On the Fermi Gas, the Sommerfeld Fine Structure Constant, and the Electron–Electron Scattering in Conductors

Electrical energy is revisited as a fundamental parameter for inclusion in Fermi gas theory, in addition to thermal energy. It is argued that electrical energy can move some electrons to above the Fermi level, providing free charges to carry the electrical current, even at absolute zero temperature. The Drude model, Ohm’s law, quantum resistance, and the electrical resistivity due to electron–electron scattering appear naturally as a consequence of the theoretical description, which is based on the quantization of the angular momentum and the Fermi–Dirac distribution, considering total energy as \(\epsilon ={k}_{B}T+{\Phi }_{0}I\) $ϵ=k_{B}T+{\mathrm{\Phi }}_{0}I$ . The electrical and magnetic forces acting on an electron are related to the ratio between the Fermi velocity and the speed of light and show that the electron motion is due to helical paths. Considering the center of mass description for the Bohr atom, it was possible to show that the magnetic force is related to the electrical force as \({F}_{M}=(\alpha /\pi ){F}_{E}\) $F_M=(\alpha /\pi )F_E$ , which demonstrates that the electrons move in helical paths along the orbit. The helical motion naturally provides for quantization of the magnetic flux, the spin of the electron, and the first correction term of the anomalous magnetic moment. Applying the model to describe the electron–electron scattering allows prediction of the behavior of the electrical resistivity of many metals at low temperatures, which is in excellent agreement with empirical observations.