Phase of transmitted wave in dynamical theory and quasi-kinematical approximation

Variation of the phase of the beam transmitted through a crystalline material as a function of the rocking angle is a well-known dynamical effect in x-ray scattering. Unfortunately, it is not so easy to directly measure these phase variations in a conventional scattering experiment. It was recently suggested that the transmitted phase can be directly measured in ptychography experiments performed on nanocrystal samples. Results of such experiment for different crystal thickness, reflections, and incoming photon energies, in principle, can be fully described in the frame of dynamical theory. However, dynamical theory does not provide a simple analytical expression for the further analysis. Here we develop a quasi-kinematical theory approach that allows one to correctly describe the phase of the transmitted beam for the crystal thickness less than extinction length that is beyond applicability of the conventional kinematical theory.

Figure 1

Schematic layout of the ptychography experiment described in Ref. [18]. Intensities of the transmitted and diffracted beams are measured simultaneously by two detectors, while the crystal is rotated near the Bragg angle. Ptychographic measurements were performed on a Au 111 crystalline nanoparticle 100 nm thick. Reconstruction of the complex amplitude of the transmitted beam allowed one to observe phase variations as a function of the rocking angle.

Figure 2

(a) Diffraction scattering experiment in Laue geometry on a single crystal of the thickness $d$. Here, ${\theta }_B$ is the Bragg angle, ${\phi }_0$ and ${\phi }_h$ are the angles between the normal $\mathbf{n}$ to the crystal entrance surface, transmitted ${\mathbf{k}}_{\mathbf{0}}$ and diffracted ${\mathbf{k}}_{\mathbf{h}}$ wave vectors, respectively. (b) Diffraction scattering experiment in Bragg geometry on a single crystal of the thickness $d$.

Figure 3

(a),(c) Reflectivity $p_R\left(\mathrm{\Delta }\theta \right)$ and (b),(d) the phase ${\phi }_{\mathrm{dyn}}\left(\mathrm{\Delta }\theta \right)$ of the transmitted beam in Laue geometry as a function of the rocking angle $\mathrm{\Delta }\theta =\theta -{\theta }_B$. Simulations were performed for Au 111 and Si 111 crystals of different thickness $d=0.2,0.4$, and $0.6L_{ex}$. Dynamical theory simulations (full lines) and simulations performed in the frame of quasi-kinematical approximation (dashed lines). Parameters of simulations are listed in Table 1.

Figure 4

Same as in Fig. 3 for crystal thickness $d=L_{ex}$ and $1.5L_{ex}$.

Figure 5

Angular dependance of the amplitude of the weakly absorbing wave $E_0^{\left(1\right)}$ (red curve) and strongly absorbing wave $E_0^{\left(2\right)}$ (black curve) for different crystal thickness (a) $d=L_{ex}$ and (b) $d=10L_{ex}$. In both cases, simulations were performed for Au 111 reflection and photon energy 8.5 keV. The difference in the amplitudes of the waves for the thick crystal is clearly seen.

Figure 6

Comparison between the quasi-kinematical approximation and dynamical theory. Here the relative error $\varepsilon$ in the phase defined in Eq. (37) is calculated using a quasi-kinematical approximation as compared to the dynamical theory as a function of the rocking angle $\mathrm{\Delta }\theta$ and relative crystal thickness $d/L_{ex}$. Dashed horizontal line corresponds to 5% difference in the two phases that appears at the crystal thickness $d\simeq 0.8L_{ex}$.

Figure 7

(a),(c) Reflectivity $p_R\left(\mathrm{\Delta }\theta \right)$ and (b),(d) the phase ${\phi }_{\mathrm{dyn}}\left(\mathrm{\Delta }\theta \right)$ of the transmitted beam in Laue geometry as a function of the rocking angle $\mathrm{\Delta }\theta =\theta -{\theta }_B$. Simulations were performed using dynamical theory approach for a Au crystal of thickness $d=300$ nm. (a),(b) Results of simulations for the incident photon energy 8.5 keV and different reflection orders 111, 220, and 222 in a Au crystal. (c),(d) Results of simulations for a Au 111 crystal and different incident photon energies of 5, 8.5, and 12 keV. Parameters of simulations are listed in Tables 2 and 3.