Mobility of an Impurity in a Fermi Liquid

Abstract

The mobility of ions in liquid He 3 is investigated by a method which avoids the assumption that ions recoil freely in collisions with Fermi quasiparticles. The temperature variation of the mobility is found to be much less than that previously predicted, and improved agreement with experiment is obtained.The damping force experienced by an impurity of atomic dimensions moving through a Fermi liquid (e.g., ions in He 3 under the influence of an electric field) is mainly due to collisions with quasiparticles if the temperature is sufficiently low. Previous theories of the mobility in this regime 1 " 3 involve a characteristic temperature T 0 = k F 2 /2Mk B (k F , M, and k B are the Fermi momentum of the liquid, the impurity effective mass, and Boltzmann's constant, respectively, and #=1), at which the mobility temperature dependence is predicted to change from T~2 at low temperatures to approximately T° at higher temperatures. This prediction is not supported by experiment. 4 With negative ions in He 3 the mobility at low temperatures (below |T 0 in the most favorable case) is only weakly temperature dependent, while with positive ions the low-temperature behavior (down to ~iT 0 ) resembles T~1 /3 rather than T~2. The above theories treat the recoil of the impurity during a collision by ascribing to it a definite effective mass. In the following, we describe a theory in which the recoil is treated in a more fundamental manner in terms of the Van Hove scattering function, and the discrepancy between theory and experiment referred to above is largely resolved.If the impurity were rigid and fixed in position, all scattering from it would be elastic and governed by a differential cross section cr(0). (The Fermi velocity is assumed to be large compared with all other relevant velocities, so that only the scattering angle 6 enters.) In fact the impurity is not fixed and the scattering is inelastic. We write the inelastic scattering cross section per unit solid angle as a(0)S^(K, co), where K and u) are the momentum and energy transfer, respectively, and v is the mean drift velocity of the impurity (under the influence of a fixed external force). S^(K, o>) is taken to be the usual scattering function 5 ' 6 which enters into the analogous problem of inelastic neutron scattering, 7 and gives the spectrum of the energy absorbed when the impurity is instantaneously given momentum K. With these assumptions, the rate at which momentum is transferred to the quasiparticles has the form

Mobility tensor of negative ions in superfluid 3He-A

“…On insertion into (11), this would result in a divergent elastic scattering cross section for the low-energy excitations. The reason for the breakdown of such perturbative approximation schemes in the superfluid can be understood as follows: due to the divergent density of excitations in the superfluid for energies approaching the gap edge, no process which involves only a finite number of intermediate normal-state quasiparticle-quasihole pairs is meaningful; only the sum total yields a convergent result.…”

“…We shall first discuss the differential scattering cross section, which was expressed in Eq. (11) in terms of (It~,~[2>, the branch-averaged squared scattering matrix element (12).…”

Superdiffusive nonequilibrium motion of an impurity in a Fermi sea

“…As pointed out in Ref. [23] in the context of heavy ion transport in He 3 , if the definite effective mass assumption is not satisfied, the temperature dependence of transport coefficients can be significantly modified. We can understand this from the observation that the mass dependence of the scattering time is τ ↓ ∼ m/m * .…”

Measurement of Ionic Mobilities in Liquid 3He by a Space Charge Method

“…3 These predictions were not borne out by experiment. 4 ,5 In a more recent theoretical investigation Josephson and Lekner 6 show that, although the T-2 law may still be expected at sufficiently low temperatures, some form of weaker dependence should be observed at higher temperatures. Since there seems to be little consensus as to the form of this weaker dependence or as to the characteristic temperature at which one regime might give way to the other, it is important that accurate experimental mobility data should be obtained.…”