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 explanation of the magic numbers for electron 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 ��. Thus with the two spin orientation there are 2m� electrons with a quantum number m.

The Nuclear Magic Numbers

The conventional magic numbers for the nucleus are {2, 8, 20, 28, 50, 82, 126}. The case is made elsewhere that 6 and 14 are also magic numbers. (A recent study also found evidence for a magic number of about 180.) Thus the magic numbers for filled shells are then {2, 6, 8, 14, 20, 28, 50, 82, 126}. This means that the shell capacities are {2, 4, 2, 6, 8, 22, 32, 44}. These are all even reflecting the tendencies of nucleons to form pairs. Thus the shell capacities in terms of the number of pairs are {1, 2, 1, 3, 11, 16, 22}.

Consider the following algorithm. Take the sequence of integers {0, 1, 2, 3, 4, 5, 6} and compute the cumulative sums; i.e., {0, 1, 3, 6, 10, 15, 21}. To each of these numbers add 1; i.e., {1, 2, 4, 7, 11, 16, 22}. Now double these numbers; i.e., {2, 4, 8, 14, 22, 32, 44}. Then compute the cumulative sums; i.e,,

{2, 6, 14, 28, 50, 82, 126}.

This is just the sequence of the nuclear magic numbers with 8 and 20 left out. There is no doubt that 8 and 20 are magic numbers. However in an analysis of the transition that takes place at magic number 28, the result which should depend upon the adjacent magic numbers is found to depend upon 14 and 50 rather than 20 and 50. It is perhaps possible that there are two classes of magic numbers and 8 and 20 are different from the other magic numbers. Also it is worth noting that 8 is the sum of the previous two magic numbers, 2 and 6, and 20 is likewise the sum of the two previous magic numbers in the above sequence, 6 and 14. It is worth noting that for electrons there are secondary peaks for the numbers 30, 48 and 80 and these numbers can be represented as sum of the electronic shell capacities. Thus for electronic shell structure there are primary magic numbers, {2, 10, 18, 36, 54, 86}, and secondary magic numbers {30, 48, 80}. If there is anything to this conjectured pattern then there should be something special about the numbers 28+6=34 and 28+14=42 for nuclides. There are five stable isotopes for element 34 (Selenium) whereas only one and two for elements 33 and 35, respectively. There are three stable nuclides with 34 neutrons but only one with 33 neutrons and none with 35 neutrons.

Here are a few of the incremental binding energy profiles involving 42.

In these cases there is a change in the amplitude of the fluctuations at the neutron number of 42. But, back to the algorithm for the magic numbers.

If the algorithm had been continued to 7 then the next magic number after 126 would be 184, not too far different from the 180 recently proposed as the next magic number. If the algorithm is extended to 8, 9 and 10 the numbers generated are 258, 350 and 462. Some interesting things occur when a shell is half filled. That would be 304 for 258 and 350. The researchers that proposed 180 as the next magic number, more recently proposed 306 as the magic number to follow 180.

In the case of electrons, each electron in a shell is identified by a quadruplet of quantum numbers (i, k, m, s). For the nucleus, each neutron would be identified by a quadruplet of quantum numbers (p, q, n, s). The algorithm indicates that there would be some quantum number p taking on all integer values from 0 to q and q ranging from 0 to some value n. The doubling of the numbers corresponds to neutrons having two spin orientations, s=��. However the algorithm implies that there is for each shell some pair of states that is not identified by the above described set of quantum numbers.

The Justification for the Above Pattern
for Nuclear Magic Numbers

Suppose there are two quantum numbers, p_θ and p_r, for a physical system. Suppose for the moment that both of these have to be non-negative integers. Let q=p_θ+p_r and call it the principal quantum number for the system. For a given q the quantum number p_θ can take on only the values 0, 1, 2, … q because p_r cannot be negative. Thus for each n there are (q+1) possible values for n_θ. This means the sequence of possible values for the quantum number pairs as function of the principal quantum number is

{1, 2, 3, 4, 5, 6, …}

Now let the rule be modified to the quantum numbers having to both be positive. Then for a given q the quantum number p_θ can only take on the values 1 through (q-1). Then the sequence of possible quantum number pairs is

{0, 1, 2, 3, 4, 5, 6, …}

Suppose for a shell of order n the quantum q can range from 0 up to (n-1). Then the number of quantum number pairs for shell n is the cumulative sum of the numbers in the above sequence; i.e.,

{0, 1, 3, 6, 10, 15, 21, …}

Now suppose there is a singular case that was left out in the above tabulation. For example, suppose p_θ can be zero only if p_r is also zero. This would mean that the numbers in the above sequence have to be incremented by unity to give:

{1, 2, 4, 7, 11, 16, 22, …}

Because there are two possible spin orientations to the particles the above numbers need to be doubled to:

{2, 4, 8, 14, 22, 32, 44, …}

These are the occupancy levels for the various shells and the cumulative sums correspond to the magic numbers

{2, 6, 14, 28, 50, 82, 126, …}

which are the magic numbers with 8 and 20 left out. There is no question but what that 8 and 20 are magic numbers, but there is the possibility that there are two categories of magic numbers: 1. The main sequence shown above, 2. A category which includes 8 and 20. In Nuclear Shell Theory Amos de-Shalit and Igal Talmi note that the magicalness of 8 and 20 can be established by a square well approximation to the nuclear potential but not the other magic numbers. Thus there is something different about 8 and 20 and there may be part of a second category of nuclear magic numbers that have to do with a sum of nuclear shell capacities which is different from a straight sequential sum. In the case of the shell model of electronic structure the magic numbers are {2, 10, 18, 36, 54, 86} corresponding to the peaks in ionization energy. However there are secondary peaks in the ionization energy at {30, 48, 80}, which make these numbers magic numbers of a second category. For more on this matter see the Shell Model of Nuclear Structure. Thus there is a precedence for a secondary category of nuclear magic numbers in the secondary category of atomic (electronic) magic numbers.

The Structure of Nuclear Shells

Historically the emphasis of attention has been on the totals of particles in filled shells; i.e., the magic numbers, but it is the shell capacities which are crucial for understanding nuclear structures. The information on nuclear shell capacities can be summarized in the form of a table.

In the table a composition refers to a possible substructure of the shell. Consider a globe. The first two particles go in at the poles. The next four particles go around the equator on the same globe. That takes care of magic number 6. The next eight particles go into two rings of four each at the Tropics of Cancer and Capricorn. That takes care of magic number 14 and complets a true shell, or perhaps supershell.

For magic number 28 there is another globe of 14 particles but with greater radius. For the shell of 22 there would be two particles at the pole and five rings of four each; one at the equator and two in each hemispher. For the shell of 32 there could be eight rings of four each without any particles at the poles or the equator. Or, there could be two at the poles and five rings of six each; one at the equator and two in each hemisphere. For the shell of 44 there could be eleven rings of four; one at the equator and five in each hemisphere with no particles at the poles. Or, there could be two particles at the poles and thirteen rings of four each with one at the equator and six in each hemisphere. Alternatively there could be five rings of eight each with one at the equator and two in each hemisphere along with two rings of two particles each near each pole.

The nuclear magic number of 8 might involve the first two rings of two each near the poles and one ring of four particles at the equator. Likewise the magic number of 20 might involve the four rings of four particles, two in each hemisphere plus two rings of two particles each near the poles.

Conclusions

Nuclear magic numbers were originally defined in terms of the number of stable isotopes or isotones. There is a certain amount of ambiguity about the magic numbers defined in this way. The magicality of certain numbers can be defined unambiguously in terms of the incremental binding energies. A certain numbers the incremental binding energies for adding a neutron or adding a proton drops sharply. This test establishes the magicality of the traditional magic numbers {2, 8, 20, 28, 50, 82, 126} but also identifies 6 and 14 as magic numbers.

There is an algorithm based upon the existence of a set of four quantum numbers for each neutron or proton that generates the set of magic numbers {2, 6, 14, 28, 50, 82, 126}. This is the magic numbers with 8 and 20 left out. It is conjectured that there are two categories of nuclear magic numbers; the main sequence {2, 6, 14, 28, 50, 82, 126} and a secondary sequence which includes 8, 20 and possibly other numbers. For the atomic (electronic) magic numbers there is such a secondary set of special numbers.

(To be continued.)