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.