Fundamentals Of Raman Spectroscopy Part1 13Nov2012 - Universiteit Twente

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13‐11‐2012 Fundamentals of Raman spectroscopy Part1 Cees Otto Vibrational Spectroscopy IR absorption spectroscopy 1‐photon effect Light scattering Stokes Raman scattering anti‐Stokes Raman scattering 2‐photon effect 1

13‐11‐2012 Nanometers, wavenumbers and relative wavenumbers Absolute wavenumbers: Example: 500 nm corresponds to 20000 cm-1 Relative wavenumbers: Example: A Raman band at 1020 cm‐1 and excited with a laser wavelength of 500 nm scatters light at a wavelength of 527 nm. A vibration that absorbs light at 1020 cm‐1 absorbs light in the infrared at a wavelength of 9.8 µm (Check everything!) The harmonic oscillator The Schrödinger equation for the harmonic oscillator describes atomic vibrations in molecules, which are also called molecular vibrations: k For a di‐atomic molecule: m2 m1 1 2 1 1 1 m1 m2 k The frequency of the vibration depends on the “spring” Constant, k, and the reduced mass, µ, of the atoms involved in the motion. The frequency is expressed in cm-1 1 cm-1 30 GigaHz Hz cm 1 c cm / s Motions of light atoms (C-H, N-H, O-H) have high frequencies: 2700-4000 cm-1 Motions of heavy atoms (S-S- bridges) have low frequencies: 600 cm-1 Motions involving C, N, O can be anywhere between 600 - 2700 cm-1 Tables relate chemical groups with vibration frequencies are available in the literature 2

13‐11‐2012 Raman spectroscopy 1: 2: 3: Band positions Band width Amplitude of a band (related to Raman cross section) A comparison between cross sections Electronic (UV-Vis) Absorption spectroscopy: Fluorescence spectroscopy: Vibrational (IR) absorption spectroscopy: Resonance Raman spectroscopy: Non-resonant Raman spectroscopy: Surface Enhanced Raman Scattering: 10-20 m2 Q x 10-20 m2 10-23 m2 10-29 m2 10-33 m2 10-? m2 Surface enhanced Raman scattering cross sections vary widely in literature reports. There seems to be consensus developing that estimates the SERS cross sections between 6 to 8 orders of magnitude larger than the “normal” non-resonant and resonant Raman cross sections. So: Surface Enhanced Raman Scattering: 10-21 to 10-27 m2 Reported literature values are even as high as 10-16 m2. This requires further research. 3

13‐11‐2012 Fluorescence “cross section” n det abs N mol fl P Q fl E det A abs 10 20 m 2 Fluorescence imaging exc 457 nm. 128 x 128 pixels image time : 8 seconds power: 5 microwatt A(rea): 2.4*10‐13 m2 Edet : 6 ‐ 13 % Intensity: 107 W/m2 Flux density: 1026 1/(m2s) n det 105 N mol fl Raman cross section nRdet R 3m 6 N mol P E det A R 10 33 m 2 (non res.) A(rea): 2.4*10‐13 m2 Edet : 6 ‐ 13 % Raman imaging exc 647 nm. 64 x 64 pixels image time : 1800 seconds power: 60 miliwatt Status:2003 Intensity: 1011 W/m2 Flux density: 1030 1/(m2s) nRdet 3m 6 10 4 N mol 10 photons/day.mol.vib 4

13‐11‐2012 The harmonic oscillator The parabolic potential for a classical oscillation The energy states of a QM harmonic oscillator form a “ladder” Ev v 1 2 h 0 Transitions can only occur between consecutive states: 1 E Ev 1 Ev h 0 From Atkins In non‐linear molecules with “n” atoms 3n‐6 vibrations exist. Each vibration has a distinct force constant and reduced mass and, therefore, frequency. Raman spectra contain information on the vibration frequencies. Chemical analysis is possible because the frequencies are unique for molecular groups. Band positions Hooke’s law: F k x with “k”, the force constant. “x” is the displacement from the equilibrium position The relation between “force” and “potential energy” is: F Follows: V 12 k x 2 a parabolic potential V x The S. eq. for the harmonic oscillator becomes: 2 2 x 1 2 2 k x x E x 2 x 2 The wave functions: x2 x 2 v x N v H v e 2 The eigenvalues: Ev v 1 2 h 0 1 0 2 v 0,1, 2, 3, . k 2 m k 1 4 The energy difference between adjacent levels is: E Ev 1 Ev h 0 5

13‐11‐2012 The wavefunctions of the harmonic oscillator x2 x 2 v x N v H v e 2 v 0,1, 2, 3, . with Hv the Hermite polynomials (See page 340, table 12.1). α is proportional to the steepness of the parabolic function, meaning 2 14 large α shallow potential well, small α steep potential well. The normalization constant Nv is: Fig. 8.19 The graph of the Gaussian function 1 N v 1 v 2 2 v! Fig. 8.20 the wavefunction and It’s square for the v 0 From Atkins 1 2 mk Fig. 8.21 the wavefunction and It’s square for the v 1 From Atkins Hermite polynomials The wave functions for the harmonic oscillator look as follows: x2 x 2 v x N v H v e 2 v 0,1, 2, 3, . The polynomial functions H v are a power series of the argument and take the following form (with y x/α): x H v y 1 Hv y 2 y Hv y 4 y 2 2 H v y 8 y 3 12 y H v y 16 y 4 48 y 2 12 H v y 32 y 5 160 y 3 120 y v 0 v 1 v 2 v 3 v 4 v 5 See page 302-306 for more details. 6

13‐11‐2012 Some properties of the harmonic oscillator x 0 is the equilibrium position of two nuclei connected with a “spring”. The expectation value for x and x2, respectively x and x 2 is: x v 0 x 2 v 12 mk Interpretation: The average value of the position is equal to the equilibrium constant, so the oscillator is equally likely to be found on either side of the equilibrium position. The mean square displacement is proportional to the vibrational quantum number. Fig. 8.23 p. 304 The probability distribution for the first 5 states of a harmonic oscillator and the state v 20. Note how the regions of highest probability move towards the turning points of the classical motion as “v” increases. From Atkins Molecular vibrations in DNA measured with Raman spectroscopy 1488 1578 1420 1530 1579 1483 1536 1426 1356 1264 1290 1093 1180 1211 1245 1577 1489 1538 1424 1328 1240 1317 1316 782 810 793 C 625 1334 1362 1177 1218 1259 1293 830 681 1093 B 597 Raman intensity 783 1181 1359 824 675 616 1094 A 1259 786 Poly(dG-dC).poly(dG-dC): C) Z-form: Watson Crick LH-helix B) B-form: Watson-Crick RH-helix A) Hoogsteen basepairing at pH 4.3 596 685 Changes in physico-chemical Parameters (pH, T, etc.) gives 400 800 1200 1600 rise to profound changes in wavenumber / cm the Raman spectrum. G.M.J. Segers‐Nolten, N.M. Sijtsema, C. Otto, Biochemistry 36(43), 13241‐13247 (1997) -1 7

13‐11‐2012 Hooke’s law F Band positions V k x x i For 3n‐6 vibrations k The force constant, ki, are obtained from : 1 2 ki i 2 V x2 Substituting Hooke’s law in the Newton equation: 2 x k x t2 And solving the equation of motion results in: x t A sin 2 i t with i 1 2 ki i How does this work out for N atoms? Band Positions The dynamic model for molecular vibrations may start from N individual atoms. Each of these atoms has a (x,y,z) Cartesian coordinate system attached. Displacement of each atom along any of its coordinates may affect other atoms along their (x,y,z) coordinates through the spring constant. For a three‐atomic z molecule nine (9) degrees of freedom exist. Three degrees of freedom are z connected to translations of the center y z x of mass of the molecule (translations of the molecule), y y Three degrees of freedom are connected x x to the rotation of the molecule around the center of mass (rotations of the molecule). 3N‐6 degrees of freedom are connected to internal motion of atoms with respect to the center of mass. These are the molecular vibrations. In a linear molecule 3N‐5 vibrations exist. The description of vibrations in cartesian coordinates of the atoms is not chemically intuitive. A coordinate systems based on internal coordinates is more suited. 8

13‐11‐2012 Band positions Let us first look at a description in Cartesian coordinates. First substitute the solution to the Newton equation back into it. We obtain: 4 2 2 x k x N atoms have 3N degrees of freedom and 3N equations of motion result. The motion of each atom in the x‐, y‐ and z‐direction is driven back to equilibrium by a spring constant and by coupling, via spring constants, to each of the other degrees of freedom. For a three‐atomic molecule we obtain 9 coupled equations of motion: 11 11 12 12 13 13 13 4 2 2 1 x1 k xx x1 k xy y1 k xz11 z1 k xx x2 k xy y2 k 12 xz z 2 k xx x3 k xy y3 k xz z 3 11 11 12 12 12 13 13 13 4 2 2 1 y1 k 11 yx x1 k yy y1 k yz z1 k yx x2 k yy y 2 k yz z 2 k yx x3 k yy y3 k yz z3 12 12 12 13 13 13 4 2 2 1 z1 k zx11 x1 k zy11 y1 k 11 zz z1 k zx x2 k zy y 2 k zz z 2 k zx x3 k zy y3 k zz z3 4 2 2 3 z3 k zx31 x1 k zy31 y1 k zz31 z1 k zx32 x2 k zy32 y2 k zz32 z 2 k zx33 x3 k zy33 y3 k zz33 z3 Band positions The notation is much more transparent in matrix form using mass‐weighted coordinates and force constants: k xy12 xi mi xi k 12 xy m1 m2 The equations become: 4 9

13‐11‐2012 Band positions Introducing the following shorthand notation for the Cartesian coordinates of the atoms: ξ the equation becomes: 4 2 2 k This is an eigenvalue equation and can be solved by standard means once one knows the matrix of force constants. The solution contains the 3N frequencies associated with the 3N degrees of freedom. The 6 degrees of freedom associated with translations and rotations will have a frequency close to “ zero”. The frequencies of the 3N‐6 (or 3N‐5 for a linear molecule) will be different from “zero” and can be associated with the frequencies of internal modes of freedom, the molecular vibartions, of a molecule. The amplitude of each of the atoms in each of the vibrations can also be obtained. This is a classical model for the nuclei. Real nuclei are not “point‐like” but need to be described by a vibrational wave function, which reflects the probability to find an atom at a certain position. The classical approach gives very reasonable results for the frequencies in spite of this approximation. Internal Coordinates Internal coordinates are defined according to “common sense” or chemical intuition. Several types of internal coordinates can be defined. In the three‐atomic molecule below, two types occur, namely bond stretch motions and a bending motion. Two stretch motions can be distinguished, one for each bond between S1 S2 O the atoms. The angle between the two OH‐groups varies around the equilibrium value θ. This angle defines a bending coordinate. S2 H θ H For more complicated molecules other internal coordinates can be conveniently S1 defined as well. The most common ones are listed below: Bond stretch coordinates In plane Bending coordinates Angle between a bond and a plane defined by two bonds Torsion coordinate Other coordinates may be defined whenever a molecule “suggests” this. 10

13‐11‐2012 Normal coordinates The internal coordinates do not form a symmetry optimized set of motions for the nuclei. For instance in the example below for “water”, it is hard to imagine that the oxygen atom will simultaneously move to the left and to the right. The symmetry optimized motions of the of the nuclei are the normal coordinates. S1 S2 One normal coordinate is associated with each normal mode of motion. The normal modes form an orthogonal system of motions. They are linear combinations of the internal coordinates (and also of the Cartesian coordinates). θ S2 S1 Properties of Normal Modes are: Each normal normal mode behaves like a harmonic oscillator A normal mode is a concerted motion of many atoms A normal mode does not translate or rotate the molecule All atoms involved in a normal mode pass simultaneously through the equilibrium position Normal modes are independent, that is they form an orthonormal set, that is they do not interact. Example of Normal Modes Normal modes can be constructed from the internal coordinates by forming symmetry‐ adapted linear combinations. For a three‐atomic system this is rather straight forward, as follows: The length of the arrows represent the amplitude of the motion of the atom in the normal mode. In the drawings the amplitude is greatly exaggerated. The amplitude of atomic vibrations in molecules is typically of the order of 1% of the inter‐atomic distance. The inter‐atomic distance is typically 0.1 nm. θ The amplitude of the vibrations is therefore typically 1 pm (1 picometer). Light atoms, like H, have somewhat larger amplitudes. 11

13‐11‐2012 OH symmetric stretch OH bend θ OH anti-symmetric stretch Raman spectrum of water Raman spectra of different compounds OH symmetric stretch OH anti-symmetric stretch OH bend Raman spectra of different compounds are different. A compound can be determined from the measured Raman spectrum. Raman spectroscopy is an analytical chemical technique!! In mixtures of compounds the Raman spectra are partially overlapping. Complicated Raman spectra result. The spectrum of the mixture can be viewed as a “pattern”. Raman pattern recognition is an important approach for medical applications concerning cell and tissue Raman spectroscopy. Multivariate analysis methods provide powerful tools to perform pattern recognition in Raman spectra. 12

13‐11‐2012 Raman cross section The previous “classical” introduction explains qualitatively all aspects but does not explain quantitative aspects of Raman scattering. A quantum-mechanical derivation of the molecular polarizability gives the right treatment. We will not do the derivation, but the result of the treatment provides the expression for the Raman polarizability for each normal mode (and for the polarizability in general). The formula explain also intensities of light scattering for normal modes and is helpful to understand the difference in non-electronic resonance Raman scattering and electronic resonance Raman scattering. The Raman scattering cross section We had: nRdet R 3m 6 N mol P E det For (3m-6)vibrations A The (dipolar) scattering depends on direction and this becomes: nRaman N PL d R d A L d For one (1) vibration This formula recognizes through the integral over the angular dependence that the scattering is not equal in all directions. The angle-dependent scattering cross section is: L vib R d R d 4 0 2 L c 4 Raman 4 2 13

13‐11‐2012 The Raman polarizability f R j j L i f L j j R i Raman jf R i jf j ji L i ji j Detuning ji L Detuning f i Larger contribution jf R Smaller contribution Understand / Explain the formula! 27 The resonance Raman polarizability f R j j L i f L j j R i Raman i jf R i jf j ji L ji 0 detuning Very large detuning The resonance Raman polarizability becomes: f R j j L i Raman j ji L i ji 28 14

13‐11‐2012 The Raman spectrum of a poly‐atomic molecule: toluene Harmonic Oscillator selection rule: v 1 39 fundamental modes are Raman active. They are all observed. The potential energy is not parabolic V 12 k x 2 The harmonic potential is an approximation to the actual potential energy A bond that can be dissociated must be described by a different potential energy. The vibrational energy levels are progressively closer when the disscociation is approached The Morse potential energy is a useful description 15

13‐11‐2012 Overtones and combination bands in a poly‐ atomic molecule: toluene Many more modes than 39 fundamental modes are observed. Actually, approx. 112 modes have been observed so far. The intermediate region improved All these modes are overtones or combination bands. 16

13‐11‐2012 Raman spectroscopy Raman spectroscopy to understand matter by probing electrons and vibrations Raman spectroscopy as a contrast method in microscopy Raman spectroscopy to determine molecular number densities Material Science Label‐free microscopy Label‐free sensing Raman Chemical Analysis nR [# / s ] R 3m 6 C L2 Local field enhancement Raman cross section Sensitivity Concentration molecules of interest Detectivity Molecular vibrations for chemical identification I l Sensitivity Optical engineering of the optimal illumination/ detection strategy Specificity 17

13‐11‐2012 Hyperspectral Confocal Raman Microscopy Confocal effect: Reduction of out-of-focus light Sharper imaging NA n High NA: arcsin Higher resolution Large collection efficiency 4 arcsin 2 2 Measurement Modes Single point measurement Single spectrum Confocal Raman imaging Rapid Scanning mode N spectra N 1024, 4096, . Not representative for the whole cell Progresses to less time consuming Single spectrum Representative and fast 18

13‐11‐2012 Confocal Raman measurement procedure Schematic view of the raster scanning for imaging of the cells Table : Raman peak assignments S.No 1 2 3 4 5 6 7 8 9 10 11 12 13 14 ** * Data obtained : Each frame 32 x 32 pix Each pix 1600 data points N 1.64 million data points M Hyper spectral data cube Rich chemical information from cell with light Peak position (cm)-1 788 853 938 1004 1032 1094 1126 1254 1304 1339 1451 1660 1045 2335 590 960 1070 Peak Assignment Nucleus: O-P-O symmetric stretch Tyrosine Protein: α - helix Phenylalanine Phenylalanine Nucleus: O-P-O symmetric stretch Protein: C-N stretch Adenine Adenine Adenine C-H deformation Protein: Amide 1 Collagen (Proline) Nitrogen (atmosphere) Phosphate PO43Phosphate PO43Carbonate CO32- 120 300 250 20 100 40 1000 40 800 150 60 1200 20 250 40 80 60 300 20 200 40 60 1452 cm-1 Proteins 1004 cm-1 Phenylalanine 788 cm-1 DNA 140 20 1092 cm-1 Nucleus Univariate imaging 200 60 60 600 100 150 40 80 80 50 80 20 100 0 80 400 100 200 100 0 100 100 50 -50 -20 120 120 0 120 120 0 -100 120 100 80 60 40 20 120 100 80 60 40 20 120 100 80 60 40 20 120 100 80 60 40 20 19

13‐11‐2012 Multivariate imaging White light micrograph of PBL cell Cluster level 2 Cluster level 3 Cluster level 4 Cluster level 5 20 Intensity (a.u.) 15 10 5 0 600 800 1000 1200 1400 1600 1800 -1 Raman Shift (cm ) Univariate and Multivariate Raman Imaging of Bovine Chondrocytes White light microscopy 9-level hierarchical cluster analysis 788 cm-1 1004 cm-1 1094 cm-1 1602 cm-1 1656 cm-1 1745 cm-1 20

13‐11‐2012 Datamining Molecular Information Example: 12-level cluster analysis Improving chemical resolution END 21

Non-resonant Raman spectroscopy: 10-33 m2 Surface Enhanced Raman Scattering: 10-? m2 Surface enhanced Raman scattering cross sec tions vary widely in literature reports. There seems to be consensus developing that estimates the SERS cross sections between 6 to 8 orders of magnitude larger than the "normal" non-resonant and resonant Raman .

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