Positron annihilation in copper

Volume 38A, n u m b e r 7

PHYSICS

LETTERS

27 M a r c h 1972

POSITRON ANNIHILATION IN COPPER H. MORINAGA *

Institute of Physics, College of General Education, University of Tokyo, Meguro-ku, Tokyo, Japan Received 28 J a n u a r y 1972

The angular c o r r e l a t i o n curve of copper was compared with the calculated one by using the phenomenological F e r m i surface of copper. The comparison suggests that the annihilation probability of positrons is enhanced near the F e r m i surface.

K a h a n a [1] h a s c a l c u l a t e d t h e e n h a n c e m e n t e f f e c t in m e t a l s a n d p r o p o s e d t h a t t h e e f f e c t d e c r e a s e d w i t h a n i n c r e a s e of t h e e l e c t r o n d e n s i t y . D o n a g h y [2] m e a s u r e d t h e a n g u l a r c o r r e l a t i o n of sodium and found the enhancement effect showing a good a g r e e m e n t w i t h K a h a n a ' s p r o p o s i t i o n . In t h e p r e s e n t p a p e r , t h e a n g u l a r c o r r e l a t i o n c u r v e o b s e r v e d by u s i n g t h e c r o s s e d - s l i t g e o m e try is compared with the curve which was calcul a t e d by u s i n g t h e p r e c i s e p h e n o m e n o l o g i c a l F e r m i s u r f a c e of c o p p e r p r o p o s e d by H a l s e [3]. Fig. 1 shows the comparison. The core-electron c o n t r i b u t i o n , i c o r e ( ~ ) , to t h e a n n i h i l a t i o n r a t e i s e s t i m a t e d a s f o l l o w s . W e d e f i n e a s I(,~): t h e t o t a l a n n i h i l a t i o n r a g e , a n d ICOnd(0): t h e c o n d u c t i o n e l e c t r o n c o n t r i b u t i o n to t h e a n n i h i l a t i o n r a t e . F u r t h e r , a i s t h e r a t i o of a n n i h i l a t i o n r a t e of c o r e - e l e c t r o n to t h e t o t a l a n n i h i l a t i o n r a t e at O = 0, n a m e l y , a = Icore(o)/I(O). Now, l e t IC°nd(o) b e t h e c a l c u l a t e d a n n i h i l a tion rate for the conduction-electron; then the c o r e c o n t r i b u t i o n in t h e f i g u r e i s g i v e n by / c ° r e ( 0 ) = Iexp(0) - ICc°nd(e), w h e r e Iexp(0) i s the observed annihilation rate and Ic°na(o) is n o r m a l i z e d to b e I e x p ( 0 ) × ( l - a ) a t 0 = 0. A s u m of two G a u s s i a n f u n c t i o n s f i t t o t h e h i g h e r m o m e n t u m t a i l i s a l s o s h o w n in t h e f i g u r e , w h i c h w a s f i t b y l e a s t s q u a r e m e t h o d . It i s e a s i l y s e e n t h a t n o n e of t h e s e I c ° r e ( 0 ) - c u r v e c a n f i t t o t h e G a u s s i a n . T h e c u r v e f o r a = 0.5 m o s t n e a r l y a g r e e s w i t h t h e G a u s s i a n at O = 0. H o w e v e r , t h e curve for a = 0.5 is seen to be larger than the G a u s s i a n e s p e c i a l l y a t O = 3 - 5 m i l l i r a d i a n s . It is easily recognized from the schematic side v i e w of t h e F e r m i s u r f a c e s h o w n in t h e f i g u r e , * P r e s e n t a d d r e s s : Department of P h y s i c s , Mei]o U n i v e r s i t y , Showa-ku, Nagoya, Japan.

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Fig. 1. Comparison of the observed angular correlation curve for copper with that calculated by using the F e r m i surface proposed by Halse [3]. The symbol a is the r a t i o of the annihilation r a t e of positron with the c o r e - e l e c t r o n to the total annihilation rate at 0 = 0. A side view of the copper F e r m i surface and the resolution curves are shown to make the existence of the enhancement effect clear. that the discrepancy is especially large where the g r e a t e r p a r t of t h e c o n d u c t i o n - e l e c t r o n s d e t e c t e d on the curve are near the Fermi surface. The a n o m a l o u s l y l a r g e e n h a n c e m e n t at t h e ( l l l > - n e c k , w h i c h w a s o b s e r v e d in t h e c a s e of t h e r o t a t i n g s p e c i m e n m e t h o d [4], m a y b e l i t t l e in t h e p r e s e n t c a s e , b e c a u s e t h e c o n t r i b u t i o n of t h e ( l l l ) - n e c k to the annihilation rate is reduced considerably

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Volume 38A, n u m b e r 7

PHYSICS

by t h e h o r i z o n t a l r e s o l u t i o n , s h o w n in fig. 1. T h u s , t h e d i s c r e p a n c y at 0 = 3 - 5 m i l l i r a d i a n s s u g g e s t s t h a t t h e a n n i h i l a t i o n p r o b a b i l i t y of p o s i t r o n s is c e r t a i n l y l a r g e n e a r t h e F e r m i s u r f a c e a l s o in copper. Up to h e r e , we a s s u m e d t h e c o r e c o n t r i b u t i o n to b e g i v e n by a G a u s s i a n f u n c t i o n , w h i c h w a s f i t to t h e h i g h e r m o m e n t u m t a i l of t h e a n g u l a r c o r r e l a t i o n c u r v e . T h e r e is no t e c h n i q u e by w h i c h t h e c o r e c o n t r i b u t i o n is d e t e c t e d a l o n e . H o w e v e r , the angular correlation curves for insulators, in w h i c h t h e p o s i t r o n a n n i h i l a t e s only w i t h c o r e e l e c t r o n s , i s s i m i l a r to a G a u s s i a n f u n c t i o n [5]. C o n s i d e r i n g t h i s s i t u a t i o n , we m a y c o n c l u d e t h a t t h e e n h a n c e m e n t e f f e c t c a n not b e i g n o r e d in c o p p e r .

514

LETTI?:RS

27 March 1972

T h e a u t h o r w i s h e s to e x p r e s s h i s t h a n k s to P r o f e s s o r K. F u j i w a r a , to P r o f e s s o r O. S u e o k a , to M r . T. A k a h a n e a n d to M r . T. H y o d o f o r t h e i r c o l l a b o r a t i o n s in a p a r t of t h i s e x p e r i m e n t .

RCf~VCI?¢.'CS [1] S. Kahana, Phys. Rev. 129 ([965) 1522. [2] J . J . Donaghy and A.T. Stewart, Phys. Rev. 164 (1967) 396. [3] M.R. Halse, Phys. T r a n s . Roy. Soc. London A265 (1969) 507. [4] H. Morinaga, J. Phys. Soc, Japan, to be published. [5] P o s i t r o n annihilation, eds. A. T. Stewart and L. O. Roelling (Academic P r e s s , New York, 1967).