Interpretation of the horizontal beam response near the third integer resonance

Interpretation of the horizontal beam response near the third integer resonance E. C. Cort´es García ,1,2,* P. Niedermayer,2R. Singh ,2R. Taylor,3,4 E. Feldmeier ,5M. Hun,5E. Benedetto,6and T. Haberer5 1Deutsches Elektronen-Synchrotron DESY, Notkestraße 85, 22607 Hamburg, Germany 2GSI Helmholtz Centre for Heavy Ion Research, Planckstraße 1, 64291 Darmstadt, Germany 3CERN, Espl. des Particules 1, 1211 Meyrin, Switzerland 4Imperial College London, Exhibition Rd., South Kensington, London SW7 2BX, United Kingdom 5Heidelberg Ion-Beam Therapy Center, Im Neuenheimer Feld, 450, 69120 Heidelberg, Germany 6SEEIIST Association, Rue de Battoirs 7, 1205 Geneva, Switzerland (Received 4 April 2024; accepted 6 December 2024; published 30 December 2024) The beam response to an external periodic excitation delivers relevant information about the optics, tune distribution, and stability of a circulating beam in a storage ring. In this contribution, the horizontal beam response to the excitation (transfer function) under conditions typical for slow extraction is presented for a coasting beam. The resulting spectrum exhibits a splitting behavior. The single particle dynamics is discussed and an interpretation based on the simulation results is presented. DOI: 10.1103/PhysRevAccelBeams.27.124001 I. INTRODUCTION Slow extraction from synchrotrons and storage rings is a well-established method to deliver a variety of beams to experiments and patients. In current versions of the method, the slow extraction is prepared by moving the horizontal working point near a third integer resonance while simultaneously reducing the dynamic aperture (DA) by driving the resonance with dedicated sextupoles. The effective horizontal dynamics under such conditions can be described by the Kobayashi Hamiltonian [1]. The beam is then extracted either by slowly reducing the DA further until it reaches zero or by slowly increasing the beam emittance beyond the (constant) DA. Different variations of this machine setup are in use [2–4]. In this contribution, the so-called radio frequency knock out (rf-KO) extraction is considered, which involves blowing-up the emittance through an external transverse excitation while keeping the DA constant [5]. In rf-KO extraction, the emittance blowup is achieved by applying an external rf electromagnetic field excitation. The amplitude of the betatron oscillations is thereby controlled and eventually increased beyond the DA, where particles become unstable and are pushed into the extraction channel. Several studies have been performed in order to engineer an optimal frequency spectrum for the rf excitation signal [6–11]; these studies have heavily relied on measured particle counts for trial-and-error optimization of the rf-signal spectrum. In this contribution, we take a step back and focus exclusively on the amplitude and phase response of the stored beam to an external sinusoidal excitation by measuring the beam transfer function (BTF) under extraction conditions. Here the sinusoidal stimulus replaces the usual rf excitation signal, but the rest of the machine setup remains the same as for rf-KO extraction. Systematic studies of BTFs close to driven resonances are not to be found in the literature, to the best knowledge of the authors. The information gained by the BTF measurements has a significant application in the context of engineering an optimal waveform for the slow extraction [12,13], where fixed tone excitations have also been proposed [13]. Our contribution is organized as follows: In Sec. II, the particle dynamics under slow extraction conditions is reviewed together with the theoretical background of the extraction, for single and multiparticle dynamics. In Secs. III and IV, measurements and simulations of the beam response of a coasting beam are presented, respectively. We conclude with a summary and a discussion of the results in Sec. V. II. THEORY A. Single particle dynamics The status-quo of the study of nonlinear dynamics relies on the Lie-algebraic formulation of transfer maps induced by Hamiltonians of different particle accelerator elements [14,15]. The workhorse for the engineering and optimization *Contact author: [email protected] Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. PHYSICAL REVIEW ACCELERATORS AND BEAMS 27, 124001 (2024) 2469-9888=24=27(12)=124001(16) 124001-1 Published by the American Physical Society of slow extraction via the third integer resonance is the Kobayashi Hamiltonian [1]. It is an approximate description of the particle dynamics when the tune Qxis close to a thirdorder resonance Qres driven by sextupole magnets. Comprehensive derivations and corollaries of this dynamics can be found in [16]. The Kobayashi Hamiltonian Hfor on-momentum particles reads H¼h0þh1;(1) with h0¼−6πðQx−QresÞJx¼:−δJx;(2) h1¼− S ffiffiffi 2 pJ3=2 xsin 3ϕx;(3) where action-angle variables ðJx;ϕxÞare used, which relate to the horizontal space coordinate xas x¼ffiffiffiffiffiffiffiffiffiffiffiffi 2βxJx psin ϕx [15–17]. Along our contribution, the fractional part of the linear (zero-amplitude) tune will be represented by qx;0, such that Qx¼nþqx;0with n∈ℕ0:Q res is the closest third integer resonance (3Qres ∈N). Moreover, Scan be interpreted as the amplitude of the resonance driving term excited by an effective single virtual sextupole. The virtual sextupole strength Sis given by S¼X N n Sne−i3μn ;(4) with Sn¼1 2Zβ3=2 xk2ds; (5) where Nis the total number of sextupoles in the lattice, μn¼Rsn 0β−1 xdsthe betatron phase advance to the nth sextupole, and k2¼1 Bρ ∂2By ∂x2the normalized sextupole field component. The integration of Eq. (5) is to be carried out along the sextupole’s length. The phase advance of the virtual sextupole with respect to the arbitrary lattice start is given by tan3μS¼PN nSnsin3μn PN nSncos 3μn :(6) Figure 1depicts the equipotential lines of Hin normalized phase space in the range where stable motion is possible. Note that the phase space is shown at the position of the virtual sextupole μSwith S>0. This phase space structure is found in any machine, where the Kobayashi Hamiltonian governs the single particle dynamics. 1. Amplitude-phase-detuning Since Hdescribes the effective motion of a particle every three consecutive turns, by using the Lie-algebraic treatment [15,18], the amplitude-phase-dependent tune averaged over three consecutive turns is given as qx¼qx;0þqx;1¼1 6π ∂H ∂Jxþqres;(7) where qx;0is the unperturbed tune given by the linear theory, and qx;1¼qx;1ðJx;ϕxÞ¼ 3S ffiffiffiffiffi 25 pπJ1=2 xsin3ϕx(8) is the amplitude-phase-dependent tune shift due to the nonlinear dynamics averaged over three turns. It is common FIG. 1. Equipotential lines in the stable region of normalized phase space induced by the Kobayashi Hamiltonian. The value of the Hamiltonian at the separatrix edge reads Hsep ¼½4πðQx− QresÞ3=S2[16]. FIG. 2. Single (top) and three turn phase advance (bottom) as function of angle ϕx, colored by the value of the Kobayashi Hamiltonian H. E. C. CORT´ ES GARCÍA et al. PHYS. REV. ACCEL. BEAMS 27, 124001 (2024) 124001-2 to apply averaging techniques over ϕxto Eq. (8) and intuitively, the contribution from the phase-amplitude detuning for a nearly constant action Jxshould vanish. However, near a resonance the phase space is deformed as the Kobayashi Hamiltonian clearly shows (Fig. 1), and the action Jxis not an invariant of motion [19]. Rather, it is a function of the angle Jx¼JxðϕxÞand as such modulated even without the effect of an external excitation. This is an important aspect: the phase advance of a single particle is not constant in time but varies as a function of the angle as the particle revolves in phase space. The resulting phaseadvance modulation is illustrated in Fig. 2. As discussed later, this aspect determines how the particle responds to a sinusoidal excitation at a certain frequency over time. Moreover, the single particle detuning can be better understood by investigating its long term average behavior [20]. The time average of a system’s quantity gis given by hgi¼Rt2 t1gdt= Rt2 t1dt. Thus, the average three-turn phase advance hμ3iyields [20] hμ3i 6πδ ¼1 2πZ2π 0 dϕx 1−ffiffiffiffiffi 2~ J pcos 3ϕx−1 ;(9) where the normalized action ~ J ~ J¼J 2Jmax ¼J 4ð4πδÞ2=S2¼(~ H=6if cos 3ϕx¼0 6þR1=3ðffiffi3 pi−1ÞþR−1=3ðffiffi3 piþ1Þð8~ Hcos2ð3ϕxÞ−9Þ 16cos2ð3ϕxÞotherwise ; R¼27 −36 ~ Hcos2ð3ϕxÞþ4~ H2cos3ð3ϕxÞ2cosð3ϕxÞ−ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2cosð6ϕxÞþ2−4=~ H q; (10) and the normalized effective energy ~ H¼H Hsep ¼H ð4πδÞ3=S2(11) were introduced. ~ J¼~ JðH;ϕxÞis expressed only asa function of the energy and the phase [20]. The right-hand side of Eq. (9) is unity for ~ J→0and thus the average three-turn phase advance is consistent with the linear theory. When the particle approaches the limit of H→Hsep then, in the corners of the stable phase space area where ϕx→0;π=3or 2π=3,~ J acquires the value of 1/2 while cos3ϕxbecomes unity. Hence, the integral in Eq. (9) diverges and the average phase advancefreezes,i.e.,theparticlehitstheresonancehμ3i→0. The behavior of the average detuning over many turns is illustrated in Fig 3. It is clearly visible that the average threeturn phase advance detunes toward the nearest third-order resonance. The effect is more drastic when the energy of the particle is closer to the edge of the separatrix. 2. External excitation A couple of archetypes of externally driven nonlinear oscillator systems have been extensively studied, e.g., the Duffing equation, the forced Van-Der Pol oscillator [21],or Chirikov’s standard map [22], to mention a few of the most prominent. In the context of particle accelerators, considerable progress on the theory of externally driven nonlinear systems has been done [23,24], where the introduction of external sinusoidal excitations provide useful schemes to either trap in nonlinear resonance islands fractions of the beam or to diffuse the beam in an rf abort case. Following the Lie-algebraic treatment, the effective time-dependent one-turn Hamiltonian of our system reads [15] HðnÞ¼−μxJx− h1 3þAxJ1=2 xsinϕxsinðΩnÞ −μx S ffiffiffi 2 pJ3=2 x1 3cos3ϕxþcotμx 2sinϕxþcosϕx; (12) where μx¼2πqx;0is the one-turn phase advance, Axis the normalized strength of the external periodic excitation, Ω represents its frequency, and nis the turn number. In this case, the system does not resemble any of the archetypes. B. Collective dynamics: Beam response The study of the collective response of an ensemble of oscillators under the influence of an external weak force fxðtÞ¼ϵcosðΩtÞcan be found in literature for the analysis FIG. 3. Average three-turn phase advance as a function of the Hamiltonian. The solid blue line shows the value of Eq. (9), while the orange dots show the result of a single particle tracking of an FODO cell with a sextupole (S¼−30) with δ¼−2.5×10−3for 3300 turns. INTERPRETATION OF THE HORIZONTAL BEAM …PHYS. REV. ACCEL. BEAMS 27, 124001 (2024) 124001-3 of beam instabilities and study of tune spectra (see, e.g., [25,26]). The amplitude of the beam centroid oscillation hxi as a function of the driving frequency (approximately) reads [27,28] hxiðtÞ¼ ϵ 2μxcosΩtP:V:ZdμρðμÞ Ω−μþπρðΩÞsin Ωt; (13) where ϵis the normalized strength of the force, ρðμÞis the distribution function of the frequencies of the oscillator ensemble, μx¼2πqx;0is the linear betatron frequency (assumed to be the mean value of the distribution), and P:V:denotes the principal value of the integral. Equation (13) is valid when ρðμÞonly peaks once and the transversal motion of the oscillators do not influence ρðμÞ itself. For the case where the oscillators in the ensemble experience, a linear detuning as a functionof their actionJxa similar result is known [29,30]. Hence an analytical expression is known for the beam transfer function (BTF) TðΩÞ TðΩÞ¼fðΩÞþigðΩÞ;(14) with the conventional definitions fðΩÞ¼P:V:ZdμρðμÞ Ω−μ;gðΩÞ¼πρðΩÞ: Note that the beam centroid oscillation amplitude [Eq. (13)] is directly related to the BTF by hxiðtÞ¼ℜϵ 2μx expð−iΩtÞTðΩÞ:(15) For a system where the nonlinear dynamics near a resonance are prevalent, there is no known analytical or semianalytical expression for the BTF. Therefore, we investigate experimentally the beam centroid oscillation amplitude to an external sinusoidal excitation under the aforementioned conditions. In the following, we will refer to hxi=ϵas the beam response to the excitation frequency Ω. III. BEAM RESPONSE MEASUREMENT Measurements of the beam response under resonant slow extraction conditions were carried out at the Heidelberg Ionbeam Therapy Center (HIT) in 2022. In this section, the measurement setup and experimental results are presented. Before we continue, we would like to introduce some nomenclature.Hereinafter,wewillrefertotheon-momentum eigenfrequency, where S¼0,as the betatron resonance. This resonance can be excited at any frequency [31] fn β¼frevðnqx;0Þ;n∈ℕ0;(16) where we refer to fn βas the nth betatron sideband. The nth betatron sideband possesses an upper fnþ β¼frevðnþqx;0Þ and a lower fn− β¼frevðn−qx;0Þfrequency. The experimental results of the beam response to excitation frequencies near two betatron sidebands—the upper ninth f9þ βand the lower eighth f8− β—will be presented. A. Experimental setup 1. Heidelberg Ion-Therapy Center synchrotron The Heidelberg Ion-Beam Therapy Center (HIT) is a medical accelerator facility dedicated to the treatment of cancer tumors in patients with light ions (pþ;He2þ;C6þ, and O6þ) using the intensity controlled raster scan [32,33]. The HIT synchrotron layout is illustrated in Fig. 4.Itisa compact machine with a circumference of 64.96 m. The ring is composed of two super cells each with three quadrupole doublet sections. Each super cell is equipped with two families of sextupole magnets, one optimally placed for chromaticity correction and the other one for the excitation of the resonance driving term. The optical functions of the synchrotron at extraction conditions are depicted in Fig. 5. The measurements are carried out with a carbon-ion beam 12C6þwith Ekin ¼124.25 MeV=nucleon. There was a dedicated measurement campaign to characterize the FIG. 4. The Heidelberg Ion-beam Therapy Center’s synchrotron. The position of the (de)focusing quadrupoles are indicated with (blue)red boxes. The position of the sextupoles are indicated with an orange ellipse. The injection channel can be seen in the top middle, whereas the extraction to the therapy rooms is visible in lower left corner. E. C. CORT´ ES GARCÍA et al. PHYS. REV. ACCEL. BEAMS 27, 124001 (2024) 124001-4 linear optics of the machine and the beam properties relevant for the study. An extended description of these measurements can be found in Appendix A. Unfortunately, the beam diagnostics at HIT do not enable the direct measurement of the transverse emittance of the coasting beam, hence this has to be kept as a free parameter. The measured machine and beam parameters are listed in Table I. 2. Beam response measurement setup The beam response to an external excitation is measured withavector networkanalyzer(VNA).Thestimulussignalis connected to the amplifier of the regular KO-exciter, providing transverse kicks to the beam. These sinusoidal kicks are of the form Δx0¼k0lsinð2πfextþφexÞwith the stimulus frequency fex ¼qexfrev and phase φex. The estimated transverse kick amplitude due to a given input stimulus power can be found in Table II. The stimulus leads finally to transverse oscillations of the beam centroid, which are recorded with regular capacitive pick-up electrodes in difference mode. This provides the amplitude and phase-dependent signal to the VNA, which calculates the system response at the stimulus frequency. The setup is depicted in Fig. 6. Each measurement was carried out in a single machine cycle, where the stimulus frequency was swept over the course of 10 s, during which 701 frequency steps were probed. All the beam response measurements presented here were performed with a coasting (unbunched) beam. B. Experimental results The beam response was measured in the vicinity of the upper ninth f9þ βand the lower eighth f8− βbetatron sidebands [see Eq. (16)]. The integrated strengths of the sextupole magnets k0 SLSwere varied in order to achieve a variation in the effective sextupole strength S[see Eq. (4)]. The results are illustrated in Fig. 7for an excitation span near f9þ β. The data shown were intentionally displaced (staggered) for better visualization of the details. Note that for the corrected chromaticity case (lower panel) for S¼32 m−1=2and 38 m−1=2the magnitude response was multiplied by a factor of two. The phase response for these measurements overlap considerably due to noise and is, therefore, not presented in Fig. 7. An example of the phase response with S¼32 m−1=2is shown in Fig. 8. The region FIG. 5. Optical functions of the HIT’s synchrotron super cell at extraction conditions. The solid curves show the transversal β-functions, whereas the dashed line illustrates the horizontal dispersion Dx. The position of the (de)focusing elements are marked with a red(blue) box on the top. The green marks indicate the position of the sextupoles. The dipoles are illustrated with a wide box. TABLE II. Estimated maximum kick for a given input stimulus power with a 12C6þcarbon-ion beam with Ekin ¼124.25 MeV= nucleon. Thevalues were estimated with an FEM simulation of the HIT KO-electrodes. Input power (dBm) Kick 01.54 μrad −10 486 nrad −20 154 nrad −30 48.7 nrad TABLE I. Measured machine and beam parameters for the beam response measurements. The model values come from a calculation from MADX [34]. The uncertainties of the measured values are listed in parenthesis. Parameter Symbol Value Model Tunes Qx=Qy1.679(1)/1.720(1) 1.679/1.755 Chromaticity ξx=ξy−2.85ð3Þ=−2.80ð3Þ−1.78=−1.35 Slip factor η0.46(3) 0.47657 Momentum spread (rms) Δp=p 0.5×10−3 Revolution frequency frev 2.1721 MHz Intensity NP8×107particles FIG. 6. Schematic of the experimental setup at HIT for the measurement of the beam response to an external excitation. INTERPRETATION OF THE HORIZONTAL BEAM …PHYS. REV. ACCEL. BEAMS 27, 124001 (2024) 124001-5 where the magnitude response splits exhibits fluctuations in the phase. For S≤19 m−1=2, the beam response is very similar. Both of the curves are in the regime where the transversal dynamics are dominated by h0from Eq. (1),the motion resembles a linear oscillator and the nonlinearity still does not play an important role. Once S≥25 m−1=2,the beam response starts splitting and exhibits two predominant maximaoronecharacteristicdip,respectively.Thehigherthe value of Sis set, the wider the gap between the two peaks becomes. It is also evident that the mean value of the beam response shifts toward the third-order resonance. This behavior comes both from the amplitude-phase detuning expectedfromtheunderlying dynamics described by Eq.(9), as well as from a contribution from the closed orbit at the sextupoles. The estimated closed orbit rms is 5 mm with a max/min excursion of 8mm=−15 mm. The same sextupole scan was performed and the beam response was measured at f8− β. The results are shown in Fig. 9. The behavior observed at this band is similar to f9− β. An exemplary measurement is included in Appendix B. Additionally, the measured phase response exhibited strong fluctuationsastheupperninthsideband.Thusthedataarenot shown. For S≤19 m−1=2, there is one single peak in the beam response, indicating that the motion is governed by the linear dynamics. Above this threshold, the nonlinearity FIG. 7. Beam response measurement as function of sextupole strength at the upper ninth betatron sideband, with natural chromaticity (top) and fully compensated horizontal chromaticity (bottom). The stimulus frequency is scanned from the resonance (red line) to higher frequencies with a strength of −30 dBm (k0l¼48.7nrad). Each graph composes the average and standard deviation over 25 cycles. The data was intentionally staggered in magnitude response. For S¼32 m−1=2and 38 m−1=2it was multiplied by a factor of two for the corrected chromaticity case for better visualization. FIG. 8. Average beam response (solid curve) measurement of 25 machine cycles with S¼32 m−1=2from the sextupole strength scan with corrected chromaticity. The rms of the measurements is shown as the shaded area. FIG. 9. Beam response measurement as function of sextupole strength at the eighth lower harmonic of the betatron resonance, with fully compensated horizontal chromaticity. The stimulus frequency is scanned from lower frequencies toward the resonance (red line) with a strength of −10 dBm (k0l¼486 nrad). Each graph composes the average and standard deviation over three cycles. The data were intentionally staggered for better visualization. E. C. CORT´ ES GARCÍA et al. PHYS. REV. ACCEL. BEAMS 27, 124001 (2024) 124001-6 induces a splitting in the measured beam response. In this case, one can recognize more than two maxima or, respectively, at least one characteristic dip. The measurement of the beam response at f8− βand f9þ βweas performed with the stimulusfrequencyspanningfromthelowerendtothehigher endofthefrequencyspancorrespondingtoeachsetting. This means that in the case of f9þ βshown in Fig. 7, the stimulus frequency went first through the third-order resonance (highlighted as a vertical line in red on the left), then through the betatron resonance and continued to thehigher end of the span. On the contrary, for the scan near f8− βshown in Fig. 9, the stimulus started with a frequency far from the resonance, then passed through the betatron resonance, and ended at the third-order resonance. This is an important aspect of the measurement that has to be considered, since the expected detuning due to the nonlinear dynamics only shifts the single particle tune toward the resonance. In other words, the direction of the excitation sweep can lead to different dynamics. Finally, the dependency on the stimulus power of the beam response was measured near f9þ βas well. The results are shown in Fig. 10, where the upper panel shows a broad scan in the stimulus frequency with the different curves showing the variation of the stimulus power (see Table II for the corresponding kick amplitude). The red line indicates the position of the nearest third-order resonance. The gray dashed lines indicate the approximate expected positions of different betatron resonance bands. The lower panel shows the phase with regard to the stimulus. The phase jumps are clear when the frequency crosses the dip in the magnitude response. Note that the phase response data was intentionally unwrapped.1By increasing the input power of the stimulus, which corresponds to an increasing strength of the kick amplitude that the beam experiences, the dip imprints deeper in the measured beam response. There is a noticeable shift of the measured response to values closer to the third-order resonance, when comparing to the results shown in Fig. 7. We attribute this shift to the influence of the excitation on the tune distribution due to the frequency sweep first going through the second order betatron sideband [36]. We note at the end of this section that the experimental results shown have been verified in various machines and with various beams. Some of these measurements are included in Appendix B. IV. BEAM RESPONSE SIMULATION The simulation of the beam response measurement is performed by launching a campaign of particle tracking computations to recover the horizontal beam oscillation amplitude. The particle tracking is performed with XSUITE [37]. The closed orbit corrector strengths are included in the model. A set with Np¼105particles is initialized, where each particle is described by an initial six-dimensional state vector z. The transverse set of conjugate pairs ðx; pxÞand ðy; pyÞis initialized such that their probability density function is described by a stationary two-dimensional Gaussian distribution. i.e., the initial distribution’s covariance matrix is given by Σx;y ¼εx;yβx;y −αx;y −αx;y γx;y ;(17) where ðβx;y;αx;y;γx;yÞrepresent the transversal optical functions from the Courant-Snyder parametrization. Both transversal planes are assumed to be uncorrelated and the transverse emittance ratio is set to unity. The initial transversal emittance was set to εx;y ¼1μmrad. The longitudinal position coordinate is distributed uniformly in the ring to resemble the coasting beam, whereas the momentum spread is modeled with a Gaussian distribution with an rms value of σp¼0.5×10−3, which is the value listed in Table I. After initialization, the particles were tracked for 104turns to let the distribution converge to a stationary distribution for a given value of S. The horizontal projection of the initial beam (before excitation) is depicted in Fig. 11. The ramping of the sextupoles was not introduced adiabatically as it was done in the experiment, nevertheless FIG. 10. Beam response with a S¼32 m−1=2for different input stimulus strengths. The dash-dotted lines indicate which resonance the peaks correspond to. The double peak structure becomes stronger with increasing excitation power. 1This means that a sudden phase jump in adjacent phase differences are never greater than πby adding 2πkfor some integer k. This signal processing routine is available in the standard numpy library [35]. INTERPRETATION OF THE HORIZONTAL BEAM …PHYS. REV. ACCEL. BEAMS 27, 124001 (2024) 124001-7 during the initial 104turns the residual coherent oscillation of the beam centroid is damped due to decoherence. Then the particles experienced the kicks from the external stimulus signal. The strength of the excitation for the simulations was set to 58 nrad. The stimulus from the measurement is modeled with a frequency chirped signal with a linear frequency sweep. The initial and end frequencies are chosen such that the range of interest shown in Figs. 7and 9are included in the sweep. This frequency range corresponds to a bandwidth of ΔqBW ¼0.026 in tune space and thus Qexc ¼nq0− ΔqBW=2to nq0þΔqBW=2with nbeing the betatron sideband under investigation (here n¼8or 9). The sampling rate of both beam oscillation amplitude and stimulus signal is set to 20 samples per turn. As described in Sec. III, the sweep time of one scan is 10 s, in which 701 frequency steps are probed. The duration of the scan is determined by the time the IF filter of the VNA (70 to 100 Hz) requires to resolve the stimulus frequency component for each measurement point. This translates to approximately 3.1×104turns per measurement point (2.17 ×107turns in total). With the available computing resources and the GPU implementation offered by XSUITE , we are able to easily track 2.17 ×107turns in total under a computing time of approximately 3.5 to 6 h for each scan. The simulated beam oscillation amplitude is then analyzed together with the input stimulus signal. The magnitude Aand phase θof a frequency component fin the signal S2with regard to the excitation signal S1is given by A¼jI2j=jI1j;(18) θ¼arctan ℑI1 ℜI1 −arctan ℑI2 ℜI2 ;(19) with I1;2¼Zt2 t1 S1;2ðtÞei2πftdt: (20) With the help of Eqs. (18) and (19), the measured curves are recovered from the simulation. The signal S2is in our experiment delivered by the capacitive pick-up electrodes and is up to a proportionality constant equivalent to the beam centroid oscillation. The signal S2used for evaluation of the simulation results is likewise given by the simulated transverse centroid oscillation. In the data analysis, the recovered phase given by Eq. (19) is intentionally unwrapped to avoid artificial discontinuities at θ¼π. A. Simulation results 1. Upper ninth betatron sideband The results from the simulation are shown in Fig. 12. The splitting behavior is visible, although one can recognize that the threshold of Sat which this happens is lower than in the measurement. The curves illustrated in Fig. 12 show qualitative agreement with the measurements depicted in Fig. 7for S≥25 m−1=2. One important addition by the simulation is the capability to fully recover the phase, which is shown in the lower panel of Fig. 12. The phase relation between driving excitation and centroid oscillations exhibits a change of πat the two visible dips, respectively. Such behavior is known to indicate resonance in the context of driven oscillators. To understand the nature of the splitting of the measured spectra, the phase space dynamics is investigated in the simulations. The final particle distribution shown in Fig. 13 reveals that the beam core is considerably depleted as consequence of the BTF excitation. From the simulation results, we can also recover the time evolution of the particle distribution. An animated version of Fig. 13 is included in Supplemental Material [[38]]. To identify the FIG. 11. Beam distribution after 104turns before excitation starts. The beam was initialized with εx;y ¼1μmrad. FIG. 12. Simulated beam response of a sextupole strength scan with chromaticity corrected. No particle loss is observed. E. C. CORT´ ES GARCÍA et al. PHYS. REV. ACCEL. BEAMS 27, 124001 (2024) 124001-8 moment at which the core is depleted, the evolution of the effective energy distribution by means of the Hamiltonian His shown in Fig. 14. When the excitation frequency crosses the visible dip, the average effective energy is increased and the low energy regime (core) starts to deplete. 2. Lower eighth betatron sideband The scan around the lower eighth betatron band was simulated as well. The results of the simulation are shown in Fig. 15 and qualitative behavior is reproduced. The magnitude of the response splits in different regions. The phase of the signal shows a similar behavior as for the previous case where the ninth upper betatron band was investigated. There are three visible phase jumps, one notably at the visible dip in the spectrum. With the results from the simulation, the particle and the effective energy distributions were closely investigated. An animation is provided in Supplemental Material [38] and reveals interesting details of the evolution of the particle distribution during the excitation process. The resulting distribution density is illustrated in Fig. 16. The distribution is shown at FIG. 15. Simulated beam response of a sextupole strength scan. The chromaticity was fully corrected for this simulation. The magnitude response for the S¼0was magnified by a factor 100. FIG. 13. Resulting particle density distribution after simulated BTF measurement for the ninth betatron upper band. The sextupole strength is S¼25 m−1=2and the simulation comprises 105particles. The black line shows the separatrix from the Kobayashi theory. Dispersive and closed orbit offsets have been corrected to center the phase space. The optical functions at the observation point s¼0are listed in Table III. FIG. 16. Resulting particle density distribution after simulated BTF measurement at the lower eighth betatron sideband. The sextupole strength is S¼25 m−1=2and the simulation comprises 105particles. The black line shows the separatrix from the Kobayashi theory. Dispersive and closed orbit offsets have been corrected to center the phase space. The optical functions at the observation point s¼0are listed in Table III. FIG. 14. Effective Hamiltonian distribution (top) and simulated BTF measurement (bottom) as a function of the excitation frequency (time of the BTF scan) for S¼25 m−1=2. The Hamiltonian distribution is recovered from a sample of 104 particles. The blue dashed line represents the average normalized effective energy. INTERPRETATION OF THE HORIZONTAL BEAM …PHYS. REV. ACCEL. BEAMS 27, 124001 (2024) 124001-9 [35] C. R. Harris, K. J. Millman, S. J. van der Walt, R. Gommers, P. Virtanen, D. Cournapeau, E. Wieser, J. Taylor, S. Berg, N. J. Smith, R. Kern, M. Picus, S. Hoyer, M. H. van Kerkwijk, M. Brett, A. Haldane, J. F. del Río, M. Wiebe, P. Peterson, P. G´erard-Marchant, K. Sheppard, T. Reddy, W. Weckesser, H. Abbasi, C. Gohlke, and T. E. Oliphant, Array programming with NumPy, Nature (London) 585, 357 (2020). [36] P. Niedermayer and R. Singh, Excitation of nonlinear second order betatron sidebands for knock-out slow extraction at the third-integer resonance, Phys. Rev. Accel. Beams 27, 082801 (2024). [37] G. Iadarola, A. Abramov, P. Belanger, X. Buffat, R. D. Maria, D. Demetriadou, L. Deniau, D. D. Croce, P. Hermes, P. Kicsiny, P. Kruyt, A. Latina, L. Mether, P. Niedermayer, K. Paraschou, T. Pieloni, M. Seidel, G. Sterbini, F. V. der Veken, L. van Riesen-Haupt, and S. Łopaciuk, Xsuite: an integrated beam physics simulation framework, in Proceedings of the 68th Advanced Beam Dynamic Workshop High-Intensity High-Brightness Hadron Beams (HB’23) (JACoW, Geneva, Switzerland, 2024), Vol. 68, pp. 73–80. [38] See Supplementak Material at http://link.aps.org/ supplemental/10.1103/PhysRevAccelBeams.27.124001 for short animations of the excitation process of the upper ninth and the lower eighth betratron sidebands under resonant slow extraction conditions. [39] F. M. Velotti, Maptrack, https://gitlab.cern.ch/abt-opticsand-code-repository/simulation-codes/maptrack. [40] R. Taylor, Slow extraction: Upgrades for next ion medical machines at FLASH timescales (2024), http://cds.cern.ch/ record/2915795/files/. [41] P. Niedermayer and R. Singh, Transverse excitation and applications for beam control, in Proceedings of the 13th International Particle Accelerator Conference, IPAC-2022 (JACoW, Geneva, Switzerland, 2022), pp. 251–253. [42] M. Beutelspacher, M. Grieser, K. Noda, D. Schwalm, and A. Wolf, Dispersive electron cooling experiments at the heavy ion storage ring TSR, Nucl. Instrum. Methods Phys. Res., Sect. A 512, 459 (2003). [43] https://github.com/hibtc/btf-simulation [44] D. Ondreka, C. Dimopoulou, H. Hüther, H. Liebermann, J. Stadlmann, and R. Steinhagen, Recommissioning of SIS18 after FAIR upgrades, in Proceedings of the 10th International Particle Accelerator Conference, IPAC-2019, Melbourne, Australia, (JACoW, Geneva, Switzerland, 2019), MOPTS035. E. C. CORT´ ES GARCÍA et al. PHYS. REV. ACCEL. BEAMS 27, 124001 (2024) 124001-16