Direct Lattice in Crystallography#

This page summarises the vector mathematics underlying the direct lattice representation used throughout ad_hoc_diffractometer. See the lattice module for the implementation.

Lattice vectors#

The unit cell of a crystal is described by three edge vectors A, B, C expressed in a Cartesian frame. With A along the \(x\)-axis and B in the \(xy\)-plane, the standard crystallographic choice is:

Vector

Cartesian components

A

\((a,\ 0,\ 0)\)

B

\((b\cos\gamma,\ b\sin\gamma,\ 0)\)

C

\((c\cos\beta,\ c\,\tfrac{\cos\alpha - \cos\beta\cos\gamma}{\sin\gamma},\ c\,v)\)

where

\[v = \frac{\sqrt{1 - \cos^2\!\alpha - \cos^2\!\beta - \cos^2\!\gamma + 2\cos\alpha\cos\beta\cos\gamma}}{\sin\gamma}\]

and \(a, b, c\) are the unit-cell edge lengths; \(\alpha, \beta, \gamma\) are the inter-edge angles (\(\alpha\) between B and C, \(\beta\) between A and C, \(\gamma\) between A and B).

Unit cell volume#

The unit cell volume is the scalar triple product of the three lattice vectors:

\[V = \mathbf{A} \cdot (\mathbf{B} \times \mathbf{C})\]

The cross product is evaluated first (standard vector-mathematics precedence), giving a vector perpendicular to B and C; the dot product with A then yields the signed volume of the parallelepiped spanned by the three vectors. With the lattice vectors defined above, this reduces to:

\[V = a\,b\,c\,v\,\sin\gamma\]

Reciprocal lattice#

The reciprocal lattice vectors \(\mathbf{b}_1, \mathbf{b}_2, \mathbf{b}_3\) satisfy the orthogonality condition

\[\mathbf{b}_i \cdot \mathbf{a}_j = 2\pi\,\delta_{ij}\]

where \(\mathbf{a}_1 = \mathbf{A}\), \(\mathbf{a}_2 = \mathbf{B}\), \(\mathbf{a}_3 = \mathbf{C}\). Each reciprocal vector includes the \(2\pi\) factor (the convention used in ad_hoc_diffractometer and by Busing & Levy 1967).

The B matrix#

The B matrix encodes the reciprocal lattice vectors as its columns:

\[\mathbf{B} = [\mathbf{b}_1\ \mathbf{b}_2\ \mathbf{b}_3]\]

It maps Miller indices \(\mathbf{h} = (h, k, l)^T\) to the scattering vector in Cartesian crystal-frame coordinates (Busing & Levy 1967, eq. 3):

\[\mathbf{Q}_c = \mathbf{B}\,\mathbf{h}\]

The magnitude \(|\mathbf{Q}_c| = |\mathbf{B}\,\mathbf{h}| = 2\pi / d_{hkl}\), where \(d_{hkl}\) is the interplanar spacing.

Explicit construction#

The B matrix is built from the reciprocal-lattice magnitudes \(a^\star\), \(b^\star\), \(c^\star\) and the reciprocal-cell angles \(\alpha^\star\), \(\beta^\star\), \(\gamma^\star\), using the explicit upper-triangular form (eq. 3 of Busing & Levy 1967):

B[0,0] = a*               B[0,1] = b* cos γ*        B[0,2] = c* cos β*
B[1,0] = 0                B[1,1] = b* sin γ*        B[1,2] = -c* sin β* cos α
B[2,0] = 0                B[2,1] = 0                B[2,2] = 2π / c

This places the reciprocal-lattice \(\mathbf{a}^\star\) along crystal-Cartesian \(\hat{x}\), \(\mathbf{b}^\star\) in the crystal-\(xy\)-plane (with positive \(y\) component), and \(\mathbf{c}^\star\) completing the right-handed orthonormal triple. The B matrix is therefore upper-triangular by construction for every crystal system.

The reciprocal-lattice magnitudes carry the \(2\pi\) factor (BL1967 convention), giving \(|\mathbf{B}\,\mathbf{h}| = 2\pi / d_{hkl}\) and \(\mathbf{b}_i \cdot \mathbf{a}_j = 2\pi\,\delta_{ij}\).

Cross-solver equivalence#

This construction is numerically identical (to machine precision) to the B matrix used by

  • SPEC (psic and other four-circle modes),

  • hkl_soleil (the libhkl library) with its default HKL_TAU = 2π compile-time setting,

  • diffcalc-core (Crystal._set_reciprocal_cell).

The equivalence holds for every crystal system, including non-orthogonal cells (hexagonal, trigonal, monoclinic, triclinic). For orthogonal cells (cubic, tetragonal, orthorhombic), the direct-lattice vectors \(\mathbf{a}\), \(\mathbf{b}\), \(\mathbf{c}\) are parallel to the reciprocal vectors \(\mathbf{a}^\star\), \(\mathbf{b}^\star\), \(\mathbf{c}^\star\) respectively, so the BL1967 crystal-Cartesian frame coincides with the “direct-lattice-\(\mathbf{a}\)-along-\(\hat{x}\)” frame. For non-orthogonal cells the two frames differ by a rotation in the crystal-Cartesian space; both encode the same physical crystal, but only the BL1967 frame is the one SPEC and hkl_soleil cross-validate against.

The no-\(2\pi\) alternative (\(\mathbf{b}_i \cdot \mathbf{a}_j = \delta_{ij}\), \(|\mathbf{B}\,\mathbf{h}| = 1/d_{hkl}\)), used by FullProf, CrysFML, parts of CCP4, and libhkl compiled with HKL_TAU = 1, is not exposed by this package.

Reference#