The impact of the resistive-wall impedance on the ILSF storage ring

The resistive contribution of the vacuum chamber represents a significant source of the collective effects and should be analysed in several aspects. Due to the NEG-coated re-designed ILSF vacuum chamber, the resistive-wall effects were carefully studied. The measurement setup of the RW test was simulated and compared with the analytical equations. Finally, the longitudinal threshold was presented here to estimate the RW effects on the single bunch instability.


Impedance calculations
The resistive wall impedances of a non-evaporable getter (NEG) coating on OFHC copper pipe are studied here as a long multi-layered chamber.The calculations are done by the CST [1] and IW2D [2] codes.It may be possible to achieve the resistive impedance by simulating the measurement setup.The simulation results of the measurement setup can be compared with analytical equations and a wakefield solver of CST.
The measurement setup was simulated by inserting two twin copper wires in the DUT (device under test).The Pin-waveguides were defined on theses wires, and the background set to PEC (Figure 1).S-parameters were obtained by the Transient Solver of CST.Finally, the S-parameters are converted into impedances by using the standard logarithmic formula [3]: Where ∆=10mm is the interval between the twin wires, 10μm wire radius, c is the velocity of light, the Zcc and Zdd are characteristic impedances for the common (cc) and the differential (dd) modes, respectively.The transmission coefficients  21  ,  21  ,  21  and  21  belong to the resistive device under test (DUT) and those for the perfectly conductive chambers (ref).S-parameters are applied to Eq. 1 and 2. The Z_(L & T) were achieved in this method and compared with the analytical equations of the RW in Figure 2. The analytical equations are [4]: The transverse impedance result of the simulation test setup (Eq.2) was compared with the analytical Eq. 4 in Figure 2. Then we used the analytical equation to verify the IW2D calculations.Figure 3 and Figure 4 are shown the longitudinal and transverse parts, respectively.The IW2D code was used to obtain a vacuum pipe with the conductivity of copper without NEG (blue), copper with NEG (orange), and a chamber with the conductivity of NEG only (purple).The IW2D results were matched with the analytical equation with the conductivity of copper (red) and NEG only (green).The analytical equation, IW2D and test setup simulation results were agreed.Therefore, the IW2D results can be applied in instability studies and tracking codes.

NEG coating thickness
The thickness study of NEG coating was discussed in CST and IW2D code.The thickness of the NEG coating was increased from 1 to 5 micrometers.The results are presented in the longitudinal (Figure 5) and transverse plane (Figure 6) for the IW2D code.The analytical equation for a chamber with the conductivity of copper and NEG were added to these figures.Then the simulations were repeated with the CST wakefield solver (Figure 8Figure 9).Finally, the IW2D and CST results were compared in Figure 10.
For the simulation configuration in CST, we employed 1 mm copper walls, normal conducting background, 1mm beam offset, and the open boundary in the z-direction.Also, 0.8 mm bunch length and 20 lines per sigma were defined as mesh properties.90 million mesh cells were applied to the simulation.The 0.1 m tube length was considered as the vacuum pipe.The geometry is shown in Figure 7.The calculation was done for 1m wake length.The results are presented in Figure 8 and Figure 9 for the longitudinal and transverse parts, respectively.The CST and IW2D results were compared in Figure 10.In addition, the analytical equation is shown here for a copper (red) and NEG (green) chamber.It should be noted that CST data needs some conversion in the real and imaginary parts.The agreement is acceptable for all the simulations.Considering the 7.9mm bunch length of the ILSF, 1μm NEG coating will be in a safe range and far from the transition frequency from NEG to the copper section of the vacuum chamber.Therefore, 1μm thickness seems proper for the storage ring of the ILSF.

Longitudinal instability
The RW portion of the impedance budget can be applied to tracking codes.It shows the contribution of the RW impedance on the threshold current.It applied to MBTRACK2 [5] and ELEGANT [5,6] tracking codes.The longitudinal single-bunch instability was studied here.The expectable increasing behavior of the bunch length is shown in Figure 11.The 10 5 macroparticles is involved in this simulation for the 30000 turns.The RW impedance for the 462m strength section was applied to the tracking codes.The variation of the energy spread is shown in Figure 12.The energy spread is averaged over all the turns.It causes a decreasing behavior at the first of the graph.
The energy spread has started to increase from 5mA in both curves of the ELEGANT and MBTRACK2 codes (Figure 12).

Current impedance budget
Up to now, the impedance budget of the ILSF is calculated for the BPMs, Insertion devices, Tapers, and Fast correctors.The tracking simulations were repeated with an impedance budget covering all these components.Figure 13 is achieved through an ELEGANT tracking code.The simulation details were the same as in the previous section.In this case, the threshold current decreased to 2mA.The final data are averaged on the last 12000 turns.Averaged data on the last 12000 turns.
Comparing Figure 13 (for the impedance budget of RW and mentioned components in the storage ring) with Figure 12 (for the impedance budget including only the RW contribution), it can be concluded that only the RW contribution of the vacuum chamber can decrease the threshold current on 5mA for the ILSF.Moreover, all the other components are responsible for the further reduction in the threshold current.

Conclusion
Resistive wall calculations for the vacuum chamber of the ILSF were calculated by the analytical equations and wakefield solvers.In addition, the test setup of RW was simulated too.All the obtained results were in good agreement.
Then, the contribution of the RW was applied to tracking codes and compared with the case in which other components were considered too.The RW impedance of the 462 m straight section leads to the threshold current of 5mA for microwave instabilities.Considering all the mentioned components to the impedance budget, the microwave instability decreased to 2mA.

Figure 1 .
Figure 1.The simulation of the test setup.

Figure 2 .
Figure 2. The transverse part of the analytical equation (Eq.4-blue curve) is compared with the simulation test setup (Eq.2-green curve).

Figure 5 .
Figure 5. Longitudinal part of the NEG thickness in IW2D code were studied for 1 to 5μm.The analytical Eq. 3 is shown for a NEG (black) and copper (pink) chamber.

Figure 6
Figure 6.Transverse part of the NEG thickness in IW2D code were studied for 1 to 5μm.The analytical Eq. 4 is shown for a NEG (blue) and copper (purple) chamber.

Figure 8 .
Figure 8.The longitudinal part of the NEG thickness in CST was studied for 1 to 5μm.The analytical Eq. 3 is shown for a NEG (green) and copper (black).chamber.

Figure 9 .
Figure 9.The transverse part of the NEG thickness in CST was studied for 1 to 5μm.The analytical Eq. 4 is shown for a NEG (brown) and copper (purple) chamber.

Figure 11 .
Figure 11.MBTRACK2 & ELEGANT comparison for the Bunch Length tracking.

Figure 12 .
Figure 12.MBTRACK2 & ELEGANT comparison for the Energy Spread tracking.The turbulent and incremental behavior of the Microwave instability is been started from 5mA in both curves.

Figure 13 .
Figure 13.ELEGANT tracking for the current impedance budget.Averaged data on the last 12000 turns.