Step response tensor

The simulated dilatations involved increasing steps of imposed dilatation on the simulation cell. To permit a detailed understanding of the dilatational response of the polymer at the atomic level the entire volume of the simulation cell was tessellated into Voronoi polyhedra at each atomic site, permitting determination of strain-increment tensor elements dcy for each site from local displacement gradients by a technique described by Mott et al. (1992). Such increments of imposed dilatation at a level of 3 x 10 were applied 100 times to obtain total system dilatations of 0.3 (Mott et al. 1993b). For eaeh dilatation increment the atomic site strain-tensor increments de were obtained for each site n. The two invariants, de", the atomic site dilatation increment, and the work-equivalent shear-strain increment, dy", were obtained from the individual increments as... 

Owing to the coherence, we need to consider the macroscopic evolution of the field in a medium that shows a macroscopic polarization induced by the field-matter interaction. This will be done in three steps. First, the polarization induced by an arbitrary field will be calculated and expanded in power series in the field, the coefficients of the expansion being the material susceptibilities (frequency domain) or response function (time domain) of wth-order. Nonlinear Raman effects appear at third order in this expansion. Second, the perturbation theory derivation of the third-order nonlinear susceptibility in terms of molecular eigenstates and transition moments will be outlined, leading to a connection with the spontaneous Raman scattering tensor components. Last, the interaction of the initial field distribution with the created polarization will be evaluated and the signal expression obtained for the relevant techniques of Table 1. 

Integral-type constitutive equations may also be written in terms of strain tensor (or relative strain tensor), instead of rate-of-strain tensor. For infinitesimally small deformations, if the response of a system can be expressed by the linear superposition of a series of separate responses at different times to a series of step changes in the input, the stress tensor a as a response can be expressed in terms of the infinitesimally small strain tensor e by... 

Now that the signal had been established as measurable and uncontaminated with cascades, the next step was to fully characterize the response. Several papers had been pubUshed examining the merits of various polarization tensor elements and the potentially unique information they might contribute. Tokmakoff et al. had pubhshed a set of two papers examining the third-order [82] and fifth-order [83] tensor elements for the nonresonant Raman case and the dipole resonant case. The result was seven nonzero tensor elements that could be measured for the fifth-order Raman response iiimii, f ii22ii 111122 ... 

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