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Using elastic and inelastic neutron scattering techniques with and withoutapplication of an external magnetic field $H$, the magnetic ground states ofZn$_x$Co$_{4-x}$(OD)$_6$Cl$_2$ ($x=0,1$) were studied. Our results show thatfor $x=0$, the ground state is a magnetic long-range ordered (LRO) state whereeach tetrahedron forms an "umbrella"-type structure. On the other hand, for$x=1$, no static ordering was observed down to 1.5 K, which resembles thebehavior found in the isostructural quantum systemZn$_x$Cu$_{4-x}$(OD)$_6$Cl$_2$. When $H$ field is applied, however the $x=1$system develops the same LRO state as $x=0$. This indicates that the $x=1$disordered state is in the vicinity of the $x=0$ ordered state.

Magnetic field-induced instability of the cooperative paramagnetic state in Zn$_x$Co$_{4-x}$(OD)$_6$Cl$_2$