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This mysterious liquid 3He
Volume
108A. number
2
PHYSICS
THIS MYSTERIOUS
LETTERS
18 March
1985
LIQUID ‘He
A.M. DYUGAEV L.D. Landau Institute for Theoretical Physics, 142432 Chernogolovka, Moscow Region, USSR Received
23 October
1984
The spin contribution to the heat capacity and entropy of 3He has been singled out. With increasing T the spin entropy reaches a constant value, different from In 2. The spin contribution - In T can be observed in the temperature dependence of the pressure.
1. Greywall [ 1] has measured the heat capacity of 3He in a wide range of temperatures (0.005-2.5 K) and ofmolar volumes(25.7-36.8 cm3/mole). Thompson et al. [Z] have determined the temperature dependence of the magnetic susceptibility x for the same range of V and T approximately. In the present paper the spin contribution to C,, the entropy S, and pressure p have been singled out on the basis of analysis and correlation of the data obtained in refs. [I ,2]. 2. The characteristic energy scale of 3He is the parameter TF determined by the value of x at T = 0: x(0) = C’,/T,, where C, is the Curie constant. At all T the dependence of x ori T and the liquid density n can be empirically described within an accuracy of 2% by x(T) = Ck(T2 + T,)-112 TJn)=
7’&ro/no)“F,
n/no = v,/v,
) TF(no) = 0.359
VF = -1.93.
K, (1)
The equilibrium density no is in conformity with the molar volume V= 36.84 cm3/mo1e. At high T> 2TF the temperature dependences of the heat capacity and entropy per particle have the form C@
= yoT+
To/T-
Sv/R
= So + yoT - ToIT - $3fT3 .
The correction flT3 for C, is small for all T < 2.5 K and the parameter /3 cannot be determined exactly. The values of So, yo, To, on the other hand, can be found precisely, since the derivatives of these quantities with respect to the volume V are known from the identity ap/aT = U/a V and the data obtained in ref.
ill: P(T) = P(O) -p.
PO = @To/a V)R 2
(2)
+ 6 T2,
fi = (ho/a
vR.
(3)
C, + S,, as a function of T, and CVT, (,a,{aT)T, as functions of T2, plotted on the basis of the data from ref. [ 11, show the dependences (2) and (3). These functions are linear for T> 2TF. The accuracy of (2) and (3) increases with growing n and T. Unfortunately Greywall stopped measuring at T = 2.5 K, while it would be interesting to find out up to which temperatures the laws (2) and (3) are valid. The parameter So in (2) displays a fascinating constancy: when the 3He density varies from no to 1.4 X no, So undergoes almost negligible oscillations in the range 0.964-0.972. The value of So noticeably differs from In 2. The parameter y. in (2) is a power function ofn: r&r) ro(no)
PT3,
In(T/Tp)
y
= ru(~u)(nln0P = 0.35 K-l
,
vr = -1.44.
(4)
This law holds for a great range, from n = no to 1.84 X no. The dependence (4) is in agreement with the data 105
PHYSICS LETTERS
Volume 108A, number 2
obtained by Pandorf et al. [3] for 3He, 4He and their solutions in the temperature range 2-4.5 K and in the molar volume range 20-23 cm3/mole. The exponent vy is almost the same for 3He and 4He. The values of the effective mass of the light quasiparticles 2 2 have been tabulated. Before the unique m; = YOP& data published in ref. [ 11, the value of rnt was almost unavailable [4,5]. Given the asymptotics of S, C, p at T > 2T, one may find the spin values SO. Co, p” at any T < 2.5 K
by P/R=C/R-yoT+/3T3.
P/R=SfR-yoT+$3T3.
pa=p-6T2.
(5)
The spin Co and So are characterised by a scaling temperature dependence. All the plots of CU and S0 versus T/TF at different values of Vlie on one curve. It should be noted that Goldstein [6] singled out the spin contribution to S, proceeding from the obvious fact that at high T the spin entropy reaches the value of In 2. Andreevwho did not take into account To/T in (2) obtained So w 0.62. Eqs. (2) (3) are typical for any system with two strongly different temperature scales. For 3He and 4He quantum effects become significant when T < I!?,where J!?- 10 K is the average kinetic energy per atom. As the degeneration temperature of 3He is small and_TF -&I?, there exists a region of average T: TF < T < E, where in terms of the zero approach we can consider T, = 0; I?‘=00. In such a situation the system possesses no energy scale, and the temperature cannot be included into the expression for S and C: C = 0; S/R = So. In the following approaches T is included in the combinations TF/T and T/I!?. For solid 3He Eis the Debye frequency of phonons 0, and the decompositions (2) start with higher degrees of T and T-‘. S/R = In 2 t T3/e3 - Tc/T”. On the other hand for the liquid, where one-particle movements of atoms are possible, the decomposition S starts with the first order terms T and T-l which are forbidden by nothing. 3. All the three coefficients in a low-temperature expansion of C have been determined in ref. [ I], C/R = C’/R + yoT, YU = Y - 70 . 106
P/R
= y@T - FT3 ln(eJ7’), (6)
18 March 1985
PHYSICS LETTERS
Volume 108A. number 2
Table 1 presents the values of the dimensionless parameters y”T,, I’T,! and 8,/T,. It is clear that they all only slightly depend on the molar volume V. The parameter T, defines the temperature scale for Ca, SU and x for the whole range 0.005-2.5 K. The parameter r in (6) is characterised by a power dependence on n: J?(n) = r(no)(n/no)ur r(n,)
)
vr = 5.02,
= 30.84 Ke3.
(7)
At all T< 2.5 K the dependence ofSa on r = T/TF can be described within an accuracy of 2% by So/R = K(r/rl)ln[l
+ (a(r,/r)]
- 5 ha3 ln(p,2 t Qz/G2).
(8)
The parameter @ is determined by Cp= xT/Ck a 7( 1 + 7)-112, and K, rl, h, ps, as are fixed by the asymptotic values of SU at T < T, and T 3- T,. The expression (8) for S can be proved by the method developed in refs. [4,7]. The first term in (8) - K is determined ambiguously, since it is sensitive to the dependence of x on the wave vector k [4]. The term -X in (8), on the other hand, is determined exactly, since it depends on x(k = 0) only. 4. If there really is a short-range order in liquid 3He and the atoms are spatially separated for most of the time then the Pauli principle should be fastly switched off with increasing density. As the residual So per atom does not tend to In 2, it becomes clear that the most simple short-range order when one atom is at a site of irregular lattice is not realised in 3He. Suppose the number of 3He atoms is less than the number of site of an irregular lattice is not realised in 3He. Suppose terpreted as the vacancy entropy. To account for the constancy of So the concentration of vacancies should not depend on the liquid density, which is impossible to our mind. Some additional disorder can be introduced by another method: two atoms with opposite spins must be allowed to exist at one site. The exper-
18 March 1985
imental data of S, and C, for 3He and 4He in the range of high T and n near the melting curve are necessary to see which of the versions has been “chosen” by nature. It is important to know whether there exists a constant of the So type besides the linear-in-T term in S, of 4He at high temperatures. Comparing the dependences for S, of 3He and 4He it is possible to elucidate whether the value So is indeed of spin nature. The experimental determination of the parameter To (2) for high n and T would make it possible to find TF since the ratio To/TF (see table 1) only slightly depends on n. Since TF is a scale for x [ 1] one may gain information about x in the range of high T, where x is small and its measurement is difficult. Extrapolation of the dependences for C and S from the ranges of high temperatures and n to small T would allow a prediction of the properties of supercooled 3He. In the paper by Andreev [8], Castaing and Nozieres [9] a model of an amorphous solid body (glass) has been considered as applied to 3He and 4He. It would be of interest to see whether this model can be used for obtaining (2), (3) and elucidating the difference of the parameter So from ln 2. The author is grateful to D.N. Khmelnitskii for discussions and to L.A. Tolochko for help in this paper. References [l] D.S. Greywall, Phys. Rev. B27 (1983) 2747. [2] J.R. Thompson, H. Ramm, J.F. Jarvis and H. Meyer, J. Low Temp. Phys. 2 (1970) 521. [3] R.S. Pandorf, L.M. Ifft and D.O. Edwards, Phys. Rev. 163 (1967) 175. [4] A.M. Dyugaev, Zh. Eksp. Teor. Fiz. 70 (1976) 2390. [5] A.A. Tolochko and A.M. Dyugaev, Zh. Eksp. Teor. Fiz. 86 (1984) 502. [6] L. Goldstein, Phys. Rev. 112 (1958) 1465. [7 ] A.M. Dyugaev, Zh. Eksp. Teor. Fiz. 83 (1982) 1005. [8] A.F. Andreev, Pis’ma Zh. Eksp. Tear. Fiz. 28 (1978) 603; PSS 127 (1979) 724. [9] B. Castaing and P. Nozieres, J. Phys. (Paris) 40 (1979) 257.
107
108A. number
2
PHYSICS
THIS MYSTERIOUS
LETTERS
18 March
1985
LIQUID ‘He
A.M. DYUGAEV L.D. Landau Institute for Theoretical Physics, 142432 Chernogolovka, Moscow Region, USSR Received
23 October
1984
The spin contribution to the heat capacity and entropy of 3He has been singled out. With increasing T the spin entropy reaches a constant value, different from In 2. The spin contribution - In T can be observed in the temperature dependence of the pressure.
1. Greywall [ 1] has measured the heat capacity of 3He in a wide range of temperatures (0.005-2.5 K) and ofmolar volumes(25.7-36.8 cm3/mole). Thompson et al. [Z] have determined the temperature dependence of the magnetic susceptibility x for the same range of V and T approximately. In the present paper the spin contribution to C,, the entropy S, and pressure p have been singled out on the basis of analysis and correlation of the data obtained in refs. [I ,2]. 2. The characteristic energy scale of 3He is the parameter TF determined by the value of x at T = 0: x(0) = C’,/T,, where C, is the Curie constant. At all T the dependence of x ori T and the liquid density n can be empirically described within an accuracy of 2% by x(T) = Ck(T2 + T,)-112 TJn)=
7’&ro/no)“F,
n/no = v,/v,
) TF(no) = 0.359
VF = -1.93.
K, (1)
The equilibrium density no is in conformity with the molar volume V= 36.84 cm3/mo1e. At high T> 2TF the temperature dependences of the heat capacity and entropy per particle have the form C@
= yoT+
To/T-
Sv/R
= So + yoT - ToIT - $3fT3 .
The correction flT3 for C, is small for all T < 2.5 K and the parameter /3 cannot be determined exactly. The values of So, yo, To, on the other hand, can be found precisely, since the derivatives of these quantities with respect to the volume V are known from the identity ap/aT = U/a V and the data obtained in ref.
ill: P(T) = P(O) -p.
PO = @To/a V)R 2
(2)
+ 6 T2,
fi = (ho/a
vR.
(3)
C, + S,, as a function of T, and CVT, (,a,{aT)T, as functions of T2, plotted on the basis of the data from ref. [ 11, show the dependences (2) and (3). These functions are linear for T> 2TF. The accuracy of (2) and (3) increases with growing n and T. Unfortunately Greywall stopped measuring at T = 2.5 K, while it would be interesting to find out up to which temperatures the laws (2) and (3) are valid. The parameter So in (2) displays a fascinating constancy: when the 3He density varies from no to 1.4 X no, So undergoes almost negligible oscillations in the range 0.964-0.972. The value of So noticeably differs from In 2. The parameter y. in (2) is a power function ofn: r&r) ro(no)
PT3,
In(T/Tp)
y
= ru(~u)(nln0P = 0.35 K-l
,
vr = -1.44.
(4)
This law holds for a great range, from n = no to 1.84 X no. The dependence (4) is in agreement with the data 105
PHYSICS LETTERS
Volume 108A, number 2
obtained by Pandorf et al. [3] for 3He, 4He and their solutions in the temperature range 2-4.5 K and in the molar volume range 20-23 cm3/mole. The exponent vy is almost the same for 3He and 4He. The values of the effective mass of the light quasiparticles 2 2 have been tabulated. Before the unique m; = YOP& data published in ref. [ 11, the value of rnt was almost unavailable [4,5]. Given the asymptotics of S, C, p at T > 2T, one may find the spin values SO. Co, p” at any T < 2.5 K
by P/R=C/R-yoT+/3T3.
P/R=SfR-yoT+$3T3.
pa=p-6T2.
(5)
The spin Co and So are characterised by a scaling temperature dependence. All the plots of CU and S0 versus T/TF at different values of Vlie on one curve. It should be noted that Goldstein [6] singled out the spin contribution to S, proceeding from the obvious fact that at high T the spin entropy reaches the value of In 2. Andreevwho did not take into account To/T in (2) obtained So w 0.62. Eqs. (2) (3) are typical for any system with two strongly different temperature scales. For 3He and 4He quantum effects become significant when T < I!?,where J!?- 10 K is the average kinetic energy per atom. As the degeneration temperature of 3He is small and_TF -&I?, there exists a region of average T: TF < T < E, where in terms of the zero approach we can consider T, = 0; I?‘=00. In such a situation the system possesses no energy scale, and the temperature cannot be included into the expression for S and C: C = 0; S/R = So. In the following approaches T is included in the combinations TF/T and T/I!?. For solid 3He Eis the Debye frequency of phonons 0, and the decompositions (2) start with higher degrees of T and T-‘. S/R = In 2 t T3/e3 - Tc/T”. On the other hand for the liquid, where one-particle movements of atoms are possible, the decomposition S starts with the first order terms T and T-l which are forbidden by nothing. 3. All the three coefficients in a low-temperature expansion of C have been determined in ref. [ I], C/R = C’/R + yoT, YU = Y - 70 . 106
P/R
= y@T - FT3 ln(eJ7’), (6)
18 March 1985
PHYSICS LETTERS
Volume 108A. number 2
Table 1 presents the values of the dimensionless parameters y”T,, I’T,! and 8,/T,. It is clear that they all only slightly depend on the molar volume V. The parameter T, defines the temperature scale for Ca, SU and x for the whole range 0.005-2.5 K. The parameter r in (6) is characterised by a power dependence on n: J?(n) = r(no)(n/no)ur r(n,)
)
vr = 5.02,
= 30.84 Ke3.
(7)
At all T< 2.5 K the dependence ofSa on r = T/TF can be described within an accuracy of 2% by So/R = K(r/rl)ln[l
+ (a(r,/r)]
- 5 ha3 ln(p,2 t Qz/G2).
(8)
The parameter @ is determined by Cp= xT/Ck a 7( 1 + 7)-112, and K, rl, h, ps, as are fixed by the asymptotic values of SU at T < T, and T 3- T,. The expression (8) for S can be proved by the method developed in refs. [4,7]. The first term in (8) - K is determined ambiguously, since it is sensitive to the dependence of x on the wave vector k [4]. The term -X in (8), on the other hand, is determined exactly, since it depends on x(k = 0) only. 4. If there really is a short-range order in liquid 3He and the atoms are spatially separated for most of the time then the Pauli principle should be fastly switched off with increasing density. As the residual So per atom does not tend to In 2, it becomes clear that the most simple short-range order when one atom is at a site of irregular lattice is not realised in 3He. Suppose the number of 3He atoms is less than the number of site of an irregular lattice is not realised in 3He. Suppose terpreted as the vacancy entropy. To account for the constancy of So the concentration of vacancies should not depend on the liquid density, which is impossible to our mind. Some additional disorder can be introduced by another method: two atoms with opposite spins must be allowed to exist at one site. The exper-
18 March 1985
imental data of S, and C, for 3He and 4He in the range of high T and n near the melting curve are necessary to see which of the versions has been “chosen” by nature. It is important to know whether there exists a constant of the So type besides the linear-in-T term in S, of 4He at high temperatures. Comparing the dependences for S, of 3He and 4He it is possible to elucidate whether the value So is indeed of spin nature. The experimental determination of the parameter To (2) for high n and T would make it possible to find TF since the ratio To/TF (see table 1) only slightly depends on n. Since TF is a scale for x [ 1] one may gain information about x in the range of high T, where x is small and its measurement is difficult. Extrapolation of the dependences for C and S from the ranges of high temperatures and n to small T would allow a prediction of the properties of supercooled 3He. In the paper by Andreev [8], Castaing and Nozieres [9] a model of an amorphous solid body (glass) has been considered as applied to 3He and 4He. It would be of interest to see whether this model can be used for obtaining (2), (3) and elucidating the difference of the parameter So from ln 2. The author is grateful to D.N. Khmelnitskii for discussions and to L.A. Tolochko for help in this paper. References [l] D.S. Greywall, Phys. Rev. B27 (1983) 2747. [2] J.R. Thompson, H. Ramm, J.F. Jarvis and H. Meyer, J. Low Temp. Phys. 2 (1970) 521. [3] R.S. Pandorf, L.M. Ifft and D.O. Edwards, Phys. Rev. 163 (1967) 175. [4] A.M. Dyugaev, Zh. Eksp. Teor. Fiz. 70 (1976) 2390. [5] A.A. Tolochko and A.M. Dyugaev, Zh. Eksp. Teor. Fiz. 86 (1984) 502. [6] L. Goldstein, Phys. Rev. 112 (1958) 1465. [7 ] A.M. Dyugaev, Zh. Eksp. Teor. Fiz. 83 (1982) 1005. [8] A.F. Andreev, Pis’ma Zh. Eksp. Tear. Fiz. 28 (1978) 603; PSS 127 (1979) 724. [9] B. Castaing and P. Nozieres, J. Phys. (Paris) 40 (1979) 257.
107