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LATTICE VIBRATION I, Thermal Energy, Classical : vibrating system harmonic…

- - - - Cv = (∂E/∂T)v - Heat Capacity at constant volume
      - Cp = (∂H/∂T)p - Heat Capacity at constant pressure
  - - - So, we can see which model can successfully explained the low T behavior in 3 models of Lattice vibrations section! lets go
  - - - U=NĒ
      - U=3/2 NkT=3/2 RT
      - Cv=3/2 R≈12.6 J/mol
      - Only suitable for monoatom because if we compare to experimental result, Cv=12.6 J/mol < 25J/K/mol (thus, the classical one is very small)
- - - - High temperature
        recovers Dulong - Petit law
      - The maximum vibration frequency is determined by Debye frequency, ωD
      - Low Temperature
        Cv = 0 T= 0 at low temperature Cv = T^3
        Debye T^3 approximation
  - - - all of the atoms(oscillators) vibrated with common frequency
      - Average thermal energy of an oscillator of frequency ω
        
        Ε=ℏω/2+ ℏω/e^(ℏω/kT)-1
        for N oscillators in 1D,
        
        Ε=N<s>ℏ ω
      - The heat capacity at constant volume is therefore
        ∁ν=3Nk (x^2 e^x)/(e^x-1)^2
        
        At the high temperature limit, x =ℏω/kT << 1 the Einstein heat capacity reduces to Cv = 3Nk, the Dulong and Petit law [prove by setting e^x ~ 1+x in the denominator]
        
        At the low temperature limit, when, x >> 1, e^x≫1 so Cv ≈ 0 **T ≈ 0** as required by the third law of thermodynamics(the exponential always win). [Prove by setting ex -1 ~ ex in the denominator for large x]
      - Summary - The high temperature behaviour is good, which T = 0 but at the low temperature not. We measure 𝐶𝑣∝𝑇^3,but at low temperature 𝐶𝑣∝𝑒^(−ℏ𝜔/𝑘𝑇)
- - - - Definition : The energy they possess as a result of zero point motion
      - lowest possible energy that a quantum mechanical system may have
      - Quantum mechanics differs from classical physics in that energy, momentum, angular momentum, and other quantities of a bound system
    - - In crystal dynamics we will use the harmonic approximation , amplitude of the lattice vibration is small.
        
        At higher amplitude some unharmonic effects occur.
        
        calculations will be restricted to lattice vibrations of small amplitude(harmonic limit)
        
        solid close to a position of stable equilibrium its motion can be calculated by generalization of the method used to analyse a simple harmonic oscillator
        
        - Linear region (the region of elastic deformation), the restoring force on each atom approximately proportional to its displacement (Hooke’s Law).
        
        The forces exerted on atoms when they are displaced from their equilibrium positions govern atomic motion.
        
        Calculate forces = necessary determine the wavefunctions and energies of the electrons within the crystal.
        Fortunately many important properties of the atomic motions can be deduced without doing these calculations.
        
        have effects of nonlinearity or ‘anharmonicity’ for larger atomic displacements. Anharmonic effects are important for interactions between phonons and photons.
        
        Elastic deformation : Temporary shape change that is self-reversing after the force is removed, the object returns to its original shape