# Comparison of Two Detector Magnetic Systems for the Future Circular Hadron-Hadron Collider

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## Abstract

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## Featured Application

**This work describes a detailed study of two possible options for the magnetic system of a Future Circular hadron-hadron Collider detector.**

## Abstract

## 1. Introduction

## 2. Modeling the Magnetic Systems

#### 2.1. Reference Geometry of the FCC-hh Detector

#### 2.2. Baseline Magnetic System Model

#### 2.3. Minimal Steel Yoke Magnetic System

## 3. Comparison of the Magnetic Systems

#### 3.1. General Comparisons

#### 3.2. Magnetic Field Double Integral Comparisons

_{max}= 1.55 m and from zero to Z

_{max}= 16 m, the magnetic field double integrals are calculated along the trajectories of the charged particles emitted in a vertical YZ plane from the origin of the coordinate system at different polar angles θ counted from the Z-axis. A pseudorapidity value η, defined as η = − ln[tan(θ/2)], corresponds to each polar angle.

_{1}= 1.88753 refers to the corner of the central tracking cylinder at R = 1.55 and Z

_{1}= 5 m. The second value η

_{2}= 2.56343 refers to the front corner of the forward tracking cylinder at R = 1.55 and Z

_{2}= 10 m. The third value η

_{c}= 3.02982 refers to the corner of the forward tracking cylinder at R = 1.55 and Z

_{max}= 16 m. The magnetic field double integrals are calculated in the pseudorapidity range from 0 to 4 with an increment of 0.05. The polar angle θ corresponding to η = 0 is equal to 90°. The polar angle θ corresponding to η = 4 is equal to 2.0986°.

_{1}, the full track length L registered in the tracking volume is equal to R

_{max}/sinθ. For the values larger than η

_{1}but smaller than η

_{2}, the length L is equal to Z

_{1}/cosθ, where R

_{max}and Z

_{1}are the radius and half a length of the central inner tracker volume. In the pseudorapidity region from η

_{2}to η

_{c}, the track length L is again determined by the ratio R

_{max}/sinθ. Finally, for pseudorapidity greater than η

_{c}, the total track length L is equal to Z

_{max}/cosθ.

_{2}is determined by Equation (1) as follows [7]:

_{(dl}

_{,B)}represents the longitudinal component of the angle between the track projection to the vertical plane and the magnetic flux density vector, i.e., both the track length and the magnetic flux density vector are considered to lie in the vertical plane [7].

_{2}is equal to B·R

_{max}

^{2}/2 in the pseudorapidity regions η < η

_{1}and η

_{2}< η < η

_{c}and drops like B·Z

_{max}

^{2}·tan

^{2}θ/2 in the pseudorapidity region η > η

_{c}. In the inhomogeneous magnetic field, the double integral I

_{2}degrades by a few percent in comparison with the ideal constant magnetic field.

_{T}[GeV/c] is dominated by the detector resolution, and the relative transverse momentum precision δ can be expressed as follows [3,14]:

_{2}[T·m

^{2}] is determined by Equation (1).

- I
_{2bh}—double integral in homogeneous constant magnetic field of 4.000174 T, that corresponds to the central magnetic flux density in the baseline configuration of the magnetic system. - I
_{2bi}—double integral in the real inhomogeneous magnetic field of the baseline configuration of the magnetic system. - I
_{2mh}—double integral in homogeneous constant magnetic field of 4.243783 T, that corresponds to the central magnetic flux density in the minimal steel yoke configuration of the magnetic system. - I
_{2mi}—double integral in the real inhomogeneous magnetic field of the minimal steel yoke configuration of the magnetic system.

_{1}, and η

_{2}< η < η

_{c}the double integral I

_{2bh}has a constant value of 4.81 T m

^{2}, the double integral I

_{2bi}drops from 4.82 to 4.69 T m

^{2}, and from 4.19 to 3.97 T m

^{2}, correspondingly. In the same pseudorapidity regions, the double integral I

_{2mh}has a constant value of 5.1 T m

^{2}, the double integral I

_{2mi}drops from 5.12 to 5.0 T m

^{2}, and from 4.52 to 4.11 T m

^{2}, accordingly. In the pseudorapidity regions η

_{1}< η < η

_{2}and η > η

_{c}, all the double integrals drop like ~tan

^{2}θ.

_{bh}= I

_{2bi}/I

_{2bh}, a ratio R

_{mh}= I

_{2mi}/I

_{2mh}, and a ratio R

_{mb}= I

_{2mi}/I

_{2bi}, where indexes h, and i stand for the homogeneous and inhomogeneous magnetic field, accordingly. The indexes b and m denote the baseline and minimal steel yoke magnetic systems, correspondingly.

_{bh}> 0.9737, 1.0004 > R

_{mh}> 0.9801, 1.0607 < R

_{mb}< 1.0678.

_{bh}> 0.8169, 0.8869 > R

_{mh}> 0.7949, 1.0808 > R

_{mb}> 1.0324.

_{2hi}and I

_{2mi}degrade with pseudorapidity comparing with the ideal magnetic field double integrals I

_{2bh}and I

_{2mh}, correspondingly, but the integral I

_{2mi}is always larger than integral I

_{2bi}by 3.2–8.1%. According to Ref. [7], larger magnetic field double integral produces larger track sagitta on the same track length L with the same transverse momentum p

_{T}.

_{h}/δ

_{i}, or R = I

_{2i}/I

_{2h}, where indexes h and i denote the homogeneous and inhomogeneous magnetic field, accordingly. The degradation of the charged particle relative transverse momentum precision is proportional to 1 − R.

_{mh}, 1 − R

_{bh}, and 1 − R

_{mb}.

_{mh}< 0.0194, − 0.0006 < 1 − R

_{bh}< 0.0256 and—0.0607 > 1 − R

_{mb}> − 0.0676.

_{mh}< 0.0199, 0.0256 < 1 − R

_{bh}< 0.0263 and − 0.0676 > 1 − R

_{mb}> − 0.0678.

_{mh}< 0.2032, 0.1294 < 1 − R

_{bh}< 0.18 and − 0.0808 < 1 − R

_{mb}< − 0.0320.

_{mh}< 0.2051, 0.18 < 1 − R

_{bh}< 0.1831 and − 0.0320 > 1 − R

_{mb}> − 0.0324.

## 4. Discussion

- 6% larger central magnetic flux density.
- 13.2% smaller compression axial force and pressure in the main coil middle plane.
- 16.2% smaller radial pressure in each forward coil.
- 12.1% shorter radial distance to the acceptable safety level of the magnetic stray field.
- 7.2% shorter axial distance to the acceptable safety level of the magnetic stray field.
- 3.2 to 8.1% larger magnetic field double integrals and, thus, better transverse momentum resolution of the charged particle registered in the inner tracker.

- 0.97% larger axial attractive force onto each forward coil.
- 9.3% lower magnetic flux density in the transition region between central and each forward coil.
- 5.8% larger stored energy in the magnetic system.
- 61.3% larger radial pressure in the main coil.
- The compression axial force to the central barrel wheel of 236.9 MN each side is 2.92 times larger than the similar axial force in the CMS magnet yoke at the 3.81 T central field.
- The attractive axial force to the yoke endcap assembly of 164 MN from each side is 2.42 times larger than the similar axial forces in the CMS magnet yoke at the 3.81 T central field.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**One quarter of the reference geometry for the FCC-hh detector with the baseline magnetic system. In black, the quarter of the main superconducting coil in the vacuum cryostat and a half of the forward coil in its own cryostat are shown. In brown, the radiation protection conical-cylinder shield made of nonmagnetic material is displayed. In red, the charged particle tracker cylinders are shown. The orange lines represent the locations of the muon detection chambers. Other colours describe electromagnetic and hadronic calorimeters, and nonmagnetic disks of the forward muon system [3]. In this study, regions of these nonmagnetic materials are considered to have free-space permeability.

**Figure 2.**Three-dimensional model of the FCC-hh detector baseline magnetic system comprising the main superconducting coil with 10.9 m inner diameter, and two superconducting forward coils with 5.6 m inner diameter. The distance between the coils is 2.823 m each side.

**Figure 3.**Distribution of the magnetic flux density on the minimal steel yoke surface of the FCC-hh detector. The color scale unit is 0.5 T. With the central magnetic flux density of 4.24 T, the minimum and maximum magnetic flux density values on the yoke surface are 0.2037 and 4.0357 T.

**Figure 4.**The main superconducting coil with an outer diameter of 11.9, the five barrel wheels measuring 3.9 m in width, the two cylindrical radiation protection shields of 10 m outer diameter and 7 m long around each forward coil, the four endcap disks of 17.5 m diameter each, and the eight endcap disks of 14 m diameter each are shown. The main solenoid coil is visible between the barrel wheels in the air gaps of 0.125 m each. The length of the barrel part around the main coil is 20 m; the total length of the yoke is 35 m. The color scale of the magnetic flux density on the yoke surface is the same as in Figure 3.

**Figure 5.**Magnetic flux density distribution in a vertical YZ plane of the FCC-hh detector baseline magnetic system. The color magnetic field map plotted with the cell size of 0.05 m has a width of 50 m and height of 23 m. The color scale unit is 0.5 T. The minimum and maximum magnetic flux density values are 0.0142 and 4.1595 T. The quarter of the tracking system used in this analysis is drawn with the white rectangles.

**Figure 6.**Magnetic flux density distribution in a vertical plane of the FCC-hh detector minimal steel yoke magnetic system. The color magnetic field map plotted with the cell size of 0.05 m has a width of 50 m and height of 23 m. The color scale unit is 0.5 T. The minimum and maximum magnetic flux density values are 0.0002 and 4.3623 T. The maximum magnetic flux density in the barrel wheel layers is 2.3 T. The magnetic flux density in the first endcap disks at the radius of 6 m is 2.4 T. The quarter of the tracking system used in the present analysis is drawn with the white rectangles.

**Figure 7.**Magnetic flux density variation along the coil axes in the baseline (smooth curve) and minimal steel yoke (dashed line) magnetic systems.

**Figure 8.**Magnetic flux density out of the coil in the main coil central plane vs. radius (smooth and dashed lines) as well as along the coil axes vs. distance from the main coil center (dash-dotted and small-dotted lines). Smooth and dash-dotted lines correspond to the baseline magnetic system. Dashed and small-dotted lines correspond to the minimal steel yoke magnetic option.

**Figure 9.**Magnetic field double integrals in a vertical plane of the inner tracker of the minimal steel yoke design (smooth—I

_{2mi}and dashed—I

_{2mh}lines) and of the baseline FCC-hh detector design (small-dotted—I

_{2bi}and dash-dotted—I

_{2bh}lines) vs. pseudorapidity.

**Figure 10.**Magnetic field double integral degradation 1 − R

_{mh}(solid line), 1 − R

_{bh}(small-dotted line), and 1 − R

_{mb}(dashed line) in the inner tracker of the FCC-hh detector vs. the pseudorapidity.

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**MDPI and ACS Style**

Klyukhin, V.; Ball, A.; Berriaud, C.P.; Curé, B.; Dudarev, A.; Gaddi, A.; Gerwig, H.; Hervé, A.; Mentink, M.; Riegler, W.;
et al. Comparison of Two Detector Magnetic Systems for the Future Circular Hadron-Hadron Collider. *Appl. Sci.* **2023**, *13*, 10387.
https://doi.org/10.3390/app131810387

**AMA Style**

Klyukhin V, Ball A, Berriaud CP, Curé B, Dudarev A, Gaddi A, Gerwig H, Hervé A, Mentink M, Riegler W,
et al. Comparison of Two Detector Magnetic Systems for the Future Circular Hadron-Hadron Collider. *Applied Sciences*. 2023; 13(18):10387.
https://doi.org/10.3390/app131810387

**Chicago/Turabian Style**

Klyukhin, Vyacheslav, Austin Ball, Christophe Paul Berriaud, Benoit Curé, Alexey Dudarev, Andrea Gaddi, Hubert Gerwig, Alain Hervé, Matthias Mentink, Werner Riegler,
and et al. 2023. "Comparison of Two Detector Magnetic Systems for the Future Circular Hadron-Hadron Collider" *Applied Sciences* 13, no. 18: 10387.
https://doi.org/10.3390/app131810387