24 - Lecture notes for Clay Mineralogy

Suggested reading:

Güven, N. (1990) Electron diffraction of clay mineral. In Electron-Optical methods in clay science Vol. 2, (ed. I. D. R. Mackinnon and F. A. Mumpton), The Clay Minerals Society, Boulder, CO.,

Electron-Optical methods in clay science

Transmission electron microscopy and electron diffraction are an extremely powerful methods for studying the atomic structure of sub-micron clay crystals.

The formation of high resolution electron images allow one to characterize the surface area, size distribution and mineral surface relationships. In addition to characterizing the crystal habit of a specimen, imaging is important for locating the area for which electron diffraction analysis is to be performed.

Electron Diffraction

Kinematic versus dynamical treatment of electron scattering.

The purpose of today's discussion is provide a framework to understand electron diffraction for those with a working knowledge of X-ray diffraction. This is most easily accomplished by comparing and contrasting the electron and X-ray scattering processes.

X-rays	Electrons
Electromagnetic waves of photons	Waves of negatively charged particles
λ is fixed and independent of excitation voltage.	λ varies with accelerating voltage (V)
where: h = Plank's constant c = speed of light ΔE = Energy of electron transition	where: mo = rest mass of an electron Plugging in the constants yields the useful relationship between wavelength and accelerating potential. Clcik here for a plot.
Scattering of photons occurs by electron clouds of the atoms	Scattering of electrons occurs by electrostatic potential fields of the atoms. The sum of the positive nuclear potential and the negative electron cloud potential.

Electron scattering by atoms of a crystal

1. Low-angle elastic scattering - no loss in energy ( λ diffracted = λ incident) Relevant to both imaging and diffraction.
2. High-angle elastic scattering
3. Inelastic scattering - due to energy loss - results in incoherence of electron waves and diffuse background intensity.
   

The electron scattering efficiency of an atom is approximated by the Mott equation.

where,

• e = electron charge
• m = relativistic mass of an electron
• Θ = Bragg angle
• Z = atomic number
• f_x = scattering contribution form the electron cloud (atomic scattering factor, i.e., the ratio of amplitude of waves scattered by the atom to that of a single electron).
  
  

The units on f_e in units of length (Å)

The atomic scattering factor for X-rays can also be expressed in units of length (Å) by factoring in the (1) charge of the electron, (2) the relativistic mass of an electron and (3) the speed of light.

By looking at the ratio of the X-ray atomic scattering factor and the electron atomic scattering factor it possible to assess their relative scattering efficiencies.

Let:

h² =  λ ² m² v²

and

c² / v²~ 4 at 100 keV

This ratio, for Θ = 1° is approximately equal to 10⁴.

The difference in scattering power of electrons, compared to X-rays makes it possible to obtain diffraction effects from a single scattering domain only 10 to 20 unit cells thick (i.e., 100's of Å).

Amplitude of scattered electron relative to X-ray

X-rays	Electrons
Amplitude of scattered X-ray relative to the incident beam is small (0.1%)	Amplitude of scattered electron relative to the incident beam is large.
Waves can be considered singly scattered	Result is large amounts of re-scattering. Therefore, must treat with dynamical theory.
Treat diffraction with kinematic theory	Can treat with kinematic theory if crystals are extremely thin.