Lecture - Day 6 - USPASuspas.fnal.gov/materials/12MSU/Lecture Day 6.pdf · Lecture - Day 6 William A. Barletta ... the beam size scales as! " x (s)= # x $ x (s) ... Incoherent tune

US Particle Accelerator School

Fundamentals of Accelerators - 2012

Lecture - Day 6

William A. BarlettaDirector, United States Particle Accelerator School

Dept. of Physics, MITFaculty of Economics, U. Ljubljana


What sets the discovery potential of colliders?

1. Energy➙ determines the scale of phenomena to be studied

2. Luminosity (collision rate)➙ determines the production rate of “interesting” events

➙ Scale L as E2 to maximize discovery potential at a given energy➙ Factor of 2 in energy worth factor of 10 in luminosity

✺ Critical limiting technologies:➙ Energy - Dipole fields, accelerating gradient, machine size➙ Current - Synchrotron radiation, wake fields➙ Focal depth - IR quadrupole gradient➙ Beam quality - Beam source, machine impedance, feedback

Luminosity =Energy ! Current

Focal depth ! Beam quality


Beams have internal (self-forces)

✺ Space charge forces➙ Like charges repel➙ Like currents attract

✺ For a long thin beam

!

Esp (V /cm) =60 Ibeam (A)

Rbeam (cm)

!

B" (gauss) = Ibeam (A)

5 Rbeam (cm)


Net force due to transverse self-fields

In vacuum:Beam’s transverse self-force scale as 1/γ2

➙ Space charge repulsion: Esp,⊥ ~ Nbeam

➙ Pinch field: Bθ ~ Ibeam ~ vz Nbeam ~ vz Esp

∴Fsp ,⊥ = q (Esp,⊥ + vz x Bθ) ~ (1-v2) Nbeam ~ Nbeam/γ2

Beams in collision are not in vacuum (beam-beam effects)


We see that ε characterizes the beam whileβ(s) characterizes the machine optics

✺ β(s) sets the physical aperture of the accelerator becausethe beam size scales as

!

"x(s) = #

x$x(s)


Example: Megagauss fieldsin linear collider

electrons positrons

At Interaction Point space charge cancels; currents add==> strong beam-beam focus

--> Luminosity enhancement--> Very strong synchrotron radiation

Consider 250 GeV beams with 1 kA focused to 100 nm

Bpeak ~ 40 Mgauss

==> Large ΔE/E


Types of tune shifts: Incoherent motion

• Center of mass does not move

• Beam environment does not “see” any motion

• Each particle is characterized by an individual amplitude &phase

From: E. Wilson Adams lectures


Incoherent collective effects

✺ Beam-gas scattering➙ Elastic scattering on nuclei => leave physical aperture➙ Bremsstrahlung➙ Elastic scattering on electrons leave rf-aperture➙ Inelastic scattering on electrons

=======> reduce beam lifetime✺ Ion trapping (also electron cloud) - scenario

➙ Beam losses + synchrotron radiation => gas in vacuum chamber➙ Beam ionizes gas➙ Beam fields trap ions➙ Pressure increases linearly with time➙ Beam -gas scattering increases

✺ Intra-beam scattering


Intensity dependent effects

✺ Types of effects➙ Space charge forces in individual beams➙ Wakefield effects➙ Beam-beam effects

✺ General approach: solve

✺ For example, a Gaussian beam has

✺ For r < σ!

" " x + K(s)x =1

#m$ 2c 2F

non% linear

!

FSC =e

2N

2"#o$2r

1% e%r2 / 2& 2

( ) where N = charge/unit length

!

FSC"

e2N

4#$o% 2r


Beam-beam tune shift

✺ For γ >> 1, Felec ≈ Fmag

✺ Therefore the tune shift is

✺ For a Gaussian beam

y w

h

l

!

"py (y) ~ Felec + Fmag( )l

2c (once the particle passes l/2 the other bunch has passed)

!

"#py (y) $ 2eE

2c =

e2NB

%ocwhy

!

" #py

po

= # $ y ~ y similar to gradient error ky#s with ky#s =# $ y

y

!

"Q = #$*

4%ky"s &

re$*N

'wh where re =

e2

4%(omc2

!

E =Nbeame

lwhy

!

"Q #re

2

$*N

%Aint


Effect of tune shift on luminosity

✺ The luminosity is

✺ Write the area in terms of emittance & β at the IR

✺ For simplicity assume that

✺ In that case

✺ And

!

L =fcollN1N2

4Aint

!

Aint

=" x" y = #x

*$x o #y

*$y

!

"x*

"y*

=#x#y$"x

*=#x#y"y* $"x

*#x =#x2

#y"y*

!

Aint

= "x#y

*

!

L =fcollN1N2

4"x#y

*~Ibeam2

"x#y

*


Increase N to the tune shift limit

✺ We saw that

or

Therefore the tune shift limited luminosity is

!

"Qy #re

2

$*N

%Aint

!

N = "Qy

2#Aint

re$*

= "Qy

2#%x$*

re$*

=2

re#%x"Qy

!

L =2

re"Qy

fcollN1#$x4$x%y

*~ "Qy

IE

%y*

&

' ( (

)

* + +


Incoherent tune shift for in a synchrotron

Assume: 1) an unbunched beam (no acceleration), & 2) uniformdensity in a circular x-y cross section (not very realistic)

!

" " x + K(s) + KSC

(s)( )x = 0 Qx0 (external) + ΔQx (space charge)

For small “gradient errors” kx

!

"Qx =1

4#kx

0

2R#

$ (s)%x (s)ds =1

4#KSC

0

2R#

$ (s)%x (s)ds

!

"Qx = #1

4$

2r0I

e% 3& 3c0

2$R

'%x (s)

a2ds = #

r0RI

e% 3& 3c

%x (s)

a2(s)

= #r0RI

e% 3& 3c(x


where

!

KSC

= "2r0I

ea2# 3$ 3c


Incoherent tune shift limits currentat injection

using I = (Νeβc)/(2πR) withN…number of particles in ringεx,y….emittance containing 100% of particles

!

"Qx,y = #r0

2$% 2& 3N

'x,y

“Direct” space charge, unbunched beam in a synchrotron Vanishes for γ » 1 Important for low-energy hadron machinesIndependent of machine size 2πR for a given N Overcome by higher energy injection ==> cost



Injection chain for a 200 TeV Collider


Beam lifetime

Based on F. Sannibale USPAS Lecture


Finite aperture of accelerator==> loss of beam particles

✺ Many processes can excite particles on orbits larger thanthe nominal.➙If new orbit displacement exceed the aperture, the

particle is lost

✺ The limiting aperture in accelerators can be either physicalor dynamic.➙Vacuum chamber defines the physical aperture➙Momentum acceptance defines the dynamical aperture

From: Sannibale USPAS Lecture


Important processes in particle loss

✺ Gas scattering, scattering with the other particles in thebeam, quantum lifetime, tune resonances, &collisions

✺ Radiation damping plays a major role for electron/positronrings➙For ions, lifetime is usually much longer

• Perturbations progressively build-up & generate losses

✺ Most applications require storing the beam as long aspossible

==> limiting the effects of the residual gas scattering==> ultra high vacuum technology



What do we mean by lifetime?

✺ Number of particles lost at time t is proportional to thenumber of particles present in the beam at time t

✺ Define the lifetime τ = 1/α; then

✺ Lifetime is the time to reduce the number of beam particlesto 1/e of the initial value

✺ Calculate the lifetime due to the individual effects (gas,Touschek, …)

!

dN = "# N t( )dt with # $ constant

!teNN"

=0

!

1

"total

=1

"1

+1

"2

+1

"3

+ ....



Is the lifetime really constant?

✺ In typical electron storage rings, lifetime depends on beamcurrent

✺ Example: the Touschek effect losses depend on current.➙ When the stored current decreases, the losses due to Touschek

decrease ==> lifetime increases

✺ Example: Synchrotron radiation radiated by the beamdesorbs gas molecules trapped in the vacuum chamber➙ The higher the stored current, the higher the synchrotron radiation

intensity and the higher the desorption from the wall.➙ Pressure in the vacuum chamber increases with current==> increased scattering between the beam and the residual gas==> reduction of the beam lifetime



Examples of beam lifetime measurements

ALSALS

DADAΦΦNENE

Electrons Positrons


Beam loss by scattering

✺ Elastic (Coulomb scattering) fromresidual background gas➙ Scattered beam particle undergoes

transverse (betatron) oscillations.➙ If the oscillation amplitude exceeds

ring acceptance the particle is lost

✺ Inelastic scattering causesparticles to lose energy➙ Bremsstrahlung or atomic excitation➙ If energy loss exceeds the

momentum acceptance the particle islost

Incident positiveparticles

nucleus

photon

Rutherford cross section

Bremsstrahlung cross section


Elastic scattering loss process

✺ Loss rate is

!

dN

dt Gas

= "#beam particlesNmolecules$*R

!

"beam particles =N

AbeamTrev=

N

Abeam

# c

Lring

!

Nmolecules = nAbeamLring

!

" *R =

d"Rutherford

d#d#

Lost$ = d%

0

2&

$d"Rutherford

d#' MAX

&

$ sin' d'

!

d"R

d#=

1

4$%0( )2

ZbeamZgase2

2& c p

'

( )

*

+ ,

2

1

sin4 - 2( )

[MKS]


Gas scattering lifetime

✺ Integrating yields

✺ For M-atomic molecules of gas

✺ For a ring with acceptance εA & for small θ

==>

( ) ( )

][

2tan

1

42

22

2

0

MKSscatteringelasticgasforrateLoss

pc

ZeZcNn

dt

dN

MAX

Inc

Gas !"#$

"$%%&

'(()

*+=

760

][

0

TorrP

nMn =

!

"MAX

=#A

$n

][4760

2

2

0

2

0

][

MKSZeZ

pc

nMcP T

A

IncTorr

Gas!

"!

!

#"$ %%

&

'(()

*+


Inelastic scattering lifetimes

✺ Beam-gas bremsstrahlung: if EA is the energy acceptance

✺ Inelastic excitation: For an average ßn

( ) ][0

][

1

ln

14.153

nTorrA

hoursBrem

PEE!"#$

!

"Gas hours[ ] #10.25

E0

2

[GeV ]

P[nTorr ]

$A [µm ]

%n [m ]


ADA - The first storage ring collider (e+e-)by B. Touschek at Frascati (1960)

The storage ring collider idea was invented by R.Wiederoe in 1943 – Collaboration with B. Touschek – Patent disclosure 1949

Completed in less than one year


Touschek effect:Intra-beam Coulomb scattering

✺ Coulomb scattering between beam particles can transfertransverse momentum to the longitudinal plane➙ If the p+Δp of the scattered particles is outside the momentum

acceptance, the particles are lost➙ First observation by Bruno Touschek at ADA e+e- ring

✺ Computation is best done in the beam frame where therelative motion of the particles is non-relativistic➙ Then boost the result to the lab frame

!

1

"Tousch.

# 1

$ 3

Nbeam

% x% y% S

1

&pA p0( )

2 #

1

$ 3

Nbeam

Abeam% S

1

ˆ V RF


Transverse quantum lifetime

✺ At a fixed s, transverse particle motion is purely sinusoidal

✺ Tunes are chosen in order to avoid resonances.➙ At a fixed azimuthal position, a particle turn after turn sweeps all

possible positions between the envelope

✺ Photon emission randomly changes the “invariant” a &consequently changes the trajectory envelope as well.

✺ Cumulative photon emission can bring the particleenvelope beyond acceptance in some azimuthal point➙ The particle is lost

!

xT = a "n sin #" nt +$( ) T = x,y


Quantum lifetime was first estimated byBruck & Sands

✺ Quantum lifetime varies very strongly with the ratiobetween acceptance & rms size.

Values for this ratio ≥6 are usually required!

"QL# "DL

exp $EA

22%E

2( )Longitudinal quantum lifetime!

where "T

2= #T$T + DT

"E

E0

%

& '

(

) *

2

T = x,y

+DT, transverse damping time

( )

lifetimequantumTransverse

yxTAA

TT

T

TDQ TT

,2exp22

2

2

=! ""

##


Lifetime summary


In colliders the beam-beam collisions also depletethe beams

This gives the luminosity lifetime


LEP-3Life gets hard very fast

William A. BarlettaDirector, US Particle Accelerator School

Dept. of Physics, MITEconomics Faculty, University of Ljubljana


Physics of a Higgs Factory

✺ Dominant decay reaction is e+ + e- => H =>W + Z

✺ MW+MZ = 125 + 91.2 GeV/c2

==> set our CM energy a little higher: ~240 GeV

✺ Higgs production cross section ~ 220 fb (2.2 x 10-37 cm2)

✺ Peak L = 1034 cm-1 s-1 = < L> ~ 1033 cm-1 s-1

✺ ~30 fb-1 / year ==> 6600 Higgs / year

✺ Total cross-section at ~ 100 pb•(100GeV/E)2

We don’t have any choice about these numbers


Tune shift limited luminosity of the collider

!

L =N

2c"

4#$n%*SB

=1

erimic2

Nri

4#$n

EI

%*

&

' (

)

* + =

1

erimic2

Nri

4#$n

Pbeam

%*

&

' (

)

* + i = e, p

Linear or Circular

Tune shift

!

L = 2.17 o1034

1+" x

" y

#

$ % %

&

' ( ( Qy

1 cm

)*

#

$ %

&

' (

E

1 GeV

#

$ %

&

' (

I

1 A

#

$ %

&

' (

Or in practical units for electrons

!

At the tune shift limit 1+" x

" y

#

$ % %

&

' ( ( Qy ) 0.1

!

L = 2.17 o1033 1 cm

"*

#

$ %

&

' (

E

1 GeV

#

$ %

&

' (

I

1 A

#

$ %

&

' (


We can only choose I(A) and ß*(cm)

✺ For the LHC tunnel with fdipole = 2/3, ρ ~ 3000 m

✺ Remember that

✺ Therefore, Bmax = 0.134 T

✺ Each beam particle will lose to synchrotron radiation

or 6.2 GeV per turn

!

" m( ) = 3.34 p

1 GeV/c

#

$ %

&

' (

1

q

#

$ % &

' (

1 T

B

#

$ %

&

' (

!

Uo(keV ) = 88.46

E4(GeV )

"(m)

Ibeam << 0.1 A


Ibeam = 7.5 mA ==> ~100 MW of radiation

✺ Then

✺ Therefore to meet the luminosity goal<ß*

xß*y>1/2

~ 0.1 cm

✺ Is this possible? Recall that is the depth of focus at the IP

!

L " 2.6 o1033 1 cm

#*

$

% &

'

( )

The “hourglass effect” lowers L==> ß* ~ σzß*


Bunch length is determined by Vrf

✺ The analysis of longitudinal dynamics gives

where αc = (ΔL/L) / (Δp/p) must be ~ 10-5 for electrons toremain in the beam pipe

✺ To know bunch length we need to know Δp/p ~ ΔE/E

✺ For electrons to a good approximation

and

!

" S =c#C

$sync

" p

p0

=c

3

2%q

p0&

0'C

h f0

2 ˆ V cos (S( )

" p

p0

!

"E # Ebeam < Ephoton >

][][665.0][

][218.2][

23

TBGeVEm

GeVEkeV

c!!==

"#


For our Higgs factory εcrit = 1.27 MeV

✺ Therefore

✺ The rf-bucket must contain this energy spread in the beam

✺ As Uo ~ 6.2 GeV,Vrf,max > 6.2 GeV + “safety margin” to contain ΔE/E

✺ Some addition analysis

✺ The greater the over-voltage, the shorter the bunch

!

"E

E

#

$ %

&

' ( max

=q

) V max

)h* c Esync

2cos+s + (2+s ,)( )sin+s

!

"S =c#C

$

"p

p0

=c

3

2%q

p0&

0#C

h f0

2 ˆ V max

cos 'S( )

"p

p0

!

"E

E#$ p

p# 0.0033


For the Higgs factory…

✺ The maximum accelerating voltage must exceed 9 GeV➙ Also yields σz = 3 mm which is okay for ß* = 1 mm

✺ A more comfortable choice is 11 GeV (it’s only money)➙ ==> CW superconducting linac for LEP 3➙ This sets the synchronous phase

✺ For the next step we need to know the beam size

✺ Therefore, we must estimate the natural emittance which isdetermined by the synchrotron radiation ΔE/E

!

" i

*= #i

*$i for i = x,y


The minimum horizontal emittancefor an achromatic transport

!

"x,min

= 3.84 #10$13 % 2

Jx

&

' (

)

* + F

min meters

, 3.84 #10$13% 2

-achromat

3

4 15

&

' (

)

* + meters

εy ~ 0.01 εx


Because αc is so small,we cannot achieve the minimum emittance

✺ For estimation purposes we will choose 20 εmin as themean of the x & y emittances

✺ For the LHC tunnel a maximum practical ipole length is15 m➙ A triple bend achromat ~ 80 meters long ==> θ = 2.67x10-2

<ε> ~ 7.6 nm-rad ==> σtransverse = 2.8 µm

How many particles are in the bunch?Or how many bunches are in the ring?


We already assumed thatthe luminosity is at the tune-shift limit

✺ We have

✺ Or

✺ So, Ne ~ 8 x 1011 per bunch

✺ Ibeam = 7.5 mA ==> there are only 5 bunches in the ring

!

L =N

2c"

4#$n%*SB

=1

erimic2

Nri

4#$n

EI

%*

&

' (

)

* + =

1

erimic2

Nri

4#$n

Pbeam

%*

&

' (

)

* + i = e, p

Linear or Circular

Tune shift

!

Q =Nr

e

4"#$ % N =

4"#$

re

Q


Space charge fields at the collision point

electrons positrons

At Interaction Point space charge cancels; currents add==> strong beam-beam focus

--> Luminosity enhancement--> Very strong synchrotron radiation

This is important in linear colliders

What about the beams in LEP-3?


At the collision point…

Ipeak = Ne /4 σz ==> Ipeak = 100 kA

✺ Therefore, at the beam edge (2σ)

B = I(A)/5r(cm) = 36 MG !

✺ When the beams collide they will emit synchrotronradiation (beamstrahlung)

✺ For LEP-3 Ecrit = 35 GeV ! (There are quantum corrections)

][][665.0][

][218.2][

23

TBGeVEm

GeVEkeV

c!!==

"#

The rf-bucket cannot contain such a big ΔE/EBeamstrahlung limits beam lifetime & energy resolution of events


There are other problems

✺ Remember the Compton scattering of photons up shifts theenergy by 4 γ2

✺ Where are the photons?➙ The beam tube is filled with thermal photons (25 meV)

✺ In LEP-3 these photons can be up-shifted as much as 2.4 GeV➙ 2% of beam energy cannot be contained➙ We need to put in the Compton cross-section and photon density to find

out how rapidly beam is lost

E=γmc2 λout

λin

Documents

Lecture - Day 6 - USPASuspas.fnal.gov/materials/12MSU/Lecture Day 6.pdf · Lecture - Day 6 William A. Barletta ... the beam size scales as! " x (s)= # x $ x (s) ... Incoherent tune