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5/24/2018 Phon Script

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SS 09 - 20 140: Experimentalphysik IV K. Franke & J.I. Pascual Lattice vibrations

Lattice vibrations


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Lattice vibrations

The motion of atoms in a linear chain is coupled, giving rise topropagating waves

The frequency of oscillation depends on the wavelength (i.e.

the wave vector) of the propagating wave

For an infinite chain, the possible frequency of oscillations is a

continuous

For a finite chain of quantum oscillator, only a discrete set offrequencies is possible

Each propagating wave with a certain frequency and hence acertain group velocity is called a phononphonon

The frequency of atomic vibrations in a phonon depends on the

phonon wave vector k: This defines the dispersion relation.

Phonon wave vectors for a 1D chain of length L are n2/L,

where n is integer. Number of phonons is

All phonon wave vectors lie between /a and /a. Therefore

the number of phonons in a 1D chain is 2/a / 2/L = L/a

Each phonon can be treated itself as a quantum oscillation. Forlow temperatures every atom can be approximated by an

harmonic oscillator, the energy of the oscillation is

RealReal spacespace: For a: For a chainchain withwith 1313 atomsatoms

separatedseparated byby a;a; lengthlengthLL isis12 a12 a

ReciprocalReciprocal spacespace:: LargestLargest frecuencyfrecuency appearsappearsatat thethe largestlargest kk valuesvalues/a and/a and /a/a

DiscreteDiscrete wavewave vectorvector valuesvalues nn22/L= n/L= n/6a/6a

In total 12In total 12 modesmodes betweenbetween/a and/a and /a/a


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Lattice vibrations

22aa

PropagatingPropagating wavewave

withwith infiniteinfinite

wavelengthwavelength ((allall

atomsatoms inin phasephase))

PropagatingPropagating

wavewave withwith

wavelengthwavelength 2a2a

((allall atomsatoms inin

antiphaseantiphase))

ItIt isis onlyonly neededneeded toto knowknow thethe dispersiondispersion relationrelation inin thethe rangerange betweenbetween andand--

aa

aa


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Two atoms per unit cell

equations of motion

two linear equations, two unknowns

(system of homogeneous linear equations)

ansatz


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this has only a solution when

coefficient matrix

two solutions for every value of k


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M1 und M2 schwingen in Phase

Akustische und optische Gitterschwingungen

M1 und M2 schwingen gegenphasig


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Periodic boundary conditions

Max Born and Theodore von Karman (1912)chain with N atoms:

longest wavelength for wave solutions

This restricts the possible k values

So there are N possible different vibrations (m=0....N-1)


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Finite chain with 10 unit cells and one atom per unit cell

N atoms give N so-called normal modes of vibration.

For long but finite chains, the points are very dense.


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Long atomic chain: quantum model

The excitations of these oscillators are called phonons.

Strong analogy with photons: both bosonic excitations Both described by quantum mechanical harmonic oscillators Wave-particle duality


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Phonons in 3D crystals: Aluminium

Results from inelastic x-ray scattering / neutron scattering.


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Phonons in 3D crystals: diamond

Results from inelastic x-ray scattering / neutron scattering.

Acoustic and optical branches present.

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