"JOURNAL OF RADIOELECTRONICS" N 1, 2002 contents discussion

Cyclotron and Synchronous Oscillations and Waves of the Electron Beam

General Relations

###### Moscow State University

General relations of the cyclotron and synchronous oscillations and

waves of the electron beam are discussed in a form of short review.

VII.     Discussion

###### I.  Introduction

Development of modern microwave communication, radio and radar facilities places additional stringent requirements upon the different microwave devices.

During last several decades mainly in Russia active and successful work is being carried on to create new electron beam microwave devices based on cyclotron and synchronous waves use [1-16] and complying with the present-day requirements.

The operation of these devices is based on the principles of transverse grouping of the electron beam in the longitudinal magnetic field. In contrast to conventional longitudinal grouping of electrons into dense bunches, this principle employs the Lorenz force as an elastic force and leads to spatial distortion of the electric beam without electron bunches being formed. In this way it is possible to considerably overcome the fundamental restrictions, which are characteristic of longitudinal grouping devices (both vacuum and solid-state ones) and are associated with non-linear influence of the space charge fields upon the process of the input signal amplification, thereby laying the basis for developing new microwave devices with essentially improved characteristics [6,9,12-16]

Author’s experience in discussions of these principles both with domestic and foreign specialists has disclosed the expediency of the preparation of this article.

# II.  A Simple Beam Model

Consider the motion of electrons in crossed electric and magnetic fields having the form

Let us represent the beam as an aggregate of flat discs of infinitesimally small thickness, moving along the z-axis with constant velocity . If the transverse dimensions of the discs and its displacements from the unperturbed position are small in comparison with wavelength, then the coupling between the discs, due to the longitudinal space charge, is also insignificant and can be neglected. The problem is reduced to an analysis of the transverse oscillation of the individual discs [17].

Let the behavior of the individual (i-th) electron ( in Fig 2.1) be characterized by a complex coordinate  (), let the coordinate of the mass center (C in Fig.2.1)

Fig. 2.1

Electron beam cross-section

of the disc be , where  is the number of electrons in the disc. Then the orbits  of the individual electron relative to the mass center of the disc are described by the coordinates  . The equation of the transverse motion in terms of the variables  assumes, in complex notation, the forms

where  are the angular cyclotron frequency and the specific charge of the electron, respectively; ,  are the external electric forces acting on the electrons numbered  by “i” and “k”.

Principal interest attaches to the last two equations. The first of them characterized the motion of the mass center of the cross section of the beam, and consequently, describes the behavior of the signal and of the noise of the beam as a whole. While the second is important in the investigations of the internal structure of the beam (expansion of the thermal orbits, their balancing, etc).

The analytical solution of the problem with Coulomb sum in Eqs. (2.2) and (2.4), entails considerable difficulties. Appreciable simplification can be obtained for a model of a round uniformly charged cylindrical beam, if one uses the approximation

(where is the plasma frequency),

which in a number of cases makes it possible to describe correctly the actual physical processes in the system [17,18].

It is important to emphasize that in the cases when the function depends linearly on the transverse coordinates and their derivations, or else entirely independent of them, the equation of motion (2.3) for the mass center of the discs coincides with the equation of motion for a single electron without allowance for the Coulomb interaction and given by

where  is the external electric force acting on an electron placed at the mass center of the disc.

III. Kinematics Analysis

Let us introduce the transit angle, and then omitting  subscripts, we can write the equation of motion of one electron

In the simplest case , i.e. when electric fields are ab­sent, the solution has the form

Now let  using the constant variation method, the solution will be sought in the form

then

Since instead of one variable, two new variables - are introduced, it is necessary to impose an additional condition to connect these variables.  A suitable condition for this purpose is [19]

then

Substituting (3.6) and (3.7) in (3.1) and using (3.5), we obtain the system of the first-order  differential equations for the synchronous  and cyclotron  radiuses (see Fig. 3.1)

Fig. 3.1

Synchronous (a) and cyclotron (b)

radiuses of an electron (-e) motion

Thus, the change in the cyclotron orbit radius and the posi­tion of the orbit center with respect to the origin are described separately, which provides an additional illustrative representation and clarity of physical interpretation of the processes involved in the interaction of the electron with external fields.

In the general case

However, the solution of the system (3.8) can be simplified significantly in two extreme cases:

1. Short lenses, when the transit angle inside the interaction re­gion is small (Q<<1), and consequently, in the right-hand parts of the system of equa­tions (3.8) one can take

2. Adiabatic action of the fields on the electron motion, when relative changes in a and b are small during the oscillation pe­riod. In this case it is possible to use the method of averaging by a rapidly oscillating phase, i.e. one can write

Calculating the integrals in the right-hand parts of these equations the values of a and b  should be taken constant [19]

The solution obtained from the smoothed (averaged) equations (3.11) takes into account only effects of first order of smallness. In those cases when this does not suffice, it is necessary to apply to Eq. (3.1) the asymptotic methods of the theory of nonlinear oscillations [20].

For many issues of the theory of cyclotron-wave devices, the structure of the external forces is such that Eq. (3.1) can be written in a bit more convenient form

where  is the angular frequency of the resonant harmonic or subharmonic of the  external force,  is the relative difference between the natural frequency of the system  and , the value of the parameter  is chosen such that it characterized the intensity of the external forces.

The character of the action of the field on the beam is most frequently adiabatic in practice, i.e., the values of a and b change little under the influence of this field within the period of the fundamental motion. This is mathematically reflected in the smallness of the parameter . In this of a resonance adiabatic field  a solution can be sought for the n-th approximation, unlike (3.3), in the form

where the functions  take into account all arbitrary harmonics in , with the exception of the zero-th and the first (the latter requirement is essentially a supplement condition similar to (3.5)).

The abbreviated equations are sought in this case also in the form of an expansion in powers of the small parameter

Substituting (3.13) and (3.14) in the initial equation (3.12) and equating the terms of equal powers of  in the right and left sides of (3.12), we obtain for the first two approximations

, ,                                                                          (3.15)

where

Accordingly

where

The first approximation of the asymptotic method does not differ from averaging method [19]. The second and succeeding approximations refine the first and make it possible to take into account successively in the solution [19] the small high frequency vibrations (harmonics of the signal and pump, combination frequencies, etc.), which appear against the background of the fundamental motion of the system [6].

# IV. Transverse (Cyclotron and Synchronous) Waves

of the Electron Beam

In the simplest case when external fields have the form (2.1), the equations of motion of an electron are given by

,   .                                                     (4.2)

Now let us assume z and t to be independent variables, thereby making a transition from the description of the motion of one electron to the description of the motion of a filamentary electron beam

Then (see Fig. 4.1)

Fig. 4.1

Model of a filamentary electron beam

The solution will be sought in the form of the sum of two run­ning waves with right and left circular polarizations. Saving terminology (notations) of [21,22,2,6] we can write

where is the electron propagation constant.

Here

Then

(4.7)

Let us represent the external transverse electrical field in the form of two oscillations with right and left circular polarizations

Substituting (4.6)-(4.8) in the initial equation (4.1) gives

where  is the cyclotron propagation constant.

Note that:  since the functions depend on the variable  only.

Therefore (4.9) can be written as

In the undisturbed case  the solution has the form

Substituting this solution in (4.4) gives

Thus the motion of a filamentary electron beam can be de­scribed by the sum of four circularly polarized waves (transverse eigenmodes of the electron beam). Two waves with the amplitudes  are cyclotron waves, they have the opposite polarizations and their phase velocities depend on the cyclotron frequency

The other two waves with the amplitudes  are called synchronous waves, because their phase velocities are equal (synchronous) to the electron beam velocity

.                                                                                                          (4.14)

The phase velocity of the wave with the amplitude  may be greater than the longitudinal velocity of the electron beam and even become infinite in the case of the cyclotron resonance (Fig. 4.2) Therefore this wave was given the name fast cyclotron wave of the electron beam. This wave can be either forward or back­ward, depending on the phase velocity direction.

The phase velocity of the wave with the amplitude  is always smaller than the longitudinal velocity of the electron beam. Therefore this wave is called a slow cyclotron wave of the electron beam. The phase velocity direction of this wave is always positive, i.e. the wave is forward.

The synchronous waves  differ by the direc­tions of their circular polarizations only, however, by analogy with cyclotron waves they are often referred to as fast  () and slow  () synchronous waves of the electron beam.

Fig. 4.2

Dispersion characteristics of transverse waves.

FCW – Fast Cyclotron Wave,  SCW- Slow Cyclotron Wave,

SW – Fast and Slow Synchronous Waves.

To clarify the structure of transverse waves it is convenient to use the system of coordinates moving at the velocity  (i.e. the electron motion velocity) along the -axis. To do so, it is suf­ficient to assume in (4.12) that  .

Then

Thus, the amplitudes of the synchronous waves (synchronous radius) describe the shift of the electron orbit centers from the -axis. The amplitudes of the cyclotron waves determine the radius of the electron rotations with the cyclotron frequency.

If we imagine the situations when in the electron beam only one kind of waves is excited consecutively, the beam configura­tions in all such cases will be represented by spirals twisted spa­tially around the - axis (Fig. 4.3). Besides, for cyclotron waves each point (an electron) of such a spiral takes part simultaneously in two motions, namely the rotation around the z-axis at the angular fre­quency  and the movement along the - axis at the velocity . For synchronous waves there is no rotational motion, the elec­trons are spatially shifted with respect to each other, thereby forming a spiral and moving along the - axis only. In all cases the beam trace (the intersection point of the beam at the plane =const) travels around a circle at the angular frequency , with the rotation direction being determined by the type of po­larization.

Thus, unlike waves of the space charge [1,23], in this case the grouping process is connected only with a spatial curving of the electron beam without forming bunches of the space charge.

Fig. 4.3

a)   - Fast forward and backward Cyclotron Waves,

b) - Slow Cyclotron Wave,

c) - Synchronous Waves with right and left polarizations.

(In all cases the internal cylinder on which the electron beam

is wound serves for the illustrative purpose only).

In the presence of an external electric field , the so­lution of the initial system of equations (4.1),(4.2) may also be sought in the form of the sum of four transverse waves having alternating amplitudes

i.e.

Here

Since instead of one pair of variables , two pairs of new variables  are introduced, it is necessary to impose some additional conditions to connect the members of this new pair of variables. It is convenient to use the following

Then

(4.21)

Substituting these expressions in (4.18) and using (4.19) gives

In the two limiting cases that of a small transit angle and that of adiabatic action of electric fields on the electron beam the solution of this system of equations can be simplified sig­nificantly.

In the first case, in the right-hand parts of the system of equations one can assume that

(4.24)

In the latter case, one can average the right-hand part of equa­tions by the period of cyclotron oscillations

Besides, when averaging, the wave amplitudes  should be taken to be constant.

Numerical integration of the system of equations (4.22), (4.23) is usually straightforward and can be made using a personal computer.

In a more general case, when

It is also possible to represent the initial equation of motion in the form

(4.29)

where

is the angular cyclotron fre­quency and the longitudinal component of the magnetic flux density at the system axis, respectively.

The solution can be readily found for the cases when the longitudinal velocity of the electron beam does not acquire modulation in time and dependents on the -coordinate only, i.e. .

Let us introduce  - the electron and cyclotron propagation constants, respectively.

The solution is sought in the form of the sum of two waves with right and left circular polarizations

As before, the right-hand part of equation (4.28) is represented as

Using a procedure similar to that employed above, we get

In the presence of the external fields  therefore the solution will be sought in the form

we obtain the system of differential equations describing the behavior of transverse wave amplitudes

.                                                (4.36)

In this case

Notes:   Applicability of the filamentary electron beam model.

The equations obtained are quite well applied for the elec­tron beam of a finite cross-section, if the external field phase changes negligibly across the beam section

where  is the electron beam radius,  is the propagation constant of the wave of the electrodynamics system.

In practice, however, the requirement  is quite sufficient for most issues.

If a filamentary electron beam with the linear charge density  is placed in the transverse electric field , the power of energy exchange between the electric field and the electron beam element of the length dz is equal to

where  , the sign * denotes the complex-conjugate value.

Besides interaction with the transverse electric field, it is necessary to take into account the interaction of the electron beam with a longitudinal electric field . To determine the value of , we use the quasi-stationary approximation [22] , for which near the -axis

, or                                                                       (5.2)

,   where  .                                    (5.3)

Accordingly, the power of energy exchange with the longitudinal field  has the form

The expression for the transverse velocity can be given in the form

where  means the transverse velocity of the electron beam trace in the plane

Summing up the powers of the longitudinal and transverse energy exchanges, we can write

Now, we can use the representations

and, accordingly

,  .                                                          (5.10)

Substituting (5.7)-(5.10) into (5.6) and averaging the power over the period , we get

Integrating (5.11) over  from 0 to , and taking into account that , we find

The expression between the braces under sing of integral depends only on the coordinate , therefore the following substitution was used

Now, choosing the integration limits outside (before and after) the interaction region, i.e. where the transverse electric field is equal to zero , we have

Let us first consider the simplest case

Then the equations for transverse wave amplitudes have the form (4.36)-(4.38), i.e.,

(5.17)

Besides

We find  from (5.17), (5.18), substitute them into the expression for energy exchange power (5.13) and take into account that

Then assuming that the electron beam has no modulation at the input into the interaction region (i.e. ) we finally get

Thus, the period-averaged power of energy exchange between the electron beam transverse waves and the external electric fields is found equal to

where  are the current and the potential of the electron beam, respectively.

The positive sign of the kinetic power means that the power is added into the electron beam at the excitation of a corresponding wave, and vice versa: the negative sign implies that when a wave is excited the power is extracted from the electron beam.

It is also possible to separate the transverse and longitudinal components of the kinetic power, which are due to interactions with transverse and longitudinal electric fields, respectively. For cyclotron waves the transverse power is associated with rotation of electrons and, hence

then

For synchronous waves the electron beam has no transverse velocity, and hence

Let us consider now a more general case when the static magnetic field varies along the  - axis. In the paraxial approximation we can write

where  is the longitudinal component of the magnetic flux density at the -axis.

In this case the equations of motion of an electron have the form

where   .

Accordingly, the equations for transverse wave amplitudes are written as

Using a similar procedure and substituting  from (5.30), (5.31) into (5.13) and then using (5.20), (5.21), we obtain the same result

of the Injecting Electron Beam

To simplify the procedure of simulation, various configurations of ‘big charged particles’ are usually used to analyze and to solve different issues of vacuum microwave electronics since the middle of 60’s.

Transverse interaction we are discussing is a 3D interaction because of the physical principles of this one and the procedure of computer simulation must be also in a 3D form. Nevertheless, in some cases an essential simplification is possible.

Let us start from the equation of motion of a single electron having number “i

or

Fields are calculated separately

where  and  are the electromagnetic fields induced by electron beam,  and  are the microwave fields and  is the external focusing magnetostatic field, respectively.

Let us assume  and  to be independent variables, thereby making a transition from the description of the motion of one electron to the description of the motion of a filamentary electron beam. Then

in (6.1) and (6.2).

Such filamentary beams can be used to describe the motion of a real beam of a finite cross-section.

** As an example, let us consider the case of the interaction of circularly polarized slow-wave having

and an electron beam of a finite and round cross-section injecting along and coaxially the axis of this wave ( i.e., the  - axes, see Fig. 6.1).

We can choose the form of each elementary (filamentary) beam as a helix inside the injecting electron beam (Fig. 6.1). Let us choose the step of this helix as

where  is the longitudinal velocity of the injecting electron beam,   is the angular frequency of the beam rotation under the inner (radial) Coulomb forces.

Fig. 6.1

Helix-type discretization of the injecting electron beam

For the uniformly charged electron beam

,

where  is the space charge density of the injecting electron beam.

In the case we are discussing, the cross-point of each elementary (filamentary) beam at the plane  is rotating at the angular frequency  and each such beam has stationary (concerning the fields of this circularly polarized wave) boundary (at ) conditions and as a result of it

for any beam and everywhere inside the interaction region ( is the linear charge density of this filamentary beam).

The boundary condition of the uniform charged electron beam are given by

(6.10)

where

,

.

- radius of the injecting electron beam,

- number of elementary beams on each ring,

- number of rings,

- total number of elementary beams ( 50-150 usually ),

,                                                (6.11)

Using continuity equation

one can write for each elementary beam

Thus, firstly:

where ,  are the space charge linear along z-axes density and  the longitudinal velocity of the elementary beam, respectively.

And secondly: the simulation may be fulfilled at any fixed moment of time.  Any changing in time is completely equal to the rotation of the system around z-axis (see Fig. 6.1).  In all equations

for transverse coordinates and  transverse velocities of any elementary beam.

It is important to emphasize that all restrictions on helix-form of each elementary beam are essential only for the injecting beam (). Inside the interaction region (z>0) the motion of each elementary beam can be absolutely arbitrary and the electron beam interaction includes both the transverse grouping and the longitudinal one.

**  It is also important to emphasize that instead of (6.6), any type of electromagnetic fields rotating around -axes at the angular frequency  can be also used here.

Coulomb fields of the electron beam can be found using approximate formulas (see Appendix)

where:

is equal to 1/2 of the min distance between partial beams at .

** As an another example, let us consider the case when the electron beam having some initial rotation around -axes is injected to the region with axially-symmetric and space-variable magnetic (or electric) static field

where is the longitudinal component of the magnetic flux density at the -axis.

Boundary conditions in such case

or

Where are the radius and the phase of the initial rotation of the electron beam, i.e. of each electron of the beam ( at  z = 0  and  t = 0 ).

Therefore, all main consequences of helix-type discretization (6.11), (6.14) and (6.15) exist in this case also.

** We could also consider the case when the electron beam having some initial rotation around -axes is injected into the region with electromagnetic fields rotating around -axes at the angular frequency and so on.

Helix-type discretization of the injecting electron beam has been carefully tested and successfully used by different authors [4-8, 25, etc.].

# VII.  Discussion

In the simplest case transverse waves of the electron beam can be analyzed by comparing them with the space charge waves in order to reveal the potential advantages that could be realized when using transverse grouping in electron beams.

Let us outline some of them briefly:

·        In contrast to space charge waves*, phase velocities of transverse waves are independent of the reduced value of the plasma fre­quency  (which is not constant along and across the electron beam and is also dependent nonlinearly on the signal amplitude) and, consequently, it is now possible to make the synchronism of the transverse wave interaction with external fields incomparably more stable. For the same reason, devices based on interaction with transverse waves must have phase-frequency characteristics with a considerably improved linearity.

·        No restrictions were imposed on the signal amplitude value when the equations for transverse waves were derived. On the other hand, for the space charge waves the restrictions of the modulation depth are essential. Hence, one can also expect a higher linearity of the amplitude characteristics in energy ex­change with transverse waves of the electron beam.

·        Transverse waves of the electron beam are circularly polarized, that provides an additional possibility to select waves taking part in energy exchange, thereby making it possible to improve the quality of interaction.

·        At the point of cyclotron resonance () the phase velocity of the fast cyclotron wave becomes infinite in contrast to phase ve­locities of other waves. Therefore just with this wave in a plane transverse and uniform electrical field of the resonator it is pos­sible to carry out an efficient prolonged (>>1) interaction to en­sure a high efficiency of signal power transmission from the resonator to the wave and vice versa.

·        A slow synchronous wave has a negative kinetic power, a phase velocity equal to the electron beam velocity and is not connected with excitation of transverse velocities of the beam electrons. The interaction with this wave must be more efficient than with the space charge slow wave and provide a single velocity character of the spent electron beam, i.e. it allows an extensive recovery of the spent beam energy. In such energy exchange of the running circularly polarized wave with the field the phase-frequency char­acteristics must have a higher linearity.

·        Some additional information [12-14] is available via Internet :

*) The phase velocities of the fast (+) and slow (-) waves of the space charge are known [1,23] to be equal to: , respectively, where  - is the reduced value of the plasma frequency in the electron beam.

# VIII.  Selected Bibliography

[1]  V.M.Lopukhin et al, “Noises and Parametric Phenomena in Electron Beams”, Nauka, Moscow, 1966. (In Russian).

[2]  V.A.Vanke et al, “Super Noiseless Cyclotron-Wave Amplifiers”, Soviet Physics Uspekhi, May-June 1970, vol. 12, No. 6, p. 743.

[3]  V.I.Yuriev et al, “Experimental Study of the Interaction of Synchronous Waves of the Electron Beam and Travelling Wave of the Electrodynamics System”, Radiotechnique & Electronics, 1972, vol. 17, No. 4, p. 830. (In Russian).

[4]  A.A.Zaitzev, “On the Efficiency of the Interaction of Transverse Waves of the Electron Beam and Electromagnetic Fields”, Ph.D. thesis, Faculty of Physics, Moscow State University, 1979. (in Russian).

[5]  V.A.Vanke et al, “The Results of the Investigations of the Circularly Polarized TWT and their Analysis”, Radiotechnique & Electronics, 1981, vol. 26, No. 11, p. 2365. (In Russian).

[6]  V.A.Vanke, “The Interaction of the Oscillations and Waves of the Electron Beam  and Electromagnetic Fields”, D.Sc. thesis, Faculty of Physics, Moscow State University, 1981. (in Russian).

[7]  V.I.Gorelikov, “The Efficiency of the Excitation and Conversion of Fast Cyclotron Waves in High Power Electron Beams”, Ph.D. thesis, Faculty of Physics, Moscow State University, 1984. (in Russian).

[8]  A.V.Konnov, “The Interaction of Circularly Polarized Electromagnetic Slow Waves and Transverse Synchronous Wave of the Electron Beam”, Ph.D. thesis, Faculty of Physics, Moscow State University, 1988. (in Russian).

[9]  Yu.A.Budzinsky, S.P.Kantyuk, “A New Class of Self-Protec­ting Low-Noise Microwave Amplifiers”, Proc. IEEE MTT-S Microwave Symposium, Atlanta, USA, June 1993, Digest,  vol. 2, p. 1123.

[10]  S.V.Bykovsky, “Theory of the Energy Exchange Processes of Transverse Waves of the Electron Beam and the Development Basing on this Theory of Cyclotron Wave Protectors with Brief Recovery Time of Around 10 ns”, Ph.D. thesis, The Istok Corporation, Fryazino, Moscow region, 1996.

[11]  I.A.Boudzinski et al, “Microwave Devices Using Fast Cyclotron Wave”, Radiotechnique, 1999, No. 4, p. 32. (In Russian).

[13]  Vanke V.A. et al,  “High Power Converter of Microwaves into DC”,  Journal of Radioelectronics, 1999, No. 9:  http://jre.cplire.ru/jre/sep99/1/text.html.

[14]  Vanke V.A. et al,  “Cyclotron Wave Electrostatic Amplifier”,  Journal of Radioelectronics, 1999, No. 10:  http://jre.cplire.ru/jre/oct99/1/text.html.

[15]  V.A.Vanke, “Microwave Electronics Based on Electron Beam Transverse Waves Using” (Invited), Technical Report of the Institute of Electronics, Information and Communication Engineers (Japan), ED99-247, 1999, p. 1.

[16]  I.A.Boudzinski, S.V.Bykovski, “Amplifying and Protective Devices Based on Electron Beam Fast Cyclotron Wave”,  Proc of the 2nd International Vacuum Electronics Conference, April 2-4, 2001, Noordwijk, The Netherlands, p. 153.

[17]  R.Adler et al, “Excitation and Amplification of Cyclotron Waves and Thermal Orbits in the   Presence of Space Charge “, J. Appl. Phys., 1961, vol. 32, No. 4, p. 672.

[18]  V.A.Vanke, V.B.Magalinsky, “On the Role of Space Charge at the Amplification of Intrinsic Noise Orbits in High-Frequency Quadrupole Field”, Higher School Reports, ser. Radiophysics, 1966, vol. 9, No. 5, p. 831. (in Russian).

[19]  P.L.Kapitza, “High-Power Electronics”, Edition of the USSR Academy of Sciences, 1962, vol. 1, ch. 1. (in Russian).

[20]  N.N.Bogolyubov, Yu.A.Mitropol’skii, Asymptotic Methods in the Theory of Nonlinear Oscillations, Fizmatgiz, 1963. (in Russian).

[21]  V.Dubravec, “Wave Theory of Cuccia Couplers”,  Arch. Elektr. Ubertrag., 1964, vol. 18, No. 10, p. 585, (in German).

[22]  V.Dubravec, “General Power Relations of Cyclotron Waves”, Electron. Commun., 1964, vol. 39, No. 4, p. 558.

[23]  W.H. Louisell, "Coupled Mode and Parametric Electronics," John Wiley & Sons, Inc., New York, London, 1960.

[24]  G.A.Sukach, “On the Calculation of the Space Charge Field”, Vestnik KPI, ser.  Radiotechnique & Electronics, 1970, No.7, p. 59. (in Russian).

[25]  A.Kita, “Study of the Conversion Region in CWC via Computer Experiment”, MS thesis, The Graduate School of Engineering, Kyoto University, 1998.  (in Japanese)

# IX. Appendix

(Coulomb Fields)

In many cases

where: ,  are the radius of the electron beam and the radiuses of curvature of the elementary (filamentary) beams, respectively.

In such cases we can simplify the task and calculate the electric field of a charged rod placing it along the tangent of the elementary beam.

Let us consider three coordinate systems (Fig. 9.1)

Fig. 9.1

The - axes is the tangent (charged rod) of the elementary beam having the cross-point coordinate  at the plane .

We can write for the electric field of such charged rod at the point

Where  are the linear charge density of the rod and the dielectric constant, respectively.

Fig. 9.2

In compliance with Fig. 9.1 and Fig. 9.2 one can write the coordinate transformation formulas

and

Using (9.2)-(9.5) one can obtain (6.16).

A small add  in (6.16) is a rational way to avoid mathematical singularities

when  [24].

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