Diffraction from point scatterers Wave: cos(kx + t)Wave: cos(kx + t) + cos(kx’ + t) max min
Diffraction from 2 points At finite distance: Fresnel diffraction At : Frauenhofer diffraction
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 )
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
Fourier transforms x -a 0 a f(x) Discrete transform ok for periodic objects Continuous transform for non-periodic objects
Box function and its transform b(x) = 1 - l <= l b(x) = 0 x l - l 0 l x
Sync function (transform of box) x x
Lattice function (and transform) s Delta fcn: (x) = , when x=0 (normalized area =1) [s(x)] = S(x) 1/s X x
Convolution Convolution: x -a 0 a c(x) a x f1f1 f2f2
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
Auto-correlation Patterson fcn Patterson function: auto-correlation Inverse transform of Product of F 1 * F 2
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
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
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
Boxing an crystal image instead of sharp reflections, get sync functions with width inversely related to box size
Image (em grid) diffraction
Smaller area (same mag) Black = zero density
Floating an image (to avoid sharp edges) b(x) f(x) b(x)·f(x) Floating: subtracts background High contrast edges diffract strongly
Boxed area - floated
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
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
Image (em grid) diffraction
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: {
Lower sampling interval (2x)
Aliasing 0,0 0,20 0, A 200 A 0,0 0,20 0, A 200 A Central transform sideband
aliasing
Aliasing cont 0 1/d
Lower sampling rate
aliasing