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Space-Charge Instabilities in Linacs

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What limits the beam current in ion linacs? This is a very important topics for anyone designing linacs. For example, it is important for beam dynamics design of RFQs. In general the beam current is limited by the focusing provided to confine the beam to within the aperture, balancing the defocusing effects of space charge and emittance. There are formulas that have been derived for this. In addition, there are some instabilities that must be avoided. That is what we will discuss.

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What instabilities are important without space charge? The equation of transverse motion for periodic focusing arrays with linear quadrupoles or solenoids is known as Hills equation. x+K(s)x=0, where K(s)=K(s+L) is periodic with period L. Hills equation gives stable motion when phase advance per focusing period is 0 <180 deg. This is an instability that affects strongly-focused beams periodic focusing arrays. The other main instability without space charge is the parametric resonance when k =2k T. This affects nonrelativistic beams. Usually these two instabilities can easily be avoided. But avoiding instability for beams with space charge imposes additional restrictions.

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The most important instability with space charge is the transverse- envelope instability This beam instability is a parametric resonance driven by the periodic-focusing lattice and mismatch oscillations of the beam. Free energy is available from the beam mismatch. See Martin Reiser, Theory and Design of Charged Particle Beams, Wiley VCH

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Transverse collective modes in RF linacs Collective modes are charge-density oscillations in the beam. They are excited by beam mismatches and by the quadrupole or solenoid periodic-focusing lens array. Pure transverse modes are described by their azimuthal symmetry, such as monopole (1 st order) or breathing mode, quadrupole (2 nd order), sextupole (3 rd order), octupole(4 th order), … These modes can be either stable or unstable, and if unstable can generate significant emittance growth or beam losses.

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Characterizing beams with space charge in linear periodic focusing arrays The importance of space charge in the beam is measured by the phase advances per focusing period and 0. These parameters are the same for every particle in the beam = the transverse phase advance per focusing period in degrees of the equivalent uniform beam* including space charge. depends on the net effect of both focusing and space charge. 0 = phase advance per focusing period of the beam without space charge. 0 depends on only the focusing. *The equivalent uniform beam is a uniform density beam with the same current and same rms properties as the real beam (which generally is not uniform).

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Another useful quantity is the tune- depression ratio 0 Range of tune-depression ratio is / 0 > 0 / 0 =1 corresponds to no space charge, only focusing affects the beam. / 0 =0 corresponds to extreme space-charge limit where space charge is large enough to exactly cancel the focusing force.

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Most important instability for beams with space charge is the envelope instability of the quadrupole mode. The stability criterion for the envelope instability depends on both 0, and Instability occurs when 0 >90 deg and <90 deg. This is more restrictive than the Hills equation instability. Simulations show that the envelope instability generates rapid and significant emittance growth and beam loss and must be avoided.

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Must avoid >90 deg for high current beam design When 0 >90 deg and <90 deg, the envelope instability gives an unstable beam. It blows up in just a couple transverse oscillation periods. The envelope instability has been confirmed experimentally at LBNL and Maryland. The general practice for conservative design is to limit the external focusing strength so that 0 <90 deg.

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Sextupole instability The sextupole instability can be excited when 0 >60 deg and <60 deg. This is a weak instability. Usually no observed emittance growth in simulations. Generally there is no design requirement from the sextupole instability to limit 0. However, at PAC09 the SNS people reported large emittance growth from the sextupole instability, attributed to their use of quadrupole lenses with dipole corrector magnets that gave a very large dodecapole component (factor of 3 higher than expected). This gave halo and beam loss in SNS a factor of 20 higher than expected. Can be corrected by reducing 0 from 60 deg to 50 deg. (See Y. Zhang, C.K.Allen, J.D.Galambos, J.Holmes, J.G.Wang, Beam Transverse Issues at the SNS Linac, PAC2009, Vancouver.)

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Conclusions about transverse envelope instabilities The quadrupole or envelope instability can be avoided by keeping 0 <90 deg. The sextupole instability can be avoided by using high quality quadrupoles especially with small dodecapole component. Sextupole is also avoided by keeping 0 <60 deg, but that restricts the focusing strength. All other transverse modes are too weak to be of concern.

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Another type of collective instability: Anisotropy instability leads to emittance transfer between transverse and longitudinal degrees of freedom Caused by space charge, and transverse-longitudinal coupling. Free energy is available when there are different temperatures for different degrees of freedom.

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Space-charge coupling Instabilities for anisotropic beams Anisotropic beams are important for linacs because the transverse and longitudinal beam parameters are usually different. Thus, beam bunches are anisotropic. Even without space charge recall that the transverse and longitudinal motions are coupled through nonlinear effects., i.e. k =2k T. a) dependence of transverse RF defocusing on the longitudinal phase in the transverse equation. b) dependence of transit-time factor on radial displacement of beam in longitudinal equation (through the I 0 Bessel function).

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Gluckstern (1966) showed that there is a parametric instability even with no space charge when k =2k t (or =2 t ) The wave number k means phase advance per unit length, and =kL where L is the period is the phase advance per focusing period. This coupling resonance is important when there is strong longitudinal focusing. You have to avoid this parametric resonance.

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Physics with space charge Note that binary collisions of the particles play no significant role in the physics. The space-charge force, a smoothed or average effect over all the particles in the bunch, is what matters. When space charge is important, the physics is controlled by collective anisotropy resonances, which causes emittance transfer between planes.* * I. Hofmann et al., Space charge resonances in two and three dimensional anisotropic beams, Phys. Rev. ST-AB, 6, (2003)

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Stability Plots of I. Hofmann, et al. The anisotropy resonancies are observed in a plot of the space-charge tune-depression ratio x / 0x (ordinate), a measure of the importance of space charge, versus the longitudinal to transverse tune ratio (abcissa) z / x. The tune ratio z / x allows identification of the anisotropy resonances. Resonances can occur when tune ratio is a ratio of integers.

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Stability plots In Hofmanns publications the symbol is sometimes replaced by the symbol k. Both represent the wave number or phase advance per unit length. The stability plots are shown for constant values of the emittance ratio z / x, which is interpreted as the emittance ratio of the initial beam if the emittances change.

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Stability plots (continued) The most prominent stop bands are located near tune ratios k z /k x = z / x =1/3, 1/2, 1, and 2, not all of which are always present. Equations of motion: Coupled equations of motion for x (transverse) and z (longitudinal) showing k x and k z phase advance per unit length in x and z. The functions f and g include space charge.

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Stability plots (cont.) The stability plots show contours of constant calculated growth rates. The contours identify calculated stop bands (regions of exponential growth) that lie near tune ratios with integer tunes.

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This shows stability plot for collective anisotropy resonances with analytically-calculated stop bands in plot of transverse tune depression versus tune ratio for z / x =2. Dashed line corresponds to equipartitioning where the ½ resonance is suppressed and no emittance transfer occurs. Resonances in this case are near 1/3, 1 and 2.

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Energy equipartitioning and the energy anisotropy parameter Define energy-anisotropy parameter T a as ratio of average kinetic energies in any two degrees of freedom x and z. T a =1 gives equipartitioning. For emittance transfer between x and z requires unequal temperatures in x and z. Energy equipartitioning means equal temperatures in all three degrees of freedom, and therefore no emittance transfer is possible Equipartitioning is introduced, not because beams necessarily evolve to equipartitioned states, which they do not, but because equipartitioned beams (T a = 1) have no free energy for emittance transfer.

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Stability plot for z / x =2 showing CERN SPL design trajectory points. Significant emittance transfer in simulation when trajectory overlaps the k z /k x =1 stop band. SPL short: Trajectory stays out of k z /k x =1 stop band. The simulation shows no emittance transfer Case 2: Trajectory overlaps with k z /k x =1 stop band. Simulation shows significant emittance transfer. (Longitudinal decreases from 0.75 to 0.56 mm-mrad. Transverse emittances increase from 0.4 to 0.5 mm-mrad.) SPL full: Very little overlap and very little emittance transfer as expected.

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Stability plot for z / x =1.4 showing the SNS linac trajectory (above) and for z / x =1.3 for the proposed ESS linac (below). SNS trajectory (above) passes through stopbands near 1/3, 1/2, and 1. Simulation shows emittance growth only for k z /k x =1. Transverse emittance growth is +27%. Longitudinal growth is +3%. Other sources of nonlinearity affect longitudinal. ESS (below) has no emittance change in simulation..

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How to minimize emittance transfer Simulations show that RMS emittance transfer is insignificant in nonequipartioned beams if the k z /k x =1 stop band is avoided. The other stop bands are too weak to matter. However, if the k z /k x =1 stopband cannot be avoided, an emittance ratio near unity (approximate equipartitioning ) would limit free energy for emittance transfer. In other words, significant emittance transfer requires overlap with k z /k x =1 stopband and an emittance ratio not near unity.

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As space charge becomes stronger As space-charge tune depression becomes stronger (smaller tune-depression ratios and stronger space charge) the stop-band widths increase and overlap. The thermodynamic picture that anisotropic beams approach energy equipartitioning applies only close to the space-charge limit where stop bands completely overlap. Then emittance transfer occurs at all tune ratios.

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A equipartitioning argument for many years has been resolved. Some argued that a nonequipartitioned beam would always equipartition at the expense of unwanted emittance transfer. They argued that the beam had to be equipartitioned to prevent emittance transfer. But, Hofmann et al. studies showed that equipartitioning is not a necessary condition to prevent emittance transfer. You can prevent emittance transfer if you can simply avoid the k z /k x =1 resonance, or minimize the time the beam is on that resonance.

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Operating near the space-charge limit without emittance growth is very difficult Near the space-charge limit, other emittance growth mechanisms with free energy from beam mismatch or nonlinear field energy will dominate, and equipartitioning will not help. Thus, maintaining a bright beam at tune depression near the space charge limit remains a significant challenge.

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Summary of space-charge instabilities in linacs To avoid the envelope instability require 0 <90 deg. To avoid the sextupole instability, minimize the dodecapole component of the quadrupole magnets, or require 0 <60 deg. To prevent emittance transfer avoid k z /k x =1 stopband or that is not possible, limit free energy for emittance transfer by approximate equipartitioning. The simple thermodynamic picture that anisotropic beams always approach energy equipartitioning applies only close to the space-charge limit where stop bands completely overlap. No one should operate there anyway!

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