research papers 890 doi:10.1107/S0021889812030762 J. Appl. Cryst. (2012). 45, 890–901 Journal of Applied Crystallography ISSN 0021-8898 Received 19 January 2012 Accepted 5 July 2012 # 2012 International Union of Crystallography Printed in Singapore – all rights reserved Source misalignment effects on crystal collection solid angle and intensity profile Sayyed Jalal Pestehe a * and Golamreza Askari Germi b a Department of Physics, University of Tabriz, 29 Bahman, Tabriz, Eastern Azerbaijan, Iran, and b Department of Physics, Parsabad Moghan Branch, Islamic Azad University Parsabad, Iran. Correspondence e-mail: sjpest@tabrizu.ac.ir The X-ray optics of singly and doubly curved crystals are studied using a vector procedure and rotation matrices and general relations for the angular deviation from the Bragg angle over the crystal surface with a source aligned or misaligned on the Rowland circle. Hence, the effective scattering area, collection solid angle and diffracted X-ray intensity profile on the crystal surface are derived. The effective areas and the diffracted X-ray intensity profiles on the crystal surface for both aligned and misaligned source cases are plotted and compared. It is argued that the introduced point-focusing crystal configuration confirms the radii that have been obtained previously by a trial and error method by optimizing the crystal collection solid angle. 1. Introduction The requirement of the satisfaction of the Bragg condition for an incident beam on a crystal surface limits the design of crystals with high collection efficiency (Wittry & Sun, 1990a, 1992). It has been shown that doubly curved X-ray diffractors provide high collection efficiencies for monochromatic X-rays from small-dimension X-ray sources (Chen et al., 2008). These diffractors are commonly used for monochromating, focusing and reflecting X-rays and are employed in many instruments for local elemental analysis, such as the electron probe microanalyser, scanning electron microscope and X-ray fluorescence microprobe (Moslemzadeh et al., 2003; Wittry & Sun, 1990a). It is essential, therefore, to study methods of optimizing the reflection properties and solid-angle collection efficiencies of curved crystals. Different aspects of the X-ray optics of the doubly curved crystal diffractors, including the effective scattering area (Wittry & Sun, 1990a, 1992), the effect of source misalignment (Chang & Wittry, 1993; Wittry & Chang, 1992), the collection solid angle and the intensity profile on the crystal surface (Chang & Wittry, 1993), the focusing properties on the image plane (Wittry & Sun, 1990b), and the X-ray penetration effects on the effective area (Wittry & Sun, 1991), have been studied by Wittry and co-workers. They have used a vector procedure to derive a general relation for the incident angle deviation from the Bragg angle on the crystal surface. In the procedure of Wittry & Sun (1992), the normal vector to a given atomic plane at an arbitrary point on a crystal surface is obtained by rotating the normal vector from the origin of the defined coordinate system and the symmetry centre (midpoint) of the crystal in the vertical plane to the focal circle, about an axis going through the centre of vertical curvature of the crystal. It is obvious that this approach is valid for crystals with very small sizes. In this paper, following the procedure of Wittry and Sun, we use rotation matrices to obtain a nearly exact normal vector to a given atomic plane at an arbitrary point on the crystal surface. Using the derived nearly exact normal vector, a general equation for the collection solid angle of crystal diffractors is derived; then the effect of the source misalign- ment on the crystal effective scattering area and diffracted X-ray intensity profiles on the crystal surface are investigated and compared with the results of Chang & Wittry (1993). It has been shown that the results for the last four doubly curved crystal geometries of Table 1, viz. the 45  point-focusing, Wittry’s general point-focusing, and the Berreman and Johann point-focusing geometries, are different from the already published results. 2. Derivations We consider a toroidally curved crystal that is defined by two toroidal surfaces with horizontal radii R 1 ; R 0 1 and vertical dimensions R 2 ; R 0 2 , which are tangent at the crystal midpoint, M. Here the primed and unprimed radii belong to the diffracting atomic plane and the crystal surface that run through the midpoint of the crystal (the point M in Fig. 1). For the sake of clarity one quadrant of the crystal is shown in the figures where it is needed. It has to be mentioned that, as shown in Fig. 1, there are several other toroidal diffracting planes between the above-mentioned toroidal surfaces with different radii that are intersected by the toroid of the crystal surface. The adapted Cartesian coordinate system, with its origin located on the Rowland circle at the centre of the horizontal curvature R 0 1 , its Y axis running through the crystal midpoint perpendicularly, and its X axis lying on the focal circle plane (or Rowland circle) normal to the Y and Z axes, is shown in