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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


Relativity
Relativity

J. J. Bisognano


Maxwell s equations
Maxwell’s Equations

J. J. Bisognano


Vector identity games
Vector Identity Games

Poynting Vector

Electromagnetic Energy

J. J. Bisognano


Propagation in conductors
Propagation in Conductors

J. J. Bisognano


Free space propagation
Free Space Propagation

J. J. Bisognano


Conductive propagation
Conductive Propagation

J. J. Bisognano


Boundary conditions
Boundary Conditions

J. J. Bisognano


Ac resistance
AC Resistance

J. J. Bisognano


Cylindrical waveguides
Cylindrical Waveguides

  • Assume a cylindrical system with axis z

  • For the electric field we have

  • And likewise for the magnetic field

J. J. Bisognano


Solving for e tangential
Solving for Etangential

J. J. Bisognano


J. J. Bisognano


  • 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

J. J. Bisognano


Te rectangular waveguide mode
TE Rectangular Waveguide Mode

x

a

y

b

J. J. Bisognano


A te mode example
a TE mode Example

J. J. Bisognano


Circular waveguide te m n modes
Circular Waveguide TEm,nModes

J. J. Bisognano


Circular waveguide te m n modes1
Circular Waveguide TEm,nModes

J. J. Bisognano


Circular waveguide tm m n modes
Circular Waveguide TMm,nModes

J. J. Bisognano


Circular waveguide modes
Circular Waveguide Modes

J. J. Bisognano


Cavities
Cavities

d

J. J. Bisognano


Cavity perturbations
Cavity Perturbations

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

J. J. Bisognano


Cavity energy and frequency
Cavity Energy and Frequency

I

++++

E

B

- - - -

-I

Attracts

Repels

J. J. Bisognano



Bead pull
Bead Pull

J. Byrd

J. J. Bisognano


Lorentz theorem
Lorentz Theorem

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

  • Start with the expression

J. J. Bisognano


Vector arithmetic
Vector Arithmetic

J. J. Bisognano


S2

S3

S1

J. J. Bisognano


Scattering matrix
Scattering Matrix

  • Consider a multiport device

  • Discussion follows Altman

S2

Sp

S1

J. J. Bisognano


S matrix
S-matrix

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

J. J. Bisognano


Two port junction
Two-Port Junction

Port X

Port Y

a2

a1

b2

b1

J. J. Bisognano



Lorentz cont
Lorentz/cont.

  • Lorentz theorem implies

  • or

J. J. Bisognano


Unitarity of s matrix
Unitarity of S-matrix

  • Dissipated power P is given by

  • For a lossless junction and arbitrary this implies

J. J. Bisognano



Powering a cavity
Powering a Cavity

b1

b2

a1

a2

J. J. Bisognano


Power flow
Power Flow

J. J. Bisognano


Power flow cont
Power Flow/cont.

J. J. Bisognano


Optimization
Optimization

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

J. J. Bisognano


At resonance
At Resonance

J. J. Bisognano


Shunt impedance
Shunt Impedance

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

  • Following P. Wilson

z2

z1

J. J. Bisognano


Shunt impedance cont
Shunt Impedance/cont

J. J. Bisognano


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

J. J. Bisognano


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 ?

J. J. Bisognano


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 

J. J. Bisognano


Beam loading cont1
Beam Loading/cont.

Ve

e

V2

q

V1 +V2

V1

J. J. Bisognano


Beam loading cont2
Beam Loading/cont.

  • With negligible loss

  • But particle loses

  • Since q is arbitrary, e =0 and

J. J. Bisognano


Beam loading cont3
Beam Loading/cont.

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

J. J. Bisognano


Beam induced voltage
Beam Induced Voltage

  • Consider a sequence of particles at

J. J. Bisognano


Summary of beam loading
Summary of Beam Loading

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

J. J. Bisognano


Vector addition of rf voltages
Vector Addition of RF Voltages

Vb

Vc

Vg

y

j

Vgr

Vbr

y

q

Vb

J. J. Bisognano


Vector algebra
Vector Algebra

J. J. Bisognano


Required generator power
Required Generator Power

  • Trig yields

J. J. Bisognano


E g assume q j
E.g, assume q=j

J. J. Bisognano


Scaling of shunt impedance
Scaling of Shunt Impedance

  • Consider a pillbox cavity of radius b & length L

J. J. Bisognano


Pillbox cavity
Pillbox Cavity

  • The energy stored and power loss are given by

J. J. Bisognano


Summary of scaling
Summary of Scaling

J. J. Bisognano


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

J. J. Bisognano


Periodic structures
Periodic Structures

J. J. Bisognano


Expansion equations
Expansion Equations

J. J. Bisognano


Holes
Holes

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

  • E.g., Bethe says

Tangential electric

field

J. J. Bisognano


Coupled equations
Coupled Equations

J. J. Bisognano



J. J. Bisognano


Pi mode
Pi Mode equations with periodic coefficients have form of periodic function times

  • Tesla Pi-mode

J. J. Bisognano


Floquet theorem
Floquet Theorem equations with periodic coefficients have form of periodic function times

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

J. J. Bisognano


Field relations for cylindrical systems
Field Relations for Cylindrical Systems equations with periodic coefficients have form of periodic function times

J. J. Bisognano


Field relations for cylindrical systems1
Field Relations for Cylindrical Systems equations with periodic coefficients have form of periodic function times

J. J. Bisognano


Integrated force at v c
Integrated Force at v=c equations with periodic coefficients have form of periodic function times

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

  • Only q=/v contributes;i.e.bn=/v

J. J. Bisognano


E g a tm mode
E.g., a TM Mode equations with periodic coefficients have form of periodic function times

J. J. Bisognano


Force on relativistic particle
Force on Relativistic Particle equations with periodic coefficients have form of periodic function times

J. J. Bisognano


Panofsky wenzel theorem
Panofsky Wenzel Theorem equations with periodic coefficients have form of periodic function times

  • 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

J. J. Bisognano


Superconductivity
Superconductivity equations with periodic coefficients have form of periodic function times

  • 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

J. J. Bisognano


Rf superconductivity
RF Superconductivity equations with periodic coefficients have form of periodic function times

  • 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

J. J. Bisognano


Cebaf rf parameters
CEBAF RF Parameters equations with periodic coefficients have form of periodic function times

  • Superconducting 5-cell cavity in CEBAF

J. J. Bisognano


Cebaf cavity assembly
CEBAF Cavity Assembly equations with periodic coefficients have form of periodic function times

J. J. Bisognano


Cavity specifications
Cavity Specifications equations with periodic coefficients have form of periodic function times

  • Frequency 1497 MHz

  • Nominal length 0.5 meters

  • Gradient >5 MeV/m

  • Accel Current 200mA  5 passes

  • Number cells 5

  • R/Q 480 ohms

  • Nominal Q0 2.4·109

  • Loaded QL 6.6 ·106

J. J. Bisognano


What s q l
What’s Q equations with periodic coefficients have form of periodic function times L ?

  • 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

J. J. Bisognano


Would copper work
Would Copper Work equations with periodic coefficients have form of periodic function times

  • 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

J. J. Bisognano


Higher order modes
Higher Order Modes equations with periodic coefficients have form of periodic function times

  • 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 equations with periodic coefficients have form of periodic function times

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

J. J. Bisognano


Loss factor for homs
Loss Factor for HOMs equations with periodic coefficients have form of periodic function times

  • 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 equations with periodic coefficients have form of periodic function times

  • 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

J. J. Bisognano


Couplers and kicks
Couplers and Kicks equations with periodic coefficients have form of periodic function times

  • 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

J. J. Bisognano


Homework topic i
Homework: Topic I equations with periodic coefficients have form of periodic function times

  • 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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