Channels Method

The coupled-channels method may be developed within the language of wave-mechanics, or more formally (and more compactly) by means of operator equations. The common feature of both approaches is that the total scattering state is expanded in internal states of reactants and products. The nature of the colliding particles and the quantum numbers of the interna] states define the reaction channel index c = a, b,. We begin with the wave-mechanical approach, some of whose features have been presented in the section on statistical theories. For the total wavefunction [pa of reactants in channel a, with relative wave vector ka, we can write 

In practice, calculations must be carried out for small basis sets that would only produce approximations to F ). To compensate for this, a possible approach is to write 

The above procedure has been followed by Wolken and Karplus (1974) for reactive H + H2 collisions. Normalizable functions were not included in the expansion. Even so, this work constitutes a very significant step towards the solution of the three-dimensional problem. The coupled equations to be solved for the matrix x /J of states ij/Jyi is, omitting the J index, 

The distorted-wave calculations referred to above have been performed by Tang and Karplus (1971). For the reaction a- b, this approximation 

Distorted waves and a linear approximation were also used in earlier (and more approximate) work by Micha (1965). His results for D + H2 - DH + H also gave backward cross sections, in agreement with recent experimental results (Geddes et a/., 1972). Distorted-wave calculations for the same system, done by Tang (1972) also agreed with experiment. 

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With the single-channel method, data are acquired in series or one channel at a time. Normally, a series of data points are established for each machine-train and data are acquired from each point in a measurement route. While this approach is more than adequate for routine monitoring of relatively simple machines, it is based on the assumption that the machine s dynamics and the resultant vibration profile are constant throughout the entire data acquisition process. This approach hinders the ability to evaluate real-time relationships between measurement points on the machine-train and variations in process parameters such as speed, load, pressure, etc. 

In the application of the two-channel method to a-glycine, use of a uniform prior density sharpens and enhances the bond peaks relative to the observed deformation density, but suppresses the lone-pair peaks to much lower levels. The use of the multipole refinement deformation density as a nonuniform prior gives better results and some increase in detail. 

This remark is associated with the amount of calculation performed and is not intended as a criticism. This work provides a valuable quantum mechanical analysis of a three-dimensional system. The artificial channel method (19,60) was employed to solve the coupled equations that arise in the fully quantum approach. A progression of resonances in the absorption cross-section was obtained. The appearance of these resonances provides an explanation of the origin of the diffuse bands found... 

M.L. Du, A. Dalgamo, Artificial-channel method for multichannel-decay-resonance energies and widths, Phys. Rev. A 43 (1991) 3474. 

In addition to the direct methods, in which one calculates first the continuum wavefunctions and subsequently the overlap integrals with the bound-state wavefunction, there are also indirect methods, which encompass the separate computation of the continuum wavefunctions the artificial channel method (Shapiro 1972 Shapiro and Bersohn 1982 Balint-Kurti and Shapiro 1985) and the driven equations method (Band, Freed, and Kouri 1981 Heather and Light 1983a,b). Kulander and Light (1980) applied another method, in which the overlap of the bound-state wavefunction with the continuum wavefunction is directly propagated. The desired photodissociation amplitudes are finally obtained by applying the correct boundary conditions for R —> oo. 

Sere J, = Jk has been assumed for simplicity. The matrix elements are computed t vith the artificial channel method [14],... 

A powerful way of achieving this goal uses the coupled-channels expansion, a method widely used in calculations of scattering cross sections [6]. In the context of quantized matter-radiation problems, the coupled-channels method amounts to expanding E, n, N ) in number states. Concentrating on the expansion in the /th mode, we write E, n, N ) as... 

The HF and SHF structure has been calculated by Bakalov and Korobov [7,8] using the best three-body wavefunctions of Korobov [9], and recently by Yamanaka et al. [10] using wavefunctions calculated by Kino et al. by the coupled rearrangement-channel method [11]. They present the HF and SHF energies in terms of the angular momentum operators as... 

Narahashi, T. et al. (1992) Overview of toxins and drugs as tools to study excitable membrane Ion channels I. Voltage-activated channels. Methods Enzymol.. 207, 620-643. 

There has also been a recent assessment of different methods for the processing of diffusion ordered spectroscopy (DOSY) NMR experiments. This included the evaluation of iterative single channel methods including single channel methods such as... 

A review is given on the application of the coupled-channel method for the calculation of the electronic energy loss of ions as well as ionization in matter. This first principle calculation, based on the solution of the time-dependent Schrodinger equation, has been apphed to evaluate the impact parameter and angular dependence of the electronic and nuclear energy losses of ions as well as the ionization due to high-power short laser pulses. The results are compared to experimental data as well as to other current theoretical models. 

The chapter is organized as follows. The principle of the coupled-channel method is reviewed in detail in Section 2. The results are discussed in connection to higher order terms in Section 3. The application to multiphoton ionization is described in Section 4. Comparisons with measurements are provided in Section 5. A simple model for the electronic energy loss is... 

Here we apply the coupled-channel method to calculate photo ionization of atomic hydrogen by short (femtosecond) laser pulses at high power densities (up to 5 X 10 " W/cm ). A classical electro-dynamical field approximates the laser/atom interaction, according to (in the Coulomb gauge)... 

A direct measurement of the electronic energy loss as a function of the impact parameter is a hard task to be performed from the experimental point of view and only a few experiments have been performed for fast light ions. Experiments in gas targets under single collision condition provide a more direct and precise comparison of the theoretical results with the experimental data. Here we compare the results of the coupled-channel method for collisions of protons with He as a function of the projectile scattering angle. 

Complex-Coordinate Coupled-Channel Methods for Predissociating Resonances in van der Waals Molecules... 

Coupled channel methods for colllnear quantum reactive calculations are sufficiently well developed that calculations can be performed routinely. Unfortunately, colllnear calculations cannot provide any Insight Into the angular distribution of reaction products, because the Impact parameter dependence of reaction probabilities Is undefined. On the other hand, the best approximate 3D methods for atom-molecule reactions are computationally very Intensive, and for this reason. It Is Impractical to use most 3D approximate methods to make a systematic study of the effects of potential surfaces on resonances, and therefore the effects of surfaces on reactive angular distributions. For this reason, we have become Interested In an approximate model of reaction dynamics which was proposed many years ago by Child (24), Connor and Child (25), and Wyatt (26). They proposed the Rotating Linear Model (RLM), which Is In some sense a 3D theory of reactions, because the line upon which reaction occurs Is allowed to tumble freely In space. A full three-dimensional theory would treat motion of the six coordinates (In the center of mass) associated with the two... 

The layered samples were analyzed with RBS/channeling method using 3 MV single stage accelerator at TLARA, JAERI/Takasaki. The analyzing beams of He ions with energy of 1.5 to 2.7 MeV were incident on samples. The size of the beam was about 1 mm in diameter and beam current was about 10 nA typically. Backscattered particles were detected by standard surface barrier detectors at 160° and 110° to the incident beam. 

The final stage in the adiabatic reduction is the solution of Eq. (4.24). Given the adiabatic potential of Eq. (4.26) this cannot be done analytically, but the resulting ordinary differential equation may be solved numerically using the finite difference method. As an example, we show in Fig. 20 a comparison between the even-parity adiabatic eigenvalues and the exact ones, obtained by solving the full coupled channels expansion, using the artificial channel method.69... 

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