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Heat capacity at constant volume

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Presentation on theme: "Heat capacity at constant volume"— Presentation transcript:

1 Heat capacity at constant volume
Thermal Properties of Crystal Lattices Remember the concept of heat capacity Temperature increase Amount of heat Heat capacity at constant volume

2 ? dV=0 constant volume Internal energy
Click for details about differentials dV=0 constant volume Internal energy How to calculate the vibrational energy of a crystal ? Classical approach In the qm description approach of independent oscillators with single frequency  is called Einstein model x In classical approach details of Epot irrelevant

3 Average thermal energy
Let us calculate the thermal average of the vibrational energy Classical: Energy can change continuously to arbitrary value defines state (point in phase space) Boltzman factor, where Average thermal energy of one oscillator




7 The same applies for the second integral

8 Classical value of the thermal average of the vibrational energy
Understanding in the framework of: Theorem of equipartition of energy every degree of freedom Example: diatomic molecule Vibration involves kinetic+pot. energy Only rotations relevant where moment of inertia

9 Solid: N atoms 3N vibrational modes with = n 24.94J/(mol K) # of moles Gas constant R=kB NA = J/(mol K) Classical limit Requires quantum mechanics

10 1D Quantum mechanical harmonic oscillator
Schrödinger equation: Solution: Quantized energy x x

Large quantum numbers: correspondence between qm an classical system x Classical probability density Classical point of reversial

12 Quantum mechanical thermal average of the vibrational energy
Einstein model: N independent 3D harmonic oscillators Energies labeled by discrete quantum number n Boltzman factor weighting every energy value Probability to find oscillator in state n

13 Let us introduce partition function therefore calculate Z

14 Bose-Einstein distribution where

15 and With In the Einstein model where for all oscillators zero point energy Heat capacity: Classical limit Note: typing error in Eq.(2-57) in J.S. Blakemore, p123

16 1 for for good news: Einstein model explains decrease of Cv for T->0 bad news: Experiments show for T->0

17 Assumption that all modes have the same frequency
unrealistic refinement Debye Model We know already: 1) 2) wave vector k labels particular phonon mode 3) total # of modes = # of translational degrees of freedom 3Nmodes in 3 dimensions N modes in 1 dimension Let us remind to dispersion relation of monatomic linear chain N atoms N phonon modes labeled by equidistant k values within the 1st Brillouin zone of width distance between adjacent k-values

18 A more detailed look to the origin of k-quantization
Quantization is always the result of the boundary conditions Let’s consider periodic boundary conditions Atom position n characterized by After N lattice constants a we end up again at atom n

19 ? In 3D we have: and One phonon mode occupies k-space volume
Volume of the crystal ? How to calculate the # of modes in a given frequency interval Density of states Blakemore calls it g(), I prefer D()

20 total # of phonon modes In a 3D crystal Let us consider dispersion of elastic isotropic medium Particular branch i: vL vT,1=vT,2=vT here

21 ? Taking into account all 3 acoustic branches
What is the density of states D(ω) good for Calculate the internal energy U # of modes in temperature independent zero point energy Energy of a mode = phonon energy # of excited phonons

22 ? How to determine the cut off frequency max
Density of states of Cu determined from neutron scattering also called Debye frequency D choose D such that both curves enclose the same area

23 v T U C ÷ ø ö ç è æ = with

24 Debye temperature energy temperature Substitution:
Click for a table of Debye temperatures

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