Angle-tunable wedge degrader for an energy-degrading RI beamline

Jongwon Hwang1,∗, Shin’ichiro Michimasa, Shinsuke Ota, Masanori Dozono, Nobuaki Imai, KoichiYoshida,YoshiyukiYanagisawa, Kensuke Kusaka, Masao Ohtake, Deuk Soon Ahn, Olga Beliuskina, Naoki Fukuda, Chihiro Iwamoto, Shoichiro Kawase, Keita Kawata, Noritaka Kitamura, Shoichiro Masuoka, Hideaki Otsu, Hiroyoshi Sakurai, Philipp Schrock, Toshiyuki Sumikama, Hiroshi Suzuki, Motonobu Takaki, Hiroyuki Takeda, Rieko Tsunoda, Kathrin Wimmer, Kentaro Yako, and Susumu Shimoura 1Center for Nuclear Study, the University of Tokyo, Wako, Saitama 351-0198, Japan 2RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, Japan 3Department ofAdvanced Energy Engineering Science, Kyushu University, Kasuga, Fukuoka 816-8560, Japan 4Department of Physics, the University of Tokyo, Bunkyo, Tokyo 113-0033, Japan ∗E-mail: jw.hwang@cns.s.u-tokyo.ac.jp


Introduction
Since a wedge-shaped foil was introduced into a spectrometer for the separation of the radioactive isotope (RI) beams at Grand Accélérateur National d'Ions Lourds (GANIL) [1], wedge degraders have emerged as ion-optical elements that can reduce the energy and its spread of ions for a variety of purposes. Optical systems for isotopic separation [2], which are one of the main applications of wedge degraders, have been used in many RI beam facilities including RIKEN [3,4], the National Superconducting Cyclotron Laboratory (NSCL) [5,6], and the GSI Helmholtz Centre for Heavy Ion Research (GSI) [7,8]. In such systems, an achromatic wedge degrader is placed at a dispersive focal plane to optimize the charge and mass resolutions without changing the energy dispersion of the beam.
Wedge degraders are also used in RI beamlines to slow down the beam for reducing its energy or to stop it. For these beamlines, a monoenergetic degrader, which is designed to eliminate the energy spread of the beam, is necessary to effectively bunch and slow down a beam. This slowing-down method was discussed for the first time by Geissel et al. [9] and they showed the advantage of a monoenergetic degrader in reducing range-straggling. Such a design of ion optics was experimentally

Design and manufacture
The angle-tunable system consists of two separate aluminum sheets each with a quadratic cross section, whose thicknesses are functions f 1 and f 2 of the horizontal positions x 1 and x 2 , respectively, from the center of the corresponding degrader as follows: In these equations, the first-order coefficients are identical and the second-order ones differ only in their signs. When the two degraders are placed away from the beamline center in opposite directions but at the same distance, e.g., the centers of the degrader 1 and 2 are at x = x 0 and x = −x 0 , respectively, the total thickness is where x is the horizontal position with respect to the beamline center. Hence, the entire system is identical to a typical wedge degrader with a wedge angle of tan −1 (4ax 0 + 2b) and a center thickness of (c 1 + c 2 ). An additional flat plate provides control of the total thickness for optimizing each experimental design. Based on the ion-optical design of the OEDO beamline, the numerical values of the parameters of the system were selected to have the proper thickness and angle.
The degrader system was fabricated by high-precision machining of the curved aluminum sheets. Figure 1 shows schematic views of the system and the mechanism of wedge angle changing. The  coefficients of f 1 and f 2 were set as a = 1/9000 mm −1 , b = 0.01, c 1 = 1 mm, and c 2 = 2 mm. In this way, the system has a fixed center thickness of 3 mm while the wedge angle varies from 0 to 40 mrad according to the variation of x 0 from −45 mm to 45 mm. The effective area is ±30 mm(H)× ± 50 mm(V), which is wide enough for the horizontal beam width, ±20 mm, as described with the dashed lines in Fig. 1. The average thickness deviations resulting from the machining precision for sheets 1 and 2 are 33 and 58 μm, respectively, which were measured using a high-precision thickness gauge; the angle uncertainty originating from these deviations is 2 mrad. By placing an additional aluminum flat-plate degrader right behind our system, we can increase the total central thickness to be more than 3 mm when necessary. In addition to the two degraders mentioned above, the system includes their guides mounted on two horizontal rails for parallel movement and two linear stepper motors, each one connected to an Al sheet and driving it along the corresponding rail, as shown in Fig. 2. The two motors operate independently by remote control. The wedge angle, therefore, can be optimized using real-time data. Furthermore, by moving both degraders in the same direction the thickness at the beam center can be precisely adjusted. Such fine tuning during the experiment is helpful to provide a beam with the expected energy.

Method
The applicable range and energy-compression performance of our system as a monoenergetic degrader were estimated by Monte Carlo-based simulations. The simulation codes include the incident beam profile, the shape of the degraders, and the energy loss in aluminum with energy and angular straggling; ATIMA [17,18], the range calculation code developed at GSI, was used for the energy loss calculation.

Range of application
Optimal wedge angle and center thickness depend on the following beam properties: the incident kinetic energy K i , the desired outgoing kinetic energy K f , and the beam nuclides. By studying these relationships, the applicable limits of the system can be estimated according to the characteristics of the RI beam. We assumed the ideal condition that the incident beam is parallel to the beam axis and has a perfect dispersion (δ ∝ x) of 5.36 mm/% in the energy deviation, which is the experimental value obtained at the dispersive focal plane in the OEDO beamline as described in Sect. 4.
The optimal center thicknesses and wedge angles were calculated for different K f and projectile atomic numbers (Z), as shown in Fig. 3. For the former, the beam nuclide was fixed as 79 Se, while, for the latter, a fixed K f of 35 MeV/u and the most abundant isotope for each Z were used. Two different K i , 170 and 345 MeV/u, were considered for both cases. The dependence on K f shows that the minimum thickness (3 mm) constrains K f while the optimal angle is determined only by K i . The dependence on Z further restricts the applicable range. Hence, based on these dependences and limits, our system can be employed for RI beams with Z = 15-84, K i = 170 MeV/u or Z = 53-84,

Energy-compression performance
The K f spread (σ out ) is an appropriate index to estimate the energy-compression performance of the proposed system as a monoenergetic degrader. We assumed that it is divided into the following factors: where σ in is originated by the position spread of the beam with the same incident energy, σ str is the straggling effect in the energy loss, and σ irr is the uncertainty from the irregular surfaces of the  degraders due to the machining precision. Unlike the others, σ in strongly depends on the ion-optical properties of the preceding beamline, in particular, the resolving power R (σ in ∝ 1/R) [13]. In this calculation, the partial and total energy spreads were estimated for different K f and Z. Since the realistic beam profile deduced from the experiment described in Sect. 4 was used, the ion-optical properties of the beamline shown in Eq. (5) and Table 1 have already been taken into account. In detail, the energy deviation and dispersion were ±1.3% (in σ ) and 5.36 mm/%, respectively, and K i was set at 170 MeV/u. The resolving power is calculated to be 231.4 using the beamline properties based on the following equation: where (x|i) are the transfer matrix elements of the beamline and x 0 is the beam width at the beamline entrance [13]. Note that δ p is a momentum deviation from the central ray because R is based on the momentum of a beam. The spreads were determined by fitting a Gaussian function in σ . To determine σ irr , we used the measured average thickness deviations, as mentioned in Sect. 2. Figure 4 shows the dependences of the partial spreads on (a) K f and (b) Z originating from the different factors of Eq. (3) together with the total one. While σ out was obtained directly from the simulation without considering the effects of straggling and irregularity, the other spreads were derived by subtracting it from the width of the simulation results including each effect. The center thicknesses and wedge angles were optimized, and Z and K f were alternatively fixed based on the same criterion cited in Sect. 3.2. In most cases, the effect of irregularity was dominant, in particular for high Z, for which the relative thickness deviation ( t/t) is large due to a thin optimal thickness but the straggling effect is modest; exceptionally, for low Z, a large optimal thickness, meaning small t/t, reduced the machining effect but increased the straggling one. Although we cannot reduce σ str because the straggling effect is intrinsic to the energy loss in the matter, the other partial spreads are adjustable. The blue dashed and dotted curves shown in Fig. 4 represent σ irr for average thickness deviations of 10 and 100 μm, respectively; machining one order more precise enormously reduces it, down to a level comparable with σ str for an average deviation of 10 μm (dashed). A high machining precision of at least < 100 μm is required for using this system as a monoenergetic degrader in slowing-down schemes. Note that σ in can also be reduced by increasing the resolving power R of the beamline.

Experimental conditions
The experiment was carried out at the OEDO beamline [16] of the RIBF, at the RIKEN Nishina Center for Accelerator-Based Science. Figure 5   (TOF), and magnetic rigidity in the BigRIPS separator [4], were transported to the degrader system located at FE9, the dispersive focus in the OEDO beamline, to obtain the 40-MeV/u beam. The degrader system had a central thickness of 6 mm by introducing an additional 3 mm thick flat-plate degrader.
Equation 5 and Table 1 show the first-order transfer matrix for the beamline from the achromatic focal plane F3 to FE9 and the parameters of the beam incident upon the degrader, respectively. The matrix elements were determined by the measured correlations, following similar procedures to those detailed in Ref. [19]. For simplicity we use the coordinates of (x, a, δ) regardless of the vertical direction and their units are mm, mrad, and % in energy deviation, respectively.
The acquired data were divided into two sets according to the beam conditions, in particular, the incident locus on the degrader. Figure 6 shows the distributions of the horizontal positions of the incident beams for two data sets with the corresponding energy deviations determined by the dispersion at FE9. The beam center for Set 2 was displaced from that for Set 1 by 6 mm, which corresponds to about 1% in energy deviation.

Experimental results
The data were acquired for several angle settings. The information about the position and angle of the beam at FE9 was obtained by using parallel-plate avalanche counters (PPACs), while K i and K f were derived from the TOF measured upstream and downstream, respectively.
The experimental and simulated wedge angle dependences of the K f distribution are shown in Fig. 7; the mean values and spreads (in σ ) were obtained by fitting a Gaussian function. Each of the two sets contains the results for four different angles. For the simulation, the experimental beam   profiles were used. The experimental results are consistent with the simulated ones. While the central thickness is insensitive to the wedge angle, the mean value depends on it due to the misalignment between the beam and the degrader center; the stronger dependence observed for Set 2 suggests larger displacement. In contrast, the dependences of the spread are similar in both sets. The spread is minimized at 20 mrad, where the beam dispersion is most reduced, and increases on both sides along with an enlarged dispersion. Detailed results for the energy distribution and dispersion of the outgoing beams are shown in Fig. 8.
For the quantitative analysis of the wedge angle optimization as a monoenergetic degrader, we reconstructed (δ|δ) total , which is the transfer matrix element for the optics of the entire beamline including the degrader system, and represents the compression capability of the relative energy spread δ of the optics [20]. For the dispersive imaging magnetic optics with a wedge degrader placed at the dispersive focal plane, we have up to the first order, where (x|δ) 1 is the dispersion of the imaging section prior to the degrader and (δ|i) d (i = x, δ) indicates the matrix elements for the degrader. The achromatic and monoenergetic conditions of a wedge degrader can be written as [2,21] (δ|δ) total = 1 (achromatic), 0 (monoenergetic), and the energy spread is compressed when |(δ|δ) total | < 1.   (7) in Ref. [21]). The reference incident and outgoing kinetic energies were assumed to be 171 and 42 MeV/u, respectively. The range information was derived by using ATIMA [17,18]. Figure 9 shows the experimental (δ|δ) total derived from the measured dispersions for the overall eight wedge angles from both data sets. The (δ|δ) total as a function of the wedge angle was calculated by using Eq. (6) for K i = 171 MeV/u, K f = 42 MeV/u, and (x|δ) 1 = 5.36 mm/%, from Table 1. The matrix elements for the degrader based on Eqs. (7) in Ref. [21] were estimated to be  in phase space of a beamline only depends on the ion-optical setting and not on the incident beam that lies within the acceptance of the beamline. The correlation loses linearity when the wedge angle moves away from the optimal value, e.g., the points at 0 and 40 mrad, because of the higher-order parts in a range or an energy loss that is not a linear function of the incident energy. Based on the experimental results, the beam energy is compressed within the wedge angle range of 20-25 mrad and the greatest energy compression can be achieved at 23 mrad in the present experimental conditions. A beam of a wider energy spread can also be obtained with an opposite dispersion ((δ|δ) total < 0). Hence, the degrader system can be employed not only for producing a monoenergetic beam but also for the momentum compression or dispersion adjusting of a beam.
The energy-compression performance was evaluated by comparing the K f deviations using the wedge angle optimized for a monoenergetic beam and a homogeneous degrader (0 mrad setting). As shown in Fig. 10, the energy spread in the homogeneous degrader case has a full width at half maximum (FWHM) of 12.6 MeV/u. By using the optimized monoenergetic degrader, the spread was reduced to 5.4 MeV/u, which is more than a factor of two, and this width is comparable with the value of 5.2 MeV/u estimated by the simulation.

Summary
We have developed an angle-tunable wedge degrader system to produce monoenergetic RI beams with an energy of a few tens of MeV/u by the slowing-down method. The wedge angle of the proposed system can be controlled by overlapping its two aluminum sheets, which have quadratic cross sections obtained by a high machining precision of about 50 μm in thickness deviations. The simulation well reproduced optimal center thicknesses and wedge angles based on the experimental beam profile. The behavior of the system was experimentally verified by using a 171-MeV/u 79 Se beam and a low-energy beam of 42 ± 2.7 MeV/u with suppressed spread was obtained. The experimental and simulation results were in agreement, and the energy-compression performance was achieved as successfully as estimated. The flexibility of the wedge angle of this system provides high versatility, useful to deal with a lot of different RI beam nuclides and energies for various purposes. This study leaves room for improvement in the energy-compression performance by applying higher-precision machining.