Polarization moments asymptotical

A new circumstance, in comparison with Eqs. (5.22) and (5.23), is the fact that the asymptotic system (5.87), (5.88) contains an infinite number of equations (0 < K < oo, 0 < k < oo). If the solution for polarization moments is found by way of expansion into a series over a small parameter, then the argument produced in Section 5.4 can be applied here without change. However, if the system of equations (5.87) and (5.88) is solved by computer, then we are deprived, in principle, of the possibility of accounting for all emerging polarization moments. What could be done in this case ... 

After determining the solution for the polarization moments observed signal. At this stage, too, one may obtain certain simplifications for molecular states with large angular momentum. Turning to the asymptotic limit J, J —> 00 we have for fluorescence intensity /(E ) in Eq. (5.34), and applying Eq. (C.12) from Appendix C [15] ... 

In just the same way as in the case of fluorescence intensity, the asymptotic equations of motion of polarization moments (5.54), (5.55) and (5.87), (5.88) must coincide with the corresponding equation of motion of classical multiple moments, as introduced by Eq. (2.16). We will show that this is indeed so in the following section. 

Summing up the above, we may conclude that the classical system of equations (5.93) and (5.94), together with the above given additional terms, coincides perfectly with the asymptotic system of equations of motion of quantum mechanical polarization moments (5.87) and (5.88). This result was actually to be expected from correspondence principle considerations. 

The second procedure, several aspects of which are reviewed in this paper, consists of directly computing the asymptotic value by employing newly-developed polymeric techniques which take advantage of the one-dimensional periodicity of these systems. Since the polarizability is either the linear response of the dipole moment to the field or the negative of the second-order term in the perturbation expansion of the energy as a power series in the field, several schemes can be proposed for its evaluation. Section 3 points out that several of these schemes are inconsistent with band theory summarized in Section 2. In Section 4, we present the main points of the polymeric polarization propagator approaches we have developed, and in Section 5, we describe some of their characteristics in applications to prototype systems. 

These treatments of periodic parts of the dipole moment operator are supported by several studies which show that, for large oligomeric chains, the perturbed electronic density exhibits a periodic potential in the middle of the chain whereas the chain end effects are related to the charge transfer through the chain [20-21]. Obviously, approaches based on truncated dipole moment operators still need to demonstrate that the global polarization effects are accounted for. In other words, one has to ensure that the polymeric value corresponds to the asymptotic limit of the oligomeric results obtained with the full operator. 

This correction is induced by the gauge invariant set of diagrams in Fig. 9.8(d) with the polarization operator insertions in the radiative photon. The two-loop anomalous magnetic moment generates correction of order a Ep to HFS and the respective leading pole term in the infrared asymptotics of the electron factor should be subtracted to avoid infrared divergence and double counting. 

Here Q is the quadrupole moment of the molecule, which has the value of 0.49 a.u. for H2 at the equilibrium internuclear separation, and / 2(cos 0) is the second-degree Legendre function, 9 being the angle between R and r. The polarization potential, also with spherical and non-spherical components, has the asymptotic form... 

The other very often considered nonadditive component is the induction energy. This component in its asymptotic form is the basis of the polarizable empirical potentials described in Section 33.3. For strongly polar systems, the second- and third-order nonadditive induction terms can indeed be expected to provide the largest nonadditive contribution except for small intermonomer separations [46] and to constitute the major part of the Hartree-Fock nonadditive contribution. The second-order terms have a very simple physical interpretation a multipole on system A induces multipole moments on B and C which interact with the permanent multipoles on C and B, respectively (see a more extensive discussion below). The second-order induction nonadditivity can be written as [85,86]... 

Its characteristic shape is presented as the solid line in fig. 20c. The polarization first drops, as time increases, from its initial value 1 to a single minimum before recovering to 1/3 at late times. As we shall see in more detail below, the 1/3 asymptote is characteristic of ZF- xSR in the case of static dipole moments (and a stationary muon). When everything is static, each muon sees a unique local field for its entire life and in the isotropic average 1/3 of the muon spins will be parallel to this field and not evolve in time. This is clearly expressed in the first term of eq. (32). 

Fig. 98. Left The Gaussian-broadened Gaussian relaxation fimction (bottom) (explanation see text). ZF pSR asymmetry spectrum in polycrystalline CeCuo2Nio.jSn at 0.08K (top). The dashed line is a fit of the static Gaussian Kubo-Toyabe relaxation function. The solid line is a fit of the static Gaussian-broadened Gaussian function. From Noakes and Kalvius (1997). Right The minimum polarization achieved by the Monte Carlo RCMMV static ZF muon spin relaxation functions, as a function of the reciprocal of the moment-magnitude correlation length, in units of the magnetic-ion nearest-neighbor separation in the model lattice. The horizontal line represents the 1/3 asymptote, above which the minimum polarization cannot rise. The dashed line is a...

The coefficients C can be computed from properties of monomers such as multipole moments and polarizabilities. The relevant formulas are obtained from the polarization series truncated at some finite order by replacing the potential V by its asymptotic expansion in powers of l/R. For the Coulomb potential 1 / r 1 —r21, such expansion has the form... 

Two characteristic times, with kinetic significance, have been observed on the anodic galvanostatic transients the peak time, Tpeak. that is, the time elapsed from the moment of switching on the rectangular current pulse until the peak value is reached and the transition time, r s, that is, the time necessary for the relaxation processes between the nonsteady state and the steady state. Because the measurement of the transition time is uncertain, due to the asymptotic character of the decay curve, it is preferable to use the time constant, a term borrowed from electrodynamics, which refers to the time necessary to reach 37% of the steady-state value. The following main features characterize the anodic polarization transients ... 

---