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Magic Numbers for Nuclear Shells and Subshells


Commemorative stamps for
Maria Goeppert Mayer

Nuclear Magic Numbers

Maria Goeppert Mayer and Hans Jensen examining the properties of the isotopes of elements discerned that isotopes in which the proton and/or the neutron numbers were particular values have notable properties such as stability. These magic numbers are

Magic Numbers2820285082126

The difference in consecutive magic numbers represent the maximum occupancy of a shell. There is no obvious rule for establishing the maximum occupancy for proton shells as there is for electron shells. Furthermore Mayer and Jensen relied primarily on the number of stable isotopes for a number compared to the number for adjacent proton numbers.

The table below shows the relationship between the number of stable isotopes and the atomic (proton) number.

Proton
Number
12345678910
Number of
Stable Isotopes
2221222313

Proton
Number
11121314151617181920
Number of
Stable Isotopes
1313142325

Proton
Number
21222324252627282930
Number of
Stable Isotopes
1513141525

Proton
Number
31323334353637383940
Number of
Stable Isotopes
2415251414

Proton
Number
41424344454647484950
Number of
Stable Isotopes
16071626110

Proton
Number
51525354555657585960
Number of
Stable Isotopes
2419161215

Proton
Number
61626364656667686970
Number of
Stable Isotopes
0416171617

Proton
Number
71727374757677787980
Number of
Stable Isotopes
1524152516

Proton
Number
81828384858687888990
Number of
Stable Isotopes
23

The odd-even alternation indicates a pairing of protons within the nucleus; perhaps the existence of alpha particle subsystems. It is not clear why some numbers were designated magic and other not. The average number of stable isotopes increases with proton number reaching a peak of 10 for proton number 50 (Tin) and declines generally thereafter. But there are 9 stable isotopes for proton number 54, Xenon. Apparently 54 was excluded because it is too close to 50 and thus would imply a proton shell of maximum occupancy 4.

There is another way to establishing the occupancy of a filled shell. Their incremental binding energy of an additional proton falls once a proton shelll is filled and the additional proton has to go into a higher shell. This test confirms the Mayer-Jensen magic numbers but it identifies 6 and 14 as magic numbers. The significance of this is that if 6 and 14 are included in the series and 8 and 20 left out there is a simple algorithm for generating the magic number. However there is no question but 8 and 20 are special. Their specialness can be interpreted in terms of the filling of proton subshells. The first two proton shells have occupancies of 2 and 4, respectively. thus the seventh and eighth protons go into the third proton shell and fill a subshell of occupancy 2. The ninth through fourteenth protons then fill out the rest of the third shell. Proton number 20 could represent the filling of two subshells in the fourth proton shell. The striking suggestion from the case of 8 and 20 is that the occupancy numbers for the subshells are the same as those for the lower shells. This proposition is tested in the following table.

The Number of Stable Isotopes
for various proposed subshells
Proton Subshell
Numbers
Proton Shell
Numbers
2 6 14 28
2 (2,1,2)
6 (2,3,1)
14 (1,4,2) (2,5,1)
28 (2,5,2) (1,5,2) (1,6,0)
50 (2,4,1) (2,4,1) (1,6,1) (2,5,1)

What is displayed in the table is the number of stable isotopes for a triplet of proton numbers in the j-th subshell after the i-th shell is fillled. For example, for the subshell of 2 in the fourth shell after the first three shells are filled with 14 protons. The middle number in the triplet (1, 4, 2) is the number of stable isotopes for proton number 16=14+2. The numbers on either side are the stable isotopes for proton numbers 15 and 17. In all but one case the middle number is impressively larger than the numbers on either side. The case of proton number 54 suggests that the can be more than one subshell of a certain size.

The real test of the magic-ness of a subshell number is whether it also appears so in terms of the neutron number. The number of stable isotopes as a function of neutron number are:

Neutron
Number
12345678910
Number of
Stable Isotopes
2211222213

Neutron
Number
11121314151617181920
Number of
Stable Isotopes
1313131305

Neutron
Number
21222324252627282930
Number of
Stable Isotopes
0323144414

Neutron
Number
31323334353637383940
Number of
Stable Isotopes
1313023714

Neutron
Number
41424344454647484950
Number of
Stable Isotopes
1534131425

Neutron
Number
51525354555657585960
Number of
Stable Isotopes
2434232313

Neutron
Number
61626364656667686970
Number of
Stable Isotopes
1513132217

Neutron
Number
71727374757677787980
Number of
Stable Isotopes
4515242433

Neutron
Number
81828384858687888990
Number of
Stable Isotopes
2841312406

Neutron
Number
919293949596979899100
Number of
Stable Isotopes
2414161524

Neutron
Number
101102103104105106107108 109110
Number of
Stable Isotopes
2225220303

Neutron
Number
111112113114115 116117118119120
Number of
Stable Isotopes
221212313 3

Neutron
Number
121122123124 125126127128129130
Number of
Stable Isotopes
1313011003

Neutron
Number
131132133134135 136137138139140
Number of
Stable Isotopes
00000311 10

The Number of Stable Isomers
for various proposed neutron subshells
Neutron Subshell
Numbers
Neutron Shell
Numbers
2 6 14 28 50
2 (1,1,2)
6 (2,2,1)
14 (1,3,1) (0,5,0)
28 (1,3,1) (1,3,0) (1,5,3)
50 (2,4,3) (2,3,2) (1,3,1) (2,4,3)
82 (4,1,3) (2,4,0) (1,6,1) (0,3,2) (0,0,0)
126 (1,0,0) (0,0,0) (1,0,0)

Generally the results confirm the expectation that the middle number is notably larger than the numbers on either side.

(To be continued.)


For more on nuclear subshells see Subshells.


For more on the nuclear shell model see Nuclear Shell Structure

For a simple explanation of the nuclear magic numbers see Algorithm for magic numbers.


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