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' 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

59-2-two-mass-matlab.png

59-3-two-mass-matlab.png Different spring constants case is mathematically equivalent with different masses caseEQUAL MASSES / SPRINGS N=8H1 = 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 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 0 0 0 0 -2 4

2 different MASSES / SPRINGS N=8

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

all the matlab codes are found in chains-2003/

99-

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)

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)

The derivation of the heat conductivity coefficient kappaFourier Law

a-1-derivation-transport.png

When the phonons are scattering from phononsand also from imperfections

The picture is plotted in any textbook (here - our online textbook)

a-5-next-

time.png

Previous lecture 2010.09.20                  Next lecture 2010.09.27