Drude expression

The first major absorption bands of alkyl halides lie in the vacuum ultraviolet (Table 6), outside the range of ordinary ORD and CD instruments. Measurements in the longer-wavelength regions of transparency often give rotations ([M] ) following a single-term Drude expression (equation 8),... 

Measurements of the polarized reflectance in the NIR have frequently been used to obtain estimates for the transfer integrals. The method consists in fitting a reflectance model based on the Drude expression [Eq. (1)] to the experimental data. The Drude expression should be considered as a tool in estimating the plasmon frequency, ftp the background dielectric constants, e0 plasma frequency, (op and so on. The validity of the Drude analysis is limited to the conducting organic materials, with the electrical conductivity not less than a few S cm-1. 

The origin of nonzero bo values in these cases is not entirely clear, but it can be formally traced to the difference between Xo and values for the simple dispersion of random coils and hence to a failure of the assumption that Xo equals Xo. If K equals Xo, then the first term of the Moffitt equation will of course be the same as the simple Drude expression known to describe the data and, there being no necessity for a second term, bo will vanish. However, if Xo differs from Xo, the Moffitt plot may still be linear but with a nonvanishing slope. Thus dispersion data that are simple when referred to one dispersion constant may appear complex when plotted against another by a form that sees matters as complex, thereby generating what may be properly suspected as pseudocomplexity. The Moffitt equation was initially intended to describe the complex dispersion of polypeptides for which the simple Drude equation is inadequate, but, as will be seen, its form is also applied to protein dispersions which can be expressed equally well by either formula. It is therefore important to examine more fully the relation of the two equations for cases in which both fit the data. 

Simple dispersion, in which only one Drude term is required, is often observed. This implies that one particular transition in the ultraviolet controls rotation at longer wavelengths, but this is usually an artefact of arithmetic since the sum of several Drude terms can often be approximated by one, in which is averaged and k is summed. In those cases where the optically active absorption bands make oppositely signed contributions, the data may fit best a two-term Drude expression ... 

For (-I-)-3-methylhexane, where measurements over the range A = 643.8 — 252.9 nm were made, a single-term Drude expression can be calculated from the data reported ... 

Absorption spectroscopy in the vacuum ultraviolet shows a strong band at 141-143 nm with normal paraffins and another at about 157 nm in isobutane and neopentane. A third band at 128-134 nm is prominent in all of these hydrocarbons as well. Cyclopropane shows bands at 162, 145 and, very strongly, at 124 nm, while cyclobutane shows end absorption up to 185 nm and peaks at 152, 142 and 126 nm. There is a reasonable probability that these are the bands indicated by the Drude expressions and, therefore, that they control the optical activity of saturated hydrocarbons in the quartz ultraviolet and visible. For trans-1,2-dimethylcyclopropane CD measurements down to 150 nm show one Cotton effect at 190 nm and two of opposite... 

Since there is no restoring force for the conduction electrons, we rightly put > 0. This provides the Drude expression widely used for metals and semiconductors. 

Compare the expression for the dielectric function due to intraband transitions, Eq. (5.49), to the Drude result, Eq. (5.39), and identify the constants in the latter in terms of fundamental constants and the microscopic properties of the solid. Provide a physical interpretation for the constants in the Drude expression. 

Kramers-Kronig analysis of the reflectance data provides the optical conductivity function cr(tu). We can obtain information about the conduction electrons in PAN-CSA by comparing this function with the Drude expression for free electrons and the localization models. The room temperature aito) is shown in Fig. 3.16 for samples A-G. In sample G there is a strong peak at —1.5 eV and a weaker peak at —0.6 eV. For... 

Figure 11.10 Complex, frequency dependent conductivities for PbS measured 10 ps after photoexcitation for two different 266 nm excitation fluences (black and grey dots). The data is described well by the Drude expression (solid lines) yielding the plasma frequency (directly related to density) and the carrier scattering time.

Einstein (f,) remarked that this point of view can be carried over to the theory of the energy content of a solid body if we suppose that the positive ions of Drude s theory ( 198) may be looked upon as the vibrating resonators, and the seat of the heat content of the body (Korperwarme). He calculated the expression ... 

The dependence on the nuclear positions is indicated by r and the dependence on the Drude positions is indicated by d. In Eq. (9-25) Ubond (r) is the intramolecular energy contribution from, typically, the bond lengths, valence angles, and dihedral angles, Ulj (r) is a Lennard-Jones 6-12 nonpolar contribution, Ueiect (r, d) represents all Coulombic interactions, atom-atom, atom-Drude, and Drude-Drude, and Useif (d) represents the atom-Drude harmonic bonds. The term Usey (d) arises from the harmonic spring separating the two charges and has the simple expression... 

The observed darkening of the indium slides results from a shift of the absorption peak because of the coating on the particles. Because of the cumbersomeness of the expressions for coated ellipsoids (Section 5.4) this shift can be understood most easily by appealing to (12.15), the condition for surface mode excitation in a coated sphere. For a small metallic sphere with dielectric function given by the Drude formula (9.26) and coated with a nonabsorbing material with dielectric function c2, the wavelength of maximum absorption is approximately... 

The optical constants of a metal are determined to a large degree by the free electrons. According to the Drude model, the contribution of the free electrons to the frequency-dependent dielectric function is expressed as follows (16) ... 

At a farther distance from the absorption region, the rotation of the molecular system is expressed as a sum of the one-term Drude formula ... 

Drude obtained the expression k/oT (3/2) (kB/e)2 = l.ll x 10 8 watt Q/K2, which is 50% of the approximate experimental k/itT values. This good fit was made possible by the fortuitous cancellation of two errors, both factors of about 100 in particular, a large value for the electronic specific heat Cv was used, which by classical equipartition arguments would equal (3/2) kB, but is 100 times smaller experimentally. Drude also estimated the thermopower as Q = —kB/2e. 

AF is related to the well-known volume contraction normally observed on dissolution of electrolytes and called electrostriction. The contraction of the solvent surrounding an ion is best expressed by the Drude-Nemst equation (5-172) ... 

This expression is analogous to the general Drude formula (10), but it suggests that, in a spectral region distant from the optically active transitions of a helical polymer, the dispersion will be complex rather than simple. 

In order to apply this general relationship to the empirical data of helical polymers, Moffitt and Yang (1956) then developed a phenomenological equation in which the above sums are replaced by single terms and thus derived an expression that is analogous to the simple Drude equation. 

The proposal to assimilate the rotatory dispersions of proteins to those of synthetic polypeptides is, in barest form, the recommendation to treat these same data by the Moffitt expression despite the fact that they can be described by the simple Drude equation. Thus one proceeds to plot [m ] -(X — Xo)/Xo against Xo/(X2 — Xo) with Xo set to 212 m/j, and thereby obtains an intercept, a , and a slope, which can be given conformational interpretations through Eqs. (35) and (36). These parameters, which are in general cited simply as Oo and foo when their experimental origin is... 

The simplest electronic theory of metals regards a metallic object as abox filled with noninteracting electrons. (A slightly more elaborate picture is the jellium model in which the free electrons are moving on the backgroimd of a continuous positive uniform charge distribution that represents the nuclei.) The Drude model, built on this picture, is characterized by two parameters The density of electrons n (number per unit volume) and the relaxation time r. The density n is sometimes expressed in terms of the radius of a sphere whose volume is the volume per electron in the metal... 

Finally, the dispersion term between molecules A and B can be approximated by the following expression due to Drude [29] ... 

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