= X-ray attenuation
{c}
{title2=$I(x)=I_0e^{-\mu x}$}
For a narrow beam of monoenergetic <photons> crossing homogeneous material of thickness $x$, the uncollided <wave intensity> obeys $I(x)=I_0e^{-\mu x}$. Here $\mu$ is the <linear attenuation coefficient>, counting removal from the incident beam by <photon absorption> and <scattering of light>. It depends on <photon> <energy> and material; it is not universally independent of <energy>.<Tables of X-Ray Mass Attenuation Coefficients and Mass Energy-Absorption Coefficients by John H. Hubbell and Stephen M. Seltzer/2. The Mass Attenuation Coefficient>[Tables of X-Ray Mass Attenuation Coefficients and Mass Energy-Absorption Coefficients by John H. Hubbell and Stephen M. Seltzer; NIST, X-Ray Mass Attenuation Coefficients, §2, Eq. (1), narrow monoenergetic beam and interaction contributions]{ref}
An original derivation makes the model explicit. If each thin layer removes fraction $\mu\,dx$ of the surviving beam, with the same $\mu$ throughout, then
$$
\frac{dI}{dx}=-\mu I,\qquad
\ln\frac{I(x)}{I_0}=-\mu x.
$$
Thus equal added thicknesses multiply transmission by the same factor. The <ordinary differential equation> describes uncollided beam survival. To test it with a detector, keep source output, distance, selected <photon> <energy> band and detector response fixed. Use a <collimator> and suitable detector acceptance to exclude <scattering of light> into the detector. Subtract background, measure an empty-path reference, and keep the detector in its linear operating range. A changing source distance adds geometric dilution and is not an attenuation measurement at fixed geometry.
For a stipulated example, take $\mu=0.40\,\mathrm{mm^{-1}}$ and $x=2.5\,\mathrm{mm}$. Transmission is $e^{-1}\simeq0.368$. If the empty-path detector rate is $1010\,\mathrm{s^{-1}}$, including background $b=10\,\mathrm{s^{-1}}$, the model predicts sample rate $10+1000e^{-1}\simeq378\,\mathrm{s^{-1}}$. Recover transmission as $(R-b)/(R_0-b)$; the raw ratio $R/R_0\simeq0.374$ is biased. These chosen values are a calculation, not measurements of a particular material or a historical apparatus.
\Image[/x-ray-narrow-beam-attenuation.png]
{title=When <X-ray> transmission is exponential, and when spectral mixing bends it}
{id=image-x-ray-narrow-beam-attenuation}
{description=Original calculated examples, with stipulated <linear attenuation coefficients> in inverse millimetres. The left panel uses one <photon> <energy> and marks $x=2.5\,\mathrm{mm}$ and the <half-value layer>. The right panel compares an equal-weight mixture of two <photon> <energies> with the exponential having the same initial slope. A logarithmic vertical axis turns a single exponential into a straight line. Neither curve is fitted to measured material data.}
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