Lorentz local field correction

N Is the number of molecules per unit volume (packing density factor), fv Is a Lorentz local field correction at frequency v(fv= [(nv)2 + 2]/3, v = u) or 2u). Although generally admitted, this type of local field correction Is an approximation vdilch certainly deserves further Investigation. IJK (resp Ijk) are axis denominations of the crystalline (resp. molecular) reference frames, n(g) Is the number of equivalent positions In the unit cell for the crystal point symmetry group g bjjj, crystalline nonlinearity per molecule, has been recently Introduced 0.4) to get general expressions, lndependant of the actual number of molecules within the unit cell (possibly a (sub) multiple of n(g)). 

The dipole strength of an induced electric dipole transition is proportional to the square of the matrix element in the dipole operator and therefore also to the square of the electric field at the lanthanide site. However, in intensity studies, the lanthanide ions are not in a vacuum, but embedded in a dielectric medium. The lanthanide ion in a dielectric medium not only feels the radiation field of the incident light, but also the field from the dipoles in the medium outside a spherical surface. The total field consisting of the electric field E of the incident light (electric field in the vacuum), plus the electric field of the dipoles is called the effective field eff> i e. the field effective in inducing the electric dipole transition. The square of the matrix element in the electric dipole operator has to be multiplied by a factor E fflEf. In a first approximation, ( efr/ = ( + 2) /9. The factor (n + 2fl9 is the Lorentz local field correction and accounts for dipole-dipole corrections. 

F(o>i) are the local field corrections for a"3wave of frequency Generally, one utilizes the Lorentz approximation for the local field in which case (1,4)... 

The measurement of x of solutions can be used to determine the microscopic nonlinearities Y of a solute, provided Y of the solvent is known. This measurement also provides information on the sign of y and (hence x of the molecules if one knows the sign of Y for the solvent (5,7) Under favorable conditions one can also use solution measurements to determine if Y is a complex quantity. The method utilizes two basic assumptions (i) the nonlinearities of the solute and the solvent molecules are additive, and (ii) Lorentz approximation can be used for the local field correction. Under these two assumptions one can write the x of the solution to be... 

When a non-centrosymmetric solvent is used, there is still hyper-Rayleigh scattering at zero solute concentration. The intercept is then determined by the number density of the pure solvent and the hyperpolarizability of the solvent. This provides a means of internal calibration, without the need for local field correction factors at optical frequencies. No dc field correction factors are necessary, since in HRS, unlike in EFISHG, no dc field is applied. By comparing intercept and slope, a hyperpolarizability value can be deduced for the solute from the one for the solvent. This is referred to as the internal reference method. Alternatively, or when the solvent is centrosymmetric, slopes can be compared directly. One slope is then for a reference molecule with an accurately known hyperpolarizability the other slope is for the unknown, with the hyperpolarizability to be determined. This is referred to as the external reference method. If the same solvent is used, then no field correction factor is necessary. When another solvent needs to be used, the different refractive index calls for a local field correction factor at optical frequencies. The usual Lorentz correction factors can be used. 

In the relations between the macroscopic susceptibilities y , y and the microscopic or molecular properties a, ft, y, local field corrections have to be considered as explained above. The molecule experiences the external electric field E altered by the polarization of the surrounding material leading to a local electric field E[oc. In the most widely used approach to approximate the local electric field the molecule sits in a spherical cavity of a homogenous media. According to Lorentz the local electric field [9] is... 

In this section, a simple description of the dielectric polarization process is provided, and later to describe dielectric relaxation processes, the polarization mechanisms of materials produced by macroscopic static electric fields are analyzed. The relation between the macroscopic electric response and microscopic properties such as electronic, ionic, orientational, and hopping charge polarizabilities is very complex and is out of the scope of this book. This problem was successfully treated by Lorentz. He established that a remarkable improvement of the obtained results can be obtained at all frequencies by proposing the existence of a local field, which diverges from the macroscopic electric field by a correction factor, the Lorentz local-field factor [27],... 

The Lorentz-Lorenz case in which the material is so microscopically inhomogeneous that the spatial distribution of OL is disjoint. The local-field correction is then L P,... 

In the above equations, /3 and y" represent the microscopic coefficients at site n which are averaged over molecular orientations 0 and 0 and summed over all sites n. The terms F(a>i) are the local field corrections for a wave of frequency Generally, one utilizes the Lorentz approximation for the local field, where ... 

Finally, (2J + 1) is the degeneracy of the initial state and the expression involving the refractive index n is known as Lorentz s local-field correction. Calculations of transition probabilities within file frame of JO theory are usually made assuming that all Stark sublevels within the ground level are equally populated and that the material under investigation is optically isotropic. The former hypothesis is only reasonable in some cases, e.g., when transitions initiate from non-degenerate states such as Eu( Fo). Otherwise, there is a Boltzmann distribution of the population among the crystal-field sublevels. The second assumption is not valid for uniaxial or biaxial crystals, but, of course, holds for solutions. 

Friedrich (1970) has measured the absolute infrared intensities of the two optic librational modes of the solids of CICN and BrCN. He also calculated the intrinsic intensities of these modes using a dipolar coupling theory referred to in Section IVB.2. Good agreement was obtained between calculated and measured values. It was found that the local field correction [Eq. (4.22)] causes an increase of about 100% in the theoretical intensities for these substances, whereas the Lorentz effective field correction gave rise to an increase of less than 50%. The samples were prepared by deposition of gas on a cold window and the sample thicknesses were measured by an interference fringe technique. 

Fig. 3.6 The effective electric field acting on a molecule in a polarizable medium (shaded rectangles) is Eiq fErtj d where is the field in the medium and / is the local-field correction factor, hi the cavity-field model (A) Ei c is the field that would be present if the molecule were replaced by an empty cavity (Ecav), in the Lorentz model (B) Ei c is the sum of E av and the reaction field (Ereaa) resulting from polarization of the medium by induced dipoles within the molecule (P)...

Figure 3.7 shows the local-field correction factors given by Eqs. (3.35) and (3.36). The Lorentz correction is somewhat larger and may tend to overestimate the contribution of the reaction field, because the cavity-field expression agrees better with experiment in some cases (Fig. 4.5). 

Fig. 4.5 Dipole strength of the long-wavelength absorption band of bacteriochlorophyll-a, calculated by Eq. (4.16a) from absorption spectra measured in solvents with various refractive indices. Three treatments of the local-field correction factor (/) were used down triangles, f= 1.0 (no correction) filled circles, f is the cavity-field factor empty circles, f is the Lorentz factor. The dashed lines are least-squares fits to the data. Spectra measured by Connolly et al. [148] were converted to dipole strengths as described by Alden et al. [4] and Knox and Spring [5]...

L is the local field correction for which we employ the Lorentz 1/3 (n + 2) prescription. 

Most of the approaches used to obtain the local field correction factor are based on the Lorentz results, which state that the internal field (i.e., the local field as experienced by a molecule E ° in a solid) is related to the applied field by... 

The crudest approximation to the density matrix for the system is obtained by assuming that there are no statistical correlations between the elementary excitations (perfect fluid), so that can be written as a simple product of molecular density matrices A. A better approximation is obtained if one does a quantum field theory calculation of the local field effects in the system which in a certain approximation gives the Lorentz-Lorenz correction L(TT) in terms of the refractive index n53). One then writes,... 

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