- Email: [email protected]
Helium drag in the zero field limit theory of impurity in liquid 4He
Volume 29A, number 1
HELIUM
PHYSICS LETTERS
DRAG
IN THE
ZERO FIELD IN LIQUID
24 March 1969
LIMIT 4He*
THEORY
OF IMPURITY
R. L O B O
Physics Department, Escola Engenharia, S. Carlos, S.P., Brazil Received 11 February 1969
The ~phonon-drag ~ effect is calculated for the mobility of charged impurities in 4He. This effect raises the conductivity and, although for low and high enough temperatures its dependence on T is the same, the coefficients of the expansions are diffferent
The p u r p o s e of the p r e s e n t p a p e r is to d e m o n s t r a t e that at sufficiently low t e m p e r a t u r e s , t h e r e e x i s t s a phonon d r a g effect in the m o b i l i t y of c h a r g e d i m p u r i t i e s in liquid 4He. The m o b i l i t y of c h a r g e d i m p u r i t i e s (of d e n s i t y Ni, charge e and m a s s m) in 4He has b e e n c a l c u l a t e d p r e v i ously [1] but this work has not c o n s i d e r e d the p o s s i b i l i t y of a " d r a g n of 4He in the f o r m of longitudinal phonons which r e t u r n to the i m p u r i t i e s , m o m e n t u m they a d q u i r e d f r o m p r e v i o u s c o l l i s i o n s . This effect i s to be expected when the phonon-phonon i n t e r a c t i o n s and the phononb o u n d a r y i n t e r a c t i o n s a r e not s t r o n g enough, i.e., at v e r y low t e m p e r a t u r e s . The theory a s s u m e s low e l e c t r i c fields so that no v o r t e x i s excited [2]. Our t r e a t m e n t follows v e r y c l o s e l y the one u s e d i n the i n v e s t i g a t i o n of s i m i l a r effect in 3He a t o m s [3]. We u s e the v a r i a t i o n a l technique to calculate the i m p u r i t y conductivity. F o r the 4He s y s t e m we use B o g o l i u b o v ' s H a m i l t o n i a n and l i m i t our work to a r a n g e i n t e m p e r a t u r e where only phonons a r e of significance. In the c o l l i s i o n t e r m of the B o l t z m a n n equation we c o n s i d e r only e l a s t i c s c a t t e r i n g , s i n c e for low enough t e m p e r a t u r e s the a b s o r p t i o n and e m i s s i o n p r o c e s s e s have a t h r e s h o l d which m a k e s t h e i r c o n t r i b u t i o n negligible [1]. A c o n s t a n t c r o s s - s e c t i o n is a s s u m e d for the p h o n o n - i m p u r i t y s c a t t e r i n g [1]. Impurity conductivity. We solve a coupled B o l t z m a n n equation for the i m p u r i t y and for the phonon s y s t e m . The phonons a r e allowed to i n t e r a c t not only with the i m p u r i t i e s but with each other and the walls. T h e s e a r e r e p l a c e d in the B o l t z m a n n equation by an a v e r a g e c o l l i sion t i m e 7. The conductivity t u r n s out to be
= a 2 / 2 d - a 2 / 2 7 = 00 + ~ d r a g .
(1)
(kBT)4N2/45m2c5h37.
w h e r e ~ -- Here c is the velocity of s o u n d in 4 H e ~ 2 3 7 m / s . a i s t h e s a m e a s in r e f . 3, a = eari / r e . In o r d e r to o b t a i n t h e t e m p e r a t u r e d e p e n d e n c e of t h e d r a g t e r m w e m u s t k n o w T(T). W e u s e f o r T t h e e x p r e s s i o n [4]
1/z = 2 x 104(l+cTn)sec -1
(2)
With this value for z, 0 can be written as
0 = [A + B/(1 + cT n)lT -4 ,
(3)
w h e r e A T -4 = 0o i s the conductivity c a l c u l a t e d by the n e g l e c t of the " d r a g effect" [1]. In o r d e r to e s t i m a t e the c o n t r i b u t i o n of the t e r m s p r e s e n t i n the i m p u r i t y conductivity we take for A the value of ref. 1, A ~ 387 cm2(OK)4/sec v o l t .
(4)
The r a t i o A / B and c which should c r i t i c a l l y d e t r m i n e the dependence of 0 with the t e m p e r a t u r e a r e given by
A/B
~
387/159
× 10 -14 N i
c = 1 . 5 × 10 3 a n d n = 9
[4].
(5)
(6)
The p r e s e n t c a l c u l a t i o n i n d i c a t e s that the i m p u r i t y conductivity should show a T -4 t e m p e r a t u r e dependence for T < 0.2°K a n d T > 0.6°K, although the coefficients of the T -4 t e r m should be different. Between 0.2°K and 0 . 6 ° K the c o n ductivity should not show a s i m p e T d e p e n d e n c e . We conclude s t r e s s i n g the role of the i m p u r i t y c o n c e n t r a t i o n dependence d r a g t e r m on the conductivity for t e m p e r a t u r e s l e s s than about 0.6OK. * Work supported by Nat. Res. Council of Brazil. 33
Volume 29A, number 1
PHYSICS LETTERS
It s t a r t s on changing the t e m p e r a t u r e dependence of O(T ~ 0.6OK), and then at T ~ 0.2OK i n c r e a s e s i t s v a l u e by about 30% for N i .~ 1014, without modifying i t s T -4 dependence.
24 March 1969
References 1. R. Abe and K. Aizu, Phys. Rev. 131 (1961) 10. 2. G.W. Rayfield and F. Reif, Phys. Rev. 136 (1964) 1194 3. M. Bailyn and R. Lobo, Phys. Rev., to be published. 4. Y. Disatnik, Phys. Rev. 158 (1967) 162.
* * * * *
INTERACTION
OF TWO
FIELD
OSCILLATORS
IN A C I R C U L A R
LASER
V. F. C H E L ' T Z O V Moscow V - 415, Leninski prospect, 128 - 3, USSR Received 21 January 1969
The correlation of two quantum oscillators interacting with the system of two-level molecules in a circular laser is investigated without the perturbation theory. The comparison of the exact formula for the beatings frequency with the experiment has shown that the stationary inversion N obeys the inequeality N<
At the p r e s e n t t i m e the p r o b l e m of the i n t e r a c t i o n of two field o s c i l l a t o r s in a c i r c u l a r l a s e r is t r e a t e d f r o m the r a d i o p h y s i c a l point of view. B e s i d e s the a t o m i c p o l a r i z a b i l i t y i s c a l c u l a t e d by m e a n s of the p e r t u r b a t i o n theory, meanwhile the c o l l e c t i v e c h a r a c t e r of t w o - l e v e l a t o m s ' coupling to the r a d i a t i o n field is n e g l e c t e d completely. However if a continuously o p e r a t i n g gas l a s e r is c o n s i d e r e d as a r e g i o n of the space occupied with c o h e r e n t photons of infinite l i f e - t i m e , the a s s u m p t i o n s m e n t i o n e d above fail. In this case the p r o b l e m should be i n v e s t i g a t e d by m e a n s of the q u a n t u m m e c h a n i c s without the p e r t u r b a t i o n theory. The a i m of the p r e s e n t p a p e r i s to d e r i v e a r i g o r o u s quantum f o r m u l a for the f r e q u e n c y v of the o s c i l l a t o r s ' b e a t i n g s a s a function of the difference (o~ 2 - ~01) of the o s c i l l a t o r s ' f r e q u e n c i e s and the i n v e r s i o n N_ in the a c t i v e m e d i u m . C o m p a r i n g the f o r m u l a for v with the e x p e r i m e n t a l data of Macek and Davis [1] we may e s t i m a t e the value of N . being r e a l i z e d in a He - Ne l a s e r . Though the e x p e r i m e n t a l a p p a r a t u s i s cons t r u c t e d in the f o r m of t r i a n g l e s or q u a d r a n g l e s , our c a l c u l a t i o n s , for the sake of s i m p l i c i t y , a r e c a r r i e d out for a l a s e r with t o r o i d a l cavity. If the r a d i u s r o of the t o r e f o r m c u r v a t u r e is much l e s s than the r a d i u s R of the c i r c u m f e r e n c e , with r e g a r d to the conditions A(r
= r e , ~o, ~9) = 0 ,
A(r,~p + 2 ~ , q J ) = A ( r , ~ o , ~),
A (r,q~,~9 + 2~)-- A ( r , ~o,~/)
(1)
we come to the s i m p l e s t e x p r e s s i o n for the v e c t o r - p o t e n t i a l A a s follows
r
t csoi sn ((n~ n~
(2)
In the eqs. ( 1 ) - (2) Cns m i s n o r m a l i z i n g factor, Y is the B e s s e l function, n , s , rn a r e i n t e g e r s , r , ~ / a r e p o l a r c o o r d i n a t e s in the p l a n e of-the t o r e f o r m c u r v a t u r e , ~ is an a z i m u t h in the plane of c i r c u m f e r e n c e When a l i n e a r p o l a r i z a t i o n i s p r e s e n t , it i s n e c e s s a r y to p r o j e c t the eq. (2) onto the v e c t o r n of p o l a r i zation. Let u s a s s u m e that two opposite r u n n i n g c i r c u l a r waves of the lowest o r d e r s t r u c t u r e (n = 0, s = 1) and with f r e q u e n c i e s co1 = ~o2 = co e x i s t in the cavity of the m o t i o n l e s s l a s e r . .Here 0~o is the d i s t a n c e b e t w e e n the l e v e l s of the unpertur°bed atom. Due to the r o t a t i o n and the conditions (1) the splitting of the f r e q u e n c i e s o c c u r s [2]
34
HELIUM
PHYSICS LETTERS
DRAG
IN THE
ZERO FIELD IN LIQUID
24 March 1969
LIMIT 4He*
THEORY
OF IMPURITY
R. L O B O
Physics Department, Escola Engenharia, S. Carlos, S.P., Brazil Received 11 February 1969
The ~phonon-drag ~ effect is calculated for the mobility of charged impurities in 4He. This effect raises the conductivity and, although for low and high enough temperatures its dependence on T is the same, the coefficients of the expansions are diffferent
The p u r p o s e of the p r e s e n t p a p e r is to d e m o n s t r a t e that at sufficiently low t e m p e r a t u r e s , t h e r e e x i s t s a phonon d r a g effect in the m o b i l i t y of c h a r g e d i m p u r i t i e s in liquid 4He. The m o b i l i t y of c h a r g e d i m p u r i t i e s (of d e n s i t y Ni, charge e and m a s s m) in 4He has b e e n c a l c u l a t e d p r e v i ously [1] but this work has not c o n s i d e r e d the p o s s i b i l i t y of a " d r a g n of 4He in the f o r m of longitudinal phonons which r e t u r n to the i m p u r i t i e s , m o m e n t u m they a d q u i r e d f r o m p r e v i o u s c o l l i s i o n s . This effect i s to be expected when the phonon-phonon i n t e r a c t i o n s and the phononb o u n d a r y i n t e r a c t i o n s a r e not s t r o n g enough, i.e., at v e r y low t e m p e r a t u r e s . The theory a s s u m e s low e l e c t r i c fields so that no v o r t e x i s excited [2]. Our t r e a t m e n t follows v e r y c l o s e l y the one u s e d i n the i n v e s t i g a t i o n of s i m i l a r effect in 3He a t o m s [3]. We u s e the v a r i a t i o n a l technique to calculate the i m p u r i t y conductivity. F o r the 4He s y s t e m we use B o g o l i u b o v ' s H a m i l t o n i a n and l i m i t our work to a r a n g e i n t e m p e r a t u r e where only phonons a r e of significance. In the c o l l i s i o n t e r m of the B o l t z m a n n equation we c o n s i d e r only e l a s t i c s c a t t e r i n g , s i n c e for low enough t e m p e r a t u r e s the a b s o r p t i o n and e m i s s i o n p r o c e s s e s have a t h r e s h o l d which m a k e s t h e i r c o n t r i b u t i o n negligible [1]. A c o n s t a n t c r o s s - s e c t i o n is a s s u m e d for the p h o n o n - i m p u r i t y s c a t t e r i n g [1]. Impurity conductivity. We solve a coupled B o l t z m a n n equation for the i m p u r i t y and for the phonon s y s t e m . The phonons a r e allowed to i n t e r a c t not only with the i m p u r i t i e s but with each other and the walls. T h e s e a r e r e p l a c e d in the B o l t z m a n n equation by an a v e r a g e c o l l i sion t i m e 7. The conductivity t u r n s out to be
= a 2 / 2 d - a 2 / 2 7 = 00 + ~ d r a g .
(1)
(kBT)4N2/45m2c5h37.
w h e r e ~ -- Here c is the velocity of s o u n d in 4 H e ~ 2 3 7 m / s . a i s t h e s a m e a s in r e f . 3, a = eari / r e . In o r d e r to o b t a i n t h e t e m p e r a t u r e d e p e n d e n c e of t h e d r a g t e r m w e m u s t k n o w T(T). W e u s e f o r T t h e e x p r e s s i o n [4]
1/z = 2 x 104(l+cTn)sec -1
(2)
With this value for z, 0 can be written as
0 = [A + B/(1 + cT n)lT -4 ,
(3)
w h e r e A T -4 = 0o i s the conductivity c a l c u l a t e d by the n e g l e c t of the " d r a g effect" [1]. In o r d e r to e s t i m a t e the c o n t r i b u t i o n of the t e r m s p r e s e n t i n the i m p u r i t y conductivity we take for A the value of ref. 1, A ~ 387 cm2(OK)4/sec v o l t .
(4)
The r a t i o A / B and c which should c r i t i c a l l y d e t r m i n e the dependence of 0 with the t e m p e r a t u r e a r e given by
A/B
~
387/159
× 10 -14 N i
c = 1 . 5 × 10 3 a n d n = 9
[4].
(5)
(6)
The p r e s e n t c a l c u l a t i o n i n d i c a t e s that the i m p u r i t y conductivity should show a T -4 t e m p e r a t u r e dependence for T < 0.2°K a n d T > 0.6°K, although the coefficients of the T -4 t e r m should be different. Between 0.2°K and 0 . 6 ° K the c o n ductivity should not show a s i m p e T d e p e n d e n c e . We conclude s t r e s s i n g the role of the i m p u r i t y c o n c e n t r a t i o n dependence d r a g t e r m on the conductivity for t e m p e r a t u r e s l e s s than about 0.6OK. * Work supported by Nat. Res. Council of Brazil. 33
Volume 29A, number 1
PHYSICS LETTERS
It s t a r t s on changing the t e m p e r a t u r e dependence of O(T ~ 0.6OK), and then at T ~ 0.2OK i n c r e a s e s i t s v a l u e by about 30% for N i .~ 1014, without modifying i t s T -4 dependence.
24 March 1969
References 1. R. Abe and K. Aizu, Phys. Rev. 131 (1961) 10. 2. G.W. Rayfield and F. Reif, Phys. Rev. 136 (1964) 1194 3. M. Bailyn and R. Lobo, Phys. Rev., to be published. 4. Y. Disatnik, Phys. Rev. 158 (1967) 162.
* * * * *
INTERACTION
OF TWO
FIELD
OSCILLATORS
IN A C I R C U L A R
LASER
V. F. C H E L ' T Z O V Moscow V - 415, Leninski prospect, 128 - 3, USSR Received 21 January 1969
The correlation of two quantum oscillators interacting with the system of two-level molecules in a circular laser is investigated without the perturbation theory. The comparison of the exact formula for the beatings frequency with the experiment has shown that the stationary inversion N obeys the inequeality N<
At the p r e s e n t t i m e the p r o b l e m of the i n t e r a c t i o n of two field o s c i l l a t o r s in a c i r c u l a r l a s e r is t r e a t e d f r o m the r a d i o p h y s i c a l point of view. B e s i d e s the a t o m i c p o l a r i z a b i l i t y i s c a l c u l a t e d by m e a n s of the p e r t u r b a t i o n theory, meanwhile the c o l l e c t i v e c h a r a c t e r of t w o - l e v e l a t o m s ' coupling to the r a d i a t i o n field is n e g l e c t e d completely. However if a continuously o p e r a t i n g gas l a s e r is c o n s i d e r e d as a r e g i o n of the space occupied with c o h e r e n t photons of infinite l i f e - t i m e , the a s s u m p t i o n s m e n t i o n e d above fail. In this case the p r o b l e m should be i n v e s t i g a t e d by m e a n s of the q u a n t u m m e c h a n i c s without the p e r t u r b a t i o n theory. The a i m of the p r e s e n t p a p e r i s to d e r i v e a r i g o r o u s quantum f o r m u l a for the f r e q u e n c y v of the o s c i l l a t o r s ' b e a t i n g s a s a function of the difference (o~ 2 - ~01) of the o s c i l l a t o r s ' f r e q u e n c i e s and the i n v e r s i o n N_ in the a c t i v e m e d i u m . C o m p a r i n g the f o r m u l a for v with the e x p e r i m e n t a l data of Macek and Davis [1] we may e s t i m a t e the value of N . being r e a l i z e d in a He - Ne l a s e r . Though the e x p e r i m e n t a l a p p a r a t u s i s cons t r u c t e d in the f o r m of t r i a n g l e s or q u a d r a n g l e s , our c a l c u l a t i o n s , for the sake of s i m p l i c i t y , a r e c a r r i e d out for a l a s e r with t o r o i d a l cavity. If the r a d i u s r o of the t o r e f o r m c u r v a t u r e is much l e s s than the r a d i u s R of the c i r c u m f e r e n c e , with r e g a r d to the conditions A(r
= r e , ~o, ~9) = 0 ,
A(r,~p + 2 ~ , q J ) = A ( r , ~ o , ~),
A (r,q~,~9 + 2~)-- A ( r , ~o,~/)
(1)
we come to the s i m p l e s t e x p r e s s i o n for the v e c t o r - p o t e n t i a l A a s follows
r
t csoi sn ((n~ n~
(2)
In the eqs. ( 1 ) - (2) Cns m i s n o r m a l i z i n g factor, Y is the B e s s e l function, n , s , rn a r e i n t e g e r s , r , ~ / a r e p o l a r c o o r d i n a t e s in the p l a n e of-the t o r e f o r m c u r v a t u r e , ~ is an a z i m u t h in the plane of c i r c u m f e r e n c e When a l i n e a r p o l a r i z a t i o n i s p r e s e n t , it i s n e c e s s a r y to p r o j e c t the eq. (2) onto the v e c t o r n of p o l a r i zation. Let u s a s s u m e that two opposite r u n n i n g c i r c u l a r waves of the lowest o r d e r s t r u c t u r e (n = 0, s = 1) and with f r e q u e n c i e s co1 = ~o2 = co e x i s t in the cavity of the m o t i o n l e s s l a s e r . .Here 0~o is the d i s t a n c e b e t w e e n the l e v e l s of the unpertur°bed atom. Due to the r o t a t i o n and the conditions (1) the splitting of the f r e q u e n c i e s o c c u r s [2]
34