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Heat conduction in spin polarized gaseous helium-3 : Recent theoretical and experimental results
Physica B 165&166 (1990) 735-736 North-Holland
HEAT CONDUCTION
IN SPIN POLARIZED
Brigitte
Franvoise
BRAMI,
GASEOUS
JOLY, Claire
HELIUM-3
: RECENT
THEORETICAL
AND EXPERIMENTAL
RESULTS
LHUILLIER
Laboratoire de Physique Theorique des Liquides, Universite Pierre et Marie Curie, 4 place Jussieu, 75252 Paris Cedex 05 and Christian LARAT, Michele LEDUC, Pierre-Jean NACHER, Genevieve TASTEVIN and GQrard VERMEULEN Laboratoire de Spectroscopic Hertzienne de 1'ENS (associ.8 au CNRS et a l'Universit8 Pierre et Marie Curie), 24 rue Lhomond, 75231 Paris Cedex 05
The transport properties of dilute spin polarized gases exhibit interesting dependences on the between colliding partipolarization M at low temperature, due to quantum indistinguishability cles. For the heat conduction K of polarized gaseous helium 3, recent calculations have checked the convergence of the moment expandonof the out of equilibrium density matrix. Four to five moments are necessary to achieve a 1% accuracy. At the same time the changes of K with M have been reliably measured in the 1.3-4.2 K range, with great care to control all the heat fluxes reaching the cell. The temperature dependence of the changes of K with M is now found in aatisfactory agreement with theory.
For dilute spin polarized gases, changes of the transport properties occur if the temperature is low enough, namely if the de Broglie wavelength is comparable to the range of the interactions between particles [II - This derives from various interference effects induced by quantum indistinguishability between colliding particles. Proper symmetrization of the wave functions leads to scattering cross-sections which depend on the spin polarization. The Boltzmann theory has to be reformulated to incorporate the Pauli principle [2]. This leads to quantitative results for all the transport coefficients and their low temperature variations as a function of the nuclear polariZatiOn M. For the heat conduction coefficient K of gaseous 3He , such variations were observed in 1987 [3]. They exhibit a strong and non intuitive dependence on temperature T. However the preliminary experimental results of [3] did not show a good quantitative agreement with theory [2]. This fact motivated us both to derive new considerations on the problem of quantum exchange in Boltzmann theory 143 and to push the previous experiment to a more advanced stage [5]. Improving the calculation of transport coefficients of polarized gases, we first examined the influence of the choice of the interatomic potentials, using a few of the more recently proposed ones. We then checked the oonvergence of the Sonine expansion commonly used for classical gases, but which may be questionable in the case Of spin polarized gases, because all
0921.4526/90/$03.50
@ 1990 - ElsevierScience
Publishers
the elements of the single particle density matrix do not relax towards equilibrium with the same collision cross-section. For the helium 3 gas at low temperature, the convergence of the expansion is rather rapid in the case of the coefficient K for the non polarized gas. On the other hand for the totally polarized gas the result is much more sensitive to the approximative form of the trial density distribution which is chosen. The reason is that the new effect is due to quantum interferences and its magnitude is the result of a summation of large terms with alternate signs: this cancellation is total at 300K, explaining interference effects do not show why in thermodynamical measurements at room temperature: it is accidental around 0.3 and 0.9K: in the temperature range 1K to 4K, K happens to increase with M. Even for slightly polarized SampleS in the range 1K to 4K, it is necessary to use as much as 3 or 4 Sonine polynomials to achieve 1% acFor l4 = 100% curacy. the first order approximation is off from the "exact" result by 20 to 25% and one has to use 4 to 5 terms in the Sonine expansion to reach the desired convergence. These results are shown in figure 2, where the dotted line corresponds to the predictions with the first approximation, the full line with the fifth These one. curves were calculated using the potential of reference [S]. Little differences were found with other potentials.
B.V. (North-Holland)
736
B. Brami, F. Joly, C. Lhuillier, C. Larat, M. Leduc, P.-J. Nacher, G. Tastevin, G. Vermeulen
The experimental values of 30%. [K(M)-K(O)]/M~ are plotted in figure 2. The uncertainty on the absolute value of Hz being large (but not on its relative value), a scaling factor for M was used to fit the theoretical curve at the arbitrarily chosen tempera ture of2X.
Thermal Cu shielding
A
Aensor
R1
i-
K(M)-M(O) M2 hW/K.m 1
f
NMR pick-up coil
I!-
FIGORE 1 Sketch
Of
the
measuring the as a function
experiment
llow pa*
of
the
heat conductivity of gaseous of spin polarization.
cell) 3Ee
As for the experiment here reported, its principle is similar to that described earlier [3], but the temperature
was
extended
down to
1.3X by
several improvements to the cryostat and to the 3We cell. Also greater attention was payed to the control of heat fluxes reaching the measurement cells, in order to avoid systematic errors on the derived value of the coefficient K. The polarization of the 3He gas is achieved by optical pumping with an arc lamp pumped WA laser. The low temperature part of the cell is shown in figure 1. The polarization of the measurement cylinder is monitored by NMR coils. Carbon resistors R, and measure the temperatures of the end p ates and electrical heating can be 9 applied to the bottom plate by means of a copper grid. The cell is shielded against radiation and spurious degasing by an aluminized coil foil screen. Additional experiments were performed to directly measure the thermal impedance of the Pyrex cell walls and substract their contribution to the measure of K. For the non polarized gas the measured values of K were in good agreement with those of reference [7]. For the polarized gas, IC was found to vary linearly with M2 at various temperatures, M ranging between 0 and
PIf3IIRB 2 2 Changes of the heat conduction coefficient of gaseous -Be with spin polariraticm U, plotted as a function of temperatux? T.
The comparison of the improved and now reliable experiments results with the new calculations is shown in figure 2. The agreement is satisfactory and even remarksbly good below 2X. The problem of heat conductivity changes in 3He with spin polarization can now be considered as well understood. 111 "Spin Polarized Quantum Systems", Proceedings of the I.S.I. Conference (Torino), Stringari Editor, World Scientific (1988). [2] C. Lbuillier and F. LaloG, J. Physique s (1982) 197 and 225; C. Ihuillier, J. Physique Q (1983) 1. [3] W. Leduc et al. Burophys. Letters 4 (1987) 59. [4] B. Brami et al. Physica (1990) to be published. [5] C. Larat et al., to be published in J. of Low Temp. Phys. [6] R.A. Aziz et al. J. Chem. Phys. x (1979) 4330. [7] D.S. Betts and J. Marshall, J. of Low Temp. Phys. ;5 (1969) 595.
HEAT CONDUCTION
IN SPIN POLARIZED
Brigitte
Franvoise
BRAMI,
GASEOUS
JOLY, Claire
HELIUM-3
: RECENT
THEORETICAL
AND EXPERIMENTAL
RESULTS
LHUILLIER
Laboratoire de Physique Theorique des Liquides, Universite Pierre et Marie Curie, 4 place Jussieu, 75252 Paris Cedex 05 and Christian LARAT, Michele LEDUC, Pierre-Jean NACHER, Genevieve TASTEVIN and GQrard VERMEULEN Laboratoire de Spectroscopic Hertzienne de 1'ENS (associ.8 au CNRS et a l'Universit8 Pierre et Marie Curie), 24 rue Lhomond, 75231 Paris Cedex 05
The transport properties of dilute spin polarized gases exhibit interesting dependences on the between colliding partipolarization M at low temperature, due to quantum indistinguishability cles. For the heat conduction K of polarized gaseous helium 3, recent calculations have checked the convergence of the moment expandonof the out of equilibrium density matrix. Four to five moments are necessary to achieve a 1% accuracy. At the same time the changes of K with M have been reliably measured in the 1.3-4.2 K range, with great care to control all the heat fluxes reaching the cell. The temperature dependence of the changes of K with M is now found in aatisfactory agreement with theory.
For dilute spin polarized gases, changes of the transport properties occur if the temperature is low enough, namely if the de Broglie wavelength is comparable to the range of the interactions between particles [II - This derives from various interference effects induced by quantum indistinguishability between colliding particles. Proper symmetrization of the wave functions leads to scattering cross-sections which depend on the spin polarization. The Boltzmann theory has to be reformulated to incorporate the Pauli principle [2]. This leads to quantitative results for all the transport coefficients and their low temperature variations as a function of the nuclear polariZatiOn M. For the heat conduction coefficient K of gaseous 3He , such variations were observed in 1987 [3]. They exhibit a strong and non intuitive dependence on temperature T. However the preliminary experimental results of [3] did not show a good quantitative agreement with theory [2]. This fact motivated us both to derive new considerations on the problem of quantum exchange in Boltzmann theory 143 and to push the previous experiment to a more advanced stage [5]. Improving the calculation of transport coefficients of polarized gases, we first examined the influence of the choice of the interatomic potentials, using a few of the more recently proposed ones. We then checked the oonvergence of the Sonine expansion commonly used for classical gases, but which may be questionable in the case Of spin polarized gases, because all
0921.4526/90/$03.50
@ 1990 - ElsevierScience
Publishers
the elements of the single particle density matrix do not relax towards equilibrium with the same collision cross-section. For the helium 3 gas at low temperature, the convergence of the expansion is rather rapid in the case of the coefficient K for the non polarized gas. On the other hand for the totally polarized gas the result is much more sensitive to the approximative form of the trial density distribution which is chosen. The reason is that the new effect is due to quantum interferences and its magnitude is the result of a summation of large terms with alternate signs: this cancellation is total at 300K, explaining interference effects do not show why in thermodynamical measurements at room temperature: it is accidental around 0.3 and 0.9K: in the temperature range 1K to 4K, K happens to increase with M. Even for slightly polarized SampleS in the range 1K to 4K, it is necessary to use as much as 3 or 4 Sonine polynomials to achieve 1% acFor l4 = 100% curacy. the first order approximation is off from the "exact" result by 20 to 25% and one has to use 4 to 5 terms in the Sonine expansion to reach the desired convergence. These results are shown in figure 2, where the dotted line corresponds to the predictions with the first approximation, the full line with the fifth These one. curves were calculated using the potential of reference [S]. Little differences were found with other potentials.
B.V. (North-Holland)
736
B. Brami, F. Joly, C. Lhuillier, C. Larat, M. Leduc, P.-J. Nacher, G. Tastevin, G. Vermeulen
The experimental values of 30%. [K(M)-K(O)]/M~ are plotted in figure 2. The uncertainty on the absolute value of Hz being large (but not on its relative value), a scaling factor for M was used to fit the theoretical curve at the arbitrarily chosen tempera ture of2X.
Thermal Cu shielding
A
Aensor
R1
i-
K(M)-M(O) M2 hW/K.m 1
f
NMR pick-up coil
I!-
FIGORE 1 Sketch
Of
the
measuring the as a function
experiment
llow pa*
of
the
heat conductivity of gaseous of spin polarization.
cell) 3Ee
As for the experiment here reported, its principle is similar to that described earlier [3], but the temperature
was
extended
down to
1.3X by
several improvements to the cryostat and to the 3We cell. Also greater attention was payed to the control of heat fluxes reaching the measurement cells, in order to avoid systematic errors on the derived value of the coefficient K. The polarization of the 3He gas is achieved by optical pumping with an arc lamp pumped WA laser. The low temperature part of the cell is shown in figure 1. The polarization of the measurement cylinder is monitored by NMR coils. Carbon resistors R, and measure the temperatures of the end p ates and electrical heating can be 9 applied to the bottom plate by means of a copper grid. The cell is shielded against radiation and spurious degasing by an aluminized coil foil screen. Additional experiments were performed to directly measure the thermal impedance of the Pyrex cell walls and substract their contribution to the measure of K. For the non polarized gas the measured values of K were in good agreement with those of reference [7]. For the polarized gas, IC was found to vary linearly with M2 at various temperatures, M ranging between 0 and
PIf3IIRB 2 2 Changes of the heat conduction coefficient of gaseous -Be with spin polariraticm U, plotted as a function of temperatux? T.
The comparison of the improved and now reliable experiments results with the new calculations is shown in figure 2. The agreement is satisfactory and even remarksbly good below 2X. The problem of heat conductivity changes in 3He with spin polarization can now be considered as well understood. 111 "Spin Polarized Quantum Systems", Proceedings of the I.S.I. Conference (Torino), Stringari Editor, World Scientific (1988). [2] C. Lbuillier and F. LaloG, J. Physique s (1982) 197 and 225; C. Ihuillier, J. Physique Q (1983) 1. [3] W. Leduc et al. Burophys. Letters 4 (1987) 59. [4] B. Brami et al. Physica (1990) to be published. [5] C. Larat et al., to be published in J. of Low Temp. Phys. [6] R.A. Aziz et al. J. Chem. Phys. x (1979) 4330. [7] D.S. Betts and J. Marshall, J. of Low Temp. Phys. ;5 (1969) 595.