CRYSTAL OF MOLECULES
Consider a crystal of molecules, each with two &alpha-helical regions.
α-helix:
side-to-side packing = 10 Å
3.6 residues / turn
1.5 Å rise / residue
5.4 Å rise / turn
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).
Download: PS-HelixSpacingUnitCell.pdf(90KB) Problem for this section
(Figure 2.17 from McPherson...)
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: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
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.
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.
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.
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).
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
The most common formula for the n^{th} atom (spherically symmetric case) is:
f_{n,θ} e^{-Bn(sin2θ)/λ2 }
bottom 4.3, Crystal Diffraction
The relationship of Real and Reciprocal space.