Quadrupolar Singlet Ground State of Praseodymium in a Modulated Pyrochlore

The complex structure and magnetism of Pr${}_{2-x}$Bi${}_x$Ru${}_2$O${}_7$ was investigated by neutron scattering and EXAFS. Pr has an approximate doublet ground-state and the first excited state is a singlet. This overall crystal field level scheme is similar to metallic Pr${}_2$Ir${}_2$O${}_7$, which is also reported here. While the B-site (Ru) is well ordered throughout, this is not the case for the A-site (Pr/Bi). A distribution of the Pr-O2 bond length indicates the Pr environment is not uniform even for $x=0$. The Bi environment is highly disordered ostensibly due to the 6s lone pairs on Bi${}^{3+}$. Correspondingly we find the non-Kramers doublet ground state degeneracy otherwise anticipated for Pr in the pyrochlore structure is lifted so as to produce a quadrupolar singlet ground state with a spatially varying energy gap. For $x=0$, below T${}_N$, the Ru sublattice orders antiferromagnetically, with propagation vector $\mathbf{\text{k}}$= (0,0,0), as for Y${}_2$Ru${}_2$O${}_7$. No ordering associated with the Pr sublattice is observed down to 100 mK. The low energy magnetic response of Pr${}_{2-x}$Bi${}_x$Ru${}_2$O${}_7$ features a broad spectrum of magnetic excitations associated with inhomogeneous splitting of the Pr quasi-doublet ground state. For $x=0$ ($x=0.97$) the spectrum is temperature dependent (independent). It appears disorder associated with Bi alloying enhances the inhomogeneous Pr crystal field level splitting so that inter-site interactions become irrelevant for $x=0.97$. The structural complexity for the A-site may be reflected in the hysteretic uniform magnetization of B-site ruthenium in the N'{e}el phase.