## The Thermal Expansion of MgO

In this exercise the structure of MgO will be optimised with
respect to the free energy at a number of temperatures. The free
energy is computed within the quasi-harmonic approximation.

A free energy optimisation is performed as follows.

Load an MgO structure and bring up the **Execute GULP** panel.

Click on **optimisation**

Click on **Optimisation opts** and select **Optimise Gibbs free energy**.

Click on **General opts** and select **Phonon DOS**.

Select suitable shrinking factors for the **k**-space sampling
based on your previous investigations.

Set the **Temperature**

Run GULP and examine the log file.

These calculations will take a little while to complete. GULP is computing
the internal energy and phonons at a sequence of geometries as it seeks
to minimise the free energy with respect to the structure.

Vary the temperature from 0K to 1000K in steps of 100K and compute the
variation in the free energy and optimal lattice constant by optimising the
structure at each temperature.

### Questions

- Plot the free energy against temperature
- Plot the lattice constant against temperature
- Comment on the shape of these curves.
- Compute the coefficient of thermal expansion for MgO
- How does this compare to that measured ? Find a measurement in
the literature or on the web - at what temperature was the
measurement made ?
- What are the main approximations in your calculation ?

### An opportunity to speculate ...

- What is the physical origin of thermal expansion ?
- As the temperature approaches the melting point of MgO how well
do the phonon modes represent the actual motions of the ions ?
- In a diatomic molecule with an exactly harmonic potential would
you expect the bond length to increase with temperature ? Why does it
increase in the solid when we are using an quasi-harmonic approximation ?

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