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Thermal conductivity measurements of polarized liquid 3 He
Physica B 329–333 (2003) 118–119
Thermal conductivity measurements of polarized liquid 3 He D. Sawkey*, V. Goudon, O. Buu1, L. Puech, P.-E. Wolf Centre de Recherches sur les Tr"es Basses Temp!eratures, CNRS, BP 166, 38042 Grenoble, Cedex 9, France
Abstract We present preliminary measurements of the thermal conductivity of spin-polarized normal liquid 3 He. Our experimental apparatus allows good thermal homogeneity of the 3 He as well as high sensitivity to the 3 He conductivity. Initial measurements show the conductivity changes by o10% for polarizations up to 70%. r 2002 Elsevier Science B.V. All rights reserved. PACS: 67.65.+z; 67.55.s; 66.60.+a Keywords: Normal liquid helium-3; Spin polarized; Thermal conductivity
Measurement of the transport properties of spinpolarized liquid 3 He provides information on the nature of quasiparticle collisions, with the view towards a microscopic understanding of liquid 3 He. Intuitively, one expects that because of the Pauli exclusion principle, polarization will increase the quasiparticle mean free path and therefore the viscosity and thermal conductivity. Such an increase in the viscosity has indeed been observed [1,2]. We present preliminary measurements of the thermal conductivity of spin-polarized liquid 3 He. Our apparatus is the same as used for the viscosity and specific heat measurements: liquids of polarizations up to 70% are produced by depressurizing, therefore melting, a polarized solid. We achieve good thermal homogeneity by confining the liquid to a silver sinter; this reduces the polarization relaxation time to 60 s; still sufficient for experiments. A vibrating wire viscometer placed in a 200 mm wide slit in the sinter is used both as a thermometer and a heater. The viscometer is driven close to its resonance frequency of 10 kHz by an AC current of several mA. To determine the viscosity, we measure the in-phase and quadrature response of the *Corresponding author. Present address: Cavendish Laboratory, University of Cambridge, Madingley Road, Cambridge CB3 0ME, UK. E-mail addresses: [email protected] (D. Sawkey), [email protected] (P.-E. Wolf). 1 Present address: Nottingham University, School of Physics and Astronomy, Nottingham NG7 2RD, UK.
viscometer at the drive frequency. Initially, we tried to heat the liquid by increasing the AC current to the viscometer, with the heat being generated by the viscous motion of the fluid, but for sufficient heating the nonlinear term in the Navier-Stokes equation made extraction of the viscosity difficult. We now use DC heating, possible because the vibrating wire is manganin. Only B1% of the Joule heat enters the 3 He, because of the long legs of the vibrating wire; the rest is absorbed by the cell walls. Because we regulate the temperature of the cell walls, it does not change, and the excess heat does not interfere with the experiment. We change the DC current from 0 to 100 mA every 6–10 s; and determine ’ where DT is the the thermal resistance as R ¼ DT=Q; temperature difference between the two states and Q’ the added heat for a nominal 1 mm length of the viscometer in the 3 He. The measured thermal resistance is composed of both that of the bulk 3 He in the slit and the resistance in the sinter, including the boundary resistance. To ensure that we are sensitive to the 3 He resistivity, we measured R in the equilibrium polarization (meq ¼ 4% at 11 T) from T ¼ 30–110 mK at 2, 10, 20, and 27 bar, shown in Fig. 1. A log–log plot of the measured R vs. the known 3 He resistivity [3] is given in Fig. 2 which shows that our sensitivity is 70%; that is, a 10% change in the 3 He resistivity changes our measured R by 7%. This is in reasonable agreement with a calculation based on a geometry of concentric cylinders, which predicted a sensitivity of 85%. Despite the presence of the sinter,
0921-4526/03/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. doi:10.1016/S0921-4526(02)01914-2
D. Sawkey et al. / Physica B 329–333 (2003) 118–119 1.1
0.18
1
0.16
0.9 R/R(meq )
Thermal resistance [mK/(nW/mm)]
0.2
119
0.14 0.12
0.7
0.1 0.08
0.8
20
40
60 80 Temperature [mK]
100
120
Fig. 1. Measured thermal resistance R for 27, 20, 10, and 2 bar (from top), for the unpolarized liquid, using the heat dissipated per mm viscometer length.
0.6
0
0.1
0.2
0.3 0.4 Polarization
0.5
0.6
0.7
Fig. 3. Relative thermal resistance of the polarized liquid Rðm; TÞ=Rðmeq ; TÞ vs. polarization at 27 bar. The temperature decreases with m from 70 to 60 mK: The line is a theoretical result [4].
Measured resistance [mK/(nW/mm)]
0.2
0.1
0.1
0.2 He-3 thermal resistivity [a.u.]
0.3
Fig. 2. Measured thermal resistance R for the unpolarized liquid vs. known 3 He resistivity on a log–log plot. Each data set is at a fixed temperature, and pressure is varied from 2 to 27 bar from left to right. Temperature increases in steps of 10 mK from 30 to 110 mK from left to right. The heavy line has slope 0.7.
therefore, we remain highly sensitive to the 3 He resistivity. This analysis assumes that the boundary resistance is independent of pressure, just as analysis of the experiment with the polarized liquid assumes the boundary resistance is independent of polarization. In the polarized liquid, measurements are complicated by the increase of viscosity with polarization m; given by ZðmÞ ð1Þ ¼ 1 þ am2 Zðmeq Þ with a ¼ 3:5 at 60 mK [2]. We measure Zðm; TÞ; and therefore a; in the Q’ ¼ 0 state using a carbon resistor thermometer close to the viscometer in the slit to determine the viscometer temperature; the viscosity of the unpolarized liquid at the same temperature is therefore given by Eq. (1) (in practice, higher order terms are kept as well). After the melting experiment,
Zðmeq ; TÞ is measured as a function of T with the sample cell in thermal equilibrium. The two relations allow the Zðm; TÞ measured during the depolarization to be converted into temperature, which in turn allows calculation of R: Because of the heat released by depolarization, the temperature of the 3 He in the slit decreases by 10 mK during a depolarization. To correct for the change in temperature, we divide by Rðm; TÞ measured during the depolarization by Rðmeq ; TÞ; measured at the same time as Zðmeq ; TÞ (this is a 5% correction). Fig. 3 shows Rðm; TÞ=Rðmeq ; TÞ vs. m at 27 bar and a final temperature of 60 mK: The change in R with m is less than 10%, therefore the 3 He resistivity changes by less than 15%. In contrast, the viscosity increases by a factor of 2.5 in the same conditions. It is necessary to quantify our systematic errors and repeat the experiment at different pressures and temperatures to compare the result to the theoretical result for an ideal Fermi gas, where the change in R is proportional to m at low m; 17% at m ¼ 0:3 [4]. Furthermore, the relaxation from one Q’ state to the other scales with RC; where C is the heat capacity. Because C is known [5], we will use the relaxation for an alternative determination of R:
References [1] G.A. Vermeulen, et al., Phys. Rev. Lett. 60 (1988) 2315; C.C. Kranenburg, et al., Phys. Rev. Lett. 61 (1988) 1372. [2] O. Buu, A.C. Forbes, L. Puech, P.E. Wolf, Phys. Rev. Lett. 83 (1999) 3466. [3] D.S. Greywall, Phys. Rev. B 29 (1984) 4933. [4] W.J. Mullin, K. Miyake, J. Low Temp. Phys. 53 (1983) 313. [5] O. Buu, L. Puech, P.E. Wolf, Phys. Rev. Lett. 85 (2000) 1278.
Thermal conductivity measurements of polarized liquid 3 He D. Sawkey*, V. Goudon, O. Buu1, L. Puech, P.-E. Wolf Centre de Recherches sur les Tr"es Basses Temp!eratures, CNRS, BP 166, 38042 Grenoble, Cedex 9, France
Abstract We present preliminary measurements of the thermal conductivity of spin-polarized normal liquid 3 He. Our experimental apparatus allows good thermal homogeneity of the 3 He as well as high sensitivity to the 3 He conductivity. Initial measurements show the conductivity changes by o10% for polarizations up to 70%. r 2002 Elsevier Science B.V. All rights reserved. PACS: 67.65.+z; 67.55.s; 66.60.+a Keywords: Normal liquid helium-3; Spin polarized; Thermal conductivity
Measurement of the transport properties of spinpolarized liquid 3 He provides information on the nature of quasiparticle collisions, with the view towards a microscopic understanding of liquid 3 He. Intuitively, one expects that because of the Pauli exclusion principle, polarization will increase the quasiparticle mean free path and therefore the viscosity and thermal conductivity. Such an increase in the viscosity has indeed been observed [1,2]. We present preliminary measurements of the thermal conductivity of spin-polarized liquid 3 He. Our apparatus is the same as used for the viscosity and specific heat measurements: liquids of polarizations up to 70% are produced by depressurizing, therefore melting, a polarized solid. We achieve good thermal homogeneity by confining the liquid to a silver sinter; this reduces the polarization relaxation time to 60 s; still sufficient for experiments. A vibrating wire viscometer placed in a 200 mm wide slit in the sinter is used both as a thermometer and a heater. The viscometer is driven close to its resonance frequency of 10 kHz by an AC current of several mA. To determine the viscosity, we measure the in-phase and quadrature response of the *Corresponding author. Present address: Cavendish Laboratory, University of Cambridge, Madingley Road, Cambridge CB3 0ME, UK. E-mail addresses: [email protected] (D. Sawkey), [email protected] (P.-E. Wolf). 1 Present address: Nottingham University, School of Physics and Astronomy, Nottingham NG7 2RD, UK.
viscometer at the drive frequency. Initially, we tried to heat the liquid by increasing the AC current to the viscometer, with the heat being generated by the viscous motion of the fluid, but for sufficient heating the nonlinear term in the Navier-Stokes equation made extraction of the viscosity difficult. We now use DC heating, possible because the vibrating wire is manganin. Only B1% of the Joule heat enters the 3 He, because of the long legs of the vibrating wire; the rest is absorbed by the cell walls. Because we regulate the temperature of the cell walls, it does not change, and the excess heat does not interfere with the experiment. We change the DC current from 0 to 100 mA every 6–10 s; and determine ’ where DT is the the thermal resistance as R ¼ DT=Q; temperature difference between the two states and Q’ the added heat for a nominal 1 mm length of the viscometer in the 3 He. The measured thermal resistance is composed of both that of the bulk 3 He in the slit and the resistance in the sinter, including the boundary resistance. To ensure that we are sensitive to the 3 He resistivity, we measured R in the equilibrium polarization (meq ¼ 4% at 11 T) from T ¼ 30–110 mK at 2, 10, 20, and 27 bar, shown in Fig. 1. A log–log plot of the measured R vs. the known 3 He resistivity [3] is given in Fig. 2 which shows that our sensitivity is 70%; that is, a 10% change in the 3 He resistivity changes our measured R by 7%. This is in reasonable agreement with a calculation based on a geometry of concentric cylinders, which predicted a sensitivity of 85%. Despite the presence of the sinter,
0921-4526/03/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. doi:10.1016/S0921-4526(02)01914-2
D. Sawkey et al. / Physica B 329–333 (2003) 118–119 1.1
0.18
1
0.16
0.9 R/R(meq )
Thermal resistance [mK/(nW/mm)]
0.2
119
0.14 0.12
0.7
0.1 0.08
0.8
20
40
60 80 Temperature [mK]
100
120
Fig. 1. Measured thermal resistance R for 27, 20, 10, and 2 bar (from top), for the unpolarized liquid, using the heat dissipated per mm viscometer length.
0.6
0
0.1
0.2
0.3 0.4 Polarization
0.5
0.6
0.7
Fig. 3. Relative thermal resistance of the polarized liquid Rðm; TÞ=Rðmeq ; TÞ vs. polarization at 27 bar. The temperature decreases with m from 70 to 60 mK: The line is a theoretical result [4].
Measured resistance [mK/(nW/mm)]
0.2
0.1
0.1
0.2 He-3 thermal resistivity [a.u.]
0.3
Fig. 2. Measured thermal resistance R for the unpolarized liquid vs. known 3 He resistivity on a log–log plot. Each data set is at a fixed temperature, and pressure is varied from 2 to 27 bar from left to right. Temperature increases in steps of 10 mK from 30 to 110 mK from left to right. The heavy line has slope 0.7.
therefore, we remain highly sensitive to the 3 He resistivity. This analysis assumes that the boundary resistance is independent of pressure, just as analysis of the experiment with the polarized liquid assumes the boundary resistance is independent of polarization. In the polarized liquid, measurements are complicated by the increase of viscosity with polarization m; given by ZðmÞ ð1Þ ¼ 1 þ am2 Zðmeq Þ with a ¼ 3:5 at 60 mK [2]. We measure Zðm; TÞ; and therefore a; in the Q’ ¼ 0 state using a carbon resistor thermometer close to the viscometer in the slit to determine the viscometer temperature; the viscosity of the unpolarized liquid at the same temperature is therefore given by Eq. (1) (in practice, higher order terms are kept as well). After the melting experiment,
Zðmeq ; TÞ is measured as a function of T with the sample cell in thermal equilibrium. The two relations allow the Zðm; TÞ measured during the depolarization to be converted into temperature, which in turn allows calculation of R: Because of the heat released by depolarization, the temperature of the 3 He in the slit decreases by 10 mK during a depolarization. To correct for the change in temperature, we divide by Rðm; TÞ measured during the depolarization by Rðmeq ; TÞ; measured at the same time as Zðmeq ; TÞ (this is a 5% correction). Fig. 3 shows Rðm; TÞ=Rðmeq ; TÞ vs. m at 27 bar and a final temperature of 60 mK: The change in R with m is less than 10%, therefore the 3 He resistivity changes by less than 15%. In contrast, the viscosity increases by a factor of 2.5 in the same conditions. It is necessary to quantify our systematic errors and repeat the experiment at different pressures and temperatures to compare the result to the theoretical result for an ideal Fermi gas, where the change in R is proportional to m at low m; 17% at m ¼ 0:3 [4]. Furthermore, the relaxation from one Q’ state to the other scales with RC; where C is the heat capacity. Because C is known [5], we will use the relaxation for an alternative determination of R:
References [1] G.A. Vermeulen, et al., Phys. Rev. Lett. 60 (1988) 2315; C.C. Kranenburg, et al., Phys. Rev. Lett. 61 (1988) 1372. [2] O. Buu, A.C. Forbes, L. Puech, P.E. Wolf, Phys. Rev. Lett. 83 (1999) 3466. [3] D.S. Greywall, Phys. Rev. B 29 (1984) 4933. [4] W.J. Mullin, K. Miyake, J. Low Temp. Phys. 53 (1983) 313. [5] O. Buu, L. Puech, P.E. Wolf, Phys. Rev. Lett. 85 (2000) 1278.