  Attempts to measure the variability of the fine structure constant alpha over
cosmological time, using spectra of high redshift quasars have produced
conflicting results. We use the Many Multiplet (MM) method with Mg II and Fe II
lines on very high signal-to-noise, high resolution (R = 72,000) Keck HIRES
spectra of eight narrow quasar absorption systems and consider both systematic
uncertainties in spectrograph wavelength calibration and also velocity offsets
introduced by internal velocity structure. We find no significant change in
alpha, Delta(alpha)/alpha =(0.43 +/- 0.34)x 10^(-5), in the redshift range z =
0.7 - 1.5 (statistical and systematic). Scatter in measurements of
Delta(alpha)/alpha arising from absorption line structure can be considerably
larger than assigned statistical errors even for apparently simple and narrow
absorption systems. We find a null result of Delta(alpha)/alpha = (-0.59 +/-
0.55) x 10^(-5) in a system at z = 1.7382 using Cr II, Zn II and Mn II, whereas
using Cr II and Zn II in a system at z = 1.6614 we find a systematic velocity
trend which, interpreted as a shift in alpha, would imply Delta(alpha)/alpha =
(1.88 +/- 0.47) x 10^(-5) (statistical plus systematic). This latter result is
almost certainly caused by varying ionic abundances in subcomponents of the
line. We conclude that spectroscopic measurements of quasar absorption lines
are not yet capable of unambiguously detecting variation in alpha using the MM
method.
