San José State University 

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The Evidence for the Formation of Proton Spin Pairs Within Nuclei 

An essential aspect of the structure of nuclei is the formation within them of nucleon spin pairs; protonproton, neutronneutron and protonneutron. Protonneutron spin pairs exist alone as deuterons but not protonproton or neutronneutron spin pairs. The evidence for the formation of protonproton spin pairs within nuclei is the oddeven fluctuation in the incremental binding energy of nuclides, examples of which are shown below.
The regularity of the sawtooth pattern demonstrates that one and only one protonproton spin pair is formed when a proton is added to a nuclide.
The same effects occur for neutronneutron spin pair formation on binding energy
The sharp dropoff after 50 protons is the effect of a shell being filled. The filled shell numbers, usually called nuclear magic numbers are 2, 8, 20, 28, 50, 82 and 126. There is also evidence of 6 and 14 being magic numbers.
The effect of protonneutron spin pairs is revealed by a sharp drop in incremental binding energy after the point where the numbers of protons and neutrons are equal.
Here is the graph for the case of the nuclides with 36 neutrons.
As shown above, there is a sharper drop in incremental binding energy when the number of protons exceeds the neutron number of 36. This illustrates that when a proton is added there is a protonneutron spin pair formed as long as there is an unpaired neutron available and none after that. This illustrates the exclusivity of protonneutron spin pair formation. It also shows that a protonneutron spin pair is formed at the same time that a protonproton spin pair is formed.
The purpose of the material which follow is to show the universality of the effects illustrated above. For a nuclide with an even number of protons the increment in the incremental binding energy of a proton is positive and for one with an odd number of protons it is negative. Allowance must be made for whether the proton number p is less than the neutron number n; or p=n+1 or n>(p+1). The increment in the incremental binding energy for an odd number of protons is strongly affected by the filled shell effect.
It must be noted that these values of the increments of the incremental binding energies include the effects of adjustments in nuclides which result from the formation of a spin pair as well as the formation of the spin pairs themselves.
Of the 2931 nuclides for which the binding energies are known the incremental binding energy of a proton can be computed for 2768. Of these 2768 there are 2605 for which the increment in incremental binding energy can be computed. Of these 1303 have an even number of protons and 1302 have an odd number of protons. Of those with an even number of protons all but ten have a positive value for the increments and of those three have negative values which are very small (0.19, 0.1 and 0.01MeV) and could be attributed to measurement errors.
Of those with an odd number of protons all have a negative value for the increment.
What are shown below are the cumulative frequency distributions for the even and odd proton number cases.
The straight portions of the cumulative frequency distribution indicate that the frequency distributions are uniform over those portions of increments in the incremental binding energies of protons.
For the cases of an odd number of protons there are no anomalies.
For the case of an even number of protons there are ten anomalies. Three of these are very small. Another two are about one quarter and one half of an MeV. There are three with values near −0.75. The most serious anomaly is one of about −2.0. Here is the graph for the data on nuclides with ten neutrons..
It appears as though the sixth proton does not form a spin pair or if it does it is not manifested as an increase in binding energy. But likely the problem is not that incremental binding energy for the sixth proton is not high enough; it is that the incremental binding energy for the fifth proton is extraordinarally high.This comes from the comparison of the binding energy of a nuclide with four protons and ten neutrons with a nuclide with five protons and ten neutrons. A nucleus with four protons and ten neutrons is very unbalanced between neutrons and protons; five protons to ten neutrons is a better balance and this would show up in terms of greater binding energy.
These cases are the ones for which the formation of protonneutron spin pairs has ceased. Thus there is a negative effect on the incremental binding energy as a result. If n is even that negative effect is offset by the positive effect of the formation of a protonproton spin pair. Here are the values for p even and equal to (n+1).
The Increments in the Incremental Binding Energies of Protons in Nuclides for which p=(n+1) and p is even 


Nuclide  Increment in IBEn 
71Kr  0.78 
63Ge  0.747 
67Se  0.74 
47Cr  0.5887 
15O  0.253587 
87Ru  1.13687E13 
59Zn  0.009 
75Sr  0.03 
79Zr  0.04 
91Pd  0.2 
43Ti  0.2155 
55Ni  0.2608 
51Fe  0.3004 
31S  0.53894 
39Ca  0.6208 
83Mo  0.7 
35Ar  0.75336 
19Ne  0.80476 
23Mg  0.84017 
7Be  1.0212 
27Si  1.15648 
11C  2.1034 
3He  3.268912 
For the cases of an odd proton number:
The Increments in the Incremental Binding Energies of Protons in Nuclides for which p=(n+1) and p is odd 


Nuclide  Increment in IBEn 
Symb  I2BEp 
5Li  21.779527 
9B  17.44012 
13N  14.013356 
17F  11.527132 
21Na  10.412158 
25Al  9.42156 
29P  8.83683 
41Sc  7.2433 
45V  7.0353 
37K  6.64809 
33Cl  6.58743 
57Cu  6.4711 
49Mn  6.015 
53Co  5.7824 
73Rb  5.4 
69Br  5.3 
65As  5.07 
85Tc  5 
81Nb  4.9 
89Rh  4.8 
77Y  4.675 
61Ga  4.6627 
There are far fewer of these cases than the ones for p<n. There are 95 cases for n even and 94 for n odd. The cumulative frequency distributions are:
For the cases of n odd the average is −4.20682 MeV and for n even it is2.14188 MeV. Thus the two esimates for the effect on binding energy due to the formation of a protonproton spin pair are 4.20682 and 2.14188 MeV. Their average is 3.17435 MeV.
For the proposition that whenever possible protons form spin pairs within nuclei there are only about ten exceptions out of 2605 cases. This is about a 99.4 percent confirmation.
The estimates of the binding energy associated with the formation of a protonproton spin pair suggest its value is in the range of 1 to 3 MeV.
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