S. Scandolo, ICTP

This lesson starts by introducing the concept of nearest neigbours in a lattice or **Coordination number **of an atom on a crystal. The nearest neigbours were obtained till the therd order in distance for the squared, triangular and **Honeycomb **lattices in two dimensions. In three dimentions the same procedure was used for the simple **cubic **the **body centered cubic **and the **face centered cubic **.

Then some experimental methods to study the crystal structure of the solids were presented starting from the **Scanning Tunneling Microscope **that use the quantum properties of electron of tunneling small potential barriers to visualize the surface of a material. Then other experimental methods that are more indirect were discussed, they can be classified in two main categories the "local" and the **"diffraction" **methods. As an example of the local method it was presented the **X-ray photoelectron spectroscopy **(XPS) method. The principle of this technique is to calculate the binding energy by subtracting the kinetic energy of the electrons that are coming out the sample from the energy of the photons that are arriving on it. Then if the environment change the binding energy also changes thus the knowledge of previous results can be used to know what the coordination number of a given atom is in an unknown lattice by measuring its binding energy and the comparing with the previous results.

At the end of this lesson the problem of **X-ray diffraction **by a lattice was considered, to solve this problem it was calculated the **electromagnetic wave **emitted by an atom due to the incident radiation at a given point called the observation point. Then the radiation of all atoms at this point was summed out. It was noticed that the intensity of the emitted radiation will be maximum in the trivial case when the position of the observer is located in the same direction of the incident radiation, but there are also some conditions at with the intensity of the emitted radiation is maximum for some specific positions of the observer if the array of atoms is ordered, in the case of **amorphous solid **just the trivial solution is possible. The case of a one dimensional lattice was considered as an example.