Raman scattering basics(很难)

Pedestrian Approach to Raman Scattering Optics Group Meeting _April 2004 S. Guha • Classical and Semiclassical Approach (including resonance Raman scattering) • Raman Microscopy • Raman Imaging/mapping • Applications (stresses in semiconductors, crystallographic orientation, chemical composition, mineralogy, biological applications, biomedical applications, art, jewelry, and forensic science) • Surface-Enhanced Raman Scattering (SERS) Later in the semester (or summer) © S. Guha Scattering Liquid Observer Sunlight (white) Violet Filter Green Filter Raman Scattering-inelastic process 1 in 108 photons get inelastically scattered Excitations can be plasmons, polaritons, magnons, phonons (thermal vibrations). Invention of Laser 1960 (Townes & Schawlow) CCD camera 1969 -in market since 1983 Holographic* early-mid 90’s Notch filters High power diode lasers Impact on Raman spectroscopy Inspector Raman: fits in your hand!!*Lippmann discovered in 1891 and was awarded the Nobel prize. These reflect 90% of the laser excitation and transmits 90% of the scattered light. Quantum mechanically......... Phonon is emitted into the lattice Phonon is absorbed from the lattice hωf hωL Rayleigh Scattering Stokes Scattering Anti-Stokes Scattering hωf hωsc hωL hωsc hωschωL Phenomenological introduction to Raman scattering qi (incident light) Eie-iωit Sample Ese-iωst qs (scattered light) ∑ −= β ω βαβα ω .),()( tiL LeEuPtM r Incident light induces a dipole moment---- Polarizability of the medium (electronic) The incident frequency (visible photons) is 2 to 3 orders of magnitude higher than the phonon frequencies, hence the massive atoms do not directly respond to light!! The induced dipole moment is electronic in origin. Pαβ will depend upon the incident frequency ωL. Expand Pαβ above the equilibrium configuration u=0: (1) ∑ +   ∂ ∂+= γ αβ αβαβ γω l tlu u P PuP ......);()0(),( 0 r u(l,α) is the displacement of the lth atom in γth direction. For simplicity assume only the fth normal mode is excited, then ( ) == tfltlu fωγχγ cos)|(; ( ) ).)(|(21; titi ff eefltlu ωωγχγ −+= Amplitude ( ) ........)|( )(2 1)0(),( 0 ++   ∂ ∂+= −∑ titi l ff eefl lu P PuP ωω γ αβ αβαβ γχγω r (2) (3) ( ) ........ )|( )(2 1)0()( (( 0 ++   ∂ ∂+= −−+−− ∑∑∑ titiL l ti L fLfLL eeEfl lu P eEPtM ωωωωβ β γ αβ β ω βαβα γχγ (4) ( ) ........)|( )(2 1 )0()( (( 0 ++   ∂ ∂ += −−+− − ∑∑ ∑ titi L l ti L fLfL L eeEfl lu P eEPtM ωωωω β β γ αβ β ω βαβα γχγ Origin of inelastic scattering----- Rayleigh Anti Stokes Stokes (5) Higher-order terms in Equation (5) gives rise to higher-order Raman scattering, where two or more phonons are involved. ∂Pαβ/∂u(lγ)|0 transforms as a third rank tensor under operation; this leads to the result that under inversion symmetry ∂Pαβ/∂u(0γ)|0=0 at an inversion site. For Raman scattering to occur, polarizability of the molecule must change during a vibration. Selection rules for Raman spectra + Polarization of a diatomic molecule in an electric field +δ -δ – hν Mx My Mz Pxx Pxy Pxz Pyx Pyy Pyz Pzx Pzy Pzz Ex Ey Ez       =             Polarizability tensor O O O C OC O O C O O C OC O O C O O C OO C O O C O +q q = 0 -q ν1- Raman active ν2 ν3 P/1 Plot Infrared absorption and Raman scattering 0 50 100 150 200 250 300 Frequency (cm-1) IR Raman -300 -200 -100 0 100 200 300 Raman shift (cm-1) )exp( TkI I Bstokes antiStokes ωh−= Energy units for optical spectroscopy hω=hck=2πhcλ k ≡ wave vector; 1/λ = ν/c = wavenumber 1/λ ∝ energy 1 eV = 8067.5 cm-1 30 GHz = 1 cm-1 Typical energies: 100 -10,000 cm-1 (1012-1014 Hz) Conservation rules for a Raman process fSL fSL kkk r h r h r h hhh += += ωωω Gkk f rrr =+∆ where G is the reciprocal lattice vector and Ls kkk rrr −=∆ Ls LfLs kk rr =∴ ≈+= ωωωω 15103.1 4880 22 −×≈°== cmAk LL π λ πr For visible light 181022 −×≈= cm a G πr 1510,0 −== cmkandG f rrHence, Raman Brillouin (cm-1) qphonon ωf π 0.5 nm optic acoustical Silicon and GaAs Selection Rules In centrosymmetric crystals, the vibrational modes must have either even (gerade) or odd parity (ungerade) under inversion. Odd parity modes: IR active Even parity modes: Raman active Diamond structure with inversion symmetry. Noncentrosymmetric zinc blende structure Phonon dispersion curves in Si and GaAs LO: LONGITUDINAL OPTICAL PHONON TO: TRANSVERSE OPTICAL PHONON LA: LONGITUDINAL ACOUSTIC PHONON TA: TRANSVERSE ACOUSTIC PHONON TO: Raman active (not IR active) TO modes: Both Raman and IR active Administrator 文本框 纵向光学声子 Administrator 文本框 横向光学声子 Administrator 文本框 纵向声学声子 Administrator 文本框 横向声学声子 Administrator 文本框 横向声学声子 Administrator 文本框 横向声学声子 Administrator 文本框 纵向声学声子 Administrator 文本框 纵向声学声子 Administrator 文本框 纵向光学声子 Administrator 文本框 纵向光学声子 Administrator 文本框 横向光学声子 Administrator 文本框 横向光学声子 C60 Raman IR What determines Raman intensities? For a classical dipole moment M(t) located at the origin and oscillating at frequency ω, the magnitude of the Poynting vector associated with radiation polarized along a unit vector n at position R is given by ∑= αβ βαβαπ ω )()( 4 32 4 tMtMnn cR S ∑= αβ βαβαπ ω )()( 4 32 4 tMtMnn cR S Average of S is (Use Eq. (1)): [ ][ ] )( )()( ** ** βλαγ γλ βλαγλγ ω βλ ω βλ γλ ω αγ ω αγλγβα PPPPEE ePePePePEEtMtM LL titititi LL ∑ ∑ += ++= −− System is at the origin, subjected to ω; for short time intervals the system’s config. u is fixed, and just the electronic system responds. (6) (7) ( )∑ += αβγλ λγβλαγβλαγβαπ ω LL EEPPPPnncR S **32 4 4 For the radiated power per unit solid angle, one needs to multiply the above equation by R2dΩ ( )∑= αβγλ λγβλαγβα ωωπ ω LL EEuPuPnnc uI ),(),( 2 )( *3 4 (8) In Raman experiments we watch for times long compared to vibrational period, so one must take a thermal average of the above expression over the vibrational levels: ,),(),( 2 )( *3 4 ∑= αβγλ λγβλαγβα ωωπ ω LL EEuPuPnnc uI (9) .|),(|''|),(|1 ' * νωννων βλ νν αγ β ν uPuPe Z E vib ∑ −= TkB/1=β The Raman tensor is defined as…… )(|),(|''|),(|1)( ' ' * ννβλ νν αγ β αβγλ ωωδνωννωνω ν −= ∑ − uPuPeZi Evib Zvib = partition functionα,β = for incident light and γ,λ=for scattered light ν= initial vibrational state ν’=final vibrational state (10) ∫ ∞+∞− −−=− dte it )(' '21)( ννωωνν πωωδ Use: ∫ ∑∞+∞− −−= ' * |),(|''|),(|1 2 1)( ' νν βλ ω αγ βω αβγλ νωννωνπω ννν uPeuPe Z dtei itE vib ti (11) A thermal average in the above equation means that the system is free to vibrate For any vibrational state---- ,νν νEH = .// νν ν hh tiEiHt ee = (12) Using the Heisenberg representation of the operator hh //)( iHtiHt AeetA −= ∫ ∞+∞− −= νωννωνπω βλαγωαβγλ |),(|''|),,(|21)( * uPtuPdtei ti (13) Eq. (13) is the common starting point for Raman calculations. It expresses the Raman scattering tensor as the Fourier transform of the time correlation function. Transform to normal coordinates ∑ = = N f fdfu 3 1 .)(χr (14) The Hamiltonian here is a harmonic oscillator problem--- f f ff ddH 222 2 1∑ += ω& (15) ∫ ∑ ∑ ∞+ ∞− −       +++ = ')( )()0()0()0()0( 2 1)( , ' * , * ,, ** fff ff f f ffff ti dtdPP tdPPdPPPP dtei βλαγ βλαγβλαγβλαγ ω αβγλ πω Using the normal coordinates---- (16) Rayleigh term The vibrational states (phonon states) are given by { } { }' 321 ...... i iN n nnnn =′ == ν ν (17) 0 ! )( 00 1 11 f n f f f ffff ffff n a n a nnna nnna + + = = −= ++= Use the creation, annihilation operators--- (18) ( )( )[ ] .1 2 )( , * , dteenen PP i ti f ti f ti f f ff ff ωωωβλαγ αβγλ ωω − +∞ ∞− −∫∑ ++= h One obtains---- Using the definition of the Dirac delta function.. ( )( )[ ].)()(1 2 )( , * ,∑ −+++= f ffff f ff nn PP i ωωδωωδωω βλαγ αβγλ h The Raman tensor is…… (19) Stokes Anti Stokes The intensity of Raman scattering per unit solid angle due to a transition from a vibrational state ν to ν′ .)( 2 )( * 2,1 3 3 λγαβγλ αβ γλ βα ωηηπ ωωω EEi c I k kkSL S ∑∑∑ = = (20) Summation over k denotes the two mutually perpendicular unit vectors (perp. to the direction of scattering) Fourier component of the electric field 1 1)exp( −    −≡ Tk n B f f ωh 100 200 300 400 500 600 700 1000 2000 3000 4000 60000 70000 80000 λ= 785 nm I n t e n s i t y Raman Shift (cm-1) λ= 514.5 nm CdS nanocrystal -1400 -1200 -1000 -800 -600 -400 -200 0 1000 1500 2000 2500 3000 3500 Y A x i s T i t l e Raman Shift (cm-1) Anti Stokes λ =514.5 nm CdS nanocrystal 2nd order 200 300 400 500 600 700 8000 10000 12000 14000 16000 18000 I n t e n s i t y Raman Shift (cm -1 ) CdS polycrystalline λ =785 nm April 04 Cds polycrystalline sample grown using pulsed laser deposition Incident λ =514.5 nm Confocal micro- Raman system ....);( 2 1 );()0(),( 2 0 2 2 0 ∑ ∑ +     ∂ ∂+ +   ∂ ∂+= γ αβ γ αβ αβαβ γ γω l l tlu u P tlu u P PuP r Hyper-Raman Effect (nonlinear Raman spectroscopy) EM P=Earlier we used only ..... 6 1 2 1 32 +++= EEEM γβP First and second hyper-polarizability CW lasers: E ~104 Vcm-1 (α>>β>>γ) Pulsed Nd-YAG: E ~109 Vcm-1 When a sample is illuminated with a giant pulse of freq. of ωL, the scattered radiation contains: 2 ωL(hyper-Rayleigh) 2 ωL± ωf (Stokes and anti-Stokes hyper-Raman) Selection rules are relaxed; mode is hyper-Raman active if the components of the hyper- polarizability tensor changes. Normal Raman: 10-8 photons scattered Hyper-Raman: 10-12 photons scattered Examples of in vivo skin Raman spectra obtained from various body locations of a healthy volunteer. (A) palm of the hand; (B) fingernail; (C) surface of the forearm; (D) volar aspect of the forearm. (Ex=785 nm) Peak position (cm-1) Protein Assignments Lipid Assignments Others 1745 ν(C=O) 1655 ν(C=O) Amide I 1445 δ(CH2), δ(CH3) δ(CH2) scissoring 1301 δ(CH2) twisting,wagging 1269 ν(CN),δ(NH) Amide III 1080 ν(CC) skeletal ν(CC),νs(PO2-) 1030 ν(CC) skeletal nucleic acids 1002 ν(CC) Phenyl ring 938 νCC) proline,valine 855 δ(CCH) aromatic,olefinic polysaccar ide 822 δ(CCH) aliphatic Near IR Raman spectroscopy for in vivo skin measurement From: Z. Huang et al. BC cancer Research center Microscopic Theory of Raman Scattering Quantum mechanical Hamiltonian of the coupled radiation field and the scattering medium is: electrons ipeppeiie HHHHHHH +++++= ions photonselectron- ion electron- photon ion- photon Treated exactly within the adiabatic approx. negligible ( )( )++ +=∑ kkkk k p aaaakH ω2 1 Photon field is treated exactly Hep between the mixed electronic vibrational states is treated by perturbation theory. Outline: How to tackle Hep? • quantize the radiation field----leads to a vector potential and electric-field operators •include the radiation field in the electronic Hamiltonian. •alternately one can express the electron-radiation in terms of electric field and magnetic field operators. •use either the A.p approach or the M.E approach )( jjj rAepp rrr +→ For periodic systems with extended states it is convenient to use the A.p formalism. Using Fermi’s golden rule, the transition rate is: Γ = − + −∑2 2π ε δh h c b b a ia bb a b | | | | ( ), H H E E E Eint int '1 kk nnfc −= 0'== kk nniaInitial state Initial matter state Initial photon state Final photon state Final State ( ) EE EE M.EM.E + EE M.EM.E iLfS Si Lik kSL if if n V S −−++− −−  =Γ ∑∑ ωωδω ωωω ππ hhh ll h llh h l l l 2 2 | |22 (22) (23) Scattering cross section ratio of the rate at which energy is removed from the incident beam by the scattered beam to the rate at which energy in the beam crosses a unit area perpendicular to its propagation direction. σ ωω≡ = h h k k k k c n V V n c Γ Γ ; Spectral differential cross section: Ωdd d Sω σ2 is the rate of removal of energy from the incident beam as a result of its scattering in volume into a solid angle element dΩ with a scattered frequency between ωs and ωs+dωs, divided by the product of dΩdωs with the incident beam intensity R. Loudon, “Quantum Theory of Light” (1983) (24) Scattering cross section cont. ( ) ( ) .22 3 3 3 2 3 Ω=Ω→ ∫ ∫∑ ∫ ∫ ddcVddkkV SSk SSS ω ω ππ Convert the summation to an integral--- ( ) ( ).||||+ |||| AV 2 ' ' 42 3 ' 2 if Si Li i SL S iMMf iMMf cdd d ωωδωω ωω ωω ω σ ηη ηη ηη −+ −=    Ω ∑ l l l ll ll h Kramers-Heisenberg formula Resonant termAnti-resonant term (25) Electric-dipole -incident pol. Electric-dipole -scattered pol. Resonant term: ∑ −−′l l ll γωω ηη i iMMf Li |||| = Res Expand the states as a product of the electronic and a vibrational function. BA= Res + For first-order Raman scattering: A u u= − −∑ ∑M M gv ev ev gvigee eg ev gv Levη η ω ω γ' ' ,'( ) ( ) ' ' | ' ' | ,0 0 B u u u u u u u u = − − + − − ∑∑ [ ( ) ( )| ' ' | ' ' | | ( )| ( ) ' ' | | ' ' | ]. ' ' ' ' ,' ' ' ' , M M gv ev ev gv i M M gv ev ev gv i ge eg ev gv Leve ge eg ev gv L η η η η ω ω γ ω ω γ 0 0 0 0 FRANCK-CONDON HERZBERG-TELLER (more imp. in solids with extended wavefunctions) 0|''|'' =gvevevgv FRANCK CONDON OVERLAP "ν ν ν 'ν 0|''|'' ≠gvevevgv A-term = 0 A-term: non-zero (totally symmetric modes) Resonance Raman profile---K6C60 0 100 200 300 400 λ=775 nm I n t e n s i t y Raman Shift (cm -1 ) 500 cm-1 0 100 200 300 400 I n t e n s i t y Raman Shift (cm-1) 500 cm -1 λ=660 nm 0 100 200 300 400 λ=560 nm I n t e n s i t y Raman Shift (cm-1) 500 cm-1 Resonance Raman profiles---K6C60 F-C term: ( ) 22 2 11 1 Γ−−+Γ−−∝ iEiE ω β ω βωα hh β1 and β1 characterize the overlap of the vibrational wave functions.