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

? 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

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

The same applies for the second integral

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

Solid: N atoms 3N vibrational modes with = n 24.94J/(mol K) # of moles Gas constant R=kB NA = 8.3145J/(mol K) Classical limit Requires quantum mechanics

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

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

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

Let us introduce partition function therefore calculate Z

Bose-Einstein distribution where

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

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

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

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

? 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()

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

? 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

? 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

v T U C ÷ ø ö ç è æ ¶ = with

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