ISSN 1062�8738, Bulletin of the Russian Academy of Sciences. Physics, 2014, Vol. 78, No. 5, pp. 466–468. © Allerton Press, Inc., 2014.Original Russian Text © M.V. Mordovskoy, I.V. Surkova, 2014, published in Izvestiya Rossiiskoi Akademii Nauk. Seriya Fizicheskaya, 2014, Vol. 78, No. 5, pp. 658–661.

466

INTRODUCTION

In [1–3], we analyzed the experimental data onneutron–nucleus interaction for even–even isotopesin the mass number range of 56 ≤ A ≤ 246 at neutronenergies of 0.04–3 MeV in terms of the coupled chan�nel optical model (CCOM). Our analysis of the neu�tron data for a large number of nuclei that differedconsiderably in their properties was aimed at creatinga unified description of the data set obtained at lowneutron energies. Another aim was to find shell effectsin the neutron cross sections.

In addition, studying the data on neutron–nucleusinteraction for even–even isotope chains in a widerange of mass numbers allowed us to find nontradi�tional magic nuclei, to determine the numbers ofvalent nucleons above a closed nuclear core, and toestablish the limits of the existence of this closed sub�shell. The experimental data under considerationincluded the integral cross sections of inelastic neu�

tron scattering with a level of excitation in an energyrange of several hundreds of keV at the reactionthreshold; total neutron cross sections; differentialcross sections of elastic and inelastic neutron scatter�ing on even–even nuclei; the s�, p�, and d�neutronstrength functions; and the radii of potential s� andp neutron scatterings.

In our investigations, we used a double�phononversion of the coupled channel optical model forspherical nuclei and a rotary variant (a rigid axiallysymmetrical rotator model) for nonspherical nuclei.The CCOM method was described in [4].

For our calculations, we used a nonspherical opti�cal potential with a real part in the Woods–Saxon formthat included the symmetry potential and spin–orbitalmember. The radial dependence of the absorption

12+

potential took the form of the derivative of real partform�factor.

The real potential included the isospin�dependentmember and took the form V = V0 – (N – Z)/V1, wherethe symmetric potential was V1 = 22 MeV. The spin–orbital interaction parameter was 8 MeV. The geomet�ric parameters of CCOM potential were constant andidentical for the real and imaginary parts: R = r0A1/3

with r0 = 1.22 fm. The diffuseness parameter was ini�tially set equal to 0.65 fm.

The free parameters of the model were the depthsof isoscalar potential V0 and imaginary potential W.The quadrupole deformation parameters defining thematrix elements of channel coupling in the CCOMwere set equal to the values obtained mainly from the

electromagnetic processes [5]. The CCOM param�eters for each nucleus were determined when the opti�mum description of neutron data under considerationwas obtained.

The optimum description of neutron data with com�mon values of V0 = 52.5 ± 1.5 MeV W = 2.5 ± 0.5 MeVwas obtained for both spherical and non�sphericalaxial nuclei in the range of 56 ≤ A ≤ 246 [1–3, 6].

In [1–3, 6], we sometimes had to adjust the diffuse�ness parameters relative to initial value a = 0.65 fm inorder to best describe the neutron data. This was nec�essary for magic and pseudomagic nuclei especially,since there were experimental data and theoreticalindications suggesting a reduction in surface layerthickness (diffuseness) for closed�shell nuclei [7–9].

Using reduced a values for such nuclei, weobtained the best description of experimental datawith V0 and W values that were identical for all nucleiof the region under consideration. To obtain the bestdescription of cross sections, we also had to lower the

el2β

Isotopic Features of the NpNn Function for Even–Even Isotope Chains of Ni and Zn

M. V. Mordovskoy and I. V. SurkovaInstitute for Nuclear Research, Russian Academy of Sciences, Moscow, 117312 Russia

e�mail: mvmordovsk@mail.ru

Abstract—Low�energy (0.04–3.0 MeV) neutron data for even–even 58–64Ni and 64–70Zn isotopes are ana�lyzed in terms of the coupled channel optical model (CCOM) as a function of NpNn, where Np(Nn) are thenumbers of valent nucleons (particles or holes), and consider the relationship between the diffuseness param�eter obtained from CCOM calculations and the value of the NpNn function. Considering the Ni and Zn iso�tope chains with the traditional magic number Z = 28 and the nontraditional N = 38 proves the existence ofN = 28–38 subshells. The results from our analysis indicate the possible existence of the nontraditional magic

nucleus

DOI: 10.3103/S1062873814050268

Zn6830 38.

BULLETIN OF THE RUSSIAN ACADEMY OF SCIENCES. PHYSICS Vol. 78 No. 5 2014

ISOTOPIC FEATURES OF THE NpNn FUNCTION 467

diffuseness parameter in the nuclei with the nontradi�tional magic number, a feature of which is lower stabil�ity of the corresponding subshells relative to the classi�cal magic numbers as the numbers of protons and neu�trons in the nuclei change.

Ni and Zn nuclei are of considerable interest, since

the isotopes display clear classic magicity at

Z = 28, while the isotopes display nontradi�tional magicity at N = 38.

We obtained the following a values for the non�spherical nuclei of Ni and Zn isotopes investigated in

[6]: a = 0.55 fm for , a = 0.60 fm for , a =

0.60 fm for , a = 0.65 fm for , a = 0.65 fm

for , a = 0.65 fm for , a = 0.60 fm for

, and a = 0.65 fm for .In [10], study of the isotope–isotone effect in total

reaction cross sections under the action of deuteronsand α particles showed that nuclear edge diffusenessincreases upon moving from a light isotope to a heavierone, due to changes in the neutron density in the

boundary layer. The isotope was also shown tohave greater diffuseness than that of the 58Ni isotope;agreement with the experimental data was achievedwhen the a value for 58Ni was equal to 0.57 fm. It wasassumed that the 64Ni nucleus is more spherical thanthat of 58Ni.

In [11], it was determined that the surface layerthickness increases with the onset of new nuclear sub�shell formation and then diminishes as the number ofnucleons in the subshell grows, reaching its minimumin the completely filled subshell.

In [6], the diffuseness parameters for magic andpseudomagic nuclei were shown to be 0.55–0.60 fm. Aminimum value of a = 0.50 fm was obtained for 114Sn(Z = 50, Z = 64). According to [3, 12], this nucleus isdouble magic and Z = N = 64 is a nontraditional magicnumber [12, 13]. This is why the a value for 114Sn wasminimal in contrast to the initial a = 0.65 fm, at whichthe neutron data are best described for most sphericalnuclei.

In [14], Casten showed that the behavior of spectralcharacteristics for the collective excitations of nucleiin all regions of the Periodic Table can be written as thefunction NpNn, where Np and Nn are the numbers ofvalence protons and neutrons (or the correspondingnumbers of hole states). In these and later works, thebehavior of different nuclear characteristics wasexplained in terms of monopole and quadrupole inter�action between valence nucleons. The behavior ofsuch characteristics associated with the developmentof collectivity in even–even nuclei as the behavior

of level energies, the E2 excitation probabilities of

these levels, the ratio of level energies,g factors, quadrupole deformation parameters, andmany others take the form of smooth functionsof NpNn.

Ni58 6428

−

Zn64 7030

−

Ni5828 30 Ni60

28 32

Ni6228 34 Zn66

30 36

Zn6430 34 Zn66

30 36

Zn6830 38 Zn70

30 40

Ni64

12+

1 1(4 ) (2 )E E+ +

In [15], the cross sections of inelastic neutron scat�tering with the excitation of first 2+ levels at neutronenergies above the reaction threshold of 300 keV werefound to depend smoothly on the products of protonNp and neutron Nn numbers above the closed shells for66 even–even isotopes of 23 elements from 56Feto 238U.

The existence of unified NpNn systematics suggeststhat the relationship between collectivity and quadru�pole nuclear polarization and the product of numbersNpNn can be taken as a measure of the collectivityassociated with the quadrupole np interaction force.

Due particularly to the presence of isotopic pecu�liarities in the inelastic cross sections, the use of suchan approach would seem to be reasonable since crosssections depend strongly on the number of neutrons ina nucleus. In such nuclei, the collective propertieschange rapidly upon a change in the number of neu�trons, due to competition between gaps in the Nilssonsingle�particle levels at different strains.

Our experimental cross sections of inelastic neu�

tron scattering with the level of excitation weretaken from [16–19]. The figure shows the experimen�

tal cross sections for the levels at neutron energies300 keV above the reaction threshold. The cross sec�tions were averaged over a range of 100 keV. For con�venience, we introduce the value 0.55 ± 0.07 b for a

12+

12+

1.0

0

1.5

8 16 24 32 40 48

0

0.5

1.0

1.5

2.0

20 40 60 80 100 120 140

0.5

σ, b

σ, b

NpNn

NpNn

Change in NpNn for with allowance for NpNn sys�tematics. The vertical and horizontal lines represent possi�ble NpNn values without and with allowance for the exist�ence of magic N = 38, respectively. The insert shows thebehavior of inelastic neutron scattering cross section with

a level of excitation at energy En = 300 keV as a functionof NpNn for the region under study. The solid line was plot�ted using the least squares method. The experimental crosssections are � for isotopes at NpNn = 0 and 4, × for Zn iso�topes, + for Ge, � for Se, � for Sr, � for Zr, � for Mo, � forRu, � for Pd, for Cd, for Te, � for Ba, for Nd,

for Sm, for W, for Os, � for Pt, and for Hg.

Zn6830 38

12+

468

BULLETIN OF THE RUSSIAN ACADEMY OF SCIENCES. PHYSICS Vol. 78 No. 5 2014

MORDOVSKOY, SURKOVA

cross section averaged over 12 isotopes instead of thecross sections for individual isotopes corresponding toNpNn = 0; for NpNn = 4, the cross section value wasaveraged over 6 isotopes.

For the Ni isotopes, the cross sections of inelastic

neutron scattering with the level of excitation are

= 0.46 ± 0.05 b for and = 0.59 ± 0.06 b for

. For the Ni isotopes, the product is NpNn = 0.The difference between the 58Ni and 60Ni cross sec�

tions could confirm the conclusion in [10] that dif�fuseness and other characteristics increase upon mov�ing from a light isotope to a heavier one. Such transi�tions could be accompanied by mutual reinforcementof the proton and neutron gaps, due to the N = 30 in58Ni being close to the magic Z = 28.

The inelastic scattering cross sections for Zn iso�

topes are = 0.58 ± 0.06 b for , = 0.60 ± 0.06 b

for , and = 0.51 ± 0.06 b for .

The product of numbers NpNn for the Zn isotopesare NpNn = 8 for 64Zn, NpNn = 4 for 66Zn, NpNn = 0 for68Zn, and NpNn = 4 for 70Zn.

These values were obtained with allowance for thediffuseness parameters of Zn isotopes, the reductionsin the inelastic scattering cross sections, and the

increase in the energy of level for 68Zn. Most impor�tant, these values were obtained with allowance for theclosing of N = 38 subshells.

Our consideration of certain spectroscopic proper�ties of nuclei and the use of different methods such asthose described in [12, 20, 21] showed that subshellclosing can affect other nontraditional magic nuclei aswell. This can be attributed with differing degrees ofconfidence to such nontraditional magic numbers ofnucleons as N = 38, 56, 64 and Z = 38, 40, 64, as hasbeen confirmed by the results of many authors.

The existence of the N = 28–38 subshell is alsoconfirmed by a low value of the cross section (n, n'γ)

for (NpNn = 0). For the neighboring isotope

(NpNn = 40), the cross section of this reactionat the same over�the�threshold energy is ~1.7 timeslarger than for the one for 70Ge, as is apparent in thedifferences between the diffuseness parameters forthese isotopes [6]. The disappearance of a closed sub�shell could be due to monopole interaction betweenvalence protons and neutrons. As was noted above,nontraditional magic numbers are characterized bylower stability upon a change in the number of protonsand neutrons in the nucleus.

CONCLUSIONS

The nucleus can be regarded as a nontradi�tional magic nucleus due to its smaller inelastic scat�tering cross section, relative to neighboring Zn iso�

topes; its lower diffuseness parameter; the increase in

its level of energy; and the behavior of other charac�teristics.

The existence of magic N = 38 is visible in all Znisotopes. If it did not exist, the product NpNn for Znisotopes would be much greater; this corresponds tothe inelastic scattering cross section of ~1.0 b in theNpNn systematics and is in absolute contradiction tothe experimental data (see Fig. 1).

REFERENCES

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Translated by K. Utegenov

12+

12+σ Ni5828 30

12+σ

Ni6028 32

12+σ Zn6430 34

Zn6630 36

12+σ Zn6830 38

12+

Ge7032 38

Ge7232 40

Zn6830 38

12+