1. Cavity-beam interaction and Longitudinal beam dynamics for CEPC DR&APDR
宫殿君

2. Outline CEPC DR Z-pole fundamental mode instabilities
DR Higgs & Z phase shift
CEPC APDR phase shift & longitudinal beam dynamics

3. Introduction
Large high current storage rings suffer from two particular beam loading effects not present in smaller storage rings:
First, the presence of phase modulations of the bunches due to gaps in the bunch train;
Second, the excitation of fast growing longitudinal coupled bunch instabilities (CBI) by the detuned fundamental RF resonance.

4. Fundamental Mode Instability of Z-pole
Fundamental mode CBI of Z-pole due to large cavity bandwidth and detuning.
For M bunches, there are M coupled-bunch modes, the phase shift between adjacent bunches for mode number n:
n = 0, 1, 2, ... M-1
CBI growth rate :
f p μ+ = (pM + μ) f f s f p μ− = [(p+1)M − μ] f 0 + f s
We refer to the mode μ=M-n as the mode μ= -n

5. CEPC DR Z machine parameters for CBI growth time calculation
Baseline
High-Lumi
Unit
Beam current
40.8
466 mA
Beam energy
45.5
GeV
Radiation loss/turn 34
MeV
Bunch number
2708
21666
Rad. damping time
0.466
0.446 s
Syn. frequency
0.106
0.204
kHz
Rf frequency
650
MHz
cavity number 16 48
Syn. phase
47.2
75.9
deg
Cavity bandwidth
0.6
5.3
Detuning frequency
-0.32
-10.9
Revolution frequency 3
Harmonic number
216660
Cavity voltage
6.3
2.9 MV
Unloaded Q
1E+10
Loaded Q
1.1E+6
1.2E+5
Coupling factor
9.23E+3
8.12E+4
R/Q
206
Ohm

6. (1) Fundamental Mode Instability of Z-pole_baseline
Longitudinal coupling-impedance
Cavity resonance frequency
𝑍 || (𝑓)= 1 𝛽 𝑅 𝑠ℎ 𝑖 𝑄 𝐿 ( 𝑓 𝑓 𝑟𝑒𝑠 − 𝑓 𝑟𝑒𝑠 𝑓
Cavity acceleration mode impedance and beam spectrum of CEPC Z_baseline mode
Growth time of CBI due to acceleration mode of CEPC Z_baseline

7. (2) Fundamental Mode Instability of Z-pole_high-lumi
Longitudinal coupling-impedance
Cavity resonance frequency
𝑍 || (𝑓)= 1 𝛽 𝑅 𝑠ℎ 𝑖 𝑄 𝐿 ( 𝑓 𝑓 𝑟𝑒𝑠 − 𝑓 𝑟𝑒𝑠 𝑓
Optimal detuning
Cavity acceleration mode impedance and beam spectrum of CEPC Z-pole mode
Growth time of CBI due to acceleration mode of CEPC Z operation (10 modes to be damped)

8. In CEPC DR Z_baseline, which detuning frequency and cavity bandwidth are small, no Acc. Mode exceed the threshold, when the beam current reach the design current.
In CEPC DR Z_high-lumi, when the beam current reaches the design current 466 mA, from the μ=-1 mode to the μ=-10 mode instability due to the Acc. Mode will be excited.
We can use the mode by mode damper system with digital filters to suppress these modes, just like SuperKEKB damper system:

9. Damper system for fundamental mode instability
Use the mode by mode damper system with digital filters to suppress (SuperKEKB)
Damper system for SuperKEKB:
Kouki Hirosawa. Development of a coupled bunch instability damper caused by the acceleration mode for SuperKEKB. PASJ2016 TPU012
Functional block diagram of digital filter for SuperKEKB

10. DR Higgs & Z phase shift H W Z_baseline Z_high lumi
Beam current I [mA]
19.2
97.1
40.8
465.8
Rev.freq [kHz] 3
Acc.phase [deg]
37.3
36.4
47.2
75.9
Vc [MV]
6.3
4.3
2.9
Voltage decrease (3% gap)
2.65%
9.82%
2.82%
69.86%
Voltage decrease (5% gap)
4.42%
16.37%
4.69%
116.44%
Phase shift (3% gap) [deg]
2.51
9.48
2.20
41.27
Phase shift (5% gap) [deg]
4.18
15.81
3.67
68.79

11. CEPC APDR longitudinal beam dynamics
(1) APDR(61km 4+4DR) RF Parameters
Based on Wangdou_ _61km parameters
APDR
H-low power
H-high lumi W Z
main ring type
8 double rings
Circumference (km) 61
revolution frequency(kHz)
4.92
bunch charge (nC) 32
13.6
9.6
bunch number/train N 18 27
100
275
bunch spacing (ns)
185
123.3
33.3
12.1
Gap time interval Tg(us)
22.08
average beam current（mA）
11.02
17.00
26.80
52.00
circulating current(mA)
172.97
259.53
408.41
793.39
SR Loss/turn(GV)
2.96
0.58
0.06
SR Power*2(MW)
65.24
100.66
31.09
6.34
RF voltage Vrf(GV)
3.51
3.48
0.70
0.12
Cavity cell number 2
rf frequency(Hz)
6.50E+08
R/Q(Ω）
213.00
Quality Factor
2.E+10
Shut impedance R(Ω)
4.26E+12
Cavity number
480
128 24
Pulse power/beam（MW）
512.00
768.21
236.88
48.40
Pulse power/cavity（MW）
1.07
1.60
1.85
2.02
Input power/cavity(kW)
135.92
209.71
242.88
264.33
Stored energy/cavity（J)
61.47
60.42
34.38
28.74

12. The phase shift is dropped by 40%
APDR
H-low power
H-high lumi W Z
Loaded quality factor QL
2.35E+05
1.54E+05
7.59E+04
5.82E+04
coupling parameter β
8.50E+04
1.30E+05
2.64E+05
3.44E+05
Loaded impedanceRL(Ω)
2.51E+07
1.64E+07
8.08E+06
6.20E+06
filling time(s)
1.15E-04
7.55E-05
3.72E-05
2.85E-05
unloaded filling time(s)
9.79
detuning frequency(Hz)
-8.19E+02
-1.24E+03
-2.58E+03
-5.49E+03
optimum detuning(Hz)
-8.80E+02
-1.30E+03
-2.89E+03
-9.46E+03
half 3db bandwidth(Hz)
1.38E+03
2.11E+03
4.28E+03
5.58E+03
detuning angle(deg)
32.51
31.73
34.05
59.45
Wilson syn phase(deg)
Cavity Voltage(MV)
7.31
7.25
5.47
5.00
Effective length(m)
0.46
Acc. Gradient(MV/m)
15.84
15.71
11.85
10.83
RF station 8
cavity number/module 6 4 3
total module number 80 32
module/station 10 1
22.5
22.1
2.5
0.4
Loss factor(V/pC)
0.31
0.25
0.24
HOM Power/Cavity(kW)
0.22
0.27
0.43
1.55
Phase shift(deg)
3.17
4.90
9.62
13.28
Voltage decrease(%)
3.26
4.69
8.99
26.60
Compared to parameters, for W and Z, the opt. detuning frequency is decreased by 60%, and cavity bandwidth is reduced to ½, both are benefit for suppressing fundamental instability.
The phase shift is dropped by 40%

13. (2) RF energy acceptance
𝜂 𝑅𝐹 =| 𝜀 max 𝐸 0 |= 𝑈 0 𝜋 𝛼 𝑝 ℎ 𝐸 0 𝐹(𝑞)
𝐹(𝑞)=2( 𝑞 2 −1 −arccos( 1 𝑞 ))
𝜂 𝑅𝐹 = 𝑈 0 𝜋 𝛼 𝑝 ℎ 𝐸 0 （ 1 tan 2 𝜑 − 𝜋 2 −𝜑）
APDR
H-LP
H-HL
W (1109)
W (0918)
Z (1109)
Z (0918)
Phase Shift [deg]
3.17
4.90
9.62
16.7
13.28
24.6
Voltage decrease
3.26%
4.69%
8.99 18
26.60 35
𝜂 𝑅𝐹 [%]
2.37
2.23
1.39
1.8
1.17
1.1
𝜂 𝑅𝐹 ′ [%]
2.01
1.71
0.81
0.6
0.70
0.3

14. (3) Nature bunch length APDR H-LP H-HL W (1109) W (0918) Z (1109)
Phase Shift [deg]
3.17
4.90
9.62
16.7
13.28
24.6
Voltage decrease
3.26%
4.69%
8.99 18
26.60 35
Nature 𝜎 𝑧 [mm]
2.7
3.23
2.95
3.9
3.78
Nature 𝜎 𝑧 ′ [mm]
2.87
3.0
3.97
5.88
4.97
9.04

15. Luminosity decrease [%]
Phase shift increases the bunch length, which decreases the Luminosity.
APDR
H-LP
H-HL
W (1109)
W (0918)
Z (1109)
Z (0918)
Phase Shift [deg]
3.17
4.90
9.62
16.7
13.28
24.6
Voltage decrease
3.26%
4.69%
8.99 18
26.60 35
Luminosity decrease [%] 12 20 34 75 39 83
Beam lifetime
The beam lifetime due to beamstrahlung effect:
APDR
H-LP
H-HL
Phase Shift [deg]
3.17
4.90
Voltage decrease
3.26%
4.69%
𝜏 𝐵𝑆 [min] 37
35.1
34.1

16. momentum compaction(10^-5)
(5) Longitudinal tune
Some parameters about longitudinal motion
APDR
H-LP
H-HL W E
momentum compaction(10^-5)
1.48
3.10
Syn. frequency 𝑓 𝑠 (kHz)
0.34
0.19
0.30
0.28
0.13
0.12
synchrotron tune
0.070
0.069
0.039
0.038
0.062
0.056
0.026
0.024
Rad. Damping time (ms)
8.24
28.03
151.61
SR damping rate (s^-1)
121.36
35.67
6.60
Longitudinal tune is decreased, frequency spread between bunches makes Landau Damping stronger, bunches instability is easier to be damped.

17. (6) Dynamic aperture H-HL： 4.69% voltage drop 4.9 degrees phase shift：
No phase shift：

18. Index CEPC DR 20170426 parameters
CEPC APDR_61km & parameters

19. Parameters for CEPC double ring （wangdou20170426-100km_2mmy）
Pre-CDR
Higgs W Z
Number of IPs 2
Energy (GeV)
120 80
45.5
Circumference (km) 54
100
SR loss/turn (GeV)
3.1
1.67
0.33
0.034
Half crossing angle (mrad)
16.5
Piwinski angle
3.19
5.69
4.29
11.77
Ne/bunch (1011)
3.79
0.968
0.365
0.455
0.307
Bunch number 50
412
5534
21300
2770
Beam current (mA)
16.6
19.2
97.1
465.8
40.8
SR power /beam (MW)
51.7 32
16.1
1.4
Bending radius (km)
6.1 11
Momentum compaction (10-5)
3.4
1.14
4.49
IP x/y (m)
0.8/0.0012
0.171/0.002
0.171 /0.002
0.16/0.002
Emittance x/y (nm)
6.12/0.018
1.31/0.004
0.57/0.0017
1.48/0.0078
0.18/0.0037
Transverse IP (um)
69.97/0.15
15.0/0.089
9.9/0.059
15.4/0.125
5.6/0.086
x/y/IP
0.118/0.083
0.013/0.083
0.0055/0.062
0.008/0.054
0.006/0.054
RF Phase (degree)
153.0
128
126.9
165.3
136.2
VRF (GV)
6.87
2.1
0.41
0.14
0.05
f RF (MHz) (harmonic)
650
650 (217800)
Nature z (mm)
2.14
2.72
3.37
3.97
3.83
Total z (mm)
2.65
2.9
4.0
HOM power/cavity (kw)
3.6 (5cell)
0.41(2cell)
0.36(2cell)
1.99(2cell)
0.12(2cell)
Energy spread (%)
0.13
0.098
0.065
0.037
Energy acceptance (%)
1.5
Energy acceptance by RF (%) 6
1.1
0.68 n
0.23
0.26
0.15
0.12
0.22
Life time due to beamstrahlung_cal (minute) 47 52
F (hour glass)
0.96
0.98
0.99
Lmax/IP (1034cm-2s-1)
2.04
2.0
5.15
11.9

20. Parameter for CEPC partial double ring （wangdou20160918-61km）
New-61km
H-high lumi.
H-low power W Z
Number of IPs 2
Energy (GeV)
120 80
45.5
Circumference (km) 61
SR loss/turn (GeV)
3.0
2.96
0.58
0.061
Half crossing angle (mrad) 15
Piwinski angle
1.88
1.84
5.2
6.4
Ne/bunch (1011)
3.91
2.0
1.98
1.16
0.78
Bunch number 54
107 70
400
1100
Beam current (mA)
16.6
16.9
11.0
36.5
67.6
SR power /beam (MW) 50
32.5
21.3
4.1
Bending radius (km)
6.1
6.2
Momentum compaction (10-5)
3.25
1.48
1.44
2.9
IP x/y (m)
0.43/
0.272/0.0013
0.275 /0.0013
0.1/0.001
Emittance x/y (nm)
6.28/0.04
2.05/0.0062
2.05 /0.0062
0.93/0.0078
0.88/0.008
Transverse IP (um)
51.7/0.2
23.7/0.09
9.7/0.088
9.4/0.089
x/IP
0.118
0.041
0.042
0.013
0.01
y/IP
0.074
0.11
0.073
0.072
VRF (GV)
6.99
3.48
3.51
0.74
f RF (MHz)
650
Nature z (mm)
2.19
2.7
2.95
3.78
Total z (mm)
2.47
3.35
4.0
HOM power/cavity (kw)
3.9
0.48
0.88
0.99
Energy spread (%)
0.13
0.087
0.05
Energy acceptance (%)
2.1
Energy acceptance by RF (%) 6
2.3
2.4
1.7
1.2 n
0.31
0.35
0.34
0.49
Lmax/IP (1034cm-2s-1)
2.02
3.1
2.01
4.3
4.48

21. Parameter for CEPC partial double ring （wangdou20161109-61km）
Pre-CDR
H-high lumi.
H-low power W Z
Z-5cell
Number of IPs 2
Energy (GeV)
120 80
45.5
Circumference (km) 54 61
SR loss/turn (GeV)
3.1
2.96
0.58
0.061
Half crossing angle (mrad) 15
Piwinski angle
1.88
1.84
4.11
5.86
5.87
Ne/bunch (1011)
3.79
2.0
1.98
0.85
0.6
Bunch number 50
107 70
400
1100
700
Beam current (mA)
16.6
16.9
11.0
26.8
52.0
33.1
SR power /beam (MW)
51.7
32.5
15.7
3.2
Bending radius (km)
6.1
6.2
Momentum compaction (10-5)
3.4
1.48
IP x/y (m)
0.8/0.0012
0.272/0.0013
0.275 /0.0013
0.16/0.001
0.12/0.001
Emittance x/y (nm)
6.12/0.018
2.05/0.0062
2.05 /0.0062
0.93/0.003
0.87/0.0046
Transverse IP (um)
69.97/0.15
23.7/0.09
12.2/0.056
10.2/0.068
x/IP
0.118
0.041
0.042
0.0145
0.0098
y/IP
0.083
0.11
0.084
0.073
VRF (GV)
6.87
3.48
3.51
0.7
0.12
f RF (MHz)
650
Nature z (mm)
2.14
2.7
3.23
3.9
Total z (mm)
2.65
2.95
2.9
3.35
4.0
HOM power/cavity (kw)
3.6
0.74
0.48
0.47
0.59
0.93
Energy spread (%)
0.13
0.087
0.05
Energy acceptance (%)
Energy acceptance by RF (%) 6
2.3
2.4
1.3
1.1 n
0.23
0.35
0.34
0.28
0.24
Life time due to beamstrahlung_cal (minute) 47 37
F (hour glass)
0.68
0.82
0.89
0.92
Lmax/IP (1034cm-2s-1)
2.04
2.01
3.5
3.44
2.2