Discrete versus continuum relaxation modes of a hard sphere gas

1984 ◽  
Vol 62 (2) ◽  
pp. 97-103 ◽  
Author(s):  
Bernard Shizgal
Keyword(s):  
Discrete Spectrum ◽  
Cross Section ◽  
Cross Sections ◽  
Hard Sphere ◽  
Expansion Method ◽  
Gas Mixture ◽  
Discrete Ordinate Method ◽  
Discrete Ordinate ◽  
Relaxation Modes ◽  
The Continuum

The nature of the discrete spectrum of the linear Boltzmann collision operators for a simple gas and for a gas mixture is studied numerically with a discrete ordinate method. The discrete ordinate method is found to give a large number of discrete eigenvalues whereas the expansion method with Burnett functions yields only a few converged eigenvalues. The hard sphere cross section is used in the present paper although the methods employed are readily applicable to other cross sections. The approach of the eigenvalues to the continuum boundary is studied in detail and a comparison with a previous asymptotic Wentzell–Kramers–Brillouin (WKB) analysis yields excellent agreement.

Dissociation and Charge Transfer of CO21+ and CO22+ Ions in Collisions with Neutral Atoms and Molecules

1977 ◽  
Vol 32 (6) ◽  
pp. 555-557 ◽  
Author(s):  
A. Müller ◽  
E. Salzborn
Keyword(s):  
Charge Transfer ◽  
Electron Capture ◽  
Cross Section ◽  
Cross Sections ◽  
Impact Energy ◽  
Linear Dependence ◽  
Hard Sphere ◽  
Single Electron Capture ◽  
Target Particle ◽  
Total Collision

AbstractCross sections for single electron capture by CO22+ ions as well as total collision cross sections σi,tot (i=1, 2) for CO21+ and CO22+ ions in He, Ne, Ar, Kr, Xe, H2, O2, N2, CH4 and CO2 gases have been measured at 10 keV impact energy. σ2,1 varies between 4.5·10-17cm2 for the He and 1.8·10·15cm2 for the CO, target. On the average, σ2,1,tot is 1.6 times larger than σ1,tot and shows a nearly linear dependence on the gaskinetic hard sphere cross section of the target particle. The largest observed value of σ2,tot is 9.4·10-15cm2 for the CO2 target.

A QUASICLASSICAL TRAJECTORY STUDY OF REACTIVE SCATTERING ON AN ANALYTICAL POTENTIAL ENERGY SURFACE FOR GeH2 SYSTEM

2011 ◽  
Vol 10 (02) ◽  
pp. 147-163
Author(s):  
LI ZHANG ◽  
CHAO-YONG ZHU ◽  
GANG JIANG ◽  
CHAOYUAN ZHU ◽  
Z. H. ZHU
Keyword(s):  
Potential Energy ◽  
Potential Energy Surface ◽  
Cross Section ◽  
Cross Sections ◽  
Collision Energy ◽  
Energy Surface ◽  
Life Time ◽  
Expansion Method ◽  
Reverse Reaction ◽  
Analytical Potential

A quasiclassical trajectory method was employed to study reaction Ge+H 2 (v=0, j=0) and reverse reaction H+GeH (v=0, j=0) on an analytical potential energy surface obtained from simplified many-body expansion method with fitting to B3P86/CC-pVTZ calculations around a global minimum and a long-range van de Waals well plus spectroscopy data for diatomic molecules GeH and H2 . Reaction probabilities from both reaction and reverse reaction were calculated. Dominant reaction is complex-forming reaction Ge+H2 (v=0, j=0) → GeH2 , and its cross section is 10 times bigger than that of complex-forming reaction from the reverse reaction. There is no threshold effect for complex-forming reaction and the cross sections for both complex-forming reactions decrease with the increase of collision energy. Life time of complex is shown to be decreasing with increase of collision energy. Dominant reverse reaction is reaction H + GeH (v=0,j=0) → Ge+H2 ; the reaction probability decreases with the increase of collision energy and differential cross section shows that this reverse reaction has almost equal angular distribution at low collision energy and mostly forward scattering at high collision energy.

A GTO basis set calculation of properties of the continuum. Photoionization and resonances of He

1992 ◽  
Vol 70 (2) ◽  
pp. 513-519 ◽  
Author(s):  
Roberto Moccia ◽  
Pietro Spizzo
Keyword(s):  
Cross Section ◽  
Cross Sections ◽  
Photoionization Cross Section ◽  
Basis Set ◽  
Basis Sets ◽  
Gaussian Basis Sets ◽  
Continuum States ◽  
K Matrix ◽  
The Continuum ◽  
Autoionizing States

By using the K-matrix technique for the continuum states that was previously employed with particularly diffuse L2 basis sets, it is shown that GTO bases are also capable of yielding accurate values for the properties belonging to the electronic continuum. The method has been tested for helium and proved of satisfactory accuracy also for the analysis of the autoionizing states. The results include the phase shifts of the continuum states of the 1Seand 1P° manifolds, the properties of the lowest resonances of these symmetries, the ground state photoionization cross section, and the S contribution to the 1s2p1P° photoionization cross section. The results obtained suggest that the proposed technique should be useful for computing molecular differential photoionization cross sections by exploiting the widely used codes that employ GTO bases. Keywords: photoionization, Gaussian basis sets, helium, autoionizing states.

Space-eigenvalue problems in the kinetic theory of gases

1970 ◽  
Vol 42 (1) ◽  
pp. 85-95 ◽  
Author(s):  
M. M. R. Williams ◽  
J. Spain
Keyword(s):  
Asymptotic Distribution ◽  
Cross Section ◽  
Hard Sphere ◽  
Eigenvalue Problems ◽  
Linear Boltzmann Equation ◽  
Hard Sphere Model ◽  
Kinetic Theory Of Gases ◽  
Sphere Model ◽  
Scattering Kernel ◽  
The Continuum

The existence of elementary, exponential solutions of the linear Boltzmann equation for gases is examined. Using the hard-sphere model of scattering, it is found that, in problems involving velocity perturbations, there are no discrete non-zero eigenvalues. Thus the relaxation to the asymptotic distribution is non-exponential and is described by the continuum eigenfunctions. For temperature perturbations, however, we find two non-zero discrete eigenvalues whose values are ±0·975 in units of the minimum scattering cross-section. Relaxation to the asymptotic distribution is therefore exponential, although still very rapid.The conclusions stated above are based upon a truncation of the scattering kernel and a subsequent numerical solution of the resulting integral equations.

Discrete relaxation modes for a hard-sphere gas

1968 ◽  
Vol 46 (21) ◽  
pp. 2463-2469 ◽  
Author(s):  
M. Rahman ◽  
M. K. Sundaresan
Keyword(s):  
Boltzmann Equation ◽  
Discrete Spectrum ◽  
Hard Sphere ◽  
Spherically Symmetric ◽  
Symmetric State ◽  
Elementary Method ◽  
Linearized Boltzmann Equation ◽  
Relaxation Modes

The discrete spectrum of a linearized Boltzmann equation for a hard-sphere gas is studied, and the results of Kuščer and Williams (1967) for a spherically symmetric state are rederived by a more elementary method. Extension is made to nonisotropic states, and it is shown that the discrete spectrum is empty for [Formula: see text].

scholarly journals VUV-absorption cross section of carbon dioxide from 150 to 800 K and applications to warm exoplanetary atmospheres

2018 ◽  
Vol 609 ◽  
pp. A34 ◽  
Author(s):  
O. Venot ◽  
Y. Bénilan ◽  
N. Fray ◽  
M.-C. Gazeau ◽  
F. Lefèvre ◽  
...  
Keyword(s):  
Carbon Dioxide ◽  
High Resolution ◽  
Cross Section ◽  
Wavelength Range ◽  
Cross Sections ◽  
Absorption Cross Section ◽  
Planetary Atmospheres ◽  
Temperature Dependency ◽  
Absorption Cross ◽  
The Continuum

Context. Most exoplanets detected so far have atmospheric temperatures significantly higher than 300 K. Often close to their star, they receive an intense UV photons flux that triggers important photodissociation processes. The temperature dependency of vacuum ultraviolet (VUV) absorption cross sections are poorly known, leading to an undefined uncertainty in atmospheric models. Similarly, data measured at low temperatures similar to those of the high atmosphere of Mars, Venus, and Titan are often lacking. Aims. Our aim is to quantify the temperature dependency of the VUV absorption cross sections of important molecules in planetary atmospheres. We want to provide high-resolution data at temperatures prevailing in these media, and a simple parameterisation of the absorption in order to simplify its use in photochemical models. This study focuses on carbon dioxide (CO2). Methods. We performed experimental measurements of CO2 absorption cross sections with synchrotron radiation for the wavelength range (115–200 nm). For longer wavelengths (195–230 nm), we used a deuterium lamp and a 1.5 m Jobin-Yvon spectrometer. We used these data in our one-dimensional (1D) thermo-photochemical model in order to study their impact on the predicted atmospheric compositions. Results. The VUV absorption cross section of CO2 increases with the temperature. The absorption we measured at 150 K seems to be close to the absorption of CO2 in the fundamental ground state. The absorption cross section can be separated into two parts: a continuum and a fine structure superimposed on the continuum. The variation in the continuum of absorption can be represented by the sum of three Gaussian functions. Using data at high temperature in thermo-photochemical models significantly modifies the abundance and the photodissociation rates of many species in addition to CO2, such as methane and ammonia. These deviations have an impact on synthetic transmission spectra, leading to variations of up to 5 ppm. Conclusions. We present a full set of high-resolution (Δλ = 0.03 nm) absorption cross sections of CO2 from 115 to 230 nm for temperatures ranging from 150 to 800 K. A parameterisation allows us to calculate the continuum of absorption in this wavelength range. Extrapolation at higher temperature has not been validated experimentally and therefore should be used with caution. Similar studies on other major species are necessary to improve our understanding of planetary atmospheres.

Notes on the Kinematics of an Octopus’s Arm

2008 ◽  
Author(s):  
Y. Levinson ◽  
R. Segev ◽  
A. Shapiro
Keyword(s):  
Cross Section ◽  
Cross Sections ◽  
Discrete Model ◽  
Continuum Model ◽  
Plane Deformation ◽  
Point Of View ◽  
Parallel Mechanisms ◽  
Rod Model ◽  
The Cross ◽  
The Continuum

In a study of the kinematics of an Octopus’s arm from the robotics point of view, we consider two models: a discrete model and a continuum model. Both models represent control of the curve in space defined by the centerline of the arm and the position of the suction elements of the arm relative to the centerline. The two models also represent the control of the arm by means of three groups of muscles: the longitudinal, transverse and helicoidal groups. Another feature of the kinematics exhibited by the two models is the use of incompressibility in the control of the arm. In the discrete model, the arm is represented as a sequence of parallel mechanisms. Each mechanism in the sequence consists of two parallel platforms of variable dimensions so that the resulting sequence of platforms represents the various cross sections of the arm. The continuum rod model that we use allows linear in-plane deformation of the cross-section and assumes that cross-sections remain perpendicular to the axis of the rod.

scholarly journals Comparison of viscosities from the Chapman-Enskog and relaxation time methods

Open Physics ◽  
2012 ◽  
Vol 10 (6) ◽  
Author(s):  
Anton Wiranata ◽  
Madappa Prakash ◽  
Purnendu Chakraborty
Keyword(s):  
Relaxation Time ◽  
Cross Section ◽  
Cross Sections ◽  
Differential Cross ◽  
Hard Sphere ◽  
Quantitative Comparison ◽  
Test Cases ◽  
Differential Cross Sections ◽  
Quantitative Results ◽  
Maxwell Gas

AbstractA quantitative comparison between the results of shear viscosities from the Chapman-Enskog and relaxation time methods is performed for selected test cases with specified elastic differential cross sections: (i) the non-relativistic, relativistic and ultra-relativistic hard sphere gas with angle and energy independent differntial cross section, (ii) the Maxwell gas, (iii) chiral pions and (iv) massive pions. Our quantitative results reveal that the extent of agreement (or disagreement) depends very sensitively on the energy dependence of the differential cross sections employed.

scholarly journals Photoionisation and Auger Electron Emission from the Lithium Molecule: Calculations using Multicentre Numerical Continuum Functions

1986 ◽  
Vol 39 (5) ◽  
pp. 809 ◽  
Author(s):  
FP Larkins ◽  
JA Richards
Keyword(s):  
Cross Section ◽  
Cross Sections ◽  
Electron Emission ◽  
First Principles ◽  
Transition Rates ◽  
Hartree Fock ◽  
Photoemission Data ◽  
Experimental Findings ◽  
Asymmetry Parameters ◽  
The Continuum

A numerical method has been used for the generation of molecular continuum wavefunctions at the relaxed Hartree-Fock level associated with the photoionisation of the lithium molecule. Exchange between the continuum electron and the ion core is included, but L coupling is neglected. Cross sections for core and valence shell photoionisation have been calculated from threshold to 6�0 a. u. The results differ significantly in detail from previous multiple scattering calculations. Continuum phase shifts and the asymmetry parameters for the various processes are also reported. The molecular cross section values are compared with atomic cross sections calculated at the relaxed Hartree-Fock level. The Li2 molecular Auger transition rates are also calculated from first principles using the appropriate two-centre continuum functions. The results provide a basis for the reinterpretation of recent experimental findings of photoemission data for the lithium vapour system.

Electron Irradiation Effects in Polyoxymethylene Crystals Grown Inside Trioxane Crystals

1971 ◽  
Vol 29 ◽  
pp. 78-79
Author(s):  
J. P. Colson ◽  
D. H. Reneker
Keyword(s):  
Cross Section ◽  
Cross Sections ◽  
Detailed Examination ◽  
The Other ◽  
Electron Micrograph ◽  
Irradiation Effects ◽  
Threefold Axis ◽  
Typical Sample ◽  
Polymer Crystals ◽  
Chain Axis

Polyoxymethylene (POM) crystals grow inside trioxane crystals which have been irradiated and heated to a temperature slightly below their melting point. Figure 1 shows a low magnification electron micrograph of a group of such POM crystals. Detailed examination at higher magnification showed that three distinct types of POM crystals grew in a typical sample. The three types of POM crystals were distinguished by the direction that the polymer chain axis in each crystal made with respect to the threefold axis of the trioxane crystal. These polyoxymethylene crystals were described previously.At low magnifications the three types of polymer crystals appeared as slender rods. One type had a hexagonal cross section and the other two types had rectangular cross sections, that is, they were ribbonlike.

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