﻿ The Binding Energies of Nuclei: Ostensibly and In Fact
San José State University

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The Binding Energies of Nuclei:
Ostensibly and In Fact

## Background

It was found that generally the mass of a nucleus is less than the sum of the masses of its constituent protons and neutrons. This is called its mass deficit. This deficit converted into energy using the Einstein formula of E=mc² is called its binding energy. Ostensibly this is what binding energy is: The energy equivalent of a mass deficit. In fact the matter is a bit more complicated because of the way the mass of a neutron is computed. The energies equivalent to the mass deficits are not the ideal quantities for the physical analysis of the nuclear structure of nuclei. What is missing is something related to the loss of potential energy when the constituent nucleons came together to form a nucleus.

The binding energy of a nucleus is also said to be the energy required to break it apart into its constituent nucleons. It should not be this because the energy required to break apart a nucleus should also incude the energy lost when it ts formed as well as the energy needed to replace the missing mass.

Consider the situation of an electron in an atom. When an electron drops to a lower energy level and its potential energy decreases by an amount E its kinetic energy increases by exactly ½E. This hold for a force which is proportional to inverse separation distance squared. The other ½E went into the spark of radiation emitted in the transition. Thus the amount of energy required to reverse the transition of the electron is ½E.

## The Deuteron

The mass of any charged particle can be meaured by injecting it into a magnetic field and observing the curvature of its path. But the mass of a neutron cannot be measured in that way. However when a proton and neutron come together to form a deuteron a gamma ray of 2.224573 million electron volts (MeV) of energy is given off. For convenience let this energy amount be denoted as γ. When gamma rays of at least this energy impinge upon deuterons the deuterons are disassociated into protons and neutrons.

Let the masses of a proton, neutron and deuteron be denoted by P, N and D, repectively. Furthermore let the mass deficit of a deuteron be denoted by ΔD and the energy lost upon the formation of a deuteron be ΔE. Thus

Properly then

#### N = D − P + ΔD = (D + ΔD) − P

But ΔD is not known. What is known is γ so one estimate of the mass of a neutron is:

#### N* = D − P + γ = (D + γ) − P

This was an early source of an estimate of the mass of neutron. It would in error but it is close to N. Actually

#### N* = D − P + ΔD + ΔE = N + ΔE

Thus this estimate of the mass of a neutron is an overestimate but it is too large only by something on the order of 1 MeV. That is 1 MeV for a quantity of approximately 1000 MeV.

## Neutron Decay

An isolated neutron decays into a proton and an electron. The mass of an electron is 0.5109906 MeV. Thus a neutron must be at least 0.5109906 MeV more massive than a proton. The half life of thia decay is about 15 minutes. Within nuclei neutrons are generally stable.

## The Current ConventionalEstimate of the Massof a Neutron

The current conventional estimate of the mass of a neutron is 939.56563 million electron volts (MeV). The difference between this value and the mass of a proton is

#### mass of neutron = 939.56563 MeV mass of proton = 938.27231 MeV difference = 1.29332 MeV

This difference is about 58.1 percent of γ. (2.224573 MeV).

## Binding Energy

For a nucleus with p protons and n neutrons of mass M(p, n) its binding energy properly is

#### BE(p, n) = Pp + nN − M(p, n)

But what is reported as the binding energy is based upon the estimate N^ of the mass of a neutron; i.e.,

#### BE^(p, n) = Pp + N^n − M(p, n)

However, if there is any error in the mass of a neutron, saymε then

#### BE^(p, n) = Pp + (N + ε)n − M(p, n) which reduces to BE^(p, n) = BE(p, n) + ε·n

Thus the conventional estimate of the binding energy of a nucleus is the energy equivalent of its mass deficit plus a term which the number of neutron times any error in the mass of a neutron.

## The Energy Required toBreak a Nucleus Apart

Consider the binding energy based upon N*, the mass of the neutron based upon γ; i.e.,

#### N* = D − P + ΔD + ΔE = N + and BE*(p, n) = Pp + (N + ½E)n − M(p, n) which reduces to BE*(p, n) = BE(p, n) + ½E·n

Thus BE*(p, n) is equal to the energy equivalence of the mass deficit of a nucleus plus a term which approximates the energy lost when the nucleus was formed from its constituent nucleons. This could be labled the total binding energy of a nucleus. For physical analysis this is a more relevant quantity than just the energy equivalent of the mass deficit.

## The Binding Energy of a Deuteron

The conventional figure for the binding energy of a deuteron is 2.224573 MeV, the value denoted above as γ. This means that the conventional value for the binding energy of the deuteron is not consistent with the conventional figure for the mass of a neutron. Thus the conventional figures for the binding energies of nuclei may or may not be the energy equivalents of their mass deficits.

The conventional mass of an alpha particle is 3,727.3 MeV. The mass of two protons and two neutrons, baed upon the conventional figures for the masses of a proton and neutron, is 3,7556.7588 MeV. The binding energy of an apha particle according to this computation is then 28.8 MeV. The conventional figure for the binding energy of an alpha particle is 28.3 MeV.

## Conclusions

The reported binding energy of a nuclide is a close approximation of the energy equivalence of its mass deficit. But a more relevant quantity than merely the energy equivalent of the mass deficit of a nucleus is the energy that would be required to break it apart. That energy includes the net loss of energy when it is formed as well as the energy equivalence of its mass deficit.

The conventional figures on binding energy may not be entirely consistent with the conventional estimate of the mass of a neutron.