An improved model of the fringing fields of a quadrupole mass filter

The fringing field region of a quadrupole mass filter is modelled using an iterative finite difference technique to solve Laplace's equation for the electrostatic potential. The results are used to formulate an expression f(z) such that the electrostatic potential of the fringing field, F(x, y, z, t) may be written in the form F(x, y, z, t) = f(z) φ(x, y, t), where φ(x, y, t) is the hyperbolic field of the quadrupole. The function f(z) is expressed in the form f(z) = 1 - exp {- az - bz²}), where a and b are constants and z is in units of r_O, where 2r_O is the spacing of the quadrupole rods. The effect of the distance from the quandrupole rods to the end plate of the quadrupole mass filter, d on f(z) is investigated and the results presented show that for d < 0.125 r_O the function f(z) does not alter significantly.     

On maximal resonance of polyomino graphs

A polyomino graph is a finite plane 2-connected bipartite graph every interior face of which is bounded by a regular square of side length one. Let k be a positive integer, a polyomino graph G is k-resonant if the deletion of any i ≤ k vertex-disjoint squares from G results in a graph either having perfect matchings or being empty. If graph G is k-resonant for any integer k ≥ 1, then it is called maximally resonant. All maximally resonant polyomino graphs are characterized in this work. As a result, the least integer k such that a k-resonant polyomino graph is maximally resonant is determined.     

A methodology to define the Cubic Equation of State of a simple fluid

A method to define the Cubic Equation of State (CES) of a simple substance is presented in this work. CES is constructed with only three parameters of the fluid, namely, the critical compressibility Z_c≡P_cv_c/RT_c, the acentric factor ω ≡ − log  (P^(sat)/P_c) − 1 (where P^(sat) is the saturated vapor pressure), and the saturated vapor volume v^(sat) at the temperature T^(sat)/T_c = 0.7 (where T_c is the critical temperature, v_c is the critical volume, and P_c is the critical pressure). The resulting CES is unique for each substance and, in general, it is different from other known CES in the literature.     

Conditional probability amplitudes and the variational principle

It is shown by counter-example, that when the theory of conditional probability amplitudes is applied to quantum mechanical systems no variational principle exists for the function U(R = fφ*(r/R)Hφ(r/R) dr, where φ(r/R) is the conditional probability amplitude and H is the total hamiltonian of the system.     

Rectangular source integral and recurrence relations

In this paper Hubbell's rectangular source integral H′(a,b), which is a double integral, is expressed as a series of many converging single integrals I_n (a,b). Recurrence relations relate these integrals. Once one integral I₁ is computed, recurrence relations are used to compute other integrals. I₁(a,b) can be computed analytically. H′(a,b) is approximated by considering the first seven terms in the series and the results are found to give good results for various values of a and b. Results are presented for the values of a and b (0.1 to 20 and to 2), respectively. The rate of convergence depends on the values of a and b.     

Theory of dynamic light scattering by polymers and gels

It is shown under very general conditions that the intermediate scattering function for the generalized Rouse—Zimm model always takes the simple form G(K, t) α exp[?K²(k_BT/f)t], when the scattering vector K becomes sufficiently large. (Here k_B is Boltzmann's constant, T is the absolute temperature and f is the individual bead friction factor.) A microscopic formulation for the bulk modulus and friction factor density of a gel network is incorporated into the viscoelastic continuum model of Tanaka et al. The resulting expression for the apparent long-wavelength diffusion coefficient of the gel is D_G = (k_BT/f)2(1 - 2/Φ), where Φ is the network functionality.     

Packing in honeycomb networks

Molecules arranging themselves into predictable patterns on silicon chips could lead to microprocessors with much smaller circuit elements. Mathematically, assembling in predictable patterns is equivalent to packing in graphs. An H-packing of a graph G is a set of vertex disjoint subgraphs of G, each of which is isomorphic to a fixed graph H. If H is the complete graph K ₂, the maximum H-packing problem becomes the familiar maximum matching problem. In this paper we give algorithms to find a perfect packing of HC(n) with P ₆ and K _1,3 when n is even and thus determines their packing numbers. Further we also study the packing of HC(n) with 1, 3-dimethyl cyclohexane.     

The structure of low temperature triplet energy traps in organic solids from pmdr spectroscopy 11. 1,3,5-Trichloro and hexachlorobenzenes

The new technique of polarized microwave PMDR spectroscopy is used to determine the structure of the low temperature (1.6 – 4.2°K) triplet energy traps. The structure of 1.3,5-trichlorobenzene (Tri-CB) and hexachlorobenzene (HCB) molecules in hexamethylbenzene (HMB) host as well as in their own neat crystals (present as x-traps) is determined from the linear polarization characteristics of their optically detected microwave zerofield (zf) transitions as well as from the analysis of their phosphorescence emission. The former technique gives the direction of the principal magnetic axes of these trap molecules in the crystal as well as the relative order of their zf levels.ln HMB host, deviation from trigonal symmetry is found to be only slight for HCB and absent for Tri-CB. ln the neat crystals, large deviations from trigonal symmetry are observed for the traps of both HCB and Tri-CB. In HCB x- traps, the HCB molecule is found to be slightly contracted along the CCl axis near parallel to the c′ crystal axes. Deviation From planarity is also strongly suggested by the large value of the zf parameter D. ln addition, the principal magnetic axis of the 2|E| moment for HCB x-traps is found to lie only 5° off the molecular N axis. The |D| + |E| and |D| — |E| moments, however, are 15° away from the A and B molecular axis, respectively.In Tri-CB neat crystal. two traps are observed optically with their phosphorescence origins 10 cm^?1 apart. The zf parameter E is found to have a non-zero value and is opposite in sign for the two traps. If the distortion is to be blamed on pseudo-Jahn-Teller forces, the results lead to the conclusion that the x-(shallow) trap is contracted while the y- (deep) trap is expanded along an in-plane axis going through the CCl bonds near parallel to the a axis of the Tri-CB crystal. The plane containing the |D| + |E| and |D| — |E| moments of the x-trap suffers a rotation around the N-molecular axis, which is almost parallel to the 2|E| moment. Th |D| + |E| moment is 10° off the A axis and the |D| — |E| moment is 10° off the B axis. The 2|E| transition moment of the y-trap lies off the molecular N axis and the plane containing the |D| + |E| and |D| — |E| moments moves upward from theThe results of these and other studies suggest that low temperature trapping in neat crystals of this type results from crystal induced geometrical and orientational changes in the molecules at point defects. The observed traps are those molecules for which the crystal field induced deformation leads to a lowering in their singlet-triplet transition energy as compared with that for the host lattice.     

Stochastic theory of direct simulation Monte Carlo method

A treatment of direct simulation Monte Carlo method as a Markov process with a master equation is given and the corresponding master equation is derived. A hierarchy of equations for the reduced probability distributions is derived from the master equation. An equation similar to the Boltzmann equation for single particle probability distribution is derived using assumption of molecular chaos. It is shown that starting from an uncorrelated state, the system remains uncorrelated always in the limit N→∞, where N is the number of particles. Simple applications of the formalism to direct simulation money games are given as examples to the formalism. The formalism is applied to the direct simulation of homogenous gases. It is shown that appropriately normalized single particle probability distribution satisfies the Boltzmann equation for simple gases and Wang Chang–Uhlenbeck equation for a mixture of molecular gases. As a consequence of this development we derive Birds no time counter algorithm. We extend the analysis to the inhomogeneous gases and define a new direct simulation algorithm for this case. We show that single particle probability distribution satisfies the Boltzmann equation in our algorithm in the limit N→∞, V _k →0, Δt→0 where V _k is the volume of kth cell. We also show that our algorithm and Bird’s algorithm approach each other in the limit N _k →∞ where N _k is the number of particles in the volume V _k .     

Synthesis of (1R,cis,αS)-cypermethrine via lipase catalyzed kinetic resolution of racemic m-phenoxybenzaldehyde cyanohydrin acetate

A technical scale preparation of optically active (1R,cis,αS)-cypermethrine 4 from racemic m-phenoxybenzaldehyde cyanohydrin acetate (RS)-1 and (1R,cis)-3-(2,2-dichlorovinyl)-2,2-dimethylcyclopropanecarboxylic acid chloride (1R,cis)-3 is described. Key steps of the new procedure are a lipase catalyzed enantioselective transesterification of (RS)-1 with n-butanol and direct acylation of the mixture of (R)-1 and (S)-cyanohydrin (S)-2 with (1R,cis)-3 to give enantiomerically pure (1R,cis,αS)-4. The unchanged (R)-1 is removed from (1R,cis,αS)-4 by distillation, and is racemized with triethylamine to give (RS)-1 which is returned to the process. The total yield of (1R,cis,αS)-4 referred to (RS)-1 is 80%.     

Approximate analytic expression for the electrophoretic mobility of a cylindrical colloidal particle. Relaxation effect

An approximate expression is derived for the electrophoretic mobility μ _⊥ of an infinitely long cylindrical particle of radius a and zeta potential ζ oriented perpendicular to an applied electric field in an electrolyte solution. The obtained expression is applicable for cylinders with large radii such that κa≥ca. 30, where κ is the Debye–Hückel parameter. It is shown that the obtained mobility expression μ _⊥, which consists of Smoluchowski’s equation and the correction term of the order of exp(ze∣ζ∣/2kT)/κa (where z is the valence of counterions in the electrolyte solution, e is the elementary electric charge, k is Boltzmann’s constant, and T is the absolute temperature), coincides with the mobility expression for a sphere of radius a and zeta potential ζ correct to the order of exp(ze∣ζ∣/2kT)/κa.     

Vertical excitation as a transient phenomenon

For a particular model with two electronic states, each with two vibrations, the dipole correlation function governing electronic absorption is e ^?iωOt cosγt with spectrum ω ₀±γ. The function starts as e ^?iωOt (γ?ω ₀), with Fourier transform peaking around ω ₀ instead of ω ₀±γ, and this is associated with vertical excitation. After a time t～2/γ the spectrum goes over into the normal one. As a generalization, a procedure is outlined for characterizing the state reached first after interaction with light is initiated. Finally it is suggested that one can understand aspects of internal conversion by analogy with the case of vertical excitation.     

Two-photon ionization of an inner shell electron of the Cl atom

We have calculated, using second order perturbation theory, the two photon ionization cross section of a K-shell electron of chlorine forE=1.6 keV incident photons. Two classes of intermediate states must be considered, those in which a 1s electron moves to an emptyp-orbital, and those in which an electron from an occupiedp-orbital moves into the continuum. The first class of intermediate states is followed by the ejection of ap-electron into the continuum. The second class of intermediate states is followed by the transfer of a 1s electron into an emptyp-state. The largest contribution comes from 3p →d-continuum followed by 1s → 3p transition. Our result is σ⁽²⁾/I=2.06×10⁴¹ cm⁴/W where σ⁽²⁾ is the two photon ionization cross-section andI is the light intensity.     

Optical potential calculations for molecular collisions

The Feshbach optical formalism is applied to elastic, nonreactive atom-diatom scattering on a single potential energy surface. The optical potential depends on GQ, the resolvent of E-QHQ, where Q projects onto open as well as closed channels. A method for generating GQ is developed which goes beyond the free-space approximation by partitioning the radial part of the intermolecular separation into a set of intervals, on each of which the projected interaction QVQ is represented by a constant diagonal form. The resulting GQ is used in calculations on a model collinear system. The calculations are carried out with various approximations on the full nonlocal optical potential equation for Pψ Emphasis is placed on two of these, one of which is characterized by a local homogeneous equation for Pψ, and the other by a local inhomogeneous equation for Pψ.     

Conductance of nano-tube junctions and its scaling law

The conductance is calculated by Landauer’s formula with a simple tight binding model for junctions connecting two different metallic carbon nanotubes. These junctions are formed by a pair of disclinations, a 5-membered ring and a 7-membered ring, without dangling bonds. Conductances of about eight hundred kinds of junctions are obtained. The conductance is determined only by the ratio i/R where i is the distance between the two disclinations and R is the circumference of the thinner tube. When i/R ? 1, the conductance is found to be almost proportional to (i/R)^-3. Even when the Fermi energy is shifted by doping, an extended scaling law holds, so far as the channel number is kept two.     

A different approach to X-ray stress analysis

A different approach to X-ray stress analysis has been developed. At the outset, it must be noted that the material to be analyzed is assumed homogeneous and isotropic. If a sphere with radius r within a specimen is subjected to a state of stress, the sphere is deformed into an ellipsoid. The semi-axes of the ellipsoid have the values of (r + ε_x), (r + ε_y), and (r + ε_z), which are replaced by d_x, d_y, and d_z, or for the cubic case, a_x, a_y, and a_z. In this technique, at a particular ϕ angle (see Fig. 1), the two-theta position of a high angle (hkl) peak is determined at ψ angles of 0, 15, 30, and 45°. These measurements are repeated for 3 to 6 ϕ angles in steps of 30°. The d_ϕψ or a_ϕψ values are then determined from the peak positions. The data is then fitted to the general quadratic equation for an ellipsoid by the method of least squares. From the coefficients of the quadratic equation, the angle between the laboratory and the specimen coordinates (direction of the principle stress) can be determined. Applying the general rotation of axes equations to the quadratic, the equation of the ellipse in the x–y plane is determined. The a_x, a_y, and a_z values for the principal axes of the lattice parameter ellipsoid are then evaluated. It is then possible to determine the unstressed a₀ value from Hooke's Law using a_x, a_y, and a_z. The magnitude of the principal strains/stresses is then determined.     

A simple sum rule in raman theory

It is shown that in the limit of large detuning energy and in the absence of coordinate dependence of the transition moment, the resonance Raman amplitude for a 0 → n transition on a harmonic potential surface is proportional to (δ²_? + δ²₊)^?1 × ?^∞_?∞ exp (?q²) · [?ⁿΔV(q)/?q¹¹] dq< where ΔV(q) is the difference between the arbitrary excited-state surface and the initial harmonic potential. The resonant and non-resonant detuning energies are given by δ_? = E ? hv and δ₊ = E + hv, where v is the incident laser frequency and E is the minimum separation between the potential surfaces.     

Molecular energy parameters by solution thermodynamics

The relationships between statistical thermodynamics and equilibrium constants, either cumulative, stepwise or specific site constants are investigated. In order to show the link between equilibrium constants and statistical thermodynamic microscopic properties, a distinction has been introduced between non-reacting and reacting systems. The non-reacting systems are those for which continuous statistical distributions of enthalpies can be assumed. The distribution function can be obtained as an integral of the interparticle potential extended to the whole ensemble. Molecular partition functions ζ_A, are used to describe the properties of the ensembles. The reacting ensemble is represented by distinct distributions of enthalpies, each distribution being grouped around a mean value. Each level is representative of one species. Around each mean level the distribution is continuous as in non-reacting ensembles. The reacting ensemble of particles is described by a grand canonical molar partition function Z_M=(1+kγ(i)[A])^t where k is the specific site constant, γ(i) is the cooperativity function, [A] is the concentration of free ligand, and the power t indicates the maximum number of i sites in one class. The specific site constant k is proportional to the affinity of binding and is related to the depth of the minimum of the potential function. The factor γ_i is the cooperativity factor given by the value of the cooperativity function γ(i) at the ith level and indicates how the depth of the potential function is affected by previous binding of a ligand. The values of the stability constant and cooperativity factor can be optimized by a computer program. The derivatives of the partition function Z_M with respect to ln[A] correspond to the formation function 〈n〉 (first derivative) and to the buffer capacity B_C (second derivative). The derivatives of the partition function Z_M with respect to temperature are reaction enthalpy ΔH (first derivative with −(1/T)) and apparent heat capacity ΔC_p,app (second derivative with ln T). The denaturation heat obtained by integration of ΔC_p,app dT for many proteins explains why the denaturation enthalpy depends linearly upon T.