Moment method, radiation

Another major advantage associated with the induced moment method is that it is easy to generalize the results for fhe radiation-induced energy shift to more than two interacting particles. From the form of the expression for the two-body potential (46), the extension to n bodies is compactly written as follows [54] ... 

The inclusion of coherent states of the radiation field in the formalism describing opfically induced forces is most conveniently carried out within the induced multipole moment method delineated in Section 5. Instead of number sfafes n(fc, 2.)) = (k)), coherent states a= =) are defined... 

Differential and Moment Methods. The radiative transfer equation can be approximated using the moments of radiation intensity instead of intensity itself. A moment is defined as the integral of intensity multiplied by a power of a direction cosine over a predetermined solid angle division. For example,... 

In order to take advantage of moment methods, the radiation intensity is expressed as a series of products of angular and spatial functions. If the angular dependence is expressed using a simple power series, the moment method (MM) is obtained if spherical harmonics are employed to express the intensity, then the method is called the spherical harmonics (SH) approximation. In principle, the first-order moment and the first-order spherical harmonics approximations are identical to each other, as well as to the lowest order discrete ordinates (DO) approximation [34]. 

From the transmission line approximation for the current distribution, the far electric fields can be computed. From this the Poynting vector, (watts per square meter), can be integrated over a large spherical surface to find the total radiated power. The radiation resistance at the feed point is then found from this radiated power and the known current at the feed point. The near fields dictate the imaginary part of the impedance. Some typical values of radiation resistance for various elements, which have small diameter, are given in Table 13.1. The section on dipole characteristics gives the input impedance of dipoles of various lengths computed from the latest moment method formulation described in Sec. 13.1.4. 

Intermediate methods include the earliest procedure based on Stein s equation [33] and one based on Samuels equation [34]. Among the direct methods is an IR spectroscopic method based on the measurement of the dichroic ratio (R), of amorphous absorption bands. In the investigations [35], the amorphous bands 898 cm" and 1368 cm", for which the angles of transition moment are a898 = 39 and aneg = 80 , respectively, were used. Other methods are spectroscopy of polarized fluorescent radiation [35,36], measurement of color di-... 

With vertically polarized exciting light, p0 — when p = 0. But when P = n/2, p0 becomes negative and is equal to —1/3. The values are +1/3 and—1/7 for unpolarized radiation. Thus negative polarization appears when 6 is small, i.e. absorption probability is high and the transition moment in emission is perpendicular to that in absorption. These observations provide a suitable method for assigning the polarization directions of transition moments in different absorption bands of a given molecule from polarization of the fluorescence excitation spectra. 

A reaction between solutes A and B in a solvent occurs at a rate k(t) [A] [B] when both reactants are distributed randomly throughout the solution. However, when A and B represent the result of bond fission (by photolysis or radiolysis), the distance to which geminate A and B pairs separate may be very small compared with the separation between pairs of A and B, unless very intense pulses of light or radiation were used. A very marked correlation in the distribution of A about B exists from the moment that recombination begins. This affects the subsequent rate of reaction and the probability that A and B will survive recombination. In Fig. 41, two initial distributions and their respective rate coefficients are shown. With the possible exception of some ESR techniques, such as 3-pulse electron spin echo, there are no methods for determining the initial distribution of reactant pairs. Indeed, as was mentioned in Chap. 6, Sect. 2 and Chap. 7, Sect. 2, the rate of reaction and survival probability of... 

More recently Christiansen 4 has made a careful study of the polythionic acids and especially of certain physical properties—K-radiations, electrical moments, and molecular weights by ebuHioscopic methods—of the p-toluoyl-tri-, -penta- and -hexa-thionates, and suggests that none of the foregoing formulae is adequate, but that the properties and reactions of the acids are best explained by the assumption that the... 

In most of the examples described in this book, the rotational angular momentum is coupled to other angular momenta within the molecule, and the selection rules for transitions are more complicated than for the simplest example described above. Spherical tensor methods, however, offer a powerftd way of determining selection rules and transition intensities. Let us consider, as an example, rotational transitions in a good case (a) molecule. The perturbation due to the oscillating electric component of the electromagnetic radiation, interacting with the permanent electric dipole moment of the molecule, is represented by the operator... 

It is evident that the adoption of CTRW method yields results that are incompatible with the adoption of a GME, corresponding to the preparation done at t = 0. This master equation would retain forever memory of the observation beginning at the moment of preparation itself. However, the corresponding trajectories would keep aging. Some of them would lose memory of this initial condition, others would remain in the initial state for a very extended time. The radiation held at time f promotes transitions on f -old trajectories, whereas the GME would retain a memory forever of the brand new initial condition at t = 0. 

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