Examination of the reliability of x-ray techniques for determining hydrogen-induced volume changes

The Bragg peaks in x-ray diffraction experiments are generally taken to be a reliable measure of the average lattice distance in crystals. We show that this assumption can be misleading when determining hydrogen-induced volume changes. Experiments based on simultaneous determination of the position of the Bragg peak and the sample volume of an Fe/V(001) superlattice while expanding the lattice by in situ hydrogen loading serve as an illustration of this counter-intuitive effect.

Figure 1

(Color online) Illustration of the accessible length scales in the described experiment. The total thickness of the superlattice is denoted by $L_{\text{s}}$, the thickness of the capping layer is denoted by $L_{\text{c}}$, the lattice parameters in the [001] direction (along the $z$ axis) are denoted by $c_{\text{V}}$ and $c_{\text{Fe}}$, and the repeat distance for the chemical modulation is denoted by $\Lambda$.

Figure 2

(Color online) Illustration of the accessible length scales in the described experiment. The step time was roughly 1.4 s/step and the wavelength was $0.15406\text{nm}$. For details, see text.

Figure 3

(Color online) A reflectivity curve of the superlattice at 348 K and hydrogen pressure below $5\times {10}^{-9}\text{mbar}$. A shows the capping layer oscillations corresponding to $L_{\text{c}}$, B marks the total thickness oscillations corresponding to the total thickness $\left(T\right)$, and C indicates the superlattice peaks corresponding to $\Lambda$. The step time was 5 s/step.

Figure 4

(Color online) The relative changes in total thickness $\left(T\right)$ from simulations of the data plotted versus changes in $\Lambda$. The thickness is given by $T=L_{\text{s}}+L_{\text{V}}+L_{\text{c}}$. The solid line is a straight-line fit through the origin with a slope of 0.96(2) with a reduced ${\chi }^2=1.4$.

Figure 5

An XRD pattern of the superlattice at 348 K and pressure below $5\times {10}^{-9}\text{mbar}$. D indicates the superlattice peaks corresponding to $\Lambda$ and E shows the Laue oscillations corresponding to the total thickness (excluding the Pd cap). The step time was 5 s/step.

Figure 6

(Color online) Relative changes of the Fe/V(002) peak position as a function of the relative volume changes in the sample. The data were taken at 348 K and in a pressure range of 0–739 mbar. At 473 K, the deviation at low $\Delta \Lambda /\Lambda$ vanishes. The red solid line represents the expected scaling of 1.0. The uncertainty in the vertical direction is smaller than the symbols.

Figure 7

(Color online) Diffraction peaks from the (002) lattice planes of the superlattice at 348 K plotted against change in the bilayer thickness $\Lambda$. The upper frame shows the normalized difference $R$ from the expected scaling defined by Eq. (8). The solid symbols in the upper panel correspond to 473 K. The angular range is converted into lattice spacing using Eq. (5). The line in the lower frame shows the volume changes corresponding to the expected scaling of slope unity [see Eq. (3)]. The hydrogen concentration scales with the volume change and the maximum expansion corresponds to almost one hydrogen per vanadium atom $\left[\left(\text{H}/V\right)\approx 1\right]$.