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Anomalous Scattering: Theory and Practice

Anomalous Scattering: Theory and Practice. 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(i k • r ) What if the wavevector k were complex? k = k r + i k i

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Anomalous Scattering: Theory and Practice

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

  2. 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!

  3. 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.

  4. 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.

  5. 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”() if”() f’() f0

  6. 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: f(h,k,l) f(-h,-k,-l)

  7. 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

  8. 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”

  9. 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

  10. What does this look like? p r e r p e

  11. 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

  12. 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

  13. 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

  14. 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

  15. 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 (6.5-13.9 KeV) • L edges are easier to experiment on:rare earths, Pt, Au, Hg, Pb

  16. 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

  17. 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!

  18. 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

  19. 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.

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