The rod-pinch diode[1] is a self-magnetically insulated electron beam diode that is capable of producing a very bright source of hard x-rays. As fielded on the Cygnus accelerator[2], the diode operates at an impedance of 50 Ohms and produces short pulse ( ~50 ns) bremsstrahlung radiation with a 2 MeV photon endpoint energy and dose of 4 rad measured at one meter, with an x-ray spot size ~ 1mm. The source can be used to image through ~ 40 g/cm 2 of material with spatial resolution of order 300 m.

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An algorithm for spectral reconstructions (unfolds) and spectrally integrated flux estimates from data obtained by a five-channel, filtered x-ray-detector array (XRD) is described in detail and characterized. This diagnostic is a broad-channel spectrometer, used primarily to measure time-dependent soft x-ray flux emitted by z-pinch plasmas at the Z pulsed-power accelerator (Sandia National Laboratories, Albuquerque, New Mexico, USA), and serves as both a plasma probe and a gauge of accelerator performance. The unfold method, suitable for online analysis, arises naturally from general assumptions about the x-ray source and spectral properties of the channel responses; a priori constraints control the illposed nature of the inversion. The unfolded spectrum is not assumed to be Planckian. This study is divided into two consecutive papers. This paper considers three major issues: (a) Formulation of the unfold method.-The mathematical background, assumptions, and procedures leading to the algorithm are described: the spectral reconstruction S unfold ðE; tÞ-five histogram x-ray bins j over the x-ray interval, 137 E 2300 eV at each time step t-depends on the shape and overlap of the calibrated channel responses and on the maximum electrical power delivered to the plasma. The x-ray flux F unfold is estimated as R S unfold ðE; tÞdE. (b) Validation with simulations.-Tests of the unfold algorithm with known static and time-varying spectra are described. These spectra included-but were not limited to-Planckian spectra S bb ðE; TÞ (25 T 250 eV), from which noise-free channel data were simulated and unfolded. For Planckian simulations with 125 T 250 eV and typical responses, the binwise unfold values S j and the corresponding binwise averages hS bb i j agreed to $20%, except where S bb ( maxfS bb g. Occasionally, unfold values S j & 0 (artifacts) were encountered. The algorithm recovered * 90% of the x-ray flux over the wider range, 75 T 250 eV. For lower T, the test and unfolded spectra increasingly diverged as larger fractions of S bb ðE; TÞ fell below the detection threshold ($ 137 eV) of the diagnostic. (c) Comparison with other analyses and diagnostics.-The results of the histogram algorithm are compared with other analyses, including a test with data acquired by the DANTE filtered-XRD array at the NOVA laser facility. Overall, the histogram algorithm is found to be most useful for x-ray flux estimates, as opposed to spectral details. The following companion paper [D. L. Fehl et al., Phys. Rev. ST Accel. Beams 13, 120403 (2010)] considers (a) uncertainties in S unfold and F unfold induced by both data noise and calibrational errors in the response functions; and (b) generalization of the algorithm to arbitrary spectra. These techniques apply to other diagnostics with analogous channel responses and supported by unfold algorithms of invertible matrix form.

In this work, two-dimensional particle-in-cell simulations are used to examine the electron physics in the rod-pinch diode, a device that can be used to produce a relatively low-energy (a few MeV) radiographic electron source. It is found that with diode parameters for which the electrons' dominant dynamics are approximated well as a magnetized fluid, the diode produces an electron source with a desired small spot size as the electrons drift to and impinge on the anode tip. However, for a large cathode-toanode radius ratio, a population of electrons that consists predominantly of electrons emitted from the downstream surface of the cathode is found to propagate in the upstream direction and the diode may perform anomalously as a consequence. A method is proposed for improving the quality of the electron source by suppressing electron emission from the downstream cathode surface to reduce the presence of unmagnetized electrons.

A five-channel, filtered-x-ray-detector (XRD) array has been used to measure time-dependent, softx-ray flux emitted by z-pinch plasmas at the Z pulsed-power accelerator (Sandia National Laboratories, Albuquerque, New Mexico, USA). The preceding, companion paper [D. L. Fehl et al., Phys. Rev. ST Accel. Beams 13, 120402 (2010)] describes an algorithm for spectral reconstructions (unfolds) and spectrally integrated flux estimates from data obtained by this instrument. The unfolded spectrum S unfold ðE; tÞ is based on (N ¼ 5) first-order B-splines (histograms) in contiguous unfold bins j ¼ 1; . . . ; N; the recovered x-ray flux F unfold ðtÞ is estimated as R S unfold ðE; tÞdE, where E is x-ray energy and t is time. This paper adds two major improvements to the preceding unfold analysis: (a) Error analysis.-Both data noise and response-function uncertainties are propagated into S unfold ðE; tÞ and F unfold ðtÞ. Noise factors are derived from simulations to quantify algorithm-induced changes in the noise-to-signal ratio (NSR) for S unfold in each unfold bin j and for F unfold ( NSR output =NSR input ): for S unfold , 1 & j & 30, an outcome that is strongly spectrally dependent; for F unfold , 0:6 & F & 1, a result that is less spectrally sensitive and corroborated independently. For nominal z-pinch experiments, the combined uncertainty (noise and calibrations) in F unfold ðtÞ at peak is estimated to be $15%.(b) Generalization of the unfold method.-Spectral sensitivities (called here passband functions) are constructed for S unfold and F unfold . Predicting how the unfold algorithm reconstructs arbitrary spectra is thereby reduced to quadratures. These tools allow one to understand and quantitatively predict algorithmic distortions (including negative artifacts), to identify potentially troublesome spectra, and to design more useful response functions.