Electron diffraction

 

Principle

 

Electron diffraction is the underlying principle of electron microscopy (EM). Electron diffraction theory is similar as that of diffraction of light and X-rays, where the wavelength is also the key-parameter.

Since electrons are charged particles they interact with matter through Coulomb forces. (The magnitude of the force on a charge q1, due to the presence of a second charge q1 at distance r, is given by Coulomb's law. Its scalar version is:

   F = (4πε0)─1q1q2r─2,                           (1)

where ε0 the dielectric constant in vacuum. The first term is the Coulomb force constant or electrostatic constant, being 9.0∙109  Nm2/F─2). This means that the incident electrons feel the influence of both the positively charged atomic nuclei and the surrounding electrons. In comparison, X-rays interact with the spatial distribution of the valence electrons, while neutrons are scattered by the atomic nuclei through the strong nuclear forces. In addition, the magnetic moment of neutrons is non-zero, and they are therefore also scattered by magnetic fields. Because of these different forms of interaction, these three types of radiation are suitable for different studies.

 

Since nowadays even single crystals can be studies, crystallographic electron diffraction techniques are increasingly applied in life sciences and medicine. After diffraction by the sample an interference pattern results. This phenomenon occurs due to the wave-particle duality, which states that a particle of matter (in this case the incident electron) can be described as a wave. For this reason, an electron can be regarded as a wave, similar as a light wave. Consequently, this technique is similar to X-ray diffraction (see Light: diffraction).

Crystallographic experiments with electron diffraction are usually performed in TEM or SEM as electron backscatter diffraction (a microstructural-crystallographic technique). 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. Working back from the observed diffraction pattern (Fig. 2 of Electron microscopy), it may be possible to deduce the structure of the crystal. However, the technique is limited by the so called phase problem (the loss of phase information. There are several estimation techniques to reconstruct the phase in order to obtain an image.)

 

 

Application

 

The main and classical field of applications of electron diffraction is in solid state physics and chemistry, in particular crystallography. Since many proteins behave as crystalline structures (e.g. in viruses), they are frequently studied with electron diffraction. However, since diffraction is the basic principle of EM, implicitly is has a wide filed of applications in life sciences and experimental medicine.

 

 

More info

 

Intensity of diffracted beams

In the kinematical approximation for electron diffraction, the intensity of a diffracted beam is given by:

  Ig =  *ψg*2  % *Fg*2                            (2)

Here ψg is the wave function of the diffracted beam and Fg is the so called structure factor of crystallography, which is the product of the scattering vector of the diffracted beam and an exponential function with the scattering power of the atom (also called the atomic form factor) and the atomic position in the exponent, summed over all atoms in the crystal. The scattering power of an element depends on the type of radiation considered. With these theory with enough a priory knowledge, it is a priory possible to have some estimate of the scattered beam intensity. 

Wavelength of electrons

The wavelength of an electron is given by the de Broglie equation:

  λ = h/p.                                                 (3)

Here h is Planck’s constant and p the momentum (mass times velocity) of the electron. The electrons are accelerated with an electric potential U to the desired velocity:

  v = (2eU/m0)0.5.                                   (4)

m0 is the mass of the electron, and e is the elementary charge. λ is then given by:

  λ = h/(2m0eU)0.5.                                 (5)

However, in an electron microscope, the accelerating potential is usually several kV causing the electron to travel at an appreciable fraction of the speed of light (c). An SEM may typically operate at an accelerating potential of 10 kV giving an electron velocity approximately 20% of the speed of light. A typical TEM can operate at 200 kV raising the velocity to 70% the speed of light, which needs a relativistic correction:

   λ = h/(2m0eU)0.5(1 + eU/2m0c2) ─0.5 (6)

The last term is a relativistic correction factor.

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 λ is 2.5 pm. This suggests a 2.5·105 better resolution than the light microscope. However, due to electromagnetic lens errors this is far from true. With apertures in the mm-range to reduce the errors, a factor 1000 is possible, which gives a best resolution of about 0.5 nm. The wavelength of X-rays usually used in X-ray diffraction is 10 -100 pm, yielding slightly worse resolutions.

 

Electron diffraction in a TEM

Specific techniques of electron diffraction in TEM are especially of importance for crystallography. Since λ of TEM is smaller than that with X-ray diffraction as a final result the diffraction experiment reveals more information about 2D-structure of crystals. 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, Convergent Beam Electron Diffraction (CBED), can reveal the full 3D-symmetry of a crystal.

In TEM, a single crystal grain or particle may be selected for the diffraction experiments. This means that the diffraction experiments can be performed on single crystals of nm-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 (EDS or EDX). This is a type of Spectroscopy, relying on the interactions between light and matter. For, each element has a unique electronic structure and, thus, a unique response to electromagnetic waves. Another technique of investigations of electronic structure and bonding is through electron energy loss spectroscopy (EELS). In EELS a material is exposed to a beam of electrons with a known, narrow range of kinetic energies. Some of the electrons will undergo inelastic scattering, which means that they lose energy and have their paths slightly and randomly deflected. The amount of energy loss can be measured via an electron spectrometer and interpreted in terms of what caused the energy loss. Finally one can study the mean inner potential through electron holography (see Holography).

 

Limitations

Electron diffraction in TEM is subject to several important limitations. First, sample thickness must be of the order of 100 nm or less to be electron transparent. Furthermore, many samples are vulnerable to radiation damage caused by the incident electrons.

The study of magnetic materials is complicated by the fact that electrons are deflected in magnetic fields by the Lorentz force. Although this phenomenon may be exploited to study the magnetic domains of materials (the smallest pieces of material maintaining its magnetic properties) by Lorentz force microscopy, it may make crystal structure determination virtually impossible.

Estimating lattice parameters and atomic positions can be done with relative errors less than 0.1%, but this is very hard to obtain and time consuming and the data are difficult to interpret. X-ray or neutron diffraction is therefore often preferred.

However, the main limitation of electron diffraction in TEM remains the comparatively high level of practical skill and theoretical knowledge of the user. This is in contrast to for instance the execution of powder X-ray (and neutron) diffraction experiments and its data analysis being highly automated and routinely performed.