### Abstract

This article presents a new formulation of the solution for fully nonlinear and unsteady planar flow of an electron beam in a diode. Using characteristic variables (i.e., variables that follow particle paths) the solution is expressed through an exact analytic, but implicit, formula for any choice of incoming velocity v
_{0}, electric field E
_{0}, and current J
_{0}. For steady solutions, this approach clarifies the origin of the maximal current J
_{max}, derived by Child and Langmuir for v
_{0}=0 and by Jaffe for v
_{0}>0. The implicit formulation is used to find (1) unsteady solutions having constant incoming flux J
_{0}>J
_{max}, which leads to formation of a virtual cathode, and (2) time-periodic solutions whose average flux exceeds the adiabatic average of J
_{max}.

Original language | English (US) |
---|---|

Article number | 056408 |

Journal | Physical Review E - Statistical, Nonlinear, and Soft Matter Physics |

Volume | 85 |

Issue number | 5 |

DOIs | |

State | Published - May 18 2012 |

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### ASJC Scopus subject areas

- Condensed Matter Physics
- Statistical and Nonlinear Physics
- Statistics and Probability

### Cite this

*Physical Review E - Statistical, Nonlinear, and Soft Matter Physics*,

*85*(5), [056408]. https://doi.org/10.1103/PhysRevE.85.056408

**Beyond the child-langmuir limit.** / Caflisch, Russel; Rosin, M. S.

Research output: Contribution to journal › Article

*Physical Review E - Statistical, Nonlinear, and Soft Matter Physics*, vol. 85, no. 5, 056408. https://doi.org/10.1103/PhysRevE.85.056408

}

TY - JOUR

T1 - Beyond the child-langmuir limit

AU - Caflisch, Russel

AU - Rosin, M. S.

PY - 2012/5/18

Y1 - 2012/5/18

N2 - This article presents a new formulation of the solution for fully nonlinear and unsteady planar flow of an electron beam in a diode. Using characteristic variables (i.e., variables that follow particle paths) the solution is expressed through an exact analytic, but implicit, formula for any choice of incoming velocity v 0, electric field E 0, and current J 0. For steady solutions, this approach clarifies the origin of the maximal current J max, derived by Child and Langmuir for v 0=0 and by Jaffe for v 0>0. The implicit formulation is used to find (1) unsteady solutions having constant incoming flux J 0>J max, which leads to formation of a virtual cathode, and (2) time-periodic solutions whose average flux exceeds the adiabatic average of J max.

AB - This article presents a new formulation of the solution for fully nonlinear and unsteady planar flow of an electron beam in a diode. Using characteristic variables (i.e., variables that follow particle paths) the solution is expressed through an exact analytic, but implicit, formula for any choice of incoming velocity v 0, electric field E 0, and current J 0. For steady solutions, this approach clarifies the origin of the maximal current J max, derived by Child and Langmuir for v 0=0 and by Jaffe for v 0>0. The implicit formulation is used to find (1) unsteady solutions having constant incoming flux J 0>J max, which leads to formation of a virtual cathode, and (2) time-periodic solutions whose average flux exceeds the adiabatic average of J max.

UR - http://www.scopus.com/inward/record.url?scp=84861935378&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84861935378&partnerID=8YFLogxK

U2 - 10.1103/PhysRevE.85.056408

DO - 10.1103/PhysRevE.85.056408

M3 - Article

VL - 85

JO - Physical review. E

JF - Physical review. E

SN - 1539-3755

IS - 5

M1 - 056408

ER -