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Thermal Properties - Coggle Diagram

- - - - .
    - - .
    - - Low temperature
        
        Cv proportional to 3R
        
        Accepted as compare to experiment
      - High temperature
        
        Cv proportional to T^-2 e^-B/T
        
        Fairly accepted since the formula derive by Einstein exponentially fall off (too rapid)
  - - - Cakculation:
        
        Derivation of specific heat
        
        Diamond atoms vibrate at a higher frequency compared to metallic substance, causes diamond to have high debye temperature and much lower specific heat at temperature
        
        U = int 0 : ωd (D(ω) n(ω) h(bar)ω)
        
        U = 9N k_B (T/Td)^3 int 0 : ωd _((x^4 e^x)/(e^x-1)^2)
        
        at low temp Cv = 3zR
        
        at high temp Cv = 234zR(T/Td)^3
      - ωd = v (6π^2 N/V)^(1/3)
      - Td = h(bar) * ωd / k_B
      - Density of state: D(ω) = dN/dω = (Vω^2)/(2(πv)^2)