San José State University

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A Formula for the Maximum
Occupancy of a Nuclear Shell
in Terms of Its Shell Number

## Introduction

In the late 1940's Maria Goeppert Mayer and Hans Jensen tried to identify filled nuclear shells by the proton and neutron numbers with the relative largest number of stable nuclides, They were awarded the Prize in Physics for 1949 for their work. The set of numbers they identified was {2, 8, 20, 28, 50, 82, 126}. They came to be called the nuclear magic numbers. They meant that the occupancies of the nuclear shells were 2, 6, 12, 8, 22, 32 and 44. The anomaly of the fourth shell having a smaller occupancy size than the third should have been an indication that something was not quite right.

Identifying a filled shell in terms of the number of stable nuclides compared to nearby arrangements is difficult to achieve definitively. The measurement of the binding energies of nuclides a sharper picture. The incremental binding energies of neutrons within a shell are about constant until its capacity is reached. When a neutron has to go into a higher shell there is a sharp drop in the incremental binding energy. The same applies for the incremental binding energy of protons.

The data on incremental binding energies identify 6 and 14 as filled shell numbers for both neutrons and protons. The numbers 8 and 20 are also special but they represent filled subshells within a shell.

This means the filled shell magic numbers are 2, 6, 14, 28, 50, 82 and 126, If S is the number of the highest shell filled then nuclear magic number M is given by

#### M = S(S² + 5)/3

Thus for the sixth shell M=6(36+5)/3=2(41)=82.

The maximum shell occupancies are 2, 4, 8, 14, 22, 32 and 44, (No anomalies.) The occupancy N for the K-th shell satisfies the following formula

#### N = K(K−1) + 2

Thus for K=6, N=6(5)+2=32. For the eighth shell N=8(7)+2=58. So the nuclear magic number beyond 126 is 126+58=184. Supposedly there is an island of stability for nuclides with neutron numbers around 184.

The ninth shell would have an occupancy of 9(8)+2=74 so the next nuclear magic number beyond 184 would be 184+74=258.

There is considerable evidence there are subshells within the nucleonic shells and the occupancies follow the same above formula as the occupancies of the shells. This would mean that in the seventh shell the nuclides with nucleonic numbers of 84, 88, 96 and 110 would have something special about them. That could be a shift in the level of incremental binding energy, a shift in a slope, a change in the odd-even fluctuation or some combination of them. See Nuclear Subshells.