The rotatory strength expressions in Eqs. 3-5 are more complicated than those for isotropic samples, which only involve the terms with the transition magnetic dipole (the first line in Eq. 3). This is because, in isotropic samples, the measured quantity is the spatial average of all possible orientations: R if = Tr(R if )/3, and the terms with the quadrupole transition moment do not contribute to this trace. However, for oriented systems like crystals and thin films, one needs the full tensor defined above.