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Beam End Erosion and Induction Cell

Brian Beaudoin

06/23/08

Institute for Research in Electronics and Applied Physics

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• Longitudinal stability of beams is important to prevent blow-up

• Designing a system to meet these specifications is already a challenge in itself

• Confine beam to a square pulse so we don’t fooltransverse measurements with an extracted beam that does not have the same current density.

Motivation

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Outline

1. Induction focusing versus RF focusing.

2. UMER System overview, parameters and operating ranges.

3. End erosion of the beam as it circulates.

a. One-dimensional theory and the cusp point. b. MATLAB code for example end-erosion

calculations.

4. Induction cell and HV Modulator operation.

5. Ear-field focusing and timing involved in setting fields. Results and comparison to one-dimension model

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RF Focusing vs. Induction Focusing

• In UMER, sinusoidal fields can not be used to

focus a square bunch.

• With RF focusing, self-fields of the beam can

distort the applied fields to the cavity

• For long pulses like in UMER, 100ns, need

very low frequency current sources to drive

an RF cavity, which means large cavities

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System Overview

Dipoles

Beam

RC3

RC5

RC6

RC9

RC11

RC12

Velocity = 0.2c

BPMs

QuadsQuads

Induction Cell

System ParametersBeam Length 20-100nsCirculation Time 197nsBeam Energy 10keVBeam current 0.6 - 100mABeam radius 0.25 - 10mm

Current Monitor

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Beam-End Expansion and

1D Model

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� Cs is the sound speed, the speed of the wave moving on the flat top region of the beam.

� γγγγ is the Lorentz factor.

� I the main beam current.

� In the linear regime, waves travel undistorted so that the shape is preserved.

� In lab, keep the amplitude of perturbation small.

Sound Speed and Perturbations

54

s

o o o

egIC

mvπε γ=

CurrentProfile

VelocityProfile

CurrentProfile

VelocityProfile

Fast Wave vo+Cs

Slow Wave vo-Cs

Density

Perturbation

Energy

Perturbation

( ),

o s o s

z zu z t Ff t Gf t

v C v C

= − + −

+ −

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Geometry Factor

� g the geometry factor is a factor determined by the ratio of the beam size to pipe size.

� α α α α is a correction factor that is 0 for a space charge dominated beam, and 0.5 for an emittance dominated beam.

� Don’t forget if quads are running at reduced currents. What happens to beam size?

a b

PipeBeam

2lnb

ga

α

= +

-Kishek / O’Shea class notes and Reiser’s Book 1st ed. p.505

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Longitudinal Expansion

• Cusp Point

– As Beam circulates in the lattice, the head and tail will eventually touch each other.

• Head/Tail Energy

– Head will accelerate from

the main beam at a ∆E and tail will decelerate at the same amount.

Velocity Profile

Cusp

Main Beam Velocity

∆v

∆v

Head

Tail

vo vo

Min Width

Max Width

Lin

e c

harg

e d

en

sit

y

Time

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Cusp Point

-Wang D. X. PhD Thesis, “Experimental Studies of Longitudinal Dynamics of Space-Charge Dominated Electron Beams", 1993

z = 0 z

ττττo

Initial Beam

vo

Reduction in linecharge density

vo

t< τ< τ< τ< τo

Cusp

t= τ= τ= τ= τo

Turn n Turn n + 1

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1D MATLAB Code to Study End-Erosion

Important Part of Code

Vary Parameters and

see what happens

Parameters to Vary

Energy,

Current,

Beam Length

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M-File Output

Velocity Profile

Current Profile

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M-File Output

Simulation of Multi-Turn

Invalid past “simple wave regime”

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My 1-D CalculationsFor 7mA Beam measured at RC10

( )2s o s

E mC v C∆ = +

• Initial beam length when formed is 5.93m.

• The rate of expansion for the 7mA beam is 5.5nsec / Turn, assuming no loss in current. So that by the 9th turn, the beam head and tail are touching each other, for 83% quads.

∆E = 549eV

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Induction Cell

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Induction Cell Operation

E�

HV Modulator

IRe- Beam

2

sLRZ

sL R s CRL=

+ +

109

010

20

Mag (

dB

)

104 105 107Freq (Hz)

106 108

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Multiple Cells and Orientations

Beam

Beam

BeamSingle Core

Cell

Two Cells back to back

Dual Core Cell

Beam

Double Core Cell

Wire

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Bs of CMD5005

Saturation Limits and Reset Pulses

2

3000

10

22.82

pulse pulse core

pulse

pulse

core

V t BA

V volts

t ns

A cm

∆ = ∆

=

∆ =

=

131B gauss∆ =

3200s

B gauss=

Flux swing

Reset pulse using a

bipolar HV modulator

0

Minimize the dB/dt

0

Accel

Reset or Decel

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High Voltage Modulator

• Pulse fields have a fixed on-time of 10-15nsec with a FWHM of ~8nsec.

• Fields are purely composed of fixed rise times so ∆t can not be changed for this pulser.

• Amplitude can vary from 0 – 3kV.

• Both positive and negative pulses must be “on” to reset the core.

Error will

show up

later

Beam Bunch

Energy Storage

Energy Storage

BehlkeSwitch

BehlkeSwitch Load

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Longitudinal Focusing

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Focusing Experiment

RC10

Current

Monitor

RC4

Induction

Cell

Time

Gap V

oltage (

volts)

0

100ns

� The compensation voltage is set

based on the one dimensional

longitudinal expansion calculations for

a rectangular bunch.

� Ear-fields are applied for a single turn.

� Pencil Beam = 176eV

Time

Curr

ent

Max Width

Min

Width