DEBEYS MODEL OF HEAT CAPACITANCE

1. Debeys Model of heat capacitance

2. Writer: Syed Ahsan Raza (syedahsanraza63@gmail.com)

3. Models For Heat Capacitance Classical Model Einstien Model Debye model Classical mechanics fails completely at quantifying the entire idea of temperature. This is because temperature is not a fundamental quantity. It is a method developed by Peter Debye in 1912[7] for estimating the phonon contribution to the specific heat (heat capacity) in a solid Einstein's law fails to account for the low temperature behavior of specific heat of solids

4. D Debeys Model

5. Assumptions Isotropic Number Of Atom N = 3N Properties of Solids Doesnt Depend on direction left-right .They Remain Same Atoms Vibrate in x,y,z direction with 3N Frequence

6. Assumptions Continuous Coupled Oscillator Atomic Sturture of Solids is continuous in solids its not discrete. Every atom has influence over every other atom e.g if one atom vibrates all the atoms near it will also vibrate

7. Assumptions Debey Frequency explains Transverse as well as longitudinal waves vibrations Debay Cut Of Frequency D ω All Atoms Dont have same frequency instead they have a fixed range of frequency from 0 to infinity ↔ 0

8. Number of Vibrational modes in finite latice µn = Asinkxx sinὼ Frequency of vibration Amplitude Wave propogation constant Vibration of n particales

9. Number of Vibrational modes in finite latice µ0 = µn = 0 n=0 0 = Asinkx(0)sinω Frequency of vibration Amplitude Wave propogation constant Vibration of n particales X=0 X=na

10. Number of Vibrational modes in finite latice Putting x =Na n=0 0 = Asinkx(Na)sinω 0 = sinkx(Na) :-sinθ=θ , θ=n Na=L nx = kx(Na) X=0 X=na =

11. Number of Vibrational modes in 3-D µx =Asinkxxsinkyysinkzzsinωt µx =Asin()xsin()ysin()zsinωt :-nx,ny,nz =1,2,3,4..... ky kx kz k = () n = ()....eq 5 k2 = k2x + k2y + k2z k2 = (n2x + n2y + n2z )() n = ()....eq 6 k2 = () This shows that we will get 1 frequency (w) for every n value every combination will give us a frequency nx,ny,nz ► n = () k = ()

12. we should only consider positive area of the sperical area as frequency can only be positive n is positive if = n3 0→ω→ = n3 ω→ω+dω→ = n= n n = () z(ω)dω=() =

13. z()d = z()d =

14. Debye Approximation z()d = Putting equation 8 into 9 z()d = = 3N [] = 3N = = [ = ....10

15. Mean Energy of ocillator Same as Einstein Model = ℏ Lattice Specific Heat = V V = = = = =

16. dx = Let = x = = 9NK Verify last equation = 9NK 0 → x = → when x = 0 and = 0 when = = 9R

17. High and low temperature High and Low Temperature LimitsThe integral in Equation (12) cannot be evaluated in closed form. But the high and low temperature limits can be assessed.

18. High and low temperature High Temperature LimitFor the high temperature case where T≫TD , the value of x is very small throughout the range of the integral. This justifies using the approximation to the exponential ex≈1+x and reduces equation

19. Short Comings Solid is not continuous it is discrete Crystal structure avoidence Sound have same velocity at all wavelength Free electron ignorance. = Constant

20. Thank you for listening