Time-dependent scattering

The wavepacket is propagated until a time where it is all scattered and is away from the interaction region. This time is short (typically 10-100 fs) for a direct reaction. Flowever, for some types of systems, e.g. for reactions with wells, the system can be trapped in resonances which are quasi-bound states (see section B3.4.7). There are eflScient ways to handle time-dependent scattering even with resonances, by propagating for a short time and then extracting the resonances and adding their contribution [69]. 

The aim is to establish the relation between the observable cross-sections and the collision dynamics. We denote the scattering state in the interaction region at t = 0 by x) and write the Hamiltonian in the form Hc.m. + Hre, i.e., the Hamiltonians associated with the center-of-mass motion and the relative motion. The propagator can be written in the form U(t) = exp(—iHc.mt/h)exp(—iHre t/h), and x(t)) = [/(f) x) describes the time-dependent scattering state at any time, i.e. (il x(f)) is the associated wave packet. 

Time-Dependence Experiments. In the fixed-angle SAXS experiments the time-dependent scattered intensity I(t) was normalized by ... 

If the scatterers are nonspherical and optically isotropic, the scattering amplitudes Ai become time dependent. If the position and orientation of single and pairs of scatterers are uncorrelated, the average of the time-dependent scattering amplitude can be separated from the position average ... 

Doom and Hupp have used preresonance Raman spectra in an analysis of the vibronic components which contribute to the intervalence absorption maximum of [(CN)5Ru -CN-Ru (NH3)5] and to the MLCT absorption maximum of [(bpy)Ru(NH3)4] ". These authors employ the time-dependent scattering approach of Heller to obtain the nuclear displacements of several vibrational modes coupled to the electronic transitions. They find in each case that several vibrational modes, spanning a wide range of frequencies, do contribute significantly to the photoinduced electron transfer processes. Hopkins and co-workers have used a two-color, ps Raman technique to investigate interligand electron transfer in Ru(II)-tn5-polypyridyl complexes, and they find vibrational relaxation of the electronically excited mole ule occurs within about 30 ps of excitation, after which interligand equilibration occurs more slowly than 5 x 10 s. 

The time-dependent scattering theory gives a description of the time evolution of the scattering process. For this purpose it is convenient to use the time-evolution operator /44t 45/... 

The time-dependent scattering theory yields a natural description of the collision process in time in a similar manner as in classical mechanics However, the stationary scattering theory has the advantage of yielding the same results in a more simple way therefore, it is to be prefered from a practical point of view /42-45/. 

All approaches discussed thus far rest on stationary-state collision theory. Time-dependent scattering theory has also been applied, making use of different potential-energy surfaces, for a description of the H + Hg reaction by BffAZUR and RUBIN /94/, Mc.GULLOUGH and WYATT /95/, and ZURT, KAMAL and ZOLIGKE /96/. The wave function in the initial state, chosen at a moment t = t far before the collision, is represented by the product (102.11) where is a trans-... 

R(t) is the relevant function to be analyzed for extraction of the kinetics. However, in micellar systems where the micelles are not fully proteated/deuterated or there is residual contrast between core and shell, ncaiUnear interference scattering contributions are present. In order to take this into account, a more accurate description of the time-dependent scattering intensity is necessary. A scattering model, where the time-dependent hyrogen/deuterium composition of the core and shell of the micelles is built into a kinetic core-shell model, is described next. 

The scattering function describing the time-dependent scattering intensity of micelles in a KZAC experiment involves a time-dependent core-shell model where the contrast is a function of the fraction of chains exchanged, /exc- Here, we shall limit the discussion to cylindrical and spherical structures using simple A-B diblock copolymers as an example. Inclusion of other structures such as vesicles could be shghtly more comphcated because the microscopic composition might be potentially different in the inner and outer shells. 

The complete time-dependent scattered intensity can be calculated by taking the average of the two types of micelles in the following way ... 

The characteristic conditions for time-dependent scattering processes are conveniently discussed by introducing the concept of time delay suffered by the photon wave packet interacting with the molecule (Goldberger and Watson, 1965a Newton, 1966). In practice this time delay may be defined as the difference between the detection times and of the scattered and nonscattered parts of a photon packet by fast detectors at the same distance from the target. A necessary condition for the time-resolved experimental observation is then that the time delay... 

For direct scattering, the detected time dependence is hence identical to that of the incoming photons, so that an experimental time separation of the scattered and unscattered light components is not possible. On the other hand, for the delayed scattering process, the defining condition Atj > At indicates that measurements of the time-dependent scattered intensity should be possible. Here, one has... 

Fig. 28. Time-dependent scattering curves for Si-TPA during crystallization of MFI. Scattering particle types include (1) primary units, (11) aggregates, (111) crystals and (BR) Bragg reflections. The inset figure shows the scattering intensity from the different particle populations. Reproduced from 194, copyright 1999, with kind permission from Wiley-VCH Verlag GmbH Co. KgaA.

Figure 8.17. Time-dependent scattering curves of silicate formation in the homogeneous phase, followed by small angle and wide angle scattering .

Figure 1. The time dependent scattering probability P, Eq. (40) for the model problem detailed in the text obtained for the incident energy of 10 kJ/mol using exact quantum dynamics (Quantum), mean-field (MF), quantized mean-field (QMF) the Bohmian quantum-classical technique (Bohmian), and the generalized surface hopping technique of Ref. [13] (SH).

The alternative approach introduces time-dependent scattering operators which depend on the internal degrees of freedom of the colliding partners. Instead of solving for a set of coupled partial... 

Figure 5 Time dependence of characteristic parameters ofSAXS profiles ofP(S-b-B) after a pressure jump from 600 to 800 bar, inverse of the normalized scattering peak maximum, (/ (l ois the peak intensity at the start of the experiment), maximum position, q, and peak width, Aq, as obtained from fits to the time dependent scattering intensity. The MST of the system is situated at approximately 600 bar with a transition range" of 200 bar. The lines are Avramiflts with an exponent of two as given in the insert and explained in the text.

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