Absence of saturation for finite injected currents in axially symmetric cavity diode.pdf

a r X i v : p h y s i c s / 0 4 0 8 0 1 4 v 1 [ p h y s i c s .p l a s m - p h ] 3 A u g 2 0 0 4 Absence of saturation for finite injected currents in axially symmetric cavity diode Debabrata Biswas, Raghwendra Kumar and R. R. Puri Theoretical Physics Division Bhabha Atomic Research Centre Mumbai 400 085, INDIA (February 2, 2008) The Child-Langmuir law is investigated numerically using a fully electromagnetic particle-in-cell code for a closed axially symmetric diode. It is found that the average current transmitted to the anode (JTR) increases with the injected current (JIN) even after the formation of virtual cathode in both the non-relativistic and relativistic cases. The increase is found to be a power law, JTR ～ J 1?β IN . In other words, the time averaged fraction f of electrons reaching the anode varies with the input current as, f ～ J?β IN where β 1. In contrast, for an infinite parallel plate diode, f ～ J?1 IN . The possibility of asymptotic saturation is also discussed. I. INTRODUCTION The Child-Langmuir law is the cornerstone of electron flow in diodes. It gives the maximum current density, JCL(1), that can be transported from an infinite planar cold cathode (velocity of emission is zero) at zero poten- tial to an infinite planar anode parallel to the cathode at a distance D and potential V . In the non-relativistic case, this is given by [1,2] JCL(1) = V 3/2 D2 1 9π ( 2e m )1/2 . (1) Here, e is the magnitude of the charge and m0 the rest mass of an electron. The Child-Langmuir law is a result of the space-charge effect. As electrons are emitted from the cathode, they gain velocity from the imposed field. At the same time, they experience a repulsive force due to the presence of other electrons. However, all the injected current is transmitted to the anode as long as the injected current, JIN, is less than a particular value, called the critical cur- rent JCR. When JIN exceeds JCR, the net force is such that some of the electrons, instead of movi