Debye formula

By replacing sin9d9 for - d(cos9), we finally arrive at the common Debye formula... 

The origin of the n2 measured using the 10 ns pulses could be electronic or molecular rotation. These can be distinguished by measuring the ratio of the critical power for self-focusing for linear and circular polarised light. The observed ratio of 2.1 is consistent with a molecular rotation (11-13.161 and relates to the anisotropic polarisability of the molecule. The rotational relaxation time, calculated from the Debye formula (H), is about 0.5-2 ns, consistent with these results. 

The tunnelling mechanism has been strongly indicated when many diffusion-controlled reactions have been examined quantitatively (Anbar and Hart, 1968). Of a large number of diffusion-controlled reactions examined, over 80% exhibited a rate in excellent agreement with that predicted by the Smoluchovsky-Debye formula (Debye, 1942)... 

At a scattering angle 20, the scattered intensity is related to the structure of the particle by the Debye formula,... 

At small values of Q, the Debye formula reduces to the Guinier approximation,... 

Conclusions that association should occur in many other cases on the ground of a departure from the simple additive Debye formula for the electrical molecular polarization are invalid, since this formula has been proved to be un-applicable to concentrated solutions (Onsager, Bottcher). With the improved formulae of these authors practically the same dipole moments as in dilute solution are found even for the pure polar liquids, as far as they are not truly associated through the formation of hydrogen bonds. 

First, consider a very dilute gas of molecules [16]. The conventional theory of the static dielectric susceptibility % of such a gas invokes the notion of polarizable molecules with permanent dipole moments that are partially aligned by the external electric field . Standard techniques of statistical thermodynamics produce the Langevin-Debye formula for x Per molecule that reads... 

In other words, the conventional Langevin-Debye formula is recovered but with the expectation value of the square of the dipole moment operator in place of the square of the permanent dipole moment. 

If the dielectric consists of Nx molecules, having mutually non-interacting electric dipole moments p, (241) reduces to the well-known Langevin-Debye formula ... 

On resorting to the Oausius-Mosotti or the Langevin-Debye formula, we obtain ... 

These assumptions and procedures are certainly not desirable. As a result his expression (4.24) does not reduce to the Debye formula. Nevertheless his intuitive method seems to be essentially correct. 

In applying the Debye formulae to the calculation of heat capacities we require values of the double integral... 

At small angles a simplified form of the Debye formula for scattering from randomly oriented systems can be used to describe the scattering pattern because the absorption factors for the incident and the scattered beam are assumed to be the same and multiple scattering can be neglected. Then,... 

Since the frequency-dependent response of the solvent is included in the kernel of the integrals collected into X , the resulting hyperpolarizabilities will also depend on the frequency spectrum of the dielectric function e(u ) of the solvent. When e(w) is described by the Debye formula (i.e. in terms of a single relaxation mode), i.e. 

These results show that the presence of the solvent leads to an increase of the (a) values with respect to the vacuum, and it reverses its frequency dependence in aqueous solution, unlike in vacuo, the molecule shows a static (a) value which is larger than that at oj 0. This effect can be related to the shape of the e(w) function. By applying Debye formula it is easy to see that, at the frequency considered in our calculations, the value of e(w) is practically equal to e(oo). which is by far smaller than e(0). K one considers the sum-over-states method for the calculation of polarizabilities with these values in minds, then it is easy to give a qualitative explanation of the behavior indicated above. In fact, when the solvent response function is described by e(oo) the actual stabilization of the excited states will be less than in a situation where the same response depends on c(0), hence smaller it will be the correspondent (a) value. ... 

Ab initio calculation of powder scattering intensity was used for the simulation of XRD patterns. The calculation was done according to the Debye formula [5]. This approach makes it possible to simulate an XRD pattern of powder of nanoparticles of any shape and any structure including defects. The only restriction is an assumption that all orientations of nanoparticles in a powder are in equal amounts. 

For the simulation of XRD pattern of A -aloinic particles the slightly modified Debye formula was used... 

Experimental study and simulation argue that the atomic structure of CdS nanoparticles prepared by wet chemical method is disordered. As it was shown by XRD and simulation using the Debye formula the nanoparticles could have the shape of a hexagonal prism with the size of 5x3 nm2. Such CdS nanoparticles consist of about 3400 atoms. 

The first theoretical estimate of solvation time was obtained by generalizing the eontinuum model of Bom and Onsager by representing the dynamie properties of the solvent through a frequeney-dependent dieleetric constant, s oi), which is sometimes approximated by the simple Debye formula... 

In order to analyze carefully the frequency-dependent ellipsometric measurements described in the previous section, a precise determination of the frequency dependence of the dielectric constant e is needed. While, the dielectric constant of nonpolar polymers is nearly constant over a wide range of frequencies, that of polar materials decreases with increasing frequency (50), In the optical range, e generally increases with the frequency and this behavior is known as normal dispersion. At these high frequencies, the origin of the polarizability is mainly electronic. However, at moderate and low frequencies the dielectric constant is enhanced compared with its optical frequency value due to the motion of the molecular dipoles. This regime is called anomalous dispersion. The orientational and electronic contributions are found in the well-known Clausius-Mossotti formula for instance. In the simplest model, the frequency dependence of the dielectric constant can be described by the Debye formula (50) ... 

Assuming that the scattering law for each block and the whole copolymer molecule can be represented by the Debye formula (Eq. (15)), then the apparent radius of gyration of the copolymer is given by ... 

Using the general Debye formula for the scattering intensity and comparing the synthesized and experimental diffraction patterns [31] confirmed the validity of the model of y-alumina described above. 

Above 1K, however, the orientation of the nuclear spins become essentially random, and this change of orientation is not included in the Debye formula. 

The use of the Lorentz field for polar molecules with permanent momentsyU and polarizabilities a leads to the familiar Langevin-Debye formula... 

Clear cut simple relations between and have been established only for the primitive Lorentz and wtb store difficulty) Onsager local field models. Simple generalization of the Langevin-Debye formula (19) to relaxation gives a formula of the form (35) vith... 

It is the structure factor (F (h)) that correlates the scattering intensity and the configuration of the molecule it is the center of the study of x-ray scattering. The structure factor can be expressed in the well-known Debye formula (1915)... 

Hence we obtain the famous Debye formula for the intensity as a function of angle. 

The viscosity j calculated by the Landau theory and the /] evaluated by using the Debye formulas are shown for comparison in Fig. 80. 

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