Crystal Structure and Crystallography of Materials Chapter 13: Diffraction Lecture No. 1

Diffraction: So far, we have discussed how the atoms arranged in crystalline structures and the different ways of analyzing the atomic arrangements in 3-D space using lattice points. But, how do we know it? Had we ever seen the atoms which is the size of angstrom scale? Nop, until we had the high resolution TEM to directly investigate the actual atomic arrangements, in projection. Si [110] lattice image (HRTEM)

Diffraction: So, how we understand the arrangement of atoms? Using, diffraction!!!! Electromagnetic wave with an angstrom scale of wavelength called X-ray Object: Crystalline object composed with angstrom scale atoms. Scattering-microscopic diffraction-macroscopic Wave Diffraction phenomena: Scattering- wave-obstacle interaction such that the dimensions of obstacles and wavelength are comparable Diffraction- wave-obstacle interaction such that the dimensions of obstacles are much larger than the wavelength of the wave motion

Diffraction: Wave : 0x ψ

Diffraction:

Transmission Function of an object: object A) Amplitude object : Aexp(2πikx) → Aexp(-μ(x))exp(2πikx) where, φ(x) = exp(-μ(x)) : transmission function B) Phase object : Aexp(2πikx) → Aexpi(2πkx+β(x)) where, φ(x) = exp(iβ(x)) : transmission function C) General object : Aexp(2πikx) → Aexp(-μ(x))expi(2πkx+β(x)) where, φ(x) = exp(-μ(x))exp(iβ(x)) : transmission function D) Opaque object : Aexp(2πikx) → 0, where, φ(x) = 0 : transmission function

Diffraction Integral: θ object k k’ ΔkΔk Diffracted beam from an object : → Fourier transformation of the transmission function “Fourier Transformation”

Simple Diffraction: Transmission function: Φ(x) Φ(x) = 1 -a/2<x<a/2 Φ(x) = 0 elsewhere θ k k’ ΔkΔk ΔkxΔkx -a/2 a/2

Simple Diffraction: Thus,

Simple Diffraction:

Diffraction Physics: Path difference: dsinθ Phase difference: If we let the wave of the center: Then, the wave of the upper side: Then, the wave of the down side:

Diffraction Physics: And if we let Φ 0 =0, and Where x is the distance from the center of the slit.

Diffraction Physics: since,

Diffraction Physics: Let A max = A when θ → 0 ※ Plot of A(θ)/A max :

Diffraction Physics: Remember that, for the 1 st minimum to occur,

Diffraction Physics: D θ θ θ The case of two scattering center,

Diffraction Physics: Use of amplitude – Phase diagram,

Diffraction Physics:

The case of three scattering center, θ D

Diffraction Physics:

Intensity max. Intensity → 0 Intensity → A 1 A 1 Phase Diagram

Diffraction Physics:

When n=4. By the simulation method, D Int. maximum Int. =0

Diffraction Physics:

When n = NWhen n → ∞

Diffraction Physics: Consider the geometry of scattering centers and the diffraction intensity distribution: D1D1 1/D 1 Diffraction Scattering centerDiffraction spot Periodic arrangement of scattering centers in real space with periodicity of D 1 : Periodic arrangement of intensity maxima in inverse space with periodicity of 1/D 1 :

Diffraction Physics: Consider the geometry of scattering centers and the diffraction intensity distribution: Diffraction Scattering centerDiffraction spot Periodic arrangement of scattering centers in real space with periodicity of D 2 : Periodic arrangement of intensity maxima in inverse space with periodicity of 1/D 2 : D2D2 1/D 2

Diffraction Physics:

Reciprocal Lattice: 1.Vector calculation 와 가 만드는 평행사변형의 면적 α

Reciprocal Lattice: p 0

※ In general,

Reciprocal Lattice: Reciprocal lattice array of points completely describes the crystal in the sense that each reciprocal lattice point is related to a set of planes in the crystal.

Reciprocal Lattice: 1. A vector drawn from the origin of the reciprocal lattice to any point in it having coordinates of hkl is perpendicular to the plane in the crystal lattice whose Miller indices are hkl. 2. The length of the vector is equal to the reciprocal of the spacing d of the (hkl) plane

Reciprocal Lattice: A 0 B C N