﻿ The Nature of the Published Figures for the Binding Energies of Nuclides
San José State University

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Thayer Watkins
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The Nature of the Published Figures
for the Binding Energies of Nuclides

The mass of a nuclues is generally less than the sum of the masses of its constituent nucleons. This is called its mass deficit. This mass deficit expressed in energy units through the Einstein formula E=mc² it is called the binding energy of the nuclide.

The problem is in the determination of the mass of a neutron. The mass of a charged partcle such as an ionized atom can be determined by ejecting it into a magnetic fleld and measuring the curvature of its path in relation to its velocity. To determine the mass of the neutral neutron another method must be used.

When a neutron and a proton come together to form a deuteron a gamma ray with energy 2.225 million electron volts (MeV) is emitted. Likewise when a deuteron absorbs a gamma ray of 2.225 MeV energy it breaks apart into a neutron and a proton. That energy is taken to be the binding energy of a deuteron and the mass of a neutron is computed from that value and the measured masses of the deuteron and the proton.

But consider the situation of an electron being absorbed by an ion. The electron would lose 27.2 electron volts (eV) of potential energy; 13.6 eV of that 27.2 eV would go into the emission of a photon of 13.6 eV energy. The other 13.6 eV would go into the kinetic energy of the electron in the atom into which it is absorbed. The potential energy loss is divided exactly equally between the emitted photon and the gain in kinetic energy because the electrostatic force is exactly a function of inverse distance squared. If one estimated the potential energy loss using only the photon energy there would be an erroneous underestimate. This is what is involved with using the energy of the gamma ray emitted when a neutron and a proton form a deuteron. The mass deficit involved in the formation of the deuteron is equivalent to the loss of potential energy. Unfortunately, because the nuclear force is probably not an inverse distance squared force the division of the mass loss between increased kinetic energy and the energy of the emitted gamma photon is not equal and ths is unknown.

Let Δm, ΔK and γ be the mass deficit, the kinetic energy and gamma ray energy, respectively, involve in the formation of a deuteron. Then

#### Δmc² = ΔK + γ

Let mP, mN and mD be the masses of the proton, neutron and deuteron, respectively. Then

Thus

#### mN = mD − mP + Δm and hence mNc² = mDc² − mPc² + ΔK + γ

Let μN be the neutron mass computed ignoring ΔK. Then

#### mNc² = μNc² + ΔK

The true binding energy BE(n, p) of a nuclide of mass M with n neutrons and p protons is

#### BE(n, p) = (nmNc² + pmPc²) − M(n, p)c²

When binding energy is computed using the erroneous μN instead of mN the result is

#### BE*(n, p) = (nμNc² + pmPc²) − M(n, p)c² and hence BE(n, p) = BE*(n, p) + nΔK

It is pretty certain that there are errors in the computed binding energies because there is one nuclide that has a negative computed binding energy. That nuclide is the berylium isotope Be5 with four protons and one neutron. Its computed binding energy is −0.768 MeV. Thus ΔK must be at least 0.768 MeV.

Since no precise estimate of ΔK is known there is no purpose served in trying to adjust the computed values of binding energies to get estimates of the energy equivalents of mass deficits. As they stand the computed binding energies are the amounts of energy required to disassociate the nuclides into their constituent nucleons.