Lecture 3 - Xray Crystallography 2 - Crystal symmetry, Reciprocal space, Data collection

Point lattice

The corners of the unit cells.

Low frequency terms

Approximate large features

Ewald sphere

Bragg's law is satisfied when a reciprocal lattice point sits on the sphere: n*lambda = 2*dhkl*sin(theta). At different crystal orientations, different plans satisfy Bragg's law.

Diffraction from a lattice

Inverse relationship between lattice and diffraction, large space in lattice --> small space in diffraction.

Higher miller indices

Less space (better), families of planes of small planar spacings

Lower miller indices

More space, families of planes of small planar spacings

Twofold

Moves half a unit cell to come to the molecule.

Lattice translations/lattice vectors

The unit cell and its contents repeated in a periodic manner along the unit cell axes.

The seven crystal systems

Triclinic, monoclinic, orthorhombic, tetragonal, rhombohedral (or ortigonal), hexagonal, cubic.

Crystal system

A classification system for crystals based on the geometry of their lattice structures. There are seven crystal systems.

Ewald sphere construction

Define a sphere with radius 1/lambda (or diameter 2/lambda), it goes through the origin of reciprocal lattice at 000 along the incident beam.

Space group

Defines the arrangement of atoms, ions or molecules within a unit cell of a crystal.

Diffraction from planes

Gives reflection. High resolution in the outer space and low resolution closer to the beam.

Asymmetric unit (ASU)

Has no self-symmetry, smallest building block. The summation of the asymmetric unit is a whole object.

Symmetrical object

If, after some operation has been carried out, the result is indistinguishable from the original object.

High frequency terms

Improve approximation by filling in the finer details.

Fourier series

Intricate periodic functions described as the sum of sin and cosin functions.

Calculate the parameters of a unit cell from the diffraction pattern

Reciprocal lattice spacings are inverse of real lattice spacings --> unit cell dimensions are inversely proportional to the spacing of reflections on planes in reciprocal space.

Intensities of the spots

Reflect the scattering density from that particular plane = reflecting the unit cell contents.

Electron density function

Rises near atomic nuclei, falls to lower values between and is otherwise zero.

Types of symmetry

Rotation, mirror plane symmetry, inversion

Target

Step function

Bragg's law and diffraction from sets of planes

The angle of diffraction, theta, is inversely related to the interlunar spacing, dhkl. Sin(theta) is proportional to 1/dhkl

Unit cell

The basic repeating unit of the crystal, repeated in all three directions, with no empty spaces allowed. It is chosen so that it encloses a full complement of the asymmetric units and reflects the symmetry properties of the space group.

Convolution in real and Reciprocal space

The crystal structure is visualized as a convolution of the real space crystal lattice with the molecular contents, and the diffraction pattern as a sampling of the molecular transform with the reciprocal lattice. The two domains of space, the real and reciprocal domains, can be transformed into each other, and back again, by the application of Fourier transformations (FTs)

Result from experiments

The electron density map telling us how the electrons are distributed in the unit cell. Based on the primary sequence of the protein we can build a model. To trace the main chain we need to see connectivity of the electron density. The higher the resolution the easier it is to identify each amino acid.

Resolution

The more orders of diffraction you observe, the higher the resolution (more details).

Miller indices (hkl)

The three integers (designated hkl) that uniquely define a given family of planes (number of times it intersects a, b, c). (230) means that the planes dissect a two times, b three times and c zero times.

The symmetry operations allowed within a unit cell.

Twofold, threefold, sixfold and screw axes.

Position and geometry of spots

Unit cell dimensions (Space group symmetry)

Matthews coefficient

Vm, a way to determine the number of molecules in the unit cell. Vm = (volume of the unit cell (Å3))/(Mw*Z*N), where Mw = molecular weight (Da), Z = multiplicity (=nr of asymmetric units in unit cell), N = number of molecules in the asymmetric unit.

Structure factors

Wave descriptions of X-ray reflections.

Problem with collecting diffraction data

We only observe the position and intensity pf spots, not their time of arrival - the phase information is lost, which is needed to be able to calculate the electron density.

Electron density

What scatters in real space in the unit cell. Intensity distribution of diffraction pattern is related to the electron density distribution in the crystal.