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Longitudinal spin relaxation in dilute, polarized Fermi gases
PHYSICA
PhysieaB 194-196 (1994) 889-890 North-Holland
Longitudinal Spin Relaxation in Dilute, Polarized Fermi Gases Erik D. Nelson and William J. Mullin Department of Physics and Astronomy, University of Massachusetts, Amherst MA 01003 We consider the calculation of the longitudinal relaxation time T 1 in a polarized Fermi system at temperatures T ranging from the degenerate statistics regime to the Boltzmann limit. We include the possibility of arbitrary polarization M of the system. While the low temperature unpolarized system behaves as T "2 as previous calculations have shown, when relaxation takes place in zero external field, polarization leads to substantially smaller values of T 1 at low T, because of the opening up of the phase space between up and down Fermi spheres. For the degenerate system T 1 is dependent on the value of M in contrast to results in the Boltzmann case. Several years ago Castaing and Nozi~res suggested [1] that in a polarized system the longitudinal relaxation time T 1 would be altered substantially by polarizing a Fermi system. There have been some calculations of T 1 in a degenerate Fermi system [2], although none have considered the situation with polarization. These find that T 1 - T -2 b e c a u s e of s t a n d a r d F e r m i - s u r f a c e momentum-space scattering restrictions. In a polarized system we find t h a t once the system is polarized the m o m e n t u m space "opens up" allowing a wider r a n g e of scattering so that the relaxation rate is much faster. Indeed, i n s t e a d of the above t e m p e r a t u r e behavior, we find t h a t T 1 approaches a constant at T=0 whose size grows smaller with increasing polarization P. A calculation [3] of T 1 for a Boltzmann gas showed that, for relaxation occurring in zero field, perhaps after the gas is polarized by optical pumping techniques, T 1 ~~T-until the temperature drops to the point that the deBroglie wave length is comparable to the scattering length at which point there is a switch over to a 1/~-dependence. The result is a minimum in T1, which will be observed as long as the gas density is low enough that the Fermi temperature is lower t h a n the temperature of the predicted minimum. At lower T, the switch over to degenerate T -2 behavior will began to occur. At still lower T, as shown here, when kT becomes smaller than the energy spacing between the up- and down-spin Fermi surfaces, T 1 will level out
and become independent of T. For the Boltzmann case, Ref. [3] shows t h a t T 1 is independent of M so t h a t there will be exponential decay starting from any magnetization. The p r e s e n t calculation shows that, in the degenerate regime, under the same conditions of zero external field, T 1 is no longer exponential but depends on P. Experimentally the minimum mentioned above has never been observed although studies in 3He gas have closely approached the predicted temperature [4]. A problem is that the gas pressure drops so sharply t h a t wall relaxation begins to dominate. Possibly experiments on dilute solutions of 3He in superfluid 4He would allow the observation of both the minimum in the Boltzmann range and the T i n d e p e n d e n c e of T 1 in the degenerate T regime. T 1 values become quite long in these experiments, but should be no more difficult to observe than those already observed in 3He gas. Our starting point is the kinetic equation for t h e d i s t r i b u t i o n f u n c t i o n n p a f o r m o m e n t u m p and spin a, which treats the t h e dipole p o t e n t i a l V d in Born approximation. For the m a g n e t i z a t i o n m=(1/~)Znp¢ ~ we find the equation pa dm_ dt - I,~,4 ~11(34;~3~4~Vd[12;~3C4)12 ~I(~2(~3a4
0921-4526/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved S S D I 0921-4526(93)E1025-H
890 where 1[I~3~410 10"2]]=
wave functions for the m i n i m u m to appear.
~t n p3o 3n p, a ,( 1-n Pl o I) ( 1-np202)hf2[
001
np,olnp202(1-np303)(1-np404) ] 0.1
In order to e v a l u a t e t h e dipole m a t r i x e l e m e n t s in t h e d e g e n e r a t e regime, it is sufficient to use plane waves states r a t h e r than the more accurate distorted-wave states of a pair potential. The reason is t h a t the dipole operator has zero matrix elements b e t w e e n singlet s t a t e s and a t low T the spatial a n t i s y m m e t r y of the triplet states produces a correlation hole t h a t removes most of the effect of a potential. This effect is easily seen in the calculation of Ref. [3]. Note t h a t in dilute solutions the effective potential is p r o b a b l y n o t k n o w n well e n o u g h to develop the distorted states anyway. In our c a l c u l a t i o n s we avoid t h e u s u a l low-T a p p r o x i m a t i o n s for a d e g e n e r a t e F e r m i system because we anticipate that there will be scattering a w a y from the Fermi surface. The techniques developed recently [5] for computing F e r m i o n collision integrals for a r b i t r a r y T can be a d a p t e d here to reduce t h e expression above to a two-dimensional integral as shown: d t ---Cff ds s f : dtsinh(2ts-A/2) 1s i n h ( 2 t s + ~ 2 )
l+en'- t+s)2l ln[ 1 l+en--( [ 1+e .q
1+ J "'"
w h e r e C is a c o n s t a n t and t h e r e are two other terms similar to the one shown. T1-1 is defined as -m "1 ddt m ;Tl° is 13 t i m e s the effective chemical potential for spin species a; and A=~+-TI_. The results of the numerical computation are shown in Figure 1, where the constant 8
-
5~h~
m3y4eF 2 with T the g y r o m a g n e t i c
ratio and e F is the Fermi energy. We indeed see t h a t , as p o l a r i z a t i o n i n c r e a s e s , T 1 b r e a k s a w a y from t h e T -2 b e h a v i o r to approach a constant at T=0. The m i n i m u m in T 1 diacussed above does not show up in these calculations because of the use of free particle states; one m u s t include interacting
0.2 0,3 1 0.9 0
.
.
.
0.03
.
.
.
.
1
O.I
Figure 1.
0'.3
T~ F
In Figure 2 we plot T 1 as a function of polarization P=ndn for fLxed T. Obviously T 1 is no longer independent of P as it is at high temperatures. At low T, T 1 is very sensitive to P. 1.2
\
i.o:
~0.1
\ ,,,,:
,
,
\002
0.8
o.~ 0.4
0.2
.
I
0.2
t
0.4
I
0.6
I
0.8
Figure 2. References
1. B. Castiang and P. Nozi~res, J. Physique 40, 239 (1979). 2. For example, A. Alhiezer and V, Aleksin, Dok. Akad. Nauk SSSR 92, 259 (1953); D. Vollhardt and P. Wolfle, Phys. Rev. Lett. 47, 190 (1981); K. S. Bedell and Meltzer, J. Low Temp. Phys. 6 3 , 2 1 5 (1986). 3. W. J. Mullin F. Lalo~, and M. G. Richards, J. Low Temp. Phys. 80, 1 (1990). 4. R. Chapman, Phys. Rev. A12, 2333 (1975). 5. J. Jeon and W. J. Mullin, J. Low Temp. Phys. 87, 421 (1987).
PhysieaB 194-196 (1994) 889-890 North-Holland
Longitudinal Spin Relaxation in Dilute, Polarized Fermi Gases Erik D. Nelson and William J. Mullin Department of Physics and Astronomy, University of Massachusetts, Amherst MA 01003 We consider the calculation of the longitudinal relaxation time T 1 in a polarized Fermi system at temperatures T ranging from the degenerate statistics regime to the Boltzmann limit. We include the possibility of arbitrary polarization M of the system. While the low temperature unpolarized system behaves as T "2 as previous calculations have shown, when relaxation takes place in zero external field, polarization leads to substantially smaller values of T 1 at low T, because of the opening up of the phase space between up and down Fermi spheres. For the degenerate system T 1 is dependent on the value of M in contrast to results in the Boltzmann case. Several years ago Castaing and Nozi~res suggested [1] that in a polarized system the longitudinal relaxation time T 1 would be altered substantially by polarizing a Fermi system. There have been some calculations of T 1 in a degenerate Fermi system [2], although none have considered the situation with polarization. These find that T 1 - T -2 b e c a u s e of s t a n d a r d F e r m i - s u r f a c e momentum-space scattering restrictions. In a polarized system we find t h a t once the system is polarized the m o m e n t u m space "opens up" allowing a wider r a n g e of scattering so that the relaxation rate is much faster. Indeed, i n s t e a d of the above t e m p e r a t u r e behavior, we find t h a t T 1 approaches a constant at T=0 whose size grows smaller with increasing polarization P. A calculation [3] of T 1 for a Boltzmann gas showed that, for relaxation occurring in zero field, perhaps after the gas is polarized by optical pumping techniques, T 1 ~~T-until the temperature drops to the point that the deBroglie wave length is comparable to the scattering length at which point there is a switch over to a 1/~-dependence. The result is a minimum in T1, which will be observed as long as the gas density is low enough that the Fermi temperature is lower t h a n the temperature of the predicted minimum. At lower T, the switch over to degenerate T -2 behavior will began to occur. At still lower T, as shown here, when kT becomes smaller than the energy spacing between the up- and down-spin Fermi surfaces, T 1 will level out
and become independent of T. For the Boltzmann case, Ref. [3] shows t h a t T 1 is independent of M so t h a t there will be exponential decay starting from any magnetization. The p r e s e n t calculation shows that, in the degenerate regime, under the same conditions of zero external field, T 1 is no longer exponential but depends on P. Experimentally the minimum mentioned above has never been observed although studies in 3He gas have closely approached the predicted temperature [4]. A problem is that the gas pressure drops so sharply t h a t wall relaxation begins to dominate. Possibly experiments on dilute solutions of 3He in superfluid 4He would allow the observation of both the minimum in the Boltzmann range and the T i n d e p e n d e n c e of T 1 in the degenerate T regime. T 1 values become quite long in these experiments, but should be no more difficult to observe than those already observed in 3He gas. Our starting point is the kinetic equation for t h e d i s t r i b u t i o n f u n c t i o n n p a f o r m o m e n t u m p and spin a, which treats the t h e dipole p o t e n t i a l V d in Born approximation. For the m a g n e t i z a t i o n m=(1/~)Znp¢ ~ we find the equation pa dm_ dt - I,~,4 ~11(34;~3~4~Vd[12;~3C4)12 ~I(~2(~3a4
0921-4526/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved S S D I 0921-4526(93)E1025-H
890 where 1[I~3~410 10"2]]=
wave functions for the m i n i m u m to appear.
~t n p3o 3n p, a ,( 1-n Pl o I) ( 1-np202)hf2[
001
np,olnp202(1-np303)(1-np404) ] 0.1
In order to e v a l u a t e t h e dipole m a t r i x e l e m e n t s in t h e d e g e n e r a t e regime, it is sufficient to use plane waves states r a t h e r than the more accurate distorted-wave states of a pair potential. The reason is t h a t the dipole operator has zero matrix elements b e t w e e n singlet s t a t e s and a t low T the spatial a n t i s y m m e t r y of the triplet states produces a correlation hole t h a t removes most of the effect of a potential. This effect is easily seen in the calculation of Ref. [3]. Note t h a t in dilute solutions the effective potential is p r o b a b l y n o t k n o w n well e n o u g h to develop the distorted states anyway. In our c a l c u l a t i o n s we avoid t h e u s u a l low-T a p p r o x i m a t i o n s for a d e g e n e r a t e F e r m i system because we anticipate that there will be scattering a w a y from the Fermi surface. The techniques developed recently [5] for computing F e r m i o n collision integrals for a r b i t r a r y T can be a d a p t e d here to reduce t h e expression above to a two-dimensional integral as shown: d t ---Cff ds s f : dtsinh(2ts-A/2) 1s i n h ( 2 t s + ~ 2 )
l+en'- t+s)2l ln[ 1 l+en--( [ 1+e .q
1+ J "'"
w h e r e C is a c o n s t a n t and t h e r e are two other terms similar to the one shown. T1-1 is defined as -m "1 ddt m ;Tl° is 13 t i m e s the effective chemical potential for spin species a; and A=~+-TI_. The results of the numerical computation are shown in Figure 1, where the constant 8
-
5~h~
m3y4eF 2 with T the g y r o m a g n e t i c
ratio and e F is the Fermi energy. We indeed see t h a t , as p o l a r i z a t i o n i n c r e a s e s , T 1 b r e a k s a w a y from t h e T -2 b e h a v i o r to approach a constant at T=0. The m i n i m u m in T 1 diacussed above does not show up in these calculations because of the use of free particle states; one m u s t include interacting
0.2 0,3 1 0.9 0
.
.
.
0.03
.
.
.
.
1
O.I
Figure 1.
0'.3
T~ F
In Figure 2 we plot T 1 as a function of polarization P=ndn for fLxed T. Obviously T 1 is no longer independent of P as it is at high temperatures. At low T, T 1 is very sensitive to P. 1.2
\
i.o:
~0.1
\ ,,,,:
,
,
\002
0.8
o.~ 0.4
0.2
.
I
0.2
t
0.4
I
0.6
I
0.8
Figure 2. References
1. B. Castiang and P. Nozi~res, J. Physique 40, 239 (1979). 2. For example, A. Alhiezer and V, Aleksin, Dok. Akad. Nauk SSSR 92, 259 (1953); D. Vollhardt and P. Wolfle, Phys. Rev. Lett. 47, 190 (1981); K. S. Bedell and Meltzer, J. Low Temp. Phys. 6 3 , 2 1 5 (1986). 3. W. J. Mullin F. Lalo~, and M. G. Richards, J. Low Temp. Phys. 80, 1 (1990). 4. R. Chapman, Phys. Rev. A12, 2333 (1975). 5. J. Jeon and W. J. Mullin, J. Low Temp. Phys. 87, 421 (1987).