Formulation#

Intersection of lattices#

We define a lattice for basis \(\mathbf{B} = (\mathbf{b}_{1}, \dots, \mathbf{b}_{n})\) as

\[ \mathcal{L}(\mathbf{B}) := \left\{ \sum_{i=1}^{n} n_{i} \mathbf{b}_{i} \mid n_{i} \in \mathbb{Z} (\forall i) \right\}.\]

For lattice basis \(\mathbf{B}\), we define the dual basis as

\[\begin{split} \mathbf{D} &:= \mathbf{B} (\mathbf{B}^{\top} \mathbf{B})^{-1} =: (\mathbf{d}_{1}, \dots, \mathbf{d}_{n}) \\ \mathbf{D}^{\top} \mathbf{B} &= \mathbf{I}.\end{split}\]

We write the dual lattice of \(\mathcal{L}(\mathbf{B})\) as

\[ \mathcal{L}(\mathbf{B})^{\ast} := \mathcal{L}(\mathbf{D}).\]

The intersection of two lattices can be obtained as follows 1:

\[ \mathcal{L}(\mathbf{B}) \cap \mathcal{L}(\mathbf{B}') = \mathcal{L}([\mathbf{D} \mid \mathbf{D}'])^{\ast} \quad (\mathcal{L}(\mathbf{D}) := \mathcal{L}(\mathbf{B})^{\ast}, \mathcal{L}(\mathbf{D}') := \mathcal{L}(\mathbf{B}')^{\ast})\]

Coincidence-site lattice (CSL)#

Set up#

Consider two lattices, \(L_{\mathbf{A}}\) and \(L_{\mathbf{A}'}\), composed of lattice vectors \(\mathbf{A} = (\mathbf{a}_{1}, \mathbf{a}_{2}, \mathbf{a}_{3})\) and \(\mathbf{A}' = (\mathbf{a}_{1}', \mathbf{a}_{2}', \mathbf{a}_{3}')\),

\[ L_{\mathbf{A}} := \left\{ \mathbf{An} \mid \mathbf{n} \in \mathbb{Z}^{3} \right\}.\]

The two lattices are related by a rotation matrix \(\mathbf{R}\),

\[ \mathbf{A}' = \mathbf{R} \mathbf{A}.\]

Rodrigue’s formula#

\[\begin{split} \mathbf{R} &= \mathbf{I} + \mathbf{K} \sin \theta + \mathbf{K}^{2} (1 - \cos \theta) \\ \mathbf{K} &= \begin{pmatrix} 0 & -k_{z} & k_{y} \\ k_{z} & 0 & -k_{x} \\ -k_{y} & k_{x} & 0 \end{pmatrix},\end{split}\]

where \(\mathbf{k}\) is a unit vector along the rotation axis.

\[\begin{split} \overline{\mathbf{R}} &:= \mathbf{A}^{-1} \mathbf{RA} \\ \mathbf{A}' &= \mathbf{RA} = \mathbf{A} \overline{\mathbf{R}}\end{split}\]

Rodrigues vector

\[\begin{split} \mathbf{\rho}^{R} &:= \tan \frac{\theta}{2} (k_{z} \, k_{y} \, k_{z})^{\top} \\ (1 + (\rho^{R})^{2}) R_{ij} &= (1 - (\rho^{R})^{2}) \delta_{ij} + 2\left( \rho^{R}_{i} \rho^{R}_{j} + \sum_{k} \epsilon_{ijk} \rho^{R}_{k} \right).\end{split}\]

CSL#

When the intersection of the two lattice forms a lattice, we call it as the the coincidence-site lattice (CSL),

\[ \mathrm{CSL}(L_{\mathbf{A}}, L_{\mathbf{A}'}) := L_{\mathbf{A}} \cap L_{\mathbf{A}'}\]

We denote the ratio of volumes of the primitive cell of CSL and the two lattices as \(\Sigma \, (\geq 1)\).

The complete pattern-shift lattice (DSCL) is defined as a maximal sublattice of \(L_{\mathbf{A}}\) such that also contains \(L_{\mathbf{A}'}\).

Grimmer showed the following [Gri76] 2:

  • (Theorem 1) If the CSL is not empty, the rotation matrix \(\overline{\mathbf{R}}\) is rational, that is, in \(\mathbb{Q}^{3 \times 3}\).

  • (Theorem 2) \(\Sigma\) is the least positive integer such that \(\Sigma \overline{\mathbf{R}}\) and \(\Sigma \overline{\mathbf{R}}^{-1}\) are integer matrices.

Theorem 2 is shown with a Smith normal form (SNF). Let \(\mathbf{E}\) be a 3x3 integer matrix and its elements are coprime, and its SNF be \(\mathrm{diag}(e_{1}, e_{2}, e_{3})\). Here, we write GCD of \(i \times i\) minors of \(\mathbf{E}\) as \(d_{i}\), we obtain \(e_{i} = d_{i}/d_{i-1}\) (\(d_{0} := 1\)). Since the elements of \(\mathbf{E}\) are coprime, we get \(d_{1} = 1\). Also, using the fact that elements of \(|\mathbf{E}| \mathbf{E}^{-1}\) are 2x2 minors of \(\mathbf{E}\), we get \(d_{2}\) as GCD of the elements of \(|\mathbf{E}| \mathbf{E}^{-1}\). Thus, we obtain \(e_{1}=1\), \(e_{2} = \) GCD of elements of \(|\mathbf{E}| \mathbf{E}^{-1}\), and \(e_{3} = |\mathbf{E}| / e_{2}\).

\(\Sigma\) values for cubic system [Gri84]#

For CSL of cubic lattices, we can take Rodrigues vector as

\[ \mathbf{\rho}^{R} = \frac{n}{m} [hkl]\]

where \(m\) and \(n\) are positive integers and \(\mathrm{GCD}(m, n) = \mathrm{GCD}(h, l, k) = 1\)

For a given rotation axis \(\hat{\mathbf{\rho}} \propto [hkl]\), the procedure to enumerate rotation angle for CSLs are as follows:

  1. Enumerate \(\Sigma\) up to some maximal value with

    \[ \alpha \Sigma = m^{2} + (h^{2} + k^{2} + l^{2}) n^{2},\]

    where \(\alpha = 1, 2, 4\).

  2. Calculate \(\theta\) for each \(\Sigma\) as

    \[ \tan \frac{\theta}{2} = \frac{n}{m} (h^{2} + k^{2} + l^{2})^{1/2}\]

References#

Gri76(1,2)

H. Grimmer. Coincidence-site lattices. Acta Crystallographica Section A, 32(5):783–785, Sep 1976. URL: https://doi.org/10.1107/S056773947601231X, doi:10.1107/S056773947601231X.

Gri84

H. Grimmer. The generating function for coincidence site lattices in the cubic system. Acta Crystallographica Section A, 40(2):108–112, Mar 1984. URL: https://doi.org/10.1107/S0108767384000246, doi:10.1107/S0108767384000246.

1

https://cseweb.ucsd.edu/classes/wi10/cse206a/lec2.pdf

2

These are simplified cases from [Gri76].