A lattice is "an infinite 1,2, or 3-D regular arrangement of points".

A 1D lattice could be polythene, which consists of the repeating unit - CH

2-D patterns: Planar lattices eg graphit, which consists of laters of carbon atoms, each atom bonded covalently to three others, with weak forces between the layers.

3-D Space Lattices. There are 7 unique unit-cell shapes that can fill all 3-D space. These are the 7 Crystal systems.

We define the unit cell using 3 vectors, {jatex options:inline}\mathbf{a},\mathbf{b}, \mathbf{c}{/jatex} called lattice paranmeters. The angles between these vectors are given by {jatex options:inline}\alpha{/jatex} (angle between {jatex options:inline}\mathbf{a}{/jatex} and {jatex options:inline}\mathbf{c}{/jatex} ), {jatex options:inline}\beta{/jatex} (angle between {jatex options:inline}\mathbf{a}{/jatex} and {jatex options:inline}\mathbf{c}{/jatex} ), and {jatex options:inline}\gamma{/jatex} (angle between {jatex options:inline}\mathbf{a}{/jatex} and {jatex options:inline}{\mathbf{b}/jatex} ).

Real crystals always possess one of these lattice types, but different crystalline compounds that have the same lattice can have different lattice parameters (these depend upon the chemical formula and the sizes of the atoms in the unit cell).

There are three types of cubic lattice.

The prmitive cubic has lattice points at the corner of a cube.

Body Centered Cubic (BCC) has lattice points at corners and centre of cube.

Face Centered Cubic (FCC) lattice points at the corners and centre of each face.

The lattice points at the corner of each cell are shared by the surrounding eight unit cells"worth 1/8 of a lattice point to each cell.

The lattice points at the centre of a face are shared between two cell so are only worth 1/2 a lattice point to each cell.

The lattice point at the centre of each cell belongs to that cell so contributes i1 lattice point to that cell.

Total number of lattice points i, for a primitive cubic 8(1/8) = 1, for a Face Centred Cubic 6x1/2 + 8(1/8) = 4, and for a Body Centred Cubic 8(1/8) + 1 = 2.

]]>

The diagonal of the unit cell is {jatex options:inline}4r{/jatex} so if so taking the unit cell to have side {jatex options:inline}a{/jatex} and using Pythagoras Theorem in three dimensions, {jatex options:inline}a^2+a^2+a^2=(4r)^2 \rightarrow a = \sqrt{4r}{\sqrt{3}}{/jatex}

The fraction of the cell that is occupied by the spheres is {jatex options:inline}\frac{8/3 \pi r^3}{(4r/ \sqrt{3}})^3 =\frac{\pi \sqrt{3}}{8}{/jatex}

]]>

The distance from A to B is {jatex options:inline}\sqrt{(2r)^2 + (2r)^2 } = 2r \sqrt{2}{/jatex}

The distance from C to D is then {jatex options:inline}\sqrt{(2r)^2 + (2r)^2 } -2r= 2r \sqrt{2} -2r{/jatex}

This face centred cubic structure has rotational symmety about the midpoint of AB, using the line AB as an axis of rotation, as well as vertical and horizontal rotational symmetry. This means that we can fit a sphere of radius {jatex options:inline}\frac{ 2r \sqrt{2} -2r}{2} = r (\sqrt{2} -1){/jatex} in this gap.]]>

The Boltzmann distribution relates the number of particles per state at in terms of the energies of the states:where is the number of particles at energywhich has degeneracyetc.

Proof: The goal is to findthat maximises

This expresses the number of ways that a particular distribution of molecules throughout a set of allowed energies can be achieved. Taking logs of this expression, and using Stirling's approximationfor largegives

We differentiate with respect to general obtaining(1).

We set this derivative equal to zero. Theare subject to the constraintis fixed, and the energyover and above the zero point energy is also fixed.

Now use Lagrange multipliers to find the

is the function(2). We differentiate with respect to eachformingorfor eachSubstitute from (1) to obtainwhich rearranges toWe can eliminate the factorby consideringat two different energies, obtaining

Consideration of (2) sincebutfor one degree of freedom so we can writeand further analysis giveshence

]]>To deal with the macroscopic behaviour of systems of particles, we must use the Boltzmann distribution expression to find how many of the particles occupy each energy state. The Boltzmann distribution must be applied to each of the very many states available to the particles. This can be done by summing over all the available states, resulting in a function called the partition function., defined as

The value of q can be calculated for any particular type of atom or molecular motion if the energies of the allowed states are known.

We can write (1) in a more convenient form. The number of particles is given by

Henceso we can write (1) as

The thermal energy of a system of particles can be expressed in terms of the partition function.

Hence

]]>