OpenMC is a CPU-based Monte Carlo particle transport simulation code recently developed in the Computational Reactor Physics Group at MIT, and which is currently being evaluated by the UK Atomic Energy Authority for use on the ITER fusion reactor project. In this paper we present a novel port of OpenMC to run on the new ray tracing (RT) cores in NVIDIA's latest Turing GPUs. We show here that the OpenMC GPU port yields up to 9.8x speedup on a single node over a 16-core CPU using the native constructive solid geometry, and up to 13x speedup using approximate triangle mesh geometry. Furthermore, since the expensive 3D geometric operations required during particle transport simulation can be formulated as a ray tracing problem, there is an opportunity to gain even higher performance on triangle meshes by exploiting the RT cores in Turing GPUs to enable hardware-accelerated ray tracing. Extending the GPU port to support RT core acceleration yields between 2x and 20x additional speedup. We note that geometric model complexity has a significant impact on performance, with RT core acceleration yielding comparatively greater speedups as complexity increases. To the best of our knowledge, this is the first work showing that exploitation of RT cores for scientific workloads is possible. We finish by drawing conclusions about RT cores in terms of wider applicability, limitations and performance portability.