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7.2 Nuclear pasta

Some theories of dense matter predict the existence of “nuclear pasta” – rods, plates, tubes, bubbles – in the bottom layer of the crust with ρ ≳ 1014 g cm–3 (see Section 3.3). In what follows we will concentrate on rods (spaghetti) and plates (lasagna). They are expected to fill most of the bottom crust layers. The matter phases containing rods and plates have properties intermediate between solids and liquids. The displacement of an element of matter parallel to the plane containing rods or plates is not opposed by restoring forces: this lack of a shear strain is typical for a liquid. On the contrary, an elastic strain opposes any bending of planes or rods: this is a property of a solid. Being intermediate between solids and liquids, these kinds of matter are usually called mesomorphic phases, or liquid crystals (see, e.g., [249]).

The elastic properties of rod and plate phases of neutron star matter were studied by Pethick & Potekhin [325Jump To The Next Citation Point]. As they stressed, the physical reasons for the forming of mesomorphic phases in neutron star crusts are very different from those relevant to liquid crystals in laboratory. For terrestrial liquid crystals it is the interaction between very nonspherical molecules, which drives them to form rods and plates. In a neutron star crust the mechanism consists in spontaneous symmetry breaking, resulting from competition between the Coulomb energy and nuclear surface energy. We will follow closely Pethick & Potekhin [325Jump To The Next Citation Point]. They calculated the energies of mesomorphic phases using the generalized liquid drop model. The plate phase has rotational symmetry about any axis perpendicular to the plates, and is therefore similar to the smectics A phase of liquid crystals [115Jump To The Next Citation Point]. Let the z-axis coincide with the symmetry axis of the equilibrium configuration. Only a displacement in the z-direction is opposed by a restoring force. Therefore, we consider only uuu = (0,0,u ). The deformation energy of a unit volume is then [115Jump To The Next Citation Point]

1 [∂u 1 ]2 1 ( )2 ɛdef = -B --- − --(∇∇∇ ⊥u)2 + --K1 ∇∇∇2⊥u , (119 ) 2 ∂z 2 2
where ∇∇∇ ⊥ ≡ (∂∕∂x, ∂∕∂y, 0). Using the generalized liquid drop model, Pethick & Potekhin [325Jump To The Next Citation Point] obtain
2 2 ( 2) B = 6ɛCoul K1 = 15-R cell ɛCoul 1 + 2w − 2w , (120 )
where ɛCoul is the Coulomb energy density (in equilibrium)
2π (1 − w)2 ɛCoul = ---(enpiRcell)2---------, (121 ) 3 w
and where Rcell is the half-distance between the plates, npi is the proton density in nuclear matter and w is the volume fraction occupied by nuclear matter (all values calculated for the relaxed system).

In the case of the rod phase, also called the columnar phase [115], the number of elastic moduli is larger. They describe the increase in energy density due to compression, dilatation, transverse shearing, and bending of the rod lattice. Elastic moduli were calculated within the liquid drop model by Pethick & Potekhin [325] and by Watanabe, Iida & Sato [419420].

At the microscopic scale (fermis), the elastic properties of the nuclear pastas are very different from those of a body-centered–cubic crystal of spherical nuclei. Nevertheless, the effects of pasta phases on the elastic properties of neutron star crusts may not be so dramatic at large scales (let’s say meters). Indeed these nuclear pastas are necessarily of finite extent since one and two-dimensional long-range crystalline orders cannot exist in infinite systems (see, for instance, [157] and references therein). How the nuclear pastas arrange themselves remains to be studied, but it is likely that the resulting configurations look more-or-less isotropic at macroscopic scales.

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