Wall-Impedance-Driven Collective Instability in Intense Charged Particle Beams

The linearized Vlasov-Maxwell equations are used to investigate detailed properties of the wallimpedance-driven instability for a long charge bunch (bunch length `b bunch radius rb) propagating through a cylindrical pipe with radius rw and wall impedance Z̃(ω). The stability analysis is carried out for perturbations with azimuthal mode number ` ≥ 1 about a cylindrical KapchinskijVladimirskij (KV) beam equilibrium with flattop density profile in the smooth-focusing approximation. Detailed stability properties are determined for dipole-mode perturbations (` = 1) assuming negligibly small axial momentum spread of the beam particles. The stability analysis is valid for general value of the normalized beam intensity sb = ω̂ pb/2γ 2 bω 2 β⊥ in the interval 0 < sb < 1, where ω̂pb = (4πn̂beb/γbmb) 1/2 is the relativistic plasma frequency and ωβ⊥ is the applied focusing frequency. PACS numbers: 29.27.Bd, 41.75.-i, 41.85.-p


I. INTRODUCTION
High energy ion accelerators, transport systems and storage rings [1][2][3][4][5][6][7][8] have a wide range of applications ranging from basic research in high energy and nuclear physics, to applications such as spallation neutron sources, heavy ion fusion, and nuclear waste transmutation.
Charged particle beams are subject to various collective instabilities that can deteriorate the beam quality.Of particular importance at the high beam currents and charge densities of practical interest are the effects of the intense self-fields produced by the beam space charge and current on determining detailed equilibrium, stability, and transport properties.
In general, a complete description of collective processes in intense charged particle beams is provided by the nonlinear Vlasov-Maxwell equations [1] for the self-consistent evolution of the beam distribution function, f b ( x, p, t), and the electric and magnetic fields, E(x, t) and B(x, t).While considerable progress has been made in analytical and numerical simulation studies of intense beam propagation , the effects of finite geometry and intense self-fields often make it difficult to obtain detailed predictions of beam equilibrium, stability, and transport properties based on the Vlasov-Maxwell equations.Nonetheless, often with the aid of numerical simulations, there has been considerable recent analytical progress in applying the Vlasov-Maxwell equations to investigate the detailed equilibrium and stability properties of intense charged particle beams.These investigations include a wide variety of diverse applications ranging from the Harris-like instability driven by large temperature anisotropy with T ⊥b T b [37], to the dipole-mode two-stream instability for an intense ion beam propagating through background electrons [38], to the resistive hose instability [39] and the sausage and hollowing instabilities [40] for intense beam propagation through background plasma, to the development of a nonlinear stability theorem [22,23] in the smooth-focusing approximation.Building on these advances [1,[37][38][39][40], in the present analysis we reexamine the classical wall-impedance-driven instability [41][42][43][44][45], also called the resistive-wall instability, making use of the linearized Vlasov-Maxwell equations [1] for perturbations about a Kapchinskij-Vladimirskij (KV) beam equilibrium f 0 b (x, p) [9][10][11] with flattop density profile.To briefly summarize, the present analysis assumes a very long charge bunch (bunch length b bunch radius r b ) with directed axial kinetic energy (γ b − 1)m b c 2 propagating in the z-direction through a cylindrical pipe with constant radius r w and (complex) wall impedance Z(ω) [2].The analysis is carried out in the smooth-focusing approximation, where the applied transverse focusing force is modeled by is the relativistic mass factor, V b = β b c is the directed axial velocity of the charge bunch, m b is the particle rest mass, ω β⊥ = const.is the applied focusing frequency, and x ⊥ = xê x + yê y is the transverse displacement of a beam particle from the cylinder axis.Denoting the number density of beam particles by nb and the particle charge by e b , it is convenient to introduce the relativistic plasma frequency ωpb defined by ωpb = (4πn b e 2 b /γ b m b ) 1/2 and the normalized (dimensionless) beam intensity s b defined by . An important feature of the present analysis of the linearized Vlasov-Maxwell equations is that it is carried out for arbitrary value of the normalized beam intensity in the interval 0 < s b < 1, assuming perturbations about a KV beam equilibrium with flattop density profile Illustrative parameters for intense beam systems are shown in Table 1 for the Tevatron [46], for coasting beam experiments in the Proton Storage Ring [47,48], and for the space-charge-dominated beams envisioned for heavy ion fusion [8].Note from Table 1 that the normalized beam intensity s b ranges from the very small value s b = 1.36 × 10 −4 in the Tevatron, where the particles are highly relativistic, to the intermediate value s b = 0.08 in the low-energy, moderate-intensity Proton Storage Ring experiment, to s b 0.98 in the low-emittance, space-charge-dominated beams for heavy ion fusion.In any case, the present kinetic analysis of the wall-impedance-driven instability is carried out for arbitrary value of normalized-beam intensity s b in the interval 0 < s b < 1, and (in principle) can be applied to the diverse range of high-intensity beam systems in Table 1.Finally, the present analysis considers the case where the axial momentum spread is negligibly small, and the corresponding Landau damping [1] by parallel kinetic effects is absent.(This gives a larger estimate of the instability growth rate than would be obtained with finite axial momentum spread.)Furthermore, the functional form of the wall impedance Z(ω) is not specified,

II. THEORETICAL MODEL AND ASSUMPTIONS
The present analysis considers a very long charge bunch with characteristic axial length b where is the relativistic mass factor, V b = β b c is the average axial velocity, and c is the speed of light in vacuo.The charge bunch propagates through a cylindrical, conducting pipe with wall radius r w , and the applied transverse focusing force on a beam particle is modeled in the smooth focusing approximation by where ω β⊥ = const.is the applied focusing frequency, and x ⊥ = xê x + yê y is the transverse displacement of a beam particle from the cylinder axis at r = 0. Furthermore, the particle motion in the beam frame is treated in the paraxial approximation with p 2 x , p 2 y , ( To describe stability properties of the charge bunch, we make use of a kinetic description based on the Vlasov-Maxwell equations, which describe the self-consistent nonlinear evolution of the distribution function f b (x, p, t) and the self-generated electric and magnetic fields, E s (x, t) and B s (x, t), in the six-dimensional phase space (x, p).For simplicity, the present analysis considers small-amplitude perturbations about the axisymmetric (∂/∂θ = 0), axially uniform (∂/∂z = 0), quasi-steady-state (∂/∂t = 0) equilibrium distribution function [38] In Eq. ( 2), nb and T⊥b are positive constants, and H ⊥ is the transverse Hamiltonian defined by where r = (x 2 + y 2 ) 1/2 is the radial distance from the cylinder axis, and p ⊥ = (p 2 x + p 2 y ) 1/2 is the transverse momentum.In Eq. (3), the equilibrium self-field potentials, φ 0 (r) and A 0 z (r), are determined self-consistently in terms of f 0 b (r, p) from the steady-state Maxwell equations.Because of the delta-function dependence on p z , note that the choice of distribution function in Eq. ( 2) is cold in the axial direction.An attractive feature of the choice of f 0 b (r, p) in Eq. ( 2) is that the corresponding equilibrium number density, n 0 b (r) = d 3 pf 0 b (r, p), has the flattop profile [38] Here, nb = const.is the number density of beam particles, and the edge radius r b is deter- where ω2 pb = 4πn b e 2 b /γ b m b is the relativistic plasma frequency-squared.Here, we have introduced the quantity ν 2 b defined by where is a convenient dimensionless measure of the normalized beam intensity.Note from Eq. ( 6) that ν b = ω β⊥ (1 − s b ) 1/2 corresponds to the (depressed) betatron frequency for transverse particle oscillations in the equilibrium field configuration.For parameters typical of the Tevatron [46], s b 1 and ν b ω β⊥ , corresponding to very weak equilibrium self fields.For parameters typical of heavy ion fusion applications [8], however, s b is in the range 0.9 < s b < 1, corresponding to very large tune depressions.On the other hand, for accelerators used in nuclear physics applications [47,48], such as the Proton Storage Ring (PSR) facility and the Spallation Neutron Source (SNS), the intensity parameter s b is in the intermediate range, An important goal of the present analysis is to develop a theoretical model that determines the effects of finite wall impedance and is valid over the entire range of normalized beam intensity, 0 < s b < 1.To this end, we express , and make use of the linearized Vlasov-Maxwell equations [1,38] to determine the self-consistent evolution of δf b (x, p, t), δE s (x, t) and δB s (x, t) for small-amplitude perturbations.For perturbations about the equilibrium distribution function f 0 b (r, p) in Eq. ( 2), the linearized Vlasov equation for δf b (x, p, t) can be expressed as where δF ⊥ = e b (δE s + v × δB s /c) ⊥ and δF z = e b (δE s + v × δB s /c) z are the perturbed transverse and longitudinal forces.We further express δE s = −∇δφ − c −1 ∂δA/∂t and δB s = ∇ × δA, and make use of the Lorentz gauge condition, ∇ • δA = −c −1 ∂δφ/∂t, to relate δA and δφ.The linearized Maxwell equations for δφ(x, t) and δA(x, t) can then be expressed as Here, v = p/γm b is the particle velocity, γ = (1 + p 2 /m 2 b c 2 ) 1/2 is the kinematic mass factor, and 2 is the perpendicular Laplacian in cylindrical polar coordinates (r, θ, z).Note that Eqs. ( 9) and (10) determine the perturbed self-field potentials, δφ(x, t) and δA(x, t), in terms of the perturbed charge and current densities, , where δf b (x, p, t) is determined self-consistently from Eq. ( 8).
In Sec. 3, Eqs.( 8)-( 10) will be analyzed for perturbations of the form where = 1, 2, . . . is the azimuthal mode number of the perturbation, k z is the axial wavenumber, and ω is the oscillation frequency.For perturbations with real ω and Imk z < 0, the perturbation is growing spatially as a function of z.On the other hand, for perturbations with real k z and Imω > 0, the perturbation is growing temporally as a function of t.For present purposes, we consider perturbations with sufficiently low frequency and long axial wavelength that Equations ( 8)-( 10) can be simplified within the context of the inequalities in Eq. ( 12).For example, making use of the Lorentz gauge condition, )δφ| over the transverse dimensions of the beam.Without presenting algebraic details [1], it therefore follows within the context of Eq. ( 12) that the δA ⊥ contributions in Eqs. ( 8)-( 10) can be neglected and that the perturbed transverse force δF ⊥ can be approximated by Similarly, for the low-frequency, long-wavelength perturbations consistent with Eq. ( 12), it can be shown that the perturbed longitudinal force term (proportional to δF z ) in Eq. ( 8) can be neglected [38].Moreover, because the axial momentum spread is negligibly small for the distribution function in Eq. ( 2), we approximate In summary, making use of the approximations outlined in the previous paragraph, the linearized Vlasov-Maxwell equations ( 8)-( 10) can be approximated by where δφ and δA z are determined from Here, V b = β b c is the average axial velocity, and δn b (x, t) is the perturbed number density of beam particles defined in terms of δf b (x, p, t) by Equations ( 14)-( 17) represent the final form of the linearized Vlasov-Maxwell equations used in the stability analysis in Sec.III, carried out for perturbations about the choice of equilibrium distribution function f 0 b (r, p) in Eq. ( 2) with flattop density profile in Eq. ( 4).Equations ( 14)-( 16) are to be solved in the beam interior (0 ≤ r < r b ) and in the vacuum region (r b < r ≤ r w ) outside the beam, enforcing the appropriate boundary conditions at the conducting wall located at radius r = r w .For present purposes, we assume that the wall impedance is described by a complex scalar function, Z(ω) = Zr + i Zi , where ω is the oscillation frequency in Eq. (11), and that the boundary condition on the perturbed tangential electric E t and magnetic Here, n = −ê r is a unit vector pointing outward from the cylindrical conducting wall surface.
In what follows we assume that the metal wall is almost perfectly conducting, implying that Assuming that perturbed quantities vary according to Eq. ( 11), and making use of (∇ × δB) r = c −1 ∂δE r /∂t in the vacuum region, the boundary conditions in Eq. ( 18) can be expressed as Neglecting contributions involving δA ⊥ (which can be done under the assumption that )δφ , and ).The boundary conditions in Eq. ( 19) then reduce to Equation ( 20) expresses the boundary conditions at the conducting wall in terms of the impedance Z(ω) and the perturbed potentials, δφ and δA z .In the limit of zero impedance, Z → 0, note that Eq. ( 20) reduces to [δφ ] r − w = 0 = [δA z ] r − w , corresponding to the boundary conditions expected for a perfectly conducting, cylindrical wall.Depending on the frequency regime, there are several models of wall impedance Z(ω) that can be used in the boundary conditions in Eq. ( 20).These range from impedance functions that depend on the wall structure and smoothness [2,42,43], to impedance functions that depend on the electrical conductivity of the wall [49].For example, a common expression for Z(ω) for a smooth-bore, cylindrical conducting wall is given by [49] where σ is the electrical conductivity of the wall.
In concluding this section, we reiterate that the inequalities  15) and ( 16).This is typically encountered in heavy ion fusion applications [8], and in some accelerators for nuclear physics applications such as the Proton Storage Ring (PSR) facility [36,47].In the general case, however, making use of Eq. ( 11), the solutions to Eqs. ( 15) and ( 16) for δφ (r) and δA z (r) in the vacuum region are linear combinations of I (κr) and K (κr), where κ(k z , ω) is defined by and I (x) and K (x) are modified Bessel functions of the first and second kinds, respectively, of order .For our purposes here, the analysis in Sec.III makes the further assumption that Whenever Eq. ( 23) is satisfied, Eqs. ( 15) and ( 16) can be approximated by ⊥ δA z in the vacuum region (r b < r ≤ r w ) where δn b = 0, and the solutions to Eqs. ( 15) and ( 16) for δφ (r) and δA z (r) are linear combinations of r and r − , where ≥ 1 is an integer.For example, if we estimate the oscillation frequency by ω k z V b , then Eq. ( 23) where Therefore, for a long, highly-relativistic charge bunch ( b r b , γ b 1), the inequality in Eq. ( 24) is relatively straightforward to satisfy, even when r w r b , provided the relativistic mass factor γ b is sufficiently large.

A. Linearized Vlasov-Maxwell Equations
We now make use of Eqs. ( 14)-( 17) and the assumptions summarized in Sec. 2 to derive a dispersion relation that describes detailed stability properties of the charge bunch.In the present analysis, the equilibrium distribution function in Eq. ( 2) can be expressed as Integrating Eq. ( 14) over p z then gives for the evolution of δF b (x, p ⊥ , t), where is the axial velocity of the beam particles.Moreover, consistent with Eqs. ( 12) and ( 23), we neglect the terms proportional to ∂ 2 /∂z 2 −c −2 ∂ 2 /∂t 2 in Eqs.(15) and ( 16), and the linearized Maxwell equations for δφ(x, t) and δA z (x, t) are approximated by and Here, δn b (x, t) = d 2 pδF b (x, p ⊥ , t) is the perturbed number density of beam particles, and . . .In the subsequent analysis of Eqs. ( 25)- (27), it is convenient to introduce the new independent variables τ and Z (replacing t and z) defined by In this case, the perturbation in Eq. ( 11) can be expressed as where = 1, 2, • • • , is the azimuthal mode number, ω is the oscillation frequency, and is the effective axial wavenumber of the perturbation in the new variables (Z, τ ).The significance of the new 'time' variable τ in Eq. ( 28) is evident.We consider the case where the head of the charge bunch passes through z = 0 at t = 0 with velocity V b > 0. Then Making use of Eq. ( 31), the linearized Vlasov equation (25) for δF b (x ⊥ , p ⊥ , Z, τ ) simplifies to become where δφ(x ⊥ , Z, τ ) and δA z (x ⊥ , Z, τ ) are determined self-consistently in terms of δF b from Eqs. ( 26) and (27).Note in Eq. ( 32) that the perturbed beam dynamics is determined in terms of the wake function δψ ≡ δφ − β b δA z .
The left-hand side of Eq. ( 32) will be recognized as the total derivative, (V b d/dZ ) × δF b (x ⊥ , p ⊥ , Z , τ ), following the particle trajectories x ⊥ and p ⊥ in the equilibrium field configuration.Here, the characteristics of the differential operator on the left-hand side of Eq. ( 32) are the particle orbit equations which can be combined to give In order to solve Eq. ( 32), the solutions of physical interest to the transverse orbit equations ( 33) and ( 34) are those that pass through the phase space point (x ⊥ , p ⊥ ) at Z = Z, i.e., Solving Eqs. ( 33) and ( 34) subject to Eq. ( 35), we readily obtain where As expected, for the flattop density profile in Eq. ( 4), the transverse orbits in Eq. ( 36) are oscillatory functions of Z − Z with wavelength is the (depressed) betatron frequency defined in Eq. ( 6).
The linearized Vlasov equation ( 32) is now formally integrated using the method of characteristics [1,[37][38][39][40].Expressing the left-hand side of Eq. ( 32) as V b (d/dZ ) ×δF b (x ⊥ , p ⊥ , Z , τ ), we assume spatially amplifying perturbations (ImΩ > 0) and integrate Eq. ( 32) from Z = −∞ (where δF b is assumed to be negligibly small) to Z = Z.This gives Here, use has been made of the fact that H ⊥ = H ⊥ = const.is a single-particle constant of the motion (dH ⊥ /dZ = 0) in the equilibrium field configuration.In the integration over Z on the right-hand side of Eq. ( 37), x ⊥ (Z ) and p ⊥ (Z ) = γ b m b v ⊥ (Z ) are the single-particle orbits in Eq. ( 6) that pass through the phase space point (x ⊥ , p ⊥ ) at Z = Z.
For the choice of equilibrium distribution F b (H ⊥ ) in Eq. ( 2), we calculate the perturbed number density δn b (x ⊥ , Z, τ ) = d 2 pδF b (x ⊥ , p ⊥ , Z, τ ) from Eq. ( 37) and substitute into Maxwell's equations ( 26) and ( 27), which gives closed equations for the perturbed potentials, δφ(x ⊥ , Z, τ ) and δA z (x ⊥ , Z, τ ).Assuming perturbations of the form in Eq. ( 24) for ImΩ > 0 and azimuthal mode number = 1, 2, • • • , and carrying out the integration over Z in Eq. ( 37), it is found that a class of solutions exists with density perturbation amplitude δn b (r) = d 2 pδF b (r, p ⊥ ) localized at the surface of the charge bunch (r = r b ).Without presenting algebraic details [1,38], we obtain Here, the response function χ b (Ω) is defined by where is the relativistic plasma frequency, and is the depressed betatron frequency.As expected, the response function in Eq. ( 39) has a rich harmonic content at harmonics of ν b .
Equations ( 41) and ( 42), derived for perturbations about the equilibrium distribution f 0 b (r, p) in Eq. ( 2) with flattop-density profile in Eq. ( 4), constitute the final forms of the eigenvalue equations used in the present stability analysis.Here, Eqs.(41) and (42) are to be solved over the interval 0 ≤ r ≤ r w for the eigenfunctions δφ (r) and δA z (r) and eigenvalue Ω, subject to the condition that δφ (r) and δA z (r) be regular at the origin (r = 0), and satisfy the boundary conditions in Eq. ( 20) at the conducting wall (r = r w ).It should be emphasized that Eqs. ( 41) and ( 42) are valid over the entire range of normalized beam intensity, 0 < s b = ω2 pb /2γ 2 b ω 2 β⊥ < 1, subject to the assumption of low-frequency, longwavelength perturbations in Eqs. ( 12) and ( 23).

B. Derivation of Dispersion Relation
We now solve Eqs. ( 41) and (42) in the beam interior (0 ≤ r < r b ) and in the vacuum region outside the charge bunch (r b < r ≤ r w ).The solutions to Eqs. ( 41) and ( 42) that are regular at r = 0 can be expressed as and where δ φ ≡ δφ (r = r b ) and δ Â z ≡ δA z (r = r b ), and A , B , A and B are constants.We enforce continuity of δφ (r) and δA z (r) at r = r b , which gives Note from Eqs. ( 41) and ( 42) that there are surface charge and current perturbations at where 46) effectively determines the discontinuity in perturbed radial electric field (azimuthal magnetic field) in terms of the perturbed surface charge density (current density), which is proportional to χ b (Ω).Solving for the coefficients A , B , A and B in terms of δ φ and δ Â z , we obtain from Eqs. ( 45) and ( 46) where χ b (Ω) is defined in Eq. ( 39), and use has been made of δ ψ = δ φ − β b δ Â z .We now enforce the boundary conditions at the conducting wall (r = r w ) given in Eq. (20).
Making use of the solutions for δφ (r) and δA z (r) in the vacuum region (r b < r ≤ r w ) given in Eqs. ( 43) and (44), the boundary conditions in Eq. ( 20) can be expressed as and where Z(ω) is the wall impedance.Equations ( 47)-( 49) can be combined to give two linear, homogeneous equations relating the perturbation amplitudes δ φ and δ Â z .The dispersion relation for the complex frequency Ω is then obtained by setting the determinant of the 2x2 coefficient matrix equal to zero.In the limit of a perfect conductor with Z → 0, note that Eqs.(48) and (49) give A → −B (r b /r w ) 2 and A → −B(r b /r w ) 2 , which correspond to the boundary conditions for a perfect conductor, δφ (r = r w ) = 0 = δA z (r = r w ), as expected.
For Z = 0, it is convenient to express and make use of Eqs. ( 48) and (49) to solve for ∆ and ∆ in terms of the impedance Z(ω).
Substituting Eq. (50) into Eqs.( 48) and (49), and making use of B = β b B [Eq. ( 47)], we obtain Equation ( 51) can be used to determine closed expressions for ∆ and ∆ in terms of the wall impedance Z(ω).For example, if then the approximate solutions to Eq. ( 51) are given correct to leading order by where V b = β b c.If we estimate ω ≈ k z V b , then the inequalities in Eq. ( 52) assure that |∆ |, |∆| 1 and that the wall impedance contributions proportional to ∆ and ∆ in Eq. ( 50) represent small corrections to the results for a perfectly conducting wall.
In any case, we now make use of Eq. ( 50) to derive the dispersion relation that determines Ω (generally complex) in terms of the oscillation frequency ω and system parameters such as the plasma frequency ωpb , depressed betatron frequency ν b , and wall impedance Z(ω).
Substituting Eq. ( 47) into Eq.( 50), where ∆ and ∆ solve Eq. ( 51), and making use of where where D b (Ω) is the dielectric function defined by and use has been made of The condition for a nontrivial solution (δ ψ = 0) to Eq. ( 56) is Equation ( 58) is the final form of the dispersion relation derived from the linearized Vlasov-Maxwell equations ( 25)-( 27) for perturbations about the choice of equilibrium distribution function in Eq. ( 2) with corresponding flattop density profile in Eq. ( 4).The dispersion relation ( 58) is valid for low-frequency long-wavelength perturbations consistent with Eqs. ( 12) and ( 23), and can be applied over a wide range of normalized beam intensity In the definition of D b (Ω) in Eq. ( 57), the response function χ b (Ω) is defined in Eq. ( 39) for general azumithal mode number = 1, 2, • • • , and the quantities ∆ and ∆ are determined in terms of the wall impedance Z(ω) from Eq. (51).
We now consider Eqs. ( 66  where we have neglected Zi in the definition of Ω 0 in Eq. (65) for δ r w .Note from Eq. (71) that the lower sideband with Ω r = −Ω 0 is unstable (ImΩ = Ω i > 0), and that the growth rate is proportional to ω2 pb /Ω 0 , which is an increasing function of the beam density nb .Moreover, the growth rate Ω i is linearly proportional to the normalized skin depth δ/r w , where δ = 1/(2πσω 0 ) 1/2 → 0 as σ → ∞.
(c) Wall with Model Impedance: The interaction of an intense beam with the induction modules of course depends on the cavity design, details of the drive circuitry, etc.This interaction is often modeled by a complex coupling impedance [2,42,43] Z(ω 0 ) = Zr (ω 0 ) + i Zi (ω 0 ), where with T ⊥b T b [37], to the dipole-mode two-stream instability for an intense ion beam propagating through background electrons [38], to the resistive hose instability [39] and the sausage and hollowing instabilities [40] for intense beam propagation through background plasma, to the development of a nonlinear stability theorem [23,24] in the smooth-focusing approximation.Building on these advances, in the present analysis we have reexamined the classical wall-impedance-driven instability [41][42][43][44][45] the case of small impedance (| Z| 1) is considered when analyzing the kinetic dispersion relation in Secs.III and IV.The organization of this paper is the following.The theoretical model and assumptions are summarized in Sec.I.In Secs.II and III, the detailed kinetic stability analysis is carried out for perturbations about a KV beam equilibrium with flattop density profile, leading to the kinetic dispersion relation (58), valid for arbitrary multipole perturbations with azimuthal mode number ≥ 1 about an axisymmetric beam equilibrium.Finally, in Sec.IV detailed properties of the wall-impedance-driven instability are calculated for dipole-mode perturbations ( = 1) and general values of the normalized beam intensity s b in the interval 0 < s b < 1.

and radius r b satisfying
b r b .The charge bunch is made up of particles with charge e b and rest mass m b propagating in the z-direction with directed axial kinetic energy (γ b − 1)m b c 2 , |ω|r b /c 1 and |k z |r b /c 1 in Eq. (12) have been used to simplify the perturbed force δF in the beam interior (0 ≤ r < r b ) in the linearized Vlasov equation (14).Insofar as the wall radius r w is not too far removed from the beam radius r b (r w /r b ∼ 2 − 3, say), then |k z |r w /c 1 and |ω|r b /c 1 are also good approximations in solving the Maxwell equations (15) and (16) in the vacuum region (r b < r ≤ r w ), and the terms proportional to ∂ 2 /∂z 2 − c −2 ∂ 2 /∂t 2 can be neglected in Eqs. (

Because f 0 b
has zero axial momentum spread about p z = γ b m b β b c, we express the perturbed distribution function in the linearized Vlasov equation (14) as δf b (x, p, t) = δF b (x, p ⊥ , t)δ(p z − γ b m b β b c).
the distance backwards from the head of the beam (at V b t) to axial position z = Z.If the charge bunch experiences a perturbation for τ > 0 with real oscillation frequency ω, it is evident from Eqs. (29) and(30) that Ω/V b represents the spatial oscillation and growth (or damping) of the perturbation as a function of axial position Z.Furthermore, in terms of the new variables Z and τ , the derivatives ∂/∂t and ∂/∂z transform according to