A simple model describing 3He-4He mixtures near a wall

2014 We describe the superfluid ordering taking place near a wall in 3He-4He mixtures by means of a semi-infinite version of the Blume-Emery-Griffiths model. A renormalization group analysis of the simplest version of this model yields a phase diagram which compares satisfactorily with available experimental data. J. Physique Lett. 45 (1984) L-591 L-596 15 JUIN 1984 Classification Physics Abstracts 67.60 64.90 The phenomenon of superfluid film formation near a wall in 3He-4He mixtures has been observed for many years [1]. It originates in an effective repulsion of 3He atoms by the wall. The van der Waals interactions between the wall and 3He or 4He atoms are equal, but 3 He atoms occupy a larger volume because of their larger zero-point motion. One observes therefore a higher 4He concentration near the wall, which may induce a local superfluid ordering [2]. This transition occurs on a line which branches from the A-line and extends to higher 3 He concentrations, where one eventually observes phase separation near the wall. Figure 1 shows the bulk 3He-4He diagram, with the available data on the location of these transitions. Although various experimental methods have been used to achieve this localization [1], little is known on the details of the transition. In particular, the thickness of the film, the order of the transition, the parameter dependence of the phase diagram have not yet been systematically investigated. In a previous paper [3] (hereafter referred to as I) we pointed out that the superfluid film formation near the ~-line is analogous to the surface and special transitions in semi-infinite magnetic systems [4]. We described this phenomenon by means of a simple semi-infinite model with short(*) GNSM-CNR, Unita di Rome. On leave of absence from : Dipartimento di Fisica, Universita « La Sapienza », Piazzale Aldo Moro 2,1-00185 Roma (Italy) (Present address). Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyslet:019840045012059100 L-592 JOURNAL DE PHYSIQUE LETTRES Fig 1. Phase diagram for 3He4He mixtures. Experimental data for surface transitions, taken from Romagnan et al. [1], were obtained by several research groups. range interactions, based on the Blume-Emery-Griffiths [5] (BEG) model for 3He-4He mixtures. We conjectured that the special transition in this model belonged to the same universality class as the « usual », semi-infinite Ising models. Our arguments relied upon a mean-field calculation of the behaviour of the semi-infinite BEG model near the special transition. We also suggested that the same model could be used to describe qualitatively the behaviour of 3He4He mixtures in the whole range of 3He concentrations. In this paper we confirm all the conjectures made in I. The simplest version of the semi-infinite BEG model has been applied to the description of the whole range of 3He concentrations. A renormalization group analysis of its behaviour (within the Migdal-Kadanoff approach) has confirmed the stability of the fixed point describing the special transition in the pure Ising model against the perturbation due to a non vanishing concentration of impurities. We have obtained a phase diagram which compares surprisingly well with the existing experimental data. 1. The semi-infinite BEG modeL The BEG model, introduced [5] to describe 3He4He mixtures, is defined as follows. To each point i of a simple cubic lattice are associated two variables : ~~, Q~. The variable ti takes the values 0 (corresponding to the presence of a 3He atom) or 1(4 He atom). The variable (fi takes the values + 1 or 1 and represents the phase of the 4 He wavefunction. This a drastic simplification, since one neglects the continuous U(I) symmetry of the superfluid order parameter. This is all the more important in surface phenomena, due to the peculiar nature of superfluid ordering in two dimensions [7] (we will return to this point in the conclusions). The partition function of the model is given by : _ where N is the total number of lattice points, the sum runs over all configurations of the variables ti, (fi and L-593 A SIMPLE MODEL DESCRIBING 3He-4He MIXTURES NEAR A WALL The last sum runs over all points in the lattice; the first two ones over all nearest-neighbour pairs. The first term represents the ordering interaction among 4 He atoms; the second describes « isotope » effects (i.e. the difference between the interactions in the 3He-3He and 4He-4He pairs); the parameter d plays the role of a chemical potential for 4He atoms. This model describes well the observed phase diagram of bulk 3He-4He mixtures already at the mean-field level, yielding superfluid ordering in homogeneous mixtures and phase separation of sufficiently low temperatures [5]. One can straightforwardly extend this model to semi-infinite geometries by taking : if i or j belong to the bulk if i and j belong to the surface if i belongs to the bulk if i belongs to the surface . Thus, even such a simple model is already described by six parameters. In the absence of more detailed experimental data, which would allow to assign values to some of them, we have preferred to analyse the simplest version. We take therefore Kij = Jip which corresponds to the Blume... Capel [8] limit of the BEG model. This amounts to neglecting the differences in the interactions among different isotopes and is quite justified experimentally [5]. We also set Jo = J, supposing no direct effect of the wall on interatomic interactions. We finally take where the parameter 0 reflects the phenomenon of effective 3He repulsion by the wall. As shown in I, the line of onset of film superfluid joins the (bulk) A-line at a point whose position depends on the value of 0. All the data available to us agree in placing this point at a 3He concentration x3 N 0.54. (This agreement is probably due to the formation of a solid helium layer on the wall of the container, what makes almost irrelevant the composition of the wall.) 2. Renormalization group analysis. We have performed a real-space renormalization group investigation of this model. The recursion relations which correspond to the doubling of the lattice constant have been derived by KadanofTs [6] bond moving approach. The one-body interactions have been evenly distributed among the bonds which end at any given lattice point and are then moved along with them [9]. One thus obtains the following recursion relations for the bulk parameters :