CER

N-A

CC

-201

4-00

7330

/06/

2014

EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH

CERN-ACC-2014-0073

CASE STUDIES IN SPACE CHARGE AND PLASMAACCELERATION OF CHARGED BEAMS

A. Bazzani1, M. Giovannozzi2, P. Londrillo3, S. Sinigardi4,G. Turchetti1

1Dipartimento di Fisica e Astronomia, Universita di Bologna and INFN Sezione di Bologna, ViaIrnerio 46, I-40126 Bologna, Italy

2BE Department, CERN, CH 1211 Geneva 23, Switzerland3INFN Sezione di Bologna, Via Irnerio 46, I-40126 Bologna, Italy

4Istituto Nazionale di Ottica, Consiglio Nazionale delle Ricerche (CNR/INO), U.O.S. ‘AdrianoGozzini’, Pisa, Italy

Abstract

Plasma acceleration with electron or proton driver beams is a challenging opportunity for high energyphysics. An energy doubling experiment with electron drivers was successfully performed at SLACand a key experiment AWAKE with proton drivers is on schedule at CERN. Simulations play an im-portant role in choosing the best experimental conditions and in interpreting the results. The Vlasovequation is the theoretical tool to describe the interaction of a driver particle beam or a driver laserpulse with a plasma. Collective effects, such as tune shift and mismatch instabilities, appear in highintensity standard accelerators and are described by the Poisson-Vlasov equation. In the paper wereview the Vlasov equation in electrostatic and fully electromagnetic case. The general frameworkof variational principles is used to derive the equation, the local form of the balance equations andrelated conservation laws. In the electrostatic case we remind the analytic Kapchinskij-Vladimirskij(K-V) model and we propose an extension of of the adiabatic theory for Hamiltonian systems, whichensures stability for perturbation of sizeǫ on times of order1/ǫ . The variational framework isused to derive the Maxwell-Vlasov equations and related conservation laws and to briefly sketch theparticle-in-cell (PIC) approximation schemes. Finally the proton-driven acceleration is examinedin the linear and quasi-linear regime. A PIC simulation with the codeALaDyn developed at theBologna University is presented to illustrate the longitudinal and transverse fields evolution whichallow a witness electron bunch to be accelerated with a gradient of a few GeV/m. We also presentsome remarks on future perspectives.

Accepted for publication in Comptes Rendus de Mecanique Academie des Sciences de Paris

Geneva, SwitzerlandJune21, 2014

Case studies in space charge and plasma acceleration of

charged beams

Armando Bazzania,b, Massimo Giovannozzic, Pasquale Londrillob, StefanoSinigardia,b,d, Giorgio Turchettia,b

aDipartimento di Fisica e Astronomia, Universit�a di Bologna, Via Irnerio 46, I-40126Bologna (BO), Italy

bINFN Sezione di Bologna, Via Irnerio 46, I-40126 Bologna (BO), ItalycBE Department, CERN, CH 1211 Geneva 23, Switzerland

dIstituto Nazionale di Ottica, Consiglio Nazionale delle Ricerche (CNR/INO), U.O.S.`Adriano Gozzini', Pisa, Italy

Abstract

Plasma acceleration with electron or proton driver beams is a challenging op-portunity for high energy physics. An energy doubling experiment with electrondrivers was successfully performed at SLAC and a key experiment AWAKE withproton drivers is on schedule at CERN. Simulations play an important role inchoosing the best experimental conditions and in interpreting the results. TheVlasov equation is the theoretical tool to describe the interaction of a driverparticle beam or a driver laser pulse with a plasma. Collective e�ects, such astune shift and mismatch instabilities, appear in high intensity standard acceler-ators and are described by the Poisson-Vlasov equation. In the paper we reviewthe Vlasov equation in electrostatic and fully electromagnetic case. The generalframework of variational principles is used to derive the equation, the local formof the balance equations and related conservation laws. In the electrostatic case,we remind the analytic Kapchinskij-Vladimirskij (K-V) model and we proposean extension of the adiabatic theory for Hamiltonian systems, which ensuresstability for perturbation of size � on times of order 1=�. The variational frame-work is used to derive the Maxwell-Vlasov equations and related conservationlaws and to brie y sketch the particle-in-cell (PIC) approximation schemes. Fi-nally, the proton-driven acceleration is examined in the linear and quasi-linearregime. A PIC simulation with the code ALaDyn developed at Bologna Uni-versity is presented to illustrate the longitudinal and transverse �elds evolutionwhich allow a witness electron bunch to be accelerated with a gradient of a fewGeV/m. We also present some remarks on future perspectives.

Email addresses: armando.bazzani@bo.infn.it (Armando Bazzani),massimo.giovannozzi@cern.ch (Massimo Giovannozzi), pasquale.londrillo@oabo.inaf.it(Pasquale Londrillo), stefano.sinigardi@gmail.com (Stefano Sinigardi),giorgio.turchetti@bo.infn.it (Giorgio Turchetti)

Preprint submitted to Elsevier June 15, 2014

1. Introduction

Particle accelerators are key instruments for the research in the domain of highenergy particle physics. To explore to even further accuracy the StandardModel, higher energies are required, thus pushing the technology of particleaccelerators to its limits. The challenge is being tackled in two di�erent do-mains, namely the design of high �eld magnets, required to control the motionof high energy charged particles, and that of high gradient radio-frequency (RF)systems, aimed at accelerating the charged particles to the energies required forprobing the intimate nature of matter and forces. Collective e�ects, such asspace charge, are relevant in the early accelerating stages and limit the inten-sity reachable in �nal ring. Their control allows to increase the �nal luminosity,a key parameter for high energy experiments. In the domain of accelerating sys-tems, the standard approach consists in generating longitudinal electric �eldsusing metallic RF cavities, either normal or superconducting. Technological lim-its arise when trying to reach accelerating gradients of the order of 100 MV/m,which would have a direct impact on the dimensions of future accelerators. Inorder to keep under control the size of the new instruments, gradients of theorder of GV/m would be needed and a plasma could be the natural solution,as it can sustain very large accelerating �elds [1, 2, 3]. It is worth recalling thee�orts devoted in the last few decades to experimental veri�cation of plasma ac-celeration, which reached very encouraging results [4, 5]. It is also important tostress the two approaches that can be used, namely laser wake�eld acceleration(LWFA) [1], whenever a laser is driving the acceleration, or plasma wake�eldacceleration (PWFA) [2, 3], whenever a bunch is used to drive the whole process.The fundamental principle behind both approaches is that a plasma acts as anenergy transformer, i.e., transferring the energy from the driver bunch to thewitness bunch. As proton bunches at the energy frontier, i.e., in the multi-TeVrange, are already available, it has been proposed the use of protons, instead ofelectrons, as driver beams opens up the possibility of generating extremely highenergy lepton beams via PWFA [6]. Furthermore, the use of protons to drivethe acceleration process would be a premi�ere, never tried before.In this paper we review the Vlasov theory and the related numerical schemes.We �rst examine the electrostatic case which allows to model the space chargee�ects in a beam, which are relevant at low-medium energies. Since spacecharge mainly a�ects the transverse motion, one and two dimensional models areadequate to describe the basic dynamical features. The analytic Kapchinskij-Vladimirskij solution [7, 8], applicable to a linear lattice, and the adiabatictheory stability results, applicable to a at beam, allow to understand manyproperties of the beam dynamics such as the mismatch oscillations. However,numerical solutions for two or three dimensional models provided by the particlein cell method (PIC) are needed to de�ne stationary solutions of Vlasov equationand to study their stability, in particular when multipolar magnetic componentsare present in the accelerator lattice.In the more general framework of laser plasma or beam plasma acceleration,the theoretical description is provided by Maxwell-Vlasov equations as long as

2

collisional e�ects can be neglected. In this case analytical results are availableonly for some one dimensional models and one has essentially to rely on thenumerical solutions provided by the PIC schemes. In spite of the remarkablecomputational power presently available, the simulation of the acceleration onlong plasma channels along thousands plasma lengths is still a challenge andreduced schemes, where the fastest scales are neglected, must be adopted. Thecontrol of accuracy is important, especially for reduced schemes, and the pres-ence of conserved variables is helpful. In this paper we present a short reviewof the theoretical framework for the electromagnetic case applicable to plasmaacceleration. As an example of numerical computations, we �nally consider theacceleration of electrons driven by proton bunches like the AWAKE experimenton schedule at CERN.The plan of the paper is the following: in section 2, we introduce the PoissonVlasov theory via the action principle, which is suitable to obtain the equationsof motion and the conservation laws. In section 3, we consider the space chargeproblem in a ring after recalling the Kapchinskij-Vladimirskij model and theprocedure to develop numerical schemes. In section 4, the stability problem fora at beam is discussed using the Hamiltonian adiabatic theory and it is provedthat the size of an instability does not overcome � � k@f=@tk for a time intervalat least 1=�. In section 5 we consider the Maxwell-Vlasov theory starting fromthe Low's Lagrangian [9]. The way to obtain the local energy balance for theelectromagnetic �eld and the charged uid is sketched. In the sections 6 weoutline the procedure to obtain the PIC schemes with an interpolating basisof linear splines and sampling the distribution functions of the charged uidwith numerical particles. In section 7 we brie y review the proposal of AWAKEexperiment discussing the acceleration of electrons driven by a proton beam ina plasma in the linear and quasi-linear regimes. The results of a simulationobtained with the code ALaDyn developed at Bologna University is presentedto illustrate the the accelerating longitudinal �eld created in the plasma. Asummary and comments on future perspectives conclude the presentation.

2. The Poisson Vlasov equation

We consider a system of N particles of charge e and mass m interacting viaCoulomb repulsion and subject to an external electrostatic �eld with potentialV (ex). Denoting with Htot the total Hamiltonian, with M = Nm and Q = Nethe total mass and charge, we write

Htot

M=

1

N

NXk=1

Hk(xk;pk)

Hk(xk;pk) =p2k

2+Qe

m

1

2N

Xj;(j 6=k)

1

rjk+

e

mV (ex)(xk)

where x;p 2 R3. Each Hk is the Hamiltonian for a single particle, with an

external potential Vk(x) generated by all the particles i 6= k. In the limitN !1

3

the N body system is replaced by a continuum, whose phase space distributionf(x;p; t) satis�es the Liouville equation for the Hamiltonian H(x;p) with apotential V , which satis�es the Poisson equation. The overall system is governedby

@f

@t+ [f;H] = 0 H =

p2

2m+Qe

mV (x) +

e

mV (ex)(x)

�V = �4� � � = e

Zf(x;p; t) dp

where [ ; ] denote the usual Poisson's bracket operator.The underlying assumption is that in this limit the collisions can be neglected, sothat statistical independence holds and the N points distribution function canbe written as the product of the single particle distribution functions f(xk;pk; t)for k = 1; : : : ; N . The ratio e=m as well as the total mass and charge are keptconstant in this limit. The plasma frequency !p = (4��e=m)1=2 is constant,since � = Q=V , whereas the Debye length �D = vth=(

p2!p) depends on the

thermal velocity. Vlasov description is correct if �D � �p where �p = (2� c)=!pwhich is ful�lled if kB T � mc2. This collisionless model is valid since, accordingto the Landau model, the frequency of binary collisions is �coll � log(N)=N ,which vanishes for N !1.In the Vlasov equation, the physical parameters enter as the ratio e=m andQ=M , and N has disappeared. However, it is customary to write the equationintroducing the physical parameters e;m;N . Of course, these parameters canbe scaled as in the PIC numerical schemes, where N numerical particles areintroduced to sample the distribution function and their charges and masses aregiven by m = mN=N and e = eN=N .

2.1. Variational formulation

We �rst consider the Poisson-Vlasov equations which are satis�ed by a systemof particles whose interaction is electrostatic. This is the case for instance of abeam in a ring once is the acceleration process is over. The Low's LagrangianAction for the system is given by [9]:

A =

Z tb

ta

ZR6

`(x;v; t) f0(x0;v0) dx0 dv0 dt +

Z tb

ta

ZR3

E2

8�(x; t) dx dt

where ` and the electric �eld E are given by

` = �m2v2 � e�� eV (ex) E = �r� :

The potential �(x; t) and the particle trajectory x(t) are kept constant at theendpoints of the time interval and (x0;v0) are the initial conditions of thetrajectory. The Action is stationary for variations of � and x, if and only if

4

m _v = eE+ eE(ex) divE = ��� = 4��

where E(ex) = �rV (ex). These equations must be self-consistently satis�ed.The variational formulation has been used also to analyze the stability of thesolutions of the Poisson-Vlasov equations [10].We �rst consider the variation of the particle Action for a given potential � andwe determine the trajectory with given endpoints x(ta) = x0 and x(tb) = xb.The initial velocity v0 is uniquely �xed by the boundary conditions. As aconsequence, by inverting the evolution with respect to x0;v0, we obtain thedistribution function in its Eulerian form f(x;v; t), which satis�es the continuityor Liouville equation. In the present case the Hamiltonian is h = p2=2m+ e�+eV (ex) and

@f

@t+ v

@f

@x+

e

m(E+Eex)

@f

@v= 0

Imposing the variation with respect to � to vanish, we obtain

Z tb

ta

ZR3

1

4�divE �� dx dt�

Z tb

ta

ZR6

f(x0;v0) e �� dx0 dv0 dt = 0 :

Changing the integration variable in the second integral from x0;v0 to x;v, wecan replace f(x0;v0) with its Eulerian form, since it is a canonical transforma-tion and the Jacobian is 1. We get:

Z tb

ta

ZR3

�1

4�divE � e�

��� dx dt = 0 � = e

ZR3

f(x;v; t) dv:

The stationary condition with respect to � imposes that the Poisson equationmust be satis�ed.The conservation laws follow immediately in this case. Indeed, de�ning thecurrent according to

j = e

ZR3

v f dv

and using the Liouville equation we have

@�

@t= �e

ZR3

�v@f

@x+eE

m

@f

@v

�dv = �div j :

Considering

HP =

ZR3

fm

2jvj2 dv HP =

ZR3

fm

2jvj2 vk dv SP = j � (E+E(ex))

where HP ;HP ;SP are the energy density, ux and source for the charged uid,and computing the second moment HP partial derivative with respect to time,

5

we obtain the local energy balance. Setting E(ex) = 0, for the electric �eld,whose energy density is HF = E2=8�, a similar balance holds:

@HP

@t+ divHP = SP @HF

@t= SF

where SF = (4�)�1E � @tE = �j �E and is opposite to the charged uid sourceif the vector (4�)�1@tE + j, (whose divergence is zero due to the continuityequation) vanishes.

3. Space charge e�ects for a 2D beam

The space charge e�ects are relevant for a beam except at extremely high ener-gies. In a ring, the beam moves with a constant longitudinal velocity v0 closeto c. If the charges are uniformly distributed on a cylinder, a test charge q ex-periences a radial force qEr due to Coulomb repulsion in the beam rest frame,which in the laboratory frame becomes qEr=

20 , 0 being (1� v20=c

2)�1=2. Thecoordinate along the beam axis is s = v0t and we de�ne the transverse momentaas p = v=v0 = dx=ds so that the energy density for the transverse motion reads

E = EF + EP =E2

4�+

ZR2

f(x;p; s)h(x;p; s) dp

where

h =p2x + p2y

2+ k0x(s)

x2

2+ k0y(s)

y2

2+�

2�(x; y; s) :

The perveance � is de�ned by

� =2eQ

mv20

�1� v20

c2

�=

2qQ

mv20

�mc2

E

�2

:

The quadratic term in the potential is due to the quadrupoles, but higher orderpolynomials, due to higher magnetic multipoles, may be present. For a briefpresentation see [11], while use [8] for an exhaustive discussion.The stationary condition implies that x(s);p(s) satisfy Hamilton's equationsand consequently f satis�es the Liouville equation, whereas the Poisson equationholds self-consistently for �.

@f

@s+ [f; h] = 0 �� = �4�e� � =

ZR2

f(x;p; s) dp :

Approximate solutions can be found using the PIC method [12, 13, 14, 15],whose key feature is to be an e�cient solver for the Poisson equation [16].Analytic solutions are known if k0x = k0y and if they are s independent (constantfocusing case), since the axial symmetry makes the problem one dimensional [8].In this case approximate analytic solutions can be found by using the Birkho�normal forms [17].

6

A family of stationary solutions is given by the distributions f = f(H) withsupport on 0 � H � E. Letting H = p2=(2m) +W (r) where W = k0 r

2=2 �12 � V (r) and W (R) = E the solution reads

� = 2�

Z W (R)

W (r)

f(E)dE1

r

d

drrd�

dr= �4� �(r)

with the normalization condition 2�R R0nrdr = 1. In the special case f(H) =

�(H �E)f0, where f0 = 1=�R2, the space density is constant � = 1=(�R2) andthe solution for the potential is � = r2=R2. The Hamiltonian becomes quadraticand the electric �eld is linear

h =p2

2+k

2r2 k = k0 � �

R2E =

2x

R2:

In the case of periodic focusing, assuming that the bunches are very elongatedalong s-coordinate, an analytic solution can be found by choosing a distributionfunction f which depends on the invariant actions, which can be determinedself consistently assuming a priori that the potential � is still quadratic. Thissolution is due to Kapchinskij and Vladimirskij, see [7, 8]. The generatingfunction F = F (x) + F (y) where F (x) = tan(� � _�x=2)x

2=(2�x) (the same for

Fy) changes h into a new Hamiltonian h depending on the actions

h =p2x + p2y

2+kx(s)

2x2 +

ky(s)

2y2 h = h+

@F

@s=

Jx�x(s)

+Jy�y(s)

:

The function �x is related to the phase of the solution for the x coordinatex = A(s) cos(�(s) + x) by �x = 1= _�x, where Ax(s); �x(x) satisfy

�Ax � J40x

A3x

+ kx(s)Ax = 01

2�x ��x + kx(s)�

2x

and similar equations hold for the y coordinate. The �rst one is the envelopeequation. The distribution function is chosen according to

f(Jx; Jy) =1

4�2J0xJ0y�

�JxJ0x

+JyJ0y

�

and since [f; h] = 0, it is stationary. The manifold on which f has its supportis the ellipsoid Jx=J0x+ Jy=J0y = 1 foliated, by invariant tori T2 de�ned as theproduct of the ellipses:

2�x Jx = x2 + (�xpx � _�x=2)2 = A2x Ax = (2�xJ0x)

1=2 and x! y :

The density �(x; y; s), obtained from f , is constant in the ellipse with semi-axisAx; Ay

7

� =1

�AxAy#

�1� x2

A2x

� y2

A2y

�:

As a consequence the electric �eld solution of the Poisson equation is linearwithin the elliptic bunch

Ex =4x

Ax(Ax +Ay)(in) Ex =

4x

A0x(A0x +A0y)

(out)

where A0x = (A2x + �)1=2; A0y = (A2

y + �)1=2 are the semi-axis of the confocal

ellipse namely � is de�ned by x2=A0x2+ y2=A0y

2= 1 and at large distance from

the origin A0x = A0y ' �.Since the �eld is linear, it is evident that the coe�cients of the quadratic termsin the Hamiltonian are given by kx = k0 x � 2�=(Ax(Ax + Ay)) (and the samefor ky) where

�Ax � 4J20x

A3x

+ k0 x(s)Ax =2�

(Ax +Ay)and x! y:

This equation admits periodic solutions �(s) and A(s) = J0�1=2(s), which are

stationary. The e�ect of nonlinearities in the lattice can be investigated using thenormal forms in the case of constant focusing [12]. There are also quasi-periodicsolutions, which are not stationary and describe the so called mismatched beam.By linearizing the envelope equation, the frequencies of the mismatched solutioncan be determined [18]. In any other case, one has to rely on the PIC numericalprocedure to determine the solution. The distribution function is sampled withnumerical particles, the density is interpolated with linear splines and the Pois-son equation is solved using a spectral or �nite di�erence scheme. The Poissonsolver is the key algorithm and an e�cient spectral algorithm capable of dealingwith any assigned boundary conditions has been developed by Bergamini et al.and discussed in [13].

4. Adiabatic invariance for a at 1D beam

As we have discussed in the previous chapter, only few explicit solutions ofthe Vlasov equation are known. Moreover, the stability problem of stationarysolution is still open in many case [19].But the possibility of describing the Vlasov dynamics as a single particle dy-namics in a self-consistent potential allows to partially extend the averagingtheorem results for Hamiltonian systems [20].We explicitly consider when the single particle Hamiltonian reduces to a 1Dintegrable system with an elliptic �xed point at the origin

H(x; p; f) =p2

2m+ V (x; t; f) : (1)

8

The potential V (x; t; f) has a functional dependence from the particle distribu-tion f(x; p; t) in the phase space, that satis�es the non linear Liouville equation

@f

@t= �p@f

@x+@f

@p

@

@xV (x; t; f) : (2)

Any stationary solution has to be a function of the invariants of the motion. Forany given phase space distribution f(x; p), one can introduce the action-anglevariables (I; �) in a neighborhood of the origin

I(E; f) =1

2�

IH(x;p;f)=E

p(x;E) dx

(3)

�(x; I; f) =@

@IF (x; I; f)

����x

where F (x; I; f) is the generatrix function computed from the de�nition

F (x;E; f) =

Z x

H(f)=E

p(x;E)dx (4)

which depends on the given distribution. Then one has to look for a self-consistent stationary distribution f(I). We remark that, in the de�nition ofF (x; I; f), one can choose the origin of the angle � in a arbitrary way: i.e., inany section of the phase space. Letting H(I; f) the new Hamiltonian, we havethe frequency of motion

(I; f) =dH(I; f)

dI:

We assume that far from separatrix curves the functional dependence of theHamiltonian and the action-angle variables from the the distribution f(x; p) areregular. Indeed in the Vlasov problem the potential V (x; f) is linearly dependenton the the particle distribution, according to

V (x; f) =

Zx0 6=x

k(jx0 � xj)f(x0; p0)dx0dp0 (5)

where k(jxj) is the kernel that de�nes the electrostatic potential. Under theseassumptions, if one considers a slowly varying distribution f(x; p; t)

@f@t = �� 1 (6)

(��1 � 1 de�nes the slow time scale), the single particle dynamics has theproperty that the action variable I(f), computed considering the distribution fconstant (frozen system), is an adiabatic invariant

jI(x(t); p(t); t; f)� I0j � O(�) for t � 1

�(7)

9

where (x(t); p(t)) is the solution of the Hamiltonian system (1). The equation(7) means that, during the time evolution, the value of the initial action can beconsidered constant for a time 1=�, which is the slow time scale. The same is truefor any regular function of the action. The proof of this result is a consequenceof the perturbation theory and it holds for a whole class of distribution functionsf(x; p; t), with the only constraint that the adiabatic parameter

�

2(f)� 1: (8)

The condition (8) points out as we cannot apply the adiabatic theory neara separatrix curve where the unperturbed frequency ! 0. The adiabaticproperty of the action variable can be understood if we choose a section � =const: in the phase space (Poincar�e section) and we denote ftngn2N, the sequenceof crossing times of an orbit (x(t); p(t)) with the section. Since the frozen energyH(x; p; f) is conserved up to an error of order O(�), the solution (x(t); p(t)) fort 2 [tn; tn+1] is close to the curve H(x; p; f(t�)) = E�, where t� 2 [tn; tn+1] andE� 2 [En; En+1]. Moreover, the dynamics of the angle variables reads:

_� = (E; f) +O(�):

Then we get the relation

dt =d�

(E; f)+O(�) (9)

and we can use the angle � as a parameter for the dynamics (x(t); p(t)). Ac-cording to the results of perturbation theory [21], we can write

In+1 � In =

Z 2�

0

1

2(E; f)

�Zx0 6=x

k(jx0 � xj)@f@tdx0dp0

(10)

��Z

x0 6=x

k(jx0 � xj)@f@tdx0dp0

��d� +O(�2)

where h i denotes the average with respect to the � variable. Since both theenergy E and the distribution f can be considered constant in time during theintegration up to an error of order O(�), the integral vanishes and the previousrelation implies

jIn+1 � Inj = O(�2) : (11)

Therefore, since the value of I varies by a quantity of order O(�) in each interval[tn; tn+1], the adiabatic invariance holds for a time of order O(1=�). The remarkon the estimate of the adiabatic parameter (8) is needed to obtain the result(11). In the adiabatic approximation for the dynamics, the angle variable reads

10

�(t) =

Z t

0

(I; f)dt+O(�)t

where I is kept constant in the integration. We will give a schematic road mapto extend the previous argument to the Vlasov problem, where one has to checkif a small perturbation of the stationary solution, which evolves according tothe non linear Liouville equation (2), satis�es the condition of the adiabaticapproximation. Let the stationary distribution perturbed be f 0(I; t) = f(I) +�f(I; �; t), then we assume the ansatz

@f0

@t

= �� 1 (12)

so that the adiabatic parameter

1

2(f)

@f0

@t

� 1

is also small. We apply the adiabatic description of motion to get an approxi-mate solution of the non linear Liouville equation (2)

f 0(�; I; t) = f(I) + �f

�� +

Z �t

0

(I; s; f)ds; I

�(13)

with an error which remains of order O(�) for a time ' 1=O(�). Then, tocorrectly apply the adiabatic approach, a consistency condition on the estimateof the slow time scale (6) should be satis�ed. Considering the leading terms weget

@f0

@t

� @f@I

j�Ij+ k�fkkk (14)

and the adiabatic invariance for the Vlasov problem requires the consistencycondition

@f@I j�Ij(�) + k�fkkk � � (15)

which can be reduced to a contraction principle if we use the equation (10). Thecondition (15) can be satis�ed if the derivative k@f=@Ik is su�ciently small:i.e. the dependence of the stationary distribution from the unperturbed actionvariable is not too steep. In such a case, the adiabatic argument holds and the

approximation (13) can be applied for a time / @f 0

@t

�1

so that we have a

stable solution up to that time scale.

11

5. The Maxwell-Vlasov equations

For simplicity, we consider the interaction of an electromagnetic �eld with onlyone species (electrons), since the extension to two or more species (electronsand ions in the plasma or in a driver bunch) is straightforward. We start againfrom the Low's Lagrangian Action [9] following from the system written as thesum of the Action for the �eld and the Action for the particle system, see [22]for a review.

A =

Z tb

ta

ZR6

f0(x0;v0) `(x;v; t) dx0 dv0 dt +

Z tb

ta

ZR3

LF dx dt

where x = x(x0;v0; t), v = dx=dt, f0(x0;v0) is the initial distribution of parti-cles in phase space, ` is the particle Lagrangian and LF is the �eld Lagrangiandensity, given by:

` = � �1(v)mc2 � e�+e

cv �A (v) =

�1� jvj2

c2

�1=2

LF =E2 �B2

8�

The electric and magnetic �elds are expressed in terms of the scalar and vectorpotentials as usually

E = �r�� 1

c

@A

@tB = rotA

and E;B are invariant with respect to the gauge transformation

�! �� = �� �

c

@g

@tA! A� = A+ �rg :

First we impose that the variation with respect to x vanishes, treating � andAk as assigned �elds, and then we impose that the variation on the �elds �;Avanishes. As a consequence, we obtain the equations of motion and we canexpress the distribution function in Eulerian form f(x;v; t), so that

md

dt( v) = eE+

e

cv �B

@f

@t+ v

@f

@x+

@

@v

hf�eE+

e

cv �B

�i= 0 :

The equations for the �elds provide the �rst two Maxwell's equations

divE = 4�� rotB =1

c

@E

@t+4�

cj

where we have de�ned

�(x; t) = e

ZR3

f(x;v; t) dv j(x; t) = e

ZR3

vf(x;v; t) dv :

12

The remaining two Maxwell's equations, divB = 0 and c rotE = �@B=@t, areexactly satis�ed thanks to the relation between the �elds and the potentials.Formulations in a di�erent functional setting where the distribution function fis itself a �eld and observables are functionals of this �eld have been introducedboth in the Lagrangian and the Hamiltonian framework [23, 24].The �rst invariance property is obtained by imposing that the Action does notchange under the gauge transformation on the �elds.The variation of the Action for the free �elds and for the interaction with thecharged uid is

@A

@�=

ZR4

�@�

@t+ div j

�g dx dt = 0 :

The arbitrariness of g imposes that the continuity equation is satis�ed.The conservation laws in the uid formulation have been derived using N�other'stheorem in [25].We obtain a balance equation in the form

@HF

@t+ divHF = SF @HP

@t+@HP

@xj= SP

where HF ;HF ;SF and HP ;HP ;SP are the energy density, ux and source forthe �elds and the charged uid.

HF =E2 +H2

8�HF =

c

4�E�B SF = �j �E

HP =

ZR3

(v)mc2 f dv HP =

ZR3

(v)mc2 v f dv SP = j �E :

The �nal form of the local energy conservation law is

@

@t(HF +HP ) + div (HF +HP ) = 0 :

In a similar way the local equation for the momentum balance, from whichthe uid equations follow, can be obtained by considering the action of spacetranslations on the Lagrangian density.

6. Particle in cell approximation schemes

In order to develop numerical schemes to solve the Poisson Vlasov equations, onecan introduce two �nite bases for the potentials and for the charged uid phasespace distribution function [26, 27]. Letting ��, where �0 = � and �j = Aj , weconsider the expansion

��(x; t) =Xg

��;g(t) g(x) (16)

13

where g = (i; j; k) and

g(x) = i(x) j(y) k(z) : (17)

We may choose linear interpolating spline functions. For the distribution func-tion the common choice is

f(x;v; t) =N

N

X�

�(x� x�(t)) �(v � v�(t)) (18)

where �(x � x�(t)) = �(x � x�(t))�(y � y�(t))�(z � z�(t)) and the functionshave compact support.In this case the functions are positive and normalized:

R�(x� x�)dx = 1.

With this basis, the Lagrangian for the particles and their interaction with the�elds becomes:

LP =

NX�=1

m

2v�

2 +Xg

���g �g + 1

cjg �Ag

�

where we have de�ned the current and the density components

�g = e

NX�=1

Sg(x�(t)) jg = e

NX�=1

v�Sg(x�(t)) :

In the equations above, e = Ne=N and m = Nm=N denote the charge andthe mass of the so called numerical particles. Sg(x�(t)) are new basis functionsde�ned by

Sg(x�(t)) =

Z g(x)�(x� x�(t)) : (19)

In a similar way one may compute the �eld Lagrangian that becomes a quadraticform in the �eld components �g and Ag.By taking the variations of the Action A =

R(LP + LF ) dt with respect to the

macro-particles coordinates x� and the potential components �g; Ag, one canobtain a set of ordinary di�erential equations for these variables. An explicitintegration scheme in time, like the fourth order Runge Kutta, allows to ob-tain an approximate solution, where the conservation laws are preserved up tointegration error.

6.1. Standard particle in cell formulation

The more traditional scheme consists in solving the exact Maxwell equations forthe �elds E; B after expanding them on the basis �g(x), according to [28, 29].

E(x; t) =Xg

Eg(t) g(x) B(x; t) =Xg

Bg(t) g(x) : (20)

14

By choosing for instance linear splines, one can obtain the discrete Maxwell'sequations projected on a staggered mesh (Yee's lattice) according to

dBg

dt= �D�Eg

dEg

dt= D�Bg � 4�

chJig

whereD is the 2 points �nite di�erence operator. The conditionD�B = 4�h�ig,true at t = 0, is satis�ed for t > 0 by centered time di�erences. The conditionD � Eg = 4��g, if true at initial time, is satis�ed for t > 0 if the continuityequation holds. It can be forced to hold following the Esirkepov's procedure.However in this case energy conservation is lost.A parallel PIC code ALaDyn, based on second and fourth order schemes, hasbeen developed at the University of Bologna group [30, 31]; a code based essen-tially on the same algorithms has been implemented on the GPU architectures[32]. Other approximations, based on a truncated moments expansion of thedistribution function, suitable also to describe the beam plasma interaction,have been proposed [33, 34].

7. Beam plasma acceleration

In recent years, the AWAKE collaboration [35] has been created, with the aimof exploring the beam-driven Plasma WakeField Acceleration (PWFA) based onprotons as driver beam. Special emphasis was put on the analysis of the severalkey processes that are needed to achieve successfully the goal of PWFA.The �rst step in this process has been the proposal of using high energy protonsfor performing PWFA of electrons [6]. Numerical simulations [36] showed that a1 TeV proton bunch, composed by 1011 particles and with an rms length of 100�m, can generate a large enough plasma wave. The key point is the generationof such a short proton bunch, which is a true challenge. However, a milestonewas achieved when numerical simulations showed [37, 38] that even much longerbunches, comparable with those that can be generated in existing facilities, canexcite a plasma wave, as long as the intensity of the proton bunch is modulatedalong its length. This opened the possibility of experimental veri�cations of theproposal, and hence the AWAKE collaboration has been set up to [35]:

� study the detail of the process of self-modulation of the long protonbunches;

� probe the acceleration process properly, by studying the properties of thegenerated wake�eld;

� study the injection mechanism of multi-GeV electron bunches;

� develop plasma cells and mechanisms to generate short proton bunches.

The issue of generating short bunches is certainly a crucial one and it has beenconsidered in detail, e.g. in Ref [39], where several options have been considered,based on the performance of existing CERN accelerators and even including new

15

techniques. In a �rst stage, the use of proton bunches extracted from the CERNProton Synchrotron (PS) at 24 GeV was considered and numerical simulations[38] con�rmed the validity of this option. Nevertheless, additional considerationsbased both on physical arguments and on available infrastructure, suggestedthat the best choice would be to use a 400 GeV proton beam extracted from theCERN Super Proton Synchrotron (SPS) and this option (see Refs. [40, 41, 42]for more detail) was retained in the o�cial proposal [35]. In the following, thekey points of the AWAKE Design Report are addressed. In the case of LWFA,the driver is a laser pulse of length �z, comparable to the plasma wavelength�pe:

�z = �e with �e =2�c

!eand !e =

�4�e �eme

�1=2

:

Under these conditions, a longitudinal accelerating �eld can be generated pro-

portional to the breaking �eld EWB [43], which in turn is proportional to �1=2e

All this favors short length for the laser pulse and high density for the plasma.Whenever the typical bunch length �z is of the order of 100�pe, then one hasto operate in the so-called self-modulated PWFA regime [37]. In this regimethe transverse bunch size �r has to satisfy kpe�r < 1, with kpe = !pe=c. Hence,the condition on a short bunch length can be translated into one on a smalltransverse bunch size, much easier to achieve even with conventional focusingdevices available in a beam transfer line.The self-modulated instability (SMI) generates a modulation of the bunch inten-sity along its length and hence it can drive resonantly the plasma so that wake-�elds are generated. Unfortunately, the instability destroys the micro-bunchingstructure soon after the peak �eld is reached [44, 45, 46]. A mitigation measurewas found in generating a plasma density gradient. It is worth mentioning thatseeding the SMI is also an important point that will be probed with AWAKE.Indeed, the natural noise in the plasma should be enough to trigger the rise ofthe SMI [36]. Nonetheless, the SMI can be more e�ciently induced by meansof a short bunch preceding the long driver bunch or by a short laser pulse. TheSMI can arise together with another instability, the hose instability (HI) [47]and in this respect seeding helps SMI to overcome HI [45, 46]. The witness elec-tron bunch will be side-injected in the plasma cell [48] at a longitudinal positionwhere the amplitude of the wake�eld generated by the long proton bunch has al-ready reached saturation. The long, �z � �pe, low-energy electron bunch shouldbe injected with an angle of few milli-radians with respect to the proton beamaxis. The electrons will approach the seeding bunch and will be trapped into thewake�eld, and later will be accelerated by the resulting buckets from the wake-�eld. In principle, a fraction of the electrons in the witness bunch correspondingto 5��40% should be trapped and accelerated to higher energies with a rathersmall relative energy spread, at the level of a percent. The scienti�c goals ofthe AWAKE collaboration rely heavily on state-of-the-art instrumentation. Thevarious devices are based on transition radiation (TR), which is emitted whencharged particles traverse media with di�erent refraction indexes. This type

16

of radiation allows for fast diagnostic device, at the level of picoseconds. Two avors of TR, namely optical transition radiation (OTR) and coherent transi-tion radiation (CTR), will be used for a complete characterization of the bunchdensity distribution. An integral part of the beam instrumentation will be theelectron spectrometer aimed at an accurate measurement of the electron bunchenergy at the end of process. The �nal choice for the location of the AWAKEexperiment has been made: it will be located in the CERN Neutrino to GranSasso (CNGS) area (see Fig. 1).

Figure 1: Layout of the CNGS transfer line including the region where the AWAKE experimentshould be installed (from Ref. [49]).

The beam line to transfer the SPS-extracted proton beams towards the CNGStarget will be re-used. However, the AWAKE experiment will be housed justupstream of the target station (see Fig. 2 for a layout of the experimental area).This will entail the modi�cation of the last section of the existing proton beamline, in order to provide the required transverse focusing of the driver bunch atthe location of the plasma cell. Moreover, an additional beam line to transportthe electron beam to the plasma cell will be built.The study and installation of the AWAKE experimental facility will be stagedbetween the proton beam line, the proper experimental area, and the electronsource and related beam line [35]. These activities will span a time period ofabout four years. The experimental activities with protons will be starting bythe end of 2016, while those with electrons one year later. Overall the datataking should take place over a period of three to four years.

7.1. PIC Simulations

As an example of applications of the Vlasov equations, we consider the case ofa driver beam of electrons or protons exiting plasma waves by which a witnesselectron beam is accelerated. In a background plasma with electron density

17

Figure 2: Layout of the CNGS transfer line including the region where the AWAKE experimentshould be installed (from Ref. [22]).

�e, an electron or proton bunch with density �b � �e excites linear plasmawaves whose longitudinal accelerating �eld is a few GV/m. In linear regimes,reached for higher densities �b � �e, the wave breaks and a bubble is formedin the plasma just like when the driver is a laser pulse. An energy doublingexperiment in a nonlinear regime was carried out with a driver beam of 42 GeVin a plasma wake�eld accelerator of 85 cm at SLAC [5] and the accelerating �eldwas of 52 GV/m. In the bubble (blow out) regime, �rst proposed in [50], theradial con�ning �eld is linear Er ' 2��er so that the electrons of the witnessbeam are accelerated just as in a conventional RF cavity. The drawbacks of thebubble regime are the instabilities (wave-breaking spike on the �rst oscillationbehind the beam driver), which do not allow a stable resonant response overlarge distances.To overcome these limitations, a quasi-nonlinear regime (QNL) has been pro-posed [51] to preserve some aspects of the linear regime such as a a constantplasma wavelength and maintain the resonance conditions.The key parameter has then identi�ed by the dimensionless charge quantityQ = Nb(kp)

3e=�e which is the beam charge normalized to the electron charge

located in a cubic plasma length k�1p = c=!p. As usual, !p =

p4�e�e=me is

the electron plasma frequency. Nb is the number of protons in the bunch.For a bi-Gaussian beam charge distribution with (�z; �r) sizes, the normalizedcharge reads:

Q = (2�)3=2�b�ek3p�z�

2r

18

where kp = 2�=�p = !p=c. A linear regime is characterized by Q� 1. A QNLregime has the peculiar condition that �b=�e > 1 for bubble formation, but ithas Q < 1 to preserve some linear features of the wake�eld.The linear response requires kp�z '

p2, so that the QNL regime thus requires

very thin beams �r � �z. In this parameter range, linear relations for thewake�eld longitudinal �eld Ez can be extended, up to a limiting value of the�b=�e ratio, where wave breaking occurs

Ez ' mc!pe

;� 100p�e=e[cm�3]Vm�1 :

Numerical simulations using 3D OSIRIS PIC code [52] or 2D cylindrical [53, 54]have already documented some aspects of QNL regimes, relevant for experimen-tal applications. For a proton bunch in the linear regime the maximum availablelongitudinal �eld [6] is

Ez = 240

�Nb

4� 1011

��0:6

�z(mm)

�2

A speci�c implementation of the standard PIC schemes has been achieved inthe code ALaDyn [55] to include the beam-driven PWFA conditions, in order toensure accuracy and e�ciency. As a �rst application, some examples of QNLregime formation on short time scale, in con�gurations with one-driver or three-driver beams, have been provides. A similar approach holds when the driver isa proton beam; the condition for linearity Q� 1 is the same.Labeling with (xe;pe) and (xb;pb) the positions and momenta for electrons(macroparticles) of the background plasma and of the bunch, we can say thatboth species have the same equations of motion. Charge densities and currentsare split according to � = �b + �e, J = Jb + Je. Each species satis�es thecorresponding set of Maxwell's equations with its proper charge density andcurrent. For initial conditions, represented by a beam freely moving in space atspeed v0 along the z axis with a density �0(x) in the comoving frame, the initialcurrent distribution is J0b = v0�0b in the rest frame. The potentials �elds inrest frame satisfy the equations

�1

20

@2

@z2+r2

?

��b = �!2

p�0b; A(b)z = �0�b Ab? = 0 :

The corresponding electric and magnetic �elds immediately follow and it is clearthat the components of the �elds orthogonal to the propagation direction aredominant and provide the leading contribution to the Lorentz force. The sourcesand the �elds evolve by pure advection on a short time-scale and on a long timescale they are modi�ed by the plasma wake�eld.On the computational side, this scale separation entails sti�ness and errors onnumerical integration. The transverse �elds decay as 1=r and involve problemswith the boundary condition. This problem was solved by separating �elds andsources for the beam into the advected part and a remainder [54].

19

Figure 3: Two-dimensional plot of the Ez wake�eld on the (z; x); y = 0 plane. The positionfor optimal acceleration of an electron witness bunch is at z ' 6:8 mm.

�b = �0b(z � v0t; x; y) + ��b Jb = J0b(z � v0t; x; y) + �Jb

Eb = E0b(z � v0t; x; y) + �Eb Bb = B0b(z � v0t; x; y) + �Bb

the residual parts can be evolved separately or can be reabsorbed in the plasma�eld components. Symbols with the zero subscript denote initial values.The application of this strategy to the case of beam proton acting as a driverdoes not imply any signi�cant change. In the speci�c case of AWAKE, wherethe plasma wavelength is 1.35 mm and the longitudinal bunch length is 0:1 mm,choosing a grid spacing �z = 10�m should allow to simulate the beam propa-gation for about 1 m using a moderate size parallel architecture.Here we shortly report on a preliminary numerical investigation of a proton-driven wake�eld, following the computational design introduced previously.We have considered a Gaussian proton beam with size �z = 0:1 mm, �x = �y =2�z; and density parameter �b=�e = 0:3, where (�e=e) = 1:0� 1015cm�3 is theelectron density of the background plasma. The driven wake�eld is in the linearregime, allowing to check the resulting maximum values of the longitudinal Ezaccelerating �eld against the available analytical formulas [56].

8. Conclusions

The space charge e�ects in high intensity accelerators and plasma accelera-tion, driven by a charged beam or a laser pulse, have a common theoreticalframework: the Vlasov equation coupled self-consistently with the Poisson or

20

Figure 4: Lineout of the Ez(z) �eld with superposed density of the proton driver, to documentthe forward shifted wake structure compared to the case of electron driven wake�eld.

Figure 5: The Ex wake �eld, documenting the focusing-defocusing action of the transverseelectric �eld.

Maxwell's equation. We have presented a brief review of this theory startingfrom the variational principles, which are the natural framework to discuss theconservation laws and develop approximation schemes, like the particle-in-cell(PIC) method. For the space charge, the analytic KV theory applicable to a2D beam and the general adiabatic theory applicable to a at 1D beam havebeen summarized. For these problems, the core PIC approximations are ane�cient Poisson solver. In the electromagnetic case, the Maxwell Vlasov equa-tions have been presented in the same variational framework. We have sketched

21

how to obtain the conservation laws and how to formulate the PIC approxi-mation schemes. The result is based on the introduction of a �nite basis forthe potentials and on sampling the distribution function with shaped numeri-cal particles. The exact formulation of the balance equations and conservationlaws for PIC schemes is still an open problem but the variational frameworkis appropriate to provide the right answers. Finally, the problem of electronaccelerators in the wake �eld of plasma waves excited by a charged driver hasbeen considered, illustrating it with a simulation, based on the code ALaDyn,in conditions similar to the AWAKE experiment, where the driver is a protonbeam. The possibility of achieving �elds of some GV/m in the linear regimeis con�rmed starting from short bunches (below 1 mm). Their generation viamodulation instabilities, even though suggested by simulations, is waiting to beexperimentally con�rmed. In spite of the challenges which include the stabilityon several meters of the driver and the control of longitudinal and transversegrowth, the AWAKE project opens a new route for electron acceleration up toTeV threshold. On the computational side, the search for more e�cient and re-liable PIC approximation is continuing. It was shown that high order schemesmay have an advantage in the description of a laser pulse since they improvethe dispersion relation and reduce the noise. A better control on the numericalheating and on the conservation laws for charge, energy and momentum mightbe achieved, within the theoretical framework we have presented. Improvementto the PIC algorithms and to the underlying hardware might allow to gain oneorder of magnitude in performances within the next few years as required toplan and interpret new challenging experiments like AWAKE.

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