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Temperature dependence of the annihilation rate of positrons in argon gas
Volume 27A, number 10
PHYSICS LETTERS
TEMPERATURE OF
DEPENDENCE POSITRONS
OF THE IN ARGON
7 October 1968
ANNIHILATION GAS *
RATE
D. B. MILLER **, P. H. R. ORTH *** and G. JONES Department of Physics, University of British Columbia, Vancouver, B. C. , Canada
Received 15 August 1968
A temperature dependence of the positron annihilation rate in argon gas has beenfound. A 28% decrease in the annihilation rate occurs when the temperature is raised from 140 to 480OK.
A velocity dependence o2 the p r o b a b i l i t y of a n n i h i l a t i o n of a slow p o s i t r o n with a noble gas atom is e x p e r i m e n t a l l y well e s t a b l i s h e d [1-3]. The dependence is not known in detail, however, s i n c e the e x p e r i m e n t a l r e s u l t s depend also on the c r o s s s e c t i o n for e l a s t i c s c a t t e r i n g , and the v e locity dependence of this c r o s s s e c t i o n is also unknown. In the e x p e r i m e n t r e p o r t e d h e r e , the m e a n l i f e t i m e of p o s i t r o n s in a r g o n gas was found to depend on the t e m p e r a t u r e of the gas. This e x p e r i m e n t is thus the f i r s t to yield i n f o r m a t i o n c o n c e r n i n g the explicit velocity dependence of the a n n i h i l a t i o n r a t e of p o s i t r o n s in argon (albeit over a v e r y s m a l l velocity range). The i n s t r u m e n t a t i o n used in this e x p e r i m e n t was e s s e n t i a l l y the s a m e as that used in e a r l i e r work [2,4,6]. In t h i s c a s e , however, the gas was contained in a s t a i n l e s s s t e e l v e s s e l , 6 in. in d i a m e t e r and 7.25 in. long. The t e m p e r a t u r e of the gas (and c h a m b e r ) could be v a r i e d over the r a n g e 140 to 480°K, with a stability of ± 3°K d u r ing a run. The c h a m b e r was filled with 99.999% a r g o n gas supplied by Matheson of Canada, Ltd., to d e n s i t i e s between 8.0 and 10.5 a m a g a t . A check that no s e r i o u s c o n t a m i n a t i o n of the gas o c c u r r e d d u r i n g the e x p e r i m e n t was afforded by continuously m o n i t o r i n g both the "shoulder width" of the d i r e c t a n n i h i l a t i o n l i f e t i m e s p e c t r u m and the o r t h o p o s i t r o n i u m l i f e t i m e , both of which have b e e n found to be v e r y s e n s i t i v e to c o n t a m i n a t i o n [2,7,8]. The s h o u l d e r - w i d t h g a s - d e n s i t y product was c o n s t a n t at 350 n s e c - a m a g a t , and the o r t h o p o s i t r o n i u m a n n i h i l a t i o n rate (at a d e n s i t y of 10.2 amagat) was (11.1 ± 0.2) × 106 sec -1, both * Research supported by the National Research Council of Canada, Grant A-1564. ** Holder of a National Research Council Fellowship. *** National Research Council Postdoctoral Fellow.
,~,o / D i
D
'l
n
i
v
i
(~ 'se;i °m°o°;'i) (O)
....
1
7
1
5
"
-
~
Za"
Z,%= 1 3 6 T
I
-Z,tt 2 = 60-2.89T
.s
i
-o.2as
~
~
I
+.056T
3.o-
~ ~
~ "~T
t
25
!
I
180
I 220
I 260
I 300
I 340
I 380
I 420
I 460
2~
TEMPERATURE (°K) POSITRON
TEMPERATURE
IO0
X%
200
I
I
_,o
(T - nlv2/2k)
400 I
600
900
I'
1200
I
I
=K
/
-.-.<__
.o/2
J
POSITRON
~I 4 VELOCITY
=
~ .06
=
i , .OS
OJO
(atomic unite)
Fig. I. a) Positron annihilation rate in argon as a function of temperature; b) Positron annihilation rate in argon as a function of velocity: The rate is given by:
lrr2oCN~(v). 1: ~l(V); II: ~2(v); III: ~(v) = 7.11 v--~; IV: Theory. 649
Volume 27A, number 10
PHYSICS
v a l u e s b e i n g in r e a s o n a b l e a g r e e m e n t w i t h p r e v i o u s w o r k [6,9]. The "free" positron annihilation rate was obt a i n e d f r o m the c o m p o s i t e p o s i t r o n d e c a y - t i m e s p e c t r u m by a m a x i m u m - l i k e l i h o o d a n a l y s i s [6,10]. F i g . l a s h o w s the r e s u l t s p l o t t e d a s a f u n c t i o n of t h e t e m p e r a t u r e . T h e r e s u l t s can b e e x p r e s s e d in t e r m s of a m e a n Z e f f of t h e s c a t t e r i n g a t o m [5,11] w h e r e Zef f
= ~a(~ ro2 cN) -1
(1)
w h e r e h a is the e x p e r i m e n t a l m e a n a n n i h i l a t i o n r a t e , r o i s the " c l a s s i c a l " e l e c t r o n r a d i u s , e2(mc2) -1, c i s t h e v e l o c i t y of light, N is t h e a t o m i c d e n s i t y of t h e h o s t gas. T h e s i m p l e s t f u n c t i o n w h i c h y i e l d e d a good fit to t h e e x p e r i m e n t a l d a t a w a s t h e p o w e r law: Zeff I ( T ) = (138 ± 17) T-( 0.285 + 0.021)
(2)
where T is the absolute temperature. A polynom i a l in p o w e r s of T IA (thus, p o w e r s of t h e p o s i t r o n v e l o c i t y ) w h i c h y i e l d e d a fit a s s a t i s f a c t o r y a s eq. (2) is; Zeff2(T) = (60.0 ± 6.6) +
(3) ± - (2.89 ± 0 . 8 0 ) T 2 + (0.056 ± 0 . 0 2 3 ) T
This function has the physically reasonable prope r t y of b e i n g f i n i t e at z e r o t e m p e r a t u r e (and h e n c e , velocity). From these temperature-dependent functions, t h e e x p l i c i t v e l o c i t y - d e p e n d e n t ~(v) f o r t h e p o s i t r o n - a r g o n i n t e r a c t i o n * is e a s i l y o b t a i n e d if one a s s u m e s t h a t the p o s i t r o n v e l o c i t y d i s t r i b u t i o n a p p r o p r i a t e to t h e r e g i o n of e x p o n e n t i a l d e c a y i s a s t a t i o n a r y d i s t r i b u t i o n c h a r a c t e r i z e d by t h e t e m p e r a t u r e , T. In s u c h a c a s e , t h e s h a p e i s w e l l a p p r o x i m a t e d by t h e M a x w e l l - B o l t z m a n n d i s t r i bution, since the annihilation rate is less than 10 . 4 of t h e e l a s t i c s c a t t e r i n g r a t e . T h u s
~ ~(v )v2 exp(mv /2kT)dv Zeff(T) =
o
(4)
f v 2 exp(-mv2/2kT)dv o 8 o l v i n g f o r ~(v) by u s i n g t h e L a p l a c e t r a n s f o r m ,
* The ~(v) r e f e r r e d to here is identical to the ~(v) defined by Massey [11], and the Zeff(v ) of Jones et al. [5,6].
650
LETTERS
7 October 1968
we obtain: ~l(V) = (4.56 ± 0.54)v (- 0.57 ± 0.04)
(5)
~2(v) = (60.0 ± 6.6) + -
(1015 ± 278)v + (5900 ± 2400)v 26) (
f r o m eqs. (2) and (3), r e s p e c t i v e l y . H e r e , t h e v e l o c i t i e s a r e in a t o m i c u n i t s (e2/h). T h e f u n c t i o n s ~1 and ~2 a r e i l l u s t r a t e d in fig. lb. T h e r e s e m b l a n c e of t h e s e c u r v e s to the f u n c t i o n ~ = 7.11 v-½ d e d u c e d f r o m f i t s of e l e c t r i c f i e l d r e s u l t s [4,5] i s s t r i k i n g . A l s o shown in fig. l b i s a t h e o r e t i c a l c u r v e o b t a i n e d by s o l v i n g a Schroedinger equation for the positron, using for t h e i n t e r a c t i o n p o t e n t i a l a s u m of t h e n o r m a l H a r t r e e - F o c k p o t e n t i a l f o r a r g o n and a p o l a r i z a t i o n t e r m of t h e H o l t s m a r k f o r m : -½ ae2/(r2 + + 0.7ao2)2 w h e r e a i s t h e p o l a r i s a b i l i t y of the n e u t r a l a t o m [5,11]. T h e m u c h g r e a t e r v e l o c i t y d e p e n d e n c e of t h e e x p e r i m e n t a l r e s u l t s at t h e s e s m a l l v e l o c i t i e s p r o b a b l y r e f l e c t s the s t r o n g s h o r t - r a n g e e l e c t r o n - p o s i t r o n i n t e r a c t i o n which is n e g l e c t e d in t h e s i m p l e m o d e l u s e d . T h e r a p i d d e c r e a s e of a n n i h i l a t i o n r a t e with i n c r e a s i n g v e l o c i t y i s e v e n s u g g e s t i v e of the e x i s t e n c e of e i t h e r a bound o r v i r t u a l e + - A r s t a t e a s s u g g e s t e d f o r n e u t r a l m o l e c u l e s by G o l ' d a n s k i i and S a y a s o v [12].
R~ferences 1. S.J. Tao, J. Bell and J. H. Green, Proe. Phys. Soc. (London) 83 (1964) 453. 2. W.R. Falk and G. Jones, Can. J. Phys. 42 (1964) 1751. 3. D . A . L . Paul, Proc. Phys. Soc. (London) 84 (1964) 563. 4. W.R. Falk, P . H . R . Orth and G. Jones, Phys. Rev. Letters 14 (1965) 447. 5. G. Jones and P. H. R. Orth in Positron annihilation, eds. A. T. Stewart and L. O. Roellig (Academic P r e s s Inc., New York, London, 1967) p.401-407. 6. P . H . R . Orth, Ph.D. Thesis, UBC, Vancouver, Canada (1967). 7. S . J . Tao and J. Bell, in Positron annihilation, eds. A. T. Stewart and L. O. Roellig (Academic P r e s s Inc., New York, London (1967) p.393-399. 8. F. F. Heymann, P. E. Osmon, J . J . Veit and W. F. Williams, Proc. Phys. Soc. 78 (1961) 1038. 9. B.G. Duff and F. F. Heymann, Proc. Roy. Soc.(London) A270 (1962) 517. 10. P . H . R . Orth, W.R. F a l k a n d G . Jones, to be published. 11. H.S.W. Massey, in Positron annihilation, eds. A.T. Stewart and L. O. Roellig (Academic P r e s s Inc., New York, London 1967) p.113-125. 12. V.I. Gol~danskii and Yu. S. Sayosov, Phys. Letters 13 (1964) 300.
PHYSICS LETTERS
TEMPERATURE OF
DEPENDENCE POSITRONS
OF THE IN ARGON
7 October 1968
ANNIHILATION GAS *
RATE
D. B. MILLER **, P. H. R. ORTH *** and G. JONES Department of Physics, University of British Columbia, Vancouver, B. C. , Canada
Received 15 August 1968
A temperature dependence of the positron annihilation rate in argon gas has beenfound. A 28% decrease in the annihilation rate occurs when the temperature is raised from 140 to 480OK.
A velocity dependence o2 the p r o b a b i l i t y of a n n i h i l a t i o n of a slow p o s i t r o n with a noble gas atom is e x p e r i m e n t a l l y well e s t a b l i s h e d [1-3]. The dependence is not known in detail, however, s i n c e the e x p e r i m e n t a l r e s u l t s depend also on the c r o s s s e c t i o n for e l a s t i c s c a t t e r i n g , and the v e locity dependence of this c r o s s s e c t i o n is also unknown. In the e x p e r i m e n t r e p o r t e d h e r e , the m e a n l i f e t i m e of p o s i t r o n s in a r g o n gas was found to depend on the t e m p e r a t u r e of the gas. This e x p e r i m e n t is thus the f i r s t to yield i n f o r m a t i o n c o n c e r n i n g the explicit velocity dependence of the a n n i h i l a t i o n r a t e of p o s i t r o n s in argon (albeit over a v e r y s m a l l velocity range). The i n s t r u m e n t a t i o n used in this e x p e r i m e n t was e s s e n t i a l l y the s a m e as that used in e a r l i e r work [2,4,6]. In t h i s c a s e , however, the gas was contained in a s t a i n l e s s s t e e l v e s s e l , 6 in. in d i a m e t e r and 7.25 in. long. The t e m p e r a t u r e of the gas (and c h a m b e r ) could be v a r i e d over the r a n g e 140 to 480°K, with a stability of ± 3°K d u r ing a run. The c h a m b e r was filled with 99.999% a r g o n gas supplied by Matheson of Canada, Ltd., to d e n s i t i e s between 8.0 and 10.5 a m a g a t . A check that no s e r i o u s c o n t a m i n a t i o n of the gas o c c u r r e d d u r i n g the e x p e r i m e n t was afforded by continuously m o n i t o r i n g both the "shoulder width" of the d i r e c t a n n i h i l a t i o n l i f e t i m e s p e c t r u m and the o r t h o p o s i t r o n i u m l i f e t i m e , both of which have b e e n found to be v e r y s e n s i t i v e to c o n t a m i n a t i o n [2,7,8]. The s h o u l d e r - w i d t h g a s - d e n s i t y product was c o n s t a n t at 350 n s e c - a m a g a t , and the o r t h o p o s i t r o n i u m a n n i h i l a t i o n rate (at a d e n s i t y of 10.2 amagat) was (11.1 ± 0.2) × 106 sec -1, both * Research supported by the National Research Council of Canada, Grant A-1564. ** Holder of a National Research Council Fellowship. *** National Research Council Postdoctoral Fellow.
,~,o / D i
D
'l
n
i
v
i
(~ 'se;i °m°o°;'i) (O)
....
1
7
1
5
"
-
~
Za"
Z,%= 1 3 6 T
I
-Z,tt 2 = 60-2.89T
.s
i
-o.2as
~
~
I
+.056T
3.o-
~ ~
~ "~T
t
25
!
I
180
I 220
I 260
I 300
I 340
I 380
I 420
I 460
2~
TEMPERATURE (°K) POSITRON
TEMPERATURE
IO0
X%
200
I
I
_,o
(T - nlv2/2k)
400 I
600
900
I'
1200
I
I
=K
/
-.-.<__
.o/2
J
POSITRON
~I 4 VELOCITY
=
~ .06
=
i , .OS
OJO
(atomic unite)
Fig. I. a) Positron annihilation rate in argon as a function of temperature; b) Positron annihilation rate in argon as a function of velocity: The rate is given by:
lrr2oCN~(v). 1: ~l(V); II: ~2(v); III: ~(v) = 7.11 v--~; IV: Theory. 649
Volume 27A, number 10
PHYSICS
v a l u e s b e i n g in r e a s o n a b l e a g r e e m e n t w i t h p r e v i o u s w o r k [6,9]. The "free" positron annihilation rate was obt a i n e d f r o m the c o m p o s i t e p o s i t r o n d e c a y - t i m e s p e c t r u m by a m a x i m u m - l i k e l i h o o d a n a l y s i s [6,10]. F i g . l a s h o w s the r e s u l t s p l o t t e d a s a f u n c t i o n of t h e t e m p e r a t u r e . T h e r e s u l t s can b e e x p r e s s e d in t e r m s of a m e a n Z e f f of t h e s c a t t e r i n g a t o m [5,11] w h e r e Zef f
= ~a(~ ro2 cN) -1
(1)
w h e r e h a is the e x p e r i m e n t a l m e a n a n n i h i l a t i o n r a t e , r o i s the " c l a s s i c a l " e l e c t r o n r a d i u s , e2(mc2) -1, c i s t h e v e l o c i t y of light, N is t h e a t o m i c d e n s i t y of t h e h o s t gas. T h e s i m p l e s t f u n c t i o n w h i c h y i e l d e d a good fit to t h e e x p e r i m e n t a l d a t a w a s t h e p o w e r law: Zeff I ( T ) = (138 ± 17) T-( 0.285 + 0.021)
(2)
where T is the absolute temperature. A polynom i a l in p o w e r s of T IA (thus, p o w e r s of t h e p o s i t r o n v e l o c i t y ) w h i c h y i e l d e d a fit a s s a t i s f a c t o r y a s eq. (2) is; Zeff2(T) = (60.0 ± 6.6) +
(3) ± - (2.89 ± 0 . 8 0 ) T 2 + (0.056 ± 0 . 0 2 3 ) T
This function has the physically reasonable prope r t y of b e i n g f i n i t e at z e r o t e m p e r a t u r e (and h e n c e , velocity). From these temperature-dependent functions, t h e e x p l i c i t v e l o c i t y - d e p e n d e n t ~(v) f o r t h e p o s i t r o n - a r g o n i n t e r a c t i o n * is e a s i l y o b t a i n e d if one a s s u m e s t h a t the p o s i t r o n v e l o c i t y d i s t r i b u t i o n a p p r o p r i a t e to t h e r e g i o n of e x p o n e n t i a l d e c a y i s a s t a t i o n a r y d i s t r i b u t i o n c h a r a c t e r i z e d by t h e t e m p e r a t u r e , T. In s u c h a c a s e , t h e s h a p e i s w e l l a p p r o x i m a t e d by t h e M a x w e l l - B o l t z m a n n d i s t r i bution, since the annihilation rate is less than 10 . 4 of t h e e l a s t i c s c a t t e r i n g r a t e . T h u s
~ ~(v )v2 exp(mv /2kT)dv Zeff(T) =
o
(4)
f v 2 exp(-mv2/2kT)dv o 8 o l v i n g f o r ~(v) by u s i n g t h e L a p l a c e t r a n s f o r m ,
* The ~(v) r e f e r r e d to here is identical to the ~(v) defined by Massey [11], and the Zeff(v ) of Jones et al. [5,6].
650
LETTERS
7 October 1968
we obtain: ~l(V) = (4.56 ± 0.54)v (- 0.57 ± 0.04)
(5)
~2(v) = (60.0 ± 6.6) + -
(1015 ± 278)v + (5900 ± 2400)v 26) (
f r o m eqs. (2) and (3), r e s p e c t i v e l y . H e r e , t h e v e l o c i t i e s a r e in a t o m i c u n i t s (e2/h). T h e f u n c t i o n s ~1 and ~2 a r e i l l u s t r a t e d in fig. lb. T h e r e s e m b l a n c e of t h e s e c u r v e s to the f u n c t i o n ~ = 7.11 v-½ d e d u c e d f r o m f i t s of e l e c t r i c f i e l d r e s u l t s [4,5] i s s t r i k i n g . A l s o shown in fig. l b i s a t h e o r e t i c a l c u r v e o b t a i n e d by s o l v i n g a Schroedinger equation for the positron, using for t h e i n t e r a c t i o n p o t e n t i a l a s u m of t h e n o r m a l H a r t r e e - F o c k p o t e n t i a l f o r a r g o n and a p o l a r i z a t i o n t e r m of t h e H o l t s m a r k f o r m : -½ ae2/(r2 + + 0.7ao2)2 w h e r e a i s t h e p o l a r i s a b i l i t y of the n e u t r a l a t o m [5,11]. T h e m u c h g r e a t e r v e l o c i t y d e p e n d e n c e of t h e e x p e r i m e n t a l r e s u l t s at t h e s e s m a l l v e l o c i t i e s p r o b a b l y r e f l e c t s the s t r o n g s h o r t - r a n g e e l e c t r o n - p o s i t r o n i n t e r a c t i o n which is n e g l e c t e d in t h e s i m p l e m o d e l u s e d . T h e r a p i d d e c r e a s e of a n n i h i l a t i o n r a t e with i n c r e a s i n g v e l o c i t y i s e v e n s u g g e s t i v e of the e x i s t e n c e of e i t h e r a bound o r v i r t u a l e + - A r s t a t e a s s u g g e s t e d f o r n e u t r a l m o l e c u l e s by G o l ' d a n s k i i and S a y a s o v [12].
R~ferences 1. S.J. Tao, J. Bell and J. H. Green, Proe. Phys. Soc. (London) 83 (1964) 453. 2. W.R. Falk and G. Jones, Can. J. Phys. 42 (1964) 1751. 3. D . A . L . Paul, Proc. Phys. Soc. (London) 84 (1964) 563. 4. W.R. Falk, P . H . R . Orth and G. Jones, Phys. Rev. Letters 14 (1965) 447. 5. G. Jones and P. H. R. Orth in Positron annihilation, eds. A. T. Stewart and L. O. Roellig (Academic P r e s s Inc., New York, London, 1967) p.401-407. 6. P . H . R . Orth, Ph.D. Thesis, UBC, Vancouver, Canada (1967). 7. S . J . Tao and J. Bell, in Positron annihilation, eds. A. T. Stewart and L. O. Roellig (Academic P r e s s Inc., New York, London (1967) p.393-399. 8. F. F. Heymann, P. E. Osmon, J . J . Veit and W. F. Williams, Proc. Phys. Soc. 78 (1961) 1038. 9. B.G. Duff and F. F. Heymann, Proc. Roy. Soc.(London) A270 (1962) 517. 10. P . H . R . Orth, W.R. F a l k a n d G . Jones, to be published. 11. H.S.W. Massey, in Positron annihilation, eds. A.T. Stewart and L. O. Roellig (Academic P r e s s Inc., New York, London 1967) p.113-125. 12. V.I. Gol~danskii and Yu. S. Sayosov, Phys. Letters 13 (1964) 300.