  We study the thermodynamic limit of the antiperiodic XXZ spin chain with the
anisotropic parameter $\eta=\frac{\pi i}{3}$. We parameterize eigenvalues of
the transfer matrix by their zero points instead of Bethe roots. We obtain
patterns of the distribution of zero points. Based on them, we calculate the
ground state energy and the elementary excitations in the thermodynamic limit.
We also obtain the two-body scattering matrix of elementary excitations. Two
types of elementary excitations and three types of scattering processes are
discussed in detailed.
