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Electron Beam Focusing by Means of a Periodic Magnetic Field

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Mathematical Models with Singularities

Part of the book series: Atlantis Briefs in Differential Equations ((ABDE,volume 1))

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Abstract

In this chapter, we consider the dynamics of an electron beam guided by an axially-symmetric periodic magnetic field.

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Notes

  1. 1.

    Busch’s Theorem can be seen as the equivalent of conservation of angular momentum. It states that the angular velocity of the electron depends only on the difference of magnetic fluxes at the initial point (at the cathode) and the point under consideration.

  2. 2.

    It seems that the mathematical community began using this expression after the 1965 seminal paper by Ding [7] (he speaks of a periodic Brillouin focusing system). As far as I know, Leon Brillouin never proposed or studied this equation. In this sense, the nomenclature can be a bit misleading from a historical point of view, but certainly has a physical meaning.

  3. 3.

    The classical way to compute numerically the eigencurves of the stability diagram of the Mathieu equation is to expand the solution as a Fourier series. Then the coefficients must verify an infinite determinant known as Hill’s determinant. Truncating this determinant, one obtains the stability intervals with arbitrary accuracy; see for instance [2].

  4. 4.

    See Definition A.3.

  5. 5.

    See Figs. 14 and 15 in [16] and the paragraph after Eq. (15) in [15].

  6. 6.

    If desingularization is not possible, one should use a numerical method with an adaptative step.

  7. 7.

    The equation was numerically integrated with the command NDsolve of Mathematica\(^{TM}\), then the orbit was drawn with the command ParametricPlot. To be sure that it is a true \(2\pi \)-periodic solution, the picture shows the curve drawn in the interval \([0,20\pi ]\) (ten periods).

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Authors and Affiliations

  1. Department of Applied Mathematics, University of Granada, Granada, Spain

    Pedro J. Torres

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  1. Pedro J. Torres
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Correspondence to Pedro J. Torres .

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Torres, P.J. (2015). Electron Beam Focusing by Means of a Periodic Magnetic Field. In: Mathematical Models with Singularities. Atlantis Briefs in Differential Equations, vol 1. Atlantis Press, Paris. https://doi.org/10.2991/978-94-6239-106-2_6

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