G. C. Johnson Sommerfeld Constantaka Fine Structure Constant aka Electron Coupling Constant
Alpha, (approximately 1/137), is the ratio of the distance between any two point charges, and the photon wavelength with the same energy as the absolute potential between the charges. In 1916 Arnold Sommerfeld identified this ratio as the fraction of lightspeed for the hydrogen atom's ground state electron. Beyond hydrogen, for all single electron ions, the ground state electron velocity/lightspeed ratio appears to increase as a multiple of the atomic number, Z. In 1924 Louis de Broglie discovered that all moving particles have a wavelike property, with a wavelength given by Planck's constant divided by the particle's momentum.
It was also proposed that electron shells are standing waves which must contain an integer number of de Broglie wavelengths. This additional constraint on the electron shell gave a new interpretation of Bohr's model in which electron orbits have an angular momentum that is an integer multiple of . This new paradigm bought atomic physics out of the realm of orbital mechanics and into the modern domain of wave mechanics. The GuessThe guess to be tested is that the ground state of a singleelectron ion is defined by two conditions: Condition I: The electron velocitytolight speed ratio is the product of alpha and the atomic number, Z.
Condition II: The shell circumference must be equal to the electron's de Broglie wavelength. The ConsequencesCondition I defines the relativistic Lorentz factor, gamma.
This allows us to write the relativistic kinetic energy of the electron as:
It can be demonstrated that condition II defines the potential energy as:
Taking the absolute value of the sum of kinetic and potential energy yields the ionization energy of singleelectron ions.
The above ionization energy is the positive energy which must be added to completely remove the groundstate electron from a nucleus of infinite mass. The experimental ionization energy is slightly less, since the kinetic energy of the nucleus is positive. (Making the total energy less negative.)
Nucleus kinetic energy can be computed from conservation of momentum. Since the net momentum of the atom cannot be changed by the internal interaction between the nucleus and the electron, their momenta must at all times be equal and opposite.
Setting the classical momentum of the nucleus equal to the electron's relativistic momentum,
the nucleus kinetic energy becomes:
The ComparisonFor hydrogen: The above model gives a hypothetical ionization energy for an infinite mass nucleus of, , and a nucleus kinetic energy of: Since, ,
From the CRC Handbook of Chemistry and Physics:
The following graph plots singleelectron (+Z1 ions) observed ionization potentials with the relativistic model.
For low atomic numbers the electron's velocity is a small fraction of the speed of light and the above model corresponds to the Rydberg equation.
For larger atoms, however, the Rydberg equation progressively underestimates the experimental values. Cu+28, for example, has a published ionization energy of 11,567.6eV, while the Rydberg model predicts 11,437.6eV; 130eV low, for a relative error of 1.1%. (Copper is the largest atom for which singleelectron ionization energy is published.) The relativistic model anticipates 11,573.3eV, a 5.7eV overshoot. (A relative error of 0.05%.)
Classical bonding energies of excited states
