PHYSICAL REVIEW SPECIAL TOPICS - ACCELERATORS AND BEAMS, VOLUME 5, 124402 (2002)

Numerical study of the photoelectron cloud in KEKB Low Energy Ringwith a three-dimensional particle in cell method

L. F. Wang,* H. Fukuma, K. Ohmi, S. Kurokawa, and K. OideHigh Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki 305-0801, Japan

F. ZimmermannEuropean Organization for Nuclear Research, CERN, Geneva, Switzerland

(Received 4 July 2002; published 6 December 2002)

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A three-dimensional particle in cell simulation code has been developed to study the photoelectroncloud instabilities in KEKB LER. In this report, the program is described in detail. In particular,typical simulation results are presented for the photoelectron motion in various kinds of magnetic fields.The simulation shows that a solenoid is very effective in confining the photoelectrons to the vicinity ofthe vacuum chamber wall and in creating a region free of photoelectrons at the vacuum pipe center. Themore uniform the solenoid field is, the more effectively does it suppress the electron-cloud buildup.Multipacting can occur both in a drift region and in a dipole magnet, and the heat load deposited on thechamber wall due to the lost electrons is important in these two cases. Electron trapping by the beamfield as well as by various magnetic fields is an important phenomenon, especially inside quadrupoleand sextupole magnets. Our numerical results qualitatively agree with the experimental studies.

DOI: 10.1103/PhysRevSTAB.5.124402 PACS numbers: 29.27.Bd, 02.60.Cb, 52.35.Qz, 29.20.Dh

distribution, heat load, multipacting, and so on. A three-dimensional irregular mesh is applied in CLOUDLAND,

The direct simulation of particle-particle interactionsis most easily implemented and it has a high accuracy.

I. INTRODUCTION

In April 1999, a blowup of the vertical size of thepositron beam was observed with the threshold beamcurrent 450 mA in the KEKB Low Energy Ring (LER)[1]. It proved to be one of the most serious luminositylimitations for KEKB. As a countermeasure, thousandsof so-called C-yoke permanent magnets were installedin the KEKB LER in May 1999 and March 2000. Theywere attached to the outside of the vacuum chambers toconfine the electrons to the vicinity of the chamber walls.A C-yoke consists of two permanent magnets and aC-shaped iron yoke. The magnets are attached every10 cm along the drift space between ring magnets. Forshort bunch trains and low beam currents, the C-yokemagnets were effective in confining the electron cloudand reducing the beam-size blowup. Therefore, inSeptember 2000 the C-yoke magnets were replaced bysolenoid magnets, in order to better remove the photo-electrons near the beam. At the beginning of January2002, the total length of the solenoids covering otherwisefield-free areas around the ring circumference reached to2180 m. This corresponds to about 95% of the total driftregion. The experimental studies demonstrated that thesolenoid magnets are very efficient in reducing the verti-cal beam size as well as the tune shift due to the electroncloud [2]. No blowup was observed up to the so far highestbeam current of 1300 mA. Thus the luminosity hasgreatly benefited from the solenoid installation. Athree-dimensional particle in cell (PIC) programCLOUDLAND has been developed to study the effects ofvarious magnetic fields on the photoelectron formation,

1098-4402=02=5(12)=124402(13)$20.00

which enables the program to solve the general three-dimensional problem. Image currents, arbitrary magneticfields, space charge of electron cloud, and secondaryemission are all taken into account in the program. Inthe following, the program and examples of simulationresults are described in detail. The results qualitativelyagree with the experimental observations.

II. METHOD

The positron bunch is longitudinally divided into anumber of slices according to a Gaussian distribution.Such slices interact with photoelectrons transverselyand are propagated according to the transfer matrix ofthe linear optics. Photoelectrons are emitted when posi-tron slices pass through a beam pipe with length L, whichis usually chosen as 1 or 2 m. In the simulation photo-electron yield of 0.1 is assumed and 30% of the photo-electrons are taken to be produced by reflected photons,i.e., randomly distributed around the chamber. The centerof the photoelectron energy distribution is assumed to be5 eV with an rms (root mean square) energy spread of 5 eV.In our simulation, the photoelectrons are represented bymacroparticles, which move in three-dimensional spaceunder the force:

Fe � Fspace � Fp � FB; (1)

where Fspace is the space charge force of the photoelec-tron, Fp is the force due to the electric field of the positronbeam, and FB is the force exerted by the external mag-netic field.

2002 The American Physical Society 124402-1

PRST-AB 5 NUMERICAL STUDY OF THE PHOTOELECTRON CLOUD . . . 124402 (2002)

However, the efficiency of such an approach is low due tothe very large number of particles and hence very longCPU time. For this reason, a mesh method seems to beapplied by most particle simulation programs. The regu-lar mesh as applied in the study of beam-beam simula-tions cannot satisfy here because of the complex shape ofthe vacuum chamber, e.g., the antechamber in PEP-II andin the future KEKB upgrade. In our code, an irregularmesh is used, so that the solver can be applied to a generalgeometry. The vacuum chamber of KEKB LER is of a

FIG. 1. Mesh example of the KEKB-LER vacuum chamber,used by the space charge solver for the photoelectron cloud.

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round shape with a radius of 50 mm. Figure 1 shows anexample mesh for the KEKB-LER vacuum chamber.

Figure 2 shows one 20-node brick element employed bythe solver. The chargeQ0 of a macroparticle is assigned toeach node i of the brick element inside which the photo-electron is located according to the shape function Ni

Qi � NiQ0; (2)

where

Ni �1

8�1� 0��1� �0��1� &0��0 � �0 � &0 � 2�; i � 1; 3; 5; 7; 13; 15; 17; 19;

Ni �1

4�1� 2��1� �0��1� &0�; i � 2; 6; 14; 18; Ni �

1

4�1� 0��1� �2��1� &0�; i � 4; 8; 16; 20;

Ni �1

4�1� 0��1� �0��1� &2�; i � 9; 10; 11; 12; (3)

where , �, and are the natural coordinates; 0 � i,�0 � �i�, &0 � &i&, and �i; �i; &i� are the coordinates ofnode i.

For the isoparametric element, the charge assignmentscheme in Eq. (2) has all the characteristics of a chargeassignment function, i.e., it fulfills the relationsX

i

Ni � 1; (4)

Xi

Niri � r; (5)

where r is the position of the photoelectron, and ri is theposition at node i. The property of the shape function inEq. (4) guarantees the charge conservation.

Figure 3 shows the distribution of the macroparticleand charge at the mesh node in one transverse section.The number of elements in this transverse section is 276,which is a small number. It already shows a good repre-sentation of the real electron-cloud distribution.

This method could be called a cloud-in-a-Cell (CIC)scheme [3]. However, it is different from the so-calledgeneral CIC schemes which are commonly applied formultiparticle simulations in two aspects. First, the gen-eral CIC method applies a regular mesh. By contrast, ourscheme uses an irregular mesh, which allows the treat-ment of complex boundary problems, such as the very flat

beam case and the antechamber. Secondly, our schemerefers to a finite element method. Many types of elementscan be considered in the finite element calculation.Higher-order elements, which means high order nonlin-ear terms of the potential, improve the accuracy of thecalculation and are far superior to the nearest-grid-pointassignment applied by the general CIC. Therefore, ourscheme offers two clear advantages: the possibility totreat arbitrary boundaries and a high accuracy. In addi-tion, an adaptive mesh can be applied if the electrons areconcentrated in a small region, in analogy to similartreatments for long-range beam-beam simulations.

The electron cloud (both the density and the distribu-tion) changes with time. However, since the electronmotion is slow, in a good approximation, we can assumea quasistatic condition. At each moment, the scalar po-tential satisfies

�� � ��="0: (6)

Equation (6) can be solved by the finite element method.

FIG. 2. 20-node brick element.

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FIG. 3. Charge assignment of the PIC method. (a) Transverse distribution of the macroparticles in a solenoid field. (b) Transversedistribution of the charge on the mesh obtained by the charge assignment.

PRST-AB 5 L. F. WANG et al. 124402 (2002)

The finite element equation is

A� � B; (7)

where the stiffness matrix A depends only on the meshand B represents the source term. The matrix A is ex-tremely sparse and there are well-known methods for

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handling such linear problems, e.g., conjugate gradientmethod, and profile or frontal techniques. If the vacuumchamber is of a round shape, as in the KEKB LER, wecan also alternatively use a Green function to compute thepotential. The potential � at R for a particle distributionf�R0� is obtained using the Green function G�R;R0�as [4]

��R� �Z L

0dz0

Z 2�

0d�0

Z a

0r0dr0f�R0�G�R;R0�; (8)

G�R;R0� �eLln�2 � r2r02=�2 � 2rr0 cos��� �0�

r2 � r02 � 2rr0 cos��� �0�

�4eL

X1n�1

cosnk�z� z0��K0�nk

�������������������������������������������������������r2 � r02 � 2rr0 cos��� �0�

q

�X1m�0

�2� �m0�Km�nk��Im�nk��

Im�nkr�Im�nkr0� cosm��� �0�

�; (9)

where L is the period length of the vacuum chamber, � is

the pipe radius, R0 is the source position, and R is thepotential position, k � 2�=L. We here use cylindricalcoordinates with the z axis oriented along the axis ofthe pipe, R � �r; �; z�, R0 � �r0; �0; z0�.

After finding the potential, the force on each particle isinterpolated using the same shape function in order toconserve the momentum. In the general PIC method, thefield at each node is first calculated using the differencescheme and the field at each particle is interpolated basedon the fields at all nodes. Unlike the general PIC method,we directly calculate the field at each particle using thepotential at the mesh nodes instead of the mesh-defined

field, namely, we compute

E � �Xi

rNi ��i; (10)

where

r �@@x

i�@@y

j�@@z

k: (11)

One example of the potential and field of the electroncloud in one transverse section is shown in Fig. 4. Thefield is automatically perpendicular to the potential

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−50 −40 −30 −20 0 10 20 30 40 50−50

−40

−30

−20

−10

0

10

20

30

40

50

X [mm]

y [m

m]

−10

FIG. 4. (Color) Potential and field interpolation of the PICmethod. The color shows the potential; the arrows denote thefield direction and strength.

PRST-AB 5 NUMERICAL STUDY OF THE PHOTOELECTRON CLOUD . . . 124402 (2002)

contour according to Eq. (10). Note that the distribution ofthe photoelectrons is three dimensional.

The positron bunch is represented by a rigid Gaussiandistribution. The kick on the photoelectrons is given bythe Bassetti-Erskine formula [5]

�#y � i�#x � Nrec

������������������������������2�

%x;y�%x � %y�

sf�x; y�;

with

f�x; y� �w�

x� iy������������������������2�%2

x � %2y�

q � exp

��

x2

2%2x�

y2

2%2y

� w�x%y=%x � iy%x=%y������������������������

2�%2x � %2

y�q

; (12)

TABLE I. KEKB-LER param

Variable

Ring circumferencerf bucket lengthBunch spacingBunch populationAverage vertical betatron functionAverage horizontal betatron functionHorizontal emittanceVertical emittancerms bunch lengthChamber diameterEnergy of the maximum secondary emission yieMaximum secondary emission yield

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where %x and %y are the positron bunch rms transversesizes, re is the classical electron radius, c the speed oflight, N the particle number in the slice, and w�x� iy�the complex error function.

The image charge effect is also included in the pro-gram. The shape of the vacuum chamber in KEKB LER isround. Therefore, the image current is easily found. Incase of an arbitrary chamber shape, a PIC calculation canbe applied to the positron bunch in the same way as to thephotoelectron cloud.

The magnetic field along the beam chamber can be anarbitrary three-dimensional field. The typical magneticfields, in which we are interested for the KEKB LER, aresolenoid, dipole, quadrupole, and sextupole.

According to Furman [6], Seiler [7], and Kirby [8], theyield of the true secondary emission can be written as

%Yts�E; �� �%max1:11x�0:35�1� e�2:3x1:35�

� exp�0:5�1� cos��; (13)

where, x � Ep�1� 0:7�1� cos��="max [6], Ep is theprimary electron energy, "max is the energy of the maxi-mum secondary emission yield, �max is the maximumsecondary emission yield for perpendicular incidence,and � is the incidence angle with respect to the surfacenormal.

The initial energy distribution of the true secondaryemission is taken to be a half-Gaussian centered at 0 withan rms spread of 5 eV.

�E�Est� � �Est exp

��

Est

2�2Est

Est>0

: (14)

The emission angles of the true secondary photoelec-trons are distributed according to dN=d� / cos� sin� ordN=d� / cos�, where � denotes the angle with respect tothe surface normal, and � is the solid angle.

The parameters used in the simulation are summarizedin Table I.

eters used in the simulation.

Symbol Value

C 3016.26 msrf 0.589 msb 4 rf bucketsN 3:3� 1010

.y 10 m

.x 10 m"x 1:8� 10�8 m"y 3:6� 10�10 m%l 4 mm2R 100 mm

ld "max 250 eV�max 1.5

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PRST-AB 5 L. F. WANG et al. 124402 (2002)

III. NUMERICAL EXAMPLES

In this section, the buildup, distribution, heat load, andtrapping of electrons in a few typical magnetic fields arediscussed.

A. Effect of C-yoke magnet and solenoid field on theconfinement of the photoelectrons

Permanent C-yoke magnets were attached to vacuumducts to sweep out the electrons in May 1999 and March2000. The magnet field in a C-yoke magnet with quadru-pole configuration is modeled as

Bx � �a� b coskz�y; (15)

By � �a� b coskz�x; (16)

Bz � �bk sin�kz�xy: (17)

Where a � 0:3 T=m, b � 0:2 T=m, 1 � 2�=k � 0:1 m,and a � 0, b � 0:5 T=m, 1 � 0:2 m for the equal polar-ity and alternating polarity, respectively. The photoelec-tron cloud density near the beam is nonzero for all C-yokemagnet configuration as shown in Fig. 5 for equal-polarity quadrupole configuration. The transverse mag-netic field is stronger than the longitudinal field in somelocations along the chamber. The relatively strongertransverse field may guide the electrons into the chambercenter, which is undesirable. This conclusion holds truefor adjacent C-yokes of both equal and opposite polarity.

Adjacent solenoids can be arranged with constant oralternating current direction in the coil. We refer to thesetwo arrangements as the equal or alternating polarity

FIG. 5. Electron-cloud distribution in the C-yoke quadrupolewith equal polarity configuration.

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configurations. The magnetic field in the equal-polarityconfiguration can approximately be expressed as

Bz�x; y; z� � Bz0 � B0 sinkz; (18)

Bx�x; y; z� � �0:5B0kx coskz; (19)

By�x; y; z� � �0:5B0ky coskz: (20)

For the opposite polarity configuration, the longitudinalfiled is

Bz�x; y; z� � B0 sinkz; (21)

and the transverse field components are the same as forthe equal-polarity case. We assume Bz0 � 30 G, B0 �20 G, and 1 � 2�=k � 1 m, so as to approximate thefield of the real machine for the equal-polarity configu-ration. There is no electron cloud at the chamber center forthe equal-polarity solenoid as shown in Fig. 3(a). Thereason is that the longitudinal magnetic field confinesthe photoelectron motion to the vicinity of the vacuumchamber wall. The more uniform the solenoid field, themore effective is the confinement. In the alternating po-larity configuration, the transverse field between twoadjacent solenoids is stronger than the local longitudinalfield and may allow some photoelectrons to penetrate intothe center of the chamber. Therefore, the equal-polarityconfiguration is better than the opposite polarity one.This is consistent with the experiment study [9].Compared with the C-yoke magnets, solenoids of equalpolarity are much preferred, in view of the vanishingelectron-cloud density at the chamber center. The elec-trons at the center are suspected to cause the beam-sizeblowup according to the Ohmi and Zimmermannmodel [10].

Figure 6 displays the photoelectron distribution at thechamber wall, i.e., the distribution of photoelectronswhich hit the wall, for the case of a solenoid field. Ifphotoelectrons impinge on the chamber wall, they areeither lost or produce secondary electrons. The solenoidfield is nonuniform in the longitudinal direction, whichrenders the lost cloud distribution dependent on the lon-gitudinal position. The radius of the gyration motion issmall in a strong field region with a z coordinate from 0 to0.5 m in Eq. (18) and Fig. 6, and hence the electron hitsthe wall near its emission point. On the other hand, theelectron at the weaker field region, z coordinate equal to�0:25 m, drifts with a big radius and hits the chamberwall far from its emission point. This is clearly shown inFig. 6. The azimuthal distribution of the lost photoelec-trons depends on the current direction in the solenoidcoil, because the deflection angle of the photoelectronsdepends on the sign of the magnetic field. As a conse-quence, the electron current measured by a photoelectronmonitor will depend on both the longitudinal position ofthe monitor and the current direction in the solenoid coil.

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FIG. 7. Electron-cloud distribution in drift region.

FIG. 6. Lost electron distribution around the chamber wall inthe solenoid field with equal polarity configuration.

PRST-AB 5 NUMERICAL STUDY OF THE PHOTOELECTRON CLOUD . . . 124402 (2002)

In a solenoid field, the photoelectrons cannot gainmuch energy from the positron bunch, because they areconfined to the vicinity of the wall, far from the chambercenter, by the solenoid magnetic field. Therefore, there islittle or no multipacting in the solenoid case. The heatload on the chamber wall due to the photoelectron bom-bardment is also small for the same reason. It can beconcluded that solenoids work well, providing both zerophotoelectron density at the chamber center and a lowerheat load on the chamber wall due to the absence ofmultipacting.

B. Multipacting in drift region and dipole magnet

In a drift region, the photoelectrons are attracted by thespace charge field of the positron bunch and, therefore, alarge photoelectron density builds up at the chambercenter. The photoelectron density at the chamber centeris 1013 m�3, which is 5 times larger than the averagevolume density due to the stronger attractive force ofthe positron beam there. The transverse distribution ofthe photoelectron cloud in a drift region is shown in Fig. 7.The photoelectrons near the chamber center cloud receive

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more energy from the passing positron bunches than thosenear the wall. Such higher energy photoelectrons maycause significant multipacting when they hit the chamberwall. Heat load is an additional serious problem in a driftregion because the photoelectrons are strongly acceler-ated by the positron bunches and the rate of photoelectronloss on the wall is large. Therefore, the drift region isconsidered to be the most dangerous case with respect toboth beam dynamics and heat load.

Figure 8 shows the photoelectron cloud distributioninside a dipole magnet. Two multipacting strips areclearly visible in the figure. There is no experimentalstudy on the electron cloud inside a dipole magnet inKEKB LER. The CERN SPS experiments [11] exhibitedtwo similar multipacting strips in a dipole magnet. Thereis little multipacting in the center region, because thephotoelectrons with horizontal coordinate close to zeroreceive further energy, exceeding the energy value atwhich the secondary emission yield is bigger than 1.The typical energy of the photoelectrons decreases fromthe horizontal center towards both sides. In other words,the energy of the photoelectrons decreases with the hor-izontal coordinate jxj as shown in Fig. 9. It is well knownthat the true secondary emission yield is smaller than 1for photoelectrons with both very high and low energy.As a result, multipacting occurs in two regions on eitherside of the chamber center. The position of the multi-pacting region depends on the energy of the photoelec-trons, which is determined by the interaction ofphotoelectrons and positron bunches. Therefore, the fill-ing pattern of the beam, including bunch current andbunch spacing, can alter the regions with strong multi-pacting. In general, when the bunch current increases, themultipacting region moves towards larger jxj on either

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FIG. 8. Electron-cloud distribution in the dipole magnet.

PRST-AB 5 L. F. WANG et al. 124402 (2002)

side of the center, and the width of the multipactingregion also increases at the same time. The exact resultsdepend on the interaction between the photoelectroncloud and positron bunches. The mechanism of the multi-pacting in a dipole magnet is clearly evident in Fig. 9.From the energy of an electron hitting the chamber wall,the secondary emission yield is calculated. Values larger

−50 −40 −30 −20 −10 0 10 20 30 40 50

10 −2

100

102

104

106

X [mm]

Ene

rgy(

eV),

Yie

ld a

nd C

harg

e

Energy [eV]Secondary emission yieldcharge distribution

FIG. 9. (Color) Mechanism of multipacting in the dipole mag-net. Blue dots show the energy of photoelectrons which hit thechamber wall. Red dots represent the secondary emission yieldof the photoelectrons which hit the wall. The black solid line isthe charge distribution of the lost photoelectrons, which isnormalized by the number 1012 in order to compare the chargedistribution with the secondary emission yield clearly.

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than 1 imply that multipacting can happen. The computedyields are consistent with the simulated charge distribu-tion of the lost electrons on the wall, as illustratedin Fig. 9.

C. Photoelectron trapping in beam electric field andmagnetic fields

Photoelectrons can be trapped by the beam field andby magnetic fields, or by a combination of fields.Experimental study in CESR found that photoelectronscan be trapped in the combined dipole magnetic field andquadrupole electrostatic leakage field from the distrib-uted ion pumps [12]. As predicted by Chao [13], the beamfield is the most effective trapping field due to its periodicfocusing force. The mirror magnetic field, such as aquadrupole or sextupole field, is the most effective mag-netic trapping field. Some electrons can also be trapped ina periodic solenoid field because it is also a type of mirrorfield. But the number of the trapped electrons is muchsmaller than that of the quadrupole and sextupole fields’cases. Electrons can also be trapped in a dipole magnet.However, in this case, the trapping in the direction of themagnetic field (typically vertical) is accomplished by thespace charge field of the positron beam and the electroncloud instead of by the magnetic field. The mechanism ofelectron trapping for different field configurations isstudied in this section.

1. Trapping in the beam field

In a drift region, the photoelectrons near the bunch canreceive linear beam kick from the bunch and can gettemporarily trapped in the beam potential, as shown inFig. 10, and will thus oscillate around the bunch [14]. Thephotoelectrons near the beam perform about one oscil-lation during the passage of a positron bunch in theKEKB-LER case. However, in the case of proton bunchesthe electrons can oscillate for many periods due to thelonger bunch length.

Photoelectrons at large amplitudes do not move muchduring the bunch passage due to their long oscillationperiod. They simply receive a nonlinear kick from thebeam. Most photoelectrons with large amplitude can stillbe trapped as shown in Fig. 11. Photoelectrons can betrapped for long or short times, which depend on its initialcondition. Photoelectrons are emitted from the chamberwall with a small initial energy, and they oscillate under

Proton/Positron bunch

FIG. 10. (Color) The electron motion during a bunch passage.

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0 500 1000 1500 2000 25000

1

2

3

4

5

6

7

8x 10

11

Time [ns]

Pho

toel

ectr

on V

olum

e D

ensi

ty

[m−3

] DipoleQuadrupolesextupole

FIG. 12. Photoelectron average volume density in differentmagnetic fields as a function of time for a train with200 bunches spaced by 7.86 ns and followed by a long gap.

50 40 30 20 10 0 10 20 30 40 5050

40

30

20

10

0

10

20

30

40

50

X [mm]

Y [m

m]

− − − − − 0 10 20 30 40 50−

−

−

−

−

0

10

20

30

40

50

−50 −40 −30 −20 −10 0 10 20 30 40 50−50

−40

−30

−20

−10

0

10

20

30

40

50

X [mm]

Y [m

m]

(a)

(b)

FIG. 11. Orbits of trapped electrons without magnetic field.(a) Long time trapped electron; (b) short time trapped electron.

PRST-AB 5 NUMERICAL STUDY OF THE PHOTOELECTRON CLOUD . . . 124402 (2002)

the influence of the focusing beam force. The electronsgain energy during the process, approaching the chambercenter. The energy and the radial position of the trappedelectron are modulated by the beam kick.

As a result, photoelectron trapping in the beam field isone important phenomenon. This mechanism leads to theuniform distribution of the photoelectron cloud in thebeam chamber although most photoelectrons are emittedin a horizontal direction (primary photoelectron), andthen such a round distribution of the cloud causes thesame coupled bunch instability in horizontal and verticaldirections. This is consistent with the experimental study[15]. The cloud distribution shown in Fig. 7 is roughly

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uniform in the azimuth angle due to the trapping effect.There are more photoelectrons along the emission direc-tion of the primary photoelectron, because these elec-trons mainly oscillate in the horizontal plane instead of ina circle around the chamber center, due to their smallinitial vertical velocity and initial coordinate. The photo-electron has a short decay time during the bunch traingap because the trapping mechanism depends on thepositron or proton bunch. Therefore, a bunch train gap isvery effective to terminate the trap.

2. Trapping in quadrupole and sextupole magnets

It is interesting that more than 90% of the photoelec-trons can remain seriously trapped in a quadrupole orsextupole magnetic field during the bunch train gap, asshown in Fig. 12 [16]. The photoelectron density is almostconstant during the train gap in these two fields, whichimplies a long electron lifetime. On the other hand, thedensity decays quickly in a dipole magnet. Figure 13shows a typical trapped-electron orbit in a normal quad-rupole field during the train gap. The drift time is about960 ns. The trapped electron spirals in an ever-tighterorbit along the magnetic field line when the field becomesstronger, converting more and more translational energyinto energy of rotation until its velocity along the fieldline vanishes. Then the electron turns around, still spi-raling in the same sense, and moves back along the samefield line. A similar phenomenon is observed in the sextu-pole magnets. The electron-trapping phenomena are quitesimilar to the plasma trapping in a mirror magnetic field.

We consider the case of no electric field, which isalmost realized for the electron cloud during the bunchtrain gap, if the space charge potential of the electroncloud is negligible compared with the magnetic potential

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−20

0

20

−42−40

−38−36

−34

−250

−249

−248

−247

−246

−245

X (mm) Y (mm)

Z (

mm

)

−50 −40 −30 −20 −10 0 10 20 30 40 50−50

−40

−30

−20

−10

0

10

20

30

40

50

X (mm)

Y (

mm

)

(a)

(b)

FIG. 13. (Color) Photoelectron trapping in a quadrupole mag-netic field during the train gap. (a) 3D orbit; (b) 2D orbit (redline); and quadrupole field (black arrows).

PRST-AB 5 L. F. WANG et al. 124402 (2002)

in a normal magnet. Since the direction of the magneticforce acting on the electron is perpendicular to the elec-tron velocity, the electron kinetic energy is conserved,

W �m#2

2� const: (22)

The motion of the electron in the magnetic field can beregarded as the superposition of the gyration motionaround the guiding center and the motion of the guidingcenter. The gyration motion of the electron is a rapidrotation around the magnetic field line. The motion ofthe guiding center is the average motion over the gyration.

Consider the case in which the magnetic field slowlyvaries in space. The variation is assumed to be sufficientlyslow so that the magnetic field at the electron positionhardly changes during the cyclotron motion. This is truefor our case where the magnetic field is strong except for

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the central region of the chamber and the electron energyis low, which means a small Larmor radius and a shortperiod. While the period of a spiraling electron changesas it moves into regions where the magnetic field isweaker or stronger, the product T � E, the period T timesthe energy E of the gyration motion, is almost a constant.It is not an exact constant, but if the rate of change is slowenough, e.g., if the field changes rather slowly, it comesvery close. A certain quality, an ‘‘adiabatic invariant,’’ isalmost kept at a constant value. In a more general way, theaction of a system with canonical variables q and p,defined by

J �Ipdq; (23)

is a constant under a slow change in an external parame-ter. Here

Hrepresents an integral over one period of the

motion. Therefore, for such a quasiperiodic motion, thereexist two adiabatic invariants given by [17]

J? �Im#?�sd’ �

4�me

7m; (24)

Jk �Im#kdl; (25)

where

7m �m#2

?

2B(26)

is the magnetic moment, v? is the gyration velocity, �s �m#?=jejB is the Larmor radius, and vk is the parallel orlongitudinal velocity which is parallel to the magneticfield. J? and Jk are called the transverse and paralleladiabatic invariants, respectively.

As the guiding center of the electron moves along thefield line, the magnetic field strength at the position of theelectron changes. Because the magnetic moment and ki-netic energy of the electron are conserved, the kineticenergy of the parallel motion varies according to therelation

1

2m#2

k�7mB � const: (27)

Equation (27) implies that the guiding center motionalong the field line behaves like a particle motion ina magnetic potential energy 7mB. In quadrupole andsextupole magnets, the magnetic field is a mirror field,in which the magnetic field is weaker at the center andstronger at both ends of the mirror field line. When theguiding center of the electron moves along the field linefrom a weaker field region to a stronger field region, theparallel velocity decreases and the gyration velocity in-creases, and the electron is heated. Therefore, the electronspirals in an ever-tighter orbit because the period ofgyration motion and parallel velocity become smaller

124402-9

FIG. 15. Photoelectron distribution during the bunch trainpassage before a positron bunch arrives.

Bmax

B0

Mirror Point

*

R B

n

FIG. 14. Motion of electron in a mirror magnetic field.

−600 −500 −400 −300 −200 −100 0 10010

15

20

25

30

35

40

45

50

55

Z (mm)

R (

mm

) an

d B

(G

auss

)

Radial coordinateSolenoid field strength

FIG. 16. Two-dimensional orbit of a trapped electron in asinusoidal solenoid field and the solenoid field strength B alongthe orbit. Here, R and Z are the radial and longitudinalcoordinates, respectively.

PRST-AB 5 NUMERICAL STUDY OF THE PHOTOELECTRON CLOUD . . . 124402 (2002)

and smaller. When the electron comes to the point wherethe parallel velocity vanishes, the electron direction ofmotion is reversed. The parallel velocity of the reflectedelectron is increased when it moves along the field lineand gets the maximum value at the weakest field point(mirror point). Then it continues a similar motion alongthe other side of the mirror point. Such kind of trap iscalled a magnetic-mirror trap. The motion of an electronin a mirror field is depicted in Fig. 14.

Figure 15 shows the electron’s cloud distribution justbefore a bunch interacts with the cloud during the bunchtrain passage. Clearly the electron density is higher nearthe mirror points. The central region is not an adiabaticregion due to the weak field there. Electrons moving alongthese diagonal field lines can receive a larger amount ofparallel energy and may be lost before the next positronbunch arrives.

3. Trapping in the periodic solenoid field

Solenoid magnets have been installed in the LER ringin order to reduce the photoelectron density near the

124402-10

beam. The periodic solenoid field can also trap a fewelectrons, but typically less than 1% of the total. Thetrapping mechanism is still the mirror magnetic fieldtrap. Figure 16 shows the trajectory of a trapped electronin a sinusoidal solenoid field as described in Eqs. (18)–(20) with Bz0 � 30 G, B0 � 20 G, and 1 � 2�=k � 1 m.The solenoid field strength along the trajectory is alsoshown in the same figure in order to compare the trajec-tory and field strength. The electron is trapped in thewake field region with the mirror point located at Z ��250 mm, which is the midpoint of the two adjacentsolenoids. Therefore, the trapped electrons stay aroundthe center point of the two adjacent solenoids.

Note that the mirror field trap strongly depends on theelectron velocity vk0 and v?0. According to Eqs. (22),(24), and (26), the trap occurs if

#2?0

#2?0 � #2

k0

>B0

Bmax; (28)

where B0 is the field at one position with velocity vk0 andv?0, and Bmax is the maximum field along the field line.Equation (28) shows that a photoelectron could be trappedif its kinetic energy of gyration motion increases.Different from the quadrupole and sextupole fields, thesolenoid field line applied here is along the longitudinaldirection, which is the beam direction. The photoelec-trons moving longitudinally along the field lines canreceive gyration motion energy from the transversebeam field. During the passage of a positron bunch, theposition of the photoelectrons does not change and thephotoelectrons simply receive energy because the posi-tron bunch length is much shorter than the period of thegyration motion. A similar statement applies to the quad-rupole and sextupole cases.

124402-10

0 0.2 0.4 0.6 0.8 110

11

1012

1013

1014

Azimuth angle [2π ]

Cha

rge

Ang

le D

istr

ibut

ion

[ar

b.un

its]

Field FreeSolenoidDipoleQuadrupoleSextupole

0 0.2 0.4 0.6 0.8 110

12

1013

1014

1015

1016

Azimuth angle [2π]

Hea

ting

Azi

mut

h A

ngle

Dis

trib

utio

n [

arb.

units

]

Field FreeSolenoidDipoleQuadpoleSextupole

0 0.2 0.4 0.6 0.8 110

12

1013

1014

1015

1016

Field FreeSolenoidDipoleQuadrupoleSextupole

(a)

(b)

FIG. 17. (Color) Charge and heat-load azimuth angle distribu-tions of photoelectrons lost to the chamber wall in differ-ent magnetic fields. (a) Charge distribution; (b) heat-loaddistribution.

PRST-AB 5 L. F. WANG et al. 124402 (2002)

However, the electron motion has very different char-acters in solenoid and quadrupole/sextupole fields. First, asolenoid field is a longitudinal field and a quadrupole fieldis a transverse field. The electron in the solenoid fieldmoves in the beam direction, which means the electronmeets the positron bunch with different time spacing. Onthe other hand, the electron in the quadrupole field meetsthe positron bunch periodically due to its very slow move-ment along the beam direction. Second, the periods forone electron to move along the mirror field line, which isthe period of the parallel motion, as shown in Figs. 13 and16, are quite different. The period depends on the mag-netic field. In quadrupole and sextupole cases, for KEKBparameters it is close to 2 times that of the bunch spacing,which indicates the resonance of the interaction betweenthe electron and positron bunch. This period is muchlonger in the solenoid case. It is about 210 times of thebunch spacing in Fig. 16. Third, the solenoid field isweaker than the quadrupole field. As a result, the trajec-tory of an electron in the solenoid field can be easilychanged by the positron beam field. Therefore, in a sole-noid field the electrons are accelerated by the passingpositron bunches in a random fashion, and, even after along time, the gyration energy acquired can be too smallto fulfill the trapping condition, Eq. (28). All these char-acters make the trapping in a solenoid more difficult thanin a quadrupole or sextupole field.

4. Trapping in dipole magnet

The electrons effectively move only along the verticalfield lines due to the strong vertical magnetic field. Theuniform dipole magnetic field cannot trap any particle.However, the space charge field of the positron beam andelectron cloud oriented in a vertical direction can trap theparticle in the vertical plane. The mechanism is similar tothe beam field trapping in a drift region, but here thetrapping refers to a one-dimensional motion instead of toa two-dimensional motion. This motion resembles a circlein the drift region as shown in Fig. 11. Simulation showsthat the photoelectrons can be easily trapped in the twomultipacting strips, where the space charge field of theelectron cloud is strong and is in the vertical direction.

D. Heat load of photoelectron cloud

The lost photoelectrons, which hit the chamber wall,can cause a temperature increment of the vacuum cham-ber. The heat load depends on the quantity and energy ofthe photoelectrons, which hit the vacuum chamber wall.Figure 17 shows the lost photoelectron charge and heat-load azimuth angle distribution for different magneticfields.

In a drift region, the photoelectrons can receive moreenergy from the beam and the quantity is larger due to themultipacting. Therefore, the electron flux on the wall ishigher and so is the heat load. The heat-load azimuth

124402-11

angle distribution is roughly uniform due to the beamtrapping effect as explained in Sec. IIIC1.

Multipacting occurs along two strips in the dipolemagnet. Both the lost charge and the heat-load distribu-tion show peaks at these two multipacting regions. Theelectron flux on the wall is not peaked at the centralregion with jxj � 0, because multipacting does not hap-pen there. However, the heat load is peaked at the cham-ber center due to the higher energy of the electrons here.

There is a lower heat load in solenoid and quadrupoleand sextupole cases where multipacting could not occur.In the solenoid case, the energy of the photoelectron islower because the photoelectron is confined away fromthe beam. Electrons can be deeply trapped by quadrupoleand sextupole magnets fields, but the electron cannoteffectively receive energy from the beam. Therefore,

124402-11

PRST-AB 5 NUMERICAL STUDY OF THE PHOTOELECTRON CLOUD . . . 124402 (2002)

both the loss rate and the energy of the photoelectrons arelower in the solenoid, quadrupole, and sextupole magnets.

The interaction of electrons with the positron beamstrongly depends on the magnetic field. Therefore, theheat-load azimuth angle distribution is also magneticfield dependent as shown in Fig. 17(b). It is always peakedat locations where the magnetic field is oriented along theradial direction, as illustrated in the figure for the quadru-pole magnet.

E. Buildup of electron cloud

Figure 18 shows the average and center volume densityin different magnetic fields as a function of time for a

0 500 1000 1500 2000 2500 30000

0.5

1

1.5

2

2.5x 10

12

Time [ns]

Pho

toel

ectr

on V

olum

e D

ensi

ty [m

−3]

Field FreeSolenoidDipoleQuadrupolesextupole

0 500 1000 1500 2000 2500 3000

1010

1011

1012

1013

Time [ns]

Pho

toel

ectr

on C

entr

al V

olum

e D

ensi

ty [m

]

Field FreeSolenoidDipoleQuadrupolesextupole

−3

(a)

(b)

FIG. 18. (Color) Photoelectron average volume densities andvolume densities at pipe center in a different magnet field as afunction of time for a train with 200 bunches spaced by 7.86 nsand followed by a long bunch train gap. (a) Average volumedensities; (b) volume densities at pipe center.

124402-12

train with 200 bunches spaced by 7.86 ns and followed bya long bunch train gap. The decay time during the bunchtrain gap in the field-free case is the shortest, becausethere is no magnetic field to confine the photoelectrons.On the other hand, the decay time is very long in quadru-pole and sextupole magnets due to the deep trapping. Theprimary photoelectrons do not much contribute to thebuildup of the photoelectron cloud in normal dipole,quadrupole, and sextupole magnets. While in quadrupoleand sextupole magnets the buildup is dominated by mag-netic trapping; in the dipole magnet multipacting is thedominant process. As a result, the average cloud densityin these three fields is almost a linear function of timeduring the cloud buildup as illustrated in Fig. 18(a).

The photoelectron density near the beam is zero in thecase of the solenoid and small for the quadrupole andsextupole. The deeply trapped photoelectrons in the quad-rupole and sextupole magnets may contribute to thecoupled bunch instabilities. On the other hand, the photo-electron densities near the beam are much larger for thedrift and the dipole field, which indicates that the photo-electrons in the drifts and dipoles are mainly responsiblefor the single-bunch blowup of the positron beam.

IV. SUMMARY AND CONCLUSIONS

A complete three-dimensional PIC program has beendeveloped. It provides a powerful tool to study the elec-tron-cloud problem. The numerical results are qualita-tively consistent with the experimental observations andwith simple theoretical models, in many aspects.

A uniform solenoid is found to be the most effectivefield for confining the photoelectrons to the vicinity of thevacuum chamber wall. Indeed, even a realistic equal-polarity periodic solenoid is more efficient than any otherkind of magnet. The solenoid reduces the photoelectroncentral density to nearly zero, and it also significantlylowers the heat load to the chamber wall. Strong multi-pacting occurs both in drift regions and in the dipolemagnet, resulting in a significant central density of elec-trons. Therefore, these two regions are thought to be themain contributors to the single-bunch beam-size blowup.In quadrupole and sextupole fields, a large numberof electrons can be trapped by a magnetic-mirror mecha-nism and survive a long gap between bunch trains.These deeply trapped electrons can cause coupled bunchinstabilities.

ACKNOWLEDGMENTS

We thank Professor A. Chao, Professor K. Nakajima,and Professor E. Perevedentsev for helpful discussions.Wealso thank the KEKB commissioning group for all oftheir help, information, and discussions.

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PRST-AB 5 L. F. WANG et al. 124402 (2002)

*On leave from IHEP, Beijing.Email address: wanglf@post.kek.jp

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