2. Wave Diffraction and Reciprocal Lattice Diffraction of Waves by Crystals Scattered Wave Amplitude Brillouin Zones Fourier Analysis of the Basis Quasicrystals.

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Presentation transcript:

2. Wave Diffraction and Reciprocal Lattice Diffraction of Waves by Crystals Scattered Wave Amplitude Brillouin Zones Fourier Analysis of the Basis Quasicrystals

Diffraction Of Waves By Crystals Bragg’s Law Reflectance of each plane is about 10  3 to 10  5.

Monochromator X-ray Diffractometer on Powdered Si 1.16A neutron beam on CaF 2 Relative intensities are due to basis.

Scattered Wave Amplitude Fourier Analysis → where  T m i integers b i is called the primitive vectors of the reciprocal lattice, and G a reciprocal lattice vector. Definethen i,j,k cyclic →

Diffraction Conditions Difference in phases between waves scattered at r and O  Scattering amplitude  Scattering vector

Diffraction condition:  (G   G)  From Problem 1: where  Diffraction condition can be written as Bragg’s law

Laue Equations Diffraction condition: →   k lies in the intersection of 3 cones about the crystal axes. Ewald construction White dots are reciprocal lattice points. Incident k drawn with end at lattice point. Scattered k obtained by drawing a circle.

Brillouin Zones Brillouin Zone  Wigner-Seitz cell of reciprocal lattice. Diffraction condition→ → k is on boundary of BZ. Square lattice

Reciprocal Lattice to SC Lattice Primitive lattice vectors: Primitive cell volume: Primitive reciprocal lattice vectors: Reciprocal lattice is also SC.

Reciprocal Lattice to BCC Lattice Primitive lattice vectors: Primitive cell volume: Primitive reciprocal lattice vectors: Reciprocal lattice is FCC. Reciprocal lattice vector: bcc 1 st BZ rhombic dodecahedron Cartesian coord

Reciprocal Lattice to FCC Lattice Primitive lattice vectors: Primitive cell volume: Primitive reciprocal lattice vectors: Reciprocal lattice is BCC. Reciprocal lattice vector: fcc 1 st BZ Cartesian coord

Fourier Analysis of the Basis Scattering amplitude Structure factor For a basis with s atoms  atomic form factor

Structure Factor of BCC Lattice With respect to the SC lattice, the BCC has a basis of 2 atoms at and → E.g., metallic Na: no (100), (300), (111), or (221) lines (cancelled by extra plane at half separation)

Structure Factor of FCC Lattice With respect to the SC lattice, the FCC has a basis of 4 atoms at →

Atomic Form Factor For a spherical distribution of electron density For For forward scattering, G  0, so that f  Z. For X-ray diffraction, f  Z. ( X-ray not sensitive to change in n(r) caused by bonding)