# DYNAMION—A Powerful Beam Dynamics Software Package for the Development of Ion Linear Accelerators and Decelerators

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Code Description

#### 2.1. RFQ Structure

_{ns}. The expression for the electrical field can be obtained by derivation of the formula for the electric potential:

_{n}, A

_{ns}—Fourier-Bessel coefficients, k = 2π/βλ, β—relative velocity, λ—wavelength of rf field and R

_{0}—average aperture of the RFQ.

#### 2.2. DTL Structure

_{z}(r,z), E

_{r}(r,z) in a gap and inside a drift tube are approximated by a series with 30 coefficients A

_{n}:

_{0}, I

_{1}—Bessel functions, l—half of cell length and k = π/l. The voltage distribution along the tank can be assumed from the design data or obtained from dedicated bead-pull measurements.

#### 2.3. Beam Transport Line

#### 2.4. Electromagnetic Fields from External Software

#### 2.5. Multiparticle Ensemble

#### 2.6. Particle Motion Equations

_{ext}, H

_{ext}—external electrical and magnetic fields, E

_{int}, H

_{int}—electrical and magnetic internal fields of the ions, t—time and c—speed of light.

#### 2.7. Space Charge Solvers

^{3}–10

^{4}particles to solve a wide range of typical problems for linacs [14,17,28,29,30]. In this case, the main and necessary condition for obtaining a reliable result is an adequate solving of the continuous equation of particle motion by discrete numerical methods, i.e., a sufficiently small timestep of integration, as well as the calculation of the Coulomb interaction between particles at each step.

_{j}—velocity and r

_{ij}—distance between two particles. Up to medium beam energies (relative velocity β < 0.5), the time of calculation can be remarkably decreased, neglecting the magnetic component. In order to prevent artificial collisions of particles after discrete steps of integration, a special routine is introduced [14]. Nevertheless, due to the relatively small size of the integration steps, the probability of such collisions is significantly low. Usually, more than 200 steps per characteristic length, corresponding to the phase length of 360°, are performed.

^{3}, SAS calculates space charge effects faster than the particle–particle method. In particular, SAS is about 24 times faster for 10

^{4}particles. This new solver allows for the calculations with a particle number of up to 10

^{6}. The analysis of the simulated beam dynamics results shows a sufficient coincidence between SAS and the existing DYNAMION solver, which is already well proven by numerous benchmarking tests and by a comparison with measured data [28]. Therefore, an advanced “two-step” scheme for the beam dynamics simulation with the DYNAMION code is proposed: initial investigations by means of the fast and reliable SAS method with the final precise and detailed proof with the more time-consuming space charge solver.

#### 2.8. Misalignments of the Linac Elements

## 3. Linear Accelerators at GSI

#### 3.1. UNILAC as a Heavy Ion High Current Linac

#### 3.1.1. Beam Matching to the RFQ

^{4+}ion beam [30]. Moreover, the corresponding DYNAMION beam dynamics simulations with the new QQ gradients showed after HSI-RFQ an about twofold increase of the transverse beam brilliance and a 60% higher longitudinal one. Mostly this result is explained by a significantly lower nonlinear deformation of the beam phase shape inside the QQ with newly found lower quadrupole gradients. Therefore, a better beam matching to the HSI-RFQ in turn leads to the correspondingly lower emittance growth along the RFQ channel.

#### 3.1.2. UNILAC Stripper Section

^{4+}pulse beam current of 15 mA rises up to 7 times during the stripping process. The U

^{28+}ions have to be purely separated from the neighboring charge states (and returned to the UNILAC axis) with the dispersion-free system of three dipoles (bending magnets) [34]. The code DYNAMION is well suited to completely simulate this specific problem, and thereby model an evolution of the particle ensemble, consisting of a mixture of many different charge states under space charge dominating condition, along the system of dipoles [14].

#### 3.1.3. Proton Acceleration at Heavy Ion UNILAC

_{3}-beam could be cracked and stripped in the UNILAC supersonic nitrogen gas jet into protons and carbon ions. During dedicated machine experiments in 2018, up to 3 mA of proton intensity (about 25% of the FAIR requirements) were measured behind the UNILAC [40,41].

^{28+}to protons, the amplifier, designed and normally operated at rf-power level at up to 2 MW, has to be adjusted to handle very low signal level of about 20 kW. However, a significant downshift of the synchronous (reference) phase in the Alvarez DTL gaps from the design value of about −25° potentially results in an accelerating regime, but with correspondingly higher rf-power. Obviously, this reduces strongly the stability problem during rf-power operation.

#### 3.2. Superconducting Continuous Wave Heavy Ion Linac

#### 3.2.1. Protons Acceleration at Heavy ion HELIAC

#### 3.2.2. Commissioning of the First HELIAC rf-Cavity

^{9+}beam energy behind the first HELIAC superconducting CH-cavity has been measured for the full range of rf-phase and compared with the corresponding DYNAMION beam dynamics simulations (Figure 3). Besides perfect coincidence of the experimental and simulated results, these data were used to calibrate the rf-power (rf-voltage) of the cavity.

#### 3.2.3. Normal Conducting HELIAC Injector Linac

## 4. RFQ-Decelerator for the GSI HITRAP Facility

^{92+}, are cooled and decelerated down to 4 MeV/u. At the HITRAP setup these ions should be further decelerated to 6 keV/u. The slow ions could be captured in a Penning trap, cooled further to cryogenic temperatures and transported to various experiments for atomic, nuclear and solid-state physics.

#### 4.1. Observed Problem

#### 4.2. RFQD Description “as Fabricated”

#### 4.3. Beam Dynamics Investigations

#### 4.4. RFQD Test at an External Facility

_{2}

^{+}beam with a variable energy, in particular inside the required range from 450 keV/u to 550 keV/u at a sufficient repetition rate to carry out the necessary investigations. Additionally, the energy-analyzing detector, usually installed behind the HITRAP IH-DTL at GSI, has been calibrated at MPI-K facility with several beams of precisely known energy. Furthermore, the data, taken during previous beam times at GSI facility, has been proved. In particular, the beam energy behind IH-DTL, operated in a nominal mode, has been confirmed as only a few keV/u below 500 keV/u, which is well consistent with the design parameters [15].

- RFQD steadily decelerates ions from 525 keV/u to the design energy of 6 keV/u;
- the acceptable input energy of ions varies from 520 to 530 keV/u;
- no deceleration was observed in the vicinity of the design energy of 500 keV/u.

#### 4.5. Further Design Issues

#### 4.6. New RFQD Design Applying the DESRFQ Code

^{92+}ions with the energy of 495 keV/u, corresponding to the optimum IH-DTL decelerator settings, down to the design beam energy of 6 ± 1 keV/u. Simultaneously, the longitudinal acceptance of the RFQD should be increased to provide for a higher yield of the decelerated ions. Due to the already fixed input- and output-energies and with the fixed overall length of the rods (to be mounted into the same tank), such requirements lead to a nontrivial designing process and require the use of an advanced and reliable beam dynamics code.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Kolomiets, A.; Pershin, V.; Vorobyov, I.; Yaramishev, S.; Klabunde, J. DYNAMION—The Code for Beam Dynamics Simulation in High Current Ion Linac. In Proceedings of the 6th European Particle Accelerator Conference (EPAC-98), Stockholm, Sweden, 22–26 June 1998; pp. 1201–1203. [Google Scholar]
- Kapchinsky, I.M. Theory of Linear Resonance Accelerators; Energoizdat: Moscow, Russia, 1982. [Google Scholar]
- Kolomiets, A.; Andreev, V.A.; Chuvilo, I.V.; Drozdovskiy, A.A.; Kozodaev, A.M.; Kuibida, R.P.; Lazarev, N.V.; Pershin, V.I. Some New ITEP Approach to Design of High Intensity Proton Linac for Transmutation. In Proceedings of the XVIII International Linear Accelerator Conference (Linac-96), Geneva, Switzerland, 26–30 August 1996. [Google Scholar] [CrossRef]
- Vengrov, R.M.; Vysotskii, S.A.; Kashinskii, D.A.; Kolomiets, A.A.; Minaev, S.A.; Pershin, V.I.; Tretyakova, T.E.; Sharkov, B.Y.; Yaramyshev, S.G. Linear Accelerator for Multiply-Charged Ions at 8 MeV/nucleon–Injector in the TWA Accelerator Complex. At. Energy
**2003**, 94, 39–44. [Google Scholar] [CrossRef] - Kolomiets, A.; Yaramishev, S. Comparative Study of Accelerating Structures Proposed for High Power Linac. In Proceedings of the 1997 Particle Accelerator Conference (PAC-97), Vancouver, BC, Canada, 12–16 May 1997; Volume 42, pp. 1387–1389. [Google Scholar] [CrossRef]
- Kolomiets, A.; Yaramishev, S. Study of Nonlinearities and Small Particle Losses in High Power Linac. In Proceedings of the XX Linac Conference LINAC-2000, Monterey, CA, USA, 21–25 August 2000; pp. 797–799. [Google Scholar] [CrossRef]
- Kashinsky, D.; Kolomiets, A.; Kulevoy, T.; Kuybida, R.; Kuzmichov, V.; Minaev, S.; Pershin, V.; Sharkov, B.; Vengrov, R.; Yaramishev, S.; et al. Commissioning of ITEP 27 MHz Heavy Ion RFQ. In Proceedings of the 7th European Particle Accelerator Conference (EPAC-2000), Vienna, Austria, 26–30 June 2000; pp. 854–856. [Google Scholar]
- Klabunde, J.; Barth, W.; Iaramychev, S.; Kolomiets, A. Beam Dynamics Simulations for the GSI High Current Injector with the New Versatile Computer Code DYNAMION. In Proceedings of the 2001 Particle Accelerator Conference (PAC-2001), Chicago, IL, USA, 18–22 June 2001; pp. 2899–3001. [Google Scholar]
- Batygin, Y.K. Particle-in-cell code BEAMPATH for beam dynamics simulations in linear accelerators and beamlines. Nucl. Instrum. Methods Phys. Res. A
**2005**, 539, 455–489. [Google Scholar] [CrossRef] - Ostroumov, P.N.; Aseev, V.N.; Mustapha, B. TRACK—A Code for Beam Dynamics Simulations in Accelerators and Transport Lines with 3D Electric and Magnetic Fields. Available online: https://www.phy.anl.gov/atlas/TRACK/Trackv39/Manuals/tv39_man_index.html (accessed on 2 April 2021).
- Uriot, D.; Pichoff, N. Status of Trace Win Code. In Proceedings of the 6th International Particle Accelerator Conference (IPAC’15), Richmond, VA, USA, 4 May 2015; pp. 92–94. [Google Scholar] [CrossRef]
- Polozov, S.; Barth, W.; Kulevoy, T.; Yaramyshev, S. Beam Dynamics Simulations and Code Comparison for a New CW rFQ Design. In Proceedings of the 57th ICFA Advanced Beam Dynamics Workshop on High-Intensity, High Brightness and High Power Hadron Beams (HB’16), Malmö, Sweden, 3–8 July 2016. [Google Scholar]
- Barth, W.; Bayer, W.; Dahl, L.; Groening, L.; Richter, S.; Yaramyshev, S. Upgrade program of the high current heavy ion UNILAC as an injector for FAIR. Nucl. Instrum. Methods Phys. Res. Sect. A
**2007**, 577, 211. [Google Scholar] [CrossRef] - Yaramyshev, S.; Barth, W.; Groening, L.; Kolomiets, A.; Tretyakova, T. Development of the versatile multiparticle code DYNAMION. Nucl. Instrum. Methods Phys. Res. Sect. A
**2006**, 558, 90. [Google Scholar] [CrossRef] - Barth, W.; Adonin, A.; Düllmann, C.E.; Heilmann, M.; Hollinger, R.; Jäger, E.; Kester, O.; Khuyagbaatar, J.; Krier, J.; Plechov, E.; et al. High brilliance uranium beams for the GSI FAIR. Phys. Rev. Accel. Beams
**2017**, 20, 050101. [Google Scholar] [CrossRef] [Green Version] - Busold, S.; Almomani, A.; Bagnoud, V.; Barth, W.; Bedacht, S.; Blažević, A.; Boine-Frankenheim, O.; Brabetz, C.; Burris-Mog, T.; Cowan, T.; et al. Shaping laser accelerated ions for future applications—The LIGHT collaboration. Nucl. Instr. Meth. Phys. Res. Sect. A
**2014**, 740, 94–98. [Google Scholar] [CrossRef] - Herfurth, F.; Andelkovic, Z.; Barth, W.; Chen, W.; Dahl, L.; Fedotova, S.; Gerhard, P.; Kaiser, M.; Kester, O.K.; Kluge, H.-J.; et al. The HITRAP facility for slow highly charged ions. Phys. Scr.
**2015**, T166, 014065. [Google Scholar] [CrossRef] - Vossberg, M.; Brodhage, R.; Kaiser, M.; Maimone, F.; Vinzenz, W.; Yaramyshev, S. Design Studies for the Proton-Linac RFQ for Fair. In Proceedings of the 6th International Particle Accelerator Conference (IPAC’15), Richmond, VA, USA, 7 May 2015. [Google Scholar] [CrossRef]
- Barth, W.; Aulenbacher, K.; Basten, M.; Dziuba, F.; Gettmann, V.; Miski-Oglu, M.; Podlech, H.; Yaramyshev, S. A superconducting CW-linac for heavy ion acceleration at GSIX. EPJ Web Conf.
**2017**, 138, 01026. [Google Scholar] [CrossRef] - Barth, W.; Aulenbacher, K.; Basten, M.; Busch, M.; Dziuba, F.; Gettmann, V.; Heilmann, M.; Kürzeder, T.; Miski-Oglu, M.; Podlech, H.; et al. First heavy ion beam tests with a superconducting multigap CH cavity. Phys. Rev. Accel. Beams
**2018**, 21, 020102. [Google Scholar] [CrossRef] [Green Version] - Ostroumov, P.; Kolomiets, A.; Kashinsky, D.; Minaev, S.; Pershin, V.; Yaramishev, S. Design of 57.5 MHz cw RFQ for medium energy heavy ion superconducting linac. Phys. Rev. ST Accel. Beams
**2002**, 5, 060101. [Google Scholar] [CrossRef] [Green Version] - Yaramyshev, S.; Barth, W.; Maier, M.; Orzhekhovskaya, A.; Schlitt, B.; Vormann, H.; Cee, D.R.; Peters, A. Upgrade of the Hit Injector Linac-Frontend. In Proceedings of the 11th International Conference on Heavy Ion Accelerator Technology (HIAT’09), Venice, Italy, 6–9 June 2009. [Google Scholar]
- Kravchuk, L.V.; Bylinsky, Y.V.; Esin, S.K.; Ostroumov, P.N.; Serov, V.L. Moscow Meson Factory Linac–Operation and Improvement. In Proceedings of the 19th International Linear Accelerator Conference (LINAC-98), Chicago, IL, USA, 23–28 August 1998; pp. 433–435. [Google Scholar]
- Hanke, K.; Lombardi, A. Design of a High-Intensity RFQ for a possible LHC Laser Ion Source. In Proceedings of the XXI International Linear Accelerator Conference, Kuongjui, Korea, 19–23 August 2002. [Google Scholar]
- Lombardi, A.; Bassato, G.; Battistella, A.; Bellato, M.; Bezzon, G.; Bertazzo, L.; Bisoffi, C.; Bissiato, E.; Canella, S.; Cavenago, M.; et al. The New Positive Ion Injector PIAVE at LNL. In Proceedings of the 1997 Particle Accelerator Conference (PAC-97), Vancouver, BC, Canada, 12–16 May 1997; Bulletin of the American Physical Society. Volume 42, pp. 139–141. [Google Scholar]
- Mardor, I.; Aviv, O.; Avrigeanu, M.; Berkovits, D.; Dahan, A.; Dickel, T.; Eliyahu, I.; Gai, M.; Gavish-Segev, I.; Halfon, S.; et al. The Soreq Applied Research Accelerator Facility (SARAF): Overview, research programs and future plans. Eur. Phys. J. A
**2018**, 54, 91. [Google Scholar] [CrossRef] - Grigorenko, L.; Sharkov, B.Y.; Fomichev, A.S.; Barabanov, A.L.; Barth, W.; Bezbakh, A.; Bogomolov, S.L.; Golovkov, M.S.; Gorshkov, A.V.; Dmitriev, S.N.; et al. Scientific program of DERICA–prospective accelerator and storage ring facility for radioactive ion beam research. UFNe
**2019**, 62, 675–690. [Google Scholar] [CrossRef] [Green Version] - Franchi, A.; Groening, L.; Gerigk, F.; Yaramyshev, S.; Bayer, W.; Sauer, A.; Yin, X.; Mutze, T.; Franchetti, G.; Hofmann, I.; et al. Linac Code Benchmarking for the UNILAC Experiment. In Proceedings of the 23rd International Conference (LINAC-06), Knoxville, TN, USA, 21 August 2006. [Google Scholar]
- Yaramyshev, S.; Vormann, H.; Adonin, A.; Barth, W.; Dahl, L.; Gerhard, P.; Groening, L.; Hollinger, R.; Maier, M.; Mickat, S.; et al. Virtual charge state separator as an advanced tool coupling measurements and simulations. Phys. Rev. ST Accel. Beams
**2015**, 18, 050103. [Google Scholar] [CrossRef] [Green Version] - Yaramyshev, S.; Barth, W.; Dahl, L.; Gerhard, P.; Groening, L.; Maier, M.; Mickat, S.; Orzhekhovskaya, A.; Schlitt, B.; Vormann, H. Advanced Beam Matching to a High Current RFQ. In Proceedings of the 27th International Linear Accelerator Conference (LINAC-14), Geneva, Switzerland, 31 August–5 September 2014. [Google Scholar]
- Orzhekhovskaya, A.; Barth, W.; Yaramyshev, S. An Effective Space Charge Solver for Dynamion Code. In Proceedings of the 46th ICFA Advanced Beam Dynamics Workshop on High-Intensity and High-Brightness Hadron Beams (HB-10), Morschach, Switzerland, 27 September–1 October 2010. [Google Scholar]
- Barth, W.; Hollinger, R.; Adonin, A.; Miski-Oglu, M.; Scheeler, U.; Vormann, H. LINAC developments for heavy ion operation at GSI and FAIR. J. Instrum.
**2020**, 15, T12012. [Google Scholar] [CrossRef] - Scharrer, P.; Jäger, E.; Barth, W.; Bevcic, M.; Düllmann, C.; Groening, L.; Horn, K.-P.; Khuyagbaatar, J.; Krier, J.; Yakushev, A. Electron stripping of Bi ions using a modified 1.4 MeV/u gas stripper with pulsed gas injection. J. Radioanal. Nucl. Chem.
**2015**, 305, 837. [Google Scholar] [CrossRef] - Barth, W.; Adonin, A.; Düllmann, C.E.; Heilmann, M.; Hollinger, R.; Jäger, E.; Khuyagbaatar, J.; Krier, J.; Scharrer, P.; Vormann, H.; et al. U28+-intensity record applying a H2-gasstripper cell. Phys. Rev. Accel. Beams
**2015**, 18, 040101. [Google Scholar] [CrossRef] [Green Version] - Yaramyshev, S.; Barth, W.; Clemente, G.; Dahl, L.; Groening, L.; Mickat, S.; Orzhekhovskaya, A.; Vormann, H.; Kolomiets, A.; Minaev, S.; et al. Advanced beam dynamics simulations with the DYNAMION code for the upgrade and optimization of the GSI-UNILAC. In Proceedings of the 46th ICFA Advanced Beam Dynamics Workshop on High-Intensity and High-Brightness Hadron Beams (HB2010), Morschach, Switzerland, 27 September–1 October 2010. [Google Scholar]
- Barth, W.; Scheeler, U.; Vormann, H.; Miski-Oglu, M.; Vossberg, M.; Yaramyshev, S. High Brilliance Beam Investigations at UNILAC. Phys. Rev. Acc. Beams
**2022**, 25, 040101. [Google Scholar] [CrossRef] - Gerhard, P.; Barth, W.; Dahl, L.; Orzhekhovskaya, A.; Tinschert, K.; Vinzenz, W.; Vormann, H.; Yaramyshev, S. Commissioning of a new CW radio frequency quadrupole at GSI. In Proceedings of the International Particle Accelerator Conference (IPAC’10), Kyoto, Japan, 23–28 May 2010; pp. 741–743. [Google Scholar]
- Ratzinger, U.; Tiede, R.; Podlech, H.; Clemente, G.; Hofmann, B.; Schempp, A.; Groening, L.; Barth, W.; Yaramishev, S.; Li, Z.; et al. The 70 MeV p-injector design for FAIR. In AIP Conference Proceedings; American Institute of Physics: College Park, MD, USA, 2005; Volume 773, pp. 249–253. [Google Scholar] [CrossRef]
- Kleffner, C.M.; Berezov, R.; Daehn, D.; Fils, J.; Forck, P.; Groening, L.; Kaiser, M.; Knie, K.; Muehle, C.; Puetz, S.; et al. Status of the FAIR pLinac. In Proceedings of the 8th International Particle Accelerator Conference (IPAC’17), TUPVA058, Copenhagen, Denmark, 14–19 May 2017. [Google Scholar]
- Barth, W.; Adonin, A.; Appel, S.; Gerhard, P.; Heilmann, M.; Heymach, F.; Hollinger, R.; Vinzenz, W.; Vormann, H.; Yaramyshev, S. Heavy ion linac as a high current proton beam injector. Phys. Rev. Accel. Beams
**2015**, 18, 050102. [Google Scholar] [CrossRef] [Green Version] - Adonin, A.; Barth, W.; Heymach, F.; Hollinger, R.; Vormann, H.; Yakushev, A. Production of high current proton beams using complex H-rich molecules at GSI. Rev. Sci. Instrum.
**2016**, 87, 02B709. [Google Scholar] [CrossRef] - Khuyagbaatar, J.; Ackermann, D.; Yakushev, A.; Khuyagbaatar, J. 48Ca + 249Bk Fusion Reaction Leading to Element Z=117: Long-Lived α-Decaying 270Db and Discovery of 266Lr. Phys. Rev. Lett.
**2014**, 112, 172501. [Google Scholar] [CrossRef] [Green Version] - Block, M.; Ackermann, D.; Blaum, K.; Droese, C.; Dworschak, M.; Eliseev, S.; Fleckenstein, T.; Haettner, E.; Herfurth, F.; Hofmann, S.; et al. Direct mass measurements above uranium bridge the gap to the island of stability. Nature
**2010**, 463, 785–788. [Google Scholar] [CrossRef] [PubMed] - Schwarz, M.; Yaramyshev, S.; Aulenbacher, K.; Barth, W.; Basten, M.; Busch, M.; Burandt, C.; Conrad, T.; Dziuba, F.; Gettmann, V.; et al. Reference beam dynamics layout for the SC CW heavy ion HELIAC at GSI. Nucl. Instr. Meth. Phys. Res. Sect. A
**2019**, 951, 163044. [Google Scholar] [CrossRef] - Lauber, S.; Aulenbacher, K.; Barth, W.; Basten, M.; Burandt, C.; Dziuba, F.; Forck, P.; Gettmann, V.; Heilmann, M.; Kürzeder, T.; et al. A dynamic collimation and alignment system for the helmholtz linear accelerator. Rev. Sci. Instrum.
**2021**, 92, 113306. [Google Scholar] [CrossRef] - Schwarz, M.; Conrad, T.; Podlech, H. Beam Dynamics Simulations for the Superconducting Heliac CW Linac at GSI. In Proceedings of the 13th International Particle Accelerator Conference (IPAC’22), Bangkok, Thailand, 12–17 June 2022. [Google Scholar]
- Lauber, S.; Aulenbacher, K.; Barth, W.; Dziuba, F.; List, J.; Burandt, C.; Gettmann, V.; Kürzeder, T.; Miski-Oglu, M.; Forck, P.; et al. Longitudinal phase space reconstruction for a heavy ion accelerator. Phys. Rev. Accel. Beams
**2020**, 23, 114201. [Google Scholar] [CrossRef] - Yaramyshev, S.; Aulenbacher, K.; Barth, W.; Basten, M.; Busch, M.; Gettmann, V.; Heilmann, M.; Kuerzeder, T.; Miski-Oglu, M.; Podlech, H.; et al. Advanced approach for beam matching along the multi-cavity SC CW linac at GSI. J. Phys. Conf. Ser.
**2018**, 1067, 052005. [Google Scholar] [CrossRef] - Lauber, S.; Yaramyshev, S.; Basten, M.; Aulenbacher, K.; Barth, W.; Burandt, C.; Droba, M.; Dziuba, F.; Forck, P.; Gettmann, V.; et al. An Alternating Phase Focusing injector for heavy ion acceleration. Nucl. Instr. Meth. Phys. Res. Sect. A
**2022**, 1040, 167099. [Google Scholar] [CrossRef] - Basten, M.; Aulenbacher, K.; Barth, W.; Burandt, C.; Dziuba, F.; Gettmann, V.; Kürzeder, T.; Lauber, S.; List, J.; Miski-Oglu, M.; et al. Continuous wave interdigital h-mode cavities for alternating phase focusing heavy ion acceleration. Rev. Sci. Instrum.
**2022**, 93, 063303. [Google Scholar] [CrossRef] [PubMed] - Adlam, J.H. A Method of Simultaneously Focusing and Accelerating a Beam of Protons; Atomic Energy Research Establishment: Berks, UK, 1953. [Google Scholar]
- Fainberg, I.B. Alternating phase focusing. In Proceedings of the Conference on High Energy Accelerators, CERN, Geneva, Switzerland, 11–23 June 1956; Available online: http://cds.cern.ch/record/1241565 (accessed on 22 November 2016).
- Iwata, Y.; Yamada, S.; Murakami, T.; Fujimoto, T.; Fujisawa, T.; Ogawa, H.; Miyahara, N.; Yamamoto, K.; Hojo, S.; Sakamoto, Y.; et al. Alternating-phase-focused IH-DTL for an injector of heavy-ion medical accelerators. Nucl. Instrum. Methods Phys. Res. A
**2006**, 569, 685–696. [Google Scholar] [CrossRef] - CST MicroWave Studio. 2021. Available online: https://www.cst.com (accessed on 6 July 2022).
- Bylinsky, Y.; Lombardi, A.M.; Pirkl, W. Rfqd—A Decelerating Radio Frequency Quadrupole for the Cern Antiproton Facility. In Proceedings of the XX International Linac Conference, Monterey, CA, USA, 21–25 August 2000. [Google Scholar]
- Hofmann, B.; Schempp, A.; Kester, O. A RFQ-Decelerator for hitrap. In Proceedings of the 2007 IEEE Particle Accelerator Conference (PAC07), Albuquerque, NM, USA, 25–29 June 2007. [Google Scholar]
- Yaramyshev, S.; Barth, W.; Clemente, G.; Dahl, L.; Gettmann, V.; Herfurth, F.; Kaiser, M.; Maier, M.; Neidherr, D.; Orzhekhovskaya, A.; et al. A New Design of the RFQ Channel for GSI HITRAP Facility. In Proceedings of the 26th International Linear Accelerator Conference (LINAC2012), Tel-Aviv, Israel, 9–14 September 2012. [Google Scholar]

**Figure 2.**Final relative velocity of each particle as a function of its input rf-phase. Green points represent design ions with charge state 28+; magenta points represent charge state 58+ (emulating about twofold rf-voltage); orange points represent charge state 88+ (emulating about threefold rf-voltage); grey points represent all other charge states.

**Figure 3.**Ar

^{9+}beam energy W

_{kin}behind the first HELIAC cavity, powered at Pt = 80 mW, as a function of the rf-phase Φ (dotted line). The corresponding simulation results are shown for 80% and 90% of the design voltage V (solid lines).

**Figure 5.**3D surface of the original RFQD rod (

**top**), reconstructed from the photometric measurements (

**bottom**).

**Figure 6.**A typical screenshot, illustrating cell-by-cell design process with the DESRFQ code; the longitudinal beam phase portrait (

**left**) and the stability diagram (

**right**) are shown.

**Figure 7.**Beam phase portraits depicting the acceptance of the original (

**top**) and new (

**bottom**) accelerating-focusing channels. The ellipses represent 90% of the beam intensity.

**Figure 8.**Measured output of the energy analyzer: the upper red spot represents heavy ions decelerated to about 6 keV/u; the lower red spot represents the non-decelerated fraction of the beam (500 keV/u); the histograms show beam intensity profiles in horizontal and vertical directions (the low energy peak is on the left).

Original | New | |
---|---|---|

Input energy (keV/u) | 525 | 495 |

Output energy (keV/u) | 6 ± 1^{(1)} | 6 ± 1 |

Voltage (kV) | 77.5 | 89.5 |

Average radius (cm) | 1.0–0.58–0.68 | 0.65 |

Rods rounding and width | variable | constant |

Max. modulation | 3.0 | 2.35 |

Min. aperture (cm) | 0.34 | 0.39 |

Frequency (MHz) | 108.408 | 108.408 |

Max. field strength (kV/cm) | 220 | 238 |

Kilpatrick factor (criterion 125 kV/cm) | 1.76 | 1.94 |

Transv. total acceptance, 90% (cm mrad) | 0.65 | 0.58 |

Long. total acceptance, 90% (keV/u⋅deg) | 177 | 357 |

Electrode length (cm) | 198.1 | 198.1 |

**No decelerated ions were observed behind the original RFQD, installed at HITRAP setup.**

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**MDPI and ACS Style**

Yaramyshev, S.; Barth, W.; Lauber, S.; Miski-Oglu, M.; Rubin, A.; Scheeler, U.; Vormann, H.; Vossberg, M.
DYNAMION—A Powerful Beam Dynamics Software Package for the Development of Ion Linear Accelerators and Decelerators. *Appl. Sci.* **2023**, *13*, 8422.
https://doi.org/10.3390/app13148422

**AMA Style**

Yaramyshev S, Barth W, Lauber S, Miski-Oglu M, Rubin A, Scheeler U, Vormann H, Vossberg M.
DYNAMION—A Powerful Beam Dynamics Software Package for the Development of Ion Linear Accelerators and Decelerators. *Applied Sciences*. 2023; 13(14):8422.
https://doi.org/10.3390/app13148422

**Chicago/Turabian Style**

Yaramyshev, Stepan, Winfried Barth, Simon Lauber, Maksym Miski-Oglu, Anna Rubin, Uwe Scheeler, Hartmut Vormann, and Markus Vossberg.
2023. "DYNAMION—A Powerful Beam Dynamics Software Package for the Development of Ion Linear Accelerators and Decelerators" *Applied Sciences* 13, no. 14: 8422.
https://doi.org/10.3390/app13148422