Scattering, Rayleigh

Rayleigh scattering techniques are divided into  

Let us now consider the time dependence of the scattered light intensity, I(q, t), under the conditions described in the preceding section. Recalling that I(q, t) is proportional to the second power of the electric field E(q, /), the time correlation function C(q, x) of I(q, t) takes the following expression (Eq. (4.5))  

The first and second terms in Eq. (4.7) describe the average intensity of scattered light and the statistical characteristics of the electric field in the light scattering phenomenon, respectively. When scattering particles smaller than the wavelength A move independently of each other, the scattered light intensity is expressed by 

As the term q2D represents the half-width of the spectra, the translational diffusion coefficient of flexible polymers in solution can in principle be obtained from Eq. (4.11). In real experiments the change in spectral half-width is very small. 

If a transparent medium contains scattering centres, the intensity of light traversing the medium in the incident direction will gradually fall as the light is scattered into other directions. The reduction in the intensity of such a beam can be written as  

The analysis of the intensity of laser light scattered by a solution of a biological macromolecule yields information about its size and shape. 

The factor occurs in the definition of the Rayleigh ratio because the hght wave spreads out over a sphere of radius r and surface area 4ltr so any sample of the radiation has its intensity 7(6) decreased by a factor proportional to Therefore, the quantity 1(6) X r, and not simply 7(6), should be compared to Ig in forming the Rayleigh ratio. 

We also note that the definition of the Rayleigh ratio given here applies only to the experimental conditions in Fig. 11.4. 

A detailed examination of the scattering shows that the Rayleigh ratio depends on the mass concentration, Cm (units kg m ), of the macromolecule and its molar mass M as  

Equation 11.7 apphes only to idecd solutions. In practice, even relatively dilute solutions of macromolecules can deviate considerably from ideality, eis we saw in In the laboratory 3.1. Being so large, macromolecules displace a large qumtity of solvent instead of replacing individual solvent molecules with negUgible disturbance. To take deviations from ideahty into account it is common to rewrite eqn 11.7 as Kcm/R(0) = IIP(6)M and to extend it to 

Dielectric particles, whose size is much less than the wavelength A, are the simplest kind of heterogeneities concerning their interaction with light. 

Let us consider a case when homogeneous isotropic particles with the dielectric constant C2 (or with the refractive index, or the index of refraction, n-i = y/TJ) are dispersed in a medium with ci = 

Assume that every particle scatters light irrespective of others and that an incident wave has the same effect on all the particles in the system. Due to its smallness, all the electrons of a particle are in the incident wave s field of equal strength at any moment, anti all the set of charges can be regarded as an induced dipole p proportional to the field strength E of the incident wave  

Let the particle be illuminated by linearly polarized light from negative x values with its plane of polarization matching the plane of scattering (horizontally polarized light) with the electric field strength vector 

The electric field strength vector of the scattered light at long distances from the oscillating dipole (in the wave zone) is given by (Rayleigh, 1881 Volkcnshtcin, 1051 Fabelinski, 1065 Kerker, 1060 Sivukhin, 1980) 

Physical Chemistry of Macromolecules Basic Principles and Issues, Second Edition. By S. F. Sun ISBN 0-471-28138-7 Copyright 2004 John Wiley Sons, Inc. 

As the incident light hits the molecule, the distribution of electrons in the molecule is distorted, resulting in the polarization of the molecule, which now acts as an oscillating dipole p. The dipole is related to the electric field Eq by 

Here a is the polarizability of the molecule. The second derivative of the oscillating dipole with respect of time dP pIdP describes the electric strength of the scattered light  

The negative sign in the above equation is dropped because we are interested only in its absolute value. 

If the incident light is plane polarized, the scattered hght can then be expressed  

The spectral width of the scattered line is related to the acoustic damping constant T. This may be obtained by substituting AP in Equation (5.77) into the acoustic equation (5.73). This gives 

The inverse of this, Xp = 1/Fg, is often referred to as the phonon lifetime. As a result, the spectrum of the Brillouin doublet is broadened by an amount 

The entropy fluctuation AS obeys a diffusion equation similar to that for the temperature fluctuation  

Following the preceding analysis, we can see that the scattering caused by the entropy fluctuation does not shift the frequency. Instead, because of the exponentially decaying dependence, it broadens the light by an amount 5co= Fj. Again, since q = 2 A i sin(0/2), we have 

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Figure Bl.2.2. Schematic representation of the polarizability of a diatomic molecule as a fimction of vibrational coordinate. Because the polarizability changes during vibration, Raman scatter will occur in addition to Rayleigh scattering.

The first temi results in Rayleigh scattering which is at the same frequency as the exciting radiation. The second temi describes Raman scattering. There will be scattered light at (Vq - and (Vq -i- v ), that is at sum and difference frequencies of the excitation field and the vibrational frequency. Since a. x is about a factor of 10 smaller than a, it is necessary to have a very efficient method for dispersing the scattered light. 

Due to the rather stringent requirements placed on the monochromator, a double or triple monocln-omator is typically employed. Because the vibrational frequencies are only several hundred to several thousand cm and the linewidths are only tens of cm it is necessary to use a monochromator with reasonably high resolution. In addition to linewidth issues, it is necessary to suppress the very intense Rayleigh scattering. If a high resolution spectrum is not needed, however, then it is possible to use narrow-band interference filters to block the excitation line, and a low resolution monocln-omator to collect the spectrum. In fact, this is the approach taken with Fourier transfonn Raman spectrometers. 

Dadap J I, Shan J, Eisenthal K B and Heinz T F 1999 Second-harmonic Rayleigh scattering from a sphere of centrosymmetric material Phys. Rev. Lett. 83 4045-8... 

Perhaps the best known and most used optical spectroscopy which relies on the use of lasers is Raman spectroscopy. Because Raman spectroscopy is based on the inelastic scattering of photons, the signals are usually weak, and are often masked by fluorescence and/or Rayleigh scattering processes. The interest in usmg Raman for the vibrational characterization of surfaces arises from the fact that the teclmique can be used in situ under non-vacuum enviromnents, and also because it follows selection rules that complement those of IR spectroscopy. 

Goldanskii V I and Krupyanskii Y F 1989 Protein and protein-bound water dynamics studied by Rayleigh scattering of Mdssbauer radiation (RSMR) Q. Rev. Biophys. 22 39-92... 

TR measurements for correspondingly increased rejection of Rayleigh scattering. The addition of each grating typically introduces a throughput efficiency factor of roughly 30%, however, so that the overall efficiency of a... 

Distribution of radiation for (a) Rayleigh scattering and (b) large-particle scattering. 

Thus Rg is a constant in any particular experiment where Rayleigh scattering is obtained, since the entire angular dependence of the light intensity is correctly contained in the 1 + cos 6 term. 

If experimental values of Rg are observed to be independent of 0, then Rayleigh scattering is established and Eq. (10.60) can be applied to the data with confidence. 

The Rayleigh scattering theory which culminates in Eq. (10.60) as its most pertinent form for our purposes is based on the explicit assumption that interference effects are absent. The objective of the present section is to correct the Rayleigh theory to allow for interference effects. There are several assumptions-limitations that are implied by our approach ... 

For Rayleigh scattering, 5j - 6, = 0-there are no phase differences-and each of the cosine terms in Eq. (10.71) equals unity. In this case, which corresponds to i Rayleigh Ee right-hand side of Eq. (10.71) equals E n, and we can write... 

Draw a plot in polar coordinates of the scattering envelope in the xy plane. How would the envelope of a Rayleigh scatterer compare with this plot By interpolation, evaluate 145, iiss, and z. Use Fig. 10.13 to estimate the value of rrms to which this dissymmetry ratio corresponds if X (in toluene) is 364 nm. What are some practical and theoretical objections to this procedure for estimating rrms ... 

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. 

The resulting spectmm is illustrated in Figure 5.15, and Figure 5.16 shows in detail the processes involved in the first Stokes and anti-Stokes transitions and in the Rayleigh scattering. 

Figure 5.16 Raman and Rayleigh scattering processes involving virtual states Fq and Fj...

The mechanism for Stokes and anti-Stokes vibrational Raman transitions is analogous to that for rotational transitions, illustrated in Figure 5.16. As shown in Figure 6.3, intense monochromatic radiation may take the molecule from the u = 0 state to a virtual state Vq. Then it may return to u = 0 in a Rayleigh scattering process or to u = 1 in a Stokes Raman transition. Alternatively, it may go from the v = state to the virtual state Fj and return to V = (Rayleigh) or to u = 0 (Raman anti-Stokes). Flowever, in many molecules at normal... 

Rawwool Raybestos g-Ray detectors Ray diagram Rayleigh number Rayleigh scatter Rayleigh scattering... 

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