Macromolecular Crystallography

Section 4.2, Diffraction Vector to Reciprocal Space

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DIFFRACTION VECTOR

λ = 2dsinθ

Refer to later Molecular Scatter chapter for a general description of scattering.
Diagrams: same examples as drawn earlier for Bragg Diffraction, but now showing the diffraction vector which is perpendicular to Bragg Planes and of a length proportional to the reciprocal distance between Bragg Planes.

Bragg Plane d = d1

Bragg Plane d = d1

Bragg Plane d = d1' = d1/2

diffVec.1.png

Parallelogram construction: s is normal to the Bragg planes (and thus is fixed with respect to the crystal). The end of s, then, is a point which by its position (direction and distance out) describes a set of Bragg planes. Note that the length of s is inversely proportional to the Bragg plane spacing d. The s of the third diagram is twice the length of the s of the first diagram, and for the third case any point on a plane halfway between the original ones will scatter in phase with points on the original planes.

Every set of Bragg Planes has its own unique diffraction vector. Just as the Bragg Planes divide up the real crystal in a regular manner, all the diffraction vectors describe the crystal. The crystal is a lattice and the ends of the diffraction vectors describe a lattice, the reciprocal lattice.


DRAWING DIFFRACTION VECTORS TO SHOW THE RECIPROCAL RELATIONSHIP

Parallelogram construction: s is normal to the Bragg planes (and thus is fixed with respect to the crystal). The end of s, then, is a point which by its position (direction and distance out) describes a set of Bragg planes. Note that the length of s is inversely proportional to the Bragg plane spacing d. The s for half-size spacing is twice the length of the s of the single spacing, and for the second case any point on a plane halfway between the original ones will scatter in phase with points on the original planes.

diffVec.1b.png
longwave10mm.jpg shortwave5mm.jpg

Diffraction vector s1 describes a set of Bragg planes spaced by d1 , s2 describes a set of Bragg planes spaced by d2 . The diffraction vectors are a way describing both the sets of Bragg planes and the diffracted rays from those planes.
   In the case where there is a square atom halfway between round atoms as shown, what is the relative intensities of the diffracted x-rays described by s1 and s2 ?

Scale of the diffraction experiment:

1cm = 100,000,000Å
unit cell size (range 50 to 200+) 100Å
crystal size (range 0.05 to 1 mm) 0.1 mm = 1,000,000Å
Spot size on detector just a little larger than the crystal cross-section. 1,000,000Å
crystal to detector distance (range 5 to 50 cm) 10 cm = 1,000,000,000 Å
0.1 mm crystal about 10,000 unit cells across. (Note: a crystal is a mosaic of crystal domains)
crystal to dectector in unit-cells: 10,000,000 uc
Unit cell on chalk-board: 1 uc = 1 ft, crystal size 10,000 ft = 2 miles (size of Duke University)
detector distance 10,000,000 ft = 2,000 miles. (somewhat beyond the chalk tray, perhaps in Arizona)


THE RECIPROCAL LATTICE

XtalToRecip.png

s is normal to the Bragg planes (and thus is fixed with respect to the crystal). The end of s, then, is a point which by its position (direction and distance out) describes a set of Bragg planes. Note that the length of s is inversely proportional to the Bragg plane spacing d. Every set of Bragg Planes has its own unique diffraction vector. Just as the Bragg Planes divide up the real crystal in a regular manner, all the diffraction vectors describe the crystal. The crystal is a lattice and the ends of the diffraction vectors describe a lattice, the reciprocal lattice.



EWALD SPHERE: CRYSTAL AND RECIPROCAL LATTICE

The crystal lattice and the reciprocal lattice are duals. That is, each one describes the other and they are logically linked together. It is convenient to make the origin of the crystal lattice and the origin of the reciprocal lattice to be the same point. Then, during data collection, as the crystal rotates, the reciprocal lattice rotates.

The Ewald sphere helps describes diffracting conditions when the crystal is oriented to the incoming x-ray beam such that particular sets of Bragg Planes can “reflect” the x-rays.

The center of the Ewald sphere is placed along the incoming x-ray beam with the center of the reciprocal lattice on the x-ray beam at the circumference of the Ewald sphere. It is convenient to make the radius of the Ewald sphere be 1/&lambda . When the crystal is rotated, the reciprocal lattice rotates, and when a reciprocal lattice point is on the surface of the Ewald sphere, the associated Bragg Planes are reflecting. This is just the conditions of our parallelogram construction!

The actual diffracted x-ray goes out from the crystal at an angle θ to the Bragg Planes and at angle 2θ to the direction of the original x-ray beam.

There are several advantages to placing the crystal at the center of reciprocal space on the circumference of the Ewald sphere besides emphasizing that the crystal lattice and the reciprocal lattice are duals that rotate tegether and are both descriptions of the real crystal.

  1. We developed the reciprocal lattice using the parallelogram construction which builds the reciprocal lattice around the crystal as a center.
  2. Simultaneous diffraction from a range of wavelengths can be shown without any rescaling.
  3. The kinemage showing diffraction conditions with the crystal and the reciprocal lattice locked together around a common center is much easier to make than the case where two rotation centers must be correlated.
xtalcombEwald.png

HOWEVER: Some textbooks show a different diagram.

e.g. the textbook by Alexander McPherson “Introduction to Macromolecular Crystallography” places the crystal at the center of the Ewald sphere, and the center of the reciprocal lattice on the circumference (where it must be). This does makes drawing the direction of the diffracted ray easier since the x-rays follow a radius of the Ewald sphere from the crystal.

xtalctrEwaldAlex.png

Ewald Sphere: Set up for HelixBearHair.kin

Diffraction from a crystal of molecules described by how the Ewald-sphere (the generalization of all possible paralellogram constructions) intersets the reciprocal lattice (the diffraction-vector description of all Bragg Planes within the crystal.)

HelixBear.Ewald4.png

Download:   download arrow HelixBearHair.kin(3.2MB)   requires Mage



Download:   download arrow PS-RecipLattEwaldDiffraction.pdf(50KB)  Problem for this section


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