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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:

\[Q_l= \hbar \mathbf{k}_i^{} - \hbar \mathbf{k}_f\]

\[\Delta E_l= \frac{\hbar^2}{2m} (k_i^2 - k_f^2)\]

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

\[\begin{split}Q= 2 \pi\left(h \mathbf{a}^* + k \mathbf{b}^* +l \mathbf{c}^* \right) = \left(\begin{array}{c}
h \\
k \\
l
\end{array}\right)\end{split}\]

The \(UB_{}^{}\) matrix formalism relates \(Q_l^{}\) and \(Q_{}^{}\) with the following equation:

\[\begin{split}Q_l = 2 \pi R \cdot U \cdot B \left(\begin{array}{c}
h \\
k \\
l
\end{array}\right)\end{split}\]

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):

\[\begin{split}B = \left( \begin{array}{ccc}
a^* & b^*\cos(\gamma^*) & c^*\cos(\beta^*) \\
0 & b^*\sin(\gamma^*) & -c^*\sin(\beta^*)\cos(\alpha) \\
0 & 0 & 1/c \end{array} \right)\end{split}\]

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:

\[\begin{split}G^* = (UB)^T UB = B^T B = \left( \begin{array}{ccc}
a^*a^* & a^*b^*\cos(\gamma^*) & a^*c^*\cos(\beta^*) \\
a^*b^*\cos(\gamma^*) & b^*b^* & b^*c^*\cos(\alpha^*) \\
a^*c^*\cos(\beta^*) & b^*c^*\cos(\alpha^*) & c^*c^* \end{array} \right)\end{split}\]

\[\begin{split}G=(G^*)^{-1}=\left( \begin{array}{ccc}
aa & ab\cos(\gamma) & ac\cos(\beta) \\
ab\cos(\gamma) & bb & bc\cos(\alpha) \\
ac\cos(\beta) & bc\cos(\alpha) & cc \end{array} \right)\end{split}\]

The distance in reciprocal space to the \(\left(h,k,l\right)\) plane is given by

\[\begin{split}d^* =\left| B \left(\begin{array}{c}
h \\
k \\
l
\end{array}\right)\right|\end{split}\]

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:

\[\begin{split}U = \left( \begin{array}{ccc}
0 & 1 & 0 \\
0 & 0 & 1 \\
1 & 0 & 0 \end{array} \right)\end{split}\]

**Category**: Concepts