In this work, scattering cross sections differential with respect to both the scattering angle and the energy of the scattered photon are derived in the relativistic impulse approximation for the light elements H, Be, and Al, and photon energies between 30 and 200 keV. The energy broadening of the scattered photons reflects the momentum distribution of the target electrons. It increases with both increasing atomic number of the scatterer and with scattering angle. Even in light elements, the energy broadening is comparable with the intrinsic energy resolution of modern Ge spectrometers. In reconstructing primary photon energy spectra by means of a Ge spectrometer and Compton scattering techniques, i.e., by measuring the photons incoherently scattered at a given angle, the energy resolution is markedly impaired compared to direct measurements in the primary beam. This is usually explained as an effect of the nonzero acceptance angle of the detector. It is shown, however, that the fundamental energy broadening of the scattered photons is alone sufficient as an explanation. The Compton scattering technique is valuable in determining energy spectra in clinical situations. Aspects of its optimal performance are discussed. The commonly used scattering angle of 90 degrees seems adequate. At small scattering angles, the incoherent-scattering cross section is badly known due to electron-electron interactions and, for photon energies less than 100 keV, coherent scattering contributes appreciably to the total scattering even in media of low atomic number. In cases where coherent scattering dominates and where the energy degradation of the incoherently scattered photons is small compared to the energy resolution of the spectrometer, the reconstruction is simplified. The double-differential cross sections derived can be used to simplify calculations of the Compton component of the mass-energy absorption coefficient.