1 Aims of this lecture The diatomic chain –final comments Next level of complexity: –quantisation – PHONONS –dispersion curves in three dimensions Measuring

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Aims of this lectureAims of this lecture• The diatomic chain – final comments

• Next level of complexity:– quantisation – PHONONS– dispersion curves in three

dimensions

• Measuring phonon dispersion – inelastic scattering

• Phonon momentum – some weird aspects

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Repeated Zone Scheme:Contains surplus info

Diatomic chain – representing dispersion curvesDiatomic chain – representing dispersion curvesπ/a2π/a–π/akω0

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Diatomic chain – representing dispersion curvesDiatomic chain – representing dispersion curvesπ/a2π/a–π/akω0 As M m....

...and when M = m...

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Diatomic chain – final commentDiatomic chain – final comment

Another common type of arrangement is the monatomic chain, with

e.g. if you look along the (111) direction of diamond you would see:

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000( ) 14

14

14( )

This produces the same form of dispersion relation as the diatomic chain – with optical and acoustic branches.

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Quantisation of atomic vibrationsQuantisation of atomic vibrationsClassically, the energy of a mode of oscillation with frequency ω will depend on the amplitude (squared) of the motion, and can take any value.

The quantum of lattice vibration is called a PHONON.

Our crystal is a system of coupled HARMONIC OSCILLATORS and quantum mechanics predicts that harmonic oscillators have their energies quantised:

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En = n + 12

⎛ ⎝ ⎜ ⎞

⎠ ⎟hω

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Lattice vibrations in three dimensionsLattice vibrations in three dimensionsAdded complications in 3D:

• one longitudinal plus two transverse branches (which have different dispersions)BUT still N modes per branch for a crystal with N primitive unit cells

• – Size of Brillouin zone

– strength of bonds, and hence

– shape of dispersion curve all depend on direction (planes of atoms have different separations for different directions)

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Lattice vibrations – orders of magnitudeLattice vibrations – orders of magnitude

vg at zone centre: 5000 ms–1

lattice spacing: 5 10–10 mhence (zone edge):

k (zone edge):p (zone edge):

ωmax:

hence Emax: (c.f. far-infrared photons)

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Experimental determination of phonon dispersion curvesExperimental determination of phonon dispersion curves

Three approaches:

1) Inelastic light scattering: extreme case, photon absorptiona far-infrared photon can be absorbed to produce a phonon

BUT note both energy and momentum are conserved– what is the momentum of such a photon?

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Three approaches:

2) Inelastic light scatteringa variant of this uses visible light, which is inelastically

scattered rather than completely absorbed

RAMAN SPECTROSCOPY

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Three approaches:

2*) Inelastic light scattering... WHAT IF we a photon with the right momentum

104 times the energy, 300eV x-rays

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Three approaches:

3) Inelastic neutron scatteringThermal neutrons have both the right energy and the right momentum:

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Phonon momentum and phonon scatteringPhonon momentum and phonon scattering

We can think of scattering of phonons by neutrons, either in terms of waves, or in terms of particles.

In terms of particles:

neutron, hk

hk'

phonon, hq

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This modification to the law of conservation of momentum has some weird consequences when we consider phonons scattering off each other:

Consider two phonons colliding, and combining to produce a third:

q1q2q3

... and suppose q3 is outside the first Brillouin zone:0π/a2π/a–π/aqωq3

q3'

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Points to note:• phonons with either q3 or q3' have negative group velocity:

The two forward-going phonons produce a third going backwards!

q1q2q3

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ω

k


Points to note:q1q2q3

• q1 and q2 cannot be co-linear. Two

phonons on the same dispersion curve cannot produce a third on the same curve:

the position that conserves both energy and momentum is above curve

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Phonon momentum and phonon scatteringPhonon momentum and phonon scatteringFinal comments:It is dangerous to take the concept of phonon momentum too literally:

• It is not normal kinematic momentum: there are no masses moving (net) distances as the phonon passes through a crystal. Indeed for a transverse wave, even the oscillatory movements that do occur are perpendicular to q.

• Phonons have a quantity associated with them which is conserved in a similar (not identical) way to momentum in collisions

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hq

• There is one phonon mode that does carry real momentum – the q = 0 mode

Documents

1 Aims of this lecture The diatomic chain –final comments Next level of complexity: –quantisation – PHONONS –dispersion curves in three dimensions Measuring