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Manuscript Title: Resonant Coherent Excitation of Hydrogen-like Ions Planar Channeled in a Crystal. Transition into the First Excited State
Authors: A. Babaev, Yu. L. Pivovarov
Program title: RCE_H-like_1
Catalogue identifier: AEKX_v1_0
Distribution format: tar.gz
Journal reference: Comput. Phys. Commun. 183(2012)705
Programming language: C++ (g++, icc compilers).
Computer: Multiprocessor systems (clusters).
Operating system: Any OS based on LINUX; program was tested under Novell SLES 10.
Has the code been vectorised or parallelized?: Yes. Contains MPI directives.
RAM: < 1MB per processor
Keywords: resonant coherent excitation, hydrogen-like ions, channeling, Schrödinger equation, fine structure, Stark effect.
PACS: 31.15.-p, 87.10-e, 02.60.Cb.
Classification: 2.1, 2.6, 7.10.

External routines: MPI library for GNU C++, Intel C++ compilers

Nature of problem:
When relativistic hydrogen-like ion moves in the crystal in the planar channeling regime, in the ion rest frame the time-periodic electric field acts on the bound electron. If the frequency of this field matches the transition frequency between electronic energy levels, the resonant coherent excitation can take place. Therefore, ions in the different states may be observed in the outgoing beam behind the crystal. To get the probabilities for the ion to be in the ground state or in the first excited state, or to be ionized, the Schrödinger equation is solved for the electron of ion. The numerical solving of the Schrödinger equation is carried out taking into account a fine structure of electronic energy levels, the Stark effect due to the influence of the crystal electric field on electronic energy levels and the ionization of ion due to the collisions with crystal electrons.

Solution method:
The wave function of the electron of ion is the superposition of the wave functions of stationary states with time-dependent coefficients. These stationary wave functions and corresponding energies are defined from the stationary Schrödinger equation. The equation is reduced to the problem of the eigen values and vectors of Hermitian matrix. The corresponding matrix equation is considered as the linear equation system. Then the time-dependent coefficients of the electron wave function are defined from the Schrödinger equation, with a time-periodic crystal field. The time-periodic field is responsible for the transitions between the stationary states. The final time-dependent Schrödinger equation represents the matrix equation which has been solved by means of the QR-algorithm.

As expected the program gives the correct results for relativistic hydrogen-like ions with the kinetic energies up to 1 GeV/u and at the crystal thicknesses of 1-100 μm. The restrictions are: first, the program might give inadequate results, when the ion kinetic energy is too large (>10 GeV/u); second, the unaccounted physical factors may be significant at specific conditions. For example, the spontaneous emission by exited highly charged ions, as well as both energy and angular spread of the incident beam, could lead to additional broadening of the resonance. The medium polarization by the electric field of ion can influence the electronic energy levels of the ion in the non-relativistic case. The role of these factors was discussed in the references. Also, the large crystal thickness may require large computational time.

Running time:
In general, the running time depends on the number of processors. In our tests we used the crystal thickness up to 100 μm and the number of 2.66 GHz processors was up to 100. The running time was about 1 hour in these conditions.