Beam Measurements and Instrumentationostroumo/MSU/Lectures...Beam halo with wire scanner...

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Steve Lidia National Superconducting Cyclotron and Facility for Rare Isotope Beams Laboratory Michigan State University PHY862 Accelerator Systems Beam Measurements and Instrumentation 1. Interactions and Sensors

Transcript of Beam Measurements and Instrumentationostroumo/MSU/Lectures...Beam halo with wire scanner...

Page 1: Beam Measurements and Instrumentationostroumo/MSU/Lectures...Beam halo with wire scanner measurements Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation

Steve LidiaNational Superconducting Cyclotron and Facility for Rare Isotope Beams Laboratory

Michigan State University

PHY862 Accelerator SystemsBeam Measurements and Instrumentation

1. Interactions and Sensors

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β€’ Measurements, diagnostics, instrumentation

β€’ Interactions and sensors

β€’ Beam generated signals

β€’ Diagnostic architecture

β€’ References and exercises

Introduction

4-Jaw Collimator

300 W Faraday Cup

Allison Scanner (x,y)

Viewer

L Profile Monitor

Chopper

Apertures

Pepper Pot

Viewer

FC100

Flapper

Profile

Monitor

L Profile

Monitor

Intensity

Reducing ScreensFC100

Viewer

S Profile

Monitor

BCM

FC100

FFCS Profile

Monitor

BCM

BPM

x/y Slits

FC100

FFCBCM

FC100

2-Jaw Collimator

Viewer

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Beams are composed of 100’s to 10**8’s of individual particles.

Getting them all from point A to point B can be a challenge

We need to perform certain operations on the beam

β€’ Tuning, optimization of beam quality

β€’ Targetry, beam collisions

β€’ Monitoring, stability, minimizing losses, machine protection

Role of diagnostics and instrumentation

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β€’ Measurementsβ€’ Incorporate (incomplete) knowledge of lattice, beam dynamics

β€’ Point vs. Distributed (lattice-dependent) measurements

β€’ Diagnostics are phenomena or techniques involved in performing a measurement

β€’ Direct vs indirect measurements

β€’ Correlations – use known dependencies

β€’ Instrumentation is the set of particular devices used in the execution of the measurement

β€’ Set of beam sensors, signal transmission lines, data acquisition and reporting systems, controls and feedback

Measurements, diagnostics, and instrumentation

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Measurement Diagnostic Instrumentation

Beam currentBeam wall return

currentAC Current Transformer + electronics

Beam positionBeam E-field

distribution @ wallsCapacitive pickups + electronics

Beam emittance (1DoF) Quad scanQuadrupole magnet + viewscreen + camera

Diagnostics vs. Instrumentation examples

Particular diagnostic/instrumentation methodologies rely on particular types of beams and beam parameters/lattice functions, available beamline space and shielding, required measurement accuracy and precision, cost, . . .

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Typical beam measurementsMeasurements and associated diagnostics and instrumentation are specific to

β€’ Geometry of beamlineβ€’ linac, synchrotron booster, storage ring,

analyzing beamline, injector, final focus, etc.)

β€’ Particle typeβ€’ Hadron, lepton, neutron, neutral atom,

rare isotope

β€’ Beam energy

β€’ Beam intensity

β€’ Beam time structure

β€’ . . .

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SIS-18 Synchrotron at GSI: diagnostic suite

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β€’ Charge/current, charge states, mass states

β€’ Beam energy

β€’ Beam position, orbit, tune

Beam parameters [1] – 1st Order Moments

LHC Day 1 Beam Position

Schottky spectra of stored and cooled rare isotopes from 197Au79+. Spectra of 16th

revolution harmonic.

(Schlitt, et al., Nucl. Phys A626 (1997), 315.)

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β€’ Profile (transverse, longitudinal), envelope, energy spread

β€’ Phase space density, emittance measures

β€’ Beam halo - transverse, longitudinal

Beam parameters [2] – 2nd Order (+ higher) Moments

Beam halo with wire scanner measurements

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β€’ Single bunch vs. many bunch measurements

β€’ Time domain vs. frequency domain

Beam parameters [3] – Bunch Trains

LHC Turn-by-turn Bunch Intensity (300 turns)

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β€’ Betatron tune Qx, Qy

Lattice parameters

β€’ Dispersion function

Large energy spread Nominal energy spread

BPMs @ A, B, C, D

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Lattice parameters Matched distributions Mismatched distributions

Power density (Gy/s) at 200 MeV

β€’ Loss distributions in SC RF modules

β€’ Beam envelope matching to lattice

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β€’ Emittance growth for mismatched beams

β€’ Instabilities

β€’ Electron cloud effects

Dynamic parameters

Direct phase measurement in resonant BPM configuration (DeSantis, et al, PAC09 TH5RFP071)

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β€’ Physics of beam-sensor interactionsβ€’ EM, Nuclear, AMO, Solid Stateβ€’ Charge and mass interceptionβ€’ Capacitive, inductive, resonant, thermal

field sensingβ€’ Secondary radiation fields

β€’ Mechanical design β€’ Thermal, structural, vacuum, actuator

β€’ Electrical designβ€’ Grounding/shieldingβ€’ HV bias and insulation

β€’ Electronicsβ€’ Signal acquisition, conditioning,

processingβ€’ Noise, bandwidth, sensitivity,

response time

Diagnostic/instrumentation design elements

System Interface Diagram

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Faraday cups

Negative HV Bias

Suppressor CollectorV

I/VIncoming ions

e- Transimpedanceamplifier

β€’ Fully intercepting charge measurement

β€’ Sensitivity to 10 pC. ~100 Hz BW (β€˜slow’, deep cup)

β€’ Beam charges impinge on Collector, are collected by electronics

β€’ Suppressor negatively biased to repel electrons

β€’ Design is to prevent escape of secondary electrons

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β€’ Based on scintillator and cameraβ€’ Coated screen (reflection mode)

β€’ Solid, thick (100 mm) scintillator (transmission mode)

β€’ Scintillator can be single- or multi-crystalline, or sintered powder

β€’ Direct 2D measurementβ€’ Direct digital output

β€’ Video out and frame grabber

β€’ Resolution depends on scintillator material (grains), CCD size, optics

β€’ Amplitude response depends on field flatness, scintillator dose and aging effects, temperature

ViewersGSI linac, 4 MeV/u, low current, YAG:Ce

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β€’ Analyzes intensity J(x or y) at first slit

β€’ Applied voltage across plates + drift + exit slit analyzes momentum

β€’ Reconstructs phase space density

Allison Scanner

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Pepperpots, slits, and pinholesβ€’ Devices scan a 1D or 2D beam distribution

β€’ Analyze intensity J(x,y)

β€’ Analyze transverse velocity over drift

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Emittance analysis

x

x’

𝛾π‘₯2 + 2𝛼π‘₯π‘₯β€² +𝛽π‘₯β€²2 = πœ€

(x0, -(a/b) x0)

2Dx0’

Dx0’=[e/b – x02/b2]1/2

2a

Ξ£4 =

π‘₯π‘₯ π‘₯π‘₯β€² π‘₯𝑦 π‘₯𝑦′

π‘₯π‘₯β€² π‘₯β€²π‘₯β€² π‘₯′𝑦 π‘₯′𝑦′

π‘₯𝑦 π‘₯′𝑦 𝑦𝑦 𝑦𝑦′

π‘₯𝑦′ π‘₯′𝑦′ 𝑦𝑦′ 𝑦′𝑦′

=Ξ£π‘₯ 𝐢

𝐢𝑇 Σ𝑦

det Ξ£π‘₯ = Ξ£π‘₯ = π‘₯π‘₯ π‘₯β€²π‘₯β€² βˆ’ π‘₯π‘₯β€² 2 = νœ€π‘₯2

𝑓𝑔 = 𝑖 πœŒπ‘–π‘“π‘–π‘”π‘– 𝑖 πœŒπ‘–

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β€’ Beam positions can be monitored using a 4-electrode array of capacitive pickups on the beampipecircumference.

β€’ Various geometries are employed for sensitivity, compactness, protection from intense radiation

Beam Position Monitors

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β€’ Positions are estimated from the normalized intensities using the difference over sum algorithm

βˆ†π‘₯ =1

𝑆π‘₯

𝑉2βˆ’π‘‰4

𝑉2+𝑉4, βˆ†π‘¦ =

1

𝑆𝑦

𝑉1βˆ’π‘‰3

𝑉1+𝑉3

β€’ Position sensitivities are proportionality constants between beam displacement and signal strength.

β€’ 𝑆π‘₯ =𝑑

𝑑π‘₯

βˆ†π‘₯

Ξ£π‘₯

%

π‘šπ‘šπ‘œπ‘Ÿ 𝑆π‘₯ =

𝑑

𝑑π‘₯π‘™π‘œπ‘”

𝑉2

𝑉4

𝑑𝐡

π‘šπ‘š

β€’ Offset displacements also occur and must be measured and calibrated.

β€’ Button-button capacitive coupling introduces frequency dependent offset and sensitivity variation.

β€’ Intensities at each button can be calculated from the transfer impedance, using the electrode surface area.

BPM Position Algorithm

Q1

Q2

Q3

Q4 x

y

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Profile monitorsβ€’ Interceptive diagnostic onto W, C, Cu-Be wires

β€’ Wide range of wire based geometries

β€’ Biased wire to discourage (or encourage) secondary e-’s

β€’ Slit + Faraday cup

Linear actuated

Rotating wire

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β€’ We observe day-to-day variation of transverse beam parametersβ€’ Two most significant factors are: ECR setting and beam center matching to the RFQ

Reconstruct beam parameters from profile data

Typical Large emittance

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Slice envelope properties from slit/slit-cup diagnostic

50 mA Li+

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β€’ Bremmstrahlung

β€’ Synchrotron

β€’ Other types found in beam diagnostics

β€’ Cerenkov

β€’ Transition

β€’ Diffraction (electrons)

Far field EM radiation for high energy charged particle beams

1947

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β€’ Relativistic beams, g >>1

β€’ Parasitic, nonintercepting

β€’ Photon image reproduces electron beam distribution

β€’ Optics, coupling

β€’ Impedances, instabilities

β€’ IR -> Hard X-rays

Synchrotron Radiation

νœ€π‘π‘Ÿπ‘–π‘‘ = β„πœ”π‘ =3ℏ𝛾3𝑐

2𝜌

= 0.665𝐸2 𝐺𝑒𝑉

𝐡 𝑇[π‘˜π‘’π‘‰] βˆ†πœ†

πœ†=

1

𝑛𝑁

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Synchrotron radiation diagnostic beamlineβ€’ Multiple/simultaneous

ways to measure relativistic beams

(Figure courtesy J. Corbett, W. Cheng, A. Fisher, W. Mok)

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Transition vs. Cerenkov Radiation

Metallic or metallized foils, platesMild to ultrarelativistic particles Fast particles in background

gas or dielectric windows

𝑑 πœ” =2 𝑐 πœ”

1 𝛾2+ πœ—2 + πœ”π‘

2

πœ”2

Coherent formation length Dielectric constant, eBeam

Beam

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Bunch Shape Monitor

A.V.Feschenko, et. al, Proc. of the XIX Int. Linear Acc. Conf., 1998, p.905-907. β€’ Used for hadron

beamlines

β€’ Scanning wire produces secondary electrons

β€’ Electrons are accelerated in DC field, sorted in RF field

β€’ Time-correlation converted to position on detector

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β€’ Beam losses provide useful information on β€’ Beam orbit deviations

β€’ Mismatches between beam distributions and lattice design; beam halo

β€’ Energy and energy spread mismatches to lattice through chromaticity

β€’ Uncontrolled beam losses are potentially harmful to the machineβ€’ Damage to sensitive components (cryomodules!)

β€’ Radioactiviation of high loss areas of the beamline – affects maintenance and access

β€’ Diagnostics employed to detect lossesβ€’ Beam current/intensity, often in a differential mode to detect changes

β€’ Secondary radiation production – gammas, neutrons, electrons

β€’ Others – halo monitors, beamline thermometry, changes to cryo loading

Beam Loss Measurements

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Gas-type ionization chambers are in wide use as x-ray and gamma detectors

Ionization chambers

133 cm3 Ar gasTypical bias 1 kV Sensitivity 70 nC/radResponse time ~1-2 ms

R. Schmidt

1.5 L volume 100 mbar overpressure N2

0.5-mm separated Al platesBias 1500 VSensitivity ~ 54 mC/GyResponse time ~300 ns e-,

80 ms ions

LHC typeSNS type

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Ionization chamber schematic

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Scintillation based detectors (gammas, neutrons)

SNS Fast Detector

Sensitivity tuned with bias

β€’ Typically employ photomultiplier tubes for high gain (105-108) with applied HV

β€’ Many types of scintillators fluoresce under gamma bombardment

β€’ Li- or B- doped plastic scintillators respond to neutronsβ€’ Additional moderation increases sensitivity at the expense

of time response.β€’ Outside Cd layer provides discrimination against gammas

SNS Neutron chamber (SBLM)

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FRIB Beam Loss Monitoring Network

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𝑬 𝒙, 𝑑 =π‘ž

4πœ‹πœ–0

π’βˆ’πœ·

𝛾2 1βˆ’πœ·βˆ™π’ 3𝑅2+

1

𝑐

𝒏× π’βˆ’πœ· Γ— 𝜷

1βˆ’πœ·βˆ™π’ 3π‘…π‘Ÿπ‘’π‘‘

𝑩 𝒙, 𝑑 =1

𝑐𝒏 Γ— 𝑬 π‘Ÿπ‘’π‘‘

Lienard-Wiechert Fields

𝒏

𝑹

𝜷

π‘‚π‘π‘ π‘’π‘Ÿπ‘£π‘’π‘Ÿ

π‘…π‘’π‘‘π‘Žπ‘Ÿπ‘‘π‘’π‘‘ π‘π‘œπ‘ π‘–π‘‘π‘–π‘œπ‘›π΄π‘π‘π‘Žπ‘Ÿπ‘’π‘›π‘‘ π‘π‘œπ‘ π‘–π‘‘π‘–π‘œπ‘›

Transverse fields ~g

Longitudinal fields ~1/g2

Uniform motion, b<<1

Velocity fieldsAcceleration fields

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β€’ Lorentz Transformation of coordinates (here 𝒏=b/b)

β€’ Lorentz Transformation of fields

β€’ Lorentz Transformation of charge and current densities

Fields of beam bunches (constant velocity)

𝑬‖′ = 𝑬‖ 𝑩‖

β€² = 𝑩‖

𝑬βŠ₯β€² = 𝛾 𝑬βŠ₯ + 𝒗 Γ— 𝑩 𝑩βŠ₯

β€² = 𝛾 𝑩βŠ₯ βˆ’πŸ

π’„πŸπ’— Γ— 𝑬

π‘πœŒβ€² = 𝛾 π‘πœŒ βˆ’ 𝛽𝒏 βˆ™ 𝑱𝑱′ = 𝑱 + 𝛾 βˆ’ 1 𝑱 βˆ™ 𝒏 𝒏 βˆ’ π›Ύπ›½π‘πœŒπ’

𝑐𝑑′ = 𝛾 𝑐𝑑 βˆ’ 𝛽𝒏 βˆ™ 𝒓𝒓′ = 𝒓 + 𝛾 βˆ’ 1 𝒓 βˆ™ 𝒏 𝒏 βˆ’ 𝛾𝛽𝑐𝒕𝒏

Effect of metallic boundaries

2b

For a point charge on axis, the extent of the pulse is approximated by (cf. Shafer)

πœŽπ‘‘ ≅𝑏

2𝛾𝛽𝑐

πœŽπ‘‘

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β€’ In the limit of neglecting longitudinal end effects (bunch length >> pipe diameter), we can solve the 2D Laplace equation in the beam rest frame with Doppler shifted spectrum

β€’ Include modulation effects (wavelength >> pipe diameter)

β€’ For long pulses that are nonrelativistic or only mildly relativistic, the fields are well approximated by electrostatics

β€’ Intense beams may require self-magnetic field corrections

Field from 2D Laplace Equation

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β€’ Assume a beam, carrying current I, of radius a centered in a pipe of radius b.

β€’ The radial electric field at the pipe surface is πΈπ‘Ÿ =πœŒπ‘Ž2

2πœ€0𝑏=

πœ†

2πœ‹πœ€0𝑏

β€’ The surface charge density induced at r=b is πœŽπ‘  = νœ€0πΈπ‘Ÿ =πœ†

2πœ‹π‘

β€’ The azimuthal magnetic field at the pipe surface π΅πœ‘ =πœ‡0𝐼

2πœ‹π‘

β€’ With surface current density (longitudinally) 𝐾𝑠 =βˆ’πΌ

2πœ‹π‘= βˆ’π‘£πœŽπ‘ 

Simple beam model

𝜌 =1

πœ‹π‘Ž2𝐼

𝑣=

πœ†

πœ‹π‘Ž2

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β€’ Off center beam in pipe creates azimuthal surface charge density distribution

Wall currents and charges

r f

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β€’ RF and beam pulse structure

β€’ Compromise between β€’ Cavity rf frequency (aperture, transit factor)

β€’ Power generation and handling (CW/pulsed)

β€’ Experimental requirements

Time/Frequency description of beam signals

1 ms/div 50 ns/div 1 ns/div

Capacitive probeWhy is the signal bipolar?

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Fourier Series and Transforms

𝑓 πœ” =1

2πœ‹ βˆ’βˆž

∞

𝑑𝑑 𝑓 𝑑 π‘’π‘—πœ”π‘‘

𝑓 𝑑 =1

2πœ‹ βˆ’βˆž

∞

π‘‘πœ” 𝑓 πœ” π‘’βˆ’π‘—πœ”π‘‘

𝑓 𝑑 =π‘Ž02+

𝑛=1

∞

π‘Žπ‘› cos π‘›πœ”0𝑑 + 𝑏𝑛 sin π‘›πœ”0𝑑

π‘Žπ‘› =2

𝑇0

𝑇0

𝑑𝑑 𝑓 𝑑 cos π‘›πœ”0𝑑

𝑏𝑛 =2

𝑇0

𝑇0

𝑑𝑑 𝑓 𝑑 sin π‘›πœ”0𝑑

We define the use of symmetrical transforms

Time domain Frequency domain

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Fourier transform pairs Function name

Function Fourier transform

Delta function 𝛿 𝑑1

2πœ‹

Impulse train (Delta function β€˜comb’), period T

ΙΙΙ 𝑑 =

π‘˜=βˆ’βˆž

∞

𝛿 𝑑 βˆ’ π‘˜π‘‡ =1

𝑇

𝑛=βˆ’βˆž

∞

𝑒2πœ‹π‘—π‘›π‘‘π‘‡ III πœ” =

1

2πœ‹

𝑛=βˆ’βˆž

∞

π‘’βˆ’π‘—πœ”π‘›π‘‡

Gaussian 𝐴 π‘’βˆ’π‘‘2

2𝜎2 π΄πœŽπ‘’βˆ’12πœ”

2𝜎2

Boxcar

π‘π‘œπ‘₯π‘π‘Žπ‘Ÿ 𝑑; 𝐴, π‘Ž, 𝑏= 𝐴 𝐻 𝑑 βˆ’ π‘Ž βˆ’ 𝐻 𝑑 βˆ’ 𝑏

H(t) is the Heaviside step function

𝐴

2πœ‹π‘ βˆ’ π‘Ž π‘’βˆ’π‘—πœ”

π‘Ž+𝑏2 𝑠𝑖𝑛𝑐 πœ”

𝑏 βˆ’ π‘Ž

2

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β€’ Beam signals reveal the bunch structure and loading or fill pattern

β€’ Multibunch dynamics are revealed in sidebands and harmonics of underlying carrier frequencies

Beam spectra examples

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 10 20 30 40 50 60 70 80

I/Ia

vg

rf harmonic (of 80.5MHz)

2˚at 80.5MHzΟƒt = 70pSecΟƒf = 2.3GHzΟƒn = 28

80.5MHz 12.4nSecΟƒ = 70pSec or 2˚Iavg = 1mA

0

1

2

3

4

0 5 10 15 20 25

mA

mp

s

nSec

80.5MHz 12.4nSecΟƒ = 70pSec or 2˚

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Assuming Cartesian geometry and variation only along n12 we can show that

Frequency dependence of wall currents, skin depth

𝒏12

s1, e1, m1

s2, e2, m2

𝒏12 Γ— 𝑬2 βˆ’ 𝑬1 = 0𝒏12 βˆ™ 𝑫2 βˆ’π‘«1 = πœŒπ‘ π’12 Γ— 𝑯2 βˆ’π‘―1 = 𝑲𝑠

𝒏12 βˆ™ 𝑩2 βˆ’ 𝑩1 = 0

πœŒπ‘ , 𝑲𝑠

𝜡 Γ— 𝑬 = π‘—πœ”π‘© = π‘—πœ”πœ‡π‘―πœ΅ ×𝑯 = 𝑱 βˆ’ π‘—πœ”π‘«

In metals:

πœ”π‘« β‰ͺ 𝑱Good conductor:

Ohm’s Law: 𝑱 = 𝑬 𝜎

𝜡 Γ— 𝑱 = π‘—πœ”πœŽπ‘©πœ΅ Γ— 𝑩 = πœ‡π‘±

𝑩βŠ₯ 𝑧 = 𝑩βŠ₯ 𝑧 = 0 π‘’βˆ’ 1+𝑗 πœ…π‘§ 𝐽βˆ₯ 𝑧 = 𝐽βˆ₯ 𝑧 = 0 π‘’βˆ’ 1+𝑗 πœ…π‘§ where πœ… = 1 𝛿𝑐=

πœ‡πœ”πœŽ

2

𝑱𝒏12

⨂𝑩

𝑧

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β€’ The tangential electric field in the conductor derives from Ohm’s Law (not present in perfect conductor)

β€’ The transverse electric field satisfies an impedance boundary condition, with surface impedance, ZS

β€’ A surface resistance (Ohms) is defined as

β€’ Power deposition (W/area) to the surface follows from

Surface resistance and Joule losses

𝑬 =1

πœŽπ›» Γ— 𝑯 β‰…

1

πœŽπ’ Γ—

πœ•π‘―

πœ•π‘§= βˆ’π‘π‘ π’ Γ— 𝑯βŠ₯

𝑍𝑠 =1 + 𝑗𝑠𝑔𝑛 πœ”

πœŽπ›Ώπ‘= 𝑅𝑠 1 + 𝑗𝑠𝑔𝑛 πœ”

𝑅𝑠 =1

πœŽπ›Ώπ‘=

πœ‡πœ”

2πœŽπ‘‘π‘ƒπ‘ π‘’π‘Ÿπ‘“

𝑑𝐴= 𝑺 βˆ™ 𝒏 =

1

2𝔑 𝑬 Γ—π‘―βˆ— βˆ™ 𝒏

=1

2𝑅𝑠 𝐾𝑠

2

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Resistive wall impedance

𝑍0βˆ₯

π‘™π‘’π‘›π‘”π‘‘β„Ž=

𝑍𝑠2πœ‹π‘

=1

2πœ‹π‘

πœ‡πœ”

2𝜎1 + 𝑗𝑠𝑔𝑛 πœ”

𝑉 πœ” = 𝐼 πœ” 𝑍0βˆ₯ πœ”

The beam senses the wall through resistive loading

β€’ The longitudinal resistive wall impedance can be defined as

β€’ The beam will experience a voltage change

β€’ Impedance: 𝑍0βˆ₯ = 𝑅𝑒𝑍0

βˆ₯ + 𝑗 πΌπ‘šπ‘0βˆ₯ (longitudinal, monopole)

What is the significance of the of the resistive and reactive components?

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Building a detector

Measured ceramic gap impedance

β€’ A nonintercepting monitor can be based on monitoring the wall return currents.

β€’ A ceramic break in the beampipe will force the wall current to seek other paths.

β€’ If nothing else is done, the wall currents will find alternative paths.

β€’ The gap impedance is a combination of the gap capacitance and all external parallel elements

β€’ At low frequencies the lowest impedance return path can be distant from the gap itself

β€’ The gap voltage π‘‰π‘”π‘Žπ‘ = πΌπ‘€π‘Žπ‘™π‘™π‘π‘”π‘Žπ‘ = πΌπ‘π‘’π‘Žπ‘šπ‘π‘”π‘Žπ‘can be generated up to the beam voltage

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β€’ We model the beam-monitor interaction with an equivalent circuit

β€’ Beam drive is modeled as a pure current source (infinite input impedance)

β€’ A gap impedance Cgap is inevitably presentβ€’ Electrodes pierce the beam wall with isolated

feedthroughs

β€’ Typically few – 100s pF

β€’ The specific signal pickup as well as the signal transmission line and passive analog components are represented by Zmon.

Impedance models and behavior

Zmon

Cgap

iw

π‘‰π‘šπ‘œπ‘› πœ” = 𝑖𝑀 πœ”π‘π‘šπ‘œπ‘›

1 βˆ’ π‘—πœ”πΆπ‘”π‘Žπ‘π‘π‘šπ‘œπ‘›

= 𝑖𝑏 πœ” 𝑍𝑑 πœ”

Zmon

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β€’ We add a network of n resistors across the gap. Rtot = Rsingle/n

β€’ Zmon = Rtot || Cgap

β€’ Broadband pickup

β€’ π‘‰π‘šπ‘œπ‘› πœ” = 𝑖𝑏 πœ”π‘…π‘‘π‘œπ‘‘

1βˆ’π‘—πœ”πΆπ‘”π‘Žπ‘π‘…π‘‘π‘œπ‘‘

β€’ Within passband π‘‰π‘šπ‘œπ‘› πœ” = 𝑅𝑠𝑖𝑛𝑔𝑙𝑒 𝑛 𝑖𝑏 πœ”

β€’ Practical implementations (eg. SPS WCM)β€’ Ceramic gap

β€’ Many resistors (30 – 100) to reduce sensitivity to beam position

β€’ Ferrite rings to tailor low frequency response ~10 kHz

β€’ High frequency response to several GHz

β€’ Shield for ground currents and noise isolation

Wall current monitor

Rtot

Cgap

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β€’ We adopt a single mode resonance to calculate the coupling impedance

β€’ The transient cavity-beam-waveguide system can be expressed as an equivalent circuit equation (cf. Whittum)

𝑑2

𝑑𝑑2+ πœ”0

2 𝑉𝑐 = βˆ’πœ”0

𝑄𝑀

𝑑

𝑑𝑑𝑉𝑐 +

πœ”0

𝑄𝑒

𝑑

𝑑𝑑𝑉𝐹 βˆ’ 𝑉𝑅 +πœ”0

π‘Ÿ

𝑄

𝑑

𝑑𝑑𝐼𝑏

Here VF=nV+, VR=nV- are the normalized forward and reverse waveguide voltages, such that VC = VF + VR. Here, n is called the transformer ratio for the mode coupling.

β€’ We can show that the longitudinal coupling impedance presented by this mode is

𝑍βˆ₯ πœ” =π‘—πœ”πœ”0 π‘Ÿ 𝑄

πœ”02 βˆ’ πœ”2 + π‘—πœ”πœ”0/𝑄𝐿

= 𝑄𝐿 π‘Ÿ 𝑄 cosπœ“ π‘’π‘—πœ“

β€’ Impedances can be expressed as Z = R +j X with the reactance 𝑋 = πœ”πΏ βˆ’1

πœ”πΆ

Resonant effects - Modal circuit equation

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Panofsky and Wenzel derived a powerful theorem connecting longitudinal electric field to transverse momentum impulses.

Relation between longitudinal and transverse effects

βˆ†π’‘ = 𝑄𝑒 π‘Ž

𝑏

𝑬 + 𝒗 Γ— 𝑩 𝑑𝑑 βˆ†π‘Š = 𝑄𝑒 π‘Ž

𝑏

𝑬 βˆ™ 𝑑𝒔

πœ•

πœ•π‘‘βˆ†π’‘ = 𝑄𝑒

π‘Ž

𝑏 πœ•π‘¬

πœ•π‘‘π‘‘π‘‘ + 𝑑𝒔 Γ—

πœ•π‘©

πœ•π‘‘= 𝑄𝑒

π‘Ž

𝑏

βˆ’π› 𝑑𝒔 βˆ™ 𝑬 + 𝑑𝑬

πœ•

πœ•π‘‘βˆ†π‘π‘  = 𝑄𝑒

πœ•

πœ•π‘‘ π‘Ž

𝑏

𝐸𝑠𝑑𝑑

We can show

πœ•

πœ•π‘‘βˆ†π’‘βŠ₯ = βˆ’π‘„π‘’

π‘Ž

𝑏

𝜡βŠ₯ 𝑑𝒔 βˆ™ 𝑬 βˆ’ 𝑑𝑬βŠ₯ = βˆ’πœ΅βŠ₯π‘Š+𝑄𝑒 𝑬βŠ₯ π‘Ž βˆ’ 𝑬βŠ₯ 𝑏

We can typically ignore the end effects by taking the integral to field free regions.

The beam trajectory is along s, and ds=vdt

Lorentz force impulse

πœ•

πœ•π‘‘βˆ†π’‘βŠ₯ = βˆ’πœ΅βŠ₯π‘Š (Panofsky-Wenzel)

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β€’ Beam impedance responseβ€’ V(w) = Ib(w) Z(w)

β€’ Everything that sees the beam can be described in terms of a beam coupling impedance

β€’ Narrowband impedances from resonant structures

β€’ Related to wake functions

β€’ Panofsky-Wenzel relates longitudinal to transverse wake/impedances for ultrarelativistic particles

Beam spectrum, impedances and beam loading

DAFNE RF cavity longitudinal impedance

Fundamental mode

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β€’ Devices may posses narrowband as well as broadband impedance characteristics.

β€’ Impedances can be characterized on benchtop measurement stands

Beam coupling impedance to broadband device

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β€’ Very high frequency response dominated by gap capacitance

β€’ Very low frequency response dominated by ferrite and shield induction

Wall Current Monitor equivalent circuit and response

Z0

Z0

Cgap

iw…R Lferrite

πŽπ’„π’–π’•π’π’‡π’‡

πŽπ’‰π’Šπ’ˆπ’‰ = 𝟏/𝑹π‘ͺ

πŽπ’π’π’˜ = 𝑹/𝑳

log

1/RCR/Lw

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β€’ Induced charge densities on the beam pipe walls can be approximated or numerically calculated.

β€’ Total induced charges on the electrodes (assuming 2D Laplace solution) for electrodes of length L

π‘„π‘π‘™π‘Žπ‘‘π‘’ = 𝐿𝑏

π‘ π‘’π‘π‘‘π‘œπ‘Ÿ

𝜎 πœƒ π‘‘πœƒ

Induced charge densities from capacitive coupling

r f

b

← 𝜎 πœƒ

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β€’ The pickup plate presents a capacitance C to ground.

β€’ The finite length of the pickup drives a differential current into the monitor.

β€’ π‘‰π‘šπ‘œπ‘› πœ” = 𝑖𝐡 πœ” 𝑍βˆ₯ πœ”

β€’ The image current on the pickup is related to the beam current

β€’ π‘–π‘–π‘š πœ” = π‘—πœ”π΄

2πœ‹π‘Žπ‘™

𝑙

𝛽𝑐𝑖𝐡 πœ”

β€’ π‘‰π‘šπ‘œπ‘› πœ” =𝑅

1βˆ’π‘—πœ”πΆπ‘…π‘–π‘–π‘š πœ” =

𝑅

1βˆ’π‘—πœ”πΆπ‘…π‘—πœ”

𝐴

2πœ‹π‘Žπ‘™

𝑙

𝛽𝑐𝑖𝐡 πœ”

β€’ 𝑍βˆ₯ πœ” =1

𝛽𝑐

1

𝐢

𝐴

2πœ‹π‘Ž

𝑗𝑅𝐢

1βˆ’π‘—πœ”π‘…πΆ

Signals from capacitive coupling

iB(t)

~ iB(t) ~ iB(t+Dt)

imon(t) = ~ [iB(t)- iB(t+Dt)]

iw(t)

l

2a A: plate area

Vmon(t)R

Ciim

R Vmon

π‘–π‘–π‘š 𝑑 = βˆ’π΄

2πœ‹π‘Žπ‘™

π‘‘π‘„π‘π‘’π‘Žπ‘šπ‘‘π‘‘

= βˆ’π΄

2πœ‹π‘Žπ‘™

𝑙

𝛽𝑐

𝑑𝑖𝐡𝑑𝑑

Dt=l/bc

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Capacitive pickup responseThe transfer impedance was found to be

𝑍βˆ₯ πœ” =1

𝛽𝑐

1

𝐢

𝐴

2πœ‹π‘Ž

𝑗𝑅𝐢

1βˆ’π‘—πœ”π‘…πΆ

Defining the cutoff frequency as πœ”π‘ = 1/𝑅𝐢

The magnitude and phase are

𝑍βˆ₯ =1

𝛽𝑐

1

𝐢

𝐴

2πœ‹π‘Ž

πœ”/πœ”π‘

1 + πœ”/πœ”π‘2, βˆ π‘βˆ₯ = tanβˆ’1 πœ”π‘/πœ”

High frequency regime, πœ”

πœ”π‘β‰« 1, 𝑍βˆ₯ β†’ 1

π‘‰π‘šπ‘œπ‘› 𝑑 =1

𝛽𝑐𝐢

𝐴

2πœ‹π‘Žπ‘–π΅ 𝑑

Direct image of beam current with no phase shift.

Low frequency regime , πœ”

πœ”π‘β‰ͺ 1, 𝑍βˆ₯ β†’ 𝑗

πœ”

πœ”π‘

π‘‰π‘šπ‘œπ‘› 𝑑 =𝑅

𝛽𝑐

𝐴

2πœ‹π‘Ž

𝑑𝑖𝐡

𝑑𝑑

Measured voltage is proportional to time derivative of current.

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Spectrum from low-b beams in capacitive pickups

β€’ Here, b is the distance from the beam to each pickup (or pipe radius for on-axis beam)β€’ Narrow bandwidth difference measurement must contend with nonlinearities in

response. β€’ Requires calibration against variation in beta for each monitoring frequency and

bandwidth.

Point particle distribution πœŽπ‘‘ ≅𝑏

2𝛾𝛽𝑐

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The magnetic field of the beam is used to measure the beam current or intensity.

Inductive coupling

𝑩 = πœ‡πΌπ΅2πœ‹π‘Ÿ

𝝋

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Current Transformers

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β€’ We will analyze this using Laplace transform pairs instead of Fourier.

β€’ Fourier (symmetrized)

β€’ β„± πœ” =1

2πœ‹ βˆ’βˆžβˆžπ‘“ 𝑑 π‘’βˆ’π‘—πœ”π‘‘

β€’ 𝑓 𝑑 =1

2πœ‹ βˆ’βˆžβˆžβ„± πœ” π‘’π‘—πœ”π‘‘

β€’ Laplace (unsymmetrized)

β€’ 𝐹 𝑠 = 𝜎 + π‘—πœ” = β„’ 𝑓 𝑑 = 0βˆ’βˆžπ‘“ 𝑑 π‘’βˆ’π‘ π‘‘

β€’ 𝑓 𝑑 = β„’βˆ’1 𝐹 𝑠 =1

2πœ‹π‘— 𝜎1βˆ’π‘—βˆžπœŽ1+π‘—βˆžπΉ 𝑠 𝑒𝑠𝑑

β€’ The Laplace transform is useful for analyzing linear, time-invariant, causal systems.

β€’ Differentiation β„’ 𝑓 𝑑 = 𝑠𝐹 𝑠 βˆ’ 𝑓 0βˆ’

β€’ Integration β„’ 0βˆ’π‘‘π‘“ 𝜏 π‘‘πœ =

𝐹 𝑠

𝑠

β€’ Linearity, Scaling, Shifting, Translation, Convolution

Current Transformer Transfer Impedance

𝑑𝐼𝑠𝑑𝑑

+𝑅𝑠𝐿𝑠𝐼𝑠 = βˆ’

1

𝑁𝑠

𝑑𝐼𝑝

𝑑𝑑𝑠 𝐼𝑠 +

𝑅𝑠

𝐿𝑠 𝐼𝑠=βˆ’

1

𝑁𝑠𝑠 𝐼𝑝

𝐻 𝑠 = 𝑉𝑠 𝐼𝑝=𝑅𝑠 𝐼𝑠 𝐼𝑝

= βˆ’π‘…π‘ π‘π‘ 

π‘ πœ

1 + π‘ πœπœ =

𝐿𝑠𝑅𝑠

𝑠 β‰ˆ π‘—πœ” ≫1

𝜏⟹ 𝐻 π‘—πœ” = βˆ’

𝑅𝑠𝑁𝑠

In the high frequency limit

𝑉𝑠 =𝑅𝑠𝐼𝑝𝑁𝑠

𝑃𝑠 = 𝐼𝑠2𝑅𝑠 =

𝑅𝑠𝐼𝑝2

𝑁𝑠2

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β€’ The low frequency response is dominated by R/L

β€’ Practical monitors contain capacitance due to ceramic wall breaks, coupling between windings, etc. This alters the response.

Inductance and Capacitance

πΏπ‘œπ‘€ π‘“π‘Ÿπ‘’π‘žπ‘’π‘’π‘›π‘π‘¦ ∢ πœ” β‰ͺ𝑅

𝐿⟹ 𝑍 = π‘—πœ”πΏ (B-dot regime)

𝑀𝑖𝑑 π‘“π‘Ÿπ‘’π‘žπ‘’π‘’π‘›π‘π‘¦ βˆΆπ‘…

𝐿β‰ͺ πœ” β‰ͺ

1

π‘…πΆβŸΉ 𝑍 β‰ˆ 𝑅

π»π‘–π‘”β„Ž π‘“π‘Ÿπ‘’π‘žπ‘’π‘’π‘›π‘π‘¦ ∢ πœ” ≫1

π‘…πΆβŸΉ 𝑍 = 1/π‘—πœ”πΆ

1

𝑍=1

𝑅+

1

π‘—πœ”πΏ+ π‘—πœ”πΆ ⟹ 𝑍 πœ” =

π‘—πœ”πΏ

1 +π‘—πœ”πΏπ‘…

βˆ’πœ”πΏπ‘…

πœ”π‘…πΆ

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Response from passive transformer

Stray cable capacitance increases risetime.

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Cavity Dipole-mode BPMs (resonant coupling)β€’ Used mainly for high energy electron beams

β€’ Resonant dipole modes have higher shunt impedance than buttons or striplinesβ€’ High sensitivity

β€’ Wakefields act back on beam

β€’ Cavity BPMs have been developed to produce sub-mm position resolution, for ~mm displacements

β€’ Monopole mode excitation is proportional to beam current

β€’ Antennae pick up combined monopole+dipole signals. Technique requires independent calibration of monopole voltage. β€’ Pillbox: fmono ~ 1.2-1.5* fdipole

β€’ Qload for both modes ~100 – 1000β€’ Mode must decay before arrival of next pulse

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β€’ For very high bandwidth impedances, modal and resonant behavior may no longer adequately capture the beam-structure interaction

β€’ Single and multi-bunch effects (coherence length > bunch length or separation)

β€’ Regenerative

β€’ Frequencies above pipe cutoff

β€’ In this class of phenomena, beam wakes fields are a more relevant descriptionβ€’ Time domain

β€’ Causal for ultrarelativistic beams

Wake fields and wake potentials

W(z)

z

π‘Ž

𝑏

𝑑𝑧 𝐹 = βˆ’π›»π‘‰

𝑉 = π‘’πΌπ‘šπ‘Šπ‘š 𝑧 π‘Ÿπ‘š cosπ‘šπœƒfor an mth multipoleπ‘Šπ‘š

β€² 𝑧 =1

2πœ‹ βˆ’βˆž

∞

π‘‘πœ” π‘’π‘—πœ”π‘§/𝑐 π‘π‘šβˆ₯ πœ”

π‘Šπ‘š 𝑧 =βˆ’π‘—

2πœ‹ βˆ’βˆž

∞

π‘‘πœ” π‘’π‘—πœ”π‘§/𝑐 π‘π‘šβŠ₯ πœ”

π‘π‘šβŠ₯ πœ” =

πœ”

π‘π‘π‘šβˆ₯ πœ”

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β€’ The complete diagnostic system starts with beamline (or nearby) sensor

β€’ Cabling to transport signals to data acquisition (DAQ) systems

β€’ Processing electronics, and controls/operator interfaces

Architecture of a diagnostic measurement

Penetrations and racks are laid out for instrumentationβ€’ Cable runs about 100 ft

β€’ Diagnostics will use ¼” superflex (Heliax)Β» solid copper jacket provides

>120 dB shielding effectiveness

No electronics in the tunnel!

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β€’ Controls and actuation for interceptive devices

β€’ Global timing and triggering

β€’ Interlocks for machine, device, and personnel protection

β€’ High level controls for data acquisition, analysis, visualization

Many interfaces are needed to deploy diagnostics

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β€’ Review of Particle Physics, http://pdg.lbl.gov

β€’ P. Strehl, Beam Instrumentation and Diagnostics (Springer, 2006).

β€’ M.G. Minty and F. Zimmerman, Measurement and Control of Charged Particle Beam (Springer, 2003).

β€’ R. Shafer, "Beam Position Monitoring", in AIP Conf. Proc. 212 (1990) 26-58.

β€’ R. Shafer, "Beam position monitor sensitivity for low-b beams", Proc. Beam Instr. Workshop (1994) 303.

β€’ J.H. Cuperas, "Monitoring of particle beams at high frequencies", Nucl. Instrum. Methods Phys. Res. 145 (1977) 219-231.

β€’ D. Whittum, "Introduction to Electrodynamics for Microwave Linear Accelerators", http://uspas.fnal.gov/materials/14Knoxville/slac-pub-7802(1).pdf

β€’ J. Byrd, et. al., http://uspas.fnal.gov/materials/13CSU/USPASMWLecture.pdf

β€’ G. Dome, CERN PAS

β€’ M. Plum, "Interceptive Beam Diagnostics - Signal Creation and Materials Interactions", BIW 2004.

β€’ J. Bosser, ed., "Beam Instrumentation", CERN-PE-ED 001-92 (rev. 1994).

β€’ D. Brandt, ed., "Beam Diagnostics", CERN-2009-005.

β€’ J. Belleman, "From analog to digital", in Digital Signal Processing (D. Brandt, ed.), CERN-2008-003, 131-166.

β€’ P. Forck, "Lecture Notes on Beam Instrumentation and Diagnostics", JUAS (2011).

β€’ R.H. Siemann, "Spectral Analysis of Relativistic Bunched Beams", SLAC-PUB-7159 (1996).

β€’ K. Wittenburg, "Specific instrumentation and diagnostics for high-intensity hadron beams"

References

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Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 69

β€’ J. Bechhoefer, "Feedback for physicists: A tutorial essay on control", Rev. Mod. Phys. 77 (2005) 783-836.

β€’ http://labs.physics.berkeley.edu/mediawiki/index.php/Introduction_to_Noise

β€’ Knoll, Radiation Detection and Measurement

β€’ Leo, Techniques for Nuclear and Particle Physics Experiments

β€’ ICFA Advanced Beam Dynamics Workshop on Beam-Halo (Halo '03)

β€’ Workshop on Halo Monitoring, SLAC, September 2014, https://portal.slac.stanford.edu/sites/conf_public/bhm_2014/Presentations/Forms/AllItems.aspx

β€’ K. Wittenburg, "Beam halo and bunch purity monitoring" in Brandt (2009).

β€’ S.W. Smith, Digital Signal Processing. A practical guide for Engineers and Scientists, (Newnes, 2003).

β€’ J.A. Hernandes and A.K.T. Assis, "Electric potential due to an infinite conducting cylinder with internal or external point charge", J. Electrostatics 63 (2005), 1115-1131.

β€’ G. Lambertson, "Dynamic Devices - Pickups and Kickers", AIP Conference Proceedings 153, 1413 (2016); doi: http://dx.doi.org/10.1063/1.36380

β€’ W. Barry, "Broad-band characteristics of circular button pickups", AIP Conference Proceedings 281, 175 (2016); doi: http://dx.doi.org/10.1063/1.44335

β€’ S. van der Meer, "An Introduction to Stochastic Cooling", AIP Conference Proceedings 153, 1628 (2016); doi: http://dx.doi.org/10.1063/1.36351.

β€’ J.D. Jackson, Classical Electrodynamics (Wiley, 1975).

β€’ A. Hirose, "Electromagnetic Theory", http://physics.usask.ca/~hirose/p812/notes.htm, Chapter 8.

β€’ H. Bichsel, "A method to improve tracking and particle identification in TPCs and silicon detectors", Nucl. Instrum. Methods Phys. Res. A 562 (2006) 154-197.

References, contd.

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β€’ Bias for a Faraday CupA 5 emA beam of Uranium ions with 12 keV/nucleon energy is collected by a Faraday cup for measurement.

The Rutherford differential cross section for scattering between slow heavy ions and free electrons defines a maximum transfer energy, Wmax = 2mec2b2g2. Further, assume that atomic ionization of the Faraday cup collector material requires ~5 eV, and escaping the work function of the cup surface requires 2 eV. Otherwise, assume that all liberated electrons exit normally from the surface of the cup.

a) What is the maximum energy of (in eV) of the emitted electrons? What bias potential needs to be present in order to trap these electrons?

b) Assume that each ion continues to ionize the cup material and produce electrons, but that each subsequent electron undergoes inelastic scattering such that its energy at the surface varies as Wmax,n =Wmax e-n. Plot the measured beam current as a function of bias in the range of -100V to 100V.

Exercise 1

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β€’ Pepperpot DesignA pepperpot is to be installed in a 100 keV proton injector beamline. Assume the hole mask is patterned with 0.250 mm diameter holes, spaced 2 mm apart (hole center to hole center) in a rectangular array, and that the mask is 2 mm thick Tungsten.

Assume the beam distribution is Gaussian in x/x’/y/y’, with rms normalized emittances of 0.1 pi mm mrad in x and 0.2 pi mm mrad in y.

If the pepperpot mask location coincides with the beam waist, what drift length between mask and screen is required for 50% intensity modulation between peak and trough on the screen?

Exercise 2