San José State University

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The Magnetic Moments of
the Even-Even Nuclides
from Helium to Tin

Background

The magnetic moment of a nucleus is due to the spinning of its charges. One part comes from the net sum of the intrinsic spins of its nucleons. The other part is due to the rotation of the positively charged protons in the nuclear structure.

However nucleons form spin pairs with other nucleons of the same type but opposite spin. Therefore there should be no magnetic moment due to the intrinsic spins of its nucleons if it has an even number of protons and an even number of neutrons.

Analysis

The magnetic moment of a nucleus μ due to the rotation of its charges is proportional to ωr²Q, where ω is the rotation rate of the nucleus, Q is its total charge and r is an average radius of the charges' orbits. The angular momentum L of a nucleus is equal to ωr²M, where M is the total mass of the nucleus. The average radii could be different but they would be correlated. Thus the magnetic moment of a nucleus could be computed by dividing its angular momentum by its mass and multiplying by it charge; i.e.,

μ = α(L/M)Q = (αQ/M)L

where α is a constant of proportionality, possibly unity. Angular momentum may be quantized. This would make μ directly proportional to Q and inversely proportional to M. But Q and M can be expected to be proportional to each other. That means that if L is quantized then μ is quantized. This means means that μ should approximately be a constant independent of the scale of the nucleus.

There could be a slight variation in μ with the neutron number n beause of its affect on the ratio (Q/M).

The Data

Here is the graph of the data.

Generally the magnetic moment is a small amount is the range of 0.3 to 1.2 magnetons independent of the number of protons. But there are a few cases outside of that range and their magnitudes appear to be proportional to the value of the proton number.

The Magnetic Moments of the Even-Even Nuclides
Proton
Number 		Neutron
Number 		Magnetic
Moment
(magnetons)
6 8 0.82
8 8 1.688
8 10 -0.57
8 12 -0.7
10 10 1.08
10 12 0.65
12 14 1.0
14 14 1.1
14 16 0.8
16 16 0.9
16 18 1.0
18 22 -0.2
20 20 1.6
20 24 -0.6
22 24 1.0
22 26 0.9
22 28 2.7
24 26 1.2
24 28 3.0
24 30 1.1
26 28 3.4
26 30 1.22
26 32 0.9
28 30 -0.1
28 32 0.2
28 34 0.68
28 36 0.9
30 34 0.8
30 36 0.5
30 38 1.0
30 40 0.6
32 36 2.4
32 38 0.94
32 40 0.8
32 42 0.87
32 44 0.84
34 42 0.8
34 44 0.8
34 46 0.8
34 48 0.9
36 42 1.08
36 48 -1.97
38 44 2.0
38 46 0.84
38 48 0.55
38 50 2.3
38 60 0.76
40 46 0.5
40 48 -1.51
40 50 6.25
40 52 -0.06
40 54 -0.52
40 60 -0.52
42 46 0.5
42 48 5.5
42 50 11.3
42 56 0.7
42 58 0.7
42 60 0.84
42 62 0.4
44 50 8.12
44 54 0.8
44 56 1.02
44 58 0.74
44 60 0.82
46 50 10.97
46 56 0.82
46 58 0.93
46 62 0.72
46 64 0.62
48 52 9.9
48 54 10.3
48 58 0.8
48 60 0.7
48 62 0.57
48 64 0.6
48 66 0.58
48 68 0.6
50 58 -0.24
50 60 0.07
50 62 0.7
50 66 -0.3
50 68 0.04
50 72 -0.1
50 74 -0.3

The Effect of the Neutron Number

The graph of magnetic moment versus neutron number reveals 50 as a critical value.

Fifty is of course a magic number in nuclear shell theory. It represent the filling of a nuclear shell. Apparently criticality only requires the neutron number to be near 50, as evidenced by the magnetic moment for the nuclide with p=42 and n=48 being 5.5 magnetons.

The graph also a couple anomalous points for 28 neutrons. Twenty eight is another nuclear magic number.

The equation μ=α(Q/M)L with L constant means that μ should be a function of (Q/M). For a nucleus with p protons and n neutrons Q is proportional to p and M is approximately proportional to (p+n). Thus Q/M is approximately proportional to p/(p+n).

The graph of μ plotted versus p/(p+n) has a couple of different clusters. The points for which μ is between 0 and 2 seem to fall near a straight line.

A regression of the magnetic moment on p/(p+n) and variables indicating whether or not p and n are equal to the critical levels of 28 or 50 gives the following results.

μ = 44.47191(p/(p+n)) −2.26033(p=50) + 0.07285(p=28) + 0.77689(n=50) −1.20370(n=28) − 18.36729
[3.5 ]    [-2.0 ]    [0.1 ]    [0.7 ]
   [-0.9 ]    [-3.3 ]    

The coefficient of determination for this equation is only 0.17, but the t-ratio of 3.5 for p/(p+n) strongly confirms the result of the analysis that μ due to nuclear rotation depends upon the Q/M ratio of the nucleus.

Conclusions

The magnetic moments of the even-even nucludes between helium and tin seem to be nearly constant except for some extreme values associated with the filling of nuclear shells. There is some slight variation but definite variation in magnetic moment with the variable p/(p+n). which represent the charge to mass ratio Q/M of a nucleus. This in turn indicates that the angular momenta of nuclei are quantized.

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