  We present a detailed study of the internal kinematics of the Galactic
Globular Cluster M 4 (NGC 6121), by deriving the radial velocities from 7250
spectra for 2771 stars distributed from the upper part of the Red Giant Branch
down to the Main Sequence. We describe new approaches to determine the
wavelength solution from day-time calibrations and to determine the radial
velocity drifts that can occur between calibration and science observations
when observing with the GIRAFFE spectrograph at VLT. Two techniques to
determine the radial velocity are compared, after a qualitative description of
their advantages with respect to other commonly used algorithm, and a new
approach to remove the sky contribution from the spectra obtained with
fibre-fed spectrograph and further improve the radial velocity precision is
presented. The average radial velocity of the cluster is $\langle v \rangle =
71.08 \pm 0.08$ km s$^{-1}$ with an average dispersion of $\mu_{v_c} = 3.97$ km
s$^{-1}$. Using the same dataset and the same statistical approach of previous
analyses, 20 additional binary candidates are found, for a total of 87
candidates. A new determination of the internal radial velocity dispersion as a
function of cluster distance is presented, resulting in a dispersion of $4.5$
km s$^{-1}$ within 2$^{\prime}$ from the center of cluster and steadily
decreasing outward. We statistically confirm the small amplitude of the cluster
rotation, as suggested in the past by several authors. This new analysis
represents a significant improvement with respect to previous results in
literature and provides a fundamental observational input for the modeling of
the cluster dynamics.

Astrophysics of Galaxies