Landau damping of transverse head-tail instabilities with a pulsed electron lens in hadron synchrotrons

A pulsed electron lens produces a betatron tune shift along a hadron bunch as a function of the longitudinal coordinates, which is a longitudinal detuning. An example of transverse detuning is the tune shifts due to octupole magnets. This paper considers a pulsed electron lens as a measure to mitigate the head-tail instabilities. Using a detailed analytical description within a Vlasov formalism, the coherent properties of the longitudinal and transverse detuning are presented. The analytical predictions are compared with the results of the particle tracking simulations. A pulsed electron lens is demonstrated to be a source of tune spread with two components: a static one, leading to Landau damping and a dynamic one, leading to an effective impedance modification, which we demonstrate analytically and in our particle tracking simulations. The effective impedance modification can be important for beam stability due to devices causing longitudinal detuning, especially for nonzero head-tail modes. The Vlasov formalism is extended to include the combination of longitudinal and transverse detuning. As a possible application at the SIS100 heavy-ion synchrotron [Facility for Antiproton and Ion Research (FAIR) at GSI Darmstadt, Germany], a combination of a pulsed electron lens with octupole magnets is considered.

Figure 1

Comparison between the betatron phase factors $B(J_z,\phi )$ for a PEL, ${\xi }^{(2)}$, and an RFQ. The PEL and the RFQ are matched to the rms bunch length, the value of ${\xi }^{(2)}$ corresponds to the same rms tune spread, $J_z={ϵ}_z$. The case of a linear chromaticity ${\xi }^{(1)}$ is shown for reference.

Figure 2

Top: head-tail mode spectra $|H_l^p(z^{(1)},z^{(2)}){|}^2$ for an airbag bunch, unperturbed (solid lines), and with the effect of a PEL (dashed lines) for $\mathrm{\Delta }{Q}_{\max }/Q_{{\mathrm{s}}_0}=0.5$ (bottom). Dependency of mode power spectrum on the PEL strength $\mathrm{\Delta }{Q}_{\max }$ at points where modes $|l|=\{0,1\}$ are near their maxima.

Figure 3

Example from an instability simulation: the beam offset evolution over several synchrotron periods (blue). The exponential fit (orange) delivers the resulting growth rate.

Figure 4

Top: incoherent tune spreads and average tune shifts (green dot). Left to right: LO (yellow), a dc EL (red), a PEL (dark-blue), an RFQ (light-blue). Bottom: stability boundaries for the head-tail mode $l=0$ from Eq. (22) compared to the results of the simulation scans with the rigid mode kick, Eq. (23).

Figure 5

dc EL stability diagrams from Eq. (22) (red lines) for the head-tail mode $l=0$ depending on the electron to ion rms beam transverse size ratio $r={\sigma }_{e_{\perp }}/{\sigma }_{x_0,y_0}$. It is compared to the results of simulation scans (green lines) with the rigid mode kick [defined in Eq. (23)]. (a) ${\sigma }_{e_{\perp }}/{\sigma }_{x_0,y_0}=0.7$ (b) ${\sigma }_{e_{\perp }}/{\sigma }_{x_0,y_0}=1.0$ (c) ${\sigma }_{e_{\perp }}/{\sigma }_{x_0,y_0}=1.4$ (d) ${\sigma }_{e_{\perp }}/{\sigma }_{x_0,y_0}=1.8$.

Figure 6

Stability diagrams for the head-tail mode $l=0$ from Eq. (22) for LO (transverse detuning $\mathrm{\Delta }{Q}_y^{\perp }(J_x,J_y)$, light-yellow line), for a PEL (longitudinal detuning $⟨\mathrm{\Delta }{Q}_y^{\parallel }(J_z){⟩}_{\phi }$, light-blue line), and for the related combination (dark-blue line). The corresponding simulation results for the device combination are shown by the green curve.

Figure 7

Spectra and the trace plots of the centroid motion for head-tail instabilities for three head-tail modes $l=0$ (right curves), $l=-1$ (middle curves), and $l=-2$ (left curves). Those are the instabilities when both the longitudinal and the transverse detuning are zero.

Figure 8

Instability growth rate dependency on the strength of Landau damping $\mathrm{\Delta }{Q}_{\mathrm{rms}}/Q_{{\mathrm{s}}_0}$ for head-tail mode $l=0$. A PEL (dark-blue), LO (yellow), a dc EL (red), and an RFQ (light-blue) results are compared with respective dispersion relations Eq. (22) (dashed lines).

Figure 9

Instability growth rate dependency on the strength of Landau damping $\mathrm{\Delta }{Q}_{\mathrm{rms}}/Q_{{\mathrm{s}}_0}$ for head-tail mode $l=-1$. A PEL (dark-blue), LO (yellow), a dc EL (red), and an RFQ (light-blue) results are compared with respective dispersion relations Eq. (22) (dashed lines).

Figure 10

Instability growth rate dependency on the strength of Landau damping $\mathrm{\Delta }{Q}_{\mathrm{rms}}/Q_{{\mathrm{s}}_0}$ for head-tail mode $l=-2$. A PEL (dark-blue), LO (yellow), a dc EL (red), and an RFQ (light-blue) results are compared with respective dispersion relations Eq. (22) (dashed lines).

Figure 11

Stability boundaries for two PEL strengths for the mode $l=-1$ (solid and dashed lines, respectively) from Eqs. (21) and (22) and respective coherent tune shifts of the instability (stars) from Eq. (20). Accelerator and beam parameters are the same as in Fig. 9. Two Landau damping strengths $\mathrm{\Delta }{Q}_{\mathrm{rms}}/Q_{{\mathrm{s}}_0}=\{0.05,0.07\}$ are considered (magenta and cyan colors, respectively).

Figure 12

Landau damping for the head-tail mode $l=-1$ due to a combination of LO and a PEL: instability growth rate dependency on the strength of a PEL. The curves correspond to four different LO strengths, given in normalized units in the plot.