Phonon propagation, anharmonicity, Heat Conductivity.docx

7/29/2019 Phonon propagation, anharmonicity, Heat Conductivity.docx

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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(departures from the 'harmonic'x

2, the parabolic shape) the average position moves outward.
http://web.ift.uib.no/AMOS/PHYS208/2010.09.20/index.htmlhttp://web.ift.uib.no/AMOS/PHYS208/2010.09.20/index.htmlhttp://web.ift.uib.no/AMOS/PHYS208/2010.09.27/index.htmlhttp://web.ift.uib.no/AMOS/PHYS208/2010.09.27/index.htmlhttp://web.ift.uib.no/AMOS/PHYS208/index.htmlhttp://web.ift.uib.no/AMOS/PHYS208/index.htmlhttp://web.ift.uib.no/AMOS/PHYS208/index.htmlhttp://web.ift.uib.no/AMOS/PHYS208/2010.09.27/index.htmlhttp://web.ift.uib.no/AMOS/PHYS208/2010.09.20/index.html


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1-anharmonic.pngAnharmonicity is also important in understanding HEAT CONDUCTIVITY.

Harmonic weves - independent - do not interact

INTERFERENCE IS NOT INTERACTION


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2-interference-not-interaction.pngAnharmonicities 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 termsx

3terms , added to the harmonic x

2, the parabolic shape .....


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3-anharmonic-a+a.png


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The Umklapp process - back to our matlab toy on waves

4-umklapp.png


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81-matlab-K.png (added after the lecture)


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82-dispersion-k-k.png (added after thelecture)

% 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


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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 ofeach 'ball' - and they appear as transversal standing waves - though in the model theyare (1 dimension) longitudinal waves.

all the matlab codes are found inchains-2003/
http://web.ift.uib.no/AMOS/PHYS208/2010.09.22/chains-2003/http://web.ift.uib.no/AMOS/PHYS208/2010.09.22/chains-2003/http://web.ift.uib.no/AMOS/PHYS208/2010.09.22/chains-2003/http://web.ift.uib.no/AMOS/PHYS208/2010.09.22/chains-2003/


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51-matri-1.png


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56-matri2.pngall the matlab codes are found inchains-2003/


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57-matri-


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proof.pngall the matlab codes are found inchains-2003/


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58-M-two-mass.png

all the matlab codes are found inchains-2003/A version with different masses - or with different spring constants - a modified matlab code matri2m.m

59-1-two-mass-matlab.pngouput from the matlab code matri2m.m aboveall the matlab codes are found inchains-2003/
http://web.ift.uib.no/AMOS/PHYS208/2010.09.22/chains-2003/http://web.ift.uib.no/AMOS/PHYS208/2010.09.22/chains-2003/http://web.ift.uib.no/AMOS/PHYS208/2010.09.22/chains-2003/http://web.ift.uib.no/AMOS/PHYS208/2010.09.22/chains-2003/http://web.ift.uib.no/AMOS/PHYS208/2010.09.22/chains-2003/http://web.ift.uib.no/AMOS/PHYS208/2010.09.22/chains-2003/http://web.ift.uib.no/AMOS/PHYS208/2010.09.22/chains-2003/http://web.ift.uib.no/AMOS/PHYS208/2010.09.22/chains-2003/


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The same output manipulated to show the earlier plotted behaviour

59-2-two-mass-matlab.png


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59-3-two-mass-matlab.pngDifferent spring constants case is mathematically equivalent with different masses caseEQUAL MASSES / SPRINGS N=8

H1 =

4 -2 0 0 0 0 0 0

-2 4 -2 0 0 0 0 0

0 -2 4 -2 0 0 0 0

0 0 -2 4 -2 0 0 00 0 0 -2 4 -2 0 0

0 0 0 0 -2 4 -2 0

0 0 0 0 0 -2 4 -2

0 0 0 0 0 0 -2 4

2 different MASSES / SPRINGS N=8


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H1 =

2.8 -2 0 0 0 0 0 0

-2 2.8 -0.8 0 0 0 0 0

0 -0.8 2.8 -2 0 0 0 0

0 0 -2 2.8 -0.8 0 0 0

0 0 0 -0.8 2.8 -2 0 0

0 0 0 0 -2 2.8 -0.8 0

0 0 0 0 0 -0.8 2.8 -2

0 0 0 0 0 0 -2 2.8



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ashcroft.png

TRANSPORT OF HEAT - phonons assumed to be the carriersThe concept of MEAN FREE PATH

croos section sigma, density of obstacles rho and the mean free path L (greek and script letters are usually used)


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60-mean-free-path.png( The mean free path used in the following derivation; the derived relation is used below - the density of scatterers-obstacles)


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The derivation of the heat conductivity coefficient kappaFourier Law


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When the phonons are scattering from phononsand also from imperfectionsThe picture is plotted in any textbook (here - our online textbook)


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a-5-next-


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time.png

Previous lecture 2010.09.20 Next lecture 2010.09.27
http://web.ift.uib.no/AMOS/PHYS208/2010.09.20/index.htmlhttp://web.ift.uib.no/AMOS/PHYS208/2010.09.20/index.htmlhttp://web.ift.uib.no/AMOS/PHYS208/2010.09.27/index.htmlhttp://web.ift.uib.no/AMOS/PHYS208/2010.09.27/index.htmlhttp://web.ift.uib.no/AMOS/PHYS208/2010.09.27/index.htmlhttp://web.ift.uib.no/AMOS/PHYS208/2010.09.20/index.html

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Phonon propagation, anharmonicity, Heat Conductivity.docx