4.6. - Sectors
Sectors correspond to different symmetry transformations of (2θ,ω,χ,φ) and may be of help in avoiding blind spots. Sectors can also be useful for samples in cryostats or ovens. All modes can have the positions chosen for the motors further influenced by the choice of sector. However, the modes azimuth-fixed , "alpha-fixed" and "beta-fixed" only allow sectors numbered zero through three, below.
For a given pair of incident and scattered X-ray beams, k [i] and k [f], there are eight orientations of the crystal in the spectrometer that give the same scattering since they present equivalent projections to the incident and scattered beams. The eight orientations are labeled as sectors 0 through 7.
Four of the orientations come from the symmetries of a pair of vectors. They correspond to the identity operation (i.e., the current diffraction angles), a rotation of 180° about k [i], a rotation of 180° about the bisector of k [i] and k [f], and a rotation about z by 180° - 2θ. The last two symmetries are based on interchanging the role of the entrance angle and the exit angle of the X-rays on the sample. These four symmetries give sectors 0, 2, 4 and 6 in fourc. For each of these four positions another position can be obtained by increasing θ by 180°, decreasing φ by -180° and changing the sign of χ. Since a rotation of 180 and of -180 are the same, both θ and φ can be increased by 180. These orientations give sectors 1, 3, 5 and 7. Studying these operations shows that sectors 2, 3, 6 and 7 have opposite signs of 2θ from the current position. Sectors 2, 3, 4 and 6 have flipped the up direction of the sample (normal to the scattering plane) to the down direction.
The value of
g_sect determines in which sector of reciprocal space the diffractometer
operates.
The actual transformations of the angles are:
 |
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | |
|
|
|||||||||
| 2θ -> | |
2θ | 2θ | -2θ | -2θ | 2θ | 2θ | -2θ | -2θ |
| ω -> | |
ω | ω - 180° | -ω | 180° - ω | -ω | 180° - ω | ω | ω - 180° |
| χ -> | |
χ | -χ | χ - 180° | 180° - χ | 180° - χ | χ - 180° | -χ | χ |
| φ -> | |
φ | φ - 180° | φ | φ - 180° | φ - 180° | φ | φ - 180° | φ |
In addition, a sector 8 is defined to minimize |χ - 90°| and |φ|. It can be used when the χ and φ circles of the diffractometer are arc segments rather than a complete circle.