Put $x={\mu }_{\mathrm{B}}B/(k_{\mathrm{B}}T)$. Summing the two Boltzmann factors gives the occupied-site internal partition functionIf the single atom can occupy any of the $M$ identical sites, its complete surface partition function isOne also commonly calls the one-site factor $z$ itself $q_{\mathrm{ads}}$. Both conventions are valid, but the combinatorial factor in part (c) must use the same convention. We use the complete one-atom surface function $q_{\mathrm{ads}}=Mz$ below and give the equivalent one-site expression.

Take the energy of an isolated, field-free gas atom as zero, and let $E_b>0$ be the field-independent binding energy at one adsorption site. The two adsorbed spin energies areHere ${\mu }_{\mathrm{B}}$ is the Bohr magneton. The lower state is stabilized by increasing $|B|$; the assignment of the two signs to names for the spin projections is a convention.

Gallium has three valence electrons, one fewer than silicon, and acts as an acceptor impurity. A neutral acceptor can be described as a negatively charged centre binding an electron hole. Thermal release of these holes gives a p-type semiconductor.

At low temperature, the bound hole can absorb far-IR photons in transitions between acceptor levels. The simplest scalar hydrogenic acceptor state model predictsso one expects a discrete bound-hole absorption series followed by a photoionization continuum. In that idealization, the first lines would be at $3R_A^{\ast }/4$, $8R_A^{\ast }/9$ and $15R_A^{\ast }/16$, with threshold $R_A^{\ast }$.

The hole effective mass is not supplied, so the electron mass $0.2m_e$ cannot be reused to give numerical Ga line positions. In a real valence band, multiple hole bands, anisotropic masses and spin-orbit effects can split and complicate the series; acceptor spectra are generally less accurately described by one scalar hydrogenic ladder.^[ref]