  The spectrum of laser-plasma generated X-rays is very important, it
characterizes electron dynamics in plasma and is basic for applications.
However, the accuracies and efficiencies of existing methods to diagnose the
spectrum of laser-plasma based X-ray pulse are not very high, especially in the
range of several hundred keV. In this study, a new method based on electron
tracks detection to measure the spectrum of laser-plasma produced X-ray pulses
is proposed and demonstrated. Laser-plasma generated X-rays are scattered in a
multi-pixel silicon tracker. Energies and scattering directions of Compton
electrons can be extracted from the response of the detector, and then the
spectrum of X-rays can be reconstructed. Simulations indicate that the energy
resolution of this method is approximately 20% for X-rays from 200 to 550 keV
for a silicon-on-insulator pixel detector with 12 $\rm \mu$m pixel pitch and
500 $\rm \mu$m depletion region thickness. The results of a proof-of-principle
experiment based on a Timepix3 detector are also shown.
