Molecular scattering intensity function

The scattering of electrons by an atom pair separated by a rigid distance, r, averaged over all possible orientations in space, gives rise to a molecular scattering intensity function which can be represented in the following way1 ... 

For polyatomic molecules the situation is somewhat more complex but essentially the same. The effect of intramolecular motion upon the scattering of fast electrons by molecular gases was first described by Debye3 for the particular case of a molecular ensemble at thermal equilibrium. The corresponding average molecular intensity function can be expressed in the following way ... 

Fig. 34. Plot of the scaled forward scattering intensity (see Eq. 89) as a function of X=A2-M c for some selected molecular architectures [165]. linear polyester O, A cross-linked polyesters near the ge-point, V weakly cross-linked polyester (see Table 5)...

FIGURE 3.17 Three-dimensional plot of scattering intensity as a function of scattering angle and elution volume for a broad molecular weight distribution (MWD) polystyrene (PS) (NITS standard reference 706). (Courtesy of Wyatt Technology Corporation, Santa Barbara, CA 93117 wyatt wyatt.com. With permission.)... 

A determination of the molecular weight by small-angle x-ray scattering requires, therefore, knowledge of the scattering intensity as a function of protein concentration and angle of observation, as well as of the electron density increment, ( — ), which replaces the refractive index... 

Now we can show the explicit relation with experiment. What is usually measured in spectroscopic or scattering experiments is the spectral density function /(to), which is the Fourier transform of some correlation function. For example, the absorption intensity in infrared spectroscopy is given by the Fourier transform of the time-dependent dipole-dipole correlation function <[/x(r), ju,(0)]>. If one expands the observables, i.e., the dipole operator in the case of infrared spectroscopy, as a Taylor series in the molecular displacement coordinates, the absorption or scattering intensity corresponding to the phonon branch r at wave vector q can be written as (Kobashi, 1978)... 

As it can be seen from (his equation, the sum of all atom pair correlation functions, weighted with electron numbers of the corresponding atomic-pair, can be determined by measuring the scattered intensity of X-rays. In case of molecular liquids, this sum, however, includes intra- and intermolec-ular contributions. The scattered intensity of an isolated molecule or a stoichiometric unit can be... 

Figure 4.5 From the result of a molecular dynamics simulation, the x-ray scattering intensity was calculated, and from it (r), given in the broken curve, was derived, by using exactly the same procedure as was used to treat experimental x-ray scattering intensities. The solid curve is the C-C atom pair distribution function calculated directly from the simulation result. (From Mondello et al.13)...

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