Cristallography - Introduction

Prévia do material em texto

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Véronique JUBERA
Veronique.jubera@u-bordeaux.fr
Inorganic chemistry of 
materials
4TPM214U
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Outline
➔Definitions in crystallography
➔Description of atoms stacking: 
› Application on metallic crystals
➔ Interstitial sites into layer stacking:
› Definition and application on metal alloys
➔ Interstitial sites into layer stacking:
› Localization and application on ionic crystals
➔Covalent and molecular crystals description
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Chapter 1 : Definition in crystallography
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I- Introduction
➔What is a crystal?
➔What is the nature of your pencil lead, of the chalk, of the 
window?
› Atoms in regular order or disorder!
➔What are the state of the matter?
› Gas: no interaction = disorder, high degree of freedom of movement
› Liquid: weak interactions = local order
› Solid: strong interaction, low degree of freedom of movement
• Small distance arrangement= vitreous solids, glassy materials
• Long distance order= crystallized solids
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A Crystal is a solid constituted by chemical species (atoms, ions, 
molecules…) regularly distributed in a long distance in order to 
create a periodic staking.
I- Introduction
➔General characteristics of matter states
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Degree of freedom
of movement
Condensed
state
Ordered
state
Proper
volume
Proper
shape
Properties
Solid
Vibrations or 
oscillations
Yes
Yes Crystallized
No
Amorphous
Yes Yes Rigid
Liquid
rotations
Yes No Yes No Visquous
Gas translations
No No No No Expandable
Compressible
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I- Introduction
➔Crystallography: Study of atoms/ions/molecule stacking in a 
crystal
› Identification of the smallest unit (volume) repeated
• In a long distance (macroscopic dimensions)
• periodically
› Study of the chemical species (atoms/ions/molecules) inside the 
smallest unit (the pattern)?
• their location?
• Their relation/multiplication considering the existence of specific 
symmetry element (axis , plane, Mirror) ?
➔Knowledge of crystal structure= mechanic, physical or 
chemical ,properties understanding.
› Graphite structure= use for writing.
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I- Introduction
➔Knowledge of crystal structure= mechanic, physical or 
chemical ,properties understanding.
› BaTiO3 oxide = capability to store electrical charges
 Microelectronic condensators manufacturing.
› LiCoO2 oxide = capability to insert Li ions.
 Li-ions battery (mobile phone or computer).
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I- Introduction
➔Main goals of this lectures
› To acquire and to control the fundamental crystallography concepts 
in order to describe simple crystal structures.
› To build structure with compact and non-compact atomic layers and 
to classify the corresponding solids as function of their nature
› To acquire basis to understand the second part of the course which 
will illustrate the X-Ray matter interaction and the construction of 
phase diagram equilibrium.
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What can be the morphology of a crystal?
➔Examples of chemical species involved in crystalline
matrices:
› Atoms
• Carbon (C) in diamant host
• Fer (Fe) in metallic iron
› Ions
• Na+ and Cl- in NaCl salt
• K+, Ni2+ and F- in K2NiF4 fluoride
› Molecules
• H2O in ice
• CO2 in carbon ice
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What can be the morphology of a crystal?
➔Crystal shape is a direct consequence of the chemical 
species stacking in relation to the symmetry of this building
› By observing the macroscopic morphology of crystals, Abbé Haüy
(French scientist), wrote basis of crystallography science in 1784, 
more than 230 ago!
› Crystallography progress are linked to the X-Ray discovery 
(Röntgen, 1895) and the resulting diffraction , consequence of 
matter-beam interaction (Bragg father and son, 1915).
 Atomic structure study
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Nowadays, crystallography science is used in chemistry, physics 
(structure-properties relation), in biology and medicine (proteins 
or virus structure) etc
II- Definitions
➔Pattern and periodicity
› A crystal is constituted by the repetition of a unique pattern, in the 
three space directions = perfect crystal.
› The crystallographic pattern is an assembling of:
• Atoms,
• Ions
• Molecules
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Crystallographic pattern/motifs/set of objects: atoms, ions, 
molecules assembling repeated within a crystal in a periodic way.
Illustration of a crystallographic pattern: triangular 
molecule containing « A » entity distributed around a 
gravity center (barycenter).
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II- Definitions
➔Pattern and periodicity
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Barycenter?
II- Definitions
➔Pattern and periodicity
› Molecule Translation along the x and y axes
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Appearance of periodicity through pattern repetition!
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II- Definitions
➔Pattern and periodicity
› The location of barycenter pattern into grating/layer constitutes a 
bidimensional array defined by elementary translation vectors
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Building of a virtual bidimentional crystal characterized by a 
unique pattern (molecule A) and two translation vectors a and b 
called elementary translation vectors.
II- Definitions
➔Pattern and periodicity
› In a real crystal, a tridimensional array, a third translation has to be
defined: c. This translation does not belong to the plane defined by 
a and b translations.
› Explanation: one generated atom through the elementary translation 
as the same nature as the initial atom. Its local environment is also
the same.
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Elementary translation: an elementary translation generates one 
point/pattern of a crystal from another point/pattern. Both patterns/points 
are equivalent
A a or b translation linked one atom of a 
pattern to an equivalent A atoms in a neighbor
patter, with the same environment. It associated
two equivalent points in the described
bidimensional array.
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Observation
➔Can you identify a pattern which is periodically repeated in 
this amphitheater?
➔What is the dimension of the cloud constituted by these
patterns (1D, 2D ou 3D)?
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II- Definitions
➔Array definition
› Bidimensional crystal: A atoms are linked to equivalent atoms by the 
elementary translations a and b.
› However, because a crystal is not limited in term of size, it remains
difficult de define an origine point for the determination of the x and y 
coordinates.
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A global lattice has to be considered!
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II- Definitions
➔Array/lattice definition
› The triangle gravity center has not a real existence but it helps to 
describe the pattern. This is a useful mathematic tool.
› It can be used to define a bidimensional punctual array built with the 
a and b elementary translation vectors.
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The bidimensional array gathering the different gravity centers of 
the patterns has no physical existence; this is a mathematical 
model, a kind of grating which possesses the same periodicity as 
the crystal! Fundamental concept!
II- Définitions
➔Lattice nodes
› Gravity centers located at the extremity of elementary translation 
vectors of the array are array nodes.
› In a crystal, it is difficult to fix an origin but one array node can be
arbitrarily selected to become the origin.
› Each node can be described from this defined origin..
› In a 3D crystal characterized by a, b and c, the location of one node
is given by a r vector:
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cwbvaur
rrrr ++=
Once we have chosen a representative lattice, appropriate to the symmetry 
of the structure, any reticular point (or lattice node) can be described by a vector 
that is a linear combination (with integer numbers) of the direct reticular axes,
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II- Definitions
➔Lattice nodes
› In a 3D crystal characterized by a, b and c, the location of one node
is given by a r vector
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cwbvaur
rrrr ++=
Each lattice nodes is defined by its u, v, w coordinates
II- Definitions
➔ The coordinates of the nodes are noted: u,v,w
➔ From this point one can defined a raw; it is cited as [u,v,w] and links the 
uvw node (the closest node from the origin) to the origin of the lattice.
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0,0 0,1 0,2 0,3
1,0 1,1 1,2 1,3
2,0 2,1 2,2 2,3 [1,2]
[2,1]
[1,1]
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II- Definitions
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O a
b
[1 1]
[1 1]
[1 1]
[1 1]
[1 0][1 0]
[1 1]
It exists an infinity of nodes in one raw
There are two potential ways to name a raw if we take into account the 
direction of the straight line !
Example: [1 1] and [-1 -1]
The node distance 
corresponds to the distance 
between two nodes which
belongs to the same raw,
II- Definitions
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A plane can be built with three nodes. A reticular line (2D) or a reticular 
plane (3D) can be used as a representative of the entire family of parallel lines 
or parallel planes.
The distance between the planes drawn on each lattice (interplanar spacing) is 
the same.
aa
b
c
Plan de la famille
(3 2 3)
A
B
C
2D 3D
Example 3D lattice: this plane contains the nodes: 200; 030; 002
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II- Definitions
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A unique plane defined by the numerical triplet known as Miller indices, 
represents and describes the whole family of parallel planes passing through every 
element of the motif. 
→ In a crystal structure, there will be as many plane families as possible numerical 
triplets exist with the condition that these numbers are primes, one to each other 
(not having a common divisor). 
The Miller indices are generically represented by the triplet of letters hkl.
The plane is identified as (hkl)
aa
b
c
Plan de la famille
(3 2 3)
A
B
C
II- Definitions
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Remark: What about the hkl value of a plane parallel to one of the axis?
→ The intersection is 
aa
b
c
Plan de la famille
(3 2 3)
A
B
C
The plane contains the nodes: 
200; 030; 002
→ This means that the intersection of 
this plane with the axis corresponds to:
2a, 3b and 2c
Thus , h k and l Miller factors are calculated as follow:
h= 1/u ×K k=1/v ×K l= 1/v ×K with K the least common multiple
→ h, k and l are primes 
u= 2: v= 3; w= 2
In this example, K= 6
Thus: h= 1/2 ×6 k=1/3 ×6 l= 1/2 ×6
h= 3 k= 2 l= 3 
(3 2 3) plane
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II- Definitions
➔Unit cell
› In a 3D lattice, le parallelepiped built by the elementary translation 
constitutes the unit cell of the lattice. 
› The unit cell is characterized by 6 parameters
• 3 length a, b and c
• 3 angles :α= (b ; c), β (a ; c), ɣ (a ; b)
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The lattice unit cell is the smallest volume which is repeated
periodically .
II- Definitions
➔Unit cell multiplicity
› Multiplicity : number of lattice nodes contains within the unit cell
• 8 nodes at the 8 edge of the parallelepiped.
• Simple unit cell (ou primitive): nodes at the edge
 Multiplicity = 1 node per unit cell. Why?
› Lattice Periodicity = each edge belongs to 8 unit neighboring unit 
cell.
Contribution of each node at a given unit cell= 1/8
Multiplicity calculation= 8 x 1/8 = 1
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II- Definitions
➔Unit cell multiplicity
› An unit cell which contains more than 1 node is a multiple unit cell
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Node at the edge = 1/8 ème
Node on a corner = 1/4 
Node on a face = 1/2
Node inside the unit cell = 1
corner
Inside the 
unit cell
Edge
II- Definitions
➔Unit cell multiplicity
› Other representation of node respective contribution.
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Edge
Face Corner
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II- Definitions
➔Crystal classes
› In addition, node location can be generated from supplementary 
translations vectors.
 Lattice modes are defined.
• No supplementary translations vectors = Primitive lattice crystal class, 
P and multiplicity equal to 1.
• A supplementary translation (½ ½ ½) locate a node in the middle of the 
unit cell= body centered lattice crystal class, I and multiplicity equal to 
2.
• 3 supplementary translations (½ ½ 0) , (½ 0 ½) et (0 ½ ½) generate 
nodes in the center of each face = face centered lattice crystal class, F
and multiplicity equal to 4.
• A supplementary translation (½ ½ 0) or (0 ½ ½ ) or (½ 0 ½) results in 
two centered faces= C, A or B base centered lattice crystal class 
respectively and multiplicity equal to 2
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II- Definitions
➔Atomic position
› To describe the location of an atom:
• Reduced coordinates (x, y, z) with 0 ≤ (x, y et z) ≤ 1.
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Ti4+ is located in the middle of the unit cell. 
Its reduced coordinates are=
x=0,5, y=0,5 et z=0,5 
→ (0,5 0,5 0,5).
O2- coordinates are (0 ½ ½ ) (1 ½ ½ )
(½ 0 ½) (½ 1 ½)
(½ ½ 0) (½ ½ 1)
The atom at the origin is Ba2+
→ Ti4+ and O2- are not generated by supplementary translations of an I or F
crystal class even if they are located in the middle of the unit cell or the faces!
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II- Definitions
➔Atomic positions
The atom at the departure point is named as the independant
crystallographic position. 
Whatever this departure node, it makes possible the construction of the 
entire unit cell
› A supplementary translation generate equivalent atomic position
from a departure node. The chemical nature is the same!
 The addition of all these positions are named general positions.
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II- Definitions
➔Coordination number
• Coordination number: number of first neighboring atoms/ions
• Coordination polyhedron : geometric figure constitutes by the 
neighboring atoms/ions (tetrahedron, octahedron, cube…)
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[A/B]= 4
The coordination polyhedron is a tetrahedron
B
B
B
B
A
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II- Definitions
➔Coordination number
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[M/M] = 8
It’s a cube
[Ti/O] = 6
It’s an octahedron
II- Definitions
➔Z : Number of motif (pattern) per unit cell
› Unit cell multiplicity of a lattice = Number of nodes contained in 
the unit cell (nodes= positions associated to the translations)
› The description of the unit cell reflects also its chemical formula: all 
the species (atoms/ions/molecules) has to be counted.
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The contained species determine the nature of the motif/pattern.
Z = the total species within the unit cell/ the motif
Z is an integer
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II- Definitions
➔Crystal systems
› Periodicity + identification of the unit cell 
 Description of the crystal.
› Geometry rules has to be followed
 It exists 7 crystalline systems defined by the unit cell parameters.
› It may involve symmetry elements (Axis, mirror, plane)
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Hexagonal hanksite crystal
Na22K(SO4)9(CO3)2Cl
Cubic pyrite crystal
FeS2
This is reflected in the cubic symmetry of its natural crystal facets.
II- Definitions
➔ 7 Crystal systems
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Cubic tetragonal Orthorhombic monoclinic
hexagonal triclinicrhomboedric
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II- Definitions
➔Bravais lattice› To describe the lattice:
• Crystal classes (7)
• Lattice system (P, I, F, A)
28 possibilities? No because of symmetry
http://ressources.univ-
lemans.fr/AccesLibre/UM/Pedago/physique/02/cristallo/bravais.html
http://www.sciences.univ-nantes.fr/sites/genevieve_tulloue/Cristallo/Bravais/
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In three-dimensional space, there are 14 Bravais lattices. These 
are obtained by combining one of the crystal families with one of 
the centering types
II- Definitions
➔Bravais lattice
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Cubic Tetragonal Orthorhombic Monoclinic
Hexagonal TriclinicRhomboedric
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II- Definitions
➔Bravais lattice
› Cubic Unit cell:
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Existence of a rotational axis in a cube (rotation 2Pi/3)
Other view
CFC cubic mode
In a cubic mode A, B or C 
base centered mode are 
impossible
P, I or F are allowed
II- Definitions
➔Compactness
› Ration between the volume of the species contained in the unit cell
by the volume of the unit cell.
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The maximum of compactness of regular spheres stacking is equal to
74%
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II- Definitions
➔Density (Masse volumique)
› Density(g.cm-3) ρ= mass/volume.
› Experimental measurement possible.
› Theoritical calculation
• Unit cell paramter → volume V
• Nature and number Z of the motif → species contained within the unit 
cell
• Molar weigth of the motif → mass M
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NV
ZM=ρ
With N or NA= Avogadro constant 
6.02×1023
II- Definitions
➔Exercice
› Calculation of ρ Polonium (Po) crystal?
› Data: 
• lattice system and symmetry: Primitive cubic unit cell
• a= 3,34Å : M= 209 g.mol-1 ,NA = 6,02.1023
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