  This is the first in a series of papers studying the astrophysics and
cosmology of massive, dynamically relaxed galaxy clusters. Here we present a
new, automated method for identifying relaxed clusters based on their
morphologies in X-ray imaging data. While broadly similar to others in the
literature, the morphological quantities that we measure are specifically
designed to provide a fair basis for comparison across a range of data quality
and cluster redshifts, to be robust against missing data due to point-source
masks and gaps between detectors, and to avoid strong assumptions about the
cosmological background and cluster masses. Based on three morphological
indicators - Symmetry, Peakiness and Alignment - we develop the SPA criterion
for relaxation. This analysis was applied to a large sample of cluster
observations from the Chandra and ROSAT archives. Of the 361 clusters which
received the SPA treatment, 57 (16 per cent) were subsequently found to be
relaxed according to our criterion. We compare our measurements to similar
estimators in the literature, as well as projected ellipticity and other image
measures, and comment on trends in the relaxed cluster fraction with redshift,
temperature, and survey selection method. Code implementing our morphological
analysis will be made available on the web.
