1. Definition of the crystalline state:
Crystals are solids (but not all solids are crystals!)‏
Crystals are the most ordered form of the matter
Crystal are 3-D (2-D) regular arrays of ions, atoms,
molecules; they have triple (double) periodicity
Crystals have long range order.
Each repeating unit (whatever it is) within a crystal
has an identical environment

2. X-ray Diffraction is the essence of the X-ray crystal structure analysis (XRA)‏
The main aim of XRA is the determination of 3-D structure of the chemical
entity = structural motive which is forming the crystal that is being
repeated periodically in the whole volume of the crystal
Crystal forming chemical entities = motives: metals, ions, atoms (e.g. diamond),
organic compounds, peptides, proteins, lipids, oligosaccharides, DNA, RNA etc.

3. X-ray analysis is the most accurate method to determine:
- structure of the crystal hence structure of the crystal motif e.g. molecule:
- bond distances and angles (C-C 1.542(2) Å, C-C-N (12)o)
- conformation of the compound
- absolute configuration of the compound/atom
X-ray crystal analysis is the best source of the above data:
they are the key components of structural databases for
further chemical (e.g. quantum) and physical calculations
Co(III)(Ph4porphyrin) (Cl)
Mn(H2O)62+
Rh(en)2Cl2+ trans

4.  phase What are the components of waves? A B  |F| wavelength
| F | = amplitude
phase

5. How to obtain the structure of the motif (compound)‏
from its crystal structure?
The foundation of recognition/”visibility” of all structures:
Scattering and Diffraction
All objects – irrelevant of their size – scatter radiation
which is shine on them.

6. Difference between scattering and diffraction:
- scattering: spherical
- diffraction: more directional as it results from
interference of scattering from many centres,
- (or it results from interference of incoming =
= source wave with new, scattered waves.)

7. How we can retrieve the information about the scattering objects?
Because we can focus back the scattered/diffracted waves again.
The image of the object
Lenses:
focussing
Scattering/Diffraction
The object

8. Why can we focus back the image of the scattering/diffracting object?
Because light travels with different media with different speed:
- different media have different refractive
index n
n is a measure how much is the speed
of light (or other waves such as sound
waves) is reduced inside the medium
> nG
> nA
Air
nAc nG
Glass
nG = 1.5
Here

9. What should be the relationship between effective
scattering and the size of the object ?
Web examples
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
Larger object – larger waves needed for an effective scattering
Smaller objects – smaller wave needed for an effective
scattering

10. Hence: X-ray radiation in X-ray analysis!
 of radiation
3.7 pm – 700 nm
Resolution
2 nm 10 nm 200nm
Typical C-C bond: Å, so
The most useful radiation to study crystal
Structure has to have  in the range Å
Hence: X-ray radiation in X-ray analysis!
1.54
~1-2
1 meter = 102 cm = 103 mm = 106 mm = 109 nm =1010 Å

11. Intensity of the wave: I ~ F 2 I = √F 2 = | F |
wave = | F | x  = F
Intensity of the wave: I ~ F 2
If we know I then:
I = √F 2 = | F |
Lenses are focussing back all information that is
contained in the scattered or diffracted waves:
- amplitudes | F | (intensities I)‏
- phases 
so they can produce back the image of the
scattering/diffracting object

12. What?!!
NO LENSES!!!!
Usually
Monochromatic:
one, well defined 
In X-ray Diffraction we do not have lenses which could
focus diffracted rays back to the crystal structure, n=1!.
We can only register:
- directions of the diffracted X-rays and their
- intensities I hence | F | amplitudes only of the diffracted rays:
!their phases are missing! = phase problem

13. 1. Intensities I of diffracted X-rays therefore | F | - amplitudes of
Information directly available from an
X-ray single crystal diffraction experiment:
1. Intensities I of diffracted X-rays
therefore | F | - amplitudes of
diffracted X-rays
2. Directions of the diffracted X-rays
The phases  must be reconstructed in
rather complex/difficult experimental
and computing methods:
phase problem = phase solution methods

14. The key-feature of XRD and XRA is the interaction between
the crystal and the incoming X-ray radiation
(l in range off 0.8 – 2 Å).
X-rays in the crystal are:
scattered by the electrons:
- Thomson scattering: the electron oscillates in the
electric field of the incoming X-ray beam and an
oscillating electric charge radiates electromagnetic waves
- this is elastic and coherent scattering:
frequencies and wavelengths of the incoming X-rays
and scattered-diffracted X-rays are the same/unchanged
this scattering is becoming very discreet in terms of directions
some scattered X-ray waves are reinforced, some weakened
as we are dealing here with the diffraction - REFLECTIONS
which is amplified by millions copies of the same atoms (electrons!)‏
in the same positions in the crystal space due to
crystal periodic, repetitive (in 3-D) unique character
But why not to measure scattering from one molecule and determine its
structure this way?…..

15. We use crystals as 3-D amplifiers of scattering
We cannot measure (yet) the X-ray scattering produced by single chemical entity (organic molecule): it is too weak.
We use crystals as 3-D amplifiers of scattering
coming from single crystal motif.
X-ray Diffraction is well welcomed “side effect” of this process due to amplifying or cancelling effect
of scattered radiation emitted by electrons
There are also other types of interactions of X-rays with electrons: e.g. excitations.
These type of high energy phenomena would damage the
single molecule almost immediately.
In crystal there are thousands of molecules – some of them survive long enough to give a measurable radiation. .

16. v

17. v

18. Crystal structure = Crystal Lattice + motif. web: http://marie. epfl
The 3-D periodicity of the crystal can be simplified
and represented by an abstract crystal lattice.

19.  X ? b a Crystal lattice is described by three translations: a, b, c
They can not be just any translations: they have to reproduce all
crystal motives (lattice points) if applied to any single lattice point
It is NOT
the unit cell
It is the right
unit cell a b ?
Two atom
compound: X 

20. They determine the unit cell, which has to be:
a lattice ‘building block’, which edges correspond to a, b, c
it should give the whole crystal lattice if moved by a, b, c
it has to be of the right handed system
it has to have the smallest possible volume
it has posses the highest possible symmetry characteristic for the lattice
(this is why some unit cells are not primitive)‏

21. a z c b   y  x The unit cell in three dimensions.
The unit cell is defined by three vectors a, b, and c, and three angles , , .    b c a y x z
Unit cells are defined in terms of the lengths of the three vectors and the three angles between them.
For example, a = 94.2 Å, b = 72.6 Å, c = 30.1 Å,  = 90.0°,  = 102.1°,  = 90.0°
a = 8.32 Å, b = Å, c = 9.28 Å, a = 90.0°, b = 90.0°, g = 90.0°

22. Content of the Unit cell c Size and the arrangement Crystal structure
(symmetry)
of the unit cell
Crystal structure
Motif = molecule, atoms
To get the structure of the motive we have to:
get the information about the unit cell size and its arrangement

23. In the crystal lattice we can distinguish: - lattice points
- lattice directions
- lattice planes
Co-ordinates of the lattice points are given in the
fractions u,v,w of the a,b,c lattice translations
(u,v,w) a b c x y z

24. Crystal Lattice directions
symbol:
examples:
[uvw]
[100] [010]
[001] : z : c b a c y x b a

25. Crystal planes The planes are “imaginary”
= inter-plane spacing measured at 90o to the planes
The planes are “imaginary”
All planes in a set of planes are identical - equivalent
The perpendicular distance between pairs of adjacent planes is called d: interplanar spacing
Need to label planes to be able to identify them…………

26. (1 3 0) plane plane (2 1 0) (h k l) (h k l) Miller index (hkl)
Find intercepts on a, b, c: 1/2, 1, (1 1/3 0)‏
Take reciprocals , 1, ( )‏
plane (2 1 0)
(1 3 0) plane
Miller index
(hkl)
(h k l)
(h k l)
General label is (h k l) which intersects at a/h, b/k, c/l
(hkl) is the MILLER INDEX of that plane (round brackets, no commas).

27. x y z 1a 1b 1c O
(111)
Standard
triangle x y z 1a 1b 1c O
(222)

28. (0 1 0) plane (0 0 1) plane Plane perpendicular to x cuts at 1,  , b c
Plane perpendicular to x cuts at 1,  , 
 (1 0 0) plane a b c a b c
(0 1 0) plane
(0 0 1) plane
NB an index 0 means that the plane is parallel to that axis

29. Planes - conclusions
Miller indices define the orientation of the plane within the unit cell
The Miller Index defines a set of planes parallel to one another (remember the unit cell is a subset of the “infinite” crystal
All possible sets of planes in a particular lattice may be described by (hkl) values
Any of these sets of planes may contain scattering electrons (atoms) (or be close to): this is crucial for scattering and diffraction.
Distance between planes is given by dhkl
Reciprocal dependence between (hkl) and dhkl :
Larger (hkl) values (finely spaced planes) then smaller dhkl.

30. Diffraction – on the optical grating
Web example!
Path difference XY between diffracted beams 1 and 2:
sin = XY/a
 XY = a sin 
For 1 and 2 to be in phase and give constructive interference, XY = , 2, 3, 4…..n
so a sin  = n where n is the order of diffraction
It is so-called grating relationship where a = is the distance between
scattering centres

31. a sin  = n a

32. Non-diffracted X-rays (97%)‏
Principles of BRAGG
X-ray diffraction
experiment:
diffracted
X-rays
Crystal
Incoming
X-ray
Non-diffracted X-rays (97%)‏
detector

33. Diffraction – on the crystal lattice “grating”
Incident X-ray
radiation
“Reflected” radiation 1
(ca. 3%)‏ 2  
dhkl   
Set of crystal planes
(h k l) X Z Y 
dhkl
dhkl
XY = YZ = dhkl sin 
Transmitted radiation
(ca. 97%!)‏
Beam 2 lags beam 1 by XY + YZ = 2d sin 
So dhkl sin  = n  Bragg Law
as inter-atomic distances are in the range of Å so  must be in the range of Å X-rays, electrons, neutrons suitable

34. Difference between light and X-ray reflections
Light:
X-ray:
Reflection of the Light:
is not coherent (multi )‏
does not depend on 
is happening only on the surface
can be focused back
almost 100% of the incident light
is reflected
X-ray Diffraction/Reflection:
coherent (usually single )‏
strictly depends on 
is happening in 3-D volume of the crystal
can not be focused back
only about 3% of the incoming X-rays
is diffracted; ~97% goes
through the crystal unchanged