The Forward Scattering

The experimentally accessible correlation function, y,j(r) = (0 (R)0j.(R + r)), can be expressed in terms of thermodynamic parameters. In order to calculate the correlation functions yj (r), we follow basically the route of de Gennes [4] and Doi [1], using the fact that the correlation function can be shown to express the proportionality constant when using linear response theory to treat energy changes. 

To find the thermodynamic relationship to the correlation function, we start to calculate the response on the total energy upon perturbating individual segments with a weak external potential. Let us assume that the potentials ua(R) and ub(R) 

Assuming that the potentials Ua(R) and ub(R) varies little over the length scale of fluctuations, the first term vanished due to mean zero of (50a (R). The fluctuation term then becomes 

The result Eq. (7.54), originally derived by de Gennes [4], is most important It expresses the linear response theory that the thermal averaged local concentration fluctuations depends linearly on the fields acting on any other sites, with proportionality constants equal the spatial correlation functions 7aa( ) defined in Eq. (7.46). Using further that = —d (R), Eq. (7.54) may be 

The correlation function y is expected to be of relative short range, that is, y is likely non-zero only for r-values up to some coil-diameters in the mixed phase. We will therefore, as above, assume that the spatial variation of Ua(R), Wb(R) is gradual, so that these potentials can be considered constant over the range where y(r) has a nonzero value. In this case, Eq. (7.55) can be approximated as follows 

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Figure Bl.24.1. Schematic diagram of the target chamber and detectors used in ion beam analysis. The backscattering detector is mounted close to the incident beam and the forward scattering detector is mounted so that, when the target is tilted, hydrogen recoils can be detected at angles of about 30° from the beam direction. The x-ray detector faces the sample and receives x-rays emitted from the sample.

The optical theorem relates the integral cross section to the unaginary part of the forward scattering amplitude by... 

Figure 1 Simplistic schematic illustration of the scattering mechanism upon which X-ray photoelectron diffraction (XPD) is based. An intensity increase is expected in the forward scattering direction, where the scattered and primary waves constructively interfere.

Figure 2. Probability density plots of the ethyl cation product, (a) from the unlabeled reaction, (b) CH2CH3 from the labeled reaction, and (c) CD3CH2 from the labeled reaction. The backward scattered ethyl cation is more probable in (b), while the forward scattered ethyl cation is more probable in (c). Reprinted from [39] with permission from Elsevier.

Leupold et al. were the first to report on coherent nuclear resonant scattering of synchrotron radiation from the 67.41 keV level of Ni. The time evolution of the forward scattering was recorded by employing the so-called nuclear lighthouse... 

In the case of resonance absorption of synchrotron radiation by an Fe nucleus in a polycrystalline sample, the frequency dependence of the electric field of the forward scattered radiation, R(oj), takes a Lorentzian lineshape. In order to gain information about the time dependence of the transmitted radiation, the expression for R(oj) has to be Fourier-transformed into R(t) [6]. 

For the NFS spectrum of [Fe(tpa)(NCS)2] recorded at 108 K, which exhibits a HS to LS ratio of about 1 1, a coherent and an incoherent superposition of the forward scattered radiation from 50% LS and 50% HS isomers was compared, each characterized by its corresponding QB pattern (Fig. 9.16) [42]. The experimental spectrum correlates much better with a purely coherent superposition of LS and HS contributions. However, this observation does not yield the unequivocal conclusion that the superposition is purely coherent, because in the 0.5 mm thick sample the longitudinal coherence predominates since many HS and LS domains lie along the forward scattering pathway. In order to arrive at a more conclusive result, the NFS measurement ought to be performed with a smaller ratio aJD on a much thinner sample. Such an experiment would require a sample with 100% eiuiched Fe and a much higher beam intensity. 

Fig. 2. A schematic diagram illustrating how a time delay, r, permits the product molecule of an A + BC reaction to rotate into the forward scattering direction. The frequency u) of the rotating complex is set by the angular momentum of the collision, J, and hence by the impact parameter, b.

Fig. 37. Reaction mechanism for the forward scattering product from the H + HD H2 + D reaction at the collision energy of 1.200 eV.

When a target crosses the baseline of a bistatic radar the RCS can be greatly enhanced. This is due to the forward scatter phenomenon or Babinets principle. Here the RCS of a target at the bistatic baseline is calculated from... 

For a sphere of radius a metres, the monostatic RCS is equal to the projected area of a sphere given by ira2. Considering a sphere with monostatic RCS equal to 0.25m2 and at a wavelength equal to 0.1m, the forward scatter RCS is ... 

The dissymmetry method is useful especially if the instrument does not afford facilities for a wide angular scan of scattered intensities. For large particles the scattering envelope is not symmetrical and, as already indicated in Fig. 1, the forward scatter is larger than that in the backward direction. Hence the dissymmetry Z is greater than unity, where... 

As demonstrated in a previous section the osmotic compressibility can be obtained from the forward scattering of light... 

The energy loss per cm per electron caused by the forward scattered electrons can then be calculated as... 

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