X-ray linear dichroism in cubic compounds: The case of ${\text{Cr}}^{3+}$ in ${\text{MgAl}}_2{\text{O}}_4$

The angular dependence (x-ray linear dichroism) of the $\text{Cr}K$ pre-edge in ${\text{MgAl}}_2{\text{O}}_4:{\text{Cr}}^{3+}$ spinel is measured by means of x-ray absorption near-edge structure spectroscopy and compared to calculations based on density functional theory (DFT) and ligand field multiplet (LFM) theory. We also present an efficient method, based on symmetry considerations, to compute the dichroism of the cubic crystal starting from the dichroism of a single substitutional site. DFT shows that the electric dipole transitions do not contribute to the features visible in the pre-edge and provides a clear vision of the assignment of the $1s\to 3d$ transitions. However, DFT is unable to reproduce quantitatively the angular dependence of the pre-edge, which is, on the other side, well reproduced by LFM calculations. The most relevant factors determining the dichroism of $\text{Cr}K$ pre-edge are identified as the site distortion and $3d\text{−}3d$ electronic repulsion. From this combined DFT, LFM approach is concluded that when the pre-edge features are more intense than 4% of the edge jump, pure quadrupole transitions cannot explain alone the origin of the pre-edge. Finally, the shape of the dichroic signal is more sensitive than the isotropic spectrum to the trigonal distortion of the substitutional site. This suggests the possibility to obtain quantitative information on site distortion from the x-ray linear dichroism by performing angular dependent measurements on single crystals.

Figure 1

(Color online) Cubic cell of the spinel structure and experimental setup. The four equivalent octahedral sites are labeled according to the coordinates given in Table . Site distortion has been slightly exaggerated for clarity. The sample is cut along the (110) plane (red) and rotated along the [110] direction, while $\hat{\mathit{\epsilon }}$ and $\hat{\mathbf{k}}$ are fixed in the laboratory system. The figure corresponds to the experimental setup taken as starting point $\left({\alpha }_{\text{rot}}=0°\right)$. For this configuration, the [001] axis of the cube is in the vertical plane, perpendicular to $\hat{\mathit{\epsilon }}=\left[010\right]$ and $\hat{\mathbf{k}}=\left[\overline{1}00\right]$.

Figure 2

(Color online) Pre-edge spectra calculated for sites A, B, C, and D in the electric quadrupole approximation. The orange lines present the spectra calculated for ${\alpha }_{\text{rot}}=0°$ $\left(\hat{\mathit{\epsilon }}=\left[010\right],\hat{\mathbf{k}}=\left[\overline{1}00\right]\right)$, while the black lines are the spectra calculated for ${\alpha }_{\text{rot}}=90°$ $\left(\hat{\mathit{\epsilon }}=\left[\frac{1}{2},\frac{1}{2},\frac{1}{\sqrt{2}}\right],\hat{\mathbf{k}}=\left[-\frac{1}{2},-\frac{1}{2},\frac{1}{\sqrt{2}}\right]\right)$.

Figure 3

(Color online) Comparison between experimental (dotted line) and calculated (solid line) isotropic XANES spectra at the $\text{Cr}K$-edge in spinel. The calculation was performed using DFT-LSDA (see Sec. 3B1). The inset presents the spectra in the pre-edge region. The dashed orange line is the calculated electric-dipole contribution.

Figure 4

(Color online) Comparison between experimental (dotted line) and calculated (solid line) $\text{Cr}K$-pre-edge spectra in spinel using DFT-LSDA. The orange lines correspond to $\hat{\mathit{\epsilon }}=\left[010\right]$, $\hat{\mathbf{k}}=\left[\overline{1}00\right]$ $\left({\alpha }_{\text{rot}}=0°\right)$. The black lines correspond to $\hat{\mathit{\epsilon }}=\left[\frac{1}{2},\frac{1}{2},\frac{1}{\sqrt{2}}\right]$, $\hat{\mathbf{k}}=\left[-\frac{1}{2},-\frac{1}{2},\frac{1}{\sqrt{2}}\right]$ $\left({\alpha }_{\text{rot}}=90°\right)$. The dichroic signal (green lines) is the difference between the black and red lines for the experimental (dotted) and the calculated spectra (solid), respectively.

Figure 5

(Color online) Electric quadrupole transitions calculated for site A using spin polarization. The upper panel (a) presents the spectrum calculated for $\left(\hat{\mathit{\epsilon }}=\left[\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},0\right],\hat{\mathbf{k}}=\left[\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}},0\right]\right)$. The middle panel (b) presents the spectrum calculated for ${\alpha }_{\text{rot}}=0°$ $\left(\hat{\mathit{\epsilon }}=\left[010\right],\hat{\mathbf{k}}=\left[\overline{1}00\right]\right)$. Lower panel (c) presents the spectrum calculated for ${\alpha }_{\text{rot}}=90°$ $\left(\hat{\mathit{\epsilon }}=\left[\frac{1}{2},\frac{1}{2},\frac{1}{\sqrt{2}}\right],\hat{\mathbf{k}}=\left[-\frac{1}{2},-\frac{1}{2},\frac{1}{\sqrt{2}}\right]\right)$. For each configuration, the respective contributions of the two spins are plotted (black solid line for spin down, orange solid line for spin up), as well as the sum (dashed line). The Fermi level is located approximately around 5990 eV.

Figure 6

(Color online) Comparison between experimental (dotted line) and calculated (solid line) $\text{Cr}K$ pre-edge spectra in spinel using ligand field multiplet theory in $D_{3d}$ symmetry. The orange lines correspond to $\hat{\mathit{\epsilon }}=\left[010\right]$, $\hat{\mathbf{k}}=\left[\overline{1}00\right]$ $\left({\alpha }_{\text{rot}}=0°\right)$. The black lines correspond to $\hat{\mathit{\epsilon }}=\left[\frac{1}{2},\frac{1}{2},\frac{1}{\sqrt{2}}\right]$, $\hat{\mathbf{k}}=\left[-\frac{1}{2},-\frac{1}{2},\frac{1}{\sqrt{2}}\right]$ $\left({\alpha }_{\text{rot}}=90°\right)$. The dichroic signal is the difference between the black and orange lines.

Figure 7

(Color online) Comparison between calculated $\text{Cr}K$ pre-edge spectra in spinel in $D_{3d}$ symmetry for increasing trigonal distortion. The orange solid line, labeled $O_h$, corresponds to a perfect $O_h$ symmetry $\left(D_{\sigma }=0,D_{\tau }=0\right)$. The black dotted line, labeled $D_{3d}$, corresponds to $\left(D_{\sigma }=-0.036\text{eV},D_{\tau }=0.089\text{eV}\right)$, which are the distortion parameters obtained from optical absorption spectroscopy (Ref. 43). The black solid line, labeled $D_{3d}-2$, was obtained for a doubled distortion parameters $\left(D_{\sigma }=-0.072\text{eV},D_{\tau }=0.178\text{eV}\right)$ and the orange dotted line, labeled $D_{3d}-3$, for tripled distortion parameters $\left(D_{\sigma }=-0.108\text{eV},D_{\tau }=0.267\text{eV}\right)$.