4.3. - Orientation Matrix
Angle calculations for the four-circle diffractometer are described in detail in Busing and Levy. ( W.R. Busing and H.A. Levy, Acta Cryst. 22 , 457 (1967). Note however, that this paper uses right-handed coordinates systems and left-handed rotations for all rotations except for χ, which is right-handed. ) You may also refer to that paper to learn how to calculate the orientation matrix. The orientation matrix, UB, describes the sample orientation with respect to the diffractometer angles. Given UB, it is possible to calculate the diffractometer angles (2θ,θ,χ,φ) necessary to rotate a particular scattering vector Q , indexed by (H,K,L), into the diffraction position. The matrix B transforms the given (H,K,L) into an orthonormal coordinate system fixed in the crystal. The matrix U is the rotation matrix that rotates the crystal's reference frame into the spectrometer's.
The first step in constructing an appropriate orientation matrix is to enter the sample crystal lattice parameters a, b, c, α, β and γ. ( The conventional symbols for the crystal lattice angles include α and β. These angles are unrelated to the orientation angles α and β defined in the introduction. The different meanings should be clear from context. ) These are real-space parameters, as might be found in Wychoff ( R.W.G. Wychoff, Crystal Structures (Wiley, New York, 1964). ) or Pearson. ( P. Villars and L.D. Calvert, Pearson's Handbook of Crystallographic Data for Intermetallic Phases (American Society for Metals, Metals Park, Ohio, 1985). ) Use the macro
setlat to assign values:
1.FOURC> setlat 3.61 3.61 3.61 90 90 90
2.FOURC>
Next, you must specify the sets of values of (2θ,θ,χ,φ) at which two Bragg reflections are in the diffracting position. One of these is called the primary reflection. Fourc ensures that the values of (H,K,L) reported for the primary reflection agree (to within a scale factor) with the values entered. However, because of experimental errors and/or uncertainties in the unit cell parameters, the values of (H,K,L) reported for the other Bragg reflection, called the secondary reflection, may not agree perfectly with the entered values (although they should be close).
You can use the
or0 and
or1 macros
to enter the parameters
for the primary and secondary reflections, respectively.
However, the
or0 and
or1 macros require that the diffractometer be moved
to the associated reflections, as
these macros use
the current angles and the entered (H,K,L)
in the calculation of the orientation matrix.
Alternatively,
you can use
the
setor0 and
setor1 macros, which prompt for both (H,K,L)
and the angles that
define the orientation
matrix, without moving the spectrometer
to the given settings.