Closed-orbit Distortion study of XiPAF upgrading project synchrotron

Xi’an 200 MeV proton application facility (XiPAF) upgrading project is now in the design phase. A six super periods synchrotron is the main component of this project. Using MADX program, a statical analysis of the relationship between closed-orbit distortion (COD) and different types of magnet errors was carried out. Because we do not know the exact error in the design phase, we use rms value to represent errors and CODs. The results show that COD is in direct proportion to magnet error in some cases while the others are square proportional. We regard all the cases as direct proportional since the errors are usually small and find that the error amplification factors differ a lot. These results can give us some meaningful guide in lattice design and magnet error control.


Introduction
Xi'an 200 MeV Proton Application Facility (XiPAF) is an accelerator-based space radiation environment simulation platform [1].It is now being upgraded to have the ability to accelerate multiple ions like helium ions, carbon ions, etc [2].In the upgrading project, the circumference of the synchrotron will change from 30.9 m to 39.96 m, with reutilization of dipoles, quadrupoles, sextupoles and the magnetic alloy loaded cavity.Third-order resonance slow extraction method is still used for beam extraction while the beam injection method will change from stripping injection to multi-turn injection.The upgrading project is now in the design phase.The synchrotron has 6 super periods, each period consists of a dipole, a focusing quadrupole and a defocusing quadrupole.The main parameters of the synchrotron are listed in table 1 and the twiss parameters are shown in figure 1.
In practical case, magnet errors are inevitable due to field imperfection and misalignments, thus causing closed-orbit distortion (COD).Theoretical analysis of COD caused by a kick has been described in detail in many textbooks [3], which can be written as where u is closed-orbit distortion, ν is tune value of the synchrotron, ψ is phase shift.Subscript z represents either horizontal or vertical coordinate.Label i and label s refer to the longitudinal location where the kick exists and the closed-orbit we observe respectively.However, before the synchrotron assembly is complete, we do not know the actual alignment errors of magnets.What we know is the approximate range of the errors according to the previous engineering experience, so the principle of statistics is a useful tool for our work.

Method
We only consider the errors of dipole and quadrupole magnets because the sextupoles work only in the extraction process and the synchrotron do not have other multipoles.For each magnet, we define the following errors: ∆BL/BL means the field error of dipole, ∆KL/KL means the field error of quadrupole, ∆x, ∆y and ∆s means the alignment error along horizontal, vertical and longitudinal directions, ∆ϕ, ∆θ and ∆ψ means rotation around horizontal, vertical and longitudinal axes.
We use MAD-X program [4] to calculate COD caused by magnet error.In engineering practice, uniform distribution is widely used to describe error distribution.Root-mean-square (rms) value is used to describe the magnitude of magnet errors and CODs.In our calculation, we set the error distribution to have a standard deviation , which means the error is uniformly distributed in (− √ 3σ, √ 3σ).It is known that the rms value of a data set equals µ 2 + σ 2 , and in this situation, the rms value of the error also equals σ because mean value µ is zero.
We pick a series of sampling points on the ring and for a specific error, the COD of each sampling point caused by this error can be calculated.Two rms value calculating process will be carried out.One is the rms value of all the sampling points and the other is the rms value of COD caused by all the errors in the data set: where N is the number of sampling points and M is the amount of data in the error data set.

Results
Figure 2 shows the relationship between COD and dipole errors, figure 3 shows that of quadrupole errors.It should be pointed out that the impact of ∆θ error of dipole is not shown in figure 2, that is because the relationship is not direct proportional but square proportional, as shown in figure 4. Just like the dipole ∆θ error, we noticed that small horizontal CODs caused by dipole ∆y error, dipole ∆ϕ error, dipole ∆ψ error, quadrupole ∆y error and quadrupole ∆ϕ error are also in square proportional to magnetic error.In figure 2 and figure 3, we use p1 to represent the direct proportional coefficient and p2 to represent the direct proportional coefficient.In XiPAF upgrading project synchrotron, the displacement and rotation errors are small, usually less than 1 mm (or mrad), so we regard these 6 relationships as direct proportional in our analysis and calculate the corresponding proportionality coefficients.The comparison of direct and square proportional coefficients is shown in table 2. The proportionality coefficient is defined as error amplification factor in [5].In our calculation, we give it a more specific define, that is the COD (in millimeters) caused by 1‰ dipole field error, or by 1 mm magnet displacement error, or by 1 mrad magnet rotation error.
The amplification factors of different type of error are listed and sorted in Table 3. From Table 3 we can see clearly that in terms of amplification factor, dipole field error and quadrupole ∆x error impact most on horizontal COD, while dipole ∆ψ and ∆y error impact most on vertical COD, this means we must control these errors strictly in manufacture and installation phase.
The ∆KL/KL, ∆s errors in quadrupole only change the optics characteristic of the synchrotron and ∆ψ error in quadrupole will cause coupling of horizontal and vertical betatron oscillations.These three types of error have no impact on COD and are not shown in figure 3.
From XiPAF synchrotron assembling and magnet measuring process, the limitation values of magnet error are listed in table 4.
Using the error amplification factors in table 3 and error data in table 4, we can estimate that the horizontal and vertical COD of upgraded synchrotron will be 2.65 mm and 3.14 mm respectively.The meaning of finding out the error amplification factors is that if we improve the magnet assembly technology in the upgrading project, we should choose these types of errors having large amplification factor, so that we can use less cost to achieve smaller CODs.

Discussion
Some interesting things can also be seen from figure 2 and figure 3. The dipole ∆y error caused horizontal COD comes from the fringe field effects at the entrance and exit of the magnet.If we consider hard edge condition, this COD will be zero, otherwise, the sketch map of fringe field in soft edge condition is shown in figure 5.The dipoles of XiPAF is rectangular type, so the existing of fringe field and ∆y error will lead to a small horizontal magnet field component, this is the reason for ∆y error of XiPAF dipoles causing a vertical COD.The small horizontal COD caused by dipole ∆y error also comes from soft edge conditions.
According to [4], if error distribution is independent of β function distribution, we have a statistic expression to describe the COD caused by dipole field errors, for XiPAF upgrading project synchrotron, the expression can be written as where N B is the number of dipoles.We plug in the parameters and calculate that the amplification factor equals 5.5, which is very close to the simulation result 5.739.The reason for the discrepancy of theoretical and simulation results is that the ring is not fully occupied by dipoles thus the independent condition of error and β function distribution is not strictly true.But when we use the same method to explain the dipole ∆x error caused horizontal COD, the theoretical amplification factor equals 2.2, while the simulation result is 0.924.This because the independent condition is no longer valid.
Though we get some meaningful conclusions to instruct synchrotron design and magnet error control, there are still several questions remain to be solved.For the error types except dipole field error, the statistic expression is an issue worth studying, but this is difficult be-cause the independent condition is not valid at all in these situations (a simple example was made in previous contents).

Concusion
We carried out a statical analysis of the impact of dif-ferent types of magnet errors on COD and found a direct proportion relationship between them.Similar study can be found in several literatures [6,7].We calculated the error amplification factors of all types of magnet errors and sorted them, finding that the factors vary a lot.This is a constructive conclusion, especially in the design phase when we do not know the actual errors.We can evaluate if the lattice design is reasonable and in the magnet manufacturing and assembling phase, we can know which errors should be restraint strictly for better synchrotron performance while which errors can be relaxed slightly for a more economic cost.

Figure 4 .
Figure 4. Dipole ∆θ error effect on COD, the quadratic term has an amplification factor of 3.408e-3.

Table 2 .
Comparison of direct and square proportional coefficients Error types Coefficient (square proportional) Coefficient (direct proportional)