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A Shell Occupancy Model of Incremental
Binding Energies of Nuclides

This is an attempt to explain the patterns of the increments in nuclear binding that result from successive additions of neutrons. The analysis starts with a model that is more easily explanable in terms of the shell model for the electrons in an atom. Therefore the explanation of the model will start with the case of electrons in atoms.

The electrons in an atom are arrayed in orbits or shells. The first shell can contain no more than two electrons; the second shell no more than eight electrons 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 by the inner shell electrons. 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 reduced by (n-1)/2 charges and the reduced charge of the nucleus is

Z'=(Z−N0-(n-1)/2) = (Z−N0+½ - n/2)

Let ri(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 ri(Z') is at least approximately inversely proportional to Z; i.e.,

ri(Z') = ν/Z'(n)

where ν is a constant. The dependence of Z' and consequently ri will be ignored in the material which immediately follows.

As stated above, from the viewpoint of an electron located at ri 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'/ri
which, if the dependence of Z'
on n is ignored, reduces to
W = (K/ri)n[Z−N0−(n-1)/2]
W = (K/ri)n([Z−N0+½] − n/2)
W = (K/ri)[(Z−N0+½)n − n²/2]

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

ΔW = (K/ri)[(Z−N0+½) − (n-½)]
which reduces to
ΔW = (K/ri)[(Z−N0+1) − n]
and further to
ΔW = (K/ri)(Z−N0+1) − (K/ri)n

This is a relationship of the form ΔW = ai − bin. The ratio ai/bi is then equal to (Z−N0+1), 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

ri(Z')/rj(Z") = bj(Z")/bi(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.

Here is the values of the ionization energies for electrons in the second electron shell as a function of the number of electrons in that shell. The first graph is for Neon.

The next graph has the same information for the three elements, Oxygen, Fluorine and Neon.

What comes out of the preliminary model presented above is that important structural information can be obtained from the statistical relation between ΔW and the number of particles in a shell, n. In particular, for a regression equation of ΔW on n, the regression equation should be to the form ΔW=a-bn. Furthermore,

Now the dependence of shell radius on the effective charge will be taken into account.

ri=ν/Z' and V=KZ'/ri
therefore
V = KνZ'and
W = nV = KνnZ'² = Kνn[(Z−N0 − ½(n-1)]²
W = Kνn[(Z−N0+½ − ½n]²

To reduce the complexity of the expressions let Z−N0+½ be denoted as Z0. Thus Z'=Z0−½n and hence

W = Kνn(Z0−½)²

The computation of ΔW is a straight forward but messy algebraic task. To make the process more succinct (∂W/∂n) will be computed first.

(∂W/∂n) = V + n(∂V/∂n)
(∂W/∂n) = KνZ'² + nKν[2Z'(−½] = KνZ'²−KνnZ'
(∂W/∂n) = KνZ'(Z'−n)

Therefore

(∂W/∂n) = Kν(Z0−½n)(Z0−3n/2)
(∂W/∂n) = Kν[Z0² −2nZ0 + ¾n²]

This is a quadratic equation in n where the value of (∂W/∂n) declines with increasing n over some range, but the relationship curves upward.

The ratio of the intercept to the absolute value of the coefficient of n is equal to one half of Z0, a quantity closely related to the effective charge on the shell as it begins to fill. Likewise the ratio of the absolute value of the coefficient of n to the coefficient of n² should be equal to (8/3)Z0.

The equation for ΔW is a bit more complex than that of (∂W/∂n) but it is manageable; i.e.,

ΔW = Kν[(Z0+½)² −2(Z0+3/8)n + ¾n²]

Thus the functional relationship between ΔW and n is a quadratic one which is concave upward, as illustrated below.

The dynamics of the nucleus with its two, and possibly three, varieties of particles is vastly more complex than the electronic shells. There are numerous reasons for a model based upon electronic shells not to apply for nuclear shells. The question of the acceptance of the model for nuclear structure will be deferred until after some preliminary empirical results are presented. Instead of potential energy nuclear binding energy will be used. There is reason to believe that binding energy may to closely related to the decrease in potential energy which results from the formation of a nucleus.

Empirical Relationships

Consider the increments in binding energy which occur as additional neutrons are added to a silver nucleus.

For silver (atomic number 47) the relationship is

The dispay starts from the nuclide with 47 protons and 47 neutrons. When the number of neutrons reaches 50, a magic number indicating a filled shell, the value drops. The alternating higher and lower is the effect of the formation of neutron pairs. The relationship is downward sloping to the right, as the model indicates. There seems to be a slight curvature involving the slope becoming less negative to the right, as the model indicates.

For indium (atomic number 49) the relationship is

For gold (atomic number 79) the relationship is

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

Again the relationship is compatible with the model presented previously. Thus the model is worth a more comprehensive testing. However, since the curvatures of the above relationships are very slight, some preliminary testing can be carried out using the following regression equation:

ΔB = c0 − c1n + cc2u(n)
where u(n) equals 1
if n is even
and 0 if n is odd.

The regression equation for silver is

ΔB = 10.46401 − 0.2057n + 2.029704u

In applying the model to nuclear binding energies the paramerters of the model have to be reinterpreted. For electrons in an atom there are protons to which they are attracted and other electrons from which they are repelled. The neutrons in the nucleus have no effect on the electrons. The effective charge is the number of protons in the nucleus less the number of electrons in lower shells less one half the number of electrons in the same shell. For neutrons the effective charge is the number of nucleons (protons as well as neutrons) in lower shells less one half of the number of nucleons in the same shell. Thus the parameter Z0 would correspond to the number of protons plus the number of neutrons in the next lower filled shell. The sum of the numbers of protons and neutrons for a nuclide is called its mass number.

For silver the ratio of the intercept to the slope is 50.87 nucleons, which is 3.87 nucleons greater than the atomic number of 47. The mass number of the nuclide with a filled shell of 50 neutrons is (47+50). One half of this value is 48.5, which is only 2.37 nucleons less than the 50.87 ratio.

For indium the regression equation is

ΔB = 10.90464 − 0.22807n + 2.200633u

The ratio of the intercept to the slope is 47.81 nucleons, which is 1.19 nucleons less than the atomic number of 49, or 1.69 nucleons less than one half of the mass number of the filled shell nuclide of indium. The slope for indium is notable close to that for silver. The model indicates that for the same shell the slopes should be the same. Silver and indium involve the shell containing 51 to 82 neutrons.

The regression equation for gold (atomic number 79) is

ΔB = 11.92086 − 0.14032n + 1.526794u

The ratio of the intercept to the slope is 84.95 neutrons, which is 5.95 neutrons greater than the atomic number of 79. The filled shell of neutrons is 82 so the mass number is 79+82=161. One half of this value is 80.5, just 4.45 nucleons less than the 84.95 ratio.

The regression equation for mercury is

ΔB = 11.44383 − 0.14051n + 2.170442u

The ratio of the intercept to the slope is 81.44 neutrons, which is 1.44 nucleons greater than the atomic number of 80. One half of the mass number of the filled neutron shell nuclide is 81, just 0.44 nucleons less than the 81.44 ratio. Gold and mercury involve the 83-to-126 neutron shell. Note that the slope for gold is very close to that for mercury. and that both are smaller in magnitude than the slopes for silver and indium.

The comprehensive testing of the model is presented elsewhere but some additional empirical results are given below.

The 83 to 126 Neutron Shell

The relationship between the incremental changes in binding energy and number of neutrons for lead (atomic number 82) 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 90.29 neutrons. This is 8.29 nucleons greater than 82, which is the atomic number and also one half of the mass number of the filled neutron shell nuclide.

For bismuth (atomic number 83) 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 10.63 neutrons greater than the atomic number of 83. The ratios of the intercepts to the slopes for bismuth and lead should, according to the model, 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.

The 29 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 34. 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 21 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 15 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 9 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.

More detailed empirical results are given in Quantitative Analysis of Incremental Binding Energies of Nuclides.

Inferences Concerning the Structure of Nuclides

Taking the results for bromine as typical of the 29 to 50 shell and those for silver for the 51 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 51 to 82 neutron shell to the radius of the 29 to 50 neutron shell.

(To be continued.)


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