Quadrupole moments in chiral material ${\mathrm{DyFe}}_3{\left({\mathrm{BO}}_3\right)}_4$ observed by resonant x-ray diffraction

By means of circularly polarized x rays at the Dy $L_3$ and Fe $K$ absorption edges, the chiral structure of the electric quadrupole was investigated for a single crystal of ${\mathrm{DyFe}}_3$(${\mathrm{BO}}_3$)${}_4$, in which both Dy and Fe ions exhibit a spiral arrangement. The integrated intensity of the resonant x-ray diffraction of space-group forbidden reflections 004 and 005 is interpreted within the electric dipole transitions from Dy $2p_{\frac{3}{2}}$ to $5d$ and Fe $1s$ to $4p$, respectively. We have confirmed that the handedness of the crystal observed at Dy $L_3$ and Fe $K$ edges is consistent with that observed at Dy $M_5$ edge reported in a previous study. The electric quadrupole moments of Dy $5d$ and Fe $4p$ are derived by analyzing the azimuth scans of the diffracted intensity. The temperature profiles of the integrated intensity of 004 at the Dy $L_3$ and the Fe $K$ edges are similar to those of Dy-O and Fe-O bond lengths, while the temperature dependence at the Dy $M_5$ edge does not match the bond-length behavior. The results indicate that the helix chiral orientations of quadrupole moments due to Dy $5d$ and Fe $4p$ electrons are more strongly coupled to the ligands states than Dy $4f$ electrons.

Figure 1

A view of the atomic configuration of the left-handed ${\mathrm{DyFe}}_3{\left({\mathrm{BO}}_3\right)}_4$ in space group $P{3}_{2}21$ (No. 154). The ${\mathrm{FeO}}_6$ octahedra and the ${\mathrm{DyO}}_6$ trigonal prisms are colored by brown and blue, respectively. The Fe(1) ions are at the $3a$ site, and the Fe(2) are at the $6c$ site. The unit cell is shown by gray lines. Yellow and gray helices represent spiral arrangements of Dy and Fe ions, respectively.

Figure 2

A schematic view of the diffraction geometry with a right-handed coordinates $x$, $y$, and $z$. The scattering vector $\mathbf{K}={\mathbf{k}}_i-{\mathbf{k}}_f$ is antiparallel to the $x$ axis. The marks $(+)$ and $(-)$ denote the positive and negative helicities of the incident beam, respectively. Here, ${\mathbf{k}}_i$ and ${\mathbf{k}}_f$ are the propagation vectors of the incident and diffracted x rays, respectively, and $\theta$ denotes the Bragg angle. The $\mathbf{\sigma }$ and $\mathbf{\pi }$ components are the unit vectors which represent the polarization of the incident beam. Here, the $\mathbf{\sigma }$ component is perpendicular to the plane of scattering $\mathbf{\sigma }=(0,0,1)$, and the $\mathbf{\pi }$ component is parallel to the plane of scattering $\mathbf{\pi }=\left(cos\theta ,sin\theta ,0\right)$. For the diffracted beam, ${\mathbf{\sigma }}^{\prime }=\mathbf{\sigma }$ and ${\mathbf{\pi }}^{\prime }=\left(cos\theta ,-sin\theta ,0\right)$.

Figure 3

Energy spectra of the x-ray absorption, the intensity of reflection 003, and that of forbidden reflection 004 around the Dy $L_3$ absorption edge. The data were observed at the beam line 3A in PF with the $\sigma$-polarized incident beam. Upper panel: the x-ray absorption spectrum obtained in the total fluorescence mode is shown by red circles and the intensity of reflection 003 is shown by blue triangles. The absorption effect is not corrected. The white line (the maximum absorption) is shown by a black arrow. Lower panel: the intensity of reflection 004 as a function of the x-ray energy was observed for the azimuth angle $\mathrm{\Psi }$ which was scanned in a range from $131.5^{\circ }$ to $133.4^{\circ }$ by every $0.1^{\circ }$ step at $T=200$ K. The resonant effect underlying in the intensity of reflection 004 is shaded around $E=7.796$ keV.

Figure 4

Energy spectra of the x-ray absorption, reflection 003, and forbidden reflection 004 around the Fe $K$ absorption edge. The data were observed at the beam line 29XUL in SPring-8 with the $\pi$-polarized incident beam. Upper panel: the x-ray absorption spectrum obtained in the total fluorescence mode is shown by red circles and the intensity of reflection 003 is shown by blue triangles. The absorption effect is not corrected. The white line (the maximum absorption) is shown by a black arrow. Lower panel: the intensity of reflection 004 as a function of the x-ray energy was observed for azimuth angles $\mathrm{\Psi }=-{100}^{\circ },-{105}^{\circ },-{110}^{\circ },-{114}^{\circ },-{115}^{\circ }$, and $-{126}^{\circ }$ at $T=50$ K. The resonant effect underlying in the intensity of reflection 004 is shaded around $E=7.13$ keV.

Figure 5

The integrated intensity of forbidden reflections 004 (top panel) and 005 (middle panel) observed at the Dy $L_3$ absorption edge ($E=7.796$ keV) and $T=100$ K as a function of $\mathrm{\Psi }$. The inset of the top panel shows rocking curves observed at $\mathrm{\Psi }=0$ for $(+)$ and $(-)$ helicity of the incident beam. Red squares (triangles) show the integrated intensity measured by the incident beam with the $(+)$ helicity for reflection 004 (005) and blue squares (triangles) show that measured by the incident beam with the $(-)$ helicity for reflection 004 (005). The bottom panel shows the difference intensity $(-)-(+)$ between the two helicity states. The cosine curves are results of fit to the data with Eq. (14) for the top and middle panels, and fit to the data with Eq. (18) for the bottom panel. The points colored faintly are strongly influenced by the multiple scattering effect, and are removed for the fitting. The dashed lines in the bottom panel are given from the result of the least-squares method described in the text.

Figure 6

The integrated intensity of forbidden reflections 004 (top panel) and 005 (middle panel) observed at the Fe $K$ absorption edge ($E=7.13$ keV) and temperature $T=100$ K as a function of $\mathrm{\Psi }$. The inset of the top panel shows rocking curves observed at $\mathrm{\Psi }=0$ for $(+)$ and $(-)$ helicity of the incident beam. Red squares (triangles) show the integrated intensity measured by the incident beam with the $(+)$ helicity for reflection 004 (005) and blue squares (triangles) show that measured by the incident beam with the $(-)$ helicity for reflection 004 (005). The bottom panel shows the difference intensity $(-)-(+)$ between the two helicity states. The cosine curves are results of fit to the data with Eq. (14) for the top and middle panels, and fit to the data with Eq. (18) for the bottom panel. The points colored faintly are strongly influenced by the multiple scattering effect, and are removed for the fitting.

Figure 7

The temperature dependence of the integrated intensity of $\omega$ scans for reflection 004. Top: the integrated intensity of reflection 004 observed at Dy $L_3$ absorption edge (circles) and Fe $K$ absorption edge (triangles), and that for reflection 001 observed at Dy $M_5$ absorption edge (open circles) [16]. The integrated intensity of reflection 001 is multiplied by a factor to compare with the others. Bottom: the square root of the integrated intensity of reflection 004 observed at Dy $L_3$ absorption edge (circles) and Fe $K$ absorption edge (triangles) together with the bond length of Dy-O(4) (diamonds) and the bond length of Fe(2)-O($2_1$) (squares) [16].

Figure 8

An image of deformation in a ${\mathrm{DyO}}_6$ trigonal prism and two ${\mathrm{FeO}}_6$ octahedra in the low-$TP{3}_{2}21$ phase due to the structural phase transition at $T_S$. This image is drawn by vesta [34]. Here, the Fe(1) ion locates at $3a$ site and the Fe(2) ion locates at $6b$ site in the low-$TP{3}_{2}21$ phase. The arrows show the direction of the elongation or the shrinkage. Pair bonds Fe(2)-O(2) lose the symmetry and that the Fe(2)-$\mathrm{O}(2_1$) bond elongates and the other Fe(2)-$\mathrm{O}(2_2$) bond shrinks.