Modeling the diffraction efficiencies of the AXAF high-energy transmission gratings

The high energy transmission grating spectrometer (HETGS) for the Advanced X-ray Astrophysics Facility (AXAF) uses transmission gratings of period 2000 and 4000 angstrom to diffract x rays in the energy range 400 - 9000 eV. The gratings are so-called phased gratings, where the phase shift of x rays through the grating bars causes constructive interference at some wavelengths, thereby increasing the efficiency (in first order) to a level higher than would be expected for an opaque grating. AXAF has a program goal of calibrating the efficiencies (or effective areas) of the various spectrometers and imaging detectors (including the contribution of the x-ray mirrors) to an accuracy of a few percent. The HETGS group has elected to attempt a calibration at the sub-assembly level (i.e., the grating elements only, independent of the detectors and mirrors) at the 1% level. Since it is impossible (in practice) to measure grating efficiencies for all of the AXAF grating elements (336 elements in the flight assembly) at all energies, we have chosen instead to develop a model of the diffraction efficiency in all orders, and determine the parameters of the model by measuring the diffraction efficiencies at a few x ray energies. We verify the model by detailed studies of a few gratings made at the National Synchrotron Light Source. To date, we believe that we have an HETG model which is nearly accurate enough in the range 2200 - 9000 eV. At the lower energies of interest (400 eV - 2200 eV), however, the best fit grating model deviates from the synchrotron data by more than a few percent on average. In this paper we describe the model, the synchrotron tests that verify it, the discrepancies that we find at lower energies, and our plan to resolve the problems.