Anomalous Scattering: Theory and Practice Andrew Howard ACA Summer School 29 July 2005 Andrew Howard ACA Summer School 29 July 2005

What is anomalous scattering?  Remember that the equation describing the spatial behavior of a wave is exp(ikr)  What if the wavevector k were complex?  k = k r + i k i  Then the wave looks like exp(-k ir)exp(ik rr): attenuation!  Remember that the equation describing the spatial behavior of a wave is exp(ikr)  What if the wavevector k were complex?  k = k r + i k i  Then the wave looks like exp(-k ir)exp(ik rr): attenuation!

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?  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?  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 say f = f 0 + 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 say f = f 0 + f’( ) + if”( ) f0f0 f’( ) if”( )

The directions depend on (h,k,l)!  The f 0 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) !  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?  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?  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? e e p r r p

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, F A 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, F A values

How do we use these?  Algebraic formulations: Hendrickson and Smith, 1980’s  FA values gave maximal (?) use of data  Numerous structures solved that way  Probabilistic formulations:  Resemble standard MIR formulations  Phase probability distributions used  Most modern packages use these  Algebraic formulations: Hendrickson and Smith, 1980’s  FA values gave maximal (?) use of data  Numerous structures solved that way  Probabilistic formulations:  Resemble standard MIR formulations  Phase probability distributions used  Most modern packages use these

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 ~ )  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 ~ )  Tuning issues at the beamline may red-shift or blue-shift the spectrum

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?  K edges are sharper than L edges  Often 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 ( KeV)  L edges are easier to experiment on: rare earths, Pt, Au, Hg, Pb  K edges are sharper than L edges  Often 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 ( KeV)  L edges are easier to experiment on: rare earths, Pt, Au, Hg, Pb

Ask your beamline people!  Some beamlines can do MAD but only for a limited range of edges  Some allow full user operation  Others 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 operation  Others require staff assistance for energy shifts  Recognize that the ultra-sharp edges (Se, As) are easy to miss

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 anomalous  Sulfur’s edge is too low to be useful  But f” is large even at 7-8KeV  Tradeoffs 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 7-8KeV  Tradeoffs between conventional absorption and anomalous scattering power  High redundancy and careful data collection help a lot

Conclusions  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.