1. PHYS208 - Lecture Thursday 4. February 2010
Topics: Concluding Debye Model – Heat Capacity
Table of Debye Temperatures
Vibrations Quanta Phonons
Comment: This is a final version of the note; Edited
Texts used
From MySpace.uib.no Exams lattice_Ex_2008.pdf
From MySpace.uib.no Band Structure by Calzaferri et al.pdf
We used matlab programs matri.m and proof.m
historical study "EnAttendentDebye.pdf"
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5. Integrand of the Debye Integral
% Simple treatment of Debye model
X=0.01:0.001:20; %% MADE SMALLER STEPS
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);
Integrand of the Debye Integral
This Debye Integral and Debye heat capacity example has been discussed last time – with «bad look-up»
Debye Integral
Debye heat capacity
Function of scaled temperature
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6. Starting Phonons Discussion
Texts used
From MySpace.uib.no Exams lattice_Ex_2008.pdf
From MySpace.uib.no Band Structure by Calzaferri et al.pdf
Phonons: Quanta of the eigenmode vibrations
.... Therefore we call a+ and a creation and annihilation operators.
They make states with one more
or one less quantum hbar omega
We used matlab programs
matri.m and proof.m
from
historical study "EnAttendentDebye.pdf"
We discussed the standing waves, and the travelling waves modes
Guitars and strings ( see the images )
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7. Standing waves have NODES, i.e.
Phonons Discussion
Standing waves have NODES,
i.e.
Stationary points where nothing happens
Travelling waves – they travel
/remember the «dragging of a pattern ...
Phonons should be travelling eigenmodes
The ring can have travelling waves.
Because of degeneracy
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11. Band Structure by Calzaferri et al.pdf
The analytic solutions
(numerical calculations
On previous slides
by matlab – numerical eigenvalues)
These examples
Are for electrons !
Note the shape of energies
ELECTRON BANDS
LATER IN THE PHYS208
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14. Phonons – Heat transporters Concept of Heat Conductivity
Next Lecture
PHYS208 - Lecture Wednesday 10. February 2010
Phonons – Heat transporters
Concept of Heat Conductivity
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