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The Shell Model of Nuclear Structure

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.

The Model of Atomic Electron Structure

If the explanation for atomic electron structure had not developed from the perspective of explanaining 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 volts, required to remove an electron from an atom of that element. For example, the ionization potential for hydrogen is about 13.6 volts. Removing an electron from an atom of helium, on the other hand, requires about 24.6 volts. To remove a second electron from that atom of helium would require about 54.4 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 alkalai metals; lithium, sodium, potassium, cesium, rubidium and francium. Hydrogen may also be consider 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 nuclie). 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 Z nd N, the total number of nucleons in a nucleus. Let  :ZmN 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  ZmN. The difference is called the mass deficit Δ


 ZΔN =  Z1m0 +  N0m1 −  ZmN
 

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


ZmN + γ → ZmN−1 + 0m1
 

Subtracting this equation from Z1m0+N0m1 gives the result


ZΔN − γ = ZΔN−1
 

Therefore the energy required to remove a neutron is


γ = ZΔNZΔN−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.


 ZεN =  ZΔN −  ZΔN−1
 

Energy Required to Dislodge a Neutron
ZNElementChange in
Binding Energy
(MeV)
11Hydrogen2.225
22Helium20.578
33Lithium5.663
44Berylium18.900
55Boron8.436
66Carbon18.722
77Nitrogen10.553
88Oxygen15.664
99Fluorine9.149
1010Neon16.865
1111Sodium11.070
1212Magnesium16.531
1313Aluminum11.365
1414Silicon17.180
1515Phosphorus11.319
1616Sulfur15.042
1717Chlorine11.508
1818Argon15.255

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 + c2c3Z
 

where the ci are constants appproximately equal to 12.4 MeV, 1.8 MeV, 10.5 MeV and 0.8, respectively.

Energy Required to Dislodge a Proton
ZNElementChange in
Binding Energy
(MeV)
11Hydrogen2.225
22Helium19.814
33Lithium4.588
44Berylium17.255
55Boron6.586
66Carbon15.957
77Nitrogen7.551
88Oxygen12.127
99Fluorine5.607
1010Neon12.844
1111Sodium6.740
1212Magnesium11.693
1313Aluminum6.306
1414Silicon11.585
1515Phosphorus5.595
1616Sulfur8.864
1717Chlorine5.143
1818Argon8.507

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

Magic Numbers2820285082126

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 Numbers2820285082126
Increments26128223244

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 Numbers2468101214
Cumulative Sum261220304256

The cumulative sum 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
26128223244
Generation from
Cumulative Sums
of Even Numbers
26122+6 2+202+302+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 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 Isomer 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.

Proton
Number
12345678910
Number of
Stable Isotopes
2221222313

Proton
Number
11121314151617181920
Number of
Stable Isotopes
1313142325

Proton
Number
21222324252627282930
Number of
Stable Isotopes
1513141525

Proton
Number
31323334353637383940
Number of
Stable Isotopes
2415251414

Proton
Number
41424344454647484950
Number of
Stable Isotopes
16071626110

Proton
Number
51525354555657585960
Number of
Stable Isotopes
2419161215

Proton
Number
61626364656667686970
Number of
Stable Isotopes
0416171617

Proton
Number
71727374757677787980
Number of
Stable Isotopes
1524152516

Proton
Number
81828384858687888990
Number of
Stable Isotopes
23

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.

Proton
Number
Range
0's10's20's30's40's50's 60's70's80's90's
Average
Number of
Stable Isotopes
1.72.32.83.03.43.73.23.31.10.0

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:

Neutron
Number
12345678910
Number of
Stable Isotopes
2211222213

Neutron
Number
11121314151617181920
Number of
Stable Isotopes
1313131305

Neutron
Number
21222324252627282930
Number of
Stable Isotopes
0323144414

Neutron
Number
31323334353637383940
Number of
Stable Isotopes
1313023714

Neutron
Number
41424344454647484950
Number of
Stable Isotopes
1534131425

Neutron
Number
51525354555657585960
Number of
Stable Isotopes
2434232313

Neutron
Number
61626364656667686970
Number of
Stable Isotopes
1513132217

Neutron
Number
71727374757677787980
Number of
Stable Isotopes
4515242433

Neutron
Number
81828384858687888990
Number of
Stable Isotopes
2841312406

Neutron
Number
919293949596979899100
Number of
Stable Isotopes
2414161524

Neutron
Number
101102103104105106107108 109110
Number of
Stable Isotopes
2225220303

Neutron
Number
111112113114115 116117118119120
Number of
Stable Isotopes
221212313 3

Neutron
Number
121122123124 125126127128129130
Number of
Stable Isotopes
1313011003

Neutron
Number
131132133134135 136137138139140
Number of
Stable Isotopes
00000311 10

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 signficance 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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