Collective Effect II Giuliano Franchetti, GSI CERN Accelerator – School Prague 11/9/14G. Franchetti1

Type of fields 11/9/14G. Franchetti2 Collective Effects Collective Effects ?

Robinson Instability 11/9/14G. Franchetti3

Robinson Instability 11/9/14G. Franchetti4 It is an instability arising from the coupling of the impendence and longitudinal motion real part control the energy loss in a bunch the revolution frequency controls the impendence R0R0 RF slower particle (if below transition) change particle energy change revolution frequency

The coupling of two effects 11/9/14G. Franchetti5 Energy loss due to impedance via the longitudinal dynamics Change of revolution frequency because of the impedance Z(ω)

Below transition 11/9/14G. Franchetti6

Below transition 11/9/14G. Franchetti7 Energy lost in one turn

Below transition 11/9/14G. Franchetti8 Where is given by the impedance

Below transition 11/9/14G. Franchetti9 energy lost per particle for non oscillating bunch

Below transition 11/9/14G. Franchetti10 In one turn energy is lost but compensated by the RF cavity frequency

Below transition 11/9/14G. Franchetti11 cavity frequency revolution frequency changes larger smaller

Below transition 11/9/14G. Franchetti12 Energy lost  increase ω  increase Z r  increase energy loss !!!

Below transition 11/9/14G. Franchetti13

Below transition 11/9/14G. Franchetti14 Stable

Below transition 11/9/14G. Franchetti15 Stable

Below transition 11/9/14G. Franchetti16 Unstable

Below transition 11/9/14G. Franchetti17 Unstable

Summary of the reasoning 11/9/14G. Franchetti18 below transition above transition stable unstable stable

More complicated 11/9/14G. Franchetti19 E < E T No Energy Loss: RF give the same energy lost by the impedance

11/9/14G. Franchetti20 set the cavity frequency here Remember that energy lost is V*I

Source of difficulty 11/9/14G. Franchetti21 E < E T Energy gain Energy lost Impedance effect 

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Still we neglect something 11/9/14G. Franchetti23 E < E T position of the bunch at turn “k” Q s is the synchrotron tune

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Current 11/9/14G. Franchetti25 The bunch current can be described by 3 components with frequency very close side band this component is out of phase (because is a sin) side band

11/9/14G. Franchetti26 That means that the energy loss due to the impedance has to be computed on the 3 currents… Voltage created by the resistive impedance Main component 1 st sideband 2 nd sideband

11/9/14G. Franchetti27 But Prosthaphaeresis formulae

11/9/14G. Franchetti28 Main component 2 nd sideband 1 st sideband Therefore the induced Voltage depends on Voltage created by the resistive impedance

Energy lost in one turn 11/9/14G. Franchetti29 energy lost per particle per turn this term can give rise to a constant loss, or a constant gain of energy

In terms of the energy of a particle 11/9/14G. Franchetti30 This is a slope in the energy, and the sign of the slope depends on and

The longitudinal motion now! 11/9/14G. Franchetti31 Ifthere is a damping Ifthere is an instability Robinson Instability

11/9/14G. Franchetti32 below transition above transition stable unstable stable Robinson Instability

Longitudinal space charge and resistive wall impedance 11/9/14G. Franchetti33

Space charge longitudinal field 11/9/14G. Franchetti34 dz RwRw ErEr ErEr EzEz EwEw

11/9/14G. Franchetti35 For a KV beam Electric Field

11/9/14G. Franchetti36 Therefore

11/9/14G. Franchetti37 Magnetic Field

11/9/14G. Franchetti38 from the equation of continuity again we find the factor! Maxwell-Faraday Law

Space charge impedance 11/9/14G. Franchetti39 Local density

11/9/14G. Franchetti40 Perfect vacuum chamber E zw = 0

Resistive Wall impedance 11/9/14G. Franchetti41 dz RwRw ErEr ErEr EzEz EwEw Do not take into account B E w = E z

11/9/14G. Franchetti42 pipe wall currents skin depth Beam on axis Wall currents are related to the electric field by Ohm’ law The thickness of the wall currents is called skin depth

11/9/14G. Franchetti43 Impedance of the surface (pipe) Longitudinal impedance (beam) J z (0) x y z μ,σ

Transverse impedance 11/9/14G. Franchetti44

Origin 11/9/14G. Franchetti45 v t Beam passing through a cavity on axis

Origin 11/9/14G. Franchetti46 t Beam passing through a cavity off-axis v

But the field transform it-self ! 11/9/14G. Franchetti47 E

11/9/14G. Franchetti48 B

11/9/14G. Franchetti49 E

11/9/14G. Franchetti50 B

Effect on the dynamics 11/9/14G. Franchetti51 The dynamics is much more affected by B, than E because this speed is high

The beam creates its own dipolar magnetic field ! 11/9/14G. Franchetti52 (dipolar errors create integer resonances…. we expect the same…)

Transverse impedance 11/9/14G. Franchetti53 Definition of longitudinal impedance (classical) Impedance System

11/9/14G. Franchetti54 source of the effect For a displaced beam It means that in the equation of motion we have to add this effect Effect this field acts on a single particle

11/9/14G. Franchetti55 In the time domain therefore for a weak effect or distributed we find

11/9/14G. Franchetti56 Now the situation is the following: System But is like a “strange” voltage it depends on frequency

Transverse beam coupling impedance 11/9/14G. Franchetti57 now the question is what is ω ?

What is it ω ? 11/9/14G. Franchetti58 It is given by the fractional tune, as this is the frequency seen in a cavity Example:Q = 2.23fractional tuneq = 0.23 beam position seen by the cavity turns

B-field induced by beam displacement 11/9/14G. Franchetti59 From electric field at the position of beam x 0 is Longitudinal impedance

11/9/14G. Franchetti60 Transverse impedance The magnetic field comes from Maxwell taking

11/9/14G. Franchetti61 pipe wall currents skin depth Beam x off axis

11/9/14G. Franchetti62 xbxb image charge Therefore the field on the beam is (for small x b /r p )

11/9/14G. Franchetti63 More charges here Less charges here beam

11/9/14G. Franchetti64 Transverse resistive Wall impedance

Transverse instability 11/9/14G. Franchetti65

Coasting beam instability 11/9/14G. Franchetti66 Force due to the impedance (in the complex notation) Equation of motion of one particle for a beam on axis Equation of motion of a beam particle when the beam is off-axis

Collective motion 11/9/14G. Franchetti67 On the other hand the beam center is If all particles have the same frequency, i.e. each particle experience a force with therefore then

11/9/14G. Franchetti68 We can define a coherent “detuning” because this is a linear equation

11/9/14G. Franchetti69 that is But now is a complex number !! Solution

11/9/14G. Franchetti70 Is the growth rate of the transverse resistive wall instability This instability always take place  Landau damping Instability suppression

An important assumption 11/9/14G. Franchetti71 We assumed that all particles have the same frequency so that This assumption means that each particle of the beam respond in the same way to a change of particle amplitude Coherent motion drive particle motion, which is again coherent

Chromaticity ? 11/9/14G. Franchetti72 What happened if the incoherent force created by the accelerator do not allow a coherent build up Momentum spread dp/p

11/9/14G. Franchetti73 chromaticity one particle with off-momentum dp/p has tune If each particle of the beam has different dp/p then the force that the lattice exert on a particle depends on the particle !

Incoherent motion damps x b 11/9/14G. Franchetti74 Equation of motion without impedances Motion of center of mass as an effect of the spread of the frequencies of oscillation The momentum compaction also provides a spread of the betatron oscillations

11/9/14G. Franchetti75 Example: N. particles = 5 dq/q = 0

11/9/14G. Franchetti76 Example: N. particles = 5 dq/q = 5E-3

11/9/14G. Franchetti77 Example: N. particles = 5 dq/q = 1E-2

11/9/14G. Franchetti78 Example: N. particles = 5 dq/q = 2.5E-2 damping of x b

But incoherent motion reduces x b 11/9/14G. Franchetti79 Example: these are 5 sinusoid with amplitude linearly growth

11/9/14G. Franchetti80 Example: now a spread dq/q of 1E-2 is added to the 5 curves The center of mass growth slower

11/9/14G. Franchetti81 Example: now a spread dq/q of 1E-2 is added to the 5 curves the spread of the particles damps the oscillations of the center of mass  the instability cannot develop

Situation 11/9/14G. Franchetti82 Coherent effect Incoherent effect Growth rateDamping rate The faster wins

instability of a single bunch 11/9/14G. Franchetti83 Q = 2.23 Q = 2.00 Q = 1.77 No oscillations  Example beam position at the cavity

11/9/14G. Franchetti84 Q = 2.23 Q = 2.00 Q = 1.77 Slow side band Fast

behavior of the field in the cavity 11/9/14G. Franchetti85 T r = time of oscillation of the field in the cavity

Cavity tuned upper sideband 11/9/14G. Franchetti86

Cavity tuned upper sideband 11/9/14G. Franchetti87

11/9/14G. Franchetti88 As for the Robinson Instability

Negative mass instability 11/9/14G. Franchetti89 x Above transition z Example with 2 particles ! Repulsion forces from Coulomb Repulsion forces from Coulomb reference frame of synchronous particle

Negative mass instability 11/9/14G. Franchetti90 x Above transition z Example with 2 particles ! Repulsion forces from Coulomb Repulsion forces from Coulomb reference frame of synchronous particle gain speed revolution time longer lose speed revolution time shorter

Negative mass instability 11/9/14G. Franchetti91 x Above transition z Example with 2 particles ! reference frame of synchronous particle

Negative mass instability 11/9/14G. Franchetti92 x Above transition z repulsive forces attract particles as if their mass were negative perturbation x z growth

Summary 11/9/14G. Franchetti93 Robinson instability Longitudinal space charge and resistive wall impedance Transverse impedance Transverse instability Landau damping Single bunch instability Negative mass instability