We have undertaken a survey of N2H+ and N2D+ towards 31 low-mass starless cores using the IRAM 30m telescope. Our main objective has been to determine the abundance ratio of N2D+ and N2H+ towards the nuclei of these cores and thus to obtain estimates of the degree of deuterium enrichment, a symptom of advanced chemical evolution according to current models. We find that the N(N2D+)/N(N2H+) ratio is larger in more centrally concentrated cores with larger peak H2 and N2H+ column density than the sample mean. The deuterium enrichment in starless cores is presently ascribed to depletion of CO in the high density (> 3*10^4 cm-3) core nucleus. To substantiate this picture, we compare our results with observations in dust emission at 1.2 mm and in two transitions of C18O. We find a good correlation between deuterium fractionation and N(C18O)/N(H2) for the nuclei of 14 starless cores. We, thus, identified a set of properties that characterize the most evolved, or pre-stellar, starless cores. These are: higher N2H+ and N2D+ column densities, higher N(N2D+)/N(N2H+), more pronounced CO depletion, broader N2H+ lines with infall asymmetry, higher central H2 column densities and a more compact density profile than in the average core. We conclude that this combination of properties gives a reliable indication of the evolutionary state of the core. Seven cores in our sample (L1521F, OphD, L429, L694, L183, L1544 and TMC2) show the majority of these features and thus are believed to be closer to forming a protostar than are the other members of our sample. Finally, we note that the subsample of Taurus cores behaves more homogeneously than the total sample, an indication that the external environment could play an important role in the core evolution.
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