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Profiles for Some Heavy Elements |
Consider the incremental binding energies for the Uranium isotopes.
It is extremely regular. There is of course the sawtooth pattern resulting from the effect of neutron pair formation. Looking at the upper edge of the relationship (the values for which all neutrons are paired) the shape is generally of the following form.
This general shape is manifested only when there is no magic number crossed; i.e., there is no neutron shell being completely filled and the filling of the next shell commenced.
The shape can be approximated by a quartic function of the number of neutrons; i.e., a fourth order polynomial of the number of neutrons. A regression equation which is quartic with an odd-even variable to capture the effect of neutron pairing fits the incremental binding energy data quite well. The coefficient of determination (R²) for the regression is 0.9891, which means 98.91 percent of the variation in the increamental binding energy is explained by the regression function. It also means that the correlation between the regression estimate and the incremental binding energy is 0.9945. The statistical fit of the regression equation estimates with the data is so good that it is hard to display graphically. The regression estimates and the data virtually coincide. What is shown below is the incremental binding energy data along with the deviations between the binding energies and the regression estimates.
The mean incremental binding energy is 6.54 MeV whereas standard error of the estimate is 0.131 MeV, a ratio of about 2 percent.
The relationship of incremental binding energy to the number of neutrons for other heaving elements typically so the break that represents the effect of filling one neutron shell and then going on to the next.
As in the case of Uranium, the isotopes of Neptunium, Plutonium and Fermium do not involve any complete filling of of a neutron shell. The same type of quartic regression equation as used for Uranium explains 99.14 percent of the variation in the incremental binding energies of Fermium. The deviations of the regression estimates from the incremental binding energies for Fermium are shown below.
There is a small break in the relationship at 152 neutrons. This is approximately at the point where the neutron shell is half filled. The half way point is 155 neutrons.
(To be continued.)
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