PX262 - K1 - electrons in crystalline solids

in many solid materials, the nuclei are arranged into ordered structures, ie. crystal lattices
they possess translational symmetry
this affects the electronic states
a perfect crystal can be generated by decorating each point in the lattice with the same basis (atoms as groups of atoms)
a lattice is an infinite array of mathematical points in space with translational symmetry

in 3D, the lattice consists of all points with position vectors, $\vec{R}$ , in the form: $\vec{R} = n_{1} \vec{a} + n_{2} \vec{b} + n_{3} \vec{c}$ , where, $\vec{a}$ , $\vec{b}$ , $\vec{c}$ are three non-coplanar vectors
the so-called 'primitive vectors' of the lattice and $n_{1}$ , $n_{2}$ and $n_{3}$ over all positive and negative integers
the unit cell of a lattice in 3D is a parallelepiped with volume:

V_{c} = | \vec{a} \cdot (\vec{b} \times \vec{c}) |

packing the unit cells together fills all space
there are just fourteen 3D crystal structures with different symmetries - the bravais lattices
they are in seven groups: cubic, tetragonal, orthorhombic, hexagonal, trigonal, monoclinic, and triclinic

wigner-seitz cell

this is a cell which is constructed around a lattice point, enclosed by the planes that bisect, and are perpendicular to the vectors connecting the point to each of the nearest neighbours