Debye dispersion relation

Equation (11.25) is called the Debye dispersion relation or the Debye equation. The complex dielectric constant is defined to be... 

For an isolated dipole rotating under an oscillating field in a viscous medium, the Debye dispersion relation is derived in terms of classical mechanics. 

The dielectric constant is a natural choice of order parameter to study freezing of dipolar liquids, because of the large change in the orientational polarizability between the liquid and solid phases. The dielectric relaxation time was calculated by fitting the dispersion spectrum of the complex permittivity near resonance to the Debye model of orientational relaxation. In the Debye dispersion relation (equation (3)), ij is the frequency of the applied potential and t is the orientational (rotational) relaxation time of a dipolar molecule. The subscript s refers to static permittivity (low frequency limit, when the dipoles have sufficient time to be in phase with the applied field). The subscript oo refers to the optical permittivity (high frequency limit) and is a measure of the induced component of the permittivity. 

Figure 2. Spectrum plot for nitrobenzene in a 7.5 nm pore at T = —4 °C this plot yields two distinct dielectric absorption regions. The solid and the dashed curves are fits to the Debye dispersion relation.

The expression for e given on the RHS of equation (9.27) is called the Debye dispersion relation. Writing the dielectric constant s = s — is", it follows immediately from equation (9.27) that... 

The fact that a quantum oscillator of frequency u> does not interact effectively with a bath of temperature smaller than hu>/kg implies that if the low temperature behavior of the solid heat capacity is associated with vibrational motions, it must be related to the low frequency phonon modes. The Debye model combines this observation with two additional physical ideas One is the fact that the low frequency (long wavelength) limit of the dispersion relation must be... 

The Debye model discussed in Section 4.2,4 rests on three physical observations The fact that an atomic system characterized by small oscillations about the point of minimum energy can be described as a system of independent harmonic oscillators, the observation that the small frequency limit of the dispersion relation stems... 

We now consider a more quantitative model of the vibrational density of states which makes a remarkable linkage between continuum and discrete lattice descriptions. In particular, we undertake the Debye model in which the vibrational density of states is built in terms of an isotropic linear elastic reckoning of the phonon dispersions. Recall from above that in order to effect an accurate calculation of the true phonon dispersion relation, one must consider the dynamical matrix. Our approach here, on the other hand, is to produce a model representation of the phonon dispersions which is valid for long wavelengths and breaks down at... 

Some of the terms appearing in Eq. (3-52) are already defined for Eq. (3-51). The new terms are the following A is the scattering cross-section of the n-th nucleus of the unit cell, the Debye-Waller factor for the n-th nucleus of mass M , mj the frequency of the j-th nonnal mode of the r-th branch in the dispersion relation. The most important quantity is Ck Cj" which gives the projection along ko of the atomic... 

Debye dodges this little problem by using the dispersion relation for a homogeneous solid which has no cutoff frequency and in which k = (ojvo and dk/do) = 1 /vq. 

Let us examine the validity of the primary assumption of the Debye model, i.e., that the dispersion relation of a chain of atoms can be represented by a linear relationship corresponding to that of a homogeneous medium. Clearly this assumption is valid in the region where the wavelength is long compared to the atomic spacing, as was shown in Chapter 16, but what about the behavior near the cutoff frequency The results are shown in Figure 17.1. 

Dispersion relation for a continuous medium assumed by Debye (dashed line) compared with that of a chain of identical atoms (solid line). The frequency cutoff frequency for the chain of atoms < o = (4)S/m) which occurs at k= nla. 

Using the Planck distribution, Debye constructed a model in which the lattice heat capacity goes to zero as 7. The Debye model assumes a linear dispersion relation characteristic of a continuous medium rather than a chain of discrete atoms and cuts off the distribution at a frequency such that the number of normal modes is equal to 3 x the number of atoms. This frequency, knovm as the Debye frequency, is given by w-d = Vo 67t N/V), where Vq is the velocity of sound in the medium and N/V is the atoms per unit volume. A Debye temperature d is defined in terms of the Debye frequency d = hco-o/k. For T D, the Debye model approaches the classical Dulong-Petit limit. 

Even though the Debye model uses an unrealistic assumption for the dispersion relation, its primary justification comes from its excellent agreement with the observed heat capacity, especially at the lower temperatures. Apparently, the heat capacity is not particularly sensitive to the exact form of the dispersion relationship. 

As we have seen in Sect.2.2.5, the Einstein model represents the optical modes well, especially if their dispersion is weak, while the Debye model is more adapted to the acoustical modes. For this reason, the Debye model is often applied only to the acoustic modes and the Einstein model only to the optic modes. The corresponding approximations for the dispersion relation are shown in Fig.3.9d. For the density of states, we write in this case... 

In addition, NSE measurements have allowed confirmation of the relation between non-Debye behaviour and Q-dispersion of r eif (Eq. 4.11) proposed from the early BS studies of Sseif(Q,t) [148]. Figure 4.14b displays for PIB the characteristic times determined from the NSE experiments. The law pro-... 

There are a number of different enthalpic interactions that can occur between polymer and packing, and in many cases multiple interactions can exist depending on the chemical structure of the polymer. Enthalpic interactions that are related to water-soluble polymers include ion exchange, ion inclusion, ion exclusion, hydrophobic interactions, and hydrogen bonding (12)- Other types of interactions commonly encountered in SEC, as well as in all other chromatographic separations, are dispersion (London) forces, dipole interactions (Keeson and Debye forces), and electron-donor-acceptor interactions (20). 

The high frequency limit of for this second process is therefore n. The result of the fit is shown in Table III where the mean values of the various parameters and their associated 95% confidence intervals are given. Considering the small amplitude of the second dispersion both in absolute t rms and in relation to the main dispersion the parameters 6m, n and Y are quite well defined, and therefore it may be concluded that the double Debye representation is an acceptable description of the dielectric behaviour of water up to around 2THz. Other alternative interpretations are clearly possible but no attempt has been made here to follow these up at this stage. What is clear is that a small subsidiary dispersion region in the far infrared is necessary to account for all the presently available permittivity data, and that such a dispersion is centred around 650GHz and has an amplitude of about 2.4 in comparison with that of the principal dispersion which is approximately 75. 

The effect of the polarisability. and of the ionization potential IP may be directly related to their impact on the energy of the dispersive forces (London, Debye and Keesom) which govern physical adsorption onto activated carbon [4]. The lower positive effect of the molar mass M may be related to the influence of the molecular overcrowding which increases the surface contact with the solid, leading to more intensive interactions. 

As illustrated in some of these figures, all the a-loss peaks are well-fitted by the one-sided Fourier transform of the KWW over the main part of the dispersion. Thus, the experimental fact of constant dispersion at constant xa can be restated as the invariance of the fractional exponent KWW (or the coupling parameter n) at constant xa. In other words, xa and (or n) are co-invariants of changing thermodynamic conditions (T and P). If w is the full width at half-maximum of the dielectric loss peak normalized to that of an ideal Debye loss peak, there is an approximate relation between w and n given by n= 1.047(1 — w-1) [112],... 

Fig. 25.2. Typical plots obtained for the main models and realistic examples, (a) Frequency dispersion of e and adsorption e" for a pure Debye model on (b) the corresponding Cole-Cole plot (e" vs. s ), (c) Cole-Cole plot commonly encountered with a real dielectric solid showing an arc of circle centred out of the e abscissa and (d) with conduction loss, (e) Charge hopping model diagrammatic representation of the potential well and corresponding plots of ct(co) vs. logo) and e" vs. e for non-interacting free charges (f) similar plots for system with trapped charges examples of a" vs. a and Z" vs. Z plots are also given. Note the relation between the slope of a(m) and the depression of the semicircle in the a" vs. a plane also note that co increases in opposite directions in the e" vs. e and a" vs. a plots ...

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