Solid State Physics 2. X-ray Diffraction 4/15/2017

Diffraction 4/15/2017

Diffraction 4/15/2017

Diffraction 4/15/2017

Diffraction using Light Diffraction Grating One Slit Two Slits http://physics.kenyon.edu/coolphys/FranklinMiller/protected/Diffdouble.html 4/15/2017

Diffraction  The diffraction pattern formed by an opaque disk consists of a small bright spot in the center of the dark shadow, circular bright fringes within the shadow, and concentric bright and dark fringes surrounding the shadow. 4/15/2017

Diffraction for Crystals Photons Electrons Neutrons Diffraction techniques exploit the scattering of radiation from large numbers of sites. We will concentrate on scattering from atoms, groups of atoms and molecules, mainly in crystals. There are various diffraction techniques currently employed which result in diffraction patterns. These patterns are records of the diffracted beams produced. 4/15/2017

What is This Diffraction? 4/15/2017

Bragg Law William Lawrence Bragg 1980 - 1971 4/15/2017

The characteristic lines involve the n = 1 or K shell electrons in a metal. If an electron with the appropriate energy strikes the metal target one of the K shell electrons is knocked out. Then, as an electron in an outer shell drops down, a x-ray photon is emitted. Mo 0.07 nm Cu 0.15 nm Co 0.18 nm Cr 0.23 nm 4/15/2017

Monochromatic Radiation 4/15/2017

Diffractometer 4/15/2017

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Nuts and Bolts The Bragg law gives us something easy to use, To determine the relationship between diffraction Angle and planar spacing (which we already know Is related to the Miller indices). But… We need a deeper analysis to determine the Scattering intensity from a basis of atoms. What we want is a set of plane waves (the incident radiation) that matches the periodicity of the lattice being studied. If we find it, we’ll see that it matches Bragg’s condition. 4/15/2017

Reciprocal Lattices Simple Cubic Lattice The volume of the simple cubic, primitive cell is a3. The reciprocal lattice is itself a simple cubic lattice with lattice constant 2/a. 4/15/2017

Reciprocal Lattices BCC Lattice The reciprocal lattice is represented by the primitive vectors of an FCC lattice. 4/15/2017

Reciprocal Lattices FCC Lattice The reciprocal lattice is represented by the primitive vectors of an BCC lattice. 4/15/2017

Drawing Brillouin Zones Wigner–Seitz cell The BZ is the fundamental unit cell in the space defined by reciprocal lattice vectors. 4/15/2017

Drawing Brillouin Zones 4/15/2017

Back to Diffraction Theorem The set of reciprocal lattice vectors Diffraction is related to the electron density. Therefore, we have a... Theorem The set of reciprocal lattice vectors determines the possible x-ray reflections. 4/15/2017

The difference in phase angle is The difference in path length of the of the incident wave at the points O and r is The difference in phase angle is For the diffracted wave, the phase difference is So, the total difference in phase angle is 4/15/2017

Diffraction Conditions Since the amplitude of the wave scattered from a volume element is proportional to the local electron density, the total amplitude in the direction k  is 4/15/2017

Diffraction Conditions When we introduce the Fourier components for the electron density as before, we get Constructive Interference 4/15/2017

Diffraction Conditions 4/15/2017

Diffraction Conditions For a crystal of N cells, we can write down 4/15/2017

Diffraction Conditions The structure factor can now be written as integrals over s atoms of a cell. Atomic form factor 4/15/2017

Diffraction Conditions Let Then, for an given h k l reflection 4/15/2017

Diffraction Conditions For a BCC lattice, the basis has identical atoms at and The structure factor for this basis is S is zero when the exponential is i × (odd integer) and S = 2f when h + k + l is even. So, the diffraction pattern will not contain lines for (100), (300), (111), or (221). 4/15/2017

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Diffraction Conditions For an FCC lattice, the basis has identical atoms at The structure factor for this basis is S = 4f when hkl are all even or all odd. S = 0 when one of hkl is either even or odd. 4/15/2017

KCl KBr 4/15/2017

Structure Determination Simple Cubic When combined with the Bragg law: 4/15/2017

X-ray powder pattern determined using Cu K radiation,  = 1.542 Å q (degrees) sin2 q ratios hkl 11.44 0.0394 1 100 16.28 0.0786 2 110 20.13 0.1184 3 111 23.38 0.1575 4 200 26.33 0.1967 5 210 29.07 0.2361 6 211 34.14 0.3151 8 220 36.53 0.3543 9 300, 221 38.88 0.3940 10 310 X-ray powder pattern determined using Cu K radiation,  = 1.542 Å 4/15/2017

Structure Determination (310) 4/15/2017

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