Accurate Electromagnetic Simulation and Design of Cyclotron Cavity Masoumeh Mohamadian, Hossein Afarideh, and Mitra Ghergherehchi

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1 TNS Accurate Electromagnetic Simulation and Design of Cyclotron Cavity Masoumeh Mohamadian, Hossein Afarideh, and Mitra Ghergherehchi Abstract This study simulated and designed a cyclotron cavity using the circuit model. Based on the full wave electromagnetic numerical simulation, this study proposes an accurate circuit model for all sections of the cavity which allows the investigation of the effects of different parts of the cavity on its characteristics. This procedure could be utilized to estimate structural parameters to avoid time consuming numerical simulations. The resonance frequency of the designed cavity is 71 MHz. A full wave electromagnetic simulation was done based on the finite difference time domain method using CST Microwave Studio. Index Terms Cyclotron RF Cavity, Electromagnetic Simulation, Circuit Modeling A I. INTRODUCTION n accelerator is an apparatus whereby electromagnetic fields are used to accelerate charged particles to high velocities. Accelerators are divided into two classes: electrostatic and oscillating field. Electrostatic accelerators use electrostatic fields to accelerate the particles, while oscillating field accelerators use radio frequency electromagnetic fields for the acceleration process. Both of these classes can be either linear or circular, depending on the factors of maximum desired particle energy and the target type. The following two categories of accelerator therefore exist: Linear particle accelerators and circular, or cyclic, ones. The second category can be further partitioned into several types, including cyclotrons, synchrocyclotrons and isochronous cyclotrons, and Betatrons [1]. Cyclotron accelerators will be discussed in the present paper. This apparatus is used as the first stage of a number of large multi stage particle accelerators to produce radiopharmaceuticals for medical applications [2]. The first cyclotrons were invented by Ernest O. Lawrence in 1932 [3]. This cyclotron accelerated the particles in circular paths using a constant magnetic field. The utilized magnetic field bent the charges in the spiral path to move them between the gaps. An electromagnetic radio frequency (RF) field in the gap was applied to accelerate the charged particles [4]. Manuscript received October 30, M. Mohamadian, is with the Department of Energy Engineering and Physics, Amirkabir University of Technology, Tehran, , Iran. ( mohamadian@aut.ac.ir). H. Afarideh (Corresponding Author) is with the Energy Engineering and Physics Department, University of Amirkabir Univercity of Technology, Tehran, ,Iran. ( Hafarideh@aut.ac.ir). M. Ghergherehchi is with College of Information and Communication Engineering, School of Electronic and Electrical Engineering, Sungkyunkwan University, Suwon, , Korea. ( mitragh@skku.edu). Several small cyclotrons with compact structures have successfully been developed since the 1990s [5, 1]. The conceptual design of the cyclotron systems is unchanged from that reviewed at the 1978 conference, but many details have been modified [6]. The RF cavity is the main cyclotron component, which couples the RF power to the charged particles for accelerating. The characteristics of this structure are very sensitive to its dimensions, and thus should be modeled and simulated accurately to find the proper dimensions for achieving the desired specific frequency [7]. Numerical methods are usually applied to the simulation of such complex geometries. Accordingly, a full wave numerical electromagnetic simulation software such as Computer Simulation Technology, Microwave Studio (CST MWS) [8], or High Frequency Structural Simulator (HFSS) [9] is typically utilized for this purpose. In most studies, an initial geometry that has been designed beforehand is typically used. Then the approximate specification is predominantly achieved through a trial and error process, and the geometry is subsequently modified [10, 11]. Both of these are time consuming procedures. In [12], based on an electromagnetic full wave simulation, a simple circuit model was used for a single part of the cavity. This model can properly demonstrate the behavior of that section. This paper proposes an accurate circuit model based on electromagnetic full wave simulation for all parts of the cyclotron cavity in the designated operating frequency range. In the first step, all parts of the cyclotron cavity structure are simulated using CST software. Based on the frequency behavior of each part, a suitable microwave circuit model is represented. Ultimately, a generalized circuit model is demonstrated in the whole structure. This circuit model facilitates considering the effects of each part of the cavity to its specifications. Therefore, it is possible to achieve the design parameters more conveniently. In addition, this method of analysis can be extended to a conceptual design of the cyclotron to identify its parameter values, whereby the time consuming numerical process will be eliminated. In Section II of this paper, the principle of the cyclotron cavity and its configuration are described. In Section III, the electromagnetic simulation and circuit model of different parts of the cavity are studied. Finally, in Section IV, the whole circuit model of the cavity structure is presented. It is shown that the model has a good agreement with the electromagnetic full wave simulation. Electromagnetic simulation was done with CST, and the circuit model was carried out using Advance Design System (ADS) software [13].

2 TNS II. CYCLOTRON ACCELERATORS constant cyclotron frequency can therefore continue to accelerate the charge [4, 12]. In the proposed designed cavity The cyclotron includes various sections that are illustrated presented in the following, the resonance frequency is 71 MHz in the block diagram of Fig. 1, wherein the RF cavity is the so as to reach the desired energy of 10 MeV. main part. A. Configuration of the Cavity With regard to the overall structure of the described cyclotron accelerator, the different parts of the RF cavity, including the Liners, Dees, Stems, couplers, and tuner sections, are illustrated in Fig. 3. Fig. 1. Block diagram of the cyclotron accelerator In the proposed cyclotron principle, a magnetostatic field which is exerted through external magnets bends the particles in a semicircle, during which time the electromagnetic field polarity is reversed to accelerate the charged particle again as it moves across the gap in the opposite direction. Accordingly, a voltage is applied across the gap between the two electrodes that can accelerate the charge to a higher energy [4]. Electromagnetic field distribution and magnetostatic field distribution are demonstrated in Figs. 2-a and 2-b respectively. Fig. 3. Overall structure of the RF cavity (a) (b) Fig 2. a) Electromagnetic field distribution with CST; b) magnetostatic field direction in the cavity with CST. The major parameters of the design cavity are the cyclotron resonance frequency and its quality factor. As previously mentioned, a charged particle in a cyclotron will move in a semicircular path under the influence of a constant magnetic field. If the time to complete one orbit is calculated as, the period is independent of the radius. Therefore, if a square wave is applied at an angular frequency (qb/m), the charge will spiral outward as its speed increases. When the square wave of an angular frequency ( ) is applied between the two sides of the magnetic poles, the charge will be boosted again precisely at the time that it is needed to accelerate it across the gap. The The Dee sections are the triangular shape plates that are parallel with each other and between which the particles revolve. The Liners are the outer sections of the cavity. The Stems are the cylindrical parts that are perpendicular to the Dees. The couplers and the tuners are the circular disks that are positioned against each other. The roles of these components are explained in this paper. The RF cavity generally works as a resonator which operates as a narrow band-pass filter [1]. The structure of the resonator is different for every application. Therefore, special requirements must be considered for the resonator used in the acceleration of charged particles. With regard to the specifications of cyclotron cavities and their geometric structures, a variety of designs have been developed. The classifications here are based on the electromagnetic and geometric properties, as follows [14]: 1. Frequency range required Fixed frequency cavities Variable energy (and variable particle type) 2. Single gap and double gap cavities Double gap ( /4 or /2 transmission line type) resonators Single gap, waveguide type resonators 3. Normal conducting and superconducting cavities A coaxial resonator of a fixed frequency that comprises a double gap and is superconducting will, therefore, be suitable for the application of the present study. The cyclotron requires a region in the center of the resonator to provide energy to the particles through an electric field because of the conditions

3 that govern charged particle rotation. The following structure is necessary to provide the required energy to the particle in each round of rotation. A horizontal plate called the Dee is in the center, and it comprises a certain angle that generates an electric field between the coaxial inner core called the Stem and its outer shell, called the Liner. III. ELECTROMAGNETIC SIMULATION AND CIRCUIT MODELING A. Simulation and Modeling of Liner and Stem Sections In terms of electromagnetism, the combination of a Liner and a Stem works as a transmission line, or a resonator, which is the main part of such a device, to provide resonance at the desired frequency. This could be the appropriate resonator for such a system. Since the Liner is divided into triangular and cylindrical shapes, it is accordingly composed of two coaxial transmission lines which are in series with each others. There are a cylindrical coaxial line and a uniquely designed, roughly triangular shaped coaxial line (Fig. 4-a). These two transmission lines become a short circuit on the one side that is connected to the chamber, while on the other side, it is physically connected to a Dee section. Hence, it could be modeled as in Fig. 4-b. TNS calculated. B. Resonator Dimensions The dimensions of the resonator including both transmission lines are calculated considering the input impedance while it is short circuited. The considered circuit model for such a transmission line is shown in Fig. 5. Thus, the input impedance is calculated as. For two sections of the transmission line, it will be. The input impedance versus the frequency is also illustrated in Fig. 5, where two frequently repeated resonance points are shown. These resonances occur at infinity and for a short circuit at zero impedances that can be modeled with parallel and series LC circuits, respectively. Since the resonance point here is near infinity in an open circuit impedance, a parallel LC equivalent circuit was used. a Fig. 4. Short circuit quarter wavelength transmission line: a) the half physical structure, and b) equivalent transmission line, l1 = 300 mm, l2 = 200 mm Moreover, because of the propagation of the Transverse Electromagnetic (TEM) mode inside the coaxial, both distinguished electrode parts have the same free space phase constant. As mentioned, the inner conductors of both lines are the same (as the Stem section) with different outer electrodes (as the Liner section). The characteristic impedances of lines are therefore distinct, and they are obtained through the characteristic impedance of the cylindrical part that is a simple coaxial relation. The second line s impedance can be obtained through electromagnetic CST simulation, depending on its geometry and the inner radius. The dimensions of the second part are determined according to the following considerations: the cyclotron magnet, the beam path dynamic, and the applied electric field. The triangular shape, however, comes from the formation of the Dee, and this is discussed later in this paper. In the next step, the dimensions of the design resonator are b Fig. 5. Short circuit transmission line behavior versus frequency, l 1 = 300 mm, l 2 = 200 mm According to the resonator circuit model discussed above, the l 1 and l 2 parameters designate the resonance frequency of the final structure. An increase in transmission line length increases the resonance frequency. These parameter values are l 1 = 300 mm and l 2 = 200 mm in this curve. As previously mentioned, the phase constants of both structures are the same as the free space phase constant, and this means that the overall structural length of these two parts will be equal to the designed coaxial structure length. In the design of the transmission line, the sum of the cylindrical and triangular coaxial lengths (rather than the individual lengths) is important. Accordingly, the height of each part is related to the other manufacturing considerations, including the location of the magnets and the mechanical implement remarks. Fig. 6 illustrates the return loss (S11) for the different Stem diameters which is an important, effective parameter of the cavity. The final value of Stem diameter is 215 mm. However, the resonance frequency of the cavity obtained with optimizing other parameters altimatly. The geometry and

4 TNS dimensions of the Dee section affect the input impedance, too. during 60 turns or 80 turns. Here, the relation is explained by This part is designed according to cyclotron requirements and the following formula: considerations regarding the charged particle s revolution path. (2) where is the total desired energy, N is the number of revolutions, V is the Dee voltage (40 kv here), [~1] is the transit time factor, and [~90º] is the phase angle. According to the above formula, the Dee angle will be approximately 42 degrees. Table 1 shows this cyclotron specifications, including design parameters and geometrical dimensions [15]. TABLE 1 DESIGN PARAMETERS AND GEOMETRICAL DIMENSIONS Cyclotron Parameters Value Energy 10 MeV Resonant Frequency 71 MHz RF Power 15 kw Harmonic No. 4 Dee angle Dee voltage 40 kv Pole Radius 45 cm Eextraction Radius 39 cm Coupling type Capacitive Fig. 6 Return loss variation versus frequency for different Stem diameters C. RF Modeling of the Dee Section With respect to the aim of the resonator here, which is to transfer RF power to the particles, a structure for coupling maximum power with the charged particles is required after the appropriate resonator has been created using the Stem and Liner sections. The Dee section does this properly. This part is typically D-shaped for the two accelerating gap design, and it is in the form of triangular plates in the four accelerating gaps design [6]. The Dee provides a sufficient electric field for the attainment of charged particles in the specific intervals according to the proper voltage magnitude in the gaps. With such a structure, the charged particles reach the desired energy in a more compact and smaller composition. A triangular construction is, therefore, most suitable for the Dee section. Furthermore, the outer shell of the Dee is also triangular for the attainment of the coaxial transmission line. For example, in the design of the special 10 MeV cyclotron, the desired structure that is discussed in [15] works on the fourth harmonic, so the approximate angle of the triangular part is obtained using the following relation: (1) where is the angle of the Dee part and h is the harmonic number. The exact value is calculated here based on the number of revolutions in which the particles, which are protons here, reach the ultimate energy. In the mentioned 10 MeV energy cyclotron, particles catch the energy in four gaps The electromagnetic modeling of this section of the cavity led to the study and consideration of a diaphragm in a waveguide transmission line. The effects of the diaphragm structures were also analyzed in the triangular transmission line. Because of the junction between the Dees in the halves of the overall structure which must be taken into account for a more accurate analysis, both top Dees were compared with each other. Fig. 8 presents the placement of the Dee structure into the triangular line. Its proper circuit model was investigated according to the scattering parameters of this physical structure with the diaphragm part (Fig. 10). The effect of this propagation between it and the outer conductor on both sides is capacitive, and an inductive effect is caused because of the junction of the diaphragm with the inner conductor of the coaxial line [16, 17, 18]. Fig. 7. Intensity of the electric field along circular path in the mid plane of the cavity. Based on the simulation and theory observations, the maximum value of the electric field exists between the diaphragm and the Liner section (Fig. 7), with an inverse direction in each gap to the opposite side. Figure 7 illustrates the electric field along the circular path in the mid plane of the

5 TNS cavity in which the maximum value is clearly depicted in the As a result of the CST simulation response and since the gaps. The electric field pattern of the TEM wave of this studied frequency interval is below 200 MHz, the resonance structure that shows the diaphragm effect and the particle frequency and its bandwidth in this interval are MHz beam trajectory is sketched in Fig. 8. In this modeling, port 1 and MHz for the band-pass and MHz and 77 MHz is located on the top side of the structure, and port 2 is on the for the stop-band, respectively. The equivalent capacitances bottom side. The magnetic field is perpendicular to the E-field and inductances are, respectively, L p = 15.7 nh and C p = 71.7 in this figure, which is shown as a cross to the inside of the pf in the band-pass, and L S = nh and C S = 35.1 pf in the plane. The phase difference of the applied voltage was stop-band. Fig. 10 shows the insertion loss (S12) parameters observed in the gaps according to its amplitude at the time the of the circuit model results that fit well with the numerical particles reached the next gap. CST simulation results. Fig. 8. Diaphragm structure in the triangular coaxial waveguide and electric field pattern between the diaphragm and the Liner in the presence of a magnetic field perpendicular to the E-field Furthermore, the diaphragm shown in Fig. 8 was modeled in CST, and Fig. 9 presents its scattering parameters versus the frequency. Several resonances, some of which are band-pass and another which is band-stop, are indicated here, and they are repeated periodically to show the diaphragm behavior in this frequency interval. According to the CST simulation, the proper circuit model of this portion is an LC in the series and another LC in the parallel that, as described in Fig. 9, are connected to each other. Fig. 9. Scattering parameters of CST simulation diaphragm considerations Fig. 10. Scattering parameters of diaphragm consideration of equivalent microwave circuit model and CST simulation: Cp = 71.7 pf, Cs = 35.1 pf, Lp = 15.7 nh, and Ls = nh. IV. SIMULATION OF WHOLE STRUCTURE A combination of the Dee and Liner sections is considered here. Also, the effect of each part is studied in the creation of the resonance frequency. Based on Fig. 3 and the circuit model of the Liner and Dee sections, the circuit model of whole structure is represented in Fig. 11-a. This final model includes the four similar transmission lines, which are shortened to model the Liner and Stem parts, and two parallel Dee section circuit models that are located symmetrically on the top and bottom of the proton beam revolution path and are connected to each other from the sides. The RF power through the transmission line the inner conductor of which is connected to the Dee section and its outer conductor to the Liner is coupled to the cavity. The scattering parameters of this structure (S12) for both the electromagnetic full wave simulation and the circuit model are depicted in Fig. 11-b and demonstrate a significant conformity. In this figure, the scattering parameter values in the CST curve are S11 = db, S12 = 0.99 in the desired resonance frequency of 71 MHz and S11 = db, S12 = 0.98 in 174 MHz. Also, in the circuit model curve, these values are S11 = db, S12 =1 in the desired resonance frequency of 71 MHz and S11 = db, S12 =1 in 177 MHz. In CST simulation, the combination of Liner and Stem as a transmission line, individually behaves such as short circuit in 130 MHz. Therefore, S21 parameter will be zero in this frequency. This

6 figure shows the band-pass behavior of the cavity. It means that the radio frequency power is coupled into the cavity at a certain frequency which depicts the operating frequency, at which the RF power is transferred to the charged particles through the gap between the Dee and Liner sections. Since the charged particle is injected into the central part of the structure and the initial energy must be caught through a strong electric field, the existence of a similar structure is necessary between the Dee and the Liner in the central part. The central part effects are considered in the diaphragm model. The circuit model demonstrates that the combination of all four sections results in a high quality factor, and the resonance frequency of the whole structure is consequently obtained. Moreover, the influences of the different parameters in this frequency were analyzed. TNS sides. In fact, in the circuit model structure, the inner conductors of the coaxial lines connect the lines to each other, and the outer conductors of all of the parts provide connections to the common ground. A. Coupling and Tuning Capacitances The two adjustable capacitors, which are constructed as two parallel disks, are used in a series with the whole structure at the beginning and at the end of the cavity. There are some reasons to utilize these parts. One is to improve the external quality factor of the overall structure. Another could be to achieve the desired accurate resonance frequency and the possibility of fine tuning. These are two parallel disks that can be seen on a physical model in Fig. 3, and the equivalent circuit in the total structure is presented in Fig. 11-a. A plate of these disks is connected to the Dees, while another plate is connected to the inner electrode of the input power transmission line. C C and C T represent the capacitive coupling and tuning, respectively, at two sides of the cavity. Fig. 12 illustrates the tuning capacitor effect in the circuit model that is comparable with the model without the capacitor. As was expected with the tuner and the coupler capacitors, the scattering parameter peaks thinned further, resulting in the achievement of a high quality factor of about (a) (b) Fig. 11. Final structure: a) Equivalent microwave circuit model, and b) CST and circuit model responses According to the requirements of the cyclotron structure, a region must be designed between the upper and lower Dees into which the charged particle is injected, whereby a revolution that reaches the desired energy is achieved. To the physical junction of the four parts, the upper and lower resonators are connected to each other by the Dees from the Fig. 12. Effects of tuner and coupler in quality factor of circuit model, and scattering parameters with and without different capacitor values of tuner and coupler disks V. CONCLUSION The investigated design and its circuit model demonstrated the effectiveness and weight of different parts in the specification of the cavity which could assist in accurately and quickly designing the cavity. This study demonstrated that the Stem and Liner determine the main operation region of the resonator, and the Dee section with the Liner part specifies resonance frequency precisely. This section couples the

7 energy to the charged particles, too. Therefore, the dimensions of this section s parameters are used to adjust the energy given to the particles. Also, after combining all parts, a high quality factor and precise resonance frequency are achieved through the input and output capacitors. All these features could be obtained with the circuit model proposed in this study. TNS REFERENCES [1] Handbook of Accelerator Physics and Engineering, 2nd ed., World Scientific Press, W. C. Alexander, et al., [2] C. J. Cleveland, C. Morris, The Dictionary of Energy, In Elsevier Science, 2nd ed., Oxford, UK, [3] E. O. Lawrence, Method and apparatus for the acceleration of ions, U.S. Patent , Feb. 20, [4] C. R. Nave. (2014). Cyclotron, Hyperphysics. Dept. of Physics and Astronomy, Georgia State Univ. [online]. Available: [5] Sabaiduc et al, New EW high intensity compact negative hydrogen Ion cyclotrons, in Proceedings of CYCLOTRONS, Lanzhou, China MOPCP017, 2010, pp [6] P. Miller, Status report on the 500MeV cyclotron, in Proceedings of 9th Int. conf. on cyclotrons and their applications, Caen, France, 1981, pp [7] M. S. Livingston, J. Blewett, Particle Accelerators, New York: McGraw-Hill, [8] CST, Computer Simulation Technology, CSTMicrowave Studio. (2016), Available: [9] HFSS Release 9.0, Ansoft Corp., (2003). [10] S. K. Jain, D. Sharma, V. K. Senecha, P. A. Naik and P. R. Hannurkar, Study of microwave components for an electron cyclotron resonance source: Simulations and performance, Sadhana, vol. 39, Part 4, pp , [11] Su Jung et al, RF cavity design for KIRAMS-430 superconducting cyclotron, Nuclear Instruments and Methods in Physics Research Section A, Vol. 777, pp , [12] T. Dong, L. Zhan, Y. Hong, M. Fan, Cyclotron cavity analysis based on field circuit model, Nucl. Inst. and Met. in Phys. Research A, vol. 513, pp , [13] Advanced Design System, (2005), Avalaible: [14] P. K. Sigg, Cyclotron cavities PART II, in Proc CAS, [15] M. Mohamadian, H. Afarideh, M. Salehi, J. S. Chai, M. Ghergherehchi, Development of Optimized RF Cavity in 10 MeV Cyclotron, in Proc IPAC2016, 2016, pp [16] D. M. Pozar, Microwave Engineering, John Wiley & Sons, [17] J. J. Vincent, Modeling and Analysis of Radio Frequency Structures using an Equivalent Circuit Methodology with Application to Charged Particle Accelerator RF Resonators, PhD dissertation, Michigan State Univ., [18] H. Bahrami, M. Hakkak, A. Pirhadi, Analysis and Design of Highly Compact Bandpass Waveguide Filter Utilizing Complementary Split Ring Resonators (CSRR), Progress In Electromagnetics Research, vol. PIER 80, pp , 2008.