The Muppet’s Guide to:The Structure and Dynamics of Solids

2. Simple Crystal Structures

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

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.

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

Hybridisation

Covalent Structures

Methane – sp3 Diamond,Si, Ge – sp3

Graphite and Graphene– sp2

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

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.

Crystal Structures How do atoms pack given their bonding?

Figures adapted from Callister, Materials science and engineering, 7 th Ed.

Packing Fraction

Nature 453, 629-632 (29 May 2008), Physics World The secrets of random packing May 29, 2008

In 2 D, each atom has 6 nearest neighbours

Dense Packed StructuresAtoms modelled as incompressible spheres

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

Crystal StructureConvolution of Basis and lattice

Basis Lattice Crystal

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

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

Simple Metals

BASISBCC LATTICE

W

W

Molecular crystals

BASISFCC LATTICE

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

SiF4

BASISLATTICE

CRYSTAL

NB: The point symmetries of the basis and lattice

MUST be compatible!

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

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

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

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

FCC (111)[111]

Body Centred Cubic

Packing Fraction=68% No. of Neighbours=8

ABABABABAB

Stacked symmetry is cubic not hexagonal

Cr

Fe

W

4

3

Ra

I

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.

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

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

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

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.

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.

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.