Hiromi Okamoto, Kiyokazu Ito,
Graduate School of Advanced Sciences of Matter, Hiroshima University, Japan.
Lucy Martin, Suzanne Sheehy, University of Oxford, United Kingdom.
David Kelliher, Shinji Machida, Christopher Prior,
STFC Rutherford Appleton Laboratory, United Kingdom.
The Fixed Field Alternating-gradient (FFA) accelerator was first invented in the 1950’s and 1960’s as an alternative to synchrotrons and cyclotrons. The machine achieves focusing using the alternating-gradient (strong) focusing principle, but the magnetic fields vary with radius rather than in time. The orbits are geometrically similar (i.e. photographic enlargements at higher energies) and the focal length scales with the orbit radius, resulting in a constant field index, k. This allows for DC magnets with low power consumption, fast acceleration limited only by the RF system and an energy reach which exceeds that available with cyclotrons. However, the scaling FFA also has a larger aperture vacuum chamber than a synchrotron and the magnet designs can be considerably more complex, with detailed pole shaping required to achieve the highly nonlinear fields.
In the 1990s a new type of FFA was invented, called a non-scaling FFA, where the magnetic guide fields consist only of dipole and quadrupole fields (i.e. linear fields). This concept was tested and demonstrated using the EMMA non-scaling FFA at Daresbury Laboratory, UK, commissioned between 2010 and 2012 . While the machine is conceptually simpler in terms of guide field, the beam dynamics are more complex than in the scaling case.
In the non-scaling type, the horizontal and vertical bare betatron tunes per unit focusing cell decrease during beam acceleration. The operating point of the machine then crosses resonance stop bands inevitably, which may seriously deteriorate the beam quality. This unique nature of non-scaling FFAs can be studied systematically with a novel tabletop system, such as a Paul ion trap, instead of relying on a large-scale machine.
A Linear Paul Trap (LPT) traps ions using an electric quadrupole field, rather than the magnetic quadrupole used to confine particles in a conventional accelerator. As briefly explained in the next section, a one-component nonneutral plasma in a linear Paul trap obeys the equations of motion physically equivalent to those governing a charged-particle beam in an alternating-gradient (AG) focusing channel [2, 3].
There are currently two LPTs dedicated to accelerator physics, the Simulator of Particle Orbit Dynamics (S-POD) [4, 5], at Hiroshima University, Japan, and the Intense Beam Experiment (IBEX) , at the Rutherford Appleton Laboratory, Oxfordshire.
The LPT consists basically of four electrode rods placed symmetrically around the trap axis, which generate a radio-frequency (rf) quadrupole field for transverse ion confinement . These rods are axially divided into several electrically isolated pieces, so that we can apply different bias voltages to form an axial potential well. Figure 1 is a picture of the LPT currently used for the S-POD system at Hiroshima University. The nominal operating rf frequency is 1 MHz. Both S-POD and IBEX systems have been optimized to confine ions, but the species of confined ions is of no essential importance in FFA studies.
In this paper, we first introduce the LPT concept, then work chronologically through beam studies relevant to FFAs, including crossing low-order space charge driven resonances and integer resonances (at low intensity) relevant to linear non-scaling FFAs. We then introduce the concept of a nonlinear Paul trap for control of higher order field components, which would be necessary to study dynamics of scaling FFAs.
All studies using Paul traps so far have kept the axial potential well static to concentrate on transverse dynamics. The equations of transverse ion motion in a regular LPT can be derived from the Hamiltonian
where the independent variable has been scaled from time to the length with being the speed of light, is proportional to the rf voltages on the quadrupole rods, is a constant parameter depending on the particle species, and is the collective Coulomb potential that satisfies the Poisson equation. Equation (1) has the same form as the well-known Hamiltonian of an AG beam transport channel. Since the particle distribution function in phase space obeys the Vlasov equation in both dynamical systems, they are physically equivalent; namely, what happens in a beam transport channel also happens in an LPT, which allows us to employ the compact LPT for beam dynamics studies [2, 3]. Of practical importance is the fact that the focusing function is highly flexible. Unlike in a real accelerator, the waveform of can readily be controlled over a very wide range simply by modifying the time structure of the rf voltages. It is thus straightforward to replicate various AG lattices and even move the operating point over an arbitrary range at arbitrary speed. We note that in future it may also be possible to excite an equivalent of synchrotron motion, if necessary, by introducing a periodic modulation to the bias voltages for axial ion confinement. However, this will not be considered further in this paper.
In the following, the sinusoidal focusing is employed because it has no essential difference from the most standard FODO focusing . We also assume for the sake of simplicity that the bare tunes per AG cell (a single sinusoidal period) are always equal; namely, .
No single-particle resonance is expected in the range under the strictly linear focusing field. In any real machines, however, lattice imperfections are unavoidable, which gives rise to additional stop bands. Even in an ideal machine without imperfections, we still have a possibility of space-charge-induced resonances when the beam density is high. The latter collective effect is inherent in high-intensity FFAs. Lee et al. have argued that the fourth-order and sixth-order space-charge resonances at the cell tunes and will impose a fundamental limitation on the non-scaling FFA performance [9, 10]. These stop bands can be identified in Fig. 2 obtained through systematic ion-loss measurements with S-POD at many different fixed tune values. The severe instability at is caused by the third-order error field due mainly to electrode misalignments. The other two regions of ion losses slightly above the cell tunes of and almost completely vanish in the low-density regime, which strongly suggests that the space-charge potential is responsible for these instabilities. It is worth noting that the instability at is particularly severe. It turns out to be more serious than the third-order resonance at in the high-density regime. This observation can be explained by the coherent resonance condition 
where represents the resonance order, is a constant depending on the order number, is the root-mean-squared tune shift induced by space charge repulsion, and is an integer. The growth rate of the coherent parametric resonance under the above condition is proportional to the beam perveance. This type of collective instability is more enhanced as the beam density becomes higher. Conversely, it disappears at the low-density limit. According to Eq. (2), the instability at should be interpreted as the linear resonance rather than the fourth-order . The former is two orders lower than the latter and, therefore, much stronger at high density, which is consistent with the experimental observation in Fig. 2. Similarly, the intensity-dependent resonance at should be understood as the third order instead of the sixth order .
We employed S-POD to see what happens when the operating tune traverses these major stop bands. The number of ions surviving after single resonance crossing is plotted in Fig. 3 as a function of the crossing speed . is defined as the ratio of (the width of cell-tune variation during acceleration) to (the number of AG periods for the cell tune to go across the width ). A vertical broken line in each panel indicates the crossing speed corresponding to the 20-turn extraction from EMMA. In the left panel, the cell tune has been changed from 0.4 down to 0.32, so that the operating point traverses the third-order stop band at in Fig. 3(a). Since the primary source of this resonance is the external error field, the ion-loss behaviour is relatively insensitive to the initial ion number. Severe ion losses have occurred even at low density. In contrast, the possible linear coherent resonance at has caused considerable ion losses in Fig. 3(b) only when the initial plasma density is high. As for the weak space-charge-driven resonance at , we detected no noticeable ion losses, at least, within the same range of the crossing speed .
A dipole perturbation can be produced easily in the LPT to investigate the effect of integer resonance crossing in FFAs. By applying pulse voltages of opposite signs to the horizontal (or vertical) pair of quadrupole rods, we can add a new driving term to the Hamiltonian in Eq. (1) :
where is proportional to the strength of the dipole perturbing field. Figure 4 shows the stop-band distribution measured in S-POD with and without the dipole perturbation. As an example, we have assumed a circular machine with 42-fold lattice symmetry (), keeping the EMMA structure in mind. The sharp drop of the ion number observed at in the absence of the dipole field (left panel) is attributed to the error-driven third-order resonance. Many additional stop bands have appeared with the dipole perturbation on (right panel). As expected, all of them are located at integer tunes.
In reference to the previous section, we note that the lowest-order (), thus strongest coherent resonance takes place at half-integer tunes as predicted by Eq. (2), which corresponds to the instability of the dipole oscillation mode . The so-called “integer resonance” is none other than this type of instability enhanced by external dipole error fields.
The rate of surviving ions after crossing a single integer stop band at was measured in S-POD over a wide range of crossing speed . The results in Fig. 5 indicate that higher is naturally required under stronger perturbation to keep ion losses to a negligible level. The measurement data are in excellent agreement with multi-particle simulations.
The controllability of low-order nonlinear fields within the LPT aperture is of crucial importance because the scaling type of FFA has a high level of inherent nonlinearities. In the conventional four-rod configuration, however, nonlinear fields come from mechanical errors, which means that we cannot control their strengths and time structure independently of the main quadrupole focusing field. All kinds of fields are excited simultaneously while rf voltages are applied to the quadrupole rods. A modified LPT was, therefore, designed and constructed for future studies of nonlinear beam dynamics. As can be seen in Fig. 6, the new trap has four extra electrodes inserted in-between the quadrupole rods . The use of these extra electrodes enables us to activate the sextupole and octupole fields at arbitrary timing. The performance test is now in progress at Hiroshima University.
Linear Paul traps such as S-POD and IBEX have already been used effectively to study a wide range of high intensity physics as the transverse Hamiltonian of a LPT is equivalent to that of an alternating gradient beam transport channel. S-POD has been used to simulate integer resonance crossing in linear non-scaling FFAs, emulating the machine EMMA. However, in a linear trap higher order fields cannot be controlled. A new trap, built at Hiroshima University and currently undergoing testing, has 4 further electrodes, allowing sextuple and octupole fields to be carefully excited. This will allow experiments relevant to resonance crossing in scaling FFAs to be conducted, as such machines are inherently nonlinear.
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