Theory of diffraction Peter Ballo. KK KK L-series Fe Cr KK KK Ni KK KK

Theory of diffraction

Peter Ballo

K

K

L-series

Fe

Cr

K

K

Ni

KK

Diffraction and destructive interference

Destructive interference

(This diagram shows the electric field of the EM radiation as it oscillates with time and distance travelled by the ‘ray’. They make the simplifying assumption that scattering from atoms does not change phase of radiation, i.e. the incoming and outgoing rays have the same value of electric field at the scattering atom.)

Outgoing wave: crests match troughs across the planar wavefront darkness

Incoming, coherent wavefront (all rays have same phase – both rays have crests at same time)

Bragg’s Law

Optical path difference = 2 (d sin )

d

Constructive interference –

optical path difference equals whole number, n , of wavelengths, .

n = 2 d sin

2

Divergence slit

Attenuator

Sample

Detector

Graphite monochromator

Anti-scatter slit

Beam knife

Receiving slit

Crystal StructuresCrystal Structures

Crystal structure: characterized by regularity of atomic arrangement.

•Atomic positions repeat periodically•“structure unit” can be specified.

Can be described by crystallographic “unit cell”

Lattices

Unit cell

All structures shown here have the same

lattice !!

N.b. identical environment

Close Packing Crystal Structures

APF = 0.74

FCC: ABCABC..

HCP: ABAB...

Diffraction Patterns in FCC and BCC

Simple cubic: All combinations of h, k and l are seen in diffraction

FCC and BCC:Systematic absences of some reflections (certain combinations of h, k, l)Centred atoms lead to destructive interference in some cases

The (100) reflection from an F-centred cubic lattice

Smart, Moore “Solid State Chemistry”

X-rays

out of phase

http://pegasus.cc.ucf.edu/~tbrueckn/2049/

Looking down on Young’s slit (interference) experiment

Diffraction gratings Diffraction patterns

Ewald Construction

First Brillouin Zone: Two Dimensional Oblique Lattice

Primitive Lattice Vectors: FCC

Brillouin Zones: FCC

First Brillouin Zone FCC