Proximity effect of pair correlation in the inner crust of neutron stars

We adopt the Skyrme-Hartree-Fock-Bogoliubov method to describe the inner crust matter. Since the method is an extension of that is used in Refs. inakura2017 ; inakura2019 , we describe it briefly with 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 the partial wave expansion. The zero temperature is assumed and the spherical symmetry of solutions is imposed. Electrons are neglected. The radial HFB equation for a given angular quantum numbers $lj$ reads

where $\phi_{1}^{qlj},\phi_{2}^{qlj}$ is the quasiparticle wave function. Index $q$ denotes neutron or proton.

We discretize the radial coordinate with an interval $h=0.2$ fm as $r_{i}=i*h-h/2=h/2,3h/2,\cdots$ $(i=1,\cdots,N)$ up to the edge $r=R_{\mathrm{box}}$ of the box, and use the nine-point formula to represent the derivatives in the Hartree-Fock Hamiltonian $h_{qlj}(r)$. We impose the Dirichlet-Neumann boundary condition NV1973 , with which even-parity wave functions vanish at the edge of the box and the first derivatives of odd-parity wave functions vanish at the same position. Equation (1) is represented as a matrix eigenvalue problem where the wave function at the discretized coordinates $\left(\phi_{1}^{qlj}(r_{1}),\cdots\phi_{1}^{qlj}(r_{N}),\phi_{2}^{qlj}(r_{1}),\cdots\phi_{2}^{qlj}(r_{N})\right)^{T}$ is a $2N$-dimensional vector. We use routine DSYEVX in the LAPACK package to solve the eigenvalue problem for the symmetric matrix. If we treat the lattice configuration of the nuclear clusters by means of the Wigner-Seitz approximation, the box radius $R_{\mathrm{box}}$ is chosen to be the size of the Wigner-Seitz cell. We shall also choose larger boxes $R_{\mathrm{box}}=100$ fm or 200 fm, as we explain below. All the quasiparticle states up to a maximal quasiparticle energy $E_{\mathrm{max}}=60$ MeV are included to calculate the number density, the pair density and all the quantities needed to calculate the selfconsistent potentials. We put also a cut-off $l_{\mathrm{max}}$ on the angular momenta of the partial waves so that $l_{\mathrm{max}}>\sqrt{E_{\mathrm{max}}/(\hbar^{2}/2m)}R_{\mathrm{box}}$: $l_{\mathrm{max}}=200\hbar$ for $R_{\mathrm{box}}=100$ fm, and $l_{\mathrm{max}}=400\hbar$ for $R_{\mathrm{box}}=200$ fm, for example. We use the parameter set SLy4 chabanat 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 $\Delta_{q}(r)$. We vary the neutron Fermi energy $\lambda_{n}$ to control the neutron density and we determine the proton Fermi energy $\lambda_{p}$ to fix the proton number $Z$ of the nuclear cluster. The other details are the same as in the previous study inakura2017 ; inakura2019 .

As the pairing interaction, we use a density-dependent delta-interaction (DDDI), given as

for neutrons. Here $V_{n}[\rho_{n}(\vec{r}),\rho_{p}(\vec{r})]$ is the density-dependent interaction strength, and $(1-P_{\sigma})/2$ is the projection operator for the spin singlet channel. The pair potential is then $\Delta_{n}(r)=V_{n}[\rho_{n}(r),\rho_{p}(r)]\tilde{\rho}_{n}(r)$ with the neutron pair density (the neutron pair condensate)

We consider the following three models for the interaction strength $V_{n}[\rho_{n}(\vec{r}),\rho_{p}(\vec{r})]$.

The first one, which we introduced in Refs. matsuo2006 ; matsuo2007 , is given as

with $\rho_{0}=0.08$ fm^-3. Here the overall constant $V_{0}=-458.4$ $\mathrm{MeV\,fm^{3}}$ is determined to reproduce the ${}^{1}S_{0}$ scattering length $a$=-18.5 fm in free space (i.e. at zero density) under the single-particle cut-off energy $e_{\mathrm{cut}}=60$ MeV. The dependence of the interaction strength $V_{n}[\rho_{n}]$ on the neutron density $\rho_{n}$ is determined so that it reproduces the neutron pairing gap in pure neutron matter which is obtained in the BCS approximation using a bare nuclear force matsuo2006 ; matsuo2007 . We denote the parameterization, Eq. (4), as “DDDI-b” since it refers to the BCS gap with the bare nuclear force. (It is the same as the parametrization DDDI-G3RS in Ref. matsuo2006 .)

In the present study we introduce more realistic modeling of the neutron pairing appropriate to the inner crust matter. Here we consider parametrizations of the DDDI that provide realistic pairing gap both in neutron matter and in finite nuclei. Concerning the neutron matter, it is known that the pairing gap is affected by medium effects beyond the BCS approximation, and many of theoretical studies trying to evaluate the medium effects predict a significant reduction from the BCS gap while the predicted values spread in a wide range Dean2003 ; Lombardo2001 ; Gandolfi-Gezerlis-Carlson ; Strinati2018 . Nevertheless, the pairing gap in the low-density limit is believed to be described reliably by a perturbative approach to the screening effect, discussed first by Gor’kov and Melik-Barkhudarov (GMB) gmb , and the pairing gap $\Delta_{\mathrm{GMB}}$ in the GMB framework gives a reduction of a factor of $(4e)^{1/3}\simeq 2.2$ from the BCS pairing gapheiselberg2000 ; Pethick-Smith ; Strinati2018 . Recently, numerical ab initio calculations based on Monte-Carlo methods have been performed for pure neutron matter in low density region $\rho_{n}\simeq 10^{-5}-10^{-2}$ fm^-3, and the predicted pairing gaps are reduced from the BCS gap by a factor of 1.5 - 2 gezerlis2010 ; abe2009 ; Gandolfi2008 ; Gandolfi-Gezerlis-Carlson . We can refer to these studies in requiring a new parametrization of the DDDI. It is also known that the pairing gap in finite nuclei cannot be described well by the BCS approximation applied to the bare nuclear force, and there is no ab initio evaluation of the gap in finite nuclei. Instead we 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 the density-dependent interaction strength:

The first term is introduced to describe the GMB gap appropriate to the low-density limit of the pure neutron matter. As discussed in Appendix A, the force strength $V_{\mathrm{GMB}}$ of the contact force which reproduces the GMB pairing gap $\Delta_{\mathrm{GMB}}$ depends on the neutron Fermi momentum $k_{F,n}$ or the density $\rho_{n}$ of neutron matter. The dependence is expressed as a linear term proportional to $k_{F,n}$ or $\rho_{n}^{1/3}$ if it is expanded in powers of $k_{F}$. Requiring that the GMB pairing gap is reproduced by the DDDI in the low-density limit $\rho_{n}\rightarrow 0,k_{F,n}\rightarrow 0$. the parameters of the first term in Eq. (5) is fixed to $\alpha_{1}=1/3$ and $\eta_{1}=0.59$.

The second and third terms are introduced to represent the pairing gap of neutron matter at finite density and that in finite nuclei. In particular, the second term together with the first term is relevant to the pairing gap in neutron matter, and we assume that the second term has a power $\alpha_{2}=2/3$, i.e., $\propto\rho_{n}^{2/3}\propto k_{F,n}^{2}$ the second power of neutron Fermi momentum $k_{F,n}$. We then require that the coefficient $\eta_{2}$ of this term is consistent with the ab initio pairing gap of neutron matter obtained for $10^{-5}$ fm${}^{-3}\lesssim\rho_{n}\lesssim 10^{-2}$ fm^-3 in the quantum Monte Carlo calculation by Gezerlis and Carlson gezerlis2010 and the determinantal lattice Monte Carlo calculation by Abe and Seki abe2009 . Note however that this requitement alone does not fix uniquely the coefficient $\eta_{2}$ since these ab initio calculations are slightly different with each other and there is no ab initio results for moderately low densities $10^{-2}$ fm${}^{-3}\lesssim\rho_{n}\lesssim 10^{-1}$ fm^-3.

The third term dependent on the proton density represents a part of medium effects associated with systems with a proton fraction. For simplicity we assume that it is proportional to the proton Fermi momentum $k_{F,p}$ or the proton density $\rho_{p}^{1/3}$. ¹¹1 A perturbative estimate of the medium effect in symmetric matter gives an attractive induced interaction proportional to $N_{0,p}\propto k_{F,p}$ heiselberg2000 . We use both the coefficient $\eta^{\prime}_{1}$ of this term and the uncertainty in $\eta_{2}$ to describe the pairing gap in finite nuclei. In practice, we require that the average neutron pairing gap $\Delta_{n,uv}=\int\Delta_{n}(r)\rho(r)d\vec{r}/\int\rho(r)d\vec{r}$ in ¹²⁰Sn obtained from our HFB model reproduces the experimental neutron gap $\Delta_{n,\mathrm{exp}}\simeq 1.3$ MeV, extracted from the 3-point odd-even mass difference Satula1998 .

In the present study we prepare two different parameter sets to represent the remaining uncertainty of the neutron pair gap. In one case (we call “DDDI-a1” below), we choose $\eta_{2}=0.06$ and $\eta^{\prime}_{1}=0$ so that the neutron pairing gap in ¹²⁰Sn is reproduced without $\eta^{\prime}_{1}$. In this case, the pairing gap of neutron matter is close to that of Abe and Seki abe2009 , and the neutron matter pairing gap at moderately low density is rather large $\Delta\sim 1-2$ MeV, as shown in Fig. 1. It is remarked that the medium effect associated with the nuclear cluster or finite nuclei is effectively included in $\eta_{2}$. In another parameter set (“DDDI-a2”), we consider a case that the neutron matter pairing gap at moderately low density is relatively small; we determine $\eta_{2}=0.255$ so as to make the neutron matter pairing gap vanish at $\rho_{n}=\rho_{0}$ as the BCS gap does. The parameter $\eta^{\prime}_{1}=-0.195$ is then determined to reproduce the neutron gap in ¹²⁰Sn. (Note that the neutron matter pairing gap reproduces approximately the result of Gezerlis and Carlson gezerlis2010 , as shown in Fig. 3.) The parameter sets of the three DDDI models are summarized in Table 1.

Figure 1 shows the neutron pairing gap in uniform neutron matter obtained from the Hartree-Fock plus BCS calculation using the DDDI models discussed above and the Skyrme functional with the parameter set SLy4. (The BCS calculation is briefly recapitulated in Appendix 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 well the gap obtained with the ab initio calculations. The DDDI-a2 reproduces approximately the result of Gezerlis and Carlson gezerlis2010 for the density range $\rho_{n}=10^{-5}-10^{-2}$ fm^-3. The gap of the DDDI-a2 at moderate density is small $\Delta<1.3$ MeV, and vanishes at $\rho_{n}\sim 0.08$ fm ($k_{F}\sim 1.4$ fm^-1) corresponding to neutrons in the saturated nuclear matter. The neutron gap of the DDDI-a1 is very close to that of the DDDI-a2 up to $\rho_{n}\mathop{\vbox{ \offinterlineskip\hbox{$<$}\hbox to7.7778pt{\hss\hbox{$\sim$}\hss}}}10^{-3}$ fm^-3, but deviate from it above $\rho_{n}\mathop{\vbox{ \offinterlineskip\hbox{$>$}\hbox to7.7778pt{\hss\hbox{$\sim$}\hss}}}10^{-3}$ fm^-3, It is rather close to the gap of Abe and Seki abe2009 , and the neutron matter pairing gap at moderately low density is rather large $\Delta\sim 1-2$ MeV. The parameter set DDDI-b gives a larger pairing gap at low and moderately low densities than DDDI-a1 and DDDI-a2, while at densities around the saturation the gap becomes small and almost vanishing. ²²2The auxiliary-field diffusion Monte Carlo method Gandolfi2008 , another ab initio calculation, predicts the pairing gap close to the BCS gap in the density range $\rho_{n}\lesssim 10^{-2}$ $\mathrm{fm^{-3}}$. Thus it may be considered that the parameter set DDDI-b represents partly this result. We consider that DDDI-a1 and DDDI-a2 are more realistic than DDDI-b while the difference between DDDI-a1 and DDDI-a2 represents the uncertainty in modeling the realistic pairing correlation. We also use the model DDDI-b since it simulates the BCS gap, which is a robust baseline common to all the models of realistic bare nuclear force Dean2003 .

Figure 2 shows the coherence length $\xi$ of superfluid uniform neutron matter, calculated as described in Appendix B. The coherence length $\xi$ depends strongly on the neutron density. The coherence length $\xi$ is as short as $\xi\mathop{\vbox{ \offinterlineskip\hbox{$<$}\hbox to7.7778pt{\hss\hbox{$\sim$}\hss}}}10$ fm at $\rho_{n}=10^{-3}-2\times 10^{-2}$ fm^-3. The coherence length becomes long gradually as the neutron density decreases less than $10^{-3}$ fm^-3, and it also does rather sharply for increasing $\rho_{n}$ more than $\sim 3\times 10^{-2}$ fm^-3. The minimum value of the coherence length is $\xi\sim 3.6$ fm for DDDI-b at neutron density corresponding to $\lambda_{n}\approx 5$ MeV, $\xi\sim 4.6$ fm for DDDI-a1 at $\lambda_{n}\approx 6$ MeV, and $\xi\sim 6.1$ fm for DDDI-a2 at $\lambda_{n}\approx 5$ MeV. The dotted curve in Fig. 2 shows the average inter-neutron distance $d=\rho_{n}^{-1/3}$. It is noted that the coherence length $\xi$ is shorter than the average inter-neutron distance $d=\rho_{n}^{-1/3}$ at wide density interval $\rho_{n}=10^{-5}-10^{-2}$ fm^-3 for DDDI-b, or comparable with $d$ at $\rho_{n}=10^{-4}-10^{-2}$ fm^-3 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-coupling pairing, characterized as the BCS-BEC crossover phenomenon Strinati2018 .

Figure 3 shows the average neutron pairing gap $\Delta_{n,uv}$ in Sn isotopes obtained in the present HFB code. The average neutron pairing gap in ¹²⁰Sn calculated with DDDI-b, DDDI-a1 and DDDI-a2 is $\Delta_{n,uv}=0.48,1.28$ and 1.28 MeV, respectively. Note that the pair gap of DDDI-a1 and DDDI-a2 reproduces the experimental gap reasonably well over the long isotope chain while DDDI-b gives only a half of the experimental value.

In order to investigate general features of the proximity effect, we shall first examine cases where the density of the surrounding neutron superfluid is systematically varied while the proton number is fixed. In the next subsection we discuss realistic configurations of the inner crust matter, for which the proton number and the density of neutron superfluid are chosen to represent various layers of the inner crust.

The proton number is $Z=28$ in all the examples in this subsection and we vary the neutron Fermi energy $\lambda_{n}$ systematically from 0.2 MeV to 6 MeV, which corresponds to the density of the uniform neutron superfluid from $\rho_{n}=4\times 10^{-5}$ fm^-3 to $1\times 10^{-2}$ fm^-3.

A typical result obtained for $\lambda_{n}=4$ MeV ($\rho_{n}=6.1\times 10^{-3}$ fm^-3) with DDDI-b is shown in Fig. 5, where plotted are the number densities of neutrons and protons, $\rho_{n}(r)$ and $\rho_{p}(r)$, and the neutron pair density $\tilde{\rho}_{n}(r)$ as a function of the radial coordinate $r$. It is seen that the nuclear cluster is well localized in a central region as seen in the profile of the neutron density $\rho_{n}(r)$ which converges rather quickly to a constant value at around $r\approx 8$ fm (the proton density $\rho_{p}(r)$ converges to zero around $r\approx 6$ fm. ). The surface of the nuclear cluster may be quantified by fitting to the neutron density with a function of the Woods-Saxon type,

where $R_{\mathrm{s}}$ defines the half-density surface, and $a$ represents the diffuseness of the surface. The constant $\rho_{n,\mathrm{M}}$ is the neutron density obtained from the uniform-BCS performed for the same value of $\lambda_{n}$. The values of $f_{0}$, $R_{\mathrm{s}}$ and $a$ are extracted from a fitting. In addition we find it useful to consider “the edge” of the nuclear cluster to evaluate the area where the cluster exists. We define the nuclear edge by $R_{\mathrm{edge}}=R_{\mathrm{s}}+4a$. The edge position $r=R_{\mathrm{edge}}$ is indicated by the black circle in Fig. 5, and it is seen that $R_{\mathrm{edge}}$ represents well the position where the neutron density $\rho_{n}(r)$ converges to $\rho_{n,\mathrm{M}}$.

A most noticeable feature in Fig. 5 is that the neutron pair density $\tilde{\rho}_{n}(r)$ exhibits behaviours different from those of the neutron number density $\rho_{n}(r)$. It is seen that the neutron pair density $\tilde{\rho}_{n}(r)$ slowly converges and reaches the uniform-BCS value at around $r\approx 12$ fm, deviating from $R_{\mathrm{edge}}$ by about 4 fm. In other words the influence of the nuclear cluster extends to the neighbour region beyond $R_{\mathrm{edge}}$. This slow convergence is nothing but the proximity effect. In this example the neutron pair density inside the cluster is significantly smaller than that outside the cluster. This reflects the characteristic density dependence of the neutron pair gap of the DDDI-b model; the gap for the density inside the cluster ($\rho_{n}\sim\rho_{0}$) is very small $\Delta\mathop{\vbox{ \offinterlineskip\hbox{$<$}\hbox to7.7778pt{\hss\hbox{$\sim$}\hss}}}0.1$ MeV whereas that for the density of neutron superfluid ($\rho_{n,\mathrm{M}}\sim 6.1\times 10^{-3}$ fm) is relatively large $\Delta\sim 2.1$ MeV.

It has been argued that the proximity effect emerges in a region adjacent to the border with its length scale characterized by the coherence length $\xi$ of the superfluid/superconducting matter gennes1964 . We here assume that the border between the neutron superfluid and the nuclear cluster is approximated by the edge radius $R_{\mathrm{edge}}$, rather than the half-density surface $R_{\mathrm{s}}$. If these considerations are reasonable, it is expected that the proximity effect is seen up to $r\approx R_{\mathrm{edge}}+\xi$. In the case shown in Fig. 5, the position where the neutron pair density converges to the uniform-BCS value corresponds well to $r=R_{\mathrm{edge}}+\xi=8.27$ 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 various neutron Fermi energies and for three different pairing interactions: the DDDI-b (panel (a) in each figure), the DDDI-a1 (b), and the DDDI-a2 (c). Figure 6(a)(b)(c) shows the results for the neutron Fermi energy $\lambda_{n}=2-6$ MeV corresponding the neutron density $\rho_{n}\sim 10^{-3}-10^{-2}$ fm^-3 (see Fig. 1), and Fig. 6(d)(e)(f) for $\lambda_{n}=0.2-1\,\mathrm{MeV}$ ($\rho_{n}\sim 10^{-4}-10^{-3}$ fm^-3).

The proximity effect is clearly visible in all the cases; the pair density converges to that of the uniform neutron superfluid at a position deviating significantly from the edge position $r=R_{\mathrm{edge}}$ of the nuclear cluster. It is also seen that the range of the proximity effect depends rather strongly on the neutron Fermi energy or the density of the neutron superfluid, especially at low neutron density $\rho_{n,\mathrm{M}}\mathop{\vbox{ \offinterlineskip\hbox{$<$}\hbox to7.7778pt{\hss\hbox{$\sim$}\hss}}}5\times 10^{-4}$ fm^-3 and $\lambda_{n}\mathop{\vbox{ \offinterlineskip\hbox{$<$}\hbox to7.7778pt{\hss\hbox{$\sim$}\hss}}}1.0$ MeV. It also depends on the three DDDI models. Despite the differences in the pairing properties, we confirm here that the range where the proximity effect reaches is described well by the position $r=R_{\mathrm{edge}}+\xi$ (marked with the square symbol), characterized by the coherence length $\xi$ measured from the edge $R_{\mathrm{edge}}$ of the nuclear cluster. (Note that the edge position $R_{\mathrm{edge}}$ of the nuclear cluster depends only weakly on the neutron Fermi energy, and there is essentially 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 $\mathop{\vbox{ \offinterlineskip\hbox{$<$}\hbox to7.7778pt{\hss\hbox{$\sim$}\hss}}}10$ fm at moderately low density $\rho_{n}=7\times 10^{-4}-2\times 10^{-2}$ fm^-3 for the three DDDI’s. This brings about the short range of the proximity effect seen for $\lambda_{n}=2-6$ MeV. This is related to the specific feature of the dilute neutron superfluid that the BCS-BEC crossover is about to occur at these densities.

A long range of the proximity effect seen for $\lambda_{n}=0.2-1.0$ MeV can be related to the monotonic and considerable increase of the coherence length $\xi$ with decreasing neutron density for very low density $\rho_{n}\mathop{\vbox{ \offinterlineskip\hbox{$<$}\hbox to7.7778pt{\hss\hbox{$\sim$}\hss}}}10^{-3}$ fm^-3. Note that for $\rho_{n}\sim 10^{-5}-10^{-4}$ fm^-3, the coherence length $\xi=5-20$ fm in the case of DDDI-b, $\xi=10-36$ fm for DDDI-a1 and $\xi=11-37$ fm for DDDI-a2. If the density of the external neutron superfluid decreases further, 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 neutron stars. Here we refer to the Wigner-Seitz cells obtained in Negele and Vauthrin NV1973 for various layers of the inner crust. We perform the HFB calculation for the cells listed in Table LABEL:nvnuc using the large-box configuration. The proton number $Z$ and the Wigner-Seitz radius $R_{\mathrm{cell}}$ of each cell is taken from Ref. NV1973 . The neutron Fermi energy $\lambda_{n}$, the control parameter of the neutron density, is chosen so that the obtained density of the external neutron superfluid reproduces approximately the density of the neutron gas in Ref. NV1973 . For simplicity we use a common value of $\lambda_{n}$ for the three DDDI models. The box size is $R_{\mathrm{box}}=100$ fm for most cells and 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 plotted radial coordinate is the Wigner-Seitz radius $R_{\mathrm{cell}}$ for each cell. A noticeable feature is that in cells 3 to 8 the pair density converges to that of the uniform-BCS at a distance shorter than the half distance of the Wigner-Seitz radius. In other words the proximity effect is restricted only in a small area nearby the nuclear cluster. The area of uniform neutron superfluid and 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 coherence length of external neutron superfluid is the smallest $\xi\approx 4-6$ fm at cells 2-6 (for DDDI-b), cells 2-5 (for DDDI-a1), and cell 3 (for DDDI-a2), which are significantly smaller than the Wigner-Seitz radius of these cells at the middle layers.

In cells 9 and 10, where the external neutron superfluid is dilute ($\rho_{n,\mathrm{M}}\mathop{\vbox{ \offinterlineskip\hbox{$<$}\hbox to7.7778pt{\hss\hbox{$\sim$}\hss}}}1\times 10^{-4}$ fm^-3), the proximity effect extends to a major area of the Wigner-Seitz cell, beyond the half length of the Wigner-Seitz radius, especially for DDDI-a1 and DDDI-a2. This reflects the long coherence length at such very low densities: $\xi\mathop{\vbox{ \offinterlineskip\hbox{$>$}\hbox to7.7778pt{\hss\hbox{$\sim$}\hss}}}20-40$ fm for DDDI-a1 and DDDI-a2, and $\xi\mathop{\vbox{ \offinterlineskip\hbox{$>$}\hbox to7.7778pt{\hss\hbox{$\sim$}\hss}}}12-20$ fm for DDDI-b. Note that the pairing gap of DDDI-a1 and DDDI-a2 in dilute neutron matter is reduced from the BCS value (corresponding to DDDI-b) by a factor of about 2, leading to a longer coherence length in these realistic gap models DDDI-a1 and DDDI-a2.

Another case where a long-range proximity effect is predicted is cell 1 at relatively high density, where the external neutron density $\rho_{n,\mathrm{M}}\sim 0.04$ fm${}^{-3}\approx\rho_{0}/2$ is about a half of that of the saturated nuclear matter. The pair density deviates from that of the uniform neutron superfluid in the whole area of the Wigner-Seitz cell. In this cell with relatively high neutron density the predicted coherence length $\xi$ varies from 7 to 30 fm depending rather strongly on the pairing models, reflecting the uncertainty of the gap at such density. However, because of the relatively high baryon density and a large N/Z ratio, the Wigner-Seitz radius $R_{\mathrm{cell}}$ becomes small ($\sim 20$ fm) and the edge position $r=R_{\mathrm{edge}}$ of the nuclear cluster becomes as large as $\sim 13$ fm due to a thick neutron skin of the cluster. Consequently the range $R_{\mathrm{edge}}+\xi$ of the proximity effect exceeds the Wigner-Seitz radius irrespective of the uncertainty of the pairing gap. Note that cell 1 corresponds to a deep layer of the inner crust, where a transition to the so called pasta phase is about to occur. The present result suggests strong proximity effect also for the pasta phase at higher baryon density. We remark also that the proximity effect in these deep layers might be even stronger than the present prediction because of the presence of adjacent nuclear clusters in the lattice configuration, but a quantitative evaluation is beyond the scope of the present study.