Asymptotic scattering function

The coefficient <, -( ) can be easily evaluated from the free asymptotic scattering function as follows ... 

For solutes comprising thin rod-like molecules the analysis of Holtzer127 for the asymptotic reciprocal scattering function yields ... 

The elastic contribution is also called elastic incoherent structure factor (EISF). It may be interpreted as the Fourier transformed of the asymptotic distribution of the hopping atom for infinite times. In an analogous way to the relaxation functions (Eq. 4.6 and Eq. 4.7), the complete scattering function is obtained by averaging Eq. 4.22 with the barrier distribution function g E) obtained, e.g. by dielectric spectroscopy (Eq. 4.5)... 

Jn the excluded volume limit similar scaling behavior is found for other observables like the scattering functions, and it turns out that all nonuni versal temperature and chemistry dependence can be absorbed into the single parameter B. Since the sealing functions typically interpolate among asymptotic limits (,s —+ 0 or s — oo, for instance), they also are known as crossover functions ... 

In the present context only the factor b/K) is of further interest. Since it contains the asymptotic phase shift Ae and the nominal wavenumber k which also is defined by its asymptotic value, the expression for b/K) holds everywhere even though it has been derived in the asymptotic limit (in contrast, the scattering function can be defined only in the asymptotic limit). Hence, the correct wavefunction P[Pg.288]

This transformed representation of the asymptotic wave functions can easily be verified for potential scattering. The asymptotic radial wave function in a given i-channel satisfies the identity... 

Nesbet, R.K., Noble, C.J., Morgan, L.A. and Weatherford, C.A. (1984). Variational R-matrix calculations of c + H2 scattering using numerical asymptotic basis functions, J. Phys. B 17, L891-L895. 

Figure 5.6 Kratky plot of the scattering function (circles) calculated16 from an atomistic model of polyethylene. The curves shown are those calculated for a Kratky-Porod chain, a Gaussian chain, and an asymptotic thin rod. (After Kirste and Oberthur.15)...

O. Goscinski and P. Lindner Asymptotic Properties of Atomic Form Factors and Incoherent Scattering Functions J. Chem. Phys. 52, 2539 (1970) Erratum 54, 826 (1971). 

By imposing the following boundary condition on E, i ) as f —> oo, the asymptotic wave function has the desired form, Eqn. (10.2), representing an incident plane wave plus scattered wave ... 

Kriste and Porod [22] and Tomita [23] worked out theories concerning the relationship between scattering and surface curvatures of the scattering objects. The asymptotic behavior of density correlation function y(r) at a small length scale r and that of scattering function I(q) at... 

Fig. 7.27 Semi-log plot of the effective exponent 7efr of the coilective scattering function iScolK - 0) (upper part)j of the order parameter exponent (middle part) and the ratio of critical amplitudes C /Cl (lower part) versus the crossover scaling variable LmjN, which is proportional to the ratio of characteristic lengths m/fcross- L is the mean size of the lattices used in the finite size scaling analysis. Asymptotic limits are shown as dashed straight lines. Open circles refer to the bond fluctuation model which has (.aa = bb = 0, ab 5 0 for effective monomers two lattice spacings apart (Fig. 7.25), full dots to the version of the bond fluctuation model with ab = aa = bb = e and Zefr rj 14 (critical temperatures of this model are included in Fig. 7.4). Curves through the points are guides to the eye only. (From Deutsch and Binder." )...

Another method is to measure the scattering function in the asymptotic regime qR 1, in which equation 3-27 reads... 

Figure 1 Zero-concentration intramolecular scattering function or form factor of a 32-arm polybutadiene star obtained by light scattering in cyclohexane with g

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