# Theory of emittance invariants

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## Keywords

Differential Rotation Moment Invariant Beam Transport Emittance Growth Dimensional Phase Space
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## References and Footnotes

- 1.For a description of emittance measurement, see the accompanying article by McDonald and Russell.Google Scholar
- 2.Erdelyi, A. et al., editors, Higher Transcendental Functions, Vol. II of Bateman Manuscript Project, p. 232, McGraw-Hill, New York (1953).Google Scholar
- 3.For a description of symplectic matrices and maps, see Dragt, A., Lectures on Nonlinear Dynamics, published in Physics of High Energy Particle Accelerators, Am. Inst. Phys. Conf. Proc. No. 87, R.A. Carrigan et al., editors, AIP, New York (1982).Google Scholar
- 4.It is often stated in the literature that space-charge forces lead to emittance growth. This may be true in present practice, but it need not be true in principle. It can be shown that in the approximation of the Vlasov equation, it is still possible to define a transfer map
*M*, and this map is symplectic.(The Vlasov approximation neglects short-range Coulomb collisions, and is simulated by Particle in Cell codes. In many cases the effect of these short-range collisions is negligible.) It follows that in the Vlasov approximation, Liouville's theorem (in six dimensions) still holds even in the presence of space-charge interactions. (See footnote 5 below.) Thus, often the only objectionable feature of space-charge forces is that they have nonlinear components. Consequently, as we learn to compute and handle nonlinear effects better, it may be possible in some cases to compensate for nonlinear space-charge effects by the use of nonlinear correctors. For a further discussion of space charge from this perspective, see Dragt, A. and R. Ryne, Proc. 1987 IEEE Part. Accel. Conf. 2, p. 1063 (1987). See also Ryne, R., Lie Algebraic Treatment of Space Charge, Ph.D. thesis, University of Maryland Physics Department (1987).Google Scholar - 5.For a proof of Liouville's theorem from this perspective, see reference 3 above.Google Scholar
- 6.The code MARYLIE is able to compute these coefficients as a standard option. For a further description of moment transport, see Ryne, R., Lie Algebraic Treatment of Space Charge, Ph.D. thesis, University of Maryland Physics Department (1987). For a description of MARYLIE, see Dragt, A. et al., MARYLIE 3.0, a Program for Charged Particle Beam Transport Based on Lie Algebraic Methods, University of Maryland Physics Department Technical Report (1988). See also Dragt, A. et al., Lie Algebraic Treatment of Linear and Nonlinear Beam Dynamics, Ann. Rev. Nucl. Part. Sci. 38, p. 455 (1988).Google Scholar
- 7.Lawson, J., P. Lapostolle, and R. Gluckstern, Emittance, Entropy, and Information, Particle Accelerators 5, p. 61 (1973). Frequently the defining relation for ε
^{2}contains an additional factor of 16. For notational simplicity in what follows, we omit this factor.Google Scholar - 8.For a mathematical discussion of equivalence relations and classes, see Bourbaki, N., Theory of Sets, Addison-Wesley, Reading, Mass. (1968).Google Scholar
- 9.The code MARYLIE is able to compute w*,
*M*, and the eigen mean square emittances as a standard option. For a description of MARYLIE, see reference 6 above.Google Scholar - 10.For a proof of the result (4.18), see reference 13. The invariance of these quantities was also discovered independently by D. Holm, C. Scovel, and J. Louck, unpublished notes (1988).Google Scholar
- 11.Note that I[w] = I[w*] since w ≈ w*.Google Scholar
- 12.The kinematic invariant I
_{2}was first discovered by W. Lysenko using a different method.See Lysenko, W. and M. Overley, Moment Invariants for Particle Beams, published in Linear Accelerator and Beam Optics Codes, Am. Inst. Phys. Conf. Proc. No. 177, C.R. Eminhizer, editor, AIP, New York (1988). See also Lysenko, W., Moment Invariants for Particle Beams, Technical Report LA-UR-88-165, Los Alamos National Laboratory (1988).Google Scholar - 13.For a more complete description and proofs of the results of this section, see Dragt, A., F. Neri, and G. Rangarajan, Lie Algebraic Treatment of Moments and Moment Invariants, paper in preparation (1989). See also Neri, F., Quadratic Invariants for Distributions of Particles Transported Through Linear Systems, University of Maryland Physics Department Technical Report (1988).Google Scholar
- 14.Rangarajan, G., Thesis research in progress.Google Scholar
- 15.The calculations for these figures were done using the code MARYLIE. For a description of MARYLIE, see reference 6 above.Google Scholar
- 16.For example, it may be possible in some cases to compensate for nonlinear space-charge effects. See footnote 4.Google Scholar

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© Springer-Verlag 1989