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X-‐Ray diffraction
Definition of the crystalline state
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Crystals are solids (but not all solids are crystals)
Crystals are the most ordered form of the matter
Crystals are 3-‐D (2-‐D) regular arrays of ions, atoms, molecules; they have triple (double) periodicity
Crystals have long-‐range order
Each repeating unit = motif (whatever it is (atom/molecule)) within a crystal has an identical
environment
X-‐Ray diffraction
This is the essence of the X-‐ray crystal structure analysis (XRA)
...
Crystal forming chemical entities = motifs: metals, ions, atoms (e
...
diamond), organic compounds, peptide
proteins, lipids, oligosaccharides, DNA, RNA etc
...
X-‐Ray analysis is the best source of the above data; they are key components of structural databases for
further chemical (e
...
quantum) and physical calculations
...
• Scattering and diffraction
These techniques are used to obtain the structure of the motif (compound) from its crystal structure
...
o All objects -‐ irrelevant of their size-‐ scatter radiation that is shined on them
...
o We can focus back the scattered / diffracted waves again
We can do this because light travels with different media with
different speed: different media have different refractive index (n)
...
o What should be the relationship between an effective scattering and the size of the object?
The power of scattering/diffraction by an object is:
-‐ Directly proportional to the similarity between the wavelength of the incident radiation
and the size of the scattering object
...
8 -‐ 2Å = X-‐ray radiation
...
This is elastic coherent scattering: frequencies and wavelengths of the incoming X-‐rays and
scattered/diffracted X-‐rays are the same/unchanged
...
We cannot measure (yet) the X-‐ray scattering produced by a single chemical entity (organic molecule)
is it is too weak, but we can use crystals as 3-‐D amplifier of scattering produced by single crystal motif
...
There are also other types of interactions of X-‐rays with
electrons, e
...
excitations
...
In the crystal there are thousands of molecules -‐ some of them survive long
enough to give a measurable radiation
...
2)
Directions of the diffracted X-‐rays
The phase α must be reconstructed in rather complex/difficult
experimental and computing methods:
Phase problem = phase solution methods
Lattices, crystal planes, hkl indices
The 3-‐D periodicity of the crystal can be
simplified and represented by an abstract crystal
lattice
...
They cannot be just any
translations, they to reproduce all crystal motives
(lattice points) if applied to any single lattice point (or
motif’s atoms)
...
To get the structure of the motive we have to first get the
information about the unit cell size and it’s arrangement
...
e
...
(hkl) is
the miller index of that plane (round
brackets no commas)
...
g
...
Distance between planes is given by d(hkl)
Reciprocal dependence between (hkl) and d(hkl): larger (hkl) values (finely spaced planes)
then smaller d(hkl)
•
Diffraction
o On the optical grating
Path difference XY between
diffracted beams 1 and 2:
sin θ = XY/a -‐> XY = a sin θ
For 1 and 2 to be in phase and give constructive interference:
XY = λ, 2λ, 3λ, 4λ…nλ
So,
a sinθ = nλ (n=order of diffraction)
This is the so-‐called grating relationship where z is the distance
between scattering centres
...
5 -‐ 2Å, λ must be in the
range of 0
...
(x, y, z ) -‐> Coordinates of any point in a unit cell
(xj, yj, zj) -‐> Coordinates of the J atom
(u, v, w) -‐> Coordinates for the lattice parameters
Principles of structure solution
Wave nature of diffraction and its relationship with electron density
•
Fourier transform
Any waveform can be represented by the
super-‐positions of sine and cosine waves
...
The Fourier transforms uses
above concepts to convert between two
different descriptions of the physical
system
...
Intensity of the reflections= Brightness
Phase of the reflections= Colour
The diffractions pattern of a crystal lattice is another lattice -‐
reciprocal lattice
...
This corresponds to sampling of molecular transform at reciprocal
lattice points
...
Spatial distribution -‐> i
...
directions = θ
•
Analytical description of the structure factors
Wave = |F| x α
Intensity of the wave: I ≈ F2
√I = √F2 = |F|
λ = wavelength, |F| = amplitude, α = phase
If we are trying to describe a physical property (a wave in our case) that has amplitude and phase, it
is convenient to represent it in form of a complex number:
Description of the wave F that results from
superposition of many (j) other waves with amplitudes
f and phases ϕ:
All above rules f super-‐position of scattered
waves apply for the scattered waves apply for the
scattering of the X-‐rays by the electron ‘clouds’ of
the atoms
...
fj is the atomic scattering factor, this is already known for a
particular atom, it is proportional to:
Atomic number (Z) x amplitude of scattering one electron
In general, some (hkl) planes will have more atoms, some less;
some scattering atoms will have more electrons (e
...
Hg, Pt, I, Br
etc
...
), hence, some reflections will be
more intense -‐ some weak depending on how many atoms = electrons contributed to them
...
hkl are miller indices of a particular reflection from a particular
(hkl) plane
...
•
The only information available after the X-‐ray experiment:
-‐ Unit cell dimensions (from Bragg law -‐ we can measure θ) (symmetry)
-‐ fj (from the tables)
-‐ |Fhkl| = √Ihkl (from the measurement of intensity of the reflection)
The above approach was based on the assumption that the scattering power of the electron cloud
surrounding each atom could be equated to the ‘atomic number’ of electrons concentrated at the
atomic centre
...
Electron density is defined as the number of electrons per unit volume, so, in any small volume dv at
the point with coordinates: (x, y, z) -‐ notes no j’s
ρ(x, y, z) dv
Benefits are that atoms are getting more real in ‘size’ -‐ we are dealing with clouds of electrons and
that we are not constrained here by atoms coordinates: (x, y, z) not (xj, yj, zj)
...
Maximum θ angles
which is experimentally achievable: nλ=2dsinθmax or the
smallest dhkl = dmin we registered the diffraction from
o Solution of the phase problem
Ø Direct methods (purely computational):
Do not require any extra information -‐ use
experimental data set |Fhkl| only
...
Ø Heavy atom methods (purely computational or
experimental):
Our (in) organic compound already has heavy atom
(e
...
Fe, Pt, Au, Hg, Ru etc
...
Good for
proteins crystals as well (lots of solvent content there -‐ 50%) -‐ heavy atoms can be
soaked into these crystals
...
o
Model building -‐ refinement of the phases/structure
Importance of the difference electron density map:
The module (|Fobs|-‐|Fcalc|) should show only the differences between the actual = observed
model: (measured I) = |Fobs|, and the model that has been built: |Fcalc|
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d
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The difference e
...
map
should show minima (negative e
...
) if the atom in
the model was modelled in a wrong place = is not
where it should be
...
d
...
Problems is refinement:
In refinement we optimise following parameters:
atomic coordinates (xj, yj, zj) (3 parameters); atomic
displacement parameters -‐ B/U factors, 3-‐isotropic
(sphere), 6-‐anisotropic (ellipsoid); occupancy -‐
atom’s movement, crystal disorder (crystal
mosaicity), symmetry -‐ special positions
...
)
-‐ The input from the chemist is crucial here: