Quantitative K$\alpha$ line spectroscopy for energy transport in ultra-intense laser plasma interaction

X-ray line spectra ranging from 17 to 77 keV were quantitatively measured with a Laue spectrometer, composed of a cylindrically curved crystal and a detector. The absolute sensitivity of the spectrometer system was calibrated using pre-characterized laser-produced x-ray sources and radioisotopes, for the detectors and crystal respectively. The integrated reflectivity for the crystal is in good agreement with predictions by an open code for x-ray diffraction. The energy transfer efficiency from incident laser beams to hot electrons, as the energy transfer agency for Au K$\alpha$ x-ray line emissions, is derived as a consequence of this work. By considering the hot electron temperature, the transfer efficiency from LFEX laser to Au plate target is about 8\% to 10\%.