research papers
Xray natural
in langasite crystal^{a}Faculty of Physics, M. V. Lomonosov Moscow State University, Leninskie Gory, Moscow 119991, Russian Federation, ^{b}European Synchrotron Radiation Facility, 71 Avenue des Martyrs, CS 40220, Grenoble 38043, France, and ^{c}A. V. Shubnikov Institute of Crystallography, Federal Scientific Research Centre `Crystallography and Photonics' of Russian Academy of Sciences, Leninskiy Prospekt 59, Moscow 119333, Russian Federation
^{*}Correspondence email: ap.oreshko@physics.msu.ru
Optical activity in the Xray range stems from the electricdipole–electricquadrupole interference terms mixing multipoles of opposite parity, and can be observed exclusively in systems with broken inversion symmetry. The gyration tensor formalism is used to describe the Xray optical activity in langasite La_{3}Ga_{5}SiO_{14} crystal with the P321 An experimental study of the Xray natural (XNCD) near the Ga Kedge in La_{3}Ga_{5}SiO_{14} single crystal was performed at ESRF beamline ID12, both along and perpendicular to the crystal optical axis. The combination of the quantum mechanical calculations and highquality experimental results has allowed us to separate the contributions into Xray absorption and XNCD spectra of Ga atoms occupying three distinct Wyckoff positions.
Keywords: optical activity; circular dichroism; Xray absorption spectra.
1. Introduction
Since the discovery by Arago (Arago, 1811) and Biot (Biot, 1812) that crystalline quartz induces a rotation of the plane of polarization of visible light, the optical activity (OA) of single crystals has fascinated many successive generations of physicists. It was, however, only at the end of the 20th century that Xray natural (XNCD) was detected in quite a few noncentrosymmetric crystals (Goulon et al., 1998, 1999; Alagna et al., 1998; Stewart et al., 1999). Such experiments, which were performed about 100 years after the discovery of Xrays (Röntgen, 1898) and of in the visible spectral range (Cotton, 1895), became feasible due to the availability of intense beams of circularly polarized Xrays at thirdgeneration synchrotron radiation sources. It has been known for many years (Nye, 1957; Landau & Lifshitz, 1960; Agranovich & Ginzburg, 1984; Yariv & Yeh, 1984) that OA in crystals is a tensor property resulting from the spatial dispersion of the dielectric response, i.e. the radiation–matter interactions depend on the space variable r and on the light wavevector k. OA refers to the firstorder terms (linear in k) and is described by a rank3 antisymmetric tensor, commonly referred to as the gyration tensor. The crystal symmetry imposes certain restrictions on the components of the gyration tensor (Jerphagnon & Chemla, 1976; Jerphagnon et al., 1978). For example, all components of the gyration tensor are zero in centrosymmetric crystals. To summarize, out of the 21 noncentrosymmetric crystal classes (crystallographic point groups), 18 have a nonzero gyration tensor and are said to be optically active or gyrotropic. Note that among them only 11 crystal classes exhibit enantiomorphism and thus are chiral. It then becomes quite obvious that enantiomorphism (or chirality) and optical activity are two distinct concepts that should not be mixed together.
The decomposition of the Cartesian rank3 gyration tensor into irreducible representations with respect to the operations of the group SO_{3} was shown to yield (Jerphagnon & Chemla, 1976; Jerphagnon et al., 1978) a pseudoscalar, a polar vector and a traceless symmetric pseudotensor of rank2, socalled pseudodeviator. It has long been recognized that only chiral crystal classes have a nonzero pseudoscalar part. On the other hand, gyrotropic crystals featuring a large pseudodeviator part are most often showing a large nonlinear susceptibility at visible wavelengths. More important for the present study is the link between the macroscopic gyration tensor and the microscopic susceptibility tensor: Buckingham & Dunn (1971) and Barron (1971, 1982) were the first to point out that the electricdipole–electricquadrupole (E1.E2) interference term contributed to the pseudodeviator part of the gyration tensor but not to the pseudoscalar which was identified with the electricdipole–magneticdipole (E1.M1) mechanism.
In the visible spectral range, both the pseudoscalar and the pseudodeviator contribute to OA, whereas, in the Xray range, the pseudodeviator part only gives rise to natural et al., 1998, 1999). The reason for this is that the E1.M1 interference terms, which dominate OA measured with visible light, become vanishingly small in The E1.M1 contribution implies a significant magnetic dipole transition probability and this is strongly forbidden in a nonrelativistic approach due to the radial orthogonality of core with valence and continuum states involved in the Xray absorption process. This restriction could be partially removed in fully relativistic formalism or due to relaxation of the corehole. In contrast, electric quadrupolar (E2) transitions, which play only a marginal role at visible wavelengths, can become significant in the Xray range since their magnitude increases with photon energy (Brouder, 1990). It is now well documented (see, for example, Goulon et al., 2003b) that the interference electricdipole–electricquadrupole (E1.E2) terms are responsible for most OArelated effects observed in the Xray range: natural (Goulon et al., 1998; Alagna et al., 1998), nonreciprocal linear dichroism (Goulon et al., 2000; van der Laan, 2001), magnetochiral dichroism (Goulon et al., 2002; Sessoli et al., 2015) and the socalled VoigtFedorov dichroism observable in scattering experiments (Goulon et al., 2007).
(GoulonThe electricdipole–electricquadrupole E1.E2 interference terms mix multipole transition moments of opposite parity, and this allows the mixed valence states to be probed by XNCD (Natoli et al., 1998; Okutani et al., 1999; Peacock & Stewart, 2001; Goulon et al., 2003a,b). Moreover, XNCD makes it possible to determine the of chiral crystals (Rogalev et al., 2008a; Sessoli et al., 2015) similarly to natural or in the visible region but with an element selectivity that is inherent to This knowledge is especially important for systems, properties of which depend on their e.g. in bio and pharmaceutical industry, in chiral liquid crystals, magnetoelectrics and other multifunctional materials.
The physical properties of a solid and its _{3}Ga_{5}SiO_{14} (hence the name), form a formidable playground for designing multifunctional materials. These crystals were discovered in the 1980s in the former Soviet Union (Belokoneva & Belov, 1981; Mill et al., 1982; Mill & Pisarevsky, 2000) and widely studied due to their piezoelectric and nonlinear optical properties (Ohsato et al., 2012; Chilla et al., 2001; Roshchupkin et al., 2003; Stade et al., 2002) related to the noncentrosymmetric The langasite structure belongs to the trigonal noncentrosymmetric P321 The general formula is A_{3}BC_{3}D_{2}O_{14}, which contains four different cationic sites. The decahedral A site and the octahedral B site form a layer at z = 0, whereas the two tetrahedral sites C and D are located on the plane z = 1/2 (see Fig. 1). Due to this atomic arrangement, the langasite structure is able to accommodate a large number of different cations with various sizes and valences, leading to a wide variety of isostructural compounds. Whenever one of the cations is substituted by magnetic ions, the crystals provide interesting examples of multiferroics with remarkable optical properties (Pikin & Lyubutin, 2012; Lee et al., 2010) and even may lead to the appearance of exotic magnetic structure (Zorko et al., 2011; Marty et al., 2010; Scagnoli et al., 2013; Ramakrishnan et al., 2017). Many other physical properties of langasites can be tailored as well by properly choosing the substituted cations and their position in the structure (Takeda et al., 2011; Sato et al., 1998; Adachi et al., 2003). That is why a detailed knowledge of the electronic properties corresponding to different crystallographic positions in langasites is very important.
are closely related and understanding this relationship is one of the greatest challenges of material science, especially in view of engineering new compounds. This is particularly true for multifunctional materials whose properties, like ferromagnetism or strongly depend on the symmetry of a substance. The langasite family of crystals, the archetype of which is LaHere, we combined elementspecific XNCD spectroscopy and ab initio quantum mechanical calculations to study local electronic properties of La_{3}Ga_{5}SiO_{14} single crystal. As we mentioned earlier, XNCD is the most appropriate technique to study the of electronic states of opposite parity, which are at the origin of nonlinear optical properties, and to give a useful insight into the electronic structure of the whole family of langasite compounds. Unfortunately, XNCD, as any Xray absorptionbased technique, measures the signal at the of a given averaged over all crystallographic positions which it occupies in the crystal. Thus, Xray dichroism is an elementspecific and edgeselective technique, but not siteselective (Brouder et al., 2008). To overcome this difficulty, one can exploit resonant Xray diffraction which is an elementspecific and siteselective method due to the phase shift between the waves scattered by different atomic sites (Hodeau et al., 2001; Dmitrienko et al., 2005). However, the symmetry of the langasite crystals does not impose any and all resonant atoms occupying nonequivalent crystallographic sites simultaneously contribute to the allowed reflections measured in Xray diffraction experiments. Being more timeconsuming, the resonant diffraction methods have no advantages in studying the electronic structure of langasites and rely on numerical simulations that are no less complicated than those used for the XNCD.
The present paper starts with a detailed analysis of the relation between the microscopic and macroscopic description of Xray optical activity in crystals with P321 symmetry, especially the langasite family of crystals. We show that analysis of the Ga Kedge XNCD in La_{3}Ga_{5}SiO_{14} single crystal based on quantum mechanical calculations allows one to disentangle the contributions from crystallographically different atomic sites.
2. Xray optical activity
Recall that, in single crystals, optical activity is a tensor property resulting from the spatial dispersion of the dielectric response, i.e. the breakdown of the usual electric dipole approximation in radiation–matter interactions. The interaction of an electromagnetic wave defined by a wavevector = with anisotropic media is usually described in terms of the dielectric permittivity tensor, which can be represented as the following sum (Agranovich & Ginzburg, 1984),
In absorbing media the components of each tensor in (1) are complex values, whose real and imaginary parts describe refraction and absorption processes, respectively.
Using Fermi's golden rule for the transition rate, the Xray absorption
in Gaussian units iswhere is the energy of the Xray photon with unit polarization vector . Labels i and f stand for initial (core) and final (empty) states with energies E_{i} and E_{f}. The conventional approach is to expand the exponent in equation (2) in powers of ,
The traditional dipole approximation corresponds to setting = 1; then the transition operator reduces to the socalled velocity form of the electric dipole (E1) operator. The above expansion is closely related to the expansion in terms of electromagnetic multipoles designated generically as EJ and MJ, J being the multipolarity. The firstorder term is related to a linear combination of electricquadrupole (E2) and magneticdipoles (M1) multipoles. The term, quadratic in k, corresponds to electricoctupole (E3) and magneticquadrupole (M2) multipoles. The multipoles behave as tensors of increasing rank (J) and alternating parity given by (−1)^{J} for electric moments and (−1)^{J–1} for magnetic ones. The validity of this approach in was discussed by Brouder (1990) and it was shown that the first two terms are sufficient to describe the Xray absorption spectra. The equation (2), is proportional to the square of the transition amplitude and can be represented as the following sum,
Let us underline the tensorial character of each contribution, which provides various kinds of Xray optical effects at photon energies close to an k; the last two are very weak in the Xray range and will not be discussed here. The pure electric quadrupolar transitions give rise to the rank4 tensor and there has been ample experimental evidence of their importance in Xray absorption spectroscopy with photon energy above 4 keV (Brouder, 1990). The symmetric part of this tensor is responsible for an optical anisotropy including cubic crystals as was observed in the visible (Pastrnak & Vedam, 1971) as well as in the Xray range (Cabaret et al., 2001). Whenever the medium is magnetic or placed in an external magnetic field, this rank4 tensor has also an antisymmetric part which leads to observation of We are most concerned in the present paper with the rank3 tensors in equation (4) which are linear in the Xray wavevector. In the absence of any external magnetic field or any spontaneous magnetic ordering in the system, these tensors are fully antisymmetric and describe effects related to natural optical activity. Let us stress again that the elements of these tensors are associated with transition probabilities that mix multipoles of opposite parity (E1.E2: electricdipole–electricquadrupole; E1.M1: electricdipole–magneticdipole), and, thus, can be nonzero only in noncentrosymmetric systems. The sum of these two tensors is closely related to the imaginary parts of the rank3 tensor (γ) in equation (1) called the gyration tensor. When a crystal is odd not only with respect to space parity but also with respect to the timereversal operator, the gyration tensor may still exhibit a symmetric part. It has been shown (Smolenskii et al., 1975) that the real and imaginary parts of the gyration tensor can be associated with four distinct classes of optical effects depending on their timereversal symmetry. These are listed in Table 1 and each caters for a unique experimental technique for investigating specific properties of the material. If the sign of the effect changes when light propagates in the forward and backward directions, the effect is called nonreciprocal. A typical example of nonreciprocal effects is Faraday rotation. As can be seen from Table 1, all nonreciprocal optical activity effects are associated with broken timereversal symmetry.
The first term is the rank2 tensor due to pure electric dipole transitions. Its symmetric part describes the linear dichroism effect. In the presence of an external magnetic field or spontaneous magnetic order, this tensor has an antisymmetric part that is at the origin of the Further terms in the symmetrical part that are quadratic in the magnetization describe the magnetic linear dichroism. There are three terms which are quadratic in

It has been shown (Agranovich & Ginzburg, 1984) that the gyration tensor is antisymmetric in the exchange of the first two indices,
Thus, in Cartesian coordinates, at most nine components out of 27 could be independent. This makes it often preferable to substitute γ_{αβη} by its dual, rank2, gyrotropy tensor G_{αβ} defined as
where δ_{αλμ} is the Levi–Civita symbol, or permutation tensor (Nye, 1957).
The gyrotropy tensor has three irreducible representations with respect to the operations of the rotation group SO_{3} (Jerphagnon & Chemla, 1976): a pseudoscalar, a polar vector and a traceless symmetric rank2 pseudotensor called a pseudodeviator. A pseudoscalar and a pseudotensor are rotational invariants but, unlike a true scalar or a true tensor, they have odd parity and therefore all their components invert the signs under spatial inversion. On the other hand, only pseudoscalar terms survive in randomly oriented samples (molecules in solution or powder) while the vector part and pseudodeviator could be detected only in systems with an orientational order. A priori the scalar part of the gyrotropy tensor could only be associated with the E1.M1 interference terms. This is because the involved transition moment tensors are of equal rank. As far as is concerned, magnetic dipole transitions (M1) are forbidden due to the radial orthogonality of the core and the valence states involved in transitions. Nevertheless, a very weak XNCD signal due to the E1.M1 contribution was detected at the Ni Kedge in αNiSO_{4}·6H_{2}O (Rogalev et al., 2008b). This observation may perhaps indicate that magnetic dipole transitions become allowed in multielectron processes involving valence electrons and which are not subject to standard selection rules.
At the same order of development as E1.M1 there is an interference term between the electric dipole and the electric quadrupole transition operators (E1.E2). Since these are tensors of unequal rank, the result cannot be a pseudoscalar and this term would not therefore contribute in random orientation. The E1.E2 interference terms appear to be responsible for the vector part of optical activity and to strongly contribute to the pseudodeviator. One should bear in mind that only 13 crystal classes (622; 32; 422; 6; 3; 4; ; ; mm2; 222; 2; m; 1) admit the pseudodeviator as a rotational invariant in SO_{3}, so exhibiting XNCD as a consequence of nonvanishing tensor . It might be worth emphasizing here that only nine out of these 13 groups are enantiomorphous and that and optical activity are two distinct physical concepts which should never be confused.
XNCD is defined as the difference in absorption cross sections for right [σ^{+}(ω)] and left [σ^{−}(ω)] circularly polarized Xray beams,
At the microscopic level the electricdipole–electricquadrupole interference term is
where c.c. stands for complex conjugate. If one chooses the ηaxis to be the Xray beam propagation direction (kr = k_{η}r_{η}), the XNCD due to E1.E2 interference terms takes the form
XNCD is linked to the imaginary part of the gyration tensor. For a deeper insight into the physical origin of this effect it is instructive to relate the XNCD i.e. to some effective operator. This is precisely the aim of the spectroscopic sum rules. In 1998, Natoli et al. (Natoli et al., 1998; Brouder et al., 1999) showed that the XNCD integral is proportional to the groundstate rank2 effective operator that probes the mixing of orbitals of different parity in final states. At the K or L_{1}edges, the degree of mixing of p and dorbitals is directly measured. Carra et al. (Carra & Benoist, 2000; Carra et al., 2003) were the first to realize and to prove that the effective operator of XNCD is the dyad product of two vectors: the timereversal odd orbital angular momentum vector and an orbital anapole moment (Khriplovich & Pospelov, 1990) which is odd with respect to both parity and timereversal.
to some groundstate observable,Experimentally, XNCD has been observed in quite a large number of noncentrosymmetric crystals (Natoli et al., 1998; Okutani et al., 1999; Peacock & Stewart, 2001; Goulon et al., 2003a,b) but it is still less widely used in comparison with its magnetic counterpart. This technique could gain renewed interest with the emergence of new molecular materials which are both chiral and magnetic (Sessoli et al., 2015).
3. Xray optical activity in langasite crystal
Langasite, which has the chemical formula La_{3}Ga_{5}SiO_{14}, is a piezoelectric crystal of the same as quartz and hence possessing similar acoustic and optical properties (Belokoneva & Belov, 1981; Mill et al., 1982). The first lanthanum gallium silicate crystals were grown in Russia in the early 1980s (Mill & Pisarevsky, 2000). The structure of langasite crystal belongs to the trigonal system with 32 [space group P321 (No.150), Z = 1, constants a = 8.1746 (6) Å, c = 5.1022 (4) Å (Maksimov et al., 2005)], being isostructural to Ca_{3}Ga_{2}Ge_{4}O_{14} (Mill et al., 1982).
There are four different cationic sites in the structure with generic chemical formula A_{3}BC_{3}D_{2}O_{14}. The A site (the 3e with 2), coordinated by eight oxygen atoms, can be represented as a distorted dodecahedron with trigonal faces, known as twisted Thomson. The octahedral B site is coordinated by six oxygen atoms and has position 1a with symmetry 32. Both C and D sites represent tetrahedral sites coordinated by four oxygen atoms, whereby the size of the C tetrahedra (position 3f, symmetry 2) is slightly larger than the D tetrahedra (2d position with symmetry 3). The structure is layered: along the c axis, tetrahedral layers alternate with layers composed of octahedra and dodecahedra (Fig. 1).
In the La_{3}Ga_{5}SiO_{14} La^{3+} ions occupy the A site, Ga^{3+} are located at the B (Ga1), C (Ga3) and half of the D (Ga2) sites, whereas Si^{4+} ions occupy another half of the D site (see Fig. 2).
The P321 is symmorphic and there are no forbidden resonant reflections. Sometimes the resonant contribution can be extracted from the weakly allowed reflections (Mukhamedzhanov et al., 2007) due to the interference between two scattering channels. But in this case the siteselective resonant contribution to the atomic factor is significantly less than the nonresonant contribution, which makes the resonant Xray scattering unsuitable for studying local anisotropy in the P321 group crystals. On the other hand, this is enantiomorphic; that is to say, there should exist two chiral atomic arrangements, left and righthanded. Crystals belonging to the 32 admit both a pseudoscalar and a pseudodeviator as rotational invariants and therefore are compatible with the observation of in the visible as well as in the Xray range. Contrary to diffraction experiments which are sensitive to the phase shift of the waves scattered by different atoms, in absorption geometry the imaginary part of gyration tensor describing in a crystal is a sum over all absorbing atoms,
where is a thirdrank tensor corresponding to the nth nonequivalent absorbing atom in a unit cell.
In general, this thirdrank tensor antisymmetric in the exchange of the first two indices contains only nine independent nonzero components, which can be written in a matrix form as (Agranovich & Ginzburg, 1984; Sirotine & Shaskolskaia, 1982)
In the P321 atoms can occupy seven nonequivalent Wyckoff positions, 1a, 1b, 2c, 2d, 3e, 3f and 6g (Hahn, 2005). Positions 1a and 1b have 32, positions 2c and 2d have 3, positions 3e and 3f have 2 and position 6g has 1. Since atoms occupying Wyckoff positions with the same have the same nonzero gyration tensor components, it is sufficient to calculate the gyration tensor for 1a, 2d, 3f and 6g.
There is only one atom with coordinates (0,0,0) and a of the P321 group, and its gyration tensor contains nonzero components (Sirotine & Shaskolskaia, 1982; Malgrange et al., 2014),
32 in position 1In position 3f there are three atoms with the coordinates (x,0,1/2), (0,x,1/2), (−x,−x,1/2) whose is 2. The gyration tensor for each atom contains five nonzero components,
The tensor components of different atoms in position 3f are related by the threefold axis parallel to the c axis of the crystal. In the orthogonal basis, their coordinates are transformed as r′ = Tr, where T is the rotation matrix
The gyration tensors for these atoms have the following forms,
The total gyration tensor corresponding to position 3f is the sum of the gyration tensors corresponding to each atom,
There are two atoms in position 2d with 3 which are connected by the twofold symmetry axis [(1/3, 2/3, z) → (2/3, 1/3, −z)]. The gyration tensor for an atom in the 2d position is
and the corresponding rotation matrix is given by
Hence, the resulting gyration tensor corresponding to position 2d is the sum of the gyration tensors corresponding to each atom,
There are six atoms, (x,y,z), (−y,x−y,z), (−x+y,−x,z), (y,x,−z), (x−y,−y,z), (−x,−x+y,z), with 1 in position 6g, and the gyration tensor for each atom contains nonzero components,
The resulting gyration tensor corresponding to position 6g is the sum of the gyration tensors corresponding to each atom,
Thus, P321 can be completely described by combining tensors (11), (15), (18) and (20). For example, Ga^{3+} ions in La_{3}Ga_{5}SiO_{14} crystal occupy three Wyckoff positions, 1a, 3f and 2d (the latter in a ratio of 1:1 with Si^{4+} ions). The resulting gyration tensor for Ga atoms is the sum of the gyration tensors corresponding to Ga atoms in each position,
in a crystal withwhere
and
Note that due to the symmetry of the 2d and 3f positions the tensor components , and do not contribute to the tensor .
If one chooses the zaxis (optical axis of the crystal) collinear with the Xray wavevector (kr = kz), the XNCD (8) due to E1.E2 transitions is given by
whereas, for the Xray beam directed along the xaxis (and similarly along the yaxis),
Here, we have neglected the pseudoscalar E1.M1 contributions, which is a well justified approximation for the Xray range and that imposes an additional symmetry restriction of the components of the gyration tensor, i.e. = 0 or = . This is manifested in the angular dependence of XNCD spectra for uniaxial crystals being proportional to 3cos^{2}θ − 1 (Natoli et al., 1998), where θ is an angle between the Xray propagation direction and the optical axis of the crystal.
4. Experiment
A highquality single crystal of La_{3}Ga_{5}SiO_{14} was grown by the Czochralski method at M. V. Lomonosov Moscow State University. For the experiment described in this paper the sample was cut in the form of a 5 mm × 5 mm × 2 mm parallelepiped with the caxis perpendicular to a large surface. It was optically polished and checked to be The XNCD spectra were recorded at ESRF beamline ID12 dedicated to polarizationdependent in the Xray range from 2 to 15 keV. The performances of this beamline are described by Rogalev et al. (2001) and we are limiting ourselves here only to details relevant to the present experiment. The source of circularly polarized Xrays is the helical undulator of APPLEII type, the fundamental harmonic of which covers the energy range 5.0–8.9 keV. For the experiments at the Kedge of Ga (∼10.37 keV), we had to use the second harmonic of the undulator emission. The Xray beam has been further monochromated using a fixedexit doublecrystal monochromator equipped with a pair of Si(111) crystals cooled down to 133 K. Given the very low emittance of the electron beam at the ESRF, the instrumental energy resolution at the Ga Kedge of 1.5 eV is better than the natural width (FWHM: full width at half maximum) of the 1s core level, ∼1.8 eV (Krause & Oliver, 1979).
The dichroic signal is proportional to the circular polarization rate of the monochromatic Xray beam which is defined as the ratio of the difference between the intensity I_{R} and I_{L} of the right and lefthanded circularly polarized beams to the sum of these intensities, (I_{R}I_{L})/(I_{R}+I_{L}). Xray polarimetry analyses have shown that, at the Ga Kedge and under the present experimental conditions, it exceeds 0.95. Xray absorption spectra were recorded at room temperature using total yield detection mode. Due to the known saturation and selfabsorption effects, the fluorescence signal is not linearly related to the Xray (Goulon et al., 1982). There is a fairly simple transformation of fluorescence data, which is generally used to reconstruct the (Loos et al., 1989). The XNCD spectra were obtained as the direct difference of two Xray absorption spectra recorded with right and leftcircularly polarized Xrays. The XNCD spectra have been recorded for two experimental configurations: (a) the Xray wavevector of incident radiation is parallel to the crystal caxis () and (b) the wavevector is parallel to the aaxis ().
5. Results and discussion
The study of langasite electronic structure by optical methods (Kitaura et al., 2004) has shown that the is mainly formed by the O 2p states, the lower part of the is dominated by the La 5d states, while the upper part is built up of the Ga 4p states with small with the Si 3s and 3p states. There is no optical data about p–d or d–f In contrast to the visible range, OA in the Xray range is observed near resonant transitions between an inner atomic shell and an of the outer electron shells, which are subjected to the action of the crystal field and interatomic interactions. In the case of the Kedge of Ga, for example, XNCD spectra will probe the onsite of p and dstates. As in optics, the outer states can be localized or delocalized. There have been some discussions about the degree of localization of excited electronic states giving the main contribution to the visible OA in langasite (Burkov et al., 2008; Qi et al., 2005; Veremeichik, 2011). could shed some light on this, since excitations to localized states give a contribution to the Xray predominantly in the preedge region, whereas transitions to delocalized states contribute at photon energies above the So, the measurement of the XNCD signal in langasite will give additional information about its electronic structure, which could be further used for understanding the unusual optical properties of this family of crystals.
The experiment has shown the existence of the XNCD spectra at the Kedge of Ga mainly above the edge with a minor contribution at the preedge (see Fig. 3). The XNCD spectra were found to be twice as large for experiments with the wavevector along the caxis and to have the opposite sign for measurements for the wavevector perpendicular to the caxis, as expected from the angular dependence of XNCD for uniaxial crystals, ∼3cos^{2}θ − 1 (Natoli et al., 1998), and as follows from the symmetry consideration (here θ is the angle between the k vector and the caxis). Up to now the XNCD was measured mainly in the structures whose was determined by a screw axis (Goulon et al., 1998; Alagna et al., 1998; Rogalev et al., 2008a). In the majority of cases, the XNCD signal has an extended spectral shape above the edge, but sometimes it is localized in the preedge (Rogalev et al., 2008a). It is clear that there are two factors determining the energy dependence and the shape of the XNCD signal: the and local chiral environment of the absorbing atom.
The experimentally observed absorption and XNCD signal at the Ga Kedge are the sum of the contributions corresponding to each nonequivalent position and are proportional to the sum of the thirdrank tensor components describing each absorbing Ga atom [see equation (21)]. There is no way to separate the XNCD signals from various positions experimentally neither in the visible nor in the Xray range. Nevertheless, it is interesting to know the XNCD signal corresponding to three different Ga positions, because it reflects the unoccupied of hybridized orbitals at each specific To disentangle the contributions from different Ga positions, we have performed ab initio calculations of XNCD spectra using the FDMNES code (Bunău & Joly, 2012). Computations performed with the FDMNES program allow us to calculate all the tensor components for Ga atoms in each position. We restrict ourselves to only those components that contribute to the XNCD signal and the sum of which can be directly compared with experimental results.
We have simultaneously simulated the Ga Kedge absorption and XNCD spectra and compared them with the experimental results. Calculations were performed for langasite structure parameters from Maksimov et al. (2005). The calculations were made using the multiplescattering model with 195 atoms inside a sphere of radius 8.6 Å, as a further increase of the sphere size did not lead to any improvement. The spectra were convolved with a Lorentzian with an energydependent width, to take into account the corehole lifetime assuming it to be the same for all Ga positions. A rather good overall agreement is obtained between calculated and experimental absorption and XNCD spectra, as illustrated in Figs. 3(a) and 3(b).
To compare the cross density of mixed states and to obtain the tensor components γ_{αβη} corresponding to each Ga position it is reasonable to calculate the XNCD per atom in each position. Certainly the largest contribution to the absorption and to the XNCD signal is expected from the 3f position which contains three times more Ga atoms compared with the 1a and 2d positions. The absorption and XNCD per Ga atom in three different crystallographic positions are shown in Figs. 4(a) and 4(b). The spectra corresponding to each Ga position are shifted along the vertical axis for clarity. The spectral shape of absorption and XNCD signals corresponding to each Ga position are different. This is not surprising taking into account the difference of their local environment.
The free ions La^{3+} and Si^{4+} have the electron shells ns^{2}np^{6} of the noble gas type ^{1}S; the outer 3d shell of Ga^{3+} is expected to be filled. In the visible region, the OA of langasite is very sensitive to the size of the structural polyhedra; first and foremost, the 3e oxygen dodecahedron and 1a Ga octahedron (Burkov et al., 2008; Qi et al., 2005; Veremeichik, 2011; Kitaura et al., 2004). In the Xray region, the shapes of the polyhedra are also essential because they influence the splitting of the electronic levels of absorbing atoms and in the multiple scattering model are described by different scattering paths. Usually the preedge peaks in Kedge XANES spectra are due to transitions into localized dstates (quadrupole peaks). Because of the absence of the preedge features in both XANES and in XNCD spectra we conclude that d or p–dhydbridized electronic states of Ga are delocalized in langasite (see Fig. 3). This conclusion confirms the delocalization of chemical bonds deduced from experiments in the visible spectral range (Burkov et al., 2008; Qi et al., 2005; Veremeichik, 2011; Kitaura et al., 2004). This result is in fact an expected one, since 3dstates of Ga are fully occupied and quadrupolar transitions are to empty 4dstates.
Figs. 4(a) and 4(b) show the partial absorption and XNCD spectra and tensor components corresponding to Ga atoms in three nonequivalent crystallographic positions. We can see that in the absorption spectra the strongest `white line' corresponds to Ga in the 1a position. Usually the white line is proportional to the number of holes in the with a symmetry defined by the selection rules for optical transitions, e.g. 4p holes at the Kedge of Ga. The 1a position possesses the highest symmetry 32 (distorted octahedra), maximal volume of surrounding oxygen polyhedra (interatomic distance Ga1–O 1.99 Å) and maximal distance for the six nextneighboring Ga (3.197 Å). Ga2 atoms in the 2d position are inside the distorted tetrahedra (1.795–1.805 Å) and three nextneighboring Ga at 3.221 Å. Ga3 3f atoms are inside the distorted tetrahedra formed by O atoms in the first sphere (1.805–1.897 Å), two O atoms at 2.653 Å and six Ga atoms between 3.198 Å and 3.33 Å. In all cases the asymmetry of the environment (without inversion center) provides the splitting of electronic p and dstates. The strength of the white line can be explained by different charges of ions corresponding to each Ga position. Calculations give the ion charges: 3.401 for the 1a position, 3.389 for the 2d position and 3.397 for the 3f position. So, the number of holes is maximal for the 1a position, followed by the strongest white line in the absorption spectrum.
As described above, Fig. 4(b) shows the calculated tensor components for Ga in each position. We can also see from Fig. 4(b) that spectral shapes are different reflecting the dependence on the local environment of the absorbing atom. These results could be correlated with an interesting explanation of the chiral properties of Nd_{3}Ga_{5}SiO_{14} (langasite family) that was given by Dudka & Mill (2013, 2014). The electron density fragments forming helixes along the caxis together with splitting of atomic environments due to statistical occupation of the 2d position were suggested as a structural basis for the of langasites. It could be expected that similar pseudohelices would exist in the other langasite compounds providing multiferroism in the presence of magnetic ordering. This emphasizes the importance of further detailed studies of local chiral atomic arrangements in the langasite family of crystals.
6. Conclusion
Natural _{3}Ga_{5}SiO_{14} single crystal was studied experimentally and theoretically at the Ga Kedge. The XMCD spectrum measured along the caxis of the crystal was found to be twice as strong as that recorded perpendicularly to it and has the opposite sign. This is in perfect agreement with the and confirms the high quality of the crystal studied. The XNCD spectra have a spectral shape extended above the without a strong feature in the preedge. Thus the hybridized p–d electronic states of Ga are mainly delocalized.
of langasite LaMicroscopic and macroscopic approaches to the description of Xray optical activity are discussed in detail. The macroscopic approach is formulated in terms of the gyration tensor. Consideration of the symmetry properties of the gyration tensor or gyrotropy tensor allows us to determine those tensor components that are responsible for the optical activity. The microscopic approach is based on multipole expansion of the transition operator. The antisymmetric part of the σ^{E1.E2} tensor is responsible for Xray natural contrary to the visible optics where the E1.M1 pseudoscalar term prevails. In the Xray region, we deal with microscopic tensors describing the properties of absorbing atoms, which depend on the symmetry of the atomic sites. For each atomic site, a thirdrank gyration tensor is invariant under the of a characteristic of a given The tensor γ_{αβη} describing XNCD of a crystal is the sum over all absorbing atoms in the The gyration tensors for each of the P321 occupied by Ga atoms have been explicitly given.
It was shown that the symmetry of La_{3}Ga_{5}SiO_{14} allows two antisymmetric thirdrank tensor components = and = , which provide XNCD signal with the wavevector perpendicular and along the threefold symmetry axis. Both components are the sum over three atomic positions (1a, 2d and 3f). Because the of langasite does not provide any forbidden reflections, Xray resonant diffraction cannot be exploited to obtain the siteselective information. So, there is no possibility of separating each contribution to the XNCD signal exclusively from experiment.
To obtain siteselective information, ab initio calculations of the absorption and XNCD spectra were carried out in the present study. A comparison of the results of calculations with the experimental data confirms their reliability and allows us to extract energy spectra corresponding to each of three nonequivalent Ga positions. The difference between the spectra corresponding to each of the Ga sites clearly reveals the different density of the p–d mixed electronic states of Ga in the 1a, 2d and 3f positions. Calculations have also shown that the strongest contribution to XNCD at the Ga Kedge arises from Ga in the 1a position. Finally, the combination of ab initio calculations with highquality experimental results is shown to be a very valuable approach to studying the local in multicomponent crystals like langasite.
Acknowledgements
The research was carried out using the equipment of the shared research facilities of HPC computing resources at Lomonosov Moscow State University.
Funding information
The following funding is acknowledged: Russian Foundation for Basic Research (grant No. 160200887). The work of VED was supported by the grant `NANO' of the Presidium of Russian Academy of Sciences.
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