isotropic compounds or $T\gg\Delta$, it reduces to the high-temperature limit: [@Kubo-SpinWave; @Eisele-Keffer-SpinWave] $$C_{M}(T)\simeq\frac{3^{\frac{3}{2}}4\pi^{2}}{15}R\left(  T/\theta\right)  ^{3}
\label{Cm-AF-highT}$$ in this case, Eq.\[Theta\] highlight the useful definition of $\left|
J_{eff}\right|  $.

A long-wave dispersion relation for an isotropic quasi-*2*$d$ case can be derived from Eq.\[dispersion-quadratic\], if we set $C_{z}\ll C_{x}$ ($\left\vert J_{\Vert}\right\vert \ll J_{\bot}$) and $H_{a}=0$: $$\hbar\omega_{k}\approx8J_{\Vert}S+J_{\bot}Sa^{2}(k_{x}^{2}+k_{y}^{2})
\label{dispersion-2d}$$ Then, to lower order in $8J_{\Vert}S/T$, one obtains $C_{M}(T)=\pi
RT/12SJ_{\bot}$ which reproduces the leading linear-in-$T$ term in the expression reported by Movshovich et al  [@Movshovich-TmNi2B2C] who (starting from a quadratic dispersion relation and including correction for the 2d and the magnon-magnon interaction) obtained for the range$\ \left\vert
J_{\Vert}\right\vert S<T<T_{N}$:$$C_{M}(T)=(\pi R/12)(T/SJ_{\bot}-6J_{\Vert}/\pi^{2}J_{\bot}+4J_{\Vert}S/3\pi^{2}J_{\bot}T) \label{Cm-2d}$$

It is worth remarking that Eqs.\[H\]-\[dispersion-quadratic\],\[Cm-bessel\] are of a more wide applicability than our above analysis might have suggested. Furthermore, depending on only $\theta$ and $\Delta$, a variety of expressions for the magnetic specific heat can be derived; each admitting different limits for its thermal evolution depending on the relations among $T$ and $\Delta$ (compare Eq.\[Cm-bessel\] with the limit Eqs.\[Cm-AF-lowT\],\[Cm-AF-highT\],\[Cm-2d\]). Based on such a scheme, one is capable of rationalizing the vast varieties of  the low-temperature thermal evolution of thermodynamical quantities (such as $C_{M}(T)$) encountered in these (and any series similar to) borocarbides. It 