﻿ The Sources of the Binding Energies of Nuclei
San José State University

applet-magic.com
Thayer Watkins
Silicon Valley
USA

The Sources of the
Binding Energies of Nuclei

## Background

The first step is to compute the incremental binding energies of the nucleons

#### IBEP(p, n) = BE(p, n) − BE(p−1, n) IBEN(p, n) = BE(p, n) − BE(p, n−1)

Here is what the graphs of IBEP and IBEN look like for p=12 and n=12 (Magnesium).  The peaks represent the effects of the formation of Proton-Proton and Neutron-Neutron spin pairs. The sharp drops at p=12 and n=12 represent the effects of Proton-Neutron spin pairs no longer being formed.

The troughs is the graphs represent the binding energy without the formation of a spin pair. These may be projected forward to give an estimate of what the binding energies would be without the spin pair formation. The difference between the actual values and the projected values give estimates of the binding energy of the spin pair formations.  Algebraically these values are

#### BEPP(p, n) = IBEP(p, n) − (IBEP(p-1, n) + ½(IBEP(p-1, n) − IBEP(p-3, n)) BENN(p, n) = IBEN(p, n) − (IBEN(p, n-1) + ½(IBEN(p, n-1) − IBEN(p, n-3))

For Mg24 these values are

#### BEPP(12, 12) = 4.17529 MeV BENN(12,12) = 4.068455 MeV

The incremental binding energies beyond p=12 and n=12 are the binding energies without the effects of the formations of proton-neutron spin pairs. These may be estimated by projecting back the peak values, as shown below.  Algebraically these values are

#### BEPN(p, n) = IBEP(p, n) − (IBEP(p+2, n) + ½(IBEP(p+4, n) − IBEP(p+2, n)) BENP(p, n) = IBEN(p, n) − (IBEN(p, n+2) + ½(IBEN(p, n+4) − IBEN(p, n+2))

For Mg24 these values are

#### BEPN = 3.12727 MeV BENP = 2.84947 MeV

The effect of the formation of a Proton-Neutron spin pair should be the same from the two different computations so the best estimate is their average, 2.98837 MeV.

## The Net Binding Energy Due to Nucleonic Interactions

The IBEP for Mg24 includes the binding energies due to the formation of a Proton-Proton spin pair and a Proton-Neutron spin pair as well as the net binding energy due to the nucleon interaction of a proton with the other 11 protons and the 12 neutrons.

The IBEP of the last proton in Mg24 is 11.69287 MeV and the combined binding energy for the two spin pairs is 7.195725 MeV. That leaves 4.497145 MeV as the net effect of the nucleonic interaction of the last proton with the other nucleons. This means that about 61.5 percent of the binding energy holding each proton in the Mg24 nucleus is due to spin pairing.

The binding energy per proton interaction is about 0.20 MeV. This is an order of magnitude smaller than that due to a spin pairing.

The IBEN of the last neutron in Mg24 is 16.53209 MeV and the combined binding energy for the two spin pairs is 7.02476 MeV. That leaves 9.50733 MeV as the net effect of the nucleonic interaction of the last neutron with the other nucleons. This means that about 42.5 percent of the binding energy holding each neutron in the Mg24 nucleus is due to spin pairing.

The binding energy per neutron interaction is about 0.41 MeV. This is also an order of magnitude smaller than that due to a spin pairing.

## Conclusions

A large proportion of the binding energy that holds a nucleus together is due to the formation of spin pairs. The binding energies per interaction are an order of magnitude smaller than those due to spin pairing.