Direct X-ray Refraction of Micro Structures

Transcription

1 11th European Conference on Non-Destructive Testing (ECNDT 2014), October 6-10, 2014, Prague, Czech Republic Direct X-ray Refraction of Micro Structures More Info at Open Access Database Andreas KUPSCH*, Manfred P. HENTSCHEL, Axel LANGE, Giovanni BRUNO, Bernd R. MÜLLER BAM Federal Institute for Materials Research and Testing, D Berlin, Germany * Phone: , Fax: ; andreas.kupsch@bam.de Abstract For the first time we present direct 2D imaging of refracted X-rays without any discrimination of the primary radiation. X-refraction works in analogy to visible light optics: X-rays are entirely deflected at interfaces where discontinuities of (electron) density occur. This is demonstrated at the example of inner and outer surfaces of model samples of well-defined geometry (fibres, capillaries, and monodisperse micro particles). The samples are scanned through a 50 µm monochromatic (20 kev) pencil beam. In order to warrant a sufficient angular resolution a 2D detector (pixel size 7 µm) is placed 3 m downstream of the sample. Scanning the sample in micron steps allows for detecting local changes of interface / surface orientation directly in two dimensions. At the actual angular resolution of about 3 seconds of arc (scattering vector increments Δk = 10-3 nm -1 ) and 50 µm spatial resolution (scanning) the material s inner surfaces (with nanometer separation) can be characterized even at sampling rates below 1 second per frame. Moreover, our technique is suited to directly determine particle diameters of up to 250 nm by means of diffraction fringes. Some potential applications to technical submicron structures are discussed. Keywords: X-ray refraction, Phase contrast, Edge artefact, Soft matter radiography, Wavelet transformation, Huygens elementary waves 1. Introduction Since Christiaan Huygens discovery of the wave character of light [1] it has been known, that condensed matter (solid bodies and fluids) deflect electro-magnetic waves because of their refractive properties. The refraction is caused by the altered speed of light inside matter, which leads to a phase shift. More than 200 years later the refraction was shown to occur for X-rays [2], as well. But it took several more decades to employ for measuring techniques. Although the deflection angles are very small (compared to visible light) the very promising aspect is the 100 % scattering factor. In 1987 Hentschel et al. [3] derived a quantitative description of X-ray refraction at cylindrical objects such as glass or polymer fibres and metal wires, which gives the (angular) distribution of intensity up to the critical angle of total reflection (at some minutes of arc). In subsequent studies the authors employed this method to develop the refraction topography as a non-destructive technique to characterize the specific surface [4,5]. This sampling technique typically detects the small angle scattering by using a conventional block camera of the Kratky type [6], usually employed for crystallographic purposes. Beyond scanning, Müller et al. [7] developed the direct refractive imaging of inner surfaces (interfaces) and edge artefacts onto 2D-detectors with the aid of a highly angular sensitive analyzer crystal for the sake of monochromatic synchrotron refraction tomography. Wilkins et al. [8] observed edge artefacts in highly resolved radiographs using polychromatic radiation and large sample-detector distances. Neglecting the deflection the authors reasoned their observation as a pure phase (shift) contrast. In other studies the edge artifacts are interpreted due to Fresnel diffraction [9]. The classical analytical coherent small angle scattering techniques (Guinier [10] and Porod [11] analysis) exhibit too low scattering intensity of dipole radiation to be relevant for the edge artefacts. As an example Fig. 1 shows a high resolution radiographic image of a cylinder edge with an intensity overshoot, caused by refraction, which is commonly named phase contrast. The experimental set-up (with a sample-detector distance of 350 mm) is drawn to the left of the 20 kev synchrotron radiograph and the intensity profile of the PMMA cylinder (diameter 75

2 mm). The intensity exceeding the I 0 level as observed outside the shadowing area cannot be explained by conventional attenuation (absorption). The graph in Fig. 1 compares the asmeasured cross section to the ideal attenuation profile (grey) and a profile obtained from simulations including refraction effects (red), which proved to be quantitatively consistent with relevant parameters (geometry, refraction index) as has been reported in previous studies [12,13]. Fig. 1: High resolution radiograph of a cylinder edge revealing the typical refraction-based overshoot of primary intensity; left: experimental set-up (sample-detector distance 350 mm); right: radiograph, intensity profile of the measurement and simulated profiles with and without the refraction effect [13]. In order to identify such radiographic artifacts as phase contrast or refraction effects, in the present study we aim at unambiguous separation of beam paths, which are superimposed in the radiographic image, by scanning the sample with mutually independent single beams. 2. Experimental proof of principle: direct refraction Here, we introduce a set-up without any secondary discrimination (Fig. 2). Experiments were performed at BAMline [14,15] at the electron storage ring for synchrotron radiation BESSY II operated by the Helmholtz Zentrum Berlin (HZB). The bandwidth of the incident, nearly parallel radiation is limited to 0.1 % by means of a double crystal monochromator (Si (111)). It is directed to a pinhole, so that the sample is hit by just a 50 µm pencil beam. The nominal mean energy was chosen as 20 kev. The entrance slits were narrowed to the pinhole in order to avoid detector backlight effects [16,17]. Fig. 2: Sketch of the experimental set-up for direct detection of refracted X-rays by means of an incident pencil beam. The sample can be moved (by translation or rotation) relative to the pencil beam.

3 After interaction with the sample the X-ray beam hits the detector without any downstream collimation (sample-detector distance 3.4 m). Together with the primary beam diameter one calculates an angular resolution of about 3 seconds of arc (Δk = 10-3 nm -1 =1 µm -1 ). A CdWO 4 scintillator screen (on quartz glass substrate converts the X-ray photons into visible light, which is then guided through objective (f = 100 mm) onto a pixel CCD array (pixel size 7.2 µm). 3. Sampling: solid cylinder As a reference sample of well-defined geometrical shape we chose a polyamide rod (diameter 1.5 mm). This rod is moved perpendicular to the pinhole beam in 10 µm increments (Fig. 3, left). When entering the cylinder edge the originally undistorted primary beam centroid is immediately broadened and radially shifted, followed by further continuous displacement. This behavior is schematically depicted by Fig. 3, right, which accumulates the vertical cross sections of the single beam exposures some of which are given at the bottom. Fig. 3: Sampling the cylinder by a pencil beam; left: sampling geometry with scanning direction indicated; right: vertical sections of scattering images according to the sampling position (top) and an enlarged detail of as measured single images at sampling positions in the edge vicinity. 4. Modeling 4.1 Geometrical optics Similar to visible light in transparent materials X-ray optical phenomena can be described by Snell s law of refraction. An incident X-ray beam (coming from vacuum) may hit a material surface under angle (relative to the surface, Fig. 4). Inside the material of refraction index n the beam alters its direction and it holds: cos /cos n (1) 2 where 2 denotes the angle of exit relative to the surface. The refraction angle (scattering angle) reads as:

4 arccos (cos ) (2) 2 n At cylinder sections (formally: radius R = 1) the incident beam position x directly correlates with the angle of incidence 1 (see Fig. 5): Since the real part of the refraction index x cos, or arccos x (3) n 1 (4) with being the refraction decrement (or dispersion correction) is (slightly) smaller than unity (for example: = for polyamide at E = 20 kev), convex objects such as cylinders act as diverging lenses (i.e., rays are bent away from the surface normal when entering matter, and consequently occurs external total reflection, Fig. 5) Fig. 4: Refractive beam deflection at plane surface: incident beam (under angle ) and refracted beam ( 2 ). Fig. 5: Sampling the cylinder by a pencil beam; left: beam deflection of two parallel incident, which hit the cylinder s surface at different sites; right: refraction angle as a function of incident position x. For cylinder sections, it follows from simple geometric considerations that refraction angles are the same when the beam enters and leaves the material. as a function of the incident beam position x is expressed as:

5 2 ( x) sign( x)[arccos x arccos( x/ n)] sign( x) 1/ x 1 for 1 x 1(5) This function is depicted in Fig. 5, right. It qualitatively reproduces the measured geometrical beam displacement. 4.2 Wavelet transformation In the following we show that wave optical approaches yield the same principal results as geometrical ray optics in the section above. Besides the absorption correction (in the imaginary part, which quantifies the true (photoelectric) absorption, = 4/), the complex-valued index of refraction n = 1-+i also comprises the dispersion correction (in the real part, is in the order of 10-6 ), which is relevant for refraction effects. An incident plane wave of wave number k (k = 2/) is denoted as (z) = exp(ikz). After penetrating an object of refraction index n and chord length distribution d(x) (along z-direction) the wave can be expressed as ( d( x)) exp( iknd ( x)) exp( kd ( x)) exp( ikd ( x)) Aexp( i) (6) at sites(x, d(x)), where the object s chord length distribution d(x) rules the (monotonic) damping term A as well as the (oscillating) phase term. In case of a cylinder cross section (i.e., a circle) d(x) can be expressed as d( x) x R (7) or d ( 1) 2sin 1 with (3) and R=1 (Fig. 5). Obviously d(x) changes most rapidly at the cylinders edges, i.e., its derivative gets large. A polygon approximation of d(x) reveals average pitches in finite sized intervals Δx and maxima near the edges. In the following we consider the Fourier transforms (FFT) of such intervals, taking into account that different pitches of the chord length (polygon) correspond to different phase ramps in the wave function. If one interprets the phase as frequency related information (in direct space), the resulting shift of the respective FFT (in dual space) is to be understood as shift of image. The FFT F(k) of an arbitrary function f(x) is defined by F ( k) f ( x) e ikx d x (8) The FFT of the product of f(x) and a phase ramp Fourier space ik 0 x e ( g 0 ik x ( x) f ( x) e ) yields a shift in ikx i ) ( ) e d ( ) e ( kk 0 x g x x f x d x F ( k k 0) (9) Analogously to the experimental sampling of the cylinder with finite increments we compute the FFT for finite intervals of the wave function (d(x)), as illustrated in Fig. 6 by differently colored regions. Outside the cylinder (grey interval) the wave is not modified; the

6 according FFT serves as a reference whose center of gravity is arbitrarily set to k = 0. Near the cylinder diameter (maximum chord length, red interval) the phase changes rather slowly when varying sample position; the center of gravity shifts slightly. Near the edge (blue interval) one yields the strongest shift, as to be expected from (9). Beyond the uniform interval pitches, also the change of slopes (i.e., the 2 nd derivative) within the intervals is largest near the edge. This leads to broader FFTs and corresponds to the experimentally observed beam broadening. The grey value image (Fig. 6, bottom right) is obtained from composition of the computed 1D profiles according to the sample position (center of the respective intervals). The composed image reproduces the experimental phenomenon (cf. Fig. 3) in a qualitatively excellent manner. Fig. 6: Schematics of wavelet transformation illustrated at the example of a cylinder; left: course of the object function (real part) as a function of position; right: wavelet as FFT of the selected intervals (marked by the respective colors); bottom right: wavelet transform from composition of grey-value converted local wavelets according to their position. 4.3 Huygens elementary wave synthesis The wavelet results of the above section implicitly assume the validity the far-field approximation (Fraunhofer approximation), i.e., given a minimal distance, the qualitative phenomena are independent of sample-detector distance (they just scale in size with changing distance). However, in the following we investigate the superposition of elementary waves, whose amplitude and phase is modified when penetrating an object (6). Intensity distorsions due to large aspect ratios are negligible because of the rather small deflection angles (resp. refraction decrements << 1). The main advantage of this procedure is the basic wave propagation without any presuppositions (i.e., without the limiting boundary conditions of Kirchhoff-Fresnel diffraction equations). The (numerical) computation of intensity patterns by Huygens elementary wave synthesis is performed as superposition of all complex wave amplitudes followed by calculating the (square) absolute of the sum. Fig. 7 depicts a sketch of the procedure at the example of a cylindrical object. In order to speed up computing the superposition of the modified object function (6) and the primary (spherical) wave at distance z 0 from the origin is performed by convolution of both complex functions. As described above the computed profiles are combined to a grey scale image according to their origin (Fig. 7, right bottom). This modeling approach reproduces the measurement in an appropriate quality, as the wavelet transform (cf. Fig. 3).

7 Fig. 7: Schematics of elementary wave synthesis by convolution; top left: real part of the complex primary (spherical) wave; center: section at distance z 0 from source; top right: sample function (real part); bottom right: composition of the convolution results according to their position. 5. Application: monodisperse powders Beyond ideally shaped single objects such as cylinders (Fig. 2), the experimental set-up was used to investigate loose bulks of monodisperse spherical particles in the micron range. The powders (melamine resin (C 3 H 6 N 6 ), = 1.57 g/cm 3 ) were filled into square shaped cuvettes, which were arranged under 45 with respect to primary beam direction. Here we introduce the results obtained from two fractions of spheres with diameters d 1 = 4 µm, d 2 =1.25 µm (identical bulk density or attenuating mass, respectively). Since the refraction index solely depends on the material both particle sizes should exhibit the identical refraction behavior. Nevertheless, the different performance in real measurements, as results from multiple interface interactions is of outstanding interest in characterizing pores or particles (incl. clustering) in material science. The multiple scattering can be exploited in order to find a direct measure for the specific surface of porous materials as well as crack density including their orientation (ODF). This refers especially to advanced materials such as ceramics, aerogels, foams, catalysts, or filters. A basic proof of principle of multiple scattering is established by pencil beam small angle scattering experiments for different path lengths in square cuvettes. Fig. 8: Photograph and sketch of the measuring arrangement of the cuvettes filled with melamine powder and single scattering images. Fig. 8 shows the cuvettes filled with powder on the left and the schematic cuvette arrangement relative to the incident beam as well as single scattering images. Starting from

8 the small primary beam spot (left and right image in the row) the powders cause significant beam broadening with increasing penetrated thickness (decreasing integral transmission intensity) and vice versa. One the one hand, the smaller particle fraction leads to a steeper rise of the broadening, evaluated by the full width of half maximum (FWHM) of the intensity distribution. On the other hand, the (integral) attenuation is very similar (Fig. 9, left). Disregarding the (known) position assignment the correlation of FWHMs and integral attenuation yields a clear assignment to the respective particle size (Fig. 9, right). Fig. 9: Plot of the absorbing mass and the full width of half maximum (FWHM) vs. sample position (left). The correlation of both quantities reveals the direct assignment to the respective particle size (right). References 1. C. Huygens: Traité de la lumière, où sont expliquées les causes de ce qui luy arrive dans la reflexion et dans la refraction, et particulierement dans l'etrange refraction du cristal d'islande; Pierre van der Aa, Leiden (1690). 2. A. Larsson, M. Siegbahn, I. Waller: Der experimentelle Nachweis der Brechung von Röntgenstrahlen, Naturwissenschaften 12 (1924) M.P. Hentschel, R. Hosemann, A. Lange, B. Uther, R. Brückner: Röntgenkleinwinkelbrechung an Metalldrähten, Glasfäden und hartelastischem Polypropylen, Acta Cryst. A 43 (1987) M.P. Hentschel, K.-W.Harbich, A. Lange: Non-destructive evaluation of single-fiber debonding in composites by x-ray refraction, NDT & E Int. 27 (1994) K.-W. Harbich, M.P. Hentschel, J. Schors: X-ray refraction characterization of nonmetallic materials, NDT&E Int. 34 (2001) O. Kratky: Small Angle X-ray Scattering, edited by O. Glatter and O. Kratky. Academic Press, London (1982). 7. B.R. Müller, A. Lange, M. Harwardt, M.P. Hentschel, B. Illerhaus, J. Goebbels, J. Bamberg, F. Heutling: Refraction computed tomography, Mater. Test. 46 (2004) S.W. Wilkins, T.E. Gureyev, D. Gao, A. Pogany, A.W. Stevenson: Phase-contrast imaging using polychromatic hard X-rays, Nature 384 (1996) P. Modregger, D. Luebbert, P. Schaefer, R. Koehler, T. Weitkamp, M. Hanke, T. Baumbach: Fresnel diffraction in the case of an inclined image plane, Optics Exp. 16 (2008) A. Guinier, G. Fournet: Small-Angle Scattering of X-Rays, Wiley, G. Porod: Die Roentgenkleinwinkelstreuung von dichtgepackten kolloiden Systemen, Kolloidzeitschrift 124 (1951) A. Lange, A.Kupsch, B.R. Müller, M.P. Hentschel: Edge Artefacts of Radiographic Images by X-ray Refraction, Proceedings 18 th World Conference on Non-Destructive Testing, Durban (2012).

9 13. M.P. Hentschel, A. Kupsch, A. Lange, B.R. Müller: Refraktions-Interface-Radiographie, Proceedings DGZfP-Jahrestagung, Dresden (2013), DGZfP-Proceedings BB 141-CD (2013). 14. W. Görner, M.P. Hentschel, B.R. Müller, H. Riesemeier, M. Krumrey, G. Ulm, W. Diete, U. Klein, R. Frahm: BAMline: The firsthard X-raybeamline at BESSY II, Nucl. Instrum. Meth.A (2001) A. Rack, S. Zabler, B.R. Müller, H. Riesemeier, G. Weidemann, A. Lange, J. Goebbels, M.P. Hentschel, W. Görner: High resolution synchrotron-based radiography and tomography using hard X-rays at the BAMline (BESSY II). Nucl. Instrum. Meth. A586 (2008) A. Lange, M.P. Hentschel, A. Kupsch, B.R. Müller: Numerical correction of X-ray detector backlighting, Int. J. Mat. Res. 103 (2012) A. Kupsch, M.P. Hentschel, A. Lange, B.R. Müller: Einfaches Korrigieren des Hinterleuchtens von Röntgendetektoren, Mater. Test. 55 (2013)