Raman and IR spectroscopy in materials science. Raman and IR - PowerPoint PPT Presentation

University of Hamburg, Institute of Mineralogy and Petrology Raman and IR spectroscopy in materials science. Raman and IR spectroscopy in materials science. Symmetry analysis of normal phonon modes Symmetry analysis of normal phonon modes Boriana Mihailova

Outline Outline 1. The dynamics of atoms in crystals. Phonons 2. 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 Atomic dynamics in crystals Visualization: UNISOFT Visualization: UNISOFT , Prof. G. Eckold et al., University of Göttingen 6 KLiSO KLiSO 4 4 , hexagonal , hexagonal

Crystal normal modes (eigenmodes eigenmodes) ) Crystal normal modes ( 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 Phonons Atomic vibrations in a periodic periodic solid � standing elastic waves ≡ normal modes ( ω S , { u i } s ) λ crystals : N atoms in the primitive unit cell vibrating in the 3 D space � 3N degrees of freedom � finite number of normal states � quantization of crystal vibrational energy N atoms × 3 dimensions ↔ 3 N phonons phonons phonon ≡ quantum of crystal vibrational energy phonon Harmonic oscillator phonons: quasi-particles (elementary excitations in solids) n=3 - E n = (n+1/2) ħ ω , n=2 - m 0 = 0, p = ħ K (quasi-momentum), K ≡ q ∈ RL n=1 - integer spin n=0 Bose-Einstein statistics: n( ω ,T)= 1/[exp( ħ ω /k B T)-1] ψ ψ 2 (equilibrium population of phonons at temperature T)

Phonon frequencies and atom vector displacements Phonon frequencies and atom vector displacements a Atomic bonds ↔ elastic springs m 1 K m 2 K = − � � ω = Hooke’s low : m x Kx m Equation of motion for a 3 D crystal with N atoms in the primitive unit cell : 1 ∑ α = 1,2,3 = i = 1,..., N ω = 2 w u ( q ) w D w α α , q , q i i α αα α , q ' , ' ' ' , q m i ii i i α ' ' i atomic vector displacements 1 = Φ ( q ) ( q ) D αα αα ' ' ' ' dynamical matrix ss ss m m ' s s second derivatives of the crystal potential ( ) in a matrix form: ω = ⋅ − ω 2 ⋅ = 2 w D ( q ) w D ( q ) δ w 0 q q q (3 N × 1) (3 N × 3 N ) (3 N × 1) � phonon ω S , { u i } s ↔ eigenvalues and eigenvectors of D = f ( m i , K ({ r i }), { r i }) � phonon ω S , { u i } s carry essential structural information !

Types of phonons Types of phonons diatomic chain chain in 1D Acoustic phonon: u 1 , u 2 , in-phase Acoustic chain in 3D Optical phonon: u 1 , u 2 , out-of-phase Optical phonon dispersion: ω ac ( q ) ≠ ω op ( q ), for q ≈ 0 , ω op > ω ac 3 D crystal with N atoms per cell : qa 3 acoustic and 3 N – 3 optical phonons induced dipole moment qa � interact with light 1 Longitudinal : wave polarization ( u ) || wave propagation ( q ) 1 Longitudinal Transverse : wave polarization ( u ) ⊥ wave propagation ( q ) 2 Transverse 2 LA LO LA LO TA TO TA TO

Phonon (Raman and IR) spectroscopy Phonon (Raman and IR) spectroscopy electromagnetic wave as a probe radiation ( photon – opt. phonon interaction ): ω = − ( ) ( ) � phonon phonon E E Infrared absorption: Infrared absorption: photon ES GS excited state ω , k ground state ≡ inelastic light scattering from optical phonons Raman scattering ≡ Raman scattering Stokes anti-Stokes ω s , k s ω ι , k i Stokes anti-Stokes ω = ω − Ω ω = ω + Ω s i s i = + = − k k K k k K s i s i

Phonon (Raman and IR) spectroscopy Phonon (Raman and IR) spectroscopy • only optical phonons near the FBZ centre are involved − = = Δ ≈ (e.g. Raman, 180 ° -scattering geometry) k k K � 2 K k k max i s i π λ i (IR, vis, UV) ~ 10 3 – 10 5 Å � k i ~ 10 -5 – 10 -3 Å ≈ K max << ( a ~ 10 Å) K � max a ≈ 0 � photon-phonon interaction only for K cm -1 � E= ħ ck = ħ c(2 π / λ ) = hc(1/ λ ) • spectroscopic units: 10 [cm -1 ] � 1.24 [meV] 10 [cm -1 ] � 0.30 [THz] [Å].[cm -1 ] = 10 8 • IR and Raman spectra are different for the same crystal different interaction phenomena � different selection rules !

Raman and IR intensities Raman and IR intensities IR activity : induced dipole moment due to the change in the atomic positions IR activity = μ μ μ μ ( , , ) x y z ∂ μ ∑ = + + μ ( ) μ ... Q k – configurational coordinate Q Q 0 ∂ k Q k ≠ 0, IR activity IR: “asymmetrical”, “one-directional” Raman activity : induced dipole moment due to deformation of the e - shell Raman activity ⎛ ⎞ α α α ⎜ ⎟ xx xy xz P = α .E = α α α ⎜ ⎟ α Polarizability tensor: xy yy yz ⎜ ⎟ α α α induced polarization ⎝ ⎠ xz yz zz ∂ α (dipole moment per unit cell) ∑ = + + α ( ) α ... Q Q 0 ∂ k Q k ≠ 0, Raman activity Raman: “symmetrical”, “two-directional” N.B.! simultaneous IR and Raman activity – only in non-centrosymmetric structures

Raman and IR activity in crystals Raman and IR activity in crystals Isolated TO 4 group Crystal: Pb 3 (PO 4 ) 2 , R 3 m Pb 2 a b Pb 2 a b O p O p c c P P Pb 1 O t Pb 1 O t Raman-active Raman-active IR-active Pb 2 Pb 2 a b a b O p O p c c P P Pb 1 Pb 1 O t O t IR-active Raman-active IR-active

Methods for normal phonon mode determination 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 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. 10 , (1981) 253-290 � � Bilbao Server, SAM, http://www.cryst.ehu.es/rep/sam.html Bilbao Server, SAM, N.B.! Tabulated information for: first-order , linear-response , non-resonance interaction processes ( ħ ω i < E ES electron ) electron - E GS ( one phonon only ) ( one photon only )

Symbols and notations Symbols and notations Symmetry element Schönflies notation International (Hermann-Mauguin) Identity 1 E Rotation axes n = 1, 2, 3, 4, 6 C n σ Mirror planes m ⊥ to n -fold axis σ h m, m z σ v || to n -fold axis m v , bisecting ∠ (2,2) σ d m d , m’ Inversion I 1 Rotoinversion axes = 1 , 2 , 3 , 4 , 6 S n n Translation t n t n Screw axes C n k k n σ g Glide planes a , b , c , n , d Point groups: Triclinic Monoclinic Trigonal Tetragonal Hexagonal Cubic (Rhombohedral) C 1 1 C 2 2 C 3 3 C 4 4 C 6 6 23 T C 3 i S 4 C 3 h 1 3 4 6 C i C S m C 2 h 2/ m C 4 h 4/ m C 6 h 6/ m T h m 3 C 2 v mm 2 C 3 v 3 m C 4 v 4 mm C 6 v 6 mm D 3 d D 2 d 42 m D 3 h 6 m 2 3 4 3 T d m m D 2 222 D 3 32 D 4 422 D 6 622 432 O D 2 h D 4 h 4/ mmm D 6 h 6/ mmm O h mmm m 3 m D n : E , C n ; nC 2 ⊥ to C n ; T : tetrahedral symmetry; O : octahedral (cubic) symmetry

Symbols and notations Symbols and notations normal phonon modes ↔ irreducible representations C 3 (3) Symmetry element: matrix representation A Reminder: ∑ Character: = σ v ( m ) Tr A ( ) A ii r 3 i r 1 Point group Symmetry elements characters r 2 C 3 v (3m) 1 3 m ⎛ ⎞ 1 0 0 ⎛ 1 0 0 ⎞ ⎜ ⎟ ⎜ ⎟ 3 0 1 1: 0 1 0 ⎜ ⎟ 0 1 0 ⎜ ⎟ reducible ⎜ ⎟ ⎜ ⎟ 0 0 1 0 0 1 ⎝ ⎠ ⎝ ⎠ ⎛ ⎞ 1 1 1 ⎜ ⎟ A 1 irreducible 1 0 0 ⎜ ⎟ ⎛ ⎞ 0 1 0 1 3 ⎜ ⎟ ⎜ ⎟ 2 -1 0 − E 0 3: ⎜ ⎟ 0 0 1 ⎜ ⎟ 2 2 ⎜ ⎟ ⎜ ⎟ 3 1 1 0 0 ⎝ ⎠ ⎜ ⎟ − − 0 3 0 1 A 1 + E ⎝ ⎠ 2 2 ⎛ ⎞ 1 0 0 ⎛ ⎞ 1 0 0 ⎜ ⎟ ⎜ ⎟ m : 0 0 1 ⎜ ⎟ 0 1 0 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 0 1 0 ⎝ ⎠ − 1 0 0 ⎝ ⎠ Mulliken symbols reducible irreducible (block-diagonal)

Mulliken symbols symbols Mulliken A, B : 1 D representations ↔ non-degenerate (single) mode only one set of atom vector displacements ( u 1 , u 2 ,…, 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: 2 D representation ↔ doubly degenerate mode two sets of atom vector displacements ( u 1 , u 2 ,…, u N ) for a given wavenumber ω T (F): 3 D representation ↔ triply degenerate mode three sets of atom vector displacements ( u 1 , u 2 ,…, u N ) for a given wavenumber ω subscripts g, u (X g , X u ) : symmetric or anti-symmetric to inversion 1 superscripts ’,” (X’, X”) : symmetric or anti-symmetric to a mirror plane m subscripts 1,2 (X 1 , X 2 ) : symmetric or anti-symmetric to add. m or C n 2 D system 3 D system Y Y X Z X T mode E mode