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the Interaction of a Nuclear Particle with Other Particles of the Same Type in the Same Shell |
There is a theorem that asserts, given a certain crucial assumption, that the second difference of the binding energy of a nuclide with respect to the number of a particular particle it contains is equal to the interaction between the last of those particles to be added to the nucleus and the next-to-last one to be added. The crucial assumption is that the interaction between the a-th particle and the j-th particle is the same as that between the (a-1)-th particle and the j-th particle for j<(a-1) and for a, (a-1) and j all in the same nucleon shell.
The investigation will be carried out using only the data for nuclides with an even number of neutrons and an even number of protons. This eliminates the distracting influence of the binding energy associated with the formation of neutron-neutron spins pairs and proton-proton spin pairs. The formation of neutron-proton spin pairs is still involved in the analysis.
First consider the following example of incremental (first difference) binding energies of neutron spin pairs.
The sharp drop at 41 neutron spin pairs (82 neutrons) occurs because a shell is filled. The graph between 25 and 41 is nearly linear. If the relationship were strictly linear then the interaction between the different spin pairs in the shell would be constant. However the relationship is not strictly linear. The second differences, shown below, become less negative with an increasing number of spin pairs.
The following schematic shows what is involved.
The theorem says that the values in the red squares should be the second differences. The crucial assumption of the theorem is that the elements in the last two columns on the right are equal. (The theorem works just as well if only the column sums on the right are equal.)
The relative values in the red and green squares for each row are relevant for the values in the last two columns. The transition from one red square to the one in the next higher row can be considered a movement to the green square above it and then a movement to the red square on the left. The change in energy between the green square and the red square to its left is about one half of the change from one red square to the another. The change in the binding energy from one red square to another is the third difference in the binding energy computed within a shell. The sum of the differences is just the difference in the value of the second difference at the beginning and end of the shell.
The difference between the value in a green square and the red square is approximately one half of the difference in the values in the red squares adjacent to the green square. This is approximately the value of the difference in the values of the squares at the right end of the row the green square is in. The difference in the values in two red squares is the difference in the values of two second difference. In other words, it is a third difference. Here are the third differences.
They are generally small and positive. Let μ be the sum of the differences in the last two columns on the right. Then the relation that holds is
Therefore if μ is positive
Thus the second difference is an upper bound on the value of the interaction. If the second difference is negative this means that the interaction term is more negative than the second difference.
(To be continued.)
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