Analysis of temperature-limited flow, space-chargelimited flow, and the transition between them using a simple planar diode with a thermionic cathode, in which the cathode surface has spatially nonuniform emission properties, is presented. Our theoretical results, which are derived from a model based on solutions to the Vlasov and Poisson equations, compare well with the results of particle-in-cell simulations. We find that the location and the shape of the knee in the anode current versus temperature characteristic (Miram or "rollover" curve) are significantly affected by non-uniformities in the space-charge density in the A-K gap, but are relatively unaffected by the electron motion parallel to the electrode surfaces. In particular, emission from an actively emitting region is strongly affected by the forces (or lack thereof) exerted by the space-charge of the electrons emitted by their neighbors. Perhaps, most remarkably, we find that the limiting current reaching the anode is approximately given by the classical 1-D Child-Langmuir law, even if a significant fraction of the cathode surface is nonemitting.

A model of a thermionic cathode in a planar diode in which the Poisson and Vlasov equations are solved in 3-D assuming an infinite magnetic field is presented. We explore how 2-D work function variations across the cathode surface may affect the transition between temperature-limited and spacecharge-limited flow, commonly known as the "knee" of the Miram curve. We study a variety of work function distributions, both realistic and idealized, and demonstrate how emission from the lowest work function regions dominates the total anode current even when such regions make up a relatively small fraction of cathode area. Our model also illustrates the ability of cathodes to reach the full Child-Langmuir current despite the presence of a sizeable nonemitting region. We find that as the length scale of these work function variations decreases, the Miram knee grows sharper, indicating improved cathode performance.

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A self-consistent model of steady-state electron flow in a planar crossed-field diode with a thermionic cathode is presented. Our formulation is a generalization of the classic work of Fry and Langmuir, to include a constant magnetic field B of arbitrary strength parallel to the electrode surfaces. Some effects of the magnetic field on the electron flow are illustrated in an example. Index Terms-Crossed-field diode, Hull cutoff, space charge limited, temperature limited, thermionic emission. I. INTRODUCTION T HERMIONIC cathodes are widely used as electron sources. In a diode, as the cathode temperature is raised, the anode current transitions from the temperature-limited value, which is described by the Richardson-Dushman law [1], [2] to the space charge-limited value, which is described by the Child-Langmuir law [3], [4]. The 1-D theory of this transition is given in the classic articles by Langmuir [4] and Fry [5], who found that, above a certain transition temperature, the anode current is controlled by the appearance of a potential minimum in front of the cathode. This potential minimum is also known as a virtual cathode [6]-[8]. We recently extended the 1-D theory of Fry and Langmuir to 2-D in order to study various effects of nonuniform emission [9]. This work was motivated by our attempt to understand the physical basis for the shape of the Miram curve [10], which is a plot of the anode current as a function of the cathode temperature at a fixed voltage. The Miram curve usually exhibits a smooth "knee" which marks the transition from the temperature-limited regime to the space charge-limited regime. Operation in the knee region is often chosen for stability and long cathode life. Much less developed is the thermionic theory of a crossed-field diode, in which an external magnetic field B is imposed parallel to the cathode surface. The anode current Manuscript