anomalous scattering: theory and practice andrew howard aca summer school 29 july 2005 andrew howard...
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Anomalous Scattering: Theory
and Practice
Andrew HowardACA Summer School
29 July 2005
What is anomalous scattering?
What is anomalous scattering?
Remember that the equation describing the spatial behavior of a wave is exp(ik•r)
What if the wavevector k were complex?
k = kr + i ki
Then the wave looks likeexp(-ki•r)exp(ikr•r): attenuation!
Remember that the equation describing the spatial behavior of a wave is exp(ik•r)
What if the wavevector k were complex?
k = kr + i ki
Then the wave looks likeexp(-ki•r)exp(ikr•r): attenuation!
So much for math.So much for math.
Can we come up with a physical explanation? Sort of:
We state that atoms absorb photons and re-emit them with a phase change.
Can we come up with a physical explanation? Sort of:
We state that atoms absorb photons and re-emit them with a phase change.
What’s the phase change?
What’s the phase change?
The phase change is in fact /2, and it’s positive; that is, the absorbed part leads the scattered part by 90º.
The phase change is in fact /2, and it’s positive; that is, the absorbed part leads the scattered part by 90º.
How do write the atomic structure
factors for this?
How do write the atomic structure
factors for this?Remember that we conventionally write the atomic structure factors as f.
(We’ve emboldened this to remind you that it’s a complex number)
We now sayf = f0 + f’() + if”()
Remember that we conventionally write the atomic structure factors as f.
(We’ve emboldened this to remind you that it’s a complex number)
We now sayf = f0 + f’() + if”()f0
f’()
if”()
The directions depend on (h,k,l)!
The directions depend on (h,k,l)!
The f0 and f’() vectors turn around by 180 degrees when we change from (h,k,l) but the if”() doesn’t, so the resultant changes size and direction:
The f0 and f’() vectors turn around by 180 degrees when we change from (h,k,l) but the if”() doesn’t, so the resultant changes size and direction:
f(h,k,l)
f(-h,-k,-l)
Thus: F(h,k,l) ≠ F(-h,-k,-l) !
Thus: F(h,k,l) ≠ F(-h,-k,-l) !
If there are few atoms with these properties, the differences will be small
But we can still look at F(h) - F(-h) as a tool in phasing
If there are few atoms with these properties, the differences will be small
But we can still look at F(h) - F(-h) as a tool in phasing
How about wavelength?How about wavelength?
Both f’ and f” are wavelength-dependent
f’ and f” are related by the Kramers-Kronig relation, which amounts to saying that the f’ is the derivative of f”
Both f’ and f” are wavelength-dependent
f’ and f” are related by the Kramers-Kronig relation, which amounts to saying that the f’ is the derivative of f”
What happens near an absorption edge?
What happens near an absorption edge?
An absorption edge is an energy at which the absorption (f”) increases dramatically as a function of energy.
It’s the energy associated with liberating an electron from a shell (typically K or L) into the vacuum
An absorption edge is an energy at which the absorption (f”) increases dramatically as a function of energy.
It’s the energy associated with liberating an electron from a shell (typically K or L) into the vacuum
What does this look like?
What does this look like?
e
e
pr
r
p
We want lots of signal:
We want lots of signal:
F(h,p) - F(-h,p) best anomalous
F(h,p) - F(h,e)
F(h,p) - F(h,r)
F(h,e) - F(h,r)Clever linear combinations of the above:Hendrickson, FA values
F(h,p) - F(-h,p) best anomalous
F(h,p) - F(h,e)
F(h,p) - F(h,r)
F(h,e) - F(h,r)Clever linear combinations of the above:Hendrickson, FA values
How do we use these?How do we use these?
Algebraic formulations: Hendrickson and Smith, 1980’sFA values gave maximal (?) use of data
Numerous structures solved that wayProbabilistic formulations:
Resemble standard MIR formulationsPhase probability distributions used
Most modern packages use these
Algebraic formulations: Hendrickson and Smith, 1980’sFA values gave maximal (?) use of data
Numerous structures solved that wayProbabilistic formulations:
Resemble standard MIR formulationsPhase probability distributions used
Most modern packages use these
Why can’t we just look the energies up in a
table?
Why can’t we just look the energies up in a
table?The exact positions of the peak and edge depend substantially on the molecular environment of the scatterer
Bonds between the anomalous scatterer and neighbors often blue-shift the energy spectrum by ~1-2 eV(E/E ~ 10-4)
Tuning issues at the beamline may red-shift or blue-shift the spectrum
The exact positions of the peak and edge depend substantially on the molecular environment of the scatterer
Bonds between the anomalous scatterer and neighbors often blue-shift the energy spectrum by ~1-2 eV(E/E ~ 10-4)
Tuning issues at the beamline may red-shift or blue-shift the spectrum
Do you have to use the real sample?
Do you have to use the real sample?
It would be nice if you didn’t have to:
Crystal decay starts with the initial irradiation
You’d hope that two crystals with the same form will have the same spectrum
Sometimes the solvent will influence the spectrum, so it would be best if you did the spectrum on the real sample
It would be nice if you didn’t have to:
Crystal decay starts with the initial irradiation
You’d hope that two crystals with the same form will have the same spectrum
Sometimes the solvent will influence the spectrum, so it would be best if you did the spectrum on the real sample
Which elements have good edges?
Which elements have good edges?
K edges are sharper than L edgesOften accompanied by a distinct “white line”, i.e. a narrow spectral peak in f”.
Some elements fit into normal beamline operations better than others:Mn, Fe, Cu, As, Se, Br (6.5-13.9 KeV)
L edges are easier to experiment on:rare earths, Pt, Au, Hg, Pb
K edges are sharper than L edgesOften accompanied by a distinct “white line”, i.e. a narrow spectral peak in f”.
Some elements fit into normal beamline operations better than others:Mn, Fe, Cu, As, Se, Br (6.5-13.9 KeV)
L edges are easier to experiment on:rare earths, Pt, Au, Hg, Pb
Ask your beamline people!
Ask your beamline people!
Some beamlines can do MAD but only for a limited range of edges
Some allow full user operationOthers require staff assistance for energy shifts
Recognize that the ultra-sharp edges (Se, As) are easy to miss
Some beamlines can do MAD but only for a limited range of edges
Some allow full user operationOthers require staff assistance for energy shifts
Recognize that the ultra-sharp edges (Se, As) are easy to miss
Why is selenium so popular?
Why is selenium so popular?
Because selenomethione is relatively easy to do in bacteria
There are even ways to do it in non-bacterial systems, but they’re trickier
Assures stoiochiometric inclusion in most cases
Check it with AA or MS if you can!
Because selenomethione is relatively easy to do in bacteria
There are even ways to do it in non-bacterial systems, but they’re trickier
Assures stoiochiometric inclusion in most cases
Check it with AA or MS if you can!
Sulfur anomalousSulfur anomalous
Sulfur’s edge is too low to be useful
But f” is large even at 7-8KeVTradeoffs between conventional absorption and anomalous scattering power
High redundancy and careful data collection help a lot
Sulfur’s edge is too low to be useful
But f” is large even at 7-8KeVTradeoffs between conventional absorption and anomalous scattering power
High redundancy and careful data collection help a lot
ConclusionsConclusions
Anomalous scatter and MAD offer a superior approach to experimental phase determination
Automated software takes a lot of the drudgery out of this approach
Try it: you’ll like it.
Anomalous scatter and MAD offer a superior approach to experimental phase determination
Automated software takes a lot of the drudgery out of this approach
Try it: you’ll like it.