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A nucleus is composed of protons and neutrons, 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 p and n be the numbers of protons and neutrons, 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 neutron (IBEn) is defined as
The incremental binding energy of a proton 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 isotopes 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 neutrons as a function of the number of neutrons in the nuclide for the isotopes of Tin (p=50).
The sharp drop after 82 neutrons 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 neutron in a higher shell is at a greater average distance from the other nucleons than one in a lower shell. It is worthwhile to consider now the data for Mercury (p=80).
This display confirms the magic number 126 but it looks as though the change might have begun at 124 neutrons rather than 126. 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 14 neutrons. The sharp drop at 8 neutrons is, at least in part, due to that being the number of protons. When the neutron number is less than the proton number an additional neutron results in the formation of a neutronproton spin pair and a corresponding effect on binding energy. When n is greater than p there is no formation of a neutronproton spin pair. Therefore the incremental binding energy] drops sharply after n equals p. This will be called the n=p effect.
It is a very remarkable fact the filled shell numbers are the same for neutrons as for protons.
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 protons and the number of neutrons 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 neutrons and protons 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 neutron number of 56. The graph below shows that the change in the pattern of the incremental binding energies prevails over proton numbers 36, 37 and 38.
There is a similar change of pattern for 42 neutrons that prevails over the three proton numbers
The conclusion is that at 56 neutrons and 42 neutrons there are subshells of the neutron shell with 51 to 82 neutrons which are filled.
The graphs for proton numbers 66, 67 and 68 include the case of neutron number 88.
There is a change in the pattern near 88 neutrons. It is a change in the slope of the relationship between incremental binding energy and the number of neutrons. This could be said to come at 90 rather than 88 neutrons but it is close to 88 neutrons. The change prevails over the proton numbers of 66, 67 and 68.
There appears to be no evidence of a change of the pattern for 12 neutrons.
This may be due to 12's proximity to 14 and the n=p effect for n=9.
Since 20 is a conventional magic number there is to be expected definitely a change in the pattern at 20 neutrons.
A subshell of size 14 in the shell with 51 to 82 corresponds to 64 neutrons. In the following display there is a change in the slope of the relationship in the neighborhood of 64 but it appears to be at 62 neutrons rather than 64.
The data for Platinum (p=78) shows a change of pattern at 96 (82+14) neutrons but none at 110 (82+28). There are such changes at 106 and 116 neutrons.
The data for Gold confirms the change of pattern at 96 neutrons. There could be a change of pattern at 110 (82+28).
The data for Mercury (p=80) cannot confirm or deny a change of pattern at 96 neutrons.
There is no evidence for a change of pattern at exactly 110 neutrons but there may be such changes at 102, 108 and 114 neutrons. As was noted previously, the definite change of shell appears as if it could occur at 124 neutrons rather than 126. This indicates that there is a margin of variation in the point where a change in shell or subshell appears to take place.
Here is what the data for the three elements look like when plotted together.
The data for Radium (p=88) nicely confirm the hypothesis. There are changes in the pattern at 132 neutrons (126+6) and at 140 (126+14) neutrons as well as the drop at 126.
This is confirmed by the data for Thorium.
The combined display of the data for Radium (p=88), Actinium (p=89) and Thorium (p=90) indicates that the changes in the pattern are a property of the neutron number rather than the proton number.
There is considerable evidence for the existence of subshells in neutron shells and that the occupancy numbers replicate those for the shells. The hypothesis concerning the structure of the subshells of the neutron shells is strongly supported. There is definitely something there.
For the corresponding analysis concerning subshells in the proton shells see Proton Subshells.
For more on nuclear subshells see Subshells.
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