Viscosity Cross Sections Research Articles

Classical energy-sudden calculations of diffusion and viscosity cross sections for atom-diatom interactions

Trajectory calculations are reported in the sudden approximation for the Maxwellian averaged diffusion and viscosity cross sections for He-N2, Ar-N2 and He-CO2 collisions. Comparison is made with classical infinite-order-sudden (IOS) calculations, also known as the Mason-Monchick approximation, and with various quantal IOS calculations. For the N2 collisions, the energy-sudden and classical IOS results agree within 1% and differ from the quantal results by less than 2%. For the He-CO2 collisions, the classical IOS results differ by about 5% from the energy-sudden calculations. An energy-corrected sudden approximation is introduced. This correction is significant only for viscosity-type cross sections, making a 6% difference for Ar-N2 and He-CO2 collisions at 300K. Criteria for the validity of the classical IOS approximation are suggested.

Interaction potentials and momentum transfer in ionic collisions: Uranium

The effect of elastic collisions of U/sup +/ with U, U/sup +/, and U/sup 2 +/ in a U/sup +/ plasma has been investigated. The required potential-energy curves were calculated by means of the electron-gas (Gordon-Kim-Rae) statistical method. An effective polarization potential was added to the ion-neutral potential curve. Collisional phase shifts were evaluated by the semiclassical JWKB method using a very efficient algorithm. Modifications necessary to take into account the long-range shielded Coulomb potential in ion-ion collisions are described. Momentum-transfer and viscosity cross sections, without charge transfer, are presented and their dependences on the long- and short-range potentials are examined.

Rotationally and vibrationally inelastic scattering in the rotational IOS approximation. Ultrasimple calculation of total (differential, integral, and transport) cross sections for nonspherical molecules

A simple, direct derivation of the rotational infinite order sudden (IOS) approximation in molecular scattering theory is given. Connections between simple scattering amplitude formulas, choice of average partial wave parameter, and magnetic transitions are reviewed. Simple procedures for calculating cross sections for specific transitions are discussed and many older model formulas are given clear derivations. Total (summed over rotation) differential, integral, and transport cross sections, useful in the analysis of many experiments involving nonspherical molecules, are shown to be exceedingly simple: They are just averages over the potential angle of cross sections calculated using simple structureless spherical particle formulas and programs. In the case of vibrationally inelastic scattering, the IOSA, without further approximation, provides a well-defined way to get fully three dimensional cross sections from calculations no more difficult than collinear calculations. Integral, differential, viscosity, and diffusion cross sections for He–CO2 obtained from the IOSA and a realistic intermolecular potential are calculated as an example and compared with experiment. Agreement is good for the complete potential but poor when only its spherical part is used, so that one should never attempt to treat this system with a spherical model. The simplicity and accuracy of the IOSA make it a viable method for routine analysis of experiments involving collisions of nonspherical molecules.

Comparison of experimental DPR and viscosity cross sections

It has previously been shown that the shape of the depolarized Rayleigh line in gases is governed by a correlation function exhibiting a spread in correlation times. Here it is shown that effective cross sections obtained from data on flow birefringence and the field dependence of the viscosity are in agreement with the DPR results. Moreover, the non-lorentzian behaviour of the H/p dependence of the viscosity, is fully interpretable in terms of the non-exponential behaviour of the DPR correlation function.

First Quantum-Mechanical Correction to the Classical Viscosity Cross Section of Hard Spheres

The classical and first quantum correction terms in a high-energy expansion of the viscosity cross section ${Q}^{(2)}$ for a Boltzmann gas of hard spheres is derived. The first correction is found to be proportional to ${(\frac{1}{k\ensuremath{\sigma}})}^{\frac{4}{3}}$, which is a term nonanalytic in $\ensuremath{\hbar}$ (i.e., ${\ensuremath{\hbar}}^{\frac{4}{3}}$), and results from scattering near the edge of the sphere. A bound is established showing the remainder of the asymptotic series to be of $O[\frac{\mathrm{ln}(k\ensuremath{\sigma})}{{(k\ensuremath{\sigma})}^{2}}]$. This asymptotic formula is compared with calculations based on the exact phase-shift expressions and its range of validity is established. The next correction terms are deduced to be proportional to $\frac{(\mathrm{ln}k\ensuremath{\sigma})}{{(k\ensuremath{\sigma})}^{2}}$ and $\frac{1}{{(k\ensuremath{\sigma})}^{2}}$ which involve ${\ensuremath{\hbar}}^{2}\mathrm{ln}\ensuremath{\hbar}$ and ${\ensuremath{\hbar}}^{2}$, respectively.

The mobility of protons and alpha particles in helium

The mobilities of alpha particles in helium at temperatures between 10 °K and 300 °K and of protons in helium at temperatures between 1 °K and 600 °K have been calculated. Both mobilities remain close to the polarization limit at all temperatures considered. A discussion is presented of the behaviour of the total and viscosity cross sections, and, for He2+ in He, of the double-charge-transfer cross section. The higher-order corrections to the mobility are evaluated and found to be negligible.

The mobility of He+ ions in He

The total, charge-transfer, diffusion and viscosity cross sections for He+ in He are calculated using numerically integrated phase shifts and term by term summations. The calculated mobility is in harmony with recent measurements. The corrections to the usual mobility expression are evaluated and found to be negligible. The average behaviour of the total cross section is satisfactorily described by the Schiff-Landau-Lifshitz approximation.

Low-Temperature Viscosity Cross Sections Measured in a Supersonic Argon Beam

An exact translational-relaxation equation is derived from the Boltzmann equation for the special case in which (a) the flow is spherically symmetric source flow, (b) the velocity distribution is ellipsoidal, and (c) the atoms are hard spheres. It is shown that, at least for small temperature differences, the appropriate cross section is the thermal-conductivity (or viscosity) transport cross section. This relaxation equation is used then to calculate values of argon cross sections from velocity distributions near the centerline of a free jet measured by molecular-beam techniques. These values (a) agree with the values obtained from viscosity measurements in the temperature range in which the two sets of data overlap, (b) continue to increase as the temperature decreases to about 25°K, and (c) decrease as the temperature decreases to about 10°K. The observed decrease in values as the temperature decreases from about 25°K was not expected and has not been explained.

Rotational Relaxation of Molecular Nitrogen

A supersonic free jet was probed to obtain a direct measure of the rotational relaxation time (τr) of nitrogen. Definition of an effective collision cross section (πσr2) for translational—rotational energy exchange in terms of τr, yields a πσr2 for nitrogen that increases with decreasing temperature faster than the viscosity cross section. Comparison of the experimental πσr2 with values obtained using thermal-conductivity data and the theory of Mason and Monchick yields good agreement.

Debye Relaxation in Symmetric-Top—Foreign-Gas Mixtures; Temperature Dependence of Collision Cross Sections

The Debye relaxation spectra of three symmetric-top gases, CH3Cl, CHF3, and SO2F2, in the pure state and in dilute mixtures with the foreign gases, He, H2, D2, Ar, N2, CH4, CO2, and C3F8, were obtained over the temperature range −20° to 145°C at a frequency of 1220 MHz. Relaxation-rate parameters and the corresponding collision cross sections are derived. The cross sections vary with the absolute temperature as T−m, where the value of m ranges from 0.3 to 0.9 for the different systems. The cross sections are considered from a classical kinetic veiwpoint and an empirical relation is obtained which accurately relates the dielectric cross section to the effective viscosity cross section (or the corresponding Lennard-Jones parameters), the internal-to-orbital-angular-momentum ratio, and one additional parameter related in some way to the shape (prolateness or oblateness) of the top. Deviations in the shape of these spectra from the simple Debye form, which are attributable to a distribution of relaxation rates, are examined in some detail and correlated with the effective collision number.

Some Aspects of the Quantal and Semiclassical Calculation of Phase Shifts and Cross Sections for Molecular Scattering and Transport

A critical study is made of the JWKB approximation for phase shifts by comparison of the approximate phases with those calculated by numerical solution of the radial wave equation. A region of E—l—Λ* space is mapped out for the Lennard-Jones (12–6) potential in which the JWKB approximation is unsatisfactory. Reasons for the failure of the JWKB approximation are noted, and a simple but effective method of predicting the region of E—l—Λ* space where failure occurs is suggested. The study is continued by a comparison of the quantal and semiclassical transport and total cross sections. It is shown that the JWKB transport cross sections are of limited practical value and that the semiclassical total cross sections are useful only at high values of E and low values of Λ*. Particular attention is centered on features associated with classical orbiting and with semiclassical rainbow and glory scattering. An apparent sign change in the quantum corrections for the viscosity cross section found by de Boer and Bird is shown to be an artifact caused by the neglect of the attractive part of the interaction potential. Finally, small-angle quantal differential cross sections for a repulsive r—12 potential are compared with some recent semiclassical calculations. The behavior as exp (—cχ2) at small angles is found to be valid to larger angles than previously thought.

Quantum Calculation of the Sensitivity of Diffusion, Viscosity, and Scattering Experiments to the Intermolecular Potential

Correlations between the form of the intermolecular potential as a function of distance and the quantum mechanical phase shifts as functions of the angular momentum quantum number have served as the basis for estimates of the sensitivity of viscosity, diffusion, and low-energy scattering experiments to the attractive and repulsive portions of the intermolecular potential. The sensitivities of the experimental techniques are approximately as follows: repulsive portion of the potential, diffusion>viscosity>scattering; to the attractive region of the potential, scattering>viscosity=diffusion. Quantum diffusion and viscosity cross sections for the Lennard-Jones potential have been calculated as functions of the reduced kinetic energy, K=μv2/2ε, and for decreasing values of the quantum parameter λ=h/σ(mε)½. The cross sections for systems with λ corresponding to T2, Ne, and CH4 are equal to the classical cross sections for K>1.2. At lower K's the deviations from classical behavior become larger as λ increases.

Quantal Study of Diffusion and Viscosity Cross Sections for a Screened Coulomb Potential

Born's approximation and the Massey-Mohr approximation are used to obtain quantum-mechanical values of the diffusion and viscosity cross sections QD and Qη for a screened Coulomb potential. The parameters are chosen to represent an ionized gas with 108 ≤ ne ≤ 1018, ne being the electron density, and 5 × 103 ≤ Te ≤ 107°K, Te being the temperature. The scattering phase shifts are also evaluated and compared with those obtained from direct numerical integration of the scattering equation. Born's approximation to the phase shift is used in a Faxén-Holtsmark type analysis to give more accurate values of the cross sections at low Te and large ne. The results are used to discuss the behavior of the electrical conductivity of a fully ionized gas.

Symmetry Effects in Gas Kinetics II: Ortho- and Para-hydrogen

The Manchester University Mk I computer has been used to calculate accurately the viscosity cross section for hydrogen molecules, and its dependence on the symmetry of the wave function for binary collisions. The calculations assume a central intermolecular potential of (exp, 6, 8) type, and cover reduced temperatures in the range 0.05 ≤ T* ≤ 2.0. Comparison is made with estimates of the relative difference in viscosity cross sections for para-para and para-ortho collisions derived from Becker and Stehl's measurements. Assuming elastic collisions only, fairly good agreement is found at 15°K, but at higher temperatures the theoretical results become increasingly too small. A central interaction is seemingly inadequate to explain the relatively large ortho-ortho cross sections observed; in this connection the effect of an increased range of interaction is briefly discussed and shown to be unimportant.

Charge Transfer Reactions in Monatomic and Diatomic Gases

Cross sections for charge transfer reactions between ions and neutral atoms and molecules have been measured in the energy range from 50 to 850 ev. Charge transfer cross sections for symmetrical reactions in monatomic and diatomic gases increase with increasing viscosity cross sections. The charge transfer cross sections decrease with increasing incident ion energy in conformity with the adiabatic criterion. For reactions in diatomic gases the charge transfer cross sections decrease as the changes in internuclear distance for the processes increase.