  We study the spin-Peierls transition in a Heisenberg spin chain coupled to
optical bond-phonons. Quantum Monte Carlo results for systems with up to N=256
spins show unambiguously that the transition occurs only when the spin-phonon
coupling alpha exceeds a critical value alpha_c. Using sum rules, we show that
the phonon spectral function has divergent (for infinite N) weight extending to
zero frequency for alpha < alpha_c. The equal-time phonon-phonon correlations
decay with distance r as 1/r. This behavior is characteristic for all 0 < alpha
< alpha_c and the q=pi phonon does not soften (to zero frequency) at the
transition.
