San José State University
Thayer Watkins
Silicon Valley
& Tornado Alley

An Investigation of What Binding
Energies of Nuclides Imply About
Their Spatial Structure

This investigation is predicated upon a number of assumptions:

The data for the nuclides made up entirely of alpha particles are likely to yield the most clear cut information on structure. Here is the display of their structural binding energies as a function of the number of alpha particles.

This pattern indicates a shell structure. When a shell is filled the slope decreases because additional alpha particles go into a higher level shell. The bend points in the relationship are at 2 alpha particles and at 14 alpha particles.

It is notable that for this case the values of the structural binding energies are independent of any possible error in the mass of a neutron. The correction for such an error in the binding energy of the nuclide is exactly offset by the correction in the binding energies of the constituent alpha particles. This makes the value near zero for two alpha particles a major puzzle.

The incremental binding energy for an alpha particle in a nuclide is the binding energy of that nuclide less the binding energy of a nuclide with one less alpha particle. Although it appears that the incremental binding energy of an alpha particle is constant over the range of 2 to 14 such is not precisely the case.

The ups and downs and slopes of this pattern have meanings. The most obvious is in relation to the magic numbers which represent filled neutron and proton shells. In the above case the proton number and the neutron number for a nuclide are exactly the same. Here is the previous graph with the magic numbers noted.

The pattern is complicated but the sharp drop after a magic number is reached distorts the pattern. Here is the graph with the data points that immediately follow a magic number left out.

The pattern that emerges is something like what is shown below.

The fact that the line segments slope upward to the right indicates that the particles are attracted to each other. For particles that repel each other the line segments for the shells slope downward to the right.

The Effect of Particles in the Same Shell on Each Other

For forces such as the electrostatic forces the effect of a spherically symmetric distribution of charge on a point outside of that charge is the same as if the total charge were concentrated at the center of the sphere. For a point strictly inside of the sphere the effect is zero. Now consider the effect on a point that is within the distribution of the charge. In this case the effect is that of the charge that is closer to the center of the sphere than the point in question as though it was concentrated at the center of the sphere. For the charge beyond the point in question there is no effect.

For particles in the same shell with a distributed charge their centers would be at the midpoint of the charge distribution so their effect on each other would be equivalent to one half of their charge concentrated at the center of the sphere.

For situations in which the charge distribution is not spherically symmetric or the force is not strictly inversely porportional to the square of the separation distance the ratio would not be one half but instead some value between zero and one.

The Incremental Effect of Particles According to Shell Theory

Suppose there are N particles in a nucleus and another particle is added. The addition creates N new interactions (bonds). Suppose the binding energy per bond is vj so the effect of that additional particle on binding energy is vjN. Now suppose another particle is added to the same shell. It creates N bonds with the previous particles plus a bond with the particle previously added. Because they are in the same shell the effect on each other is not vj but something more of the order ½vj, say ρvj where ρ is between 0 and 1. Likewise for the third particle. The incremental binding energy for the n-th particle added to the same shell is then

ISBE = vj[N + ρ(n-1)]

This suggest a regression equation of the form

ISBE = Dj + cj*(n-1)
where j denotes the shell

This is implemented by creating variables, usually called dummy variables, di,j, such that di,j=1 if the i-th datum is in the j-th shell and 0 otherwise. Also variables of the form ui=di,j*(n-1) are created to capture the effect of the shell on the slope of the relationship between ISBE and the number of alpha particles.

The results of the regression are displayed in terms of the graph of the regression estimate of the ISBE along with the actual value.

The coefficient of determination (R²) for the regression is 0.98770. Although the data points which do not comfortably fit the pattern were left out of the regression the results of the regression are quite satisfactory. They confirm the basic validity of the shell occupancy model for nuclides.

The equations

ISBE = vj[N + ρ(n-1)]
ISBE = Dj + cj*(n-1)
imply that
vjN = Dj
ρvj = cj
and hence
cj/[Dj/N] = ρ

Thus the shell occupancy theory implies that the ratio cj/[Dj/N] should be in the range 0 to 1 and should be the same for every shell.

These implications can be tested using regression analysis. Simple linear regression lines of the form ISBE = D + c(n-1) for obtained for the three different shells. (There were only two data points for the first shell.) The results were:

Alpha ShellDND/Ncc/[D/N]
4 to 77.98041932.6601396670.6679690.251102981
8 to 146.4454813497 0.920783050.282159070.306433823
15 to 252.404192667140.1717280480.052896970.30802755

The ratios are of the right order of magnitude and very close to the same, (only a 0.5 of 1% differenc), for the third and fourth shells and not too far off for the second shell even though there are only two data points to estimate the relationship.

(To be continued.)

The Incremental Binding Energies
of Nuclides with Extra Neutrons

Below is the graph of the incremental binding energies of alpha particles in nuclides which would consist of alpha particles plus one extra neutron.

The similarity of this pattern to the one for nuclides consisting entirely of alpha particles is shown by superimposing the two graphs, as shown below.

The similarity is more dramatic if the line for the alpha plus one neutron nuclides is shifted backward by one alpha particle.

In some places the shift made the fit of the two curves better and in other places it disturbed an existing fit.

(To be continued.)

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