Phasing by Anomalous Dispersion Methods

BCH681 2012: There can be different effective scattering power at certain atom postions under certain circumstances, and as such Anomalous Phasing is just like having another "derivative".

Oct 10 2012 update: this section now has pages that refer back to our discussion last Thursday Oct. 4 about the physics that cause the anomalous effect, and a few pages that show the phase triangle relationships similar to, and useable with, the phase triangle relationships developed for Isomorphous Phasing.

Anomalous Scattering

Usually the x-ray wave which drives the electrons into forced vibrations is of a much higher frequency than the resonant frequency of the electrons. In this case, the forced damped simple harmonic oscillations of the electrons will be 180° out of phase with the driving wave. This behavior is a completely general characteristic of forced simple harmonic motion. It depends on the interaction of the driving force and the natural restoring force of the oscillator, and it can easily be demonstrated with a pendulum or spring. The relationship of phase lag to frequency is shown on the diagram:

Sometimes the binding of some inner electrons of an atom is of such a strength that the resonant frequency is shifted from that usual for free electrons toward the frequency of the x-ray wave. These electrons will oscillate nearer to 90° out of phase (and also more violently, since they are nearer resonance). For a given atom, then, most of the scattered ray has a 180° phase lag and there will be a few electrons-worth of scattering with a 90° phase lag. Relative to the usual diffracted ray, therefore, the anomalous part of the scattering has a 90° phase advance.

The amount of anomalous scattering for a given atomic species is known and can be looked up in tables (International Tables for Crystallography, Vol. III). For a given x-ray wavelength an atom’s scattering will be described as:

f_{total} = f_{0} + f' + if"

f_{0} is the usual scattering at that 2θ

f' is a correction to usual scattering

i means 90° out of phase

f" is the number of electrons worth of anomalous scattering

This is what a vector diagram looks like with anomalous scattering, for a reflection of phase φ:

This turns out to have consequences for the diffraction pattern, if we consider a crystal in which only a few of the atoms (usually, only the heavy ones) exhibit anomalous scattering.

Friedel pairs are similar reflections that are taken on opposite sides of a crystal. Normally their intensities are exactly equal (as we will show on the next page); it is this equivalence that introduces a center of symmetry into all diffraction patterns, whether the crystals have a center of symmetry or not.

Now let us consider the phase relationships within each triatomic molecule of the crystal lattice on the last page. The geometry is the same for reflections from opposite sides of the crystal, except that for the +θ case the rays from the triangle and circle atoms are ahead of the reference ray (square atoms) and for the -θ case they are behind.

|F(+θ)| = |F(-θ)| because the vector diagrams would superimpose if the -θ one were reflected up onto the +θ one; so the Friedel pair reflections have the same intensity. But suppose one of the atoms (say, the triangles) has anomalous scattering: then an f" term with a 90° phase advance is added onto its f vector. Now when we superimpose the two diagrams as at the right the f" parts do not fall on top of each other and |F(+θ)| < |F(-θ)|.

These intensity differences between |F_{hkl}|^{+} and |F_{hkl}|^{-} are small, but measurable.

Uses of Anomalous Scattering

*Place holder page for equations that relate observed differences of intensity to the geometry and known ratio of anomalous to usual scattering of a particular atom at a particular wavelength.*

Uses of Anomalous Scattering, continued...

In Single Isomorphous Replacement with Anomalous Scattering (SIRAS) experiment:

As in the diagram at the left, anomalous scattering resolves the twofold ambiguity in each protein phase determination, because if |F_{PH}|^{+} < |F_{PH}|^{-} the triangle on the right must be the correct one. This means that theoretically one heavy-atom derivative is sufficient to determine protein phases and make an electron density map (this SIRAS is now done quite routinely with the help of a few additional tricks); or, if you have several isomorphous derivatives then anomalous scattering measurements will improve the accuracy of your φ_{P} values.

The best way to handle the anomalous scattering to get φ_{P} is illustrated below. Instead of just using the direction of the difference between |F_{PH}|^{+} and |F_{PH}|^{-} it makes use of the magnitude of that difference. That way you get an accurate phase if the measurements are exact, and you go wrong less spectacularly if they are inexact.

After laying out the known F_{H} vector with its +θ and -θ anomalous terms, you make a series of trials at each φ_{P} direction. At each trial there will be a trial difference between |F_{PH}|^{+} and |F_{PH}|^{-} which can be compared with the difference you measured experimentally.

If Δ = trial difference - experimental difference, then

probability P = e^{-( Δ2 / ε2 )}

(where ε is an estimate of the error in the anomalous difference measurements) will give a phase probability distribution that can be combined with the ones derived in the Phasing Isomorphous section.

PhaseSIRAS.kin Isomorphous part

PhaseSIRAS.kin Anomalous Scattering part

PhaseSIRAS.kin Isomorphous part combined with Anomalous Scattering part

*Place holder pages for more example of specialized use of anomalous dispersion, geometry and equations of general use, with an introduction to the physics and strategy of Multiple wavelength Anomalous Dispersion (MAD) experiments.*

Download: PhaseSIRAS.kin(30KB)

Download: PhaseMADasSIRAS.kin(130KB)

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