a brief introduction of radio frequency cavity in particle ... · in accelerator physics (and many...

A Brief Introduction of Radio Frequency Cavity

in Particle Accelerator

Nawin JuntongAccelerator Physicist

July 30, 2014

ATD seminarSLRI, Nakhon Ratchasima, Thailand

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Contents outline

• A brief history of DC and RF acceleration

• Principle of RF cavity

• RF cavity classification

• RF cavity properties

• RF cavity design criterion

• Effects of RF cavity to particle beam and suppression method

• RF cavity fabrication

• 2nd RF cavity at SLRI

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Lorentz Force

• Force that keeps a charged particle circulate in a ring or travel along a line.

• In the presence of E-field, a charged particle will experience a force in direction of E-field.

• In the presence of B-field, a charged particle will experience a force in a perpendicular direction of B-field.

𝐹 = 𝑞𝐸 + 𝑞( 𝑣 × 𝐵)

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DC Acceleration

• Use high DC voltage to accelerate a charged particle.

• Limitations:– High energy = High voltage DC source

(1 MeV = 1 MV DC source)

– Impossible for circular machine

𝐸 ∙ 𝑑 𝑠 = 0

– Breakdown limit of material

Cockcroft-Walton’s electrostatic accelerator

(1932)

Protons were accelerated and hit the lithium target producing helium and energy

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DC Acceleration

• Energy calculation

N. Juntong ([email protected]) February 7, 2014

Siam Photon Laboratory

V = 100 kVUE = 100 keV, 1 eV = 1.602×10−19 JUE = 1.6 x 10-14 J

Special relativity theory

E = E0 + Ek = mc2 = (𝑝𝑐)2+𝐸02 = γ𝐸0

𝛾 =1

1 − 𝛽2, 𝛽 =

𝑣

𝑐

𝛽 = 1 −1

𝛾2

Electrostatic potential energy

𝑈𝐸 = − 𝑞𝐸 ∙ 𝑑 𝑠 = qV Kinetic energy UE = Ek = mν2/2 = 1.6 x 10-14 J

ν = 2𝑈𝐸/𝑚 , me= 9.1 x 10-31 kg

ν = 1.8 x 108 m/sν = 0.6 c , c = 2.998 x 108 m/s

γ = E0 + Ek /E0

β = 1 − (𝐸0 (𝐸0+𝐸𝑘))2

, E0= 0.511 MeV

β = 0.548

ν = 0.548 c

40 MeV -> v = 0.99992 c

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RF Acceleration (time-varying field)

• Rolf Wideroe applied a 25-kV, 1 MHz AC voltage to the drift tube between two grounded electrodes. The beam experienced an accelerating voltage in both gaps.

• He accelerated Na and K beams to 50 keV kinetic energy equal to twice the applied voltage.

• This is not possible using electrostatic voltages

1928 - World’s first RF accelerator

Drift tube linac (DTL)

For slow particles- Proton @ few MeV

Drift tube length vary according to the velocity of particle and the frequency of a source

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RF Acceleration (time-varying field)

Image courtesy of F. Gerigk Image courtesy of F. Gerigk

Wideroe linac (1928) Louis Alvarez linac (1946)Energy gain

∆𝐸 = 𝑞𝑁𝑔𝑎𝑝𝑉𝑅𝐹

𝑙𝑛 =𝑣

2𝑓

synchronism condition

- The length of the drift tubes become too long for higher velocities.

- The drift tube become antennas when operation at higher frequencies (> 10MHz) - very poor efficiency

- Put Wideroe linac in a conducting cylinder.

- Each gap has the same resonant frequency, which is now determined by the diameter of the cylinder, the distance between the drift tubes, and their diameter

- Frequency up to 200MHz

Π -mode 2Π -mode

This is not the RF cavity This is the RF cavity

Yes, we can accelerate without cavities, but reachinghigher energies becomes very inefficient

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Principle of RF cavity

Maxwell’s equations

𝛻 × 𝐻 = 𝐽 +𝜕𝐷

𝜕𝑡

𝛻 × 𝐸 = −𝜕𝐵

𝜕𝑡

𝛻 ∙ 𝐷 = 𝜌

𝛻 ∙ 𝐵 = 0

𝐷 = 𝜀𝐸, 𝐵 = 𝜇𝐻J- current density, ρ – charge density

The total flux entering a bounded region equals the total flux leaving the same region

The total flux of electric displacement crossing a closed surface equals the total electric charge enclosed by that surface

A time-dependent magnetic field generates an electric field

The magnetic field H integrated around a closed loop equals thetotal current passing through that loop

Maxwell’s equations are of fundamental importance in electromagnetism, because they tell us the fields that exist in the presence of various charges and materials.

In accelerator physics (and many other branches of appliedphysics), there are two basic problems:

• Find the electric and magnetic fields in a system of chargesand materials of specified size, shape and electromagneticcharacteristics.

• Find a system of charges and materials to generate electricand magnetic fields with specified properties.

- Andy WOLSKI

Maxwell’s equations are linear- This means that if two fields satisfy Maxwell’s equations, so

does their sum.- As a result, we can apply the principle of superposition to

construct complicated electric and magnetic fields just by adding together sets of simpler fields

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Pill-box cavitySolving in cylindrical coordinates (r,φ,z)

𝐸𝑟 = 𝑗𝑘𝑧𝑘𝑐𝐸0𝐽1(𝑘𝑐𝑟)𝑒

−𝑗𝑘𝑧𝑧𝑒𝑗𝜔𝑡

𝐸𝑧 = 𝐸0𝐽0(𝑘𝑐𝑟)𝑒−𝑗𝑘𝑧𝑧𝑒𝑗𝜔𝑡

𝐵𝜑 = 𝑗𝑘

𝑍0𝑘𝑐𝐸0𝐽1(𝑘𝑐𝑟)𝑒

−𝑗𝑘𝑧𝑧𝑒𝑗𝜔𝑡

J0 and J1 are Bessel functions of the first kind of zeroth and first order

ω = 2πf is the angular frequency

k is the wave number

kz is the wave propagation constant

Z0 is the wave-impedance in free space

E0 is the amplitude of the electric field

j indicates imaginary parts

r = a

Ez = 0 at r = a cut off wave length λc

𝑘 =2𝜋

𝜆=𝜔

𝑐𝑍0 =

𝜇0𝜀0

= 377 Ω

Only waves with shorter wavelength, or a frequency above the cut off frequency ωc can propagate in the cavity undamped 𝑘𝑧

2 = 𝑘2 − 𝑘𝑐2

𝑘𝑐 =2𝜋

𝜆𝑐=𝜔𝑐𝑐

𝜆𝑐 ≈ 2.61 𝑎

𝑣𝑝ℎ =𝜔

𝑘𝑧

Phase velocity of wave

𝑘𝑧2 =

𝜔2

𝑣𝑝ℎ2 =

𝜔

𝑐

2

−𝜔𝑐𝑐

2

Dispersion relation (Brioullin diagram)

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Image courtesy of F. Gerigk

Dispersion relation (Brioullin diagram) of a cylindrical pipe

- Each frequency corresponds to a certain phase velocity

- The phase velocity is always larger than the speed of light

- At ω = ωc the propagation constant kz goes to zero and the phase velocity becomes infinite

- It is impossible to accelerate particles in a circular waveguide because synchronism between the particles and the RF is impossible (particles would have to travel faster than light to be synchronous with the RF)

- Information and therefore energy travels at the speed of the group velocity vg = dω/dkz, and is always slower than the speed of light

𝑘𝑧2 =

𝜔2

𝑣𝑝ℎ2 =

𝜔

𝑐

2

−𝜔𝑐𝑐

2

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Principle of RF cavity (TW)Slowing down phase velocity, in order to accelerate particles

- Put obstacles into waveguide, i.e. discs.

𝜔 =2.405𝑐

𝑏1 + 𝜅(1 − cos(𝑘𝑧𝐿)𝑒

−𝛼ℎ)

𝜅 =4𝑎3

3𝜋𝐽12(2.405)𝑏2𝐿

≪ 1, α ≈2.405

𝑎

Image courtesy of F. Gerigk Thomas P. Wangler, RF Linear Accelerator

Dispersion curve of disc-loaded TW structure

𝑣𝑝ℎ ≤ 𝑐, 2𝜋 3 < 𝑘𝑧𝐿 < 𝜋When a structure operates in the 2π/3 mode it means that the RF phase shifts by 2π/3 per cell, or in other words one RF period stretches over three cells. The particles then move in synchronism with the RF phase from cell to cell.

Constant impedance each cell’s bore radii are kept constant, i.e. uniform structure. The electromagnetic wave becomes more and more damped along the structure.

Constant gradient bore radius are changing from cell to cell. The maximum possible accelerating gradient in each cell.

R. B. Neal, M Report No. 259 (1961), Hansen Laboratories of Physics, Stanford University

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Principle of RF cavity (SW)

http://www.boost.org/doc/libs/1_54_0_beta1/libs/math/doc/html/math_toolkit/bessel/bessel_root.html

beam

Closing both ends of a circular wave guide with electric walls. This will yield multiple reflections on the end walls until a standing wave pattern is established.

Only with discrete frequencies and discrete phase changes can exist in the cavity. If one feeds RF power at a different frequency, then the excited fields will be damped exponentially.

𝜔𝑛 =𝜔0

1 + 𝑘 cos( 𝑛𝜋 𝑁), 𝑛 = 0,1,… ,𝑁

Dispersion curve of half-cell terminated SW structure, magnetic coupling

𝑘 =𝜔𝜋 − 𝜔0

𝜔0

Thomas P. Wangler, RF Linear Accelerator

Image courtesy of F. Gerigk

The frequency bandwidth |ωπ-ω0| is independent of the number of cells, which means that we can determine the cell-to-cell coupling constant by measuring the complete structure

For electric coupling the 0-mode has the lowest frequency and the π-mode has the highest frequency. In case of magnetic coupling this behavior is reversed.

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RF cavity classification



Constant velocity, constant RF frequency Changing velocity, constant RF frequency

Non-accelerating cavities Changing velocity, variable RF frequency

Application

EM wave EM mode

Material

Travelling wave (TW)- Constant impedance- Constant gradient

Standing wave (SW)

Normal conducting (NC)

Superconducting (SC)

TEM

TM TE

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RF cavity classification Application

Constant velocity, constant RF frequency- Relativistic particles (β = v/c > 0.99)- Synchrotron, no need to adapt RF frequency to the

revolution frequency of the particles- Linac, no need to change the distance between

accelerating gaps- Electron accelerators energy > 1 MeV- High energy proton accelerators (GeV)- Ion accelerators 10s-100s of GeV (depending on

ion species)

Changing velocity, constant RF frequency- The velocity of the particles is increased along the

acceleration cycle but where the RF frequency can be kept constant

- Cyclotrons, use single-cell cavities to avoid any problems with synchronicity between subsequent gaps of multi-cell structures

- Low-β ion and proton linacs, use multi-cell structures in order to keep the overall linac length compact, it is necessary to adapt the gap distance to the velocity of the particles

Non-accelerating cavities- Mainly used with relativistic particles- RF deflection

- Chopper cavities, Low energy beam chopping systems, JPARC- CRAB cavities, to maximize the collision rate of colliding particle

beams, which are brought to collision under a certain angle- Funneling cavities, combination of two beams arriving at a certain

angle into one common beam line, if interleaved one by one the bunch repetition frequency will doubles during this process

- RF bunching- Used to keep bunches longitudinally confined during beam

transport- Form bunches out of CW beams coming from a particle source

- Beam monitoring cavities

Changing velocity, variable RF frequency- These are the most demanding operating conditions

for RF cavities, in- Low-β (ion/proton) synchrotrons and FFAGs- A change in RF frequency can be achieved by putting

a material with an adjustable permeability (typically ferrites) into a cavity

- Small accelerating voltages (10s of keV) can be achieved in these cavities at rather poor efficiencies

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RF cavity examples

Constant velocity, constant RF frequency

Application

MAX-IV cavityhttp://www.lightsources.org/

http://www.esrf.eu/ ESRF cavity

ILC cavity

CEBAF cavityhttp://wwwold.jlab.org/

http://anim3d.web.cern.ch/

LHC cavityhttp://hep.phys.sfu.ca/

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RF cavity examples

Changing velocity, constant RF frequency

Application

CAS, RF for cyclotron (materials)Sytze Brandenburg (KVI), P. K. Sigg(PSI)

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RF cavity examples

Changing velocity, variable RF frequency

Application

ISIS ferrite loaded cavity

EMMA ns-FFAG cavity

Ian Gardner (ISIS, RAL)

http://www.hep.manchester.ac.uk/g/accelerators/conform/news/guide/

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RF cavity examplesNon-accelerating cavities

Application

http://pbpl.physics.ucla.edu/

Deflecting cavity

Hans-H. Braun / PSI

Graeme Burt ,Lancaster University/ CICarlo J. Bocchetta (ELETTRA (Trieste)

Harmonic cavity can be operated in ACTIVE mode (RF external power supply) or in PASSIVE mode (voltage induced by beam current). The choice of several Synchrotron Light Sources nowadays has been the Passive Mode

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RF cavity classification EM wave

Travelling wave (TW)- Cavities are filled in space (cell after

cell) – sub- µs

- Constant impedance- Each cell has the same shape (iris

radius)- Constant gradient

- Each cell has different iris radius

Standing wave (SW)- Cavities are filled in time (Slowly

build up SW pattern at desired amplitude) - 10s µs to ms

NLC cavity

MAX-IV cavityMAX-IV linac cavityLars Malmgren, ESLS RF 2012

F. Gerigk (CERN/BE/RF)

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Material

Normal conducting (NC)- Need water cooling- Higher gradient < 150 MV/m

Superconducting (SC)- Need Cryogenic plant- Low gradient < 45 MV/m

ILC cavity

Hitachi

Toshiba

LHC cavityhttp://hep.phys.sfu.ca/

MAX-IV cavityhttp://www.lightsources.org/

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EM Mode

TM- Transverse Magnetic field on beam

path- Mainly use for accelerating particle

TE- Transverse Electric field on beam

path- Low surface losses- Can be adapted by adding drift tube

to make axial E field on beam path for acceleration (H-mode cavity)

TEM- Low frequency < 100 MHz- Frequency determined by the length- QWR (Quarter wave resonator)- HWR (Half wave resonator)- Spoke cavity

Crossbar H-mode cavityTM-mode cavity (50.6 MHz)

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RF cavity classification EM Mode

F. Gerigk (CERN/BE/RF)

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RF cavity properties

• Accelerating voltage (Va)

– 𝑉𝑎 = 𝐸𝑧𝑒𝑖𝜔𝑧/𝑐𝑑𝑧

• Accelerating gradient (Ea)

– 𝐸𝑎 =𝑉𝑎

𝐿, L – effective length of cavity

• Store Energy (U)

– 𝑈 =1

2𝜇0 𝑉 𝐻

2𝑑𝑉 =

1

2𝜀0 𝑉 𝐸

2𝑑𝑉

• Power dissipate on cavity wall (Pc)

– 𝑃𝑐 =1

2𝑅𝑠 𝑆 𝐻

2𝑑𝑆 , Rs – Surface

resistance

• Unloaded Quality factor (Q0)

– 𝑄0 =𝜔𝑈

𝑃𝑐

• Shunt impedance (Ra)

– 𝑅𝑎 =𝑉𝑎2

𝑃𝑐

• R/Q

–𝑅

𝑄=

𝑉𝑎2

𝜔𝑈

• External Q (Qex), Loaded Q (QL)

– 𝑄𝑒𝑥 =𝜔𝑈

𝑃𝑒𝑥, 𝑄𝐿 =

𝜔𝑈

𝑃

–1

𝑄𝐿=

1

𝑄0+

1

𝑄𝑒𝑥

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RF cavity design criterion

• Efficiency of cavity

– 𝜂 = 𝑃𝑏𝑒𝑎𝑚𝑃𝑔𝑒𝑛

=𝑃𝑏𝑒𝑎𝑚

𝑃𝑏𝑒𝑎𝑚+𝑃𝑐



NC SC

NC – increase peak fields in the area of nose to maximize shunt impedance, but it’s limit by Kilpatrick (maximum achievable field in vacuum as a function of frequency)

• Kilpatrick limit (1957)

– 𝑓 𝑀𝐻𝑧 = 1.64𝐸𝑘2𝑒

−8.5

𝐸𝑘 , Ek in MV/m

Modern cavities, Esurf = bEk, b is the bravery factor, 1<b<2.- Better vacuum and cleaner inner surfaces

SC – Pb >> Pc, no longer limit by losses- Optimize shape for a low ratios of Epeak,surf/Ea and Bpeak,surf/Ea , Ep/Ea ~ 2- Nose cone is not suitable, elliptical shape is basically used

F= 118 MHz, Ek =12 MV/m, Esm = 24 MV/mF= 500 MHz, Ek = 21 MV/m, Esm = 42 MV/m

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Thomas P. Wangler, RF Linear Accelerator

Cost - Cost of building cavity, cost of RF station, cost of Cooling sytem, etc..

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Overall efficiency of project J.P.Delahaye, CLIC-ILC LET Beam Dynamics Workshop 23/06/09

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Effects to particle beam

Roger JONES (Manchester University), CAS 2010

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Effects to particle beam


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HOM suppression


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HOM suppression


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HOM suppression

Jörn Jacob, ESRF

V. Serrière, ESRF

Pascal SCHUMMER, SDMS

V. Serrière, ESRF, IPAC2011

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Cavity fabrication (ESRF cavity)

V. Serrière, ESRF

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Cavity fabrication (SCRF cavity)• From Sheets to Fields Hanspeter Vogel, RI

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SLRI 2nd RF Cavity

Source: Ake Andersson, MAX-LabSource: Soren Pape Moller, ASTRID2

Source: Ake Andersson, MAX-Lab

Source: Ake Andersson, MAX-Lab

Based on MAX-IV Cavity

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Proposed Cavity 3D shape

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Accelerating mode fields (118 MHz)

Electric field Magnetic field

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Higher Order Modes (HOM) in RF Cavity

Monopole Dipole Quadrupole

Sextupole Tenth-pole Twelfth-pole

15/11/2012

Thank you

Synchrotron Light Research Institute

Sirindhornwitchothai Building

111 University Avenue

Suranaree Sub-district, Muang District

Nakhon Ratchasima 30000 THAILAND

Website: http://www.slri.or.th

Tel: +66-44-217040

Fax: +66-44-217047

http://www.slri.or.th/

a brief introduction of radio frequency cavity in particle ... · in accelerator physics (and many...

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