Macromolecular Crystallography

Section 3.2, Molecular Scatter

301


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SCATTERING OF LIGHT (X-RAYS) BY A MOLECULE

illustrative diagram of an arbitrary SPECIAL CASE mlsct.1a.png mlsct.1b.png


The combination of rays scattered by any two (or more) atoms depends on the amplitude and phase of each scattered ray. The amplitude depends on the [oscillator strength | number of electrons] in that atom (since it is electrons that scatter x-rays). The relative phase of the rays depends on the relative path lengths they have travelled. Wave 2 travels (δ1 + δ2) further than wave 1, so the phase change in that length is (δ1 + δ2)/λ .

The scattering pattern is distributed in all directions, and varies fairly smoothly.

mlsct.1c.png

Intensity in any particular direction is a function of the [strength of the oscillators | number of electrons in the atoms] and of their relative positions. So if the molecules were randomly positioned, e.g. by free rotation, one could only find out some general property concerning distance such as how big the diameter (radius of gyration) is of the region where the oscillators are distributed. e.g. low angle x-ray scattering, light scattering experiments. This is of mathematical interest only, unless we can relate the pattern seen (or imagined) here to the pattern of structure in the molecule.


For x-rays the scattered ray has 180° phase shift from the incident ray - but this is true for all scattered rays so it makes no difference to how they combine.



SCATTERING FROM 2 OSCILLATORS IN A PLANE

mlsct.2.png


SCATTERING FROM MULTIPLE OSCILLATORS: GENERAL CASE

mlsct.3.png

Rule:
h ⊥ y, z
k ⊥ x, z
l ⊥ x, y
If x,y,z orthogonal,
then h,k,l directions
coincident with x,y,z


    Point P is an oscillator
    O is an arbitrary origin, which may or may not be an oscillator
    r is the vector from O to P, with components x, y, z
    a and b are standard vectors in the directions of the incident and scattered rays: s = b - a
    a, b scaled from unit vectors by 1/λ ; so all these related dimensions will be in Å-1 and express distance as fractions of wavelength.
Path length difference δ = d2 - d1 where,
    d1 is projection of r on a direction : d1  =  ra
    d2 is projection of r on b direction : d2  =  rb
So, δ  =  d2 - d1  =  r  •  b  -  r  •  a  =  r  •  ( b  -  a )  =  r  •  s
    d1 and d2 are unitless as is δ. i.e. δ = δ'/λ where δ' is the real measurement in Å and λ is wavelength in Å so δ is unitless fractional length in terms of wavelength.
           δ        =        r        •        s
(components):     x, y, z    •    h, k, l    (dot product) =     hx + ky + lz
δ is dimensionless and x, y, z are in Å;  h, k, l are therefore in Å-1 (reciprocal Å):  they give us a convenient way of describing a scattered ray.
Relative phase φ = 2π ( hx + ky + lz ) as in |F|eiφ = |F| ei2π( hx + ky + lz ) , expression for a light ray scattered from P.
In this general case h, k, l are not constrained to be integers.

General Equation, simplest form:

Fhkl = |Fhkl| ehkl = n fn ei2π(hx + ky + lz)    fn :scattering power of atom n at given λ and scattering angle.



Special Case redrawn with general case vectors:

For any scattering direction b the set of three vectors a, b, s make a triangle, so we can draw the scattering diagram conveniently in the plane of our paper or screen. In the general case of scattering from a single object even though it can contain many scattering points (atoms), the h,k,l of the scattering vector s are not constrained to be integers.

mlsct.1b.vectors.png

NOTE that the a, b, s relationship is NOT dependent on the location of scattering point P.
Since a and b are "unit" vectors, the magnitude and direction of s is set by the angle of b (the direction in which the scattered ray is observed) to a (the direction of the incident ray).



Download:   download arrow PS-2-interPlanePhases.pdf(120KB)  Problem for this section



A point to come back to...

The vector s can be considered as the normal to a set of planes of spacing d which is proportional to 1/|s|. d then can be thought of as a mark on a "Molecular Ruler" normal to those planes such that if two scattering points were separated in that direction by d, then their scattering would be in phase for that s described direction; i.e. for the rays scattered at the angle b makes to a. This point is implied in the math, and can be appreciated by carefully thinking about relative phase, but is much easier to see geometrically for the case where molecules are arranged in a crystal lattice!

mlsct.Xtal.vectors.png



More wresting with units, dimensions:

We need    δ     =       r        •        s   as unitless fractions of wavelength,
so components:     x, y, z    •    h, k, l    must multiply to cancel units:
if   x, y, z are in Å;   then h, k, l in reciprocal Å,:  and
if   x, y, z are in fractions of something;   then h, k, l in related reciprocal fractions.
(Macromolecular coordinates are standardly given in Å.
Spacing in a crystal is traditionally in fractions of a unit cell.)

The pair: Real Space -- Reciprocal Space is a "dual"
There are definable ways, and general mathematical rules, to transform from one set of such coordinate systems to the other. For crystals, the algebra is relatively straight forward to figure out, though a bit tedious for those with non-orthogonal coordinate systems.

The equations we use for this imply repeating functions:

mlscatTwoEquation.png

    So the idea of a box around the molecule of interest in which we map ρxyz is not only a convenient enclosure, it can be thought of as one box of many identical ones, i.e., a repeating function, a crystal. Each such box would then be called a unit cell.

    If we express x, y, z as fractional coordinates in terms of the box dimensions, then h, k, l must be in terms of reciprocal fractions relating to the box dimensions.

    If the data is in terms of unitless h, k, l we can get back to the molecule in terms of unitless x, y, z. We would need to know the wavelength to predict where the scattered rays would appear in our experiment, and to get the size of the box in real dimensions for describing the distances between atoms in real dimensions.



SCATTERING FROM A MOLECULE OF MULTIPLE-ATOMS USING VECTOR NOTATION

mlsct.5.png

Intensity in any given direction (if molecule is in a crystal then only the directions given by the Bragg Law are allowed) is a function of the number of electrons in the atoms (oscillators) and the relative positions of the atoms.


Let h, k, l index the direction of the diffracted (scattered) rays (reflections).
By definition, Intensity = (Amplitude)2 or I = F2 and | Fhkl | = √Ihkl

mlscatStructFacEqPhiN.png

Where fn is the scattering power of the nth atom (oscillator), and
    φn is the phase of the ray scattered by the nth atom
    (φn is a function of the x,y,z coordinates of that atom)

webimages/301-web-images/mlscatStructFacEqhkl.png" alt="mlscatStructFacEqhkl.png" width="400" height="50"/> mlscatt-FmolSumFatoms.png

If we knew or guessed all the atom positions we could calculate |F| and φ and so could check a trial structure against the real one by comparing |Fcalculated| vs |Fobserved| for each diffracted ray (reflection). But protein molecules have too many atoms for which to guess positions!

NB: actually fn is dependent on θ, the scattering angle because of polarization effect, size of atom, etc.



EQUATIONS AND THE FOURIER TRANSFORM RELATIONSHIP

The Fourier transform is a general concept applicable to all kinds of light scattering, but it really comes into its own with x-ray crystallography where there is sufficient information to apply the details of the equations in a very simple and direct way.

Since it is electrons that are scattering x-rays, the scattering factor can be thought of in terms of the electron density, ρ, throughout the molecule. Rewriting the formula from the previous page in terms of ρ:

Equation-FfromRho.png

where the sum is over a volume that contains the scattering points, for a crystal this is the entire unit cell. So the Volume in these equations is a scaling number but calling it the volume of the repeating unit is a provincialism.

The F’s are the Fourier transform of the electron density, and it conveniently happens to be true that the inverse relation also holds:

Equation-RhofromF.png

where in theory, but not in practice, the h, k, l sums are from -∞ to +∞. So the electron density (which is our desired final result) is the Fourier transform of the F’s. Everything on the right-hand side of the equation is known except, still, for φhkl !

We assume that the molecule stays still, oriented in a definite position. For stationary molecules in a crystal, one must consider the consequences of a regular array. (e.g. crystal diffraction)

If however, the molecules tumble in solution, then one must average over all positions.
e.g. Small Angle X-ray Scattering, SAXS, in the earlier light scattering discussion.



THE EQUATIONS

EquationsThree.png EquationSymbols.png

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