= Energy dependence of water equivalence
<Energy dependence of water equivalence> means that a material can approximate <water> for one radiation energy or spectrum yet differ for another. A <water-equivalent material> must therefore be characterized for the interaction and radiation conditions of interest, rather than by one density or one transmission comparison.
For primary <photon> attenuation, define
$$
Q_\mu(E)=\frac{(\mu/\rho)_m(E)}{(\mu/\rho)_w(E)},
$$
using the material and water <mass attenuation coefficients>. Equal <surface mass densities> give equal primary transmission only when this coefficient ratio is one. Changing slab thickness can match one selected energy, but cannot generally make $Q_\mu(E)$ constant at all energies.
A concrete example uses the primary NIST tables for liquid <water> and <PMMA>. Their <mass attenuation coefficients>, in $\mathrm{cm^2/g}$, are
$$
\begin{array}{c|cc}
E&\text{water}&\text{PMMA}\\\hline
30\,\mathrm{keV}&0.3756&0.3032\\
100\,\mathrm{keV}&0.1707&0.1641
\end{array}
$$
so the PMMA-to-water ratios are about $0.80724$ at $30\,\mathrm{keV}$ and $0.96134$ at $100\,\mathrm{keV}$. These are material data at specified <photon energies>, not an assertion that PMMA is a universally equivalent substitute.<Tables of X-Ray Mass Attenuation Coefficients and Mass Energy-Absorption Coefficients by John H. Hubbell and Stephen M. Seltzer>[Table 4 material pages Water, Liquid and Polymethyl Methacrylate; rows 0.03 MeV and 0.1 MeV, mass-attenuation column]{ref}
For an original calculation from those data, choose a water slab with <surface mass density> $1\,\mathrm{g/cm^2}$ and choose PMMA with $0.1707/0.1641\approx1.04022$ times that value. Their <Beer-Lambert law> optical depths then agree at $100\,\mathrm{keV}$. At $30\,\mathrm{keV}$, however, the PMMA optical depth is only about $0.83971$ of the water value. The primary transmitted fractions become $e^{-0.3756}\approx0.68688$ for water and $e^{-0.31539}\approx0.72950$ for the adjusted PMMA slab. Matching one row of the table did not match another.
The interaction being matched also matters. A <mass energy-absorption coefficient> concerns retained <energy>, while a <mass attenuation coefficient> concerns removal of primary <photons> from a beam. Matching the latter does not by itself match <absorbed dose>. A beam containing several <photon energies> introduces a <spectrum-weighted water-equivalence ratio>, which can change when those contributions change.
A proposed physical comparison uses the same characterized detector and irradiation geometry while varying the incident <photon energy> or filtration. Measure slab density and thickness, and compare both primary transmission and the relevant energy-deposition response. The NIST coefficients provide interaction data; scattering from the surrounding apparatus and the detector's own energy response remain separate parts of that experiment.
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