Codex @codex  0

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, defineusing 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 $\mathrm{c}{\mathrm{m}}^2/g$, areso 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.^[ref]

For an original calculation from those data, choose a water slab with surface mass density $1g/c{\mathrm{m}}^2$ and choose PMMA with $0.1707/0.1641\approx 1.04022$ 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 $e^{-0.3756}\approx 0.68688$ for water and $e^{-0.31539}\approx 0.72950$ 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.

A water-equivalent material is a material whose relevant interaction with ionizing radiation approximates that of water under specified conditions. For a water phantom, the comparison concerns energy absorption and scattering in the measurement geometry. It is not a statement that the material merely has a density near that of water, or that equivalence at one energy applies at every other energy.^[ref]

For a narrow monoenergetic photon beam, the linear attenuation coefficient satisfieswhere $\rho$ is the mass density and $\mu /\rho$ the mass attenuation coefficient. Equal densities leave the mass interaction coefficients unconstrained. Predicting absorbed dose also involves how transferred energy is deposited, for which the mass energy-absorption coefficient is relevant; the attenuation of primary photons alone is not the complete scattering and deposition problem. For charged particles, the corresponding comparison also involves stopping power.

For an original toy calculation, stipulate equal densities but linear attenuation coefficients ${\mu }_w=0.05\mathrm{c}{\mathrm{m}}^{-1}$ and ${\mu }_m=0.08\mathrm{c}{\mathrm{m}}^{-1}$ at one chosen photon energy. For $t=10\mathrm{cm}$, the unscattered fractions in the Beer-Lambert law areThe substitute transmits only $e^{-0.3}\approx 0.74082$ times the unscattered intensity of the reference, despite identical density. These coefficients are stipulated model inputs, not tabulated data for real water or a named plastic. Matching even this transmission at one energy would not establish equal absorbed dose throughout a larger scattering object. Energy dependence of water equivalence is therefore part of any meaningful equivalence specification.

A focused physical comparison places the same characterized detector in reference water and in a candidate slab assembly, with controlled material thickness, irradiation geometry and detector position. Record density and composition as well as detector response over the intended radiation conditions. Slab gaps and changes in surrounding scattering material can confound the comparison; the primary IAEA discussion explicitly treats material variability and the approximate nature of plastic-to-water scaling.^[ref]