University of Hamburg, Institute of Mineralogy and Petrology Raman and IR spectroscopy in materials science. Symmetry analysis of normal phonon modes Boriana Mihailova
Outline. The dynamics of atoms in crystals. Phonons. Raman and IR spectroscopy : most commonly used methods to study atomic dynamics 3. Group theory analysis : phonon modes allowed to be observed in IR and Raman spectra
Atomic dynamics in crystals Visualization: UNISOFT, Prof. G. Eckold et al., University of Göttingen 6 KLiSO 4, hexagonal
Crystal normal modes (eigenmodes( eigenmodes) Atomic vibrations in crystals = Superposition of normal modes (eigenmodes) e.g., a mode involving mainly S-O t bond stretching a mode involving SO 4 translations and Li motions vs K atoms
Phonons Atomic vibrations in a periodic solid standing elastic waves normal modes (ω S, {u i } s ) λ crystals : N atoms in the primitive unit cell vibrating in the 3D space 3N degrees of freedom finite number of normal states quantization of crystal vibrational energy N atoms 3 dimensions 3N phonons phonon quantum of crystal vibrational energy phonons: quasi-particles (elementary excitations in solids) -E n = (n+/)ħω, - m =, p = ħk (quasi-momentum), K q RL - integer spin Bose-Einstein statistics: n(ω,t)= /[exp(ħω/k B T)-] (equilibrium population of phonons at temperature T) Harmonic oscillator n=3 n= n= n= ψ ψ
Phonon frequencies and atom vector displacements a m K m Atomic bonds elastic springs Hooke s low : m x = Kx ω = K m Equation of motion for a 3D crystal with N atoms in the primitive unit cell : ω w = D in a matrix form: ( ) w dynamical matrix ω w (3N ) = D( q) q w q (3N 3N) D (3N ) w = ss' αα ' ( q) = Φ ss' αα ' ( q) m m phonon ω S, {u i } s eigenvalues and eigenvectors of D = f (m i, K({r i }), {r i }) s s' α =,,3 iα, q ii', αα ' q iα,q iα, q i' α ', q mi i' α ' atomic vector displacements u ( D( q) ω δ) w = phonon ω S, {u i } s carry essential structural information! i =,..., N second derivatives of the crystal potential q
Types of phonons diatomic chain chain in D Acoustic phonon: u, u, in-phase chain in 3D Optical phonon: u, u, out-of-phase phonon dispersion: ω ac (q) ω op (q), for q, ω op > ω ac 3D crystal with N atoms per cell : 3 acoustic and 3N 3optical phonons induced dipole moment qa interact with light qa Longitudinal: wave polarization (u) wave propagation (q) Transverse: wave polarization (u) wave propagation (q) LA LO TA TO
Phonon (Raman and IR) spectroscopy electromagnetic wave as a probe radiation (photon opt. phonon interaction): Infrared absorption: ω photon = E E ( phonon) ( phonon) ES GS ω, k excited state ground state Raman scattering inelastic light scattering from optical phonons anti-stokes Stokes ω ι, k i ω s, k s ω s k = ω i s = i Stokes Ω k K anti-stokes = i + Ω k k + K ω s ω s = i
Phonon (Raman and IR) spectroscopy only optical phonons near the FBZ centre are involved k i k s = K Kmax = Δk ki (e.g. Raman, 8 -scattering geometry) π λ i (IR, vis, UV) ~ 3 5 Å k i ~ -5-3 Å K max Kmax << (a ~ Å) a photon-phonon interaction only for K spectroscopic units: cm - E= ħck = ħc(π/λ) = hc(/λ) [cm - ].4 [mev] [cm - ].3 [THz] [Å].[cm - ] = 8 IR and Raman spectra are different for the same crystal different interaction phenomena different selection rules!
Raman and IR intensities IR activity: induced dipole moment due to the change in the atomic positions μ = ( x y z μ, μ, μ ) μ μ Q) = μ + Q +... Q ( k k Q k configurational coordinate, IR activity IR: asymmetrical, one-directional Raman activity: induced dipole moment due to deformation of the e - shell α Polarizability tensor: α = α α α α( Q) = α + Qk +... Q k xx xy xz α α α xy yy yz α α α xz yz zz, Raman activity Raman: symmetrical, two-directional P = α.e induced polarization (dipole moment per unit cell) N.B.! simultaneous IR and Raman activity only in non-centrosymmetric structures
Raman and IR activity in crystals Isolated TO 4 group Crystal: Pb 3 (PO 4 ), R3m a b Pb a b Pb c P O p c P O p Pb O t Pb O t Raman-active Raman-active IR-active a b Pb a b Pb c O p c O p P P Pb O t Pb O t IR-active Raman-active IR-active
Methods for normal phonon mode determination Three techniques of selection rule determination at the Brillouin zone centre: Factor group analysis the effect of each symmetry operation in the factor group on each type of atom in the unit cell Molecular site group analysis symmetry analysis of the ionic group (molecule) site symmetry of the central atom + factor group symmetry Nuclear site group analysis site symmetry analysis is carried out on every atom in the unit cell set of tables ensuring a great ease in selection rule determination preliminary info required: space group and occupied Wyckoff positions Rousseau, Bauman & Porto, J. Raman Spectrosc., (98) 53-9 Bilbao Server, SAM, http://www.cryst.ehu.es/rep/sam.html N.B.! Tabulated information for: first-order, linear-response, non-resonance interaction processes (one phonon only) (one photon only) (ħω i < E ES electron -E GS electron )
Symbols and notations Point groups: Symmetry element Schönflies notation International (Hermann-Mauguin) Identity E Rotation axes C n n =,, 3, 4, 6 Mirror planes σ m to n-fold axis to n-fold axis bisecting (,) σ h σ v σ d m, m z m v, m d, m Inversion I Rotoinversion axes S n n =,,3,4, 6 Translation t n t n Screw axes k C n n k Glide planes σ g a, b, c, n, d Triclinic Monoclinic Trigonal (Rhombohedral) Tetragonal Hexagonal Cubic C C C 3 3 C 4 4 C 6 6 T 3 C i C S m C 3i 3 S 4 4 C 3h 6 C h /m C 4h 4/m C 6h 6/m T h m 3 C v mm C 3v 3m C 4v 4mm C 6v 6mm D 3d 3 m D d 4m D 3h 6m T d 43m D D 3 3 D 4 4 D 6 6 O 43 D h mmm D 4h 4/mmm D 6h 6/mmm O h m3 m D n : E, C n ; nc to C n ; T: tetrahedral symmetry; O: octahedral (cubic) symmetry
Symbols and notations Symbols and notations normal phonon modes irreducible representations Symmetry element: matrix representation A C 3v (3m) r 3 r r C 3 (3) σ v (m) Point group 3 m 3: Character: = i A ii ) ( Tr A Symmetry elements : m: reducible irreducible (block-diagonal) 3 reducible characters 3 3 irreducible A E - A + E 3 Mulliken symbols Reminder:
Mulliken symbols A, B : D representations non-degenerate (single) mode only one set of atom vector displacements (u, u,,u N ) for a given wavenumber ω A: symmetric with respect to the principle rotation axis n (C n ) B: anti-symmetric with respect to the principle rotation axis n (C n ) E: D representation doubly degenerate mode two sets of atom vector displacements (u, u,,u N ) for a given wavenumber ω T (F): 3D representation triply degenerate mode three sets of atom vector displacements (u, u,,u N ) for a given wavenumber ω subscripts g, u (X g, X u ) : symmetric or anti-symmetric to inversion superscripts, (X, X ) : symmetric or anti-symmetric to a mirror plane m subscripts, (X, X ) : symmetric or anti-symmetric to add. m or C n D system Y 3D system Y E mode X Z X T mode
Bilbao Crystallographic Server, SAM Working example: CaCO 3, calcite, R3c (67) D 6 3d (,,)+ (/3,/3,/3)+ (/3,/3,/3)+ Ca: (6b) C : (6a).5 O : (8e).568.5 http://www.cryst.ehu.es/ or google Bilbao server or input, then click
Bilbao Crystallographic Server, SAM Calcite Ca: (6b) C : (6a).5 O : (8e).568.5
Bilbao Crystallographic Server, SAM Calcite Point group number of operation of each class symmetry operations selection rules Raman-active IR-active μ z μ x, μ y α xx = α yy α zz non-zero components + acoustic normal modes (xz,yz) α xx = -α yy α xy α xz α yz rotation (inactive) characters Ca: (6b) : C : (6a) : O : (8a) : A u + A u + E u acoustic: A u +E u (the heaviest atom) A g + A u + E g + E u A g + A u + A g + A u + 3E g + 3E u Total: A + E = 3 3N = 3 (N = 6:3 +6:3+8:3 = ) Γ opt = A g (R) + A u (ina) + 3A g (ina) + 3A u (IR)+ 4E g (R) + 5E u (IR) 5 Raman peaks and 8 IR peaks are expected
Spectra from 5 4 3
Spectra of Calcite from Hamburg, Γ Raman-active = A g + 4E g Γ IR-active = 3A u + 5E u
Bilbao server, SAM Practical exercise: number of expected Raman and IR peaks of aragonite CaCO 3, aragonite, Pnma (6) D h 6 Ca : (4c).446.5.45 C : (4c).858.5.76 O : (4c).9557.5.94 O : (8d).876.47347.6865 Solution: Γ opt = 9A g (R) + 6A u (ina) + 6B g (R) + 8B u (IR) + 9B g (R) + 5B u (IR) + 6B 3g (R) + 8B 3u (IR) 3 Raman peaks and IR peaks are expected
Spectra of CaCO 3 Calcite: Γ Raman-active = A g + 4E g Aragonite: Γ Raman-active = 9A g + 6B g + 9B g + 6B 3g observed
Bilbao server, SAM Perovskite-type type structure ABO 3 double perovskite-type single perovskite-type chemical B-site disorder chemical : B-site order 5 A(B,B )O 3 Pm3 m () AB.5 B.5 O 3 Fm3m (5) O h O h A: (b):.5.5.5 B /B : (a): O: (3d):.5 (,,)+ (,/,/)+ (/,,/)+ (/,/,)+ A : (8c) :.5.5.5 B : (4a) : B : (4b) :.5 O : (4e):.55
Bilbao server, SAM A(B,B )O 3 A: (b): B: (a): O: (3d): Pm3 m () T u T u T u + T u Total: 5T = 5 3N (5 atoms) Γ opt = 3T u (IR) + T u (ina) acoustic O h z y x BO 6 stretching (3 sets) BO 6 bending (3 sets) B-c. transl. (3 sets)
Bilbao server, SAM 5 AB.5 B.5 O 3 Fm3m (5) O h Exercise: determine the atom vector displacements for A g, E g, T g, and add. T u A : (8c): B : (4a): B : (4b): O: (4e): T u + T g T u acoustic T u T u A g + E g + T g + T u + T g + T u Total: A+E+ 9T = 3 3N (N=8:4+4:4+4:4+4:4) Γ opt = A g + E g + T g (ina) + 3T u (IR) + T u (ina) A g E g T g T g T u z y x
Bilbao server, SAM Perovskite-type type structure ABO 3 : ferroelectric phases BaTiO 3 Pm3 m () O h cubic Ba: (a):... Ti: (b):.5.5.595 O: (3c):.5.5 tetragonal orthorhombic rhombohedral P4mm (99) C 4v Ba: (a):... Ti: (b):.5.5.595 O: (b):.5.5 -.5 O: (c):.5..489 4 Amm (38) C v (,,)+ (,/,/)+ Ba: (a):... Ti: (b):.5.5.55 O: (b):.5.5.9 O: (4c):.5.64.48 5 R3m (6) C 3v Ba: (a):... Ti: (a):.4853.4853.4853 O: (b):.588.588.85
Bilbao server, SAM Perovskite-type type structure ABO 3 : ferroelectric phases BaTiO 3 Pm3m P4mm Amm R3m Ba: Ti: O: T u T u T u T u T u Ba: Ti: O: O: A + E A + E A + E A + E B + E Ba: Ti: O: O: A + B + B A + B + B A + B + B A + B + B A + A + B Ba: Ti: O: A + E A + E A + E A + E A + E Polar modes: simultaneously Raman and IR active mode polarization along μ (~ u) LO: q μ TO: q μ α xxz, α yyz, α zz z
Experimental geometry Infrared transmission (only TO are detectible) I IR μ α I inc I trans I meas or Z Y X polarizer μ z μ y Raman scattering I Raman α αβ back-scattering geometry (k i = k s +q, E is always to k) right-angle geometry Z Z Y Y X X k i Porto s notation: A(BC)D A, D - directions of the propagation of incident (k i ) and scattered (k s ) light, B, C directions of the polarization incident (E i ) and scattered (E s ) light k s k i k s (q x,q y,) (q x,,) X ( YY) X X ( ZZ ) X X ( YZ) X E i α yy E s E i E s E i E s α yz α zz n = x LO n = y, z TO α yy n α zz n α yz n Y ( XY ) X Y ( XZ ) X Y ( ZY ) X Y ( ZZ ) X α xy α xz α zy α n xy α n xz α n zy α zz α zz n n = x,y LO+TO n = z TO
Experimental geometry cubic system e.g., T d ( ) 43m Y X Z k s k i k i = k s + q q = (,,q z) E i E s E i E s Z ( YY) Z Z ( YX) Z α yy z α xy μ = (,,μ z) q μ A, E T (LO) X k s X ( ZY ) Z Y Z E i E s x α yz μ = ( μ x,,) q x μ q z μ T (LO+TO) k i q = ( q x,,q z ) E i E s X ( ZX ) Z y α xz μ = (,μ y,) q μ T (TO)
Experimental geometry non-cubic system e.g., trigonal, C 3v (3m) (hexagonal setting) Y X Z k i k s E i E s Z ( YY) Z z α yy μ = (,,μ z) q = (,,q z) α μ = ( μ x,,) x yy A (LO) E(TO) Z Y E i E s y α xy Z ( YX) Z E(TO) z α zz μ = (,μ y,) q = (,,q z) k s Y ( ZZ) Y μ = (,,μ ) A X k z (TO) i q = (,q y,) E i E s Y Y Z k s X ( Y ' Y ') X X k i E i E s contribution from x α yy α μ = (,,μ z) z yy μ = ( μ x,,) q = ( q x,,) E(LO) A (TO)
LO-TO splitting More peaks than predicted by GTA may be observed (info is tabulated) Cubic systems: LO-TO splitting of T modes: T(LO) + T(TO) Non-cubic systems: {A(LO),A(TO)}; {B(LO),B(TO)}; {E(LO),E(TO)} ω ( LO) > ω( TO) general rule: (the potential for LO: U+E; for TO: U) Cubic crystals: LO-TO splitting covalency of atomic bonding Δω LO-TO : larger in ionic crystals, smaller in covalent crystals Uniaxial crystals: if short-range forces dominate: TO A LO TO E LO if long-range forces dominate : TO LO A E A E Raman shift (cm - ) Raman shift (cm - ) under certain propagation and polarization conditions quasi-lo and quasi-to phonons of mixed A-E character
LO-TO splitting LO-TO splitting: sensitive to local polarization fields induced by point defects Bi 4 Ge 3 O :Mn I3 (T 3 ) I 43 Bi SiO :X Bi(4f), Si(a), O(7f), O(8c), O3(8c).899 8 cm -3 Bi(6c), Ge(a), O(48e) (T d6 ).476.94 undoped Raman shift / cm - Raman shift / cm - a change in I LO /I TO depending on type and concentration of dopant
One-mode / two-mode behaviour in solid solutions different types of atoms in the same crystallogr. position, e.g. (B -x B x )O y two-mode behaviour: two peaks corresponding to pure B -O and B -O phonon modes intensity ratio I(B -O )/I(B -O ) depends on x covalent character of chemical bonding : short correlation length relatively large difference in f(b /B -O) and/or m(b /B ) one-mode behaviour: one peak corresponding to the mixed B -O/B -O phonon mode ~ lineal dependence of the peak position ω on dopant concentration x ionic character of chemical bonding : long correlation length similarity in r i (B /B ), f(b /B -O) and m(b /B ) intermediate classes intermediate classes of materials: two-mode over x (, x m ) and one-mode over x (x m, )
One-mode / two-mode behaviour in solid solutions one-mode behaviour PbSc.5 (Ta -x Nb x ).5 O 3 two-mode behaviour Pb 3 [(P -x As x )O 4 ] T > T C ω = f m
Non-centrosymmetric crystals with compositional disorder LO-TO splitting + one-mode/two-mode behaviour: we may observe four peaks instead of one!
a a a b b b 3 3 b b d d d d d d d d d d a a a b b b d d d d d d 3 3 b b X Y Z X Y Z rot Z (ϕ=45 ) α = U T αu 3 4 5 6 7 8 9...4.6.8...4 Intensity / a.u. Raman shift / cm - PSTcpp deg, p-pol. PSTdpp 45 deg, p-pol. 3 4 5 6 7 8 9..5..5. Intensity / a.u. Raman shift / cm - PSTccp deg, c-pol. PSTdcp 45 deg, c-pol. Transformation of Transformation of polarizability polarizability tensors tensors A g (O) + E g (O) I(F g ) F g (O) + F g (Pb) I(E g )
Conclusions Group theory: predicts the number of expected IR and Raman peaks one needs to know: crystal space symmetry + occupied Wyckoff positions Deviations from the predictions of the group-theory analysis: LO-TO splitting if no centre of inversion (info included in the tables) one-mode two-mode behaviour in solid solutions local structural distortions (length scale ~ -3 nm, time scale ~ - s) Experimental difficulties (low-intensity peaks, hardly resolved peaks) What should we do before performing a Raman or IR experiment?