# The Design of a High-Intensity Deuteron Radio Frequency Quadrupole Accelerator
## 7 Vacuum Design
### Vacuum Gas Load Calculation  
The gas load Q of the RFQ is divided into three variables [36]:  
1. The surface outgassing rate of the vacuum material (Q1) is defined as the product of the outgassing rate of the material and the surface area of the system. For an RFQ made of oxygen-free copper vacuum material, the outgassing rate is \(7.98\times 10^{-9}\) Pa \(\cdot\) L \(\cdot\) s\({}^{-1}\) cm\({}^{-2}\). For a total surface area of about 94,199 cm\({}^{2}\), the Q1 value is then 7.52 \(\times\) 10\({}^{-4}\) Pa \(\cdot\) L \(\cdot\) s\({}^{-1}\).
2. The system leakage and permeability (Q2) is typically 5-10% of the total gas load Q1 in an ultra-high vacuum system. For a conservative estimate, Q2/Q1 = 10%.
3. The outgassing capacity of other components added to the system (Q3) mainly consists of vacuum gauges outgassing. We considered Q3/Q1 = 10%.  
Summing over all source variables, the total gas output of the RFQ system is approximately 9.02 \(\times 10-4\) Pa \(\cdot\) L \(\cdot\) s\({}^{-1}\).  
During the passage of the beam, the continuous release of hydrogen gas from the ion source into the LEBT section contributes to a larger gas concentration into the RFQ cavity. Here, the beam pipes in both the low-energy and medium-energy segments are cylindrical, as shown in Figure 33. The capacity for the gas to pass through the vacuum pipe is referred to as its conductance and is defined as the amount of gas that flows through the pipeline under a unit pressure difference. For conductance C and pressures at either end of the vacuum pipe p1 and p2, the flow rate is  
\[\text{Q}=\text{C}\Delta\text{p} \tag{16}\]  
The mean free path of the gas is larger than the inner diameter of the tube. As a result, gas particle collisions with the inner wall of the pipe are common and dominated by free molecular flow. When the length of the pipe is more than 20 times its own diameter, the pipe conductance can be calculated with the long flow conductance equation [36]:  
\[\text{C}=\frac{1}{6}\sqrt{\frac{2\pi RT}{M}}.\frac{d^{3}}{L} \tag{17}\]  
Figure 32: Vacuum system.  
where \(d\) is the pipe diameter (m), \(L\) is the pipe length (m), \(R\) is the molar gas constant (8.3143 J/(K \(\cdot\) mol)), and \(T\) is the gas temperature (K). In 1932, [PERSON] gave a simplified flow conductance formula C (L/s) for short, cylindrical tubes (L/d \(<\) 20) at 20 \({}^{\circ}\)C,  
\[C_{20^{\circ}\mathrm{C}}=11.6~{}A_{0}\alpha \tag{18}\]  
where \(A_{0}\) is the area cross-section of the pipe (cm\({}^{2}\)), and \(\alpha\) is the Clausing coefficient. The smaller the L/d ratio, the closer \(\alpha\) is to unity. In order to guarantee the vacuum pressure, \(\alpha\) was chosen to be one. The vacuum pressures of the LEBT and MEBT are about 3 \(\times\) 10\({}^{-5}\) Pa and 1 \(\times\) 10\({}^{-5}\) Pa, respectively. The diameter of the beam input pipe was designed to be 1 cm, and the exit diameter was set to 2.2 cm. Details of the conductance parameters and air intake for both the low-energy and high-energy sections are provided in Table 12. The ion source employs hydrogen gas, for which the conductance with air is:  
\[C_{H_{2}}=3.87\mathrm{C} \tag{19}\]  
The gas loads amount to 1.022 \(\times\) 10\({}^{-3}\) Pa \(\cdot\) L/s for the LEBT and 1.536 \(\times\) 10\({}^{-3}\) Pa \(\cdot\) L/s for the MEBT. The gas load between sections is determined using the formula for the H\({}_{2}\) flow conductance.