Properties of α-monoclinic selenium

The growth of bulky and platelet shaped α-monoclinic crystals is discussed. A simple method is devised for identifying and orienting them.

The density, previously in disagreement with the value calculated from x-ray studies, is carefully redetermined, and found to be in good agreement with the latter.

The relative dielectric constant is determined, an effort being made to eliminate errors inherent in previous measurements, which have not been in agreement. A two parameter model is derived which explains the anisotropy in the relative dielectric constant of orthorhombic sulfur, which is also composed of 8-atom puckered ring molecules. The model works less well for α-monoclinic selenium. The relative dielectric constant anisotropy is quite noticeable, being 6.06 along the crystal b axis, and 8.52-8.93 normal to the axis.

Thin crystal platelets of α-monoclinic selenium (less than 1µ thick) are used to extend optical transmission measurements up to 4.5eV. Previously the measurements extended up to 2.1 eV, limited by the thickness of the available crystals. The absorption edge is at 2.20 eV, with changes in slope of the absorption coefficient occurring at 2.85 eV and 3.8 eV. Measurement of transmission through solutions of selenium in CS_2 and trichlorethylene yield an absorption edge of 2.75 eV. There is evidence the selenium exists in solution partly as Se_8 rings, the building block of monoclinic selenium. Transmission is measured at low temperatures (80°K and 10°K) using the platelets. The absorption edge is at 2.38 eV and 2.39 eV, respectively, for the two temperatures. Measurements at low temperatures with polarized and unpolarized light reveal interesting absorption anisotropy near 2.65 eV.

Oxidation of Volatile Organic Compounds in Aqueous Solution and at the Air-water Interface of Aqueous Microdroplets

Isoprene (ISO),the most abundant non-methane VOC, is the major contributor to secondary organic aerosols (SOA) formation. The mechanisms involved in such transformation, however, are not fully understood. Current mechanisms, which are based on the oxidation of ISO in the gas-phase, underestimate SOA yields. The heightened awareness that ISO is only partially processed in the gas-phase has turned attention to heterogeneous processes as alternative pathways toward SOA.

During my research project, I investigated the photochemical oxidation of isoprene in bulk water. Below, I will report on the λ > 305 nm photolysis of H₂O₂ in dilute ISO solutions. This process yields C₁₀H₁₅OH species as primary products, whose formation both requires and is inhibited by O₂. Several isomers of C₁₀H₁₅OH were resolved by reverse-phase high-performance liquid chromatography and detected as MH⁺ (m/z = 153) and MH⁺-18 (m/z = 135) signals by electrospray ionization mass spectrometry. This finding is consistent with the addition of ·OH to ISO, followed by HO-ISO· reactions with ISO (in competition with O₂) leading to second generation HO(ISO)₂· radicals that terminate as C₁₀H₁₅OH via β-H abstraction by O₂.

It is not generally realized that chemistry on the surface of water cannot be deduced, extrapolated or translated to those in bulk gas and liquid phases. The water density drops a thousand-fold within a few Angstroms through the gas-liquid interfacial region and therefore hydrophobic VOCs such as ISO will likely remain in these relatively 'dry' interfacial water layers rather than proceed into bulk water. In previous experiments from our laboratory, it was found that gas-phase olefins can be protonated on the surface of pH < 4 water. This phenomenon increases the residence time of gases at the interface, an event that makes them increasingly susceptible to interaction with gaseous atmospheric oxidants such as ozone and hydroxyl radicals.

In order to test this hypothesis, I carried out experiments in which ISO(g) collides with the surface of aqueous microdroplets of various compositions. Herein I report that ISO(g) is oxidized into soluble species via Fenton chemistry on the surface of aqueous Fe(II)Cl₂ solutions simultaneously exposed to H₂O₂(g). Monomer and oligomeric species (ISO)1-8H⁺ were detected via online electrospray ionization mass spectrometry (ESI-MS) on the surface of pH ~ 2 water, and were then oxidized into a suite of products whose combined yields exceed ~ 5% of (ISO)1-8H⁺. MS/MS analysis revealed that products mainly consisted of alcohols, ketones, epoxides and acids. Our experiments demonstrated that olefins in ambient air may be oxidized upon impact on the surface of Fe-containing aqueous acidic media, such as those of typical to tropospheric aerosols.

Related experiments involving the reaction of ISO(g) with ·OH radicals from the photolysis of dissolved H₂O₂ were also carried out to test the surface oxidation of ISO(g) by photolyzing H₂O₂(aq) at 266 nm at various pH. The products were analyzed via online electrospray ionization mass spectrometry. Similar to our Fenton experiments, we detected (ISO)1-7H⁺ at pH < 4, and new m/z⁺ = 271 and m/z^- = 76 products at pH > 5.

Extended fractional Fourier transforms

The concept of an extended fractional Fourier transform (FRT) is suggested. Previous PBT's and complex FRT's are only its subclasses. Then, through this concept and its method, we explain the physical meaning of any optical Fresnel diffraction through a lens: It is just an extended FRT; a lens-cascaded system can equivalently be simplified to a simple analyzer of the FRT; the two-independent-parameter FRT of an object illuminated with a plane wave can be readily implemented by a lens of arbitrary focal length; when cascading, the Function of each lens unit and the relationship between the adjacent ones are clear and simple; and more parameters and fewer restrictions on cascading make the optical design easy. (C) 1997 Optical Society of America.

I. Numerical solution of exact pair equations. II. Numerical solution of first-order pair equations

Part I

Numerical solutions to the S-limit equations for the helium ground state and excited triplet state and the hydride ion ground state are obtained with the second and fourth difference approximations. The results for the ground states are superior to previously reported values. The coupled equations resulting from the partial wave expansion of the exact helium atom wavefunction were solved giving accurate S-, P-, D-, F-, and G-limits. The G-limit is -2.90351 a.u. compared to the exact value of the energy of -2.90372 a.u.

Part II

The pair functions which determine the exact first-order wavefunction for the ground state of the three-electron atom are found with the matrix finite difference method. The second- and third-order energies for the (1s1s)¹S, (1s2s)³S, and (1s2s)¹S states of the two-electron atom are presented along with contour and perspective plots of the pair functions. The total energy for the three-electron atom with a nuclear charge Z is found to be E(Z) = -1.125•Z² +1.022805•Z-0.408138-0.025515•(1/Z)+O(1/Z²)a.u.

Analysis of optical systems with extended depth of field using the Wigner distribution function

We describe the use of a Wigner distribution function approach for exploring the problem of extending the depth of field in a hybrid imaging system. The Wigner distribution function, in connection with the phase-space curve that formulates a joint phase-space description of an optical field, is employed as a tool to display and characterize the evolving behavior of the amplitude point spread function as a wave propagating along the optical axis. It provides a comprehensive exhibition of the characteristics for the hybrid imaging system in extending the depth of field from both wave optics and geometrical optics. We use it to analyze several well-known optical designs in extending the depth of field from a new viewpoint. The relationships between this approach and the earlier ambiguity function approach are also briefly investigated. (c) 2006 Optical Society of America.

Superresolution, extended focal depth or focal shift with electrically controllable radial birefringent filter

In this paper an electrically controllable radial birefringent pupil filter is proposed. It consists of two polarizers and an improved electrically controllable optical azimuth rotator which has two lambda/4 retarders, one electro-optical crystal and one radial birefringent crystal. The evolution and distribution of polarization states of this pupil filter are discussed. The most interesting and useful advantage of such a structure is that the characteristic of transverse superresolution and axial extended focal depth or focal shift can be obtained merely by controlling the applied voltage on the electro-optical crystal. The radial birefringent crystal azimuth angle cooperating with different electrical inductive phase differences will determine the transverse and axial intensity distribution. It is shown that for particular ranges of electrical inductive phase difference it is possible to obtain transverse superresolution along with extended focal depth or with a focal shift.

I. Quantum mechanics of one-dimensional two-particle models. II. A quantum mechanical treatment of inelastic collisions

Part I

Solutions of Schrödinger’s equation for system of two particles bound in various stationary one-dimensional potential wells and repelling each other with a Coulomb force are obtained by the method of finite differences. The general properties of such systems are worked out in detail for the case of two electrons in an infinite square well. For small well widths (1-10 a.u.) the energy levels lie above those of the noninteresting particle model by as much as a factor of 4, although excitation energies are only half again as great. The analytical form of the solutions is obtained and it is shown that every eigenstate is doubly degenerate due to the “pathological” nature of the one-dimensional Coulomb potential. This degeneracy is verified numerically by the finite-difference method. The properties of the square-well system are compared with those of the free-electron and hard-sphere models; perturbation and variational treatments are also carried out using the hard-sphere Hamiltonian as a zeroth-order approximation. The lowest several finite-difference eigenvalues converge from below with decreasing mesh size to energies below those of the “best” linear variational function consisting of hard-sphere eigenfunctions. The finite-difference solutions in general yield expectation values and matrix elements as accurate as those obtained using the “best” variational function.

The system of two electrons in a parabolic well is also treated by finite differences. In this system it is possible to separate the center-of-mass motion and hence to effect a considerable numerical simplification. It is shown that the pathological one-dimensional Coulomb potential gives rise to doubly degenerate eigenstates for the parabolic well in exactly the same manner as for the infinite square well.

Part II

A general method of treating inelastic collisions quantum mechanically is developed and applied to several one-dimensional models. The formalism is first developed for nonreactive “vibrational” excitations of a bound system by an incident free particle. It is then extended to treat simple exchange reactions of the form A + BC →AB + C. The method consists essentially of finding a set of linearly independent solutions of the Schrödinger equation such that each solution of the set satisfies a distinct, yet arbitrary boundary condition specified in the asymptotic region. These linearly independent solutions are then combined to form a total scattering wavefunction having the correct asymptotic form. The method of finite differences is used to determine the linearly independent functions.

The theory is applied to the impulsive collision of a free particle with a particle bound in (1) an infinite square well and (2) a parabolic well. Calculated transition probabilities agree well with previously obtained values.

Several models for the exchange reaction involving three identical particles are also treated: (1) infinite-square-well potential surface, in which all three particles interact as hard spheres and each two-particle subsystem (i.e. BC and AB) is bound by an attractive infinite-square-well potential; (2) truncated parabolic potential surface, in which the two-particle subsystems are bound by a harmonic oscillator potential which becomes infinite for interparticle separations greater than a certain value; (3) parabolic (untruncated) surface. Although there are no published values with which to compare our reaction probabilities, several independent checks on internal consistency indicate that the results are reliable.

Vectorial solution of Kukhtarev equations for doubly doped crystals and optimal choice of recording directions in nonvolatile holographic storage

Vectorial Kukhtarev equations modified for the nonvolatile holographic recording in doubly doped crystals are analyzed, in which the bulk photovoltaic effect and the external electrical field are both considered. On the basis of small modulation approximation, both the analytic solution to the space-charge field with time in the recording phase and in the readout phase are deduced. The analytic solutions can be easily simplified to adapt the one-center model, and they have the same analytic expressions given those when the grating vector is along the optical axis. Based on the vectorial analyses of the band transport model an optimal recording direction is given to maximize the refractive index change in doubly doped LiNbO3:Fe: Mn crystals. (c) 2007 Optical Society of America.

Optimized phase pupil masks for extended depths of field

By properly designing a phase pupil mask to modulate or encode the optical images and then digitally restoring them, one can greatly extend the depth of field and improve image quality. The original works done by Dowski and Cathey introduce the use of a cubic phase pupil mask to extend the depth of field. The theoretical and experimental researches all verified its effectiveness. In this paper, we suggest the use of an exponential phase pupil mask to extend the depth of field. This phase mask has two variable parameters for designing to control the shape of the mask so as to modulate the wave-front more flexible. We employ an optimization procedure based on the Fisher information metric to obtain the optimum values of the parameters for the exponential and the cubic masks, respectively. A series of performance comparisons between these two optimized phase masks in extending the depth of field are then done. The results show that the exponential phase mask provide slight advantage to the cubic one in several aspects. (c) 2006 Elsevier B.V. All rights reserved.

Some theorems in classical elastodynamic

This investigation is concerned with various fundamental aspects of the linearized dynamical theory for mechanically homogeneous and isotropic elastic solids. First, the uniqueness and reciprocal theorems of dynamic elasticity are extended to unbounded domains with the aid of a generalized energy identity and a lemma on the prolonged quiescence of the far field, which are established for this purpose. Next, the basic singular solutions of elastodynamics are studied and used to generate systematically Love's integral identity for the displacement field, as well as an associated identity for the field of stress. These results, in conjunction with suitably defined Green's functions, are applied to the construction of integral representations for the solution of the first and second boundary-initial value problem. Finally, a uniqueness theorem for dynamic concentrated-load problems is obtained.