  We study vacuum birefringence and x-ray photon scattering in the head-on
collision of x-ray free electron and high-intensity laser pulses. Resorting to
analytical approximations for the numbers of attainable signal photons, we
analyze the behavior of the phenomenon under the variation of various
experimental key-parameters and provide new analytical scalings. Our optimized
approximations allow for quantitatively accurate results on the one-percent
level. We in particular demonstrate that an appropriate choice of the x-ray
focus and pulse duration can significantly improve the signal for given laser
parameters, using the experimental parameters to be available at the Helmholtz
International Beamline for Extreme Fields at the European XFEL as example. Our
results are essential for the identification of the optimal choice of
parameters in a discovery experiment of vacuum birefringence at the
high-intensity frontier.
