The distorted-wave Born approximation formalism for the description of the $(e,{e}^{\ensuremath{'}}\ensuremath{\gamma})$ reaction, in which emitted photons and scattered electrons are simultaneously detected, is outlined. Both the Coulomb and the magnetic scattering are fully taken into account. The influence of electron bremsstrahlung is estimated within the plane-wave Born approximation. Recoil effects are also discussed. The formalism is applied for the low-energy $(e,{e}^{\ensuremath{'}}\ensuremath{\gamma})^{92}\mathrm{Zr}$ reaction with excitation of the first collective (${2}_{1}^{+}$) and mixed-symmetry (${2}_{2}^{+}$) states. The corresponding transition charge and current densities are taken from a random-phase approximation (RPA) calculation within the quasiparticle phonon model. It is shown, by this example, in which way the magnetic subshell population of the excited state influences the angular distribution of the decay photon. For these quadrupole excitations the influence of magnetic scattering is only prominent at the backmost scattering angles, where a clear distinction of the photon pattern pertaining to the two states is predicted.
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