Field-Gradient Effect in Quantum Beamstrahlung

The correction terms to the Sokolov-Ternov radiation formula due to variation of the magnetic field strength along the electron trajectory are calculated up to the second order in the power expansion of $\frac{B_{\tau }}{B}$, where $\tau$ is the formation time of radiation. It is found that the field-gradient effect reduces radiation intensity in the classical regime, and enhances it in the quantum regime. This is then applied to quantum beamstrahlung with Gaussian variation in $e^{+}{e}^{-}$ bunch currents. The correction is shown to be substantial for beam parameters suggested by Himel and Siegrest.