UBC Theses and Dissertations

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UBC Theses and Dissertations

Atomic hydrogen on the surface of superfluid helium: the sticking probability and polaronic behavior Zimmerman, Dan Simon 1982

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ATOMIC HYDROGEN ON THE SURFACE OF SUPERFLUID HELIUM: STICKING PROBABILITY AND POLARONIC BEHAVIOR. by DAN SIMON ZIMMERMAN B . S c , The Hebrew U n i v e r s i t y Of J e r u s a l e m , 1980 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES Department Of P h y s i c s We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA September 1982 © Dan Simon Zimmerman, 1982 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my D e p a r t m e n t o r by h i s o r h e r r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f P h y s i c s The U n i v e r s i t y o f B r i t i s h C o l u m b i a 2075 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1W5 D a t e : 12 Sep 82 i i A b s t r a c t A s t u d y i s made of t h e i n t e r a c t i o n between h y d r o g e n atoms and t h e s u r f a c e e l e m e n t a r y e x c i t a t i o n s o f s u p e r f l u i d "He. C a l c u l a t i o n s f o r t h e s t i c k i n g p r o b a b i l i t y and f o r t h e e n e r g y and e f f e c t i v e mass of a h y d r o g e n atom bound t o t h e s u r f a c e a r e p r e s e n t e d . As a f i r s t s t e p i n t h e c a l c u l a t i o n s , we f o r m u l a t e t h e H a m i l t o n i a n d e s c r i b i n g a h y d r o g e n atom i n t e r a c t i n g w i t h t h e s u r f a c e o f "He i n i t s g r o u n d s t a t e t o g e t h e r w i t h t h e i n t e r a c t i o n c o u p l i n g t o t h e s u r f a c e e l e m e n t a r y e x c i t a t i o n s ( t h e r i p p l o n s ) . The d e r i v a t i o n i s b a s e d on t h e a s s o c i a t i o n o f t h e s u r f a c e g r o u n d s t a t e w i t h a f l a t s u r f a c e and t h e e x c i t e d s t a t e s w i t h a s i n u s o i d a l l y v a r y i n g h e i g h t of t h e s u r f a c e . The i n t e r a c t i o n p o t e n t i a l i s d e r i v e d by summing a *He-H atom-atom p a i r p o t e n t i a l o v e r h e l i u m atoms below t h e s u r f a c e . The atom-atom p a i r p o t e n t i a l i s c h o s e n so t h a t t h e d e r i v e d s u r f a c e p o t e n t i a l i s a Morse p o t e n t i a l w i t h p a r a m e t e r s w h i c h a r e f i t t e d t o t h e e f f e c t i v e s u r f a c e p o t e n t i a l d e r i v e d by Mantz and Edwards f o r a h y d r o g e n atom i n t e r a c t i n g w i t h N-1 h e l i u m atoms. The e n e r g y and a n g l e d e p e n d e n t s t i c k i n g p r o b a b i l i t y , S(E , 0 ) , and t h e t h e r m a l l y a v e r a g e d s t i c k i n g p r o b a b i l i t y , S ( T ) , a r e c a l c u l a t e d . R e s u l t s a r e compared w i t h t h e e x p e r i m e n t a l v a l u e o f S ( T ) , S ( T ) = 0 . 0 3 5 ± 0 . 0 0 5 , m e a s u r e d f o r t h e t e m p e r a t u r e r a n g e 0.18<T<0.27K. The p o s s i b i l i t y t h a t a h y d r o g e n atom bound above t h e s u r f a c e may e x h i b i t p o l a r o n i c b e h a v i o u r i s i n v e s t i g a t e d . The e n e r g y o f t h e h y d r o g e n s u r f a c e " p o l a r o n " and i t s e f f e c t i v e mass a r e c a l c u l a t e d u s i n g p e r t u r b a t i o n t h e o r y . The h y d r o g e n atom i s f o u n d t o be w e a k l y c o u p l e d t o t h e s u r f a c e e l e m e n t a r y e x c i t a t i o n s , and t h e r e f o r e t h e p o l a r o n i c e f f e c t s a r e weak. The c o n t r i b u t i o n t o t h e s e r e s u l t s f r o m v i r t u a l t r a n s i t i o n t o f r e e s t a t e s i s e m p h a s i z e d . i v T a b l e of C o n t e n t s A b s t r a c t i i L i s t o f T a b l e s y L i s t o f F i g u r e s ..yi Acknowledgement v i i I . INTRODUCTION 1 I I . ATOMIC HYDROGEN AND THE SURFACE OF "HE 3 2.1 S p i n P o l a r i z e d H y drogen 3 2.2 The UBC E x p e r i m e n t 6 2.3 S u r f a c e P o l a r o n s 10 2.4 The Model F o r The S u p e r f l u i d H e l i u m S u r f a c e 11 I I I . THE HAMILTONIAN 14 3.1 The Atom S u r f a c e C o u p l i n g 14 3.2 The System H a m i l t o n i a n 17 3.3 The Morse P o t e n t i a l 19 3.4 P r o p e r t i e s Of The M a t r i x E l e m e n t s 23 IV. THE STICKING PROBABILTY 24 4.1 T h e o r y 24 4.2 The Low T e m p e r a t u r e A p p r o x i m a t i o n 26 4.3 RESULTS 27 4.3.1 S(E , e ) 27 4.3.2 S ( T ) 30 V. THE HYDROGEN SURFACE POLARON 3 3 5.1 I n t r o d u c t i o n 33 5.2 The H y d r o g e n - R i p p l o n I n t e r a c t i o n E n e r g y 35 5.2.1 T h e o r y 35 5.2.2 The Long W a v e l e n g t h L i m i t 38 5.2.3. T=0 k = 0 39 5.3 Hydrogen P o l a r o n ? 42 5.3.1 GMY L o c a l i z a t i o n 43 5.3.2 The E f f e c t i v e Mass At T=0K 44 5.3.3 The R i p p l o n C l o u d 46 5.4 C o n c l u s i o n 47 BIBLIOGRAPHY 49 APPENDIX A - NUMERICAL METHODS FOR INTEGRATION 52 L i s t o f T a b l e s The measured t e m p e r a t u r e d e p e n d e n c e o f t h e r e l a x a t i o n r a t e l / T ^ and t h e f r e q u e n c y s h i f t A w, a l o n g w i t h t h e r e s u l t i n g v a l u e s of t h e a v e r a g e d s t i c k i n g p r o b a b i l i t y v i L i s t of F i g u r e s The Morse s u r f a c e p o t e n t i a l ( t h e s o l i d l i n e ) compared t o t h e ME p o t e n t i a l ( t h e c i r c l e s ) 22 The s t i c k i n g p r o b a b i l i t y a t 0 = 4 5 ° a s a f u n c t i o n o f t h e atom e n e r g y f o r v a r i o u s v a l u e s o f t h e f i l m t e m p e r a t u r e T 29 The t h e r m a l l y a v e r a g e d s t i c k i n g p r o b a b i l i t y S ( T ) v e r s u s t h e common t e m p e r a t u r e T o f t h e f i l m and t h e gas 32 v i i A cknowledgement I would l i k e t o thank my s u p e r v i s o r A . J . B e r l i n s k y f o r t h e tremendous amount of h e l p , a d v i c e and knowledge w h i c h I g o t from him d u r i n g t h e l a s t 24 months. S p e c i a l t h a n k s a r e due f o r t a k i n g me w i t h him t o t h e p h y s i c s d e p a r t m e n t of M.I.T where he s t a y e d d u r i n g h i s s a b b a t i c a l . I am p l e a s e d t o a c k n o w l e d g e h e l p and u s e f u l d i s c u s s i o n s w i t h members of t h e A t o m i c H y d r o g e n g r o u p o f UBC: W.N. Hardy, B.W. S t a t t , M. Morrow, R. Jochemsen, and J . de B r u y n . Thanks a r e due t o t h e h o s p i t a l i t y and s u p p o r t o f t h e M.I.T p h y s i c s d e p a r t m e n t , where p a r t of t h i s work was done, s p e c i a l l y t o T . J . G r e y t a k , D. K l e p p n e r and R. C l i n e . T hanks a r e a l s o due t o M.A. P o t t s , B. S u l l i v a n and B. S t a t t f o r a d v i c e c o n c e r n i n g t h e use o f c o m p u t e r s . I thank D. Edwards f o r s e n d i n g us t h e Mantz and Edwards p o t e n t i a l . I am p l e a s e d t o t h ank t h e N a t i o n a l S c i e n c e F o u n d a t i o n of t h e USA, t h e N a t u r a l S c i e n c e s and E n g i n e e r i n g R e s e a r c h C o u n c i l of Canada and t h e A l f r e d P. S l o a n F o u n d a t i o n f o r f i n a n c i a l s u p p o r t . F i n a l l y , I thank t h e p e o p l e of Canada and t h e p e o p l e of t h e U n i t e d S t a t e s o f A m e r i c a f o r a l l o w i n g me t o p u r s u e my e d u c a t i o n i n t h i s c o n t i n e n t of f r e e d o m , p e a c e and p r o s p e r i t y . 1 I . INTRODUCTION E x p e r i m e n t a l e f f o r t s t o s t a b i l i z e h i g h d e n s i t i e s of s p i n p o l a r i z e d a t o m i c h y d r o g e n a t t e m p e r a t u r e s below 1K have l e d t o a g r e a t d e a l o f i n t e r e s t i n t h e p r o p e r t i e s o f a t o m i c h y d r o g e n . i n t e r a c t i n g w i t h t h e f r e e s u r f a c e o f s u p e r f l u i d "He. 1 In t h e e x p e r i m e n t s p e r f o r m e d t h e w a l l s of t h e c e l l i n w h i c h t h e atoms a r e c o n f i n e d a r e c o a t e d w i t h s u p e r f l u i d "He. The r e a s o n i s t h a t t h e w a l l s n e c e s s a r i l y a t t r a c t t h e h y d r o g e n atom t h r o u g h Van d e r Waals f o r c e s , and t h e r e f o r e some f r a c t i o n o f t h e atoms a r e t r a p p e d i n a bound s t a t e above t h e s u r f a c e . A t a g i v e n t e m p e r a t u r e t h e d e n s i t y o f atoms on t h e s u r f a c e i s a s t r o n g l y i n c r e a s i n g f u n c t i o n of t h e b i n d i n g e n e r g y and t h e r e f o r e one d e s i r e s t o m i n i m i z e t h e s u r f a c e d e n s i t y by c h o o s i n g a s u r f a c e w i t h t h e s m a l l e s t p o s s i b l e b i n d i n g e n e r g y . The l o w e s t p o s s i b l e v a l u e s f o r t h e b i n d i n g e n e r g y a r e a c h i e v e d by c o a t i n g t h e w a l l s of t h e c e l l w i t h s u p e r f l u i d "He, w i t h 3He o r w i t h 3He-"He m i x t u r e s . At low d e n s i t i e s t h e atoms i n t h e bound s t a t e move as f r e e p a r t i c l e s , on t h e s u r f a c e , r e s e m b l i n g a two d i m e n s i o n a l i d e a l g a s . A h y d r o g e n atom, i n t h e bound s t a t e , d i s t o r t s t h e s u r f a c e b e n e a t h i t . W h i l e moving p a r a l l e l t o t h e s u r f a c e , t h e atom c a r r i e s t h e d i s t o r t i o n a l o n g w i t h i t s m o t i o n , and t h u s i t a c q u i r e s an e f f e c t i v e mass m* g r e a t e r t h a n t h e h y d r o g e n mass m. A h y d r o g e n atom i n t h e g a s , w i t h e n e r g y E, i n c i d e n t on t h e s u r f a c e f r o m an a n g l e 0 t o t h e n o r m a l t o t h e s u r f a c e has a p r o b a b i l i t y S(E,0) o f s t i c k i n g t o t h e s u r f a c e . The t h e r m a l l y a v e r a g e d s t i c k i n g p r o b a b i l i t y S(T)=<S(E,9)> has been measured by 2 Jochemesen e t a l . 5 and a v a l u e of S=0.035±0.005 was found i n the te m p e r a t u r e range 0.18<T<0.27K. In Chapter 2 we e x p l a i n t h e m o t i v a t i o n f o r p e r f o r m i n g t h e s e e x p e r i m e n t s and the reas o n s f o r u s i n g l i q u i d h e l i u m s u r f a c e s . We d e s c r i b e the e x p e r i m e n t s performed by the UBC atomic hydrogen group i n which the b i n d i n g energy and the s t i c k i n g p r o b a b i l i t y were measured.'" 5 Then the model which we used t o d e s c r i b e the s u r f a c e i s p r e s e n t e d and f i n a l l y the work of Guyer, M i l l e r and Y a p l e , which suggested t h a t a hydrogen atom becomes " l o c a l i z e d " above the s u r f a c e , i s i n t r o d u c e d . 1 " In Chapter 3 the h y d r o g e n - s u r f a c e p o t e n t i a l i s d e r i v e d a l o n g w i t h the c o u p l i n g t o the elementary s u r f a c e modes. A model H a m i l t o n i a n d e s c r i b i n g the system i s i n t r o d u c e d , and i t s p r o p e r t i e s a re d e s c r i b e d . In Chapter 4 the s t i c k i n g p r o b a b i l i t y and the t h e r m a l l y averaged s t i c k i n g p r o b a b i l i t y a r e c a l c u l a t e d . T h e o r e t i c a l r e s u l t s f o r the t h e r m a l l y averaged s t i c k i n g p r o b a b i l i t y a r e compared w i t h e x p e r i m e n t a l v a l u e s . The dependence of the r e s u l t s on the c o u p l i n g f u n c t i o n t o the r i p p l o n s i s emphasized. In Chapter 5, the ground s t a t e energy of the system i s c a l c u l a t e d u s i n g p e r t u r b a t i o n t h e o r y and t h e e f f e c t i v e mass, m*, f o r an atom moving on the s u r f a c e i s e s t i m a t e d . The importance of p o l a r o n i c e f f e c t s f o r the bound hydrogen atom i s e x p l o r e d . 3 II . ATOMIC HYDROGEN AND THE SURFACE OF "HE 2.1 Spin Polarized Hydrogen The interaction between two hydrogen atoms depends on the t o t a l electronic spin, S, of the atoms. 1 5 For S=0, the interaction potential has a large well depth, and therefore two hydrogen atoms interacting through this singlet potential form a bound state, i . e . , recombine into a hydrogen molecule with a binding energy of 4.5ev (50,000K). For S=1, the potential i s mainly repulsive except for a weak Van der Waals at t r a c t i o n whose depth well i s 6.5K at a separation of 4.15A. 1 6" 1 8 The t r i p l e t potential resembles the *He-*He atom-atom potential which has a deeper well (of about 1 0 K ) . 1 9 Applying a magnetic f i e l d w i l l align the spins and force the atoms to interact through the t r i p l e t p o t e n t i a l , suggesting that a magnetic f i e l d and low temperatures would be enough to prevent recombination of hydrogen atoms into molecules. Unfortunately the nuclear spin perturbs the electronic spin states and therefore one of the two lowest hyperfine energy level s of a hydrogen atom in a large magnetic f i e l d contains a small admixture of elec t r o n i c spin which i s not aligned with the magnetic f i e l d . Thus a slow recombination rate s t i l l remains even after the magnetic f i e l d i s applied. Hydrogen atoms, l i k e helium atoms, obey Bose-Einstein s t a t i s t i c s . Due to the s i m i l a r i t y of the atom-atom interaction potential and the same s t a t i s t i c s , the low temperature phase of spin polarized hydrogen atoms i s expected to show quantum 4 p r o p e r t i e s s i m i l a r to "He at low temperature. C a l c u l a t i o n s f o r the low temperature phase of a system of hydrogen atoms suggest t h a t , f o r pressures below 100 bars, the system w i l l remain gaseous even at T=0K. 2 0 - 2 2 At a c e r t a i n temperature T Q , which depends on the d e n s i t y , n«, the gas i s expected to undergo a phase t r a n s i t i o n i n t o a B o s e - E i n s t e i n condensed s t a t e , i . e . a s t a t e where a macroscopic f r a c t i o n of the hydrogen atoms occupy the zero momentum s t a t e . T h i s phase with i t s e x o t i c quantum p r o p e r t i e s i s the u l t i m a t e goal of the experimental e f f o r t s . A s t i l l u nresolved q u e s t i o n i s the r e l a t i o n between the s u p e r f l u i d phase t r a n s i t i o n observed e x p e r i m e n t a l l y f o r "He and B o s e - E i n s t e i n condensation. A m i c r o s c o p i c c a l c u l a t i o n which shows a ' phase t r a n s i t i o n of the B o s e - E i n s t e i n type has not yet been performed. The d i f f i c u l t y i s a consequence of the strong i n t e r a c t i o n between "He atoms which make i t impossible to f i n d a small expansion parameter to sol v e the system p e r t u r b a t i v e l y . Hydrogen atoms at the d e n s i t i e s needed to observe B o s e - E i n s t e i n condensation are weakly i n t e r a c t i n g and t h e r e f o r e a p e r t u r b a t i v e approach i s j u s t i f i e d . Comparison between theory and experiment may shed a new l i g h t on the r e l a t i o n between the s u p e r f l u i d phase t r a n s i t i o n and B o s e - E i n s t e i n condensation. The temperature T Q i n which the phase t r a n s i t i o n i s expected to occur i s r e l a t e d to the d e n s i t y through the r e l a t i o n : (1) T0 = UMcr 1 4 m.H^ j< 5 w h i c h means, f o r example, t h a t a t T=0.1K a d e n s i t y o f 1 . 6 X 1 0 1 9 atoms/cm 3 i s needed t o o b s e r v e c o n d e n s a t i o n . The h i g h e s t d e n s i t i e s a l r e a d y a c h i e v e d i n l a b o r a t o r i e s a r e 2-3 o r d e r s of m a g n i t u d e s m a l l e r . The main d i f f i c u l t y i n a c h i e v i n g t h e s e d e n s i t i e s i s r e c o m b i n a t i o n o f h y d r o g e n atoms i n t o h y d r o g e n m o l e c u l e s . R e c o m b i n a t i o n o c c u r s as a 3 body p r o c e s s i n t h e gas and a s a 2 body p r o c e s s on t h e s u r f a c e o f t h e c o n t a i n e r i n w h i c h t h e atoms a r e c o n t a i n e d , s i n c e t h e s u r f a c e a c t s as t h e t h i r d body needed t o c o n s e r v e e n e r g y and momentum. Ass u m i n g t h a t t h e h y d r o g e n atoms, t r a p p e d above t h e s u r f a c e , a r e i n t h e r m a l e q u i l i b r i u m w i t h h y d r o g e n atoms i n t h e b u l k , t h e d e n s i t y of t h e atoms on t h e s u r f a c e i s g i v e n i n t h e low d e n s i t y l i m i t by: where n s and n f e a r e s u r f a c e and b u l k d e n s i t i e s , Eg i s t h e b i n d i n g e n e r g y , A i s t h e t h e r m a l de B r o g l i e w a v e l e n g t h and k i s B o l t z m a n n c o n s t a n t . S o l i d s u r f a c e s b i n d a t o m i c h y d r o g e n w i t h l a r g e b i n d i n g e n e r g i e s ( t h e l e a s t b i n d i n g i s m o l e c u l a r h y d r o g e n w i t h a b i n d i n g e n e r g y o f 3 8 K 2 2 ) , i m p l y i n g t h a t a t t e m p e r a t u r e s below 1K e s s e n t i a l l y a l l t h e atoms w i l l s t i c k t o t h e s o l i d s u r f a c e and r e c o m b i n e . The o n l y a l t e r n a t i v e i s t o use l i q u i d h e l i u m t o c o a t t h e w a l l s o f t h e c o n t a i n e r . F o r h e l i u m s u r f a c e s t h e b i n d i n g e n e r g y was measured by s e v e r a l g r o u p s g i v i n g t h e f o l l o w i n g 6 r e s u l t s : a) "He s u r f a c e : 1.15±0.05K Morrow e t a l . " 1.01±0.06K C l i n e e t a l . 7 0.89±0.07K M a t t h e y e t a l . 1 2 b) 3He s u r f a c e : 0.42±0.05K Jochemsen e t a l . 5 c ) 3He-"He s u r f a c e : 0.34±0.03K van Y p e r e n e t a l . 1 3 In a d d i t i o n t o t h e low b i n d i n g e n e r g y needed t o m i n i m i z e s u r f a c e d e n s i t i e s , t h i c k h e l i u m f i l m s a r e a l s o e x p e c t e d t o s h i e l d a t o m i c h y d r o g e n from p a r a m a g n e t i c i m p u r i t i e s on t h e s u b s t r a t e below t h e l i q u i d . 2.2 The UBC E x p e r i m e n t The s t i c k i n g p r o b a b i l i t y and t h e b i n d i n g e n e r g y were measured f o r "He and 3He s u r f a c e s by t h e UBC a t o m i c h y d r o g e n g r o u p . * " 5 A c y l i n d r i c a l p y r e x c e l l of d i a m e t e r 1cm and l e n g t h 7cm i s f i l l e d a t room t e m p e r a t u r e w i t h h y d r o g e n m o l e c u l e s and h e l i u m atoms. A f t e r t h e c e l l i s c o o l e d , t h e h y d r o g e n m o l e c u l e s form a s o l i d and t h e h e l i u m atoms form a l i q u i d on t h e w a l l s o f t h e c e l l . An r f d i s c h a r g e p u l s e i s u s e d t o c a u s e h y d r o g e n m o l e c u l e s t o e v a p o r a t e from t h e w a l l s and d i s s o c i a t e i n t o h y d r o g e n atoms. A f t e r t h e t e r m i n a t i o n of t h e d i s c h a r g e p u l s e s , t h e r e m a i n i n g h y d r o g e n m o l e c u l e s and h e l i u m atoms c o n d e n s e a g a i n on t h e w a l l s w h i l e some amount of h y d r o g e n atoms r e m a i n t r a p p e d i n s i d e t h e c e l l i n a g a s e o u s s t a t e . Maximum d e n s i t i e s o f 0.5x10' 3 atoms/cm 3 were o b t a i n e d i n t h i s way. A p u l s e d m a g n e t i c r e s o n a n c e t e c h n i q u e was u s e d t o o b s e r v e 7 t r a n s i t i o n s between t h e a t o m i c h y d r o g e n h y p e r f i n e l e v e l s |00> and |10> . The atoms were o b s e r v e d a t t e m p e r a t u r e s l e s s t h a n 1K, w i t h z e r o m a g n e t i c f i e l d , a t a d e n s i t y o f t h e o r d e r o f 1 0 1 0 t o 1 0 1 1 a t o m s / c m 3 . The h y p e r f i n e f r e q u e n c y o b s e r v e d i s s h i f t e d f r o m i t s v a l u e f o r f r e e h y d r o g e n atoms f 0 = 1 4 2 0 4 0 5 7 5 1 . 7 6 8 ( 2 ) H Z 2 4 , due t o t h e i n t e r a c t i o n o f t h e atoms w i t h t h e s u r f a c e and due t o a " p r e s s u r e s h i f t " p r o p o r t i o n a l t o t h e d e n s i t y o f "He atoms i n t h e c e l l . The p r e s s u r e s h i f t i s n e g l i g i b l e below 1K and t h e r e f o r e most o f t h e c o n t r i b u t i o n t o t h e f r e q u e n c y s h i f t comes f r o m t h e s u r f a c e atoms. The f r e q u e n c y s h i f t d epends on t h e s u r f a c e d e n s i t y and t h e r e f o r e i s s t r o n g l y t e m p e r a t u r e d e p e n d e n t . I f A s i s t h e change i n t h e h y p e r f i n e f r e q u e n c y f o r a s u r f a c e atom, t h e a v e r a g e phase s h i f t p e r s t i c k i n g e v e n t i s g i v e n by Jf = 3i1TTs A s where Xs i s t h e a v e r a g e t i m e an atom spends on t h e w a l l . A s s u m i n g a P o i s s o n d i s t r i b u t i o n of t i m e s on t h e s u r f a c e ( w i t h as a v e r a g e ) and a P o i s s o n d i s t r i b u t i o n of t i m e s i n t h e g a s , w i t h an a v e r a g e t i m e t ^ , t h e f r e e i n d u c t i o n d e c a y c u r v e i s g i v e n by: (3) s e t ; = s(o> e. t where: (4) (5) (6) _&U) and 1/T-. were measured a t d i f f e r e n t t e m p e r a t u r e s and t h e 8 v a l u e s o b t a i n e d a r e shown i n T a b l e 1. By e l i m i n a t i n g >^ , E q u a t i o n s 5 and 6 a r e s o l v e d f o r Xy i n t e r m s o f t h e measured q u a n t i t i e s Au> and 1/T^ g i v i n g : The s t i c k i n g p r o b a b i l i t y S i s g i v e n by S= " ^ c / t ^ where T c i s t h e a v e r a g e t i m e between w a l l c o l l i s i o n s . t e was e s t i m a t e d u s i n g a computer s i m u l a t i o n f o r atoms i n s i d e a c e l l whose d i m e n s i o n s a r e s i m i l a r t o t h o s e o f t h e c e l l u s e d i n t h e e x p e r i m e n t . A v a l u e of t c = 7 . 4x 1 0" 5 T ~ °-5sec was t h e r e b y o b t a i n e d . U s i n g E q u a t i o n 7 and t h e v a l u e of t t , S was c a l c u l a t e d a t v a r i o u s t e m p e r a t u r e s . An e x p e r i m e n t a l e r r o r e q u a l t o 10% o f 1/T^ was a t t r i b u t e d t o A L O and t o 1/T A. T h e s e r e s u l t s a r e shown i n T a b l e 1. The b i n d i n g e n e r g y was o b t a i n e d from v a l u e s of AU) i n t h e t e m p e r a t u r e r a n g e where t h e f r a c t i o n of t h e atoms on t h e w a l l s i s s m a l l . Then AlO i s g i v e n by: A E a / f t T (9) AOO = ZTTAsf\~^ where A i s t h e s u r f a c e a r e a o f t h e c e l l and V i s t h e volume. The b i n d i n g e n e r g y i s o b t a i n e d from t h e s l o p e of XVI(AU?T^) a s a f u n c t i o n o f 1/T. The v a l u e E ^ = 1 . 1 5 ± 0 . 0 5 K was o b t a i n e d f o r "He and E = 0 . 4 2 ± 0 . 0 5 K f o r 3He. The v a l u e S = 0 . 0 1 6 ± 0 . 0 0 5 was o b t a i n e d f o r t h e s t i c k i n g p r o b a b i l i t y on 3He. 9 T 1/T* A CO S K e l v i n s e c " 1 s e c " 1 0 . 2 7 4 3 . 3 2 6 . 14 0 . 0 3 0 ± 0 . 0 1 0 0 . 2 2 5 1 8 . 2 6 9 . 6 8 0 . 0 4 4 ± 0 . 0 0 6 0 . 2 0 6 41 . 7 1 0 7 . 1 3 0 . 0 5 1 ± 0 . 0 0 7 0 . 1 9 8 7 7 1 0 5 . 6 0 . 0 3 7 ± 0 . 0 0 5 0 . 1 8 5 1 0 5 1 0 4 . 3 0 . 0 3 6 ± 0 . 0 0 4 0 . 1 8 0 1 5 2 8 5 . 7 7 0 . 0 3 5 ± O . 0 0 5 0 . 1 7 1 1 8 9 5 0 . 8 9 0 . 0 3 6 ± 0 . 0 0 5 T a b l e I - The measured temperature dependence of the r e l a x a t i o n r a t e 1/T^ and the fr e q u e n c y s h i f t Aw, a l o n g w i t h the r e s u l t i n g v a l u e s of the averaged s t i c k i n g p r o b a b i l i t y S. 10 2.3 S u r f a c e P o l a r o n s G u y e r , M i l l e r and Y a p l e (GMY) s u g g e s t e d t h e p o s s i b i l i t y t h a t a h y d r o g e n atom, bound t o t h e s u r f a c e , w o u l d r e d u c e i t s e n e r g y by becoming " l o c a l i z e d " i . e . i t s e f f e c t i v e mass would become g r e a t l y e n h a n c e d . A l t h o u g h a p o s i t i v e k i n e t i c e n e r g y i s a s s o c i a t e d w i t h l o c a l i z a t i o n , a r e d u c t i o n i n t h e t o t a l e n e r g y may be o b t a i n e d b e c a u s e t h e n e g a t i v e a t o m - s u r f a c e i n t e r a c t i o n p o t e n t i a l may i n c r e a s e i n m a g n i t u d e , t h u s becoming more n e g a t i v e , a s t h e t h e atom becomes l o c a l i z e d . GMY r e f e r r e d t o t h e l o c a l i z e d atom a s a " s u r f a c e p o l a r o n " . GMY assumed a G a u s s i a n wave p a c k e t w i t h h a l f w i d t h # f o r t h e two d i m e n s i o n a l ( t h e p a r a l l e l ) wave f u n c t i o n of t h e bound h y d r o g e n . They assumed an ad-hoc ^ - f u n c t i o n c o n t a c t p o t e n t i a l and o b t a i n e d m a t r i x e l e m e n t s f o r t h e h y d r o g e n - s u r f a c e c o u p l i n g w h i c h were i n d e p e n d e n t of r i p p l o n wave v e c t o r s . GMY c a l c u l a t e d t h e p o s i t i v e k i n e t i c e n e r g y t e r m and t h e n e g a t i v e i n t e r a c t i o n t e r m and l o o k e d f o r t h e c o n d i t i o n s w h i c h would g i v e a t o t a l n e g a t i v e e n e r g y . They a s s o c i a t e d a " l o c a l i z e d " h y d r o g e n atom w i t h t h i s n e g a t i v e e n e r g y . The GMY work i s not b a s e d on a r e a l i s t i c s u r f a c e - a t o m p o t e n t i a l and t h e r e f o r e s h o u l d be r e g a r d e d as b e i n g o n l y q u a l i t a t i v e . A major s h o r t c o m i n g i s t h a t , t h e h a l f w i d t h , c a n n o t be e s t i m a t e d from GMY c a l c u l a t i o n s and t h e r e f o r e t h e p o l a r o n s i z e o r i t s e n e r g y a r e n o t g i v e n . 11 2.4 The Model For The S u p e r f l u i d Helium Surface Throughout t h i s work we w i l l use the f o l l o w i n g model, in t r o d u c e d by F r e n k e l 2 5 and A t k i n s 2 6 , to d e s c r i b e the p r o p e r t i e s of the s u r f a c e of "He. The helium l i q u i d i s t r e a t e d as an i n c o m p r e s s i b l e , non-v i s c o u s c l a s s i c a l l i q u i d i n which the only p o s s i b l e d e n s i t y f l u c t u a t i o n s are s u r f a c e waves r e s t o r e d by the s u r f a c e t e n s i o n and by the presence of the ea r t h ' s g r a v i t a t i o n a l f i e l d . F r e n k e l proposed t h a t the elementary e x c i t a t i o n s , of the f r e e s u r f a c e of "He are q u a n t i z e d s u r f a c e waves, c a l l e d r i p p l o n s , and A t k i n s e x p l a i n e d the temperature dependence of the s u r f a c e t e n s i o n by assuming a c o n t r i b u t i o n to the s u r f a c e energy from t h e r m a l l y e x c i t e d r i p p l o n s . A quantum hydrodynamic f o r m u l a t i o n of the problem g i v e s the f o l l o w i n g q u a n t i z a t i o n f o r the modes: 2 7 where h(r*) i s the displacement of the s u r f a c e , at p o s i t i o n r , from i t s e q u i l i b r i u m height at z=0, <^  =0.378 erg/cm 2 i s the surf a c e t e n s i o n at T=0K, J? o=0.145 g/cm3 i s the d e n s i t y at zero (10) 12 p r e s s u r e , g i s t h e e a r t h ' s g r a v i t a t i o n a l a c c e l e r a t i o n and r ^ i s a c r e a t i o n o p e r a t o r f o r a s u r f a c e e l e m e n t a r y e x c i t a t i o n (a r i p p l o n ) w i t h momentum q and e n e r g y tW^ . T h i s m o d e l d o e s n o t t a k e i n t o a c c o u n t t h e c o m p r e s s i b i l i t y o f t h e l i q u i d , w h i c h g i v e s r i s e t o o t h e r modes ( p h o n o n s , r o t o n s . . . ) . A l s o b e i n g an h y d r o d y n a m i c t r e a t m e n t , i t becomes i n a c c u r a t e f o r r i p p l o n s w i t h wave v e c t o r s c o m p a r a b l e i n m a g n i t u d e t o t h e i n v e r s e i n t e r a t o m i c d i s t a n c e . F o r t h i n h e l i u m f i l m s t h e Van d e r W a a l s a t t r a c t i o n b e t w e e n t h e s u b s t r a t e a n d t h e l i q u i d becomes i m p o r t a n t 2 8 a n d t h e c o n s t a n t g i n E q u a t i o n 10 h a s an a d d e d t e r m e q u a l t o t h e f o r c e p e r u n i t mass a t t h e f r e e s u r f a c e due t o t h e s u b s t r a t e . A l s o t h e d i s p e r s i o n r e l a t i o n and h(r*) a r e m o d i f i e d t o a l l o w f o r t h e f i n i t e t h i c k n e s s o f t h e f i l m : 2 9 ( 1 1 ) where d i s t h e t h i c k n e s s o f t h e f i l m . F o r v a l u e s o f q > l O ~ 2 A ~ 1 t h e g r a v i t a t i o n a l t e r m i s n e g l i b l e c o m p a r e d t o t h e s u r f a c e t e n s i o n t e r m and c a n be i g n o r e d . A q u a n t i t y w h i c h w i l l become i m p o r t a n t i n what f o l l o w s , i s t h e t h e r m a l l y a v e r a g e d mean s q u a r e d i s p l a c e m e n t of t h e s u r f a c e <h 2>. Widom n o t i c e d , t h a t w i t h g = 0 , <h 2> i s d i v e r g e n t due t o t h e l o n g w a v e l e n g t h r i p p l o n s . 3 0 C o l e o b s e r v e d t h a t t h e d i v e r g e n c e i s removed f o r n o n - z e r o v a l u e s o f g . 3 1 C o l e f o u n d 13 t h a t <h 2> c a n be w r i t t e n a s a sum o f a t e m p e r a t u r e d e p e n d e n t t e r m a n d a t e m p e r a t u r e i n d e p e n d e n t t e r m : The f i r s t t e r m , w h i c h d o e s n o t d e p e n d on g o r on t h e t e m p e r a t u r e T o f t h e s u r f a c e , c o r r e s p o n d s t o t h e c o n t r i b u t i o n f r o m t h e z e r o p o i n t m o t i o n o f t h e r i p p l o n s . I t d e p e n d s on t h e r i p p l o n c u t - o f f q ^ a s q ^ 5 w h i c h makes t h e v a l u e o f t h i s t e r m r a t h e r u n c e r t a i n . The v a l u e q^=1.0A _ 1 was o b t a i n e d by r e q u i r i n g t h a t t h e t o t a l number o f s u r f a c e modes be e q u a l t o t h e number o f atoms i n a m o n o l a y e r a t t h e s u r f a c e . 2 5 The s e c o n d t e r m , w h i c h i s t e m p e r a t u r e d e p e n d e n t , d i v e r g e s l o g a r i t h m i c a l l y a s g v a n i s h e s . Most o f t h e c o n t r i b u t i o n s t o t h e s e c o n d t e r m r e s u l t f r o m l o n g w a v e l e n g t h f l u c t u a t i o n s . (12) * T T (So * 14 I I I . THE HAMILTONIAN 3.1 The Atom S u r f a c e C o u p l i n g The i n t e r a c t i o n between a h y d r o g e n atom and t h e s u r f a c e o f "He r e s u l t s from t h e i n t e r a c t i o n between 4He and h y d r o g e n atoms. The "He-H atom-atom p o t e n t i a l i s a t t r a c t i v e f o r l a r g e enough v a l u e s o f i n t e r a t o m i c s e p a r a t i o n and r e p u l s i v e f o r s m a l l e r d i s t a n c e s . T h e r e f o r e t h e s u r f a c e - a t o m p o t e n t i a l a l s o has an a t t r a c t i v e and a r e p u l s i v e p a r t . A h y d r o g e n atom a p p r o a c h i n g t h e s u r f a c e w i l l f i r s t be a t t r a c t e d and t h e n , f o r s m a l l e r d i s t a n c e s , i t w i l l be r e p e l l e d . The *He-H atom-atom p a i r p o t e n t i a l can be p a r a m e t r i z e d by a s s u m i n g a f u n c t i o n a l f o r m f o r t h e d i f f e r e n t p a r t s of t h e p o t e n t i a l w i t h p a r a m e t e r s w h i c h a r e f i t t e d t o e x p e r i m e n t a l d a t a and t h e o r e t i c a l p r e d i c t i o n s . The a t t r a c t i v e p a r t , w h i c h r e s u l t s from Van d e r Waals f o r c e s , i s a sum of i n v e r s e powers w i t h a l e a d i n g t e r m p r o p o r t i o n a l t o X" 6, where X i s t h e i n t e r a t o m i c d i s t a n c e . When i n t e g r a t e d o v e r h e l i u m atoms below t h e s u r f a c e t h i s t e r m g i v e r i s e t o a t e r m p r o p o r t i o n a l t o Z" 3 i n t h e s u r f a c e - h y d r o g e n p o t e n t i a l , where Z i s t h e d i s t a n c e of t h e h y d r o g e n atom from t h e s u r f a c e . The r e p u l s i v e p a r t c a n be modeled by an e x p o n e n t i a l o r by a power law. The a t o m - s u r f a c e p o t e n t i a l c an be g e n e r a t e d by summing t h e atom-atom p o t e n t i a l o v e r t h e l i q u i d . T h i s p r o c e d u r e u s u a l l y d oes n o t g e n e r a t e an a c c u r a t e s u r f a c e p o t e n t i a l . One o f t h e r e a s o n s i s t h a t s h o r t range c o r r e l a t i o n s between t h e gas and t h e s u r f a c e atoms a r e not p r o p e r l y t a k e n i n t o a c c o u n t . An e x a c t a p p r o a c h w o u l d be t o t r y t o s o l v e t h e N-body S c h r o d i n g e r 15 e q u a t i o n f o r N-1 h e l i u m atoms and one h y d r o g e n atom. C l e a r l y t h i s i s n o t an e x a c t l y s o l u b l e p r o b l e m . Mantz and E d w a r d s 3 2 u s e d a v a r i a t i o n a l p r o c e d u r e b a s e d on t h e F e y n m a n - L e k n e r 3 3 " 3 * v a r i a t i o n a l method t o r e l a t e t h e N-body p r o b l e m t o a one body p r o b l e m f o r a h y d r o g e n atom i n an e f f e c t i v e s u r f a c e p o t e n t i a l U 0 ( Z ) . (The r e a s o n f o r t h e s u b s c r i p t z e r o w i l l become a p p a r e n t i n t h e f o l l o w i n g . ) The d e r i v a t i o n t a k e s i n t o a c c o u n t t h e l i q u i d s u r f a c e d e n s i t y p r o f i l e a nd i n c l u d e s t h e e f f e c t s of "He-'He and "He-H . c o r r e l a t i o n s . S o l v i n g t h e one d i m e n s i o n a l S c h r o d i n g e r e q u a t i o n f o r t h e ME e f f e c t i v e p o t e n t i a l , a b i n d i n g e n e r g y E B=0.63K i s o b t a i n e d . T h i s v a l u e i s s m a l l e r by 0.3K t o 0.5K t h a n t h e e x p e r i m e n t a l l y m e asured b i n d i n g e n e r g y . The r e a s o n f o r t h e d i s c r e p a n c y i s t h e v a r i a t i o n a l n a t u r e of t h e c a l c u l a t i o n and t h e u n c e r t a i n t y i n t h e "He-H atom-atom p o t e n t i a l . ME e m p h a s i z e d t h a t t h e a p p r o x i m a t i o n s u s e d t e n d e d t o u n d e r e s t i m a t e t h e m a g n i t u d e of t h e b i n d i n g e n e r g y . ME assumed t h a t t h e l i q u i d i s i n i t s g r o u n d s t a t e , i . e . not d i s t o r t e d by e l e m e n t a r y e x c i t a t i o n s , and t h e r e f o r e t h e ME e f f e c t i v e p o t e n t i a l c o r r e s p o n d s t o t h e c o u p l i n g between a h y d r o g e n atom and t h e l i q u i d s u r f a c e i n t h e g r o u n d s t a t e . To o b t a i n t h e c o u p l i n g t o t h e s u r f a c e e l e m e n t a r y e x c i t a t i o n s i t would be n e c e s s a r y t o p e r f o r m a c a l c u l a t i o n i n t h e same s p i r i t a s t h e ME c a l c u l a t i o n , a s s u m i n g t h a t t h e s u r f a c e i s i n i t s e x c i t e d s t a t e s . We a d o p t a s i m p l e r p r o c e d u r e w h i c h i s b a s e d on t h e a s s o c i a t i o n o f t h e g r o u n d s t a t e w i t h a f l a t s u r f a c e and t h e 16 e x c i t e d s t a t e s w i t h a s i n u s o i d a l l y v a r y i n g h e i g h t o f t h e s u r f a c e . We f i r s t c o n s t r u c t an e f f e c t i v e "He-H atom-atom p o t e n t i a l w h i c h , when summed o v e r *He below a f l a t s u r f a c e (and t h e r e f o r e a s u r f a c e i n t h e g r o u n d s t a t e ) y i e l d s t h e ME p o t e n t i a l . Then we sum t h a t e f f e c t i v e "He-H p o t e n t i a l o v e r "He atoms below a s i n u s o i d a l l y v a r y i n g s u r f a c e t o o b t a i n t h e c o u p l i n g t o t h e r i p p l o n s . We t r e a t t h e "He l i q u i d a s an i n c o m p r e s s i b l e c l a s s i c a l f l u i d w i t h a s h a r p d e n s i t y p r o f i l e S(^|-)= ^ 9(^-+ Kv5)) > where §o =0.145g/cm 3 i s t h e "He b u l k d e n s i t y , r i s a two d i m e n s i o n a l v e c t o r on t h e s u r f a c e , and Q ( x ) i s a u n i t s t e p f u n c t i o n w h i c h i s z e r o f o r x>0. F o r a h y d r o g e n atom l o c a t e d a t (R,Z) t h e e f f e c t i v e s u r f a c e -atom i n t e r a c t i o n i s g i v e n by: (13) m $ = nm\l^j'l}r V^P-RIM--*) 1)*]^*- 1^ n M E i s t h e "He number d e n s i t y and v i s t h e e f f e c t i v e atom-atom p o t e n t i a l . E x p a n d i n g U(R*,Z) i n powers o f h ( r ) , u s i n g t h e n o r m a l mode d e c o m p o s i t i o n of h ( ? ) and k e e p i n g o n l y t h e l i n e a r and t h e q u a d r a t i c t e r m s i n t h e e x p a n s i o n i n powers o f h ( r ) we o b t a i n : 17 + J_ Y L I Pl<%+%'>-*- * *K+vW + F Z — 5 ? — where The f i r s t t e r m i s t h e s t a t i c p o t e n t i a l w h i c h c o u p l e s h y d r o g e n atoms t o a f l a t s u r f a c e . The s e c o n d t e r m , w h i c h i s l i n e a r i n t h e r i p p l o n v a r i a b l e s , w i l l be r e f e r r e d t o as t h e l i n e a r c o u p l i n g term . The t h i r d t e r m i s t h e q u a d r a t i c c o u p l i n g t o t h e r i p p l o n s . 3.2 The System H a m i l t o n i a n The t o t a l H a m i l t o n i a n of t h e s y s t e m i s w r i t t e n a s a sum of t h r e e t e r m s : ( 1 6 ) H= H v + H„ + Hz where Hy. d e s c r i b e s t h e s u r f a c e of "He t h r o u g h i t s e l e m e n t a r y e x c i t a t i o n s , H H d e s c r i b e s t h e f r e e h y d r o g e n atoms i n t h e gas and t h e atoms bound t o t h e s u r f a c e , and Rx d e s c r i b e s t h e i n t e r a c t i o n between t h e atom and t h e s u r f a c e . The s y s t e m i s assumed t o be c o n f i n e d w i t h i n a box w i t h volume V, V=AxL, where A i s t h e s u r f a c e a r e a and L i s t h e l e n g t h 18 o f t h e box i n t h e d i r e c t i o n n o r m a l t o t h e s u r f a c e . The f r e e r i p p l o n p a r t o f t h e H a m i l t o n i a n i s g i v e n by: ( . 7 ) H,= £ - k ^ y - ^ ? The f r e e h y d r o g e n p a r t o f t h e H a m i l t o n i a n i s g i v e n by: < , 8 ) H-= + £(t)) a V a v where - s a c r e a t i o n o p e r a t o r f o r a h y d r o g e n atom i n t h e s t a t e l£M<r> it i s a two d i m e n s i o n a l wave v e c t o r c o r r e s p o n d i n g t o a p l a n e wave on t h e s u r f a c e , and i s t h e s o l u t i o n of t h e S c h r o d i n g e r e q u a t i o n , l a b e l e d by t h e quantum number , f o r t h e s u r f a c e p o t e n t i a l U Q ( Z ) and w i t h e n e r g y E ( ^ r ) . The p o t e n t i a l U Q (Z) has o n l y one bound s t a t e . F o r t h i s s t a t e we a s s i g n t o t h e symbol B and E ( B ) = - E B . S o l u t i o n s w i t h p o s i t i v e s e n e r g i e s a r e a s s o c i a t e d w i t h f r e e atoms i n t h e gas T h e s e s o l u t i o n s become p l a n e waves f a r above t h e s u r f a c e and <T c a n be i d e n t i f i e d w i t h k 4 , t h e wave v e c t o r f o r v e r t i c a l m o t i o n . The i n t e r a c t i o n t e r m i s t h e sum o f t h e l i n e a r c o u p l i n g and t h e q u a d r a t i c c o u p l i n g t e r m s . The l i n e a r t e r m i s g i v e n by: 19 a n d t h e q u a d r a t i c t e r m i s g i v e n by: A s i m i l a r c o u p l i n g t e r m s were u s e d by C o l e 2 7 t o c a l c u l a t e e s c a p e rates., f o r e l e c t r o n s t r a p p e d above t h e s u r f a c e o f "He. 3.3 The Morse P o t e n t i a l A s i g n i f i c a n t amount o f n u m e r i c a l c a l c u l a t i o n s c a n be a v o i d e d by u s i n g a Morse p o t e n t i a l t o model U Q ( Z ) . ,22, ^ ^ [ ^ ^ - H - ^ ] F o r a Morse p o t e n t i a l t h e g r o u n d s t a t e wave f u n c t i o n , t h e f r e e s t a t e s lcr> and t h e m a t r i x e l e m e n t s ^ ^ L ? | T ) > c a n be c a l c u l a t e d a n a l y t i c a l l y . The e f f e c t i v e p a i r p o t e n t i a l w h i c h g i v e s a Morse p o t e n t i a l f o r U G ( Z ) i s g i v e n b y : 3 5 (23) The c o u p l i n g f u n c t i o n i s g i v e n by: ( 4 ) "^""^ *P 6 L (i+ <f/+^)t (i+ <^»t J 20 The g r o u n d s t a t e wave f u n c t i o n and t h e b i n d i n g e n e r g y a r e g i v e n where ^ 1 *T ' ^ The n o r m a l i z e d f r e e s o l u t i o n w i t h e n e r g y U i s g i v e n by: «»»• ^ = i F r R ^ p w v ^ - e r e i _ - ( ^ J - = and W/ji.^ ^ ) i s W h i t t a k e r ' s f u n c t i o n . The n o r m a l i z a t i o n of t h e f r e e waves i s a box n o r m a l i z a t i o n , and t h e r e f o r e t h e a s y m p t o t i c b e h a v i o r o f t h e p e r p e n d i c u l a r p a r t o f t h e f r e e wave f u n c t i o n i s ~ siu (<r% - S~) . The m a t r i x e l e m e n t ^ B j i l i - t ^ J g ^ i s g i v e n by: ( < 2 ? > _ e . M T ( ^ / ^ ) e T ^ 3 ( " ^ K * * J iTE/V) The m a t r i x e l e m e n t ^ 61 2__±112|°^is g i v e n by: 21 (28) ^ _ L € ^ ^ V r / H ^ r - H L - o . 5 + | x The p a r a m e t e r s p , £ and Z Q a r e o b t a i n e d by f i t t i n g t h e p o s i t i o n o f t h e miniumum, t h e w e l l d e p t h and t h e z e r o c r o s s i n g of t h e ME p o t e n t i a l t o p , £ and Z„ r e s p e c t i v e l y . The v a l u e s o b t a i n e d a r e S=4.48K, p = 0 . 5 2 A - 1 and Z 0=4.2A. The s o l i d l i n e i n F i g u r e 1 shows t h e Morse p o t e n t i a l w i t h t h e s e p a r a m e t e r s . The c i r c l e s show t h e v a l u e of t h e ME p o t e n t i a l . S o l v i n g t h e S c h r o d i n g e r e q u a t i o n , f o r t h e Morse p o t e n t i a l w i t h t h e s e p a r a m e t e r s , we g e t E B=0.70K compared w i t h t h e v a l u e E B=0.63K o b t a i n e d by u s i n g t h e e x a c t ME p o t e n t i a l . B o t h of t h e s e v a l u e s a r e s i g n i f i c a n t l y s m a l l e r t h a n t h e e x p e r i m e n t a l v a l u e s and t h e r e f o r e i n some of t h e c a l c u l a t i o n s we w i l l compare t h e r e s u l t s o b t a i n e d w i t h t h e above p a r a m e t e r s w i t h r e s u l t s o b t a i n e d w i t h p a r a m e t e r s a d j u s t e d t o g i v e t h e e x p e r i m e n t a l v a l u e s f o r t h e b i n d i n g e n e r g y . By making t h e d e p t h of t h e w e l l £ d e e p e r , h i g h e r b i n d i n g e n e r g i e s a r e o b t a i n e d . F o r example t h e p a r a m e t e r s p=0.52A" 1, £ = 5.52K and x Zo=4.2A* g i v e t h e e x p e r i m e n t a l v a l u e of t h e b i n d i n g e n e r g y E B=1.15K a s measured by t h e UBC g r o u p . 22 F i g u r e 1 - The Morse s u r f a c e p o t e n t i a l ( t h e s o l i d l i n e ) compared to the ME p o t e n t i a l (the c i r c l e s ) . 23 3.4 P r o p e r t i e s Of The M a t r i x E l e m e n t s We d e s c r i b e t h e q—»0 and t h e 0 l i m i t s o f t h e m a t r i x e l e m e n t s . The f o l l o w i n g p r o p e r t i e s h o l d r e g a r d l e s s o f t h e p o t e n t i a l d^e^D u s e d . F i r s t we n o t i c e t h a t t h e bound s t a t e e x p e c t a t i o n v a l u e o f 3 U 0 ( V ) ^ s z e r 0 f w h i c h i s a g e n e r a l p r o p e r t y o f any p o t e n t i a l and i t s bound s t a t e s . F o r t h e q—->0 l i m i t t h e m a t r i x e l e m e n t <B| 11-0> v a n i s h e s as q 2 , and t h e m a t r i x e l e m e n t s q u a r e d , a q u a n t i t y w h i c h a p p e a r s o f t e n i n p e r t u r b a t i v e c a l c u l a t i o n s , v a n i s h e s as q " . T h i s power law b e h a v i o r i s l a r g e enough t o e l i m i n a t e i n f r a r e d d i v e r g e n c e s i n a l l t h e c a l c u l a t i o n s w h i c h we p e r f o r m e d . The e x p e c t a t i o n v a l u e o f ^^^.^^ between t h e g r o u n d s t a t e and t h e f r e e s t a t e s does not v a n i s h as q v a n i s h e s and t h e r e f o r e i n f r a r e d s i n g u l a r i t i e s a r e more l i k e l y t o o c c u r i n c a l c u l a t i o n s w h i c h i s a c o n s e q u e n c e of t h e box n o r m a l i z a t i o n o f t h e f r e e waves, and t h e r e f o r e t h e m a t r i x e l e m e n t s q u a r e d i s p r o p o r t i o n a l t o t h e p e r p e n d i c u l a r e n e r g y . 3^ i n v o l v i n g The s m a l l • e n e r g y l i m i t of 24 IV. THE STICKING PROBABILTY 4.1 T h e o r y We c a l c u l a t e t h e p r o b a b i l t y p e r c o l l i s i o n S(E , 6 ) f o r a f r e e h y d r o g e n atom, i n c i d e n t on t h e s u r f a c e from an a n g l e © and w i t h e n e r g y E, t o s c a t t e r i n t o t h e bound s t a t e . The c a l c u l a t i o n i s p e r f o r m e d u s i n g f i r s t o r d e r t i m e d e p e n d e n t p e r t u r b a t i o n t h e o r y , where t h e u n p e r t u r b e d H a m i l t o n i a n i s Hy_+HH and t h e p e r t u r b a t i o n i s t h e l i n e a r term i n H1 . T h i s a p p r o x i m a t i o n c o r r e s p o n d s t o s t i c k i n g p r o c e s s e s i n w h i c h t h e number o f r i p p l o n s on t h e s u r f a c e i s c h a n g e d by one. Two c h a n n e l s a r e a v a i l a b l e : a r i p p l o n c a n e i t h e r be c r e a t e d o r be d e s t r o y e d w h i l e a h y d r o g e n atom s t i c k s t o t h e s u r f a c e . The most p r o b a b l e e v e n t i s s t i c k i n g w h i l e c r e a t i n g a r i p p l o n . The u n p e r t u r b e d s t a t e s , c o r r e s p o n d i n g t o t h e u n p e r t u r b e d H a m i l t o n i a n , a r e t w o - d i m e n s i o n a l p l a n e waves m u l t i p l i e d by s o l u t i o n s t o t h e s u r f a c e p o t e n t i a l U 0 ( Z ) f o r t h e h y d r o g e n wave f u n c t i o n and r i p p l o n s t a t e s i n t h e number o c c u p a t i o n r e p r e s e n t a t i o n f o r t h e s u r f a c e s t a t e s . D u r i n g a s t i c k i n g e v e n t w i t h c r e a t i o n o f a r i p p l o n on t h e s u r f a c e , t h e s t a t e of t h e s y s t e m c h a n g e s from i t s i n i t i a l s t a t e l O l l * > | / r»£) t o a f i n a l s t a t e l8> | £ > \ Y ) ^ + 1 ) , where £ i s a two d i m e n s i o n a l wave v e c t o r f o r t h e h y d r o g e n on t h e s u r f a c e . The i n i t i a l and f i n a l e n e r g i e s a r e g i v e n by: 25 k and V a r e g i v e n by: g-= (a^t/-^)* cose The r a t e o f t r a n s i t i o n i s g i v e n by: where rtu<k) = v,e. -'J and N(q-k) i s t h e a n a l o g o u s f u n c t i o n f o r h y d r o g e n atoms. The s t i c k i n g p r o b a b i l i t y S ( E , Q ) i s g i v e n by n o r m a l i z i n g t h e r a t e of t r a n s i t i o n t o a u n i t i n c i d e n t f l u x . (32) s ( E j e ) . - ^ ( - t j e > The t h e r m a l l y a v e r a g e d s t i c k i n g p r o b a b i l i t y i s o b t a i n e d by a v e r a g i n g S(E,fc) o v e r a d i s t r i b u t i o n P ( E , © ) of i n c i d e n t p a r t i c l e s . oo n/a. ( 3 3 , - f . ) 5ftt0\ We u s e d £ ( E j 9 ; ~ fi, .Slfce (oia w h i c h i s t h e d i s t r i b u t i o n of i n c i d e n t p a r t i c e l s f o r a B o l t z m a n gas a t t e m p e r a t u r e T. The a n g u l a r i n t e g r a t i o n i n E q u a t i o n 33 c a n be p e r f o r m e d a n a l y t i c a l l y g i v i n g t h e f o l l o w i n g r e s u l t : (34) S - L a l r 26 (E . /am)* cos© *fc where X = ( «fc fe/*0* - C ° and C = - f e t O ^ + - t ? ^ / ^ - E B - fc^/a*, 0 ( X ) r e s t r i c t t h e q i n t e g r a t i o n i n t e r v a l t o be f r o m t h e minimum t o t h e maximum r i p p l o n wave v e c t o r a l l o w e d by c o n s e r v a t i o n o f e n e r g y and c o n s e r v a t i o n of s u r f a c e momentum . The i n t e g r a n d d i v e r g e s a s ( q - q . , ) " 0 - 5 a t t h e l i m i t s q. i = 1 ,2 of t h e i n t e g r a t i o n . A p p e n d i x A e x p l a i n s t h e n u m e r i c a l methods u s e d t o p e r f o r m t h e n u m e r i c a l i n t e g r a t i o n . 4.2 The Low T e m p e r a t u r e A p p r o x i m a t i o n We c a l c u l a t e t h e low t e m p e r a t u r e d ependence (kT<<E B) of t h e s t i c k i n g p r o b a b i l i t y . At low t e m p e r a t u r e s t h e i n c i d e n t p a r t i c l e s have v e r y l i t t l e e n e r g y and t h e r e f o r e t h e i n c i d e n t f r e e waves have s m a l l v a l u e s of . I n t h e low e n e r g y l i m i t t h e m a t r i x e l e m e n t < s|~^~^)cr> i s p r o p o r t i o n a l t o E 0 - 5 and t h e r e f o r e we d e f i n e M(q) t h r o u g h t h e e q u a t i o n : (35) <8|2^1|T->= J _ K ^ E i o s e A s s u m i n g t h a t t h e bound s t a t e i s n e a r l y empty, N ( q - k ) = 0 , u s i n g E q u a t i o n 31 and i g n o r i n g t e r m s c o n t a i n i n g E i n <T(EJ-EJ) we o b t a i n : 27 (36) S(Ej6) = S0 COS© where: (37) and q f t s a t i s f i e s the r e l a t i o n : (38) q D i s the wave vecto r of the r i p p l o n c r e a t e d when a hydrogen atom with n e g l i g i b l e k i n e t i c energy i s trapped by the sur f a c e p o t e n t i a l . The r i p p l o n c r e a t e d and the bound hydrogen atom have the same wave vecto r magnitude q e with opposite d i r e c t i o n s . For the th e r m a l l y averaged s t i c k i n g p r o b a b i l i t y we o b t a i n : which shows that in the low temperature l i m i t the s t i c k i n g p r o b a b i l t y i s p r o p o r t i o n a l to T 0 , 5. 4.3 RESULTS 4.3.1 S(E,6) For a given i n c i d e n t energy E and angle 0 the s t i c k i n g p r o b a b i l i t y i s a f u n c t i o n of the temperature T of the helium f i l m through the t h e r m a l l y averaged occupation number f o r the (39) S ( T ) = s 0 T * 28 r i p p l o n s 1+n(q) i n E q u a t i o n 31. We c a l c u l a t e S ( E , 6 ) a t z e r o t e m p e r a t u r e a s s u m i n g t h a t t h e bound s t a t e i s empty and u s i n g t h e p a r a m e t e r s £ =4.48K, f =0.52A' _ 1 and Z 0 = 4.2A> w h i c h g i v e a b i n d i n g e n e r g y o f 0.70K. We f i n d t h a t , f o r t h e v a l u e s 0<E<1K and 0 < 6 < 8 5 ° , S(E,Q) c a n be f i t t e d w i t h an a c c u r a c y b e t t e r t h a n 7% t o (40) S ( E j 0 ) = 0 .0£S(l+ U & £ ) C O * e We n o t e t h a t S(-_,©)/( Cos Q) i s v e r y weakly d e p e n d e n t on © and t h e r e f o r e E q u a t i o n 40, where c°se) i s i n d e p e n d e n t o f © , i s a good a p p r o x i m a t i o n . F i g u r e 2 shows S " ( E j e ) ' ( E "•^s©) a s a f u n c t i o n of E, a t 9 = 4 5 ° f o r v a r i o u s f i l m t e m p e r a t u r e s . T h e r e i s a l a r g e d e v i a t i o n from t h e low e n e r g y l i m i t S(E.J9)~ E^OO$Q > f o r 0<E<1K, i m p l y i n g l a r g e d e v i a t i o n s from t h e low t e m p e r a t u r e l i m i t f o r t h e t e m p e r a t u r e r a n g e i n w h i c h t h e e x p e r i m e n t was p e r f o r m e d . At h i g h e r t e m p e r a t u r e s S ( E , 6 ) i n c r e a s e s due t o t h e h i g h e r number of t h e r m a l l y e x c i t e d r i p p l o n s on t h e s u r f a c e . At t h e s e t e m p e r a t u r e s SC-ij©) C o i e ) c o n t i n u e s t o be m a i n l y d e p e n d e n t on t h e e n e r g y w i t h o n l y a s m a l l a n g u l a r d e p e n d e n c e . 29 F i g u r e 2 - The s t i c k i n g p r o b a b i l i t y a t 6 = 4 5 ° a s a f u n c t i o n o f t h e atom e n e r g y f o r v a r i o u s v a l u e s o f t h e f i l m t e m p e r a t u r e T. i i i l I 0.2 0.4 0-6 0.8 1.0 E(K) 30 4.3.2 S ( T ) The t h e r m a l l y a v e r a g e d s t i c k i n g p r o b a b i l i t y S ( T ) was c a l c u l a t e d f o r d i f f e r e n t v a l u e s o f common s u r f a c e and gas t e m p e r a t u r e T by p e r f o r m i n g t h e d o u b l e i n t e g r a t i o n ( s e e A p p e n d i x A) i n E q u a t i o n 33 and u s i n g S ( E , 9 ) o b t a i n e d w i t h t h e p a r a m e t e r s s p e c i f i e d i n t h e p r e v i o u s s e c t i o n . The s o l i d l i n e i n F i g u r e 3 shows t h e r e s u l t s o b t a i n e d f o r S ( T ) . The e x p e r i m e n t a l b a r s i n F i g u r e 3 c o r r e s p o n d t o t h e v a l u e s o f t h e s t i c k i n g p r o b a b i l i t y measured a t UBC ( s e e T a b l e 1 ) . F o r t h e t e m p e r a t u r e range i n w h i c h t h e e x p e r i m e n t was p e r f o r m e d t h e r e i s good agreement between our r e s u l t s and t h e d a t a . U n f o r t u n a t e l y , t h e e x p e r i m e n t a l d a t a a r e not s u f f i c i e n t l y a c c u r a t e o v e r a a wide enough r a n g e o f t e m p e r a t u r e t o r e v e a l t h e t e m p e r a t u r e d e pendence o f t h e s t i c k i n g p r o b a b i l i t y , and t h e r e f o r e we c a n n o t t e s t t h i s f e a t u r e of our r e s u l t s . The p a r a m e t e r s u s e d t o c a l c u l a t e t h e p r e v i o u s r e s u l t s g i v e a b i n d i n g e n e r g y o f 0.70K wh i c h i s s m a l l e r t h a n t h e measured b i n d i n g e n e r g y . We have a l s o c a l c u l a t e d S ( T ) u s i n g t h e same v a l u e s o f p and Z 0 but c h a n g i n g & t o g e t h i g h e r b i n d i n g e n e r g i e s . I n c r e a s i n g £ w i l l make t h e w e l l d e p t h l a r g e r and t h e b i n d i n g e n e r g y w i l l i n c r e a s e . The s t i c k i n g p r o b a b i l t y a l s o i n c r e a s e s w i t h t h e d e e p e n i n g o f t h e w e l l . We v a r i e d £ t o g e t t h e d i f f e r e n t v a l u e s f o r t h e b i n d i n g e n e r g y m easured by t h e d i f f e r e n t g r o u p s . The symbols i n F i g u r e 3, a t T=0.2K, c o r r e s p o n d t o t h e v a l u e s o f S ( T ) f o r t h e d i f f e r e n t v a l u e s of E g . Th e s e r e s u l t s a r e n o t i c e a b l y h i g h e r t h a n t h e e x p e r i m e n t a l v a l u e s o f S ( T ) . V a r i a t i o n of t h e o t h e r 31 p a r a m e t e r s shows t h a t t h e s t i c k i n g p r o b a b i l i t y i s m a i n l y d e p e n d e n t on t h e v a l u e o f t h e b i n d i n g e n e r g y . Thus we a r e l e f t w i t h a f a c t o r o f between 1.5 and 2 d i s c r e p a n c y between t h e s i z e o f t h e measured s t i c k i n g p r o b a b i l t y and t h e v a l u e w h i c h we c a l c u l a t e d f o r S ( T ) , when t h e p a r a m e t e r s a r e f i t t e d t o t h e measured b i n d i n g e n e r g i e s . The d i s c r e p a n c y most l i k e l y i s due t o an i n a c c u r a c y i n t h e f u n c t i o n w h i c h we use t o model t h e a t o m - r i p p l o n c o u p l i n g . We have a l s o c a l c u l a t e d S ( T ) w i t h a m a t r i x e l e m e n t w h i c h c o r r e s p o n d s t o a c o u p l i n g i n d e p e n d e n t of q and e q u a l t o . T h i s c o u p l i n g can be d e r i v e d a s s u m i n g t h a t t h e h y d r o g e n atom i n t e r a c t s o n l y w i t h an e l e m e n t of l i q u i d j u s t below t h e p o s i t i o n of t h e atom. The d a s h e d l i n e i n F i g u r e 3 , c a l c u l a t e d w i t h p a r a m e t e r s w h i c h y i e l d E g = 1 . l 5 K , shows a much l o w e r S ( T ) , i n b e t t e r agreement w i t h t h e e x p e r i m e n t a l d a t a . T h i s shows t h a t t h e f orm of t h e q-dependent c o u p l i n g can have a marked e f f e c t on t h e s t i c k i n g p r o b a b i l i t y even f o r c o u p l i n g f u n c t i o n s w h i c h have t h e same q=0 l i m i t . We b e l i e v e t h a t a t h e c o r r e c t q dependence l i e s somewhere i n between t h e two forms c o n s i d e r e d a b o v e . 32 F i g u r e 3 - The t h e r m a l l y a v e r a g e d s t i c k i n g p r o b a b i l i t y S ( T ) v e r s u s t h e common t e m p e r a t u r e T o f t h e f i l m and t h e g a s . The s o l i d l i n e r e p e r e s e n t S ( T ) f o r E 6 = 0.70K. The v e r t i c a l b a r s r e p r e s e n t t h e e x p e r i m e n t a l d a t a shown i n T a b l e I w h i l e t h e sy m b o l s O , • and 4- c o r r e s p o n d s t o t h e t h e o r e t i c a l r e s u l t s f o r E B=0.89K, E B=1.01K and E B=1.15K r e s p e c t i v e l y . ' The d a s h e d l i n e c o r r e s p o n d s t o a d i f f e r e n t r i p p l o n atom c o u p l i n g and E B=1.15K as e x p l a i n e d i n t h e t e x t . Temperature (K) 33 V. THE HYDROGEN SURFACE POLARON 5.1 I n t r o d u c t i o n In t h i s c h a p t e r we w i l l e x p l o r e t h e p o s s i b i l i t y t h a t a h y d r o g e n atom bound t o t h e h e l i u m s u r f a c e may e x h i b i t s i g n i f i c a n t p o l a r o n i c b e h a v i o u r . The c o n c e p t of a p o l a r o n was i n t r o d u c e d t o d e s c r i b e a c o n d u c t i o n e l e c t r o n moving i n an i o n i c c r y s t a l 3 6 . The e l e c t r o n p o l a r i z e s t h e l a t t i c e i n i t s v i c i n i t y , and w h i l e moving i n t h e c r y s t a l i t c a r r i e s w i t h i t t h e p o l a r i z a t i o n . The e l e c t r o n i s s a i d t o be moving w i t h an a c c o m p a n y i n g "phonon c l o u d " . Such a s y s t e m i s r e f e r r e d as a p o l a r o n . A H a m i l t o n i a n c o n t a i n i n g a f r e e phonon p a r t , a B l o c h e l e c t r o n t e r m , and an e l e c t r o n - p h o n o n i n t e r a c t i o n t e r m was i n t r o d u c e d by F r o h l i c h t o d e s c r i b e t h i s p r o b l e m . A s t r a i g h t f o r w a r d p e r t u r b a t i v e c a l c u l a t i o n , i n w h i c h t h e i n t e r a c t i o n term p e r t u r b s t h e f r e e phonons and B l o c h e l e c t r o n s t a t e s , i s not a d e q u a t e f o r c e r t a i n c r y s t a l s , t h e r e a s o n b e i n g t h a t t h e c o u p l i n g c o n s t a n t m u l t i p l y i n g t h e i n t e r a c t i o n term i s l a r g e . T h e r e f o r e d i f f e r e n t methods were i n t r o d u c e d t o c a l c u l a t e t h e p o l a r o n e n e r g y and i t s mass f o r weak, i n t e r m e d i a t e and s t r o n g c o u p l i n g . An i n g e n i o u s method, w h i c h c a n be u s e d f o r a l l v a l u e s of t h e c o u p l i n g c o n s t a n t , was i n t r o d u c e d by F e y n m a n . 3 7 The method c o n s i s t s o f e l i m i n a t i n g t h e phonon c o o r d i n a t e s e x a c t l y from t h e H a m i l t o n i a n , t h u s o b t a i n i n g an e f f e c t i v e but e x a c t f o r m u l a t i o n of t h e p r o b l e m c o n t a i n i n g o n l y t h e e l e c t r o n v a r i a b l e s . Next t h e p r o b l e m i s t r e a t e d v a r i a t i o n a l l y w i t h r e s u l t s w h i c h a r e l o w e r i n 34 e n e r g y , and t h e r e f o r e more a c c u r a t e , t h a n r e s u l t s o b t a i n e d by any o t h e r method. A s i m i l a r p r o b l e m i s an e l e c t r o n bound above t h e s u r f a c e o f "He i n t e r a c t i n g w i t h t h e r i p p l o n s i n t h e p r e s e n c e o f an e x t e r n a l e l e c t r i c f i e l d . J a c k s o n and P l a t z m a n n , ( J P ) , s u g g e s t e d t h a t t h e bound e l e c t r o n w i l l show p o l a r o n i c p r o p e r t i e s . 3 8 " 3 9 They p r o p o s e d t h a t t h e s t r e n g t h of t h e c o u p l i n g between t h e e l e c t r o n and t h e s u r f a c e c o u l d be c h a n g e d c o n t i n o u s l y f r o m t h e weak t o t h e s t r o n g l i m i t by v a r y i n g t h e s t r e n g t h o f t h e e l e c t r i c f i e l d o r t h e t h i c k n e s s o f t h e f i l m o r by c h o o s i n g d i f f e r e n t s u b s t r a t e s on w h i c h t h e l i q u i d f i l m r e s i d e s . J P u s e d Feynman's a p p r o a c h t o c a l c u l a t e t h e e l e c t r o n e f f e c t i v e mass as a f u n c t i o n of t h e c o u p l i n g . I n b o t h p r o b l e m s , d i s c u s s e d above, some d e g r e e s o f f r e e d o m a r e not t a k e n i n t o a c c o u n t . F o r t h e e l e c t r o n - p h o n o n p r o b l e m , t h e e l e c t r o n i s assumed t o be i n a c e r t a i n band, and t h e r e f o r e no i n t e r - b a n d v i r t u a l t r a n s i t i o n s a r e c o n s i d e r e d . F o r t h e e l e c t r o n - r i p p l o n p r o b l e m t h e e l e c t r o n i s assumed t o be i n t h e g r o u n d s t a t e of t h e s u r f a c e p o t e n t i a l , and t h u s e x c i t e d p e r p e n d i c u l a r s t a t e s a r e n o t c o n s i d e r e d . In our c a s e a h y d r o g e n atom i n t h e bound s t a t e above t h e s u r f a c e i n t e r a c t s w i t h t h e s u r f a c e and d i s t o r t s i t . W h i l e moving p a r a l l e l t o t h e s u r f a c e t h e atom d r a g s w i t h i t t h e d i s t o r t i o n o f t h e s u r f a c e t h u s a c q u i r i n g an e f f e c t i v e mass m*, d i f f e r e n t f r o m i t s f r e e mass m. In t h i s C h a p t e r we c a l c u l a t e t h e e n e r g y o f t h e h y d r o g e n - r i p p l o n s y s t e m , t h e e f f e c t i v e mass m* and t h e number of r i p p l o n s i n t h e " r i p p l o n c l o u d " moving w i t h 35 t h e atom. We c o n s i d e r t h e e f f e c t o f t h e f r e e s u r f a c e s t a t e s on t h e s e r e s u l t s . We a l s o l o o k a t t h e l o c a l i z a t i o n p r o p o s e d by GMY u s i n g o u r c o u p l i n g f u n c t i o n . 5.2 The H y d r o g e n - R i p p l o n I n t e r a c t i o n E n e r g y 5.2.1 T h e o r y F o r t h e p u r p o s e o f t h i s c h a p t e r we w i l l w r i t e t h e t o t a l H a m i l t o n i a n i n t h e f o l l o w i n g one p a r t i c l e n o t a t i o n : ( 41 ) £ o The e n e r g y of t h e h y d r o g e n p o l a r o n i s o b t a i n e d by c a l c u l a t i n g t h e g r o u n d s t a t e e n e r g y o f t h e above H a m i l t o n i a n . The c a l c u l a t i o n i s b a s e d on p e r t u r b a t i o n t h e o r y where t h e u n p e r t u r b e d H a m i l t o n i a n i s g i v e n by: and t h e p e r t u r b a t i o n t e r m s a r e t h e c o u p l i n g t o t h e r i p p l o n s t e r m s . The u n p e r t u r b e d H a m i l t o n i a n i s s e p a r a b l e . I t s e i g e n f u n c t i o n s a r e r i p p l o n s t a t e s i n t h e o c c u p a t i o n number r e p r e s e n t a t i o n m u l t i p l i e d by h y d r o g e n atom s u r f a c e s t a t e s . The h y d r o g e n s t a t e i s a 2-D f r e e wave m u l t i p l i e d by a s o l u t i o n o f 36 U 0(Z). The energy i s the sum of the p e r p e n d i c u l a r energy (-Eg fo r the bound s t a t e or •fc^ TVa.m. f o r the s t a t e |<r> ) , the p a r a l l e l f r e e p a r t i c l e k i n e t i c energy and r i p p l o n e n e r g i e s . The p e r t u r b a t i v e part of the Hamiltonian c o n s i s t s of a term which i s l i n e a r i n the r i p p l o n v a r i a b l e s and a second term which i s q u a d r a t i c . The e f f e c t of these terms i s to mix the p a r a l l e l , the p e r p e n d i c u l a r , and the r i p p l o n s s t a t e s . In order to take i n t o account a l l the c o n t r i b u t i o n s to the energy, to second order i n the r i p p l o n v a r i a b l e s , we must c a l c u l a t e the c o n t r i b u t i o n from the l i n e a r c o u p l i n g term in second order p e r t u r b a t i o n theory and the c o n t r i b u t i o n from the q u a d r a t i c term in f i r s t order p e r t u r b a t i o n theory. The hydrogen-ripplon i n t e r a c t i o n energy AE ( f e ) i s d e f i n e d as the c o r r e c t i o n to the unperturbed energy, of a hydrogen atom with s u r f a c e wave ve c t o r of magnitude k, due to the c o u p l i n g terms. The t o t a l energy of such an atom w i l l be given by: ( 4 3 , E W = + ~ ~ + * E ( t ° We d e f i n e A t ( t e ) = "+ A b * ( k ) where i s the c o n t r i b u t i o n to & & ( f c ) from f i r s t order p e r t u r b a t i o n theory and the c o n t r i b u t i o n from second order p e r t u r b a t i o n theory. The r e s u l t s from the p e r t u r b a t i v e c a l c u l a t i o n s a r e : 37 U 4 ) AE' = i^ I ^Jl+3 l^<el# l^«>=i-<B|^ l^ ) where <h2> i s the thermally averaged mean square displacement from equilibrium of the surface. (45) _ j_y v t y w w 1<«-| Is-)!3- ^ T " P V clvvi 3Hn ' - B Ty >* tJ/ The sum over <r includes a term corresponding to the single bound state and a sum over a continuum of free states: (46) JTtfO—> ^ - [ ^ M V t ffs) cr 0 where f ( cr) i s an arb i t r a r y function of <r. Therefore we w i l l def ine: (47) where AE Clb) and AE g(fe)are the contributions to AE^k)from the free states and the bound states respectively. Thus AE(fe) i s given as a sum of three terms. (48) A E ( k ) = A E * 0 ) + A E g ( «0 + & E ' These results d i f f e r from the usual results obtained in polaronic calculations for the electron-phonon or the electron-38 r i p p l o n s y s t e m , by t h e i n c l u s i o n o f t h e terms AE.' and AE.^(|e). F o r t h o s e probems o n l y t h e l i n e a r c o u p l i n g t e r m was c o n s i d e r e d and n o n - a d i a b a t i c c o r r e c t i o n s were n e g l e c t e d . I n o u r c a s e k & c ( o ) i s c a l c u l a t e d and i t s m a g n i t u d e shows t h a t i t c a n n o t be n e g l e c t e d . A l s o AE. ' i s i n c l u d e d t o a v o i d a non p h y s i c a l r e s u l t f o r T>0 a s w i l l be e x p l a i n e d i n t h e f o l l o w i n g s u b c h a p t e r . We n o t i c e t h a t A E . 1 and AE*(fe) have o p p o s i t e s i g n s , i . e A E 1 i s p o s i t i v e and AE?(|e) i s n e g a t i v e . 5.2.2 The Long W a v e l e n g t h L i m i t The term w h i c h i s q u a d r a t i c i n t h e r i p p l o n s v a r i a b l e s g i v e s a c o n t r i b u t i o n t o AE 1 w h i c h i s p r o p o r t i o n a l t o <h 2>. The t e m p e r a t u r e d e p e n d e n t t e r m d i v e r g e s l o g a r i t h m i c a l l y a s g, t h e c o u p l i n g t o t h e g r a v i t a t i o n a l a n d / o r s u b s t r a t e p o t e n t i a l , v a n i s h e s . F o r n o n - z e r o g most o f t h e c o n t r i b u t i o n t o <h 2>, and t h e r e f o r e t o Ak. , come from t h e l o n g w a v e l e n g t h r i p p l o n s . T h i s i s c l e a r l y an u n p h y s i c a l r e s u l t b e c a u s e t h e l o n g w a v e l e n g t h r i p p l o n s o n l y s h i f t t h e p o s i t i o n of t h e s u r f a c e and t h e r e f o r e c a n n o t s i g n i f i c a n t l y change t h e b i n d i n g e n e r g y . T h i s u n p h y s i c a l r e s u l t does n ot o c c u r when b o t h t e r m s a r e t a k e n i n t o a c c o u n t as m e n t i o n e d a b o v e . U s i n g t h e sum r u l e : ( 4 9 ) g * U o f c ) i R \ _ Y |<<ri 3M=t)/3i|e>|* and l o o k i n g a t t h e q—»0 c o n t r i b u t i o n t o we n o t i c e t h a t t h e z e r o q l i m i t s o f t h e c o n t r i b u t i o n t o AE.a(°) and t h e same l i m i t f o r A E 1 e x a c t l y c a n c e l e a c h o t h e r . T h i s means t h a t , t a k i n g 39 i n t o account both terms, there i s no s i g n i f i c a n t c o n t r i b u t i o n to the b i n d i n g energy from the long wavelength r i p p l o n s . 5.2.3 T=0 k=0 For the f i l m temperature T=0 and approximating a slow hydrogen atom by t a k i n g k=0, we o b t a i n : where i s d e f i n e d i n Equation 12. From Equations 45 and 47 we o b t a i n : (51 ) (52) Ignoring the g r a v i t a t i o n a l term in the r i p p l o n spectrum we ob t a i n f o r <h 2>: and using a cut o f f q = l A " 1 we o b t a i n : <h 2>=2.39A 2. Next we c a l c u l a t e <^ gj ^ ^ ^ j ^ For a Morse p o t e n t i a l : 40 (54) ± £ £ _ > = 3 ^ [ j £ ' i f " " , t ) - ^ P C - W j ' and u s i n g : ,55, <B|£-1"-*-MB>= i M g J ^ l L . we o b t a i n : ( 5 6 ) a n ' l w - l B > - ? L t*s rM (teji r (.o J U s i n g t h e p a r a m e t e r s p = 0.52A~' £ = 4 . 4 8 K and Z 0=4.2A we o b t a i n f r o m E q u a t i o n 56 : and t h e r e f o r e from E q u a t i o n 53 and 57 we o b t a i n : A E ' - 1.15 K We n o t e t h a t t h i s r e s u l t i s s e n s i t i v e t o t h e wave v e c t o r c u t - o f f u s e d as q 1 , 5 . AE- ' (° ) i s c a l c u l a t e d n u m e r i c a l l y u s i n g t w o - d i m e n s i o n a l and o n e - d i m e n s i o n a l G a u s s i a n n u m e r i c a l i n t e g r a t i o n f o r A^J^Oand AEj(») r e s p e c t i v e l y ( s e e A p p e n d i x A ) . We o b t a i n A & o ^ 5 5 *" K a n d & E B * ( ° ) = - O . 0 4 K g i v i n g A E ( o " ) = o.fcfc K T h i s r e s u l t i s a l m o s t e q u a l i n i t s a b s o l u t e m a g n i t u d e t o t h e b i n d i n g e n e r g y E^=0.70K o b t a i n e d f o r t h e s u r f a c e p o t e n t i a l w i t h 41 the above pa r a m e t e r s . Thus the t o t a l energy f o r a hydrogen atom bound t o the s u r f a c e would be -0.70K+0.66K=-0.04K, i n complete disagreement w i t h the measured v a l u e of the b i n d i n g energy. A p o s s i b l e e x p l a n a t i o n i s the l a r g e c o n t r i b u t i o n t o from r i p p l o n wave v e c t o r s 0.5<q<1A' 1 , which i s the p a r t of the spectrum which i s not g i v e n a c c u r a t e l y by the model we used f o r the s u r f a c e . The l a r g e c o n t r i b u t i o n t o the energy from t h i s p a r t of the r i p p l o n spectrum i s m a n i f e s t e d by the s e n s i t i v i t y of the r e s u l t s t o the v a l u e of q^. A d i f f e r e n t model f o r the s u r f a c e e l e m e n t a r y e x c i t a t i o n , which t a k e s i n t o account the c o m p r e s s i b i l t y of the l i q u i d may g i v e a d i f f e r e n t r e s u l t , l e s s s e n s i t i v e t o the a r b i t r a r y c u t - o f f used. The s e n s i t i v i t y of thes e r e s u l t s t o the r i p p l o n spectrum c u t - o f f can be demonstrated by c h o o s i n g an ad-hoc c u t - o f f v a l u e . For example c h o o s i n g q^=0.75A"1 g i v e s : At' = 0.1-5 k and t h e r e f o r e AE(o) = o.30)<-Another p o s s i b l e e x p l a n a t i o n i s a breakdown of p e r t u r b a t i o n t h e o r y . A n o n - p e r t u r b a t i v e approach t o the p e r p e n d i c u l a r problem may be needed. Due t o the f a c t t h a t the c o n t r i b u t i o n from the f r e e s t a t e s i s the dominant c o n t r i b u t i o n t o the change i n the energy, an a p p l i c a t i o n of Feynman's method would r e q u i r e t a k i n g these v a r i a b l e s i n t o a c c o u n t . An e x a c t e l i m i n a t i o n of 42 t h e r i p p l o n v a r i a b l e s c a n be done e a s i l y b u t t h e l a c k o f a s u i t a b l e and e a s y - t o - u s e v a r i a t i o n a l method f o r t h e r e s t of t h e p r o b l e m i s t h e main o b s t a c l e w h i c h p r e v e n t s t h e c a l c u l a t i o n f r o m from b e i n g done. We have u s e d t h e Feynman method t o c a l c u l a t e AE.g(°) • The d i f f e r e n c e between r e s u l t s o b t a i n e d i n t h i s way and t h e p r e v i o u s v a l u e s f o r ML^(o) o b t a i n e d from p e r t u r b a t i o n t h e o r y i s n e g l i b l e . T h i s i s a s e x p e c t e d , s i n c e t h e s m a l l s i z e o f AEgXOin^icates t h a t t h e c o u p l i n g between s u r f a c e atoms and t h e r i p p l o n s i s s m a l l . In t h e weak c o u p l i n g l i m i t , p e r t u r b a t i o n t h e o r y i s p e r f e c t l y a d e q u a t e , g i v i n g t h e same r e s u l t s as Feynman's method. 5 . 3 Hydrogen P o l a r o n ? The p o s s i b l e breakdown of p e r t u r b a t i o n t h e o r y does not n e c e s s a r i l y i m p l y p o l a r o n i c b e h a v i o r , i t o n l y shows t h a t t h e p e r p e n d i c u l a r p r o b l e m and t h e p a r a l l e l p r o b l e m a r e s t r o n g l y c o u p l e d and c a n n o t be s e p a r a t e d . 1 * P o l a r o n i c b e h a v i o r w o u l d emerge f o r s t r o n g c o u p l i n g between t h e h y d r o g e n and t h e r i p p l o n s . In t h i s s u b c h a p t e r we w i l l i n v e s t i g a t e t h e p a r a l l e l c o u p l i n g . I n t h e f o l l o w i n g s e c t i o n , we c h e c k w h e t h e r t h e l o c a l i z a t i o n p r e d i c t e d by GMY o c c u r s . We c a l c u l a t e t h e e f f e c t i v e mass on t h e s u r f a c e and t h e number of r i p p l o n s moving a l o n g w i t h t h e atom i n t h e bound s t a t e . 43 5.3.1 GMY L o c a l i z a t i o n We r e p e a t t h e c a l c u l a t i o n o f GMY w h i c h s u g g e s t e d t h a t , i n t h e g r o u n d s t a t e o f t h e h y d r o g e n - r i p p l o n s y s t e m , t h e h y d r o g e n p a r a l l e l wave p a c k e t may be s i g n i f i c a n t l l y l o c a l i z e d . The c r u c i a l d i f f e r e n c e between t h e f o l l o w i n g c a l c u l a t i o n s and t h e one p e r f o r m e d by GMY i s t h e c o u p l i n g t o t h e s u r f a c e u s e d . GMY assumed a <J~ f u n c t i o n c o u p l i n g between an e l e m e n t o f l i q u i d and t h e atom and o b t a i n e d m a t r i x e l e m e n t s i n d e p e n d e n t o f t h e r i p p l o n wave v e c t o r q . We w i l l use t h e c o u p l i n g d e r i v e d i n C h a p t e r 2. A d o p t i n g t h e a p p r o a c h o f GMY,- we g e t f o r t h e p a r a l l e l e n e r g y : 1* r*K . a <r* (58) E(*)= ^ 0 The f i r s t t e r m i s t h e k i n e t i c e n e r g y t e r m c o r r e s p o n d i n g t o a two d i m e n s i o n a l G a u s s i a n wave p a c k e t w i t h h a l f w i d t h K , w h i l e t h e s e c o n d t e r m i s t h e n e g a t i v e p o t e n t i a l e n e r g y r e s u l t i n g f r o m t h e l i n e a r c o u p l i n g t o t h e r i p p l o n s . In GMY, t h e m a t r i x e l e m e n t s were i n d e p e n d e n t of q and t h e r e f o r e c o u l d be t a k e n o u t o f t h e i n t e g r a n d . Then most o f t h e ft c o n t r i b u t i o n t o t h e i n t e g r a l i s p r o p o r t i o n a l t o 1/g and comes from t h e l o n g w a v e l e n g t h r i p p l o n s . Thus GMY o b t a i n e d a r e s u l t w h i c h c o u l d be made as n e g a t i v e as one w i s h e s by making g as s m a l l as needed, f o r example, by making t h e f i l m v e r y t h i c k . In our c a l c u l a t i o n we do not o b t a i n any c o n t r i b u t i o n f r o m t h e l o n g w a v e l e n g t h l i m i t b e c a u s e t h e m a t r i x e l e m e n t <(E( I B)> v a n i s h e s as q 2 f o r s m a l l q. 44 U s i n g t h e p a r a m e t e r s £ = 4 . 4 8 K , p = 0 . 5 2 A _ 1 and Z 0 = 4.2A°we f i n d t h a t f o r a l l E( "6 )>0 i m p l y i n g t h a t t h e l o c a l i z a t i o n e f f e c t s u g g e s t e d by GMY i s an a r t i f a c t o f t h e model t h e y u s e d t o d e s c r i b e t h e c o u p l i n g . The k i n e t i c e n e r g y t e r m i s a l s o a l w a y s l a r g e r t h a n t h e p o t e n t i a l e n e r g y term, g i v i n g a p o s i t i v e e n e r g y , when t h e p a r a m e t e r s w h i c h y i e l d a b i n d i n g e n e r g y o f 1.15K a r e u s e d . 5.3.2 The E f f e c t i v e Mass At T=0K Next we c a l c u l a t e t h e e f f e c t i v e mass m* f o r a bound h y d r o g e n atom moving above t h e s u r f a c e . We d e f i n e R g and R c t h r o u g h t h e f o l l o w i n g e q u a t i o n s , w h i c h a r e o b t a i n e d by e x p a n d i n g A E L B ( b ) and A f c G ( k ) i n powers of k 2 . (59) A E c * ( k ) = A E > > + R c ^ ^ F o r s m a l l enough v a l u e s o f k we c a n i g n o r e a l l powers of k e x c e p t t h e q u a d r a t i c t e r m . Then we g e t f o r t h e t h e t o t a l p a r a l l e l e n e r g y of a s l o w h y d r o g e n atom moving above th e s u r f a c e : 45 (6D E(»0* - E B + A E ( o ) + aha* VYL where m* i s t h e r e n o r m a l i z e d h y d r o g e n mass. U s i n g : (62) , e x p a n d i n g t h e i n t e g r a n d i n t h e a n g u l a r i n t e g r a t i o n i n powers o f (63) 1v* k % co* e and m a i n t a i n i n g o n l y t h e f i r s t and s e c o n d t e r m s we o b t a i n : <64) D = *4 L 00 The i n t e g r a l i s c a l c u l a t e d n u m e r i c a l l y u s i n g two d i m e n s i o n a l G a u s s i a n i n t e g r a t i o n . A E g O O ^ s g i v e n by: r1 m (65) o 0 cr U T T n ^ V $ In t h i s c a s e t h e a n g u l a r i n t e g r a l has a r e a l and an i m a g i n a r y p a r t . The i m a g i n a r y p a r t c o r r e s p o n d s t o damping, i . e . t o t h e l o s s of e n e r g y t o t h e s u r f a c e by c r e a t i n g r i p p l o n s . We a r e i n t e r e s t e d i n t h e r e a l p a r t w h i c h i s g i v e n by: (66) — 46 for)* J% * ^^[(^V^T^^fl^ where q. s a t i s f i e s ^ l A , •+- ft* ^ — "^k o . We n o t i c e t h a t i n t h e v i c i n i t y o f q, t h e i n t e g r a n d d i v e r g e s a s (q~C[ | ) ~ 0 , 5 and t h e r e f o r e a s p e c i a l t y p e of G a u s s i a n i n t e g r a t i o n i s needed t o c a l c u l a t e t h i s i n t e g r a l ( s e e a p p e n d i x A ) . AEg"(b.) i s c a l c u l a t e d n u m e r i c a l l y f o r s m a l l v a l u e s o f k, and R g i s o b t a i n e d by f i t t i n g t h e r e s u l t s t o E q u a t i o n 60. U s i n g t h e c u t o f f q=1A _ 1 and t h e p a r a m e t e r s £ = 4 . 4 8 K p=0.52A" 1 a n d Zo=4.2A we o b t a i n R c = - 0 . 8X 1 0 " 2 and R b = - 0 . 9X 1 0 " 2 w h i c h g i v e , u s i n g E q u a t i o n 61, m*=1.0l7m. F o r t h e p a r a m e t e r s w h i c h c o r r e s p o n d t o t h e measured b i n d i n g e n e r g y R c = - 1 . 1X 1 0 " 2 and R b = - 1 . 7X 1 0 " 2 g i v i n g m*=1.029m . T h e s e r e s u l t s show t h a t t h e r e n o r m a l i z e d mass i s o n l y s l i g h t l y h i g h e r t h a n t h e f r e e h y d r o g e n mass. Edwards e s t i m a t e d t h e e f f e c t i v e mass on t h e s u r f a c e t o be 1.1m."1 5.3.3 The R i p p l o n C l o u d We c a l c u l a t e t h e a v e r a g e number o f v i r t u a l r i p p l o n s moving w i t h a h y d r o g e n atom on t h e s u r f a c e . F o l l o w i n g P i n e s " 1 we c a l c u l a t e t h e e x p e c t a t i o n v a l u e o f t h e r i p p l o n number o p e r a t o r (67) i n t h e p e r t u r b e d s t a t e o f t h e s y s t e m . 47 I ? > | B > /o> (68) For slow atoms, approximated by t a k i n g k=0, we get: (69> <j>= _L_ y y K-I 3^1 Performing the i n t e g r a t i o n n u m e r i c a l l y we o b t a i n : (70) <5>=r (o.JS + a , 3 > l o ~ * » 3.^ -10 - a where the f i r s t term i s the c o n t r i b u t i o n from the f r e e s t a t e s and the second from the bound s t a t e . The perturbed wave f u n c t i o n i s a mixture of a zero r i p p l o n A s t a t e and one r i p p l o n s t a t e s . Values for <N> of the order of one or more would not be compatible with such a mixture of s t a t e s , and would i n d i c a t e that s t a t e s c o n t a i n i n g more r i p p l o n s should be i n c l u d e d in the s t a t e d e s c r i b i n g the system. However the value of <N> which we obtained, <N>= 0.03, i s c o n s i s t e n t with a one r i p p l o n approximation. 5.4 C o n c l u s i o n In t h i s chapter we showed that no important p o l a r o n i c e f f e c t s occur f o r the bound hydrogen. The e f f e c t i v e mass of a hydrogen atom on the su r f a c e i s only s l i g h t l y m o d i f i e d by the i n t e r a c t i o n with the r i p p l o n s . T h e r e f o r e f r e e s t a t e s with an e f f e c t i v e mass m* are a good approximation f o r the bound 48 h y d r o g e n p a r a l l e l wave f u n c t i o n . S i m i l a r l y we h a v e shown t h a t a m i x t u r e o f a z e r o r i p p l o n s t a t e a n d one r i p p l o n s t a t e s i s an a d e q u a t e d e s c r i p t i o n f o r t h e s u r f a c e s t a t e . The s m a l l v a l u e o f <N> a n d t h e s m a l l d i f f e r e n c e b e t w e e n t h e e f f e c t i v e mass o f a h y d r o g e n a t o m bound t o t h e s u r f a c e c o m p a r e d w i t h t h e mass o f a f r e e h y d r o g e n atom i n d i c a t e s t h a t t h e r i p p l o n - h y d r o g e n c o u p l i n g s h o u l d be c o n s i d e r e d a s b e i n g weak. We a l s o saw t h a t t h e two l e a d i n g t e r m s i n t h e p e r t u r b a t i v e c a l c u l a t i o n s f o r t h e r i p p l o n -a t o m i n t e r a c t i o n e n e r g y h a v e o p p o s i t e s i g n s a n d a r e o f t h e same o r d e r o f m a g n i t u d e a s t h e b i n d i n g e n e r g y , i n d i c a t i n g a p o s s i b l e b r e a k d o w n o f p e r t u r b a t i o n t h e o r y . The m a i n c o n t r i b u t i o n s t o t h e s e l a r g e s h i f t s comes f r o m t h e t e r m q u a d r a t i c i n t h e r i p p l o n v a r i a b l e s a n d f r o m t h e l i n e a r t e r m w h i c h c o u p l e s t h e v i r t u a l s t a t e s w h i c h a r e n o t bound t o t h e s u r f a c e . 49 BIBLIOGRAPHY 1. F o r r e v i e w see A . J . B e r l i n s k y , J . A p p l . P h y s . 5_2, 2309 (1981) and p a p e r s by W.N. Hard y e t a l . and I . F . S i l v e r a i n t h e p r o c e d i n g s o f t h e 16th i n t e r n a t i o n a l c o n f e r e n c e on Low T e m p e r a t u r e P h y s i c s , P h y s i c a 108 B+C No.3 ( 1 9 8 1 ) . 2. W.N. Hardy, A . J . B e r l i n s k y , a nd L.A. W h i t e h e a d Phys.Rev. L e t t . 42, 1042 ( 1 9 7 9 ) . 3. W.N. Hardy, M. Morrow, R. Jochemsen, B.W. S t a t t , P.R. K u b i k , R.M. M a r s o l a i s , A . J . B e r l i n s k y and A. Landesman P h y s . R e v . L e t t . 45, 453 ( 1 9 8 0 ) . 4. M. Morrow, R. Jochemsen, A . J . B e r l i n s k y and W.N. Hardy, P h y s . R e v . L e t t . 46, 195 (1980),'Erratum P h y s . R e v . L e t t . 45 453 ( 1 9 8 0 ) . 5. R. Jochemsen, M. Morrow, A . J . B e r l i n s k y , and W.N. Hardy, P h y s . R e v . L e t t . 47, 852 ( 1 9 8 1 ) . 6. R.W. C l i n e , D.A. S m i t h , T . J . G r e y t a k , and D. K l e p p n e r , P h y s . R e v . L e t t . 45, 2117 ( 1 9 8 0 ) . 7. R.W. C l i n e , T . J . G r e y t a k , and D. K l e p p n e r , P h y s . R e v . L e t t . 47, 1195 ( 1 9 8 1 ) . 8. I . F . S i l v e r a and J.T.M. W a l r a v e n , P h y s . R e v . L e t t . 44, 164 ( 1980). 9. J.T.M. W a l r a v e n and I . F . S i l v e r a , P h y s . R e v . L e t t . 44, 168 ( 1980). 10. J.T.M. W a l r a v e n , I . F . S i l v e r a , and A.P.M. M a t t h e y P hys.Rev. L e t t . 45, 449 ( 1 9 8 0 ) . 11. I . F . S i l v e r a and J.T.M. W a l r a v e n P h y s . R e v . L e t t . 45, 1268 ( 1 9 8 0 ) . 12. A.P.M. M a t t h e y , J.T.M. W a l r a v e n and I . F . S i l v e r a , P h y s . R e v . L e t t . 46, 668 ( 1 9 8 1 ) . . • 13. G.H. van Y p e r e n , A.P.M. M a t t h e y , J.T.M. W a l r a v e n , and I . F . S i l v e r a , P h y s . R e v . L e t t £ 7 , 800 ( 1 9 8 1 ) . 14. R.A. G u y e r , M.D. M i l l e r and J . Y a p l e , Phys.Rev.B 25, 4570 (1982) . — 15. W. H e i t l e r and F. London, Z . P h y s i k 44, 455 ( 1 9 2 7 ) . 16. W. K o l o s and L. W o l n i e w i c z , J.Chem.Phys. 4_3, 2429 ( 1 9 6 5 ) . 17. W. K o l o s and L. W o l n i e w i c z , C h e m . P h y s . L e t t . 24, 457 50 ( 1 9 7 4 ) . 18. W. K o l o s and L. W o l n i e w i c z , J . M d . S p e c t r . 54, 303 ( 1 9 7 5 ) . 19. I . F . S i l v e r a Rev.Mod.Phys. 52,393 ( 1 9 8 0 ) . 20. C.E. H e c h t , P h y s i c a 2 5 , 1159 ( 1 9 5 9 ) . 21. J . V . Dugan and R.D. E t t e r s , J.Chem.Phys. 59, 6171 ( 1 9 7 3 ) ; R.D. E t t e r s , J.V. Dugan and R.W. P a l m e r , J.Chem.Phys. 62, 313 ( 1 9 7 5 ) ; R.D. E t t e r s , P h y s . L e t t . 42A, 439 ( 1 9 7 3 ) ; R.L. D a n i l o w i c z , J.V. Dugan and R.D. E t t e r s , J.Chem.Phys. 65, 498 ( 1 9 7 6 ) . 22. W.C. S t w a l l e y and L.H. Nosanow, P h y s . R e v . L e t t . 36, 910 (1976) . 23. S. Crampton, J o u r n a l de P h y s i q u e C7 249 ( 1 9 8 1 ) . 24. H. H e l l w i g , R.F.C. V e s s o t , M.S. L e c i n e , P.W. Z i t z e w i t z , D.W. A l l e n , and J.W. G l a z e , I E E E T r a n s . I n s t r u m . Meas. IM-19, 200 ( 1 9 7 0 ) . 25. F r e n k e l , J . J . P h y s U.S.S.R 3:355 1940. 26. K.R. A t k i n s , C a n . J . P h y s . 3J_, 1165 ( 1 9 5 3 ) . 27. M. W. C o l e , Phys.Rev.B 2, 4239 ( 1 9 7 0 ) . 28. C G . Kuper, P h y s i c a 22, 1291 ( 1 9 5 6 ) . 24, 1009 ( 1 9 5 6 ) . 29. H. I k e z i and P.M. P l a t z m a n , Phys.Rev.B 23, 1145 ( 1 9 8 0 ) . 30. A. Widom, Phys.Rev.A 216 ( 1 9 7 0 ) . 31. M. W. C o l e , Phys.Rev.A J_, 1840 ( 1 9 7 0 ) . 32. I.B. Mantz and D. Edwards, Phys.Rev.B 20, 4518 ( 1 9 7 9 ) . 33. R.P. Feynman, Phys.Rev. 9_4, 262 ( 1 9 5 4 ) . 34. J . L e k n e r , P h i l . M a g . 22, 669 ( 1 9 7 0 ) . 35. M. W. C o l e and F. T o i g o p r e p r i n t . 36. F o r Review see J . A p p e l , In S o l i d S t a t e P h y s i c s , V o l . 2 1 , ed.by F. S e i t z and D. T u r n b u l l page 193. 37. R.P. Feynman, Phys.Rev. 97, 660 ( 1 9 5 5 ) . 38. S.A. J a c k s o n and P.M. P l a t z m a n Phys.Rev.B 2A, 449 (19 81). 51 39. S.A. J a c k s o n and P.M. P l a t z m a n Phys.Rev.B 25, 4886 ( 1 9 8 1 ) . 40. P r i v a t e c o m m u n i c a t i o n . 41. D. P i n e s , i n P o l a r o n s and E x c i t o n s , P r o c e e d i n g s of t h e t h i r d S c o t t i s h U n i v e r s i t i e s Summer S c h o o l , e d i t e d by C.G.Kuper and G . D . W h i t f i e l d (Plenum, New Y o r k , 1963). 42. Handbook o f M a t h e m a t i c a l F u n c t i o n s w i t h F o r m u l a s G r a p h s and M a t h e m a t i c a l T a b l e s . E d i t e d by M. A b r a m o w i t z and I.A. S t e g u n ( D o v e r , New Y o r k ) . 52 APPENDIX A - NUMERICAL METHODS FOR.INTEGRATION i . One d i m e n s i o n a l i n t e g r a l s were c a l c u l a t e d by d i v i d i n g t h e i n t e g r a t i o n i n t e r v a l i n t o N e q u a l segments and c a l c u l a t i n g t h e i n t e g r a l u s i n g 8 p o i n t G a u s s i a n i n t e g r a t i o n i n e a c h o f t h e s e g e m e n t s . The v a l u e o f N was f i x e d so t h a t t h e d e s i r e d a c c u r a c y would be o b t a i n e d . i i . In t h e same s p i r i t two d i m e n s i o n a l i n t e g r a l s were c a l c u l a t e d by d i v i d i n g t h e i n t e g r a t i o n a r e a i n t o MxN p a r a l l e l o g r a m s and p e r f o r m i n g 8x8 G a u s s i a n I n t e g r a t i o n i n e a c h o f them. i i i . I n t e g r a l s where t h e i n t e g r a n d f ( y ) d i v e r g e s a s ( y ~ y i ) " 0 ' 5 a t t h e l i m i t o f t h e i n t e g r a t i o n y x were c a l c u l a t e d u s i n g a G a u s s i a n - t y p e i n t e g r a t i o n d e s c r i b e d i n R e f . 42 S e c t i o n 25.4.39 . The i n t e g r a t i o n i n t e r v a l was f o u n d n u m e r i c a l l y by s o l v i n g t h e a l g e b r a i c e q u a t i o n s d e f i n i n g t h e l i m i t s of t h e i n t e g r a t i o n . i v . The t h e r m a l a v e r a g e i n E q u a t i o n 33 was c a l c u l a t e d by c h a n g i n g t h e v a r i a b l e E t o k 2 and © t o c o s e . Then t h e d ( k 2 ) i n t e g r a l was c a l c u l a t e d , f r o m z e r o t o i n f i n i t y , by u s i n g t h e i n t e g r a t i o n method d e s c r i b e d i n R e f . 42 S e c t i o n 25.4.46 and u s i n g n = 8. The d f c o s e ) i n t e g r a t i o n was c a l c u l a t e d u s i n g t h e method d e s c r i b e d S e c t i o n 25.4.33 i n R e f . 42 w i t h k=2 and n=4. 

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