**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. 41-68.

In reciprocal space, the size of the reflection sphere or Ewald sphere is related to the wavelength of radiation. Recall that the wavelength (λ) of electrons is proportional to the accelerating potential.

Previously, we let the constant *k* equal unity. In fact,
*k* is related to the radius of the Ewald sphere. This figure
below shows a plot of Ewald size (in reciprocal Å or Å^{-1})
as
a function of voltage.

Elastically scattered electrons from a thin crystallite with
interplaner spacings *hkl* give rise to diffracted beams
defined by Bragg's law.

Recall that this relationship can be represented vectorially
with the components of the incident beam, diffracted beam (for
a particular *hkl*) and the reciprocal lattice projection
on the Ewald sphere (sphere of reflection).

We have previously shown that

Combining Bragg's law yields athe length of the reciprocal vector,

As shown graphically in the figure above,
The wavelength of electrons accelerated through a voltage
of 100 kV is 0.037Å. This gives an Ewald sphere radius of
27Å^{-1}.

Typical interplaner d-values are about 4Å, which produces
a reciprocal lattice vector of 0.25Å^{-1}.

Thus, to a first approximation, the first plane section through the reciprocal lattice is nearly normal to the incident wave vector.

The length of the reciprocal lattice vector for any particular
*hkl* plane can be given as,

- P = Photographic plane
- L = distance of specimen from P
- T = Forward scattered beam
- O = point where T strikes P
- S = Bragg diffracted beam
- G = point where S strikes P
**R**= vector distance from O to G

From the above figure is the following geometric relationships can be seen that

Recall Bragg's law,

for small values of Θ,

therefore,

Since

then,

The vector **R** is therefore, a direct measure of the
reciprocal
lattice vector (* ρ*)
.

λL is the **camera constant**.

This is typically determined by using a known material such as gold.
Gold sputtering equipment is commonly found in electron microscopy labs
(to enhance sample conductivity). Gold coating deposit a thin layer of
randomly oriented crystals, which results in powder-like diffraction
pattern. Instead of discreet diffraction from a single crystal to
specific locations on the Ewald sphere, a set of powder rings develop.
Gold is isometric, therefore the {hkl}
are of the same form. This gives a few strong set of lines with known
indices and d-spacings. The distance from the center of the photo
below (collected at a known magnification and kV) to the first line is
1/2.3469 Å^{-1 }or 0.4261 Å^{-1}.

Gold

a = 4.0650 Å SPACE GROUP: Fm3m

2.3469 1 1 1

2.0325 2 0 0

1.4372 2 2 0

1.2256 3 1 1

1.1735
2
2 2

Constructing a reciprocal lattice

The diagram below contains an orthogonal view of a monoclinic
lattice with lattice parameters: *a* = 3 Å, *b*
= 4 Å, *c* = 8 Å and *b* = 105°. The
view contains the *a* and *c* axes on the paper/screen
plane and the common *b** and *b*** **axis is
perpendicular
to paper/screen plane. The scale is set to 1Å = 2 cm.

The reciprocal lattice points for the first view (*i.e.,
*the *ac* face) over the range of reiprocal indices +/- 303
are plotted. The *a** and *c** axes are also labeled.
The distance from the origin to the reciprocal lattice point are
be proportional to λ, where

Interpreting an electron diffraction pattern (reciprocal lattice)

The figure below is adapted from Güven, N. (1990). The
pattern is from a muscovite crystal oriented perpendiular to the
real *a* axis and *b* axis. With calibration, the camera
constant, λL can be
determined.
Commonly a TEM grid is gold coated. The length (radius) of gold
diffraction rings serve as an external calibration. Obviously,
the pattern must be properly indexed in order to proceed.

In the image below, if the distance from the origin to a diffraction
spot is measured in mm, then the λL
= 72.9 mm per Å^{-1}. Note: Your image size may
appear at a different scale.

For example,

If the distance from the origin to the 200 spot is 28 mm, then *ρ*_{(200)
}= 0.385 Å^{-1}.

This translate to a d_{}_{(200)
} = 2.60Å

Therefore,

*a*
= 5.19Å

With TEM and electron diffraction don't expect to get resolution much better than +/- 0.01Å