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. 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 | 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 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

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