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

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 chemicalentity = 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.

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) , C-C-N 123.72(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 forfurther chemical (e.g. quantum) and physical calculationsMn(H2O)62+Co(III)(Ph4porphyrin) (Cl) Rh(en)2Cl2+ trans

How to obtain the structure of the motif (compound)from its crystal structure?The foundation of recognition/visibility of all structures:

Scattering and DiffractionAll objects irrelevant of their size scatter radiationwhich is shine on them.http://www.acoustics.salford.ac.uk/feschools/waves/super.htm#phasehttp://www.numathics.com/arens/scattering/Scattering.html

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.)

How we can retrieve the information about the scattering objects?Because we can focus back the scattered/diffracted waves again.The objectScattering/DiffractionLenses:focussingThe image of the object

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 AirGlassnAcnGnG nA >Here nG = 1.5

What should be the relationship between effectivescattering 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

of radiation 3.7 pm 400 700 nmResolution 2 nm10 nm200nm

1 meter = 102 cm = 103 mm = 106 mm = 109 nm =1010

wave = | F | x = FIntensity of the wave: I ~ F 2 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 objectIf we know I then:

What?!! NO LENSES!!!! 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 problemUsually Monochromatic:one, well defined

Information directly available from an X-ray single crystal diffraction experiment:

1. Intensities I of diffracted X-rays therefore | F | - amplitudes of diffracted X-rays2. Directions of the diffracted X-rays

The phases must be reconstructed in rather complex/difficult experimental and computing methods:phase problem = phase solution methods

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 ).X-rays in the crystal are:

scattered by the electrons:- Thomson scattering: the electron oscillates in theelectric field of the incoming X-ray beam and anoscillating 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 tocrystal periodic, repetitive (in 3-D) unique character

But why not to measure scattering from one molecule and determine its structure this way?..

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 thesingle molecule almost immediately. In crystal there are thousands of molecules some of them survive long enough to give a measurable radiation..

Crystal structure = Crystal Lattice + motif web: http://marie.epfl.ch/x-ray/rlattice/

=+The 3-D periodicity of the crystal can be simplified and represented by an abstract crystal lattice.

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

X

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)

The unit cell in three dimensions. The unit cell is defined by three vectors a, b, and c, and three angles , , .Unit cells are defined in terms of the lengths of the three vectors and the three angles between them.

For example, a = 94.2 , b = 72.6 , c = 30.1 , = 90.0, = 102.1, = 90.0 a = 8.32 , b = 15.23 , c = 9.28 , a = 90.0, b = 90.0, g = 90.0

Content of the Unit cell cMotif = molecule, atomsSize andthe arrangement (symmetry) of the unit cell Crystal structure To get the structure of the motive we have to:get the information about the unit cell size and its arrangement

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

Crystal Lattice directionssymbol:

examples:[uvw]

[100][010]

[001]: abcabcxyz:

Crystal planes= inter-plane spacing measured at 90o to the planesThe planes are imaginaryAll planes in a set of planes are identical - equivalentThe perpendicular distance between pairs of adjacent planes is called d: interplanar spacing Need to label planes to be able to identify them

Find intercepts on a, b, c: 1/2, 1, 0 (1 1/3 0)

Take reciprocals 2, 1, 0 (1 3 0) (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).

Plane perpendicular to x cuts at 1, , (1 0 0) planeNB an index 0 means that the plane is parallel to that axis(0 0 1) plane

(0 1 0) plane

Planes - conclusions Miller indices define the orientation of the plane within the unit cellThe Miller Index defines a set of planes parallel to one another (remember the unit cell is a subset of the infinite crystalAll possible sets of planes in a particular lattice may be described by (hkl) valuesAny 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 dhklReciprocal dependence between (hkl) and dhkl : Larger (hkl) values (finely spaced planes) then smaller dhkl.

Diffraction on the optical gratingPath 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..nso a sin = n where n is the order of diffractionWeb example!It is so-called grating relationship where a = is the distance between scattering centres

Diffracted light

Coherent incident light

X

Y

a

(

(

1

2

a sin = n

Diffracted light

Coherent incident light

X

Y

a

(

(

1

2

CrystalIncoming X-raydiffracted X-raysPrinciples of BRAGG X-ray diffraction experiment: Non-diffracted X-rays (97%)detector

XY = YZ = dhkl sin (ca. 97%!)(ca. 3%)Diffraction on the crystal lattice gratingBeam 2 lags beam 1 by XY + YZ = 2d sin So 2dhkl sin = n Bragg LawXYZdhklIncident X-ray radiation

Reflected radiationTransmitted radiation12dhkldhklSet of crystal planes

(h k l) as inter-atomic distances are in the range of 0.5 - 2 so must be in the range of 0.5 - 2 X-rays, electrons, neutrons suitable

Diffracted light

Coherent incident light

X

Y

a

(

(

1

2

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-raysis diffracted; ~97% goes through the crystal unchangedDifference between light and X-ray reflectionsLight:X-ray:

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