Sound Propagation and Transport Properties of Liquid ³He in Aerogel

Superfluid ³He confined in aerogel offers a unique chance to study the effects of a short mean free path on the properties of a well defined superfluid Fermi liquid with anisotropic pairing. Transport coefficients and collective excitations, e.g. longitudinal sound, are expected to react sensitively to a short mean free path and to offer the possibility for testing recently developed models for quasiparticle scattering at aerogel strands. Sound experiments, together with a theoretical analysis based on Fermi liquid theory for systems with short mean free paths, should give valuable insights into the interaction between superfluid ³He and aerogel.

A model for liquid ³He in aerogel based on a random distribution of short-ranged potentials acting on ³He quasiparticles has been shown to account semi-quantitatively for the reduction of the transition temperature, T_c, and the suppression of the superfluid density, r_s(T).[1] Although the order parameter for ³He in aerogel is not firmly identified, measurements of the magnetization indicate that the low pressure phase is an ESP state.[2] A transverse NMR shift is observed and is roughly consistent with that of an axial state even though the B-phase is stable in pure ³He at these pressures. The addition of a small concentration of ⁴He, which coats the aerogel strands, induces a suppression of the magnetization for T < T_c^aerogel, indicative of a non-ESP state like the BW phase.[2] Thus, the stability of the superfluid state of ³He in aerogel is quite sensitive to the detailed interaction between ³He and the aerogel strands. Calculations of the free energy for ³He in aerogel which include anisotropic and magnetic scattering, and the effects of orientational correlations of the aerogel strands, confirm the sensitivity of the superfluid phases to the interaction between ³He and the aerogel. [1,3]

The high porosity of the aerogel implies that the silica structure does not significantly modify the bulk properties of normal ³He. The dominant effect of the aerogel structure is to scatter ³He quasiparticles moving at the bulk Fermi velocity. If the coherence length, x₀ = (^h/_2p) v_f/2 pT_c0, is sufficiently long compared to the average distance between scattering centers then a reasonable starting point is to treat the aerogel as a homogenous scattering medium (HSM) described by a mean-free path l.[1] In this article we discuss the effects of a short mean free path on some of the transport properties of liquid ³He. The calculations presented below for sound propagation assume the BW phase is stable; however, the propagation and damping of low-frequency sound are expected to be qualitatively similar for other phases.
 
 
 

The superfluid transition temperature for ³He in aerogel is suppressed by quasiparticle scattering off the aerogel structure. In the HSM model the suppression of T_c is given by the Abrikosov-Gorkov formula,

The transport properties of ³He should also be strongly affected by scattering from the aerogel. In the normal state the quasiparticle distribution function, f([^(p)],R;e,t), satisfies the Boltzmann-Landau transport equation,

At low frequencies a compressible aerogel will move nearly in phase with the ³He density and longitudinal current mode; sound propagates but it is damped by the viscous coupling of the ³He to the aerogel, a₁ = [(w²)/(rc₁³)]h, where c₁ is the hydrodyanmic sound velocity and r is the mass density of ³He. The viscous damping of hydrodynamic sound saturates for T < T^* at a₁/q @ [(3)/(5)](wt_el)/(1+F^s₀). At higher frequencies the impedance mismatch between ³He and the aerogel sound mode leads to an increasingly out-of-phase motion of ³He excitations and aerogel. Hydrodynamic sound may become overdamped and reemerge as a diffusive mode. The frequency at which the cross-over from damped hydrodynamic sound to an overdamped diffusive mode occurs depends on the elastic compliance of the aerogel and the microscopic details of the coupling between ³He and the aerogel strands. Here we assume the frequency is above this cross-over, in which case the hydrodynamic mode is an overdamped diffusive mode, w = -iD_s q², where D_s = c₁² t is the acoustic diffusion constant with 1/t = (1+F^s₁/3)/t_el. At still higher frequencies, wt >> 1, zero sound can propagate, albeit with relatively high attenuation, a₀ @ 1/c₀t ~ 10⁴ cm^-1 at p = 10 bar. The regime for propagating zero sound is also pushed to higher frequencies, w/2p > v_f/2pl @ 42 MHz at p = 10 bar.
 
 
 

The collective mode spectrum of superfluid ³He in aerogel is also expected to show significant changes compared to bulk ³He. [6,7,8] A typical example is low frequency sound in superfluid ³He (w << D/(^h/_2p)). In bulk ³He one has collisionless zero sound and hydrodynamic first sound with nearly the same velocities, c²₀ » c²₁ = ¹/₃(1+F^s₀)(1+¹/₃F₁^s)v_f², and small damping by quasiparticle-quasiparticle scattering (for reviews on sound and collective modes in ³He see e.g. Refs. ). In aerogel, on the other hand, one expects a behavior which is similar to sound in ³He confined to small channels.[12] Sound will be weakly damped for ql >> 1, ceases to be a well defined mode for ql » 1, and reappears for ql << 1 as fourth sound with a temperature dependent velocity, c₄ @ Ö{r_s(T)/r} c₁. Fig. 2 shows the effects of elastic scattering on the superfluid density.[13] Note the reduction in r_s/r at T = 0. For a mean-free path of l = 1800 Å less than 50 % of the ³He mass density contributes to r_s(T = 0). The inset shows the fourth sound velocity neglecting damping.

Measurements of sound propagation in ³He-aerogel should provide a sensitive test of the HSM model. We study the spectrum of longitudinal sound in this model by calculating the linear response of the ³He density, r(q;w), to a driving force of wavevector q and frequency w. Our driving force will be a scalar field described by a potential, dU_ext(q;w), which couples to the ³He density. More realistic ``experimental driving forces'' require more elaborate calculations, which are not called for given the present experimental status. The calculation follows the quasiclassical version[14] of the method of Betbeder-Matibet and Nozières.[15] In the low frequency limit one has to solve Boltzmann-Landau transport equations for the branches of particle-like and hole-like excitations with distribution functions df_B1,B2([^(p)],R;e,t). We keep the dominant Landau parameters, F₀^s and F₁^s, and obtain an effective scalar potential, d[u\tilde](q;w), and longitudinal vector potential, d[a\tilde](q;w) = v_f·dA(q;w). The collision terms in the HMS model have the form, I_B1,B2 = -[1/(t(e))]( df_B1,B2 - ádf_B1,B2 ñ_FS), where t(e) = l/v(e), and v(e) = v_fÖ{e²- \midD\mid²}/e is the energy dependent quasiparticle velocity in the superfluid state. In the low frequency limit the transport equations have to be supplemented by Landau's self-consistency equations for the effective potentials, d[u\tilde] and d[a\tilde], and the particle conservation law, [(r)\dot]+Ñ·j = 0. In this limit we can ignore the less important self-consistency equation for the amplitude, d\midD\mid, of the order parameter. The five coupled linear equations for the distribution functions, the effective potentials, and the phase, dY(q;w), of the order parameter can be solved, and will be described elsewhere.

The calculated crossover from weakly damped zero sound to weakly damped fourth sound is shown in Fig. 3. We display the frequency dependent spectral function, -Ámc(q,w), for various wavelengths at fixed temperature and elastic scattering time. One can see clearly the transition from zero sound at v_fq >> 1/t_el to fourth sound at v_fq << 1/t_el. Fig. 4 shows the real and imaginary parts of the response function, c(q,w), at various temperatures and fixed t_el. Because of the short mean free path, ql = 0.1, the zero sound resonance with wavevector q is overdamped in the normal state and just below T_c. The damping decreases exponentially in the superfluid state for T << D because of the freezing out of thermally excited quasiparticles; and, as can be seen from Fig. 4, a well defined zero sound mode develops.

We thank the Alexander von Humboldt-Stiftung, the Deutsche Forschungsgemeinschaft (SFB279) and the STC for Superconductivity (NSF 91-20000) for their support.

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