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The First and Second
Differences in Binding Energy


The binding energy of a nucleus in general means the amount of energy that must be supplied to break it up into its constituent components. There are two parts to it. One is the amount of potential energy lost when the constituents are brought together. The other is the energy value of the mass deficit of the nucleus. The mass of a nucleus is less than the sum of the masses of its constituents. Only the second type of binding energy is known for nuclei but it is quite plausible that this type of binding energy is strongly correlated with the total binding energy.

Here is an illustration of how such a correlation might arise. When an electron in an atom drops to a lower energy orbit it loses potential energy but gains kinetic energy. Because the potential energy is based upon the electron being attracted to the nucleus by a force that is proportional to the reciprocal of its orbit radius squared the loss of potential energy is exactly evenly divided between the gain in kinetic energy and the emission of a photon. Thus the loss in the potential energy of the electron is twice the energy of the emitted photon.

The Binding Energy
of a Nucleus

The mass deficit binding energy of a nucleus is made up of two components. One is due to the formation of spin pairs of nucleons and the other is due to the interaction of those nucleons. The spin pairs are proton-proton, proton-neutron and neutron-neutron. Such spin pairing is exclusive in that a proton can spin pair with one other proton and one neutron, but no more. It is likewise for a neutron. The interaction of nucleons however is not exclusive. This component of binding energy is proportional to the number of interactions.

Hereafter binding energy will refer only to the mass deficit binding energy of a nucleus.

The Components
of Binding Energy

Let p and n be the numbers of protons and neutrons, respectively, in a nucleus. The numbers of proton-proton and neutron-neutron spin pairs can be denoted as [p/2] and [n/2], where [z] stands for the largest integer in z. The number of proton-neutron pairs is equal to min(p, n).

The number of proton-proton interactions is equal to p(p-1)/2. Likewise the number of neutron-neutron interactions is n(n-1)/2 and the number of proton-neutron interactions is pn.

The Equation for
Binding Energy

The binding energy BE can be expressed as

BE(p, n) = SPP[p/2] + SPNmin(p, n) + SNN[n/2]
+ cppp(p-1)/2 + cpnpn + cnnn(n-1)/2 + c0

where Sx and cy stand for numerical coefficients.

The model indicates that nuclear binding energy of nuclides is a linear function of these variables. This equation will be referred to later as the unmodified equation. First consideration must be made of special substructures within nuclei.

Alpha Modules

When spin pairs form there will be sequences of the form -p-n-n-p- (or equivalently -n-p-p-n-) and these may close to form rings. The smallest such module is the alpha particle. There may be something special in terms of binding energy about such alpha modules. The number of alpha modules in a nucleus is equal to the minimum of the numbers of proton and neutron pairs; i.e., α = min([p/2], [n/2]).

The equation for binding energy now takes the form

BE(p, n) = Sαα + SPP*[p/2] + SPN*min(p, n) + SNN*[n/2]
+ cppp(p-1)/2 + cpn + cnnn(n-1)/2 + c0

where PP*, PN* and NN* now stand for the spin pairs not in an alpha module.

Regression Estimates

The mass deficit binding energies are known for 2931 nuclei. From these observations the regression equation coefficients can be obtained. Here are those coefficients and their t-ratios (the ratios of the coefficients to their standard deviations).

The Results of Regression Analysis
of the Model of Nuclear Structure
Number of
Alpha Modules
Number of
Proton-Proton Spin Pairs
Not in an Alpha Module
Number of
Neutron-Proton Spin Pairs
Not in an Alpha Module
Number of
Neutron-Neutron Spin Pairs
Not in an Alpha Module

Concerning Results

The coefficient of determination (R²) for this equation is 0.9998825 and the standard error of the estimate is 5.47 MeV. The average binding energy for the nuclides included in the analysis is 1072.6 MeV so the coefficient of variation for the regression equation is 5.47/1072.6=0.0051. Most impressive are the t-ratios. A t-ratio of about 2 is considered statistically significant at the 95 percent level of confidence. The level of confidence for a t-ratio of 923 is beyond imagining.

It is notable that the coefficients for all three of the spin pair formations are roughly equal. They all are larger from what one would expect from the binding energies of small nuclides.

This model does not take into account the shell structures of the nucleons. When a two-way classification of shell structure is incorporated into the model the coefficient of determination is raised to 0.999923. The two-way classification for proton and neutron numbers is 50 or below and above 50.

A three-way classification raises it to 0.9999492, but the regression coefficients for some of the variables are not computed. The three-way classiffication for both proton and neutron numbers is 28 or below, greater than 28 but less than or equal to 82, greater than 82.

For more details on this alpha module ring model of nuclear structure see Nucleus.

First Differences
in Binding Energy

The incremental binding energy of a nucleon is the increase in binding energy resulting from increasing the number of that nucleon by one; i.e.,

Δp(p, n) = BE(p+1, n) − BE(p, n)
Δn(p, n) = BE(p, n+1) − BE(p, n)

These quantities could also be called the forward first differences for protons and neutrons.

Let odd(k) stand for the oddness function; i.e., odd(k)=1 if k is odd and odd(k)=0 if k is even. Let g(z)=1 if z>0 and 0 otherwise. From the unmodified equation it is found that

Δp(p, n) = SPPodd(p) + SPNg(n-p) + cppp + cpnn
Δn(p, n) = SPNg(p-n) + SNNodd(n) + cpnp + cnnn

Second Differences

The second differences are defined as

Δpp(p, n) = Δp(p+1, n) − Δp(p, n)
Δpn(p, n) = Δp(p, n+1) − Δp(p, n)
Δnp(p, n) = Δn(p+1, n) − Δn(p, n)
Δnn(p, n) = Δn(p, n+1) − Δn(p, n)

Later Δpn(p, n) and Δnp(p, n) will be referred to as cross differences.

When the number of neutrons is greater than the number of protons an increase in the number of neutrons does not produce a proton-neutron spin pair. But an increase in the number of protons does. Therefore for the unmodified equation and n>p the above equations reduce to:

Δpp(p, n) = SPP(odd(p+1)−odd(p)) + cpp
Δpn(p, n) = SPN + cpn
Δnp(p, n) = cpn
Δnn(p, n) = SNN(odd(n+1)−odd(n)) + cnn

The function (odd(k+1)−odd(k)) evaluates to −1 if k is odd and +1 if k is even. Let this function be denoted as σ(k). Then the above equations can be expressed as

Δpp(p, n) = SPPσ(p)+ cpp
Δpn(p, n) = SPN + cpn
Δnp(p, n) = cpn
Δnn(p, n) = SNNσ(n) + cnn

Note that Δpn(p, n) is not equal to Δnp(p, n) as might be expected.

When n<p the above equations instead take the form

Δpp(p, n) = SPPσ(p)+ cpp
Δpn(p, n) = cpn
Δnp(p, n) = SPN + cpn
Δnn(p, n) = SNNσ(n) + cnn


The illustrations clearly indicate that the model coefficients should be a function of the shell of the nucleons involved. The plus/minus dependence indicated by the analysis is clearly found in the empirical results. But the analysis indicated that the cross differences both should be constant and that is not the case. While random variation about a constant value would be a constant would be an acceptable approximation the cross differences do not involve random variation. It is all systematic. Something is missing in the model and/or its analysis.

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