Theory of diffraction Peter Ballo

Ka L-series Ka Fe Cr Kb Kb Ka Kb Ni

Diffraction and destructive interference Incoming, coherent wavefront (all rays have same phase – both rays have crests at same time) Outgoing wave: crests match troughs across the planar wavefront darkness 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.)

Bragg’s Law n  = 2 d sin   d Constructive interference – Optical path difference = 2 (d sin  ) Constructive interference – optical path difference equals whole number, n , of wavelengths, . n  = 2 d sin 

Detector Divergence Receiving slit Anti-scatter Beam Attenuator knife q 2 Divergence slit Attenuator Sample Detector Graphite monochromator Anti-scatter Beam knife Receiving

Crystal Structures Can be described by crystallographic “unit cell” Crystal structure: characterized by regularity of atomic arrangement. Atomic positions repeat periodically “structure unit” can be specified. Can be described by crystallographic “unit cell”

All structures shown here have the same lattice !! Lattices All structures shown here have the same lattice !! Unit cell N.b. identical environment

Close Packing Crystal Structures APF = 0.74 FCC: ABCABC.. HCP: ABAB...

Diffraction Patterns in FCC and BCC X-rays 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 out of phase The (100) reflection from an F-centred cubic lattice Smart, Moore “Solid State Chemistry”

Looking down on Young’s slit (interference) experiment Learning outcomes: Appreciation of importance and prevalence of metallic bonding in elemental solids. Apart from intermediate cases (which in general behave as large covalent molecules), everything else is a gas H atom is exception http://pegasus.cc.ucf.edu/~tbrueckn/2049/

Diffraction gratings Diffraction patterns

Ewald Construction

First Brillouin Zone: Two Dimensional Oblique Lattice

Primitive Lattice Vectors: FCC

Brillouin Zones: FCC

First Brillouin Zone FCC