An interference effect involving electrons from an incoming beam scattering from different layers of atoms in a solid, giving distinctive intensity patterns which can be used to determine the solid's structure. It is especially useful for surface studies, since electrons (being charged) do not penetrate far into the material. The original observation of electron diffraction was made in 1927 by US physicists Clinton Davisson and Lester Germer, and was crucial in establishing the dual wave-particle nature of electrons.
Electron diffraction is a technique used to study matter by firing electrons at a sample and observing the resulting interference pattern.
Electron diffraction is most frequently used in solid state physics and chemistry to study the crystal structure of solids. These experiments are usually performed in a transmission electron microscope (TEM), or a scanning electron microscope (SEM) as electron backscatter diffraction. In these instruments, the electrons are accelerated by an electrostatic potential in order to gain the desired energy and wavelength before they interact with the sample to be studied.
The periodic structure of a crystalline solid acts as a diffraction grating, scattering the electrons in a predictable manner.
Apart from the study of crystals, electron diffraction is also a useful technique to study the short range order of amorphous solids, and the geometry of gaseous molecules. De Broglie's formula was confirmed three years later for electrons (which have a rest-mass) with the observation of electron diffraction in two independent experiments.
Theory
Electron interaction with matter
Unlike other types of radiation used in diffraction studies of materials, such as X-rays and neutrons, electrons are charged particles and interact with matter through the Coulomb forces. This means that the incident electrons feel the influence of both the positively charged atomic nuclei and the surrounding electrons.
Intensity of diffracted beams
In the kinematical approximation for electron diffraction, the intensity of a diffracted beam is given by:
Here is the wavefunction of the diffracted beam and is the so called structure factor which is given by:
where is the scattering vector of the diffracted beam, is the position of an atom i in the unit cell, and fi is the scattering power of the atom, also called the atomic form factor.
The structure factor describes the way in which an incident beam of electrons is scattered by the atoms of a crystal unit cell, taking into account the different scattering power of the elements through the term fi.
Wavelength of electrons
The wavelength of an electron is given by the de Broglie equation
Here h is Planck's constant and p the momentum of the electron. The electrons are accelerated in an electric potential U to the desired velocity:
m0 is the mass of the electron, and e is the elementary charge.The electron wavelength is then given by:
However, in an electron microscope, the accelerating potential is usually several thousand volts causing the electron to travel at an appreciable fraction of the speed of light. An SEM may typically operate at an accelerating potential of 10,000 volts (10 kV) giving an electron velocity approximately 20% of the speed of light, while a typical TEM can operate at 200 kV raising the electron velocity to 70% the speed of light. It can be shown that the electron wavelength is then modified according to:
c is the speed of light. The wavelength of the electrons in a 10 kV SEM is then 12.3 x 10-12 m (12.3 pm) while in a 200 kV TEM the wavelength is 2.5 pm.
Electron diffraction in a TEM
Electron diffraction of solids is usually performed in a Transmission Electron Microscope (TEM) where the electrons pass through a thin film of the material to be studied.
Benefits
As mentioned above, the wavelength of electron accelerated in a TEM is much smaller than that of the radiation usually used for X-ray diffraction experiments. A consequence of this is that the radius of the Ewald sphere is much larger in electron diffraction experiments than in X-ray diffraction.
Furthermore, the electron lenses allows the geometry of the diffraction experiment to be varied. However, by converging the electrons in a cone onto the specimen, one can in effect perform a diffraction experiment over several incident angles simultaneously. This technique is called Convergent Beam Electron Diffraction (CBED) and can reveal the full three dimensional symmetry of the crystal. This means that the diffraction experiments can be performed on single crystals of nanometer size, whereas other diffraction techniques would be limited to studying the diffraction from a multicrystalline or powder sample. Furthermore, electron diffraction in TEM can be combined with direct imaging of the sample, including high resolution imaging of the crystal lattice, and a range of other techniques. These include chemical analysis of the sample composition through energy-dispersive X-ray spectroscopy, investigations of electronic structure and bonding through electron energy loss spectroscopy, and studies of the mean inner potential through electron holography.
Practical aspects
Figure 1 to the right is a simple sketch of the path of a parallel beam of electrons in a TEM from just above the sample and down the column to the fluorescent screen. This lens acts to collect all electrons scattered from one point of the sample in one point on the fluorescent screen, causing an image of the sample to be formed.
If the sample is tilted with respect to the incident electron beam, one can obtain diffraction patterns from several crystal orientations.
Limitations
Electron diffraction in TEM is subject to several important limitations.
The study of magnetic materials is complicated by the fact that electrons are deflected in magnetic fields by the Lorentz force.
Furthermore, electron diffraction is often regarded as a qualitative technique suitable for symmetry determination, but too inaccurate for determination of lattice parameters and atomic positions. In principle, this is not quite the case: lattice parameters of high accuracy can in fact be obtained from electron diffraction, relative errors less than 0.1% have been demonstrated.
However, the main limitation of electron diffraction in TEM remains the comparatively high level of user interaction needed. Whereas both the execution of powder X-ray (and neutron) diffraction experiments and the data analysis are highly automated and routinely performed, electron diffraction requires a much higher level of user input.
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