Comparison of longitudinal emittance of various RFQs

In various projects a large variety of RFQs has been developed, for different application, with different average current, frequency, and energy range. In this article, a comparison, in a scaled way, will be done, using the build RFQs of IFMIF, ESS, SPES, ANTHEM. In particular, the characteristics of the beam dynamics will be analyzed, such as transmission, longitudinal output emittance, and real performance versus simulation.


Introduction
Longitudinal emittance at the RFQ output is a very important parameter that defines the beam quality in the subsequent accelerator.A low value of longitudinal emittance permits higher real state energy gain, so a more compact Linac.Achieving a small longitudinal output emittance is difficult because the initial bunching of the injected dc beam, initially emittance dominated for low current beam and space charge dominated for high current beam, tends to fill the evolving bucket separatrix in both cases.On the other hand, an external multi-harmonic buncher does not guarantee high particle capture efficiency, especially for a high current beam.
A standard strategy is the use of slow of quasi-adiabatic bunching process that is highly nonlinear, this process requires tens of RFQ cells.The result from this slow adiabatic bunching process would require an unrealistically long RFQ to accommodate the several longitudinal phase space rotations needed.
Why do this RFQs comparison?Essentially the longitudinal emittance formation process is not fully understood, and this induces different choices in the design method.So how is it possible to design a small longitudinal emittance RFQ to be compliant with a small longitudinal linac acceptance?Is possible to considered a RFQ design evolution, i.e. a further optimization on the parameters choice?

RFQ main formulas
The main parameters that characterize an RFQ are reported in figure 1, where  is the minimal aperture between the electrodes,  is the modulation factor,  *  is the maximum aperture, and  is the applied voltage.As reported in [1], the formula that define the acceleration factor  10 is: where  = 2   =    with  =   the speed  respect to  the speed of light,  the RFQ wavelength and   the cell length.The transverse focusing force in a RFQ is express in the factor , with formula: with  the RFQ frequency, / the charge to mass ratio,  is the voltage and  0 the aperture at half cell (  4 ).In same case, the so call "2term" modulation, the average aperture  0 can be express as: In some other case, the so call "sin" modulation, the average aperture  0 can be express as: The longitudinal and transverse phase advance at zero current are defined as: where   ,   is the synchronous energy and phase.
The separatrix area, i.e. the stable area in the longitudinal phase space can be defined as: . Where Ψ(  ) ≈ −3  + 0.27  3 − 0.252347  5 is the stable phase zone, around |3  | for small |  |.The maximum surface field in a RFQ can be estimate by using: where the  is a field enhancement factor that depends on the modulation and the average aperture of the RFQ, typically, the factor goes from 1 to 1.6.The values of  are coming from the Crandall tables [2].
Due to the fact that at different frequencies the RF breakdown occurs at different values of surface field, a convenient way to compare cavities with different frequencies is to normalize the surface field versus the Kilpatrick limit [3].In all the considered RFQs is possible to consider 3 sections, a shaper section, a gentle-buncher section, and an accelerator section.The bunching is started in the shaper section, and the adiabatic bunching is done in the gentle-buncher section.The final accelerator section is used to accelerate the beam to the final energy; typically this is the longest section.

RFQ selections
In this context, the analyzed 4-vane type RFQs are already built; each of them is the result of several optimization processes which involve practical considerations such as realistic length, RF power, and manufacturing process.The selected RFQs for the comparison are: the TRASCO RFQ [6] that now is used for the ANTHEM BNCT project [7]; the IFMIF/EVEDA RFQ is high current CW RFQ now under commissioning in Rokkasho (Japan) [8]; the RFQ for the LNL RIBs beams project SPES also CW but for low current and able to handle a range of 3 < A/q < 7 [9] as ions mass over charge; the SPIRAL2 RFQ is CW for ions [10].The ESS proton RFQ is a pulsed machine with 14 Hz as rep.rate [11].In table 1 is reported in summary the main parameters of the RFQs.Each RFQ has been designed with a specific goal in terms of performance, reliability, and cost optimization.
The TRASCO RFQ is the longest respect to the wavelength and this requires the use of 2 coupling RF cells.The coupling cells has been take into account in the simulations.
The IFMIF and SPES RFQ are quite long in the absolute way, so mechanical is made by various segments, with RF continuity but with a small electrode separation, this is kept into account in the simulations.

Design consideration
The RFQs have been designed to match the specification in terms of energy and current; the design method has been similar for the TRASCO/IFMIF RFQ, with the LANL code chain (CURLI-RFQUICK-PARI-PARMTEQM) for high current [4].The SPES RFQ has been designed with a homemade code based on the program used for the design of CERN linac3 RFQ.The SPIRAL2 and ESS RFQs have been designed with the help of CEA RFQ Designer code [12].There are no general rules about the design, and this produces different laws for the voltage shape, trans-verse, and longitudinal phase advance, focusing force and modulation.In a general way, in each RFQs it is possible to define a shaper section where the parameters like phase and modulation are changed smoothly, a Gentle Buncher where the almost adiabatic process is done to capture as mush as possible particles and the Acceleration section, where the focusing force is typically reduced and the acceleration rate increases as much as possible.In figure 2 is reported the various voltage laws used in the mentioned RFQs.
To compare RFQs at various frequency the various graphs reported as abscissa the cell number /2.This permit to consider the beam dynamics evolution inside the RFQs in a scaled way.In the TRASCO case, the voltage is constant along the structure, due to the classic design choice done in 2000.The voltage is linearly increasing with respect to the beam axis (SPES) and slowly increases along the RFQ (IFMIF, SPIRAL2, ESS), but in the latter cases with a different and nonlinear voltage law.In the case of SPES RFQ, a large modulation has been used (m > 3.2); this is almost at the limit of what has been typically done on RFQs and also included on the Crandall table [2].Such modulation value is used to get a value of A10, the acceleration coefficient, of about 0.81 (i.e., more than 80% of the voltage is used for the acceleration) 3.In figure 4 is reported the modulation used on the RFQs.Before the gentle buncher, the modulation is changed smoothly to obtain a lower longitudinal emittance and avoid losses.In the accelerator region the change of m is very fast, obtaining the maximum value of m at the RFQs end.The law for m(z) in the shaper does not influence the efficiency.The transverse focus force  in the RFQs is typically increased from the beginning to the end of Gentle Buncher (where a maximum is reached).After that point,  is reduced to obtain more voltage for acceleration in the accelerator section.The various focusing forces B are reported in figure 3. The higher value of B is on the IFMIF RFQ; this is due to the high value of design current that must consider the effect of the large space charge effects.At the start of the RFQs, the focusing force is typically reduced to allow an easy way to inject the beam into the RFQ from the LEBT.For a smooth injection into the MEBT the focusing force is also reduced at the RFQ end.However, the reduction of  at the beginning and end of the RFQ is not always done.The variation of  along the RFQ can also be done in a fast way without compromising the beam quality.
-4 - A way to handle a higher emittance and obtain a higher transmission is to increase  0 see figure 4, like the SPIRAL2 RFQ, but this can reduce the global RFQ efficiency, reducing the energy gain per meter as reported on figure 5.The minimum of  0 is on the TRASCO RFQ of about 3 mm.
All the considered RFQs get a maximum of surface field below 1.9 kp, for the cw cases the limit is reduced to 1.8 kp; see figure 4.
The   in the RFQs ranges from −90 • to the final values in the range of −30 • −20 • .In the cases of SPIRAL2 and ESS RFQ, the phase in the gentle buncher remains −90 • for about 100 cells, as reported in figure 4.This is done to reduce the longitudinal emittance.In the other RFQs the   is changed linearly with the cell period in the shaper.In the Gentle Buncher the phase is changed quite fast, but slowly in the ESS and IFMIF RFQs.In the last accelerator section (cell > 300) the   is changed linearly in the IFMIF RFQ and not linearly for the ESS RFQ, for the other RFQs the phase is kept constant; see figure 4. A small separatrix area is very important for defining a low longitudinal emittance, as reported in figure 5 the area is very small on SPES and SPIRAL2 RFQ, larger on the other RFQs, to handle the high current value with a large longitudinal defocusing force.The separatrix area growth fast on the RFQs for high current design and slowly on low current RFQs, getting its maximum value at RFQ end.The longitudinal phase advance at zero current follows a similar trend with a fast growth on the RFQs for high current and a very slow increase on the SPES RFQ.The   after a maximum on the Gentle Buncher section remains constant for some cells and after that decrease in the accelerator section of the RFQ, the decrease of   is due to the increase of the energy in the final part of the RFQ.
-5 - The transverse phase advance at zero current follow a similar trend of  0 , with a small initial value a fast growth up to the end of gentle buncher and a decrease in the accelerator section, see figure 5.The highest value is on the IFMIF RFQ to handle the high-space-charge effects.A small value of   is on the ESS and SPES RFQs.
In general, there is no need to change smoothly   ,   along the RFQ.

JINST 19 P01024 5 Comparison method
The comparison was facilitated using the TraceWin/Toutatis program [12].For each RFQ under consideration, a multiparticle Particle-in-Cell (PIC) simulation was executed with 100,000 macroparticles, utilizing the nominal design input file with the description cell by cell of voltage, average aperture  0 , modulation  and  10 to fully define each RFQ cell.This approach allowed a comprehensive control over all simulation parameters, input conditions, and program options, including steps and longitudinal cutting.
To maintain consistency and eliminate variations associated with the source and Low-Energy Beam Transport (LEBT) injection line, a meticulously selected set of perfectly matched input beam Twiss parameters was employed for each RFQ.This strategy ensured that the comparison focused solely on the RFQ performance.The chosen beam input distribution adopted a Gaussian profile cut at 3  with 20 steps per cell period on Toutatis, and no energy spread on the input beam was introduced.
In order to isolate and assess only the accelerated particles, a defined longitudinal limit for particle rejection was set at +/-0.2 MeV/u relative to the synchronous energy.This criterion allowed for a targeted analysis of the particles directly influenced by the RFQ acceleration process.
The specific input values utilized in the simulation are detailed in table 2, providing a transparent reference for the parameters and conditions employed in the comparative study.This meticulous approach ensures a rigorous and systematic evaluation of the RFQ performance under controlled and standardized conditions, laying the foundation for a robust and meaningful comparison across different RFQ designs.

Comparison results
The calculated longitudinal emittances at full nominal current are reported in figure 6, with units of MeVdeg/u to handle the various masses, as a function of cell number.The cell number allows us to do the comparison for various frequencies.After the emittance formation process, typically in 50 cells, there is an increase of the emittance due to the not accelerated particles, that are cut off by the longitudinal energy limit.After the cutoff point, the longitudinal emittance is almost preserved up to the end of the RFQs.The RFQs designed for high current i.e., > 20 mA shows larger emittances, with the TRASCO RFQ that has a longitudinal emittance of 0.2 MeVdeg/u.On the other hand, the RFQs designed for low current and for ions show a lower longitudinal emittance, with a minimum on the SPES RFQ of 0.045 MeVdeg/u.This is due to a long shaper region in the SPES RFQ, about 1.2 meters, and a small separatrix area.
-8 - The RFQ of SPES show a fast transmission drop of 5% at cell 200, due to the longitudinal cut.The longitudinal output emittances obtained are also reported in table 3: also the emittance inside the 95% of the particles and the relative rms emittance are reported.From the ratio of total particles and rms, it is possible to calculate the longitudinal halo; TRASCO result to have the larger halo with respect to the RFQs here analysed.The measured longitudinal emittance is for the SPIRAL2 RFQ of 0.74 MeVdeg (= 0.046 MeVdeg/u) for 18O6+, in agreement with the relative TraceWin simulations [13]; the SPES RFQ presents the lowest longitudinal emittance, at the cost of a reduced transmission (93% of the accelerated particles).In figure 7 is reported the transmission of the accelerated particles; the best transmission is obtained by the SPIRAL2 RFQ, with almost all the particles accelerated, for A/q = 3.This result has also been confirmed experimentally [14].
-9 - The IFMIF RFQ obtained a real transmission of more than 90%, in a pulsed d.c. of 0.1%, with a total of 125 mA of deuteron in output at the beam dump [15].
The ESS RFQ gets a measured transmission of about 95.5% [16].
The RFQs of TRASCO and SPES has not yet been tested with beam.
The measured RFQs transmission performance impact with reality in terms of not perfect input matching beam, like the actual voltage shape with RF tuning versus ideal one, the mechanical construction tolerances on electrode positions and diagnostics calibrations.By taking into account that in the simulation program, and recalculating the transmission with the actual voltage shape, vanes positioning, and beam input, the RFQs of IFMIF, SPIRAL2, and ESS shows a transmission that agree with the measurements.
The impact of the beam current on the longitudinal emittance, whit units of  , is reported on figure 8, as final longitudinal phase space at RFQ exit.When the space charge effects are missing, the longitudinal phase space creates a filamentation on the RFQ designed for high current, like IFMIF,ESS, TRASCO and this produces a larger longitudinal emittance.
The number of turns in the phase space filamentation can be express as: getting a value of about 14, 11, 4, 9, 10 for respectively the RFQs of IFMIF, ESS, SPES, SPIRAL2, TRASCO.The count of turns can also be done directly on the plots of figure 8.A larger number of turns is also connect to the increase of longitudinal emittance at zero current.The rms longitudinal -10 - emittance increase at zero current goes from 5% in the case of SPIRAL2 to 75% in the case of IFMIF, due to the high current design.
Is possible to see the resonance effects, at zero current, of the condition   =   for the SPES and SPIRAL2 RFQ, this effect induce a larger beam size on the halo, i.e. the outside particles with density less than 1%, blue zone on figure 9, between the red dashed lines.Anyhow this effect is not dangerous, because is not inducing any losses or emittance growth.
In the other RFQs the distance of   and   is quite large to handle the tune depression.-11 -

JINST 19 P01024 7 Conclusion
A direct comparison of the nominal RFQs for various project have shown the following common points: • The simulation code can well define the beam dynamics inside any RFQs.
• The design can be optimized to reduce the longitudinal emittance.
• There are no general common rules on how to design an RFQ.
The simulation code, TraceWin [12], has demonstrated successful comparison with experimental results, particularly in terms of RFQ transmission [14][15][16].This success is attributed to its ability to handle the RF tuned voltage shape and mechanical electrode positioning, even in cases where the constructed RFQ configuration breaks the vanes quadrupole symmetry; this is done by loading the realized vanes geometry into the simulation program.
Regarding RFQ design, adjustments in voltage and modulation, along with oscillations in parameters like the average aperture  0 play a crucial role.Specifically,  0 tends to be larger at the beginning and end of the RFQ but smaller in the middle, near the end of the Gentle Buncher.A precise definition of the RFQ parameters at this stage is essential for achieving optimal longitudinal capture.
To ensure a favorable RFQ transmission, optimization of the transverse focusing force  is necessary.A too-small value can lead to losses, especially in the presence of high space charge effects.Conversely, a large value of  requires a small  0 or a higher voltage, resulting in an increased surface electrical field.
Typically, about 50 RFQ cells are involved in longitudinal emittance formation.However, to achieve very low longitudinal emittance in the SPES RFQ, approximately 100 cells are employed.Different projects adopt varying approaches to designing the shaper section of the RFQ, with modulation kept at almost 1 in some cases (e.g., SPES RFQ) and   in the shaper kept at almost −90 • in others (e.g., SPIRAL2 and ESS).All these approaches produce low longitudinal emittance in simulations, with measurements that confirm this, up to now, only for the SPIRAL2 RFQ [13,14].
Illustrating the limits of optimization in longitudinal emittance, figure 10 presents complete simulations of multiple particles obtained through the LANL sequence of programs [4] and a swarm optimization algorithm [5].In a case study involving the TRASCO RFQ [6], redesigned with the IFMIF voltage shape law, the lower left part of the figure indicates a lower peak RF power and longitudinal emittance compared to the original TRASCO RFQ if the dot color is near blue.The orange dashed lines represent the actual characteristics of the TRASCO RFQ.Achieving a factor 2 lower longitudinal emittance requires increasing the RFQ length by about 1 meter, resulting in a factor 2 increase in peak RF power.
Another aspect to analyze the trade-off on longitudinal emittance is the shaper length.Figure 10 displays the shaper length for the same TRASCO RFQ cases.To achieve a factor of 2 less longitudinal emittance, increasing the shaper length by a factor of 3 is necessary, leading to a total increase in RFQ length of more than 1 meter.
In conclusion, the application of new methods, such as swarm optimization, enables improvement in RFQ design for various parameters.In the case of the TRASCO RFQ redesign, a 30% reduction in RF power was achieved with a shorter RFQ (from 7.1 to 6.9 m), maintaining the same longitudinal emittance of 0.2 MeVdeg and reducing the surface field from 1.77 to 1.7 kp.This was accomplished -12 -by employing a voltage increase law and  0 as in the IFMIF RFQ, leveraging significant computing power to simulate over 10,000 RFQs in parallel with a broad variation in parameters.
In summary, this comparison of RFQs illustrates the various parameter choices made for RFQ optimization, providing a valuable starting point for consideration in future RFQ designs.

Figure 2 .
Figure 2. Voltage along the RFQs as function of cell number.

Figure 3 .
Figure 3. Focusing force B and accelerating factor A10 as function of cell number.

Figure 4 .
Figure 4. ,  0 ,   ,   in the RFQs as function of cell number.

Figure 5 .
Figure 5. Separatrix Area, Energy gain and   ,   as function of cell number.

Figure 6 .
Figure 6.rms longitudinal Emittance as function of cell number.

Figure 7 .
Figure 7. RFQs transmission as function of cell number.

Figure 9 .
Figure 9. SPES and SPIRAL 2 RFQ with   =   Resonance at zero current.

Figure 10 .
Figure 10.Longitudinal emittance as function of Transmission, Shaper Length, RF power, GB phase and total RFQ Length.

Table 2 .
RFQs used input conditions.