Third-Order Response

Second order response Third order response Fourth order response Fifth order response... 

The simplest solid-state membranes are designed to measure test ions, which are also the mobile ions of the crystal (first-order response) and are usually single-substance crystals (Figure 4.11). Alternatively, the test substance may be involved in one or two chemical reactions on the surface of the electrode which alter the activity of the mobile ion in the membrane (Figures 4.12 and 4.13). Such membranes, which are often mixtures of substances, are said to show second- and third-order responses. While only a limited number of ions can gain access to a particular membrane, a greater number of substances will be able to react at the surface of the membrane. As a result, the selectivity of electrodes showing second- and third-order responses is reduced. 

Figure 4.13 A solid-state electrode showing a third-order response. An alternative modification to the electrode described in Figure 4.11 will permit the measurement of cadmium ions in solution. The membrane is composed of a mixture of silver and cadmium sulphides. The surface reaction between the cadmium ions in the test solution and the sulphide ions in the membrane will affect the equilibrium between the sulphide ions and the silver ions in the membrane.

For the dynamical distribution it will in general be necessary to consider both the auto and cross time correlation functions of the 0-1 and the 1-2 frequencies (117). For example, if the fluctuations, <5A(t), in the anhar-monicity are statistically independent of the fluctuations in the fundamental frequency, the oscillating term (1 — elAt3) in Equation (18) would be damped. In a Bloch model the fluctuations in anharmonicity translate into different dephasing rates for the 0-1 and 1-2 transitions that were discussed previously for two pulse echoes of harmonic oscillators. Thus we see that even if A vanishes, the third-order response can be finite (94). 

An alternative perspective on third-order responses of N coupled vibrators, which will be particularly helpful to describe spectral diffusion processes in such coupled systems (see Section IV. D), can be developed by assuming that Ri and R2 are the same and writing the total response function as ... 

In resonant infrared multidimensional spectroscopies the excitation pulses couple directly to the transition dipoles. The lowest order possible technique in noncentrosymmetrical media involves three-pulses, and is, in general, three dimensional (Fig. 1A). Simulating the signal requires calculation of the third-order response function. In a small molecule this can be done by applying the sum-over-states expressions (see Appendix A), taking into account all possible Liouville space pathways described by the Feynman diagrams shown in Fig. IB. The third-order response of coupled anharmonic vibrations depends on the complete set of one- and two-exciton states coupled to thermal bath (18), and the sum-over-states approach rapidly becomes computationally more expensive as the molecule size is increased. 

Eq. (5) has eight terms coming from the three commutators. For each choice of a wavevector, only some of these terms survive the RWA. Let us consider an impulsive technique involving short pulses and denote the relevant terms of 1Z by RJ. Various detection modes that probe different projections of the third-order response function TZ(h, t2, ti) may be employed. The... 

In this chapter we surveyed the theoretical analysis of resonant multidimensional spectroscopies generated by the interaction of 3 fs pulses with a Frenkel exciton system. Closed expressions for the time-domain third-order response function derived by solving the NEE are given in terms of various exciton Green functions. Alternatively, the multidimensional time-domain signal can be calculated starting from the frequency domain the third-order... 

Finally, we note that the time scale for the PE experiment is determined by the dephasing times, which are very short in proteins ( 300 fs) (41). Other complementary 2D techniques were proposed in Ref. 17. In particular, energy relaxation, which occurs in proteins and polypeptides on a large time scale ( 2-15 ps) (15,41), can be studied by utilizing the transient grating and the three pulse PE techniques. These can be calculated as well using the third-order response function presented here. 

APPENDIX A SUM-OVER-STATE REPRESENTATION OF THE THIRD-ORDER RESPONSE... 

In this appendix we present the sum-over-one- and two-exciton state expressions for the third-order response function. Double-sided Feynman diagrams representing the Liouville space pathways contributing to the four wave mixing in the RWA are given in Fig. IB. The response function is... 

Thus, the ratio is dependent on experimental parameters such as the optical path length, sample number density, and the phase matching conditions for the intermediate third-order processes, as well as the ratio of the third-and fifth-order response functions. The ratio of the response functions is directly related to the magnitude of the nonlinearity in the system, which is reflected by the magnitude of the potential anharmonicity, g(3), and the nonlinearity in the polarizability, a,2>. For example, let us consider only the NP contribution to the direct fifth-order response [Equation (21)]. For simplicity we will consider a system represented by a single mode, in other words the response is isotropic. If we express the third-order response functions in term of the coordinate [Equation (17)] and ignore all higher order terms,... 

It is important to note that the two electric fields that lead to a Raman transition can have different polarizations. Information about how the transition probability is affected by these polarizations is contained within the elements of the many-body polarizability tensor. Since all of the Raman spectroscopies considered here involve two Raman transitions, we must consider the effects of four polarizations overall. In time-domain experiments we are thus interested in the symmetry properties of the third-order response function, R (or equivalently in frequency-domain experiments... 

The different tensor elements of the third-order response can all be described in terms of the isotropic and anisotropic components of FF The... 

The second- and third-order response functions of the interacting system are formally given by the functional derivatives... 

The dephasing processes of low frequency modes have been observed by femtosecond optical Kerr and impulsive stimulated light scattering (ISS) experiments using pulses with durations that are short compared to the period of vibrational oscillation.2 For example, the third order response from CS2 has been extensively studied by femtosecond optical Kerr and... 

Kobko et al.200 have used a third order response function formalism with TDHF and TDDFT to assess different levels of theory for calculations of excited state structure and nonlinear optical responses in donor-donor and donor-acceptor Ji-conjugated molecules. They make suggestions for numerically efficient approximations. 

Repeating the procedure once more, we obtain the third order response as (40) rfp)(r) = -ir... 

Table 2. Molecular properties described by the first-, second-, and third-order response functions...

We have reviewed the principles of operation of the most relevant techniques employed to characterize the second- and third-order response of nonlinear media. 

While, in principle, the second-order response should have a higher strength than the third-order response, a strong geometrical condition (noncentrosymmetry at the atomic/molecular and at the bulk levels) limits the availability of second-order nonlinear materials. Experimentally, one has to ensure that a noncentrosymmetric configuration is used if one desires to measure the strength of the second-order nonlinear response, characterized by... 

Here and below the permutation symbols (bed) and (cab) refer to all terms in the large parentheses. We need the first-order perturbed coefficients Q the determination of these requires the solution of m sets of response equations. The third-order response coefficients can be eliminated by using the Handy-Schaefer device for the corresponding response equation... 

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