Field Quality Improvement of Septum Magnets for SuperKEKB Injection System

The SuperKEKB accelerator is a electron-positron collider consisting of the 7 GeV electron ring (high energy ring or HER) and the 4 GeV positron ring (low energy ring or LER). The commissioning of the SuperKEKB accelerator is underway, aiming to supply a great number of interaction events of electrons and positrons to the Belle II detector which is used for discovering the new physics beyond the standard model. The important milestone is to obtain integrated luminosity of 15 ab–1 in the next decade, so that the luminosity should exceed 2 × 1035 cm–2s–1 in several years. To achieve the goal, both rings have to be filled with high current beam of a few amperes, where the high injection efficiency is vitally important because lifetime is expected to be very short. One of the key components of the injection system is the septum magnet. It has been found that a transverse fringe field near the septum plate has sizable multipole components. A tracking simulation shows such fringe fields generate a vertical non-Gaussian tail, which could cause a beam background as well as a bad injection efficiency. Though quadrupole component in the multipoles could be, in principle, cancelled by adjusting the quadrupole magnets in the upstream of the beam line, it turned out to be difficult in actual operation. Even the quadrupole component was perfectly cancelled, components higher than quadrupole would be still harmful, since it generates a beam halo in the ring after injection, which increases the beam backgrounds to the Belle II detector. This paper describes about improvement of the field quality of the septum magnet.


Introduction
Each injection system for the SuperKEKB [1] main rings (MRs) consists of the septum, kicker and quadrupole magnets.The drive current pulse shape for the septum magnet is the full sinusoid.So far the betatron injection has been adopted for both rings; as for the HER, the injection system has been designed to be capable of the synchrotron injection [2].To store the higher current, increasing the injection efficiencies has been one of the most important issues.Recently, it has been found that the nonuniformity of the transverse edge of the septum field causes the degradation of the injection efficiency.

Transverse edge of Septum field
The septum field has been measured and the distortion of the injection beam by the horizontal kick in the septum magnet has been estimated and concluded to be negligible for injection [3].Recently, it has been found that the vertical kick by the septum transverse fringe field affects the injection efficiency.The result of the tracking simulations with and without the septum fringe field is shown in Figure 1, where the real coordinates u ≡ (x, p x , y, p y ) ⊤ are transformed to the normal coordinates U ≡ (X, P X , Y, P Y ) ⊤ with the transformation matrix M as U = M u.The definition of M is given by: where the symbols α x , β x and α y , β y denote Twiss parameters for the horizontal and vertical planes for an optical model that includes linear part of the fringe field.Even though the beta mismatch of BT-line can be reduced, the septum fringe field induces irrecoverable vertical tail of the injection beam.The nonuniformity of septum fringe field implies increase of effective septum width.In other words, reducing nonuniformity derives the reduction of injection amplitude, thus the increase of injection efficiency.The reduction of radiation background during injection are also expected.

Analytic one-dimensional calculation on magnetic shielding
To understand septum field structure, it is important to analyze the relationship between the magnetic flux permeation and the eddy current in the septum conductor.Generally, the more effective the confinement of the magnetic flux in the septum gap is, the higher the uniformity of the density of that is.Supposing the copper plate 1 mm thick is located in the region 0 ≤ x ≤ 1 mm, the shielding effect for the region x ≥ 1 mm can be calculated if the region x ≤ 0 is filled with the magnetic flux density ⃗ B 0 (x < 0, t) = (0, B 0 sin ω 0 t, 0), where the symbol B 0 is constant, the symbols ω 0 and t denote the angular frequency and time, respectively.From the reference [4], the magnetic field and eddy current in the septum conductor are derived as follows: where symbols s, µ 0 , σ, δ, B 0 , B 1 , j z denote the following: s Laplace variable, µ 0 the permeability in vacuum, µ 0 = 4π × 10 −7 NA −2 , σ the electric conductivity, σ = 5.76 × 10 7 Sm −1 , δ the skin depth with quasi-static regime assumed, B 0 the maximum magnetic flux density at inner (core side) surface x = 0 of septum conductor at the time t = τ /4 for the pulse period τ = 2π/ω 0 , B 1 the magnetic flux density permeating through the septum conductor, j z the real current density in the septum conductor.The symbols L −1 and Res(f (s), s 0 ) stand for the inverse Laplace transformation and residue of function f (s) at the pole s = s 0 , respectively.In this case, the poles lie at s = ±iω 0 from Eq. ( 4).The results of the numerical calculations of B 1 and j z are plotted in Figure 2. The integral of the current density over the septum conductor width (0 ≤ x ≤ 1 mm) as a function of τ is shown in Figure 3.It has been found that the total eddy current takes the maximum absolute value at τ = 40 µs, thus the magnetic flux is most effectively shielded with that pulse width for 1 mm thickness septum.

Possible upgrade
To generate a pulse of τ = 40 µs, production of new power supply is needed, which costs too much of the SuperKEKB budget.Moreover, the remodeling of septum magnet structure is required to be more robust to an electric discharge because of shorter pulse.From these external conditions, we have decided to make use of the present power supply by changing parameters to get as smaller τ as possible while keeping output voltage within discharge threshold.The result of compromise was that τ = 257.2µs.

Simulation study on septum field
The septum field calculation has been performed to optimize the shim shape using Opera-2D [5].

Shim shape study
Several simulation and measurement results on the peak field in the core gap are shown in Figure 4 hence the field has been simulated with every 10 µm of h and w top .

Quadrupole component of each shim shape
The path integral of the quadrupole component B ′ along the trajectory in the septum field with each shim shape, which is denoted as B ′ L, has been calculated as a function of the injection position x inj , which is defined as the distance between the center of MR beam duct and the beam orbit at the exit of the septum magnet [2].
In the optimization of the shim shape, three criteria has been adopted that B ′ L (i) should be as small as possible, (ii) should cross zero at the absolute value of x inj as small as possible within the operation region, (iii) should be less dependent on x inj as possible.
The second criterion is necessary to obtain the best operation point where both of the absolute value of B ′ L and the injection amplitude get their minimums.Almost all of parameters were rejected by the first criterion, then the latter two criteria helped to select one from several remaining candidates, e.g.shown in Figure 5.

Final decision of shim shape
It was decided that the drive current pulse period is reduced to τ = 257.2µs from 300 µs, and the shim shape is changed as drawn in Figure 6.As for τ = 40 µs option, even though it performs the ideal uniformity of the septum field, it was rejected from viewpoints of the electric specification and of the cost, as described in the previous section.The shim edges have round forms to keep from getting a damage in manufacturing processes.

Summary and conclusion
To increase the injection efficiencies for MRs, the field quality improvement of septum has been studied.According to the analytic calculation, the drive pulse period τ = 40 µs shows the best shielding performance for the septum conductor of 1 mm thick.Even though it has been given up to fabricate the new septum magnet with τ = 40 µs at this time, the realistic parameters to improve the uniformity of septum field have been found.Fabrication of a new septum magnet for HER with τ = 257.2µs will be completed in FY2023.After confirming the performance of the new septum magnet, we plan to build a new septum for LER.

Figure 1 .
Figure 1.The effect of nonuniformity of transverse fringe field in septum core has been investigated with a tracking simulation of the beam transportation (BT) line.The upper(lower) row represents the horizontal(vertical) plane.The first (or the left) column represents the distributions in real coordinates, including the 1σ-ellipses of the lattices with (magenta) and without (lightgray) including just the linear part of the fringe field.The second column represent the distributions in normal coordinates, which are converted with Twiss parameters of the magenta ellipse.

Figure 2 .
Figure 2. The calculated distributions of B 1 (left) and j z (right) in the 1 mm thickness septum conductor for several τ values.The legend in right figure is common to both figures.

Figure 3 .
Figure 3.The integral of the current density over the septum conductor width as a function of τ is shown.The absolute current peaks at about τ = 40 µs.

Figure 4 .
Figure 4.The magnetic field distributions with different shim shapes for entire gap (left) and close-up near to the septum (right) are shown.The blue open circle stands for the scaled measurement result, colored lines stands for the simulation results.The three numbers in the legend show the values of τ [µs], h [mm] and w top [mm].

Figure 5 .
Figure 5.The plotted curves indicate B ′ L as functions of x inj with the different shim shapes.The horizontal and vertical axes represent x inj and B ′ L. The three numbers in the legend show the values of τ [µs], h [mm] and w top [mm].The blue curve in the whole view (left) is for the present configuration.The right figure shows the detail of B ′ L around zero, where the color definition is same as that in left figure.

Figure 6 .
Figure 6.Decided shim shape (blue), the septum conductor (lightblue) and the gap of core (gray) are shown.