Recent investigations by Fuchs et al. [Nat. Phys. 11, 964 (2015)] revealed an anomalous frequency shift in non-linear Compton scattering of high-intensity X-rays by electrons in solid beryllium. This frequency shift was at least 800 eV to the red of the values predicted by analytical free-electron models for the same process. In this paper, we describe a method for simulating non-linear Compton scattering. The method is applied to the case of bound electrons in a local, spherical potential to explore the role of binding energy in the frequency shift of scattered X-rays for different scattered angles. The results of the calculation do not exhibit an additional redshift for the scattered X-rays beyond the non-linear Compton shift predicted by the free-electron model. However, it does reveal a small blue-shift relative to the free electron prediction for non-linear Compton scattering. The effect of electron-electron correlation effects is calculated and determined to be unlikely to be the source of the redshift. The case of linear Compton scattering from a photoionized electron followed by electron recapture is examined as a possible source of the redshift and ruled out.

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