The Muppet’s Guide to: The Structure and Dynamics of Solids Thomas Hase Room MAS4.02 E-mail: [email protected].

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

ICE

Cubic

Ice VII

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