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Theory of regular electrostatic beams of charged particles

Abstract

In some papers concerned with the exact solution of the equations of a nonrelativistic single-energy beam of charged particles (e.g., [1, 2]), the opinion has been expressed that, while the method of separation of the variables has possibilities, serious difficulties can arise in obtaining the actual systems with separated variables. In particular, it has become popular, when investigating regular electrostatic flows, to transform to a coordinate system connected with the trajectory. In this system the velocity vector only has one component, say v={ix t, 0}, so that flow only occurs in the x1 direction (x1 flow). We shall also refer to a single-component flow, as in [3], This method is thought (e.g., [4]) to be effective for a wide class of flows. The question of the coordinate systems that allow flows in the1 direction is more specialized than the general problem of separation of variables.

The concept of an x1 flow is discussed in §1 from the point of view of its utility for obtaining solutions of the regular electrostatic beam equations. Transformation to a coordinate system connected with the trajectory is found to be only justifiable for four orthogonal systems: cartesian, cylindrical, helical-cylindrical, and spherical. It is shown that, in the case of two-dimensional systems on a plane with conformal metric, the condition required for an x1 flow and the conditions for the space to be euclidean can be used effectively to establish the existence in the given class of coordinate systems of an x1 flow starting from a fictitious emitter (§2). The usual tensor notation is employed.

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References

  1. 1.

    P. T. Kirstein and G. S. Kino, “Solution to the equations of space-charge flow by the method of the separation of variables,” J. Appl. Phys., vol. 29, no. 12, 1958.

  2. 2.

    G. S. Kino and K. J. Harker, “Space-charge theory for ion beams,” Electrostatic Propulsion Academic Press, New York-London, 1964.

  3. 3.

    B. Meltzer, “Single-component stationary electron flow under space-charge conditions,” J. Electr., vol. 2, no. 2, 1956.

  4. 4.

    B. Meltzer, “Dense electron beams,” Brit. J. Appl. Phys., vol. 10, no. 9, 1959.

  5. 5.

    D. Gabor, “Dynamics of electron beams,” Proc. IRE, vol. 33, no. 11, 1945.

  6. 6.

    K. Spangenberg, “Use of the action function to obtain the general differential equations of space charge flow in more than one dimension,” J. Franklin Inst., vol. 232, no. 4, 1941.

  7. 7.

    A. R. Lucas, B. Meltzer, and G. A. Stuart, “A general theorem for dense electron beams,” J. Electr. Contr., vol. 4, no. 2, 1958.

  8. 8.

    B. Meltzer and A. R. Lucas, “Sufficient and necessary trajectory conditions for dense electron beams,”J. Electr. Contr., vol. 4, no. 5, 1958.

  9. 9.

    P. T. Kirstein, “Comments on ‘A general theorem for dense electron beams’ by A. R. Lucas, B. Meltzer, and G. A. Stuart,” J. Electr. Contr., vol. 4, no. 5, 1958.

  10. 10.

    W. M. Mueller, “Necessary and sufficient trajectory conditions for dense electron beams,” J. Electr. Contr., vol. 5, no. 6, 1959.

  11. 11.

    W. M. Mueller, “Comments on ‘Necessary and sufficient trajectory conditions for dense electron beams,’” J. Electr. Contr., vol. 8, no. 2, 1960.

  12. 12.

    J. Rosenblatt, “Three-dimensional space-charge flow, ” J. Appl. Phys., vol. 31, no. 8, 1960.

  13. 13.

    B. Meltzer, “Electron flow in curved paths under space-charge conditions,” Proc. Phys. Soc., B, vol. 62, no. 355, 1949.

  14. 14.

    P. T. Kirstein, “The complex formulation of the equations of two-dimensional space-charge flow,” J. Electr. Contr., vol. 4, no. 5, 1958.

  15. 15.

    V. A. Syrovoi, “Group-invariant solutions of the equations of a plane stationary beam of charged particles,” PMTF, no. 4, 1962.

  16. 16.

    V. T. Ovcharov, “On the potential motion of charged particles,” Radiotekhn. i elektr., vol. 4, no. 10, 1959.

  17. 17.

    V. A. Syrovoi, “On single-component beams of particles of like charge,” PMTF, no. 3, 1964.

  18. 18.

    V. A. Syrovoi, “Group-invariant solutions of the equations of a three-dimensional stationary beam of charged particles,” PMTF, no. 3, 1963.

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Syrovoi, V.A. Theory of regular electrostatic beams of charged particles. J Appl Mech Tech Phys 7, 1–3 (1966). https://doi.org/10.1007/BF00912822

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Keywords

  • Coordinate System
  • Mechanical Engineer
  • Exact Solution
  • Charged Particle
  • Velocity Vector