The second-order effects

4 ALTERNATING CURRENT PERTURBATION. SECOND-ORDER RESPONSES 2.4.1 The second-order effects 

Section 2.3 was exclusively dedicated to the first-order response to a sinusoidal perturbation that is filtered from the total response by tuning the detection device to the fundamental frequency, oj. Due to the non- 

A non-linear object will produce the intermodulation response to this signal, namely terms in (c + co2)t and (co, — to2)t. The classification intermodulation is preferred when col and co2 have considerably different values, whereas demodulation applies to the case where Od — co2 = 2col is relatively small. 

As all the effects mentioned here are essentially of the second order, it is predictable that their mathematical description relies on a common basic theory. This will be the subject of the next section. 

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FIGURE 5.1 Isotropic hyperfine pattern for 51VIV in S-band. The spectrum is from V0S04 in aqueous solution. Use of the low frequency enhances the second-order effect of unequal splitting between the eight hyperfine lines. 

For the latter two ions, the presence of low-lying excited states makes the inclusion of both the first-order contribution of the excited states and the second-order effects due to coupling of the ground / state with the excited states crucial in a correct estimation of the room-temperature values of xT. Indeed, second-order contribution in Van Vleck [9] expansion of the susceptibility is inversely proportional to the energy difference between the ground and the excited states ... 

Now, as in the case of the energy, up to this point, we have worked with the nonsmooth expression for the electronic density. However, in order to incorporate the second-order effects associated with the charge transfer processes, one can make use of a smooth quadratic interpolation. That is, with the two definitions given in Equations 2.23 and 2.24, the electronic density change Ap(r) due to the electron transfer AN, when the external potential v(r) is kept fixed, may be approximated through a second-order Taylor series expansion of the electronic density as a function of the number of electrons,... 

In the first-order effect ajU is always finite the second-order effect arises from the... 

It may be emphasized here that such an elaboration is possible for any small amplitude perturbation technique. It is only necessary to explicitize either the first-order current or the first-order interfacial potential, corresponding to the type of perturbation, to be able to derive expressions for 7q, 1 and AEl. So, the treatment is also useful to estimate the error due to second-order non-linearity in the step methods. However, a separate measurement of the second-order effect can only be done with (sinusoidal) a.c. perturbation. In Table 5, the explicit expressions for SF pertaining to the four methods mentioned in Sect. 2.4.1 are given in such a way that the connection between them is clearly shown. 

In Equation 6, n (a>.) is the intensity independent refractive index at frequency u).,.0 Tlie sum in Equation 5 is over all the sites (n) the bracket, < >, represents an orientational averaging over angles 0 and . Unlike for the second-order effect, this orientational average for the third-order coefficient is nonzero even for an isotropic medium because it is a fourth rank tensor. Therefore, the first step to enhance third order optical nonlinearities in organic bulk systems is to use molecular structures with large Y. For this reason, a sound theoretical understanding of microscopic nonlinearities is of paramount importance. 

The second order effect of uniting R to S is therefore not only small, but also much the same for different pairs of hydrocarbons R, S. If we estimate bond energies empirically, the major part of this perturbation energy will be tacitly included in our value for the bond energy of a single bond between sp2 carbon atoms in a series of similar reactions, contributions of this kind will be... 

In conformity with the significance of the terms employed by investigators of anisotropy (Tsvetkov et al. 1964), the effects associated with the first-order terms in equation (10.6) may be called the effects of intrinsic anisotropy, while the second-order effects may be referred to as the effects of mutual interaction. In the second approximation, the principal axes of the relative permittivity tensor do not coincide, generally speaking, with the principal axes of the orientation tensor. It is readily seen that interesting situations may arise when Aa < 0 in this case, the coefficients of the first- and second-order terms have different signs. 

The first order effect of L (t) is obtained exactly as in linear response theory and generates the counterpart of terms in (Aa) in eqs 11 and 12. The new problem comes in obtaining the second order effect of the dipole torque operator L (t) for terms in E, as one must then evaluate the consequences of operations of the form... 

The second-order effect of spin-orbit coupling on the angular dependence of the zero-field splitting has been investigated by Hall and Hameka.457... 

The second-order effect of the mixing between the 2 n and 2 + states is... 

The direction of the magnetic field defines the space-fixed p = 0 (or Z) direction. Equation (8.239) represents a very simplified version, in that it neglects the nuclear and rotational Zeeman effects, as well as the second-order effects of spin-orbit coupling, none of which are negligible. Nevertheless (8.239) will allow us to derive theoretical values for the first-order effective g-factors, for comparison with the experimental spectra [43]. The required matrix elements of (8.239) in a case (b) hyperfine-coupled basis are as follows ... 

For most materials the higher-order non-linear optical effects are small and extremely difficult to detect. So the NLO effects investigated most are the second-order effects. 

The dipole polarizability also expresses the second-order effect of an electric field on the energy levels of an atom or molecule. We can write, for an atom (which has no dipole moment and therefore, in general, no first-order effect) ... 

In ESR spectroscopy, the second-order effect of the spin-orbit coupling causes the variation of the g-factor from the free electron value and the elimination of the degeneracy of the triplet state of transition-metal ions. 

One of the results obtained for tetrahedral centers formed by 3d ions is that one for Mn " " (3d -configuration) in ZnS [47]. The splitting of the " Ti orbital triplet of Mn + ion was analyzed using the second-order effective spin-Hamiltonian and comparing the calculated splittings with the observed ones. The lowest estimate for the JT energy in ZnSMn " " was obtained to be 750 cm [47]. 

MAS of second-order quadrupole effects. The same type of rotations using Wigner rotation matrices can be applied to the second-order effects as for the first-order interaction which leads to a second-order quadrupole energy of interaction such that... 

Using a perturbation satisfying V-e = Ve (such as perturbation proportional to cos 0), we obtain for the second-order effective Floquet Hamiltonian... 

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