Theoretical study of the dual harmonic system and its application on the CSNS/RCS

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1 Theoretical tudy of the dual harmonic ytem and it application on the CSNS/RCS Yao-Shuo Yuan, Na Wang, Shou-Yan Xu, Yue Yuan, and Sheng Wang Dongguan branch, Intitute of High Energy Phyic, CAS, Guangdong Province, Dongguan 53803, China Correponding Author: Sheng Wang addre: Potal addre: No.1, Zhongziyuan Road, 53803, Dalang Town, Dongguan, Guangdong Province, China Telephone: Supported by National Natural Science Foundation of China ( ) Abtract The dual harmonic ytem ha been widely ued in high intenity proton ynchrotron to uppre the pace charge effect, a well a reduce the beam lo. To invetigate the longitudinal beam dynamic in the dual rf ytem, the potential well, the ub-bucket in the bunch and the multi-olution of the phae equation have been tudied theoretically. Baed on thee theoretical tudi, the optimization of bunching factor and rf voltage waveform are made for the dual harmonic rf ytem in the upgrade phae of the CSNS/RCS. In the optimization proce, the imulation with pace charge effect i done by uing a newly developed code C-SCSIM. 1 Introduction Space charge effect i the principal caue of emittance growth and beam lo in high intenity accelerator. In high intenity ynchrotron, the pace charge effect i the limit of the maximum number of particle that can be accumulated. The pace charge induced tune hift i alway ued a a tandard of pace charge, and in a ynchrotron, the relation between the bunching factor B f and the tune hift Δν i [1] r n, (1) p t - 3 B where r p i the claic radiu of proton, n t i the bunch population, ε i the tranvere emittance, β and γ are the relativitic factor. To decreae the pace charge tune hift, increaing the bunching f

2 factor by uing dual harmonic rf ytem i a common method in the high intenity proton ynchrotron [-5]. China Spallation Neutron Source (CSNS) [6-7] i a puled neutron ource with beam power of 100 kw, which i now under contruction. It accelerator conit of an 80 MeV H- linac and a 1.6 GeV proton Rapid Cycling Synchrotron (RCS). In the future upgrade, the beam power will upgrade to 500 kw. A an important way of depreing the pace charge effect, the dual harmonic rf ytem will be employed in RCS in the upgrade tage. In thi paper, ome important beam dynamic iue in the dual rf ytem, uch a the potential well, the multi-olution of the phae equation and the ub-bucket in the bunch ha been theoretically tudied. Baed on thee theoretical tudy, the optimization of bunching factor and rf voltage waveform are made for the dual harmonic rf ytem in the upgrade tage of the CSNS/RCS. In the optimization proce, the imulation with pace charge effect i done by uing a newly developed code C-SCSIM [8]. The relation between bunching factor and the rf voltage ha been tudied and the correponding beam ditribution in the phae pace i preented.. The theoretical tudy.1 The potential well To obtain the formula for the potential well conciely, coordinate (ф, P) i ued [9]. Here, ф i the phae relative to the fundamental rf cavity and P -h / () where h i the ratio between the harmonic number in fundamental rf ytem and higher harmonic rf ytem (h= for dual rf ytem), η the phae lip factor, δ the momentum deviation, ν the ynchrotron tune at zero amplitude for the fundamental rf ytem. The Hamiltonian in thi coordinate ytem can be written a 1 ( P) P U1( ) U ( ) (3) H U1( ) co 1 co ( 1 ) in1 (4) r U ( ) co co[ h( 1 )] h( 1 ) in h (5) where U 1 and ф 1 are the potential energy and ynchronou phae in the fundamental cavity, while U and ф are for the econd-harmonic cavity, r i the ratio of the econd-harmonic voltage and

3 fundamental voltage, i.e. r =-V /V 1. The total voltage can be expreed a V ) V in( ) V in( ) (6) ( 1 where ф i the rf phae angle of the econd-harmonic cavity relative to the fundamental one, (7) 1 - Subtituting Eq.(7) into Eq.(5), the potential energy in the dual harmonic rf ytem can be expreed a 1 ( ) ( co in ) r[ co( ) in( )] (8) U.The effect of r on the potential wall For implicity, we aume ν equal 1. The curve of the function U(ф) in Eq.(8) are plotted in Fig.(1), which how that the hape of the potential well change a r varie from 0.4 to 0.8. From the figure we can ee that when r become larger, the bottom of the potential well become higher. Figure 1: The variation of the potential energy with different r. (a) ф =ф =0; (b) enlarged view of (a) in the bottom; (c) ф =30 deg; ф =30deg; (d) ф =45deg; ф =15deg In order to explain the reult, we take the firt-order derivative of the potential formula in

4 Eq.(8): du( ) (in in ) - rin( ) in( d ) (9) which give, du( ) d 0 (10) Therefore, the lope at ф=ф identically equal to zero, and i independent of the value of ф and ф. Moreover, the econd derivative i d U( ) d co r co( ) (11) Let r equal to 0.5, we have d U ( ) d which mean the potential well i alway flat at ф=ф..3 Multi-olution and two ub-bucket 0 (1) If we aume ν =1, ф =0 and ф =0, Eq.(8) can be implified to du( ) ev1 d E in - rin( ) 0 0, (13) which give r co 1 0 (14) The relation between r and ф in Eq.(14) are lited in Table 1. It can be een that when r>0.5, Eq.(8) ha two olution, correponding to the two bottom of potential well, a hown in Fig.(1)(a)(b). Table 1: The relation between r and ф r ф(deg) 0.4 Nan ± ± ±51.3

5 On the other hand, the total rf voltage in the dual harmonic ytem i V ) V [in( ) rv in( )] (15) ( 1 In the accelerating proce, the rf voltage hould be db(t) V ( ) L (16) dt where ρ and L denote the bending radiu and the circumtance of the ring, B i the magnetic field of the dipole. A hown in Fig.(), when r<0.5, Eq.(16) ha only one olution, i.e. the ynchronou phae ф but when r>0.5, there are three olution: one i the ynchronou phae ф, and other two olution are ф 1 and ф. When r>0.5, the voltage function i no longer monotonic. Figure : Schematic drawing of the voltage wave and the potential at (a) r=0.4; (b) r=0.5 and (c) r=0.8 In which, ф 1 and ф are correponding to the two bottom of the potential well. In thi cae, particle in the bucket can be divided into two part by their energy deviation ΔE, thoe with mall ΔE are trapped within the two depreion in the potential well, and can only ocillate within the range of the two ub-bucket, a marked by rectangular dot in Fig.(3); thoe with larger ΔE can extend to all the whole bucket, a marked by round dot. The ub-bucket plotted a a chematic drawing in Fig.(3), can be oberved by beam imulation, a hown in Fig.(5-6).

6 Figure 3: The chematic drawing of particle with variou ΔE in the potential well and in the two ub-bucket.4. The effect of ф and ф on the potential well Let r=0.5, the curve of the potential well varie with different ф and ф are hown in Fig.(4). Figure 4: the curve of the potential well with r=0.5: (a) ф v. U, with ф =0; (b) ф v. U, with ф =0; (c) ф v U,with ф =30º; (d) ф v. U, with ф =30º To um up, the r affect the extent of the flatne in the bottom of the potential well, which i the reaon of the formation of the ub-bucket in a bucket. The bottom i ymmetrical about the line ф=ф when ф =ф =0, and become uneven when either ф 0 or ф 0. The uneven depend on

7 the value of ф or ф. 3 The application on the CSNS/RCS 3.1 The optimization of bunching factor In ection, we have obtained theoretically the formula of potential well and how the four coefficient ф, ф, V 1 and V influence the hape of the potential well. In fact, thee parameter are not independent. Alo it i difficult to find out an analytic olution for the bunching factor, which often can be conidered a a crucial parameter in the dual harmonic ytem and taken a a key criterion in the rf voltage optimization proce. In the upgrade tage of CSNS/RCS, the dual harmonic ytem will be employed to achieve uniform longitudinal beam ditribution (i.e. the flat potential well) and much larger bunching factor. Early tudy ha been performed for the upgrade cae of a 00 kw beam power [10]. Here, by uing the imulation code C-SCSIM, the relation between the potential well and the bunching factor for the cae of 500 kw beam power ha been invetigated. Table lit the input parameter ued in the imulation. Table : Main input parameter ued in the imulation Circumference(m) 7.9 Bending radiu(m) 8.01 Repetition rate(hz) 5 Harmonic number Chopping factor 50% Injection turn 500 Injection energy(mev) 50 Extraction energy(gev) 1.6 Number of proton(10 13 ) 7.8 Starting time at injection(m) -0.3 Macro-particle for injection 0000 The imulation ha been performed in two tep. Firt, during 0-m in an accelerating proce, the value of r (r=-v /V 1 ) i canned from 0.4 to 0.9 by increaing V while keeping V 1 a a contant in each time point, a lited in Table 3. The imulation reult are hown in Fig.(5). One

8 can ee that a r increae, both the top and the bottom of the bucket hrink while the bunching factor increae at firt and then decreae. Moreover, when r>0.5, the bunch divide into two ub-bucket, and they trend to become more and more clear a r increae. We can alo oberve that when r equal 0.5, the bucket become flat, but the bunching factor i not at it maximum. Table 3: The fundamental voltage waveform ued in tep 1 of the imulation Time(m) Fundamental voltage(kv) Secondly, let the V 1 and V change, but keep r a a contant. Let the value of V 1 and V ued in the Fig.5(f) be a reference, change the value to it 0.6, 0.8,1. and 1.4 time. The imulation reult are hown in Fig.(6), from which we can oberve that a V 1 and V increae, the area of the bucket i enlarged but the area of the bunch and the bunching factor doe not change much compared with the cae in Fig.(5).

9 Figure 5: the particle ditribution with different r at the end of the injection proce (500 th turn) (ф =8.deg,ф =14.0deg, V 1 =77.8kV) Figure 6: The particle ditribution with different V 1 and V at the end of the injection proce 3. The optimization of rf voltage waveform According to the property of the bunching factor obtained above, the voltage waveform can be optimized with an iteration method uing the code C-SCSIM. Generally, the optimization procedure conit of two tep. At firt, the fundamental rf voltage are calculated uing the iteration method [3]. Secondly, the econd rf voltage waveform are optimized according to the beam ditribution and the bunching factor calculated by uing the code. The calculated fundamental rf voltage waveform and the optimized econd-harmonic rf

10 voltage waveform are hown in Fig.7. Fig.8 give the particle ditribution at 500 th turn for the upgrading CSNS/RCS, uing only the calculated fundamental rf voltage waveform and the dual harmonic ytem, repectively. From Fig.8(b) it can be een that the longitudinal beam emittance and the bunching factor i 1.5 ev and 0.51 repectively, which meet the deign requirement. Figure 7: (a) the optimized voltage waveform; (b) the comparion of the bunching factor in dual and ingle rf ytem Figure 8: the comparion of the beam ditribution at 500 th turn for upgrading phae with ingle rf 4 Summary ytem and dual rf ytem

11 Beam dynamic for the dual harmonic rf ytem ha been invetigated by theoretical analyi. The influence of the rf voltage, the relative phae ф and the ynchronou phae ф on the hape of the potential well ha been introduced. The bunching factor and the formation and the characteritic of the ub-bucket in the bunch have been illutrated. Thee theoretical reult are applied on the optimization deign of dual harmonic rf ytem for the future upgrade of CSNS/RCS, and the reult i important for the dual rf ytem deign in the upgrade tage. Reference [1] Lalett L J., On Intenity Limitation Impoed by Tranvere Space-Charge Effect in Circular Particle Accelerator. Proc Summer Sch. BNL, [] Baillod J M, Magnani L, Naibian G et al. IEEE Tran. Nucl. Sci., 1983, 30: [3] Brennan J M, Roer T. Proc. of PAC1999. New York, [4] Middendorf M E, Brumwell F R, Dooling J C et al. Proc. of PAC007. Albuquerque, New Mexico, [5] M. Yamamoto, et al. Nucl. Intr. and Meth. A, 61(010): [6] CSNS Feaibility Study Report, IHEP, 009. [7] J. Wei et al., Chinee Phyic C (HEP & NP), 33(11): , 009. [8] Y. S. Yuan, et al. Nucl. Intr. and Meth. A, 013, 79: [9] S.Y Lee, Accelerator Phyic, World Scientific, Singapore, p.37-p.334, 004. [10] J. F. Chen et al., Chinee Phyic C (HEP & NP), 34(10): , 010.