## 2 Design and Fabrication of the Current Leads
### Vapor-Cooled Current Lead  
Vapor-cooled current leads are generally used in a superconducting magnet system that uses liquid helium, and they are cooled by the evaporated helium gas to minimize heat inflow [17]. Typically, copper and brass are used for the fabrication of current lead. The total heat input \(Q_{in}\) flowing into and generated within a differential volume \(A\Delta z\) of vapor-cooled current leads carrying the current lead's rated current \(I_{0}\) is expressed as Equation (1) [18]:  
\[Q_{in}=\left[Ak(T)\frac{dT}{dz}\right]_{z+\Delta z}+\dot{m}_{I}C_{p}(T)T+ \frac{\rho(T)I_{0}^{\ 2}}{A}\Delta z \tag{1}\]  
where \(z\) is the axial distance along the current lead, and \(z=0\) at the current lead's cold end. The symbols \(k\left(T\right)\), \(A\), and \(\rho\left(T\right)\), are the thermal conductivity, active cross-sectional area, and electrical resistivity of the current lead, respectively. The symbols \(\dot{m}_{I}\) and \(C_{p}(T)\) denote the helium mass flow rate and specific heat, respectively. It is assumed that the heat transfer rate between liquid helium and current lead is perfect. The symbol T is the temperature of both at \(z\). The total heat output \(Q_{out}\) flowing out of the differential volume of \(A\Delta z\) is expressed as Equation (2) [18]:  
\[Q_{out}=\left[Ak(T)\frac{dT}{dz}\right]_{z}+\dot{m}_{I}C_{p}(T)(T+\Delta T) \tag{2}\]  
In the steady state condition, The \(Q_{in}\) and \(Q_{out}\) are perfectly equal. Thus, Equation (3) [18] can be obtained as:  
\[\left[Ak(T)\frac{dT}{dz}\right]_{z+\Delta z}-\left[Ak(T)\frac{dT}{dz}\right] _{z}-\dot{m}_{I}C_{p}(T)(T+\Delta T)+\frac{\rho(T)I_{0}^{\ 2}}{A}\Delta z \tag{3}\]  
dividing Equation (3) by \(\Delta z\) and replace \(\Delta z\) to 0, Equation (4) can be obtained as:  
\[\frac{d}{dz}\left[Ak(T)\frac{dT}{dz}\right]-\dot{m}_{I}C_{p}(T)\frac{dT}{dz} +\frac{\rho(T)I_{0}^{\ 2}}{A} \tag{4}\]  
The heat input \(Q\) at the cold end for a given electrical current \(I\), and the optimized shape ratio of vapor-cooled current leads are obtained from Equations (5) and (6), respectively [18] as follows:  
\[\frac{Q}{I}=\sqrt{\frac{h_{L}k_{0}\rho_{0}}{cp_{0}}} \tag{5}\]  
\[\left(\frac{I_{0}\ L}{A}\right)=\int_{T_{0}}^{T_{I}}\frac{dT}{\rho(T)}\sqrt{ \frac{Cp_{0}k_{0}\rho_{0}}{h_{L}}} \tag{6}\]  
where \(h_{L}\), \(k_{0}\), \(\rho_{0}\), and \(Cp_{0}\) are latent heat of vaporization of liquid helium, thermal conductivity of metal current lead at cold end, electrical resistivity at cold end, and specific heat of liquid helium, respectively. Figure 3 shows the thermal properties of the copper and brass. The solid line indicates the variations of the integral value of the thermal conductivity. The dashed line indicates the values obtained by multiplication of the thermal conductivity and electrical resistivity of brass and copper in the temperature range 4-300 K [19; 20]. As indicated by Equation (2), the heat input value is related to the thermal properties. The \(Q\)/\(l\) value, at the liquid helium temperature for brass (Cu90:Zn10) obtained through Equation (5), as well as for brass (Cu60:Zn40), copper RRR10, and copper RRR30 are 0.7332, 0.7560, 0.6153, and 0.6155 mW/A, respectively. The optimum shape ratio (\(I\)-\(L\)/\(A\)) values calculated through Equation (6) for brass (Cu90: Zn10), brass (Cu60:Zn40), copper RRR10, and copper RRR30 are \(1.967\times 10^{6}\), \(1.036\times 10^{6}\), \(9.008\times 10^{6}\), and \(1.880\times 10^{7}\) A/m, respectively. For a more accurate design, the elemental analysis of commercially available brass was conducted with a glow discharge mass spectrometer (GDMS). Correspondingly, the ratio of copper to zinc was approximately 60:40 [21]. Considering the height of the cryostat for a 5 T superconducting magnet, the total length of the current lead is 0.98 m. For a length of 0.98 m, the cross-sectional areas of the brass (Cu60:Zn40) and copper RRR10 vapor-cooled current leads are 42.56 and 4.90 mm\({}^{2}\), respectively.