# Research and Design of the RF Cavity for an 11 MeV Superconducting Cyclotron

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## Abstract

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## Featured Application

**The focus of this paper is on the design of the RF (Radio Frequency) resonant cavities required for small modular superconducting cyclotrons. The design of these RF cavities aims to match the smaller extraction radius of superconducting cyclotrons and make easier of installation and maintenance. To this purpose, one has to adopt a design that allows for easy assembly and disassembly with the magnet system that has been adopted. Through the study of the circuitry of such RF cavities, along with calculations of electromagnetic fields and methods of frequency tuning, we provide a set of solutions that can serve as a reference for the design of smaller superconducting accelerators in the future. This not only helps in advancing the technology of superconducting cyclotrons but also can provide a method reference for related technical fields.**

## Abstract

## 1. Introduction

^{11}C and

^{18}F stand out as the most effective PET isotopes, characterized by their short half-lives of 20 and 108 min, respectively. Currently, the predominant method of producing

^{11}C and

^{18}F involves the use of low-energy cyclotrons (with energies below 15 MeV). A prevailing trend in the development of these cyclotrons is the emphasis on creating more compact, lightweight, and cost-efficient models [4,5,6]. Furthermore, there is a growing preference for cyclotrons that can operate in smaller settings and produce isotopes in proximity to patients. This approach aims to mitigate transportation challenges, minimize isotope activity loss, substantially reduce production costs, decrease the demand for associated infrastructure, and curtail expenses related to subsequent auxiliary equipment [7].

## 2. Miniaturization of RF Cavity for an 11 MeV Superconducting Cyclotron

#### 2.1. Structure of the Cyclotron’s RF Cavity Using the Stepped Impedance Resonator (SIR) Approach

_{1}and Z

_{2}, respectively. Their electrical lengths are represented by θ

_{1}and θ

_{2}, where θ = βl. In this equation, β stands for the propagation constant, and l signifies the length of the transmission line. For simplicity in analysis, the characteristic impedances Z

_{1}and Z

_{2}of the structure are assumed to be uniformly distributed [10,14,15].

_{left}is the input impedance on the left side of the reference surface, and Z

_{right}is the input impedance on the right side of the reference surface. According to the input impedance formula [16]:

_{0}is the characteristic impedance of the transmission line and Z

_{L}is the characteristic impedance of the end load. The left and right input impedances can be calculated by selecting the step junction surface as the input reference surface. When the step junction discontinuity and the edge capacitance effect at the open-circuit end are neglected, the total susceptance at the reference surface is zero, and the impedance ratio R

_{Z0}is calculated as:

_{1}, θ

_{2}, and the impedance ratio R

_{Z0}. Before the introduction of the SIR structure, the resonance condition of a typical quarter-wavelength coaxial cavity depended only on the length of the transmission line, which had one less degree of design freedom compared to the SIR structure.

_{n}at 0 < R

_{Z0}< 1:

_{Z0}critically affects the length of the normalized resonator. Thus, theoretically, a smaller impedance ratio can be utilized to reduce the cavity’s length. The design strategy involves increasing the transmission line’s transverse dimension to compensate for a reduction in its longitudinal dimension. This adjustment aligns with the design needs of a superconducting cyclotron that requires a smaller radial dimension. Consequently, cavity miniaturization can be achieved by optimizing the shape and structural parameters of the RF cavity’s Dee and Stem.

#### 2.2. Effects of Discontinuity

_{O.C}), at the short-circuit end (Y

_{S.C}), and the admittance due to the impedance discontinuity (Y

_{S}) are, respectively:

_{d}is the step capacitance. The resonance condition can be obtained according to the susceptance method as:

_{Z}is the impedance ratio after considering the effects of discontinuity.

## 3. Equivalent Circuit of the RF Cavity

#### 3.1. Equivalent Circuit Model of RF Cavity

_{0}of a plane wave in free space, and the transverse component is zero [12,13].

_{d}. In addition, the distributed capacitance C

_{a}between Dee and Dummy Dee needs to be considered. The equivalent circuit of the cavity is shown in Figure 2.

_{in}is the input admittance, and Y

_{0}and Y

_{L}are the characteristic admittance and load admittance of the transmission line, respectively. Thus, for a two-segment transmission line, when the load end is the short-circuit end, its input admittance is [19],

_{1}and Y

_{2}are the characteristic admittance of transmission line 1 and transmission line 2, respectively. When the load end is the open-circuit end, its input admittance is,

_{3}and Y

_{4}are the characteristic admittance of transmission line 3 and transmission line 4, respectively.

#### 3.2. Determination of Equivalent Circuit Parameters

_{c0}), the rectangular coaxial transmission line (Z

_{r0}) [20], and the transition coaxial transmission line (Z

_{cr}):

_{cr}.

_{0}= (μ

_{0}/ε

_{0})

^{1/2}= 376.73 Ω, ε

_{r}is the relative dielectric constant, w and t are the length and width of the inner conductor of the rectangular coaxial transmission line, a and b are the length and width of the outer conductor of the rectangular coaxial transmission line, and n

_{ac}is the approximation coefficient of Z

_{cr}.

_{v}) and horizontal (C

_{h}) directions can be calculated separately:

## 4. Design and Simulation of the RF Cavity

#### 4.1. Design of the RF Cavity

#### 4.2. Structural Analysis of the RF Cavities

_{c}= 138 lg(D/d) Ω, where d is the diameter of the inner conductor and D is the diameter of the outer conductor. When the diameter of the circular Stem increases, the radius ratio of the circular coaxial line decreases, the characteristic impedance becomes smaller, and the resonant frequency of the cavity increases. Therefore, the designed target resonant frequency value can be achieved when the diameter of the circular Stem inner conductor is adequate. However, increasing the diameter leads to an increase in the impedance discontinuity at the connection between the Dee and the Stem inner conductor, resulting in an increase in the step capacitance, which leads to a decrease in the resonant frequency of the RF cavity. It is calculated that the RF cavity resonant frequency decreases from 45.224 MHz to 42.001 MHz when the step capacitance is increased from 1 pF to 10 pF. However, when the step capacitance is increased from 10 pF to 100 pF, the RF cavity resonant frequency decreases rapidly to 26.145 MHz. In addition, the increase in the diameter of the conductor inside the circular Stem increases the step junction surface loss and the RF cavity surface loss. After comprehensive consideration, a rectangular Stem is used to connect to the Dee, making the step capacitance at the contact surface as small as possible.

#### 4.3. Optimized Design of the RF Cavity

_{ic}, the outer conductor radius r

_{oc}, and the length L. The effect of these three structural factors on the resonant frequency of the cavity computed using 3D electromagnetic field numerical simulation software is shown in Figure 6. The RF cavity resonant frequency is smaller when r

_{ic}is larger, r

_{oc}is smaller, and L is shorter. When r

_{ic}is certain, r

_{oc}is roughly inversely proportional to L at the same resonant frequency. Therefore, it should be avoided that the length L of the round coaxial resonator is too short, which leads to a sharp increase in the radius r

_{oc}of the outer conductor of the round coaxial resonator.

_{0}of the cavity is calculated. Figure 7 shows the effect of r

_{oc}and L on the magnitude of the unloaded quality factor of the RF cavity for r

_{ic}= 20 mm. It can be determined that the larger r

_{oc}and L are, the larger the Q

_{0}value of the RF cavity is, and different Q

_{0}values can be taken at the same resonant frequency.

_{oc}at r

_{ic}= 20 mm and the same resonant frequency, and the surface area size of the round coaxial cavity corresponding to each set of data and the unloaded quality factor are calculated by using 3D electromagnetic field simulation software. According to Figure 8, it is observed that at the same resonant frequency, the smaller L is, the larger the surface area of the cavity, and the larger the unloaded quality factor Q

_{0}. The surface area of the round coaxial resonator is related to the vacuum degree of the cavity; thus, the cavity design needs to follow the principle of minimizing the surface area of the cavity. This requirement works in the opposite way to the selection of the quality factor. Furthermore, it is established that L is inversely related to r

_{oc}. In summary, choosing a smaller L will result in a larger Q

_{0}but also lead to an overly large r

_{oc}and surface area of the cavity. Conversely, selecting a larger L, although it minimizes the surface area of the cavity as much as possible and achieves a smaller r

_{oc}, necessitates a reduction in the Q

_{0}of the cavity. Consequently, the above influencing factors are considered in the actual design, and an appropriate value is chosen as a compromise based on demand.

#### 4.4. Numerical Calculation of Electromagnetic Fields of the RF Cavity

_{Z}will fluctuate compared to the theoretically calculated value 0.117.

## 5. Calculation and Verification of the RF Cavity for 11 MeV Superconducting Cyclotrons

#### 5.1. Calculation and Verification of the RF Cavity with Different Structural Parameters

#### 5.2. Analysis of the RF Cavity Based on Multiple Linear Regression Models

_{ic}, the outer conductor inner radius r

_{oc}, and the cavity length L. The effect of each of these three parameters on the resonant frequency of the RF cavity is shown in Figure 10. In order to determine the relationship between these three structural parameters and the cavity resonance frequency, and to obtain the importance of the influence of these three variables on the cavity resonance frequency, a multiple linear regression model can be introduced.

_{1}, X

_{2}, …, X

_{k}represent k explanatory variables; and ε is a random error term that explains the variation in the observation Y caused by the random variable, where E(ε) = 0 and V(ε) = σ

^{2}and ε~N(0, σ

^{2}). In this paper, there are three main structural parameter variables (k = 3), where Y is the RF cavity resonance frequency f, X

_{1}corresponds to r

_{ic}, X

_{2}corresponds to r

_{oc}, and X

_{3}corresponds to L.

^{2}, were chosen to test and assess the effectiveness and accuracy of the model. Where RMSE is used to assess the deviation between actual and predicted values and R

^{2}is used to measure the degree of fit of the regression model, the respective expressions are [27]:

_{i}is the true value, ${\widehat{y}}_{i}$ is the predicted value, and ${\overline{y}}_{i}$ is the sample mean.

^{2}, and regression equations, where the smaller the RMSE and the closer the R

^{2}is to 1, the better the model’s predictive accuracy. The regression equation shows that r

_{ic}and L have a significant impact on the resonance frequency. Conversely, r

_{oc}’s effect is less pronounced. If r

_{ic}increases significantly, widening the magnet yoke opening becomes necessary. Thus, changes to r

_{ic}are kept within the rectangular cavity’s height limits. By adjusting r

_{ic}and L, the resonant frequency of the RF cavity can vary widely. Altering r

_{oc}leads to a narrower frequency adjustment. Therefore, in further cavity design and optimization, the cavity resonant frequency can be roughly adjusted by adjusting the length of the round coaxial cavity L. After determining the resonant frequency adjustment range, the regression equation can be used to accurately calculate the available range for moving the short-circuit end. In addition, the resonant frequency fine-tuning structure design of the RF cavity can refer to the influence of the outer conductor inner radius r

_{oc}on the resonant frequency.

_{ic}= 20 mm and r

_{oc}= 240 mm. It illustrates a high degree of fit between numerical simulation and equivalent circuit calculations. The errors in the prediction results at the target frequency for both models do not surpass 1%. They remain under 0.3%, satisfying the engineering design’s predictive needs.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 3.**The RF cavity with different shapes of Dee and Stem structures: (

**a**) rectangular Stem and (

**b**) cylindrical Stem.

**Figure 6.**Influence of each parameter of a round coaxial resonator on the resonant frequency of the RF cavity.

**Figure 7.**Influence of each parameter of the round coaxial resonator on the unloaded quality factor of the RF cavity.

**Figure 8.**Effects of different round coaxial resonator lengths on the surface area of the round coaxial resonator and unloaded quality factor at the same resonant frequency.

**Figure 10.**Round coaxial line parameters versus resonant frequency of cavity. (

**a**) Line length, (

**b**) radius of line outer conductor, (

**c**) radius of line inner conductor radius.

**Figure 11.**Impact of transition coaxial line parameters on the resonant frequency of the cavity. (

**a**) Line length and (

**b**) radius of line outer conductor.

**Figure 12.**Impact of rectangular coaxial line parameters on the resonant frequency of cavity. (

**a**) Line length and (

**b**) line height.

**Figure 13.**Performance of the RF cavity resonant frequency prediction based on multiple linear regression models.

Cyclotron Parameters | Value |
---|---|

Energy | 11 MeV |

Resonant Frequency | 45.026 MHz |

Harmonic No. | 1 |

Extraction Radius | 161 mm |

Dee Voltage | 30 kV |

Coupling Type | Inductive |

**Table 2.**Performance of the RF cavity resonant frequency prediction based on multiple linear regression models.

Method | RMSE | R^{2} | Regression Equation |
---|---|---|---|

Numerical simulation | 0.040 | 0.998 | f = 57.573 + 0.3172 × r_{ic} − 0.0174 × r_{oc} − 0.1005 × L |

Equivalent circuit | 0.052 | 0.998 | f = 56.856 + 0.3995 × r_{ic} − 0.0183 × r_{oc} − 0.1054 × L |

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**MDPI and ACS Style**

Wu, Y.; He, Z.-F.; Wan, W.-S.; Zheng, P.-P.; Yu, H.-F.
Research and Design of the RF Cavity for an 11 MeV Superconducting Cyclotron. *Appl. Sci.* **2024**, *14*, 3549.
https://doi.org/10.3390/app14093549

**AMA Style**

Wu Y, He Z-F, Wan W-S, Zheng P-P, Yu H-F.
Research and Design of the RF Cavity for an 11 MeV Superconducting Cyclotron. *Applied Sciences*. 2024; 14(9):3549.
https://doi.org/10.3390/app14093549

**Chicago/Turabian Style**

Wu, Yue, Zi-Feng He, Wei-Shi Wan, Pan-Pan Zheng, and Hua-Fei Yu.
2024. "Research and Design of the RF Cavity for an 11 MeV Superconducting Cyclotron" *Applied Sciences* 14, no. 9: 3549.
https://doi.org/10.3390/app14093549