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The longitudinal “trapping” phenomenon in the simultaneous acceleration of intense H+ and H− in an RFQ
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Nuclear Instruments and Methods in Physics Research A 538 (2005) 143–153 www.elsevier.com/locate/nima
The longitudinal ‘‘trapping’’ phenomenon in the simultaneous acceleration of intense H+ and H in an RFQ Q.Z. Xinga,, S.N. Fub, Y.Z. Lina, S.X. Fangb a
Department of Engineering Physics, Accelerator Laboratory, Tsinghua University, Beijing 100084, PR China b Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100039, PR China Received 10 August 2004; accepted 14 September 2004 Available online 14 October 2004
Abstract The phenomenon of longitudinal ‘‘trapping’’ in the simultaneous acceleration of intense H+ and H beams in an RFQ accelerator is described in this paper. A new code named PARMTEQVC is compiled based on an old version of tcode PARMTEQ for the computer simulation. In this code, the 2D PIC method is applied to calculate the space charge forces. Multi-bunch effects are also taken into account. In our simulation the RFQ structure parameters which were designed only for single beam (H+ or H) acceleration are adopted. The simulation result shows that more ions near the ends of buckets are ‘‘trapped’’ into the bunches with opposite charge state. These ions leave their own buckets gradually and become lost in either longitudinal or transverse direction, leading to a descent of the total transmission rate. Further studies reveal that the depth of longitudinal potential well decreases in the case of the simultaneous acceleration, which is attributed to the weaker bunching forces because of the longitudinal space charge interaction between the positive and negative ion bunches. It seems that a new RFQ structure should be specially designed for the simultaneous acceleration in order to neutralize the transverse space charge effect and increase the beam transmission rate at the same time. The design ideas are then presented. r 2004 Elsevier B.V. All rights reserved. PACS: 29.27.Bd; 29.27.Eg; 29.27.Fh; 41.75 i Keywords: Computer simulation; Simultaneous acceleration; RFQ; PARMTEQ; PIC method
1. Introduction The high-intensity proton linacs (with peak currents more than tens of mA) have become Corresponding author. Fax: +86 10 62782658.
E-mail address: [email protected] (Q.Z. Xing).
increasingly attractive for the spallation neutron source [1] and the accelerator-driven energy production and transmutation of waste system [2] in the development and application of neutron science. Beam loss is a major problem in the high intensity proton linacs, which can cause activation of the accelerator components and
0168-9002/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2004.09.011
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make unconstrained maintenance difficult. One of the reasons for beam loss is the formation of beam halo [3]. Under certain conditions, a small fraction of particles can acquire enough transverse energy from the repulsive space charge forces within the beam to form a halo [4]. The simultaneous acceleration of two intense beams with opposite charge state is then considered to compensate the large nonlinear space charge effect and restrain the beam halo evolution in the low-energy section of the linac, where the large Cockcroft–Walton DC preaccelerator has been usually replaced by an RFQ [5]. The positive and negative CW ion beams extracted from respective sources are mixed and then sent into an RFQ. The two beams are focused, bunched and accelerated in the RFQ and then entered into the following accelerating structures, such as a DTL or CCDTL section, where the two beams are then separated longitudinally. The mechanism of simultaneous acceleration is most complicated in the RFQ since the strongest interaction between the two beams happens here. This interaction must be carefully considered and clearly understood in order to achieve a high transmission rate. High-intensity experiments of simultaneous acceleration in the RFQ have never ever been performed to the best of the authors’ knowledge. The proton linac of LANSCE can simultaneously accelerate H+ and H ion beams to 800 MeV with peak currents of 17 and 12 mA, respectively, but no RFQ is used [6]. Simultaneous acceleration of O+ and O ion beams in an Integrated Split Ring RFQ has been carried out at low intensity (peak currents are less than 1 mA) in Peking University [7], which proves experimentally that in the RFQ both the positive and negative half periods of the RF cycle can be used to accelerate both signs of ions at the same time. Other project of simultaneous acceleration using RFQ has only been proposed [8]. Prior to deliberate experiments, computer simulation [9] should be chosen to study the simultaneous acceleration since the highly nonlinear and time-dependent behavior of this process in the RFQ probably precludes the accurate analytic solution. Simulations of simultaneous acceleration in the RFQ including the space charge effect have
been accomplished independently by Crandall [10], Oguri [11] and LIDOS team of Moscow Radiotechnical Institute [12]. Similar results have been reported by Crandall and Oguri that positive and negative beams can be accelerated simultaneously in an RFQ linac, but in the dual beam operation the transmission efficiency decreases rapidly with the input beam intensity compared with that in the single beam mode. It has also been found by Oguri that the attractive forces between the positive and negative ions disturb the bunch formation in the structure, leading to an increase in the longitudinal beam loss. Further studies of LIDOS team reveal that with current growth in the bunched beam a small part of oppositely charged particles are trapped inside the bunch in the beginning of the RFQ. Crandall gives a simplified analysis where only the positive particles are traced and the effect of the negative beam on the positive one is calculated by assuming that the two charge distributions are identical but offset by 1801. Moreover, it can only deal with two beams having equal currents. In the study of Oguri, the space charge effect is calculated by summing up the Coulomb forces from all other macroparticles, for which calculation with macroparticles more than 29 is not routinely possible owing to the extremely long CPU time. LIDOS team only reveals the phenomenon of ‘‘trapping’’ but the detailed dynamic process is not presented. In this paper, the simultaneous acceleration of intense H+ and H beams in the RFQ is studied in more detail. To this end, the dynamic simulation of dual ion beams is added into one old version of PARMTEQ [13,14], a common RFQ design and dynamic simulation code. The modified code is written in VC++ language to maintain a more user friendly interface, named as PARMTEQVC. One positive bunch and one negative bunch are traced in the dynamic simulation. Calculation of the space charge forces is accomplished with the 2D Particle-In-Cell (PIC) method, and up to order of 104 macroparticles can be traced keeping the computational time tolerable. The numbers of longitudinal and radial meshes are unlimited and can only be restricted by the computer capacity. The input function of the RFQ cell parameters is added into PARMTEQVC by the means of
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reading a file generated by other design codes. Based on the cell parameters of an RFQ designed for single beam acceleration in the Clean Nuclear Power Project of Major State Basic Research Development Program in China, the detailed dynamic simulation result of simultaneous acceleration is given and discussed, and the analysis of the phenomenon of longitudinal ‘‘trapping’’ is also presented. The paper is organized in the following way. Section 2, describes the code revision for the dynamic simulation of two beams with opposite charge states. Results and discussion are given in Section 3. Conclusions drawn from this study are presented in Section 4.
2. Revision of code PARMTEQ To modify the PARMTEQ code and simulate the dynamics of positive and negative ions simultaneously, two initial distributions of the two beams must be produced separately. Then, the motion of the two oppositely charged ion beams in the external field should be considered. Especial attention should be focused on the space charge interaction of the two kinds of ions. This section introduces our work on these aspects. The time-dependent simulation (t-code) is adopted in PARMTEQVC to avoid errors due to the z-to-t transformation in the space charge calculation of the position-dependent simulation (z-code) [15]. However, the initial six-dimensional distribution ðx0 ; x00 ; y0 ; y00 ; f0 ; W 0 Þ of the two beams at the entrance of RFQ in the z-code is produced first, then the corresponding distribution ðxt0 ; x_ t0 ; yt0 ; y_ t0 ; zt0 ; z_t0 Þ in the t-code is derived from it pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi z_t0 ¼ W 0 ð2m0 þ W 0 Þ c=ðm0 þ W 0 Þ zt0 ¼ z_t0 T ðf0 3p=2Þ=2p y_ t0 ¼ y00 z_t0 yt0 ¼ y0 þ y00 zt0 x_ t0 ¼ x00 z_t0 xt0 ¼ x0 þ x00 zt0
ð1Þ
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where a dot or prime denotes the derivative with respect to time t or longitudinal position z, and T is the RF period, m0 the rest mass of one ion and W 0 the initial kinetic energy of the ion. The acceptance phase space of the positive ion beam is the same as the negative one in both x–x0 and y–y0 at the entrance of RFQ. The two initial beams are focused at the RFQ entrance. They have an initial uniform distribution in the four-dimensional transverse phase ellipsoid, and a uniform longitudinal phase distribution and random energy spread. Although positive and negative ions occupy the cells continuously at the beginning of RFQ, only the particles in 2p phase for each beam are analyzed. Because the interval of adjacent positive and negative buckets is p; the initial phase distribution is from 3p=2 to p=2 for positive ions and from p=2 to 3p=2 for negative ions in PARMTEQVC. One negative synchronous particle with initial phase p=2 is appended into the code (the positive one has initial phase p=2). The derived longitudinal distribution in the t-code is shown in Fig. 1. The motion of both the positive and negative synchronous macroparticles is traced in the dynamic calculation except in the space charge calculation. The external electric fields can be obtained from the two-term potential function [13]. More terms corresponding to the nonlinear
Fig. 1. Initial longitudinal distribution of positive and negative particles traced in the t-code and z-code. The distribution in tcode is derived from that in z-code through Eq. (1).
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field in the quadrupole channel with non-hyperbolic vane profiles should be used [11] for more precise study, but in the present work, results from the two-term expression still make sense. The trajectory of each macroparticle is traced by the drift/thin-lens/drift method, and the space charge calculation is carried out with the 2D PIC method each time the positive synchronous particle reaches the middle of cells. Space charge forces are calculated assuming that the charge distribution is periodic, that is, the adjacent pairs of positive and negative bunches are assumed to have the same charge distribution as that of the positive and negative bunches being traced. This is reasonable provided that the structure parameters change gradually along the cells of RFQ. The longitudinal midpoint between the positive and negative synchronous particles is chosen as the longitudinal center of the field grid (as shown in Fig. 2). The charge (with its sign) of each macroparticle is distributed to the corresponding charge grid nodes according to the transformed position of the macroparticle. The positions of positive and negative macroparticles are calculated separately because positive and negative bunches have different RMS sizes. The electric fields at the field
Fig. 2. Sketch of field grid and charge grid in the calculation of space charge forces. Total charge contained within each cell of field grid is q placed at the location of the centroid of that cell, ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi that is, r ¼ ðr21 þ r22 Þ=2:
grid nodes produced by the two kinds of macroparticles are calculated and then preserved separately. After interpolation the electric fields at the position of each macroparticle can be derived. At the beginning of RFQ, the positive and negative macroparticles to be traced occupy 3-cell length in the longitudinal direction, so the number of longitudinal meshes is increased from 40 to 400 in the dual beam calculation. The longitudinal mesh interval is changeable to keep the total longitudinal mesh length as 3-cell length. The number of adjacent pairs of positive and negative bunches considered is increased from 5 to 10. The population ratio between the positive and negative macroparticles is held to be equal to the current ratio between them. Up to order of 104 macroparticles can be traced keeping the computational time tolerable. Using a PC with two 1.2 GHz Pentium CPUs and 512 M memory, the time cost is less than 10 min for one run.
3. Results and discussion The function of acquiring the external cell parameters is added into PARMTEQVC, so as to do simulations with the cell structures designed by other codes. The matched input beam parameters are calculated before each run. Simulations of simultaneous acceleration (H+ and H) with different mixing ratios and total currents are carried out based on the structure parameters (see Fig. 3) of the RFQ accelerator in the Clean Nuclear Power Project of Major State Basic Research Development Program in China, which was designed for single beam acceleration (nominal current is 50 mA). Table 1 gives the main design parameters of this RFQ. Fig. 4 shows the longitudinal phase spaces at times when the positive synchronous particle reaches the centers of some cells along the RFQ for both I ¼ 0 and I=100 mA (50 mA H+ and 50 mA H), starting with initial DC H+ and H ion beams. Each point represents one positive or negative macroparticle. The region inside the closed curves indicate the longitudinal stable areas. The negative bunch being traced precedes the positive bunch by one cell, that is, the negative
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Fig. 3. Structure parameters of the RFQ in the Clean Nuclear Power Project. B is the focusing factor, X and A are the focusing and accelerating parameters, respectively, W is the kinetic energy of the synchronous particle, f is the synchronous phase, m is the modulation factor, R0 and a are the mean and minimum apertures, respectively.
Table 1 Main design parameters of the RFQ in the Clean Nuclear Power Project Parameter
Value
Beam species Frequency RFQ type Input energy Output energy Nominal current RFQ total length Number of cells
H+ 352.2 MHz Four-vane 75 KeV 3.5 MeV 50 mA 468.22 cm 342
bunch experiences one more cell’s acceleration. Therefore, from Fig. 4(d) to (f) the mean longitudinal velocity of the negative macroparticles are a little larger than that of the positive macroparticles apparently. The numbers of the positive and negative macroparticles are both 104. The
unnormalized total emittances at the entrance of RFQ are both 95.2 mm mrad. The initial energy spreads are set to zero. The matched Twiss parameters for the dual beams are obtained from that of a zero-current beam because the positive and negative beams are initially space–charge neutralized. The longitudinal dynamic behavior of simultaneous acceleration can be observed from Fig. 4 as follows: (1) From the Radial Matching Section (RMS) to the beginning stage of Shaper (Fig. 4(a) and (b)), the longitudinal phase space has little change. The positive and negative particles are mixed together so that space charge forces are neutralized sufficiently. (2) In the Shaper (from Fig. 4(b) to (c)), the positive and negative particles begin to bunch in their own buckets. The phase interval between the adjacent positive and negative
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Fig. 5. Phase space at the entrance and exit of the RFQ. (a) Phase space at the entrance of the RFQ; (b) Phase space at the exit of the RFQ.
bunches is 1801. The two bunches are not completely separated at the end of the Shaper, where a proportion of particles with opposite charge state ‘‘trapped’’ in the positive and negative bunches exists. (This will be discussed later.) (3) In the Gentle Buncher (from Fig. 4(c) to (d)), the positive and negative bunches separate gradually until they become separated completely at the end of it. The ‘‘trapped’’ particles at the end of the Shaper have disappeared. (4) After the Gentle Buncher (from Fig. 4(d) to (f)), the longitudinal internal between positive and negative bunches become larger while the length of cells increases along the RFQ. The interaction between positive and negative particles becomes negligible. The two bunches are accelerated until they reach the end of the RFQ.
The transverse phase space distribution at the entrance and exit of the RFQ is given in Fig. 5. Alternating longitudinal fields in the RFQ cause the two beams to be separated longitudinally by one half of the RF cycle. Therefore, positive and negative ion bunches have almost the same
distribution in the x–x0 and y–y0 phase space at the exit of the RFQ which is due to different phases of positive and negative bunches. More simulation results are given in Fig. 6 to illustrate the change of transmission rate with the proportion of positive ion beam current while the total current keeps constant. Results of different total currents are given on different curves. As we can see, curves are almost symmetric about the center where the positive and negative beams have the same current (the proportion of the positive ion beam current is 50%). That is to say, for example, the transmission rate of accelerating 20 mA H+ and 80 mA H is the same as that of accelerating 80 mA H+ and 20 mA H. This is reasonable because positive and negative beams are DC beams at the entrance of the RFQ and the acceleration field is RF field. The symmetry suggests validity of the results, and the small deviation of symmetry mainly results from the random initial distribution of macroparticles. The following conclusions can be drawn from Fig. 6: (1) When the total current I t o20 mA; high transmission rates can be achieved by both single beam acceleration and simultaneous
Fig. 4. Longitudinal phase spaces when the positive synchronous particle reach the centers of some cells, starting with initial CW H+ and H ion beams: (a) cell 1 (the beginning of the RFQ), (b) cell 7 (start of the Shaper), (c) cell 94 (start of the Gentle Buncher), (d) cell 215 (end of the Gentle Buncher), (e) cell 272 (start of the Accelerating section), (f) cell 342 (end of RFQ). The negative bunch being traced precedes the positive bunch by one cell.
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100%
Transmission rate
95% 90% 85% 80% 75% 70% 0%
10%
20%
It =20mA It =100mA
30% 40% 50% 60% 70% 80% Proportion of positive ion beam current It =40mA It =120mA
It =60mA It =140mA
90%
100%
It =80mA
Fig. 6. The change of transmission rate with the proportion of positive ion beam current for different total currents. The total current keeps constant on each curve.
acceleration. The interaction between the two beams is negligible. (2) When 20 mAoI t o100 mA; higher transmission rates can be achieved by single beam acceleration. The transmission rate is lowest when the current of positive and negative beams are the same. (3) When It4100 mA, higher transmission rates can be achieved by simultaneous acceleration. The transmission rate is highest when the currents of positive and negative beams are the same. Comparisons of the transmission rate and emittance of single beam acceleration (50 mA H+) with simultaneous acceleration (25 mA H+ and 25 mA H) is given in Fig. 7, which is expected to illustrate why the transmission rate of simultaneous acceleration is lower when 20 mAoI t o100 mA: Fig. 7(a) gives the transverse and longitudinal transmission rate along longitudinal position Z. As can be seen, for simultaneous acceleration, the loss occurs in the Gentle Buncher (about Z=100 cm), where the transmission rate drops quickly in both transverse and longitudinal directions. Fig. 7(b) and (c) give the transverse and
longitudinal RMS emittance of H+ along longitudinal position Z. In distinct contrast to (b), Fig. 7(c) reveals that both transverse and longitudinal emittances for simultaneous acceleration reach maximum in the Gentle Buncher (about Z=100 cm). Due to the longitudinal space charge effect of the other beam with opposite charge state, more ions near the ends of their own buckets are attracted (or ‘‘trapped’’) into the oppositely charged bunches (refer to Fig. 4(c)). These ions leave their own buckets gradually and become lost in either longitudinal or transverse direction, leading to a quick descent of the total transmission rate. Similar result has been reported in Ref. [12]. The emittances diminish after the lost particles are discarded by the code. Potential functions are adopted to demonstrate the characteristics of longitudinal motion of ions in the RFQ, which is expected to explain why the transmission rate of simultaneous acceleration is higher when I t 4100 mA: These functions are obtained at the beginning of the Gentle Buncher (the 94th cell, Z 50 cm), where the phenomenon of the aforementioned ‘‘trapping’’ is prominent. In the calculation, positive and negative bunches are represented by uniformly charged ellipsoids with
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Transmission rate R
100% single beam acceleration
98% Rt
96%
simultaneous acceleration
94% Rl
92% 90% 0
40
80
120
(a)
160 200
240
280 320
360
εt
0.6
0.6 εl
0.4
0.8
0.4 0.2
0.2
0
40
(b)
Longitudinal emittance εl (MeV.Deg)
0.8
0 80 120 160 200 240 280 320 360 400 440 480 Longitudinal position Z (cm)
1
1
εl
simultaneous acceleration
0.8
0.8
0.6
0.6 εt
0.4
0.4 0.2
0.2
Longitudinal emittance εl (MeV.Deg)
Transverse emittance εt (mm.mrad)
single beam acceleration
0
0
0 0 (c)
440 480
1
1
Transverse emittance εt (mm.mrad)
400
Longitudinal position Z (cm)
40
80 120 160 200 240 280 320 360 400 440 480 Longitudinal position Z (cm)
Fig. 7. Simulation results of single beam acceleration (50 mA H+) and simultaneous acceleration (25 mA H+ and 25 mA H). (a) Transverse and longitudinal transmission rates for single beam acceleration and simultaneous acceleration along longitudinal position Z. (b) Transverse and longitudinal RMS emittances of H+ for single beam acceleration along longitudinal position Z. (c) Transverse and longitudinal RMS emittances of H+ for simultaneous acceleration along longitudinal position Z.
azimuthal symmetry. Furthermore, two approximations are made: (1) the width of the bucket is 2p and (2) the charge distribution of adjacent bunches are considered to be point charges. Fig. 8(a)–(c) give the potential function curves for the positive ions of single beam acceleration considering only the external fields (I t ¼ 0 mA; Fig. 8(a)), and for both the external fields and the space charge effect (Fig. 8(b) for I t ¼ 50 mA and Fig.
8(c) for I t ¼ 100 mA). We can see that when the current increases from 0 to 100 mA, the bucket width of single beam acceleration becomes narrower (a descent of 37%) due to the space charge effect of the beam itself, which will account for the rapid increase of the longitudinal loss, resulting in the decrease of transmission rate. Fig. 8(d) gives the potential function curve for the positive ions of simultaneous acceleration considering both the
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Fig. 8. Potential functions in the longitudinal motion of single beam acceleration and simultaneous acceleration. (a) Potential function curve for the positive ions of single beam acceleration considering only external fields (I=0 mA). (b) Potential function curve for the positive ions of single beam acceleration considering both external fields and space charge effect (I=50 mA). (c) Potential function curve for the positive ions of single beam acceleration considering both external fields and space charge effect (I=100 mA). (d) Potential function curve for the positive ions of simultaneous acceleration considering both external fields and space charge effect (I=50 mA(H+)+50 mA(H)).
external fields and the space charge effect (It=50 mA(H+)+50 mA(H)). In contrast to single beam acceleration of I t ¼ 100 mA (Fig. 8(c)), the bucket width is larger (an increase of 15%) for simultaneous acceleration (It=50 mA(H+)+50 mA(H)), which is mainly because the total charge in one bunch of simultaneous acceleration (It=50 mA(H+)+50 mA(H)) is halved. Therefore, simultaneous acceleration has less longitudinal loss when I t 4100 mA: In contrast to single beam acceleration of I t ¼ 50 mA (Fig. 8(b)), the bucket depth (corresponding to the depth of longitudinal potential well) is much less (a descent of 70%) for simultaneous acceleration (It=50mA(H+)+50 mA(H)). This is mainly due to the longitudinal debunching effect of bunches on other bunches with opposite charge state. Therefore, ions near the ends of their own buckets
are ‘‘trapped’’ more easily into the bunches with opposite charge state.
4. Conclusions We have studied in detail the simultaneous acceleration of intense H+ and H with the modified code of PARMTEQ, based on one RFQ structure designed for single beam acceleration (the nominal current is 50 mA). Simulation results with different mixing ratios and total currents are presented. The typical dynamic characteristics is described for the simultaneous acceleration of 50 mA H+ and 50 mA H. ‘‘Trapping’’ of particles is illustrated by comparing the transmission rate and emittances of single beam acceleration (50 mA H+) with simultaneous accel-
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eration (25 mA H+ and 25 mA H), which results in the quick descent of the total transmission rate for simultaneous acceleration when the total current is larger than 20 mA. Similar result has also been reported [12]. Further studies on potential functions at the beginning of Gentle Buncher reveal that the depth of longitudinal potential well is much less for simultaneous acceleration (50 mA H+ and 50 mA H) due to the longitudinal debunching effect on the other bunches with opposite charge state, comparing with the single beam acceleration of 50 mA H+. Therefore, the ions near the ends of their own buckets are ‘‘trapped’’ more easily into the oppositely charged bunches. However, simultaneous acceleration has less longitudinal loss than single beam acceleration when the total current is larger than 100 mA, which is due to the halved total charge in one bunch and larger bucket width. So far it seems that a new RFQ accelerator should be specially designed for simultaneous acceleration to achieve a high transmission rate. Based on our simulation result, some suggestions are presented for the design of RFQ accelerators for simultaneous acceleration: (1) The increment of synchronous phase should be slowed down in the Shaper. That is, to slow down the shrink speed of bucket in the Shaper, so that the ions to be ‘‘trapped’’ can return back more easily to their own buckets. (2) The growth of modulation factor should be faster in the Shaper. That is to enlarge the bunching forces, which can also shorten the length of the RFQ. (3) The mean aperture should be increased provided that the intervane voltage keeps unchanged. That is to reduce the focusing strength in the Shaper. Because of the neutralization of the transverse space charge effect, less focusing strength is needed at the beginning of the RFQ. At the same time, the envelop of matched beam will enlarge for the weaker focusing strength, which will relax the parameter design of the LEBT section before the RFQ. Sometimes the scheme of simultaneous acceleration of two beams with opposite charge states is still advantageous for the design of the whole accelerator facility, though new RFQ accelerators
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need to be constructed specially and the matching problems need to be handled properly. The neutralization of transverse space charge forces has the advantage of suppressing the transverse emittance growth for simultaneous acceleration, and the total charge in one bunch is halved, which will relax the parameter design of the DTL and CCL sections following RFQ accelerators. Therefore, simultaneous acceleration may be attractive to reduce the total economic cost, which will promote the continuous study of it.
Acknowledgments This work is supported by Major State Basic Research Development Program (G1999022600). The authors would like to thank Kenneth R. Crandall for great help in PIC method, Zihua Luo, Xialing Guan, Qingchang Yu, Huashun Zhang, Huafu Ouyang, and Taoguang Xu for helpful suggestions, and Haotong Zhao, Yading Yuan, and Guofeng Xie for fruitful discussions in VC++ programming. References [1] T.E. Mason, et al., Appl. Phys. A 74 (Suppl.) (2002) S11. [2] R.A. Jameson, et al., Nucl. Instr. and Meth. B 68 (1992) 474. [3] M. Pabst, et al. Proceedings of 1998 EPAC Conference, 1998, p. 146. [4] T.P. Wangler, et al., Proceedings of 2000 LINAC Conference, 2000, p. 341. [5] J.W. Staples, AIP (1992) 1485. [6] Los Alamos Neutron Science Center Division Annual SelfAssessment, LA-UR-00-1701, April 2000. [7] C.E. Chen, et al., Proceedings of 1996 EPAC Conference, 1996, p. 702. [8] H.E. Ahn, et al., Nucl. Instr. and Meth. A 474 (2001) 1. [9] R.W. Hockney, et al.,Computer Simulation Using Particles, 1988. [10] K.R. Crandall, Proceedings of 1991 PAC, 1991, p. 401. [11] Y. Oguri, Nucl. Instr. and Meth. A 373 (1996) 175. [12] B. Bondarev, et al., Proceedings of APAC01, 2001, p. 400. [13] K.R. Crandall, et al., Proceedings of 1979 LINAC Conference, 1979, p. 205. [14] K.R. Crandall, AIP (1988) 22. [15] J. Qiang, et al., Phys. Rev. Spec. Topo. Accel. Beams 5 (2002) 064201.
Nuclear Instruments and Methods in Physics Research A 538 (2005) 143–153 www.elsevier.com/locate/nima
The longitudinal ‘‘trapping’’ phenomenon in the simultaneous acceleration of intense H+ and H in an RFQ Q.Z. Xinga,, S.N. Fub, Y.Z. Lina, S.X. Fangb a
Department of Engineering Physics, Accelerator Laboratory, Tsinghua University, Beijing 100084, PR China b Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100039, PR China Received 10 August 2004; accepted 14 September 2004 Available online 14 October 2004
Abstract The phenomenon of longitudinal ‘‘trapping’’ in the simultaneous acceleration of intense H+ and H beams in an RFQ accelerator is described in this paper. A new code named PARMTEQVC is compiled based on an old version of tcode PARMTEQ for the computer simulation. In this code, the 2D PIC method is applied to calculate the space charge forces. Multi-bunch effects are also taken into account. In our simulation the RFQ structure parameters which were designed only for single beam (H+ or H) acceleration are adopted. The simulation result shows that more ions near the ends of buckets are ‘‘trapped’’ into the bunches with opposite charge state. These ions leave their own buckets gradually and become lost in either longitudinal or transverse direction, leading to a descent of the total transmission rate. Further studies reveal that the depth of longitudinal potential well decreases in the case of the simultaneous acceleration, which is attributed to the weaker bunching forces because of the longitudinal space charge interaction between the positive and negative ion bunches. It seems that a new RFQ structure should be specially designed for the simultaneous acceleration in order to neutralize the transverse space charge effect and increase the beam transmission rate at the same time. The design ideas are then presented. r 2004 Elsevier B.V. All rights reserved. PACS: 29.27.Bd; 29.27.Eg; 29.27.Fh; 41.75 i Keywords: Computer simulation; Simultaneous acceleration; RFQ; PARMTEQ; PIC method
1. Introduction The high-intensity proton linacs (with peak currents more than tens of mA) have become Corresponding author. Fax: +86 10 62782658.
E-mail address: [email protected] (Q.Z. Xing).
increasingly attractive for the spallation neutron source [1] and the accelerator-driven energy production and transmutation of waste system [2] in the development and application of neutron science. Beam loss is a major problem in the high intensity proton linacs, which can cause activation of the accelerator components and
0168-9002/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2004.09.011
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make unconstrained maintenance difficult. One of the reasons for beam loss is the formation of beam halo [3]. Under certain conditions, a small fraction of particles can acquire enough transverse energy from the repulsive space charge forces within the beam to form a halo [4]. The simultaneous acceleration of two intense beams with opposite charge state is then considered to compensate the large nonlinear space charge effect and restrain the beam halo evolution in the low-energy section of the linac, where the large Cockcroft–Walton DC preaccelerator has been usually replaced by an RFQ [5]. The positive and negative CW ion beams extracted from respective sources are mixed and then sent into an RFQ. The two beams are focused, bunched and accelerated in the RFQ and then entered into the following accelerating structures, such as a DTL or CCDTL section, where the two beams are then separated longitudinally. The mechanism of simultaneous acceleration is most complicated in the RFQ since the strongest interaction between the two beams happens here. This interaction must be carefully considered and clearly understood in order to achieve a high transmission rate. High-intensity experiments of simultaneous acceleration in the RFQ have never ever been performed to the best of the authors’ knowledge. The proton linac of LANSCE can simultaneously accelerate H+ and H ion beams to 800 MeV with peak currents of 17 and 12 mA, respectively, but no RFQ is used [6]. Simultaneous acceleration of O+ and O ion beams in an Integrated Split Ring RFQ has been carried out at low intensity (peak currents are less than 1 mA) in Peking University [7], which proves experimentally that in the RFQ both the positive and negative half periods of the RF cycle can be used to accelerate both signs of ions at the same time. Other project of simultaneous acceleration using RFQ has only been proposed [8]. Prior to deliberate experiments, computer simulation [9] should be chosen to study the simultaneous acceleration since the highly nonlinear and time-dependent behavior of this process in the RFQ probably precludes the accurate analytic solution. Simulations of simultaneous acceleration in the RFQ including the space charge effect have
been accomplished independently by Crandall [10], Oguri [11] and LIDOS team of Moscow Radiotechnical Institute [12]. Similar results have been reported by Crandall and Oguri that positive and negative beams can be accelerated simultaneously in an RFQ linac, but in the dual beam operation the transmission efficiency decreases rapidly with the input beam intensity compared with that in the single beam mode. It has also been found by Oguri that the attractive forces between the positive and negative ions disturb the bunch formation in the structure, leading to an increase in the longitudinal beam loss. Further studies of LIDOS team reveal that with current growth in the bunched beam a small part of oppositely charged particles are trapped inside the bunch in the beginning of the RFQ. Crandall gives a simplified analysis where only the positive particles are traced and the effect of the negative beam on the positive one is calculated by assuming that the two charge distributions are identical but offset by 1801. Moreover, it can only deal with two beams having equal currents. In the study of Oguri, the space charge effect is calculated by summing up the Coulomb forces from all other macroparticles, for which calculation with macroparticles more than 29 is not routinely possible owing to the extremely long CPU time. LIDOS team only reveals the phenomenon of ‘‘trapping’’ but the detailed dynamic process is not presented. In this paper, the simultaneous acceleration of intense H+ and H beams in the RFQ is studied in more detail. To this end, the dynamic simulation of dual ion beams is added into one old version of PARMTEQ [13,14], a common RFQ design and dynamic simulation code. The modified code is written in VC++ language to maintain a more user friendly interface, named as PARMTEQVC. One positive bunch and one negative bunch are traced in the dynamic simulation. Calculation of the space charge forces is accomplished with the 2D Particle-In-Cell (PIC) method, and up to order of 104 macroparticles can be traced keeping the computational time tolerable. The numbers of longitudinal and radial meshes are unlimited and can only be restricted by the computer capacity. The input function of the RFQ cell parameters is added into PARMTEQVC by the means of
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reading a file generated by other design codes. Based on the cell parameters of an RFQ designed for single beam acceleration in the Clean Nuclear Power Project of Major State Basic Research Development Program in China, the detailed dynamic simulation result of simultaneous acceleration is given and discussed, and the analysis of the phenomenon of longitudinal ‘‘trapping’’ is also presented. The paper is organized in the following way. Section 2, describes the code revision for the dynamic simulation of two beams with opposite charge states. Results and discussion are given in Section 3. Conclusions drawn from this study are presented in Section 4.
2. Revision of code PARMTEQ To modify the PARMTEQ code and simulate the dynamics of positive and negative ions simultaneously, two initial distributions of the two beams must be produced separately. Then, the motion of the two oppositely charged ion beams in the external field should be considered. Especial attention should be focused on the space charge interaction of the two kinds of ions. This section introduces our work on these aspects. The time-dependent simulation (t-code) is adopted in PARMTEQVC to avoid errors due to the z-to-t transformation in the space charge calculation of the position-dependent simulation (z-code) [15]. However, the initial six-dimensional distribution ðx0 ; x00 ; y0 ; y00 ; f0 ; W 0 Þ of the two beams at the entrance of RFQ in the z-code is produced first, then the corresponding distribution ðxt0 ; x_ t0 ; yt0 ; y_ t0 ; zt0 ; z_t0 Þ in the t-code is derived from it pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi z_t0 ¼ W 0 ð2m0 þ W 0 Þ c=ðm0 þ W 0 Þ zt0 ¼ z_t0 T ðf0 3p=2Þ=2p y_ t0 ¼ y00 z_t0 yt0 ¼ y0 þ y00 zt0 x_ t0 ¼ x00 z_t0 xt0 ¼ x0 þ x00 zt0
ð1Þ
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where a dot or prime denotes the derivative with respect to time t or longitudinal position z, and T is the RF period, m0 the rest mass of one ion and W 0 the initial kinetic energy of the ion. The acceptance phase space of the positive ion beam is the same as the negative one in both x–x0 and y–y0 at the entrance of RFQ. The two initial beams are focused at the RFQ entrance. They have an initial uniform distribution in the four-dimensional transverse phase ellipsoid, and a uniform longitudinal phase distribution and random energy spread. Although positive and negative ions occupy the cells continuously at the beginning of RFQ, only the particles in 2p phase for each beam are analyzed. Because the interval of adjacent positive and negative buckets is p; the initial phase distribution is from 3p=2 to p=2 for positive ions and from p=2 to 3p=2 for negative ions in PARMTEQVC. One negative synchronous particle with initial phase p=2 is appended into the code (the positive one has initial phase p=2). The derived longitudinal distribution in the t-code is shown in Fig. 1. The motion of both the positive and negative synchronous macroparticles is traced in the dynamic calculation except in the space charge calculation. The external electric fields can be obtained from the two-term potential function [13]. More terms corresponding to the nonlinear
Fig. 1. Initial longitudinal distribution of positive and negative particles traced in the t-code and z-code. The distribution in tcode is derived from that in z-code through Eq. (1).
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field in the quadrupole channel with non-hyperbolic vane profiles should be used [11] for more precise study, but in the present work, results from the two-term expression still make sense. The trajectory of each macroparticle is traced by the drift/thin-lens/drift method, and the space charge calculation is carried out with the 2D PIC method each time the positive synchronous particle reaches the middle of cells. Space charge forces are calculated assuming that the charge distribution is periodic, that is, the adjacent pairs of positive and negative bunches are assumed to have the same charge distribution as that of the positive and negative bunches being traced. This is reasonable provided that the structure parameters change gradually along the cells of RFQ. The longitudinal midpoint between the positive and negative synchronous particles is chosen as the longitudinal center of the field grid (as shown in Fig. 2). The charge (with its sign) of each macroparticle is distributed to the corresponding charge grid nodes according to the transformed position of the macroparticle. The positions of positive and negative macroparticles are calculated separately because positive and negative bunches have different RMS sizes. The electric fields at the field
Fig. 2. Sketch of field grid and charge grid in the calculation of space charge forces. Total charge contained within each cell of field grid is q placed at the location of the centroid of that cell, ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi that is, r ¼ ðr21 þ r22 Þ=2:
grid nodes produced by the two kinds of macroparticles are calculated and then preserved separately. After interpolation the electric fields at the position of each macroparticle can be derived. At the beginning of RFQ, the positive and negative macroparticles to be traced occupy 3-cell length in the longitudinal direction, so the number of longitudinal meshes is increased from 40 to 400 in the dual beam calculation. The longitudinal mesh interval is changeable to keep the total longitudinal mesh length as 3-cell length. The number of adjacent pairs of positive and negative bunches considered is increased from 5 to 10. The population ratio between the positive and negative macroparticles is held to be equal to the current ratio between them. Up to order of 104 macroparticles can be traced keeping the computational time tolerable. Using a PC with two 1.2 GHz Pentium CPUs and 512 M memory, the time cost is less than 10 min for one run.
3. Results and discussion The function of acquiring the external cell parameters is added into PARMTEQVC, so as to do simulations with the cell structures designed by other codes. The matched input beam parameters are calculated before each run. Simulations of simultaneous acceleration (H+ and H) with different mixing ratios and total currents are carried out based on the structure parameters (see Fig. 3) of the RFQ accelerator in the Clean Nuclear Power Project of Major State Basic Research Development Program in China, which was designed for single beam acceleration (nominal current is 50 mA). Table 1 gives the main design parameters of this RFQ. Fig. 4 shows the longitudinal phase spaces at times when the positive synchronous particle reaches the centers of some cells along the RFQ for both I ¼ 0 and I=100 mA (50 mA H+ and 50 mA H), starting with initial DC H+ and H ion beams. Each point represents one positive or negative macroparticle. The region inside the closed curves indicate the longitudinal stable areas. The negative bunch being traced precedes the positive bunch by one cell, that is, the negative
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Fig. 3. Structure parameters of the RFQ in the Clean Nuclear Power Project. B is the focusing factor, X and A are the focusing and accelerating parameters, respectively, W is the kinetic energy of the synchronous particle, f is the synchronous phase, m is the modulation factor, R0 and a are the mean and minimum apertures, respectively.
Table 1 Main design parameters of the RFQ in the Clean Nuclear Power Project Parameter
Value
Beam species Frequency RFQ type Input energy Output energy Nominal current RFQ total length Number of cells
H+ 352.2 MHz Four-vane 75 KeV 3.5 MeV 50 mA 468.22 cm 342
bunch experiences one more cell’s acceleration. Therefore, from Fig. 4(d) to (f) the mean longitudinal velocity of the negative macroparticles are a little larger than that of the positive macroparticles apparently. The numbers of the positive and negative macroparticles are both 104. The
unnormalized total emittances at the entrance of RFQ are both 95.2 mm mrad. The initial energy spreads are set to zero. The matched Twiss parameters for the dual beams are obtained from that of a zero-current beam because the positive and negative beams are initially space–charge neutralized. The longitudinal dynamic behavior of simultaneous acceleration can be observed from Fig. 4 as follows: (1) From the Radial Matching Section (RMS) to the beginning stage of Shaper (Fig. 4(a) and (b)), the longitudinal phase space has little change. The positive and negative particles are mixed together so that space charge forces are neutralized sufficiently. (2) In the Shaper (from Fig. 4(b) to (c)), the positive and negative particles begin to bunch in their own buckets. The phase interval between the adjacent positive and negative
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Fig. 5. Phase space at the entrance and exit of the RFQ. (a) Phase space at the entrance of the RFQ; (b) Phase space at the exit of the RFQ.
bunches is 1801. The two bunches are not completely separated at the end of the Shaper, where a proportion of particles with opposite charge state ‘‘trapped’’ in the positive and negative bunches exists. (This will be discussed later.) (3) In the Gentle Buncher (from Fig. 4(c) to (d)), the positive and negative bunches separate gradually until they become separated completely at the end of it. The ‘‘trapped’’ particles at the end of the Shaper have disappeared. (4) After the Gentle Buncher (from Fig. 4(d) to (f)), the longitudinal internal between positive and negative bunches become larger while the length of cells increases along the RFQ. The interaction between positive and negative particles becomes negligible. The two bunches are accelerated until they reach the end of the RFQ.
The transverse phase space distribution at the entrance and exit of the RFQ is given in Fig. 5. Alternating longitudinal fields in the RFQ cause the two beams to be separated longitudinally by one half of the RF cycle. Therefore, positive and negative ion bunches have almost the same
distribution in the x–x0 and y–y0 phase space at the exit of the RFQ which is due to different phases of positive and negative bunches. More simulation results are given in Fig. 6 to illustrate the change of transmission rate with the proportion of positive ion beam current while the total current keeps constant. Results of different total currents are given on different curves. As we can see, curves are almost symmetric about the center where the positive and negative beams have the same current (the proportion of the positive ion beam current is 50%). That is to say, for example, the transmission rate of accelerating 20 mA H+ and 80 mA H is the same as that of accelerating 80 mA H+ and 20 mA H. This is reasonable because positive and negative beams are DC beams at the entrance of the RFQ and the acceleration field is RF field. The symmetry suggests validity of the results, and the small deviation of symmetry mainly results from the random initial distribution of macroparticles. The following conclusions can be drawn from Fig. 6: (1) When the total current I t o20 mA; high transmission rates can be achieved by both single beam acceleration and simultaneous
Fig. 4. Longitudinal phase spaces when the positive synchronous particle reach the centers of some cells, starting with initial CW H+ and H ion beams: (a) cell 1 (the beginning of the RFQ), (b) cell 7 (start of the Shaper), (c) cell 94 (start of the Gentle Buncher), (d) cell 215 (end of the Gentle Buncher), (e) cell 272 (start of the Accelerating section), (f) cell 342 (end of RFQ). The negative bunch being traced precedes the positive bunch by one cell.
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100%
Transmission rate
95% 90% 85% 80% 75% 70% 0%
10%
20%
It =20mA It =100mA
30% 40% 50% 60% 70% 80% Proportion of positive ion beam current It =40mA It =120mA
It =60mA It =140mA
90%
100%
It =80mA
Fig. 6. The change of transmission rate with the proportion of positive ion beam current for different total currents. The total current keeps constant on each curve.
acceleration. The interaction between the two beams is negligible. (2) When 20 mAoI t o100 mA; higher transmission rates can be achieved by single beam acceleration. The transmission rate is lowest when the current of positive and negative beams are the same. (3) When It4100 mA, higher transmission rates can be achieved by simultaneous acceleration. The transmission rate is highest when the currents of positive and negative beams are the same. Comparisons of the transmission rate and emittance of single beam acceleration (50 mA H+) with simultaneous acceleration (25 mA H+ and 25 mA H) is given in Fig. 7, which is expected to illustrate why the transmission rate of simultaneous acceleration is lower when 20 mAoI t o100 mA: Fig. 7(a) gives the transverse and longitudinal transmission rate along longitudinal position Z. As can be seen, for simultaneous acceleration, the loss occurs in the Gentle Buncher (about Z=100 cm), where the transmission rate drops quickly in both transverse and longitudinal directions. Fig. 7(b) and (c) give the transverse and
longitudinal RMS emittance of H+ along longitudinal position Z. In distinct contrast to (b), Fig. 7(c) reveals that both transverse and longitudinal emittances for simultaneous acceleration reach maximum in the Gentle Buncher (about Z=100 cm). Due to the longitudinal space charge effect of the other beam with opposite charge state, more ions near the ends of their own buckets are attracted (or ‘‘trapped’’) into the oppositely charged bunches (refer to Fig. 4(c)). These ions leave their own buckets gradually and become lost in either longitudinal or transverse direction, leading to a quick descent of the total transmission rate. Similar result has been reported in Ref. [12]. The emittances diminish after the lost particles are discarded by the code. Potential functions are adopted to demonstrate the characteristics of longitudinal motion of ions in the RFQ, which is expected to explain why the transmission rate of simultaneous acceleration is higher when I t 4100 mA: These functions are obtained at the beginning of the Gentle Buncher (the 94th cell, Z 50 cm), where the phenomenon of the aforementioned ‘‘trapping’’ is prominent. In the calculation, positive and negative bunches are represented by uniformly charged ellipsoids with
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Transmission rate R
100% single beam acceleration
98% Rt
96%
simultaneous acceleration
94% Rl
92% 90% 0
40
80
120
(a)
160 200
240
280 320
360
εt
0.6
0.6 εl
0.4
0.8
0.4 0.2
0.2
0
40
(b)
Longitudinal emittance εl (MeV.Deg)
0.8
0 80 120 160 200 240 280 320 360 400 440 480 Longitudinal position Z (cm)
1
1
εl
simultaneous acceleration
0.8
0.8
0.6
0.6 εt
0.4
0.4 0.2
0.2
Longitudinal emittance εl (MeV.Deg)
Transverse emittance εt (mm.mrad)
single beam acceleration
0
0
0 0 (c)
440 480
1
1
Transverse emittance εt (mm.mrad)
400
Longitudinal position Z (cm)
40
80 120 160 200 240 280 320 360 400 440 480 Longitudinal position Z (cm)
Fig. 7. Simulation results of single beam acceleration (50 mA H+) and simultaneous acceleration (25 mA H+ and 25 mA H). (a) Transverse and longitudinal transmission rates for single beam acceleration and simultaneous acceleration along longitudinal position Z. (b) Transverse and longitudinal RMS emittances of H+ for single beam acceleration along longitudinal position Z. (c) Transverse and longitudinal RMS emittances of H+ for simultaneous acceleration along longitudinal position Z.
azimuthal symmetry. Furthermore, two approximations are made: (1) the width of the bucket is 2p and (2) the charge distribution of adjacent bunches are considered to be point charges. Fig. 8(a)–(c) give the potential function curves for the positive ions of single beam acceleration considering only the external fields (I t ¼ 0 mA; Fig. 8(a)), and for both the external fields and the space charge effect (Fig. 8(b) for I t ¼ 50 mA and Fig.
8(c) for I t ¼ 100 mA). We can see that when the current increases from 0 to 100 mA, the bucket width of single beam acceleration becomes narrower (a descent of 37%) due to the space charge effect of the beam itself, which will account for the rapid increase of the longitudinal loss, resulting in the decrease of transmission rate. Fig. 8(d) gives the potential function curve for the positive ions of simultaneous acceleration considering both the
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Fig. 8. Potential functions in the longitudinal motion of single beam acceleration and simultaneous acceleration. (a) Potential function curve for the positive ions of single beam acceleration considering only external fields (I=0 mA). (b) Potential function curve for the positive ions of single beam acceleration considering both external fields and space charge effect (I=50 mA). (c) Potential function curve for the positive ions of single beam acceleration considering both external fields and space charge effect (I=100 mA). (d) Potential function curve for the positive ions of simultaneous acceleration considering both external fields and space charge effect (I=50 mA(H+)+50 mA(H)).
external fields and the space charge effect (It=50 mA(H+)+50 mA(H)). In contrast to single beam acceleration of I t ¼ 100 mA (Fig. 8(c)), the bucket width is larger (an increase of 15%) for simultaneous acceleration (It=50 mA(H+)+50 mA(H)), which is mainly because the total charge in one bunch of simultaneous acceleration (It=50 mA(H+)+50 mA(H)) is halved. Therefore, simultaneous acceleration has less longitudinal loss when I t 4100 mA: In contrast to single beam acceleration of I t ¼ 50 mA (Fig. 8(b)), the bucket depth (corresponding to the depth of longitudinal potential well) is much less (a descent of 70%) for simultaneous acceleration (It=50mA(H+)+50 mA(H)). This is mainly due to the longitudinal debunching effect of bunches on other bunches with opposite charge state. Therefore, ions near the ends of their own buckets
are ‘‘trapped’’ more easily into the bunches with opposite charge state.
4. Conclusions We have studied in detail the simultaneous acceleration of intense H+ and H with the modified code of PARMTEQ, based on one RFQ structure designed for single beam acceleration (the nominal current is 50 mA). Simulation results with different mixing ratios and total currents are presented. The typical dynamic characteristics is described for the simultaneous acceleration of 50 mA H+ and 50 mA H. ‘‘Trapping’’ of particles is illustrated by comparing the transmission rate and emittances of single beam acceleration (50 mA H+) with simultaneous accel-
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eration (25 mA H+ and 25 mA H), which results in the quick descent of the total transmission rate for simultaneous acceleration when the total current is larger than 20 mA. Similar result has also been reported [12]. Further studies on potential functions at the beginning of Gentle Buncher reveal that the depth of longitudinal potential well is much less for simultaneous acceleration (50 mA H+ and 50 mA H) due to the longitudinal debunching effect on the other bunches with opposite charge state, comparing with the single beam acceleration of 50 mA H+. Therefore, the ions near the ends of their own buckets are ‘‘trapped’’ more easily into the oppositely charged bunches. However, simultaneous acceleration has less longitudinal loss than single beam acceleration when the total current is larger than 100 mA, which is due to the halved total charge in one bunch and larger bucket width. So far it seems that a new RFQ accelerator should be specially designed for simultaneous acceleration to achieve a high transmission rate. Based on our simulation result, some suggestions are presented for the design of RFQ accelerators for simultaneous acceleration: (1) The increment of synchronous phase should be slowed down in the Shaper. That is, to slow down the shrink speed of bucket in the Shaper, so that the ions to be ‘‘trapped’’ can return back more easily to their own buckets. (2) The growth of modulation factor should be faster in the Shaper. That is to enlarge the bunching forces, which can also shorten the length of the RFQ. (3) The mean aperture should be increased provided that the intervane voltage keeps unchanged. That is to reduce the focusing strength in the Shaper. Because of the neutralization of the transverse space charge effect, less focusing strength is needed at the beginning of the RFQ. At the same time, the envelop of matched beam will enlarge for the weaker focusing strength, which will relax the parameter design of the LEBT section before the RFQ. Sometimes the scheme of simultaneous acceleration of two beams with opposite charge states is still advantageous for the design of the whole accelerator facility, though new RFQ accelerators
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need to be constructed specially and the matching problems need to be handled properly. The neutralization of transverse space charge forces has the advantage of suppressing the transverse emittance growth for simultaneous acceleration, and the total charge in one bunch is halved, which will relax the parameter design of the DTL and CCL sections following RFQ accelerators. Therefore, simultaneous acceleration may be attractive to reduce the total economic cost, which will promote the continuous study of it.
Acknowledgments This work is supported by Major State Basic Research Development Program (G1999022600). The authors would like to thank Kenneth R. Crandall for great help in PIC method, Zihua Luo, Xialing Guan, Qingchang Yu, Huashun Zhang, Huafu Ouyang, and Taoguang Xu for helpful suggestions, and Haotong Zhao, Yading Yuan, and Guofeng Xie for fruitful discussions in VC++ programming. References [1] T.E. Mason, et al., Appl. Phys. A 74 (Suppl.) (2002) S11. [2] R.A. Jameson, et al., Nucl. Instr. and Meth. B 68 (1992) 474. [3] M. Pabst, et al. Proceedings of 1998 EPAC Conference, 1998, p. 146. [4] T.P. Wangler, et al., Proceedings of 2000 LINAC Conference, 2000, p. 341. [5] J.W. Staples, AIP (1992) 1485. [6] Los Alamos Neutron Science Center Division Annual SelfAssessment, LA-UR-00-1701, April 2000. [7] C.E. Chen, et al., Proceedings of 1996 EPAC Conference, 1996, p. 702. [8] H.E. Ahn, et al., Nucl. Instr. and Meth. A 474 (2001) 1. [9] R.W. Hockney, et al.,Computer Simulation Using Particles, 1988. [10] K.R. Crandall, Proceedings of 1991 PAC, 1991, p. 401. [11] Y. Oguri, Nucl. Instr. and Meth. A 373 (1996) 175. [12] B. Bondarev, et al., Proceedings of APAC01, 2001, p. 400. [13] K.R. Crandall, et al., Proceedings of 1979 LINAC Conference, 1979, p. 205. [14] K.R. Crandall, AIP (1988) 22. [15] J. Qiang, et al., Phys. Rev. Spec. Topo. Accel. Beams 5 (2002) 064201.