The hexagonal lattice

For a hexagonal lattice with base vectors:

$$\ensuremath{\mathbf{a}}_1 = a\ensuremath{\mathbf{\hat{x}}}$$ (6.1)

$$\ensuremath{\mathbf{a}}_2 = {1 \over 2}a\ensuremath{\mathbf{\hat{x}}} + {\sqrt{3} \over 2}a\ensuremath{\mathbf{\hat{y}}}$$ (6.2)

$$\ensuremath{\mathbf{a}}_3 = {\sqrt{8 \over 3}}a\ensuremath{\mathbf{\hat{z}}}$$ (6.3)

the lattice points are:

$$\ensuremath{\mathbf{P}} = n_1\ensuremath{\mathbf{a}}_1 + n_2\ensuremath{\mathbf{a}}_2 + n_3\ensuremath{\mathbf{a}}_3$$ (6.4)

$$(n1, n2, n3) \in \mathbb{N}^3$$ (6.5)

$\ensuremath{\mathbf{P}}$The positional vector for lattice geometries Shown in figure 6.4 on the right is a hexagonal structure which exhibits this property. However, one should note that there are two ways of packing this layer as there are two positions in which the first sphere can be placed. The two structures are called hexagonal close-packed and face-centred cubic. The hexagonal close-packed structure, or hcp, has the third layer in $c$ having the same $x$ and $y$ coordinates as the first layer, the second layer has the same $x$ and $y$ coordinates as the fourth layer (ABABAB...) and so on (Kittel, 1996). The face-centred cubic structure has an alternative arrangement of spheres in the third layer where the spheres share the same $x$ coordinate with the first layer but have different $y$ coordinates (ABCABC...). Although this arrangement appears at the outset to be hexagonal, by rotating its primitive cell the vectors can be shown to be a variant of a cubic lattice.

Figure 6.4: Two layers of spheres packed cubically (left) and hexagonally (right)

Figure 6.5: 600x600x150nm cut of simple cubic nickel antispheres in zero applied field. The colouring represents the angle between $x$ and $y$ in radians; the lower left inset shows an $x$-$y$ cut-plane through the centre of the sample, the lower right shows a cut-plane through a lower part of the same sample