Chapter 4Phonons I. crystal vibrationKey points: Crystal vibrations and lattice wave PhononsPhonon momentumPhonon is one kind of elementary excitations in solidsNameFieldElectronPhotonElectromagnetic wavePhononElastic wavePlasmonCollective electron waveMagnonMagnetization (spin) wavePolaronElectron + elastic deformationExcitonPolarization waveImportant elementary excitations in solidsVibrations of crystals with monoatomic basisConsider the elastic vibrations of a crystal with one atom in the primitive cell.Describe with a single coordinate us the displacement of the plane s from its equilibrium position.Three mode for each wavevector, one of longitudinal polarization and two of transverse polarization.longitudinal modetransverse modeNote: for N atoms there are 3N mode, N L-modes and 2N T-modes.Assume that the elastic response of the crystal in linear function of the force. (Hookes law)i.e. the elastic energy is a quadratic function of the relative displacement of any two points in the crystal.Neglect cubic and higher order terms.From Hookes law, the force on the s plane caused by the displacement of the plane s+p is proportional to the difference of their displacement,i.e. Fsp = Csp (us+p us)Consider only the nearest neighbor interactions. The total force on s comes from the plane s1Fs = C (us+1 us) + C (us-1 us) = C (us+1 + us-1 2us)The equation of the motion is:a traveling wave solution:us = u expi(sKa t),where a is the spacing between planes and K is the wavevector.Then we have:the dispersion relation (色散关系):orFor Ka), KAt the boundary of first Brillouin zone (K = /a),d/dK = 0First Brillouin zoneOnly K in the first Brillouin zone is physically significant for the elastic waves.The range of independent values of K is specified by:We may treat a value of K outside the 1st Brillouin zone by subtracting a appropriate reciprocal lattice vector in order to obtain an equivalent wavevector in the 1st Brillouin zone.At the Brillouin zone boundaries, K = /a,us = u expi( s t) = (1)s u exp(it) (standing wave)The critical value, K = /a, satisfies the Bragg condition.Note: for x-ray, it is possible to have other n which cause the wavevector K outside 1st Brillouin zone.Group velocityThe group velocity is the transmission velocity of a wave packet, which is also the velocity of energy propagation in the medium.With the particular dispersion relationLong wavelength limitWhen a i.e. Ka 1Derivation of force constants from experimentIn metals the effective forces may be of quite long range, In metals the effective forces may be of quite long range, carried from ion to ion through the conduction electron sea.carried from ion to ion through the conduction electron sea.Considering p nearest planes, the force on s planeThe equation of motion isThe dispersion relationTwo atoms per primitive basisThe dispersion relation shows new features in crystals with two or more atoms per primitive basis.acoustical and optical branches (声学支和光学支)With p atoms in the primitive cell and N primitive cells, there are pN atoms, 3pN degrees of freedom, N LA branches, 2N TA branches, (p1)N LO branches and (2p2)N LO branches.If there are p atoms in the primitive cell, there are 3p branches to the dispersion relation: 3 acoustical branches (1 LA and 2 TA) and 3p3 optical branches (p1 LO and 2p2 LO ).Assume that each plane interacts only with the nearest-neighbor planes. The equations of motion isthe solution in a form of a traveling wave:substitute the solution in the equationsThe homogenous linear equations have a solution only if the determinant of the coefficients of the unknown u, v vanishes.orthe dispersion relationat long wavelength limit (Ka 1)For the optical branchThe atoms vibrate against each other, but their center of mass is fixed.For the acoustical branchThe atoms and their center of mass move together.at the 1st Brillouin zone boundary (Ka = )There is a frequency gap at boundary of the 1st Brillouin zone.Quantization of the elastic wavesThe energy of a lattice vibration is quantized. The quantum of the energy is called phonon in analogy with the photon of the electromagnetic wave.Thermal vibrations in crystals are thermally excited phonons.The energy of the elastic modethis mode occupied by n phonons with a frequency the zero point energyThe amplitude of the elastic vibrationConsider a standing wave modeThe time average kinetic energy isThe kinetic energy of an atom isWhat is the sign of ?The energy of a phonon must be positive.It is conventional and suitable to view as positive.A mode with imaginary (negative 2) will be unstable.Phonon momentumA phonon of wavevector K will interact with particles such as photons, neutrons, and electrons as if it has a momentum .However a phonon does not carry physical momentum.For most practical purpose a phonon acts as if its momentum were , which is called the crystal momentum.In crystal there exists wavevector selection rules for allowed transition between quantum states.The true momentum of the whole system always is rigorously conserved.For the elastic scattering of a photon by a crystal, the wavevector selection rule isIf the scattering of the photon is inelastic, with the creation of a phonon , the wavevector selection rule becomesIf a phonon is absorbed in the inelastic scattering, the wavevector selection rule becomesInelastic scattering by phononsPhonon dispersion relations (K) are most often determined by inelastic scattering of neutrons with the emission or absorption of a phonon.A neutron sees the crystal lattice chiefly by interaction with the nuclei of the atoms.The wavelength and the energy of the slow neutron is just within the order of phonons.The general wavevector selection rule is:The statement of conservation of energy is:Where are the wavevector and the energy of a phonon created (+) or absorbed () in the process.PbSiPhysical processes related to neutron scatteringA new neutron source (CSNC) will be set up at DongGuan in the next five years. Repetition rate ： 25 Hz Average proton current： 62.5 A Proton kinetic energy： 1.6 GeV Average beam power： 100 kW Target: Tungsten Moderators: H2O,LCH4 ,LH2 Spectrometers: HRPD HIPD Reflectometer,SANS Chopper spectrometer