The Muppet’s Guide to:
The Structure and Dynamics of Solids
2. Simple Crystal Structures
Bonding
 rij 
ER  B exp  




or
 B
ER   12
 rij





EA is bonding dependent
Already looked
at vdW
Figure adapted from Callister, Materials science and engineering, 7th Ed.
Ionic Bonds
EA  
 
E rij
Small lattice parameters
Figure adapted from Callister, Materials science and engineering, 7th Ed.
e2
4 0rij
12 

2
B
e


  
 4 0 rij  rij  
  

EA  
e2
4 0rij
Ionic Materials
Range of materials such as NaCl, CsCl, MgO etc.
Moderate
fall-off (1/r)
Both first and second nearest
neighbours attract
Strong bond
High melting temperatures
Non-directional – bond strength same in all directions
brittle and hard materials
All positive ions surrounded by negative ions and we want
to maximise the number of nearest neighbours
Simple Cubic Structures
Covalent Bond
Short range interaction between pairs
of atoms
Highly directional in space to minimise
Coulomb repulsion of nuclei
Number of bonds proportional to
number of valence electrons
Conduction band
(semi-conductors or insulators)
E A  ECov
Figure adapted from Callister, Materials science and engineering, 7th Ed.
Valence band
Covalent Bond
Relies on orbital overlap (hybridisation)
Total wavefunction must be anti-symmetric
s - bond
 - bond
Bonding orbital formed from overlap of symmetric wavefunctions,
Electrons must be anti-symmetric
Figure adapted from hyperphysics
Hybridisation
sp2
sp3
Covalent Structures
Graphite and Graphene– sp2
Methane – sp3
Diamond,
Si, Ge – sp3
Covalent Materials
Si, Ge, Diamond, Organic molecules and
Polymers, SiH4, CH4, H2O, HNO3, HF..
Range of bond
energies
Diamond: >3550°C
Bismuth: 270°C
Strong angular preference of bonds due to overlap
sp2 hybridisation – trigonal planar structure
sp3 hybridisation – tetragonal tetrahedra
Low density materials
Open structures, polymorphs
Large lattice parameters
Metallic Bonds
H

i
2
2

ze
i 2 

 2m
4 0


l
1
ri  Rl

e2

 4 0

 r r
1
j
i
j
Complex bonding
mechanism between the
degenerate electrons and
the ion cores but also
between electrons.
Not all electrons involved in bonding –
good electrical and thermal conductors
Range of bond
energies
Figure adapted from Callister, Materials science and engineering, 7th Ed.
Tungsten: 3410°C
Mercury: -39°C
Crystal Structures
How do atoms pack given their bonding?
Figures adapted from Callister, Materials science and engineering, 7th Ed.
Packing Fraction
Nature 453, 629-632 (29 May 2008),
Physics World The secrets of random packing May 29, 2008
Dense Packed Structures
Atoms modelled as incompressible spheres
In 2 D, the highest packing
fraction occurs when each
sphere has 6 nearest
neighbours forming a
hexagon.
Unit Cell, Lattice and Basis
A 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 (,,).
A primitive unit cell contains 1 lattice point. Other unit cells
(lattice points > 1) can be used. These highlight the underlying
symmetry.
TRANSLATIONAL SYMMETRY then maps the unit cell
across the entire volume of the crystal
Crystal Structure
Convolution of Basis and lattice
Basis  Lattice  Crystal
2D Bravais Lattices
A lattice is an infinite periodic set of points defined by the three
basis vectors, a,b and c.
T
In 2D total
of 5 distinct
lattices
Lattice vector:
T  Ua  Vb  Wc 
Bravais Lattices – 14 possible in 3D
F
P
I
P
I
P
R
T – trigonal
R- rhombohedral
I
P
P
C
F
C
T
All lattices have translational symmetry
Simple Metals
W
W
BCC LATTICE
BASIS
Molecular crystals
FCC LATTICE
BASIS
Lattice and Basis
(b)
(a)
Cl
Na
(c)
The basis can be convolved with the lattice in different ways
due to the symmetry of the basis and lattice
SiF4
BASIS
LATTICE
NB: The point
symmetries of the
basis and lattice
MUST be
compatible!
CRYSTAL
Dense Packed Structures
Atoms modelled as incompressible spheres
In 2 D, each atom has 6
nearest neighbours
Extend to three
dimensions by layering
sheets on top of each
other
Repeat Patterns:
ABABAB…. Hexagonal close packed
ABCABCABC… Face centred cubic
Simple Centred Cubic
AAAAAAAAAA
Stacked symmetry is cubic
P
Polonium
Centre of 4 unit cells is an octahedral site
Packing Fraction=52.4%
a  2R
Figure adapted from Callister, Materials science and engineering, 7th Ed.
No. of Neighbours=6
Hexagonal Close Packed
ABABABABAB
Cd,
P
Mg,
Zn
Co
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
Figure adapted from Callister, Materials science and engineering, 7th Ed.
c/a=1.663
Face Centred Cubic
Noble
Gases
[111]
ABCABCABCABC
F
Cu,
Ag,
Au,
Ni,
Al,
Pb
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%
a  2R 2
Figure adapted from Callister, Materials science and engineering, 7th Ed.
No. of Neighbours=12
[111]
FCC (111)
Body Centred Cubic
Cr
ABABABABAB
Fe
Stacked symmetry is cubic
not hexagonal
W
I
a
Packing Fraction=68%
4R
3
No. of Neighbours=8
Tetragonal Distortions
c
a
1
In such cases the
structure is usually
written as bct or fct
Figure adapted from Callister, Materials science and engineering, 7th Ed.
Covalent Elements - Diamond
Group VI elements such as C, Si and Ge
sp3 hybridisation - tetrahedra
2 FCC lattices
http://cwx.prenhall.com
http://www.ipap.jp/jpsj/news/jpsj-nc_17-fig1.gif
Packing Fraction=37%
Number of neighbours=12
Diatomic, AX type structures
• The three most common AX type structures are cubic
and named after the representative examples:
• Rocksalt – NaCl
• Caesium Chloride – CsCl
Ionic
• Zinc blende or sphalerite - ZnS
Covalent
Diatomic, AX type structures
• The three most common AX type structures are cubic
and named after the representative examples:
• Rocksalt – NaCl
• Caesium Chloride – CsCl
Ionic
• Zinc blende or sphalerite - ZnS
Covalent
The Rocksalt Structure
Structure adopted for materials with strong ionic bonds. Maximises
the number of nearest neighbours and ensures charge neutrality.
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, 7th Ed.
Caesium Chloride
Resembles bcc lattice but it is not. The atom at the centre is
different. Thus the centre is not a lattice point.
Primitive lattice
Each cation/anion is
surrounded by 8
neighbours of the
opposite kind.
Figure adapted from Callister, Materials science and engineering, 7th Ed.
Zinc Blende
III-V and I-VII as well as ZnS (l=18%), SiC (l=12%), CdTe, ZnTe, MnTe
A structure that resembles the
Diamond structure of 2 interlocking fcc structures. In this
case not Diamond as the
elements on the 2 sites different.
Common in materials which
exhibit low ionic character and
thus favour sp3 hybridised bonds
and tetragonal bond angles
No. of Neighbours=12
Figure adapted from Callister, Materials science and engineering, 7th Ed.