the muppet’s guide to: the structure and dynamics of solids 2. simple crystal structures
TRANSCRIPT
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The Muppet’s Guide to:The Structure and Dynamics of Solids
2. Simple Crystal Structures
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Bonding
12Rij
BE
r
exp ijR
rE B
or
EA is bonding dependent
Figure adapted from Callister, Materials science and engineering, 7 th Ed.
Already looked at vdW and
ionic
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Covalent Bond
Short range interaction between pairs of atoms
Highly directional in space
Number of bonds proportional to number of valence electrons
Conduction band
Valence bandA CovE E
(semi-conductors or insulators)
Figure adapted from Callister, Materials science and engineering, 7 th Ed.
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Covalent BondRelies on orbital overlap (hybridisation)
Total wavefunction must be anti-symmetric
Figure adapted from hyperphysics
Bonding orbital formed from overlap of symmetric wavefunctions,
Electrons must be anti-symmetric
s - bond
p - bond
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Hybridisation
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Covalent Structures
Methane – sp3 Diamond,Si, Ge – sp3
Graphite and Graphene– sp2
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Si, Ge, Diamond, Organic molecules and Polymers, SiH4, CH4, H2O, HNO3, HF..
Strong angular preference of bonds due to overlap
Low density materials
Open structures, polymorphs
Range of bond energies
Diamond – >3550°C
Bismuth – 270°C
Covalent Materials
sp2 hybridisation – trigonal planar structure
sp3 hybridisation – tetragonal tetrahedra
Large lattice parameters
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Metallic Bonds
Complex bonding mechanism between the degenerate electrons and
the ion cores but also between electrons.
2 2 22
0 0
1 1
2 4 4ii l i ji l j
ze eH
m r R r r
Range of bond energies
Tungsten: 3410°C Mercury: -39°C
Not all electrons involved in bonding – good electrical and thermal conductors
Figure adapted from Callister, Materials science and engineering, 7 th Ed.
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Crystal Structures How do atoms pack given their bonding?
Figures adapted from Callister, Materials science and engineering, 7 th Ed.
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Packing Fraction
Nature 453, 629-632 (29 May 2008), Physics World The secrets of random packing May 29, 2008
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In 2 D, each atom has 6 nearest neighbours
Dense Packed StructuresAtoms modelled as incompressible spheres
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Unit Cell, Lattice and BasisA crystal is a parallelepiped that is made up of a regular repeat of some representative unit, called the unit cell.
Unit Cell: A volume of space bounded by lattice points which describe the symmetry. It is defined in terms of their axial lengths (a,b,c) and the inter-axial angles (,,).
TRANSLATIONAL SYMMETRY maps the unit cells across the entire volume of the crystal
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Crystal StructureConvolution of Basis and lattice
Basis Lattice Crystal
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A lattice is an infinite periodic set of points defined by the three basis vectors, a,b and c.
Lattice vector:
T Ua Vb Wc
T
2D Bravais Lattices
In 2D total of 5 distinct
lattices
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P
P
P
P
I
I
F
F
I
C
C
T – trigonalR- rhombohedral
T
P
Bravais Lattices – 14 possible in 3D
R
All lattices have translational symmetry
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Simple Metals
BASISBCC LATTICE
W
W
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Molecular crystals
BASISFCC LATTICE
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Lattice and Basis
(a) (b)
(c)
C l
N a
The basis can be convolved with the lattice in different ways due to the symmetry of the basis and lattice
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SiF4
BASISLATTICE
CRYSTAL
NB: The point symmetries of the basis and lattice
MUST be compatible!
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In 2 D, each atom has 6 nearest neighbours
Dense Packed StructuresAtoms modelled as incompressible spheres
Extend to three dimensions by layering sheets on top of each
other
Repeat Patterns:
ABABAB…. Hexagonal close packed
ABCABCABC… Face centred cubic
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Simple Centred Cubic
Packing Fraction=52.4% No. of Neighbours=6
AAAAAAAAAA
Stacked symmetry is cubic
Polonium
2a RFigure adapted from Callister, Materials science and engineering, 7 th Ed.
P
Centre of 4 unit cells is an octahedral site
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Hexagonal Close Packed
The second layer (B) is translated with respect to the first (A) such that the atoms in layer B sit in the dimples between the atoms in layer A
Packing Fraction=74% No. of Neighbours=12 c/a=1.663
ABABABABAB
Cd,
Mg,
Zn
Co
Figure adapted from Callister, Materials science and engineering, 7 th Ed.
P
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Face Centred Cubic
Initial stacking is the same as hcp. Then the third layer (C) is translated with respect to both the first and second such that the atoms in layer C
sit in the dimples between the atoms in layer B.
Packing Fraction=74% No. of Neighbours=12
ABCABCABCABCNoble Gases
Cu,
Ag,
Au,
Ni,
Al,
Pb
2 2a R
[111]
Figure adapted from Callister, Materials science and engineering, 7 th Ed.
F
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FCC (111)[111]
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Body Centred Cubic
Packing Fraction=68% No. of Neighbours=8
ABABABABAB
Stacked symmetry is cubic not hexagonal
Cr
Fe
W
4
3
Ra
I
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Tetragonal Distortions
1ca
In such cases the structure is usually written as bct or fct
Figure adapted from Callister, Materials science and engineering, 7 th Ed.
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Covalent Elements - DiamondGroup VI elements such as C, Si and Ge
sp3 hybridisation - tetrahedra
http://cwx.prenhall.comhttp://www.ipap.jp/jpsj/news/jpsj-nc_17-fig1.gif
Packing Fraction=37%Number of neighbours=12
2 FCC lattices
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Diatomic, AX type structures
• The three most common AX type structures are cubic and named after the representative examples:
• Rocksalt – NaCl• Caesium Chloride – CsCl
• Zinc blende or sphalerite - ZnS
Ionic
Covalent
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Diatomic, AX type structures
• The three most common AX type structures are cubic and named after the representative examples:
• Rocksalt – NaCl• Caesium Chloride – CsCl
• Zinc blende or sphalerite - ZnS
Ionic
Covalent
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The Rocksalt StructureStructure adopted for materials with strong ionic bonds
MgO, MnS, LiF, FeO, Alkali halides and hydrides and II-VI compounds
fcc lattice
Each cation/anion is surrounded by 6 neighbours of the opposite kind in a perfect octahedral arrangement.
Figure adapted from Callister, Materials science and engineering, 7 th Ed.
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Caesium Chloride
Primitive lattice
Each cation/anion is surrounded by 8 neighbours of the opposite kind.
Resembles bcc lattice but it is not because the atom at the centre is different and so it is not a lattice point.
Figure adapted from Callister, Materials science and engineering, 7 th Ed.
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Zinc Blende
A structure that resembles the Diamond structure.
Common in materials which exhibit low ionic character and thus favour sp3 hybridised bonds and tetragonal bond angles
III-V and I-VII as well as ZnS (l=18%), SiC (l=12%), CdTe, ZnTe, MnTe
No. of Neighbours=12Figure adapted from Callister, Materials science and engineering, 7 th Ed.