Evolution of nuclear properties in the long chain of Sn isotopes from A=100 to A=132

. This paper presents results of both microscopical and semi-empirical calculations of single-particle characteristics of nuclei and nuclear binding energies, as well as their root-mean-square radii, excitation energies and transition rates in the long chain of Sn isotopes, from the extremely neutron deﬁcient 100 Sn up to neutron excess 136 Sn, where the experimental information is available by now. The comprehensive comparison with the experimental data is carried out.

Nuclear binding energies B, as well as one-nucleon separation energies S , are the major nuclear characteristics that define borders of nuclear stability and the decay modes of nuclei. Study of evolution of nuclear properties in the long chains of isotopes or isotones, from the extremely proton-excess up to the extremely neutron-excess nuclei is of special theoretical interest, as here one can check the adequacy of the used theoretical models in the broad interval of (N − Z)/A. Formerly, [1] we studied in details the chain of the N = 82 isotones. Different long chain is represented by the sequence of the nickel isotopes from 48 Ni up to 78 Ni, where all nuclei, except for 78 Ni, turn out to be discovered by the present time. The mentioned chain is of special interest as it includes doubly-magical nucleus 48 Ni (which is an extremely proton-excess one), and 78 Ni (the last one is strongly neutron-excess, and also the doublymagical nucleus). This chain also includes doubly magical 56 Ni and semi-magical 68 Ni. The mentioned series of isotopes was theoretically analyzed by us in [2]. Another long isotopical chain is offered by the succession of tin isotopes having Z = 50. The experimental data are available from 102 Sn (N = 52) up to 136 Sn (N = 86). These nuclei are theoretically studied by us here.
For description of global properties of these isotopes we apply the Hartree-Fock-Bogoliubov method. We also use the approach based on the phenomenological mean field potential that was defined by us before [3]. This potential correctly takes into account isovector terms, which is very important when we consider long isotopical chains of nuclei. Here, we examine global properties of nuclei, such as their masses, root-mean-square radii of nucleon distributions, one-and two-nucleon separation energies, as well as single-particle energies of protons and neutrons. We also study excitation energies of nuclei, as well as the reduced transition rates as respect to the electromagnetic and weak transitions.
To define nuclear binding energies B in the self-consistent approach, we performed calculations based on the HF+BCS procedure that employs the Skyrme interaction and constant pairing with the corresponding proton and neutron a e-mail: visakov@thd.pnpi.spb.ru pairing constants G p and G n to account for pairing correlations. In this case, one can represent the total energy of an even-even nuclei in its ground state in the following form, see the details in [4,5]: Here and below, "n" refers to neutrons, while "p" to protons. Pairing correlations in the Hartree-Fock-Bogoliubov energy density H HFBCS (r) were considered by introducing the occupancies v 2 i into the single-particle density of matter, as well as into the kinetic energy and spin densities. In this way, the Hartree-Fock problem with modified densities was solved in coordinate representation, while the iteration procedure was applied for joint solution of the HF+BCS equations.
In our self-consistent calculations we used parameters of the Skyrme 3 interaction, while the pairing constants G were close to those from [6]. The exchange Coulomb terms were treated in the Slater approximation. In our calculations we considered all bounded single-particle levels as well as the quasi-stationary states.
One can see the values of the obtained binding energies for even isotopes of Sn in figure 1. In accordance with the experiment, the maximal binding energy per one nucleon happens for stable isotopes having A ∼ 124.
In figure 2, we show both theoretical and empirical values of the two-neutron separation energies S 2n for the sequence of the even Sn isotopes. One can see the good agreement with the experiment. In particular, we observe rapid fractures of the values S 2n at N = 82 and N = 50, these fractures correspond to the closure of the corresponding neutron shells. Unfortunately, the mass of the Z = 50, N = 48 nucleus is unknown yet in the experiment. Note that we observe very small singularity of the curve at N = 64 (if at all). This is in contrast with the situation characteristic for isotones having N = 82, where one can see the visible fracture at Z = 64. This peculiarity, in aggregate with other experimental data, affords ground to consider 146 Gd as a nucleus having the features of the magical one [7]. All the experimental data concerning nuclear masses are from the Atomic Mass Evaluation (AME 03), see [8]. Together with the self-consistent potential, we used also the phenomenological one. We defined this potential in [3], and it has the form .
, R = r 0 A 1/3 , t z =1/2 for neutrons and t z = −1/2 for protons. In the case of protons we added to (2) the potential of a uniformly charged sphere with R c = r c A 1/3 . The parameters of potential used here were as follows: V 0 = −51.0 MeV, V ls = 32.4 MeV·fm 2 , r 0 =1.27 fm, r c =1.25 fm, β=1.31, β ls = −0.6, and a = 0.6 fm for both neutrons and protons. These parameters are very close to those from the paper [3].
In figure 3, we show evolution of the proton singleparticle energies in the sequence of even tin isotopes as calculated by using the phenomenological and the selfconsistent approaches. One can observe the similar behavior of the proton levels calculated in the framework of both schemes. We see that the energies of the proton singleparticle levels decrease with the increase of the neutron number N when we approach the neutron drip line, this   The comparison of the empirical and theoretical (WS) values of the one-proton separation energies S p for the chains of isotopes 50 Sn N and 51 Sb N is shown in figure 4. The same comparison but for the HFBCS approach is represented in figure 5. We see that in both cases one can observe a good agreement with the experiment, this agree- In figure 6, we show evolution of the neutron singleparticle levels for the succession of even isotopes of Sn. Here, one can see the striking difference between the forecasts as given by the WS and the HFBCS (Skyrme 3 + pairing) schemes. The neutron levels obtained in the framework of the self-consistent method do not follow the isotopical dependence of the central phenomenological potential (2). Self-consistent calculations show that by the approach to the neutron drip line bounded neutron levels close to the Fermi surface resist their extrusion into continuum. This gives a chance for existing very neutronexcess even Sn isotopes that are stable as respect to the neutron emission. Single-particle levels are not the observable ones, especially if we are far from the filled shells. So, let us compare the observable values, for example the oneneutron separation energies obtained in the frameworks of two methods. We show in figure 7 results of calculations of single-neutron separation energies S n (nl j) for a sequence of Sn isotopes in the framework of the phenomenological potential by using the ansatz: ; N is even, while λ is chemical potential. Results of concurrent calculations, but performed in the HFBCS approach are shown in figure 8. Here, separation energies are found as differences of the corresponding binding energies. It is remarkably that both calculations lead to similar results and agree with the experiment.
By now, there exist large experimental information on the values of the root-mean-square charge radii of tin isotopes, the majority of information is obtained by means of methods of the laser spectroscopy. Thus, we performed corresponding calculations, both for protons and for neutrons. In Table 1 we show results of our calculations that were carried out in the self-consistent procedure by using both standard and modified (see below) Skyrme 3 schemes. Both calculations lead to close results. One can observe an agreement with experiment accurate within 0.3 %.  We also calculated proton and neutron density distributions for even isotopes of tin both in the standard Skyrme 3 + pairing, and in the WS schemes. The pattern of these distributions for the nuclei 120 Sn, that is close to the stability line is shown in figure 9. Both methods lead to close results.  For description of excited states and transition rates we used the QRPA approach [2], [9] with the phenomenological mean field potential shown by us before, as well as the effective interaction, the same in the particle-particle, particle-hole and pairing channels. For a system of only 11002-p.3 Table 1. Root-mean-square radii R p and R n of the proton and the neutron distributions in the even isotopes of Sn, in the units of fm. Experimental data are borrowed from the electronic database [11] and from [12]. Calculations are performed by using the HFBCS method with the Skyrme 3 interaction and constant pairing having the standard value of the pairing constant [6], G n = 21/A MeV. Results of calculations performed by using the modified meanfield spin-orbital term are shown in square brackets. like particles and in a particle-particle channel this interaction coincides with the interaction used in the paper [10].
Our effective interactionθ has the form ) .
Using the standard procedure, we can pass to the quasiparticle basis, a + → ξ + : Supposing the presence of correlations in the true ground state |0⟩ of an even-even nuclei, we define the creation operator Q + n,JM of the one-phonon excited state |ω n , JM⟩ with |ω n , JM⟩ = Q + n,JM |0⟩ in the following way: where X n,J j a j b = ⟨ω n ; JM| In Eqs. (6), (7) and below δ j a j b = δ j a j b δ l a l b δ n a n b .
One may obtain the set of the QRPA equations which define the amplitudes "X" and "Y" of the states |ω n , JM⟩ and the eigenvalues ω n . These equations have the form Here, E = E ab = E j a + E j b , I cd,ab = δ j a j c δ j b j d , while the matrix elements of the sub-matrices A and B in the case of even-even nuclei are as follows: In Eqs. (10) and (11), a ⟨ j c j d ; J|θ| j a j b ; J⟩ a and a ⟨ j cjd ; J|θ| j ajb ; J⟩ a are the antisymmetric matrix elements of the effective interactionθ in the particle-particle and particle-hole channels with a given spin. They have the form Using an explicit form of the matrix equation (9), we obtain the orthonormality relation ∑ a≥b which in terms of the QRPA bosons corresponds to the condition ⟨ω n , JM| ω m , JM⟩ = ⟨0|Q n,JM · Q + m,JM |0⟩ = δ mn . (15) Considering the transition rates, we must distinguish between two different types of transitions, namely, transitions between the phonon states (i.e., between the two excited states), and the phonon-ground-state transitions. The latter transition is described by the reduced matrix element ⟨0∥M(λ)∥ω n , J⟩ = (−1) λ δ(J, λ)δ(π n π λ ) × where the upper signs refer to T-even (Eλ), while the lower ones to T-odd (Mλ) transitions. One can show that the "phonon-phonon" matrix element has the form where the upper signs refer to Eλ, while the lower ones -to Mλ transitions. In figure 10, we show systematics of ex-  perimental and theoretical energies of the first quadrupole and octupole states in the sequence of N-even isotopes of Sn, while figure 11 demonstrates the comparison of theoretical and experimental values of B(E2). For magical nuclei we also show here fractions of the corresponding transition rates as compared to the total sum rule. One can find results of alternative calculations in the paper [13]. Here we used values of corresponding effective charges defined by us before in description of the E2 transitions in the vicinities of 208 Pb and 132 Sn. Similar comparison, but for the E3 transitions is shown in figure 12. One should mention here previous calculations [14][15][16] performed for nuclei where the experimental information is available. Results of the more detailed calculations for 130 Sn are shown in figure 13. Here we also show results of our RPA calculations obtained by solution of the Bethe-Salpeter equation in media. Results of the analogous calculations, but for 102 Sn are shown in figure 14. By now, there exist two sets of experimental data pertinent to 102 Sn, namely [17], and [18]. Being strongly different from each other, they both lead to anomalously large values of the effective neutron charge as defined from the 6 + to 4 + transition. The structure of the lowest states in 102 Sn strongly depends on the mutual disposition of the neutron d 5/2 and g 7/2 states, and is sensitive to the spin-orbit splitting. In the Skyrme scheme this splitting is generated by the parameter W, that defines the strength of the two-body spin-orbit interaction. At the same time, some spin-orbit splitting arises also from the spin-density term, which is usually ignored in the Skyrme single-particle equations. The contribution of this term into splitting is proportional to where t 1 and t 2 are the Skyrme force parameters.
As the radial wave functions of the spin-orbital doublet are very close to each other, one can easily see that contributions from filled levels of the spin-orbital doublet mutually contract each other, if the core is the spin-saturated one (for example, 40 Ca). In the case of nuclei just above 100 Sn the non-zero contribution to splitting arises only from the 11002-p.5   filled 1g 9/2 proton and neutron sub-shells, this contribution is small and of the opposite sign as compared to that mediated by the spin-orbit two-body interaction. However, we also performed calculations with inclusion of the spindensity term into the corresponding single-particle equations. The results are shown in figure 15, from which we see that here the spin-orbital splitting really decreases a little, with the main effect of variation of the mutual disposition of the neutron d 5/2 and g 7/2 levels in even isotopes of Sn close to A = 100. Here, the neutron g 7/2 level proves to be lower than the d 5/2 one, this difference may be important for description of the electromagnetic transition rates in these nuclei. One can observe the same pattern for the proton levels. It is very important that such modification of the Skyrme single-particle potential in practice does not influence the results of calculations shown in figures 1, 2, and figure 8. The influence of this term on the values of the root-mean-square radii is also small, as one can see from  Authors of the recent paper [29] carefully measured Q EC and T 1/2 for the very strong Gamow-Teller transition from 0 + (g.s., 100 Sn) to 1 + in 100 In. This gives us the opportunity to define the |G A /G V | ratio for this transition. The comparison of the obtained result with the results of our our recent calculations [30] performed for other N ∼ Z nuclei are shown in figure 16. We see that the renormalization of the G A constant in nuclei as compared to that in the decay of a free neutron is very small. Note that in our calculation for the decay of 100 Sn we did not take into consideration fragmentation of the πg 9/2 and νg 7/2 states, this fragmentation is not known in the experiment in nuclei close to 100 Sn. However, we considered similar effects in other nuclei. The account of this fragmentation will lead to a further small increase of the |G A /G V | ratio in the decay of 100 Sn.
In our calculations, in the framework of the spherical scheme we considered global properties of tin isotopes in the large interval of the neutron number N, from the 11002-p.6 NSRT12 neutron-deficient 100 Sn up to the neutron-excess 132 Sn. For description of different nuclear properties we used moreor-less standard methods, as well as previously defined entering parameters, and obtained results that are mainly in agreement with the experiment.
Our results show that the neutron d 5/2 and g 7/2 states of the 50 − 82 shell are separated from the h 11/2 , d 3/2 and s 1/2 ones. It exhibits in the downfall of the B(E2; 2 + 1 → 0 + 1 ) values and in a small raising of the E(2 + 1 ) energies at N ∼ 64, both effects are seen in the experiment. At once we do not see the fracture of the S 2n values at N = 64.
We notice that the value of the neutron effective charge defined from the 6 + 1 → 4 + 1 transition in 102 Sn is abnormally large. At the same time, it was shown in [31] that by using the experimental value [32] of B(E2; 8 + 1 → 6 + 1 ) in 98 Cd, we obtain the anomalously small value of the proton effective charge. One may conventionally assume the presence of strong superfluid correlations in nuclei just close to 100 Sn. This results in suppression of the transition matrix elements between the two-quasiparticle states in 98 Cd due to the factor (u 1 u 2 − v 1 v 2 ), and thus leads to the increase of the proton effective charge. However, we are unable to explain in this way experimental data in 102 Sn. For a while we do not have satisfactory quantitative explanation of abnormal values of the effective charges.
Our calculations denote that the quenching of the M1 transition rates in nuclei against those calculated with the bare value of G A is small.