Helium atom phase transition

The mysteries of the helium phase diagram further deepen at the strange A-line that divides the two liquid phases. In certain respects, this coexistence curve (dashed line) exhibits characteristics of a line of critical points, with divergences of heat capacity and other properties that are normally associated with critical-point limits (so-called second-order transitions, in Ehrenfest s classification). Sidebar 7.5 explains some aspects of the Ehrenfest classification of phase transitions and the distinctive features of A-transitions (such as the characteristic lambda-shaped heat-capacity curve that gives the transition its name) that defy classification as either first-order or second-order. Such anomalies suggest that microscopic understanding of phase behavior remains woefully incomplete, even for the simplest imaginable atomic components. 

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... 

Using the finite-size scaling method, study of the analytical behavior of the energy near the critical point shows that the open-shell system, such as the lithium-like atoms, is completely different from that of a closed-shell system, such as the helium-like atoms. The transition in the closed-shell systems from a bound state to a continuum resemble a first-order phase transition, while for the open-shell system the transition of the valence electron to the continuum is a continuous phase transition [9]. 

From the present calculations, the expectation value of the operator r 2 may provide a direct physical picture about the thermodynamic stability and dissociation of Hj-like molecules. As shown in Fig. 16, there is a vertical jump of the mean value ru at Xc. We note that there are similarities and differences between helium-like atoms and Hj-like molecules. In Section V.A of heliumlike systems, based on an infinite mass assumption, we show that the electron at the critical point leaves the atom with zero kinetic energy in a first-order phase transition. This limit corresponds to the ionization of an electron as the nuclear charge varies. For the Hj-like molecules, the two protons move in an electronic potential with a mass-polarization term. They move apart as X approaches its critical point and the system approaches its dissociation limit through a first-order phase transition. 

Hartree-Fock Version. As with the analogous mean field approximation in statistical mechanics, the error in the Hartree-Fock approximation is expected to diminish as D increases. This is because fluctuations decrease in proportion to D, as illustrated in Fig. 2 for the hydrogenic atom. However, whereas the mean field approximation for critical exponents of phase transitions becomes exact [86] for sufficiently large D, for the Hartree-Fock approximation the correlation energy remains nonzero and relatively large even for D oo. As a function of the total energy, AEd for the helium atom varies from 2.3% at the D — 1 limit to 1.5% for D = 3 to 0.99% at the D oo limit. 

Helium is unusual in that it is the only known substance for which there is no triple point (i.e., no combination of pressure and temperature at which all three phases can co-exist in equilibrium). This is because the interatomic forces, which normally participate in the formation of solids, are so weak in helium that they are of the same order as the zero-point energy. At 2.186 K helium undergoes a transition from liquid helium 1 to liquid helium 11, the latter being a true liquid but exhibiting superconductivity and an immeasurably low viscosity (superfluidity). This low viscosity allows the liquid to spread in layers a few atoms thick, described by some as an action like flowing uphill . 

Experimental data represented by heavy solid lines in Fig. 6.6 show that the solubility of helium in the denser phase ( ) decreases monoton-ically as the density increases along the metal-nonmetal transition line of mercury. The dotted lines are extrapolations based on a calculated point at about 11 g cm . The solubility at that density was estimated from the effective immersion energy for a helium atom in a free electron gas whose density corresponds to divalent mercury at 11 g cm (Stephenson, 1983). 

There are some phase transitions that do not fall either into the first-order or second-order category. These include paramagnetic-to-ferromagnetic transitions in some magnetic materials and a type of transition that occurs in certain solid metal alloys that is called an order-disorder transition. Beta brass, which is a nearly equimolar mixture of copper and zinc, has a low-temperature equilibrium state in which every copper atom in the crystal lattice is located at the center of a cubic unit cell, surrounded by eight zinc atoms at the corners of the cell. At 742 K, an order-disorder transition occurs from the ordered low-temperature state to a disordered high-temperature state in which the atoms are randomly mixed in a single crystal lattice. The phase transition between normal liquid helium and liquid helium II was once said to be a second-order transition. Later experiments indicated that the heat capacity of liquid helium appears to approach infinity at the transition, so that the transition is not second... 

In the process of surface formation, there may exist positions k in reciprocal space where the frequency of a vibration is so deeply modified, as a consequence of bond-breaking, that it vanishes. The mode becomes unstable and one speaks of a soft phonon. The atoms are no longer pulled back to their equilibrium positions. A new periodicity, characterized by the wave vector k, shows up in low-energy electron or helium atom diffraction experiments. The mechanism of phonon softening was suggested to explain some reconstructions observed on metal surfaces (Blandin et al, 1973 Fasolino et al, 1980 Pick, 1990). When a true structural phase transition occurs at a temperature Tc, the frequency of the soft mode decreases as the temperature decreases in the fluctuative regime (T > Tc). Below the critical temperature Tc, superstructure satellites appear. Such phenomena were, for example, observed by Ernst et al. (1987) on W(OOl), by diffraction of heUum atoms. They are also present in quasi-one-dimensional conductors which display a Peierls transition. One should nevertheless remember that there exist structural phase transitions that are not driven by soft phonons. 

One of the problems with VMC is that it favors simple states over more complicated states. As an example, consider the liquid-solid transition in helium at zero temperature. The solid wave function is simpler than the liquid wave function because in the solid the particles are localized so that the phase space that the atoms explore is much reduced. This biases the difference between the liquid and solid variational energies for the same type of trial function, (e.g. a pair product form, see below) since the solid energy will be closer to the exact result than the liquid. Hence, the transition density will be systematically lower than the experimental value. Another illustration is the calculation of the polarization energy of liquid He. The wave function for fully polarized helium is simpler than for unpolarized helium because antisymmetry requirements are higher in the polarized phase so that the spin susceptibility computed at the pair product level has the wrong sign ... 

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