Solid State Physics

1. Solid State Physics 2. X-ray Diffraction

2. Diffraction

3. Diffraction

4. Diffraction

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

6. 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.

7. 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.

8. What is This Diffraction?

9. Bragg Law William Lawrence Bragg 1980 - 1971

10. Mo 0.07 nm Cu 0.15 nm Co 0.18 nm Cr 0.23 nm

11. Monochromatic Radiation

12. Diffractometer

14. 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.

15. Reciprocal Lattices • Simple Cubic Lattice The reciprocal lattice is itself a simple cubic lattice with lattice constant 2/a.

16. Reciprocal Lattices • BCC Lattice The reciprocal lattice is represented by the primitive vectors of an FCC lattice.

17. Reciprocal Lattices • FCC Lattice The reciprocal lattice is represented by the primitive vectors of an BCC lattice.

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

19. Drawing Brillouin Zones

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

21. The difference in path length of the of the incident wave at the points O and r is So, the total difference in phase angle is The difference in phase angle is For the diffracted wave, the phase difference is

22. 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

23. Diffraction Conditions • When we introduce the Fourier components for the electron density as before, we get Constructive Interference

24. Diffraction Conditions

25. Diffraction Conditions • For a crystal of N cells, we can write down

26. Diffraction Conditions • The structure factor can now be written as integrals over s atoms of a cell. Atomic form factor

27. Diffraction Conditions • Let • Then, for an given h k l reflection

28. 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).

30. 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.

31. KCl KBr

32. Structure Determination Simple Cubic When combined with the Bragg law:

33. X-ray powder pattern determined using Cu K radiation,  = 1.542 Å

34. Structure Determination (310)