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arXiv:cond-mat/0006415 · Physical review. B, Condensed matter

Superfluid transition temperature from the Lindemann-like criterion

Sergey M. Apenko

Published Oct 1, 2000• 0 citations
arXivHTMLPDF

Abstract

We further analyze the recently proposed criterion, according to which a superfluid transition occurs whenever the r.m.s. displacement of a particle from its initial position in imaginary time reaches a certain fraction of the interparticle distance. The critical temperature ${T}_{\ensuremath{\lambda}}$ is expressed in terms of the single particle mobility. Within the Drude approximation for the mobility ${T}_{\ensuremath{\lambda}}$ is determined by the friction coefficient (or by the viscosity for dense liquids). To check the validity of the criterion the resulting formula is applied to nonideal Bose gas, to liquid helium and to ${}^{3}\mathrm{He}{\ensuremath{-}}^{4}\mathrm{He}$ mixtures. A reasonable qualitative agreement with available experiments is obtained in all cases. In the case of the Bose gas the present approach results also in an upper bound on the peak value of ${T}_{\ensuremath{\lambda}}{/T}_{\ensuremath{\lambda}}^{0},$ where ${T}_{\ensuremath{\lambda}}^{0}$ is the critical temperature for the ideal gas.

DOI
10.1103/physrevb.62.9041

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We further analyze the recently proposed criterion, according to which a superfluid transition occurs whenever the r.m.s. displacement of a particle from its initial position in imaginary time reaches a certain fraction of the interparticle distance. The critical temperature T\ensuremathλ{T}_{\ensuremath{\lambda}}T\ensuremathλ​ is expressed in terms of the single particle mobility. Within the Drude approximation for the mobility T\ensuremathλ{T}_{\ensuremath{\lambda}}T\ensuremathλ​ is determined by the friction coefficient (or by the viscosity for dense liquids). To check the validity of the criterion the resulting formula is applied to nonideal Bose gas, to liquid helium and to 3He\ensuremath−4He{}^{3}\mathrm{He}{\ensuremath{-}}^{4}\mathrm{He}3He\ensuremath−4He mixtures. A reasonable qualitative agreement with available experiments is obtained in all cases. In the case of the Bose gas the present approach results also in an upper bound on the peak value of T\ensuremathλ/T\ensuremathλ0,{T}_{\ensuremath{\lambda}}{/T}_{\ensuremath{\lambda}}^{0},T\ensuremathλ​/T\ensuremathλ0​, where T\ensuremathλ0{T}_{\ensuremath{\lambda}}^{0}T\ensuremathλ0​ is the critical temperature for the ideal gas.
A phenomenological criterion for the superfluid transition is proposed, which is similar to the Lindemann criterion for the crystal melting. Then we derive a formula for the critical temperature, relating ${T}_{\ensuremath{\lambda}}$ to the mean kinetic energy per particle above the transition. The suppression of the critical temperature in a sufficiently dense liquid is described as a result of the quantum decoherence phenomenon. The theory can account for the observed dependence of ${T}_{\ensuremath{\lambda}}$ on density in liquid helium and results in an estimate ${T}_{\ensuremath{\lambda}}\ensuremath{\sim}1.1$ K for molecular hydrogen.
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.
The goal of this chapter will be to briefly describe the remarkable properties of helium at low temperatures. After stating some of these properties we will see how they can be understood in terms of the phenomenon of Bose–Einstein condensation described in Chapter 7. We will give the main argument in two different formulations, once using the quasi-particle method of Bogoliubov, and then using a Green function approach. We start with some experimental facts. Helium is a remarkable element. It was predicted to exist from observations of the Sun before it was found on Earth. It is the only element which remains a liquid at zero temperature and atmospheric pressure. Experimentally the phase diagram of 4 He is shown in Figure 10.1. Helium I is a normal fluid and has a normal gas-liquid critical point. Helium II is a mixture of a normal fluid and a superfluid. The superfluid is characterized by the vanishing of its viscosity. Helium I and helium II are separated by a line known as the λ -transition line. At T λ = 2.18 K, P λ = 2.29 Pa, helium I, helium II, and helium gas coexist. The specific heat of liquid helium along the vapor transition line forms a logarithmic discontinuity shown in Figure 10.2. The form of this diagram resembles the Greek letter λ and is the reason for calling the transition a λ -transition. The lack of viscosity of helium II leads to some remarkable experimental consequences, one of which we briefly describe. Let two containers A and B be linked by a thin capillary through which only a fluid with zero (or very low) viscosity can pass freely.
A method for calculation of the critical temperature of transition of a many-particle Fermi system into a superfluid or a superconducting state at an arbitrary pair potential of the interparticle interaction is proposed. An original homogeneous integral equation, that determines the critical temperature, is transformed into a homogeneous integral equation with a symmetric kernel that enables application of a general theory of integral equations for calculation. Examples are given of calculation of the superconducting transition critical temperature for the BCS potential with the Coulomb repulsion and the critical temperature of the superfluid transition of liquid helium-3 into ppp-wave pairing state for the Morse potential.
First, a condition for the Bose-Einstein transition in the superfluid liquid helium is presented due to the creation of a free neutron spinless pairs in a liquid helium and a dilute neutron gas mixture. We proposed a new model of dilute Bose gas, where presence atoms in the condensate is a suppressor for the collective modes as well as a creator for single-particle excitations On other hand, it is shown that the terms, of the interaction between the excitations of the Bose gas and the density modes of the neutron, meditate an attractive interaction via the neutron modes, which in turn leads to a bound state on a spinless neutron pair. The lambda transition is defined by a condition for the Bose-Einstein transition, which transforms reflected neutron pair modes to single neutron modes.
Higher-order perturbation-theory corrections to the superfluid transition temperature in a weakly nonideal Fermi gas with repulsion are determined. This involves calculating the contribution of third-and fourth-order diagrams in terms of the gas parameter ap F to the effective interaction which determines the super-fluid transition temperature and also allowing for effects associated with retardation and renormalization of the polar part of the Green’s function. The expressions obtained provide evidence in support of attraction in the effective interaction in the second, third, and fourth orders of perturbation theory. It is shown that the critical temperature is mainly determined by second-and third-order terms of perturbation theory. Calculations are made of the superfluid transition temperature for a gas comprising neutral Fermi particles in a magnetic field. The limits of validity of the theory are analyzed and the possibility of applying the results to dilute solutions of 3 He in 4 He and neutral-particle Fermi gases in magnetic traps is discussed.