A General Deduction of Compton Scattering Effect Formula

Compton scattering is a fundamental physical phenomenon that is widely present in nature. It plays an important role in the conversion and transfer of matter energy and has a key impact on the evolution of celestial bodies. In the Compton scattering process of free electrons and photons, the wavelength of the scattered photon increases, and energy is transferred from the photon to the electron. In contrast, when high-speed electrons undergo inverse Compton scattering with photons, the electrons decelerate, and the radiation energy is significantly enhanced. However, existing textbooks usually lack a complete derivation of the inverse Compton scattering process and fail to clearly present its scattering formula. This paper considers the general case where particles have arbitrary initial velocities. Based on the law of conservation of momentum and energy, a universal relationship between the scattered photon frequency, the initial photon frequency, and the initial velocity vector of the particle is strictly derived. This paper employs two methods to derive the Compton scattering formula: one is based on the discrete form of the conservation equations and the energy-momentum relationship; the other utilizes the four-dimensional covariant conservation equations, combined with the properties of four-momentum, resulting in a more concise derivation process. In the special case where the electron is initially at rest, the universal relationship can be reduced to the common Compton scattering formula. For the case of inverse Compton scattering of high energy electrons and low energy photons, the scattered photon energy can be up to four times the Lorentz factor squared of the initial photon. This paper provides a physical interpretation of the energy enhancement mechanism of scattered photons in the inverse Compton scattering process. This result can be used as a reference for the teaching of college physics and special relativity.