This is an attempt to take nuclear properities such as binding energy and determine other properties such as dimensions of nuclear structures. The analysis starts with a model that is more easily explanable in terms of the shell model for the electrons in an atom.

The electrons in an atom are arrayed in orbits or shells. The first shell can contain no more than 2 electrons; the second no more than eight and the third no more than 18. The numbers of electrons corresponding to filled shells are 2, 10, 28 and so on. These are the magic numbers for electron shells corresponding to the magic numbers for nuclear structure, 2, 6, 8, 14, 20, 28, 50, 82 and 126.

Consider an atom with a nucleus of Z protons and N neutrons. Suppose that for the i-th shell there are N₀ neutrons in the shells which precede the i-th and N₁ in the shells which folow the i-th shell. The inner N₀ electrons completely shield the effect of N₀ protons in the nucleus on any electrons in the i-th shell. The N₁ outer electrons have no net effect on the electrons in the i-th shell. Therefore the effective charge of the nucleus is reduced to Z−N₀. But this is not the only adjustment to the effective charge of the nucleus. If there are n electrons in the i-th shell then the charge distribution of each can be considered to be half inside and half outside the radius of an electron in the i-th shell. Thus the effective charge experienced by one electron in the i-th shell is Z'=(Z−N₀-(n-1)/2)

Let r_i(Z') be the midpoint radius of the i-th shell for the case in which the nucleus has an effective charge of Z'. From previous work it is known that r_i(Z') is at least approximately inversely proportional to Z; i.e.,

r_i(Z') = ν/Z'

where ν is a constant.

As stated above, from the viewpoint of an electron located at r_i another electron in the i-th shell is half inside and half outside the midpoint radius of the i-th shell. Thus an electron located in the i-th shell shields one half of a unit charge. Therefore if there are n electrons in the i-th shell each one experiences an effective charge in the nucleus of Z'=Z−N₀−(n-1)/2.

The potential energy V of one electron at a distance r from a positive charge of Z' is

V = KZ'/r

The total potential energy W of n neutrons in the i-th shell is

W = nV = nKZ'/r_i
which reduces to
W = (K/r_i)n[Z−N₀−(n-1)/2]
W = (K/r_i)n([Z−N₀+1/2] − n/2)
W = (K/r_i)[(Z−N₀+1/2)n − n²/2]

The incremental change in the potential energy ΔW=W(n)−W(n-1) is given by

ΔW = (K/r_i)[(Z−N₀+1/2) − (n+1/2)]
which reduces to
ΔW = (K/r_i)[(Z−N₀) − n]
and further to
ΔW = (K/r_i)(Z−N₀) − (K/r_i)n

This is a relationship of the form ΔW = a_i − b_in. The ratio a_i/b_i is then equal to (Z−N₀), which can be considered to be the effective charge of the nucleus at the beginning of the filling of the i-th shell. Also shell radii should be inversely proportional to the slopes. However since the radii depend upon the effective charge

r_i(Z')/r_j(Z") = b_j(Z")/b_i(Z')

When a shell is filled up and the next electron goes into an outer shell then the effect of all the electrons which were felt only at half their value becomes full value and there is a corresponding drop in potential energy.

The dynamics of the nucleus with its two, and possibly three, varieties of particles is vastly more complex than the electronic shells. The presentation of the nuclear version of the model will be deferred until after some preliminary empirical investigations are made.

Empirical Estimates

The 82 to 126 Neutron Shell

The relationship between the incremental changes in binding energy and number of neutrons for lead is:

The regression equation for lead for the range of no more than 126 neutrons is

ΔB = 11.66121 − 0.12915n + 2.133599u

where u is a variable that is equal to 1 if the total number of neutrons is even and 0 if not. Thus the enhancement of binding energy due to the formation of a neutron pair is 2.133599 MeV. The ratio of the intercept to the slope is 90.2932 neutrons.

For bismuth the relation ship is

The regression equation is

ΔB = 12.26606 − 0.13101n + 1.47034u

The ratio of the intercept to the slope is 93.629 neutrons, which is 10.629 neutrons greater than the atomic number of 83. The ratios of the intercepts to the slopes for bismuth and lead should differ by one neutron. They differ by 2.3358 neutrons.

According to the model the slopes should be inversely proportional to radii and radii are inversely proportional to the effective charge of the nucleus. Therefore the ratio of the slopes should be directly proportional to the ratio of the effective charges of the nuclei. The ratio of the slope for bismuth to the slope for lead is 1.014. The ratio of the atomic numbers is 83/82 which is 1.0122. The ratio of the estimated effective charge of bismuth, 93.629, to that of lead, 90.2932, is 1.037. This is not a confirmation of the model but it is also not a denial of the model.

For mercury (atomic number 80) the relationship between incremental increase in binding energy and the number of neutrons is

The regression equation for mercury is

ΔB = 11.44383 − 0.14051n + 2.170442u

The ratio of the intercept to the slope is 81.4428 neutrons, which is 1.44281 neutrons greater than the atomic number of 80. The magnitude of the slope for mercury is larger than that for lead whereas the model says it should be smaller.

For gold (atomic number 79) the relationship and regression equation are

ΔB = 11.92086 − 0.14032n + 1.526794u

The ratio of the intercept to the slope is 84.9531 neutrons, which is 5.95308 neutrons greater than the atomic number of 79. The slope for gold is very close to that for mercury.

The 50 to 82 Neutron Shell

For indium (atomic number 49) the relationship and regression equation are

ΔB = 10.90464 − 0.22807n + 2.200633u

The ratio of the intercept to the slope is 47.8123 neutrons, which is 1.187737 neutrons less than the atomic number of 49.

For silver (atomic number 47) the relationship and regression equation are

ΔB = 10.46401 − 0.2057n + 2.029704u

The ratio of the intercept to the slope is 50.8692 neutrons, which is 3.86916 neutrons greater than the atomic number of 47. The slope for silver is close to that for indium.

The 28 to 50 Neutron Shell

Bromine is of atomic number 35. The relationship and regression results are

ΔB = 10.99744 − 0.29832n + 2.365964u

The ratio of the intercept to the slope is 36.8647 neutrons, which is 1.86466 neutrons greater than the atomic number of 35.

Selenium is of atomic number 35. The relationship and regression results are

ΔB = 10.43328 − 0.32336n + 3.498238u

The ratio of the intercept to the slope is 32.2649 neutrons, which is 1.735 neutrons less than its atomic number of 34.

The 20 to 28 Neutron Shell

Calcium is of atomic number 20. The relationship and regression results are

Δ = 8.65859 − 0.22798n + 3.219435u

The ratio of the intercept to the slope is 37.9788 neutrons, which is nearly 18 neutrons greater than the atomic number of 20.

The 14 to 20 Neutron Shell

Sulfur is of atomic number 16. The relationship and regression results are

Δ = 9.40556 − 0.79593n + 3.63517u

The ratio of the intercept to the slope is 11.8171 neutrons, which is about 4 neutrons less than the atomic number of 16.

Silicon is of atomic number 14. The relationship and regression results are

Δ = 9.163565 − 0.88301n + 3.484508u

The ratio of the intercept to the slope is 10.3777 neutrons, which is about 3.6 neutrons less than the atomic number of 14.

The 8 to 14 Neutron Shell

Neon is of atomic number 10. The relationship and regression results are

Δ = 7.510346 − 0.76474n + 4.39858u

The ratio of the intercept to the slope is 9.8208 neutrons, which is 0.179 neutrons less than the atomic number of 10.

Inferences Concerning the Structure of Nuclides

Taking the results for bromine as typical of the 28 to 50 shell and those for silver for the 50 to 82 shell the ratio of the slopes is 0.29832/0.2057=1.45. This should be the approximate value of the ratio of the radius of the 50 to 82 neutron shell to the radius of the 28 to 50 neutron shell.

If the results for bismuth are taken to be typical for the 82 to 126 neutron shell then the ratio of the slopes is 1.57.

(To be continued.)