  Using the width of emission lines in XMM-Newton Reflection Grating
Spectrometer spectra, we place direct constraints on the turbulent velocities
of the X-ray emitting medium in the cores of 62 galaxy clusters, groups and
elliptical galaxies. We find five objects where we can place an upper limit on
the line-of-sight broadening of 500 km/s (90 per cent confidence level), using
a single thermal component model. Two other objects are lower than this limit
when two thermal components are used. Half of the objects examined have an
upper limit on the velocity broadening of less than 700 km/s. To look for
objects which have significant turbulent broadening, we use Chandra spectral
maps to compute the expected broadening caused by the spatial extent of the
source. Comparing these with our observed results, we find that Klemola 44 has
extra broadening at the level of 1500 km/s. RX J1347.5-1145 shows weak evidence
for turbulent velocities at 800 km/s. In addition we obtain limits on
turbulence for Zw3146, Abell 496, Abell 1795, Abell 2204 and HCG 62 of less
than 200 km/s. After subtraction of the spatial contribution and including a 50
km/s systematic uncertainty, we find at least 15 sources with less than 20 per
cent of the thermal energy density in turbulence.
