direction. Considering these and the above observations together with entropy arguments, one concludes that the low-temperature structure is either a *squared-up*, *equal-amplitude*, and collinear state or - as supported by Mossbauer analysis- a strongly bunched, *squared-up*, and *equal-amplitude* state. As far as the magnon specific heat is concerned, the interactions in both structures can be represented by the Hamiltonian of Eq.\[H\] (allowing for a unimportant difference in the direction of the axis of quantization).

Figs.1-6 and Tab.I demonstrated convincingly that, based on only $\Delta$ and $\theta,$ the diverse functional form of the measured $C_{M}(T)$ can be systematized: when both $\Delta$ and $\theta$ are large, $C_{M}(T<\Delta)$ reflects a magnon contribution from an anisotropic magnetic structure as in $R=$Er, Ho, Dy, Tb. When $\Delta$ is relatively small but $\theta$ is large, $C_{M}(T>\Delta)$ reflects a magnetic contribution from a quasi-isotropic magnetic structure as in GdNi$_{2}$B$_{2}$C. For a weak $\Delta$ and $J_{\Vert}$, $C_{M}(T)$ reflects a contribution from a quasi-2$d$ structure as in TmNi$_{2}$B$_{2}$C.

The evolution of $\Delta$ and $\theta$ across the studied compounds is reasonable. $\theta$, on the one hand, reflects predominately the evolution of the de Gennes factor (see Eqs.5,7 and Tab.I) as can be appreciated on observing that $\theta$ scales very well 