The development of a directional, small-divergence, and short-duration picosecond x-ray probe beam with an energy greater than 50 keV is desirable for high energy density science experiments. We therefore explore through particle-in-cell (PIC) computer simulations the possibility of using x-rays radiated by betatron-like motion of electrons from a self-modulated laser wakefield accelerator as a possible candidate to meet this need. Two OSIRIS 2D PIC simulations with mobile ions are presented, one with a normalized vector potential a0 = 1.5 and the other with an a0 = 3. We find that in both cases direct laser acceleration (DLA) is an important additional acceleration mechanism in addition to the longitudinal electric field of the plasma wave. Together these mechanisms produce electrons with a continuous energy spectrum with a maximum energy of 300 MeV for a0 = 3 case and 180 MeV in the a0 = 1.5 case. Forward-directed x-ray radiation with a photon energy up to 100 keV was calculated for the a0 = 3 case and up to 12 keV for the a0 = 1.5 case. The x-ray spectrum can be fitted with a sum of two synchrotron spectra with critical photon energy of 13 and 45 keV for the a0 of 3 and critical photon energy of 0.3 and 1.4 keV for a0 of 1.5 in the plane of polarization of the laser. The full width at half maximum divergence angle of the x-rays was 62 x 1.9 mrad for a0 = 3 and 77 x 3.8 mrad for a0 = 1.5.
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