Practical KK transformation and anisotropy

Transcription

1 Optical spectroscopy in materials science 6. Practical KK transformation and anisotropy Kamarás Katalin MTA Wigner FK Optical spectroscopy in materials science 6. 1

2 Optical anisotropy χ, ε, n are tensors if the medium is not isotropic, there is different response to excitations of different polarization Polarization Quantity Notation linear n (linear) birefringence (lineáris) kettőstörés linear n (linear) dichroism (lineáris) dikroizmus circular n optical rotation circular n optikai forgatás Or: circular birefringence cirkuláris kettőstörés circular dichroism optical activity optikai aktivitás cirkuláris dikroizmus Optical spectroscopy in materials science 6. 22

3 Optical activity Optical rotatory dispersion (ORD): different index of refraction for left and right circularly polarized light polarization plane of linearly polarized light will rotate Circular dichroism: extinction coefficient different linearly polarized light changes into elliptical Optical spectroscopy in materials science 6. 3

4 Optical rotation angle of rotation: θ = ( n L ' n θ = α = ( n l R L ωl ') = αl 2c ω ' nr' ) = ( n 2c L ' n R ') π λ specific rotation (for solutions) : fajlagos forgatóképesség [ ] α = θ lc' Important! λ = 589 nm (Na D-line) T = 25 o C solvent: water [c ]: g/100 cm 3 [l]: dm Application: sugar concentration (health system, food industry) polarimetry, sacharimetry Optical spectroscopy in materials science 6. 4

5 Natural optical rotation Kézsmárki István, BME Fizika Tsz tartaric acid Pasteur (1849) Optical rotation Levo Dextro Borkősav Levo (+) Dextro (-) Levo Dextro Optical spectroscopy in materials science 6. 5

6 Circular dichroism Ellipticity: n different absorption different Δ ε ~ Ψ φ π = Ψ = ( n L " nr") l λ 100φ [ Ψ] = lc' specific ellipticity fajlagos ellipticitás: also used: difference in extinction (absorption) coefficient: (specific, molar) φ ellipticity, Ψ ellipticity created on 1 cm pathlength Δε = ε L ε R nl" nr" Summarized in one quantity: Φ = α + iψ = n L n ) Definition of R, L and sign is arbitrary! Δε( ω) = c π λ 0 4 log10 ( R Ψ( ω) Optical spectroscopy in materials science 6. 6

7 Cotton effect π Φ = α + iψ = n~ ~ L n λ ( R ) α and Ψ are related by Kramers-Kronig relations Positive Cotton effect: Ψ > 0 Negative: Ψ < 0 Optical spectroscopy in materials science 6. 7

8 Applications information about optical isomers (identical structure, except for rotation) resolution can be increased in some cases(sign!) magnetic dipole transitions can increase intensity ORD: in the whole frequency range CD: only around excitations Optical spectroscopy in materials science 6. 8

9 Measurement: dichrograph piezoelectric modulator Measured quantity: T log T ΔI/I ΔA Incident light linearly polarized (45 ) modulator changes index of refraction in one direction on applied voltage Garab Győző, SZBK Optical spectroscopy in materials science 6. 9

10 Optical isomerism Structural condition: chirality ( lack of symmetry element S n ) enantiomers Optical isomerism: enantiomers: can be transformed into each other by reflection across a plane racemic mixture (racém keverék): mixture of enantiomers in a 1:1 ratio diastereomers: isomers formed by the reaction of two chiral molecules (not enantiomers) Optical spectroscopy in materials science 6. 10

11 Optical isomerism Three chirality centers menthol Kajtár Márton: Változatok négy elemre Optical spectroscopy in materials science 6. 11

12 Separation of optical isomers Separation of optical isomers: symmetric synthesis diastereomers Natural substances (enzymes, amino acids): only one enantiomer exists (nature is chiral) Application: age determination (amino acid racemization) Optical spectroscopy in materials science 6. 12

13 Reflectance spectroscopy αd I 0 = RI0 + (1 R) I0e + I T 0 I A Sample Fresnel equations for normal incidence: E r n' + in' ' 1 r = = = E n' + in' ' + 1 i iθ Re R = I I ( n' 1) + n" 2 2 R = ( n' + 1) + n" I 0 Incident light Absorption Transmission I T I R Reflectance Reflectance spectroscopy Optical spectroscopy in materials science 6. 13

14 Dispersion relations - reflectance E r n' + in' ' 1 r = = = E n' + in' ' + 1 i iθ Re ln R ln r = + iθ egyenlet hibás! (-1-szerese) ln r is also a good transfer function, dispersion relations hold measured quantity: R(ω) Kramers-Kronig (KK) transformation: θ ( ω ) ω ln R ( ξ ) 1 ξ + ω d ln R( ξ ) = P dξ ln π = P 2 2 ξ ω 2π ξ ω dξ 0 0 dξ Contribution to θ: ξ-ω small or dlnr/dξ large Optical spectroscopy in materials science 6. 14

15 ω 0 : KK analysis: extrapolations known σ(dc): fitting, smoothing semiconductors, bound electrons: constant R ε =0, ε can be calculated from R ω : asymptotic limit (free electrons): ~ ω R ω p 4 intermediate region: reflectance of plasma (ω>10 6 cm -1, all electrons excited) S R ω ~ S (and even ω p ) arbitrary, because we only want ω p to approximate the integral, R ~ ω -S Optical spectroscopy in materials science 6. 15

16 ω 0 : KK analysis: extrapolations known σ(dc): fitting, smoothing semiconductors, bound electrons: constant R ε =0, ε can be calculated from R ω : asymptotic limit (free electrons): ~ ω R ω p 4 intermediate region: reflectance of plasma (ω>10 6 cm -1, all electrons excited) S R ω ~ S (and even ω p ) arbitrary, because we only want ω p to approximate the integral, R ~ ω -S Optical spectroscopy in materials science 6. 16

17 KK analysis: errors R 0, R n 1+ r = 1 r 1 Semiconductor Superconductor Optical spectroscopy in materials science 6. 17

18 KK analysis: procedure 1. ε or σ (any extrapolation) 2. starting parameters (Drude-Lorentz ω p, ω 0, γ) 3. fit to measured quantity (reflectance) 4. KK transformation with extrapolation from model frequency range: only measured range Optical spectroscopy in materials science 6. 18