Design and Prototype Development of a Combined-Function Quadrupole-Sextupole Magnet for the SPS-II Booster Synchrotron

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The manuscript reports on a simulation and prototyping work aimed at designing the combined quadrupole sextupole magnets for the SPS-II synchrotron booster to be built in Thailand. The authors use Opera-3D code to perform their simulation work.

The manuscript is well written and documents quite accurately all the steps (including the requirements and the concept and technical design). The approach chosen in that of a pole shaping. With this respect, I would kindly ask the authors to be more precise on the final solution, with reporting the shape of the poles.

Provided this is done by the authors, I can recommend the publication.

Author Response

First of all, we would like to thank the reviewer for taking your valuable time to read and give us your thoughtful comments and suggestions. We sincerely appreciate the time and effort that the reviewer has dedicated to reviewing our paper. The valuable feedback has helped us improve the overall quality of our work.

Comments and suggested points

The manuscript reports on a simulation and prototyping work aimed at designing the combined quadrupole sextupole magnets for the SPS-II synchrotron booster to be built in Thailand. The authors use Opera-3D code to perform their simulation work.

The manuscript is well written and documents quite accurately all the steps (including the requirements and the concept and technical design). The approach chosen in that of a pole shaping. With this respect, I would kindly ask the authors to be more precise on the final solution, with reporting the shape of the poles.

Provided this is done by the authors, I can recommend the publication.

 

Comment:
“… The approach chosen in that of a pole shaping. With this respect, I would kindly ask the authors to be more precise on the final solution, with reporting the shape of the poles. Provided this is done by the authors, I can recommend the publication…”

Response:

We appreciate the reviewer’s suggestion regarding the clarity of the final pole shape in our combined quadrupole-sextupole magnet design. In response, we have revised the manuscript to include a detailed description of the pole shape in section 3.1 line 161 and have revised the figure for adding the detail on shimming pole in figure 2.

“   The magnets are designed to have external dimensions not exceeding 400 mm × 400 mm to ensure sufficient clearance between the booster synchrotron and the storage ring, which are co-located in the same tunnel. A 3D model of the combined-function quadrupole–sextupole magnet used in the magnetic field calculations is shown in Figure 2(left), where the magnet yoke is rendered in green and the coils in red.

For fabrication purposes, the pole is shaped at a 65° angle with respect to the horizontal axis, while the pole tip is precisely maintained at 45° to ensure the desired field profile. The pole profile of this combined-quadrupole magnet is illustrated in Figure 2(right). Initially, the pole shape is calculated using xy=R²/2, where R is the bore radius of 23 mm. The calculation is performed over the range x from -25 to 0 mm for the left pole and from 0 to 25 mm for the right pole. To achieve the desired sextupole field strength, the quadrupole pole profiles in the top two quadrants are rotated clockwise by 0.6°. To reduce the dipole field generated, the bore radius is adjusted to 23 mm for the right poles and 23.153 mm for the left poles. At the end of each pole, a shimming plate is added to locally enlarge the GFR. As shown in Figure 2 (right) in green, the left pole side is shimmed with a thickness of 1.2 mm, while the right pole side is shimmed with 0.6 mm. The top and bottom poles are shimmed symmetrically. To ensure precise magnet positioning, a reference surface along x-axis at the pole should extend at least 5 mm. For this magnet, a 10 mm long reference surface, machined along the x-axis, has been provided to serve as a fiducial for accurate alignment.”

We hope this revision addresses the reviewer’s request adequately. Thank you once again for your valuable input.

Reviewer 2 Report

Comments and Suggestions for Authors

The authors present the design, simulation, and measurements related to a combined function magnet for the booster upgrade to the Siam Photon Source.  The material is clearly presented, and represents efforts to comprehensively model and measure a new magnet.  Although the project has since moved to another quadrupole design, this paper offers a clear summary of recent progress.  After making a few edits to improve some of their discussion I think that the paper can be published.  I offer these below.   Figure 9: It is impossible to differentiate the measurements at 210 A from the Opera simulations.  Perhaps the measurements could be plotted at points while the simulations as lines?  Or at least mention that the 2 are on top of each other.   Table 3 uses B1, B2, and B3 for the dipole, quadrupole, and sectupole components, while everywhere else B0, B1, and B3 are used.  Please make these consistent.   It is unclear to me how the authors defined the normalized multipoles plotted on the left of Figure 12.  What is the reference field that was used?  It seems odd that the quadrupole term is only 0.005 and the sextupole term is ~0.0023 when this is a combined quadrupole-sextupole magnet, and that these terms are only a factor <6 larger than the unwanted octupole term.  The authors should define how these multipoles are defined, which will hopefully clear this up.  Also, if these normalized multipoles are the same as those that are required to be < 10^{-3} then the authors should state it, since it would then appear that this magnet meets specifications.  Finally, it would be useful if the authors made some estimate of the positional accuracy of the Hall probe set-up, and how this might affect the measured dipole and quadrupole components via feed-down.

Author Response

We thank the reviewer for the constructive and insightful feedback, as well as for their positive assessment of our manuscript. We have carefully considered each comment and made revisions accordingly.

 

Comments and suggested points

The authors present the design, simulation, and measurements related to a combined function magnet for the booster upgrade to the Siam Photon Source.  The material is clearly presented, and represents efforts to comprehensively model and measure a new magnet.  Although the project has since moved to another quadrupole design, this paper offers a clear summary of recent progress.  After making a few edits to improve some of their discussion I think that the paper can be published. 

I offer these below.   Figure 9: It is impossible to differentiate the measurements at 210 A from the Opera simulations.  Perhaps the measurements could be plotted at points while the simulations as lines?  Or at least mention that the 2 are on top of each other.   Table 3 uses B1, B2, and B3 for the dipole, quadrupole, and sectupole components, while everywhere else B0, B1, and B3 are used.  Please make these consistent.   It is unclear to me how the authors defined the normalized multipoles plotted on the left of Figure 12.  What is the reference field that was used?  It seems odd that the quadrupole term is only 0.005 and the sextupole term is ~0.0023 when this is a combined quadrupole-sextupole magnet, and that these terms are only a factor <6 larger than the unwanted octupole term.  The authors should define how these multipoles are defined, which will hopefully clear this up.  Also, if these normalized multipoles are the same as those that are required to be < 10^{-3} then the authors should state it, since it would then appear that this magnet meets specifications.  Finally, it would be useful if the authors made some estimate of the positional accuracy of the Hall probe set-up, and how this might affect the measured dipole and quadrupole components via feed-down.

 

Comment 1: “Figure 9: It is impossible to differentiate the measurements at 210 A from the Opera simulations. Perhaps the measurements could be plotted at points while the simulations as lines? Or at least mention that the 2 are on top of each other.”

Response: We appreciate this suggestion and have revised the caption of Figure 9 accordingly to improve clarity. We separated to Figure 9 and Figure 10. We have revised the caption “Figure 9. Vertical magnetic field along the x-axis (y=z=0). Blue symbols denote the measured magnetic field using Hall probe technique. Gray and black symbols represent the magnetic field calculated using Opera-3D for coil currents of 210 A and 300A, respectively.

Figure 10. Vertical magnetic field along the z-axis (x=10 mm, y=0). Blue symbols denote the measured magnetic field using Hall probe technique. Gray and black symbols represent the magnetic field calculated using Opera-3D for coil currents of 210 A and 300A, respectively.”

Comment 2: “Table 3 uses B1, B2, and B3 for the dipole, quadrupole, and sextupole components, while everywhere else B0, B1, and B3 are used. Please make these consistent.”

Response: Thank you for pointing out this inconsistency. We have updated Table 3 to use B0, B1, and B3 for the dipole, quadrupole, and sextupole components, respectively, to ensure consistency throughout the manuscript.

Comment 3: “It is unclear to me how the authors defined the normalized multipoles plotted on the left of Figure 12. What is the reference field that was used? It seems odd that the quadrupole term is only 0.005 and the sextupole term is ~0.0023 when this is a combined quadrupole-sextupole magnet, and that these terms are only a factor <6 larger than the unwanted octupole term. The authors should define how these multipoles are defined, which will hopefully clear this up. Also, if these normalized multipoles are the same as those that are required to be <10^{-3} then the authors should state it, since it would then appear that this magnet meets specifications.”

Response: We thank the reviewer for this important observation. The quadrupole component (n=2) is used as the reference field for normalization. We have added a clear definition of the normalized multipoles used in Figure 12 (left) to the manuscript text. In addition, the figure has been revised to use a logarithmic scale to improve clarity.

In section 5.3 line 432, we revised the paragraph to “…The normalized magnetic field error is defined relative to the main quadrupole component (n = 2), allowing higher-order multipole errors to be expressed as a fraction of the primary quadrupole field. Accordingly, the main quadrupole component (n = 2) was used as the reference for calculating the normalized field error. Figure 13(left) presents a comparison of the normalized normal multipole components at an excitation current of 210 A, as obtained from both simulation and measurements. For the dipole term and higher-order components (n=4, 5, 6, …), the measured multipole strengths tend to exceed the simulated values. The measured deviation is likely due to mechanical misalignments of the magnet poles, which can introduce field distortions. Any residual misalignment is expected to manifest primarily as a dipole component rather than as higher-order multipoles. Higher-order field errors can be further minimized by chamfering the ends of the magnet poles [8]. In the actual machine, the dipole term affecting the beam orbit in the booster can be corrected by applying appropriate magnetic fields using the corrector magnets”

 

Comment 4: “Finally, it would be useful if the authors made some estimate of the positional accuracy of the Hall probe set-up, and how this might affect the measured dipole and quadrupole components via feed-down.”

Response: We thank the reviewer for the helpful suggestion. To address the potential probe offset, we implemented a magnetic field scanning procedure to accurately identify the probe position. This explanation has been added to the manuscript in the following sentence:

In section 4.2 line 296, we added “To address probe positional discrepancies, a centering procedure was performed prior to the magnetic field measurements to accurately determine the position of the Hall probe. This procedure involved scanning the probe along both the horizontal (x) and vertical (y) axes to locate the magnetic center of the vertical magnetic field. Based on this method, we are confident that the Hall probe was well-centered during the measurements.”

 

We hope this revision addresses the reviewer’s request adequately. Thank you once again for your valuable input.

Reviewer 3 Report

Comments and Suggestions for Authors

The authors reported their efforts to develop a prototype of combined function magnet for SPS-II booster synchrotron.

I'm bit worrying about the influence of the geometrical offset of the hole sensor on the 3-axis hole probe which they used in this research since there is no description about that. 

Please check this point. This offset cause some error in measurement.

The multipole analysis mentioned in line 393-396 has no reference. If authors developed this analysis, more detailed description should be given. If other researcher developed this analysis and reported in elsewhere, the authors must add a citation.

The hole sensor has certain width. In the case of the quadrupole, the width effect can be cancelled due to linear nature of the magnetic field. However, in the case of sextupole, there is no cancellation mechanism as like quadrupole. This perhaps cause too large dipole component observed in Figure 12 (left). To mimic this effect in the simulation, the authors are advised to take into accout the width of the hole sensor in the simulation side.

In addition to comments made above, I have many comments on the manuscript. Those comments are directly added to PDF file attached. Please check the attached file.

Comments for author File: Comments.pdf

Author Response

We sincerely thank the reviewer for the detailed and constructive feedback, which has significantly contributed to improving the quality of our manuscript. Below we provide point-by-point responses to each comment.

 

Comments and suggested points

The authors reported their efforts to develop a prototype of combined function magnet for SPS-II booster synchrotron.

I'm bit worrying about the influence of the geometrical offset of the hole sensor on the 3-axis hole probe which they used in this research since there is no description about that. 

Please check this point. This offset cause some error in measurement.

The multipole analysis mentioned in line 393-396 has no reference. If authors developed this analysis, more detailed description should be given. If other researcher developed this analysis and reported in elsewhere, the authors must add a citation.

The hole sensor has certain width. In the case of the quadrupole, the width effect can be cancelled due to linear nature of the magnetic field. However, in the case of sextupole, there is no cancellation mechanism as like quadrupole. This perhaps cause too large dipole component observed in Figure 12 (left). To mimic this effect in the simulation, the authors are advised to take into accout the width of the hole sensor in the simulation side.

In addition to comments made above, I have many comments on the manuscript. Those comments are directly added to PDF file attached. Please check the attached file.

 

Comment 1: “I'm bit worrying about the influence of the geometrical offset of the hole sensor on the 3-axis hole probe which they used in this research since there is no description about that. Please check this point. This offset causes some error in measurement.”

Response: We thank the reviewer for raising this important point.

We are aware that the 3-axis Hall probe may have a small internal geometrical offset between the X, Y, and Z sensing elements. To minimize the influence of this effect, we performed a centering scan before the main measurements. By scanning the probe in both horizontal (x) and vertical (y) directions, we identified the magnetic center based on the vertical field component and aligned the average sensing point of the probe accordingly. We add more detail in manuscript in section 4.2 line 296 “ To address probe positional discrepancies, a centering procedure was performed prior to the magnetic field measurements to accurately determine the position of the Hall probe. This procedure involved scanning the probe along both the horizontal (x) and vertical (y) axes to locate the magnetic center of the vertical magnetic field. Based on this method, we are confident that the Hall probe was well-centered during the measurements.”

 

Comment 2: “The multipole analysis mentioned in line 393–396 has no reference. If authors developed this analysis, more detailed description should be given. If other researcher developed this analysis and reported elsewhere, the authors must add a citation.”

Response: Thank you for pointing this out. The multipole analysis method we used is based on established techniques. We have now revised the manuscript to include appropriate references “…The angular position was varied from 0° to 360° around the center point at y = z = 0, and the multipole components were extracted using Fourier series fitting [17]. ..”.

Comment 3: “The hole sensor has certain width. In the case of the quadrupole, the width effect can be cancelled due to linear nature of the magnetic field. However, in the case of sextupole, there is no cancellation mechanism as like quadrupole. This perhaps causes too large dipole component observed in Figure 12 (left). To mimic this effect in the simulation, the authors are advised to take into account the width of the hole sensor in the simulation side.”

Response: We thank the reviewer for this insightful observation. In the simulation, the magnetic field was collected over a range similar to that used in the measurements to ensure a consistent comparison. The dipole component arises when the pole shape is rotated during the generation of the sextupole component. This effect can be mitigated by optimizing the bore radius of the magnet. In our design, the bore radius was optimized to 23.132 mm on the left side and 23.00 mm on the right side.

For clarity, we have included additional discussion on the origin and mitigation of the dipole component in Section 5.3, line 440 of the manuscript “…The measured deviation is likely due to mechanical misalignments of the magnet poles, which can introduce field distortions. Any residual misalignment is expected to manifest primarily as a dipole component rather than as higher-order multipoles. High-er-order field errors can be further minimized by chamfering the ends of the magnet poles [8]. In the actual machine, the dipole term affecting the beam orbit in the booster can be corrected by applying appropriate magnetic fields using the corrector magnets”

 

Comment 4: “In addition to comments made above, I have many comments on the manuscript. Those comments are directly added to PDF file attached. Please check the attached file.”

Response: We gratefully acknowledge the additional comments provided in the annotated PDF. We have carefully reviewed each suggestion and incorporated all necessary corrections and clarifications in the revised manuscript.

Comments4.1:  Wrong citation in Figure 1.

Response: Thank you for your careful review. The citation has been corrected in the revised manuscript.

Comments4.2: The authors mentioned that there are five main sections but only four are listed in here.

Response: We revised to “This paper is organized into five main sections: Introduction, Requirement and design parameters, Magnet Design and Simulation, Magnet Prototype, Results and Discussion, and Conclusion.”

Comments4.3: In Table 1, the sectupole field is listed as 22.327 T/m2.

Response: We revised from 22.326 T/m² to “22.327 T/m²”

 

Comments4.4:If possible, could you indicate the shimming plate part and the reference surface in Figure 2 (right)?

Response: We revised the figure 2 and added the sentence in section 3.1 line 166 “…At the end of each pole, a shimming plate is added to locally enlarge the GFR. As shown in Figure 2 (right) in green, the left pole side is shimmed with a thickness of 1.2 mm, while the right pole side is shimmed with 0.6 mm. The top and bottom poles are shimmed symmetrically. To ensure precise magnet positioning, a reference surface along x-axis at the pole should extend at least 5 mm. For this magnet, a 10 mm long reference surface, machined along the x-axis, has been provided to serve as a fiducial for accurate alignment.”

Comments4.5:Is the unit (m) correct? It should be (mm).

Response: The unit has been corrected to mm for the effective length.

Comments4.6: Is there no unit for the mesh size? The description can be understood by readers who have never used Opera-3D. This comments applies to following description and Figure 4.

Response: The unit for the mesh size has been added, and Figure 4 has been revised accordingly.

Comments4.7: Three hole sensors are not mounted exactly same place. There are 2-mm mismatches in horizontal and vertical direction. Did the authors compensate those mismatches of the position? If so, it is better to describe.

Response: To account for probe positional discrepancies, a centering procedure was performed prior to the magnetic field measurements to accurately determine the position of each Hall probe. We added the paragraph to explain in section 4.2 line 296 “ To address probe positional discrepancies, a centering procedure was performed prior to the magnetic field measurements to accurately determine the position of the Hall probe. This procedure involved scanning the probe along both the horizontal (x) and vertical (y) axes to locate the magnetic center of the vertical magnetic field. Based on this method, we are confident that the Hall probe was well-centered during the measurements”

 

Comments4.8: Figure 8 Text shown in color bar is not visible. Too many digits in the color bar axis label.

Response: The figure has been revised to improve visibility and simplify the color bar axis labels.

Comments4.9: Since the simulated and experimental results overlaps and the dots in plot is large, it is difficult to see the difference between the simulated and measured results. I recommend authors to also show the difference between the measured and simulated results (B_meas - B_sim). This comment also applied to Figure 9 (right).

Response: The figures have been revised for clarity by adjusting the scale to better visualize the differences. Additionally, the following sentence has been added in Section 5.3, line 353 “…The field difference between the two data set is approximately 1%, which can be at-tributed to the resolution limit of the Hall probe.” and in Section 5.3, line 361 “Based on this analysis, the physical length of the combined-function quadrupole magnet was determined to be 223.5 mm, approximately 10% shorter than the effective length. The magnetic field profile along the z-axis, as shown in Figure 10, was used to estimate the effective length. Magnetic fields were carried out 10 mm along the x-axis at the center of the y-axis. The vertical magnetic field data from simulation and measurement show good agreement at the magnet center; however, a larger deviation is observed near the end of the pole. This variation results in a difference in the effective magnetic length between simulation and measurement. It was found that the simulated effective length is approximately 250 mm, whereas the measured effective length is about 0.8% greater than the simulation.”

 

Comments4.10: How large is the fitting errors? Please add error bars due to fitting error to FIgure 10.

Response: We revised to include error bars representing the fitting errors.

Comments4.11: Definition of the normalized magnetic field is unclear for me. Please clearly define this quantity.

Response: We added the sentence on normalized magnetic field in section 5.3 line 404

“For a combined-function quadrupole magnet, it is essential to minimize the presence of unwanted higher-order multipole components that can adversely affect beam dynamics. This can be assessed through the normalized error distribution, defined as the magnetic field profile scaled with respect to the reference components up to the sextupole term, as expressed in the equation below.”

Comment4.12: About this technique, please add reference. If you developed this technique, more description why you can measure the multipole field by this method should be written.

Response: We added a reference for the multipole measurement technique (Reference 17).

“17. Jan, J.C.; Kuo, C.Y.; Chang, C.H.; Chu, Y.L.; Yu, Y.T.; Lin, F.Y.; Chen, H.H.; Huang, H.M.; Yang, C.S.; Hwang, C.S. Magnetic field character of TPS booster magnets, In Proceedings of the 4th International Particle Accelerator Conference, Shanghai, China, 12–17 May 2013; pp. 3594–3596, ISBN 978-3-95450-122-9.”

 

Comments4.13: The transverse probe of lake shore 3 axis probe is not on the center of the probe itself. Do authors corrected this? If the probe is rotated around the center of the probe itself, the position of the x-probe moves around.

Response: We added the explanation of how to align the center of probe in section 4.2 line 296 “To address probe positional discrepancies, a centering procedure was performed prior to the magnetic field measurements to accurately determine the position of the Hall probe. This procedure involved scanning the probe along both the horizontal (x) and vertical (y) axes to locate the magnetic center of the vertical magnetic field. Based on this method, we are confident that the Hall probe was well-centered during the measurements.”

Comments4.14: Please add reference for stretch-wire technique.

Response: We added the references 18 and 19

“17.      Chen, C.; Yang, C.; Yang, C.; Chen, H.; Huang, J. Integrated Hall probe and stretched wire measurement system for an in vacuum undulator, In Proceedings of the 15th International Particle Accelerator Conference (IPAC 2025), Nashville, TN, USA, May 19–24, 2024, pp. 1398–1401; doi: 10.18429/JACoW IPAC2024 TUPG64.

18. Le Bec, G.; Chavanne, J.; Pennel, C. Stretched wire measurement of multipole accelerator magnets, Phys. Rev. Accel. Beams 2012, 15, 22401.”

 

Comments4.15: If possible please add reference “ Although the most recent SPS-II booster design has replaced the combined-function 428 quadrupole magnet with a pure quadrupole magnet to provide greater flexibility in mag-429 netic field tuning and to reduce multipole field errors”

Response: No reference is available; however, we have revised the paragraph as follows:

“Although the most recent SPS-II booster design has replaced the combined-function quadrupole magnet with a pure quadrupole magnet to provide greater flexibility in magnetic field tuning and to reduce multipole field errors, the comparison results will be reported elsewhere. Nevertheless, the experience gained from this prototype remains valuable.”

 

We hope this revision addresses the reviewer’s request adequately. Thank you once again for your valuable input.

Author Response File: Author Response.docx

Reviewer 4 Report

Comments and Suggestions for Authors

The manuscript presents the design, simulation, fabrication, and initial testing of a combined-function quadrupole–sextupole magnet for the SPS-II. The work is important to the local accelerator community, especially in the context of localizing magnet production capabilities. The authors demonstrate the magnet’s performance through both numerical simulation and experimental validation. Therefore, I recommend publication after minor revision.

 

Major comments:

Many of the figures are blurry, please consider improving them.

Will the linear permeability of 3,048 oversimplify the actual transient response?

 

Editorial comments:

Figure 1 is blurry, please consider improving it.

Table 1: “GFR (mm)”->"Good Field Region (GFR) (mm)" (abbreviation should be spelled out the first time it appears in the manuscript).

Line 94: “such as their ability to operate within their cost-effectiveness.”->“such as cost-effectiveness.”

Line 117: “22.326 T/m²”->“22.327 T/m²” (please unify the numbers for consistency)

Line 173: “York magnet”-> should it be “yoke magnet”?

Line 290: “while y and z coordinate were set at the center”->“while y and z coordinates were set at the center”

Line 355: “represent to dipole field”->“represent the dipole field”

Line 386: “misalignment of between”->“misalignment between”

Line 416: “confirms its precision”->“confirms the precision”

Author Response

We sincerely thank the reviewer for the thorough reading of our manuscript and for the constructive comments and suggestions. We have revised the manuscript accordingly, and our detailed responses are as follows:

Comments and suggested points

Major comments:

Many of the figures are blurry, please consider improving them.

Will the linear permeability of 3,048 oversimplify the actual transient response? 

Editorial comments:

Figure 1 is blurry, please consider improving it.

Table 1: “GFR (mm)”->"Good Field Region (GFR) (mm)" (abbreviation should be spelled out the first time it appears in the manuscript).

Line 94: “such as their ability to operate within their cost-effectiveness.”->“such as cost-effectiveness.”

Line 117: “22.326 T/m²”->“22.327 T/m²” (please unify the numbers for consistency)

Line 173: “York magnet”-> should it be “yoke magnet”?

Line 290: “while y and z coordinate were set at the center”->“while y and z coordinates were set at the center”

Line 355: “represent to dipole field”->“represent the dipole field”

Line 386: “misalignment of between”->“misalignment between”

Line 416: “confirms its precision”->“confirms the precision”

 

Comment 1: “Many of the figures are blurry, please consider improving them.”

Response: We thank the reviewer for pointing this out. All figures in the revised manuscript have been replaced with high-resolution versions to improve clarity and readability.

Comment 2: “Will the linear permeability of 3,048 oversimplify the actual transient response?”

Response: We acknowledge that using a linear relative permeability of 3,048 is a simplification. This value is generated from our model. Neglecting hysteresis effects is considered an acceptable approximation when the magnetic field remains within a range where the material behaves approximately linearly, and the primary focus is on the overall transient magnetic behavior rather than detailed core loss estimation. For clarity, we have added the statement, “Neglecting hysteresis effects is considered an acceptable approximation when the magnetic field remains within a range where the material behaves approximately linearly” in section 3.5 line 241.

 

Editorial Comments
Comment1: “Figure 1 is blurry, please consider improving it.”

Response: We thank the reviewer for this suggestion. Figure 1 has been replaced with a high-resolution version in the revised manuscript.

Comment2: Table 1: “GFR (mm)”->"Good Field Region (GFR) (mm)" (abbreviation should be spelled out the first time it appears in the manuscript).

Response: We revised. The abbreviation has been spelled out in Table 1 as suggested.

Comment3: Line 94: “such as their ability to operate within their cost-effectiveness.”->“such as cost-effectiveness.”

Response: The sentence has been revised accordingly.

Comment4: Line 117: “22.326 T/m²”->“22.327 T/m²” (please unify the numbers for consistency)

Response: The value has been corrected for consistency.

Comment5: Line 173: “York magnet”-> should it be “yoke magnet”?

Response: We thank the reviewer for identifying this typo. The term has been corrected.

Comment6: Line 290: “while y and z coordinate were set at the center”->“while y and z coordinates were set at the center”

Response: We thank the reviewer for pointing out this grammatical issue. The sentence has been corrected as suggested.

Comment7: Line 355: “represent to dipole field”->“represent the dipole field”

Response: The wording has been revised accordingly.

Comment8: Line 386: “misalignment of between”->“misalignment between”

Response: The wording has been corrected as suggested.

Comment9: Line 416: “confirms its precision”->“confirms the precision”

Response: The wording has been corrected as suggested.

 

We hope this revision addresses the reviewer’s request adequately. Thank you once again for your valuable input.

Author Response File: Author Response.docx

Round 2

Reviewer 3 Report

Comments and Suggestions for Authors

The authors answered all of my comment. However, there are some remained problems listed below.

1. The authors added a reference [17] about the Fourier series fitting (line 432). I checked the reference but there was no description about the Fourier series fitting. What performed in the reference is the polynominal fitting and not Fourier series fitting. 
2. The description of measurement method in line 428-432 is still unclear for me. The authors set the probe position at x=4mm and set the y=z=0. Then changed the angular position. Is it possible? To change the angular position of the probe with fixing the radius (respect to the center of the magnet), the vertical position should also be varied. Do I misunderstand the meaning of the angular position?
3. Figure 8: The color scale only have range from 1.5E-2 to 4.4E-2. Color scale should cover entire range of the plot.
4. Table 3: B0, B1 and B2 should be B₀, B₁, and B₂, respectively.
5. Table 3: In the table, so precise values are listed for the experimental results. Please consider the measurement accuracy and fitting errors. The overall errror information should be added to the table like 13.73 ± 0.05.

Author Response

We sincerely thank the reviewer for carefully reading our manuscript and providing valuable comments, which have greatly contributed to improving the quality of our work. Below we provide point-by-point responses to each comment.

Comments and suggested points

The authors answered all of my comment. However, there are some remained problems listed below.

1. The authors added a reference [17] about the Fourier series fitting (line 432). I checked the reference but there was no description about the Fourier series fitting. What performed in the reference is the polynominal fitting and not Fourier series fitting. 
2. The description of measurement method in line 428-432 is still unclear for me. The authors set the probe position at x=4mm and set the y=z=0. Then changed the angular position. Is it possible? To change the angular position of the probe with fixing the radius (respect to the center of the magnet), the vertical position should also be varied. Do I misunderstand the meaning of the angular position?
3. Figure 8: The color scale only have range from 1.5E-2 to 4.4E-2. Color scale should cover entire range of the plot.
4. Table 3: B0, B1 and B2 should be B₀, B₁, and B₂, respectively.
5. Table 3: In the table, so precise values are listed for the experimental results. Please consider the measurement accuracy and fitting errors. The overall errror information should be added to the table like 13.73 ± 0.05.

 

Comment 1: “The authors added a reference [17] about the Fourier series fitting (line 432). I checked the reference but there was no description about the Fourier series fitting. What performed in the reference is the polynominal fitting and not Fourier series fitting.”

Response: We sincerely apologize for the mistake in our previous revision. You are correct that reference [17] describes polynomial fitting and not Fourier series fitting. We appreciate your careful reading and pointing this out. In the revised manuscript, we have corrected this error and replaced the incorrect reference with an appropriate one that properly describes the Fourier series fitting method.

“ 17. Sunwong, P.; Prawanta, S.; Jummunt, S.; Numanoy, P.; Leetha, T.; Phimsen, T.; Pruekthaisong, P. Effects of Eddy Current and Permeability of Vacuum Chamber on Magnetic Field in Booster Synchrotron of Siam Photon Source II, J. Phys.: Conf. Ser. 2025, 2934, 012013; doi: 10.1088/1742-6596/2934/1/012013.”

Comment 2: “The description of measurement method in line 428-432 is still unclear for me. The authors set the probe position at x=4mm and set the y=z=0. Then changed the angular position. Is it possible? To change the angular position of the probe with fixing the radius (respect to the center of the magnet), the vertical position should also be varied. Do I misunderstand the meaning of the angular position?”

Response: We apologize for the unclear explanation in our previous version. You are correct in your understanding. The description “the probe position was set at x = 4 mm and y = z = 0” refers only to the starting point of the measurement. To change the angular position of the probe while keeping the radius fixed at 4 mm with respect to the magnet center, both the horizontal and vertical positions were varied accordingly. In order to clarify, we have revised the text in lines 428–432 to clarify this point

“In this study, multipole analysis was performed using both Opera-3D simulations and Hall-probe measurements within a circular region of 4 mm radius from the magnet center (z=0). The probe was rotated from 0° to 360° in 5° steps, with the horizontal and vertical coordinates adjusted simultaneously at each angular position to maintain a constant radius.”

Comment 3: “Figure 8: The color scale only have range from 1.5E-2 to 4.4E-2. Color scale should cover entire range of the plot.”

Response: Thank you for pointing this out. We have revised Figure 8 to adjust the color scale so that it now covers the entire range of the plot.

Comment 4: “Table 3: B0, B1 and B2 should be B₀, B₁, and B₂, respectively.”

Response: Thank you for your suggestion. We have revised Table 3 to correctly present B₀, B₁, and B₂, respectively.

 

Comment 5: “Table 3: In the table, so precise values are listed for the experimental results. Please consider the measurement accuracy and fitting errors. The overall errror information should be added to the table like 13.73 ± 0.05.”

Response: We have revised Table 3 to include the fitting errors for the quadrupole and sextupole terms, which quantify the uncertainty in the fitting procedure, while the dipole term exhibits a negligible error (very small value).

Author Response File: Author Response.docx