Third order

The distortion model (equations 3) describes the mapping of virtual, undistorted image points Pi = xi,yiY to distorted image points P = (i , y Y the real image using third order (2-dimensional) polynomals. 

In the third order of long-range perturbation theory for a system of tluee atoms A, B and C, the leading nonadditive dispersion temi is the Axilrod-Teller-Mutd triple-dipole interaction [58, 59]... 

Bell R J 1970 Multipolar expansion for the non-additive third-order interaction energy of three atoms J. 

Joo T, Jia Y, Yu J-Y, Lang M J and Fleming G R 1996 Third-order nonlinear time domain probes of solvation dynamics J. Chem. Phys. 104 6089... 

The initial classification of phase transitions made by Ehrenfest (1933) was extended and clarified by Pippard [1], who illustrated the distmctions with schematic heat capacity curves. Pippard distinguished different kinds of second- and third-order transitions and examples of some of his second-order transitions will appear in subsequent sections some of his types are unknown experimentally. Theoretical models exist for third-order transitions, but whether tiiese have ever been found is unclear. 

That analyticity was the source of the problem should have been obvious from the work of Onsager (1944) [16] who obtained an exact solution for the two-dimensional Ising model in zero field and found that the heat capacity goes to infinity at the transition, a logarithmic singularity tiiat yields a = 0, but not the a = 0 of the analytic theory, which corresponds to a finite discontinuity. (Wliile diverging at the critical point, the heat capacity is synnnetrical without an actual discontinuity, so perhaps should be called third-order.)... 

We have seen in previous sections that the two-dimensional Ising model yields a syimnetrical heat capacity curve tliat is divergent, but with no discontinuity, and that the experimental heat capacity at the k-transition of helium is finite without a discontinuity. Thus, according to the Elirenfest-Pippard criterion these transitions might be called third-order. 

It has long been known from statistical mechanical theory that a Bose-Einstein ideal gas, which at low temperatures would show condensation of molecules into die ground translational state (a condensation in momentum space rather than in position space), should show a third-order phase transition at the temperature at which this condensation starts. Nonnal helium ( He) is a Bose-Einstein substance, but is far from ideal at low temperatures, and the very real forces between molecules make the >L-transition to He II very different from that predicted for a Bose-Einstein gas. 

Neglecting derivatives of the third order and higher, we obtain Fourier s law of thennal conduction... 

It is clear from figure A3.4.3 that the second-order law is well followed. Flowever, in particular for recombination reactions at low pressures, a transition to a third-order rate law (second order in the recombining species and first order in some collision partner) must be considered. If the non-reactive collision partner M is present in excess and its concentration [M] is time-independent, the rate law still is pseudo-second order with an effective second-order rate coefficient proportional to [Mj. 

Empirically, one indeed finds a third-order rate law... 

However, the postulated trimolecular mechanism is highly questionable. The third-order rate law would also be consistent with mechanisms arising from consecutive bimolecular elementary reactions, such as... 

All nonlinear (electric field) spectroscopies are to be found in all temis of equation (B 1.3.1) except for the first. The latter exclusively accounts for the standard linear spectroscopies—one-photon absorption and emission (Class I) and linear dispersion (Class II). For example, the temi at third order contains by far the majority of the modem Raman spectroscopies (table B 1.3.1 and tableBl.3.2). 

Consider all of the spectroscopies at third order s = 3). To be as general as possible, suppose the total incident field consists of the combination of three experimentally distinct fields (/ = 1, 2, 3). These can differ in any combination of their frequency, polarization and direction of incidence (wavevector). Thus the total field is written as... 

B1.3.2.2 THE GENERATORS FOR ALL THIRD ORDER SPECTROSCOPIES FROM THE COMPLEX REPRESENTATION OF THE FIELD... 

At third order, one must expand + ) Ur=i enumerate the generators ... 

B1.3.2.3 THE FIELD GENERATORS FOR ALL THIRD ORDER RAMAN SPECTROSCOPIES... 

B1.3.2.4 TIME EVOLUTION OF THE THIRD ORDER POLARIZATION BY WAVE MIXING ENERGY LEVEL (WMEL) DIAGRAMS. THE RAMAN SPECTROSCOPIES CLASSIFIED... 

The general task is to trace the evolution of the third order polarization of the material created by each of the above 12 Raman field operators. For brevity, we choose to select only the subset of eight that is based on two colours only—a situation that is connnon to almost all of the Raman spectroscopies. Tliree-coloiir Raman studies are rather rare, but are most interesting, as demonstrated at both third and fifth order by the work in Wright s laboratory [21, 22, 23 and 24]- That work anticipates variations that include infrared resonances and the birth of doubly resonant vibrational spectroscopy (DOVE) and its two-dimensional Fourier transfomi representations analogous to 2D NMR [25]. 

Interestingly, tliree-coloiir spectroscopies at third order can only be of Class II, since the generators cannot possibly contain any quadrature. Maximal quadrature is necessary for Class I. 

If we neglect pure dephasing, the general tensor element of the third order hyperpolarizability relates to those of the first order polarizability tensor according to... 

Flere, the linear polarizability, a (oip 2), corresponds to the doorway stage of the 4WM process while to the window stage. We also see the (complex) Raman resonant energy denominator exposed. Of the tliree energy denominator factors required at third order, the remaining two appear, one each, m the two Imear polarizability tensor elements. 

Wliatever the deteetion teehnique, the window stage of the 4WM event must eonvert these evolved vibrational wavepaekets into the third order polarization field that oseillates at an ensemble distribution of optieal frequeneies. One must be alert to the possibility that the window event after doorway ehaimel B may involve resonanees from eleetronie state manifold e to some higher manifold, say r. Thus ehaimel B followed by an e (ket) or a (bra) event might be enlianeed by an e-to-r resonanee. However, it is nonnal to eonfine the... 

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