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:
n1a + n2b + n3c
n1, n2, n3: integers
a, b, c: vectors
Crystals
What defines a crystal?
crystal
primary building block:
lattice:
the unit cell
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, 200-5000Å3
NB: 1 Å = 10-10 m
= 0.1 nm
Crystals
some terminology
* solvates: crystalline mixtures of a compound plus solvent
c
a
b
- 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)
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:
rotation axes ()
mirror planes
inversion centers
rotation-inversion axes
----------------------------- +
point groups
crystals
rotation axes (1,2,3,4,6)
mirror planes
inversion centers
rotation-inversion axes
screw axes
glide 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)
equivalent positions
Set of symmetry-elements present in a crystal: space group
examples: P1; P1; P21; P21/c; C2/c
Asymmetric unit: smallest part of the unit cell from which
the whole crystal can be constructed, given the space group.
Crystals
X-ray diffraction
diffraction: scattering of X-rays by periodic
electron density
diffraction ~ reflection against lattice planes,
if: 2dhklsin = n
X
 ~ 0.5--2.0Å
Cu: 1.54Å
Data set:
list of intensities I
and angles 
dhkl

path: 2dhklsin
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 =  | Fhklobserved - Fhklcalculated |
 Fhklobserved
OK if 0.02 < R < 0.06 (small molecules)
Fhkl =  Ihkl
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: 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
Crystals - doing calculations on a structure from the CSD
exporting from ConQuest/importing into Cerius
2
Cerius
CSD
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: ~0.9xRVdW
Crystals - doing calculations on a structure from the CSD
Optimizing the geometry
CSD optimized*)
a 7.86 7.76
b 3.94
4.36
c 15.75 15.12
 90
90
 102.6 107.4
 90
90
!
* space-group symmetry
imposed
Crystals - doing calculations on a structure from the CSD
Optimizing the geometry
CSD opt/spgr opt*)
a 7.86 7.76 7.69
b 3.94
4.36 4.66
c 15.75 15.12 15.93
 90
90
90
 102.6 107.4 106.8
 90
90 90
* space-group symmetry
not imposed;
Is it retained?
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