On the properties of N = 50 isotones from 78 Ni to 100 Sn

By using the self-consistent approach based on the Skyrme functional and with account of pairing correlations, we calculate global properties of nuclei, such as binding energies, one and two-neutron separation energies, density distributions of nucleons, root-mean-square radii of nucleon distributions. For description of excitation energies and transition rates we apply the RPA+BCS approach based on the finite range effective interaction defined by us before. Everywhere, where it is possible, we perform comparison of theoretical results with the experimental data. Study of evolution of nuclear properties in the long chains of isotopes or isotones, from the extremely protonexcess 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 48Ni up to 78Ni, where all nuclei, except for 78Ni, turn out to be discovered by the present time. The mentioned chain is of special interest as it includes doubly-magical nucleus 48Ni (which is an extremely proton-excess one), and 78Ni (the last one is strongly neutron-excess, and also the doublymagical nucleus). This chain also includes doubly magical 56Ni and semi-magical 68Ni. 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 102Sn (N = 52) up to 136Sn (N = 86). These nuclei were theoretically studied by us in [3]. Here, we study chain of isotones from 78Ni up to 100Sn having N = 50, this chain includes nuclei 88Sr and 90Zr which also demonstrate some properties of magicity (Z = 38 and Z = 40, correspondingly). For description of global properties of nuclei such as their masses, root-mean-square radii of nucleon distributions, oneand two-nucleon separation energies, as well as single-particle energies of protons and neutrons, we apply the Hartree–Fock+BCS method. We also study in the framework of the QRPA approach excitation energies of nuclei and reduced electromagnetic transition rates. To this aim, we use the phenomenological mean field potential that was defined by us before [4]. This potential correctly takes into account isovector terms, which is very important when we consider long isotopic or isotonic chains ae-mail: visakov@thd.pnpi.spb.ru of nuclei. In our QRPA calculations we also employ effective two-body interaction used by us in our previous papers [2, 3]. To determine nuclear binding energies B in the selfconsistent approach, we performed calculations based on the HF+BCS procedure that employs the Skyrme interaction and constant pairing with the corresponding proton and neutron pairing constants Gp and Gn 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 E = −B = ∫ HHF+BCS(r) dr − Δ 2 n

Study of evolution of nuclear properties in the long chains of isotopes or isotones, from the extremely protonexcess 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 were theoretically studied by us in [3].Here, we study chain of isotones from 78 Ni up to 100 Sn having N = 50, this chain includes nuclei 88 Sr and 90 Zr which also demonstrate some properties of magicity (Z = 38 and Z = 40, correspondingly).
For description of 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 apply the Hartree-Fock+BCS method.We also study in the framework of the QRPA approach excitation energies of nuclei and reduced electromagnetic transition rates.To this aim, we use the phenomenological mean field potential that was defined by us before [4].This potential correctly takes into account isovector terms, which is very important when we consider long isotopic or isotonic chains a e-mail: visakov@thd.pnpi.spb.ru of nuclei.In our QRPA calculations we also employ effective two-body interaction used by us in our previous papers [2,3].
To determine nuclear binding energies B in the selfconsistent approach, we performed calculations based on the HF+BCS procedure that employs the Skyrme interaction and constant pairing with the corresponding proton and neutron 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 Here and below, "n" refers to neutrons, while "p" to protons.Pairing correlations in the Hartree-Fock-Bogoliubov energy density H HF+BCS (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 cases of odd, or odd-odd nuclei one should add to (1) the quasiparticle energy (energies) E j 0 of an odd particle (particles), In our self-consistent calculations we used parameters of the Skyrme 3 interaction, while the pairing constants G were close to those from [5].The exchange Coulomb terms were treated in the Slater approximation.In our calculations of binding and single-particle energies as well as of the quasiparticle characteristics, we considered all bounded and quasistationary single-particle levels.
One can see the values of the obtained binding energies for even isotones having N = 50 in Fig. with the experiment, the maximal binding energy per one nucleon happens for isotones having A ∼ 90(88).In Fig. 2, we show both theoretical and empirical values of the two-proton separation energies S 2p for the sequence of even isotones having N = 50, where calculations were performed in the framework of the HF+BCS method.One can see a good agreement with the experiment.The comparison of theoretical (HF+BCS method) and empirical values of the one-proton separation energies S p for the chains of isotones having N = 50 and even and odd values of Z is shown in Fig. 3. Analogous calculations, but in the framework of phenomenological (WS+BCS) scheme were performed by using the ansatz: ; Z and N are even.In our phenomenological calculations we used the mean-field potential of the form . (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 = −51.0MeV, V ls = 32.4MeV•fm 2 , r 0 =1.27 fm, r c =1.25 fm, β=1.31, β ls = −0.6, a = 0.67 fm for protons and a = 0.55 fm for neutrons.These parameters are very close to those from the paper [4].
The comparison of the empirical and theoretical (WS) values of the one-neutron separation energies S p for the chains of isotones with N = 50 and N = 51 is shown in Fig. 4.
For description of excited states and transition rates we used the QRPA approach, see details in [2,3], 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.This interaction was also used in our previous paper and has the form The entering parameters are as follows: V = −16.65,V σ = 2.33, V T = −3.00,V τ = 3.35, V τσ = 4.33, V τT = 3.00 (all these values are in MeV), while r 00 = 1.75 fm.
Using the standard procedure we can pass to the quasiparticle basis, a + → ξ + : (4) We also define the creation operator Q + n,JM of the onephonon excited state |ω n , JM with |ω n , JM = Q + n,JM | 0 in the following way: Then we 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 [2,3]: In Eqs. ( 7) and ( 8), a j c j d ; J| θ| j a j b ; J a and a j c jd ; J| θ| j a jb ; J a are the antisymmetric matrix elements of the effective interaction θ in the particle-particle and particle-hole channels with a given spin.They have the form × j c j d ; J| θ| j a j b ; J + (−1) j a + j b +J+1 j c j d ; J| θ| j b j a ; J , a j c jd ; J| θ| j a jb ; J a = − × j b j c ; J | θ| j d j a ; J + (−1) j d + j a +J +1 j b j c ; J | θ| j a j d ; J .
Using an explicit form of the matrix equation (6), we obtain the orthonormality relation which in terms of the QRPA bosons corresponds to the condition The transition rate between the one-phonon and the ground state is described by the reduced matrix element, see also [2,3] 0 M(λ) ω n , J = (−1) λ δ(J, λ)δ(π n π λ ) × (13) where the upper signs refer to T-even (Eλ), while the lower ones to T-odd (Mλ) transitions.In Fig. 5, we show systematics of 2 + 1 levels and reduced transition rates B(E2, 2 + 1 → gr.st.) in even N = 50 isotones.One can see a good agreement with the experiment.
Table 1 shows the comparison of the empirical and theoretical (HF+BCS) root-mean-square radii for N = 50 isotones.The deviation from the experiment is no more than 0.2 %.
Figures 6-8 demonstrate proton and neutron densities for 78 Ni, 88 Sr, and 100 Sn calculated both in (HF+BCS) and (WS+BCS) approaches.Here, we see the characteristic behavior of densities by moving from neutron-excess to neutron-deficient nuclei.In particular, HF+BCS calculations, as compared to WS+BCS ones, give the more smooth variation of both proton and neutron densities as functions of radii.

Figure 1 .
Figure 1.(Color online) Binding energies per nucleon,B/A in the chain of N = 50 isotones.All experimental data are from [6].

Figure 3 .
Figure 3. (Color online) Proton separation energies in the chain of N = 50 isotones, calculated in the HF+BCS scheme.Experimental data are from [6].

Figure 4 .
Figure 4. (Color online) Neutron separation energies for the chains of N = 50 and N = 51 isotones; WS scheme.The experimental data are from [6].