Single-ion properties of the $S_{eff}$ = 1/2 XY antiferromagnetic pyrochlores, Na$A^{\prime}$Co$_2$F$_7$ ($A^{\prime} = $ Ca$^{2+}$, Sr$^{2+}$)

The antiferromagnetic pyrochlore material NaCaCo$_2$F$_7$ is a thermal spin liquid over a broad temperature range (~ 140 K down to $T_F = 2.4 $K), in which magnetic correlations between Co$^{2+}$ dipole moments explore a continuous manifold of antiferromagnetic XY states. The thermal spin liquid is interrupted by spin freezing at a temperature that is ~ 2 % of the mean field interaction strength, leading to short range static XY clusters with distinctive relaxation dynamics. Here we report the low energy inelastic neutron scattering response from the related compound NaSrCo$_2$F$_7$, confirming that it hosts the same static and dynamic spin correlations as NaCaCo$_2$F$_7$. We then present the single-ion levels of Co$^{2+}$ in these materials as measured by inelastic neutron scattering. An intermediate spin orbit coupling model applied to an ensemble of trigonally distorted octahedral crystal fields accounts for the observed transitions. The single-ion ground state of Co$^{2+}$ is a Kramers doublet with a strongly XY-like $g$-tensor ($g_{xy}/g_{z}$ ~ 3). The local disorder inherent from the mixed pyrochlore $A$ sites (Na$^{+}$/Ca$^{2+}$ and Na$^{+}$/Sr$^{2+}$) is evident in these measurements as exaggerated broadening of some of the levels. A simple model that reproduces the salient features of the single-ion spectrum produces approximately 8.4 % and 4.1 % variation in the $z$ and $xy$ components of the $g$-tensor, respectively. This study confirms that an $S_{eff} =1/2$ model with XY antiferromagnetic exchange and weak exchange disorder serves as a basic starting point in understanding the low temperature magnetic behavior of these strongly frustrated magnets.