Confocal energy-dispersive X-ray diffraction tomography employing a conical shell beam

: We introduce a new high-energy X-ray diffraction tomography technique for volumetric materials characterization. In this method, a conical shell beam is raster scanned through the samples. A central aperture optically couples the diffracted flux from the samples onto a pixelated energy-resolving detector. Snapshot measurements taken during the scan enable the construction of depth-resolved dark-field section images. The calculation of d-spacing values enables the mapping of material phase in a volumetric image. We demonstrate our technique using five ~15 mm thick, axially separated samples placed within a polymer tray of the type used routinely in airport security stations. Our method has broad analytical utility due to scalability in both scan size and X-ray energy. Additional application areas include medical diagnostics, materials science, and process control.


Introduction
The high penetration power of X-rays is the basis for projection radiography and X-ray computed tomography. These modalities are highly developed and deployed routinely within security screening, industrial inspection, and medical diagnostics. Within this broad application space, there are many critical spatial imaging tasks, which would also benefit from the identification of material phase attributed to components within a volume. The spectroscopic analysis of transmitted X-rays can provide some useful materials discrimination information [1]. However, such methods are limited fundamentally, as the image forming Xrays incident on a detector have propagated along linear paths without interacting with the sample. Conversely, crystallography deals with the collection of coherently scattered or diffracted X-rays from a sample to enable structural analysis or 'molecular fingerprinting'. Traditional XRD instruments or diffractometers may be categorized into either angular [2] or energy dispersive [3][4][5][6] modalities. The former employs monochromatic radiation to measure the diffraction angle, 2θ, subtended by the diffracted flux (from a sample) and the primary beam, while the latter measures the energy or wavelength, λ, at a fixed, known diffraction angle. Bragg's condition, λ = 2d sin θ, enables lattice parameters, e.g. d-spacings, to be calculated in each case. Laboratory scale instruments employ typically bright X-ray sources, e.g. 40 mA at ~40 kV. The amount of diffracted flux from a sample is relatively small and <<1% in comparison with the incident primary flux. Even with the use of a bright source and carefully prepared samples, the measurement time can range from minutes to hours.
Ultimately, the driver for our work is the detection and identification of homemade explosives (HMEs) and narcotics within the carry-on and checked luggage security at airports. A practical scanner requires exposure times of the order of seconds or less per measurement. There has been a considerable effort in developing high-energy methods utilizing novel X-ray beam topologies [2,3,[7][8][9][10][11][12] and/or post-sample encoders [4,5,[12][13][14][15]. Focal construct geometry (FCG) is an example of the former, which exploits the 'lensing' of diffracted flux by extended gauge volumes produced by a conical shell beam of radiation incident on a sample. This technology is capable of providing depth-resolved material specific signatures without prior knowledge of the sample location [12]. We demonstrate an augmented version of this imaging architecture that produces a volumetric reconstruction of a heterogeneous phantom to validate our theoretical treatments.
The organization of our paper is as follows. Section 2 presents the methods and includes the theory background; our new tomographic approach and describes the experiment conditions. Section 3 presents our experiment results and discussion. Section 4 summarizes our conclusions, discusses the implications of our findings and the future direction of the work.

Theory background
FCG employs a conical shell X-ray beam to produce bright material specific patterns in the diffracted flux from samples. The extended annular gauge volumes provide a relative increase in the total number of crystallites of the correct orientation that satisfy Bragg's condition. The result is a significant increase in the diffracted intensity, resulting in reduced exposure times and/or a lower X-ray power burden. This beam topology has been implemented in energy [3] and angular dispersive modes [2,[8][9][10][11], used to identify liquid samples [8] and shown to deal favorably with crystallographic textures [2,3,9,10] (i.e. preferred orientation and large grain size) that are known to be problematic [16,17]. Other implementations include the production of absorption tomography [18] and angular dispersive tomography [19] via raster scanning of annular projections over two orthogonal axes. Recently, we have demonstrated snapshot FCG [12] providing depth-resolved XRD patterns from a single stationary exposure. This paper describes a tomographic method in which a raster scanning snapshot FCG probe directly measures XRD sections to enable material specific volumetric visualizations.

New imaging technique
The FCG snapshot probe shown in Fig. 1 employs a post sample encoding-aperture optically coupled to a pixelated energy resolving detector. No prior positional information is required to provide depth-resolved XRD patterns. Diffracted rays propagate from within the shell beam's intersection with a sample (or gauge volume) onto a spatially resolving detection surface via a circular aperture. The linear distances x D and y D specify a position on the detection surface with respect to an origin defined by the piercing point of the shell beam symmetry axis, as shown in Fig. 1. The polar coordinates are given by where α is the polar angle subtended at the center of the circular beam footprint and r is the polar radius. The separation between the detector and aperture is specified as a focal length f = D-A, as shown in Fig. 1. The diffraction angle 2θ is a function of the focal length f, the radius of the detected photon r and the half-opening angle of the conical shell beam φ as given by gurations [12] which is given l z perture distanc racted is given l x = l y = ing aperture an et x t ,y t of the snapshot is dis l coordinates o xel provides the ssociated d-spa y substituting fo 1 1 tan 2 ion (x l ,y l ) . g l z z = The global Cartesian coordinate system (x g ,y g ,z g ) can describe the spatial distribution of coherent scatter measurements and or d-spacing information via Eq. (4) so enabling the construction of a volumetric data set.

Experiment conditions
Experiments were conducted using an IXS series VGA X-ray source operating at 160 kV accelerating voltage and 5 mA current. A conical shell beam was produced with the aid of a bespoke tungsten optic with a mean half-opening angle φ = 2.5° where φ max = 2.55°, φ min = 2.45°. A pinhole aperture of radius 0.75 mm in a 4 mm thick lead sheet, was placed 785 mm from the X-ray source. Scattered X-rays were detected using a 250 μm pitch (80 2 pixels) 20x20x1 mm 3 cadmium telluride (CdTe) energy resolving detector placed at 895 mm from the X-ray source. The aperture-to-detector separation or focal length f = 110 mm is fixed throughout the experiments. The energy resolution of the detector ΔE at FWHM was estimated to be ~850 eV at 60 keV [20]. The phantom consisted of a polymer 'security' tray (see Fig. 2.) containing five ~15 mm thick, 90 mm diameter Petri dish samples (detailed in Table 1) positioned ~500 mm (z-axis) from the X-ray focus. The incident beam diameter is ~44 mm with a wall thickness of ~0.9 mm. The symmetry axis of the shell beam is orthogonal to the (x g ,y g ) plane. A two-axis raster scan comprises stepwise translation of the polymer tray along the x-axis with synchronized translation of the snapshot probe back and forth along the y-axis. In this way diffracted flux measurements from each detector pixel were integrated for 1 second over successive axial intervals of Δy t = 25 mm. At the end of each (y-axis) linear scan the polymer tray was stepped by Δx t = 25 mm before scanning the probe along the reverse (y-axis) direction. This sampling regime enabled the collection of 576 measurements/pixel over a total inspection area in the translation plane of ~600x600 mm 2 . To estima envelop of th 4(b), located analysis predi well with obs finite diamete indicates that step sizes Δx t during the ras and d-spacin coordinates. T size (i.e. 25 m o be calculated diffraction 2θ g can be calcula ed in Table 1) corresponding terns obtained e spectral flux rystallites pro of measuremen hese have bee example of rep agreement betw chloride) sho findings prese and the angula he 2θ measure determine the p Ǻ (220) plane i grams as illustr elsewhere [12] 4. (a) Conventiona ated by integrating oherent scattering z ate the spatial e diffracted flu at a mean dista icted a reconst servations of t er of the enco t the z-axis sp and Δy t . This ster scan as a ri ng resolution The transverse mm) for optical d. The centrall given by Eq. (3 ated for each v ) are presented g regions of in from the ICD measurements viding enhanc nts in the prese en shown to b peatability in m ween the diffr own in Fig. 4 ented elsewher ar spread of th ement of Δ2θ, precision of the in sodium chlo rated in Fig. 4(a ].
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Conclusions and future work
We have demonstrated coherent scattering tomography by raster scanning a conical shell beam through five spatially distributed samples contained on a polymer inspection tray. Measuring the angle of incidence and the energy of the diffracted photons from the samples, via a pinhole aperture optically coupled to a pixelated energy-resolving detector, enabled the calculation of material phase. Successive, depth-resolved snapshot exposures of 1 second enabled the (x,y,z) coordinate positions of the diffracted photons to be calculated and a volumetric image to be presented. Prior knowledge of sample position(s) was not required to calculate material specific lattice spacing information or d-spacing values in our experiment. Our imaging architecture can be setup to provide much greater spatial detail by reducing the snapshot pitch. By hypothesis, we anticipate the staring mode resolution [12] of the probe will limit spatial resolving power.
We believe our approach is potentially beneficial for fields including medicine and industrial process control. However, its application to checkpoint security scenarios is of immediate relevancy where ongoing work is optimizing our method for speed of operation. The materials information provided by our probe is orthogonal to the conventional Z-effective and density data provided by dual-energy computed tomography, as employed in checkpoint screening systems. Combining our XRD tomographic probe with dual-energy CT will provide new opportunities to further improve probability of detection and reduce false alarms in the presence of stream-of-commerce clutter.