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Thermal Properties 1, ADIBAH HANISAH BINTI SAMSUDIN 991022-04-5198 …

- - - - assumed that the
        atoms are vibrating as harmonic oscillators
        
        classical expression for the
        energy of an oscillator
        
        assumed Planck’s (1990) quantization rule for each oscillator (vibrating oscillators (atoms) in a solid have quantized energies)
      - Occupation of energy level n: (probability of oscillator
        being in level n)
        classical physics (Boltzmn factor)
        
        average total energy of solid:
      - the prediction:
      - Limiting Behavior of CV (T)
        
        high T limit
        
        low T limit
        
        Cv --> 3R for large T
        
        Cv --> 0 as T --> 0:
        
        heat capacity equation
        
        , CV → 0 when T → 0 is true
        
        At low T, Einstein’s formula is essentially an
        exponential falloff
    - - i) Developed a more sophisticated (but still approximate) treatment of
        atomic vibrations in solids:
        
        Atoms are considered as harmonic oscillators that produce elastic
        waves with varying frequencies from
        
        Treat solid as continuous elastic medium
        
        3N normal modes (patterns) of oscillations
        
        ii) relations between the energy of a phonon E, the angular
        frequency ω and the wave vector q
        iii) Debye temperature TD
      - Heat Capacity
      - At high temperature limit, T>>TD
        
        value of x is very small throughout the range of the
        integral
        
        At low temperature limit, T<<TD
        
        Only long wavelength acoustic modes are thermally excited
        
        approximate XD≡TD/T to infinity to obtain
      - Debye model successfully describes the temperature dependence of the heat capacity not only at high temperatures but also at low temperatures.
        
        T3 Debye law is known for the contribution of the phonon to the heat capacity at low temperatures
        
        for z=1
      - Debye model work better at low
        T than the Einstein model?
        
        Debye model gives a better representation for the
        very low energy vibrations
        
        At low temperatures, these vibrations matter most
      - Limits of the Debye model