  In bismuth ferrite (BiFeO3), antiferromagnetic and ferroelectric order
coexist at room temperature, making it of particular interest for studying
magnetoelectric coupling. The mutual control of magnetic and electric
properties is very useful for a wide variety of applications. This has led to
an enormous amount of research into the properties of BiFeO$_3$. Nonetheless,
one of the most fundamental aspects of this material, namely the symmetries of
the lattice vibrations, remains controversial.We present a comprehensive Raman
study of BiFeO$_3$ single crystals with the approach of monitoring the Raman
spectra while rotating the polarization direction of the excitation laser. Our
method results in unambiguous assignment of the phonon symmetries and explains
the origin of the controversy in the literature. Furthermore, it provides
access to the Raman tensor elements enabling direct comparison with theoretical
calculations. Hence, this allows the study of symmetry breaking and coupling
mechanisms in a wide range of complex materials and may lead to a noninvasive,
all-optical method to determine the orientation and magnitude of the
ferroelectric polarization.
