lecture 7 - rf linacs - uspas 7_rf...us particle accelerator school lecture 7 rf linacs william a....

US Particle Accelerator School

Lecture 7RF linacs

William A. BarlettaDirector, US Particle Accelerator School

Dept. of Physics, MITEconomics Faculty, University of Ljubljana


S-band (~3 GHz) RF linac

RF-input

RF-cavities


Translate circuit model to a cavity model:Directly driven, re-entrant RF cavity

Outer region: Large, single turn Inductor

Central region: Large plate Capacitor

Beam (Load) current

I

B

EDisplacement current

Wall current

a

Rd

!

L =µo"a

2

2"(R + a)

!

C = "o#R2

d

!

"o = 1LC

= c 2((R + a)d#R2a2

$

% & '

( )

12

Q – set by resistance in outer region

!

Q =LCR

Expanding outer region raises Q

Narrowing gap raises shunt impedance

Source: Humphries, Charged ParticleAccelerators


Properties of the RF pillbox cavity

We want lowest mode: with only Ez & Bθ

Maxwell’s equations are:

and

Take derivatives

==>

!

1r""r

rB#( ) =1c 2

""tEz

!

""rEz =

""tB#

!

""t

1r""r

rB#( )$

% & '

( ) =""t

"B#"r

+B#r

$

% & '

( ) =1c 2

" 2Ez"t 2

!

""r"Ez"r

=""r"B#"t

!

" 2Ez"r 2

+1r"Ez"r

=1c 2

" 2Ez"t 2

d

Ez

b

Bθ

!

" walls =#


For a mode with frequency ω

Therefore,

(Bessel’s equation, 0 order)

Hence,

For conducting walls, Ez(R) = 0, therefore

!

Ez r, t( ) = Ez (r) ei"t

!

" " E z +" E z

r+#c$

% &

'

( ) 2

Ez = 0

!

Ez (r) = Eo Jo"cr

#

$ %

&

' (

!

2"fcb = 2.405


E-fields & equivalent circuit: Ton1o mode

Ez

Bθ

Rel

ativ

e in

tens

ity

r/R

T010

C L


E-fields & equivalent circuitsfor To2o modes

T020


E-fields & equivalent circuits for Tono modes

T030

T0n0 has n coupled, resonant

circuits; each L & C reduced by 1/n


Simple consequences of pillbox model

L

Ez

R

Bθ

Increasing R lowers frequency ==> Stored Energy, E ~ ω-2

E ~ Ez2

Beam loading lowers Ez for thenext bunch

Lowering ω lowers the fractionalbeam loading

Raising ω lowers Q ~ ω -1/2

If time between beam pulses,Ts ~ Q/ω

almost all E is lost in the walls


Keeping energy out of higher order modes

d/b

ωb/c

0 1 2 3 4

TM020

TM010

TE011

TE111Source bandwidth

(green)

Dependence of mode frequency on cavity geometry

Choose cavity dimensions to stay far from crossovers

10

5

1

TE111 modeEnd view

Side view


The beam tube makes the fieldmodes (& cell design) morecomplicated

Ez

Bθ

Peak E no longer on axis Epk ~ 2 - 3 x Eacc

FOM = Epk/Eacc

ωo more sensitive to cavitydimensions Mechanical tuning & detuning

Beam tubes add length & €’sw/o acceleration

Beam induced voltages ~ a-3

Instabilities

a


Cavity figures of merit


Figure of Merit: Accelerating voltage

The voltage varies during time that bunch takes to cross gap reduction of the peak voltage by Γ (transt time factor)

!

" =sin # 2( )#

2 where # =$d%c

2TrfFor maximum acceleration ==> Γ = 2/π

dV

t

Epk


Compute the voltage gain correctly

The voltage gain seen by the beam can computed in the co-moving frame, or we can use the transit-time factor, Γ & compute V at fixed time

!

Vo2 = " E(z)dz

z1

z2#


Figure of merit from circuits - Q

!

E =µo

2H

v"

2dv = 1

2L IoIo

*

!

P = Rsurf

2H

s"

2ds = 1

2IoIo

*Rsurf

!

" Q = LC

Rsurf

= #$$o

%

& '

(

) *

+1

!

Q = "o o Energy storedTime average power loss

= 2# o Energy storedEnergy lost per cycle

!

!

Rsurf =1

Conductivity o Skin depth~ "1/ 2


Measuring the energy stored in the cavityallows us to measure Q

We have computed the field in the fundamental mode

To measure Q we excite the cavity and measure the E fieldas a function of time

Energy lost per half cycle = UπQ

Note: energy can be stored in the higher order modes thatdeflect the beam

!

U = dz0

d" dr2#r

0

b" $Eo

2

2%

& '

(

) * J1

2(2.405r /b)

= b2d $Eo2 /2( ) J1

2(2.405)


Measuring the energy stored in the cavityallows us to measure

We have computed the field in the fundamental mode

To measure Q we excite the cavity and measure the E fieldas a function of time

Energy lost per half cycle = UπQ

Note: energy can be stored in the higher order modes thatdeflect the beam

!

U = dz0

d" dr2#r

0

b" $Eo

2

2%

& '

(

) * J1

2(2.405r /b)

= b2d $Eo2 /2( ) J1

2(2.405)


Figure of merit for accelerating cavity:Power to produce the accelerating field

Resistive input (shunt) impedance at ωο relates power dissipated in walls toaccelerating voltage

Linac literature commonly defines “shunt impedance” without the “2”

Typical values 25 - 50 MΩ

!

Rin = V 2(t)P

= Vo2

2P = Q L

C

!

Rin = Vo2

P ~ 1

Rsurf


Computing shunt impedance

!

P = Rsurf

2H

s"

2ds

!

Rin = Vo2

P

!

Rsurf =µ"

2# dc

= $Zo%skin&rf

where Zo = µo'o

= 377(

The on-axis field E and surface H are generally computed with acomputer code such as SUPERFISH for a complicated cavity shape


Make the linac out of many pillbox cavities

Power the cavities so that Ez(z,t) = Ez(z)eiωt


How can we improve onan array of pillboxes?

Return to the picture of the re-entrant cavity

Nose cones concentrate Ez near beam for fixed stored energy

Optimize nose cone to maximize V2; I.e., maximize Rsh/Q

Make H-field region nearly spherical; raises Q & minimizesP for given stored energy


In warm linacs “nose cones” optimize the voltageper cell with respect to resistive dissipation

Thus, linacs can be considered to bean array of distorted pillbox cavities…

!

Q =LC

R surface

Rf-power in

a

Usually cells are feed in groups not individually…. and


Linacs cells are linked to minimize cost

==> coupled oscillators ==>multiple modes

Zero mode π mode


Modes of a two-cell cavity

. ~ .

.. .. .

f

~

ttl I

f t

ft.

t t

t &

•

t t

t •

t t

, t

• ~

'\ , ,

.. -.

. • ~

J

I I

. l

~ ~

f •

f J

, •

• ,

~ ~

.. .

l 4

~ ~

~ , ~

\-\

" ~ . .. •

•

~ ' .

& ~

f 1

t t

, ,

f t

t t

f •

l f t f

f.

· 1

t t

.. t

1 t

t ..

t ,

, ,

t "

t t

t t

t t

,

f t t

t f

• t

t t

t •

t ,

t t

• 1

t t

t ,

~

t ~

, "

t " .. ..

• •

- -- -I ,... ~ .;; '" ~ ~

§


9-cavity TESLA cell


Example of 3 coupled cavities

!

x j = i j 2Lo and " = normal mode frequency


Write the coupled circuit equationsin matrix form

Compute eigenvalues & eigenvectors to find the three normal modes

!

Lxq =1"q

2 xq where

!

L =

1/"o2 k /"o

2 0k /2"o

2 1/"o2 k /2"o

2

0 k /"o2 1/"o

2

#

$

% % %

&

'

( ( ( and xq =

x1

x2

x3

#

$

% % %

&

'

( ( (

!

Mode q = 0 : zero mode "0 =#o

1+ k x0 =

111

$

%

& & &

'

(

) ) )

!

Mode q =1: "/2 mode #1 =$o x1 =

10%1

&

'

( ( (

)

*

+ + +

!

Mode q = 2 : " mode #2 =$o

1% k x2 =

1%11

&

'

( ( (

)

*

+ + +


For a structure with N coupled cavities

==> Set of N coupled oscillators N normal modes, N frequencies

From the equivalent circuit with magnetic coupling

where B= bandwidth (frequency difference between lowest & highfrequency mode)

Typically accelerators run in the π-mode!

"m ="o

1# Bcosm$N

%

& '

(

) * 1/ 2 +"o 1+ Bcosm$

N%

& '

(

) *


Magnetically coupled pillbox cavities

Cavity 1 C1vrty 2 fa l

~\I- yl CaVIty 2

----~

Coupling hole

E

(e)

CouS)ling SIGt

c

(d)

L

L'ni\'(.'fhll} of lJubljll/1lI

I ..

c


5-cell π-mode cell with magnetic coupling

The tuners change the frequencies by perturbing wall currents ==> changes the inductance==> changes the energy stored in the magnetic field

!

"#o

#o

="UU


Dispersion diagram for 5-cell structure

"'0 (1-

E ( II ) A . (mn(2n-l)) m

ce n = SIn ----n 2N

Phase shift per cell



Power exchange with resonant cavities

Ibeam

Rf power in

Beam power out

Al

R t

Vrf


Effects of wall-losses & external couplingon stored energy, U

Define “wall quality factor”, Qw, & “external” qualityfactor, Qe

Power into the walls is Pw = ωU/ Qw.

If Pin is turned off, then the power flowing out Pe = ωU/Qe

Net rate of energy loss = ωU/ Qw + ωU/Qe = ωU/ Qloaded


Till time & coupling

Loaded fill timeTfill = 2QL/ω

Critically coupled cavity: Pin = Pw ==> 1/Qe = 1/Qw

In general, the coupling parameter ß = Qw /Qe

Ztr

ibeam


Effects of the rf source & beam at resonance

Voltage produced by the generator is

The voltage produced by the beam is!

Vgr =2 ß1+ ß

" RshuntPgen

!

Vb,r =ibeam

Ztr (1+ ß)"IdcRshunt

(1+ ß)


Effects of the rf source & beam at resonance

The accelerating voltage is the sum of these effects

==> Vacc decreases linearly with increasing beam current

!

Vaccel = RshuntPgen2 ß1+ ß

1" Kß

#

$ %

&

' (

)

* + +

,

- . .

= RshuntPwall

where K = Idc2

RshuntPgen

is the "loading factor"


Power flow in standing wave linac

P(t)

Equilibrium value with Ibeam

TimeInject beam at this time


Efficiency of the standing wave linac

!

" =IdcVacc

Pgen=2 ß1+ ß

2K 1# Kß

$

% &

'

( )

*

+ , ,

-

. / /


Schematic of energy flowin a standing wave structure

F[ .. IJ,JJS

~,,1J:,r

~,,1,P • • • • • • •


What makes SC RF attractive?


Comparison of SC and NC RF

Superconducting RF High gradient

==> 1 GHz, meticulous care

Mid-frequencies==> Large stored energy, Es

Large Es

==> very small ΔE/E

Large Q==> high efficiency

Normal Conductivity RF High gradient

==> high frequency (5 - 17 GHz)

High frequency==> low stored energy

Low Es

==> ~10x larger ΔE/E

Low Q==> reduced efficiency


Recall the circuit analog

V(t)

I(t)

CL

Rsurf

As Rsurf ==> 0, the Q ==>∞.

In practice,

Qnc ~ 104 Qsc ~ 1011


Figure of merit for accelerating cavity:Power to produce the accelerating field

Resistive input (shunt) impedance at ωο relates power dissipated in walls toaccelerating voltage

Linac literature more commonly defines “shunt impedance” without the “2”

For SC-rf P is reduced by orders of magnitude

BUT, it is deposited @ 2K

!

Rin = V 2(t)P

= Vo2

2P = Q L

C

!

Rin = Vo2

P ~ 1

Rsurf


Traveling wave linacs


Electromagnetic waves

From Maxwell equations, we can derive

for electromagnetic waves in free space (no charge or current distributionspresent).

The plane wave is a particular solution of the EM wave equation

whenω = c k

Phase of the wave =φ


Dispersion (Brillouin) diagram:monochromatic plane wave

k

ω (k)

The phase of this plane wave is constant for

or

!

d"dt

=# $ k dsdt%# $ kvph = 0

!

vph ="k

= c


Plane wave representation of EM waves

In more generality, we can represent an arbitrary wave as asum of plane waves:

Periodic Case Non-periodic Case


Exercise: Can the plane wave acceleratethe particle in the x-direction?

kz

E

B = (1/ω) k x Evx

!

S = E "H =1µE "B =

1µc

E 2 =#µE 2

!

W ="2E 2 +

1"µ

B2#

$ %

&

' ( =

"2E 2 + c 2B2( ) = "E 2


Can accelerating structures besmooth waveguides?

Assume the answer is “yes”

Then E = E(r,θ) ei(ωt-kz) with ω/k = vph < c

Transform to the frame co-moving at vph < c

Then, The structure is unchanged (by hypothesis) E is static (vph is zero in this frame)==> By Maxwell’s equations, H =0==> ∇E = 0 and E = -∇φ But φ is constant at the walls (metallic boundary conditions)==> E = 0

The assumption is false, smooth structures have vph > c


We need a longitudinal E-fieldto accelerate particles in vacuum

Example: the standing wave structure in a pillbox cavity

What about traveling waves? Waves guided by perfectly conducting walls can have Elong

But first, think back to phase stability To get continual acceleration the wave & the particle must stay in

phase Therefore, we can accelerate a charge with a wave with a

synchronous phase velocity, vph ≈ vparticle < c

EE E


Propagating modes & equivalent circuits

(a)

{b ,

(c

All frequencies can propagate

Propagation is cut-off at low frequencies



To slow the wave, add irises

In a transmission line the irises

a) Increase capacitance, C

b) Leave inductance ~ constant

c) ==> lower impedance, Z

d) ==> lower vph

Similar for TM01 mode in the waveguide

!

"k

=1LC

Z =LC


Fields in waveguides

Figure source: www.opamp-electronics.com/tutorials/waveguide Lessons In Electric Circuits copyright (C) 2000-2002 Tony R. Kuphaldt

Magnetic field

.... ___ .,-- Electric field

Magnetic field

Wave propagation

Magnetic flux lines appear as continuous loops Electric flux lines appear with beginning and end points



Dispersion diagrams

Transmission lineTEM mode

Weakly coupled pillboxesTM0n0 modes

Smooth waveguide,TM0n modes


Dispersion relation for SLAC structure

Small changes in alead to large

reductionin vg


Notation

βg = vg/c = Relative group velocity

Ea = Accelerating field (MV/m)

Es = Peak surface field (MV/m)

Pd = Power dissipated per length (MW/m)

Pt = Power transmitted (MW/m)

w = Stored energy per length (J/m)


Structure parameters for TW linacs

!

rshunt =Ea

2

dPtdz

(M"/m)

Q =w#dPt

dz

rshuntQ

=Ea

2

w#

!

s =Ea

w

w= Elastance (M"/m/µs)

Wacc = emergy/length for acceleration


Variation of shunt impedancewith cell length

Calculations by D. Farkas (SLAC)

I

! .- reflection plane I

~ -6- ~ -------.----------~-------.- - ---.-----.--------~

t:niq'fSlll oj lJllbljmltl

1I 1t ~ 3x - - - 1C 3 2- q.

___ e-.~ ~

t .......

bO ./' "-

"'" /' SLAC Structure " ~ a ! lfo '-' \..

o~------------~--~--------~ (IO ill) PHASE/CELL (degrees)

190


The constant geometry structure

In a structure with a constant geometry,the inductance & capacitance per unit length are constant

==> constant impedance structure

RF- in = Po

CL

δ

z

a

b

RF-out= PL

L


Filling the traveling wave linacL'ni\'(.'fhll} of lJubljll/1lI

P(z)

z


Why do we need beams?

Collide beams

FOMs: Collision rate, energystability, Accelerating field

Examples: LHC, ILC, RHIC


In LHC storage rings…

Limited space & Large rf trapping of particles V/cavity must be high

Bunch length must be large (≤ 1 event/cm in luminous region)

RF frequency must be low

Energy lost in walls must be small Rsurf must be small

SC cavities were the only practical choiceSC cavities were the only practical choice


For ILC, SC rf provides high power,low ε beams at high efficiency

• To deliver required luminosity (500 fb-1 in 4 years) ==>

– powerful polarized electron & positron beams (11 MW /beam)

– tiny beams at collision point ==> minimizing beam-structure interaction

• To limit power consumption ==> high “wall plug” to beam power efficiency

– Even with SC rf, the site power is still 230 MW !


Why do we need beams?

Intense secondary beams

FOM: Secondaries/primaryExamples: spallation neutrons,

neutrino beams

1 MW target at SNS1 MW target at SNS


The Spallation Neutron Source

1 MW @ 1GeV (compare with ILC 11 MW at 500 GeV(upgradeable to 4 MW)

==> miniscule beam loss into accelerator

==> large aperture in cavities ==> large cavities

==> low frequency

==> high energy stability

==>large stored energy

==> high efficiency at Ez

==> SC RF SNS


Matter to energy:Synchrotron radiation science

Synchrotron light source(pulsed incoherent X-ray emission)

FOM: Brilliance v. λ

B = ph/s/mm2/mrad2/0.1%BWPulse duration

Science with X-rays Imaging Spectroscopy


Matter to energy: Energy Recovery Linacs Hard X-rays ==> ~5 GeVHard X-rays ==> ~5 GeV

Synchrotron light source(pulsed incoherent X-ray emission)

Pulse rates Pulse rates –– kHz => MHz kHz => MHzX-ray pulse durationX-ray pulse duration ≤≤ 1 1 pspsHigh average High average e-beam e-beam brilliancebrilliance&& e-beam e-beam duration duration ≤≤ 1 1 psps

⇒⇒ One pass throughOne pass through ringring⇒⇒ Recover beam energyRecover beam energy

⇒⇒ High efficiencyHigh efficiency

⇒⇒ SC RFSC RF


Even higher brightness requirescoherent emission ==> FEL

Free electron laser

FOM: Brightness v. λ Time structure


Full range of FEL-based science requires…

Pulses rates 10 Hz to 10 MHz (NC limited to ~ 100 Hz) High efficiency

Pulse duration 10 fs - 1 ps

High gain Excellent beam emittance

==> Minimize wakefield effect==> large aperture==> low frequency

Stable beam energy & intensity==> large stored energy in cavities==> high Q

===> SC RF

lecture 7 - rf linacs - uspas 7_rf...us particle accelerator school lecture 7 rf linacs william a....

Documents