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# Lattice¶

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.

## Theory¶

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 \(\cos( Q_1^{}, Q_2^{})=\frac{(BQ_1)(BQ_2)}{|(BQ_1)| |(BQ_2)|}\)

## Unit cells¶

The `UnitCell`

class provides the functions to access direct and
reciprocal lattices.

## Oriented lattices¶

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.

## Note about orientation¶

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