2.1 Crystallization of White Dwarfs

Following the seminal investigation by Brush et al.4, Hansen8 and his coworkers9 have made the OCP one of the best-studied models in modern statistical physics. It has now become possible to compute tens of millions of configurations for systems containing thousands of particles, so that thermodynamic averages can be calculated with precision, accurate interparticle correlation functions can be obtained, and various transport processes can be evaluated. The most recent and accurate calculations for the OCP of which I am aware are those by Slattery, Doolen, and DeWitt10,11, who find that the OCP freezes into a bcc lattice at Γ = 178 ± 1. If crystallization of the plasma is prevented entirely, the OCP can undergo a transition to a “Coulomb glass” at still higher values of Γ12 (cf. §2.2. below).

It is important to recognize that a real dense plasma, such as that found in the core of a white dwarf or the crust of a neutron star, differs from an OCP in one very significant way. The OCP by definition has rigid, uniform background without any physical properties. In a real dense plasma, however, the neutralizing background in which the ions move consists of electrons with very definite physical properties. In particular, the Fermi energy EF of the electrons completely dominates the pressure of the matter, while the ions dominate the thermal properties. The latter are strongly affected by the Coulomb energy ECoul, particularly at low temperatures, where ECoul >> kBT or equivalently Γ >> 1. At sufficiently high densities, where EF >> Ecoul, the electron density is very nearly uniform, so that the OCP provides a good approximation for the properties of the ionic component of the plasma. This is the case in the deep interior of a white dwarf. At lower densities, however, the polarization of the electron background by the ions must be taken into account, and this complicates the calculation of the phase diagram of dense plasmas considerably13.

The first calculations of the evolution of white dwarfs which included the full effects of the fluid/solid phase transition were carried out by Lamb14,15. For a 1 M⨀, pure12C white dwarf model, they found that core crystallization begins at an age ∼ 109 years, when the star has cooled to a luminosity L ≈ 1.6 × 10−3L⨀ and an effective temperature Teff ≈ 13,000 K. This is indeed sufficiently hot and bright to permit direct observational study. Real white dwarfs are observed to have luminosities as faint as ≈ 10−4.5L⨀ and temperatures as low as ∼ 4500 K. However, they are believed to have cores consisting of a mixture of C and O, rather than pure C, and they generally have masses closer to 0.6M⨀ than to 1.0M⨀. Nevertheless, more recent calculations16 continue to indicate that crystallizing white dwarfs are in principle observable.

Crystallization releases the latent heat of fusion associated with the formation of the solid lattice. This is a significant fraction of the thermal energy of the plasma and initially slows the cooling of the star appreciably. Eventually, however, the core temperature T of the white dwarf falls below the Debye temperature ΘD of the lattice. The heat capacity subsequently falls as (T/Θd)3, and the star begins to cool increasingly rapidly. Both of these effects are evident in the theoretical white dwarf “luminosity function,” the number density of stars (per pc−3) per unit interval in log(L/L⨀). The difficulty of determining the observational luminosity function at very low luminosities has prevented a sufficiently accurate determination of this quantity to test these theoretical predictions with present data, but we expect that data from the Hubble Space Telescope will make this possible in the near future. (See also §||| below).

2.2 Transition to a Glassy State

Ichimaru et al.12 have used an improved HNC scheme to study the OCP at large values of Γ. They found that the second peak in pair correlation function g(r) broadened near Γ ≈ 200 and subsequent split in two for Γ ≥ 300. As such splitting is common to glassy substances, they therefore concluded that the supercooled, metastable state is indeed a “Coulomb glass.” They estimated the transition time to the stable, crystalline phase to be ∼ 105 years and thus suggested that astrophysical dense matter with 171 Γ 210, may be in such an amorphous, glassy state, rather than a crystalline state. More recently, Ogata and Ichimaru17,18 have investigated the dynamic evolution of the microstructure in supercooled OCPs using Monte Carlo simulations. They studied four cases of rapid quenches from an equilibrium fluid at Γ = 160 to Γ = 200,300, or 400. Except for the quench to Γ = 200, which formed a supercooled fluid rather than glass, all of these cases relaxed to metastable states (Coulomb glasses) with internal energies lying distinctly below that of the fluid, but above the bcc lattice energy.

These results are consistent with experience in terrestrial laboratories, where the preparation of a metastable state requires either (i) “splat cooling,” which provides such a rapid quench that the system has no chance to reach the state of lowest internal energy or else (ii) very gradual cooling through the...