Simulation of operation modes of isochronous cyclotron by a new interative method

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1 NUKLEONIKA 27;52(1):29 34 ORIGINAL PAPER Simultion of opertion modes of isochronous cyclotron y new intertive method Ryszrd Trszkiewicz, Mrek Tlch, Jcek Sulikowski, Henryk Doruch, Tdeusz Norys, Artur Srok, Igor N. Kiyn Astrct. Opertion mode simultion methods re sed on selection of trim coil currents in the isochronous cyclotron for formtion of the required mgnetic field t certin level of the min coil current. The trditionl current selection method is sed on finding solution for ll trim coils simultneously. After setting the clculted opertion mode, it is usully necessry to perform control mesurement of the mgnetic field mp nd to repet the clcultion for more ccurte solution. The new current selection method is sed on successively finding solutions for ech prticulr trim coil. The trim coils re tken one y one in reverse order from the edge to the center of the isochronous cyclotron. The new opertion mode simultion method is sed on the new current selection method. The new method, s ginst the trditionl one, includes itertive clcultion of the kinetic energy t the extrction rdius. A series of experiments on proton em formtion within the rnge of working ccelertion rdii t extrction energies from 32 to 59 MeV, which were crried out t the AIC144 multipurpose isochronous cyclotron (designed minly for the eye melnom tretment nd production of rdioisotopes) t the INP PAS (Krków), showed tht the new method mkes unnecessry ny control mesurements of mgnetic fields for getting the desired opertion mode, which indictes high ccurcy of the clcultion. Key words: cyclotron opertion mode mthemticl model Introduction R. Trszkiewicz, M. Tlch, J. Sulikowski, H. Doruch, T. Norys, A. Srok The Henryk Niewodniczński Institute of Nucler Physics, Polish Acdemy of Sciences, 152 Rdzikowskiego Str., Krków, Polnd, Tel.: , Fx: , E-mil: I. N. Kiyn Joint Institute for Nucler Reserch (JINR), Dun, Moscow Region, Russi Received: 25 July 26 Accepted: 2 Novemer 26 The opertion mode of multipurpose isochronous cyclotron designed for ccelertion of prticles with different A/Z rtio comprises set of currents in the min coil nd trim coils nd clculted vlue of the RF oscilltor frequency nd dee potentil. The opertion mode is simulted for prticulr prticle ccelertion tsk nd llows the desired mgnetic field to e formed with given ccurcy within the rnge of working ccelertion rdii from the injection system to the ion em extrction system. The initil dt include the type of prticles to e ccelerted, the extrction rdius, the kinetic energy of prticles t the extrction rdius, the hrmonic numer, the mgnetic structure periodicity, the working point selection rdius nd the mesured or simulted mgnetic field mps (for the flutter ccounting). The working point is the cross point of the isochronous mgnetic field nd the men mgnetic field from the min coil current. The isochronous mgnetic field t the working point selection rdius is used for the min coil current clcultion. The opertion mode

2 3 R. Trszkiewicz et l. Fig. 1. Block digrm of the simultion of the isochronous cyclotron opertion modes. 1 Input of initil dt: the type of prticles to e ccelerted, the extrction rdius, the kinetic energy of prticles t the extrction rdius, the hrmonic numer, the mgnetic structure periodicity, the working point selection rdius nd the mesured or simulted mgnetic field mps (for the flutter ccounting); 2 clcultion of the prticle circultion frequency s function of the kinetic energy t the given extrction rdius; 3 clcultion of the min coil current for the isochronous mgnetic field t the working point selection rdius; 4 ccurcy evlution criterion for the clcultion of the min coil current I mc < ε I ; 5 in the trditionl method: simultneous clcultion of set of employed trim coil currents t the clculted level of the min coil current. In the new method: successive clcultion of current in ech employed trim coil t the clculted level of the min coil current; 6 in the trditionl method: criterion for evlution of the isochronous mgnetic field shift vlue depending upon the influence of the contriutions from the trim coils on the flutter B is < ε B (optionl). In the new method: criterion for evlution of the mximum phse motion vlue depending upon the error of the men mgnetic field: ϕ < ε ϕ ; 7 clcultion of the kinetic energy t the given extrction rdius; 8 ccurcy evlution criterion for clcultion of the kinetic energy t the given extrction rdius E < ε E ; 9 correction of the kinetic energy t the given extrction rdius; 1 setting of the clculted trim coil currents with susequent mesurement of the mgnetic field; 11 criterion for evlution of the mximum phse motion vlue depending upon the error of men mgnetic field: ϕ < ε ϕ ; 12 clcultion of the RF oscilltor frequency nd formtion of the isochronous cyclotron opertion mode. clcultion is crried out using set of mesured or simulted mgnetic field mps. The pointers to corresponding files re stored in the isochronous cyclotron dtse. Simultion of the opertion mode is sed on selection of trim coil currents t the clculted level of the min coil current. The trditionl current selection method is sed on finding solution for ll trim coils simultneously. The new current selection method is sed on successively finding solutions for ech prticulr trim coil. The trim coils re tken one y one in reverse order from the edge to the center of the isochronous cyclotron. Opertion mode simultion lgorithm Figure 1 shows generl lock digrm of the simultion lgorithm for the opertion mode of n isochronous cyclotron. Mrked in dsh lines is tht prt of the lgorithm which is used only with the trditionl method

3 Simultion of opertion modes of isochronous cyclotron y new intertive method 31 nd mrked in point lines is tht prt of the lgorithm which is used only with the new method of selecting trim coil currents t the clculted level of the min coil current. In the trditionl method, fter setting the simulted opertion mode, it is usully necessry to perform control mesurement of the mgnetic field mp nd to repet the clcultion for receiving corrections to the previous solution. A series of experiments on proton em formtion within the rnge of working ccelertion rdii t extrction energies from 32 to 59 MeV, which were crried out t the AIC144 isochronous cyclotron t the INP PAS (Krków), showed tht the new method mkes unnecessry ny control mesurements of mgnetic fields for getting the desired opertion mode, which indictes high ccurcy of the clcultion. Mthemticl models In the trditionl s well s in the new trim coil current selection method ll clcultions re crried out t the clculted level of the min coil current determined on the sis of the working point selection. Using the trditionl trim coil current selection method, one first tkes the difference etween the isochronous mgnetic field nd the men mgnetic field from the min coil current (1) B s,i = B is,i, (i = ~ n, I mc ) where: i is the index of the rdius vlue; B is,i is the isochronous mgnetic field t the i-th rdius; B s,i is the verged y zimuth men mgnetic field from the min coil current t the i-th rdius; I mc is the min coil current. The rdius vlues re tken from to mximum vlue ccording to the rnge nd step of the mgnetic field mp mesurements. To compenste for the ove difference, one tkes, with some residul, sum of the men mgnetic fields of the contriutions from the trim coils: (2) where: B i,j is the men mgnetic field of the contriution from the j-th trim coil t the i-th rdius; I j is the chnge of the j-th trim coil current for B i,j; I j is the current of the j-th trim coil. The dependence of the trim coil contriution upon its current t the clculted level of the min coil current is ssumed to e directly proportionl for ll vlues of the trim coil current. When using the point lest-squres method, one constructs the following functionl to minimize the sum of squres of residues over ll i-th rdii: (3) m Bi, j i j mc j= I j B = I, ( i = ~ n, I ) (,,..., ) = ( i s, i), ( ) 1 m i= where: I, I 1,, I m re the trim coil currents. n S I I I B B I 2 mc The lest-squres method yields the following inhomogeneous system of liner lgeric equtions: (4) where: i is the index of the rdius vlue; j is the index of the trim coil, nd k is the index of the lgeric eqution in the system. A solution of the inhomogeneous system of lgeric equtions cn e found y the direct Gussin method nd represented s (5) I j, (j = ~ m, I mc ) where I j re the trim coil currents t the clculted level of the min coil current. In the new trim coil current selection method, the difference etween the men mgnetic field, depending upon the set of currents in ll coils of the isochronous cyclotron nd the isochronous mgnetic field, is tken for ech trim coil: (6) B mn,i(i l ) = B mn,i(i l ) B is,i, (i = ~ n, l = ~ s, I mc ) where: i is the index of the rdius vlue; l is the index of the trim coil current vlue; B mn,i(i l ) is the men mgnetic field t the i-th rdius depending upon the l-th current vlue in the trim coil under considertion when min field nd contriutions from the considered erlier trim coils re tken into ccount; B is,i is the isochronous mgnetic field t the i-th rdius; I mc is the min coil current. The forementioned difference is tken s mny times s it is prescried ccording to the required step in the current in the trim coil under considertion nd the rnge of its permissile vlues. Then, one constructs discrete function of the current in the trim coil under considertion t the clculted level of the min coil current for the rdius-verged squre of the devition of the men mgnetic field from the isochronous mgnetic field: (7), m n n Bi, j Bik, Bik, I j = Bs, i j= i= I j I k i= Ik ( k = ~ m, I ) mc ( Bmn, i ( Il )) where n is the mximum index vlue for the lst rdius vlue. The necessry interpoltion of the vlues of the discrete function is crried out y the cuic-splines method. Then, n optimum vlue of the current of the trim coil under considertion t the clculted level of the min coil current is found with the required ccurcy y the Newton method: (8) I opt, (I mc ) n 2 ( ) i= l( l) =, = ~, mc F I l s I n + 1

4 32 R. Trszkiewicz et l. Isochronous mgnetic field is successively clculted ccording to the procedure descried in [1], using the softwre descried in kind of the seprte module in [2] (see references therein). Optimum currents in ech prticulr trim coil re successively clculted one y one in reverse order from the edge to the center of the isochronous cyclotron. The minimum nd mximum min coil current vlues, the minimum nd mximum trim coil current vlues nd the low limit nd upper limit vlues of the isochroniztion rnge re used s oundry conditions. The min difference etween two mthemticl models is conditioned y the methods of finding the solution. The trditionl trim coil current selection method implies finding the solution from the min field y one step. The new trim coil current selection method implies finding the solution from the min field step y step. Results On 16 Decemer 24, 2 June 25 nd 15 Decemer 25, experiments on proton em formtion within the rnge of working ccelertion rdii for the clculted middle, high nd low kinetic energies of the prticles respectively were crried out t the AIC144 multipurpose isochronous cyclotron of the INP PAS (Krków). All the ion ems were successfully ccelerted without empiricl trimming of the trim coil currents nd without empiricl trimming of the RF oscilltor frequency within the rnge of the working ccelertion rdii. Tle 1 shows the results of the clcultions. The extrction rdius is the men rdius of the closed equilirium orit. The extrction rdius vlue is used for clcultion of the rottion frequency nd the kinetic energy of the prticles. Conditions for the optiml phse motion within the rnge of the working Tle 1. Isochronous cyclotron opertion modes Item Mode 1 Mode 2 Mode 3 Mode 4 ( ) ( /I) ( /II) (2.6.25) Type of prticles protons protons protons protons Working point selection rdius (cm) Isochroniztion rnge (cm) 8 ~ ~ ~ ~ 57 Extrction rdius (cm) Extrction energy (MeV) RF oscilltor frequency (MHz) Dee voltge (KV) Min coil current (clculted) (A) Min coil current (A) Trim coil current N1 (A) Trim coil current N2 (A) Trim coil current N3 (A) Trim coil current N4 (A) Trim coil current N5 (A) Trim coil current N6 (A) Trim coil current N7 (A) Trim coil current N8 (A) Trim coil current N9 (A) Trim coil current N1 (A) Trim coil current N11 (A) Trim coil current N12 (A) Trim coil current N13 (A) Trim coil current N14 (A) Trim coil current N15 (A) Trim coil current N16 (A) Trim coil current N17 (A) Trim coil current N18 (A) Trim coil current N19 (A) Trim coil current N2 (A)

5 Simultion of opertion modes of isochronous cyclotron y new intertive method 33 ccelertion rdii re chieved ccording to the following rtio: the working point selection rdius vlue the upper limit vlue of the isochroniztion rnge the extrction rdius vlue. The est result cn e chieved when the working point selection rdius vlue is equl to the extrction rdius vlue. The phse motion cn e corrected y chnging the ump prmeters or y chnging the edge mgnetic field grdient vlue. The phse motion correction is used for optiml ion em extrction through the electrosttic deflector. Figures 2 nd 2 show the clculted phse motion nd the experimentl proton em current vlue in mode 1. The significnt phse losses of the ion em current in the opertion mode 1 re due to the repir of the mgnetic system of the isochronous cyclotron efore the experiment. The experiment showed the necessity for new mesurements of the min mgnetic field mps used in clcultions. The following mesurements confirmed the vlidity of this conclusion. Figures 3 nd 3 show the clculted phse motion nd the experimentl proton em current vlue in mode 2. Figures 4 nd 4 show the clculted phse motion nd the experimentl proton em current vlue in mode 3. Fig. 2. Phse motion () nd em current () in mode 1. Fig. 3. Phse motion () nd em current () in mode 2. Fig. 4. Phse motion () nd em current () in mode 3.

6 34 R. Trszkiewicz et l. Fig. 5. Phse motion () nd em current () in mode 4. Opertion modes 2 nd 3 were clculted for the sme vlue of the RF oscilltor frequency ut for the two different vlues of the working point selection rdii. Aprt from tht, two different isochroniztion rnges were set, which led to two different solutions. Figures 5 nd 5 show the clculted phse motion nd the experimentl proton em current vlue in mode 4. The working point selection rdius vlue nd the upper limit vlue of the isochoniztion rnge in opertion mode 4 re significntly lower thn in the previous opertion modes. The selection of the rdius vlue for determintion of the working point is conditioned y the significnt sturtion of the min mgnet iron t lrge vlues of the min coil current. The selection of the upper limit vlue of the isochroniztion rnge is conditioned y the smll vlue of the verticl free oscilltion frequency in front of the extrction rdius. An opertion mode simultion progrm sed on the new method of selecting trim coil currents t the clculted level of the min coil current ws used for the clcultions. This work ws performed without tking into ccount the trim coil mutul influence, ut the new pproch provides such possiility. Conclusion A series of experiments on proton em formtion within the rnge of working ccelertion rdii t kinetic energies of prticles t the extrction rdius from 32 to 59 MeV, which were crried out t the AIC144 multipurpose isochronous cyclotron t the INP PAS (Krków), showed tht the new method of opertion mode simultion does not require control mesurements of mgnetic fields for getting the desired result. The new method, s ginst the trditionl one, llows the time of ringing the ion em to the finl ccelertion rdius fter setting the simulted isochronous cyclotron opertion mode to e considerly reduced (to out 1 min), ecuse the ion em is ccelerted without empiricl trimming of trim coil currents nd without empiricl trimming of the RF oscilltor frequency. The min coil current virtully does not differ from the clculted vlue nd its insignificnt djustment rises only from temperture vrition in the permeility of the min mgnet iron. In ddition, phse losses of the ion em current within the rnge of working ccelertion rdii considerly re decresed ecuse the men mgnetic field more closely pproches to the isochronous mgnetic field. All the foresid indictes high ccurcy of the clcultions y the new method, which leds to pprecile sving in the running time. This is especilly importnt for the opertion of multipurpose fcility which should e switched to new, not yet clculted opertion mode within period of 2 4 h etween the tsk setting nd the formtion of ion em t the finl ccelertion rdius. Another dvntge of the new method over the trditionl one is simpler mthemticl tools, which is in line with the min gol of mthemtics to simplify the existing solution of the prolem. On the other hnd, the new method is fr more complicted in reliztion thn the trditionl method ecuse it implies processing lrge dt rrys. The progrm for simultion of opertion modes of the isochronous cyclotron ws developed in C++ using Visul C++ NET. The isochronous cyclotron dtse ws developed in Trnsct-SQL for MS SQL Server 2. The MS Windows Server 23 opertionl system is used for running this softwre. References 1. Gordon MM (1983) Clcultion of isochronous fields for sector-focused cyclotrons. Prticle Accelertors 13: Kiyn IN, Vorojtsov SB, Trszkiewicz R (22) The men mgnetic field clcultion of the isochronous cyclotron with llownce for flutter progrm description. Communiction of the Joint Institute for Nucler Reserch P JINR, Dun

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