  We have mapped from the quantum to the classical limit the spin excitation
spectrum of the antiferromagnetic spin-1 Heisenberg chain system CsNiCl3 in its
paramagnetic phase from T=5 to 200K. Neutron scattering shows that the
excitations are resonant and dispersive up to at least T=70K, but broaden
considerably with increasing temperature. The dispersion flattens out with
increasing temperature as the resonance energy Delta at the antiferromagnetic
wave-vector increases and the maximum in the dispersion decreases. The
correlation length xi between T=12 and 50K is in agreement with quantum Monte
Carlo calculations. xi is also consistent with the single mode approximation,
suggesting that the excitations are short-lived single particle excitations.
Below T=12K where three-dimensional spin correlations are important, xi is
shorter than predicted and the experiment is not consistent with the random
phase approximation for coupled quantum chains. At T=200K, the structure factor
and second energy moment of the excitation spectrum are in excellent agreement
with the high-temperature series expansion.
