Kinetic theory of solitary waves on coasting beams in synchrotrons Hans Schamel and Renato Fedele Citation: Physics of Plasmas 7, 3421 (2000); doi: 10.1063/1.874206 View online: http://dx.doi.org/10.1063/1.874206 View Table of Contents: http://scitation.aip.org/content/aip/journal/pop/7/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Studies of beam injection with a compensated bump and uncompensated bump in a synchrotron Rev. Sci. Instrum. 84, 083303 (2013); 10.1063/1.4817676 Strip injection for carbon ion synchrotrons Rev. Sci. Instrum. 78, 096104 (2007); 10.1063/1.2786933 Beam Loss Control on the ISIS Synchrotron: Simulations, Measurements, Upgrades AIP Conf. Proc. 693, 154 (2003); 10.1063/1.1638344 Simulation of space charge effects in a synchrotron AIP Conf. Proc. 448, 73 (1998); 10.1063/1.56782 Micromap approach to space charge in a synchrotron AIP Conf. Proc. 448, 233 (1998); 10.1063/1.56776 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 132.180.92.90 On: Wed, 15 Jul 2015 12:38:30
Transcript
Kinetic theory of solitary waves on coasting beams in synchrotronsHans Schamel and Renato Fedele Citation: Physics of Plasmas 7, 3421 (2000); doi: 10.1063/1.874206 View online: http://dx.doi.org/10.1063/1.874206 View Table of Contents: http://scitation.aip.org/content/aip/journal/pop/7/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Studies of beam injection with a compensated bump and uncompensated bump in a synchrotron Rev. Sci. Instrum. 84, 083303 (2013); 10.1063/1.4817676 Strip injection for carbon ion synchrotrons Rev. Sci. Instrum. 78, 096104 (2007); 10.1063/1.2786933 Beam Loss Control on the ISIS Synchrotron: Simulations, Measurements, Upgrades AIP Conf. Proc. 693, 154 (2003); 10.1063/1.1638344 Simulation of space charge effects in a synchrotron AIP Conf. Proc. 448, 73 (1998); 10.1063/1.56782 Micromap approach to space charge in a synchrotron AIP Conf. Proc. 448, 233 (1998); 10.1063/1.56776
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
Kinetic theory of solitary waves on coasting beams in synchrotronsHans SchamelPhysikalisches Institut, Universita¨t Bayreuth, D-95440 Bayreuth, Germany
Renato FedeleDipartimento di Scienze Fisiche, Universita` ‘‘Federico II’’ and INFN, Complesso Universitario di M.S.Angelo, I-80126 Napoli, Italy
~Received 20 January 2000; accepted 19 April 2000!
Observations1,2 as well as numerical simulations2,3 of acoasting ion~hadron! beam in circular accelerators and stor-age rings have shown the excitation of long-lived coherentstructures superimposed on the beam. Such a beam, which isinteracting with the electromagnetic fields induced by thering environment, can be unstable at high intensities and candevelop holes~or notches! in the longitudinal distributionfunction and in the associated line density. Another way ofexcitation is an external forcing, e.g., by means of an appliedvoltage impressed on the beam.1 Analytically, these holeshave been described4,5 in steady state by equilibria of theVlasov–Poisson system for large resistive coupling imped-ances. A typical scenario of this structure formation processis that of an initially linear resistive wall instability, whichsaturates nonlinearly by particle trapping. The trapping ofparticles in the potential trough of the electrostatic structurehas rather dramatic consequences. It not only affects thephase space topology in the vicinity of the structure, but alsochanges more or less the wave characteristics. In general,there is no relation anymore to standard linear wave theoriesand to their nonlinear descendants4,5 and hence a fluid ap-proach for its description is no longer appropriate. It is akinetic approach which has to be invoked.
The change in the wave characteristics is exemplarilyseen by the spatial shape of a solitary wave which possessesa sech4(z) dependence rather than a Korteweg–de Vries-type~KdV-type! sech2(z) dependence. The latter typically resultsfrom a nonlinear wave analysis which rests in the small am-plitude limit on linear hydrodynamic waves, such as slowspace charge waves. Solitary waves on coasting and bunched
beams based on these latter linear wave branches have beenexamined in Refs. 6 and 7.
In the present analysis we generalize the kinetic theoryof solitary waves by taking into account a more general cou-pling impedance and evaluate the various possibilities ofstructure formation. We extend the description of hole equi-libria and, by taking appropriate limits, we also recover thehydrodynamic KdV-type solitons. The plan of the paper is asfollows.
In Sec. II we present the governing system of equationsand derive two well-known expressions for the generalizedPoisson’s equation for two limiting cases. In Sec. III wediscuss qualitatively the possible existence of hole and humpsolutions in the zero resistivity limit. In Sec. IV the varioussolitary potential structures are discussed more quantitativelyfor the two extremal limits of resistivity,R→` and R→0,respectively, and a first necessary condition for the existenceof solutions is presented. The second condition, the nonlineardispersion relation~NDR!, is evaluated in detail in Sec. V forthree different regimes of the phase velocity. In Sec. VI amacroscopic fluid description is chosen as a basis to deriveKdV soliton solutions and a comparison is made with thecorresponding class of solutions of Sec. V. Another type of ararefaction KdV soliton is also derived. The paper finisheswith a summary and conclusions in Sec. VII.
II. BASIC EQUATIONS
Using the notation of Ref. 4@see also the Appendix,where a derivation of~2! is presented#, the basic equationsare
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
132.180.92.90 On: Wed, 15 Jul 2015 12:38:30
] t f 1 z]z f 2hR0
E0
c
bv0
2qEz] zf 50, ~1!
Ez1S b
2g D 2
]zL1S b
Ag0
,zD1
bb
2c F R2e0bbc3
R0L]zGL1~b,z!50, ~2!
where
L1~r ,z!5ql1~z!
pe0r 2 2Ez8~z!. ~3!
Equation~1! is the Vlasov equation for the particle distribu-tion in the labframe8 and ~2! is the equation describing theself-fieldEz(z) in the frame comoving with the perturbation,as derived in the Appendix. Except for the factorq/pe0r 2,the quantityL1 can be regarded as the perturbed line densitycorrected by the space charge effect associated with the per-turbation.
Two known expressions are easily recovered as limitingcases of~2!. In the limit of a strong resistivity~R→`! it isthe third term in~2! that survives, givingL1(b,z)50 or
Ez8~z!5ql1~z!
pe0b2 , ~4!
which is the well-known Poisson equation governing thefield in the presence of a resistive instability.
In the limit of a purely reactive coupling impedance~R→0!, ~2! instead becomes
Ez1S b
2g D 2
]zL1S b
Ag0
,zD 2e0~bbc!2
2R0L]zL1~b,z!50.
~5!
If we, in addition, ignore inL1 the space charge correctiontermEz8(z) and determine the total voltage a particle sees perturn, given byU52pR0Ez , we get withv5bc,
U52qvR0F g0
2bg2 Z02vR0
LGl18~z!, ~6!
an expression that coincides with Hofmann’s Eq.~5! in Ref.8. In ~6! the first term on the right-hand side~RHS! repre-sents the capacitive space charge effect (Z051/e0c5377V), and the second term is the inductive contributionto the imaginary impedance (n51, Ref. 9!
Zi5g0Z0
2bg22v0L,
wherev0R05v5bc.Adopting the normalization
z
2pR0→z,
v0t
2p→t,
hq
b2E0f→f,
~7!
eªhqEz
2pR0
b2E0, uª
Dv
v0,
2pR0l1
N→l1
and introducing, furthermore, the following dimensionlessparameters:
aª2hNR0
e0E0S q
bbD 2
, mªS 4pgR0
b D 2
,
~8!
RªR 4pR0bg2
bc, LªL 2e0
R0~bgc!2,
we get as the governing system
@] t1u]z2e]u# f ~z,u,t !50, ~9!
~12L !e91Re82me5a@Rl11~g02L !l18#, ~10!
where the dash stands for]z .Note thata carries the sign of the slip factorh and that
m@1. R and L are the dimensionless resistance and the in-ductivity, respectively. The capacitive space charge effect isrepresented in~10! by the termg0l18 , where the geometryfactor g0 , with g0.1 is defined in~A14!.
III. A PRELIMINARY QUALITATIVE DISCUSSION
To get a first impression about the possible structureshidden in~9! and~10! we take the limit of a purely reactiveimpedance (R→0) in which case Eq.~10! reduces to
e95me1al18 , ~11!
where we defined
mª
m
12L, aªa
g02L
12L. ~12!
Assuming first a not too large inductance,L,1, we then seethat m and a carry the sign ofm anda, respectively. By anintegration, usinge52f8, we find from~11!
f952al11mf. ~13!
In the following we look for a positive solitary potentialhump, i.e., we assumef>0. ~Later, in Sec. VI,f<0 willalso be admitted.! This is suggested by~9!, which equals anelectron Vlasov equation and has, when coupled with Pois-son’s equation, electron hole solutions10 with f>0.
Note that according to~7!, it holds thatf;hqf, whichconsequently determines the sign of the dimensional electricpotentialf. Furthermore, the structure of~9! and ~13! tellsus that below transition energy,a,0, the system resemblesthat of a neutral plasma~later called a positive mass system!because the first term on the RHS of~13! becomesuau@* f du21#, which is the electron density corrected bythe unperturbed ion density. We hence have an electron holesolution forq,0 ~antiproton beam! and an ion hole situationfor q.0 ~proton beam!. For a beam above transition energy~a.0! the situation is quite different, as the dimensionalelectric potentialf changes sign. However, if in addition thesign of the mass is changed too, the physics of the Vlasovequation for a given species can be interpreted as before,which means that the potential again can trap particles. Fur-thermore, since the curvature off at maximum excursionhas changed its sign too, we now need a hump instead of ahole in the distribution function to be consistent with Pois-son’s equation. This is what Dory’s mass-conjugationtheorem11 essentially predicts. This preliminary discussion issketched qualitatively in Fig. 1 for a potential structure that
3422 Phys. Plasmas, Vol. 7, No. 8, August 2000 H. Schamel and R. Fedele
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
132.180.92.90 On: Wed, 15 Jul 2015 12:38:30
is moving more or less with the nominal beam velocity. Itwill be substantiated and refined in the following paragraphs,noting that the termmf in ~13!, of course, contributes to thesolubility of the problem, and, furthermore, that in the purelyresistive case (R→`) a different scenario may be valid.
From a linear point of view, a beam above transitionenergy,h.0, is known in the space charge dominated re-gime to be subject to the negative mass instability8,12 and,hence, the clump formation in the right column of Fig. 1 canbe viewed as a nonlinear saturated state of this instability.For a discussion of the necessity of a linear instability as aprecursor of nonlinear structure formation see, however,Ref. 4.
IV. A MORE GENERAL EVALUATION OF SOLITARYWAVE SOLUTIONS
As before, we keep the conditionf>0, which implies anO-type separatrix in phase space corresponding to an ellipticfix point in the phase portrait of a simple pendulum~see alsoFig. 1!. This comes from the fact that the dimensionless par-ticle energy in~9! is given bye5u2/22f, and, hence, theseparatrix~e50! corresponds tous(z)56A2f, showing anO-type ~elliptic! topology. We could, of course, also investi-gate solutions havingf<0, in which case anX-type ~hyper-bolic! separatrix would prevail, yielding further classes ofequilibria. This situation, however, deserves a separate in-vestigation~see, however, Secs. V A and VI!. In the positivemass case, a beam in the unperturbed asymptotic region~f50! would then be double-humped and therefore two-stream like.
A stationary solution of~9! is now found, as in Ref. 4,by (z2Dut) ansatz, whereDu stands for the yet unknown
phase velocity of the structure in the comoving frame, i.e., inthe frame moving with the nominal beam velocity.
We switch now to the frame moving with the phase ve-locity, as implied by the ansatz, and assume that the unper-turbed beam is represented by a shifted Maxwellian in thiswave frame. A stationary solution of the Vlasov equation~9!is then provided by the distribution~14! of Ref. 4, whichdepends on the two constants of motione and sgnv ~forpassing particles! and which is a continuous, non-negativedistribution function. The parameterb, which controls thestate of trapped particles, is left open. Its influence on thesolution and determination will be part of the following in-vestigation.
Taking the velocity integral over this distribution, we getin the low amplitude limit,c!1, wherec represents theamplitude of the wave and 0<f(z)<c the following per-turbed line density:13
l1[l21
521
2Zr8S Du
A2D f2
4b
3f3/21
1
16Zr-S Du
A2D f21¯ ,
~14!
whereZr is the real part of the plasma dispersion function14
and b is defined as
b5p21/2~12b2Du2!exp~2Du2/2!. ~15!
Expression~14! inserted into~10! gives a third-order ordi-nary differential equation forf(z) which has to be solved,subject to appropriate boundary conditions.
However, to keep the analysis as simple as possiblewithout losing too much physics, we concentrate on two lim-iting cases:
~1! the Ohmic case (R@1),~2! the purely reactive case (R50).
For R@1 the generalized Poisson’s equation~10! reduces tof952al1 where we usede52f8. Inserting~14!, we canwrite it as
f95Af1Bf3/21Cf2[2V8~f!, ~16!
with constantsA, B, C depending onDu ~andb! given by
A5a
2Zr8~Du/A2![A1 , ~17a!
B54ab
3[B1 , ~17b!
C52a
16Zr-~Du/A2![C1 . ~17c!
For R50, on the other hand, Eq.~10! reduces to
2~12L !f-1mf85a~g02L !l18 .
By integrating this equation with respect tox, using theboundary conditionf505f9 ~at uzu5`), we again get anequation of the form~16! but with a different set of con-stants, given this time by
A5m1a 12Zr8~Du/A2![A2 , ~18a!
FIG. 1. The dimensional electric potentialf as a function of space and thecorresponding phase space pattern for four different cases~q is the charge ofa beam particle,h is the slip factor!. In brighter regions of phase space thedistribution function is less dense. In all cases the dimensionless potentialf;qhf is bell-shaped, satisfying 0<f<c.
3423Phys. Plasmas, Vol. 7, No. 8, August 2000 Kinetic theory of solitary waves on coasting beams in . . .
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
132.180.92.90 On: Wed, 15 Jul 2015 12:38:30
B54ab
3[B2 , ~18b!
C52a
16Zr-~Du/A2![C2 , ~18c!
wherem and a are defined in~12!.The procedure of getting a solution is now straightfor-
ward. Multiplying ~16!, in which the ‘‘classical’’ potentialV(f) has already been introduced, byf8 and integrating wefind
f8~z!2
21V~f!50, ~19!
whereV(f) is given by
2V~f!5Af2
21
2
5Bf5/21
C
3f3 ~20!
and whereV(0)50 is assumed. A quadrature of~19! imme-diately gives
z5E0
f df
A22V~f![z~f!, ~21!
which by inversion yieldsf(z), the desired potential shapeof the solution. Two constraints must, however, be satisfiedin order to get a localized solution
V~c!50, ~22a!
V~f!,0, 0,f,c. ~22b!
Equation~22a! determinesDu in terms of the five param-etersc, b, a, m, L and hence represents a nonlinear disper-sion relation~NDR!. It becomes
A14
5BAc1
2C
3c50. ~23!
With the use of~23! the ‘‘classical’’ potential can be writtenas
2V~f!522B
5f2@Ac2Af#2
C
3f2~c2f!, ~24!
and inserting this expression into~21!, we obtain byinversion15
f~z!5c@11~11Q!21tanh2 y#22sech4 y ~25!
with
yªA2Cc~11Q!
24, Q5
6B
5CAc. ~26!
Expression~25! describes the shape of the solitary wave andhas two interesting limits.
WhenQ5B50, ~25! reduces to
f~z!5c sech2SA2Cc
6zD , ~27!
which is the hydrodynamic KdV soliton. Its existence obvi-ously requires
C,0. ~28!
On the other hand, whenuQu@1, i.e., uBu@uCuAc, then weget
f~z!5c sech4SA2Bc1/2
20zD , ~29!
an expression found earlier for nonisothermal ion acousticsolitons15,16and for electron17 and ion holes,18 see also Refs.10 and 19. Again, the existence requires
B,0. ~30!
Two questions remain:~1! Under these circumstances is theNDR ~23! solvable?~2! What can we learn from its solution?
V. THE NONLINEAR DISPERSION RELATION „NDR…
AND ITS SOLUTIONS
To answer these questions we now discuss in some de-tail the first condition@Eq. ~22a!# which is manifest in~23!.ReplacingC by Q through~26! it can be written as
2A54
5BS 11
1
QDAc ~31!
or, by using expressions~17! and ~18!,
21
2Zr8S Du
A2D 5D, ~32!
whereD is given in the Ohmic case~being characterized bythe index 1! by
D5 1615b~11Q1
21!Ac[D1 ~33!
and in the purely reactive case by
D5m/a1 1615b~11Q2
21!Ac[D2 . ~34!
Figure 2 shows the function2 12Zr8 in ~32! as a function
of x5Du/A2. We see that a solution of~32!, with D giveneither by~33! or by ~34!, exists ifD satisfies
20.285<D<1. ~35!
FIG. 2. The real part of the plasma dispersion function, more precisely
212Zr8(x), as a function ofx. Two branches, distinguished by its slope are
labeled. IfD.0, only one solution of the NDR~32! exists; if D5D8,0 asecond solution~on the hydrodynamic branch! becomes possible.
3424 Phys. Plasmas, Vol. 7, No. 8, August 2000 H. Schamel and R. Fedele
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
132.180.92.90 On: Wed, 15 Jul 2015 12:38:30
If D is in addition positive, 0<D<1, a solution exists, how-ever, only on the so-called kinetic branch, which impliesx<0.924 orDu<1.307. In this case the phase velocityDulies in the thermal range of the distribution function. If, onthe other hand,D is negative and satisfies20.285<D,0, afurther solution becomes possible, lying on the so-called hy-drodynamic branch, which typically implies phase velocitiesin the tail of the distribution function. In the latter case, afluid description can equally well be used, as will be shownlater.
In Secs. V A–V C we discuss these possibilities sepa-rately. For this reason we first quote three expansions of2 1
2Zr8(x) valid in different regions ofx:
21
2Zr8~x!55
122x2~122x2/31... , uxu!1
21
x0~x2x0!1~x2x0!2, ux2x0u!1
x050.924
21
2x2 S 113
2x2 1...D , uxu@1.
~36a!
~36b!
~36c!
A. Nonpropagating solitary wave structures
First, we evaluate the existence of solitary structureswhich are resting in the bulk of the distribution. For this casethe left-hand side~LHS! of ~32! can be replaced by unity byvirtue of ~36a!, valid in the limit x5Du/A2→1, and we getthe relation
D51, ~37!
whereD stands forD1 ~33! in the Ohmic and forD2 ~34! inthe purely reactive case. In both expressionsb simplifies to
b5p21/2~12b!, ~38!
according to~15! in the limit Du→0. Furthermore, using2 1
2Zr-(0)524, which follows from~36a!, we first get in theOhmic case forD1 by evaluating~33!
D15 1615bAc2c/3. ~39!
Inserting~38! and ~39! in ~37! we then obtain
~12b!5 1516Ap
c~11c/3!, ~40!
which fixes the trapping parameterb for a given amplitudec. Since c!1, b'2 15
16Ap/c, i.e., it is strongly negative.The distribution at zero velocity is depressed. The ratio be-tween the distribution at the center of the structure,u50, andthe distribution at the separatrix,u5A2c, becomes
f ~0!
f ~A2c!5ebc'11bc512
15
16Apc. ~41!
In the small amplitude limit, a small depression~notch! ofthe distribution function at the resonant velocity,Du50, isalready sufficient to let a nonpropagating solitary wave struc-ture exist, which is of the type~29!. Using B154ab/3'5a/4Ac, we get from~29!
f~z!5c sech4SA2a
4zD . ~42!
This means the width of the solitary structure is independentof the amplitude and dependent of (2a)21/2. The widththerefore broadens as the beam approaches the transition en-ergy. In the strongly resistive case (R@1), a nonpropagatingsolitary wave structure exists, provided that2a.0, thatmeans, in view of~8!, that h,0. A hole-type solitary wavestructure requires a beam below transition energy in theOhmic regime. Since, according to~7! the dimensional elec-tric potentialf is related tof by f;hqf, we see that foran antiproton beam (q,0) f is a positive hump~e-holecase!, and for a proton beam (q.0) f is a negative dip~i-hole case!, as depicted also in the left column of Fig. 1. Incontrast, for an ‘‘Ohmic’’ beam above transition energy anonpropagating solitary wave structure, as proposed in theright column of Fig. 1, does not exist. The picture of Fig. 1,which was deduced for the purely reactive case, cannot sim-ply be transferred to the resistive case.
Hence, we come to the somewhat surprising result thatabove transition energy a resistive wall instability cannotsaturate in a standing solitary hole structure by particle trap-ping, valid in the small amplitude limit, and under the as-sumption off>0.
However, a quick look at the other type of solitarywaves having anX-type separatrix shows that a saturation byparticle trapping can indeed occur in this situation. The ar-gument goes as follows.
For this other type of solitary waves the line density~14!must be modified and becomes
l5S 12k0
2
2c D F12
1
2Zr8S Du
A2D ~f1c!
24b
3~c1f!3/21¯G . ~148!
Inserting this in f952a(l21)[2V8(f), the Poissonequation for the Ohmic case, we see that asf→2c thecurvature off is positive for a beam above transition energy~a.0! (f95ak0
2c/2.0 at f52c!, as it should.Since the curvature should vanish asf→0 ~i.e., at uzu
→`), we get for a standing structureDu50 (2 12Zr8(0)
51):
k02/2512
4b
3Ac
and the ‘‘classical potential’’V(f) becomes afterf integra-tion with V(2c)50,
2V~f!5aH c~c1f!21
2~c1f!2
24b
3~c1f!Fc3/22
2
5~c1f!3/2G1¯J .
The NDR,V(0)50 in this case, becomes
3425Phys. Plasmas, Vol. 7, No. 8, August 2000 Kinetic theory of solitary waves on coasting beams in . . .
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
132.180.92.90 On: Wed, 15 Jul 2015 12:38:30
8b
5Ac51,
so thatV(f) simplifies to
2V~f!5aH ~2f!~c1f!
22
1
3
~c1f!
Ac
3@c3/22~c1f!3/2#1¯J ,
which has the desired properties for a solitary wave associ-ated with anX-type separatrix, as one can see by inspection.
Now we turn to the purely reactive case.With b given by~38!, B25(4a/3)b andC252a/2 and,
therefore,Q25216b/15Ac. D2 from ~34! then becomesD25m/a1(16/15)bAc2c/3 and the NDL ~37!: D251yields
~12b!515
16Ap
c S m2a
2a D , ~43!
where the last term inD2 has already been ignored becauseof its smallness. The shape of the structure is given by~29!,with B given byB2 , and becomes
f~z!5c sech4SAm2a
4zD . ~44!
We hence arrive at the condition
a,m ~45!
for the existence of the solution. Sincem according to~12!and~8! is positive~we assumedL,1), a can be both, nega-tive and positive.
For a negativea5a(g02L)/(12L) alsoa is negative~because ofL,1,g0) and from~43! and~45! b is stronglynegative. Below transition energy we have a nonpropagatingsolitary wave solution~44! with a distribution having a notchin its center as depicted the left column of Fig. 1. On theother hand, ifa, and thereforea, is positive@and satisfies~45!# b from ~43! is strongly positive and hence the distribu-tion function has an additional hump at the center as depictedin Fig. 1 right column.
Hence, in the purely reactive case we indeed confirm thepreliminary investigation of Sec. III for sufficiently low in-ductances. A negative mass instability may therefore saturateby particle trapping, giving rise to a solitary wave structureassociated with an overpopulation of trapped particles.
B. Solitary wave structures with phase velocities inthe thermal range
Next, we allow a propagating structure and select for thesake of demonstration the velocity range satisfying~36b!. Din ~32! must therefore be small and the NDR~32! has thesolution (A2x051.307)
Du51.307@12D#, ~46!
with D given either by~33! or by ~34!. In the case ofQ21
negligible the structure is given by~29! with B either given
by ~17b! ~resistive case! or by ~18! ~purely reactive case!.With Du given by ~45!, b from ~15! becomes
b5p21/2~2b20.71!exp~20.854!'0.24~2b20.71!.~47!
The existence condition is in the resistive case
B154a
3b,0 ~48!
and in the purely reactive case
B254a
3b,0. ~49!
Hence, below transition energy~a,0! b must be positive,and from~46! we find
2b.0.71, ~50!
i.e.,b must again be negative corresponding to a notch in thedistribution function at the resonant velocity~45!.
On the other hand, above transition energy~a.0! bmust be negative, and it must hold from~46! that
b.20.71. ~51!
A plateau-like trapped particle distribution~b50! suffices tosatisfy this condition, and a hump-like distribution~b.0! is,of course, also admissible. In other words, the situation assketched in Fig. 1, which was approved for standing solitarywaves in the purely reactive case, can be transformed topropagating structures as well.
The smallness ofD is justified in the Ohmic case by~33!and becomes in the purely reactive case
m!uau~g02L !. ~52!
In the latter case one should not be too close to the transitionenergy.
C. Hydrodynamic solitons
To investigate the possible existence of solitary wavesolutions propagating with velocities large compared withthermal beam velocity we make use of~36c!.
The NDR ~32! then becomes
21
~Du!2 S 113
~Du!2D5D, ~53!
which means thatD must be negative and its modulus small
1
~Du!2 .2D!1. ~54!
From ~36c! with ~53! we find to lowest order for the thirdderivative of2 1
2Zr ,
21
2Zr-S Du
A2D '212D2 ~55!
and b from ~15! then turns out to be exponentially small(b;exp(1/2D)) and negligible. This means that in bothcasesB1505B2 and we have the situation of~27!, whichimplies C,0. The resistive case yields from~17c!
3426 Phys. Plasmas, Vol. 7, No. 8, August 2000 H. Schamel and R. Fedele
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
132.180.92.90 On: Wed, 15 Jul 2015 12:38:30
C152 32aD1
2 ~56!
and the purely reactive case gives from~18c!
C252 32aD2
2. ~57!
In both cases a solution can at most exist above transitionenergy,a.0 anda.0, respectively. The questions remainswhether the inequality~54! can be satisfied. With a negli-gible b, the resistiveD1 becomes from~33! and ~26! D1
52D12c, which impliesD1521/c. D1 is in fact negative,
as required, its modulus, however, is by no means small.To conclude, a strongly resistive beam does not admit
hydrodynamic solitons of the type~27!.To see whether hydrodynamic solitons can exist and
propagate in a purely reactive beam, we have to evaluate~34!. Neglectingb and evaluatingQ2
21 we get
m/a2D22c5D2 ~58!
and hence to lowest order
D25m/a5m
a~g02L !, ~59!
which must be negative and its modulus small. A negativeD2 and a positivea are achieved in two cases. Either thebeam is above transition energy~a.0! and the inductivity issufficiently large (1,g0,L) or the beam is below transitionenergy~a,0! in which case the inductivity has to satisfy 1,L,g0 . Furthermore, the smallness ofuD2u implies a suf-ficiently large number of beam particlesN, since um/au;N21.
We conclude, that KdV-type solitary waves of the kind~27! propagating with large phase velocities can exist in apurely reactive beam situation only. This supports the pictureof a saturation of the negative-mass instability by dispersionand hydrodynamic nonlinearity, leading to KdV-type struc-tures, provided thatg0,L.
VI. HYDRODYNAMIC SOLITONS IN THE FLUIDDESCRIPTION
The latter result can of course be obtained more simplyby directly applying the fluid description to the beam dynam-ics. For this reason we start with the continuity and momen-tum equation assuming an adiabatic pressure law,p;l3:
] tl1]z~lu!50, ~60a!
] tu1u]zu5]zf23l]zl. ~60b!
Assuming a traveling waveg(z2Dut) for g5l, u, andf,and adopting the boundary conditionsf50, l51, and u50 at infinity, we get
lu52Du, ~61a!
u2/22f1 32l
25Du2/213/2, ~61b!
where we have setuªu2Du. Replacingu by 2Du/l in~61b!, which follows from~61a!, we obtain
S Du
l D 2
13l25~Du!21312f. ~62!
This equation can be easily resolved forl using f!1 and1!Du. We get up to second order in the smallness quantityf/(Du)2:
l512f
~Du!2 13
2~Du!4 f21¯ , ~63!
which, inserted into~13!, which holds for zero resistivity,yields
f95@m1a/~Du!2#f23af2
2~Du!4 [2V8~f!. ~64!
The ‘‘classical’’ potentialV(f) then becomes after integra-tion with V(0)50:
2V~f!5@m1a/~Du!2#f2/22af3
2~Du!4 . ~65!
The NLD, V(c)50, therefore reads
m1a/~Du!25ac
~Du!4 . ~66!
Its solution is in lowest order given by
1
~Du!2 52m
a5
m
a~L2g0!.0, ~67!
which yields the phase velocityDu. It coincides with~54!and ~59! in the case of a purely reactive impedance and de-mandsa(L2g0).0, as before.
If we, on the other hand, make use of~66! in ~65! we get
2V~f!5a
2~Du!4 f2~c2f!, ~68!
which requiresa[a(L2g0)/(L21).0, i.e., either a beamabove transition energy andL.g0 or a beam below transi-tion energy and 1,L,g0 , as we had again before. Thesolitary wave pulse is finally given by
f~z!5c sech2S m
2A c
a~L2g0!~L21!zD . ~69!
We conclude that both approaches, the kinetic and the hy-drodynamic one, lead in the purely reactive regime to theexistence of hydrodynamic solitons of KdV-form~69! trav-eling along a coasting beam. These solitary waves preferen-tially exist when the beam is above transition energy andwhen the imaginary impedance is negative, i.e., when theinductive contribution toZi exceeds the space charge one.The phase velocity of this structure follows from~67! andbecomesDu5@a(L2g0#)/m] 1/2;(uhuN)1/2, i.e., the num-ber of beam particles must be sufficiently high. In view of0,f,c and~63!, we recognize that the soliton is a rarefac-tion soliton.
In the latter solitary wave solution all dynamical quanti-ties, l, m, and f, take part in the evolution and are spacedependent.
There exists, however, another distinct solitary wave so-lution of the system~13!, ~60a!, and ~60b! of similar shapebut with opposite polarity and the restriction that the fluidvelocity is strictly constant and undetermined. To prove this,we setu5u05const in ~60a! and get as a solutionl(z,t)
3427Phys. Plasmas, Vol. 7, No. 8, August 2000 Kinetic theory of solitary waves on coasting beams in . . .
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
132.180.92.90 On: Wed, 15 Jul 2015 12:38:30
5l(j) with j5z2u0t. Since the left-hand side of~60b! iszero, there must be force balance, which implies
f5 32~l221!, ~70!
where we again usedf→0, l→1 asuju→`. Solving~70! forl and restricting ourselves to a weak nonlinearity, we obtain
l511f/32f2/181¯ . ~71!
Poisson’s equation~13!, valid for zero resistivity, then be-comes
f9~j!5~m2a/3!f1a
18f2[2V8~f!, ~72!
from which follows by integration
2V~f!5~m2a/3!f2
21
a
54f3. ~73!
A self-consistent solution requires, as can be seen by inspec-tion, a negativef. Denoting the maximum depth off by 2cwe obtain from2V(2c)50 the NLD
m2a/35ac/27, ~74!
and the ‘‘classical’’ potentialV(f) reduces to
2V~f!5a
54f2~c1f!. ~75!
A negativeV then immediately implies
a5ag02L
12L.0, ~76!
and the solution of~74!, remember thatc!1, requires
m.a
3~g02L !.0. ~77!
Both inequalities can be satisfied if it holdsL,1 anda.0.We hence proved the existence of another rarefaction
~again l<1! KdV soliton with an opposite polarity of theelectric potential. The beam must be above transition energyand the imaginary impedance has to be positive according toa dominance of space charge effect~which is just the oppo-site limit in comparison with our earlier solution!. The speedof this rarefaction soliton remains, however, an open param-eter, and the solution only exists, if the beam parametersmore or less exactly satisfya53m, which is a rather strin-gent requirement. In the previous case—see, e.g.,~66! and~67!—a disparity betweena andm could be balanced by anappropriate speedDu of the soliton~provided, of course, thatit holdsDu@1). The previous rarefaction soliton, hence, hasa larger range of validity and applicability than the presentone.
Solitary waves withf,0 could, of course, also be ob-tained in the kinetic regime for lower phase velocities. We,however, restricted our analysis to 0,f in order to keep theanalysis comprehensible to a certain extent at least.
It can be shown20 that one can keep the condition0<f<c if one looks for the more general class of cnoidalwaves which are periodic in nature and have solitary wavesas limiting cases. One limit corresponds to a potential hump
with f~0!5c andf→0 asuzu→`, the other one to a poten-tial well with f~0!50 andf→c asuzu→`. The transforma-tion f2c→f would then correspond to the second class ofsolitary waves just mentioned.
VII. SUMMARY AND CONCLUSIONS
In the present paper we have studied by an explicit con-struction the existence of solitary waves traveling along acoasting ion beam in circular accelerators. The analysis wasbased on a new coupled system of equations, the Vlasovequation and a generalized form of Poisson’s equation. Thelatter equation was derived in the Appendix from Maxwellequations and by using appropriate boundary conditions. Theeffect of a coherent longitudinal structureEz(r ,z) was takeninto account up toO(g22) in the derivation of the feedbackbetween a perturbed line density and the self-fields. The con-struction of a solution made use of a formalism developedoriginally for plasmas, termed the potential method. In thismethod a complete solution of Vlasov’s equation, namely theone for passing particles and the one for particles trapped inthe electrostatic potential trough, was inserted into the gen-eralized Poisson’s equation, which was then solved for twolimiting cases of the coupling impedance. In the purely reac-tive case, a variety of solitary wave solutions could be found,the properties of which were more or less strongly parameterdependent. A typical example was a solitary wave movingwith approximately thermal velocity of the beam. The corre-sponding distribution function was depressed in the region ofresonant~trapped! particles when the beam was below tran-sition energy, but hump-like when it was above, provingaposteriori Dory’s conjecture about mass conjugation. In thestrongly resistive case, solitary wave solutions could also befound, but the situation was less transparent than in the caseof zero resistivity. A hole-type solution for beams belowtransition energy did not have a hump-type solution forbeams above transition energy as a counterpart, in violationof Dory’s mass conjugation ‘‘theorem.’’
Solitary waves propagating with velocities that are largecompared with the thermal velocity, could also be found,kinetically and hydrodynamically. Both procedures resultedin identical expressions, namely in KdV solitons. Further-more, another kind of KdV soliton could be derived. Onlysolitary wave solutions corresponding to anO-type separa-trix ~like an elliptic fix point in the phase portrait for a simplependulum! have been investigated in detail, i.e., solutionscorresponding to anX-type separatrix~hyperbolic fixpoint!have only been considered in two special cases.
Many questions remain open, reflecting among othersthe huge parameter space involved. One of the open prob-lems is the search for solitary wave solutions for arbitraryimpedances, in which case a third-order differential equationfor f arises. Another one is the connection of this approachwith the thermal wave model~and the corresponding Made-lung fluid, respectively! where the complex beam wave func-tion together with a Wigner-type transformation can be usedto investigate quantum-like corrections on the kineticlevel.21,22Linearly, the new coupled system can be evaluatedto update the stability regimes for uniform beams, such as inRef. 23. Some answers will be given in near future.
3428 Phys. Plasmas, Vol. 7, No. 8, August 2000 H. Schamel and R. Fedele
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
132.180.92.90 On: Wed, 15 Jul 2015 12:38:30
APPENDIX
In the Appendix we calculate the fields and induced wallcurrents due to small deviations from the nominal distribu-tion and determine Eq.~2! describing the self-force actingback on the beam. We assume that the guide field whichdetermines the longitudinal behavior of a single particle ofgiven momentum~and chargeq! and the nominal particledistribution in longitudinal position and energy are known. Afurther assumption is that the behavior of the beam is stillmainly determined by the guide field and that the self-fieldrepresents only a relatively small perturbation. Only effectsoccurring under normal operating conditions in circular ac-celerators and storage rings are considered.8
We assume for simplicity a radially uniform beam ofcircular cross section with radiusa moving with nominalvelocity v along the axis of a circular pipe of radiusb. Weallow for space charge effects in the longitudinalz directionand assume that the particle density%5* f (r ,z,u,t)d3u,wheref is the actual distribution function, the beam currentdensityJ.q%v z, and the fields vary in longitudinal space–time asz2vwt, wherevw is the relativistic phase velocity ofthe perturbation withgwª(12vw
2 /c2)21/2@1. We assume aTM mode and solve Maxwell’s equations in the wave frameby perturbation theory up toO(gw
22).To show that an iterative procedure is involved in the
presence of a longitudinal electric field perturbation, we ex-emplarily derive the fields forr<a. Sincevw'v'c, we canneglect effects of (v2vw). According to the TM mode incylindrical geometry we shall determine
B5Bu~r ,z!u, E5Er~r ,z! r1Ez~r ,z!z ~A1!
in the stationary wave frame (z2vwt→z).Eliminating Er from theu-component of Faraday’s law
and from thez-integratedr component of Ampe`re’s law, weget
] rEz5S c2
v221D v]zBu[c
bg22]zBu , ~A2!
with b5v/c andg'gw . ~The latter equality might be cor-rected in cases where the wave velocity strongly deviatesfrom the nominal beam velocity.! On the other hand, thezcomponent of Ampe`re’s law becomes
1
r] r~rBu!1
b
c]zEz5m0Jz , ~A3!
whereJz.qv% andm0 is the vacuum permeability.z differ-entiation of~A3! andBu elimination through~A2! yields
g21
r] r~r ] rEz!1]z
2Ez5cm0
b]zJz . ~A4!
Integration of~A4! multiplied by r from 0 to r gives
g2r ] rEz~r ,z!1]z2E
0
r
dr8 r 8Ez~r 8,z!
5q
e0]zE
0
r
dr8 r 8%~r 8,z!, ~A5!
where e0 is the vacuum permittivity and it holdse0m0
5c22. Since for r ,a the beam density is assumed to beconstant, we can set%(r ,z)5l(z)/pa2, wherel(z) is theline density, and we can integrate the RHS of~A5! to get
g2r ] rEz1]z2E
0
r
dr8 r 8Ez~r 8,z!5qr2
2pe0a2 l8~z!. ~A6!
Given at this level isl(z) and, hence, this integral equationdeterminesEz(r ,z). Sinceg2 is assumed to be large,Ez(r ,z)must to lowest order ber independent. To get the correctionin Ez correct up toO(g22) we therefore make the ansatz
Ez~r ,z!5Ez~z!1g22Ez~1! ~A7!
and determineEz(1) . The result is
Ez~r ,z!5Ez~z!1g22r 2
4]zL~a,z!, ~A8!
with
L~r ,z!ªql~z!
pe0r 22Ez8~z![qlc~r ,z!
pe0r 2 . ~A9!
Note, thatL can be considered as the space charge correctedline density ~except for the factorq/pe0r 2). Hence, therdependence ofEz(r ,z) not only involves the line density butthe lowest order electric field atr 50 as well, a fact whichwill influence the longitudinal behavior of the collective per-turbation. KnowingEz(r ,z), we can easily get the remainingfields for r<a:
Bu~r ,z!5br
2cL~a,z!, Er~r ,z!5
r
2L~a,z!. ~A10!
In the purely electrostatic field response to the beam, thesefields andL(a,z) vanish. Nonzero values of these quantities,hence, correspond to the next order correction which is elec-tromagnetic in nature. A similar analysis, valid for thevacuum regiona<r<b, gives
Ez~r ,z!5Ez~z!1S r
2g D 2
]zL~r ,z!1q ln r /a
2pe0g2 l8~z!,
~A11!
Bu~r ,z!5br
2cL~r ,z!, Er~r ,z!5
r
2L~r ,z!. ~A12!
We see from~A8!, ~A10!, ~A11!, and ~A12! that the fieldsare continuous atr 5a. Also, if no longitudinal correction istaken into account, the usual field expressions of electrody-namics for a constant cylindrical charge and current distribu-tions are obtained.
What remains to be calculated is a relation betweenEz(z) and the wall electric fieldEw(z). Performing a loopintegral over Faraday’s law,8 we find
Ez~z!52S b
2g D 2
]zLS b
Ag0
,zD 1Ew~z!, ~A13!
where
g0ª112 lnb/a. ~A14!
3429Phys. Plasmas, Vol. 7, No. 8, August 2000 Kinetic theory of solitary waves on coasting beams in . . .
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
132.180.92.90 On: Wed, 15 Jul 2015 12:38:30
~A13! couples the beam dynamics with the image currentdynamics in the wall. In the case of a uniform coasting beamall terms in~A13! vanish.
Since we are interested in the low frequency response ofthe cavity only, we can set9
Ew~z!52R@Bu~b,z!2Bu0~b!#1
L2pR0
d
dtI w
c , ~A15!
whereI wc 52qvlw
c is the wall current corrected by the spacecharge effect. Note that in the wave framed/dt52vd/dz,and thereforeEw , vanishes for an unperturbed uniformbeam, as it should. Inserting in~A15! Bu from ~A12! andlc
from ~A9! for r b(lc(b,z)[lwc ) we get
Ew~z!52R bb
2cL1~b,z!1
Le0~bcb!2
2R0
d
dzL1~b,z!, ~A16!
where L1(r ,z) equalsL(r ,z) except thatl in ~A9! is re-placed byl1[l2l0 , i.e., by the perturbed line density. In~A15! and ~A16! R is a measure of the Ohmic resistance~i.e., R/2pe0c2b is the Ohmic resistance of the wall per unitlength! and L/2pR0 is the wall resistance per unit length.Finally, insertingEw(z) from ~A16! in ~A13! we obtain
Ez~z!1R bb
2cL1~b,z!1S b
2g D 2
]zL1S b
Ag0
,zD2L e0~bcb!2
2R0]zL1~b,z!50, ~A17!
the desired equation.
1P. L. Colestock and L. K. Spentzouris, inThe Tamura Symposium Pro-ceedings, Austin, TX, 1994@AIP Conf. Proc.356 ~1996!#.
2O. Boine-Frankenheim, I. Hofmann, and G. Rumolo, Phys. Rev. Lett.82,3256 ~1999!.
3I. Hofmann, Laser Part. Beams3, 1 ~1985!.4H Schamel, Phys. Rev. Lett.79, 2811~1997!.5H. Schamel, Phys. Scr., T75, 23 ~1998!; H. Schamel, Nonlinear andStochastic Beam Dynamics in Accelerators—A Challenge to Theoreticaland Computational Physics, Lu¨neburg, Germany, 1997~DeutschesElektronen-Synchrotron DESY, Hamburg, 1998!, p. 224.
6J. J. Bisognano,Solitons and Particle Beams, Particles and Field Series,Vol. 47, High Brightness Beams for Advanced Accelerator Applications@AIP Conf. Proc.253 ~1991!#.
7R. Fedele, G. Miele, L. Palumbo, and V. G. Vaccaro, Phys. Lett. A179,407 ~1993!.
8A. Hofmann, inProceedings of the International School of Particle Ac-celerator, Erice, Italy, 10–22 November 1976, edited by H. W. Blewett~CERN, Geneva, 1977!, CERN Publication No. 77-13, p. 139.
9V. K. Neil and R. J. Briggs, Plasma Phys.9, 631 ~1967!.10H. Schamel, Phys. Rep.140, 161 ~1986!.11R. A. Dory, Plasma Phys.6, 511 ~1964!.12R. J. Briggs and V. K. Neil, Plasma Phys.8, 255 ~1966!.13J. Korn and H. Schamel, J. Plasma Phys.56, 307 ~1996!; 56, 339 ~1996!,
references therein.14B. D. Fried and S. D. Conte,The Plasma Dispersion Function~Academic,
New York, 1961!.15H. Schamel, Plasma Phys.14, 905 ~1972!.16H. Schamel, J. Plasma Phys.9, 377 ~1973!.17H. Schamel, Phys. Scr.20, 336 ~1979!.18H. Schamel and S. Bujarbarua, Phys. Fluids23, 2498~1980!.19S. Bujarbarua and H. Schamel, J. Plasma Phys.25, 515 ~1981!.20H. Schamel~unpublished!.21R. Fedele and V. I. Manko, Phys. Rev. E58, 992 ~1998!.22R. Fedele and H. Schamel~unpublished!.23G. Rumolo, G. Miano, C. Serpico, and I. Hofmann, Phys. Plasmas6, 4349
~1999!.
3430 Phys. Plasmas, Vol. 7, No. 8, August 2000 H. Schamel and R. Fedele
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
