Two-level system ($E_0=0,E_1=\Delta >0$)

For N independent units with nondegenerate levels 0 and $\Delta >0$, the canonical partition function is $(1+e^{-\beta \Delta }{)}^N$. Its energy is $N\Delta /(1+e^{\beta \Delta })$ and heat capacity is $N{k}_B{x}^2{e}^x/(1+e^x{)}^2$, with $x=\Delta /(k_{B}T)$. Both are activated at low temperature. At high temperature the energy saturates at $N\Delta /2$ and heat capacity decays as $T^{-2}$. The equal-degeneracy Schottky peak is at $x\simeq 2.39936$ and height $0.43923N{k}_B$. A common energy shift leaves heat capacity unchanged; unequal level degeneracies require a different partition function.

For independently labelled particles with nondegenerate levels, choosing which $N_{+}$ particles occupy the upper level gives a binomial multiplicity. With $x=(N_{+}-N_{-})/N$, its large-number entropy is $N{k}_B[\ln 2-(1+x)\ln (1+x)/2-(1-x)\ln (1-x)/2]$. This counting assumes distinguishable or localized degrees of freedom, not indistinguishable particles sharing one pair of quantum orbitals.

A model in which each excitation changes volume by a fixed increment $v$. If the all-ground configuration has volume $V_0$, then $V=V_0+v⟨N_{\mathrm{ex}}⟩$. Energies and statistical weights must be specified under the relevant mechanical constraint; volume endpoints alone do not define arbitrary intermediate configurations.

For a specified optical transition, the on-resonance saturation parameter is the light intensity divided by the saturation intensity. The independent-beam Doppler cooling force model assumes weak total excitation and defines $s$ separately for each beam.

The saturation intensity is the characteristic light intensity used to normalize the on-resonance saturation parameter of a specified optical transition. Its value depends on the transition and the stated illumination convention.