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**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

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**? 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

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**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

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**The same applies for the second integral**

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**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

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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

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**1D Quantum mechanical harmonic oscillator**

Schrödinger equation: Solution: Quantized energy x x

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**CORRESPONDENCE PRINCIPLE**

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

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**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

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Let us introduce partition function therefore calculate Z

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Bose-Einstein distribution where

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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

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

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**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

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**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

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**? 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()

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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

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**? 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

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**? 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

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v T U C ÷ ø ö ç è æ = with

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**Debye temperature energy temperature Substitution:**

Click for a table of Debye temperatures

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Phonon Energy quantization of lattice vibration l=0,1,2,3 Bose distribution function for phonon number: for :zero point oscillation.

Phonon Energy quantization of lattice vibration l=0,1,2,3 Bose distribution function for phonon number: for :zero point oscillation.

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