**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.,

Klug and Alexander (1974) pages 38-41, 132-135.

When a beam is diffracted by a crystal structure, the locations of the diffracted beam provide a map of the reciprocal lattice of the crystal. This can be shown by starting with a rearrangement of Bragg's Law:

The reciprocal lattice is initially a difficult concept to
comprehend from a physical
standpoint, because it is an imaginary construct used for the
convenience
of crystallography (the units are in inverse-Å or Å^{-1}).
Recall
that real space lattices are defined
by translations about the crystallographic axes *a, b* and
*c* and their respective inter axial angles *α, β, *and*
γ*. It is possible
to construct an imaginary lattice that has points hkl defined
by vectors perpendicular to the real lattice planes (*hkl*).
The point hkl in the reciprocal lattice lies normal to the origin
of the (hkl) plane at a distance *ρ* from
the origin, where

and *k* is constant (we can take value
of k to be unity for the
moment).

Perhaps it is best to show this graphically using the figure below.

A triangle inscribed in :

- A circle of unit radius (diameter is distance from the source to the sample AO) center B. This is the sphere of reflection.
- 2Θ is the Bragg condition
for the reflected beam of a particular
*hkl*. The direction is OR. *tt'*represents the trace of the plane*hkl*.- the angle AO
*t*is Θ. is the reciprocal space vector for the plane*ρ**hkl*. It stands perpendicular to*hkl*and intersects the sphere of reflection at point P- The direction of line BP is parallel to line OR, where R is the intercept of the reflected beam and larger limiting sphere.
(*ρ**hkl*) = 2 sinΘ means thatcan not exceed a value of 2. Therefore, no point outside a sphere of radius 2 can ever give a reflection for a given wavelength. Hence the term limiting sphere.*ρ*- the angle PAO is also Θ .
- for any X-ray or electron beam that passes along the diameter AO, all points along the surface of the sphere of reflection satisfy the condition for Bragg reflection.
- The sphere is now considered to be placed in the reciprocal
lattice for a crystal at O.

If we let k^{2} = λ*,*

then

*d _{(hkl) }
= *λ

By substitution in to Bragg's Law:

The particulars for the reciprocal lattice by a simple cubic lattice can be demostrated with the series of figures below.

Features to note in the first figure are:

- Source A
- Sample O
- Recipocal lattice parameter
*a** - Reciprocal lattice point for the (100) plane
- Sphere of radius
*ρ*defined by the wavelength of radiation (λ) and_{100}*a*. - Intersection of sphere of reflection and sphere of
*ρ*(B_{100}_{1}).

The properties of a reciprocal lattice are such that : *a*.a
= b*.b = c*.c = *1, and * **α***
+ **α**= 180°, **β*** + **β**=
180°
and
**γ*** + **γ**= 180° and **ρ** _{(hkl)
} = *λ

The sphere of reflection is also known as the "Ewald sphere", which
we will see next, its size is related to the wavelength of radiation.

_{
}