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Accelerator Physics Topic I Acceleration. Joseph Bisognano Synchrotron Radiation Center University of Wisconsin. Relativity. Maxwell’s Equations. Vector Identity Games. Poynting Vector. Electromagnetic Energy. Propagation in Conductors. Free Space Propagation. Conductive Propagation.

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Accelerator physics topic i acceleration

Accelerator PhysicsTopic IAcceleration

Joseph Bisognano

Synchrotron Radiation Center

University of Wisconsin

J. J. Bisognano



J. J. Bisognano

Maxwell s equations

Maxwell’s Equations

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Vector identity games

Vector Identity Games

Poynting Vector

Electromagnetic Energy

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Propagation in conductors

Propagation in Conductors

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Free space propagation

Free Space Propagation

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Conductive propagation

Conductive Propagation

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Boundary conditions

Boundary Conditions

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Ac resistance

AC Resistance

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Cylindrical waveguides

Cylindrical Waveguides

  • Assume a cylindrical system with axis z

  • For the electric field we have

  • And likewise for the magnetic field

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Solving for e tangential

Solving for Etangential

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J j bisognano topic one acceleration

  • Maxwell’s equations then imply (k=/c)

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J j bisognano topic one acceleration

  • All this implies that E0zandB0z tell it all with their equations

  • For simple waveguides, there are separate solutions with one or other zero (TM or TE)

  • For complicated geometries (periodic structures, dielectric boundaries), can be hybrid modes

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Te rectangular waveguide mode

TE Rectangular Waveguide Mode





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A te mode example

a TE mode Example

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Circular waveguide te m n modes

Circular Waveguide TEm,nModes

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Circular waveguide te m n modes1

Circular Waveguide TEm,nModes

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Circular waveguide tm m n modes

Circular Waveguide TMm,nModes

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Circular waveguide modes

Circular Waveguide Modes

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Cavity perturbations

Cavity Perturbations

Now following C.C. Johnson, Field and Wave Dynamics

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Cavity energy and frequency

Cavity Energy and Frequency





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Energy change of wall movement

Energy Change of Wall Movement

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Bead pull

Bead Pull

J. Byrd

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Lorentz theorem

Lorentz Theorem

  • Let and be two distinct solutions to Maxwell’s equations, but at the same frequency

  • Start with the expression

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Vector arithmetic

Vector Arithmetic

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J j bisognano topic one acceleration

  • Using curl relations for non-tensor m, e one can show that expression is zero

  • So, in particular, for waveguide junctions with an isotropic medium we have




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Scattering matrix

Scattering Matrix

  • Consider a multiport device

  • Discussion follows Altman




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S matrix


  • Let apamplitude of incident electric field normalize so that ap2 = 2(incident power) and bp2 = 2(scattered power)

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Two port junction

Two-Port Junction

Port X

Port Y





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Implication of lorentz theorem

Implication of Lorentz Theorem

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Lorentz cont


  • Lorentz theorem implies

  • or

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Unitarity of s matrix

Unitarity of S-matrix

  • Dissipated power P is given by

  • For a lossless junction and arbitrary this implies

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Symmetrical two port junction

Symmetrical Two-Port Junction

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Powering a cavity

Powering a Cavity





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Power flow

Power Flow

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Power flow cont

Power Flow/cont.

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  • With no beam, best circumstance is ; I.e., no reflected power

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At resonance

At Resonance

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Shunt impedance

Shunt Impedance

  • Consider a cavity with a longitudinal electric field along the particle trajectory

  • Following P. Wilson



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Shunt impedance cont

Shunt Impedance/cont

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Shunt impedance cont1

Shunt Impedance/cont.

  • Define

  • where P is the power dissipated in the wall (the term)

  • From the analysis of the coupling “b”

  • where is the generator power

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Beam loading

Beam Loading

  • When a point charge is accelerated by a cavity, the loss of cavity field energy can be described by a charge induced field partially canceling the existing field

  • By superposition, when a point charge crosses an empty cavity, a beam induced voltage appears

  • To fully describe acceleration, we need to include this voltage

  • Consider a cavity with an excitation V and a stored energy

  • What is ?

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Beam loading cont

Beam Loading/cont.

  • Let a charge pass through the cavity inducing and experiencing on itself.

  • Assume a relative phase

  • Let charge be bend around for another pass after a phase delay of 

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Beam loading cont1

Beam Loading/cont.





V1 +V2


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Beam loading cont2

Beam Loading/cont.

  • With negligible loss

  • But particle loses

  • Since q is arbitrary, e =0 and

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Beam loading cont3

Beam Loading/cont.

  • Note: we have same constant (R/Q) determining both required power and charge-cavity coupling

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Beam induced voltage

Beam Induced Voltage

  • Consider a sequence of particles at

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Summary of beam loading

Summary of Beam Loading

  • References: Microwave Circuits (Altman); HE Electron Linacs (Wilson, 1981 Fermilab Summer School)

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Vector addition of rf voltages

Vector Addition of RF Voltages











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Vector algebra

Vector Algebra

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Required generator power

Required Generator Power

  • Trig yields

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E g assume q j

E.g, assume q=j

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Scaling of shunt impedance

Scaling of Shunt Impedance

  • Consider a pillbox cavity of radius b & length L

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Pillbox cavity

Pillbox Cavity

  • The energy stored and power loss are given by

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Summary of scaling

Summary of Scaling

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More multicell cavities

More Multicell Cavities

  • Given a solution of a single cell cavity, one can consider the coupling of multiple cells in that mode by an expansion where

    • the expansion coefficients give the strength of excitation of each cell in that mode

    • coupling comes from perturbation of purely conductive boundary by holes communicating field between cells

  • This is a recondite subject, with all sorts of dangers from “conditional” covergence of Fourier series; see Slater (and Gluckstern) for a complete picture of this

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Periodic structures

Periodic Structures

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Expansion equations

Expansion Equations

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  • But if there are holes in the cavity talking to neighboring cells, we have

  • E.g., Bethe says

Tangential electric


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Coupled equations

Coupled Equations

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J j bisognano topic one acceleration

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J j bisognano topic one acceleration

  • Result of Floquet Theorem that solutions of differential equations with periodic coefficients have form of periodic function times exp(jb0z)

  • Phase velocities less than c  particle acceleration possible

  • From Slater, RMP 20,473

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Pi mode

Pi Mode

  • Tesla Pi-mode

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Floquet theorem

Floquet Theorem

  • For example, for a disk loaded circular cylindrical structure, the TM01 is of the form

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Field relations for cylindrical systems

Field Relations for Cylindrical Systems

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Field relations for cylindrical systems1

Field Relations for Cylindrical Systems

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Integrated force at v c

Integrated Force at v=c

  • Let be longitudinal force seen by a particle. Consider a trajectory z=vt, r=r0. The integrated force is then=

  • Only q=/v contributes;/v

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E g a tm mode

E.g., a TM Mode

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Force on relativistic particle

Force on Relativistic Particle

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Panofsky wenzel theorem

Panofsky Wenzel Theorem

  • Pure TE mode doesn’t kick; pure TM mode, as in previous example, behaves to cancel denominator, so falls off as -2 ; hybrid modes don’t cancel denominator, so finite kick may obtain even when v=c

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  • Basic mechanism

    • Condensation of charge carriers into Cooper pairs, coupled by lattice vibrations

    • Bandgap arises, limiting response to small perturbations (e.g., scattering)

    • No DC resistance

  • At temperatures above 0 K, some of the Cooper pairs are “ionized”

  • But for DC, these ionized pairs are “shorted out” and bulk resistance remains zero

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Rf superconductivity

RF Superconductivity

  • But pairs exhibit some inertia to changing electromagnetic fields, and there are some residual AC fields (sort of a reactance)

  • These residual fields can act on the ionized, normal conducting carriers and cause dissipation

  • But it’s very small at microwave frequencies (getting worse as f2)

  • At 1.3 GHz, copper has Rs~10 milli-ohm

  • At 1.3 GHz, niobium has Rs~800 nano-ohm at 4.2 K

  • At 1.3 GHz, niobium has Rs~15 nano-ohm at 2 K

  • Q’s of 1010 vs. 104

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Cebaf rf parameters

CEBAF RF Parameters

  • Superconducting 5-cell cavity in CEBAF

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Cebaf cavity assembly

CEBAF Cavity Assembly

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Cavity specifications

Cavity Specifications

  • Frequency1497 MHz

  • Nominal length0.5 meters

  • Gradient>5 MeV/m

  • Accel Current200mA  5 passes

  • Number cells5

  • R/Q480 ohms

  • Nominal Q02.4·109

  • Loaded QL6.6 ·106

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What s q l

What’s QL ?

  • Run in linac mode, essentially on crest, j = 0

  • In storage rings, j  0 for longitudinal focusing

  • Wall losses V2/R=(2.5 · 106volts)2 /(480W ·2.4·109 ) are 5.4 watts

  • Power to beam: (200mA  5 passes)·(2.5 · 106 volts) is 2500 watts

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Would copper work

Would Copper Work

  • Typical Q is now  104

  • Wall losses (2.5 · 106volts)2 /(480W ·104 ) are now 1.3 MW vs. beam power of 2500 watts

  • Some optimizations could yields “2’s” of improvement

  • More importantly, SRF losses are at 2 K, which requires cryogenic refrigeration.

  • Efficiencies are order 10-3

  • So, 5 watts at 2 K is 5 kW at room temperature, but still factor of 100 to the good

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Higher order modes

Higher Order Modes

  • There are higher order modes, which can be excited by the beam

  • These can generate wall losses, and fields can act on beam to generate destructive collective effects

  • First question is whether wall losses are large or small compared to fundamental wall losses

J. J. Bisognano

Loss factors

Loss Factors

  • When a bunch passes through a cavity, it loses energy of

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Loss factor for homs

Loss Factor for HOMs

  • Bunch spectrum extends out to

  • For a typical 1 ps linac bunch,

  • For for a 1.5 GHz fundamental, there are many tens of longitudinal HOMs for the beam to couple; coupling is weaker than to fundamental because of more rapid temporal and spatial variation

  • From codes,

J. J. Bisognano

Power estimates

Power Estimates

  • For 1 mA CEBAF 5-pass beam, with 0.5 pC/superbunch, we have only10mW of loss

  • But in, say FEL application, with 100 pC bunches at 5mA, we have 10 watts in the wall, more than the power dissipated by fundamental!

  • So extraction of HOM power is issue for high current applications with short bunches

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Couplers and kicks

Couplers and Kicks

  • Waveguide couplers break cylindrical symmetry

  • Result is that the nominal TM01mode now has TE content and m0; hybridized

  • By Panofsky-Wenzel, introduces steering and skew quad fields that require compensation

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Homework topic i

Homework: Topic I

  • From Accelerator Physics, S.Y. Lee

    • Reading: Chapter 3, VIII, p. 352 & onward

    • Problems:3.8.1 (really table 3.10), 3.8.2, 3.8.4

  • The next generation CEBAF cavity can achieve 20 MV/m gradients with a 7 cell structure. a) Assuming R/Q is 7/5 higher, what would be the heat load generated per cavity if the Q is unchanged? b) How much higher a Q is necessary to main the heat load at the levels from the first generation cavity discussed in the lecture?

  • Consider a single cell RF system operating at 500 MHz with an effective length of 0.3 meters, which is to be operated at a gradient of 2 MV/m with beam current of 100 mA. Assume the R/Q of the cavity is 100 ohms and that the Q from resistive wall losses is 40,000.

    A) Calculate the optimal coupling coefficient for powering the cavity with the 100 a mA beam passing through it.

    B) Calculate the power necessary with 100 mA beam to power the cavity.

    C) Calculate the power received by the beam and the power dissipated in wall losses.

    D) Describe what will happen to the power requirements and reflected power from the cavity if the beam is lost, but the feedback system attempts to maintain the 2 MV/meter gradient.

  • A CEBAF problem: Use the nominal specs given in lecture

    • a) For a 1 mA at 5 MV/m calculate total power and reflected power on resonance and 10 degrees off resonance with bunch on crest.

    • b) If this beam is allowed to pass for a second time through the cavity for energy recovery at 170 degrees off accelerating crest, simultaneously with the first pass beam on crest, calculate the total and reflected power when the cavity is tuned on resonance.

J. J. Bisognano

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