An experimental search for variation in the fundamental coupling constants is strongly motivated by modern high-energy physics theories. Comparison of quasar absorption line spectra with laboratory spectra provides a sensitive probe for variability of the fine structure constant, alpha, over cosmological time-scales. We have previously developed and applied a new method providing an order of magnitude gain in precision over previous optical astrophysical constraints. Here we extend that work by including new quasar spectra of damped Lyman-alpha absorption systems. We also re-analyse our previous lower redshift data and confirm our initial results. The constraints on alpha come from simultaneous fitting of absorption lines of subsets of the following species: Mg I, Mg II, Al II, Al III, Si II, Cr II, Fe II, Ni II and Zn II. We present a detailed description of our methods and results based on an analysis of 49 quasar absorption systems (towards 28 QSOs) covering the redshift range 0.5 < z < 3.5. There is statistical evidence for a smaller alpha at earlier epochs: da/a = (-0.72 +/- 0.18) * 10^{-5}. The new and original samples are independent but separately yield consistent and significant non-zero values of da/a. We summarise the results of a thorough investigation of systematic effects published in a companion paper. The value we quote above is the raw value, not corrected for any of these systematic effects. The only significant systematic effects so far identified, if removed from our data, would lead to a more significant deviation of da/a from zero.
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