Lecture 4 Andrei Sirenko, NJIT 1Phys 446: Solid State Physics / Optical PropertiesLattice vibrations: Thermal, acoustic, and optical propertiesFall 20072Solid State Physics Lecture 4 (Ch. 3)Last week: • Diffraction from crystals• Scattering factors and selection rules for diffraction• Today:• Lattice vibrations: Thermal, acoustic, and optical properties• Start with crystal lattice vibrations. • Elastic constants. Elastic waves. • Simple model of lattice vibrations – linear atomic chain• HW1 discussion3Material to be included in the testOct. 12th2007• Crystalline structures. 7 crystal systems and 14 Bravais lattices• Crystallographic directions and Miller indices • Definition of reciprocal lattice vectors: • What is Brillouin zone• Bragg formula: 2d·sinθ = mλ ; ∆k = G21222222⎟⎟⎠⎞⎜⎜⎝⎛++=clbkahndhkl4•Factors affecting the diffraction amplitude:Atomic scattering factor (form factor): reflects distribution of electronic cloud.In case of spherical distribution •Structure factor•Elastic stiffness and compliance. Strain and stress: definitions and relation between them in a linear regime (Hooke's law):•Elastic wave equation: rdenflia3)(rkr⋅∆∫=()∫⋅⋅=002∆∆sin)(4radrrkrkrnrfπ∑++=jlwkvhuiajjjjefF)(2πklklijklijCεσ∑=klklijklijSσε∑=2222xuCtuxeff∂∂=∂∂ρsound velocityρeffCv =5• Lattice vibrations: acoustic and optical branches In three-dimensional lattice with s atoms per unit cell there are 3s phonon branches: 3 acoustic, 3s - 3 optical• Phonon - the quantum of lattice vibration. Energy ħω; momentum ħq• Concept of the phonon density of states • Einstein and Debye models for lattice heat capacity.Debye temperatureLow and high temperatures limits of Debye and Einstein models • Formula for thermal conductivity • Be able to obtain scattering wave vector or frequency from geometry and data for incident beam (x-rays, neutrons or light)CvlK31=6The Bragg LawConditions for a sharp peak in the intensity of the scattered radiationBragg formula: 2d·sinθ = mλ (m=1)Diffraction amplitude is determined by a product of several factors: atomic form factor, structural factorAtomic scattering factor (form factor): reflects distribution of electronic cloud.In case of spherical distribution Atomic factor decreases with increasing scattering angleStructure factorwhere the summation is over all atoms in unit cellrdenflia3)(rkr⋅∆∫=()∫⋅⋅=002∆∆sin)(4radrrkrkrnrfπ∑++=jlwkvhuiajjjjefF)(2π7 8Elastic propertiesElastic properties are determined by forces acting on atoms when they are displaced from the equilibrium positionsTaylor series expansion of the energy near the minimum (equilibrium position): ...)(21)()(0220000+−∂∂+−∂∂+= RRRURRRUURURRFor small displacements, neglect higher terms. At equilibrium, 00=∂∂RRUSo, 2)(20kuURU +=where 022RRUk∂∂=u = R - R0- displacement of an atom from equilibrium position9force F acting on an atom:kuRUF −=∂∂−=k - interatomic force constant. This is Hooke's law in simplest form.Valid only for small displacements. Characterizes a linear region in which the restoring force is linear with respect to the displacement of atoms. Elastic properties are described by considering a crystal as a homogeneous continuum medium rather than a periodic array of atoms In a general case the problem is formulated as follows:• Applied forces are described in terms of stressσ, • Displacements of atoms are described in terms of strainε.• Elastic constants C relate stressσand strainε, so thatσ= Cε.In a general case of a 3D crystal the stress and the strain are tensors10xyyxAF=σStress has the meaning of local applied “pressure”. Applied force F(Fx, Fy, Fz) → Stress componentsσij(i,j = 1, 2, 3)x ≡ 1, y ≡ 2, z ≡ 3Compression stress: i = j, i.eσxx, σyy, σzzxxxxAF=σShear stress: i ≠ j, i.eσxy, σyx, σxzσzx, σyz, σzyShear forces must come in pairs: σij= σji(no angular acceleration) ⇒ stress tensor is diagonal, generally has 6 components11In 3D case, introduce the displacement vector as u = uxx + uyy + uzzStrain tensor components are defined as jiijxu∂∂=εyuxxy∂∂=εxuxxx∂∂=εSince σijand σjialways applied together, we can define shear strains symmetrically:⎟⎟⎠⎞⎜⎜⎝⎛∂∂+∂∂==ijjiijijxuxu21εεSo, the strain tensor is also diagonal and has 6 components12Elastic stiffness (C) and compliance (S) constantsklklijklijCεσ∑=relate the strain and the stress in a linear fashion:This is a general form of the Hooke’s law.6 components σij, 6 εij→ 36 elastic constantsNotations: Cmnwhere 1 = xx, 2 = yy, 3 = zz, 4 = yz, 5 = zx, 6 = xyFor example, C11= Cxxxx, C12= Cxxyy, C44= CyzyzTherefore, the general form of the Hooke’s law is given byklklijklijSσε∑=13Elastic constants in cubic crystalsDue to the symmetry (x, y, and z axes are equivalent) C11= C22= C33 ;C12= C21= C13 = C31= C23= C32; C44= C55= C66 Also, the off diagonal shear components are zero: C45= C54= C46 = C64= C56= C65 and mixed compression/shear coupling does not occur:C45= C54= C46 = C64= C56= C65⇒ the cubic elastic stiffness tensor has the form:only 3 independent constants14Elastic constants in cubic crystalsLongitudinal compression (Young’s modulus): LuAFCxxxx==εσ11Transverse expansion:LyyxxCεσ=12φεσAFCxyxy==44Shear modulus:15Elastic wavesConsidering lattice vibrations three major approximations are made:• atomic displacements are small: u << a , where a is a lattice parameter• forces acting on atoms are assumed to be harmonic, i.e. proportional to the displacements: F = - Cu(same approximation used to describe a harmonic oscillator)• adiabatic approximation is valid – electrons follow atoms, so that the nature of bond is not affected by vibrationsThe discreteness of the lattice must be taken into account For long waves λ>> a, one may disregard the atomic nature –solid is treated as a continuous medium. Such vibrations are referred to as elastic (or acoustic) waves.16Elastic wavesFirst, consider a longitudinal wave of compression/expansionmass density ρsegment of width dx at the point x; elastic displacement u→⇒xxFAtuxx∂∂=∂∂=∂∂σρ122where F/A = σxxAssuming that the wave propagates along the [100] direction, can write the Hooke’s law in the formxxxxCεσ11=Since xuxxx∂∂=εget wave equation: 221122xuCtux∂∂=∂∂ρ17A solution of the wave equation - longitudinal