Pseudopotential calculation of positron annihilation in beryllium

Volume 30A. number 8

PHYSICS

LETTERS

8 December 1969

PSEUDOPOTENTIAL CALCULATION OF ANNIHILATION IN BERYLLIUM*

POSITRON

J. B. SHAND Jr. ** Savannah Ri~,er L a b o r a t o r y , E.l.du Pont de N e m o u r s and Conzpany. A i k e n , South Carolina 29801, USA

Received

i0 October

1969

A local pseudopotential model for conduction electrons in beryllium, chosen to fit the F e r m i surface, has been used to calculate the angular correlation of photon pairs from annihilation of positrons in beryllium. T h e a n g u l a r d i s t r i b u t i o n of a n n i h i l a t i o n r a d i a tion f r o m a m e t a l can be r e l a t e d to the m o m e n t u m d i s t r i b u t i o n of c o n d u c t i o n e l e c t r o n s in the m e t a l [1]. M e a s u r e m e n t s on o r i e n t e d s i n g l e c r y s t a l s of b e r y l l i u m p e r f o r m e d by S t e w a r t et a1.[2] and B e r k o [3] s h o w e d a p p r e c i a b l e a n i s o t r o p y . T h e a n g u l a r d i s t r i b u t i o n c o n s i s t s of a b r o a d , s l o w l y v a r y i n g p a r t and a m u c h n a r r o w e r , r o u g h l y p a r a bolic central part. The experiment has been rep e a t e d by Shand and S t e w a r t [1] w i t h h i g h e r r e s o l u t i o n f o r the (0001) d i r e c t i o n [4]. D a t a a r e p r e s e n t e d in fig. 1. In the s i m p l e s t a p p r o x i m a t i o n , the photon p a i r d e t e c t i o n r a t e at a p a r t i c u l a r a n g l e is p r o p o r t i o n a l to the a r e a of a s l i c e t h r o u g h the F e r m i s e a [1]. T h i s t r e a t m e n t i s i n a d e q u a t e for b e r y l l i u m , w h i c h has l a r g e e n e r g y g a p s at B r i l l o u i n z o n e f a c e s i n t e r s e c t i n g the F e r m i s u r f a c e and associated large higher momentum components of the e l e c t r o n w a v e f u n c t i o n . The F e r m i s u r f a c e is c o n s i d e r a b l y d i s t o r t e d f r o m a f r e e e l e c t r o n s p h e r e [4]. T h e c o n d u c t i o n e l e c t r o n w a v e f u n c t i o n may be r e p r e s e n t e d , in O P W t e r m s , a s a s m o o t h p a r t p l u s a s u m of c o r e s t a t e s w h i c h g u a r a n t e e s c o n d u c t i o n - c o r e o r t h o g o n a l i t y . T h e c e n t r a l p a r t of the p o s i t r o n d a t a is i n t e r p r e t e d h e r e a s due to the s m o o t h p a r t of the e l e c t r o n i c w a v e f u n c t i o n and the b r o a d p a r t to the o r t h o g o n a l i t y p l u s a n n i h i l a t i o n with o r d i n a r y c o r e e l e c t r o n s . In this

t r e a t m e n t only the c e n t r a l p a r t i s c o n s i d e r e d . T h e s m o o t h p a r t of the w a v e f u n c t i o n ¢bk is the s o l u t i o n to a m o d i f i e d S c h r S d i n g e r e q u a t i o n (- V 2 + Vp ) q~ = Ek~b k (in a t o m i c units) w h e r e V_ is the p s e u d o p o t e n t i a l , potentialP[ 5]. ~ k m a y be e x p a n d e d ~ble = -~aac(k) e x p [ i ( k -

478

or effective

(~. r ]

(2)

w h e r e e a c h G is a r e c i p r o c a l l a t t i c e v e c t o r . T h e 23 s h o r t e s t r e c i p r o c a l l a t t i c e v e c t o r s , a c o n v e n i e n t n u m b e r for the h e x a g o n a l s t r u c t u r e , w e r e u s e d . Vp w a s t r e a t e d a s an o r d i n a r y l o c a l p o t e n tial. Substitution into e q u a t i o n (1) g i v e s

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* T h e i n f o r m a t i o n c o n t a i n e d in t h i s a r t i c l e w a s d e v e l o p e d d u r i n g t h e c o u r s e of w o r k u n d e r C o n t r a c l A T ( 0 7 - 2 ) - I w i t h t h e U. S, A t o m i c E n e r g y C o m m i s s i o n , while the author was an Oak Ridge Associated Universities Research l)articipant at the Savannah River I.aboratory. ** P r e s e n t a d ( I r e s s : B e r r y C o l l e g e , [{()me, G e o r g i a 3O 149.

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I 2 3 4 5 6 7 S AnGle Between Annihilation Photons in Millirodions J

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Total Photon Wave Vector

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t 1.0

k z in Atomic

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Units

Fig. 1. C-deulated and experimental angular c o r r e l ation of annihilation photons for (0001) direction in beryllium

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Volume 30A, n u m b e r 8

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PHYSICS

Data (Low

Resolution)

g: Backgr?und Subtracted From Doto

2-

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8 D e c e m b e r 1969

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E

LETTERS

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Angle Between Annihilation Photons in Milliradians

Fig.2. Calculated and e x p e r i m e n t a l angular c o r r e l a t i o n of annihilation photons for (11E0) direction in beryllium

aG,[(k-G')2

- Ek] + ~ a G V G , _ G = O

(3)

T h e p o t e n t i a l c o e f f i c i e n t s V G, _ G w e r e s e t equal to-zero, except for (/'-(/set e q u a l to 1010, 0 0 0 2 , 1011, 1012 a n d v e c t o r s e q u i v a l e n t to t h e m . These four coefficients were used as disposable p a r a m e t e r s i n f i t t i n g t h e d i m e n s i o n s of t h e F e r m i s u r f a c e . A b s o ] u t e v a l u e s of t h e f i n a l p o t e n t i a l c o e f f i c i e n t s a r e : (10]'0), 0 . 1 1 0 0 Ry; (0002), 0 . 3 5 0 2 Ry; (1011), 0 . 2 5 8 8 R y ; (1012), 0 . 0 5 1 5 Ry; w i t h F e r m i e n e r g y 0.84 Ry. T h e s e v a l u e s a r e not unique and may be regarded as parameters in an interpolation scheme. The positron wave function was approximated as constant. The wave vector distribution for ann i h i l a t i o n of s u c h a p o s i t r o n [8] w i t h ~bb i s

(41

G E q u a t i o n s (3) w e r e s o l v e d f o r t h e c o e f f i c i e n t s a t t h e e q u i v a l e n t of 6000 p o i n t s i n t h e f i r s t z o n e and the distribution P was summed for all occup i e d s t a t e s . R e s u l t s a r e s h o w n i n fig.1 a n d f i g . 2 . The broad, slowly varying background was subtracted from the data. Data were plotted for one s i d e of t h e l i n e of s y m m e t r y . T h e l o w e r r e s o l u t i o n e x p e r i m e n t w a s d e s c r i b e d in r e f . 1 . T h e h i g h resolution experiment was conducted similarly. T h e d e t e c t o r s l i t s s u b t e n t e d 96 × 0.2 m i l l i r a d i a n s a t t h e s a m p l e . T h e s a m p l e w a s k e p t a t 1 0 0 ° K [7]. A n e a r l i e r , s i m p l e t r e a t m e n t of t h e (0001) d a t a a p p e a r e d p r e v i o u s l y [1]. W h e n o n l y two p l a n e w a v e s f o r ~b w e r e u s e d , t h e s i m p l e t r e a t m e n t did not adequately represent the data near zone b o u n d a r i e s , but t h e p r e s e n t t r e a t m e n t d o e s . T h e c l o s e n e s s of fit i n d i c a t e s t h a t t h e p o s i t r o n d a t a c a n b e i n t e r p r e t e d i n t e r m s of t h e e l e c t r o n w a v e function in a pseudopotential treatment. The author is grateful for helpful discussions with Dr. A.T. Stewart.

References 1. P o s i t r o n annihilation, eds. A. T. Stewart and L.O. Roellig {Academic P r e s s , Inc., New York, 1967). 2. A. T. Stewart, J. B. Shand, J. J. Donaghy and J. H. K u s m i s s , Phys. Rev. 128 {1962) 118. 3. S. Berko, Phys. Rev. 128 {1962) 2166. 4. B. R. Watts. Proc. Roy. Soc. A282 (1964) 521; J. H. Tripp, P. M. E v e r e t t , W. L. Gordon and R.W. Stark. Phys. Rev. 180 {1969} 669. 5. W.A. Harrison, Pseudopotentials in the theory of m e t a l s (W. A. Benjamin. Inc.. New York, 1966} and r e f e r e n c e s included. 6. S. Berko and J. S. Plaskctt, ]Phys. Rev. 112 {1958~ 1877. 7. A. T. Stewart, J. B. Shand and S. M. Kim, Proc. Phys. Soc. London) 88 {1966) 1001.

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