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X-ray Diffraction

/hcE

Typical interatomic distances in solid are of the order of an angstrom.Thus the typical wavelength of an electromagnetic probe of such distances Must be of the order of an angstrom.

Upon substituting this value for the wavelength into the energy equation,We find that E is of the order of 12 thousand eV, which is a typical X-rayEnergy. Thus X-ray diffraction of crystals is a standard probe.

Wavelength vs particle energy

Bragg Diffraction: Bragg’s Law

Bragg’s Law

The integer n is known as the order of the corresponding Reflection. The composition of the basis determines the relativeIntensity of the various orders of diffraction.

Many sets of lattice planes produce Bragg diffraction

Bragg Spectrometer

Characteristic X-Rays

Brehmsstrahlung X-Rays

Bragg Peaks

X-Ray Diffraction Recording

von Laue Formulation of X-Ray Diffraction

Condition for Constructive Interference

Bragg Scattering

=K

The Laue Condition

Ewald Construction

Crystal and reciprocal lattice in one dimension

First Brillouin Zone: Two Dimensional Oblique Lattice

Primitive Lattice Vectors: BCC Lattice

First Brillouin Zone: BCC

Primitive Lattice Vectors: FCC

Brillouin Zones: FCC

Near Neighbors and Bragg Lines: Square

First Four Brillouin Zones: Square Lattice

All Brillouin Zones: Square Lattice

First Brillouin Zone BCC

First Brillouin Zone FCC

Experimental Atomic Form Factors

Reciprocal Lattice 1

Reciprocal Lattice 2

Reciprocal Lattice 3

Reciprocal Lattice 5

Real and Reciprocal Lattices

von Laue Formulation of X-Ray Diffraction by Crystal

Reciprocal Lattice Vectors

• The reciprocal lattice is defined as the set of all wave vectors K that yield plane waves with the periodicity of a given Bravais lattice.

• Let R denotes the Bravais lattice points;consider a plane wave exp(ik.r). This will have the periodicity of the lattice if the wave vector k=K, such that

exp(iK.(r+R)=exp(iK.r)

for any r and all R Bravais lattice.

Reciprocal Lattice Vectors

• Thus the reciprocal lattice vectors K must satisfy

• exp(iK.R)=1

Brillouin construction