Experimental evidence of thermal fluctuations on the x-ray absorption near-edge structure at the aluminum $K$ edge

After a review of temperature-dependent experimental x-ray absorption near-edge structure (XANES) and related theoretical developments, we present the Al $K$-edge XANES spectra of corundum and beryl for temperature ranging from 300 to 930 K. These experimental results provide a first evidence of the role of thermal fluctuation in XANES at the Al $K$ edge, especially, in the pre-edge region. The study is carried out by polarized XANES measurements of single crystals. For any orientation of the sample with respect to the x-ray beam, the pre-edge peak grows and shifts to lower energy with temperature. In addition, temperature induces modifications in the position and intensity of the main XANES features. First-principles DFT calculations are performed for both compounds. They show that the pre-edge peak originates from forbidden 1$s$$\to$$3s$ transitions induced by vibrations. Three existing theoretical models are used to take vibrations into account in the absorption cross-section calculations: (i) an average of the XANES spectra over the thermal displacements of the absorbing atom around its equilibrium position, (ii) a method based on the crude Born-Oppenheimer approximation where only the initial state is averaged over thermal displacements, and (iii) a convolution of the spectra obtained for the atoms at the equilibrium positions with an approximate phonon spectral function. The theoretical spectra so obtained permit to qualitatively understand the origin of the spectral modifications induced by temperature. However, the correct treatment of thermal fluctuation in XANES spectroscopy requires more sophisticated theoretical tools.

Figure 1

A sketch of the 21-point cubic grid used in Eq. (2).

Figure 2

Experimental x-ray absorption polarized spectra of corundum (left) and beryl (right) at the Al K edge for different temperatures along with difference of each spectrum with respect to the 300 K reference. The top and bottom panels display the ${\sigma }_{\parallel }$ and ${\sigma }_{\perp }$ spectra, respectively. The upper-right insets use the same legend code and present a zoom on the main peak(s).

Figure 3

Temperature dependence of the x-ray natural linear dichroism (XNLD) measured at the Al $K$ edge in corundum (top panel) and beryl (bottom panel). The cross section ${\sigma }_{\mathrm{XNLD}}$ is equal to the difference ${\sigma }_{\parallel }$-${\sigma }_{\perp }$. Note that the $y$-axis range is twice larger for beryl than for corundum.

Figure 4

Pre-edge intensity and energy position for both orientations and for both samples as a function of temperature. The $y$-axis units of the upper panels are consistent with those of Fig. 2. The error bars correspond to an uncertainty of 10% on the best fit value (R${}^2\simeq$ 0.9).

Figure 5

Top: calculated and experimental isotropic (2${\sigma }_{\perp }$+${\sigma }_{\parallel }$)/3 spectra at the Al $K$ edge of corundum (left) and beryl(right). Bottom: partial and local density of states of corundum (left) and beryl (right) calculated for the supercell including a 1$s$ core hole. The role of the Al 3$s$ of the absorbing atom is prominent in the pre-edge region.

Figure 6

Calculated ${\sigma }_{\parallel }$ and ${\sigma }_{\perp }$ spectra at the Al K edge of corundum (left) and beryl (right) along with difference of each spectrum with respect to the equil. reference. The “equil.” spectra correspond to DFT calculations performed with the atoms at the equilibrium positions. The spectra labeled “meth. 1,” “meth. 2,” and “meth. 3” refer to the three methods of vibration modeling described in Sec. 2. The upper-right insets use the same legend code and present a zoom on the main peak(s).