On the surface nature of the nuclear pairing

Available online at www.sciencedirect.com

Physics Reports 391 (2004) 261 – 310 www.elsevier.com/locate/physrep

On the surface nature of the nuclear pairing M. Baldoa , U. Lombardob; d , E.E. Sapersteinc;∗ , M.V. Zverevc a

INFN, Sezione di Catania, 57 Corso Italia, I-95129 Catania, Italy b INFN-LNS, 44 Via S.-Soa, I-95123 Catania, Italy c Russian Research Centre, Kurchatov Institute, 123182 Moscow, Russia d Dipartimento di Fisica, 57 Corso Italia, I-95129 Catania, Italy Editor: G.E. Brown

Abstract The surface nature of nuclear pairing is con1rmed microscopically. A two-step approach is used in which the full Hilbert space S is split into the model subspace S0 and the complementary one, S
. The gap equation is solved in the model space in terms of the e6ective interaction Vpe6 which obeys the Bethe–Goldstone-type equation in the complementary subspace. The simplest nuclear systems with one-dimensional inhomogeneity are considered, i.e. semi-in1nite nuclear matter and the nuclear slab. Numerical solution is carried out for the separable representation of the Paris NN-potential. The equation for the e6ective pairing interaction is solved directly, without use of any form of local approximation. A version of the local approximation, the local potential approximation, is suggested which works su:ciently well even in the surface region. The e6ective pairing interaction obtained in our calculations reveals a strong variation in the surface region changing from a strong attraction outside the nuclear matter to almost zero value inside. The e6ective interaction is found to be dependent on the chemical potential . At  = −8 MeV, it reproduces qualitatively the phenomenological density-dependent e6ective pairing interaction, with the surface dominance, which was found previously in the self-consistent 1nite Fermi systems theory and in the new version of the energy functional method by Fayans et al. As || decreases, the surface attraction becomes stronger. The gap equation was solved in semi-in1nite matter and in the slab system with the help of the method by Khodel, Khodel and Clark which was suggested recently for nuclear matter. This method extended to non-homogeneous systems turned out to be very e:cient in this case. The gap  found for both the systems exhibits a strong variation in the surface region with a pronounced maximum near the surface. The surface e6ect in  turned out to be -dependent being enhanced at small ||. c 2003 Elsevier B.V. All rights reserved.  PACS: 21.30.−x; 21.60.−n; 21.65.+f Keywords: Nuclear pairing; Surface e6ects

∗

Corresponding author. Russian Research Centre, Kurchatov Institute, 123182 Moscow, Russia. E-mail addresses: [email protected] (E.E. Saperstein), [email protected] (M.V. Zverev).

c 2003 Elsevier B.V. All rights reserved. 0370-1573/$ - see front matter  doi:10.1016/j.physrep.2003.10.007

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Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Pairing in nuclear matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Gap equation for 1D-inhomogeneous nuclear matter with separable NN-force in terms of the e6ective pairing interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Equation for the e6ective pairing interaction in terms of the free o6-shell T -matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 5. E6ective pairing interaction in semi-in1nite nuclear matter. The local potential approximation . . . . . . . . . . . . . . . . . 6. E6ective pairing interaction for a slab of nuclear matter. A simple microscopic model for Vpe6 . . . . . . . . . . . . . . . 7. Surface behavior of the pairing gap in semi-in1nite nuclear matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. The pairing gap in a slab of nuclear matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. Conclusions and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A. The mixed representation of the form factors for the singlet 1 S channel . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix B. Evaluation of the free o6-shell T -matrix in the coordinate representation . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

262 267 273 277 279 291 294 298 304 305 305 307 309

1. Introduction Belyaev is one of the pioneers in the exploration of nuclear pairing. In his famous paper [1] of 1959, he outlined several lines along which this nuclear phenomenon was examined later by a lot of people for a long period. It is su:cient to mention the interplay between pairing and low-lying collective nuclear excitations or the inLuence of pairing on the nuclear moments of inertia. Later, in collaboration with Zelevinsky, Belyaev [2] predicted a new Goldstone-type collective mode in nuclei named coherent pairing Luctuations. In 1987, in collaboration with Smirnov, Tolokonnikov and Fayans [3], Belyaev developed a new method for the treatment of pairing in nuclei. A technique was elaborated for the exact treatment of the Gor’kov equations with local gap  for spherical nuclei in coordinate-space representation, in the case of the local gap . In this approach, no approximation is made in the treatment of the particle continuum. This rigorous method led to several new predictions, e.g. to appearance of a width of deep hole states due to the coupling to the continuum. It was shown also that this method could be helpful to distinguish the volume pairing from the surface one. Thus, the problem of pairing in nuclei has a long standing history. However, up to now no consensus has been reached on the nature of this phenomenon. The majority of practical calculations adopts the simplest BCS approximation in which the pairing Hamiltonian contains only one e6ective constant G which represents the matrix element of the e6ective pairing interaction Vpe6 . All the information about Vpe6 is embodied inside the value of G. Usually the quantity G, after separating a trivial 1=A dependence on the mass number A, is supposed to be a constant, thereby the volume character of pairing in nuclei is implicitly assumed. On the other hand, already at the dawn of the theory of nuclear pairing an alternative surface nature of this phenomenon was discussed. In particular, the model of purely surface pairing with a delta-function e6ective interaction localized near the surface was suggested by Green and Moszkowski [4] as far back as 1965. Simultaneously, the possibility of surface pairing in nuclei was considered in [5], where an attempt was made to 1nd out what type of pairing leads to a closer agreement with experimental nuclear masses. For this

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purpose, the double even–odd mass di6erences for near-magic nuclei were analyzed on the basis of the 1nite Fermi system (FFS) theory [6]. In this case, the e6ective interaction Vpe6 introduced in the treatment is identical to that in the pairing problem, but the magnitude of the e6ect is more sensitive to the individual quantum numbers of two particles (holes) added to the double-magic nucleus than in the case of nuclei with developed pairing. A two-parameter ansatz for the e6ective interaction was used, in the zero range form with the strength depending on the density (r) [5]. Speci1cally, the pairing interaction was taken in the form
  (r) p in (r) ex Ve6 (r) = C0 0 ; (1.1) + 0 1 − 0 0 where 0 = (r = 0), and the normalization factor C0 is equal to the inverse level density at the Fermi surface: C0 = (dn=dF )−1 = 300 MeV fm3 . This simple ansatz makes possible to choose beex in tween the volume pairing (in 0 = 0 = 0 ¡ 0) and the surface pairing (0  0 or positive). An intermediate situation is also possible. The analysis performed in [5] gave some evidences in favor in of the surface pairing with strong surface attraction (ex 0  −(3 : 4)), the internal parameter 0 being close to zero. However, the model of volume pairing can hardly be rejected conclusively on the basis of a purely phenomenological approach. Approximately 20 years later, this conclusion was con1rmed by a series of calculations [7,8] performed within the self-consistent FFS theory [9,10] with a two-parameter ansatz for Vpe6 which was close to Eq. (1.1). Again the model with a dominant surface contribution to matrix elements of Vpe6 looked preferable, but the model of volume pairing worked also su:ciently well with a constant 0 weakly dependent on A. In fact, this smooth A-dependence of the coupling constant was the only evidence against the volume pairing found in [7,8]. More recently, the group headed by Fayans attacked the problem within a new version of the energy functional method developed in [11,12]. The exact treatment of the Gor’kov equations in coordinate space [3] is an essential ingredient of this approach. Much more complicated form of the density-dependent e6ective pairing interaction, as compared to Eq. (1.1), was used, but also mainly ex determined by two parameters similar to in 0 and 0 . Very high accuracy was reached in these calculations in the description of nuclear binding energies and radii, approximately of the same quality as in the best HF calculations, but again no direct evidence pro or contra a surface pairing was found. To this aim, Fayans et al. [13] analyzed in detail odd–even e6ects in masses and radii of several long isotopic chains. It was found that a version of Vpe6 , with strong density dependence and surface dominance, reproduces the evolution of charge radii and nucleon separation energies su:ciently well, including kinks at magic neutron numbers and sizes of odd–even staggering. At the same time, a version of volume pairing could reasonably describe the odd–even staggering in separation energies but fails in reproducing sizes of the staggering in radii. Thus, a decisive evidence was found in [13] against the volume character of pairing in nuclei. To con1rm this phenomenological observation on the basis of general principles, one must inevitably invoke ab initio approaches. Apart from few-nucleon systems, ab initio approaches to nuclear theory deal mainly with in1nite nuclear matter. Although the analysis of pairing in nuclear matter in the framework of Brueckner theory predicts the existence of pairing in in1nite matter, it supports indirectly the idea of surface pairing in 1nite nuclei. Indeed, though quantitative predictions depend on the particular

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version of the realistic NN-potential used in the calculations and the approximation adopted for the nucleon self-energy, most of mean-1eld predictions are qualitatively similar and show that the value of the gap  at the saturation density 0 is rather small (0.2–0:3 MeV [14,15]) in comparison with the experimental value,   1 MeV, in heavy nuclei. Furthermore, the critical density c at which the pairing vanishes exceeds 0 by a very small amount and, on the other side, for ¡ 0 , the quantity ( ) increases rapidly with decreasing [15]). These results provide additional evidence in favor of a crucial role of the surface in nuclear paring. However, the accuracy of the mean-1eld approximation is questionable. For the problem of pairing in neutron matter, vertex corrections due to medium polarization were investigated in [16]. It turned out that these corrections change the gap at 1xed value of signi1cantly, reducing  in a broad range of and increasing it near the critical density of neutron matter nc found without taking into account the polarization corrections. Concurrently, the nc value itself increases as a result of the polarization e6ects. Self-energy corrections also contribute to sizably suppress the gap in neutron matter [17]. But these results of [16] cannot be directly applied to the nuclear matter, where the medium corrections are quite di6erent from neutron matter. In nuclear matter the vertex corrections include additional contributions from the tensor force with the isospin I =0 which cancel largely the previous e6ects [18]. In view of the sharp density dependence of  in nuclear matter in the vicinity of = 0 , additional uncertainties arise in applying the nuclear matter results to the interior of 1nite nuclei. First, parameters entering the Brueckner equations for nuclear matter di6er from those of 1nite nuclei. They are the two-particle energy E = 2, where  is the chemical potential, and the Fermi momentum kF . The conventional nuclear matter parameters are 0 = −16 MeV and kF0 = (1:5 0 =2 )1=3 = 1:35 fm−1 , the latter corresponding to 0  0:16 fm−3 . On the other hand, for stable 1nite nuclei we have   −8 MeV and kF  1:42 fm−1 . We see that the di6erence is not so big but not negligible. Second, the local density approximation (LDA) which is commonly used as a bridge from in1nite matter to 1nite nuclei fails in the case of pairing problem, even for the interior, because of the large value of the pairing correlation length P . Indeed, if estimated at  = 1 MeV, P exceeds the nuclear radius even for heavy nuclei. At last, it is evident that the LDA fails in the surface region because there is a density range in which the nuclear matter is unstable. Therefore, the LDA cannot be used as a tool to investigate the role of the surface in nuclear pairing. To clarify the role of the surface in nuclear pairing, we applied the Brueckner-type approach to semi-in1nite nuclear matter embedded in the Saxon–Woods potential with realistic nuclear parameters without LDA or any other local approximation [19–21]. The choice of the system was motivated by the following considerations. First, semi-in1nite nuclear matter is the simplest system that makes possible to consider surface phenomena in nuclei. Second, one can hope that, in the surface low-density region, the approximations made in Brueckner theory lead to smaller errors than in nuclear interior. In any case, this method trivially becomes legitimate in the vacuum at asymptotically large distances from the boundary because only ladder diagrams summed in the Brueckner theory survive there. It can be therefore conjectured that the parameter ex 0 in (1.1) is evaluated correctly within this approach. In [19] a method of direct solution of the Bethe–Goldstone equation for the case of the semi-in1nite geometry was developed. It was chosen the simplest model of  force as NN-interaction with the momentum cuto6 which correctly reproduces the NN-scattering data in the low-energy limit. In [20] study was extended to the case of realistic NN-interaction

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of a separable form. In particular, the separable representation [22,23] of the Paris potential [24] was used. The problem of solving the gap equation was treated in [19,20] with the help of a two-step approach by introducing the e6ective pairing interaction Vpe6 . In this approach, the full Hilbert space S is split into two parts. The 1rst one is the model subspace S0 , comparatively small, in which the gap equation is considered in terms of Vpe6 . In the complementary subspace S  the equation for Vpe6 is de1ned in terms of the bare NN-interaction. The main advantage of such a separation is the possibility of disregarding the pairing e6ects in the subspace S  provided the model space is wide enough. As a result, the equation for Vpe6 is reduced to the form similar to the Bethe–Goldstone equation in the singlet 1 S channel. This equation was solved in [20] for semi-in1nite nuclear matter without additional approximations with respect to the Brueckner theory itself. The mixed coordinate-momentum representation was used in which the Bethe–Goldstone-type equation for Vpe6 , for the case of the separable NN-interaction, is reduced to a set of one-dimensional integral equations which can be solved numerically. The model space was chosen in such a way that it includes all single-particle negative-energy states. This choice of the model space is typical for practical calculations for 1nite nuclei with the inclusion of pairing. Therefore the corresponding e6ective interaction, in principle, can be used in such calculations. It is worth to mention that a new type of local approximation was found in [20] which was named as the local potential approximation (LPA) to distinguish it from the LDA. It turned out that LPA, contrary to LDA, works su:ciently well, even at the surface. It was shown in [25] that LPA can be used also in the Bethe–Goldstone equation for the G-matrix in non-uniform systems. The singlet 1 S and triplet 3 S + 3 D channels were considered. A small modi1cation of the LPA recipe is necessary in this case. The e6ective pairing interaction found in [20] is a rather complicate non-local operator. After the application of some averaging procedure for calculating matrix elements of the states nearby the Fermi surface, it can be expressed as a -function with the coordinate-dependent strength which can be then approximately reproduced by Eq. (1.1). It was found in [20] that Vpe6 undergoes a sharp variation in the surface region, from almost zero in the bulk to very strong attraction in vacuum. The physical origin of this attraction is clear. Indeed, as it can be readily shown, in the asymptotic region, Vpe6 coincides with the o6-shell T -matrix of free NN-scattering T (E = 2) which exhibits a resonant behavior at small E in the 1 S-channel under consideration. Thus, the idea of surface nature of nuclear pairing is con1rmed. It is worth to point out that another mechanism of the surface e6ect in nuclear pairing was suggested in [26] and developed in detail for a slab model in [27]. It is related to the contribution to the e6ective pairing interaction of the virtual exchange by collective surface vibrations. We think that there is some overestimation of this contribution because the so-called local diagrams of vibration exchange were not taken into account. Indeed, the analysis [9] of the surface vibration e6ects in di6erent nuclear observables showed that, as a rule, the local contributions compensate, to a large extent, those of usual, non-local, vibration diagrams. Of course, the compensation is not complete, so that the surface vibration contribution should be taken into consideration in a consistent description of pairing in nuclei. Analysis of the e6ective pairing interaction found in [20] showed [28] that Vpe6 can be approximated by the o6-shell T -matrix not only in the asymptotic region. It turned out that a simple model of Vpe6 in terms of T (E = 2) works reasonably well in all the space. The reason is

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the occurrence of an occasional cancellation of di6erent in-medium contributions to Vpe6 which takes place only for a speci1c choice of the model space in [20]. It turned out that this approximation for Vpe6 is valid for di6erent values of  and can be used also in the vicinity of the drip line. In [21] the gap equation for semi-in1nite nuclear matter was solved with the e6ective interaction found in [20]. Instead of directly solving the gap equation with non-local Vpe6 which is a rather complicated job, we used a method proposed by Khodel, Khodel and Clark (hereafter referred to as KKC) developed in Refs. [29,30] for in1nite nuclear matter. To be more precise, in [21], we extended the KKC method to non-uniform systems. The application of this method simpli1es the solution of the gap equation signi1cantly. All the calculations were made for two values of the chemical potential:  =−16 and −8 MeV. A surface enhancement of the gap  was found, the e6ect being more pronounced for  = −8 MeV. The latter 1nding con1rms the above-mentioned necessity to be cautious with applying results obtained for nuclear matter to 1nite nuclei. The analogous e6ect of surface enhancement was found in [31] for a more simpli1ed model in which the nuclear matter was embedded in a semi-in1nite hard wall potential and the pairing problem was considered in the BCS approximation with the e6ective Gogny force. In [32] an analogous analysis was carried out for a slab of nuclear matter within the two approaches in the hope that a direct comparison of results can help to clarify the general features of the phenomenon under consideration. The slab system is much closer to a real atomic nucleus than the semi-in1nite one and a lot of results can be qualitatively related to it. In both models, a noticeable surface e6ect for the pairing gap was obtained of the same order of magnitude as it was previously found in semi-in1nite nuclear matter. The shape of the gap in coordinate space turned out to be qualitatively similar in both cases, with a signi1cant surface enhancement. For the value of the chemical potential  = −8 MeV which simulates stable atomic nuclei, the enhancement is of the order of 30% for the model with Gogny force and the box potential and almost 100% for the Paris interaction and Saxon–Woods potential. In both cases, a -dependence of the surface e6ect was found, the enhancement coe:cient increasing as the absolute value of the chemical potential || decreases. The latter e6ect is more pronounced for the second case reaching 30% with diminishing || from 8 to 4 MeV. For the 1rst model, the corresponding -e6ect is three times smaller. This review is organized as follows. In Section 2 a short discussion of the present status of the theory of pairing in nuclear matter within the mean-1eld approach with realistic forces is made. In Section 3 the gap equation for a nuclear system with one-dimensional (1D) inhomogeneity is formulated in terms of the e6ective pairing interaction for the case of separable NN-force. In Section 4 the equation for the e6ective pairing interaction is renormalized in terms of the free o6-shell T -matrix. In Section 5 the calculation and the results for the e6ective pairing interaction are presented and analyzed. A new version of the local approximation, the LPA, is discussed either. In Section 6 the e6ective pairing interaction for a slab of nuclear matter is considered and a simple model for Vpe6 is suggested. In Section 7 the gap equation for semi-in1nite nuclear matter is solved with the help of KKC method. In Section 8 the analogous solution for a slab of nuclear matter is presented. In Section 9 the conclusions are drown. Some technical details are presented in the appendices: Appendix A contains the explicit form of the form-factors of the separable Paris force of [22] in the mixed coordinate-momentum representation; a convenient method for evaluating the free o6-shell T -matrix in the coordinate representation is given in Appendix B.

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Fig. 1. Illustration of the notation for the momenta of interacting particles.

2. Pairing in nuclear matter In many-body theory, the general form of the gap equation is as follows [6,33]:  = VGGs  ;

(2.1)

where V is the interaction block irreducible in the particle–particle channel, Gs is the normal Green’s function in the superLuid system, and G is the Green’s function in which pairing e6ects are disregarded. Within the Brueckner theory, the block V coincides with the free NN-potential because all the ladder diagrams, taken into account in this approach, are summed up in the gap equation itself. For simplicity, we retain the same notation for the NN-potential. A scheme for solving the Brueckner problem for in1nite nuclear matter was developed in [15] for a separable representation of the NN-potential V. Let us recall this procedure. If the density of nuclear matter is close to the normal nuclear density 0 , the inequality 
F holds true and the weak coupling scheme for treatment of pairing is applicable. In this case, the gap equation (2.1) can be reduced to that of the BCS theory. The notation for the momenta of interacting particles for non-homogeneous systems is illustrated in Fig. 1. It corresponds to the case of a non-homogeneous system (P = P ) which is the main subject of our consideration. We take into account the fact that the pairing in nuclear matter occurs with the zero total momentum of the Cooper pairs. Then, for the case of zero temperature T = 0, the explicit form of Eq. (2.1) reads  (k  )  (k) = − V(k; k ) d ; (2.2) 2Ek
 where d = d 3 k=(2)3 , Ek = 2k + 2 (k), k = k − F , and k = k 2 =(2m∗ ), m∗ being the e6ective mass. A characteristic feature of the pairing in nuclear matter is the relevance of the high momentum components of the NN interaction. These components are essential for handling the strong repulsive core and to realize the correct delicate balance between repulsion and attraction in the pairing channel. To establish a connection with the usual treatment of pairing, it is reasonable, as advocated by Anderson and Morel [36], to split the momentum space into two domains separated by a cuto6 kc ¿ kF . Correspondingly, the new gap equation takes the form  (k  )  Vpe6 (k; k ) d ; (2.3) (k) = − 2Ek
k
¡kc

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where the free interaction is replaced by the e6ective pairing interaction Vpe6 acting only at momenta close to the Fermi surface. The latter is given by  1 V(k; k1 ) Vpe6 (k1 ; k ) d1 : (2.4) Vpe6 (k; k ) = V(k; k ) − 2E k1 k1 ¿kc Eqs. (2.3) and (2.4) are completely equivalent to Eq. (2.2). However, the separation of the momentum space on small momenta and large ones permits to fasten the convergence of the iteration procedure for the gap equation as far as the inequality 2k1 2 (k1 ) is valid in the denominator of (2.4). Therefore the gap  can be neglected (or taken into account approximately) in this equation. As it was discussed in the Introduction, the analogous splitting of the two-particle Hilbert space into the model subspace and the complementary one, together with the corresponding e6ective interaction, is especially useful for 1nite systems. Another method to take care of the high momentum components and to calculate the e6ective pairing interaction close to the Fermi surface, is the renormalization group method, which has been pursued in [37]. Eqs. (2.3), (2.4) and analogous equations for T = 0 were solved in [15] with the interaction V taken in the separable (3 × 3) form of the Paris potential [24] proposed in [22]:  2 V(k; k ) = ij gi (k 2 )gj (k  ) : (2.5) ij

The form factors gi (k 2 ) and coe:cients ij can be found in Appendix A. This interaction reproduces the NN-scattering data up to an energy of 300 MeV with high accuracy. In principle, any non-singular interaction can be represented in a separable form with su:cient accuracy, provided the rank is high enough. Substituting this expression into Eq. (2.4), we obtain  2 Vpe6 (k; k ) = ij gi (k 2 )gj (k  ) ; (2.6) ij

where the coe:cients ij obey the following set of equations:  ij = ij + il Blm mj ; lm

 Blm =

(2.7)

k¿kc

gl (k 2 )

1 gm (k 2 ) d : 2Ek

Substitution of Eq. (2.6) into the gap equation (2.3) yields the gap function in the form  i gi (k 2 ) ; (k) =

(2.8)

(2.9)

i

where the coe:cients i obey the set of equations  i = Kij ((k))j : j

(2.10)

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Fig. 2. 1 S0 neutron pairing gap at the Fermi surface (kF ) and the corresponding critical temperature Tc in symmetrical nuclear matter versus the Fermi momentum kF .

Here the matrix kernel Kij is de1ned in terms of the e6ective interaction:  0 Kij ((k)) = il Blj ((k)) ;

(2.11)

l

0 ((k)) Blm

 =

k¡kc

gl (k 2 )

1 gm (k 2 ) d : 2Ek

(2.12)

The set of Eqs. (2.7)–(2.12) can be solved numerically. Such a solution was found in [15] for various values of the density of nuclear matter. The results for the pairing functions and strength, obtained with the separable form of Eq. (2.5), turned out to be very close to those obtained with the original Paris potential. The gap equation (2.2), both for symmetrical and pure neutron matter, has been extensively examined with di6erent bare NN-interactions. Typical results for the 1 S0 neutron pairing gap at the Fermi surface and the critical temperature Tc in symmetrical nuclear matter are reported in Fig. 2 [14]. These results were obtained with the separable form of the Paris potential, but they are almost identical to those for the original Paris potential. The pairing gap, as a function of density, has a maximum around kF = 0:8 fm−1 , with a value at the maximum as large as 3 MeV. At higher density the value of the gap decreases rapidly, and it turns out that at saturation density its value is quite small, about 0.2–0:3 MeV. In a naive local density scheme, this would imply that in 1nite nuclei the main contribution to pairing is concentrated at the surface, where the density decreases and goes through the value at which the pairing gap has its maximum in nuclear matter. Of course, this is a too simpli1ed description, and a more accurate and microscopic calculation must be performed to assess the possible surface nature of the pairing gap in 1nite nuclei. This is one of the main issue

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in the microscopic theory of pairing in nuclei, and it will be discussed extensively in the sequel of the paper. Beyond the framework of the Brueckner theory, the e6ective pairing interaction is also determined by the e6ects of the medium, since the irreducible two-body interaction in Eq. (2.1) is a complex many-body quantity, which can be identi1ed with the bare NN interaction only at very small density. In particular, the screening of the interaction due to the exchange of particle–hole excitations has been considered by many authors [37–39,16]. Furthermore, the reduction of the quasiparticle strength due to dispersive self-energy e6ects has been also considered [40]. In nuclear matter both these e6ects seem to produce a reduction by a large factor of the values of the pairing gap. In 1nite nuclei, evidence has been presented [26] that the exchange by collective surface excitations can, on the contrary, produce an enhancement of the pairing gap. Both dispersive and screening e6ects will not be considered in this paper, since our main goal is to present results of a consistent consideration of pairing in non-uniform nuclear systems within the simplest version of the Brueckner theory, but without use of the LDA or similar local approximations. In the general case of the non-separable form of the NN-interaction, the solution of the gap equation turns out to be rather complicated even for in1nite nuclear matter because of the strong non-locality of the NN-force. The main computational problem originates from the strongly non-linear form of the integral gap equation with a singular behavior of the kernel at (kF ) → 0. As the result, in a e.g. iteration scheme, a lot of iterations is necessary to obtain the solution. In Ref. [29] a method was devised which simpli1es the solution of the gap equation signi1cantly. The main idea of the KKC method is that the non-linear form is important only for the magnitude of the gap and not for its momentum dependence. The latter is determined by integrals over a wide momentum range which do not practically depend on , provided the parameter (kF )=F is small enough. The gap (k) was identically represented in [29] as a product (k) = F #(k)

(2.13)

of the constant F = (kF ) and the gap-shape function #(k) normalized to #(kF ) = 1. Then the initial non-linear gap equation was split into two equations. The 1rst one, for the gap-shape function #(k), is a linear integral equation which does not practically depend on the value of F and can be readily solved. If the function #(k) is known, the equation for the gap amplitude F is just an algebraic non-linear equation which can be solved straightforwardly by means of standard methods. The KKC method is valid also for the non-zero temperature [29,30]: (k; T ) = F (T )#(k; T ) :

(2.14)

It is remarkable that the gap-shape function turns out to be almost T -independent: #(k; T )  #(k). Below we shall apply the KKC method for T = 0 to non-uniform nuclear systems. Here we demonstrate a good convergence of this method in in1nite matter for the case of the separable NN-interaction. We use a simpli1ed version of the Brueckner theory with the e6ective nucleon mass m∗ equal to the bare one. Therefore some results are di6erent a little from those of [15]. Let us consider 1rst the case of T = 0. The explicit KKC procedure, in notation of [30], is as follows. First, the potential V is identically represented as a sum V(k; k ) = VF $(k)$(k  ) + W (k; k ) ;

(2.15)

M. Baldo et al. / Physics Reports 391 (2004) 261 – 310

271

where VF = V(kF ; kF )

(2.16)

$(k) = V(k; kF )=VF :

(2.17)

and In accordance with de1nition (2.15), the identity W (kF ; k  ) ≡ W (k; kF ) ≡ 0 is valid. Representation of (k) in the form of Eq. (2.13) results in a set of two coupled equations [29,30] for the value of the gap and the gap-shape function:  #(k) d ; (2.18) 1 = −VF $(k)  2 2 k + 2F #2 (k)  #(k  ) #(k) = $(k) − W (k; k )  2 d : (2.19) 2 k
+ 2F #2 (k  ) Let us now substitute potential (2.5) into the above relations. We 1nd  $(k) = $i gi (k 2 ) ;

(2.20)

i

where $i =



ij gj (kF2 )=VF

(2.21)

j

and VF =



ij gi (kF2 )gj (kF2 ) :

(2.22)

ij

Then we get W (k; k ) =



2

ij gi (k 2 )gj (k  ) ;

(2.23)

ij

where ij = ij − VF $i $j : The gap-shape function can be found as a sum  #(k) = #i gi (k 2 ) :

(2.24) (2.25)

i

Then Eqs. (2.18) and (2.19) are reduced to the following simple equations:  1 = −VF $i bij #j ;

(2.26)

ij

# i = $i −

 jl

ij bjl #l ;

(2.27)

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where

 bij =

and

gi (k 2 )gj (k 2 ) d 2Ek

  2  Ek = 2k + 2F #l gl (k 2 ) :

(2.28)

(2.29)

l

The advantage of the KKC equations, in comparison with the initial BCS form of the gap equation, consists of the regular behavior of the integral equation (2.19) at F → 0 due to vanishing at the Fermi surface of the “residual interaction” W . For the separable force considered above, it

2 corresponds to the following identity: i ij gi (kF ) = 0. As a result, the singular part of the integral of Eq. (2.28) in the sum of Eq. (2.27) vanishes. The iteration method for integral equations with non-singular kernels works perfectly well. As to the singular equation (2.18) (or Eq. (2.26)), it is, in fact, an algebraic equation which can be solved with standard methods (e.g., with Newton’s method). At the zero step of the KKC method, one can put F = 0 in the denominator of Eq. (2.19) (or Eq. (2.28)). Then Eq. (2.27) becomes just a set of liner equations for #i0 with known coe:cients as far as, within this approximation, the integrals  gi (k 2 )gj (k 2 ) (0) bij = d (2.30) 2k do not contain any unknown quantities. As it was shown in [29,30], already the gap-shape function of zero approximation #0 (k) describes the momentum dependence of the gap (k) su:ciently well. For the separable force, #0 (k) is found by substitution of the coe:cients #i0 into Eq. (2.25). Then the coe:cients #i0 are substituted into denominator (2.29) of Eq. (2.28) yielding the integrals b(1) ij (F ) (1) 0 which contain F as a parameter. Substitution of bij (F ) and #i to (2.26) results into the equation for F which solution gives the gap amplitude 0F in the zero approximation of the KKC method. 0 At the next step of the KKC method, the integrals b(1) ij (F ) are substituted in Eqs. (2.26) and (2.27) resulting in the gap-shape function of next approximation #(1) (k). The iteration procedure can be repeated until the necessary accuracy is obtained. Very fast convergence of the KKC method is demonstrated in Table 1 where the solutions of the KKC equations are given as a function of the iteration number N , for several values of the chemical potential . One sees that already the zero KKC iteration leads to a percent accuracy. The second or third iteration guarantees the constancy of the 1rst 8 1gures. It should be mentioned that, contrary to the standard method of solving the BCS equation, the smallness of F (at small ||), makes the convergence of the iteration procedure faster. It makes the KKC method especially e6ective for T = 0, as far as F is tending to zero in the vicinity of T = Tc . In the case of T = 0, all the above equations remain valid with only one exception: the temperature factor appears in Eq. (2.28),    gi (k 2 )gj (k 2 ) Ek bij (T ) = d : (2.31) th 2T 2Ek

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273

Table 1 Convergence of the KKC method at T = 0; the gap F and coe:cients #i in Eq. (2.25), in dependence on the iteration number N N

F (MeV)

#1

#2

#3

 = −16 MeV 0 1 2 3 4

0.98508579 0.97535836 0.97551858 0.97551858 0.97551858

0.51496182 0.51247844 0.51252118 0.51252047 0.51252047

1.7332633 1.7354232 1.7353864 1.7353870 1.7353870

−8:3303282 × 10−4 −9:0958849 × 10−4 −9:0834607 × 10−4 −9:0836649 × 10−4 −9:0836649 × 10−4

 = −8 MeV 0 1 2 3

0.19878907 0.19863648 0.19863648 0.19863648

0.30019636 0.30006468 0.30006487 0.30006487

2.6898012 2.6899770 2.6899768 2.6899768

−1:8056271 × 10−2 −1:8068021 × 10−2 −1:8068005 × 10−2 −1:8068005 × 10−2

 = −2 MeV 0 1 2

0.011648583 0.011648583 0.011648583

0.083704097 0.083703371 0.083703371

4.1089851 4.1089871 4.1089871

−5:6122312 × 10−2 −5:6122463 × 10−2 −5:6122463 × 10−2

Table 2 shows the fast convergence of the KKC method for non-zero temperature for  = −16 MeV. To make the table less cumbersome, we show the results of the third iteration only once (for T = 0), as far as they coincide practically with those of the second iteration. It is seen that the coe:cients #i of the gap-shape function, indeed, are almost T -independent. The last temperature presented in the table is very close to the critical one, Tc = 0:5562. It is worth to stress that the usual iteration procedure needs several thousands of iterations for 1nding (T ) in vicinity of Tc . Below we shall use an extension of the KKC method for non-uniform systems to solve the gap equation for semi-in1nite nuclear matter and for nuclear slab systems as well. 3. Gap equation for 1D-inhomogeneous nuclear matter with separable NN-force in terms of the e"ective pairing interaction The explicit form of the gap equation (2.1) in the coordinate representation, which is convenient for non-uniform systems, is as follows:  (r1 ; r2 ; ) = V(r1 ; r2 ; r3 ; r4 ; E; ;  ) 

   E E d  s  dr3 dr4 dr5 dr6 : × G r3 ; r5 ; +  G r4 ; r6 ; −  (r5 ; r6 ;  ) 2 2 2i

(3.1)

The total two-particle energy E in the right side of this equation is equal to E = 2 ;

(3.2)

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Table 2 The same as in Table 1, but for T = 0 ( = −16 MeV) T (MeV)

N

F (MeV)

#1

#2

#3

0

0 1 2 3

0.985085 0.975358 0.975518 0.975518

0.514961 0.512478 0.512521 0.512520

1.73326 1.73542 1.73538 1.73538

−8:33032 × 10−4 −9:09588 × 10−4 −9:08346 × 10−4 −9:08366 × 10−4

0.200

0 1 2

0.982225 0.966927 0.967046

0.513799 0.512528 0.512559

1.73418 1.73537 1.73535

−8:48253 × 10−4 −9:07982 × 10−4 −9:07095 × 10−4

0.400

0 1 2

0.923231 0.788421 0.788558

0.513805 0.513239 0.513261

1.73418 1.73475 1.73473

−8:48160 × 10−4 −8:84127 × 10−4 −8:83438 × 10−4

0.540

0 1 2

0.517052 0.284619 0.284686

0.514033 0.514439 0.514443

1.73399 1.73369 1.73368

−8:44951 × 10−4 −8:43175 × 10−4 −8:43071 × 10−4

0.550

0 1 2

0.178203 0.177372 0.177395

0.514540 0.514549 0.514550

1.73359 1.73359 1.73359

−8:38096 × 10−4 −8:39412 × 10−4 −8:39389 × 10−4

0.554

0 1 2

0.147605 0.106174 0.106202

0.514595 0.514593 0.514593

1.73355 1.73355 1.73355

−8:37362 × 10−4 −8:37903 × 10−4 −8:37888 × 10−4

0.556

0 1 2

3:24171 × 10−2 3:26736 × 10−2 3:26660 × 10−2

0.514611 0.514615 0.514615

1.73353 1.73353 1.73353

−8:37134 × 10−4 −8:37129 × 10−4 −8:37130 × 10−4

( Tc )

where  is the chemical potential of the system under consideration. In Section 2, dealing with nuclear matter, we set  = 0  −16 MeV (the leading term in the Weizsaecker mass formula), whereas we have   −8 MeV for 1nite nuclei. For a while, we treat the quantity E in Eq. (3.1) as a parameter. For brevity we omitted the spin variables in Eq. (3.1). Just as in Section 2, we consider the singlet pairing (the channel with quantum numbers I = 1 and S = 0). Hereafter, we limit ourselves with the simplest version of the Brueckner approach in which the quantity V coincides with the free NN-potential which is independent of energy. In this case, the gap  is also independent of energy; hence, the product of the two Green’s functions in Eq. (3.1) can be integrated with respect to  : s

A (r1 ; r2 ; r3 ; r4 ; E) =



    d E E  s  G r1 ; r2 ; +  G r3 ; r4 ; −  : 2i 2 2

(3.3)

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275

Now the gap equation (3.1) can be written in a compact form as  = VAs  :

(3.4)

Within the same approximation of the Brueckner theory, we substitute for G the pole term only, thereby disregarding the regular part of the Green function. Let us now expand Green’s functions G and Gs in Eq. (3.3) in the basis of the functions $ (r) which diagonalize G (corresponding eigenenergies are  ). These functions also approximately diagonalize the function G s . Small non-diagonal terms of (G s )
can arise only in the equation for , rather than in the equation for the e6ective interaction considered below. For this reason and for the sake of simplicity, we disregard these terms and set (Gs )
= (Gs ) 
. As a result, we obtain  As
(E)$ (r1 )$∗ (r3 )$
(r2 )$∗
(r4 ) ; (3.5) As (r1 ; r2 ; r3 ; r4 ; E) = 


where As
(E)

 =

d G 2i



   E E s +  G
− : 2 2

(3.6)

It is well known that the sum over ( ) in Eq. (3.5) diverges. This divergence in Eq. (3.1) is
removed by matrix elements V 1 2 of the interaction V. However, the realistic NN-interaction is a short-range one, therefore the convergence is extremely slow [20]. On the other hand, for states distant from the Fermi level, | − |, the di6erence between two Green’s functions, Gs and G , in Eq. (3.6) is negligible. But the gap function does not enter to the de1nition of the latter one that makes the calculation procedure much simpler. For this reason, it is convenient to use some renormalization operation in Eq. (3.1) with breaking down the sum over ( ) in Eq. (3.5) into two parts: As = As0 + A :

(3.7)

The term As0 involves only states close to the Fermi level (the “model space”), while all the rest is included into the term A . For any state entering A , the inequality |1 + 2 − 2| should be satis1ed. As a result, the di6erence between Gs and G in A can be neglected (for this reason, we omitted the upperscript “s” on A ). We then obtain 1 − n  − n
; (3.8) A
(E) = E −   − 
where n and n
are the quasiparticle occupation numbers equal to 1 for occupied states and equal to 0 for unoccupied states. Eq. (3.4) can now be written as  = Vpe6 As0  ;

(3.9)

where the e6ective interaction Vpe6

= V + VA



Vpe6

:

Vpe6

obeys the following equation: (3.10)

In other words, the complete Hilbert space S is split into two domains. The 1rst one is the model subspace S0 , in which the gap equation is written down in terms of the e6ective interaction Vpe6 . The second one is the complementary subspace S  , in which the equation for Vpe6 is obtained in

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terms of the free NN-interaction V. In this subspace, the pairing e6ects are not signi1cant, therefore the equation for Vpe6 (3.10) has the form of the Bethe–Goldstone equation. In [20] the last equation was analyzed and a direct method was developed for solving this equation for semi-in1nite nuclear matter without use of the LDA or any other local approximation. We de1ne the model space S0 (E0 ) including all the single particle states with the energies  ¡ E0 . In this case, the part of the sum of Eq. (3.5) corresponding to S0 contains the two-hole states which are allowed by the Pauli projection operator of Eq. (3.8) and the two-particle states with energies ( ¡  ; 
¡ E0 ). In the case of in1nite nuclear matter, see Section 2, this splitting of the Hilbert space corresponds to representation of the momentum integral in Eq. (2.2) as a sum of two ones separated by the momentum kc . In modern self-consistent nuclear calculations, both in the FFS theory and based on the HFB method, all states ( ) of negative energy are usually included in the model space which corresponds to E0 = 0. The same model space was chosen in [20]. In this case, the complementary subspace S  involves the two-particle states with positive energies  ; 
and such two-particle states when one energy is positive but the second one is negative. In calculations by Fayans et al. [13] within the energy functional method a wider model space was used with E0 = 40 MeV. In this paper, we deal with semi-in1nite or slab-shape nuclear matter imbedded in a potential well U (x), where the x-direction of the 1D-inhomogeneity. All general formulae are the same for the both systems as well as for any 1D-inhomogeneous system. In this case, it is natural to use the mixed representation, i.e., the coordinate representation for the x-direction and the momentum representation for the (y; z)-plane (or s-plane). In this representation, the quasiparticle wave functions can be written as $ (r) = eip⊥ s yn (x) ;

(3.11)

where p⊥ is the two-dimensional momentum vector in the s-plane. For these vectors, we use notation similar to that introduced in Section 2 for the three-dimensional momenta in in1nite matter, for example, p1⊥ = P⊥ =2 + k⊥ , p2⊥ = P⊥ =2 − k⊥ , etc. In the s-plane, the momentum is conserved and  =0 for the pairing problem. The functions y (x) in Eq. (3.11) stand for solutions of we have P⊥ =P⊥ n the one-dimensional SchrSodinger equation in the potential U (x). For the case of semi-in1nite matter, we consider the potential U (x) vanishing at large positive x and being a constant at x → −∞. For the slab geometry, U (x) vanishes at x → ±∞. Let us introduce the CM and relative coordinates in the x-direction: X = (x1 + x2 )=2, x = x1 − x2 , X  =(x3 +x4 )=2, x =x3 −x4 . For the free NN-potential, the separable form [22,23] of the Paris potential [24] was used, as it was discussed in Section 2. After substitution of Eq. (2.5) into Eq. (3.10) and making the inverse Fourier transformation in the x-direction, we obtain  2 2 2 2  Vpe6 (k⊥ ; k⊥ ij (X; X  ; E)gi (k⊥ ; x)gj (k⊥ (3.12)
; x1 ; x2 ; x3 ; x4 ; E) =
; x ) ; ij

where the form factors in the mixed representation have the form  ∞ d kx 2 2 : (3.13) gl (k⊥ ; x) = gl (k⊥ + kx2 )e−ikx x 2 −∞ This form of the e6ective interaction is a natural generalization of expansion (2.6) for in1nite matter. The coe:cients ij satisfy the set of integral equations   ij (X; X  ; E) = ij (X − X  ) + il dX1 Blm (X; X1 ; E)mj (X1 ; X  ; E) ; (3.14) lm

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277

where Blm are given by the convolution integrals of the type of Eq. (2.8) which explicit form is as follows:   d 2 k⊥ (1 − nk ;n − nk ;n
) ⊥ ⊥  2 2 ; X )gnm
∗n (k⊥ ; X ) ; (3.15) Blm (X; X ; E) = gl
(k⊥ 2 E −  − 
− k 2 =m nn (2) n n
⊥ nn  l 2 2 gn; n
(k⊥ ; X ) = d x gl (k⊥ ; x)yn (X + x=2)yn
(X − x=2) : (3.16) The prime in the sum of Eq. (3.15) shows that the summation is carried out over (nn ) which are not included in the model space of Eq. (3.9) for . In terms of the short notation  = (n; k⊥ ), the summation and integration in Eq. (3.15) is restricted with the condition ( ) ∈ S  . For the semi-in1nite case, the symbolic summation over (nn ) in Eq. (3.15) denotes the integration over dp dp =(2)2 . For the slab case, it means the summation over discrete spectrum for negative energies and the same integration, for positive ones. All the integrals and sums in Eq. (3.15) for the propagators Blm (X; X  ) can be found numerically. Then set (3.14) of the integral equations for the coe:cients ij (X; X  ) of the separable expansion (3.12) of the e6ective pairing interaction Vpe6 can be solved with standard methods. 4. Equation for the e"ective pairing interaction in terms of the free o"-shell T-matrix In solving the set of integral equations (3.14), the main computational problem is the calculation of kernels (3.15). In principle, the integrals of Eqs. (3.15) and (3.16) can be evaluated numerically, provided the wave functions yn (x) are known. The form factors (3.13) in the mixed representation for the speci1c form of separable (3×3) version [22,23] of the Paris potential can be found analytically. They are given in Appendix A. The functions gl (k 2 ) fall at large k 2 very slowly, which results in rather poor convergence of the integrals in Eq. (3.15). To make the convergence faster, it is useful to transform Eq. (3.10) with the help of the analogous equation for the o6-shell free T -matrix: T = V + VA0 T ;

(4.1)

where A0 is the propagator of two free particles. It should be written down in the mixed coordinatemomentum representation via the same procedure as Eq. (3.10) with the change of functions yn (x) by plane waves. Obviously, the coe:cients of the expansion  2 2 2 2  ; k⊥ Tij (X; X  ; E)gi (k⊥ ; x)gj (k⊥ (4.2) T (k⊥
; x1 ; x2 ; x3 ; x4 ; E) =
; x ) ij

depend only on the di6erence of (X − X  ):     0 Tij (X − X ; E) = ij (X − X ) + il dX1 Blm (X − X1 ; E)Tmj (X1 − X  ; E) ;

(4.3)

lm

where B0 is determined in terms of the integrals of the form of Eqs. (3.15) and (3.16) with substitution of exp(ipx) as yn (x).

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The standard renormalization of Eq. (3.10) results in the following equation: Vpe6 = T + T (A − A0 )Vpe6 :

(4.4)

The kernel of this equation converges much faster than that of the original equation. Of course, the problem of bad convergence of the integral at large momenta does not disappear. It passes to Eq. (4.3). However, in this case, this di:culty can be overcome much easier. Indeed, we deal now with the one-dimensional (in the X -space) vector Tij (X − X  ) instead of the two-dimensional matrix ij (X; X  ). It is convenient 1rst to 1nd the T -matrix in the momentum representation by solution of the following set of equations:  0 Tij (Px ; E) = ij + il Blm (Px ; E)Tmj (Px ; E) ; (4.5) lm

where 0 Blm (Px ; E) =



gl (k 2 )gm (k 2 ) d3 k : (2)3 E − Px2 =4m − k 2 =m

Then Tij (X ) can be found from Tij (Px ) with the help of the inverse Fourier transformation:  ∞ dPx Tij (Px ; E) exp(−iPx t) : Tij (t; E) = −∞ 2

(4.6)

(4.7)

The form factors gi in Eq. (4.6) are rational functions on k 2 [22,23], namely, combinations of the Yukawa function and their derivatives with di6erent masses 2in (n = 1; : : : ; 4) (see Appendix A). This integral can be evaluated analytically, but such calculation is very cumbersome because of a huge number ( 70) of particular terms appearing in the integrand. We prefer, following to [19], to integrate it numerically, with the cut-o6 momentum kc = 60 fm−1 which guarantees an accuracy better than 1%. The Fourier integral Eq. (4.7), after separating the constant term ij , was calculated in [20] by direct integration along the real Px -axis. This method works well at small t, but for t ¿ (2–3) fm the integrand contains rapidly oscillating factors multiplied by a slowly falling function (Tij (Px2 ) − ij )  1=Px2 , which makes the convergence very poor. To obtain a reasonable accuracy at t = 4–5 fm, the large value of the cut-o6 momentum Pxc = 3000 fm−1 was taken in [20] with very small integration step. More simple and elegant method to integrate Eq. (4.7) was suggested in [28]. It is based on performing the integration in Eq. (4.7) in the complex plane of Px . This method is explained in detail in Appendix B. The set of renormalized equations for components of the e6ective interaction reads:    dX1 dX2 Til (X − X1 ; E) ij (X; X ; E) = Tij (X − X ; E) + lm 0 × [Blm (X1 ; X2 ; E) − Blm (X1 ; X2 ; E)]mj (X2 ; X  ; E) :

(4.8)

The unrenormalized equation (3.14) contains one additional calculation problem connected with the singular (˙ (X − X  )) form of the non-uniform term. One can easily see that the renormalized equation (4.8) contains the same problem. Indeed, the T -matrix involves the same -form term as the initial NN-potential V. It can be easily seen from the analysis of Eq. (4.5) in the momentum space where the (X )-function corresponds to a constant. Indeed, the integral term of Eq. (4.5) tends

M. Baldo et al. / Physics Reports 391 (2004) 261 – 310

279

to zero at Px → ∞, due to the denominator of the propagator. Hence the T -matrix can be written as the sum of Tij (X ; E) = ij (X ) + Tmj (X ; E) ;

(4.9)

where the terms Tmj (X ; E) are non-singular. To avoid the singularity of the integral equation (4.8), let us write ij as a sum of ij = Tij + #ij :

(4.10)

Substituting Eq. (4.10) into Eq. (4.8) we obtain   0  #ij (X; X ; E) = #ij (X; X ; E) + dX1 dX2 Til (X − X1 ; E) lm 0 × [Blm (X1 ; X2 ; E) − Blm (X1 ; X2 ; E)]#mj (X2 ; X  ; E) ;

where #ij0 (X; X  ; E) =



(4.11)

dX1 dX2 Til (X − X1 ; E)

lm 0 × [Blm (X1 ; X2 ; E) − Blm (X1 ; X2 ; E)]Tmj (X2 − X  ; E) :

(4.12)

Eq. (4.11) is rather convenient for a numerical solution. It was solved in [20] directly, without any additional approximations. 5. E"ective pairing interaction in semi-in(nite nuclear matter. The local potential approximation Let us consider semi-in1nite nuclear matter which is the simplest system with well de1ned surface. Following to [20], we calculate the e6ective pairing interaction for the model space S0 (E0 ) with E0 = 0. Such a choice of S0 is typical for practical nuclear calculations. Up to now, the two-particle energy E = 2 in all the above equations was considered as a free parameter. In the explicit calculations of this section we 1rst 1x the value of  = −16 MeV. This is the “self-consistent” chemical potential of semi-in1nite nuclear matter which, obviously, coincides with that of in1nite matter. Then we repeat all the calculations for  = −8 MeV which is a typical value for stable nuclei. As far as our main goal is to clarify e6ects of the semi-in1nite con1guration on appearance of the surface structure of the e6ective pairing interaction, we do not consider the problem of completely self-consistent calculation which involves evaluation of the mass operator. Therefore we approach the mass operator with an energy-independent mean 1eld U (x) of a realistic shape. In particular, we take U (x) as the Saxon–Woods potential U (x) = −

U0 1 + exp(x=d)

(5.1)

with typical for 1nite nuclei values of the potential well depth U0 =50 MeV and di6useness parameter of d = 0:65 fm.

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Let us 1rst describe the problem of calculating kernels of the set of integral equations (3.14) for the e6ective pairing interaction. This calculation is the most complicated part of the problem under consideration and takes the overwhelming portion of the total computational time. It remains essentially the same for the di6erential kernels of the renormalized equations (4.11) and (4.12) which are really solved. The integrals Blm de1ned by Eq. (3.15) are the most important ingredients of these kernels. Eq. (3.15) de1nes the folding integrals of the two-particle propagator A with two 2 2 matrix elements gn;l n
(k⊥ ; X ), gn;m n
(k⊥ ; X  ) of the form factors gl , gm . For the sake of brevity, we name these integrals as propagators. Let us write down the integral of (3.15) in the explicit form. By integrating over the azimuthal angle of the vector k⊥ , going from summation over n; n to integration over dp dp and taking into account an evident symmetry between p; p we 1nd Blm (X; X  ; E) = 2

 0

pmax

dp 2



pmax

p

dp 2

 0

max k⊥

l 2 m∗ 2  gpp
(k ; X )gpp
(k ; X ) k⊥ d k ⊥ ⊥ ⊥ : 2 2 E − p2 =2m − (p )2 =2m − k⊥ =m

(5.2)

2 2 ¡ pmax , Here the integration limits were found using the following relations: p ¡ p and (p )2 + k⊥  max 2  2 i.e., k⊥ = pmax − (p ) . 0 A few words about calculating the free propagators Blm (X − X  ) in the di6erence kernels of Eqs. (4.11) and (4.12). In principle, they can be easily found by the inverse Fourier transformation 0 of the corresponding propagators Blm (Px ), as it is described in Appendix B. We do in such a way when calculating the free T -matrix which is also an ingredient of these kernels. However, dealing with the di6erence of the propagators B − B0 , we prefer to use for B0 the same recipe as for B, but with vanishing potential well, i.e., with U (x) ≡ 0. In this case, numerical errors are minimized. In [20] the very slow convergence of momentum integrals of Eq. (5.2) was demonstrated and the faster one for the di6erence of propagators B − B0 . It turned out that the di6erence converges within the accuracy of 1% at the cuto6 momentum pmax = 60 fm−1 . On the other hand, the same accuracy is achieved for the propagator B itself at pmax = 400–500 fm−1 only. This explains why the renormalized equation (4.11) for the e6ective pairing interaction is more suitable for solution than the initial one, Eq. (3.14). The propagators Blm (X; X  ) were calculated in [20] at a 1xed grid X4 = −L + Hx (4 − 1), X2 = −L + Hx (2 − 1), 4; 2 = 1; : : : ; 2N + 1, L = NHx , with L = 8 fm and Hx = 0:1 fm. The set of functions {yn (xi )} was calculated at the wider interval of [ − 12; 12] fm for a correct calculation of integrals (3.16) at L = 8 fm. All numerical details can be found in [20]. Before discussing the calculation results, one remark should be made. As it is explained in Appendix A, we changed the original normalization [22,23] of expansion (2.5) in such a way that the identity gi (0) = 1 holds true. Therefore the absolute values of the ij -coe:cients give direct information on the strength of the corresponding terms of the force. Their values are given in Table 3 of Appendix A. As it is seen, the strengths of all the components containing only the indices i = 1; 2 are much stronger than those with the index i=3. Therefore the “small” components are not essential for a qualitative analysis, and we, as a rule, shall limit ourselves with “large” components in the graphical presentation of the results. Of course, in the calculations all the terms ik are considered on the equal foot.

M. Baldo et al. / Physics Reports 391 (2004) 261 – 310

281

Fig. 3. The propagators Blm (X; X  ) for lm = 11 (solid line), lm = 12 (dashed line) and lm = 22 (dotted line) at 1xed X  = −4 fm. The sign of all the propagators is changed for the graphical convenience.

The propagators Blm (X; X  ) are very sharp functions of the di6erence t = X − X  at t → 0. It is shown in Fig. 3 for the large components (lm), X  = −4 fm, the peaks being cut at small t. This property of B originates from the discussed above slow convergence of the integrals in Eq. (3.15) in the momentum space due to short range character of the free NN-force [24]. 1 The same is true 0 for the free propagators Blm (X − X  ). These sharp peaks are partially suppressed in the di6erences 0 of Blm − Blm , therefore such di6erences are more suitable for graphical representation than the propagators themselves. These pro1le functions are depicted in Fig. 4 for the same values of (lm) and three values of X  . These functions have sharp maxima at X = X  for X  = −4 fm (inside nuclear matter) and for X  = 0 (at the border) and fall rapidly with increase of |X − X  |. The maximum 0 at X = 4 fm is very weak because the values of Blm (X; X  ) and Blm (X; X  ) almost coincide in this region. However, any of these functions taken separately has a sharp maximum at X  = 4 fm, like that in the internal region. In [20] a new version of the local approximation was suggested which was named as the LPA. LPA Within the LPA, for a given value of X , the set of propagators Blm (X + t=2; X − t=2) are de1ned as nm the set of functions Blm (t) calculated for “nuclear matter” with a constant potential U0 being equal to a 1xed value U0 = U (X ). The physical reason for such approximation is as follows. We divided the complete Hilbert space S = S0 + S  in such a way that the model subspace S0 (E0 ) contains all the two-particle states (;  ) with both single-particle energies  ; 
smaller than E0 . In the complementary subspace, S  (E0 ), one of these energies or both of them are large, max( ; 
) ¿ E0 . 2 The contribution of an individual state (;  ) 3 to the integral of Eq. (5.2) could be important only if the corresponding energy denominator is small. Such contributions produce the long-range terms of the propagator Blm and must be calculated in a direct way. If the value of E0 is large enough, no individual state (;  ) in the complementary subspace is important and only wide intervals of 1 Let us remind that the “propagators” Blm are, in fact, the folding integrals of the two-particle propagator A with two form factors, gl and gm , of the free NN-potential. 2 In fact, the di6erence  −  is small or large. Just such di6erences enter the denominator of Eq. (3.15) or Eq. (5.2) at E = 2. 3 The “individual” state means a small interval of integration over p, p and k⊥ .

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0 Fig. 4. The di6erences Blm (X; X  ) − Blm (X; X  ) for the same set of lm as in Fig. 3 as functions of X at X  = −4 fm  (solid lines), X = 0 (dashed lines) and X  = 4 fm (dotted lines). The sign of the propagators is changed.

the integration over all the momenta contribute to Blm signi1cantly. The corresponding term of the propagator is sharply peaked and is mainly determined by the local properties of the system [20]. Therefore it is natural to use for it some kind of local approximation. For the problem under consideration, it seems to be more natural to use LPA rather than LDA because the propagator Blm in the vicinity of the point X is determined directly by the potential well U (X ) but not by the density, (X ). At the same time, in the surface region there is no simple local relation between (X ) and U (X ). We use now the separation energy E0 = 0. Is this value of E0 su:ciently big for validity of LPA, or not, should be checked. nm The propagators Blm (U0 ) are calculated as follows. As was mentioned above, one can calculate 0 the free propagators Blm by using Eqs. (3.15) and (3.16) with change of n by p2 =2m  and yn (x) by nm exp(ipx). The propagators Blm (U0 ) are de1ned analogously with a substitution of q = p2 − 2mU0  2 for p and q = p − 2mU0 for p . In Fig. 5 the propagators Blm (X; X  ) are compared with those found within LPA for the point X  =0, which is the most “dangerous” from the point of view of validity of any local approximation. It is seen that, for all three values of (lm), the di6erence between the exact propagator and the LPA one is rather small. Indeed, in this comparison, the large scale of the 1gure should be taken into account, with cutting of the maxima, which magni1es all the deviations.

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283

X, fm -6

-4

-2

0

2

4

6

0.2 0.0

-0.4

l,m = 1,1

-0.6

0.0

-0.5

4

Blm(X,X') X 10 , MeV

-1

[X'=-4; 0; 4 fm]

-0.2

l,m = 1,2

-1.0

0.0

-1.0

l,m = 2,2

-2.0 -6

-4

-2

0 X, fm

2

4

6

Fig. 5. The propagators Blm (X; X  ) for the same set of lm as in Fig. 3 (solid lines) together with the corresponding LPA propagators (dashed lines) as functions of X at X  = 0. The sign of the propagators is changed.

In order to characterize the obtained propagators on average and to analyze the di6erence between the exact propagators and the LPA ones quantitatively, we calculated the zero moments  U Blm (X ) = dt Blm (X + t=2; X − t=2) : (5.3) The average values BU lm (X ) of the three “large” components of the propagator are displayed in Fig. 6 together with corresponding LPA predictions. One sees that in the surface region the deviation reaches 10–15%. Of course, such agreement looks not so bad for a local approximation, but it is much worse than the accuracy of the LPA found in [20] for the “simpli1ed” complementary subspace S1 which includes only the two-particle states (;  ) with both positive single-particle energies  ; 
. Fig. 7 demonstrates accuracy of the LPA for this simpli1ed subspace S1 . One can see that in this case the typical deviation is not bigger than 5%. 0 Once the propagator di6erences Blm (X; X  ) − Blm (X; X  ) are calculated, we need the free T -matrix in the coordinate representation for calculating the kernels of the renormalized equations (4.11) and (4.12) for the e6ective interaction Vpe6 . A simple and very accurate method of calculating the components Tij (X − X  ) of the free T -matrix is given in Appendix B. The kernel of Eq. (4.11) is

284

M. Baldo et al. / Physics Reports 391 (2004) 261 – 310 X, fm -0.5

-6

-4

-2

0

2

4

-2

0 X, fm

2

4

-0.6

-0.7

3

Blm(X) X 10 , MeV

-1

X

fm

-3

l,m = 1,1

-1.1 -1.2

l,m = 1,2

-1.3

-2.0 -2.1 -2.2

l,m = 2,2

-2.3 -6

-4

-0.4

3

B11(X) X 10 , MeV

-1

fm

-3

Fig. 6. The averaged values BU lm (X ) for the same set of lm as in Fig. 3 (solid lines) together with the corresponding LPA averaged values (dashed lines).

-0.5 -0.6 -0.7 -4

-2

0 X, fm

2

4

Fig. 7. The averaged value BU 11 (X ) together with the LPA one calculated for the “simpli1ed” complementary subspace.

found by the convolution integral  0 Kim (X1 ; X2 ) = dXTil (X1 − X ; E)[Blm (X; X2 ; E) − Blm (X; X2 ; E)] : l

(5.4)

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285

The non-uniform terms (4.12) are given by analogous integrals with double convolution of the propagator di6erence with the T -matrix. At the 1xed grid {X4 }, after transformation of the integration over X2 in Eq. (4.11) to the step summation, we obtain the following set of linear equations:  #ij (X4 ; X2 ) = #ij0 (X4 ; X2 ) + Hx Kil (X4 ; X )#lj (X ; X2 ) : (5.5) l



It can be solved by standard methods (see [20]). Then the components ij (X; X  ) of the complete e6ective interaction Vpe6 are readily found, in accordance with Eq. (4.10) as the sum of #ij (X; X  ) and Tij (X; X  ) which were found earlier. The set of 6 independent functions ij (X; X  ), which de1ne the e6ective interaction in the form of Eq. (3.12), is the output of the computational procedure for Vpe6 and the input for the gap equation.  For calculating the e6ective interaction LPA lm (X; X ) within the framework of LPA, one should LPA repeat the procedure described above with substituting the obtained earlier propagators Blm (X; X  )  instead of the exact ones, Blm (X; X ). As it was discussed above, the complete e6ective interaction Vpe6 (X; X  ) contains the singular -term V0 (X − X  ). It is more convenient to display the components of the correlation part of the e6ective interaction Vpe6 = Vpe6 − V ;

(5.6)

   Ve6 ij (X; X ) = ij (X; X ) − ij (X − X ) ;

(5.7)

which does not contain this singularity. Fig. 8 shows the exact and LPA pro1le functions Ve6 ij of the correlation part of the e6ective interaction for several values of ij taken at X  = 0. Just as in the case of propagators, deviations of the exact results from the LPA ones are rather small, taking into account the “large-scale e6ect”. To demonstrate a high accuracy of LPA in the simpli1ed complimentary subspace S1 described above, we present Fig. 9 [20] for the components #ij (X; X  = 0) of the Vpe6 with extracted T -matrix. As it is seen, in this case the agreement between the exact result and the LPA one is really perfect. For a qualitative analysis of the CM coordinate dependence of separate components of the e6ective interaction we again calculate the zero moments,  U (5.8) ij (X ) = dt ij (X − t=2; X + t=2) : To have an idea of the overall characteristics of the e6ective interaction it is reasonable to consider the volume integrals over the relative coordinates. The total strength of the free NN-interaction for the case of the separable representation can be de1ned as follows:  2 0 V = dr dr V(r; r ) = V(k 2 = 0; k  = 0) : (5.9) With the normalization of the form factors of Eq. (2.5) to unit (see Appendix A) we get  V0 = ij = −720 MeV fm3 : (5.10) ij

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 Fig. 8. Pro1le functions Ve6 ij (X; X ) (solid lines) together with the LPA ones (dashed lines) of the correlation part of the e6ective interaction for di6erent ij, taken at X  = 0.

Similar quantity can be naturally de1ned for the e6ective interaction. In this case, the total “strength” of the e6ective interaction will be dependent on X; X  :    t t t t p0   p 2 2   : (5.11) Ve6 (X; X ) = dt dt Ve6 k⊥ = 0; k⊥
= 0; X + ; X − ; X + ; X − 2 2 2 2 As it can be easily seen, we get   Vp0 ij (X; X  ) : e6 (X; X ) =

(5.12)

The zero moment of the total “strength” of the e6ective interaction is equal to  U 0e6 (X ) = U ij (X ) : V

(5.13)

ij

ij

Similar quantities (evidently, X -independent) can be also de1ned for the T -matrix. Fig. 10 shows the zero moment of the “strength” of the complete e6ective interaction in comparison with the LPA predictions. We see that in the bulk and in the asymptotic region both quantities are very close to each other, as to the surface region a deviation is of order of ten percent.

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287

Fig. 9. Pro1le functions of the di6erence #ij (X; X  = 0) of the e6ective interaction for the simpli1ed S  and of the free T -matrix. The solid and dotted lines show results of the exact and LPA calculations, respectively. The sign is changed for ij = 12.

U 0e6 (X ) of the complete “strength” of the e6ective interaction. The solid and dashed lines show Fig. 10. Averaged value V results of the exact and LPA calculations, respectively. The dotted line corresponds to the T -matrix and the dashed–dotted line, to the bare interaction.

The “strength” of the e6ective pairing interaction introduced above has a small relevance to the pairing gap itself. Indeed, the gap equation is governed mainly by the e6ective interaction taken nearby the Fermi surface rather than that at k = k  = 0. To present approximately the components of the e6ective interaction relevant to the gap, let us make the next step in the spirit of the LPA and replace the operators k 2 ; k  2 in Eq. (3.12) by the local values of kF2 (X ) = k  2F (X ) = 2m( − U (X )). Then, instead of Eq. (5.13), we obtain the “Fermi averaged” e6ective interaction,  VFe6 (X ) = (5.14) U ij (X )gi (kF2 (X ))gj (kF2 (X )) : ij

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Fig. 11. The Fermi-averaged e6ective pairing interaction VFe6 (X ) (solid lines) together with those within the LPA for  = −16 and −8 MeV.

Such a recipe is not valid in the classically forbidden region where the inequality 2m( − U (X )) ¡ 0 takes place. In this case, we shall use the same Eq. (5.14), but with kF2 (X )=0. In fact, such a simple prescription for kF (X ) is correct only far from the reLection point where  − U (X ) = 0. However, we use this rough approximation everywhere, because our aim in this point is just to present qualitatively general features of the e6ective pairing interaction which really is calculated exactly. An approval of the representation of Vpe6 in the form of Eq. (5.14), including the above recipe for kF (X ) in the classically forbidden region, and the analysis of accuracy of this representation may be found in [20,34]. The Fermi-averaged e6ective pairing interaction VFe6 is drawn in Fig. 11. The strong e6ect of going from k = 0 to kF takes place inside the nuclear matter where the value of VFe6 turns out to be rather small. This should lead to a small value of the gap  inside nuclear matter, in qualitative agreement with [15]. Again, the maximum deviation of the exact calculation from the LPA one does not exceed 10%. Thus, for the model space under consideration which corresponds to E0 = 0, the direct use of LPA for the two-particle propagator in the complete complementary space S  leads to an accuracy in the calculation of the e6ective interaction of the order of 10%. Obviously, it is possible to improve the LPA procedure using the “simpli1ed” subspace S1 for which the disagreement does not exceed several percent. For this aim, one could split the complementary space as S  = S1 + S  and, correspondingly, the propagator in Eq. (3.10) as A = A1 + A . As we have seen, the propagator A1 can be calculated with the help of LPA within a su:cient accuracy. At the same time, in the component A , there are terms with “small denominators” for which accuracy of LPA is less. It is better to calculate this part of the propagator directly, without any local approximation. Such a calculation is much simpler than the direct calculation of the complete propagator A . We do not carry out here this improved LPA calculation of Vpe6 because we have results of direct calculation of this quantity [20]. In fact, there is a more convenient way of improving LPA which was developed in [25] for the problem of solving the Bethe–Goldstone equation for the G-matrix in 1D-inhomogeneous nuclear systems.

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289

In this method, a procedure of choosing the optimal value of the separating energy E0 was suggested which results in achievement of necessary accuracy if LPA is used in the complimentary space. It turned out, that for the 1 S-channel under consideration the value of E0 = 10–20 MeV is enough for accuracy of the order of several percent. All the calculation of the e6ective pairing interaction was repeated for the “realistic” value of the nuclear chemical potential  = −8 MeV. All the general features of Vpe6 and of the procedure itself discussed earlier remain valid in this case, too. In particular, the LPA can be applied within approximately the same accuracy as for  = −16 MeV. The Fermi-averaged e6ective interaction VFe6 for  = −8 MeV is also shown in Fig. 11. It can be seen, that the latter curve corresponds to strong attraction in the surface region and that the value of VFe6 in the bulk of nuclear matter is an order of magnitude less than that near the surface. If an interpolation formula of type (1.1) is used for this function the values of the parameters appearing in it prove to be ex 0 = −4:60;

in 0 = −0:46 :

(5.15)

This e6ective pairing interaction found within the Brueckner approach for semi-in1nite nuclear matter agrees qualitatively with the phenomenological density-dependent e6ective interactions of [8] or [13], thus con1rming the idea of surface pairing in nuclei. It should be stressed that any correction to ex the Brueckner theory could inLuence the value of in 0 but not 0 . Indeed, in the asymptotic region p X → ∞ all the polarization and other corrections to Ve6 vanish and Vpe6 should coincide with the free T -matrix taken at E = 2. In the case of  = −16 MeV, qualitatively, the situation is similar, but the dominance of the surface attraction is less, as far as the parameters of Eq. (1.1) are as follows: ex 0 = −4:04;

in 0 = −0:64 :

(5.16)

We see that the e6ective pairing interaction is rather sensitive to the value of . The reason of increasing of the absolute value of the parameter ex 0 with decreasing of || is quite obvious. Indeed, 1 the free T -matrix in the S-channel exhibits a pole-like behavior at small E corresponding to the virtual pole near the point E = 0. The external interaction should become greater and greater with || approaching zero. This enhancement of the surface attraction should stop only at ||  . Indeed, at such small values of || the equation for Vpe6 should be modi1ed with pairing e6ects taken into account. As estimations show, in this case the asymptotic behavior of Vpe6 at big positive X changes and the free T -matrix should be replaced by an analogous function with a virtual pole moved in the complex plain of the energy on the value of the order of . The increase of |ex 0 | at small || is important for consideration of the pairing e6ects in the drip-line vicinity. Some decrease of |in 0 | with decreasing || is explained with falling of the form factors gi (kF ) in Eq. (5.14) with increase of the Fermi momentum value in the bulk, kF2 = 2m(U0 − ||). This e6ect is analyzed in detail in [34]. Recently, in a series of papers, inLuence of exchange by surface vibrations on nuclear pairing was examined [26,27]. From a heuristic point of view, this investigation could be considered as a development of the main idea of the old paper by Green and Mozskowski [4]. Calculations of [27] resulted in a signi1cant contribution to the e6ective pairing interaction which leads to a gap  which is only a little less than the experimental one. On the other hand, our calculations of Vpe6 also qualitatively reproduce the phenomenological pairing interaction of [8,13], leaving no room for a signi1cant additional contribution. Thus, there is a contradiction between our results and predictions

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Fig. 12. The diagrams for the L-multipole vibration correction L 9 to the mass operator.

+

=

+ ...

Fig. 13. Examples of diagrams for the local contribution loc L 9 to the mass operator.

+

Fig. 14. The diagrams for the vibration corrections to the e6ective NN-interaction.

of [26,27]. As it was mentioned in Introduction, we think that, in some extent, the contribution of surface vibrations to pairing is overestimated in the latter calculation due to not accounting for so-called “local” diagrams. Indeed, the analysis of contribution of low-lying collective vibrations to di6erent nuclear observables showed [9] that the local diagrams are, as a rule, important and partially compensate the contribution of the usual (non-local) diagrams. Let us explain this point in more detail limiting ourselves, just as in [26,27], to the lowest (second) order in nucleon-vibration coupling. Consider 1rst the L-multipole vibration corrections L 9 to the mass operator. They are drawn in Fig. 12 where the 1rst diagram, which is commonly used, is essentially non-local whereas the second one represents a sum of all the more complicated diagrams. In the coordinate space, they are much more local than the 1rst one [9], that explains the term “local” diagrams. The wavy line denotes the D-function of a vibration of the multipolarity L, the bold dot stands for the nucleon-vibration vertex, the open circle, for the local contribution loc L 9. Some examples of local diagrams are given in Fig. 13. Here the big circle with four legs denotes the NN-interaction amplitude. The vibration corrections to the e6ective NN-interaction are shown in Figs. 14 and 15. The direction of the nucleon ends corresponds to the e6ective pairing interaction under consideration. It should be noted that analogous diagrams are relevant to the Landau–Migdal amplitude provided the direction of the nucleon ends is changed in a proper way. A method of accounting for the local diagrams was elaborated by Khodel [42] on the base of the self-consistency relation for arbitrary 1nite Fermi system between gradients of the mass operator and single-particle Green function [43]. A concept was developed for consideration of the surface vibrations, low-laying collective excitations of natural parity, as the Goldstone mode associated with spontaneous breaking of the translation symmetry in nuclei. The spurious dipole state is the head

M. Baldo et al. / Physics Reports 391 (2004) 261 – 310

=

+

291

+ ...

Fig. 15. Examples of local diagrams for the vibration corrections to the e6ective NN-interaction.

of this Goldstone branch, its properties being de1ned by the self-consistency relation mentioned above. The local diagrams do contribute for the case of the spurious mode, being essential for obtaining correct results which follow from conservation laws. For example, the local spurious 1− -state contribution is necessary for obtaining the zero splitting of the particle+ghost vibration triplet. For a heavy nucleus, after separation of trivial angular factors, the local diagrams almost do not depend on the vibration multipolarity L. This helps to evaluate the local contributions for arbitrary multipolarity. In [44] the surface vibration corrections to the Landau–Migdal amplitude F were evaluated. They were found to be rather small, the local contribution signi1cantly compensating the non-local one (the 1rst diagram of Fig. 14). That is why we think that accounting for the local diagrams is important also in the pairing problem. After evaluating the two contributions, the non-local and local ones, the sum should be added to the e6ective pairing interaction found above. p

6. E"ective pairing interaction for a slab of nuclear matter. A simple microscopic model for Ve" In the previous section, the microscopic e6ective pairing interactionVpe6 in the 1 S0 -channel for non-uniform nuclear systems was found within Brueckner theory for semi-in1nite nuclear matter with the separable 3 × 3 form [22] of the Paris NN-potential. In this case, the Bethe–Goldstone-type equation for Vpe6 can be solved directly, without any additional approximation. The output of such a solution is a set of six independent coordinate functions ij (X; X  ) which de1ne Vpe6 according to Eq. (3.12). However, the procedure is rather cumbersome, and possible approximations are advisable. One approximation, the LPA, was suggested in [20] and is discussed in Section 5. The LPA highly simpli1es the calculation procedure for 1nding two-particle propagators which are the most essential ingredients of the equation for Vpe6 . However, the representation of Vpe6 in terms of ij (X; X  ) remains the same as in the case of the exact solution and looks rather cumbersome. In [28] another approximation for Vpe6 was found which makes the procedure of 1nding the e6ective pairing interaction quite simple. The analysis of the direct solution of the equation for Vpe6 has shown that, within a good accuracy, it can be approximated by the o6-shell free T -matrix taken at the negative energy E = 2. As to the T -matrix itself, a very simple and accurate method for 1nding this quantity in the coordinate representation was suggested in [28] (see Appendix B). Following [28], we go from semi-in1nite nuclear matter to a more realistic 1D-inhomogeneous nuclear system. We consider a nuclear-matter slab of thickness 2L placed into the one-dimensional Saxon–Woods potential U (x) symmetrical with respect to the point x = 0, U0 U (x) = (6.1) 1 + exp((x − L)=d) + exp(−(x + L)=d)

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with the same parameters U0 = 50 MeV and d = 0:65 fm, as for semi-in1nite matter. The thickness parameter was chosen as L = 8 fm to imitate heavy nuclei. All the formulae of the Sections 3 and 4 remain valid. As to the numerical procedure, the main di6erence, in comparison with the semi-in1nite geometry case, is in the explicit form of Eq. (3.15). In Section 5, the symbolic summation over (n; n ) in Eq. (3.15) was transformed to the integration over dp dp =(2)2 . In the slab case, there are discrete states with negative energies n , and therefore there is a real summation over these discrete states and the integration over continuum. A simpli1cation of the numerical procedure for solving Eq. (3.10) in the slab system under consideration can be made using the parity conservation which follows from the symmetry of the Hamiltonian under the axis reLection x → −x. As a result, the eigenfunctions yn can be separated into even, yn+ , and odd, yn− , functions. Then the two-particle propagator in Eq. (3.10) splits into the sum A = A+ + A −

(6.2)

of the even and odd components. The 1rst one, A+ , originates from the terms of the sum in Eq. (3.15) containing states (;  ) with the same parity, and the second one, A− , from those with opposite parity. So long as the NN-potential V does conserve the parity, the propagators A+ and A− do not mix in Eq. (3.10). Let us transform Eq. (3.10) to the equation for the correlation part of the e6ective pairing interaction Vpe6 de1ned by Eq. (5.6). Obviously, Vpe6 is also a sum of the even and odd components, − Vpe6 = V+ e6 + Ve6 ;

(6.3)

which obey the separated equations Ve6 = VA V + VA Ve6 ;

(6.4)

 is the parity. It is obvious that the integral equation (6.4) can be reduced to the form containing positive x values only which simpli1es all the calculations. This equation should be solved for both values of  separately, then the complete Vpe6 could be found from Eqs. (6.3) and (5.6). With this exclusion, the calculation scheme repeats mainly that for semi-in1nite matter case and we omit details. Let us 1rst consider the “realistic” value of the chemical potential  = −8 MeV which is relevant to stable nuclei. In [28,41] the e6ective pairing interaction Vpe6 calculated with the use of LPA was compared with the free o6-shell T -matrix in detail. Here we present only some average characteristics of these two interactions which, as we have seen in Section 5, contain more information for quantitative conclusions. Fig. 16 shows the zero-order moments U ij (X ), Eq. (5.8), of the e6ective interaction for all three “large” components. They are compared with the zero moments of the T -matrix analogously de1ned:  (6.5) TU ij = dt Tij (t) : As it is seen, the di6erence between these averages is rather small, of the order of 5 –10%. For comparison, similar quantities multiplied by the factor 10 are drawn for a small 13 component. Here the relative di6erence is larger, but the quantity is very small itself.

M. Baldo et al. / Physics Reports 391 (2004) 261 – 310

293

Fig. 16. The zero moments U ij (X ) (solid lines) and the values TU ij (dashed lines) for =−8 MeV. The small 13 component is multiplied by factor 10 for the graphical convenience.

Fig. 17. The Fermi-averaged e6ective interaction VFe6 (X ) (solid lines) and T F (X ) (dashed lines) for two values of the chemical potential .

As it was discussed in Section 5, the overall contribution of all the components of Vpe6 can be characterized by the Fermi averaged form VFe6 de1ned in Eq. (5.14). Indeed, this Fermi average comes, in fact, to the gap equation which will be considered in the next section. Let us consider the Fermi averaged T -matrix de1ned analogously:  T F (X ) = TU ij gi (kF2 (X ))gj (kF2 (X )) ; (6.6) ij

kF2 (X )

where is the local Fermi momentum which explicit form is given in Section 5. In Fig. 17 the Fermi averaged e6ective interaction and T -matrix are depicted. It is seen that the both curves are very close to each other. Indeed, the maximum di6erence is of the order of 15% in the inner region where the both quantities are very small in comparison with the surface values at X  8 fm. Dealing with applications to 1nite nuclei, the surface contribution to the matrix elements of the e6ective interaction must dominate. But in this region the di6erence between the e6ective interaction and the free T -matrix is only about 5%. Thus, for the chemical potential

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 = −8 MeV corresponding to stable nuclei, the o6-shell T -matrix taken at the energy E = 2 is a good approximation for the e6ective interaction. To imitate the situation in the vicinity of drip-lines, all the calculations were repeated for  = −4 MeV. In this case, we limit ourselves to the analysis of the Fermi averaged values of Vpe6 and T -matrix. Results are shown in the same Fig. 16. They are similar to those for  = −8 MeV. Thus, the free o6-shell T -matrix taken at the energy E = 2, indeed, is a good approximation for the microscopic e6ective pairing interaction. This simple approximation of Vpe6 with the free T -matrix can be used not only for stable nuclei but also for predicting properties of nuclei distant from the 2-stability valley. For the separable representation of the Paris NN-potential it can be readily found following to the recipe of Appendix B. As far as we deal with comparatively small shift from the mass shell, all the realistic NN-potentials must produce approximately the same T -matrix. A question may be put why such a simple approximation for Vpe6 takes place. The 1rst reason is that, as it was discussed above, Vpe6 should coincide with T (E = 2) in the asymptotic region of |X | ¿ L. We have seen that an approximate equality Vpe6  T is valid also in the bulk of nuclear matter. The analysis of [21,28] showed that the latter equality is, in some sense, occasional. It appears due to a compensation of several in-medium e6ects in the case of the model space S0 (E0 = 0). Another choice of the separating energy E0 could make this approximation worse or completely invalid. 7. Surface behavior of the pairing gap in semi-in(nite nuclear matter In this section, we use the e6ective pairing interaction found above for semi-in1nite nuclear matter to solve the gap equation in this system. For this aim, we consider Eq. (3.9) with the separable NN-potential used above. In notation of Section 3, the gap function reads:  2 2 (k⊥ ; x1 ; x2 ) = i (X )gi (k⊥ ; x) ; (7.1) i

where the form factors gi in the mixed representation are given in Appendix A. For the coe:cients i of expansion (7.1) we have a set of three integral equations similar to Eq. (3.14). However, now the kernels should be found with taking into account pairing e6ects. Instead of the expansion of Eq. (3.6) one obtains a similar one in terms of u- and v-functions, which obey the Bogolyubov equations with non-local gap . These equations have the integro-di6erential form that makes their solution rather complicate. In [34] a method was elaborated of solving such equations, but it turned out to be rather cumbersome. As it is well known, a rather slow convergence is inherent to the usual iterative method of solving the gap equation. Since each iteration requires to solve these integro-di6erential Bogolyubov equations, a huge cpu time is necessary to obtain a reliable solution in this way. To simplify the solution of the gap equation, the KKC method [29,30] for non-zero temperature was used in [21]. For in1nite nuclear matter, this method was described in Section 2. In semi-in1nite matter an additional coordinate dependence appears, but all physical reasons of Eq. (2.14)-type representation of the gap  remain valid. In a non-homogeneous system, the gap-shape function # is only dependent on the coordinate and momentum variables, whereas the T -dependence is localized in the factor F (T ). This latter can be found as the asymptotic value inside nuclear matter which

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coincides with that of in1nite system. Therefore it can be obtained by a much simpler calculation. To 1nd the shape function we go to the limit T → Tc where the gap equation becomes linear. For low temperature, T 6 , the gap equation can be written in the form similar to Eq. (3.9): (T ) = Vpe6 As0 (T ) (T ) ;

(7.2)

where the e6ective pairing interaction Vpe6 can be considered to be T -independent. For semi-in1nite nuclear matter, we assume that the temperature dependence of the gap function can be separated in the form similar to Eq. (2.14): 2 2 (x1 ; x2 ; k⊥ ; T ) = F (T )#(x1 ; x2 ; k⊥ ) :

It is obvious that the gap-shape factor can be also written as a sum  2 2 )= #i (X ) gi (k⊥ ; x) : #(x1 ; x2 ; k⊥

(7.3)

(7.4)

i

After substitution of Eqs. (7.3), (3.12) and (7.4) into Eq. (7.2) at T =Tc we arrive at the following equation for the components #i :  0 dX1 dX2 il (X; X1 )Blm #i (X ) = (X1 ; X2 ; Tc )#m (X2 ) ; (7.5) lm

where



dk⊥ 2 2 d x1 d x2 gl∗ (k⊥ ; x1 )gm (k⊥ ; x2 ) (2)2  x1 x1 x2 x2 2  ;T : (7.6) ×A0 X1 + ; X1 − ; X2 + ; X2 − ; k⊥ 2 2 2 2 As is known [35], for the two-particle propagator A0 (T ) at T ¿ 0 the Matsubara technique leads to the same expression as at T = 0,  1 − N  − N
A0 (r1 ; r2 ; r3 ; r4 ; E = 2; T ) = ’ (r1 )’∗ (r3 )’
(r2 )’∗
(r4 ) (7.7)
2 −  −   
0 (X1 ; X2 ; T ) = Blm

;

with T -dependent occupation numbers N (T ) =

1 : 1 + e( −)=T

(7.8)

The summation in Eq. (7.7) is carried out over the single-particle states belonging to the model space,  = (n; k⊥ ) ∈ S0 . After simple manipulations we 1nally obtain   dk⊥ 1 − Nn (k⊥ ; T ) − Nn (k⊥ ; T ) 1 2 0 (X; X  ; T ) = − Blm 2 2 (2)  +  + k =m − 2 n1 n2 ⊥ n n 1 2

2 2 ×gnl 1 n2 (k⊥ ; X )gnm1 n2 (k⊥ ; X ) ; 2 where the matrix elements of the form factors gnl 1 n2 (k⊥ ; X ) are de1ned by Eq. (3.16).

(7.9)

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The e6ective interaction Vpe6 was de1ned in Sections 3–5, following to [20], in such a way that the model space involves only the single-particle states with negative energies. The symbolic summation over n1 ; n2 , as usual for the semi-in1nite system, denotes the integration over dp1 dp2 =(2)2 . 2 The above formulae determine the kernels of Eq. (7.5) for the gap-shape function #(x1 ; x2 ; k⊥ ). Solution of this equation should be substituted into Eq. (7.3) together with the factor of F (T = 0) which can be found at X → −∞. Therefore it coincides with that of in1nite nuclear matter. Eq. (7.5) can be considered as a particular case of a more general set of homogeneous integral equations  dX  Kij (X; X  ; T )#j (X  ) ; (7.10) #i (X ) = (T ) j

where Kij (X; X  ; T ) are the kernels of this equation for arbitrary T and (T ) stands for the corresponding eigenvalue. The critical temperature can be found from the condition that the minimum eigenvalue 1 (Tc ) is equal to unit. In principle, such procedure which requires solving Eq. (7.10) for a number of values of T is possible but rather cumbersome. For the case of semi-in1nite nuclear matter under consideration, the procedure can be simpli1ed. Indeed, at large negative X , the gap  in such a system tends to its value in homogeneous matter. Hence, for the case under consideration, when the pairing in in1nite nuclear matter exists, the value of Tc in semi-in1nite matter is the same as in in1nite one. Therefore it is much simpler to 1nd Tc for in1nite matter and use it in Eq. (7.5) (or Eq. (7.10) at  = 1; T = Tc ) as an input. Let us now describe brieLy how to solve the set of integral equations, Eq. (7.5). The kernels Kim (X; X  ) of these equations are given by folding the coe:cients il (X; X1 ) of the e6ective inter0 action Vpe6 with those, Blm (X1 ; X  ; Tc ), of the propagator A0 (T = Tc ). The vectors il (X; X1 ) were calculated in Section 5 for two values of the chemical potential,  = −8 and −16 MeV. Let us consider the 1rst one which simulates the situation in 1nite nuclei. In this case the critical temperature is Tc = 0:113 MeV. In Section 5 the e6ective interaction was found only within the interval {X } = (−8 fm; 8 fm) because the properties of Vpe6 are trivial outside this interval. So, at X; X  ¡ − 4 fm the magnitude of Vpe6 coincides practically with that calculated for in1nite matter, whereas at X; X  ¿ 4 fm it tends rapidly to the free T -matrix taken at negative two-particle energy E = 2. As it is shown in [25], at a 1xed value of X0 = (X + X  )=2 the e6ective interaction Vpe6 (X; X  ) vanishes rapidly when the relative distance t = X − X  is growing, so that the range of integrals involving Vpe6 (X; X  ) can be cut at |t| ¿ 4 fm. That is why the interval (−8 fm; 8 fm) for the e6ective interaction is su:cient for any calculations. As to Eq. (7.5) for the gap-form function, a much wider X -space should be considered, {X } = (−Lin ; Lex ) with a minimum value of Lin  40 fm (the value of Lex = 8 fm is large enough). The reason is that at small value of F . 1 MeV the correlation length  ∼ vF =F is very large ( & 10 fm). The distance between the left cut-o6 −Lin and the point under consideration (say, X & −10 fm) should include several correlation lengths for the e6ects of the left boundary not to distort the asymptotic behavior of the gap-shape function inside nuclear matter. For X; X  ¡ − 8 fm we substitute for the e6ective interaction its value Vpe6 ∞ (t) calculated for in1nite nuclear matter. It is obvious that it depends only on the di6erence t = X − X  . 0 The propagators Blm (T ) are the new ingredients of the problem. They can be readily calculated in accordance with Eq. (7.7) using the technique elaborated in Section 5 for zero temperature.

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297

1.5 F

(X), MeV

1.0

0.5

0.0 -8

0

-4

4

X, fm

Fig. 18. The function F (X ) for  = −8 MeV (dashed line) and for  = −16 MeV (solid line).

Such a large X -space makes it di:cult to solve Eq. (7.5) in the coordinate space directly along the way which was used above for a similar equation for Vpe6 . It is more convenient to use the Fourier expansion of the gap-shape function in the interval (−Lin ; Lex ):  #in fn (X ) ; (7.11) #i (X ) = n

where fn (x) are sin(2n(X − Xc )=L) and cos(2n(X − Xc )=L), L = Lin + Lex , Xc = (Lex − Lin )=2. The kernels Kij (X; X  ) are also expanded in the double Fourier series. Finally, we are left with a set of homogeneous linear equations for the coe:cients #in : #in =

3  N 




Kijnn #jn




(7.12)

j=1 n
=1


with the matrix Kijnn having not very high dimension. In this calculation we used the interval of {X }= (−40 fm; 10 fm) (Xc =−15 fm), the value of N =101 being enough to obtain an accuracy better than 2 1%. When the coe:cients #in are found, the gap-shape function #(X; x; k⊥ ) can be readily obtained 2 with use of Eqs. (7.11) and (7.4). To obtain the complete gap at zero temperature (X; t; k⊥ ; T = 0), one has just to multiply # by a constant F = 0:200 MeV found for in1nite nuclear matter. To present the results in a more transparent way, it is convenient to consider the function  #F (X ) = #i (X )gi (k 2 = kF2 (X )) ; (7.13) i

where kF (X ) is the local Fermi momentum de1ned in Section 5. Correspondingly, we may introduce the “Fermi averaged” gap function F (X ) = F #F (X ). It gives approximately matrix elements of  for the single-particle states taken at the Fermi surface. Let us now repeat all the calculations for the second value of the chemical potential, =−16 MeV, which is the “self-consistent” value for semi-in1nite nuclear matter corresponding to the saturation point of in1nite matter. In this case we have Tc = 0:556 MeV and F = 0:975 MeV. The Fermi averaged gap function, F (X ), is drawn in Fig. 18 for both values of . As it is seen, all curves exhibit pronounced maxima at the surface. Such a surface behavior is due to a big surface value of the e6ective pairing interaction (see Fig. 11). To make the -dependence of the surface e6ect more obvious, the Fermi averaged gap-shape functions #F (X ), for both values of , are drawn

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1.5 1.0 0.5 0.0

-8

-4

0

4

X, fm

Fig. 19. The gap-shape function #F (X ) in semi-in1nite nuclear matter for  =−8 MeV (dashed line) and for  =−16 MeV (solid line).

together in Fig. 19. As it is seen, the surface e6ect in  is more pronounced for  = −8 MeV. Indeed, in this case the ratio of the maximum surface value to the asymptotic one inside the matter is  1:8, whereas it is  1:5 for  = −16 MeV. Besides, the position of the maximum is closer to the surface (X = 0) for  = −8 MeV than for  = −16 MeV. It should be noted that the surface e6ect for  turns out to be less pronounced than the one for the e6ective interaction itself. This is a consequence of a strong surface–volume coupling which takes place in the gap equation due to a big value of the pairing correlation length which is signi1cantly larger than the width of the surface layer. Such a coupling is inherent to a pure quantum calculation and can be partially lost when a local approximation is used. It suppresses partially the surface maximum of . The -dependence of the surface e6ect in the gap originates from two reasons. The 1rst one is a direct energy dependence of the e6ective interaction Vpe6 taken at E = 2 (see Section 5). The second one is the strong momentum dependence of the form-factors gi (k 2 ) taken at k 2 = kF2 (X ) = 2m( − U (X )). Both e6ects cooperate in enhancing the surface values of the e6ective interaction and of  for smaller values of . However, there is an e6ect which works in the opposite direction making the -dependence of the surface maximum of  less pronounced. This is a decrease of F at smaller values of || which results in growing the correlation length. The latter smooths the surface e6ect itself and its -dependence.

8. The pairing gap in a slab of nuclear matter In this section we consider the solution of the gap equation for a slab of nuclear matter, which is much closer to 1nite nuclei than the semi-in1nite system, so that many results can be qualitatively related to them. We consider nuclear matter embedded in the one-dimensional Saxon–Woods potential well U (X ), Eq. (6.1), symmetrical with respect to the point x = 0, with the width of 2L and the same parameters U0 = −50 MeV, and d = 0:65 fm as used in Section 6. The half-width parameter L will be changed to examine the size dependence of the surface e6ect under consideration.

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299

For evaluating the gap-shape function we again use the KC method [29]. In the case of slab geometry, we assume for the gap function the same ansatz (7.3) as that in the case of semi-in1nite matter. Unfortunately, in the slab case no direct relation with in1nite nuclear matter exists. For evaluating the normalization factor, we will use some version of the local approximation of the gap equation to be derived below. The entire calculation scheme is similar to that for semi-in1nite matter with the exception of some details. First, in the slab case we deal with the discrete spectrum n in Eq. (7.9). Second, due to above discussed reLection symmetry of the slab system in the x-direction, all the integral equations under consideration can be readily reduced to a form including positive X only. Just as in Section 7, instead of the direct solution of Eq. (7.10) in the coordinate space, we employ the Fourier expansion,  #in fn (X ) ; (8.1) #i (X ) = n

where the symmetrical Fourier interval (−L0 ; L0 ) is used. Hence, only the X -even functions, fn (X ) = cos(n(X − Xc )=L0 ), must be retained in Eq. (8.1) The kernels Kij (X; X  ) of Eq. (7.10) are also expanded in a double Fourier series. Finally we arrive at a set of homogeneous linear equations for the coe:cients #in : #in =

3  N 







Kijnn #jn ;

(8.2)

j=1 n
=1

which can be solved by standard numerical methods. Then the components of the gap-shape function are found from Eq. (8.1). We calculated the gap-shape function #F (X ), normalized to unit at the center of the slab, for four values of the half-width of the slab, L = 4; 6; 8 and 10 fm, and two values of the chemical potential,  = −8 and −4 MeV. The 1rst one is typical of stable nuclei whereas the second one simulates the approach to the nucleon drip line. The results are shown in Fig. 20. One can observe a well pronounced surface bump. The enhancement is around 80 –100%, which is quite similar to the semi-in1nite matter case. Going from =−8 to −4 MeV, a noticeable increase of the enhancement of the order of 30% is observed. On the other hand, there is a little variation with the thickness of the slab, the maximum of  being only a couple of percent larger for a smaller slab size. As in the semi-in1nite case, the surface enhancement in  originates from the peculiarities of the surface behavior of the e6ective interaction Vpe6 . Let us remind that Vpe6 undergoes a sharp variation in the surface region, from almost zero in the bulk to very strong attraction in vacuum. In the asymptotic region, the latter coincides with the o6-shell T -matrix of free NN-scattering T (E =2) which exhibits a resonant behavior at small E. The strong surface attraction and the sharp variation in the surface region are mostly responsible for the surface e6ect of the gap. ex The -dependence of the surface e6ect is explained by the increase of the jump Vpe6 =Vin e6 −Ve6 with decreasing ||. This increase is caused by two reasons. The 1rst one is the strong k 2 -dependence of the form factors in Eq. (2.5) leading to a reduction of Vin e6 for the larger values of kF in the bulk. The second one is an increase of Vex with decreasing || caused by the pole-like behavior of e6 the T -matrix at small E. Thus, both reasons jointly work towards making the surface e6ect stronger at small values of ||. Finally, it should be mentioned that a large value of the coherence length of pairing which is comparable with the size of the slab makes all the e6ects under consideration

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Fig. 20. The gap-shape function #F (X ) in the slab of nuclear matter for  = −8 MeV (panel a) and  = −4 MeV (panel b). The half-width L of the slab is given by the numbers near the curves (in fm).

rather smooth. In particular, it results in the weak dependence of the surface enhancement of the pairing gap on the slab thickness. As it was mentioned above, contrary to the semi-in1nite case, there is no direct relation to in1nite matter in the slab case. To calculate the normalization of the gap, we elaborate a kind of the local approximation for the gap equation. We start from the coordinate form of the gap equation (r1 ; r2 ) = V(r1 − r2 )?(r1 ; r2 ) ;

(8.3)

where V(r1 − r2 ) is the pairing interaction, and the abnormal density matrix ?(r1 ; r2 ) is calculated through the pairing gap operator by means of the integral relation,   d d 3 r1 d 3 r2 G(r1 ; r1 ;  + )Gs (r2 ; r2 ;  − )(r1 ; r2 ) : ?(r1 ; r2 ) = − (8.4) 2i Here, Gs is the single-particle Green function in a superLuid system, while G is the one-body propagator without pairing correlations, and  is the chemical potential. Instead of using the Fourier transformation,  (p1 ; p2 ) = e−ip1 r1 −ip2 r2 (r1 ; r2 ) d 3 r1 d 3 r2 ; (8.5)

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301

useful for homogeneous systems, where due to the condition P = p1 + p2 = 0 the pairing gap is a function of a single variable p = p1 − p2 , in a non-homogeneous system, it is convenient to use the Wigner transformation of all the terms of Eqs. (8.3) and (8.4). The Wigner transform of the gap operator, as well as of the abnormal density matrix, is given by  (8.6) (R; p) = e−ips (R + s=2; R − s=2) d 3 s ; 

e−ips ?(R + s=2; R − s=2) d 3 s :

?(R; p) =

The single-particle propagators are transformed in the same way,  G(R; k; !) = e−iks G(R + s=2; R − s=2; !) d 3 s ;

(8.7)

(8.8)

while the momentum argument has di6erent meaning: k = (p1 + p1 )=2, where p1 and p1 are input and output momentum arguments of the Green functions. In the Wigner representation, the gap equation takes the form,  d 3p (R; q) = V(q; p)?(R; p) ; (8.9) (2)3 where

 ?(R; p) =

d 3 R d 3 S

d 3 K d 3p iK(R−R
)−i(p−p
)S e (R ; p ) (2)6

×A((R + R + S)=2; (R + R − S)=2; (K + p + p )=2; (K − p − p )=2) and A is the two-body propagator  d A(R1 ; R2 ; k1 ; k2 ) = − G(R1 ; k1 ;  + )Gs (R2 ; k2 ;  − ) : 2i

(8.10)

(8.11)

Let us use the generalized local ansatz [45] for the Green functions, G(R; k; !) =

1 ! − H (R; k) + 

(8.12)

with the Hamiltonian being the sum of the kinetic term and the mean 1eld, H (R; k) =

k2 + U (R) 2m

(8.13)

and Gs (R; k; !) = where E(R; k) =



v2 (R; k) u2 (R; k) + ; ! − E(R; k) ! + E(R; k)

[H (R; k) − ]2 + 2 (R; k)

(8.14)

(8.15)

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and u2 (R; k) v2 (R; k)




 H (R; k) −  1 1± : = 2 E(R; k)

(8.16)

Upon substitution of the Green functions (8.12) and (8.14) into Eq. (8.11) we obtain the local approximation for the two-particle propagator,   (1 − n(R1 ; k1 ))u2 (R2 ; k2 ) n(R1 ; k1 ) v2 (R2 ; k2 ) + ; (8.17) A(R1 ; R2 ; k1 ; k2 ) = − E(R2 ; k2 ) − H (R1 ; k1 ) +  E(R2 ; k2 ) + H (R1 ; k1 ) −  where n(R; k) = B( − H (R; k)). There are two types of non-localities, so-called longitudinal and transverse, in the two-body propagator (8.17). According to the usual LDA recipe, one saturates both of them and obtains the algebraic relation ?(R; p) =

(R; p) ; 2E(R; p)

(8.18)

where E(R; p) is given by Eq. (8.15). To obtain a more general local approximation, we keep one of the two non-localities which catches better the physics of pairing correlations in the systems under consideration. To make a choice, we note that the transverse non-locality is not speci1c for non-uniform systems. It plays in a full force in homogeneous matter as well, where it is responsible for the momentum dependence of the abnormal density matrix. On the contrary, it is the longitudinal non-locality which allows the center-of-mass of a Cooper pair to propagate over the system. Due to this non-locality the gap function does feel the non-uniform geometry. If the transverse non-locality is integrated over then two of four 3D integrations remain in Eq. (8.10) and we obtain the generalized local approximation we are searching for. In this case we put S = 0 in the 1rst two arguments of A in Eq. (8.10). Integrating then over S and p we obtain  d 3 K iK(R−R
) e (R ; p) ?(R; p) = d 3 R (2)3 ×A((R + R )=2; (R + R )=2; K=2 + p; K=2 − p) :

(8.19)

It can be easily seen that closing also the remaining non-locality results in the usual LDA approximation: ?(R; p) = A(R; R; p; −p)(R; p) :

(8.20)

We used the developed approximation for the calculation of the gap value inside the slab. In the slab case, Eq. (8.19) is reduced to the equation  dKx ?(X; px ; p⊥ ) = dX  cos(Kx (X − X  ))(X  ; px ; p⊥ ) 2 ×A((X + X  )=2; (X + X  )=2; Kx =2 + px ; Kx =2 − px ) ; 2 where px2 + p⊥ = p2 .

(8.21)

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303

Fig. 21. The function F (X ) in the slab of nuclear matter for  = −8 MeV. The half-width L of the slab is shown as in Fig. 20.

Then the components i (X ) of the gap are calculated as follows:    3  dpx p⊥ dp⊥ 2 2 ij gj px + p⊥ ?(X; px ; p⊥ ) : i (X ) = (2)2 j=1

(8.22)

Having in hand the shape functions #i (X ) one needs only to 1nd the normalization factor 0 for evaluation of the gap functions i (X ) = 0 #i (X ) :

(8.23) 0

Upon substitution of expression (8.23) with a 1xed value of the normalization factor  into the propagator in the r.h.s. of Eq. (8.21) one obtains the gap function ˜ i (X ) in the l.h.s. of Eq. (8.22). These output functions, evaluated within the generalized local approximation, are not exactly proportional to the shape functions #i (X ) on the input, calculated within the exact scheme. To obtain the algebraic equation for the normalization factor 0 we calculate the e6ective volume factor of proportionality 0i for each component:
 X0 −1  X0 0 2 ˜ i = #i (X ) dX (8.24) i (X )#i (X ) dX 0

0

with X0 = L − 2 fm and then evaluate the complete factor 0 =

3 

wi 0i

(8.25)

i=1

with the weights wi = #i (0)gi (kF (0))



3 

− 1 #i (0)gi (kF (0))

:

(8.26)

i=1

The above algebraic equation was used for the evaluation of the normalization factor 0 of the gap functions (8.23) for the slab of nuclear matter with half-widths L = 4; 6; 8; 10 fm and  = −8 MeV. The functions F (X ) = 0 #F (X ) are shown in Fig. 21. One sees that the gap value at the center of the slab is very smooth function of the slab width, and F (0)  0:21 MeV for L = 10 fm is rather close to the value 0:18 MeV of the gap in the bulk of semi-in1nite nuclear matter (see Section 7).

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The former may serve as a proof of the robustness of the elaborated approximate scheme for evaluation of the normalization of the gap function. 9. Conclusions and discussion This review paper is devoted to the microscopic approach to the pairing problem in nuclei based on the Brueckner theory for nuclear systems with 1D inhomogeneity. It was developed by the authors in a series of papers [20,21,28,34,32]. In particular, semi-in1nite nuclear matter and the slab nuclear system are considered and a separable version [22] of the Paris potential is used. The pairing problem is formulated with the help of a two-step approach in which the full Hilbert space S is split into the model subspace S0 and the complementary one, S  . The model space was chosen in a form suitable for practical nuclear calculations, including all the two-particle states (;  ) with negative energies  ; 
¡ 0. The gap equation is formulated in the model space in terms of the e6ective interaction Vpe6 . The latter obeys an equation in the complementary subspace in which the pairing e6ects can be disregarded. In this approximation, this equation coincides with the Bethe–Goldstone one. In 1D-inhomogeneous systems with the separable NN-potential under consideration, the Bethe–Goldstone equation for Vpe6 reduces to a set of several integral equations in the coordinate space which can be solved directly, without using any form of the local approximation. Such solution was found for semi-in1nite nuclear matter for several values of the chemical potential  in the case of semi-in1nite nuclear matter. The obtained e6ective interaction exhibits a sharp variation in the surface region changing from a strong attraction outside nuclear matter to almost zero value inside. For =−8 MeV, which corresponds to stable nuclei, the absolute value of Vpe6 at the surface is approximately 10 times greater than the one in the bulk. Such a space behavior of Vpe6 is rather close to that of the phenomenological density-dependent e6ective pairing interaction, with the surface dominance, of [8,13]. Thereby, the surface nature of nuclear pairing is con1rmed microscopically. The e6ective interaction is found to be dependent on the chemical potential , the surface attraction being enhanced with the decrease of ||. The e6ective interaction obtained in this calculation can be used for solving the gap equation not only in semi-in1nite matter but for the slab geometry as well and can be approximately adopted for 1nite nuclei. A new version of the local approximation was suggested for the approximate calculation of Vpe6 which was named local potential approximation, to distinguish it from commonly used LDA. It turned out that LPA works su:ciently well even in the surface region where LDA fails. The LPA, in a slightly modi1ed form, can be used also for 1nding the Brueckner G-matrix [28]. The gap equation was solved in semi-in1nite matter and in the slab system with the help of the KKC method [29,30]. This method was suggested recently to simplify the solution of the gap equation in nuclear matter. In our investigation it was extended to non-homogeneous systems and actually it turned out to be very e:cient in this case. The gap  found for both the systems exhibits a signi1cant variation in the surface region with a pronounced maximum near the surface. However, the surface e6ect in  turned out to be less pronounced than that in the e6ective pairing interaction which induces . This is a consequence of a large value of the pairing correlation length resulting in a strong surface–volume coupling for the pairing problem. This e6ect suppresses partially the surface maximum in  and makes it, in the case of the slab, almost independent of the slab size. The surface e6ect in  is -dependent, being enhanced at smaller values of ||.

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305

Recently, another mechanism of the surface e6ect in nuclear pairing, in the e6ective pairing interaction and in  as well, was suggested in [26,27]. In this approach, a contribution to Vpe6 induced by the virtual exchange of collective surface vibrations is evaluated. According to [27], this induced pairing interaction yields approximately 70% of the empirical . This does not agree with results of our calculations of Vpe6 within the Brueckner approach to non-uniform systems. Indeed, as we have seen, Vpe6 presented in Section 5 is rather close to the empirical pairing interaction of [8,13]. In any case, there seems to be no room for any signi1cant additional contribution to Vpe6 . We think that the surface vibration contribution to Vpe6 is overestimated in the papers cited above due to the neglect of the so-called local diagrams [9]. In Section 5, arguments are given in favour of some compensation of the local diagrams and those taken into account in [26,27]. Of course, the compensation is not complete, so that a consistent description of pairing in nuclei should take into account the surface vibration contribution, with local diagrams taken into consideration in a proper way. Acknowledgements We are greatly indebted to P. Schuck and M. Farine who are the coauthors of one paper observed in the present article. We are highly thankful to S.T. Belyaev, M. Di Toro, S.A. Fayans, V.A. Khodel, P. Schuck and S.V. Tolokonnikov for helpful discussions of various problems during the process of our work on papers creating the basing material of this review. This research was partially supported by the Grant NS-1885.2003.2 of The Russian Ministry for Industry and Science. Appendix A. The mixed representation of the form factors for the singlet 1 S channel In notation similar to those of Ref. [22], the form factors gi (k 2 ) of Eq. (2.5) have the form 2

gi (k ) =

4  Cin 22 k 2(n−1) in

n=1

2 n (k 2 + 2in )

:

(A.1)

Here we use the modi1cation of Ref. [23] of the NN-potential of Ref. [22]. For convenience we rede1ne the coe:cients il in Eq. (2.5) and Cin in Eq. (A.1), as compared with [22], in such a way that the relations of gi (k 2 = 0) = 1 take place. The relation between the “renormalized” coe:cients and the corresponding coe:cients of Ref. [22] (we denote them with the help of the tilde note) is as follows: Cin = fi C˜ in ;

(A.2)

il = 22 fi fl ˜il ;

(A.3)

fi =

2 2i1 : 2 2in Ci1

(A.4)

306

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Table 3 Renormalized parameters of the separable representation [23] of the Paris neutron–neutron potential 2in (fm−1 )

Cin

ij (MeV fm3 )

211 = 1:11157 212 = 2:02123 213 = 2:64343 214 = 4:08779 221 = 1:02979 222 = 1:54360 223 = 2:61292 224 = 4:08379 231 = 1:04166 232 = 1:87418 233 = 3:05501 234 = 3:84792

C11 = 1. C12 = 0:16574 C13 = −3:95926 C14 = 3:67948 C21 = 1: C22 = 1:29056 C23 = −7:77217 C24 = 7:01385 C31 = 1: C32 = 2:36399 C33 = 60:82470 C34 = −61:13654

11 = −3659:18784 12 = 2169:28620 13 = −23:61143 22 = −1484:65023 23 = 57:60745 33 = 17:17908 ij = ji

The factor of 22 in Eq. (A.3) originates due to the use of the standard Fourier transform de1nition, contrary to the “symmetrical” Fourier transform of Refs. [22,23]. Table 3 contains the coe:cients of Eqs. (2.5) and (A.1) which are calculated according the above formulae. The values of the parameters of 2in are the same as in [23]. The inverse Fourier transformation (3.13) of the form factors (A.1) can be readily made leading to the following expressions: 2 ; x) = gi (k⊥

4 

2 Cin fin (k⊥ ; x) ;

n=1

where 2 fi1 (k⊥ ; x) = exp(−si1 x)=(2si1 ) ;


 2 2i2 exp(−si2 x) 1 − 2 (1 + si2 x) ; fi2 (k⊥ ; x) = 2si2 2si2
 2 4 2i3 exp(−si3 x) 2i3 2 2 1 − 2 (1 + si3 x) + 4 [3 + 3si3 x + (si3 x) ] ; fi3 (k⊥ ; x) = 2si3 2si3 8si3
322 exp(−si4 x) 324 2 1 − 2i4 (1 + si4 x) + 4i4 [3 + 3si4 x + (si4 x)2 ] ; x) = fi4 (k⊥ 2si4 2si4 8si4  6 2i4 2 3 + [15 + 15s x + 6(s x) + (s x) ] ; i4 i4 i4 6 48si4  2 2 sin = 2in + k⊥ : 2

(A.5)

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307

Appendix B. Evaluation of the free o"-shell T-matrix in the coordinate representation As it was discussed in the Section 4, calculation of integral (4.7) of the inverse Fourier transform, even after separating the constant term ij , by direct integration along the real Px -axis is rather complicated. The problem arises because for large distances t ¿(2–3) fm the integrand contains rapidly oscillating factors multiplied by a slowly falling function. Now we examine the asymptotic behavior of the integrand at large Px . Let us begin from the free propagator (4.6). It can readily be shown that, for Px → ∞, the asymptotic behavior in question is given by 0 (Px ) → − Blm

where

 blm = 4m

blm ; Px2

d 3p gl (p2 )gm (p2 ) : (2)3

(B.1)

(B.2)

In order to regularize the asymptotic term at the point Px = 0, we rede1ne it as a =− Blm

blm ; + 2

Px2

(B.3)

where  is an arbitrary constant. Then we have 0 a  Blm = Blm + Blm :

(B.4)

The inverse Fourier transformation of expression (B.3) can be performed straightforwardly. The result is a (t) = − Blm

blm exp(−|t|) : 2

Upon the substraction of the asymptotic term, the propagator assumes the form 
 d 3p 1 1  2 2 ; gl (p )gm (p ) 2 − Blm (Px ) = −4m (2)3 Px + 4p2 + 20 Px2 + 2

(B.5)

(B.6)

0   1=P 4 ). where 20 = −8m. This di6erence converges for Px → ∞ faster than Blm (namely, Blm x Let us now address directly to the T -matrix. It can be seen from Eq. (4.5) that, for Px → ∞, it involves the constant ij -term corresponding to a -function in the x space. Therefore it is reasonable to analyze the di6erence

Tij (Px ; E) = Tij (Px ; E) − ij : It can be easily seen from Eq. (4.5) that, for Px → ∞, we have 4ij Tij (Px ) → Tija (Px ) = − 2 ; Px +  2 where 4ij =

 lm

il b0lm mj :

(B.7) (B.8)

(B.9)

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M. Baldo et al. / Physics Reports 391 (2004) 261 – 310

We again separate the T -matrix on two terms: Tij = Tija + Tij ;

(B.10)

where Tija (t) = −

4ij exp(−|t|) : 2

(B.11)

For Px → ∞, the convergence of Tij is again much faster than that of Tij ( 1=Px4 instead of  1=Px2 ). In this case, the integration in Eq. (4.7) along the real axis of Px with the same cuto6 parameter and the same step of integration as previously leads to a correct behavior of Tij (X ) at X  5 fm, but unphysical oscillations again arise at X  6 fm. In order to avoid them completely, it would be better, in accordance with general prescription for integrating quickly oscillating functions [46], to go over to integration in the complex plane of Px . We must modify the integration contour to arrive at a form convenient for the calculations with taking into consideration the singularities of the integrand. As was indicated above, the form factors gi (k 2 ) in integral (4.6) are rational functions of k 2 . Therefore, the entire integrand is also a rational function of k 2 . We will show that, in this case, the free propagator Bij0 (Px ; E) at negative energy E = 2 has no singularities other than two cuts going along the imaginary axis symmetrically with √ respect to the origin and issuing from the branching points Px = ±i0 , where 0 = −8m. For this purpose, we 1rst consider the explicit form of one of the typical terms in the sum which arises upon substituting into Eq. (4.6) the rational form factors of [22,23] given in Eq. (A.1):  ∞ 1 1 1 k2 d k 2 ; (B.12) I1 = 2 2 2 2 k + 4 k + 21 k + 222 0 where 42 = (Px2 + 20 )=4 ;

(B.13)

while 21 ; 22 are the masses of the corresponding form-factor terms. Expression (B.11) emerges from those form-factor terms that are Fourier transforms of the Yukawa functions. Upon simple algebra, we obtain I1 =

1  : 2(21 + 22 ) (4 + 21 )(4 + 22 )

(B.14)

By 4, we mean the arithmetic value of the square root of expression (B.13). It is obvious that the above branching points are the only singularities of expression (B.14) considered as a function of the complex variable Px . Let us now consider the term that is next in the order of complexity and which arises upon taking the derivative of the Fourier transform of the Yukawa function in one of the form factors:  ∞ k2 1 1 I2 = k2 d k 2 : (B.15) 2 2 2 2 2 k + 4 (k + 21 ) k + 222 0

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309

Fig. 22. The integration contour for the inverse Fourier transformation of the T -matrix.

It can easily be expressed in terms of the integral in Eq. (B.11) as I2 = I 1 +

21 9I1 : 2 921

(B.16)

From here, it becomes clear what is to be done to construct the algorithm for obtaining other terms in Eq. (4.6), which involve higher derivatives of Yukawa function. It can also be seen that, in just the same way as Eq. (B.15), these terms do not have singularities other than those of expression (B.14). Thus, we have shown that the two cuts going along the imaginary axis symmetrically with respect 0 to the origin (see Fig. 22) exhaust the list of singularities of the functions Blm (Px ; E ¡ 0) in the complex plane of Px . It is convenient to deform the integration contour in such a way that it embraces the upper cut (in Fig. 22, b → ∞). The convergence of the integral in the parameter b is much faster than the convergence in the parameter Pxc in the case of integration along the real axis. In practice, we have used the contour depicted in Fig. 22 with the following values of the parameters: a = 2 fm−1 and b = 130 fm−1 . In this case, it lies su:ciently far o6 the cut, and the integral in Eq. (4.6) is calculated numerically without any di:culties. This calculation leads to a nearly precise exponentially decaying result at any t of interest. As can be seen from Eq. (4.7), the singularities of the propagator B0 are present in the T -matrix. Apart from this, new poles can appear in the T -matrix on the imaginary axis which correspond to a real or a virtual level. In the case of the singlet S = 0 channel under consideration, this level is virtual, and the relevant poles occur on the cuts. Therefore, in order to calculate the T -matrix, we can use the same contour as that for the propagator B0 . References [1] [2] [3] [4] [5] [6] [7]

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