Editing Raman spectroscopy (section)

== Polarization dependence of Raman scattering ==
Raman scattering is polarization sensitive and can provide detailed information on symmetry of Raman active modes. While conventional Raman spectroscopy identifies chemical composition, polarization effects on Raman spectra can reveal information on the orientation of molecules in single crystals and anisotropic materials, e.g. strained plastic sheets, as well as the symmetry of vibrational modes.

Polarization–dependent Raman spectroscopy uses (plane) polarized laser excitation from a [[polarizer]]. The Raman scattered light collected is passed through a second polarizer (called the analyzer) before entering the detector. The analyzer is oriented either parallel or perpendicular to the polarization of the laser. Spectra acquired with the analyzer set at both perpendicular and parallel to the excitation plane can be used to calculate the [[depolarization ratio]]. Typically a [[polarization scrambling|polarization scrambler]] is placed between the analyzer and detector also.{{citation needed|date=April 2024}}It is convenient in polarized Raman spectroscopy to describe the propagation and polarization directions using Porto's notation,<ref name="Porto's notation">{{Cite web|url=http://www.cryst.ehu.es/cgi-bin/cryst/programs/nph-doc-raman|title=Raman scattering|website=cryst.ehu.es|access-date=2019-07-04}}</ref> described by and named after Brazilian physicist [[Sergio Pereira da Silva Porto]].

For isotropic solutions, the Raman scattering from each mode either retains the polarization of the laser or becomes partly or fully depolarized.  If the vibrational mode involved in the Raman scattering process is totally symmetric then the polarization of the Raman scattering will be the same as that of the incoming laser beam. In the case that the vibrational mode is not totally symmetric then the polarization will be lost (scrambled) partially or totally, which is referred to as depolarization. Hence polarized Raman spectroscopy can provide detailed information as to the symmetry labels of vibrational modes.

In the solid state, polarized Raman spectroscopy can be useful in the study of oriented samples such as single crystals. The polarizability of a vibrational mode is not equal along and across the bond. Therefore the intensity of the Raman scattering will be different when the laser's polarization is along and orthogonal to a particular bond axis. This effect can provide information on the orientation of molecules with a single crystal or material. The spectral information arising from this analysis is often used to understand macro-molecular orientation in crystal lattices, [[liquid crystal]]s or polymer samples.<ref name="Joseph M. Grzybowski, R. K. Khanna, E. R. Lippincott">{{cite journal| last = Khanna| first = R.K. | title = Evidence of ion-pairing in the polarized Raman spectra of a Ba<sup>2+</sup>—CrO<sub>4</sub><sup>2-</sup> doped KI single crystal| journal = Journal of Raman Spectroscopy | volume = 4 | issue = 1 | pages = 25–30 | date = 1957| doi = 10.1002/jrs.1250040104| bibcode = 1975JRSp....4...25G}}</ref>

=== Characterization of the symmetry of a vibrational mode ===
The polarization technique is useful in understanding the connections between [[molecular symmetry]], Raman activity, and peaks in the corresponding Raman spectra.<ref>{{cite journal|last1=Itoh|first1=Yuki|last2=Hasegawa|first2=Takeshi|title=Polarization Dependence of Raman Scattering from a Thin Film Involving Optical Anisotropy Theorized for Molecular Orientation Analysis| journal=The Journal of Physical Chemistry A |date=May 2, 2012|doi=10.1021/jp301070a| pmid=22551093|volume=116|issue=23 | pages=5560–5570| bibcode=2012JPCA..116.5560I}}</ref> Polarized light in one direction only gives access to some Raman–active modes, but rotating the polarization gives access to other modes. Each mode is separated according to its symmetry.<ref>{{cite journal|display-authors=4| last1=Iliev|first1=M. N.| last2=Abrashev|first2=M. V.| last3=Laverdiere|first3=J.| last4=Jandi|first4=S.| last5=Gispadinov|first5=M.M.| last6=Wang|first6=QY.-Q| last7=Sun|first7=Y.-Y| title=Distortion-dependent Raman spectra and mode mixing in RMnO<sub>3</sub> perovskites (R=La,Pr,Nd,Sm,Eu,Gd,Tb,Dy,Ho,Y)| journal=Physical Review B | volume=73| issue=6| pages=064302|date=February 16, 2006 |doi=10.1103/physrevb.73.064302 |bibcode=2006PhRvB..73f4302I |s2cid=117290748}}</ref>

The symmetry of a vibrational mode is deduced from the depolarization ratio ρ, which is the ratio of the Raman scattering with polarization orthogonal to the incident laser and the Raman scattering with the same polarization as the incident laser: <math>\rho = \frac{I_r}{I_u}</math> Here <math>I_r</math> is the intensity of Raman scattering when the analyzer is rotated 90 degrees with respect to the incident light's polarization axis, and <math>I_u</math> the intensity of Raman scattering when the analyzer is aligned with the polarization of the incident laser.<ref name=Banwell>{{cite book |last1=Banwell |first1=Colin N. |last2=McCash |first2=Elaine M. |date=1994 |title=Fundamentals of Molecular Spectroscopy |url=https://archive.org/details/isbn_9780077079765 |url-access=registration |edition=4th |publisher=McGraw–Hill |pages=[https://archive.org/details/isbn_9780077079765/page/117 117]–8 |isbn=978-0-07-707976-5 }}</ref> When polarized light interacts with a molecule, it distorts the molecule which induces an equal and opposite effect in the plane-wave, causing it to be rotated by the difference between the orientation of the molecule and the angle of polarization of the light wave. If <math display="inline">\rho \geq \frac{3}{4}</math>, then the vibrations at that frequency are ''depolarized''; meaning they are not totally symmetric.<ref>{{cite web|url=http://www.horiba.com/us/en/scientific/products/Raman-spectroscopy/Raman-academy/Raman-faqs/what-is-polarised-Raman-spectroscopy/|title=What is polarised Raman spectroscopy? - HORIBA| website=horiba.com}}</ref><ref name=Banwell/>

=== Raman Excitation Profile Analysis ===
Resonance Raman [[selection rule]]s can be explained by the [[Kramers–Heisenberg formula|Kramers–Heisenberg equation]] using the Albrecht A and B terms, as demonstrated.<ref>{{Cite journal |last=Albrecht |first=Andreas C. |date=1961-05-01 |title=On the Theory of Raman Intensities |url=http://dx.doi.org/10.1063/1.1701032 |journal=The Journal of Chemical Physics |volume=34 |issue=5 |pages=1476–1484 |doi=10.1063/1.1701032 |bibcode=1961JChPh..34.1476A |issn=0021-9606}}</ref> The Kramers–Heisenberg expression is conveniently linked to the polarizability of the molecule within its frame of reference:<ref name=":0">{{Cite book |last=McHale |first=Jeanne L. |date=2017-07-06 |title=Molecular Spectroscopy |url=http://dx.doi.org/10.1201/9781315115214 |doi=10.1201/9781315115214|isbn=978-1-4665-8659-8 }}</ref>

<math display="block">(\alpha_{\rho\sigma})_{if} = \frac{1}{\hbar} \sum_n \left[ \frac{\langle i \vert \mu_\rho \vert n \rangle \langle n \vert \mu_\sigma \vert f \rangle}{\omega_0 + \omega_{nf} + i \Gamma_n} - \frac{\langle i \vert \mu_\sigma \vert n \rangle \langle n \vert \mu_\rho \vert f \rangle}{\omega_0 - \omega_{nf} - i \Gamma_n} \right] \equiv \langle i \vert \hat{\alpha}_{\rho\sigma} \vert f \rangle</math>

The [[polarizability]] operator connecting the initial and final states expresses the transition polarizability as a [[Matrix element (physics)|matrix element]], as a function of the incidence frequency ω<sub>0</sub>.<ref name=":0" /> The directions x, y, and z in the molecular frame are represented by the [[Cartesian tensor]] ρ and σ here. Analyzing Raman excitation patterns requires the use of this equation, which is a sum-over-states expression for polarizability. This series of profiles illustrates the connection between a Raman active vibration's excitation [[frequency]] and [[Intensity (physics)|intensity]].<ref name=":0" />

This method takes into account sums over [[Franck–Condon principle|Franck-Condon's]] active vibrational states and provides insight into electronic [[Absorption spectroscopy|absorption]] and [[emission spectrum|emission spectra]]. Nevertheless, the work highlights a flaw in the sum-over-states method, especially for large molecules like visible [[chromophore]]s, which are commonly studied in Raman spectroscopy.<ref name=":0" /> The difficulty arises from the potentially infinite number of intermediary steps needed. While lowering the sum at higher vibrational states can help tiny molecules get over this issue, larger molecules find it more challenging when there are more terms in the sum, particularly in the condensed phase when individual [[Quantum state|eigenstates]] cannot be resolved spectrally.<ref name=":0" />

To overcome this, two substitute techniques that do not require adding eigenstates can be considered. Among these two methods are available: the transform method.<ref>{{Cite journal |last1=Hizhnyakov |first1=V.V. |last2=Tehver |first2=I.J. |date=March 1980 |title=Resonance Raman profile with consideration for quadratic vibronic coupling |url=http://dx.doi.org/10.1016/0030-4018(80)90274-6 |journal=Optics Communications |volume=32 |issue=3 |pages=419–421 |doi=10.1016/0030-4018(80)90274-6 |bibcode=1980OptCo..32..419H |issn=0030-4018}}</ref><ref>{{Cite journal |last1=Shreve |first1=Andrew P. |last2=Haroz |first2=Erik H. |last3=Bachilo |first3=Sergei M. |last4=Weisman |first4=R. Bruce |last5=Tretiak |first5=Sergei |last6=Kilina |first6=Svetlana |last7=Doorn |first7=Stephen K. |date=2007-01-19 |title=Determination of Exciton-Phonon Coupling Elements in Single-Walled Carbon Nanotubes by Raman Overtone Analysis |url=http://dx.doi.org/10.1103/physrevlett.98.037405 |journal=Physical Review Letters |volume=98 |issue=3 |page=037405 |doi=10.1103/physrevlett.98.037405 |pmid=17358727 |bibcode=2007PhRvL..98c7405S |issn=0031-9007}}</ref><ref>{{Cite journal |last1=Blazej |first1=Daniel C. |last2=Peticolas |first2=Warner L. |date=1980-03-01 |title=Ultraviolet resonance Raman excitation profiles of pyrimidine nucleotides |url=http://dx.doi.org/10.1063/1.439547 |journal=The Journal of Chemical Physics |volume=72 |issue=5 |pages=3134–3142 |doi=10.1063/1.439547 |bibcode=1980JChPh..72.3134B |issn=0021-9606}}</ref> and Heller's time-dependent approach.<ref>{{Cite journal |last1=Lee |first1=Soo-Y. |last2=Heller |first2=E. J. |date=1979-12-15 |title=Time-dependent theory of Raman scattering |url=http://dx.doi.org/10.1063/1.438316 |journal=The Journal of Chemical Physics |volume=71 |issue=12 |pages=4777–4788 |doi=10.1063/1.438316 |bibcode=1979JChPh..71.4777L |issn=0021-9606}}</ref><ref>{{Cite journal |last1=Heller |first1=Eric J. |last2=Sundberg |first2=Robert |last3=Tannor |first3=David |date=May 1982 |title=Simple aspects of Raman scattering |url=http://dx.doi.org/10.1021/j100207a018 |journal=The Journal of Physical Chemistry |volume=86 |issue=10 |pages=1822–1833 |doi=10.1021/j100207a018 |issn=0022-3654}}</ref><ref>{{Cite journal |last=Heller |first=Eric J. |date=1981-12-01 |title=The semiclassical way to molecular spectroscopy |url=http://dx.doi.org/10.1021/ar00072a002 |journal=Accounts of Chemical Research |volume=14 |issue=12 |pages=368–375 |doi=10.1021/ar00072a002 |issn=0001-4842}}</ref><ref>{{Cite journal |last1=Tannor |first1=David J. |last2=Heller |first2=Eric J. |date=1982-07-01 |title=Polyatomic Raman scattering for general harmonic potentials |url=http://dx.doi.org/10.1063/1.443643 |journal=The Journal of Chemical Physics |volume=77 |issue=1 |pages=202–218 |doi=10.1063/1.443643 |bibcode=1982JChPh..77..202T |issn=0021-9606}}</ref> The goal of both approaches is to take into consideration the frequency-dependent Raman cross-section σ<sub>R</sub>(ω<sub>0</sub>) of a particular normal mode.<ref name=":0" />