Bragg coherent diffractive imaging is a powerful technique that can be used to explore the internal structure and strain of nanoscale crystalline objects. During the data collection process, the Bragg peak typically stays within a small range of pixels on the x-ray sensitive area detector. Here, we report abrupt and irreversible Bragg peak movement during the coherent x-ray data collection process for both Pd nanocubes and a Ni nanowire. We report that this phenomenon can be attributed to x-ray momentum transfer, also known as radiation pressure, to the nanocrystals. Understanding this effect is crucial given the anticipated coherent flux increases at next-generation synchrotron sources.
Third-generation synchrotron facilities provide high brilliance x-ray beams with a very high degree of coherence, allowing researchers to make significant advances in the study of nanostructures. Bragg coherent x-ray diffraction, combined with computational phase retrieval techniques, has proven useful for understanding the structure of nanocrystalline materials and mapping strain in single particles during a variety of interesting processes.1–7 These types of characterizations are possible because during exposure to monochromatic x-ray radiation, there is a direct relationship between the object's shape and strain, and the diffraction intensity measured in the far field.
Three-dimensional diffraction patterns are formed from an appropriately collected set of 2D diffraction patterns in a process known as a rocking scan. During a rocking scan, the sample is rotated over a very small angular range (typically less than 0.5°) relative to the incident x-ray beam. Inversion of the 3D diffraction pattern, after appropriate phase retrieval,8,9 typically results in a complex-valued density function in real space. The amplitude of this complex function is proportional to the Bragg electron density in the crystal, while the phase can be interpreted as the deformation of the lattice projected onto the momentum transfer vector.10,11 This technique relies on the fact that the x-ray beam is non-destructive so that both the sample location and its properties are not altered during the measurement. Some biological samples and protein crystals, however, can be damaged or even destroyed by an intense x-ray beam.12,13 This radiation damage has been acknowledged as an issue in x-ray science, resulting in rigorous studies of the minimum dose required to produce an image with a given resolution.14–16 Typically, inorganic materials have higher radiation tolerance and are largely immune to damage induced by synchrotron x-ray radiation.
However, it occasionally occurs that the Bragg peak shifts irreversibly during a rocking scan while performing Bragg coherent x-ray diffraction on single nanocrystals.17,18 We do not observe this effect in nanocrystals that have a large area in contact with the substrate relative to their height.19 The unusual Bragg peak movement was obtained during data collection for Pd nanocrystals.20 In addition, we carried out a rocking scan with a Ni nanowire attached to a substrate21 and analyzed the behavior of the sample under these circumstances. It is necessary to understand and quantify this phenomenon to determine appropriate experimental and computational corrections for Bragg coherent x-ray diffractive imaging, especially in light of the anticipated increases in coherent flux at major synchrotron sources. Here, we report the investigation of the radiation pressure effect on both Pd nanocubes and Ni nanowires to demonstrate the effect across a wide range of samples.
Figure 1(a) is a schematic image of the experimental setup for a Bragg coherent x-ray diffraction experiment. The measurements were performed at beamline 34-ID-C of the Advanced Photon Source at Argonne National Laboratory, with an x-ray energy of 9 keV. Figure 1(b) shows a series of measured 2D diffraction patterns of a single-crystal Pd nanocube during a rocking scan. The form of the 2D diffraction pattern changes generally during the rocking scan and is determined by the intersection between the area detector and the 3D intensity distribution. In addition, the diffraction intensity drops to the level of noise at the end of the rocking scan. The isosurface rendering of the 3D diffraction pattern, formed by stacking up the 2D diffraction data collected during rocking scans, is shown in Fig. 1(c). The intensity should have a maximum value at the Bragg peak center. The fringes, seen as spatial modulations in the diffraction pattern, originate from the interference between waves diffracted from pairs of sharply terminated crystalline facets. So, three pairs of fringe modulations imply the presence of six facets.
In contrast to the expected results shown in Fig. 1, we have observed a shift of the Bragg peak during rocking scans for a Pd nanocube, in addition to another Pd nanocube (see supplementary material, Figure 1). This phenomenon is defined as an abnormal rocking scan in this report. Figures 2(a) and 2(b) show the isosurface of the 3D diffraction patterns for the Pd nanocubes. As previously discussed, Fig. 2(a) shows a 3D diffraction pattern of a Pd nanocube, obtained by a normal rocking scan because the fringes are clearly seen and the diffraction pattern is nearly centro-symmetric. However, the 3D diffraction pattern in Fig. 2(b) is very unusual. Figures 2(c) and 2(d) show several contours extracted from the corresponding 2D diffraction patterns. In a normal rocking scan, the Bragg peak splits in certain directions and moves away from the center as shown in Fig. 2(c). In contrast, for Fig. 2(d), it is observed that the center of the Bragg peak starts to shift along a particular direction in the middle of the rocking scan. To analyze the Bragg peak movement, the center of the mass of the intensity at each 2D diffraction pattern was extracted. The position of the center of mass relative to the initial frame was plotted as a function of rotation θ in Fig. 2(e). In addition, the curves at 2D diffraction patterns of the abnormal rocking scan were fitted to a Gaussian and the peak positions were extracted. These quantities are plotted versus rotation angle θ in Fig. 2(f).
In the abnormal case, after some initial variation, there is a roughly linear increase in the Bragg peak shift after a certain angle, which is denoted as . The Bragg peak shifts by approximately 0.2 nm−1 shift in reciprocal space, which is much larger than the normal variation, in the 0.04 nm−1 range shown in Fig. 2(e). This indicates that the Pd nanocube moved during the rocking scan. However, due to the degrees of freedom (translation plus rotation) for a particle resting on but not adhered to a substrate, it is hard to determine what specific motion caused the diffraction peak change. When we carried out a rocking scan with a Ni nanowire, which is attached to a Si substrate,22 we found the signature for the motion of the sample.
While collecting the data from the Ni nanowire, it was observed that the Bragg peak moved in the middle of rocking scan as shown in Fig. 3(a), which is similar to Fig. 2(b). A series of contours show the movement of the Bragg peak clearly in Fig. 3(b). 2D diffraction patterns seen on a detector are cross sections of 3D diffraction patterns and generally very sensitive to the rotation θ as shown in Fig. 1(b). However, the 2D diffraction patterns of the Ni nanowire on the Debye-Scherrer (D-S) ring are constantly the same during the second half of rocking scan as shown in Figs. 3(b) and 3(c). The normalized cross correlation of 2D diffraction intensities in Fig. 3(c) represents the similarity of the diffraction pattern with the one measured at θ = 0°. During the second half, the plateau indicates that the shapes of 2D diffraction patterns are constant. The blue dots are the reflected image of the first half of rocking scan, which should be seen in a normal rocking scan.
RESULTS AND DISCUSSION
We studied the motion of the sample based on the movement of the 3D diffraction pattern. The Ni nanowire has a 4 μm height and a 200 nm width and the base is tapered (see supplementary material, Figure 3) so that the bottom portion can be a pivot point for the 3D motion.23 Even if the external force is applied on the Ni nanowire, the deformation is concentrated on the taper so that the Ni nanowire can be considered as a rigid body, except for the bottom portion. In addition, the nanowire has three degrees of rotational motion, but the three degrees of translational motion are eliminated because it is attached to the Si substrate.
If a rotation axis and a rotating angle are given in a rigid body, a rotation matrix can be computed with Rodrigues' rotation formula24
where is a rotation angle of the Bragg reflection planes. The rotation angle is determined by two normal vectors of the Bragg reflection planes at θi and . pi is a rotation axis vector. (see supplementary material, Figure 4) The normal vector of the Bragg reflection plane at each θi can be identified based on the fact that the 3D diffraction pattern moved on the detector plane during the second half of the rocking scan. Consequently, the rotation angle and rotation axis vector pi can be found. Therefore, the rotation matrix Ri finally can be obtained.
The rotation matrix can be decomposed into components with respect to a global coordinate system. The y-axis is defined parallel to the incident x-ray beam ki and the z-axis is normal to the ground, while the x-axis is orthogonal to both the y and z-axes. In addition, the rotation angles α, β, and γ are the rotations around z-, y-, and x-axes, respectively (see Figure 4(a)). The rotational angles α, β, and γ can be calculated by equating each element in Ri with the corresponding element in the matrix product (see supplementary material) as shown in Fig. 4(c). The rotational angle α corresponds to a twist of the sample, but β and γ are responsible for a tilt of the sample.
We determined that the sample was twisted and tilted simultaneously. The tilting motion of the sample does not coincide with the x-ray beam. Instead, it is opposite to the momentum transfer vector Q111. If this was a normal rocking scan, the rotation α was the same as the change of θ during the 2nd half of a rocking scan, which is 0.2°, but β and γ were 0°. However, the rotation β, which is responsible for the transverse tilt of the sample, is dominant and the rotation α increased by 0.07° in the global coordinates. It indicates that the sample was twisted by 0.13° with respect to the sample stage, which is rotated by the θ motor, implying that the motion of the sample is irrelevant to the gravitational force direction. The photoelectric effect and x-ray absorption, which can cause the sample to move along the x-ray beam direction, are excluded from the main causes because the tilt of the sample is not in the direction of the x-ray beam.
Figure 4(d) shows a top view of the motion of the Ni nanowire and the tracking of the top of the Ni nanowire is denoted by the dots as a function of θ. The distance that the top of the Ni nanowire moved is 20 nm. The motion of the sample is consistent with the direction of x-ray radiation pressure, which is exerted on the Bragg reflection plane and opposite to the momentum transfer vector as shown in Fig. 4(d). When data were collected at the Bragg reflections such as (002) whose momentum transfer points upwards, it has never been observed that the Bragg peak moved because the radiation pressure would be exerted towards the substrate. Radiation pressure effects are also supported by the fact that the Bragg peak was not in the same location after the rocking scan. The lack of the Bragg peak after the scan indicates that the nanocrystal samples no longer existed in their original spatial location and/or orientation.
The x-ray radiation pressure force due to the diffraction on the Bragg reflection plane Fdiffraction can be calculated as 2npħω/c,25 where ħ is the reduced Planck's constant, ω is a frequency of the x-ray beam so ħω is the energy of a single photon and np is the flux of x-ray photons that are involved in the diffraction event. In this study, the photon energy is 9 keV with a flux of 109 photons/s (Ref. 26) within a 2.25 μm2 area. The single photon energy at 9 keV equals the value ħω9keV, 2.3 × 10−16 J, but np varies with the rotation angle θ so the radiation pressure force has the maximum value when the angle θ is at zero or close to it. The estimated flux of x-ray photons on the Bragg reflection plane np is 1.2 × 106 photons/s at the Bragg angle (see supplementary material). Therefore, the radiation pressure force exerted on the (111) Bragg reflection plane of the Ni nanowire is 1.84 aN at 9 keV. It implies that the tapered Ni nanowire, which is attached to the Si substrate, withstands the external force, up to 1.84 aN. Again, the fact that nearly all movement is seen when θ is at or near the exact Bragg angle, corresponding to maximum momentum transfer, and the 3D motion of sample illuminated by the x-ray beam points to x-ray radiation pressure as the primary cause of the observed effects.
It is important to rule out other potential causes of sample motion, including thermal effects. As the temperature of nanocrystals increases, the vibration of atoms in a lattice around their equilibrium position increases so that the overall diffraction intensity decreases by the Debye-Waller factor without a spatial shift of the Bragg peak.27 The increase of temperature can cause the expansion of lattice spacing so that the Bragg peak moves out of the D-S ring, whose radius depends on the lattice constant. In our experiment, however, it was observed that the Bragg peak consistently remains on the D-S ring, indicating that the heating of x-ray radiation on the sample is not the cause of the Bragg peak shift. Furthermore, this mechanism would be expected to be reversible, which we do not observe.
If the substrate was deformed by a thermal strain due to x-ray radiation, the nanocrystals can tilt, and thereby lead to a shift of the Bragg peak. We employed a one-dimensional finite difference scheme for solving the governing heat transfer equation to estimate the contribution of x-ray photons to heating the Si substrate28
where T is a temperature. ρ, k, and c are the density, thermal conductivity, and specific heat, material properties of the Si substrate (Table I). The heat source is the energy deposited by photons and can be calculated as
where Io is the intensity of the x-ray beam and μ is the attenuation coefficient of Si for 0.13 nm wavelength x-rays.29 The free convection and radiative transfer at the outer surface are implemented as boundary conditions. The calculated temperature increase due to the x-ray photon energy in the substrate is 0.3 K for a 41 min exposure time, even without taking into account heat dissipation in the lateral direction, which would decrease the rise in temperature. It can be concluded that the x-ray radiation plays a negligible role in raising the temperature of the substrate and thereby, the thermal effect on the both sample and substrate is insignificant.
|Density ρ (kg m−3) .||Thermal conductivity k (W m−1 K−1) .||Specific heat c (J kg−1 K−1) .||Attenuation coefficient μ (m−1) .|
|Density ρ (kg m−3) .||Thermal conductivity k (W m−1 K−1) .||Specific heat c (J kg−1 K−1) .||Attenuation coefficient μ (m−1) .|
In conclusion, we have observed abnormal 3D diffraction patterns of Pd nanocubes and a Ni nanowire during the course of a rocking scan. Our analysis suggests that these intriguing data result from the effect of x-ray radiation pressure on the samples. X-ray radiation does not affect the motion of the sample in most cases, even though it inherently carries energy and exerts a pressure. As a sample size gets smaller and photon flux increases, it becomes increasingly important to take into account the x-ray radiation pressure on a sample for hard x-ray scattering at synchrotron sources. In order to avoid the situation where the sample moves during a rocking scan, we can either reduce the power flux density (i.e., photon energy) or choose a Bragg reflection plane whose momentum transfer direction is upwards so that the radiation pressure effects would be applied towards the substrate.
Pd nanocubes synthesis
Pd nanocubes are prepared according to Ref. 19. To make the initial Pd seed crystals, A 10 mM H2PdCl4 solution was prepared. A measure of 1 ml of 10 mM H2PdCl4 solution was added to 20 ml of 12.5 mM cetyltrimethylammonium bromide (Unilab, 98%) solution heated at 95 °C under stirring (700 rpm) in a 20 ml round bottom flask. After 5 min, 160 of freshly prepared 100 mM L-ascorbic acid (BDH Chemicals, 98.7%) solution was added and a 160 aliquot of this as-synthesized nanocube seed solution and 500 portion of 10 mM H2PdCl4 solution were added to 20 ml of 100 mM cetyltrimethylammonium bromide in a separate 50 ml round bottom flask. Freshly prepared 100 mM ascorbic acid solution (200 ) was added following this, and the solution was mixed thoroughly. The resulting solution was placed in a water bath at 60 °C for 1 h. Then, a further 500 of 10 mM H2PdCl4 solution was added, followed by 200 ml of freshly prepared 100 mM ascorbic acid solution, and the solution was well mixed. The flask was returned to the water bath at 60 °C and the reaction was stopped 1 h later by centrifugation (6000 rpm, 10 min). Two more centrifugations (6000 rpm, 10 min) were applied.
Ni nanowires synthesis
Ni nanowires (NW) were formed via a thermal CVD method as described in Ref. 20. Briefly, approximately 1.0 g of NiCl2·6H2O precursor is loaded into a Ni boat and placed within the quartz tubing at the center of the furnace. A series of 0.7 cm × 1.0 cm substrates are then placed downstream of the precursor in-line at distances of 3–15 cm. SiO2||Si is the substrate material used in successful NW growth. Ar is flowed through the CVD system at a typical rate of 10 sccm during NW growth. Preceding the NW growth is a baking step at 200 °C to alter the precursor moisture content. The subsequent growth period proceeds for approximately 30 min at 650 °C. The chamber is subsequently allowed to cool down naturally to room temperature while still sealed and maintaining inert gas flow.
Coherent diffraction experiment details
A double crystal monochromator was used to select E = 9 keV x-rays with 1 eV bandwidth and longitudinal coherence length of 0.7 . The rocking curve around the (111) Bragg peak was collected by recording 2D coherent diffraction patterns with an x-ray sensitive area detector (each pixel 22.5 μm × 22.5 μm) around (). It was placed at a distance of 0.5 m away from the sample. Coherent diffraction patterns were recorded for the rocking curves of the (111) Bragg reflections by rotating the sample through the Bragg condition in increments of 0.01°. 41 frames of 2D diffraction patterns were collected over a time period of 41 min.
See supplementary material for the additional abnormal rocking scan, SEM image of Pd nanocube, TEM image of Ni nanowire, the details of the Rodrigues's rotation formula, and the calculation of the number of photons that are involved in scattering event.
The coherent x-ray imaging work at UCSD was supported by U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Contract No. DE-SC0001805. Crystal growth was supported by NSF Award Nos. DMR-0906957 and DMR-1411335. Use of the Advanced Photon Source, an Office of Science User Facility operated for the U.S. Department of Energy (DOE) Office of Science by Argonne National Laboratory, was supported by the U.S. D.O.E. under Contract No. DE-AC02-06CH11357.