Thermodynamic Properties of Three-Dimensional Orthogonal Dimer Model for SrCu${}_2$(BO${}_3$)${}_2$

Effects of the inter-layer couplings for the orthogonal dimer system SrCu${}_2$(BO${}_3$)${}_2$ are discussed. The spin-gap $\mathrm{\Delta }$ of the three-dimensional model is independent of the inter-layer couplings when they are small. Therefore at low temperatures ($T<\mathrm{\Delta }$) thermodynamic properties are described well by the two-dimensional model. On the other hand at high temperatures the mean-field type scaling ansatz is useful to discuss the magnetic susceptibility for week inter-layer couplings. From fit of the magnetic susceptibility, the estimated coupling constants are J = 85 K for the nearest-neighbor couplings, $J^{{}^{\mathrm{\prime }}}=54$ K for the next-nearest-neighbor couplings, and $J^{{}^{\mathrm{\prime }\mathrm{\prime }}}=8$ K for the inter-layer couplings. These parameters are consistent with the temperature dependence of the specific heat at low temperatures.