Although the rough surface of a material can be observed with the bare eye or, more delicately, with a microscope, learning something about the atomic structure is something of a more complicated matter.
If we look at homogeneous crystalline solids, we can derive information about the crystal structure by applying the so-called LEED method.
LEED is short for “Low-Energy Electron Diffraction”, and – as the name already reveals – describes the diffraction of electrons with a relatively low energy (20 – 200eV) on a surface.
After diffraction, the electrons will move towards a screen made of a fluorescent material, where glowing points will be visible, if the electrons interfere constructively towards this point.
To describe this process a little bit more in detail, we will first look at the behaviour of the electron.
Given the wave-particle duality, we know that all particles – in our case, the electron – can also
behave like waves (and vice versa).
This gives us the wave-equation for the electron:
with k: wave vector and
Furthermore, we can derive the wavelength of the electron by using the de Brogie wavelength
and using gives us .
So far, we know that an electron with a defined energy, and thus a defined k is moving towards the surface and then will interact with the atomic structure of the material.
To describe the process, we will use the kinematic theory, which states, that all electrons will only be scattered elastically and only once, as well as that they only interact with the top (surface) layer of atoms.
Taking this into account, we can take a deeper look into the diffraction process itself.
The material consists of a single crystal with a given crystal structure, in which the atoms will be placed in a way, where they can be located via lattice vectors .
For the observation, we will furthermore introduce the reciprocal lattice: It is described by reciprocal lattice vectors , which are defined by
  with for and  for .
The electron that has approached the surface, will now be scattered on places with a high electron density and the intensity of the interference pattern will be given by a Fourier-transformed.
To make predictions about where reflexes will be visible on the fluorescent screen, we will use the fact that, as we assumed, all electrons are scattered elastically.
Using this, we know, that , which means, that the absolute value of the k (wave vector) remains the same throughout the process. To achieve constructive interference (so that we will be able to see an interference/diffraction pattern on the fluorescent screen) we will use the “Laue-condition”, which gives us constructive interference, if (G is a composition of fundamental reciprocal lattice vectors ).

Ewalds sphere construction in the case of a 2D lattice Src:

The obtained LEED-pattern on the screen can be interpreted as a depiction of the reciprocal lattice.
An example of a LEED-analysis is given by the observation of a Wolfram single crystal.
Although it is constantly stored in a ultra-high vacuum for observations, layers of not-wanted atoms have deposited onto its surface, which is visible in the picture

with electrons with an energy of 56 eV: contrary to a pattern of defined, clear points, one can observe cross-like shapes above the interference pattern,
which hints to a layer of adsorbates, which must be removed for further observations.
When a clean/sharp image is created, a further examination of the image can be made. Due to the fact, that the image obtained contains of Fourier-transformed data with loss of phase-information, it is not possible to directly calculate back to the original lattice. Information that can be derived is the hexagonal crystal structure.
To gather other information about the material, a simulation program is used. In picture

a simulation of the diffraction of electrons with 76 eV is displayed
The real image is clearly tilted towards the bottom
from which we can conclude that our crystal must be tilted in the same direction.

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