Diffraction from point scatterers

Wave: cos(kx + t) Wave: cos(kx + t) + cos(kx’ + t)

max

min

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

A1cos(2x/) + A2cos(2x/ + 2)2 = 2dsin()/If dsin() = n, get max because two cosine terms are in phase

A 1cos(2x/)

A 2cos(2x/ + 2

)

Frauenhofer diffraction: sum of sin terms

Sum of 2 point scatterers: A1cos(2x/) + A2cos(2x/ + 2)

Sum of n point scatterers(cosine transform):

n

iii xA

1

)/2cos(

n

iii ixBixAxf

1

)/2sin()/2cos()(

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

i

ii ixBixAxf )2sin()2cos()(

x-a 0 a

f(x)

dxexFxf ixX2)()(

Discrete transform ok for periodic objects

Continuous transform for non-periodic objects

xixeix sincos

Box function and its transform

b(x) = 1 - l <= l b(x) = 0 x < -l or x > l

-l 0 l x

)()()( 2 XBdxexbboxfcn ixX

iX

edxe

ixXixX

2

22

X

X

iX

Xi

iX

ee XiXi

2

2sin2

2

2sin2

2

22

Sync function (transform of box)

-0.5 0 0.5 x

-2.0 0 2.0 x

Lattice function (and transform)

n

nsxxs )()(

s

Delta fcn: (x) = , when x=0 (normalized area =1)

[s(x)] = S(x)

n

snXXS )/()(

1/s X

x

Convolution

)()()( 2121 xcdttftxfffConvolution:

x-a 0 a

c(x)

a

x

f1

f2

Cross-Correlation

dttxftfxffC )()()]([ 2121Correlation of f1&f2:

x-a 0 a

c(x)

x

f1

f2

x

C(f1f2)

f1(x)* = f1(x) when real fcn

Auto-correlation Patterson fcn

h

ihuehFuP 22)()(Patterson function:

dxuxfxfc )()( 1*

112auto-correlation

][)()( 2*

12

2*

1 FFehFhFh

ihu Inverse transform ofProduct of F1

*F2

Convolution Theorem2121 ][ FFff

-l 0 l x

-1/2l 1/2l 2/2l-2/2l-4/2l -4/2l

a x

1/a X

F1·F2

xl < a then 1/l > 1/a

)()( xsxb

- 4/a -3/a -2/a -1/a 0 1/a 2/a 3/a 4/a

1D crystal

Truncating the crystal (finite size)

3

3

)()(n

naxxs

-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 contrastedgesdiffract strongly

Boxed area - floated

Image sampling(for digital FT)

Shannon-Nyquist sampling limit:Finest spatial period must be sampled >2xOtherwise aliasing (jaggies)

Must see peaks and valleys of a feature2d

d

Fast Fourier Transform

N x N imageReal numbers

N/2 x N transform(complex numbers)

orig

0,0 N/2,0

0,N/2

0,-N/2 Spatial frequency correspondingTo 2 pixels in orig image

Reciprocal pixelsIn transform1/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 resolution20 pixels in recip space 10 A resolution (this is max in transform, consistent with Shannon-Nyquist sampling limit1 pixel in recip space 40x5=200 A resol (i.e., frame size of image – max spatial freq)

10 A200 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,-20

10 A200 A

0,0

0,20

0,-20

10 A 200 A

Central transform sideband

alia

sing

Aliasing cont

0 1/d

Lower sampling rate

alia

sing

aliasing