fundamental concepts of particle accelerators

Fundamental Concepts of Particle Accelerators

Koji TAKATA

KEK

[email protected]://research.kek.jp/people/takata/home.html

Accelerator Course, Sokendai,Second Term, JFY2010

Oct. 28, 2010

The Dawn of Particle Accelerator TechnologyBasic Concepts

Accelerators in FutureLivingston Chart

ContentsI The Dawn of Particle Accelerator Technology

I DC high voltage generatorsI Use of magnetic induction: betatronI Drift tube linac and cyclotronI Great progress just after world war II

I Basic ConceptsI Principle of RF phase stabilityI Strong focusingI Synchrotron radiation (SR)I ColliderI Technical issues

I Accelerators in FutureI ERL (Energy Recovery Linac) : SR source of new typeI LC : Linear ColliderI µ-µ Collider and/or µ-FactoryI Laser-plasma acceleration

I Livingston ChartFundamental Concepts of Particle Accelerators



DC high voltage generatorsUse of magnetic induction: betatronDrift tube linac and cyclotronGreat progress just after world war II

The Dawn of Particle Accelerator Technology

I Artificial disintegration of atomic nuclei

I First Accelerators

I from DC Acceleration to RF Acceleration

I Problems in RF Acceleration

I Rapid Development of Electronics aroundWorld War II (1941 - 1945) or after





First artificial disintegration of atomic nuclei (1)

I Ernest Rutherford’s discovery of nuclear disintegration(1917 - 1919)I He confirmed that protons were produced in a nitrogen-gas

filled container in which a radioactive source emitting alphaparticles was placed.

α + 147N → p + 16

8O

I This provoked strong demand for artificially generate highenergy beams to study the nuclear disintegration phenomenain more detail.

I Thus started the race for developing high energy accelerators,and Rutherford himself was a great advocator.





First artificial disintegration of atomic nuclei (2)

I The first disintegration of atomic nuclei with acceleratorbeams was achieved at the Cavendish Laboratory in 1932 byJohn D. Cockcroft and Ernest T. S. Walton, who used 800 kVproton beams from a DC voltage-multiplier.

p + 73Li → α + α





DC HV Accelerators

I DC Generators：two major methodsI Cockcroft & Walton’s 800 kV voltage-multiplier circuit with

capacitors and rectifier tubes

I Van de Graaff’s 1.5 MV belt-charged generator (1931)

I Electrostatic accelerators are still in use for the massspectroscopy, because of their fine and stable tunability of theacceleration voltage.

I analysis of the ratio 14C/12C : an important tool forarchaeology





Cockcroft & Walton’s voltage-multiplier circuit

6V 0

V(3+cos ωt)V(1+cos ωt)V cos ωt

AC

V(5+cos ωt)

4V2V0





Cockcroft around 1932

See the picture in From X-rays to Quarks, page 227 by Segre, E.(W. H. Freeman and Company, 1980) .





Glass Tube with Beam Acceleration Gaps

Visit the home page :http://www.daviddarling.info/encyclopedia/C/Cockcroft.html





750 keV Cockcroft-Walton Accelerator Used at KEK





Van de Graaff’s 1.5MV Belt-charged Generator





Limitations in Electrostatic Accelerators

I DC acceleration is limited by high-voltage breakdown (BD).I typical BD voltages for a 1cm gap of parallel metal plates

Ambience Typical BD Voltagesair (1 atm) ≈ 30 kVSF6 (1 atm) ≈ 80 kVSF6 (7 atm) ≈ 360 kVtransformer oil ≈ 150 kV

UHV ≈ 220 kV

I no drastic increase in BD limits for much larger plate gaps.





High Voltage Breakdown of a Van de Graaff generator

A demonstration of BD to housing walls.

Search for the key word ”van der graaf generator” athttp://en.wikipedia.org/wiki/





Intermediate stage towards RF Acceleration

Use of Faraday’s law of induction

I Irrotational electric field due to magnetic flux change,a prelude to RF acceleration [Donald W. Kerst’s betatron(1940)]:

∇×E = −∂B∂t,

then ∮CEsds = − ∂

∂t

∫∫SB · n dxdy = − ∂

∂tΦ





Kerst’s Betatron





Start of Real RF Accelerators

Linear and/or CircularI Linear accelerator (linac) :

I Gustaf Ising’s proposal (1925)

I Rolf Wideroe made a prototype of the Ising linac (1928)

I Multiple RF acceleration in a magnetic fieldI Ernest Lawrence’s cyclotron (1931)：

the first circular accelerator

I repeated acceleration at the cyclotron frequency ：

ωc = eB⊥/m





The first linac by Wideroe

I 25 kV per gap ×2 with the drift tubeI he convinced the scheme can be repeated indefinitely many

times to reach higher beam energies

RF

Beam

Ion

So urce





First Cyclotrons

See the picture in From X-rays toQuarks, page 229 by Segre, E.(W. H. Freeman and Company,

1980) .

A Riken cyclotron acceleratedprotons to 9 MeV and deuterons to

14 MeV (1939)





Circular Orbit of Charged Particles in Magnetic Field

Search for the key word ”Cyclotron” inhttp://en.wikipedia.org/wiki/





Principle of Cyclotron Operation

RF Generator

rn rn+1(> rn)

Electric FieldMagnetic Field

dee

dee

dee

dee

beam





Problems in RF Acceleration

I Linacs：I poor RF sources; electron tube technology was yet in its

infancy.

I Cyclotrons：I relativistic increase of particle mass→ decrease of ωc

→ asynchronism with RF

I Betatrons：I intensity of trapped beam depends critically on the injected

beam’s positions and angles.I analysis of transverse oscillations of particles led to the theory

of betatron oscillations of today.





Advances during World War II (1941 - 1945)

I High power microwave tubes for the radars were put topractical use

I magnetrons and klystrons

I Discovery of the phase stability principle in RF acceleration

I Vladimir Veksler (1944) and Edwin M. McMillan (1945)

I cyclotron → synchrocyclotron → synchrotron




Principle of RF phase stabilityStrong focusingcollidersynchrotron radiation (SR)Technical issues

The Principle of Phase Stability

I Particles of different energies haveI differences in velocity and in orbit length;I then, particles may be asynchronous with the RF frequency.

I The RF field, however, may have a restoring force at a certainphase, around which asynchronous particles be captured, thatis to say bunched.

I This enables a stable, continuous acceleration of the wholeparticles in a bunch to high energies.

I Circular accelerators based on this principle are called“synchrotron.”

I This principle is also applicable to linacs, particularly in lowenergy range, to bunch continuous beams emitted from asource and to lead bunches to downstream acceleratorsections.





Synchrotron Oscillation (1)

I Assume a sinusoidal RF electric field in an RF cavity gap:

V = V0 sinωt

.

I Assume a synchronous particle pass the gap center at

ωt = 0, 2π, 4π, . . .

and its acceleration voltage be Va(< V0).

I Then in one RF period, there are there are two ϕ’s whichsatisfy

Va = V0 sinϕ.





Sinusoidal RF Wave

-V0

V0

φπ/2 π0

Va





Synchrotron Oscillation (2)

I Only one of the two ϕ’s can capture particles, which makeoscillations around the phase.

I These oscillations are called synchrotron oscillation and thephase is the synchronous phase ϕs.

I Which one is the ϕs depends on that the revolution time islonger or shorter for a energy deviation ∆E(> 0) from thesynchronous energy.





Synchrotron Oscillation in an RF Bucket (1)

For the case of ϕs = 30

abscissa : ∆ϕ = ϕbeam − ϕs, ordinate : ∆E = Ebeam − Es





Synchrotron Oscillation in an RF Bucket (2)

For the case of ϕs = 0

abscissa : ∆ϕ = ϕbeam − ϕs, ordinate : ∆E = Ebeam − Es

-3 -2 -1 1 2 3

-3

-2

-1

1

2

3

-3 -2 -1 1 2 3

-2

-1

1

2

time sequence of motion ofparticles initially on the abscissa(particles of a larger ∆ϕ moveslower or have a smaller ωs)





Advances in Beam Focusing Technique

I Magnetic, not electric, focusing for high energy particles

I Weak focusing in early cyclotrons and betatrons

I Strong focusingI Nicholas C. Christofilos (1950)

I Ernest D. Courant, M. Stanley Livingston,and Hartland S. Snyder (1952)





Equation of Motion

I In electric field E and magnetic field B, the equation motionof a particle is

dp

dt= e (E+ v ×B)

wherep = mv = γm0v

withm0 : rest mass

γ = 1/√

1− β2 : Lorentz factor

β = |v| /c = v/c





Coordinate System

I In the analysis of beam focusing, it is usually important todescribe the equation of motion of particles only for smalldeviations x and y along the path s of the reference orbit of asynchronous particle

I Thus, a Frenet-Serret frame with respect to the referenceorbit is preferred:

I unit vector tangent to the curve,I unit vector in the direction of curvature,I and the cross product of the them.

ρ

x

y

reference orbit

tangent at s

particle

s





Typical Weak-Focusing Magnetic Field

Cylindrically symmetric magnet poles and magnetic fields of theearly cyclotrons

z

0r





Betatron Oscillations : Weak FocusingI First order approximation of the field pattern of the previous

page

By = B0

(1− n

x

ρ+ . . .

)and Bx = B0

(−ny

ρ+ . . .

),

where n =dBy

By/dρρ : the n value

I Equation of motion

d2x

ds2+

1− n

ρ2x = 0 and

d2y

ds2+n

ρ2y = 0

I Focusing both horizontally and vertically → 0 < n < 1I Betatron wavelength

λβ,x = 2πρ/√1− n

λβ,y = 2πρ/√n





Quadrupole Magnetic Fields for Stronger Focusing

I No limitations for the n value.I Focusing in one direction, defocusing in the other.I Later we will see the focusing is superior to the defocusing

-2 -1 1 2x

-2

-1

1

2

y





Q magnets and B magnets of JPARC RCS synchrotron (1)

http://j-parc.jp/Acc/en/index.html





Q magnets and B magnets of JPARC RCS synchrotron (2)






Q magnets and B magnets of JPARC Main Ring (1)






Q magnets and B magnets of JPARC Main Ring (2)

Sextupole magnets are sometimes used.

http://j-parc.jp/Acc/en/index.htmlFundamental Concepts of Particle Accelerators




Optical Lens Equivalent of a Quadrupole Magnet

convex lens in one direction and concave lens in the perpendiculardirection

beam





Strong Focusing with a Standard FODO Array

F : focusing Q, D : defocusing Q, O : drift section vspace3mm

I In the following figure, convex lenses are for horizontalfocusing and concave lenses for vertical focusing.

I The red curves are beam envelopes for a unit emittance.

0.5 1 1.5 2 2.5 3

-1

-0.5

0.5

1





Emittance Ellipse for a periodic sequence of Q magnets

-1 0.5 1

0

0.5

PSfrag replacements

#0

#1

#2

#3

#4

#5

#6

x/x0

x!/k

1

x2 +x′2

k2= x20 where k =

1

L

√L

f

(1− L

4f

)f : focal length, L : length between neighboring focusing Q’sFundamental Concepts of Particle Accelerators




Betatron Oscillations : Strong Focusing (1)

I Use quadrupole magnets with |n| ≫ 1, but with changing thesign of n alternatively

I Equation of motion

d2x

ds2+Kx (s)x = 0

d2y

ds2+Ky (s) y = 0

I Focusing/Defocusing forces Kx (s) and Ky (s) are periodicfunctions for the ring circumference L.

I They are Mathieu-Hill type functions






I General solution：x = Ax

√β(s) cos (ψ(s)− ψ0) (similar too

for y)I Ax and Ay are constants proper to each particles and are

independent of the position s on the orbit

A2x =

4 + β′2

4βx2 − β′βxx′ + βx′2

I measure a particular particle’s (x, x′) or (y, y′) for many turnsat a position s, the points trace an ellipse on thecorresponding phase space.

I ellipse’s direction and eccentricity are functions of s,but area= πA2

x (y) is conservedI the largest area is, roughly speaking, called the emittance of

the bunch






I Beta function βx (y)(s) is defined as the betatron amplitudefor Ax (y) = 1 :

2ββ′′ − β′2 + 4β2K (s) = 4.

I Phase ψ of betatron oscillation :

ψ =

∫ s

ds/β.

I Wavelength λβ of the betatron oscillation : the lengthcorresponding the phase advance ∆ψ = 2π.

I Betatron tune νβ ≡ L/λβ .





Colliders (1)

I In order to observe the high energy particle reactions : targetsin laboratory frame were solely used (fixed target experiment).

I The reaction, however, depends not on the laboratory energyof the projectile from an accelerator, but on the center ofmass energy of the projectile and target.

I Touschek’ idea to use colliding beams (1960)

I The first collider：AdA (Frascati, 1961)

200MeV e− ⇒⇐ 200MeV e+

I The collider has become a paradigm of high energyaccelerators of today.





Colliders (2) : ECM

I Consider collision of particles of the same rest mass m0.

I In a fixed target case with the projectile accelerated to γm0c2

I the total energy: ET /m0c2 = (γ + 1)

I the total momentum: pT /m0c = βγ =√γ2 − 1

I since E2 − c2p2 is a Lorentz invariant,

ECM/m0c2 =

√2γ + 2 ≈

√2γ

I In a collision of two particles of the same energy γm0c2

ECM/m0c2 = ET /m0c

2 = 2γ





Colliders (3) : Luminosity

I For reaction cross section σ and beam cross section at thecollision point S, the probability of reaction for a pair ofparticles is,

σ

S

I Hence the probability for N+ and N− particles at a rate off times per second

f ×N+ ×N− × σ

S

I Coefficient of σ is the luminosity L

L = f × N+ ×N−S





Synchrotron Radiation (1)

I The synchrotro radiation, SR, is an electric dipole radiationfrom a charged particle in acceleration v

I Radiation power in the rest frame is given by Larmour’sformula

P =2reme

3c

(dv

dt

)2

=2re3mec

(dp

dt

)2

where re ≡ e2/(4πε0mec2) = 2.82× 10−15m is the electron’s

classical radius.





Electric Dipole Radiation : Electric Field Pattern

Radiation pattern (cylindrically symmetric) ofan electric dipole at rest

0 1 2 3 4 5

-4

-2

0

2

4

PSfrag replacements

#0

#1

#2

#3

#4

#5

#6

x/x0

xe/x0

x!/k

! (m)

s = 0

s = 0 + nLs = L

2

s = L

4

s = 0+ + nL

s = 0+

s = 0+

s = L" + nL = 0" + (n + 1)L

s/L

s (m)

f = 1.6L

" = 50 m

L = 2 m

z/#

r/#

1






I Since P is the ratio of radiated energy to elapsed time, bothof which transform in the same manner under Lorentztransformations, P must be an invariant.

I Then ( )2 in the right hand side of the equation should havethe following invariant form

(dp/ds)2 − (dE/ds)2 /c2

where ds is the differential of proper time

ds =√dt2 − (dx2 + dy2 + dz2) /c2 = dt/γ.






I Hence in laboratory frame the radiated power is

P =2reme

3cγ2

[d (γv)

dt

]2−

[d (γc)

dt

]2I The radiated energy per turn ∆E for a ring with radius ρ

∆E

mec2=

4π

3

reρβ3γ4

I A practical formula for ∆E(keV), E(GeV) and ρ(m)

∆E(keV) ≈ 88.5 [E(GeV)]4 /ρ(m).






I Pattern of electric dipole radiation in electron’s rest frame（x′, y′, z′, ct′）

dP/dΩ ∝ sin2 θ

where Ω being the solid angle and θ the angle from z′ axis.I Transformation to laboratory frame

x′ = x, y′ = y, z′ = γ (z − vt) ,

ct′ = γ (ct− vz/c) .

I Angles of axes x′ and y′ with respect to z axis are ∼ 1/γ.I forward radiation power is within a cone of

a full angle of ∼ 2/γ.I electron is observable for an arc length of ∼ 2ρ/γ.I doppler effect shortens the wavelength by (1− v/c) ∼ 1/2γ2.

I Critical wavelength (Schwinger-Jackson’s definition)

λc ≡ 4πρ/3γ3





Accelerating Cavity (1)

There are many types of accelerating cavity, which, however,basically are variations of a cylindrical cavity (or pillbox cavity),

I operating on the fundamental TM010 mode.

Hθ Ez

2b

r=b

0

d

0.5 1 1.5 2

0.2

0.4

0.6

0.8

1

Hθ

Ez

χ01r/b

0

arbit

rary

sca

le






Single-Cell Accelerating Cavity for Photon Factory Storage Ring(fRF = 500MHz, Vpeak = 0.7MV )

R234.69mm

R91.375mm

R50mm

220mm

300mm

R130mm

R10mm

Ez (r=0)

z

r






Global behavior of a resonant cavity is well describedby an equivalent circuit comprising three parameters

L, C, R.

L

C

R






I first of all, the resonant frequency and the Q value are derivedfrom the following two equations :

ω0 = 1/√LC and Q = ω0RC.

I one more independent relation is required to determine thethree parameters L, C, andR.

I for this sake, we choose the peak acceleration voltage alongthe beam orbit

I this choice is reasonable, because it satisfies the energyconservation of the（EM fields + beam）system.∫∫∫

VJ ·EdV +

∫∫S(E×H) · ndS = 0.

(J : beam current distribution, E×H : Poynting vector,V : cavity volume, S : cavity surface)





High Gradient Electric Fields and Breakdown

I Kilpatrick’s empirical rule

I Fowler-Nordheim’s theory for field emission

I Surface damage on an X-band copper structure

I Weak discharge: multipacting





Kilpatrick Criterion

W. D. Kilpatrick, Rev. Sci. Instr. 28 (1957) 824





Fowler-Nordheim’s LawR. H. Fowler and L. Nordheim, Proc. Roy. Soc. A 119 (1928) 173

J. W. Wang and G. A. Loew, SLAC-PUB-7684 (1997)

I DC field emission current density jF [A/m2] :

jF =1.54× 10−6 × 104.52ϕ

−0.5E2

ϕexp

(−6.53× 109ϕ1.5

E

)I microscopic surface gradient E [V/m]

I metal work function ϕ [eV]

I Averaged over one RF cycle, jF is modified as:

jF =5.7× 10−12 × 104.52ϕ

−0.5E2.5

ϕ1.75exp

(−6.53× 109ϕ1.5

E

)I Field enhancement factor β : E = βEmacro





Surface Damage on the Iris of an X-band Linac Structure

R. E. Kirby, SLAC-PEL, 2000





Multipacting : a weak discharge phenomenon

A. J. Hatch and H. B. Williams, Phys. Rev. 112 (1958) 581





Superconductor and RF

I Niobium is mostly used, which is a Type II superconductorI critical temperature Tc = 9.2K

I critical field Hc = 2× 103Oe

I in meissner state for H ≤ Hc1 = 1.7× 103Oe

I in normal state for H ≥ Hc2 = 2.3× 103Oe

I Maxwell equations + London equationsI London’s penetration depth λL

• about 50 nm for niobium

I coherent length ξ0• about 40 nm for niobium

I wall losses do still exist, although very small, which are causedby normal electrons





Equations for Superconducting State

I Maxwell’s equations:

∇×E+∂B

∂t= 0 and ∇×H− ∂D

∂t= J

I London equations:

Js = −j nse2

ωmeE and ∇× Js = −nse

2

meµ0H

I Field equations:

∇2 (J,E,H) =(λ−2L + jωσµ0 − ω2ε0µ0

)(J,E,H)

I London’s penetration depth: λL =√me/nse2µ0





Klystron (1)

I Also an accelerator with decelerating electric fields

I Perveance µpI Child-Langumuir law for space-charge limited flow

• µp ∝ I/V 3/2

• cf. M. Reiser:Theory and Design of Charged Particle Beams,John Wiley & Sons, 1994.

I Efficiency vs. perveanceI cf. R. B. Palmer and R. Miller: SLAC-PUB-4706, September

1988.





Klystron (2)





500 MHz-1 MW CW Klystron for KEKB




Future Accelerators

I ERL: Energy Recovery Linac

I LC : Linear Collider

I µ-µ Collider and/or µ-Factory

I Laser-plasma acceleration




KEK-PF-ERL : A Future Plan

I An SR source with a superconducting linac energy-recoveredby returned electron beams




Linear Collider: schematic layout




µ-µ Collider




Laser Plasma Acceleration (1)

cf. C. Joshi and T. Katsouleas’s article in Physics Today, June2003, p.47.




Laser Plasma Acceleration (2)




Livingston Chart

Proton Synchrotron

Collider

(Equivalent Energy)

Proton Linac

Electrostatic Accelerator

Betatron

Electron Linac

Electron SynchrotronSynchro-cyclotron

Cyclotron

DC Generator

1MeV

1GeV

1TeV

1930 1940 1950 1960 1970 1980 1990 2000 2010

1017

1016

1015

1014

1013

1012

1011

1010

109

108

107

106

Ac

ce

ler

ato

r E

ne

rg

y (

eV

)

1PeV

I Originally given by M. S. Livingston &J. P. Blewett: ”Particle Accelerators,p.6”, MacGraw Hill, 1962

I Energies for the colliders are equivalentvalues for the fixed target system

I Maximum beam energy ever achievedI Electron Synchrotron : 2 × 100 GeV

(2000, CERN LEP)I Proton Synchrotron : 2 × 7 TeV

(2010, CERN LHC)http://lhc.web.cern.ch/lhc/

I Electron-positron linear collider2 × 500 GeV? (2025 or later?)




References (1)

I Segre, E. : From X-rays to Quarks (W. H. Freeman andCompany, 1980).

I Historical introduction to the evolution of high energy physicsand accelerator science

I Chao, A. W. and Tigner, M. (ed.) : Handbook of AcceleratorPhysics and Engineering (World Scientific, 1999).

I Compact encyclopedia of accelerator science and technology

I Wiedemann, H. : Particle Accelerator Physics I, II (Springer,1999).

I Text book on accelerator physics

I Courant, E. D. and Snyder, H. S.: Annals of Physics, 3(1958) p.1.

I A classical paper on the theory of the strong focusing




References (2)

I Schwinger, J. : Physical Review, 75 (1949) p.1912.I A classical paper on the theory of the synchrotron radiation

I Gilmour, A. S. : Microwave Tubes (Artech House, 1986).I Text book on the electron tube technology

I Padamsee, H., Knobloch, J. and Hays, T. :RF Superconductivity for Accelerators (John Wiley & Sons,1998).

I Text book on RF superconductivity and its application toenergy accelerators


fundamental concepts of particle accelerators

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