Diffraction and Crystal Structure

1. Diffraction and Crystal Structure NANO Workshop 2006

2. Outline • Light and X-rays • Diffraction basics • Determining crystal structures • Activities!

3. What is Light? • Particle or wave? Both! • Photons: particles or “quanta” of light E = hf • Many photons → wavelike • A disturbance that propagates • Usually requires a medium

4. Anatomy of a Wave • Wavelength  • Frequency f • # of cycles per unit time • Speed v = f • Amplitude A

5. Electromagnetic Waves • Speed = 3  105 km/sec (about 186,000 mi/sec)

6. The Electromagnetic Spectrum

7. Visible Light • The color of visible light is determined by its wavelength • White light is a mixture of all colors • We can separate out individual colors with a prism

8. Visible Light 400–440 nm Violet 440–480 nm Blue 480–530 nm Green 530–590 nm Yellow 590–630 nm Orange 630–700 nm Red

9. Superposition and Interference • When multiple waves pass through the same place, the total wave is obtained by adding together the individual wave displacements (Principle of Superposition)

10. X-Ray Diffraction • X-rays have wavelengths comparable to atomic sizes and spacings, about 10–10 m • Crystals and molecules reflect X-rays in specific patterns depending on their structures X-ray diffraction pattern of myoglobin

11. Interaction of X-Rays with Atoms • Involves the electrons, primarily

12. 1 2 d Fraunhofer Diffraction • “Fraunhofer” = incident, outgoing rays parallel • Start with single plane of atoms (a grating) • Wave “in phase” – incident crests and troughs aligned

13. Fraunhofer Diffraction • Extra distance traveled by 2 is d sin • If this is an integer number of wavelengths, we get constructive interference 1 d  2

14. Condition for Maxima where n = 0, 1, 2, … • If  is known, measuring tells us d • Or if d is known we can get , etc.

15. Diffraction Maxima

16. Many Diffraction Centers • All combine constructively at the specified angles • Nice sharp maxima!

17. Bragg Diffraction

18. Bragg’s Law • W. H. Bragg and W. L. Bragg, 1913 (Nobel 1915) • Condition for constructive interference: • Diffraction from different sets of planes in the crystal gives a picture of the overall structure

19. More Information • Intensities of diffraction maxima can vary – more information about detailed structure • Symmetry of the crystal structure is reflected in the diffraction pattern

20. Electron Diffraction • Can be done with particles too, due to their wave nature! • Direct test of the De Broglie relation  = h/p (Davisson and Germer, Thomson) Electron diffraction image

21. Optical Transform Exercises • Uses visible light,  ~ 400-700 nm • Diffracting patterns created with features on this scale • Illuminate with laser pointer – coherent light source

22. Discovery Slide • Horizontal line patterns (a and c) • Diffraction pattern is a set of vertical dots – why? • Check spacing of dots for different line spacings • “Reciprocal lattice effect” • Check spacing of dots for different wavelengths

23. Group Exercise • Given a known wavelength of light (it’s shown on the laser pointer), determine the line spacing of the Discovery Slide pattern a.

24. Other Exercises • Explore the symmetry relations between diffraction patterns and the “atomic” arrangements • Look for variations in intensity for diffraction maxima – when does this occur?