Debye’s function

Now Debye s function decreases with temperature much more slowly than Einstein s this has already been mentioned above, but may be further illustrated by the subjoined short table —... 

Usually, however, we shall make use of Einstein s or Debye s functions to represent Cp. 

The above numbers were calculated by Fischer by assuming the formula proposed by Lindemann and myself. When Debye s paper on specific heats appeared I went through the corresponding calculation on the assumption of Debye s function for the specific heats as was to be expected, practically identical values were obtained. 

The atomic heat of solid mercury has been measured, by the two methods worked out in our laboratory, by Koref (loc. cit,), and then down to 31° abs. by Pollitzer (46) Koref s value fits perfectly into Pollitzer s series of observations. Pollitzer represented the measurements available by the formula of Lindemann and myself fiv — 97), plus a small additional term (= Cp — Ct). We now know that Debye s function is to be preferred, although the difference is very slight. It may easily be shown that when using this function one must put... 

Debye s function D is the characteristic size of nonuniform particles. Qp =... 

The coefliciont. h n(u) determines the contribution to a spectrum depending on the tran.s-lational diffusion coefficient of the coil as a whole only. The coefficient Si v) determines the first, most significant summand depending on the intramolecular motion with (he mode relaxational tirru with k = 1 5i is absent in liquation 61 aa. S l C -S u, and A l . ST Stibstitution of t = 0 into Equation 64 leads to Debye s function R d. R ) for Gau.ssian coils ... 

Akcasu and Sanchez (1988) have written potential 95 as a series in the vertex functions which, in their turn, are expressed via Debye s functions giAq Rj). 

We often posit the function 0(770 ), known as Debye s function, defined by ... 

Although long-time Debye relaxation proceeds exponentially, short-time deviations are detectable which represent inertial effects (free rotation between collisions) as well as interparticle interaction during collisions. In Debye s limit the spectra have already collapsed and their Lorentzian centre has a width proportional to the rotational diffusion coefficient. In fact this result is model-independent. Only shape analysis of the far wings can discriminate between different models of molecular reorientation and explain the high-frequency pecularities of IR and FIR spectra (like Poley absorption). In the conclusion of Chapter 2 we attract the readers attention to the solution of the inverse problem which is the extraction of the angular momentum correlation function from optical spectra of liquids. 

The solution of eqn. (44) for a coulomb potential with boundary conditions (45) and (46) for either initial conditions (48) or (49) has only been developed in recent years. Hong and Noolandi [72] showed that the solution of the Debye—Smoluchowski equation is related to the Mathieu equation. Many of the details of their analysis are discussed in the Appendix A, Sect. 4, and the Appendix eqn. (A.21) is the Green s function (fundamental solution), which is the probability that a reactant B is at r given that it was initially at r0. This equation is developed as the Laplace transform. To obtain the density of interest p(r, ), with either condition, the Green s function has to be averaged over the initial distribution, as in eqn. (A.12), and the Laplace transform inverted. Alternatively, the density p(r, ) can be found from the inverse Laplace transform of the linear combination of independent solutions (A.17) which satisfy the boundary and initial conditions. This is shown in Fig. 10. For a Boltzmann initial condition, Hong and Noolandi [72] found... 

In order to treat the combined effects of added salt and dielectric boundaries on a manageable level, we use screened Debye-Hiickel (DH) interactions between all charges. In the presence of a dielectric interface, the Green s function can in general not be calculated in closed form [114] except for (i) a metallic substrate (with a substrate dielectric constant e =oo) and (ii) for e =0 (which is a fairly accurate approximation for a substrate with a low dielectric constant). For two unit charges at positions r and r one obtains for the total electrostatic interaction including screening and dielectric boundary effects... 

Fig. XIV-3.—Specific heat of a solid as a function of the temperature, according to Debye s...

All the results confirm the fact (required by Debye s law) that the dielectric constant is a linear function of the reciprocal of the temperature. The electric moments (/x) of the molecules calculated from the slopes of the lines are in each case given under the table. 

Moreover, an equation for /as a function of absorptive crystal temperature, was obtained from the Debye s model for solids ... 

Taylor and Smith (10] reported S (298.15 K) = 32.29 cal K mol" based on S (60 K) 6.68 cal K" mol" obtained from Debye-Einstein functions which represented their C data to only 1.8 percent from 60 to 100 K. A comparison of their extrapolated C data with those which have been measured for SrCl, BaCl, and Cal, (4] indicates that the values for SrBr decrease much more rapidly with temperature below 50 K than would be expected. We have made our own extrapolation to 0 K for... 

For the intermediate region of medium temperatures expansions may be obtained similar to those used in the evaluation of Debye s atomic heat function.J To bring out this similarity we form the expression... 

Fig. 4.—Graph of specific heats at low temperatures according to Debye the small circles show observed points, the continuous curves correspond to Debye s theory. (-) is a temperature characteristic of the substance, such that C(= Cv) is a function of...

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