International
Tables for
Crystallography
Volume H
Powder diffraction
Edited by C. J. Gilmore, J. A. Kaduk and H. Schenk

International Tables for Crystallography (2018). Vol. H, ch. 2.5, pp. 121-122

Section 2.5.2.2. Detector space and pixel position

B. B. Hea*

aBruker AXS Inc., 5465 E. Cheryl Parkway, Madison, WI 53711, USA
Correspondence e-mail: bob.he@bruker.com

2.5.2.2. Detector space and pixel position

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A typical 2D detector has a limited detection surface, and the detection surface can be spherical, cylindrical or flat. Spherical or cylindrical detectors are normally designed for a fixed sample-to-detector distance, while a flat detector has the flexibility to be used at different sample-to-detector distances so as to choose either high resolution at a large distance or large angular coverage at a short distance.

2.5.2.2.1. Detector position in the laboratory system

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The position of a flat detector is defined by the sample-to-detector distance D and the detector swing angle α. D and α are referred to as the detector-space parameters. D is the perpendicular distance from the goniometer centre to the detection plane and α is a right-handed rotation angle about the ZL axis. Detectors at different positions in the laboratory coordinates XL, YL, ZL are shown in Fig. 2.5.4[link]. The centre of detector 1 is right on the positive side of the XL axis (on-axis), α = 0. Both detectors 2 and 3 are rotated away from the XL axis with negative swing angles (α2 < 0 and α3 < 0). The detection surface of a flat 2D detector can be considered as a plane, which intersects the diffraction cone to form a conic section. Depending on the swing angle α and the 2θ angle, the conic section can appear as a circle, an ellipse, a parabola or a hyperbola.

[Figure 2.5.4]

Figure 2.5.4 | top | pdf |

Detector positions in the laboratory-system coordinates.

2.5.2.2.2. Pixel position in diffraction space for a flat detector

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The values of 2θ and γ can be calculated for each pixel in the frame. The calculation is based on the detector-space parameters and the pixel position in the detector. Fig. 2.5.5[link] shows the relationship of a pixel P(x, y) to the laboratory coordinates XL, YL, ZL. The position of a pixel in the detector is defined by the (x, y) coordinates, where the detector centre is defined as x = y = 0. The diffraction-space coordinates (2θ, γ) for a pixel at P(x, y) are given by[\eqalignno{2\theta &= \arccos {{x\sin \alpha + D\cos \alpha } \over {( {{D^2} + {x^2} + {y^2}})^{1/2}}}\quad(0 \,\lt\, 2\theta \,\lt\, \pi), &(2.5.6)\cr \gamma &= {{x\cos \alpha - D\sin \alpha } \over {\left| {x\cos \alpha - D\sin \alpha } \right|}}\arccos {{ - y} \over {[{{y^2} + {{(x\cos \alpha - D\sin \alpha)}^2}})]^{1/2} }}&\cr &\quad\quad(- \pi \,\lt\, \gamma \le \pi). &(2.5.7)}]

[Figure 2.5.5]

Figure 2.5.5 | top | pdf |

Relationship between a pixel P and detector position in the laboratory coordinates.

2.5.2.2.3. Pixel position in diffraction space for a curved detector

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The conic sections of the diffraction cones with a curved detector depend on the shape of the detector. The most common curved detectors are cylinder-shaped detectors. The diffraction frame measured by a cylindrical detector can be displayed as a flat frame, typically a rectangle. Fig. 2.5.6[link](a) shows a cylindrical detector in the vertical direction and the corresponding laboratory coordinates XL, YL, ZL. The sample is located at the origin of the laboratory coordinates inside the cylinder. The incident X-rays strike the detector at a point O if there is no sample or beam stop to block the direct beam. The radius of the cylinder is R. Fig. 2.5.6[link](b) illustrates the 2D diffraction image collected with the cylindrical detector. We take the point O as the origin of the pixel position (0, 0). The diffraction-space coordinates (2θ, γ) for a pixel at P(x, y) are given by[\eqalignno{2\theta &= \arccos \left [R\cos \left({x\over R}\right) \big/({R^2} + {y^2})^{1/2} \right],&(2.5.8)\cr \gamma &= {x\over |x|}\arccos \left \{ - y\big/\left[ y^2 + R^2 \sin ^2\left({x \over R}\right)\right]^{1/2} \right\}\quad(-\pi \,\lt\, \gamma\leq \pi).&\cr &&(2.5.9)}]The pixel-position-to-(2θ, γ) conversion for detectors of other shapes can also be derived. Once the diffraction-space coordinates (2θ, γ) of each pixel in the curved 2D detector are determined, most data-analysis algorithms developed for flat detectors are applicable to a curved detector as well.

[Figure 2.5.6]

Figure 2.5.6 | top | pdf |

Cylinder-shaped detector in vertical direction: (a) detector position in the laboratory coordinates; (b) pixel position in the flattened image.








































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