3. Crystals

What defines a crystal?Atoms, lattice points, symmetry, space groupsDiffractionB-factorsR-factorsResolutionRefinementModeling!

CrystalsWhat defines a crystal?

3D periodicity: anything (atom/molecule/void) presentat some point in space, repeats at regular intervals,in three dimensions.

X-rays ‘see’ electrons (r) = (r+X)

(r): electron density at position rX: n1a + n2b + n3cn1, n2, n3: integersa, b, c: vectors

CrystalsWhat defines a crystal?

crystalprimary building block:

the unit celllattice:

set of points withidentical environment

CrystalsWhich is the unit cell?

primitivevs.

centered lattice

primitive cell:smallest possiblevolume 1 lattice point

Crystalsorganic versus inorganic

* lattice points need not coincide with atoms

* symmetry can be ‘low’

* unit cell dimensions:ca. 5-50Å, 200-5000Å3

NB: 1 Å = 10-10 m = 0.1 nm

Crystalssome terminology

* solvates: crystalline mixtures of a compound plus solvent

- hydrate: solvent = aq - hemi-hydrate: 0.5 aq per molecule* polymorphs: different crystal packings of the same compound* lattice planes (h,k,l): series of planes that cut a, b, c into h, k, l parts respectively, e.g (0 2 0), (0 1 2), (0 1 –2)

c

ba

Crystalscoordinate systems

Coordinates: positions of the atoms in the unit cell

‘carthesian: using Ångstrøms, and an ortho-normalsystem of axes. Practical e.g. when calculating distances.

example: (5.02, 9.21, 3.89) = the middle of the unit cell of estrone

‘fractional’: in fractions of the unit cell axes.Practical e.g. when calculating symmetry-related positions.

examples: (½, ½, ½) = the middle of any unit cell(0.1, 0.2, 0.3) and (-0.2, 0.1, 0.3): symmetry related positionsvia axis of rotation along z-axis.

Crystalssymmetry

Why use it?- efficiency (fewer numbers, faster computation etc.)- less ‘noise’ (averaging)

finite objects: crystalsrotation axes () rotation axes (1,2,3,4,6)mirror planes mirror planesinversion centers inversion centersrotation-inversion axes rotation-inversion axes----------------------------- + screw axespoint groups glide planes

translations--------------------------- +space groups

Crystalssymmetry and space groups

symmetry elements * translation vector * rotation axis * screw axis * mirror plane * glide plane * inversion center

Crystalssymmetry and space groups

symmetry elements * translation vector * rotation axis * screw axis * mirror plane * glide plane * inversion center

examples(x, y, z) (x+½, y+½, z)(x, y, z) (-y, x, z)(x, y, z) (-y, x, z+½)(x, y, z) (x, y, -z)(x, y, z) (x+½, y, -z)(x, y, z) (-x, -y, -z)

Set of symmetry elements present in a crystal: space groupexamples: P1; P1; P21; P21/c; C2/c

Asymmetric unit: smallest part of the unit cell from whichthe whole crystal can be constructed, given the space group.

-

equivalent positions

CrystalsX-ray diffractiondiffraction: scattering of X-rays by periodic electron densitydiffraction ~ reflection against lattice planes, if: 2dhklsin = n

~ 0.5--2.0ÅCu: 1.54Å

path: 2dhklsin

dhkl

X

Data set:list of intensities I

and angles

Crystalsinformation contained in diffraction data

* lattice parameters (a, b, c, , , ) obtained from the directions of the diffracted X-ray beams.*electron densities in the unit cell, obtained from the intensities of the diffracted X-ray beams.Electron densities atomic coordinates (x, y, z) Average over time and space

• Influence of movement due to temperature: atoms appear ‘smeared out’compared to the static model ADP’s (‘B-factors’).• Some atoms (e.g. solvent) not present in all cells occupancy factors.• Molecular conformation/orientation may differ between cells disorder information.

Crystalsinformation contained in diffraction data

* How well does the proposed structure correspond to the experimental data? R-factor

consider all (typically 5000) reflections, and comparecalculated structure factors to observed ones.

R = | Fhklobserved - Fhkl

calculated | Fhkl = Ihkl

Fhklobserved

OK if 0.02 < R < 0.06 (small molecules)

Crystals - doing calculations on a structure from the CSD

We can search on e.g. compound name

Crystals - doing calculations on a structure from the CSD

We can specify filters!

Crystals - doing calculations on a structure from the CSD

• ‘refcodes’• re-determinations• polymorphs• *anthraquinone*

Crystals - doing calculations on a structure from the CSD

Crystals - doing calculations on a structure from the CSD

Crystals - doing calculations on a structure from the CSD

Z: moleculesper cell

Z’: molecules perasymmetric unit

Crystals - doing calculations on a structure from the CSD

Crystals - doing calculations on a structure from the CSD

Crystals - doing calculations on a structure from the CSDexporting from ConQuest/importing into Cerius

CSD Cerius2

cif

cssrfdatpdb

Not all bond (-type) information in CSD data add that first!

Crystals - doing calculations on a structure from the CSDChecking for close contacts and voids

minimal ‘void size’how close is ‘too close’

default: ~0.9xRVdW

Crystals - doing calculations on a structure from the CSDOptimizing the geometry

* space-group symmetry imposed

CSD optimized*)

a 7.86 7.76b 3.94 4.36c 15.75 15.12 90 90 102.6 107.4 90 90

!

Crystals - doing calculations on a structure from the CSDOptimizing the geometry

* space-group symmetry not imposed;Is it retained?

CSD opt/spgr opt*)

a 7.86 7.76 7.69b 3.94 4.36 4.66c 15.75 15.12 15.93 90 90 90 102.6 107.4 106.8 90 90 90

Crystals - doing calculations on a structure from the CSDOptimizing the geometry

Application of constraints during optimization:• space-group symmetry -- if assumed to be known• cell angles and/or axes -- e.g. from powder diffraction• positions of individual atoms -- e.g non-H, from diffraction• rigid bodies -- if molecule is rigid, or if it is too flexible...

Crystalssingle crystal versus powder diffraction

Powder: large collection of small single crystals, in many orientations

Single crystal all reflections (h,k,l) can be observed individually, leading to thousands of data points.Powder all reflections with the same overlap, leading to tens of data points.

Diffraction data can easily becomputed verification ofproposed model, or refinement(Rietveld refinement)

Next week….

Modeling crystals: how does it differ from small systems?

Applications: predicting morphologypredicting crystal packing