1

LONGITUDINAL DYNAMICSIN PARTICLE ACCELERATORS

by

Joël Le DuFF

(retired from LAL-IN2P3-CNRS)

Cockroft Institute, Spring 2006

2

Bibliography : Old Books

M. Stanley Livingston High Energy Accelerators (Interscience Publishers, 1954)

J.J. Livingood Principles of cyclic Particle Accelerators (D. Van Nostrand Co Ltd , 1961)

M. Stanley Livingston and J. B. Blewett Particle Accelerators (Mc Graw Hill Book Company, Inc 1962)

K.G. Steffen High Energy optics (Interscience Publisher, J. Wiley & sons, 1965)

H. Bruck Accelerateurs circulaires de particules (PUF, Paris 1966)

M. Stanley Livingston (editor) The development of High Energy Accelerators (Dover Publications, Inc, N. Y. 1966)

A.A. Kolomensky & A.W. Lebedev Theory of cyclic Accelerators (North Holland Publihers Company, Amst. 1966)

E. Persico, E. Ferrari, S.E. Segre Principles of Particles Accelerators (W.A. Benjamin, Inc. 1968)

P.M. Lapostolle & A.L. Septier Linear Accelerators (North Holland Publihers Company, Amst. 1970)

A.D. Vlasov Theory of Linear Accelerators (Programm for scientific translations, Jerusalem 1968)

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Bibliography : New Books

M. Conte, W.W. Mac Kay An Introduction to the Physics of particle Accelerators (World Scientific, 1991)

P. J. Bryant and K. Johnsen The Principles of Circular Accelerators and Storage Rings

(Cambridge University Press, 1993)D. A. Edwards, M. J. Syphers An Introduction to the Physics of High Energy Accelerators

(J. Wiley & sons, Inc, 1993)H. Wiedemann Particle Accelerator Physics

(Springer-Verlag, Berlin, 1993)M. Reiser Theory and Design of Charged Particles Beams

(J. Wiley & sons, 1994)A. Chao, M. Tigner Handbook of Accelerator Physics and Engineering

(World Scientific 1998)K. Wille The Physics of Particle Accelerators: An Introduction

(Oxford University Press, 2000)E.J.N. Wilson An introduction to Particle Accelerators

(Oxford University Press, 2001)

And CERN Accelerator Schools (CAS) Proceedings

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Types of accelerators

Kinetic energy W

Electrons Protons/ ions

Electrostatic Van de Graaf &Tandems

20- 35 MeV (Vivitron)

Betraton

10- 300 MeV Microtron 25- 150 MeV

Cyclotron

10- 100 MeV

Synchro- cyclotron

100- 750 MeV

Synchrotron

1- 10 GeV 1- 1000 GeV

Storage ring

1- 7 GeV (ESRF)

Collider ring

10- 100 GeV (LEP) 1- 7 TeV (LHC)

Linacs

20 MeV- 50 GeV (SLC) 50- 800 MeV(LAMPF) MeV(LAMPF) (LAMPF) Linear collider

50- 1000 GeV (TESLA)

Total energy = Rest energy + Kinetic energy

E0 = m0c2

= E0+ W electron E0=0,511 MeVprotons E0=938 MeV

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Brief history of accelerators

1919 Rutherford gets the first nuclear reactions using natural alpha rays (radio activity) of some MeV).

« He notes already that he will need many MeV to study the atomic nucleus »

1932 Cockcroft & Walton build a 700 KV electrostatic generator and break Lithium nucleus with 400 KeV protons.

(Nobel Price in 1951)

1924 Ising proposes the acceleration using a variable electric field between drift tubes ( the father of the Linac).

1928 Wideroe uses Ising principle with an RF generator, 1MHz, 25 kV and accelerate potassium ions up to 50 keV.

1929 Lauwrence driven by Wideroe & Ising ideas invents the cyclotron.

1931 Livingston demonstrates the cyclotron principle by accelerating hydrogen ions up to 80 KeV.

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Brief history of accelerators (2)

1932 The cyclotron of Lawrence produces protons at 1.25 MeV and « breaks atoms » a few weeks after Cockcroft & Walton

(Nobel Prize in 1939)

1923 Wideroe invents the concept of betatron

1927 Wideroe builds a model of betatron but fails

1940 Kerst re-invents the betatron which produces 2.2 MeV electrons

1950 Kerst builds a 300 MeV betatron

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Main Characteristics of an Accelerator

ACCELERATION is the main job of an accelerator.•The accelerator provides kinetic energy to charged particles, hence increasing their momentum. •In order to do so, it is necessary to have an electric field , preferably along the direction of the initial momentum.

eEdtdp

BENDING is generated by a magnetic field perpendicular to the plane of the particle trajectory. The bending radius obeys to the relation :

Be

p

FOCUSING is a second way of using a magnetic field, in which the bending effect is used to bring the particles trajectory closer to the axis, hence to increase the beam density.

E

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Acceleration & Curvature

o

zx, r

s

Within the assumption:

EE

zBB

the Newton-Lorentz force:

BveEedtpd

becomes:

rzr uBevueEuv

mudtmvd

2

leading to:

zBep

eEdtdp

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Energy Gain

2220

2 cpEE

zeEdtdp

dzdpv

dzdE

eVdzEeWdzeEdEdW zz

In relativistic dynamics, energy and momentum satisfy the relation:

Hence: vdpdE

The rate of energy gain per unit length of acceleration (along z) is then:

and the kinetic energy gained from the field along the z path is:

where V is just a potential

WEE 0

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Methods of Acceleration

1_ Electrostatic Field

Energy gain : W=n.e(V2-V1)

limitation : Vgenerator = Vi

2_ Radio-frequency Field

Synchronism : L=vT/2

v=particle velocity T= RF periodWideroe structure

Electrostatic accelerator

220

TvLalso :

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3_ Acceleration by induction

From MAXWELL EQUATIONS :

AHB

tAE

The electric field is derived from a scalar potential and a vector potential A The time variation of the magnetic field H generates an electric field E

Methods of Acceleration (2)

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Accélérateur colonne

d.c. high voltage generator

HV system used by Cockcroft & Walton to break the lithium nucleus

Electrostatic accelerator

Accelerating column

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Van de Graaf type electrostatic accelerator

Electrostatic accelerator (2)

An insulated belt is used to transport electric charges to a HV terminal .

The charges are generated by field effect from a comb on the belt . At the terminal they are extracted in a similar way.

The HV is distributed along the column through a resistor.

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Betatron

The betatron uses a variable magnetic field with time. The pole shaping gives a magnetic field Bo at the location of the trajectory, smaller than the average magnetic field.

A constant trajectory also requires :

dtBd

Rdtd

ER z22

Induction law

Newton-Lorentz force

dt

BdReeE

dt

dp z

2

1

dt

dBRe

dt

dp

BRep

0

0

zo BB2

1

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At each radius r corresponds a velocity v for the accelerated particle. The half circle corresponds to half a revolution period T/2 and B is constant:

eB

mT

eB

mv

eB

pr

2

The corresponding angular frequency is :

m

eB

Trfr

2

2

Synchronism if : rRF

m = m0 (constant) if W <<

E0

v=Vsint

Cyclotron

If so the cyclotron is isochronous

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Cyclotron (2)

Cyclotron of M.S.Livingstone (1931)On the left the 4-inch vacuum chamberUsed to validate the concept.On the right the 11-inch vacuum chamberof the Berkeley cyclotron that produced1,2 MeV protons.In both cases one single electrode (dee).

Here below the 27-inch cyclotron,Berkeley (1932). The magnet was originally part of the resonant circuit of an RF current generator used in telecommunications.

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Cyclotron (3)

Cyclotron SPIRAL at GANIL

Here above is the magnet and its coils

Here below is an artist view of the spiral shaped poles and the radio-frequency system.

SPIRAL accelerates radio-active ions

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Cyclotron (4)

Cyclotrons at GANIL, Caen

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Cyclotron (5)

Energy-phase equation:

Energy gain at each gap transit:Particle RF phase versus time:

sinV̂eE

tRF

where is the azimuthal angle of trajectory

Differentiating with respect to time gives: EBecRFrRF

2

Smooth approximation allows:

r

rT 2/

12Bec

ERF

r

Relative phase change at ½

revolution

1sinˆ 2Bec

E

VeEdEd RF

And smooth approximation

again:

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Cyclotron (6)

2000

00

0 ˆ21ˆcoscos EE

EVeEE

Ve r

RF

r

RF

with :

dEBecE

Ved RF

1ˆcos 2

Integrating:

Separating:

0

0

0

r

E

Rest energy

Injection phase

Starting revolution frequency

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The expression shows that if the mass

increases, the frequency decreases :

m r

Synchronism condition:

m

eBr

mTr

If the first turn is synchronous :

electrons 0.511 MeV

Energy gain per turn protons 0.938 GeV !!!

Since required energy gains are large the concept is essentially valid for electrons.

)1(intint 0 egeregerT

T turnRF

r

Microtron (Veksler, 1954)

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Microtron « Racetrack »

Synchronism is obtained when the energy gain per turn is a multiple of the rest energy:

()/turn = integer

Allows to increase the energy gain per turn by using several accelerating cavities (ex : linac section)

Carefull !!!! This is not a « recirculating » linac

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The advantage of Resonant Cavities

- Considering RF acceleration, it is obvious that when particles get high velocities the drift spaces get longer and one loses on the efficiency. The solution consists of using a higher operating frequency.

- The power lost by radiation, due to circulating currents on the electrodes, is proportional to the RF frequency. The solution consists of enclosing the system in a cavity which resonant frequency matches the RF generator frequency.

-Each such cavity can be independently powered from the RF generator.

- The electromagnetic power is now constrained in the resonant volume.

- Note however that joule losses will occur in the cavity walls (unless made of superconducting materials)

RF

Ez

JouH

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The Pill Box Cavity

02

2

002

tAA )( HouEA

krJEz 0

krJZj

H 10

37762,220Za

ck

e tj

Ez H

From Maxwell’s equations one can derive the wave equations :

Solutions for E and H are oscillating modes, at discrete frequencies, of types TM ou TE. For l<2a the most simple mode, TM010, has the lowest frequency ,and has only two field components:

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The Pill Box Cavity (2)

The design of a pill-box cavity can be sophisticated in order to improve its performances:

-A nose cone can be introduced in order to concentrate the electric field around the axis,

-Round shaping of the corners allows a better distribution of the magnetic field on the surface and a reduction of the Joule losses. It also prevent from multipactoring effects.

A good cavity is a cavity which efficiently transforms the RF power into accelerating voltage.

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Energy Gain with RF field

RF acceleration

cosˆ

cosˆcosˆˆˆ

VeW

tzEtzEzEVdzzE RF

In this case the electric field is oscillating. So it is for the potential. The energy gain will depend on the RF phase experienced by the particle.

Neglecting the transit time in the gap.

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Transit Time Factor

tgVtEEz coscos0

vtz

Oscillating field at frequency and which amplitude is assumed to be constant all along the gap:

Consider a particle passing through the middle of the gap at time t=0 :

The total energy gain is: dzvz

geVW

g

g

2/

2/

cos

eVTeVW 2/

2/sin

angletransitvg

factortimetransitT( 0 < T < 1 )

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Transit Time Factor (2)

dzezEeE tjgze

0

dzezEeeE v

zjgz

j

ep

0

g v

zj

zjj

e dzezEeeeE ip

0

cos0 g vzj

z dzezEeE

g

z

g tjz

dzzE

dzezET

0

0

pvzt

ip

Consider the most general case and make use of complex notations:

p is the phase of the particle entering the gap with respect to the RF.

Introducing:

and considering the phase which yields the maximum energy gain:

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Important Parameters of Accelerating Cavities

Shunt Impedance

Filling Time

RVPd

2

PWQ

d

sW

VQR

s2

QWQdtWd

P ss

d

Quality Factor

Relationship between gap voltage and wall losses.

Relationship between stored energy in the volume and dissipated power on the walls.

Exponential decay of the stored energy due to losses.

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Shunt Impedance and Q Factor

The shunt impedance R is defined as the parameter which relates the accelerating voltage V in the gap to the power dissipated in the cavity walls (Joule losses).

The Q factor is the parameter which compares the stored energy, Ws, inside the cavity to the energy dissipated in the walls during an RF period (2/). A high Q is a measure of a good RF efficiency

PWQ

d

s

WV

QR

s2

RVPd

2

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Filling Time of a SW Cavity

From the definition of the Q factor one can see that the energy is dissipated at a rate which is directly proportional to the stored energy:

WQdtWd

P ss

d

leading to an exponential decay of the stored energy:

eWWt

ss 0 avec Q

Since the stored energy is proportional to the square of the electric field, the latter decay with a time constant 2 .

If the cavity is fed from an RF power source, the stored energy increases as follows:

eWWt

ss 212

0

(filling time)

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Equivalent Circuit of a Cavity

RF cavity: on the average, the stored energy in the magnetic field equal the stored energy in the electric field, Wse=Wsm

dVHdVE VV 2020

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RLC circuit: the previous statement is true for this circuit, where the electric energy is stored in C and the magnetic energy is stored in L:

*

0*

21

0*

21

4141

CVVWWW

LIVavecIILW

LCCVVW

smses

lLlsm

se

Leading to:

LRRCQ

RVV

Pd0

0

*

21

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Input Impedance of a Cavity

The circuit impedance as seen from the input is:

0

111 avecCj

LjRZ e

0

20

20

212

Qj

RRjL

RLZ e

Within the approximation «0 the impedance becomes:

When satisfies the relation Q=0/2 one has Ze= 0,707 |Ze|

max , with |Ze|max= R. The quantity 2/0 is called the bandwidth (BW) :

BWQ 1

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Loaded Q

If R represents the losses of the equivalent resonant circuit of the cavity, then the Q factor is generally called Q0.

Introducing additional losses, for instance through a coupling loop connected to an external load, corresponding to a parallel resistor RL , then the total Q factor becomes Ql ( loaded Q ):

Defining an external Q as, Qe=RL/0L, one gets:

L

Lt

tl RR

RRRavec

LRQ 0

QQQ el

1110

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Let’s consider a succession of accelerating gaps, operating in the 2π mode, for which the synchronism condition is fulfilled for a phase s .

For a 2π mode, the electric field is the same in all gaps at any given time.

sVeseV sinˆ is the energy gain in one gap for the particle to reach the next gap with the same RF phase: P1 ,P2, …… are fixed points.

Principle of Phase Stability

If an increase in energy is transferred into an increase in velocity, M1

& N1 will move towards P1(stable), while M2 & N2 will go away from P2 (unstable).

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Transverse Instability00

z

zE

t

VLongitudinal phase stability means :

The divergence of the field is zero according to Maxwell : 000.

x

E

z

E

x

EE xzx

defocusing RF force

External focusing (solenoid, quadrupole) is then necessary

A Consequence of Phase Stability

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Focusing

Accelerating section, of an electron linac, equipped with

solenoids

Accelerating section, of an electron linac, equipped with

quadrupoles

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Focusing (2)

For protons & ions linacs, small

quadrupoles are generally placed inside the drift tubes. Those quadrupoles can be

either electro-magnets or permanent magnets.

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The Traveling Wave Case

0

0 cos

ttvz

vk

kztEE

RF

RFz

velocityparticlev

velocityphasev

The particle travels along with the wave, and k represents the wave propagation factor.

00 cos

tvvtEE RFRFz

If synchronism satisfied:

00 cos EEandvv z

where 0 is the RF phase seen by the particle.

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Multi-gaps Accelerating Structures:A- Low Kinetic Energy Linac (protons,ions)

Mode L= vT/2 Mode 2 L= vT =

In « WIDEROE » structure radiated power CV In order to reduce the

radiated power the gap is enclosed in a resonant volume at the operating frequency. A common wall can be suppressed if no circulating current in it for the chosen mode. ALVAREZ structure