Vibrations in molecules and solids

Transcription

1 University of Hamburg, Department of Earth Sciences Applications of representations theory in solid-state state physics and chemistry Vibrations in molecules and solids Boriana Mihailova

2 Goals Group theory analysis of vibrational spectra fun easy absolutely necessary! User-friendly approach to group-theory analysis without forgetting that Human being Clicking species = Thinking species

3 Outline. Atomic dynamics. Phonons 2. Brillouin zone centre spectroscopy. Raman and Infrared the most commonly used methods to study atomic dynamics 3. Group theory analysis (GTA) : - selection rules for conventional IR and Raman spectra by hand and using on-line tools - deviations from the GTA predictions - second order phonon processes; compatibility relations; hyper-raman scattering, a note about resonance Raman spectra 4. Electron states. Selection rules for optical transitions

4 References Rousseau, Bauman & Porto, Normal mode determination in crystals. J. Raman Spectrosc., (98) Fateley, McDevitt & Bentley, Infrared and Raman selection rules for lattice vibrations: the Correlation method. Appl. Spectrosc. 25, (97) M. Dresselhaus, G. Dresselhaus, A. Jorio, Group Theory. Application to the Physics of Condensed Matter. Springer

5 Atomic dynamics in crystals Visualization: UNISOFT, Prof. G. Eckold et al., University of Göttingen 6 KLiSO 4, hexagonal

6 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

7 Phonons Atomic vibrations in a periodic solid standing elastic waves normal modes ( S, {u i } s ) 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+/2)ħ, - m =, p = ħk (quasi-momentum), K q RL - integer spin Bose-Einstein statistics: n(,t)= /[exp(ħ/k B T)-] (equilibrium population of phonons at temperature T) Harmonic oscillator n=3 n=2 n= n= 2

8 Phonon frequencies and atom vector displacements a m K m 2 Atomic bonds elastic springs Hooke s low : m x Kx K m Equation of motion for a 3D crystal with N atoms in the primitive unit cell : 2 w D in a matrix form: ( ) w dynamical matrix 2 w (3N) D( q) q w q (3N3N) w D (3N) ss' ' ( q) ss' ' ( q) m m phonon S, {u i } s eigenvalues and eigenvectors of D = f (m i, K({r i }), {r i }) s s' =,2,3 i, q ii', ' q i,q i, q i' ', q mi i' ' atomic vector displacements u 2 D( q) δ w phonon S, {u i } s carry essential structural information! i =,..., N second derivatives of the crystal potential q

9 Types of phonons diatomic chain chain in D Acoustic phonon: u, u 2, in-phase Optical phonon: u, u 2, out-of-phase phonon dispersion: ac (q) op (q), 3D crystal with N atoms per cell : 3 acoustic and 3N 3optical phonons for q, op > ac induced dipole moment interact with light chain in 3D Brillouin zone Longitudinal: wave polarization (u) wave propagation (q) 2 Transverse: wave polarization (u) wave propagation (q) LA LO TA TO

10 Phonon (Raman and IR) spectroscopy electromagnetic wave as a probe radiation (photon opt. phonon interaction): Infrared absorption: photon E E ( phonon) ( phonon) ES GS, k excited state ground state Raman scattering inelastic light scattering from optical phonons anti-stokes Stokes, k i s, k s s k s Stokes i k K i anti-stokes i k k K s s i

11 Phonon (Raman and IR) spectroscopy only optical phonons near the BZ centre are involved k i k s K Kmax k 2ki (e.g. Raman, 8-scattering geometry) 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(2/) = hc(/) [cm - ].24 [mev] [cm - ].3 [THz] [Å].[cm - ] = 8 IR and Raman spectra are different for the same crystal different interaction phenomena different selection rules!

12 Raman and IR intensities IR activity: induced dipole moment due to the change in the atomic positions μ ( x y z,, ) Q k configurational coordinate μ μ( Q) μ Qk... Qk, IR activity I IR μ Q k 2 IR: asymmetrical, one-directional Raman activity: induced dipole moment due to deformation of the e - shell Polarizability tensor: α α α( Q) α Qk... Qk xx xy xz xy yy yz xz yz zz P =.E Raman induced polarization (dipole moment per unit cell) I e S α Q k e i 2, Raman activity Raman: symmetrical, two-directional N.B.! simultaneous IR and Raman activity only in non-centrosymmetric structures

13 Raman and IR activity in molecules sym. stretching bending anti-sym. stretching H 2 O 2 3 IR : change in Raman : change in pol. ellips. size (Tr()), shape ( ) and/or orientation ( ij, ij)

14 Raman and IR activity in crystals Isolated TO 4 group Crystal: Pb 3 (PO 4 ) 2, R3m a b Pb 2 a b Pb 2 c P O p c P O p Pb O t Pb O t Raman-active Raman-active IR-active a b Pb 2 a b Pb 2 c O p c O p P P Pb O t Pb O t IR-active Raman-active IR-active

15 Methods for normal phonon mode determination Three techniques of selection rule determination at the Brillouin zone centre: Factor group analysis (Bhagavantum & Venkatarayudu) the effect of each symmetry operation in the factor group on each type of atom in the unit cell Molecular site group analysis (Correlation method) 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 primitive 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) Bilbao Server, SAM, N.B.! Tabulated information for: first-order, linear-response, non-resonance interaction processes (one phonon only) (one photon only) (ħ i < E ES electron -E GS electron )

16 Spectroscopy and Group theory. The Philosophy normal phonon modes irreducible representations Lattice (molecular) dynamics = normal phonon modes obey the symmetry restrictions from all symmetry operations in the group symmetry-specific transformation upon different symmetry operations Crystals: Space groups specific point symmetry at each k Raman and IR k = (-point) Crystals, molecules: Point group Irreducible representations

17 Dynamical matrix in a diagonal form D ij 2 U u u i j u w i i (eigenvectors) 2 s (eigenvalues) single single doubly degenerate triply degenerate D D D D2 D xx xy xz xx nxz D D D D xy yy yz 2xy D D D D xz yz zz 2xz D D D D 2xx 2xy 2xz 22xx D D nnxz nnzz n u x u y uz u 2x u2 y u2z u nx uny unz u u u u u u u u u x y z x y 3 z 3 x y atomic displacement vectors for a given normal mode (symmetry of irrep) z symmetryallowed set of a.d.v. another symmetryallowed set 2 symmetryequivalent sets of a.d.v. 3 symmetryequivalent sets of a.d.v.

18 Dynamical matrix and Group Theory D commutates with all matrix forms of the mechanical representation m m D = D m m block-diag. form via irreps, i.e. S: S - m S = m = S S - S S - D = D S S - S- ( )S S - S S - DS = S - DS S - S S - DS = S - DS block-diagonalized form of D! D has two parts force-field F and geometry G F peak positions (exact values of eigenfrequencies) and intensities (exact values of the atomic displacement amplitudes) G mode degeneracy (relative values of eigen frequencies) mode symmetry (sets of relative atomic amplitudes)

19 Character tables normal phonon modes irreducible representations Reminder: Symmetry element: matrix representation A C 3 (3) ( 23) Character: Tr( A) i A ii r 3 r v (m) (232) reducible Point group C 3v (3m) irreducible A E Symmetry elements 3 m A + E 3 characters : 3: r Mulliken symbols m: reducible irreducible (block-diagonal)

20 Mulliken symbols A, B : D representations non-degenerate (single) mode only one set of atom vector displacements (u, u 2,,u N ) for a given wavenumber 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: 2D representation doubly degenerate mode two sets of atom vector displacements (u, u 2,,u N ) for a given wavenumber T (F): 3D representation triply degenerate mode three sets of atom vector displacements (u, u 2,,u N ) for a given wavenumber 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,2 (X, X 2 ) : symmetric or anti-symmetric to add. m or n 2D system Y X E mode 3D system Y Z X T mode biaxial crystals (tricl, monocl, orth) : only A, B uniaxial crystals (tetra, rhombo, hexa): A,B, E cubic: A, E, T

21 Symbols and notations Point groups: Symmetry element Schönflies notation International (Hermann-Mauguin) Identity E Rotation axes C n n =, 2, 3, 4, 6 Mirror planes m to n-fold axis to n-fold axis bisecting (2,2) h v d Triclinic Monoclinic Trigonal (Rhombohedral) m, m z m v, m d, m Inversion I Rotoinversion axes S n n,2,3,4, 6 Translation t n t n Screw axes k C n n k Glide planes g a, b, c, n, d (on the Bilbao Server: POINT) Tetragonal Hexagonal Cubic C C 2 2 C 3 3 C 4 4 C 6 6 T 23 C i C S m C 3i 3 S 4 4 C 3h 6 C 2h 2/m C 4h 4/m C 6h 6/m T h m 3 C 2v mm2 C 3v 3m C 4v 4mm C 6v 6mm D 3d 3 m D 2d 42m D 3h 6m2 T d 43m D D 3 32 D D O 432 D 2h mmm D 4h 4/mmm D 6h 6/mmm O h m3 m D n : E, C n ; nc 2 to C n ; T: tetrahedral symmetry; O: octahedral (cubic) symmetry

22 Spectroscopy and Group theory. General outlines Identify the symmetry operations defining the point group G in equilibrium Find the irrep decomposition of equivalence rep eq (atoms remaining invariant under the sym. operations of G); ( eq ) for each SO is the number of inv. atoms vibrations transformation properties of a radial vector r (x,y,z) mol.vibr. = eq vec trans rot different degrees of freedom latt.vibr. = eq vec, opt.latt.vibr. = eq vec acous ( ac (k=) = ) trans : simple translation rot : simple rotation about the centre of mass irreps of polar vector, e.g. r (x,y,z) irreps of axial vector, e.g. r p () C i () C i Infrared activity: transformation properties of a vector ( st -rank tensor) final vec initial totally sym irrep G x,y,z remain positive Raman activity: transformation properties of a 2 nd -rank symmetric tensor initial ' irrep x 2, y 2, z 2 Raman final, xy, zy, zx remain positive Lattice vibrations for k : GTA of the point group the corresponding k-point (N.B. ac (k) ) compatibility relations between neighbouring k-points to form a phonon branch

23 Constructing character tables Working example for simple point groups, C s (m), Point group sym. operations selection rules Z ( x, y, z) C s (m) A E () h (m) functions x, y, x 2, y 2, z 2, xy, J z X Y ( x, y, z) A z, xz, yz, J x, J y small rot. about Z does not change the symmetry change in sign irreps Two single irreps: totally sym A, antisym to reflection A (symmetry types of modes) IR dipole moments μ μ ( x y A') (,,) ( z A'') (,, ) Raman tensors α( A ') α( A '') a d e d b f c e f

24 Point group D 2 (222) A B B 2 B 3 irreps (modes) Constructing character tables Exercise: construct the character table for D 2 (222) sym. operations E () C 2x (2 x ) C 2y (2 y ) C 2z (2 z ) Dipole moments A : IR-inactive Polarizability tensors selection rules functions x 2, y 2, z 2 z, J z, xy y, J y, xz x, J x, yz ( x, y, z) C 2x (2 x ) μ B ) (,, ) μ B ) (,,) μ B ) (,,) ( z ( 2 y ( 3 x C 2z (2 z ) ( x, y, z) ( x, y, z) ( x, y, z) : 2 x : 2 y : 2 z : C 2y (2 y ) ( x, y, z) ( x, y, z) ( x, y, z) ( x, y, z) a α(a) b c B ) α( d d B ) α( 2 e e B ) α( 3 f f

25 Normal modes in molecules. H 2 O Exercise: construct the character table for H 2 O C 2 (2) y (m y ) Point group irreps sym. operations selection rules C 2 (mm2) E () C 2 (2) y (m y ) x (m x ) functions A A 2 B B 2 z, x 2, y 2, z 2 J z, xy x, J y, xz y, J x, yz E (): C 2 (2): y (m y ): x (m x ): x (m x ) Z Y X ( x, y, z) ( x, y, z) ( x, y, z) ( x, y, z) Characters for atomic site transformations 2 m y m x eq (H 2 O) 3 3 2A + B 2 atoms remaining invariant 3 irreps O H H O O O HH only 2A + B 2 either 2A + B 2 or A + 2B 2 mol.vibr. = eq vec trans rot mol.vibr. = (2A +B 2 ) (A +B +B 2 ) (A +B +B 2 ) (A 2 +B +B 2 ) = = 2A +2B +2B 2 +B 2 +A 2 +A A B B 2 A 2 B B 2 H2O = 2A (Raman, IR)+B 2 (Raman, IR)

26 Correlation method. Example H 2 O Using tabulated info: Character tables Correlation tables e.g. C 2 (2) y (m y ) x (m x ) X Z Y H 2 O: Point group C 2 O: site symmetry C 2 H: site symmetry C S = C h

27 Correlation method. Example H 2 O H 2 O: Point group C 2 O: site symmetry C 2 C 2 (2) y (m y ) x (m x ) H: site symmetry C S = C h X Z Y Oxygen Hydrogen Identify the translation modes: (vibration translation) O (C 2 ) : A + B + B 2 2H (C s ) : 2(2A + A )= 4A + 2A

28 Correlation method. Example H 2 O (C 2 ) Correlate the site group irreps with the point group irrep O (C 2 ) : A + B + B 2 the same 2H (C s ) : 4A + 2A 2(A + B 2 ) + A 2 + B H2O = A +B +B 2 + 2A +A 2 +B + 2B 2 (A +B +B 2 ) (A 2 +B +B 2 ) = 2A + B 2 O atom H atoms translation rotation

29 Normal modes in molecules. Bilbao Server: SAM O: (a) (,,z) H: (2g) (,y,z) (,-y,z) C 2 (2) y (m y ) x (m x )

30 Normal modes in molecules. Bilbao Server: SAM H 2 O C 2 (mm2) O: (a) (,,z) H: (2g) (,y,z) (,-y,z) Pmm2

31 Normal modes in molecules. Bilbao Server: SAM Point group symmetry operations selection rules H 2 O C 2 (mm2) O: (a) (,,z) H: (2g) (,y,z) (,-y,z) H O normal modes characters H2O : A +B +B 2 + 2A +A 2 +B +2B 2 (A +B +B 2 ) (A 2 +B +B 2 ) = 2A + B 2 O atom H atoms H 2 O translation H 2 O rotation sym. stretching bending anti-sym. stretching A A B 2

32 Normal modes in Crystals. Bilbao Server: SAM Working example: CaCO 3, calcite, R3c (67) D 6 3d (,,)+ (2/3,/3,/3)+ (/3,2/3,2/3)+ Ca: (6b) C : (6a).25 O : (8e) Structural information on

33 Normal modes in Crystals. Bilbao Server: SAM Calcite or input, then click Ca: (6b) C : (6a).25 O : (8e)

34 Normal modes in Crystals. Bilbao Server: SAM Calcite Point group number of operation of each class symmetry operations Raman-active selection rules silent (inactive) IR-active z x, y + acoustic rotation (inactive) xx = yy zz non-zero components xx = - yy xy xz yz normal modes characters Ca: (6b) : C : (6a) : O : (8a) : A u + A 2u + 2E u acoustic: A 2u +E u (the heaviest atom) A 2g + A 2u + E g + E u A g + A u + 2A 2g + 2A 2u + 3E g + 3E u Total: A + E = 3 3N = 3 (N = 6:3 +6:3+8:3 = ) opt = A g (R) + 2A u (ina) + 3A 2g (ina) + 3A 2u (IR)+ 4E g (R) + 5E u (IR) 5 Raman peaks and 8 IR peaks are expected

35 Spectra from

36 Normal modes in Crystals. Bilbao Server: SAM Practical exercise: number of expected Raman and IR peaks of aragonite CaCO 3, aragonite, Pnma (62) D 2h 6 Ca : (4c) C : (4c) O : (4c) O2 : (8d) Solution: opt = 9A g (R) + 6A u (ina) + 6B g (R) + 8B u (IR) + 9B 2g (R) + 5B 2u (IR) + 6B 3g (R) + 8B 3u (IR) 3 Raman peaks and 2 IR peaks are expected

37 Spectra of CaCO 3 Calcite: Raman-active = A g + 4E g Aragonite: Raman-active = 9A g + 6B g + 9B 2g + 6B 3g 22 observed

38 Normal modes in Crystals. Bhagavantam-Venkatarayudu Method Working example: Pm3 m O h A: (b): B: (a): O: (3d):.5 The perovskite structure type ABO 3 Magic formula n ( ) ( ) gr R g R crystal g ( R) number of modes for each symmetry species g R order of the point group G R g R number of elements for each class R crystal ( ) R ( R) w ( 2cos) the character for R and irreducible representation R character for a certain crystal cos + for proper rotation at = 36/n R sin - for improper rotation (rotation + reflection in h ) (E is proper rot. at, I improper rot at 8, h improper rot. at ) w R number of atoms remaining invariant under operation R sin cos

39 Normal modes in Crystals. Bhagavantam-Venkatarayudu Method. Determine w R (number of atoms invariant under R) and crystal (R) Number of atoms per unit cell: A: ; B: 8 /8 = ; O: 2 /4 = 3; total = 5; Z = ABO 3 Identity E (), g R = : 3 C 4 w R = 5, cr = 5(++2cos)=5(+2)=53= 5 4 C 4 2 C C C 4 four-fold axis C 4 (4), g R = 6 : N.B. we consider only one axis from all 6 equivalent as well as all axes parallel to this certain axis, which go through the unit cell w R = A + 4(2 /8 B + /4 O) = 3 cr = 3(++2cos9)=3(+2)= 3 = 3 4 C 3 3 C 3 C 3 2 C 3 three-fold axis C 3 (3), g R = 8 : 6 C 3 w R = A + 2 /8B + 6 /8 B = + =2 cr = 2(++2cos2)=3(-2/2)= 3 = 7 C 3 5 C 3

40 Normal modes in Crystals. Bhagavantam-Venkatarayudu Method 4 C 2 5 C 2 3 C 2 2 C C 2 2 two-fold axis C 2 (2), g R = 3 : w R = A + 4(2/8B+/4O) + 4( 2 /4O) = +2+2=5 cr = 5(++2cos8)=5(-2)= 5(-) = -5 9 C 2 8 C 2 6 C 2 7 C 2 2 C 2 3 C C 5 2 C C C 2 8 C 2 7 C 2 C 2 two-fold axis C 2 (2 ), g R = 8 : w R = ( A + 2/4O) + 2( 2 /8B) + +4(/8B) + 2(/4O) = +/2+/2+/2+/2=3 cr = 3(++2cos8)=3(-2)= 3(-) = -3 centre of inversion I ( ), g R = : w R = 5 (the same as for identity operation) cr = 5(-+2cos8)=5(--2)= 5(-3) = -5

41 Normal modes in Crystals. Bhagavantam-Venkatarayudu Method improper four-fold axis S 4 ( 4), g R = 6 : w R = 3 (the same as for proper four-fold rotation C 4 ) cr = 3(-+2cos9)=3(-+)= 3(-) = -3 improper three-fold axis S 6 ( 3 ), g R = 8 : w R = 2 (the same as for proper three-fold rotation C 3 ) cr = 2(-+2cos6)=2(-+)= 2 = 2 h h 3 h mirror plane h (m), g R = 3 : w R = 3(A+8/8B+2/4O) = 5 (all atoms invariant) cr = 5(-+2cos)=5(-+2)= 5 = 5 3 d d 2 d mirror plane d (m ), g R = 6 : w R = A+8/8B+4/4O = 3 cr = 3(-+2cos)=3(-+2)= 3 = 3

42 Normal modes in Crystals. Bhagavantam-Venkatarayudu Method 2. Apply the magic formula n ( ) ( ) gr R g R g = 48 ABO 3 crystal ( R) O h ( m3 m ) E () C 4 (4) C 2 (2) C 3 (3) C 2 (2 ) I ( ) S 4 ( 4 ) S 6 ( 3 ) h (m) d (m ) g R A g A u A 2g A 2u E g E u T g T u T 2g T 2u cr (R) A g : n = ( )/48 = no such mode A u : n = ( )/48 = no such mode likewise A 2g, A 2u, E g, E u, T g, T 2u : n () = no such modes T u : n = 4 T 2u : n = latt.vibr. (ABO 3 ) = 4T u + T 2u opt.vibr. (ABO 3 ) = 3T u + T 2u

43 Normal modes in Crystals. 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 (22) AB.5 B.5 O 3 Fm3m (225) O h O h A: (b): B /B : (a): O: (3d):.5 (,,)+ (,/2,/2)+ (/2,,/2)+ (/2,/2,)+ A : (8c) : B : (4a) : B : (4b) :.5 O : (24e):.255

44 Normal modes in Crystals. Bilbao Server: SAM A(B,B )O 3 A: (b): B: (a): O: (3d): Pm3 m (22) T u T u 2T u + T 2u Total: 5T = 5 3N (5 atoms) opt = 3T u (IR) + T 2u (ina) acoustic O h z y x BO 6 stretching (3 sets) BO 6 bending (3 sets) B-c. transl. (3 sets)

45 Normal modes in Crystals. Bilbao Server: SAM 5 AB.5 B.5 O 3 Fm3m (225) O h Exercise: determine the atom vector displacements for A g, E g, T 2g, and add. T u A : (8c): B : (4a): B : (4b): O: (24e): T u + T 2g T u acoustic T u T u A g + E g + T g + 2T u + T 2g + T 2u Total: A+E+ 9T = 3 3N (N=8:4+4:4+4:4+24:4) opt = A g + E g + T g (ina) + 3T u (IR) + T 2u (ina) A g E g T 2g T 2g T u z y x

46 Normal modes in Crystals. Bilbao Server: SAM Perovskite-type type structure ABO 3 : ferroelectric phases BaTiO 3 Pm3 m (22) O h cubic Ba: (a):... Ti: (b): O: (3c):.5.5 tetragonal orthorhombic rhombohedral P4mm (99) C 4v Ba: (a):... Ti: (b): O: (b): O2: (2c): Amm2 (38) C 2v (,,)+ (,/2,/2)+ Ba: (2a):... Ti: (2b): O: (2b): O2: (4c): R3m (6) C 3v Ba: (a):... Ti: (a): O: (b):

47 Normal modes in Crystals. Bilbao Server: SAM Perovskite-type type structure ABO 3 : ferroelectric phases BaTiO 3 Pm3m P4mm Amm2 R3m Ba: Ti: O: T u T u T u T u T 2u Ba: Ti: O: O2: A + E A + E A + E A + E B + E Ba: Ti: O: O2: A + B + B 2 A + B + B 2 A + B + B 2 A + B + B 2 A + A 2 + B 2 Ba: Ti: O: A + E A + E A + E A + E A 2 + E Polar modes: simultaneously Raman and IR active mode polarization along (~ u) LO: q TO: q xxz, yyz, zz z

48 Experimental geometry Infrared transmission (only TO are detectible) I IR 2 I inc I trans I meas or Z Y X polarizer z y Raman scattering I Raman 2 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 of the 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 yy E s E i E s E i E s yz zz n = x LO n = y, z TO yy n zz n yz n Y ( XY ) X Y ( XZ ) X Y ( ZY ) X Y ( ZZ ) X xy xz zy n xy n xz n zy zz zz n n = x,y LO+TO n = z TO

49 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 yy z xy μ (,,μ z) q μ A, E T 2 (LO) X k s X ( ZY ) Z Y Z E i E s x yz μ ( μ x,,) q x μ q z μ T 2 (LO+TO) k i q ( q x,,q z ) E i E s X ( ZX ) Z y xz μ (,μ y,) q μ T 2 (TO)

50 Experimental geometry non-cubic system e.g., rhombohedral, C 3v (3m) (hexagonal setting) Y X Z k i k s E i E s Z ( YY) Z z yy μ (,,μ z) q (,,q z) μ ( μ x,,) x yy A (LO) E(TO) Z Y E i E s y xy Z ( YX) Z E(TO) z 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 yy μ (,,μ z) z yy μ ( μ x,,) q ( q x,,) E(LO) A (TO)

51 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)} ( LO) ( TO) general rule: (the potential for LO: U+E; for TO: U) Cubic crystals: LO-TO splitting covalency of atomic bonding 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

52 LO-TO splitting LO-TO splitting: sensitive to local polarization fields induced by point defects Bi 4 Ge 3 O 2 :Mn I23 (T 3 ) I 432 Bi 2 SiO 2 :X Bi(24f), Si(2a), O(24f), O2(8c), O3(8c) cm -3 Bi(6c), Ge(2a), O(48e) (T d6 ) undoped Raman shift / cm - Raman shift / cm - a change in I LO /I TO depending on the type and concentration of dopant

53 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 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, )

54 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 ] 2 T > T C f m

55 Non-centrosymmetric crystals with compositional disorder LO-TO splitting + one-mode/two-mode behaviour: we may observe four peaks instead of one!

56 a a a b b b b b d d d d d d d d d d a a a b b b 2 d d d d d d 3 3 b b X Y Z X Y Z rot Z ( = U T U Intensity / a.u. Raman shift / cm - PSTcpp deg, p-pol. PSTdpp 45 deg, p-pol 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 2g ) F 2g (O) + F 2g (Pb) I(E g )

57 How GTA helps in complex systems Perovskite-type relaxor ferroelectrics (A,A )(B,B )O 3 chemical B-site order/disorder ferro-/paraelectric B B + dynamic PNR Diffraction methods: Pm3m average structure over time and space

58 GTA of Relaxors.. Peak assignment prototype structure Fm3m Pb : F 2g (R) B-cations : F u (IR) + F u (IR) O : A g (R) + E g (R) + F g (s) + 2F u (IR) + F 2g (R) + F 2u (s) Pb-localised BO 3 transl. vs Pb B-cation localised Pb-O bond str. BO 6 bending BO 6 stretching Pb-localised BO 3 transl. vs Pb B-localised Pb-O bond str. BO 6 bending BO 6 str. A-site B-site O F u p-pol E g A g F 2g c-pol bold: E i [] thin: E i [] F u F u F u F 2u F 2g PST PSN m(ta) > m(nb) f(ta-o) > f(nb-o)

59 GTA of Relaxors.. Temperature as variable X-ray diffraction hk Raman scattering F 2g phonon modes F u F u F 2u

60 GTA of Relaxors.. Temperature as variable allowed T a T m anomalous Pb-localised mode ~ 53 cm - T B T* T* T m intra-cluster inter-cluster B-localised mode ~ 24 cm -

61 GTA of Relaxors.. Pressure as variable X-ray diffraction 6 64 Raman scattering phonon modes F 2g F u F u F 2u

62 Second-order order phonon processes one photon two phonons, k excited state ground state low probability low intensity overtones: 2, 2 2, 2 3, combinational modes: 2, 3, 2 3, slight -deviation due to mode-mode interactions General rules: the symmetry of the II-order state = direct product of the I-order states overtones: i i combinations: i i (high-order states : further multiplication) modes away from the BZ-centre may contribute to II-order spectra e.g. phonon(-k) + phonon(+k) = state (k=) N.B.! acoustic (k), i.e. non- acoustic modes may also participate Benefits: inactive -point modes may be observed in IR and Raman spectra non--point modes may be observed in IR and Raman spectra

63 Second-order order phonon processes Reminder about the direct product rules: Product tables:

64 Second-order order phonon processes. Bilbao Server e.g. A A 2 = A 2 inactive; EE = A +A 2 +E Raman-active; IR-inactive

65 Second-order order phonon processes. Bilbao Server Raman Raman Raman, IR Raman, IR T d (43m) Dresselhaus et al, Springer, 28

66 Second-order order phonon processes. k Determine the point group at the corresponding k-point Find the corresponding normal modes and overtones and combinations e.g. overtones and combinations at R are the same as at

67 Second-order order phonon processes. k In some cases, the determination of non- overtones and combination modes may be far from straightforward, this holds especially for nonsymmorphic groups (having glide plane or screw axis) Bilbao Server, DIRPRO can help (Special Thanks to Alexander Litvinchuk, University of Houston)

68 Second-order order phonon processes. Bilbao Server

69 Second-order order phonon processes. Bilbao Server Scroll down to the end of the resulting file (the most important info) The phase Different nomenclature is confusing consider the matrices of S to recognize the symmetrical type, compare with matrix representation in POINT/SAM; be patient!

70 Compatibility relations Going away from -point point (applied to phonon and electron states): Symmetry of the wave-vector group may change - fewer elements of symmetry at k (if no glide planes or screw axes ) - more elements of symmetry at k (glide planes or screw axes ) The effect of a given symmetry element on the wave function must be continuous between the centre and the boundary of the Brillouin zone (the associated characters cannot jump ) k-branches must follow the compatibility relation principle Pb Kuzmany, Springer 998

71 Compatibility relations Working example: Consider the compatibility relations along the line X for Pm3m. Compare the character tables of (m-3m) and (4mm) 2. Compare the character tables of (4mm) and X (4/mmm) 3. Find which modes can develop along the X-direction

72 Compatibility relations Find the normal modes having the same characters for the corresponding symmetry elements in the two point groups Fully compatible modes: m-3m 4mm A g A A u A 2 A 2g B A g B 2 For degenerate modes possible combinations (sums of characters) must be considered e.g. E in 4mm is not compatible with modes in m-3m ((2) for 4mm for m-3m)

73 Compatibility relations X Fully compatible modes: 4mm 4/mmm A A 2 B B 2 E A g and A 2u A 2g and A u B g and B 2u B 2g and B u Eg and Eu

74 Compatibility relations Some modes split following the compatibility relations along the k-line Pm3m Dresselhaus et al, Springer, 28 When phonons (e - levels) of different symmetry approach one another: crossing: retain their original symmetry after crossing When phonons (e - levels) of the same symmetry approach one another: anticrossing: admixed wave functions in an appropriate linear combination phonon-phonon (e - -e - ) interaction

75 Hyper-Raman Scattering Non-linear process P i E ij j 2 ijk E jek 6 ijkle jekel..., k i ) Stokes 2 s k s 2k i i K anti-stokes s 2i k k K s 2 s, k s HR spectra of a perovskite-type relaxor i Hehlen et al, PRB 27 ijk -3 rd -rank tensor β β( Q) β Qk... Qk ijk = ikj If E j E k crystallographic axis ijk = ijj ijj, hyper-raman activity 33 non-symmetric matrix xxx yxx zxx xyy yyy zyy xzz yzz zzz hyperpolarizability tensor

76 Hyper-Raman Scattering Selection rules: according to the transformation properties of a 3 rd -rank tensor Outlines: IR-active modes are always HR-active, no matter of the crystal symmetry In centrosymmetric systems only the odd modes ( u ) may be HR-active Ramanactive modes are not hyper-raman-active and vice versa in non-centrosymmtrical systems some types of modes may be both Raman and HR-active modes inactive in both Raman and IR spectra may be HR-active the nonsymmetrical part of the HP tensor which is symmetrical in two indices gives rise to additional HR-active modes as compared to the completely symmetrical tensor Denisov et al., Phys.Rep. 5 (987) -92

77 Hyper-Raman Scattering V.N.Denisov, B.N. Mavrin, V.P. Podobedov, Phys.Rep. 5 (987) -92

78 Hyper-Raman Scattering V.N.Denisov, B.N. Mavrin, V.P. Podobedov, Phys.Rep. 5 (987) -92

79 Hyper-Raman Scattering V.N.Denisov, B.N. Mavrin, V.P. Podobedov, Phys.Rep. 5 (987) -92

80 Resonance Raman Scattering, k i s k s Stokes i k K i anti-stokes i k k K s s i e - ES s, k s dielectric constant Intensity / a.u conduction E g valence r i 2.7eV (457nm) 3.eV (43nm) E g (PST) parallel-polarised Energy / ev 3.55eV (35nm) 3.2eV (387.5nm) 457 nm 43 nm 35 nm 334 nm 3.7eV (334nm) RRS Raman shift / cm - Selection rules for both e - and phonon states () Fundamental and II-order harmonics may be enhanced LO are strongly enhanced

81 Electron states GTA concept : the same for phonon and electron states D s 2 E n u s n Atomic orbitals: spherical harmonics: m m Y, ) N P (cos ) e l ( l, m l im m r l Yl (, ) Normalization coefficient Legendre polynomial linear combination of simple Cartesian coordinate functions l = : ; l = : x, y, z; l = 2 : xy, yz, zx, x 2 -y 2, 3z 2 -r 2, s-like p-like d-like, 3d s 2p 4f

82 Electron states Cubic crystal field environment for a paramagnetic transition cation Crystal field theory: how the atomic surroundings affects the electron energy levels 3d d x 2 -y 2 d z 2 two-fold level d xy d xz d yz m Y (, ) 2 five-fold level three-fold level

83 Electron states d x 2 -y 2 d z 2 the electron lobes are toward the anions add. repulsive forces destabilization d xy d xz d yz o : c : d : t = : -8/9 : -/2 : -4/9 vice versa octahedron (6-fold polyhedron): t 2g stabilized, e g destabilized; cube (8-fold), dodecahedron (2-fold), tetrahedron (4-fold): opposite

84 Electron states Character tables: e.g. basic functions for e - states irreducible representations splitting of levels: according to transformation rules of irreps Optical absorption processes, k electric dipole transitions IR selection rules transformation properties of a polar vector final vec initial totally sym irrep G dipole operator

85 Optical transitions. Symmetry selection rules Working example: Determine the allowed electric dipole transitions in O h from T 2g irrep for the dipole operator: T u T u T 2g = A 2u + E u + T u + T 2u (see product tables) check for each irrep if contains A g A g (A 2u +E u +T u +T 2u ) = (A 2u +E u +T u +T 2u ) A u (A 2u +E u +T u +T 2u ) = (A 2g +E g +T g +T 2g ) A 2g (A 2u +E u +T u +T 2u ) = (A u +E g +T 2u +T u ) A 2u (A 2u +E u +T u +T 2u ) = (A g +E g +T 2g +T g ) E g (A 2u +E u +T u +T 2u ) = (E u +A u +A 2u +E u +T 2u +T u +T 2u +T u ) E u (A 2u +E u +T u +T 2u ) = (E g +A g +A 2g +E g +T 2g +T g +T 2g +T g ) T g (A 2u +E u +T u +T 2u ) = (T 2u +T u +T 2u +A u +E u +T 2u +T u +A 2u +E u +T 2u +T u ) T u (A 2u +E u +T u +T 2u ) = (T 2g +T g +T 2g +A g +E g +T 2g +T g +A 2g +E g +T 2g +T g ) T 2g (A 2u +E u +T u +T 2u ) = (T u +T u +T 2u +A 2u +E u +T 2u +T u +A u +E u +T 2u +T u ) T 2u (A 2u +E u +T u +T 2u ) = (T g +T g +T 2u +A 2g +E g +T 2g +T g +A g +E g +T 2g +T g ) no no no yes no yes no yes no yes

86 Optical transitions. Symmetry selection rules Tabulated info on transition products

87 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 ~ 2-3 nm, time scale ~ -2 s) Experimental difficulties (low-intensity peaks, hardly resolved peaks) What should we do before performing a Raman or IR experiment? Do group-theory analysis