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Transcript of The Muppet’s Guide to: The Structure and Dynamics of Solids Thomas Hase Room MAS4.02 E-mail:...
The Muppet’s Guide to:The Structure and Dynamics of Solids
Thomas Hase
Room MAS4.02
E-mail: [email protected]
My Research
X-ray (and neutron) scattering from thin films and multilayers….
Objectives for part 11. Understand how thermodynamics and bonding are responsible for crystal structure and material
properties2. 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, strain and particle size from scattering data.
Recommended Books – Part 1
The Physics and Chemistry of SolidsStephen ElliottISBN: 978-0-471-98195-4, Wiley, Paperback June 1998£47.50 / €71.30 (Wiley)
Materials Science and Engineering - An Introduction, 7th EditionWilliam D. Callister, Jr., ISBN: 978-0-471-73696-7, Wiley, Hardback 2007£37.95 / €58.50 (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
Exam Structure
For this course:– 1 compulsory question:
• covering all 3 parts (“bite size”)
– 2 from 3 questions• one each from the three sections (detailed)
The Muppet’s Guide to:The Structure and Dynamics of Solids
1. Structure from a Thermodynamic Viewpoint
Objectives1. Understand how thermodynamics and bonding are responsible for crystal structure and material
properties2. 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, strain and particle size from scattering data.
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….
• Entropy: A measure of disorder or randomness in a system:
• 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 J
Thermodynamics
1QS JK
T
U is the Internal energy of crystal
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, DS=DQ/T, becomes
• Consider small change in Gibbs free energy (G= H-TS)
Q U P V
T S U P V
H
U PV
U
G
G
TS
T S S
P V V
T
T S S TTP S
PV
G H TS
Entropy, S
• Disorder in atomic positions / moments
• Thermal vibrations about average position
Enthalpy, H
• Largest contribution comes from the internal energy U:
• Potential due to bonding
• Lattice vibrations (phonons and magnons)
H U PV
Initially we will assume that the lattice vibrations do not contribute to H
V P SG T
Gibbs free energy
“Every system seeks to achieve a minimum of free energy.”
- +Displacement
G -
Fre
e e
negy
Stable at x=0 Unstable at x=0
Stable Phases
phase phase
Tc
Temperature
Fre
e E
ner
gy
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
ORDERED DISORDERED
Solid
Liquid
G H TS
Solid Phase Minimum G when H is at optimum valueU stabilised by bonding - ORDERED
Liquid Phase: Disorder becomes more importantBonding requirements loosened
Gas Phase: No-longer any real bonding requirementsdominated by S and high disorder -
DISORDERED
Low Temp, TS < H Minimise enthalpy
High Temp, TS > H Maximise entropy
G H
G TS
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.
,
, ij iji j
U P V r Expect fij to depend on the type of bonding and to be different for ionic, covalent, etc.
S. Klotz et al. Europhys. Lett., 72 (4), p. 576 (2005)
L. Pauling considered the packing of water molecules in ice in 1923.
Developed the ‘Ice rules’
U(P,V) - Ice Structures
Bonding
For stable bonds balance attractive and repulsive forces:
0A NRFF F
Bond energy:
E F dr
Figure adapted from Callister, Materials science and engineering, 7 th Ed.
Stable bonds have a minimum in the total energy
Bonding
12Rij
BE
r
exp ijR
rE B
or
EA is bonding dependent
Figure adapted from Callister, Materials science and engineering, 7 th Ed.
van der Waals
Temporary time varying electric dipole created in neutral materials through zero point motion of the electrons.
6Aij
AE
r
6 12
4ijij ij
E rr r
e=Potential energy at equilibrium separation
s=distance between the two atoms at zero energyFigure adapted from Callister, Materials science and engineering, 7 th Ed.
Lennard-Jones
van der Waals Materials
Inert gases and Molecular crystals
He, Ar, Ne, Xe, N2, O2, H2, Cl2, Graphitic Carbon
No restriction on bond angles
Want to maximise the number of nearest neighbours
Maximise Packing Density:
Simple Cubic Structures
6Aij
AE
r
Fast fall-off Only nearest neighbours attract
Weak bond Low melting temperatures
Ionic Bonds2
04Aij
eE
r
122
04ijij ij
e BE r
r r
Small lattice parameters
Ionic Materials
Range of materials such as NaCl, CsCl, MgO etc.
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
Moderate fall-off
First and second nearest neighbours attract
2
04Aij
eE
r
Strong bond High melting temperatures