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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
( )2 1- 2 1- ɛdef = 2 K (∇∇∇ ⋅uuu) + μ uik − 3 δik (∇∇∇ ⋅uuu) . (110 )
Here, μ is the shear modulus and K is the compression modulus. The elastic stress tensor is, therefore,
( ) ∂ɛdef 1 σik = ∂u--- = K (∇∇∇ ⋅ u)δik + 2μ uik − 3-(∇∇∇ ⋅ u)δik . (111 ) ik
Considering a small pure uniform compression, one finds
∂P K = nb---- = γP , (112 ) ∂nb
where P is the total pressure given by Equation (24View Equation) and γ is the adiabatic index defined by Equation (80View Equation).

Monte Carlo calculations of the effective shear modulus of a polycrystalline bcc Coulomb solid were performed by Ogata & Ichimaru [309Jump To The Next Citation Point]. The deformation energy, resulting from the application of a specific strain uik, 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 c11, c12 and c44 (see Chapter 3, pp. 80 – 87 of the book by Kittel [241Jump To The Next Citation Point]). For pure shear deformation, only two independent elastic moduli are relevant,

( 2 2 2 ) ( 2 2 2 ) ɛdef = b11 uxx + uyy + uzz + 2c44 uxy + uxz + uyz for (∇∇∇ ⋅uuu) = 0 , (113 )
because 1 b11 = 2(c11 − c12). At T = 0, Ogata & Ichimaru [309Jump To The Next Citation Point] find
(Ze )2 b11 = 0.0245 nN ------, (114 ) Rcell
2 c = 0.1827 n (Ze)--. (115 ) 44 N Rcell
These values agree with the classical result of Fuchs [151].
View Image

Figure 43: Effective shear modulus μ versus density, for a bcc lattice. Solid line – cold catalyzed matter (Haensel and Pichon 1994 model [183Jump To The Next Citation Point] for the outer crust (Section 3.1), and that of Negele and Vautherin 1973 [303Jump To The Next Citation Point] for the inner crust (Section 3.2.3)). Dash-dotted line – cold catalyzed matter calculated by Douchin and Haensel 2000 [126] (compressible liquid drop model, based on SLy effective N-N interaction, Section  3.2.1). Dotted line – accreted crust model of Haensel and Zdunik 1990 [185Jump To The Next Citation Point] (Section 4). Figure made by A.Y. Potekhin.

The definition of an “effective” shear modulus of a bcc polycrystal deserves a comment. In numerous papers, a standard preferred choice was μ = c44 ([43319285Jump To The Next Citation Point], and references therein). However, replacing μ by a single maximal elastic modulus of a strongly anisotropic bcc lattice is not correct. An effective value of μ was calculated by Ogata & Ichimaru [309]. They performed directional averages over rotations of the Cartesian axes. At T = 0, they obtained

1 n (Ze )2 μ = --(2b11 + 3c44) = 0.1194--N------, (116 ) 5 Rcell
some 30% smaller than μ = c44 used in previous papers. Dependence of μ 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 μ, Equation (116View Equation), can be rewritten as

( )2∕3 μ = 0.0159 Z-- Pe , (117 ) 26
where Pe is the pressure of ultrarelativistic degenerate electrons. Therefore,
μ ( Z )2 ∕3 P --= 0.016 --- --e- ≪ 1. (118 ) K 26 γP
The Poisson coefficient σ ≃ 1∕2, while the Young modulus E ≃ 3μ.

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 P ≃ Pe. For the inner crust these formulae are only approximate.

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