Magnetic Excitations of the Spin Gap System TlCuCl$_3$

TlCuCl$_3$ has a singlet ground state with the excitation gap $\Delta =0.65$ meV. The magnetic excitations in TlCuCl$_3$ have been investigated by means of neutron inelastic scattering experiments. The constant-$\it{\pmb Q}$ energy scan profiles were collected in the $a^*-c^*$ plane. A well-defined single magnetic excitation mode was observed. The dispersion relations along four different directions were determined. The lowest excitation occurs at ${\it{\pmb Q}}=(h, 0, l)$ with integer $h$ and odd $l$, as observed in KCuCl$_3$. A cluster series expansion up to the sixth order was applied to analyze the dispersion relations, and the individual exchange interactions were evaluated. It was demonstrated that TlCuCl$_3$ is a strongly coupled spin-dimer system.