Kinetic description of intense beam propagation through a periodic focusing field for uniform phase-space density

The Vlasov-Maxwell equations are used to investigate the nonlinear evolution of an intense sheet beam with distribution function fb x, x0, s propagating through a periodic focusing lattice kx s 1 S kx s , where S const is the lattice period. The analysis considers the special class of distribution functions with uniform phase-space density fb x, x0, s A const inside of the simply connected boundary curves, x0 1 x, s and x0 2 x, s , in the two-dimensional phase space x, x0 . Coupled nonlinear equations are derived describing the self-consistent evolution of the boundary curves, x0 1 x, s and x0 2 x, s , and the self-field potential c x, s ebf x, s gbmbb bc. The resulting model is shown to be exactly equivalent to a (truncated) warm-fluid description with zero heat flow and triple-adiabatic equation of state with scalar pressure Pb x, s const nb x, s 3. Such a fluid model is amenable to direct analysis by transforming to Lagrangian variables following the motion of a fluid element. Specific examples of periodically focused beam equilibria are presented, ranging from a finite-emittance beam in which the boundary curves in phase space x, x0 correspond to a pulsating parallelogram, to a cold beam in which the number density of beam particles, nb x, s , exhibits large-amplitude periodic oscillations. For the case of a sheet beam with uniform phase-space density, the present analysis clearly demonstrates the existence of periodically focused beam equilibria without the undesirable feature of an inverted population in phase space that is characteristic of the Kapchinskij-Vladimirskij beam distribution.


I. INTRODUCTION
Periodic focusing accelerators, transport systems, and storage rings [1][2][3][4][5] have a wide range of applications ranging from basic research in high energy and nuclear physics, to applications such as heavy ion fusion, spallation neutron sources, tritium production, and nuclear waste transmutation.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, t͒, and the electric and magnetic fields, E͑x, t͒ and B͑x, t͒.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.For example, the only known fully self-consistent equilibrium solution (including electric and magnetic self-fields) to the nonlinear Vlasov-Maxwell equations for an intense beam propagating through a periodic focusing field configuration is the so-called Kapchinskij-Vladimirskij (KV) distribution function f KV b [1,[6][7][8][9].Such a distribution, due to its highly inverted population in phase space, of course is of very limited practical interest.While Hamiltonian averaging techniques have been developed [31][32][33][34] that justify the smooth-focusing approximation and thereby permit investigation of a whole class of (approximate) beam equilibria, these averaging techniques typically require sufficiently small vacuum phase advance (s vac , 60 ± , say) and other approximations for their validity.Therefore, whether or not there exist periodically focused non-KV solutions to the Vlasov-Maxwell equations remains a question of continued fundamental importance, which we examine in this paper for an intense sheet beam propagating through a periodic focusing field.
To briefly summarize, the present analysis considers an intense sheet beam which is infinite in the y dimension and propagates in the z direction with average axial velocity V b b b c const and directed kinetic energy ͑g b 2 1͒m b c 2 , where g b ͑1 2 b 2 b ͒ 21͞2 is the relativistic mass factor, m b is the rest mass of a beam particle, and c is the speed of light in vacuo.The beam propagates through a periodic focusing lattice k x ͑s 1 S͒ k x ͑s͒, where S const is the lattice period, which provides transverse confinement of the beam particles in the x direction.The self-consistent evolution of the system is described by the nonlinear Vlasov-Poisson equations [Eqs.(1) and (2)] for the beam distribution function, f b ͑x, x 0 , s͒, and the normalized self-field potential, c͑x, s͒ e b f͑x, s͒͞g b m b b 2 b c 2 .Here x 0 dx͞ds is the dimensionless velocity, e b is the charge of a beam particle, and f͑x, s͒ is the space-charge potential.In the present analysis, we consider the special class of distribution functions with uniform phase-space density in which f b ͑x, x 0 , s͒ A const [Eq.(23)] inside of the simply connected boundaries, x 0 1 ͑x, s͒ and x 0 2 ͑x, s͒, in the phase space ͑x, x 0 ͒.Coupled equations are derived describing the self-consistent evolution of the boundary curves, x 0 1 ͑x, s͒ and x 0 2 ͑x, s͒, and the self-field potential c͑x, s͒ [Eqs.( 27)- (29)].Quite remarkably, the resulting model is found to be exactly equivalent to a (truncated) warm-fluid description with zero heat flow and triple-adiabatic equation of state [Eqs.(33), (35), (36), and (37)].Such a fluid model is amenable to direct analysis by transforming to Lagrangian variables following the motion of a fluid element [35,36].
The organization of the paper is the following.The theoretical model based on the Vlasov-Maxwell equations is summarized in Sec.II, including a derivation of the statistical rate equations describing the general nonlinear evolution of the rms beam thickness, centroid motion, and the unnormalized beam emittance.In Sec.III, the Vlasov-Maxwell equations are simplified for the case of a sheet beam with uniform phase-space density; the dynamical equations are derived for the boundary curves, x 0 1 ͑x, s͒ and x 0 2 ͑x, s͒, in phase space (Sec.III A); and the equivalence of the model to a (truncated) warm-fluid description with triple-adiabatic equation of state is demonstrated (Sec.III B).For the specific example of a pulsating parallelogram with uniform phase-space density, closed dynamical equations for the self-consistent evolution of the system are derived in Sec.IV.Finally, in Sec.V, the (closed) warm-fluid model derived in Sec.III is transformed to Lagrangian variables ͑x 0 , t͒ following the motion of a fluid element.This leads to a single nonlinear partial differential equation [Eq.( 76)] for the number density n b ͑x 0 , t͒ of beam particles.Specific numerical examples corresponding to large-amplitude collective oscillations in the cold-beam limit are also considered in Sec.V, including back transformation to the laboratory frame.
For the case of a sheet beam with uniform phase-space density, the present analysis clearly demonstrates the existence of periodically focused beam equilibria without the undesirable feature of an inverted population in phase space that is characteristic of the Kapchinskij-Vladimirskij beam distribution.It should be emphasized that the existence of periodically focused beam equilibrium for a non-KV distribution with uniform density in the twodimensional phase space ͑x, x 0 ͒ does not imply that periodically focused beam equilibria exist for non-KV beam distributions in four and six dimensions.

II. THEORETICAL MODEL
In the present analysis, we consider an intense sheet beam, made up of particles with charge e b and rest mass m b , which is infinite in the y dimension ͑≠͞≠y 0͒, and propagates in the z direction with average axial velocity b ͒ 21͞2 is the relativistic mass factor, and c is the speed of light in vacuo.The beam propagates through a periodic focusing lattice k x ͑s 1 S͒ k x ͑s͒, where S const is the lattice period, which provides transverse confinement of the beam particles in the x direction.
Introducing the (dimensionless) self-field potential c͑x, s͒ ϵ e b f͑x, s͒͞g b m b b 2 b c 2 , where f͑x, s͒ is the electrostatic (space-charge) potential, the nonlinear Vlasov-Maxwell equations describing the self-consistent nonlinear beam dynamics and collective processes in the paraxial approximation are given by [37] ≠f b ≠s 1 In Eqs. ( 1) and ( 2), f b ͑x, x 0 , s͒ is the distribution of particles in the two-dimensional phase space ͑x, x 0 ͒, x 0 dx͞ds is the (dimensionless) velocity in the x direction, s s 0 1 b b ct is a normalized time variable, the focusing coefficient k x ͑s 1 S͒ k x ͑s͒ has dimensions ͑length͒ 22 , and is the number density of beam particles (number of beam particles per unit volume).For present purposes, we assume that perfectly conducting walls are located at x 6x w (the case where x w !`is not excluded), and enforce the boundary conditions in solving Eqs. ( 1) and (2).Furthermore, the constants K b and N b occurring in Eqs. ( 1) and ( 2) are defined by and Here, K b is the normalized self-field perveance, with dimensions of ͑length͒ 21 , and N b R dx n b ͑x, s͒ is the area number density of beam particles, with dimensions of ͑length͒ 22 .The validity of Eqs. ( 1) and ( 2) assumes negligibly small axial momentum spread in the z direction and that the particle motions are nonrelativistic in a frame of reference moving with the beam (axial velocity in the z direction b b c const).We further assume that f͑x, x 0 6`, s͒ 0 , f͑jxj .jx 0 j, x 0 , s͒ 0 , (7) such that there are no beam particles beyond some transverse x dimension jxj jx 0 j , x w .
084402-2 084402-2 The Vlasov-Maxwell equations ( 1) and ( 2), subject to the boundary conditions in Eqs. ( 4) and (7), are valid for an intense sheet beam in the paraxial approximation and can be used to describe the nonlinear beam dynamics and collective processes in the phase space ͑x, x 0 ͒ over a wide range of system parameters and applied focusing field configurations k x ͑s 1 S͒ k x ͑s͒.While the full solutions for f b ͑x, x 0 , s͒ and c͑x, s͒ are of special interest, it is also possible to derive exact equations for the evolution of statistical averages ͗x͘ ͑s͒.Here, the statistical average of a general phase function x͑x, x 0 , s͒ over the phase-space distribution f b ͑x, x 0 , s͒ is defined by where is the area number density of beam particles in the sheet beam.For x x and x x 0 , taking the appropriate moments of Eq. ( 1), readily gives [38] and where use has been made of Eq. ( 7).Substituting Eq. ( 9) into Eq.( 10) readily gives for the evolution of the centroid location ͗x͘ ͑s͒, In Eq. ( 11), the average self-field force, 2͗≠c͞≠x͘, is determined self-consistently from Eqs. (1), (2), and (8).
Similarly, it can be shown (exactly) from Eqs. ( 1) and ( 8) that and or equivalently, In addition, it can be shown from Eqs. ( 1) and (8) that or equivalently, where use is made of Eq. (12).Equations ( 11), (14), and (16), derived from the Vlasov equation ( 1), are exact equations describing the nonlinear evolution of the statistical averages ͗x͘ ͑s͒, ͗x 2 ͘ ͑s͒, and ͗x 02 ͘ ͑s͒.In a similar manner, making use of Eq. ( 1), dynamical equations can be derived for the evolution of statistical averages for higherorder moments, ͗x m ͘, ͗x 0 n ͘, and ͗x p x 0 q ͘, for m .2, n .2, etc.
For future reference, it is convenient to rewrite Eq. ( 14) in terms of the mean-square beam dimension X 2 b ͑s͒ defined by We further make use of the identities Substituting Eqs. ( 17) and ( 18) into Eq.( 14), and making use of Eq. ( 11) to eliminate ͑d 2 ͞ds 2 ͒ ͗x͘, we readily obtain Subtracting out the centroid motion, it is convenient to introduce the unnormalized beam emittance e x ͑s͒ defined by where X 2 b ͗͑x 2 ͗x͒͘ 2 ͘, and use has been made of Eq. (18).Using Eq. ( 20) to eliminate ͗͑x 0 2 ͗x 0 ͒͘ 2 ͘ in favor of e 2 x ͑s͒ in Eq. (19) gives directly the dynamical equation Equation ( 21) is fully equivalent to Eq. ( 14) and describes the nonlinear evolution of the rms beam thickness X b ͑s͒.
In a similar manner, Eq. ( 16) for ͗x 0 2 ͘ ͑s͒ can be replaced by a dynamical equation for the unnormalized beam emittance.Without presenting algebraic details, making use of Eqs. ( 11), ( 14), (16), and (18), we obtain 084402-3 084402-3 Equation (22) shows clearly that space-charge effects (proportional to ≠c͞≠x) generally cause a variation in the beam emittance e x ͑s͒.Only in the limit of very low beam intensity ͑j≠c͞≠xj !0͒, or very special choices of distribution function f b ͑x, x 0 , s͒ (see Sec. IV), is the emittance a conserved quantity.

III. NONLINEAR VLASOV-MAXWELL EQUATIONS FOR UNIFORM PHASE-SPACE DENSITY
We now return to the Vlasov-Maxwell equations (1) and (2) for the distribution function f b ͑x, x 0 , s͒ and self-field potential c͑x, s͒.For specified applied focusing field k x ͑s 1 S͒ k x ͑s͒ and initial distribution function f b ͑x, x 0 , s 0͒, obtaining the solutions to Eqs. (1) and (2) is generally difficult analytically, although solutions to Eqs. (1) and (2) are accessible using nonlinear df simulation techniques [22][23][24][25].For present purposes, we consider a special case where considerable analytical simplification occurs in the analysis of Eqs. ( 1) and (2).In particular, as illustrated in Fig. 1, we consider the case where the distribution function f b ͑x, x 0 , s͒ has constant phase-space density (independent of x, x 0 , and s) within the simply connected boundary curves x 0 1 ͑x, s͒ and x 0 2 ͑x, s͒, and zero phase-space density outside.That is, we take If f b ͑x, x 0 , s͒ satisfies Eq. ( 23) initially at s 0, then the nonlinear Vlasov equation ( 1) assures that the phase-space density remains constant at subsequent values of s as the boundary curves x 0 1 ͑x, s͒ and x 0 2 ͑x, s͒ distort and evolve Phase-space plot of distribution function with uniform phase-space density f b ͑x, x 0 , s͒ A const in the region nonlinearly in response to the applied focusing field and the self-generated fields.Of course as the system evolves, Eq. ( 1) assures that or equivalently from Eq. ( 23) and Fig. 1, That is, no matter how complicated the evolution of the boundary curves, x 0 1 ͑x, s͒ and x 0 2 ͑x, s͒ in Fig. 1, the total area within the phase-space boundary remains constant.
In the subsequent analysis, we assume that the boundary curves, x 0 1 ͑x, s͒ and x 0 2 ͑x, s͒, in Fig. 1 remain singlevalued functions of x as the system evolves.
In summary, for the case of constant phase-space density in Eq. ( 23) and Fig. 1, there has been an enormous simplification in the kinetic description based on the Vlasov-Maxwell equations ( 1) and ( 2).In particular, Eqs. ( 1) and (2), which are partial differential equations in the three variables ͑x, x 0 , s͒, are replaced exactly by Eqs. ( 27)- (29), which are partial differential equations in the two variables ͑x, s͒.Equations ( 27)-( 29) can be solved analytically in at least one case of special interest (Sec.IV), and can be integrated numerically for a wide range of initial phase-space boundaries, x 0 1 ͑x, s 0͒ and x 0 2 ͑x, s 0͒.

B. Equivalence to a warm-fluid model
It is instructive to recast the basic equations derived in Sec.III A in a form familiar in macroscopic warm-fluid descriptions widely used in plasma physics [35].In this regard, we introduce the number density n b ͑x, s͒, and the (normalized) macroscopic flow velocity, V xb ͑x, s͒, pressure, P b ͑x, s͒, and heat flow, Q b ͑x, s͒, defined by Referring to Eq. ( 23) and Fig. 1, some straightforward algebra gives Note from Eq. ( 31) that the transverse flow velocity V xb ͑x, s͒ is given by and the pressure P b ͑x, s͒ can be expressed as where Pb0 ͞ n3 b0 ϵ 1͞12A 2 const is a constant coefficient.Furthermore, because there is no "skew" in the x 0 dependence of f b ͑x, x 0 , s͒ in Eq. ( 23), the heat flow Q b is identically zero ͑Q b 0͒.Making use of the expressions for n b V xb , P b , and V xb in Eqs.(31) and (32) gives the useful identity We now return to the basic dynamical equations for x 0 1 and x 0 2 derived in Eqs. ( 25) and (26) in Sec.III A. Substituting Eqs. ( 31)- (34) into Eqs.(25) and (26) gives directly the familiar macroscopic continuity and force balance equations, and where P b ͑x, s͒ ͑ Pb0 ͞ n3 b0 ͒n 3 b ͑x, s͒ satisfies the tripleadiabatic pressure relation in Eq. (33).Furthermore, Poisson's equation ( 27) is simply expressed as Finally, Eq. ( 36) can be further simplified by making use of Eqs.(33) and (35) to eliminate ≠n b ͞≠s.We readily obtain 084402-5 084402-5 in the region where n b fi 0. Equation (38) shows clearly that the transverse acceleration of a beam fluid element is produced by the combined effects of the pressure-gradient force ͑2n 21 b ≠P b ͞≠x͒, the applied focusing force ͑2k x x͒, and the self-field force ͑2≠c͞≠x͒.
To summarize, for the case of constant phase-space density in Eq. ( 23) and Fig. 1, the macroscopic fluid description provided by Eqs. ( 33), ( 35), (36), and (37), or equivalently, Eqs. ( 35), (37), and (38), is fully equivalent to the nonlinear Vlasov-Maxwell equations ( 1) and ( 2).This remarkable simplification, i.e., closure of the macroscopic fluid equations with the first two velocity moments for n b ͑x, s͒ and V xb ͑x, s͒, is a consequence of the fact that the heat flow satisfies Q b ͑x, s͒ 0 exactly for the class of beam distribution functions in Eq. ( 23) and Fig. 1 [35].Similar to Sec.III A, the Vlasov-Maxwell equations ( 1) and ( 2), which are nonlinear partial differential equations in the three variables ͑x, x 0 , s͒, have been replaced by the macroscopic fluid-Maxwell equations ( 35), (37), and (38), which are nonlinear partial differential equations in the two variables ͑x, s͒.It should be pointed out that Eqs. ( 35), (37), and (38) are readily amenable to numerical solution, and can also be investigated analytically, e.g., by introducing a Lagrangian transformation to a frame of reference moving with velocity V xb ͑x, s͒ of a beam fluid element.
Equations ( 35), (37), and (38) can be used to investigate detailed beam propagation properties over a wide range of focusing field configurations k x ͑s 1 S͒ k x ͑s͒ and system parameters.In this regard, it should be noted that the dynamical equations for the beam centroid ͗x͘ ͑s͒ [Eq.(11)], the rms beam thickness X b ͑s͒ [Eq.( 21)], and the unnormalized beam emittance e x ͑s͒ [Eq.( 22)], derived in Sec.II, also apply to the class of distribution functions described by Eq. ( 23) and Fig. 1.

IV. PULSATING PARALLELOGRAM WITH UNIFORM PHASE-SPACE DENSITY
As an application of the analysis in Sec.III, we now consider the uniform phase-space distribution illustrated in Fig. 2, where the boundary curves x 0 1 ͑x, s͒, x 0 2 ͑x, s͒, x 1 b ͑s͒, x 2 b ͑s͒ corresponds instantaneously to a parallelogram centered at the phase-space point (͗x͘, ͗x 0 ͘), where ͗x͘ ͑s͒ is the centroid position, and ͗x 0 ͘ ͑s͒ ͑d͞ds͒ ͗x͘ ͑s͒ is the centroid velocity.Here, the boundary curves are defined by where a͑s͒ is a (yet unspecified) s-dependent coefficient.The distribution function f b ͑x, x 0 , s͒ with constant phasespace density consistent with Eq. (39) and Fig. 2 is given by It is clear from Eq. (40) and Fig. 2 that as required by the definition of statistical averages in Eq. ( 8).In the subsequent analysis in Sec.IV, we consider the class of solutions in which the phase-space boundaries x 2 b ͑s͒ and x 1 b ͑s͒ in Fig. 2 remain vertical, i.e., there is not an initial perturbation corresponding to a tilt (relative to the vertical) of the boundaries x 2 b and x 1 b in Fig. 2.

A. Evaluation of macroscopic quantities and statistical averages
The simple shape of the boundary curves in Fig. 2 and Eq.(40) makes it straightforward to calculate the various macroscopic properties and statistical averages of physical interest.For example, it follows directly from Eq. (40) and Fig. 2 that the density profile n b ͑x, s͒ R dx 0 f b ͑x, x 0 , s͒ corresponds to the simple step-function profile and the area number density N b R dx dx 0 f b ͑x, x 0 , s͒ is given by Here, 2x b ͑s͒ and 2x 0 b ͑s͒ are the thicknesses of the parallelogram in Fig. 2 in the x direction and the x 0 direction, respectively.While both x b ͑s͒ and x 0 b ͑s͒ depend on s, it is clear from Eq. (43) that the product x b ͑s͒x 0 b ͑s͒ is constant.Furthermore, combining Eqs. ( 42) and (43), the 084402-6 084402-6 The average flow velocity V xb ͑x, s͒ n 21 b R dx 0 3 x 0 f b ͑x, x 0 , s͒ is also straightforward to calculate from Eq. (40) and Fig. 2. Some algebraic manipulation gives where ͗x͘ ͑s͒ and ͗x 0 ͘ ͑s͒ ͑d͞ds͒ ͗x͘ ͑s͒ are the centroid position and velocity.
To evaluate the statistical averages ͗x 2 ͘, ͗x 0 2 ͘ and ͗xx 0 ͘, we make use of the identities (see Sec. II) where statistical averages are defined by ͗x͘ N 21 b R dx dx 0 xf b .Some straightforward algebraic manipulation that makes use of Eqs. ( 40) and (43) and Fig. 2 gives the simple expressions Therefore, from Eq. ( 47), the mean-square beam thickness X 2 b ͑s͒ ͗͑x 2 ͗x͒͘ 2 ͘ is given by where x b ͑s͒ is the half thickness of the beam.Substituting ͗͑x 2 ͗x͒͘ ͑x 0 2 ͗x 0 ͒͘͘ a͑s͒X 2 b ͑s͒ [Eq.( 47)] into Eq.( 46) gives d ds Making use of ͗x 2 ͘ X 2 b 1 ͗x͘ 2 [Eq.( 46)] and ͑d͞ds͒ ͗x͘ ͗x 0 ͘, Eq. (49) gives directly where X b ͑s͒ ͑1͞ p 3 ͒x b ͑s͒ is the rms beam thickness.Equation (50) determines a͑s͒ in terms of X b ͑s͒.Making use of Eqs. ( 48) and (50), the expressions in Eq. ( 47) reduce exactly to for the choice of distribution function in Eq. (40) and Fig. 2. The unnormalized beam emittance (subtracting out the centroid motion) defined in Eq. ( 20) is a quantity of considerable physical interest.Making use of Eqs. ( 20) and (51), we obtain where N b const is the area number density of beam particles, and A const is the phase-space density of beam particles.The fact that e 2 x ͑s͒ e 2 x0 const for the choice of distribution function in Eq. (40) and Fig. 2 leads to enormous simplification in the analysis of the dynamical equation (21) for the evolution of the rms beam thickness X b ͑s͒ ͑1͞ p 3 ͒x b ͑s͒.
We now turn to a determination of the self-field potential c͑x, s͒ required in evaluating the statistical averages ͗≠c͞≠x͘ and ͗͑x 2 ͗x͒͘≠c͞≠x͘ occurring in the rate equations (11) and (21) for the centroid position ͗x͘ ͑s͒ and rms beam thickness X b ͑s͒.

B. Solution for self-field potential c͑ ͑ ͑x, s͒ ͒ ͒
We now solve Poisson's equation (37) for the choice of step-function density profile n b ͑x, s͒ in Eq. ( 42), which corresponds to the distribution function f b ͑x, x 0 , s͒ in Eq. (40) and Fig. 2. Making use of Eqs. ( 42)-( 44), Poisson's equation (37) becomes where n b ͑s͒ N b ͞2x b ͑s͒ is the particle number density.Referring to Fig. 3, Eq. ( 54) can be solved for c͑x, s͒, enforcing the boundary conditions c͑x 6x w , s͒ 0 at the conducting walls, and continuity of c͑x, s͒ and ≠c͑x, s͒͞≠x at the leftmost boundary of the density profile, x 2 b ͑s͒ ͗x͘ 2 x b ͑s͒, and at the rightmost boundary of the density profile, x 1 b ͑s͒ ͗x͘ 1 x b ͑s͒, in Fig. 3.For the three regions in Fig. 3, making use of Eq. ( 54) and n b ͑s͒ N b ͞2x b ͑s͒, the solution for c͑x, s͒ can be expressed as c͑x, s͒ The solution for c͑x, s͒ in Eq. ( 55) is valid even in the case where the centroid ͗x͘ ͑s͒ undergoes nonlinear motion, provided the beam surfaces do not come in contact with the conducting walls at x 6x w .Of course Eq. (55) simplifies for the case of a centered beam with ͗x͘ 0.
Of particular interest in the rate equations (11) and ( 21) for the centroid position ͗x͘ ͑s͒ and rms beam thickness where n b ͑x, s͒ R dx 0 f b ͑x, x 0 , s͒ is the number density of beam particles.For the choice of distribution function in Eq. (40) and Fig. 2, the density profile n b ͑x, s͒ has the step-function profile in Eq. (42) and Fig. 3, and the expression for ͗≠c͞≠x͘ in Eq. (56) reduces to ø ≠c ≠x where c II ͑x, s͒ is defined in Eq. ( 55 In a similar manner, it can be shown from Eqs. ( 55) and (56) that ø where use is made of X b ͑s͒ x b ͑s͒͞ p 3. The expressions for the statistical averages in Eqs. ( 58) and (59), valid for the choice of distribution function in Eq. (40) and Fig. 2, can be used directly in the rate equations for ͗x͘ ͑s͒ and X b ͑s͒ in Eqs.(11) and (21).

C. Rate equations for ͗ ͗ ͗x͘ ͘ ͘ ͑ ͑ ͑s͒ ͒ ͒ and X b ͑ ͑ ͑s͒ ͒ ͒
We substitute Eqs. ( 58) and (59) into the rate equations (11) and ( 21) for the motion of the beam centroid ͗x͘ ͑s͒ and rms beam thickness X b ͑s͒ ͗͑x 2 ͗x͒͘ 2 ͘ 1͞2 .This gives and where e x0 is the constant emittance defined in Eq. ( 53).Note from Eqs. ( 58) and ( 60) that the self-field force on the beam centroid, 2͗≠c͞≠x͘ ͑pK b ͞x w ͒ ͗x͘, is always defocusing and is proportional to the displacement ͗x͘ from the center position ͑x 0͒, and the constant factor pK b ͞x w .Furthermore, the self-field force in Eq. ( 60) is associated with image charges in the conducting wall.In particular, at fixed beam intensity ͑K b ͒, the self-field force in Eq. (60) becomes negligibly small as x w !`.On the other hand, from Eqs. ( 59) and (61), the self-field force term is 2X 21 b ͗͑x 2 ͗x͒͘≠c͞≠x͘ pK b ͞ p 3 const, which is also defocusing, but is independent of X b and ͗x͘.
Equations ( 60) and (61) constitute closed dynamical equations for the motion of the beam centroid ͗x͘ ͑s͒ and rms beam thickness X b ͑s͒.Moreover, Eqs. ( 60) and ( 61) can be integrated numerically for a wide range of choices of lattice function k x ͑s͒, beam emittance e x0 , beam intensity K b , and conducting wall location x w .Note from Eq. (60) that if the beam is initially centered with ͗x͘ ͑s 0͒ 0 ͓d͗x͘͞ds͔ s0 , then ͗x͘ 0 ͑d͞ds͒ ͗x͘ at all subsequent s.Most importantly, solving Eqs. ( 60) and (61) for ͗x͘ ͑s͒ and X b ͑s͒ is fully equivalent to solving the nonlinear Vlasov-Maxwell equations ( 1) and ( 2 Therefore, a determination of ͗x͘ ͑s͒ and X b ͑s͒ from Eqs. ( 60) and (61) fully specifies the distribution function in Eq. ( 40) and Fig. 2. For matched-beam solutions X b ͑s 1 S͒ X b ͑s͒ to the nonlinear rms envelope equation ( 61), the shape of the parallelogram in Fig. 2 pulsates with period S in a frame of reference centered at ͑͗x͘, ͗x 0 ͒͘.Depending on the initial conditions for ͗x͘ ͑s 0͒ and ͓͑d͞ds͒ ͗x͔͘ s0 , however, the motion of the centroid ͑͗x͘, ͗x 0 ͒͘ in Fig. 2 can be more irregular, with both fast-oscillatory and slow-oscillatory components.
For present purposes, we examine Eqs. ( 60) and (61) for the choice of a periodic step-function lattice k x ͑s 1 S͒ k x ͑s͒ illustrated in Fig. 4. Here, kx const is the lattice amplitude, and h is the filling factor.For simplicity we consider here a centered beam with ͓͗x͔͘ s0 ͓d͗x͘͞ds͔ s0 0, so that ͗x͘ d͗x͘͞ ds 0 at all subsequent s.Typical numerical results obtained from Eq. ( 61) for the rms beam thickness X b ͑s͒ are shown in Fig. 5, where X b ͑s͒͞ p e x0 S is plotted versus s͞S for a matched beam with moderate space-charge intensity propagating through the periodic step-function lattice in Fig. 4. Here, the dimensionless system parameters correspond to kx S 2 14.92, filling factor h 0. and the corresponding phase-space boundaries in Eq. ( 62) correspond to a periodically focused beam equilibrium with x 0 6 ͑x, s 1 S͒ x 0 6 ͑x, s͒ and x 6 b ͑s 1 S͒ x 6 b ͑s͒.

V. LAGRANGIAN DESCRIPTION OF NONLINEAR BEAM DYNAMICS
For the case of uniform phase-space density, it was shown in Sec.III B that the basic dynamical equations for a planar sheet beam propagating through a periodic focusing field could be cast into the form of the macroscopic warm-fluid equations ( 35 where E x ͑x, s͒ solves Here, y 2 Tb ϵ Pb ͞ n3 b0 const is the normalized thermal speed.As noted in Sec.III, for uniform phase-space density, Eqs. ( 63)-( 65) are exactly equivalent to the Vlasov-Maxwell equations ( 1) and ( 2), and Eqs. ( 27)-( 29) for c͑x, s͒ and the phase-space boundary curves, x 0 1 ͑x, s͒ and x 0 2 ͑x, s͒.

A. Dynamical equations in Lagrangian variables
The fluid-Maxwell equations ( 63)-( 65) are particularly amenable to analysis in Lagrangian variables following the motion of a fluid element [35,36].We introduce the Lagrangian variables ͑x 0 , t͒ following a fluid element defined by t s , Here, t is a (normalized) time variable, and it follows exactly from Eq. (66) that derivatives transform according to From Eq. ( 67) we obtain and the continuity equation (63) in Lagrangian variables becomes Equation ( 69) can be integrated exactly with respect to t to give Note that Eq. (70) gives a closed expression for n b ͑x 0 , t͒ in Lagrangian variables in terms of n b ͑x 0 , 0͒ and V xb ͑x 0 , t͒.
Poisson's equation (65) also simplifies in Lagrangian variables.Making use of Eq. (67) we obtain where n b ͑x 0 , t͒ is given by Eq. (70).Equations ( 70) and (71) readily give A very important consequence of transforming to Lagrangian variables is evident from Eq. (72).In particular, E x ͑x 0 , t͒ E x ͑x 0 , 0͒ is independent of t, and depends only on the initial density profile n b ͑x 0 , 0͒.We now return to the force balance equation (64), transforming to Lagrangian variables according to (67), (68), and (70).This gives (exactly) where E x ͑x 0 , t͒ is determined self-consistently in terms of n b ͑x 0 , 0͒ from Eq. ( 72).Operating on Eq. ( 73) with ≠͞≠x 0 and rearranging terms, we readily obtain Equation ( 74) is a closed, partial differential equation for the density compression factor in Lagrangian variables, which is fully equivalent to the original dynamical equations ( 63)-(65) in laboratoryframe variables.Substituting Eq. (75) into Eq.(74) gives directly in the region where n b ͑x 0 , 0͒ is nonzero.Equation (76) [or equivalently, Eq. ( 74)] constitutes the final dynamical equation in Lagrangian variables and can be used to investigate the detailed nonlinear dynamics of intense beam propagation for a wide variety of input density profiles n b ͑x 0 , 0͒, lattice functions k x ͑t͒, normalized beam intensity ͑K b ͒, and beam emittance (proportional to y 2 Tb ).Furthermore, Eq. ( 76) is well posed as an initialvalue problem.For specified n b ͑x 0 , 0͒ and V xb ͑x 0 , 0͒, it follows from Eq. (75 and Eq. ( 76) can generally be integrated numerically to determine n b ͑x 0 , t͒ once the initial profiles for n b ͑x 0 , 0͒ and V xb ͑x 0 , 0͒ and beam parameters are specified.
Once n b ͑x 0 , t͒ is determined from Eq. ( 76), the expression for n b ͑x 0 , 0͒͞n 0 ͑x 0 , t͒ can be used to formally determine the inverse transformation to laboratory-frame variables ͑x, s͒ defined in Eq. (66).For example, consider the case where the initial density profile is an even function of x 0 with n b ͑2x 0 , 0͒ n b ͑x 0 , 0͒, and the initial flow velocity profile is an odd function of x 0 with V xb ͑2x 0 , 0͒ 2V xb ͑x 0 , 0͒ and V xb ͑x 0 0, 0͒ 0. It readily follows from Eqs. (74)-(76) that where V xb ͑x 0 0, t͒ 0 for all values of t.Integrating Eq. (75) with respect to x 0 then gives which is required to determine the inverse transformation from Lagrangian variables ͑x 0 , t͒ to laboratory-frame variables ͑x, s͒ in Eq. (66).

B. Cold-beam limit
As noted earlier, Eq. ( 76) [or equivalently, Eq. ( 74)] can be used to describe in Lagrangian variables the beam dynamics for a wide variety of initial profiles and system parameters.For present purposes, we consider the special case of a cold beam with negligible transverse emittance, i.e., In this case, Eq. ( 76) simplifies to become 084402-11 084402-11 Note that Eq. ( 81) is an inhomogeneous, linear, ordinary differential equation for 1͞n b ͑x 0 , t͒, with x 0 occurring as a continuous parameter.Indeed, Eq. ( 81) can be integrated numerically with respect to t for a wide variety of periodic lattice functions k x ͑t 1 S͒ k x ͑t͒, and initial density profiles n b ͑x 0 , 0͒ consistent with Eq. (77).

C. Examples in cold-beam limit
The Lagrangian formulation developed in Secs.V A and V B can be applied to a wide variety of intense beam profiles n b ͑x 0 , 0͒ and V xb ͑x 0 , 0͒.To illustrate the power of the Lagrangian formalism in analyzing the beam dynamics, we consider here two simple examples in the cold-beam limit.
The first example corresponds to a periodic stepfunction lattice in which k x ͑t 1 s͒ k x ͑t͒ has the waveform illustrated in Fig. 4 with filling factor h and constant amplitude kx .In this case, the periodic solutions w x ͑t 1 s͒ w x ͑t͒ obtained (numerically) from Eq. ( 83) have vacuum phase advance s vac determined from Moreover, the corresponding value of the (approximate) smooth-focusing coefficient k sf is given by [1] In the first example, we consider the choice of initial density profile at t 0 corresponding to Here, nb n b ͑x 0 6x b0 , 0͒ is the initial edge density of the sheet beam, and D is a (dimensionless) measure of the amplitude of the initial density perturbation at x 0 0, with n b ͑x 0 0, 0͒ nb ͑1 1 D͒.It is further assumed that V xb ͑x 0 , 0͒ 0, and therefore n b ͑x 0 , t͒ and V xb ͑x 0 , t͒ evolve according to the symmetries in Eq. (78).Finally, we introduce the (dimensionless) measure of normalized beam intensity s b defined by  5), the dynamical equation (81) for n b ͑x 0 , t͒ in Lagrangian variables can be expressed (exactly) in the equivalent form (92) As a numerical example corresponding to the stepfunction lattice in Fig. 4 and the initial density profile in Eq. (90), we consider the choice of system parameters kq S 2  Note from Eq. ( 93) that the beam has large intensity close to the space-charge limit ͑s b 1͒, and that the density perturbation (relative to the uniform beam density) has moderately large amplitude ͑D 20.2͒.The numerical results corresponding to Eq. ( 93) are illustrated in Figs.6-8.In Fig. 6, the profile for n b ͑x 0 , t͒ obtained numerically from Eqs. (82)-( 86) is plotted versus x 0 and t.Note that n b ͑x 0 , t͒ n b ͑2x 0 , t͒ evolves symmetrically, as expected from Eqs. ( 78) and (90).In Lagrangian variables, it is evident from Fig. 6 that the layer maintains constant thickness 2x b0 , but undergoes strong oscillatory modulation as a function of t͞S, and over the layer cross section as a function of x 0 ͞x b0 .From Eqs. (66) and ( 79), the corresponding backtransformation to laboratory-frame variables ͑x, s͒ consistent with Eq. ( 92) and Fig. 6 is determined from For example, at the layer edge 6x b ͑s͒, Fig. 7 shows a plot of x b ͑s͒ ϵ x͑x b0 , s͒ obtained numerically from Eqs. ( 82)-(86) and Eq. ( 93).Note from Fig. 7 that the layer edge in the laboratory frame, 6x b ͑s͒, has a fast oscillatory modulation with period equal to the lattice period S, plus a slow oscillatory modulation with period approximately equal to 2p͞ p k sf 2pS.Finally, making use of Eq. ( 70), or equivalently, the numerical solution for n b ͑x 0 , t͒ obtained from Eqs. ( 82)-( 86), with t s and x 0 x 0 ͑x, s͒, the density profile n b ͑x, t͒ in laboratory-frame variables ͑x, s͒ is illustrated in Fig. 8, which clearly shows the large-amplitude modulation of the density profile in the laboratory frame.As a second example, we adopt a smooth-focusing model in which the lattice function k x ͑t͒ is replaced by the constant value k sf const in Eq. ( 81), or equivalently, in (92).In this case, Eq. ( 92) is exactly integrable for general initial density profile n b ͑x 0 , 0͒.Assuming initial conditions with n b ͑2x 0 , 0͒ n b ͑x 0 , 0͒ and   94) for the initial density profile in Eq. ( 90) and choice of system parameters in Eq. ( 93).
V xb ͑2x 0 , 0͒ 2V xb ͑x 0 , 0͒, it follows exactly from Eqs. ( 75) and (92) that the solutions for n b ͑x 0 , t͒ and V xb ͑x 0 , t͒ can be expressed as and where we have introduced the notation k b p k sf .
Furthermore, it follows from Eqs. (66) and (96) that the laboratory-frame variables ͑x, s͒ and Lagrangian variables ͑x 0 , s͒ are related by The condition for the backtransformation (97) to remain single valued is given by ≠x͞≠x 0 .0, which is equivalent to the requirement that the solution for n b ͑x 0 , t͒ in Eq. ( 95) remain non-negative.Evaluating Eq. (95) or Eq.(97) at k b t p, some straightforward algebra shows that the condition for the transformation to remain single valued is given by which assures that wave breaking does not occur.As a particular choice of initial density profile, we consider the case where which is illustrated in Fig. 9 for k 0 x b0 5p͞2 and D 0.45.Note from Eq. (99) and Fig. 9 that n b ͑x 0 , 0͒ corresponds to a sinusoidal density perturbation with amplitude D nb superimposed on a flattop density profile with constant density nb .The corresponding transformation of variables consistent with Eqs. ( 97) and ( 99) is given by s t , and the inequality in Eq. (98) gives the requirement  which assures that the transformation in Eq. (100) remains single valued.As a numerical example, we consider the choice of parameters Shown in Fig. 10 is a plot of the inverse transformation x͑x 0 , s͒ versus x 0 obtained from Eq. (100) at k b s 0, p͞2, p, 3p͞2, 2p for the choice of system parameters in Eq. ( 102), over the interval k 0 jx 0 j , k 0 x b0 5p͞2.Note from Fig. 10

VI. CONCLUSIONS
In this paper, the Vlasov-Maxwell equations ( 1) and ( 2) were used to investigate the evolution of an intense sheet beam with distribution function f b ͑x, x 0 , s͒ propagating through a periodic focusing lattice k x ͑s 1 S͒ k x ͑s͒, where S const is the lattice period.The analysis considered the special class of distribution functions with uniform phase-space density f b ͑x, x 0 , s͒ A const inside of the simply connected boundary curves, x 0 1 ͑x, s͒ and x 0 2 ͑x, s͒, in the two-dimensional phase space ͑x, x 0 ͒ [Eq.( 23 Such a model is amenable to direct analysis by transforming to Lagrangian variables ͑x 0 , t͒ following the motion of a fluid element.This resulted in the single nonlinear partial differential equation (76) for the number density n b ͑x 0 , t͒ of beam particles in Lagrangian variables, with backtransformation to the laboratory-frame variables ͑x, s͒ specified by Eq. (66).Specific examples of periodically focused beam equilibria were presented, ranging from a finite-emittance beam in which the boundary curves in phase space correspond to a pulsating parallelogram (Sec.IV), to a cold beam in which the number density of beam particles exhibits large-amplitude periodic oscillations (Sec.V).For the case of a sheet beam with uniform phase-space density [Eq.( 23)], the present analysis clearly demonstrates the existence of periodically focused beam equilibria without the undesirable feature of an inverted population in phase space that is characteristic of a KV beam distribution.In future work, the warm-fluid model developed in Secs.III B and V I will be used to derive a nonlinear Schrödinger equation describing the evolution of perturbations about a uniform density beam, including soliton solutions.It should be emphasized that the existence of periodically focused beam equilibrium for a non-KV distribution with uniform density in the two-dimensional phase space ͑x, x 0 ͒ does not imply that periodically focused beam equilibria exist for non-KV beam distributions in four and six dimensions.
V b b b c and directed kinetic energy ͑g b 2 1͒m b c 2 , where g b ͑1 2 b 2

FIG. 3 .
FIG. 3. Plot versus x of the step-function density profile n b ͑x, s͒ in Eq. (42) corresponding to the parallelogram distribution in Eq. (40) and Fig. 2. Here, n b ͑s͒ N b ͞2x b ͑s͒, and the density profile is centered at x ͗x͘ ͑s͒.

FIG. 4 . 0 FIG. 5 .
FIG. 4. Plot of lattice function k x ͑s͒ versus s͞S for a periodic step-function lattice with amplitude kx const and filling factor h.

where v2 pb 4p nb e 2 b
͞g b m b is the relativistic plasma frequency squared, and v bЌ p k sf b b c is the (smoothfocusing) betatron frequency.Then, making use of the definition of K b in Eq. (