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Inert gas and molecular crystals: Van der Waals forces (physics)
Crystallographic structure Physical vs Chemical bonding in solids
Water and organic chemistry H bonds (physics)
Quartz crystal SiO2: covalent bonds (chemistry)
Stronger bonds
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Ionic bond : NaCl
For most solids bonds are a partly partly ionic and partly covalent
- +
fcc corresponds to packing in an “as close packed as possible” structure
CsCl bcc structure (bonds are more directional)
Neutral building block
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Chemical bonding in solids
Covalent bond Bonding and antibonding levels form due to orbital hybridization. The bond is stable if only (or mostly) bonding orbitals are filled
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Hybridization Goup IV elements
Diamond Structure sp3
Graphite and graphene sp2
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Metallic bond For Ni bonds occurs through s orbitals Fcc structure
If directional d orbitals matter bcc or hcp structures (e.g. Fe and W)
Interaction beyond first nearest neighbors
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a2
a1
R
R’
T
R’=R+n1a1+ n2a2+ n3a3
T=n1a1+ n2a2+ n3a3
n1, n2, n3 arbitrary integers a1, a2, a3 fundamental translation vectors
The set of points R’ defines a lattice while spanning all over ni
3D Crystals
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R’=R+n1a1+ n2a2+ n3a3
T=n1a1+ n2a2+ n3a3 n1, n2, n3 arbitrary integers a1, a2, a3 fundamental translation vectors
The set of points R’ defines a lattice while spanning over all ni The atomic arrangement looks the same in every respect (including orientation) when viewed from any point R of the lattice The lattice is the regular periodic arrays of points in space The crystal structure is formed when a basis is attached identically to every lattice point
3D Crystals and Lattice
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The lattice and the translation vectors a1, a2, a3 are said to be primitive if with a suitable choice of the integers n1, n2, n3 any two points R and R’ always satisfy R’=R+n1a1+ n2a2+ n3a3 In this case the vectors a1, a2, a3 are primitive translation vectors and no cell of smaller volume can serve as a building block for the crystal structure A lattice translation operation is defined as the displacement of a crystal by a crystal translation vector T=n1a1+ n2a2+ n3a3
3D Crystals and Translation Operation
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Often, though not always, primitive translation vectors a1, a2, a3 are used to define the crystal axes.
More than one lattice is always possible for any given structure
More than one set of axes is always possible for a given lattice
The basis is chosen with an arbitrary choice.
3D Crystals
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a2’’’’
a2’
a1’
a2’’
a1’’
a1’’’’
a1’’’
a2’’’
All pairs of vectors a1, a2 are translation vectors, but
(a1’’’’ , a2’’’’) is not a primitive translation vector pair All other pairs can be taken as primitive translation vectors and the related unit volume is the same
Primitive Cell and Axes
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3D Crystals and Symmetry Operations
A symmetry operation of a crystal carries the crystal structure into itself
Translation operations (T=n1a1+ n2a2+ n3a3) are symmetry operations
Rotations, reflections and inversions can be symmetry operations, called point symmetry operations
Compound operations can be symmetry operations
The collection of symmetry operations is a group
The simultaneous fulfillment of the translation operations with the point group symmetry operations leads to 14 special lattice types (Bravais lattices)
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a2
a1
R
R’
T
x
x x x
x x
x x x
A rotation of π radians around any point marked x is a symmetry operation since it carries the crystal structure into itself
Point Operations
3D Crystals
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Group of the octahedron Cubic symmetry
Possible symmetry operations: Rotations Reflections Inversions Combinations of them
Group of the tetrahedron
Different symmetry operations are possible!
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The Seven crystal systems: a) Cubic (simple cubic, body centered cubic, face centered cubic)
b) Tetragonal , cubic symmetry is reduced by pulling on two opposite bases one obtains a square base and a non equivalent height or c-axis (simple and body centered tetragonal)
Distortion of fcc and bcc lattices leads to the same tetragonal symmetry
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The Seven crystal systems:
c) Orthorombic , the square faces are deformed into mutially perpendicular rectangles: i) Simple orthorombic ii) Base centered orthorombic iii) Body centered orthormbic iv)Face centered orthorombic
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The Seven crystal systems: d) Monoclinic , the rectangular face othogonal to
the c axis is distorted into a rhombus: simple base centered
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The Seven crystal systems: e) Triclinic, the c axis is tilted and no longer
perpendicular to the other two. The crystal has only inversion symmetry f) Trigonal or rhombohedral, this lattice is
obtained by stretching a cube along a body diagonal. The lattice vectors have identical length and make equal angles with one another
g) Hexagonal, it is the symmetry group of a right
prism with a hexagon at the base
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3D Bravais Lattices
There are 14 special lattice types (Bravais lattices) in 3D space (hkl) Indices of a plane {hkl} Planes equivalent by symmetry [uvw] Indices of a direction
Combining point symmetry and translations one gets 230 possible 3D lattices
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Classification of the point groups
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Number of point groups
Number of space groups
Bra
vais
la
ttic
es
(bas
is o
f sp
her
ical
sym
met
ry)
Cry
stal
la
ttic
es
(bas
is o
f ar
bit
rary
sym
met
ry)
7 32
14 Bravais lattices
230 space groups
C cyclic; D dihedral; S Spiegel (mirror) The subscripts h, v and d stand for horizontal, vertical and diagonal mirror planes
Combining rotational and translational symmetry:
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3D Crystals and Primitive Lattice Cell The unit volume defined by the primitive a1, a2, a3 axes is called primitive cell A unit cell will fill all space by the repetition of suitable translation operations A primitive cell is a minimum-volume cell There are many possible choices for the primitive axes and for the cell for a given lattice The number of atoms in a primitive cell or primitive basis is always the same for a given crystal structure
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Compact structures: fcc ABCABC hcp ABABAB
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fcc, different possible choices of unit vectors
The primitive choice is complicated, better working with a non-primitive unit cell like the face centered cube
The primitive cell has one fourth of the volume of the conventional unit cell
Packing fraction 0,74
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bcc
Elements crystallizing in bcc structure have mostly directional bonds determined by d states. However, also alkali metals are bcc, the reason being that the bcc lattice optimizes the superposition of the orbitals corresponding to the 2nd and 3rd nearest neighbors as illustrated in the figure
Packing fraction 0,68
Different choices are possible for the unit cell. The primitive cell has half the volume of the conventional unit cell
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Fcc and bcc lattices are Bravais lattices since, even if they have four and two atoms in the unit cell the surroundings look alike when viewed from any lattice point
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hcp structure: ABAB packing
Hexagonal layer : graphene Layered crystals: graphite
More complicated ABCABABCAB packing possible, too
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Packing fraction = 0,34
Lattice with two atom basis: the diamond structure
It consists of two identical fcc lattices displaced by (¼¼¼) In total we therefore have 8 atoms in the unit cell. It is not a Bravais lattice since the surroundings do not look the same from the two atoms of the basis.
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Primitive Cell and Basis
There is always one lattice point per primitive cell The basis associated with a primitive cell is called primitive basis No basis contains fewer atoms than a primitive basis Wigner-Seitz cell: defined as follows
1) Draw lines to connect a given lattice point to all nearby points 2) Draw new lines at the midpoint and normal to the above lines 3) The smallest volume enclosed in this way is the Wigner-Seitz primitive cell
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Wigner Seitz cells
bcc
fcc
Complicated geometry, used rarely
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NaCl structure Two fcc lattices shifted by (½,0,0) filled by two different atoms 4 atoms per unit cell
Compounds
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CsCl structure Two simple cubic lattices consisting of different atoms 2 atoms per unit cell
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zincblende structure Like diamond but with two different atomic species, 4 atoms per unit cell
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NaCl CsCl
Fluorite CaF2
Perovskite BaTiO3
Laves phase Cu2Mg A15 or
β tungsten structure Nb3Sn