Macromolecular Crystallography

Section 4.3, Crystal Diffraction

402


Link to Contents...
link to previous page


CRYSTAL OF MOLECULES

Consider a crystal of molecules, each with two &alpha-helical regions.

XtalSct.1a.png

α-helix:
side-to-side packing = 10 Å
3.6 residues / turn
1.5 Å rise / residue
5.4 Å rise / turn

XtalSct.1b.png

View ⊥ to ab planes
10,0,0 planes shown across unit cell,
detail shows scattering from, e.g. α carbons, nearly in phase over all the unit cell (of course, exactly in phase with corresponding one in next unit cell).



XtalSct.1c.png

Download:   download arrow PS-HelixSpacingUnitCell.pdf(90KB)  Problem for this section


braggplanesCrop.png

(Figure 2.17 from McPherson...)


2DcompressionWaves.png

Concepts: Reciprocal Lattice vs Bragg Planes

We go on to further development of diffraction from a lattice by using the parallelogram construction to show the relationship of diffraction vectors to specific distances and directions in the real crystal. Two concepts emerge from this approach:

One concept is that the reciprocal lattice is a description of the Bragg planes within the real space lattice of the crystal. The reciprocal lattice points are “named” (indexed) by the same “names” (indices) that describe the Bragg planes. The reciprocal lattice points are in a direction from the origin that is normal to their corresponding Bragg planes, and at a distance inversely proportional to the “d” spacing (the perpendicular displacement) of those Bragg Planes. These indices are integers, and describe how the Bragg planes divide the Unit Cells of the crystal. Each reciprocal lattice point corresponds to a possible diffracted ray from the crystal -- it is the parallelogram construction that illustrates that correspondence -- and the Ewald sphere construction (describes when diffraction will occur from a set of Bragg planes) is a general summary of all the possible parallelogram constructions.

Another concept is that a value can be assigned to each reciprocal lattice point that is the relative intensity of the corresponding diffracted ray. Of course, each diffracted ray also has a relative phase which can be assigned to its reciprocal lattice point, but we can not measure that phase directly. The intensity of a diffracted ray is determined by the relative offsets of the scattering points, the atoms, from the diffracting Bragg plane. We will explore that by filling in a few atoms in a crystal and seeing how their perpendicular position with respect to the Bragg planes affects the phase of their scattering in the direction of the diffracted ray, and consequently, affects the intensity of that diffracted ray. This phase as a function of the distance between Bragg planes can be described as “clock” turns around the circle that describes the magnitude and phase of the wave from an atom.

Bragg Plane definition:
Bragg Planes for crystals of few atoms (and as W. L. Bragg seems to have first described them) are conviently thought of as the planes of atoms making up the crystal. However, as you have seen already, sometimes only some atoms line up on the Bragg planes even for relatively simple molecules, and for proteins few if any atoms are actually exactly on a given Bragg plane.
    The more robust way of describing Bragg Planes is in terms of the unit cell, however you happen to define it. Then the Bragg Planes cut the unit cell edges into integer fractions, and the strength of diffraction from a Bragg Plane set has to do with the spacing between atoms in the direction of that Bragg Plane normal where atoms separated by that “d” scatter in phase with each other.

The Unit Cell is the small parallelepiped built upon the three translations selected as unit translations. The unit cell repeats by those 3 translations to fill all space.


GEOMETRICAL FACTORS THAT AFFECT MEASURED INTENSITIES

XtalSct.2a.png
  1. Polarization of the reflected x-rays:
    The component of the incoming oscillation that is parallel to the plane of reflection is undiminished.
       But the perpendicular component is only “seen” by the reflected ray in projection (since the wave cannot oscillate parallel to its direction of travel;) thus some energy is lost, and the amount of intensity lost is a function of θ.

  2. Lorentz factor:
    To collect the full scattering pattern the crystal must be moved. If, because of this motion, some reflections get more time than others their relative measured intensities will be increased. This is known as the Lorentz factor and is a function of almost every variable involved in the motion, including θ.

  3. Absorption:
    Matter (electrons) absorb x-rays as well as scattering them. If the crystal is not a sphere some paths through it are longer than others and will absorb relatively more of the x‑ray beam. This absorption correction is sometimes measured empirically and sometimes calculated geometrically from the known shape of the crystal and its position during each reflection.

XtalSct.2b.png

None of these geometrical effects contains any information we are interested in: they are simply factors we must correct for before using the intensities to find out about the structure of our protein molecule.


SIZE OF THE SPOT

4 main factors contribute the the actual size of the observed spot that the diffracted ray makes on the detection device (e.g. film).

  1. Size of the crystal.
  2. Mosaic spread of the crystal.
  3. Wavelength dispersion of the source.
  4. Size of the x-ray source
    (Home source e.g. Copper emmission: each point of the actual source radiates in all directions and the crystal "sees" the source through a beam tunnel)
    Synchrotron: initial beam is more collimated)
    The issue is how parallel the rays are coming from the effective source.
    Either of these might be further affected by, e.g., a monochromator crystal or glancing angle mirrors.)
  5. (How thick the detector is and the angle the x-rays hit it. For instance, a multi-wire area-sensitive detector has to have a finite depth in which to catch the x-ray photons -- if the x-ray beam comes in at an angle, then there is a latteral uncertainty as to when the photon ionizes the detector gas. Old fashioned film and many modern detector-surfaces are relatively quite thin.)

SIZE OF MOSAIC BLOCKS

The trick is for a crystal to be perfectly-imperfect for the wave-length/atom-types such that the mosaic blocks are large enough to make sharp diffraction rays, yet small enough that the chance of a scattered ray to diffract again within the mosaic block is relatively small.

Effective interaction probability, i.e. efficiency of an X-ray interacting with electrons in a crystal according to James p 53 is 10-4 for a “strong” reflection (perhaps from NaCl, which perhaps would interact more strongly with xrays than CNO molecules). Apparently a multiplier to the atomic scattering factor.
REALLY NEED A DRAWING OF MOSAIC CRYSTAL...
google mosaic block protein crystal:
(need IUCR code to download pdf’s) Albrecht Messerschmidt X-ray Crystallography of Biomolecules page 77: mosaic block of size ca. 0.1 micrometer with average tilt angle 0.1 - 0.5 degrees for protein crystals.
0.1u = 0.0001 mm= 0.00001 cm = 1000Å


HOW AN INDIVIDUAL ATOM AFFECTS THE STRUCTURE FACTOR

  1. f, the atomic scattering factor:
    Atoms (i.e., the distribution of electrons) have a real, finite size, and so it is possible for various parts of an atom to scatter out of phase with one another. The same sort of path-Iength arguments apply as when we considered two separate atoms scattering out of phase. But for a given atom, the electron distribution is approximately spherically symmetric, and the contribution of this factor can be calculated as a function of θ or looked up in tables.
  2. XtalSct.3.png
  3. the B factor :
    All uncertainties in the position of an atom is put into the B factor. Thermal motion is only a minor part of the B-factor for atoms of a macromolecule. Atoms seem to vibrate, so their effective scattering is spread over a larger volume. (It may not be a spherically symmetric volume, such as when vibration is along the direction of a bond.) This effect is also a function of θ.

The most common formula for the nth atom (spherically symmetric case) is:
fn,θ e-Bn(sin2θ)/λ2


bottom 4.3, Crystal Diffraction

Next:    5. Fourier Transforms    

The relationship of Real and Reciprocal space.


   or: Back to Table of contents