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, α, β, γ. The reciprocal lattice will be described by parameters a∗, b∗, c∗, α∗, β∗, γ∗. Note that Mantid uses a∗=1a 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 Ql and Q with the following equation:
The B matrix transforms the h,k,l triplet into a Cartesian system, with the first axis along a∗, the second in the plane defined by a∗ and 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 (h,k,l) plane is given by
The distance in real space to the (h,k,l) plane is given by d=1d∗
The angle between Q1 and Q2 is given by cos(Q1,Q2)=(BQ1)(BQ2)|(BQ1)||(BQ2)|
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.