For numerous applications, it is desirable to develop electron beam driven efficient sources of electromagnetic radiation that are capable of producing the required power at beam voltages as low as possible. This trend is limited by space charge effects that cause the reduction of electron kinetic energy and can lead to electron reflection. So far, this effect was analyzed for intense beams propagating in uniform metallic pipes. In the present study, the limiting currents of intense electron beams are analyzed for the case of beam propagation in the tubes with gaps. A general treatment is illustrated by an example evaluating the limiting current in a high-power, tunable 1–10 MHz inductive output tube (IOT), which is currently under development for ionospheric modification. Results of the analytical theory are compared to results of numerical simulations. The results obtained allow one to estimate the interaction efficiency of IOTs.

## I. INTRODUCTION

The development of high current accelerators in the late 1960s and early 1970s initiated an active study of the limiting currents for charged particle beams transported through metallic pipes in vacuum and/or plasma (see, for example, review papers,^{1,2} textbooks^{3} and^{4} and references therein). Subsequently, a novel device operating at currents exceeding the limiting current was proposed. In this device, named the “*vircator*” (this term originates from a “virtual-cathode oscillator”), after the electrons pass a certain distance from the electron gun, they enter a region of a larger diameter where the current exceeds the limiting value, and hence, the electrons do not propagate any longer. Then, these reflected electrons start bouncing between the real and virtual cathodes, and these electrons excite oscillating electromagnetic fields within the region. The operation of various versions of the vircators is described in Refs. 5–7 (see also references therein).

The simplest case which allows for an analytical study of limitations in beam propagation due to space charge effects is the case when a beam of charged particles (electrons or ions) propagates through a uniform metallic tube being guided by an infinitely strong external focusing magnetic field. (This is the case analyzed by many authors, see, e.g., Refs. 1–4.) In real vacuum RF and microwave tubes, the geometry of metallic pipes is, however, more complicated. In particular, in many tubes, there are possible gaps in the pipes, which can serve as parts of the cavities in low frequency klystron like devices. The propagation of electron beams through the gaps in relativistic klystrons was considered in Refs. 8 and 9. Consideration of the effect of a gap on the limiting currents can also be important for accurate design of such high-power, low-frequency RF sources as inductive output tubes (IOTs^{10,11}) also known as klystrodes^{12} operating in the MHz frequency range.

In the present paper, we analyze the space charge effect of intense electron beams on the limiting current in pipes with gaps. The analytical theory describing this effect is developed. Its results are compared to results of numerical simulations performed by using the 2D electron gun code Michelle.^{13} This comparison is done for a high-power, 1–10 MHz IOT, which is currently under development for ionospheric modification.^{14} The paper is organized as follows. Section II contains formulation of the problem. In Section III, results of the analytical theory are presented. In Section IV, the beam propagation in a preliminary designed electron gun of the IOT for ionospheric modification is shown. This beam propagation was calculated by using the 2D electron gun code Michelle.^{13} Results of numerical simulations are compared to the results of the analytical theory. Section V contains a brief discussion of the results of the study and conclusions.

## II. LIMITING CURRENT OF AN ELECTRON BEAM

The final goal of this work is to determine the limiting current of an electron beam in a system schematically shown in Fig. 1. Here, a cylindrical electron beam propagates through a pipe being guided by a strong magnetic field. In this section, first, we reproduce for completeness known results, i.e., we consider an ideal annular electron beam propagating in a uniform pipe (no gap) and performing a 1D motion in an infinitely strong magnetic field. Then, we include into consideration the effect of electron deceleration in the gap.

### A. Propagation in uniform pipes

In the case of propagation in a uniform tunnel, the limiting current can be found from the 1D Laplace equation for the electrostatic potential $\Phi $ that applies between the beam and the wall

with the boundary conditions at the grounded wall $\Phi (Rw)=0$ and at the outer surface of a beam having the outer radius $Rb$

which follows from Gauss' law. Taking into account the boundary condition at the wall, the difference between the potential at the wall and the potential at the outer surface of the beam can be determined to be

It is important to note in Eqs. (2) and (3) the presence of the electron velocity in the denominator, which depends on the potential $\Delta \Phi $. The relation between this velocity (normalized to the speed of light), $\beta =v/c$, and the total electron energy normalized to the rest energy is

In (4), $V$ is the accelerating voltage applied between the cathode and the anode, and normally represents the kinetic energy an electron would have in the absence of a space charge potential.

Equations (2)–(4) represent a set of nonlinear equations. To analyze these, we introduce a normalized potential

and the relativistic factor

characterizing the electron energy in the absence of the space charge depression. The results of previous studies^{1–4,15} show that the quantity $\beta \Phi \u0302$ has a maximum value

that occurs when

Thus, using (3), the maximum beam current that can be propagated through the tube steadily is

where $IA=mc3/e=17.04\xd7103A$ is related to the Alfven current.

In the presence of finite guiding magnetic field and imperfect electron optics, electrons may exhibit some transverse oscillation in addition to their axial motion. Then, as shown in Ref. 16, the limiting current is determined by

where $\gamma \u22a52=1+p\u22a52/m2c2$ and $p\u22a5=mc\u2009\gamma \u2009\beta \u22a5$ is the transverse oscillation momentum. When the transverse electron momentum is negligibly small, Eq. (8) reduces to (7). Similar expressions for the maximum current can also be derived for solid beams^{1} and for beams of arbitrary cross-sections.

### B. Propagation through a decelerating gap

Consider now a drift tube with a decelerating gap shown in Fig. 1. The left side of the gap, $z=0$, is held at the full accelerating potential. The right side of the gap, $z=L$, is held at a lower potential $VRF$, where $VRF$ is the deceleration RF potential. This potential is applied by a circuit or a cavity at a larger radius, $r=Rw2\u226bRw$. There are now two effects that will cause the beam to slow down and possibly exceed the limiting current. First is the deceleration potential $VRF$ intended to extract the kinetic energy of the beam that will slow the beam and increase the local charge density. Second, the presence of the gap and the absence of conductor near the beam will enhance the space charge depression for a given density. We will assume in what follows that the transit time of electrons through the gap region is much shorter than the period of the RF, and thus the RF potential can be considered to be static. Even with this assumption, a rigorous treatment of this problem still requires solving for the self-consistent motion of the electrons in the presence of their space charge field. Below, we by making further approximations give an estimate as to the maximum current that can pass through the gap.

We begin an assessment of these effects by writing an expression for the electrostatic potential in the region of the gap, *0 < z < L*, $Rw<r<Rw2$

In (9), $\Phi n$ are the Fourier coefficients of the potential at the opening of the gap, $kn=n\pi /L$, and $K0(x)$ is the modified Bessel function of zero order which goes to zero at large values of the argument. Here, we have also assumed $Rw2\u226bRw$, so that only the RF field is present at $Rw2$.

The amplitudes $\Phi n$ will be determined by matching the potential and its derivatives at $r=Rw$ to a solution in the beam tunnel. We will do this approximately here by assuming that Eq. (2) can be applied at $r=Rw$ with $Rb$ replaced by $Rw$, and with the velocity $v$ treated as a constant. Taking the radial derivative of (9), we thus have

We can then extract the amplitudes $\Phi n$ by regarding the right-hand side to be a constant. We further assume that we can evaluate the Bessel functions in the large argument limit, in which case $K\u20320(x)/K0(x)=\u22121$. (This approximation underestimates the first term in (10) by only 14% when $L=Rw$. Accuracy is improved for higher order terms, or if $L<Rw$.) The result is $\Phi n=8IL/(\pi 2n2vRw)$ for n odd and zero for n even. Substituting back into Eq. (9), we find the potential at the wall

where

The function g(x) is related to “so called” polylogarithms.^{17} A plot of this function is shown in Fig. 2.

With the expression (11) for the potential at the wall radius $Rw$, we can follow through the steps leading to (7) and obtain a sufficient condition on the beam current

where

## III. RESULTS

Results presented in Fig. 3 illustrate the axial dependencies of the left- and right-hand sides of Eq. (13). Calculations were done for the beam voltage $V=70kV$, the beam-to-wall radii ratio $Rb/Rw=0.9$, and the gap width-to-wall radius ratio $L/Rw=1.0$. The left-hand side of (13) denoted as $f\gamma (z)$ is shown by black curves for several values of the RF voltage $VRF$. The right-hand side denoted as $fI(z)$ is shown by blue lines for several values of the beam current.

Let us discuss the results shown in Fig. 3 bearing in mind that the device efficiency approaches 100% when the RF voltage $VRF$ approaches the beam voltage at the injection to the gap $V$. Consider, as an example, the case when the beam current is equal to 12 A. As follows from results shown in Fig. 3, when the RF voltage is 67 kV or less, the RHS of Eq. (13) is larger than its LHS at all z's, so Eq. (13) holds and the beam can propagate through the gap. However, when $VRF$ approaches 68 kV, in the output cross-section, the RHS first becomes equal and then smaller than the LHS. The points of intersection of these two curves for a 12 A beam current are shown in Fig. 3 by small circles. So, when $VRF=68kV$, the beam stops at $z/L=0.997$, while at $VRF=69kV$, it occurs at $z/L=0.97$ and so on.

Resulting dependence of the maximum $VRF$, at which the beam propagates through the gap, on the beam current is shown in Fig. 4. Calculations were done for the same parameters as in Fig. 3. Note that at the beam current equal to 20 A, the maximum RF voltage is 67 kV. So, if we define the efficiency as the ratio of this voltage to the beam voltage, the maximum efficiency in this case exceeds 95%.

Another important dependence is the dependence of the maximum RF voltage on the gap width. This dependence is illustrated by Fig. 5, where it is shown for the beam current equal to 15 A and the same beam-to-wall radii ratio $Rb/Rw=0.9$.

Results shown in Fig. 5 also indicate that the maximum efficiency can exceed 95%.

Note that so far we consider a thin annular electron beam whose radius is close to the radius of a uniform pipe ($Rb/Rw=0.9$).

## IV. MICHELLE SIMULATIONS

In addition to the analytical studies, a few simulations of the electron optics and electron beam propagation through the gap region were performed by using the 2D electron gun code Michelle.^{13} Calculations were performed for parameters of the design of a tunable 1–10 MHz IOT for ionospheric modification described in Ref. 14. The peak on-axis solenoidal field was 2.41 kG. The number of relaxation cycles used in the simulation was 100, and the averaging factor was 0.05.

In Fig. 6, the electron beam propagation through the gap region is shown for three values of the voltage across the gap: (a) $VRF=0.0kV$, (b) $VRF=66.8kV$, and (c) $VRF=66.9kV$ for $Rb/Rw=0.42$ and $L/Rw=2$. As one can see, while in the case (a), all the beam goes through; in the case (c), all electrons stop at the gap and travel back towards the emitter.

As a matter of fact, however, the first reflected electrons appear much earlier, at lower RF voltages. In order to detect these reflected electrons, it makes sense to show the phase space of the electron beam as it is done in Fig. 7.

Results shown in Fig. 7 reveal the following sequence of events. In the absence of the decelerating voltage (Fig. 7(a)), the electron normalized momentum $p\u0302=p/mc=\gamma \beta $ increases in the accelerating region. Then, the beam propagates through the gap located in the region between 0.298 and 0.336 m. This figure shows the presence of some ripples that can be attributed to transverse oscillations in an electron beam and the spread in electron momenta. In the case of the decelerating RF voltage equal to 54 kV (Fig. 7(b)), electrons lose significant part of their energy in the gap, but still all electrons move through the gap. Finally, in the case of 55 kV (Fig. 7(c)), the first reflected electrons appear. Note that, when reflected electrons return to the emitter region, they are slightly displaced. Therefore, such electrons will hit not the cathode, but a focusing anode that may cause some deterioration of this tiny element of the gun.

The maximum RF voltage found in Michelle simulations presented in Fig. 7 is close to 55 kV that is much lower than the voltage shown in Figures 4 and 5, where the analytical results were presented. This discrepancy can easily be explained by the fact that simulations whose results are presented in Figures 4 and 5 were performed for a thin annular electron beam located close to the wall ($Rb/Rw=0.9$). As shown in Fig. 6, the outer radius of the beam in Michelle simulations is equal to 7 mm, i.e., it is 0.42 of the pipe radius ($Rb/Rw=0.42$). For this ratio of radii and a chosen value of the normalized gap width $L/Rw=2$, the analytical theory predicts the maximum value of the voltage across the gap equal to 59.8 kV that still is a little higher than the critical voltage of 55 kV which can be found in Fig. 7. This discrepancy on the order of a few kV can be explained by a finite radial width of an electron beam obtained in Michelle simulations and also by the presence of some transverse oscillations visible in Figures 6 and 7.

## V. SUMMARY

The theory describing the effect of space charge forces on the propagation of electron beams through a gap in a pipe has been developed. This theory allows one to estimate the efficiency of high-power IOTs. If, based on simulation results, we take 55 kV as the maximum value of the RF voltage for 70 kV beams, this will correspond to the maximum efficiency in the range of 78%–80%. However, these numbers are very sensitive to the geometry of cylindrical electron beams. Optimization of electrode configuration in the electron gun and beam propagation regions, which lies beyond the scope of our present study, can result in location of an electron beam much closer to the wall. Then, in line with our theoretical results, one may expect the IOT efficiency to be close to 90% and above.

## ACKNOWLEDGMENTS

This work was supported by the Air Force Office of Scientific Research under Grant No. FA95501410019.