High precision computer simulation of cyclotrons KARAMYSHEVA T., AMIRKHANOV I. MALININ V., POPOV D.

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1 High pecision compute simulation of cyclotons KARAMYSHEVA T., AMIRKHANOV I. MALININ V., POPOV D.

2 Abstact Effective and accuate compute simulations ae highly impotant in acceleatos design and poduction. The most difficult and impotant task in cycloton development is the magnetic field simulations. It is necessay to achieve the accuacy of the model that is highe than the toleance fo the magnetic field in the eal magnet. An accuate model of the magnet and othe systems of the cycloton allows us to pefom beam tacking though the whole acceleato fom the ion souce to the extaction. While high accuacy is necessay in the late stages of eseach and development woks, high pefomance of the simulations and ability to swiftly analye and apply changes to the poject play a key ole in the ealy stages of the poject. Techniques and algoithms fo high accuacy and pefomance of the magnet simulations have been ceated and used fo development of the SC0 cycloton fo poton theapy, which is unde poduction by collaboation between JINR (Dubna, Russia) and ASIPP (Hefei, China).

3 JINR MEDICAL TECHNICAL COMPLEX with the futue SC0 cycloton It is planned to manufactue two cyclotons in China : one will opeate in Hefei cycloton medical cente, the othe will eplace Phasoton in Medico-technical complex JINR Dubna and will be used to teat cance with potons. SC0 Dubna, Phasoton. Oiginally built in 1949, modenied in 1984

4 SC0 magnet design Compute model of the magnet 3D magnet simulation is the most impotant pat when it comes to the development of an isochonous supeconducting cycloton as it defines the paticlemotion in the acceleato. 0.6 Woking diagam Q= Q 0. 01MeV Q-Q= Magnet modeling goes hand in hand with beam dynamics modeling. Q Meshed model

5 Beam dynamics simulations Beam dynamics simulation solves the system of diffeential equations of paticle motion with field maps fom FEM (finite element method) codes o field measuements with diffeent initial coodinatesand velocities. Runge Kutta fouth-ode method

6 Motion of chaged paticles in electomagnetic fields 1 1 ( ) dv q V E VB V VE dt m c c = Systems of diffeential equations of motion of chaged paticles in an electomagnetic field ae deived fom the Newton-Loent equations: = = = = 1 1 ) ( 1 ) ( 1 ) ( ) ( ) ( ) ( V c t V m q B B B B V mc q V m q B B B B mcv q β ε ε ε ε ϕ ϕ ϕ ϕ In cyclotons, it is most convenient to use the aimuth angle as a vaiable. So the epesentation of the equations of motion with an independent vaiable aimuth angle is widely used in the design of cyclotons.

7 Repesentation of the electomagnetic field D D, 3D When calculating paticle tajectoies, an electomagnetic field can be specified analytically, in the fom of a field map obtained as a esult of measuements (D maps) o as a esult of model calculations (3D field maps).

8 Motion of chaged paticles in an electomagnetic field. E-Field B-Field Up to 30 GB The epesentation in the fom of a 3D field map is obviously the most acceptable fo calculations of beam dynamics, but maps can be huge when high accuacy is needed.

9 About the accuacy of the simulations Also the accuacy of the simulations has been inceased and the numeical eo is lowe than 0.1 Gs. This accuacy, of couse, cannot be achieved in poduction, howeve when the accuacy of the model is much highe than the accuacy of manufactuing, it becomes possible to simulate toleances and eos and the majo weak spots that would helpon the poductionstage. B, T B, T Aimuth angle Aimuth angle

10 Radial component of the magnetic field in the median plane as a function of the adius (diffeent cuves coespond to diffeent values of the aimuth angle) B, G R,mm Theoetically, the components Bϕ and B ae equal to eo in the median plane, the pesence of components indicates an eo value that is ±0.03 Gs in the extaction one.

11 Black cuves - the deviation of the magnetic field fom the aveage value in valleys (minimum points), geen lines - deviation fom the aveage value in sectos (maximum points) DeltaB,G The fou-secto stuctue of the magnet makes it possible to estimate the accuacy of the model calculation by compaing the magnetic field fo identical pats of the magnet sectos (calculation in the 360-degee model). R,mm

12 Magnet system design The majo poblem of R&D phase of the SC00 poject was that 3D simulations of the magnet wee vey time-consuming and it was difficult to achieve an acceptable accuacy of the simulated field. In ode to fix this issue CST studio macos and quick analysis of the magnet field map in MATLAB have been developed. Scipts fo poducing 3D magnetic field maps in CST studio and fo eading them into the Matlab wokspace wee witten. This technique allowed us to incease the numbe of simulated magnet geometies to ove 5000 in half a yea. Such fine tuning allowed us to optimie the magnet in vey shot time.

13 B B ϕ = = Using a D field map in the median plane fo beam dynamics simulations. In ode to study non-linea esonances and define the conditions of thei appeaance we have pefomed simulations with a D field map in the median plane used to pefom paticle tacking. This is a taditional method and it is widely used in cycloton physics. Only a vetical component exists in the median plane of the cycloton (due to the symmety) and the othe components of the magnetic field can be calculated by: B B ϕ 3 f3 3 g3 B = B f 4 f 4 f f 1 B 1 B 1 B (, ϕ) = ϕ f g B 1 B 1 B 1 B B (, ϕ) = 6 ϕ ϕ B 1 B 1 B (, ϕ) = 6 ϕ ϕ ϕ B 1 B B B B (, ϕ) = ( 4 ϕ ϕ ϕ B 4 B 1 B 4 3 ϕ The fomulas ae pesented fo the cylindical coodinate system, which is most suitable fo cyclotons. It is impotant to be able to calculate all deivatives fo coect beam dynamics simulations duing eseach and development and, especially on commissioning stage, when eal magnetic field will be measued only on the median plane. )

14 Deivatives of the magnetic field in the median plane taken without smoothing algoithms. The biggest poblem of these calculation is that the deivatives must be taken using the measued o calculated field map, which aleady contains eo. Special mathematical algoithms wee developed in ode to obtain smooth and ealistic deivatives of the magnetic field on the median plane.

15 Deivatives of the magnetic field in the median plane taken with smoothing algoithms. Such esults wee obtained by combining fitting of the field map by spline suface togethe with smoothing algoithms. The developed softwae ceates3d fieldmaps which can be used fo beam dynamic simulations.

16 Acceleating RF system design It is also impotant fo beam dynamics simulations to use an adequate epesentation of the acceleating field. We simulate the acceleating RF cavity in CST Studio. Both electic and magnetic fields of RF cavity fom 3D simulations wee used in paticle tacking fom the ion souce to the extaction point of the SC0. High accuacy of the simulations ae equied in ode to pefom such tacking. Scipts fo poducing 3D field maps in CST Studio and fo eading them into the Matlab wokspace wee witten. The code fo voltage distibution calculation was also ceated. The voltage value was obtained by integating the electic field in the median plane of the esonant cavity. Voltage distibution along the adius

17 Conclusion Codes fo beam dynamics simulations wee upgaded by new algoithms fo calculations of the magnetic field outside the median plane. Scipts fo poducing 3D magnetic field maps, 3D electic and magnetic field maps fom RF cavity simulations in CST Studio and fo eading them into the Matlab wokspace wee witten. High accuacy and high efficiency of simulations will help on commissioning stage duing shimming of the magnet.