We present the results of a high-resolution X-ray-scattering study of the o
rder parameter critical exponent, beta, at the spin-Peierls phase transitio
n in CuGeO3 As expected from previous X-ray data, there is a crossover from
a 3D to a 2D regime, and the critical exponents in both regimes agree well
with the values predicted for an n = 2, XY order parameter. The consequenc
e of the presence of new topological magnetic excitations onto the spin exc
itation spectrum is discussed.