Zero-temperature dilute-limit thermodynamics of liquid ³He–⁴He mixtures under pressure: the non-analytic low-density expansion, its coefficients, and their microscopic basis

Liquid ³He–⁴He mixtures are the archetype of a dilute degenerate Fermi gas embedded in a quantum Bose liquid, and their complete thermodynamic behavior at absolute zero is encoded in a single surface: the energy density $\varepsilon(n_3, n_4)$ as a function of the two number densities. A 1987 reconstruction by De Bruyn Ouboter and Yang assembled the $T = 0$ surface for pressures from 0 to 20 atm from molar-volume, heat-of-mixing, osmotic-pressure and solubility data, and found that an ordinary power series in the ³He density $n_3$ is inconsistent with the data, while a non-analytic fractional-power form $\varepsilon(p, n_3) = \eta_0 + \eta_1 n_3 + \eta_a n_3^{5/3} + \eta_2 n_3^2 + \eta_b n_3^{8/3}$ — the fingerprint of Fermi statistics of the ³He quasiparticles — reproduces the constraints at 0, 10 and 20 atm. The reconstruction was phenomenological: the paper did not derive the surface from microscopic theory, and the data then available did not reach the lowest $n_3$ at finite pressure, so the low-density expansion coefficients rested on extrapolation. Subsequent work has established this leading non-analytic structure in outline through Fermi-liquid theory and extended the millikelvin data set to saturated vapor pressure and to 25.3 bar (the crystallization pressure), but no complete quantitative microscopic derivation exists; the surviving open core is the determination of the $T = 0$ dilute-limit coefficients — the zero-concentration ³He effective mass $m_3^*(p)$ and the effective ³He–³He quasiparticle interaction, whose concentration dependence remains disputed — as functions of pressure over the reconstructed range 0–20 atm, together with their reconciliation with the reconstructed surface.

Background

Helium has two stable isotopes: ⁴He, a boson that becomes superfluid below 2.17 K, and ³He, a fermion that becomes superfluid only below a few millikelvin. When the two are mixed, the lighter ³He atoms remain soluble in liquid ⁴He only up to a saturation concentration $x_s$ (about 6.6% at saturated vapor pressure, rising to about 9.5% near 10 bar), where $x$ is the ³He concentration, i.e. the fraction of ³He atoms among all atoms; below about 0.87 K the mixture separates into a ³He-rich (concentrated) phase and a dilute phase. This phase separation is the working principle of the dilution refrigerator — the standard laboratory apparatus for reaching temperatures of a few millikelvin — so the thermodynamics of the dilute phase is not only of fundamental interest but also a matter of practical low-temperature engineering.

At absolute zero the entire equilibrium thermodynamics of the homogeneous (one-phase, dilute) region is fixed by the energy density $\varepsilon(n_3, n_4)$ — the internal energy per unit volume as a function of the number densities $n_3$ and $n_4$ of the two isotopes. Its partial derivatives give the chemical potentials $\mu_3$ and $\mu_4$, and the pressure follows from the tangent-plane (envelope) relation $\varepsilon = -p + n_3\mu_3 + n_4\mu_4$. Experimentally one does not measure $\varepsilon$ directly but observables that constrain it: the osmotic pressure (the pressure difference that develops across a membrane permeable only to superfluid ⁴He, driven by the ³He content), the heat of mixing (the enthalpy change on mixing the two isotopes), the partial molar volumes $V_3$ and $V_4$ (the volume change per added atom of each species), and the saturation curve $x_s(p)$ at which the dilute phase coexists with the concentrated phase.

The physical reason the dilute phase is special is that a ³He atom moving through superfluid ⁴He behaves, to a very good approximation, as a single fermionic quasiparticle — a ³He atom dressed by a backflow of the surrounding ⁴He — with an effective mass $m_3^*$ larger than the bare mass (the mass of a free ³He atom; about 2.2–2.4 bare masses at vapor pressure, rising with pressure). Because these quasiparticles obey Fermi statistics, their zero-point kinetic energy density is proportional to $n_3^{5/3}$, so the chemical potential $\mu_3$ contains a term proportional to $n_3^{2/3}$. An ordinary Taylor series in integer powers of $n_3$ therefore cannot describe the $n_3 \to 0$ limit: the expansion is intrinsically non-analytic, with fractional powers. This is the physical content of the Landau–Pomeranchuk picture — the late-1940s recognition that ³He dissolved in superfluid ⁴He forms a degenerate Fermi gas of such quasiparticles — made quantitative by Bardeen, Baym and Pines, whose effective-interaction theory of ³He–⁴He mixtures relates the expansion coefficients to the quasiparticle effective mass and the effective ³He–³He interaction.

In this context, De Bruyn Ouboter and Yang (1987) reconstructed the $T = 0$ surface $\varepsilon(n_3, n_4)$ between 0 and 20 atm from all available data. Their key finding was quantitative: the linear coefficient of an integer-power expansion inferred from osmotic-pressure fits disagreed with the heat-of-mixing extrapolation at low $n_3$, whereas the fractional-power form given above reproduced osmotic-pressure, heat-of-mixing and phase-matching constraints simultaneously at 0, 10 and 20 atm, with the partial molar volume $V_3$ essentially independent of concentration. They presented the fractional-power form as an inferred asymptotic description motivated by Fermi-quasiparticle considerations, and derived the dilute-limit offsets (such as the difference $\mu_3 - \mu_4^0$ between the ³He chemical potential in the dilute phase and in pure ⁴He at the same pressure) from the measured pressure dependence of $V_3$.

The reconstruction, however, was phenomenological: the paper did not derive the surface or its coefficients from microscopic theory, and the data then available did not reach low enough $n_3$ at finite pressure to isolate the $n_3^{5/3}$ behavior directly. Three intertwined questions therefore arise: whether the leading non-analytic structure implied by Fermi statistics is borne out quantitatively as $n_3 \to 0$; what the precise low-density expansion coefficients are — equivalently $m_3^*(p)$ at vanishing concentration and the effective ³He–³He quasiparticle interaction — as functions of pressure; and whether the whole $T = 0$ surface can be derived microscopically from the quasiparticle physics, rather than reconstructed from data?

Problem Statement

Establish the absolute-zero dilute-limit thermodynamics of liquid ³He–⁴He mixtures as a function of pressure, in the formulation left by the 1987 reconstruction of the $T = 0$ energy-density surface $\varepsilon(n_3, n_4)$ between 0 and 20 atm. The leading non-analytic structure of the homogeneous dilute phase as $n_3 \to 0$ — the fractional-power expansion $\varepsilon(p, n_3) = \eta_0 + \eta_1 n_3 + \eta_a n_3^{5/3} + \eta_2 n_3^2 + \eta_b n_3^{8/3}$, whose $n_3^{5/3}$ term is a structural consequence of Fermi statistics of the ³He quasiparticles — is established in outline; the open content of the problem is its quantitative completion. Concretely: determine the low-density expansion coefficients — equivalently the zero-concentration ³He quasiparticle effective mass $m_3^*(p)$ and the effective ³He–³He quasiparticle interaction, including its concentration dependence — as functions of pressure, and reconcile the disputed zero-concentration extrapolation of $m_3^*$ and the conflicting determinations of the ³He–³He interaction, consistently with the reconstructed phenomenology and with all measured observables (osmotic pressure, heat of mixing, solubility, and second sound — an entropy-temperature wave that propagates in superfluids and their mixtures, distinct from ordinary sound). Millikelvin osmotic-pressure data as a function of concentration currently exist only at saturated vapor pressure and at the crystallization pressure of about 25.3 bar, so at intermediate pressures the coefficients still rest on extrapolation. On the theoretical side, a complete microscopic derivation of the $T = 0$ surface and its non-analytic dilute-limit structure from the many-body theory of the quasiparticle interactions, consistent with the determined coefficients, remains open as a longer-range goal that would supply the microscopic basis of the result; the present problem is the determination and reconciliation of the coefficients above.

References

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