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Short-range correlations and shell structure of medium-mass nuclei

The single-particle spectrum of the two nuclei 133Sb and 101Sn is studied within the framework of the time-dependent degenerate linked-diagram perturbation theory starting from a class of onshell-equivalent realistic nucleon-nucleon potentials. These potentials are derived from the CD-Bonn interaction by using the so-called V-low-k approach with various cutoff momenta. The results obtained evidence the crucial role of short-range correlations in producing the correct 2s1d0g0h shell structure.


I. INTRODUCTION
Shell structure plays a key role in explaining many features of atomic nuclei. In particular, the energy gap between groups of levels is crucial in determining their stability properties. However, what is the microscopic origin of the shell structure is still a subject of great theoretical interest.
In recent years special attention has been focused on the role of the various components of the nuclear force in the formation and evolution of the shell structure. This is mainly related to the advent of new advanced techniques and facilities, and in particular to the development of radioactive ion beams, which allow to study how the properties of nuclei may change when going toward the proton or neutron drip line. It is for instance a current matter of investigation the influence of the tensor force on the single-nucleon energies, especially in the case of exotic nuclei [1,2,3], where it is expected to produce a reduction of the spin-orbit splitting [4]. This aspect is also discussed in Ref. [5] along with the influence of many-body correlations and of states unbound to particle emission on the single-particle spectrum of neutron-rich nuclei.
In this work, we have studied the structure of the 2s1d0g0h-shell for protons and neutrons, calculating the single-particle spectrum of 133 Sb and 101 Sn with realistic nucleon-nucleon (N N ) potentials. Both these nuclei are far from stability, 133 Sb having a large neutron excess and 101 Sn lying very close to the proton drip line. On the other hand, 132 Sn shows strong shell closures for both protons and neutrons while this is not the case for 100 Sn, which may imply different core polarization effects on 133 Sb and 101 Sn. We shall see that the softness of the core plays an important role in the perturbative description of the single-particle structure of these nuclei.
The two nuclei are described as a doubly-closed core plus one valence nucleon, whose correlations with the core are taken into account by means of the timedependent degenerate linked-diagram perturbation theory [6]. In previous works, single-particle valence nuclei of the sd-shell have been extensively studied within the same framework by other authors [7,8,9]. However, the presence of the intruder state 0h 11/2 makes a more challenging problem to reproduce the structure of the 2s1d0g0h-shell.
We have employed a class of onshell-equivalent realistic N N potentials, derived from the CD-Bonn interaction [10] by way of the so-called V low−k approach [11]. This approach is a very valuable tool since, by varying the cutoff momentum Λ, it provides different N N potentials defined up to Λ which are all onshell equivalent. This allows to study how the short-range (high-momentum) components of the N N potential contribute to determine the nuclear shell structure.
The paper is organized as follows. In Sec. II we give an outline of our calculations, focusing attention on the time-dependent degenerate linked-diagram expansion. Sec. III is devoted to the presentation and discussion of our results for the 133 Sb and 101 Sn single-particle spectra. Some concluding remarks are given in Sec. IV.

II. OUTLINE OF CALCULATIONS
As mentioned in the Introduction, we calculate the single-particle energies of 133 Sb and 101 Sn within the framework of the time-dependent degenerate linkeddiagram expansion [6,12]. This means that we describe the wave function of each nucleus as a singlenucleon state |j , whose energy ǫ j is calculated taking into account perturbatively the interaction of the odd nucleon with the closed core. As a starting point, an auxiliary one-body potential U is introduced to break up the hamiltonian as the sum of an unperturbed term H 0 , which describes the independent motion of the nucleons, and a residual interaction H 1 : The effective single-particle energies are expressed by way of the folded-diagram expansion in terms of the so-calledQ box: ǫ j 0 being the unperturbed energy of the single-particle state |j , i.e. Eq.
(2) represents the well known Kuo-Lee-Ratcliff (KLR) folded-diagram expansion [6,12]. This expansion is written in terms of the vertex functionQ-box, which is composed of one-body irreducible valence-linked Goldstone diagrams. Once theQ-box has been calculated through a certain order in the input interaction V N N , the series of the folded diagrams is summed up to all orders using the Lee-Suzuki iteration method [13].  Our calculations have been performed including all the one-body irreducible valence-linked Goldstone diagrams up to third order in V N N . In Fig. 1 the first-order diagram, which is composed of the so-called self-energy diagram minus the auxiliary potential U -insertion, is reported while the second-and third-order diagrams are shown in Figs. 2 and 3, respectively. The U -insertion diagrams arise in the perturbative expansion owing to the presence of the −U term in H 1 (see Eq. (1)). In our calculation we have used as auxiliary potential the harmonic oscillator (HO) one, which is the most common choice.
Let us now focus our attention on the input interaction V N N . Of course, it would be very desirable to use directly a realistic N N potential that reproduces with high precision the two-body scattering data and deuteron properties. However, to perform nuclear structure calculations in a perturbative framework with realistic N N potentials one has first to deal with the strong repulsive behavior of such potentials in the high-momentum regime. An advantageous method to renormalize the bare N N interaction has been proposed in [11,14]. It consists in deriving from V N N a low-momentum potential V low−k defined within a cutoff momentum Λ. This is a smooth potential which preserves exactly the onshell properties of the original V N N and is suitable for being used directly in nuclear structure calculations.
Let us now outline briefly the derivation of V low−k [14]. The repulsive core contained in V N N is smoothed by integrating out the high-momentum modes of V N N down to Λ. This integration is carried out with the requirement that the deuteron binding energy and low-energy phase shifts of V N N are preserved by V low−k . This is achieved by the following T -matrix equivalence approach.
We start from the half-on-shell T matrix for V N N where P denotes the principal value and k, k ′ , and q stand for the relative momenta. The effective lowmomentum T matrix is then defined by where the intermediate state momentum q is integrated from 0 to the momentum space cutoff Λ and (p ′ , p) ≤ Λ. The above T matrices are required to satisfy the condition The above equations define the effective lowmomentum interaction V low−k , and it has been shown [14] that their solution is provided by the same KLR folded-diagram expansion [6,12]     In addition to the preservation of the half-on-shell T matrix, which implies preservation of the phase shifts, this V low−k preserves the deuteron binding energy, since eigenvalues are preserved by the KLR effective interaction. For any value of Λ, the V low−k can be calculated very accurately using iteration methods. Our calculation of V low−k is performed by employing the iteration method proposed in [15], which is based on the Lee-Suzuki similarity transformation [13].
We have derived from the high-precision CD-Bonn N N potential [10] a class of V low−k 's corresponding to values of the cutoff Λ ranging from 2.1 to 2.6 fm −1 . These V low−k 's are all onshell equivalent, more precisely they all yield the same deuteron binding energy and N N scattering data up to the anelastic threshold, which corresponds to a relative momentum k ≃ 2.1 fm −1 , as the CD-Bonn potential. However, this does not imply that these V low−k 's give the same results when employed in the calculation of the single-particle spectra, since the larger the value of the cutoff the more the repulsive highmomentum components are explicitly included in V low−k . In other words, V low−k 's corresponding to different cutoffs have different offshell behaviors (see for istance Coraggio et al. [16]). To illustrate this, let us consider the offshell tensor force strength. This is related to the Dstate probability of the deuteron P D , which implies that, when comparing V low−k 's with different cutoffs, offshell differences are seen in P D differences. From inspection of Table I, where the P D 's for various V low−k 's are reported, it can be seen that the larger the value of the cutoff the stronger the offshell tensor force.

III. RESULTS
Calculated and experimental single-particle spacings for the one-proton valence nucleus 133 Sb are reported in Fig. 5 as a function of the cutoff Λ. We have found it appropriate for the perturbative expansion to limit the value of Λ to 2.6 fm −1 , since larger values of the cutoff lead to a rapid deterioration of the the convergence properties. The calculations of diagrams have been performed using intermediate states whose excitation energy is up to N max = 15 harmonic-oscillator quanta, so to obtain the stability the stability of the results within few tens of keV when increasing the number of intermediate particle states. This is shown in Fig. 4, where we report the relative SP energies calculated at third order in H 1 , with a cutoff value Λ =2.6 fm −1 , as a function of N max . The oscillator parameter used is ω = 7.88 MeV, accord-ing to the expression ω = 45A −1/3 − 25A −2/3 [17] for A = 132, and for protons the Coulomb force has been explicitly added to the V low−k potential. It is important to point out that our model space is spanned by the 0g 7/2 , 1d 5/2 , 1d 3/2 , 2s 1/2 , and 0h 11/2 states above the 132 Sn core, so we have downshifted the unperturbed energies of the 0h 11/2 and 0g 9/2 levels by a quantity equal to ω. In this way the 2s1d0g0h shell is degenerate, as required by the theory of the timedependent degenerate linked-diagram expansion, and the 0g 9/2 acts as a hole state below the Fermi surface. Anyway, the inclusion of diagrams with (V − U )-insertions in theQ box assures that our results are perturbatively independent of the choice of U . In fact, to sum up the (V − U )-insertion diagrams to all orders is equivalent to employ a Hartree-Fock self-consistent basis. The results shown in Fig. 5 (blue lines) are normalized with respect to the 0d 5/2 state (straight black line) in order to evidence the changes in the spacings when increasing the cutoff Λ. In this figure, we have also reported (red dashed lines) the results obtained by calculating the Padé approximant [2|1] [2|1] =Q 0 +Q 1 +Q Q i being the the vertex functionQ-box calculated through the ith-order in the input interaction V N N . Using Padé approximants [18,19] one may obtain an estimate of the value to which the perturbative series should converge. So the comparison between the third-order results and those obtained by means of the Padé approximant [2|1] is an indicator of the dependence of our results on the higher-order perturbative terms. From inspection of Fig. 5, it can be seen that the calculated spin-orbit splitting between 1d 3/2 and 1d 5/2 levels is almost independent from Λ, the experimental separation being reasonably well reproduced. The distance between the 2s 1/2 energy and the centroid of the 1d levels is rather independent from the cutoff too, but there is no experimental counterpart. It is worth to point out that the 1d 3/2 and 2s 1/2 relative energies do not exhibit a significant dependence on higher-order contributions, the difference between the third-order results and those obtained with the [2|1] Padé approximant being very small.
The situation is very different for the calculated relative energies of the 0g 7/2 and 0h 11/2 levels, whose spin-orbit partners have been placed outside the chosen model space. The calculated 0g 7/2 , 0h 11/2 relative energies show a strong dependence on the cutoff Λ and a slightly worse perturbative behavior. The most striking feature is that for small values of the cutoff the discrepancies with respect to the experimental data are quite large, while a better agreement is obtained increasing Λ. In particular, it should be pointed out that with small values of the cutoff the 0h 11/2 level is far away in energy from the other levels, thus implying a shell closure at Z = 70. A much more reasonable spectrum of 133 Sb is obtained when increasing the value of the cutoff which cause the intruder 0h 11/2 proton state to join the 50-82 shell. This evidences the non-negligible role of the high-momentum components of V N N in the formation of the observed shell structure.
However, after these remarks one may wonder whether when increasing Λ the model space should be enlarged to include the 2p1f 0h levels. More precisely, for large value of the cutoff it might occur a sort of "shell-quenching" when calculating the single-particle spectrum above the closed-shell 132 Sn. As a consequence, the 0h 11/2 state as well as its parity partners (0h 9/2 , 1f 7/2 , 1f 5/2 , 2p 3/2 , 2p 1/2 ) would approach in energy the 2s1d0g states, leading to the birth of a sort of "macro"-shell 2s1d0g2p1f 0h. In order to investigate this question, we have performed a kind of theoretical "experiment". We have carried out calculations where the 2p1f 0h proton-particle unperturbed energies have been considered degenerate with the 2s1d0g ones, and then we have turned on the interaction H 1 . The calculated energies relative to the 0d 5/2 level are reported in Fig. 6. It can be seen that the spin-orbit splittings of the 2p1f 0h levels are practically independent of the cutoff Λ, while the 0h 11/2 level is pushed down from its parity partners. In particular, for large value of the cutoff the 0h 11/2 state is much closer in energy to the 2s1d0g levels, confirming the reliability of a model space spanned only by the five 2s1d0g0h states.
We have found it interesting to investigate the counterpart of the 2s1d0g0h shell for neutrons, namely the single-particle spectrum of 101 Sn. In such a case, the closed-shell core is the exotic N = Z nucleus 100 Sn. To this end, we have performed the same calculation as for 133 Sb, except that the harmonic-oscillator parameter is ω = 8.55 MeV. In Fig. 7 we report the calculated and experimental single-neutron energies for 101 Sn relative to the 1d 5/2 state as a function of the cutoff Λ. As can be seen, the main difference with respect to the results shown in Fig.  5 relies in the convergence properties of the 0g 7/2 − 1d 5/2 spacing, that is the only spacing observed in 101 Sn. More precisely, for the 0g 7/2 energy level in 101 Sn we have found that the third-order contributions of diagrams (27), (32), and (33) of Fig. 3, with respect to the second order diagram (3) of Fig. 2, are larger than what has been found for 133 Sb. All these graphs represent the so-called Tamm-Dancoff plus Random-Phase Approximation diagrams. The stronger role of higher-order perturbative terms in the linked-diagram expansion is probably due to the fact that 100 Sn is a much "softer" core than 132 Sn.
Apart these concernings about the perturbative behavior of the linked-diagram expansion, our results confirm the need of the high-momentum components of the nuclear force in order to improve the description of the neutron shell structure.

IV. SUMMARY AND CONCLUSIONS
In this paper, we have calculated the single-particle spectra of 133 Sb and 101 Sn within the framework of the time-dependent degenerate linked-diagram perturbation theory. To investigate how the high-momentum components of the N N potential influence the structure of the proton/neutron-2s1d0g0h shell, we have employed a class of onshell-equivalent realistic nucleon-nucleon potentials. These potentials have been derived from the high-precision CD-Bonn interaction [10] by way of the so-called V low−k approach, varying the cutoff momentum from 2.1 up to 2.6 fm −1 .
The perturbative behavior of the calculated singleparticle relative energies is quite good for 133 Sb, while is less satisfactory for 101 Sn. This may be traced to the higher degree of exoticity of 101 Sn, which is very close to the proton drip line.
Our results show that, while the calculated spin-orbit splittings are scarcely dependent on the cutoff value, the relative energy of the intruder 0h 11/2 state is strongly sensitive to the explicit inclusion of the high-momentum components of the N N potential. We have found that for small values of the cutoff the 0h 11/2 level lies far away in energy from the other single-particle levels, while the correct shell structure is restored when increasing the cutoff.
However, it should be noted that the lack of the highmomentum components could be compensated by the inclusion of an effective three-body force [20,21], which, for small cutoffs, may play a non-negligible role in determining the single-particle spectrum [22].