The dipole approximation

We have mentioned that the small-K limit is unrealistic when discussing inelastic events because the minimum value of k is always finite in such experiments. A useful guide to the K-dependence of the cross-section is obtained from the so-called dipole approximation to the magnetic interaction operator. Experience shows that this approximation is the next best thing to use beyond the small-K limit when building up a picture of a class of scattering events. To implement this approximation for lanthanide ions, there is next to no extra work involved beyond what we have discussed already. 

In the dipole approximation, the magnetic moment operator is replaced by a weighted sum of L and S, and the weighting factors depend also on the magnitude, k, of k. The dependence is expressed in terms of spherical Bessel functions averaged over the electron radial density I 4((/ ). It is convenient to define the quantities 

The modification to the cross-section, brought about by employing the dipole approximation, is more or less obvious. For an inelastic event in which a transition occurs between states J = J the matrix element of J is zero, i.e., 

We conclude that the dipole approximation to the cross-section is obtained from our first approximation by multiplying the corresponding expression for the cross-section (21) by ((/q) - 

This new factor can be regarded as an inelastic structure factor for the 

---

As discussed in section A 1.6.1. on a microscopic quantum mechanical level, within the dipole approximation, the polarization, P(t), is given by... 

The basic Hamiltonian describing the motion of atoms and molecules under a strong laser is simple in the dipole approximation,... 

Using the time-dependent perturbation method and the dipole approximation [53,66]... 

The dipole approximation is valid only for point dipoles, i.e. when the donor-acceptor separation is much larger than the molecular dimensions. At short distances or when the dipole moments are large, it should be replaced by a monopole-monopole expansion. Higher multipole terms should also be included in the calculations. 

Another feature of particular importance in the study of large, spatially-extended molecular systems such as those examined here, is the use of the dipole approximation in assessing solute-solvent interactions. 

Within the semiclassical, perturbational treatment of the interaction of radiation with matter [77,78] and within the dipole approximation [79], the total energy absorption cross section may be written in the form [11,12,20,80]... 

Second harmonic generation (SHG) is one of the most intensively studied nonlinear optical effects that have ever been combined with near-held scanning optical microscopy (Shen et al. 2000 Zayats and Sandoghdar 2000 Zayats and Sandoghdar 2001 Takahashi and Zayats 2002). SHG, which is an even-order nonlinear process, is forbidden in centrosymmetric media under the dipole approximation (Shen 1984). Non-centrosymmetric molecules and lattices are allowed to exhibit SHG light. The second-order nonlinear polarization for SHG (T shg) is given in a scalar form by... 

The total Hamiltonian reads 7i(0 = Hq + V t), where Hq is the time-independent Hamiltonian of the unperturbed system and V(t) is the interaction describing the coupling between field and electron. In the dipole approximation. 

If we make the dipole approximation the following expression is obtained for the difference of absorption coefficients of left and right circularly polarized light ... 

The electronic optical cross section a(hv) can be expressed in the dipole approximation as... 

The first term in Eq. (1) describes the density matrix evolution under dissipation and field free conditions. The system-field interaction in the dipole approximation is... 

Now suppose (a) the potential V is quadratic and (b) the dipole approximation k R < 1 may be used. Then (2.31) decomposes into three terms of the form... 

To illustrate how the preceding formalism is generally used, we apply it to the solution of a well-known problem. Let. us derive an analytic expression for the Doppler broadening in the dipole approximation. The Hamiltonian which describes the interaction between radiation of polarization, e, and matter in the dipole approximation was discussed in the first section of this review article. 

Eq. (15.29) is only valid in the strong field regime. It is clear that iiJm(KEmJco) is small, using Eq. (15.26) we compute a small value of bm(Emv/). When this value approaches the size of the atom, Eq. (15.29) is no longer valid, for the integral of Eq. (15.27) is dominated by contributions for b bm(Emvi), where neither the assumption of the energies being independent of r nor the dipole approximation... 

These may be generated by irradiating an atom with a beam of monochromatic X-rays or ultraviolet rays. X-ray and electron spectroscopy is one of the main methods used for studying the structure of atomic electronic shells, particularly inner ones, as well as the role of relativistic and correlation effects. A wealth of such information may also be obtained from the studies of angular distribution of photoelectrons. It is interesting to notice that with increase of the energy of X-rays the dipole approximation fails to correctly describe the angular distribution of electrons. 

Here 0) is the frequency of the radiation, i denotes the initial state of the molecule, and f labels the final state of the photofragments, dv = Pj dE, where is the density of final states. Usually, the wavelength of the radiation considerably exceeds the size of the molecule, and one can use the dipole approximation (see, e.g., ref. 17). Then Hf d. fi (d is the component of the dipole moment along the external electric field) and the problem reduces to the analysis of the dipole matrix element... 

---