﻿ The Binding Energy Associated with the Formation of a Neutron-Proton Spin Pair
San José State University

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Thayer Watkins
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The Binding Energy Associated
with the Formation of a
Neutron-Proton Spin Pair

The incremental binding energy of a neutron (IBEN) is the effect on binding energy of having one more neutron in a nuclide. It includes the effect of the interaction through the nuclear strong force of that additional neutron with the other nucleon in the nucleus. It also include the effects of the formation of a neutron-neutron spin pair and/or a neutron-proton spin pair. A neutron-neutron spin pair is formed only if the additional neutron is added to a nuclide with an odd number of neutrons. A neutron-proton spin pair is formed only if the additonal neutron is added to a nuclide in which there are fewer neutrons than protons. The effect of the formation of a neutron-neutron spin pair has been investigated elsewhere. This material concerns the estimation of effect of the formation of a neutron-proton spin pair.

The effect of the formation of a neutron-proton spin pair can be estimated by comparing the IBEN for the case in which the number of neutrons is one less than the number of proton with the case in which the number of neutrons is one more than the number of protons; i.e.,

#### ΔNP = IBEN(n=p-1, p) − IBEN(n=p+1, p)

The results of this computation are shown in the graph below.

The value of ΔNP appears to be a function of the shell for the number of protons. A previous study based upon one observation per shell concluded that the relationship was of the form

#### ΔNP = c0 + c1/z

where z is a variable such as number of neutrons. From the above graph it applears that the relationship is of the form

#### ΔNP = c0 + c1/S(z) + uS

where S is the shell number and uS is a random variable whose variance depends upon the shell number S.

The variation in the values indicates that something more than the effect of the formation of a neutron-proton spin pair is involved. When such a pair is formed the other nucleons may undergo a rearrangement that involves a change in energy. For smaller nuclei such a rearrangement may be more substantial than for larger nuclei and thus the binding energy effects larger.

If one wanted an estimate of the binding energy due to the formation of a neutron-proton spin pair the value in the 51 to 82 shell appears to be the most relevant; i.e., about 2.7 MeV.

The estimate of ΔNP might be refined by projecting the IBEN forward one unit from n=p-1 to n=p and projecting the IBEN backwards one unit from n=p+1 to n=p. The amounts of the projection is based upon the slopes of the relationships. The slopes are computed for a two unit change to avoid the effect of a neutron-neutron spin pair formation. Therefore

#### IBEN0 = IBEN(n=p-1, p) + ½[IBEN(n=p-1, p) − IBBEN(n=p-3, p)] = 1.5IBEN(n=p-1, p) − 0.5IBEN(n=p-3, p) and IBEN1 = IBEN(n=p+1, p) − ½[IBEN(n=p+3, p) − IBBEN(n=p+1, p)] = 1.5IBEN(n=p+1, p) − 0.5IBEN(n=p+3, p) and ΔNP = IBEN0 − IBEN1

The graph of the results of this computation are shown below

The extreme value for n=24 appears to be a result of 24 being four more and four less than the magic numbers of 20 and 28.