Angle calculations for a six-circle surface X-ray diffractometer

Abstract

Angle calculations are presented for a new type of diffractometer suitable for surface X-ray diffraction. The new geometry results from combining the four-circle and the z-axis geometries and involves six circles. Instruments based on this concept can be operated in different diffraction geometries. Compared to the four-circle and z-axis geometries, a larger fraction of the reciprocal space perpendicular to the sample is available to experiments when all six circles are used. This results in an improved out-of-plane resolution in any X-ray structure study. In each geometry, either the angle of incidence or the outgoing angle of the X-rays can be chosen. At the same time, the surface normal can be aligned in the horizontal diffractometer plane, providing optimum resolution and sample illumination. All equations are given in closed form. The different modes of operation are compared and operation schemes are discussed. Journal of Applied Crystallography

A (2+3)-Type Surface Diffractometer: Mergence of the z-Axis and (2+2)-Type Geometries

“…In addition, a large range of perpendicular momentum transfer is needed. Many diffractometer types are in use that fulfil these requirements in different ways: z-axis (Bloch, 1985), five-circle (Vlieg et al, 1987), six-circle (Lohmeier & Vlieg, 1993) and (2+2) types (Evans-Lutterodt & Tang, 1995;Renaud et al, 1995). In this paper we show that two common geometries, z-axis and (2 + 2)-circle, can be made fully equivalent by the addition of an azimuthal degree of freedom to the detector.…”

“…These can either be two tilt stages, or one tilt stage together with a rotation axis. When using a z-axis geometry the two options are equivalent, but the rotation/tilt combination does allow the diffractometer to be used in a six-circle mode, which enhances the area of reciprocal space that can be accessed (Lohmeier & Vlieg, 1993). A complete horizontal-axis diffractometer would thus in total require seven degrees of freedom, in a (4 + 3) geometry, of which only (2 + 3) are needed after the initial sample alignment.…”

“…The reciprocal-lattice point is given by H = hbl + kb2 + /b3, where (hkl) are the Miller indices and bi the reciprocal-lattice vectors. As a convention, the Miller index l will be used to denote the direction of perpendicular momentum transfer Q±, while h and k define the in-plane momentum transfer QII" The solution of (1) with the appropriate boundary condition of fixed incoming or outgoing angle has been given by Bloch (1985) and Lohmeier & Vlieg (1993), for the z-axis diffractometer, and by Evans-Lutterodt & Tang (1995) for the (2 + 2) geometry.…”

DESY NanoLab

“…The most common one used is for samples with a well-de ned face, such as polished single crystal substrates or thin lms, with their surface mounted horizontally in the lab frame (Lohmeier & Vlieg, 1993). In particular, the di ractometer is suited for measurements with a xed angle of incidence with respect to the sample surface, which is bene cial for the signal-to-background ratio when measuring surface sensitive crystal truncation rods.…”

A new x-ray diffraction method for structural investigations of solid-liquid interfaces

“…37 Due to its high brilliance, this beamline is well suited for these studies. The UHV chamber is mounted onto the horizontal diffractometer, which is operated in the z-axis mode 38 or in the reflectivity mode. 39 In the z-axis mode different values of the perpendicular momentum transfer along the CTRs are selected by varying the exit angle ␣ f , while keeping the angle of incidence ␣ i fixed below the critical angle for total reflection ͑typically 0.07°-0.12°͒.…”