Double stop-band structure near half-integer tunes in high-intensity rings

This paper addresses a detailed experimental study of collective instability bands generated near every half-integer tune per lattice period by coherent dipole and quadrupole resonances. Both instabilities appear side by side or overlap each other but are mostly separable because the dipole resonance often creates a narrower stop band accompanied by more severe particle losses. The separation of these low-order resonance bands becomes greater as the beam intensity increases. In principle, the double stop-band structure can be formed even without machine imperfections when the beam’s initial phase-space profile is deviated from the ideal stationary distribution. The tabletop ion-trap system called “S-POD” is employed to experimentally demonstrate the parameter dependence of the double stop-band structure. Numerical simulations are also performed for comparison with experimental observations.

Figure 1

A typical linear Paul trap for S-POD.

Figure 2

Electrode excitation patterns employed for the present experiment. The four circles in each picture represent the cross sections of the quadrupole electrodes aligned along the LPT axis. (a) Quadrupole excitation for transverse ion confinement. The periodic voltage $V_{\mathrm{AG}}(t)$ generates a strong AG focusing force. (b) Superperiodic pulse excitation. $V_{\mathrm{sp}}$ is superimposed on $V_{\mathrm{AG}}$ in order to introduce a specific superperiodic modulation to the focusing function $K(t)$. (c) Horizontal dipole excitation. A pulse voltage $V_{\mathrm{kick}}$ is switched on once at the beginning, if necessary, to kick an ion bunch horizontally.

Figure 3

Radio-frequency waveforms employed for the present experiment. We have assumed an AG lattice consisting of three superperiodic structures ($N_{\mathrm{sp}}=3$), each of which includes nine unit focusing cells ($N_{\mathrm{cell}}=9$). The discrete FODO waveform is approximated by a sine curve for simplicity. The upper panel (a) thus emulates an AG beam transport channel composed of 27 identical FODO cells. We superimpose the periodic pulses in the lower panel (b) to generate superperiodic driving harmonics in the focusing function $K(t)$ in Eq. (6). Note that the superperiodic number $N_{\mathrm{sp}}$ is of no physical importance as long as we disregard lattice imperfections.

Figure 4

Stop-band distribution in the absence of the superperiodic perturbation. The number of ${{}^{40}\mathrm{Ar}}^{+}$ ions in the LPT is measured with the MCP after a storage period of 10 ms. The horizontal and vertical bare tunes have been set equal, i.e., $Q_{0x}=Q_{0y}(\equiv Q_0)$, in all measurements carried out over the range $2/3\le Q_0\le 4$. Three different initial ion numbers, i.e., approximately $10^5$ (blue), $10^6$ (red), and $10^7$ (black), are considered in this example. Rapid drops in ion number near both ends of the abscissa are simply because the operating point is too close to the stability boundary of Hill’s equation. Each curve is composed of 979 discrete data points obtained at 979 different bare tune values.

Figure 5

Stop-band distributions in the presence of the superperiodic perturbation. The number of ${{}^{40}\mathrm{Ar}}^{+}$ ions surviving after 1 ms is plotted as a function of the bare tune $Q_0$ evaluated in disregard of the perturbing pulses. The actual horizontal and vertical tunes ($Q_{0x}$, $Q_{0y}$) are slightly different from $Q_0$. The initial ion number is set roughly at $10^7$ in all measurements. The perturbation voltage relative to the sinusoidal focusing amplitude is chosen to be $\alpha =1\%$ in the upper picture (a) and 2% in the lower picture (b).

Figure 6

Intensity dependence of the double stop-band structure near $Q_0=1$. The perturbation pulse height $V_{\mathrm{sp}}$ is the same as the case in Fig. 5; namely, $\alpha =1\%$. The ion storage period is fixed at 1 ms in all experiments performed at different initial plasma densities.

Figure 7

Schematic drawing of the tune diagram in the vicinity of $Q_0=1$. The coherent dipole resonance occurs under the conditions $Q_{0x(0y)}\approx 1$ indicated by the thick horizontal and vertical lines. The hatched areas represent the stop bands within which the quadrupole mode becomes resonantly unstable. The ion-loss measurements in Fig. 6 have been performed along the broken line slightly shifted downward from $Q_{0x}-Q_{0y}=0$ due to the superperiodic perturbation. If we start the measurements from the low tune side, the horizontal dipole resonance ($Q_{0x}\approx 1$) is encountered first and then the vertical ($Q_{0y}\approx 1$). The horizontal and vertical stop bands of the quadrupole instability are too wide and, therefore, not separable in the present case where the perturbation-induced difference between $Q_{0x}$ and $Q_{0y}$ is small.

Figure 8

Time evolution of the dipole and quadrupole instability bands near $Q_0=2$. The ordinate represents the number of surviving ions after a certain storage period, normalized with the initial ion number ($\approx 10^6$ in this measurement). The perturbation strength has been fixed at $\alpha =1\%$.

Figure 9

Initial transverse profile of an ion bunch at $Q_0=1.99$. The right panel shows the density profile on the horizontal $x$ axis derived from the phosphor image in the left panel.

Figure 10

Transverse profiles of ion bunches projected onto the phosphor screen. The left panel (a) and right panel (b) are the phosphor images measured after a 2.55-ms storage of ions at $Q_0=1.99$ and after a 0.7-ms storage of ions at $Q_0=2.02$, respectively. Sharp ion losses occur at these tunes, as indicated in Fig. 8. Note that individual ions hit the screen at different timings, depending on their axial positions and velocities when the potential barrier on the MCP side is removed for plasma extraction. The phosphor images shown here thus reflect the axial integrations of transversely oscillating ion distributions over a certain extraction period.

Figure 11

Effect of an initial transverse displacement of the bunch centroid on the dipole and quadrupole instabilities near $Q_0=2$. An ion bunch in the LPT has been kicked once at $t=0$ [ms] in the horizontal direction by a dipole electric pulse in Fig. 2. Three symbols correspond to three different kick voltages, namely, $V_{\mathrm{kick}}=0[\mathrm{V}]$ (circle), 1 [V] (triangle), and 2 [V] (cross). The number of ions initially stored in the LPT is about $10^6$ in all three cases. The operating tune in each panel is adjusted to the point at which the ion loss rate becomes maximum within a particular stop band. For the dipole instability, the adjusted tune stays close to (a) $Q_0=1.99$ or (b) $Q_0=2.02$ throughout the ion-loss measurements. In case (c), the operating tune has been reduced from $\sim 2.05$ down to $\sim 2.03$ in 4 ms because the quadrupole stop band moves to the lower tune side with a gradual decrease in the ion density.

Figure 12

Intensity dependence of ion-loss behavior under coherent dipole resonance. The time evolution of the fraction of surviving ions was measured with the MCP, starting from several different plasma densities. The initial ion numbers in the LPT are $1.0\times 10^6$ (circle), $2.5\times 10^6$ (triangle), $4.0\times 10^6$ (cross), and $8.0\times 10^6$ (plus). The horizontal kick voltage of $V_{\mathrm{kick}}=1[\mathrm{V}]$ has been used in all four cases. The operating point is the same as that in Fig. 11, namely, $Q_0\approx 1.99$ where the ion loss rate becomes maximum due to the horizontal coherent dipole instability.

Figure 13

Warp simulation results assuming the AG lattice waveforms in Fig. 3. The rate of rms emittance growth (red broken line) and the position of the plasma centroid (blue solid line) after a storage period of 0.5 ms are numerically evaluated at different operating tunes. The plasma centroid is displaced initially by 0.1 mm in both transverse directions to activate the coherent dipole oscillation. The tune depression has been fixed at 0.9 in all three cases where the perturbation strength parameter $\alpha$ is chosen to be (a) 0%, (b) 1%, and (c) 2%.

Figure 14

Transverse oscillations of the bunch centroid at slightly different operating points about $Q_0=2$. The perturbation strength $\alpha$ is 1% in both examples. The tune depression is fixed at 0.9. The effect of the quadrupole rods (perfect conductor) is included in these simulations. The bunch centroid has been shifted by 0.1 mm at the beginning to excite the coherent dipole motion. Without this intentional initial shift, the bunch stays on axis for a very long period. Macroparticles are removed when their transverse distances from the axis exceed the LPT aperture size, i.e., 5 mm in radius. In the upper case (a), all particles have been lost at $t\approx 0.9[\mathrm{ms}]$ due to a large horizontal dipole oscillation.

Figure 15

Transverse profiles of an ion bunch at (a) $t=0.5[\mathrm{ms}]$ and (b) $t=0.7[\mathrm{ms}]$. The simulation parameters are identical to those assumed in Fig. 14. The four hatched areas indicate the quadrupole rods.

Figure 16

Ion loss simulations by the Warp code. The simulation parameters are identical to those assumed in Fig. 14 except for the size of the initial bunch displacement. The bunch is only horizontally displaced by 0, 0.1, 0.2, and 0.3 mm at $t=0[\mathrm{ms}]$.