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AXISYMMETRIC ELECTRIC FIELD CALCULATION WITH ZONAL HARMONIC EXPANSION

By F. Gluck

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Abstract:
The electric potential and field of an axially symmetric electric system can be computed by expansion of the central and remote zonal harmonics, using the Legendre polynomials. Garrett showed the usefulness of the zonal harmonic expansion for magnetic field calculations, and the similar radial series expansion has been widely used in electron optics. In this paper, we summarize our experience of using the zonal harmonic expansion for practically interesting axisymmetric electric field computations. This method provides very accurate potential and field values, and it is much faster than calculations with elliptic integrals. We present formulas for the central and remote expansions and for the coefficients of the zonal harmonics (source constants) in the case of general axisymmetric electrodes and dielectrics. We also discuss the general convergence properties of the zonal harmonic series (proof, rate of convergence, and connection with complex series). Practical considerations about the computation method are given at the end. In our appendix, one can find many useful formulas about properties of the Legendre polynomials, various derivatives of the zonal harmonic functions, and a simple numerical integration algorithm.

Citation:
F. Gluck, "Axisymmetric electric field calculation with zonal harmonic expansion," Progress In Electromagnetics Research B, Vol. 32, 319-350, 2011.
doi:10.2528/PIERB11042106
http://www.jpier.org/pierb/pier.php?paper=11042106

References:
1. Paszkowski, B., Electron Optics, London Eliffe Books, American Elsevier, New York, 1968.

2. Szilágyi, M., Electron and Ion Optics, Plenum Press, New York and London, 1988.

3. Hawkes, P. W. and E. Kasper, Principles of Electron Optics, Vol. 1, Academic Press, Harcourt Brace Jovanovich, 1989.

4. Kraus, C., et al., "Final results from phase II of the Mainz neutrino mass search in tritium β decay," Eur. Phys. J. C, Vol. 40, 447, 2005.
doi:10.1140/epjc/s2005-02139-7

5. Lobashev, V. M., "The search for the neutrino mass by direct method in the tritium beta-decay and perspectives of study it in the project KATRIN," Nucl. Phys. A, Vol. 719, 153c, 2003.
doi:10.1016/S0375-9474(03)00985-0

6. Glück, F., et al., "The neutron decay retardation spectrometer aSPECT: Electromagnetic design and systematic effects," Eur. Phys. J. A, Vol. 23, 135, 2005.
doi:10.1140/epja/i2004-10057-1

7. Baessler, S., et al., "First measurements with the neutron decay spectrometer aSPECT," Eur. Phys. J. A, Vol. 38, 17, 2008.
doi:10.1140/epja/i2008-10660-0

8. Beck, M., et al., "WITCH: A recoil spectrometer for weak interaction and nuclear physics studies," Nucl. Instrum. Methods A, Vol. 503, 567, 2003.
doi:10.1016/S0168-9002(03)00994-X

9. Friedag, P., Bahnverfolgungssimulationen für das WITCH-experiment, Diploma thesis, University of MÄunster, 2008.

10. Angrik, J., et al., KATRIN design report 2004, FZKA Scientific Report 7090, Forschungszentrum Karlsruhe, 2005, http://bibliothek.fzk.de/zb/berichte/FZKA7090.pdf.

11. Chari, M. V. K. and S. J. Salon, Numerical Methods in Electromagnetism, Academic Press, San Diego, 2000.

12. Maxwell, J. C., A Treatise on Electricity and Magnetism, Vol. 1, Clarendon Press, 1873.

13. Durand, E., Electrostatique, Vol. 1, Masson et Cie, Paris, 1964.

14. Jackson, J. D., Classical Electrodynamics, John Wiley & Sons, New York, 1999.

15. Smythe, W. R., Static and Dynamic Electricity, McGraw Hill Book Company, New York, 1968.

16. Kellogg, O. D., Foundations of Potential Theory, Springer Verlag, Berlin, 1967.

17. Garrett, M. W., "Axially symmetric systems for generating and measuring magnetic fields," J. Appl. Phys., Vol. 22, 1091-1107, 1951.
doi:10.1063/1.1700115

18. Garrett, M. W., "The method of zonal harmonics," High Magnetic Fields, H. Kolm, et al. (eds.), John Wiley and Sons, 1962.

19. Garrett, M. W., Computer programs using zonal harmonics for magnetic properties of current systems with special reference to the IBM 7090, ORNL-3318, Oak Ridge National Laboratory, USA, 1962.

20. Garrett, M. W., "Thick cylindrical coil systems for strong magnetic ¯elds with field or gradient homogeneities of the 6th to 20th order," J. Appl. Phys., Vol. 38, 2563, 1967.
doi:10.1063/1.1709950

21. Thümmler, T., Präzisionsüberwachung und Kalibration der Hochspannung für das KATRIN-experiment, Dissertation, University of Münster, 2007.

22. Pocanic, D., et al., "Nab: Measurement principles, apparatus and uncertainties," Nucl. Instrum. Methods A, Vol. 611, 211, 2009.
doi:10.1016/j.nima.2009.07.065

23. Valerius, K., Elektromagnetisches Design für das Hauptspektrometer des KATRIN Experiments, Diploma thesis, University of Bonn, 2004.

24. Valerius, K., Spectrometer-related background processes and their suppression in the KATRIN experiment, Dissertation, University of Münster, 2009.

25. Hugenberg, K., Design of the electrode system for the KATRIN main spectrometer, Diploma thesis, University of Münster, 2008.

26. Zacher, M., Electromagnetic design and field emission studies for the inner electrode system of the KATRIN main spectrometer, Diploma thesis, University of Münster, 2009.

27. Wandkowsky, N., Design and background simulations for the KATRIN main spectrometer and air coil system, Diploma thesis, Karlsruhe Institute of Technology, 2009.

28. Fränkle, F., Background investigations of the KATRIN prespectrometer, Dissertation, Karlsruhe Institute of Technology, 2010.

29. Groh, S., Untersuchung von UV-laser induziertem Untergrund am KATRIN Vorspektrometer, Diploma thesis, Karlsruhe Institute of Technology, 2010.

30. Hein, J. H. C., Angular defined photo-electron sources for the KATRIN experiment, Diploma thesis, University of Münster, 2010.

31. Valerius, K., et al., "Prototype of an angular-selective photoelectron calibration source for the KATRIN experiment," J. Inst., Vol. 6, P01002, 2011.
doi:10.1088/1748-0221/6/01/P01002

32. Lukic, S., et al., "Ion source for tests of ion behavior in the KATRIN beam line," Rev. Sci. Instrum., Vol. 82, 013303, 2011.
doi:10.1063/1.3504372

33. KATRIN homepage, Talks and Publications, Diploma and Ph.D. theses, http://www-ik.fzk.de/~katrin/publications/thesis.html.

34. Westfälische Wilhelms-Universität Münster, Institut für Kernphysik, AG Prof. Dr. C. Weinheimer, http://www.unimuenster.de/Physik.KP/AGWeinheimer/Arbeiten-de.html.

35. Vöcking, S. Implementierung der multipole boundary element methode für das KATRIN-experiment, Diploma thesis, University of Münster, 2008.

36. Corona, T. J. Tools for electromagnetic field simulation in the KATRIN experiment, master thesis, MIT, 2009.

37. Leiber, B., "Non-axially symmetric field and trajectory calculations for the KATRIN experiment," Diploma thesis, 2010.

38. Babutzka, M., et al., The comprehensive guide to KASSIOPEIA, version 1.00.00, KATRIN internal report.

39. Böttcher, C. J. F., Theory of Electric Polarization, Vol. 1, Elsevier Sci. Publ. Comp., Amsterdam, 1973.

40. Griffiths, D. J., Introduction to Electrodynamics, Prentice Hall, Upper Saddle River, New Jersey, 1999.

41. Cowan, E. W., Basic Electromagnetism, Academic Press, New York, 1968.

42. Panofsky, W. K. H. and M. Phillips, Classical Electricity and Magnetism, Dover Publications, Mineola, New York, 1962.

43. Ravaud, R., G. Lemarquand, and S. I. Babic, "Introducing fictitious currents for calculating analytically the electric field in cylindrical capacitors," Progress In Electromagnetics Research M, Vol. 9, 139-150, 2009.
doi:10.2528/PIERM09101509

44. Gamelin, T. W., Complex Analysis, Springer-Verlag, New York, 2001.

45. Needham, T., Visual Complex Analysis, Clarendon Press, Oxford, 1997.

46. Szegö, G., "On the singularities of zonal harmonic expansions," J. Rat. Mech. Anal., Vol. 3, 561, 1954.

47. Sneddon, I. N., Special Functions of Mathematical Physics and Chemistry, Longman, London and New York, 1980.

48. Rainville, E. D., Special Functions, The MacMillan Company, New York, 1960.

49. Bell, W. W., Special Functions for Scientists and Engineers, D. Van Nostrand Company Ltd., London, 1965.

50. Andrews, L. C., Special Functions of Mathematics for Engineers, Oxford University Press, Oxford, 1998.

51. Boas, M. L., Mathematical Methods in the Physical Sciences, John Wiley & Sons, New York, 2006.

52. Frerichs, V., W. G. Kaenders, and D. Meschede, "Analytic construction of magnetic multipoles from cylindric permanent magnets," Appl. Phys. A, Vol. 55, 242, 1992.
doi:10.1007/BF00348392

53. Evans, G., Practical Numerical Integration, John Wiley & Sons, Chichester, 1993.

54. Kythe, P. K. and M. R. Schäferkotter, Handbook of Computational Methods for Integration, Chapman & Hall/CRC, London, 2005.


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