Abstract

1 INTRODUCTION
Strong coupling between the transverse planes of a particle beam leads to an "equipartition" of the oscillation energy, including the instability growth rates in the case of coherent instability. In the presence of a frequency spread, there can also be a partition of Landau damping for "optimum" coupling [1]. Linear coupling can therefore become very effective for stabilisation when a dissymmetry, e.g. in the frequency spreads or coupling impedances, is present. In Ref. [2], a formula for transverse coherent instabilities in the presence of linear coupling (near the coupling resonance  Q x - Q y - l ) was given in the form of a 4×4 determinant. In the case of mode coupling, this formula takes into account the coupling between two adjacent modes, m and m+1. The general two- dimensional dispersion relation [3], including all the head-tail modes, is given below. It is expressed in the form of a determinant of infinite matrices, which have to be truncated in practice. The results in the absence/presence of linear coupling, frequency spread and mode coupling have already been treated in Ref. [2]. In the present paper, the general formula is discussed in Section 2, and Section 3 is devoted to the application for the case of the PS beam for the future LHC [4].
2 THEORY
2.1  General 2D Dispersion Relation
In the presence of linear coupling (near the coupling resonance  Q x - Q x - l ), the stability of intense beams can be discussed using the following infinite determinant
Here, ∆ω ⋆ , γ are matrices whose elements are given by Eq. (2), I- 1 , are matrices whose (inverse) elements are given by Eqs. (9-10), and I  is the identity matrix.
The elements of the "dispersion matrices" I x,y entering into Eq. (1) are
Notice that I and I , are the horizontal and vertical dispersion integrals, ω  is the coherent frequency to be determined, ω,( ˆ x, ˆ y ) and ω,( ˆ x, ˆ y ) are the transverse incoherent betatron frequencies of the particles, f ( x ) and  f ｡ ( y ) are the uncorrelated distribution functions of the incoherent betatron amplitudes, ω  is the synchrotron frequency and m - ...,- 1,0,1,... is the head-tail mode number. Furthermore, ˆ κ ( i ) is the l th Fourier coefficient of the skew gradient κ 0 - ( e / ρ 0 )( ∂ B x / ∂ x ), with e  the elementary charge, p o the design momentum and B j the horizontal magnetic field, R is the average radius of the machine, Ω o is the average revolution frequency of the particles,  ω xo,,o - Q xo,,o   are the unperturbed betatron frequencies, ∆ω π, π are the complex betatron frequency shifts given by Sacherer's formula [5], j- √-1 is the imaginary unit, β and γ are the relativistic velocity and mass factors, I , - N , e Ω , / (2 π ) is the current in one bunch, m o is the proton rest mass, L - β c τ b  is the total bunch length (in metres), Z , , are the coupling impedances, ωξ ′′ -(k + Ω xo , o ) Ω o + mω , with −∞ ≤ k≤ +∞ ,    ω ,  = ( ξ x , / η )Ω , , , , Ω     are   the   transverse chromatic frequencies, with ξ x, -( d Q x, / d p)( p o / Q xo, o ) the chromaticities, and η − γ In the case of coasting beams, Eq. (1) still applies with m=0, the impedance removed from the summation and the bunch length L equal to 2πR . The dispersion integrals for a coasting beam are given by Eqs. (9-10) with m =0, adding also the contribution of the momentum spread. The incoherent betatron frequency spread around the harmonic k 2 v  corresponding to the frequency ω , of the driving impedance is given by | ω, + ωξ, | × | η | × Δρ / ρ o . Note that for head-tail instabilities below the mode- coupling threshold, the momentum spread is not effective for Landau damping since several synchrotron periods occur during the instability rise-time, and on average the spread is zero. In the case of mode coupling, the momentum spread helps to stabilise the beam, and it has thus to be taken into account in the dispersion integral.
2.2  Situation in the Absence of both Frequency
Spread and Mode Coupling
In the absence of linear coupling, the determinant of Eq. (1) is the product of the two one-dimensional determinants, each of which equals zero. Below the mode-coupling threshold, each mode m can be treated separately, and the following dispersion equations are obtained, I - 1 - A ω π , and I , 1 - A ω π , . In the absence of Landau damping, the stability condition for the m th mode is Im ( ∀ω π ) ≥ 0, where Im ( ) stands for imaginary part. In the presence of linear coupling, but without frequency the  mode-coupling and  below spread threshold, Eq. (1) leads to the following necessary condition for stability of the m th mode,
where V α = - Im ( △ω = γ ) are the transverse instability growth rates. If Eq. (11) is true, it is possible to stabilise this mode by increasing the skew gradient and/or by working closer to the coupling resonance Q x - Q y - l . The stabilising values of the modulus of the /th Fourier coefficient of the skew gradient are given by
where  Q x x = ( ω x 0 , γ 0 + U π x , γ ) / Ω o  are the horizontal and vertical coherent tunes in the presence of wake fields ( U n = Re ( ∀ω π , π ), where Re ( ) stands for real part), but in the absence of coupling. Furthermore, in the case coupled-bunch instabilities of M bunches, of k − n , , + k' M  in Eq. (3) with  − ∞ ≤ k'≤ + ∞ , and the coupled-bunch mode numbers are related by  n, -  n, - l . Notice that, when Eq. (11) is verified, it is verified for "any" intensity.
3  EXPERIMENTS
The plot of the transverse single-bunch head-tail instability growth rates as functions of the head-tail mode number is represented in Fig. 1(a) for the PS beam for LHC [6]. In fact the initial scheme has been modified [4]  and  slightly  different  parameters  are  now used to cope with instabilities. The theory, based on the above model (considering both resistive-wall and broad- band impedances), predicts horizontal single-bunch instabilities with the most critical head-tail mode number |m|=6. To test the validity of the one-dimensional theory, the skew quadrupole current was set to have the minimum of linear coupling between the transverse planes, which turned out to be I skew ≈ 0.33 A , due to the "natural" coupling present in the PS (see Fig. 1(b) where κ = ˆ κ ( ο ) ). In this situation, a head-tail instability develops, and in accordance with theory the mode | m |=6 is observed (see Fig. 2, showing 6 nodes). The beam losses due to this instability are shown in Fig. 3(a). One notes that the beam from the PSBooster (PSB) is injected into the PS in 2 batches of 3 bunches each, to overcome space charge effects in the PSB [4]. The drawback of this filling scheme is that the first batch has to wait 1.2 s at 1.4 GeV kinetic energy, and during that time about 2/3 of the beam is lost due to the above instability, if no counter measures are taken. From Fig. 1(a), one can deduce that Eq. (11) is verified for each mode, and that it will therefore be possible to stabilise the beam by linear coupling. The fact that the mode | m |=6 is unstable in the horizontal plane and stable in the vertical one, is due to the different natural chromaticities ( ξ x - - 0.9  and ξ x --1.3 ). In the presence of coupling, the transfer of instability growth rate is essentially here a chromaticity sharing [7]. Setting a skew quadrupole current of about –0.4 A, with the working point ( Qx - 6.22 ,  Qy - 6.25 ), the instability is damped without any loss (see Fig. 3(b)). Furthermore, this method is reproducible and no emittance blow-up
4 CONCLUSION
A general formula for the transverse coherent instabilities in the presence of linear coupling has been given. The beneficial effect due to the transfer of the head-tail instability growth rates, which critically depend on chromaticities, has then been emphasised. This result has been verified experimentally on the PS beam for the future LHC, where a head-tail instability |m|=6 is damped using linear coupling, in agreement with theory. This method can be profitably used in the design of new machines, to find optimum values for the transverse tunes, the skew quadrupole and octupole currents, and the chromaticities.
REFERENCES
[1] E. Métral, "Theory of Coupled Landau Damping", Part. Accelerators, 62(3-4), p. 259, 1999. [2] R. Cappi, E. Métral, D. Möhl, "Transverse Coherent Instabilities in the Presence of Linear Coupling", Proc. 7° EPAC Conf., Vienna, Austria, June 2000. [3] E. Métral, Thesis, Grenoble (France), 1999. M. Benedikt et al., "Performance of the LHC Pre- [4] Injectors", this Conference. [5] F.J. Sacherer, Proc. 9 h Int. Conf. on High Energy Acc., Stanford 1974, p. 347. R. Cappi, R. Garoby and E. Métral, "Collective Effects in [6] the CERN-PS Beam for LHC", CERN/PS 99-049 (CA), 1999. R. Talman, "Theory of Head-Tail Chromaticity Sharing", [7]