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A localized magnetic moment can polarize a metal's conduction electrons, whose response changes the energy of a second moment. This indirect exchange interaction can be long ranged and oscillate in sign with separation because of the Fermi surface. In the three-dimensional isotropic free-electron asymptotic limit its range dependence is proportional to $\cos (2k_{F}r)/r^3$, with coupling-dependent prefactor and sign convention. The original calculation for nuclear moments used their contact hyperfine interaction with conduction electrons; it was motivated by nuclear magnetic resonance line widths in metallic silver.^[ref]

Heating promotes electrons across the band gap, creating mobile conduction electrons and electron holes. Their contributions add:where $u_e,u_h$ are the electron mobility and hole mobility, and $e$ is the positive elementary-charge magnitude. The increase in carrier density gives the rising electrical conductivity in the intrinsic activation regime. Mobility need not itself increase with temperature; the model attributes the dominant increase to carrier generation.

The finite-temperature Fermi level lies at the middle of the gap in the stipulated symmetric intrinsic model:Here $E_v$ is the valence-band maximum and $E_c$ the conduction-band minimum. Thermal promotion creates one conduction electron and one electron hole, so charge neutrality requires $n=p$. Equal constant densities of states in broad, symmetric bands give equal populations when the distances $E_c-\mu$ and $\mu -E_v$ are equal.

This is a statement about the chemical potential, not an available electron state in the gap. Unequal band densities of states would shift the intrinsic Fermi level away from exact midgap; the equality follows from the symmetry assumed here.

The ambient-temperature Fermi level is raised into the upper part of the gap, toward the conduction-band edge, relative to intrinsic silicon. The sample is an n-type semiconductor: ionized donors supply many more conduction electrons than the intrinsic population.

For a nondegenerate, substantially ionized sample with negligible acceptors, $n\simeq N_D$. With effective density of states $N_c$, the conduction-band relation givesBecause a few ppm normally gives $N_D\ll N_c$ at ambient temperature, the chemical potential remains below the conduction-band edge rather than entering it as in a strongly degenerate semiconductor. Its precise position requires the dopant density, density of states and degree of ionization; the broad qualitative shift is fixed by donor doping.

It is not necessary for $\mu$ to lie between the donor level and the conduction edge. In a substantially ionized nondegenerate sample, the chemical potential can lie below the shallow donor level, as in the illustrative diagram. Donor ionization is determined by the level's occupation relative to $\mu$; a finite-temperature Fermi level is not the highest filled one-electron level of every individual configuration.

Let $z_c$ be the number of conduction electrons per atom. Then $n=z_c{n}_a$ and the Drude relations give $\sigma =ne\mu =n{e}^2\tau /m_e$. ConsequentlyUnder an explicit one-electron-per-atom assumption, $\mu =5.24\times 10^{-3}{\mathrm{m}}^2{\mathrm{V}}^{-1}{\mathrm{s}}^{-1}$ and $\tau =2.98\times 10^{-14}\mathrm{s}$. The carrier count is not specified: monatomic does not by itself imply monovalent, so the boxed dependence on $z_c$ is the general result supported by the data. The collision time here is the transport relaxation time inferred within the free-electron model, not a measurement of every microscopic scattering event.

Use thermal de Broglie wavelength ${\lambda }_T=h/\sqrt{2\pi m{k}_{B}T}$ and spacing $a=n^{-1/3}$. Degeneracy onset from thermal wavelength occurs atThe numerical factor depends on the convention for a typical momentum, but the scale ${\hslash }^2{n}^{2/3}/(m{k}_B)$ does not. For fermions the Fermi temperature $T_F={\hslash }^2(3{\pi }^{2}n{)}^{2/3}/(2m{k}_B)$ gives a comparable scale for spin degeneracy two.

Representative estimates, with the assumed densities stated explicitly, are:

Here $u$ is the atomic mass constant. Thus the characteristic orders are $10^5\mathrm{K}$ for electrons, sub-microkelvin for a dilute heavy atomic gas, and $10^{12}\mathrm{K}$ for neutrons. There is no density-independent quantum-degeneracy temperature for atomic gases: for example, taking $n=2.5\times 10^{25}{\mathrm{m}}^{-3}$ and mass $4u$ instead gives about $0.065\mathrm{K}$. These are illustrative input estimates, not values supplied by the paper. Denser neutron-star cores require relativistic and interaction corrections; the estimate identifies the degeneracy scale rather than a precise stellar temperature.