spec

Contents → FOUR-CIRCLE REFERENCE → Introduction

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When invoked by the name fourc, spec runs with code appropriate for a four-circle diffractometer. This section of the Reference Manual focuses on the features of spec unique to the fourc version.

The four circles of the standard four-circle diffractometer are: 2θ, the angle through which the beam is scattered, and θ, χ, and φ, the three Euler angles, which orient the sample. Of these three, θ is the outermost circle with its axis of rotation coincident with that of 2θ. The χ circle is mounted on the θ circle, with its axis of rotation perpendicular to the θ axis. The φ circle is mounted on the χ circle such that its axis of rotation lies in the plane of the χ circle.

From the keyboard and on the screen, the angles are named tth, th, chi and phi, respectively, and conventionally referred to in that order. For fourc to work properly, angles with these names must be configured.

In describing the operation of a four-circle diffractometer, it is convenient to consider three coordinate systems: 1) a frame fixed in the laboratory, 2) a frame fixed on the spectrometer and 3) the natural axes of the sample. Note that fourc uses right-handed coordinate systems. All rotations are right-handed except for the χ rotation.

The x-y plane of the laboratory coordinate system is called the scattering plane and contains the sample and the points reached by the detector as it rotates on the 2θ arm. A counter-clockwise rotation of the 2θ axis corresponds to increasing 2θ, with the 2θ rotation axis defining the positive z direction in the laboratory. The zero of 2θ is defined as the setting at which the undeflected X-ray beam hits the detector.

The positive y axis is along the line from the sample to the X-ray source. The position at which θ rotates the χ circle to put the χ rotation axis along the y axis defines the zero of θ. A clockwise rotation of χ corresponds to increasing χ.

The zero of χ is the position which puts the φ rotation axis along the positive z axis. The positive x axis direction is determined by the cross product of the y and z axes ( x=y×z ). x^{^}=y^{^}×z^{^} ), which completes the definition of the right-handed coordinate system.

It is important to note that the zeroes of 2θ, θ and χ and the direction of positive rotation of all the circles must be set as described above and cannot be freely redefined.

The spectrometer coordinate system is defined as a right-handed system fixed on the φ rotation stage at the sample position such the coordinate system is aligned with the laboratory coordinate system when all four spectrometer angles are zero. This definition determines the zero of φ.

The third coordinate system is aligned with specific directions in the sample. A common and useful example are coordinates defined as the lattice vectors of a crystalline sample. When placing a sample in the spectrometer, it is unlikely that its axes will line up with the spectrometer axes. Nevertheless, fourc allows the sample orientation to be fully specified by finding the angles at which two Bragg peaks are detected and giving the corresponding reciprocal lattice indices. This process is described fully in the section on the Orientation Matrix.

To orient a sample so as to measure the intensity at a particular reciprocal lattice position requires that the reciprocal lattice vector of interest is aligned with the scattering vector of the spectrometer. Since any rotation about the scattering vector does not change the diffraction condition, there is a high degree of degeneracy that must be resolved in order for fourc to determine unique angle settings. How the degeneracy is lifted described in the section on Four-Circle Modes.

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