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An Estimate of the Binding Energy Due to
the Formation of a Neutron-Proton Spin Pair

One component of the binding energy of a nuclide is that associated with the formation of neutron-proton (np) spin pairs. When a neutron is added to a nuclide and the number of neutrons is less than the number of protons then a neutron-proton spin pair is created. After the number of neutrons matches the number of proton the addition of another neutron does not create an np spin pair and the incremental binding energy is reduced accordingly. This is illustrated for a simple case in the following graph.

Let the binding energy of a nuclide with n neutrons and p protons be denoted as BE(n, p). The incremental binding energy for the the nuclide with n neutrons and p protons is given by

IBE(n, p) = BE(n, p) − BE(n-1, p)

The incremental binding energies for the three cases are shown below.

A simple estimate of the binding energy associated with the formation of an np spin pair can be obtained by comparing IBEn(n+1, n) with IBE(n-1, n); i.e.,

BEnp(n) = IBEn(n-1, n) − IBE(n+1, n)

The values of the estimates are given below.

Estimates of the Binding Energies
Associated with the Formation of
Neutron-Proton Spin Pairs as a
Function of the Number of
Protons in the Nuclide
Number
of Protons
BE of np
(MeV)
4 9.01101
6 8.173065
8 9.079046
10 4.865889
12 5.81653
14 4.83765
16 4.41174
18 3.9523
20 4.9261
22 2.74
24 2.571
26 3.1308
28 3.95
30 2.799
32 2.68
34 2.65
36 2.65
38 2.37

The graph of these data is

The values appear to approach a lower limit of about 2.65 MeV. This value is independent of any error in the mass of the neutron. Since the conventional estimate of the binding energy of a deuteron is about 2.2 MeV this means there is an error the binding energy of the deuteron and hence in the computed mass of the neutron.

The same sort of analysis may be carried out for the incremental binding energies of protons.

Estimates of the Binding Energies
Associated with the Formation of
Neutron-Proton Spin Pairs as a
Function of the Number of
Neutrons in the Nuclide
Number
of Neutrons
BE of np
(MeV)
2 11.215495
4 10.16051
6 9.284028
8 9.607148
10 5.563019
12 6.5228
14 5.52321
16 5.02052
18 4.51285
20 5.296
22 3.3155
24 3.0826
26 3.6725
28 4.3702
30 2.9595
32 2.861
34 2.81
36 2.57
40 3.092
42 3
44 3

When the data for the two estimates of the binding energies associated with the formation of neutron-proton spin pairs are displayed in the same graph there is a near coincidence of values.

The average of the near coincidence of the two curves at the bottom is 2.7035 MeV. Thus the binding energy due strictly to the formation of a neutron-proton spin pair is about 2.70 MeV.

A bit more refined estimate of that binding energy is obtained by fitting functions of the forms

BEnp = c0 + c1/n
BEnp = d0 + d1/p

The results of the regressions are

BEnp = 3.30794 + 20.87419/n
BEnp = 2.06870 + 34.08697/p

The average of the estimates of c0 and d0 is 2.68832 MeV.


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