Impact parameter definition

Fig. 2.2. Definition of scattering angle x and impact parameter b of classical trajectories. The point of closest approach is at the intersection of the trajectory with the dotted line.

When we study the effect of charged particles on a substance, we often need to estimate the probabilities of excitation or ionization as functions of the distance from the axis of the track (i.e., of the impact parameter b). This is done using the quasi-classical approach, within which we assume that the charged particle moves along a definite trajectory. In the... 

From a very general point of view every ion-atom collision system has to be treated as a correlated many-body time-dependent quantum system. To solve this from an ab initio point of view is still impossible. So, one has to rely on various approximations. Nowadays the best method which can be applied to realistic collision systems (which we discuss here) is on the level of the non-selfconsistent time-dependent Hartree-Fock-Slater or, in the relativistic case, the Dirac-Fock-Slater method. Up-to-now no correlation beyond this approximation can be taken into account in the case of 3 or more electrons. (This is in accordance with the definition of correlation given by Lowdin [1] in 1956) In addition no QED contributions, i.e. no correction to the 1/r Coulomb interaction between the electrons, ever have been taken into account, although in very heavy collision systems this effect may become important. This will be discussed in section 5. A short survey of the theory used is followed by our results on impact parameter dependent electron transfer and excitation calculations of ion-atom and ion-solid collisions as well as first results of an ab initio calculation of MO X-rays in such complicated many particle scattering systems. 

In Vol. 1, Sect. 3.3. we discussed how elastic and inelastic collisions contribute to the broadening and shifts of spectral lines. In a semiclassical model of a collision between partners A and B, the particle B travels along a definite path r(t) in a coordinate system with its origin at the location of A. The path r t) is completely determined by the initial conditions r(0) and (dr/df)o and by the interaction potential V(r, Ex, E-b), which may depend on the internal energies Ex and b of the collision partners. In most models a spherically symmetric potential V r) is assumed, which may have a minimum at r = ro (Fig. 8.1). If the impact parameter b is large compared to tq the collision is classified as a soft collision, while for b hard collisions occur. 

Fig. 6.1 Schematics of collision ontcomes for viscous liquids in the We-B colUsion map [6] (/ ), and definition of the impact parameter B (right). B is the sine of the angle between the relative veloeity vectOT and the position vectOT whieh connects the eentroids of the two drops. B = 0 stands for head-on eoUision and B > 0 indieates off-eenter eollisions...

A situation where molecules are possibly even more tightly aligned is when they are adsorbed on a surface. When the precursor and the coadsorbed reactant have a definite location wifli respect to one another, both the orientation and file impact parameter are restricted. The precxusor is photodissociated to initiate the reaction. See Figure 12.9. 

In Section 3.3 we discussed how elastic and inelastic collisions contribute to the broadening of spectral lines. In a semi classical model, where the colliding particles travel along definite paths, an impact parameter b can be defined (see Fig.12.1) and the collisions may be classified as soft collisions (impact parameter b large compared to the minimum location r of the interaction potential) and hard collisions (bsoft collisions probe the... 

Keilson-Storer kernel 17-19 Fourier transform 18 Gaussian distribution 18 impact theory 102. /-diffusion model 199 non-adiabatic relaxation 19-23 parameter T 22, 48 Q-branch band shape 116-22 Keilson-Storer model definition of kernel 201 general kinetic equation 118 one-dimensional 15 weak collision limit 108 kinetic equations 128 appendix 273-4 Markovian simplification 96 Kubo, spectral narrowing 152... 

In order to evaluate the extent of attrition and its impact on the particle size distribution, there is a need of a qualitative and quantitative characterization. This, however, is not as simple as it may seem at first. There are many different properties, parameters and effects that manifest themselves and could be measured. In addition, as will be shown, the choice of the assessment procedure is strongly connected with the definition of attrition which, on its part, depends on the degradation mechanism that is considered to be relevant to the process. Hence there are a lot of procedures and indices to characterize the process of particle attrition. Section 3 deals with those which are relevant to fluidized beds and pneumatic conveying lines. 

Because of the close similarity in shape of the profiles shown in Fig. 16-27 (as well as likely variations in parameters e.g., concentration-dependent surface diffusion coefficient), a controlling mechanism cannot be reliably determined from transition shape. If reliable correlations are not available and rate parameters cannot be measured in independent experiments, then particle diameters, velocities, and other factors should be varied and the observed impact considered in relation to the definitions of the numbers of transfer units. 

The value objective function is oriented at the company s profit and loss definitions. Guiding principle is to only use value parameters that can be found in the cost controlling of the company signed by controlling. Penalty costs and without currency and weighting factors being applied to steer optimization results but having no actual financial impact - as it can be often found in supply chain optimization models - do not meet this requirement. 

The selection of variables is of central importance for the outcome of a system comparison on environmental and resource use impacts. The ideal variable or set of variables respectively provides information and describes the state of environmental phenomena with certain significance. Thus, applying a set of variables should make it possible to monitor and assess the state of the environment, to identify changes and trends, to transmit scientific data to become relevant for policy, and to evaluate already implemented policy measures. The concept of environmental indicators is broadly accepted as an adequate tool. Accordingly, an indicator is defined as a parameter or a value derived from parameters, which indicates the state of the environment with significance extending beyond that which is directly associated with a parameter value. A parameter s definition in this context is a property that is measured or observed (OECD 1994). Fieri et al. (1996) states that the purposes of indicators are as follows ... 

In addition to the somewhat empirical and difficult development of NIR applications, thorough documentation must be produced. NIR methods have to comply with the current good manufacturing practice (cGMP) requirements used in the pharmaceutical industry. Various regulatory aspects have to be carefully considered. For example, NIR applications in classification, identification, or quantification require extensive model development and validation, a study of the risk impact of possible errors, a definition of model variables and measurement parameters, and... 

To reduce the number of parameters in the kinetic equations that are to be determined from experimental data, we used the following considerations. The values klt k2, and k4 that enter into the definition of the constant L, (236), are of analogous nature they indicate the fraction of the number of impacts of gas molecules upon a surface site resulting in the reaction. So the corresponding preexponential factors should be approximately the same (if these elementary reactions are adiabatic). Then, since k1, k2, and k4 are of the same order of magnitude, their activation energies should be almost identical. It follows that L can be considered temperature independent. 

---