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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.
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