3-D X-ray Diffraction Imaging at the Nanoscale Andrei Y. Nikulin*, Ruben A. Dilanian*, Brian M. Gable‡, and Barrington C. Muddle‡

*School of Physics, Monash University, Clayton, Victoria, Australia 3800 Email: Andrei.Nikulin@sci.monash.edu.au

†Department of Material Engineering, Monash University, Clayton, Victoria, Australia 3800 Email: Barry.Muddle@eng.monash.edu.au

Abstract—We report a “momentum transfer” x-ray diffraction data analysis approach to nondestructive determination of the shape of an average nano-particle in three dimensions. The possibility of using incoherent x-ray sources allows the method to be implemented in any laboratory. Our results demonstrate that this approach to x-ray diffraction data analysis provides the potential for 3-D reconstruction without a tomographic synthesis of 2-D images, as in conventional coherent radiation imaging experiments. A simple and robust quantitative technique to reconstruct an average nano-particle randomly dispersed over a large volume within a sample, such as presented here, should enable researchers to study nano-particles using conventional laboratory x-ray diffraction equipment with resolution up to a few nanometers.

Nano-particles, 3-D imaging, x-ray diffraction

I. INTRODUCTION

A feature of emerging nanotechnology is the development of novel materials based on the dispersion of nano-particles in a broad variety of matrices. Applications of such materials range from novel paints and coatings, to a new generation of light alloys, advanced photonics and telecommunication devices, cosmetics and novel biomedical devices and sensors, etc. In order to effectively control the processing, structure and performance of these nano-materials, there is a priority on the development of robust non-destructive techniques for imaging nano-particles. The detailed and high-resolution study of the microstructure of the nano-particles put in the forefront new special requirements to experimental conditions, data acquisition, and data analysis. The destructive and generally two-dimensional nature of electron microscopy often limits the nano-particle imaging to qualitative or semi-quantitative analysis.

From this point of view, x-ray diffraction technique offer unique opportunity for non-destructive characterization of nano-scale objects, such as nano-particles, nano-clasters, and nanotubes, because of the greater penetration length of x-rays. X-ray small-angle scattering can estimate the average scale of nano-particles over a large area/volume without destroying the specimen, however it does not resolve the shape of the particles. X-ray diffraction microscopy has recently been demonstrated capable of 2-D and 3-D image reconstruction of nano-scale materials [1-6]. This technique adopts so-called direct or “real space” imaging approach where the data is

collected in real space and the intensities are measured as a function of real space coordinates, Fig. 1 (a). The direct imaging method is strongly correlated with the spatial and temporal coherence of the incident beam. Moreover, this method can only be implemented using a mechanically stable x-ray source (source-optics-sample-detector arrangement).

Here we suggest an alternative solution to the aforementioned problems which are intrinsic to the direct imaging experiments. The solution is to record the diffracted intensity as a function of the angular (momentum transfer) space coordinates [7, 8]. The methodology is based on the measurement of a high angular resolution x-ray Fraunhofer diffraction patterns with further application of a phase-retrieval formalism. The advantage of experimental data collection in momentum transfer (angular) space is that the diffraction pattern of the object is not susceptible to any of its linear translation. Consequently, the intensity distribution measured as a function of the angular direction in reciprocal space does not require coherent radiation and/or extraordinary stability of the radiation source and/or sample, [7]. Such measurements can be done by using a crystal-analyzer and counting detector instead of complicated optics and pixel-structured detector systems. The only principal restriction on achievable resolution is the physical limit of the order of sub-nanometer.

Unlike the microscopy technique, the method to be presented does not allow one to distinguish the shape and size of individual nano-particle within a distribution. This limits the application of the method to studies of structures with mono-dispersed nano-particles. This very same limitation, however, can prove a significant advantage of the method for materials science as it does allow non-destructive, in-situ analysis over large volumes (several mm3) of material, and the determination of the average shape-size of the dispersed particles and a close evaluation of the distribution of particle sizes.

Recently, the method has been used to image the shape of ~50 nm diameter Al2O3 nano-particles dispersed in a polymer matrix with a spatial resolution of 2 nm, [9]; ~350 nm intermetallic nano-particles in age-hardenable Al-Cu alloys with a spatial resolution of 8 nm, [10]; and ~10 nm diameter carbon nanotubes with spatial resolution of 1 nm, [10].

II. EXPERIMENTS

The experimental arrangement is shown schematically in Fig. 1 (b). The required x-ray radiation energy is selected using

606 ICONN 20061-4244-0453-3/06/$20.00 2006 IEEE

the primary double-crystal Si beamline monochromator. Further angular collimation can be performed by a double-crystal channel-cut Si monochromator. Then the beam spatially collimated by a pair of slits. The sample is placed downstream just after the slits. The Si crystal analyzer and a detector with a pair of collimated slits just before it is placed downstream from the sample to collect diffracted from the sample intensity as a 2-D function of the analyzer and detector-slit arrangement angular positions. In such experiments the spatial resolution is determined by conventional Fraunhofer diffraction relationship: z = / , where is the total angular aperture of the experimental data [7, 9].

III. RESULTS AND DISCUSSION

The proposed method exploits a 3-D geometrical construction, known as the Ewald sphere in reciprocal space [11], in order to record a set of 2-D diffraction images. The resulting 2-D map of diffracted intensity from the sample is the essence of intersection of the 3-D diffraction image of the sample and the equatorial plane of the Ewald sphere. In this case the rotation of the sample around the main optical axis (incident beam direction) will change the intersection between the 3-D intensity profile and the Ewald sphere, and as a result change the recorded 2-D map of the diffracted intensity from the sample. Each of the 2-D maps recorded in the angular space will be corresponded to the Fourier transforms of the different projections of the same 3-D object. In the case of spherical symmetrical objects these 2-D maps are virtually indistinguishable and only one map will be necessary for the reconstruction of the object. However, in the case of highly asymmetric objects the resulting 2-D maps will change noticeably with rotation of the sample around the main optical axis. In this case several maps will be necessary for the final 3-D reconstruction. In addition varying the radiation energy (and thus the curvature of the Ewald sphere) permits independent diffraction images to be recorded and the production of a unique solution for the 3-D image of the average nano-scale object.

Figure 1. Sketch of the setup for an image reconstruction experiment: (a) direct imaging method; (b) momentum transfer method.

Since the 2-D maps of the sample are measured as functions of the angular direction in momentum transfer space, the proposed technique is virtually immune to sample vibrations and does not require coherent radiation and unusual stability of the radiation source and/or sample, unlike the direct imaging technique. Moreover, there is no need for a beamstopto block the direct beam. Use of a single photon counting detector (e.g. APD instead of e.g. CCD) allows one to record data within a linear dynamic range of 8-9 orders of magnitude. The crucial central peak of the diffraction pattern is therefore readily measurable in this case.

We used the Gerchberg-Saxton (G-S) formalism to reconstruct the shape of an average nano-particle from the experimental diffraction data [12, 13]. In general, the G-S algorithm iteratively finds a solution consistent with measured data and known constraints.

The reconstruction algorithm consisted of the following iterative steps:

Step (1): To take a Fourier transform of the real-space function gk(r) to obtain a simulated diffraction pattern, Gk(q).

Step (2): To generate an intermediate “experimental” diffraction pattern by replacing the modulus of the Fourier transform from step (1) with the experimentally measured modulus (reciprocal-space constraint).

Step (3): To take an inverse Fourier transform G’ to obtain a new g’(r).

Step (4): To generate a new real-space function of the object using the real-space constraints.

The following equation represents the real-space constraint:

otherwisergrgrgSrrg

rgk

k ),()(]0)([),(

)(1

Here q = (qx, qy) is the scattering vector, r = (x, y) is the radius vector, is a constant feedback parameter, and S is a support function.

Unfortunately the limitation of the G-S approach is that it is suited only to reconstruct a real positive function, according to the real-space constraints, [12]. The function g(r), however, is a complex function and the reconstruction algorithm can not be applied in form presented in this paragraph. Thus, in order to use the G-S algorithm some additional assumptions must be made.

In case of small scatters such as nano-particles the diffracted intensity profiles can be well approximated as a modulus squared of a Fourier transform of the complex transmission function g(x, y), [14]:

g(x, y) = exp(i ) = exp{i(2 / )nT(x,y)},

where is the wavelength, n is the complex refractive index, and T(x, y) is a function of the object thickness. If the magnitude of is small compared to unity, << 1, then one can simplify:

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g(x, y) 1 + i

In this case, the complex transmission function, g(x, y), may be expressed by the atomic scattering factor (or electron density), of the object

g(x, y) = g1(x, y) + ig2(x, y).

For a homogenous object we can also assume that the complex refractive index, n, is a constant and its value depends only on chemical composition of the object and radiation energy, [15].

Consequently the “shape” of the real and imaginary parts of the expanded complex transmission function should be the same the real and positive function, T(x, y). This means that the positivity constraints, [1], can be used for reconstruction of scattering objects. In addition, since in case of hard x-rays the imaginary part of a complex refractive index is generally much smaller than the real part, it is also very useful to deliberately use the x-rays energy just above the absorption edge of the dominated in the scattered object chemical element to be able to compare the results of reconstruction focused on either of the parts of the generally complex function.

Figure 2. Image reconstruction of an Al2O3 nano-particle from the experimental diffraction data: (a) experimentally recorded x-ray diffraction

intensity (contour map, logarithmic scale); (b) 3-D representation of an average Al2O3 nano-particle.

It should be noted, however, that substitution of the complex transmission function by the complex atomic scattering factor is possible only if the following condition is satisfied:

|n|T << /(2 ).

For example, for the = 1.0 Å the average size of Al2O3particles should be less than 3.0 m and for = 2.0 Å should be less than 1.5 m. Clearly, this approximation worked well in the present experiments and analysis.

Fig. 2 (a) represents experimentally recorded 2D map of the diffraction intensities for Al2O3 nano-particles dispersed in a polymer matrix. The central vertical line in the maps is the main peak due to Bragg reflection of the monochromator. The initial complex transmition function go(r) of the nano-particles was estimated using experimental data for the size of the average nano-particle and a randomly distributed phase of the diffracted wave. The support function was selected as a circle with a diameter D 60 nm with the following conditions:

S = 1, if R2 = (x – x0)2 + (y – y0)2 < (D/2)2

S = 0 – otherwise,

where (x0, y0) is the position of the origin. We also performed several reconstructions using different initial sizes of the input function, D, ranging from 10 to 60 nm. The shape of the resulting function g(r) yielded the average size of the nano-particles to be approximately 50 nm, which is in good agreement with the expected value. Then we used the a prioriinformation about the nearly spherical symmetry of the particles to also produce a quasi 3-D image of the average particle (Fig. 2 (b)).

Fig.3 (a) represents experimentally recorded 2D maps of the diffraction intensities for two azymuthal orientations of the Al2Cu nano-particles. The central vertical line in the maps is the main peak due to Bragg reflection of the monochromator. In this case we recorded two sets of 2D maps of the diffracted intensities, defined as 0 degrees and 45 degrees, because of highly asymmetric behavior of the shape of Al2Cu nano-particles. It is visible from these experimental observations that the diffracted intensity profiles are sensitive to the azimuthal orientation of the sample, unlike in real space imaging.

The support function was selected as a rectangle with a size D 400 nm. The results of the reconstruction are presented in Fig.3 (b). The two orthogonal variants of the reconstructed particle represent an average particle in two perpendicular planes within the given aperture of the incident beam. To reconstruct the third variant of the Al2Cu platelet nano-particles, a rotation of the sample around an axis which is parallel to the sample surface is required.

The shape of the resulting particle yielded the average size of the nano-particles to be approximately 320 nm. Fluctuations in size are reflected in slightly blurred boundaries of the reconstructed 2-D images and the surrounding background.

Fig. 4 represents 3D rendering of a straight section of a carbon nanotube dispersed in a polymer matrix. The angular range of the analyzer where chosen to allow the reconstruction of the average nanotube with spatial resolution of 1.0 nm.

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Figure 3. Image reconstruction of an Al2Cu nano-particle from the experimental diffraction data: (a) experimentally recorded x-ray diffraction

intensity (logarithmic scale); (b) 3-D rendering of two variants of Al2Cunano-particles embedded in an aluminum matrix.

In this case we reconstructed a shape and an average size of the cross-section of nanotubes. The length of the straight section of the reconstructed image, shown on Fig.4, does not have any physical meaning. The shape of the resulting cross-section yielded the average size (diameter in this case) of the carbon nanotube to be approximately 10 nm, which is in good agreement with the expected value.

IV. CONCLUSION In summary, we have reconstructed 3D-images of the nano-

scale particles embedded in an amorphous/crystalline matrix from experimental x-ray diffraction 2D-data collected using momentum transfer method. This method is limited to the study of structures with dispersed nano-particles. The method does not allow one to reconstruct shape/size of an individual particle.

Figure 4. 3D rendering of a straight section of a carbon nanotube dispersed in a polymer matrix.

However, this simple and robust technique has a dramatic advantage for material science as it does allow non-destructive and in-situ analysis over large volumes of material (several mm3) and determination of the average shape and size of the dispersed particles. It should also be emphasised that presented method does not require coherent x-rays and can be implemented in a laboratory or at a “low-end” synchrotron.

ACKNOWLEDGMENT

The work was supported by the Australian Synchrotron Research Program (ASRP) and the Australian Research Council (ARC) Discovery grants.

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