  Barium is a key element in constraining the evolution of the (not well
understood) r-process in the first galactic stars and currently the Ba
abundances in these very metal-poor stars were mostly measured under the Local
Thermodynamical Equilibrium (LTE) assumption, which may lead in general to an
underestimation of Ba. We present here determinations of the barium abundance
taking into account the non-LTE (NLTE) effects in a sample of extremely
metal-poor stars (EMP stars): 6 turnoff stars and 35 giants. The NLTE profiles
of the three unblended Ba II lines (455.4, 585.3, 649.6nm) have been computed.
The computations were made with a modified version of the MULTI code, applied
to an atomic model of the Ba atom with 31 levels of Ba I, 101 levels of Ba II,
and compared to the observations. The ratios of the NLTE abundances of barium
relative to Fe are slightly shifted towards the solar ratio. In the plot of
[Ba/Fe] versus [Fe/H], the slope of the regression line is slightly reduced as
is the scatter. In the interval -3.3 <[Fe/H] < -2.6, [Ba/Fe] decreases with a
slope of about 1.4 and a scatter close to 0.44. For [Fe/H] <-3.3 the number of
stars is not sufficient to decide whether [Ba/Fe] keeps decreasing (and then
CD-38:245 should be considered as a peculiar "barium-rich star") or if a
plateau is reached as soon as [Ba/Fe] ~ -1. In both cases the scatter remains
quite large, larger than what can be accounted for by the measurement and
determination errors, suggesting the influence of a complex process of Ba
production, and/or inefficient mixing in the early Galaxy.
