When the binding energies of the isotopes of an element are compiled and the incremental increases in binding energy per additional neutron are computed the relationships are of the form shown below for lead (atomic number 82).

The display shows an alternating higher/lower fluctuation probably due to the formation of neutron pairs. There is a smooth, regular downward trend until the number of neutrons reaches 126. Above 126 the relationship is different from what is below 126. There is a drop in the level and a reduction of the amplitude of the fluctuations when the number of neutrons goes beyond 126.

The relationship of incremental increases in binding energy to number of neutrons for mercury (atomic number 80) is of the same form.

The break in the relationship comes at the same magic number of 126. This indicates that there is a shell that is being filled as neutron are added. At 126 neutrons an arrangement of neutrons is completed and beyond 126 the neutrons go into a different arrangement (shell). The completion points are at 2, 6, 8, 14, 20, 28, 50, 82 and 126, indicating that the successive shells have capacities of 2, 4, 2, 6, 8, 22, 32 and 44 neutrons.

This is an examination of the magnitude of the drops in incremental binding energy that occur at the completion points of the nuclear shells. The other aspects of the relationships, such as the slope and the amplitude of the even/odd fluctuations, are examined elsewhere at Neutron Magic I, Neutron Magic II, and Neutron Magic III. <

Analysis Based Upon an Electronic Shell Model

Consider an atom with a nucleus of Z protons and N neutrons. Suppose that for the i-th shell there are N₀ electrons which are in the shells which precede the i-th and N₂ 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 N₂ 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 ofthe nucleus.

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') = νi²/Z'

where ν is a constant and i is the quantum number for the i-th shell.

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 electrons 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')

The incremental potential energy ΔW is thus a function of the number of electrons in the atom so the previous formula for ΔW is more properly expressed as

ΔW(N₀+n) = (K/r_i)(Z−N₀) − (K/r_i)n

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. Let N₁ be the capacity of the i-th shell. Then the incremental potential energy for the electron which exactly fills the i-th shell is

ΔW(N₀+N₀) = (K/r_i)[Z − N₀ − N₁]
or, equivalently
ΔW(N₀+N₁) = (K/r_i)[Z − (N₀+N₁)]

The potential energy when there the i-th shell is filled and there is one electron in the (i+1) shell is the potential energy of the filled i-the shell, W₁ plus the potential energy of the one electron in the (i+1) shell. Therefore the incremental potential energy is just the potential energy of that single electron in the (i+1) shell, which is

ΔW(N₀+N₁+1) = (K/r_i+1)[Z − (N₀+N₁)]

D = ΔW(N₀+N₁) − ΔW(N₀+N₁+1)

From the previous formulas this means that

D = K[Z − (N₀+N₁)][1/r_i − 1/r_i+1]

From a previous formula it is known that

1/r_i = [Z − (N₀+N₁)]/(νi²)

where ν is a constant.

Thus

D = [Z − (N₀+N₁)]²(1/ν)[1/i² − 1/(i+1)²]
which reduces to
D = [Z − (N₀+N₁)]²(1/ν)[(2i+1)/(i²(i+1)²)]

The term (N₀+N₁) represents the number for completely filled shells; in effect a magic number. Thus the above formula says the drop in incremental potential energy is a quadratic function of the deviation of the atomic number Z from the magic number, a parabolic function.

Thus the relationship between D and Z could look something like this.

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 magnitude of the drop in incremental binding energy can be computed in a variety of ways. The method used here is to average the drops from the high-to-high and the low-to-low points. That is to say, the drop from the high point of the fluctuation at or below the magic number to the high point above the magic number is computed and likewise the analogous drop from the low-to-low. If N is the magic number and ΔB is the incremental binding energy then the drop D is computed as:

D = [(ΔB(N)−ΔB(N+2)) + (ΔB(N-1)−ΔB(N+1))]/2
which is the same as
D = [((ΔB(N)+ΔB(N-1))/2 − (ΔB(N+2)+ΔB(N+1))/2]

This method slightly overestimates the magnitude of the drop, but the overestimate is relatively small.

For the breaks at 50 neutrons the magnitude of the drop as a function of the atomic number (number of protons) of elements is as follows.

There is a slight even/odd fluctuation but generally the form is of a decrease to a minimum then the rise to a maximum followed by a decrease. The maximum occurs at atomic number 39 (yttrium), which is exactly half way through the shell of capacity 22; i.e., 39=28+0.5(22).

For the breaks at 28 neutrons the functional form is generally the same as that for 50 neutrons.

The maximum drop occurs at atomic number 21 (scandium) which is exactly half way between the magic numbers of 14 and and 28.

For the breaks at 20 neutrons again there is a decline to a minimum and then a rise to a maximum. The maximum drop occurs at atomic number 17 (chlorine). This value is half way between the magic numbers of 14 and 20.

For the breaks at 14 neutrons there is a severely limited number of observations.

The maximum drop occurs for atomic number 9 (fluorine), which is not at the midpoint of the 8-to-14 shell, but the functional form can be construed to be a truncated version of the form for the higher shells shown previously.

Interpretation of the Results

Reasoning from the model of electronic shells indicated that the drop in the incremental binding energy should be of the form

D = [Z − (N₀+N₁)]²(1/ν)[(2i+1)/(i²(i+1)²)]

The relationship between D and Z for the 50 neutron transition was found to be

Based upon the indicated functional form quadratic regression equations were fitted to the data. One for the data for Z=32 to Z=39 and the other was for Z=39 to Z=48. The regression equations were of the form

D = c₀ + c₁Z + c₂Z² + c₃u

where u=1 if Z is even and 0 if Z is odd.

The regression results for the curve on the left are

D = 10.02593 - 0.62145Z + 0.011351Z² - 0.12361u
R² = 0.9914

This equation can be algebraically rearranged to

D = 1.520451 + 0.011351(Z − 27.373)² − 0.12361

The value 27.373 is notably close to the shell transition point (magic number) of 28.

For the curve on the right the regression results are

D = 19.6932 - 0.71892Z + 0.0074592Z² - 0.1237u
R² = 0.9835

The regression equation can be put into the form

D = 2.37004 + 0.007459(Z − 48.1921)² − 0.1237u

Again the value of 48.1921 is notably close to the shell transition point (magic number) of 50.

Instead of the relationship between D and Z looking like this

it looks more this this

The model for electronic shells says the coefficient of (Z-N)² should be

(1/ν)[(2i+1)/(i²(i+1)²)]

where ν is a constant and i is the quantum number for the shell. The quantum number for the shell should be related to the order number of the shell. The quantum number for a shell would be the angular momentum number. There is evidence from nuclear magnetic moments that the angular momenta of the nucleons in the deuteron are zero. Thus the angular momentum number for the neutrons in the first shell would be zero. Consider the shells and their orders (first, second, etc.).

The constant ν is unknown but if the ratio of coefficients of the (Z-N)² term for the 50-82 shell to that for the 28-50 shell the result should be

[(2i+1)/(2i-1)][(i-1)/(i+1)]²

where i is the quantum (order) number of the higher shell. For the comparison the higher shell is the 50-82 shell and its quantum number i is 7. The ratio of the coefficients from the regression equations is

0.007459/0.011351 = 0.657086
The value of the ratio computed
from the above formula for i=7 is
0.649038.

The computed value differs from the empirical value by just a little over 1 percent.

The value of ν can be computed from the formula

ν = (1/c₂)(2i+1)/[i²(i+1)²]

with c₂=0.007459 and i=7. Thus

ν = (15/7²8²)/0.007459 = 0.641269

With c₂=0.011351 and i=6

ν = (13/6²7²)/0.011351 = 0.649248

Conclusion

There is a pattern to the magnitudes of the drops in incremental binding energy that occur at the completion of the nuclear shells. The pattern is there even though it is yet to be fully explained. The analysis suggested that there would be a quadratic dependence of the drop in incremental binding energy on the atomic number of the element and that quadratic dependence is observed. The quadratic dependence appears to be in terms of the deviation of the atomic number from a magic number.

The analysis suggests that the ratio of shell radii should be inversely proportional to the slopes of the relations between incremental binding energy and the number of neutrons. Previous work gave estimates of those slopes. 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. Here the analysis suggests that the radius of a shell depends upon the square of its quantum number, which is the same as its order number. If the magic numbers are numbered 1,2,3,… such that 28-50 and 50-82 are the seventh and eighth shells, respectively, then their quantum numbers are 6 and 7. Therefore (7/6)=1.167 and 1.167 squared is 1.361, only 6.5 percent lower than the above estimate.