A unified formulation of dichroic signals using the Borrmann effect and twisted photon beams

Dichroic X-ray signals derived from the Borrmann effect and a twisted photon beam with topological charge l = 1 are formulated with an effective wavevector. The unification applies for non-magnetic and magnetic materials. Electronic degrees of freedom associated with an ion are encapsulated in multipoles previously used to interpret conventional dichroism and Bragg diffraction enhanced by an atomic resonance. A dichroic signal exploiting the Borrmann effect with a linearly polarized beam presents charge-like multipoles that include a hexadecapole. A difference between dichroic signals obtained with a twisted beam carrying spin polarization (circular polarization) and opposite winding numbers presents charge-like atomic multipoles, whereas a twisted beam carrying linear polarization alone presents magnetic (time-odd) multipoles. Charge-like multipoles include a quadrupole, and magnetic multipoles include a dipole and an octupole. We discuss the practicalities and relative merits of spectroscopy exploiting the two remarkably closely-related processes. Signals using beams with topological charges l ≥ 2 present additional atomic multipoles.

We calculate the the value of F produced by the interaction of twisted radiation with ions, and adopt the standard assumptions. A dipole matrix element of the type needed in F has been calculated by Alexandrescu et al. with the same assumptions [11]. Radiation is treated classically in the paraxial approximation. The spatial spread of electronic states is assumed to be small compared to the waist w of the twisted beam. In these circumstances the electric field E can be expressed in terms of solid spherical-harmonics ℜ l n(b) with an argument b proportional to the transverse component r⊥ of the position of an electron. The angular orientation of b is carried by a spherical harmonic in ℜ l n(b). For a transverse component r⊥ the topological charge and its projection must satisfy l + n even and, with n = ±l and b = r⊥/w. The polarization vector ε ε ε ε and r⊥ are confined to the plane normal to the direction of propagation of the beam, which is taken to be the z-axis in Fig. 1 of the main text. The proportionality factor in (A4) is purely real. The corresponding dipole interaction operators are, with the electron position r ∝ ℜ ℜ ℜ ℜ 1 (r) measured relative to an origin at R, giving wb = R⊥ + r⊥.
For a topological charge l = 1 the interaction V is evidently a sum of (rα R⊥) and (rα r⊥).
Application of the triangle-rule for the product of two dipoles, (rα r⊥) say, tells us that the it can be represented by the sum of a scalar, dipole and a quadrupole ℜ 2 µ(r). An expansion of facilitates the evaluation of matrix elements for l ≥ 2.
Returning to the amplitude, we consider a typical term in F that is diagonal with respect to the topological charge. The product of the interesting matrix elements is, where Π K' Q' = (ln l−nK' Q') that is different from zero when Q' = 0. We assume that the intermediate state is spatially isotropic, to a good approximation, leaving it characterized solely by total angular momentum Jc that resides in the atomic tensor ϒ K Q(k', k). This simplification of the product of matrix elements is not necessary, however. A general result, with all quantum labels of the intermediate state, is given by Balcar and Lovesey together with steps in its reduction to (A6) [22]. The spherical tensor ϒ K Q(k', k) is also a function of quantum labels in |λ and |λ' that belong to the ground-sate of an ion, whereas intermediate states |η are virtual and do not obeys Hund's rules. Not shown explicitly in (A6) is a product of reduced matrix elements (RMEs) for spherical harmonics [(lv||C(k')||lc)(lc||C(k)||lv)], where lv and lc are angular momenta for the valence and core states, respectively. An RME of this type is different from zero for lv + lc + k even, say, so the aforementioned product is different from zero for (k + k') even. The 3-j symbols in (A6) are different from zero for (l + k') and (l + k) odd integers, which leads to the same condition on (k + k'). Variables in each row and each column of the 9-j symbol are subject to a triangular condition.
The Clebsch-Gordan coefficient Π K' 0 = (ln l−nK' 0) = (− 1) K' (l−n lnK' 0), i.e., Π K' 0 is an odd function of n for K' odd and an even function of n for K' even. In an experiment this finding translates to a powerful selection rule on atomic information available from a difference ∆F of dichroic signals produced with opposite handedness in the photon beam. The selection rule becomes even more influential when it is combined with specific polarization in the primary beam, e.g., K" = 1 for circular polarization.
The photon tensor for a twisted beam (l = 1) and circular polarization can be different from zero for zero projection (Q = 0), and we write it as H K 0(n, P2). One finds, and, is different from zero for K even. Specific values of H K 0(n, P2) appear in Table 4. The result H 4 0(+,+) = H 4 0(+,−) accounts for the absence of a hexadecapole in the difference signal listed in Table 1.