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The Complete Explanation of the Nuclear Magic Numbers
Which Indicate the Filling of Nucleonic Shells and
the Revelation of Special Numbers Indicating
the Filling of Subshells Within Those Shells

Dedicated to: Hjørdis
                       forever

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. A higher shell involves a greater separation from the other nucleons and lower interaction energy. The conventional magic numbers are {2, 8, 20, 28, 50, 82, 126}. These numbers were found in the case of protons by comparing the number of stable isotopes for different proton numbers. For neutrons the magic numbers were found by comparing the number of stable nuclides with the same neutron numbers.

A stronger indication of magicality of a number is in terms of the incremental binding energies (IBE).

Let BE(n, p) indicate the binding energy of a nuclide with n neutrons and p protons. The incremental binding energy of the n-th neutron in that nuclide is

IBEn = BE(n, p) − BE(n-1, p)

For example, the IBEn for the isotopes of Strontium are:

The sharp drop after 50 neutrons is evidence of a shell being filled with 50 neutrons. The odd-even sawtooth pattern is an indication of the formation of neutron-neutron spin pairs. The amplitude of the fluctuation associated with the formation of neutron-neutron spin pairs also includes the effect of the adjustment to the spin pair. The sharp drop after 38, 38 being the atomic number of Strontium, is a result of there not being any additional formation of neutron-proton spin pairs after 38 neutrons.

The examination of the incremental binding energies reveals the magicality of the conventional nuclear magic numbers, but it also reveals that 6 and 14 are magic numbers.

It is a very remarkable fact the filled shell numbers are the same for protons as for neutrons. The data for protons is not included here simply in order to keep the details of this topic manageable.

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 not quite right with the conventional sequence of magic numbers.

Before going on with the matter of the nuclear shell structure let us consider the structure of the electron shells of atoms.

The Magic Numbers for Electrons in Atoms

The noble gases are helium, neon, argon, xenon and radon. The inertness of these elements is a consequence of the stability of the filled shells of electrons. The atomic numbers of the noble gases are 2, 10, 18, 36, 54 and 86. These can be considered magic numbers for electron structure stability. The differences in these numbers are: 8, 8, 18, 18, 32. These differences are twice the value of the squares of integers; i.e., 2(22), 2(22), 2(32), 2(32), 2(42). The first number, 2, in the series {2, 10, 18, 36, 54, 86} is also of the form of twice the square of an integer, 2(12).

The accepted explanation of the magic numbers for electron shell structures is that there are shells for 2(n2) electrons where n=1, 2, 3, 4... The reason for the coefficient 2 in the formula is that there are two spin orientations of an electron. Pauli's exclusion principle operates and so electrons fill the states sequentially with no two electrons of an atom in the same state.

The reason for the squared integer is that the electrons have four quantum numbers (i, k, m, s). The first quantum number i can take on all integral values from −k to +k. That totals (2k+1) possible states, an odd number. The quantum number k can take on all integral values from 0 to m. The sum of the first m odd numbers is m. The quantum number s is the spin and s can take on only the values ±½ . Thus with the two spin orientation there are 2m electrons with a quantum number m.

There are the anomalies of the second and third electron shells both having capacities of 8, rather than 8 and 18, respectively; and the fourth and fifth shells both having capacities of 18 rather than 32 and 50, respectively. This is explained in terms of the detailed energies of the electron states.

The Nuclear 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}. These are doubled 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.

This algorithm can be justified in terms of there being nucleonic states characterized by sets of four quantum numbers, say (j, l, n, s). The quantum number j can take on integer values from 1 to l and the quantum number l can take on integer values for 0 to n. The spin quantum number s can take on values of ±½. The quantum number n is the principal quantum number.

In the following display the shell number is not the same as the principal quantum number. Instead n is one less than the order number of the shell.

SHELL
NUMBER
Quantum Number lNumber
of States
 0123456
101
2012
30124
401237
50123411
601234516
7012345622

For a Quantum Number l of k greater than zero the Quantum Number j runs from 1 to k. Therefore the number of states for Quantum Number l of k is k if k>0. For a Quantum Number l of zero there is just the single state. The column on the right is the number of states not counting spin. The conversion to the total number of states is shown below.

Translation of the Number of States
to the Nuclear Magic Numbers
Shell
Number
Number
of States
(not counting
spin)
Number
of States
(counting
spin)
Cumulative
Sum
1122
2246
34814
471428
5112250
6163282
72244126

The numbers in the column on the right are just the magic numbers with 8 and 20 left out.

Subshells Within Shells

It is perfectly plausible that there could be substructures within shells. The table in yellow shown previously indicates why there might be subshells within each Shell. Each shell is the sum of the states for the previous shells in which the quantum numbers j and l are the same. Thus it is clear from the above display why for a given nuclear shell why the magic numbers for prior shells should show up as special numbers with the shell. That is to say, for example that for the sixth shell, which contains the 51st through 82nd nucleons, that there may be subshells filled with 2, 6, 14 and/or 28 nucleons.

The natural breakdown of the occupancies of the shells is as follows:

Shell
Number
OccupancyComposition
122
242+2
382+2+4
4142+2+4+6
5222+2+4+6+8
6322+2+4+6+8+10
7442+2+4+6+8+10+12
8582+2+4+6+8+10+12+14

Note that the last portion to be filled in a shell does not always correspond to the number in a lower level shell. Instead the numbers in the last portions are successively larger even numbers; i.e., 2, 4, 6, 8, 10, 12, 14. The subtraction of these numbers from the filled shell totals gives the filled subshell totals. For example, the magic number for the completion of the 7th shell is 126. The last portion of this filling is 12 nucleons. This means the completion of the subshells in the 7th shell involves 126−12 or 114 nucleons. Thus the numbers associated with the filling of the 6th shell are 84, 88, 96 and 114. These should be special numbers.

Special Numbers

A change in the pattern is a difference between the value a point and what a continuation of the trend of the lower number of neutrons. This may involve a change in the upper or lower levels of the pattern or a change in the amplitude of the fluctuations. Thus the previous analysis indicates there may be a change in the pattern of the incremental binding energies for the sixth shell after 52, 56, 64 and/or 72 nucleons. Consider the IBE data for Molybdenum (atomic number 42) and Iodine (atomic number 53).

For Molybdenum, after 56 neutrons and 64 neutrons there is a change in the slope of the pattern and a change in the amplitude of the odd-even fluctuations. For Iodine the change in pattern is an increase in the amplitude of the odd-even fluctuations after 64 neutrons.

The next display shows the persistence of the change in the pattern after 56 neutrons.

The next display shows the persistence of the pattern over the range 50 through 66 neutrons.

Ignoring the odd-even fluctuations the nature of the changes are as indicated below.

The possibility of 72 neutrons being special is illustrated in the data for Silver (atomic number 47).

When a subshell is filled the energetics of the next subshell may or may not be sufficiently different so as to produce a change in pattern. Thus there may be a change in pattern under some circumstances and not under other circumstances. The crucial thing in the matter of identifying subshells is whether there is some circumstance in which a change in pattern shows up.

For the shell containing the 83rd through 126th nucleons, as noted previously, the special numbers would be 84, 88, 96 and 114. For an illustration of the change in the pattern after 88 neutrons consider the IBEn data for Gadolinium (atomic number 64).

Another striking example of the change in the pattern at 88 and 96 is the data for Terbium (atomic number 65).

For Terbium the numbers of neutrons in the shell containing 83 through 126 neutrons after which there is a change in the pattern are 6 and 14, both magic numbers.

The data for Mercury (atomic number 80) illustrates a change in pattern after 114 neutrons.

For the shell containing the 127th through the 184th nucleons the special numbers would be 128, 132, 140, 154 and 170. The data for Thorium (atomic number 90) provides evidence for 132 and 140 as special numbers.

There is a previous study which identified 152 as being a magic number. This is not 154 but it is very close to it. However, the evidence indicates that definitely it is 152 which is the special number.

The data for Mendelevium (atomic number 101), Nobelium (atomic number 102), Lawrencium (atomic number 103) and Rutherfordium (atomic number 104) show a similar change of pattern after 152 neutrons. One hundred fifty two represents 76 neutron pairs as opposed to 77 for 154 neutrons.

In the lower shells the changes of pattern get mixed up with the n=p effect. However the special numbers would be as follows. For the shell containing the 29th through 50th nucleon the special numbers would be 30, 34 and/or 42. The next special number would be 56, which is in the next shell. The data for Manganese (atomic number 25) shows a change in the pattern after 30 neutrons but virtually nothing for 34 and 42 neutrons.

For Iron (atomic number 26) there is something happening to the pattern after 41 neutrons.

For Cobalt (atomic number 27) and Nickel (atomic number 28) there are some things happening to the pattern after 30 neutrons and 39 neutrons.

For the shell containing the 3rd through 6th nucleons the only possible special number is 4.

For the shell containing the 7th through 14th nucleons the special numbers would be 8 and 12.

For the shell containing the 15th through the 28th nucleons the special numbers would be 16, 20 and 28. The numbers 8, 20 and 28 are magic numbers. The number 28 could not be a special number for a subshell because it is the number for a fully filled shell.

Definitely there is a change in the pattern of the incremental binding energies for 8 and 20.

Conclusions

The nuclear magic numbers can be explained in terms of there being states identified by four quantum numbers. The magic numbers correspond to filled shells. Within these shells there are subshells involving the same numbers as lower level filled shells. These subshells are identified by changes in the pattern of incremental binding energies. The change of pattern do not always come after exactly the number specified by the theory, but close to it. In the case of 152, the theory says 154, but the data clearly indicate that the special number is 152.


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