Small oscillations and the Heisenberg Lie algebra

The Adler Kostant Symes [A-K-S] scheme is used to describe mechanical systems for quadratic Hamiltonians of $\mathbb R^{2n}$ on coadjoint orbits of the Heisenberg Lie group. The coadjoint orbits are realized in a solvable Lie algebra $\mathfrak g$ that admits an ad-invariant metric. Its quadratic induces the Hamiltonian on the orbits, whose Hamiltonian system is equivalent to that one on $\mathbb R^{2n}$. This system is a Lax pair equation whose solution can be computed with help of the Adjoint representation. For a certain class of functions, the Poisson commutativity on the coadjoint orbits in $\mathfrak g$ is related to the commutativity of a family of derivations of the 2n+1-dimensional Heisenberg Lie algebra $\mathfrak h_n$. Therefore the complete integrability is related to the existence of an n-dimensional abelian subalgebra of certain derivations in $\mathfrak h_n$. For instance, the motion of n-uncoupled harmonic oscillators near an equilibrium position can be described with this setting.Comment: 17 pages, it contains a theory about small oscillations in terms of the AKS schem

Facile synthesis of low band gap ZnO Microstructures

Abstract In this work a simple chemical route was employed to synthesize ZnO microparticles by precipitation from aqueous solution of ZnCl2 as precursor, NaOH as oxidizing and sodium dodecyl sulfate (SDS) as surfactant. Samples of ZnO microparticles were analyzed by scanning electron microscopy (SEM), FTIR spectroscopy, Raman spectroscopy, X-Ray diffraction, UV/Vis-NIR diffusereflectance,highresolutiontransmissionelectronmicroscopy(HR-TEM),andN2 adsorption-desorption.Itwasobserved from SEM analysis that ZnO microparticles with morphologies resembling six-blade impeller with diameters in the range of 500 nm to 1 m, and sheet-like (approximately 200 nm×300 nm) were obtained through this technique. X-Ray diffraction and Raman analyses confirmed the obtaining of hexagonal wurtzite ZnO crystal structure. The calculated band gap energy was 3.19 eV, which is slightly lower than the average value reported in the literature. Specific BET area of ZnO microparticles was 26.5 m2/g. Keywords: ZnO, microstructures, band gap energy, SEM, morphology. Resumen En este trabajo se empleó un a ruta química sencilla para sintetizar micropartículas de ZnO mediante precipitación en solución acuosa de ZnCl2 como precursor, NaOH como oxidante y dodecil sulfato de sodio (SDS) como tensoactivo. Las muestras de ZnO fueron analizadas mediante microscopía de barrido electrónico (SEM), espectroscopía de FTIR, espectroscopía Raman, difracción de rayos-X, reflectancia difusa de UV/Vis-NIR, microscopía de transmisión de electrones de alta resolución (HRTEM), y mediante adsorción-desorción de N2. Se observó mediante análisis de SEM que mediante esta técnica se obtienen micropartículas de ZnO con morfologías similares a impulsores de seis-aspas con diámetros entre 500 nm y 1 m, morfologías tipo-hojas(deaproximadamente200nm×300nm).Losanálisisdedifracciónderayos-XydeRamanconfirmaronlaobtención de ZnO con estructura cristalina wurtzita hexagonal. La energía de band gap calculada fue de 3.19 eV, la cual es ligeramente menor que el valor promedio reportado en la literatura. El área superficial BET de las nanopartículas de ZnO fue de 26.5 m2/g Palabras clave: ZnO, microestructuras, energía de band gap, SEM, morfología