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The Magic Numbers for Nuclear Isotope Stability

The noble gases; helium, neon, argon, xenon and radon; are chemical inert; i.e., they are very stable chemically. The interpretation is that the electrons form shells and when a shell is filled the configuration is exceptionally stable and requires a lot of energy to knock an electron out of a filled shell. On the other hand, an electron in excess of a filled shell is very easy to remove from the atom. Thus the inertness of these elements is a consequence of the stability of the filled shells. The elements one electron beyond a filled shell are the highly reactive alkalai metals; lithium, sodium, potassium, cesium, rubidium and francium. The elements with one electron less than a filled shell are the highly reactive halogens; flourine, chlorine, bromine and isodine. Hydrogen may also be consider a member of either one of these groups.

The atomic numbers of the noble gases are 2, 10, 18, 36, 54 and 86. These can be considered magic numbers for electron structure stability. The differences in these numbers are: 8, 8, 18, 18, 32. These differences are twice the value of the squares of integers; i.e., 2(22), 2(22), 2(32), 2(32), 2(42). The first number 2 in the series {2, 10, 18, 36, 54, 86} is also of the form of twice the square of an integer, 2(12).

The explanation of the magic numbers for electron structures is that there are shells for 2(n2) electrons where n=1, 2, 3, 4... The reason for the coefficient 2 in the formula is that there are two spin orientations of an electron. Pauli's exclusion principle operates and so electrons fill the states sequentially with no two electrons of an atom in the same state.


Commemorative stamps for
Maria Goeppert Mayer

Nuclear Magic Numbers

Maria Goeppert Mayer and other physicists 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

It is profoundly significant that there are magic numbers for the proton and neutron numbers separately because that indicates that the protons and neutrons are organized in separate shells. In some respects a proton and a neutron appear to be simply different states of a basic particle called a neucleon. However there apparently is a distinct differentiation in the nucleus of the the proton and neutron nucleons.

The magic numbers for atomic electron structure are perfectly explainable in terms of a formula. The increments in the particular stable atomic numbers are equal to twice the square of an integer. These increments correspond to the maximum occupancy levels of shells. Therefore in looking for an explanation of the nuclear magic numbers it is reasonable to look at the increments in the magic numbers; i.e.,

Magic Numbers2820285082126
Increments26128223244

The difference in consecutive magic numbers could be sums of shell occupancy levels. For example, the value of 44 might correspond to two shells of 22 each. Likewise a value of 22 might correspond to two shells, one of occupancy 20 and one of occupancy 2.

The maximum occupancy levels for the atomic electron shells being the square of an integer n corresponds to the sum of the first n odd numbers. Let us consider the sum of the first even numbers.

Even Numbers2468101214
Cumulative Sum261220304256

The cumulative sum sum values show up in the set of increments of nuclear magic numbers and the increments can be represented as simple sum of these cumulative sum values, as is shown below.

Increments of
Magic Numbers
26128223244
Generation from
Cumulative Sums
of Even Numbers
26122+6 2+202+302+42

If there is anything to this pattern the number 14=2+12 should be something like a magic number increment (rather than a magic number). This would mean that 22 and 34 should be nearly magic numbers. Titanium, atomic number 22, has five stable isotopes but scandium, the element with atomic number 21, has only one and vanadium, the element with atomic number 23 has only two. Selenium, atomic number 34, has six stable isotopes but bromine, the element with atomic number 35, has only two and arsenic, the element with atomic number 33 has only one. This is suggestive of the magic-ness of 22 and 34 and thus that 14=2+12 is a magic number increment.

These results look impressive but, as the table below shows, there is an even-odd alternation in 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.

The average number of stable isotopes increases with proton number reaching a peak for proton number 50 (Tin) and declines generally thereafter, as shown in the table below.

Proton
Number
Range
0's10's20's30's40's50's 60's70's80's90's
Average
Number of
Stable Isotopes
1.72.32.83.03.43.73.23.31.10.0

The real test of the magic-ness of a number is whether it 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

Note that Silicon is the element with atomic number 14 and it has three stable isotopes whereas the elements Aluminum and Phosphorus having atomic numbers 13 and 15, respectively, have only one stable isotope each. The isotope of silicon which is most prevalent has atomic weight 28=14+14. The binding energy per nucleon for silicon is however not exceptionally high, but it is the end result of a fusion process in stars called silicon burning, indicating a significant degree of stability.

(To be continued.)


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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