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3.2 Structure of the inner crust

With increasing density, the ground-state value of Z ∕A decreases and neutrons become less and less bound. Let us define the “net neutron chemical potential” of a neutron in a nucleus
( ) μ′ ≡ μ − m c2 = ∂E--{A,Z-} − m c2. (28 ) n n n ∂N Z n
As long as ′ μn < 0, all neutrons are bound within nuclei. The neutron drip point corresponds to ′ μn = 0; beyond this point neutrons “drip out of nuclei”, i.e. they begin to fill states in the continuous part of the energy spectrum. We can roughly localize the neutron drip point using the approximate mass formula for E ′{A, Z } = E{A, Z } − Amc2, where, for simplicity, we neglect the neutron-proton mass difference, putting 2 mn ≈ mp ≈ m = 939 MeV ∕c. Neglecting surface and Coulomb terms, we have
E ′{A, Z } ≈ A (E + E δ2) , (29 ) vol sym
where δ ≡ (N − Z)∕A, and Evol and Esym are nuclear volume and symmetry energies, respectively. Experimentally, Evol ≃ − 16 MeV and Esym ≃ 32 MeV. Using Equation (29View Equation), we can easily show that the value of δ at which ′ μn = 0 is
∘ -------------- δND = 1 − Evol∕Esym − 1. (30 )
Using experimental values of Evol and Esym, we find δND = 0.225.

Neglecting neutron-proton mass difference, the beta equilibrium condition

p + e → n + νe (31 )
reads
μe = μn − μp ≈ 4Esym δ. (32 )
At the density under consideration, electrons are ultrarelativistic, so that μe ≃ 5.16 (ρ9Z∕A )1∕3 MeV, where ρ = ρ∕109 g cm −3 9. It is now easy to show that the neutron drip density is roughly 11 − 3 ρND ≈ 2 × 10 g cm, which is about half of the value obtained in complete calculations [42Jump To The Next Citation Point183Jump To The Next Citation Point357Jump To The Next Citation Point]. For ρ > ρND ≈ 4 × 1011 g cm −3 a fraction of neutrons thus forms a gas outside the nuclei.

The 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.

3.2.1 Liquid drop models

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 ɛn of the neutron gas

ɛtot = nNE {A, Z } + ɛe + ɛL + ɛn . (33 )
The nuclear clusters are treated as liquid drops of nuclear matter whose energy can be decomposed into volume, surface and Coulomb terms
E {A,Z } = EN,vol + EN,surf + EN,Coul . (34 )
In the simplest version, the drop is supposed to be incompressible with a density on the order of ρ 0 corresponding to the density inside heavy nuclei. This implies that the volume and surface terms in Equation (34View Equation) are proportional to A and A2∕3, respectively. Each contribution to the energy, Equation (34View Equation), can then be parameterized in terms of the numbers A and Z. The parameters are adjusted to the known experimental masses of nuclei, with Z ∕A ∼ 0.5. The first models of neutron star crust were based on such semi-empirical mass formulae, see for instance the book by Haensel, Potekhin and Yakovlev [184Jump To The Next Citation Point]. However, such formulae can not be reliably extrapolated to the neutron rich nuclear clusters in neutron star crusts, where Z ∕A varies from ∼ 0.3 at the neutron drip threshold to ∼ 0.1 at the bottom of the crust. Besides, the presence of the neutron liquid has a profound effect on the clusters. First, it reduces the surface energy of the clusters as compared to that of isolated nuclei. Second, it exerts a pressure on the clusters. A major breakthrough was reached by Baym, Bethe and Pethick [39Jump To The Next Citation Point], who applied a compressible liquid-drop model, which included the results of microscopic many-body calculations, to describe consistently both the nucleons in the clusters and the “free” neutrons. The energy Equation (34View Equation) of the cluster then depends not only on A and Z but on a few additional parameters, such as, for instance, the size of the cluster and the density of the neutrons and protons inside it.

The volume contribution in Equation (34View Equation) is given by

E = ɛ {n ,n }𝒱 , (35 ) N,vol ni pi N
where ɛ{nn,np } is the energy density of homogeneous nuclear matter and nni and npi are respectively the neutron and proton densities inside the clusters. 𝒱N is the volume of the cluster. For consistency the energy density of the surrounding neutron gas is expressed in terms of the same function ɛ{n ,n } n p as
ɛ = ɛ{n ,0} (1 − w ), (36 ) n no
where nno is the number density of free neutrons outside the clusters and
( ) 𝒱N rp 3 w = 𝒱--- = R---- (37 ) cell cell
is the volume fraction of the cluster. Let us define the surface thermodynamic potential per unit area σ and the chemical potential μns of neutrons adsorbed on the surface of the drop (forming a neutron skin) by
∂E || ∂E || σ = ---N,surf|| , μns = --N,surf|| , (38 ) ∂𝒜 Ns ∂Ns 𝒜
where 𝒜 is the surface area of the cluster and N s is the number of adsorbed neutrons. Since energy is an extensive thermodynamic variable, it follows from Euler’s theorem about homogeneous functions that
EN,surf = σ 𝒜 + Ns μns. (39 )
The Coulomb energy of a uniformly-charged spherical drop of radius rp is given by
3 Z2e2 EN,Coul = -------. (40 ) 5 rp
Corrections due to the diffuseness of the cluster surface and due to quantum exchange can be found, for instance, in reference [277Jump To The Next Citation Point]. The electron energy density is approximately given by Equation (22View Equation). In the Wigner–Seitz approximation, assuming uniformly-charged spherical clusters of radius rp, the lattice energy is given by [39Jump To The Next Citation Point]
9 ( 4π )1∕3 ( 1 ) ɛL = − --- --- Z2 ∕3e2n4e∕3 1 − --w2∕3 . (41 ) 10 3 3
The physical input of the liquid drop model is the energy density of homogeneous nuclear matter ɛ{nn, np}, the surface potential σ and the chemical potential μns. For consistency, these ingredients should be calculated from the same microscopic nuclear model. The surface properties are usually determined by considering two semi-infinite phases in equilibrium (nucleons in clusters and free neutrons outside) separated by a plane interface (for curvature corrections, see, for instance, [273126Jump To The Next Citation Point]). In this approximation, the surface energy is proportional to the surface area
( ) EN,surf = σs + sn (nni − nno)μns 𝒜 , (42 )
where σs is the surface tension and sn the neutron skin thickness. Unlike the surface area 𝒜, the proportionality coefficient does not depend on the actual shape of the nuclear clusters.

The structure of the inner crust is determined by minimizing the total energy density ɛtot for a given baryon density nb imposing electric charge neutrality np = ne. The conditions of equilibrium are obtained by taking the partial derivative of the energy density ɛtot 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 σ s is independent of the shape and size of the drop. The variation of the surface tension with the neutron excess is illustrated in Figure 6View Image.

View Image

Figure 6: Surface tension of the nuclei in neutron star crusts versus neutron excess parameter δi inside the nuclear cluster, in the plane interface approximation with different Skyrme models [123].

The conditions of chemical equilibrium are

μbulk= μbulk = μns , (43 ) ni no
bulk bulk 8π-2 2 μni − μpi − μe = 5 e npirpf3{w }, (44 )
where
μbulk≡ -∂ɛ--, (45 ) X ∂nX
and X = ni,no,ns, ...
f3{w } ≡ 1 − 3-w1∕3 + 1w . (46 ) 2 2
The mechanical equilibrium of the cluster is expressed by
Pibulk− Pboulk= 2σs-− 4π-e2n2pir2p(1 − w ), (47 ) rp 15
which can be easily recognized as a generalization of Laplace’s law for an isolated drop. We have introduced bulk nuclear pressures by
bulk 2 ∂(ɛ∕nX-)- bulk PX ≡ nX ∂n = nX μX − ɛ . (48 ) X
Note that P biulk means the total pressure inside the drop, P biulk ≡ P bnuilk + Ppbiulk. The last term in Equation (47View Equation) comes from the pressure due to Coulomb forces between protons inside the cluster and between clusters of the lattice.

The mechanical equilibrium of the crystal lattice can be expressed as

EN,surf = 2ECoul, (49 )
where ECoul is the total Coulomb energy
ECoul = EN,Coul + 𝒱cellɛL . (50 )
Relation (49View Equation) is referred to as a “virial” theorem [39Jump To The Next Citation Point].

Combining Equations (40View Equation) and (41View Equation), the total Coulomb energy can be written as

ECoul = EN,Coulf3{w }, (51 )
where the dimensionless function f3{w} is given by Equation (46View Equation). Let us emphasize that the above equilibrium conditions are only valid if nuclear surface curvature corrections are neglected.

Equation (49View Equation) 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 [125Jump To The Next Citation Point], is illustrated in Figures 7View Image and 8View Image. One remarkable feature, which is confirmed by more realistic models, is that the number Z ∼ 40 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 rp varies very little. Let us also notice that at the bottom of the crust the number Ns 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 9View Image.

View Image

Figure 7: Structure of the ground state of the inner crust. Radius Rcell of the Wigner–Seitz cell, proton radius rp of spherical nuclei, and fraction w of volume filled by nuclear clusters (in percent), versus average baryon number density nb as calculated by Douchin & Haensel [124Jump To The Next Citation Point].
View Image

Figure 8: Composition of nuclear clusters in the ground state of the inner crust. Baryon number A of spherical clusters and their proton number Z, versus average baryon number density nb as calculated by Douchin & Haensel [124Jump To The Next Citation Point]. Ns is the number of neutrons adsorbed on the surface of the clusters.
View Image

Figure 9: Proton number Z of the nuclear clusters vs. density ρ in the ground state of the inner crust of neutron stars, calculated by various authors from different liquid drop models based on many-body theories with effective interactions: RBP(Ravenhall, Bennett and Pethick) [344], Douchin & Haensel [124Jump To The Next Citation Point], Pethick & Ravenhall [326Jump To The Next Citation Point]. For comparison, the results of the quantum calculations of Negele & Vautherin [303Jump To The Next Citation Point] (diamonds) are also shown.

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.

3.2.2 Semi-classical models

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

r r r r r r ɛ(rr) = ɛN{nn(rr),np (rr)} + ɛe{ne(rr)} + ɛCoul{ne(rr),np(rr)}, (52 )
where ɛN, ɛe and ɛCoul are the nuclear, electron and Coulomb contributions, respectively.

The 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 (22View Equation).

The Coulomb part in Equation (52View Equation) can be decomposed into a classical and a quantum contribution. The classical contribution is given by

1 ɛcClaosusl{ne (rrr),np (rrr)} = -e (np(rrr) − ne (rrr )) φ(rrr), (53 ) 2
where e is the proton electric charge and φ is the electrostatic potential, which obeys Poisson’s equation,
Δ φ = − 4πe (n (rrr) − n (rrr)) . (54 ) p e
The quantum contribution ɛcCoorrul accounts for the quantum correlations. For instance, the local part of the Coulomb exchange correlations induced by the Pauli exclusion principle is given by the Slater–Kohn–Sham functional [243Jump To The Next Citation Point]
3( 3 )1∕3 ( ) ɛcCororul{ne (rrr),np (rrr)} = − -- -- e2 np(rrr)4∕3 + ne(rrr)4∕3 . (55 ) 4 π
The nuclear functional ɛN {nn(rrr),np(rrr)} is less certain. Its local part is just a function of nn and np, and can, in principle, be inferred using the results of many-body calculations of the ground state of uniform asymmetric nuclear matter. However, the many-body calculations for the nonlocal part of the nuclear functional are much more difficult and have never been done in a fully satisfactory way. A simpler procedure is to postulate a purely phenomenological expression of the nonlocal part of the nuclear functional. The free parameters are then determined to reproduce some nuclear properties, for instance, the experimental atomic masses. Alternatively, the nuclear functional can be obtained from effective theories. In this case, the bare nucleon-nucleon interaction is replaced by an effective phenomenological interaction. It is then possible to deduce the nuclear functional in a systematic way using the extended Thomas–Fermi approximation (see for instance [58]). This approach has been developed for neutron star crust matter by Onsi and collaborators [312311Jump To The Next Citation Point].

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 nb and global electro-neutrality

∫ ∫ nb = -1-- d3r (nn(rrr) + np(rrr)) , d3r (np(rrr) − ne (rrr )) = 0 , (56 ) 𝒱cell cell cell
leads to Euler–Lagrange equations for the nucleon densities. In practice the unit cell is usually approximated by a sphere of the same volume 𝒱cell. The boundary conditions are that the gradients of the densities and of the Coulomb potential vanish at the origin r = 0 and on the surface of the sphere r = Rcell. Instead of solving the Euler–Lagrange equations, the nucleon densities are usually parameterized by some simple analytic functions with correct boundary behavior. Free parameters are then determined by minimizing the energy as in the compressible liquid drop models discussed in Section 3.2.1. The proton number of the nuclear clusters in the inner crust vs. mass density is shown in Figure 10View Image for different models. In those semiclassical models (as well as in liquid drop models discussed in Section 3.2.1), the number of bound nucleons inside the clusters in neutron star crusts varies continuously with depth. However, the nuclear clusters are expected to exhibit specific magic numbers of nucleons as similarly observed in isolated terrestrial nuclei, due to the clustering of quantum single-particle energy levels. The scattering of the unbound neutrons on the nuclear inhomogeneities leads also to “shell” (Casimir or band) effects [62278Jump To The Next Citation Point90Jump To The Next Citation Point]. The energy corrections due to shell effects have been studied perturbatively in semiclassical models [316130] and in Hartree–Fock calculations [278Jump To The Next Citation Point]. They have been found to be small. However, since the energy differences between different nuclear configurations are small, especially at high densities, these shell effects are important for determining the equilibrium structure of the crust. Calculations of the ground state structure of the crust, including proton shell effects, have recently been carried out by Onsi et al. [311Jump To The Next Citation Point]. As can be seen in Figure 11View Image, these shell effects significantly change the composition of the clusters predicting proton magic numbers Z = 20,40,50.
View Image

Figure 10: Proton number Z of the nuclear clusters vs. the mass density ρ in the ground state of the inner crust, calculated by different semi-classical models: Buchler & Barkat [61], Ogasawara & Sato [308Jump To The Next Citation Point], Oyamatsu [314], Sumiyoshi et al. [400], Goriely et al. [171]. For comparison the results of the quantum calculations of Negele & Vautherin [303Jump To The Next Citation Point] (diamonds) are also shown.

3.2.3 Quantum calculations

Quantum calculations of the structure of the inner crust were pioneered by Negele & Vautherin [303Jump To The Next Citation Point]. These types of calculations have been improved only recently by Baldo and collaborators [29Jump To The Next Citation Point30Jump To The Next Citation Point]. Following the Wigner–Seitz approximation [422Jump To The Next Citation Point], the inner crust is decomposed into independent spheres, each of them centered at a nuclear cluster, whose radius is defined by Equation (16View Equation), as illustrated in Figure 5View Image. 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 [302Jump To The Next Citation Point] derived a set of nonlinear equations for the single particle wave functions of the nucleons, ϕ (qα)(rrr), where q = n,p for neutron and proton, respectively, and α is the set of quantum numbers characterizing each single particle state. Inside the Wigner–Seitz sphere, these equations take the form

ℏ2 Wq (r ) − ∇∇∇ ⋅----⊕---∇∇∇ ϕ(αq)(rrr) + Uq(r)ϕ (qα)(rrr) + ------ℓℓℓ ⋅σσσ ϕ(αq)(rrr) = ε(αq)ϕ (qα)(rrr), (57 2m q (r) r
where ε(qα) is the single particle energy, ℓℓℓ ≡ − irrr × ∇∇∇ is the dimensionless orbital–angular-momentum operator and σσσ is a vector composed of Pauli spin matrices. The effective masses ⊕ m q (r), the mean fields Uq (r) and the spin-orbit potentials Wq (r) depend on wave functions of all nucleons inside the sphere through the particle number densities
∑ (q) 2 nq(rrr) = |ϕα (rrr)| , (58 ) α
the kinetic energy densities (in units of ℏ2∕2m, where m is the nucleon mass)
∑ (q) 2 τq(rrr) = |∇∇∇ ϕ α (rrr)| , (59 ) α
and the spin-orbit densities
∑ JJJqqq(rrr) = rrr ϕ(q)(rrr )∗ℓℓℓ ⋅σσσ-ϕ(q)(rrr). (60 ) α α r2 α

Equations (57View Equation) 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 eφ (rrr ), where φ (rrr) is the electrostatic potential, which obeys Poisson’s Equation (54View Equation), 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 3∕2 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 (55View Equation) with respect to the proton density n (r) p. 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 (57View Equation) have to be solved self-consistently. For a given number N of neutrons and Z of protons and some initial guess of the effective masses and potentials, the equations are solved for the wave functions of N neutrons and Z protons, which correspond to the lowest energies (q) εα. These wave functions are then used to recalculate the effective masses and potentials. The process is iterated until the convergence is achieved.

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Figure 11: Proton number Z of the nuclear clusters vs. the mass density ρ in the ground state of the inner crust, calculated by different quantum models: Negele & Vautherin [303Jump To The Next Citation Point], the model P2 from Baldo et al. [31Jump To The Next Citation Point32Jump To The Next Citation Point] and the recent calculations of Onsi et al [311Jump To The Next Citation Point].

Negele & Vautherin [303Jump To The Next Citation Point] 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 Z and the electron energy is easily evaluated from Equation (22View Equation) with ne = Z∕𝒱cell, where 𝒱cell is the volume of the sphere. As for the choice of boundary conditions, Negele & Vautherin imposed that wave functions with even parity (even ℓ) and the radial derivatives of wave functions with odd parity (odd ℓ) vanish on the sphere r = Rcell. 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 9View Image in Section 3.2.1) and semiclassical models (see Figure 10View Image 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 Z = 40 and Z = 50. The same magic numbers have been recently found by Onsi et al. [311Jump To The Next Citation Point] 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 11View Image. 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 12View Image. 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 Z = 40 are strongly favored in neutron star crusts, while Z = 40 is not a magic number in ordinary nuclei (however, it corresponds to a filled proton subshell).


Table 3: Sequence of nuclear clusters in the ground state of the inner crust calculated by Negele & Vautherin [303Jump To The Next Citation Point]. Here N is the total number of neutrons in the Wigner–Seitz sphere (i.e., it is a sum of the number of neutrons bound in nuclei and of those forming a neutron gas, per nucleus). Isotopes are labelled with the total number of nucleons in the Wigner–Seitz sphere.
ρ [g cm–3] Element Z N Rcell [fm]
4.67 × 1011 180Zr 40 140 53.60
6.69 × 1011 200Zr 40 160 49.24
1.00 × 1012 250Zr 40 210 46.33
1.47 × 1012 320Zr 40 280 44.30
2.66 × 1012 500Zr 40 460 42.16
6.24 × 1012 950Sn 50 900 39.32
9.65 × 1012 1100Sn 50 1050 35.70
1.49 × 1013 1350Sn 50 1300 33.07
3.41 × 1013 1800Sn 50 1750 27.61
7.94 × 1013 1500Zr 40 1460 19.61
1.32 × 1014 982Ge 32 950 14.38


Table 4: Sequence of nuclear clusters in the ground state of the inner crust calculated by Baldo et al. [31Jump To The Next Citation Point32Jump To The Next Citation Point] including pairing correlations (their P2 model). The boundary conditions are the same as those of Negele and Vautherin [303Jump To The Next Citation Point]. Similarly, N is the total number of neutrons in the Wigner–Seitz sphere. The isotopes are labelled with the total number of nucleons in the Wigner–Seitz sphere, as in Table 3.
ρ [g cm–3] Element Z N R cell [fm]
4.52 × 1011 212Te 52 160 57.19
1.53 × 1012 562Xe 54 508 52.79
3.62 × 1012 830Sn 50 780 45.09
7.06 × 1012 1020Pd 46 974 38.64
1.22 × 1013 1529Ba 56 1473 36.85
1.94 × 1013 1351Pd 46 1305 30.31
2.89 × 1013 1269Zr 40 1229 25.97
4.12 × 1013 636Cr 20 616 18.34
5.65 × 1013 642Ca 20 622 16.56
7.52 × 1013 642Ca 20 622 15.05
9.76 × 1013 633Ca 20 613 13.73

View Image

Figure 12: Nucleon number densities (in fm–3) along the axis joining two adjacent Wigner–Seitz cells of the ground state of the inner crust, for a few baryon densities n b (in cm–3), as calculated by Negele & Vautherin [303Jump To The Next Citation Point].

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 [2832Jump To The Next Citation Point29Jump To The Next Citation Point30Jump To The Next Citation Point31Jump To The Next Citation Point] 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 [303Jump To The Next Citation Point], as can be seen from Table 4 and Figure 11View Image. 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).

3.2.4 Going further: nuclear band theory

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 [78Jump To The Next Citation Point] for the application to the pasta phases and Chamel [90Jump To The Next Citation Point91Jump To The Next Citation Point] 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 [241Jump To The Next Citation Point]. Single particle wave functions of nucleon species q = n,p in the crust are characterized by a wave vector kkk and obey the Floquet–Bloch theorem

ϕ (q)(rrr + TTT ) = eikkk⋅TTTϕ (q)(rrr), (61 ) kkk kkk
where TTT is any lattice translation vector (which transforms the lattice into itself). This theorem implies that the wave functions are modulated plane waves, called simply Bloch waves
(q) ikkk⋅rrr (q) ϕkkk (rrr) = e ukkk (rrr), (62 )
with u (q)(rrr) kkk having the full periodicity of the lattice, u(q)(rrr + TTT) = u(q)(rrr ) kkk kkk.

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 [47391], single particle states are solutions of the equations

---ℏ2--- (q) (q) (q) (q) (q) − ∇ ⋅2m ⊕ (rrr )∇ ϕkkk (rrr) + Uq(rrr)ϕkkk (rrr) − iWWWqqq (rrr) ⋅ ∇ × σσσϕ kkk (rrr) = ε (kkk)ϕkkk (rrr), (63 ) q
neglecting pairing correlations (the application of band theory including pairing correlations has been discussed in [77Jump To The Next Citation Point]). Despite their apparent simplicity, these equations are highly nonlinear, since the various quantities depend on the wave functions (see Section 3.2.3).

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 13View Image, is a truncated octahedron.

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Figure 13: Wigner–Seitz cell of a body-centered cubic lattice.

Equations (63View Equation) 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 (61View Equation). This theorem also determines the boundary conditions to be imposed at the cell boundary.

For each wave vector kkk, there exists only a discrete set of single particle energies (q) εα (kkk), labeled by the principal quantum number α, for which the boundary conditions (61View Equation) 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 kkk (bands are labelled by increasing values of energy, so that (q) (q) εα (kkk) ≤ εβ (kkk) if α < β). The band index α is associated with the rotational symmetry of the nuclear clusters around each lattice site, while the wave vector kkk 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 (63View Equation) have to be solved for all wave vectors kkk. Nevertheless, it can be shown by symmetry that the single particle states (and, therefore, the single particle energies) are periodic in kkk-space

ϕ(q) (rrr) = ϕ(q)(rrr), (64 ) kkk+GGG kkk
where the reciprocal lattice vectors GGG are defined by
GGG ⋅TTT = 2πN , (65 )
N being any positive or negative integer. The discrete set of all possible reciprocal vectors GGG defines a reciprocal lattice in kkk-space. Equation (64View Equation) entails that only the wave vectors kkk lying inside the first Brillouin zone (i.e. Wigner–Seitz cell of the reciprocal lattice) are relevant. The first Brillouin zone of a body-centered cubic lattice is shown in Figure 14View Image.
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Figure 14: First Brillouin zone of the body-centered cubic lattice (whose Wigner–Seitz is shown in Figure 13View Image). The directions x, y and z denote the Cartesian axis in kkk-space.

An example of neutron band structure is shown in the right panel of Figure 15View Image from Chamel et al.[96Jump To The Next Citation Point]. 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 ε(kkk) = ℏ2k2∕(2m ⊕ ) + Un n, 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 kkk lying on Bragg planes (i.e., Brillouin zone faces, see Figure 14View Image).

The (nonlinear) three-dimensional partial differential Equations (63View Equation) are numerically very difficult to solve (see Chamel [90Jump To The Next Citation Point91Jump To The Next Citation Point] for a review of some numerical methods that are applicable to neutron star crusts). Since the work of Negele & Vautherin [303Jump To The Next Citation Point], the usual approach has been to apply the Wigner–Seitz approximation [422]. The complicated Wigner–Seitz cell (shown in Figure 13View Image) is replaced by a sphere of equal volume. It is also assumed that the clusters are spherical so that Equations (63View Equation) reduce to ordinary differential Equations (57View Equation). 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  [303Jump To The Next Citation Point55Jump To The Next Citation Point360Jump To The Next Citation Point237Jump To The Next Citation Point413Jump To The Next Citation Point31Jump To The Next Citation Point294Jump To The Next Citation Point].

However, the Wigner–Seitz approximation overestimates the importance of neutron shell effects, as can be clearly seen in Figure 15View Image. The energy spectrum is discrete in the Wigner–Seitz approximation (due to the neglect of the kkk-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. [96Jump To The Next Citation Point]. Negele & Vautherin [303Jump To The Next Citation Point] 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 15View Image, shell energy gaps are on the order of Δ ε ∼ 100 keV, at 11 − 3 ρ ≃ 7 × 10 g cm. Since these gaps scale approximately like 2 2 Δ ε ∝ ℏ ∕(2mnR cell) (where mn is the neutron mass), they increase with density ρ 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 [30Jump To The Next Citation Point96Jump To The Next Citation Point]. 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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Figure 15: Single particle energy spectrum of unbound neutrons in the ground state of the inner crust composed of 200Zr, at ρ ≃ 7 × 1011 g cm–3, obtained by Chamel et al. [96]. Left: calculation in the Wigner–Seitz approximation. Right: full band structure calculation (reduced zone scheme) assuming that the crust is a body-centered cubic lattice with nuclear clusters (solid line) and without (dashed line). The capital letters on the horizontal axis refer to lines or points in k-space, as indicated in Figure 14View Image.

In recent calculations [278304168] the Wigner–Seitz cell has been replaced by a cube with periodic boundary conditions instead of Bloch boundary conditions (61View Equation). 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 k kk = 0. 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 kkk-dependence of the states). Carter, Chamel & Haensel [78Jump To The Next Citation Point] 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 [78Jump To The Next Citation Point] in the pasta phases of rod and slab-like clusters (discussed in Section 3.3) and by Chamel [90Jump To The Next Citation Point91Jump To The Next Citation Point] 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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