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On the Extention of the Bohr Model to Nuclides

The Bohr model performs marvelously well in explaining the spectrum of hydrogen and hydrogen-like atoms and ions. It works equally as well in explaining the ionization energies of atoms and ions with multiple electrons in the outer shell. The Bohr model of electrons in atoms is based upon the electrons being held in orbits about nuclei by the electrostatic force which obeys the inverse distance squared law.

The neutrons and protons in a nucleus are held together by an attractive force which is not a strictly inverse distance squared law. This means that all of the wonderful simplifications of an inverse distance squared law do not hold. These simplifications are:

The Bohr model implies that the ionization energy of an electron is proportional to the square of the net charge Z exerted on that electron. This net charge Z is the number of protons in the nucleus less the shielding by electrons in shells within the shell of the electron and electrons in the same shell. In the nucleus there are concentric shells of protons and neutrons. A neutron in a shell is subject to an attraction by the protons in inner shells but a repulsion by neutrons in inner shells and the same shell. On this latter point see neutron repulsion. This means that Z is a weighted sum of the effects of not only the protons and neutrons in various inner shells, but also those in outer shells. Furthermore the incremental binding energy due to a neutron is not necessarily proportional to Z². The functional relationship is more complicated. Thus the energy relationships in nuclei are enormously more complicated than those of electrons in atoms. But some order can be discerned.

Consider the relationships between incremental binding energies for the first through fourth neutrons in the second neutron shell and the proton number.

When the above relationships are displayed in the same graph a degree of order can be discerned.

The curves for the second and fourth neutrons appear to be related for higher proton numbers and also, to a lesser degree, the first and third. This suggests that spin pairings of the neutrons are involved. However the curves for the first and second neutrons start at the same level for hydrogen (proton number equal to one) and likewise for the third and fourth neutrons.

It notable that except for the curve for the third neutron in the shell the curves have an initial segment that appears to be quadratic. It is tempting to characterize the relationships as cubic, but the declining slope for higher proton numbers is probably due to there being protons in shells beyond the second shell. The attraction of such protons might in some way counterbalances the attraction of the protons in the inner shell thus effectively reducing the potential energy of the neutron from what it otherwise would be.

As indicated above the simple proton number is not likely the relevant variable for determining the incremental binding energies. Below are the data for incremental binding energies, where #n and #p stand for the number of neutrons and protons, respectively.

The Incremental Binding Energies of the
First Neutron in the Second Neutron Shell as a
Function of the Occupancy of the Proton Shells
Incremental
Binding
Energy
(MeV)
#n
1st shell
#n
2nd shell
#p
1st shell
#p
2nd shell
#p
3rd shell
-2.901821 2 1 1 0 0
-0.885674 2 1 2 0 0
5.6646 2 1 2 1 0
10.6764 2 1 2 2 0
13.0178 2 1 2 3 0
14.2521 2 1 2 4 0

Because the relationship between incremental binding energy IBE) and the proton shell occupancies is nonlinear, possibly quadratic, all of the cross-product terms should be used in a regression equation. However there are not enough observations to do so. In lieu of such a regression, a regression involving the square of the proton number was carried out. The results were:

IBE = -0.950167714(#p)² + 4.871284314(#p1st)
+ 11.36421651(#p2nd) − 6.8229376
[-14.4] [15.2]
[21.3] [-12.9]
R² = 0.99953

The figures in square brackets below the array of regression coefficients are the t-ratios for the corresponding regression coefficient; i.e., the regression coefficients divided by their standard deviation.

The Incremental Binding Energies of the
Second Neutron in the Second Neutron Shell as a
Function of the Occupancy of the Proton Shells
Incremental
Binding
Energy
(MeV)
#n
1st shell
#n
2nd shell
#p
1st shell
#p
2nd shell
#p
3rd shell
-2.88 2 2 1 0 0
1.8591 2 2 2 0 0
7.2499 2 2 2 1 0
18.89911 2 2 2 2 0
18.5766 2 2 2 3 0
21.2864 2 2 2 4 0
22.85 2 2 2 4 1

IBE = -1.238122857(#p)² + 7.656084857(#p1st)
+ 14.92311286(#p2nd) + 17.81136086(#p3rd)
− 9.297962
[-1.6] [2.0]
[2.4] [1.5]
[-1.5]
R² = 0.97370

The Incremental Binding Energies of the
Third Neutron in the Second Neutron Shell as a
Function of the Occupancy of the Proton Shells
Incremental
Binding
Energy
(MeV)
#n
1st shell
#n
2nd shell
#p
1st shell
#p
2nd shell
#p
3rd shell
3.1 2 3 1 0 0
-0.4491 2 3 2 0 0
2.0328 2 3 2 1 0
1.66539 2 3 2 2 0
8.4363 2 3 2 3 0
13.1194 2 3 2 4 0
15.6913 2 3 2 4 1
17.009 2 3 2 4 2

IBE = 0.824337143(#p)² - 5.768069929(#p1st)
- 3.144056357(#p2nd) - 9.598495357(#p3rd)
+ 8.043732786
[2.1] [-3.0]
[-1.0] [-1.6]
[2.5]
R² = 0.98027

The Incremental Binding Energies of the
Fourth Neutron in the Second Neutron Shell as a
Function of the Occupancy of the Proton Shells
Incremental
Binding
Energy
(MeV)
#n
1st shell
#n
2nd shell
#p
1st shell
#p
2nd shell
#p
3rd shell
2.588 2 4 1 0 0
4.0636 2 4 2 0 0
6.8123 2 4 2 1 0
11.4541 2 4 2 2 0
18.721828 4 3 2 3 0
20.064 2 4 2 4 0
23.17523 4 3 2 4 1
24.95 4 3 2 4 2

IBE = 0.818034702(#p)² - 2.423357964(#p2nd)
- 9.870034964(#p3rd) - 0.99483906
[3.3] [-1.2]
[-2.6] [1.1]
R² = 0.99325

Conclusions

Although there are hints of a quadratic relationship at low values of the proton number such as prevails for the ionization energies of electrons at higher proton numbers the relationships are ones of declining slopes. Statisticcal relationships show some promise but the small number of observations prevent any full testing of the Bohr model extended to nuclides.


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