research papers
Helicitydependent resonant Xray scattering in CuB_{2}O_{4}
^{a}Physics Department, Moscow State University, Moscow 119899, Russia, ^{b}European Synchrotron Radiation Facility, 38000 Grenoble, France, and ^{c}A. V. Shubnikov Institute of Crystallography, FSRC Crystallography and Photonics RAS, Moscow 119333, Russia
^{*}Correspondence email: en_ovchinnikova@physics.msu.ru, kozlovskaya@physics.msu.ru
Exploitation of Xray circular polarized beams to study forbidden Bragg reflections and new information that could be obtained in these experiments are discussed. It is shown that the intensities of such reflections can be different for the right and leftcircular polarizations (i.e. exhibiting circular dichroism) even for the dipole–dipole resonant transitions involved in the scattering process. This difference can be observed only in crystals having no center of inversion. Here, this approach is used to study helicitydependent resonant diffraction in copper metaborate CuB_{2}O_{4} single crystal, which is noncentrosymmetric but achiral. Nonetheless, a strong has been observed for hh0 forbidden reflections in the vicinity of the Cu Kedge. This effect is shown to originate from dipolar transitions in Cu atoms occupying the 8(d) only.
1. Introduction
The physical properties of a solid and its et al., 1998; Goulon et al., 1998, 1999, 2003; Natoli et al., 1998; Carra & Benoist, 2000; Peacock & Stewart, 2001; Izumi et al., 2013; Platunov et al., 2021). XNCD occurs at photon energies close to an due to the electricdipole–electricquadrupole E1.E2 interference terms in absorption whereas electricdipole–magneticdipole E1.M1 interference terms, which dominate optical activity in visible optics (Buckingham, 1967; Barron, 1982), become vanishingly small in Owing to its E1.E2 origin, the main drawback of XNCD is that it is observable only in single crystals that belong to 13 out of 21 noncentrosymmetric crystal classes (Jerphagnon & Chemla, 1976; Jerphagnon et al., 1978; Goulon et al., 1998, 1999, 2003). However, XNCD can be applied to opaque samples, whereas in the visible spectral range can be measured only in transparent substances.
are closely related and understanding this relationship is one of the greatest challenges of material science, especially in view of engineering materials with new functionalities. Materials with no inversion symmetry attract special attention because they could exhibit intriguing physical properties like multiferroicity and of matter (the existence of the right and lefthanded forms) is believed to be an internal property determined by atomic arrangement featuring no inversion center. This property plays a crucial role in biology and pharmaceutics but is also very important in physical phenomena and technological applications. Thanks to the development of at thirdgeneration synchrotrons, it became possible to probe on a microscopic level using Xray natural (XNCD) (AlagnaComplementary to ; Dmitrienko & Ovchinnikova, 2003; Dmitrienko et al., 2005; Collins et al., 2007; Lovesey & Collins, 1996), thermal displacements (Dmitrienko et al., 1999; Dmitrienko & Ovchinnikova, 2000; Kokubun et al., 2001; Kirfel et al., 2002; Collins et al., 2003) and local of scattering atoms (Dmitrienko & Ovchinnikova, 2001; Di Matteo et al., 2003). This is because RXS is sensitive to phase shifts of the waves scattered by different atoms in a as well as to multipole contributions to atomic scattering factors. Unique information related to local can be further obtained from resonant Xray diffraction using circularly polarized photons. Unfortunately, use of this technique is very scarce.
measurements, resonant Xray scattering (RXS) techniques offer site selectivity to study properties of crystals, among them (Belyakov & Dmitrienko, 1989On the contrary, linear dichroism of Bragg reflections, i.e. the difference of their intensities for σ and π polarization of incident radiation, is well known, because it exists even in conventional nonresonant Xray scattering (James, 1950). In a resonant regime σ and π polarizations can be mixed, leading to a change of polarization during the scattering process. This effect was observed many times in RXS experiments, especially in studies of socalled forbidden reflections. Those are absent when the atomic scattering amplitudes are added in antiphase, i.e. the contribution from Thomson scattering vanishes. However, when the photon energies are in the vicinity of an (in resonance with transitions from a core state) in a crystal belonging to a nonsymmorphic the forbidden reflections could become allowed. Since forbidden reflections result from an anisotropy of electron density of scattering atoms, they appear to be highly sensitive to the (Dmitrienko, 1983, 1984; Dmitrienko et al., 2005; Templeton & Templeton, 1985a,b, 1986).
The resonant Xray scattering arises from multipole electronic transitions (Blume, 1994; Dmitrienko & Ovchinnikova, 2003; Dmitrienko et al., 2005; Lovesey & Collins, 1996; Lovesey et al., 2005; Collins et al., 2007) and is usually described in terms of scattering tensors of various ranks (Blume, 1994; Hannon et al., 1988). The strongest contribution to forbidden reflections is due to the dipole–dipole E1.E1 scattering whereas higherorder contributions, such as dipole–quadrupole E1.E2 (Templeton & Templeton, 1994; Dmitrienko & Ovchinnikova, 2001; Di Matteo et al., 2003) or quadruple–quadrupole E2.E2 (Carra & Thole, 1994; Finkelstein et al., 1992, 1994), become detectable when the former is weak or absent for symmetry reasons. Interestingly, forbidden reflections that originate from interference between different scattering channels (e.g. E1.E1 with E1.E2) have been experimentally observed (Kirfel et al., 2002; Collins et al., 2003; Beutier et al., 2014, 2015; Richter et al., 2014; Kokubun et al., 2001; Oreshko et al., 2012). However, local could not be fully investigated in these experiments since linearly polarized Xrays were used. In particular, interference with magnetic scattering allows the sign of the DzyaloshinskiiMoria interaction in canted antiferromagnets to be related to local crystal (Dmitrienko et al., 2010, 2014).
To gain a deeper insight into the local ). It was shown that the intensities of forbidden reflections in certain chiral symmetries are different for the right and leftcircular polarized Xrays. This was confirmed experimentally for SiO_{2} and Te (Tanaka et al., 2008, 2010), in which the 3_{1} or 3_{2} screw axes give rise to the righthanded or lefthanded crystals. Recently the difference between the integrated intensities of the forbidden reflections measured with right and leftcircular polarization was used for the chiral domains imaging in DyFe_{3}(BO_{3})_{4} (Usui et al., 2014). There was another RXS experiment using circularly polarized radiation that deserves to be mentioned. It concerns observation of the vector part of optical activity in ZnO (Goulon et al., 2007), which crystallizes in the P6_{3}mc. Even though this is achiral, the difference in the intensity of the allowed 300 Bragg reflection for the right and leftcircularly polarized beams was observed at energies close to the Zn Kedge. It was suggested that this peculiar signal is proportional to an of the scattering atomic sites (Carra et al., 2003). This experiment has shown the possibility to separate resonant from Thomson scattering in a study of allowed reflections without analyzing the polarization of scattered Xrays, as was recently proposed by Ovchinnikova et al. (2019). It is especially important taking into account that many functional materials are described by in which there are no forbidden Bragg reflections.
of a crystal, use of circularly polarized Xrays in resonant diffraction experiments is mandatory. For the first time, the `chirality' of forbidden reflections in RXS was predicted by Dmitrienko (1983In this paper, we present a thorough experimental and theoretical study of _{2}O_{4}) single crystal. This crystal recently attracted a lot of attention because of its unique physical properties, like complex magnetic structure with a series of commensurate/incommensurate phase transitions at low temperatures of 21 K, 10 K and 2 K (Boehm et al., 2003), magnetoelectric properties (Khanh et al., 2013; Nénert et al., 2007; Mero et al., 2021), the gigantic optical magnetoelectric effect (Saito et al., 2008a), and magneticinduced second harmonic generation (Pisarev et al., 2004). Moreover, Saito et al. (Saito et al., 2008b; Arima & Saito; 2009) claimed that it would be possible `to control the crystal by magnetic field' in copper metaborate. This statement violates the fundamental symmetry principle [see, for example, Lovesey & Staub (2009)] and it was shown that the observed effect can be explained by sublatticesensitive antiferromagnetic linear dichroism (Boldyrev et al., 2015). Copper metaborate crystallizes in which is nonenantiomorphous (righthanded and lefthanded crystals do not exist), because its contains lefthanded and righthanded screws, formed by two nonequivalent Cu atoms (Figure 1). The electronic and magnetic properties of these two distinct Cu atoms need to be studied separately. Recently the neutron diffraction and in magnetically ordered copper metaborate was considered by Lovesey (2016) within a minimal model of copper atomic sites. The Cu Kedge XNCD signal was also measured on this crystal (Ovchinnikova et al., 2016) but all Cu atoms in the were contributing to it. in resonant Xray diffraction offers a unique possibility to perform element and siteselective study of the electronic properties of crystals. The main goal of the present work is to show that the local electronic structure of one of the two nonequivalent Cu sites in a CuB_{2}O_{4} single crystal can be investigated using in `forbidden' 〈hh0〉 reflections at the Cu Kedge.
in forbidden reflections in copper metaborate (CuB2. Symmetry analysis of resonant Xray scattering
Whenever the energy of Xrays is close to an f^{ s}_{ij} corresponding to the satom contains nine elements out of which only six are nonequivalent since f_{ij} = f_{ji}. But if resonant atoms are in a special its symmetry restricts the number of independent elements since some of them turn out to be zero.
of an atom, resonant Xray scattering becomes anisotropic. Indeed, various multipolar transitions between an inner shell and the crystalfieldsplit excited multiplets of a resonant atom contribute to the which is a tensor. The dipole–dipole contribution to the in nonmagnetic crystals is described by a rank2 tensor whose structure is fully determined by the symmetry of the atomic site. If resonant atoms are in the general the dipole–dipole tensorTo identify these elements one can use the invariance of the atomic tensor factor under the symmetry operations for a given ),
(Hahn, 1996where is a
of a of the of the crystal.The
is the sum of atomic scattering factors of all atoms in a taking into account the rotation of the tensor axes under the symmetry elements of the = ,where is the rotational part of the τ_{s} is the corresponding fractional translation, and is the tensor of the atom with coordinates within a (Dmitrienko et al., 2013).
The scattering amplitude is obtained through the threedimensional tensor taken between polarization vectors of the incident α and scattered β radiation. Since those are actually twodimensional vectors, is commonly represented by 2 × 2 matrices^{1},
If the polarizations are linear we have in the basis σ, π,
The nondiagonal terms in (4) may be nonzero in contrary to the Thomson scattering case. Let us determine the circular polarization vectors as = , = . The change of the basis from linear as , to circular as , is obtained through
where and are unitary matrices
One can express then the scattering amplitude components for circularly + (right) and − (left) polarized radiation,
The intensity of the diffracted radiation is a function of photon energy and proportional to
This formulation is valid in kinematic diffraction theory for an infinitely thick sample with H perpendicular to the sample surface. The μ_{α}(ℏω) can be represented as a sum of the nonresonant, dipole and quadrupole terms (Brouder, 1990). Even though linear and circular dichroisms have been measured at the Cu Kedge spectra in copper metaborate crystal (Ovchinnikova et al., 2016), they are rather weak and we therefore neglect the polarization dependence of absorption, i.e. we use the isotropic spectrum μ(ℏω).
At energies close to an , 1984). Polarization properties of these reflections were also considered by Belyakov & Dmitrienko (1989) and the intensities of some of them were shown to be different for incident right and leftcircularly polarized radiation, i.e. they exhibit `circular dichroism'. We would like to underline that this `circular dichroism' is fundamentally different from in absorption and in diffraction for allowed reflections (Goulon et al., 2007). The latter is observable only when the electronic states with mixed parity are excited via the E1.E2 resonant channel. The in the diffraction experiment could be expressed in the following manner,
the anisotropy of removes some of Bragg reflections corresponding to the screw axes and glide planes, even in the dipole–dipole approximation (Dmitrienko, 1983We can also rewrite (9) in linear polarizations basis,
where F′ and F′′ are the real and imaginary parts of the complex scattering amplitudes. One can see from (10) that the necessary condition for in Bragg diffraction is the intermixing of σ and π polarizations in the scattering process. Indeed, this of a nonmagnetic reflection is absent in the E1.E1 approximation when a crystal has a center of inversion. The considered to be nonmagnetic, is invariant in the E1.E1 approximation under the space inversion; the incident and diffracted Xray wavevectors as well as the atomic positions are unchanged whereas the circular polarization is reversed, so that the diffracted intensities for right and leftcircularly polarized Xrays should be strictly identical.
Given the fact that copper metaborate crystal has no inversion center, one could therefore observe E1.E1 approximation. Below we consider of 〈hh0〉 `forbidden' reflections in copper metaborate crystal.
in Bragg diffraction even in the2.1. of the forbidden reflections in copper metaborate: dipole–dipole scattering
At room temperature, CuB_{2}O_{4} crystallizes in the tetragonal noncentrosymmetric (No. 122) (MartinezRipoll et al., 1971) with 12 formula units in a cell of dimensions a = 11.484 Å and c = 5.620 Å. The is shown in Figure 1. There are two nonequivalent sites occupied by Cu^{2+} ions. The Cu^{2+} ions in 4(b) Wyckoff positions (Asites with coordinates (0,0,1/2) and (1/2,0,1/4)) are surrounded by four oxygen atoms in planar quadratic coordination so that the is . The Cu^{2+} ions in the 8(d) Wyckoff positions [Bsites with coordinates (x,1/4,1/8), , , (3/4,x,7/8) where x = 0.0815] are located in the center of a distorted octahedron with an exceptionally large distance of 3.069 Å to the two apical oxygen ions. As a consequence the of this site is 2.
The presence of two substantially different copper sublattices leads to a number of interesting and unusual optical and magnetic properties. Despite the fact that both Cu ions have the same Cu^{2+} only the Cu(A) spontaneously orders at the Neel temperature T_{N} = 21 K into an antiferromagnetic commensurate structure with magnetic moments lying in the (ab)plane. These moments are slightly canted due to the DzyaloshinskiiMoriya interaction yielding a weak ferromagnetic moment along the [110] direction (Petrakovskii et al., 1999). Whereas the Cu(A) magnetic moments reach the value 0.86μ_{B} at 12 K, the Cu(B) ions acquire a of only 0.2μ_{B} at the same temperature. Below the into an incommensurate helicoidal phase at 10 K, this moment starts to increase, reaching 0.54μ_{B} at 2 K. The Cu(B) is not fully ordered and the moments fluctuate even at the lowest temperature. The precise determination of its magnetic behavior is consequently very difficult (Boehm et al., 2003; Nénert et al., 2007; Fiebig et al., 2003). To obtain a deeper insight into the electronic structure of the Cu(B) site only is highly desirable. As we show below, the resonant Xray diffraction using forbidden hh0 reflections allows us to obtain this information.
Using equation (1) and given the of the Cu(A) site, the corresponding atomic scattering tensor in the dipole–dipole approximation has only two nonequivalent components: f_{xx}(A) and f_{zz}(A). The symmetry of the imposes that
where f_{ij}(An) describes the scattering factors of copper atoms occupying Asites. For the 4(b) n = 1, 2, 3, 4, but translation ( 1/2, 1/2, 1/2) does not change f_{ij}(An), therefore there are only two different scattering factors corresponding to Cu(A) sites. Coefficients f_{ij} are complex values and in the dipole–dipole approximation can be calculated by the following,
where R = is a sum over all electrons in the atom, E_{g} and E_{f} are the energies of the ground and excited states, and Γ is the width of the also depending on E.
The B) site implies that there are only four independent components (see Sirotine & Chaskolskaia, 1982); the symmetry operations of the crystal allows us to express the corresponding atomic scattering tensor as
2 of the Cu(where f_{ij}(Bn) corresponds to the copper atoms of Bsites. For the Bposition, n = 1,…, 8. However, there are only four different scattering factors f_{ij}(Bn), because translation ( 1/2, 1/2, 1/2) does not change their value. Let us note that components f_{ij}(An) and f_{ij}(Bn) are energydependent and complex.
In the hhl, 2h + l = 4n + 2, are forbidden off resonance due to a glideplane symmetry (Hahn, 1996). In the particular case of hh0, h = 2n + 1, the Cartesian components of the scattering tensor defined in equation (2) are given by
of copper metaborate the reflectionsThe factor of two in equation (18) takes into account the fact that there are only two different atomic factors for copper atoms in 4(b) and four atomic factors in the position 8(d). Taking into account that f_{ij}(A1) = f_{ij}(A2) [see (11)], the contribution of the Cu(A) atoms is canceled and only copper atoms occupying B sites contribute to the hh0 dipole–dipole resonant Xray scattering near the Cu Kedge. In fact, Cu(A) atoms can contribute to this forbidden reflection but only in the dipole–quadrupole approximation, which is much weaker and, therefore, neglected here. Finally, the tensor for the hh0 (h = 2n + 1) reflections in the dipole–dipole approximation for the CuB_{2}O_{4} crystal takes the following form,
Let us now calculate the hh0 (h = 2n + 1) forbidden reflections and its azimuthal dependence. The corresponding diffraction geometry is shown in Figure 2. The lattice vectors define the reference frame. The incident and diffracted wavevectors are then given by
forwhere θ is the and ϕ is an angle around the scattering vector. The corresponding polarization vectors are
and
Both wavevectors k, and polarization vectors , , , are unit vectors.
Using equations (3) and (19) the components of the scattering amplitude tensor in linear polarization basis are expressed as follows,
The polarizationaveraged intensity of the hh0 reflections is then given by
Inserting the components of the scattering amplitude tensor into equation (10) we find that the of the forbidden hh0 reflection in the dipole–dipole approximation is equal to
where Re(f) and Im(f) are the real and imaginary parts of the atomic tensor factor. We can see from (28) that the dichroism ΔI_{±}(hh0) has the same azimuthal dependence for all hh0 reflections (see Figure 3) and only its amplitude is varying with h. The values of the sin(4πhx) and of the for incident photon energies of 8.99 keV and for different hh0 reflections as well as the relative amplitudes of with respect to that of the 110 reflection are given in Table 1.

The largest dichroic signal is expected for the 110 and 550 reflections, whereas it is more than one order of magnitude weaker for the 330 reflection.
3. Experimental
The resonant Xray scattering experiments were performed at ESRF beamline ID12 which offers full control of the polarization state of incident Xrays in the photon energy range from 2 to 15 keV. For the experiments at the Cu Kedge (∼9 keV) we used the second harmonic of radiation emitted by a helical undulator of HeliosII type with magnetic period of 52 mm. A fixedexit monochromator was equipped with a pair of Si 〈111〉 crystals diffracting in the vertical plane. The crystals were cooled down to 140 K by refrigerated helium gas. The degree of circular polarization of the monochromatic Xrays was estimated to be more than 0.95. Higherorder harmonics of the undulator emission on axis in pure helical mode were well below the and therefore the use of mirrors was not necessary. The experiments were performed using an Xray scattering endstation, shown in Figure 4. It is essentially a highvacuum threecircle diffractometer operating in horizontal scattering geometry with a sample and detector arm rotations driven from outside the vacuum by Huber goniometers with a very high accuracy and an excellent reproducibility of 5 µrad.
The sample holder (or eventually a cold finger of a cryostat) is mounted on the axis of a goniometer allowing rotation of a crystal around the scattering vector (azimuthal scan). The diffractometer is connected to the beamline using flexible bellows of diameter 100 mm so that the angular range of the θ scans are ±18°. There are three interchangeable chambers covering the θ angular ranges from 0° ± 18°, 12°–48° and 27°–63°, respectively. We paid special attention to keep the rotation axis of the chamber (θ axis) to be strictly perpendicular to the beam direction and to coincide with the sample surface. A peculiar feature of this diffractometer is the possibility to record simultaneously the intensity of the diffracted beam and the total using two independent Si photodiodes. The first one behind the slits with its width adjustable from 50 µm to 2 mm is mounted on a 1 mlong detection arm and measures the diffracted beam; the second photodiode collecting the fluorescence photons is located inside the chamber at ∼4 cm from the sample (see Figure 4). Note that there is no window between the monochromator, the sample and the detectors which avoids the generation of the parasitic scattered Xrays. This is very important whenever detection of the weak signals is concerned, e.g. dichroism and/or forbidden reflections. Excellent signaltonoise ratio in both detection channels is achieved exploiting a digital lockin detection based on a square wave modulation of the incident Xray beam at ca. 67 Hz using a mechanical chopper. and any lowfrequency instabilities of the Xray beam are efficiently eliminated. Moreover, being installed upstream with respect to the monochromator the chopper reduces the heat load on the first crystal by a factor of two and makes the whole setup more stable.
Highquality single crystals of copper metaborate were grown by the Kyropulos method from the melt of oxides B_{2}O_{3}, CuO, Li_{2}O and MoO_{3} (Aleksandrov et al., 2003). A planeparallel plate of (110) orientation with a size of 10 mm × 10 mm × 0.5 mm was cut from an Xray oriented singlecrystal and its surface was polished.
The diffracted intensity is measured as the integral over the rocking curve for a given reflection. We have recorded the intensities of the forbidden 330 and 110 reflections as a function of the photon energies in the vicinity of the Cu Kedge or at a fixed energy as a function of azimuthal angle. signal was obtained as a direct difference of the diffracted intensities measured with left and rightcircular polarized incident Xrays. In the present experiments the total diffracted intensities were measured without any polarization analysis.
4. Results and discussion
In this section we present the results of Kedge which are complemented with quantum mechanical calculations using the FDMNES code (Joly, 2001; see also https://fdmnes.neel.cnrs.fr). We have measured the series of the rocking curves for the 330 and 110 reflections for the right and leftcircularly polarized incident Xrays across the Integration over the rocking curves gives the integrated intensity. The polarization averaged integrated intensities I(hh0)_{av} and the ΔI(h00)_{±} for both reflections were calculated using the results of two measurements for right and leftpolarized beam. They are shown in Figure 5 as a function of photon energy. The Xray at the Cu Kedge measured using total detection mode is also reproduced in the figure. As one can see, both 〈110〉 and 〈330〉 forbidden reflections show very similar energy dependence with two maxima at ∼8991 eV and ∼8998 eV corresponding to dipolar 1s → 4p transitions in copper atoms. A small peak centered at 8981.5 eV in the and I(hh0)_{av}(E) is due to 1s → nd scattering events, but it is absent in the ΔI(h00)_{±}(E) spectrum. The polarizationaveraged intensity of the 330 reflection was found, however, to be nearly three times stronger than that of the 110 reflection, which is in good agreement with theoretical predictions (see Section 2.1). On the other hand, the for the 〈330〉 reflection measured at the same azimuthal angle is found to be more than 20 times weaker than for the 〈110〉 reflection. The latter signal is surprisingly high and is comparable with the polarization averaged intensity of 〈110〉 diffracted intensities at the azimuthal angle ϕ = 30°.
in Xray for the 110 and 330 reflections measured at the CuThe measurements of the 330 reflections were made at three different azimuthal angles revealing the maximum at about ϕ = 40°. The signal, i.e. the difference between the integrated intensities corresponding to the left and rightcircular polarizations of the incident beam, was observed for both reflections, but it was much stronger for the 110 reflection confirming the predictions based on symmetry considerations (see Section 2.1).
The energy spectra of the 330 and 110 reflections near the Cu Kedge was calculated using two methods realized in the FDMNES code: multiple scattering and the finite difference methods (FDM) taking into account the E1E1, E1.E2 and E2.E2 multipole contributions corresponding to both copper sites. In the multiple scattering option, all the atoms inside a sphere of radius 7.5 Å were involved in calculations while in the FDM method a radius equal to 5.8 Å gave satisfactory results. Some other parameters, for example the energy dependence of the full width of the Γ(E) and the shift between the spectra corresponding to the different copper positions, were taken from the previous studies of the absorption and XNCD in copper metaborate (Ovchinnikova et al., 2016). Figure 6 demonstrates the calculated energy spectrum of the 110 reflection in comparison with the experimentally measured spectrum at the azimuthal angle ϕ = 45°. The calculations of the hh0 reflections energy spectra have shown that the E1.E1 contribution to the close to the Cu Kedge is predominant, which shows the fairness of the approach described above and the adequacy of the expressions describing the polarization properties of these reflections.
Azimuthal dependencies of the 110 reflection I_{+}(100) and I_{−}(100) measured with right and leftpolarizations of the incident radiation are shown in Figure 7 together with the fit using the expression (28) given in Section 2.1. The azimuthal dependencies of the 100 reflection average intensity and of its are shown in Figure 8 together with calculations based on the use of the expressions (27) and (28). In calculations the values f_{yz} and f_{xx} − f_{yy} are supposed to be variable parameters.
We see that the ϕ = 0° and ϕ = 180°. At those angles the wavevectors k and lie in the mirror plane close to the zaxis (the for the 110 reflection is small, 4.88°). At the angles ϕ = 90° and ϕ = 270° the wavevectors lie in the plane ab close to the fourfold axis, and the does not change its sign.
changes its sign atIt follows from expression (27) that it is possible to determine the value f_{yz}/f_{xx} − f_{yy} just from the experimental data as follows,
The substitution of the experimental data with the values given in Table 1 gives the value f_{yz}/f_{xx} − f_{yy} = 0.572.
Equation (28) can be rewritten as follows,
where Δα is a relative phase describing tensor components: Δα = α_{xx, yy} − α_{yz}, = , = ), and Ref and Imf are the real and imaginary parts of the complex tensor components.
This relative phase can be calculated using the experimental data for the 110 reflection. It is equal to
Substitution of the experimental data into equation (31) gives Δα = 78°.
Let us discuss the physical meaning of the obtained quantities. The dipole–dipole atomic factor is a tensor of the second rank, which can be reduced to the system of principal axes that characterize the crystal field at the location of the Cu(B) atoms. In this system of principal axes, the tensor is diagonal. In the crystal field, p electronic states split into three sublevels usually described as p_{x}, p_{y} and p_{z} orbitals, where x, y and z are the main axes of the local anisotropy. In the considered case, the xaxis of the second rank tensor f_{ij} at the Cu(B1) atom coincides with the a crystal axis, but the main axes y and z do not coincide with crystal axes b and c. The axes of local anisotropy can be aligned with the axes of the crystal by rotation around the a axis by an angle equal to = . Unfortunately, some additional information about the tensor components f_{xx} or f_{zz} is necessary to determine the absolute value of δ for the Cu(B) site in copper metaborate. Usually only the relative values can be determined from the experimental data. Absolute values of f_{ij} can be obtained only in comparison with the known value as shown by Mukhamedzhanov et al. (2007).
Taking into account that f_{yz} = , it also follows from equation (30) that f_{xx} ≠ f_{yy} and f_{zz} ≠ f_{yy}, hence all p electron states at the Cu(B) site are split. Each diagonal tensor component describes through a certain sublevel. Because D_{ij} in equation (13) for the E1.E1 scattering is a real function depending on energy, the imaginary parts of the tensor components appear exclusively due to the resonant denominator in (12). So, the phases corresponding to all tensor components f_{ii} are equal to
If the excited electronic levels are degenerate, then the energies E_{f} are the same for all electronic transitions, hence α_{ii} is the same for all tensor components, Δα_{ii} = 0. However, there is no in this case, because the diagonal tensor components are equal to each other. When the excited states split in a crystal field, α_{ii} may be different providing a phase shift between the tensor components. So, both the value obtained from equation (29) and the value from equation (31) characterize the degree of the levels splitting. However, we need more information to draw more specific conclusions about the tensor components f_{ij} corresponding to the Cu(B) sites. Those data could be obtained, for example, from the study of of other reflections, including allowed reflections. However, in copper metaborate both copper sites contribute to the allowed reflections which strongly complicates the interpretation of the data.
5. Conclusions
We have proposed and experimentally realized a new method based on measurements of the E1.E2 multipoles but also by other kinds of multipoles, even by E1.E1, and therefore the dichroism can be very strong. However, similarly to XNCD, in Bragg diffraction can exist only in crystals without an inversion center. The great advantage of this method is that it allows to study separately by different Wyckoff sites, while all atoms together contribute to absorption and XNCD. This was demonstrated in a study of the 330 and 110 forbidden reflections in copper metaborate carried out at beamline ID12 of the ESRF. These forbidden reflections at incident radiation energies close the Cu Kedge exist exclusively due to Cu(B) atomic sites. The measurements of the XNCD signal in copper metaborate (Ovchinnikova et al., 2016) have shown that it exists at almost the same energies as of the 330 and 110 reflections observed in the present study, but both copper sites contribute to the XNCD. It means that in the vicinity of the Cu Kedge there are many kinds of various electronic states, which may provide the interference between the waves in various multipole transitions, which can be revealed due to the various kinds of However, the contributions of higherorder multipoles is negligible for the hh0 reflections in copper metaborate, so their energy and azimuthal dependencies are well described by only the E1.E1 transition.
of Bragg reflections. It differs from XNCD in absorption because it is caused not only by the mixedIn this paper we show that use of circularly polarized radiation in Bragg diffraction is useful not only to study et al. (2008) and Usui et al. (2014), but being further developed it will allow us to obtain some quantitative physical information about splitting of electronic states in crystals. The proposed method may be realized not only for forbidden reflections but also for allowed reflections. However, in the latter case, it needs very high quality circularly polarized beams because even a small amount of linear polarization can essentially spoil the results. Further development of the method requires consideration of other reflections, higherorder transitions, taking into account the magnetic contribution to the scattering amplitude, as well as interference of different scattering channels.
of matter, as carried out by TanakaFootnotes
^{1}In this representation the matrix (F_{αβ}) is no longer symmetric.
Funding information
Work was partly supported by the Ministry of Science and Higher Education of the Russian Federation (state task for the Federal Research Center Crystallography and Photonics, Russian Academy of Sciences). This work was supported by the Russian Foundation for Basic Research (project No. 195212029, study of the forbidden Bragg reflections, and project No. 190200483, study of the use of circular polarization and the effect of Renninger reflections).
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