Exchange of angular momentum in EMCD experiments

In energy loss magnetic chiral dichroism (EMCD) experiments a chiral electronic transition is induced that obeys the dipole selection rule for the magnetic quantum number Δ m = ± 1 or Δ L z = ± ℏ . The incident plane electron wave is inelastically scattered and is detected in the diffraction plane, i.e. again in a plane wave state. Naïve reasoning suggests that the angular momentum L z of the probe electron has not changed in the interaction since plane waves have 〈 L z 〉 = 0 . This leads to the seeming contradiction that angular momentum is not conserved in the interaction. A closer inspection shows that the density matrix of the probe has indeed 〈 L z 〉 = ± ℏ after a chiral interaction. However, 〈 L z 〉 is not conserved when the probe electron propagates further to the exit surface of the specimen because the rigid lattice breaks rotational symmetry. Thus, the angular momentum of the photo electron that is created in a chiral electronic transition stems from both the probing electron and the crystal lattice.