Proximity effect of pair correlation in the inner crust of neutron stars
PPreprint number:
Proximity effect of pair correlation in the inner crustof neutron stars
Toshiyuki Okihashi and Masayuki Matsuo Graduate School of Science and Technology, Niigata University, Niigata950-2181, Japan Department of Physics, Niigata University, Niigata 950-2181, Japan ∗ E-mail: [email protected] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
We study proximity effect of pair correlation in the inner crust of neutron stars by means ofthe Skyrme-Hartree-Fock-Bogoliubov theory formulated in the coordinate space. We describea system composed of a nuclear cluster immersed in neutron superfluid, which is confined ina spherical box. Using a density-dependent effective pairing interaction which reproduces boththe pair gap of neutron matter obtained in ab initio calculations and that of finite nuclei, weanalyze how the pair condensate in neutron superfluid is affected by the presence of the nuclearcluster. It is found that the proximity effect is characterized by the coherence length of neutronsuperfluid measured from the edge position of the nuclear cluster. The calculation predicts thatthe proximity effect has a strong density dependence. In the middle layers of the inner crustwith baryon density 5 × − fm − < ∼ ρ b < ∼ × − fm − , the proximity effect is well limitedin the vicinity of the nuclear cluster, i.e. in a sufficiently smaller area than the Wigner-Seitzcell. On the contrary, the proximity effect is predicted to extend to the whole volume of theWigner-Seitz cell in shallow layers of the inner crust with ρ b < ∼ × − fm − , and in deep layerswith ρ b > ∼ × − fm − . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Subject Index D10,D11,D13,D41 PTP
TEX.cls a r X i v : . [ nu c l - t h ] S e p Introduction
The inner crust of neutron stars is an exotic inhomogeneous matter consisting of a latticeof neutron-rich nuclear clusters which is immersed in neutron superfluid [1]. One of thecentral issues of the physics of the inner crust is interplay between the superfluidity andthe inhomogeneity, which influences various properties of the inner crust such as the specificheat, the thermal conductivity, and the pinning and unpinning of vortices. These are essentialfactors to understand astrophysical issues, such as the cooling and the glitch phenomenonof the neutron stars.Microscopic many-body approaches to these phenomena have been pursued in the frame-work of the Hartree-Fock-Bogoliubov (HFB) theory, which has a capability to describemicroscopically the inhomogeneous pair-correlated system. It has been argued for instancethat the presence of the nuclear cluster modifies the quasiparticle excitation spectrum andthe average pair gap, leading to a sizable difference in the specific heat of the inner crustfrom that of the uniform neutron superfluid [2–6]. evaluate the pinning energy of superfluidvortices [7–9]. Recent interest also concerns with a dynamical aspect of the issues, i.e. theinteraction between the vibrational motion of the nuclear cluster and the phonon excitation(the Anderson-Bogoliubov collective mode) of the neutron superfluid. This is one of the keyingredients which influence the thermal conductivity of the inner crust in magnetars [10–14].In an attempt to analyze this dynamical coupling from a microscopic viewpoint, we haveinvestigated the collective excitation of the inner crust matter by means of the quasipar-ticle random phase approximation based on the HFB theory [15, 16]. We found that thedynamical coupling between the collective motions of the nuclear cluster and of the neutronsuperfluid is weak.In the present study, we intend to reveal the interplay between the nuclear cluster andthe neutron superfluid but from a different viewpoint, i.e. the proximity effect of the pairingcorrelation [17, 18]. The proximity effect is a general phenomenon which emerges around aborder region of the system of a superconducting/superfluid matter in contact with normalmatter (or matter with different pairing property). The pairing correlations in both matterare affected mutually in the border region since the Cooper pairs penetrate the border. Theproximity effect in the inner crust matter is discussed in a few preceding works [2, 3, 27] , butonly in a qualitative manner. In the present study, we aim at characterizing the proximityeffect quantitatively in order to reveal basic features of the pair correlation arising from theinhomogeneous structure of the inner crust matter.As a theoretical framework to perform this study, we adopt the HFB theory using theSkyrme functional with a implementation of a few new features. One of the key elements2n the HFB approach is the effective pairing interaction or the effective pairing functional,which generates the pair correlation in the system under study, and a density-dependentcontact force, called the density-dependent delta interaction (DDDI), is often adopted. Notehowever that the inner crust matter consists of the neutron superfluid, whose density variesin a wide range from zero to that of the nuclear saturation, and the nuclear clusters, whichresemble to isolated neutron-rich nuclei. In order to take into account this feature, we preparea new parameter set of DDDI, which is required to describe the pairing gap of neutronsuperfluid obtained in ab initio calculations [28, 29] as well as the experimental pairing gapin finite nuclei. Secondly, we quantify the range of the proximity effect by identifying thedistance where the presence of the nuclear cluster influences the pairing property in neutronsuperfluid. Using this measure, we discuss in detail the dependence of the proximity effecton the density of the neutron superfluid, and clarify how large the proximity effect is indifferent layers of the inner crust.In Section 2, we explain the adopted Skyrme-HFB model and the new parameter setof DDDI. In the present HFB all the nucleons are described as quasiparticles confined in aspherical box. If we adopt the box size equal to the Wigner-Seitz radius of the lattice cell, it isthe same as the Wigner-Seitz approximation often adopted in the preceding works. However,the box truncation causes so called finite-size effect, and it make difficult to analyze theproximity effect. In Section 3, we examine the finite-size effect, and propose a different settingof the analysis using a large box truncation in place of the Wigner-Seitz approximation.Section 4 is devoted to a systematic analysis of the proximity effect. In subsection 4.1 wedescribe our scheme of the analysis that quantifies the range of the proximity effect, andjustify the scheme with a systematic variation of the density of neutron superfluid immersingthe nuclear cluster. In subsection 4.2, we apply the same analysis to various layers of a realisticconfiguration of the inner crust of neutron stars. Section 5 is devoted to the conclusions.
We adopt the Skyrme-Hartree-Fock-Bogoliubov method to describe the inner crust mat-ter. Since the method is an extension of that is used in Refs. [15, 16], we describe it brieflywith emphasis on new aspects which are introduced in the present study.We solve the HFB equation in a spherical box using the radial coordinate space and thepartial wave expansion. The zero temperature is assumed and the spherical symmetry ofsolutions is imposed. Electrons are neglected. The radial HFB equation for a given angular3uantum numbers lj reads (cid:34) h qlj ( r ) − λ q ∆ q ( r )∆ q ( r ) − h qlj ( r ) + λ q (cid:35) (cid:34) φ qlj ( r ) φ qlj ( r ) (cid:35) = E (cid:34) φ qlj ( r ) φ qlj ( r ) (cid:35) , (1)where φ qlj , φ qlj is the quasiparticle wave function. Index q denotes neutron or proton.We discretize the radial coordinate with an interval h = 0 . r i = i ∗ h − h/ h/ , h/ , · · · ( i = 1 , · · · , N ) up to the edge r = R box of the box, and use the nine-pointformula to represent the derivatives in the Hartree-Fock Hamiltonian h qlj ( r ). We impose theDirichlet-Neumann boundary condition [30], with which even-parity wave functions vanishat the edge of the box and the first derivatives of odd-parity wave functions vanish atthe same position. Equation (1) is represented as a matrix eigenvalue problem where thewave function at the discretized coordinates (cid:16) φ qlj ( r ) , · · · φ qlj ( r N ) , φ qlj ( r ) , · · · φ qlj ( r N ) (cid:17) T isa 2 N -dimensional vector. We use routine DSYEVX in the LAPACK package to solve theeigenvalue problem for the symmetric matrix. If we treat the lattice configuration of thenuclear clusters by means of the Wigner-Seitz approximation, the box radius R box is chosento be the size of the Wigner-Seitz cell. We shall also choose larger boxes R box = 100 fmor 200 fm, as we explain below. All the quasiparticle states up to a maximal quasiparticleenergy E max = 60 MeV are included to calculate the number density, the pair density and allthe quantities needed to calculate the selfconsistent potentials. We put also a cut-off l max onthe angular momenta of the partial waves so that l max > (cid:112) E max / ( (cid:126) / m ) R box : l max = 200 (cid:126) for R box = 100 fm, and l max = 400 (cid:126) for R box = 200 fm, for example. We use the parameterset SLy4 [19] for the selfconsistent Hartree-Fock potential in h q ( r ). We adopt the density-dependent delta interaction, as described below, to derive the pair potential ∆ q ( r ). We varythe neutron Fermi energy λ n to control the neutron density and we determine the protonFermi energy λ p to fix the proton number Z of the nuclear cluster. The other details are thesame as in the previous study [15, 16]. As the pairing interaction, we use a density-dependent delta-interaction (DDDI), givenas v pair ,n ( (cid:126)r , (cid:126)r ) = V n [ ρ n ( (cid:126)r ) , ρ p ( (cid:126)r )] (cid:18) − P σ (cid:19) δ ( (cid:126)r − (cid:126)r ) , (cid:126)r = (cid:126)r (= (cid:126)r ) , (2)for neutrons. Here V n [ ρ n ( (cid:126)r ) , ρ p ( (cid:126)r )] is the density-dependent interaction strength, and (1 − P σ ) / able 1 DDDI parameters adopted in the present study. For the definition, see Eq. (5)and the text. The parameters are appropriate for the cut-off energy e cut = 60 MeV. V (MeVfm ) ρ (fm − ) η α η α η (cid:48) α (cid:48) DDDI-a1 -458.4 0.08 0.59 1/3 0.06 2/3 0 1/3DDDI-a2 -458.4 0.08 0.59 1/3 0.255 2/3 -0.195 1/3DDDI-b -458.4 0.08 0.845 0.59 - - - -∆ n ( r ) = V n [ ρ n ( r ) , ρ p ( r )] ˜ ρ n ( r ) with the neutron pair density (the neutron pair condensate)˜ ρ n ( (cid:126)r ) = (cid:104) ψ n ( (cid:126)r ↑ ) ψ n ( (cid:126)r ↓ ) (cid:105) . (3)We consider the following three models for the interaction strength V n [ ρ n ( (cid:126)r ) , ρ p ( (cid:126)r )].The first one, which we introduced in Refs. [20, 21], is given as V n [ ρ n ( (cid:126)r )] = V (cid:40) − . (cid:18) ρ n ( r ) ρ (cid:19) . (cid:41) (4)with ρ = 0 .
08 fm − . Here the overall constant V = − . is determined to repro-duce the S scattering length a =-18.5 fm in free space (i.e. at zero density) under thesingle-particle cut-off energy e cut = 60 MeV. The dependence of the interaction strength V n [ ρ n ] on the neutron density ρ n is determined so that it reproduces the neutron pairing gapin pure neutron matter which is obtained in the BCS approximation using a bare nuclearforce [20, 21]. We denote the parameterization, Eq. (4), as “DDDI-b” since it refers to the B CS gap with the b are nuclear force. (It is the same as the parametrization DDDI-G3RS inRef. [20].)In the present study we introduce more realistic modeling of the neutron pairing appropri-ate to the inner crust matter. Here we consider parametrizations of the DDDI that providerealistic pairing gap both in neutron matter and in finite nuclei. Concerning the neutronmatter, it is known that the pairing gap is affected by medium effects beyond the BCSapproximation, and many of theoretical studies trying to evaluate the medium effects predicta significant reduction from the BCS gap while the predicted values spread in a wide range[22–25]. Nevertheless, the pairing gap in the low-density limit is believed to be describedreliably by a perturbative approach to the screening effect, discussed first by Gor’kov andMelik-Barkhudarov (GMB) [26], and the pairing gap ∆ GMB in the GMB framework givesa reduction of a factor of (4 e ) / (cid:39) . ρ n (cid:39) − − − fm − , and the predicted pair-ing gaps are reduced from the BCS gap by a factor of 1.5 - 2 [24, 28, 29, 33]. We can refer5o these studies in requiring a new parametrization of the DDDI. It is also known that thepairing gap in finite nuclei cannot be described well by the BCS approximation applied tothe bare nuclear force, and there is no ab initio evaluation of the gap in finite nuclei. Insteadwe will refer to experimental information on the pairing gap in finite nuclei.In order to satisfy these conditions we introduce the following extended form of thedensity-dependent interaction strength: V n [ ρ n ( (cid:126)r ) , ρ p ( (cid:126)r )] = V (cid:40) − η (cid:18) ρ n ( r ) ρ (cid:19) α − η (cid:18) ρ n ( r ) ρ (cid:19) α − η (cid:48) (cid:18) ρ p ( r ) ρ (cid:19) α (cid:48) (cid:41) . (5)The first term is introduced to describe the GMB gap appropriate to the low-densitylimit of the pure neutron matter. As discussed in Appendix A, the force strength V GMB ofthe contact force which reproduces the GMB pairing gap ∆
GMB depends on the neutronFermi momentum k F,n or the density ρ n of neutron matter. The dependence is expressed asa linear term proportional to k F,n or ρ / n if it is expanded in powers of k F . Requiring thatthe GMB pairing gap is reproduced by the DDDI in the low-density limit ρ n → , k F,n → α = 1 / η = 0 . α = 2 /
3, i.e., ∝ ρ / n ∝ k F,n the second power of neutron Fermi momentum k F,n . We then require that the coefficient η of this term is consistent with the ab initiopairing gap of neutron matter obtained for 10 − fm − (cid:46) ρ n (cid:46) − fm − in the quantumMonte Carlo calculation by Gezerlis and Carlson [28] and the determinantal lattice MonteCarlo calculation by Abe and Seki [29]. Note however that this requitement alone does notfix uniquely the coefficient η since these ab initio calculations are slightly different witheach other and there is no ab initio results for moderately low densities 10 − fm − (cid:46) ρ n (cid:46) − fm − .The third term dependent on the proton density represents a part of medium effects asso-ciated with systems with a proton fraction. For simplicity we assume that it is proportionalto the proton Fermi momentum k F,p or the proton density ρ / p . We use both the coefficient η (cid:48) of this term and the uncertainty in η to describe the pairing gap in finite nuclei. In prac-tice, we require that the average neutron pairing gap ∆ n,uv = (cid:82) ∆ n ( r ) ρ ( r ) d(cid:126)r/ (cid:82) ρ ( r ) d(cid:126)r in Sn obtained from our HFB model reproduces the experimental neutron gap ∆ n, exp (cid:39) . A perturbative estimate of the medium effect in symmetric matter gives an attractive induced interactionproportional to N ,p ∝ k F,p [31]. -5 -4 -3 -2 -1 Δ ( M e V ) ρ n (fm -3 )DDDI-b DDDI-a1DDDI-a2Abe 09Gezerlis 10 Fig. 1
The pair gap ∆ of superfluid neutron matter obtained in the uniform-BCS calcu-lation using the three DDDI models, plotted as a function of neutron density. Solid, dashedand dot-dashed curves represent results for DDDI-b, DDDI-a1 and DDDI-a2, respectively.The ab initio Monte-Carlo results by Gezerlis and Carlson [28] and Abe and Seki [29] areshown for comparison.In the present study we prepare two different parameter sets to represent the remaininguncertainty of the neutron pair gap. In one case (we call “DDDI-a1” below), we choose η = 0 .
06 and η (cid:48) = 0 so that the neutron pairing gap in Sn is reproduced without η (cid:48) . Inthis case, the pairing gap of neutron matter is close to that of Abe and Seki [29], and theneutron matter pairing gap at moderately low density is rather large ∆ ∼ − η . In another parameter set (“DDDI-a2”), we considera case that the neutron matter pairing gap at moderately low density is relatively small; wedetermine η = 0 .
255 so as to make the neutron matter pairing gap vanish at ρ n = ρ as theBCS gap does. The parameter η (cid:48) = − .
195 is then determined to reproduce the neutron gapin
Sn. (Note that the neutron matter pairing gap reproduces approximately the result ofGezerlis and Carlson [28], as shown in Fig. 3.) The parameter sets of the three DDDI modelsare summarized in Table 1.Figure 1 shows the neutron pairing gap in uniform neutron matter obtained from theHartree-Fock plus BCS calculation using the DDDI models discussed above and the Skyrmefunctional with the parameter set SLy4. (The BCS calculation is briefly recapitulated inAppendix B, and we call it the uniform-BCS calculation in the following.) By construction,the pairing gaps obtained with the DDDI-a1 and the DDDI-a2 reproduce reasonably well7 -5 -4 -3 -2 -1 ξ , d (f m ) ρ n (fm -3 )DDDI-b DDDI-a1DDDI-a2d= ρ n-1/3 Fig. 2
The coherence length ξ of superfluid neutron matter obtained in the uniform-BCScalculation. The dotted curve is the average inter-neutron distance d = ρ − / n . See also thecaption of Fig. 1.the gap obtained with the ab initio calculations. The DDDI-a2 reproduces approximatelythe result of Gezerlis and Carlson [28] for the density range ρ n = 10 − − − fm − . Thegap of the DDDI-a2 at moderate density is small ∆ < . ρ n ∼ . k F ∼ . − ) corresponding to neutrons in the saturated nuclear matter. The neutrongap of the DDDI-a1 is very close to that of the DDDI-a2 up to ρ n < ∼ − fm − , but deviatefrom it above ρ n > ∼ − fm − , It is rather close to the gap of Abe and Seki [29], and theneutron matter pairing gap at moderately low density is rather large ∆ ∼ − We consider that DDDI-a1 and DDDI-a2 are more realistic than DDDI-bwhile the difference between DDDI-a1 and DDDI-a2 represents the uncertainty in modelingthe realistic pairing correlation. We also use the model DDDI-b since it simulates the BCSgap, which is a robust baseline common to all the models of realistic bare nuclear force [22].Figure 2 shows the coherence length ξ of superfluid uniform neutron matter, calculated asdescribed in Appendix B. The coherence length ξ depends strongly on the neutron density.The coherence length ξ is as short as ξ < ∼
10 fm at ρ n = 10 − − × − fm − . The coherencelength becomes long gradually as the neutron density decreases less than 10 − fm − , and The auxiliary-field diffusion Monte Carlo method [33], another ab initio calculation, predicts the pairinggap close to the BCS gap in the density range ρ n (cid:46) − fm − . Thus it may be considered that the parameterset DDDI-b represents partly this result. .00.51.01.52.0 100 110 120 130 140 150 Δ ( M e V ) A DDDI-b DDDI-a1DDDI-a2exp AME2016
Fig. 3
The neutron pair gap in Sn isotopes obtained with the Skyrme-Hartree-Fock-Bogoliubov method using the three DDDI pairing interaction models. Solid, dashed anddot-dashed curves correspond to DDDI-b, DDDI-a1 and DDDI-a2, respectively. The Skyrmeparameter SLy4 is adopted. The open circle is the experimental neutron pair gap derivedusing the odd-even mass difference [34] and AME2016 [39]. See text for details.it also does rather sharply for increasing ρ n more than ∼ × − fm − . The minimumvalue of the coherence length is ξ ∼ . λ n ≈ ξ ∼ . λ n ≈ ξ ∼ . λ n ≈ d = ρ − / n .It is noted that the coherence length ξ is shorter than the average inter-neutron distance d = ρ − / n at wide density interval ρ n = 10 − − − fm − for DDDI-b, or comparable with d at ρ n = 10 − − − fm − for DDDI-a1 and DDDI-a2. The coherence length shorter than d implies that the pair correlation at these densities is in the domain of the strong-couplingpairing, characterized as the BCS-BEC crossover phenomenon [23].Figure 3 shows the average neutron pairing gap ∆ n,uv in Sn isotopes obtained in thepresent HFB code. The average neutron pairing gap in Sn calculated with DDDI-b, DDDI-a1 and DDDI-a2 is ∆ n,uv = 0 . , .
28 and 1.28 MeV, respectively. Note that the pair gapof DDDI-a1 and DDDI-a2 reproduces the experimental gap reasonably well over the longisotope chain while DDDI-b gives only a half of the experimental value.9 λ n =7.2 MeV (cell 2) 0.005 0.0055 0.006 0 10 20 30 40 50 p a i r d e n s ti y (f m - ) (b) λ n =2.9 MeV (cell 5) 0.00015 0.0002 0.00025 0 10 20 30 40 50r (fm) (c) λ n =0.20 MeV(cell 10) 0.015 0.016 0.017 0.018 0.019 0.02 0 10 20 30 40 50(d) λ n =7.2 MeV (cell 2) 0.0025 0.003 0.0035 0 10 20 30 40 50 d e n s ti y (f m - ) (e) λ n =2.9 MeV (cell 5)0.000020.000030.000040.00005 0 10 20 30 40 50r (fm)(f) λ n =0.20 MeV (cell 10) Fig. 4
Panels (a)(b) and (c): The pair density ˜ ρ n ( r ) of pure neutron matter obtainedwith the HFB calculation with a spherical box for the neutron Fermi energy; (a) λ n = 7 . R box = R cell = 28 fm, 39 fm, and 54 fm of the respective cells (indicated bythe triangle symbol). The red dot-dashed curve is the result obtained with a large box size R box = 100 fm for (a) and (b) and R box = 200 fm for (c). The horizontal line is the result ofthe uniform-BCS calculation obtained with the same Fermi energy. The arrows in the rightvertical axis indicate the deviation of ±
5% from the uniform-BCS result. Panels (d)(e) and(f): the same as (a)(b) and (c), but for the number density ρ n ( r ). Since the present HFB calculation is performed in the radial coordinate space truncatedwith a finite box radius R box , obtained results depend on the box radius R box especiallywhen R box is not large. This kind of dependence is often called the finite-size effect. If weadopt the Wigner-Seitz approximation, where the box size is chosen equal to the Wigner-Seitz radius R cell of the lattice cell of the inner crust, results also include the finite-size effect.10e shall examine how the Wigner-Seitz approximation is affected by the finite size effect.For this purpose, we here describe pure neutron matter using the same HFB code. For pureneutron matter, we can obtain an accurate numerical result by means of the uniform-BCScalculation, which corresponds to the limit of infinite size R box → ∞ . Comparison with theuniform-BCS result makes it possible to evaluate the finite size effect.We have applied the present HFB model to the pure neutron systems by simply neglectingthe proton contributions. Figure 4 shows a few example of the results, in which the neutronFermi energy is chosen as λ n = 7 .
2, 2 . . × − , 3 . × − and 3 . × − fm − , respectively.The pairing interaction DDDI-a1 is adopted.Dashed curves are the results for the calculation in which the box radius R box is set to theradius R cell = 28, 39 and 54 fm of the corresponding Wigner-Seitz cells. It is seen that boththe number density and the pair density of neutrons deviate from the uniform-BCS results;the finite size effect in the pair density is not negligible and much larger than that for thenumber density. The deviation from the uniform-BCS result (horizontal lines) is more than20% in cell 10 although it is less than about 5% in the other cells 2 and 5. The boundarycondition with the finite box causes discretization of the energy spectrum of the quasiparticlestates, and the pairing property is influenced by the discretization if the pair gap is not largeenough than the energy spacing. It is also seen that the deviation from the uniform-BCS isworse at positions close to the origin than at far positions. A possible explanation is thatthe influence of the discretization of the quasiparticle energy spectrum may be stronger atsmall r than at larger r ; the number of contributing quasiparticle states is effectively smallsince the wave function of high- (cid:96) partial waves is suppressed at small r .The above results indicate that the Wigner-Seitz approximation to the inner crust mattermay not be accurate enough to discuss the proximity effect. One needs to control the finitesize effect in a better way. A desirable approach may be to take into account the latticestructure of the inner crust matter using the band theory method and the Bloch waves,where the continuity of the neutron quasiparticle spectrum is kept. However the band theoryapplied to the HFB calculation is presently quite limited [2], and a calculation with a largequasiparticle space is too demanding and difficult to be performed. Instead we adopt asimpler approach where a nuclear cluster is placed in a neutron superfluid confined in a largebox, where the box size is chosen sufficiently large in order to reduce the finite-size effect asmuch as possible.We find that R box > ∼
100 fm gives the pair density convergent to the uniform-BCS withaccuracy of around 1% for densities ρ n > ∼ × − fm − as shown in Figure 2(a)(b), where weplot the results obtained with R box = 100 fm. In very-low-density cases ρ n < ∼ × − fm − ,11 D e n s it y (f m - ) (a)DDDI-b, λ n = 4 MeVUnifrom-BCSNeutronProton 0 0.005 0.01 0.015 0.02 0 5 10 15 20 25 30 P a i r d e n s it y (f m - ) r (fm) (b)Neutron Fig. 5
Result of the HFB calculation for the configuration with the proton number Z = 28,the neutron Fermi energy λ n = 4 MeV, and the box size R box = 100 fm. The parameter setDDDI-b is used. Panel (a) shows the number densities ρ n ( r ) and ρ p ( r ) of neutrons andprotons, respectively. Panel (b) shows the neutron pair density ˜ ρ n ( r ). The horizontal linesare the uniform-BCS results with the same λ n . The triangle and circle symbols indicate thehalf-density surface R s and the edge radius R edge of the cluster whereas the square symbolpoints to R edge + ξ . See the text for the definitions of these quantities.the pairing gap becomes very small ∆ < ∼ .
01 MeV. In this case, influence of the discretizationin quasiparticle levels is less negligible, and hence a larger box is required. For cell 10 (Fig.2(c)), we obtained the agreement to the required accuracy with R box = 200 fm.In the following we adopt this large-box configuration to discuss the proximity effectassociated with the presence of the nuclear cluster. We shall now discuss the pair correlation in the inner crust matter. As discussed above weconsider the system confined in a large box, at the center of which a nuclear cluster is placed.Using this setup, we shall investigate how the presence of the nuclear cluster influences thepair correlation of neutron superfluid in the neighborhood region around the cluster.
In order to investigate general features of the proximity effect, we shall first examinecases where the density of the surrounding neutron superfluid is systematically varied whilethe proton number is fixed. In the next subsection we discuss realistic configurations of the12nner crust matter, for which the proton number and the density of neutron superfluid arechosen to represent various layers of the inner crust.The proton number is Z = 28 in all the examples in this subsection and we vary theneutron Fermi energy λ n systematically from 0.2 MeV to 6 MeV, which corresponds to thedensity of the uniform neutron superfluid from ρ n = 4 × − fm − to 1 × − fm − .A typical result obtained for λ n = 4 MeV ( ρ n = 6 . × − fm − ) with DDDI-b is shownin Fig. 5, where plotted are the number densities of neutrons and protons, ρ n ( r ) and ρ p ( r ),and the neutron pair density ˜ ρ n ( r ) as a function of the radial coordinate r . It is seen thatthe nuclear cluster is well localized in a central region as seen in the profile of the neutrondensity ρ n ( r ) which converges rather quickly to a constant value at around r ≈ ρ p ( r ) converges to zero around r ≈ f ws ( r ) = ρ n, M + f (cid:0) r − R s a (cid:1) , (6)where R s defines the half-density surface, and a represents the diffuseness of the surface.The constant ρ n, M is the neutron density obtained from the uniform-BCS performed for thesame value of λ n . The values of f , R s and a are extracted from a fitting. In addition wefind it useful to consider “the edge” of the nuclear cluster to evaluate the area where thecluster exists. We define the nuclear edge by R edge = R s + 4 a . The edge position r = R edge is indicated by the black circle in Fig. 5, and it is seen that R edge represents well the positionwhere the neutron density ρ n ( r ) converges to ρ n, M .A most noticeable feature in Fig. 5 is that the neutron pair density ˜ ρ n ( r ) exhibitsbehaviours different from those of the neutron number density ρ n ( r ). It is seen that theneutron pair density ˜ ρ n ( r ) slowly converges and reaches the uniform-BCS value at around r ≈
12 fm, deviating from R edge by about 4 fm. In other words the influence of the nuclearcluster extends to the neighbour region beyond R edge . This slow convergence is nothingbut the proximity effect. In this example the neutron pair density inside the cluster issignificantly smaller than that outside the cluster. This reflects the characteristic densitydependence of the neutron pair gap of the DDDI-b model; the gap for the density insidethe cluster ( ρ n ∼ ρ ) is very small ∆ < ∼ . ρ n, M ∼ . × − fm) is relatively large ∆ ∼ . ξ of the super-fluid/superconducting matter [17]. We here assume that the border between the neutron13uperfluid and the nuclear cluster is approximated by the edge radius R edge , rather thanthe half-density surface R s . If these considerations are reasonable, it is expected that theproximity effect is seen up to r ≈ R edge + ξ . In the case shown in Fig. 5, the positionwhere the neutron pair density converges to the uniform-BCS value corresponds well to r = R edge + ξ = 8 .
27 fm+3 .
63 fm = 11.9 fm, and the above argument appears to hold.Figure 6 show systematic behaviours of the neutron pair densities calculated for variousneutron Fermi energies and for three different pairing interactions: the DDDI-b (panel (a) ineach figure), the DDDI-a1 (b), and the DDDI-a2 (c). Figure 6(a)(b)(c) shows the results forthe neutron Fermi energy λ n = 2 − ρ n ∼ − − − fm − (see Fig. 1), and Fig. 6(d)(e)(f) for λ n = 0 . − ρ n ∼ − − − fm − ).The proximity effect is clearly visible in all the cases; the pair density converges tothat of the uniform neutron superfluid at a position deviating significantly from the edgeposition r = R edge of the nuclear cluster. It is also seen that the range of the proximityeffect depends rather strongly on the neutron Fermi energy or the density of the neutronsuperfluid, especially at low neutron density ρ n, M < ∼ × − fm − and λ n < ∼ . r = R edge + ξ (marked with the square symbol), characterized by the coherencelength ξ measured from the edge R edge of the nuclear cluster. (Note that the edge position R edge of the nuclear cluster depends only weakly on the neutron Fermi energy, and there isessentially no dependence on the three choices of the pairing interaction. )We here recall Fig. 2 where the coherence length is shown to become as small as < ∼
10 fmat moderately low density ρ n = 7 × − − × − fm − for the three DDDI’s. This bringsabout the short range of the proximity effect seen for λ n = 2 − λ n = 0 . − . ξ with decreasing neutrondensity for very low density ρ n < ∼ − fm − . Note that for ρ n ∼ − − − fm − , thecoherence length ξ = 5 −
20 fm in the case of DDDI-b, ξ = 10 −
36 fm for DDDI-a1 and ξ = 11 −
37 fm for DDDI-a2. If the density of the external neutron superfluid decreasesfurther, the range of the proximity effect is expected to exceed far beyond 50 fm.
Finally, we discuss the proximity effect for realistic situations of the inner crust of neutronstars. Here we refer to the Wigner-Seitz cells obtained in Negele and Vauthrin [30] for various14 (a) DDDI-b λ n = 2 MeV λ n = 3 MeV λ n = 4 MeV λ n = 5 MeV λ n = 6 MeV 0 0.005 0.01 0.015 0 5 10 15 20 25 30 N e u t r on p a i r d e n s it y (f m - ) (b) DDDI-a1 λ n = 2 MeV λ n = 3 MeV λ n = 4 MeV λ n = 5 MeV λ n = 6 MeV 0 0.005 0.01 0.015 0 5 10 15 20 25 30 r (fm)(c) DDDI-a2 λ n = 2 MeV λ n = 3 MeV λ n = 4 MeV λ n = 5 MeV λ n = 6 MeV 0 0.002 0.004 0 5 10 15 20 25 30 (d) DDDI-b λ n = 0.2 MeV λ n = 0.4 MeV λ n = 0.6 MeV λ n = 0.8 MeV λ n = 1.0 MeV 0 0.01 0.02 0 5 10 15 20 25 30 0 0.001 0.002 (e) DDDI-a1 λ n = 0.2 MeV λ n = 0.4 MeV λ n = 0.6 MeV λ n = 0.8 MeV λ n = 1.0 MeV 0 0.01 0.02 0 5 10 15 20 25 30 0 0.001 0.002 r (fm) (f) DDDI-a2 λ n = 0.2 MeV λ n = 0.4 MeV λ n = 0.6 MeV λ n = 0.8 MeV λ n = 1.0 MeV Fig. 6
Calculated neutron pair density ˜ ρ n ( r ) for the neutron Fermi energy λ n = 0 . − . Z and the Wigner-Seitz radius R cell of eachcell is taken from Ref. [30]. The neutron Fermi energy λ n , the control parameter of theneutron density, is chosen so that the obtained density of the external neutron superfluidreproduces approximately the density of the neutron gas in Ref. [30]. For simplicity we use acommon value of λ n for the three DDDI models. The box size is R box = 100 fm for most cellsand 200 fm only for cell 1 with DDDI-b2, cell 10 with DDDI-a1 and cell 10 with DDDI-a2.The calculated neutron pair density is shown in Fig. 8. The maximum of the plottedradial coordinate is the Wigner-Seitz radius R cell for each cell. A noticeable feature is that incells 3 to 8 the pair density converges to that of the uniform-BCS at a distance shorter than15 able 2 The proton number Z and the neutron Fermi energy λ n employed in the presentcalculation to represent realistic configurations of the inner crust cells [30]. The next columnsare the density ρ n, M , the pairing gap ∆ and the coherence length ξ , obtained from theuniform-BCS calculation for the corresponding neutron matter. Results of the three DDDImodels, DDDI-b, DDDI-a1, and DDDI-a2, are listed for ∆ and ξ while ρ n, M is shown only forDDDI-a1. The third last column is the edge radius R edge of the nuclear cluster extracted fromthe Hartree-Fock-Bogoliubov calculation with DDDI-a1. The second last is the Wigner-Seitzradius R cell of the cells[30] while the last is the average baryon density ρ b = (cid:82) R cell ( ρ n ( r ) + ρ p ( r )) r dr/ ( R /
3) (with DDDI-a1) evaluated using the same Wigner-Seitz radius. See textfor details.
Cell
Z λ n ρ a1 n, M ∆ b ∆ a1 ∆ a2 ξ b ξ a1 ξ a2 R a1edge R cell ρ a1b (MeV) (fm − ) (MeV) (MeV) (MeV) (fm) (fm) (fm) (fm) (fm) (fm − )1 Zr 11.00 4 . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × −
10 Zr 0.20 3 . × − . × − the half distance of the Wigner-Seitz radius. In other words the proximity effect is restrictedonly in a small area nearby the nuclear cluster. The area of uniform neutron superfluidand that of the nuclear cluster are well separated in these middle layers of the inner crust.This feature is common to the three DDDI pairing models. It is noted that the coherencelength of external neutron superfluid is the smallest ξ ≈ − ρ n, M < ∼ × − fm − ),the proximity effect extends to a major area of the Wigner-Seitz cell, beyond the half lengthof the Wigner-Seitz radius, especially for DDDI-a1 and DDDI-a2. This reflects the longcoherence length at such very low densities: ξ > ∼ −
40 fm for DDDI-a1 and DDDI-a2, and ξ > ∼ −
20 fm for DDDI-b. Note that the pairing gap of DDDI-a1 and DDDI-a2 in diluteneutron matter is reduced from the BCS value (corresponding to DDDI-b) by a factor ofabout 2, leading to a longer coherence length in these realistic gap models DDDI-a1 andDDDI-a2. 16 -4 -3 -2 -1 ξ , R ce ll (f m ) ρ b (fm -3 ) cell 1cell 2cell 3cell 4cell 5cell 6cell 7cell 8cell 9cell 10 R cell DDDI-b DDDI-a1DDDI-a2
Fig. 7
The coherence length ξ of uniform neutron superfluid corresponding to cells 1 to10 listed in Table 2. Results of the three gap models, DDDI-b, DDDI-a1 and DDDI-a2, areshown. The horizontal axis is the baryon density ρ b of the cells. The Wigner-Seitz radius R cell of the cells, taken from Ref. [30], is also plotted for comparison.Another case where a long-range proximity effect is predicted is cell 1 at relatively highdensity, where the external neutron density ρ n, M ∼ .
04 fm − ≈ ρ / ξ varies from 7 to 30 fm depending rather strongly onthe pairing models, reflecting the uncertainty of the gap at such density. However, becauseof the relatively high baryon density and a large N/Z ratio, the Wigner-Seitz radius R cell becomes small ( ∼
20 fm) and the edge position r = R edge of the nuclear cluster becomes aslarge as ∼
13 fm due to a thick neutron skin of the cluster. Consequently the range R edge + ξ of the proximity effect exceeds the Wigner-Seitz radius irrespective of the uncertainty of thepairing gap. Note that cell 1 corresponds to a deep layer of the inner crust, where a transitionto the so called pasta phase is about to occur. The present result suggests strong proximityeffect also for the pasta phase at higher baryon density. We remark also that the proximityeffect in these deep layers might be even stronger than the present prediction because ofthe presence of adjacent nuclear clusters in the lattice configuration, but a quantitativeevaluation is beyond the scope of the present study.17 N e u t r on p a i r d e n s it y (f m - ) r (fm) cell 1 cell 2 cell 3 cell 4 cell 5 DDDI-b DDDI-a1DDDI-a210 -3 -2
0 10 20 30 40 N e u t r on p a i r d e n s it y (f m - ) r (fm) cell 6 -3 -2
0 10 20 30 40r (fm) cell 7 -4 -3 -2
0 10 20 30 40r (fm) cell 8 -4 -3 -2
0 10 20 30 40r (fm) cell 9 -4 -3 -2
0 10 20 30 40 50r (fm) cell 10
Fig. 8
Calculated neutron pair densities in various cells in the inner crust of neutronstars, listed in Table 2, obtained with three DDDI models, DDDI-b (dotted curve), DDDI-a1(dot-dashed), and DDDI-a2 (dashed). The horizontal line is the results of the uniform-BCScalculation. For the symbols, see the caption of Fig. 5.
We have studied in detail the proximity effect of neutron pair correlation in the innercrust of neutron stars by applying the Skyrme-Hartree-Fock-Bogoliubov theory formulatedin the coordinate representation. We describe a many-nucleon system consisting of Z protons(which form a nuclear cluster) and neutrons with a given positive Fermi energy, confined ina spherical box. If we choose the box radius R box equal to the Wigner-Seitz radius of thelattice cell, the calculation corresponds to the Wigner-Seitz approximation often adopted inpreceding studies. We found however that for the realistic Wigner-Seitz radius R cell ∼ − R box ≥
100 fm. In other words,we considered a simplified model of the inner crust matter in which a single nuclear clusteris immersed in a uniform neutron superfluid, prepared in a sufficiently large box. As the18ffective interaction causing the pairing correlation, we introduced new parameterizations ofthe density-dependent delta interaction (DDDI-a1 and DDDI-a2) so that they reproduce theab initio evaluations of the pair gap in low-density neutron matter as well as the experimentalpair gap in finite nuclei.Focusing on the neutron pair density ˜ ρ n ( r ) (i.e. a locally defined pair condensate), we haveexamined how ˜ ρ n ( r ) is affected by the presence of the nuclear cluster and how this quantityaround the cluster converges to the limiting value of the immersing neutron superfluid. Itis found from a systematic analysis that range of the proximity effect is characterized bythe coherence length of neutron superfluid measured from the edge position of the cluster.An important feature is that the coherence length ξ depends strongly on the density ρ n ofneutron superfluid. The coherence length is as short as ξ ∼ − × − fm − < ∼ ρ n < ∼ × − fm − while it increases gradually at lower density ρ n < ∼ × − fm − and rather quickly at higher density ρ n > ∼ × − fm − .Applying the above result to the realistic configurations of the inner crust, we predictthat the proximity effect is well limited in the vicinity of the nuclear cluster, i.e. in a suffi-ciently smaller area than the Wigner-Seitz cell in the middle layers of the inner crust withbaryon density 5 × − fm − < ∼ ρ b < ∼ × − fm − . On the contrary, the proximity effectis predicted to extend to the whole volume of the Wigner-Seitz cell in the shallow layersof the inner crust with ρ b < ∼ × − fm − . Another region where the range of the proxim-ity effect is expected to cover the whole Wigner-Seitz cell is deep layers of the inner crustwith ρ b > ∼ × − fm − , where the Wigner-Seitz radius becomes small R cell < ∼
20 fm whilethe coherence length may becomes comparable or larger than R cell . This observation indi-cates that in these layers there is no clear separation between the nuclear cluster and theimmersing neutron superfluid as far as the pairing correlation is concerned. It implies thatthe phenomena originating from the pair correlation and superfluidity, such as the vortexpinning and the superfluid phonon excitations may also be affected by the proximity effect.It is noted also that theoretical approaches taking into account the lattice configuration ispreferred for such cases. It is a subject to be pursued in future study. Acknowledgement
We thank T. Inakura, K. Sekizawa, and K. Yoshida for valuable discussions. We alsothank A. Ohnishi for a critical comment on the DDDI models. This work was supported bythe JSPS KAKENHI (Grants Nos. 17K05436 and 20K03945).19 ppendix A: Effective contact interaction for the GMB gap
Here we discuss the parameter set of DDDI which reproduces the pairing gap of Gor’kovMelik-Barkuhudarov (GMB) in the dilute limit of neutron matter. This is introduced bycombining the known arguments on the GMB pairing gap [31, 32] and on the effectivestrength of the contact interaction [35, 36].Let us first outline the relation between the strength of the contact interaction and thepairing gap in the BCS approximation. For the pairing interaction of the contact two-bodyforce v ( (cid:126)r − (cid:126)r ) = V δ ( (cid:126)r − (cid:126)r ) , the gap equation in the weak-coupling BCS approximationreads 1 V = − (cid:88) (cid:126)k (cid:112) ( e k − λ ) + ∆ (7)where e k = (cid:126) k m , λ = e F = (cid:126) k F m , and ∆ is the single-particle energy, the Fermi energy (withthe Fermi momentum k F ) and the pairing gap, respectively. To avoid the divergence inherentto the contact interaction, the sum (cid:80) (cid:126)k ≡ π ) (cid:82) k cut πk dk is performed with a cut-offmomentum k c or a cut-off single-particle energy e cut = (cid:126) k / m . The force strength v can be chosen so that the same interaction reproduces the zero-energy T-matrix T = π (cid:126) am ,and the scattering length a of the nucleon scattering in the S channel. This requirementis expressed in terms of the Lippmann-Schwinger equation for the T-matrix, which can bewritten as 1 V = 1 T + 12 (cid:88) (cid:126)k e k , (8)which determines the force strength V as [35, 36] V = − π (cid:126) m k cut − π a . (9)The gap equation (7) combined with the T-matrix equation (8) is written as1 T = − (cid:88) (cid:126)k (cid:32) (cid:112) ( e k − e F ) + ∆ − e k (cid:33) . (10)The gap equation (10) is known to be solved analytically in the low-density limit k F → k F | a | (cid:28) k F (cid:28) k c [37, 38]. The right hand side of Eq. (10) is evaluated as (cid:39) N log (cid:16) e ∆8 e F (cid:17) , where N = mk F π (cid:126) is the single-particle level density at the Fermi energy.The paring gap in this limit is then given [23, 32, 37, 38] as∆ BCS = 8 e F e exp (cid:18) T N (cid:19) = 8 e F e exp (cid:18) π k F a (cid:19) . (11)Note that the T-matrix T plays a role of a renormalized interaction strength of the contactforce. 20t is known that the medium effect in the low-density limit can be evaluated per-turbatively as originally discussed by Gor’kov and Melik-Barkhudarov [26]. The effect isrepresented as an induced interaction [31, 32] U ind = N T (1 + 2 log 2) / T → T + U ind , where the numerical factor (1 + 2 log 2) / T → T + U ind (cid:39) T − N , (12)and hence the GMB pairing gap ∆ GMB valid in the low-density limit is given as∆
GMB = 8 e F e exp (cid:18) T N − (cid:19) = 1(4 e ) / ∆ BCS (13)with a reduction of a factor of (cid:39) / . V GMB of the contact force which reproduces the GMB pairing gap is given by1 V GMB = 1 T − N + 12 (cid:88) (cid:126)k e k , (14)which determines V GMB as V GMB = V (cid:26) − k F k cut − π a + O (cid:0) k F (cid:1)(cid:27) . (15)We note that the force strength V GMB depends on the Fermi momentum k F . Expanded inpowers of k F , relevant to the low-density limit k F → k F . It can beexpressed also in terms of the density ρ = k F / π as V GMB = V (cid:40) − η (cid:18) ρρ (cid:19) / + O (cid:32)(cid:18) ρρ (cid:19) / (cid:33)(cid:41) (16)with η = 1 + 2 log 23 k F k cut − π a , k F = (3 π ρ ) / . (17) Appendix B: BCS calculation for uniform neutron matter
Here we describe the selfconsistent Hartree-Fock plus BCS approximation which isadopted to describe the pairing property of uniform neutron matter.21or a given value of the neutron Fermi energy λ n , we numerically solve the coupledequations ∆( λ n ) = − V n [ ρ n ]4 π (cid:90) k (cid:48) c dkk ∆( λ n ) E ( k ) , (18) E ( k ) = (cid:113) ( e ( k ) − λ n ) + ∆ , e ( k ) = (cid:126) k m ∗ n ( ρ n ) + U n ( ρ n ) , (19) ρ n ( λ n ) = 12 π (cid:90) k (cid:48) c dkk (cid:26) e ( k ) − λ n E ( k ) (cid:27) , (20)where U n ( ρ n ) and m ∗ n ( ρ n ) are the Hartree-Fock potential and the effective mass of neutrons,obtained from the SLy4 functional. The cut-off momentum k (cid:48) c is determined by e ( k (cid:48) c ) − λ = E cut so that it corresponds to the cut-off energy in the coordinate-space HFB calculation.The above scheme is called the uniform-BCS calculation in this paper.The coherence length ξ can be calculated by evaluating the size of the Cooper pair andis given ξ = (cid:113) (cid:104) r (cid:105) (21) (cid:10) r (cid:11) = (cid:90) d(cid:126)rr | Ψ pair ( r ) | = (cid:82) ∞ dkk ( ∂∂k u k v k ) (cid:82) ∞ dkk ( u k v k ) . (22) References [1] N. Chamel, and P. Haensel, Living Rev. Relativity, , 10 (2008). https://doi.org/10.12942/lrr-2008-10[2] N. Chamel, S. Goriely, J. M. Pearson, and M. Onsi, Phys. Rev. C , 045804 (2010).https://doi.org/10.1103/PhysRevC.81.045804[3] N. Sandulescu, N. V. Giai, and R. J. Liotta, Phys. Rev. C , 045802 (2004).https://doi.org/10.1103/PhysRevC.69.045802[4] C. Monrozeau, J. Margueron, N. Sandulescu, Phys, Rev. C , 065807 (2007).https://doi.org/10.1103/PhysRevC.75.065807[5] A. Pastore, Phys. Rev. C , 015809 (2015). https://doi.org/10.1103/PhysRevC.91.015809[6] P. M. Pizzochero, F. Barranco, E. Vigezzi, R. A. Broglia, Astrophys. J. , 381 (2002).https://doi.org/10.1086/339284[7] A. Avogadro, F. Barranco, R. A. Broglia, and E. Vigezzi, Nucl. Phys. A811 , 378 (2008).https://doi.org/10.1016/j.nuclphysa.2008.07.010[8] G. Wlaz(cid:32)lowski, K. Sekizawa, P. Magierksi, A. Bulgac, and M. N. Forbes, Phys. Rev. Lett. , 232701 (2016).https://doi.org/10.1103/PhysRevLett.117.232701[9] S. Jin, A. Bulgac, K. Roche, and G. Wlaz(cid:32)lowski, Phys. Rev. C , 044302 (2017).https://doi.org/10.1103/PhysRevC.95.044302[10] D. N. Aguilera, V. Cirigliano, J. A. Pons, S. Reddy, and R. Sharma, Phys. Rev. Lett. , 091101 (2009).https://doi.org/10.1103/PhysRevLett.102.091101 In our previous publication [15], we made an approximation to Eq. (20) using a Fermi gas relation ρ n = π (cid:0) mλ n (cid:126) (cid:1) / . Hence the result for neutron matter shown in Fig. 2 of Ref. [15] is slightly different fromthat of the present study.
11] C. J. Pethick, N. Chamel, and S. Reddy, Prog. Theor. Phys. Supple. , 9 (2010).https://doi.org/10.1143/PTPS.186.9[12] V. Cirigliano, S. Reddy, and R. Sharma, Phys. Rev. C , 045809 (2011).https://doi.org/10.1103/PhysRevC.84.045809[13] D. Page and S. Reddy, in Neutron Star Crust , ed. by C. Bertulani and J. Piekarewicz (Nova Science, 2012),p.281.[14] N. Chamel, D. Page, and S. Reddy, Phys. Rev. C , 035803 (2013).https://doi.org/10.1103/PhysRevC.87.035803[15] T. Inakura, and M. Matsuo, Phys. Rev. C , 025806 (2017). http://doi.org/10.1103/PhysRevC.96.025806[16] T. Inakura, and M. Matsuo, Phys. Rev. C , 045801 (2019). http://doi.org/10.1103/PhysRevC.99.045801[17] P. D. De Gennes, Rev. Mod. Phys. , 225 (1964). https://doi.org/10.1103/RevModPhys.36.225[18] P.G. de Gennes, Superconductivity of Metals and Alloys , (Advanced Book Program, Perseus Books, New York,N.Y., 1999).[19] E. Chabanat, P. Bonche, P. Heenen, J. Meyer, and R. Schaeffer, Nucl. Phys. A , 231 (1998).https://doi.org/10.1016/S0375-9474(98)00180-8[20] M. Matsuo, Phys. Rev. C , 044309 (2006). https://doi.org/10.1103/PhysRevC.73.044309[21] M. Matsuo, Y. Serizawa, and K. Mizuyama, Nucl. Phys. A , 307c (2007).https://doi.org/10.1016/j.nuclphysa.2007.01.017[22] D. J. Dean and M. Hjorth-Jensen, Rev. Mod. Phys. , 607 (2003).https://doi.org/10.1103/RevModPhys.75.607[23] G. C. Strinati, P. Pieri, G. R¨opke, P. Schuck, M. Urban, Phys. Rep. , 1 (2018).https://doi.org/10.1016/j.physrep.2018.02.004[24] S. Gandolfi, A. Gezerlis, J. Carlson, Ann. Rev. Nucl. Part. Sci. , 303 (2015). https://doi.org/10.1146/annurev-nucl-102014-021957[25] U. Lombardo and H.-J. Schulze, in Physics of Neutron Star Interiors , edited by D. Blaschke, N. K. Glendenning,and A. Sedrakian, Vol. 578 of Lecture Notes in Physics (Springer, New York, 2001), p.30.[26] L. P. Gor‘kov and T. K. Melik-Barkhudarov, Sov. Phys. JETP , 1018 (1961).[27] F. Barranco, R. A. Broglia, H. Esbensen, and E. Vigezzi, Phys. Rev. C , 1257 (1998).https://doi.org/10.1103/PhysRevC.58.1257[28] A. Gezerlis, and J. Carlson, Phys. Rev. C , 025803 (2010). https://doi.org/10.1103/PhysRevC.81.025803[29] T. Abe, and R. Seki, Phys. Rev. C , 054002 (2009). https://doi.org/10.1103/PhysRevC.79.054002[30] J. W. Negele, and D. Vautherin, Nucl. Phys. A , 298 (1973). https://doi.org/10.1016/0375-9474(73)90349-7[31] H. Heiselberg, C. J. Pethick, H. Smith, and L. Viverit, Phys. Rev. Lett. , 2418 (2000).https://doi.org/10.1103/PhysRevLett.85.2418[32] C. J. Pethick and H. Smith, Bose-Einstein Condensation in Dilute Gasses , (Cambridge Univ. Press, Cambridge,2002).[33] S. Gandolfi, A. Yu. Illarionov, S. Fantoni, F.Pederiva, K.E.Schmidt, Phys. Rev. Lett. , 132501 (2008).https://doi.org/10.1103/PhysRevLett.101.132501[34] W. Satula, J. Dobaczewski, and W. Nazarewicz, Phys. Rev. Lett. , 3599 (1998).https://doi.org/10.1103/PhysRevLett.81.3599[35] G. F. Bertsch and H. Esbensen, Ann. Phys. (N.Y.) , 327 (1991). https://doi.org/10.1016/0003-4916(91)90033-5[36] E. Garrido, P. Sarriguren, E. Moya de Guerra, and P. Schuck, Phys. Rev. C , 064312 (1999).https://doi.org/10.1103/PhysRevC.60.064312[37] T. Papenbrock and G. F. Bertsch, Phys. Rev. C , 2052 (1999). https://doi.org/10.1103/PhysRevC.59.2052[38] M. Marini, F. Pistolesi, and G. C. Strinati, Eur. Phys. J. B , 151 (1998).https://doi.org/10.1007/s100510050165[39] The 2016 Atomic Mass Evaluation, http://amdc.in2p3.fr/web/masseval.htmlhttp://amdc.in2p3.fr/web/masseval.html, 151 (1998).https://doi.org/10.1007/s100510050165[39] The 2016 Atomic Mass Evaluation, http://amdc.in2p3.fr/web/masseval.htmlhttp://amdc.in2p3.fr/web/masseval.html