When protons and neutrons form a nucleus the mass of the nucleus is less than the sum of the masses of the particles which make it up. Gamma rays are given off in the process of formation and the nucleus is broken up only if the nucleus absorbs a gamma ray of the same or greater energy than the one given off in its formation. The phenomenon of mass deficits has been known and accepted for many decades without much comment on what an enigma it is.

The case of the deuteron, the combination of a proton and a neutron, is easy to visualize. It was first noticed in the 1930's that if gamma rays of at least 2.22457 MeV irradiated deuterium then deuterons would dissociate into free neutrons and protons. Later it was noticed that if slow neutrons were brought in contact with protons deuterons would form, accompanied by the emission of gamma rays of 2.22457 MeV energy.

When particles come together they lose potential energy which is transformed into an increase in kinetic energy. When electrons move into a lower energy state they lose more potential energy than they gain in kinetic energy and the difference goes into the emission of photons. In nuclear processes it is a puzzle as to why the system must lose mass as well as potential energy.

According to Einstein's General Theory of Relativity mass warps space in its vicinity. Creatures of a three dimensional world cannot envision this warping, but it is easy to envision it for a two dimensional world. In a plane the effect of mass would be the creation of a dimple. One could say that the dimple is the mass. If there was a set of straight coordinate lines in the plane before the creation of a dimple the coordinate lines would be distorted by the creation of the dimple. Light rays traveling along those coordinate lines would deviate when passing through the dimple. The dimple would appear to have attracted those light rays.

The opposite of a dimple in the plane would be a bump. The bump would appear to distort the coordinate lines in the opposite way leading to the appearance that light rays are repelled by the bump.

If two dimples were moved into close proximity they would appear to create a small bump between them. This is better illustrated in the one dimensional world of a line. Below is depicted a particle of positive mass in that world.

A particle of negative mass is shown for comparison.

There are several conceptions of negative mass and a few physical models. For more on the concepts and models of negative mass see Negative Mass.

Now consider two particles of positive mass separated some distance from each other.

If they are moved closer together there appears between them something in the nature of a particle of negative mass; i.e.,

When the particles are still closer together the space between looks even more like a particle of negative mass. It is smaller in magnitude however.

Thus the mass of the two-particle combination is the sum of the masses of the two particles less the magnitude of the negative mass particle created between them by their proximity. The mass of the negative particle is the mass deficit of the combination of particles. If the particles are separated the negative mass particle disappears.

In the two dimensional world of the plane when two dimples are brought close together they create an ambiguous entity between them. It is in the nature of a mountain pass, which is a peak in one direction and a valley in the other direction. However if three dimples are brought together there is something more in the nature of a bump that is created between them. This suggests that if in a three dimensional world four masses are brought together there is created something in the nature of a negative mass between them. It is notable that when a deuteron is formed from two particles the mass deficit is small, 0.1 of 1 percent of the mass of the constituent particles. When three particles are brought together to form tritium H-3 or He-3 the mass deficit is larger but still only 0.2 of 1 percent. However when an alpha particle is formed from four particles the mass deficit increases to about 0.7 of 1 percent, which is the approximate mass deficit per nucleon for the heavier nuclides.

This construction of the mass deficit through the formation of negative mass created by bringing particles into proximity with one another suggests a number of things. First it suggests that there is a close tie between proximity of the particles and the mass deficit. But there is also a close tie between proximity and the potential energy of the combination. Perhaps it could be said that the mass deficit and the loss of potential energy due to proximity are simply two ways of the measuring the same thing. It also suggests that the formation of negative mass at the center of mass of the particles does not much affect the dynamics of the system created by the particles.

The Energy Accounting of a Particle System

A system has kinetic and potential energy. It also has the energy incorporated in the masses of its particles. Leaving aside the mass energy for the moment, when a system changes there are changes in its kinetic energy K and its potential energy V. If these energies are not exactly offsetting then the difference has to be made up of the emission or absorption of a quantum of radiation. Let γ denote the energy of this radiation. Conservation of energy then requires

γ + ΔK + ΔV = 0
or, equivalently
γ = −(ΔK + ΔV)
or
−ΔV = γ + ΔK

In the formation of a deuteron the usual assumption is that

mass deficit = γ

This is unjustified. The previous material suggests that the proper accounting is

mass deficit = loss in potential energy = −V
and thus
mass deficit = γ + ΔK

Thus the mass deficit of the deuteron is larger than the 2.22457 MeV of the gamma ray emitted upon its formation. The mass of the neutron m_n is computed from the mass of the deuteron m_d and the mass of the proton m_n by

m_n = m_d + γ − m_p
when it should be computed as
m_n = m_d + γ + ΔK − m_p

Thus the mass of the neutron is underestimated and consequently the mass deficits of all nuclides are similarly underestimated. The crucial bit of information is how much does the kinetic energy of the nucleons of the deuteron increase upon the formation of a deuteron.

For material on this topic see Two Particle Spectrum.

(To be continued.)