Xray diffraction

1
Crystals
Electron Density Map
Data analysis and phase determination
Model building and refinement
X-ray diffraction data collection
Structural Model
Diffraction Image
2
How do we get expression for desired unknown in
terms of measurable F?
?(x y z)e 2? i (hx ky lz) dxdydz
F(h k l)
Summation of a periodic function. Physically,
summation of scattered waves satisfying Braggs
law
3
Fourier Series and Transforms
?(x y z)e 2? i (hx ky lz) dxdydz
F(h k l)
Fourier Theorem any function can be written as
f(x)e 2? i(hx) dx
F(h)
F(h) is the fourier transform of f(x)
h and x have inverse units
4
Fourier Series and Transforms
f(x)e 2? i(hx) dx
F(h)
?
?
F(h)e-2? i(hx) dh
Back transform
f(x)
-?
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Fourier Series and Transforms
?
?
?
F(h k l)
V ?(x y z)e2? i (hx ky lz) dxdydz
x
y
z
?
?
?
1/V F(h k l) e-2? i (hx ky lz) dhdkdl
?(x y z)
h
k
l
?
?
?
1/V F(h k l) e-2? i (hx ky lz)
?(x y z)
k
h
l
If we measure F(h k l) we can calculate ?(x y z)
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Do we measure F(h k l)?
F(h k l) is a wave it has amplitude, frequency,
phase.
I (h k l) F(h k l)2
We effectively measure F(h k l) not F(h k l)
7
Phase problem
?
?
?
1/V F(h k l) e-2?i(hx ky lz)
?(x y z)
k
h
l
?
?
?
1/VF(h k l) e-2?i(hx ky lz)i?(hkl)
?(x y z)
h
k
l
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Fourier Series and Transforms
?
?
?
F(h k l)
V ?(x y z)e2?i(hx ky lz) dxdydz
x
y
z
?
?
?
1/V F(h k l) e-2?i(hx ky lz)
?(x y z)
h
k
l
Note If we know ?(x y z) we can calculate F(h k
l)
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Diffraction Image
Electron Density Map
I(h k l)
a(h k l)
1) Isomorphous Replacement 2) Anomalous
Scattering 3) Molecular Replacement
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Isomorphous Replacement
FP
FPH
Native Data
Derivative Data
What does isomorphous mean?
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Fhlk
FPH FP FH
FP
FH
FPH
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Quantities we want
FP FPH - FH
FP
Quantities we know
FP
Intensities of native diffraction
FPH
Intensities of derivative diffraction
FH
Structure factors w/phases of heavy atoms
?
?
?
1/V F(h k l) e-2?i(hx ky lz)
?(x y z)
k
h
l
?
?
?
F(h k l)
V ?(x y z)e2?i(hx ky lz) dxdydz
x
y
z
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FP FPH-FH
FPa
-FH
FPH
FP
FPb
How do we choose between FPa and FPb?
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Make a second derivative!
FP FPH-FH
FPa
FPH
-FH
FPb
?
?
?
1/V FP(h k l) e-2?i(hx ky lz)
?(x y z)
h
k
l
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?
?
?
FH(h k l)
V ?H(x y z)e2?i(hx ky lz) dxdydz
x
y
z
?H(x y z) requires knowing the position of heavy
atoms Determine using Patterson function
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Patterson Function
?(x, y, z)
?
P(u,v,w)
?(x, y, z) ?(xu,yv,zw) dxdydz
vol unit cell
(u v w)
(u v w)
y
v
x
u
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Patterson Function
?
P(u,v,w)
?(x, y, z) ?(xu,yv,zw) dxdydz
vol unit cell
Patterson map will contain points corresponding
to vectors between atoms in the real cell
Real Cell
Patterson Cell
v
y
u
x
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Patterson Function
Real Cell
Patterson Cell
1) Patterson is symmetric about origin
(centrosymmetry)
2) Can see pattern of real cell in patterson cell
repeated N times
3) Contains N(N-1) peaks (not counting origin) ?
gets complicated!
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Patterson Function
?(x, y, z)
?
P(u,v,w)
?(x, y, z) ?(xu,yv,zw) dxdydz
vol unit cell
Key point can calculate P(u,v,w) from
experimental data
?
?
?
F(h k l)2 cos 2?(hu kv lw)
P(u,v,w)
1/V2
k
h
l
but complicated map
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Heavy Atom Patterson Function
?
?
?
P(u,v,w)
F(h k l)2 cos 2?(hu kv lw)
1/V2
k
h
l
Patterson analysis is simplified for heavy atoms
1) Use (FPH(h k l)- FP(h k l))2 as
coefficients ? difference map reflects heavy
atom contribution
2) High Z ? strong peaks
3) Calculate (x,y,z) of heavy atoms directly from
Harker section peaks
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Harker Peaks
Symmetry related atoms give rise to peaks in
Patterson map in specific locations
2 fold
y
(x, y, z)
(u,v,w) (x--x, y--y, z-z)
(u,v,w) (2x,2y, 0)
z
x
(-x, -y, z)
w0
(2x,2y, 0)
v
w0 called Harker plane (section)
u
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Harker Peaks
How many Harker sections are there for P222? What
are they?
Harker Planes u0 v0 w0
Equivalent Positions 1) (x, y, z) 2) (x, -y,
-z) 3) (-x, y, -z) 4) (x, y, z)
Patterson Peaks (1) (2) (0,2y,2z) (1)
(3) (2x,0,2z) (1) (4) (2x,2y,0)
Note If more than one heavy atom, will have
multiple peaks but in same Harker sections
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Data! Harker section at w1/2
What kind of symmetry gives Harker section at
w1/2?
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• Isomorphorous Replacement Step-by-step

1) Collect data for native crystal and
derivitized crystal
FPH(h k l)
FP(h k l)
2) Calculate difference patterson map using
intensity differences as coefficients
?
?
?
P(u,v,w)
(FPH(h k l)- FP(h k l))2 cos 2?(hu kv lw)
1/V2
k
h
l
3) Determine heavy atom positions from Harker
peaks
4) Use heavy atom (x,y,z) to calculate FH from
back fourier transform
?
?
?
FH(h k l)
V ?H(x y z)e2?i(hx ky lz) dxdydz
x
y
z
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• Isomorphorous Replacement Step-by-step

5) Repeat with second derivative to get FH
6) Use FH and FH with FP and FPH to
calculate FP (h k l)
7) Use FP (h k l) to calculate ?(x, y, z) with
forward transform
?
?
?
1/V F(h k l) e2?i(hx ky lz)
?(x y z)
k
h
l
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• Anomalous scattering

At absorption edge there is an additional
contribution to the structure factor anomalous
scattering factor
Fano F DFano
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• Anomalous scattering

At absorption edge Friedels law no longer holds
Fano(h k l) ? Fano(-h -k -l)
(2 1 0)
b

• Need for anomalous signal
• ? close to absorption edge
• heavy atoms

a
(-2 -1 0)
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• Anomalous scattering

At absorption edge Friedels law no longer holds
Fano (h k l) ? Fano (-h -k -l)
-

Fano F DFano
DFano ? DF-ano
DFano
F
Fano
F-
F-ano
DF-ano
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F Fano - DFano
Quantities we know
F
Intensities of diffraction at l far from
absorption edge
Fano
Intensities of diffraction at l near absorption
edge
DFano
Structure factors w/phases of anomalous
contribution
DFano only from heavy atoms Get intensity
DFano from table Phase from known positions of
heavy atoms
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F Fano - DFano
Fa
-DFano
Fano
F
Fb
How do we choose between Fa and Fb?
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F Fano-DFano
F- F-ano-DF-ano
Fa
F-ano
-DF-ano
Fb
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• Other Notes on Anomalous Diffraction

F
DFano
F Fano - DFano
Fano

• Requires data collection at multiple wavelengths
• MAD Multiple anomalous dispersion

2) Only change wavelength ? Data sets are
isomorphous
3) Common to use selenomethionine 4) Synchrotron
required!!