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Abstract
X-ray free-electron lasers (XFELs) with megahertz repetition rates enable X-ray photon correlation spectroscopy (XPCS) studies of fast dynamics on microsecond and sub-microsecond time scales. Beam-induced sample heating is one of the central concerns in these studies, as the interval time is often insufficient for heat dissipation. Despite the great efforts devoted to this issue, few have evaluated the thermal effects of X-ray beam profiles. This work compares the effective dynamics of three common beam profiles using numerical methods. Results show that under the same fluence, the effective temperatures increase with the nonuniformity of the beam, such that the Gaussian beam profile yields a higher effective temperature than the donut-like and uniform profiles. Moreover, decreasing the beam sizes is found to reduce beam-induced thermal effects, in particular the effects of beam profiles.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
X-ray photon correlation spectroscopy (XPCS) is an emerging technique to probe nanoscale and microscale dynamics [1–4]. It has been applied to various sample systems including amorphous materials [5–8], colloidal dispersions [9–11], and strongly correlated materials [12,13]. XPCS is typically performed in the sequential mode, where the sample is continuously illuminated by a coherent X-ray beam to produce a series of speckle patterns at desired delays. The dynamic properties of the sample are then extracted from the patterns via correlation analysis and model fitting.
High repetition-rate X-ray free-electron lasers (XFELs) such as the European XFEL, LCLS-II, and SHINE can deliver XFEL pulses at megahertz (MHz) repetition rates, enabling XPCS studies on microsecond and sub-microsecond time scales. In such studies, the time interval is often insufficient for heat dissipation [14], such that beam-induced sample heating becomes one of the central concerns. Pioneer studies conducted by Lehmkühler et al., Dallari et al., and Reiser et al. have carefully evaluated the thermal effects of fluences and dose rates on MHz XPCS [15–17]. Meanwhile, although the beam profiles of XFEL pulses can vary [18–20], few have evaluated the thermal effects of beam profiles. It is possibly because of the difficulty in facilitating beams of different profiles and comparable fluences. However, as the thermal patterns of sample heating at MHz XFELs largely depend on beam profiles, they would presumably affect the effective sample dynamics.
In this work, we performed a numerical study to evaluate the thermal effects of beam profiles on XPCS at MHz XFELs, using parameters in a recent experiment performed at the European XFEL [15]. We first simulated the Brownian motion of particles dispersed in water, which were illuminated by MHz XFEL pulses of three common beam profiles. We then retrieved the dynamics via XPCS correlation analysis and model fitting. Results show that nonuniform beam profiles lead to faster effective dynamics, such that the Gaussian profile yields higher effective temperatures than the donut-like and uniform ones. Moreover, decreasing the beam sizes is found to reduce beam-induced thermal effects, in particular the effects of beam profiles.
2. Methods
2.1 Initial conditions
Brownian motion simulations were performed on spherical particles dispersed in water, following the recent experiment performed at the European XFEL [15]. Key parameters were almost identical to the ones therein, including the hydrodynamic radius (69 nm), the capillary diameter (0.7 mm), the capillary wall thickness (10 µm), the photon energy (9.3 keV), the maximum pulse number (120), the repetition rate (1.128 MHz), and the size of the Gaussian beam (4 µm). A transverse circular region of 8 µm diameter was taken as the probing region, which corresponds to 4.71σ acceptance of the Gaussian beam. Four fluences therein (H = 3.9, 10.5, 27.7, and 56.8 mJ/mm2) as well as H = 0 were simulated.
Initially, 400 particles were randomly distributed in the probing region (the dashed circle). Figure 1(a) shows the typical initial particle distribution, where the X- and Z-positions were separately generated using the numpy.random.random function of Python. X and Z are the two transverse directions perpendicular to the beam and orthogonal to each other. The environmental temperature was set to 293 K for all conditions. Three beam profiles were used, including the uniform profile (flat-top), the donut-like profile (TEM01*), and the Gaussian profile (TEM00).
Fig. 1. (a) The typical particle distribution in the transverse probing region of 8 µm diameter, as indicated by the red dashed circle. Every blue sphere denotes one particle. X and Z are the two transverse directions perpendicular to the beam. (b)(c)(d) The adiabatic temperature increases of water induced by each uniform, donut-like, and Gaussian pulse. The fluence H is 10.5 mJ/mm2. The color bar indicates the temperature increase in K.
2.2 Modeling beam-induced temperature increases
In this work, photon absorption by particles is ignored. Week-absorbing materials like water mainly dissipate the heat in the radial directions (transversely from the beam center outward). We thus integrated the heat in the beam direction (Y) and obtained the temperature increase on the XZ plane.
The effective fluences Heff at the sample depend on the original fluences H and the transmission of the beam through quartz and water, yielding
where Ts = 0.95 is the transmission of the front capillary wall, W is the capillary diameter, and L is the attenuation length of water at 9.3 keV.The adiabatic temperature increase on the XZ plane induced by each uniform pulse is shown in Fig. 1(b). It was calculated with
where the area term gets canceled, ρ is the water density, and cp is the specific heat capacity of water. The reduction factor 4 results from the fact that the probing region diameter is twice the beam size in the Ref. [15]. It ensures that the applied pulse energy is the same as the one of the same H therein.The adiabatic temperature increase on the XZ plane induced by each donut-like pulse is shown in Fig. 1(c). It was calculated with
The adiabatic temperature increase on the XZ plane induced by each Gaussian pulse is shown in Fig. 1(d). It was calculated with
The temperature increase after N pulses was obtained by multiplying the right sides of Eqs. (2) (3) and (4) by an accumulation factor $\sum\nolimits_{n = 0}^N {\{ 1 - \exp [\frac{{ - 1}}{{2(1 + n\Delta t/{t_0})}}]} \}$, where $\Delta t = \; 0.886\;\mathrm{\mu s}$ is the time interval and t0 is the thermal relaxation time. The t0 for Gaussian beams was adopted from the literature [15], whereas the ones for non-Gaussian beams were calculated according to the proportions of deposited pulse energies as shown in the first section of the Supplement.
2.3 Brownian motion simulation
Brownian motion simulations were performed in 2D on the XZ plane. The particles motion was driven by diffusion with
where D is the diffusion coefficient, kb is the Boltzmann constant, T is the solution temperature, and η is the T-dependent dynamic viscosity [21]. After each time interval Δt, every particle moved by two random normal steps, one in the X-direction and one in the Z-direction. The step sizes were generated independently for each particle and each direction with a standard deviation of $\sqrt {2D\Delta t} $ and a mean value of 0, using the numpy.random.randn function of Python. If some of the particles were to escape the probing region, the moving directions were reversed. As the particle movement was minimal compared with the size of the probing region, these events were rather rare, and the dynamics were almost unchanged by reversals.2.4 Correlation analysis and model fitting
In this section, intermediate scattering functions (ISFs) were obtained by autocorrelating the scattering amplitudes, and the dynamics parameters (the effective temperature Teff and the compression factor γ) were then obtained by fitting the ISFs.
First, the scattering amplitudes were calculated with
where the form factor of particles is ignored, Np = 400 is the total particle number in the probing region, q is the scattering vector, and ri is the position of the ith particle. A(ri) is the position-dependent amplitude factor, which equals unity for uniform pulses, $\sqrt {\exp [ - \frac{{{{({r_i} - \mu )}^2}}}{{2{\sigma ^2}}}]} $ for donut-like pulses, and $\sqrt {\exp ( - \frac{{{r_i}^2}}{{2{\sigma ^2}}})} $ for Gaussian pulses.Then, the ISFs were extracted from E(q) via [22]
For each condition, 30’000 simulations and autocorrelations were performed, yielding 30’000 ISFs of rather high statistic errors. They were first grouped into 3 datasets (3 × 10’000). Then, the ISFs of each dataset were averaged to yield 3 statistically reliable ISFs. They were all fitted by the Kohlrausch–Williams–Watts (KWW) function where Γ(q) is the relaxation rate and γ is the compression factor. For diffusion, Γ(q) depends on the effective temperature Teff, with Finally, Teff and γ were obtained by averaging the fitting results of 3 ISFs. All Teff were normalized, such that the uniform Teff of H = 0 equals 293 K.3. Results and discussion
3.1 Intermediate scattering functions and fits
Figure 2 shows the simulated intermediate scattering functions (symbols) and fitting curves (solid lines) of all three beam profiles at different fluences H. Firstly, at H = 0, the ISFs of different beam profiles almost overlap. Despite different statistic weights applied by nonuniform profiles to particles at different positions, no differences are observable when no heating takes place. It validates our simulation protocol. Secondly, at H > 0, ISFs of different profiles (colored symbols and lines) separate from each other. At the same H, the ISFs of donut-like beams decay faster than the ones of uniform beams and slower than the ones of Gaussian beams. It confirms our presumption about the thermal effects of beam profiles. As the heat accumulates, nonuniform beams apply higher statistic weights to particles in the hotter regions (see Fig. 1(b) and Fig. 1(c)). The higher the nonuniformity is, the faster the decays of the ISFs become. Moreover, the ISFs of H = 27.7 mJ/mm2 decay slightly slower than the ones in the reference because of the neglect of photon absorption by particles [15]. Below, we will further analyze the results by inspecting the effective temperatures and the compression factors obtained from model fitting.
Fig. 2. Simulated intermediate scattering functions (symbols) and their fitting curves (solid lines) at q = 0.125 nm-1. The fluences and beam profiles are shown in the legend.
Figure 3(a) shows the effective temperatures Teff as a function of the fluences H. At H = 0, Teff of different beam profiles are almost the same, in line with the overlapping ISFs above. At H > 0, the Gaussian profile always yields the highest Teff, whereas the uniform profile always yields the lowest Teff, again indicating the significant effects of beam nonuniformity on the effective dynamics. Moreover, all curves are highly linear, suggesting that increasing H only enhances these effects. As expected, our simulations result in lower Teff than the experimental ones because of the neglect of photon absorption by particles [15]. Figure 3(b) shows the compression factors γ plotted as a function of H. At H = 0, all γ are close to unity. At H > 0, γ decrease with the increase of H because of the nonequilibrium induced by continuous heating and averaging over pulses [15], which is confirmed by a pulse-dependent analysis shown in Fig. S1 of the Supplement. More importantly, the decrease of γ is more profound for the nonuniform profiles, again suggesting strong effects of beam profiles on the effective dynamics. All Teff and γ are summarized in Table S1 of the Supplement.
Fig. 3. (a) Effective temperatures at q = 0.125 nm-1 obtained from KWW model fitting plotted as a function of fluences. Dashed lines serve as visual indicators. All temperatures are normalized such that the uniform ones equal 293 K. All standard deviations are less than 0.10 K. (b) Compression factors at q = 0.125 nm-1 obtained from KWW model fitting plotted as a function of fluences. Dashed lines serve as visual indicators. All standard deviations are less than or equal to 0.01.
The above results confirm our presumption about the thermal effects of beam profiles. Nonuniform profiles apply different local fluences to different regions, such that the statistical weights of particles in hotter regions are higher in the ISFs. Therefore, results of nonuniform beams show faster dynamics and higher degrees of nonequilibrium. The dynamics parameters Teff and γ scale not only with H but also with the nonuniformity of the beams. The Gaussian profile brings about the highest Teff and the lowest γ compared with the uniform and donut-like profiles.
3.2 Γ-q2 dependence
Figure 4 shows the q2-dependence of the relaxation rates Γ for all beam profiles at different fluences H. Firstly and expectedly, Γ increase with H owing to stronger heating. They are slightly smaller than the experimental ones of the same H because of the neglect of photon absorption by particles [15]. In addition, they also increase with the nonuniformity of the beam profiles, such that the Gaussian ones are always the largest at the same H. These results agree with the findings above that the effective temperatures increase with H and the beam nonuniformity. Moreover, small deviations from Eq. (9) appear at small and large q for all fluences including H = 0. The result of H = 0 suggests that these deviations stem neither from heating-induced nonequilibrium nor from averaging over pulses. Further simulations demonstrate that they result from the spread of particle radial positions on the XZ plane (see Fig. S2 in the Supplement). Similar deviations were also observed experimentally [15]. However, the experimental Γ of H = 56.8 mJ/mm2 therein deviates significantly from Eq. (9). It could be attributed to the high degree of nonequilibrium induced by the strong photon absorption of particles, which is not included here. The numerical studies in this work are thus more instructive for soft-matter particle dispersions, where the photon absorption by particles is on the same level as water.
Fig. 4. The q2-dependence of the relaxation rates Γ for all beam profiles at different fluences H. The fluences and beam profiles are listed in the legend. Dashed lines are the Γ = Dq2 fits using the effective temperatures of q = 0.125 nm-1 (q2 = 0.016 nm-2). The highest and lowest effective temperatures are given in black. Standard deviations of all data points are less than 100 s-1.
3.3 Thermal effects at different beam sizes
The above simulations and analysis were performed at the Gaussian beam size of d = 4 µm, where beam nonuniformity induces faster dynamics and higher degrees of nonequilibrium. It would be interesting to see how they change with the beam sizes. To this end, the Gaussian beam size d was varied. The other sizes were also varied proportionally to d. The thermal relaxation times of different beam sizes and profiles were calculated using the value and the dependence in the Refs. [15,23]. The pulse energies were kept constant, which is the common situation in experiments.
Figure 5 shows the effective temperatures Teff of different beam profiles at H = 57.7 mJ/mm2 plotted as a function of d. In the tested d range, Teff increase with d for all beam profiles. The increase is significant at smaller d but minimal at larger d. It indicates that the thermal effects induced by photon absorption could be reduced by using smaller beams. Regarding different beam profiles, the finding that nonuniform beams yield higher Teff remains valid. Teff of the Gaussian beams are always the highest, whereas Teff of the uniform beams are always the lowest. However, for smaller beams, the differences in Teff of different beam profiles are also smaller. For $d = \; 8\;\mathrm{\mu m}$, the differences are about twice as large as those of $d = \; 4\;\mathrm{\mu m}$. For $d = 1\;\mathrm{\mu m}$, the differences are almost negligible. Therefore, decreasing the beam size could be a practical strategy for MHz XPCS experiments to reduce beam-induced thermal effects, in particular the effects of beam profiles.
Fig. 5. The effective temperatures of different beam profiles plotted as a function of the Gaussian beam size d. All the other sizes are proportional to d. Simulations and analysis were performed at q = 0.125 nm-1. When varying d, the pulse energies were kept constant using the value of H = 57.7 mJ/mm2 in [15]. Dashed lines serve as visual indicators. The thermal relaxation time was calculated using the one in [15] and its dependence on d in [23]. All standard deviations are smaller than 0.15 K. Corresponding compression factors are between 0.89 and 0.96.
4. Conclusion
XPCS at high repetition-rate XFELs such as the European XFEL offers unique capabilities and is thus under rapid development. However, critical issues such as the heating problem must first be better understood. The motivation of this work is to assess the thermal effects of different beam profiles, which can be altered by optics. The heat dissipation of water is relatively slow at high repetition rates. For the uniform beam, all particles are subject to nearly the same temperature, such that the statistic weights are equal for all particles. For nonuniform beams, however, heating is stronger in the regions with higher local fluences, such that ‘hotter’ particles obtain higher statistic weights in the correlation functions. Therefore, nonuniform beams could presumably facilitate faster effective dynamics. To verify this presumption, we performed Brownian motion simulations of particle dispersions under the illumination of differently-shaped beams at several fluences, where photon absorption by particles was neglected. Particle dynamics was revealed by correlation analysis and model fitting.
Results show that the effective temperatures of nonuniform beam profiles are indeed significantly higher than the uniform ones. Further analyses suggest that the dynamics of all beam profiles follows the Γ-q2 dependence. Additional studies of various beam sizes show that decreasing the beam size reduces the thermal effects induced by photon absorption of the beam profiles. Moreover, it diminishes the differences in the effective dynamics of different beam profiles.
For most of the simulations in this work, the differences in the effective dynamics of different profiles are only significant at high H, where radiation damage and particle heating are of greater concern. The thermal effects of nonuniform profiles would be critical only if significant differences occur at low H. Solutions with higher absorption or higher viscosity-temperature dependence are likely to facilitate such significant differences. Should proper thermal relaxation time be available, similar simulations will be performed for these two scenarios as well.
The numerical studies in this work cannot fully describe experimental conditions. The main limitation of this work is the neglect of photon absorption by particles. It depends on many factors such as absorbance, heat capacity, particle geometry, and convective heat transfer between particles and solvent [15,16]. While considering the photon absorption by particles would better describe the experiments, it would also include a great amount of complexity. In this first study, we have focused on the thermal effects of photon absorption by the solvent. Another limitation of our numerical studies is that heat mainly dissipates in the radial directions, such that the temperature distribution of the uniform profile is not entirely uniform. Nonetheless, this work offers a glimpse into the thermal effects of beam profiles on XPCS at high repetition-rate XFELs. Moreover, similar simulations could also be performed for XPCS studies at 4th generation synchrotrons. As they provide highly brilliant and coherent X-ray beams, they are also attractive sources for XPCS studies on short time scales. The average brilliance of 4th generation synchrotrons is only a few orders of magnitude lower than MHz XFELs [24], such that the heating problem is also worth noting. This work investigates the thermal effects of three common beam profiles. The findings could help select proper optics and beam profiles to reduce beam-induced sample heating in high repetition-rate XPCS experiments at these two highly brilliant X-ray facilities.
Funding
Natural Science Foundation of Shanghai (22ZR1442100).
Acknowledgments
The authors thank Dr. Felix Lehmkühler, Dr. Yajun Tong, Dr. Fan Yang, and Dr. Menglu Hu for their helpful discussions. This work was sponsored by Natural Science Foundation of Shanghai (Grant No. 22ZR1442100).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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