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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

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

slide13
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

a te mode example
a TE mode Example

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

slide27
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

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

slide63
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

J. J. Bisognano

pi mode
Pi Mode
  • Tesla Pi-mode

J. J. Bisognano

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

J. J. Bisognano

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;i.e.bn=/v

J. J. Bisognano

e g a tm mode
E.g., a TM Mode

J. J. Bisognano

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

J. J. Bisognano

superconductivity
Superconductivity
  • 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
  • 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
  • Superconducting 5-cell cavity in CEBAF

J. J. Bisognano

cebaf cavity assembly
CEBAF Cavity Assembly

J. J. Bisognano

cavity specifications
Cavity Specifications
  • 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 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

J. J. Bisognano

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

J. J. Bisognano

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

J. J. Bisognano

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

J. J. Bisognano

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

J. J. Bisognano

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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