Wavenumber, scattering

Figure 4. TiOj absorption band for 100 nm spheres and ellipsoids. A) Distinct values of Lj yield three distinct absorption maxima. B) Increasing the standard deviation, a = 0.02, 0.05, 0.1 and 0.2. broadens the absorption and shifts it to lower wavenumbers. Scattering = 0 in this spectral region.

Since typical scattering experiments probe the system fluctuations in the frequency-wavenumber space, the Fourier transfonn v )is closer to measurements, which is in fact the imagmary (dissipative) part of the... 

The time-dependent structure factor S k,t), which is proportional to the intensity I k,t) measured in an elastic scattering experiment, is a measure of the strength of the spatial correlations in the ordering system with wavenumber k at time t. It exliibits a peak whose position is inversely proportional to the average domain size. As the system phase separates (orders) the peak moves towards increasingly smaller wavenumbers (see figure A3.3.3. 

The inverse scattering function of dilute polymer solutions for small scattering wavenumber qR- 1) obeys tire so-called Zimm relation [22] ... 

It was predicted in 1923 by Smekal and shown experimentally in 1928 by Raman and Krishnan that a small amount of radiation scattered by a gas, liquid or solid is of increased or decreased wavelength (or wavenumber). This is called the Raman effect and the scattered radiation with decreased or increased wavenumber is referred to as Stokes or anti-Stokes Raman scattering, respectively. 

All three terms in this equation represent scattering of the radiation. The first term corresponds to Rayleigh scattering of unchanged wavenumber v, and the second and third terms correspond to anti-Stokes and Stokes Raman scattering, with wavenumbers of (v + 2v () and (v — 2v () respectively. 

Molecules initially in the J = 0 state encounter intense, monochromatic radiation of wavenumber v. Provided the energy hcv does not correspond to the difference in energy between J = 0 and any other state (electronic, vibrational or rotational) of the molecule it is not absorbed but produces an induced dipole in the molecule, as expressed by Equation (5.43). The molecule is said to be in a virtual state which, in the case shown in Figure 5.16, is Vq. When scattering occurs the molecule may return, according to the selection mles, to J = 0 (Rayleigh) or J = 2 (Stokes). Similarly a molecule initially in the J = 2 state goes to... 

Hyper Raman scattering is at a wavenumber 2vq v r, where Vq is the wavenumber of the exciting radiation and —v r and +V[jr are the Stokes and anti-Stokes hyper Raman displacements, respectively. The hyper Raman scattering is well separated from the Raman scattering, which is centred on Vq, but is extremely weak, even with a 0-switched laser. 

The reason why the spacings are equal, and not the 1-0, 2-1, 3-2,... anharmonic intervals, is explained in Figure 9.21. The laser radiation of wavenumber Vg takes benzene molecules into the virtual state Fj from which they may drop down to the v = level. The resulting Stokes scattering is, as mentioned above, extremely intense in the forward direction with about 50 per cent of the incident radiation scattered at a wavenumber of Vg — Vj. This radiation is sufficiently intense to take other molecules into the virtual state V2, resulting in intense scattering at Vg — 2vj, and so on. 

The scattered radiation V3 is to high wavenumber of Vj (i.e. on the anti-Stokes side) and is coherent, unlike spontaneous Raman scattering hence the name CARS. As a consequence of the coherence of the scattering and the very high conversion efficiency to V3, the CARS radiation forms a collimated, laser-like beam. 

The similarity between a two-photon absorption and a Raman scattering process is even closer. Figure 9.27(a) shows that a Raman transition between states 1 and 2 is really a two-photon process. The first photon is absorbed at a wavenumber to take the molecule from state 1 to the virtual state V and the second photon is emitted at a wavenumber Vj,. 

For a more complete discussion of critical phenomena, we consider the scattering intensity S(q) from concentration fluctuations (q is the wavenumber of the scattering) which can be derived when one supplements Eq. (1) by a gradient-square term " ... 

However, mathematics is essential to explain how structural data are derived from EXAFS. The EXAFS function, x(k), is extracted from the X-ray absorption spectrum in Fig. 4.10 by removing the approximately parabolic background and the step, i.e. the spectrum of the free atom. As in any scattering experiment, it is customary to express the signal as a function of the wavenumber, k, rather than of energy. The relation between k and the kinetic energy of the photoelectron is ... 

Since the Raman scattering is not very efficient (only one photon in 107 gives rise to the Raman effect), a high power excitation source such as a laser is needed. Also, since we are interested in the energy (wavenumber) difference between the excitation and the Stokes lines, the excitation source should be monochromatic, which is another property of many laser systems. 

Because the time scale of the Raman scattering event ( 3.3 x 10-14s for a vibration with wavenumber shift 1000 cm-1 excited in the visible) is much shorter than that of the fastest conformational fluctuations in biomolecules, the ROA spectrum is a superposition of snapshot spectra from all the distinct chiral conformers present in the sample. Together with the dependence of ROA on chirality, this leads to an enhanced sensitivity to the dynamic aspects of biomolecular structure. The two-group model provides a qualitative explanation since it predicts ROA intensities that depend on absolute chirality in the form of a sin x dependence... 

In either case, the information on the vibrational transition is contained in the energy difference between the excitation radiation and the inelastically scattered Raman photons. Consequently, the parameters of interest are the intensities of the lines and their position relative to the Rayleigh line, usually expressed in wavenumbers (cm 1). As the actually recorded emissions all are in the spectral range determined by the excitation radiation, Raman spectroscopy facilitates the acquisition of vibrational spectra through standard VIS and/or NIR spectroscopy. 

When a compound is irradiated with monochromatic radiation, most of the radiation is transmitted unchanged, but a small portion is scattered. If the scattered radiation is passed into a spectrometer, we detect a strong Rayleigh line at the unmodified frequency of radiation used to excite the sample. In addition, the scattered radiation also contains frequencies arrayed above and below the frequency of the Rayleigh line. The differences between the Rayleigh line and these weaker Raman line frequencies correspond to the vibrational frequencies present in the molecules of the sample. For example, we may obtain a Raman line at 1640 cm-1 on either side of the Rayleigh line, and the sample thus possesses a vibrational mode of this frequency. The frequencies of molecular vibrations are typically 1012—1014 Hz. A more convenient unit, which is proportional to frequency, is wavenumber (cm-1), since fundamental vibrational modes lie between 4000 and 50 cm-1. 

Figure 8.16. KM-scattering coefficients of aerosils as a function of the wavenumber packing density = 0.107 spec, surface areas in m2g l are given as parameters. The slopes of the double-logarithmic straight lines are (from top to bottom) 2.6, 3.0, 3.2, 3.5, 3.6. (From Ref. 40.)...

Figure 8.18. The phase of the scattered field, evaluated at the position of the source dipole, as a function of free-space wavenumber (cm-1).

Experimental spectra are usually presented as plots of the intensity of (absorbed, emitted, reflected, or scattered) radiation versus the photon energy (in eV), the wavelength (in nm) or the wavenumber (in cm ). Using Equation (1.1), useful interconversion equations between these different units can be obtained ... 

Thus, if the monochromator is set to a wavenumber of 650cm", the formation of the C—Br bond can be recorded as a function of time during a chemical reaction between bromine and an olefin. Buehler used this method to follow the bromine/octadecene reaction. It should be pointed out that the Raman scattering is a function of droplet size as well as composition, and unlike bulk Raman it is complicated by the morphological resonances... 

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