Dipole approximation, electric

Equation (A3.13.17) is a simple, usefiil fomuila relating the integrated cross section and the electric dipole transition moment as dimensionless quantities, in the electric dipole approximation [10, 100] ... 

Not only can electronic wavefiinctions tell us about the average values of all the physical properties for any particular state (i.e. above), but they also allow us to tell us how a specific perturbation (e.g. an electric field in the Stark effect, a magnetic field in the Zeeman effect and light s electromagnetic fields in spectroscopy) can alter the specific state of interest. For example, the perturbation arising from the electric field of a photon interacting with the electrons in a molecule is given within die so-called electric dipole approximation [12] by ... 

To see how this result is used, consider the integral that arises in formulating the interaction of electromagnetic radiation with a molecule within the electric-dipole approximation ... 

Less widely appreciated is the fact that the angular distribution functions Eqs. (2 and 3) are actually subcases of a more general form. It was first proposed by Ritchie [34] that, even in the pure electric-dipole approximation, another term was required for completeness, and that hence the general photoionization angular distribution function, normalized over the surface of a unit sphere, should be written as [35] ... 

If the isotropic coefficient is specified to be unity, a is just the total (integrated) cross-section. In Appendix A, an alternative quantum mechanical expression for this cross-section is obtained in the electric dipole approximation. By comparing the two expressions, it can be seen that the Legendre polynomial coefficients in Eq. (11) may be obtained from the inner summation terms in Eq. (A.15). Hence, the Legendre polynomial coefficients are... 

It may be worthwhile to compare briefly the PECD phenomenon discussed here, which relates to randomly oriented chiral molecular targets, with the likely more familiar Circular Dichroism in the Angular Distribution (CDAD) that is observed with oriented, achiral species [44 7]. Both approaches measure a photoemission circular dichroism brought about by an asymmetry in the lab frame electron angular distribution. Both phenomena arise in the electric dipole approximation and so create exceptionally large asymmetries, but these similarities are perhaps a little superficial. 

In accordance with the predictions that can be made on the basis of just the electric dipole approximation (see Section III.A) the observed dichroism is equal, but of opposite sign for the two enantiomers. This could be seen also in the valence shell ionization results for glycidol presented in Fig. 2. The added significance here is that a contribution to the angular distribution by higher order... 

We consider the expression of the lab frame photoelectron angular distribution for a randomly oriented molecular sample. The frozen core, electric dipole approximation for the differential cross-section for electron emission into a solid angle about a direction k can be written as... 

Second-order NLO processes, including SFG, are strictly forbidden in media with inversion symmetry under the electric dipole approximation and are allowed only at the interface between these media where the inversion symmetry is necessarily broken. In the IR-Visible SFG measurement, a visible laser beam (covis) and a tunable infrared laser beam (cOir) are overlapped at an interface and the SFG signal is measured by scanning cOir while keeping cOvis constant. The SFG intensity (Isfg) is enhanced when coir becomes equal to the vibration levels of the molecules at the interface. Thus, one can obtain surface-specific vibrational spectra at an interface... 

The theoretical framework developed above is valid in the electric dipole approximation. In this context, it is assumed that the nonlinear polarization Ps 2a)) is reduced to the electric dipole contribution as given in Eq. (1). This assumption is only valid if the surface susceptibility tensor co, m) is large enough to dwarf the contribution from higher... 

Direct Evidence of Failure of Electric Dipole Approximation in Second-Harmonic Generation from Chiral Polymer Film... 

The requirements for even-order nonlinear phenomena are particularly stringent because such processes are forbidden in centrosymmetric materials in the electric dipole approximation of the field-matter interaction. For this reason,... 

Any symmetry operation is required to leave the sign and magnitude of physical properties unchanged and therefore y xxx = 0. The same line of reasoning can be used to show that all tensor components will vanish under inversion. Hence, second-order nonlinear optical properties are not allowed in centrosymmetric media using the electric dipole approximation. The presence of noncentrosymmetry is one of the most stringent requirements in... 

Nonvanishing Components of Second-Order Susceptibility Tensor for Second-Harmonic Generation in Electric-Dipole Approximation for Achiral and Chiral Isotropic (i.e. isotropic in the plane of the film) Films0... 

It is important to note that the coefficients fp, gp, and hs are always nonvanishing, for both achiral and chiral isotropic films. On the other hand, fs, gs, and hp can only be nonvanishing if the isotropic film is chiral (nonracemic) because they completely depend on the chiral susceptibility components. Note that gs is always equal to zero within the electric dipole approximation. The sign of the chiral expansion coefficients changes between enantiomers, while that of the achiral expansion coefficients stays the same. Experimental determination of all expansion coefficients fully characterizes the nonlinearity and nonlinear optical activity of the sample. Once all expansion coefficients are... 

This effect, which is in a loose sense the nonlinear analog of linear optical rotation, is based on using linearly polarized fundamental light and measuring the direction of the major axis of the ellipse that describes the state of polarization of the second-harmonic light. For a simple description of the effect, we assume that the expansion coefficients are real, as would be the case for nonresonant excitation within the electric dipole approximation.22 In this case, the second-harmonic light will also be linearly polarized in a direction characterized by the angle... 

We first have attempted to explain the results in the electric dipole approximation. However, the four independent solutions (obtained from different experimental configurations) to the components of jeee (Table 9.5) are mutually incompatible, and we conclude that the results cannot be explained in the electric dipole approximation. 

Tensor Components Determined Within the Electric Dipole Approximation and Normalized to ye 7 = 1 ... 

In the electric dipole approximation, the polarization can in general (for any second-order nonlinear optical process) be written as9... 

The probability of finding the system in the state vRjn> has an oscillatory time dependence. For off-resonance conditions, the system presents a line width at half maximum equal to 4 ll/fi. This matrix element can be expanded in a multipolar expansion, the first term being the electric dipole approximation [45, 152, 154],... 

The above-mentioned nonlinear optical effects can be described by the perturbation of the electromagnetic held intensity under the electric dipole approximation. Actually, this approximation is broken in optical near-helds. Hence, a perturbation effect of multipole such as electric quadrupole or magnetic dipole should also be considered, although such a higher-order effect is normally negligible. Indeed, electric quadrupole contributions can be comparable with electric dipole contributions... 

The discontinuity of the interface leads to two contributions to the second order nonlinear polarizability, the electric dipole effect due to the structural discontinuity and the quadrupole type contribution arising from the large electric field gradient at the surface. Under the electric dipole approximation, the nonlinear susceptibility of the centrosymmetric bulk medium 2 is zero. If the higher order magnetic dipole... 

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