25 - Lecture notes for Clay Mineralogy

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.

The Reciprocal lattice

When a beam is diffracted by a crystal structure, the positions 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 difficult to comprehend from a physical standpoint. It is an imaginary construct used for the convenience of crystallography. 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 ρfrom the origin, where

and k is constant (we can take its value 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 AOt is Θ.
• ρ is the reciprocal space vector for the plane hkl. It stands perpendicular to hkl and intersects the sphere of reflection at point P
• The direction of BP is the same as OR, where R is the intercept of the reflected beam and larger limiting sphere.
• ρ(hkl) = 2 sinΘ means that ρ can 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.
• the angle PAO is also Θ .
• for any X-ray or electron beam that passes along the diameter, 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 k² = λ,

then

d_(hkl) = λ / ρ_(hkl)

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 ρ₁₀₀ defined by the wavelength of radiation (λ) and a.
• Intersection of sphere of reflection and sphere of ρ₁₀₀ (B₁).
  

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/ d_(hkl)