San José State University
Thayer Watkins
Silicon Valley
& Tornado Alley

A Comparison of the Binding Energies Due to the
Formation of the Three Types of Nucleon Spin Pairs

The formation of spin pairs is a crucial element of the structure of nuclei. There are neutron-neutron, proton-proton and neutron-proton spin pairs. The binding energy involved in the formation of any one of them is on the order of 3 million electron volts (MeV), which is far greater than the interaction of two nucleons through the so-called nuclear strong force, which is on the order of 0.5 MeV. However spin pair formation is exclusive in the sense that a neutron can form a spin pair\ with only one other neutron and with one proton. The same applies to a proton. Interaction through the strong force is not exclusive so the interaction of a nucleon with a number of other nucleons in a nucleus may outweigh the effect of spin pairing.

Estimates of the binding energy due to spin pairings can be obtained by looking at incremental binding energies. However those incremental binding energies may include that due to any structural rearrangement due to the formation of the spin pair as well as that due to the formation itself. Previous studies attempted to estimate the binding energies of the three types of spin pairs due strictly to the pair formations, Those values and their sources are:

Binding Energies Due to Spin Pair Formations
Spin PairBinding
St. Dev.
of Estimate

The estimate for a neutron-proton spin pair is an average of two methods of estimation, one value of which is 3.3 MeV. Therefore the binding energies for all three spin pairs could be on the order of 3.5 MeV.

There is some speculation that all three values might be equal. There may be a way to estimate the differences. When a neutron is added to a nuclide with an odd number of neutrons there is an increment due to the formation of a neutron-neutron spin pair. When a neutron is added to a nuclide in which there are fewer neutrons than protons a neutron-proton spin pair is formed. But if the number of neutrons is greater than or equal to the number of protons then no neutron-proton spin pair is formed. So when the number of neutrons is equal to the number of protons plus one there is an increment in binding energy due to the formation of a neutron-neutron spin pair and a decrement due to the non-formation of a neutron-proton spin pair. Here is an illustration for the the isotopes of Bromine (p=35) .

The incremental binding energy for the 36th neutron is almost the same as that for the 35th neutron because of the two nearly offsetting effects. A similar effect is found for Rubidium (p=37)

Below is the table of the differences in incremental bindinding energy n=p+1 and n=p.

Estimates of the Difference in the Binding Energies
Due to the Formations of Neutron-Neutron and
Neutron-Proton Spin Pairs
Nuclide protons neutrons Binding
11B 5 6 4.2042
15N 7 8 -0.620795
19F 9 10 -0.401145
23Na 11 12 1.98656
27Al 13 14 0.63909
31P 15 16 -0.74602
35Cl 17 18 0.33249
39K 19 20 0.43212
43Sc 21 22 -0.9384
47V 23 24 0.8633
51Mn 25 26 0.6855
55Co 27 28 0.4029
59Cu 29 30 -1.3264
63Ga 31 32 -0.0033
67As 33 34 0.13
71Br 35 36 0.21
75Rb 37 38 0.425
79Y 39 40 0.275
83Nb 41 42 0.3
87Tc 43 44 -0.1
91Rh 45 46 0
95Ag 47 48 0.9
99In 49 50 0.2

Here is a graph of the data.

There are a number of outliers which may have no explanations. If the outliers of negative value and greater than 1 in magnitude are eliminated the average for those that are left is 0.33 MeV and they have a standard deviation of 0.32 MeV. This is consistent with a value of 3.19 MeV for neutron-neutron spin pairs and about 2.88 MeV for a neutron-proton spin pair.

A similar type of analysis might lead to an estimate in the difference in the binding energy between that due the formation of a proton-proton spin pair and a neutron-proton spin pair. However the situation is a bit more complicated.

The increment for the 18th proton is lower than for the other protons but not very much lower and the value seems to fit into some systematic variation rather being something special for p=n+1.

(To be continued.)

HOME PAGE OF applet-magic
HOME PAGE OF Thayer Watkins