RF design and beam tracking in a compact racetrack CW microtron boosted with a tabletop Rhodotron

Nuclear Inst. and Methods in Physics Research, A 953 (2020) 163160

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RF design and beam tracking in a compact racetrack CW microtron boosted with a tabletop Rhodotron A.V. Smirnov a ,∗, R. Agustsson b , R. Berry b , S. Boucher b , Y. Chen b , S. Kutsaev b , F. O’Shea c a

ViewRay Inc., 815 E. Middlefield Rd., Mountain View, CA 94043, United States RadiaBeam Technologies LLC, 1735 Stewart St., Santa Monica, CA 90404, United States c Elettra-Sincrotrone Trieste, Basovizza, Trieste, 34149, Italy b

ARTICLE Keywords: Microtron Rhodotron Orbit Magnet Inspection Electron beam

INFO

ABSTRACT A conceptual design of a compact accelerator for active X-ray interrogation is presented. The light-weight design allows sub-millisecond beam energy and current modulation with beam power up to 10 kW and dynamic control across more than four orders of magnitude range. The main accelerator uses a C-band racetrack microtron equipped with two stripline kickers for fast beam outcoupling at three different energies. Injector employs an L-band Rhodotron operating at 8th sub-harmonic in a correspondingly optimized and designed cavity. RF design simulations coupled with thermal simulations as well beam dynamics simulations are presented.

1. Introduction Non-intrusive inspection (NII) systems are a critical element of the Global Nuclear Detection Architecture (GNDA) in the Defense of the Homeland [1], as they provide the ability to detect shielded radiation and nuclear threats. However, the accelerators that are currently available have been designed and optimized for medical and industrial applications, and thus their performance (especially duty factor) and physical attributes (i.e. size and weight) are not well matched for NII applications. Front edge NII system requires quickly switchable (within about 0.1 ms) energy between few MeV and 10 MeV with 2–3 MeV steps, CW beam current also switchable across wide range (0.1 μA– 1 mA), beam power variable in the range 0.2 W–10 kW, and extremely light weight (∼5000 lbs) [2]. Almost all accelerators currently used for NII systems are pulsed. However, cargo inspection techniques could benefit from continuous wave (CW) operation. For example, both the EZ-3D and NRF techniques used by Passport Systems [3] require CW sources. The current CW accelerator used by Passport Systems, classical Rhodotron cannot provide that fast switching and too large to be portable. Normal conducting CW linacs have been developed previously for research and industrial applications. However, they are also quite large and usually need sophisticated system with special probes for frequency stabilization to withstand RF heating. Superconducting RF linacs have been proposed for this application. With a cryogen-free refrigeration system (in batch mode operation mode), they can be made relatively compact. However, for NII portable applications and on-field deployment they are not robust enough.

A racetrack microtron (RTM) is usually considered as the first candidate for a CW low-energy compact electron accelerator. Other cyclic accelerators, such as betatron or synchrotron, are not suitable because of their discontinued (in terms of NII scale) time structure. There is only a few CW RTMs have been built. Three-stage, cascaded Mainz microtron in Germany provides a continuous wave, high intensity, polarized electron beam with energy up to 1.6 GeV. Its first stage ‘‘MAMI A’’ is probably the oldest CW RTM. It has been operating since 1979 at 14 MeV energy, 1.66 m distance between magnets and 9 kW RF power fed into 11 cells structure [4]. Initially it was operating at 2.1 MeV injection energy, later the energy was increased to 3.5 MeV. CW RTM of Moscow State University [5] delivered (24–175) MeV energy and 100 μA beam current at 6 MeV injection energy. Construction of three CW RTMs was started long time ago, but, to our knowledge, not commissioned to date. The first one is NBS-LANL CW Microtron [6] for 185 MeV energy, 500 μA current. It is designed for 5 MeV injection energy. Another CW RTM under construction is IASA 246 MeV in Greece [7] with 10 MeV injector linac. And another one is ∼38 MeV IFUSP RTM in Brazil [8], 45 kW total RF power comprising main RTM 13 kW (17 cells structure, 2.57 m distance between magnets) with injection energy 5 MeV (that can be reduced down to 2.5 MeV [8]). The IFUSP RTM design uses five-turn RTM booster (7 kW, 14 cells) and 1.8 MeV injector (2 × 9 kW RF power). The RTMs above (with exception of NBS LANL) are designed to operate 2.45 GHz frequency (𝜆 = 12.24 cm) using standard, on-axis coupled, biperiodic, standing wave structure. The NBS LANL Microtron is designed for 2.38 GHz frequency and uses side coupled, 40 cells long structure.

∗ Corresponding author. E-mail address: [email protected] (A.V. Smirnov).

https://doi.org/10.1016/j.nima.2019.163160 Received 25 June 2019; Received in revised form 10 November 2019; Accepted 20 November 2019 Available online 27 November 2019 0168-9002/© 2019 Elsevier B.V. All rights reserved.

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Nuclear Inst. and Methods in Physics Research, A 953 (2020) 163160

Table 1 Performance metrics of CW portable accelerator. Output beam energy System weight Average beam current Average beam power Output beam emittance Output beam ∅ rms DC electron gun voltage Injection energy into RTM Rhodotron frequency RTM RF frequency Rhodotron # of passes Number of orbits RTM magnet field Reverse RTM dipole field Distance between magnets RTM magnets weight RTM quadrupole strength Rhodotron bending field Rhodotron RF power RTM RF power

4–10 MeV, switchable within 0.1 ms with ∼3 MeV step. 6000 kG (with local shielding) 0.1 μA–1 mA, variable 0.2 W–10 kW <10 mm mrad ∼2 mm 25 kV 2.4 MeV minimum 898.75 MHz 7.19 GHz 16 23 0.158 T 0.0746 T 343 mm <165 kG <3.3 T/m 0.25 T (last bend) 16 kW maximum 13 kW maximum

availability. The scaling law of the energy (W) attainable in Rhodotron employing standard coaxial cavity vs. RF power dissipated √ in the cavity walls (PRhW ) is given by√J. Pottier [10]: W ≈ 2.3𝑛𝜆1∕4 PRhW for R2 = 0.27𝜆 and W ≈ 3.1𝑛𝜆1∕4 PRhW for R2 = 0.5𝜆, where R2 is the larger radius of a coaxial Rhodotron cavity, n is the number of beam passes across the cavity, W in mega-electronvolts, PRhW in megawatts, and the wavelength 𝜆 is in meters.

Reduction of footprint and weight of the accelerating systems is a key to portable RTM-based accelerator system for active interrogation with X-rays using up to 10 MeV output beam energy. Important breakthrough has been made relatively recently in RTM technology due to using of permanent magnets (PMs). Iron-dominated PM-based dipoles forming two 180◦ magnetic mirrors for a compact, light-weight RTM are developed [9]. The novel PM end magnets provide also proper focusing addressed by separating the functions of beam mirroring and beam focusing by introducing an undulator-like end field matching. This matching is enabled by additional poles for each magnet with fields having opposite sign. The problem of field accuracy and homogeneity is solved in the PM design by using of multiple tuners and also pre-sorting of the PM blocks. In this novel coil-free system the main poles and the poles with reverse field provide beam bending and focusing to compensate vertical fringe field defocusing. A similar magnet was assumed in the conceptual CW RTM design considered here (at lower field and larger dimensions). In this paper we evaluate a compact and portable CW accelerator featuring GHz-range Rhodotron as an injector, and a C-band racetrack microtron for its 2.4–10 MeV relativistic part. Unlike conventional Sband systems, the 7.19 GHz frequency chosen enables to use a relatively compact 25 kW VA-876 CW klystron from CPI. Main parameters of the accelerator considered in this paper are given in Table 1. The conceptual design foresees extraction of the beam from different orbits using pulsed kickers at sub-millisecond resolution, and also addresses the frequency drift caused by heating by implementing a regenerative feedback loop using matched bandwidths of the klystron and the resonant structure. To limit the paper length we did not include here magnetic design of the key magnets, which presents a subject of a separate study.

2.1. Design of microwave Rhodotron cavity In practice the maximum frequency for microwave Rhodotron can be limited by the following factors: (i) maximum power deposition density in the cavity (related to cooling capability); (ii) minimum apertures for the beams as well as wall thickness of the copper wall separating the apertures (for a standard coaxial Rhodotron cavity); (iii) power available from a CW RF source (an amplifier). Direct scaling of a conventional Rhodotron cavity to a microwave cavity also is not possible because usually the latter one requires larger√aperture-towavelength ratio at higher Q-factor that scaled directly (∼ 𝜆). Besides, the re-designed cavity requires in-depth customization of cooling and RF coupling designs. Taking into account the factors above we have chosen here 8th harmonic, which corresponds to 898.75 MHz frequency. For instance, for 2.5 MeV output beam energy, 16 passes, and 33.4 cm internal cavity diameter we need only 4.7 kW RF power PRhW (at negligible beam loading) according to the Pottier’s formula for that frequency. In Fig. 1 we illustrate design of a microwave Rhodotron cavity that we adapted to the 898.75 MHz frequency, 12 beam passes and included cooling channels. The design is developed for beam aperture ∅5 mm, cavity radii R1,2 = (2.4, 16.7) cm, beam bending radius in end magnets Rbend = 3.6 cm, bending field in last magnet Bmax = 0.24 T. One can see very moderate maximum temperature (43 ◦ C) at also moderate (3 kW/◦ Km2 ) heat transfer coefficient enabling laminar flow at moderate flow rates. As it can be seen from Fig. 2, the analytical and numerical results for shunt impedance are very close validating the Pottier’s formula used in characterizing performance of the Rhodotron type injector. Slightly enhanced efficiency of the optimized cavity of Fig. 1 (see Fig. 2) enables 6.6 MΩ shunt impedance. We have further reduced the Rhodotron cavity dimensions and increased its efficiency as shown in Figs. 3 and 4. Note for the optimized cavity shapes of Fig. 4 the shunt impedance achieved is ∼7 MΩ at substantial beam aperture (∅5.5 mm). Important to note the shunt impedance approaches the maximum possible impedance 7.9 MΩ of classical toroidal cavity at that frequency. Note one of the factors limiting the Rhodotron booster frequency and/or number of passes can be mitigated or even eliminated (at

2. Microwave Rhodotron as an injector for CW RTM Note all the RTMs considered above use linacs for injection. However, for CW RTM operating at relatively moderate beam energies (∼10 MeV) linac injector would dominate footprint of the total accelerator system even without corresponding RF power source and its supply. That means a compact portable system should use exceptionally efficient accelerating structure and topology not only for the main accelerator, but also for the injector. Rhodotron is among the most effective CW accelerating schemes known for low energy electrons including non-relativistic ones. As a trade-off between efficiency determined by RF losses and dimensions (weight) determined by frequency we found, that 7th–10th sub-harmonic of the RTM frequency chosen (7.19 GHz) would be about optimal in terms of balance between compactness, losses, heat management as well as RF source power cost and 2

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Fig. 1. LEFT: Conventional Rhodotron cavity design adapted to ∼33 cm wavelength with 12 beam passes and cooling channels. RIGHT: Temperature map for 3 kW/m2 C◦ heat transfer coefficient in the water channels and 6.3 kW power deposition in the walls. The ambient temperature is 22 ◦ C. Peak power density deposition is ∼110 W/cm2 .

Fig. 2. LEFT: Analytical and numerical calculations of shunt impedance [MΩ] vs. beam energy [MeV] for classical coaxial shape and that given in Fig. 1. RIGHT: Radial component of RF electric field simulated along the radius.

Fig. 3. Geometry of modified cavity (for reduced dimensions and 12 passes) and temperature map (right) at 3 kW/m2 C◦ heat transfer coefficient in the water channels and 6.3 kW power deposition in the walls. The ambient temperature is 22 ◦ C.

Fig. 4. Advanced Rhodotron cavity shapes enabling ∼7 MΩ shunt impedance, 12 passes, and 5.9 kW power deposition for ∅5.5 mm (a) and ∅5 mm (b) apertures. (c): Analytical and numerical calculations of shunt impedance [MΩ] vs. beam energy [MeV] for the advanced shape (a, b) (red solid) and classical coaxial (blue dotted).

3

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Fig. 5. Modification of Fig. 4b toroidal cavity without vertical walls of the apertures between beamlets in the central conductor (Rsh = 5 MΩ).

Fig. 6. Rhodtron cavity model for 12 rectangular loop couplers (left; 30◦ , 1/12th sector), 14 circular loop couplers (middle) employed as power combiner and S11-curve (right) simulated in transient (time) domain with simultaneous excitation. The coaxial ports correspond to N-type connector. Enlarged bandwidth in part is due to numerical asymmetry of the hexahedral mesh employed in transient simulation.

scheme reported earlier (i.e., with 12–14 ways combining) or single stage combining with 21-ways using the 750 W MRF13750H-915 MHz pallets [13].

slightly reduced shunt impedance). In Fig. 5 we simulated the Rhodotron cavity having no walls between the adjacent beamlets forming an evanescent (not capacitive) gap. The gap did not change noticeably the operating mode as well as did not introduce any new modes that could potentially affect operation of the accelerating mode. The microwave version of a Rhodtron cavity may employ a single or multiple couplers of magnetic or electrical type. The RF design results shown in Fig. 6 are produced for the variant in which the cavity is employed as a power combiner. The coaxial ports correspond to N-type connector, whereas the overcoupling seen in Fig. 6c addresses beam loading. An external L-band power combiner can be employed (see, e.g., Ref. [11]) as a part of a solid state power amplifier (see, e.g., Ref. [12]). A single high power coupler has been designed and optimized for critical and slightly overcoupled feeder using a 1-5/8′′ EIA connector. Note scaling of conventional loop-type coupler together with a Rhodotron cavity results in too small loop radius and correspondingly poor geometry control and too high RF current densities. Besides, that type of coupler may interfere with cooling channels of the downsized cavity. Therefore we have applied a pin-type (i.e., electrical) high power coupler presented in Fig. 7 and characterized in Fig. 8. One can see the coupler design is capable of suitable power feeding (including beam loading) with low return and insertion losses. Advantages over standard Rhodotron coupling is remoteness from beam trajectories as well as ‘‘hot’’ areas seen in Fig. 7b (and so no overlapping with cooling channels). Assuming conservatively we need a ∼4 mA current for injection (see GPT simulations below), it implies 9.6 kW power of the beam injected from the Rhodotron at 2.4 MeV. That means we need ∼16 kW total maximum RF power at ∼900 MHz instead of ∼9 kW planned earlier. That power can be addressed with two stage combining for the

2.2. Beam dynamics in L-band Rhodotron First we simulated beam capture from the electron gun and only two passes through the cavity in 3D fields imported from RF designs given in Fig. 4a at 5.9 kW RF power dissipated in the cavity walls. We used GPT code [14] for the most critical (in terms of beam capture and bunching), initial part of the beam trajectory (see Fig. 9). We used sector magnet model built-in the GPT code with 8.35◦ edge focusing angle with respect to the beam entrance and 25 keV injection energy at a few μm transverse emittance from the electron gun. After adjustment of radial position of the sector magnet we obtained ∼310 keV kinetic energy for about 7% of injected beam, 6.6% energy spread, normalized horizontal and vertical emittances 75 μm and 1.8 μm respectively. Programming a 12-magnet Rhodotron topology in GPT code turned out is a not trivial task. Therefore next we have used the output beam parameters found from the ‘‘single-leaf’’ GPT simulations as the input for PARMELA to simulate the remaining part of beam dynamics in the small Rhodotron with 12-magnets. Thus, we use here a combination of GPT and Parmela codes despite slight asymmetry of the fields seen by the beamlets in the Rhodotron, which becomes less significant as the beam becomes relativistic. We believe this approach is more accurate than the full (i.e., from the very injection) PARMELA simulation of a Rhodotron [15]. To provide proper phasing we adjusted again radial position of each end magnet to the center found preliminary analytically. The distances found allow sufficient azimuthal distance (engineering gap) between the adjacent magnets. Focusing was adjusted by changing the angle between magnet edges and the i/o 4

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Nuclear Inst. and Methods in Physics Research, A 953 (2020) 163160

Fig. 7. Pin-type (i.e., electrical) coupler with 1-5/8′′ EIA connector in scaled and optimized Rhodotron cavity (a) with RF field pattern of magnetic (b) and electric (c) components.

Fig. 8. Smith chart (left) and S11 curve (right) simulated in frequency domain for the Rhodotron mono-coupled cavity of Fig. 7.

Fig. 9. YZ (top) and XZ (bottom) views for non-collimated beam trajectories in the 1st leaf of the microwave Rhodotron.

3. C-band accelerating structure of the CW racetrack microtron (RTM)

trajectories by few degrees. The main results of the Rhodtron beam dynamics are shown in Figs. 10–12. One can see that output bunch is rather short (∼4 mm) at extremely low energy spread (6–17 keV) enabling low loss injection into the RTM resulting in ∼20◦ phase spread that matches very well microtron longitudinal acceptance simulated below. Such a microwave (frequency up-scaled) Rhodotron can also be a nearly perfect candidate for a benchtop accelerator for charged particle detector calibration. Such a high precision, ultra-low current (pA-sub-pA), CW beam sources can be designed with ultra-low dark current. High quality energetic filtering out of the dark current can be accomplished by placing correspondingly calibrated collimators (beam scrapers) in every bend.

Choosing of C-band RF structure configuration for CW RTM should satisfy simultaneously a number of challenging requirements: (i) capability to withstand high heat deposition density and related stresses caused by CW RF power losses; (ii) high shunt impedance combined with wide bandwidth addressing field stability at frequency detuning caused by thermal deformations; (iii) manufacturability and ease of tunability at affordable cost. Thus, the structure should provide also maximum efficiency at minimum number of orbits (at limited klystron power). Number of orbits defines magnet dimension, accuracy of magnet fields (which scales as ∼1∕N2 ), and also beam stability. 5

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Fig. 10. Kinetic energy (left) and energy spread (right) simulated with Parmela at 4◦ input phase spread along the trajectory in the 898.75 MHz Rhodotron.

Fig. 11. Kinetic energy spread at 18◦ input phase spread (left) and transverse normalized emittances along trajectory in the Rhodotron.

Fig. 12. RMS beam dimensions along the Rhodotron trajectory: Transverse (left) and longitudinal (right).

Besides, RTM phase stability and capture strongly depend on phase slippage effect, which limits not only accelerating section length and free space between the section and the magnet, but also variation of the transit time factor of the accelerating section, which, in turn, depends on the injection energy and structure type. The preliminary analysis has been performed on the base of combination of analytical 1D modeling of RTM beam dynamics (including beam loading) and electromagnetic modeling of the representative accelerating cells. For the C-band CW RTM we have analyzed and compared the following standing wave (SW) bi-periodic structures: on-axis coupled, side-coupled, and cross-bar [16]. We also analyzed several traveling wave structures employed as a resonant ring. In addition to the structures above designed for traveling wave (TW) mode, including 3𝜋/4 mode side-coupled and on-axis coupled structures [17], we also considered structures with parallel feed (or distributed coupling [18, 19]). We compared minimum power deposition per cell, minimum

number of orbits to achieve 10 MeV energy, and maximum electronic efficiency of the RTM assuming injection energy about 2 MeV and structure length less than 5𝜆. In terms of this combined criterion the clam shell (split) variant of side coupled SW structure having 124 MΩ/m shunt impedance and 2.5% coupling between cells. The latter variant demonstrated nearly best performance being, however, very competitive with the standard shape of on-axis coupled bi-periodic structure (scaled to 7.19 GHz). Most promising among TW resonator structures considered we found are on-axis coupled structure (most effective 4𝜋/3 structure [17]) and ‘‘split’’ variant of a side coupled TW structure that features low group velocity (fraction of percentage) and high shunt impedance (about 140 MΩ/m). The latter TW variant slightly exceeds again the on-axis coupled TW variant due to smaller number of orbits and lower power deposition per unit length. Note the best TW variant did not exceed performance of the best SW variant in spite of presence in SW pattern of a partial backward wave that does not participate in acceleration. That happened because the equivalent 6

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Nuclear Inst. and Methods in Physics Research, A 953 (2020) 163160

Fig. 13. 3D maps resulted from structural static analysis performed with ANSYS of a single cell of a split-type side coupled structure. Thermal map (a), equivalent stress (b), total deformation (c), and equivalent elastic strain (d). Ambient temperature is 22 ◦ C, RF heat deposition per cell is 700 W, and heat transfer coefficient to the water cooling channels is 6000 W/cm2 K.

Fig. 14. Maximum stress (a), maximum deformation (b) and maximum elastic strain (c) simulated with ANSYS using structural static analysis as a function of heat transfer coefficient.

Q-factor of corresponding TW resonator is noticeably less than that for SW structure (resulted from elevated attenuation at low group velocity as the downside of high shunt impedance). Besides, the SW pattern enables some RF focusing unlike TW variant. Therefore, a standing wave variant of a split-type side coupled structure has been chosen for further analysis due to potentially better combined performance and simplified fabrication (vs. on-axis coupled).

for on-axis coupled cell revealed more than factor of eight higher maximum stress (occurring on the coupling slot edge) for comparable conditions. That dramatic difference may also be explained by the internal gap in the split structure. We have also performed transient thermal-mechanical ANSYS simulations of the plastic deformations. That was done with multi-linear isotropic hardening model [20] in Fig. 15. As expected, the plastic model indicates lower maximum stress (∼47 MPa) within the cavity, though higher maximum stress shifted from the RF surface towards one of the cooling channels. Since the maximum stress is outside the RF structure, it does not present direct threat to the structure performance dominated by skin depth. The ANSYS analysis above suggests reduction of the heat load below the 700 W heat deposition per cell to ensure reliable operation well below the maximum yield stress, which is 69 MPa [21]. Besides, presence of the coupler operating at substantial overcoupling reduces access to cooling channels as well as introduces non-uniformity of the field distribution. That may require further reduction of the average power deposited per cell.

3.1. Thermal-mechanical analysis for single, side-coupled, split cell Next important step of the analysis is single cell temperature and stress analysis performed for both static and dynamic modes using CST Suite and ANSYS codes. We imported the CST model of a single cell into HFSS code followed by import of the RF losses into the ANSYS. As it can be seen from Fig. 13a, the peak temperature simulated by ANSYS code is 84 ◦ C at 700 W heat load per cell (at ambient temperature 22 ◦ C and heat transfer coefficient to the water cooling channels 6000 W/cm2 K). Temperature map was exported to ANSYS Static Structural model constrained by one boundary at each plane. The material used is OFHC copper. Interesting, there is an explicit minimum of stress and elastic strain in the vicinity of ∼5000–7000 W/cm2 K heat transfer coefficient (see Fig. 14). That can be explained by presence of ∼1.3 mm gap in the middle plane of the structure, which is evanescent for RF fields and serves as local open structural boundary allowing to relief additional stress within some range of deformations. Besides, simulation performed

3.2. Transverse kick in a multi-cell, side-coupled split structure with power coupler In general, bi-periodic structures having couplers for power input and/or asymmetric side cells (or coupling slots) may impose transverse kick. In a multi-orbit system such as a racetrack microtron that kick 7

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Fig. 15. ANSYS steady-state results for total stress, deformation, and plastic strain obtained from transient structural plastic model using multi-linear isotropic hardening model [20].

Fig. 16. Transverse kick voltage (red) and kick (blue) for horizontal (left) and vertical (right) planes as a function of pass number for 11-cell split structure with interdigital side cells and symmetrized coupler. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

may build-up and result in beam loss. Therefore, in this task we calculated the transverse kick for different structure variants for realistic power levels (determined by power loss in the walls or energy gain per orbit 𝛥U). We considered RF section configurations with different number of cells (even/odd), presence of coupler (symmetric/asymmetric), positioning of side coupling cells (interdigital/one-side) and coupling strength (coefficient 𝛽𝑐 = 1.3–3.2).

3.3. RF performance of a 9-cell CW RTM structure with coupler From analytical model based on 1D dynamics as well as calculations of minimum transverse kick, considered above, we have selected two suitable variants of the RTM accelerating section: 9 and 11 cells. Nine cells variant of the RTM section is illustrated in Fig. 17. The structure assembly we engineered has ∼4.6 cm thickness and 10 cm width. That means that the width of the accelerating section assembly to be oriented perpendicularly to the plane of RTM orbits (see Fig. 25). That enables sufficient space for both the structure width and bended WR137 waveguide feeder to be placed between the assembly surface and the 1st orbit (∼13 cm apart from the structure axis). Moreover this configuration also addresses multi-orbit beam outcoupling (see Figs. 25, 27). The structure is designed to handle up to P = 12.7 kW input RF power, N = 25 passes, averaged shunt impedance rsh = 105 MΩ/m, coupling coefficient 𝛽c = 2.47, minimum injection energy Einj = 1.9 MeV, and Pcell = 514 W averaged heat deposition per cell. For 11-cell variant we have found P = 11.5 kW, N = 22, rsh = 102 MΩ/m, 𝛽c = 2.49, Einj = 2.5 MeV, and Pcell = 420 W respectively. Under these conditions the structures provide energy gain 318 keV and 328 keV respectively at ∼60% electronic efficiency. Note the RF assemblies we designed use symmetrized coupling to substantially reduce (by more than an order) transverse kick (unlike some of conventional RTM designs). S11 Parameter and field profile along the 9-cell structure shown in Fig. 17 are simulated in frequency domain in Fig. 18. The structure provides close to the optimal coupling

The comparison showed significant advantage of symmetrized coupler and interdigital positioning of the side cells. Balancing of side cells in terms of horizontal kick requires even number of them. That, in turn, implies an odd number of main cells. And finally, larger coupling coefficient results in slightly higher kick when other conditions are equal. Effective kick voltage and deflection angles are determined by the RF Lorentz force calculated on the base of integration of 3D electric and magnetic fields extracted from full-wave simulations. For 11-cell split structure with interdigital side cells and symmetrized coupler (𝛽𝑐 = 3.2, coupling area 1.05 cm2 ) we plotted the kick voltage and deflection in Fig. 16 as a function of orbit number. As expected, the deflection force decreases as electron energy decreases [22]. The transverse kicks per orbit calculated are noticeably lower than the 0.08 mrad kick calculated for the 12 MeV C-band RTM [23] (dominated by coupling slots). In practice accumulation of this kick can be suppressed by combination of magnetic correctors, quads and edge focusing of the end magnets. 8

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Nuclear Inst. and Methods in Physics Research, A 953 (2020) 163160

Fig. 17. Two halves of nine-cell, clam-shell, side-coupled RTM accelerating section: 3D model (a) and RF structure mockup fabricated before final cleaning (b).

Fig. 18. S11 parameter [dB] curve vs. frequency [MHz] (left) and Ez accelerating field profile [V/m] simulated along the 9-cell structure (right) shown in Fig. 17 (at 1 W excitation power).

Fig. 19. Surface temperature distribution of RTM accelerating section having 20 channels ∅6.35 mm operating with laminar flow Re = 2365.

(2.7 instead of 2.5) that might be beneficial for the case of slightly

Note both structure variants provide very substantial frequency

reduced Q-factor caused by fabrication imperfections. Reduced field

separation between operational and adjacent modes (25 and 20 MHz

in the coupling cell seen in Fig. 18 is also useful to reduce local

respectively). That separation is large enough to avoid any substantial

overheating caused by power flow conformal to reduced cooling in the

distortion of the field uniformity along the structure due to strong inter-

vicinity of the coupling waveguide.

cell coupling as well as to avoid confusion of the AFC system with 9

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Table 2 Comparison of thermal performance of different cooling configurations (Reynolds numbers Re; flow rates G gallons per minute; flow velocities V, m/s; pressure drop 𝛥P, Pa; averaged copper, water flow, channel surface and peak copper temperatures Tav, 𝛥T, Tcylsurf., and TCu, max◦ C) for RTM section simulated with COMSOL at 30 ◦ C inlet temperature and 5 kW dissipation power. Cooling channel length is 104 mm. N channels

∅, mm

Re

G, Gal/min

V, m/s

Tav, ◦ C, Cu

Tav, ◦ C H2 O

𝛥P, Pa

𝛥T, ◦ C

Tcylsurf. ◦ C

TCu, max ◦ C

36

5.16

2000 6000

4.1 12.3

0.33 1.03

47.34 45.868

32.706 30.767

38 406

4.6 1.54

46.717 38.404

69.91 70.1

2353 6000

2.33 8.44

0.331 0.841

52.045 49.66

33.31 30.95

43 215

5.69 2.25

51.54 40.572

75.9 71

2356 4700 9400

1.66 3.3 6.6

0.33 0.66 1.32

55.45 57.5 46.5

35.51 32.4 31.2

165.2 471 1450

11.4 5.7 2.87

55 45.5 39.16

78.7 78.67 67

20 6.35 10 (with bends)

Fig. 20. The focusing scheme used for RTM simulations.

adjacent mode, because the estimated frequency detuning caused by the heating is much lower (∼1 MHz at 514 W heat deposition per cell).

with focusing elements is given in Fig. 20. The distance between the end magnets is limited by phase slippage at low energies and related to stable acceleration and capture in longitudinal phase space. The distance was preliminary chosen from a 1D semi-analytical model with shunt impedance of the accelerating section imported as a function of beam energy from the 3D model above. Beam defocusing by the end magnet fringe field is compensated by reverse field magnets [25] as shown schematically in Fig. 20. The injected beam parameters are taken from the PARMELA results for the scaled Rhododtron output (normalized beam emittance ∼4𝜋 mm mrad and 20◦ phase length at the 7.19 GHz RTM frequency). 3D maps of the fields have been imported directly into the GPT model from 3D simulations for the custom magnets (permanent magnet quadrupoles, PMQs) as well as for the RF fields of the 9-cells RTM accelerating section (in frequency domain with coupler). For the main RTM magnet we used model of an equivalent rectangular magnet (built into the GPT code) with the same 15 mm gap. Beam Lorentz factor 𝛾 and 23-orbit trajectories are depicted in Fig. 21 for horizontal plane. Beam transverse emittances and rms beam sizes are given in Fig. 22 as functions of time. Longitudinal bunch size and capture temporal profiles are presented in Fig. 23. Note the capture coefficient calculated here requires some correction for the injected current and hence RF power required for the microwave Rhodotron booster operating at 8th subharmonic. One can see the RTM provides stable operation with significant capture (∼30%). The GPT modeling leads us to the following conclusions:

3.4. Thermal performance of a 9-cell CW RTM structure with coupler We have completed thermal simulations with fluid heat transfer in the RF section to include different modes (turbulent and laminar) as well as number of parallel channels. The purpose is to validate best cooling configuration consistent with the optimal heat transfer coefficient (∼6 kW/m2 K◦ ) found previously at more practical 30 ◦ C inlet temperature. In Fig. 19 we illustrate the simulations results for temperature distribution in configuration with 20 parallel channels ∅6.35 mm for laminar and turbulent flows determined by Reynolds number (Re). The results of the coupled heat transfer and fluid simulations are summarized in Table 2, in which we outlined the preferable variants (bolded). For these variants the maximum and minimum temperatures — correlate very well with that analyzed in-depth earlier in terms of minimum stress and maximum temperature gain. The heat transfer coefficient estimated from the simulations exceeds the corresponding standard values (e.g., calculated via Nusselt and Prandtl numbers) for the given parameters by up to ∼50%. The reasons for this discrepancy are the following: (i) the estimate does not use classical definition but ‘‘local’’ value (i.e. through averaged volumetric and surface temperatures of the channels); and (ii) the length of inlet/outlet ports is likely insufficient. To achieve the optimal heat transfer (∼6000 W/cm2 K◦ ) one can introduce surface roughness (using corresponding bits and/or sand blasting) that significantly enhances heat transfer coefficient due to corresponding increase of friction factor at relative roughness ≥0.01 [24]. Thus, the configuration chosen (the third line in Table 2) with 20 parallel channels has been validated for moderately turbulent flows.

(i) The analytical model and predictions made for the conceptual and base design have been validated including main dimensions and sensitivity to the RF and magnetic field accuracy. (ii) The RF design of the real accelerating section is validated including influence of RF fields asymmetry and related transverse kicks. Both horizontal and vertical orientations of the split-type RF section are found suitable in terms of the beam dynamics. (iii) The fringe fields of PMQ as well as end-magnets (dominated by the 15 mm gap employed both in the magnet designed and GPT model) are admissible and not critical for the performance.

4. Beam dynamics in C-band CW racetrack microtron We have performed core simulations and analysis of beam dynamics in RTM part of the system using GPT code. The RTM layout scheme 10

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Fig. 21. Gamma-factor and trajectories in horizontal plane of RTM simulated with GPT code. Main magnet field is 0.158 T, reverse dipole field is 0.0746 T.

Fig. 22. Normalized beam emittances (top) and rms beam sizes (bottom) in horizontal (left) and vertical (right) planes simulated with GPT code as a function of time [s] spent when orbiting within the RTM.

Fig. 23. Beam FWHM dimension projected onto the OZ-axis (left) and fraction of injected beam accelerated (percentages, right) as a function of time [s].

(iv) The PMQ strengths (<0.084 T field integral for ∼1 inch long PMQ) found here are very moderate and easy to implement with low grade adjustable, iron-free permanent magnets.

or mechanically insertable dipoles for beam outcoupling. Therefore, for the maximum beam energy we designed a static, in-vacuum, small magnet, whereas for lower energies we apply very thin quasi-electrostatic pulsed kicker. The goal for the extracting magnet was to design a specialized, finetunable magnet that provides a ∼2.2◦ bend in the horizontal plane at sufficient acceptance of beam delivery transportation. The bend angle is simply a ratio of RTM trajectory straight length to the orbit separation.

4.1. Fast beam outcoupling at three levels of energies Relatively small orbit separation (𝛽𝜆∕𝜋 ≈ 13 mm) as well as ultrashort switching time (∼0.1 ms) makes rules out usage of electromagnets 11

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Fig. 24. LEFT: Magnetic design of adjustable magnet for last orbit beam outcoupling. The clear gap is 5 mm, interaction length along the beam 2.5’’, pole width 1’’ (along OX, perpendicular to the beam in horizontal plane). The magnet is ended by thin (∼0.75 mm) iron screen backed with thin (∼0.75 mm) permanent magnet. Note the axis OY is oriented along the beam, whereas the mid-plane of the magnet coincides with the RTM mid-plane (see Fig. 24, right).

In the magnet design we minimized the magnet fringe field to enable the bend without disturbing the adjacent trajectory. The septum-like performance was accomplished by using a thin (∼0.75 mm) iron screen backed with thin (∼0.75 mm) permanent magnet at the magnet end as shown in Fig. 24, left. Upstream surface of the narrow side (∼1.5 mm thick) of the return yoke (placed between the last two orbits) to be shielded from the beam halo by Molybdenum block. To get the ∼2.2◦ bend at 3.5’’ interaction length the field required is ∼0.2 T, which is attained with PMs only. The bend can be adjusted in a wide range (up to ±35%) and also fine-tuned with the same trimming coil. We found from simulations fringe field transition region is ∼8 mm vs. the ∼13 mm beam separation enabling the bend at beam horizontal dimension as large as 4 mm for the two last orbits. The relatively small beam bend in the outcoupling magnet allows elegant beam out-coupling as simulated in Fig. 25, left. Next, we designed remaining two fast quasi-electrostatic kickers to make it consistent with the full RTM beam tracking simulated with GPT code. That resulted in a short kicker shown in Fig. 26. In spite of the short 62 mm length of the deflector the maximum voltage required is still within the ratings (i.e. <25 kV) of the BEHLKE switches. The 3D fields we simulated for the kicker of Fig. 26 have been imported again into the GPT code. The resulting trajectories related to 15th and 5th orbits are shown in Fig. 27. The strips of the electrostatic kicker can be made rather thin to avoid interception with the beam. The static charge produced by the low energy part of the beam halo is suppressed by shunting resistor. The strips do not require cooling as the high energy (relativistic) electrons of the halo have insignificant energy loss. Thus, the outcoupling parameters (maximum voltages or magnetic fields) are within hardware capabilities (electronics or PM-iron magnets), whereas the kicker lengths provide the beam outcoupling at three energies: 9.71 MeV (when both kickers are off), 7.15 MeV, or 4.1 MeV (when one of the HV switches is on). As it can be seen from GPT simulations of Figs. 25 and 27, the out-coupled beam has ∼50– 58 mrad tilt, which is well within the acceptance of typical beam delivery transport channel and does not interfere with cooling jacket of the RF section. The beam outcoupling allows the RF coupler with the bend to be inside the RTM orbits.

signal from a coupling loop installed in one of the accelerating cells of the structure is directed to the amplifier input through a feedback circuit, which includes adjustable attenuator and phase shifter, as well as a ferrite isolator. Conditions for self-oscillation at operating mode frequency are met through selection of the feedback loop attenuation and phase shift values. Self-oscillation frequency automatically follows the resonance frequency of the structure, which changes due to thermal processes. If RF discharges occur in the structure or in the waveguide, the conditions for self-oscillations are broken, and they discontinue. Thus, when operating in CW self-oscillation mode, the amount of RF power reflected from the cavity is determined by its matching with the waveguide; if matching is optimal, klystron can operate with high Q loads without circulator between klystron and the section. Elimination of the circulator makes the accelerator cheaper and smaller in size, and allows connecting flanges of the klystron and of the section directly, using vacuum window of the klystron for vacuum insulation of the structure. The self-oscillation method can also be used for a multisection accelerator. In particular, such an arrangement was used for a two-section accelerator with beam energy of 1.2 MeV and average beam power 60 kW [26]. If accelerated beam current is high, fields of separate sections can be automatically synchronized with the field of the first section, where accelerated bunches are formed, with phases required for acceleration, provided there is certain correlation between beam current and frequency deviation [27]. For the two-cavity system we applied here a slightly modified singleklystron feedback system [24] enhanced with AFC-driven tuner of the Rhodotron booster section as shown in Fig. 28. Thus the large resonant frequency deviations of the RTM section are addressed by positive selfoscillating regeneration serving as floating-frequency master oscillator (MO) for the AFC contour of the injector (Rhodotron) section. The low-level and high power RF network conceptual schematic with AFC shown in Fig. 28 maintains automatically resonances in both RTM and Rhodotron cavities: ‘‘master’’ contour at 7.19 GHz provides the resonance with self-oscillation using positive feedback loop and provides reference signal for the ‘‘slave’’ booster cavity operating at 8th sub-harmonic. The Rhodotron cavity is routinely maintained at resonance due to another contour of the AFC based on a standard phase control at the resonance and cavity frequency tuners. High speed of the tuner is enabled by fast (fraction of millisecond) step motors similar to that applied in [28]. Frequency and phase of the Rhodotron cavity can be tuned even faster using advanced ferroelectric tuners controlled directly by electric field [29]. Note positive feedback employed in the master contour can be unstable. Therefore, below we validate the AFC scheme by demonstrating presence stable regime of that feedback operation through simulations as well as determine pre-amp gain and group delay required. We apply the noise-triggered process [30] to the RTM using a typical gain vs. power plot at ∼50 dB maximum gain of the loop (may require preamplifier) and duration limited by ∼60 μs. The results are given in Figs. 29 and 30 for low and full beam loading respectively.

5. Automatic frequency control In the conceptual design of RTM CW RF system we consider selfoscillation method for automatic frequency control (AFC). This method uses an amplifier (klystron) as a RF source and feedback loop and proved to be effective for accelerators with high beam power and with high accelerating gradients, with significant changes of the operating frequency usually caused by RF heating and beam loading. With additional stabilization circuits, the method can be used also to get high-precision electron beams. In this method, a low-power RF 12

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Fig. 25. TOP: RTM trajectories in horizontal plane simulated with GPT code with beam outcoupling at the last 23rd orbit (∼9.71 MeV). MIDDLE and BOTTOM: Outcoupled beam with respect to the PMQ and RF section assembly.

Fig. 26. Modified quasi-electrostatic kicker inserted into beam dynamics simulations. Kicker length is 62 mm, maximum voltage is 24.7 kV. Effective thickness of each strip with bend ∼ <1 mm, strip thickness <0.55 mm.

One can see from Fig. 29, that for low beam loading at 47.7−41.7 = 6

and moderate reflection. Note the group delay also includes that one

dB total gain in the feedback loop and <0.1 μs group delay, the self-

occurring within the klystron itself (determined by its phase curve vs.

oscillation is stable and occurs at maximum power (close to saturation)

frequency). 13

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Fig. 27. Beam trajectories in horizontal plane with outcoupling at lower energies: 7.15 MeV (left) & 4.1 MeV (right).

Fig. 28. RF low-level and high power network schematics with automatic control of resonant frequency in both contours: ‘‘master’’ contour at 7.19 GHz (right) and ‘‘slave’’ contour at 8th sub-harmonic (left). Analog to digital converters (ADC) and related auxiliary electronic blocks are not shown.

Usually the wider the flat top of the gain curve the shorter the delay. According to our discussions [31] the wideband VA-876P klystron has significantly less than 100 ns group delay. In case of SSPA the group delay can be made negligible (<20 ns) by means of using a broadband combiner such as the ‘‘bottle’’ type design we tested [11]. The higher the gain and delay of the feedback loop (in excess of certain limits seen in Fig. 29) the less stable is the self-oscillating generation at nominal frequency resulting in ringing and parasitic modulations of the RF envelop. We found from simulations that the feedback loop provides stable and reliable self-oscillation at the self-adjusted frequency of the accelerating RTM cavity when the loop attenuation is about 5–9 dB less than the klystron gain (including its pre-amp for VA-876P) whereas the group delay to be less than 150 ns. If the loop attenuation is too high, then the oscillations become more stable and the envelope is smoothed until the RF power is quenched: the delay increases and when the attenuation approaches the klystron gain the generation may not start at all. In that case external RF source with corresponding RF switch are required to trigger the self-oscillation.

We found that ∼47.7 dB gain is required between the klystron output and LLRF feedback that provides the klystron input. That means that for the VA-876P with only 40 dB gain we need about 8 dB from a pre-amplifier, whereas no pre-amp is required for VA-876J due to its 50 dB gain. Finally, in Fig. 30 we simulated performance of the feedback with full beam loading (23 orbits 0.95 mA each gives 22 mA beam loading). Numerous runs indicate that the presence of beam loading makes the feedback system even more stable due to significant excess over the noise at startup and enlarged cavity bandwidth due to overcoupling. We found that the minimal gain required for the power amplifier is 43 dB in the presence of the beam (instead of 47.7 dB at negligible current and critical coupling). As a result, the self-oscillation starts with the beam (if it is injected on the flat top of the HV). Another benefit of the overcoupled cavity of the self-oscillation start-up considered here (and also with beam loading) is significantly reduced peak reflections. All other features (i.e. influence of phase, attenuation, and delay) of the self-oscillation stability with beam loading are similar to that without beam loading. Thus even if the system is over-stabilized (too high attenuation or dephasing in the loop) and does not self-trigger 14

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Fig. 29. TOP: Waveforms for accelerating voltage in the RTM section with feedback loop (red on top) and HV envelope normalized to max energy gain (dashed blue). BOTTOM: Forward (dashed blue) and reflected (red solid) power waveforms at 𝛥trise = 3 μs, tHVpulse = 40 μs. LEFT: Kprobe = −41.7 dB, 𝛥tdelay = 100 ns. MIDDLE: Kprobe = −33.7 dB, 𝛥tdelay = 100 ns. RIGHT: Kprobe = −37.7 dB, 𝛥tdelay = 500 ns. Kprobe is the section transmission from the main coupler to the pick-up probe.

Fig. 30. LEFT: Stable waveforms for self-oscillated accelerating voltage in the RTM section with feedback loop (red, top) and HV envelope normalized to max energy gain without beam loading (dashed blue). RIGHT: Forward (dashed blue) and reflected (red solid) power waveforms at full beam loading (22 mA). Kprobe = −40 dB, 𝛥tdelay = 250 ns.

RF oscillations, an auxiliary RF source can be connected for a short time (∼μs) to the amplifier input to trigger self-oscillation. This was already implemented in an industrial 1 MeV CW linac [32]. While it is not required in our setup, it can be used for additional control of the stability at expense of external triggering instead of noise (when HV is applied). We also found that the admissible group delays determined above (250 ns and 150 ns with and without beam loading) are consistent with the broad bandwidth of both VA-883 and VA-876P klystrons. Adjustment of the loop gain and fine tuning of the phase at LLRF are very beneficial for the RTM to ensure accurate resonant microtron condition between the magnetic field and energy gain per orbit. Maintaining of the resonance can be easily made electronically at LLRF level.

(i) A compact microwave Rhodotron operating at subharmonic employed as an injector and capable to serve simultaneously as a combiner for a solid state amplifier. (ii) Ultrafast, multi-energy beam outcoupling using high voltage pulsed kickers fed by fast solid state switches. (iii) Self-oscillating RF system enabling automatic frequency tuning due to usage of positive feedback without circulator for main accelerating cavity and AFC operating in slavery mode for the injector fed by solid state amplifier. (iv) A split-type side-coupled biperiodic accelerating structure combining high shunt impedance and sustainability to high heat deposition density. The design uses innovative iron-free compact PM magnets we conceived for the outcoupled beam line, variable PM-electromagnets for

6. Summary

Rhodotron as well as PM-iron main magnet with array of tuners for the main magnet. Some of the key RF components (e.g., the accelerating

Feasibility of a portable, C-band, CW electron accelerator for active X-ray interrogation is demonstrated with basic simulations of RF accelerating cavities as well as beam dynamics. The conceptual design presented here is based on several key novel solutions not implemented so far in CW racetrack microtrons:

structure seen in Fig. 17b) and magnetic elements (e.g., PMQ parts) have been fabricated, but not tested yet. The key RF and beam optics components need further engineering development and validation but is currently limited by funding. 15

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Declaration of competing interest

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The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. CRediT authorship contribution statement A.V. Smirnov: Methodology, Supervision, Conceptualization, Software, Writing - review & editing. R. Agustsson: Supervision. R. Berry: Data curation, Validation, Resources. S. Boucher: Supervision. Y. Chen: Visualization, Data curation, Methodology. S. Kutsaev: Formal analysis, Investigation, Data curation. F. O’Shea: Formal analysis, Investigation. Acknowledgments The authors deeply appreciate Dr. S.B. van der Geer and Dr. M.J. de Loos for their help and ingenuous efforts to customize GPT code for beam tracking and post-processing. One of the authors (AVS) is very grateful to Dr. Y. A. Kubyshin, Dr. Hans-Joachim Kreidel, Robert Heine, and Dr. Klaus Floettmann for interest and discussions related to RTM modeling. This work has been supported by the U.S. Department of Homeland Security (DHS), Domestic Nuclear Detection Office (DNDO), under a competitively awarded contract No. 70RDND18C00000002. This support does not constitute an express or implied endorsement on the part of the Government. References [1] K.G. Hart, The role of global nuclear detection architecture in the defense of the homeland, in: Presentation at the Baltimore-Washington Health Physics Society Annual Meeting, May 11, 2012. [2] Exploratory research in preventing nuclear and radiological terrorism. DHS broad agency announcement No. HSHQDN-16-R-0002, 2016. [3] S. Korbly, Non-intrusive inspection using CW photon beams, in: Talk at DNDO Accelerator Requirements Workshop, Fermi National Accelerator Laboratory, 2015, pp. 5–7. [4] H. Herminghaus, A. Feder, K.H. Kaiser, W. Manz, H.V.D. Schmitt, Nucl. Instrum. Methods 138 (1976) 1–12. [5] V.I. Shvedunov, A.S. Alimov, A.S. Chepurnov, O.V. Chubarov, I.V. Gribov, B.S. Ishkhanov, I.V. Surma, A.V. Tiunov, Proc. of 1993 Particle Accelerator Conf, 1993, pp. 2059–2061.

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