Superfluid transitions

Obviously 9 =0 corresponds to the SmA phase. This transition is analogous to the nonnal-superfluid transition in liquid helium and the critical behaviour is described by the AT model. Further details can be found elsewhere [18, 19 and 20]. 

Besides being used as a tool for scientific research, siUca aerogels can be the cause for new scientific phenomena. For example, the long-range correlations of the disorder in siUca aerogels are beheved to be responsible for the intriguing observations of the superfluid transitions in He and He and on the ordering of He— He mixtures (75). 

Helium vapour pressure and latent heat of evaporation The latent heat of evaporation L and the vapour pressure />vap are fundamental parameters when using these two cryoliquids in the refrigeration process. Figure 2.5 shows L of 3He and 4He as a function of temperature. Note that L ( 20.9 J/g for 4He) is very small in comparison, for example, with that of hydrogen (445 J/g) or of nitrogen (200 J/g). Note also the minimum at 2.2 K in the graph for 4He, in correspondence with the superfluid transition. 

The idea that new phenomena could be present in 3He at very low temperatures arose from thermal measurements. The first observation was the anomaly in the specific heat at the normal superfluid transition which reminded the behaviour of specific heat at the superconductive transition in metals (Fig. 2.11) [34-36]. 

We note that even short-range interactions may, however, allow a mean-field scenario, if the system has a tricritical point, where three phases are in equilibrium. A well-known example is the 3He-4He system, where a line of critical points of the fluid-superfluid transition meets the coexistence curve of the 3He-4He liquid-liquid transition at its critical point [33]. In D = 3, tricriticality implies that mean-field theory is exact [11], independently from the range of interactions. Such a mechanism is quite natural in ternary systems. For one or two components it would require a further line of hidden phase transitions that meets the coexistence curve at or near its critical point. 

Helium-4 Normal-Superfluid Transition Liquid helium has some unique and interesting properties, including a transition into a phase described as a superfluid. Unlike most materials where the isotopic nature of the atoms has little influence on the phase behavior, 4He and 3He have a very different phase behavior at low temperatures, and so we will consider them separately Figure 13.11 shows the phase diagram for 4He at low temperatures. The normal liquid phase of 4He is called liquid I. Line ab is the vapor pressure line along which (gas + liquid I) equilibrium is maintained, and the (liquid + gas) phase transition is first order. Point a is the critical point of 4He at T= 5.20 K and p — 0.229 MPa. At this point, the (liquid + gas) transition has become continuous. Line be represents the transition between normal liquid (liquid I) and a superfluid phase referred to as liquid II. Along this line the transition... 

Not only do the thermodynamic properties follow similar power laws near the critical temperatures, but the exponents measured for a given property, such as heat capacity or the order parameter, are found to be the same within experimental error in a wide variety of substances. This can be seen in Table 13.3. It has been shown that the same set of exponents (a, (3, 7, v, etc.) are obtained for phase transitions that have the same spatial (d) and order parameter (n) dimensionalities. For example, (order + disorder) transitions, magnetic transitions with a single axis about which the magnetization orients, and the (liquid + gas) critical point have d= 3 and n — 1, and all have the same values for the critical exponents. Superconductors and the superfluid transition in 4He have d= 3 and n = 2, and they show different values for the set of exponents. Phase transitions are said to belong to different universality classes when their critical exponents belong to different sets. 

Chapters 13 and 14 use thermodynamics to describe and predict phase equilibria. Chapter 13 limits the discussion to pure substances. Distinctions are made between first-order and continuous phase transitions, and examples are given of different types of continuous transitions, including the (liquid + gas) critical phase transition, order-disorder transitions involving position disorder, rotational disorder, and magnetic effects the helium normal-superfluid transition and conductor-superconductor transitions. Modem theories of phase transitions are described that show the parallel properties of the different types of continuous transitions, and demonstrate how these properties can be described with a general set of critical exponents. This discussion is an attempt to present to chemists the exciting advances made in the area of theories of phase transitions that is often relegated to physics tests. 

E. Energetics, Thermodynamics, Response, and Dynamics of Ultracold Finite Systems Size Effects on the Superfluid Transition in ( He) Finite Systems... 

C. Finite Size Scaling of the Superfluid Transition Temperature and Density... 

II. SIZE EFFECTS ON THE SUPERFLUID TRANSITION IN ( He) FINITE SYSTEMS... 

The Onset of the Superfluid Transition in the Finite System [50, 65, 66, 155]. This transition is referred to as the X point in the bulk system. What is the analogy in a finite system Pioneering quantum path integral Monte-Carlo simulations [65, 66] established the appearance of a rounded-off (smeared) X transition in finite (He) (N = 64 and 128) clusters. This was manifested by a maximum in the temperature dependence of the specific heat (Fig. 6), which... 

Finite size scaling of the order parameter v[/p = p /p, Eqs. (11) and (23), for the superfluid transition provides signiflcant information on the size dependence... 

The short correlation length for superfluidity in bulk He implies that threshold cluster sizes are small—that is, of the order of interatomic distance. A simple-minded argument will imply that the minimal cluster size for the realization of a superfluidity transition is Using Eq. (40b),... 

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