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Lifetime of positrons in an electron gas
Volume 30A, number 6
PHYSICS
w i t h the h y p e r f i n e i n t e r a c t i o n c o n t r i b u t i n g l e s s than originally predicted.
1. D. E. Barnaa!, R. G. Barnes, B.R. McCart, C.W. Mohn and D.R. Torgeson, Phys. Rev. 157 (1967) 510. 2. W. T. Anderson, M. Ruhilig and R. R. Hewitt, Phys. Rev. 161 (1967) 293. 3. C. H. Townes, C. Herring and W. D. Knight, Phys. Rev. 77 (1950) 852. 4. C. Herring and A. G. Hill, Phys. Rev. 58 (1940) 132. 5. V. E. Wood and F. J. Milford, J.Phys. Chem. Solids 23 (1962) 160. 6. R. Jacques, Cahiers de Phys. 70 (1956) 1. 71-72 (1956) 23; 75-76 (1956) 17.
LIFETIME
OF
POSITRONS
LETTERS
17 November 1969
7. M. Pomerantz and T. P. Das, Phys. Rev. 119 (1960)
70. 8. W. Schneider, L. Jansen and L. Etienne-Amberg, Physics 30 (1964) 84. 9. P. Jena, S.D. Mahanti and T. P. Das, Phys. Rev. Letters 20 (1968) 544. I0. G. F e h r e r and A. F. Kip, Phys. Rev. 98 (1955} 337. 11. Wei-Mei Shyu, G.D. Gaspari and T. P. Das, Phys. Rev. 141 (1966) 603. G. D. Gaspari, Wei-Mei Shyu and T. P. Das, Phys. Rev. 134 (1964) A852. 12. B. J. Austin, V. Heine and L. J. Shaw, Phys. Rev. 127 (1962) 276. 13. T. L. Loucks and P. H. Cutler, Phys. Rev. 133 (1964) A819; 134 (1964) A1618. 14. J. H. T e r r e l , Phys. Rev. 149 (1966) 526.
IN
AN
ELECTRON
GAS
J . G. G A R G and B. L. S A R A F
Department of Physics, University of Rajasthan, Jaipur, India Received 10 October 1969
The positron lifetimes in an electron gas are calculated using the bound state concept of Bergersen. In o r d e r to explain the results of two- T angular distribution, a fast exchange between the electron in the bound s t a t e and the free electrons outside it, is assumed. The calculated annihilation rates are found in good agreement with the experimental and theoretical results. T h e l i f e t i m e s of p o s i t r o n s in the e l e c t r o n g a s h a v e b e e n s u b j e c t of d e t a i l e d i n v e s t i g a t i o n s [1,2]. Since a one-electron model predicts lifetimes l a r g e r t h a n the m e a s u r e d v a l u e s [3] m a n y - b o d y t e c h n i q u e s h a v e o f t e n b e e n u s e d . It can be s h o w n t h a t a p o s i t r o n ~ u m - l i k e bound s t a t e d o e s not e x i s t at m e t a l l $ c d e n s i t i e s and the c o n c e p t of the bound s t a t e in m e t a l s c a n n o t be d e f i n e d p r o p e r l y [4]. H o w e v e r , B e r g e r s e n [5] h a s s h o w n that a bound s t a t e in the e l e c t r o n g a s can be e x p l i c i t l y i n t r o d u c e d by m a k i n g an a n s a t z f o r t h e p o l a r i z a t i o n c l o u d a r o u n d the p o s i t r o n . But h i s r e s u l t s f o r the a n n i h i l a t i o n r a t e s do not a g r e e w i t h the e x p e r i m e n t a l r e s u l t s , and he d o e s not g i v e any e x p l a n a t i o n f o r t h e o b s e r v e d two T - a n g u l a r c o r relation results. In the p r e s e n t n o t e we h a v e c a l c u l a t e d t h e p o s i t r o n a n n i h i l a t i o n r a t e s u s i n g t h e bound s t a t e c o n c e p t of B e r g e r s e n . In a d d i t i o n , we a s s u m e a f a s t e x c h a n g e b e t w e e n t h e e l e c t r o n in the bound s t a t e and t h e f r e e e l e c t r o n s o u t s i d e it. T h i s e x p l a i n s t h e o b s e r v e d two T - a n g u l a r c o r r e l a t i o n r e s u l t s in m e t a l s . B e l o w we g i v e t h e r e a s o n i n g for this calculation.
T h e o b s e r v e d a n n i h i l a t i o n r a t e f o r low d e n s i t y m e t a l s a p p r o a c h e s that of s p i n a v e r a g e d p o s i t r o n i u m [3]; we a s s u m e that the bound s t a t e d e f i n e d by B e r g e r s e n [5], a n n i h i l a t e s at t h e s a m e r a t e a s that of s p i n a v e r a g e d p o s i t r o n i u m , i . e . , ~ p o s = 2 x 10 9 s e c -1. W h e n one c o n s i d e r s t h e bound s t a t e to be p e r f e c t l y c h a r g e n e u t r a l , one c a n n o t o b t a i n a s a t i s f a c t o r y d e s c r i p t i o n of the a n g u l a r c o r r e l a t i o n e x p e r i m e n t s , f o r it i m p a r t s a d e f i n i t e c e n t r e of m a s s m o m e n t u m to the bound s t a t e and g i v e s r i s e to a n a r r o w p e a k in two g a m m a a n g u l a r d i s t r i b u t i o n , w h i c h i s n e v e r o b s e r v e d [6]. T o a v o i d t h i s , the p o s i t r o n m u s t be correlated in the s a m e w a y with all the electron states and no electron state must be singled out and treated in a different w a y from others. O n e m a y probably a c c o m m o d a t e this by considering a very fast exchange between the electron in the bound state and the free electrons outside it. The annihilation rate would be expected to be
[s],
X = X p o s + Xp.
(I)
where ;~p stands for the "pick off" rate and is
369
Volume 30A, number 6
PHYSICS
given by ~p = T r r o2 c p
.
(2)
H e r e r o i s t h e c l a s s i c a l e l e c t r o n r a d i u s , c i s the v e l o c i t y of l i g h t and p i s t h e e f f e c t i v e e l e c t r o n d e n s i t y o u t s i d e the bound s t a t e . To c a l c u l a t e p, w e a s s u m e that in g e n e r a l , the bound s t a t e d i s c u s s e d a b o v e n e e d not n e c e s s a r i l y b e c h a r g e n e u t r a l ; it is n e u t r a l only at e x t r e m e ly low e l e c t r o n d e n s i t i e s . The m e a n a n n i h i l a t i o n r a t e kv would t h e n b e ~v = f ~ p o s + (1 - f ) (~tpo s + ~tp) ,
(3)
w h e r e f , s t a n d s f o r the f r a c t i o n of the t i m e bound s t a t e o c c u r s a s n e u t r a l and (1 - f ) c o r r e s p o n d s to the t i m e in w h i c h r a p i d e x c h a n g e t a k e s p l a c e . T h e d e n s i t y of t h e e l e c t r o n g a s i s g i v e n by a 3 r 3 ]-1
(4)
w h e r e a o i s the B o h r r a d i u s . T h e p r e c i s e d e p e n d e n c e o f f on r s i s not known: We p r o p o s e the f o l l o w i n g r e l a t i o n ; f : (r3s/X 3 - 1) 7
n =
O
r s
1
r s >~x
t
(r3s < 2x 3)
(5)
(7)
F o r r s > x, a s the p r o b a b i l i t y of c h a r g e n e u t r a l bound s t a t e o c c u r r e n c e i s f i n i t e , t h e e n h a n c e m e n t in d e n s i t y of the e l e c t r o n gas w i l l be v e r y s m a l l .
3'70
17 November 1969
We m a y t h e r e f o r e t a k e p = n. Eq. (7) t h e n r e d u c e s to
I v]rs>X For r s
4xvl
--
pos + (2
r 3 / x 3)
(8)
eq. (7) h o w e v e r , a s s u m e s the f o r m
s<,: Xpos +
co
¢9)
In a F e r m i g a s s y s t e m a s a w h o l e , the t r a n s i t i o n f r o m " s c r e e n i n g " to " b i n d i n g " is s m o o t h [7], t h e r e f o r e kv should not h a v e a s i n g u l a r i t y a t r s = x. T h u s (8) and (9) b e c o m e i d e n t i a l . Eq. (4) and (8) g i v e s ~v = hpos + 12.05 r s 3 (2 -
r3/x 3) × 109
(10)
If we t a k e We n o t e that L¢ ~ ~pos at r s = 23x. i x ~ 6, the r s v a l u e - a t w h i c h e l e c t r o n - p o s i t r o n p r o p a g a t o r d i v e r g e s [8], the n e u t r a l bound s t a t e w i l l a p p e a r at r s ~ 7.6, w h i c h i s c l o s e t o t s ~ 8, p r e d i c t e d by H e l d and K a b a n a [4]. One m a y e a s i l y s e e that v a r i a t i o n of ~v w i t h r s u s i n g eq. (10) i s in good a g r e e m e n t with the e x p e r i m e n t a l [3] and t h e o r e t i c a l r e s u l t s [2].
(6)
In eq. (5), x c o r r e s p o n d s to the v a l u e of r s at w h i c h bound s t a t e a n n i h i l a t e s w i t h a r a t e g i v e n by eq. (1). On c o m b i n i n g e q s . (2) to (6), we get ~v = kpos + (1 + ~ - 77. r3s/X3) 7rro2C p
LETTERS
1. S. Kahana, Phys. Rev. 129 (1963) 1622. 2. J. P. Carbotte, Phys. Rev. 155 (1967) 197.
3. H. Weisberg and S. Berko, Phys. Rev. 154 (1967) 249. 4. A. Held and S. Kahana, Can. J. Phys. 42 (1964) 1908. 5. B. Bergersen, Phys. Rev. 181 (1969) 499. 6. P. G. Varlashkin and A. T. Stewart, Phys. Rev. 148 (1966) 459. 7. W. Kohn and C. Majumdar, Phys. Rev. 138 (1965) A1617. 8. J. Arponen and P. Jauho, Phys. Rev. 167 (1968) 239.
PHYSICS
w i t h the h y p e r f i n e i n t e r a c t i o n c o n t r i b u t i n g l e s s than originally predicted.
1. D. E. Barnaa!, R. G. Barnes, B.R. McCart, C.W. Mohn and D.R. Torgeson, Phys. Rev. 157 (1967) 510. 2. W. T. Anderson, M. Ruhilig and R. R. Hewitt, Phys. Rev. 161 (1967) 293. 3. C. H. Townes, C. Herring and W. D. Knight, Phys. Rev. 77 (1950) 852. 4. C. Herring and A. G. Hill, Phys. Rev. 58 (1940) 132. 5. V. E. Wood and F. J. Milford, J.Phys. Chem. Solids 23 (1962) 160. 6. R. Jacques, Cahiers de Phys. 70 (1956) 1. 71-72 (1956) 23; 75-76 (1956) 17.
LIFETIME
OF
POSITRONS
LETTERS
17 November 1969
7. M. Pomerantz and T. P. Das, Phys. Rev. 119 (1960)
70. 8. W. Schneider, L. Jansen and L. Etienne-Amberg, Physics 30 (1964) 84. 9. P. Jena, S.D. Mahanti and T. P. Das, Phys. Rev. Letters 20 (1968) 544. I0. G. F e h r e r and A. F. Kip, Phys. Rev. 98 (1955} 337. 11. Wei-Mei Shyu, G.D. Gaspari and T. P. Das, Phys. Rev. 141 (1966) 603. G. D. Gaspari, Wei-Mei Shyu and T. P. Das, Phys. Rev. 134 (1964) A852. 12. B. J. Austin, V. Heine and L. J. Shaw, Phys. Rev. 127 (1962) 276. 13. T. L. Loucks and P. H. Cutler, Phys. Rev. 133 (1964) A819; 134 (1964) A1618. 14. J. H. T e r r e l , Phys. Rev. 149 (1966) 526.
IN
AN
ELECTRON
GAS
J . G. G A R G and B. L. S A R A F
Department of Physics, University of Rajasthan, Jaipur, India Received 10 October 1969
The positron lifetimes in an electron gas are calculated using the bound state concept of Bergersen. In o r d e r to explain the results of two- T angular distribution, a fast exchange between the electron in the bound s t a t e and the free electrons outside it, is assumed. The calculated annihilation rates are found in good agreement with the experimental and theoretical results. T h e l i f e t i m e s of p o s i t r o n s in the e l e c t r o n g a s h a v e b e e n s u b j e c t of d e t a i l e d i n v e s t i g a t i o n s [1,2]. Since a one-electron model predicts lifetimes l a r g e r t h a n the m e a s u r e d v a l u e s [3] m a n y - b o d y t e c h n i q u e s h a v e o f t e n b e e n u s e d . It can be s h o w n t h a t a p o s i t r o n ~ u m - l i k e bound s t a t e d o e s not e x i s t at m e t a l l $ c d e n s i t i e s and the c o n c e p t of the bound s t a t e in m e t a l s c a n n o t be d e f i n e d p r o p e r l y [4]. H o w e v e r , B e r g e r s e n [5] h a s s h o w n that a bound s t a t e in the e l e c t r o n g a s can be e x p l i c i t l y i n t r o d u c e d by m a k i n g an a n s a t z f o r t h e p o l a r i z a t i o n c l o u d a r o u n d the p o s i t r o n . But h i s r e s u l t s f o r the a n n i h i l a t i o n r a t e s do not a g r e e w i t h the e x p e r i m e n t a l r e s u l t s , and he d o e s not g i v e any e x p l a n a t i o n f o r t h e o b s e r v e d two T - a n g u l a r c o r relation results. In the p r e s e n t n o t e we h a v e c a l c u l a t e d t h e p o s i t r o n a n n i h i l a t i o n r a t e s u s i n g t h e bound s t a t e c o n c e p t of B e r g e r s e n . In a d d i t i o n , we a s s u m e a f a s t e x c h a n g e b e t w e e n t h e e l e c t r o n in the bound s t a t e and t h e f r e e e l e c t r o n s o u t s i d e it. T h i s e x p l a i n s t h e o b s e r v e d two T - a n g u l a r c o r r e l a t i o n r e s u l t s in m e t a l s . B e l o w we g i v e t h e r e a s o n i n g for this calculation.
T h e o b s e r v e d a n n i h i l a t i o n r a t e f o r low d e n s i t y m e t a l s a p p r o a c h e s that of s p i n a v e r a g e d p o s i t r o n i u m [3]; we a s s u m e that the bound s t a t e d e f i n e d by B e r g e r s e n [5], a n n i h i l a t e s at t h e s a m e r a t e a s that of s p i n a v e r a g e d p o s i t r o n i u m , i . e . , ~ p o s = 2 x 10 9 s e c -1. W h e n one c o n s i d e r s t h e bound s t a t e to be p e r f e c t l y c h a r g e n e u t r a l , one c a n n o t o b t a i n a s a t i s f a c t o r y d e s c r i p t i o n of the a n g u l a r c o r r e l a t i o n e x p e r i m e n t s , f o r it i m p a r t s a d e f i n i t e c e n t r e of m a s s m o m e n t u m to the bound s t a t e and g i v e s r i s e to a n a r r o w p e a k in two g a m m a a n g u l a r d i s t r i b u t i o n , w h i c h i s n e v e r o b s e r v e d [6]. T o a v o i d t h i s , the p o s i t r o n m u s t be correlated in the s a m e w a y with all the electron states and no electron state must be singled out and treated in a different w a y from others. O n e m a y probably a c c o m m o d a t e this by considering a very fast exchange between the electron in the bound state and the free electrons outside it. The annihilation rate would be expected to be
[s],
X = X p o s + Xp.
(I)
where ;~p stands for the "pick off" rate and is
369
Volume 30A, number 6
PHYSICS
given by ~p = T r r o2 c p
.
(2)
H e r e r o i s t h e c l a s s i c a l e l e c t r o n r a d i u s , c i s the v e l o c i t y of l i g h t and p i s t h e e f f e c t i v e e l e c t r o n d e n s i t y o u t s i d e the bound s t a t e . To c a l c u l a t e p, w e a s s u m e that in g e n e r a l , the bound s t a t e d i s c u s s e d a b o v e n e e d not n e c e s s a r i l y b e c h a r g e n e u t r a l ; it is n e u t r a l only at e x t r e m e ly low e l e c t r o n d e n s i t i e s . The m e a n a n n i h i l a t i o n r a t e kv would t h e n b e ~v = f ~ p o s + (1 - f ) (~tpo s + ~tp) ,
(3)
w h e r e f , s t a n d s f o r the f r a c t i o n of the t i m e bound s t a t e o c c u r s a s n e u t r a l and (1 - f ) c o r r e s p o n d s to the t i m e in w h i c h r a p i d e x c h a n g e t a k e s p l a c e . T h e d e n s i t y of t h e e l e c t r o n g a s i s g i v e n by a 3 r 3 ]-1
(4)
w h e r e a o i s the B o h r r a d i u s . T h e p r e c i s e d e p e n d e n c e o f f on r s i s not known: We p r o p o s e the f o l l o w i n g r e l a t i o n ; f : (r3s/X 3 - 1) 7
n =
O
r s
1
r s >~x
t
(r3s < 2x 3)
(5)
(7)
F o r r s > x, a s the p r o b a b i l i t y of c h a r g e n e u t r a l bound s t a t e o c c u r r e n c e i s f i n i t e , t h e e n h a n c e m e n t in d e n s i t y of the e l e c t r o n gas w i l l be v e r y s m a l l .
3'70
17 November 1969
We m a y t h e r e f o r e t a k e p = n. Eq. (7) t h e n r e d u c e s to
I v]rs>X For r s
4xvl
--
pos + (2
r 3 / x 3)
(8)
eq. (7) h o w e v e r , a s s u m e s the f o r m
s<,: Xpos +
co
¢9)
In a F e r m i g a s s y s t e m a s a w h o l e , the t r a n s i t i o n f r o m " s c r e e n i n g " to " b i n d i n g " is s m o o t h [7], t h e r e f o r e kv should not h a v e a s i n g u l a r i t y a t r s = x. T h u s (8) and (9) b e c o m e i d e n t i a l . Eq. (4) and (8) g i v e s ~v = hpos + 12.05 r s 3 (2 -
r3/x 3) × 109
(10)
If we t a k e We n o t e that L¢ ~ ~pos at r s = 23x. i x ~ 6, the r s v a l u e - a t w h i c h e l e c t r o n - p o s i t r o n p r o p a g a t o r d i v e r g e s [8], the n e u t r a l bound s t a t e w i l l a p p e a r at r s ~ 7.6, w h i c h i s c l o s e t o t s ~ 8, p r e d i c t e d by H e l d and K a b a n a [4]. One m a y e a s i l y s e e that v a r i a t i o n of ~v w i t h r s u s i n g eq. (10) i s in good a g r e e m e n t with the e x p e r i m e n t a l [3] and t h e o r e t i c a l r e s u l t s [2].
(6)
In eq. (5), x c o r r e s p o n d s to the v a l u e of r s at w h i c h bound s t a t e a n n i h i l a t e s w i t h a r a t e g i v e n by eq. (1). On c o m b i n i n g e q s . (2) to (6), we get ~v = kpos + (1 + ~ - 77. r3s/X3) 7rro2C p
LETTERS
1. S. Kahana, Phys. Rev. 129 (1963) 1622. 2. J. P. Carbotte, Phys. Rev. 155 (1967) 197.
3. H. Weisberg and S. Berko, Phys. Rev. 154 (1967) 249. 4. A. Held and S. Kahana, Can. J. Phys. 42 (1964) 1908. 5. B. Bergersen, Phys. Rev. 181 (1969) 499. 6. P. G. Varlashkin and A. T. Stewart, Phys. Rev. 148 (1966) 459. 7. W. Kohn and C. Majumdar, Phys. Rev. 138 (1965) A1617. 8. J. Arponen and P. Jauho, Phys. Rev. 167 (1968) 239.