X-ray free electron lasers (XFELs) are capable of producing high-brightness, ultra-short, fully coherent pulses, and provide an unparalleled tool to explore fundamental questions in chemistry, biology, atomic and molecular science, and materials science. Several hard X-ray free electron laser (FEL) user facilities are currently in operation or under construction, including LCLS^[1^], SACLA^[2^], PAL-XFEL^[3^], SwissFEL^[4^], European XFEL^[5^] and SHINE^[6^]. However, the high peak power of XFELs creates a severe thermal load on the X-ray optics, such as crystal monochromators, crystal splitters and mirrors. This thermal load issue has been a concern for several decades with the development of synchrotron radiation light sources (SRLSs) and XFELs. The optical beamline system aims to focus the beam, preserve the wavefront and purify the spectrum, but the thermal load results in wavefront disturbance, efficiency degradation and crystal lattice distortion. More generally, the thermal load is non-uniform, which brings distortion to the rocking curve, decreases the efficiency, shifts the central photon energy and broadens the bandwidth. For XFELs, the brightness is about 10 orders of magnitude higher than that of SRLSs, and the thermal load effect is severe. The high brightness brings challenges to the design of X-ray optics, such as the hard X-ray self-seeding monochromator^[7^], X-ray free electron laser oscillator (XFELO) cavity^[8^,9^], crystal compressor^[10^] and crystal beam splitter^[11^,12^]. The difficulties arising from a non-uniform thermal load urgently call for the investigation of X-ray diffraction by distorted crystals.

The distortion and central wavelength shift of rocking curves have attracted much attention for decades^[13^,14^]. Researchers have realized that the central wavelength shift can be estimated by the maximum thermal strain, and the rocking curve distortion can be attributed to the slope error caused by the non-uniform thermal load. However, there is still a lack of theoretical descriptions of X-ray diffraction by thermally loaded crystals. Bushuev^[15^] developed a theoretical model to calculate the rocking curves of non-uniform thermally loaded crystals by analytically solving the equation of thermal conduction under spatiotemporal dependences of the distribution of temperature. In this model, Bushuev^[16^] first treated the thermal properties of the crystals as constant, and then expanded to temperature-dependent thermal properties. Bushuev applied his model to estimate the performance of the self-seeding crystal monochromator of the European XFEL. Yang et al.^[17^] investigated the transient thermal stress wave and vibration to describe the thermomechanical issues for thin diamond crystals under high-repetition-rate operation of an XFEL, but no further analysis was carried out about the distortion of the rocking curve induced by the non-uniform thermal load. Based on Bushuev’s work, Qu et al.^[18^] further formulated the factors (such as the maximum strain and non-uniform thermal load) that affect the central wavelength shift and the distortion of the rocking curve, and they discussed these effects separately and quantitatively by an analytical method. They applied their method to estimate the thermal load on the self-seeding monochromator^[19^,20^]. The above theoretical methods have successfully illustrated the central wavelength shift and the distortion of rocking curves. However, these methods are developed under the assumption that the Debye–Waller factor and electric susceptibility are temperature independent.

In this study, we focus the investigation on the temperature dependence in X-ray diffraction via the Debye–Waller factor and electric susceptibility. For a thermally loaded crystal, the vibration of atoms at different temperatures contributes to the Debye–Waller factor and further affects the form factor of atoms and the structure factor of crystals. Besides, thermal expansion changes the volume of the unit cell, which results in the perturbation of the structure factor and eventually impacts the electric susceptibility. This paper is organized as follows. In Section 2, we briefly review the dynamical theory of X-ray diffraction, the electric susceptibility and the Debye–Waller factor. Then, we establish a numerical model to estimate the rocking curves of thermally loaded crystals while considering the temperature dependence of the Debye–Waller factor and electric susceptibility. We also consider the thermal strain and incident angle deviation. These factors have a combined impact on the central photon energy shift and distortion of the rocking curves. In Section 3, under the assumption of uniform thermal load, the impact of temperature dependence on the electric susceptibility and the Debye–Waller factor in X-ray diffraction is analyzed. In Section 4, we make a comparison between our method and Bushuev’s approach. Finally, we apply this model to estimate the pulse-to-pulse performance of a thermally loaded diamond crystal. This work can provide guidance to the design work of crystal monochromators, crystal splitters, crystal compressors and XFELO cavities.

In this section, we first briefly review the dynamical theory of X-ray diffraction^[21^–23^]. Then, we review the electric susceptibility and Debye–Waller factor^[24^]. Finally, we establish a model to estimate the rocking curves of thermal load crystals by combining the Debye–Waller factor, electric susceptibility, thermal strain and slope error.

The dynamical theory of X-ray diffraction can accurately describe X-ray diffraction from a perfect crystal^[21^–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:

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

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

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

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

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

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

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

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

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

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

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:

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

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

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

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:

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

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

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

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

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

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:

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

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

In this section, we study the influence of the Debye–Waller factor and electric susceptibility on X-ray diffraction.

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^[25^]. The thermal expansion coefficient is extracted and fitted from Refs. [26,27]. The thermal conductivity is fitted from Kazan et al.’s work^[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).

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^[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.

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.

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. [15,16,18]. 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:

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

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^[30^], and the thermal conductivity of diamond crystal is extracted and fitted from Wei et al.’s work^[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.

In this section, a numerical validation of the proposed model is performed and compared with the model by Bushuev. Then, we applied the model to estimate X-ray diffraction performance in the dynamic pulse-to-pulse process.

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

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

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

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

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.

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.^[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.

This study investigates X-ray diffraction by a crystal under thermal load and develops a comprehensive model that not only considers several key factors, including thermal strain, incident angle deviation and slope error, but also includes the temperature dependence of the Debye–Waller factor and electric susceptibility. Our findings suggest that the influence of the Debye–Waller factor on efficiency degradation cannot be neglected, particularly in the case of high-order Miller index reflections at high temperatures. Although the Debye–Waller factor can potentially narrow the bandwidth of the rocking curve, the non-uniform thermal load still dominates the bandwidth broadening. The central photon energy shift is primarily attributed to thermal strain and slope error, while the shift induced by the temperature dependence of electric susceptibility can be neglected. The non-uniform thermal load causes variations in the local Debye–Waller factor, thermal strain and slope error, which further results in the distortion and degradation of the rocking curve. Proper cooling strategies, minimizing slope error and tuning the incident angle can effectively mitigate the effects of the thermal load.