X-ray flux from filtered arrays of detectors without unfolding

A simple computational method is proposed for estimating the time-dependent flux F[ΔE](t) of an x-ray spectrum S(E,t) over domain [ΔE] from data Dk(t)(k=1,...,N) obtained by an N-channel array of filtered detectors. It is assumed that the data are related to the spectrum by a discrete, inhomogeneous, first-kind Fredholm integral equation Dk=∫S(E,t)Rk(E)dE, where Rk(E) is the known response function for each detector channel of the diagnostic. The proposed method constructs a spectral sensitivity HLS(E) for the diagnostic array as a linear combination Σk=1NakRk(E) of the responses, where the coefficients ak are obtained by a least-squares criterion plus a constraint. The ak values, once determined, apply as long as the responses are valid. The flux estimate is then simply FLS(t)=Σk=1NakDk(t), without a spectral unfold of the data. The method is useful for quick analyses of time-dependent data, for comparisons with other flux-measuring diagnostics, and for the experimental design of filtered-detector arrays. The method is applied to a five-channel array of filtered photoemissive x-ray detectors [G. A. Chandler et al., Rev. Sci. Instrum. 70, 561 (1999)], used for z-pinch measurements at the Z-accelerator facility [R. B. Spielman et al., Phys. Plasmas 5, 2105 (1998)]. Comparisons with unfold results are made, and a first-order analysis of error propagation into FLS(t) is presented