Spectrum-weighted water-equivalence ratio

A spectrum-weighted water-equivalence ratio compares a specified material response with the corresponding water response after accounting for the radiation's energy distribution. Its definition must identify the interaction coefficient and the spectral weighting. An attenuation comparison and an energy-absorption comparison are different quantities.

For an energy-absorption comparison, let ${\Psi }_E(E)dE$ be the photon energy fluence in an energy interval, and write $a_j(E)=({\mu }_{\mathrm{en}}/\rho {)}_j(E)$ for material $j$. Define the coefficient-weighted response and its ratio bywith $A_w>0$. The same prescribed local photon energy fluence is used for both materials in this comparison. NIST distinguishes the mass energy-absorption coefficient from the mass attenuation coefficient and relates energy fluence to photon fluence through the additional photon-energy factor.^[ref]

The ratio is a weighted mean of the monoenergetic ratios $q(E)=a_m(E)/a_w(E)$, but its normalized weights areThus the weights depend on the water response as well as the incident spectrum. Averaging the ratios with energy-fluence weights alone generally gives a different result.

For an original two-line calculation, use the NIST water and PMMA energy-absorption data: at $30\mathrm{keV}$ the coefficients are $0.1557$ and $0.09645\mathrm{c}{\mathrm{m}}^2/g$ respectively; at $100\mathrm{keV}$ they are $0.02546$ and $0.02368\mathrm{c}{\mathrm{m}}^2/g$.^[ref] Equal energy fluences in the two lines giveEqual photon counts instead give energy-fluence weights proportional to $30$ and $100$, producing $Q_{\Psi }\approx 0.72904$. The different answers reflect different physical spectra, not a discrepancy in the tabulated coefficients.

An absorbed dose interpretation needs additional conditions such as charged-particle equilibrium. A finite object can also change the local spectrum through beam hardening and scattering. The ratio above therefore specifies a local coefficient-weighted comparison; it is not a universal conversion factor for every phantom size or irradiation geometry. A physical validation should characterize the local spectrum and detector energy response rather than assume that equal total fluence means equal spectral weighting.