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
• ‘refcodes’• re-determinations• polymorphs• *anthraquinone*
Crystals - doing calculations on a structure from the CSD
Z: moleculesper cell
Z’: molecules perasymmetric unit
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)