In the X-ray region, novel dichroic phenomena have been investigated at the ESRF by Goulon and his collaborators, who reported the observation of two effects:
- X-ray natural circular dichroism (XNCD), probed in Na3Nd(digly)3 x 2NaBF4 x 6H2O [1] and in -LiIO3 [2]. (The effect was observed at the Nd L3 edge and at the iodine L edges.)
- X-ray nonreciprocal linear dichroism (XNLD), detected at vanadium K edge in the low-temperature insulating phase of a Cr-doped V2O3 crystal [3].

We remind the reader that XNCD measures the difference in absorption between right and left circularly-polarised radiation. XNLD implies a difference in absorption between radiation with linear polarisation parallel or perpendicular to a local symmetry axis.

It is crucial to observe that both phenomena stem from the interference between electric-dipole (E1) and electric-quadrupole (E2) transitions that raise an inner-shell electron to empty valence orbitals. Detecting a nonvanishing signal thus requires an ordered structure (crystal) and the breaking of space inversion.

The work of Goulon and his collaborators is of particular importance as it identifies new directions in the microscopic analysis of materials using X-ray absorption spectroscopy. In fact, simple symmetry considerations indicate that XNCD and XNLD are, respectively, sensitive to the electric and magnetoelectric properties of crystals.

It is known that X-ray dichroism probes crystalline orderings, which are described by magnetic (orbital momentum and spin) and centrosymmetric charge order parameters, in the case of pure electric multipole transitions.

Recent work by scientists in the ESRF's Theory Group [4] has shown that E1-E2 dichroism is described by space-inversion-odd electron operators, revealing the presence of parity nonconserving interactions. XNCD and XNLD are thus sensitive to additional order parameters, which can be obtained from the following fundametal operators: the familiar orbital angular momentum L, an electric dipole n = r/r, and = (n x L ­ L x n)/2, a magnetoelectric vector. Two classes of order parameter are identified:
- Space-odd and time-odd operators, obviously invariant under the combined symmetry Spaceinversio . Time-reversal and thus describing magnetoelectric properties of crystals.
- Time-even and space-odd operators corresponding to one-electron polar properties, which arise from a noncentrosymmetric distribution of charge.

These afford a microscopic interpretation of XNCD and XNLD experiments. (For the readers convenience, a summary of X-ray dichroic effects is provided in Table 3).

Table 3: Order parameters and X-ray dichroic effects.


As a concluding remark, we would like to observe that the foregoing ideas are readily extended to E1-E2 X-ray resonant scattering. In this case, the form of the scattering amplitude indicates that ferroelectric and antiferroelectric structures can be studied using X-rays at resonance. It is believed that a confirmation that this is indeed the case is provided by the change with temperature of the (0,0,3)H reflection in (V0.972 Cr0.028)2O3 measured by Paolasini et al. [5].

[1] L. Alagna, T. Prosperi, S. Turchini, J. Goulon, A. Rogalev, C. Goulon-Ginet, C.R. Natoli, R.D. Peacock and B. Stewart, Phys. Rev. Lett. 80, 4799 (1998).
[2] J. Goulon, C. Goulon-Ginet, A. Rogalev, V. Gotte, C. Malgrange, C. Brouder and C.R. Natoli, J. Chem. Phys. 108, 6394 (1998).
[3] J. Goulon, A. Rogalev, C. Goulon-Ginet, G. Benayoun, L. Paolasini, C. Brouder, C. Malgrange, and P.A. Metcalf, Phys. Rev. Lett., 85, 4385 (2000).
[4] P. Carra, A. Jerez and I. Marri, cond-mat/0104582.
[5] L. Paolasini, S. Di Matteo, C. Vettier, F. de Bergevin, A. Sollier, W. Neubeck, F. Yakhou, P.A. Metcalf and J.M. Honig, J. Elec. Spec. Rel. Phen. 120, 1 (2001).

P. Carra