CLOSED ORBIT INSTABILITY*

V. Danilov, S. Henderson, J.A. Holmes, ORNL, Oak Ridge A. Burov, FNAL, Batavia Abstract Due to a long-range resistive wake, the closed orbit in high-intensity circular accelerators may experience an unstable drift. Unlike the conventional betatron instabilities, this closed orbit instability is not sensitive to the spread of the betatron frequencies. For bunched beams, feedback appears to be the only way to stabilize the closed orbit above threshold. This new instability can be significant for long, high intensity rings.


BASIC PHYSICS
To begin, consider the simple model of a single macroparticle inside a homogeneous chamber.Taking into account the accumulation of the wake field, the transverse equation of motion can be written (see [1]): where N is the number of particles, r 0 is the particle's classical radius, W(z) is the wake function,  .Let us suppose for simplicity that the only singularity is a simple pole at In this case, the dispersion equation (3) yields where b ν is the betatron tune.There are several interesting features of this solution.First of all, it describes a term in an infinite sequence of eigenfrequencies, indexed by the arbitrary integer n.In fact, all these modes describe the multi-turn dynamics of the wake fields, and the number n is associated with the n-th spatial harmonic of the wake field expansion over the ring azimuth . When the number of particles is small enough, all these modes decay as the field decrement λ , some of them being slightly faster, and some slightly slower.The value λ i − can be considered as a single-particle "frequency" of the drift modes, while the intensity-dependent part in the right hand side of Eq. ( 5) gives their coherent "tune shifts".When the beam intensity increases, the decrement of the most long-lived mode approaches zero, and at a certain current this mode becomes unstable.This threshold intensity is calculated from Eq. ( 5).To determine the threshold, assume, as is usual, that the wake amplitude 0 where Note that the threshold is determined by the amplitude of the integral wake reflects the specific nature of this instability that it is caused by the long-term accumulation of the wake fields, leading to a drift of the closed orbit.Hence, it follows that this closed orbit instability is determined by the total number of particles in the ring, N, and is insensitive to their bunching, so long as . In other words, the same dispersion equation ( 5) and threshold condition (6) describe single-bunch, multi-bunch, and coasting beams.
One further important point is the independence of these drift modes on the spread of the betatron frequencies.For bunched beams, neither the spread of the revolution frequencies nor synchrotron motion matter.For a coasting beam, however, the spread of the revolution frequencies where the wake decrement was assumed to be small enough, . The betatron tune shift (7) is almost exactly opposite to the coherent tune shift of the closed orbit mode of the same number, given by Eq. ( 5).This means that the stability conditions for these two types of modes are opposite: where the betatron mode is most stable, the drift mode is most unstable and vice versa.Note that this equality of the coherent parts of the betatron and drift modes is specific to the exponential wake function, introduced above as a simple model.In fact, however, this exponential function is more than just a simple model: it describes a resistive wall wake in the long-range limit, as we discuss below.

DRIFT EQUATION IN GENERAL CASE
In this section, we derive an equation for drift modes with arbitrary impedance and lattice functions.As was pointed out above, the behaviour of the drift modes does not depend on the beam bunching.Here, the calculations are carried out for a coasting beam.Then, for the specific case of a homogeneous wake function and smooth focusing, the result is shown to be identical to the macroparticle beam of the previous section, Eq. (3).
To begin, consider the case of a long-range wake function generated by a localized element at a specific azimuth of the storage ring.This is opposite to the previous section, where the wake function was assumed to be homogeneous around the orbit.At the location s′ , where the wake field is located, each particle is kicked by an angle Note that for the drift modes, the dispersion relation depends on both the impedance and the local β-function, as it does for the conventional betatron modes.In distinction from the homogeneously distributed impedance of Eq. ( 3), this localized impedance gives only a finite number of solutions, namely one in the case of the exponential wake treated above.The reason for this is that, for the localized wake source, the fields have a finite number of degrees of freedom.
It is now simple to write an equation for an arbitrary impedance distribution around the ring.Because of linearity, all the closed orbit perturbations can be directly summed.Thus, if ) , ( ~Ω ′ ⊥ s Z is defined to be a linear density of impedance at the azimuth s′ , so that the impedance of the length , then the equation for the periodic eigenfunctions ) (s y and eigenvalues Ω is: It is straightforward to show that Eq. (12) reduces to Eq. (3) in the homogeneous case of the previous section with

IMPEDANCE OF RESISTIVE WALL
Conventionally, the resistive wall impedance is calculated assuming that the skin depth is much smaller than the wall thickness d.This assumption can be incorrect for slow drift modes, and Proceedings of EPAC 2002, Paris, France especially for a thin metallic coating, where the skin depth can be larger than the metal thickness.For infinite vacuum outside the metal chamber, the impedance per unit length is found to be (see Ref. [3,5]) where the material factor by definition, it follows that the impedance can be taken out of the integral equation ( 12) for threshold calculations.In the smooth approximation, this gives the following formula for the threshold intensity (see Eq. ( 6)): Note that this result is independent of the wall conductivity.The reason is that at threshold, only the zero-frequency impedance matters, which describes static image charges, independent of the conductivity, and the image currents of the surrounding magnet enter through the parameter 1 ≅ g .

PRACTICAL EXAMPLES
In this section, the closed orbit instability is examined for several high-intensity rings.
For the LER of the PEP-II B-factory with  [5]), the smooth-approximation threshold is found to be an order of magnitude higher than the design intensity: For the present design values of the Very Large Hadron Collider [6] with the circumference 233 km, the betatron tune 214, the fractional betatron tune 0.5 (taken as an example), the vacuum chamber radius 0.9 cm, and the relativistic factor γ=1000 at injection, the threshold occurs at 4.9*10 14 protons, i.e. two times less than the design number of protons.
In reality, the threshold could be smaller than the smooth approximation result of Eq. ( 15), depending on beta function beating and the wake localization, see Eq. (12); this is considered in detail in Ref. [4].

CONCLUSION
The drift instability of the closed orbit described here can be dangerous for high-intensity rings.The instability rate is highest when the working point is chosen slightly above the integer resonance, where the conventional betatron modes for the resistive wall impedance are most stable.There are no chromaticity effects or Landau damping for the drift modes, because they are not coupled to the betatron oscillations.High intensity machines can be examined theoretically for the closed orbit instability by solving the dispersion equation derived here for arbitrary lattice functions and impedance distribution.
circumference, ω b is the betatron frequency, and the remaining constants correspond to the standard physical notation.With

..*
Far more important for the present discussion is the fact that Eq. (2) has additional solutions that can be significant for long-range wakes, by which we Let us assume that Eq. (2) has a solution in the vicinity of a multiple of the revolution frequency, sum on the right hand side of Eq. (2), and the remaining slow variation can be approximated by an integral, Research sponsored by UT-Batelle, LLC, under contract no.DE-where the slow variation Ω is dropped from the left hand side this case, the least stable mode has integer part of the betatron tune ] to compare the closed orbit and the betatron mode increments.For a coasting beam, the coherent betatron tune shift b Ω of the n-th azimuthal mode driven by the same exponential wake (4) is: outside.If the chamber can be considered thin (d > δ), the impedance is single-pole, which corresponds to the exponential wake function considered as a model in section II, tilde means that the value is taken per unit length.Note that in this case the amplitude of the integral wake λ / 0 W does not depend on the chamber conductivity or thickness.Taking into account that at the threshold of instability