Angular divergence-bandwidth coupling and DuMond diagram of a monochromatic x-ray

According to X-ray dynamical diffraction theory [33], a coupling relationship exists between the beam divergence and bandwidth. Even when the incident white beam is perfectly parallel, the diffracted beam exhibits a certain bandwidth. Similarly, even when the incident divergent beam is perfectly monochromatic, the diffraction beam has an angular divergence limited by the Darwin width. For an SR beam with vertical and horizontal angular divergences, the beam’s cross-section from the DCM forms a rectangle with a specific energy bandwidth. For example, for a crystal placed horizontally in DCM, photons of different energies and angles do not have a specific distribution in the horizontal direction, and the photon distribution in the vertical direction is characterized by the top region comprising lower-energy photons with high diffraction angles and the bottom region comprising higher-energy photons with low diffraction angles. However, because of the dynamic diffraction theory, the exact values of the two coupling quantities, energy, and angle of the photon cannot be quantitatively determined simultaneously. This phenomenon impedes further improvement in measurement accuracy with rocking curves, typically obtained in the presence of divergence-bandwidth coupling.

To decouple the angular divergence and beam bandwidth, a DuMond diagram is used to replace the direct beam after the monochromator [29, 32]. The DuMond diagram was transformed from phase space to real space using a new experimental setup containing an analyzer orthogonal to the DCM and a 2D detector. A DuMond diagram can be imaged directly on a detector by diffracting the beam after DCM horizontally, resulting in a specific location distribution of photons with different energies and horizontal divergence angles [27]. In the rectangular cross-section of the beam, the vertical axis allows accurate determination of the vertical divergence angle, and the horizontal axis is used to determine the energy accurately. After decoupling the divergence bandwidth, a DuMond diagram imaged in real space can characterize a monochromator with high precision.

The diffraction lattice planes and probe energies were considered. The DuMond diagram is based on the relationship between the diffraction angle θ and wavelength λ in the Bragg formula for X-rays. At equal wavelengths, higher indices, such as 333 and 555, had narrower Darwin widths and gentler slopes than the lower index of 111. For example, for the diffractions of Si(111), (333), and (555) with energies of 18 keV, the Darwin widths were ${\omega }_{\text{D}}^{111}=2.95^{″}$, ${\omega }_{\text{D}}^{333}=0.64^{″}$, and ${\omega }_{\text{D}}^{555}=0.25^{″}$, respectively. This implies that a higher diffraction index can achieve better energy and angular resolution but at the expense of the beam intensity. DCM typically has the lowest index of 111 to achieve a higher beam flux. However, the resolution of 111 diffraction is insufficient for measuring the thermal deformation of monochromator crystals, which requires a resolution of less than 1″.

Considering the integrated aspects of the experimental implementation, detection, and precision, we chose an index of 333 and an energy of 18 keV. The 333 diffraction has an energy resolution of 1/14.5 of the 111 diffraction, which can improve the resolution by more than one order of magnitude. Because synchrotron radiation has a broad spectrum, the λ and λ/3 components corresponding to the indices 111 and 333 exist simultaneously in the DCM beam. The measurements were obtained with 333 diffractions at 18 keV, whereas the monochromator crystal Si(111) was operated at 6 keV.

DuMond diagram of an ideal working DCM

DCM with two identical Si crystals is commonly used in synchrotron radiation beamlines. The first crystal diffracts the incident white beam at a certain wavelength and bandwidth to the second crystal. The second crystal diffracts the monochromatic beam to maintain the original direction of the incoming beam, which is insusceptible to diffraction angle variation. Ideally, the two perfect crystals with equal indices are flat and symmetric, and the DuMond diagrams of the DCM are parallelograms (whether detuned or not).

In the next section, we briefly describe the detuning phenomenon of two flat crystals to distinguish it from the thermal deformation case. The intensity of the beam exiting the DCM was maximized when the diffraction planes of the two crystals were perfectly parallel. Any deviation from the parallel state is known as detuning and is quantified by the deviation angle from the parallel position, the detuning angle. Assuming a slight detuning angle α between the two crystals without thermal deformation, the DCM beam intensity at a particular energy E is proportional to the following factors [34]: $I\left(E\mathrm{,}\alpha \right)\propto N\left(E\right)R\left(\alpha \right)\cot {\theta }_{\text{B}}\sqrt{{\text{Ω}}^2+{\omega }^2\left(\alpha \right)}$ (1) where N(E) is the incident beam intensity at spectral energy E with a bandwidth of Δ E, θ_B is the Bragg angle, R(α) is the total reflectivity of the two crystals at a certain detuning angle α, Ω is the divergence angle of the incident beam (which is related to the synchrotron radiation source and opening of the front slit), and ω(α) is the acceptance angle width of the double-crystal diffraction. The rocking curve of the first crystal is identical to that of the second $R_1\left(\theta \right)=R_2\left(\theta \right)$. Therefore, R(α) is only a function of the detuning angle α: $R\left(\alpha \right)={\displaystyle {\int }_{-1/2\gamma }^{1/2\gamma }}{R}_1\left(\theta -\alpha \right)R_2\left(\theta \right)\text{d}\theta .$ (2) The integration interval of the variate θ is the angular divergence of the synchrotron radiation source, which is generally larger than the Darwin width of the crystal diffraction. The acceptance angle ω(α) of the double-crystal diffraction is given by Eq. (3): $\frac{1}{{\omega }^2\left(\alpha \right)}=\frac{1}{{\left({\omega }_{\text{s}}-\alpha \right)}^2}+\frac{1}{{\omega }_{\text{s}}^2}.$ (3) Here, ωs is the common intrinsic angular width (Darwin width) of the first and second crystals (Figs. 1(a)). The relative intensity of the exit beam of the DCM at the detuning angle α is given by $\frac{I\left(\alpha \right)}{I\left(0\right)}=\frac{R\left(\alpha \right)\sqrt{{\text{Ω}}^2+{\omega }^2\left(\alpha \right)}}{R\left(0\right)\sqrt{{\text{Ω}}^2+{\omega }^2\left(0\right)}}.$ (4) In this study, the 333 diffraction has a Darwin width of 0.64″ at 18 keV, and the incidence angle Ω is generally tens or tens of arcseconds.

Fig. 1Full-screen View

(Color online) (a) DuMond diagrams of DCM at different detuning angles. For two perfect flat crystals without deformation, the shape of the overlap of their DuMond diagrams is always a parallelogram. (b) Variation in the overlap degree of two crystal rocking curves at different detuning angles. The area where the rocking curves of the two crystals overlap represents the beam flux. (c) Relative beam intensity is calculated by the overlapping areas (The parameters are energy of 18 keV and diffraction index of 333). The ordinate of beam intensity ratio is normalized by the overlapping area at the detuning angle α=0″

In addition to the total beam intensity, geometrically representing the diffraction process using DuMond diagrams is more intuitive. We began with a condition without thermal deformation, as illustrated in Fig. 1(a), which shows a DuMond diagram of an ideal DCM. The two vertical black dotted lines indicate the angular divergence range of the incident synchrotron radiation (the divergence angle of the bending magnet light source was approximately 1/γ). The red and black oblique lines indicate the dynamic diffraction bands (DuMond diagram) of the first and second crystals, respectively, and α is the detuning angle between the two crystals. The colored area is the overlapping part of the DuMond diagrams of the two crystals, the DuMond diagram of DCM. It depicts how the diffraction beam varies with the detuning angle: as the detuning angle α increases, the horizontal width of the DuMond diagram narrows, and the photon intensity decreases. Figure 1(b) shows the θ-I sectional profile of the DuMond diagram (with the intensity axis I perpendicular to the surface of the paper), that is, the rocking curves of the first and second crystals. Here, the colored area represents the overlapping part of the two rocking curves corresponding to the beam intensity. α is the detuning angle between the two crystals, which varies as the rocking curve of the first crystal translates, whereas the rocking curve of the second crystal remains fixed. Figure 1(c) shows the relationship between the beam intensity and detuning angle α, which is obtained from the overlapping areas of the two rocking curves at different detuning angles in Fig. 1(b). When the two crystals were parallel (α=0), their rocking curves completely overlapped when the beam intensity reached its maximum value. As the absolute value of the detuning angle α increased, the overlapping area of the two rocking curves and the beam intensity decreased.

Figure 1 illustrates the primary principles of the measurement method reported in this study. It is essential to emphasize that although this may appear similar to scanning rocking curves, there are distinctions between the two methods owing to the experiments. Compared with scanning rocking curve, the θ-I-sectional profile of the DuMond diagram was obtained by removing the divergence-bandwidth coupled photons from the beam. In addition, the latter is directly obtained in one image without rotating the monochromator; therefore, the incident power density and thermal deformation do not vary.

DuMond diagram of a DCM with deformed crystal

Under ideal conditions without thermal deformation, regardless of the detuning angle, the lattice planes of the two crystals were flat. In real operation, the first crystal of the DCM is irradiated with a high-power white beam, which causes lattice deformation and expansion. The beam incident on the second crystal is already monochromatic; thus, its heat power is negligible [7]. The deformation can be interpreted as follows. The crystal surface is divided into numerous flat minute-area cells along the beam footprint, with one microfacet of the first crystal and one microfacet of the second crystal constituting a microfacet pair. The total intensity of the beam exiting the DCM is the sum of the intensities of all microfacet pairs. The detuning angles α are identical for every microfacet pair without thermal deformation. When the first crystal surface undergoes thermal lattice deformation, the normals of microfacets at different positions x on the first crystal are tilted by different angles Δα, whereas the normal direction of each microfacet on the second crystal remains constant. The detuning angle between a pair of microfacets is $\alpha ={\alpha }_0+\text{Δ}\alpha$, where α₀ is the macroscopic detuning angle between the two crystals before the thermal deformation.

Then Eq. (2), and Eq. (3) can be rewritten as follows: $\begin{array}{l}R\left({\alpha }_0\right)={\displaystyle {\int }_{\text{Δ}{\alpha }_1}^{\text{Δ}{\alpha }_2}}{\displaystyle {\int }_{-1/2\gamma }^{1/2\gamma }}{R}_1\left(\theta -{\alpha }_0-\text{Δ}\alpha \right)R_2\left(\theta \right)\\ \text{d}\theta \text{d}\text{Δ}\alpha ,\end{array}$ (5) $\frac{1}{{\omega }^2\left({\alpha }_0+\text{Δ}\alpha \right)}=\frac{1}{{\left({\omega }_{\text{s}}-{\alpha }_0-\text{Δ}\alpha \right)}^2}+\frac{1}{{\omega }_{\text{s}}^2}.$ (6) The integration interval of θ is the angular divergence of the synchrotron radiation source, and the integration interval of Δα is the range of normal tilt angles of microfacets within the light footprint on the crystal surface.

As shown in Fig. 1(a), DuMond diagrams of ideal DCM are always parallelograms. The sequence of parallelogram DuMond diagrams shows the detuning phenomenon on the timescale introduced by rotating the adjustable crystal. This is significantly different from the distorted DuMond diagrams that show the spatial distribution of the detuning phenomenon induced by thermal deformation. Similarly, Fig. 2(a) shows DuMond diagrams representing the diffraction process of a deformed crystal. For a point light source, the diffraction-angle range of the flat crystal is limited only by the angular divergence of the light source, as shown in the upper right diagram in Fig. 2(a). However, the diffraction angle range of the deformed crystal is bounded by both the source angular divergence and angular deformation of the crystal surface, as shown in the upper left diagram of Fig. 2(a). It is necessary to consider the physical meaning when two such crystals comprise a DCM. To unify the angle ranges of the two DuMond diagrams, the transverse axis of the actual diffraction angle θ is replaced by the light-source divergence angle θ′, as illustrated in the lower diagrams of Fig. 2(a). Beams with different divergence angles were incident on the microfacets of the first crystal, which had different local deformation angles Δα. These deformation angles change the actual diffraction angle of the first crystal and introduce local offsets into the DuMond diagram. The DuMond diagram of the DCM with a macroscopic detuning angle α₀=0 is shown on the lower left and the DuMond diagram of the DCM with a small macroscopic detuning angle α₀is shown on the lower right. Comparing to Fig. 1(a), the DuMond diagram of the DCM with a deformed crystal undergoes a significant distortion in shape and shifts its position as the macroscopic detuning angle of the DCM varies.

Fig. 2Full-screen View

(Color online) (a) Schematic diagram illustrating DuMond diagrams of the DCM with a deformed first crystal and a flat second crystal. The transverse coordinates in the upper two figures represent the actual diffraction angle, while transverse coordinates in the lower two figures depict the divergence angle of the light source. (b) Schematic diagram of DuMond diagrams obtained in real space. In the upper case of two flat crystals, the only effect caused by the different α₀ is the different widths of the parallelograms, and their normalized integral intensity curves remain the same

The experimental configuration allowed us to directly observe the DuMond diagrams of the monochromator under the operational conditions. As shown in Fig. 2(b), the upper figure illustrates a DCM crystal without thermal deformation, whereas the lower figure depicts a DCM crystal with thermal deformation. The beam underwent sequential longitudinal diffraction by the two crystals and subsequent horizontal diffraction by a downstream analyzer (omitted in the figure). The resulting image on the 2D detector was a divergence-bandwidth-decoupled light spot corresponding to the DuMond diagram of the DCM. The ordinate represents the vertical divergent angle of the beam exiting the DCM, the abscissa represents the energy distribution of the beam decoupled by the analyzer crystal, and the color shades represent the photon intensity. Without thermal deformation, the DuMond diagram appears as a parallelogram of uniform color. With thermal deformation, the DuMond diagram assumed a willow-leaf shape with an uneven color distribution. The distortion of the DuMond diagram can be quantified by the intensity loss, as illustrated by the rightmost integral intensity curve in Fig. 2(b). After eliminating the energy effects through horizontal integration, only the two-dimensional correspondence between the longitudinal divergence angle and the intensity was retained. The rightmost integration curve corresponds to the intensities of microfacet pairs at different longitudinal positions on the crystal surface. For a certain α₀, the output intensity of a pair of microfacets is directly related to the deformation angle Δα. The central width of the willow-leaf-like DuMond diagram was comparable to that of the parallelogram DuMond diagram and progressively diminished as the position moved away from the central region. Meanwhile, both ends of the rightmost integral intensity curve show a decline.

Experimental configuration, instruments, and parameters

An energy of 6 keV was selected as the operating energy of DCM Si(111), where the Bragg angle was 19.24, and the energy of the Si 333 diffraction was 18 keV. The experimental setup is illustrated in Fig. 3. With the light source as the origin point, the white light slit, DCM, and beryllium window were located at locations of 18.2 m, 21 m, and 38 m, respectively. The analyzer crystal Si(333) was at a distance of 39 m from the light source and was orthogonal to the crystals inside the DCM. The 2D detector is at a distance of 0.5 m from the analyzer crystal, and the dynamic range of the detector is 12 bits. The white beam is radiated to the DCM, and the monochromatic beam after the DCM is diffracted horizontally and decoupled by the analyzer Si(333), removing the energy-angle coupled photons that cannot satisfy both the energy and angle criteria. The initial beam with a rectangular cross-section was transformed into a tilted DuMond spot on the 2D detector monitor. The tilt angle, fixed at 45, is consistent because of the fixed and identical index planes for the horizontal and vertical diffraction.

Fig. 3Full-screen View

(Color online) Schematic diagram of experimental setup. After being diffracted by the analyzer placed vertically, the original rectangular output beam of the DCM manifests as a light spot inclined at an angle of 45° on the imaging detector, representing the DuMond diagram of the DCM

The opening size of the white light slit is 3.15 mm × 1.75 mm (H × V). The horizontal and vertical dimensions of the incident white beam arriving at the front of the DCM are 3.63 mm and 2.02 mm, respectively. The first crystal had dimensions of 50 mm × 70 mm × 15 mm (L × W × H) and was indirectly water-cooled at a temperature of 25 ℃ at a flow rate of 0.9 L/min. After exiting the beryllium window and traveling 1.5 m in the atmosphere, the 6 keV component of the beam decayed to 1.7% of its initial intensity, whereas the 18 keV component remains at 85.7% of its initial intensity. Therefore, it can be concluded that the photons arriving at the 2D detector were 18 keV reflected by the 333.

In this study, a series of DuMond diagrams were obtained under different ring currents using the above experimental configuration. The correspondence between the ring current and the power density of the light source and that of the first crystal surface is shown in Table 1. The calculation was performed using Spectral [37].

Experimental procedure and results

To compare and demonstrate the distortion of the DuMond diagram, we observed DuMond diagrams at a low storage ring current (9.9 mA) measured with 333 diffraction and 18 keV and DuMond diagrams at a high storage ring current (220.3 mA) measured with 111 diffraction and 6 keV. The DCM of BL09B is equipped with a precision rotation mechanism consisting of a motor and piezoelectric device on the first crystal, which can alter the macroscopic detuning angle α₀ between the two crystals by rotating the first crystal.

Figure 4 shows the DuMond diagrams of the DCM decoupled by an orthogonal analyzer, sampled at an exposure time of 500 ms. The ordinate and abscissa represent the original vertical angle distribution and the wavelength of the beam from the DCM, respectively. Both axes are expressed in detector pixels, and the brightness of the spot indicates the beam intensity. Fig. 4(a) and 4(b) show the measurement results for indices 111 and 333, respectively, at different macroscopic detuning angles α₀. Figure 4(a) and 4(b) show that when the detuning angle α₀=0, the DuMond diagram has the maximum intensity and full width at half maximum (FWHM), regardless of whether the indexes are 111 or 333. The light spot intensity and transverse size decreased as the detuning angle increased. It is also worth emphasizing that all the measured spots had the basic shape of a parallelogram in either case. This result can be interpreted as follows: at a low current, the heat load on the crystal is marginal, and the detected thermal deformation is not significant, even if we used a Si 333 analyzer. At a high current, the heat load on the crystal was high, and the thermal deformation on the crystal surface was significant. However, the basic shape of the observed DuMond diagram remains a parallelogram owing to the low sensitivity of the diffraction index 111.

Fig. 4Full-screen View

(a) DuMond diagrams at various macroscopic detuning angles α₀, measured with the low sensitivity index 111 and 6 keV under high heat load (220.3 mA). (b) DuMond diagrams at various macroscopic detuning angles α₀, measured with the high sensitivity index 333 and 18 keV under low heat load (9.9 mA). (c) DuMond diagrams at various macroscopic detuning angles α₀, measured with the high sensitivity index 333 and 18 keV under high heat load (220.3 mA). (d) DuMond diagrams at different storage ring currents with a fixed macroscopic detuning angle α₀, measured with the index of 333 and 18 keV

To verify whether the willow-leaf-like distortion of the DuMond diagram was due to thermal lattice deformation, we maintained a higher ring current of 220.3 mA and measured DuMond diagrams with 333 diffractions. Figure 4(c) shows the willow-leaf-like DuMond diagrams at different macroscopic detuning angles α₀, which were altered by rotating the first crystal forward and reverse. The brightest position in the spot corresponds to the minimum detuning angle $\alpha =\text{Δ}\alpha +{\alpha }_0=0$. As the first crystal rotates, the brightest position in the spot gradually shifts from the lower-right corner, with a low diffraction angle and high energy, to the upper-left corner, with a high diffraction angle and low energy. However, irrespective of how the crystal was rotated, the DuMond diagrams did not form parallelograms with uniform brightness and width at all positions, as shown in Fig. 4(a) and 4(b). This is consistent with the detuning phenomenon of DCM with a deformed crystal, as shown in Fig. 2(a), indicating the distribution of local deformation angles Δα(x) between the two crystals.

Figure 4(d) shows the DuMond diagrams obtained when DCM was operated under different storage ring currents. The beam was held in each state for a period to ensure that all samples were in an equivalent operating condition, except for the ring current. As the ring current increased to 85.7 mA, the DuMond diagram showed a distinct nonuniform intensity distribution: the spot intensity was higher in the middle and lower at both ends in both the energy and angle directions. As the ring current increases to 148.2 mA, the DuMond diagram transforms from a uniform parallelogram to a willow-leaf shape with narrow ends. This suggests that the spatial detuning phenomenon characterized by the local deformation angle Δα(x) is a consequence of the heat load on the DCM. In addition, the presence of local light or dark points in the figure can be attributed to crystal defects. These points do not vary with increasing current but can be eliminated by varying the position of the crystal surface receiving light during the measurement.