San José State University

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 The Specification of a Statistical Function for the Binding Energies of Nuclides

There are almost three thousand different nuclides stable enough for their masses to be measured. The difference between the mass of a nuclide and the masses of its constituent protons and neutrons is called its mass deficit. The energy equivalent of this mass deficit is called the binding energy. The average binding energy of the 2931 nuclides is 1072 Mev. The average numbers of protons and neutrons is 56 and 78, respectively. This means that the average nuclide is an isotope of barium, but the binding energy of Ba-134 is not equal to the binding energy of 1072 because of the nonlinear nature of the relationship between the number of protons and neutrons and the binding energy.

The variance of the binding energies of the nuclides is 254,825.6 (MeV)² and the standard deviation is 504.8 MeV. That is a coefficient of variation of 47.1 percent.

The protons and neutrons in a nucleus do not exist as separate particles. They form pairs; proton-neutron pairs (deuterons), neutoron-neutron pairs and, conceivably, proton-proton pairs. There is considerable evidence that the protons and neutrons form alpha particles within a nucleus whenever possible. It is also possible that the neutrons in a nucleus may disassociate into protons and electrons.

The binding energy of an alpha particle is 28.295674 MeV. Suppose this amount is deducted from the binding energy of a nuclide for every alpha particle it could contain. This could be called the nuclides excess binding energy. The variance of this excess binding energy is 19,306.7 MeV. Thus the binding energy arising from the formation of alpha particles accounts for 0.924236 of the variation in the binding energies of nuclides.

The relevant binding energy for a nucleus is its binding energy in excess of the binding energies of the substructures which exist within it. For the nuclides which could contain an integral number of alpha particles, hereafter these are called alpha nuclides, the excess binding energy is easily computed. The graph of this excess binding energy versus the number of alpha particles it contains is displayed below. This indicates a shell structure. There is virtually zero excess binding energy for the case of 0, 1, 2 alpha particles. The first two are zero by definition, the third is a small but nonzero amount. This means that the first shell can contain two alpha particles. The third up to the fourteenth go into a second shell. Beyond fourteen the alpha particles go into a third shell. There are only 25 alpha nuclides sufficiently stable to have their masses measured. We do not know for sure if the third shell is completely filled with eleven alpha particles.

## The Use of Regression Analysis to Estimate the Parameter of a Broken Line Function

Let #α be the number of alpha particles. A ramp function u(x) is defined as

#### u(x) = 0 if x<0 u(0) = 0 u(x)= x if x> 0

For the estimation of the slopes of the line segment define

#### u1 = u(#α−2) u2 = u(#α−14)

The regression equation then of the form

#### δBE = c0#α + c2u1 + c2u2

where δBE is the binding energy less the binding energy of the constituent alpha particles; i.e., BE − #α(28.295674 MeV).

The value of c1 is the increase in the slope when going from the first shell to the second. Likewise c2 is the net increase in the slope when going from the second to the third shell.

The regression results are

#### δBE = −0.06887#α + 7.375948u(#α-2) −4.62492u(#α-14)        [-0.06]          [6.5]          [-39.5]     R² = 0.992411 df = 22

These results mean that the binding energy added per alpha particle in the first shell is -0.06887 MeV (a number not significantly different from zero. For the second shell the number is (7.375948−0.06887)=7.307078 Mev. For the third shell it is (7.307078−4.62492)=2.682162 MeV.

## Going Beyond 25 Alpha Particles

As mentioned above, we do not know for sure if the third shell is completely filled with eleven alpha particles. To answer this question the binding energies of all nuclides which contain an integral number of alpha particles plus four additional neutrons were computed and the results plotted. This indicates that the full occupancy of the third shell of alpha particles is indeed eleven.

The same technique for estimating the relationship between excess binding energy and the number of alpha particles can be applied, in principle, to the data for the excess binding energies of the alpha-plus-4-neutrons nuclides. There is however an additional complication. The effect on binding energy of the four neutrons varies with the number of alpha particles. Below is shown the differences in binding energies of the alpha+4n and the alpha nuclides. This relationship appears to have a bendpoint at 5 alpha particles and another one at 23 alpha particles. The bump over the range of 10 to 14 alpha particles shows up in other relationships and is apparently associated with some structural feature of nuclei.

The regression equation is

#### δBE = −5.44334 + 8.555668#α + 5.4663u(#α-2) −5.29715u(#α-5) [-3.9]       [4.7]          [-9.3]          [2.6] −5.30692u(#α-14) − 2.69494u(#α-23) −3.90285u(#α-25) [-23.6]          [-3.6]         [-3.9]          R² = 0.999393 df = 22

In order to investigate possible critical levels beyond 25 alpha particles the excess binding energies of the alpha-plus-24-neutron nuclides were computed. The graph of the values were: The excess binding energy reaches a peak at 29 alpha particles. For this to be a bendpoint the incremental binding energy needs to drop in value at that number of alpha particles. There are two points at which the incremental binding energy drops sharply. One is at 25 alpha particles and the other is at 29 alpha particles.

## Interpretation of the Critical Levels, the Bendpoints

The critical number of alpha particles for the change in the slope of the relationship between excess binding energy and the number of alpha particles was at 2, 14 and 25 for the alpha nuclides. For the alpha+4n nuclides there was a critical point at 23 alpha particles. For the alpha+24n nuclides there was a critical point at 29 alpha particles. The number of neutrons in the nuclide at the critical points is then 4, 28 and 50. The critical point for the alpha+4n nuclides is at 2(23)+4=50. For the critical point at 29 alpha particles for the alpha=24n nuclides the number of neutrons is 2(29)+24=82. These are all magic numbers of neutrons. The bendpoint for the alpha+4n nuclides at 5 is probably a misinterpretation of where the critical point is. The bump on the linear segment corresponds to an upward shift at 20 neutrons and a downward shift at 28. It is easier to discern critical points in terms of changes in the incremental binding energy, as shown below. Clearly the critical point came at 3 alpha particle so the number of neutrons at the critical point is 2(3)+4=10. There is also a critical point at 8 alpha particle which corresponds to 2(8)+4=20, a magic number. There is rise after the decline, shifting the relationship upwards until 12 alpha particles are reached. This corresponds to 2(12)+4=28 neutrons, a magic number. The decline continues until 16 alpha particles is reached. This corresponds to 36 neutrons, which is not a magic number but might represent something significant. The incremental excess binding energy is about constant until 23 alpha particles (50 neutrons) is reached.

It is now worthwhile to look at the incremental binding energies of the alpha nuclides. The obvious big drops come at 2 and 14 alpha particles or 4 and 28 neutrons but there are minor critical points within the interval of 2 to 14 alpha particles. Something critical happens at 4 and 5 alpha particles and also at 7. Four and seven alpha particles correspond to 8 and 14 neutrons. Likewise there is a drop and rise at 10 alpha particles (20 neutrons) and a drop at 14 alpha particles (28 neutrons). Thus there are bends and breaks when the number of neutrons reaches a magic number level. However, in some cases the shifts take place over such a small interval they are not quantitatively important.

The binding energy (BE) of a nuclide is a function only of the number of its protons P and the number of its neutrons N; i.e., BE(N, P). These number can be reduced to an equivalent set of numbers. The most important is the number of alpha particles #α.

#### #α = [min(N,P)/2]

[z] is the whole number contained in z; i.e., the integer part of z.

As state previously the binding energy due solely to the formation of alpha particles explains 92.42 percent of the variation in binding energies of nearly three thousand nuclides. Therefore the problem is reduced to explaining the binding energy in excess of that which is accounted for by the formatio of the alpha particles.

A regression of excess binding energies on #α with bend points at #α=2, #α=14; #α=25 and #α=41 with no intercept or jumps at the critical points explains 74.35 percent of the variation in the excess binding energy. This means that 94.36 percent of the variation in binding energy is explained by the bent line function of #α.

The regression just mentioned does not take into account all of the information on the numbers of neutrons and protons. A regression on the number of neutrons and protons with critical points at 2, 28, 50, 82 and 126 (no nuclide reaches a proton number level of 126) with the possibility of jumps as well as bends at the critical points explains 85.65 percent of the variation in excess binding energy and 99.9449 percent of the variation in binding energy. The standard error of the estimate for this last regression is 11.9 MeV. The ratio of this value to the average binding energy of 1071.9 is 0.0111.

If critical points are included for 6, 8, 14 and 20 as well as 2, 28, 50, 82 and 126 the regression explains 99.9456 percent of the variation in binding energy, a very marginal increase in the coefficient of determination over the above value of 99.9449. The standard error of the estimate is 11.85 MeV.

To go above an R² value of 0.999456 some more detailed analysis is needed. The incremental binding energy function for neutron pairs can be specified on the basis of the shell occupancy model. Let #nn be the number of neutron pairs in a particular shell. Then

#### ΔBE = c0 −c1#nn + c2(#nn)²

Here are illustrations of this phenomenon.  The sawtooth character of the relationships has to do with the formation of neutron pairs. Let #n be the number of neutrons in a shell and hence the number of pairs #nn is [#n/2]. Let U(#n) be a variable that is +1 if #n is odd and 0 if it is even. The form of the relationship for ΔBE is then

#### ΔBE = d0 −d1#n + d2(#n)² + d3U(#n)

The binding energy is then the sum of the values of ΔBE from 0 up to a particular level #nn. This means that

#### BE = BE(Next lower full shell) + e0#n − e1(#n)² + e2(#nn)³ + e3U(#n)

The magnitude of the coefficients vary with the shell. Since all of the critical levels are even numbers the oddness or evenness (parity) of the number of neutrons or protons in a shell will depend only upon the oddness or evenness of N or P of the nuclide.

The inclusion of two variables, U(N) and U(P) raises the coefficient of determination only slightly, from 0.999456 to 0.999457.

The inclusion of quadratic terms for the number of neutrons in the shells for 28, 50, 82 and 126 and for the number of protons in the shells for 28, 50 and 82 raises the coefficient of determination to 0.999498. The standard error of the estimate is reduced to 11.4 MeV. The inclusion of cubic terms as well as quadratic raises the coefficient of determination for the regression only marginally to 0.999503 and reduces the standard error of the estimate to 11.358 MeV.

(To be continued.)

## Thoughts Toward a Possible Alternate Formulation

The above formulation is a purely shell occupancy model of nuclei. An alternative approach would be to define variables on the basis of the neutron and proton numbers. The number of alpha particles #α would be the first such structural variable. Some other such variables follow.

Whether or not the nuclide contains a proton-neutron (p-n) pair in addition to the alpha particles is expressed as a {0, 1} value #d by

#### #d = 1 if N>2(#α) and P>2(#α) #d = 0 otherwise

It cannot be that both N>2(#α)+1 and P>2(#α)+1 because if that were the case there would be another alpha particle.

Let #nn and #pp be the additional neutron pairs or proton pairs beyond those contained in the alpha particles of the nuclide. At least one of the two quantities #nn and #pp must be zero. No deduction is made for a proton-neutron pair because the evidence is that the formation of a proton-neutron pair does not preclude those particles from forming a neutron pair or a proton pair. The values of the additional neutrons n and the additional protons p beyond those in the alpha particles are given by

#### n = N − 2(#α) and p = P − 2(#α)

The number of additional neutron and proton pairs are then given by

#### #nn = [n/2] and #pp = [p/2]

There might be a singleton neutron or a single proton (but not both). Let ##n and ##p be {0, 1} variables indicating the existence of a singleton neutron or singleton proton where

#### ##n = n − 2#nn and ##p = p − 2#pp

The binding energy of a nuclide would be the sum of four functions

• The alpha particle component A(#α), which in turn is composed of the binding energy due to the formation of alpha particles and the excess binding energy function. The excess binding energy function displays a shell structure.

The excess binding energy function is empirically known for the value from 0 to 25. It must be extended up to a value of about 50 alpha particles.

• The neutron pair component Nn that is also a function of the number of alpha particles in the nuclide and the number of neutron pairs; i.e., Nn = f(#α, #nn).
• The proton pair component Pp that is also a function of the number of alpha particles in the nuclide; i.e., Pp = g(#α, #pp).
• The neutron-proton (deuteron) pair component Pd that is a function of the number of alpha particles in the nuclide; i.e., Pd = h(#α, #nn, #pp).
• The singleton neutron component Sn = q(#α, #nn, #pp, #d)
• The singleton proton component Sp = r(#α, #nn, #pp, #d) )

The binding energy is then

#### B = A(#α) + Nn + Pp + D + Sn + Sp

(To be continued.)