more appropriate. However, for the present discussion, the above two-level approximation is sufficient. It is highly possible that such a contribution is due to 6% defect/impurity which is beyond the limit of detection of our X-ray structural characterization. Coincidently, anomalous hystereses effects were observed in the magnetostriction curves that were measured within the same temperature range.[@Peng-Dy-hystresis-2K] At any case, for $T>\delta$, both $C_{sch}(T)$ and $C_{N}(T)$ are smaller than $C_{M}(T)$ (see inset of Fig.3). Nevertheless, we considered $C_{M}(T)=C_{tot}(T)-C_{S}(T)-C_{D}(T)-C_{N}(T)-C_{sch}(T)$ (see Fig.3).

The thermal evolution of $C_{M}(T)$ is shown in a log-log plot in Fig.3 and as ln($C_{M}(T)$) $versus$ $1/T$ plot in Fig.4. In both figures, the comparison with Eq.\[Cm-bessel\] (solid line) is also presented. Evidently over a wide range of temperatures, $C_{M}(T\leq T_{c})$ follows excellently Eq.\[Cm-bessel\] with $\theta$=19.3$\pm$0.2K and $\Delta$=8.3$\pm$0.3 K.

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[magnon-Fig3.eps]{}

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[magnon-Fig4.eps]{}

HoNi$_{2}$B$_{2}$C                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
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$C_{tot}(T)$ of single crystal HoNi$_{2}$B$_{2}$C (shown in Fig.5) reveals a cascade of three transitions that are usually attributed to magnetic transformations.[@Canfield-HoNi2B2C-M-Cp; @Elhagary-HoNi2B2C-Cp] The signature of the onset of superconductivity is too weak to be observable in our present measurements. On the other hand, for $T<$5 K, $C_{M}(T)$ follows the description of Eq.\[Cm-bessel\] with $\theta$=9.7$\pm$0.2K and $\Delta
$=8.3$\pm$0.3 K.

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[magnon-Fig5.eps]{}

ErNi$_{2}$B$_{2}$C
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Two intriguing features 