Effective collision cross section

Here p is the radius of the effective cross-section, (v) is the average velocity of colliding particles, and p is their reduced mass. When rotational relaxation of heavy molecules in a solution of light particles is considered, the above criterion is well satisfied. In the opposite case the situation is quite different. Even if the relaxation is induced by collisions of similar particles (as in a one-component system), the fraction of molecules which remain adiabatically isolated from the heat reservoir is fairly large. For such molecules energy relaxation is much slower than that of angular momentum, i.e. xe/xj > 1. 

The important fact is that the number of collisions Zr increases with temperature. It may be attributed to the effect of attraction forces. They accelerate the molecule motion along the classical trajectories favouring more effective R-T relaxation. This effect becomes relatively weaker with increase of temperature. As a result the effective cross-section decreases monotonically [199], as was predicted for the quantum J-diffusion model in [186] (solid line) but by classical trajectory calculations (dotted and broken lines) as well. At temperatures above 300 K both theoretical approaches are in satisfactory mutual agreement whereas some other approaches used in [224, 225] as well as SCS with attraction forces neglected [191] were shown to have the opposite temperature dependence for Zr [191]. Thus SCS results with a... 

The first complication to be considered is the presence of an electrostatic field during the mass spectrometric study of the reaction. Only few quantitative studies have allowed for the possible contribution of hard collisions to cross-section (25), and the possibility that competitive reactions of the same ion may depend on ion energy is generally neglected in assigning ion-molecule reaction sequences. These effects, however, do not preclude qualitative application of mass spectrometric results to radiation chemistry. 

The effective cross section s for ionizing collisions depends on the type of gas. According to (2.25), the discharge current i is a function of the number particle density ng, as in a Penning gauge, and it can be used as a measure of the pressure in the range from 10 to 10 mbar. At lower pressures the measurements are not reproducible due to interferences from field emission effects. 

These measurements have been carried out in collaboration with de Maeyek[4]). The rate constant was found to be (1 3 0-2)-10n litres/ mol-sec thus the neutralization reaction is the fastest known bi-molecular reaction in aqueous solution. Molecular-kinetic considerations show that the velocity of recombination is solely determined by the collision frequency of the ions. Furthermore, the effective cross section of the proton is so large that the reaction already proceeds spontaneously when ions approach each other within a distance of two to three H-bonds. This means that the motion of the proton within the hydration complex (the diameter of which corresponds to about two to three H-bonds) proceeds rapidly compared to the actual movement of the ions towards each other. 

For a molecule in the excited state, the effective cross-section for coil os can be much greater than those for kinetic collision. The optical collisions nay be defined as the minimum distance of approach over which the excited mole-cule can interact with another molecule to bring about a physical or chemical change. 

Dubois, A., Nielsen, S.E. and Hansen, J.P. (1993) State selectivity in H+-Na(3s/3p) collisions differential cross sections, alignment and orientation effects for electron capture J. Phys. B, 26, pp. 705-721. 

For a spherical particle, the effective cross section for collision, a, is... 

The details of the relaxation channels contributing towards the total effective cross-section of relaxation of the ground state a requires measurements with fixed vibrational and rotational quantum numbers v", J" and v J, J" of the reaction (3.1). Data on such measurements, e.g. in Na2/Na beams in collisions with noble gases can be found in the monograph [116], and those on Li2-containing vapour can be found in [306]. 

Table 3.8. Total effective cross-section of relaxation in the electronic ground state of the dimers (-X E, 1,73) [11, 13] and Na2(X1E+, 3,43) [239], in collisions with admixture atoms X and with own atoms A...

It is noted that the collision cross-sectional area, also referred to as the total scattering cross section (2.163), can be approximated in different ways too. In one approach the target particles are approximated as point-like particles. Thus, the cross section of the target particles are not considered in calculating the effective cross-sectional area, so oap Trdg/d. To achieve an improved estimate of the effective cross-sectional area the size of the target particles should be taken into account as well, thus oap =... 

Fig. 7.—Gas-kinetic effective cross-section in a collision the centres of mass of the two equal molecules cannot come nearer than the distance G (G = diameter of a molecule).

The collision efficiency E(Dp,dp) is by definition equal to the ratio of the total number of collisions occurring between droplets and particles to the total number of particles in an area equal to the droplet s effective cross-sectional area. A value of E = 1 implies that all particles in the geometric volume swept out by a falling drop will be collected. Usually < 1, although E can exceed unity under certain conditions (charged particles). Experimental data suggest that all particles that hit a hydrometeor stick, and therefore, a sticking efficiency of unity is assumed. 

In these and the above equations, the a are cross sections per imit volume, the a in (8) is scattering cross section, the average loss in r per collision. The are used because the material may contain different types of atoms. The (Ta is the thermal absorption cross section r(r) the resonance absorption cross section per unit volume. The = qef is the multiplication constant divided by the resonance escape probability. The product of thermal utilization / and (Ta is the effective cross section of uranium per unit volume, i.e., its cross section per unit volume multiplied by the thermal neutron density in it and divided by the average thermal neutron density. One can write, therefore, (Tu for f(Ta- If one multiplies this with rj the result is the same as crfU where fission cross section for thermal neutrons per unit volume, p the number of fast neutrons per fission. As a result, the third term in (7) can be written also as e is the multiplication by fast effect)... 

The effective lifetime of an excited molecular level is tesip = 5 mbar) = 8 X 10" s and Tesip = 1 mbar) = 12 x 10" s for molecules with the mass M = 43 AMU in a gas cell with argon buffer gas at T = 500 K. Calculate the radiative lifetime, the collision-quenching cross section, and the homogeneous linewidth Av( >). 

The cross-section, as its name suggests, is the effective area for collision. The cross-section of a spherical target is cr = Trr. In aiming a beam of particles at a target (which is much smaller than the beam), as in the Rutherford scattering experiment, the scattering process is treated statistically in terms of the cross-section for interaction with a nucleus. 

Usually, measured effective cross-sections are reported as a function of the mean collision energy, Ep). For two well-collimated monoenergetic beams crossing at an angle A, Ep is given by... 

The dilute-gas theory is presented here for the first time in terms of effective collision cross sections in a comprehensive readily usable form which applies to both polyatomic fluids and monatomic fluids. This description should now be used exclusively but, because it is relatively new, expressions are given for the macroscopic quantities in terms of these effective cross sections, and certain simple relationships between these effective cross sections and the previously used collision integrals are also described. 

The effective cross section 6(p0j/ 9) exactly the same as the cross section (pOst) introduced in Section 4.2 for a pure gas. All of the other cross sections introduced depend upon the binary interaction of two molecular species alone, as is made clear by the definition of the Wang Chang-Uhlenbeck or Boltzmann collision operators. From the point of view of the present volume this is a very important result because it means that if the collision cross sections can be determined for each binary interaction, then it becomes possible to evaluate the contribution of that interaction to the transport properties of a multicomponent mixture immediately. If repeated for all possible binary interactions in the mixture then the prediction of the transport properties of any multicomponent mixture containing them is possible. Furthermore, if the effective cross sections for the pure components of a binary mixture are known then those characteristic of the single, unlike binary interaction may be deduced from the properties of a mixture. It is worthy of note that this result remains valid even if higher orders of kinetic theory approximation are invoked. 

It has been argued that in special cases, such as the interactions between strongly polar substances, the representation of the effective cross sections by means of the extended law of corresponding states is inadequate. In such circumstances each interaction will have to be treated individually, in an effort to determine the most appropriate inter-molecular pair potential model to represent it, so that a suitable estimate of the cross section can be obtained. Effective cross sections or, more often, collision integrals, for a number of potential models are available. 

To derive these relationships is not a simple task therefore, a complete list cannot be found in the literamre. But, in order to apply, for instance, the Extended Law of Corresponding States with the coefficients and parameters given by Maitland et al. (1987), just three relationships are needed to be able to calculate effective cross sections ftom expressions for collision integrals. These formulas can be derived from the appropriate equations defining p, D and A and read... 

In Figure 14.1, a comparison is presented between three older correlations for (Hanley 1974 Kestin et al. 1984 Rabinovich et al. 1988) and a correlation recently derived from the work of Bich etal. 990). The latter correlation serves as the baseline in the temperature range 100-2000 K. In each of the four cases the correlations, which originated from equation (14.4) for were not expressed in terms of the reduced effective collision cross section 6j[ but instead in terms of a reduced collision integral S2(2,2) Furthermore, for the higher-order correction the second-order Kihara approximation (Maitland et al. 1987) was applied. As detailed in the Appendix to Chapter 4, the conversion from collision integrals to effective cross sections is simple for isotropic interatomic potentials, namely. 

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