Stability study of intense hadron bunches in linear accelerators using a Paul ion trap

The collective behavior of an ion cloud confined in a linear Paul trap is physically almost equivalent to that of a relativistic beam traveling in a particle accelerator. This fact provides a new possibility for experimental beam-dynamics studies that can be done without relying on large-scale machines. We have constructed a compact ion trap to explore various accelerator-physics issues in a local tabletop environment. The present trap system is designed particularly for the study of high-intensity short hadron bunches typical in linear accelerators. Resonance-induced ion losses are measured and plotted as a function of the betatron and synchrotron phase advances over a wide range, which offers a piece of indisputable experimental evidence for the presence of various betatron and synchrobetatron resonance stop bands. We confirm that these instability bands shift in the stability map depending on the ion density. The recently proposed stop-band diagram, free from the conventional concept of incoherent tune spread, is employed to explain the experimental observations.

Figure 1

Schematic drawing of the LPT for short-bunch experiment. The LPT consists of three quadrupole sections electrically isolated from each other. The transverse rf quadrupole field of the same strength is always excited in all three sections. The central section, in which ions are stored, is 8.9 mm long; this length is determined so as to make the axial potential well nearly parabolic when an equal DC bias voltage is applied to the two end sections of 30 mm long. An electron gun is placed above the central section to ionize neutral Ar atoms. The radii of the aperture and quadrupole rods are $r_0=5\mathrm{mm}$ and $\rho =5.75\mathrm{mm}$. The DC bias $U_{\parallel }$ on the MCP side is dropped to extract remaining ions and count them with the detector.

Figure 2

Long-term ion-loss distribution in the low-density regime. The number of ${\text{Ar}}^{+}$ ions remaining in the LPT after 1 s (${10}^6$ sinusoidal periods) is measured at many different values of ${\mu }_{0\perp }$ with the axial confinement bias fixed at $U_{\parallel }=3.14\mathrm{V}$ and $U_{\parallel }=10.89\mathrm{V}$. Each curve consists of 360 data points. Three vertical lines indicate the locations of the resonances expected from Eq. (2) with $(n_{\perp },n_{\parallel },n)=(3,0,1)$ (red), (4,0,1) (green), and (2,2,1) (blue). The ideal structural parameters, i.e., $r_0=5.00\mathrm{mm}$ and ${\ell }_z=8.23\mathrm{mm}$, are assumed in the upper panels (a) to determine the bare phase advances. In the lower panels (b), these parameters are modified to $r_0=5.022\mathrm{mm}$ and ${\ell }_z=8.397\mathrm{mm}$. In addition to the three resonances above, we notice several other resonances that should be of the fourth and fifth orders according to Eq. (2). For instance, the cause of the two small dips between the green and red lines in the left panels can be understood as the nonlinear resonances with $(n_{\perp },n_{\parallel },n)=(3,2,1)$ and (3,1,1). These high-order resonances have only little impact on bunch stability in a short timescale corresponding to the length of a typical hadron linac, unless they are driven by strong error fields.

Figure 3

Number of confineable ions with the betatron and synchrotron phase advances fixed through the whole experiment process from the ion production to the extraction. The map contains 6000 independent data points. The final ion number at each operating point was measured after the routine ionization procedure of 1 s long, followed by 51.2-ms storage.

Figure 4

Long-term decay of stored ions at the operating point $({\mu }_{0\perp },{\sigma }_{0\parallel })=(50.0°,26.4°)$. The initial ion number $N_{\mathrm{in}}$ is adjusted to about $1\times {10}^6$ (solid), $2\times {10}^6$ (dashed), and $3\times {10}^6$ (dotted). The three curves are obtained from the fitting function in Eq. (3) with the decay constants listed in Table 1.

Figure 5

Ion losses due to the transverse overfocusing effect caused by the removal of an axial potential barrier. The transverse phase advance ${\mu }_{0\perp }$ is varied over a wide range with the synchrotron phase advance adjusted to ${\sigma }_{0\parallel }=26.4°$, $70.7°$, and $88.3°$. An ion bunch is generated initially at $({\mu }_{0\perp },{\sigma }_{0\parallel })=(50.0°,26.4°)$ in all three cases. After about $2\times {10}^6$ ions are accumulated there, the operating point is quickly moved onto a fixed-${\sigma }_{0\parallel }$ line and stay there for $150\mu \mathrm{s}$ before ion extraction.

Figure 6

Stability tune diagram for the ion storage period of 0.5 ms (500 AG cells). The number of ions confined initially in the LPT is adjusted to about (a) $1.0\times {10}^6$, (b) $2.0\times {10}^6$, and (c) $3.5\times {10}^6$.

Figure 7

Stability tune diagram for the ion storage period of 10 ms (10000 AG cells). The number of ions confined initially in the LPT is adjusted to about (a) $1.0\times {10}^6$, (b) $2.0\times {10}^6$, and (c) $3.5\times {10}^6$.

Figure 8

Stability tune diagram obtained from the coherent resonance condition in Eq. (4) and the bandwidth formula in Eq. (a1). The coherent core instabilities of the second-order or third-order modes could be excited in gray shaded areas. The number of particles in a bunch is fixed everywhere at the value that corresponds to the two cases where (a) ${\eta }_{\perp }=0.8$ and (b) ${\eta }_{\perp }=0.9$ at the operating point $({\mu }_{0\perp },{\sigma }_{0\parallel })=(50.0°,26.4°)$. The bunch is assumed to be equipartitioned at this initial ionization point. Three numbers in the bracket written on each instability band represent $(n_{\perp },n_{\parallel },n^{\prime })$ in Eq. (4) used to draw the band. The same solid and dashed lines as indicated in Figs. 6 and 7 are replotted for reference.