Levy-Student Distributions for Halos in Accelerator Beams

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by  N. Cufaro Petroni

L´evy–Student Distributions for Halos in Accelerator Beams
Nicola Cufaro Petroni∗
Dipartimento di Matematica dell’Universit`a di Bari,
and INFN Sezione di Bari,
via E. Orabona 4, 70125 Bari, Italy
Salvatore De Martino,† Silvio De Siena,‡ and Fabrizio Illuminati§
Dipartimento di Fisica dell’Universit`a di Salerno,
6
0 INFM Unit`a di Salerno and INFN Sezione di Napoli – Gruppo collegato di Salerno,
0 Via S. Allende, I–84081 Baronissi (SA), Italy
2 (Dated: October28, 2005)

n We describe the transverse beam distribution in particle accelerators within the controlled,
a stochastic dynamical scheme of the Stochastic Mechanics (SM) which produces time reversal in-
J variant diffusion processes. This leads to a linearized theory summarized in a Shchr¨odinger–like

7 (S–ℓ)equation. Thespacechargeeffectshavebeenintroducedinarecentpaper[1]bycouplingthis
1 S–ℓ equation with the Maxwell equations. We analyze the space charge effects to understand how
thedynamicsproducestheactualbeamdistributions,andinparticularweshowhowthestationary,

] self–consistent solutions are related to the (external, and space–charge) potentials both when we
h
supposethattheexternalfieldisharmonic(constant focusing),andwhen wea priori prescribethe
p
shapeofthestationarysolution. Wethenproceed todiscussafewnewideas[2]byintroducingthe
-
c generalized Student distributions, namely non–Gaussian, L´evy infinitely divisible (but not stable)
c distributions. We will discuss this idea from two different standpoints: (a) first by supposing that
a the stationary distribution of our (Wiener powered) SM model is a Student distribution; (b) by
.
s supposingthatourmodelisbasedona(non–Gaussian)L´evyprocesswhoseincrementsareStudent
c distributed. We show that in the case (a) the longer tails of the power decay of the Student laws,
i
s and in the case (b) the discontinuities of the L´evy–Student process can well account for the rare
y escape of particles from thebeam core, and hence for theformation of a halo in intensebeams.
h
p PACSnumbers: 02.50.Ey,05.40.Fb,29.27.Bd,41.75.Lx
[

2
I. INTRODUCTION terms of classical,deterministic dynamicalsystems. The
v
standardmodelisthatofacollisionlessplasmawherethe
9
corresponding dynamics is embodied in a suitable phase
0
In high intensity beams of chargedparticles,proposed
1 space (see for example [8]). In this framework the beam
in recent years for a wide variety of accelerator–related
0 is studied by means of the particle–in–core (pic) model
1 applications,itis veryimportantto keepatlowlevelthe and the simulations show that the instabilities due to a
5 beam loss to the wall of the beam pipe, since even small parametric resonance can allow the particles to escape
0 fractionallosses ina high–currentmachine cancause ex-
from the core with consequent halo formation [5, 6, 7].
/ ceedingly high levels of radioactivation. It is now widely
s The present paper takes a different approach: it follows
c believed that one of the relevant mechanisms for these
the idea that the particle trajectories are samples of a
i lossesis the formationofalowintensitybeamhalomore
s stochastic process, rather than usual deterministic (dif-
y orlessfarfromthecore. Thesehaloshavebeenobserved
ferentiable) trajectories. In the usual dynamical mod-
h [3]orstudiedinexperiments[4],andhavealsobeensub-
els there is a particle probability distribution obeying
p
jected to an extensive simulation analysis [5]. For the
: the Vlasov equation, and its evolution is Liouvillian in
v next generation of high intensity machines it is however
the sense that the origin of the randomness is just in
i stillnecessarytoobtainamorequantitativeunderstand-
X the initial conditions: along the time evolution, which is
ing not only of the physics of the halo, but also of the
supposed to be deterministic, there is no new source of
r
beam transverse distribution in general [6]. In fact “be-
a uncertainty. It is the non linear character of the equa-
cause there is not a consensus about its definition, halo
tions which produces the possible unpredictable charac-
remains an imprecise term” [7] so that several proposals
ter of the trajectories. On the other hand in our model
have been put forward for its description.
thetrajectoriesarereplacedbystochasticprocessessince
The charged particle beams are usually described in thetimeevolutionissupposedtoberandomlyperturbed
evenafter the initial time. It is open to discussionwhich
oneofthesetwodescriptionismorerealistic;inparticular
weshouldaskifthemutualinteractionsamongthebeam
∗Electronicaddress: [email protected] particles look like random collisions, or rather like con-
†Electronicaddress: [email protected] tinuous deterministic interactions. In the opinion of the
‡Electronicaddress: [email protected] authors,however,a plasma(with collisions)describedin
§Electronicaddress: [email protected]
2
terms of controlled stochastic processes is a good candi- (Q-ℓ) systems, in analogy with other recent researches
date toexplainthe rareescapeofparticlesfroma quasi– on this subject [15, 16]. In fact the SM can be used
stable beam core by statistically taking into account the to describe every stochastic dynamical system satisfying
random inter–particle interactions that can not be de- fairly generalconditions: it is knownsince longtime [17],
scribed in detail. Of course the idea of a stochastic ap- for example, that for any given diffusion there is a cor-
proachishardlynew[8,9],butthereareseveraldifferent respondencebetweendiffusionprocessesandsolutionsof
ways to implement it. S–ℓ equations where the Hamiltonians come in general
First of all let us remark that the system we want to from suitable vector potentials. Under some regularity
describeisendowedwithsomemeasureofinvarianceun- conditions this correspondence is seen to be one-to-one.
der time reversal, since the external fields act to keep it TheusualSchr¨odingerequation,andhenceQM,isrecov-
in a quasi–stationary non diffusive state despite the re- ered when the diffusion coefficient coincides with ~/2m,
pulsive electro–magnetic (e.m.) interactions among the namelyisconnectedtothePlanckconstant. Howeverwe
constituent particles. However,a widespreadmisconcep- are interested here not in a stochastic model of QM, but
tionnotwithstanding,atheoryofstochasticprocessesnot in the description of particle beams.
always describe irreversible systems: the addition of a
In the present paper we intend to widen the scope
dynamics to a stochastic kinematics can in fact ascribe
of our SM model by introducing the idea that an im-
a measure of time reversal invariance also to a stochas-
portant role for the beam dynamics can be played by
tic system [10]. The standard way to build a stochastic
non–GaussianL´evy distributions. In fact these distribu-
dynamicalsystemistomodifythephasespacedynamics
tionsenjoyedawidespreadpopularityintherecentyears
by adding a Wiener noise B(t) to the momentum equa-
because of their multifaceted possible applications to a
tion only, so that the usual relations between position
large set of problems from the statistical mechanics to
and velocity is preserved:
the mathematical finance (see for example [10, 18] and
mdQ(t) = P(t)dt, references quoted therein). In particular the so called
stable laws (see Section A) are used ina largenumber of
dP(t) = F(t)dt+βdB(t).
instances,asforexampleinthedefinitionoftheso–called
In this way we get a derivable, but not Markovian po- L´evy flights. Our researchis instead focused on a family
sition process Q(t). The standard example of this ap- of non–Gaussian L´evy laws which are infinitely divisible
proach is that of a Brownian motion in a force field de- but not stable: the generalize Student laws. As will be
scribed by an Ornstein–Uhlenbeck system of stochastic discussed later this will allow us to overcome – without
differential equations (SDE) [11]. Alternatively we can resorting to the trick of the truncated laws – the prob-
add a Wiener noise W(t) with diffusion coefficient D to lemsraisedbythefactthatthestablenon–Gaussianlaws
the position equation: always have divergent variances: a feature which is not
realistic to ascribe to most real systems. It is possible
dQ(t) = v (Q(t),t)dt+√DdW(t).
(+) to show indeed that by suitably choosing the parame-
andgetaMarkovian,butnotderivableQ(t). Inthisway ters of the Student laws we can have distributions with
finite variance, and approximating the Gaussian law as
the stochastic system is also reduced to a single SDE
well as we want. On the other hand the infinitely divisi-
since we are obliged to drop the second (momentum)
equation: in fact now Q(t) is no more derivable. The ble characterof these laws is all that is requiredto build
a stationary, stochastically continuous Markov process
standard example of this reduction is the Smoluchowski
with independent increments, namely the L´evy process
approximationof the Ornstein–Uhlenbeck process in the
that we propose to use to represent the evolution of our
overdamped case [11]. As a consequence we will work
particle beam.
only in a configuration, and not in a phase space; but
this does not prevent us from introducing a dynamics – Of course it is not alwaysmathematically easy to deal
aswewillshowintheSectionIIA–eitherbygeneralizing with the infinitely divisible processes, but we will show
the Newton equations[11, 12], or by means of a stochas- thatatleastintworespectstheywillhelpustohavesome
ticvariationalprinciple[13]. Remarkthatinthisscheme further insightin the beamdynamics. Firstofallwe use
theforwardvelocityv (r,t)cannomorebeanapriori the Student distributions in the framework of the tradi-
(+)
given field: rather it now plays the role of a new dy- tionalSMwheretherandomnessofprocessissuppliedby
namical variable of our system. This second scheme, the a Gaussian Wiener noise: here we examine the features
Stochastic mechanics (SM), is universally known for its oftheself–consistentpotentialswhichcanproduceaStu-
originalapplicationtotheproblemofbuildingaclassical dent distribution as stationarytransverse distribution of
stochastic model for Quantum Mechanics (QM), but in aparticlebeam. Inthisinstancethefocusofourresearch
factitisaverygeneralmodelwhichissuitableforalarge is on the increase of the probability of finding the par-
number of stochastic dynamical systems [10, 14]. We ticles at a great distance from the beam core. Then we
will also see in the Section IIA that from the stochastic passtothedefinitionofatrueL´evy–Student process,and
variational principles two coupled equations are derived we show with a few simulations that these processes can
whichareequivalenttoaSchr¨odinger–like(S–ℓ)differen- help to explain how a particle can be expelled from the
tialequation: inthissensewewillspeakofquantum–like bunchbecause ofsomekind ofhardcollision. In factthe
3
trajectories of our L´evy–Student process show the typi- process is gaussian with law (0,Idt), where I is the
N
caljumps ofthe non–GaussianL´evyprocesses: a feature 3 3 identity matrix. Finally the diffusion coefficient
×
thatweproposetouseasamodelforthehaloformation. D is supposed to be constant: the quantity α = 2mD,
Itisworthremarkingthat,albeitthe morerecentempir- which has the dimensions of an action, will be later con-
icaldataabouthalos[19]arestillnotaccurateenoughto nected to the characteristic transverse emittance of the
distinguish between the suggested distributions and the beam. The equation (1) defines the random kinematics
usual Gaussian ones, our conjecture on the role of Stu- performed by the particle, and replaces the usual deter-
dent laws in the transverse beam dynamics has recently ministic kinematics
found a first confirmation [20] in numerical simulations
showing how these laws are well suited to describe the dq(t)=v(q(t),t)dt (2)
statistics of the random features of the particle paths.
where q(t) is just the trajectory in the 3–DIM space.
In a few previous papers [21] we connected the (trans-
To counteract the dissipation due to this stochastic
verse) r.m.s. emittance to the characteristic microscopic
kinematics, a dynamics must be independently added.
scaleandtothe totalnumberofthe particlesinabunch,
In SM we do not have a phase space: our description
and implemented a few techniques of active control for
is entirely in a 3-DIM configuration space. This means
the dynamics of the beam. In this paper we first of all
in particular that the dynamics is not introduced in a
reviewthetheoreticalbasis[1,21]oftheproposedmodel:
Hamiltonian way, but by means of a suitable stochastic
in the Section II we define our SM model with emphasis
least action principle [13] obtained as a generalizationof
added on the potentials which control the beam dynam-
the variational principle of classical mechanics. In the
ics and on the possible non stationary solutions of this
following we will briefly review the main results, refer-
model [22]. In the Section III we review our analysis of
ring for details to the references [10, 11, 13]. Given the
the self–consistent, space charge effects due to the e.m.
SDE (1), we consider the probability density function
interactionamongtheparticles,addingafewnewresults
(pdf) ρ(r,t) associated to the diffusion Q(t) so that, be-
and comments. In the Section IV we then discuss the
sides the forwardvelocity v (r,t), we cannow define a
idea [2] that the laws ruling the transverse distribution (+)
backwardvelocity
of particle beams are non–Gaussian, infinitely divisible,
L´evy laws as the generalized Student laws. In particular
ρ(r,t)
we analyze the behavior of our usual SM model under v (r,t)=v (r,t) 2D∇ . (3)
(−) (+) − ρ(r,t)
the hypothesis that the stationary transverse distribu-
tion is a Student law. Finally in the Section V we study
We can then introduce also the current and the osmotic
the possibility of extending our SM model to L´evy pro-
velocity fields, defined as:
cesses whose increments are distributed accordingto the
Student law. We think in particular that the presence v +v v v ρ
of isolated jumps in the trajectories can help to build a v= (+) (−) ; u= (+)− (−) = D∇ . (4)
2 2 ρ
realistic model for the possible formation of halos in the
particle beams. We end the paper with a few conclusive Here v represents the velocity field of the density, while
remarks. uisofintrinsicstochasticnatureandisa measureofthe
non differentiability of the stochastic trajectories.
A first consequence of the stochastic generalization of
II. STOCHASTIC BEAM DYNAMICS the least action principle [11, 13] is that the current ve-
locity takes the following irrotational form:
A. Stochastic mechanics
mv(r,t)= S(r,t), (5)
∇
First of all we introduce the stochastic process per-
while the Lagrangeequationsofmotionforthe densityρ
formedbyarepresentativeparticlethatoscillatesaround
andforthecurrentvelocityvarethecontinuityequation
the closed ideal orbit in a particle accelerator. We con-
associated to every stochastic process
sider the 3–dimensional (3–DIM) diffusion process Q(t),
taking the values r, which describes the position of the ∂ ρ= (ρv), (6)
t
representative particle and whose probability density is −∇·
proportional to the particle density of the bunch. As and a dynamical equation
stated in the Section I the evolution of this process is
ruled by the Itoˆ stochastic differential equation (SDE) m 2√ρ
∂ S+ v2 2mD2∇ +V(r,t)=0, (7)
t
2 − √ρ
dQ(t)=v (Q(t),t)dt+√DdW(t), (1)
(+)
which characterizes our particular class of time–reversal
where v (r,t) is the forward velocity, and dW(t) invariant diffusions (Nelson processes). The last equa-
(+)
≡
W(t+dt) W(t) is the incrementprocessof astandard tion has the same form of the Hamilton–Jacobi–
−
Wiener noise W(t); as it is well known this increment Madelung (HJM) equation, originally introduced in the
4
hydrodynamic description of quantum mechanics by We will refer to it as a Schr¨odinger–like (S–ℓ) equa-
Madelung [23]. Since (5) holds, the two equations (6) tion: clearly (16) has not the same meaning as the usual
and (7) can be put in the following form Schr¨odinger equation; this would be true only if α = ~,
while in general α is not an universal constant, and it
1
∂ ρ = (ρ S) (8) is rather a quantity characteristic of the system under
t
−m∇· ∇ consideration (in our case the particle beam). In fact α
1 2√ρ turns out to be of the order of magnitude of the beam
∂ S = S2+2mD2∇ V(r,t) (9)
t −2m∇ √ρ − emittance, a quantity which – in formal analogy with ~
– hasthe dimensions ofanactionandgivesa measureof
which now constitutes a coupled, non linear system of theposition/momentumuncertaintyproductforthesys-
partial differential equations for the pair (ρ,S) which tem. Thus the SM model of our beam, as incorporated
completely determines the state of our beam. On the inthephenomenologicalSchr¨odingerequation(16),while
otherhand,becauseof(5),thisstateisequivalentlygiven keeping a few features reminiscent of the QM, is in fact
by the pair(ρ,v). a deeply different theory.
It can also be shown by simple substitution from (4)
that (6) is equivalent to the standard Fokker–Planck
(FP) equation
∂ ρ= [v ρ]+D 2ρ (10)
t (+)
−∇· ∇
B. Controlled distributions
formally associated to the Itoˆ equation (1). In fact also
the HJM equation (7) can be cast in a form based on
v rather than on v, namely We have introduced the equations that in the SM
(+)
model are supposed to describe the dynamical behavior
m
∂ S = v2 +mDv lnf ofthebeam: wenowbrieflysumupageneralprocedure,
t − 2 (+) (+)∇
already exploited in previous papers [21, 24], to control
+mD2 2lnf V (11)
the dynamics of our systems. Let us suppose that the
∇ −
pdf ρ(r,t) be given all along its time evolution: think
where f is a dimensionless density defined by
in particular either to a stationary state, or to an en-
ρ(r,t)=Cf(r,t) (12) gineered evolution from some initial pdf toward a final
state with suitable characteristics. We know that the
where C is a dimensional constant. On the other hand, FP equation (10) must be satisfied, for the given ρ, by
from(3) and (4), we knowthat alsothe forwardvelocity someforwardvelocityfieldv (r,t). Sincealsotheequa-
(+)
v is irrotational: tion(13)musthold,wearefirstofallrequiredtofindan
(+)
irrotationalv whichsatisfiesthe FP equation(10) for
(+)
v (r,t)= W(r,t), (13)
(+) the givenρ. We then take into accountalso the dynami-
∇
calequation(11): sinceρandv (andhencef andW)
andthatbytaking(5)intoaccountthe functionsW and (+)
are now fixed and satisfy (10), the equation (11) plays
S are connected by the relation
theroleofaconstraintdefiningacontrollingpotentialV
S(r,t)=mW(r,t) mDlnf(r,t) θ(t) (14) when we also take into account the equation (14). We
− − list here the potentials associatedto the three particular
where θ is an arbitrary function of t only. cases analyzed in the previous papers.
Thetime–reversalinvarianceisnowmadepossible[12]
Inthe1-DIMcasewithgivendimensionlesspdff(x,t)
bythe factthatthe forwarddriftvelocityv (r,t) is no
(+) and a<x<b (a and b can be infinite) we easily get
more an a priori given field, as is usual for the diffusion
processesof the Langevintype; insteadit is dynamically
determined at any instant of time, starting by an initial ∂ ρ(x,t) 1 x
condition,throughthe HJMevolutionequation(7). Itis v (x,t)=D x ∂ ρ(x′,t)dx′ (17)
(+) ρ(x,t) − ρ(x,t) t
finally important to remark that, introducing the repre- Za
sentation [23] V(x,t)=mD2∂x2lnf +mD(∂tlnf +v(+)∂xlnf)
m x
Ψ(r,t)= ρ(r,t)eiS(r,t)/α, (15) − 2 v(2+)−m ∂tv(+)(x′,t)dx′+θ˙ (18)
Za
(with α = 2mD) the copupled equations (8) and (9) are
made equivalent to a single linear equation of the form
For a 3-DIM system with cylindrical symmetry around
ofthe Schr¨odingerequation,with the Planckactioncon-
the z-axis (the beam axis), if we denote with (r,ϕ,z)
stant replaced by α:
thecylindricalcoordinates,andifwesupposethatρ(r,t)
iα∂tΨ= α2 2ψ+VΨ. (16) rdaedpieanlldysdoinrelycteodnwriathndmto,daunludstdheaptevn(d+in)g=onv(l+y)o(rn,tr)aˆrnids
−2m∇
5
t, we have A different non stationary problem also discussed in
previous papers [21, 22] consists in the analysis of some
∂ ρ(r,t) 1 r
v (r,t)=D r ∂ ρ(r′,t)r′dr′ (19) particular time evolution of the process with the aim of
(+) ρ(r,t) − rρ(r,t) Z0 t finding the dynamics that control it. For instance we
mD2 studied the possible evolutions which start from a pdf
V(r,t)= ∂ (r∂ lnf)+mD(∂ lnf +v ∂ lnf)
r r r t (+) r with halo and evolve toward a halo–free pdf: this would
m r allow us to find the dynamics that we are requested to
− 2 v(2+)−m ∂tv(+)(r′,t)dr′+θ˙ (20) apply in orderto achieve this result. If for simplicity the
Z0 overallprocessis supposedto be anOrnstein–Uhlenbeck
Finally in the 3-DIM stationary case the pdf ρ(r) is in- process, the transition pdf would be completely known
dependent from t. This greatly simplifies our formulas and all the result can be exactly calculated through the
and, by requiring that θ˙(t) = E be constant, namely Chapman–Kolmogorov equation by supposing suitable
that θ(t)=Et, we get shapes for the initial and final distributions. Then a di-
rectapplicationof(18) allowsus to calculate the control
ρ(r)
v (r) = D∇ (21) potential corresponding to this evolution. For the sake
(+) ρ(r) of brevity we do not give the analytical form of this po-
2√ρ tentialandreferto the quotedpapers forfurther details.
V(r) = E+2mD2 ∇ . (22)
√ρ
Of course in this context the constant E will be chosen III. SELF-CONSISTENT EQUATIONS
by fixing the zero of the potential energy. Let us remark
finally that in this stationary case the phenomenological A. Space charge interaction
wave function (15) takes the form
In QM a system of N particles is described by a wave
Ψ(r,t)=√ρe−iEt/α
functionina3N–DIMconfigurationspace. Ontheother
typical of the stationary states. hand in our SM scheme a normalized Ψ(r,t)2, func-
| |
tion of only three space coordinates r = x,y,z , plays
{ }
the role of the pdf of a Nelson process. In a first ap-
C. Non stationary distributions proximation we will consider this N–particle system as
a pure ensemble: as a consequence we will not introduce
a 3N–DIM configuration space, since N Ψ(r,t)2d3r in
In the following we will be mainly concernedwith sta-
| |
the 3–DIMspace willplay the roleof the number ofpar-
tionary distributions, but in a few previous paper we
ticles in a small neighborhood of r. However, since our
treated also non stationary problems. For instance, if
system of N charged particles is not a pure ensemble
we consider the stationary, ground state pdf (without
due to their mutual e.m. interaction, in a further mean
nodes) ρ (r) of a suitable potential, and if we calculate
0
fieldapproximation wewilltakeintoaccountthesocalled
v (r) and write down the corresponding FP equation,
(+)
space charge effects: morepreciselywewillcoupleourS–
it is possible to show (see the generalproof in a few pre-
ℓ equation with the Maxwell equations describing both
vious papers [24, 25, 26]) that, ρ (r) will play the roleof
0
theexternalandthespacechargee.m.fields,andwewill
an attractor for every other distribution (non extremal
get in the end a non linear system of coupleddifferential
with respectto a stochasticminimalactionprinciple). If
equations.
the accelerator beam is ruled by such an equation, this
In our model a single, charged particle embedded in a
would imply that the halo can not simply be wiped out
beam and feeling both an external, and a space charge
byscrapingawaytheparticlesthatcomeoutofthebunch
potential is first of all described by a S–ℓ equation
core: in fact they simply will keep going out in the halo
until the equilibrium is reached again since the distribu-
iα∂ Ψ(r,t)=HΨ(r,t),
t
tion ρ (r) is a stable attractor.
0
In a recent paper [22] we gave an estimate of the time whereΨ(r,t)isourwavefunction,αacoefficientwiththe
b
required for the relaxation of non extremal pdf’s toward dimensionsofanactionwhichisaconstantdependingon
the equilibrium distribution. This is an interesting test thebeamcharacteristics,andH isasuitableHamiltonian
for our model since this relaxationtime is fixed once the operator. If Ψ is properly normalized and if N is the
form of the forward velocity field is given; this is in turn number of particles with indivbidual charge q0, the space
fixed when the form of the halo distribution is given as charge density and the electrical current density are
in the reference [1], and one could check if the estimate
ρ (r,t) = Nq Ψ(r,t)2, (23)
is in agreement with possible observed times. In par- sc 0| |
α
ticular we estimated that in typical conditions all the j (r,t) = Nq Ψ∗(r,t) Ψ(r,t) . (24)
sc 0
non–stationary solutions of this FP equation will be at- mℑ{ ∇ }
tracted toward ρ with a relaxation time of the order of Hence our particles in the beam will feel both an electri-
0
τ 2mσ2/α 10−8 10−7sec. cal and a magnetic interaction and we will be obliged to
≈ ≈ ÷
6
couple the S–ℓ equationwith the equations of the vector As a consequence our wave functions will take the form
and scalar potentials associatedto this electro–magnetic
field. eipzz/α
ψ(r)=χ(x,y) (32)
Thee.m.potentials(A ,Φ )ofthespacechargefields √L
sc sc
obeying the gauge condition
and our equations (30) and (31) become
1
∇·Asc(r,t)+ c2 ∂tΦsc(r,t)=0, (25) α2 (∂2+∂2)χ=(V +V E ) χ (33)
2m x y e sc− T
must satisfy the wave equations Nq2 q2
(∂2+∂2)V = 0 χ2 = N 0 χ2 (34)
x y sc −Lǫ | | − ǫ | |
1 0 0
2A (r,t) ∂2A (r,t) = µ j (r,t) (26)
∇ sc − c2 t sc − 0 sc where = N/L is the number of particles per unit
N
1 ρ (r,t) length, and E =E p2/2m is the energy of the trans-
∇2Φsc(r,t)− c2 ∂t2Φsc(r,t) = − scǫ (27) versemotion.TIffinal−lyozursystemhasacylindricalsym-
0
metry around the z axis, namely if – in the cylindrical
On the other hand, for our particle in the beam the coordinate system r,ϕ,z (r2 = x2 +y2) – our poten-
e.m. field is the superposition of the space chargepoten- tialsdependonlyon{r,then} wecanseparatethevariables
tial (Asc,Φsc), and of the external potentials (Ae,Φe). with χ(x,y)=u(r)Φ(ϕ), the angular eigenfunctions are
Hence (see for example [27], chapter XV) our S–ℓ equa-
tion takes the form eiℓϕ
Φ (ϕ)= , ℓ=0, 1, 2,... (35)
ℓ
√2π ± ±
1 q 2
iα∂ Ψ = iα 0(A +A ) Ψ
t sc e
2m ∇− c and for ℓ=0 the equations become
h +q (Φ i+Φ )Ψ (28)
0 sc e α2 u′
u′′+ = (V +V E )u (36)
e sc T
It is apparent now that (25), (26), (27) and (28) con- 2m r −
(cid:18) (cid:19)
stitute a self–consistent system of non linear differen- V′ q2
tial equations for the fields Ψ, A and Φ coupled V′′ + sc = N 0 u2 (37)
sc sc sc r −2πǫ
through (23) and (24). 0
If we then consider stationary wave functions with the following radial normalization
Ψ(r,t)=ψ(r)e−iEt/α (29) +∞
ru2(r)dr =1.
whereE istheenergyoftheparticle,andtakeA =0for Z0
e
theexternalinteraction,passingtothepotentialenergies Remarkthatnowwearereducedtoasystemofordinary
differential equations.
V (r)=q Φ (r), V (r)=q Φ (r),
e 0 e sc 0 sc
our system is reduced to only two coupled, non linear C. Dimensionless formulation
equations for the pair (ψ,V ), namely
sc
To eliminate the physical dimensions one introduces
α2 2ψ = (V +V E)ψ, (30) two quantities η and λ which are respectively an energy
e sc
2m∇ − andalength. Then,bymeansofthedimensionlessquan-
Nq2 tities
2V = 0 ψ 2 (31)
sc
∇ − ǫ0 | | r E q2
s= , β = T , ξ = N 0 (perveance)
λ η 2πǫ η
0
B. Cylindrical symmetry w(s)=λu(λs)
V (λs) V (λs)
sc e
v(s)= , v (s)=
e
We suppose now that the longitudinal motion along η η
the z–axis is both decoupled from the transverse motion
the equations (36) and (37) take the form
in the x,y–plane, and free with constant momentum p ,
z
andvelocitybz =b0 ≫bx,by. Moreoverwesuppose that sw′′(s)+w′(s) = [ve(s)+v(s) β]sw(s) (38)
thebeamparticleswillbeconfinedinacylindricalpacket −
sv′′(s)+v′(s) = ξsw2(s) (39)
of length L, so that by the imposing periodic boundary −
conditions we will quantize the longitudinal momentum
The usual choice for the dimensional constants is
pz = 2kLπα, k =0,±1,±2,... η =mb20, λ= mbα0√2, (40)
7
whereb isthelongitudinalvelocityofthebeam. Wecan withaproperfrequencyω(constantfocusing),andletwe
0
now look at our equations in two different ways. Firstof also introduce the characteristic length
all we can suppose that v is a given external potential:
e α
in this case our aim is to solve the equations for the two σ2 =
2mω
unknowns w (radial particle distribution) and v (space
charge potential energy). However in general no simple which will representsa measure of the transversedisper-
analyticalsolutionofthis problemis atpresentavailable sionof the beam. In cylindricalcoordinates r,ϕ in the
fortheusualformsoftheexternalpotentialve: thereare transverse plane our potential energy is { }
not even solutions playing the same role played by the
Kapchinskij–Vladimirskij (KV) distribution in the usual mω2 α2
V (r)= r2 = r2 (44)
models. This phase space distribution – which is simple e 2 8mσ4
andself-consistentintheusualdynamicalmodels–leads
so that the corresponding 2–DIM S–ℓ equation without
to an uniform transversespace distribution of the beam,
spacecharge(zeroperveance)wouldhaveaslowesteigen-
andisastationarysolutionoftheVlasovequationwitha
value E =αω, and as ground state wave function
harmonic potential. Moreover its space charge potential 0
calculated from the Poisson equation is still harmonic. u (r) e−r2/4σ2
Instead in the SM model the uniform distributions are χ (r,ϕ)= 0 = . (45)
00
notsolutionsofthestationarySchr¨odingerequation,and √2π σ√2π
we know no simple stationary distribution connected to
Of course the self–consistent solution would be different
theharmonicpotentialastheKV.Eventhegaussiandis-
if there is a space charge (non zero perveance). To find
tributions – later discussed in this paper – can not play
this solution one introduces the so called phase advance
thesamerole: theyaresolutionsconnectedwithanexter-
nal harmonic potential, but their space charge potential 1 ω α
= =
calculated from the Poisson equation is not harmonic. λ b 2mb σ2
0 0 0
Alternativelywecanassumeasknownagivendistribu-
tionw, andsolveour equationsto findboth the external (λ0 is a length) and, with the constants (40), the dimen-
andthespacechargeself–consistentpotentialenergiesv sionless form of the harmonic potential (44)
e
and v. In this second form the problem is more simple,
V (r) ω2 r2 α2
and analytical solutions are available. We adopted the v (s) = e = r2 = = s2 =γ2s2
first standpoint in a few previous papers [1] where we e mb2 b2 2λ2 4λ2m2b2
0 0 0 0 0
numerically solved the equations (38) and (39); here we α αω σ2
will rather elaborate a few new ideas about the second γ = = = .
2λ mb 2mb2 λ2
one. Tothisenditisimportanttoremarkthatthespace 0 0 0 0
charge potential energy As a consequence the equations (38) and (39) become
v(s)= ξ s dy yxw2(x)dx (41) sw′′(s)+w′(s) = γ2s2+v(s) β sw(s) (46)
− y −
Z0 Z0 sv′′(s)+v′(s) = (cid:2) ξsw2(s) (cid:3) (47)
−
is always a solution of the Poisson equation (39) satis-
fying the conditions v(0+) = v′(0+) = 0. On the other These equations are now a coupled, non linear system
which must be numerically solved since we do not know
hand, by substituting (41) in the first equation (38) we
simpleself–consistentsolutionsoftheformoftheKVdis-
readily obtain also the self–consistent form of the exter-
tribution. In reference [1] we extensively analyzed these
nal potential energy
numericalsolutionsandwerefertothispaperfordetails.
s dy y In fact in [1] there was a small difference with respect to
v (s) = v (s)+ξ xw2(x)dx, (42)
e 0 y whathasbeenpresentedhere. Theformoftheequations
w′′(s) Z10w′(s)Z0 to solve is the same, but the dimensionless formulation
v (s) = + +β (43) wasachievedbymeansoftwonumericalconstantsdiffer-
0
w(s) s w(s)
ent from (40) and drawn from the characteristics of the
where v (s) is the potential that we would have with- transverse harmonic oscillator force:
0
out space charge (ξ = 0), while the second part in the α2 αω
external potential (42) exactly compensate for the space η = = , λ=σ√2 (48)
4mσ2 2
charge potential.
Thenthedimensionlessquantitieshaveadifferentnumer-
ical value and the dimensionless equations (36) and (37)
D. Constant focusing take the form
Letussupposenowthatthetransverseexternalpoten- sw′′(s)+w′(s) = [s2+v(s) β]sw(s) (49)
−
tialV (r)isacylindricallysymmetric,harmonicpotential sv′′(s)+v′(s) = ξsw2(s) (50)
e
−
8
sincenowγ =1. Inanycasethe equations(46)and(47)
150
caneasilybeturnedintotheequations(49)and(50),and
vice versa, by means of simple transformations through
the parameter γ which turns out to be at the same time 100
theratiooftheenergyconstants,andthatofthesquared
length constants. As a consequence in the following we
50
willalwaysusethesystem(49),(50),withtheadvantage
of simply putting γ =1 in the model.
s
2 4 6 8 10
IV. SELF–CONSISTENT POTENTIALS
-50
A. Gaussian transverse distributions
Inthe SMmodelitispossibletonumericallyintegrate FIG.1: Thedimensionlesspotentialsv(s)(thinline),v0(s)=
the Schr¨odinger–Poisson system (38) and (39) with a s2 (dashed line) and ve(s) (thick line). They reproduce re-
spectively equations(52),(53) and (54) for ξ=20 (see refer-
givenexternalpotentialandcalculate the self–consistent
ence [1] for this value). When the external potential is ve(s)
distributions and their space charge potentials [1]. On
theself–consistentwavefunctioncoincideswiththatofasim-
the other hand, if we fix a particular distribution, it is
ple harmonic oscillator for zero perveance(51).
alwayspossibletoexactlycalculatefromtheseequations
theexternalandspacechargepotentialgivingrisetothat
distribution. When we adopt this second alternative ap- ξ
proachand we take as giventhe form of the distribution v(s) = log(s2)+C Ei( s2) (52)
−2 − −
w(s), the unknowns in the equations (38) and (39) are
v (s) = s2 (cid:2) (cid:3) (53)
the two potential energies v(s) and v (s). In this case 0
e
ξ
we only need to calculate the expressions (41) and (42) v (s) = s2+ log(s2)+C Ei( s2) (54)
e
in terms of the given distribution w(s). Of course if we 2 − −
take an arbitrary w(s) we will not get any simple and (cid:2) (cid:3)
In a sense the meaning of the equations (41), (42)
meaningful form for the external potential v (s); and on
e
and (43) is rather simple: if we want to get a self-
the other hand to guess the right form of w(s) giving
consistent distribution which coincides with a solution
rise,forinstance,exactlytoaharmonicpotential(44)as
of the S–ℓ equation for a given zero perveance poten-
external potential would be tantamount to solve (42) as
tial, the simplest way it is to calculate the space charge
an integro-differential equation for a given external po-
potentialforthisfrozen distributionthroughthePoisson
tential. However in a few explicit cases the results are
equation,andthencompensatetheexternalpotentialex-
quite simple and interesting.
actlyforthat. This is whatwe didinourexample where
Let us take as firstexample ofa stationarywavefunc-
the gaussian solution is the fundamental state of a har-
tionthatofthegroundstateu (r) oftheharmonicoscil-
0
monic oscillator: we finally gota total potentialwhichis
latorwithzeroperveancegivenin(45). Itsdimensionless
v (s) = v(s)+v (s) = s2 (namely that of a simple har-
representation is: 0 e
monic oscillator), and an energy value which coincides
w(s)=√2e−s2/2, β =2 (E =αω); (51) with the first eigenvalue. In other words, if you want a
T
gaussian transverse distribution you should not simply
whichisalsoapparentlynormalized. Wenowwanttocal- turn on a bare harmonic potential s2: you should rather
culate both the external and the space charge potentials teleologically compensate for the space charge by using
that produce (51) as stationary wave function for (49) the potential ve(s).
and (50). From (41), (42) and (51) we then have
w′′(s) 1 w′(s)
+ +β = v (s)=s2 B. Student transverse distributions
0
w(s) s w(s)
s dy y 1
w2(x)xdx = log(s2)+C Ei( s2) If the halo consists in the fact that large deviations
y 2 − − fromthebeamaxisarepossible,anewideaistosuppose
Z0 Z0
(cid:2) (cid:3) thatthethestationarytransversedistributionisdifferent
where C 0.577 is the Euler constant and
≈ fromthegaussiandistribution(51)introducedintheSec-
x et tion IVA. To this end we will introduce in the following
Ei(x)= dt, x<0
a family of distributions which decay with the distance
t
Z−∞ from the axis only with a power law.
istheexponential–integralfunction,andhenceweimme- Letusconsiderthefollowingfamilyofunivariate,two–
diately get (see also FIG. 1) parametersprobabilitylawsΣ(ν,a2)characterizedbythe
9
0.4 has variance σ2, and the standard (with unit variance)
generalized Student laws are Σ(ν,ν 2).
−
Inordertodescribethebeamwewillalsointroducethe
0.3
bivariate, circularly symmetric Student laws Σ (ν,a2)
2
with pdf
0.2
ν aν
f(x,y)= . (57)
2π(x2+y2+a2)ν+22
0.1
Its marginal laws are both Σ(ν,a2) and non–correlated,
x albeit not independent (as in the case of the circularly
-4 -2 2 4 symmetric gaussianbivariate laws). The total beam dis-
tribution will then be
1 νaν L
FIG. 2: The Gauss pdf N(0,1) (dashed line) compared with ρ(x,y,z)= H z (58)
the Σ(2,2) (thick line) and the Σ(10,12) (thin line). The 2πL (x2+y2+a2)ν+22 (cid:18)2 −| |(cid:19)
flexesofthethreecurvescoincide. Apparentlythetailsofthe
where H(z) is the Heaviside function. In the descrip-
Studentlaws are much longer.
tion of a beam in an accelerator it is realistic to suppose
that the transverse distribution is endowed with a finite
following pdf’s variance. Hence we will look for distributions (58) with
ν >2. Ontheotherhandthiswillcorrespondtosuppose
Γ ν+1 aν thatinourmodelthetransverseStudentlawsshouldnot
f(x)= 2 (55)
Γ 21(cid:0) Γ (cid:1)ν2 (x2+a2)ν+21 biseinrasdoimcaellysendsiffeearnenetfffercotmwhaicGhaiussssmianal:liwnhfeanctcotmhepahraeldo
(cid:0) (cid:1) (cid:0) (cid:1) with the total beam. From this standpoint the family
whichapparentlyaresymmetricfunctionswiththemode
inx=0andtwoflexesinx= a/√ν+2. Alltheselaws of laws Σ(ν,a2) has also the advantage that we can fine
± tune the parameters ν,a in order to get the right dis-
are centered at the median. In particular a plays just
tance from the gaussianlaws(this wouldnot be possible
the role of a scale parameter, while ν rules the power
decayofthetails: forlargexthetailsgoasx−(ν+1) with if we adopted stable laws; see subsequent Section VA).
ν+1 > 1. For a comparison with a Gauss law (0,σ2) With this hypothesis in mind we will limit our present
N considerations to the case ν > 2 so that the transverse
seeFIG.2. Remarkthatwhenν growslargerandlarger,
marginals of (58) will have a finite variance σ2. Then
thedifferencebetweenthetwopdf’sbecomessmallerand
smaller. It is typical of the laws Σ(ν,a2) that they have from (56) we choose a2 =(ν 2)σ2 and write (58) as
−
i(sfinveitrei)fiemd;omheenncteafoofroνrder2kthoenrelyisifntohveacrioanndciet,iownhkile<foνr ρ(x,y,z)= ν [(ν−2)σ2]ν2 H L z
νha≤nd1wnhoetnevνe>n t2hetheexvp≤aercitaantcioenofisΣd(eνfi,nae2d).exOisntsthaendotihser 2πL [x2+y2+(ν−2)σ2]ν+22 (cid:18)2 −|(5|(cid:19)9)
Passing to cylindrical random variables we then have
a2
σ2 = ν 2. (56) ρ(r,ϕ,z)=r ν [(ν−2)σ2]ν2 H L z
It will be useful to remark tha−t the laws Σ(1,a2) are the 2πL [r2+(ν−2)σ2]ν+22 (cid:18)2 −| |(cid:19)
well–known Cauchy laws (a) with pdf namely
C
1 a 1 r 2ν
f(x)= , ρ(r,ϕ,z) =
πx2+a2 σ√2 σ√2 ν 2
−
r2 −ν+22 H L z
while the laws Σ(n,n) with n=1,2,... are the classical 1+ 2 −| |
t–Student laws (n) with pdf × (ν 2)σ2 2πL
S (cid:20) − (cid:21) (cid:0) (cid:1)
f(x)= Γ n+21 (n+x2)−n+21 . so that finally with the shorthand notation
√π(cid:0)Γ n2(cid:1) s√2
z = (60)
WewillthenrefertoΣ(ν,a(cid:0)2)(cid:1)asgeneralizedStudentlaws √ν−2
sincetheyarejustStudentlawswithacontinuousparam- the dimensionless, normalized radial distribution is
eter ν >0 and a scale parameter a. For ν >2 variances
exist and we are then entitled to standardize our laws: w2(s)= 2ν 1 . (61)
indeedfrom(56)everyΣ(ν,(ν−2)σ2)witha2 =(ν−2)σ2 ν−2(1+z2)ν+22
10
Here we adopt the dimensional constants 7
α2 6
η = , λ=σ√2 (62)
4mσ2 5
L
where σ2 is the variance of our Student laws. We can Hs 4
0
now use the relations (41), (42) and (43) in order to get v
3
thepotentialswhichhave(59)asstationarydistribution:
first of all the space charge potential produced by (59) 2
has the form
1
ξ 2z−ν ν ν ν+2 1 s
v(s)= F , ; ;
−2 ν 2 1 2 2 2 −z2 5 10 15 20
(cid:20) (cid:18) (cid:19)
ν
+logz2+C+ψ (63)
2
(cid:16) (cid:17)i FIG. 3: The control potential v0(s) (64) for a Studenttrans-
where 2F1(a,b;c;w) is a hypergeometric function and verse distribution Σ2(22,20σ2). Also displayed are the value
ψ(w) = Γ′(w)/Γ(w) is the logarithmic derivative of the ofβ =2.4(thelimitvalueofv0 forlarges,thinline)andthe
Euler Gamma function (digamma function). On the behaviors for small and large s (66) and (67) (dashed lines).
other hand, by choosing β = 2 + 8 to put the po-
ν−2
tential energies to zero in the origin, we get the control
values (ν = 22, ξ = 20) the two potentials look particu-
potential for zero perveance
larly similar. In fact, given the asymptotic behavior of
ν+2 z2(4z2+ν+10) the hypergeometric function in (63) and of the exponen-
v (s)= (64)
0 ν 2 2(1+z2)2 tial integral in (52), for s + both potentials behave
− as ξlogs. On the other→hand∞we immediately see from
−
andhencetheexternalpotentialrequiredtokeepatrans- FIG.5thatthecontrolpotentialsforzeroperveancev (s)
0
verse student distribution Σ2(ν,(ν 2)σ2) with a given behave differently when we move away from the beam
−
variance σ2 is axis; beyond a distance of about r 2σ the two curves
≃
are different: while in the Gaussian case the potential
ν+2 z2(4z2+ν+10)
v (s) = diverges as s2, in the Student case it goes to the con-
e ν−2 2(1+z2)2 stantvalue β as quickly as s−2. Of coursethis difference
ξ 2z−ν ν ν ν+2 1 fades away when ν grows larger and larger; that points
+ F , ; ;
2 ν 2 1 2 2 2 −z2 to the fact that the principal difference between the two
(cid:20) (cid:18) (cid:19)
ν casescanbe confinedinaregionthatcanbemadeasfar
+logz2+C+ψ (65)
2 removed from the beam core as we want by a suitable
(cid:16) (cid:17)i choice of ν. Finally in the FIG. 6 we compare the total
Formulas (63), (64) and (65) give the self–consistent po- external potentials needed to keep the transverse beam
tentialsassociatedwiththebeamdistribution(59)which
respectivelyinaStudentandinaGaussdistribution. We
istransversallyaStudentΣ2(ν,(ν 2)σ2). IntheFIG.3 then see that for large s (far away from the beam core,
−
we can see an example of the control potential v (s) for
0 whileintheGausscasethetotalexternalpotentialgrows
a particular value of the parameter ν, together with its with s as s2+ξlogs, in the Student case this potential
limit behaviors
only grows as ξlogs. In any case, even if the potential
(ν+2)(ν+10) near the beam axis is harmonic,deviations from this be-
v0(s) ∼ (ν 2)2 s2, (s→0+) (66) havior in a region removed form the core can produce a
− deformation of the distribution from the gaussianto the
(ν+2)2 8
Student.
v (s) +2+ , (s + ) (67)
0 ∼ 4s2 ν 2 → ∞
−
Now this results must be compared with the similar re-
C. Estimating the emittance
sults (52), (53) and (54) associated to a transversally
gaussian distribution. We will choose the gaussian pa-
If u(r) is a self–consistent, cylindrically symmetric so-
rameters in such a way that the behavior near the beam
lution of (36) and (37) the position probability density
axis be similar to (66), namely (with β =2γ)
ρ(r,ϕ,z) in cylindrical coordinates will have the form
w(s)= 2γe−2γs2/2, γ2 = (ν+2)(ν+10). 1 u2(r) 0 ϕ<2π, L z L,
(ν 2)2 ρ(r,ϕ,z)= ≤ −2 ≤ ≤ 2
p − 2πL(0 otherwise.
First of all in the FIG. 4 we compare the space charge
potential produced by both a Student and Gauss trans- In order to estimate the emittance we need to calculate
verse distribution: remark as for the chosen parameter mean values of positions x and momenta p along one
x