Masses and Pairing Energies of Deformed Nuclei
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es and Pairing Energies of Deformed Nuclei A. K. Vlasnikova, *, A. I. Zippaa, †, and V. M. Mikhajlova aSt.
Petersburg State University, St. Petersburg, Russia *e-mail: [email protected]
Received May 11, 2020; revised June 2, 2020; accepted June 26, 2020
Abstract—A description is considered of the mass surfaces and pairing energies of a number of even–even and odd deformed nuclei with mass numbers in the range of 150 to 190 using polynomials no higher than the second order. An approach in which pairing energy depends on the mass of one odd nucleus is applied. It is shown that the main features of the behavior of pairing energies are preserved in this approach. DOI: 10.3103/S1062873820100287
INTRODUCTION The effects of pairing correlations [1–4] play an important role in the properties of atomic nuclei. The main information on these correlations is derived from the values of pairing energies: proton Pp and neutron Pn. The energy of a neutron-odd nucleus is usually compared to those of neighboring even–even isotopes to determine Pn, and the energies of neighboring even–even isotones are used to determine Pp. However, the validity of choosing certain even–even isotopes or isotones can be questioned. It is possible that the energies of other even–even nuclei would be better for determining pairing energies. Other even–even nuclei can also be considered if the energies of nuclei E ( N + s, Z + t ) with number of neutrons N + s and number of protons Z + t (where s and t take both positive and negative integer values) can be presented as an n-th order surface in the immediate neighborhood t s of the given values of N and Z ∼ < 0.05 : N Z N +s E ( N + s, Z + t ) = 1 1 − ( −1) Pn 2 Z +t 1 + 1 − ( −1) Pp + % ( N , Z ) 2 + din,kp s i t k (i ! k !).
(1)
i + k >0
Equation (1) is valid for odd nuclei and even–even nuclei; odd–odd nuclei are not considered here. In an ideal case, Pτ , % ( N , Z ) and parameters din,kp , determined using different groups of neighboring nuclei, are similar. It was shown in [5, 6], however, that the describing the mass or energy surface to the fourth and sixth orders does not result in closer † Deceased.
% ( N , Z ) , or first and second order parameters calculated using different groups of nuclei. In this work, we therefore consider first and second order surfaces only: (i + k ≤ 2) . The energies of two neighboring odd atomic nuclei are normally used in calculating pairing energies [1, 2], which results in their averaging. Indeed, an odd nucleon is usually in a different quantum state in deformed nuclei with Nodd and Nodd + 2 or Zodd and Zodd + 2. It was therefore proposed in [7] that pairing energies be determined in a way more appropriate for deformed atomic nuclei, i.e., by including the energy of one odd nucleus only: Pn ( N odd , Z even ) = E ( N odd , Z even )
− 9 [E ( N odd + 1, Z even ) + E ( N odd − 1, Z even )] (2) 16 + 1 [E ( N odd + 3, Z even ) + E ( N odd − 3, Z even )] . 16 To obtain Pp , we must make the following substitution: n ↔ p; N ↔ Z and put N i
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