fourier transform and its applications. fourier transforms are used in x-ray diffraction electron...
Post on 19-Dec-2015
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Fourier Transforms are used in
• X-ray diffraction
• Electron microscopy (and diffraction)
• NMR spectroscopy
• IR spectroscopy
• Fluorescence spectroscopy
• Image processing
• etc. etc. etc. etc.
Fourier Transforms
• Different representation of a function – time vs. frequency– position (meters) vs. inverse wavelength
• In our case:– electron density vs. diffraction pattern
What is a Fourier transform?
• A function can be described by a summation of waves with different amplitudes and phases.
Discrete Fourier Transforms
• Function sampled at N discrete points– sampling at evenly spaced intervals– Fourier transform estimated at discrete values:
– e.g. Images
• Almost the same symmetry properties as the continuous Fourier transform
,...3,2,1,0,1,2,3...,
)(
n
nhhn
N
nfn2
,...,2
NNn
DFT formulas
Niknh
tifhdttifthfH
N
kk
nn
N
kknn
/2exp
2exp2exp)()(
1
0
1
0
1
0
/2expN
kkn NiknhH nn HfH )(
1
0
/2exp1 N
nnk NiknH
Nh
Properties of Fourier Transforms
• Convolution Theorem
• Correlation Theorem
• Wiener-Khinchin Theorem (autocorrelation)
• Parseval’s Theorem
Convolution Theorem
•The Fourier transform of a convolution is the product of the Fourier transforms•The Fourier transform of a product is the convolution of the Fourier transforms
Special Convolutions
Convolution with a Gauss function
Gauss function:
Fourier transform of a Gauss function:
Calculation of the electron density
j
jj if SrSF 2exp)(
dvicell
j SrrSF 2exp)()(
x,y and z are fractional coordinates in the unit cell
0 < x < 1
Calculation of the electron density
1
0
1
0
1
0
)(2exp)()(x y z
dxdydzlzkyhxixyzVhkl F
dv icell
j S r r S F 2 exp ) ( ) (
dxdydz V dv
yz kl hx
z y x z y x
S c S b S a S c b a S r) (
Calculation of the electron density
1
0
1
0
1
0
)(2exp)()(x y z
dxdydzlzkyhxixyzVhkl F
)(2exp)(1
)( lzkyhxihklV
xyzh k l
F
This describes F(S), but we want the electron densityWe need Fourier transformation!!!!!F(hkl) is the Fourier transform of the electron density
But the reverse is also true:
Calculation of the electron density
)(2exp)(1
)( lzkyhxihklV
xyzh k l
F
)()(2exp)(1
)( hklilzkyhxihklFV
xyzh k l
Because F=|F|exp(ia):
I(hkl) is related to |F(hkl)| not the phase angle alpha
===> The crystallographic phase problem
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