Abstract
In this article, we have set ourselves the objective of summarizing the results of the principal published works concerning the influence of space charge on the propagation of intense beams of charged particles, and at the same time providing a summary of formulas and information that may be useful in work with intense beams. The following will be considered: the spreading of beams under the action of their own space charge, some methods for forming intense beams, current limitation in beams, neutralization of space charge in beams by the action of particles of the opposite sign, and related issues.
Full Text
Influence of Space Charge in the Propagation of Intense Beams of Charged Particles
M. D. Gabovich
In recent years, technology employing intense electron and ion beams has acquired ever greater importance, and in electron optics it has become increasingly essential to take into account the action of space charge.
It should be noted that, alongside the universally recognized major contribution of Soviet scientists to the creation of the theoretical foundations of general electron optics, in the development of the particular branch considered here a substantial and fundamental role was played by the works of Boguslavsky and Bursian¹, ², ³, ⁴ on the study of the influence of space charge on the motion of streams of charged particles; by Lukoshkov⁵ on the study of the propagation of intense beams; and by a number of Soviet scientists⁶–¹¹, who were the first and most rigorous to develop the questions of space-charge neutralization in beams, gaseous self-concentration of beams, etc.
Despite the fact that by the present time a considerable number of works devoted to intense beams has already accumulated in the literature, there are almost no survey works on this question¹², ¹³.
Therefore, in the present article we have set as our aim to present the results of the principal published works concerning the influence of space charge on the propagation of intense beams of charged particles, and at the same time to give a compendium of formulas and information that may be useful in working with intense beams.
Here the following will be considered: the spreading of beams under the action of their own space charge, certain methods of forming intense beams, limitation of currents in beams, neutralization of space charge in beams by the action of particles of the opposite sign, and questions connected with all this.
M. D. GABOVICH
1. SPREADING OF BEAMS OF CHARGED PARTICLES UNDER THE ACTION OF THEIR OWN SPACE CHARGE
When a beam of charged particles propagates, the space charge of these particles leads to a change in the field or, if the space before the introduction of the beam was equipotential, to the appearance of a certain electric field. Finding the shape of the beam under specified boundary conditions encounters the well-known difficulties associated with solving Poisson’s equation (in fact, one must solve a system—the Poisson equation, the continuity equation, the equation of motion), and even in solving the simplest problems of this type one has to make substantial assumptions.
The first question of interest to us will be the solution of a number of the simplest problems in which the spreading of beams is considered as a consequence of the action of the field of space charges or, in other words, as a consequence of the electrostatic repulsion of like-charged particles of the beam.
Fig. 1.
A beam of charged particles, possessing charge \(e\) and mass \(M\), emerging from the plane \(z=0\) and having a circular cross section, propagates in the direction of the \(z\)-axis (Fig. 1) in an equipotential space (not counting the field formed by the charges of the beam). In the more general case, the particles of the beam, in addition to the velocity \(v_z\), possess an initial radial component of velocity \(v_r(r)\), which they may have acquired, for example, after passing through a certain lens (the latter will henceforth be assumed “thin”). It is natural to assume that particles passing through the lens at different distances from the axis will possess initial radial velocities satisfying the relation
\[ v_r(r)_i=-v_z\frac{r}{f}, \tag{1} \]
i.e., that the beam is homocentric; in this expression \(r\) is the distance of the particles from the axis in the plane \(z=0\), and \(f\) is the focal length of the lens.
The Poisson equation for an axially symmetric system is:
\[ \frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial U}{\partial r}\right)+\frac{\partial^2 U}{\partial z^2}=-4\pi\rho . \tag{2} \]
If the length of the beam considerably exceeds its diameter and pre-
space in the absence of the beam is equipotential, one may neglect the second term of the left-hand side and write the equation in the form
\[ \frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial U}{\partial r}\right)=-4\pi\rho . \tag{3} \]
The first integration of (3), under the condition that \(\rho\) is independent of \(r\) (constancy of the current density and of the velocity \(v_z\) over the cross section of the beam, which, strictly speaking, does not hold), gives the expression for the radial field at the surface of the beam:
\[ E_r=2\pi\rho r=\frac{2I}{rv_z}, \tag{4} \]
where \(I\) is the total current of the beam, and \(r\) is the radius of the beam at the given point. The electric force acting on a peripheral particle of the beam and causing the latter to diverge is equal to
\[ F_e=eE=\frac{2Ie}{rv_z}. \tag{5} \]
Determining the shape of the diverging beam reduces to integrating the equation of radial motion of a peripheral particle
\[ M\ddot r=F_e=\frac{2Ie}{rv_z} \tag{6} \]
under the initial conditions \(t=0,\ r=r_0,\ \dot r=-v_r\), where \(v_r\) is the initial radial velocity of the particle, emitted in the plane \(z=0\) at a distance \(r_0\) from the axis, and \(r_0\) is the initial radius of the beam; in this case the relation holds
\[ v_r=v_z\frac{r_0}{f}. \tag{7} \]
The first integration of (6) leads to the expression
\[ \dot r^{\,2}=\frac{4Ie}{v_zM}\ln\frac{r}{r_0}+v_r^2 . \tag{8} \]
Using the condition \(r=r_{\min}\) when \(\dot r=0\), we obtain the expression for the minimum radius of the beam
\[ r_{\min}=r_0\exp\left(-\frac{Mv_zv_r^2}{4Ie}\right). \tag{9} \]
At \(r=r_{\min}\), the radial motion of the particle toward the axis ceases, and its motion away from the axis begins under the action of electrostatic repulsion forces.
The second integration gives the form of the converging part of the beam
\[ f\left(\frac{r_0}{r_{\min}}\right)-f\left(\frac{r}{r_{\min}}\right) = \sqrt{\frac{Ie}{M}}\, \frac{z}{v_z^{3/2}r_{\min}} = K\frac{r_0}{r_{\min}} . \tag{10} \]
where \(z=tv_z\), \(f(x)\) is a known tabulated function,
\[ f(x)=\frac{1}{2}\int_{1}^{x}\frac{dy}{\sqrt{\ln y}} \]
(see Table 1), and \(K\) is a dimensionless parameter.
Table 1
| \(x\) | \(f(x)\) | \(x\) | \(f(x)\) |
|---|---|---|---|
| 1.01 | 0.1 | 1.9 | 1.01 |
| 1.04 | 0.203 | 2.25 | 1.22 |
| 1.09 | 0.309 | 2.72 | 1.46 |
| 1.17 | 0.422 | 4.22 | 2.14 |
| 1.28 | 0.545 | 7.10 | 3.24 |
| 1.43 | 0.681 | 12.9 | 5.17 |
| 1.63 | 0.833 | 22.5 | 8.86 |
| 54.6 | 16.4 |
The problem set forth above was considered by Lukoshkov\(^5\), and its solution serves as the basis for solving other, more complicated problems.
In the special case the initial radial component of the velocity may be absent \((\dot r=0\) at \(z=0)\), and the shape of the expanding beam is described by the equation
\[ f\left(\frac{r}{r_0}\right) = \sqrt{\frac{\bar I e}{M}}\, \frac{z}{v_z^{3/2}r_0} = 2^{-3/4}I^{1/2}\left(\frac{M}{e}\right)^{1/4}U_0^{-3/4}r_0^{-1}z = K. \tag{11} \]
This follows directly from (10) upon replacing \(r_{\min}=r_0\) and changing the sign.
On passing to practical units, bearing in mind that
\[ v_z=\sqrt{2\frac{e}{M}U_0}, \]
where \(U_0\) is the particle energy in volts, we obtain:
\[ f\left(\frac{r}{r_0}\right) = 88\left(\frac{M}{m}\right)^{1/4} I^{1/2}\,(\text{ampere})\cdot U_0^{-3/4}\,(\text{volt})\cdot r_0^{-1}z, \tag{12} \]
where \(\dfrac{M}{m}\) is the ratio of the particle mass to the electron mass.
Above, no account was taken of the circumstance that, in addition to the force \(F_e\), the particle is acted upon also by the magnetic force \(F_H\), due to the existence of the magnetic field of the beam current and leading to the contraction of the latter toward the axis. The intensity of the magnetic field around a cylindrical beam is
\[ H=\frac{2I}{rc}. \tag{13} \]
and, consequently, the magnitude of the magnetic force is equal to
\[ F_H=\frac{2 I e v_z}{r c^2}. \tag{14} \]
Thus, the resultant force acting on the particle is
\[ F=F_e-F_H=\frac{2 I e}{r v_z}\left[1-\left(\frac{v_z}{c}\right)^2\right]. \tag{15} \]
As can be seen, allowance for this correction is advisable only at sufficiently high particle velocities. For example, for electrons with an energy of 10,000 volts, the attractive force arising from the action of the magnetic field amounts to only 4% of the force of mutual repulsion of the particles.
Returning again to the case when the particles of the spreading beam have no initial radial velocity, on the basis of (11) one can write an expression determining the shape of the beam for \(z>z_{\min}\) (Fig. 1):
\[ f\left(\frac{r}{r_{\min}}\right) = \sqrt{\frac{I e}{M}\, \frac{(z-z_{\min})}{v_z^{3/2} r_{\min}}}. \tag{16} \]
From (10) it follows (substitution: \(r=r_{\min}\), \(f(1)=0\))
\[ z_{\min}=f\left(\frac{r_0}{r_{\min}}\right) \sqrt{\frac{M}{I e}}\, v_z^{3/2} r_{\min}. \tag{17} \]
Finally, instead of (16), one obtains the expression
\[ f\left(\frac{r}{r_{\min}}\right) + f\left(\frac{r_0}{r_{\min}}\right) = K\frac{r_0}{r_{\min}}. \tag{18} \]
This last equation makes it possible to solve the following problem. A beam, having initial radius \(r_0\), emerges from the plane \(z=0\), where a certain “thin” lens is located. It is necessary to determine for what focal length of the lens, or, in other words, for what initial radial velocity \(v_r\), the cross section of the beam on a screen located in the plane \(z\) will be the smallest, and what the magnitude of this smallest beam cross section is. The existence of an optimal value of \(v_r\), at which the beam on the screen has the smallest cross section, was shown\({}^{14}\) by calculation carried out for some particular cases; for larger or smaller values of the radial component, the beam cross section on the screen increases (curves 1 and 3 in Fig. 2).
Fig. 2.
It proved possible\(^{15}\) to solve this problem in the general form as well. The beam is characterized by the assigned values of the dimensionless parameter \(K\) (see above) and of the initial radius \(r_0\). Taking into account equation (18) and using the extremum condition, which may be represented in the form
\[ \frac{dr}{dr_{\min}}=0, \]
it was possible to determine the dependence of the quantities \(\dfrac{r_{\min}}{r_0}\), \(\dfrac{z_{\min}}{z_0}\), \(\dfrac{r_k}{r_0}\) on the parameter \(K\), where \(r_k\) denotes the radius of the smallest cross section of the beam in the plane of the screen for a given value of \(K\), and \(r_{\min}\) and \(z_{\min}\) are the radius and the location of the smallest cross section of the beam. All these dependences are presented in Fig. 3. Having determined the value of \(r_{\min}\) and
Fig. 3.
using (9), one can determine the optimal value of \(v_r\) and the quantity
\[ \frac{f}{z}= \frac{1}{2K\sqrt{\ln \dfrac{r_0}{r_{\min}}}}, \tag{19} \]
where \(f\) is the optimal focal length of the lens.
Previously, the spreading of homocentric beams was considered. In reality, however, even with a small space charge, there are greater or smaller deviations from homocentricity; and instead of a point focus, the intersection of the rays forms a certain minimum cross section, located at a distance \(L\) from the lens and having radius \(a\). With a substantial space charge, spreading leads to an increase of the minimum cross section and to a change in its position. Analysis of this case\(^{16}\) led to the conclusion that, for a sufficiently intense space charge, the radius of this increased minimum cross section is determined by the course of the so-called
PROPAGATION OF BEAMS OF CHARGED PARTICLES
... of the so-called “outer ray,” i.e., the trajectory of the peripheral electron, and can be calculated from the formula
\[ r_{\min}=r_0 \exp\left[-3.3\cdot 10^{-5}\frac{r_0^2}{L^2}\,U^{3/2} I^{-1}\left(\frac{a}{r_0}-1\right)^2\right]. \tag{20} \]
If here one sets \(a\to 0\), \(L=f\) (a homocentric beam), and takes (1) into account, then the expression for \(r_{\min}\), as was to be expected, coincides with (9).
For a comparatively small space charge, the minimum cross section can be found from the intersection of the “outer” ray with the “inner” one, going from the edge of the beam in the plane of the lens exit aperture to the opposite side of the minimum cross section. For this case Wendt \(^{16}\) gives graphs that make it possible to carry out such calculations.
Determination of the shape of a spreading beam becomes more complicated if the latter propagates in a non-equipotential space. The problem was solved by Moss \(^{17}\) for the case in which along the axis of propagation of the beam there is a constant potential gradient \(E\). The equation of motion for this case has the form
\[ M\ddot r=\frac{2Ie}{r\left(\frac{eE}{M}t+v_{z_0}\right)}. \tag{21} \]
The solution of this equation in the case when the peripheral particles possess an initial radial component of velocity \(v_r\) leads to the expression
\[ \lambda-\lambda_0= \int_{\beta_0}^{\beta} \frac{d\beta}{ \sqrt{\frac{\beta^2-\beta_0^2}{4}+\frac{2B}{A^3}\ln\frac{\beta}{\beta_0}+\mu_0^2} }, \tag{22} \]
where \(\mu_0=-\dfrac{v_r}{A}\sqrt{v_{z_0}}-\dfrac{\beta_0}{2}\), \(\lambda=\ln(At+v_{z_0})\), \(r=\beta e^{\lambda/2}\), \(\lambda_0=\ln v_{z_0}\),
\[ A=\frac{eE}{M}, \qquad B=\frac{2Ie}{M}, \qquad \beta_0=\frac{r_0}{\sqrt{v_{z_0}}}, \]
and \(v_{z_0}\) is the component of the initial velocity directed along the \(z\) axis. In solving specific problems, from the known values \(v_{z_0}\), \(v_r\), \(r_0\), \(\dfrac{e}{M}\), \(E\), \(I\), the quantities \(\lambda_0\), \(\beta_0\), \(\mu_0\), \(B\), \(A\) are determined, and with the help of (22) \(\lambda=\Phi(\beta)\) is determined. After calculating \(r=\varphi(\lambda)=f(t)\), with the aid of the relation
\[ z=\frac{At^2}{2}+v_{z_0}t \]
one obtains \(r=F(z)\), which gives the shape of the beam.
If the particles have no initial radial velocity \((v_r = 0)\), expression (22) is simplified:
\[ \lambda - \lambda_0 = \int_{\beta_0}^{\beta} \frac{d\beta}{\sqrt{\dfrac{\beta^2}{4} + \dfrac{2B}{A^2}\ln \dfrac{\beta}{\beta_0}}}. \tag{23} \]
In the work of Guille and Gratiot1, the spreading of a beam of circular cross section, propagating between two plane electrodes to which a certain potential difference is applied, was considered by means of the method of electric images.
Fig. 4.
In many cases beams of rectangular cross section2 (ribbon beams) are used, for which the width \(\omega\) is considerably greater than the thickness \(2y\) (Fig. 4). The starting point in this case is Poisson’s equation:
\[ \frac{\partial^2 U}{\partial z^2} + \frac{\partial^2 U}{\partial y^2} = -4\pi \rho . \tag{24} \]
Assuming, on the same grounds as before, that
\[ \frac{\partial^2 U}{\partial z^2} \ll \frac{\partial^2 U}{\partial y^2}, \]
and that \(\rho\) does not depend on \(y\), one can obtain from (24) the field strength at the surface of the beam:
\[ E_y = \frac{2\pi I}{v_z \omega}. \tag{25} \]
The magnitude of the electric force stretching the particles is
\[ F_e = \frac{2\pi I e}{v_z \omega}. \tag{26} \]
Here \(I\) is the total beam current, and \(I/\omega\) is the current calculated per unit width of the beam. Neglecting the magnitude of the magnetic force, which in this case is equal to
\[ F_H = \frac{2\pi I e v_z}{\omega c^2}, \tag{27} \]
we obtain the equation of motion of the particle in the form
\[ \ddot y = \frac{2\pi I e}{v_z \omega M}. \tag{28} \]
Integrating this equation under the initial conditions: \(t=0\), \(y=y_0\) (half-thickness), \(\dot y=0\), leads to the expression
\[ \frac{y-y_0}{y_0}=\frac{\pi I e z^2}{y_0 v_z^3 M\omega}, \tag{29} \]
or, on passing to practical units,
\[ \frac{y-y_0}{y_0}=2.4\cdot 10^4\left(\frac{M}{m}\right)^{1/2} I\;(\text{amperes})\; U_0^{-3/2}\;(\text{volts})\;\frac{z^2}{\omega y_0}. \tag{30} \]
If the ribbon beam under consideration has passed through a lens of focal length \(f\), then the peripheral particle, on leaving the lens, acquires a velocity along the axis equal to \(v_y=-\dfrac{y_0}{f}v_z\). Integration of (28) with the new initial conditions
\[ t=0,\quad y=y_0,\quad \dot y=-\frac{y_0}{f}v_z \]
leads to the expression
\[ y=y_0-\frac{y_0}{f}z+\frac{\pi I e z^2}{v_z^3 dM}. \tag{31} \]
Putting \(y=0\), one can determine the conditions under which focusing of the beam to the \(z\)-axis is possible:
\[ z=f'=\frac{y_0}{2\delta f}-\frac{1}{2\delta}\sqrt{\frac{y_0^2}{f^2}-4y_0\delta}, \tag{32} \]
where \(f'\) is the focal distance taking space charge into account, and
\[ \delta=\frac{\pi I e}{v_z^2 M d}. \]
Focusing of the beam to the axis is possible only in the case when
\[ 4\delta<\frac{y_0}{f^2}. \tag{33} \]
The maximum focal distance is obtained by substituting \(\delta=\dfrac{y_0}{4f^2}\) into equation (32): \(f'_{\max}=2f\) (see Fig. 8, б). For a larger value of \(\delta\), the beam is not focused to the axis at all.
The spreading of beams was considered by the hydrodynamic method\(^{24}\).
Attention should be drawn to the fundamental difference in the behavior of beams of circular and rectangular cross section. As was shown, a ribbon beam can have a line of intersection even in the presence of a substantial space charge. A beam of circular cross section cannot have a point of intersection unless only the charge density (formed by moving charged particles) is reduced to such a degree that its discreteness becomes substantial. This difference is connected with the fact that, for
for a beam of circular cross section the repelling force \(F_e \to \infty\) as \(r \to 0\), whereas for a ribbon beam the force remains finite as \(y \to 0\), since the magnitude of the force (26) does not depend on the thickness of the beam.
It is also of interest to compare the degree of spreading of these beams. The parameter \(K\) introduced by us earlier is a function of the initial current density \(j_0\):
\[ K=\pi^{1/2}\left(\frac{e}{M}\right)^{1/2}\frac{j_0 z}{v_z^{3/2}} . \tag{34} \]
Taking into account (29) and the fact that \(J=2y_0\omega j_0\), one may write the relation
\[ \frac{S}{S_0}=\frac{y}{y_0}=1+2K^2, \tag{35} \]
where \(\dfrac{S}{S_0}\) is the ratio of the final cross section \(S\) of the ribbon beam to its initial cross section \(S_0\). Using (11), one can calculate the analogous ratio
\[ \left(\frac{S}{S_0}\right)'=\frac{r^2}{r_0^2} \]
for a beam of circular cross section, also as a function of the parameter \(K\). In Fig. 5 these two dependences are given, and from their comparison it is seen that, for the same initial current density and particle velocity (the same values of \(K\)), the beam of circular cross section spreads more strongly; moreover, the greater the degree of spreading, the more it differs for the two beams under consideration. In the work of Wax \(^{19}\) it was shown that a hollow beam of annular cross section spreads less than a solid beam of circular cross section; however, in contrast to the comparison considered above, the distribution of particle velocities over the beam cross section was also taken into account there.
Fig. 5.
The simplified theories of beam spreading given above are based on a number of assumptions and undoubtedly require experimental verification.
In Fig. 6 are shown dependences \(^{20}\) of the ratio of the final beam radius to the initial one on the gas pressure in the volume where the beam propagates; the parameter is the energy of the electrons moving in equipotential space. Arrows indicate-
the calculated values of \(\dfrac{r}{r_0}\). Comparison of the experimental curves with the calculated values shows that the true spreading is less than the calculated one, but as the gas pressure is decreased the agreement improves substantially. The dependence on pressure may be connected with the action of the space charge of positive ions formed during the motion of the electron beam; this effect will be considered below.
\(T=50\,\mathrm{m\mu};\ z=36\,\mathrm{cm};\ j_n=0.1\,\mathrm{a/cm^2}\)
Fig. 6.
For checking the theory of spreading of a ribbon beam, Klemperer\(^{21}\) carried out experiments, the arrangement of which is shown in Fig. 7. The electron beam was passed through chamber \(A\) and could enter the Faraday cylinder \(F\), after passing through a slit \(S\) of adjustable width. The dependence was measured of the aperture angle of the slit \(\Theta\), at which one half of the cathode emission current enters \(F\), on the space-charge coefficient
\[ \eta=\frac{I}{U_a^{3/2}}, \]
which characterizes the intensity of the beam; the latter quantity was varied by regulating the cathode temperature.
Fig. 7.
The observed dependence (Fig. 7) can be explained from the point of view of the theory presented earlier as follows. At small values of \(\eta\), the electrons approximately follow electron-optical trajectories; the character of the field at the cathode leads to a sharp crossing of the electron paths. In this case a strong divergence of the beam is observed (a large value of \(\Theta\))—see Fig. 8, c.
Fig. 8.
With increasing space charge \((\eta)\), the repulsion of the electrons on the cathode side of the intersection reduces the initial convergence of the paths and, consequently, reduces the beam divergence. The left-hand part of the curve in Fig. 7 corresponds to the transition from the conditions characterized by Fig. 8, c, to the conditions of Fig. 8, b. The curve in Fig. 7 passes through a minimum and increases again when, with increasing space charge, the form of the beam approaches that shown in Fig. 8, a.
To study the spreading of electron beams, a photographic method was used\(^ {31}\).
The calculations presented are not strict in a quantitative sense, although they correctly convey the principal features of the beam-spreading process. One of the reasons for the errors in the formulas given above is the neglect of the distribution of potential over the cross section of the beam.
II. DISTRIBUTION OF POTENTIAL IN THE BEAM
The presence of space charge leads not only to an overall change in the potential in the region of the beam, but also to the appearance of potential gradients within the beam. In this, the boundary conditions naturally play an essential role. Thus, for example, if the beam propagates inside an equipotential metallic tube (Fig. 9), then the action of charges of opposite sign induced on its walls leads to a reduction of the “sagging” of the potential in the region of the beam.
The distribution of potential in a tube of finite length was considered by V. Brodskii\(^ {25}\). As in all similar calculations, here [[unclear: sentence continues on next page]]
it is assumed that the spreading of the beam is limited by the action of a magnetic field (see below) of unlimited strength.
For the case of a beam unlimited in length (tube length \(l \gg R\)), the potential may be considered a function of only one radius, and the problem reduces to the integration of one-dimensional Poisson equations for the region of the beam and Laplace equations for the region between the beam and the tube.
Fig. 9.
The magnitude of the potential difference \((U_0 - U_{\text{п}})\) between the tube (potential relative to the particle source \(U_0\)) and the axis of a long beam (potential \(U_{\text{п}}\)) can be calculated3 by means of the equation
\[ 6.07 \cdot 10^4 \left( \frac{M}{m} \right)^{1/2} I \;(\text{amperes})\; U_0^{-3/2}\;(\text{volts}) = (1 - x)^{3/2} S, \tag{36} \]
where \(I\) is the beam current,
\[ x = \frac{|U_0 - U_{\text{п}}|}{U_0}, \]
and
\[ S = f\left(x,\frac{R}{\rho}\right). \]
In the first approximation this function may be represented in the form
\[ S_1 = \frac{4x}{1-x}\cdot \frac{1}{1 + 2\ln \dfrac{R}{\rho}}. \tag{37} \]
The potential distribution directly in the beam, for
\[ \frac{R}{\rho} \gg 1, \]
may be represented by the formula
\[ \frac{|U(r)-U_{\text{п}}|}{U_0} = \frac{1-x}{4}\cdot S \cdot \frac{r^2}{\rho^2}, \tag{38} \]
and, consequently, the total potential drop in the beam itself is equal to
\[ |U(\rho)-U_{\text{п}}| = \frac{|U_0-U_{\text{п}}|}{1+2\ln \dfrac{R}{\rho}}. \tag{39} \]
For a very thin beam
\[ \left(\frac{R}{\rho} \gg 1\right) \]
and for \(x \ll 1\), as follows from (36) and (37),
\[ |U_0-U_{\text{п}}| = 3\cdot 10^4 \left(\frac{M}{m}\right)^{1/2} I\;(\text{amperes})\; U_0^{-1/2} \ln\frac{R}{\rho}. \tag{40} \]
This formula can be derived simply and clearly in the following way.4 The charge contained in \(1\ \text{cm}\) of length of such a beam is equal to
\[ q = \frac{I}{\sqrt{2\dfrac{e}{M}U_{\text{п}}}}. \tag{41} \]
On the other hand, the charge induced on the tube must be equal to
\[ q=C(U_0-U_{\mathrm p}), \tag{42} \]
where \(C\) is the capacitance of the cylindrical capacitor formed by the beam and the tube. Equating these two expressions, we obtain formula (40).
The magnitude of the radial potential drop in a freely propagating beam, but one with limited spreading, can be estimated very simply (assuming \(j\) and \(v_z\) to be constant over the beam cross section):
\[ |U(\rho)-U_{\mathrm p}|=\int_0^\rho E\,dr =\int_0^\rho \frac{2I(r)}{rv_z}\,dr =\int_0^\rho \frac{2\pi r^2 j_0}{rv_z}\,dr =\frac{I}{v_z}. \tag{43} \]
If a beam of finite length \(l\) passes perpendicularly through two parallel grids having potential \(U_0\), then the potential at the center of the beam can be represented by the formula\({}^{28}\)
\[ U_C=U_0[1-\mu\psi(0,0)], \tag{44} \]
where
\[ \mu=\frac{4lU_0^{-3/2}}{\sqrt{2\frac{e}{M}}} \]
or
\[ \mu=1.92\left(\frac{M}{m}\right)^{1/2}l\;(\text{ampere})\,U_0^{-3/2}\;(\text{kilovolt}), \]
\(\psi\) is a quantity varying within the limits from \(1\div 3\) (for small values of \(|U_0-U_C|\), the quantity \(\psi\to 1\)); \(I(0,0)\) is the value of the function \(I(z,r)\) at the center of the beam. Table II gives values of \(I(0,0)\) for different values
\[ L=\frac{l}{2\rho}. \]
Table II
| \(L\) | \(I(0,0)\) |
|---|---|
| Small | \(\dfrac{L^2}{2}\) |
| 0.5 | 0.110 |
| 1.0 | 0.326 |
| 2.0 | 0.553 |
| 4.0 | 0.893 |
| 10.0 | 1.343 |
| 20.0 | 1.690 |
Thus, for a very short beam (\(L\) small) the potential difference between the grid and the center of the beam is equal to
\[ |U_0-U_C|=\frac{0.5\pi jl^2}{\sqrt{2\frac{e}{M}U_0}}, \tag{45} \]
where \(j\) is the current density in the beam.
As follows from the formulas given, the potential drop in a long, thin beam is determined by the total beam current, whereas in the case of a short and wide beam the drop is determined by the current density.
The potential distribution in beams of rectangular cross section was considered by Haeff\({}^{27}\). To characterize this case, let us introduce
consider only Fig. 10, which shows the dependence of the ratio of the potential on the beam axis to the potential of the box in which the beam propagates on the value \(F\). The latter, for electrons, is determined by the formula
\[ F=0.107\cdot 10^6 \frac{I}{U_0^{3/2}\left(\frac{\omega}{d}\right)}. \tag{46} \]
Here \(U_0\) is the potential of the box in volts, \(I\) is the beam current in amperes, \(\omega\) and \(t\) are, respectively, the width and thickness of the beam, and \(d\) is the thickness of the box. The parameter in Fig. 10 is the value of the ratio of the beam and box thicknesses, \(\frac{t}{d}\).
Fig. 10.
The drop of potential in the beam leads to a decrease in the velocities of the particles and also to the appearance of a certain distribution of particle velocities in the beam. This circumstance should cause more intense spreading of beams. Smith and Hartman\(^{26}\) compared the spreading of a beam with and without taking account of the potential distribution. As the calculation showed, at the maximum beam current (see below) in the first case the beam spreads over a specified relative distance by approximately a factor of two less than when the effect under consideration is ignored (the calculation is given for \(\frac{R}{\rho}=10\)).
A method for direct experimental study of the potential distribution in beams has not been developed. Let us note that Klemperer\(^{21}\) attempted to investigate the potential distribution in an intense electron beam, using as a probe a thin electron beam of low intensity.
The redistribution of potential in a beam caused by the action of space charge also entails a change in the distribution of current density over its cross section. An initial nonuniformity of current density is amplified owing to this effect. Experimental investigation of the current distribution in a beam was carried out, for example, by Klemperer\(^{21}\) and Roys\(^{29}\). Roys’s method consists in the fact that, under the action of the alternating field of deflector plates, the beam periodically crosses a metallic filament collector. The current in the circuit of this filament is amplified and observed oscillographically as a function of time; the resulting pattern gives a clear representation of the distribution of current density in the beam. An anomalous, nonmonotonic distribution of potential in an electron beam with a small electron energy, \(\sim 20\) volts, was observed by Clarke and Jacoby\(^{30}\).
III. MODELING FIELDS WITH SPACE CHARGE
Long ago, in connection with the investigation of the gun of an electron-beam tube, a method was described[^31] for taking space charge into account by the method of successive approximations. The starting point was the potential distribution in a gun of a given design in the absence of a beam. From this given field, which can be determined, for example, by means of an electrolytic tank,[^34] electron trajectories[^35] emerging from the cathode are constructed. Knowing the trajectories, the field, and the initial velocities of the electrons, one can determine the electron velocities and then the concentration of space charge \(\rho\) as a function of the coordinates. Knowing \(\rho\), one can obtain a new potential distribution which, in the first approximation, takes the space charge into account. After this the procedure may be repeated the desired number of times, depending on the required accuracy. In doing so, methods of numerical integration of partial differential equations[^36],[^70] may prove useful.
Attempts are known to apply an electrolytic tank to obtain the potential distribution in the presence of continuously distributed space charge.[^32],[^33],[^37] Poisson’s equation for an electron flow has the form
\[ \frac{\partial^2 U}{\partial x^2}+\frac{\partial^2 U}{\partial y^2} = C\,\frac{j}{\sqrt{U}} . \tag{47} \]
If \(h=h(x,y)\) is the depth of the electrolyte in the tank, then \(\sigma=ah\) and \(\operatorname{div}\sigma \mathbf{E}=0\), whence
\[ \operatorname{div}\mathbf{E}+\frac{\mathbf{E}\operatorname{grad}h}{h}=0 . \tag{48} \]
Putting
\[ \frac{\mathbf{E}\operatorname{grad}h}{h} = \mathbf{E}\operatorname{grad}\ln h = -C\,\frac{j}{\sqrt{U}}, \tag{49} \]
we obtain an equation identical with (47).
If the distribution of the electrolyte depth is specified (the bottom of the tank is made of rubber, which makes it possible to change the depth) so that (49) is satisfied, the potential distribution on the surface of the electrolyte gives \(U(x,y)\). Initially one sets \(h=\mathrm{const}\) and determines \(U_0(x,y)\) and \(E_0(x,y)\). Then, using the approximate expression
\[ h=h_0\exp\left[-C\int_{S_0}^{S}\frac{j}{E_0\sqrt{U}}\,ds\right], \tag{50} \]
one can, by numerical integration, determine \(h(x,y)\) and then, in accordance with the obtained \(h(x,y)\), set the depth of the electro-
of the sheet and determine anew \(U(x,y)\), \(E(x,y)\), etc. After each approximation it is necessary to construct trajectories in order to obtain a new distribution of current density.
A very interesting method is the modeling of the motion of charged particles with allowance for space charge, using the properties of a rubber membrane. The method was proposed by Bobykin, Kelman, and Kaminskii \(^{38}\) and later repeated in work \(^{39}\).
The magnitude of the displacement of the membrane elements from the equilibrium position \(h=h(x,y)\) satisfies the following equation:
\[ \frac{\partial^{2} h}{\partial x^{2}}+\frac{\partial^{2} h}{\partial y^{2}}=-\frac{P}{T}. \tag{51} \]
Here \(T\) is the stretching force applied per unit length of the edge of the membrane, \(P=P(x,y)\) is the force normal to the membrane, referred to unit surface area. The equation given coincides in form with equation (24), and if one sets
\[ 4\pi \rho(x,y)=C\frac{P}{T} \tag{52} \]
and imposes the corresponding boundary conditions, then
\[ U(x,y)=Ch(x,y). \tag{53} \]
It follows from (53) that the deflection of the membrane is proportional to the value of the potential at the corresponding point of the field, including the effect of space charge.
On the membrane, along certain lines corresponding to the electrodes, deflections proportional to the potentials of the electrodes are prescribed, while to the surface of the rubber free from electrodes there is applied a distributed load proportional at each point of the field to the space-charge density. First, with the aid of rolling balls and stroboscopic illumination, the trajectories and velocities of the particles are determined in the zeroth approximation. This makes it possible in the first approximation to compute the distribution of space charge. A load corresponding to this distribution is applied to the membrane, and again, with the aid of balls, the trajectories, velocities, and distribution of space charge are determined. As follows from specially arranged experiments \(^{38,39}\), already in the second approximation results close to reality are obtained.
For the approximate construction of particle trajectories in an electric field with allowance for space charge, the following method was also proposed \(^{71}\).
The starting point, as usual, is the field of the given electron-optical system without allowance for space charge. The true field,
represented by equipotential lines (Fig. 11) \(U_1, U_2, U_3\), is replaced by a field consisting of potential jumps localized on the former equipotential lines \(\dfrac{U_1}{U_2}, \dfrac{U_2}{U_3}, \dfrac{U_3}{U_4}\), and of equipotential regions situated between these lines.
Fig. 11.
To construct the shape of the beam with account taken of its space charge, one proceeds as follows. The initial segment of the trajectory of an edge particle of the beam approaches the point \(A\), lying in the region of a potential jump. Here the continuation of the trajectory is constructed according to the known rules for the refraction of an electron beam. It is assumed that in the interval between two jumps the beam moves only under the action of the field of the space charge, and, by the known formulas given in the first section, one can determine the vertical displacement of the trajectory. Thus, for a beam of circular cross section this displacement is determined with the aid of the formula \(\Delta r=\dfrac{\ddot r t^2}{2}=\dfrac{Iez^2}{Mrv_0^3}\). Since the curve \(AB\) is a parabola, the tangent to the curve at the terminal point divides the tangent drawn at the point \(A\) (distance \(x\)) in half, and this circumstance makes it possible to simplify the approximate construction.
A graphical method for constructing trajectories with account taken of space charge is considered in Broer’s work \(^{73}\).
IV. CURRENT LIMITATION IN BEAMS
Above, the spreading of beams under the action of their own space charge was considered. However, even if the spreading is limited in some way (see below), the harmful action of the space charge continues to manifest itself in that it limits the magnitude of the current in the beam.
Clarification of this question was initially carried out as applied to particle flows not bounded in cross section. Thus, for example, in the simplest case of a plane two-electrode system, the maximum current density, limited by space charge, is determined by the formula
\[ j_{\max}=\frac{1}{9\pi}\sqrt{2\frac{e}{M}}\,\frac{U^{3/2}}{z^2}, \tag{54} \]
where \(U\) is the potential difference between two unbounded planes, one of which is the source of slow particles, and \(z\) is the distance between the planes.
An approximate expression of the “three-halves law” for a bounded cathode was obtained by Levintov \(^{47}\).
A somewhat more complicated case is considered in the theory of the four-electrode tube \(^{40,41,42}\). The source of particles is the plane \(k\) (Fig. 12). By varying the distance \(z_1\) (with current limited by space charge) or the emission of \(k\), it is possible, at a constant value of \(U_1\), to change the magnitude of the current density in the space between the grids \(g_1\) and \(g_2\). Analysis shows that as this current density \(j\) increases, a potential minimum appears in the region \(g_1-g_2\) (\(U_1\gg0,\ U_2\gg0\)). When \(j\) reaches the value
\[ j_{\max}=2.33\cdot10^{-6}\frac{\left(U_1^{\prime 1/2}+U_2^{\prime 1/2}\right)^3}{z^2}, \tag{55} \]
the magnitude of the minimum potential drops abruptly from the value
\[ U'_{\min}=\frac{U_1}{\left(\sqrt{\frac{U_1}{U_2}}+1\right)^2} \tag{56} \]
to zero and remains zero for all larger values of \(j\) (a virtual cathode is formed). With a subsequent decrease of \(j\), the zero value of the minimum is preserved until \(j\) has fallen to the value
\[ j'_{\max}=2.33\cdot10^{-6}\frac{\left(U_1^{3/4}+U_2^{3/4}\right)^2}{z^2}. \tag{57} \]
The magnitude of the maximum current density in this and similar cases can be determined \(^{41}\) with the aid of the equation
\[ \frac{\partial j}{\partial U_{\min}}=0. \tag{58} \]
Fig. 12.
Turning directly to the limitation of the current in beams, let us consider a cylindrical beam of radius \(\rho\), unlimited in length, propagating along the axis of a conducting tube of radius \(R\). Suppose that the spreading of the beam is limited, for example, by means of an axial magnetic field. In the case \(R \gg \rho\), an approximate solution of the problem can be carried out quite explicitly and very simply \(^{27}\).
The charge contained in the beam per one centimeter of its length is equal to
\[ q=\frac{I}{\sqrt{2\frac{e}{M}U_m}}, \tag{59} \]
where \(I\) is the total beam current, \(U_m\) is the potential in the region of the beam (in reality there is a certain distribution of potential in the beam). If the potential of the tube relative to the source is equal to \(U_0\), then
\[ q=C(U_0-U_m), \tag{60} \]
where \(C\) is the capacitance of the cylindrical capacitor formed by the beam and the tube, referred to unit length. Combining equations (59) and (60) gives
\[ I=C(U_0-U_m)\sqrt{2\frac{e}{M}U_m}. \tag{61} \]
Using condition (58), we obtain from (61) the value of the potential in the beam region at which the current will be maximal:
\[ U_m=\frac{U_0}{3}. \tag{62} \]
The magnitude of the maximum beam current is therefore equal to
\[ I_{\max}=\frac{1}{3\sqrt{3}}\sqrt{2\frac{e}{M}}\,\frac{U_0^{3/2}}{\ln \frac{R}{\rho}}, \tag{63} \]
or, in practical units,
\[ I_{\max}=1.3\cdot 10^{-5}\left(\frac{m}{M}\right)^{1/2}\frac{U_0^{3/2}}{\ln \frac{R}{\rho}}. \tag{64} \]
The same problem was solved by Smith and Hartman \(^{26}\), taking into account the distribution of potential over the cross section of the beam. This more exact solution makes it possible to consider also the other limiting case: \(\frac{R}{\rho}=1\).
For \(R=\rho\) the magnitude of the maximum beam current does not depend on the cross-section of the tube and is equal to
\[ I_{\max}=3.2\cdot 10^{-5}\left(\frac{m}{M}\right)^{1/2} U_0^{3/2}. \tag{65} \]
By the approximate method given earlier, one can also obtain\({}^{27}\) an expression for the maximum current of a ribbon beam:
\[ I_{\max}=8.1\cdot 10^{-6}\left(\frac{m}{M}\right)^{1/2} U_0^{3/2}\frac{\omega}{d}. \tag{66} \]
Here \(\omega\) is the width of the beam, \(d\) is the thickness of the metal box through which the beam passes (it is assumed that \(t\ll d\), where \(t\) is the thickness of the beam).
If the ribbon beam completely fills the box \((t=d)\), the magnitude of the maximum current is
\[ I_{\max}=18.7\cdot 10^{-6}\left(\frac{m}{M}\right)^{1/2} U_0^{3/2}\frac{\omega}{d}. \tag{67} \]
G. A. Grinberg and M. I. Pevzner\({}^{43}\) considered the passage of an electron beam inside a metal sphere. Defining the potential inside the sphere as the sum of the potentials produced by the beam and by the induced charges, the authors obtained a number of formulas that make it possible to determine the magnitude of the maximum beam current.
In all the cases cited above it is assumed that spreading is limited. For a beam passing through a region of limited cross-section, one can calculate the magnitude of the maximum current limited by spreading. Thus, if the passage of a focused beam is restricted to a region of length \(z\) and radius \(r_0\), then, with optimum focusing of the beam (see above), a current not exceeding the value
\[ I_{\max}=1.5\cdot 10^{-4}\left(\frac{m}{M}\right)^{1/2} U_0^{3/2}\left(\frac{r_0}{z}\right)^2 \tag{68} \]
can pass through this region.
This expression can be obtained if, from Fig. 3, one determines the value of the parameter \(K\) for which \(\dfrac{r_k}{r_0}=1\).
The conditions that occur in the beam at maximum current can be characterized as follows. Following V. Kalinin\({}^{44}\), let us estimate the potential energy in some section of the beam having length \(l\):
\[ \mathcal{E}_{\mathrm{pot}}\simeq \frac{q^2}{l}. \tag{69} \]
where \(q\) is the charge contained in this volume. On the other hand, the kinetic energy of the particles in this volume is
\[ \mathscr{E}_{\text{kin}} \simeq qU_{\text{p}} . \tag{70} \]
The ratio of the potential energy to the kinetic energy is
\[ \frac{\mathscr{E}_{\text{pot}}}{\mathscr{E}_{\text{kin}}} \simeq \frac{q}{lU_{\text{p}}} \simeq \frac{I}{\sqrt{2\frac{e}{M}\,U_{\text{p}}^{3/2}}}. \tag{71} \]
Taking relation (43) into account, from (71) we obtain:
\[ \frac{\mathscr{E}_{\text{pot}}}{\mathscr{E}_{\text{kin}}} \simeq \frac{U(\rho)-U_{\text{p}}}{U_{\text{p}}} = \frac{\Delta U}{U_{\text{p}}}, \]
\[ \left( \frac{\mathscr{E}_{\text{pot}}}{\mathscr{E}_{\text{kin}}} \right)_{\max I} \simeq 1 \quad \text{for } I=I_{\max}. \tag{72} \]
Thus, at the maximum beam current, when the radial drop of the potential \(\Delta U\) tends to the value \(U_{\text{p}}\), the potential energy of the particles associated with their electrostatic repulsion becomes, in order of magnitude, equal to their kinetic energy.
Fig. 13.
The formulas for the maximum current of a ribbon beam were checked experimentally \(^{27}\). Electrons from the source \(K\) (Fig. 13) were accelerated by the potential \(U_1\) of grid \(g_1\) and passed inside the metallic box \(S\), which was closed at the ends by slit grids \(g_2\) and \(g_3\) (potentials, respectively, \(U_2\) and \(U_3\)). The collec-
tor was the plate \(C\) (potential \(U_C\)). The entire system was placed inside a solenoid, and an approximately uniform magnetic field was created along the direction of the beam. Figure 13 gives the dependence of the collector current on the potential \(U_1\), obtained under the following conditions: \(H = 500\ \text{oe}\), \(U_2 = U_3 = 240\ \text{V}\), \(U_C = 300\ \text{V}\). The parameter for the curves shown in this figure is the value of the box potential \(U_S\). As follows from this figure, with an increase in the injected current (increase of \(U_1\)) the current first increases, reaches a maximum value, and then decreases. Table III compares the calculated and experimentally obtained values of the maximum currents.
Table III
| \(U_1\) (V) | \(I_{\max}\) (mA) for \(\dfrac{t}{d} = 0.4\) (calculated) | \(I_{\max}\) (mA) for \(\dfrac{t}{d} = 0.05\) (calculated) | \(I_{\max}\) (mA) (experimental) |
|---|---|---|---|
| 50 | 17.2 | 13.4 | 15.2 |
| 75 | 31.7 | 24.6 | 25.5 |
| 100 | 48.8 | 37.9 | 38.5 |
| 125 | 68.5 | 53.2 | 52.8 |
| 150 | 90.0 | 69.8 | 68.0 |
The calculations were carried out with the aid of the formula for the maximum current:
\[ I_{\max}=9.35\cdot 10^{-6}\left(\frac{m}{M}\right)^{1/2}\left(\frac{\omega}{d}\right)F_{\max}, \tag{73} \]
where \(F_{\max}=f\left(\dfrac{t}{d}\right)\), so that for \(\dfrac{t}{d}=1\), \(F_{\max}=2\), and formula (73) becomes (67). Since the quantity \(\dfrac{t}{d}\) is not known exactly, the calculations were carried out for two values: \(\dfrac{t}{d}=0.4\) and \(\dfrac{t}{d}=0.05\) (the first value was determined from the geometrical dimensions). As is seen from the table, satisfactory agreement is observed between the experimental and calculated values of the maximum current.
V. NEUTRALIZATION OF SPACE CHARGE BY THE ACTION OF CHARGED PARTICLES OF THE OPPOSITE SIGN
To overcome the harmful action of space charge in beams, the neutralizing action of charges of the opposite sign can be used effectively. It is therefore of interest to consider the action of charges of the opposite sign as a means
for preventing the spreading of beams, improving their focusing, and also increasing the maximum beam current.
The question of such neutralization has repeatedly, in various aspects, been subjected to theoretical and experimental investigation. We shall not consider here works in which this question was analyzed as applied to a gas discharge (for example, \(^{45}\)), and shall only mention the works of Morgulis \(^{10}\), Ptitsyn and Tsukkerman \(^{9}\), Gurtovoi and Kovalenko \(^{11}\), devoted to the neutralization of volume charge in a two-electrode system in which one electrode served as the source of particles of both signs. In one of these works \(^{9}\) it was shown directly, experimentally, with the aid of probes, that the potential maximum arising due to the action of the ionic volume charge is smoothed out under the influence of oppositely charged particles—electrons.
Of interest for neutralization in beams of limited cross section is the consideration of the well-known problem of bipolar current \(^{72}\).
If two infinite planes, separated by a distance \(d\), are sources of particles of different signs moving toward one another, then the corresponding Poisson equation may be represented in the form
\[ \frac{d^2 U}{d x^2} = 4\pi \left( \frac{j_e}{\sqrt{2\frac{e}{m}U}} - \frac{j_p}{\sqrt{2\frac{e}{M}(U_0-U)}} \right), \tag{74} \]
where \(j_e\) is the electron-current density, and \(j_p\) is the ion-current density.
If as boundary conditions one takes \(x=0,\ \dfrac{dU}{dx}=0,\ U=0\) and \(x=d,\ U=U_0\), i.e. assumes that the electron current is limited by the volume charge, then the first integration gives
\[ \left(\frac{dU}{dx}\right)^2 = \frac{16\pi j_e}{\sqrt{2\frac{e}{m}}} \left( \sqrt{U} + a\sqrt{U_0-U} - a\sqrt{U_0} \right), \tag{75} \]
where
\[ a=\frac{j_p}{j_e}\sqrt{\frac{M}{m}} . \tag{76} \]
Integration of (75) leads to the following expression for the ratio of the electron-current density in the presence of a counter ion flow to the density of the electronic unipolar current:
\[ \frac{j_e}{j_{e_0}}=f(a), \]
where \(f(a)\) is a certain complicated function. In the case when also the ion-
...the current is limited by the space charge \(\left(x=d,\ \dfrac{dU}{dz}=0\right)\), the quantity \(a=1\), and \(f(a)\) has a maximum value equal to \(1.85\). Thus, the counter ion current, whose magnitude is \(\sqrt{\dfrac{M}{m}}\) times smaller than the electron current, increases the electron current by approximately a factor of 2.
Equation (55) makes it possible to determine the maximum density of the current passing through a system consisting of two parallel grids, at which a virtual cathode is formed. For the particular case when the grids are at the same potential \(U_0\), the value of the maximum current density is
\[ j_{\max}=18.6\cdot 10^{-6}\frac{U_0^{3/2}}{d^2}. \tag{77} \]
This makes it possible to pose the problem of determining the maximum current density \(j'_{\max}\) that can pass through the system if, in the space between the grids, there is an ionic space charge with a concentration equal to the concentration of electrons. A consideration of periodic oscillations that may occur in the electron stream under these conditions (the ions are immobile), using the Poisson equation, the continuity equation, and the equation of motion, led Pierce to the following conclusion\(^{46}\). If the density of the electron stream is less than a certain limiting value \(j'_{\max}\), then perturbations arising in the stream decay with time; but if the current density exceeds this value, the perturbations grow. The magnitude of this maximum stationary current density turns out to be
\[ j'_{\max}=104\cdot 10^{-6}\frac{U_0^{3/2}}{d^2}. \tag{78} \]
Comparison of (77) and (78) shows that the neutralizing action of the ions increases the maximum current density by approximately a factor of six.
For the maximum electron current that can pass inside a tube (whose radius \(R\ll l\)) in the presence of neutralizing ions, the expression obtained is
\[ I'_{\max}=190\cdot 10^{-6}U_0^{3/2}. \tag{79} \]
It should be noted that these calculations have not yet been confirmed experimentally, and to what extent equations (78) and (79) actually represent the maximum value of the density of an electron stream under neutralization by positive ions remains unclear.
A volume charge of the opposite sign has long been used for focusing electron beams. From works devoted to the so-called “gas concentration” of beams, it is known that a beam of electrons passing through a gas at a pressure of \(10^{-2}\)—\(10^{-3}\) mm Hg, or even lower, is concentrated and assumes the form of a thread or of a standing wave (nodal beams). This fact is a consequence of ionization of the gas and the formation, along the path of the electron beam, of a positive ionic charge. A theoretical investigation of this phenomenon was carried out by Frenkel and Bobkovsky\({}^{6}\), Scherzer\({}^{47}\), and others, and also recently by Bredov\({}^{7}\) and Davydov and Braginsky\({}^{8}\).
Bredov developed a theory of gas concentration of beams of rectangular cross section, having in mind the following basic assumptions: the free path length of the particles is large in comparison with the thickness of the beam; electrons that have undergone collisions, and electrons arising in ionization, do not have an appreciable effect on the distribution of potential in the beam; thermal motion and recombination in the beam are ignored. In the theory, which considers a linear source of electrons, there appears the parameter
\[ \omega^2 = 6.9\,p\varepsilon(U)\,U^{3/4}\sqrt{\frac{A}{j}}, \tag{80} \]
where \(A\) is the atomic weight, \(j\) is the current density in the beam in \(\mathrm{ma}/\mathrm{cm}^2\), \(U\) is the electron energy, \(p\) is the pressure in mm Hg, and \(\varepsilon(U)\) is the number of ionization events made by an electron over a path of \(1\ \mathrm{cm}\). For example, for the case \(\omega^2 \ll 1\), the maximum beam thickness in the bunch is found to be
\[ 2y_{\max}=3\cdot10^{-2}\frac{\gamma_m}{p\varepsilon\sqrt{A}} =\frac{4\gamma_m}{\pi}\frac{1}{\nu\varepsilon}\sqrt{\frac{m}{M}}, \tag{81} \]
and the distance between nodes is
\[ a=4.65\cdot10^{-2}\frac{1}{p\varepsilon\sqrt{A}} =\frac{2}{\nu\varepsilon}\sqrt{\frac{m}{M}}. \tag{82} \]
Here \(\gamma_m=\dfrac{dy}{dx}\) at \(x=0,\ y=0\) (\(x\) is the direction of propagation of the beam).
For example, at \(U=1000\) V, \(\gamma_m=0.05\), and \(p=10^{-3}\) mm Hg, \(2y_{\max}=0.094\ \mathrm{cm}\), \(a=2.9\ \mathrm{cm}\), while at \(p=10^{-4}\) mm Hg, \(2y_{\max}=0.94\), \(a=29\ \mathrm{cm}\).
The author points out that the formation of a ribbon beam is more probable, since the nonuniformity of the angular distribution of intensity at the source, the finite dimensions of the latter, and the aberrations of the electron-optical system favor the formation of beams precisely of this type. Since the beam can turn into
PROPAGATION OF BEAMS OF CHARGED PARTICLES
into a ribbon beam only after several pulsations, then the so-called “nodal beam” apparently represents several of the first pulsations of a ribbon beam.
Davydov and Braginskii \(^{8}\) drew attention to the fact that the shape and dimensions of a gas-focused beam depend to a considerable degree on the initial distribution of electron velocities. The Maxwellian distribution of initial velocities is superposed on the velocity of the convective motion of the electrons, and if the longitudinal components associated with the initial velocities may be neglected, then the transverse components must be taken into account, since they may be of the same order as the focusing potential difference in the beam.
The application of hydrodynamic equations to the motion of electrons in a beam made it possible to consider the formation of threadlike and nodal beams and to derive formulas for calculating the radii and the distances between nodes.
To illustrate the formulas derived, Davydov and Braginskii \(^{8}\) give the following example: a beam of electrons with a current of \(10\ \mu\text{A}\) and an energy of \(300\ \text{eV}\), with an effective angle of divergence \(\sim 2^\circ\), passes through argon at a pressure \(p = 5 \cdot 10^{-3}\ \text{mm Hg}\). The effective beam radius at the cathode is \(R_0 = 1\ \text{mm}\), and the initial energy of the electrons is \(\varphi = 0.1\ \text{eV}\). Under these conditions a nodal beam is formed: the equilibrium radius is \(R_1 = 0.7\ \text{mm}\), the maximum radius is \(R_{\max} = 1.75\ \text{mm}\), the ratio of the maximum radius to the minimum is
\[ \frac{R_{\max}}{R_{\min}} = 4.4, \]
and the distance between nodes is \(11.7\ \text{cm}\).
Zaitsev and Reichrudel, in their experiments on focusing electron beams in pulsed X-ray tubes, established that when the distance between the electrodes is varied, a periodic change is observed in the dimensions of the luminous spot on the anode, which is coated with a phosphor. This fact is explained by the action of the positive space charge of ions, which has time to form within several microseconds at residual-gas pressures of \(10^{-5}\)–\(10^{-4}\ \text{mm Hg}\). It is assumed that the positive space charge determines the nodal form of the beam and that the observed experimental facts are thus explained. The influence of the space charge of ions had also been assumed in other analogous cases \(^{49}\). The contraction phenomenon observed under conditions of a glow discharge \(^{50}\) was likewise explained by the effect of gas focusing.
Of practical importance is the question of whether it is possible to prevent the spreading of beams propagating under high-vacuum conditions by the action of a space charge of the opposite sign.
If a long electron beam propagates along the axis of a metallic tube \(^{51}\), the potential inside the tube is lowered and
a certain potential channel is formed (Fig. 14). Even under conditions of high vacuum, ion formation will occur. On any ion that has formed inside the tube there will act a force directed toward the center of the beam, and thus this “channel”
Fig. 14.
will act as a trap for ions. The ions accumulating here raise the potential in the region of the “channel” and thereby eliminate the field causing the beam to spread.
The number of ions formed during 1 sec on one centimeter of beam length is equal to
\[ \left(\frac{dN_p}{dt}\right)_1=\pi r^2 \varepsilon n_e v_e p, \tag{83} \]
where \(r\) is the beam radius, \(n_e\) and \(v_e\) are the concentration and velocity of the electrons, \(\varepsilon\) is the specific ionization, and \(p\) is the pressure of the residual gases. Assuming that the ions inside the “channel” form an ionic gas
and the radial distribution of the concentration is determined by the Boltzmann formula, Linder and Hernqvist\(^{51}\) write the number of ions going to the wall in the form
\[ \left(\frac{dN_p}{dt}\right)_2 =2\pi R n_{p0}\left(\frac{kT_p}{2\pi M}\right)^{1/2} e^{\frac{e\Delta U}{kT_p}}, \tag{84} \]
where \(R\) is the radius of the tube, \(n_{p0}\) is the ion concentration on the axis, \(\Delta U\) is the potential difference axis—tube, and \(T_p\) is the “temperature” of the ion gas in the trap. From the equilibrium condition there follows the expression
\[ p= \frac{R n_{p0}}{\sqrt{\pi r^2\varepsilon n_e}} \left(\frac{mU_p}{MU_e}\right)^{1/2} e^{\frac{\Delta U}{U_p}} . \tag{85} \]
Here \(U_p\) is the temperature of the ion gas, expressed in equivalent volts, and \(U_e\) is the energy of the electrons in the beam. From this expression, putting \(n_{p0}=n_e\), one can estimate the magnitude \(\Delta U\), or, assuming in addition \(\Delta U=0\), estimate the magnitude of the pressure at which complete neutralization of the space charge will occur.
If the pressure is greater than this latter value, it is not impossible that \(\Delta U\) will change its sign and the potential distribution will become such that the “channel” will turn into a trap for slow electrons formed in the beam.
Bearing in mind the case of pulsed switching-on of the electron beam, one should estimate the time of accumulation of the neutralizing ionic charge \(\tau\). The rate of accumulation is determined by the quantity
\[ \frac{dN_p}{dt} = \left(\frac{dN_p}{dt}\right)_1 - \left(\frac{dN_p}{dt}\right)_2 , \]
but for a rough estimate one may neglect the second term, and then one can obtain the minimum time required for accumulation of the charge \(\tau_{\min}\). Understanding by \(\tau_{\min}\) the time during which the concentrations of ions and electrons are equalized, we obtain
\[ \int_0^{\tau_{\min}} \left(\frac{dN_p}{dt}\right)_1\,dt = \pi r^2 n_e, \]
whence, if \(\tau_{\min}\) is measured in microseconds, \(p\) in mm Hg, and \(\varepsilon\) is the number of ionizations per centimeter (at a pressure of 1 mm Hg),
\[ \tau_{\min}=\frac{0.0169}{p\varepsilon\sqrt{U_e}} . \tag{86} \]
The above considerations were subjected to experimental verification\(^{51}\) using a pulsed method. The cathode \(K\)—grid \(C\) system made it possible to obtain a converging beam of electrons, which, through an aperture in the diaphragm \(A\), could pass into the Faraday cylinder \(\Phi\) (see Fig. 14, a). During the time \(a-b\) (Fig. 14, b) the space \(C-A\) is equipotential; between \(K\) and \(C\) there exists
accelerating field; under these conditions ionization of the gas is possible in the region \(C — A\), and, consequently, neutralization. At the moment \(b\) the electrode \(C\) acquires the potential of the cathode, and during the time \(b — c\) the beam is blocked, as can be seen in the schematically represented oscillogram (Fig. 14, б). During the time \(b — c\), the ions formed earlier may be neutralized on the walls and electrodes. At the moment \(c\) the potential \(C\) is restored to its former value, but the current, as follows from the oscillograms, does not immediately increase to its previous value (time \(c — d\)). This is explained by the fact that during the time \(c — d\) positive charges accumulate, which neutralize the electron charge, reduce the spreading of the focused beam, and increase the current of particles penetrating into the Faraday cylinder. At currents of tens of milliamperes and gun voltages of hundreds of volts, neutralization by ions increases the current density in the aperture of diaphragm \(A\) by tens of times. The experimentally measured time for a pressure of \(10^{-5}\) mm Hg and electron energies of \(200—300\) V amounted to several microseconds, in agreement with the calculated data obtained from equation (86).
The finite accumulation time of the neutralizing charges is an important circumstance, since it accounts for the absence of neutralization in devices operating in a pulsed regime\({}^{69}\) with short pulse durations.
Figure 6 shows that, in the case of an electron beam with electron energy \(3—7\) keV and a current density in the beam \(\sim 0.1\) A/cm\({}^{2}\), with a beam length of 36 cm, no spreading occurred at residual-gas pressures above \(10^{-6}\) mm; when the pressure was reduced below this value, appreciable spreading appeared. These experimental facts are likewise explained by the effect of neutralization of the space charge by ions.
These same effects of neutralization of the volume electron charge by ions were also considered as applied to another system (with a broad flux of particles)—the tetrode\({}^{52}\). By means of the method described earlier it was found what influence ion neutralization exerts on the anode characteristic of the tetrode. If the characteristic is recorded in the usual way, it exhibits the dip characteristic of a tetrode, indicating a dynatron effect of the anode. If, however, the characteristic is recorded in pulses, measuring the anode current at the moment the anode potential appears, it reveals a monotonic increase of the anode current. The absence of the dynatron effect in the latter case is explained by a sagging of the potential in the grid–anode region due to the action of the electron space charge; during the short time there is no accumulation of a significant number of neutralizing ions. In the first case this potential “channel” is neutralized by ions, and secondary electrons can go to the screen grid—the dynatron effect is observed.
In both cases considered (a beam of small cross section and a stream of large cross section), the accumulation of neutralizing ions can be hindered by adversely acting fields.
A substantial role in the accumulation of a neutralizing ionic charge may be played by the field penetrating through the exit aperture into the equipotential space through which the beam passes[^53]. Usually, in the region of this aperture there is a potential gradient that carries ions away, and neutralization may therefore be impeded. A change in the direction of this gradient should favor the neutralization of the space charge. As experiment has shown, the introduction of an additional potential difference (Fig. 14, в) does indeed lead to better neutralization and to a reduction in beam spreading.
Let us note that the presence of ions in an electron beam may be one of the causes of the occurrence of fluctuations[^54]. Linder and Hernqvist[^51] observed oscillations whose frequency was close to the ionic oscillations of a plasma.
Fig. 15.
Experimental confirmation of the neutralization of space charge by ions made it possible to understand better the processes occurring in a number of electron devices, in particular in microwave devices[^55]. It turned out that analogous phenomena also occur in ionic devices, for example in mass spectrographs. Here, in an ion beam, a potential well for electrons is formed; the latter can be introduced by ionization of atoms by ions or by means of...
correspondingly arranged sources of electrons^56. In this case as well it is necessary to create such conditions that the neutralizing particles (in the present case electrons) cannot freely leave the potential trap. In one case^57 (Fig. 15, a) a beam of \(A^+\) ions with a current of 10 mA and an energy up to 30,000 eV passed from the source \(S\) through an aperture in the accelerating electrode \(E\) and entered the equipotential chamber \(C\). The ratio of the number of ions striking the collector \(P_2\) to the total number of ions striking \(P_1\) and \(P_2\) characterizes the degree of divergence of the ion beam. Fig. 15, b shows that, by applying the appropriate potential difference between \(E\) and \(C\), one can prevent the electrons from leaving the ion beam and reduce the divergence of the latter. Experiment showed that under these conditions the intensity of the x rays produced by fast electrons is indeed reduced (see Fig. 15). The time required for neutralization of an intense ion beam by electrons was determined in work^77 and is equal to \(\tau=\dfrac{1}{N\sigma}\left(\dfrac{M}{2eU}\right)^{1/2}\), where \(eU\) is the energy of the ions, \(N\) is the number of molecules in \(1\ \mathrm{cm}^3\), and \(\sigma\) is the effective ionization cross section.
VI. FORMATION OF A BEAM AND LIMITATION OF ITS FURTHER SPREADING
One of the possible methods of forming a rectilinearly propagating beam is the creation along its path of a definite potential distribution. Let a beam of circular cross section have as its origin the plane \(z=0\), where there is located a plane source of particles possessing small initial energy. In order that the beam preserve its cross section, the potential must be a function only of the coordinate \(z\) and must not depend on \(r\). From equation (2) follows the condition under which the potential is not a function of the radius and, consequently, the beam will not spread:
\[ \frac{d^2U}{dz^2}+4\pi\rho = \frac{d^2U}{dz^2} + \frac{4\pi j_0}{\sqrt{2\frac{e}{M}U}} =0, \tag{87} \]
where \(j\) is the current density in the beam.
For the particular case when the beam has as its origin the surface \(z=0\), where \(U=0\) and \(\dfrac{dU}{dz}=0\) (the source current is limited by space charge), the required potential distribution along the axis can be obtained by integrating (87):
\[ U(z)=5680\,j^{2/3}(a/c\mathcal{M}^2)\left(\frac{M}{m}\right)^{1/3}z^{4/3}. \tag{88} \]
Such a potential distribution can approximately be realized, for example^61, by passing the beam through a series of diaphragms whose potential increases in accordance with (88).
Instead of employing a multielectrode system, which makes it possible to create only a rough approximation to the necessary potential distribution and is structurally complicated, it proved possible to use, with great success, a simpler two-electrode system. The method for finding the shape of the electrodes of such a two-electrode system was given by Pierce[^60]. The case of a sheet beam is solved analytically.
Fig. 16.
In the region outside the beam (Fig. 16) the Laplace equation is indeed valid,
\[ \frac{\partial^{2}U}{\partial z^{2}}+\frac{\partial^{2}U}{\partial y^{2}}=0. \tag{89} \]
In the region of the beam, equation (87) is valid. If the current is limited by space charge, the solution of this equation has the form \(U=Az^{4/3}\), where
\[ A=\left(\frac{9\pi j}{\sqrt{2\frac{e}{M}}}\right)^{2/3}. \]
In order that the beam not spread out, the field at its boundary must be equal to zero \(\left(\dfrac{\partial U}{\partial y}=0\right)\). Since the real and imaginary parts of an arbitrary analytic function are solutions of Laplace’s equation, and the solution of the latter for \(y=0\) must have the form \(U=Az^{4/3}\), we must have
\[ A(z+iy)^{4/3}=U+iV=Ar^{4/3}e^{\,i\frac{4}{3}\theta} =Ar^{4/3}\left(\cos\frac{4}{3}\theta+i\sin\frac{4}{3}\theta\right). \]
and, consequently, the equation
\[ U = Ar^{4/3}\cos \frac{4}{3}\theta \tag{90} \]
represents the required distribution of the potential outside the beam. By placing the electrodes on the equipotential surfaces \(U=0\) and \(U=U_a\), we obtain a two-electrode system (Fig. 16, right) ensuring rectilinearity of beam propagation between these two electrodes for any values of the potential \(U_a\), provided only that the current is limited by the space charge. The electrode with potential \(U=0\) turns out to be a plane; the angle \(\theta\) is determined from the condition
\[ \cos \frac{4}{3}\theta = 0 \]
and is equal to \(\theta=67.5^\circ\). This value of the angle is also retained for other “guns” of similar type, which serve to form a rectilinear beam of circular cross section, a conically converging beam.
Fig. 17.
The shape of the electrodes of a gun intended for the formation of a rectilinearly propagating beam of circular cross section (Fig. 17) is determined with the aid of a special electrolytic bath (Fig. 18). The electrolyte in the bath is in the form of a wedge, the edge of which represents the axis of symmetry of the beam. The two electrodes are
Fig. 18.
parts of figures of revolution. The boundary of the beam is simulated by a dielectric plate parallel to the beam axis and located at the corresponding distance (the beam radius) from the edge of the tank. This automatically ensures fulfillment of the condition \(\frac{\partial U}{\partial n}\) at the boundary of the beam (absence of current in the plate).
The distribution of the potential \(U(z)\) along the dielectric plate is determined experimentally, and such a shape of the electrodes is sought for which \(U(z)=\mathrm{const}\, z^{4/3}\).
Harrison\({}^{48}\), with the aid of certain approximations, obtained a simple equation for the shape of the electrodes of this system.
The shape of the electrodes of a gun intended to form a conical, converging beam is likewise determined with the aid of an electrolytic tank. The beam current is determined by the formula
\[ I_0 = 14.67 \cdot 10^{-6}\,\frac{1-\cos\theta}{a^2}\,U_a^{3/2}, \tag{91} \]
where \(\theta\) is half the cone angle, \(U_a\) is the potential of the anode (radius of curvature \(r_a\)) with respect to the cathode (radius of curvature \(r_k\)), and \(a^2\) is a certain function of the ratio \(\frac{r_k}{r_a}\) (see Table IV).
Table IV
| \(\frac{r_k}{r_a}\) | \(a^2\) | \(\frac{r_k}{r_a}\) | \(a^2\) | \(\frac{r_k}{r_a}\) | \(a^2\) |
|---|---|---|---|---|---|
| 1.0 | 0.0000 | 1.6 | 0.2968 | 2.6 | 1.712 |
| 1.05 | 0.0024 | 1.7 | 0.394 | 2.7 | 1.901 |
| 1.1 | 0.0096 | 1.8 | 0.502 | 2.8 | 2.098 |
| 1.15 | 0.0213 | 1.9 | 0.691 | 2.9 | 2.302 |
| 1.2 | 0.0372 | 2.0 | 0.750 | 3.0 | 2.512 |
| 1.25 | 0.0571 | 2.1 | 0.888 | 3.2 | 2.954 |
| 1.3 | 0.0809 | 2.2 | 1.036 | 3.4 | 3.421 |
| 1.35 | 0.1084 | 2.3 | 1.193 | 3.6 | 3.913 |
| 1.4 | 0.1396 | 2.4 | 1.358 | 3.8 | 4.429 |
| 1.45 | 0.1740 | 2.5 | 1.131 | 4.0 | 4.968 |
| 1.5 | 0.2118 |
Formula (91) is a consequence of the solution of Poisson’s equation for a spherical system of electrodes.
In the case of guns of this type, the effect of the anode aperture is not taken into account.\({}^{62}\) When the latter is sufficiently large (in comparison with \(d=r_k-r_a\)), the field becomes substantially different from
assumed in the theory, and the actual current will be less than the value following from equation (91). In addition, this aperture is a lens acting in a defocusing manner on the beam emerging from the gun. Defocusing of the beam at the exit from the gun will also occur because of the space charge of the beam itself. If it is assumed that outside the gun there is no electric field, except for the fields formed by space charges,
Fig. 19.
then this circumstance can be taken into account and the value of the minimum cross section of the beam (Fig. 19) and its location can be determined\({}^{12}\). This makes it possible to calculate the maximum current density
\[ j_{\max}=j_0\left(\frac{r_k}{r_{\min}}\right)^2, \tag{92} \]
assuming, for example, that the entire current taken from the cathode passes through the anode aperture.
It is known\({}^{12}\) that, with negligibly small space charge and with an ideal electron-optical system, the maximum current density that can be obtained in the beam depends on the magnitude of the initial velocities of the particles and is determined by the formula
\[ j'_{\max}=j_0\left(1+\frac{eU}{kT}\right)\sin^2\theta, \tag{93} \]
where \(j_0\) is the current density at the source (cathode), \(T\) is the cathode temperature, and \(U\) is the potential at the point under consideration. As the anode potential \(U_a\) of the gun under consideration is increased, the influence of the initial velocities becomes less and less significant, and with suffi-
exactly large \(U_a\), the maximum attainable current density is determined by the influence of the space charge. At small potentials, on the contrary, the influence of the initial velocities predominates over the influence of the space charge, and the maximum attainable current density is determined by equation (93).
Along with guns of this type, there exist various empirical systems\(^{12,21}\) which, for particular applications, prove more convenient or eliminate the shortcomings of theoretically substantiated systems associated with deviations from the boundary conditions prescribed by the theory.
Limitation of spreading appears possible not only by the use of an electric field, but also of a magnetic field\(^{12}\). It is known that an electron beam tends to follow magnetic lines of force. If, for example, a beam is placed in a uniform magnetic field whose lines are parallel to the direction of propagation of the beam, then when the latter spreads the electrons will cross the magnetic flux, and the magnetic field will rotate the electrons back toward the original line of force. The stronger the magnetic field, the smaller the deviation of the electron from the original line of force. In such a case the motion of the particle occurs in a combined field—a radial electric field of the beam space charge and a magnetic field.
Since, with the exception of motion along the axis, the conditions here are the same as in a magnetron, it was possible\(^{56}\) to use the known solution of the magnetron problem and obtain an equation that makes it possible to determine the maximum beam radius \(r_{\max}\):
\[ \frac{2Ic^2}{v_z}\ln\frac{r_{\max}}{r_0} = \frac{H_z^2 e r_0^2}{2M}\operatorname{sh}^2\left(\ln\frac{r_{\max}}{r_0}\right) - \frac{Mv_r^2}{2e}. \tag{94} \]
Here \(I\) is the total beam current, \(r_0\) its initial radius, \(v_r\) the initial radial velocity of the particles, and \(v_z\) the velocity of the particles along the \(z\) axis. If one neglects the quantity \(v_r\) and seeks the value of \(H\) at which
\[ \frac{r_{\max}-r_0}{r_0}=\alpha\ll 1, \]
then from (94) it follows that
\[ H=\left(2\frac{M}{e}\right)^{3/4}\frac{I^{1/2}c}{U_0^{1/4}r_0\sqrt{\alpha}}, \tag{95} \]
where
\[ eU_0=\frac{Mv_z^2}{2}. \]
Pierce\(^{12}\) carried out the following experiment. An electron beam with a current of \(10\ \mathrm{ma}\) passed through the aperture of an accelerating diaphragm and entered a long tube (length \(50\ \mathrm{mm}\), diameter \(1.2\ \mathrm{mm}\)), which had a potential of \(150\ \mathrm{V}\) relative to the particle source.
In the presence of a longitudinal magnetic field (300 G), an electron current of 7 mA went to the collector, and a current of 3 mA struck the walls of the tube. If, by formula (12), one estimates the spreading of the beam that would occur in free motion ($d_0 = 1.2$ mm, $I = 10$ mA, $U_0 = 150$ V) in the absence of a magnetic field, then the current to the collector turns out to be insignificant $\left(\dfrac{r}{r_0} \simeq 7\right)$. Consequently, in the experiment indicated, the magnetic field substantially limited the spreading of the beam. The required magnitude of the magnetic-field strength in these experiments corresponds to the value following from formula (95).
In considering the question of the formation of a beam in a magnetic field, attention should be paid to the character of the distribution of the magnetic flux. The equation of motion of a particle in the presence of the radial electric field of the space charges and of the longitudinal magnetic field $H = H_z$ has the form
\[ M\ddot r = e\,\frac{dU}{dr} - e r \dot\theta H_z + M r \dot\theta^2 = F + e\,\frac{dU}{dr}. \tag{96} \]
In this expression the quantity $F$ may be interpreted[^64] as a force counteracting the spreading of the beam caused by the radial field $\dfrac{dU}{dr}$. From (96) it follows that
\[ \frac{F}{M r \omega_H^2} = -\,\frac{2\dot\theta}{\omega_H} + \left(\frac{\dot\theta}{\omega_H}\right)^2, \tag{97} \]
where $\omega_H = \dfrac{1}{2}\dfrac{e}{M}H_z$.
The maximum negative value of $F$ for a given value of $\omega_H$ occurs at $\dot\theta = \omega_H$ and, consequently, the latter represents the condition for the existence of the given value of the force $F$ at the minimum magnitude of $H_z$.
Considering the motion of a charged particle in an axially symmetric field (having no component in the $\theta$-direction), one can obtain the expression
\[ \dot\theta = \frac{e}{2\pi r^2 M}\,(\psi - \psi_0). \tag{98} \]
Here $\psi$ and $\psi_0$ are, respectively, the flux passing inside the surface of revolution containing the trajectory of the particle at the point $z$ and at the cathode, where $\dot\theta = 0$. If, for the time being, the field is assumed homogeneous, then from (98) one can obtain the expression
\[ \dot\theta = \omega_H\left[1 - \left(\frac{r_0}{r}\right)^2\right], \tag{99} \]
from which it follows that $\dot\theta = \omega_H$ when $r_0 = 0$, i.e., under such conditions when the flux through the cathode is equal to zero.
This proposition also holds^65 in the more general case when the magnetic field is nonuniform, and also for beams of different shape. Figure 20 shows what the spatial distribution of the magnetic flux should be in the case of a hollow beam. This is one of the conditions under which it is possible to balance the forces of electrostatic repulsion and magnetic focusing, so that the beam can travel a long distance without change in its transverse cross section.
An example of the use of these considerations for obtaining a parallel beam is a device^66 in which the electron beam, after leaving the gun, entered a sharply increasing and then uniform magnetic field.
Fig. 20.
In Fig. 21 a schematic is given of a device, based on an analogous principle, for obtaining a conical converging beam^67. Here an ordinary Pierce system, shielded from the influence of the magnetic field, gives an electron beam that passes through
Fig. 21.
the anode aperture into the region of a paraboloidal magnetic field. In the region of the sharp increase of the magnetic field the particles acquire the necessary rotation, and subsequently the forces connected with the influence of the space charge are balanced by the action of the magnetic field. As in the preceding case, the region of increase
of the magnetic field must have a minimum length, since here the forces are not balanced and spreading takes place.
In the case of ion beams, magnetic fields, owing to the large mass of the ions, become ineffective; however, another possibility arises[^56]. If electrons are introduced into the ion beam, then the magnetic field can act effectively on them, and they, through the field of their space charge, can act on the ions. This principle is used, for example, to create a lens for focusing high-energy ions[^68].
In the path of the ion beam there is placed a device which is a magnetron of special design. The volume-charge density in the magnetron and, consequently, the radial field are respectively equal to
\[ \left. \begin{aligned} \rho &= \frac{eH^{2}}{8\pi mc^{2}},\\[4pt] E_r &= -2\pi\rho r = -\frac{eH^{2}r}{4mc^{2}} = -0.044H^{2}r \ \text{V/cm}. \end{aligned} \right\} \tag{100} \]
This field, increasing with radius, is focusing for ions. With the aid of a coil 20 cm long with a field of 500 gauss, this magnetron lens makes it possible to obtain, for 100-MeV protons, a focal distance of the order of 9 m. If a similar coil is used directly as a focusing lens for 100-MeV electrons, the focal distance will be 900 m.
As is clear from what has been set forth above, at the present time the basic principles have been developed for the formation and preservation of intense beams of charged particles, which constitute an essential component of electron and ion optics.
The practical importance of the question, and the necessity of taking space charge into account in the most diverse physical investigations connected with the use of intense beams, testify to the need both for further experimental research in this field and for the creation of a more rigorous theory of the formation and propagation of such beams.
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