Time-delay matrix

Figure 5.19 Collision lifetime in picoseconds from the time delay matrix versus collision energy in kcal/mol. The resonance states are seen to be long lived.

An important perspective of multichannel resonance scattering may be gained from the lifetime matrix Q(E), introduced by Smith [46] as a generalization of the time delay At of Eq. (24) in single-channel scattering. It is now more often called the time-delay matrix or the delay-time matrix. After extending At for matrix elements Qy(E) for multichannel scattering, he proved that... 

The time-delay matrix Q(E) can be diagonalized by a unitary transformation... 

Figure 4.5 Eigenvalues of the time-delay matrix Q( ) for e -He 2S (left) and 2P (right) scattering near the He(n = 2) excitation thresholds based on the same 5-matrix data as were used in more complicated Figure 4.4. The energy is measured from the He (23S) threshold. Symbol 0 indicates the eigenvalue corresponding to elastic scattering from the ground state. Figure from Ref. [43].

The relation (50) between the time-delay matrix Q and the eigenphase sum 5 is quite general and applies to the Simonius Q matrix, too. Thus, it follows from Eq. (64) that... 

There seems no simple unitary transform of the time-delay matrix QSim, such as Eq. (72), if more than two resonances overlap each other. Instead, we may take advantage of the rank 1 of the projection matrix Pv of Eq. (61) in Qv of Eq. (69). Thus, for the eigenvalue equation... 

Figure 4.9 The eigenvalues g/(E)l (black solid curves) of the time-delay matrix Q(E) for He( P°) and their sum, Tr Q(E) (red broken curves), obtained from hyperspherical closecoupling calculations and plotted against the total energy E. The resonance peaks are numbered from the lower to higher energy positions. Most eigenvalues are too small to be distinguishable from the base line q,(E) = 0. Figure from Ref. [69],...

In terms of the time-delay matrix Q, Eq. (84) implies a convenient fitting formula... 

Eigenvalues of the time-delay matrix, calculable from the S matrix, are a powerful tool for analyzing complicated resonances, such as a few overlapping resonances and resonances obscured by another much stronger resonance or by a strong background. Illustrative applications have been presented. The eigenvectors of the time-delay matrix provide the concept of completely separate resonance channel space and nonresonance channel space. Only the former is associated with resonances, and the latter is irrelevant to any resonances. 

A. Igarashi, I. Shimamura, Time-delay matrix analysis of resonances Application to the positronium negative ion, J. Phys. B At. Mol. Opt. Phys. 37 (2004) 4221. 

The density of states so defined can be shown to be connected to the time delay that an electron experiences when scattering off the parent ion. The connection between time delay and energy derivative of the phase shift in the single channel case was first discussed by Eisenbud and Wigner [71,72], and later generalized and extended, with the concept of "dwell time," to the multichannel case by Smith [73]. Smith introduced a time-delay matrix Q, defined as... 

Lifetime Matrix. In the analysis of our RIOS results for both FH and FD., we shall employ the lifetime matrix analysis due to Smith.T22) In this approach, the lifetimes are given by the diagonal elements of the time delay matrix... 

N. Abu-Salbi, D. J. Kouri, M. Baer, and E. Poliak, A study of the quantal time delay matrix in collinear reactive scattering, /. Chem. Phys. 82 4500 (1985). 

The energy derivatives of 5 are usefiil to diaracterize the position and width of resonances. They also allow constnicti(xi of a time delay matrix for the collision process. 

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