The Nature of Spin Pairing

The term spin pairing is used freely with respect to particle interactions, but it is misleading because it is not the spins per se that pair. It is the fields generated by the spinning of the charges of the particles that pair the particles together.

The spinning of electrostatically charged particles generates magnetic moments. Spinning nuclei have magnetic moments due in part to the spinning of their nucleons. The magnetic moment of a nucleon pair appears to be zero. Their spins are said said to be opposite and cancel out. Their spins are the same but their magnetic fields cause them to line up side by side with the north pole of one attracted to the south pole of the other. The south pole of the first is then attracted to the north pole of the other. Their mutual electrostatic repulsion keeps them separated.

Ionization Energies

The ionization energy IE, or as it is usually called the ionization potential, for an electron in an atom or ion is the amount of energy required to dislodge it. The Bohr model of a hydrogen-like atom indicates that the energy required to remove an electron, called the ionization potential, should follow the form

IE = 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, effectively the shell number.

Here is the graph of the ionization potential of the innermost electron of the first five elements.

Clearly the relationship is very regular and quadratic. If IE is regressed upon (#p)², as the above equation indicates, the result is

IE = 13.60828 (#p)²

The Rydberg constant is 13.60569 eV.

The coefficient of determination for this equation is 0.999999993 and the standard error of the estimate is 0.01794 eV. The statistical fit is excellent, but it will be shown later that it can be improved upon.

Charge Shielding

The Bohr model is strictly for a hydrogen-like atom or ion; i.e., one in which there is a single electron in the outermost shell. However the regression equation also fits very well the cases of multiple electrons in the outer shells if charge shielding is taken into account. That shielding is for the electrons in the inner shells and also in the same shell. But the shielding by electrons in the same shell is only a fraction of their charge. As it turns out, shielding even for electrons in the inner shells the shielding is less than the full value of their charge.

If inner shell electron 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 their charges are concentrated at the center of the atom 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.

The value of Z in the above Bohr formula is the number of protons in the nucleus #p less the shielding ε by the electrons in inner shells or in the same shell. Thus the ionization energy would be

IE = (R/n²)(#p−ε)²
which can be put in the form
IE = (R/n²)((#p)² − 2(#p)ε + ε²)

This is equivalent to a regression equation of the form

IE = c₂(#p)² + c₁(#p) + c₀

Such a form gives a very good fit to the data. The value of ε is found as

ε = −½c₁/c₂

However, according to the equation, it also should be that c₀/c₂ is equal to ε² and thus equal to the square of the value found from c₁ and c₂. The regression coefficients are not constrained to achieve that equality. Thus effectively the form assumed for the relationship for ionization potential is

IE = (R/n²)[(#p−ε)² + ζ]

where R is an empirical value, rather than necessarily being the Rydberg constant, and ζ is a constant. However the values found for R by regression analysis are notably close to the Rydberg constant.

For the second electron the quadratic regression that fits the data is

IE = 13.61003(#p)² − 17.00677(#p) + 4.16385        [7774.7]     [-1201.3]     [159.9]

The figures in square brackets are the t-ratios for the regression coefficients 1above them; i.e., the ratios of the regression coefficients to their standard deviations.

The coefficient of determination for this equation is 0.999999999 and the standard error of the estimate is 0.00574 eV.

The estimate of ε which comes from this equation

ε = −½(-17.00677)/13.61003 = 0.62479.

Thus the shielding by another electron in the same shell is approximately 0.5.

For the third electron (the first in the second shell) the regression equation is

IE = 3.42839(#p)² − -11.15594(#p) + 7.99664

The coefficient of determination for this equation is 0.999999913 and the standard error of the estimate is 0.01547 eV.

The estimate of ε which comes from this equation

ε = −½(-11.15594)/3.42839 = 1.62699.

Full shielding by the two electrons in the first would give ε=2.0.

The coefficient of (#p)² is R/n² where n for the second shell is 2. Thus the R value for this case is 2²(3.42839)=13.71357.

Now it is worthwhile to apply the methodology to the case of the first electron. There is no shielding in this case so ε should be zero.

The regression equation for the first electron is

IE = 13.61453(#p)² − 0.03161(#p) + 0.01798        [8874.2]     [-3.4]     [1.5]

The coefficient of determination for this equation is 0.999999999 and the standard error of the estimate is 0.00574 eV.

The estimate of ε which comes from this equation

ε = −½(-0.0316067)/13.61453 = 0.00116

This is essentially zero, thus confirming the methodology.

For the fourth electron ε is equal to 2.19001 rather than an expected 2.5. For the fifth electron it is 3.16743 rather an expected 3.0. For the sixth electron it is 3.83669 rather an expected 3.5.

(To be continued.)