Rigidity of three-dimensional lattices and dimension reduction in heterogeneous nanowires

Figure 1.  In the face-centred cubic lattice the nearest-neighbour structure of the atoms provides a subdivision of the space into tetrahedra (A) and octahedra (B). Figure (C) shows a quarter of an octahedron in the same unit cell. Grey dots denote points lying on the hidden facets

Figure 2.  The hexagonal close-packed lattice is associated with a tessellation of tetrahedra and octahedra as the ones in the figure. Only some of the bonds and some of the polyhedra of the pretriangulation are displayed

Figure 3.  The body-centred cubic lattice is associated with a tessellation of irregular tetrahedra as the one in the figure

Figure 4.  Cubic cell in the diamond lattice $\mathcal{L}^{\rm{D}}$. Atoms from the sublattice $\mathcal{L}^{\rm{D}}_1$ are represented in black/grey, while white atoms are from the sublattice $\mathcal{L}^{\rm{D}}_2$. Nearest-neighbour bonds are displayed by solid thick lines. Moreover, the picture shows a tetrahedron from the Delaunay pretriangulation of $\mathcal{L}^{\rm{D}}_1$: its edges (solid and dashed thin lines) correspond to next-to-nearest neighbours in $\mathcal{L}^{\rm{D}}$. A white atom lies at the barycentre of the tetrahedron, which is further divided into four irregular tetrahedra by the bonds between the barycentre and each vertex

Figure 5.  Bonds and triangulation in a honeycomb lattice. The lattice is given by $\mathcal{L}^✡:=\mathcal{L}^✡_1 \cup \mathcal{L}^✡_2$, where $\mathcal{L}^✡_i:={\rm{u}}_i^✡ + \xi_1{\rm{v}}^✡_1+\xi_2{\rm{v}}^✡_2 \colon \ \xi_1,\xi_2 \in\mathbb{Z}\}$, ${\rm{v}}^✡_1:=(1,0)$, ${\rm{v}}^✡_2:=(\frac12,\frac{\sqrt3}2)$, ${\rm{u}}^✡_1:=(0,0)$, ${\rm{u}}^✡_2:=(0,\frac{\sqrt3}3)$. This results into two interpenetrating sublattices $\mathcal{L}^✡_1$ and $\mathcal{L}^✡_2$, both being hexagonal (i.e., equilateral triangular). Atoms from $\mathcal{L}^✡_1$ and $\mathcal{L}^✡_2$ are displayed in different colors in the picture, respectively in black and in white. In the left part of the figure we indicate nearest neighbour (solid) and next-to-nearest neighbour bonds (dashed lines). The right part of the figure shows a possible triangulation, that is the natural triangulation of $\mathcal{L}^✡_1$ enriched by considering the nearest-neighbour bonds between atoms $x\in\mathcal{L}^✡_1$ and $y\in\mathcal{L}^✡_2$. This corresponds to ignoring the bonds between atoms of $\mathcal{L}^✡_2$, cf. Section 2.3

Figure 6.  By cutting a cubic lattice along certain transverse planes, one finds two-dimensional hexagonal Bravais lattices. (A) face-centred; (B) body-centred

Figure 7.  Dislocations in a honeycomb-type lattice. The bonds at the interface are chosen in the following way: First one considers only black atoms and finds a Delaunay pretriangulation, which is then refined to a triangulation (dashed lines); the same is done for white atoms (dotted lines). The dashed and dotted lines thus obtained give the bonds between next-to-nearest neighbours. Finally, each white (resp. black) atom lying inside a triangle formed by three black (resp. white) atoms is connected to the vertices of that triangle by nearest-neighbour bonds (solid lines)

Figure 8.  The tetrahedron $\mathcal S$ and its image $F(\mathcal S)$

Figure 9.  The octahedron $\mathcal O$

Figure 10.  (A) The image of $\mathcal{O}$ through a piece-wise affine map $u$ such that $l_i=1$ for each $i\neq 3$. (B) The projection of $u(\mathcal{O})$ on the plane $p$, where $O=\Pi(Q_1)=\Pi(Q_4)$