# Comparison of Two Detector Magnetic Systems for the Future Circular Hadron-Hadron Collider
## 3 Comparison of the Magnetic Systems
### Magnetic Field Double Integral Comparisons  
To quantify the quality of the magnetic field inside the inner tracker, the magnetic field double integrals method is applied. This method is fully described in Ref. [7] and was used to characterize the quality of the magnetic field in different options [12,13] of the magnetic system for the FCC-hh detector [3]. As shown in Figure 1, the considered inner tracker of the detector consists of three cylinders with a radius of 1.55 m each sitting on the coil axes. The central cylinder has a length of 10 m and is located inside the main superconducting coil. The other two cylinders are 6 m long and are located inside the forward coils at the distances of 5 m from the central tracker cylinder. In the following study, only one quarter of the tracking system in the vertical plane is considered. This quarter is shown in Figures 5 and 6 as white rectangles of 1.55 m radius propagated along the \(Y\) axis. A half length of 16 m is directed along the \(Z\) axes in the same figures. In this region from zero to \(R_{max}\) = 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 \(\theta\) counted from the \(Z\)-axis. A pseudorapidity value \(\eta\), defined as \(\eta=-ln[tan(\theta/2)]\), corresponds to each polar angle. Three specific pseudorapidity values are referred to the corners of the inner tracker volumes in the vertical plane. The first value \(\eta_{1}\) = 1.88753 refers to the corner of the central tracking cylinder at \(R\) = 1.55 and \(Z_{1}\) = 5 m. The second value \(\eta_{2}\) = 2.56343 refers to the front corner of the forward tracking cylinder at \(R\) = 1.55 and \(Z_{2}\) = 10 m. The third value \(\eta_{c}\) = 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 \(\theta\) corresponding to \(\eta\) = 0 is equal to 90\({}^{\circ}\). The polar angle \(\theta\) corresponding to \(\eta\) = 4 is equal to 2.0986\({}^{\circ}\). Inside the central tracking cylinder while the pseudorapidity values are smaller than \(\eta_{1}\), the full track length \(L\) registered in the tracking volume is equal to \(R_{max}\)/\(sin\theta\). For the  
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. values larger than \(\eta_{1}\) but smaller than \(\eta_{2}\), the length \(L\) is equal to \(Z_{1}/cos\theta\), where \(R_{max}\) and \(Z_{1}\) are the radius and half a length of the central inner tracker volume. In the pseudorapidity region from \(\eta_{2}\) to \(\eta_{c}\), the track length \(L\) is again determined by the ratio \(R_{max}/sin\theta\). Finally, for pseudorapidity greater than \(\eta_{c}\), the total track length \(L\) is equal to \(Z_{max}/cos\theta\). The magnetic field double integral \(I_{2}\) is determined by Equation (1) as follows [7]:  
\[I_{2}=\int\limits_{0}^{L\ sin\theta}\int\limits_{0}^{\pi/sin\theta}B\ sin\theta_{( dI,\mathbf{B})}\ dl\ dr. \tag{1}\]  
Here, \(r\) is a running transverse radius and \(l\) is a running track length along the track length \(L\) during the double integration of a product of the total magnetic flux density \(B\) and the track direction. The polar angle \(\theta_{(dI,\mathbf{B})}\) 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].