### 7.1 Isotropic solid (polycrystal)

Microscopically, the ground state corresponds to a body-centered cubic (bcc) crystal lattice. However,
one usually assumes that macroscopically, the neutron star crust is an isotropic bcc polycrystal. Elastic
properties of an isotropic solid are described by two elastic moduli. The deformation energy can be
expressed as
Here, is the shear modulus and is the compression modulus. The elastic stress tensor is, therefore,
Considering a small pure uniform compression, one finds
where is the total pressure given by Equation (24) and is the adiabatic index defined by
Equation (80).
Monte Carlo calculations of the effective shear modulus of a polycrystalline bcc Coulomb solid were
performed by Ogata & Ichimaru [309]. The deformation energy, resulting from the application of a specific
strain , was evaluated through Monte Carlo sampling.

As we have already mentioned, for an ideal cubic crystal lattice there are only three independent elastic
moduli, denoted traditionally as , and (see Chapter 3, pp. 80 – 87 of the book by
Kittel [241]). For pure shear deformation, only two independent elastic moduli are relevant,

because . At , Ogata & Ichimaru [309] find
These values agree with the classical result of Fuchs [151].
The definition of an “effective” shear modulus of a bcc polycrystal deserves a comment. In
numerous papers, a standard preferred choice was ([43, 319, 285], and references
therein). However, replacing by a single maximal elastic modulus of a strongly anisotropic bcc
lattice is not correct. An effective value of was calculated by Ogata & Ichimaru [309]. They
performed directional averages over rotations of the Cartesian axes. At , they obtained

some 30% smaller than used in previous papers. Dependence of on temperature was studied,
using the Monte Carlo method, by Strohmayer et al. [393]. As expected, the effective shear modulus
decreases with increasing temperature.
The formula for , Equation (116), can be rewritten as

where is the pressure of ultrarelativistic degenerate electrons. Therefore,
The Poisson coefficient , while the Young modulus .
Let us remember that the formulae given above hold for the outer crust, where the size of the nuclei is
very small compared to the lattice spacing and . For the inner crust these formulae are only
approximate.