In 1969 Josephson and Lekner showed that the mobility of ions in the degenerate Fermi liquid 3^3He is far more weakly temperature-dependent at low temperatures than effective-mass theories predicted, by describing the ion dynamics through the Van Hove scattering function instead of free recoil, and they left three linked questions open: the self-consistent solution of their closed set of mobility equations, a microscopic description of the frequency-dependent mobility μ(ω)\mu(\omega) for positive and negative ions, and the ultimate low-temperature behavior of the static mobility (they conjectured a possible reversion to the T−2T^{-2} law at microkelvin temperatures). The open core is a parameter-free microscopic theory of μ(ω)\mu(\omega) for both ion species and a controlled determination of the asymptotic low-temperature law.

Background

Liquid helium-3 (3^3He) is a Fermi liquid: below about 0.1 K its atoms, which obey Fermi-Dirac statistics, form a degenerate Fermi system whose low-energy excitations are long-lived quasiparticles, each carrying the mass, spin, and charge quantum numbers of a 3^3He atom. An electrically charged impurity can be injected into the liquid and dragged by an electric field, and its drift velocity per unit field defines the mobility μ\mu. A positive ion (a bare He2+_2^+ seed that accretes a solid-like icosahedral shell of helium atoms, the so-called snowball, effective mass of order tens of 3^3He masses) and a negative ion (an electron that excavates a spherical cavity, the electron bubble, of mass of order one hundred 3^3He masses) probe the quasiparticle gas as heavy mesoscopic scatterers. Because the host is a Fermi liquid, the ion mobility at low temperature is governed by the scattering of quasiparticles off the ion, restricted by the Pauli principle (transitions only into unoccupied states near the Fermi surface) and by energy conservation in the ion-quasiparticle collision.

Effective-mass theories (Abe and Aizu 1961; Clark 1963; Schappert 1968) treated the ion as a free particle of effective mass MM undergoing elastic binary collisions with quasiparticles. They predicted that below the recoil temperature T0=ℏ2kF2/(2MkB)T_0=\hbar^2 k_F^2/(2Mk_B), where kFk_F is the Fermi wave number and kBk_B the Boltzmann constant, the mobility should grow as T−2T^{-2}: energy conservation strongly suppresses large-angle scattering once the recoil energy transferred to the ion exceeds the thermal energy. (The Abe and Aizu paper is primarily a Boltzmann-equation calculation for ions in a Bose gas, liquid 4^4He; its closing section extends the same treatment to free Fermi particles and obtains the μ∝T−2\mu\propto T^{-2} law that Josephson and Lekner cite it for.) Experiments (Anderson, Kuchnir, and Wheatley 1968) instead found a much weaker temperature dependence. Gould and Ma (1968) computed the drag on a large hard-sphere impurity in an interacting Fermi liquid and found 1/μ=A+BT2ln⁡(1/T)+O(T2)1/\mu = A + B T^2\ln(1/T) + O(T^2), a saturating inverse mobility.

Josephson and Lekner (1969) attacked the problem without assuming that the ion recoils freely in collisions. They expressed the mobility through the Van Hove scattering function, which encodes the spatio-temporal correlations of the fluid, and specifically through the self scattering function F(K,t)F(K,t) of the ion, i.e. the probability that the ion propagates by momentum ℏK\hbar K over the time tt; the fluctuation-dissipation theorem (the general linear-response relation linking fluctuations to dissipation) connects this to a frequency-dependent mobility μ(ω)\mu(\omega). (They define the mobility as drift velocity per unit force rather than per unit electric field; the two conventions differ by the ion charge ee.) In a Gaussian approximation F(K,t)=exp⁡[−K2γ(t)/2]F(K,t)=\exp[-K^2\gamma(t)/2] with a kernel γ(t)\gamma(t) built from μ(ω)\mu(\omega), they obtained a closed set of equations (their Eqs. (2)-(4)) for the static mobility μs\mu_s, provided μ(ω)\mu(\omega) is characterized by the single relaxation time μsM\mu_s M, corresponding to the frequency (μsM)−1≈1010 s−1(\mu_s M)^{-1}\approx 10^{10}\,\mathrm{s}^{-1}. They did not solve these equations self-consistently; an upper-limit estimate gave a small negative temperature exponent for negative ions (magnitude below 0.1, versus the value near 2 predicted by the effective-mass theories). They noted that terms of order T2T^2 dominate the temperature coefficient in the experimental range, that the positive-ion assumptions are already violated between 30 and 100 mK, and that at still lower temperatures, of the order of microdegrees, the derivation assumptions break down and the mobility "may well be T−2T^{-2} in agreement with" the earlier theories. A fully microscopic description of μ(ω)\mu(\omega) for both ion signs was left open. (Note that 3^3He itself becomes superfluid below Tc≈1T_c\approx 1-2.52.5 mK, a transition discovered in 1971-72, two years after the paper, so the microkelvin regime it conjectured about lies inside the superfluid phase; the ultimate low-temperature behavior of the normal state can therefore be established only as a theoretical continuation below TcT_c, untestable by normal-phase experiment.) Subsequent measurements established that the negative-ion mobility in the normal phase is essentially temperature independent, μN≈1.7×10−6 m2/Vs\mu_N\approx 1.7\times 10^{-6}\,\mathrm{m^2/Vs} from just above the superfluid transition up to about 30 mK (Shevtsov and Sauls 2017); this observed plateau is the empirical context for any microscopic theory of μ(ω)\mu(\omega).

Problem Statement

For the charged ions in normal, degenerate liquid 3^3He — the positive ion (a solid-like helium cluster, the snowball) and the negative ion (an electron trapped in a spherical cavity, the electron bubble) — derive the frequency-dependent mobility μ(ω)\mu(\omega) and the static mobility μs(T)\mu_s(T) microscopically for both ion species, whose effective masses and microscopic structures differ, from the quasiparticle-ion interaction and the Fermi-liquid dynamics of the host, going beyond the single-relaxation-time characterization of μ(ω)\mu(\omega) assumed in 1969, and treating the ion's own diffusive motion self-consistently as in the Josephson-Lekner program (their Eqs. (2)-(4) close the mobility through the ion self scattering function F(K,t)F(K,t), the probability that the ion propagates by momentum ℏK\hbar K over a time tt, once μ(ω)\mu(\omega) is known). Determine the sign and magnitude of the temperature coefficient through the region between the recoil temperature T0=ℏ2kF2/(2MkB)T_0=\hbar^2 k_F^2/(2Mk_B) (where kFk_F is the Fermi wave number of the liquid, MM the ion effective mass, and kBk_B Boltzmann's constant) and the superfluid transition temperature Tc≈1T_c\approx 1-2.52.5 mK, where terms of order T2T^2 and the positive-ion anomalies enter, and settle the ultimate low-temperature behavior of the normal state continued below TcT_c (a theoretical question, since the real liquid becomes superfluid there): whether the mobility would revert to the effective-mass T−2T^{-2} law at the lowest temperatures, as conjectured by Josephson and Lekner for the microkelvin range, remains finite with weak temperature dependence as observed down to about 1 mK, or crosses over to a free-propagation law μ∝1/T\mu\propto 1/T as predicted by mode-coupling theory (a memory-function treatment expressing the ion's friction kernel through coupled density and current fluctuations of the liquid), together with the crossover temperature scales.

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