1. SPPC Longitudinal Dynamics
Linhao Zhang, Jingyu Tang
Nov. 13, 2018
Good morning, everyone! It's my honor to be here to make a presentation. My talk is SPPC longitudinal beam dynamics.

2. Outline Parameter design of SPPC longitudinal dynamics for 400 MHz
800 MHz RF system
MHz dual harmonic RF system
Results and comparison
Matching between injectors
Summary

3. Collisions at crossing angle
Parameter design of SPPC longitudinal dynamics at 400 MHz
The basic starting point
According to the basic requirements for luminosity, and meanwhile taking luminosity upgrade into account.
Besides, leaving a margin of certain parameter space for RF system to upgrade.
Luminosity
Luminosity: (head-on)
Two effects lowering luminosity:
beam current :
beam-beam parameter :
Collisions at crossing angle
ρ1(x,y,s,-s0)
ρ2(x,y,s,s0) θc

4. Parameter design of SPPC longitudinal dynamics at 400 MHz
Luminosity
Φ is Piwinski Angle
σz=0.0755
hourglss effect
The larger β*/σz, the higher the luminosity.
Only a little effect on luminosity by shortening bunch length.
β*/σz>>1, it can be neglected.
Luminosity upgrade measures:
decreasing β*, meanwhile shortening bunch length;
increasing beam-beam parameter;
increasing bunch numeber in order to increase single beam current

5. The requirements for SPPC longitudinal dynamics
Parameter design of SPPC longitudinal dynamics at 400 MHz
The requirements for SPPC longitudinal dynamics
Input parameters
Value
Circumference (km)
100
Energy (TeV)
37.5
Transition gamma γtr
99.21
Bunch intensity
1.5 x1011
Number of bunches
10080
Bunch spacing (ns) 25
Bunch length during physics (m)
0.0755
RF frequency (MHz)
400
Two constraints:
Intra-beam Scattering
Beam instability limits
The goals:
How to achieve this goal of bunch length 7.55cm.
The variable range of bunch length for luminosity upgrade.
The design of longitudinal dynamics
Determination of longitudinal emittance (IBS and beam instabilities need to be considered, pivotal issue);
Given bunch length, we can get momentum spread;
Determination of RF cavity voltage ( frf = 400MHz )

6. Intra-beam Scattering (IBS)
Parameter design of SPPC longitudinal dynamics at 400 MHz
Intra-beam Scattering (IBS)
Intrabeam Scattering (IBS):
Multiple small-angle Coulomb scattering of charged particle beams
An increase in both transverse and longitudinal emittance.
The effects of IBS:
Redistribution of beam momenta
Beam diffusion with impact on the beam quality (Brightness, luminosity, etc)
IBS growth rate:
Theoretical models and their approximations
Classical models of Piwinski (P) and Bjorken - Mtingwa (BM)
Complicated integrals averaged around the rings.
Depend on optics and beam properties.
High energy approximations Bane, CIMP, J.Wei, etc
Integrals with analytic solutions
Tools: MADX, BETACOOL, etc

7. Intra-beam Scattering (IBS)
Parameter design of SPPC longitudinal dynamics at 400 MHz
Intra-beam Scattering (IBS)
Injection
Collision
At injection, longitudinal emittance between 1.5 eVs and 2.5 eVs may be a good choice; (injection time 840s)
In collision, longitudinal emittance between 6 eVs and 9 eVs may be a good choice; (time during physics 14.2h)

8. Beam instability limits
Parameter design of SPPC longitudinal dynamics at 400 MHz
Beam instability limits
Two instability bottlenecks:
Loss of Landau damping
Transverse mode coupling instability (TMCI)
Landau damping of the instability can come from the spread in the synchrotron frequency due to the nonlinearity of the synchrotron force in the bunch.
The coherent tune shift induced by the broad-band impedance for a longitudinal mode of order n can be approximate to
The synchrotron tune spread
Requiring
the impedance
threshold

9. Parameter design of SPPC longitudinal dynamics at 400 MHz
σz≥8 cm, (Z/n)th≥0.1Ω
σz=7.55 cm, Vrf ≥ 32MV (Z/n)th≥0.1Ω
easy to bunch-receiving
from SS
Shorter bunch length (luminosity upgrade)
smaller longitudinal impedance.
These two pictures show the longitudinal impedance threshold varies with bunch length at injection and collision, respectively. At injection, RMS bunch length is at least 8cm in order that longitudinal impedance threshold is lager than 0.1Ω, and it's easy to bunch-receiving from SS. To reach the bunch length 7.55cm, the RF voltage at least 32MV may be a good choice, and it sets an impedance threshold of 0.1Ω approximately. But for a shorter bunch length, it will set a more stringent requirement for longitudinal impedance. Therefore, longitudinal impedance of less than 0.1Ω needs to be studied. Besides, one way to increase the longitudinal impedance threshold is to increase the synchrotron tune spread, such as installing a higher harmonic cavity, so 800MHz RF cavity needs to be studied.
Injection: σz=9 cm, MV; Collision: σz=7.55 cm, MV
Longitudinal impedance less than 0.1 Ω needs to be studied!
In order to Landau damp longitudinal coupled-bunch instability, a large spread in synchrotron frequency inside the bunch is required.
One way: install a higher harmonic cavity (800MHz RF cavity needs to be studied)

10. Beam instability limits
Parameter design of SPPC longitudinal dynamics at 400 MHz
Beam instability limits
Transverse mode coupling instability (TMCI)
This instability arises when the relative tune shift of two adjacent head-tail modes equals the synchrotron tune.
The corresponding impedance threshold :
injection
collision
βav=290m
βav=208m
σz≥8 cm, Vrf≥20MV ZT≥ 6MΩ/m
σz=7.55 cm, Vrf ≥ 32MV ZT≥ 80MΩ/m
Shorter bunch length (luminosity upgrade)
more stringent requirement for transverse impedance.
Transverse impedance of less than 80MΩ/m needs to be studied !

11. Parameter design of SPPC longitudinal dynamics at 400 MHz
The momentum spread and longitudinal emittance dependence on RF voltage at given σz=7.55cm and frf=400MHz.
SPPC total RF voltage variable range: 20—60 MV (in collision)
The design values: 8.4eVs, 40MV
Momentum spread will be always less than 1x10-4!
Filling factor in momentum
Wbk Wb
-2π 2π
(small amplitude)
(accurate)

12. 800 MHz RF system
One way to relieve the restriction of beam instabilities is to use a higher harmonic RF system (800MHz).
MHz, qp > 0.9, i.e. significant beam loss. So the rms. bunch length will have to be reduced if using 800 MHz RF cavity.
qp ≤ > σz ≤ 5.56 cm;
the goal bunch length is set as σz =5.2 cm, then qp=76.2%;
Correspondingly: εs=6.4eVs, VRF=52MV, σδ=0.79x10-4
The corresponding impedance threshold:
Both transverse and longitudinal impedance thresholds have been improved to some extent at 800 MHz.

13. 400 MHz + 800 MHz dual harmonic RF system
Applied RF voltage is:
in (φ, ΔE/ω0) phase space, the Hamiltonian is
The potential well is
In normalised phase space , the Hamiltonian is
Bunch lengthening mode (BLM):
n=2, k=0.5, 2=0, for >0
Bunch shortening mode (BSM):
n=2, k=0.5, 2=, for >0
ABLM/Asingle~1.1478, ABSM/Asingle~0.7854;
The bucket height keeps the same.

14. 400 MHz + 800 MHz dual harmonic RF system
The BLM is used much more often:
For the same voltage and harmonic ratios it gives larger synchrotron frequency spread.
provides larger bucket area as well as reduced peak line density and therefore reduced space charge effects.
For SPPC, MHz dual harmonic RF system (BSM) can be considered.

15. Results and Comparison

16. Results and Comparison
Comparison among 400MHz, 800MHz, MHz(BSM)
Both transverse and longitudinal impedance thresholds have been improved to some extent at 800MHz.
Compared with 400 MHz, both the bucket area and bucket height for 800 MHz are reduced, but bunch filling factor is still within rational range. RMS bunch length shorter, Luminosity increased by 7%.
Beam stability can also be improved using 400MHz+800MHz dual harmonic RF system working on BSM mode because the average synchrotron tune and tune spread are increased.
Similarly, the bucket area is reduced to some extent and momentum filling factor increased, but bunch length is shorter.
RF power requirements and costs are not considered in detail at present.

17. Matching between injector chain
Matched bunch-to-bucket transfer
Requirements:
stationary bucket;
the physical aspect ratio of the bunch/bucket remains unchanged in sending and receiving accelerators.
Benifits:
More stable;
Time structure preserved;
H. Damerau. Timing, Synchronization & Longitudinal Aspects, CAS 2017.
Phase space aspect ratio:
the voltage ratio between RF systems:
Other considerations:
Longitudinal emittance is conserved during transfer to the next;
Furthermore, bunch length is conserved during transfer.
Longitudinal emittance may blow up during acceleration at each ring, but keep constant as far as possible;

18. SS ring RF consideration: Injection: 200MHz, extraction: 200/400MHz;
Or dual harmonic RF system

19. MSS ring
injection into SPPC
beam application

20. Summary Summary Next to do
SPPC longitudinal dynamics parameters for MHz+ MHz, + + dual harmonic RF system (BSM) have been considered and compared to achieve the required bunch length and stability.
Both transverse and longitudinal impedance threshold have been improved to some extent with 800 MHz RF system or 400MHz+800MHz dual harmonic RF system, meanwhile the bunch length becomes shorter, which is beneficial to luminosity.
IBS is not an extremely severe problem for SPPC;
Preliminary parameters design of SS and MSS have been given taking matching between injectors and IBS into consideration.
Finally, I make a summary. SPPC longitudinal dynamics parameters for these three RF cases have been considered and compared. Both transverse and longitudinal impedance threshold have been improved to some extent with 800 MHz RF system or 400MHz+800MHz dual harmonic RF system, meanwhile the bunch length becomes shorter, which is beneficial to luminosity. IBS is not an extremely severe problem for SPPC;
Next, I will study Beam instability criteria for dual harmonic RF system. Longitudinal dynamics of injector chain considering space charge and beam instability;
Next to do
Beam instability criteria for dual harmonic RF system.
Longitudinal dynamics of injector chain considering space charge and beam instability;
Special study for p-RCS is needed;

21. Thanks for your attention !