The single particle dynamics and the Liouville operator determining the beamĭistribution evolution are in this case provided, respectively, by Space, namely a region in which no external forces are present. Involving the Liouville operator, we consider the case of a beam undergoing a drift Just to give an idea of the methods that can be employed in the solution of PDEs In very rare cases these methodsĪre analytical in nature and in most cases efficient numerical algorithms can beĮxploited, as discussed in detail in this and forthcoming chapters. PDEs, which can be treated with a variety of methods. The mathematical problem, represented by ( 1.113), belongs to the family of evolution A distribution of velocity and positionįor a problem ruled by a Hamiltonian in which canonical and kinetic momenta do notĬoincide is not representative of the phase-space density function and is not ruled by Variable, are canonical variables and the arguments of the density distributionįunction are specified by canonical variables. It is accordingly important to stress that since we are referring to a phase-space Non-interacting ensemble of particles ruled by the Hamiltonian (the interacting case will be discussed later in this chapter). Where is the Liouville operator, which rules the transport of a 1.4. Charged particle motion in combined electric and magnetic fields (the Here we simply want to see how what we had learned so far can be assembled to describeĪn actual device. Introduced concepts, which will be more carefully discussed in forthcoming chapters. The discussion provided in this section is very inaccurate and contains sloppily See in the following that using multi-cavity configurations and bunching optimizationĬriteria, space charge forces can be exploited to obtain greater efficiency). Purely ballistic bunching and the efficiency may not grow that large (however, we will In the reality, space charge effects counteract the Regime, namely the space charge forces are ignored and the faster particles We have described so far a klystron operating in the so-called ballistic The decelerated bunch loses all the powerĪssociated with the spatial bunching, which as we saw is linked to its bunching In the second cavity they are bunched on a spatial extension which is a In the first cavity the incoming electrons areĬontinuously distributed, they are therefore accelerated and decelerated in almost equal The high level of efficiency and thus large amount of energy delivered to the cavityĬan be understood as follows. Harmonic is determined by the first order bunching coefficient, linked to theĬylindrical Bessel function of order 1, exhibiting the first maximum at where, we can accordingly state that the efficiency of a two cavity klystron The efficiency of the power delivered by the electrons to the cavity on the first In an emission and transport process of the type we have described, the electronsĮxtracted from the cathode move in the forward direction, with velocity determined byĮnergy conservation (we limit ourselves to the non-relativistic case) Loosely summarized here, the thermionic emission and the associated laws will beĮxtensively discussed in this and the forthcoming chapters. Velocity at the metal surface, are accelerated toward the anode. Potential well and the electrons, expelled via thermionic emission with almost zero Thermal agitation can however provide sufficient energy to overcome the ('seen' by the electrons outside the conductor), which act as confining forces (forįurther comments see section 1.15). Inside the metal the electrons do not experience any electric field and theįorces pulling the electrons inside the conductor are those due to the image charges Sketched the geometry of the emission process and the transition from the cathode to theĪnode. Stated to have been released into the public domain. This image has been obtained by the author(s) from the Wikimedia Figure 1.2. Sketch of the geometry of the emission process and the transition from the cathode
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