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Steve LidiaNational Superconducting Cyclotron and Facility for Rare Isotope Beams Laboratory
Michigan State University
PHY862 Accelerator SystemsBeam Measurements and Instrumentation
1. Interactions and Sensors
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 2
• Measurements, diagnostics, instrumentation
• Interactions and sensors
• Beam generated signals
• Diagnostic architecture
• 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
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 3
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
• Tuning, optimization of beam quality
• Targetry, beam collisions
• Monitoring, stability, minimizing losses, machine protection
Role of diagnostics and instrumentation
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 4
• Measurements• Incorporate (incomplete) knowledge of lattice, beam dynamics
• Point vs. Distributed (lattice-dependent) measurements
• Diagnostics are phenomena or techniques involved in performing a measurement
• Direct vs indirect measurements
• Correlations – use known dependencies
• Instrumentation is the set of particular devices used in the execution of the measurement
• Set of beam sensors, signal transmission lines, data acquisition and reporting systems, controls and feedback
Measurements, diagnostics, and instrumentation
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 5
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, . . .
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 6
Typical beam measurementsMeasurements and associated diagnostics and instrumentation are specific to
• Geometry of beamline• linac, synchrotron booster, storage ring,
analyzing beamline, injector, final focus, etc.)
• Particle type• Hadron, lepton, neutron, neutral atom,
rare isotope
• Beam energy
• Beam intensity
• Beam time structure
• . . .
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 7
SIS-18 Synchrotron at GSI: diagnostic suite
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 8
• Charge/current, charge states, mass states
• Beam energy
• 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.)
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 9
• Profile (transverse, longitudinal), envelope, energy spread
• Phase space density, emittance measures
• Beam halo - transverse, longitudinal
Beam parameters [2] – 2nd Order (+ higher) Moments
Beam halo with wire scanner measurements
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 10
• Single bunch vs. many bunch measurements
• Time domain vs. frequency domain
Beam parameters [3] – Bunch Trains
LHC Turn-by-turn Bunch Intensity (300 turns)
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 11
• Betatron tune Qx, Qy
Lattice parameters
• Dispersion function
Large energy spread Nominal energy spread
BPMs @ A, B, C, D
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 12
Lattice parameters Matched distributions Mismatched distributions
Power density (Gy/s) at 200 MeV
• Loss distributions in SC RF modules
• Beam envelope matching to lattice
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 13
• Emittance growth for mismatched beams
• Instabilities
• Electron cloud effects
Dynamic parameters
Direct phase measurement in resonant BPM configuration (DeSantis, et al, PAC09 TH5RFP071)
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 14
• Physics of beam-sensor interactions• EM, Nuclear, AMO, Solid State• Charge and mass interception• Capacitive, inductive, resonant, thermal
field sensing• Secondary radiation fields
• Mechanical design • Thermal, structural, vacuum, actuator
• Electrical design• Grounding/shielding• HV bias and insulation
• Electronics• Signal acquisition, conditioning,
processing• Noise, bandwidth, sensitivity,
response time
Diagnostic/instrumentation design elements
System Interface Diagram
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 15
Faraday cups
Negative HV Bias
Suppressor CollectorV
I/VIncoming ions
e- Transimpedanceamplifier
• Fully intercepting charge measurement
• Sensitivity to 10 pC. ~100 Hz BW (‘slow’, deep cup)
• Beam charges impinge on Collector, are collected by electronics
• Suppressor negatively biased to repel electrons
• Design is to prevent escape of secondary electrons
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 16
• Based on scintillator and camera• Coated screen (reflection mode)
• Solid, thick (100 mm) scintillator (transmission mode)
• Scintillator can be single- or multi-crystalline, or sintered powder
• Direct 2D measurement• Direct digital output
• Video out and frame grabber
• Resolution depends on scintillator material (grains), CCD size, optics
• Amplitude response depends on field flatness, scintillator dose and aging effects, temperature
ViewersGSI linac, 4 MeV/u, low current, YAG:Ce
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 17
• Analyzes intensity J(x or y) at first slit
• Applied voltage across plates + drift + exit slit analyzes momentum
• Reconstructs phase space density
Allison Scanner
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 18
Pepperpots, slits, and pinholes• Devices scan a 1D or 2D beam distribution
• Analyze intensity J(x,y)
• Analyze transverse velocity over drift
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 19
Emittance analysis
x
x’
𝛾𝑥2 + 2𝛼𝑥𝑥′ +𝛽𝑥′2 = 𝜀
(x0, -(a/b) x0)
2Dx0’
Dx0’=[e/b – x02/b2]1/2
2a
Σ4 =
𝑥𝑥 𝑥𝑥′ 𝑥𝑦 𝑥𝑦′
𝑥𝑥′ 𝑥′𝑥′ 𝑥′𝑦 𝑥′𝑦′
𝑥𝑦 𝑥′𝑦 𝑦𝑦 𝑦𝑦′
𝑥𝑦′ 𝑥′𝑦′ 𝑦𝑦′ 𝑦′𝑦′
=Σ𝑥 𝐶
𝐶𝑇 Σ𝑦
det Σ𝑥 = Σ𝑥 = 𝑥𝑥 𝑥′𝑥′ − 𝑥𝑥′ 2 = 휀𝑥2
𝑓𝑔 = 𝑖 𝜌𝑖𝑓𝑖𝑔𝑖 𝑖 𝜌𝑖
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 20
• Beam positions can be monitored using a 4-electrode array of capacitive pickups on the beampipecircumference.
• Various geometries are employed for sensitivity, compactness, protection from intense radiation
Beam Position Monitors
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 21
• Positions are estimated from the normalized intensities using the difference over sum algorithm
∆𝑥 =1
𝑆𝑥
𝑉2−𝑉4
𝑉2+𝑉4, ∆𝑦 =
1
𝑆𝑦
𝑉1−𝑉3
𝑉1+𝑉3
• Position sensitivities are proportionality constants between beam displacement and signal strength.
• 𝑆𝑥 =𝑑
𝑑𝑥
∆𝑥
Σ𝑥
%
𝑚𝑚𝑜𝑟 𝑆𝑥 =
𝑑
𝑑𝑥𝑙𝑜𝑔
𝑉2
𝑉4
𝑑𝐵
𝑚𝑚
• Offset displacements also occur and must be measured and calibrated.
• Button-button capacitive coupling introduces frequency dependent offset and sensitivity variation.
• 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
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 22
Profile monitors• Interceptive diagnostic onto W, C, Cu-Be wires
• Wide range of wire based geometries
• Biased wire to discourage (or encourage) secondary e-’s
• Slit + Faraday cup
Linear actuated
Rotating wire
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 23
• We observe day-to-day variation of transverse beam parameters• Two most significant factors are: ECR setting and beam center matching to the RFQ
Reconstruct beam parameters from profile data
Typical Large emittance
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 24
Slice envelope properties from slit/slit-cup diagnostic
50 mA Li+
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 25
• Bremmstrahlung
• Synchrotron
• Other types found in beam diagnostics
• Cerenkov
• Transition
• Diffraction (electrons)
Far field EM radiation for high energy charged particle beams
1947
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 26
• Relativistic beams, g >>1
• Parasitic, nonintercepting
• Photon image reproduces electron beam distribution
• Optics, coupling
• Impedances, instabilities
• IR -> Hard X-rays
Synchrotron Radiation
휀𝑐𝑟𝑖𝑡 = ℏ𝜔𝑐 =3ℏ𝛾3𝑐
2𝜌
= 0.665𝐸2 𝐺𝑒𝑉
𝐵 𝑇[𝑘𝑒𝑉] ∆𝜆
𝜆=
1
𝑛𝑁
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 27
Synchrotron radiation diagnostic beamline• Multiple/simultaneous
ways to measure relativistic beams
(Figure courtesy J. Corbett, W. Cheng, A. Fisher, W. Mok)
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 28
Transition vs. Cerenkov Radiation
Metallic or metallized foils, platesMild to ultrarelativistic particles Fast particles in background
gas or dielectric windows
𝑑 𝜔 =2 𝑐 𝜔
1 𝛾2+ 𝜗2 + 𝜔𝑝
2
𝜔2
Coherent formation length Dielectric constant, eBeam
Beam
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 29
Bunch Shape Monitor
A.V.Feschenko, et. al, Proc. of the XIX Int. Linear Acc. Conf., 1998, p.905-907. • Used for hadron
beamlines
• Scanning wire produces secondary electrons
• Electrons are accelerated in DC field, sorted in RF field
• Time-correlation converted to position on detector
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 30
• Beam losses provide useful information on • Beam orbit deviations
• Mismatches between beam distributions and lattice design; beam halo
• Energy and energy spread mismatches to lattice through chromaticity
• Uncontrolled beam losses are potentially harmful to the machine• Damage to sensitive components (cryomodules!)
• Radioactiviation of high loss areas of the beamline – affects maintenance and access
• Diagnostics employed to detect losses• Beam current/intensity, often in a differential mode to detect changes
• Secondary radiation production – gammas, neutrons, electrons
• Others – halo monitors, beamline thermometry, changes to cryo loading
Beam Loss Measurements
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 31
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
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 32
Ionization chamber schematic
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 33
Scintillation based detectors (gammas, neutrons)
SNS Fast Detector
Sensitivity tuned with bias
• Typically employ photomultiplier tubes for high gain (105-108) with applied HV
• Many types of scintillators fluoresce under gamma bombardment
• Li- or B- doped plastic scintillators respond to neutrons• Additional moderation increases sensitivity at the expense
of time response.• Outside Cd layer provides discrimination against gammas
SNS Neutron chamber (SBLM)
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 34
FRIB Beam Loss Monitoring Network
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 35
𝑬 𝒙, 𝑡 =𝑞
4𝜋𝜖0
𝒏−𝜷
𝛾2 1−𝜷∙𝒏 3𝑅2+
1
𝑐
𝒏× 𝒏−𝜷 × 𝜷
1−𝜷∙𝒏 3𝑅𝑟𝑒𝑡
𝑩 𝒙, 𝑡 =1
𝑐𝒏 × 𝑬 𝑟𝑒𝑡
Lienard-Wiechert Fields
𝒏
𝑹
𝜷
𝑂𝑏𝑠𝑒𝑟𝑣𝑒𝑟
𝑅𝑒𝑡𝑎𝑟𝑑𝑒𝑑 𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛𝐴𝑝𝑝𝑎𝑟𝑒𝑛𝑡 𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛
Transverse fields ~g
Longitudinal fields ~1/g2
Uniform motion, b<<1
Velocity fieldsAcceleration fields
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 36
• Lorentz Transformation of coordinates (here 𝒏=b/b)
• Lorentz Transformation of fields
• Lorentz Transformation of charge and current densities
Fields of beam bunches (constant velocity)
𝑬‖′ = 𝑬‖ 𝑩‖
′ = 𝑩‖
𝑬⊥′ = 𝛾 𝑬⊥ + 𝒗 × 𝑩 𝑩⊥
′ = 𝛾 𝑩⊥ −𝟏
𝒄𝟐𝒗 × 𝑬
𝑐𝜌′ = 𝛾 𝑐𝜌 − 𝛽𝒏 ∙ 𝑱𝑱′ = 𝑱 + 𝛾 − 1 𝑱 ∙ 𝒏 𝒏 − 𝛾𝛽𝑐𝜌𝒏
𝑐𝑡′ = 𝛾 𝑐𝑡 − 𝛽𝒏 ∙ 𝒓𝒓′ = 𝒓 + 𝛾 − 1 𝒓 ∙ 𝒏 𝒏 − 𝛾𝛽𝑐𝒕𝒏
Effect of metallic boundaries
2b
For a point charge on axis, the extent of the pulse is approximated by (cf. Shafer)
𝜎𝑡 ≅𝑏
2𝛾𝛽𝑐
𝜎𝑡
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 37
• 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
• Include modulation effects (wavelength >> pipe diameter)
• For long pulses that are nonrelativistic or only mildly relativistic, the fields are well approximated by electrostatics
• Intense beams may require self-magnetic field corrections
Field from 2D Laplace Equation
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 38
• Assume a beam, carrying current I, of radius a centered in a pipe of radius b.
• The radial electric field at the pipe surface is 𝐸𝑟 =𝜌𝑎2
2𝜀0𝑏=
𝜆
2𝜋𝜀0𝑏
• The surface charge density induced at r=b is 𝜎𝑠 = 휀0𝐸𝑟 =𝜆
2𝜋𝑏
• The azimuthal magnetic field at the pipe surface 𝐵𝜑 =𝜇0𝐼
2𝜋𝑏
• With surface current density (longitudinally) 𝐾𝑠 =−𝐼
2𝜋𝑏= −𝑣𝜎𝑠
Simple beam model
𝜌 =1
𝜋𝑎2𝐼
𝑣=
𝜆
𝜋𝑎2
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 39
• Off center beam in pipe creates azimuthal surface charge density distribution
Wall currents and charges
r f
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 40
• RF and beam pulse structure
• Compromise between • Cavity rf frequency (aperture, transit factor)
• Power generation and handling (CW/pulsed)
• Experimental requirements
Time/Frequency description of beam signals
1 ms/div 50 ns/div 1 ns/div
Capacitive probeWhy is the signal bipolar?
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 41
Fourier Series and Transforms
𝑓 𝜔 =1
2𝜋 −∞
∞
𝑑𝑡 𝑓 𝑡 𝑒𝑗𝜔𝑡
𝑓 𝑡 =1
2𝜋 −∞
∞
𝑑𝜔 𝑓 𝜔 𝑒−𝑗𝜔𝑡
𝑓 𝑡 =𝑎02+
𝑛=1
∞
𝑎𝑛 cos 𝑛𝜔0𝑡 + 𝑏𝑛 sin 𝑛𝜔0𝑡
𝑎𝑛 =2
𝑇0
𝑇0
𝑑𝑡 𝑓 𝑡 cos 𝑛𝜔0𝑡
𝑏𝑛 =2
𝑇0
𝑇0
𝑑𝑡 𝑓 𝑡 sin 𝑛𝜔0𝑡
We define the use of symmetrical transforms
Time domain Frequency domain
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 42
Fourier transform pairs Function name
Function Fourier transform
Delta function 𝛿 𝑡1
2𝜋
Impulse train (Delta function ‘comb’), period T
ΙΙΙ 𝑡 =
𝑘=−∞
∞
𝛿 𝑡 − 𝑘𝑇 =1
𝑇
𝑛=−∞
∞
𝑒2𝜋𝑗𝑛𝑡𝑇 III 𝜔 =
1
2𝜋
𝑛=−∞
∞
𝑒−𝑗𝜔𝑛𝑇
Gaussian 𝐴 𝑒−𝑡2
2𝜎2 𝐴𝜎𝑒−12𝜔
2𝜎2
Boxcar
𝑏𝑜𝑥𝑐𝑎𝑟 𝑡; 𝐴, 𝑎, 𝑏= 𝐴 𝐻 𝑡 − 𝑎 − 𝐻 𝑡 − 𝑏
H(t) is the Heaviside step function
𝐴
2𝜋𝑏 − 𝑎 𝑒−𝑗𝜔
𝑎+𝑏2 𝑠𝑖𝑛𝑐 𝜔
𝑏 − 𝑎
2
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 43
• Beam signals reveal the bunch structure and loading or fill pattern
• 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 80.5MHzσt = 70pSecσf = 2.3GHzσn = 28
80.5MHz 12.4nSecσ = 70pSec or 2˚Iavg = 1mA
0
1
2
3
4
0 5 10 15 20 25
mA
mp
s
nSec
80.5MHz 12.4nSecσ = 70pSec or 2˚
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 44
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 × 𝑬2 − 𝑬1 = 0𝒏12 ∙ 𝑫2 −𝑫1 = 𝜌𝑠𝒏12 × 𝑯2 −𝑯1 = 𝑲𝑠
𝒏12 ∙ 𝑩2 − 𝑩1 = 0
𝜌𝑠, 𝑲𝑠
𝜵 × 𝑬 = 𝑗𝜔𝑩 = 𝑗𝜔𝜇𝑯𝜵 ×𝑯 = 𝑱 − 𝑗𝜔𝑫
In metals:
𝜔𝑫 ≪ 𝑱Good conductor:
Ohm’s Law: 𝑱 = 𝑬 𝜎
𝜵 × 𝑱 = 𝑗𝜔𝜎𝑩𝜵 × 𝑩 = 𝜇𝑱
𝑩⊥ 𝑧 = 𝑩⊥ 𝑧 = 0 𝑒− 1+𝑗 𝜅𝑧 𝐽∥ 𝑧 = 𝐽∥ 𝑧 = 0 𝑒− 1+𝑗 𝜅𝑧 where 𝜅 = 1 𝛿𝑐=
𝜇𝜔𝜎
2
𝑱𝒏12
⨂𝑩
𝑧
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 45
• The tangential electric field in the conductor derives from Ohm’s Law (not present in perfect conductor)
• The transverse electric field satisfies an impedance boundary condition, with surface impedance, ZS
• A surface resistance (Ohms) is defined as
• Power deposition (W/area) to the surface follows from
Surface resistance and Joule losses
𝑬 =1
𝜎𝛻 × 𝑯 ≅
1
𝜎𝒏 ×
𝜕𝑯
𝜕𝑧= −𝑍𝑠𝒏 × 𝑯⊥
𝑍𝑠 =1 + 𝑗𝑠𝑔𝑛 𝜔
𝜎𝛿𝑐= 𝑅𝑠 1 + 𝑗𝑠𝑔𝑛 𝜔
𝑅𝑠 =1
𝜎𝛿𝑐=
𝜇𝜔
2𝜎𝑑𝑃𝑠𝑢𝑟𝑓
𝑑𝐴= 𝑺 ∙ 𝒏 =
1
2𝔑 𝑬 ×𝑯∗ ∙ 𝒏
=1
2𝑅𝑠 𝐾𝑠
2
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 46
Resistive wall impedance
𝑍0∥
𝑙𝑒𝑛𝑔𝑡ℎ=
𝑍𝑠2𝜋𝑏
=1
2𝜋𝑏
𝜇𝜔
2𝜎1 + 𝑗𝑠𝑔𝑛 𝜔
𝑉 𝜔 = 𝐼 𝜔 𝑍0∥ 𝜔
The beam senses the wall through resistive loading
• The longitudinal resistive wall impedance can be defined as
• The beam will experience a voltage change
• Impedance: 𝑍0∥ = 𝑅𝑒𝑍0
∥ + 𝑗 𝐼𝑚𝑍0∥ (longitudinal, monopole)
What is the significance of the of the resistive and reactive components?
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 47
Building a detector
Measured ceramic gap impedance
• A nonintercepting monitor can be based on monitoring the wall return currents.
• A ceramic break in the beampipe will force the wall current to seek other paths.
• If nothing else is done, the wall currents will find alternative paths.
• The gap impedance is a combination of the gap capacitance and all external parallel elements
• At low frequencies the lowest impedance return path can be distant from the gap itself
• The gap voltage 𝑉𝑔𝑎𝑝 = 𝐼𝑤𝑎𝑙𝑙𝑍𝑔𝑎𝑝 = 𝐼𝑏𝑒𝑎𝑚𝑍𝑔𝑎𝑝can be generated up to the beam voltage
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 48
• We model the beam-monitor interaction with an equivalent circuit
• Beam drive is modeled as a pure current source (infinite input impedance)
• A gap impedance Cgap is inevitably present• Electrodes pierce the beam wall with isolated
feedthroughs
• Typically few – 100s pF
• 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
𝑉𝑚𝑜𝑛 𝜔 = 𝑖𝑤 𝜔𝑍𝑚𝑜𝑛
1 − 𝑗𝜔𝐶𝑔𝑎𝑝𝑍𝑚𝑜𝑛
= 𝑖𝑏 𝜔 𝑍𝑡 𝜔
Zmon
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 49
• We add a network of n resistors across the gap. Rtot = Rsingle/n
• Zmon = Rtot || Cgap
• Broadband pickup
• 𝑉𝑚𝑜𝑛 𝜔 = 𝑖𝑏 𝜔𝑅𝑡𝑜𝑡
1−𝑗𝜔𝐶𝑔𝑎𝑝𝑅𝑡𝑜𝑡
• Within passband 𝑉𝑚𝑜𝑛 𝜔 = 𝑅𝑠𝑖𝑛𝑔𝑙𝑒 𝑛 𝑖𝑏 𝜔
• Practical implementations (eg. SPS WCM)• Ceramic gap
• Many resistors (30 – 100) to reduce sensitivity to beam position
• Ferrite rings to tailor low frequency response ~10 kHz
• High frequency response to several GHz
• Shield for ground currents and noise isolation
Wall current monitor
Rtot
Cgap
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 50
• We adopt a single mode resonance to calculate the coupling impedance
• The transient cavity-beam-waveguide system can be expressed as an equivalent circuit equation (cf. Whittum)
𝑑2
𝑑𝑡2+ 𝜔0
2 𝑉𝑐 = −𝜔0
𝑄𝑤
𝑑
𝑑𝑡𝑉𝑐 +
𝜔0
𝑄𝑒
𝑑
𝑑𝑡𝑉𝐹 − 𝑉𝑅 +𝜔0
𝑟
𝑄
𝑑
𝑑𝑡𝐼𝑏
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.
• We can show that the longitudinal coupling impedance presented by this mode is
𝑍∥ 𝜔 =𝑗𝜔𝜔0 𝑟 𝑄
𝜔02 − 𝜔2 + 𝑗𝜔𝜔0/𝑄𝐿
= 𝑄𝐿 𝑟 𝑄 cos𝜓 𝑒𝑗𝜓
• Impedances can be expressed as Z = R +j X with the reactance 𝑋 = 𝜔𝐿 −1
𝜔𝐶
Resonant effects - Modal circuit equation
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 51
Panofsky and Wenzel derived a powerful theorem connecting longitudinal electric field to transverse momentum impulses.
Relation between longitudinal and transverse effects
∆𝒑 = 𝑄𝑒 𝑎
𝑏
𝑬 + 𝒗 × 𝑩 𝑑𝑡 ∆𝑊 = 𝑄𝑒 𝑎
𝑏
𝑬 ∙ 𝑑𝒔
𝜕
𝜕𝑡∆𝒑 = 𝑄𝑒
𝑎
𝑏 𝜕𝑬
𝜕𝑡𝑑𝑡 + 𝑑𝒔 ×
𝜕𝑩
𝜕𝑡= 𝑄𝑒
𝑎
𝑏
−𝛁 𝑑𝒔 ∙ 𝑬 + 𝑑𝑬
𝜕
𝜕𝑡∆𝑝𝑠 = 𝑄𝑒
𝜕
𝜕𝑡 𝑎
𝑏
𝐸𝑠𝑑𝑡
We can show
𝜕
𝜕𝑡∆𝒑⊥ = −𝑄𝑒
𝑎
𝑏
𝜵⊥ 𝑑𝒔 ∙ 𝑬 − 𝑑𝑬⊥ = −𝜵⊥𝑊+𝑄𝑒 𝑬⊥ 𝑎 − 𝑬⊥ 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
𝜕
𝜕𝑡∆𝒑⊥ = −𝜵⊥𝑊 (Panofsky-Wenzel)
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 52
• Beam impedance response• V(w) = Ib(w) Z(w)
• Everything that sees the beam can be described in terms of a beam coupling impedance
• Narrowband impedances from resonant structures
• Related to wake functions
• Panofsky-Wenzel relates longitudinal to transverse wake/impedances for ultrarelativistic particles
Beam spectrum, impedances and beam loading
DAFNE RF cavity longitudinal impedance
Fundamental mode
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 53
• Devices may posses narrowband as well as broadband impedance characteristics.
• Impedances can be characterized on benchtop measurement stands
Beam coupling impedance to broadband device
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 54
• Very high frequency response dominated by gap capacitance
• Very low frequency response dominated by ferrite and shield induction
Wall Current Monitor equivalent circuit and response
Z0
Z0
Cgap
iw…R Lferrite
𝝎𝒄𝒖𝒕𝒐𝒇𝒇
𝝎𝒉𝒊𝒈𝒉 = 𝟏/𝑹𝑪
𝝎𝒍𝒐𝒘 = 𝑹/𝑳
log
1/RCR/Lw
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 55
• Induced charge densities on the beam pipe walls can be approximated or numerically calculated.
• Total induced charges on the electrodes (assuming 2D Laplace solution) for electrodes of length L
𝑄𝑝𝑙𝑎𝑡𝑒 = 𝐿𝑏
𝑠𝑒𝑐𝑡𝑜𝑟
𝜎 𝜃 𝑑𝜃
Induced charge densities from capacitive coupling
r f
b
← 𝜎 𝜃
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 56
• The pickup plate presents a capacitance C to ground.
• The finite length of the pickup drives a differential current into the monitor.
• 𝑉𝑚𝑜𝑛 𝜔 = 𝑖𝐵 𝜔 𝑍∥ 𝜔
• The image current on the pickup is related to the beam current
• 𝑖𝑖𝑚 𝜔 = 𝑗𝜔𝐴
2𝜋𝑎𝑙
𝑙
𝛽𝑐𝑖𝐵 𝜔
• 𝑉𝑚𝑜𝑛 𝜔 =𝑅
1−𝑗𝜔𝐶𝑅𝑖𝑖𝑚 𝜔 =
𝑅
1−𝑗𝜔𝐶𝑅𝑗𝜔
𝐴
2𝜋𝑎𝑙
𝑙
𝛽𝑐𝑖𝐵 𝜔
• 𝑍∥ 𝜔 =1
𝛽𝑐
1
𝐶
𝐴
2𝜋𝑎
𝑗𝑅𝐶
1−𝑗𝜔𝑅𝐶
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
𝑖𝑖𝑚 𝑡 = −𝐴
2𝜋𝑎𝑙
𝑑𝑄𝑏𝑒𝑎𝑚𝑑𝑡
= −𝐴
2𝜋𝑎𝑙
𝑙
𝛽𝑐
𝑑𝑖𝐵𝑑𝑡
Dt=l/bc
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 57
Capacitive pickup responseThe transfer impedance was found to be
𝑍∥ 𝜔 =1
𝛽𝑐
1
𝐶
𝐴
2𝜋𝑎
𝑗𝑅𝐶
1−𝑗𝜔𝑅𝐶
Defining the cutoff frequency as 𝜔𝑐 = 1/𝑅𝐶
The magnitude and phase are
𝑍∥ =1
𝛽𝑐
1
𝐶
𝐴
2𝜋𝑎
𝜔/𝜔𝑐
1 + 𝜔/𝜔𝑐2, ∠𝑍∥ = tan−1 𝜔𝑐/𝜔
High frequency regime, 𝜔
𝜔𝑐≫ 1, 𝑍∥ → 1
𝑉𝑚𝑜𝑛 𝑡 =1
𝛽𝑐𝐶
𝐴
2𝜋𝑎𝑖𝐵 𝑡
Direct image of beam current with no phase shift.
Low frequency regime , 𝜔
𝜔𝑐≪ 1, 𝑍∥ → 𝑗
𝜔
𝜔𝑐
𝑉𝑚𝑜𝑛 𝑡 =𝑅
𝛽𝑐
𝐴
2𝜋𝑎
𝑑𝑖𝐵
𝑑𝑡
Measured voltage is proportional to time derivative of current.
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 58
Spectrum from low-b beams in capacitive pickups
• Here, b is the distance from the beam to each pickup (or pipe radius for on-axis beam)• Narrow bandwidth difference measurement must contend with nonlinearities in
response. • Requires calibration against variation in beta for each monitoring frequency and
bandwidth.
Point particle distribution 𝜎𝑡 ≅𝑏
2𝛾𝛽𝑐
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 59
The magnetic field of the beam is used to measure the beam current or intensity.
Inductive coupling
𝑩 = 𝜇𝐼𝐵2𝜋𝑟
𝝋
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 60
Current Transformers
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 61
• We will analyze this using Laplace transform pairs instead of Fourier.
• Fourier (symmetrized)
• ℱ 𝜔 =1
2𝜋 −∞∞𝑓 𝑡 𝑒−𝑗𝜔𝑡
• 𝑓 𝑡 =1
2𝜋 −∞∞ℱ 𝜔 𝑒𝑗𝜔𝑡
• Laplace (unsymmetrized)
• 𝐹 𝑠 = 𝜎 + 𝑗𝜔 = ℒ 𝑓 𝑡 = 0−∞𝑓 𝑡 𝑒−𝑠𝑡
• 𝑓 𝑡 = ℒ−1 𝐹 𝑠 =1
2𝜋𝑗 𝜎1−𝑗∞𝜎1+𝑗∞𝐹 𝑠 𝑒𝑠𝑡
• The Laplace transform is useful for analyzing linear, time-invariant, causal systems.
• Differentiation ℒ 𝑓 𝑡 = 𝑠𝐹 𝑠 − 𝑓 0−
• Integration ℒ 0−𝑡𝑓 𝜏 𝑑𝜏 =
𝐹 𝑠
𝑠
• Linearity, Scaling, Shifting, Translation, Convolution
Current Transformer Transfer Impedance
𝑑𝐼𝑠𝑑𝑡
+𝑅𝑠𝐿𝑠𝐼𝑠 = −
1
𝑁𝑠
𝑑𝐼𝑝
𝑑𝑡𝑠 𝐼𝑠 +
𝑅𝑠
𝐿𝑠 𝐼𝑠=−
1
𝑁𝑠𝑠 𝐼𝑝
𝐻 𝑠 = 𝑉𝑠 𝐼𝑝=𝑅𝑠 𝐼𝑠 𝐼𝑝
= −𝑅𝑠𝑁𝑠
𝑠𝜏
1 + 𝑠𝜏𝜏 =
𝐿𝑠𝑅𝑠
𝑠 ≈ 𝑗𝜔 ≫1
𝜏⟹ 𝐻 𝑗𝜔 = −
𝑅𝑠𝑁𝑠
In the high frequency limit
𝑉𝑠 =𝑅𝑠𝐼𝑝𝑁𝑠
𝑃𝑠 = 𝐼𝑠2𝑅𝑠 =
𝑅𝑠𝐼𝑝2
𝑁𝑠2
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 62
• The low frequency response is dominated by R/L
• Practical monitors contain capacitance due to ceramic wall breaks, coupling between windings, etc. This alters the response.
Inductance and Capacitance
𝐿𝑜𝑤 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 ∶ 𝜔 ≪𝑅
𝐿⟹ 𝑍 = 𝑗𝜔𝐿 (B-dot regime)
𝑀𝑖𝑑 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 ∶𝑅
𝐿≪ 𝜔 ≪
1
𝑅𝐶⟹ 𝑍 ≈ 𝑅
𝐻𝑖𝑔ℎ 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 ∶ 𝜔 ≫1
𝑅𝐶⟹ 𝑍 = 1/𝑗𝜔𝐶
1
𝑍=1
𝑅+
1
𝑗𝜔𝐿+ 𝑗𝜔𝐶 ⟹ 𝑍 𝜔 =
𝑗𝜔𝐿
1 +𝑗𝜔𝐿𝑅
−𝜔𝐿𝑅
𝜔𝑅𝐶
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 63
Response from passive transformer
Stray cable capacitance increases risetime.
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 64
Cavity Dipole-mode BPMs (resonant coupling)• Used mainly for high energy electron beams
• Resonant dipole modes have higher shunt impedance than buttons or striplines• High sensitivity
• Wakefields act back on beam
• Cavity BPMs have been developed to produce sub-mm position resolution, for ~mm displacements
• Monopole mode excitation is proportional to beam current
• Antennae pick up combined monopole+dipole signals. Technique requires independent calibration of monopole voltage. • Pillbox: fmono ~ 1.2-1.5* fdipole
• Qload for both modes ~100 – 1000• Mode must decay before arrival of next pulse
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 65
• For very high bandwidth impedances, modal and resonant behavior may no longer adequately capture the beam-structure interaction
• Single and multi-bunch effects (coherence length > bunch length or separation)
• Regenerative
• Frequencies above pipe cutoff
• In this class of phenomena, beam wakes fields are a more relevant description• Time domain
• Causal for ultrarelativistic beams
Wake fields and wake potentials
W(z)
z
𝑎
𝑏
𝑑𝑧 𝐹 = −𝛻𝑉
𝑉 = 𝑒𝐼𝑚𝑊𝑚 𝑧 𝑟𝑚 cos𝑚𝜃for an mth multipole𝑊𝑚
′ 𝑧 =1
2𝜋 −∞
∞
𝑑𝜔 𝑒𝑗𝜔𝑧/𝑐 𝑍𝑚∥ 𝜔
𝑊𝑚 𝑧 =−𝑗
2𝜋 −∞
∞
𝑑𝜔 𝑒𝑗𝜔𝑧/𝑐 𝑍𝑚⊥ 𝜔
𝑍𝑚⊥ 𝜔 =
𝜔
𝑐𝑍𝑚∥ 𝜔
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 66
• The complete diagnostic system starts with beamline (or nearby) sensor
• Cabling to transport signals to data acquisition (DAQ) systems
• Processing electronics, and controls/operator interfaces
Architecture of a diagnostic measurement
Penetrations and racks are laid out for instrumentation• Cable runs about 100 ft
• Diagnostics will use ¼” superflex (Heliax)» solid copper jacket provides
>120 dB shielding effectiveness
No electronics in the tunnel!
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 67
• Controls and actuation for interceptive devices
• Global timing and triggering
• Interlocks for machine, device, and personnel protection
• High level controls for data acquisition, analysis, visualization
Many interfaces are needed to deploy diagnostics
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 68
• Review of Particle Physics, http://pdg.lbl.gov
• P. Strehl, Beam Instrumentation and Diagnostics (Springer, 2006).
• M.G. Minty and F. Zimmerman, Measurement and Control of Charged Particle Beam (Springer, 2003).
• R. Shafer, "Beam Position Monitoring", in AIP Conf. Proc. 212 (1990) 26-58.
• R. Shafer, "Beam position monitor sensitivity for low-b beams", Proc. Beam Instr. Workshop (1994) 303.
• J.H. Cuperas, "Monitoring of particle beams at high frequencies", Nucl. Instrum. Methods Phys. Res. 145 (1977) 219-231.
• D. Whittum, "Introduction to Electrodynamics for Microwave Linear Accelerators", http://uspas.fnal.gov/materials/14Knoxville/slac-pub-7802(1).pdf
• J. Byrd, et. al., http://uspas.fnal.gov/materials/13CSU/USPASMWLecture.pdf
• G. Dome, CERN PAS
• M. Plum, "Interceptive Beam Diagnostics - Signal Creation and Materials Interactions", BIW 2004.
• J. Bosser, ed., "Beam Instrumentation", CERN-PE-ED 001-92 (rev. 1994).
• D. Brandt, ed., "Beam Diagnostics", CERN-2009-005.
• J. Belleman, "From analog to digital", in Digital Signal Processing (D. Brandt, ed.), CERN-2008-003, 131-166.
• P. Forck, "Lecture Notes on Beam Instrumentation and Diagnostics", JUAS (2011).
• R.H. Siemann, "Spectral Analysis of Relativistic Bunched Beams", SLAC-PUB-7159 (1996).
• K. Wittenburg, "Specific instrumentation and diagnostics for high-intensity hadron beams"
References
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 69
• J. Bechhoefer, "Feedback for physicists: A tutorial essay on control", Rev. Mod. Phys. 77 (2005) 783-836.
• http://labs.physics.berkeley.edu/mediawiki/index.php/Introduction_to_Noise
• Knoll, Radiation Detection and Measurement
• Leo, Techniques for Nuclear and Particle Physics Experiments
• ICFA Advanced Beam Dynamics Workshop on Beam-Halo (Halo '03)
• Workshop on Halo Monitoring, SLAC, September 2014, https://portal.slac.stanford.edu/sites/conf_public/bhm_2014/Presentations/Forms/AllItems.aspx
• K. Wittenburg, "Beam halo and bunch purity monitoring" in Brandt (2009).
• S.W. Smith, Digital Signal Processing. A practical guide for Engineers and Scientists, (Newnes, 2003).
• 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.
• G. Lambertson, "Dynamic Devices - Pickups and Kickers", AIP Conference Proceedings 153, 1413 (2016); doi: http://dx.doi.org/10.1063/1.36380
• W. Barry, "Broad-band characteristics of circular button pickups", AIP Conference Proceedings 281, 175 (2016); doi: http://dx.doi.org/10.1063/1.44335
• S. van der Meer, "An Introduction to Stochastic Cooling", AIP Conference Proceedings 153, 1628 (2016); doi: http://dx.doi.org/10.1063/1.36351.
• J.D. Jackson, Classical Electrodynamics (Wiley, 1975).
• A. Hirose, "Electromagnetic Theory", http://physics.usask.ca/~hirose/p812/notes.htm, Chapter 8.
• 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.
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 70
• 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
Physics 862 Accelerator Systems, Fall 2018 Beam Measurements and Instrumentation I 71
• 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