Classical k-ratio models, e.g. ZAF and phi(rho z), used in electron probe microanalysis (EPMA) assume a homogeneous or multi-layered material structure, which essentially limits the spatial resolution of EPMA to the size of the interaction volume where characteristic x-rays are produced. We present a new model for characteristic x-ray emission that avoids assumptions on the material structure to not restrict the resolution of EPMA a-priori. Our model bases on the spherical harmonic (PN) approximation of the Boltzmann equation for electron transport in continuous slowing down approximation. PN models have a simple structure, are hierarchical in accuracy and well-suited for efficient adjoint-based gradient computation, which makes our model a promising alternative to classical models in terms of improving the resolution of EPMA in the future. We present results of various test cases including a comparison of the PN model to a minimum entropy moment model as well as Monte-Carlo (MC) trajectory sampling, a comparison of PN-based k-ratios to k-ratios obtained with MC, a comparison with experimental data of electron backscattering yields as well as a comparison of PN and Monte-Carlo based on characteristic X-ray generation in a three-dimensional material probe with fine structures.