3. Crystals What defines a crystal? Atoms, lattice points, symmetry, space groups Diffraction B-factors R-factors Resolution Refinement Modeling!

Crystals What defines a crystal? 3D periodicity: anything (atom/molecule/void) present at some point in space, repeats at regular intervals, in three dimensions. X-rays ‘see’ electrons   (r) =  (r+X)  (r): electron density at position r X: n 1 a + n 2 b + n 3 c n 1, n 2, n 3 : integers a, b, c: vectors

Crystals What defines a crystal? crystal primary building block: the unit cell lattice: set of points with identical environment

Crystals Which is the unit cell? primitive vs. centered lattice primitive cell: smallest possible volume  1 lattice point

Crystals organic versus inorganic * lattice points need not coincide with atoms * symmetry can be ‘low’ * unit cell dimensions: ca. 5-50Å, Å 3 NB: 1 Å = m = 0.1 nm

Crystals some 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 b a

Crystals coordinate systems Coordinates: positions of the atoms in the unit cell ‘carthesian: using Ångstrøms, and an ortho-normal system 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 positions via axis of rotation along z-axis.

Crystals symmetry Why use it? - efficiency (fewer numbers, faster computation etc.) - less ‘noise’ (averaging) finite objects: crystals rotation axes (  )rotation axes (1,2,3,4,6)mirror planesinversion centersrotation-inversion axes screw axes point groupsglide planes translations space groups

Crystals symmetry and space groups symmetry elements * translation vector * rotation axis * screw axis * mirror plane * glide plane * inversion center

Crystals symmetry 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 group examples: P1; P1; P2 1; P2 1 /c; C2/c Asymmetric unit: smallest part of the unit cell from which the whole crystal can be constructed, given the space group. - equivalent positions

Crystals X-ray diffraction diffraction: scattering of X-rays by periodic electron density diffraction ~ reflection against lattice planes, if: 2d hkl sin  = n  ~ Å Cu: 1.54Å  path : 2d hkl sin  d hkl X Data set: list of intensities I and angles 

Crystals information 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.

Crystals information contained in diffraction data * How well does the proposed structure correspond to the experimental data?  R-factor consider all (typically 5000) reflections, and compare calculated structure factors to observed ones. R =  | F hkl observed - F hkl calculated | F hkl =  I hkl  F hkl observed 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

Z: molecules per cell Z’: molecules per asymmetric unit

Crystals - doing calculations on a structure from the CSD

Crystals - doing calculations on a structure from the CSD exporting from ConQuest/importing into Cerius CSD Cerius 2 cif cssr fdat pdb Not all bond (-type) information in CSD data  add that first!

Crystals - doing calculations on a structure from the CSD Checking for close contacts and voids minimal ‘void size’ how close is ‘too close’ default: ~0.9 x R VdW

Crystals - doing calculations on a structure from the CSD Optimizing the geometry * space-group symmetry imposed CSD optimized *) a b c    !

Crystals - doing calculations on a structure from the CSD Optimizing the geometry * space-group symmetry not imposed; Is it retained? CSD opt/spgr opt *) a b c   

Crystals - doing calculations on a structure from the CSD Optimizing 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...

Crystals single 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 be computed  verification of proposed model, or refinement (Rietveld refinement)

Next week…. Modeling crystals: how does it differ from small systems? Applications:predicting morphology predicting crystal packing