Excitation modes and pairing interaction in the inner crust of a neutron stars

ELSEVIER

Nuclear Physics A73 1 (2004) 401-408 www.elsevier.comllocate/npe

Excitation stars.

modes and pairing

G. Goriaatl, F. Ramponi and E. Vigezzi ”

a,b, F. Barranco

B Dipartimento Milano, Italy

di Fisica,

“INFN,

di Milano,

Sezione

“Departamento Descubrimientos

interaction

Universit6

via Celoria

‘, R.A. Broglia

degli Studi

University

di Milano,

Superior

of Copenhagen,

crust

of a neutron

a,b,d, G.L. Co16 B;b, D. Sarchi a.b via Celoria

16, 20133, Milano,

de Fisica Aplicada III, Escuela s/n, 41092 Seville, Spain.

“The Niels Bohr Institute, 0, Denmark

in the inner

16, 20133,

Italy de Ingenieros,

Blegdamsvej

Camino

de 10s

17, 2100 Copenhagen

We perform a detailed calculation of the pairing properties of the inner crust of a neutron star. We focus on a specific density, and take into account the coexistence of a nuclear lattice with a superfluid gas of neutrons. The Hartree-Fock-Bogoliubov (HFB) equations are solved adding the matrix elements of the direct nucleon-nucleon interaction and those of the interaction induced by medium-polarization effects. The latter are obtained after a detailed calculation of the excited states of the system, within the random phase approximation. It is concluded that for the density considered, polarization processes reduce the pairing gap of about a factor of 2.

1. THE INNER

CRUST

OF A NEUTRON

STAR.

In the interior of a neutron star there exists a region, the so called inner crust, where nuclei are placed at the vertices of a Coulomb lattice, in order to minimize the Coulomb repulsion. These nuclei are surrounded by a sea of free neutrons that, due to the low temperature (T M 0.1 MeV), are superfluid. The properties of this lattice were firstly investigated in detail by Negele and Vautherin [l]: they treated the lattice within the Wigner-Seitz method, namely they studied the properties of an elementary cell which represents the building block of the whole lattice. For different regions of the inner crust, each characterized by a given density, they determined the lattice step and the number of protons and neutrons in the elementary cell which minimize the energy of the system. The energy was calculated within the framework of the Hartree-Fock theory by using an effective interaction. Approaching the center of the star, the step of the lattice reduces while the density of free neutrons increases. The inner crust plays an important role in many models which try to connect the electromagnetic radiation emitted from neutron stars with their internal structure. One 0375-9474/$ - see front matter 0 2004 Published by Elsevier B.\! doi:10.1016/j.nuc1physa.2003.11.052

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Figure 1. In the first row, the Feynman graphs associated to the bare nucleon-nucleon interaction (a) and to the interaction induced by exchange of phonons (b) are depicted. In the second row it is drawn the Feynman graph associated to the vertex entering in the calculation of equation (1) produced by the interaction of a particle state with the RPA phonon expanded on the particle-hole basis.

can mention the phenomenon of the glitches (cf. e.g. [2] and references therein), namely, the sudden acceleration observed in the rotational frequency of the star. One of the explanations proposed for this acceleration is based on the dynamics of vortices within the Coulomb lattice. The cooling process is another potential source of information about the interior of a neutron star. In fact, the low-density region of the inner crust represents a bottleneck in the heat diffusion inside a neutron star and the diffusion time in this region is strongly influenced by the pairing properties (cf. [3]). The pairing properties of nuclear and neutron matter arise from the combined effect of the bare nucleon-nucleon interaction as well as of the interaction induced by medium polarization effects (cf. Fig. la, b and also the contribution by Prof. Lombard0 in these Proceedings). The aim of our work is thus to make a calculation of the pairing properties of the inner crust taking into account, on one hand, the coexistence of the nuclear lattice with the gas of free neutrons and their proximity effects [4]; and, on the other hand, the combined effect of the direct and of the induced interaction (51.

2. THE LATTICE

AND

THE WIGNER-SEITZ

METHOD.

In order to account for the presence of the lattice, we adopt the Wigner-Seitz method, namely we study a cell represented by a spherical infinite well whose radius is equal to half the lattice step. In the following, we shall focus on a particular density (pi =1.8~10~~ cm -3, corresponindg to k p = 0.33 fm-I). We th en rely on the calculations of Negele and

G. Gori et al. /Nuclear Physics A731 (2004) 401-408

403

0.08 -

0.06 -

Figure 2. The proton (dashed line) and neutron (solid line) density distributions in the Wigner-Seitz cell studied in this contribution. Although the density of the external neutron gas is one order of magnitude smaller than the density at the center of the nucleus, there are about 400 free neutrons in the cell, and only 130 bound neutrons.

Vautherin, who found that at this density the radius of the Wigner-Seitz cell is equal to 42 fm, and that it contains 530 neutrons and 50 protons. The nucleons in this box feel a self-consistent potential obtained performing a Hartree-Fock calculation by using a Skm’ Skyrme force. In Fig. 2, the density profiles of protons and neutrons obtained after a self-consistant calculation are shown. 3. POLARIZATION THEORY.

EFFECTS

AND

BLOCH-HOROWITZ

PERTURBATION

When two neutrons inside a neutron star interact, their mutual force my be rather different to that in free space. In fact the presence of the other neutrons may produce a screening, which in general will depend on the neutron density and also on the detailed structure of the star at the nuclear scale (Coulomb lattice). A fully quanta1 description of such a scenario may be obtained by a generalization to positive Fermi energies of the methods developed for the study of the induced particle-particle interaction in nuclei [5]. In fact the Wigner-Seitz cell may be viewed as a single nucleus for which the neutron Fermi energy has been raised to positive values, the neutrons occupying positive energy levels corresponding to the free neutrons of the star. In order to assess the effects of polarization processes, the matrix elements associated to the exch.ange of phonons are thus calculated within the Bloch-Horowitz perturbation theory [B]. The matrix element that describes the scattering of two particles coupled to zero angular momentum, is given

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Physics A731 (2004) 401-408

where a, b, c, d, p, and q are single-particle states characterized by the set of quantum numbers n, 1, j, mj, while JM represents the angular momentum of the phonon and its projection, Eint is the energy of the intermediate states given by two particles and one phonon and Eo is the energy of the correlated ground state. A central role in the calculation of the matrix element (1) is played by the vertex connecting the single-particle states to the excited states of the system (cf. Fig. lc). In order to calculate the vertices, we need the energies of the single-particle states, the interaction which couples the single-particle states to the phonons and the properties of the excited states. The effective interaction is calculated as the second derivative of the Hartree-Fock energy with respect to the density (cf. [7] and reference therein):

For a Skyrme-force,

this effective

interaction

reduces to a S-force

(2) In this effective interaction we may disentangle a part which depends on the spin variable and another part independent from the spin variable; we will see how the former induces graphs of Fig. lb which are repulsive, while those associated to the latter are attractive. In order to describe the properties of the excited states in the spherical cell, we solved the coupled RPA equation (cf. [8], formula 14.12) for ph onons with defined total angular momentum J and parity 7r. Fig. 3 (top) shows the 2+ response of the system to the operator F(r’) = r2Y2(fi) and to F(f) = r2[Y2(s2) @ CT]. Th e b ars and the dashed lines correspond to the unperturbed response, while the solid lines show the correlated response obtained after the RPA calculation. The response is much more similar to that of a uniform system than to that of atomic nucleus; in particular, the shell structure that in ordinary nuclei gives rise to the low-lying 2+ state or to the Giant Resonance is completely washed out. In the case of the 2+ response, we need to consider excitations with energies up to 2 MeV in order to satisfy the EWSR. This cut-off has to be increased when we consider higher multipolarity as in the case of the lO+ response where we need to include states with energy of 10 MeV (cf. Fig. 3, bottom) . 4. INDUCED

INTERACTION

MATRIX

ELEMENTS.

Making use of the calculated excited states of the system, one can calculate the induced interaction matrix elements. In order to discuss some features of these matrix elements and hence of the interaction induced by polarization effects, we will analyze in some detail the values of the diagonal matrix elements. In this case, only phonons with positive parity may be exchanged between the particles in time reversal states. Phonons with natural

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Physics A731 (2004) 401-408

405

?

kphoniMeVj

Figure 3. These four pictures depicts the response of the cell after the action of different external fields. Starting from the top left and moving clockwise the external fields are respectively r2Y2, r2[Ys@a12, rl”Yio and rlo[Yr~@~]lo. In each picture the bars correspond to the descrete solutions we obtained from the diagonalization of the RPA matrix while the continues lines are obtaine by enalrging each discrete solution with a lorentian of width equalt to 0.7 MeV. The dashed lines and bars are associated to the uncorrelated Hartree-Fock response while the solid ones are associated to the correlated RPA one.

(non-natural) parity are coupled to the single-particle states by the spin-independent (spin-dependent) part of the interaction. In Fig. 4 the empty circles denote the matrix elements associated to the exchange of non-natural parity phonons: the exchange of these phonons lead to a repulsive correlations. The filled circles are associated to the exchange of natural parity phonons: these correlations are attractive. Due to the energy denominator entering Eq. (l), both contributions are important only for states which lie close to the Fermi energy. The two contributions are of the same order of magnitude and therefore they partially cancel out. In Fig. 5, the induced interaction matrix elements are compared with the matrix elements of the Gogny interaction. From that graph, one can see how the order of magnitude of the induced interaction is comparable with that of the nucleonnucleon interaction but with an opposite sign. We therefore expect a reduction of the pairing gap.

G. Gori et al. /Nuclear Physics A731 (2004) 401-408

406

I

I

I

I 2

I 4

I 6

I

3 Oi’ F2 -0.004 V I -0.008 t I 0

EaWV1 Figure 4. Matrix elements of the interaction induced by polarization effects associated to diagonal configurations (nnolajomo, n,l,j,m,IV,,,In,l,j,m,, n,l,j,6,). The filled circles are the matrix elements associated to the exchange of phonons with natural parity while the empty circles are those associated to the exchange of phonons with non-natural parity. In both cases, the exchanged phonons have J up to 18.

5. CALCULATION

OF THE

PAIRING

GAP.

In order to estimate the influence of these polarization processes on the pairing properties of the system, we solved the generalized Bogoliubov-Valatin quasi-particle equations as discussed in [4]. The pairing matrix elements are obtained by summing the contributions of the nucleon-nucleon bare interaction - here simulated by a Gogny force - and of the induced interaction matrix element. In Fig. 6 we show the diagonal elements of the pairing gap: the filled circles are associated to the calculation performed by using only the Gogny force, while the empty circles are obtained adding the induced interaction. As we argued before, at the density of this cell, the polarization effects turn out to be globally repulsive and therefore lead to a reduction of the pairing gap. We compare the average of the state-dependent gaps associated to levels 5 which lie within two MeV from the Fermi energy. For the calculation with only the bare nucleon-nucleon interaction, we obtain a value of 1. MeV while for the complete calculation bare plus induced interaction, we obtain a value of 0.6 MeV, corresponding to a reduction of about a factor two. This factor is in reasonable agreeement with the factor 2.2 predicted by Gorkov and Melik-Barkhudarov [9] in their analytic study of polarization effects in a system of dilute fermions interacting through a delta force. Note that in [lo] a complete annihilation of the pairing gap due to the exchange of spin modes is predicted. around ICF = 0.3fm-l. Th en calculation is quite different from ours

G. Gori et al. /Nuclear

Physics A731 (2004) 401-408

Figure 5. Diagonal matrix elements (n,l,j,m,, interaction induced by polarization effects (filled to the Gogny interaction (empty triangles).

407

~,l,j,m,lVl~n,l,j,m,, circles)

compared

n,l,j,fia) with

of the those associated

and a comparison is not easy. A possible origin of the difference can be found in the fact that they not only renormalize the particle-particle but also the particle-hole interaction, which according to their results strongly reduce, at this density, the attractive contribution associated to the exchange of density modes. With respect to [ll] it must be noted that they calculate the effects of the self-energy plus the screening due to the insertion of a bubble. In our calculation we neglect the self-energy contribution, but at this low density it does not appear to be the source of discrepancy since, as they find, its effect is negligible. It would be thus interesting to establish if our stronger effect arises from the fact that we consider the exchange of correlated particle-hole excitations and not only of single bubbles. 6. CONCLUSION

AND

PERSPECTIVES.

We have proposed a procedure to incorporate the medium polarization effects in a Wigner-Seitz cell in order to describe the pairing properties of the inner crust of a neutron star, considering both the free neutron sea and the nuclei at the vertices of the Coulomb lattice. We have applied it to a cell in the low density region and have found a reduction of the pairing gap compatible with analytical results for uniform neutron matter, that is a reduction of about a factor 2. Calculations corresponding to higher density regions are in progress. It is expected that the results will apart from the uniform system predictions due to the more relevant

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Physics A731 (2004) 401-405

Figure 6. The values of the &k in the canonical energy of the level Ic. The filled circles correspond only the Gogny forces, while the empty circles interaction.

basis as a function of the single-particle to the values of the gaps obtained using are those obtained adding the induced

presence of the lattice nuclei in each single cell. The calculations presented in this contribution were carried out on machines at CILEA, Consorzio Interuniuersitario Lombard0 per 1’Elaborazione Automatica: the assistance of its staff is gratefully aknowledged.

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

J. Negele and D. Vautherin, NuclPhysA207 (1973)298; P.M. Pizzochero et al. Phys.Rev.Lett 79(1997)3347; P.M. Pizzochero et al. Astrophys. J 569 (2002)381; F. Barranco et al. Phys.Lett. B390 (1997)13; F. Barranco et al., Phys.Rev.Lett. 83(1999)2147; G.G. Dussel and R.J. Liotta, Phys.Lett. 37B(1971)477; V. Bernard and N. Van Giai, Nucl. Phys. A348(1980)75; D.J. Rowe, Nuclear Collective Motion, Methuen and Co., London (1970); L.P. Gorkov and T.K. Melik-Barkhudarov, Sov.Phys. JETP 13(1961)1018; H.-J. Schulze et al., Phys.Lett.B375 (1996)l; C. Shen et al., nucl-thf0212027.