Open access peer-reviewed chapter

Concept and Numerical Simulations of Multi-Beam Linear Accelerators EVT with Depressed Collector for Drive Beam

Written By

Vladimir E. Teryaev

Submitted: 23 May 2017 Reviewed: 13 September 2017 Published: 20 December 2017

DOI: 10.5772/intechopen.70956

From the Edited Volume

Accelerator Physics - Radiation Safety and Applications

Edited by Ishaq Ahmad and Maaza Malek

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Abstract

The concept of the electron multi-beam linear accelerator electron voltage transformer (EVT) invokes aspects of a two-beam accelerator. Here, effective transformation of energy of several drive beams to energy of a high voltage accelerated beam takes place. It combines an RF generator and essentially an accelerator within the same vacuum envelope. Acceleration occurs in inductively-tuned cavities. The ratios between values of RF beam currents and their voltages are similar to ratios for an electric transformer that is leaded to dub this device an electron voltage transformer (EVT). Properties and high efficiency of the EVT accelerator proves to be true by the example of numerical simulations of three accelerator configurations generating electron beams with energies 1, 2 and 4.3 MeV, respectively. The submitted results of numerical simulations show a resource for the further increase in efficiency of the EVT accelerator up to 70–75% by using a depressed collector for a spent dive beam.

Keywords

  • two-beam accelerator
  • electron gun
  • drive beam
  • accelerated beam
  • RF buncher
  • accelerating cavity
  • depressed collector

1. Introduction

Two-beam linear collider can serve as the most known example use of two-beam concept in the field of high energy physics [1]. Derbenev Ya et al. [2] can serve as an example of consideration of the two-beam concept in the range of low energy (1–10 MeV). Specificity of low energy when the relativistic factor is low is that a change of velocity of particles for conservation of the appropriate phase ratio between drive bunches and accelerated bunches, i.e. for keeping of synchronism in process of beam-beam interactions, shall be taken into account. Derbenev Ya et al. assumed that the use of the H 020 mode in a pill-box cavity where acceleration occurs on an axis of the cavity and the annular drive beam changes its radius during movement [2]. Synchronization is carried out due to change in a focusing magnetic field along the axis of the device and this implies changing of the radius of the annular drive beam. However, this concept has never been developed any further.

Here, a new concept of the electron accelerator briefly named electron voltage transformer (EVT) is considered, where the multi-beam approach is of a crucial importance. The approach to use several beam-lets interacting with the common cavity is well-known and is implemented successfully in multi-beam klystrons (MBK’s) [3, 4]. Such design is also inherent to the greatest degree in the concept of EVT. All beams in EVT are generated in a common electron gun. Then, they are bunched in RF cavities as in a klystron. The energy is transferred from the drive bunches to the accelerated bunch in the accelerating structure. Each cavity of the structure is coupled to all beams, but the cavities are not coupled one to another.

Basic difference from the concept quoted above [2] is that axes of all beams are parallel straight lines and that for beams focusing a homogeneous magnetic field of a solenoid is used. The cavities in EVT are inductively detuned, having the resonant frequencies of their working modes being higher than the operating frequency. As a result, the phase angle between the gap voltage of the cavities and the fundamental harmonic of the drive beam current is close to 90°, as against conditions for the accelerated beam where the phase angle between accelerating gap voltage and a harmonic of a current is close to zero. This operating regime allows one to achieve the desired value of the accelerating field. The most important advantage of using inductively tuned cavities is effective longitudinal bunch focusing, which is exerted by the cavity RF fields of the drive bunches. The drive beam-lets in EVT are weakly relativistic. Therefore, synchronous energy transfer from the drive beams to the accelerated beam is achieved by correct choices of distances ∆z between the adjacent cavity gaps. A similar condition, of two-beam acceleration in inductive tuned cavity, is considered by Kazakov et al. [5], where, in contrast to EVT, the authors have placed the drive beam and the accelerated beam on the same axis. Dolbilov G has discussed a two-beam accelerator of protons, where the use of inductive tuned cavities for an electron drive beam is considered [6, 7]. The accelerated beams and the drive beams move there in opposite directions in order to maintain the condition of synchronization.

Preliminary results of numerical simulations using the known and reliable code MAGIC [8] are presented for an EVT prototype, which operates in S-band, where acceleration up to 1 MeV was predicted at efficiency of 66% [9]. A project version of an EVT accelerator operating at L-band, in which a 6-A, 110-kV beam is accelerated to 2 MeV with an overall efficiency of 60%, is presented below. The further development of the EVT concept is based on a principle of two-stage acceleration, where an ensemble of the drive beam-lets is consisting of two groups. At the first stage, the first group of drive beam-lets transfers their energy to the second group of drive beam-lets. At the second stage, the second group of drive beam-lets transfers their energy to the accelerated beam. Such configuration allows a transformation ratio and energy of the accelerated beam to be increased considerably. Numerical simulations predict acceleration of an initial 1 A, 60 keV beam up to energy of 4.3 MeV at efficiency of 22%. The use of a novel view on a Depressed Collector (DC) for a spent drive beam, where the DC is considered as a separator of a spectrum of particles, increases efficiency in the first two EVT projects up to 74 and 68%, respectively. Numerical results are shown in Figures and Tables.

A positive feature of the EVT concept is that the accelerator does not require an external high power RF-source, but only a 60–110-kV pulsed source for operation. The duty cycle for the device will be easily adjusted via the pulse duty cycle of the power supply. The absence of an RF-source, utilization of a relatively low-voltage pulsed power supply to drive the machine, and the compact size of the device would ensure a relatively light weight and highly versatile applied accelerators.

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2. EVT configuration

The schematic representation of an EVT accelerator is shown in Figure 1 , where the accelerating structure is pop up.

Figure 1.

Schematic representation of an EVT (left); shape of the accelerating cavity and electric field E z along x-axis (right).

An electron gun, a magnetic focusing system and a RF buncher will be provided below in the description of specific projects of EVTs. Figure 1 (right) provides an example of the accelerating cavity. The operating mode is TM 020. An accelerated beam is located in the centre, and a drive beam is located on the periphery, at a peak value of an electric field. The drive beam consists of six beam-lets in the case shown. An additional passive ring-shaped bunching cavity with the fundamental TM 010 mode is located in a long drift tube. Synchronous energy transfer from the drive beams to the accelerated beam can be achieved through correct choices of distances ∆z1 and ∆z2 between gaps of adjacent accelerating cavities.

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3. Scientific and technical justification of the EVT concept

It is known that a field E r t in a volume Ω of a cavity generated by a beam with a current density J r t can be presented as eigenmode expansion: E r t = λ e λ t E λ r , where eigen vectors E λ r are functions of coordinate r that are normalized by energy of the cavity as follows:

W λ = ε 0 2 Ω E λ r 2 d Ω . E1

A time-dependent multiplier e λ t is the solution of the equation of oscillator with the second member, Eq. (2):

d 2 e λ d t 2 + ω λ Q λ d e λ dt + ω λ 2 e λ = 1 2 W λ d dt Ω J ̇ E λ E2

Let’s express Eq. (2) in a complex form and consider one of the eigenmodes only, having frequency ω 0 , replacing: e λ t e ̇ 0 exp iωt , J r t J r exp iωt . It gives

e ̇ 0 = i ω ω 2 ω 0 2 i ω ω 0 / Q 0 1 2 W 0 Ω J ̇ r E 0 r E3

Other terms of expansion generated by the beam current will have small values at high quality Q 0 of the cavity and at ω ω O , and they will not be taken into account. Eqs. (2) and (3) are the most general expressions for determination of fields in cavities [10]. As it takes place ω ω O , the equation Eq. (3) can be rewritten as:

e ̇ 0 = exp Q 0 1 + ξ 0 2 1 2 ω W 0 Ω J ̇ E 0 E4

where ξ 0 is a relative detuning:

ξ 0 = Q 0 ω 2 ω 0 2 ω ω 0 2 Q 0 ω 0 ω ω 0 = ω ω 0 φ = tan 1 ξ 0 E5

Practically, the cavity is described via integral characteristics, such as geometric impedance of drive gap (index 1) and accelerated gap (index 2):

ρ 1 = R / Q 1 , ρ 2 = R / Q 2 ; E6

circuit current I ̇ 1 of the drive gap and circuit current I ̇ 2 of the accelerating gap equal to

I ̇ 1 = exp 1 T G 1 2 F B 1 I 01 , I ̇ 2 = exp 2 T G 2 2 F B 2 I 02 ; E7

RF beam currents:

I RF 1 = 2 F B 1 I 01 , I RF 2 = 2 F B 2 I 02 ; E8

voltage V ̇ 1 of the drive gap along axis z 1 and voltage V ̇ 2 of the accelerating gap along axis z 2 :

V ̇ 1 = e ̇ 0 V 1 = e ̇ 0 z 1 E 0 z dz , V ̇ 2 = e ̇ 0 V 2 = e ̇ 0 z 2 E 0 z dz E9

where: φ 1 and φ 2 are phase angles; T G 1 , T G 2 are transit time factors of the drive and accelerating gaps, respectively; F B 1 , F B 2 are form-factors of bunches; I 01 , I 02 are average drive beam currents of the drive and accelerated beams.

For relativistic beams, when it is possible to neglect relative moving of particles inside a bunch, and taking into account that the integral on the second member of Eq. (2) is not equal to zero in gap 1 and gap 2 only where beam currents exist, the Eq. (2) is rewritten as follows:

e ̇ 0 = exp Q 0 1 + ξ 0 2 1 2 ω W 0 I ̇ 1 V 1 + I ̇ 2 V 2 . E10

Let us add in relationships under consideration

K V = V 2 / V 1 = ρ 2 / ρ 1 , E11

following from: 2 ω W 0 = V 1 2 / ρ 1 = V 2 2 / ρ 2 .

The description of the equivalent circuit for a case of interaction of relativistic beams with the cavity is

V ̇ 1 = exp Q 0 ρ 1 1 + ξ 0 2 I ̇ 1 + K V I ̇ 2 = Z C I ̇ E12
V ̇ 2 = K V V ̇ 1 E13

where Z C is impedance of the drive gap, φ is a phase angle between gap voltage V ̇ 1 and equivalent circuit current I ̇ . Eq. (13) shows properties of the chosen operating eigenmode of a cavity when phases of fields in accelerating and drive gaps are shifted by angle π .

At interaction of a cavity with low relativistic beam it is necessary to consider also beam-loading effect or a beam admittance Y B = G B + i B B . Therefore, Eq. (12) is revised as

V ̇ 1 = 1 1 / Z C + Y B I ̇ 1 + K V I ̇ 2 E14

Let’s set out well-known analytical formulas for a rectangular bunch with angular length ψ B and for the flat cavity gap having a transit time angle ψ C :

T C = sin ψ C / 2 ψ C / 2 ; F B = sin ψ B / 2 ψ B / 2 ; E15
G B = I 0 V B 1 2 T G 2 1 ψ C 2 ctan ψ C 2 ; B B = I 0 V B sin ψ C 1 + cos ψ C ψ C / 2 ψ C 2 , E16

where I 0 is an average beam current, V B is an average beam voltage in the corresponding gap (drive or accelerating), T G is the corresponding transit time factor and F B is a form-factor of the corresponding beam.

3.1. Conditions for preset drive bunching maintenance

As it is known, a bunching cavity has inductive detuning, i.e. ω < ω 0 , at which the phase angle between a gap voltage and an equivalent current of a cavity circuit becomes close to π / 2 . Figure 2 shows mutual position of bunches, currents and gap voltages on a phase plane in an accelerating cavity. The phase position of a head H ̇ head and tail T ̇ tail of a drive bunch is shown here, too. Let’s notice, that complex RF beam current I ̇ RF 1 is displaced at angle π relative to the circuit current I ̇ 1 .

Figure 2.

Electric field in cavity gaps (left) and position of complex values on the phase plane (right), providing phase focusing of a drive bunch and acceleration of the beam.

3.2. An impedance of the drive gap loaded with the accelerated beam

As it is evidently from Figure 2 , suitable conditions for an accelerating cavity shall be as follows: φ π / 2 , exp i . It is the chosen phase of drive current: φ 1 = 0 . The phase of the accelerated beam current equal to φ 2 π / 2 provides a bunch being in the electric field close to the peak value, i.e. I ̇ 1 = I 1 and I ̇ 2 = i I 2 . Thus, using Eqs. (12) and (13), the required impedance of the loaded drive gap is

Z 1 = V ̇ 1 I ̇ 1 = Q 0 ρ 1 1 + ξ 0 2 K V K Tr + i E17

where K Tr = I 1 / I 2 is a transformation ratio. An impedance of the accelerating gap is similarly found:

Z 2 = V ̇ 2 I ̇ 2 = Q 0 ρ 1 1 + ξ 0 2 K V K Tr i K V 2 E18

On the other hand for the equivalent circuit of the cavity the impedance Z 1 is given as:

Z 1 = r + i x = Q load ρ 1 1 + ξ load 2 1 i ξ load E19

where Q load and

ξ load 2 Q load ω load / ω E20

are parameters of the equivalent loaded cavity circuit. Equating the real and imaginary parts of Eqs. (18) and (19), the loading quality Q load and relative detuning ξ load can be represented as follows:

Q load = Q 0 1 + ξ 0 2 K Tr K V 1 + K V 2 K Tr 2 = Q 0 1 + ξ 0 2 K Tr K V ξ load = K Tr K V E21

Considering Eq. (21) together with Eq. (20) leads to:

ω load ω = 1 + ξ 0 2 2 Q 0 1 1 + K V 2 / K Tr 2 ω 0 ω E22

where a high value of detuning ω 0 is suggested, which results in 1 + ξ 0 2 ξ 0 . In addition, it is assumed that the transformation ratio is large enough, i.e., K V 2 / K Tr 2 1 . For example, K V 2 , K Tr 20 , ω 0 / ω 10 3 , Q 0 8000 , ξ 0 16 .

Eq. (22) shows that loading of the accelerated beam practically does not change the imaginary part of the impedance of the cavity, and, thus, it is possible to model loading as a certain equivalent external load quality. This feature is used below in the description of numerical model, see Section 4.

3.3. Voltage transformation ratio

Let’s find the voltage V B 1 lost by the drive beam and the voltage V B 2 added by the accelerated beam, which are connected to a voltage V 1 of the drive gap and a voltage V 2 of an accelerating gap as assigned at the corresponding moment of time (see Figure 2 ):

V B 1 T G 1 V 1 , V B 2 T G 2 V 2 E23

where transit-time factors T G 1 , T G 2 are determined the same as in Eq. (7). Let us find voltages V 1 and V 2 using Eqs. (12) and (13) as follows:

V 1 = R e Z 1 I 1 = Q 0 ρ 1 ρ 2 1 + ξ 0 2 I 2 V 2 = R e Z 2 I 2 = Q 0 ρ 1 ρ 2 1 + ξ 0 2 I 1 E24

where the transformation ratio K Tr is:

K Tr = I 1 I 2 = V 2 V 1 E25

In view of Eqs. (7), (8), and (23), the transformation ratio can be written through the RF beam currents as:

K Tr = I RF 1 I RF 2 V B 2 V B 1 , E26

or, at a high value of transit time factors T G 1 , Eq. (26) can be written as

K Tr = I 01 I 02 , E27

where I 0 is the average current of the corresponding beam. It reflects the law of conservation of energy for the idealized conditions applied in the beginning of section 3.3 and corresponding to small resistive loss in a cavity. In view of losses it is resulted:

K Tr = I 1 I 2 = V 2 V 1 1 η B E28

where η B is efficiency of power transfer, defined in the Section 3.4.

3.4. Efficiency of power transfer between beams and other parameters of the EVT

From conservation of energy, the efficiency of power transfer from one beam to another can be estimated as:

η B = V 2 I 2 V 2 I 2 + V 2 2 / 2 R s 2 E29

where R s 2 = Q 0 ρ 2 is the shunt impedance of the accelerating gap. For achieving a high efficiency of energy transfer, it is important to have a large enough current I 2 of the accelerated beam, i.e. I 2 V 2 / 2 R s 2 .

Based on the above consideration and Figure 2 , it can see that in order to reach the effective bunching of drive beam, with an angle between the current I ̇ 1 and the voltage V ̇ 1 close to π / 2 , two conditions shall be satisfied. The first is to have a large enough value of the cavity detuning ω 0 ; the second is I 1 / K V I 2 . The angle between I ̇ 1 and I ̇ 2 should be close to π / 2 for effective acceleration since the accelerated bunch should pass through an accelerating gap at the time when the voltage on the gap is near its maximum. Thus, conditions

I 1 / K V I 2 V 2 / 2 R s E30

are the necessary factors of effective performance of the EVT, which are readily achieved when multiple drive beam-lets are used. For example, I 1 = 60 A , K V = 2 , I 2 = 3 A , V 2 = 80 kV , V 2 = 100 kV , Q 0 8000 , ρ 2 = 60 ohm , η B = 0.95 .

We define the efficiency η of the accelerator as a ratio of the power of the accelerated beam at the EVT output to the total beam power in the gun, namely:

η = P Accel I Gun V Gun E31

where I Gun is the total current of the gun, and V Gun is the gun voltage. The high EVT efficiency is confirmed by the results of numerical simulations, as shown in Table 1 .

Multi-beam linear accelerator S-band L-band
Operating frequency, F 2856 1300 MHz
Voltage of common electron gun 60 110 kV
Output energy of accelerated beam 1.1 2 MeV
Total gun current of drive beam 72 180 A
Gun current of accelerated beam 3 6 A
Focusing magnetic field in solenoid 800 1000 G
Energy spread of accelerated beam ±10 ±10 %
Micro-perveance of one drive beam-let 0.816 0.82 μA/V3/2
Transmission of drive beam 0.99 0.99
Transmission of accelerated beam 0.90 0.87
Transformation ratio 24 30
Voltage gain factor ≈18 ≈18
Typical R/Q factor of accelerating gap 58 54 Ω
Typical cavity detuning, ∆F = F0 - F 8 ÷ 12 7 MHz
Typical KV factor of gaps of cavity (Eq. 11) 4.5 5
Efficiency of transfer of power (Eq. 29) 94 97 %
Number of beam-lets of drive beam 6 6
Number of cavities of RF bunchers 4 4
Number of accelerating cavities 14 12
Number of passive bunching cavities 1 7
Length of RF part of accelerator 0.79 2.4 m
Efficiency of accelerator EVT ≈66 ≈60 %
Efficiency of EVT with DC for drive beam 72 66 %

Table 1.

The simulated parameters of the S-band and L-band linear accelerators EVT.

We define a voltage gain factor to be a ratio of the voltage of the accelerated beam at the EVT output (the final beam energy) to the gun voltage. It is:

K EVT = V Accel / V Gun 1 + K Tr η E32

where η and K EVT are shown in the Table 1 , as a result of numerical simulations.

3.5. Condition of synchronization and phasing of bunches

The drive beam-lets in the EVT are essentially low relativistic. Therefore, a synchronous energy transfer from the drive beams to the accelerated beam is achieved through correct choices of distances ∆z between the adjacent accelerating cavities gaps. In case of equality of distances between gaps shown in the Figure 1 , a condition of synchronism is as follows:

z = z 1 = z 2 = 2 πc ω β 1 β 2 β 1 β 2 E33

where β 1 c and β 2 c are velocities of the drive and accelerated bunches in space z between cavities gaps. Phasing capabilities considerably extend if distances z 1 and z 2 between a gaps of adjacent cavities are not equal to each other. More general Eq. (34) is corresponding to this case:

z 1 β 1 z 2 β 2 = N λ ; N = 0 ; 1 ; 2 ; ; λ = 2 πc / ω E34

It is supposed to use RF bunchers of klystron type for generation of drive bunches. Further bunches should be decelerated in cavities, gradually giving the energy to the accelerated bunches. There is a reasonable question for how long the drive bunches can keep their form and phase length. Results of 2D simulations show long enough conservation of the generated bunch (see below projects of EVT). The similar research confirming these results can be found in [6, 7].

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4. Representation of 3D objects via equivalent 2D objects

3D codes have been used for simulation of elements of the EVT (cavities, magnetic system), but for the analysis and optimization of dynamics of ensemble of the interacting beams moving in numerous drift tubes the direct and slow 3D analysis is too intricate problem. Difficulty is aggravated also with necessity of carrying out numerous optimizations.

The drive beam-lets are located on a circle, see Figure 1 (right), and it can be quite replaced with single equivalent 2D annular beam. It is possible to choose the corresponding width of the ring filled with a beam, providing close values of plasma wave-lengths of 2D and 3D models of the beam. In the same way it is replaced real 3D cavities with equivalent 2D cavities. There are chose parameters (gap, R/Q) of interaction of a 2D beam with equivalent 2D cavities similar to interaction of a 3D beam with 3D cavities. Certainly, such equivalent model does not describe exact dynamics of the drive beam, but reflects quite well the properties of the EVT accelerator and can serve as initial approach. The results of parameter optimization for the EVT accelerators using the 2D code MAGIC [8], are provided in Sections 5–7.

It is obvious that, on the one hand, implementation of a pure annular concept of the drive beam is embarrassing technically, and, on the other hand, the annular model is rather rough to describe dynamics of a set of cylindrical drive beam-lets. This simulation is provided here to illustrate a method of acceleration used in the EVT as it correctly represents interaction of two electron beams without any additional assumptions.

The following, more adequate model is appropriate for description of a 3D problem, which can be named 2 × 2D model. This model is based on thesis about an impedance of the loaded drive gap considered in Section 3.3 and on the following circumstances:

  1. Each of beam-lets and electric RF fields of cavities near to a beam-let can be considered as axially symmetric relative to a local axis of the given beam-let;

  2. Parameters of drive beam-lets are identical;

  3. Accelerating cavities are weakly loaded, and, therefore, RF fields of cavities are rather close to a field of an operating RF eigenmode. I.e. the loaded quality of a cavity remains high enough. In other words, the RF power circulating in a cavity P RF = V 1 2 2 R / Q 1 is much higher than power P 2 = P output 2 P input 2 transferred to the accelerated beam, for example, Q load = P RF / P 2 100 .

According to these points, it is considered a 3D problem as two connected 2D problems, see Figure 3 . The drive beam goes through the first equivalent 2D cavity, having the R/Q of drive gap equal to R/Q of corresponding 3D cavity. This cavity has an RF load in which RF power P load = P drive is absorbed.

Figure 3.

Two 2D cavities, the parameters of which are related via the shown relationships, model processes in a 3D cavity.

The accelerated beam goes through the second cavity having a gap voltage V gap 2 and a phase angle φ gap 2 , which are calculated based on a voltage V gap 1 on the gap of the cavity 1 for the drive beam and where power of the accelerated beam increases at the value P 2 = P output 2 P input 2 . Positions of gaps of these two equivalent cavities correspond to the position of gaps of the 3D cavity. Combination of these two processes will consist in maintenance of equations:

P 2 = P accel = 6 P drive P loss V gap 2 = K V V gap 1 φ gap 2 = φ gap 1 + π E35

where N = 6 is the number of drive beam-lets, P loss is resistive loss in the 3D cavity. Eq. (35) is carried out by choosing the corresponding conductivity of an external RF load of the first (drive) cavity; the value K V = V gap 2 V gap 1 3 D is defined (see Eq. 11) upon the results of the 3D cavity eigenmode analysis. Numerical simulations using the code MAGIC show fast convergence of the Eq. (35) process. 2–3 iterations are practically enough.

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5. Design of S-band 1 MeV linear accelerator EVT

The design of a linear accelerator EVT operating in S-band (F = 2856 MHz) and generating 1 MeV electron beam is presented in this section. One accelerated beam and 6 drive beam-lets are generated by a common electron gun fed from a 60 kV power supply. The magnetic lenses around the gun and two matching lenses ensure formation of beams without ripples in the same way, as it is described in Ref. [3]. Bunches are formed in the bunching section energized by two coherent microwaves via RF couplers. The generated drive beam-lets interact with the accelerated beam in the cavities of the energy transfer section (accelerating structure), with the cavities working at the TM020 mode. For a layout of the S-band EVT, please see Figure 4 . The results of the initial numerical analysis using code MAGIC are given below in Table 1 . Two models described above in Section 4 have been applied for the analysis. The models have shown close results. Beam dynamics of the annular model is more evident and results in Figure 5 . Figure 6 shows energy distribution of particles of beams in the 1st accelerating cavity (z > 314 mm) and at the output of the accelerating structure (z > 787 mm). Energy of 1.1 MeV, averaged on the size of a bunch, is achieved. You can see disruption of the spent drive bunches at the accelerator output. The achieved 66% efficiency is twice the best efficiency accessible for the facility consisting of a high power klystron and a high efficient linear accelerator [11].

Figure 4.

Lay-out of the accelerator EVT.

Figure 5.

The geometry of a compound RF buncher, accelerating structure and beams dynamics are shown for the S-band EVT.

Figure 6.

Particle energy distribution in the 1st accelerating cavity (left) and at the output of the accelerating structure (right).

5.1. Features of the multi-beam electron gun

The radius of a ring on which drive beam-lets are located is defined by a frequency and configuration of the cavities (see Figure 1 (right)). Six cathodes with a current of 12 A are freely located on this radius at the gun voltage 60 kV, Figure 7 . Clearly, a drive beam-let perveance should not be too high to not interfere with phase stability and synchronization of bunches.

Figure 7.

The multi-beam cathode of the electron gun. The peripheral cathodes generate the drive beam-lets, and the central cathode generates the accelerated beam.

Distance between the drive beam-let cathodes is big enough in comparison with distance between the cathode and anode therefore transverse electrostatic fields on an axis of drive beam-lets are insignificant. Thus the electron optics of each of beam-lets can be quite considered as axial symmetric. Figure 8 shows the 2D geometry and an example of simulation of a drive beam-let. The double lens around the gun allows obtain the desired profile of the magnetic field to focus the beam-lets and minimize their ripples.

Figure 8.

The geometry and an example of 2D simulation of the drive beam-let gun.

Cells of magnetic lenses are formed by flat iron pole-pieces and system of apertures (see Figure 4 ). Properties of cells for drive beam-lets are identical. The sizes of apertures and distances between them are such that there are no appreciable transverse fields generated on an axis of beam-lets. Thus, the magnetic field quite possesses axial symmetry near the beam-let axis.

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6. Specification of L-band 2 MeV linear accelerator EVT

The S-band EVT project has shown achievement of a high efficiency, but has simultaneously shown limited capabilities for an acceleration rate and for a factor of transformation as decay of bunches after N cav = 15 cavities has been observed. Numerical results show difficulty of bunch confinement at a voltage of a drive gap V gap 1 > 1 2 V gun . It results in the following limiting estimated condition:

V Accel < 1 2 K V N cav V gun , E36

where K V is from Eq. (11). This estimation is close to observable result V Accel = 1.1 MeV achieved in the S-band EVT.

Thus, an advance in a path of higher voltage of the gun and a low frequency is meant to increase a voltage of an accelerated beam in the described L-band EVT. Its layout does not differ from Figure 4 . Numerical simulations have shown feasibility of reaching the energy of the accelerated beam about 2 MeV with efficiency about 60% at a gun voltage equal to 110 kV. More detailed data of the project are submitted in Table 1 . The accelerating structure includes 12 accelerating cavities and 7 passive cavities (see Figure 9 ). The code MAGIC 2D and both models described in Section 4 have been applied for the analysis. For the results of beam dynamics of the annular model, please see Figure 9 , too. Figure 10 shows particle energy distribution.

Figure 9.

Geometry of the RF structure and the result of numerical simulations are shown for L-band EVT. For more details on the dynamics of the beams, please see below.

Figure 10.

Particle energy distribution at the output of the 1st accelerating cavity (left) and at the output of the accelerator section (right).

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7. Two-stage multi-beam linear accelerator EVT

The further increase in energy of the accelerated beam will consist in increase in a voltage of a drive beam. It is suggested to consider two-stage acceleration. Let’s assume that the set of the drive beam-lets consists of two groups. At the first stage, the first group of drive beam-lets that form drive beam 1 transfer their energy to the second group of beam-lets that form drive beam 2. At the second stage, the second group of drive beam-lets transfers their energy to the accelerated beam. Such configuration is allowed to increase a transformation ratio and energy of the accelerated beam considerably. Preliminary results of numerical simulations of two-stage EVT operating in S-band with a 60 kV gun and generating a 1 A, 4.3 MV beam at its output, with an efficiency of 22%, are presented. See Figure 12 and Table 2 .

Operating frequency 2856 MHz
Voltage of common electron gun 60 kV
Output energy of accelerated bunch 4.3 MeV
Average energy of accelerated beam 3.6 MeV
Gun current of accelerated beam 1 A
Total gun current of drive beams 252 A
Solenoid magnetic field 1000 G
Length of RF part of accelerator 1.9 m
Transmission of accelerated beam 90 %
Total efficiency of two-stage EVT ≈22 %
Stage 1st 2nd
Gun current of drive beam 216 36 A
Perveance of drive beam-let 0.41 0.41 μA/V3/2
Transformation ratio 6 36
Voltage gain factor of stage ≈3.5 ≈21
Number of drive beam-lets 36 6
Transmission of drive beam 98 90 %
Efficiency of stage ≈55 ≈45 %

Table 2.

Simulated parameters of the two-stage EVT.

Figure 11 (left) shows a multi-beam cathode system. 36 peripheral cathodes generate the drive 1 beam-lets, and 6 cathodes generate the drive 2 beam-lets. The central cathode generates the accelerated beam. A rather low perveance (0.41 μA/V3/2) of drive beam-lets is chosen. It is allowed to simplify the magnetic system and electron optics of the gun. The drive beam is frozen in a homogeneous magnetic field of the focusing solenoid. You can see the presence of ripples having an allowable value in Figure 11 (right).

Figure 11.

Multi-beam cathode system of the two-stage EVT (left) and an example of 2D simulation of the drive beam-let gun (right).

MAGIC 2D simulations of beam dynamics were carried out using an annular model of beams and cavities. Please see the results of numerical simulations in Figures 12 14 and in Table 2 .

Figure 12.

MAGIC 2D results of simulation of beam dynamics in the bunching part and at the 1st stage.

Figure 13.

Accelerating structure of 2nd stage. Dynamics of drive 2 and accelerated beams.

Figure 14.

Energy distribution at the output of the two-stage EVT.

Figure 14 shows particle energy distribution at the accelerator output. For the change in beams energy vs. distance, please see Figure 15 .

Figure 15.

Change of beams voltage along the accelerator axis.

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8. Depressed collector (DC) for spent drive beam

The efficiency of 66% achieved in preliminary numerical simulations of the S-band EVT is a rather high value. The further increase in efficiency of the EVT accelerators is possible at the use of a depressed collector. It is suggested considering a novel concept of a DC, which allows theoretical estimation of DC properties. The given concept is based on the thesis of existence of an extremum of the DC efficiency versus DC voltage. The concept does not struggle with the energy distribution of electrons having a place after the device output, but uses properties of energy distribution. Let us name this concept a depressed collector with a spectra separator, DC&SS. The schematic circuit of the tube with a one-stage DC is shown in Figure 16 (left). Table 3 shows the basic relationships describing the circuit of the EVT with a DC.

Figure 16.

Schematic circuit of a tube with depressed collector. One-stage DC (left); two-stage DC (right).

Beam voltage VB
Voltage of stage VC , V1 , V 2
Current of potential part of stage IC , I1 , I2
Beam power PB = IBVB
RF power, transfer to accelerated beam PRF
Loss power in ground part of collector PG
Loss power in potential part PC , PC1 , PC2 ,
EVT electron efficiency η e = PRF/PB
Beam power at output of tube POut = (1- η e)PB
Recovery power PR = VC IC
Total power generating by high-voltage source Pt = VB IB - VC IC
Efficiency of depressed collector η c = PR/POut = VC IC /POut
Total tube efficiency with DC η t = PRF /Pt  = PRF /(PB - VC IC )

Table 3.

Notations and relationships describing a DC.

Substituting the above resulted relationships, it is derived the total tube efficiency with a DC:

η t = η e 1 1 η e η c E37

Eq. (36) is well known (see, for example [12]), but we shall regard it now jointly with the presence of an energy spectrum of particles leaving a tube. Variable I c = I c V c is remarkable since it is a function of a collector voltage as a consequence of the presence of energy distribution. Our subsequent problem is to study the function η c V c in order to find an extremum at some assumptions concerning the energy distribution j V , which is normalized on the beam current I B :

0 j V dV = I B E38

In the gap of the separator, the beam is divided into low-voltage and high-voltage fractions. The current

I c V c = V c j V dV E39

that has reached the potential part of the collector is the fraction of the current at which V > V C . In the same way, it is written down the equation for the beam power leaving the accelerating structure:

P out = 0 V j V dV E40

Thus, it is evaluated the fundamental formula Eq. (41) for efficiency η c of a depressed collector included in the DC&SS concept:

η c V c = V c V c j V dV 0 V j V dV E41

Figure 6 (right) above shows a numerical example of the energy distribution for the spent drive beam of the S-band EVT versus distance along beam axes. Figure 17 shows the energy distribution of the same beam. The gamma distribution x e x (curved line) can serve as a close approximation for numerical distribution.

Figure 17.

A numerical simulated histogram of particle distribution vs. voltage and an approximating gamma distribution are shown for the S-band EVT.

Thus, let us approximate j V by the gamma distribution:

j V = I B V 0 x e x E42

where x = V / V 0 , V 0 = V B 1 η e / 2 . Further, it is defined I c according to Eq. (39):

I c x c = I B e x C x C + 1 , where x c = V c / V 0 , and it is defined P out according to Eq. (40):

P out = 2 I B V 0 E43

Substituting the found expressions in Eq. (41), it is obtained:

η c x c = 1 2 x c e x c x c + 1 E44

The function η c x c has the maximum equal to 0.42 at x c = 1 + 5 2 = 1.618 .

Thus, it is derived the collector efficiency and other parameters at the optimum point:

η c = 0.42 ; I c = 0,519 I B ; I G = 0,481 I B ; V c = 1.618 V 0 . E45

As an example of the S-band EVT ( Table 1 ), one have:

V B = 6 10 4 ; η e = 0.66 ; V 0 = V B 1 η e / 2 = 1.02 10 4 ; V c = 1.65 10 4 . E46

The total efficiency (according to Eq. 37) at the optimum point is η t = 0.77 , which is the upper theoretical limit!

For a DC having m stages i.e. m gaps and consequently m gap voltages (see Figure 16 ), efficiency is as follows:

η t = η e 1 1 η e η c η c = n = 1 m η n E47

where η c is DC efficiency in multi-stage case. The collector efficiency η n of the nth stage is calculated according to the Eq. (41), where the lower limit of integration is equal to potential U n = i = n m V i of the given stage concerning the ground. This Eq. (48) defines the concept of a DC&SS in multi-stage case:

η n V 1 V 2 V n = V n U n j V dV 0 V j V dV E48

Using the approximation (see Eq. 42) for beam distribution j V , leads to:

η n = 1 2 x n 1 + i = n m x i e i = n m x i η c = n = 1 m η n E49

where: x i = V i / V 0 , x n = V n / V 0 , V 0 = V B 1 η e / 2 , V i is gap voltage of the i th stage, V B is beam voltage. For a schematic circuit of a tube with a two-stage DC, Figure 17 (right), the total collector efficiency will be as follows:

η c x 1 x 2 = η 1 + η 2 = 1 2 x 1 1 + x 1 e x 1 + x 2 1 + x 1 + x 2 e x 1 + x 2 E50

The extremum of this function is equal to η c = 0.586 at: x 1 = 1.118 ; x 2 = 1.398 .

At η e = 0.66 , according to Eq. (37), it will correspond to the total tube efficiency η t = 0.824 .

Table 4 shows the results of efficiency η c optimization for a collector that contains one to six stages.

m x 1 x 2 x 3 x 4 x 5 x 6 η c
1 1.618 0.420
2 1.118 1.398 0.586
3 0.884 0.941 1.319 0.676
4 0.743 0.732 0.876 1.276 0.734
5 0.648 0.608 0.675 0.841 1.249 0.773
6 0.578 0.526 0.558 0.645 0.819 1.229 0.803

Table 4.

The optimization result for the function η c x 1 x 2 via its parameters x 1 , x 2 , .

The theoretical premise of an increase in efficiency of a device with a multi-stage DC are rather optimistic. However, simulations of beam dynamics in conditions close to real do not show any noticeable increase in the device’s efficiency while engineering complexity of such multi-stage DC tangibly increases. Thus, use of a one-stage DC is, perhaps, more preferable in practice.

The analytical treatment described above is rather idealized as it does not include the beam particle movement dynamics; however, it shows parameters to which there can try to approach. The result of initial simulation of a one-stage DC for a spent drive beam of the S-band EVT is shown in Figure 18 . No special optimization of the DC geometry and the magnetic field shape was carried out. Secondary emission was not taken into consideration either. The annular 2D model of the drive beam has been used.

Figure 18.

Geometry of a one-stage DC for the S-band EVT. A picture of particles energy is added at the bottom of the figure.

The results of initial simulations have shown a DC efficiency about 25%, which increases the total efficiency of the S-band EVT up to 72%. It certainly is substantially lower than the idealized theoretical results of the previous section and allows further optimization.

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9. Conclusion

Theoretical and numerical research of properties of the linear accelerator EVT has shown that there is sufficient comprehension about it properties. There is no visible vagueness in technical problems. Certainly, the EVT accelerator does not show a record rate of acceleration; however, it shows that to reach a high efficiency of 70% or more is possible. It is much higher than the total efficiency of a klystron loaded with an ordinary linear accelerator. The two-stage configuration of the accelerator considerably increases the transformation ratio, but the accelerator’s efficiency decreases; therefore, further optimization of this configuration is preferable. Thus, there are all preconditions in place to proceed with the engineering design.

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Written By

Vladimir E. Teryaev

Submitted: 23 May 2017 Reviewed: 13 September 2017 Published: 20 December 2017