Lecture 25 - E. Wilson - 04/21/23 - Slide 1

Lecture 6

ACCELERATOR PHYSICS

HT8 2010

E. J. N. Wilson

http://cas.web.cern.ch/cas/Loutraki-Proc/PDF-files/I-Schindl/paper2.pdf

Lecture 25 - E. Wilson - 04/21/23 - Slide 2

Summary of last lecture – Instabilities I

1. General Comment on Instabilities 2. Negative Mass Instability 3. Driving terms (second cornerstone) 4. A cavity-like object is excited 5. Equivalent circuit 6. Above and below resonance 7. Laying the bricks in the wall (row 1) 8. Laying the bricks in the wall (row 2) 9. By analogy with the negative mass

Lecture 25 - E. Wilson - 04/21/23 - Slide 3

1. A short cut to solving the instability 2. An imaginative leap 3. The effect of frequency shift 4. Square root of a complex Z 5. Contours of constant growth 6. Landau damping 7. Stability diagram 8. Robinson instability 9. Coupled bunch modes 10 Microwave instability

Instabilities II

Lecture 25 - E. Wilson - 04/21/23 - Slide 4

A short cut to solving the instability

From theory of synchrotron motion: Recall the effect of a voltage of a cavity

Assume the particles have initially a small phase excursion about s = 0

or

where

is the synchrotron frequency and is the revolution frequency.

s2

hV0

2E02

0

2

0

0sinsin2 02

20

sV

fE

dtd

02 02

20

eV

f

E

02 s

Lecture 25 - E. Wilson - 04/21/23 - Slide 5

s2

hV0

2E02

0

2

An imaginative leap

Put in the volts induced by the beam in the cavity instad of the volts imposed from outside

i reflects the fact that, unlike the RF wave the volts induced by a resistive load cross zero 90 degrees after the passage of the particle

This bypasses much analysis and gives the right formula for the frequency shift.

Voh inZI0

h n / 0

2 i0

2nI02 2E

Z iZ

Lecture 25 - E. Wilson - 04/21/23 - Slide 6

The effect of frequency shift

Remember that a force driving an oscillator may be written on the right hand side:

Alternatively it can be assimilated into the frequency

where:

if is positive and Z pure imaginary (reactive) is real and there is just a change in frequency. if Z has a resistive component this gives an imaginary part to Imaginary frequencies can signal exponential growth

2 i0

2nI02 2E

Z iZ

2

iZ

tF 20

020

Lecture 25 - E. Wilson - 04/21/23 - Slide 7

Square root of a complex Z

Be careful to first multiply Z by i and then take the square root

There will be a locus in (X,Y) space where the imaginary part is constant which will be a contour of constant growth rate Suppose the solution to the differential equation is

We must solve for constant

Eliminate

0 e it 0 e i i t

0 et e it

iZ i X iY

2 i 2

2 2 2 2i

2 iZ Y iX

X 2

X2

Y 2X2

2 2 2

X 2 Y / 2 / 2X 2 Y / 2 / 2

Lecture 25 - E. Wilson - 04/21/23 - Slide 8

Contours of constant growth

Changing the growth rate parameter we have a set of parabolas

X 2 Y / 2 / 2X 2 Y / 2 / 2

Lecture 25 - E. Wilson - 04/21/23 - Slide 9

Landau damping – the idea

Two oscillators excited together become incoherent and give zero centre of charge motion after a number of turns comparable to the reciprocal of their frequency difference

Lecture 25 - E. Wilson - 04/21/23 - Slide 10

Landau Damping – the maths

N particles (oscillators), each resonating at a frequency between 1 and 2 with a density g()

g()d 11

2

Response X of an individual oscillator with frequency to an external excitation with

S N

20

idg

d

deit

1

2

Coherent response of the beam obtained by summing up the single-particle responses of the n oscillators

normalization

External excitation is inside the frequency range of the oscillators The integral has a pole at

ωΩωΩ

1

ωΩ

1X

22

eit eit

External excitation is outside the frequency range of the oscillators

No damping

Landau damping

Lecture 25 - E. Wilson - 04/21/23 - Slide 11

Stability diagram

Keil Schnell stability criterion:

222

0

FWHHo pp

IcFm

nZ

Lecture 25 - E. Wilson - 04/21/23 - Slide 12

Single Bunch + Resonator: “Robinson” Instability

A single bunch rotates in longitudinal phase plane with s:

its phase and energy E also vary with s

“Dipole” mode or “Rigid Bunch”

mode

Bunch sees resonator impedance at r 0

Whenever E>0:• increases (below transition)• sees larger real impedance R+

• more energy taken from beam STABILIZATION

Whenever E>0:• decreases (above transition)• sees smaller real impedance R+

• less energy taken from beam INSTABILITY

r

r

UNSTABLE STABLE

Lecture 25 - E. Wilson - 04/21/23 - Slide 13

Longitudinal Instabilities with Many Bunches

2n

M, 0 n M 1 M modes

Fields induced in resonator remain long enough to influence subsequent bunches Assume M = 4 bunches performing synchrotron oscillations

Four possible phase shifts between four bunches

M bunches: phase shift of coupled-bunch mode n:

Coupled-Bunch

Modes n

Lecture 25 - E. Wilson - 04/21/23 - Slide 14

Longitudinal Microwave Instability

• High-frequency density modulation along the bunch• wave length « bunch length (frequencies 0.1-1 GHz)• Fast growth rates – even leptons concerned • Generated by “BROAD-BAND” IMPEDANCE

Z() Rs

1 iQ2 r

2

r

1 Q2 r

2

r

2

Q 1

r 1 GHz

All elements in a ring are “lumped” into a low-Q resonator yielding the impedance

r

0s

r

0

0

s

r

s

ω

nω

Q

Ri

ω

ω

ω

ω

Q

Ri

Qω

ωRiωZ

For small and

n

Lω

RQ

r

s

00

Lωn

Z

“Impedance” of a synchrotron in •This inductive impedance is caused

mainly by discontinuities in the beam pipe • If high, the machine is prone to instabilities• Typically 20…50 for old machines• < 1 for modern synchrotrons

Lecture 25 - E. Wilson - 04/21/23 - Slide 15

1. A short cut to solving the instability 2. An imaginative leap 3. The effect of frequency shift 4. Square root of a complex Z 5. Contours of constant growth 6. Landau damping 7. Stability diagram 8. Robinson instability 9. Coupled bunch modes 10 Microwave instability

Summary of Instabilities II