FCC-ee High Energy Booster

Collective effects studies updates

Adnan Ghribi¹
Barbara Dalena⁴

¹CNRS

Mauro Migliorati²
Quentin Bruant^3,4

²Univ. La Sapienza

Ali Rajabi³
Rainer Wanzenberg³

³DESY

Antoine Chance⁴
~

⁴CEA

November 13, 2023

Outline

Introduction
Input parameters
Booster parameters table, injection parameters, imepdance budget
Constraints on mismatched beams
Injecting from the LINAC and the SPS
Instabilities
Wakefield, bunch population, momentum compaction
Coming up next
Dev. tools, IBS, ramp-up

Introduction

Overview

FCC booster studies started in 2019 with only two people¹ ;
Since then, several people have joined the team ;
But we remain not enough FTE² to tackle all required tasks.

usage

\(\Rightarrow\) access to source codes and inputs/outputs :
.madx .seq .ele .json .py
\(\Rightarrow\) Access to the corresponding wiki entry.

Purpose

Explore effects usually hidden in classical optics design ;
Optimize the design of the high energy booster ;
Provide inputs to the different working-groups
ex. injectors ; RF ; collider ; vacuum ; costing, …

Input parameters

Booster parameters table, extraction, injection, hypothesis

Parameters table

Booster Baseline - Table 1.

Modes		\(z\)	\(h\)
\(\psi\)	deg	60	90	Phase advance
\(I5\)	\(10^{-11}\)	5.21	1.79	5th synchrotron integral
\(\alpha_c\)	\(10^{-6}\)	14.9	7.34	Momentum compaction
\(\delta_{p}\)	%	1.63	3.63	Momentum acceptance
\(Q_{x/y}\)	.225/.29	278/277	415/416	Horizontal tune
C	km	91.174		Circumference
\(\nu\)	MHz	800		RF Frequency
\(R_p\)	mm	25		Pipe radius

Parameters table

Extraction - Table 2.

Modes		\(z\)	\(w\)	\(h\)	\(t\bar{t}\)
\(E\)	GeV	45.6	80	120	182.5	Ext. energy
\(\varepsilon_{nx}\)	nm	0.29	0.81	0.63	1.45	Ext. horizontal eq. emittance
\(\varepsilon_{ny}\)	pm	0.53	1.62	1.25	2.9	Ext. vertical eq. emittance
\(\sigma_{z}\)	mm	4.38	3.55	3.34	1.94	Ext. bunch length
\(\sigma_{e}\)	\(10^{-3}\)	0.38	0.67	1.01	1.53	Ext. energy spread
\(V_{rf}\)	MV	49.48	458.6	2015	11533	RF voltage LINAC/SPS
\(N_{top/b}\)	\(10^{10}\)	2.14	0.87	0.69	0.93	Top-up particles/bunch

Parameters table

Injection - Table 3.

		LINAC	SPS
\(E\)	GeV	20	16	Injection energy
\(\varepsilon_{nx}\)	\(\mu\)m	10	190 \(\rightarrow\) 1000	Normalised horizontal emittance
\(\varepsilon_{ny}\)	\(\mu m\)	2 \(\rightarrow\) 15	4 \(\rightarrow\) 20	Normalised vertical emittance
\(\sigma_{z}\)	mm	1 \(\rightarrow\) 10	4	bunch length
\(\sigma_{e}\)	%	0.1 \(\rightarrow\) 0.5	0.4	energy spread
\(V_{rf}\)	MV	104.9/52.85	82.97/41.36	RF voltage 60º/90º
\(N_{max/b}\)	\(10^{10}\)	2.45		Maximum particles per bunch

Parameters sweep

While \(\sigma_z\) sets the cavities RF voltage at extraction and \(\varepsilon\) is a fixed target to achieve, many input parameters remain variables and need(-ed) to be explored :

Injection : Energy, emittance, bunch length, energy acceptance
Booster optics : Momentum compaction, Tune, I5 ;
- Baseline : PA31 collider optics
- HFD Alternative optics ¹
Booster elements and geometries : impedance budget, beam pipe diameter/material, number of cavities.

Spoiler

The driving case

While the the \(z\) mode might seem like the worst case scenario for collective effects in the collider (\(z\) mode)¹, The \(t\bar{t}\) mode at injection seems to be the driving case for the booster.

Assumptions

Only resistive wall is included (impedance budget ongoing)²
Longitudinal impedance and wake potential of a 0.4 mm Gaussian bunch used as Green function in beam dynamics simulations ;
Single bunch instabilities.

Constraints on mismatched beams

Injecting from the LINAC and the SPS

Injection from the LINAC

Results

Copper beam pipe with a radius of 25 mm ;
90º phase advance booster lattice ;
Other parameters as show in Table 3
LINAC : E=20Gev, … ;
Variying \(\sigma_z\) and \(\varepsilon_{y}\) at injection.

\(\Rightarrow\) Booster parameters seem robust to mutliple injection of \(\sigma_z\) and \(\varepsilon_{n,y}\).

Injection from the SPS

Results

Copper beam pipe with a radius of 25 mm ;
90º phase advance booster lattice ;
Other parameters as show in Table 3
SPS : E=16Gev, … ;

\(\Rightarrow\) No issues at lower number of turns and transverse blow of the beam appears.

Instabilities

Wakefield, bunch population, momentum compaction

Beam pipe material : Copper vs Stainless Steel

Hypothesis

Only resistive wall impedance contribution ;
Pipe diameter = 50 mm ;
Injection from the LINAC¹
with \(\sigma_z\) = 1 mm and \(\varepsilon_{ny}\) = 8 \(\mu m\). ;
90 deg phase advance booster lattice².

Beam pipe material : Copper vs Stainless Steel

Results : RF bucket

Beam pipe material : Copper vs Stainless Steel

Results :Bunch length

Let us focus on the copper case.

\(\Rightarrow\) More than x3 larger bunch length after SR damping.

TMCI and Microwave Instabilities

Results : at nominal parameters for the 90 deg phase advance lattice

\(\Rightarrow\) Tranverse Mode Coupling Instabilities.

\(\Rightarrow\) Microwave Instabilities.

TMCI and Microwave Instabilities

Bunch population scan : Analysis approach

Explore momenta¹

Sweep bunch population ;
FFT with hamming window ;
2D stack spectra.

TMCI and Microwave Instabilities

Bunch population scan : injecting from the LINAC

90 deg phase advance, copper, d=50mm

\(\Rightarrow\) Threshold below the nominal 2.4e10 particles.

TMCI and Microwave Instabilities

Bunch population scan : injecting from the SPS

90 deg phase advance, copper, d=50mm

\(\Rightarrow\) Threshold is even lower than for the LINAC.

TMCI and Microwave Instabilities

Bunch population scan : injecting from the LINAC

60 deg phase advance, copper, d=50mm

\(\Rightarrow\) TMCI cured ! Momentum compaction ?

TMCI and Microwave Instabilities

Momentum compaction scan : What if we scan \(\alpha\) ?

90 deg phase advance, copper, d=50mm

\(\Rightarrow\) We need to increase the momentum compaction.

TMCI and Microwave Instabilities

What if we increase the pipe diameter for a stainless steel chamber ?

Applying a simple scaling law

\[ \left.\begin{matrix} Z_{\parallel}(\omega_n)_{res} = (1-i)\frac{\overline{R}}{r_w\sigma\delta_{skin}}|_{res} \\ Z_{\perp}(\omega)_{res} = \frac{2c}{\omega r^2_w}Z_{\parallel}(\omega)|_{res} \end{matrix}\right\} \Rightarrow \begin{matrix} Z_{\parallel}(r_2) = Z_{\parallel}(r_1)\frac{r_1}{r_2} \\ Z_{\perp}(r_2) = Z_{\perp}(r_1)\left(\frac{r_1}{r_2}\right)^3 \end{matrix} \]

TMCI and Microwave Instabilities

What if we increase the pipe diameter for a stainless steel chamber ?

Hypothesis :

We inject from the LINAC¹ with :
- \(\sigma_z = 4~mm\), \(\varepsilon_{ny} = 10~\mu m\) and \(N_p=2.43~10^{10}\) particles.
We consider only wake potential with transverse and longitudinal components scaled according to the previous scaling law.

TMCI and Microwave Instabilities

What if we increase the pipe diameter for a stainless steel chamber ?

\(\Rightarrow\) We would need to increase the pipe diameter from 50 mm to 84 mm.
What would be the real cost/performance compromise ?

Takeaway on instabilities

Booster seems robust to beam mismatch at injection,
However, instabilities at higher number of turns are critical :
- Beam pipe size and material ;
- Lattice parameters (momentum compaction, …).
These instabilities are under-estimated because :
- Impedance budget is incomplete ;
- Interplay between several collective effects is not yet taken into account.

Coming up next

Dev. tools, IBS, ramp-up

Todos

Adding other impedance contributors
(ex. collimators, bellows, cavities, …) ;
Improving impedance model for the beam pipe ;
Simulating alternative optics¹ ;
Cycling, energy ramp-up and constraints of the interplay of collective effects on equilibrium emittances.

Cooking at the moment

Moving from PyHEADTAIL to XSuite :
- Benchmarking with synchrotron radiation ;
- Adding wakefield ;
- Adding IBS ;
- Dynamic aperture estimation with collective elements.

Going beyond state of the art

Preparing the active learning framework :
- Smart/Fast parametric sweep ;
- Interface layer - compatible with PyHEADTAIL, XSuite, PTC, …
- Intersects the artifact¹ project in preparation for a 2024 Horizon Europe call.

Conclusion/Summary

The stage is set but a lot of things are still to be done :

Injection constraints to the booster ;
Effects of optics and beam pipe material/dimension on instabilities ;
Realistic impedance budget ;
Interplay of wakefield, SR and IBS ;
Cycling and energy ramp-up.

Thank you

Questions ?

References

Métral, E, and M Migliorati. 2020. “Longitudinal and Transverse Mode Coupling Instability: Vlasov Solvers and Tracking Codes.” Physical Review Accelerators and Beams 23 (7): 071001.

Wiedemann, Helmut. 2015. Particle Accelerator Physics. Springer Nature.