Second-order phenomena

2zzz macroscopic third-order susceptibility, which is related to the 

Commonly, the evaluation of the susceptibility Xzzzz related to a reference standard. A detailed description of both experimental techniques and data evaluation is given in the article by Singer et al. [20]. 

In contrast to the EFISH method, the hyperpolarizability P can be measured directly by means of the HRS method developed by Clays and Persoons [21, 22]. This method involves measuring the intensity of the incoherently scattered, frequency-doubled light from isotropic solutions. As shown in Fig. 3.2, an infrared laser beam is focused on the center of a cell containing a solution of the NLO-active compound. 

The intensity of the scattered light, ha, is proportional to the square of the intensity of the incident hght, la, as given by Eq. (3-22). 

g is a set-up dependent factor Ni and N2 are the number densities of solvent and solute molecules, respectively and is the mean value of the square of 

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Comparison of these two polarizations shows that P2 Pi- Hence, in an isotropic medium such as a gas or a liquid x " = 0 and second order phenomena are not observable. Thus, only anisotropic media such as certain crystals are suitable for three-wave mixing processes. A consequence of a crystal being anisotropic is that it exhibits birefringence. However, the crystal birefringence enables phase matching to be achieved resulting in efficient generation of the new wave. 

Capacity parameters are not often used as primary optimization parameters in chromatography. Therefore, they are only included in table 3.10 in those cases in which they are used with some frequency. It should be noted, however, that changing one of the capacity factors usually involves the use of a completely different column and is therefore unattractive. Although changing the capacity parameters affects retention in an essentially predictable way, changing the column (packing material, film thickness, etc.) may give rise to unexpected second order phenomena. This is a second reason for which capacity parameters should not be recommended as primary optimization parameters. 

Similar conclusions can be drawn from the study of NMR spectra given by unsaturated closed-shell molecules (see 18 21>). However, the theoretical analysis of the hyperfine structure is more involved for NMR spectra than for ESR spectra, because the nuclear spin-spin coupling constants are second-order phenomena as compared with the electron-nucleus coupling constants. 

Thus, when fluorine and/or proton NMR spectra do not appear as simple as you might think they should, it is generally because of a second order phenomenon resulting from one of those factors described above. 

Semiclassical perturbation theory [56, 59] is applicable to describe TPA as a second-order phenomenon. Light can be seen as an electromagnetic wave perturbing the stationary wavefunctions of a molecule. Quantities related to these wavefunctions are the absorption cross sections rxgi and <7if (Fig. 3.1). Furthermore, high excitation power/photon density requires additional higher order terms according to perturbation theory. The spatial (Cartesian coordinates, r) and time (f) dependent wavefunction i//g2) (r, t) is shown for second-order perturbation in Eq. (2) [23] ... 

In the preceding section it was shown that differential ion transfer takes place at the phase boundary. This process can be modified by using a shunt resistance or by changing the solution pH or by altering the freezing rate. Differential ion transfer is a second-order phenomenon superimposed on the rejection, from the sohd, of the major portion of all the impurities, anions and cations alike. Thus, the solute becomes distributed unequally between the phases. The parameter governing the distribution is the solute partition coeflBcient, interface distribution co-eflScient, or simply distribution coeflBcient, defined as... 

Its N value is 1.4 (measured with l p=2.5, 5 and 10 C/min). The second peak at 135 C is the nematic-isotropic transition peak. Its N value is 1.3 (same heating rates). These two N values are typically the ones of first order phase transitions. The heat capacity of PAA is small compared to the height of the peak. Even if an anomalous second order phenomenon occurs, increasing the heat capacity jump under the nematic-isotropic peak by 100% or 200%, it cannot shift N towards two in a detectable way. 

The second category of actuators is based on electrostriction as exhibited by PMN [Pb(Mg 3Nb2/3)03] based ceramics. Although it is a second-order phenomenon of electromechanical coupling (x = ME, where M is called the electrostrictive coefficient), the induced strain can be extraordinarily large (more than 0.1%) [33], An attractive feature of these materials is the near absence of hysteresis (Fig. 4.1.19b). The superiority of PMN to PZT was demonstrated in a scanning tunneling microscope (STM) [34]. The STM probe was... 

More recent development of complex perovskite structures has resulted in a new class of electromechanical materials, electmstrictors. Purely electrostrictive materials are para-electric and centrosymmetric that is, they do not possess a polar axis and are typically cubic. The electrostrictive materials of most interest are ferroelectrics that are operated above or near their transition temperatures. The electrostrictive effect is a second-order phenomenon whereby an applied electric field results in a lattice distortion and mechanical distortion in the material. 

A related phenomenon with electric dipoles is ferroelectricity where there is long-range ordermg (nonzero values of the polarization P even at zero electric field E) below a second-order transition at a kind of critical temperature. 

The phenomenon was established firmly by determining the rates of reaction in 68-3 % sulphuric acid and 61-05 % perchloric acid of a series of compounds which, from their behaviour in other reactions, and from predictions made using the additivity principle ( 9.2), might be expected to be very reactive in nitration. The second-order rate coefficients for nitration of these compounds, their rates relative to that of benzene and, where possible, an estimate of their expected relative rates are listed in table 2.6. 

The first major pole is contributed by the output L-C filter. It represents a second order pole which exhibits a Q phenomenon, which is typically ignored, and a -40dB/decade rolloff above its corner frequency. The phase plot will quickly begin to lag starting at a frequency of 1/lOth the corner frequency, and will reach the full 180 degrees of lag at 10 times the corner frequency. The location of this double pole is found from... 

We have seen that 10" M s is about the fastest second-order rate constant that we might expect to measure this corresponds to a lifetime of about 10 " s at unit reactant concentration. Yet there is evidence, discussed by Grunwald, that certain proton transfers have lifetimes of the order 10 s. These ultrafast reactions are believed to take place via quantum mechanical tunneling through the energy barrier. This phenomenon will only be significant for very small particles, such as protons and electrons. 

An attempt was made by Unsold (33) to evaluate to the second-order the interaction of a proton and a hydrogen atom. He found, neglecting the resonance phenomenon, that the second-order perturbation energy is given approximately by the expression... 

It will be seen that the second-order treatment leads to results which deviate more from the correct values than do those given by the first-order treatment alone. This is due in part to the fact that the second-order energy was derived without considerar-tion of the resonance phenomenon, and is probably in error for that reason. The third-order energy is also no doubt appreciable. It can be concluded from table 3 that the first-order perturbation calculation in problems of this type will usually lead to rather good results, and that in general the second-order term need not be evaluated. 

For semicrystalline isotropic materials a qualitative measure of crystallinity is directly obtained from the respective WAXS curve. Figure 8.2 demonstrates the phenomenon for polyethylene terephthalate) (PET). The curve in bold, solid line shows a WAXS curve with many reflections. The material is a PET with high crystallinity. The thin solid line at the bottom shows a compressed image of the corresponding scattering curve from a completely amorphous sample. Compared to the semicrystalline material it only shows two very broad peaks - the so-called first and second order of the amorphous halo. 

One of the more interesting applications of non-linear optical effects is the generation of the second harmonic. This phenomenon results when a laser beam passes through a material having second-order NLO properties (hence, composed by non-centrosymmetric molecules) the light emitted has a frequency double that of the incident radiation (or the wavelength has been halved). 

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