Cohesion in crystals

Cohesion energy is the energy required to separate (bring at distance) the atoms of a crystal.

We will describe two kinds of solids based on different linking characteristics:

1. Van der Waals bonding (crystals of inert gases)
2. Ionic bonding

Crystals of inert gases

These crystals can be formed at extremely low temperatures only and we expect them to be transparent and electrical insulators. Except for the atoms crystallize in the closest possible form (FCC) and their interaction is based on van der Waals forces.

The potential can be determined empirically, we will use the Lennard-Jones potential:

where and are empirical parameters.

The repulsive term can be also written as

where, again, and are empirical parameters.

The repulsion is due to the Pauli exclusion principle.

We want to find the coordinates of the minimum of the potential which corresponds to the equilibrium point (i.e. the point where repulsion and attraction balance out).

We can do that by solving

where is the equilibrium distance. We get:

The point can be calculated as

Equilibrium reticular constant

If we have atoms, the total potential energy is given by

Where is the distance between the atoms and .

If we define the geometrical factor as

we can rewrite

where the two summations depend only on the kind of crystalline structure and thus can always be calculated.

The equilibrium distance for FCC can be calculated as

Since the summations can be calculated, the only unknown left is the ratio , which can be calculated as well. For the FCC structure, for example, .

Ionic crystals

Ionic crystals are made up of positive and negative ions. The ionic bond results from the electrostatic interaction of oppositely charged ions.

For ionic crystals (such as ) the energy will come from Coulombic interactions (attractive and repulsive) and from short range repulsion (Pauli exclusion principle). There are also van der Waals interaction which can be neglected.

Since the i-th atom will interact with all the others, we will get

Where is the interaction between ion and ion . In the sum () the positive sign is taken when the charges are the same and vice versa.

If we have ions .

As before, we can introduce the geometrical factor

So

And, since the Pauli repulsion only affects first neighbours, we get:

If we have first neighbours:

Where the is a sign that takes into account the attraction or repulsion and is the Madelung constant (the summation is a converging series).

If the crystal is stable.

Similarly to before, the equilibrium constant can be calculated by imposing

getting the condition

which can be substituted in to calculate :

Since usually the Madelung energy contains 90% of the overall cohesion energy.

Example: Madelung constant

Kittel, page 64

Let’s consider the example of a 1D crystal like the one in the image

From the definition of the Madelung constant and from we get

By looking at the drawing we can calculate the summation as follows:

The series converges to