Chapter 3
Beam Dynamics issues
3.1

Beam-beam interaction

3.1.1

Introduction

SuperKEKB is designed with the strategy of so-called the nano beam scheme. Electronpositron beams collide with a horizontal crossing angle of 2ϕc = 2 × 41.5 mrad. The
beam sizes at the interaction point (IP) are σx+ = 10 µm and σx− = 11 µm for positronand electron beams, respectively. The overlap area of the two beam is ∆s = σx /ϕc ≃0.25 mm. The area is 1/20-1/25 of the bunch length, since the bunch lengths are
σz+ = 6 mm and σz− = 5 mm. Figure 3.1 sketches out the collision, when the central
parts of a pair of bunches collide. The beams travel along s to the opposite direction
for electron and positron. The parallel translation along the x direction for the two
beams is neglected in this picture.

S

Figure 3.1: Schematic view of nano-beam collision.
Another characteristic of SuperKEKB is the small vertical beta function (βy∗ ) at
the IP. The beta function is squeezed to be βy+ = 0.27 mm and βy− = 0.30 mm since40

the overlap area is very small size of 0.25 mm.
The vertical emittances are assumed to be εy+ = 8.64 pm, εy− = 12.9 pm withtaking into account of the beam-beam interaction. The design luminosity for 2500
bunches is 8 × 1035 cm−2 s−1 .
The tune shifts due to the beam-beam collision are ξx/y+ = 0.0028/0.0881, ξx/y− =0.0012/0.0807 for positrons and electrons, respectively, with z √= 0. The horizontal tune
shift is small, because of the large projected beam size Σx = (σz− ϕc )2 + (σz+ ϕc )2 .
The luminosity performance is evaluated by a weak-strong beam-beam simulation. In the weak-strong simulation, one (weak) beam is expressed by many (macro)particles, and other (strong) beam is assumed to be a fixed Gaussian charge distribution. As shown in later, the positron beam easily enlarges compared to the electron
beam. The positron and electron beams are treated as the weak and strong beams,
respectively, in the most of this report.
The weak beam, which is Gaussian distribution initially, is generated by randomvariables. The horizontal emittances are εx+ = 3.2 nm and εx− = 4.6 nm with taking
into account of the intrabeam eﬀect.
The vertical emittance of the weak beam are initialized with εy = 4.7pm(e+ ) or 1.6
pm(e− ), since x-y coupling at zero current is κ+ = 0.05% and κ− = 0.23%.
Macro particles in the weak beam are tracked during many turns with a transformation along arc section and the beam-beam interaction with the strong beam. The
luminosity performance is evaluated after the equilibrium for the beam-beam interaction, radiation damping and excitation.
The revolution of the beam along arc section is expressed by a transformation
using 6 × 6 matrix (M ) in this subsection. The revolution matrix is represented by the
eigenvalues and eigenvectors.
M = V U V −1

V = Rη RB(3.1)

The block diagonalized matrix (U ) is represented by the tunes (νx , νy , νz ).
Ux 0 0
U =  0 Uy 0 
0 0 Uzcos µi sin µi 
Ui = 
− sin µi cos µi(3.2)

where µi = 2πνi .
The eigenvectors are parametrized by the matrices, Rη , R and B.
T
S2 /(1 + rη,0 ) Rη,x{1 − |Rη,x |/(1 + rη,0 )}I2 Rη,x S2 Rη,yTRη =  Rη,y S2 Rη,x S2 /(1 + rη,0 ) {1 − |Rη,y |/(1 + rη,0 )}I2 Rη,y ,TTrη,0 I2
S2 Rη,y S2
S2 Rη,x S2

41

(3.3)

where Rη,x , Rη,y , S2 , and I2 are 2 × 2 matrices:




ζi ηi
Rη,i =  ′ ′ ζi ηii = x, y,





0 1 
S2 = 
−1 0

(3.4)


1 0 
I2 = 
,
0 1

(3.5)

√

rη,0 = 1 − |Rη,x | − |Rη,y |. ζ’s are induced by crab cavities. Such a transverse deflecting component is very weak in ordinary accelerators, thus ζ’s = 0, rη,0 = 1 in thisreport.


r0 I2 −S2 R2T S2 0


R=
r0 I2
0 
 −R2

0
0
I2





r1 r2 R2 = 
,
r3 r4

(3.6)

√

where r0 =1 − |R2 |.


Bx 0 0B =  0 By 0 

0
0 Bz

Bi = 


√1

βi
α
x
√
βi



0

√ 
βi

(3.7)

The weak beam particles are transferred before and after beam-beam interactioneﬀectively due to the crossing angle. The particles are transferred before collision as
follows,
(

∗

x

=

y∗ =
z∗ =
p∗x =
p∗y =
p∗z =
where

H = (1 + pz ) −
√

ps =

)

p∗
tan θz + 1 + x∗ sin θ x
psp∗yy + sin θ ∗ x
psz
H∗− ∗ sin θxcos θpspx − tan θH
cos θ
py
cos θ
pz − tan θpx + tan2 θH,√(1 + pz )2 − p2x − p2y(1 + pz )2 − p2x − p2y .

The particles are transferred after the collision with the inverse of Eq.(3.8).
42

(3.8)

The linear part of Eq.(3.8) is expressed by a matrix
10 0 1/ cos θ


 0
0

TL = 
 0
0


 0
0

0 − tan θ

0
0
tan θ
0
0
0
1
0
0
0 1/ cos θ0001/ cos θ
0
0
0



0

0 


0 

.
0 


0 

1

(3.9)

This matrix transformation performs x − z tilt as shown in Figure 3.1.
The strong beam is assumed to have a rigid Gaussian distribution in real 3 dimensional space. The electro-magnetic field of a point charge traveling the light speed
is formed in the perpendicular plane to the traveling direction. The strong beam issliced along the bunch length z. The electro-magnetic field of each slice depends on the
charge in a slice of δz and the distribution (Gaussian in x-y plane). The weak beam
particles travel with the kick of electro-magnetic field and drift between slices. The
kick, which a positron particle with a deviation of (x, y) from the center of distribution
experiences, is expressed by
2n− (z)δzre
dpy + idpx =
γ





x + iyw  √22(σx − σy2 )( − exp −

)

√

 σy

π
2(σx2 − σy2 )
x+

×

σx 
y

σy 


 σx
y
x
− 2 w
√ ,2
2σx 2σy2(σx2 − σy2 )22(3.10)

where n− (z− ) is the electron line density at z− and w(x) is the complex error function.
In the revolution matrix (3.1), ν and β is certain number but other 15 parameters
are 0 in the design of SuperKEKB. βx,y is determined for the design luminosity. Choiceof tune is discussed in 3.1.2 in detail.
Figure 3.2 shows the simulation result for the tune νx = 0.53, νy = 0.58, which
was promised area in KEKB. Red curve plots luminosity and beam size evolution, in
which the positron and electron beams are weak and strong, respectively. Blue curveplots them, in which the role of weak/strong is exchanged; weak electron and strong
positron. Blue curve is higher than the design luminosity, while red curve just achieves
the design luminosity in the equilibrium (after 2000 turns). The positron beam easily
enlarges due to the interaction with the electron beam. The red curve is realistic case,because electron beam does not blow up. Emittance evolution, which is evaluated by
the second order moment of the weak particles, is plotted in the right picture. The
design luminosity is achieved even though emittance of positron beam is twice of thedesign. This fact means the core distribution is not enlarged so strongly. Hereafter
43

the case of the positron weak and electron strong beams is studied in the weak-strong
simulation.

14

25
20
Symbol ey (pm)

L (1035 cm-2m-1)

12
10
8
6
4

0
0

5000

10000
turn

15000

10
5

e+
e-

2

15

e+
e-

0
20000

0

5000

10000
turn

15000

20000

Figure 3.2: Luminosity and emittance evolution in the weak-strong simulation. Redline is given for weak positron and strong electron beams. Blue line is given for weakelectron and strong positron beams. Cyan and magenta curves in the right picture are
the design emittance of electron and positron beams, respectively.

3.1.2

Tune Scan

It is well known that high luminosity can be achieved at electron-positron colliders
with working point chosen to be chose to be the half integer. To search for the best
working point in the tune space, tune scans are performed for both LER and HER, withfractional tunes in the range of [0.50, 0.75] and the beam currents set to design values.
The results of tune scans for the LER and HER are demonstrated in Fig. 3.3-3.5 withscaled colors. Figure 3.3 shows the results of luminosity scan, and Fig. 3.4 and 3.5
are the relevant scans of horizontal and vertical beam sizes. It is seen that the strong
synchro-betatron resonances of 2νx − N νs = Integer exist in the nano-beam scheme.
This is due to the large crossing angle chosen for the purpose of mitigating hourglass
eﬀects. Furthermore, the resonances of νx + 2νy + N νs = Integer, 2νy − νs = Integer,and νx −νy −νs = Integer also restrict the choice of working point. The working points
have to be kept far enough from these strong resonances. In general, the luminosity
is very sensitive to the vertical beam size. It is also observed that the electron beamin HER is more robust than the positron beam in LER with respect to the beambeam driven synchro-betatron resonances. At present, both the main rings of theSuperKEKB are optimized with fractional tunes of (0.53, 0.57).
The eﬀects of synchro-betatron resonances are more clearly seen in Fig. 3.6, where
the scans of horizontal tune are done with fixed fractional vertical tune at 0.57. The
red solid lines correspond to the results using the pure weak-strong model. The widths
44

(a) LER(b) HER

Figure 3.3: Luminosity tune scans for the LER and HER.

(a) LER(b) HER

Figure 3.4: Horizontal beam sizes with beam-beam for the LER and HER.
of resonances are narrower in the case of HER than in the case of LER. It implies thatthe high energy beam is relatively immune from beam-beam perturbation.

3.1.3

Errors of orbit at collision point

The luminosity degradation is caused by collision with an oﬀset of the two beams.The
geometrical luminosity loss with orbit oﬀsets is given by)(∆x2 ∆y 2∆L,= exp − 2 −
L0
4Σx
4Σ2y
√

(3.11)

where the projected beam size Σx = σx2 + (σz ϕc )2 , Σy = σy . Actually the luminosity
degradation is stronger due to the dynamic eﬀect of the beam-beam interaction. The
luminosity degradation depends on whether the collision oﬀset is static or fast (turnby-turn) fluctuation.
45

(a) LER(b) HER

Figure 3.5: Vertical beam sizes with beam-beam for the LER and HER.

(a) LER(b) HER

Figure 3.6: Total luminosity as a function of horizontal tune for LER and HER, withfractional vertical tune of 0.57.
Static oﬀset error
To evaluate the eﬀect of a static oﬀset, the weak-strong simulation is performed withthe strong beam shifted by (∆x, ∆y). For the nano-beam scheme, the contribution ofbunch length is dominant in the horizontal projected beam size, Σx ≫ σx . While thehorizontal oﬀset ∆x results a shift of overlap area ∆s as shown in Figure 3.7. Twobeams collide with a shift of vertical waist. Therefore the sensitivity for horizontal
oﬀset is σx , but not Σx .
The equilibrium luminosity and vertical emittance are calculated for several oﬀset
values. Figure 3.8 shows the equilibrium luminosity and vertical beam size normalizedby L0 = 8 × 1035 cm−2 s−1 and the design beam size σy,0 . On the whole the luminosity
degradation is quadratic for small errors. For 20% degradation, tolerable oﬀset is 0.8
σx and 0.8 σy in horizontal and vertical, respectively.

46

1

6

1

0.8

5

0.9

0.6

4

0.8

0.4

3

0.2

2

0.6

1

0.5

0
0

0.5

1
δx/σx

1.5

2

2.6
2.4

2
0.7

σy/σy,0L/L02.2

σy/σy,0L/L0

Figure 3.7: nano-beam collision scheme.

1.8
1.6
1.4
0

0.5

1

1.5

2

2.5

δy/σy

Figure 3.8: Liminosity degradation for static collision oﬀset. Left and right pictures
depict the degradation for horizontal and vertical oﬀset, respectively. Red and bluelines are luminosity and beam size, respectively.

47

Turn-by-turn oﬀset fluctuation
In the weak-strong simulation, particles move in a given potential formed by colliding
beam; Hamiltonian system. Fluctuation of collision oﬀset results in fluctuation of
the potential, which enhances emittance growth. Figure 3.9 shows the luminosity
degradation for turn-by-turn oﬀset fluctuation. For 20% degradation, tolerable oﬀset
is 0.08 σx and 0.09 σy in horizontal and vertical, respectively. The sensitivity is oneorder severer than that of static oﬀset.

4

1

4

1

0.9

0.9
3

2

0.8

σy/σy,0

0.7

L/L0

0.8

σy/σy,0

L/L0

3

0.7

2

0.6

0.6
0.5

1
0

0.02

0.04
0.06
δx/σx

0.08

0.1

0.5

1
0

0.05

0.1
δy/σy

0.15

0.2

Figure 3.9: Liminosity and emittance degradations for turn-by-turn collision oﬀset.
Left and right pictures depict the degradations for horizontal and vertical oﬀset, respectively. Red and blue lines are luminosity and beam size, respectively.

3.1.4

Error of linear optics parameters

Optics parameters at the IP which are zero in the design have nonzero values due tostrength and alignment errors of magnets in a real accelerator. The nonzero values ofthe parameters break symmetry of the beam-beam force and cause emittance growthand luminosity degradation.
βx is larger than βy and σz . Therefore, the errors for βx and αx should not besensitive to the luminosity performance. Errors for βy and αy are treated as a vertical
waist shift in this report. The minimum beta is evaluated accurately by a measurementof βy at IR magnets, while it is ambiguous how the waist of βy shifts from the IP. Figure
3.10 shows the luminosity and emittance degradation due to the waist shift. For 20%
degradation, tolerable waist shift is 0.07 mm. As is mentioned in Sec. 3.1.3, the waist
error gives similar eﬀect as the horizontal oﬀset; ds = δx/2ϕc is satisfied.
Luminosity degradation due to x-y coupling and dispersion at the IP is evaluatedas shown in Figures 3.11 and 3.12. Small horizontal dispersion (ηx δ ≪ σx ) should not
48

6

0.8

5

0.6

4

0.4

3

σy/σy,0

L/L0

1

2

0.2
σx/φc=0.2mm
0

1
0

0.05

0.1
∆s (mm)0.15

0.2

Figure 3.10: Luminosity and emittance degradations due to vertical waist error. Red
and blue lines are luminosity and beam size respectively.
be serious.
Chromatic aberrations for linear optics parameters
Chromatic aberrations of the optics parameters, dνxy /dδ, dβx,y /dδ and dri /dδ, aﬀect
the luminosity performance. The weak-strong simulations were performed to evaluate
tolerance for the chromatic aberrations.

3.1.5

Summary for the error tolerance

The tolerance for the optics errors are summarized in Table 3.1.

3.1.6

Lattice nonlinearity

Beam-beam interaction may interplay with lattice nonlinearity and cause luminosity
loss. To evaluate its eﬀect, the SAD code [1] is utilized to do element-by-element
tracking simulations with the beam-beam element inserted at the IP. The total oneturn map used in the simulations can be represented by
M = Mrad ◦ Mbb ◦ M0 ,(3.12)

where Mbb and Mrad are maps for the beam-beam interaction and radiation damping/quantum excitation, respectively. And M0 indicates the transfer map felt by aparticle when it travels through normal magnetic and electromagnetic components
along the ring. The lattice nonlinearity is naturally included in M0 when a realistic
lattice is loaded in the SAD code.
Another method for the simulations with momentum-dependent lattice nonlinearity
was discussed in Refs. [2, 3], where a symplectic formalism was developed to describe
49

4

1
0.9

1

5

0.9

4

0.8
3
0.7

2

2

0.6

0.6
0.5

0.5

1
0

1

2

3
4
5
r1 (mrad)

6

7

4

1

1
0

8

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
r2 (mm)
4

1

0.9

0.9

3

3
0.8

σy/σy,0

0.7

L/L0

0.8

σy/σy,0L/L0σy/σy,0

0.7

L/L0

0.8

σy/σy,0

L/L0

3

0.7

2

2

0.6

0.6
0.5

0.5

1
0

5

10
15
r3 (m-1)

20

1
0

25

0.1

0.2

0.3

0.4
r4

0.5

0.6

0.7

0.8

Figure 3.11: Luminosity and emittance degradations due to x-y coupling. Four pictures
depict the degradations for r1 − r4 . Red and blue lines are luminosity and beam size
respectively.

4

1

4

1

0.9

0.9
3

2

0.8

σy/σy,0

0.7

L/L0

0.8

σy/σy,0

L/L0

3

0.7

2

0.6

0.6
0.5

1
0

5

10

15

20 25
ηy (µm)

30

35

40

0.5

1
0

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
η’yFigure 3.12: Luminosity degradation due to vertical dispersion. Left and right pictures
depict the degradations for ηy and ηy′ , respectively. Red and blue lines are luminosityand beam size respectively.

50

Table 3.1: Summary for error tolerancedx/σx (static)
dy/σy (static)
dx/σx (fast)dy/σy (fast)
ds
r1
r2
r3
r4
dr1 /dδ
dr2 /dδdr3 /dδ
dr4 /dδηyηy′SuperKEKB
0.8
0.8
0.08
0.09
0.07 mm
3.0 mrad0.1 mm10 m−1
0.4 rad2.1
0.074 m
8400 m−1
290
31 µm
0.23

KEKB
0.4
0.025
2 mm
2.1 mrad0.4 mm0.35 m−10.07 rad6.1
2.5
1100 m−1
440
500 µm0.6

the perturbation maps for the chromatic aberrations. In that method, the momentumdependent lattice nonlinearities are lumped to the IP.
The green dashed lines in Fig. 3.6 indicate the results using the SAD code. It isseen that the lattice nonlinearity enhances the synchro-betatron resonances by widening their widths. On the other hand, direct emittance growth and significant luminosity
degradation are also observed from simulations. Due to these synchro-betatron resonances, the horizontal tunes of SuperKEKB are hard to be more closer to the half
integer, as has been achieved in KEKB.
The luminosity performance as a function of bunch current products are shownin Fig. 3.13. In the figure, the red solid lines indicate results using the pure weakstrong model. The blue dashed lines indicates results using the weak-strong modelplus perturbations of chromatic aberrations. The green dashed lines indicates results
using the SAD code with the weak-strong model. The cyan lines represent the design
values of luminosity and beam current products. It is seen that remarkable loss ofluminosity appears at high bunch currents due to interplay of beam-beam and latticenonlinearity in the LER. Especially, the specific luminosity drops quickly at very low
beam currents. These phenomena can not be explained by the momentum-dependent51

lattice nonlinearity. Dedicated analysis of the lattice nonlinearity in the LER arenecessary to understand the mechanism. One possibility is that amplitude-dependent
nonlinearities plays an important role in the LER. On the other hand, the luminosity
loss due to interplay of beam-beam and lattice nonlinearity in the HER is not as serious
as in the LER, and can be well attributed to the chromatic aberrations in the HER
lattice.

Figure 3.13: Specific luminosity as a function of bunch current products. Left picture
is for LER, and right picture is for HER.
To further illustrate how lattice nonlinearity interplays with beam-beam interaction,
frequency map analysis (FMA) is performed for three cases: bare lattice, pure beambeam, and beam-beam with lattice nonlineairity. The initial conditions are taken overa mesh in the horizontal (x) and vertical direction (y) inside an area of 10σx ×10σy , andthe corresponding tune footprints are plotted in the tune plane. The color indicates thediﬀusion rate of the particle motion. Figures 3.14 and 3.15 show the FMA results for
LER and HER, respectively. In each figure, the blue dots extended from the origin (.53,.57) indicate footprints for a bare lattice; the black dots indicate footprints for purebeam-beam; the rest dots represent footprints for beam-beam with lattice nonlinearity.
In the same figures, resonance lines up to eighth order are also plotted for reference.
From the frequency maps, the footprints in tune space with the bare lattice show
strong dependence on initial amplitude. This is the results of strong amplitudedependent lattice nonlinearity in both rings. The pure beam-beam interaction causes
large spread in the vertical tunes while very small spread in the horizontal tune. The
footprints are strongly deteriorated by the interplay between lattice nonlinearity andbeam-beam. Particle with initial amplitudes of several sigmas performs very chaotic
motion with large diﬀusion rate. The resonances driven by beam-beam are not clearlyseen when lattice nonlinearity is included, because the particle motions become strongly
chaotic.

52

Figure 3.14: FMA plot for the LER in the plane of betatron tunes.

Figure 3.15: FMA plot for the HER in the plane of betatron tunes.

3.1.7

Space charge

The first-order space-charge tune shift experienced by particles performing small oscillations around the beam centroid in a uncoupled lattice and for a gaussian bunch canbe estimated by
λβi
1 2re ∫ C
∆νi = −
ds
(3.13)
4π β 2 γ 3 0 σi (σx + σy )with βx , βy are the beta functions, σx , σy are the horizontal and vertical rms beam√
sizes, and i = x, y. The the longitudinal peak density is λ(s) = N/ 2πσz (s) withgaussian bunch profile assumed. In the case of SuperKEKB, the bunch length σz isalmost constant in the rings and we can ignore s-dependence of λ. In the absenceof linear coupling, the horizontal beam sizes are calculated from emittance via σx2 =ϵx βx + ⟨δ 2 ⟩D2 with D the dispersion.
With design bunch populations, the estimated linear space-charge tune shifts are∆νx = −0.003, ∆νy = −0.087 for LER, and ∆νx = −0.0004, ∆νy = −0.012 for LER.
The tune shifts are rather small in HER, but the vertical tune shift in LER has the53

same order as beam-beam tune shift with the opposite signs. The rms beam sizes andtune shift in the vertical plane along the LER lattice are plotted in Fig.3.16.

Figure 3.16: rms beam sizes and space charge tune shift in the vertical plane along the
LER.
Similar to the beam-beam interaction, the space-charge force is highly nonlinear,and depends on the betatron amplitudes. Consequently, the footprints in the tune
space may be distorted with the interplay of beam-beam, lattice nonlinearity andspace charge. The influence of space-charge on luminosity performance is to be studied
carefully via tracking simulations.

3.2

Electron Cloud eﬀects

3.2.1

Electron cloud build up

Electrons, which is produced by photo-emission and secondary emission, are buit up in
the positron ring, LER. To protect electrons, antechamber is adopted for the vacuum
pipe. Synchrotron light hits the antechamber slot. Most of electrons do not comethe central part of the chamber, where positron beam passes through. The electron
cloud build up is evaluated by a simulation [4]. Primary electron is produced with thequantum eﬃciency 0.1 per photon incidence. The secondary electron is produced withthe formula [5],E
1.44
δ2 (E) = δ2,max ×,(3.14)
Emax 0.44 + (E/Emax )1.44
where δmax = 1.2 at Emax = 300 eV.
Figure 3.17 shows the electron cloud distribution and density in the antechamber.
The electron density near the beam is obtained as 2.2 × 1011 m−3 . As is shown later,
the density is critical for both of the coupled and single bunch instabilities. The density
is reduced one order magnitude by groove structure and weak solenoid magnets.
54

Figure 3.17: Electron cloud distribution and buildup in antechamber.

3.2.2

Coupled bunch instability

Bunch train experiences a long range (∼ 10 ns) wake field due to the electron cloud.
A coupled bunch instability is caused by the wake field [6]. Figure 3.18 shows thewake field and growth rate of the instability. The fastest growth is the mode number∼ 2000 and its rate is 0.025 (40 turns). This growth rate is comparable with the
feedback damping time. Since the electron cloud is reduced further 1/2-1/10, thiscoupled bunch instability is manageable.

100

0.03

η=0.001dy=0.002
0.02
0.01

0

T0/τgr

W y0 (m-1)

50

-50

-0.01

-100
-150
100

0

-0.02
-0.03

110

120
130
bunch

140

150

0

500

1000 1500
mode

2000

2500

Figure 3.18: Wake field and growth rate for electron cloud induced coupled bunch
instability.

55

3.2.3Single bunch instability

constant beta
As the first step, the estimation of single bunch instability is performed in the condition that electon cloud distributes uniformly along the ring and the beta function isconstant. Electrons oscillate in the electric potential formed by the positron bunch.
The frequency is expressed byvuuωe = tλp r e c 2,
σy (σx + σy )(3.15)

where λp is the local line density of the positron bunch. The frequency is 10-100 GHz
in typical positron rings. The electrons oscillate several period during the passage ofa positron bunch. The oscillation induces a short range wake field with the frequency,
ωe . Fast head tail instability is caused by the wake field. Horizontal motion is slower
than vertical one, thus vertical motion is more serious for the instability. The thresholdof the vertical single bunch (fast head-tail) instability is given by
2γνs ωe σz /c
ρe,th = √,3KQre βy L

(3.16)

where K = ωe σz /c and Q = min(ωe σz /c, 10) are used.
Tables 3.2 and 3.3 show parameters and threshold electron density of positron rings,
in which electron cloud eﬀect is/should be observed. The threshold is 2.7 × 1011 m−3
for SuperKEKB.
The single bunch instability is simulated by solving beam-electron cloud interactions. The simulation is performed by the same method as the strong-strong beambeam simulation. A bunch is sliced into many pieces (> ωe σz /c). Interaction between
a bunch slice and electron cloud is evaluated by potential solver based on the particle
in cell method. The synchrotron radiation damping and excitation are not taken into
account to be clear the instability threshold in this subsection. The radiation damping/excitation is taken into account in Sec. 3.2.4 to evaluate equilibrium emittance.
Figure 3.19 shows evolution of the vertical beam size and a snapshot of beamelectron motion. The threshold density is 3.8×1011 m−3 as shown in left picture. Right
picture depicts motion of bunch slices and electron centroid during the interaction forρe = 4.2 × 1011 m−3 at 4000-th turn. Coherent motion of bunch slices and electron
cloud is seen.
In the threshold estimation, βy = 10 m is used as a typical value. Eq.(3.17) informs∫
threshold of ρe βy ds: i.e. lower threshold for higher beta. This is true if the dynamics

56

Table 3.2: Basic parameters of the positron rings

Lattice
Circumference
EnergyBunch population
Beam current
Emittance

KEKB Cesr-TA PETRA-III SuperKEKB Super B
L (m)
3,016
768
2304
3016
1260
E (GeV)
3.5
2-5
6
4.0
6.7
10
N+ (10 )
8
2
0.5
9
5
I+ (A)
1.7
0.1
3.6
1.9
εx (nm)
18
2.3
1
3.2
2
εy (nm)
0.18
0.023
0.01
0.01
0.005
−4
Momentum compaction α(10 )
3.4
68
12.2
3.5
Bunch lengthσz (mm)
6
6.8
12
6
5
−3
RMS energy spread
σE /E(10 )
0.73
0.8
0.8
0.64
Synchrotron tune
νs
0.025
0.067
0.049
0.02560.0126
Damping timeτx (ms)
40
56.4
16
43
26

8e-05

5e-05
4e-05
3e-05

20
15
10

y (µm)

6e-05

σy

25

3.0x1011
3.2x1011
3.4x1011
3.6x1011
3.8x1011
4.0x1011
4.2x1011
4.4x1011

7e-05

5
0
-5

2e-05

-10

1e-05

-15

0
0

2000

4000
turn

6000

8000

y(z)
σy(z)
ye(t)

-20
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5z/σz

Figure 3.19: Evolution of the vertical beam size for various electron cloud density (left)
and motions of bunch slices and electron centroid (right).

57

Table 3.3: Threshold of electron cloud density in positron rings.

KEKB
KEKB
Cesr-TA PETRA-III SuperKEKB SuperB(no sol.) (50 G sol.)
N+ (1010 )I+ (A)ℓsp (ns)ωe /2π(GHz)ωe σz /cρe (1012 m−3 )Bunch population
Beam current
Bunch spacing
Electron frequency
Phase angle
Threshold

3
0.5
8
28
3.6
0.63

8
1.7
7
40
5.9
0.38

2
4-14
43
11.0
1.7

0.1
8
35
8.8
1.2

8
3.6
4
150
18.8
0.27

5
1.9
4
175
18.3
0.54

of the beam-electron cloud do not change. The threshold is obtained asIρe βy ds = 27(formula) − 38(simulation) × 1012 m−1 .

(3.17)

These threshold values are critical for the electron build-up estimation without special
cures in Figure 3.17.
realistic beta and cloud density
The beam-electron cloud interaction depends on the local electron density and beam
size. The frequency of electrons ωe in Eq.(3.15) depends on s, because of the beam size
variation. βy gives coupling between beam-electron interaction as shown in Eq.(3.16).
The tune shift caused by the electron clouds is expressed as
∆νy =

re
ρ e βy ,kγ(3.18)

where k = 1 or 2 for flat or round electron distributions, respectively.
The vacuum group estimates electron density profile along s. Figure 3.20 shows thevariation of βx,y and ρe . The density is around 0.2 × 1011 m−3 in most of the sections asshown in left picture. It is high at the symmetric point for IR. The density near IR is
depicted in the right picture. Very high β and dispersive sections are located near IR(s = 27 and 68 m) for the local chromaticity correction. Bending magnets are locatedin the sections. Single bunch instability and nonlinear emittance growth are studied
in two cases; (1) electron is suppressed (green) and (2) is not suppressed (cyan) at thehigh beta sections.
∫Figure 3.21 shows 0s ρe βds and the vertical tune shift along the ring for the twocases. Tune shift, which arises in arc the section, is 0.0003, while that in IR is 0.0012 or
0.0009 for case (1) or (2), respectively. In either case, the tune shift in IR is dominant.
58

Figure 3.22 shows electron oscillation phase ωe σz /c. The electron oscillation is 20 inarc, while is 5 or less in IR. The variation of ωe reduces the quality factor of the wakeforce induced by electron cloud.
180

βx
βy

160

ρe,1
ρe,2

140

3000

1

0.8

0.4

0.6
βx,y

80

ρe/1011 m-3

0.6
100

1500
0.4

ρe/1011 m-3

2000

120
βx,y

ρe,1
ρe,2

2500

0.8

1

βx
βy

1000

60
40

0.2

0.2

500

20
0

0
500

1000

1500
s (m)

2000

0

2500

0

20

40

60

80

0
100

s (m)

Figure 3.20: βy and electron central density ρe along s. Left and right pictures depict
whole ring and IR area, respectively.

140.00161
local
integral 3
integral 6120.00012local
integral 3
integral 60.0014
0.8

6

0.4

4

8e-05

0.001

0.6

8

∆νy

ρeβy (1011 m-2)0.00010.0012100.00086e-05

0.00064e-05

0.0004
0.2

2

2e-05

0.0002

0
0

500

1000

1500
s (m)

2000

Figure 3.21:

2500

0
3000

0
0

500

1000

1500
s (m)

2000

2500

0
3000

∫s

0 ρe βds and the vertical tune shift along the ring.

Figure 3.23 shows the evolution of the vertical beam size for various electron clouddensity. The density at symmetry point (s = 1500 m) is written in the pictures. The
design cloud density is 1011 m−3 at s = 1500 m. The density distributes along s asshown in Fig. 3.20.
Left and right pictures depict beam size evolution for the cases (1) and (2), respectively. The threshold is 6× or 4× of the design for the cases (1) and (2), respectively.
We investigate which electron clouds in arc or IR is dominant for the instability.
Figure 3.24 shows instability threshold caused by electron cloud in IR (left picture) orarc (right picture). The threshold density is 4 − 5× or 20× of the design for electrons
only in IR or only in arc, respectively. The electron cloud distribution of the case (2)∫is examined. The contribution of ρβy ds is 4:1 as shown in Fig.3.21. The threshold∫densities are consistent with the ratio of ρβy ds.

59

50

ωσz/c

40
30
20
10
0
0

500

1000 1500 2000 2500 3000s (m)Figure 3.22: Variation of electron oscillation phase ωe σz /c.

8

62x10113x10114x10115x10116x10117x10118x10119x1011108σy/σy05σy/σy012

2x10113x10114x10115x10116x10117x10118x10119x1011

7

4

6

3

4

2

2

1
0

0
0

1000

2000

3000

4000
turn

5000

6000

7000

0

1000

2000

3000

4000
turn

5000

6000

7000

6
5.5
5
4.5
4
3.5
3
2.5
2
1.5
1

3x10114x10115x10116x10117x101116

10x101115x101120x101125x101130x101114
12σy/σy0σy/σy0Figure 3.23: Evolution of the vertical beam size for various electron cloud density. The
relative density distribution is given in Fig. 3.20.

10
8
6
4
2
0

0 1000 2000 3000 4000 5000 6000 7000 8000
turn01000 2000 3000 4000 5000 6000 7000 8000
turn

Figure 3.24: Evolution of the vertical beam size for various electron cloud density. In
the left picture, electrons distribute only in IR, s < 75 and s > 2940 m. In the right
picture, electrons distribute only in arc section, 75 < s < 2940 m. The electron cloud
distribution is the case (2).

60

3.2.4

Nonlinear emittance growth

Emittance growth is seen bellow the threshold of the fast head-tail instability in Figures
3.23 and 3.24. The emittance growth is caused by nonlinear force due to the electron
cloud charge distribution. In the simulation, particles are tracked with a large time
step ∆s > βy in arc section, while with a short time step ∆s < βy . Emittance growthcaused by electron in IR can be predicted accurately. The tune shift contribution isdominant in IR. Figure 3.25 shows the variation of βy and its phase in IR section.
Locations with very high βy are separated with the phase π. Since electron transverse
distribution is considered to be symmetric for y, the nonlinear eﬀect is accumurated
every high βy locations.
3000

βy
φy

1.75

2500

1.25

2000

0.75

φy

βy

0.25
1500
-0.25
1000

-0.75

500

-1.25
-1.75

0
-100

-50

0
s (m)

50

100

Figure 3.25: Vertical beta function and betatron phase variations in IR section.
To evaluate equilibrium emittance for the nonlinear growth, the radiation damping
and excitation is taken into account in the simulation. Figure 3.26 shows evolution of
the vertical beam size with (green) and without radiation damping/excitation. The
vertical radiation damping time is 4300 turns. The equilibrium emittance is realized
after around 9000 turns.
Figure 3.27 shows the evolutions of the vertical beam size for various electron density. Left and right pictures depict the evolutions for the cases (1) and (2). The
equilibrium beam size is obtained as functions of the electron density. In the right
picture, beam size evolutions above the threshold (ρe = 3× design) with and without synchrotron damping/excitation are plotted. We find that the radiation damping
somewhat suppresses the coherent single bunch instability.
Figure 3.28 shows the equilibrium emittance as function of electron density. The
emittance increase is not serious for the design cloud density, but become serious 3
times or more.

61

1.8

rad,3x1011
norad,3x1011

1.7

σy/σy0

1.6
1.5
1.4
1.3
1.2
1.1
1
0

2000

4000
turn

6000

8000

2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0

4.5

2x10113x10114x10115x1011
norad,5x1011

4
3.5
σy/σy0

3

0

2000

4000
turn

6000

2.5
2
1.5

2x10113x10114x10115x1011

1
0.5
0
8000

0

2000

4000
turn

6000

8000

Figure 3.27: Evolution of the vertical beam size for various electron density. Left andright pictures depict the evolutions for the cases (1) and (2).

εy (pm)σy/σy0Figure 3.26: Evolution of the vertical beam size with (green) and without (cyan)radiation damping/excitation.

55
50
45
40
35
30
25
20
15
10
5

1
2

0

1

2
3
ρe (1011 m-3)

4

5

Figure 3.28: Equilibrium emittance as functions of the electron cloud density. Twolines, red and blue, are drawn for the cases (1) and (2), respectively.

62

3.2.5

Summary

Conclusions for the electron cloud eﬀects are summarized in the following.
• The vacuum system is designed so that electron density near the beam is suppressed around ρe = 1 × 1011 m−3 at the peak (s = 1500 m) and 2 × 1010 m−3 inaverage of whole ring.
• The growth time of the coupled bunch instability is about 400 turn and is manageable.
• The threshold electron density is 4-6 times higher than that of the design value.
• Electrons near IR are dominant sources for the instability.
• The emittance growth is negligible for ρe ≤ 2× design value.

63

Bibliography[1] http://acc-physics.kek.jp/SAD/sad.html.
[2] D. Zhou, K. Ohmi, Y. Seimiya, Y. Ohnishi, A. Morita and H. Koiso, Phys. Rev. ST
Accel. Beams 13, 021001 (2010).
[3] Y. Seimiya, K. Ohmi, D. Zhou, J. W. Flanagan and Y. Ohnishi, Progress of Theoretical
Physics 127, 1099 (2012).
[4] K. Ohmi, Phys Rev. Lett. 75, 1526 (1995).
[5] M.A. Furman, G.R.Lambertson, Proceedings of MBI97 (KEK-report No.97-17), p.170.
[6] K. Ohmi, F. Zimmermann, Phys. Rev. Lett. 85, 3821 (2000).

64