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Pairing correlations in the inner crust of neutron stars
Nuclear Physics A 752 (2005) 600c–603c
Pairing correlations in the inner crust of neutron stars E. Vigezzia , F. Barrancob , R.A. Brogliaa,c , G. Col`oa , G. Goria , F. Ramponia a
Dipartimento di Fisica, Universit`a di Milano and INFN Sez. Milano, Via Celoria 16, 20123 Milano b Departamento de Fisica Aplicada III, Escuela Superior de Ingenieros, University of Sevilla, Camino de los Descubrimientos s/n, 41092 Sevilla, Spain c The Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, 20100 Copenhagen, Denmark We calculate the pairing gap in the inner crust of neutron stars, taking into account both the proximity effects associated with the coexistence of the nuclear lattice with the sea of free neutrons, and the polarization effects associated with the exchange of density and spin fluctuations. It is found that the presence of the nuclei attenuates the quenching of the gap produced by the polarization. It is generally accepted that in the density range 0.001ρ0 < ρ < 0.5ρ0 , where ρ0 is the nuclear saturation density, neutron stars display an inner crust, consisting of a Coulomb lattice of nuclei immersed in a gas of free neutrons and relativistic electrons [1]. In analogy to the case of neutron matter in the same density range, the crust is expected to be superfluid in the 1 S0 channel, and this has important consequences on the thermal properties of the star [2]. Many studies have shown that it is essential to include polarization effects in the calculation of the pairing gap in infinite neutron matter [3]. In fact, such effects reduce the gap substantially, as compared to the values obtained in a mean-field calculation with the bare nucleon-nucleon force. Furthermore, in the present case one needs to determine the microscopic spatial dependence of the pairing gap, in order to have an accurate description of the superfluidity in the crust, which is affected by the proximity effects associated with the coexistence of two different phases (finite nuclei and neutron gas). This is particularly important for an accurate study of vortex dynamics, which according to theoretical models is at the origin of the phenomenon of glitches [4]. Recently, proximity effects have been calculated at the mean field level in the Hartree-Fock-Bogoliubov (HFB) approximation [5,6]. In this contribution we extend such studies, including the effects associated with the exchange of density and spin vibrations (cf. Fig. 1), which at low density are the most important processes renormalizing the pairing gap. We shall perform our calculation treating the nuclear lattice in the Wigner-Seitz approximation, considering a single Wigner cell with a spherical nucleus at the center, surrounded by a sea of free neutrons. In the following we shall present results for a cell of radius Rcell = 0375-9474/$ – see front matter © 2005 Published by Elsevier B.V. doi:10.1016/j.nuclphysa.2005.02.062
E. Vigezzi et al. / Nuclear Physics A 752 (2005) 600c–603c
601c
Figure 1. Diagrams depicting (a) the particle-vibration coupling vertex and (b) the pairing interaction induced by the exchange of phonons.
42 fm containing 50 protons and about 500 neutrons. These parameters are appropriate for the low-density part of the crust, and have been taken from the study of Negele and Vautherin, who determined the optimal lattice step and the ratio between neutrons and protons [7]. We first determine the mean-field potential and the single-particle levels of energy ν (ν ≡ nlj) with the associated radial wavefunctions φν , performing an HartreeFock calculation with a two-body effective interaction of the Skyrme type (SkM∗ ). The nucleus accomodates 130 bound neutrons, while the other neutrons are found at positive energy, and the Fermi energy F lies at about 1.4 MeV. Making use of the associated particle-hole basis states |jp jh−1 , JM we then calculate the vibrational states (phonons) in the Random Phase Approximation (RPA) [8]. The phonons are classified according to their angular momentum and projection J, M and to their parity π, while i =1,2,. . . labels the states with increasing energy. The spin-orbit interaction has been included in the calculation but it has a very small effect on the free neutrons, which dominate the RPA response. Consequently, the phonons have essentially good spin and can be characterized as density modes (S = 0) or spin modes (S = 1). From the RPA calculation one gets the transition densities δρ (we denote by δρn and δρp the neutron and proton contributions) of these excited states, which are used to calculate the particle-vibration coupling vertices (cf. Fig. 1(a)). The matrix elements of the interaction induced by the exchange of the phonons between pairs of neutrons in time reversal are given by the expression [9] ¯ |vind |νmν m < ν m ν m ¯ >= 2
m νm (f + g)ννm;J π M i (f − g)νm;J π M i E0 − Eint JπM i
(1)
where the factor of 2 arises from the two possible time ordering associated with graph (b) of Fig. 1. The particle-vibration matrix elements (cf. Fig. 1(a)) f and g in Eq.(1) refer respectively to the coupling to density modes and spin modes, and are given by
f = il−l jν m |iJ YJM |jν m g=
J+1
drϕν [(F0 + F0 )δρiJ π n + (F0 − F0 )δρiJ π p ]ϕν ,
il−l jν m |iL [YL × σ]JM |jν m
(2)
drϕν [(G0 + G0 )δρiJ π Ln + (G0 − G0 )δρiJ π Lp ]ϕν .(3)
L=J−1
The functions F0 (r), F0 (r), G0 (r) and G0 (r) are the generalized Landau-Migdal parameters associated with the Skyrme interaction [10]. The matrix elements of vind should then
602c
E. Vigezzi et al. / Nuclear Physics A 752 (2005) 600c–603c
< νν | vind | νν> [MeV]
0.008
εF
0.004
0
-0.004
-0.008 -6
-4
-2
0 εν [MeV]
2
4
6
Figure 2. Matrix elements of the interaction induced by polarization effects associated with diagonal configurations νmν m|v ¯ ind |νmν m. ¯ The filled and empty circles are the matrix elements associated with the exchange of density and spin phonons respectively. In both cases, the exchanged phonons have J up to 18.
be added to those of the bare interaction vbare . For convenience, in the present calculations we use the Gogny interaction, which at low density yields gaps very close to those of vbare [11]. The total interaction v = vGogny + vind , is inserted in the HFB equations in order to determine the (state dependent) pairing gap [12]. In the energy denominator ¯ ωi ) denotes the energy of the intermediate of Eq. (1), Eint = (|ν − F | + |ν − F | + h state of the process shown in Fig. 1(b). The denominator is always negative, because E0 denotes the pairing correlation energy of a Cooper pair, which should be determined selfconsistently from the HFB calculation. In the present calculation we have used a value ¯ = −1 MeV, where ∆ ¯ is the average value of the pairing gap close to the Fermi E0 ≈ −2∆ energy (cf. Fig. 3). The diagonal matrix elements of the induced interaction are shown in Fig. 2. It can be seen that they are peaked at the Fermi energy, and that the repulsive spin modes prevail in magnitude over the attractive density modes. The magnitude of the total matrix element is about one third of the value of the matrix elements of the Gogny interaction around the Fermi energy. The state-dependent pairing gaps calculated with and without the induced interaction are shown in Fig. 3 (left), where they are compared to those obtained without the presence of the nucleus (right). In the latter case, the result is essentially the same as one would obtain in neutron matter, with small fluctuations due to the finite size of the cell. The ¯ considering only the Gogny force, is close to 1 MeV in both cases, but the value of ∆ presence of the nucleus determines a much stronger state dependence of the gap [5,13]. In both cases, the exchange of vibrations is dominated by the repulsive spin modes (cf. ¯ when the induced interaction is taken into account; Fig. 2) leading to a reduction of ∆ ¯ is reduced to about 0.4 MeV. It is seen that the presence without the nuclear potential, ∆ ¯ ≈ 0.6 MeV. of the nucleus acts against this suppression, as in this case ∆ We conclude that polarization effects associated with the exchange of spin modes
603c
E. Vigezzi et al. / Nuclear Physics A 752 (2005) 600c–603c
5
εF
5
3 2 1 0 -30
εF
4 ∆νν [MeV]
∆νν [MeV]
4
3 2 1
-20
-10
0 εν [MeV]
10
20
30
0 0
10
20 εν [MeV]
30
40
Figure 3. Values of the state-dependent pairing gap as a function of the single-particle energies with (left) and without (right) the nucleus. The Fermi energy is adjusted in the two cases in order to have the same number of neutrons in the cell. The filled circles correspond to the values of the gaps obtained using only the Gogny forces, while the empty circles are those obtained adding the induced interaction. strongly reduce the attractive nature of the 1 S0 nucleon-nucleon interaction, leading to a conspicuous suppression of the associated pairing gap. However, the presence of nuclear impurities in the crust attenuates this quenching in an important way. REFERENCES 1. C.J. Pethick and D.G. Ravenhall, Ann. Rev. Nucl. Part. Sci. 45 (1995)429. 2. D.G. Yakovlev and C.J. Pethick, Ann. Rev. Astron. Astrophys. 42 (2004) 169. 3. U. Lombardo and H.J. Schultze, in Physics of neutron star interiors, D. Blaschke, N.K. Glendenning and A. Sedrakian eds., Springer-Verlag (2001), p.30, and refs. therein. 4. P.W. Anderson and N. Itoh, Nature (London) 256 (1975)25 5. P.M. Pizzochero, F. Barranco, E. Vigezzi, R.A. Broglia, ApJ 569(2002) 381. 6. N. Sandulescu, N. Van Giai and R.J. Liotta, Phys. Rev. C69 (2004) 045802 7. J. Negele and D. Vautherin, Nucl.Phys. A207 (1973) 298. 8. D.J. Rowe, Nuclear Collective Motion, Methuen and Co., London (1970) 9. J. Terasaki et al., Nucl. Phys. A697(2002)127; F. Barranco et al., Phys. Rev. Lett. 83, 2147 (1999), and to be published. 10. V. Bernard and N. Van Giai, Nucl. Phys. 327 (1979) 397. 11. C. Shen et al., Phys. Rev. C67 (2003) 061302. 12. More precisely, the HFB equations are solved self-consistently only in the pairing channel (cf. ref. [13]). Note that in these equations also matrix elements of the type ¯ |vind |ν1 mν2 m < ν1 m ν2 m ¯ > are included, while for simplicity Eq. (1) refers to the particular case of single-particle wavefunction having the same number of nodes. 13. F. Barranco, R.A. Broglia, H. Esbensen, E. Vigezzi, Phys. Rev. C58(1998) 1257.
Pairing correlations in the inner crust of neutron stars E. Vigezzia , F. Barrancob , R.A. Brogliaa,c , G. Col`oa , G. Goria , F. Ramponia a
Dipartimento di Fisica, Universit`a di Milano and INFN Sez. Milano, Via Celoria 16, 20123 Milano b Departamento de Fisica Aplicada III, Escuela Superior de Ingenieros, University of Sevilla, Camino de los Descubrimientos s/n, 41092 Sevilla, Spain c The Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, 20100 Copenhagen, Denmark We calculate the pairing gap in the inner crust of neutron stars, taking into account both the proximity effects associated with the coexistence of the nuclear lattice with the sea of free neutrons, and the polarization effects associated with the exchange of density and spin fluctuations. It is found that the presence of the nuclei attenuates the quenching of the gap produced by the polarization. It is generally accepted that in the density range 0.001ρ0 < ρ < 0.5ρ0 , where ρ0 is the nuclear saturation density, neutron stars display an inner crust, consisting of a Coulomb lattice of nuclei immersed in a gas of free neutrons and relativistic electrons [1]. In analogy to the case of neutron matter in the same density range, the crust is expected to be superfluid in the 1 S0 channel, and this has important consequences on the thermal properties of the star [2]. Many studies have shown that it is essential to include polarization effects in the calculation of the pairing gap in infinite neutron matter [3]. In fact, such effects reduce the gap substantially, as compared to the values obtained in a mean-field calculation with the bare nucleon-nucleon force. Furthermore, in the present case one needs to determine the microscopic spatial dependence of the pairing gap, in order to have an accurate description of the superfluidity in the crust, which is affected by the proximity effects associated with the coexistence of two different phases (finite nuclei and neutron gas). This is particularly important for an accurate study of vortex dynamics, which according to theoretical models is at the origin of the phenomenon of glitches [4]. Recently, proximity effects have been calculated at the mean field level in the Hartree-Fock-Bogoliubov (HFB) approximation [5,6]. In this contribution we extend such studies, including the effects associated with the exchange of density and spin vibrations (cf. Fig. 1), which at low density are the most important processes renormalizing the pairing gap. We shall perform our calculation treating the nuclear lattice in the Wigner-Seitz approximation, considering a single Wigner cell with a spherical nucleus at the center, surrounded by a sea of free neutrons. In the following we shall present results for a cell of radius Rcell = 0375-9474/$ – see front matter © 2005 Published by Elsevier B.V. doi:10.1016/j.nuclphysa.2005.02.062
E. Vigezzi et al. / Nuclear Physics A 752 (2005) 600c–603c
601c
Figure 1. Diagrams depicting (a) the particle-vibration coupling vertex and (b) the pairing interaction induced by the exchange of phonons.
42 fm containing 50 protons and about 500 neutrons. These parameters are appropriate for the low-density part of the crust, and have been taken from the study of Negele and Vautherin, who determined the optimal lattice step and the ratio between neutrons and protons [7]. We first determine the mean-field potential and the single-particle levels of energy ν (ν ≡ nlj) with the associated radial wavefunctions φν , performing an HartreeFock calculation with a two-body effective interaction of the Skyrme type (SkM∗ ). The nucleus accomodates 130 bound neutrons, while the other neutrons are found at positive energy, and the Fermi energy F lies at about 1.4 MeV. Making use of the associated particle-hole basis states |jp jh−1 , JM we then calculate the vibrational states (phonons) in the Random Phase Approximation (RPA) [8]. The phonons are classified according to their angular momentum and projection J, M and to their parity π, while i =1,2,. . . labels the states with increasing energy. The spin-orbit interaction has been included in the calculation but it has a very small effect on the free neutrons, which dominate the RPA response. Consequently, the phonons have essentially good spin and can be characterized as density modes (S = 0) or spin modes (S = 1). From the RPA calculation one gets the transition densities δρ (we denote by δρn and δρp the neutron and proton contributions) of these excited states, which are used to calculate the particle-vibration coupling vertices (cf. Fig. 1(a)). The matrix elements of the interaction induced by the exchange of the phonons between pairs of neutrons in time reversal are given by the expression [9] ¯ |vind |νmν m < ν m ν m ¯ >= 2
m νm (f + g)ννm;J π M i (f − g)νm;J π M i E0 − Eint JπM i
(1)
where the factor of 2 arises from the two possible time ordering associated with graph (b) of Fig. 1. The particle-vibration matrix elements (cf. Fig. 1(a)) f and g in Eq.(1) refer respectively to the coupling to density modes and spin modes, and are given by
f = il−l jν m |iJ YJM |jν m g=
J+1
drϕν [(F0 + F0 )δρiJ π n + (F0 − F0 )δρiJ π p ]ϕν ,
il−l jν m |iL [YL × σ]JM |jν m
(2)
drϕν [(G0 + G0 )δρiJ π Ln + (G0 − G0 )δρiJ π Lp ]ϕν .(3)
L=J−1
The functions F0 (r), F0 (r), G0 (r) and G0 (r) are the generalized Landau-Migdal parameters associated with the Skyrme interaction [10]. The matrix elements of vind should then
602c
E. Vigezzi et al. / Nuclear Physics A 752 (2005) 600c–603c
< νν | vind | νν> [MeV]
0.008
εF
0.004
0
-0.004
-0.008 -6
-4
-2
0 εν [MeV]
2
4
6
Figure 2. Matrix elements of the interaction induced by polarization effects associated with diagonal configurations νmν m|v ¯ ind |νmν m. ¯ The filled and empty circles are the matrix elements associated with the exchange of density and spin phonons respectively. In both cases, the exchanged phonons have J up to 18.
be added to those of the bare interaction vbare . For convenience, in the present calculations we use the Gogny interaction, which at low density yields gaps very close to those of vbare [11]. The total interaction v = vGogny + vind , is inserted in the HFB equations in order to determine the (state dependent) pairing gap [12]. In the energy denominator ¯ ωi ) denotes the energy of the intermediate of Eq. (1), Eint = (|ν − F | + |ν − F | + h state of the process shown in Fig. 1(b). The denominator is always negative, because E0 denotes the pairing correlation energy of a Cooper pair, which should be determined selfconsistently from the HFB calculation. In the present calculation we have used a value ¯ = −1 MeV, where ∆ ¯ is the average value of the pairing gap close to the Fermi E0 ≈ −2∆ energy (cf. Fig. 3). The diagonal matrix elements of the induced interaction are shown in Fig. 2. It can be seen that they are peaked at the Fermi energy, and that the repulsive spin modes prevail in magnitude over the attractive density modes. The magnitude of the total matrix element is about one third of the value of the matrix elements of the Gogny interaction around the Fermi energy. The state-dependent pairing gaps calculated with and without the induced interaction are shown in Fig. 3 (left), where they are compared to those obtained without the presence of the nucleus (right). In the latter case, the result is essentially the same as one would obtain in neutron matter, with small fluctuations due to the finite size of the cell. The ¯ considering only the Gogny force, is close to 1 MeV in both cases, but the value of ∆ presence of the nucleus determines a much stronger state dependence of the gap [5,13]. In both cases, the exchange of vibrations is dominated by the repulsive spin modes (cf. ¯ when the induced interaction is taken into account; Fig. 2) leading to a reduction of ∆ ¯ is reduced to about 0.4 MeV. It is seen that the presence without the nuclear potential, ∆ ¯ ≈ 0.6 MeV. of the nucleus acts against this suppression, as in this case ∆ We conclude that polarization effects associated with the exchange of spin modes
603c
E. Vigezzi et al. / Nuclear Physics A 752 (2005) 600c–603c
5
εF
5
3 2 1 0 -30
εF
4 ∆νν [MeV]
∆νν [MeV]
4
3 2 1
-20
-10
0 εν [MeV]
10
20
30
0 0
10
20 εν [MeV]
30
40
Figure 3. Values of the state-dependent pairing gap as a function of the single-particle energies with (left) and without (right) the nucleus. The Fermi energy is adjusted in the two cases in order to have the same number of neutrons in the cell. The filled circles correspond to the values of the gaps obtained using only the Gogny forces, while the empty circles are those obtained adding the induced interaction. strongly reduce the attractive nature of the 1 S0 nucleon-nucleon interaction, leading to a conspicuous suppression of the associated pairing gap. However, the presence of nuclear impurities in the crust attenuates this quenching in an important way. REFERENCES 1. C.J. Pethick and D.G. Ravenhall, Ann. Rev. Nucl. Part. Sci. 45 (1995)429. 2. D.G. Yakovlev and C.J. Pethick, Ann. Rev. Astron. Astrophys. 42 (2004) 169. 3. U. Lombardo and H.J. Schultze, in Physics of neutron star interiors, D. Blaschke, N.K. Glendenning and A. Sedrakian eds., Springer-Verlag (2001), p.30, and refs. therein. 4. P.W. Anderson and N. Itoh, Nature (London) 256 (1975)25 5. P.M. Pizzochero, F. Barranco, E. Vigezzi, R.A. Broglia, ApJ 569(2002) 381. 6. N. Sandulescu, N. Van Giai and R.J. Liotta, Phys. Rev. C69 (2004) 045802 7. J. Negele and D. Vautherin, Nucl.Phys. A207 (1973) 298. 8. D.J. Rowe, Nuclear Collective Motion, Methuen and Co., London (1970) 9. J. Terasaki et al., Nucl. Phys. A697(2002)127; F. Barranco et al., Phys. Rev. Lett. 83, 2147 (1999), and to be published. 10. V. Bernard and N. Van Giai, Nucl. Phys. 327 (1979) 397. 11. C. Shen et al., Phys. Rev. C67 (2003) 061302. 12. More precisely, the HFB equations are solved self-consistently only in the pairing channel (cf. ref. [13]). Note that in these equations also matrix elements of the type ¯ |vind |ν1 mν2 m < ν1 m ν2 m ¯ > are included, while for simplicity Eq. (1) refers to the particular case of single-particle wavefunction having the same number of nodes. 13. F. Barranco, R.A. Broglia, H. Esbensen, E. Vigezzi, Phys. Rev. C58(1998) 1257.