Fundamental Concepts of Particle Accelerators

Koji TAKATA

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

Accelerator Course, Sokendai, Second Term, JFY2010

Oct. 28, 2010 The Dawn of Particle Accelerator Technology Basic Concepts Accelerators in Future Livingston Chart Contents I The Dawn of Particle Accelerator Technology I DC high voltage generators I Use of magnetic induction: betatron I Drift tube linac and cyclotron I Great progress just after world war II I Basic Concepts I Principle of RF phase stability I Strong focusing I Synchrotron radiation (SR) I Collider I Technical issues I Accelerators in Future I ERL (Energy Recovery Linac) : SR source of new type I LC : Linear Collider I µ-µ Collider and/or µ-Factory I Laser-plasma acceleration I Livingston Chart Fundamental Concepts of Particle Accelerators The Dawn of Particle Accelerator Technology DC high voltage generators Basic Concepts Use of magnetic induction: betatron Accelerators in Future Drift tube linac and cyclotron Livingston Chart Great 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 around World War II (1941 - 1945) or after

Fundamental Concepts of Particle Accelerators The Dawn of Particle Accelerator Technology DC high voltage generators Basic Concepts Use of magnetic induction: betatron Accelerators in Future Drift tube linac and cyclotron Livingston Chart Great progress just after world war II 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 alpha particles was placed. 14 → 16 α + 7N p + 8O I This provoked strong demand for artificially generate high energy beams to study the nuclear disintegration phenomena in more detail. I Thus started the race for developing high energy accelerators, and Rutherford himself was a great advocator.

I The ﬁrst disintegration of atomic nuclei with accelerator beams was achieved at the Cavendish Laboratory in 1932 by John D. Cockcroft and Ernest T. S. Walton, who used 800 kV proton beams from a DC voltage-multiplier. 7 → p + 3Li α + α

I DC Generators：two major methods I 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 mass spectroscopy, because of their fine and stable tunability of the acceleration voltage. I analysis of the ratio 14C/12C : an important tool for archaeology

V cos ωt V(1+cos ωt) V(3+cos ωt) V(5+cos ωt)

0 2V 4V 6V 0

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

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

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 Voltages air (1 atm) ≈ 30 kV SF6 (1 atm) ≈ 80 kV SF6 (7 atm) ≈ 360 kV transformer oil ≈ 150 kV UHV ≈ 220 kV

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

Fundamental Concepts of Particle Accelerators The Dawn of Particle Accelerator Technology DC high voltage generators Basic Concepts Use of magnetic induction: betatron Accelerators in Future Drift tube linac and cyclotron Livingston Chart Great progress just after world war II 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” at http://en.wikipedia.org/wiki/

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)]:

∂B ∇ × E = − , ∂t

then I ∫∫ ∂ ∂ Esds = − B · n dxdy = − Φ C ∂t S ∂t

Linear and/or Circular I Linear accelerator (linac): I Gustaf Ising’s proposal (1925) I Rolf Wider¨oe made a prototype of the Ising linac (1928)

I Multiple RF acceleration in a magnetic ﬁeld I Ernest Lawrence’s cyclotron (1931)： the ﬁrst circular accelerator I repeated acceleration at the cyclotron frequency ：

ωc = eB⊥/m

I 25 kV per gap ×2 with the drift tube I he convinced the scheme can be repeated indeﬁnitely many times to reach higher beam energies

RF Ion Source Beam

See the picture in From X-rays to Quarks, page 229 by Segr`e,E. (W. H. Freeman and Company, 1980) .

A Riken cyclotron accelerated protons to 9 MeV and deuterons to 14 MeV (1939)

Fundamental Concepts of Particle Accelerators The Dawn of Particle Accelerator Technology DC high voltage generators Basic Concepts Use of magnetic induction: betatron Accelerators in Future Drift tube linac and cyclotron Livingston Chart Great progress just after world war II Circular Orbit of Charged Particles in Magnetic Field

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

RF Generator

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r n rn+1(> rn)

Magnetic Field Electric Field

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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.

I High power microwave tubes for the radars were put to practical 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

Fundamental Concepts of Particle Accelerators Principle of RF phase stability The Dawn of Particle Accelerator Technology Strong focusing Basic Concepts collider Accelerators in Future synchrotron radiation (SR) Livingston Chart Technical issues The Principle of Phase Stability I Particles of different energies have I 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 certain phase, around which asynchronous particles be captured, that is to say bunched. I This enables a stable, continuous acceleration of the whole particles 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 low energy range, to bunch continuous beams emitted from a source and to lead bunches to downstream accelerator sections.

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

V = V0 sin ωt

. 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 which satisfy Va = V0 sin ϕ.

V0 Va

0 π/2 π φ

-V0

I Only one of the two ϕ’s can capture particles, which make oscillations around the phase. I These oscillations are called synchrotron oscillation and the phase is the synchronous phase ϕs. I Which one is the ϕs depends on that the revolution time is longer or shorter for a energy deviation ∆E(> 0) from the synchronous energy.

◦ For the case of ϕs = 30 abscissa : ∆ϕ = ϕbeam − ϕs, ordinate : ∆E = Ebeam − Es

◦ For the case of ϕs = 0 abscissa : ∆ϕ = ϕbeam − ϕs, ordinate : ∆E = Ebeam − Es

1 3

2 -3 -2 -1 1 2 3

1 -1

-3 -2 -1 1 2 3 -2 -1 time sequence of motion of -2 particles initially on the abscissa -3 (particles of a larger ∆ϕ move slower or have a smaller ωs)

I Magnetic, not electric, focusing for high energy particles

I Weak focusing in early cyclotrons and betatrons I Strong focusing I Nicholas C. Christoﬁlos (1950) I Ernest D. Courant, M. Stanley Livingston, and Hartland S. Snyder (1952)

I In electric ﬁeld E and magnetic ﬁeld B, the equation motion of a particle is dp = e (E + v × B) dt where p = mv = γm0v with m : rest mass 0 √ γ = 1/ 1 − β2 : Lorentz factor β = |v| /c = v/c

I In the analysis of beam focusing, it is usually important to describe the equation of motion of particles only for small deviations x and y along the path s of the reference orbit of a synchronous particle I Thus, a Frenet-Serret frame with respect to the reference orbit 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. y

x particle

ρ s

tangent at s reference orbit Fundamental Concepts of Particle Accelerators Principle of RF phase stability The Dawn of Particle Accelerator Technology Strong focusing Basic Concepts collider Accelerators in Future synchrotron radiation (SR) Livingston Chart Technical issues Typical Weak-Focusing Magnetic Field

Cylindrically symmetric magnet poles and magnetic ﬁelds of the early cyclotrons z

0 r

Fundamental Concepts of Particle Accelerators Principle of RF phase stability The Dawn of Particle Accelerator Technology Strong focusing Basic Concepts collider Accelerators in Future synchrotron radiation (SR) Livingston Chart Technical issues Betatron Oscillations : Weak Focusing I First order approximation of the ﬁeld pattern of the previous page ( ) ( ) x y B = B 1 − n + ... and B = B −n + ... , y 0 ρ x 0 ρ where n = dBy / dρ : the n value By ρ I Equation of motion d2x 1 − n d2y n + x = 0 and + y = 0 ds2 ρ2 ds2 ρ2 I Focusing both horizontally and vertically → 0 < n < 1 I Betatron wavelength √ − λβ,x = 2πρ/√1 n λβ,y = 2πρ/ n

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

y 2

-2 -1 1 2 x

-1

-2

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

Sextupole magnets are sometimes used.

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

convex lens in one direction and concave lens in the perpendicular direction

beam

F : focusing Q, D : defocusing Q, O : drift section vspace3mm I In the following ﬁgure, convex lenses are for horizontal focusing and concave lenses for vertical focusing. I The red curves are beam envelopes for a unit emittance.

1 0.5

0.5 1 1.5 2 2.5 3 -0.5 -1

x!/k

PSfrag replacements #0

#1 0 x/x0 -1 0.5 1 #2 0.5 #3 #4 #5 #6

√ ( ) x′2 1 L L x2 + = x2 where k = 1 − k2 0 L f 4f f : focal length, L : length betweenFundamental neighboring Concepts focusingof Particle Accelerators Q’s

1 Principle of RF phase stability The Dawn of Particle Accelerator Technology Strong focusing Basic Concepts collider Accelerators in Future synchrotron radiation (SR) Livingston Chart Technical issues Betatron Oscillations : Strong Focusing (1)

I Use quadrupole magnets with |n| ≫ 1, but with changing the sign of n alternatively I Equation of motion d2x + K (s) x = 0 ds2 x d2y + K (s) y = 0 ds2 y

I Focusing/Defocusing forces Kx (s) and Ky (s) are periodic functions 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 4 + β′2 A2 = x2 − β′βxx′ + βx′2 x 4β I measure a particular particle’s (x, x′) or (y, y′) for many turns at a position s, the points trace an ellipse on the corresponding phase space. I ellipse’s direction and eccentricity are functions of s, 2 but area= πAx (y) is conserved I the largest area is, roughly speaking, called the emittance of the bunch

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

2ββ′′ − β′2 + 4β2K (s) = 4.

I Phase ψ of betatron oscillation : ∫ s ψ = ds/β.

I Wavelength λβ of the betatron oscillation : the length corresponding the phase advance ∆ψ = 2π. I Betatron tune νβ ≡ L/λβ.

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

I The reaction, however, depends not on the laboratory energy of the projectile from an accelerator, but on the center of mass energy of the projectile and target. I Touschek’ idea to use colliding beams (1960) I The ﬁrst collider：AdA (Frascati, 1961) 200 MeV e− ⇒⇐ 200 MeV e+

I The collider has become a paradigm of high energy accelerators of today.

Colliders (2) : ECM

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

I 2 In a ﬁxed target case with the projectile accelerated to γm0c I 2 the total energy: ET /m0c = (γ + 1) √ I 2 the total momentum: pT /m0c = βγ = γ − 1 I since E2 − c2p2 is a Lorentz invariant, √ √ 2 ECM /m0c = 2γ + 2 ≈ 2γ

I 2 In a collision of two particles of the same energy γm0c

2 2 ECM /m0c = ET /m0c = 2γ

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

I Hence the probability for N+ and N− particles at a rate of f times per second σ f × N × N− × + S

I Coeﬃcient of σ is the luminosity L N × N− L = f × + S

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

I Radiation power in the rest frame is given by Larmour’s formula ( ) ( ) 2r m dv 2 2r dp 2 P = e e = e 3c dt 3mec dt 2 2 −15 where re ≡ e /(4πε0mec ) = 2.82 × 10 m is the electron’s classical radius.

xe/x0 ! x /k 4 η (m) s = 0 s = 0 + nL L 2 s = 2 L s = 4 s = 0+ + nL 0 s = 0+ r/λ s = 0+ s = L− + nL = 0− + (n + 1)L s/L-2 s (m) f = 1.6L ρ = 50 m -4 L = 2 m

0 1 2 3 4 5

Fundamental Concepts of Particle Accelerators

I Since P is the ratio of radiated energy to elapsed time, both of which transform in the same manner under Lorentz transformations, P must be an invariant. I Then ()2 in the right hand side of the equation should have the 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 {[ ] [ ] } 2r m d (γv) 2 d (γc) 2 P = e e γ2 − 3c dt dt

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

∆E 4π re 3 4 2 = β γ mec 3 ρ I A practical formula for ∆E(keV), E(GeV) and ρ(m)

∆E(keV) ≈ 88.5 [E(GeV)]4 /ρ(m).

Fundamental Concepts of Particle Accelerators Principle of RF phase stability The Dawn of Particle Accelerator Technology Strong focusing Basic Concepts collider Accelerators in Future synchrotron radiation (SR) Livingston Chart Technical issues Synchrotron Radiation (4) 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) 3 λc ≡ 4πρ/3γ

I operating on the fundamental TM010 mode.

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Global behavior of a resonant cavity is well described by an equivalent circuit comprising three parameters L, C, R. L

I first of all, the resonant frequency and the Q value are derived from the following two√ equations : ω0 = 1/ LC and Q = ω0RC. I one more independent relation is required to determine the three parameters L, C, andR. I for this sake, we choose the peak acceleration voltage along the beam orbit I this choice is reasonable, because it satisfies the energy conservation∫∫∫ of the（EM∫∫ fields + beam）system. J · EdV + (E × H) · ndS = 0. V S (J : beam current distribution, E × H : Poynting vector, V : cavity volume, S : cavity surface)

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

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

R. H. Fowler and L. Nordheim, Proc. Roy. Soc. A 119 (1928) 173 J. W. Wang and G. A. Loew, SLAC-PUB-7684 (1997) I 2 DC ﬁeld emission current density jF [A/m ] : − ( ) 1.54 × 10−6 × 104.52ϕ 0.5 E2 6.53 × 109ϕ1.5 j = exp − F ϕ E

I microscopic surface gradient E [V/m] I metal work function ϕ [eV]

I Averaged over one RF cycle, jF is modiﬁed as: − ( ) 5.7 × 10−12 × 104.52ϕ 0.5 E2.5 6.53 × 109ϕ1.5 j = exp − F ϕ1.75 E

I Field enhancement factor β : E = βEmacro

R. E. Kirby, SLAC-PEL, 2000

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

I Niobium is mostly used, which is a Type II superconductor I critical temperature Tc = 9.2K I 3 critical ﬁeld Hc = 2 × 10 Oe I 3 in meissner state for H ≤ Hc1 = 1.7 × 10 Oe I 3 in normal state for H ≥ Hc2 = 2.3 × 10 Oe

I Maxwell equations + London equations I 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 caused by normal electrons

I Maxwell’s equations: ∂B ∂D ∇ × E + = 0 and ∇ × H − = J ∂t ∂t I London equations:

2 2 nse nse Js = −j E and ∇ × Js = − µ0H ωme me I Field equations: ( ) ∇2 −2 − 2 (J, E, H) = λL + jωσµ0 ω ε0µ0 (J, E, H) √ I 2 London’s penetration depth: λL = me/nse µ0

I Also an accelerator with decelerating electric ﬁelds

I Perveance µp I Child-Langumuir law for space-charge limited ﬂow 3/2 • µp ∝ I/V • cf. M. Reiser: Theory and Design of Charged Particle Beams, John Wiley & Sons, 1994. I Eﬃciency vs. perveance I cf. R. B. Palmer and R. Miller: SLAC-PUB-4706, September 1988.

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Fundamental Concepts of Particle Accelerators The Dawn of Particle Accelerator Technology Basic Concepts Accelerators in Future Livingston Chart Future Accelerators

I ERL: Energy Recovery Linac

I LC : Linear Collider

I µ-µ Collider and/or µ-Factory

I Laser-plasma acceleration

Fundamental Concepts of Particle Accelerators The Dawn of Particle Accelerator Technology Basic Concepts Accelerators in Future Livingston Chart KEK-PF-ERL : A Future Plan

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

Fundamental Concepts of Particle Accelerators The Dawn of Particle Accelerator Technology Basic Concepts Accelerators in Future Livingston Chart Linear Collider: schematic layout

Fundamental Concepts of Particle Accelerators The Dawn of Particle Accelerator Technology Basic Concepts Accelerators in Future Livingston Chart µ-µ Collider

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Fundamental Concepts of Particle Accelerators The Dawn of Particle Accelerator Technology Basic Concepts Accelerators in Future Livingston Chart Laser Plasma Acceleration (1)

cf. C. Joshi and T. Katsouleas’s article in Physics Today, June 2003, p.47.

Fundamental Concepts of Particle Accelerators The Dawn of Particle Accelerator Technology Basic Concepts Accelerators in Future Livingston Chart Laser Plasma Acceleration (2)

Fundamental Concepts of Particle Accelerators The Dawn of Particle Accelerator Technology Basic Concepts Accelerators in Future Livingston Chart Livingston Chart

I Originally given by M. S. Livingston & 1017 J. P. Blewett: ”Particle Accelerators, 16 10 p.6”, MacGraw Hill, 1962 1015 1PeV Collider

14 (Equivalent Energy) I 10 Energies for the colliders are equivalent 1013 values for the ﬁxed target system

12 10 1TeV Proton Synchrotron I Maximum beam energy ever achieved 1011 Electron Linac I × 1010 Electron Synchrotron : 2 100 GeV Electron Synchrotron 10 9 1GeV Synchro-cyclotron (2000, CERN LEP)

Accelerator Energy (eV) Accelerator I 10 8 Proton Linac Proton Synchrotron : 2 × 7 TeV 1MeV 7 Electrostatic Accelerator 10 (2010, CERN LHC) Betatron Cyclotron 10 6 DC Generator http://lhc.web.cern.ch/lhc/

1930 1940 1950 1960 1970 1980 1990 2000 2010 I Electron-positron linear collider 2 × 500 GeV? (2025 or later?)

Fundamental Concepts of Particle Accelerators The Dawn of Particle Accelerator Technology Basic Concepts Accelerators in Future Livingston Chart References (1)

I Segr`e,E. : From X-rays to Quarks (W. H. Freeman and Company, 1980). I Historical introduction to the evolution of high energy physics and accelerator science I Chao, A. W. and Tigner, M. (ed.) : Handbook of Accelerator Physics 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

Fundamental Concepts of Particle Accelerators The Dawn of Particle Accelerator Technology Basic Concepts Accelerators in Future Livingston Chart 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 to energy accelerators

Fundamental Concepts of Particle Accelerators