1. Diffraction from point scatterers Wave: cos(kx +  t)Wave: cos(kx +  t) + cos(kx’ +  t) max min

2. Diffraction from 2 points At finite distance: Fresnel diffraction At  : Frauenhofer diffraction

3. Frauenhofer diffraction  Bragg’s Law d dsin(  ) – difference in path for lower ray A 1 cos(2  x/  ) + A 2 cos(2  x/  +  2 )  2 = 2  dsin(  )/ If dsin(  ) = n, get max because two cosine terms are in phase A 1 cos(2  x/  ) A 2 cos(2  x/  +  2 )

4. Frauenhofer diffraction: sum of sin terms Sum of 2 point scatterers:A 1 cos(2  x/  ) + A 2 cos(2  x/  +  2 ) Sum of n point scatterers (cosine transform): Any periodic function can be broken down into sum of sines and cosines with same fundamental period Fourier transform: sum of sin terms

5. Fourier transforms x -a 0 a f(x) Discrete transform ok for periodic objects Continuous transform for non-periodic objects

6. Box function and its transform b(x) = 1 - l <= l b(x) = 0 x l - l 0 l x

7. Sync function (transform of box) -0.5 0 0.5 x -2.0 0 2.0 x

8. Lattice function (and transform) s Delta fcn:  (x) = , when x=0 (normalized area =1)  [s(x)] = S(x) 1/s X x

9. Convolution Convolution: x -a 0 a c(x) a x f1f1 f2f2

10. Cross-Correlation Correlation of f 1 &f 2 : x -a 0 a c(x) x f1f1 f2f2 x C(f 1 f 2 ) f 1 (x) * = f 1 (x) when real fcn

11. Auto-correlation  Patterson fcn Patterson function: auto-correlation Inverse transform of Product of F 1 * F 2

12. Convolution Theorem - l 0 l x -1/2 l 1/2 l 2/2 l -2/2 l -4/2 l  a x  1/a X

13. F 1 ·F 2 x l 1/a - 4/a -3/a -2/a -1/a 0 1/a 2/a 3/a 4/a 1D crystal

14. Truncating the crystal (finite size) -3a -2a -1a 0 1a 2a 3a x b(x) = 1, when -3 > x < 3 -3a -2a -1a 0 1a 2a 3a  - 4/a -3/a -2/a -1/a 0 1/a 2/a 3/a 4/a

15. Boxing an crystal image instead of sharp reflections, get sync functions with width inversely related to box size

16. Image (em grid) diffraction

17. Smaller area (same mag) Black = zero density

18. Floating an image (to avoid sharp edges) b(x) f(x) b(x)·f(x) Floating: subtracts background High contrast edges diffract strongly

19. Boxed area - floated

20. Image sampling (for digital FT) Shannon-Nyquist sampling limit: Finest spatial period must be sampled >2x Otherwise  aliasing (jaggies) Must see peaks and valleys of a feature 2d d

21. Fast Fourier Transform N x N image Real numbers N/2 x N transform (complex numbers) orig 0,0N/2,0 0,N/2 0,-N/2 Spatial frequency corresponding To 2 pixels in orig image Reciprocal pixels In transform 1/size-of-image-pixels

22. Image (em grid) diffraction

23. Say 5 Å pixel size in image and 40 x 40 pixels in image 0,0 0,20 0,-20 20,0 40 pixels in recip space  5 A resolution 20 pixels in recip space  10 A resolution (this is max in transform, consistent with Shannon-Nyquist sampling limit 1 pixel in recip space  40x5=200 A resol (i.e., frame size of image – max spatial freq) 10 A 200 A 0,0 0,10 0,-10 10,0 20 A200 A 2x reduced sampling: {

24. Lower sampling interval (2x)

25. Aliasing 0,0 0,20 0,-20 10 A 200 A 0,0 0,20 0,-20 10 A 200 A  Central transform sideband

26. aliasing

27. Aliasing cont 0 1/d

28. Lower sampling rate

29. aliasing