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Physical Review B

Theoretical modeling of ion mobility in superfluid 4 He

Steven L. Fiedler, David Mateo, Tatevik Aleksanyan, Jussi Eloranta

Published Oct 19, 2012• 26 citations
Publisher

Abstract

A method for calculating hydrodynamic added mass within the framework of time-dependent bosonic density functional theory (DFT) for superfluid ${}^{4}$He is developed. As a calibration of the model, it is shown to reproduce the classical hydrodynamic limit for purely repulsive interactions. To model real systems for which experimental data are available, the following ions were considered: Be${}^{+}$, K${}^{+}$, Ca${}^{+}$, Sr${}^{+}$, and Ba${}^{+}$ cations as well as the F${}^{\ensuremath{-}}$, Cl${}^{\ensuremath{-}}$, I${}^{\ensuremath{-}}$, and Br${}^{\ensuremath{-}}$ anions. The DFT model requires the ion-helium pair potential data as input, which were obtained from electron structure calculations by employing coupled clusters theory. The resultant static liquid density profiles as calculated by DFT were found to be in good agreement with previously published quantum Monte Carlo data. The calculated added masses for the positive ions correlated directly with the experimentally observed ion mobility data, by which the ions could be separated into two different categories based on the degree of the first solvent shell following the ion. The calculated added masses for the negative ions were found to be in disagreement with the existing experimental data, suggesting the possibility that other negatively charged species were observed in previous experiments. The negatively charged ions are predicted to have mobilities ($\ensuremath{\mu}$) within the range 0.8--1.0 cm${}^{2}$ V${}^{\ensuremath{-}1}$ s${}^{\ensuremath{-}1}$ in superfluid helium at 1.3 K with the order $\ensuremath{\mu}$(I${}^{\ensuremath{-}}$) $>$ $\ensuremath{\mu}$(Br${}^{\ensuremath{-}}$) $>$ $\ensuremath{\mu}$(Cl${}^{\ensuremath{-}}$) $>$ $\ensuremath{\mu}$(F${}^{\ensuremath{-}}$).

DOI
10.1103/physrevb.86.144522

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This article reviews recent experimental and theoretical work on the mobility of negative ions in the superfluid A and B phases of liquid 3He. In the normal Fermi liquid at temperatures below ∼ 50 mK and also in the superfluid close to the superfluid transition temperature, Tc , the mobility of a negative ion may simply be considered as limited by the elastic scattering of 3He quasiparticles. This explains the constancy of the ion mobility in the normal phase. However, underlying the rapid increase of the measured mobility in the superfluid phases there is a subtle quantum-mechanical scattering effect. Detailed solutions of the 3He quasiparticle—negative ion scattering process in the pair-correlated state provide a simple physical picture of an energy-dependent forward-peaking phenomenon. This yields quantitative theoretical results for the ion mobility in the quasi-isotropic B phase and for the ion mobility tensor in the anisotropic A phase which agree with the experimental data.
We calculate the roton-limited mobility of ions in superfluid He4{\mathrm{He}}^{4}He4 for temperatures below \ensuremath{\sim} 1.7\ifmmode^\circ\else\textdegree\fi{}K. Taking, as a first approximation, a constant ion-roton transition-matrix element, which we treat as an adjustable parameter, we find good agreement with experiment for both positive and negative ions. This model calculation indicates that the most mobile of the recently discovered "exotic" negative carriers is a very light ion.
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The crossover from ideal-Bose-gas behavior to nonclassical, XY-like criticality in a dilute, interacting Bose fluid is considered in detail with emphasis on the superfluid density, ${\ensuremath{\rho}}_{s}$(T), below the transition. A general discussion of crossover scaling in constrained systems shows that ${\ensuremath{\rho}}_{s}$ and other temperature-dependent thermodynamic functions at controlled overall density can be expressed in terms of universal scaling functions. The finite-temperature Green's function formalism is used to calculate the associated linear scaling fields and the crossover exponent exactly for all dimensionalities d>2, scaling predictions, including logarithmic singularities just at d=3, being checked via second-order computations. A mapping of the Bose problem onto the classical s4{s}^{4}s4 spin model is achieved which is asymptotically exact in the critical region. In this way, previously known renormalization-group results are used to calculate the scaling functions for ${\ensuremath{\rho}}_{s}$, for the off-diagonal susceptibility, etc., to first order in \ensuremath{\epsilon}=4-d. Good agreement is found with the experimental measurements by Reppy and co-workers of ${\ensuremath{\rho}}_{s}$(T) for helium absorbed at low coverages in Vycor glass (of pore size 50--80 A\r{}). An effective-mass ratio m∗{m}^{\mathrm{*}}m∗/m\ensuremath{\simeq}1.5, and an effective-scattering-length ratio a∗{a}^{\mathrm{*}}a∗/a\ensuremath{\simeq}1.3 are indicated for mobile helium in the Vycor (although the irregular character of Vycor has not, at this point, been fully allowed for in the theory). Other approaches in the literature to computing the critical behavior of quantum-mechanical systems are reviewed and related to the present analysis.
We present a simple numerical procedure for calculating the irrotational hydrodynamic flow around a spherical solute in superfluid helium. The results for alkali cations are compared to recent many-body variational Monte Carlo (VMC) calculations. The helium contribution to the effective masses calculated by the VMC and hydrodynamic methods are 12.9(4.6) versus 9.4 u for Li+, 48.2(5.6) versus 52.1 u for Na+, 69.6(4.8) versus 70.1 u for K+, and 6.4(8.8) versus 6.8 for Cs+. In all cases, the agreement of the two results are in within the error estimate of the VMC calculation, demonstrating the accuracy of hydrodynamic treatment of helium motion even on the atomic length scale.