  We study both theoretically and experimentally the free induction decay (FID)
of the electron spin associated with a single nitrogen-vacancy defect in high
purity diamond, where the main source of decoherence is the hyperfine
interaction with a bath of $^{13}$C nuclear spins. In particular, we report a
systematic study of the FID signal as a function of the strength of a magnetic
field oriented along the symmetry axis of the defect. On average, an increment
of the coherence time by a factor of $\sqrt{5/2}$ is observed at high magnetic
field in diamond samples with a natural abundance of $^{13}$C nuclear spins, in
agreement with numerical simulations and theoretical studies. Further
theoretical analysis shows that this enhancement is independent of the
concentration of nuclear spin impurities. By dividing the nuclear spin bath
into shells and cones, we theoretically identify which nuclear spins are
responsible for the observed dynamics.
