Drude Equation Research Articles

Using Guided Inquiry to Study Optical Activity and Optical Rotatory Dispersion in a Cross-Disciplinary Chemistry Lab

We propose a new lab for the determination of the optical activity and optical rotatory dispersion of a compound. First our students determine the optical activity of S-(+)-carvone using the standard polarimeter (λ = 589 nm) . Then they are given a He-Ne laser with a detector and a polarizer and are asked to design the most accurate method of determining the angle of rotation for a second wavelength (λ = 633 nm). The method of guided inquiry is used throughout this part of the lab. The students ultimately should conclude that the most accurate data are obtained by using the orientation of polarizer where the intensity I of light exiting the polarizer is half of the sum of the minimal and maximal values, the reason being that this is the point where the slope of the I (θ) curve is maximal (θ is the angle between the plane of polarization of light and the axis of the polarizer). They then analyze the results and determine whether they are better described by the Drude equation or its modified version.

Nonequilibrium thermodynamics formulation for electric transients. (I). The generalized Drude equation

The application of the nonequilibrium relaxation theory to describe the transient of electric charged system in a material out of equilibrium is presented. We show that in order to generalize the Drude equation, an internal variable has to be introduced to accommodate the influence of a possible internal mechanism, which modifies the natural evolution of a charged system. This variable could represent one of the following effects: (a) Trapped charges, (b) collective behavior of the electric charges with interactions and (c) resonance processes. The final description of the free charge evolution, affected by the internal process in the material, is expressed by means of a second-order differential relation, the so-called generalized Drude's equation. The application of this last result to different physical situations is discussed.

Carrier concentration and lattice absorption in bulk and epitaxial silicon carbide determined using infrared ellipsometry

We have measured the dielectric function of bulk nitrogen-doped 4H and 6H SiC substrates from 700 to 4000 ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$ using Fourier-transform infrared spectroscopic ellipsometry. Photon absorption by transverse optical phonons produces a strong reststrahlen band between 797 and 1000 ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$ with the effects of phonon anisotropy being observed in the region of the longitudinal phonon energy (960 to 100 ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$). The shape of this band is influenced by plasma oscillations of free electrons, which we describe with a classical Drude equation. For the 6H-SiC samples, we modify the Drude equation to account for the strong effective mass anisotropy. Detailed numerical regression analysis yields the free-electron concentrations, which range from $7\ifmmode\times\else\texttimes\fi{}{10}^{17}$ to ${10}^{19}{\mathrm{cm}}^{\mathrm{\ensuremath{-}}3},$ in good agreement with electrical and secondary ion mass spectrometry measurements. Finally, we observe the Berreman effect near the longitudinal optical phonon energy in $n\ensuremath{-}/n+$ homoepitaxial 4H SiC and hydrogen implanted samples, and we are able to determine the thickness of these surface layers.

Electric potential energy of the incident light and the Hamiltonian of the induced oscillators in non-absorbing isotropic dielectrics

By considering an energy balance between the electric potential energy of the incident light and the Hamiltonian of the induced oscillators in a non-absorbing isotropic medium, the Newton–Drude equation, the Sellmeier dispersion formula, and the wavelength-dependent mean polarizability were derived. The Newton–Drude formula and the Lorentz–Lorenz equation were used to calculate the density-dependent mean polarizability of the densified silica glasses. The evaluated mean polarizability was compared with the band shifts in the UV- and IR-spectra with the density of glasses, and the validity of the Newton–Drude formula was discussed.

Review and analysis of refractive index temperature dependence in amorphous SiO 2

Combining previously proposed relationship describing the density-dependent mean polarizability with the Newton–Drude formula and Lorentz–Lorenz equation, two relations were obtained to study of the temperature dependence of refractive index of glasses. These relations were used to study the temperature-dependent refractive index of silica glass for different wavelengths. The validity of the Newton–Drude equation is discussed in terms of an energy balance between the electric potential energy of the incident light and the Hamilton function of the induced oscillators.

Pulse Electrical Current Resonance: Classical and Quantum Echoes

We discuss the theoretical basis of a recent proposal for a new experimental technique in Solid State Physics: Electrical Current Resonance (ECR); continuous wave (CW) and pulsed. In the first part of the paper we show that the Bloch equations for magnetic moments in magnetic fields are formally identical to the Drude equation of motion for the electrical current in electric and magnetic fields. Then, applying a matrix method to solve the current equation of motion, we explore the consequences of this similarity. First we consider continuous magnetic fields; various examples of applications are worked out. In the pulsed case we deduce expressions for the current echoes and free current decays, transient effects similar to spin echoes and free induction decays in the magnetic case. We briefly discuss the application of the technique to the study of electron scattering rates and ECR spectroscopy. On the second part we show that the quantum problem of a charged particle in a oscillating magnetic field has an exact solution, both for continuous and pulsed fields. In the continuous case we show that the particle energy states in the field (the Landau tubes) are turned by 90° as the resonance frequency is approached. In the pulsed case we calculate the expected value of the particle momentum [Formula: see text] after a sequence of one and two magnetic pulses, and show that this quantity corresponds to the free current decay and current echo, deduced classically.

Linear and nonlinear optical properties of sol-gel-derived Au nanometer-particle-doped alumina

The precursor Au-doped alumina gel film was prepared by a sol-gel method using HAuCl4⋅4H2O and alumina sol derived from AlCl3⋅6H2O. The precursor gel was heat treated in a H2 gas flow at 300–800 °C. Average particle diameters of the Au particle of doped films were in the range of 4.6–12.7 nm. In the absorption spectra of the film, plasmon resonance absorption was strongly dependent on the heat treatment temperature of the alumina matrix, showing more sharp spectra and significant red shift with decreasing particle size. It was interpreted by a modified Mie–Drude equation by taking into consideration of a spilling out of electrons from the Au particle to the alumina matrix. The measured values of χm(3) of the doped films are in the range of 9.2×10−6–4.0×10−5 esu in the neighborhood of plasmon resonance wavelength of 530–570 nm. These are larger than those reported for the other doped systems. A dominant nonlinear response on the 1–3 ps time scale was obtained, reflecting higher thermoconductivity of alumina compared to that of conventional matrix such as SiO2.

Optical properties of plane and rough interfaces in the electrostatic approximation

The optical properties of an interface with small-scale roughness (i.e. the wavelengths and the amplitudes of the modes of this roughness being smaller than the wavelength of the light) are calculated, without restriction on the local interfacial slope. The ellipticity of the reflected light (which can be measured by ellipsometry) on a rough interface with small slope is rederived; it is proportional to Σqζ 1 2 (where ζ q is the amplitude of the mode q). In this case the ellipticity is sensitive to the amplitude and the slope of the roughness. It is shown that when the interfacial slope and the deepness of the roughness increase, the interface becomes equivalent to a flat but diffuse interface, the optical properties of which can be calculated using a generalized Drude equation, taking into account the interfacial optical anisotropy.

Optical and Electronic Transport Properties of Electrodeposited Thallium(III) Oxide Films

Thallium oxide is a degenerate n-type semiconductor which can be electrodeposited from aqueous solution at room temperature. Thin films were characterized by transmission and specular reflectance spectroscopy and by four-point resistivity and Hall measurements. Optical parameters were determined by fitting the observed specular reflectance to the Drude equation. Due to the high free carrier concentration, the material reflects strongly in the near-infrared, and the band-to-band optical transitions are shifted by up to 1.1 eV by the Moss-Burstein effect. The optical and electrical properties of the films were a function of the deposition overpotential. Films grown at 44 mV had an intrinsic bandgap of 0.66 eV, resistivity of 2.8 x 10{sup -4} ohm-cm, mobility of 27 cm{sup 2}Vs, and conduction band effective mass of 0.43m{sub o}. Films grown at 300 mV had an intrinsic band gap of 0.51 eV, resistivity of 7.8 x 10{sup -5} ohm cm, mobility of 93 cm{sup 2}V s, and conduction band effective mass of 0.29m{sub o}. Mobilities measured by contact and optical methods are similar, which shows the optical technique may be used for conditions in which contact methods might fail. 20 refs., 11 figs., 2 tabs.

Optical rotatory dispersion (ORD) of glycogen

The ORD spectra of three types of glycogen in salt solutions at different concentrations, pH and ionic strength values were determined to investigate the influence of these variables on the rotatory power. The ORD curves obtained with the samples recently prepared correspond to positive monotone dispersion curves, similar to available data for the other polysaccharides. Once the mutarotation equilibrium is attained a maximum is observed at 210±10 nm. The rotatory power decreases with the elapsed time and in some cases, it is possible to find a change in the sign of these curves. Such a phenomenon may be attributed to conformational changes in the monomer units constituting the macromolecule. A considerable increase in the rotatory power is found at pH extreme values. The dispersion constant,λ c , was calculated by means of a modified Drude equation and results indicate that theλ c value is higher for muscle glycogen in the mussel species, suggesting a higher degree of asymmetry in this macromolecule.

Dihydrozeatin: An improved synthesis and resolution of both isomers

(±)-Dihydrozeatin was synthesized in a 3-step synthesis by: (1) a Michael condensation of methyl methacrylate with nitromethane to give (±) - methyl 2 - methyl - 4 - nitrobutyrate, which was (2) reduced to (±) - 4 -amino - 2 - methylbutan - 1 - ol and (3) reaction of the aminoalcohol with 6-chloropurine. Hydrolysis of racemic nitroester gave (±) - 2 - methyl - 4 - nitrobutyric acid, which was resolved by means of (+) - and (-) -α -methylbenzylamine salts. Conversion of the salts to the corresponding methyl esters and subsequent reductions yielded optically active 4 - amino 2 - methylbutan - 1 - ols. Examination of the NMR spectra of the resolved methyl 2 - methyl - 4 - nitrobutyrates in the presence of a chiral shift reagent established their optical purities to be greater than 98%. The specific rotations at 589 nm of theS- (−) andR- (+)- dihydrozeatins derived from optically active butanols were appreciably lower than previously reported. Application of the Drude equation to ORD values from 320 to 589 nm verified the low 589 nm rotations of the dihydrozeatin enantiomers. The biological activities of (R), (S) and (R,S) dihydrozeatins in the betacyanin stimulation assay withAmaranthus parallel the activities found in other cytokinin bioassays.

Thermal radiative properties of some cryogenic materials

The normal reflectivities of Al and Al alloys which are important as cryogenic materials are measured mainly in the infrared region below room temperature down to 16 K. A new model is proposed which assumes that in the Drude equation, additional scattering of conduction electrons occur at the surface layer which explains the temperature dependence of the radiative properties. This model can predict the radiative properties of metals at low temperatures better than the anomalous skin effect theory.

Extension of the ring oxygen helicity rule to phenyl 1-thioglycopyranosides.

A strong Cotton effect, which practically govern the sign of the optical rotation at 589 nm ([M]D), was studied in phenyl 1-thio-α (and β)-D-glycopyranosides with our new chiroptical technique. The proposal optical rotatory dispersion (ORD) method, with calculations based on a one term Drude equation, showed the presence of a strong Cotton effect at 200-210nm. Circular dichroism (CD), with accumulation technique, also gave the same Cotton effect. Agreement in these two methods suggests the usefulness of the proposed ORD calculation method. The rotational strengths and the signs were shown to reflect the anomeric configurations and conformations (α-anomer gave positive and β-anomer gave negative signs; axial gave strong and equatorial gave weak bands). This result is an extension of the ring oxygen helicity rule of alkyl and alkyl thioglycosides to phenyl 1-thioglycopyranosides, and probably to other aromatic glycosides.

Form of the memory function for a degenerate electron gas with elastic scatterers

The high frequency response of an electron gas with elastic scatterers can be described by a Drude equation with a memory term, which is an integral over a kernel times the square of the scattering potential. In the momentum (and frequency or time) representation, this kernel is discontinuous at the diameter of the Fermi sphere.

Brillouin scattering and pockels coefficients in silicate glasses

The Brillouin spectra of binary and ternary silicate glasses were investigated and absolute values for the Pockets coefficients p 12 and p 44 and ( p 11 – p 12) were calculated from the measured spectra. The results were interpreted as to the density and refractive index dependence of the Pockels coefficients in terms of the theoretical treatments by Carleton, Mueller and Sipe. From the measured p 12 and p 44 values the quantity ( ϱ∂ϵ/ ∂p), the change of the dielectric constant with respect to density, was calculated and compared with ( ϱ∂ϵ/ ∂ϱ) evaluated from the Lorentz-Lorenz equation, Drude equation and the Carleton model. The Lorentz-Lorenz and Drude equations overestimated the magnitude of ( ϱ∂ϵ/ ∂ϱ), while the Carleton model with its two limiting cases set upper and lowe bounds to ( ϱ∂ϵ/ ∂ϱ). From the ( ϱ∂ϵ/ ∂ϱ) results the strain polarizability constant ( λ 0) was determined and its physical significance discussed. In general, the alkali-oxide concentration dependence of the Pockels coefficients of various silicate glasses has been measured and qualitatively explained by several existing theoretical formulations.

Far UV Optical Activity of Sodium Chlorate

Abstract The optical rotatory dispersion (ORD) and circular dichroism CD) of sodium chlorate is presented in the far UV at and near the UV absorption peak. Although previous data published on sodium chlorate showed the ORD in the visible and near UV, the far UV was less accessible due to the large absorption in the region of interest. This obstacle has been overcome by using thin plates of single crystal sodium chlorate. The most significant aspect of this work is that it confirms previous predictions of an anomaly near the absorption peak based upon a Drude equation.

Optical properties and viscosity of hyaluronic acid in mixed solvents: Evidence of conformational transition

AbstractCircular dichroism, optical rotatory dispersion, and viscosity of hyaluronic acid at various solvents compositions, concentrations, and pH values have been studied. The data show a large change in the molecular properties in organic/water solvents such as ethanol, p‐dioxane, or acetonitrile/water at pH ⩽ pKa. At this pH range of aqueous solution, hyaluronic acid shows a CD minimum near 210 nm whereas in the presence of organic solvent it exhibits a strong negative dichroism (below 200 nm) and a positive band near 226 nm. It undergoes a sharp, cooperative transition with respect to pH and solvent. The observed CD features are assigned to the π‐π* and n‐π* transitions of the amide and carboxyl chromophores. The ORD results show a gradual blue shift of trough at 220 nm with increasing magnitude of rotation when the organic solvents and hydrogen ion concentrations are increased. A one‐term Drude's equation was used to analyze the ORD data, and the result show a variation of dispersion parameters with different solvents in accordance with the observed CD changes. The intrinsic viscosity of hyaluronic acid in mixed solvent at pH 2.6 is lower than that of aqueous solution. All the observed property changes of hyaluronic acid are reversed on addition of foramide in mixed solvents indicating that the hydrogen bonds are involved in this transition. The observed spectroscopic and hydrodynamic features are attributed to a conformational change of hyaluronic acid in a mixed solvent involving intramolecular hydrogen bonding between the acetamido and carboxyl groups. The possible conformational state of hyaluronic acid in solution under various conditions is discussed in terms of the reported helical structure of hyaluronic acid from x‐ray diffraction studies.

Polypeptide models of collagen: Properties of (Pro-Pro-betaAla)n.

AbstractWe have synthesized (Pro‐Pro‐βAla)n as a model for collagen. The synthetic polytripeptide, mol wt 6500, exhibits a large negative optical rotation with a very strong negative Cotton effect centered at 216 nm. The optical rotatory dispersion of (Pro‐Pro‐βAla)n followed a single‐term Drude equation and the λc was 195 nm. The rotation decreased markedly on heating with the midpoint of the broad transition at 55°C. Preliminary studies also showed loss of structure in guadinine HCl. The circular dichroism spectrum of the polymer exhibited a deep trough at 190 nm. The marked similarities of solution properties of (Pro‐Pro‐βAla)n to (Pro‐Pro‐Gly)n suggest that β‐alanine can replace glycine in generating collagen‐like helix in solution.

Magneto optical rotatory dispersion studies of simple electrolyte solutions. Part 3.—Measurements on solutions of chlorides and bromides of some group II metals

The m.o.r.d. of concentrated aqueous solutions of MgCl2, CaCl2, SrCl2, BaCl2, BaCl2 CaBr2, SrBr2 and BaBr2 were measured in the wavelength range 350–600 nm. The variation of the specific rotations in terms of the effects of the solutions with concentration of solute and the molar magnetic rotations could be interpreted in terms of the effects of the solutes upon the competing processes of solvent structure-making and structure-breaking. There was a correlation between molar magnetic rotation and total ionic volume. The changes in rotation with wavelength were fitted to a one-term Drude equation and the constants of this equation, K and λ0, were calculated by the Biot–Lowry method. The values of λ0, however, could not be identified with the peak position of the halide ion charge-transfer-to-solvent spectrum and the Drude equation is merely a convenient empirical relationship for the m.o.r.d. results in the wavelength range studied. The variation of K with solute concentration paralleled the relative effects of cations and anions on the structure of water.

Conformational change of the triple-helical structure. II. Conformation of (Pro-Pro-Gly)n and (Pro-Pro-Gly)n (Ala-Pro-Gly)m(Pro-Pro-Pro-Gly)n in an aqueous solution.

AbstractMeasurements of the molecular weight of (Pro‐Pro‐Gly)n and (Pro‐Pro‐Gly)n(Ala‐Pro‐Gly)m(Pro‐Pro‐Gly)n, which were synthesized by the solid‐phase method, revealed that they formed a trimer in an aqueous solution, and dissociated into single‐stranded chains on warming. Accompanying the transition, a large decrease of optical rotation was observed, like the collagen–gelatin transition. The shape of the trimeric molecule was rodlike, and the dimensions were 12 Å in diameter and 2.8 Å per residue in length, regardless of the length of Ala‐Pro‐Gly sequences in a peptide chain. The data indicate that both Pro‐Pro‐Gly sequences and Ala‐Pro‐Gly sequences from the triple‐helical structure similar to that of collagen in aqueous solution. All optical rotational dispersion (ORD) curves of solutions of the peptides were represented by a single‐term Drude equation, and the Drude constant λc was 200 nm for all peptides regardless of the length of Ala‐Pro‐Gly sequences. The resemblance between the helical structure formed by Pro‐Pro‐Gly sequences and that by Ala‐Pro‐Gly sequences was also suggested by the formation of the hybrid triple helix from two kinds of peptide chains with different lengths of Ala‐Pro‐Gly sequences.