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Although the topic of this page is nuclear structure it is conveniently instructive to introduce the problems of nuclear structure by looking at the model of atomic electron structure. Atomic electron structure is a theory that is well developed and well understood and, in fact, is the basis for looking for a shell structure for nuclei.
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(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 explanation of the magic numbers for electron 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.
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.
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 _{Z}m_{N} denote the mass of a nucleus composed of Z protons and N neutrons. The mass of a proton in this notation is :_{1}m_{0} and that of a neutron :_{0}m_{1}.
One might expect the mass of a (Z, N) nucleus to be Z _{1}m_{0}+N _{0}m_{1}, but generally that is not the value of _{Z}m_{N}. The difference is called the mass deficit Δ
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
Subtracting this equation from Z_{1}m_{0}+N_{0}m_{1} gives the result
Therefore the energy required to remove a neutron is
Thus for the deuteron _{1}H_{1}, 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, _{3}Li_{3}, γ = 31.9941− 26.3307 = 5.6634 MeV.
Energy Required to Dislodge a Neutron  

Z  N  Element  Change in Binding Energy (MeV) 
1  1  Hydrogen  2.225 
2  2  Helium  20.578 
3  3  Lithium  5.663 
4  4  Beryllium  18.900 
5  5  Boron  8.436 
6  6  Carbon  18.722 
7  7  Nitrogen  10.553 
8  8  Oxygen  15.664 
9  9  Fluorine  9.149 
10  10  Neon  16.865 
11  11  Sodium  11.070 
12  12  Magnesium  16.531 
13  13  Aluminum  11.365 
14  14  Silicon  17.180 
15  15  Phosphorus  11.319 
16  16  Sulfur  15.042 
17  17  Chlorine  11.508 
18  18  Argon  15.255 
The graph of the above data indicates a dependence on the oddevenness 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
where the c_{i} are constants approximately equal to 12.4 MeV, 1.8 MeV, 10.5 MeV and 0.8, respectively.
Energy Required to Dislodge a Proton  

Z  N  Element  Change in Binding Energy (MeV) 
1  1  Hydrogen  2.225 
2  2  Helium  19.814 
3  3  Lithium  4.588 
4  4  Berylium  17.255 
5  5  Boron  6.586 
6  6  Carbon  15.957 
7  7  Nitrogen  7.551 
8  8  Oxygen  12.127 
9  9  Fluorine  5.607 
10  10  Neon  12.844 
11  11  Sodium  6.740 
12  12  Magnesium  11.693 
13  13  Aluminum  6.306 
14  14  Silicon  11.585 
15  15  Phosphorus  5.595 
16  16  Sulfur  8.864 
17  17  Chlorine  5.143 
18  18  Argon  8.507 
There are a variety of other models of nuclear structure. The most prominent is the liquiddrop 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 _{2}He^{4} 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.
Maria GoeppertMayer 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
Magic Numbers  2  8  20  28  50  82  126 
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.,
Magic Numbers  2  8  20  28  50  82  126 
Increments  2  6  12  8  22  32  44 
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.
Even Numbers  2  4  6  8  10  12  14 
Cumulative Sum  2  6  12  20  30  42  56 
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.
Increments of Magic Numbers  2  6  12  8  22  32  44 
Generation from Cumulative Sums of Even Numbers  2  6  12  2+6  2+20  2+30  2+42 
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 magicness 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 number of stable nuclides for magic numbers compared to the adjacent numbers look impressive but, as the table below shows, there is an evenodd alternation in the relationship between the number of stable isotopes and the atomic (proton) number.
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 
Proton Number  31  32  33  34  35  36  37  38  39  40 
Number of Stable Isotopes 
2  4  1  5  2  5  1  4  1  4 
Proton Number  41  42  43  44  45  46  47  48  49  50 
Number of Stable Isotopes 
1  6  0  7  1  6  2  6  1  10 
Proton Number  51  52  53  54  55  56  57  58  59  60 
Number of Stable Isotopes 
2  4  1  9  1  6  1  2  1  5 
Proton Number  61  62  63  64  65  66  67  68  69  70 
Number of Stable Isotopes 
0  4  1  6  1  7  1  6  1  7 
Proton Number  71  72  73  74  75  76  77  78  79  80 
Number of Stable Isotopes 
1  5  2  4  1  5  2  5  1  6 
Proton Number  81  82  83  84  85  86  87  88  89  90 
Number of Stable Isotopes 
2  3 
The oddeven 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.
Proton Number Range  0's  10's  20's  30's  40's  50's  60's  70's  80's  90's 
Average Number of Stable Isotopes 
1.7  2.3  2.8  3.0  3.4  3.7  3.2  3.3  1.1  0.0 
The real test of the magicness 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:
Neutron Number  1  2  3  4  5  6  7  8  9  10 
Number of Stable Isotopes 
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 Isotopes 
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 Isotopes 
0  3  2  3  1  4  4  4  1  4 
Neutron Number  31  32  33  34  35  36  37  38  39  40 
Number of Stable Isotopes 
1  3  1  3  0  2  3  7  1  4 
Neutron Number  41  42  43  44  45  46  47  48  49  50 
Number of Stable Isotopes 
1  5  3  4  1  3  1  4  2  5 
Neutron Number  51  52  53  54  55  56  57  58  59  60 
Number of Stable Isotopes 
2  4  3  4  2  3  2  3  1  3 
Neutron Number  61  62  63  64  65  66  67  68  69  70 
Number of Stable Isotopes 
1  5  1  3  1  3  2  2  1  7 
Neutron Number  71  72  73  74  75  76  77  78  79  80 
Number of Stable Isotopes 
4  5  1  5  2  4  2  4  3  3 
Neutron Number  81  82  83  84  85  86  87  88  89  90 
Number of Stable Isotopes 
2  8  4  1  3  1  2  4  0  6 
Neutron Number  91  92  93  94  95  96  97  98  99  100 
Number of Stable Isotopes 
2  4  1  4  1  6  1  5  2  4 
Neutron Number  101  102  103  104  105  106  107  108  109  110 
Number of Stable Isotopes 
2  2  2  5  2  2  0  3  0  3 
Neutron Number  111  112  113  114  115  116  117  118  119  120 
Number of Stable Isotopes 
2  2  1  2  1  2  3  1  3  3 
Neutron Number  121  122  123  124  125  126  127  128  129  130 
Number of Stable Isotopes 
1  3  1  3  0  1  1  0  0  3 
Neutron Number  131  132  133  134  135  136  137  138  139  140 
Number of Stable Isotopes 
0  0  0  0  0  3  1  1  1  0 
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
For other models of nuclear structure see Nuclear Structure
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