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The Statistical Estimation of the Parmeters
of the Binding Energy Functions for 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 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 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.

To answer this latter question the binding energies of all nuclides which contain an integral number of alpha particles plus four additional neutrons was computed and the results plotted.

This indicates that the full occupancy of the third shell of alpha particles is indeed eleven.

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(28.29567 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.

The same technique, in principle, can be applied to the data for the excess binding energies of the alpha plus 4 neutrons. 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 represents the effect of the four neutron combination on the binding energy of nuclides.This relationship has a bendpoint at 5 alpha particles and another one at 23 alpha particles. The regression equation of the above relation would have the form

ΔBE = c0 + c1#α + c3u(#α) + c5u(#α-23)

This would be combined wtih the binding energy of the alpha nuclides The bump over the range of 11 to 13 alpha particles shows up in other relationships and is apparently associated with some structural feature of nuclei. The nuclides associated with the bump are isotopes of Titanium, Chromium and Iron.

Linear regression cannot possibily explain the bump and the inclusion of the data points for the bump would distort the estimates of the parameters of the linear portions of the relationship. Therefore the data points for the bump are left out of the regression analysis. The results of the regression are

ΔBE = −3.33795 + 7.160709#α −6.04507u(#α-5) −3.76789u(#α-23)
       [-4.2]          [38.9]          [-29.9]         [-8.9]    
R² = 0.99703
df = 18

This means that for each additional alpha particle in the range of 0 to 5 the binding energy increases by 7.160709 MeV. From 6 to 23 alpha particles the increase is (7.160709−6.04507)=1.115639 MeV. Beyond 23 each additional alpha particle decreases binding energy by -(1.115639-3.76789)=2.652251 MeV.

In the graph below the data for the difference in binding energies of the alpha+4neutrons nuclides and the alpha nuclides are displayed along with the regression estimates for the data points. The actual data is displayed as the red squares and the regression estimates are displayed as the upper edge of the yellow area. The match is not perfect but it is very close except for the data points for the bump. The correlation between the data points and their regression estimates is 0.9985.

The regression equation for the excess binding energy of the alpha plus 4 neutron nuclides could be estimated as the sum of the excess binding energy of the alpha nuclides plus the binding energy difference for the alpha plus 4 neutrons nuclides. The results of such a regression are

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

The coefficient of determination is impressive but it is impossible to separate the influence of an additional alpha particle into the part associated with filling the shells and the effect of the additional four neutrons.

The results of the previous regression can be used to extend the information for the the excess binding energies of the alpha nuclides beyond the range of stable alpha nuclides. The regression estimates of ΔBE can be substracted from the δBE for the alpha plus 4 neutron nuclides to give δBE for the alpha nuclides for the range above 25 alpha particles.

The above type of analysis can be applied to other sets of nuclides, such as those for the alpha plus 8 neutron nuclides. The graphs for the alpha plus 8 neutrons nuclides corresponding corresponding to those shown above for the alpha plus 4 neutrons are as follows.

The range of the data is affected by the limits of the stability of the nuclides. For example, for the alpha plus 24 neutrons the graphs are:

Thus, in this case, the data reveals only a portion of one linear segment.

Conclusions

Linear regression analysis can easily be applied to obtain values for the the parameters of bent-line relationships. It gives values for the different impacts of additional particles in different nuclear shells.

(To be continued.)


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