Previous lecture 2010.09.20 Next lecture 2010.09.27 to index wednesday 22.09.2010 - Anharmonic effects, Phonon transport, matlab One of the usual arguments for existence of anharmonicity - THERMAL EXPANSION. When the oscillators are in higher energetic states, due to the real shape of the potential
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Previous lecture 2010.09.20 Next lecture 2010.09.27 to index
One of the usual arguments for existence of anharmonicity - THERMAL EXPANSION.When the oscillators are in higher energetic states, due to the real shape of the potential
(departures from the 'harmonic' x2, the parabolic shape) the average position moves outward.
1-anharmonic.png Anharmonicity is also important in understanding HEAT CONDUCTIVITY.
Harmonic weves - independent - do not interact
INTERFERENCE IS NOT INTERACTION
2-interference-not-interaction.png Anharmonicities are known from non-linear crystalsFrequency doubling - Frequency changing optical elements
The understanding of anharmonic effects on the transport:phonons - quanta - change their number (3. order terms x3terms , added to the harmonic x2, the parabolic shape .....
3-anharmonic-a+a.png
The Umklapp process - back to our matlab toy on waves
4-umklapp.png
81-matlab-K.png (added after the lecture)
82-dispersion-k-k.png (added after the lecture) % to make the picture: figure (3); x=(-1.5:0.01:1.5); plot (x, abs(sin(pi*x)) ); hold on; plot( [-0.5 -0.5 ], [0 1 ],'-', [0.5 0.5 ] ,[0 1],'-');set(gcf,'color','white') line( [-1.4 1.4 ], [0.3 0.3 ])
Neutron scattering; EigenmodesA problem - homework - think about:
How to the the diagonalization for a 2-dimensional structure
5-neutron+2-dim-matrix.png
Matlab programs - inspection of eigenmodesWhat are the programs doing:They perform a simple diagonalization of a matrix.The eigenvalues provide the allowed value of Omega2, to get the shape resembling the dispersion relation,we must take a square root.
The values of Omega are plotted by number, there is no value of wavenumber in the model.But plotting the EIGENVECTORS, i.e. the modes - in the following way:... plots display the value of (maximum, amplitude) of the displacement at position of each 'ball' - and they appear as transversal standing waves - though in the model theyare (1 dimension) longitudinal waves.
all the matlab codes are found in chains-2003/
51-matri-1.png all the matlab codes are found in chains-2003/
56-matri2.png all the matlab codes are found in chains-2003/
57-matri-
proof.png all the matlab codes are found in chains-2003/
58-M-two-mass.png
all the matlab codes are found in chains-2003/A version with different masses - or with different spring constants - a modified matlab code matri2m.m
59-1-two-mass-matlab.png ouput from the matlab code matri2m.m aboveall the matlab codes are found in chains-2003/
The same output manipulated to show the earlier plotted behaviour