Old open clusters are very useful targets to investigate mechanisms responsible for lithium (Li) depletion during the main sequence. Comparison of the Li abundances in clusters of different age allows us to understand the efficiency of the Li destruction process. Our goal is the determination of membership and Li abundance in a sample of candidate members of the open cluster NGC 3960 (age ~1 Gyr), with the aim to fill the gap between 0.6 and 2 Gyr in the empirical description of the behavior of the average Li abundance as a function of the stellar age. We use VLT/FLAMES Giraffe spectra to determine the radial velocities and thus the membership of a sample of 113 photometrically selected candidate cluster members. From the analysis of the Li line we derive Li abundances for both cluster members and non-members. 39 stars have radial velocity consistent with membership, with an expected fraction of contaminating field stars of about 20%. Li is detected in 29 of the RV members; we consider these stars as cluster members, while we make the reasonable assumption that the remaining 10 RV members without Li, are among the contaminating stars. Li abundances of the stars hotter than about 6000 K are similar to those of stars in the Hyades, while they are slightly smaller for cooler stars. This confirms that NGC 3960 is older than the Hyades. The average Li abundance of stars cooler than about 6000 K indicates that the Li Pop. I plateau might start already at ~1 Gyr rather than 2 Gyr that is the upper limit previously derived in the literature. We also find that the fraction of field stars with high Li abundance (>1.5) is about one third of the whole sample, which is in agreement with previous estimates. The fraction of contaminating field stars is consistent with that previously derived by us from photometry.
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