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On the magnetic transition in superfluid 3He
Volume 47A, number 6
PHYSICS LETTERS
ON THE MAGNETIC
TRANSITION
6 May 1974
IN SUPERFLUID
3 He
B.R. PATTON Department o f Physics, Massachusetts, Institute o f Technology, Cambridge, Mass. 02139, USA
Received 26 March 1974 Comparison of recent measurements of the magnetic transition in superfluid 3He A with a fluctuation theory of the transition yield an estimate of the coherence length. Recent experiments [1 ] have revealed an interesting fine structure in the magnetization at the transition from the A phase to the normal state in superfluid 3He. With increasing temperature the magnetization first drops abruptly, then more slowly, the whole change being of order 0.5% of the magnetization M N of the normal state. The small but finite change in the magnetization at the A - N transition results from the variation of the density of states with energy and is directly related to the splitting in a magnetic field of the transition temperatures of the up and down spin populations [2]. The magnetization has been calcu!,ted in the meanfield approximation [3] using the Ginzburg-Landau form for the free energy [2] with paramagnon effects [4]. We calculate here fluctuation corrections to the magnetization which turn out to be large and to depend sensitively on the coherence length. For l = 1 the free energy functional has the form (setting the component A 0 = At~ = 0)
j=lN(O)fdSri=,,t(tilAi 1 2 ~
+t2IV'AiI2+t2TIVXAil2 )
+J4(/3,6),
(1)
where t i = tT-rlh, t = ( T - Tc)/Tc, h = 3,H/k B T¢, and the two coherence lengths [5] are given by 5t2/9 = 5 t 2 / 3 = ~2 _ 7 f ( 3 ) h 2 V 2 [ 4 8 7 r 2 k 2 T 2 .
(2)
For t i> 0 the gradient terms may be obtained by a simple expansion of the pair t-matrix equation. The order parameters At~ and A, are vectors in m o m e n t u m space. The free energy F is the functional integral exp ( - F / k B T) = n 1
fd2 % exp ( - J / k B T ) ,
and the magnetization is M = - a F / ~ H or
(3)
M-M
N =N(O)Trl((IA¢I2)--([A,12))/2kB T .
(4)
Replacing A i by its equilibrium value Aio , determined froth OJ/bA i = 0, gives the mean field result. Above the transition, eq. (4) gives a fluctuation contribution. We find 1/2
M2MN
t\ (
ec ] t
1/2
\
ec !
~, 1 / 2
---._ ec/l+ t+'Th -ec MA-MN i;,h t , e c ,
-- ]
I
(q] t~]
t~-~h
k-
where e c is the Ginzburg temperature for critical fluctuations ec = 7 f(3)TF/8rrk
ffTc,
(6)
and the effective coherence length teff is t e 2 = tL 2 + 2 t ~ 2 = 4 . 7 2 t ~ 2 ,
(7)
The explicit form (5) results from the use of the Hartree approximation for the fourth order terms, which interpolates between the mean t'mld regions above and below Tc. Corrections occur for It +-1?hl <~ e c and appear to be insignificant. A value of 5 = 0.6 [4] narrows the mean-field transition width slightly, but has little effect on the fluctuation tail. The data of [1 ] are com. pared with eq. (5) in fig. 1 for a value e c = 6 X 10 -5 and an experimental splitting of the A 1 - A 2 transitions of 5.7 pK/kG. From the normal state Fermi liquid data [6] at 30 bar, T F = 1.05K and k F = 0.88A -1. Eqs. (6) and (7) then yield t0 = 50A. In comparison, eq. (2) gives t0 = 124A (o F = 3.1 X 103 cm/s). The value of t0 inferred from e c is probably a minimum since effects which broaden the transition decrease the value of %. The role of thermal inhomogeneity would appear to be 459
,
Volume 47A, number 6 I.2 ~ - - " T " ~ I
PHYSICS LETTERS T-'--""I
l°r~-a°~e°% I 0 8'1-
't%llbl
M-MN 06 ~ MAM-------~
• 3~.2Bor
-
o 29.5
_
~J"~_
oJ
-2
r
I
\'o -I
0
I
2 3 T - T c (/zK)
4
5
6
7
Fig. 1. The reduced magnetization as a function of temperature at the transition between superfluid aHe A and the normal Fermi liquid for two pressures. MN is the magnetization of the normal liquid, and MA that of 3He A. The solid line is eq. (5) for ec = 6 × 10-5, and the dotted line is mean field theory (ec~ 0). The data are from the experiments of ref. [1].
small, since it would remove the sharp initial decline observed in the magnetization. Other effects, such as uncertainty in the baseline M N or the splitting parameter 71may increase the inferred value of the coherence length, but in view of the extreme sensitivity of the fluctuation effects to the coherence length, it seems unlikely that a value as large as that obtained from (2) is possible. An estimate of the various uncertainties would indicate 50A < GO< 70A.
460
6 May 1974
The magnetization due to fluctuations in the A0 component of the order parameter may be similarly calculated
M 0 - M N = - 7 ~ h { C - (t+2~h2)Z/2}/Tr~3ff,
(8)
where C is a constant of order unity. The temperature dependent part o f M 0 is estimated to be an order of magnitude smaller than (4) in the temperature region of interest. The rounding of the specific heat discontinuity is also small, less than 0.1/xK from the above value of e c. I thank Professor J.C. Wheatley for simulating discussions and the use of their data.
References [1 ] D.N. Paulson, H. Kojima, and J.C, Wheatley, submitted to Physics Letters. [2] V. Ambegaokar and N.D. Mermin, Phys. Rev. Lett. 30 (1973) 81. [3] S. Takagi, preprint. [4] W.F. Brinkman and P.W. Anderson, Phys. Rev. A8 (1973) 2732; W.F. Brinkman, J. Serene, and P.W. Anderson, preprint. [5] V. Ambegaokar,P.G. de Gennes and D. Rainer, preprint. [6] J.C. Wheatley, Physica 69 (1973) 218.
PHYSICS LETTERS
ON THE MAGNETIC
TRANSITION
6 May 1974
IN SUPERFLUID
3 He
B.R. PATTON Department o f Physics, Massachusetts, Institute o f Technology, Cambridge, Mass. 02139, USA
Received 26 March 1974 Comparison of recent measurements of the magnetic transition in superfluid 3He A with a fluctuation theory of the transition yield an estimate of the coherence length. Recent experiments [1 ] have revealed an interesting fine structure in the magnetization at the transition from the A phase to the normal state in superfluid 3He. With increasing temperature the magnetization first drops abruptly, then more slowly, the whole change being of order 0.5% of the magnetization M N of the normal state. The small but finite change in the magnetization at the A - N transition results from the variation of the density of states with energy and is directly related to the splitting in a magnetic field of the transition temperatures of the up and down spin populations [2]. The magnetization has been calcu!,ted in the meanfield approximation [3] using the Ginzburg-Landau form for the free energy [2] with paramagnon effects [4]. We calculate here fluctuation corrections to the magnetization which turn out to be large and to depend sensitively on the coherence length. For l = 1 the free energy functional has the form (setting the component A 0 = At~ = 0)
j=lN(O)fdSri=,,t(tilAi 1 2 ~
+t2IV'AiI2+t2TIVXAil2 )
+J4(/3,6),
(1)
where t i = tT-rlh, t = ( T - Tc)/Tc, h = 3,H/k B T¢, and the two coherence lengths [5] are given by 5t2/9 = 5 t 2 / 3 = ~2 _ 7 f ( 3 ) h 2 V 2 [ 4 8 7 r 2 k 2 T 2 .
(2)
For t i> 0 the gradient terms may be obtained by a simple expansion of the pair t-matrix equation. The order parameters At~ and A, are vectors in m o m e n t u m space. The free energy F is the functional integral exp ( - F / k B T) = n 1
fd2 % exp ( - J / k B T ) ,
and the magnetization is M = - a F / ~ H or
(3)
M-M
N =N(O)Trl((IA¢I2)--([A,12))/2kB T .
(4)
Replacing A i by its equilibrium value Aio , determined froth OJ/bA i = 0, gives the mean field result. Above the transition, eq. (4) gives a fluctuation contribution. We find 1/2
M2MN
t\ (
ec ] t
1/2
\
ec !
~, 1 / 2
---._ ec/l+ t+'Th -ec MA-MN i;,h t , e c ,
-- ]
I
(q] t~]
t~-~h
k-
where e c is the Ginzburg temperature for critical fluctuations ec = 7 f(3)TF/8rrk
ffTc,
(6)
and the effective coherence length teff is t e 2 = tL 2 + 2 t ~ 2 = 4 . 7 2 t ~ 2 ,
(7)
The explicit form (5) results from the use of the Hartree approximation for the fourth order terms, which interpolates between the mean t'mld regions above and below Tc. Corrections occur for It +-1?hl <~ e c and appear to be insignificant. A value of 5 = 0.6 [4] narrows the mean-field transition width slightly, but has little effect on the fluctuation tail. The data of [1 ] are com. pared with eq. (5) in fig. 1 for a value e c = 6 X 10 -5 and an experimental splitting of the A 1 - A 2 transitions of 5.7 pK/kG. From the normal state Fermi liquid data [6] at 30 bar, T F = 1.05K and k F = 0.88A -1. Eqs. (6) and (7) then yield t0 = 50A. In comparison, eq. (2) gives t0 = 124A (o F = 3.1 X 103 cm/s). The value of t0 inferred from e c is probably a minimum since effects which broaden the transition decrease the value of %. The role of thermal inhomogeneity would appear to be 459
,
Volume 47A, number 6 I.2 ~ - - " T " ~ I
PHYSICS LETTERS T-'--""I
l°r~-a°~e°% I 0 8'1-
't%llbl
M-MN 06 ~ MAM-------~
• 3~.2Bor
-
o 29.5
_
~J"~_
oJ
-2
r
I
\'o -I
0
I
2 3 T - T c (/zK)
4
5
6
7
Fig. 1. The reduced magnetization as a function of temperature at the transition between superfluid aHe A and the normal Fermi liquid for two pressures. MN is the magnetization of the normal liquid, and MA that of 3He A. The solid line is eq. (5) for ec = 6 × 10-5, and the dotted line is mean field theory (ec~ 0). The data are from the experiments of ref. [1].
small, since it would remove the sharp initial decline observed in the magnetization. Other effects, such as uncertainty in the baseline M N or the splitting parameter 71may increase the inferred value of the coherence length, but in view of the extreme sensitivity of the fluctuation effects to the coherence length, it seems unlikely that a value as large as that obtained from (2) is possible. An estimate of the various uncertainties would indicate 50A < GO< 70A.
460
6 May 1974
The magnetization due to fluctuations in the A0 component of the order parameter may be similarly calculated
M 0 - M N = - 7 ~ h { C - (t+2~h2)Z/2}/Tr~3ff,
(8)
where C is a constant of order unity. The temperature dependent part o f M 0 is estimated to be an order of magnitude smaller than (4) in the temperature region of interest. The rounding of the specific heat discontinuity is also small, less than 0.1/xK from the above value of e c. I thank Professor J.C. Wheatley for simulating discussions and the use of their data.
References [1 ] D.N. Paulson, H. Kojima, and J.C, Wheatley, submitted to Physics Letters. [2] V. Ambegaokar and N.D. Mermin, Phys. Rev. Lett. 30 (1973) 81. [3] S. Takagi, preprint. [4] W.F. Brinkman and P.W. Anderson, Phys. Rev. A8 (1973) 2732; W.F. Brinkman, J. Serene, and P.W. Anderson, preprint. [5] V. Ambegaokar,P.G. de Gennes and D. Rainer, preprint. [6] J.C. Wheatley, Physica 69 (1973) 218.