LatticeCopiesΒΆ

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Many geometries like photonic crystals are constructed by a replication of a unit cell geometry at certain positions of a lattice. The following figure shows such a geometry with several circles copied to the lattice positions of a square lattice while omitting some lattice points (see the example in the GeoTutorial Lattice Copies):

_images/lattice_copies_square.png

A lattice is defined by two lattice vectors \pvec{a}_1 and \pvec{a}_2, the structures is replicated at positions displaced by

\begin{eqnarray*}
\pvec{d}_{l_1, l_2} & = & \pvec{a}_1\cdot l_1 + \pvec{a}_2 \cdot l_2
\end{eqnarray*}

with integers l_1 and l_2. The lattice vectors are defined by their lengths [a_1, a_2] (LatticeVectorLengths) and the angle \alpha between them (LatticeAngle), that is

\begin{eqnarray*}
\pvec{a}_1 & = & a_1\cdot[1; 0] \\
\pvec{a}_2 & = & a_2\cdot [\cos(\alpha); \sin(\alpha)].
\end{eqnarray*}

The lattice positions where the replicated geometry is actually placed can be defined in sub-sections Positions.