Neglecting neutron-proton mass difference, the beta equilibrium condition
reads At the density under consideration, electrons are ultrarelativistic, so thatThe inner crust of a neutron star is a unique system, which is not accessible in the laboratory due to the presence of this neutron gas. In the following we shall thus refer to the “nuclei” in the inner crust as “clusters” in order to emphasize these peculiarities. The description of the crust beyond neutron drip therefore relies on theoretical models only. Many-body calculations starting from the realistic nucleon-nucleon interaction are out of reach at present due to the presence of spatial inhomogeneities of nuclear matter. Even in the simpler case of homogeneous nuclear matter, these calculations are complicated by the fact that nucleons are strongly interacting via two-body, as well as three-body, forces, which contain about twenty different operators. As a result, the inner crust of a neutron star has been studied with phenomenological models. Most of the calculations carried out in the inner crust rely on purely classical (compressible liquid drop) and semi-classical models (Thomas–Fermi approximation and its extensions). The state-of-the-art calculations performed so far are based on self-consistent mean field methods, which have been very successful in predicting the properties of heavy laboratory nuclei.
We will present in detail the liquid drop model because this approach provides very useful insight despite its
simplicity. As in Section 3.1, we first start by writing the total energy density including the contribution
of the neutron gas
The volume contribution in Equation (34) is given by
The structure of the inner crust is determined by minimizing the total energy density for a given
baryon density
imposing electric charge neutrality
. The conditions of equilibrium
are obtained by taking the partial derivative of the energy density
with respect to the
free parameters of the model. In the following we shall neglect curvature corrections to the
surface energy. In this approximation, the surface potential
is independent of the shape and
size of the drop. The variation of the surface tension with the neutron excess is illustrated in
Figure 6
.
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The conditions of chemical equilibrium are
where andThe mechanical equilibrium of the crystal lattice can be expressed as
where Combining Equations (40) and (41
), the total Coulomb energy can be written as
Equation (49) shows that the equilibrium composition of the cluster is a result of the competition
between Coulomb effects, which favor small clusters, and surface effects, which favor large clusters. This also
shows that the lattice energy is very important for determining the equilibrium shape of the cluster,
especially at the bottom of the crust, where the size of the cluster is of the same order as the lattice
spacing. Even at the neutron drip, the lattice energy reduces the total Coulomb energy by about
15%.
The structure of the inner crust, as calculated from a compressible liquid drop model by
Douchin & Haensel [125], is illustrated in Figures 7
and 8
. One remarkable feature, which is
confirmed by more realistic models, is that the number
of protons in the clusters is almost
constant throughout the inner crust. It can also be seen that, as the density increases, the
clusters get closer and closer, while their size
varies very little. Let us also notice that at
the bottom of the crust the number
of neutrons, adsorbed on the surface of the clusters,
decreases with increasing density, because the properties of the matter inside and outside the
clusters become more and more alike. The results of different liquid drop models are compared in
Figure 9
.
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The liquid drop model is very instructive for understanding the contribution of different physical effects to the structure of the crust. However this model is purely classical and consequently neglects quantum effects. Besides, the assumption of clusters with a sharp cut surface is questionable, especially in the high density layers where the nuclei are very neutron rich.
Semi-classical models have been widely applied to study the structure of neutron star crusts. These models assume that the number of particles is so large that the quantum numbers describing the system vary continuously and instead of wave functions one can use the number densities of the various constituent particles. In this approach, the total energy density is written as a functional of the number densities of the different particle species
whereThe idea for obtaining the energy functional is to assume that the matter is locally homogeneous: this is known as the Thomas–Fermi or local density approximation. This approximation is valid when the characteristic length scales of the density variations are much larger than the corresponding interparticle spacings. The Thomas–Fermi approximation can be improved by including density gradients in the energy functional.
As discussed in Section 3.1, the electron density is almost constant so that the local density
approximation is very good with the electron energy functional given by Equation (22).
The Coulomb part in Equation (52) can be decomposed into a classical and a quantum contribution.
The classical contribution is given by
The total energy density is equal to the energy density of one unit cell of the lattice times the number of
cells. The minimization of the total energy density under the constraints of a fixed total baryon density
and global electro-neutrality
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Quantum calculations of the structure of the inner crust were pioneered by Negele & Vautherin [303]. These
types of calculations have been improved only recently by Baldo and collaborators [29
, 30
].
Following the Wigner–Seitz approximation [422
], the inner crust is decomposed into independent
spheres, each of them centered at a nuclear cluster, whose radius is defined by Equation (16
),
as illustrated in Figure 5
. The determination of the equilibrium structure of the crust thus
reduces to calculating the composition of one of the spheres. Each sphere can be seen as an exotic
“nucleus”. The methods developed in nuclear physics for treating isolated nuclei can then be directly
applied.
Starting from many-body calculations of uniform nuclear matter with realistic nucleon-nucleon
interaction, and expanding the nucleon density matrix in relative and center of mass coordinates, Negele &
Vautherin [302] derived a set of nonlinear equations for the single particle wave functions of the nucleons,
, where
for neutron and proton, respectively, and
is the set of quantum numbers
characterizing each single particle state. Inside the Wigner–Seitz sphere, these equations take the form
Equations (57) reduce to ordinary differential equations by expanding a wave function on the basis of
the total angular momentum. Apart from the nuclear central and spin-orbit potentials, the protons also feel
a Coulomb potential. In the Hartree–Fock approximation, the Coulomb potential is the sum of a direct part
, where
is the electrostatic potential, which obeys Poisson’s Equation (54
), and an exchange
part, which is nonlocal in general. Negele & Vautherin adopted the Slater approximation for
the Coulomb exchange, which leads to a local proton Coulomb potential. As a remark, the
expression of the Coulomb exchange potential used nowadays was actually suggested by Kohn &
Sham [243]. It is smaller by a factor
compared to that initially proposed by Slater [380]
before the formulation of the density functional theory. It is obtained by taking the derivative of
Equation (55
) with respect to the proton density
. Since the clusters in the crust are
expected to have a very diffuse surface and a thick neutron skin (see Section 3.2.2), the spin-orbit
coupling term for the neutrons (which is proportional to the gradient of the neutron density) was
neglected.
Equations (57) have to be solved self-consistently. For a given number
of neutrons and
of
protons and some initial guess of the effective masses and potentials, the equations are solved for the wave
functions of
neutrons and
protons, which correspond to the lowest energies
. These wave
functions are then used to recalculate the effective masses and potentials. The process is iterated until the
convergence is achieved.
Negele & Vautherin [303] determined the structure of the inner crust by minimizing the total energy per
nucleon in a Wigner–Seitz sphere, and thus treating the electrons as a relativistic Fermi gas. Since the
sphere is electrically neutral, the number of electrons is equal to
and the electron energy is easily
evaluated from Equation (22
) with
, where
is the volume of the sphere. As for the
choice of boundary conditions, Negele & Vautherin imposed that wave functions with even parity (even
)
and the radial derivatives of wave functions with odd parity (odd
) vanish on the sphere
. This prescription leads to a roughly constant neutron density outside the nuclear clusters.
However, the densities had still to be averaged in the vicinity of the cell edge in order to remove
unphysical fluctuations. The structure and the composition of the inner crust is shown in Table 3.
These results are qualitatively similar to those obtained with liquid drop models (see Figure 9
in Section 3.2.1) and semiclassical models (see Figure 10
in Section 3.2.2). The remarkable
distinctive feature is the existence of strong proton quantum-shell effects with a predominance of
nuclear clusters with
and
. The same magic numbers have been recently found
by Onsi et al. [311
] using a high-speed approximation to the Hartree–Fock method with an
effective Skyrme force that was adjusted on essentially all nuclear data. Note however that
the predicted sequence of magic numbers differs from that obtained by Negele and Vautherin
as can be seen in Figure 11
. Neutron quantum effects are also important (while not obvious
from the table) as can be inferred from the spatial density fluctuations inside the clusters in
Figure 12
. This figure also shows that these quantum effects disappear at high densities, where
the matter becomes nearly homogeneous. The quantum shell structure of nuclear clusters in
neutron star crusts is very different from that of ordinary nuclei owing to a large number of
neutrons (for a recent review on the shell structure of very neutron-rich nuclei, see, for instance,
[119]). For instance, clusters with
are strongly favored in neutron star crusts, while
is not a magic number in ordinary nuclei (however, it corresponds to a filled proton
subshell).
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Negele & Vautherin assume that nucleons can be described as independent particles in a mean field
induced by all other particles. However, neutrons and protons are expected to form bound
pairs due to the long-range attractive part of the nucleon-nucleon interaction, giving rise to the
property of superfluidity (Section 8). Baldo and collaborators [28, 32, 29
, 30
, 31
] have recently
studied the effects of these pairing correlations on the structure of neutron star crusts, applying
the generalized energy-density–functional theory in the Wigner–Seitz approximation. They
found that the composition of the clusters differs significantly from that obtained by Negele &
Vautherin [303
], as can be seen from Table 4 and Figure 11
. However, Baldo et al. stressed that
these results depend on the more-or-less arbitrary choice of boundary conditions imposed on
the Wigner–Seitz sphere, especially in the deepest layers of the inner crust. Therefore, above
2
1013 g cm–3 results of Baldo et al. (and those of Negele & Vautherin) should be taken with a grain
of salt. Another limitation of the Wigner–Seitz approximation is that it does not allow the calculation of
transport properties, since neutrons are artificially confined inside the sphere. A more realistic
treatment has been recently proposed by applying the band theory of solids (see [92] and references
therein).
The unbound neutrons in the inner crust of a neutron star are closely analogous
to the “free” electrons in an ordinary (i.e. under terrestrial conditions)
metal2.
Assuming that the ground state of cold dense matter below saturation density possesses the symmetry of a
perfect crystal, which is usually taken for granted, it is therefore natural to apply the band
theory of solids to neutron star crusts (see Carter, Chamel & Haensel [78] for the application to
the pasta phases and Chamel [90
, 91
] for the application to the general case of 3D crystal
structures).
The band theory is explained in standard solid-state physics textbooks, for instance in the book by
Kittel [241]. Single particle wave functions of nucleon species
in the crust are characterized by a
wave vector
and obey the Floquet–Bloch theorem
In the approach of Negele & Vautherin [302] (see Section 3.2.3), or in the more popular mean field method with effective Skyrme nucleon-nucleon interactions [47, 391], single particle states are solutions of the equations
neglecting pairing correlations (the application of band theory including pairing correlations has been discussed in [77 As a result of the lattice symmetry, the crystal can be partitioned into identical primitive cells, each
containing exactly one lattice site. The specification of the primitive cell is not unique. A particularly
useful choice is the Wigner–Seitz or Voronoi cell, defined by the set of points that are closer to
a given lattice site than to any other. This cell is very convenient since it reflects the local
symmetry of the crystal. The Wigner–Seitz cell of a crystal lattice is a complicated polyhedron
in general. For instance, the Wigner–Seitz cell of a body-centered cubic lattice (which is the
expected ground state structure of neutron star crusts), shown in Figure 13, is a truncated
octahedron.
Equations (63) need to be solved inside only one such cell. Indeed once the wave function in
one cell is known, the wave function in any other cell can be deduced from the Floquet–Bloch
theorem (61
). This theorem also determines the boundary conditions to be imposed at the cell
boundary.
For each wave vector , there exists only a discrete set of single particle energies
, labeled by
the principal quantum number
, for which the boundary conditions (61
) are fulfilled. The
energy spectrum is thus formed of “bands”, each of them being a continuous (but in general not
analytic) function of the wave vector
(bands are labelled by increasing values of energy, so that
if
). The band index
is associated with the rotational symmetry
of the nuclear clusters around each lattice site, while the wave vector
accounts for the
translational symmetry. Both local and global symmetries are therefore properly taken into account.
Let us remark that the band theory includes uniform matter as a limiting case of an “empty”
crystal.
In principle, Equations (63) have to be solved for all wave vectors
. Nevertheless, it can be shown by
symmetry that the single particle states (and, therefore, the single particle energies) are periodic in
-space
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An example of neutron band structure is shown in the right panel of Figure 15 from Chamel et al.[96
].
The figure also shows the energy spectrum obtained by removing the nuclear clusters (empty lattice),
considering a uniform gas of unbound neutrons. For comparison, the single particle energies, given in this
limiting case by an expression of the form
, have been folded into the first
Brillouin zone (reduced zone scheme). It can, thus, be seen that the presence of the nuclear clusters leads to
distortions of the parabolic energy spectrum, especially at wave vectors
lying on Bragg planes (i.e.,
Brillouin zone faces, see Figure 14
).
The (nonlinear) three-dimensional partial differential Equations (63) are numerically very difficult to
solve (see Chamel [90
, 91
] for a review of some numerical methods that are applicable to neutron star
crusts). Since the work of Negele & Vautherin [303
], the usual approach has been to apply the Wigner–Seitz
approximation [422]. The complicated Wigner–Seitz cell (shown in Figure 13
) is replaced by a sphere of
equal volume. It is also assumed that the clusters are spherical so that Equations (63
) reduce to ordinary
differential Equations (57
). The Wigner-Seitz approximation has been used to predict the structure of the
crust, the pairing properties, the thermal effects, and the low-lying energy-excitation spectrum of the
clusters [303
, 55
, 360
, 237
, 413
, 31
, 294
].
However, the Wigner–Seitz approximation overestimates the importance of neutron shell
effects, as can be clearly seen in Figure 15. The energy spectrum is discrete in the Wigner–Seitz
approximation (due to the neglect of the
-dependence of the states), while it is continuous
in the full band theory. The spurious shell effects depend on a particular choice of boundary
conditions, which are not unique. Indeed as pointed out by Bonche & Vautherin [54], two types of
boundary conditions are physically plausible yielding a more-or-less constant neutron density
outside the cluster: either the wave function or its radial derivative vanishes at the cell edge,
depending on its parity. Less physical boundary conditions have also been applied, like the
vanishing of the wave functions. Whichever boundary conditions are adopted, they lead to
unphysical spatial fluctuations of the neutron density, as discussed in detail by Chamel et al. [96
].
Negele & Vautherin [303
] average the neutron density in the vicinity of the cell edge in order to
remove these fluctuations, but it is not clear whether this ad hoc procedure did remove all the
spurious contributions to the total energy. As shown in Figure 15
, shell energy gaps are on the
order of
, at
. Since these gaps scale approximately like
(where
is the neutron mass), they increase with density
and eventually
become comparable to the total energy difference between neighboring configurations. As a
consequence, the predicted equilibrium structure of the crust becomes very sensitive to the choice
of boundary conditions in the bottom layers [30
, 96
]. One way of eliminating the boundary
condition problem without carrying out full band structure calculations, is to perform semi-classical
calculations including only proton shell effects with the Strutinsky method, as discussed by Onsi et
al.[311].
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In recent calculations [278, 304, 168] the Wigner–Seitz cell has been replaced by a cube
with periodic boundary conditions instead of Bloch boundary conditions (61). Although such
calculations allow for possible deformations of the nuclear clusters, the lattice periodicity is still not
properly taken into account, since such boundary conditions are associated with only one kind of
solutions with
. Besides the Wigner–Seitz cell is only cubic for a simple cubic lattice and
it is very unlikely that the equilibrium structure of the crust is of this type (the structure of
the crust is expected to be a body centered cubic lattice as discussed in Section 3.1). Let us
also remember that a simple cubic lattice is unstable. It is, therefore, not clear whether these
calculations, which require much more computational time than those carried out in the spherical
approximation, are more realistic. This point should be clarified in future work by a detailed
comparison with full band theory. Let us also mention that recently Bürvenich et al. [64] have
considered axially-deformed spheroidal W–S cells to account for deformations of the nuclear
clusters.
Whereas the Wigner–Seitz approximation is reasonable at not too high densities for determining the
equilibrium crust structure, full band theory is indispensable for studying transport properties (which
involve obviously translational symmetry and, hence, the -dependence of the states). Carter, Chamel &
Haensel [78
] using this novel approach have shown that the unbound neutrons move in the crust as if they
had an effective mass much larger than the bare mass (see Sections 8.3.6 and 8.3.7). This dynamic effective
neutron mass has been calculated by Carter, Chamel & Haensel [78
] in the pasta phases of rod and slab-like
clusters (discussed in Section 3.3) and by Chamel [90
, 91
] in the general case of spherical clusters. By
taking consistently into account both nuclear clusters, which form a solid lattice, and the neutron liquid,
band theory provides a unified scheme for studying the structure and properties of neutron star
crusts.
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