Impact collinear

At the time the experiments were perfomied (1984), this discrepancy between theory and experiment was attributed to quantum mechanical resonances drat led to enhanced reaction probability in the FlF(u = 3) chaimel for high impact parameter collisions. Flowever, since 1984, several new potential energy surfaces using a combination of ab initio calculations and empirical corrections were developed in which the bend potential near the barrier was found to be very flat or even non-collinear [49, M], in contrast to the Muckennan V surface. In 1988, Sato [ ] showed that classical trajectory calculations on a surface with a bent transition-state geometry produced angular distributions in which the FIF(u = 3) product was peaked at 0 = 0°, while the FIF(u = 2) product was predominantly scattered into the backward hemisphere (0 > 90°), thereby qualitatively reproducing the most important features in figure A3.7.5. 

There has been a long history in theoretical efforts to understand H + H/Cu(lll) and its isotopic analogs because it represents the best studied prototype of an ER/HA reaction. These have evolved from simple 2D collinear quantum dynamics on model PES [386] to 6D quasi-classical dynamics on PES fit to DFT calculations [380,387,388], and even attempts to include lattice motion on ER/HA reactions [389]. These studies show that there is little reflection of incident H because of the deep well and energy scrambling upon impact, i.e., a % 1. Although some of the... 

In this section, collisions among rigid spherical particles are studied. Two simple cases, collinear and planar collisions, are described. For the general theory of stereomechanical impact of irregular-shaped rigid bodies in arbitrary motion, readers may refer to Goldsmith (1960). 

For elastic spheres, the maximum collisional force in a collinear impact between two particles, Fc (= fm in Eq. (2.132)), is given by... 

Whilst semi-classical theories have been successful in describing non-reactive processes such as elastic and inelastic scattering, they have had less impact on calculations for reactions. The majority of calculations have been for simple reactions such as H + H2, F + H2 or H + Cl2 which have been treated collinearly for the most part. The most successful use of semi-classical theory is for treating tunnelling and it has found use extending the scope of purely classical calculations by describing classically forbidden processes and threshold effects [156]. 

The collinear models are also useful close to the two-electron break-up threshold. In 1994 Rost was able to obtain the correct Wannier exponent by a semiclassical treatment of electron impact ionization of hydrogen, another important quantum problem which involves nonintegrable three-body dynamics (see also Rost (1995)). 

Fig. 38. The probability to exit on the reactive side (of the upper electronic potential energy surface as a function of the relative velocity in a near collinear CH3I + CH3I collision, see inset. The results are shown for three impact parameters as indicated. The arrow indicates the nominal energy threshold for accessing the upper electronic surface. Computed by the quantal FMS method. The reactive side includes both the formation of molecular products (CH3CH3 +12 as well as CH3 + CH3 + I2 etc.).

The collinear model (Eq. (15)) has been successfully used in the semiclassical description of many bound and resonant states in the quantum mechanical spectrum of real helium [49-52] and plays an important role for the study of states of real helium in which both electrons are close to the continuum threshold [53, 54]. The quantum mechanical version of the spherical or s-wave model (Eq. (16)) describes the Isns bound states of real helium quite well [55]. The energy dependence of experimental total cross sections for electron impact ionization is reproduced qualitatively in the classical version of the s-wave model [56] and surprisingly well quantitatively in a quantum mechanical calculation [57]. The s-wave model is less realistic close to the break-up threshold = 0, where motion along the Wannier ridge, = T2, is important. 

Early attempts to extend collinear calculations have allowed for rotation of the axis on which the three atoms are positioned, so that velocities may point in other directions and impact parameters do not need to be zero. More recently, the first computed results on coplanar motion have appeared. Studies on spatial motion are being developed at present, but equations have only been solved for schematic potentials. 

Another modified collinear model was developed by Connor and Child (1970). They constrained atoms A, B and C to a line, but allowed this to rotate. In this way impact parameters could be different from zero, although velocities remained always in the same plane. The two coordinates (X, x) in centre of mass were replaced by natural ones, (s, p). The wavefunction F was taken as a product of an adiabatic vibrational state (s, p) times a translational function i6), with 0 the axial angle. Then... 

Paulsen et al. (1972) developed an optical model for vibrational relaxation in reactive systems. Only collinear atom-diatom collisions were considered, i.e. impact parameter dependencies were omitted. The model was applied to vibrational relaxation of electronically excited I2 in inert gases, in which case dissociation of I2 is responsible for flux loss. Olson (1972) used an absorbing-sphere model for calculating integral cross sections of ion-ion recombination processes A++B ->A + B + AE, with A or B atoms or molecules. He employed the Landau-Zener formula to obtain a critical crossing distance Rc, and assumed the opacity to be unity for distances... 

Simon, S.D., and Lesage, J.P. The impact of collinearity involving the intercept term on the numerical accuracy of regression. Computer Science in Economics and Management 1988 1 137-152. 

To show the impact of laser spectroscopy on nuclear physics we discuss selected results from the aspect of information about the nuclear structure. Owing to collinear-beam spectroscopy it has now been possible to study a number of interesting regions all over the chart of nuclides. 

Planer collisions] Let us collide H+ with LiH with no impact parameter but different relative angles other than 180° (collinear collision). LiH is placed on the x-coordinate (Li at left and H at right). The collision angle 6 is measured by an angle between the x-coordinate and the line connecting H+ and the centroid of LiH [(LiHH) is 0 = 0°, while for (HLiH) 0 = 180°]. Three configurations, 0 = 30°, 60°, and 90° are examined. In... 

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