San José State University

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Thayer Watkins
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The Magnetic Moments of
the Odd p Even n Nuclides

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 for an even n, odd p nucleus there should be the net magnetic moment due to the intrinsic spins of one proton. The magnetic dipole moment of a proton, measured in magneton units, is 2.79285.

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 = α(Q/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 approximately proportional to each other. That means that if L is quantized then μ is quantized. This would mean that μ should approximately be a constant independent of the scale of the nucleus.

More precisly Q is proportional the proton number p. The mass of a nucleus is proportional to p+γn, where γ is the ratio of the mass of a neutron to that of a proton. Thus (Q/M)≅p/(p+γn).

There could be a slight variation in μ with proton and neutron number n because of their effects on the ratio (Q/M).

The Data

Here are the graphs of the data for the magnetic moments of the even n, odd p nuclides .

The data themselves are:

Magnetic Moments of the Even n, Odd p Nuclides
protons →
neutrons ↓ 1 3 5 7
2 2.97896244
4 3.256427
6 3.4391 2.6886489 0.3222
8 3.668 3.1778 -0.28318884
10 -0.352
12 2.55

For proton number 3 the magnetic moments for the different neutron numbers are roughly equal. For proton number 5 the value for neutrron number 8 is out of line but this may be a nuclear magic number effect.

Magnetic Moments of the Even n, Odd p Nuclides
protons →
neutrons ↓ 9 11 13 15
8 4.7213
10 2.628868 2.8363
12 3.93 2.217522 3.6455
14 3.683 3.6415069 1.2349
16 3.895 1.1316
18 2.449
20 2.305

For proton number 9 it is the value neutron number 10 that is out of line by being too low. For proton number 11 it is the vajues for14 and 16 which are out of line by being too high.

Magnetic Moments of the Even n, Odd p Nuclides
protons →
neutrons ↓ 17 19 21 23
14 0.752
16 0.8218743
18 0.6841236 0.20321
20 0.39147 5.431
22 0.2148701 4.62
24 0.1633 4.756487
26 0.1734 5.34 3.5863
28 1.933 5.024
30 3.4532

Except for the nuclear magic number of 28 neutrons the values are roughly constant for the different neutron numbers associated with each proton number. Neutron number 20 for proton number 19 is also out of line but this may be a magic number effect.

Magnetic Moments of the Even n, Odd p Nuclides
protons →
neutrons ↓ 25 27 29 31
26 3.5863
28 5.024 4.822
30 3.4532 4.72
32 4.627 2.14
34 0.2148701 2.227206
36 0.1633 2.3817 1.8507
38 0.1734 5.34 2.01659
40 1.933 2.56227

Magnetic Moments of the Even n, Odd p Nuclides
protons →
neutrons ↓ 33 35 37 39
36 1.58
38 1.674 1.97
40 1.63 0.76 0.654468
42 1.43948 0.92 3.3579
44 0.74 2.1064 1.4249 2.1
46 2.270562 1.35298 6.2
48 0.1734 2.75131 6.06
50 1.933 2.3836 -0.1374154
52 2.1815 0.1641

Magnetic Moments of the Even n, Odd p Nuclides
protons →
neutrons ↓ 41 43 45 47
46 7
48 6.216
50 9.14 6.32 10.9
52 6.1705 5.94 5.62
54 6.141 5.6847 5.43 5.7
56 6.153 -0.884 4.47
58 4.41 0.1014
60 -0.11357
62 0.13056
64 -0.146
66 0.159

Magnetic Moments of the Even n, Odd p Nuclides
protons →
neutrons ↓ 49 51 53 55
56 5.675
58 5.585
60 5.538
62 5.503
64 5.5289 3.46 3.1 5.46
66 5.5408 3.43 2.9 0.77
68 5.519 3.45 2.3 1.377
70 5.515 3.3634 2.818 1.409
72 5.502 2.5498 2.821 1.459
74 5.491 2.63 2.81327 1.491
78 2.79 2.742 2.582025
80 2.89 2.856 2.7324
82 3.00 2.8513
84 2.696
86 2.438
88 0.87
90 0.784

Magnetic Moments of the Even n, Odd p Nuclides
protons →
neutrons ↓ 57 59 61 63
76 7.5
78 0 6.6 3.673
80 2.695 4.2754 3.8 3.999
82 2.7830455 2.701 3.8 3.736
84 2.58 3.576
86 3.3 3.4717
88 1.8 1.5324
90 1.52
92 1.5
94 1.38

Magnetic Moments of the Even n, Odd p Nuclides
protons →
neutrons ↓ 65 67 69 71
80 1.7
82 1.35
84 0.919 6.81
86 3.44 3.51 0.476
88 2.01 4.35 3.42
90 2.01 4.28 2.4
92 2.014 4.25 -0.082
94 4.23 -0.139
96 4.17 -0.197 2.297
98 -0.231 2.305
100 -0.228 2.28
102 2.2323
104 2.239

Magnetic Moments of the Even n, Odd p Nuclides
protons →
neutrons ↓ 73 75 77 79
98 1.7
100 2.27
102 2.25 2.8 1.97
104 2.289 3.19 2.36 2.17
106 2.3705 3.168 2.605 0.535
108 2.36 3.187 0.13 0.494
110 0.1507 0.1369
112 0.1637 0.1396
114 0.1487
116 0.145746
118 0.261

Magnetic Moments of the Even n, Odd p Nuclides
protons →
neutrons ↓ 81 83 85 87
104 1.55
106 3.878
108 1.588
110 1.591
112 1.58
114 1.58 4.6
116 1.6 4.8
118 1.605 4.017 3.89
120 1.62225787 4.605 3.75 3.95
122 1.63821461 4.081 10 4
124 1.876 4.1103 9.56 4.02
126 4.5 3.8 1.58
128 3.89 1.17
130 1.07
132 1.5

neutrons ↓
Magnetic Moments of the Even n, Odd p Nuclides
protons →
89 91 93 95
124 7.82
126 3.83
128 1.1
130
132
134
136
138 2.01
140 4
142 3.14
144 2 2.6
146 1.58
148 0.951

(To be continued.)

The Overall Effects

As noted above, the average magnetic moment was computed for each proton and the results were plotted versus the proton number:

There appears to be two sinusoidal curves involved in this relationship.

(To be continued.)


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