Debye series

To understand better the connection between the geometrical optics approach and wave equation solutions, we will discuss in this section the basic equations describing high frequency scalar wavefield propagation. Following Bleistein (1984) and Bleistein et al. (2001), we represent the solution of the scalar wave equation (13.56) outside of the source in the form of the Debye series... 

We now substitute the Debye series into the left-hand side of the scalar wave-field equation (13.56) ... 

The situation for electrolyte solutions is more complex theory confimis the limiting expressions (originally from Debye-Htickel theory), but, because of the long-range interactions, the resulting equations are non-analytic rather than simple power series.) It is evident that electrolyte solutions are ideally dilute only at extremely low concentrations. Further details about these activity coefficients will be found in other articles. 

Equation (10.82) is a correct but unwieldy form of the Debye scattering theory. The result benefits considerably from some additional manipulation which converts it into a useful form. Toward this end we assume that the quantity srj, is not too large, in which case sin (srj, ) can be expanded as a power series. Retaining only the first two terms of the series, we obtain... 

D. Brydges, P. Federbush. Debye screening in classical Coulomb systems. In G. Velo, A. S. Wightman, eds. NATO Advanced Science Institutes. Series No. B 74. New York Plenum Press, 1981, pp. 371-385. 

Equation (7.25) can be substituted into equation (7.20) to give a second order differential equation in ijj. In theory, the resulting equation can be solved to give ip as a function of r. However, it has an exponential term in -ip, that makes it impossible to solve analytically. In the Debye-Hiickel approximation, the exponential is expanded in a power series to give... 

In this one-dimensional flat case the Laplace operator is simpler than in the case with spherical symmetry arising when deriving the Debye-Huckel limiting law. Therefore, the differential equation (B.5) can be solved without the simplification (of replacing the exponential factors by two terms of their series expansion) that would reduce its accuracy. We shall employ the mathematical identity... 

Whereas the latter expression must be solved numerically for low temperatures, the entropy at high temperatures can be derived by a series expansion [4], For the Debye or Einstein models the entropy is essentially given in terms of a single parameter at high temperature ... 

Figure 8.18 Entropy Debye temperature, , for (a) alkali earth dihalides [10] and(b) first series transition metal carbides [11].

Figure 5.8 A Debye-Scherrer powder camera for X-ray diffraction. The camera (a) consists of a long strip of photographic film fitted inside a disk. The sample (usually contained within a quartz capillary tube) is mounted vertically at the center of the camera and rotated slowly around its vertical axis. X-rays enter from the left, are scattered by the sample, and the undeflected part of the beam exits at the right. After about 24 hours the film is removed (b), and, following development, shows the diffraction pattern as a series of pairs of dark lines, symmetric about the exit slit. The diffraction angle (20) is measured from the film, and used to calculate the d spacings of the crystal from Bragg s law.

We then show that this series, which is formally meaningless for small k, may be explicitly summed and gives precisely the Debye-Huckel result (157). 

The mean activity coefficient is the standard form of expressing electrolyte data either in compilations of evaluated experimental data such as Hamer and Wu (2) or in predictions based on extensions to the Debye-Huckel model of electrolyte behavior. Recently several advances in the prediction and correlation of mean activity coefficients have been presented in a series of papers starting in 1972 by Pitzer (3, Meissner 04), and Bromley (5) among others. 

Friedman (1962) has used the cluster theory of Mayer (1950) to derive equations which give the thermodynamic properties of electrolyte solutions as the sum of convergent series. The first term in these series is identical to and thus confirms the Debye-Huckel limiting law. The second term is an I2.nl term whose coefficient is, like the coefficient in the Debye-Huckel limiting law equation, a function of the charge type of the salt and the properties of the solvent. From this theory, as well as from others referred to above, a higher order limiting law can be written as... 

An important series of papers by Professor Pitzer and colleagues (26, 27, 28, 29), beginning in 1912, has laid the ground work for what appears to be the "most comprehensive and theoretically founded treatment to date. This treatment is based on the ion interaction model using the Debye-Huckel ion distribution and establishes the concept that the effect of short range forces, that is the second virial coefficient, should also depend on the ionic strength. Interaction parameters for a large number of electrolytes have been determined. 

For a range of potential in which the interfacial charge is relatively small, the reciprocal of the interfacial electric capacity, C, of metal electrodes has conventionally been represented by a Laurent series with respect to the Debye length L-o of aqueous solution as shown in Eqn. 5-25 [Schmickler, 1993] ... 

Of course, when the volume concentration of mobile charges is sufficiently high that the Debye length is comparable with the ionic radius of the mobile ion(s), a combination of the Helmholtz and Gouy-Chapman models is required. This is achieved by assuming that the measured Cdi value is a series combination of that due to the Gouy-Chapman model (Cgc) and that due to the Helmholtz model (Ch), i.e. 

Experimenters would do well to avoid any unnecessary changes in the ionic composition of reaction samples within a series of experiments. If possible, chose a standard set of reaction conditions, because one cannot readily correct data from one set of experimental conditions in any reliable manner that reveals the reactivity under a different set of conditions. Maintenance of ionic strength and solvent composition is desirable, and correction to constant ionic strength often effectively minimizes or ehminates electrostatic effects. Even so, remember that Debye-Hiickel theory only applies to reasonably dilute electrolyte solutions. Another important fact is that ion effects and solvent effects on the activity coefficients of polar transition states may be more significant than more modest effects on reactants. 

The function on the right hand side of Eq. (34) consists of a series of elliptic integrals, which depend not only on the unknown electrostatic force but also on the surface charge densities, q and on the interface and protein surface, respectively, and on the inverse Debye screening length (1/K). 

Fig. 8.1.8 Electron micrograph of SijN4 UFPs and the electron diffraction pattern, where two series of Debye rings, (A) and (B), correspond to a- and P-S13N4, respectively. (From Ref. 38. Reprinted with permission of the Society of Materials Science, Japan.)...

Another member of this series is bis(cyclopentadienylnickel carbonyl), (CsHaNiCO it is dimeric, diamagnetic, and must, therefore, contain a nickel-nickel bond (79). The dipole moment, quoted (79) as 0 0.38 Debye unit, indicates that the molecule must be very nearly centro-sym-metric in benzene. The infrared spectrum, however, shows two carbonyl stretching frequencies in the solid state and in solution, but the vapor at 100°C shows only one band (173, 199). The wave-numbers are shown in Table IV. 

Wada 109, 110) pioneered studies of polypeptide conformation by the dielectric method. He found 110) a linear dependence of (ft2)1 2 on Mw for a series of PBLG samples (ranging from 7 x 104 to 18 x 104 in Mw) in EDC at 25° C and obtained 3.5 D for gh, where D stands for debye units. He computed (ft2) by the use of an approximate equation derived by himself 109) for rigid-rod molecules, which for very dilute solutions may be written... 

Mie wrote the scattering and absorption cross sections as power series in the size parameter 0, restricting the series to the first few terms. This truncation of the series restricts the Mie theory to particles with dimensions less than the wavelength of light but, unlike the Rayleigh and Debye approximations, applies to absorbing and nonabsorbing particles. 

We introduce the first of the Debye-Hiickel approximations by considering only those situations for which < kBT). In this case the exponentials in Equation (28) may be expanded (see Appendix A) as a power series. If only first-order terms in z,eyp/kBT) are... 

As a check on the consistency of our mathematics, it is profitable to verify that Equation (63) reduces to Equation (37) in the limit of low potentials. Expanding the exponentials in T and truncating the series so that only one term survives in both the numerator and denominator results in the Debye-Hiickel expression, Equation (37). 

Figure 1.1 A plot of relaxation times versus molecular volume for a series of closely related alcohols demonstrating the overall validity of the Debye equation. The outlying point refers to benzyl alcohol (see text for further discussion).

---