1 Aims 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

2 Repeated Zone Scheme: Contains surplus info Diatomic chain – representing dispersion curves

3 As M  m and when M = m...

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

5 Quantisation of atomic vibrations Classically, the energy of a mode of oscillation with frequency  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:

6 Lattice vibrations in three dimensions Added 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)

7 Lattice vibrations – orders of magnitude v g at zone centre:5000 ms –1 lattice spacing:5  10 –10 m hence (zone edge): k (zone edge): p (zone edge):  max : hence E max : (c.f. far-infrared photons)

8 Experimental determination of phonon dispersion curves Three approaches: 1) Inelastic light scattering: extreme case, photon absorption a 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?

9 Experimental determination of phonon dispersion curves Three approaches: 2) Inelastic light scattering a variant of this uses visible light, which is inelastically scattered rather than completely absorbed RAMAN SPECTROSCOPY

10 Experimental determination of phonon dispersion curves Three approaches: 2*) Inelastic light scattering... WHAT IF we a photon with the right momentum  10 4 times the energy, 300eV  x-rays

11 Experimental determination of phonon dispersion curves Three approaches: 3) Inelastic neutron scattering Thermal neutrons have both the right energy and the right momentum:

12 Phonon 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,  k  k' phonon,  q

13 Phonon momentum and phonon scattering 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:...and suppose q 3 is outside the first Brillouin zone:

14 Phonon momentum and phonon scattering Points to note: phonons with either q 3 or q 3 ' have negative group velocity: The two forward-going phonons produce a third going backwards!

15  k Phonon momentum and phonon scattering Points to note: q 1 and q 2 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

16 Phonon momentum and phonon scattering Final 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 There is one phonon mode that does carry real momentum – the q = 0 mode