Magnetic properties and crystal field in Pr$_2$ Zr$_2$ O$_7$

Abstract : In this work, we revisit the crystal field acting on the non-Kramers Pr$^{3+}$ ion (4f$^2$) in the quantum spin-ice candidate Pr$_2$ Zr$_2$ O$_7$ using both a standard calculation restricted to the ground spin-orbit multiplet and intermediate coupling states in the full basis of the f$^2$ configuration. Analysis of the thermal variation of the polycrystal magnetic susceptibility and of the local susceptibilities $\chi$⊥ and $\chi _\parallel$ determined by means of polarized neutron diffraction experiments reveals that the effective antiferromagnetic exchange is strongly depleted at low temperature with respect to its high-temperature value. We then discuss the influence of crystal field imperfections arising from residual strains, which are especially important for a non-Kramers ion. We find that they are an essential ingredient to account for the very low temperature $M(H)$ magnetization curves, showing that the saturation is not achieved even at 8 T. Furthermore, as possible candidates to qualitatively understand the Curie-like behavior observed below 0.5 K, we discuss the influence of the magnetic hyperfine interaction.