1. Thermal Properties of Materials
Thermal conductivity (electrons, phonons)
Heat capacity (specific heat) at constant volume
Heat capacity (specific heat) at constant pressure
Thermal expansion

2. Thermal Conductivity (Survey)
Contribution to thermal conductivity:
Phonons (lattice vibrations)  low contribution to thermal conductivity
Electrons (connected to electrical conductivity)  high contribution to thermal conductivity

3. Specific Heat (Heat Capacity)
Einstein and Debye models – quantum mechanical description of transport phenomena

4. Definition of the Heat Assumption, that: 𝑊=0 Δ𝐸 = 𝑄
… change in energy of a thermodynamic system (𝑊 is the work performed by the system, 𝑄 is the heat)
Assumption, that:
𝑊=0
Δ𝐸 = 𝑄

5. Heat Capacity
The amount of heat (energy) required to raise the systems temperature by one degree (usually expressed in Kelvin)
… heat capacity at a constant volume
… heat capacity at a constant pressure (𝐻 is the enthalpy)
𝛼 … coefficient of thermal expansion
𝑇 … (absolute) temperature
𝑉 … volume of material
𝜅 … compressibility

6. Specific Heat
… per unit of mass:
… per mole:
Temperature dependency

7. Temperature Dependency of Specific Heat
CV = 25 J mol-1 K-1 = 5.98 cal mol-1 K-1
Experimental results:
Specific heat of materials with one atom per unit cell is about 25 J mol-1 K-1 at room temperature.
At low temperatures the specific heat decreases. Metals: 𝐶 v ≈𝑇 Insulators: 𝐶 v ≈ 𝑇 3
In magnetic materials specific heat increases if ordering of magnetic moments increases
Fig a
Specific heat capacity 𝐶 v 𝑇 of different materials

8. Specific Heat at Phase Transition
Specific heat capacity of KH2PO4, which has a first-order phase transition at 120 K
The material needs additional energy (heat) for the phase transition

9. Structure Transition in KH2PO4: paraelectric  ferroelectric
RG: Fdd2 (orthorhombic)
a = Å,
b = Å,
c = 6.967Å
… K
… P
… O
… H
Paraelectric
RG: I -42d (tetragonal)
a = 7.444Å, c = 6.967Å

10. Magnetic Phase Transition of CePtSn
Antiferromagnetic with 𝑇N = 7.5 K
Change in ordering of magnetic moments

11. Ideal Gas
Na = x 1023 mol-1
R = kB Na = J mol-1 K-1 = cal mol-1 K-1
Kinetic energy of ideal gas
𝑝 … pressure
𝑝 ∗ … impulse
𝐴 … area
𝑁 … number of atoms
𝑇 … temperature

12. Classical Theory of Heat Capacity (Ideal Gas)
CV = 25 J mol-1 K-1 = 5.98 cal mol-1 K-1
𝐸mol … energy per mole
Good compliance with experiments at higher temperatures

13. Quantum Theory of Heat Capacity
1903: Einstein postulated the quantum behavior of lattice vibrations (lattice oscillations) analogous to the quantum behavior of electrons.
The quanta of lattice vibrations are called phonons.
… impulse (de Broglie)
… energy
Longitudinal oscillations
Transversal oscillations

14. Dispersion Relation (Phonon Dispersion)
Analogy to the band structure
Optical phonons (optical branch)
Frequency
Acoustic phonons (acoustic branch)
Frequency of phonons (THz)
Wave vector
Optical phonons… higher energy (frequency)
Acoustic phonons… lower energy (frequency)
𝐾/𝐾max in [111]-direction

15. Phonon dispersion as obtained from the neutron diffraction experiments
𝐸′=ℏ𝜔′
𝑝′ =ℏ 𝑘′
𝐸=ℏ𝜔
𝑝 =ℏ 𝑘
𝐸 𝑃ℎ =ℏΩ
𝑝 𝑃ℎ =ℏ 𝐾
Fig. 5.17
Acoustic and optical branches determined by neutron diffraction a) for aluminum b) for potassium bromide A – acoustic; O – optical; T – transversal; L – longitudinal

16. Acoustic and Optical Branches of a Linear Atomic Chain
Acoustic branch
Optical branch
Fig Typical movement pattern of atoms in a linear chain

17. Energy of a Quantum Mechanical Oscillator
… quantum energies
… Bose-Einstein distribution
… Fermi function (distribution) of electrons

18. Heat Capacity – The Einstein Model
E = 0.01 eV KP QM
Number of phonons
Temperature [K]

19. Heat Capacity – The Einstein Model
Classical approximation
CV = 3R
CV  exp(-/kBT)
Temperature [K]
Extreme case:

20. Comparison of Theoretical and Experimental Results
Specific heat of materials with one atom per unit cell is about 25 J mol-1 K-1 at room temperature.
At low temperatures the specific heat decreases. Metals: 𝐶 v ≈𝑇 Insulators: 𝐶 v ≈ 𝑇 3
Theory (Einstein model):
Specific heat is about 25 J mol-1 K-1 at high temperatures.
At low temperatures the specific heat decreases exponentially 𝐶 v ≈exp⁡(−𝜔/ 𝑘 B 𝑇)
The Einstein model considers only phonons with particular (discrete) frequencies.

21. Heat Capacity – The Debye Model
Phonons with different energies
… number of (acoustic) phonons
… distribution (density) of oscillation frequency [DOS* of electrons]
vs … speed of sound
* Density of states

22. Heat Capacity – The Debye Model

23. Debye Temperatures Fig. 5.26 a
Specific heat capacity 𝐶 v 𝑇 of different materials

24. Heat Capacity at High and Low Temperatures (Debye Model)
𝐶v  𝑇3: Better compliance with experimental values at low temperatures
!!! For insulators !!!

25. Total Heat Capacity Phonons (Debye model) T < QD Electrons 𝐶V/𝑇
𝛽 … contribution of phonons
𝛾 … contribution of electrons 𝑇2

26. for the investigation of lattice vibrations
Experimental Methods
for the investigation of lattice vibrations
X-ray diffraction
Profile change of electron density (thermal vibrations of electrons)
Influence on the intensities of diffraction lines
Neutron diffraction
Interaction between low-energy (slow) neutrons and the Phonons

27. Heat Conductivity Thermal conductivity: K
Partial differential equation:
Solution for definite initial and boundary conditions
Temperature dependency – similar to the change in concentration at diffusion processes
J = 0
J = 0
𝑇 = const.
𝑇 = const.

28. Thermal Conductivity 𝑛 … number of electrons
𝑙 … free path between two collisions (electron-phonon)
𝑣 … velocity of electrons

29. Thermal Conductivity Metals Dielectrics Wiedemann-Franz law:
Thermal conductivity, W/cm/K
Temperature, K
Wiedemann-Franz law:
Materials with high electrical conductivity exhibits a high thermal conductivity
Material K [W/cm/K]
SiO2 0,13 – 0,50 (at 273K or 80K)
NaCl 0,07 – 0,27 (at 273K or 80K)
Al2O at 30K
Cu at 20K
Ga at 1.8 K

30. Inharmonic vibrations:
Thermal Expansion
Intramolecular force
Harmonic vibrations:
Inharmonic vibrations:
Thermal expansion

31. Thermal Expansion Argon (kfz)
Change of mean interatomic distance with temperature:
Temperature dependency of lattice parameters:
Argon (kfz)
Lattice parameter increases quadratically with the temperature
The thermal expansion at 𝑇 = 0K is zero
Lattice parameter [Å]
Density [g/cm³]
Temperature [K]

32. Thermal Expansion in GdNiAl