Self-consistent Vlasov-Maxwell description of the longitudinal dynamics of intense charged particle beams

024401-1 This paper describes a self-consistent kinetic model for the longitudinal dynamics of a long, coasting beam propagating in straight (linear) geometry in the z direction in the smooth-focusing approximation. Starting with the three-dimensional Vlasov-Maxwell equations, and integrating over the phasespace x?;p? transverse to beam propagation, a closed system of equations is obtained for the nonlinear evolution of the longitudinal distribution function Fb z; pz; t and average axial electric field hEzi z; t . The primary assumptions in the present analysis are that the dependence on axial momentum pz of the distribution function fb x;p; t is factorable, and that the transverse beam dynamics remains relatively quiescent (absence of transverse instability or beam mismatch). The analysis is carried out correct to order kzrw assuming slow axial spatial variations with kzrw 1, where kz @=@z is the inverse length scale of axial variation in the line density b z; t R dpzFb z; pz; t , and rw is the radius of the conducting wall (assumed perfectly conducting). A closed expression for the average longitudinal electric field hEzi z; t in terms of geometric factors, the line density b, and its derivatives @ b=@z; . . . is obtained for the class of bell-shaped density profiles nb r; z; t b= rb f r=rb , where the shape function f r=rb has the form specified by f r=rb n 1 1 r=rb n for 0 r < rb, and f r=rb 0 for rb < r rw, where n 0; 1; 2; . . . . The general kinetic formalism developed here is valid for the entire range of beam intensities (proportional to b) ranging from low-intensity, emittance-dominated beams, to very-high-intensity, low-emittance beams.


I. INTRODUCTION
High energy 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, to mention a few examples.Charged particle beams are subject to various collective processes that can affect 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 three-dimensional Vlasov-Maxwell equations [1] for the self-consistent nonlinear evolution of the beam distribution function, f b x; p; t, and the self-generated electric and magnetic fields, E s x; t and B s 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 applications ranging from the Harris-type instability driven by large temperature anisotropy with T ?b T kb [37], to the dipole-mode two-stream instability for an intense ion beam propagating through background electrons [29], to the resistive hose instability [34] and the sausage and hollowing instabilities [35] for intense beam propagation through background plasma, to the development of a nonlinear stability theorem [22,23] in the smooth-focusing approximation.
While the collective processes described in the previous paragraph are three dimensional in nature, considerable theoretical progress has also been made in the development and application of one-dimensional Vlasov-Maxwell models [40 -47] to describe the longitudinal beam dynamics for a long coasting beam, with applications ranging from plasma echo excitations to the investigation of coherent soliton structures, both compressional and rarefactive (holelike).Such one-dimensional Vlasov descriptions rely heavily on using a quasi-selfconsistent geometric-factor (g-factor) model [46 -55] to incorporate the average effects of the transverse beam geometry and the surrounding wall structure.Several limitations are evident in existing g-factor models.First, the models [40 -55] typically assume a flattop (step-function) density profile in the plane perpendicular to beam propagation.While this is a good approximation for very-low-emittance, space-charge-dominated beams [1,[49][50][51][52][53], the case typically encountered in beams with low-to-moderate intensity is one in which the transverse density profile is bell shaped.Second, with the exception of the g-factor model for a step-function density profile developed by Reiser et al. [50 -53], other g-factor models neglect the dependence of the beam edge radius r b and root-mean-square radius R b on the line density b of beam particles, an effect that becomes increasingly important at moderate and high beam intensities.Finally, some one-dimensional Vlasov treatments Taylor expand the field perturbations about the beam axis at r 0, an approximation which is not necessary and has questionable validity, particularly at moderate and high beam intensities.
The purpose of the present paper is to develop an improved one-dimensional kinetic model describing the self-consistent nonlinear evolution of the longitudinal distribution function F b z; p z ; t and average axial electric field hE s z iz; t for a very long charge bunch (coasting beam) propagating through a cylindrical conducting pipe with radius r w , and confined in the transverse direction by an applied focusing force F tr foc described in the smoothfocusing approximation.For simplicity, to illustrate the basic approach, in the present analysis the cylindrical pipe is assumed to be perfectly conducting, and the beam transport geometry is assumed to be straight (linear).The analysis can be extended in a straightforward manner to include the effects of a conducting wall with finite resistive and capacitive impedance, as well as the effects of a slip factor in large-aspect-ratio circular geometry.Extension of the present analysis to incorporate these effects will be the subject of a future publication.In the present analysis, the coasting beam propagates in the z direction with directed axial kinetic energy b ÿ 1m b c 2 , where b 1 ÿ 2 b ÿ1=2 is the relativistic mass factor, V b b c is the average axial velocity of the beam particles, m b is the rest mass of a beam particle, and c is the speed of light in vacuo.Furthermore, the beam dynamics is treated in the thinbeam (paraxial) approximation, and the particle motions in the beam frame are assumed to be nonrelativistic.
For simplicity, the present analysis is carried out in the beam frame where the particle motions are nonrelativistic and the self-generated fields are assumed to have electrostatic polarization (r E s 0 and B s 0).The final results for the longitudinal Vlasov-Maxwell equations are then Lorentz transformed back to the laboratory frame, moving with axial velocity ÿV b ÿ b c relative to the average motion of the particles in the beam frame.
In Sec.II, the starting point is the fully nonlinear, three-dimensional Vlasov-Maxwell equations for the distribution function f b x; p; t and self-generated fields in the beam frame (unprimed variables).A reduced Vlasov equation for the longitudinal distribution function F b z; p z ; t R dxdy R dp x dp y f b x; p; t is obtained by projecting out (integrating over) the transverse phasespace variables x ?; p ? .Making the single ansatz so that the dependence of the distribution function f b x; p; t on axial momentum p z is factorable [Eq.(15)] leads to a closed system of equations describing the selfconsistent evolution of the longitudinal distribution function F b z; p z ; t and the average axial electric field hE s z iz; t.Here, assuming axisymmetry in the transverse plane @=@ 0, the average h i denotes the weighted transverse spatial average over n b r; z; t= b z; t defined in Eq. ( 26 b n for 0 r < r b , and fr=r b 0 for r b < r r w .Here, n 0; 1; 2; . . . is an integer, with n 0 corresponding to a step-function density profile, and r b is the edge radius of the beam.Moreover, the root-meansquare beam radius R b hr 2 i 1=2 and edge radius r b generally depend on the line density through the radial force-balance condition in Eqs.(33) and (38), respectively.Of course, this dependence is weak (r b ' const) for a low-intensity, emittance-dominated beam, whereas r 2 b / b for a very-low-emittance, space-chargedominated beam.Denoting @=@z L ÿ1 z k z , we assume slow axial variations of F b z; p z ; t and b z; t with k 2 z r 2 w 1.The average electric field hE s z iz; t, expressed in terms of @ b =@z and higher-order derivatives, together with closed forms for the corresponding geometric factors are calculated self-consistently correct to 0k 2 z r 2 w for the class of bell-shaped density profiles with n 0; 1; 2; . . .described above.The results show a strong dependence of the geometric factors on profile shape and beam intensity b .
In Sec.IV, as a simple application, the resulting coupled equations for the longitudinal distribution function F b z; p z ; t and the average electric field hE s z iz; t are solved in the linearization approximation for the case of low-to-moderate beam intensity treating r b ' const (independent of b ).As expected, the analysis leads to collective oscillations with sound-wave-like characteristics modified by cubic dispersion.The detailed oscillation and damping properties of the perturbation of course depends on the choice of equilibrium distribution function F 0 b p z about which the system is perturbed.Finally, in Sec.V, the key results derived in the beam frame are transformed back to the laboratory frame to facilitate practical applications of the improved kinetic model of the longitudinal beam dynamics developed here.

II. THEORETICAL MODEL AND ONE-DIMENSIONAL VLASOV EQUATION
The present analysis considers a very long charge bunch (coasting beam) with characteristic axial length is the relativistic mass factor, V b b c is the average axial velocity of the beam particles, and c is the speed of light in vacuo.It is assumed that the beam propagates through a straight, perfectly conducting cylindrical pipe with wall radius r w , and the applied transverse focusing force F tr foc is modeled in the smooth-focusing approximation.For example, one simple model is F tr foc ÿ b m b ! 2 ?x ?, where !?const is the average focusing frequency associated with the applied focusing field, and x ? is the transverse displacement from the cylinder axis.Finally, the nonlinear dynamics of the beam particles is treated in the thin-beam (paraxial) approximation, and the particle motions in the beam frame are assumed to be nonrelativistic [1].
The most general description of the nonlinear collective interactions in an intense charged particle beam is provided by the nonlinear Vlasov-Maxwell equations [1], which is the theoretical framework adopted in the present analysis.For simplicity, the analysis is carried out in the beam frame (V b b c 0 and b 1), and the kinetic description is 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.Here, the (unprimed) variables x; p; t denote beam-frame variables.For present purposes, the self-generated fields in the beam frame are assumed to have longitudinal polarization (electrostatic approximation) with r E s 0 and B s 0. In this case, the nonlinear Vlasov-Maxwell equations describing the self-consistent nonlinear evolution of f b x; p; t and E s x; t can be expressed as [1] and where v p=m b is the (nonrelativistic) particle velocity in the beam frame.Of course, once the solutions for f b x; p; t and E s x; t are determined from Eqs. ( 1) and (2) in the beam frame, the corresponding solutions in the laboratory frame (moving with axial velocity ÿV b ê e z relative to the beam frame) can be obtained by transforming the phase-space variables and field components to the laboratory frame (primed variables) according to [22] x 0 x; y 0 y; (3) and In Eq. (3), 0 1 p 02 =m 2 b c 2 1=2 and 1 p 2 =m 2 b c 2 1=2 are the kinematic mass factors, and ' 1 p 2 =2m 2 b c 2 because the particle motions in the beam frame are assumed to be nonrelativistic.
Returning to the beam frame, the Vlasov-Poisson equations ( 1) and ( 2 ( Here, R dxdy R r w 0 drr R 2 0 d denotes integration over the accessible transverse configuration space extending to the conducting wall located at radius r r w , R dp z denotes integration over momentum space, and r; ; z are cylindrical polar coordinates with x r cos and y r sin, where r x 2 y 2 1=2 is the radial distance from the beam axis at x; y 0; 0. Because the cylindrical wall located at r r w is perfectly conducting, the nonlinear Vlasov-Poisson equations are to be solved subject to the boundary conditions E z rr w 0 E rr w : (6) Moreover, the present analysis, carried out for a long coasting beam, assumes that the distribution function f b x; p; t is such that the transverse beam dynamics is relatively quiescent (e.g., no transverse instability), and that there are no particles extending beyond some outer radius r 0 < r w , i.e., f b x; p; t 0; for r 0 < r r w : (7) That is, there is a well-defined vacuum region outside the beam with zero number density, n b 0 in the region r 0 < r r w [see Eqs. ( 5) and (7)].Finally, it is assumed that f b x; p; t 0; for p x !1; which is consistent with the existence of the density integral defined in Eq. ( 5).
For purposes of deriving a one-dimensional Vlasov equation describing the beam dynamics in the longitudinal phase space z; p z , it is convenient to rewrite the three-dimensional Vlasov equation ( 1 Here, x ?xê e x yê e y and p ? p x ê e x p y ê e y denote the transverse phase-space variables, and v ?p ? =m b is the transverse particle velocity.We introduce the transverse projection operator h i ?defined by where Note that the operator defined in Eq. ( 10) has the effect of projecting out the transverse phase-space coordinates x ?; p ? .Introducing the longitudinal distribution function F b z; p z ; t defined by we operate on Eq. ( 9) with R dxdy R dp x dp y .Some straightforward integration by parts that makes use of the boundary conditions in Eqs. ( 7) and ( 8) readily gives for the nonlinear evolution of Note that E s z x; t occurs in the average in Eq. ( 12).We operate on Poisson's equation r 2) with @=@z, and make use of r E s 0 in cylindrical polar coordinates, which gives @E s z =@r @E s r =@z and @E s =@z r ÿ1 @E s z =@.Poisson's equation for E s z r; ; z; t readily becomes where n b r; ; z; t R d 3 pf b r; ; z; p; t is the number density of beam particles.In Eqs. ( 12) and ( 13) we note from Eqs. ( 5) and ( 11) that the axial line density The Vlasov equation (12) for the evolution of F b z; p z ; t contains the average e b hE s z @f b @p z i ?over the three-dimensional distribution function f b x; p; t.Of course the total distribution function f b x; p; t evolves according to the nonlinear Vlasov equation ( 1), which generally couples the longitudinal and transverse particle dynamics.As a simplifying ansatz for closure, consistent with the assumption that the transverse beam dynamics remains relatively quiescent, we make the assumption that the dependence of f b x; p; t on axial momentum p z is factorable according to f b x; p; t G b x ?; p ? ; z; tF b z; p z ; t; (15) where G b x ?; p ? ; z; t and F b z; p z ; t are the transverse and longitudinal distribution functions, respectively.From Eqs. ( 5), (11), and ( 15), we readily obtain Substituting Eqs. ( 15) and ( 16) into the definition of e b hE s z @f b =@p z i ?then gives where the average axial electric field hE s z iz; t is defined by Here, use has been made of the definition of the perpendicular projection operator h i ? in Eq. ( 10).From Eqs. ( 17) and ( 18), an important consequence of the factorability ansatz in Eq. ( 15) is that the average electric field hE z iz; t occurring in the longitudinal Vlasov equation ( 12) corresponds to a transverse spatial average of E s z x; t appropriately weighted by the density profile of the beam particles.
In the remainder of this paper we specialize to the case where all beam and field quantities are assumed to be axisymmetric @=@ 0. Making use of Eqs. ( 12), (13), and (18), the final set of equations describing the nonlinear evolution of F b z; p z ; t and hE s z iz; t are given by and Here, E s z r; z; t is determined self-consistently in terms of the density profile n b r; z; t from Poisson's equation (13), which can be expressed as for @=@ 0. In addition, from Eqs. ( 5) and ( 14 Equations ( 19) -( 22) constitute the final form of the nonlinear Vlasov-Maxwell equations describing the evolution of the longitudinal distribution function F b z; p z ; t and line density b z; t in a long coasting beam.An important feature is that Eqs. ( 19) -( 22) have been derived for general radial dependence of the density profile n b r; z; t.The procedure for the solution is the following.Once the radial dependence of n b r; z; t is specified, E s z r; z; t is calculated from Poisson's equation (21).The resulting expression for E s z r; z; t is then substituted into Eq.( 20) to determine the average electric field hE s z iz; t, which in turn is used in the Vlasov equation (19) to determine the self-consistent nonlinear evolution of F b z; p z ; t.Most importantly, the average electric field hE s z iz; t occurring in Eqs. ( 19) and ( 20) is appropriately weighted by a transverse spatial average over the density profile of the beam particles.No a priori assumption has been made that the density profile corresponds to a flattop (step-function) distribution in the radial direction [40 -53].Nor have we Taylor expanded Poisson's equation ( 21) about r 0 to estimate the average electric field hE s z iz; t [46,47].Indeed, Eqs. ( 19) -( 22) are valid for the general choice of the radial density profile and can be applied for arbitrary beam intensity ranging from low-intensity, emittance-dominated beams to very-high-intensity, space-charge-dominated beams.

III. EVALUATION OF AVERAGE LONGITUDINAL
FIELD hE S Z i Equations ( 19) -( 22) constitute the final form of the Vlasov-Maxwell equations describing the nonlinear evolution of the distribution function F b z; p z ; t and line density b z; t.In this section, we make use of Eqs.(20) and (21) to derive a simplified expression for the average longitudinal electric field hE s z iz; t, valid for a wide range of choices of bell-shaped density profiles.For a specified functional form of n b r; z; t, Eq. ( 21) can, of course, be formally solved for E s z r; z; t using a Green's function method [50,51].For our purposes here, however, we specialize to the case where the z variation in E s z r; z; t is slow in comparison with the r variation.In particular, denoting @=@z L ÿ1 z k z , it is assumed that where r w is the conducting wall radius.
A. Average electric field hE s z i for k 2 z r 2 w 1 Imposing the boundary condition E s z r r w ; z; t 0 at the perfecting conducting wall [Eq.( 6)], the formal solution to Eq. ( 21) can be expressed as where n b r; z; t is the density profile.The second term on the right-hand side of Eq. ( 24) is small in comparison with the first term by virtue of the inequality in Eq. ( 23).
Solving iteratively for E s z r; z; t then gives correct to 0k 2 z r 2 w .Equation ( 25) is the degree of accuracy used for the longitudinal electric field E s z r; z; t in the subsequent analysis.Substituting Eq. ( 25) into Eq.( 20) and performing a straightforward integration by parts gives Equation ( 26) can be expressed in the equivalent form (27) where hE s z i 1 and hE s z i 2 are defined by and Equations ( 27) -( 29) can be used to evaluate a closed expression for hE s z i hE

B. Fixed-shape density profiles n b r; z; t
A wide variety of functional forms for the density profile n b r; z; t can be used to evaluate hE s z i from Eqs. ( 27) - (29).For example, one approach is to take the transverse distribution function to correspond to the class of matched-beam quasiequilibrium distributions of the form G b H ? [1], where H ? p 2 ?=2m b m b ! 2 ?r 2 =2 e b is the Hamiltonian for transverse particle motion, and is the space-charge potential determined from Poisson's equation.In this case, the density profile is determined from 16)].For specified functional form of G b H ? , the radial dependence of the density profile can be determined self-consistently from Poisson's equation r ÿ1 @=@rr@=@r ÿ4e b n b for weak variations with axial coordinate z.Depending on the choice of G b H ? , the radial density profile ranges from a step-function profile for a Kapchinskij-Vladimirskij distribution [1,[9][10][11], to a bell-shaped profile with diffuse radial boundary for a thermal equilibrium distribution [1,2], to a bell-shaped profile with a sharp radial boundary for a waterbag distribution [1,14,15,38].Whatever the choice of G b H ? , the mean-square beam radius G b H ? is determined exactly for a matched beam from the radial force balance equation [1] Here, the scaled emittance in the beam frame is defined by The radial force balance equation ( 30) can be expressed as where The physically acceptable solution to Eq. ( 31) is given by Note from Eqs. ( 32) and ( 33) that the mean-square beam radius R 2 b depends on the line density b .For a lowintensity emittance-dominated beam with R 2 2R 2 , Eq. ( 33) reduces to the familiar result R 2 b ' R 2 =2! ?, which is independent of b .On the other hand, for a very-low-emittance, space-charge-dominated beam with R 2 2R 2 , Eq. (33 illustrating this behavior is shown in Fig. 1. To obtain a closed expression for hE s z i from Eqs. ( 27) -( 29), for our purposes here we specialize to the class of fixed-shape density profiles n b r; z; t of the form where fr=r b is a smooth function that depends on the scaled radial variable r=r b , and b z; t 2 R r w 0 drrn b r; z; t is the line density of beam particles.Note from Eq. ( 34) that the density profile n b r; z; t has a sharp radial edge at r r b , and that there is a distinct vacuum region outside the beam r b < r r w .From b 2 R r w 0 drrn b , we obtain from Eq. ( 34) the normalization condition Similarly, from the definition of mean-square beam radius where the constant b is defined by For specified shape function fr=r b , we note from Eqs. ( 33) and ( 36) that the edge radius r b can be expressed as obtained from the radial force balance Eq.(33).
where R 2  and R 4 are defined in Eq. ( 32), and the constant b is defined in Eq. (37).There are many practical choices of density shape function fr=r b in Eq. (34).One form of particular interest is the profile where n 0; 1; 2; . . . is a positive integer, and the normalization of f n r=r b in Eq. ( 39) satisfies Eq. (35).For the profile in Eq. ( 39), we readily obtain from Eq. ( 37) where n 0; 1; 2; . . . .Here, the root-mean-square beam radius R b is related to the edge radius r b by R 2 b b r 2 b .The profile shape function fr=r b defined in Eq. (39) gives a wide range of density profile peakedness, ranging from a step-function density profile (for n 0) to increasingly peaked profiles (for n 1; 2; . . .).The two cases corresponding to n 0 and n 2 are illustrated in Fig. 2. For n 2, note from Fig. 2 and Eqs. ( 34) and ( 39) that the bell-shaped density profile n b approaches zero continuously at r r b with n b rr b 0 @n b =@r rr b .
We reiterate that the validity of the one-dimensional kinetic model consisting of Eqs. ( 19), (22), and (26) requires that the beam dynamics remains relatively quiescent in the transverse plane (no transverse instability or beam mismatch).In this regard, it is important to recognize that a sufficient condition for transverse stability of matched-beam quasiequilibrium distributions of the form G b H ? is given by @G b H ? =@H ?0 [see, for example, Chapters 4 and 7 of Ref. [1]].Here G b H ? is the transverse distribution function, H ? p 2 ?=2m b m b ! 2 ?r 2 =2 e b r is the Hamiltonian for transverse particle motion, and is the space-charge potential.The condition @G b H ? =@H ?0 assures that there is not free energy available to drive a collective transverse instability.Finally, for the class of density profiles in Eq. ( 34), it should be noted that the number density n b r; z; t depends on axial coordinate z through the line density b z; t and the edge radius r b z; t, where r b z; t depends parametrically on b z; t through Eqs. ( 32) and (38).We emphasize, however, that the present analysis has been carried out for a long coasting beam and has not been developed for a finite-length charge bunch.Therefore, perturbations in b z; t are envisioned to be about a constant value of line density b0 const (independent of z and t).

C. Evaluation of hE s
z i 1 for fixed-shape density profile We now make use of Eq. ( 28) to evaluate the leadingorder longitudinal electric field hE s z i 1 for the class of fixed-shape density profiles described by Eq. (34).Because the beam radius r b generally depends on the line density b [e.g., see Eq. ( 38)], it is evident from Eqs. ( 28) and ( 34) that the radial integrations over n b = b required in Eq. ( 28) will also depend on b .Referring to Eq. ( 28), it is useful to define It then follows from Eqs. ( 28) and (41) that Equation ( 42) can be expressed in the compact form where the g 0 factor is defined by and h 0 b is defined in Eq. (41).We now evaluate h 0 b for the class of density profiles described by Eq. ( 34).Substituting Eq. ( 34) into Eq.( 41) and introducing the dimensionless radial variable X r=r b readily gives For specified profile shape function fr=r b , note that the second term on the right-hand side of Eq. ( 45) is constant (independent of b ), whereas the logarithmic term depends on b through the beam radius r b b [see Eq.Note from Eq. ( 47) the sensitive dependence of the constant n on the profile shape, ranging from 0 1=2 for a step-function density profile n 0 to 1 11=12 for a parabolic profile n 1, to 2 73=60 for n 2, and so on.This, of course, affects the precise values of h 0 [Eq.( 46)] and g 0 [Eq.( 44)].Finally, substituting Eq. ( 46) into Eq.( 44), we obtain for the g 0 factor Substituting the expression for r 2 b in Eq. ( 38) into Eq.( 48), we readily obtain where R 2 b e 2 b =m b ! 2 ?and R 2 =2! ?are defined in Eq. (32).Equation ( 49) clearly displays the dependence of g 0 on the line density b .For a low-intensity, emittance-dominated beam with R 2 2R 2 e , the final term in Eq. ( 49) is approximately ÿR 2 =4R 2 , which represents a negligibly small correction to 'nr 2 w =r 2 b n .On the other hand, for a space-charge-dominated beam with R 2 2R 2 the final term in Eq. ( 49) is approximately ÿ1=2, representing a sizable contribution to the g 0 factor.Indeed, for the special case of a step-function density profile (n 0 and 0 1=2) the second and third terms on the right-hand side of Eq. (49) exactly cancel in the limit of a space-charge-dominated beam R 2 2R 2 , and the g 0 factor is given approximately by g 0 ' 'nr 2  w =r 2 b as previously obtained by Reiser et al. [52,53].The expression for g 0 b in Eq. ( 49), of course, is valid for arbitrary beam intensity and the entire class of fixed-shape density profiles consistent with Eqs.(34) and (39) for n 0; 1; 2; . . . .

D. Evaluation of hE s
z i 2 for fixed-shape density profile We now evaluate the second-order electric field hE s z i 2 defined in Eq. ( 29) for the class of fixed-shape density profiles described by Eq. ( 34).As a first example, we consider the case of a low-to-moderate intensity beam R 2 2R 2 where the variation of beam radius r b with line density b is treated as negligibly small [see Eq. ( 38)].
In this case, treating n b = b as independent of b , Eq. ( 29) simplifies directly to give where the g 2 factor occurring in Eq. ( 50) is defined by We substitute Eq. ( 34) into Eq.( 51) with shape function f n r=r b n 11 ÿ r 2 =r 2 b n specified by Eq. ( 39) for 0 r < r b and n 0; 1; 2; . . . .Some straightforward but tedious integration over r in Eq. ( 51) gives the compact result Here, the constant n is defined by and 36), (38), and ( 40)].It is evident from Eq. ( 52) that the precise value of the geometric factor g 2 exhibits a sensitive dependence on profile shape.For example, it follows from Eq. ( 53) that 0 1=6 for a step-function density profile n 0, whereas 1 19=60 for a parabolic density profile n 1.Furthermore, it follows from Eq. ( 52) that g 2 > 0, with g 2 ' 1=2 for r 2 b =r 2 w 1 and g 2 ' 1=2 n for r 2 b =r 2 w ! 1.For the special case of a step-function density profile n 0, note from Eq. ( 52) that the g 2 factor is given by Evaluation of hE z i 2 from Eq. ( 29) for arbitrary beam intensity is somewhat more complicated.For present purposes, we specialize to the case of a step-function density profile n 0. Substituting Eqs. ( 34) and (39) into Eq.( 29), and carrying out the integrations over r for n 0, we obtain z.Indeed, Eq. ( 55) can generally be expressed in the form where the coefficients g g 2 , g g 3 , and g g 4 depend on b , @r 2 b =@ b , etc.
In the limit of a low-intensity beam R 2 2R 2 , where r 2 b ' 2R 2 const (independent of b ), Eqs. ( 55) and ( 56) give the expected result g g 2 g 2 ; g g 3 0; g g 4 0; where the geometric factor g 2 is defined in Eq. ( 54) for a step-function density profile.In this case, Eq. ( 56) reduces to hE s z i 2 ÿe b r 2 w g 2 @ 3 b =@z 3 .On the other hand, for a high-intensity, space-charge dominated beam R 2 2R 2 , it follows that r 2 b ' 2R 2 , and the coefficients g g 2 , g g 3 , and g g 4 in Eq. ( 56) can be approximated by Equations ( 55) -( 58) clearly display the strong dependence of hE s z i 2 on line density b .To conclude this section, Table I shows the values of the geometric factors g 0 and g 2 obtained from Eqs. ( 49) and ( 52) for several density profile shapes.Here, a low-intensity beam R 2 2R 2 has been assumed with r 2 b ' const (independent of b ).
Finally, it is important to recognize that the longitudinal dynamics considered in the present paper is assumed to be much slower than the transverse dynamics.Indeed, the longitudinal dynamics in the long-wavelength approximation considered here has characteristic frequency !jj k z r b !pb (in the beam frame), where !pb 4n b e 2 b =m b 1=2 is the plasma frequency, whereas the transverse dynamics has characteristic frequency !?! ?!pb .Since, k 2 x r 2 b 1 is assumed [see Eq. ( 23)], it therefore follows that !jj !? .Hence, in the long-wavelength approximation we can separate the two time scales.The transverse dynamics has enough time to come to a quasiequilibrium which depends parametrically on longitudinal quantities.In this case there is a self-consistent separation of the longitudinal dynamics from the transverse beam dynamics.

IV. LONGITUDINAL VLASOV-MAXWELL EQUATIONS FOR A LOW-INTENSITY BEAM
In Secs.II and III, we provided a systematic derivation of the longitudinal Vlasov-Maxwell equations and associated g factors, valid for a wide range of beam intensity and choice of transverse density profiles.In this section, we specialize to the low-intensity regime R 2 2R 2 where the dependence of the beam radius r b on line density b can be neglected.In this case, making use of Eqs. ( 19), ( 43), ( 49), (50), and (52), the Vlasov-Maxwell equations describing the evolution of the longitudinal beam distribution F b z; p z ; t and line density b z; t R 1 ÿ1 dp z F b z; p z ; t can be expressed as where Here, the geometric factors g 0 and g 2 are defined by The constants n and n are defined in Eqs. ( 47) and (53) for n 0; 1; 2; . . ., and the transverse density profile assumed in deriving Eqs. ( 59) -( 62) is specified by Eq. ( 43) ).Here, 0:5772 is Euler's constant.

Profile
Normalized profile 0 r < r b Geometric factor Geometric factor Index n  59) -(62) can be used to investigate detailed linear and nonlinear properties of the self-consistent evolution of the beam distribution function F b z; p z ; t and average self-generated electric field hE s z iz; t for a wide range of system parameters and choices of initial distribution function.For our purposes here, we summarize the key stability properties obtained from a linear stability analysis of Eqs. ( 59) and (60).Perturbations are assumed to be about a spatially uniform @=@z 0 coasting beam with equilibrium distribution function , where b0 const is the unperturbed line density.Substituting F b z; p z ; t F 0 b p z F b z; p z ; t and b z; t b0 b z; t into Eqs.( 59) and (60), and retaining terms linear in the perturbation amplitude, we obtain For present purposes, the perturbations in Eqs. ( 63) -(65) are taken to be of the form z; t ^ expik z z ÿ i!t; (66) where k z is the axial wave number, and ! is the complex oscillation frequency, with Im! > 0 corresponding to instability (temporal growth).Making use of @=@t !ÿi! and @=@z !ik z , Eqs. ( 63) and (64) can be combined to give where the dielectric function Dk z ; ! is defined by for a nontrivial solution to Eq. ( 69) with b Þ 0; note that Dk z ; !0 plays the role of a dispersion relation that determines the complex oscillation frequency ! in terms of k z and properties of the equilibrium distribution function F 0 b p z .Introducing the distribution function f 0 b p z ÿ1 b0 F 0 b p z normalized according to R 1 ÿ1 dp z f 0 b p z 1, and integrating by parts with respect to p z in Eq. (70) using @v z =@p z 1=m b , the dispersion relation Dk z ; !0 is readily expressed in the compact form The dispersion relation (71) can be used to calculate the complex oscillation frequency ! for a wide range of choices of beam distribution function f 0 b p z [1].It is convenient to introduce the effective susceptibility k z ; ! with dimensions frequency ÿ2 defined by (72) The resulting expressions for k z ; ! are displayed in Table II for various choices of distribution function f 0 b p z ranging from a cold distribution function (entry No. 1) to a Maxwellian distribution (entry No. 5).In Table II, the constant u T is the effective thermal speed, which is a measure of the velocity spread of the distribution function f 0 b p z .For the Maxwellian distribution in Table II PRST-AB 7 SELF-CONSISTENT VLASOV-MAXWELL DESCRIPTION . . .

(2004)
024401-10 024401-10 Because k 2 z r 2 w 1 has been assumed [Eq.( 23)], the final term in Eq. (74) represents a small correction.Representing !!r i! i where !r Re! and !i Im!, it is clear from Eq. ( 74) that the solutions are purely oscillatory with The solution in Eq. ( 75) corresponds to forward-moving !r =k z > 0 and backward-moving !r =k z < 0 soundlike waves propagating with constant speed u 2 b0 u 2 T 1=2 in the beam frame, with weak cubic dispersive corrections (the term proportional to u 2 b2 k 3 z ).Although the cubic dispersive corrections in Eq. ( 75) are small, it is precisely this effect that can lead to Korteveg-deVries-like solitons in a weakly nonlinear treatment of the nonlinear beam dynamics [56].
As a second example, we consider the Lorentzian distribution in entry No. 3 of Table II.Substituting the corresponding expression for k z ; !into Eq.( 71), we obtain the dispersion relation In this case, the solutions to Eq. ( 76) for k 2 z r 2 w 1 can be expressed as Because Im! ÿjk z ju T < 0, we note from Eq. (77) that the wave perturbation is damped due to resonant waveparticle interactions (classical Landau damping) for the choice of Lorentzian distribution function in Table II.This damping is weak j! i =! r j 1 whenever the effective thermal speed u T is small in comparison with the signal speed u b0 , and the damping is strong j! i =! r j * 1 whenever u T =u b0 * 1.
Detailed linear stability properties can, of course, be calculated for other choices of the distribution function in Table II.The main point of this section is that the longitudinal Vlasov-Maxwell equations (59) and (60) are rich in physics content, even at the linearization (small-signal) level.We defer further discussion of Eqs. ( 59) and (60) to a future publication in which detailed nonlinear properties are discussed.

V. TRANSFORMED EQUATIONS IN THE LABORATORY FRAME
The analysis in Secs.II, III, and IV was carried out in the beam frame (unprimed variables) where r E s 0.
The transformation of the key results back to the laboratory frame (primed variables) moving with axial velocity ÿV b ÿ b c relative to the beam frame is readily accomplished according to the transformation in Eqs. ( 3) and (4).Without presenting algebraic details, the resulting Vlasov equation for the longitudinal distribution function F 0 b z 0 ; p 0 z ; t 0 in the laboratory frame is given by Here, the average longitudinal electric field hE s 0 z i 0 z 0 ; t 0 in the laboratory frame is also readily expressed in terms of geometric factors @ 0 b =@z 0 and higher-order derivatives of the laboratory-frame line density 0 b z 0 ; t 0 R dp 0 z F 0 b z 0 ; p 0 z ; t 0 .One key modification in the laboratory frame occurs in the radial force balance equation (30), which is replaced by where 02 =4R 02 b hv 02 ?i 0 .In Eq. ( 79), b m b is the transverse mass, and the additional factor 1= 2 b 1 ÿ 2 b corresponds to a reduction in the space-charge force (the term proportional to ÿ 2 b ) due to the self-magnetic field B s 0 ÿ b E s 0 r in the laboratory frame.Equation (79) readily gives [1] [compare with Eqs. ( 32) and ( 33 for the class of fixed-shape density profiles in Eqs.(34) and (39), where b 1=n 2 is defined in Eq. ( 40) for n 0; 1; 2; . . . .Finally, in the laboratory frame, some straightforward algebra shows that the formal expressions for the average longitudinal electric field hE s 0 z i 0 z 0 ; t 0 in Eqs. ( 9), (26), and (28) are modified with the replacements n b !n 0 b = b , b ! 0b = b , and @=@z !ÿ1 b @=@z 0 .For example, Eq. ( 43) is modified to become where the laboratory-frame geometric factor g 0 0 0 b is defined by Here, g 0 0 b is defined in Eq. ( 44) [or Eq. ( 49 Similarly, in the laboratory frame, for low-to-moderate beam intensity, the second-order electric field in Eq. ( 50) is modified to become where the geometric factor g 0 2 is defined by and g 2 is defined in Eqs. ( 51) - (53).Similarly, for arbitrary beam intensity, the second-order electric field in the laboratory frame has the same form as in Eq. ( 56) with the right-hand side of Eq. ( 56) scaled by 1= 4 b , and b and @=@z replaced by primed variables.
In concluding this section, it should be emphasized that the detailed theoretical analysis was relatively straightforward to carry out in the beam frame.The key results in the laboratory frame were then readily obtained by Lorentz transformation back to the laboratory frame.

VI. CONCLUSIONS
The purpose of the present paper was to develop an improved one-dimensional kinetic model describing the self-consistent nonlinear evolution of the longitudinal distribution function and average axial electric field hE s z iz; t for a very long charge bunch (coasting beam) propagating through a cylindrical conducting pipe with radius r w , and confined in the transverse direction by an applied focusing force F tr foc described in the smooth-focusing approximation.For simplicity, to illustrate the basic approach, in the present analysis the cylindrical pipe was assumed to be perfectly conducting, and the beam transport geometry was assumed to be straight (linear).As discussed in Sec.II, the starting point was the fully nonlinear, three-dimensional Vlasov-Maxwell equations for the distribution function f b x; p; t and selfgenerated fields in the beam frame (unprimed variables).A reduced Vlasov equation for the longitudinal distribution function F b z; p z ; t R dxdy R dp x dp y f b x; p; t was obtained by integrating over the transverse phasespace variables x ?; p ? .Making the single ansatz that the dependence of the distribution function f b x; p; t on axial momentum p z is factorable, led to a closed system of equations describing the self-consistent evolution of the longitudinal distribution function F b z; p z ; t and the average axial electric field hE s z iz; t.Here, assuming axisymmetry in the transverse plane @=@ 0, the average h i denotes the weighted transverse spatial average over n b r; z; t= b z; t defined in Eq. ( 26 b n for 0 r < r b , and fr=r b 0 for r b < r r w .Here, n 0; 1; 2; . . . is an integer, with n 0 corresponding to a step-function density profile, and r b is the edge radius of the beam.Moreover, the root-mean-square beam radius R b hr 2 i 1=2 and edge radius r b generally depend on the line density through the radial force-balance condition in Eqs.(33) and (38), respectively.Of course, this dependence is weak (r b ' const) for a low-intensity, emittancedominated beam, whereas r 2 b / b for a very-low-emittance, space-charge-dominated beam.Denoting @=@z L ÿ1 z k z , we assumed slow axial variations of F b z; p z ; t and b z; t with k 2 z r 2 w 1.The average electric field hE s z iz; t, expressed in terms of @ b =@z and higher-order derivatives, together with closed forms for the corresponding geometric factors were then calculated selfconsistently in Sec.IV correct to 0k 2 z r 2 w for the class of bell-shaped density profiles with n 0; 1; 2; . . ., described above.The results showed a strong dependence of the geometric factors on profile shape and beam intensity b .
In Sec.IV, as a simple application, the resulting coupled equations for the longitudinal distribution function f b z; p z ; t and the average electric field hE s z iz; t were solved in the linearization approximation for the case of low-to-moderate beam intensity treating r b ' const (independent of b ).As expected, the analysis led to collective oscillations with sound-wave-like characteristics modified by cubic dispersion.The detailed oscillation and damping properties of the perturbation, of course, depended on the choice of equilibrium distribution function F 0 b p z about which the system is perturbed.Finally, in Sec.V, the key results derived in the beam frame were transformed back to the laboratory frame to facilitate practical applications of the kinetic model of the longitudinal beam dynamics developed here.
In conclusion, the present analysis has developed an improved one-dimensional kinetic model describing the nonlinear evolution of the longitudinal distribution function F b z; p z ; t and average electric field hE s z iz; t.The analysis has been carried out for arbitrary beam intensity and general density profile shape, leading to important generalizations of previous g-factor models [see Eqs. ( 43) -( 45), (50), and (51)].Since the longitudinal beam dynamics typically depends in detail on the precise values and degree of nonlinearity of the g factors, it is expected that the new results presented here will have several key applications.
In future investigations, the kinetic model for the longitudinal beam dynamics developed here will be applied to various investigations, including the nonlinear Korteweg-deVries-like evolution of axial disturbances and the self-consistent generation of compressional solitons [56].
), where n b R d 3 pf b is the number density of beam particles, and b R dp z F b R dxdyn b is the axial line density of beam particles.In Sec.III, we assume that the beam dynamics is relatively quiescent in the transverse plane (no transverse instability or beam mismatch), and take the transverse density profile to have the fixed-shape form n b b =r 2 b fr=r b where the shape function fr=r b n 11 ÿ r 2 =r 2

PRST
radius r b satisfying l b r b .The coasting beam is made up of particles with charge e b and rest mass m b propagating in the z direction with directed axial kinetic energy b ÿ 1m b c 2 , where b 1 ÿ 2 b ÿ1=2 f b x; p; t in the equivalent form

FIG. 2 .
FIG.2.Shape function profile f n r=r b defined in Eq. (39) plotted versus r=r b for n 0 and n 2.
a step-function density profile, r 2 b depends on b through the force-balance constraint r 2 b b R 2 R 4 4R 4 1=2 , where R 2 b e 2 b =m b ! 2 ?and R 2 =2! ?[see Eqs.(32) and (38) for n 0].For a general value of beam intensity, note from Eq. (55) that hE s z i 2 generally has a nonlinear dependence on b and derivatives of b with respect to PRST-AB 7 SELF-CONSISTENT VLASOV-MAXWELL DESCRIPTION . . .

with profile shape function f n r=r b n 11 ÿ r 2
=r 2 b n for 0 r < r b .Equations ( which relates the perturbed distribution function F b to the perturbed line density b .It is convenient to introduce effective sound speeds u b0 and u b2 associated with the geometric factors g 0 and g 1 defined by (67) and (68) into Eq.(65) then gives Dk z ; !b 0; r b ; R ; R ; b !r 0 ), where n b R d 3 pf b is the number density of beam particles, and b R dp z F b R dxdyn b is the axial line density of beam particles.In Sec.III, we assumed that the beam dynamics was relatively quiescent in the transverse plane (no transverse instability or beam mismatch), and took the transverse density profile to have the fixed-shape form n b b =r 2 b fr=r b , where the shape function fr=r b n 11 ÿ r 2 =r 2

TABLE I .
Values of the geometric factors g 0 and g 2 for several density profile shapes and low beam intensity (R 2 2R 2

TABLE II .
Susceptibility k z ; ![Eq.(72)] for various choices of f 0 b p z .