Spatial descriptions

Single crystal diffraction software often refers to several different spatial coordinate systems. Common conventions are summarized below.

Laboratory reference frame

As the name suggests, the laboratory reference frame refers to the coordinate system attached to the instrument. By convention, the \(z\)-axis is defined as the direction corresponding to the incident beam and the \(y\)-axis is the verical upward direction. The \(x\)-axis forms the right-handed set.

• \(x=\mathrm{horizontal}\)

• \(y=\mathrm{vertical}\)

• \(z=\mathrm{beam}\)

Coordinate system used for the incident and scattered beam.

It sometimes shortened to Q-lab. This frame can be convenientally described using a spherical coordinate system with polar angle corresponding to the \(2\theta\) scattering angle and \(\phi\) called the azimuthal angle. Alternatively, it can be described by a second set of spherical coordinates with \(\gamma\) as the in-plane angle and \(\nu\) the out of plane angle.

\[\begin{split}\begin{bmatrix} Q_x \\ Q_y \\ Q_z \end{bmatrix} = \frac{2\pi}{\lambda} \begin{bmatrix} \sin{2\theta}\cos{\phi} \\ \sin{2\theta}\sin{\phi} \\ \cos{2\theta}-1 \end{bmatrix} = \frac{2\pi}{\lambda} \begin{bmatrix} \cos\nu\sin\gamma \\ \sin\nu \\ \cos\nu\cos\gamma-1 \end{bmatrix}\end{split}\]

The above definition of the momentum transfer is in the crystallographic convention. That is, \(\boldsymbol{Q}=\boldsymbol{k}_f-\boldsymbol{k}_i\). In spectroscopy experiments, the vector is often reversed which is referred to as the inelastic convention. In terms of \(hkl\), the momentum transfer in this frame can be expressedin terms of the goniometer matrix \(R\) and \(UB\) matrix.

\[\begin{split}\begin{bmatrix} Q_x \\ Q_y \\ Q_z \end{bmatrix} = 2\pi \begin{bmatrix} R_{11} & R_{12} & R_{13} \\ R_{21} & R_{22} & R_{23} \\ R_{31} & R_{32} & R_{33} \end{bmatrix} \begin{bmatrix} UB_{11} & UB_{12} & UB_{13} \\ UB_{21} & UB_{22} & UB_{23} \\ UB_{31} & UB_{32} & UB_{33} \end{bmatrix} \begin{bmatrix} h \\ k \\ l \end{bmatrix}\end{split}\]

Goniometer conventions

The goniometer rotation matrix is calculated according the the motion and settings of the motors. The matrix can be decomposed into elementary rotations about certain axes using Euler angles. A common convention is \(yzy\) with angles \((\omega,\chi,\varphi)\) as intrinsic rotations.

• \(R_y(\omega)=\mathrm{outermost}\)

• \(R_y(\varphi)=\mathrm{innermost}\)

The counterclockwise rotations about each axis are considered positive. This is the right-hand convention.

\[R(\omega,\chi,\varphi)=R_y(\omega)R_z(\chi)R_y(\varphi)=\]

\[\begin{split}= \begin{bmatrix} \cos\omega & 0 & \sin\omega \\ 0 & 1 & 0 \\ -\sin\omega & 0 & \cos\omega \end{bmatrix} \begin{bmatrix} \cos\chi & -\sin\chi & 0 \\ \sin\chi & \cos\chi & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} \cos\varphi & 0 & \sin\varphi \\ 0 & 1 & 0 \\ -\sin\varphi & 0 & \cos\varphi \end{bmatrix}\end{split}\]

Goniometer rotations \(yzy\) convention.

Sample reference frame

To cover large regions of reciprocal space, it is necessary to re-orient the sample using the goniometer motors. A reference frame independent of the orientation is one attached directly to the sample called the sample reference frame or Q-sample. This is the frame the data is projected into that can be used determine the \(U\) matrix which describes how the sample is mounted relative to the goniometer and the \(B\) matrix which converts the possibly non-orthogonal crystal axes to a Cartesian system.

\[\begin{split}\begin{bmatrix} Q_1 \\ Q_2 \\ Q_3 \end{bmatrix} = 2\pi \begin{bmatrix} U_{11} & U_{12} & U_{13} \\ U_{21} & U_{22} & U_{23} \\ U_{31} & U_{32} & U_{33} \end{bmatrix} \begin{bmatrix} B_{11} & B_{12} & B_{13} \\ 0 & B_{22} & B_{23} \\ 0 & 0 & B_{33} \end{bmatrix} \begin{bmatrix} h \\ k \\ l \end{bmatrix}\end{split}\]

Crystallographic frame

The crystallographic axes are described by lattice parameters with angles that are not necessarily right angles (hexagonal, rhombohedral, monoclinc, or triclinic lattice system). It is convenient to use a metric tensor convenient for describing distances and angles. If each crystallographic axis \(\boldsymbol{a}\), \(\boldsymbol{b}\), and \(\boldsymbol{c}\), then the dot product of each vector with all other vectors including itself can be used to construct the metric tensor, a 3x3 matrix.

\[\begin{split}G = \begin{bmatrix} \boldsymbol{a}\cdot\boldsymbol{a} & \boldsymbol{a}\cdot\boldsymbol{b} & \boldsymbol{a}\cdot\boldsymbol{c} \\ \boldsymbol{b}\cdot\boldsymbol{a} & \boldsymbol{b}\cdot\boldsymbol{b} & \boldsymbol{b}\cdot\boldsymbol{c}\\ \boldsymbol{c}\cdot\boldsymbol{a} & \boldsymbol{c}\cdot\boldsymbol{b} & \boldsymbol{c}\cdot\boldsymbol{c}\end{bmatrix}= \begin{bmatrix} a^2 & ab\cos{\gamma} & ac\cos{\beta} \\ ba\cos{\gamma}& b^2 & bc\cos{\alpha} \\ ca\cos{\beta} & cb\cos{\alpha} & c^2 \end{bmatrix}\end{split}\]

This matrix has useful properties. For example, multiply by any fractional coordinate \((x,y,z)\) to obtain

Also, its determinant is the squared unit cell volume. That is, \(V^2=\det{G}\).

An analagous matrix exists for the reciprocal lattice with the intuitive property \(G^\ast=G^{-1}\).

\[\begin{split}G^\ast = \begin{bmatrix} \boldsymbol{a}^\ast\cdot\boldsymbol{a}^\ast & \boldsymbol{a}^\ast\cdot\boldsymbol{b}^\ast & \boldsymbol{a}^\ast\cdot\boldsymbol{c}^\ast \\ \boldsymbol{b}^\ast\cdot\boldsymbol{a}^\ast & \boldsymbol{b}^\ast\cdot\boldsymbol{b}^\ast & \boldsymbol{b}^\ast\cdot\boldsymbol{c}^\ast\\ \boldsymbol{c}^\ast\cdot\boldsymbol{a}^\ast & \boldsymbol{c}^\ast\cdot\boldsymbol{b}^\ast & \boldsymbol{c}^\ast\cdot\boldsymbol{c}^\ast\end{bmatrix}= \begin{bmatrix} (a^\ast)^2 & a^\ast b^\ast\cos{\gamma^\ast} & a^\ast c^\ast\cos{\beta^\ast} \\ b^\ast a^\ast\cos{\gamma^\ast}& (b^\ast)^2 & b^\ast c^\ast\cos{\alpha^\ast} \\ c^\ast a^\ast\cos{\beta^\ast} & c^\ast b^\ast\cos{\alpha^\ast} & (c^\ast)^2 \end{bmatrix}\end{split}\]

Illustration of real and reciprocal lattice vectors with Cartesian system.

Reciprocal space coverage

The extent to which reciprocal space is coverged by a particular instrument is determined by several factors. Not only is the the detector coverage of the instrument involved but also the incident wavelength. When the beam is white, that is, the incident beam contains a finite bandwith of netruons with various energies or neutrons, it is referred to as the Laue technique. Conversely, when the beam is monochromatic, the incident beam essentially contains a single wavelength. The coverage can be illustrated through the Ewald sphere with detector coverage spanning from \(\theta_{\text{min}}\) to \(\theta_{\text{max}}\).

Laue (white beam)

Given the incident bandwidth \(\lambda_{\text{min}}\) to \(\lambda_{\text{max}}\), a single orientation produces many instances of the Bragg condition. A finite rotation of the crystal results in movement of the peaks across the detector with diffraction condition satistfied at a different wavelength than the previous within the wavelength band.

Laue: two-dimensional representation of the Ewald sphere showing the volume coverage from a single crystal orientation with a finite bandwidth of incident neutrons.

Although the Laue technique is more efficient, the presence of an incident beam with a spectrum composed of various wavelengths means that appopriate normalization is performed based on the incident flux. In addition, any correction with wavelength dependence (Lorentz correction, absorption correction, primary and secondary extinction correction) become more difficult.

Rotating crystal (monochromatic beam)

In the monochromatic case, the incident wavelength \(\lambda\) is essentially fixed and it is necessary to rotate the crystal through some scan range. Thus, rotation of the crystal brings peaks at fixed locations on the detector in and out of the diffraction condition from a single wavelength.

Rotating crystal: two-dimensional representation of the Ewald sphere showing the volume coverage from a single incident wavelength and finite rotation of the crystal (detector).

The rotating crystal technique with monchromatic beam is less efficient at mapping the reciprocal space volume for the same detector coverage, however, the correction of intensities is much more straightforward.

Modulation vectors

The reciprocal lattice vector (3+3) is denoted with six integer \((h,k,l,m,n,p)\) indices. The first three \((h,k,l)\) index peaks of the parent while the last three \((m,n,p)\) index satellite peaks away from the main peak.

\[\frac{\boldsymbol{Q}}{2\pi} = h\boldsymbol{a}^\ast + k\boldsymbol{b}^\ast + l\boldsymbol{c}^\ast + m\boldsymbol{k}_1 + n\boldsymbol{k}_2 + p\boldsymbol{k}_3\]

Satellite peak offset from a main peak.

The three \(\boldsymbol{k}_i\)-vectors, \(i=1,2,3\) are referenced to the parent reciprocal lattice vectors using fractional offsets \((\Delta{h}_i,\Delta{k}_i,\Delta{l}_i)\).

\[\begin{split}\boldsymbol{k}_1& = \Delta{h}_1\boldsymbol{a}^\ast + \Delta{k}_1\boldsymbol{b}^\ast + \Delta{l}_1\boldsymbol{c}^\ast \\ \boldsymbol{k}_2& = \Delta{h}_2\boldsymbol{a}^\ast + \Delta{k}_2\boldsymbol{b}^\ast + \Delta{l}_2\boldsymbol{c}^\ast \\ \boldsymbol{k}_3& = \Delta{h}_3\boldsymbol{a}^\ast + \Delta{k}_3\boldsymbol{b}^\ast + \Delta{l}_3\boldsymbol{c}^\ast\end{split}\]

The fractional indices are calculated using the six integer parameters and modulated matrix \(\Delta{(h,k,l)}\) formed by the offsets of its columns.

\[\begin{split}\begin{bmatrix} h \\ k \\ l \end{bmatrix}+ \begin{bmatrix} \Delta{h}_1 & \Delta{h}_2 & \Delta{h}_3 \\ \Delta{k}_1 & \Delta{k}_2 & \Delta{k}_3 \\ \Delta{l}_1 & \Delta{l}_2 & \Delta{l}_3 \end{bmatrix} \begin{bmatrix} m \\ n \\ p \end{bmatrix}\end{split}\]

Without cross terms, fractional indices are indexed with \((m,0,0)\), \((0,n,0)\), and \((0,0,p)\) where integers \(m\), \(n\), and \(p\) range from negative to positive maximum order excluding zero. With cross terms, they are indexed with \((m,n,p)\) where \(m\), \(n\), and \(p\) integers range from negative to positive maximum order including zero except \(m=n=p=0\). The wavevector in the sample frame can be written in terms of the \(UB\) matrix with modulated \(\tilde{UB}\) matrix extension along columns.

\[\begin{split}\frac{Q}{2\pi}= \begin{bmatrix} UB_{11} & UB_{12} & UB_{13} & \tilde{UB}_{11} & \tilde{UB}_{12} & \tilde{UB}_{13}\\ UB_{21} & UB_{22} & UB_{23} & \tilde{UB}_{21} & \tilde{UB}_{22} & \tilde{UB}_{23}\\ UB_{31} & UB_{32} & UB_{33} & \tilde{UB}_{31} & \tilde{UB}_{32} & \tilde{UB}_{33} \end{bmatrix} \begin{bmatrix} h \\ k \\ l \\ m \\ n \\ p \end{bmatrix}\end{split}\]

The modulated \(\tilde{UB}\) matrix is the multiplication of the \(UB\) matrix and modulated offset matrix \(\Delta{(h,k,l)}\).

\[\begin{split}\tilde{UB}= \begin{bmatrix} UB_{11} & UB_{12} & UB_{13} \\ UB_{21} & UB_{22} & UB_{23} \\ UB_{31} & UB_{32} & UB_{33} \\ \end{bmatrix} \begin{bmatrix} \Delta{h}_1 & \Delta{h}_2 & \Delta{h}_3 \\ \Delta{k}_1 & \Delta{k}_2 & \Delta{k}_3 \\ \Delta{l}_1 & \Delta{l}_2 & \Delta{l}_3 \end{bmatrix}\end{split}\]

It is possible to apply a cell transformation to the indexing. In this case, the \((m,n,p)\) remain fixed after the transform and the integer \((h,k,l)\) and \(\Delta{(h,k,l)}\) are multiplied by a transformation matrix.

\[\begin{split}\begin{align} \begin{bmatrix} h^\prime \\ k^\prime \\ l^\prime \end{bmatrix}&= \begin{bmatrix} T_{11} & T_{12} & T_{13} \\ T_{21} & T_{22} & T_{23} \\ T_{31} & T_{32} & T_{33} \\ \end{bmatrix} \begin{bmatrix} h \\ k \\ l \end{bmatrix}\\ \begin{bmatrix} \Delta{h}_1^\prime & \Delta{h}_2^\prime & \Delta{h}_3^\prime \\ \Delta{k}_1^\prime & \Delta{k}_2^\prime & \Delta{k}_3^\prime \\ \Delta{l}_1^\prime & \Delta{l}_2^\prime & \Delta{l}_3^\prime \end{bmatrix}&= \begin{bmatrix} T_{11} & T_{12} & T_{13} \\ T_{21} & T_{22} & T_{23} \\ T_{31} & T_{32} & T_{33} \\ \end{bmatrix} \begin{bmatrix} \Delta{h}_1 & \Delta{h}_2 & \Delta{h}_3 \\ \Delta{k}_1 & \Delta{k}_2 & \Delta{k}_3 \\ \Delta{l}_1 & \Delta{l}_2 & \Delta{l}_3 \end{bmatrix} \end{align}\end{split}\]

Illustration of cell transformation with modulation vectors.