In recent years, near-edge dichroism in crystals, i.e. the dependence of X-ray absorption on crystal and/or magnetic orientations with respect to the polarisation of the photon, has been thoroughly investigated at synchrotron radiation sources. The phenomenon occurs when the X-ray energy approaches the value required to excite an inner-shell electron to an empty orbital of its atom's valence shell. Particular attention has been given to magnetic circular X-ray dichroism, namely the difference in absorption between right- and left-circularly polarised photons in a system with a net magnetisation. Various authors have demonstrated the effect, which requires the breaking of time-reversal symmetry and the presence of a spin-orbit interaction. Electric-dipole (E1) and, in some cases, electric-quadrupole (E2) transitions accounts for the inner-shell excitations.

Much theoretical effort has gone towards the identication of the crystalline microscopic properties revealed by the observed spectra. Atomic theory has provided a number of results including a set of sum rules, which relate integrated dichroic intensities to the ground-state expectation values of effective one-electron operators. These operators coincide with the spin and orbital contributions to the magnetic moment, in the case of magnetic circular X-ray dichroism, thus providing experimentalists with a simple interpretative paradigm.

More recently a novel phenomenon, termed X-ray natural circular dichroism (XNCD), was observed in an organic non-centrosymmetric single crystal and in a stereogenic organometallic complex. The effect stems from the interference between E1 and E1 transitions, thus requiring the breaking of space inversion. These experiments were performed at the ESRF by Goulon et al. [1,2]. Building on previous work of Natoli et al. [3], the ESRF theory group has carried out a theoretical investigation of this new experimental work and found an XNCD sum rule, relating the integrated spectra to the ground-state expectation value of an orbital pseudodeviator [4], namely a rank-two tensor which is even under time reversal and odd under space inversion. The theoretical analysis turned out to be particularly challenging, as it demanded the use of an extended symmetry group to account for interference effects. The extended symmetry was identified with a de Sitter group, O(3,2), a non-compact version of O(5), the rotation group in five dimensions. The theory also provides a microscopic expression for the long-sought vector term in gyrotropy. This term has recently been observed by Goulon and Rogalev through the detection of E1-E2 linear X-ray dichroism in V2O3. In the O(3,2) framework an exhaustive picture of one-electron effects accessible to X-ray dichroism is obtained. Four cases are distinguished:

  1. Time-odd electronic properties in centrosymmetric crystals. They are detected by X-ray magnetic circular dichroism. For E1 transitions, the familiar orbital sum rule (LZ) is recovered.
  2. Time-even electronic properties in centrosymmetric crystals. They are detected by X-ray linear dichroism. For E1 transitions, the quadrupolar orbital sum rule is obtained.
  3. Time-even electronic properties in non-centrosymmetric crystals. They are detected by X-ray circular dichroism. For E1-E2 interference, the orbital-pseudodeviator sum rule is obtained. (If ferromagnetism is present, pure electric multipole transitions will also contribute yielding time-odd orbital tensors, usually detected by X-ray magnetic circular dichroism. These terms vanish when the magnetisation direction is perpendicular to the photon vawe vector.)
  4. Time-odd electronic properties in non-centrosymmetric crystals. They are detected by X-ray linear dichroism. For E1-E2 interference, the gyrotropic vector term and other electronic properties will be observed. (In the case of a magnetic crystal, pure electric transitions will also contribute yielding time-even effects. These terms can by distinguished by full angular-dependence analysis.)

From the foregoing theoretical considerations it is clear that interference dichroism, by probing intrinsic hybridisation moments, appears as a novel powerful experimental technique for investigating electronic properties of centro- and non-centrosymmetric crystals.

References
[1] J. Goulon, C. Goulon-Ginet, A. Rogalev, V. Gotte, C. Malgrange, C. Brouder, C.R. Natoli, J. Chem. Phys., 108, 6394 (1998).
[2] L. Alagna, T. Prosperi, S. Turchini, J. Goulon, A. Rogalev, C, Goulon-Ginet, C. R. Natoli, R.D. Peacock, B. Stewart, Phys. Rev. Lett., 80, 4799 (1998).
[3] C.R. Natoli, C. Brouder, P. Sanctavit, J. Goulon, C. Goulon-Ginet, A. Rogalev, Eur. Phys. J., B 4, 1 (1998).
[4] P. Carra and R. Benoist, unpublished.

Author
P. Carra.
ESRF