Ionization Energy and Charge Shielding of Electrons in Atoms and Ions (Part III)
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Ionization Energy and Charge
Shielding of Electrons in
Atoms and Ions (Part III)

This is a continuation of a study on the ionization energy for electrons in different positions within atoms and ions. Ionization energy, or as it is usually called the ionization potential, for an electron is the amount of energy required to dislodge it. The previous study focused on the variation in ionization as a function of the number of protons in the nucleus. This one focuses on the variation wirh respect to electron position.

Here is a graph of the ionization energies of the different electrons in an Argon atom.

Clearly the relationship depends on the shell number and within a shell ionization energy depends upon the number of electrons in the shell. It is difficult to detect any variation with respect to the subshell. It is also difficult to detect any quadratic dependence.

Here is the rationale for the shieldings. If inner shell electrons execute trajectories that take them over a spherical shell it is as though their charges are smeared over a spherical shell and their effect on outer shell electrons is the same as though the charges of the inner shell electrons are concentrated at the center of the atom or ion and thus cancel out an equal number of positive charges.

The shielding by electrons in the same shell is a bit more complicated. The effect of a charge distributed over a spherical shell on an electron entirely within that spherical shell is zero. If the electron is entirely outside of the spherical shell the effect is the same as if the charge were concentrated at the center of the spherical shell. But if the center of the electron is located exactly on the shell then roughly half of the electron is inside of the spherical shell and is unaffected by its charge. Thus an electron is shielded by an amount approximately equal to one half of the charges in the same shell. That is the rough theory. It needs to be tested empirically.

Ionization Energies

Electrons are organized in shells and subshells within the shells. In each shell the first subshell can contain two electrons. Where there is a second subshell it can contain at most six electrons. For the third subshell the capacity is ten. The capacities 2, 6 and 10 are twice the first three odd numbers 1, 3 and 5. The total capacities of the first three shells are 2, 2+6=8, and 2+6+10=18, respectively.

The Bohr model of a hydrogen-like atom or ion indicates that the energy I required to remove an electron should follow the formula

I = RZ²/n²

where R is the Rydberg constant (approximately 13.6 electron Volts (eV), Z is the net charge experienced by the electron and n is the principal quantum number of the electron, effectively the shell number. The net charge Z is equal to the number of protons P in the nucleus less the number charges V shielded by the electrons which are in inner shells or lower subshells or in the same subshell.

The generalization of the Bohr formula is then

I = (β/n²)(P−V)²

where β is a constant closely related to R.

A little algebraic manipulation of the above formula gives

I½n = β½(P−V) .

Let N be the number of electrons in inner shells or subshells, L the number in lower subshells and S the number in the same subshell. The data for N, L and S are given in the appendix. The relationship of N, L, S and E is

N + L + S = E − 1

The number of charges shielded V is related linearly to N, L and S, say

V = ρNN + ρLL + ρSS

Thus the equation to be determined is

I½n = β½(P − ρNN − ρLL − ρSS)

A previous study found that the shielding ratios ρN and ρLare approximately equal to 1.0 and ρS to 0.5.

Empirical Results

The regression equation estimated is

I½n = cPP + cNN + cLL + cS

The data used is one published by the Physics Department of Ohio State University. It only extends to nuclei of atomic number 47. The results are:

I½n = 4.02783P − 3.8534N − 3.91675L − 3.20805S
       [279,0]    [-191.2]     [-85.6]     [-52.9]

Those results can put into the form

I½n = 4.02783(P − 0.95671N − 0.97296L − 0.79647S )
thus revealing that
rN = 0.95671
rL = 0.97296
rS = 0.79647

The value of cP² is 16.22345 eV, substantially above the Rydberg constant of 13.6 eV.

The coefficient of determination for the regression equation is 0.99619.

The following graph has the actual data values plotted versus their regression estimates.

Conclusions

A generalization of the Bohr model fits the ionization data very well.

The shielding ratio for electrons in a lower subshell in the same shell is not any less than the shielding ratio for electrons in a lower shell. The shielding ratios are both essentially one,

Electrons in the same subshell do definitely shield but at a ratio of about 0.79647, or roughly 0.8.

Appendix

Number N of Electrons in Inner Shells,
Number L in Lower Subshells and the
Number S in the Same Subshell for
Each Electron Position E
E N L S
1 0 0 0
2 0 0 1
3 2 0 0
4 2 0 1
5 2 2 0
6 2 2 1
7 2 2 2
8 2 2 3
9 2 2 4
10 2 2 5
11 10 0 0
12 10 0 1
13 10 2 0
14 10 2 1
15 10 2 2
16 10 2 3
17 10 2 4
18 10 2 5
19 10 4 0
20 10 4 1
21 10 4 2
22 10 4 3
23 10 4 4
24 10 4 5
25 10 4 6
26 10 4 7
27 10 4 8
28 10 4 9
29 28 0 0
30 28 0 1
31 28 2 0
32 28 2 1
33 28 2 2
34 28 2 3
35 28 2 4
36 28 2 5
37 28 8 0
38 28 8 1
39 28 8 2
40 28 8 3
41 28 8 4
42 28 8 5
43 28 8 6
44 28 8 7
45 28 8 8
46 28 8 9
47 28 18 0

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