Ordering field

In addition to the fourth-order response field Tfourth, the probe light generates two SH fields of the same frequency 211, the pump-free SH field Eq(2 Q), and the pump-induced non-modulated SH field non(td> 211). The ground-state population is reduced by the pump irradiation and the SH field is thereby weakened. The latter term non(td, 211) is a virtual electric field to represent the weakened SH field. Time-resolved second harmonic generation (TRSHG) has been applied to observe E on (td, 211) with a picosecond time resolution [20-25]. The fourth-order field interferes with the two SH fields to be detected in a heterodyned form. 

Hirose et al. [26] proposed a homodyne scheme to achieve the background-free detection of the fourth-order field. With pump irradiation in a transient grating configuration, the fourth-order field propagates in a direction different from that of the second-order field because of different phase match conditions. The fourth-order field is homodyned to make ffourth(td. 2 D) and spatially filtered from the second-order response hecond td, 2 D). 

Ifourth(fd, 2 Q) was multiplied with a window function and then converted to a frequency-domain spectrum via Fourier transformation. The window function determined the wavenumber resolution of the transformed spectrum. Figure 6.3c presents the spectrum transformed with a resolution of 6cm as the fwhm. Negative, symmetrically shaped bands are present at 534, 558, 594, 620, and 683 cm in the real part, together with dispersive shaped bands in the imaginary part at the corresponding wavenumbers. The band shapes indicate the phase of the fourth-order field c() to be n. Cosine-like coherence was generated in the five vibrational modes by an impulsive stimulated Raman transition resonant to an electronic excitation. 

We can assign a formal order to each term in equation (81) assuming B and y y as ordering parameters. First, let us consider the first-order field independent terms in equation (81)... 

A set of first-order field equations was proposed by Hertz [53-55], who substituted the partial time derivatives in Maxwell s equations by total time derivatives... 

Equation (96) is known as the linear diffusion equation since the lowest-order field dependence is linear. Thus we have a microscopic derivation of the Einstein relation, eqn. (98). This relation is normally derived from quite different considerations based on setting the current equal to zero in the linear diffusion equation and comparing the concentration profile C (x) with that predicted by equilibrium thermodynamics. 

The IS — 2S transition obeys the selection rule AF = 0, Am = 0 and is almost field-independent However, the g-factor for the bound electron is slightly less than for free space due to relativistic effects, and this gives the transition a small first-order field dependence. In the IS state the g-factor is g(lS) = g0 (1 - a2/3) [10]. The relativistic term is proportional to the binding energy so that g(2S) = ge (1 - a2/12). Thus, the field-dependence of the transition IS- 2S, (F=l,m=l AF=0,Am=0) leads to a frequency shift... 

In a self-heterodyne experiment, however, there is no independent control over the phase of the local oscillator field, so that the complete information on the complex third-order polarization of Equation (32) cannot be obtained. It is necessary to analyze in more detail the measurement process in order to determine the accessible information. In the actual experiment the spectrometer performs the Fourier transform of the generated third-order field of Equation (31) with respect to time coordinate t2, generating the field components of El3,(ti oy, ) given by ... 

Unlike diffusion charging, field charging takes place in an ordered field of unipolar ions, i.e., in a region where the ions are in an electric... 

With the general definition of the electric multipole (40) and n-th order field (53), the potential energies of electrostatic interaction between electric multipoles and fields take the general form ... 

Energy of Electrostatic Interaction Between Two Multipole Systems.— Consider two arbitrary electric charge systems qt and with multipoles and mutually distant by r (Figmre 5). The potential energy of their interaction can be determined from equation (54) on repladng therein the -th order field by the field F " of order n produced by the... 

The method of deriving the stress and displacement fields is not difficult, however, the algebra involved in deriving the 1-st order fields is involved. We then show only the important steps in deriving these fields, which are slightly different from the conventional method [6], Deriving the 1-st order stress fields, we obtain the following relations from the boundary conditions, = 0, and = 0, at the crack surfaces, 6 = Jt, respectively as... 

When the shear waves propagate through the elastic layer, or the elastic plate and reach the steady state, the type of the wave, SH wave for example, and it s dispersion relation are determined by the boundary conditions at the plate surfaces [7]. We have assumed that the sound waves modulate the stress fields at the tip of the crack, and then solved the wave equations with the boundary conditions at the surfaces of the crack and the plate. If the analysis is extended to derive the higher order fields and the dispersion relation of the wave is then obtained, such a wave do exist in the steady state. In this case we could confirm the existence of such "new wave" associated with the crack. Much algebra is required to obtain the higher order fields, however, it is not difficult to see the structure of the fields with the boundary condition at the plate surfaces. We find the boundary conditions at the plate surfaces for the second order stress fields are satisfied by the factor, cos /5 z, in the similar manner to Eq. 

When the sound wave generated from the branched microcrack or reflected from the boundary of the plate interact with the crack, the singular stress fields at the tip of the crack are modulated in such form as given by Eqs. (25), which are proportional to cos((wr - P x). On the other hand the 1-st order fields consist of two terms which are proportional either to cos(o>r- p x) or sin(o>r-/5 jX). The latter term, for example, comes from the real part of such term. 

Moments and polarizabilities can also be obtained by the fixed-charge method [76]. This technique allows for the single-step incorporation of the nonuniform electric field contributions due to gradients and higher order field derivatives. One or more charges are placed around the molecule in regions where the molecular wavefunctions are negligible. It is important that the basis set used for the field-free molecule be the same as that used in the presence of the field and that the molecule basis be adequate to describe any... 

The existence of the LC ordering affects the anisotropic NPs as an effective ordering field. To emphasize this feature the average free energy of the system is expressed as... 

We consider systems that can exist in several thermodynamic phases, depending on external thermodynamic variables which we take as intensive variables here (independent of the volume), such as temperature T, pressure p, external fields, etc. Assuming that an extensive thermodynamic variable (i.e., one which is proportional to the volume) can be identified that distinguishes between these phases, namely the order parameter ij>, we introduce the conjugate thermodynamic variable, the ordering field H, such that... 

It is clear that thermodynamic relations as written in eq. (5) apply to any material 4> qualifies as an order parameter when a particular value of the ordering field exists where the order parameter exhibits a jump singularity between two distinct values (fig. 9). This means that for these values of the ordering field a first-order phase transition occurs, where a first derivative of the thermodynamic potential F exhibits a singularity. At this transition, two phases can coexist i.e. at the liquid-gas transition... 

As a final topic of this section, we return to the simple Landau theory, eq. (14), and consider the special case that by the variation of a non-ordering field h one can reach a special point ht, Tt where the coefficient u(ht) = 0 while for h < one has u(h) > 0 and thus a standard second-order... 

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