San José State University

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Thayer Watkins
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& Tornado Alley
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Nuclear Subshells and Numbers
of Stable Isotopes and Isotones

The existences of shells for protons and neutrons were discerned by examing the number of stable isotopes and isotones. The filled shells were manifested by a relative large number of stable isotopes or isotones compared to adjacent values. These filled shell numbers were called nuclear magic numbers. These magic numbers are confirmed by the incremental binding energies. After a filled shell the incremental binding energy drop significantly. Incremental binding energies reveal 6 and 14 as magic numbers as well as the conventional magic numbers of 2, 8, 20, 28, 50, 82 and 126. It appears that 8 and 20 represent filled subshells of 2 and 6.

Changes in the character of the relationship between incremental binding energy and the number of nucleons indicate a system of subshells. The filled subshell numbers may replicate the shell magic numbers. This material is to investigate this hypothesis in terms of the number of stable isotopes and isotones. s

Number of Stable Isotopes as a Function of Shell Number and Occupancy of the Shell
Number of Protons in Shell
Shell
Number
Range 1 2 3 4 5 6 7 8 9 10
1 1→2 2 2
2 3→6 2 1 2 2 2 3
3 7→14 2 3 1 3 1 3
4 15→24 1 4 2 3 2 5 1 5 1 3
5 29→38 2 5 2 4 1 5 2 5 1 4
6 51→60 2 4 1 9 1 6 1 2 1 5

Number of Stable Isotopes as a Function of Shell Number and Occupancy of the Shell
Number of Protons in Shell
Shell
Number
Range 11 12 13 14 15 16 17 18 19 20
4 25→28 1 4 1 5 2 5 1 5 1 3
5 39→48 1 4 1 6 0 7 1 6 2 6
6 61→70 0 4 1 6 1 7 1 6 1 7

Number of Stable Isotopes as a Function of Shell Number and Occupancy of the Shell
Number of Protons in Shell
Shell
Number
Range 21 22 23 24 25 26 27 28 29 30 31 32
5 49→50 1 1
6 71→82 0 4 1 6 1 7 1 6 1 71 5

Number of Stable Isotones as a Function of Shell Number and Occupancy of the Shell
Number of Neutrons in Shell
Shell
Number
Range 1 2 3 4 5 6 7 8 9 10
1 1→2 2 2
2 3→6 1 1 2 2 2 2
3 7→14 2 2 1 3 1 3 1 3
4 15→24 1 3 1 3 0 5 0 3 2 3
5 29→38 1 4 1 3 0 2 0 2 3 7
6 51→60 2 4 3 4 2 3 2 3 1 3
7 83→92 4 1 3 1 2 4 0 6 2 4
8 127→136 1 0 0 3 0 0 0 0 0 3

Number of Stable Isotones as a Function of Shell Number and Occupancy of the Shell
Number of Neutrons in Shell
Shell
Number
Range 11 12 13 14 15 16 17 18 19 20
4 25→28 1 4 4 4
5 39→48 1 4 1 5 3 4 1 3 1 4
6 61→70 1 5 1 3 1 3 2 2 1 7
7 93→102 1 4 1 6 1 5 2 4 2 2
8 137→146 1 1 1 1 0 0 0 0 0 0

Number of Stable Isotones as a Function of Shell Number and Occupancy of the Shell
Number of Neutrons in Shell
Shell
Number
Range 21 22 23 24 25 26 27 28 29 30 31 32
5 49→50 2 5
6 71→82 4 5 1 5 2 4 2 4 3 328
7 102→114 2 2 2 5 2 2 2 0 3 032

Number of Stable Isotones as a Function of Shell Number and Occupancy of the Shell
Number of Neutrons in Shell
Shell
Number
Range 31 32 33 34 35 36 37 38 39 40 41 42 43 44
7 115→126 2 1 2 3 1 3 3 1 3 130 11

The columns for the occurpancies of the shells which correspond to magic numbers are marked in white. There does seem to be an indication that those cases involve greater numbers of stable isotopes or isotones. The situation is obscured by the odd-even fluctuations in the data. Below is the data for the first table with the odd number cases left out.

Number of Stable Isotopes as a Function of Shell Number and Occupancy of the Shell
Number of Protons in Shell
Shell
Nuber
Range 2 4 6 8 10 12 14
1 1→2 2
2 3→6 1 2 3
3 7→14 3 3 3
4 15→28 4 3 5 5 3 4 5
5 29→50 5 4 5 5 4 4 6
6 51→82 4 9 6 2 5 4 6

There is a tendency for larger numbers of stable isotopes for the shell occupancies corresponding to the magic numbers for the shells, but it is not strong.

(To be continued.)



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