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Figure 3.17. Melting temperature (left) and electron and ion plasma temperatures (right) of the ground-state matter in the crust. Solid lines are based on the models of Haensel & Pichon (1994) and Negele & Vautherin (1973) for the outer and inner crusts, respectively. Jumps are associated with changes of nuclides. Dot-and-dashed lines are based on the CLDM of Douchin & Haensel (2000); their smooth behavior is an approximation inherent to the model. Thick vertical dashes indicate the neutron drip.

Here, / is the shear modulus and K is the compression modulus. Then the stress tensor is where y is the adiabatic index.

Detailed calculations of the direction-averaged effective shear modulus of a bcc Coulomb solid, appropriate for the polycrystalline crusts, were performed by Ogata & Ichimaru (1990). These authors considered a one component bcc Coulomb crystal, neglecting screening by degenerate electrons, as well as quantum zero-point motion of ions.

For an ideal bcc lattice there are only three independent elastic moduli denoted traditionally as en, c12, and c44 (Chapter 6 of Kittel 1986). A compres-sional deformation (V- u = 0) is determined only by two independent elastic moduli, because

Edef = bn [u2xx + u2yy + u2zz) + 2C44 [u2xy + v2xz + u2yz) , (3.52)

where bn = \(en — ci2). For T = 0, Ogata & Ichimaru (1990) find bn = 0.0245 nN (Ze)2/rc, c44 = 0.1827 nN (Ze)2/rc. These values agree with the

Figure 3.17. Melting temperature (left) and electron and ion plasma temperatures (right) of the ground-state matter in the crust. Solid lines are based on the models of Haensel & Pichon (1994) and Negele & Vautherin (1973) for the outer and inner crusts, respectively. Jumps are associated with changes of nuclides. Dot-and-dashed lines are based on the CLDM of Douchin & Haensel (2000); their smooth behavior is an approximation inherent to the model. Thick vertical dashes indicate the neutron drip.

For a pure uniform compression

Figure 3.18. Effective shear modulus p versus density at T = 0 for bcc lattice. The solid line is for the models of Haensel & Pichon (1994) and Negele & Vautherin (1973) (in the outer and inner crust, respectively; §§ 3.2 and § 3.3). The dot-and-dashed line is for the model of Douchin & Haensel (2000) (§ 3.3).

Figure 3.18. Effective shear modulus p versus density at T = 0 for bcc lattice. The solid line is for the models of Haensel & Pichon (1994) and Negele & Vautherin (1973) (in the outer and inner crust, respectively; §§ 3.2 and § 3.3). The dot-and-dashed line is for the model of Douchin & Haensel (2000) (§ 3.3).

classical result of Fuchs (1936). A significant difference between bu and c44 indicates a strong elastic anisotropy of an ideal bcc monocrystal.

While treating the crust as an isotropic solid is a reasonable approximation (we most probably deal with a bcc polycrystal), the choice of an "effective" shear modulus deserves a comment. In numerous papers treating the elastic aspects of neutron star dynamics, a standard choice was i = c44 (Baym & Pines 1971;Pandharipande etal. 1976; McDermott etal. 1988b, and references therein). It is clear, that replacing i by a single maximal elastic modulus of a strongly anisotropic lattice is not accurate. The correct effective value of i was calculated by Ogata & Ichimaru (1990) by direct averaging over rotations of Cartesian axes. For T = 0 their result is

5 rc about 1.5 times smaller than i = c44 used in previous papers.

Now return to the isotropic solid. Equation (3.53) can be rewritten as

where Pe is the pressure of ultra-relativistic degenerate electrons. Therefore, j/K = 0.016 (Z/26)2/3 (Pe/YP) < 1 • (3.55)

The crust is much more susceptible to shear than to compression; its Poisson coefficient a ~ 1/2, while its Young modulus E ~ 3j.

Strictly speaking, the above formulae hold for the outer crust, where rN ^ rc and P ~ Pe. In the inner crust they are only approximate.

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