= Beam hardening
{wiki}
An original two-<energy> model demonstrates the failure of a single constant <linear attenuation coefficient> for a mixed beam. Suppose the two components contribute equally to the empty-path detector signal and have stipulated $\mu_1=0.40\,\mathrm{mm^{-1}}$ and $\mu_2=0.10\,\mathrm{mm^{-1}}$. With fixed linear detector response and no accepted <scattering of light>,
$$
T(x)=\frac12e^{-\mu_1x}+\frac12e^{-\mu_2x},\qquad
-\frac{d\ln T}{dx}
=\frac{\mu_1e^{-\mu_1x}+\mu_2e^{-\mu_2x}}
{e^{-\mu_1x}+e^{-\mu_2x}}.
$$
The initial slope corresponds to $\mu=0.25\,\mathrm{mm^{-1}}$, but at $x=5\,\mathrm{mm}$, $T\simeq0.371$, rather than $e^{-0.25(5)}\simeq0.287$. The more strongly attenuated component disappears faster. If that component has lower <photon> <energy>, the surviving beam has a higher mean <energy>: <beam hardening>. The formula is explicitly a two-component example, not a universal rule that attenuation decreases monotonically with <photon> <energy>. Detector weighting also matters when interpreting a broadband transmission signal.
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