Raman Spectroscopy

• Lasers
  • Lexel 3000 Ar/ion (single line output, tuneable). 514 nm, 502 nm, 496 nm, 488 nm, 477 nm, 473 nm, 466 nm, 458 nm, and 455 nm lines accessible with > 10 mW power, 50 mW typical output in TEM 00.
  • Uniphase 17 mW He-Ne TEM 00 (633 nm line).
  • Spectra Physics 125 He-Ne TEM 00 (633 nm @45 mW, 611 nm possible; tuneable).
  • Coherent 750 nm diode laser (45 mW max).
  • Coherent Kr/ion (single line output, tuneable). Currently not working.
• Spectrometers
  • Jobin-Yvonne 800 triple-grating spectrometer (double-grating filter plus spectrometer) coupled to a SPEX integrating CCD (nitrogen-cooled). 800 mm focal length; resolution as fine as one wavenumber, to within one wavenumber of the excitation (depending on sample surface smoothness).
• Cryostats (Image 56 k)
  • Janis UHV optical cryostat with linear position adjustment, usable range 475 K-2.5 K. Requires liquid nitrogen (cools to 77 K) or liquid helium (cools to 2.5 K).
  • National Industries magneto-optical cryostat, usable range 325 K to 2.5 K and 0-7 T. Requires liquid nitrogen and liquid helium.

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• Condensed Matter
  • Spin and charge ordering in manganites and related perovskites
  • Characterization of ferroelectric films
  • Stripe charge/magnetic order in the copper oxide plane of cuprate HTSC's
• Biophysics
  • Characterization of semiconductor nanocrystals as tags for biomolecules

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Raman scattering is a powerful light scattering technique used to diagnose the internal structure of molecules and crystals. In a light scattering experiment, light of a known frequency and polarization is scattered from a sample. The scattered light is then analyzed for frequency and polarization. Raman scattered light is frequency-shifted with respect to the excitation frequency, but the magnitude of the shift is independent of the excitation frequency. This "Raman shift" is therefore an intrinsic property of the sample.

Because Raman scattered light changes in frequency, the rule of conservation of energy dictates that some energy is deposited in the sample. A definite Raman shift corresponds to an excitation energy of the sample (such as the energy of a free vibration of a molecule). In general, only some excitations of a given sample are "Raman active," that is, only some may take part in the Raman scattering process. Hence the frequency spectrum of the Raman scattered light maps out part of the excitation spectrum. Other spectroscopic techniques, such as IR absorption, are used to map out the non-Raman active excitations.

Additional information, related to the spatial form of the excitation, derives from the polarization dependence of the Raman scattered light. The shape of an excitation in a material, for example a vibration pattern of the atoms in a molecule, and the polarization dependence of the scattering, are determined by the equilibrium structure of the material through the rules of group theory. By this route one gleans valuable and unambiguous structural information from the Raman polarization dependence.

As an example of some actual Raman data taken here, consider the intensity/frequency spectra presented in the figure below.

Here, the frequency is plotted relative to the laser frequency, so the frequency scale represents the Raman shift. The peaks in the intensity occur at the frequencies of the Raman active modes. The spectra differ because of the different polarization conditions enforced on the incident and scattered light. Different polarization conditions select different sets of Raman active excitations.

Experimental Considerations: the NIST Raman Apparatus
Raman scattering is, as a rule, much weaker than Rayleigh scattering (in which there is no frequency shift) because the interactions which produce Raman scattering are higher order. Therefore most experiments require an intense source which is as monochromatic as possible--a laser with a narrow linewidth is usually used--and the collected light must be carefully filtered to avoid the potentially overwhelming Rayleigh signal. Other potentially large sources of non-Raman signal include fluorescence (the decay of long-lived electronic excitations) and of course light from ambient sources. Fluorescence can be particularly pernicious to a Raman measurement because the fluorescence signal is also shifted from the laser frequency, and so can be much more difficult to avoid. (Note that although the fluorescence spectrum is shifted from the laser frequency, the fluorescence shift depends on the laser frequency whereas the Raman shift does not).

So the relative weakness of the Raman signal dictates the organization of the data-taking apparatus. The original Raman setup here at NIST is fairly standard. The photo below shows the basic Raman configuration.

The laser beam from the Argon-ion laser is filtered for monochromaticity and directed by a system of mirrors to a focussing/collecting lens. The beam is focussed onto the sample; the scattered light which passes back through the same lens is then passed through a second lens into the first stage of the spectrometer. One point to notice is that a sample should be oriented such that the specular reflection from the sample passes outside of the collection lens--otherwise, the laser signal might damage the detector which is sensitive enough to see the weak Raman signal.

The Spectrometer and Detector
The spectrometer itself is a commercial "triple-grating" system. Physically, it is separated into two stages which are shown schematically here.

The first stage is called a monochromator, but is really used as a filter. Its structure is basically two diffraction gratings, separated by a slit, with input and output focussing mirrors. The incoming signal from the collecting lenses is focussed on the first grating, which separates the different wavelengths. This spread-out light is then passed through a slit. Because light of different wavelengths is now travelling in different directions, the slit width can be tuned to reject wavelengths outside of a user-defined range. This rejection is often used to eliminate the light at the laser frequency. The light which makes it through the slit is then refocussed on the second grating, whose purpose is only to compensate for any wavelength-dependence in the dispersion of the first grating. This grating is oriented such that its dispersion pattern is the mirror image of that from the first grating. Finally the light is refocussed and sent out to the second stage.

The second stage focusses the filtered light on the final grating. The dispersed light is now analyzed as a function of position, which corresponds to wavelength. The signal as a function of position is read by the system detector. In the present case the detector is a multichannel charge-coupled device array (CCD) in which the different positions (wavelengths) are read simultaneously. The wavelength/intensity information is then read to a computer and converted in software to frequency/intensity. This is the Raman spectrum which appears as the raw data.

Frequency Resolution; Intensity Limits
The Raman data comes out as an intensity/frequency plot. To resolve a Raman peak of a certain width, the resolution of the spectrometer should be smaller than the peak width. In the system described above, the resolution is determined by a final slit between the third grating and the CCD array. The final dispersed image of the sample spot is focussed in the plane of the CCD array; the slit width determines the extent to which the image may shift along the face of the CCD array, and hence the frequency resolution. When the apparatus is properly aligned, the intensity is a function of four factors: the applied laser power, the sample properties (how absorptive/reflective the sample is, and the intrinsic strength of the Raman modes), the width of the spectrometer's admission slit, and the width of the resolution slit. There is a tradeoff between resolving power and intensity. As for signal noise, statistically speaking, Raman is like a random decay process, so the noise in the Raman spectrum follows Poisson statistics. Finally, the CCD array has a certain dark current which is a function of the detector temperature. Typically, reducing the CCD array temperature to about 150 K with liquid nitrogen as a cryogen reduces the variation in dark current to about 20 counts per CCD pixel. Hence 20 counts is the practical limit of a measured signal.

Spectrometer Response; Raman Intensity Standards
Ideally, first stage of the spectrometer filters out the laser frequency, while leaving the rest of the frequencies unaffected, and the second stage spreads the filtered light onto the CCD array, which then reponds uniformly to each frequency. Of course, nearly the opposite is true: every spectrometer in its parts and as a whole has a wavelength (or frequency) dependent transmittance. The actual spectrum displayed by the software is the product of the spectrometer frequency response with the actual spectrum of the scattered light. To know not only the energies of the Raman-active excitations, but also the relative magnitudes of the scattering at different frequencies, one needs a calibration of the spectrometer response to a source with a known spectrum. Typically, one uses a NIST-traceable standard lamp; a recent NIST project concentrated on using the spectrum from a well-characterized piece of luminescent glass, and this is one of the calibrations we use with our instrument.

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This is a slightly more advanced introduction to the basic ideas of Raman scattering in crystalline solids. Raman scattering from noncrystalline solids, liquids, and gasses is quite similar although the selection rules are those of point groups rather than space groups. For the sake of definiteness, this treatment assumes that the scattering occurs from phonons in the crystal, although the general results apply to any sort of excitation.

Derivation of Raman Selection Rules
The term "selection rule" is shorthand for the set of conditions under which a given transition matrix element is nonzero. The term "Raman scattering experimen" might well be shorthand for the determination of the selection rules for light-scattering from a sample, in the event that the incident and scattered light have different frequencies.

Consider the following model Raman matrix element:

in which the violet state is the "in" state of the vector potential and the phonon field of the crystal, the green state is the "out" state. Here they are connected by an interaction which is second-order in the photon creation/annihilation operators. This matrix element corresponds to the absorption of a photon by the crystal, an internal transition of the phonon field, and the reemission of a (different frequency) photon. By examining the way the different parts of the matrix element transform under the symmetry operations of the total Hamiltonian (crystal plus vector potential), we may determine the selection rules for Raman scattering.

The vector potential part of the Hamiltonian must be invariant with respect to general three-dimensional rotations (because Maxwell's equations are invariant under general rotations and translations). Hence the symmetry of H₍₂₎ is limited to the symmetry of the crystal, that is, the symmetry of the crystal's space group. The crystal Hamiltonian transforms according to the identity representation of the space group, that is, it always has a phase of +1 under the space group symmetry operations.

Suppose now that we rotate the crystal through the symmetry operations of the crystal's space group, while keeping the rest of the apparatus fixed. Clearly, the result of the experiment should not change: the crystal should behave the same way when considered in any equivalent orientation. In this case we are effectively rotating the "partial matrix element"

Unless the "in" and "out" phonon states are the same, that is, unless

the partial matrix element will pick up a nonzero phase corresponding to the relative phase of the different phonon states under the transformation.

If we now rotate the entire experiment, crystal plus apparatus, corresponding to the entire matrix element, then there can be no additional phase. Therefore the transformation properties of the "in" and "out" states of the vector potential must exactly compensate for the phase which we get by rotating the crystal alone, e.g., the change in phase due to the "in" and "out" phonon states. In other words, the product of the phonon states must transform in the same way as the product of the vector potential states. But the vector potential states transform like vectors, so the operator connecting them must transform like a rank-two tensor.

Hence we find the Raman selection rule: the partial matrix element

is a rank-two tensor which belongs to a tensor representation of the crystal space group.

Splitting the crystal space group into its irreducible representations, we also split the partial matrix element into pieces which correspond to specific irreducible representations of the crystal space group.

In a typical experiment, one controls the polarization of the incident light, and selects different polarizations of the scattered light. In this manner the Raman experiment probes different components of < n_phonons |  H₍₂₎  | n_phonons >. From now on, we shall refer to the partial matrix element as the "Raman tensor".

The Raman Shift and Classification of Phonons
The Raman shift is the change in the frequency of the scattered light. A shift in the frequency means that energy is deposited in the sample. Comparing the Raman tensor to the complete matrix element (which will be time-independent) shows that the Raman tensor must have a frequency equal to the negative of the Raman shift. The origin of the frequency difference is the energy deposited in the crystal in the form of phonons. We may therefore identify each frequency component of the whole Raman tensor with a corresponding excitation of the phonon field. Because the Raman tensor also decomposes into the basis of irreducible representations of the crystal space group, phonons are also classified by their irreducible representation. Correlating the polarization and energy dependence of the scattered light gives the energy spectrum and symmetry species of the excitations of the phonon field which contribute to 231 H₍₂₎.

Enumerating the Modes: "Diagonal Ions"
I wrote above that the Raman-active phonon modes can be classified according to the irreducible representations of the crystal space group. The actual procedure for enumerating the modes hinges on the orthogonality of the characters of irreducible representations. The character table of the space group is tabulated in many places (see, for example, the group theory references).

For visible light, Raman phonons live near the center of the Brillouin zone (zero wavevector) because the wavelength of visible light is large compared to the lattice spacing. We can therefore concentrate on the subgroup of the crystal space group corresponding to zero wavevector.

For this subgroup, the representation of all zero-wavevector phonons is the one which simultaneously depicts the vector displacements of all of the atoms in the unit cell--it is 3N dimensional, if N is the number of atoms in the unit cell. Under a symmetry operation of the subgroup, two tiers of transformations occur. Firstly, sets of equivalent ions are shuttled among themselves. For example, if we rotate the tetragonal crystal shown above about its X axis by 180 degrees, the equivalent La/Sr ions switch places. Secondly, the vector displacements of each ion are transformed according to the rotation and/or reflection.

If we use, as a basis of the 3N dimensional representation, the displacement coordinates of each ion, we see that, for a given element of the group, only those ions whose identities are unchanged in the transformation can contribute to the character. The others will be by definition off-diagonal, and so cannot contribute to the trace. For each one of these special "diagonal" ions, the contribution to the trace is just the contribution of a vector displacement of unit length under the given transformation. The sum of the contributions for the diagonal ions gives the trace of the 3N-dimensional representation for the transformation; the collection of these traces for the entire group gives the character of this representation for the group.

Once the character of the 3N dimensional representation is known, we can decompose the representation into irreducible pieces by using the orthogonality of characters of irreducible representations.

Example: Disappearing Phonons, Persistent Phonons
The data below provide an example of selection rules at work. These Raman spectra were taken on a single crystal sample, with the orientation of the crystal axes known. The label for each denotes the polarization of the incident and scattered light. So, "ZZ" means that the incident and scattered polarizations are parallel to the crystal's Z axis; XY means that the incident polarization is parallel to the crystal's X axis while the scattered polarization is parallel to the crystal's Y axis. A prime indicates an axis at an angle of 45 degrees with respect to the crystal axis.

The narrow peak near 240 wavenumbers' Raman shift, due to the vibration of the La/Sr ions, appears in ZZ, XX, and X'X' scattering conditions, but not in the XY or X'Y' conditions. This selection rule reflects the symmetry of the tensor which represents this particular excitation. It is

                    A  0  0

                    0  A  0

                    0  0  C

              (X)             (X)
               1    A  0  0    1

               0    0  A  0    0   =  A;

               0    0  0  C    0


              (Y)             (X)
               0    A  0  0    1

               1    0  A  0    0  =  0.

               0    0  0  C    0

On Determining the Actual Modes and Frequencies
One part of the process is the part I have left out so far is determining the actual displacement pattern of the ions corresponding to a given peak. For the present case, simple inspection is sufficient to show that the vertical displacement of the La/Sr ions has the full tetragonal symmetry of the above tensor.

However, a similar vertical displacement of the apical oxygen ions also shares the full tetragonal symmetry. A priori, there is no way to know which combination of these modes contributes to a given Raman peak. This question brings the analysis beyond group theory, into the realm of lattice dynamics--the modelling of the forces between the ions in the unit cell. In the present example, the oxygen and La/Sr displacements are believed to separate completely into two different modes.

Beyond Zero Wavevector: Brillouin Scattering

To a good approximation Raman scattering occurs from zero-wavevector phonons. However, to the extent that the phonon wavevector differs from zero, phonon selection rules will deviate from the zero-wavevector rules and will depend on the angle between the direction of propagation of the incident and scattered light. For "optical phonons," which have zero dispersion at the zone center, any direction dependence in the Raman shift is quite small. On the other hand, for the "acoustic phonons," which have a linear dispersion near the zone center, the angular dependence of the Raman shift is more pronounced. Because the phonon wavevector is still quite small the acoustic phonon will have a small energy. Raman scattering from low-energy acoustic phonons is known as Brillouin Scattering. Brillouin scattering manifests as extra phonons, at low energy. The essential difference between Raman and Brillouin scattering is the sensitivity of the "Brillouin shift" to the relative angle of scattering.

The principles of determining the selection rules are the same, but the subgroup of the crystal space group one uses to enumerate the Brillouin phonons is no longer the group of the zone center, but rather the "group of the wavevector" of the phonon doing the scattering. For example, if the phonon wavevector occurs along one of the fourfold-symmetric directions in a tetragonal crystal, then the point P in the Brillouin zone corresponding to the phonon momentum has only twofold symmetry corresponding to reflection across the line between P and the zone center. When we enumerate the modes, we include only those symmetry elements which preserve the wavevector P. Otherwise, the procedure is identical to that described above for the zone center.

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Introductory Raman Spectroscopy by John R. Ferraro and Kazuo Nakamoto (Academic Press, 1994). (Explanations are somewhat recipe-book, but nice up-to-date section on techniques.)

Introduction to the Theory of the Raman Effect by J.A. Koningstein (D. Reidel, 1972). (Quantum mechanical treatment based on the Placek polarizability theory, a little stilted.)

Raman Spectroscopy by D.A. Long (McGraw-Hill, 1977). (Classical, semiclassical, and basic quantum-mechanical formulations of Placek's theory. A good place to start.)

Group Theory and Its Applications in Physics (Study Edition) by T. Inui, Y. Tanabe, and Y. Onodera (Springer Series in Solid State Sciences #78, 1996). (The best course you never had in group theory. Their commentary on the Raman effect is a bit cursory but they get the idea across well.)

The Analytical Expression of the Results of the Theory of Space Groups (2nd Edition) by Ralph W.G. Wyckoff (Carnegie Institution of Washington, 1930). (Indispensable aid to visualizing the transformations of crystal lattices, with exhaustive treatment of all 230 space groups and, most importantly, graphic representations of the symmetry elements.)