Transverse mode coupling instability threshold with space charge and diﬀerent wake ﬁelds

Transverse mode coupling instability of a single bunch with space charge (SC) and a wake ﬁeld is considered within the frameworks of the boxcar model. Eigenfunctions of the bunch without wake are used as a basis for the solution of the equations with the wake ﬁeld included. A dispersion equation for constant wake is presented in the form of an inﬁnite continued fraction and also as the recursive relation with an arbitrary number of the basis functions. Realistic wake ﬁelds are considered as well including resistive wall, square, and oscillating wakes. It is shown that the TMCI threshold of the negative wake grows up in absolute value when the SC tune shift increases. Threshold of positive wake goes down at the increasing SC tune shift. The explanation is developed by an analysis of the bunch spectrum.

Transverse mode coupling instability of a single bunch with space charge (SC) and a wake field is considered within the frameworks of the boxcar model. Eigenfunctions of the bunch without wake are used as a basis for the solution of the equations with the wake field included. A dispersion equation for constant wake is presented in the form of an infinite continued fraction and also as the recursive relation with an arbitrary number of the basis functions. Realistic wake fields are considered as well including resistive wall, square, and oscillating wakes. It is shown that the TMCI threshold of the negative wake grows up in absolute value when the SC tune shift increases. Threshold of positive wake goes down at the increasing SC tune shift. The explanation is developed by an analysis of the bunch spectrum.

I. INTRODUCTION
Transverse mode coupling instability (TMCI) has been observed first in PETRA and explained by Kohaupt on the base of the two-particle model [1]. A lot of papers on this subject have been published later, including handbooks and surveys (see e.g. [2]). It is established that the instability occurs as a result of a coalescence of the neighboring head-tail tunes caused by the bunch wake field.
TMCI with space charge has been considered first by Blaskiewicz [3]. The main point of this paper is that the SC pushes up the TMCI threshold that is improves the beam stability. However, a non-monotonic dependence of the TMCI threshold and rate on the SC tune shift has been sometimes demonstrated in the paper. It followed from several examples that the stability and instability areas can change each other when the SC tune shift increases. The results have been confirmed later by the same author with help of numerical simulation of the instability at modest magnitude of the SC tune shift [4] So-called three-mode model has been developed in Ref. [7] for analytical description of the TMCI with space charge, chromaticity, and arbitrary wake. This simple model confirms that the TMCI threshold of negative wakes goes up in modulus when the SC tune shift increases. However, only the case of modest SC has been investigated in [7] though the proposed equations allow to suggest that a sudden kink of the threshold curve is possible at the higher shift. Therefore, field of application of the three-mode model is still an open question.
The case of very high space charge has been considered in Ref. [5,6]. It was confirmed in both papers that the space charge heightens the TMCI threshold until the * Electronic address: balbekov@fnal.gov ratio of the SC tunes shift to synchrotron tune reaches the border in several tens or a hundred units. However, the authors have expressed the different opinions about further behavior of the threshold. As it follows from [6], the threshold growth should continue at higher SC as well. On the contrary, it was suggested in Ref. [5] that the threshold growth can cease and turn back over the mention border.
The last statement has been supported in my recent preprint [8]. I have used the known eigenfunctions of the boxcar bunch [9] to get a convenient basis for investigation of the TMCI problem in depth. However, disclosure of some errors at numerical solutions of obtained equations forces me to revise the conclusions. The equations are recomputed in presented paper at any value of the SC tune shift and different wakes including the resistive wall, square, the oscillating ones. The increase of the TMCI threshold by the SC is observed in all the cases.

II. BASIC EQUATIONS AND ASSUMPTIONS
The terms, basic symbols and equations of Ref. [7] are used in this paper. In particular, linear synchrotron oscillations are considered here being characterized by amplitude A and phase φ, or by corresponding Cartesian coordinates: Thus θ is the azimuthal deviation of a particle from the bunch center in the rest frame, and variable u is proportional to the momentum deviation about the bunch central momentum (the proportionality coefficient plays no part in the paper). A coherent transverse displacement of the particles in some point of the longitudinal phase space will be presented as the real part of the function where Ω 0 is the revolution frequency, Q 0 is the central betatron tune, and ν is the tune addition produced by space charge and wake field. Generally, ζ is the normalized chromaticity; however, only the case ζ = 0 will be investigated in this paper. Then the function Y satisfies the equation [6,7]: where F (θ, u) and ρ(θ) are the normalized distribution function and corresponding linear density of the bunch, Q s is the synchrotron tune, ∆Q(θ) ∝ ρ(θ) is the space charge tune shift, andȲ (θ) is the bunch displacement in the real space which can be found by meant of the formula The function q(θ) is proportional to the usual transverse wake field function W 1 (z) with r 0 = e 2 /mc 2 as the classic radius of the particle, R as the accelerator radius, N b as the bunch population, β and γ as the normalized velocity and energy [2]. A solution of Eq. (3) can be found by its expansion in terms of the eigenfunctions of corresponding homogeneous equation which is It is easy to check that the functions form an orthogonal basis with the weight function F (θ, u) . Besides, we will impose the normalization condition: where the star denotes complex conjugation. Then, looking for the solution of Eq. (3) in the form one can get the expression for the unknown coefficients C j : whereȲ j and Y j are also connected by Eq. (4). Multiplying Eq. (9) by factor F (θ, u) Y * J (θ, u) , integrating over θ and u, and using normalization condition (7), one can get the series of equations for the coefficients C j :

III. BOXCAR MODEL
The boxcar model is characterized by following expressions for the bunch distribution function and its linear density: Because the eigenfunctions depend on two variables (θ-u) (or A-φ), it is more convenient to represent j as a pair of the indexes: An analytical solutions of Eq. (6) for the boxcar bunch have been found by Sacherer [9]. The most important point is that the averaged eigenfunctionsȲ n,m do not depend on second index being proportional to the Legendre polynomials:Ȳ n,m (θ) =Ȳ n (θ) ∝ P n (θ), n = 0, 1, 2, . . .  Fig. 1. It is seen that all of them take start at ∆Q = 0 from the points ν n,m (0) = mQ s . It is the commonly accepted convention to use the term "multipole" for the collective synchrotron oscillations of such frequency, that is the index m should be treated here as the multipole number. Another index n characterizes the eigenfunction power. This feature is normally associated with a radial mode number, the lower power corresponding to the lower number. Because n ≥ |m| in this case, the mode {|m|, m} should be treated as the lowest radial mode of m-th multipole. At ∆Q = 0, the multipoles mix together, and the eigentunes split on 2 groups. In the first of them, It is e as y t o v erif y t h at, at w = 1 a n d N = 0, t h e m atri x R N, n is . T h e first st e p i n t his w a y is a n i n v esti g ati o n of t h e e q u ati o n D 2 (ν ) = 0. A c c or di n g t o E q. ( 2 3), its e x p a n d e d f or m is w h er e W 1 (ν ) is gi v e n b y E q. ( 2 4), a n d   [ 2]. It cr eat es a n os cill ati n g w a k e ∝ c os( 2 π z / λ ) h a vi n g t h e p h as e a d v a n c e φ = 2 π z b / λ wit hi n t h e b u n c h.
We c o nsi d er t h e c as e φ < 2 π , a n d r e pr es e nt t h e w a k e i n t h e f or m