Dispersion formulas

The total interaction between two slabs of infinite extent and depth can be obtained by a summation over all atom-atom interactions if pairwise additivity of forces can be assumed. While definitely not exact for a condensed phase, this conventional approach is quite useful for many purposes [1,3]. This summation, expressed as an integral, has been done by de Boer [8] using the simple dispersion formula, Eq. VI-15, and following the nomenclature in Eq. VI-19 ... 

Color from Dispersive Refraction. This mechanism involves the variation of the refractive index n with wavelength X, given by the SeUmeier dispersion formula ... 

Maximum Ground-Level Concentrations The effective height of an emission having been determined, the next step is to study its path downward by using the appropriate atmospheric-dispersion formula. Some of the more popular atmospheric-dispersion calculational procedures have been summarized by Buonicore and Theodore (op. cit.) and include ... 

If the compound causing the odor is known and can be chemically analyzed, it may be possible to get valid quantitative data by direct gas sampling. An example would be a plant producing formaldehyde. If the effluent were sampled for formaldehyde vapor, this could be related, through proper dispersion formulas, to indicate whether the odor would cause any problems in residential neighborhoods adjacent to the plant. 

The generality of a simple power series ansatz and an open-ended formulation of the dispersion formulas facilitate an alternative approach to the calculation of dispersion curves for hyperpolarizabilities complementary to the point-wise calculation of the frequency-dependent property. In particular, if dispersion curves are needed over a wide range of frequencies and for several optical proccesses, the calculation of the dispersion coefficients can provide a cost-efficient alternative to repeated calculations for different optical proccesses and different frequencies. The open-ended formulation allows to investigate the convergence of the dispersion expansion and to reduce the truncation error to what is considered tolerable. 

To obtain for 71 and jk compact dispersion formulas similar as Eq. (79) for 7, these hyperpolarizability components must be written as sums of tensor components which are irreducibel with respect to the permu-tational symmetry of the operator indices and frequency arguments ... 

A PRACTICAL ILLUSTRATION OF THE APPLICATION OF DISPERSION FORMULAE TO ODOUR PROBLEMS... 

This is very crude measurement. Nevertheless, it is sufficient to discriminate against some of the dispersion formulae that have been proposed. This matter is considered in the next paper where it is shown that the formulae which are satisfactory in that they put the anomalous dispersion at the observed frequencies yield values for (jun 1) which differ from that given by experiment, by several times the margin indicated in (8). 

Migration of Plutonium in Rock Incorrect Dispersion Formula," H. Krugmann, Science, 200, 88 (1978). 

The general dispersion formula obtained for the coupling of the vibrational equations with the Maxwell field can be brought into the form of Fresnel s wellknown equation for the wave normal from crystal optics. It is usually written in the form... 

When used as the dispersion formula for the phonons and polaritons in orthorhombic crystals, the symbols in Eq. (11.22) have the following meaning r)= 1,2,3 designates the three directions of the principal orthogonal axes. sv are the direction cosines of the normalized wave vector s = k/k with respect to the three principal axes of the crystal. If the unit vectors in the directions of these three principal axes are designated eue2,e3, one can write... 

Schroeder also derived an approach to optimally flatten a harmonic series [Schroeder, 1986, Schroeder, 1970a]. This method however requires exact harmonicity and can be shown to be a special case of the KFH phase dispersion formula. 

D. M. Bishop,/. Chem. Phys., 90, 3192 (1989). General Dispersion Formulae for Molecular Third-Order Nonlinear Optical Properties. 

If this result is put into the dispersion formula, one gets... 

A dispersion formula results, based on frequency dependency on masses, force constant and distance between the two masses, such as... 

For gases which deviate significantly from ideality, one may either derive a dispersion formula for a given equation of state, or else correct the observed sound velocity or absorption to ideality before fitting data to equations (79), (81), (82), or (83). 

With the possibility of an experimental verification at hand it becomes interesting to make this calculation. And here we appreciate the strength of Kramers dispersion formula— for not only is it difficult to check experimentally but the numerical computation is none too simple. For the sums (6) and (6 ) and not the same as those appearing in (4) and (5). 

For many simple gas molecules [e.g. the rare gases, Hg, Nj, Og, CH4), the empirical dispersion curve has been found to be representable, in a large frequency interval, by a dispersion formula of the type (14) consisting of one single term only. That means that for these molecules the oscillator strength for frequencies of a small interval so far exceed the others that the latter can entirely be neglected. In this case, and for the limiting case v -> O (polarisability in a static field) the formula (14) can simply be written ... 

This formula is identical with 13) in the case of two molecules of the same kind. It can, of course, only be applied if one already knows that the dispersion formula has the above-mentioned special form. But in any case, if the dispersion formulae of the molecules involved are empirically known, their data can be used and are sufiScient to build up the attractive force (15). No further details of the molecular structure need be known. 

We give, in Table I., a list of theoretical values for the attractive constant c [i.e. the factor of — i /i in the above interaction law) for rare gases and some other simple gases where the refractive index can fairly well be represented by a dispersion formula of one term only. The characteristic frequency vj) multiplied by h is in all these cases very nearly equal to the ionisation energy hvj. This may, to a first approximation, justify using the latter quantity in similar cases where a dispersion formula has not yet been determined. It is seen that the values of... 

Original author calculated on a different basis— used Slater-Kirkwood dispersion formula instead of isoelectronic compound. 

Table 5 shows the experimental specific refractivities, K X) = n(l) l]/ p, and the average polarizability as calculated from equation (1) at a number of frequencies for liquid and vapour phases. The values of the specific refractivity of the vapour have been obtained from the Cauchy dispersion formula of Zeiss and Meath.39 In this paper the authors assess the results of a number of experimental determinations of the refractive index of water vapour and its variation with frequency. Even after some normalization of the data to harmonize the absolute values from different determinations there is a one or two percent spread of results at any one wavelength. Extrapolation of the renormalized data for five independent sets of data leads to zero frequency values of K(7.) within the range (2.985-3.013) x 10-4 m3 kg 1, giving, via equation (1), LL — 9.63 0.10 au. Extrapolation of the earlier refractive index data of Cuthbertson and Cuthbertson40 by Russell and Spackman41 from 8 values of frequency between 0.068 and 0.095 au, leads to a zero frequency value, of y.i, 1,(0) = 9.83 au. While the considerable variation between the raw experimental data reported in different determinations is cause for some uncertainty, it appears that the most convincing analysis to date is that of... 

Curves A and B are alternative interpretations of the experimental situation. Curve B is a plot of the 6 term Cauchy dispersion formula derived by Zeiss and Meath, while curve A is a simple quadratic interpolation (2-term Cauchy formula) between the static value of Cuthbertson40 and the Zeiss-Meath39 value at 514.5 nm (the only point where the polarizability anisotropy has been measured). Theoreticians appear to have taken these two values to heart. Curves C and D are plots of similar formulae [a(co) = 4(1 + Bofi) derived theoretically by Christiansen et al.44 and Kongsted et al.45 respectively, using the methods shown in Table 6 with suitable time-dependent procedures. The points obtained from the MCSCF46 work and the DFT/SAOP method48 are also plotted. The ZPVA correction of 0.29 au has been added at all theoretical points at all frequencies. 

The polarisability tensor is given by a second-order perturbation expression known as the Kramers-Heisenberg dispersion formula (4)... 

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