Fig. 1: Comparing free-electron-driven coherent X-ray radiation (CXR) in the classical and quantum formalisms.

CXR is excited by the coherent interaction of free electrons traversing a crystal, such as a silicon film. a In the classical formalism, the electron is modeled as a point charge, and the radiation as classical waves. b In the quantum formalism, the electron is represented by a wave, while the radiation by discrete photons. c Mapping the recoil effects in free-electron radiation. The axes represent the inelastic and transverse recoil on the electron, defined by the change in its energy and transverse momentum, respectively. The electron-photon joint states are defined by the energy and transverse wavevector of both electrons and photons. In the classical domain, the inelastic recoil (\(\Delta E={{\hslash }}\omega\)) and the transverse recoil (\(\Delta {q}_{\perp }\)) experienced by the electron are minor relative to the phase space uncertainties of the electron. Quantum corrections to electron dynamics become significant when the recoils approach the electron uncertainties. Yet, inelastic recoil effects in radiation require an even larger inelastic recoil, comparable to the electron kinetic energy⁷. The strongest influence of transverse recoil on the radiation appears for photon emission that is accompanied by different transverse wavevector changes (\(\Delta {q}_{\perp }\) and \(\Delta {q}^{ \, \prime} _{\perp }\)), with the difference exceeding the electron uncertainty \({{\rm{\delta }}}{q}_{\perp }\).