If the explanation for atomic electron structure had not developed from the perspective of explaining spectroscopic regularities it might have emerged from investigating such phenomena has ionization. The ionization potential of an element is the amount of energy, usually measured in electron volts (eV), required to remove an electron from an atom of that element. For example, the ionization potential for hydrogen is about 13.6 electron volts. Removing an electron from an atom of helium, on the other hand, requires about 24.6 electron volts. To remove a second electron from that atom of helium would require about 54.4 electron volts.

The data on ionization potential for all the elements are shown in the graph below. These data are for removing only one electron in each case. The source of the data is the Handbook of Physics and Chemistry. A value is not given for element Astatine (At), atomic number 85, and an interpolated value is used in order to not create a problem in the display of the data.

As is seen in the above graph, the ionization potential reaches a peak for certain levels of the atomic number and then abruptly falls to a minimal level. The interpretation is that the electrons form shells and when a shell is filled the configuration is exceptionally stable and requires a lot of energy to knock an electron out of a filled shell. On the other hand, an electron in excess of a filled shell is very easy to remove from the atom. The marked peaks in ionization potential are those of the noble gases; helium, neon, argon, xenon and radon. The inertness of these elements is a consequence of the stability of the filled shells. The elements one electron beyond a filled shell are the alkali metals; lithium, sodium, potassium, cesium, rubidium and francium. Hydrogen may also be considered a member of this group.

The atomic numbers of the noble gases are marked in the graph. There values 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.

There is a slight problem with the above presentation. Element 80, mercury, also represents a local peak in ionization potential. The explanation for mercury would require too much attention to the problem of electron structure for the purpose of this introduction. The purpose of this introductory material is to show what the evidence for a shell structure looks like; i.e., characteristics reaching one extreme for some atomic numbers and then immediately shifting to the opposite extreme for the next larger atomic numbers.

However it is notable that secondary peaks occur for atomic numbers 30 and 48 as well as 80. The numbers 30 and 48 are the sums of three previous peak numbers; i.e., 30=18+10+2 and 48=36+10+2. The number 80 does not quite fit this scheme, but the crucial quantities are the shell capacities rather than the magic numbers of electronic structure. These numbers are merely the sum of the shell capacities, {2, 8, 18, 32, 50, 72}. Thus the number 80 is 50+18+8+2+2. The numbers 30 and 48 are 18+8+2+2 and 36+8+2+2, respectively. Thus there is pattern to the secondary peaks.

There are other points of relative maxima at {4, 12, 26, 46, 64, 70}. Four is of course 2+2. Twelve is 8+2+2. Twenty six is 18+8 and 46 is 36+8+2. Sixty four is 32+32 and 70 is 50+18+2. What this suggests is that when complete shells cannot be filled there is an advantage to filling a lower degree shell rather than having a partially filled higher degree shell.

Models of Nuclear Structure

It is generally believed that nuclei are composed of protons and neutrons, although there is some question as to whether a nucleus contains protons and neutrons as separate substructures. It could be that a nucleus is just a bag of the quarks that compose the protons and neutrons. At one time some thought that a nucleus was composed of substructures of alpha particles (helium nuclei). For the analysis here a nucleus is considered as being composed of protons and neutrons.

Let Z represent the number of protons in a nucleus and N the number of neutrons. The atomic number A is the sum of nd N, the total number of nucleons in a nucleus. Let  _Zm_N denote the mass of a nucleus composed of Z protons and N neutrons. The mass of a proton in this notation is  :₁m₀ and that of a neutron  :₀m₁.

One might expect the mass of a (Z, N) nucleus to be Z ₁m₀+N ₀m₁, but generally that is not the value of  _Zm_N. The difference is called the mass deficit Δ

 _ZΔ_N =  Z₁m₀ +  N₀m₁ −  _Zm_N

When the mass deficit is expressed in energy units it is called the binding energy of the nucleus. Usually this binding energy is expressed in millions of electron volts (MeV). The binding energy for a nucleus is the amount of energy that would be required to disintegrate it into its component nucleons. It represents the amount of energy gained by putting the component nucleons into that nucleus' particular structure.

The study of nuclear shell structure began with an investigation of the changes in the binding energy of nuclei as the neutron number is changed by one.

Information on how strongly a particular nucleon is bound can be gained by looking at the generic reaction

_Zm_N + γ → _Zm_N−1 + ₀m₁

Subtracting this equation from Z₁m₀+N₀m₁ gives the result

_ZΔ_N − γ = _ZΔ_N−1

Therefore the energy required to remove a neutron is

γ = _ZΔ_N − _ZΔ_N−1

Thus for the deuteron ₁H₁, γ = 2.2245 MeV − 0 = 2.2245 MeV. For the Helium 4 nucleus the energy required to remove a neutron is γ = 28.2957 − 7.7180 = 20.5777 MeV. For Lithium 6, ₃Li₃, γ = 31.9941− 26.3307 = 5.6634 MeV.

 _Zε_N =  _ZΔ_N −  _ZΔ_N−1

The graph of the above data indicates a dependence on the odd-even-ness of the proton and neutron number. This is not a dependence on the equality of the proton and neutron number which is equal for all the cases considered.

The form of the dependence is of the nature

B = c₀ + (−1)^Zc₁ + c₂c₃^Z

where the c_i are constants approximately equal to 12.4 MeV, 1.8 MeV, 10.5 MeV and 0.8, respectively.

There are a variety of other models of nuclear structure. The most prominent is the liquid-drop model. This model is embodied in the Semiempirical Formula for Nuclear Masses, which gives good explanation for nuclear masses except for isolated cases such as the alpha particle, the ₂He⁴ nucleus. It appears that there is some sort of shell structure that accounts for the special cases that are not properly predicted by the semiempirical mass formula.

Magic Numbers

Maria Goeppert-Mayer and other physicists examining the properties of the isotopes of elements discerned that isotopes in which the proton and/or the neutron numbers were particular values have notable properties such as stability. These magic numbers are

The magic numbers for atomic electron structure are perfectly explainable in terms of a formula. The increments in the particular stable atomic numbers are equal to twice the square of an integer. These increments correspond to the maximum occupancy levels of shells. Therefore in looking for an explanation of the nuclear magic numbers it is reasonable to look at the increments in the magic numbers; i.e.,

The difference in consecutive magic numbers could be sums of shell occupancy levels. For example, the value of 44 might correspond to two shells of 22 each. Likewise a value of 22 might correspond to two shells, one of occupancy 20 and one of occupancy 2.

The maximum occupancy levels for the atomic electron shells being the square of an integer n corresponds to the sum of the first n odd numbers. Let look at the sum of the first even numbers.

The cumulative sum values show up in the set of increments of nuclear magic numbers and the increments can be represented as simple sum of these cumulative sum values, as is shown below.

If there is anything to this pattern the number 14=2+12 should be something like a magic number increment (rather than a magic number). This would mean that 22 and 34 should be nearly magic numbers. Titanium, atomic number 22, has five stable isotopes but scandium, the element with atomic number 21, has only one and vanadium, the element with atomic number 23 has only two. Selenium, atomic number 34, has six stable isotopes but bromine, the element with atomic number 35, has only two and arsenic, the element with atomic number 33 has only one. This is suggestive of the magic-ness of 22 and 34 and thus that 14=2+12 is a magic number increment. The numbers 6 and 14 would represent the maximum occupancy of some shell structure. If higher shell structures can appear without all the lower shell structures being filled then 6 and 14 could themselves be considered to be magic numbers.

The Stable Isotope and Isotone Data

The number of stable nuclides for magic numbers compared to the adjacent numbers look impressive but, as the table below shows, there is an even-odd alternation in the relationship between the number of stable isotopes and the atomic (proton) number.

The odd-even alternation indicates a pairing of protons within the nucleus; perhaps the existence of alpha particle subsystems.

The average number of stable isotopes increases with proton number reaching a peak for proton number 50 (Tin) and declines generally thereafter, as shown in the table below.

The real test of the magic-ness of a number is whether it appears so in terms of the neutron number. The numbers of stable isotopes as a function of neutron number are:

Note that Silicon is the element with atomic number 14 and it has three stable isotopes whereas the elements Aluminum and Phosphorus having atomic numbers 13 and 15, respectively, have only one stable isotope each. The isotope of silicon which is most prevalent has atomic weight 28=14+14. The binding energy per nucleon for silicon is however not exceptionally high, but it is the end result of a fusion process in stars called silicon burning, indicating a significant degree of stability.

The case for the magicalness of the magic numbers is much less impressive in terms of neutron number than for the proton numbers. In particular, there is as much of a case for 14 and 22 being magic in terms of neutron number as for 20. Likewise the case for 8 is not notable in terms of neutron number. Fifty two seems as much of a magic neutron number as 50. Likewise 70 would appear to be as much of a magic number as 82.

The magicality of neutron and proton numbers can be established unambiguously by looking at the incremental binding energies of nuclides as a function of neutron number or proton number. The incremental binding energy drops significantly at the magic numbers. For example,

The sharp drop occurs when the neutron number is 82. The sawtooth pattern is due to the formation of neutron pairs. For more on this topic where the significance of the magic numbers is clearly established and 14 is revealed to also be a magic number see

• The Analysis of the Impact of Additional Neutrons on the Binding Energy of Nuclides and the Matter of Magic Numbers
• The Analysis of the Impact of Additional Neutrons on the Binding Energy of Nuclides and the Matter of Magic Numbers, Part 2
• The Number 126 as a Magic Number for Neutrons
• The Binding Energies of Integral Alpha Particle Nuclides
• The Drop in the Incremental Increases in Binding Energy of Nuclides at the Magic Numbers of Neutrons
• Quantitative Analysis of Relationships Concerning Binding Energy and Number of Neutrons in Nuclides for the Various Nuclear Shells
• Quantitative Analysis of Relationships Concerning Binding Energy and Number of Neutrons for the 50-t0-82 Neutron Shell
• A Preliminary Investigation of Nuclear Properties Based Upon a Model of Shell Occupancy
• The Relationship Between the Incremental Binding Energy and the Number of Additional Neutrons in a Nuclide
• The Pattern of the Impact of Additional Neutrons on Binding Energies of Nuclides
• The Differential Impact of Neutrons and Protons on the Binding Energy of Nuclides
• The Differential Impact of Additional Protons and Neutrons on the Binding Energy of Nuclides
• Evidence for Alpha Particle Substructures in Nuclei
• The Net Effect on Binding Energy of the Addition or Subtraction of a Proton or Neutron from Integral Alpha Particle Nuclides
• The Effect on Binding Energy of the Addition of a Proton/Neutron Pair to Integral Alpha Particle Nuclides

For other models of nuclear structure see Nuclear Structure