Channel coupling

The status bar displays information about the current status of the acquisition system the position of each of the four axes of the probe position monitor the maximum amplitude of the signal within the gate for both the coupling channel and the signal (flaw detection) channel and the current operating mode of the system, which may be record-... 

Rectified/Unrectified. The format of the A-scan signal displayed is selected using the appropriate button. The coupling channel signal is only available as a rectified signal. 

In matrix-vector fonn these coupled-channel equations are... 

The present derivation can easily be generalized to systems with an arbitrary number of internal degrees of freedom, and it leads to coupled channel equations identical with equation (A3.11.63). where the coupling temis (A3.11.62) are expressed as matrix elements of the interaction potential using states which depend on these internal degrees of... 

Although the Sclirodinger equation associated witii the A + BC reactive collision has the same fonn as for the nonreactive scattering problem that we considered previously, it cannot he. solved by the coupled-channel expansion used then, as the reagent vibrational basis functions caimot directly describe the product region (for an expansion in a finite number of tenns). So instead we need to use alternative schemes of which there are many. 

This equation may be solved by the same methods as used with the nonreactive coupled-channel equations (discussed later in section A3.11.4.2). Flowever, because F(p, p) changes rapidly with p, it is desirable to periodically change the expansion basis set ip. To do this we divide the range of p to be integrated into sectors and within each sector choose a (usually the midpoint) to define local eigenfimctions. The coiipled-chaimel equations just given then apply withm each sector, but at sector boundaries we change basis sets. Let y and 2 be the associated with adjacent sectors. Then, at the sector boundary p we require... 

We had earlier vindicated treating one of the OH bonds in the H2O molecule as a spectator bond in studying the abstraction reaction. Another key assumption that needed to be checked was the centrifugal sudden (CS) approximation which was invoked to reduce the number of rotational basis functions used in the computations.28 Under the CS approximation and using only the K = 0 rotational basis functions, there was a total of 220 million basis functions for J = 15 alone. Relaxing the CS approximation, for example, with K = 0,1 and J = 15 led to 650 million basis functions. To approach the fully coupled-channel (CC) results, i.e. without... 

Mizutani T. and Koltun D. S. Coupled channel theory of pion-deuteron reaction applied to threshold scattering, Annals Phys. 109, 1, (1977)... 

A little more complicated system is the de-excitation of He(2 P) by Ne, where the deexcitation is dominated by the excitation transfer and only a minor contribution from the Penning ionization is involved. The experimental cross section obtained by the pulse radiolysis method, together with the numerical calculation for the coupled-channel radial Schrodinger equation, has clearly provided the major contribution of the following excitation transfer processes to the absolute de-excitation cross sections [151] (Fig. 15) ... 

Following the well-established coupled-channel theory [17], we represent the eigenstates of the full Hamiltonian by an expansion in terms of some orthonormal states. 

Coupled Channel Formalism - Fully Uncoupled Basis... 

The main conclusion of this section is that the matrix elements of all terms in the collision Hamiltonian in the fully uncoupled space-fixed representation can be reduced to simple products of integrals of the type (8.46). Such matrix elements are very easy to evaluate numerically. The fiilly uncoupled representation is therefore very convenient for the development of the coupled channel codes for collision problems involving open-shell molecules with many angular momenta that need to be accounted for. The price for simplicity is a very large number of basis states that need to be included in the expansion of the eigenstates of the full Hamiltonian to achieve full basis set convergence (see Section 8.3.4). 

No one wants to work with 18 -y symbols. There are two work-arounds for the numerical implementation of the coupled channel theory in the total angular basis. First - as suggested by Tscherbul and Dalgarno [21] - one can use a basis of angular momentum states defined in the body-fixed coordinate frame. For example, for two molecules in a S electronic state with nonzero electron spin, the eigenstates of the full Hamiltonian can be written as... 

The numerical computations of the cross sections for molecular collisions in fields are demanding. Part of the complexity is due to the fact that most molecules of interest are open-shell radicals, so the basis for the coupled channel calculations must include multiple angular momentum states. In addition, the interactions of the molecules with an external field couple states with different total angular momenta. Note that in the presence of a single, axially symmetric field, the... 

In order to appreciate the size of the basis sets required for fully converged calculations, consider the interaction of the simplest radical, a molecule in a electronic state, with He. The helium atom, being structureless, does not contribute any angular momentum states to the coupled channel basis. If the molecule is treated as a rigid rotor and the hyperfine structure of the molecule is ignored, the uncoupled basis for the collision problem is comprised of the direct products NMf ) SMg) lnii), where N = is the quantum number... 

Most of the recent literature on molecular collisions in external fields [1-3, 9, 10, 15, 16, 18, 19, 21, 23, 25, 27-84, 92] is focused on collisions of molecules at low and ultralow temperatures. As mentioned in the introduction, it is at temperatures < 10 K that strong electromagnetic fields are expected to have a noticeable effect on the scattering properties of molecules. The coupled-channel calculations... 

First of all, the theory presented is based on a few assumptions, which, while valid for the molecular systems considered in the literature so far, need to be care-fidly examined in every specific case. As mentioned in Section 8.3, we assume that the effects of external fields on the kinetic energy operator for the relative motion are negligible and that the interactions with electromagnetic fields are independent of the relative separation of the colliding particles. In addition, we ignore the nonadiabatic interactions that may be induced by external fields and that, at present, cannot be rigorously accounted for in the coupled channel calculations. 

If we consider the collisions of two molecules (rather than atom + molecule, as above), the number of coupled channels is approximately the square of the number of accessible internal states of either molecule separately. Thus for rotational excitation of two hydrogen molecules near room temperature, Nc (jmax/2 + = 81 for Jmax = 4> and quantum calculations are... 

Takatsuka and Gordon (21a) have developed a "full collision" formulation of photodissociation which describes a multichannel process on the repulsive surface for both direct and indirect events. The scattering wavefunctions that are used to generate the T-matrix and the FC overlaps are not zeroth-order uncoupled functions, but solutions of the coupled-channel problem. 

Coupling channels. In multichannel systems, a coupling channel is used as the equivalent to an n-channel intensity stereo system. This system is also known under the names dynamic crosstalk or generalized intensity coding. Instead of n different channels, for part of the spectrum only one channel with added intensity information is transmitted. Coupling channels are used in AC-3 ([Fielder et al., 1996]) and MPEG-2 AAC ([Johnston et ah, 1996]). 

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