Introduction to Accelerator Physics Beam Dynamics for „Summer Students“ Bernhard Holzer, CERN-LHC

Home , Dipole magnet, Quadrupole magnet, Strong focusing

IP5The Ideal World I.) Magnetic Fields and Particle Trajectories

IP8

IP2 IP1

* Luminosity Run of a typical storage ring:

LHC Storage Ring: Protons accelerated and stored for 12 hours distance of particles travelling at about v ≈ c L = 1010-1011 km ... several times Sun - Pluto and back intensity (1011)

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2 -

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 guide the particles on a well defined orbit („design orbit“)  focus the particles to keep each single particle trajectory within the vacuum chamber of the storage ring, i.e. close to the design orbit. 1.) Introduction and Basic Ideas

„ ... in the end and after all it should be a kind of circular machine“  need transverse deflecting force

Lorentz force F q*() E v B

8 m typical velocity in high energy machines: vc3 *10 s

Example:

m Vs B 1 T F q 3 10 8 1 2 s m technical limit for el. field: MV F q 300 MV m E 1 m equivalent el. field ... E old greek dictum of wisdom: if you are clever, you use magnetic fields in an accelerator wherever it is possible.

The ideal circular orbit ρ θ ● s circular coordinate system condition for circular orbit:

Lorentz force F L e v B p m v 2 B centrifugal force F 0 centr e

B ρ = "beam rigidity" m v 2 0 e v B 2.) The Magnetic Guide Field

Dipole Magnets: n I define the ideal orbit B 0 homogeneous field created h by two flat pole shoes

Normalise magnetic field to momentum: convenient units:

p 1 e B Vs GeV B B T p e p m 2 c

Example LHC: 8.3 Vs 8 m 1 2 8.3 s * 3 *10 e m s B 8.3 T 9 2 7000 *10 9 eV 7000 *10 m c GeV p 7000 1 8.3 c 0.333 1 7000 m The Magnetic Guide Field ρ α

field map of a storage ring dipole magnet

2.53 km 2πρ = 17.6 km B 1 ... 8 T ≈ 66%

rule of thumb: 1 B T „normalised bending strength“ 0.3 p GeV / c 2.) Focusing Properties – Transverse Beam Optics

classical mechanics: there is a restoring force, proportional pendulum to the elongation x:

dx2 m** c x dt2

general solution: free harmonic oszillation x( t ) A * cos( t )

Storage Ring: we need a Lorentz force that rises as a function of the distance to ...... ? ...... the design orbit

F()**() x q v B x Quadrupole Magnets: required: focusing forces to keep trajectories in vicinity of the ideal orbit linear increasing Lorentz force

linear increasing magnetic field B y g x B x g y normalised quadrupole field:

g k p / e

g (T / m ) simple rule: k 0.3 p(GeV / c ) LHC main quadrupole magnet

g 25 ... 220 T / m

B what about the vertical plane: E y B x B =j 0 g ... Maxwell t x y Focusing forces and particle trajectories: normalise magnet fields to momentum (remember: B*ρ = p / q )

Dipole Magnet Quadrupole Magnet

BB1 g k : p/ q B pq/ 3.) The Equation of Motion:

B ( x ) 1 1 1 k x m x 2 n x 3 ... p / e 2! 3!

only terms linear in x, y taken into account dipole fields quadrupole fields

Separate Function Machines:

Split the magnets and optimise them according to their job:

bending, focusing etc

Example: heavy ion storage ring TSR * man sieht nur dipole und quads  linear yˆ The Equation of Motion:

● Equation for the horizontal motion: * ρ y θ ● 1 x x ( k ) 0 x 2 s

* Equation for the vertical motion:

1 y 0 no dipoles … in general … 2

k k quadrupole field changes sign x

y k y 0 4.) Solution of Trajectory Equations

Define … hor. plane: Kk1 2 x K x 0 … vert. Plane: Kk

Differential Equation of harmonic oscillator … with spring constant K

Ansatz: Hor. Focusing Quadrupole K > 0:

1 x() s x00 cos( K s ) x sin( K s ) K

x() s x00 K sin( K s ) x cos( K s )

For convenience expressed in matrix formalism:

1 x x cos K l sin K l M K M foc * foc x x s1 s 0 K sin K l cos K l s = 0 s = s1 hor. defocusing quadrupole:

x K x 0

Ansatz: Remember from school

1 coshK l sinh K l K x(s) a1 cosh( s) a 2 sinh( s) M defoc Ksinh K l cosh K l

drift space: K = 0

x(s) x 0 * s 1 l M drift 01

! with the assumptions made, the motion in the horizontal and vertical planes are independent „ ... the particle motion in x & y is uncoupled“ Transformation through a system of lattice elements

combine the single element solutions by multiplication of the matrices

focusing lens MMMMMM**** total QF D QD Bnde D *..... dipole magnet

defocusing lens x x M (s , s ) * x ' 2 1 x ' s2 s1 court. K. Wille

in each accelerator element the particle trajectory corresponds to the movement of a harmonic oscillator „

x(s)

0 typical values in a strong foc. machine: s x ≈ mm, x´ ≤ mrad 5.) Orbit & Tune:

Tune: number of oscillations per turn

64.31 59.32

Relevant for beam stability: non integer part

LHC revolution frequency: 11.3 kHz 0.31 *11 .3 3.5kHz LHC Operation: Beam Commissioning

First turn steering "by sector:" One beam at the time Beam through 1 sector (1/8 ring), correct trajectory, open collimator and move on. Question: what will happen, if the particle performs a second turn ?

... or a third one or ... 1010 turns

s II.) The Ideal World: Particle Trajectories, Beams & Bunches

Bunch in a Storage Ring

(Z X Y) Astronomer Hill: differential equation for motions with periodic focusing properties „Hill„s equation“

Example: particle motion with periodic coefficient

equation of motion: x( s ) k ( s ) x ( s ) 0

restoring force ≠ const, we expect a kind of quasi harmonic k(s) = depending on the position s oscillation: amplitude & phase will depend k(s+L) = k(s), periodic function on the position s in the ring. 6.) The Beta Function

„it is convenient to see“ ... after some beer ... general solution of Mr Hill can be written in the form:

Ansatz: ε, Φ = integration constants x() s * ()*cos(() s s ) determined by initial conditions

β(s) periodic function given by focusing properties of the lattice ↔ quadrupoles

()()s L s

ε beam emittance = woozilycity of the particle ensemble, intrinsic beam parameter, cannot be changed by the foc. properties. scientifiquely spoken: area covered in transverse x, x´ phase space … and it is constant !!! Ψ(s) = „phase advance“ of the oscillation between point „0“ and „s“ in the lattice. For one complete revolution: number of oscillations per turn „Tune“

1 ds Q y 2 (s ) 7.) Beam Emittance and Phase Space Ellipse

()*sxs22 () 2()()() sxsxs ()() sxs

x´ Liouville: in reasonable storage rings area in phase space is constant.

● A = π*ε=const ● ●

● x ● ● x(s)

ε beam emittance = woozilycity of the particle ensemble, intrinsic beam parameter, cannot be changed by the foc. properties. Scientifiquely spoken: area covered in transverse x, x´ phase space … and it is constant !!! Particle Tracking in a Storage Ring

Calculate x, x´ for each linear accelerator element according to matrix formalism plot x, x´as a function of „s“

● … and now the ellipse:

note for each turn x, x´at a given position „s1“ and plot in the phase space diagram Emittance of the Particle Ensemble:

(Z X Y) Emittance of the Particle Ensemble:

x(s) (s) cos( (s) ) xˆ(s) (s)

1 x 2 Gauß N e 2 2 x Particle Distribution: ( x ) e 2 x

particle at distance 1 σ from centre ↔ 68.3 % of all beam particles single particle trajectories, N ≈ 10 11 per bunch

LHC: 180 m 5 * 10 10 m rad

* 5 *10 10 m *180 m 0.3 mm

aperture requirements: r 0 = 12 * σ III.) The „not so ideal“ World Lattice Design in Particle Accelerators

x , y

1952: Courant, Livingston, Snyder: Theory of strong focusing in particle beams Recapitulation: ...the story with the matrices !!!

Equation of Motion: Solution of Trajectory Equations

x K x 0 Kk1 2 … hor. plane: x x M * x x Kk … vert. Plane: s1 s 0

1 l M drift 0 1

1 cos( K l ) sin( K l ) K M foc K sin( K l ) cos( K l )

1 cosh( K l ) sinh( K l ) K M defoc K sinh( K l ) cosh( K l )

M total M QF * M D * M B * M D * M QD * M D * ... 8.) Lattice Design: „… how to build a storage ring“

Geometry of the ring: B*/ p e p = momentum of the particle, ρ = curvature radius

Bρ= beam rigidity Circular Orbit: bending angle of one dipole

ds dl Bdl B

The angle run out in one revolution must be 2π, so for a full circle

Bdl 2 B

p Bdl 2 … defines the integrated dipole field around the machine. q Example LHC:

7000 GeV Proton storage ring dipole magnets N = 1232 B dl N l B 2 p / e l = 15 m q = +1 e 2 7000 10 9 eV B 8.3 Tesla m 1232 15 m 3 10 8 e s FoDo-Lattice A magnet structure consisting of focusing and defocusing quadrupole lenses in alternating order with nothing in between. (Nothing = elements that can be neglected on first sight: drift, bending magnets, RF structures ... and especially experiments...)

Starting point for the calculation: in the middle of a focusing quadrupole Phase advance per cell μ = 45°,  calculate the twiss parameters for a periodic solution 9.) Insertions

x , y

D β-Function in a Drift:

2 l l

() 0 0 * β0

At the end of a long symmetric drift space the beta function reaches its maximum value in the complete lattice. -> here we get the largest beam dimension.

-> keep l as small as possible

7 sima beam size iside a mini beta quadrupole ... unfortunately ... in general ... clearly there is another problem !!! high energy detectors that are Example:installed Luminosity in that optics drift at spaces LHC: β * = 55 cm are a little bit bigger than a few centimeters ... for smallest βmax we have to limit the overall length and keep the distance “s” as small as possible. The Mini-β Insertion:

R L * react production rate of events is determined by the cross section Σreact and a parameter L that is given by the design of the accelerator: … the luminosity

1 II* L * 12 2 * * 4ef0 bxy * 10.) Luminosity

p2-Bunch

10 11 particles p1-Bunch IP ± σ 10 11 particles

Example: Luminosity run at LHC

x , y 0.55 m f 0 11 .245 kHz 10 n 2808 x , y 5 10 rad m b 1 I p1 I p 2 x , y 17 m L * 2 4 e f 0 n b x y

I p 584 mA

L 1.0 10 34 1 cm 2 s Mini-β Insertions: Betafunctions

A mini-β insertion is always a kind of special symmetric drift space. greetings from Liouville

/ x´ the smaller the beam size the larger the bam divergence ● ● ●

● x ● ● Mini-β Insertions: some guide lines

* calculate the periodic solution in the arc * introduce the drift space needed for the insertion device (detector ...) * put a quadrupole doublet (triplet ?) as close as possible * introduce additional quadrupole lenses to match the beam parameters to the values at the beginning of the arc structure

x,, xDD x x

parameters to be optimised & matched to the periodic solution: y,, yQQ x y

8 individually powered quad magnets are needed to match the insertion ( ... at least) IV) … let´s talk about acceleration Electrostatic Machines (Tandem –) van de Graaff Accelerator

creating high voltages by mechanical transport of charges

* Terminal Potential: U ≈ 12 ...28 MV

using high pressure gas to suppress discharge ( SF6 )

Problems: * Particle energy limited by high voltage discharges * high voltage can only be applied once per particle ...... or twice ? * The „Tandem principle“: Apply the accelerating voltage twice ...... by working with negative ions (e.g. H-) and stripping the electrons in the centre of the structure

Example for such a „steam engine“: 12 MV-Tandem van de Graaff Accelerator at MPI Heidelberg 12.) Linear Accelerator 1928, Wideroe + -̶ + -̶ + -̶ + Energy Gain per „Gap“:

W q U 0 sin RF t drift tube structure at a proton linac (GSI Unilac)

500 MHz cavities in an electron storage ring

* RF Acceleration: multiple application of the same acceleration voltage; brillant idea to gain higher energies 13.) The Acceleration

Where is the acceleration? Install an RF accelerating structure in the ring:  E 

z B. Salvant N. Biancacci 14.) The Acceleration for Δp/p≠0 “Phase Focusing” below transition ideal particle  particle with Δp/p > 0  faster    particle with Δp/p < 0  slower

Focussing effect in the  longitudinal direction  keeping the particles  close together ... forming a “bunch”

h s qU 0 cos s oscillation frequency: f s f rev * ≈ some Hz 2 E s ... so sorry, here we need help from Albert:

2 E total 1 v mc 2 2 1 mc v c E 2 1 c 2 v/c

0.9

0.8

0.7

0.6

0.5

0.4

0.3 ... some when the particles 0.2 do not get faster anymore 0.1 0 .... but heavier ! 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

kinetic energy of a proton 15.) The Acceleration for Δp/p≠0 “Phase Focusing” above transition

ideal particle  particle with Δp/p > 0  heavier    particle with Δp/p < 0  lighter

  

Focussing effect in the longitudinal direction keeping the particles close together ... forming a “bunch”

... and how do we accelerate now ??? with the dipole magnets ! The RF system: IR4 S34 S45 ADT D3 D4 Q5 Q6 Q7 ACS ACS B2 420 mm 194 mm B1 ACS ACS

Bunch length (4 ) ns 1.06 Energy spread (2 ) 10-3 0.22 Synchr. rad. loss/turn keV 7

Synchr. rad. power kW 3.6 RF frequency M 400 Hz Harmonic number 35640 RF voltage/beam MV 16

4xFour-cavity cryo module 400 MHz, 16 MV/beam Energy gain/turn keV 485 Nb on Cu cavities @4.5 K (=LEP2) Synchrotron Hz 23.0 Beam pipe diam.=300mm frequency LHC Commissioning: RF

(Z X Y) a proton bunch: focused longitudinal by the RF field

RF off Bunch~ length200 ~ 1turns.5 ns  ~ 45 cm

RF on, phase optimisation

RF on, phase adjusted, beam captured Problem: panta rhei !!! (Heraklit: 540-480 v. Chr.)

(Z X Y) just a stupid (and nearly wrong) example) Bunch length of Electrons ≈ 1cm

U 0 500 MHz 60 cm t c

o sin( 90 ) 1 U 6.0 10 3 sin( 84 o ) 0.994 U

p 1.0 10 3 typical momentum spread of an electron bunch: p 17.) Dispersion and Chromaticity: Magnet Errors for Δp/p ≠ 0

Influence of external fields on the beam: prop. to magn. field & prop. zu 1/p

B dl p dipole magnet x D ( s) D ( s) p / e p

g focusing lens k p e

particle having ... to high energy to low energy ideal energy Dispersion Example: homogeneous dipole field x β Closed orbit for Δp/p > 0

xβ

. ρ p xi ()() s D s p

Matrix formalism:

p x()()() s x s D s p x C S x p D x C S x p D p s 0 0 x()()()() s C s x00 S s x D s p or expressed as 3x3 matrix

x C S D x x C S D x pp pp0 0 1 s 0

Example

x1 ... 2 m m

D( s ) 1... 2 m Amplitude of Orbit oscillation contribution due to Dispersion ≈ beam size p 3 1 10  Dispersion must vanish at the collision point p !

Calculate D, D´: ... takes a couple of sunny Sunday evenings ! V.) Are there Any Problems ???

sure there are

Some Golden Rules to Avoid Trouble I.) Golden Rule number one: do not focus the beam ! 1 (s) * s1 * cos( s1 s Q) ds s1 x co (s) Problem: Resonances 2 sin Q

Assume: Tune = integer Q 1 0

Integer tunes lead to a resonant increase Qualitatively spoken: of the closed orbit amplitude in presence of the smallest dipole field error. Tune and Resonances

m*Qx+n*Qy+l*Qs = integer

Tune diagram up to 3rd order

Qy =1.5

… and up to 7th order Qy =1.3

Homework for the operateurs: find a nice place for the tune Qy =1.0 where against all probability the beam will survive Qx =1.0 Qx =1.3 Qx =1.5 II.) Golden Rule number two: Never accelerate charged particles !

Fdef

Transport line with quadrupoles Transport line with quadrupoles and space charge

x K(s)x 0 x (K(s) K SC (s))x 0

2r I x K(s) 0 x 0 ea 2 β 3 γ 3 c

KSC Golden Rule number two: Never accelerate charged particles !

r N Tune Shift due to Space Charge Effect Q 0 x, y 2 Problem at low energies 2 x, y

v/c

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 ... at low speed the particles 0.1 repel each other 0 0 1000 2000 3000 4000 5000

Linac 2 Ekin=60 MeV Ekin of a proton Linac 4 Ekin=150 MeV III.) Golden Rule number three:

Never Collide the Beams !

the colliding bunches influence each other 25 ns  change the focusing properties of the ring !! most simple case: linear beam beam tune shift

* x * rp * N p Q x 2 p ( x y ) * x Qx

and again the resonances !!!

Courtesy W. Herr Qx IV.) Golden Rule Number four: Never use Magnets

“effective magnetic length”

lmag

B * leff Bds 0 Clearly there is another problem ...... if it were easy everybody could do it

Again: the phase space ellipse for each turn write down – at a given position „s“ in the ring – the ● single partilce amplitude x x x M turn * x x and the angle x´... and plot it. s1 s 0

A beam of 4 particles – each having a slightly different emittance: Installation of a weak ( !!! ) sextupole magnet

The good news: sextupole fields in accelerators cannot be treated analytically anymore.  no equatiuons; instead: Computer simulation „ particle tracking “

● Effect of a strong ( !!! ) Sextupole …

 Catastrophy !

●

„dynamic aperture“ Golden Rule XXL: COURAGE

and with a lot of effort from Bachelor / Master / Diploma / PhD and Summer-Students the machine is running !!!

thank‟x for your help and have a lot of fun