Second Sound Velocity Research Articles

Hydrodynamic heat transport in dielectric crystals in the collective limit and the drifting/driftless velocity conundrum

We apply a recently developed method for solving the linearized phonon Boltzmann equation to study the hydrodynamic thermal transport in dielectrics in the collective limit, i.e., when normal collisions dominate resistive ones. The method recovers Guyer and Krumhansl results for a single Debye branch and extends them to general dispersion relations and branches. Specifically, we obtain explicit microscopic expressions for the phonon distribution and for the transport coefficients in this limit. We find that the phonon distribution differs from the commonly used displaced distribution in two terms: one accounting for viscous flow and another one which allows us to solve a long-standing issue on drifting and driftless second-sound velocities. Thus, the new method allows us to generalize previous results and fill some gaps on fundamental aspects of the collective limit through a simple mathematical formalism. We compare the hydrodynamic framework with previous models and discuss its limitations.

Experiments on the rapid mechanical expansion of liquidHe4through its superfluid transition

Phenomena following a rapid mechanical quench of liquid 4He from its normal to its superfluid phase are reported and discussed. The mechanical expansion apparatus is an improved version of that described previously. It uses a double-cell geometry to effect a partial separation of the sample from the convolutions of the bellows that form the outer wall of the cell. Consistent with earlier work, no evidence is found for the production of quantized vortices via the Kibble-Zurek (KZ) mechanism. Although the expansion is complete within 15ms , the second-sound velocity and attenuation continue to increase for a further approximately 60ms ; correspondingly the temperature decreases. Subsequently, the temperature rises again toward its final value as the second-sound velocity and attenuation decrease. It is shown that this unexpected behavior is apparently associated with a large-amplitude second-sound oscillation produced by the expansion, and it is suggested that the observed vortices are created by the normal fluid-superfluid counterflow that constitutes the second-sound wave. If production of large-amplitude second sound is inherent to the mechanical expansion of liquid 4He through the superfluid transition, as appears to be the case for final temperatures more than 3mK from the lambda transition, the phenomenon sets a lower bound on the density of KZ vortices that can be detected in this type of experiment.

Emission of the second sound with an expanding3He-concentrated droplet and phase-separation kinetics in a superfluid3He−4Hemixture

We study the growth kinetics of a droplet of the ${}^{3}\mathrm{He}$-concentrated phase in a superfluid ${}^{3}{\mathrm{H}\mathrm{e}\ensuremath{-}}^{4}\mathrm{He}$ supersaturated mixture. The growth equation, which generalizes the Rayleigh-Plesset equation for a radial expansion of bubbles in the normal fluids, is derived under the assumption of an arbitrary boundary condition for the normal velocity. The total intensity of the first- and second-sound emissions for a droplet expanding in the superfluid mixture is calculated. The emission of the second-sound mode is found to be predominant due to the smallness of the second-sound velocity compared with the velocity of the first sound. In contrast to demixing normal mixtures the diffusion and heat conduction processes play a minor role in the phase-separation kinetics of the supersaturated ${}^{3}{\mathrm{H}\mathrm{e}\ensuremath{-}}^{4}\mathrm{He}$ superfluid mixtures.

Testing critical point universality along the λ-line

We are currently building a prototype for a new test of critical-point universality at the lambda transition in 4He, which is to be performed in microgravity conditions. The flight experiment will measure the second-sound velocity as a function of temperature at pressures from 1 to 30 bars in the region close to the lambda line. The critical exponents and other parameters characterizing the behavior of the superfluid density will be determined from the measurements. The microgravity measurements will be quite extensive, probably taking 30 days to complete. In addition to the superfluid density, some measurements of the specific heat will be made using the low-g simulator at the Jet Propulsion Laboratory. The results of the superfluid density and specific heat measurements will be used to compare the asymptotic exponents and other universal aspects of the superfluid density with the theoretical predictions currently established by renormalization group techniques.

Second sound very near T?

The results of an experimental investigation of the evolution of planar, non-linear, second-sound pulses in superfluid4He, to within 650 nK of the λ-transition, are presented. A new method for extracting the second-sound velocity and damping is demonstrated. As predicted from two-fluid hydrodynamics, the pulses are well modeled by the solutions of Burgers' equation. The second-sound velocity (u 20) and damping (D 2) are extracted from fits of the model to the data. Damping data are obtained in this fashion to 3×10−7 in reduced temperature at saturated vapor pressure; nearly two decades closer to Tλ then any previous measurements. The superfluid density is extracted from theu 20 measurements and the critical exponent, ζ, is determined. A study of very large amplitude pulses near Tλ is also presented. These pulses extend well beyond the range of validity of Burgers' equation. The amplitude of the shock that forms at the trailing edge of the pulse is observed to saturate as a function of heater power and then decrease suddenly, as has been previously observed away from Tλ. However, the pulse shapes are quite different from any previously observed.

Superfluid fraction of 4He very close to T lambda.

Measurements of the superfluid fraction ${\mathrm{\ensuremath{\rho}}}_{\mathit{s}}$/\ensuremath{\rho} of $^{4}\mathrm{He}$ are presented for the reduced-temperature range 3\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}7}$\ensuremath{\lesssim}t\ensuremath{\equiv}1-T/${\mathit{T}}_{\ensuremath{\lambda}}$\ensuremath{\lesssim}${10}^{\mathrm{\ensuremath{-}}2}$. The data were obtained from the second-sound velocity ${\mathit{u}}_{20}$, which in turn was determined from nonlinear-pulse transit-time measurements. The exponent \ensuremath{\zeta}, which describes the leading singularity of ${\mathrm{\ensuremath{\rho}}}_{\mathit{s}}$/\ensuremath{\rho}, is found to be 0.6705\ifmmode\pm\else\textpm\fi{}0.0006. This result is in good agreement with existing theories and previous data much further away from ${\mathit{T}}_{\ensuremath{\lambda}}$.

Quantitative studies of nonlinear second sound in superfluid 4He.

Experimental studies of the nonlinear evolution in time of planar second-sound pulses in superfluid $^{4}\mathrm{He}$ have been carried out at temperatures 2 to 25 mK below ${\mathit{T}}_{\ensuremath{\lambda}}$. The pulse evolution in helium is compared to solutions of the Burgers equation, and corrections for the effect of the heater and bolometer substrates are made by numerical integration of the heat-diffusion equation with appropriate boundary conditions. Agreement between measured and calculated pulse shapes is found to be excellent. Fits to the data of the model are used to obtain the linear second-sound velocity. A new method for studying the critical behavior of second sound very close to ${\mathit{T}}_{\ensuremath{\lambda}}$, where nonlinear effects are always important, is suggested.

Second sound in metastable and phase-separated 3He-4He mixtures.

Propagation of second sound has been used to study superfluid $^{3}\mathrm{\ensuremath{-}}^{4}$He mixtures quenched into nonequilibrium states inside the miscibility gap. From the results for the second-sound velocity we conclude that the superfluid density in the metastable state is well described by extrapolation from equilibrium values. The attenuation of the second-sound amplitude displays a sharp increase as nucleation sets in at the cloud point, and during the subsequent coarse-graining process it varies in proportion to the total surface area of the droplets.

Quantitative studies of nonlinear second sound near Tλ

Abstract Experimental studies of the nonlinear evolution in time of planar second-sound pulses in superfluid 4 He have been carried out a few milliKelvin below T λ . When thermal effects in the second-sound generator and detector are taken into consideration, the data can be quantitatively understood using Burgers' equation to describe sound propagation in the superfluid. A method for extracting the second-sound velocity and damping very near T λ , where nonlinear effects are always important, is described.

Thermodynamics of liquid 3He− 4He mixtures

The osmotic pressure of 3He− 4He solutions is shown to follow in a natural way from the separation of the specific heat in a quasiparticle contribution and a 4He-contribution. Expressions for important thermodynamic properties such as the chemical potentials, the internal energy, the entropy, specific heat, osmotic pressure, etc. are given for the case the quasiparticle gas can be treated as a nearly-ideal Fermi gas. The zero kelvin and the high-temperature limits are tabulated. A first-order approximation is given for the effect of the 3He quasiparticle contribution to the second-sound velocity in the high-temperature limit.

High-resolution second-sound velocity measurements near the lambda point of helium.

We report measurements of the second-sound velocity in superfluid helium in the region very close to the \ensuremath{\lambda} point. The sound was detected using a powdered-paramagnetic-salt thermometer with a superconducting readout system. Large gravitational and power effects were observed near the transition. The corrected data yield the value 0.6740\ifmmode\pm\else\textpm\fi{}0.0005 for the exponent characterizing the divergence of the superfluid density at the transition.

Precision second-sound velocity measurements in helium II

The velocity of second soundu2 in helium II under its saturated vapor pressure was measured in an acoustic resonator with a typical precision of 0.07% and was found to agree with existing measurements. Data were taken from 1.135 to 2.09 K, with special attention given to the local maximum near 1.65 K, which can be used as a temperature reference point. A least squares cubic spline fit is given to the present data and selected data of other authors givingu2 from 0.55 K to within 60 µK of Tλ.

Second sound, osmotic pressure, and Fermi-liquid parameters inHe3-He4solutions

Second-sound velocities and osmotic pressures are analyzed to obtain the first experimental values for the Landau compressibility parameter F/sub 0//sup s/ in /sup 3/He-/sup 4/He solutions. Data are presented as a function of pressure and /sup 3/He concentration, and are compared to theoretical predictions. The square of the second-sound velocity at finite temperature is found to be accurately proportional to the internal energy of a perfect Fermi gas. Using inertial effective masses given by the Landau-Pomeranchuk theory, the square of the velocity is found to separate into two parts: a temperature-dependent part characterized completely by ideal Fermi-gas behavior and a temperature-independent part containing all the Fermi-liquid corrections. This is related to a similar separation found in the osmotic pressure.

Entropy of He II from 1.6 K to theλline

The fountain pressure of He II was measured and the results were used to obtain the entropy from 1.6 K to near ${T}_{\ensuremath{\lambda}}$ at vapor pressure and from 1.7 K to near ${T}_{\ensuremath{\lambda}}$ at various pressures up to 29 bar. The data have a precision of \ifmmode\pm\else\textpm\fi{}0.1% and an accuracy of \ifmmode\pm\else\textpm\fi{}0.5%. There is excellent agreement with recent calorimetric measurements at vapor pressure and good agreement within the combined error estimates with some, but not all, previous fountain-pressure measurements. The small quantity $\ensuremath{\Delta}S=\ensuremath{\int}{{T}_{0}}^{{T}_{\ensuremath{\lambda}}}(\frac{{C}_{p}}{T)}\mathrm{dT}$ was added to our measurements near ${T}_{\ensuremath{\lambda}}$ to yield values for ${S}_{\ensuremath{\lambda}}(P)$ which were fitted to an empirical equation with a scatter of \ifmmode\pm\else\textpm\fi{}0.1%. The increased accuracy of our new measurements will reduce considerably the experimental errors in the determination of the superfluid fraction from the second-sound velocity near the superfluid transition line.

On the velocity and absorption of second sound in dilute 3He-4He mixtures under pressure

Abstract The velocity and absorption of second sound in dilute 3 He- 4 He mixtures under pressure have been measured by a heat-pulse, time-of-flight technique. In order to eliminate path-length effects, the velocity of second sound was determined by subtracting the simultaneously measured times of flight of single second-sound pulses over two different path lengths. At overall pressures between p = 0.0 MPa and p = 2.0 MPa, tabulated values of the second-sound velocity are presented, in pure 4 He in the temperature range between T = 0.7 K and T = 1.1 K, and in dilute 3 He- 4 He mixtures in the temperature range between T = 0.2 K and T = 1.1 K at molar 3 He concentrations between X = 5.5 × 10 -4 and X = 1.73 × 10 -3 . Values of the absorption coefficient, deduced from the broadening of the transmitted second-sound pulses, are given, in pure 4 He in the temperature range between T = 0.7 K and T = 1.1 K, and in dilute 3 He- 4 He mixtures in the temperature range between T = 0.35 K and T = 1.1 K at a zero-pressure molar 3 He concentration of X = 9.2 × 10 -4 . From the data in pure 4 He, values of the phonon and roton parameters were calculated. From the data in dilute 3 He- 4 He mixtures, values of the 3 He contribution to the normal-fluid density were calculated. The values of the effective mass of the 3 He quasiparticles, determined from these values, are in excellent agreement with results of other experiments.

Technique for determining second-sound attenuation near the superfluid transition in 4He

A procedure has been developed to measure the attenuation of second sound in liquid helium at temperatures near Tλ. The decay rate of a second-sound resonance in a high-Q cavity is determined, and data for more than one harmonic is obtained allowing the separation of the attenuation due to bulk fluid from losses at the cavity walls. This method has proven particularly successful near Tλ, where the second-sound velocity is strongly dependent on temperature and a high degree of temperature stability is needed. Results are given for the temperature range 40 μK<Tλ-T<40 mK at saturated vapor pressure. With minor modifications the apparatus can be used for measurements at elevated pressure and over a much wider range of temperatures.

Second-sound velocities in hexagonal crystals

The velocities of "drifting" and "driftless" second sound at very low temperatures are calculated for 12 dielectrics having hexagonal crystal structure. The variation of the two second-sound velocities with density is studied for hcp $^{4}\mathrm{He}$. It is shown that the discrepancy pointed out by Maris between the experimental and theoretical values of the drifting second-sound velocities in hcp $^{4}\mathrm{He}$ can be reduced if another set of elastic constants is used in the calculations.

Measurements of Second Sound in Partially Spin-PolarizedHe3-He4Solutions

The second-sound velocity of three dilute $^{3}\mathrm{He}$-$^{4}\mathrm{He}$ mixtures was measured with high precision down to a temperature of 10 mK and in a magnetic field of 93 kOe. For the most dilute sample, which had a molar concentration of 0.001, the maximum spin polarization of the $^{3}\mathrm{He}$ atoms was estimated to be 36%. The measured relative change in the secondsound velocity was 3%. This is considerably larger than the change predicted recently by Bashkin and Meyerovich. An explanation for this discrepancy is presented.

Calculation of the 3He effective mass from experimental results on the second-sound velocity in dilute 3He- 4He mixtures

From our experimental results on the velocity of second sound in dilute 3He- 4He mixtures [1] information is obtained about the 3He quasiparticle spectrum and the interaction between the 3He quasiparticles. The method of calculation is presented, together with the results on the 3He effective mass and other relevant quantities. The ratio between the effective mass of a single 3He quasiparticle in pure 4He and the bare 3He atomic mass is found to be 2.20 ± 0.04.

A dilute hard-sphere bose-gas model calculation of low-density atomic hydrogen gas properties

The isolated pair wave equation is solved for two isolated hydrogen atoms interacting via the 3Σ u + potential, from which a scattering length a = 0.72 » is determined and used as an effective hard-core diameter in a dilute hard-sphere Bose-gas model calculation. In the low-density limit, these results for the ground state energy compare closely with earlier Monte Carlo data. Using established criteria for the values of density and temperature over which the hard-sphere model is valid, we have calculated the energy, first- and secondsound velocities, depletion factor, and superfluid transition temperature, and expressions are given for the specific heat and normal fluid fraction. The scattering length of atomic hydrogen interacting via the singlet potential is found to be 0.12 » and similar calculations for helium give a scattering length much too large to justify application of the hard-sphere model.