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The purpose of this document is to explain how Mantid is using information about unit cells and their orientation with respect to the laboratory frame. For a detailed description, see the UB matrix implementation notes.
The physics of a system studied by neutron scattering is described by the conservation of energy and momentum. In the laboratory frame:
Note that the left side in the above equations refer to what is happening to the lattice, not to the neutron.
Let’s assume that we have a periodic lattice, described by lattice parameters a,\ b,\ c,\ \alpha,\ \beta,\ \gamma. The reciprocal lattice will be described by parameters a^*,\ b^*,\ c^*,\ \alpha^*,\ \beta^*,\ \gamma^*. Note that Mantid uses a^*=\frac{1}{a} type of notation, like in crystallography.
For such a lattice, the physics will be described in terms of reciprocal lattice parameters by
The UB_{}^{} matrix formalism relates Q_l^{} and Q_{}^{} with the following equation:
The B_{}^{} matrix transforms the h^{}_{}, k, l triplet into a Cartesian system, with the first axis along \ \mathbf{a}^*, the second in the plane defined by \ \mathbf{a}^* and \ \mathbf{b}^*, and the third axis perpendicular to this plane. In the Busing and Levi convention (W. R. Busing and H. A. Levy, Angle calculations for 3- and 4-circle X-ray and neutron diffractometers - Acta Cryst. (1967). 22, 457-464):
The U_{}^{} matrix represents the rotation from this Cartesian coordinate frame to the Cartesian coordinate frame attached to the innermost axis of the goniometer that holds the sample.
The R_{}^{} matrix is the rotation matrix of the goniometer
Other useful equations:
The distance in reciprocal space to the \left(h,k,l\right) plane is given by
The distance in real space to the \left(h,k,l\right) plane is given by d=\frac{1}{d^*}
The angle between Q_1^{} and Q_2^{} is given by \cos( Q_1^{}, Q_2^{})=\frac{(BQ_1)(BQ_2)}{|(BQ_1)| |(BQ_2)|}
All the functions defined for UnitCell
are inherited by the
OrientedLattice
objects. In addition, functions for manipulating
the U and UB matricies are also provided.
Most of the instruments have incident beam along the \mathbf{z} direction. For an orthogonal lattice with \mathbf{a}^* along \mathbf{z}, \mathbf{b}^* along \mathbf{x}, and \mathbf{c}^* along \mathbf{y}, the U^{}_{} matrix has the form:
Category: Concepts