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of Nuclear Magic Numbers |
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 hence lower interaction energy. The conventional magic numbers are {2, 8, 20, 28, 50, 82, 126}. These numbers were found in the case of neutrons by comparing the number of stable isomers for different neutron numbers.
Neutron Number | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
Number of Stable Isomers |
2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 3 |
Neutron Number | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
Number of Stable Isomers |
1 | 3 | 1 | 3 | 1 | 3 | 1 | 3 | 0 | 5 |
Neutron Number | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 |
Number of Stable Isomers |
0 | 3 | 2 | 3 | 1 | 4 | 4 | 4 | 1 | 4 |
It is hard to distinguish between many neutron numbers such as 6, 8, 10, 12 and 14 on the basis of the criterion of the number of stable isomers in comparison to the neutron numbers one higher and one lower. The magicality of neutron number 28 is hard to justify.
For protons the magic numbers were found by comparing the number of stable isotopes, but the proton number never gets to 126.
Proton Number | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
Number of Stable Isotopes |
2 | 2 | 2 | 1 | 2 | 2 | 2 | 3 | 1 | 3 |
Proton Number | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
Number of Stable Isotopes |
1 | 3 | 1 | 3 | 1 | 4 | 2 | 3 | 2 | 5 |
Proton Number | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 |
Number of Stable Isotopes |
1 | 5 | 1 | 3 | 1 | 4 | 1 | 5 | 2 | 5 |
A stronger indication of magicality of a nucleon 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
For example, the IBEn for the isotopes of Strontium are:
The sharp drop and change in the pattern of the IBEn 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 formation of a spin pair. The sharp drop after 38, 38 being the atomic number of Strontium, is a result in at least in part of there not being any additional formation of neutron-proton spin pairs after 38 neutrons.
The same pattern prevails for the incremental binding energies of protons. For example
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. See Magicality of 6 and 14.
It is a very remarkable fact the filled shell numbers are the same for protons as for neutrons. The data for protons are not included more extensively 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 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(2^{2}), 2(2^{2}), 2(3^{2}), 2(3^{2}), 2(4^{2}). 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(1^{2}).
The accepted explanation of the magic numbers for electron shell structures is that there are shells for 2(n^{2}) 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 m². The quantum number s is the spin and s can take on only the values ±½ . Thus with the two spin orientation there are 2m² electrons with a principal Squantum 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.
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, 28}. 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 replaced by 6 and 14.
This algorithm can be justified in terms of there being nucleonic states characterized by sets of four quantum numbers, say (n, j, l,s). The quantum number n is the principal quantum number which can take on any integer value from 1 to 7. The spin quantum number s can take on values of ±½. The nature of the other two quantum numbers will dealt with later.
The above algorithm generates the modified nuclear magic numbers. The table below gives an alternate derivation of the relationship. In effect it is a derivation of the above algorithm.
Shell Number | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
Magic Numbers | 2 | 6 | 14 | 28 | 50 | 82 | 126 |
Shell Capacities | 2 | 4 | 8 | 14 | 22 | 32 | 44 |
Pairs Capacities | 1 | 2 | 4 | 7 | 11 | 16 | 22 |
Sequential Differences | 1 | 2 | 3 | 4 | 5 | 6 | |
Above it was noted that there are four quantum numbers for a nucleon. One is the principal one, essentially the order number of the shell it is in. A second is the order number of the subshell it is in within that shell. The third is the order number of the nucleon within that subshell. The fourth quantum number for a nucleon is its spin. For material on nuclear subshells see Subshells.
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