# Comparison of Two Detector Magnetic Systems for the Future Circular Hadron-Hadron Collider
## 3 Comparison of the Magnetic Systems
### Magnetic Field Double Integral Comparisons
In the constant homogeneous magnetic field, the integral \(I_{2}\) is equal to \(B\cdot R_{max}{}^{2}/2\) in the pseudorapidity regions \(\eta<\eta_{1}\) and \(\eta_{2}<\eta<\eta_{c}\) and drops like \(B\cdot Z_{max}{}^{2}\cdot tan^{2}\theta/2\) in the pseudorapidity region \(\eta>\eta_{c}\). In the inhomogeneous magnetic field, the double integral \(I_{2}\) degrades by a few percent in comparison with the ideal constant magnetic field. In the silicon tracker with \(N\) equidistant detector planes and a resolution \(\sigma\) [m] per layer, the momentum resolution at the large transverse momenta \(p_{T}\) [GeV/\(c\)] is dominated by the detector resolution, and the relative transverse momentum precision \(\delta\) can be expressed as follows [3; 14]:  
\[\delta=\frac{dp_{T}}{p_{T}}\approx\frac{\sigma p_{T}}{0.3I_{2}}\sqrt{\frac{720 }{N+4}}, \tag{2}\]  
where \(I_{2}\) [T\(\cdot\)m\({}^{2}\)] is determined by Equation (1). In Figure 9, the dependence of the magnetic field double integrals vs. pseudorapidity is presented for four cases:  
* \(I_{2 ht}\)--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. * \(I_{2 bi}\)--double integral in the real inhomogeneous magnetic field of the baseline configuration of the magnetic system. * \(I_{2 mh}\)--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. * \(I_{2 mi}\)--double integral in the real inhomogeneous magnetic field of the minimal steel yoke configuration of the magnetic system. In the pseudorapidity regions \(\eta<\eta_{1}\), and \(\eta_{2}<\eta<\eta_{c}\) the double integral \(I_{2 hh}\) has a constant value of 4.81 T m\({}^{2}\), the double integral \(I_{2 hi}\) 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 \(I_{2 mh}\) has a constant value of 5.1 T m\({}^{2}\), the double integral \(I_{2 mi}\) 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 \(\eta_{1}<\eta<\eta_{2}\) and \(\eta>\eta_{c}\) all the double integrals drop like \(\sim\)_tan\({}^{2}\theta\)_. Further comparisons are performed for the variables as follows: a ratio \(R_{bh}=I_{2 hi}/I_{2 ht}\), a ratio \(R_{mh}=I_{2 mi}/I_{2 mh}\), and a ratio \(R_{mh}=I_{2 mi}/I_{2 hi}\), 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. As reflected in Figure 9, in the pseudorapidity range from 0 to 2.55 the ratios \(R\) stay in the limits as follows: 1.0006 \(>R_{bh}>\) 0.9737, 1.0004 \(>R_{mh}>\) 0.9801, 1.0607 \(<R_{mb}<\) 1.0678. In the pseudorapidity range from 2.6 to 4, the ratios \(R\) stay in the limits as follows: 0.8706 \(>R_{bh}>\) 0.8169, 0.8869 \(>R_{mb}>\) 0.7949, 1.0808 \(>R_{mb}>\) 1.0324. These results show that both integrals \(I_{2 hi}\) and \(I_{2 mi}\) degrade with pseudorapidity comparing with the ideal magnetic field double integrals \(I_{2 hh}\) and \(I_{2 mh}\), correspondingly, but the integral \(I_{2 mi}\) is always larger than integral \(I_{2 hi}\) 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 \(p_{T}\). From Equation (2), the relative momentum precision for a real versus ideal solenoid field is given by a ratio \(R=\delta_{h}/\delta_{i}\), 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 \(1-R\).