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

Lattice vibrations

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

Lattice vibrations

►The motion of atoms in a linear chain is coupled, giving rise to propagating 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  phonon phonon

►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 n·2 π  /L,

where n is integer. Number of phonons is

►All phonon wave vectors lie between – π  /a and π  /a. Therefore 

the number of phonons in a 1D chain is 2 π  /a / 2 π  /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; length length LL is is 12 a 12 a 

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

DiscreteDiscrete wavewave vectorvector valuesvalues nn··22ππ/L= n/L= n··ππ/6a/6a

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

8/19/2019 EPIV 09SS SolSt K3 Lattice Vibrations

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

Lattice vibrations

22ππaa

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--ππ

aaππ

aa

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

Two atoms per unit cell

equations of motion

two linear equations, two unknowns

(system of homogeneous linear equations)

ansatz

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

Two atoms per unit cell

this has only a solution when

coefficient matrix

two solutions for every value of k

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

Two atoms per unit cell

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

M1 und M2 schwingen in Phase

Akustische und optische Gitterschwingungen

M1 und M2 schwingen gegenphasig

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

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

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

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

Phonons in 3D crystals: Aluminium

• Results from inelastic x-ray scattering / neutron scattering.

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

Phonons in 3D crystals: diamond

• Results from inelastic x-ray scattering / neutron scattering.

• Acoustic and optical branches present.