The Coulomb pressure Pii is exactly one third of the Coulomb energy density, because Uii/V depends on the single parameter r a V1/3/T (for a classical Coulomb plasma, §2.3.2a). The Coulomb pressure and energy are negative, because Coulomb interactions introduce an additional binding of ions.

In the gaseous regime the Coulomb energy is small, |Uii| ^ NkBT, so that the Coulomb binding is lower than the thermal energy of ions, and the ions are nearly free. In the strongly coupled liquid (r > 1) one has |Uii| > NkBT and the Coulomb energy exceeds the thermal one. In this regime the ions are bound in Coulomb potential wells (whose depth is ~ Z2e2/ai) and slightly oscillate there due to thermal motion. In a Coulomb liquid there is no exact order and the wells slowly migrate within the liquid. In the limit of r ^ 1 from Eq. (2.86) one obtains the linear r dependence, Uii/NkBT & A1 r. In this limit Uii & NA1Z2e2/ai and Pii & A1 nNZ2e2/(3ai) are nearly independent of temperature. The numerical factor A1 is very close to the value A1S = —0.9 predicted by the ion-sphere model (§ 2.1.3). In that model the Coulomb energy of one ion sphere is equal to —0.9Z2e2/ai, which includes the electron-electron interaction, 0.6Z2e2/ai, and the electron-ion interaction, —1.5Z2e2/ai (e.g., Shapiro & Teukolsky 1983).

Below the melting temperature, at T < Tm, an infinite ion motion is replaced by oscillations near equilibrium positions, which means that a crystal is formed. Let us recall the basic concepts of the physics of crystals (e.g., Kittel 1986). A three-dimensional crystal lattice can be mapped onto itself through a translation by a vector T = l1a1 + l2a2 + l3a3, where aj are called primitive translation vectors and lj are arbitrary integers. The lattice can be figured out as composed of primitive cells that fill all space by repetition of translations T. A primitive cell centered at an equilibrium ion position is a Wigner-Seitz cell. It is often convenient to consider a reciprocal lattice, defined by the primitive vectors bl such that bl ■ aj = 2n5lj. The reciprocal lattice is mapped on itself by a reciprocal lattice vector G = l1b1 + l2b2 + l3b3, where lj are again some integers. A Wigner-Seitz cell for the reciprocal lattice is known as a Brillouin zone. Its volume equals (2n)3/nN.

The ground state of the OCP of ions corresponds to the body-centered cubic (bcc) lattice. In a real crystal, this happens at high densities (at rs ^ 1), whereas other types of lattices (face-centered cubic, fcc; hexagonal close-packed, hcp) may form the ground state at rs > 1 (e.g., Kohanoff & Hansen 1996). In the following, we focus on the cubic lattices (bcc and fcc). Note that the simple cubic Coulomb lattice is unstable (as any simple cubic lattice of particles interacting via central forces; Born 1940).

In this section we describe thermodynamic functions of Coulomb crystals. Another important aspect is the elasticity of crystallized matter. Although elastic properties of the neutron star crust are fully determined by Coulomb interactions, they will be considered in § 3.7 of Chapter 3.

While studying strongly coupled Coulomb systems, it is often very useful to approximate the actual Wigner-Seitz cell by a sphere of the same volume. In simple lattices, the radius of the Wigner-Seitz cell is equal to the ion-sphere radius ai [see Eq. (2.23)]. The same approximation in reciprocal lattice results in a spherical Brillouin zone with an equivalent radius qBZ = (6nuN )1/3. (2.90)

Let us outline the standard method to study ion vibrations (e.g., Kittel 1963, 1986). Let the ions oscillate near equilibrium positions Rj. An ion position can be written as rj = Rj + Uj, where Uj is a small ion displacement. Since the ions are considered as pointlike, their local number density can be written as n(x,t) = ^2 S(x - Rj - Uj(t)). (2.91)

Expanding the Coulomb energy of an ion (say, j = 1) in Taylor series with respect to Uj and keeping the terms up to u,, we get the Coulomb energy in the harmonic approximation. In particular, the potential energy increase due to the ion displacements is

j — 2 j where u1j = u1 - uj. The sum comes from the ion-ion interactions, while the last term comes from the ion interaction with the electron background. Taking ion displacements in the form uj (t) = Aeks exp(ikRj - iwkst) (2.93)

and using the Newtonian equation of ion motion, one obtains the dynamical equations (e.g., Coldwell-Horsfall & Maradudin 1960),

where a and ß are Cartesian indices. These equations determine the eigen-frequencies uks and polarization vectors eks. Here, s enumerates branches of crystal oscillation modes,

is the dynamical matrix, and l is a vector with integer components (either all odd or all even for the bcc lattice). The convergence of sums over lattice sites can be improved using the Ewald summation technique (Ewald, 1921). In the OCP model of a simple Coulomb crystal (OCP of ions on the rigid electron background), there are three oscillation branches, two transverse acoustic ones (s = 1, 2, with a linear dispersion relation w(k) a k at small wave numbers k) and one longitudinal optical branch (s = 3, whose frequency goes to wpi at k — 0). It is easy to see from Eqs. (2.94) and (2.95) that the eigenfrequencies obey the Kohn's sum rule

Taking into account a finite compressibility of the ambient electron gas, one finds that this rule is not exact at small k. When k decreases below kTF, the frequency of the vibration branch s = 3 tends to zero with a limiting dispersion law w — wpi k/kTF (Pollock & Hansen, 1973). Thus the branch s = 3 is not the true optical branch. It becomes acoustic at large wavelengths owing to a finite electron screening.

In order to quantize ion motion, which is important at T < Tpi, one has to consider lattice dynamics in terms of elementary quantum excitations - phonons (e.g., Landau & Lifshitz 1993; Kittel 1986). Accordingly, the local ion density given by Eq. (2.91) should be replaced by the density operator p, which acts in space of quantum states characterized by phonon numbers; analogously, Uj must be replaced by the operator Uj of ion displacement. Here we restrict ourselves to the most important case in which the effects of quantum statistics of ions (exchange corrections) can be neglected. In this case one should take into account the ion spin degeneracy but, otherwise, one can treat the ions as spinless particles 3.

The harmonic approximation proves to be very useful to describe quantum crystals. In this approximation (e.g., Landau & Lifshitz 1993) the free energy of the ion lattice is s

3Such particles are sometimes called "boltzmannons".

and U0 is the zero-point lattice energy. The averaging <.. .)ph is performed over phonon wave vectors k in the first Brillouin zone and over phonon polarizations s:

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