Boltzmann`s factor --> Heat Capacity

Heat Capacity- Einstein Model
Builds on: Boltzmann's factor
Heat Capacity- Debye Model
Lecture 02.05.2007
Boltzmann's factor -->
Eaverage
-->
kT
Heat Capacity- Einstein Model _______________________________________
http://web.ift.uib.no/AMOS/fys208/debye/einstein-note30.jpg
+
The meaning of average < n > : what is it when kT >> hbar omega (added later)
kT -> Average energy per osc.
E0 = hbar omega, Energy quantum;
< n > = Average energy / Energy quantum, i.e. < n > = kT/ E0 when kT >> E0
http://web.ift.uib.no/AMOS/fys208/debye/einstein-note30.jpg (lower part)
Heat Capacity- Debye Model
Old slides - density of modes quadratic in omega - and cut-off:
This explains why in three dimensions the "mode density" is different
from one dimension case.
Today's whiteboard:------------------------------------------------------------------------
Heat Capacity- Debye Model Note:
http://web.ift.uib.no/AMOS/fys208/debye/debye30.jpg
and one more picture:
Debye's model in octave
GNU-Octave is a free program, partly compatible with MATLAB
The program:
% Simple treatment of Debye model
%
x=0.01:0.01:20;
y=(x.^4.*exp(x))./(exp(x)-1).^2;
figure(1);
plot(x,y)
% Showing the integrand of Debye function;
%
% evaluate the integral by simple summation formula
%
Iy=y*0;
% Fill Iy by zeros
for k=2:max(size(y));Iy(k)=Iy(k-1)+y(k);end % perform the summations
Iy=Iy*(x(2)-x(1));
% multiply by Delta x
% plot Debye integral function
figure(2);plot(x,Iy)
ipos=0;
for tau=0.01:0.01:2;
[val,ind]=sort(abs(x-1/tau));
ipos=ipos+1;HeatCap(ipos)=Iy(ind(1))*tau^3;
T(ipos)=tau;
end
figure(3);
plot(T,HeatCap);