2.1 Dynamical theory of X-ray diffraction

The dynamical theory of X-ray diffraction can accurately describe X-ray diffraction from a perfect crystal^[ Reference Shvyd’ko ^{21 –} Reference Authier ^{23 ]}. According to this theory, the Bragg diffraction ${R}_{0\mathrm{H}}$ and forward Bragg diffraction ${R}_{00}$ amplitudes by a perfect thin crystal can be expressed as follows:

(1a) $$\begin{align}{R}_{0\mathrm{H}}={R}_1{R}_2\frac{1-{e}^{-i\left({\aleph}_1-{\aleph}_2\right)l}}{R_2-{R}_1{e}^{-i\left({\aleph}_1-{\aleph}_2\right)l}},\end{align}$$

(1b) $$\begin{align}{R}_{00}={e}^{-i{\aleph}_1l}\frac{R_2-{R}_1}{R_2-{R}_1{e}^{-i\left({\aleph}_1-{\aleph}_2\right)l}},\end{align}$$

where

(2a) $$\begin{align}{R}_j=\frac{S\left({\gamma}_\mathrm{h}\right)}{\sqrt{\left|\gamma \right|}}\cdot \frac{\sqrt{\chi_{\mathrm{h}}{\chi}_{\overline{h}}}}{\chi_{\overline{h}}}\left[y\pm \sqrt{y^2+S\left({\gamma}_\mathrm{h}\right)}\right],\ j=1,2,\end{align}$$

(2b) $$\begin{align}{\aleph}_j=\frac{k{\chi}_0}{2{\gamma}_0}+\frac{S\left({\gamma}_\mathrm{h}\right)}{2{\Lambda}_0}\left[y\pm \sqrt{y^2+S\left({\gamma}_\mathrm{h}\right)}\right],\ j=1,2, \!\quad\end{align}$$

(2c) $$\begin{align}y=\frac{k{\Lambda}_0}{2{\gamma}_\mathrm{h}}\left(-\alpha +{\alpha}^{\prime}\right),{\Lambda}_0=\frac{\lambda \sqrt{\gamma_0\left|{\gamma}_\mathrm{h}\right|}}{P\sqrt{\chi_{\mathrm{h}}{\chi}_{\overline{h}}}}, \qquad\qquad\end{align}$$

and

(3a) $$\begin{align}\alpha =2\sin 2{\theta}_\mathrm{B}\left[\Delta \theta +\left(\frac{\Omega}{\omega_0}+\frac{\Delta d}{d}\right)\tan {\theta}_\mathrm{B}\right],\end{align}$$

(3b) $$\begin{align}{\alpha}^{\prime}={\chi}_0\left(\gamma -1\right), \ \ \ \qquad\qquad\qquad\qquad\qquad\end{align}$$

and the intensities of Bragg diffraction (reflectivity) and forward Bragg diffraction (transmissivity) can be written as follows:

(4) $$\begin{align}{I}_{0\mathrm{H}}=\gamma {\left|{R}_{0\mathrm{H}}\right|}^2,\quad {I}_{00}={\left|{R}_{00}\right|}^2.\end{align}$$

Here, the parameter $\alpha$ is introduced to account for a small perturbation that causes a deviation from the Bragg condition. Such perturbation may stem from various sources, such as deviations in the incident angle ( $\Delta \theta$ ), angular frequency ( $\Omega$ ) and lattice spacing ( $\Delta d$ ). The parameter ${\alpha}'$ specifically characterizes the deviation from the Bragg diffraction, which arises from the refraction of the crystal. Here, $\gamma$ is the asymmetry ratio, and is given by the following:

(5a) $$\begin{align}\gamma =\frac{\gamma_{\mathrm{h}}}{\gamma_0}=\frac{\cos {\varPsi}_\mathrm{h}}{\cos {\varPsi}_0}, \qquad\qquad\qquad\end{align}$$

(5b) $$\begin{align}{\varPsi}_0=\eta +{\theta}_\mathrm{B}-\frac{\pi }{2},\quad {\varPsi}_\mathrm{h}=\eta -{\theta}_\mathrm{B}-\frac{\pi }{2},\end{align}$$

where $\eta$ is the asymmetry angle that describes the angle between the crystal surface and the parallel reflecting atomic planes and $S\left({\gamma}_\mathrm{h}\right)$ denotes the sign of ${\gamma}_\mathrm{h}$ .

2.2 Electric susceptibility and Debye–Waller factor

In a crystal, both the charge density function $\rho \left(\boldsymbol{r}\right)$ and the electric susceptibility $\chi \left(\boldsymbol{r}\right)$ are triply periodic functions of the space coordinates $\boldsymbol{r}$ , and the electric susceptibility can be expressed as follows:

(6) $$\begin{align}\chi \left(\boldsymbol{r}\right)=-\frac{r_{\mathrm{e}}{\lambda}^2}{\pi}\rho \left(\boldsymbol{r}\right)=\sum \limits_{\boldsymbol{h}}{\chi}_\mathrm{h}\exp \left(i\boldsymbol{h}\cdot \boldsymbol{r}\right).\end{align}$$

The coefficients ${\chi}_\mathrm{h}$ of the Fourier expansion can be related to the structure factor ${F}_\mathrm{h}$ of the crystal, and then we have the following:

(7a) $$\begin{align}{\chi}_\mathrm{h}=-\frac{r_{\mathrm{e}}{\lambda}^2}{\pi {V}_\mathrm{c}}{F}_\mathrm{h}, \qquad\quad\qquad\end{align}$$

(7b) $$\begin{align}{F}_\mathrm{h}=\int \rho \left(\boldsymbol{r}\right)\exp \left(-i\boldsymbol{h}\cdot \boldsymbol{r}\right)\mathrm{d}\boldsymbol{r}.\end{align}$$

It is easy to have a picture that atoms are arranged in a periodic lattice in a crystal. The vibration of the atoms can be thermally excited inside a crystal. Therefore, the vibration is temperature dependent, and can result in the decrease of the intensity of the scattered wave. The structure factor of a vibratory crystal can be expressed as follows:

(8) $$\begin{align}{F}_\mathrm{h}=\sum \limits_j\left[{f}_j\left(\boldsymbol{h}\right)+{f}_j^{\prime }+{f}_j^{{\prime\prime}}\right]\exp \left(-{M}_j\right)\exp \left(-i\boldsymbol{h}\cdot \boldsymbol{r}\right),\end{align}$$

where ${f}_j\left(\boldsymbol{h}\right)$ is the form factor of atom $j$ , ${f}^{\prime }$ and ${f}^{{\prime\prime} }$ are the dispersion corrections, $\exp \left(-{M}_j\right)$ is the Debye–Waller factor^[ Reference Jens ^{24 ]} and ${M}_j$ is given by the following:

(9) $$\begin{align}{M}_j={B}_\mathrm{T}^j{\left(\frac{\sin \theta }{\lambda}\right)}^2.\end{align}$$

If the atom vibrates isotropically and only one type of atom is considered, the thermal factor ${B}_\mathrm{T}$ can be written as follows:

(10a) $$\begin{align}{B}_\mathrm{T}=\frac{6{h}^2}{m_{\mathrm{A}}{k}_\mathrm{B}\Theta}\left[\frac{\phi \left(\Theta /T\right)}{\Theta /T}+\frac{1}{4}\right],\end{align}$$

(10b) $$\begin{align}\phi (x)=\frac{1}{x}{\int}_{{\kern-4pt}0}^{x}\frac{\xi }{e^{\xi }-1}\mathrm{d}\xi, \!\qquad\qquad\quad\end{align}$$

where ${m}_\mathrm{A}$ , ${k}_\mathrm{B}$ , $\Theta$ and $h$ are the mass of the atom, Boltzmann constant, Debye temperature and Planck constant, respectively. The detailed derivation of Equations (9) and (10) can be found in Ref. [Reference Jens24].

2.3 Model of X-ray diffraction by thermally loaded crystals

In this section, we establish a numerical model to estimate the X-ray diffraction performance of thermally loaded crystals. In this model, we take the Debye–Waller factor, electric susceptibility, thermal strain, incident angle deviation and slope error into consideration.

When a series of XFEL pulses are incident on a crystal, the thermal characteristics, such as the thermal conductivity, thermal expansion coefficient, specific heat capacity and Debye–Waller factor, vary at different spatial locations due to non-uniform thermal loading. To estimate the overall reflectivity and transmissivity, we employ a weighted average method similar to that utilized in Bushuev’s research:

(11a) $$\begin{align}{\overline{I}}_{0\mathrm{H}}=\frac{\int {\int}_{-\infty}^{+\infty }{I}_{0\mathrm{H}}\left(\alpha, {\alpha}^{\prime },T,x,y\right)I\left(x,y\right)\mathrm{d}x\mathrm{d}y}{\int {\int}_{-\infty}^{+\infty }I\left(x,y\right)\mathrm{d}x\mathrm{d}y},\end{align}$$

(11b) $$\begin{align}{\overline{I}}_{00}=\frac{\int {\int}_{-\infty}^{+\infty }{I}_{00}\left(\alpha, {\alpha}^{\prime },T,x,y\right)I\left(x,y\right)\mathrm{d}x\mathrm{d}y}{\int {\int}_{-\infty}^{+\infty }I\left(x,y\right)\mathrm{d}x\mathrm{d}y},\end{align}$$

where $I\left(x,y\right)$ is the intensity distribution. To establish the local reflectivity ${I}_{0\mathrm{H}}\left(\alpha, {\alpha}^{\prime },T,x,y\right)$ and transmissivity ${I}_{00}\left(\alpha, {\alpha}^{\prime },T,x,y\right)$ , we have devised a methodological framework as depicted in Figure 1. Firstly, we obtain the thermal properties of the crystal at different temperatures, including the thermal expansion coefficient, thermal conductivity and specific heat. Secondly, we calculate the temperature distribution $T\left(x,y,t\right)$ through either a theoretical model^[ Reference Bushuev ^{15 ,} Reference Bushuev ^{16 ]} or thermal analysis software such as COMSOL. Thirdly, we establish the distribution of the Debye–Waller factor and the lattice spacing at the given temperature $T\left(x,y,t\right)$ , which also allows one to derive the slope error distribution. Then, we can calculate the local volume of the unit cell and the local atomic form factor. Afterwards, we construct the local structure factor ${F}_\mathrm{h}$ and the local Fourier coefficient of the electric susceptibility ${\chi}_\mathrm{h}$ . Next, by substituting the local ${F}_\mathrm{h}$ and ${\chi}_\mathrm{h}$ into Equation (1), we can derive the local reflectivity and transmissivity. Finally, we obtain the overall reflectivity ${\overline{I}}_{0\mathrm{H}}$ and transmissivity ${\overline{I}}_{00}$ by using Equation (11).

Figure 1 The calculation framework of the local reflectivity and transmissivity.

3.1 Efficiency degradation induced by the Debye–Waller factor

In this section, we discuss the impact of the Debye–Waller factor in dynamical X-ray diffraction. Figure 2 shows examples of calculations by taking the Debye–Waller factor into account in X-ray diffraction. In order to include the effect of thermal expansion, we use the data by interpolating the thermal expansion coefficient of silicon in Table 1. The specific heat capacity of a silicon crystal is fitted from Okhotin et al.’s work^[ Reference Okhotin, Pushkarskii and Gorbachev ^{25 ]}. The thermal expansion coefficient is extracted and fitted from Refs. [Reference Lyon, Salinger, Swenson and White26,Reference Yasumasa and Yozo27]. The thermal conductivity is fitted from Kazan et al.’s work^[ Reference Kazan, Guisbiers, Pereira, Correia, Masri, Bruyant, Volz and Royer ^{28 ]}. The X-ray photon energy is 15 keV and the thickness of the crystal is 0.5 mm. The reflectivity curves of Si(111), Si(333), Si(555) and Si(777) are calculated at different temperatures in Figure 2(a). The rocking curves of Si(777) at temperatures of 100, 300, 500 and 700 K are shown in Figure 2(b).

Figure 2 (a) Reflectivity as a function of temperature for Si(111), Si(333), Si(555) and Si(777). (b) Rocking curve calculations of Si(777) at different temperatures (100, 300, 500 and 700 K).

It is evident that as the temperature increases, the Debye–Waller factor exhibits a decline in the height of the rocking curves and a narrowing of the bandwidth of the rocking curves. In addition, as illustrated in Figure 2(a), a more significant reduction in reflectivity is observed at higher Miller index reflections. Our findings are consistent with the study conducted by Chung^[ Reference Chung ^{29 ]}. Thus far, we can infer that the Debye–Waller factor plays a crucial role in X-ray diffraction. We have integrated this effect into our method of X-ray diffraction by non-uniform thermally loaded crystals.

3.2 Photon energy shift caused by electric susceptibility

According to the dynamical theory of X-ray diffraction, the central photon energy shift of the rocking curve arises from two terms, $\alpha$ and ${\alpha}^{\prime }$ . The parameter $\alpha$ can describe a small perturbation that results in a deviation from the Bragg condition. The perturbation could derive from incident angle deviation $\Delta \theta$ , angular frequency deviation $\Omega$ and lattice deviation $\Delta d$ . The parameter ${\alpha}^{\prime }$ is the deviation from the Bragg condition due to the refraction of crystals. The refractive index is temperature dependent, which is due to the electric susceptibility change with the temperature. In this section, we investigate the temperature dependence of electric susceptibility, which may contribute to the central photon energy shift of the rocking curve.

Table 1 Thermal properties of silicon crystal.

The coefficients ${\chi}_\mathrm{h}$ are related to the volume of the unit cell ${V}_\mathrm{c}$ and the structure factor ${F}_\mathrm{h}$ . There is no doubt that ${\chi}_\mathrm{h}$ is temperature dependent, as ${V}_\mathrm{c}$ changes with the thermal expansion and ${F}_\mathrm{h}$ changes with the Debye–Waller factor. Electric susceptibility ${\chi}_\mathrm{h}$ is associated with several physical quantities, such as the refractive index $n$ , crystal detuning effect ${\alpha}^{\prime }$ , Darwin width and extinction length. The central photon energy shift caused by quantity $\alpha$ has been investigated in Refs. [Reference Bushuev15,Reference Bushuev16,Reference Qu, Ma, Zhou and Wu18]. These studies show that the shift is mainly attributed to the maximum thermal strain ( $\Delta d/d$ ). Here, we discuss the central photon energy shift arising from the quantity ${\alpha}^{\prime }$ . The shift can be expressed as follows:

(12) $$\begin{align}\frac{\Omega_{\mathrm{detuning}}}{\omega_0}=\frac{\alpha^{\prime }}{4\;{\sin}^2{\theta}_\mathrm{B}},\quad {\alpha}^{\prime}={\chi}_0\left(\gamma -1\right),\end{align}$$

where ${\chi}_0$ is temperature dependent. In general, ${\chi}_0$ is a very small complex parameter. The imaginary part $\operatorname{Im}\left({\chi}_0\right)$ and the real part Re( ${\chi}_0$ ) are related to the cross-section of photon absorption and the atomic Thomson scattering amplitude, respectively. Typically, for diamond and silicon crystals, $\operatorname{Re}\left({\chi}_0\right)\gg \operatorname{Im}\left({\chi}_0\right)$ , and $\operatorname{Re}\left({\chi}_0\right)$ is in the range of 10^–4–10^–7 for 3–25 keV.

Figure 3 illustrates the variation of the photon energy shift that corresponds to ${\alpha}^{\prime }$ and the thermal strain ( $\Delta d/d$ ) as a function of temperature. The thermal expansion data can be acquired by interpolating the data presented in Tables 1 and 2. The thermal expansion coefficient ${\alpha}_\mathrm{L}$ and specific capacity ${c}_\mathrm{p}$ of diamond crystal are extracted from Reeber and Wang’s work^[ Reference Reeber and Wang ^{30 ]}, and the thermal conductivity of diamond crystal is extracted and fitted from Wei et al.’s work^[ Reference Wei, Kuo, Thomas, Anthony and Banholzer ^{31 ]}. Subsequently, the photon energy shift caused by ${\alpha}^{\prime }$ at varying temperatures can be calculated by modifying the volume of the unit cell and the Debye–Waller factor. The lattice constants corresponding to different temperatures are calculated based on the thermal expansion coefficient and the lattice constant at the reference temperature. In Figure 3, the reference temperatures of diamond and silicon are respectively specified as 293.15°C and 298.15°C. The numerical method utilized in this analysis is elaborated in Figure 1. The computations presented in Figures 3(a) and 3(b) pertain to C ${}^{\ast }$ (400) and Si(400) at 9 keV, respectively. For a crystal with a specific Miller index and a given photon energy, the photon energy shift attributable to ${\alpha}^{\prime }$ is roughly three orders of magnitude lower than the shift caused by thermal strain. As a result, thermal strain predominantly governs the central photon energy shift of the rocking curve, and the influence of ${\alpha}^{\prime }$ can be disregarded.

Figure 3 (a) Temperature dependence of the photon energy shift induced by ${\alpha}^{\prime }$ and lattice expansion in C ${}^{\ast }$ (400). (b) Temperature dependence of the photon energy shift induced by ${\alpha}^{\prime }$ and lattice expansion in Si(400). The calculation is performed under the assumption of a uniform thermal load.

4.1 Comparison

Here, the analytical model developed by Bushuev^[ Reference Bushuev ^{15 ,} Reference Bushuev ^{16 ]} is employed to compute the temperature distribution under the assumption of a 2D infinite domain:

(13) $$\begin{align}T\left(x,y,t\right)=\sum \limits_{j=1}^n\frac{\Delta {T}_j}{\sqrt{\beta_x{\beta}_y}}\exp \left(-\frac{x^2}{r_x^2{\beta}_x}-\frac{y^2}{r_y^2{\beta}_y}\right).\end{align}$$

Here,

(14a) $$\begin{align}{\beta}_x=1+\frac{\sin^2{\theta}_\mathrm{B}\left(t-{t}_j\right)}{\tau_{\mathrm{T}}},\quad {\beta}_y=1+\frac{\left(t-{t}_j\right)}{\tau_{\mathrm{T}}},\end{align}$$

(14b) $$\begin{align}\Delta {T}_j=\frac{\mu {Q}_\mathrm{p}}{\pi {c}_\mathrm{p}\rho {r}^2},\quad {\tau}_\mathrm{T}=\frac{r^2{c}_\mathrm{p}\rho }{4\kappa }, \!\!\!\qquad\qquad\quad\qquad\end{align}$$

where $\mu$ , ${Q}_\mathrm{p}$ , ${c}_\mathrm{p}$ , $\rho$ , $r$ and $\kappa$ are the absorption factor, pulse energy, specific heat capacity, density, transverse pulse radius and thermal conductivity, respectively. Here, $n$ is the number of pulse impingements on the crystal and ${t}_j$ is the arrival time of the jth pulse.

Our study focuses on analyzing the rocking curve of a thermally loaded Si(444) with a thickness of 50 μm. The initial temperature of the crystal is set to 300 K. We consider the pulse repetition rate is 1 MHz and the pulse energy is 200 μJ. The radius of the transverse beam profile is 800 μm. It is assumed that the bandwidth of the incident pulse is significantly wider than the width of the rocking curve. The photon energy is 10 keV, and with an absorption length of 134 μm. To conduct numerical simulations, we calculate the temperature-dependent spatial distribution of the Debye–Waller factor at $T\left(x,y,t\right)$ by implementing the method outlined in Section 2. The temperature-dependent parameters of silicon, such as the specific heat capacity, thermal conductivity and thermal expansion coefficient, are obtained by interpolating the data provided in Table 1.

When heat is deposited into the crystal, non-uniform distortion occurs, leading to changes in the overall reflectivity and transmissivity curves at different time intervals, as illustrated in Figures 4(a) and 4(b). The black curves represent diffraction by a perfect Si(444) crystal at 300 K. The solid curves were obtained using our proposed method, while the circles represent the results obtained using Bushuev’s method, which does not take into account the temperature dependence of the Debye–Waller factor. Both methods reveal that the rocking curves shift and distort with heat accumulation. Compared to the dashed lines, the height of the rocking curve decreases, and the width becomes slightly narrower, which indicates the evident influence of the Debye–Waller factor. Specifically, when the heat accumulation time is 10 μs, the reflectivities of our proposed model and Bushuev’s model are 40% and 48%, respectively. As the heat accumulation time increases, the differences in reflectivity and transmissivity between the two models are expected to become more significant. The numerical simulation results demonstrate that for a non-uniform thermally distorted crystal, the Debye–Waller factor plays a critical role in reducing the height and width of the rocking curves.

Figure 4 Comparative analysis of results obtained at 2, 6 and 10 μs using Bushuev’s method (circles) and our proposed method (solid lines). The black curve refers to the undeformed crystal at 300 K.

4.2 Dynamic pulse-to-pulse process

So far, we have developed a numerical model to describe X-ray diffraction from a thermally loaded crystal and have gained an understanding of the distortion of its rocking curve. To facilitate a more intuitive understanding of the properties of a thermally loaded crystal, we investigate the case of dynamic pulse-to-pulse thermal conduction processes in this section. Specifically, we investigate C ${}^{\ast }$ (400) with an initial temperature of 200 K and a thickness of 110 μm. The input pulse energy, repetition rate, photon energy and transverse beam radius are set at 500 μJ, 1 MHz, 9 keV and 800 μm, respectively. The absorption length is 977 μm. The temperature distribution of a diamond crystal is estimated using Equation (13). The specific heat capacity, thermal conductivity and thermal expansion coefficient of diamond can be obtained by interpolating the data in Table 2.

In the simulation, the pulse interval is 1 μs and the thermal runaway characteristic time ${\tau}_\mathrm{T}$ (thermal exchange time) corresponding to the crystal at 200 K is 42.3 μs. Therefore, under these conditions, the temperature of the crystal will continue to accumulate, and the performance of the crystal will become increasingly worse. Figures 5(a) and 5(b) present the reflectivity and transmissivity records of C ${}^{\ast }$ (400). The results indicate a notable widening of the bandwidth and a shift in the central photon energy over time. Moreover, a decrease in reflection efficiency is observed as a result of heat accumulation. The temperature evolution at the center of the crystal is depicted in Figure 5(c), indicating a rapid temperature increase. Apparently, for this case, there is not enough time for the crystal to cool down due to the high repetition rate. Consequently, the reflectivity decreases rapidly due to distortion and the shift of the rocking curve for a fixed diffraction geometry. To optimize the reflectivity, a strategy of tuning the pitch angle of the crystal can be employed, as shown in Figure 5(c) for both tuned and untuned reflectivity curves. This strategy is implemented by controlling the proper angular speed of the motor to mitigate the shift of the central photon energy, as shown in Figure 5(e), and Qu et al. ^[ Reference Qu, Ma, Zhou and Wu ^{20 ]} reported the strategy in their work. Furthermore, Figure 5(d) exhibits that the bandwidth and central photon energy shift increase with heat accumulation, where the latter can be compensated for by the tuning strategy, while the former can be reduced by decreasing the slope error.

Figure 5 (a) Time evolution of the reflectivity of C ${}^{\ast }$ (400). (b) Time evolution of the transmissivity of C ${}^{\ast }$ (400). (c) Time evolution of the temperature at the crystal’s center (green curve). The reflectivity curves for untuned and tuned cases are represented by circles and stars, respectively. (d) Time evolution of the normalized bandwidth and photon energy shift. The data have been normalized based on the rocking curve bandwidth (0.78 eV) of C ${}^{\ast }$ (400) at 200 K. (e) Time evolution of the tuning angle and the angular speed.