# Additive Manufacturing of Side-Coupled Cavity Linac Structures from Pure Copper: A First Concept

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## Abstract

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^{®}. A 3 GHz SCCL concept consisting of 31 SCs was designed. The effective shunt impedance $Z{T}^{2}$ simulated by CST corresponds to $60.13\phantom{\rule{3.33333pt}{0ex}}\frac{\mathrm{M}\mathrm{\Omega}}{\mathrm{m}}$ and is comparable to the $Z{T}^{2}$ of SCCL in use. The reduction in $Z{T}^{2}$ expected after Hirtisation (R) can be justified in practice by up to 70% lower manufacturing costs. However, future studies will be conducted to further increase ${Q}_{0}$.

## 1. Introduction

^{®}(CST) simulations, which can be manufactured by AM. Using green laser L-PBF, two sets of SCs were fabricated with different downskin angles. Studies of geometric accuracy, surface roughness, and RF properties were performed on set 1 to evaluate what angle is feasible for our target SC with a resonance frequency ${f}_{R}$ of 3 GHz. To reduce the surface roughness and thus increase the surface conductivity, set 2 was post-processed by the electrochemical process Hirtisation (R) [24]. Subsequently, geometrical accuracy, surface roughness, and RF properties of set 2 were evaluated to study the influence of post-processing. Finally, a rudimentary concept of a 3 GHz biperiodic side-coupled linac (SCCL) composite of 32 of the elaborate SCs is presented. The SCCL is characterized by CST simulations and compared with a conventionally manufactured SCCL.

## 2. Principle of a Side-Coupled Cavity Linac

## 3. Materials and Methods

#### 3.1. Cavity Design and Electromagnetic (EM) Simulation

^{®}[29]. In terms of material, we always assume the annealed copper included in CST with an electrical conductivity of $5.8\times {10}^{7}\phantom{\rule{4pt}{0ex}}\frac{\mathrm{S}}{\mathrm{m}}$ (International Annealed Copper Standard, IACS).

#### 3.2. Additive Manufacturing

#### 3.3. Post-Processing Procedure

#### 3.4. RF Measurements

#### 3.5. Evaluation of Dimensional Accuracy

#### 3.6. Evaluation of Surface Roughness

## 4. Results

#### 4.1. Single-Cell EM Simulations

#### 4.2. Dimensional Accuracy after Printing

#### 4.3. Surface Roughness after Printing

#### 4.4. Dimensional Accuracy after Post-Processing

#### 4.5. ${Q}_{0}$ and Surface Roughness after Post-Processing

#### 4.6. Self-Supporting 3 GHz Biperiodic Side-Coupled Linac Concept

## 5. Discussion

#### Outlook

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Illustration of four printed layers (brown) forming an overhang with an angle $\alpha $ between it and the building platform (black). (

**a**) A large $\alpha $ allows for high-quality prints. (

**b**) Layer deformation increases as $\alpha $ decreases. (

**c**) Support structures (blue) are needed to ensure print quality with a small $\alpha $.

**Figure 2.**(

**a**) Periodic on-axis coupled structure of five SCs with the five specific modes. In green, the E-field distribution of the modes is displayed. (

**b**) Dispersion relation between eigenfrequencies and phase shift per SC.

**Figure 3.**(

**a**) Periodic on-axis coupled structure in $\frac{\pi}{2}$ mode. (

**b**) Side-coupled cavity structure (SCCL) in which the field-free cells were removed from the beam axis, thus increasing the effective shunt impedance per unit of length $Z{T}^{2}$.

**Figure 4.**Quadrant of the longitudinal section through an AC of an SCCL structure as (

**a**) traditional geometry and (

**b**) L-PBF optimized geometry. Table (

**c**) summarizes the relevant geometric parameters.

**Figure 5.**(

**a**) Positions for ${S}_{q}$ measurement on the nose cone (red) and on the side wall of the cavity (green). (

**b**) Measurement principle for h.

**Figure 6.**Distribution of electric (E) and magnetic (B) fields in the longitudinal section of the (

**a**) ${25}^{\circ}$ and (

**b**) ${45}^{\circ}$ single cavity.

**Figure 7.**$Z{T}^{2}$ normalized to the $Z{T}^{2}$ of the 25° SC as a function of ${C}_{A}$ or ${O}_{A}$.

**Figure 8.**Cell pairs of the respective angles ${C}_{A}$ and ${O}_{A}$ of ${30}^{\circ}$, ${35}^{\circ}$, ${40}^{\circ}$, and ${45}^{\circ}$.

**Figure 9.**Circles and triangles indicate $\mathsf{\Delta}{f}_{R}$ of the measured and simulated ${f}_{R}$ for the first (blue) and second (green) SC set, respectively. The asterisks (black) correspond to the $\mathsf{\Delta}{f}_{R}$ between the two SC sets. The crosses represent the maximum possible frequency tuning.

**Figure 10.**One half of each cell with the respective angles ${C}_{A}$ and ${O}_{A}$ of ${30}^{\circ}$, ${35}^{\circ}$, ${40}^{\circ}$, and ${45}^{\circ}$.

**Figure 11.**Difference Z between measured surface and CAD data of the individual SC halves (halve a and halve b).

**Figure 12.**The prominence h for different angles ${C}_{A}$ and ${O}_{A}$ and sides of the cut-open cells. The x-coordinates are located on the yellow circle shown in Figure 5b. x = 0 corresponds to the yellow arrow, and the measurement direction corresponds to the direction of the arrow.

**Figure 14.**One half of each cell with the respective angles ${C}_{A}$ and ${O}_{A}$ of ${30}^{\circ}$, ${35}^{\circ}$, ${40}^{\circ}$, and ${45}^{\circ}$ after Hirtisation (R).

**Figure 15.**Difference Z between measured surface and CAD data of the individual SC halves (halve a and halve b) after Hirtisation (R).

**Figure 16.**The prominence h for different angles ${C}_{A}$/${O}_{A}$ and sides of the cut-open SCs after Hirtisation (R). The x-coordinates are located on the yellow circle shown in Figure 5b. x = 0 corresponds to the yellow arrow. The measurement direction corresponds to the direction of the arrow.

**Figure 17.**Frequency shift $\mathsf{\Delta}{f}_{R}$ relative to the resonant frequency ${f}_{R}$ after printing of the individual SCs of set 2 as a function of the material removal by Hirtisation (R).

**Figure 19.**Measured quality factor (${Q}_{0M}$) normalized to the quality factor simulated by CST (${Q}_{0S}$) as a function of the material removal $MR$ for the different SCs.

**Figure 20.**Simulated ${Q}_{0S}$ and measured ${Q}_{0M}/{Q}_{0S}$ in dependence of ${S}_{q\lambda}$. In blue and black, ${Q}_{0S}$ predicted by the gradient model and Hammerstad model are displayed, respectively. The single points show the measurements for SC set 1 and set 2 after and before Hirtisation (R), respectively.

**Figure 22.**Unit cell of the SCCL concept as used for the CST simulations; (

**a**) perspective, (

**b**) a section of the perspective, (

**c**) the half CC side, and (

**d**) the half AC side.

**Figure 24.**(

**a**) E-field distribution of the $\frac{\pi}{2}$ mode. (

**b**) E-field distribution of the $\frac{\pi}{2}$ mode on the z axis.

L | 19.0 mm | ${\mathit{R}}_{\mathit{n}\mathit{i}}$ | 1.0 mm | ${\mathit{C}}_{\mathit{A}}$ | ${25}^{\circ},{30}^{\circ},\cdots ,{45}^{\circ}$ |

D | 46.5 mm | ${R}_{no}$ | 1.0 mm | ${O}_{A}$ | ${25}^{\circ},{30}^{\circ},\cdots ,{45}^{\circ}$ |

g | 4.0 mm | ${R}_{ci}$ | 2.0 mm | ${R}_{o}$ | 1.0 mm |

S | 3.0 mm | ${R}_{co}$ | 2.0 mm | ||

${R}_{b}$ | 3.0 mm | F | 0.0 mm |

${\mathit{C}}_{\mathit{A}}$ and ${\mathit{O}}_{\mathit{A}}$ | ${25}^{\circ}$ | ${30}^{\circ}$ | ${35}^{\circ}$ | ${40}^{\circ}$ | ${45}^{\circ}$ |

${f}_{R}\phantom{\rule{4pt}{0ex}}\left[\mathrm{MHz}\right]$ | 3931 | 3992 | 4064 | 4150 | 4260 |

${Q}_{0}$ | 8574 | 8550 | 8481 | 8331 | 8100 |

$Z{T}^{2}\phantom{\rule{4pt}{0ex}}\left[\frac{\mathrm{M}\mathrm{\Omega}}{\mathrm{m}}\right]$ | 63.4 | 61.8 | 59.7 | 56.9 | 53.3 |

$\frac{{E}_{max}}{{E}_{0}}$ | 3.07 | 3.14 | 2.99 | 2.92 | 2.97 |

$\frac{{H}_{max}}{{E}_{0}}\phantom{\rule{4pt}{0ex}}\left[\frac{\mathrm{A}}{\mathrm{kV}}\right]$ | $1.63$ | $1.77$ | $1.53$ | $1.95$ | $1.95$ |

$\frac{{S}_{c,max}}{{E}_{0}}\phantom{\rule{4pt}{0ex}}\left[\frac{\mathrm{A}}{\mathrm{kV}}\right]$ | $0.56$ | $0.55$ | $0.55$ | $0.54$ | $1.95$ |

${\mathit{C}}_{\mathit{A}}/{\mathit{O}}_{\mathit{A}}$ | $\overline{\mathit{h}}$-Side a | $\overline{\mathit{h}}$-Side b | $\mathsf{\Delta}\mathit{g}\phantom{\rule{4pt}{0ex}}$ |
---|---|---|---|

${45}^{\circ}$ | $6.00$ mm | $5.98$ mm | $0.02$ |

${40}^{\circ}$ | $5.95$ mm | $5.98$ mm | $0.07$ |

${35}^{\circ}$ | $5.87$ mm | $5.57$ mm | $0.56$ |

${30}^{\circ}$ | $5.69$ mm | $5.39$ mm | $0.92$ |

${\mathit{C}}_{\mathit{A}}$ and ${\mathit{O}}_{\mathit{A}}$ | ${30}^{\circ}$ | ${35}^{\circ}$ | ${40}^{\circ}$ | ${45}^{\circ}$ |
---|---|---|---|---|

${S}_{q}$ [µm]-Loc. 1 | 21.57 | 19.49 | 23.79 | 29.60 |

${S}_{q}$ [µm]-Loc. 2 | 24.60 | 26.51 | 23.89 | 24.96 |

${S}_{q}$ [µm]-Loc. 3 | 18.54 | 21.90 | 20.94 | 28.70 |

${S}_{q}$ [µm]-Loc. 4 | 27.46 | 21.72 | 23.38 | 26.33 |

${S}_{q}$ [µm]-Loc. 5 | 23.75 | 21.44 | 23.96 | 25.43 |

${S}_{q\lambda}$ [µm]-Loc. 1 | 5.25 | 5.91 | 3.69 | 4.84 |

${S}_{q\lambda}$ [µm]-Loc. 2 | 5.58 | 4.84 | 5.07 | 4.20 |

${S}_{q\lambda}$ [µm]-Loc. 3 | 3.47 | 4.20 | 3.94 | 4.08 |

${S}_{q\lambda}$ [µm]-Loc. 4 | 3.63 | 3.80 | 4.30 | 5.24 |

${S}_{q\lambda}$ [µm]-Loc. 5 | 4.17 | 3.74 | 4.66 | 5.33 |

${\mathit{C}}_{\mathit{A}}/{\mathit{O}}_{\mathit{A}}$ | $\overline{\mathit{h}}$-Side a | $\overline{\mathit{h}}$-Side b | $\mathsf{\Delta}\mathit{g}\phantom{\rule{4pt}{0ex}}$ |
---|---|---|---|

${45}^{\circ}$ | $5.96$ mm | $5.93$ mm | 0.11 |

${40}^{\circ}$ | $5.94$ mm | $5.93$ mm | 0.13 |

${35}^{\circ}$ | $5.83$ mm | $5.67$ mm | 0.50 |

${30}^{\circ}$ | $5.64$ mm | $5.15$ mm | 1.21 |

${\mathit{C}}_{\mathit{A}}/{\mathit{O}}_{\mathit{A}}$ | ${30}^{\circ}$ | ${35}^{\circ}$ | ${40}^{\circ}$ | ${45}^{\circ}$ |
---|---|---|---|---|

${S}_{q}$ [µm]-Loc. 1 | 1.31 | 1.01 | 0.83 | 1.23 |

${S}_{q}$ [µm]-Loc. 2 | 1.14 | 0.84 | 1.00 | 1.99 |

${S}_{q}$ [µm]-Loc. 3 | 1.83 | 1.02 | 0.98 | 1.85 |

${S}_{q}$ [µm]-Loc. 4 | 0.94 | 0.68 | 0.75 | 1.85 |

${S}_{q}$ [µm]-Loc. 5 | 1.11 | 0.91 | 0.76 | 1.14 |

${S}_{q\lambda}$ [µm]-Loc. 1 | 0.30 | 0.38 | 0.34 | 0.34 |

${S}_{q\lambda}$ [µm]-Loc. 2 | 0.40 | 0.27 | 0.31 | 0.36 |

${S}_{q\lambda}$ [µm]-Loc. 3 | 0.60 | 0.38 | 0.33 | 0.37 |

${S}_{q\lambda}$ [µm]-Loc. 4 | 0.26 | 0.29 | 0.22 | 0.21 |

${S}_{q\lambda}$ [µm]-Loc. 5 | 0.31 | 0.20 | 0.27 | 0.20 |

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## Share and Cite

**MDPI and ACS Style**

Mayerhofer, M.; Brenner, S.; Helm, R.; Gruber, S.; Lopez, E.; Stepien, L.; Gold, G.; Dollinger, G.
Additive Manufacturing of Side-Coupled Cavity Linac Structures from Pure Copper: A First Concept. *Instruments* **2023**, *7*, 56.
https://doi.org/10.3390/instruments7040056

**AMA Style**

Mayerhofer M, Brenner S, Helm R, Gruber S, Lopez E, Stepien L, Gold G, Dollinger G.
Additive Manufacturing of Side-Coupled Cavity Linac Structures from Pure Copper: A First Concept. *Instruments*. 2023; 7(4):56.
https://doi.org/10.3390/instruments7040056

**Chicago/Turabian Style**

Mayerhofer, Michael, Stefan Brenner, Ricardo Helm, Samira Gruber, Elena Lopez, Lukas Stepien, Gerald Gold, and Günther Dollinger.
2023. "Additive Manufacturing of Side-Coupled Cavity Linac Structures from Pure Copper: A First Concept" *Instruments* 7, no. 4: 56.
https://doi.org/10.3390/instruments7040056