Transverse Instabilities of Coasting Beams with Space Charge

Transverse beam stability is strongly affected by the beam space charge. Usually it is analyzed with the rigid-beam model. However this model is only valid when a bare (not affected by the space charge) tune spread is small compared to the space charge tune shift. This condition specifies a relatively small area of parameters which, however, is the most interesting for practical applications. The Landau damping rate and the beam Schottky spectra are computed assuming that validity condition is satisfied. The results are applied to a round Gaussian beam. The stability thresholds are described by simple fits for the cases of chromatic and octupole tune spreads.


INTRODUCTION
Particle interaction via the walls of the vacuum chamber is conventionally described by the wake functions and impedances. In absence of damping, this interaction leads to beam coherent instabilities. However, if there are particles in resonance with coherent motion, they effectively exchange their incoherent energy with the energy of coherent oscillations. If the phase space density of the resonant particles is sufficiently large, the instability is stabilized by this mechanism, called the Landau damping. Contrary to the wake fields, Coulomb interaction does not drive the instability by itself, since it preserves the total energy and momentum. However, the collective Coulomb field can strongly affect the beam stability because it separates coherent and incoherent frequencies. Indeed, when the beam oscillates as a whole, its collective motion does not see the space charge, while an individual particle does. Thus, if the coherent and incoherent frequencies are separated, there are no resonant particles, and no Landau damping.
To analyze the beam stability with space charge, an effective method was suggested by D. Möhl Here i x is the offset of i-th particle, are its revolution frequency, the tune and the direct space charge tune shift, 0 0 , Q Ω are the average revolution frequency and tune, x is the offset of beam center and c Q ∆ is the impedance-driven coherent tune shift. Although perturbation of a particle motion depends on its amplitudes, this equation assumes that the beam oscillates as a rigid body when the coherent beam fields are computed. Consequently, the beam coherent motion is completely described by the dipole offset x . This assumption is correct if all lattice frequencies identical. In this case, all particles respond identically to the coherent field, x x i = δ , consequently, the beam oscillates as a rigid body, and the spread of the space charge tune shifts does not matter. However, a spread of the lattice frequencies generally makes the rigid-body model of Eq. (1) incorrect. Indeed, an individual response to the coherent field is determined by the separation of the individual lattice frequency from the coherent frequency, which varies from particle to particle. Since individual responses are not identical, the beam shape is not preserved in the dipole oscillations, so the rigid-body model of Eq. (1) is not self-consistent and generally cannot be justified.
In 2001, M. Blaskiewicz showed a way to analyze the problem, avoiding the rigid-beam assumption [2]. Within a one-dimensional model, he developed an integral equation on the phase space density perturbation. He found two cases when his equation gives the same result as the rigid-beam approach. The first case was the Lorentz momentum distribution, and the second one was the water-bag distribution over the transverse actions. With some additional model simplifications, he plotted several stability diagrams for distributions close to Gaussian. The same problem of self-consistent beam stability analysis was recently examined by D. Pestrikov [3]. Considering a two-dimensional model, he came to a general integral equation and found it "too complicated even for a numerical solving." To proceed, he considered a singledimensional problem, came to the same integral equation as M. Blaskiewicz, and reproduced his Lorentz and waterbag results. For a Gaussian distribution, he plotted additional stability diagrams, and found no anti-damping, found earlier in his rigid-beam model studies [4]. Indeed, Landau anti-damping cannot exist at all if the distribution is close to Gaussian: this is a mere consequence of the second law of thermodynamics. A Hamiltonian system in thermal equilibrium is always stable. Appearance of Landau anti-damping in the rigid-beam model is a striking example of how wrong the results of this model can be. Rigid-beam stability diagrams were presented in several ___________________________________________ papers [4][5][6]; however, the range of their applicability was not clarified.

MODEL JUSTIFICATION
As mentioned above, the rigid-beam model is correct if all the lattice frequencies are identical, This case is simple, but not so interesting, since there is no Landau damping, and any impedance with non-zero real part makes the beam unstable. Now let us assume that the lattice frequency spread is sufficiently small so that the rigid-beam model would still be a good approximation. That requires the rms spread of the lattice frequencies ) ( i i Q Ω σ to be small compared with the separation frequency, which is a difference of the coherent frequency from an average incoherent one: a relatively small frequency spread is sufficient to stabilize the beam. Near the threshold, the small frequency spread is not significant for a bulk of the beam, which oscillates almost the same way as for zero tune spread. The tiny amount of the resonant particles has almost no influence on the coherent motion, except a slow transfer of the coherent energy into incoherent one, and thus, a slow collective mode damping. In other words, when Eq. (3) is satisfied, the rigid-beam model is applicable for calculation of Landau damping required for beam stabilization. This energy-based calculation of Landau damping leads to the same result as a formal solution of the dispersion equation [8]. In this paper, we limit ourselves to a case of thin tail, or small frequency spread approximation of Eq. (2), where the rigid-beam model is applicable. This allows us to calculate the Landau damping and the threshold parameters of the beam for a relatively small growth rate (3). Our primary interest is the threshold calculation. This is additionally simplified due to exponentially small phase space density of resonant particles, and consequently the Landau damping. When the damping rate is a steep function of the dimensionless frequency separation 1 ) , the threshold condition mostly determines this big ratio, being only slightly dependent on the coherent growth rate.
In practice, the far tails of the distributions are not wellmeasured or well-reproducible, so that even exact formulas cannot produce reliable results for the instability growth rates. On the contrary, the stability threshold for the beam intensity, being an inverse function of the Landau damping rate, depends much weaker on specific behavior of the distribution tails, and therefore can be predicted much better. Note also that the condition of small growth rate of Eq. (3) is typically well-satisfied for low and medium energy hadron machines, mainly addressed by this paper.

DISPERSION EQUATION
After validity limits of the rigid-beam model are specified, a solution of Eq. (1) can be considered in more details.
Here, all the notations are rather conventional: x J and y J are the transverse actions; In this paper we will only consider the first two terms contributing to the total lattice-related tune shift: the chromatic contribution and the contribution due to octupole non-linearity so that: with the wall. This interaction produces both the dipole and quadrupole forces, or, in other words, driving and detuning wakes [9]. Thus, the entire force acting on i-th particle can be expressed as

LANDAU DAMPING
When the condition (2) Note that the sign of the damping rate Λ is always determined by the sign of the derivative of the distribution function to the real part of the eigenvalue play a role when the distribution function drops exponentially; in this case even a small correction to the tune of the resonant particles significantly affects the damping rate Λ .
Let us first assume that the tune spread is purely chromatic. In this case the first-order correction is equal to zero, 0 Note that this result is e times smaller than a simpleminded formula neglecting the second-order term . Another possibility to stabilize the beam is an introduction of octupole non-linearity. Contrary to the chromatic spread, the first-order correction to the eigenvalue is non-zero here. For the Gaussian transverse distribution, including this first-order correction reduces the rate Λ by a constant factor ~2-3, similar to the role of the second-order term for the chromatic tune spread. For the octupole tune spread the second-order term makes only a small correction to the damping rate and can be neglected.
As was pointed out above, the rigid-beam model is valid only if the frequency spread is small compared with the separation frequency, 1 / within the rigid-beam approximation assumes that the inaccuracy of the model is smaller than these corrections. The correctness of this assumption is a subject of separate study. Presently, we can only refer to a specific example of chromatic tune spread for a Gaussian beam, considered in Ref. [3] within a framework of one-dimensional self-consistent model, compared with the rigid-beam result. As it is clearly seen from a presented stability diagram, the discrepancy between the two results is rather small, ~ 10-20% in the area of rigid-beam model validity. This suggests that accounting the eigenvalue corrections ) 2 ( ) 1 ( , Q Q δ δ is within the model accuracy, and thus it is legitimate. Finally, it should be noted that although the corrections ) 2 ( ) 1 ( , Q Q δ δ change the damping rate Λ by 2-3 times, their influence on the threshold space charge over the tune spread value is relatively small, since the Landau damping exponentially depends on beam parameters (like in Eq. (6)), and an error in the pre-exponential factor (~2-3) only slightly modifies the threshold.

THRESHOLD LINES
As it was stated above, rigid-beam stability diagrams are mostly invalid if the space charge is present. A small correct part of them lies typically so close to zero that it is hard to resolve details on the pictures usually presented in the literature (Ref. [4][5][6]). Therefore we do not draw these diagrams here and present the stability threshold in a different way. Indeed, Eq. (6) shows that the stability condition depends on two dimensionless parameters. The first parameter determines to what extent the coherent and incoherent frequencies are separated; obviously, it is defined by the ratio of the separation frequency over the lattice frequency spread. The second parameter shows how strong is the instability to be suppressed by the Landau damping; it can be described by the coherent growth rate c Q ∆ Ω Im 0 in units of the separation frequency. A dependence of the threshold dimensionless separation over dimensionless coherent growth can be called the threshold line. In this section we present it for round Gaussian beams. The problem is solved both for a pure chromatic tune spread, and for an axially-symmetric octupole-induced spread, The results are presented in Figures 1, 2. Here we additionally assume that . We do not consider negative sign of the octupoles, since they would detune even more incoherent frequencies from the coherent line, making the beam more unstable.
The dots ar e numer ical r esults, and the line is a fit with the for mula highlighted in yellow.
Above we used a following presentation for the space charge tune shift of a round Gaussian beam as a function of the transverse actions J x ,J y [10]: is the maximal space charge tune shift with λ as the linear density, C as the orbit circumference, r p as the classical radius of the beam particles, ε as the unnormalized rms emittance, and β, γ as relativistic factors. For numerical calculations, we approximated the exact result (7) by the following fit: which is accurate within a few percent for 6 , ≤ y x a a ; it has the right Tailor expansion at small amplitudes and the right asymptotic behavior at large amplitudes.
Figur e 2: Thr eshold line for the octupole tune spr ead.
Notations ar e similar to Fig.1.
As it is seen from the plots, the suggested fits for the threshold lines for the chromatic spread, and (10) for the symmetric octupole spread are accurate within 10% or better.
Note that the stabilizing rms tune spread is 3-4 times smaller for the octupoles than for the chromatic case. The reason is that the octupole-driven tune shift goes quadratically with amplitudes, while the chromatic tune shift is a linear function of the momentum offset.
It is instructive to compare thresholds for dimensionless growth rates the Gaussian threshold growth rate is almost 3 orders of magnitude higher than that for the KV beam. The reason for this advantage of the Gaussian beam is that its resonant particles may have not so high momentum offset in expense of higher transverse amplitudes, where the space charge tune shift goes down. For the KV beam this is impossible, and that is why the KV threshold growth rate is always lower.

SCHOTTKY NOISE
Particle interaction affects the spectrum of beam Schottky noise. In the application to the beam with significant space charge, this problem was solved in Ref. [4] in a framework of the rigid-beam model. Comparison of this analytic solution with a particle tracking code and with real beam measurements was considered in Ref. [7]. In this section we apply the results obtained above to the problem of beam Schottky noise.
In the rigid-beam approximation, the spectral power of the transverse Schottky noise ) ( 2 ν x is [4]: where N x , β are the beta-function at the pickup and the number of particles. Note that this result assumes the validity of the rigid-beam model, As above, it is true in vicinity of the coherent peak due to the high value of the tune separation. Since the model is not generally correct at the incoherent frequency range, , the above result is not justified there. However, the noise power (11) reaches its maximum at the coherent peak, where the model is valid, and, consequently, Eq. (11) can be used. Expansion of the denominator at ν . When the beam is being cooled, its total Schottky noise (13) almost does not change, being equal to its zero-impedance limit, until the very threshold of the instability, where it immediately jumps to infinity. That is why measuring the Schottky noise can hardly help to see a real part of impedance responsible for the coherent rate c Q ∆ Im : the rate is either invisible or fatal.

SUMMARY
The applicability of the rigid-beam model is considered for the case when the space charge plays significant role in beam dynamics. The results prove that the stability diagrams obtained with this model are not valid for most of the complex plane of the coherent shift. However, the small area of its validity typically covers the entire area of practical interest. Based on the rigid-beam model, rather simple formulas for the Landau damping were calculated. These formulas are used for computation of the threshold space charge tune shift versus coherent growth time. Convenient analytical fits for the threshold lines are presented for round Gaussian beams.
The results obtained here do not undermine results of Refs [4][5][6], based on the analysis of the dispersion equation at the rigid-beam approximation, as soon as this approximation is valid. What is new in this paper is, first, a delimitation of validity area for the rigid-beam approximation, and, second, solutions of the dispersion equation are presented in more details, important for practical analysis.
At the end, the results are applied for the Schottky noise. For strong space charge case studied here, ν σ ν >> ∆ − | | sc Q , the Schottky spectrum is dominated by narrow resonant peaks at coherent frequencies.