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A nucleus is composed of neutrons and protons, collectively called nucleons. The mass of a nucleus is generally less than the combined masses of its nucleons. This mass deficit when expressed in energy units via the Einstein formula of E=mc² is called binding energy. The Binding Energy (BE) of a nuclide represents the energy required to break a nucleus up into its constituent nucleons. Binding energy is usually expressed in terms of millions of electron volts (MeV).
Let n and p be the numbers of neutrons and protons, respectively, in a nucleus. Then the binding energy of a nucleus BE is a function of only its p and n. The incremental binding energy of a proton (IBEp) is defined as
The incremental binding energy of a neutron is defined analogously.
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 isotones and isotones. They can also be established in terms of sharp drops in the incremental binding energies. For example, consider the incremental binding energies of protons as a function of the number of protons in the nuclide for the isotones of n=110.
The sharp drop after 82 protons establishes that 82 is a magic number; i.e. that a shell is completely full. It was also found to be a magic number by Goepert Mayer and Jensen. The reason for the drop is that a proton in a higher shell is at a greater average distance from the other nucleons than one in a lower shell. There is also a change in the pattern that comes at 76 or 78 protons. This indicates that there is a margin of variation with respect to changes in the pattern of the incremental binding energies.
The incremental binding energy test however also establishes that 6 and 14 are magic numbers which are not conventional magic numbers.
Here there are sharp drops in incremental binding energy at 6, 8 and 10 protons. The sharp drop at 10 protons is, at least in part, due to that being the number of neutrons. When the proton number is less than the neutron number an additional proton results in the formation of a protonneutron spin pair and a corresponding effect on binding energy. When p is greater than n there is no formation of a protonneutron spin pair. Therefore the incremental binding energy] drops sharply after n equals p. This will be called the p=n effect.
It is a very remarkable fact the filled shell numbers are the same for protons as for neutrons.
This hypothesis came from empirical evidence and the possibility that same values and constraints on quantum numbers that generate the shell sizes might apply for the subshells. If this is so then the relevant nucleon numbers are those given the following table.



0  2  6  14  28  50  82  
2  4  
6  8  12  
14  16  20  
28  30  34  42  
50  52  56  64  78  
82  84  88  96  110  
126  128  132  140  154  176 
The numbers in this table are predictions of the numbers of neutrons and the number of protons at which subshells are filled and consequently there will be changes in the pattern of incremental binding energies.
Maria Goeppert Mayer and Hans Jensen established that there are certain numbers of protons and neutrons for which there are an unusually large number of stable nuclides. This was taken to be evidence of the complete filling of nucleonic shells. Eugene Wigner coined the term magic number for them and unfortunately he name stuck.
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, 7} and generate the cumulative sums; i.e., {0, 1, 3, 6, 10, 15, 21, 28}. Now add 1 to each of these numbers to get {1, 2, 4, 7, 11, 16, 22, 29}. Now take the cumulative sums of that sequence to get {1, 3, 7, 14, 25, 41, 63, 92}. Double these because there are two spin orientations for each nucleon. The result is {2, 6, 14, 28, 50, 82, 126, 184} which is just the magic numbers with 8 and 20 left out but 6 and 14 included and 184 given as the next magic number beyond 126. Magic numbers 8 and 20 are the sums of the two previous magic numbers in the sequence. This suggests that within a shell there are subshells and the filling of a subshell might also result in a change in the pattern of the incremental binding energies.
It is not possible to find evidence for subshells of size 2 unless such a subshell results in a drop in the incremental binding energy. Therefore the investigation will start with a subshell of size 4 which means the nucleon number is 6 larger than the next lower nucleon shell.
First consider the case of a proton number of 56. The graph below shows that the changes in the pattern of the incremental binding energies which prevails for neutron numbers 67 and 69.
The change seems to come at 54 neutrons rather than 56,
There is no similar change of pattern for 42 protons that prevails over the two neutron numbers.
The data for 68 shows no change at either 42 or 54/56.
The conclusion is that at 54 or 56 protons a subshell of the proton shell with 51 to 82 protons is filled.
The graphs for neutron number 133 include the case of proton number 88.
There is definitely a change in the pattern at 88 protons. It is a change in the level and the slope of the relationship between incremental binding energy and the number of protons.
It is difficult to establish evidence of a change of the pattern for 12 protons because the changes at 6, 8 14 disrupt the patterns below and above 12. Here is the data for the case of the isotones of 10 neutrons.
The sharp drop at p=10 is due to the p=n effect.
Since 20 is a conventional magic number there is to be expected definitely a change in the pattern at 20 protons.
A subshell of size 14 in the shell with 51 to 82 corresponds to 64 protons. In the following display there is a change in the amplitude of the oddeven fluctuations in the neighborhood of 64 but it is perhaps at 62 protons rather than 64.
The data for protons does not provide tests for changes of pattern at 96 (82+14) protons and 110 (82+28) protons.
(To be continued.)
There is considerable evidence for the existence of subshells in proton shells and that the occupancy numbers replicate those for the shells. When the data does not provide a confirmation of a change in the pattern of the incremental binding energies it does not means that a subshell does not exist. It could just mean that the subshell is not manifested in terms of a change in pattern. The hypothesis concerning the structure of the subshells of the proton shells is strongly supported. There is definitely something there.
For the corresponding analysis concerning subshells in the neutron shells see Neutron Subshells.
For more on nuclear subshells see Subshells.
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