in terms of  small-amplitude spin-wave excitations and, in addition, to elucidate the character and dimensionality of  these excitations. These magnetic excitations can be probed by various techniques, among which is the magnetic specific heat. We carried out extensive zero-field specific heat measurements on five single crystals $R$Ni$_{2}$B$_{2}$C ($R$= Er, Ho, Dy, Tb, Gd) covering at least the range $0.1$ K$<T<T_{N}.$ These specific heats, together with  that of TmNi$_{2}$B$_{2}$C (Ref.[@Movshovich-TmNi2B2C]), reveal a diversified and wide varieties of thermal evolutions. None the less, based on a phenomenological model, all of the specific heat curves can be systematized in terms of only two parameters, namely an effective exchange coupling and a magnetic anisotropy interaction.

The format of this paper is as follows: In Sec.II, we derive an approximate, but of wide applicability, expression for the magnon specific heat. Experimental techniques and procedures are described in Sec.III. Results and their analysis are described in Sec.IV and discussed in Sec.V.

Approximate expressions for magnon specific heat of $R$Ni$_{2}$B$_{2}$C.
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The magnetic structures of $R$Ni$_{2}$B$_{2}$C ($R$=Tm-Gd) can be visualized as magnetic layers that are stacked along the c-axis.[@Lynn-RNi2B2C-mag-xrd] The most  dominant interactions are the RKKY and anisotropic couplings. The former can be approximated by effective isotropic 