A shell is a configuration in which a number of particles all have the same quantum number. The quantum number corresponds to the quanta of angular momenta. This material investigates the radii of the shells of electrons in atoms and the shells of nucleons in nuclei. The material for electronic shells is of interest in its own right but it also provides insights into the much more complex matter of nuclear shells.

The Shells of Electrons in Atoms

From Quantum Theory it is known that the occupancy of electronic shells is 2k² where k is an integer. The 2 arises from there being two spin orientations of the electrons and the k² from the number of electrons in the shell being the sum of odd numbers. Thus the shell occupancies are 2, 8, 18, 32. However this does not mean that the numbers of electrons in completed shells are 2, 10, 28 and 60. Instead there are repeated shell occupancy numbers.

The noble gases; helium, neon, argon, krypton, xenon and radon; represent filled electron shells. Their atomic numbers are 2, 10, 18, 36, 54 and 86. This means that first a shell of two electrons is filled then one of eight. But then another shell of eight is filled. Next a shell of 18 and then a second shell of 18. And finally there is a shell of 32 to be filled. The electron shells correspond to the rows of the Periodic Table.

Let the shells be labeled by their angular momentum quantum number i and 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 inversely proportional to Z; i.e.,

r_i(Z') = νi²/Z'

where ν is a constant.

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, where N₀ is the number of electrons interior to the i-th shell. The relevant parameter is Z−N₀, the effective charge on the i-th shell as it begins to be filled. This quantity will be denoted as Z₀. Note that Z'=Z₀+½−n/2.

The dependence of the shell radius on n is then

r_i = νi²/(Z₀+½−n/2)

The graph illustrates the dependence for the hypothetical data of two shells with capacities of 8 and i=3 for the first and i=4 for the second. The value of Z₀ is 20.

The graph illustrates the notion that as the shell fills its radius increases. For a higher level shell the radius increases both due to the higher value of the square of the quantum number and the lower effective charge reduction as the inner shell shields the outer shell from the charge of the nucleus.

Potential Energies

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_i(n) of n electrons in the i-th shell is

W(_i)(n) = nV = nKZ'/r_i
which reduces to
W_i(n) = (K/r_i)nZ'
W_i(n) = (K/r_i)n([Z₀+½] − n/2)
W_i(n) = (K/r_i)[(Z₀+½)n − n²/2]

The incremental change in the potential energy is defined by

ΔW_i(n) = W_i(n) − W_i(n-1)

If the dependence of the radius r_i on Z' is ignored then

ΔW_i(n) = (K/r_i)[(Z₀+½) − (n+½)]
which reduces to
ΔW_i(n) = (K/r_i)[Z₀ − n]
and further to
ΔW_i(n) = (K/r_i)Z₀ − (K/r_i)n

This is a relationship of the form ΔW = a_i − b_in. The ratio a_i/b_i would then be equal to Z₀, the effective charge of the nucleus at the beginning of the filling of the i-th shell. Also the shell radii for should be inversely proportional to the slopes.

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

But the shell radius does depend upon the effective charge and thus upon the number of electrons in the shell. By the previously given relationship

1/r_i = (1/(νi²))Z'

Thus

Δ(1/r_i) = (1/(νi²))ΔZ' = (1/(νi²))(−1/2)

Therefore

ΔW_i(n) = (K/r_i)[Z₀ − n] + Δ(1/r_i)KnZ'
which reduces to
ΔW_i(n) = (K/(νi²)){Z'[Z₀ − n] − nZ'/2}
which further reduces to
ΔW_i(n) = (K/(νi²))Z'[Z₀ − 3n/2]
or, equivalently
ΔW_i(n) = (K/(νi²))[Z₀+½ − n/2][Z₀ − 3n/2]

This is a quadratic dependence upon n. The change in ΔW_i(n) with respect to a unit increase in n , Δ⁽²⁾W_i(n), is

Δ⁽²⁾W_i(n) = (K/(νi²))[(−1/2)(Z₀ − 3n/2) + (−3/2)(Z₀+½ − n/2)]
which reduces to
Δ⁽²⁾W_i(n) = (K/(νi²))[−2Z₀ − 3/4 + n]

This suggest that the curvature of the relation of ΔB to n is negative if n<(2Z₀+3/4) and positive if n>(2Z₀+3/4); and of course zero if n=(2Z₀+3/4).

Nuclear Shells

Consider the incremental increases in binding energy for the isotopes of uranium as the number of neutrons increases.

This relationship of incremental binding energy with the number of neutrons displays a near linearity but an actual quadratic dependence. In this case the slope of the relation becomes more negative as the number of neutrons increases. (The sawtooth shape comes from the effect of the formation of neutron pairs.)

For gold the relationship is more nearly linear.

For silver the relationship curves slightly upward.

For nickel the upward curvature is somewhat more pronounced.

For palladium the upward curvature is definite.

Many more examples could be given but the point seems to be well illustrated that the relationship of incremental binding energy with the number of neutrons is a downward sloping, nearly linear curve but with a curvature that can be positive or negative.

(To be continued.)