  We study the nuclear magnetic relaxation rate and Knight shift in the
presence of the orbital and quadrupole interactions for three-dimensional Dirac
electron systems (e.g., bismuth-antimony alloys). By using recent results of
the dynamic magnetic susceptibility and permittivity, we obtain rigorous
results of the relaxation rates $(1/T_1)_{\rm orb}$ and $(1/T_1)_{\rm Q}$,
which are due to the orbital and quadrupole interactions, respectively, and
show that $(1/T_1)_{\rm Q}$ gives a negligible contribution compared with
$(1/T_1)_{\rm orb}$. It is found that $(1/T_1)_{\rm orb}$ exhibits anomalous
dependences on temperature $T$ and chemical potential $\mu$. When $\mu$ is
inside the band gap, $(1/T_1)_{\rm orb} \sim T ^3 \log (2 T/\omega_0)$ for
temperatures above the band gap, where $\omega_0$ is the nuclear Larmor
frequency. When $\mu$ lies in the conduction or valence bands, $(1/T_1)_{\rm
orb} \propto T k_{\rm F}^2 \log (2 |v_{\rm F}| k_{\rm F}/\omega_0)$ for low
temperatures, where $k_{\rm F}$ and $v_{\rm F}$ are the Fermi momentum and
Fermi velocity, respectively. The Knight shift $K_{\rm orb}$ due to the orbital
interaction also shows anomalous dependences on $T$ and $\mu$. It is shown that
$K_{\rm orb}$ is negative and its magnitude significantly increases with
decreasing temperature when $\mu$ is located in the band gap. Because the
anomalous dependences in $K_{\rm orb}$ is caused by the interband particle-hole
excitations across the small band gap while $\left( 1/T_1 \right)_{\rm orb}$ is
governed by the intraband excitations, the Korringa relation does not hold in
the Dirac electron systems.
