Evaporative cooling energy balance ($\alpha =(\eta +\kappa -3)/3$)

For a classical three-dimensional harmonic trap of fixed frequencies, $E=3N{k}_{B}T$. If each lost atom removes mean energy ${ϵ}_{\mathrm{ev}}=(\eta +\kappa )k_{B}T$, differential energy balance gives $d\ln T/d\ln N=(\eta +\kappa -3)/3$. The result assumes rapid elastic equilibration and excludes separate work from changing trap frequencies.

For a classical three-dimensional harmonic trap of fixed frequencies, $E=3N{k}_{B}T$ and $\mathcal{D}\propto N{T}^{-3}$. Combining $dE={ϵ}_{\mathrm{ev}}dN$ with these equations gives $d\ln \mathcal{D}/d\ln N=4-{ϵ}_{\mathrm{ev}}/(k_{B}T)$. Since evaporation decreases $N$, phase-space density of an atomic gas increases only when the removed mean energy exceeds $4k_{B}T$. Merely lowering temperature requires only ${ϵ}_{\mathrm{ev}}>3k_{B}T$. Changing confinement adds trap work and changes these formulas.