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

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

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

4. 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

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

6. 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

7. 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

8. 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

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

10. 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
z is number of particles per unit time and unit volume that hit the end face

11. 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

12. Classical Theory of Heat Capacity
Each atom in a crystal is bound to its site by a harmonic force. Absorbing thermal energy atom stats to vibrate about its point of rest. The average energy of the oscillator
E=kBT
The solids are three-dimensional. Therefore the average energy per atom is
E=3kBT
There are Na atoms in mole and total energy per mole is

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. Einstein postulated a particle-wave duality. Phonon waves propagate through the crystal with the speed of sound. Phonon waves are elastic waves vibrating in longitudinal and/or transversal mode.
… 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
Phonons are created by raising of temperature and eliminated by lowering it. Each phonon has the same energy 𝐸=ℏ𝜔 , or the same frequency of vibration, .

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:
kBθE=ℏ𝜔
θE is Einstein temperature; can be obtained by Cp fitting (at T where Cp≈CV)

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
Atoms in a crystal interact with each other and oscillators vibrate interdependently
… number of (acoustic) phonons
… distribution (density) of oscillation frequency w (density of vibrational modes for phonons) [DOS* of electrons]
vs … speed of sound
* Density of states
D(w) for a continuous medium
The cutoff frequency is quantity characterizing the Debye spectrum of crystal usually given in terms of the Debye temperature θD. Debye temperature is the highest temperature that can be achieved due to a single normal vibration.
Density of states for a Debye solid

22. Heat Capacity – The Debye Model
Debye modified Einstein equation by replacing 3Na oscillators of single frequency with number of modes in a frequency interval dw and by summing up over all allowed frequencies. The total energy of vibration for solid is then
Eosc is the same as in Einstein model

23. Heat Capacity – The Debye Model

24. Heat Capacity – The Debye Model
Debye temperature – calculation from elastic constants
The Debye model assumes that the solid is an elastic continuum in which all the sound waves travel at the same velocity independent of their wave length. Thus the phonon density of state becomes parabolic and Debye cutoff frequency, wD, can be determined by normalization condition that the total number of frequencies should be equal the 3N (N number of atoms). The Debye temperature is a measure of cutoff frequency and it is proportional to Debye sound velocity vD:
The vD is independent of crystallographic directions, but different for the longitudinal and transverse branches
L and S are longitudinal and transverse moduli
Voigt-Reus-Hill approximation:
n is Poisson ratio of polycrystalline material, B is bulk modulus

25. Heat Capacity – The Debye Model
Poisson ratio is negative of the ratio of transverse strain to axial strain
r0 is equilibrium Wigner-Seitz radius defined as
Alternative route is to calculate qD is to fit measured Cp data is the low temperature range, where Cp≈CV
using qD as fitting parameters

26. Debye temperature – calculation from XRD
Diffraction measurements provide two convenient methods to determine Debye temperature: in the first method the variation of Bragg intensity of a particular reflection (or reflections) with the temperature is measured, while in the second variation of Bragg intensity with reflection angle at fixed temperature is measured. The second method require a knowledge of theoretical atomic scattering factor. The measured structure factor F(hkl) is related to the atomic scattering factor f(hkl) by equation
where B is Debye-Waller temperature factor and K is the conversion factor. In terms of the Debye theory B is given by
where y (x) is Debye integral, qM is Debye temperature appropriate to diffraction measurements.

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

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

29. Total Heat Capacity Electrons Phonons (Debye model) T < QD 𝐶V/𝑇
Only the kinetic energy of the free electrons can be raised with increasing temperature. Only those electrons which lie within energy interval kBT of the Fermi energy can be exited in sufficient number into higher state.
𝐶V/𝑇
𝛽 … contribution of phonons
𝛾 … contribution of electrons 𝑇2

30. The applicability of Debye Model
Applicability of the Debye model to real substances is governed by the extent to which the actual vibrational spectrum by g(w) with the assumed sharp upper cutoff frequency wD. The monatomic substances in which there is only one atom per unit cell the vibrational spectrum is reasonably well represented by Debye model. The separate spectral contribution from P and S vibrations (P waves are compressional, longitudinal, S are shear transverse waves) cause deviation from Debye spectrum at higher frequencies; however the form of dispersion relation for the individual branches departs from linearity assumed by the Debye model and tends to compensate the effect of separate P and S waves.
A similar conclusion can be made for polyatomic solids (n>1) if the different atoms play nearly equivalent mechanical roles in the vibrational process: (1) the various atoms have nearly equal masses (2) coordination environment of different atoms are nearly identical (3) environments are essentially isotropic and (4) the various near-neighbor atomic force constants are nearly equal.
Linear monatomic chain (a) Dispersion relation for monatomic chain (solid) compared with dispersion relation for Debye solid (dashed) (b), Vibrational spectrum for monatomic chain compared to Debye spectrum (c) Vibrational spectrum of isotropic monatomic substance with separate P and S velocities (d) Vibrational spectrum of monatomic anisotropic solid with two shear velocity (e).

31. 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

32. 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

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

34. Thermal Conductivity 𝑛 … number of electrons
𝑙 … free path between two collisions (electron-phonon)
𝑣 … velocity of electrons
The heat conductivity is larger the more electrons, nv, are involved, the larger is their velocity v and the larger the mean path between two consecutive electron-atom collisions, 𝑙 .

35. Thermal Conductivity Metals Dielectrics
𝑙 is phonon free path decreasing with temperature raise, v is phonon velocity which is relatively temperature independent.
Celv increases with temperature, 𝑙 and, to a small degree , v are decreasing
Metals
Drifting phonons interact with lattice imperfections, external boundaries and other phonons, which constitute thermal resistivity
Dielectrics
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
L is Lorentz number, e is change of electron, s is electrical conductivity

36. Experimental methods to measure thermal conductivity
Steady state and non-steady state methods
Guarded hot plate. A solid sample is places between two plates: one is heated and the other is cooled. Temperature of plates is monitored until they are constant. The steady state temperatures, thickness of sample and heat input of hot plate are used to calculate thermal conductivity K
q is quantity of heat passing through a unit area, distance between two sides of the sample T1 and T2 are temperatures on warmer side and lower side
Hot wire and modified hot wire. The heated wire is inserted into material
Laser flash diffusivity. A laser flash deliver short pulse of heat to the front of sample and infrared scanner observes the temperature change at the rare face as function of time. K=a(T)r(T)Cp(T), where
a is thermal diffusivity [m2s-1], Cp is specific heat [Jkg-1K-1], r is density [kgm-3]

37. Inharmonic vibrations:
Thermal Expansion
Intramolecular force
repulsion attraction
Harmonic vibrations:
Inharmonic vibrations:
Thermal expansion

38. 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]

39. Thermal Expansion in GdNiAl
av is volume thermal expansion coefficient determined by XRD
Hexagonal structure

40. Thermal expansion coefficient
Dilatometry: The length of rod, L, increases with increasing temperature.
Coefficient of linear expansion, aL
The thermal expansion coefficient aL is proportional to the molar heat capacity, i.e. temperature dependence of aL is similar to the temperature dependence of CV. The aL approaches constant at T>qD and vanishes as T3 for
At low temperatures atom may rest in equilibrium position r0 at minimum of potential energy. At temperature increase amplitude of vibrating atom increase. Amplitude of vibrating atom is symmetric about a median position, but potential curve is not symmetric and a given atom moves father apart from its neighbors to larger r. Thermal expansion is a direct consequence of the asymmetry of potential energy curve.