Autofocusing drift tube linac envelopes

To date, beam dynamics studies and design of combined zero degree drift tube linac (DTL) structures (Kombinierte Null Grad Struktur; KONUS) have only been carried out in multiparticle codes. Quantities such as the beam envelopes are obtained by averaging over particles, whose tracking is computationally intensive for large bunch populations. Tune computations, which depend on this average, are burdensome to obtain. This has motivated the implementation of a simulation of KONUS DTLs in the code transoptr, whose Hamiltonian treatment of beam dynamics enables the integration of energy gain from the longitudinal electric field on axis while simultaneously elaborating the field in transverse directions to obtain the linear optics. The code also features an in-built space charge capability. The evolution of the beam matrix including longitudinal optics is computed in a reference Frenet-Serret frame through the time-dependent DTL cavity fields. This enables fast envelope simulations for DTLs, resulting in a variable energy sequential tune optimization capability. The implementation methodology and optimization techniques, applicable for any combination of DTL tanks and bunchers, is outlined. Comparisons with the code lorasr, in addition to beam-based $E/A$ measurements of a DTL are presented.

Figure 1

Overview of the ISAC-I linear accelerator complex. The offLine ion source (OLIS) provides stable pilot beams. Beam energy is measured at the Prague (HEBT1) high energy analyzing magnet station, a 90° dipole which provides a reading of energy and energy spread.

Figure 2

The separated function ISAC-DTL. Quadrupole triplets along the lattice define the transverse optics. Courtesy of M. Marchetto, TRIUMF.

Figure 3

opera2d computed normalized on axis electric fields ($E_Z/E_{\max }≔{\mathcal{E}}_{\mathrm{n}}(s)$) for Tanks 1–5 of the ISAC-DTL. Note the nonflatness of the peak electric field intensity across the cavities. Gap effective voltages for an ${{}^{14}\mathrm{N}}^{3+}$ beam are shown as histograms.

Figure 4

transoptr computed reference particle $E/A$ for IH Tank-3, using the field shown in Fig. 3, center. The horizontal axis is the $s$-coordinate along the cavity’s central optical axis, where the spatial component of the field is ${\mathcal{E}}_{\mathrm{n}}(s)$. Contours (white) delineate constant reference particle energy. The rf scaling parameter has been kept constant for each cavity phase $\varphi$. The zero-degree phase causes monotonous energy increase along Tank-3. At the end of the cavity, the resulting $E/A$ vs $\varphi$ response is characteristically nonsinusoidal, instead featuring a sharp dropoff in $E/A$ to phases negative of zero-degree. Phases above zero-degree cause a more gentle decrease in energy, until the output falls below the injection $E/A$ of $0.461\mathrm{MeV}/u$. This characteristic profile is useful to establish a calibration between beam-based measurements and model computations, relating the field scaling in Eq. (8) and the rf amplitude parameter in the control system [38], given a computed field profile ${\mathcal{E}}_{\mathrm{n}}(s)$.

Figure 5

Comparison of transoptr (solid, 2 rms) and lorasr (dashed, 96% containment) beam envelopes for ${{}^{14}\mathrm{N}}^{3+}$ through the ISAC-DTL, for an initial distribution listed in Table 2. In this simulation, the ISAC-DTL is configured for an $E/A=1.530\mathrm{MeV}/\mathrm{u}$ output, with tanks at negative synchronous phase. Longitudinal envelopes are presented in coordinate-5 units of 20° for plotting clarity, with $\varphi =2\pi z/(\beta \lambda )$.

Figure 6

Coordinate 5, 6 phase space distributions: lorasr normalized density for 5000 particles (points) and transoptr 4 rms containment ellipses (purple) at injection and through the ISAC-DTL, for the envelopes shown in Fig. 5, with input beam parameters shown in Table 2. Both graph scales are with respect to the synchronous particle through each cavity, computed in lorasr. Normalized color density scale shown on the right of each plot. The intermediate distributions are shown at the exit of each triplet, with the energy per nucleon shown in each graph. As the bunch progresses through the linac, second-order components of the accelerating fields cause rf filamentation, which is represented in the multiparticle code. On the other hand, transoptr’s ellipse arises only from the linear component of the forces, which nevertheless agrees with the core of the multiparticle beam simulation. The DTL output distribution consists of 10 000 particles, highlighting the core of the distribution, where higher order effects are minimized.

Figure 7

Top: lorasr (blue) and transoptr (red) computed ISAC-DTL Tank-1 $E/A$ vs $\varphi$ response and associated lorasr particle transmission (pink), which sharply drops at phases far from maximum acceleration. The near optimum distribution ($-6°$, bottom left) becomes severely broadened in time as the phase is detuned ($-71°$, bottom right), causing particle losses into adjacent buckets which also shifts the average energy.

Figure 8

transoptr (red) and lorasr (blue) simulations of ISAC-DTL IH-Tank output $E/A$ at variable rf phase but fixed on axis electric field scaling $V_0$. ISAC-DTL measured ${{}^{16}\mathrm{O}}^{4+}$ $E/A$ response superimposed in green. The lorasr model assumes perfect field flatness, while the transoptr model features a sagging in the maximum peak amplitude (Fig. 3). The setpoint of each tank at maximum operational acceleration (Table 1) is shown as a black cross on each figure. Note the $x$ axis phases are absolute, meaning optimum acceleration is not necessarily at 0°.

Figure 9

transoptr computed elements ${\mathbf{M}}_{21}$, ${\mathbf{M}}_{43}$, and ${\mathbf{M}}_{65}$ (solid lines) across DTL Tank-1. Optimum acceleration to $E/A=0.251\mathrm{MeV}/\mathrm{u}$ is shown, where the energy gain (blue, right $y$ axis) is monotonous, minimizing longitudinal restoring forces, which in turn reduces transverse rf focal effects. The plot also shows the ($x$, $y$, $z$) 2 rms envelopes, each as dotted lines. The scaled value of Tank-1 ${\mathcal{E}}_{\mathbf{n}}(s)$ is shown in grey.

Figure 10

Graphical representation of tuning sequences from Table 3 for various ISAC-DTL configurations, corresponding to Tank 1 to 5 $E/A$ cases (Table 1). Colors represent optimization at a given energy, e.g., dark blue is up to 0.254 MeV/u. The optimizer is called upon in the sequence $a$ to $e$, 1 to 5. Step $a$ corresponds to longitudinal tank configuration, while steps $\{b,c,d,e\}$ accomplish DTL triplet tuning, in each case using transoptr’s optimizer. Tanks 1 to 3 are optimized with their bunchers.

Figure 11

transoptr computations of 2 rms envelopes and reference particle energy for a ${{}^{20}\mathrm{Ne}}^{4+}$ beam in the ISAC-I linac, from OLIS extraction at $E/A=2.04\mathrm{keV}/\mathrm{u}$ up to the Prague Magnet station (Fig. 1). The dip in $E/A$ near $s=0$ is from a decelerating Einzel lens [53]. The OLIS slits are shown as vertical dashed lines. The code is run in 6D bunched mode, with initial longitudinal beam matrix values chosen to mimic the acceptance of the prebuncher, which is represented as a single discrete 11 MHz rf kick (arrows), though OLIS beams are continuous. The starting energy spread is equal to the prebuncher’s three harmonic waveform deviation from a perfect sawtooth. The top shows the ISAC-DTL configured for acceleration up to 0.803 MeV/u. The DTL triplets and HEBT optics have been reoptimized, along with Tanks 4 and 5, to retune the linac to 1.53 MeV/u acceleration in the bottom figure. The longitudinal mismatch into the RFQ is corrected by optimizing the prebuncher and low-energy optics. Additionally, the MEBT time focus (top) has been retuned to debunch the beam longitudinally (bottom), requiring a change in MEBT & DTL rf settings. The high-energy analyzing magnet (Fig. 1) is located between $s=4200$ and 4600 cm, upstream of the parallel wire detector.