Crystal Structure and Crystallography of Materials Chapter 14: Diffraction Lecture No. 2

Review of Diffraction Lecture 1: We learned that diffraction is the results of constructive and deconstructive interaction of the scattered waves. The interaction of scattered waves depends on the path difference (or phase difference) of the waves scattered to a certain direction. The periodic nature of the scattering centers in real space resulted in periodic nature of the intensity maxima (or minima) in reciprocal space. More importantly, we noted that the periodic nature of intensity maxima (or minima) is related to the periodic nature of the planes of the scattering center arrangement. Scattering center arrangement in an unit cell is defined either by noting the arrangement of scattering centers in 3-D space or by the arrangement of planes (namely, by the vertical direction to the planes and the spacing between planes). This resulted in the derivation of Reciprocal Lattice:

Bragg’s Law: As we know, a diffraction beam may be defined as a beam composed of a large number of scattered rays mutually reinforcing one another. So, why the diffraction phenomena is related to the plane spacing in a reciprocal way, not by the scattering centers – Bragg’s law where, QK-PR = PKcosθ-PKcosθ = 0 rays 1 and 2, scattered by atoms K and L ML+LN = d’sinθ+d’sin θ =2d’sinθ

Scattering by a Unit Cell: Assuming that Bragg’s law is satisfied, wish to find out the intensity of the beam diffracted by a crystal as a function of atom position We do need this in order to analyze all the different crystal structure by diffraction.

Scattering by a Unit Cell: · A : origin · Diffraction occur from the (h00) plane If the position of atom B is specified by its fractional coordinate u=x/a, then the phase difference becomes

Scattering by a Unit Cell: Extend to 3-D : B atom has actual coordinates xyz fractional coordinates then Express any scattered wave in the complex exponent

Diffraction from Crystal Structure: out of phase d 10 λ difference (10) Plane → no scattering (out of phase) d 11 λ difference (11) Plane → scattering (in phase)

Diffraction from Crystal Structure: There is only one lattice point in each unit cell.

Diffraction from Crystal Structure: (a)Primitive lattice: all the reciprocal lattice point survives with the same intensity since there is only one lattice point in an unit cell (b) Body-centered lattice: there are two lattice points in an unit cell a x y z reciprocal lattice

Diffraction from Crystal Structure: (c) Face-centered lattice: there are four lattice points in an unit cell reciprocal lattice a x y z

Diffraction from Crystal Structure: CsCl Crystal Structure: 1.cubic crystal system: 2.primitive cubic bravais lattice: 3.one lattice point – two atoms (Cl (000), Cs (1/2,1/2,1/2)) Cl Cs +

Diffraction from Crystal Structure: NaCl Crystal Structure: 1.cubic crystal system: 2.Face-centered cubic bravais lattice: 3.one lattice point – two atoms (Na (000), Cl (1/2,1/2,1/2)) NaCl 의 atom position: Na 000 ½½0 ½0½ 0½½ Cl ½½½ 00½ 0½0 ½00

Diffraction from Crystal Structure:

Reciprocal lattices: (1) diamond crystal structure (2) zinc blende crystal structure (3) hexagonal crystal structure (4) wurtzeit crystal structure (5) CaF 2 crystal structure Etc…

Diffraction from Crystal Structure: So far, we have discussed the reciprocal lattice points arrangement in an aspect of position and its relative intensities. Once crystal structure is known, we can draw the reciprocal lattice points in a reciprocal space. Still, it does not have any relationship with respect to the wavelength, namely, with an experimental condition.

Diffraction from Crystal Structure:

A diffracted beam will be formed only if reinforcement occurs, and this requires that  be an integral multiple of 2π. → this can happen only if h, k, and l are integers

Diffraction from Crystal Structure: Laue eq’n

Diffraction from Crystal Structure: c P(h,k,l) incident beam 0 Origin of the reciprocal lattice “Ewald sphere” or called “sphere of reflection” : Condition for diffraction from the (hkl) planes is that the point hkl in the reciprocal lattice touch the surface of the sphere.

Summary (Diffraction): Crystal structure (atom position in an unit cell) Reciprocal lattice (the position and intensity variation in a reciprocal space) Diffraction condition (Laue condition, Bragg’s law) Ewald sphere construction and 2  information for (hkl) reciprocal lattice point which is related to the plane spacing, 1/d hkl