Average Number Of Collisions Research Articles

Statistical treatment of collision-induced relaxation of electronically excited benzene and hexafluorobenzene in gases

We show that the efficiency of a single collision on the relaxation of an excited molecule cannot be derived adequately by accounting only for an average number of collisions during the excited-state lifetime. A more detailed treatment of the collision statistics is presented and introduced into two theoretical models, differing in the way in which the velocity distribution is taken into account.Application of the theoretical results to the pressure dependence of the fluorescence quantum yield of C6H6 and C6F6 requires the adjustment of a single parameter to obtain agreement with experiment, instead of multiparameter variation as needed in previous work.

Collisional relaxation of transient vibrational energy distributions in a thermal unimolecular system. The variable encounter method

A method has been developed, called the Variable Encounter Method, for the study of the relaxation of an initial vibrationally cold ensemble of molecules into a vibrationally hot distribution by a known and variable number of successive collisions with a hot wall. The theory of the experiment is presented. The system studied was the isomerization of 1,1-cyclopropane- d 2 with a fused quartz wall temperature of 800 K to 1175 K, and average number of collisions from 2.3 to 22.3. Various modified gaussian and exponential models of energy transfer were found to give agreement with the data. The average down-step size was found to decline from ≤ 3500 cm −1 at the lowest temperature to ≈ 2500 cm − at the highest on the basis of a gaussian model. A mathematical analysis of the relation between mean first passage times and incubation times is given. Incubation times increase from ≈ 7 to ≈ 12 collisions with increasing temperature. Transient population distributions and the sequential reaction probabilities as a function of collision number are calculated.

Stationary Random Vibration of a Nonlinear System with Collision

This paper deals with the response of a nonlinear system with collision subjected to a stationary random excitation. The system consists of an oscillator and reflectors at both sides. Various stochastic properties of the response are determined analytically, including the joint probability density function and the root-mean-square values for the displacement and the velocity of the oscillator, the average number of collisions per unit time, and the probability density functions, the mean values and the standard deviations of the impact velocity, the impulse and the impact acceleration. Experiments are also carried out to examine the analytical results, and good agreement between the analytical and the experimental results is obtained. Numerical results are presented to show effects of the clearance gap between the oscillator and each of the reflectors on the stochastic properties of the response. The width of the gap for snubbers is evaluated, employing the average number of collisions whose impact velocities are beyond a certain level.

Monte Carlo calculation of electron back diffusion in a Townsend discharge

The back diffusion of electrons in a Townsend discharge has been analyzed by a simple Monte Carlo calculation. The problem was formulated in terms of three parameters: the ratio of molecular mass to electronic mass, the average number of collisions of an electron, and v0/vd, the ratio of emission speed to drift speed. In all cases, the back diffusion is found to be small at low values of the third parameter. It rises to a maximum and then decreases at larger values of v0/vd. For an absorbing cathode, the back diffusion is found to be predicted by the Thompson-Loeb formula with surprising accuracy, considering the approximate nature of its theoretical basis. Results are presented in a form that permits reasonable estimation of the back diffusion to be expected in experimental situations for which the parameter E/N is sufficiently high that the thermal gas motion may be neglected and sufficiently low that ionization is unimportant.

Reactions of rare-gas hydride ions with ethylene

Product distributions for reaction of HeH +, NeH +, ArH +, KrH + and XeH + with ethylene are reported as a function of the average number of collisions the reactant ion undergoes in an ion source. C 2H 4 + and C 2H 3 + products dominate for the more energetic species and C 2H 5 + becomes most important for the lower-energy XeH + reactant ions.

A tandem-ICR study of internal energy effects in the D 3+ + Ar and ArD + + H 2(D 2) reactions

The reactions of D 3 + with Ar and ArD +(ArH +) with H 2(D 2) are examined in a Tandem—Ion Cyclotron Resonance mass spectrometer. Product distributions and rate constants are reported as a function of the average number of collisions with D 2(H 2) which the reactant ion undergoes in the source prior to reaction. The D 3 +(Ar, D 2)ArD + reaction is endothermic and the rate constant drops rapidly as the D 3 + ions are collisionally relaxed. The reaction of ArD + with H 2 yields both H 2D + and ArH + products with a rate constant for the first channel of less than a quarter of the collision rate constant. This result is rationalized in terms of a simple polarization-stripping model.

Transmission of Gamma Radiation through Finite Cylindrical Barriers

The properties of energy distributions resulting from transmission of gamma rays through finite cylindrical barriers were studied using a modified version of the MORSE Monte Carlo transport code. Systematic studies were carried out to investigate the effects of density, composition, source energy, and cylinder diameter on the scattered and unscattered flux. In addition the average number of collisions and the contribution of the single-, double-, and triple- scattered photons were calculated for each run. It was found that for the same energy group, the intensity of the scattered flux reaches a maximum at a density that depended directly on the leakage probability but only slightly on composition. The correspondence between the Monte Carlo simulation and experimental results was found to be good. (auth)

Radiolysis Free Ion Yields and Electron Ranges in Polar and Nonpolar Liquids: Fluoromethanes

The free ion yields at zero applied electric field strength in the liquid fluoromethanes are Gfi0 in CH3F, 1.9 in CH2F2 and 1.1 in CHF3, all at 183 °K, and 0.07 in CF4 at 143 °K. Median ranges ymcd of the secondary electrons were estimated by fitting the electric field dependence of Gfi to a model equation: ymcd = 41 Å in CH3F, 39 Å in CH2F2 and CHF3, and 132 Å in CF4, at the above-mentioned temperatures. The average number of collisions, n, required to thermalize a secondary electron is about 64 000 in liquid Ar, 7 700 in CH4, 660 in C(CH3)4, 360 in CF4, 110 in C3H8, 44 in CH3F, 39 in CH2F2, 35 in CHF3, 17 in H2O, and 12 in CH3OH. The value of n with nonpolar molecules (up to neopentane in size) tends to decrease with increasing molecular complexity and with decreasing sphericity. Presence of a large dipole moment in the molecule decreases n further, and hydrogen bonds reduce n still more. Energy transfer between the electrons and rotational modes of the molecules appears to be important.

Monte Carlo methods of simulating micromixing in chemical reactors

The random coalescence model, simulated by Monte Carlo techniques, has been extended to the case of reactors for which the two separate reactant feed streams have differing residence time distributions. The parameter characterising the micromixing is the average number of collisions each fluid element experiences in its sojourn in the reactor vessel. This parameter has been varied for a simple second order, isothermal, homogeneous reaction for two arbitrarily chosen sets of residence time distributions, and it has been shown that an infinite collision rate corresponds to one of the maximum mixedness models presented by the authors in a previous paper. Methods by which the micromixing parameter can be measured, and the value subsequently used for predicting the outcome of any given reaction are discussed. Comments are also offered on the reliability of Monte Carlo techniques, in terms of the parameters affecting the averaged output conversions, and on possible ways of reducing computation times.

The dielectric breakdown of low density gases I. Theoretical

Although the Townsend theory of dielectric breakdown of gases is generally accepted, it is also realized that it does not apply at low gas density. This embraces the range of gas density between a true gas breakdown and vacuum breakdown. Under such conditions, the concept of an average number of collisions per unit path length, fundamental in the Townsend theory, can not be applied. A review of previous theoretical work covering this region is given and it is shown that it is essential to calculate the number of ionizing collisions made by an electron moving from the cathode to the anode before any model of breakdown can be developed. A method of making such a calculation using Monte Carlo computer techniques is described with special reference to mercury vapour. This gas shows are-entrant from of Paschen curve, similar to that exhibited by helium . It is shown that the theory developed for pre-breakdown current growth can explain why some gases exhibit the re­mentrant form and others do not. However, complete verification is not possible because of lack of experimental data.

Rotational temperature profiles of shock waves in diatomic gases

The variation of the translational temperature, rotational temperature, and density through shock waves in oxygen and nitrogen was studied using classical laws of mechanics and a Monte Carlo scheme. The collision dynamics were calculated using an intermolecular potential by Parker with both a two-dimensional approximation and the full three-dimensional calculations. The rotational velocity frequency distributions were also calculated. The average number of collisions a molecule will experience a t various stages passing through a shock wave were found and plotted with the temperature and density profiles. The nitrogen results were compared with experimental results and good agreement was found. This also provided a method for giving a first approximation to the three-dimensional intermolecular potential.

Vibration—Vibration Energy Exchange between Carbon Monoxide and Methane

The quenching of the vibrational fluorescence of CO by CH4 has been measured between 303° and 363°K. The quenching arises from the vibration—vibration exchange: CO*+CH4=CO+CH4* where all molecules are in ground electronic states. The average number of collisions between CO* and CH4 required for a V–V transfer to occur varies from 33 000 at 303°K to 24 000 at 363°K.

Statistics of Particle Diffusion in Spherical Geometry

The diffusion of particles from the center of a sphere of radius R is considered theoretically. An expression is obtained for the probability that a particle will be collected at a small probe within the sphere rather than at the surface. This result forms the basis for an experiment to determine the momentum transfer cross section Q/sub m/ for electrons in a gas. The only parameter which must be known in order to obtain absolute cross sections by the technique proposed is the acceptance coefficient for electrons at the probe. The probability A/sub c/ that a particle will have exactly c collisions characterized by a constant cross section Q before striking the surface of the sphere for the first time is evaluated. It is assumed that these collisions are not correlated with the diffusion process. In contrast with previous work the present method takes into account the variation of total diffusion time as well as the probability of a particular number of collisions during a given time. The probabilities A/sub 0/, A/sub 1/, and A/sub 2/ are given in terms of elementary functions and a recursion relationship is given for the other A's. The expression A/sub 0/ = y cschmore » y (where y = RN(3QQm)/sup 1/2/ and N is the gas density) is useful in an experiment to determine the cross section for inelastic collisions between electrons and gases. The probability of more than any given number of collisions, the average number of collisions, and the mean-square deviation from this average are also evaluated. (auth)« less

On the slowing down of neutrons in an homogeneous infinite medium

The object of the present paper is the study of the neutrons slowing down in an infinite homogeneous medium. The study is based on the discussion of the integral and differential equations obeyed by a characteristic differential quantity, i.e., the average number of collisions a neutron experiences in an infinitesimal interval of lethargy. The discussion is divided into three sections. The slowing down without capture is treated in Section I, where both the basic equations and the fundamental questions connected with the problem are discussed. The effects of constant capture are considered in Section II, where the moderating properties of different media will be compared. In Section III the slowing down with varying capture and in particular the case of 1/ v-capture is treated.

The Number of Elastic Collisions in a Finite Lethargy Interval

An integral equation is derived for the function giving the average number of collisions required to thermalize neutrons by elastic collisions with nuclei of relative mass M. An asymptotic solution is obtained which corrects the customary result by between and 1 collision (depending on M). Numerical results for small lethargy increments are presented for the cases M = 2 and M = 235.

Theory of Sputtering by High-Speed Ions

The theoretical treatment of ion sputtering at energies above about 50 kev is simplified by the fact that the emergent particles originate at depths in the material which are small compared to the range of the incident particles. The displacement rate is nearly constant over this region and this enables one to obtain relatively simple solutions of the diffusion problem for emission.The emission problem is reduced to an effective one-velocity diffusion calculation by an artifice. The volume displacement rate is increased by the factor $\overline{\ensuremath{\nu}}$, where $\overline{\ensuremath{\nu}}$ is the average number of secondaries, and the macroscopic absorption and scattering cross sections are adjusted to make the average number of collisions of each particle equal to the actual average over the secondaries. The resultant sputtering ratio, $\mathfrak{R}$, varies with incident energy $E$, incident angle $\ensuremath{\psi}$, and mass ratio $\ensuremath{\mu}=\frac{{M}_{1}}{{M}_{2}}$ (where ${M}_{1}=\mathrm{mass}\mathrm{of}\mathrm{incident}\mathrm{particle}$, ${M}_{2}=\mathrm{mass}\mathrm{of}\mathrm{target}\mathrm{particle}$), as $\mathfrak{R}\ensuremath{\propto}\ensuremath{\mu}(\frac{\mathrm{ln}E}{E})sec\ensuremath{\psi}$.

On the passage of neutrons through paraffin wax

The velocity-distribution of neutrons passing through paraffin wax is investigated on the assumption that the collisions made before thermal velocity has been reached are elastic collisions with protons. Within the body of the wax, and provided the variation of free path with velocity is not rapid in the neighbourhood of thermal velocities, thermal neutrons outnumber all others in the ratio of about P to 1, where P is the average number of collisions made by a thermal neutron before it is absorbed. According to Bjerge and Westcott, P is of the order of 80. The number of neutrons crossing unit area is estimated; in this case also the majority have thermal velocities. The velocity-distribution of neutrons emerging from the surface differs from that in the interior, thermal neutrons, though probably still in the majority, being less prominent under the former conditions. The ratio of the flux of thermal neutrons in the interior to that at the surface is calculated and shows satisfactory agreement with the experimental value of about 13 to 1 obtained when the neutrons are estimated from the artificial radioactivity of silver.

Velocity Distributions for Elastically Colliding Electrons

The form of the function giving the distribution in velocity of electrons in a gas is determined by a pair of equations which correspond to the detailed balancing of energy and momentum, and take into account the variation of collision cross section with velocity. The kinetic energy $\ensuremath{\epsilon}$ of the electrons is supposed to be larger than that of the gas atoms, yet small enough so that the majority of the energy lost is by elastic collisions with the atoms. The equations are solved in detail for two cases. One case is that of electrons in a uniform electric field, where the distribution is independent of position. The distribution function is found to be proportional to $\mathrm{exp}(\frac{{\ensuremath{-}\ensuremath{\epsilon}}^{2}}{{a}^{2}})$, instead of to $\mathrm{exp}(\frac{\ensuremath{-}\ensuremath{\epsilon}}{a})$, as is the case for the Maxwell distribution, when the electrons are in temperature equilibrium with the gas atoms. The average energy of the electrons, the drift current, etc., are computed as a function of the field strength. The other case considered is that of a homogeneous beam of electrons of energy ${\ensuremath{\epsilon}}_{0}$, shot into a field free space, where they lose energy to the gas atoms by collisions. The distribution function depends on $z$, the distance along the beam in mean free paths, and on $t$, the average number of collisions the electron has had before its energy decreases from ${\ensuremath{\epsilon}}_{0}$ to $\ensuremath{\epsilon}$. This distribution is also not Maxwellian, but depends on a solution of an equation in $z$ and $t$ having the form of the heat flow equation. The solution has been tested experimentally, and a quantitative check is obtained.

4. On the Mean Free Path, and the average Number of Collisions per particle per second in a Group of Equal Spheres

There is general agreement as to the value of the quantity nv, which expresses the fraction of the whole group whose members have speeds from v to v + dv; and as to the corresponding Mean Path pv,. The “mean,” in this case, is found according to the definition given by De Morgan:—