San José State University

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 The Structural Binding Energies of Nuclides and the Effect of an Error in the Mass of a Neutron on Their Computation

Nuclei are apparently composed of neutrons and protons, but their measured masses are less than the combined masses of their constituent neutrons and protons. These differences are called their mass deficits and when expressed in energy units via the Einstein equation of E=mc² they are called their binding energies.

Let E stand for the mass of a nucleus or a constituent expressed in energy units and let #n and #p stand for the numbers of neutrons and protons, respectively, in the nucleus. Then the binding energy BE of the nucleus is given by

BE = (#n·En + #p·Ep) − E

The constituent nucleons of a nucleus may be combined into substructures such as alpha particles. This would mean that its binding energy is composed of two parts: that due to the formation of substructures and that due to the combination of those substructures into the nucleus. This latter component can be called the structural binding energy of the nucleus.

Consider the nuclides that could be composed entirely of alpha particles. Hereafter these are called the alpha nuclides. Let #a stand for the numbers of alpha particles in any such nuclide and let SBE denote its structural binding energy. Then

SBE = BE − #a·BEa

where BEa is the binding energy of an alpha particle; i.e., that of the Helium 4 nuclide 28.29567 million electron volts (MeV).

Now suppose the mass of the neutron has been underestimated by an amount Δ and should be En+Δ rather than En. Then the corrected structural binding energy SBE' would be

SBE' = [(#n·(En+Δ + #p·Ep) − E] − #a·(Ea+2Δ) which reduces to SBE' = (#n·En + #p·Ep) − E + #nΔ − #a·Ea − 2#aΔ which, since #n=2#a, reduces to SBE' = SBE

Thus the computed structural binding energies of alpha nuclides are independent of any error in the mass of the neutron.

The Measurment of Nuclide Masses and the Deduction of the Mass of the Neutron

The mass of a nuclide is measured by ejecting large numbers of it into a magnetic field and measuring the radius of the circular trajectories they follow. That radius depends upon the electrostatic charge of the nuclide, the velocities, the magnetic field strength normal to path of the nuclei and the mass of the nuclide. This can be done for all nuclides with a charge. That leaves out the neutron. The mass of the neutron must be deduced rather than measured.

The deuteron is formed from a neutron and a proton. When a deuteron is formed a gamma ray with energy of 2.22457 million electron volts (MeV) is emitted. On the other hand, if deuterons are subjected to gamma rays of energy of at least 2.22457 MeV they disintegrate. Without much of a theoretical basis physicists long ago took the binding energy of the deuteron to be equal to the energy of the gamma rays involved in the formation or disassociation of deuterons. Since

BEd = En + Ep − Edit follows that En = Ed − Ep + BEd

According to the conventional estimate of the mass of the neutron

BEd = γ and hence En = Ed − Ep + γ

where γ is the energy of the gamma ray emitted upon the formation of an alpha particle, 2.22457 MeV.

The flaw in this line of reasoning is that there is no reason to believe that the binding energy of the deuteron should be equal to the energy of the emitted gamma ray. Generally the binding energy of a nuclide is the loss of potential energy involved in its formation. When particles lose potential energy through a change in quantum state parts of the energy goes into an increase in kinetic energy and the rest is emitted as a photon. For an electron in an atom exactly half of loss of potential energy in a change of state goes into increased kinetic energy and half into the emitted photon. This is as a result of the electrostatic force obeying a strictly inverse squared law. The nuclear strong force does not have such a force form but there is no reason to believe that none of the loss of potential energy goes into an increase in kinetic energy. The deduced mass of the neutron is in error by the amount of kinetic energy, rotational and radial, in a deuteron. An estimate of the amount of kinetic energy in a deuteron was made using the Virial Theorem. This estimate was slightly less than 1 MeV.

However, although there is good reason to believe the mass of the neutron is in error, the computed structural binding energies of the alpha nuclides are unaffected by that error.

The Structural Binding Energies of the Alpha Nuclides

Here are the structural binding energies of the alpha nuclides along with the incremental structural binding energies.

 #Alphas SBE(MeV) ISBE(MeV) 1 0 0 2 -0.091838 -0.091838 3 7.274706 7.366544 4 14.43664 7.161934 5 19.166489 4.729849 6 28.482846 9.316357 7 38.467172 9.984326 8 45.415268 6.948096 9 52.054634 6.639366 10 59.09526 7.040626 11 64.22229 5.12703 12 71.913912 7.691622 13 79.853238 7.939326 14 87.848564 7.995326 15 90.55689 2.708326 16 93.219216 2.662326 17 95.373542 2.154326 18 97.777868 2.404326 19 100.482194 2.704326 20 103.88652 3.404326 21 106.690846 2.804326 22 108.895172 2.204326 23 111.299498 2.404326 24 114.303824 3.004326 25 117.50815 3.204326

The graphs of the data are shown below.

This graph displays an evident shell structure. The graph of the incremental structural binding energies of the alpha particles shows that the shell structure is more complex than the previous graph suggests.

Other Substructures in Nuclei

Although alpha particles are the most important substructures there can also be nucleonic pairs; i.e., neutron pairs, proton pairs or a neutron-proton pair. A neutron-proton pair is a deuteron. There can be only one deuteron because two would constitute an alpha particle.

If a nuclide could have a deuteron in addition to alpha particles and the binding energy of a deuteron is deducted from the binding energy of nuclide in computing its structural binding energy this value is independent of any error in the mass of the neutron for the same reason as for the structural binding energies of the alpha nuclides.

The graphs for the structural binding energies of the alpha plus one deuteron nuclides are shown below.

Again there is a shell structure.

There is a similar pattern here as for the alpha nuclides.

Neutron-Neutron Pairs and Proton-Proton Pairs

There are no obvious measurements of the binding energies of neutron pairs or proton pairs but these would need to be estimated to compute the structural binding energies of nuclides which contain them. Any error in their estimation would result in a corresponding error in the computed structural binding energies. Suppose suitable values are found for these pairs. The matter of the effect of any error in the mass of a neutron could then be reduced to a simple matter.

A nuclide is composed of alpha particles, one or none deuteron, neutron pairs or proton pairs (but not both) and possibly a singleton neutron or proton (but not both). Let #n' stand for the number of neutrons in the substructures (alpha particles and nucleonic pairs) in a nuclide. The effect of an error in the mass of the neutron, Δ, on the structural binding energy is then as follows:

SBE' = SBE + #n·Δ − #n'·Δ or equivalently SBE' = SBE + (#n−#n')Δ

Since (#n−#n') is either zero or one depending on whether or not there is a singleton neutron in the nuclide the computed structural binding energy of a nuclide is at most in error by Δ.