _ E X P T QD MS - i — i — i 1—i r — i — i 1 / \ f ( 0 2 ) beam iJ K.J V..--'~-/- ^ P T 40 80 120 160 200 240 Electron energy (eV) - 1 0 % - i — i — i — i — ' — r — r n 1 1 1 1 r—I l I - i 1 1—I 1 1 1 1 1 ,x (02) beam i i — i — r 40 80 120 160 200 240 Electron energy (eV) Figure 6.1: Comparison of experimental 1(E) curves f o r normal incidence on Rh(llO) with those calculated with the quasidynamical method and the f u l l m u l t i p l e - s c a t t e r i n g method when the topmost i n t e r -layer spacing equals the bulk value (0%) and when i t i s contracted by 10%. -153-1(E) curves for d i f f e r e n t models of the Rh(110)-c(2x2)-S surface calcu-lated by the quasidynamical (QD) method were compared, by d i r e c t observation, with the experimental 1(E) curves and also with the corresponding curves calculated with the mu l t i p l e - s c a t t e r i n g (MS) method for the 4F model with the o topmost Rh-S i n t e r l a y e r spacing (d R h_g) equal to 0.75 A. Overall i t was d i f f i c u l t to pin-point the s t r u c t u r a l model from the quasidynamical c a l c u l a t i o n which gives the best agreement with the experimental curves; i n part t h i s was because of the e f f e c t s of errors i n r e l a t i v e i n t e n s i t i e s f o r successive por-tions of the calculated 1(E) curves. Also there are systematic s h i f t s i n peak positions f or the quasidynamically-calculated 1(E) curves. However i t did seem possible to conclude, from the v i s u a l a n a l y s i s , that the best match with the 1(E) curves from the f u l l m u l t i p l e - s c a t t e r i n g c a l c u l a t i o n s occurred f o r o the 4F model i n the quasidynamical c a l c u l a t i o n s with d R ^ _ g - 1.15 A. Conclusions on conditions for correspondence between quasidynamically-calculated and experimental 1(E) curves were aided with the r e l i a b i l i t y index of Zanazzi and Jona. Two-dimensional contour plots of r versus d n, and . r r Rh-S V0T> f o r each s t r u c t u r a l model, are shown i n fi g u r e 6.2. Comparisons of con-tour plots suggest that the 4F model gives the lowest r ^ value (0.23). No lo c a l minima are found for the 2SB model whereas, for the 2LB and IF models, minima i n T occur with rather high values (>0.42) which suggests that these models are less probable. The contour pl o t s of r ^ for the 4F and 2LB models show the common feature of e x h i b i t i n g more than one l o c a l minimum (figure 6.2). o For the 4F model, the f i r s t minimum (with r ^ = 0.23) occurs for d R ^ g = 0.83 A and Vnr_ = -24.4 eV, the second (with a s l i g h t l y higher value of r , v i z . 0.28) -154-Figure 6.2: Contour plo t s of r r for Rh(110)-c(2x2)-S versus V Q r and the Rh-S i n t e r l a y e r spacing f or four d i f f e r e n t s t r u c t u r a l models calculated with the quasidynamical method. -155-o _ occurs with d_, „ = 1.02 A and V = -18.0 eV and the t h i r d (r = 0.30) occurs Rh-S or r at d,,, „ = 0.72 A and V = -16.4 eV (Table 6.1). This s i t u a t i o n i s to be Rh-S or "compared with a s i n g l e minimum f o r the corresponding contour p l o t of r ^ f o r the same system when the cal c u l a t i o n s u t i l i z e the f u l l m u l t i p l e - s c a t t e r i n g o procedures (figure 5.7); i n t h i s case d^ ^ = 0.77 A, V = -12.2 eV and « r r = 0.17 (Table 6.1). In p r i n c i p l e the existence of more than one l o c a l minimum could r e l a t e to multiple-coincidences i n adsorbate-substrate spacings as discussed by Andersson and Pendry [ l 5 6 j . However, against t h i s p o s s i b i l i t y are the f o l -lowing observations: i ) no such e f f e c t was detected i n the previous analysis with the multiple-sc a t t e r i n g c a l c u l a t i o n (figure 5.7), and i i ) v i s u a l analysis of the i n d i v i d u a l 1(E) curves calculated with the quasi-o dynamical method for the spacings 0.75, 0.85 and 1.05 A are on balance less o s a t i s f a c t o r y than those calculated f or 1.15 A. Two e f f e c t s seem to be involved here. The f i r s t concerns the incomplete nature of the quasidynamical method, and the second appears to be associated with the r e l i a b i l i t y - i n d e x analysis being less r e l i a b l e for assessing i n t e r l a y e r spacings when the r e l a t i v e i n t e n s i t i e s of successive portions of i n d i v i d u a l 1(E) curves are not calculated c o r r e c t l y , even though a reasonable match i n positions of structure may s t i l l be recognized between the experimental and calculated 1(E) curves. -156-For quasidynamical c a l c u l a t i o n s for the 4F model, minima i n r ^ are associated with values of V i n the range -16.4 to -24.4 eV. These values or b are s u b s t a n t i a l l y changed from the value of -12.2 eV reported from the mu 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 . This shows up in the v i s u a l analysis of the i n d i v i d u a l 1(E) curves; features from the quasidynamical c a l c u l a t i o n s occur on average at about 6 eV lower i n energy than do the corresponding features from 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 . This need f o r a systematic s h i f t i n the 1(E) curves must be associated with the neglect of i n t r a l a y e r m u l t i p l e - s c a t t e r i n g i n the quasidynamical c a l c u l a t i o n s . Similar changes i n have also been observed for the quasidynamical c a l c u l a t i o n s of Rh(llO) (Table 6.1) and of S i (100) [47]. 33 Figure 6.3 compares experimental 1(E) curves f o r the (01) and (--) d i f f -racted beams with the corresponding quasidynamically-calculated 1(E) curves f o r p a r t i c u l a r geometries of the four d i f f e r e n t s t r u c t u r a l models. Also shown are the corresponding 1(E) curves calculated by the m u l t i p l e - s c a t t e r i n g method for d ^ g = 0.75 A. For these two representative beams, quasidynamical calcu-l a t i o n f o r the 2SB and 2LB models do not show any agreement with the experi-mental 1(E) curves. S i g n i f i c a n t l e v e l s of agreement are apparent f o r both beams for the 4F model, whereas f or the IF model the quasidynamical c a l c u l a t i o n 33 gives some reasonable agreement f o r the (yr) beam but l i t t l e agreement f or the (01) beam. These comparisons emphasize the matching of peak p o s i t i o n s ; when a l l a v a i l a b l e data from the quasidynamical c a l c u l a t i o n s are considered the 4F model appears to give the best correspondence with 1(E) curves from both experiment and from the reference 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 . -157-fth (IIO)-c(2*2)-S • ........ EXPT t : j ; • *• A ; * ' • • \ ' • \.-v : OD 2LB.I5 A \ f: i • \ ; \ OD . \ / \ \ ''• 2SB ,1 6A "A \.. i \ •-. •. 1 : \ \ 1 \ • '• / IF , 2.2A '. *\ / \ •"' / \ * OD 4F,I 15 A \ •"• MS 4F 0.75 A 120 •ncrgy («V) t o o ( 3 | ) b « a m EXPT A l y * OD . \ ,2LB,I.5A i\ / \!\ {'• .*> » ^ I / v v \ oD ; / \2SB,I.6A QD \ IF, 2.2A] 40 •ntrgy («V ) 33 Figure 6.3: Comparison of 1(E) curves measured for the (01) and (-^j) d i f f r a c t e d beams f o r normal incidence on Rh(110)-c(2x2)-S with those ca l c u l a t e d by the quasidynamical method and by the f u l l multiple-s c a t t e r i n g method for the four s t r u c t u r a l models described i n tex t . -158-Table 6.2: A demonstration o f the correspondence between peak positions i n 1(E) curves calculated with the quasidynamical method for the four models of Rh(110)-c(2*2)-S at the s p e c i f i e d S-Rh i n t e r l a y e r spacing and those given by experiment and by the corresponding f u l l m u l t i p l e - s c a t t e r i n g c a l c u l a t i o n s . In the entries for each beam, the denominator s p e c i f i e s the number of s i g n i f i c a n t peaks i n the relevant 1(E) curve from experiment or from the f u l l m u l t i p l e - s c a t t e r i n g c a l c u l a t i o n s , and the numerator gives the number of those peaks that are matched to within 7 eV by the quasidynamical c a l c u l a t i o n s . Beam 4F Expt S-Rh=1.15A F u l l MS IF Expt S-Rh=2.2A F u l l MS 2SB Expt S-Rh=1.6A F u l l MS 2 LB Expt S-Rh=1.5A F u l l MS (01) 2/4 3/5 2/4 3/5 2/4 1/5 1/4 1/5 (02) 2/3 3/4 2/3 3/4 1/3 3/4 1/3 2/4 (03) 0/1 1/3 0/1 1/3 0/1 1/3 1/1 1/3 (10) 3/5 4/5 3/5 5/5 1/5 2/5 3/5 4/5 (11) 3/4 2/2 2/4 2/2 2/4 2/2 2/4 1/2 (12) 3/4 4/4 1/4 2/4 2/4 2/4 1/4 3/4 (13) • 0/2 0/1 0/2 1/1 0/2 0/1 0/2 0/1 (20) 1/2 2/4 0/2 2/4 1/2 3/4 0/2 1/4 (21) 1/1 2/2 0/1 1/2 0/1 1/2 1/1 1/2 (hh) 2/2 2/2 1/2 2/2 1/2 1/2 2/2 2/2 (h 3/2) 2/4 1/4 3/4 2/4 3/4 2/4 1/4 1/4 (h 5/2) 2/2 4/4 0/2 2/4 2/2 3/4 2/2 3/4 (3/2 h) 2/2 2/3 1/2 1/3 2/2 1/3 1/2 1/3 (3/2 3/2)1/2 4/4 0/2 1/4 1/2 2/4 1/2 2/4 T o t a l 24/38 34/47 15/38 28/47 18/38 24/47 17/38 23/47 - 1 5 9 -Figure 6.4: Comparisons of some experimental 1(E) curves for f r a c t i o n a l -order beams for normal incidence on Rh ( 1 1 0)-c ( 2 x 2)-S and Rh ( 1 0 0)-p ( 2 x 2)-S with those calculated for the centre adsorption s i t e s with the quasidynamical method and with the f u l l m u l t i p l e - s c a t t e r i n g method. The topmost Rh-S i n t e r -o layer spacings i n the quasidynamical cal c u l a t i o n s are 1 . 1 5 A and 1 . 3 A for Rh ( 1 1 0)-c ( 2 * 2)-S and Rh ( 1 0 0 ) - p ( 2 x 2 ) - S r e s p e c t i v e l y ; the corresponding values f o r the mu l t i p l e - s c a t t e r i n g c a l c u l a t i o n s o o are 0 . 7 5 A and 1 . 3 A. -160-Rh(HO)-c(2x2)-S EXPT A ( ^ | ) b e o m £ | ) b e o m ( f l ) b e o m Rh(100)-p(2x2)-S (O^)beam V A - V ' ^ E X P T 40 80 120 160 200 ( l ^ ) b e a m v E X P T ( ^ ) b e a m V' - EXPT ( O l ) b e o m \ /v EXPT v " i — i — i — i i 40 80 120 160 200 Electron energy (eV) Electron energy (eV) -161-Evidence i s provided i n Table 6.2, where the d e t a i l s i n matching of peak positions f or each beam and f o r each model are summarized. A spread i n peak positions of up to 7 eV was allowed i n t h i s matching i n order to accom-modate v a r i a t i o n s of V for the d i f f e r e n t surface models. or From the comparisons indicated i n Table 6.2, i t appears for the 4F model that the l e v e l of agreement, between the quasidynamical c a l c u l a t i o n s and either experiment or f u l l m u l t i p l e - s c a t t e r i n g c a l c u l a t i o n s , i s better for the f r a c t i o n a l - o r d e r beams than for the integral-order beams. A s i m i l a r observa-t i o n was also reported by Tong and Maldonado f o r the Si(100) surface [ 4 7 ] , Figure 6.4 (a) d e t a i l s some s p e c i f i c 1(E) curves f o r the f r a c t i o n a l - o r d e r beams calculated with the quasidynamical method for Rh(110)-c(2x2)-S, and compares with those from experiment and from m u l t i p l e - s c a t t e r i n g c a l c u l a t i o n s . 6.3 (b) Rh(100) and Rh (100)-p(2x2)-S The previous analysis of LEED i n t e n s i t i e s from Rh(100), based on multiple-s c a t t e r i n g c a l c u l a t i o n s and the use of the r e l i a b i l i t y - i n d e x r , indicated that the topmost i n t e r l a y e r spacing i s very close to the bulk value, there being a surface layer contraction o f about 1% [43,150]. A s i m i l a r analysis made here with beam i n t e n s i t i e s c a l c ulated with the quasidynamical method also suggests a small contraction, t h i s time by 3% (Table 6.1). Figure 6.5 indicates for clean unreconstructed Rh(100) appreciable correspondence between peaks i n 1(E) curves calculated with the quasidynamical method and those from e i t h e r experiment or m u l t i p l e - s c a t t e r i n g c a l c u l a t i o n s . In matching with the experi-mental 1(E) curves, the quasidynamically-calculated 1(E) curves needed s h i f t i n g to loweii energy by approximately 6 eV. This i s consistent with r ^ being mini-mized at V = -18.0 eV. or energy (eV ) e nerg y (eV ) Comparisons of some experimental 1(E) curves for normal incidence on Rh(100) with those calculated with the quasidynamical method and with the f u l l m u l t i p l e - s c a t t e r i n g method. -163-For Rh(100)-p(2x2)-S, previous analysis with 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 (section 4.4) pointed to the conclusion that the 4F model with dRh S = ^ § i v e s tbe best correspondence with the experimental 1(E) curves (Table 6.1). In t h i s e a r l i e r analysis we noted that the 2F model also produced a minimum r ^ which i s comparable with that from the 4F model. Similar analyses here with the quasidynamical c a l c u l a t i o n s h i g h l i g h t corres-ponding features; both 4F and 2F models give l o c a l minima with comparable r ^ values (figure 6.6a) although no minimum i s found for the IF model. With o quasidynamical c a l c u l a t i o n s , r i s minimized at d„. „ = 1.32 A and V = -21.0 eV n r Rh-S or o for the 4F model, whereas for the 2F model the corresponding values are 1.70 A and -12.2 eV r e s p e c t i v e l y . To assess t h i s further we made a v i s u a l evaluation of the i n d i v i d u a l I(E) curves and evaluated r just f o r the f r a c t i o n a l - o r d e r beams. The l a t t e r beams r J are expected to be e s p e c i a l l y associated with the adsorbate layer and Table 6.2 notes for Rh(110)-c(2x2)-S that the quasidynamical method appears to work better f or the f r a c t i o n a l - o r d e r beams than f o r the integral-order beams. Figure 6.6(b) shows contour plots of r ^ f o r Rh(100)-p(2x2)-S, from quasi-dynamical c a l c u l a t i o n s , where only the f r a c t i o n a l order beams are included i n the comparison with.experiment. Both the 4F and 2F models give d e f i n i t e minima i n the contour p l o t s , although the minimum value of r ^ for the 4F model (with dn. „ = 1.34 A, V = -21.0 eV) i s now c l e a r l y better than that from v Rh-S or the 2F model (with d n, = 1.91 A and V = -27.2 eV). Support f o r the 4F Rh-S or r r model from the quasidynamical c a l c u l a t i o n i s provided by the observation that the values o f dr,, „ and V which give minimum r from the f r a c t i o n a l - o r d e r Rh-S or b r -164-Figure 6.6: Contour pl o t s of r ^ for Rh (100) -p (2x2)-S versus and the Rh-S i n t e r l a y e r spacing f or the 4F and 2F s t r u c t u r a l models calculated by the quasidynamical method: (a) comparisons with a l l i n t e g r a l - and f r a c t i o n a l - o r d e r beams; (b) compari-sons with f r a c t i o n a l - o r d e r beams only. -165-R h l 1 0 0 ) - P ( 2 x 2 J - S ; Ouaatdynamical calculation (•) intagralffractional C°) fractional only -166-Rh HOO)-p(2x2)-S 3 >» w CO 40 (o 1 ) beam A '• t EXPT : A • * ; • » •: ; \ /•: / \ " V . / '•• QD 2F , 1.8 A QD IF , 2.2A QD 4F , 1.3 A MS 4 F , 1.3 A 120 energy ( eV ) 2ffo I J L i beam ( i i ) EjXPT • • QD i \ 2 F, 1.8 A QD IF , 2.2 A • 4 F J . 3 A 4 F J 3 A 200 energy (eV ) 13 Figure 6.7: Comparisons of 1(E) curves measured for the (01) and (^j) d i f f r a c t e d beams for normal incidence on Rh(100)-p(2*2)-S wi-th those calculated by the quasidynamical method and by the f u l l m u l t i p l e - s c a t t e r i n g method for three possible s t r u c t u r a l models. - 1 6 7 -Tab1e 6 . 3 : A demonstration of the correspondence between peak posit i o n s i n 1(E) curves calculated with the quasidynamical method for the four models of Rh ( 1 0 0 )-p ( 2 x 2 )-S at the s p e c i f i e d S-Rh i n t e r l a y e r spacing and those given by experiment and by the corresponding f u l l m u l t i p l e - s c a t t e r i n g c a l c u l a t i o n s . In the entries f o r each beam, the denominator s p e c i f i e s the number of s i g n i f i c a n t peaks i n the relevant 1(E) curve from experiment or from the f u l l m u l t i p l e - s c a t t e r i n g c a l c u l a t i o n s , and the numerator gives the number of those peaks that are matched to within 7 eV by the quasidynamical c a l c u l a t i o n s . Beam AF Expt O S-Rh=l.3A F u l l MS IF S-Expt O -Rh=2.2A F u l l MS 2F Expt O S-Rh=1.8A F u l l MS (01) 2/2 1/2 2/2 1/2 1/2 2/2 ( I D 2/2 1/3 2/2 2/3 2/2 2/3 (02) 1/1 1/1 0/1 0/1 1/1 1/1 (12) 1/1 1/1 1/1 1/1 1/1 1/1 (hh) 2/3 2/4 1/3 1/4 1/3 1/4 (h 3/2) 2/3 3/4 2/3 3/4 1/3 1/4 (0 h) 4/5 4/5 2/5 1/5 2/5 1/5 ( l h) 2/4 2/4 2/A 1/4 3/4 2/4 (0 3/2) 3/3 A/5 2/3 0/5 2/3 3/5 (3/2 3/2) - 2/2 - 1/2 ' 0/2 Total 19/24 21/31 1A/2A 14/31 14/24 14/31 -168-beams alone are very s i m i l a r to those from the combination of f r a c t i o n a l -order and integral-order beams. By contrast, the conditions for minimum r are very d i f f e r e n t i n these two s i t u a t i o n s from the 2F model. O v e r a l l , then, we be l i e v e that the quasidynamical c a l c u l a t i o n indicates that the 4F model gives the best correspondence with experimental 1(E) curves for Rh(100)-p(2x2)-S with d n L r = 1.32 A and V = -21.0 eV. r Rh-S or Figure 6.7 compares quasidynamically-calculated 1(E) curves for the 13 (01) and (^j) beams of Rh(100)-p(2x2)-S with those from experiment and from m u l t i p l e - s c a t t e r i n g c a l c u l a t i o n s . Correspondences i n peak po s i t i o n s are apparent for a l l models with the (11) beam, but the 4F model shows the 13 best match for the (^j) beam. Deta i l s of comparisons of i n d i v i d u a l 1(E) curves are summarized i n Table 6.3. Again t h i s table shows that the best matching f o r the f r a c t i o n a l - o r d e r beams i s provided by the 4F model. (Some actual 1(E) curves are i l l u s t r a t e d i n fi g u r e 6.4(b)). 6.4 Concluding Remarks The r e s u l t s presented i n Table 6.2 and 6.3 for the quasidynamical method indica t e that adsorption occurs i n the 4F s i t e s f o r both Rh(110)-c(2x2)-S and Rh(100)-p(2x2)-S; comfortingly these are just the adsorption s i t e s i n d i -cated by the f u l l m u l t i p l e - s c a t t e r i n g c a l c u l a t i o n s . For Rh(100)-p(2x2)-S the quasidynamical c a l c u l a t i o n , i n conjunction with the Zanazzi-Jona r e l i -o a b i l i t y - i n d e x r , indicates a topmost i n t e r l a y e r spacing of 1.32 A, i n very o close agreement with that (1.30 A) from 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 (Table 6.1); however the s i g n i f i c a n c e of t h i s close correspondence must be -169-tempered by the appreciable discrepancies found for both clean Rh(llO) and Rh(110)-c(2*2)-S. In general, the index r ^ seems less r e l i a b l e f o r assessing i n t e r l a y e r spacings and V Q r from the quasdynamica'l c a l c u l a t i o n s , e s p e c i a l l y since t h i s method can be erroneous for c a l c u l a t i n g r e l a t i v e peak i n t e n s i t i e s over successive portions of 1(E) curves. Comparisons i n f i g u r e 6.4 show that some peaks i n experimental 1(E) curves are either absent i n the quasidynamically-calculated curves or are represented only by shoulders. In part the l a t t e r may represent a consequence of the r e l a t i v e l y large values of V that are needed i n our quasidynamical c a l c u l a t i o n s to avoid occasional d i f f i c u l t i e s i n convergence. Although the quasidynamical method c l e a r l y i s not exact, we are neverthe-less encouraged by our observations for Rh(110)-c(2x2)-S and Rh(100)-p(2x2)-S that i t i s able to s e l e c t the correct adsorption s i t e s as providing the most l i k e l y models for these surfaces. Moreover the c a l c u l a t i o n s here were made for a metal which i s a r e l a t i v e l y strong s c a t t e r e r and therefore does not correspond to the s i t u a t i o n s f or which the quasidynamical method was i n i t -i a l l y judged to be most h e l p f u l . These observations support the p o s s i b i l i t y of using the quasidynamical method for making preliminary assessments of those t r i a l models that need more d e t a i l e d analyses with f u l l m u l t i p l e - s c a t t e r i n g methods, although further t e s t s are needed to delineate the ranges of scat-t e r i n g strengths and geometrical types for which t h i s conclusion may be applica b l e . I f such ranges can be obtained, then t h i s would c l e a r l y provide a most s i g n i f i c a n t r o l e f o r the'quasidynamical method i n LEED crystallography. In any event t h i s method should have value i n making preliminary assessments -170-of adsorption systems which involve weakly-scattering adsorbates at low coverage, p a r t i c u l a r l y where the number of f r a c t i o n a l - o r d e r beams i s large and the conventional m u l t i p l e - s c a t t e r i n g procedures r a p i d l y become i n t r a c t a b l e . -171-REFERENCES 1. G.A. Somorjai, " P r i n c i p l e s o f Surface Chemistry", Prentice H a l l , Englewood C l i f f , New Jersey (1972). 2. Abdus Salam, ed. "Surface Science", Lectures Presented at an Inter-national Course at T r i e s t e organized by the International Centre for Theoretical Physics, T r i e s t e , International Atomic Energy Agency, Vienna (1975). 3. J.M. Blakely, "Introduction to the Properties of Cr y s t a l Surfaces" Pergamon, New York (1973). 4. S. Andersson, Surface S c i . , 18, 325 (1969). 5. R. Vanselow and S.Y. Tong, "Chemistry and Physics o f S o l i d Surface" CRC Press, Inc. Cleveland, Ohio (1977). 6. E.W. Plummer and T. Gustafsson, Science 198, 165 (1977); J.R. S c h r i e f f e r and P. Soven, Physics Today 28(4), 24 (1975). 7. S. Ino, Japanese J . Appl. Phys. 16, 891 (1977). 8. H.H. Brongersma and J.B. Theeten, Surface S c i . 54, 519 (1976); J.F. Van der Veen, R.G. Smeenk, R. M. Tromp and F. S a r i s , Surface S c i . 79, 219 (1979) . 9. M.J. C a r d i l l o and G.E. Becker, Phys. Rev. Lett. 42, 508 (1979). 10. H.P. Bonzel, Surface S c i . 68, 236 (1977). 11. K. Baron, D.W. Blakely and G.A. Somorjai; Surface S c i . H , 45 (1974). 12. C.J. Davisson and L.H. Germer, Phys. Rev. 3J, 705 (1927). 13. P. Auger, J . Phys. Radium 6, 205 (1925). 14. J . J . Lander, Phys. Rev. A l , 1382 (1953). 15. L.N. Tharp and E.J. Scheibner, J . Appl. Phys. 38, 3320 (1967). 16. R.E. Weber and W.T. Pe r i a , J . Appl. Phys. 38, 4355 (1967). 17. P.W. Palmberg and T.N. Rhodin, J . Appl. Phys. J59, 2425 (1968). 18. C.R. Brundle, J . Vac. S c i . Techn. 11, 212 (1974). 19. H. Ibach i n "Electron Spectroscopy for Surface Analysis", Topics i n Current Physics Vol. 4, ed. H. Ibach, Springer-Verlag (1974). -172-20. C.J. Powell, Surface S c i . J4, 29 (1974). 21. H. Raether, Surface S c i . £, 233 (1967). "22. C.B. Duke, Adv. Chem. Phys. 27, 1 (1974). 23. T.A. Carlson, "Photoelectron and Auger Spectroscopy", Plenum, New York (1975). 24. J.B. Pendry, "Low Energy Electron D i f f r a c t i o n " , Academic Press, New York (1974). 25. M.B. Webb and M.E. Lagally, S o l i d State Physics, 28, 301 (1973). 26. G.E. Rhead, Surface S c i . 68, 20 (1977). 27. N.F.M. Henry and K. Lonsdale, eds. "International Tables for X-ray Crystallography", Vol. 1, The Kynoch Press, Birmingham (1952). 28. E.A. Wood, J . Appl. Phys. 35, 1306 (1964). 29. R.L. Park and H.H. Madden, Surface S c i . 11, 188 (1968). 30. P.J. Estrup and E.G. McRae, Surface S c i . 25, 1 (1971). 31. C C . Chang, Surface S c i . 25, 53 (1971). 32. T.W. Haas and J.T. Grant, Phys. Rev. Lett. 30A, 272 (1969)-, J . Vac. S c i . Technol. 2, 43 (1970). 33. F.J. Szalkowski and G.A. Somorjai, J . Chem. Phys. 61, 2065 (1974). 34. K. Siegbahn et a l . , "ESCA: Atomic, Molecular, and S o l i d State Structures Studied by the Means of Electron Spectroscopy", Almquist and Wiksells, Uppsala (1967). 35. Y. Strausser and J . J . Uebbing, "Varian Chart of Auger Electron Energies", Varian Corp., Palo A l t o (USA) (1970). 36. P.W. Palmberg, G.E. Riach, R.E. Weber, and N.C. MacDonald, "Handbook of Auger Electron Spectroscopy", Phys. Elec. Ind. Inc., Edina, Minnesota (1972). 37. P.W. Palmberg, G.K. Bohn and J.C. Tracy, Appl. Phys. Lett. 15, 254 (1969). .38. C C . Chang, Surface S c i . 48, 9 (1975). 39. M.P. Seah, Surface S c i . 32, 703 (1972). -173-40. H.P. Bonzel, Surf. S c i . 27, 387 (1971). 41. W.M. Mularie and W.T. Pe r i a , Surface S c i . 26, 125 (1971). -42. A.E. Rae and M. Bebbington, "An Annotated Bibliography of Ruthenium, Rhodium and Iridium as Catalysts", Int. Nickel Co. Inc., New York (1959). 43. P.R. Watson, F.R. Shepherd, D.C. Frost, and K.A.R. M i t c h e l l , Surface S c i . 72, 562 (1978). 44. F.R. Shepherd, P.R. Watson, D.C. Frost, and K.A.R. M i t c h e l l , J. Phys. C 11, 4591 (1978). 45. E. Zanazzi and F. Jona, Surface S c i . 62, 61 (1977). til 46. S.Y. Tong, M.A. Van Hove, and B.J. Mrstik, Proc. 7 Intern. Vacuum Congr. and t h i r d Intern. Conf. on S o l i d Surfaces, Vienna, p. 2407 (1977). 47. S.Y. Tong and A.L. Maldonado, Surface S c i . 78., 459 (1978). 48. S. Andersson and B. Kasemo, Surface S c i . 25, 2?3 (1971). 49. R.W. James, "The Optical P r i n c i p l e s of D i f f r a c t i o n of X-ray", Cornell U n i v e r s i t y Press, Ithaca (1965). 50. T.B. Rymer, "Electron D i f f r a c t i o n " , Methuen (1970). 51. N.F. Mott and H.S.W. Massey, "The Theory of Atomic C o l l i s i o n s " , Oxford U n i v e r s i t y Press (1965). 52. R.M. Stern and F. Balibar, Phys. Rev. Lett. 25, 1338 (1970). 53. R.L. Dennis and M.B. Webb, J . Vac. S c i . Technol. IQ, 192 (1973). 54. D. Tabor, J.M. Wilson, and T.J. Bastow, Surface S c i . 20, 471 (1971). 55. L. Hedin and S. Lundqvist, S o l i d State Phys. 23, 1 (1969). 56. J.E, Demuth, P.M. Marcus and D.W. Jepsen, Phys. Rev. B 11, 1460 (1975). 57. D.W. Jepsen, P.M. Marcus and F. Jona, Phys. Rev. B5, 3933 (1972). 58. D.P. Jepsen, P.M. Marcus and F. Jona, Phys. Rev. B8, 5523 (1973). 59. P.M. Marcus, J.E. Demuth, and D.W. Jepsen, Surface S c i . 53, 501 (1975). 60. T.N. Rhodin and S.Y. Tong, Physics Today, 28(10), 23 (1975). -174-61. S.Y. Tong, J.B. Pendry and L.L. Kesmodel, Surface S c i . 54, 21 (1976). 62. J.C. S l a t e r , Phys. Rev. 81, 385 (1951). 63. L.J. S c h i f f , "Quantum Mechanics", McGraw-Hill, New York (1968). 64. R.G. Newton, "Scattering Theory of Waves and P a r t i c l e s " , McGraw-Hill, New York (1966). 65. S.Y. Tong, Prog, i n Surf. S c i . 7, 1 (1975). 66. N. Stoner, M.A. Van Hove, and S.Y. Tong, i n "Characterization of Metal and Polymer Surfaces", ed. L.H. Lee, Academic Press, New York (1976). 67. E.G. McRae, J . Chem. Phys. 45., 3258 (1966). 68. C.B. Duke and C.W. Tucker, Surface S c i . 15, 231 (1969). 69. B.I. Lundqvist, Phys. State. Sol. 32, 273 (1969). 70. J.L. Beeby, J . Phys. C l , 82 (1968). 71. S.Y. Tong and T.N. Rhodin, Phys. Rev. Lett. 26, 711 (1971). 72. S.Y. Tong, T.N. Rhodin, and R.H. T a i t , Phys. Rev. B8, 421; 430 (1973). 73. E.G. McRae, Surface S c i . i l , 479 (1968). 74. J.B. Pendry, J . Phys. C4, 2501; 2514 (1971). 75. K. Kambe, Z. Naturforsch, 22a, 332 (1967). 76. K. Kambe, Z. Naturforsch, 23a, 1280 (1968). 77. D.W. Jepsen, P.M. Marcus and F. Jona, Phys. Rev. Lett. 26, 1365 (1971). 78. M.A. Van Hove and S.Y. Tong, J . Vac. S c i . Technol. 12, 230 (1975). 79. S.Y. Tong and M.A. Van Hove, Phys. Rev. B16, 1459 (1977). 80. R.S. Zimmer and B.W. Holland, J . Phys. C8, 2395 (1975). 81. M.A. Van Hove and S.Y. Tong, "Surface Crystallography by LEED", Springer-Verlag (1979). 82. M.A. Van Hove and J.B. Pendry, J . Phys. C8, 1362 (1975). -175-83. J.E. Demuth, D.W. Jepsen and P.M. Marcus, S o l i d State Comm. 13, 1311 (1973) 84. M.A. Van Hove and S.Y. Tong, Phys. Rev. Lett. 35, 1092 (1975). 85. M.A. Van Hove, S.Y. Tong and E. Elconin, Surface S c i . 64, 85 (1977). 86. D.G. Fedak and N.A. Gjostein, Surface S c i . 77 (1967). 87. 88 A. Dulong, i n "LEED-Surface Structure of S o l i d s " ed. M. Laznicka, Union of Czechoslovak Mathematicians and P h y s i c i s t s , Prague (1972). J.P. Hobson, Adv. C o l l o i d Interface Sci._4_, 79 (1974). 89. W.J. Lange, Physics Today 25, 40 (1972). 90. T. Tom, Physics Today 25, 32 (1972). 91. F. Rosebury, "Handbook of Electron Tube and Vacuum Techniques", Addison-Wesley, Messachusettes (1965). 92. W.H. Kohl, "Handbook of Material and Technique for Vacuum Devices", Reinhold, New York (1967). 93. Research Organic / Inorganic Chemicals Corp. USA. 94. Courtesy of Dr. C.W. Tucker, General E l e c t r i c Research and Development Centre, Schenectady, New York. N.F.M. Henry, H. Lipson and W.A. Wooster, "The Interpretation of X-ray D i f f r a c t i o n Photographs", MacMillan, London (1960). 95, 96. D.G. Castner, B.A. Sexton and G.A. Somorjai, Surface S c i 7J, 519 (1978). 97. R.A. Marbrow and R.M. Lambert.Surface S c i . 67, 489 (1977). 98. H.E. Farnsworth, i n "The Solid-Gas Interfaces" ed. E.A. Flood, Marcel Dekker, New York (1967). 99. E. Bauer, Tech. Metal Res. 2, 502 (1969). 100. F. Jona, J . Phys. Chem. 11, 4271 (1978). 101. P.W. Palmberg, G.K. Bohn, and J.C.Tracy, Appl. Phys. Lett. 15, 524 (1964). 102. J.T. Grant and T.W. Haas, Surface S c i . 21, 76 (1970). 103. W.A. Coghlan and R.E. Clausing, "A Catalog of Calculated Auger Transitions for the Elements", USAEC Report 0RNL-TM-3576, Oak Ridge National Laboratory (1971); Atomic Data j i , 317 (1973). -176-104. J.E. Demuth and T.N. Rhodin, Surface S c i . 42, 261 (1974). 105. L. McDonnell and D.P. Woodruff, Surface S c i . 46, 505 (1974). 106. P.C. S t a i r , T.J. Kaminska, L.L. Kesmodel and G.A. Somorjai, Phys. Rev. B l l , 623 (1975). 107. D.C. Frost, K.A.R. M i t c h e l l , F.R. Shepherd and P.R. Watson, J . Vacuum S c i . Technol. J_2, 1196 (1976). 108. K.A.R. M i t c h e l l , F.R. Shepherd, P.R. Watson and D.C. Frost, Surface S c i . 64, 737 (1977). 109. D.C. Frost, S. Hengrasmee, K.A.R. M i t c h e l l , F.R. Shepherd and P.R.Watson, Surface S c i . Z£, L585 (1978). 110. V.L. Moruzzi, J.F. Janak and A.R. Williams, "Calculations of E l e c t r o n i c properties of metals", Plenum Press, New York (1978). 111. P.R. Watson, Ph.D. Thesis, U n i v e r s i t y of B r i t i s h Columbia (1978). 112. M.A. Van Hove and S.Y. Tong, Surface S c i . 54, 91 (1976). 113. J.A. S t r o z i e r , D.W. Jepsen and F. Jona, i n "Surface Physics of M a t e r i a l s " v o l . 1 ed. J.M. Blakely, Academic Press, New York (1975). 114. M.G. Lagally, i n "Surface Physics of M a t e r i a l s " v o l . I I . ed. J.M. Blakely, Academic Press, New York (1975). 115. K.A. Gschneider, S o l i d State Phys. 16, 275 (1964). 116. L.A. Har r i s , J . Appl. Phys. 38, 1419 (1968). 117. K.O. Legg, M. Prutton and C. Kinniburgh, J . Phys. Chem. 7, 4236 (1974). 118. CM. Chan, P.A. T h i e l , J.T. Yates and W.H. Weinberg, Surface S c i . 76, 296 (1978) 119. C.W. Tucker, J r . , J . Appl. Phys. 37, 3013 (1966). 120. C.W. Tucker, J r . , J . Appl. Phys. 3J, 4147 (1966). 121. C.W. Tucker, J r . , J . Appl. Phys. 38, 2696 (1967). 122. C.W. Tucker, J r . , Acta Met. 15, 1465 (1967). 123. S. Hengrasmee, P.R. Watson, D.C. Frost and K.A.R. M i t c h e l l , Surface S c i . , 87, L249 (1979). -177-124. S. Hengrasmee, P.R. Watson, D.C. Frost and K.A.R. M i t c h e l l , Surface S c i . 92, 71 (1980). 125. L. McDonnell, Ph.D. Thesis, U n i v e r s i t y of Warwick, 1974. 126. M. Salmeron and G.A. Somorjai, Surface S c i . 91, 373 (1980). 127. P.A. T h i e l , J.T. Yates, J r . , and W.H. Weinberg, Surface S c i . 82, 22 (1979). 128. H. Froitzheim, i n "Electron Spectroscopy for surface a n a l y s i s " ed. H. Ibach, Topics i n Current Physics, v o l . 4, Springer-Verlag, B e r l i n Heidelberg, New York (1977). 129. Y. Gauthier, D. Aberdam and R.R. Baudoing, Surface S c i , 78, 339 (1978). 130. J.E. Demuth, D.W. Jepsen, and P.M. Marcus, Surface S c i . 45, 733 (1974). 131. J.E. Demuth, D.W. Jepsen, and P.M. Marcus, Phys. Rev. Lett. £1, 540 (1973). 132. S.R. Keleman and T.E. Fischer, Surface S c i . 8_Z, 53 (1979). 133. L. Pauling, "The Nature of The Chemical Bond" Cornell U n i v e r s i t y Press, Ithaca, New York (1960). 134. S. G e l l e r , Acta Crys. 15, 1198 (1962). 135. E. Parthe\ D. Mohnke and F. H u l l i g e r , Acta Cryst. 23, 832 (1967). 136. P. Colamanno and P. O r i o l i , J . Chem. Soc. Dalton Trans. 845 (1976). 137. R.J. Butcher and E. Sinn, J . Am. Chem. Soc. 98, 2440 (1976). 138. R.H. Morris, Ph.D. Thesis, U n i v e r s i t y of B r i t i s h Columbia (1978). 139. F. Jona, Surface S c i . 68, 204 (1977). 140. J.E. Demuth, D.W. Jepsen and P.M. Marcus, Phys. Rev. Lett. 32, 1182 (1974). 141. CM. Chan and W.H. Weinberg, J . Chem. Phys. 21, 5988 (1979). 142. K.O. Legg, F. Jona, D.W. Jepsen and P.M. Marcus, Surface S c i . 6J>, 25 (1977) 143. K.A.R. M i t c h e l l , Surface S c i . 9_2, 79 (1980). "144. K.A.R. M i t c h e l l , Surface S c i . , (in press). 145. S.L. Altmann, C A . Coulson and W. Hume-Rothery, Proc. Roy. Soc. (London) A240, 145 (1957). -178-146. M.A. Van Hove and S.Y. Tong, J . Vacuum S c i . Technol. 12, 230 (1975). 147. CM. Chan and W.H. Weinberg, J . Chem. Phys. 71, 3988 (1979). 148. A. Salwen and J . Rundgren, Surface S c i . 53, 523 (1975). 149. J.E. Demuth, D.W. Jepsen and P.M. Marcus, J . Phys. Chem. _6, L307 (1973). 150. S. Hengrasmee, K.A.R. M i t c h e l l , P.R. Watson and S.J. White, Canadian Journal of Physics. 58 (2), 200 (1980). 151. CM. Chan, K.L. Luke, M.A. Van Hove, W.H. Weinberg and S;P. Withrow, Surface S c i . 78, 386 (1978). 152. J.F. Van der Veen, R.M. Tromp, R.C Smeenk and F.W. S a r i s , Surface S c i . 82, 468 (1979). 153. K.O. Legg, F. Jona, D.W. Jepsen and P.M. Marcus, Phys. Rev. B 16, 5271 (1977). 154. C.W. Tucker and C.B. Duke, Surface S c i . 23, 411 (1970); 29, 237 (1972). 155. F. Jona, H.D. Shih, D.W. Jepsen and P.M. Marcus, J . Phys. Chem. 12, L455 (1979). 1 5 6 _ A. Andersson and J.B. Pendry, S o l i d State Comm. 16., 563 (1975). 157. D.L. Adams and U. Landman, Phys. Rev. B 15, 3775 (1977). -179-Appendices The following appendices contain a l l the experimental data from rhodium surfaces c o l l e c t e d during t h i s work. In a l l cases, the data i s as c o l l e c t e d and has not been smoothed. Appendix Al A3 A4 A5 A6 Surface Angle Rh (100)-(3xl)-0 6=0, <j>=0 2 equal domains A 2 Rh (100)-(3xl)-0 9=0, <f>=0 s i n g l e domain Rh(100)-p(2x2)-S 6=0, ty=0 Expt. 1 Rh(100)-p(2x2)-S 6=0, <$>=0 Expt. 2 Rh(110)-c(2x2)-S 6=0, c|,=0 Expt. 1 Rh(110)-c(2x2)-S 6=10, c}>=135 Expt. 2 -181--182--184-n ergy (ev ) -186-

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

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

Comment

Related Items