US Particle Accelerator School
Unit 4 - Lecture 10
RF-accelerators: Standing wave linacs
William A. Barletta
Director, United States Particle Accelerator School
Dept. of Physics, MIT
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Q of the lumped circuit analogy
The width is0
=R
L /.C
Z( ) ~ 12
o2
2
+ ( RC)2
1
Converting the denominator of Z to a real number we see that
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Measuring the energy stored in the cavity
allows us to measure
We have computed the field in the fundamental mode
To measure Q we excite the cavity and measure the E field
as a function of time
Energy lost per half cycle = U Q
Note: energy can be stored in the higher order modes that
deflect the beam
U = dz0
ddr2 r
0
b Eo2
2
J1
2(2.405r /b)
= b2d Eo2 /2( ) J1
2(2.405)
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Figure of Merit: Accelerating voltage
The voltage varies during time that bunch takes to cross gap
reduction of the peak voltage by (transt time factor)
=sin 2( )
2 where = d
c
2
TrfFor maximum acceleration ==> = 2/
d
Vt
Epk
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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, & compute V at fixed time
Vo2
= E(z)dzz1
z2
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Make the linac with a
series of pillbox cavities
Power the cavities so that Ez(z,t) = Ez(z)ei t
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How can we improve on
an 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 & minimizes
P for given stored energy
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In warm linacs “nose cones” optimize the voltage
per cell with respect to resistive dissipation
Thus, linacs can be considered to be
an array of distorted pillbox cavities…
Q =L
CR surface
Rf-power in
a
Usually cells are feed in groups not individually…. and
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Linacs cells are linked to minimize cost
==> coupled oscillators ==>multiple modes
Zero mode mode
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Modes of a two-cell cavity
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9-cavity TESLA cell
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Example of 3 coupled cavities
x j = i j 2Lo and = normal mode frequency
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Write the coupled circuit equationsin matrix form
Compute eigenvalues & eigenvectors to find the three normal modes
Lxq =1
q2 xq where L =
1/ o2 k / o
2 0
k /2 o2 1/ o
2 k /2 o2
0 k / o2 1/ o
2
and xq =
x1
x2
x3
Mode q = 0 : zero mode 0 = o
1+ k x0 =
1
1
1
Mode q =1: /2 mode 1 = o x1 =
1
0
1
Mode q = 2 : mode 2 = o
1 k x2 =
1
1
1
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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& high frequency mode)
Typically accelerators run in the -mode
m = o
1 Bcosm
N
1/ 2 o 1+ Bcosm
N
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Magnetically coupled pillbox cavities
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5-cell -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
=U
U
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Dispersion diagram for 5-cell structure
Em cell n( ) = An sinm (2n 1)
2N
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Evolution of the Los Alamos structure
/2-mode has high vg & good frequency stability, BUT low Rsh
Bi-periodic structure raises Rsh by shrinking the unexcited cells
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Side-coupled cavity
Side-Coupled Structure shrinks the unexcited cells to zero.
Rsh is the same as the -mode
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Power exchange with resonant cavities
Rf power in
Beam power out
Al
R t
Vrf
Ibeam
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Effects of wall-losses & external coupling
on stored energy, U
Define “wall quality factor”, Qw, & “external” quality
factor, Qe
Power into the walls is Pw = U/ Qw.
If Pin is turned off, then the power flowing out Pe = U/Qe
Net rate of energy loss = U/ Qw + U/Qe = U/ Qloaded
Pin
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Till time & coupling
Loaded fill time
Tfill = 2QL/
Critically coupled cavity: Pin = Pw ==> 1/Qe = 1/Qw
In general, the coupling parameter ß = Qw /Qe
Ztr
ibeam
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At resonance, the rf source & the beam
have the following effects
Voltage produced by the generator is
The voltage produced by the beam is
Vgr =2
1+RshuntPgen
Vb,r =ibeam
Ztr (1+ )
IdcRshunt
(1+ )
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At resonance, the rf source & the beam
have the following effects
The accelerating voltage is the sum of these effects
==> Vacc decreases linearly with increasing beam current
Vaccel = RshuntPgen
2
1+1
K
= RshuntPwall
where K =Idc
2Rshunt
Pgen is the "loading factor"
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Power flow in standing wave linac
P(t)
Equilibrium value with Ibeam
TimeInject beam
at this time
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Efficiency of the standing wave linac
=IdcVacc
Pgen
=2
1+2K 1
K
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Schematic of energy flow
in a standing wave structure
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What makes SC RF attractive?
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Comparison of SC and NC RF
Superconducting RF
High gradient
==> 1 GHz, meticulous care
Mid-frequencies
==> Large stored energy, Es
Large Es
==> very small E/E
Large Q
==> high efficiency
Normal Conductivity RF
High gradient
==> high frequency (5 - 17 GHz)
High frequency
==> low stored energy
Low Es
==> ~10x larger E/E
Low Q
==> reduced efficiency
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Recall the circuit analog
V(t)
I(t)
C
L
Rsurf
As Rsurf ==> 0, the Q ==>
In practice,
Qnc ~ 104 Qsc ~ 1011
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Figure of merit for accelerating cavity:power to produce the accelerating field
Resistive input (shunt) impedance at relates power dissipated in walls to
accelerating 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 = Vo
2
P ~
1
Rsurf
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Why do we need beams?
Collide beams
FOMs: Collision rate, energy
stability, Accelerating field
Examples: LHC, ILC, RHIC
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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
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For ILC SC rf provides high power, highquality beams at high efficiency
~4.5 km ~13 km~13 km
• 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 !
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Why do we need beams?
Intense secondary beams
FOM: Secondaries/primary
Examples: spallation neutrons,
neutrino beams
1 MW target at SNS1 MW target at SNS
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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
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Matter to energy:Synchrotron radiation science
Synchrotron light source
(pulsed incoherent X-ray emission)
FOM: Brilliance v.
B = ph/s/mm2/mrad2/0.1%BW
Pulse duration
Science with X-rays Imaging
Spectroscopy
ERLsERLs
FELs
3rd Generation Sources
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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 => MHz
X-ray pulse durationX-ray pulse duration 1 1 psps
High 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
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Even higher brightness requires
coherent emission ==> FEL
Free electron laser
FOM: Brightness v.
Time structure
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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
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End of unit
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Field measurement
Nylon line
Motor
Rf generator
R t
V rf
Frequencydetector