Asymptotics beyond All Orders pp 267-273 | Cite as

# Existence and Stability of Particle Channeling in Crystals on Timescales Beyond all Orders

## Abstract

Particle channeling in crystals is an important aspect of particle-solid interactions for which I have developed a mathematical theory using modern techniques in Hamiltonian perturbation theory. This article describes that theory in broadest terms; ample discussion and detailed proofs have been or will be shortly published elsewhere (cf. Dumas 1988, 1989, 1991, and Dumas and Ellison 1991). A number of people helped me develop these ideas, but I am especially grateful to my thesis advisor, Jim Ellison, whose extensive work on mathematical aspects of channeling was the starting point for what is described here. I am also indebted to Pierre Lochak and to other members of the CMA at the École Normale Supérieure (Paris) for their assistance during the year 1986–87. It was there that I learned many of the mathematical tools used to formulate the channeling theory: a collection of results comprising KAM theory and the related but less familiar theory due to Nekhoroshev (1971, 1977, 1979) and, more recently, to Bennetin, Galgani, and Giorgilli (1985) and to Bennetin and Gallavotti (1986). The latter group of mathematical physicists first recognized the advantages—in principle at least—of Nekhoroshev’s approach in applications to physical systems where rigorous results on long-time stability of motion are desired; this approach is closely related to some of the techniques discussed in Professor Marsden’s article (these proceedings) on adiabatic invariance.

## Keywords

Transverse Energy Integrable Hamiltonian System Close Encounter Adiabatic Invariance Continuum Potential## Preview

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## References

- Benettin, G., Galgani, L., and Giorgilli, A., 1985, A proof of Nekhoroshev’s theorem for the stability times in nearly integrable Hamiltonian systems,
*Celestial Mech.*37: 1–25.MathSciNetADSMATHCrossRefGoogle Scholar - Benettin, G. and Gallavotti, G., 1986, Stability of motion near resonances in quasi-integrable Hamiltonian systems,
*J. Stat. Phys.***44**: 293–338.MathSciNetADSMATHCrossRefGoogle Scholar - Carrigan, R. A. and Ellison, J. A., Eds., 1987,
*Relativistic Channeling*, Plenum, New-York.Google Scholar - Carrigan, R. A., Toohig, T. A., and Tsyganov, E. N., 1990, Beam extraction from
*TeV*accelerators using channeling in bent crystals,*Nucl. Instr. Meth.***B 48**: 167–170.ADSGoogle Scholar - Dumas, H. S., 1988, A mathematical theory of classical particle channeling in perfect crystals, Ph. D. thesis (University of New Mexico, Albuquerque), UMI, Ann Arbor, MI.Google Scholar
- Dumas, H. S., 1989, Nekhoroshev’s theorem and particle channeling in crystals, in
*Integrable Systems and Applications*(M. Balabane, P. Lochak, and C. Sulem, Eds.) Lecture Notes in Physics No. 342: 87–94, Springer-Verlag, New York.CrossRefGoogle Scholar - Dumas, H. S., 1991, to appear in
*Dynamics Reported*.Google Scholar - Dumas, H. S. and Ellison, J. A., 1991, Nekhoroshev’s theorem, ergodicity, and the motion of energetic charged particles in crystals, IMA Preprint Series # 775, to appear in
*Foundations of Quantum and Classical Dynamics*(J. A. Ellison and H. Überall, Eds.).Google Scholar - Gemmell, D. S., 1974, Channeling and related effects in the motion of charged particles through crystals,
*Rev. Mod. Phys.***46**: 129–227.ADSCrossRefGoogle Scholar - Kumakhov, M. A., and Komarov, F. F., 1989,
*Radiation from Charged Particles in Solids*, American Institute of Physics, Translation Series, New York.Google Scholar - Lindhard, J., 1965, Influence of crystal lattice on motion of energetic charged particles,
*Mat. Fys. Medd. Dan. Vid. Selsk.***34**, no. 14.Google Scholar - Nekhoroshev, N. N., 1971, English translation: Behavior of Hamiltonian systems close to integrable,
*Funct. Anal.***5**: 338–339 (1971).Google Scholar - Nekhoroshev, N. N., 1977, English translation: An exponential estimate of the time of stability of nearly integrable Hamiltonian systems,
*Russian Math. Surveys***32**(6): 1–65 (1977).ADSMATHCrossRefGoogle Scholar - Nekhoroshev, N. N., 1979, English translation: An exponential estimate of the time of stability of nearly integrable Hamiltonian systems II, in
*Topics in Modern Mathematics, Petrovskii Seminar No.5*(O. A. Oleinik, Ed.), Consultants Bureau, London, (1980).Google Scholar