|San José State University|
& Tornado Alley
the Nucleons of Nuclei are, Where Possible,
Organized into Alpha Particles
An alpha particle, composed of two neutrons and two protons, is an amazing structure. It is relatively compact and has an extraordinary level of binding energy compared with smaller nuclides such as a deuteron or triteron. Binding energy is like, and perhaps is identical with, potential energy. Energetically it would be difficult for nucleons in a nucleus not to come together and form alpha particles wherever possible. This suggests that the binding energies of larger nuclei are composed of that due to the formation of alpha particles and that due to the arrangement of alpha particles and the extra nucleons. This latter binding energy will be called the excess binding energy. It is computed for a nuclide by subtracting from its binding energy the possible number of alpha particles it could contain times 28.29567 million electron volts (MeV), the binding energy of an alpha particle. The plot of this excess binding energy for the nuclides which could contain exactly an integral number of alpha particles is shown below.
It might appear that the graph above indicates the existence of only three shells: 1 to 2, 3 to 14 and 15 to 25. The upper limits of those shells correspond to filled shells. Fourteen alpha particles means there are 28 neutrons and 28 protons. Twenty five alpha particles correspond to 50 neutrons and 50 protons. Fifty and 28 are nuclear magic numbers.
The incremental binding energy of a nuclide with a alpha particles is the excess binding energy of that nuclide less the excess binding energy of the nuclide with (a-1) alpha particles. An inspection of the graph for the incremental excess binding energies of the alpha nuclides, shown below, reveals that the 3 to 14 shell is composed of subshells. The end points of those subshells are levels of neutrons and protons that correspond with the nuclear magic numbers.
The numbers of alpha particles where there is a sharp drop; 4, 7, 10 and 14 correspond to 8, 14, 20 and 28 neutrons and protons, all magic numbers.
There are no alpha nuclides beyond those containing 50 neutrons and 50 protons. If two extra neutrons are included then the range of alpha particles is extended; as shown below.
Again the incremental values more clearly show where the transitions between shells occur, as illustrated below.
As indicated above the transitions generally correspond to a magic number of neutrons rather than the number of alpha particles.
The case of the alpha-plus-four-neutrons nuclides the picture is similar. Again the transitions occur at the magic numbers of 28 and 50 neutrons, but transitions take place of a range of several alpha particles rather than as sharply as in the case of the alpha nuclides. However, for the smaller nuclides the transitions are occurring not at the magic numbers of 6, 8 or 14 neutrons but at 3 and 7 alpha particles.
For the alpha-plus-eight-neutrons nuclides the picture again one in which there are transitions at 28 and 50 neutrons, but there are also apparently subshells where transitions occur at 14 and 25 alpha particles. Those numbers of alpha particles were transition points in the alpha nuclides case.
In the case of the alpha-plus-sixteen-neutrons nuclides the picture becomes definite. Transitions occur where the numbers of neutrons equal one of the magic numbers. Transitions also occur where the numbers of alpha particles are equal to special numbers, but those special numbers of alpha particles correspond to magic numbers of protons.
What is different for this case is that there is no longer a range of near-constant values for the incremental excess binding energy within the shell that extends from 14 to 25 alpha particles.
The final illustration is for the case of the alpha-plus-thirty-two-neutrons nuclides. The lowest data point is for 25 alpha particles which corresponds to 82 neutrons and 50 protons. This is a doubly magic nuclide and the incremental excess binding energy is exceptionally high. The next magic number above 82 is 126 and that number is not reached. The number of protons does reach the next magic number above 50, which is 82, but only minor deviation in the pattern occurs that point. It is notable that the there appear to be subshells in which the IXSBE of the alpha particles is linear, generally declining but some instances of increases.
According to the theory developed previously the increments in the incremental excess binding energies of alpha particles (the second differences in excess binding energy) should be negative, reflecting the net repulsion of alpha particles for each other, and constant within an alpha particle shell. The graph of the data for the last case considered above is shown below.
Except where the spikes occur for transitions from one shell to another the values are generally negative and roughly constant.
The case for the alpha+16neutrons nuclides is much the same; i.e., the increments in the incremental excess binding energies of alpha particles are negative except for spikes where there are transitions between shells.
Only in the last two graphs, the cases of the alpha nuclides and the alpha+2neutrons nuclides, are the increments not predominantly negative. In those case, except for the spikes at the shell transitions, the positive values are near zero. The interaction of a composite particle with other particles reduces to that of its net nucleonic charge only as the ratio of its spatial scale to the separation distance of the particle it is interacting with goes to zero. This means that the theory works better for large nuclides than for small nuclides.
In contrast, here is the graph for the cross differences in the incremental excess binding energies which the analysis says are the interactive binding energies for the last alpha particle with the last extra neutron. The theory says these should be positive, reflect the attraction of the alpha particles and neutrons for each other. Such an attraction stems from the nucleonic charge of a neutron being smaller in magnitude as well as opposite in sign.
The incremental excess binding energy of alpha particles for various numbers of extra neutrons display sharp drops at particular numbers of alpha particles. These drops occur at the numbers of neutrons or protons correspond to the number of neutrons or the number of protons reach a level where a nucleonic shell is filled and additional nucleons must go to a higher shell.
The previous theoretical analysis that the increments in the incremental excess binding energies of alpha particles should be negative and constant within a shell is confirmed.
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