The simulated parameters of the S-band and L-band linear accelerators EVT.
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.
- two-beam accelerator
- electron gun
- drive beam
- accelerated beam
- RF buncher
- accelerating cavity
- depressed collector
Two-beam linear collider can serve as the most known example use of two-beam concept in the field of high energy physics . Derbenev Ya et al.  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
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  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. , 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  are presented for an EVT prototype, which operates in S-band, where acceleration up to 1 MeV was predicted at efficiency of 66% . 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.
2. EVT configuration
The schematic representation of an EVT accelerator is shown in Figure 1 , where the accelerating structure is pop up.
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
3. Scientific and technical justification of the EVT concept
It is known that a field in a volume of a cavity generated by a beam with a current density can be presented as eigenmode expansion: , where eigen vectors are functions of coordinate that are normalized by energy of the cavity as follows:
A time-dependent multiplier is the solution of the equation of oscillator with the second member, Eq. (2):
Let’s express Eq. (2) in a complex form and consider one of the eigenmodes only, having frequency , replacing: , . It gives
Other terms of expansion generated by the beam current will have small values at high quality of the cavity and at , and they will not be taken into account. Eqs. (2) and (3) are the most general expressions for determination of fields in cavities . As it takes place , the equation Eq. (3) can be rewritten as:
where is a relative detuning:
Practically, the cavity is described via integral characteristics, such as geometric impedance of drive gap (index 1) and accelerated gap (index 2):
circuit current of the drive gap and circuit current of the accelerating gap equal to
RF beam currents:
voltage of the drive gap along axis and voltage of the accelerating gap along axis :
where: and are phase angles; , are transit time factors of the drive and accelerating gaps, respectively; , are form-factors of bunches; , 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:
Let us add in relationships under consideration
following from: .
The description of the equivalent circuit for a case of interaction of relativistic beams with the cavity is
where is impedance of the drive gap, is a phase angle between gap voltage and equivalent circuit current . 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 .
At interaction of a cavity with low relativistic beam it is necessary to consider also beam-loading effect or a beam admittance . Therefore, Eq. (12) is revised as
Let’s set out well-known analytical formulas for a rectangular bunch with angular length and for the flat cavity gap having a transit time angle :
where is an average beam current, is an average beam voltage in the corresponding gap (drive or accelerating), is the corresponding transit time factor and 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. , at which the phase angle between a gap voltage and an equivalent current of a cavity circuit becomes close to . 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 and tail of a drive bunch is shown here, too. Let’s notice, that complex RF beam current is displaced at angle relative to the circuit current .
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: , . It is the chosen phase of drive current: . The phase of the accelerated beam current equal to provides a bunch being in the electric field close to the peak value, i.e. and . Thus, using Eqs. (12) and (13), the required impedance of the loaded drive gap is
where is a transformation ratio. An impedance of the accelerating gap is similarly found:
On the other hand for the equivalent circuit of the cavity the impedance is given as:
where a high value of detuning is suggested, which results in . In addition, it is assumed that the transformation ratio is large enough, i.e., . For example, , , , , .
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 lost by the drive beam and the voltage added by the accelerated beam, which are connected to a voltage of the drive gap and a voltage of an accelerating gap as assigned at the corresponding moment of time (see Figure 2 ):
where the transformation ratio is:
or, at a high value of transit time factors , Eq. (26) can be written as
where 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:
where 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:
where 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 of the accelerated beam, i.e. .
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 and the voltage close to , two conditions shall be satisfied. The first is to have a large enough value of the cavity detuning ; the second is . The angle between and should be close to 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
are the necessary factors of effective performance of the EVT, which are readily achieved when multiple drive beam-lets are used. For example, , , , , , , ,
We define the efficiency 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:
where is the total current of the gun, and 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, ||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|
|Voltage gain factor||≈18||≈18|
|Typical R/Q factor of accelerating gap||58||54||Ω|
|Typical cavity detuning, ||8 ÷ 12||7||MHz|
|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||%|
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:
where and 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:
where and are velocities of the drive and accelerated bunches in space between cavities gaps. Phasing capabilities considerably extend if distances and between a gaps of adjacent cavities are not equal to each other. More general Eq. (34) is corresponding to this case:
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].
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,
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
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;
Parameters of drive beam-lets are identical;
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 is much higher than power transferred to the accelerated beam, for example, .
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
The accelerated beam goes through the second cavity having a gap voltage and a phase angle , which are calculated based on a voltage on the gap of the cavity 1 for the drive beam and where power of the accelerated beam increases at the value . 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:
where is the number of drive beam-lets, 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 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.
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. . 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
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.
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.
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.
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 cavities has been observed. Numerical results show difficulty of bunch confinement at a voltage of a drive gap . It results in the following limiting estimated condition:
where is from Eq. (11). This estimation is close to observable result 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.
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 .
|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||%|
|Gun current of drive beam||216||36||A|
|Perveance of drive beam-let||0.41||0.41||μA/V3/2|
|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||%|
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).
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.
|Voltage of stage|
|Current of potential part of stage|
|RF power, transfer to accelerated beam|
|Loss power in ground part of collector|
|Loss power in potential part|
|EVT electron efficiency|
|Beam power at output of tube|
|Total power generating by high-voltage source|
|Efficiency of depressed collector|
|Total tube efficiency with DC|
Substituting the above resulted relationships, it is derived the total tube efficiency with a DC:
Eq. (36) is well known (see, for example ), but we shall regard it now jointly with the presence of an energy spectrum of particles leaving a tube. Variable 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 in order to find an extremum at some assumptions concerning the energy distribution , which is normalized on the beam current :
In the gap of the separator, the beam is divided into low-voltage and high-voltage fractions. The current
that has reached the potential part of the collector is the fraction of the current at which . In the same way, it is written down the equation for the beam power leaving the accelerating structure:
Thus, it is evaluated the fundamental formula Eq. (41) for efficiency of a depressed collector included in the DC&SS concept:
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 (curved line) can serve as a close approximation for numerical distribution.
Thus, let us approximate by the gamma distribution:
where , . Further, it is defined according to Eq. (39):
, where , and it is defined according to Eq. (40):
Substituting the found expressions in Eq. (41), it is obtained:
The function has the maximum equal to
Thus, it is derived the collector efficiency and other parameters at the optimum point:
As an example of the S-band EVT ( Table 1 ), one have:
The total efficiency (according to Eq. 37) at the optimum point is , which is the upper theoretical limit!
For a DC having
where is DC efficiency in multi-stage case. The collector efficiency of the
Using the approximation (see Eq. 42) for beam distribution , leads to:
where: , , , is gap voltage of the
The extremum of this function is equal to at: ; .
At , according to Eq. (37), it will correspond to the total tube efficiency .
Table 4 shows the results of efficiency optimization for a collector that contains one to six stages.
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.
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.
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.