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Longitudinal Relaxation Time in a Degenerate, Polarized Fermi Gas

Erik D. Nelson

Published Jan 1, 1993• 0 citations
OpenAlex

Abstract

The longitudinal relaxation time T$\\sb1$ is calculated for a dilute Fermi gas. We include the possibility of arbitrary degrees of polarization P, and uniform external magnetic fields. At low temperatures T$\\sb1$ is proportional to T$\\sp{-2}$ for the unpolarized gas in agreement with earlier calculations. For a polarized system in zero applied field the separation between up and down Fermi spheres provides an open channel in phase space for spin flipping processes. As temperature is reduced to zero T$\\sb1$ levels off and approaches a constant value which depends sensitively on P, leading to much smaller values of T$\\sb1$ than in the unpolarized case. For relaxation in a uniform external field H, Zeeman energy conservation during spin flipping interactions essentially shifts the location of the T$\\sp{-2}$ behavior from zero polarization to the polarization P(H) which would be produced in equilibrium with the external field H. When the initial polarization of the system is different from P(H), T$\\sb1$ again approaches a constant value which is equally sensitive to the initial polarization as temperature is reduced to zero. We apply our calculations to the case of dilute $\\sp3$He-HeII solutions. We also consider the calculation of density dependent corrections to T$\\sb1$ for unpolarized, or weakly polarized $\\sp3$He-HeII using the long wavelength limit of the Bardeen-Baym-Pines interaction as a form for the spin independent part of the $\\sp3$He-$\\sp3$He interaction. Owing to the positive sign of the antisymmetric Landau parameter $\\rm F\\sbsp{o}{a}$ in dilute $\\sp3$He-HeII Fermi Liquid interactions act to screen the magnetic dipole interactions which cause relaxation. To lowest order in density we find that the relaxation rate 1/T$\\sb1$ is reduced by factor $\\rm (1 - 3F\\sbsp{o}{a})$ which compares favorably with recent calculations of T$\\sb1$ for pure $\\sp3$He.

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The longitudinal relaxation time Tsb1\\sb1sb1 is calculated for a dilute Fermi gas. We include the possibility of arbitrary degrees of polarization P, and uniform external magnetic fields. At low temperatures Tsb1\\sb1sb1 is proportional to Tsp−2\\sp{-2}sp−2 for the unpolarized gas in agreement with earlier calculations. For a polarized system in zero applied field the separation between up and down Fermi spheres provides an open channel in phase space for spin flipping processes. As temperature is reduced to zero Tsb1\\sb1sb1 levels off and approaches a constant value which depends sensitively on P, leading to much smaller values of Tsb1\\sb1sb1 than in the unpolarized case. For relaxation in a uniform external field H, Zeeman energy conservation during spin flipping interactions essentially shifts the location of the Tsp−2\\sp{-2}sp−2 behavior from zero polarization to the polarization P(H) which would be produced in equilibrium with the external field H. When the initial polarization of the system is different from P(H), Tsb1\\sb1sb1 again approaches a constant value which is equally sensitive to the initial polarization as temperature is reduced to zero. We apply our calculations to the case of dilute sp3\\sp3sp3He-HeII solutions. We also consider the calculation of density dependent corrections to Tsb1\\sb1sb1 for unpolarized, or weakly polarized sp3\\sp3sp3He-HeII using the long wavelength limit of the Bardeen-Baym-Pines interaction as a form for the spin independent part of the sp3\\sp3sp3He-sp3\\sp3sp3He interaction. Owing to the positive sign of the antisymmetric Landau parameter rmFsbspoa\\rm F\\sbsp{o}{a}rmFsbspoa in dilute sp3\\sp3sp3He-HeII Fermi Liquid interactions act to screen the magnetic dipole interactions which cause relaxation. To lowest order in density we find that the relaxation rate 1/Tsb1\\sb1sb1 is reduced by factor rm(1−3Fsbspoa)\\rm (1 - 3F\\sbsp{o}{a})rm(1−3Fsbspoa) which compares favorably with recent calculations of Tsb1\\sb1sb1 for pure sp3\\sp3sp3He.
We present a finite temperature calculation of the transverse spin-diffusion coefficient ${D}_{\ensuremath{\perp}}$ in a dilute degenerate Fermi gas, in the presence of a small external magnetic field HHH. While the longitudinal diffusion coefficient displays the conventional Fermi-liquid dependence, ${D}_{\ensuremath{\parallel}}\ensuremath{\propto}{T}^{\ensuremath{-}2}$, the behavior of ${D}_{\ensuremath{\perp}}$ shows three separate regimes: (a) ${D}_{\ensuremath{\perp}}\ensuremath{\sim}{H}^{\ensuremath{-}2}$ for $T\ensuremath{\ll}H$, (b) ${D}_{\ensuremath{\perp}}\ensuremath{\sim}{T}^{\ensuremath{-}2}$, $\frac{{D}_{\ensuremath{\perp}}}{{D}_{\ensuremath{\parallel}}}\ensuremath{\ne}1$ for $T\ensuremath{\gg}H$ and large spin-rotation parameter $\ensuremath{\xi}\ensuremath{\gg}1$, and (c) ${D}_{\ensuremath{\perp}}={D}_{\ensuremath{\parallel}}\ensuremath{\propto}{T}^{\ensuremath{-}2}$ for $T\ensuremath{\gg}H$ and $\ensuremath{\xi}\ensuremath{\ll}1$. Our results are qualitatively consistent with the experimental data in weakly spin-polarized 3He^{3}\mathrm{He}3He and 3He^{3}\mathrm{He}3He- 4He^{4}\mathrm{He}4He mixtures.
A calculation is presented of the relaxation rate ${{T}_{1}}^{\ensuremath{-}1}$ between the lowest two hyperfine states of a dilute gas of H atoms in high magnetic field at low temperatures. Dipole-dipole interactions dominate ${{T}_{1}}^{\ensuremath{-}1}$, leading to long relaxation times at low temperatures (${{T}_{1}}^{\ensuremath{-}1}\ensuremath{\approx}{10}^{\ensuremath{-}2}{n}_{\mathrm{H}}{T}^{\frac{1}{2}}$ ${\mathrm{sec}}^{\ensuremath{-}1}$ at $H\ensuremath{\approx}100$ kG). The recombination rate due to wall collisions is much greater than ${{T}_{1}}^{\ensuremath{-}1}$ and can lead to a depletion of the lowest hyperfine state from thermal equilibrium and an effective recombination rate equal to ${{T}_{1}}^{\ensuremath{-}1}$.
We compute spin diffusion in a dilute Fermi gas at arbitrary temperature, polarization, and strong interaction in the normal phase using kinetic theory. While the longitudinal spin diffusivity ${D}_{\ensuremath{\parallel}}$ depends weakly on polarization and diverges for small temperatures, the transverse spin diffusivity ${D}_{\ensuremath{\perp}}$ has a strong polarization dependence and approaches a finite value for $T\ensuremath{\rightarrow}0$ in the Fermi liquid phase. For a 3D unitary Fermi gas at infinite scattering length, the diffusivities reach a minimum near the quantum limit of diffusion $\ensuremath{\hbar}/m$ in the quantum degenerate regime and are strongly suppressed by medium scattering, and we discuss the importance of the spin-rotation effect. In two dimensions, ${D}_{\ensuremath{\perp}}$ attains a minimum at strong coupling $\ensuremath{-}1\ensuremath{\lesssim}\mathrm{ln}({k}_{F}{a}_{2D})\ensuremath{\lesssim}1$ and reaches ${D}_{\ensuremath{\perp}}\ensuremath{\sim}0.2...\phantom{\rule{0.16em}{0ex}}0.3\phantom{\rule{0.16em}{0ex}}\ensuremath{\hbar}/m$ at large polarization. These values are consistent with recent measurements of two-dimensional ultracold atomic gases in the strong coupling regime.
The aim of this work is to study a dilute Fermi gas, whether it has a spin 1/2 or it has a high degeneracy. The starting point is to calculate the energy of the system. However, as we are dealing with a many body system, a perturbation series method will be applied. R. F. Bishop in reference [1] wrote the terms of the expansion up to fourth order in the gas parameter, which is defined as the Fermi momentum times the S-wave scattering length, but only for the case of a non-polarized gas. What we do in this present work is the generalization of that expansion up to third order including any possible polarization. The universal expansion, that is up to second order, have been computed analytically. For the third order terms was not possible, but thanks to numerical calculation we have obtained the energy for different polarizations. Therefore, this work will contain a combination of both analytical and numerical calculus. The final expression obtained for the energy depends on the polarization, the gas parameter and the degeneracy and stands for low density gases. Afterwards, thanks to knowing the relation between the energy and the polarization, the study of the ferromagnetic phase transition is presented. We see how the transition point changes due to the degeneracy and what kind of phase transition we obtain. For spin 1/2 a continuous transition is observed, while a discontinuous transition happens for higher degeneracy. To complement the phase transition study, different physical properties have been computed, such as, the magnetic susceptibility, and the Tan?s constant. Our study is extended to highly degenerate Fermi gases which are now realized in cold Fermi labs.
A kinetic equation, which includes the effects of degeneracy, is derived for dilute, polarized systems by the Green's function method of Kadanoff and Baym. When the Born approximation is used for the self-energy, the equation reduces to a result due to Silin. In the Boltzmann limit our result is equivalent to the equation of Lhuillier and Laloë, with the addition of a mean-field drift term analogous to that appearing in the Landau-Silin equation. Our kinetic equation is used to derive an expression for the transverse spin-diffusion relaxation time, τ>, for a Fermi system. In the Boltzmann and low-polarization limits τ> reduces to τ ∥, the longitudinal relaxation time. However, in a highly polarized degenerate system τ> can be very much shorter than τ ∥.
The linear equations for transverse spin dynamics in a weakly polarized degenerate Fermi liquid with arbitrary relationship between temperature T and polarization $\ensuremath{\gamma}H$ are derived from Landau-Silin phenomenological kinetic equation with general form of two-particle collision integral. The temperature and polarization dependence of the spin current relaxation time is established. It is found in particular that at finite polarization transverse spin wave damping has a finite value at T=0.T=0.T=0. The analogy between temperature dependences of spin wave attenuation and ultrasound absorption in degenerate Fermi liquid at arbitrary temperature is presented. We also discuss spin-polarized Fermi-liquid in the general context of the Fermi-liquid theory and compare it with ``Fermi liquid'' with spontaneous magnetization.