Laue Photography . Mathematics Structures time-resolved crystallography neutron crystallography electron crystallography. Laue Method. Uses polychromatic (“white”) X-rays. (generally l <2.0Å)
Polychromatic X-rays should have a flat intensity profile.
Note: Very small separation between spots. This means the crystals must be small, the beam must be small, and the crystals must be well-ordered.
visible part of transform
crystal resolution limit
Ewald sphere has radius 1/l. Longer wavelength X-rays cause the Ewald sphere to shrink, picking up a different part of the reciprocal lattice.
Bragg’s Law for multiple wavelengths
In monochromatic crystallography, one q angle translates to one d (one resolution). In Laue crystallography, one q angle covers a range of d.
All reflections (nh,nk,nl) have the same S direction, but the length is inversely proportional to l. So (h,k,l) at l=2.0Å and (2h,2k,2l) at l=1.0Å diffract to exactly the same spot on the film.
Summing harmonics is like viewing a lattice from the origin.Every line of sight represents one spot on the film.
Standing in the middle of a circular orchard, how many tree trunks you see are hiding other tree trunks (“multiples”, red)? And how many are not (“singles”, green)?
range of Ewald spheres (fixed beam)
19 refl (18 spots)2 multiples (1 spot)17 singles=89%
Now cut down all the trees except the ones that are on the Ewald sphere for one of the wavelengths (range lmin-lmax). Only the trees in the grey region remain. How many are multiples?
n=1,2,3 etc. within “Cruickshank range”
Scale factors. A function of wavelength, polarization, etc.
total intensity for all harmonics in range
Scale factors (f) may be found for singles.Then, each multiple is a linear equation of the unknown F2’s,which can be solved during least squares refinement.
If a reaction can be initiated in a crystal, simultaneously throughout the crystal, then Laue photography can capture the structural changes at the ns (10-9s) to ms (10-3s) timescale.
Crystals must withstand ultrahigh fluxes of X-rays (or are destroyed in the process)
Light-initiated reactions can be studied using the Laue method.
Photoactive yellow protein (from the phototrophic bacterium Ectothiorhodospira halophila)
Genick et al. Science, 275 (5305): 1471