The Nature of Nuclear Shells

One of the elements of the physics of nuclei is the matter of magic numbers. They represent a shell being completely filled so additional nucleons have to go into a higher shell. The conventional magic numbers are {2, 8, 20, 28, 50, 82, 126}. These values were established by examining the relative numbers of stable isotopes and isotones. They can also be established in terms of sharp drops in the incremental binding energies. This test however also establishes that 6 and 14 are magic numbers.

It is a very remarkable fact the filled shell numbers, usually called magic numbers are the same for protons as for neutrons.

If only the conventional magic numbers {2, 8, 20, 28, 50, 82, 126} are considered the shell capacities are {2, 6, 12, 8, 22, 32, 44}. Thus there is the anomaly of the shell capacity decreasing from 12 to 8 rather than increasing for each higher shell number as occurs for all of the other cases. This suggests that there may be something wrong with the conventional sequence of magic numbers.

Consider the following algorithm. Take the number sequence {0, 1, 2, 3, 4, 5, 6} and generate the cumulative sums; i.e., {0, 1, 3, 6, 10, 15, 21}. Now add 1 to each of these numbers to get {1, 2, 4, 7, 11, 16, 22}. Now take the cumulative sums of that sequence to get {1, 3, 7, 14, 25, 41, 63}. Double these because there are two spin orientations for each nucleon. The result is {2, 6, 14, 28, 50, 82, 126} which is just the magic numbers with 8 and 20 left out. Magic numbers 8 and 20 are the sums of the two previous magic numbers in the sequence. This suggest that within a shell there are subshells and the filling of a subshell results in a change in the pattern of the incremental binding energies.

The subject of this webpage is the investigation of the existence of subshells within the proton shells of nuclei. The principles involved in this investigation are:

• The filling of a subshell may be manifested by a change in the pattern of the incremental binding energies. Such a change in the pattern is a difference between the value a point and what a continuation of the trend of the lower or higher number of protons. This may involve a change in the upper or lower levels of the pattern or the slope of the pattern or a change in the amplitude of the fluctuations.
• Changes in the pattern come from differences in energy levels. In some situations the energy levels may be such as to result in a change in the pattern, but in other situations the levels may be such that no change in the pattern is detectable. Thus the absence of a change in the pattern in some situations is not necessarily evidence for the non-existence of subshells.
• When the change of pattern is subtle the display of the data for several consecutive elements may establish that such a change in pattern is real.
• The sharp drop in the incremental binding energy after the number of protons equals the number of neutrons (the p=n effect) is not evidence of a change in the pattern. It is evidence of the formation of neutron-proton spin pairs for the the cases in which the number of protons is less than or equal to the number of neutrons and no such pair being formed otherwise.

The Shell Containing 51 through 82 Protons

As an illustration of what is involved consider the data for the incremental binding energies of the protons in isotones with 41 neutrons.

The sharp drop after 28 protons indicates that a shell is filled at 28 protons and any additional protons must go into a higher level shell. The odd-even fluctuations are due to the formation of proton-proton spin pairs.

Ignoring the odd-even fluctuations the pattern is as below.

The pattern of the data for the isotones with 39 neutrons repeats the pattern observed for n=41, but with a change in the pattern after p=32. The change is an increase in the amplitude of the odd-even fluctuations.

The data for the isotones with 38 neutrons shows a similar pattern.

The sharp drop after 38 protons is the p=n effect. This did not show up in the case of n=39 because effect of the formation of a proton-proton pair nearly offset the non-formation of a neutron-proton pair.

The data for the isotones with 40 neutrons more or less duplicates the graphs for the above cases.

The data for the elements with neutron numbers 38 through 41 are shown below.

The irregularity on the right is just the p=n effect. However the graph shows something similar occurring on the left near p=25. The nature of this phenomenon is as yet unknown.

There is some change of pattern after 32 or 34 protons that is likely due to the filling of a subshell.

The same phenomena is revealed in the data for elements with neutron numbers 33 through 37.

The changes in the pattern are more dramatic for odd number of neutrons.

There are slight changes of pattern after 34 protons as is illustrated in the data for the nuclei with 47 protons.

The Shell Containing 83 through 126 Protons

The evidence for subshells is available for the next higher shell.

Thus, for the isotones with 110 neutrons the numbers of protons in the shell where there is a change in the pattern are 74 and 78, and, of course 82. Seventy eight neutrons indicate subshells having a total occupancy of 28, a magic number.

For the isotones with 128 neutrons there is no indication of a change in the pattern except for the sharp drop after 82 protons.

For the higher levels of neutron numbers there are just a small range of proton numbers available. Often this range is not long enough to include a possible subshell.

The Shell Containing 29 through 50 Protons

There appears to be something in the nature of a subshell being filled at 36 or 38 protons. The consistency of the pattern is revealed when the data for the four elements are plotted in the same graph.

The Shell Containing 15 through 28 Protons

In the lower shells the pattern is sometimes distorted by the p=n effect but some of the graphs are quite striking.

The sharp drops after 20 protons would represent the filling of a subshell.

The Shell Containing 7 through 14 Protons

In the this shell as well as the one above the pattern is sometimes distorted by the p=n effect but the graphs are still quite striking.

It is clear that 8 is the significant number in the 7 through 14 shell and represents the formation of a subshell of two protons.

The Shell Containing 3 through 6 Protons

Even the data for the next-to-smallest shell has some bearing on the matter of subshells.

It is clear that 4 protons is a significant number in the shell and that 4 protons represents the formation of a subshell of two.

Conclusions

The results indicate that subshells are formed within shells. When a subshell is filled there often results in a change in the pattern of the incremental binding energy of protons.