SUMMER STUDENT LECTURES ON SUMMER STUDENT LECTURES ON

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SUMMER STUDENT LECTURES ON SUMMER STUDENT LECTURES ON

ACCELERATORS (2, 3 & 4/5)

Elias

Elias Métral (Métral (3 3 ââ 45 min = 135 min, 92 slides45 min = 135 min, 92 slides)) 1 & 5/5

⇒ Simone

Introduction and references (4 slides)

Transverse beam dynamics (27) + Discussion Sessions 22

Longitudinal beam dynamics (12)

Figure of merit for a synchrotron/collider = Brightness/Luminosity (6)

B t l (8) 33

Beam control (8)

Limiting factors for a synchrotron/collider ⇒ Collective effects (34)

Space chargep g (8)( )

Wake field and impedance (12)

Beam-beam interaction (8)

Electron cloud (6)

44 [email protected]@cern.ch T l 72560 164809 T l 72560 164809

Elias Métral, Summer Student Lectures, CERN, 5-6-9-10-11/07/07 1/92

Electron cloud (6) Tel.: 72560 or 164809Tel.: 72560 or 164809

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PURPOSE OF THIS COURSE

Try to give you an overview of the basic concepts &y g y vocabulary ⇒ No mathematics, just physics

…But I would also like you to catch a glimpse of what accelerator physicists are “really” doing, showing you pictures of some recent/current studies

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Duoplasmatron = Source î 90 keV (kinetic energy) LINAC2 = Linear acceleratorî 50 MeV LINAC2 = Linear accelerator î 50 MeV

PSBooster = Proton Synchrotron Booster î 1.4 GeV PS = Proton Synchrotron î 25 GeV

SPS = Super Proton Synchrotron î 450 GeV LHC = Large Hadron Collider î 7 TeV

Linear accelerator

Transfer line

Injection Ejection

Circular accelerator (Synchrotron)

Elias Métral, Summer Student Lectures, CERN, 5-6-9-10-11/07/07

LHC beam in the injector chain

3/92

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PS Booster

(after the wall) PS

Linac2

SPS tunnel PS tunnel

Linac2

What happens to the particles inside the vacuum chamber?

~10^-10 Torr in the LHC = ~3 million molecules / cm³

Reminder : 1 atm = 760 Torr and 1 mbar = 0.75 Torr = 100 Pa

the vacuum chamber?

Vacuum chamber (f = 13 cm here)

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REFERENCES

[0] 2005 Summer Student Lectures of O. Bruning,

+ our lectures in 2006

[ ] g,

[http://agenda.cern.ch/askArchive.php?base=agenda&categ=a054021&id=a054021/transparencies]

[1] M. Martini, An Introduction to Transverse Beam Dynamics in Accelerators, CERN/PS 96-11 (PA), 1996, [http://doc.cern.ch/archive/electronic/cern/preprints/ps/ps-96-011.pdf]

[2] L. Rinolfi, Longitudinal Beam Dynamics (Application to synchrotron), CERN/PS 2000-008 (LP), 2000, [http://doc.cern.ch/archive/electronic/cern/preprints/ps/ps-2000-008.pdf]

[3] Theoretical Aspects of the Behaviour of Beams in Accelerators and Storage [3] Theoretical Aspects of the Behaviour of Beams in Accelerators and Storage Rings: International School of Particle Accelerators of the ‘Ettore Majorana’

Centre for Scientific Culture, 10–22 November 1976, Erice, Italy, M.H. Blewett (ed.), CERN report 77-13 (1977)

[http://preprints.cern.ch/cgi-bin/setlink?base=cernrep&categ=Yellow_Report&id=77-13] [4] CERN Accelerator Schools [http://cas.web.cern.ch/cas/]

[5] K Schindl Space Charge CERN-PS-99-012-DI 1999 [5] K. Schindl, Space Charge, CERN PS 99 012 DI, 1999

[http://doc.cern.ch/archive/electronic/cern/preprints/ps/ps-99-012.pdf]

[6] A.W. Chao, Physics of Collective Beam Instabilities in High Energy Accelerators New York: Wiley 371 p 1993 [http://www slac stanford edu/ achao/wileybook html] Accelerators, New York: Wiley, 371 p, 1993 [http://www.slac.stanford.edu/~achao/wileybook.html] [7] Web site on LHC Beam-Beam Studies [http://wwwslap.cern.ch/collective/zwe/lhcbb/]

[8] Web site on Electron Cloud Effects in the LHC [http://ab-abp-rlc.web.cern.ch/ab-abp-rlc-ecloud/]

[9] LHC Design Report [http://ab-div.web.cern.ch/ab-div/Publications/LHC-DesignReport.html]

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TRANSVERSE BEAM DYNAMICS (1/27)

Si l ti l t j t Single-particle trajectory

In the middle of the vacuum chamber Circular design orbit

One particle

ϑ

The motion of a charged particle (proton) in a beam transport channel or a circular accelerator is governed by the LORENTZ FORCE

( ^r ^r )

r ^e ( ^E ^v ^B )

F r r r r

× +

=

The motion of particle beams under the influence of the Lorentz force

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The Lorentz force is applied as a

BENDING FORCE (using DIPOLES) to guide the particles along a predefined ideal path the DESIGN ORBIT on which along a predefined ideal path, the DESIGN ORBIT, on which – ideally – all particles should move

FOCUSING FORCE (using QUADRUPOLES) to confine the

FOCUSING FORCE (using QUADRUPOLES) to confine the particles in the vicinity of the ideal path, from which most particles will unavoidably deviate

LATTICE = Arrangement of magnets along the design orbit

The ACCELERATOR DESIGN is made considering the beam as a collection of non-interacting single particles

(8)

DIPOLE = Bending magnetg g Constant force in x

S-pole

y

Bend

B

and 0 force in

v

Designy

g B g

x

^magnet

Design orbit

s

Beam

N-pole

F ^ρ ( ) ^s

⇒ A particle, with a constant energy, describes a circle in equilibrium between the centripetal magnetic force and the centrifugal force

BEAM RIGIDITY

^B ^ρ [ ^T ^m ] ⁼ ³ ^. ³³⁵⁶ ^p

⁰

[ ^GeV ^/ ^c ]

Magnetic field Curvature radius Beam momentum

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LEP vs LHC magnets (in same tunnel) î A change in technology

LEP LHC

r [m] 3096 175 2803 95 r [m] 3096.175 2803.95 p

₀

[GeV/c] 104 7000

B [T] 0.11 8.33

Room-temperature coils

Superconducting coils

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TRANSVERSE BEAM DYNAMICS (5/27) protons per

l

PS magnetic field for the LHC beam

^{1 Tesla}

= 10⁴ Gauss pulse

8 10

pp] 12000

14000

6

Y [1012 p

8000 10000

auss]

2 4

TENSITY

4000 6000 B [Ga 2^nd Injection

(1370)

0

INT 2

0 2000

( )

150 650 1150 1650 2150

TIME IN THE CYCLE [ms]

1^st Injection

(170) Ejection

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TRANSVERSE BEAM DYNAMICS (6/27) QUADRUPOLE = Focusing magnet QUADRUPOLE Focusing magnet

In x (and Defocusing in y) ⇒ F type Permutating

S l

N l

y

in y) ⇒ F-type. Permutating the N- and S- poles

gives a D-type

S-pole N-pole

Beam

x

Linear force in x&y

N-pole S-pole

x

_Beam

( ) ⁺ ( ) ⁼ ⁰

′′ s K x s x

⇒

^x ( ) ^s ⁺ ^K ^x ( ) ^s ⁼ ⁰

: Equation of a harmonic oscillator

⇒ : Equation of a harmonic oscillator

From this equation, one can already anticipate the elliptical shape of the particle trajectory in the phase space (x, x’) by integration

( )

²

( ) ^Constant

2

+ =

′ s K x s x

( ) ^s ⁺ ^K ^x ( ) ^s ⁼ ^Constant

x

(12)

TRANSVERSE BEAM DYNAMICS (7/27) Analogy with light optics

Principle of focusing for light

f θ x

tan = − f

(

^D ^F

)

2

F

0 when f f f = f > = −

Principle of STRONG FOCUSING for light

(

^D ^F

)

d

î Focusing in both g planes

(13)

Along the accelerator K is not constant and depends on s (and is periodic) ⇒ The equation of motion is then called HILL’S EQUATION

The solution of the Hill’s equation is a pseudo-harmonic oscillation with varying amplitude and frequency called BETATRON with varying amplitude and frequency called BETATRON OSCILLATION

Betatron function

( ) ^s ⁼ ( ) ^s [ ^μ ( ) ^s ⁺ ^ϕ ]

x a β cos

An invariant, i.e. a constant of motion, (called COURANT-SNYDER INVARIANT) can be found from the solution of the Hill’s equation

⇒ Equation of an ellipse (motion for one particle) in the phase space plane (x x’) with area π ^a²

phase space plane (x, x ), with area π ^a²

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( )

^s

x′

( )

′

( )

^s ^x

( )

^s

x

γ(s) β(s)

(s)

−aα

β(s) (s) aα

−

( )

^s

x

) ( a

β(s)

Courtesy M. Martini

) s ( γ

are called TWISS PARAMETERS

( ) ( )

2 s s β ′

−

=

( ) ^s α

β ( ) ( )

( )

β s 1

γ

2

s + α

=

The shape and orientation of the phase plane ellipse evolve along

( ) ₂

( )

β ^β ( ) ^s

(15)

Stroboscopic representation or POINCARÉ MAPPING Stroboscopic representation or POINCARÉ MAPPING

( ) ^s

x ^x ′ ^′ ( ) ^s

⁰

( ) ^s

x ( ) ^s

⁰

x

Depends on the TUNEepe ds o t e U

= Q_x = Number of betatron oscillations per machine revolution

( ^π )

μ ² ^R

= Q

π

x

2 Q

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MATRIX FORMALISM: The previous (linear) equipments of the

MATRIX FORMALISM: The previous (linear) equipments of the accelerator (extending from s₀ to s) can be described by a matrix, M (s / s₀), called TRANSFER MATRIX, which relates (x, x’) at s₀ and (x x’) at s

(x, x ) at s

( ) _⎥ ^⎤ ⁼ ( ) ( ) _⎢ ^⎡ _⎥ ^⎤

⎢ ⎡

₀

/

0

x s s

s s M

x

( ) ^⎥ ⎦ ( ) ( ) ^⎢ _{⎣ ′} ^⎥ _⎦

⎢ ⎣ ′ /

⁰ ⁰

s s x

s s M

x

The transfer matrix over one revolution period is then the product of the individual matrices composing the machine

The transfer matrix over one period is called the TWISS MATRIX

Once the Twiss matrix has been derived the Twiss parameters can be obtained at any point along the machine

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In practice, particle beams have a finite dispersion of momenta about the ideal momentum p₀. A particle with momentum p π p₀ will perform betatron oscillations around A DIFFERENT CLOSED ORBIT from that of the reference particle

( ) ( ) ( )

0 0

0

p s p

p D p s p

D s

x

_Δ

=

_x

− =

_x

Δ

⇒ Displacement of

0

p

( )

is called the DISPERSION FUNCTION

( ) ^s

D

_x

(18)

LHC optics for the Interaction Point (IP) 5 (CMS) in collision

β β D D _M _t _{ff t} ₀

MAD program

β_x β_y D_x D_y Momentum offset = 0

m] m]

β x,y[m D x,y[m

(19)

BEAM EMITTANCE = Measure of the spread in phase space of the

BEAM EMITTANCE Measure of the spread in phase space of the points representing beam particles ⇒ 3 definitions

1) In terms of the phase plane “amplitude” a_q enclosing q % of 1) In terms of the phase plane amplitude a_q enclosing q % of

the particles ( ^% )

f lli

q

x

d

dx ′ = π ε

∫∫

2) In terms of the 2^nd moments of the particle distribution

a

"

"amplitude ellipseof

q [mm mrad] or [μm]

2) In terms of the 2 moments of the particle distribution

(

^stat

) ≡ < x

²

> < x ′

²

> − < x x ′ >

²

ε

x Determinant of the

covariance matrix 3) In terms of σx the standard deviation of the particle

distribution in real space (= projection onto the x axis)

x covariance matrix

distribution in real space (= projection onto the x-axis)

( ) ^x

x ^x

σ

ε

^σ

≡

²

Elias Métral, Summer Student Lectures, CERN, 5-6-9-10-11/07/07 x 19/92

x

β

ε

(20)

( ) ^s

⁰

x ′

particle with

“amplitude”^amplitude

a a ε ε

_q

q

= ε

a

( ^%)

a

_q

= ε

_x^q

( ) ^s

⁰ ^x⁽^q^% ⁾

x

ε

γ

( ) ^s

⁰

x

( )

0

( ) ^s

⁰⁰

A

_x

x x

= γ

Emittance

( ) ^s

⁰

x

Beam divergence

particle distribution ₀

( ) ^s

⁰ ^x⁽^q^% ⁾

^E

^x

( ) ^s

⁰

x

ε =

β

Beam envelope The β-function reflects the size of

the beam and depends only on the lattice the beam and depends only on the lattice

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MACHINE mechanical (i e from the vacuum chamber) ACCEPTANCE

NORMALIZED BEAM EMITTANCE Relativistic factors

MACHINE mechanical (i.e. from the vacuum chamber) ACCEPTANCE or APERTURE = Maximum beam emittance

NORMALIZED BEAM EMITTANCE

( )

x

( )

x

x r r norm

x

σ β γ ε σ

ε

_x_,_norm

= β

_r

γ

_r _x

⇒ The normalized emittance is conserved during acceleration

(i th b f ll ti ff t )

(in the absence of collective effects…)

ADIABATIC DAMPING: As β_r g_r increases proportionally to the particle momentum p the (physical) emittance decreases as 1 / p particle momentum p, the (physical) emittance decreases as 1 / p

However, many phenomena may affect (increase) the emittance

An important challenge in accelerator technology is to preserve

beam emittance and even to reduce it (by COOLING)

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CHROMATICITY = Variation of the tune with the momentum

/ p p

Q

_x

Q

^x

Δ

= Δ

′

/ p

0

Δ p

The control of the chromaticity (using a SEXTUPOLE magnet) is very important for 2 reasons

very important for 2 reasons

Avoid shifting the beam on resonances due to changes induced by chromatic effects (see later)

by chromatic effects (see later)

Prevent some transverse coherent (head-tail) instabilities (see also later)

SEXTUPOLE = 1^st nonlinear magnet

Particles approach

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Multipole magnetic FIELD EXPANSION (used for the LHC magnets)

[ ]

⁻¹

∞

⎟ ⎞

⎜ ⎛ +

∑ ^[ ^] ^x ^j ^y

ⁿ

=1

⎟⎟

⎠

⎜⎜ ⎞

⎝

⎛ +

+

=

+ ∑

n r

n n

ref x

y

R

y j a x

j b

B B

j B

^B_ref = magnetic field at the reference radius R_r = 17 mm

^{n = 1} î dipole; n = 2 î quadrupole; n = 3 î sextupole…

b_n î normal harmonics

a_n î skew harmonics (the skew magnets differ from the regular magnets only by a rotation about the s-axis by an angle p / 2n where n is the order of the multipole)

angle p / 2n, where n is the order of the multipole)

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In the presence of extra (NONLINEAR) FORCES the Hill’s equation

In the presence of extra (NONLINEAR) FORCES, the Hill s equation takes the general form

( ) ^s ^K ( ) ( ) ^s ^x ^s ^P ( ^x ^y ^s )

x ′′ ( ) ^s + ^K ( ) ( ) ^s ^x ^s = ^P ( ^x ^y ^s )

x +

_x

=

_x

, ,

Any perturbation Any perturbation

P t b ti t i th ti f ti l d t UNSTABLE

Perturbation terms in the equation of motion may lead to UNSTABLE motion, called RESONANCES, when the perturbating field acts in synchronism with the particle oscillations

A multipole of nth order is said to generate resonances of order n.

Resonances below the 3^rd order (i e due to dipole and quadrupole Resonances below the 3 order (i.e. due to dipole and quadrupole field errors for instance) are called LINEAR RESONANCES. The NONLINEAR RESONANCES are those of 3^rd order and above

(25)

General RESONANCE CONDITIONS

M Q + N Q = P

General RESONANCE CONDITIONS

M Q

_x

+ N Q

_y

P

where M, N and P are integers, P being non-negative, |M| + |N| is the order of the resonance and P is the order of the perturbation order of the resonance and P is the order of the perturbation harmonic

Plotting the resonance lines for different values of M N and P in the

Plotting the resonance lines for different values of M, N, and P in the (Q_x, Q_y) plane yields the so-called RESONANCE or TUNE DIAGRAM

6.5 Qy_y

Q

6.4 6.5

6 2

This dot in the tune 6.3

diagram is called the WORKING POINT

6.1

WORKING POINT 6.2

(case of the PS, here)

Elias Métral, Summer Student Lectures, CERN, 5-6-9-10-11/07/07 6.1 6.2 6.3 6.4 6.5QQx_x 25/92

(26)

RESONANCE WIDTH = Band with some thickness around every

RESONANCE WIDTH Band with some thickness around every resonance line in the resonance diagram, in which the motion may be unstable, depending on the oscillation amplitude

STOPBAND = Resonance width when the resonance is linear (i.e.

below the 3^rd order), because the entire beam becomes unstable if), the operating point (Q_x, Q_y) reaches this region of tune values

DYNAMIC APERTURE L t ill ti lit d hi h i

DYNAMIC APERTURE = Largest oscillation amplitude which is stable in the presence of nonlinearities

TRACKING: In general, the equations of motion in the presence of nonlinear fields are untractable for any but the simplest situations.

Tracking consists to simulate (user computer programs such as Tracking consists to simulate (user computer programs such as MAD) particle motion in circular accelerators in the presence of nonlinear fields

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KICK MODEL: Any nonlinear magnet is treated in the “point-like”

KICK MODEL: Any nonlinear magnet is treated in the point like approximation (i.e. the particle position is assumed not to vary as the particle traverses the field), the motion in all other elements of the lattice is assumed to be linear Thus at each turn the local the lattice is assumed to be linear. Thus, at each turn the local magnetic field gives a “kick” to the particle, deflecting it from its unperturbed trajectory

Close to 1/3

HENON MAPPING ^Q^x ⁼ ⁰^.³²⁴ ^Q^x ⁼ ⁰^.³²⁰ Close to 1/3

xnorm′

= Stroboscopic repre- sentation of phase-space

trajectories (normalised x^norm

j (

⇒ circles instead of ellipses for linear

motion) on every ^Q^x ⁼ ⁰^.²⁵²

211 .

= 0 Qx

motion) on every machine turn at the fixed azimuthal position of the perturbation

perturbation

Close to 1/4 Close to 1/5

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SEPARATRICES define boundaries between stable motion (bounded oscillations) and unstable motion (expanding oscillations)) ( g )

The 3^rd order resonance is a drastic (unstable) one because the particles which go onto this resonance are lost

particles which go onto this resonance are lost

(STABLE) ISLANDS: For the higher order resonances (e.g. 4^th and

(STABLE) ISLANDS: For the higher order resonances (e.g. 4 and 5^th) stable motions are also possible in (stable) islands. There are 4 stable islands when the tune is closed to a 4^th order resonance and 5 when it is closed to a 5^th order resonance

when it is closed to a 5 order resonance

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(fast) EXTRACTION( ) from a ringg

KICKER

SEPTUM KICKER

(fast dipole < 1 turn) SEPTUM

(field π 0 between the 2 blades)

(fast) INJECTION into a ring î Reverse process

(30)

Many other injection & ejection schemes î Example of another extractiony j j p

Tune variation Tune variation

′

Simulation

Simulation forfor thethe NEW

NEW PSPS MULTIMULTI-- xnorm

NEW

NEW PSPS MULTIMULTI TURN

TURN EXTRACTIONEXTRACTION (Under

(Under study)study) Phase space

Phase space portrait

portrait portrait portrait

Courtesy

(31)

Final stage after 20000 turns (about 42 ms for the PS) Final stage after 20000 turns (about 42 ms for the PS)

The particles here are extracted

Sl

Sl ((ff thth dd At the extraction septum location

At the extraction septum location

The particles here are extracted through the TT2 transfer line

Slow

Slow ((fewfew thousandthousand turns

turns)) bumpbump firstfirst (closed(closed distortion

distortion ofof thethe periodicperiodic B

B_field_field ≠ 0≠ 0 B

B_field_field == 00 xnorm′

orbit) orbit) F t

F t ((ll thth tt )) xnorm

Fast

Fast ((lessless thanthan oneone turnturn)) bump

bump afterwardsafterwards (closed(closed distortion

distortion ofof thethe periodicperiodic orbit)

orbit) Courtesy About 6 cm in physical space

About 6 cm in physical space

M. Giovannozzi

(32)

BETATRON MATCHING = The phase space ellipses at the injection

BETATRON MATCHING The phase space ellipses at the injection (ejection) point of the circular machine, and the exit (entrance) of the beam transport line, should be homothetic. To do this, the Twiss parameters are modified using quadrupoles If the ellipses are not parameters are modified using quadrupoles. If the ellipses are not homothetic, there will be a dilution (i.e. a BLOW-UP) of the emittance

x ′ x ′ x ′

Courtesy D. Möhl

x ′

x x

x

x x

Form of Unmatched beam Filamenting beam Fully filamented

DISPERSION MATCHING = D_x and D’_x should be the same at the

matched ellipse Unmatched beam Filamenting beam

beam

injection (ejection) point of the circular machine, and the exit (entrance) of the beam transport line. If there are different, there will be also a BLOW-UP, but due to a missteering (because the beam is, g (

(33)

LONGITUDINAL BEAM DYNAMICS (1/12)

The electric field is used to accelerate or decelerate the particles

The electric field is used to accelerate or decelerate the particles, and is produced by one or more RF (Radio-Frequency) CAVITIES

] m [

r [ m ] r

2 /

g

∫ ( )

−

= Δ

2 /

,

g

s

s t ds

E e

e^-

E

] m [

s

^g ^/ ²

[eV] [V/m]

Courtesy

L. Rinolfi

(34)

2 ferrite loaded cylinders that permit the cavity to be tuned

10 MHz RF cavity in Straight

Section (SS) 11 of the PS Final power amplifier Accelerating gap

Six 200 MHz in SS6 RF gymnastics

Section (SS) 11 of the PS

⇒ For the acceleration

(35)

TRANSITION ENERGY: The increase of energy has 2 contradictory

TRANSITION ENERGY: The increase of energy has 2 contradictory effects

^An increase of the particle’s velocity

^An increase of the length of the particle’s trajectory

According to the variations of these 2 parameters, the revolution frequency evolves differentlyq y y

Below transition energy: The velocity increases faster than the

l th Th l ti f i

length ⇒ The revolution frequency increases

Above transition energy: It is the opposite case ⇒ The revolution frequency decreasesq y

At transition energy: The variation of the velocity is compensated by the variation of the trajectory ⇒ A variation of energy does not modify the frequency

modify the frequency

(36)

( )

V

V ˆ i φ ^ω ^h ^ω

Sinusoidal voltage applied

^V

^RF

⁼ ^V

^RF

^sin ^φ

^RF

( ) ^t ^ω

^RF

⁼ ^h ^ω

^rev

1 RF

1

^e ^V ˆ sin φ

E =

⇒ Δ ^Δ ^E

₁

^e ^V

_RF

sin φ

₁ Harmonic number ^Δ^E

⇒

Harmonic number

SYNCHROTRON OSCILLATION (here, below transition)

BUNCHED beam in a stationary BUCKET

V

RF

φRF

ˆ

RF

V

⇒ Harmonic oscillator

TUNE Q_z (<< 1)

RF

t ω φ

RF

=

φ

RF

φ

_RF

= ω

_RF

t

φ

1

for the small amplitudes

φ

1RF

φ

Synchronous Courtesy

(37)

≠ 0 φ φ

Synchronous phase

Synchrotron oscillation during acceleration (below transition)

s

0

1

= φ ≠

φ

Above transition, the stable phase is

π − φ

_s

V

RF

V ˆ

_RF

V

φ

RF

Courtesy L Ri lfi

L. Rinolfi

(38)

V

RF_RF ^Courtesy

L. Rinolfi

φ

RF

SEPARATRIX

= Limit between the stable and unstable regions ⇒ Determines

Accelerating bucket

φ

regions ⇒ Determines the RF BUCKET

φ

RF

The number

of buckets is given by hg y

(39)

( ) ^t ⁺ ^ω

^s²

^τ ( ) ^t ⁼ ⁰

τ &&

⇒ : Equation of a harmonic oscillator

t = Time interval between the passage of the synchronous particle and the

ˆ

of the synchronous particle and the

particle under consideration = Momentum compaction factor a_p

( )

^rev

2

RF

/ 2

cos ˆ β ω π

φ ω η

e E

h V

r s

s

= η ⁼ γ

^tr⁻²

⁻ γ

^r⁻²

⁼ ⁽ ^Δ ^T ^/ ^T

⁰

⁾ ^/ ⁽ ^Δ ^p ^/ ^p

⁰

⁾

ω

s

Q

S h

Slip factor (sometimes defined with a negative sign…)

ω

rev s

Q

z

=

⇒ : Synchrotron tune

Number of synchrotron

oscillations per machine revolution

(40)

LONGITUDINAL BEAM DYNAMICS (8/12) TOMOSCOPEO OSCO (developed(de e oped

by S. Hancock, CERN/AB/

RF)

Th i f TOMOGRAPHY

Longitudinal The aim of TOMOGRAPHY

is to estimate an unknown distribution (here the 2D

bunch profile

longitudinal distribution) using only the information in the bunch profiles (see

Projection

P Loene

p (

Beam control)

[MeV]

Surface =

L it di l EMITTANCE

Projectio ongitudinergy prof

Longitudinal EMITTANCE of the bunch

= ε_L[eV.s]

on nal file

Surface = Longitudinal [ns]

ACCEPTANCE of the

(41)

LONGITUDINAL BEAM DYNAMICS (9/12) Examples of RF gymnastics

[MeV] Courtesy S. Hancock

p gy

Longitudinal BUNCH SPLITTING

[ns]

(42)

Triple bunch Triple bunch

splitting _Courtesy R. Garoby

00 μs)tions (~ 8056 revolut 35

(43)

% 8 Δp 3

Parabolic distribution

% 8 . 3

0

p = p

ns

=50

τ

b

BUNCH ROTATION

ns 2 . 9 deg

1 =

ROTATION (with ESME)

(44)

EXTRACTION AND LONGITUDINAL MATCHING

î The RF buckets (expressed in energy vs time) of the 2 rings î The RF buckets (expressed in energy vs. time) of the 2 rings should be homothetic (DE / Dt conserved), otherwise longitudinal BLOW-UP

(45)

FIGURE OF MERIT FOR A SYNCHROTRON / COLLIDER:

BRIGHTNESS / LUMINOSITY (1/6)

(2D) BEAM BRIGHTNESS

y x

B I

ε ε π

²

= Beam current

Transverse emittances

MACHINE LUMINOSITY

ond events

L N

σ

sec

= / Number of events per second

generated in the collisions LUMINOSITY

σ

_event

Cross-section for the event under study

[cm^-2 s^-1] under study

The Luminosity depends only on the beam parameters [cm^-2 s^-1]

The Luminosity depends only on the beam parameters

⇒ It is independent of the physical reaction

Reliable procedures to compute and measure

(46)

⇒ For a Gaussian (round) beam distribution

Number of Revolution

f M N²

Number of particles per bunch

Number of bunches per beam

Revolution frequency

Relativistic velocity factor

f F M L N

n

r rev b

* 2

4

π ε β

=

γ

Normalized transverse beam

Geometric reduction

factor due to the crossing angle at the IP

emittance

β-function at the collision pointp

PEAK LUMINOSITY for ATLAS&CMS in the LHC =

L = 10

³⁴

cm

^-²

s

^-¹

(47)

Number of particles per bunch N 1 15 10¹¹

Number of particles per bunch N_b 1.15 â 10¹¹

Number of bunches per beam M 2808

Revolution frequency f 11245 Hz

Revolution frequency f_rev 11245 Hz

Relativistic velocity factor g_r 7461 (î E = 7 TeV)

b-function at the collision point b* 55 cm

b p b 55 cm

Normalised rms transverse beam emittance e_n 3.75 â 10^-4 cm

Geometric reduction factor F 0.84

F ll i l t th IP 285 d

Full crossing angle at the IP q_c 285 mrad

Rms bunch length s_z 7.55 cm

Transverse rms beam size at the IP * 16 7

Transverse rms beam size at the IP s* 16.7 mm

(48)

INTEGRATED LUMINOSITY

^L ⁼

^T

∫ ^L ( ) ^t ^dt

0 int

0

⇒ The real figure of merit =

L

_{i t}

σ

t

= number of events

⇒ The real figure of merit

L

_int

σ

_event

number of events

LHC integrated Luminosity expected per year: [80-120] fb^-1

Reminder: 1 barn = 10^-24 cm² and femto = 10^-15

and femto = 10

(49)

The total proton-proton cross section at 7 TeV is ~ 110 mbarns:

I l ti 60 b

Inelastic î s_in = 60 mbarns

Single diffractive î s_sd = 12 mbarns

^ElasticElastic îî ss_el_l = 40 mbarns40 mbarns

The cross section from elastic scattering of the protons and

diff ti t ill t b b th d t t it i l th

diffractive events will not be seen by the detectors as it is only the inelastic scatterings that give rise to particles at sufficient high angles with respect to the beam axis

Inelastic event rate at nominal luminosity = 10³⁴ â 60 â 10^-3 â 10^-24 = 600 millions / second per high-luminosity experiment

600 millions / second per high luminosity experiment

(50)

The bunch spacing in the LHC is 25 ns î Crossing rate of 40 MHz

However there are bigger gaps (for the kickers) î Average

However, there are bigger gaps (for the kickers) î Average crossing rate = number of bunches â revolution frequency = 2808 â 11245 = 31.6 MHz

(600 millions inelastic events / second) / (31.6 â 10⁶) = 19 inelastic events per crossing

events per crossing

Total inelastic events per year (~10⁷ s) = 600 millions â 10⁷= 6 â 10¹⁵~ 10¹⁶

10¹⁶

The LHC experimental challenge is to find rare events at levels of 1

The LHC experimental challenge is to find rare events at levels of 1 in 10¹³ or more î ~ 1000 Higgs events in each of the ATLAS and CMS experiments expected per year

(51)

BEAM CONTROL (1/8)

New CERN Control Centre (CCC) at Prevessin since March 2006

Island for the PS complex

Island for the Technical Infrastructure + LHC Island for the PS complex cryogenics

Island for the LHC

ENTRANCE

Elias Métral, Summer Student Lectures, CERN, 5-6-9-10-11/07/07 Island for the SPS 51/92

(52)

WALL CURRENT MONITOR = Device used to measure the instantaneous value of the beam current

instantaneous value of the beam current Load

For the vacuum + EM shielding

Ferrite rings

Output

Beam Induced or

wall current

⇒ High- frequency signals do not Longitudinal Gap

bunch profiles for

a LHC-type beam Courtesy

signals do not see the short

circuit

(53)

WALL CURRENT MONITOR IN SS3 OF THE PS

IN SS3 OF THE PS

(54)

(Transverse) beam POSITION PICK-UP MONITOR( )

Beam

s

_Beam

L R

U U

x w

+

= − 2

U

L

U

_R

L

R

U

x U

0

w

Courtesy H. Koziol

⇒ Horizontal beam orbit measurement in the PS

4 6 8

4 -2 0 2

[mm]

6 spikes

-8 -6 -4

3 13 23 33 43 53 63 73 83 93

6 spikes observed as

25 .

≈ 6

Q

x

(55)

BEAM LOSS MONITOR

Correction QUADRUPOLE

Correction DIPOLE POSITION PICK-UP IN SS5 OF THE PS

QUADRUPOLE DIPOLE

(56)

FAST WIRE SCANNER

⇒ Measures the transverse beam profiles by detecting the particles scattered from a thin wire swept

HORIZONTAL PROFILE

p rapidly through the beam

800 1000

Courtesy

600 800

Gaussian fit

y S. Gilardoni

400

fit

( ) x

x ^x

β ε

^σ

≡ σ

²

@ D

200

β

x

@

(57)

SCINTILLATORS

⇒ Producing lightg g

Light conductors

PHOTOMULTIPLIER

⇒ Converting light into and electrical signal Motor of the

wire scanner

FAST WIRE SCANNER IN SS54 OF THE PS

SUMMER STUDENT LECTURES ON SUMMER STUDENT LECTURES ON