Collisions isotropic

All the theory developed up to this point has been limited in the sense that translational motion (the continuum degree of freedom) has been restricted to one dimension. In this section we discuss the generalization of this to three dimensions for collision processes where space is isotropic (i.e., collisions in homogeneous phases, such as in a... 

During adiabatic collision the molecule completes many rotations. Consequently AJ is oriented isotropically, and the operator (1.9) should be uniformly averaged over F ... 

Fig. 3.4. The isotropic Q-branch width (a) and rotational shift (b) calculated in the models of strong (1) and weak (2) collisions as a function of r E = ojqte and T = 1/tj correspondingly. The straight lines are perturbation theory estimates of spectral width and shift...

Fig. 3.9. (a) Dependence of the experimental half-width of the isotropic Q-branch of N2 and CO on the density ( ) CO, 295 K (+) CO in CF4, 273 K (A) CO in C02> 323 K (O) N2> 295 K (A) N2 in C02> 323 K. The error in the measurements of half-width is +0.2 cm-1, (b) The same data as in (a) but in relation to measured T = I/ojqXj. Theoretical curves for strong (curve 1) and weak (curve 2) collision limits are identical to curves A and B in Fig. 3.8. The upside-down triangles near the broken line present the difference between the actual half-width of CO in CO2 (A) and curve 1. The error in all cases is approximately the same as that indicated for CO.

In quantum theory as in classical theory the isotropic Raman spectrum is expressed in terms of the average value of the polarizibility tensor a(0) = (1/3) Sp a randomly changing in time due to collisions ... 

The rotational phase shift 5, which cannot exceed a mean angle of a molecular rotation during collisional time (anc), is certainly small in the case of non-adiabatic collisions. This condition is exactly that needed for anisotropic scattering (or IR absorption) spectrum narrowing, just as vibrational dephasing must be weak for an isotropic spectrum to narrow. 

As can be seen, the difference in behaviour of orientational relaxation times Te,2 in models of weak and strong collisions is manifested more strongly than in the case of isotropic scattering. Relation (6.26) is... 

The lower boundary corresponds to strong collisions, and the upper one to weak collisions. This conclusion can be confirmed by experiment. According to [259], nitrogen dissolved in SF6 has a symmetrical spectrum of isotropic scattering, indicating that collapse of the spectrum has already occurred. At the same densities, the Q-branch of the anisotropic spectrum is still well separated from the side branches, and in [259] the lower bound for its half-width is estimated as 5 cm-1. So,... 

Maxwellian distribution 129 infinite-order sudden (IOS) approximation 155-6 semi-classical calculation 136-7 Sack s model rotational relaxation 19 strong collision model 219 scattering see isotropic scattering spectra ... 

The number of elastic collisions needed to give a nearly isotropic distribution increases with energy. For example, about 15 collisions are needed for an initial 1-KeV electron in water, whereas that number is about 74 for an initial 10-KeV electron. 

Fluorescence quenching may be dynamic, if the photochemical process is the result of a collision between the photoexcited indicator dye and the quencher species, or static, when the luminophore and the quencher are preassociated before photoexcitation of the former20. It may be easily demonstrated that dynamic quenching in isotropic 3-D medium obeys the so-called Stem-Volmer equation (2)21 ... 

We consider a collection of molecular dipoles in thermal equilibrium. It is assumed that all the molecules are identical and they can take on any orientation. Because of thermal energy each molecule undergoes successive collisions with the surrounding molecules. In the absence of an applied electric field, the collisions tend to maintain a perfectly isotropic statistical orientation of the molecules. This means that for each dipole pointing in one direction there is statistically a corresponding dipole pointing in the opposite direction, as described by Fig. 1.2. 

The angular dependence of the secondary ion intensity is expected to follow a simple cosine law, in particular for randomly oriented polycrystalline surfaces. The explanation for this is that upon impact the collision cascade takes care of an isotropic distribution of the energy through the sample. Hence the intensity of collision... 

An explanation was offered by van Kranendonk many years after the experimental discovery. Van Kranendonk argued that anticorrelations exist between the dipoles induced in subsequent collisions [404], Fig. 3.4. If one assumed that the induced dipole function is proportional to the intermolecular force - an assumption that is certainly correct for the directions of the isotropic dipole component and the force, and it was then thought, perhaps even for the dipole strength - an interference is to be expected. The force pulses on individual molecules are correlated in... 

The total number of collisions with the dead comers is proportional to the total number of drops in the surface layer opposite the dead comers and—when the theory of local isotropic turbulence holds here also—proportional to the turbulent fluctuation frequency u/d. Near the wall, however, the theory of local isotropic turbulence certainly will not hold and —more or less—stationary large scale eddies will occur. Therefore, centrifugal effects will strongly increase the collision rate when the dispersed phase... 

Thus Q = K — K implies that Q = k — k. In other words, Q is the momentum transfer to the electron. If we assume that there is no particular orientation of the perturber during the collision we can replace U (fi, fi, p) by the isotropic potential U(fi, fi, p). With this approximation in Eq. (11.5) we recognize the Bom approximation to the electron scattering amplitude... 

We shall start with the Boltzmann collision model, which belongs to the last group. The Boltzmann-induced distribution FB. established at the instant t = 0 of the strong collision, in the case of an isotropic medium, is given by... 

In the case of a parabolic well the period is independent on the phase variables, the anharmonicity vanishes, and the bandwidth is nonzero only due to strong collisions. The more a potential profile differs from the parabolic one, the larger the anharmonicity and the wider the absorption band. The intensity of the absorption peak should then decrease since in accord with the Gordon rules (see, e.g., GT, Section III.G or see Section VIIA.4 in the present chapter) in an isotropic medium the integrated absorption does not depend on parameters of the model. 

The simplest method consists of investigating the collisional depopulation of a laser excited rovibronic level, i.e. of measuring the rates and cross-sections of the collisional relaxation of its population bPo- The relaxation rate Tk of polarization moments bPQ of various rank K may be represented, in the case of isotropic collisions, as follows ... 

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