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Magnetization measurements were performed to investigate the criticalbehavior of the field-induced magnetic ordering in gapped spin systemTlCuCl$_3$. The critical density of the magnons was obtained as a function oftemperature and the magnon-magnon interaction constant was evaluated. Theexperimental phase boundary for $T < 5$ K agrees almost perfectly with themagnon BEC theory based on the Hartree-Fock approximation with realisticdispersion relations. The phase boundary can be described by the power law$[H_{N}(T)-H_{c}] propto T^{phi}$. With decreasing fitting temperature range,the critical exponent ${phi}$ decreases and converges at $phi_{BEC} =3/2$predicted by the magnon BEC theory.

Magnetic-Field Induced Bose-Einstein Condensation of Magnons and Critical Behavior in Interacting Spin Dimer System TlCuCl$_3$