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Ionization Energy, Charge
Shielding and Spin Pairing
of Electrons in Atoms and Ions

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 with the specific goal of discerning whether the ionization energergy is depends upon whether or not the electron is a member of a pair

Ionization Energies

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 subshells or in the same subshell.

The generalization of the Bohr formula is then

I = βn(P−V)² + ζJ

where βn is a constant closely related to R//n² and J is 1 if the electron number E is even and 0 otherwise.

Let N be the number of electrons in inner shells or subshells and S the number in the same subshell. The relationship of N, S and E is

N + S = E − 1

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

V = ρNN + ρSS

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

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 shells are 2, 2+6=8, and 2+6+10=18.

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.

Let Z1 be the net charge experienced by an electron in a subshell. Then

I = βn(Z1−ρ2S)² + ζJ
which expanded is
I = βn(Z1² −2Z1ρ2S + ρ2²S²) + ζJ

If Z1 is kept constant the dependence of I on S should be of the form

I = c0 + c1S + c2S² + c4J
where
J = mod (S+1)

This regression equation for the electrons in the second subshell of the second shell for Argon (P=18) is

I = 758.51786 − 76.27679S + 1.73786S² − 0.80250J
       [121.0]     [-13.3]     [1.6]     [-0.1]

The t-ratio for c4 indicate that there is no statistically significant effect on ioinization energy due to spin pairing.

The ratio c1/c2 should be

−2Z1ρ2/(ρ2²) = −2Z12

The actual value of this ratio is −43.89129. Thus Z12 should be 21.94564. The value of Z1 is approximately 14 and that of ρ2 approximately 0.5. The ratio of 28 is the same order of magnitude as the value of about 22 from the regression results.

The ratio c0/c1 should be −Z1/(2ρ2); its actual value is −9.94428. Using the approximate value of 14 and 0.5 that ratio should be 14. Again the values are the same order of magnitude.

Since the coefficient for J is not statistically significant it is appropriate to eliminate it from the regression. The results are:

I = 758.28857 − 76.34557S + 1.73786S²
       [152.7]     [-16.3]     [1.9]

The ratio c1/c2 from these results is −43.93087, essentially the same as for the regression including J. The ratio c0/c1 is −9.93232, again essentially the same.

If the regression is applied to the figures for the electron in the second subshell of the third shell for Argon the results are:

I = 91.86681 − 17.91746S + 0.5175S² − 0.3124
       [46.6]     [-10.0]     [1.5]     [-0.2]

Again there is no statistically significant effect on ioinization energy due to spin pairing.

The approximate value of Z1 is 6. With an approximate value of ρ2 of 0.5 the value of c1/c2 should be −24. Its actual value for this case is −34.6.

The ratio c0/c1 should be −Z1/(2ρ2)=−6; its actual value is −5.12722. This is no too bad of a correspondence.

For Potassium (P=19) the value Z1 is approximately 15 for the electrons in the second subshell of the second shell. The ratio c1/c2 should be −60; whereas its actual value is −44.75879.

The ratio c0/c1 should be −15. It's actual value is −10.57069.

Conclusions

An extension of the Bohr model for the ionization energies of the outer electron in Hydrogen-like ions which takes into account shielding by electrons in the same subshell gives a resonably good statistical explanation of ionization energies for electrons in general.

There is no evidence in terms of ionization energies for an effect of spin pairing of electrons.


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