Relaxing Kondo screened Kramers-doublets in CeRhSi$_{3}$

CeRhSi$_{3}$ is a superconductor under pressure coexisting with a weakly antiferromagnetic phase characterized by a Bragg peak at $\vec{q}_{0}$=($\sim$ 0.2, 0, 0.5) (N. Aso et al. J. Magn. Magn. Mater. 310, 602 (2007)). The compound is also a heavy fermion material with a large specific heat coefficient $\gamma$=110 mJ $\cdot$ mol$^{-1}$ $\cdot$ K$^{-2}$ and a high Kondo temperature of $T_{K}$=50 K indicative that CeRhSi$_{3}$ is in a strongly Kondo screened state. We apply high resolution neutron spectroscopy to investigate the magnetic fluctuations in the normal phase, at ambient pressures, and at low temperatures. We measure a commensurate dynamic response centered around the $\vec{Q}$=(0, 0, 2) position that gradually evolves to H$\sim$ 0.2 with decreasing temperature and/or energy transfers. The response is broadened both in momentum and energy and not reminiscent of sharp spin wave excitations found in insulating magnets where the electrons are localized. We parameterize the excitation spectrum and temperature dependence using a heuristic model utilizing the random phase approximation to couple relaxing Ce$^{3+}$ ground state Kramers doublets with a Kondo-like dynamic response. With a Ruderman-Kittel-Kasuya-Yosida (RKKY) exchange interaction within the $ab$ plane and an increasing single site susceptibility, we can qualitatively reproduce the neutron spectroscopic results in CeRhSi$_{3}$ and namely the trade-off between scattering at commensurate and incommensurate positions. We suggest that the antiferromagnetic phase in CeRhSi$_{3}$ is driven by weakly correlated relaxing localized Kramers doublets and that CeRhSi$_{3}$ at ambient pressures is on the border between a Rudderman-Kittel-Yosida antiferromagnetic state and a Kondo screened phase where static magnetism is predominately absent.