Abstract
We report microscopic calculations of the thermal conductivity, diffusion constant, and thermal diffusion constant for classical solutions of \(^3\)He in superfluid \(^4\)He at temperatures \(T \lesssim 0.6\) K, where phonons are the dominant excitations of the \(^4\)He. We focus on solutions with \(^3\)He concentrations \(\lesssim \) \(10^{-3}\), for which the main scattering mechanisms are phonon–phonon scattering via 3-phonon Landau and Beliaev processes, which maintain the phonons in a drifting equilibrium distribution, and the slower process of \(^3\)He–phonon scattering, which is crucial for determining the \(^3\)He distribution function in transport. We use the fact that the relative changes in the energy and momentum of a \(^3\)He atom in a collision with a phonon are small to derive a Fokker–Planck equation for the \(^3\)He distribution function, which we show has an analytical solution in terms of Sonine polynomials. We also calculate the corrections to the Fokker–Planck results for the transport coefficients.
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Notes
The expansion in the parameter, \(k/\sqrt{m^*T}\) is essentially one in small angle scattering. The \(^3\)He–phonon collision term is thus similar to a Fokker–Planck collision term, to which Sonine polynomials have previous been applied.
Here, and in experiments determining heat transport in superfluid helium, the \(\varvec{v}_{ph} -\varvec{v}_s\) achieved depends on the ambient temperature gradient.
Compare with Eq. (59.6) of Ref. [12], with the identifications \(\varvec{i} = m_3n_3(1-y)\varvec{u}\) and \(\varvec{q} - \mu \varvec{i} = \varvec{Q}_3 -T\sigma \varvec{u}\).
This uncertainty is directly reflected in an uncertainty in the calculated diffusion constant of 25 %.
In Ref. [7] we wrote the effective \(^3\)He–phonon scattering rate as \(\varGamma /p^2\); we emphasize that when the energy transfer in \(^3\)He–phonon collisions is taken into account, the scattering rate is independent of \(p\) and has the value \(\varGamma /3m^*T\), which is the value of \(\varGamma /p^2\) for \(p^2\) replaced by its thermal average \(3m^*T\).
Three-phonon Landau damping and the Beliaev process, although they involve phonons alone, can affect the rate at which momentum is transferred from phonons to \(^3\)He. These processes conserve the total momentum and energy flux of the phonons. However, the cross section for scattering of phonons by \(^3\)He atoms is strongly dependent on the phonon momentum and therefore the total rate at which momentum is transferred from phonons to \(^3\)He depends on the details of the phonon distribution, not just the total momentum of the phonons.
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Acknowledgments
This research was supported in part by NSF Grants PHY08-55569, PHY09-69790, PHY-1205671, and PHY13-05891. Author GB is grateful to the Aspen Center for Physics, supported in part by NSF Grant PHY-1066292, and the Niels Bohr International Academy, where parts of this research were carried out.
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Baym, G., Beck, D.H. & Pethick, C.J. Low-Temperature Transport Properties of Very Dilute Classical Solutions of \(^3\)He in Superfluid \(^4\)He. J Low Temp Phys 178, 200–228 (2015). https://doi.org/10.1007/s10909-014-1235-0
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DOI: https://doi.org/10.1007/s10909-014-1235-0