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Published byFaith Parlett Modified over 2 years ago

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

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

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

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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 Only rotations relevant where moment of inertia Vibration involves kinetic+pot. energy

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with # of moles Gas constant R=k B N A = J/(mol K) = n 24.94J/(mol K) Classical limit Solid:N atoms 3N vibrational modes 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 point of reversial Classical probability density

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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 Quantum mechanical thermal average of the vibrational energy

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Let us introduce partition function therefore calculate Z

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where Bose-Einstein distribution

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

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1 for for good news: Einstein model explains decrease of C v 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: wave vector k labels particular phonon mode 1) 2) 3) total # of modes = # of translational degrees of freedom 3Nmodes in 3 dimensionsN modes in 1 dimension Let us remind to dispersion relation of monatomic linear chain N atomsN phonon modes labeled by equidistant k values within the 1 st Brillouin zone of width distance between adjacent k-values

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A more detailed look to the origin of k-quantization D a Quantization is always the result of the boundary conditions Lets 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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vLvL v T,1 =v T,2 =v T Let us consider dispersion of elastic isotropic medium total # of phonon modesIn a 3D crystal Particular branch i: 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 Energy of a mode # of excited phonons temperature independent zero point energy = phonon energy

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

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with v T U v C

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

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