Introduction to particle accelerators and their ... · Introduction to accelerators : How they work /49 G. Chiodini - May 2015 How a particle is accelerated The relativistic mass

Introduction to particle accelerators and their applications - Part I:

how they work Gabriele Chiodini

Istituto Nazionale di Fisica Nucleare Sezione di Lecce

PhD lessons in Physics for Università del Salento

2015-16 (20 hours, 4 CFD)

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/49Introduction to accelerators : How they work G. Chiodini - May 2015

Introduction

• You are new in the field of accelerators

• Only basic concepts are going to be introduced

• Completely intuitive approach

• Clarify the concepts of physics, no mathematics, no rigorous scientific derivation

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What is an accelerator• In the first half of the '900 accelerating structures are

built to increase the kinetic energy of charged atoms ( ions ) and induce new nuclear reactions by collision with a target ( artificial radioisotopes ). ⇒ Applied Physics

• In the second half of the ‘900 accelerator complex are built to reach energies higher and higher ( ultra-relativistic energy) and study the infinitesimal properties of the matter ( subatomic particles and fundamental interactions ). ⇒ Fundamental Physics

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How a particle is accelerated

Classic mechanics (Newton)

• Force = mass x acceleration: F=ma (It is false for speed near the speed of light)

• momentum = mass x velocity: p=mv

Relativistic mechanics (Einstein)

• increase of p = force x time: Δp=FT (always true)

• p=mv: where m is the relativistic mass

• Δp=pfinal-pinitial ~ mΔv + Δmv: increase of p due to an increase of velocity (v) or to an increase of relativistic mass (m)

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How a particle is accelerated

The relativistic mass ( than time and energy ) tends to infinity for speed near to the speed of light c = 300,000 km/s → the speed of light is an insurmountable limit.

In relativistic regime is more correct to speak of increase of energy than acceleration because the speed saturates at β → 1 , γ → ∞

m = γm0

γ =11− β 2

β =vc

mass at rest

Relativistic factor of dilatation

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speed relative to speed of light (c)

β = 1− 1γ 2

relativistic mass


Kinetic and relativistic energyClassic mechanics (Newton)

• The kinetic energy T of a particle is quadratic in the velocityand proportional to the rest mass.

!

Relativistic mechanics (Einstein)

• The relativistic energy E of a particle is the hypotenuse of a right triangle having as cathets the rest mass and the moment relativistic.

• The kinetic energy T is defined ad the difference between the relativistic energy and the energy at rest E0=m0c2

T = m0v2

2= E

T = E −m0c2 = E − E0

E = (m0c2 )2 + (pc)2m0c

2

pc

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/49 G. Chiodini - May 2015Introduction to accelerators : How they work

Potential energy• The gravitational force on earth is F=mg

where g=9.8m/s2 is e constant.

• The work L is the product between Force and Displacement L=Fh=mgh and it corresponds to the potential energy U of the gravitational field.

• The total energy E=U+T is constant

• The potential energy U=mgh is converted in kinetic energy T=1/2mv

2 during the body

fall-down

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U=mgh,T=0

U=mgh/2=T

U=0,T=mgh v = 2gh


Electric forces and fields

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Attractive and repulsive electrical forces occur easily by rubbing

Matter electrifies when atoms of matter loose electrons ( + ) or gain electrons ( - ) electrons

The electric field E acts on the charge q with the electric force F=qE The electron charge is 1.6E-19 Coulomb


Electrostatic energy and elettronvolt (eV)

• The electric potential V[=Energy/Charge] is the work done on the unit charge q=1C

• V=L/q=Fh/q=qEh/q=Eh

• V is measured in Volt = Joule/Coulomb

• The potential energy is U=qV

• It is practical to use as energy unit the elettronvolt, which is equal to the energy acquired by an electron moving in a potential difference of 1 Volt: 1eV= (e) (1V) = (1.6E-19 C) x (J / C) 1eV=1.6E-19 J

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1 Volt battery


Electron and proton mass

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me=9.1E-31 kg mec2=9.1E-30 kg (3E8)2(m/s)2=81E-14 J

mec2/e=81E-14 J / 1.6E-19 C = 50E5 eV = 0.5MeV

Mp=1.7E-27kg 1870xme=0.94GeV

The mass can be measured in energy by multiplying for c2

Often you write m(GeV/c2)

1GeV=1000MeV, 1MeV=1000keV, 1keV=1000eV


Electron and proton momentum

• The electron energy increase above 1 MeV is mostly due to relativistic mass increase

• The proton velocity increase is important up to thousand of MeV

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T=E-E 100keV 1MeV 10MeV 100MeV

β=v/c 0,55 0,943 0,9975 0,999987

γ 1,2 3 20 200

T=E-E 1MeV 10MeV 100MeV 1GeV

β=v/c 0,0447 0,0197 0,416 0,866

γ 1,001 1,01 1,1 2

elettrone me=0.5MeV/c2

protone Mp=1GeV/c2

Use formula of slide 5,6 and 7 to calculate the 2nd and 3rd columns from the 1st one.

The momentum can be measured in energy by multiplying for c Often you write p(GeV/c)


Pre-accelerators era

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Particle sources

• In 1895 Lenard builds the C RT f o r s c a t t e r i n g experiments on gas by accelerating electrons .

• The CRT is sold to Rontgen who discovers that X-rays were produced

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A hot filament ( K ) which acts as a cathode in vacuum emits electrons ( - ) that are accelerated by a potential difference ( Ua ) and strike a metal target ( A) which acts as an anode emitting X-rays. The target is maintained at a temperature lower than the melting point by means of a liquid coolant ( W ) .


Natural radioactivity

• In 1906 Rutherford bombards mica and gold sheets with natural alpha radiation of a few MeV energy

• In 1919 Rutherford i n d u c e s a n u c l e a r reaction with natural alpha radiation

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The alpha radiation emitted by a natural source natural is transformed into a collimated beam and directed towards a thin sheet of gold through a hole of the lead shield. A fluorescence screen reveals the deflection at large angles of alpha radiation discovering in the atom the presence of a small nucleus that contains almost all the mass ( the atomic nucleus ).


Electrostatic accelerators era

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Electrostatic accelerator

• Final energy = Initial energy + (ion charge) x V

• It is necessary to have a DC (direct current) high voltage generator:

• Cockcroft-Walton’s voltage multiplier

• Van De Graaff ’s generator

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Dome at high electric voltage V

Accelerating tube

Grounded mechanical base

Source of ions extracted from a discharge tube

Target


Cockcroft-Walton multiplier

• Rutherford pushes for accelerators exceeding MeV but this was not in those times reach

• In 1928 Gamov predicts that 0.5 MeV may be enough to induce nuclear reactions thanks to the tunnelling effect.

• In 1932 Cockcroft and Walton reach 0.7 MeV and split lithium atom with protons accelerated to 0.4 MeV: ( Li7 + p → He4 + He4 )17


Voltage multiplier

• Diodes D1 e D2 conduct c u r re n t i n t h e a r ro w direction only.

• The capacitors C1 and C2 charge-up to maximum voltage by the diodes

• The output voltage is the sum of the capacitor voltages: Vout=VC1+VC2=2VINP

• n stages: Vout=2nVINP

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Cockroft-Walton at FNAL in Chicago

• The AC transformer of a few kV is not shown

• In the cubic structure the electrons are added to hydrogen atoms to form negative ions

• Negative ions are passed into the top left tube towards the 0.75 MV Cockroft - Walton generator

• The Cockroft-Walton generator is on the left with a dome on the top

• The capacitors are most of the vertical blue cylinders

• The diodes are the diagonal cylinders diagonal

• The metallic balls and toroids prevent the formation of corona and/or arc discharges between connection points

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Van de Graaff ’s generator

• In the early 30's Van de Graaff builds its high- voltage generator of up to 1.5 MV.

• These generators can operate up to 10 MV, provide stable beams, highly directional and w i t h l o w e n e r g y dispersion.

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Tandem Van de Graaff

• The doubling of energy is achieved with a very clever idea : change the sign of the accelerated particles charge and use a second generator with opposite polarity

• These generators can operate up to 10 MV, provide stable beams, highly directional and w i t h l o w e n e r g y dispersion.

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E = V + zV

Negative ions gain energy V at the HV terminal where they are transformed in neutral atoms (z=0) and positive ions (z=1,2,…) by the stripping gas.


The limit of electrostatic accelerators

• The limit of electrostatic generators is of about 10 MV beyond which electrostatic breakdown of electrical insulation occurs and you can not increase energy by putting more generators in cascade

• The electrostatic field is conservative and energy gain can not be boost through multiple passes

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“True” accelerators era

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Linear and circular accelerators

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Linear

Circular

… but time variable fields must be used


1. Cascade of identical accelerating structure. 2. Time variable electric fields to avoid an increase of voltage going from one accelerating structure to the next one.

Linear accelerators

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Wideroe’s linac

• In 1924 Ising proposes to use variable electric fields between consecutive cylindrical conductor (drift tubes) to boost the energy beyond the maximum vo l tage o f the sys tem ( " t rue" accelerator).

• In 1928 Wideroe demonstrates the Ising’s principle by a 1MHz radio frequency oscillator of 25 kV amplitude accelerating potassium ions at 50 keV.

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E=0E=0 E=0E=0

+→− !E<0

−→+ !E>0

+→− !E<0

+ +- -

+ -

The beam is extracted in bunches. Only synchronous particles are accelerated ( next slide )


Synchronous condition

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L0 =v0T2

+ - + -

π mode

+- +-

+ - + -

E(t) = E0 cos(2πft)

t0

t2=t0+T/2

t2=t0+T

L = vT2E = E0

L1 =v1T2

L2 =v2T2

E = E0

E = E0


Alvarez’s linac• The drift tubes are limited to 10 MHz

becoming antennas and dissipating energy in space. With this frequency upper limit and at high energy the length of the tubes becomes prohibitive

• In 1946 Alvarez surrounds the drift tubes with a RF Resonant Cavity supplied by an external High Power-High Frequency RF source that generates electromagnetic waves at 2 0 0 M H z f r e q u e n c y ( R a d a r Technology of the 2nd World War)

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synchronous condition: 2π modeL = vT


Electromagnetic waves

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f = 1T

f=10MHz → T=100ns → λ=30m f=200MHz → T=5ns → λ=1.5m f=3GHz → T=0.33ns → λ=0.1m

E(t,z) = E0 cos(2πtT− 2π x

λ)

λ =cf= Tc

Frequency

Wavelength

An electromagnetic wave in vacuum is constituted by mutually orthogonal electric and magnetic fields varying sinusoidally in time and space and orthogonal to the propagation direction . NB : In a resonant cavity the electric field acquires a component parallel to the propagation direction and can accelerate charged particles.


Phase velocity

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E(t,z) = E0 cos(2πtT− 2π x

λ)

E(t,z) = E0 cos(2πt + ΔtT

− 2π xλ)

The phase velocity is determined by the apparent motion of the wave crest.

vfase =ΔxΔt

= c

NB : In a resonant cavity the phase velocity vfase of an electromagnetic wave is less than

the speed of light and can accelerate the particles satisfying the synchronous (or resonant) condition vparticella = vfase .

What happen in a waveguide?

tt

t

fase = cos tan te = ϕ =2π tT

−2πxλ

2π tT

−2πxλ

=2π (t + Δt)

T−2π (x + Δx)

λΔtT=Δxλ→

ΔxΔt

=λT


Dispersion relation of: vacuum, wave guide, cavities

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The limit of Linacs

• The use of radio frequency allows to have zero potential at both accelerator ends avoiding the system breakdown

• An unlimited number of drift tubes spaced by acceleration gaps can be cascaded.

• The linac becomes impractical when the energies is too high because the length becomes unrealistic

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Circular accelerators

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1. Use time-varying fields to increase energy along closed orbits. 2. Deflection fields in several regions needed to keep particles in closed orbits.


Magnetic fields• The magnetic field B is generated by a macroscopic

electric current (coil) or by a microscopic electric current (material ferromagnetic domains) of the materials ) and it is orthogonal to it.

• The magnetic field generated by a coil ( or by a magnet ) generate a force acting on other coil ( or other magnet ).

• The magnetic poles of a coil ( or of a magnet ) repel ( attract ) each other if generated by currents having the same direction ( opposite ).

• The magnetic poles of a coil ( or of a magnet) can not be separated (there are no magnetic monopoles ) then the magnetic force acting on a coil ( or on a magnet) tends to make it rotate ( like the needle of a compass ).

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Current loop is equivalent to magnetic compass needle

The Earth is a large magnet interacting with the magnetic compass needle and orienting the needle from north to south


Electric and magnetic force acting on a charged particle

• The electric force is parallel to the electric field and only the component parallel to particle velocity can accelerate the particle

• The magnetic force is orthogonal to particle speed and to the magnetic field, than it doesn’t accelerate the particle

• In relativistic regime v ~ c and the magnetic force becomes very efficient in deflecting the particle

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Felettrica = qE

Fmagnetica = qvBB

v

v

E


Uniform circular motion

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Fcentripeta = macentripeta = mv2

ρa = v

2

ρ

A particle of mass m during a uniform circular motion of radius ρ with tangential velocity v is subject to an acceleration towards the center equal to:

a=v2/ρ (centripetal acceleration).

The Newton’s law of force implies that to a centripetal acceleration corresponds to a centripetal force:

centripetal acceleration = mass x centripetal acceleration

v

mρ

Fcentriguga = −macentripetaFcentripeta + Fcentriguga = 0

Fcentripeta


Uniform circular motion in magnetic field

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−Fcentrifuga = ma = mv2

ρ=pvρ

Fcentripeta = qvB

pvρ= qvB

Centripetal force = Magnetic force

pq= ρB p(GeV / c)

z= 0.3ρ(m)B(T)magnetic rigidity

pq= ρB→ cp / e

q / e= cρB→ p(eV / c)

z= cρB→ p(GeV / c)

z= cρB ⋅10−9 → p(GeV / c)

z= 3 ⋅108 ⋅ ρB ⋅10−9

q=ze where z is the charge in unit of the electron charge

Fcentripetal


The cyclotron

• In 1929 Lawrence designs the famous cyclotron : a linac wrapped on itself

• In 1931 his student Livingston builds a demonstrator acce lerat ing hydrogen ions up to 80 keV

• In1932 Lawrence builds a cyclotron accelerating protons up to 1.25 MeV and splits atoms

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The Lawrence’s cyclotron

• An electromagnet generates a magnetic field which rotates the charged particles.

• On two D-shaped hollow containers an alternating voltage is applied synchronised with the charged particles arrival.

• At each charged particle passing between the two D’s the particles are accelerated.

• The charged particles released from the source at the centre between the two D’s spiral-out until they are extracted and sent to the target .

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Cyclotron synchronous condition

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T = 2πρv

=2πρp / m

=2πρ

qρB / m= 2π m

qB

f = 1T=12πqBm

Revolution period

Revolution frequency

In the non-relativistic regime f is constant and the accelerated particle remains synchronous with the radio frequency ( suitable for protons and not for electrons)

The beam is extracted in bunches. Only synchronous particles are accelerated.

(isochronous not synchronous, see later)


The betatrons• In 1923 Wideroe designs

the betatrons discovering the famous rule 2 to 1, but his prototype does not work (vacuum problems)

• In 1940 Kerst reinvents the betatron and builds one for electrons up to 2.2 MeV

• In 1950 Kerst builds the largest betatron in the world for electrons up to 300 MeV

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The betatron

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• A pulsed electromagnet generates a variable magnetic field which rotates the charged particles .

• The charged particles circulate in a circular tube and concatenate the variable vertical magnetic field

• The “guide” magnetic field keeps the particles in a circular orbit and the “mean” magnetic field accelerates the particles by magnetic induction

• The “mean” field and the “guide” field must satisfy the rule 2 : 1 to keep the synchronous particles (Wideroe’s principle)


Lenz’s law of magnetic induction

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B= EXTERNAL MAGNETIC FIELD changing in time

E=induced electric field

The work of the electric field along a closed curve C of length L is equal to the rate change of the magnetic flux Φ crossing the surface S.

V = EL = − Φmax

T= −

BmaxST

The electric field induced by the variable magnetic field is orthogonal to it and capable of accelerating a charged particle along the closed trajectory

S=surface with boundary C

C=closed curve

The a l ternator conver ts mechan i c a l e ne r g y i n to electrical energy

V = −ΔΦmax

Δt= −2πfBmaxSsin(2πft)

NB: The direction of the electric field is such to induce a current which generates a magnetic field opposite to the field variation.

E(t) = E0 cos(2πft)B(t)=Bmax


Ratio 2:1

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Epoloidale 2πρ =Φmax

T=Bmaxmedioπρ2

T

p = FpoloidaleT = eEpoloidaleT =eBmax

medioρ2

p = eρBguida

Bguida =12Bmaxmedio

T

• The extracted beam is continuos (during the spill) • T is of the order of milliseconds (spill period)

Acceleration

Rotation

/49 G. Chiodini - May 2015Introduction to accelerators : How they work 45

The synchrotron• In 1943 Oliphant combines three concepts :

acceleration with RF resonators, variable frequency, pulsed guide magnetic fields.

• In 1944 McMillan and Veksler independently propose the synchrotron with Phase Stability

• In 1946 Goward and Barnes are the first to build a synchrotron in UK

• In 1952 several groups invent the Strong Focusing

• In 1956 MURA in US proposes to increase the beam intensity by Stacking

• In 1961 Touschek creates the first single-loop electron-positron collider (e+-e- )

Plenty of room for injection, experiments, extraction, RF cavities ... thanks to the "strong focusing" of the quadrupole ( next lesson ).


Deflection of charge particles

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Magnetic deflection C-shaped magnetic dipoles employed as particle field guide in synchrotron (excellent in relativistic regime)

Electrostatic deflection Parallel plates ( V ~ 200kV ) used to inject beams in the synchrotron ( excellent for low energy )

E = Vh

B = µ0nIh

where μ0=4π10-7H/m, n=number of turns, I=coil current, h=pole gap V=voltage and E=vertical electric field

p(GeV / c)z

= 0.3ρ(m)B(T) magnetic rigidity θ =vhv=mvhmv

=qETmv

=qVLmv2h

θ

L

hv

vh << v


Synchrotron cycle

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p(GeV / c)z

= 0.3ρ(m)B(T)

Acceleration phase

Synchrotron condition: the radio frequency f must be an integer of the rotation frequency fr

• The beam is in bunches • T is of the order of hours or days

The name synchrotron is due to the fact that the frequency f of the RF cavities should be adjusted during the acceleration to meet the synchrony condition and the magnetic field B has to be increased during acceleration to keep the particle in orbit.

RF ↑ B ↑

RF=OFF B=constant

Circular orbit

Syncrotron cycle

( s y n c h ro n o u s , n o t isochronous)

fr =2πvρ

=fn


Electro-syncrotron in Frascati (1959-1975 electrons at 0.4-1 GeV)

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4 dipoles→“week focusing” !Very little space for injection, exper iments , ex trac t ion , radiofrequency, … It needs a "strong focusing" → quadrupole ( next lesson ) .


Classification• Electrostatic accelerators

• Cockcroft-Walton

• Van De Graaff

• Tandem

• Accelerators with time-varing electric field

• Induction accelerators

• betatrons

• Radiofrequency accelerators

• Linac

• Cyclotron

• Synchrotron

EL = V

V = EL = − BmaxST

Betatron: Unbunched E poloidal

Resonator: Bunched E axial

Conservative electrostatic force

Electromagnetic induction force

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Introduction to particle accelerators and their ... · Introduction to accelerators : How they work /49 G. Chiodini - May 2015 How a particle is accelerated The relativistic mass