= Solution
An <electron> accelerated in an <ion>'s Coulomb field emits <bremsstrahlung>. Both <electron> states are unbound, giving free-free radiation rather than a discrete bound-state line. Averaging over a Maxwellian <electron> distribution gives an optically thin spectral emissivity with dependence
$$
j_\nu=C_{\rm ff}Z^2n_en_iT^{-1/2}e^{-h\nu/(k_BT)}\bar g_{\rm ff}(\nu,T).
$$
Here $C_{\rm ff}$ depends on the emissivity convention, and the <Gaunt factor> supplies the quantum correction. No numerical emissivity normalization is needed for these diagnostics. For a mixture, sum $Z_i^2n_i$ over <ion> species. The bolometric emissivity is proportional to $n_e\sum Z_i^2n_i\,T^{1/2}$ with a frequency-averaged <Gaunt factor>. Thermal <electron> speeds control the spectrum while the encounter rate supplies its density-squared dependence.
<temperature> can be measured from the high-energy exponential cutoff at <photon> energy of order $k_BT$, provided the spectrum spans that turnover and foreground absorption and nonthermal contamination are modelled. At radio <frequencies> $h\nu\ll k_BT$, the exponential is nearly one, so the optically thin continuum is a poor thermometer by slope alone. Kirchhoff's law, $\alpha_\nu=j_\nu/B_\nu$, and the <Rayleigh-Jeans law> give $\alpha_\nu\propto n_en_iT^{-3/2}\nu^{-2}\bar g_{\rm ff}$ for an isothermal thermal <plasma>. The transfer solution is
$$
I_\nu=B_\nu(T)(1-e^{-\tau_\nu}),\qquad T_b\simeq T(1-e^{-\tau_\nu})
$$
in the radio limit. A resolved optically thick region therefore measures its <electron> <temperature> through <brightness temperature>, if it fills the beam. The turnover between thick and thin emission constrains the line-of-sight <emission measure> $\mathrm{EM}=\int n_e^2dl$ for a hydrogen-dominated fully ionized <plasma>. In the thin regime $T_b\simeq T\tau_\nu$, rather than $T_b=T$. The slowly varying <Gaunt factor> gives a nearly flat optically thin radio flux-density spectrum, in contrast to the $\nu^2$ thick limit at fixed projected area.<Essential Radio Astronomy by James J. Condon and Scott M. Ransom>[chapter 4, §4.3, especially equations (4.57)–(4.58): emission measure, absorption and optically thick/thin diagnostics]{ref}
To obtain <density>, one also needs geometry. An angular extent and independent distance give a transverse size; an assumed shape provides a depth or volume. For approximately uniform emitting gas with <astrophysical volume filling factor> $f_V$, $L_{\rm ff}\propto n_e^2 f_VVT^{1/2}$, so $n_e\propto[L_{\rm ff}/(f_VVT^{1/2})]^{1/2}$. Alternatively a measured column <emission measure> and assumed path length give an rms <density>. Clumping raises the mean squared <density> relative to the square of the mean: the inferred value is not automatically the mean gas <density>. \b[Free-free brightness alone does not uniquely determine <temperature>, <density> and physical size.] Resolved imaging, a distance, optical-depth information or an independent column-density diagnostic is needed to remove those degeneracies. Cluster SZ data are complementary because their leading signal depends on $\int n_eT_e\,dl$, whereas free-free emission depends on <density> squared. Metal lines, unresolved <temperature> structure, foreground absorption, synchrotron emission and <background subtraction> must be controlled before attributing the continuum to a single thermal component.
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