Structure Factors & Fourier Transforms How to get more than unit cell sizes from your diffraction data
Chem 4850/6850/8850 X-ray Crystallography The University of Toledo
http://homepages.utoledo.edu/clind/ Yet again… expanding convenient concepts
First concept introduced: Reflection from lattice planes - facilitates derivation of Bragg’s law - explains the location of spots on the observed diffraction pattern
Second concept introduced: Scattering from atoms, all the electron density was assumed to be concentrated in a “lattice point” (e.g., origin of the unit cell) - gave the correct locations for spots in the diffraction pattern
But what about atoms that are located “between lattice planes”? - any atom can scatter an X-ray that hits it - what does this imply for the overall scattering amplitude? Atoms “between lattice planes”
“Atoms between lattice planes” will always scatter X-rays so that the resulting wave is partially out of phase with X-rays scattered from the “lattice planes” - the “lattice planes” can be envisioned as atoms located in the plane described
“Structure Determination by X-ray Crystallography”, Ladd and Palmer, Plenum, 1994. Description of a plane
Point P can be described by the vector p, which is perpendicular to the plane LMN, or by the coordinates (X,Y,Z) - the intercept equation of the plane is given by
X/ap+Y/bp+Z/cp = 1 - this can be rearranged to “Structure Determination by X-ray Crystallography”, X cos + Y cos + Z cos = p Ladd and Palmer, Plenum, 1994.
- describes family of parallel planes that are located at distance |p| from 0 Path difference
X cos + Y cos + Z cos = p
For plane O’: p = dhkl; ap = a/h; bp = b/k; cp = c/l
cos = dhkl/(a/h); cos = dhkl/(b/k); cos = dhkl/(c/l)
For plane A: pA = dA; atom at (XA, YA, ZA) d d d ⇒d =X ⋅ hkl +Y ⋅ hkl +Z ⋅ hkl a/h b/k c/l
d = (hx + ky + lz )dhkl using fractional coordinates
According to Bragg’s law, the path difference between O and A is
δ = 2d sinθhkl = 2dhklsinθhkl(hx +ky + lz )
Using wave equations instead
We can describe a wave as
W0 = f·cos(2X/) or W0 = f·cos(t)
where W0 is the transverse displacement of a wave moving in the X direction, the maximum amplitude is f at t=0 and X=0
For other waves with a maximum at t=tn and X=Xn, we can write
Wn = fn·cos(2X/ - n) where n = 2Xn/
Two arbitrary waves of same frequency can then be represented as
W1 = f1·cos(t- 1) = f1·[cos(t) cos(1) + sin(t) sin(1)]
W2 = f2·cos(t- 2) = f2·[cos(t) cos(2) + sin(t) sin(2)]
Sum: W = W1+W2 = cos(t)·( f1·cos(1)+ f2·cos(2))+sin(t)·( f1·sin(1)+ f2·sin(2)) Wave equations continued
We know that we can write any wave as
W = F cos(t - ) = F [cos(t) cos() + sin(t) sin()]
Comparison with the equations on the last slide gives
F cos() = f1 cos(1) + f2 cos(2)
F sin() = f1 sin(1) + f2 sin(2)
F can be calculated as
2 2 1/2 F = [(f1 cos(1) + f2 cos(2)) + (f1 sin(1) + f2 sin(2)) ] Argand diagrams
Waves can be represented as vectors in an Argand diagram - represented with real and imaginary components - allows for straightforward wave addition - For any given wave, we can write F = |F|·(cos(t) + i·sin(t)) = |F|·eit (compare to slide 9 of handout 6!)
“Crystal Structure Analysis for Chemists and Biologists”, Glusker, Lewis and Rossi, VCH, 1994. Wave addition using Argand diagrams
Any number of waves can be added using an Argand diagram
- For N waves, 퐹 cos( 휑)= 푓 cos( 휑 ); 퐹 sin( 휑)= 푓 sin( 휑 )
- Alternatively, we can write 퐅 = 푓 퐞 =|퐹|퐞
“Structure Determination by X-ray Crystallography”, Ladd and Palmer, Plenum, 1994. Phase difference
From our treatment of path differences of waves scattered by arbitrary atoms, we obtained
훿 = 휆(ℎ푥 + 푘푦 + 푙푧 )
The corresponding phase difference can be written as
This implies that our expression for F depends on the diffraction direction as described by the Miller indices hkl (or the diffraction angle, if you prefer) Structure factors
F should be more properly written as F(hkl) or Fhkl to express its dependence on the diffraction angle
There will be a unique F(hkl) corresponding to each reflection in a diffraction pattern - note – and remember! – that all atoms in a crystal contribute to every F(hkl)!
F(hkl) is called the structure factor for the (hkl) reflection
Can be broken down in its real and imaginary components
퐅(ℎ푘푙)= 퐴(ℎ푘푙)+ 푖퐵(ℎ푘푙) where
퐴(ℎ푘푙) = 푓 cos( 2휋(ℎ푥 + 푘푦 + 푙푧 )) 퐵(ℎ푘푙) = 푓 sin( 2휋(ℎ푥 + 푘푦 + 푙푧 )) Atomic scattering factors
The individual atomic components, fj, are called atomic scattering factors
- depend on nature of atom, direction of scattering, and X-ray wavelength
- provide a measure of how efficiently an atom scatters X-rays compared to an electron
- usually written as fj, although they should be written as fj,, - listed as a function of sin()/ for each atom in the International Tables
th - maximum value of fj is Zj, the number of electrons of the j atom Intensity and structure factors
We cannot measure structure factors, instead, we will measure the intensity of diffracted beams
The phase is given by
Problem: We lose the phase information when measuring intensities! Factors between I and F
There are several factors between I and F that will change the measured intensity:
2 2 - thermal vibration of the atoms: exp[-Bj(sin ()/ ] - Lorentz factor, for normal 4 circle diffractometers: 1/sin(2)
- absorption: e-t, where is the linear absorption coefficient and t the thickness of the specimen
- polarization: p = (1+cos2 (2))/2
- scale factor Thermal vibration
Thermal vibration “smears out” the electron cloud surrounding an atom
- we see time average over all configurations
- displacement of atoms leads to slight phase difference for atoms in neighboring unit cells
2 2 - f = f0·exp[-Bj(sin ()/ ]
-
- is the mean square
amplitude of displacement Systematic absences
Some reflections will be systematically absent from your diffraction pattern
These absences are a result of
- lattice centering
- glide planes
- screw axes
Note that simple rotations and mirror planes to not give systematic absences!
Space groups can be assigned based on systematic absences Systematic absences (2)
“Structure Determination by X-ray Crystallography”, Ladd and Palmer, Plenum, 1994. Fourier transforms and crystals
“Crystal Structure Analysis for Chemists and Biologists”, Glusker, Lewis and Rossi, VCH, 1994. The electron density
The electron density in a crystal can be described as a periodic function - same contents in each unit cell - represented by a sum of cosine and sine functions - wavelength is related to lattice periodicity
The construction of a periodic function by the summation of sine and cosine waves is called a Fourier synthesis
The decomposition of a periodic function into sine and cosine waves is called a Fourier analysis Fourier synthesis and analysis
“Crystal Structure Analysis for Chemists and Biologists”, Glusker, Lewis and Rossi, VCH, 1994. Fourier series
The complete set of components that are necessary to describe a periodic function is called a Fourier series
The intensity of a Bragg reflection is related to the amplitude of an electron density wave in the crystal in hkl direction - all Bragg peaks: Fourier series describing the electron density of a crystal - scattering of X-rays by atoms can be regarded as a Fourier analysis,
producing the Fhkl structure factor waves The electron density can be determined from the diffraction pattern by a Fourier synthesis: 1 휌(푥푦푧)= 퐹(ℎ푘푙) exp( − 2휋푖(ℎ푥 + 푘푦 + 푙푧)− 휑 ) 푉 Electron density waves
In order to obtain the electron density, we need to sum all contributing electron density waves (Fourier synthesis)
Note that the electron density waves do not have the same frequencies (wavelengths)! - periodicity depends on hkl
- wavelength is dhkl “Crystal Structure Analysis for Chemists and Biologists”, Glusker, Lewis and Rossi, VCH, 1994. Summing electron density waves
“Crystal Structure Analysis for Chemists and Biologists”, Glusker, Lewis and Rossi, VCH, 1994. A 1D example of a Fourier synthesis
“Crystal Structure Analysis for Chemists and Biologists”, Glusker, Lewis and Rossi, VCH, 1994.
(b) Effects of successive addition of terms in (a). Note how the inclusion of more terms leads the the formation of distinct electron density peaks. Fourier transforms
Generic definition: FT FT-1
Convolution* of two Fourier transforms:
* A convolution is an integral which expresses the amount of overlap of one function g as it is shifted over another function f. It therefore "blends" one function with another. The relationship between F and
We have shown that the structure factors can be described by a Fourier synthesis from waves scattered by a crystal – A diffraction pattern is the Fourier transform of an “object” (e.g., the crystal for crystallographic experiments)
We have also shown that the electron density can be obtained by a Fourier synthesis of electron density waves (structure factors) – The inverse Fourier transform of the diffraction pattern gives the original object again ( )
( ) ( )
The FT relationship between f and atomic
“Crystal Structure Analysis for Chemists and Biologists”, Glusker, Lewis and Rossi, VCH, 1994. Diffraction from holes – a FT perspective
Diffraction from a single hole gives a ring pattern
What happens when we have a second, identical hole at a distance a from it? For one hole, we can write: 퐒⋅퐫
퐒⋅퐫 퐒⋅(퐫 퐚) For two holes as described above:
퐒⋅퐚
The second hole superimposes a fringe function that will double the amplitude of
A0 when a·S is integral, and will lead to extinction halfway in between Fringe patterns
“Structure Determination by X-ray Crystallography”, Ladd and Palmer, Plenum, 1994. What’s a hole got to do with a crystal?
The hole scatters the light rays
An electron scatters X- rays
Imagine a crystal as a “hole pattern” of electrons! “Structure Determination by X-ray Crystallography”, Ladd and Palmer, Plenum, 1994. The opposite approach…
…transforming your diffraction pattern into an electron density map
“Structure Determination by X-ray Crystallography”, Ladd and Palmer, Plenum, 1994. The resolution of your electron density map will strongly depend on how many reflections you use in creating it! Fourier transforms of crystals
“Crystal Structure Analysis for Chemists and Biologists”, Glusker, Lewis and Rossi, VCH, 1994.