A basis-set time-independent method to calculate tunneling rates in bound systems through a potential barrier is presented. The tunneling decay rates are associated with the imaginary parts of the complex eigenvalues of the Schrödinger equation where the reaction coordinate r′ is complex scaled such that, dr = dr′[1/cos θ(r′)]exp (iθ(r′)), where tan θ(r′) = tan θ∞g(r′). The function g(r′) fulfills 0 ≤ g(r′) ≤ 1 and shows a smooth transition from 0 to 1 near r′ = r0 which is the location of the top of the barrier. The value of θ∞ should be larger than a critical value for which a sharp transition from a real eigenvalue spectrum to a complex one is obtained. Illustrative numerical applications to two isomerization reaction models are given.
Time-dependent spin correlations in the Heisenberg magnet at infinite temperature
The relaxation rate for depolarization of a positive muon implanted in an isotropic magnetic salt with ferromagnetic exchange interactions is studied theoretically, on the basis of the coupled-mode theory of critical and paramagnetic spin fluctuations and a full numerical evaluation of the dipole field experienced by the muon. The main findings from studies of realistic models of EuO and EuS are (a) a significant dependence of the relaxation rate, A, on the assumed position of the implanted muon and (b) a monotonic temperature dependence, with A -~3 12 in the approach to the critical temperature at which the correlation length, ~, diverges. In contrast, previous results for a model of an isotropic magnet with an antiferromagnetic exchange, RbMnF 3 , show that A for this magnet is not a monotonic function of the temperature, and in the precursor region to Tc A increases with decreasing temperature with a power law behaviour A.-~1 12• The calculated values of A for EuO are consistent with data from preliminary experiments on the same salt.(1.) Introduction Several experimental studies have demonstrated that measurements of the depolarization of positive muons implanted in magnetic materials have the potential to provide useful information, at an atomic level of detail, on the fluctuations of the magnetic moments; see, for example, Cox (1987) and Dalmas de Reotier et al. (1994).In a previous paper we provided a comprehensive theoretical investigation of relaxation in the paramagnetic phase of an antiferromagnetically coupled material (Lovesey et al. 1994). The present paper reports findings from a similar, comprehensive investigation of muon relaxation in isotropic, ferromagnetically coupled systems. Results for two materials, EuO and EuS, are provided.The overall plan of the work is the same as that used to study the antiferromagnetically coupled salt, RbMnF 3 , namely, a complete numerical evaluation of the spatial Fourier transform of the dipole field between the muon moment and the atomic moments, and a description of critical and paramagnetic fluctuations of the atomic moments from a full version of the coupled-mode theory. In view of the strong similarities in the work for the two types of magnetic salts the background to the methods given here is very brief; the reader interested in these matters is referred to Lovesey et al. (1994).Turning to the results of our work based on realistic models of EuO and EuS -an isotropic Heisenberg magnet with exchange interactions out to the second shell of neighbouring spins -we find that the magnitude and temperature dependence of the relaxation rate, A, depend on the position assumed for the implanted muon. A similar finding was obtained in the study of RbMnF 3 • For the latter material, "A is not a monotonic function of the temperature. In contrast, the relaxation rates for EuO and EuS are found to be monotonic functions of the temperature, cf. Table (1). In the approach to the critical temperature ' A increases, and the temperature dependence, expressed in term...
An analytic representation of the full Green's function including bound states, resonances, and remaining contributions has been obtained for a class of dilatation analytic potentials, including the superimposed Coulomb potential. It is demonstrated how to obtain the locations and residues of the poles of the Green's function as well as the associated generalized spectral density. For a model potential which has a barrier and decreases exponentially at infinity we have found a certain delation property of the generalized spectral density. A qualitative explanation of this phenomenon is suggested. This constitutes the motivation for an approximation that explicitly shows a decomposition of the (real) continuum, corresponding to scattering data, into resonances and background contributions. The present representation is also shown to incorporate the appropriate pole-background interferences. Numerical residue strings are computed and analyzed. Results for the Coulomb potential plus the above-mentioned model potential are reported and compared with the previous non-Coulomb case. A similar delation e8'ect is seen to occur, as well as basically the same poleand residue-string behavior. The relevance of the present analysis in relation to recently planned experiments with electron-cooled beams of highly charged ions is briefly discussed.
Cumulative reaction probability by the complex coordinate scattering theory J. Chem. Phys. 98, 6327 (1993); 10.1063/1.464826The complex coordinate scattering theory and the Kohn variational method: A general formulation and application to long range potentials J. Chem. Phys. 97, 6443 (1992); 10.1063/1.463702The complex coordinate scattering theory: Broken inversion symmetry of corrugated surfaces in helium diffraction from Cu (115)
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