diffraction from point scatterers wave: cos(kx + t)wave: cos(kx + t) + cos(kx’ + t) max min
TRANSCRIPT
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
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
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
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
Floating an image(to avoid sharp edges)b(x)
f(x)
b(x)·f(x)
Floating:subtracts background
High contrastedgesdiffract strongly
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
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:
{