The Muppet’s Guide to:
The Structure and Dynamics of Solids
Thomas Hase
Room MAS4.02
E-mail: T.P.A.Hase@warwick.ac.uk
My Research
X-ray (and neutron)
scattering from thin
films and
multilayers….
Not much diffraction!
Objectives for part 1
1.
Understand how thermodynamics and bonding are responsible for crystal structure and material
properties
2. Have knowledge of simple crystal structures and how they relate to bonding requirements. Be able to
reproduce the bcc, fcc, diamond and perovskite structures and be able to explain the concepts of the
lattice and the basis.
3. Understand the structural origin of the ferroelectric effect.
4. Be able to explain the difference between a first and second order phase transition and justify why they
occur from a thermodynamic point of view.
5. Be able to read and use a unary and binary phase diagram and calculate phase compositions and relative
amounts. Understand the diffusive mechanisms in determining structure and predict the evolution of
structure using a eutectic phase diagram.
6. Have an understanding of different crystal growth methods and the types of materials they produce. Be
able to describe simple defect structures and how these relate to material properties.
7. Understand how x-rays, neutrons and electrons interact with matter. Appreciate the differences between
probes and be able to compare x-ray and neutron scattering .
8. Be able to apply basic scattering concepts to the study of single crystal and powder materials. Be able to
predict the likely scattering from simple structures and describe the changes it scattering as a material
undergoes a phase transition.
9. Understand how scattering is related to reciprocal space. Know how to determine material properties
such as lattice parameters, layer thickness, roughness, particle size and shape. Understand how sample
shape affects the scattering. Determine strain and particle size from scattering data.
10. Understand the complementarity between x-rays and neutrons
11. Understand x-ray spectroscopy and spectroscopic probes
Recommended Books – Part 1
Materials Science and Engineering - An Introduction,
7th Edition
William D. Callister, Jr.,
ISBN: 978-0-471-73696-7, Wiley, Hardback 2007
£37.95 / €58.50 (Wiley)
The Physics and Chemistry of Solids
Stephen Elliott
ISBN: 978-0-471-98195-4, Wiley, Paperback June 1998
£47.50 / €71.30 (Wiley)
Extended notes which cover all the course content and more are
available on-line and a printed copy will be handed out soon
Feedback
• Please give honest and critical feedback
during the course:
– During the lectures
– After the lectures
The Muppet’s Guide to:
The Structure and Dynamics of Solids
1. Structure from a Thermodynamic
Viewpoint
Bonding in Solids
Arrangement of atoms in a solid (on a local atomic
level) is driven by energy considerations
Require a thermodynamical
treatment of atomic positions
and bonding….
Thermodynamics
• Entropy: A measure of disorder or randomness in a
system:
Q
S 
T
 JK 
1
• Enthalpy: The total energy of a thermodynamic system. It
is the sum of the internal energy of a body and the
energy associated with displacing it from its
environment:
H  U  PV
U is the Internal energy of crystal
 J 
The Gibbs Free Energy
• Is a measure of the energy that depends on
both enthalpy and entropy:
G  H  TS
J 
G  U  PV  TS
G  U  P, V   PV  TS
Also known as the free enthalpy
Re-expression of 1st Law
• Recall 1st law of
thermodynamics:
• Which using the definition of
entropy, S=Q/T, becomes
• Consider small change in
Gibbs free energy (G= H-TS)
Q  U  PV
T S  U  PV
G  H   TS 
 U    PV   T S  S T
 U  PV  V P  T S  S T
G  V P  S T
G  H TS
G  V P  S T
Enthalpy, H
H  U  PV
•Largest contribution comes
from the internal energy U:
Entropy, S
• Disorder in atomic
positions / moments
• Thermal vibrations about
average position
•Potential due to bonding
•Lattice vibrations (phonons
and magnons)
Initially we will assume that the lattice vibrations do not contribute to H
Gibbs free energy
G - Free enegy
“Every system seeks to achieve a minimum of free energy.”
-
Displacement
Stable at x=0
+
Unstable at x=0
Stable Phases
Free Energy
G  H  TS
Solid
 phase
 phase
Liquid
Tc
ORDERED
Temperature
DISORDERED
At low temperatures the Gibbs free energy is lowered by minimising the
enthalpy (H=U-PV) and this is associated with an ordered ground state. As the
temperature rises it becomes more important to maximise the entropy (S).
G  H TS
Low Temp, TS < H
G H
Minimise enthalpy
High Temp, TS > H
G  TS
Maximise entropy
Solid Phase
Minimum G when H is at optimum value
U stabilised by bonding - ORDERED
Liquid Phase:
Disorder becomes more important
Bonding requirements loosened
Gas Phase:
No-longer any real bonding requirements
dominated by S and high disorder DISORDERED
SOLIDS - Internal Energy
• Assume that the contribution to the internal energy of the
lattice arising from vibrations is low (meV)
Energy, U proportional to position of atoms in unit cell
Position determined by bonding potential, fij
which only depends on separation, rij.
U  P, V  
  
fij rij
i, j
Expect fij to depend on the type
of bonding and to be different for
ionic, covalent, etc.
G  V P  ST
U(P,V) - Ice Structures
L. Pauling considered the
packing of water molecules in
ice in 1923.
Developed the ‘Ice rules’
S. Klotz et al. Europhys. Lett., 72 (4), p. 576 (2005)
ICE
Ice VII
Cubic
Bonding
For stable bonds balance
attractive and repulsive
forces:
FA  FR  FN  0
Bond energy:
E 
 F dr
Stable bonds have a
minimum in the total energy
Figure adapted from Callister, Materials science and engineering, 7th Ed.
Bonding
 rij 
ER  B exp  




or
 B
ER   12
 rij





EA is bonding dependent
Figure adapted from Callister, Materials science and engineering, 7th Ed.
van der Waals
Temporary time varying electric dipole created in neutral
materials through zero point motion of the electrons.



E rij  4 
EA
 rij

=Potential energy at equilibrium separation
 A
  6
 rij





 
=distance between the two atoms at zero energy
Figure adapted from Callister, Materials science and engineering, 7th Ed.
6
12 

 
  

 rij 

 



Lennard-Jones
EA
 A
  6
 rij





van der Waals Materials
Inert gases and Molecular crystals
He, Ar, Ne, Xe, N2, O2, H2, Cl2, Graphitic Carbon
Fast fall-off
Only nearest neighbours attract
Weak bond
Low melting temperatures
No restriction on bond angles
Want to maximise the number of nearest neighbours
Maximise Packing Density:
Simple Cubic Structures
Ionic Bonds
EA  
 
E rij
Small lattice parameters
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
Strong bond
First and second nearest
neighbours attract
High melting temperatures
Non-directional – bond strength same in all directions
brittle and hard materials
All positive ions surrounded by negative ions and want to
maximise the number of nearest neighbours
Simple Cubic Structures