Fundamentals of Accelerators - 2012 Lecture - Day 6
William A. Barletta Director, United States Particle Accelerator School Dept. of Physics, MIT Faculty of Economics, U. Ljubljana
US Particle Accelerator School 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 Energy ! Current Luminosity = Focal depth ! Beam quality ➙ 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
US Particle Accelerator School Beams have internal (self-forces)
✺ Space charge forces ➙ Like charges repel ➙ Like currents attract ✺ For a long thin beam
60 Ibeam (A) E sp (V /cm) = Rbeam (cm)
I (A) B (gauss) = beam " 5 R (cm) ! beam
! US Particle Accelerator School 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 2 2 ∴Fsp ,⊥ = q (Esp,⊥ + vz x Bθ) ~ (1-v ) Nbeam ~ Nbeam/γ
Beams in collision are not in vacuum (beam-beam effects)
US Particle Accelerator School We see that ε characterizes the beam while β(s) characterizes the machine optics
✺ β(s) sets the physical aperture of the accelerator because
the beam size scales as " x (s) = #x$x (s)
!
US Particle Accelerator School Example: Megagauss fields in linear collider
electrons positrons
At Interaction Point space charge cancels; currents add ==> strong beam-beam focus --> Luminosity enhancement --> Very strong synchrotron radiation
Consider 250 GeV beams with 1 kA focused to 100 nm
Bpeak ~ 40 Mgauss ==> Large ΔE/E
US Particle Accelerator School Types of tune shifts: Incoherent motion
• Center of mass does not move • Beam environment does not “see” any motion • Each particle is characterized by an individual amplitude & phase
US Particle Accelerator School From: E. Wilson Adams lectures Incoherent collective effects
✺ Beam-gas scattering ➙ Elastic scattering on nuclei => leave physical aperture ➙ Bremsstrahlung ➙ Elastic scattering on electrons leave rf-aperture ➙ Inelastic scattering on electrons ======> reduce beam lifetime ✺ Ion trapping (also electron cloud) - scenario ➙ Beam losses + synchrotron radiation => gas in vacuum chamber ➙ Beam ionizes gas ➙ Beam fields trap ions ➙ Pressure increases linearly with time ➙ Beam -gas scattering increases ✺ Intra-beam scattering
US Particle Accelerator School Intensity dependent effects
✺ Types of effects ➙ Space charge forces in individual beams ➙ Wakefield effects ➙ Beam-beam effects ✺ General approach: solve 1 x "" + K(s)x = F #m$ 2c 2 non%linear ✺ For example, a Gaussian beam has 2 e N 2 2 F = 1% e%r / 2& where N = charge/unit length SC ! 2 ( ) 2"#o$ r ✺ For r < σ e2N F " r SC 4 2 ! #$o% US Particle Accelerator School
! Beam-beam tune shift
l "py (y) ~ (Felec + Fmag ) (once the particle passes l/2 the other bunch has passed) 2c l Nbeame ✺ For γ >> 1, Felec ≈ Fmag E = y lwh h eE e2N ! "#p (y) $ 2 = B y y w y 2c cwh %o p # y ! #y $ " = #y $ ~ y similar to gradient error ky#s with ky#s = po y ! ✺ Therefore the tune shift is * * 2 $ re$ N e ! "Q = # ky"s & where re = 2 4% 'wh 4%(omc ✺ For a Gaussian beam r $ *N "Q # e ! 2 %Aint US Particle Accelerator School
! Effect of tune shift on luminosity
✺ The luminosity is f N N L = coll 1 2 4A int ✺ Write the area in terms of emittance & β at the IR
* * Aint = " x" y = #x$x o #y$y ! ✺ For simplicity assume that * 2 "x #x * #x * * #x * * = $ "x = "y $ "x#x = "y ! "y #y #y #y ✺ In that case * Aint = "x#y ✺ And ! f N N I 2 L = coll 1 2 ~ beam 4" # * " # * ! x y x y US Particle Accelerator School
! Increase N to the tune shift limit
✺ We saw that * re $ N "Qy # 2 %Aint or * 2#Aint 2#%x$ 2 N!= "Qy * = "Qy * = #%x"Qy re$ re$ re Therefore the tune shift limited luminosity is
! 2 f N #$ & IE ) L = "Q coll 1 x ~ "Q ( + r y 4$ % * y( % * + e x y ' y *
US Particle Accelerator School ! Incoherent tune shift for in a synchrotron
Assume: 1) an unbunched beam (no acceleration), & 2) uniform density 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 1 2R# 1 2R# ! "Qx = $ k x (s)%x (s)ds = $ KSC(s)%x (s)ds 4# 0 4# 0
where 2r I K = " 0 SC ea2 3 3c ! # $
1 2$R 2r I % (s) r RI % (s) r RI "Q = # 0 x ds = # 0 x = # 0 x 4$ ' e% 3& 3c a2 e% 3& 3c a2 (s) e% 3& 3c( !0 x
US Particle Accelerator School From: E. Wilson Adams lectures
! Incoherent tune shift limits current at injection
r0 N "Qx,y = # 2 3 2$% & 'x,y using I = (Νeβc)/(2πR) with N…number of particles in ring
εx,y….emittance containing 100% of particles ! “Direct” space charge, unbunched beam in a synchrotron Vanishes for γ » 1 Important for low-energy hadron machines Independent of machine size 2πR for a given N Overcome by higher energy injection ==> cost
US Particle Accelerator School From: E. Wilson Adams lectures Injection chain for a 200 TeV Collider
US Particle Accelerator School Beam lifetime
Based on F. Sannibale USPAS Lecture
US Particle Accelerator School Finite aperture of accelerator ==> loss of beam particles
✺ Many processes can excite particles on orbits larger than the nominal. ➙If new orbit displacement exceed the aperture, the particle is lost
✺ The limiting aperture in accelerators can be either physical or dynamic. ➙Vacuum chamber defines the physical aperture ➙Momentum acceptance defines the dynamical aperture
US Particle Accelerator School From: Sannibale USPAS Lecture Important processes in particle loss
✺ Gas scattering, scattering with the other particles in the beam, quantum lifetime, tune resonances, &collisions
✺ Radiation damping plays a major role for electron/positron rings ➙For ions, lifetime is usually much longer • Perturbations progressively build-up & generate losses
✺ Most applications require storing the beam as long as possible
==> limiting the effects of the residual gas scattering ==> ultra high vacuum technology
US Particle Accelerator School From: Sannibale USPAS Lecture What do we mean by lifetime?
✺ Number of particles lost at time t is proportional to the number of particles present in the beam at time t
dN = "# N(t)dt with # $ constant
"t ! ✺ Define the lifetime τ = 1/α; then N = N0e ✺ Lifetime! is the time to reduce the number of beam particles to 1/e of the initial value ✺ Calculate the lifetime due to the individual effects (gas, Touschek, …) 1 1 1 1 = + + + .... " total "1 " 2 " 3
US Particle Accelerator School From: Sannibale USPAS Lecture
! Is the lifetime really constant?
✺ In typical electron storage rings, lifetime depends on beam current ✺ 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 beam desorbs 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
US Particle Accelerator School From: Sannibale USPAS Lecture Examples of beam lifetime measurements
ALS
DAΦNE
Electrons Positrons
US Particle Accelerator School Beam loss by scattering
✺ Elastic (Coulomb scattering) from Incident positive particles residual background gas ➙ Scattered beam particle undergoes transverse (betatron) oscillations. ➙ If the oscillation amplitude exceeds ring acceptance the particle is lost Rutherford cross section ✺ Inelastic scattering causes
particles to lose energy photon ➙ Bremsstrahlung or atomic excitation ➙ If energy loss exceeds the momentum acceptance the particle is lost nucleus Bremsstrahlung cross section
US Particle Accelerator School Elastic scattering loss process
dN * ✺ Loss rate is = "#beam particlesNmolecules$ R dt Gas N N # c " = = beam particles A T A L ! beam rev beam ring
Nmolecules = nAbeam Lring
! 2& & * d" Rutherford d" Rutherford " R = d# = d% sin' d' $Lost d# $ $ d# 0 ' MAX ! ' 2 * 2 d" R 1 Zbeam Zgase 1 = 2 ) , 4 [MKS] ! d# (4$%0 ) ( 2& c p + sin (- 2)
US Particle Accelerator School
! Gas scattering lifetime
✺ Integrating yields
2 dN $ n N" c * Z Ze 2 ' 1 = + ( Inc % dt 2 ( c p % 2 Gas (4$ # 0 ) ) " & tan (! MAX 2) Loss rate for gas elastic scattering [MKS]
P ✺ For M-atomic molecules of gas n = M n [Torr] 0 760
#A ✺ For a ring with acceptance εA & for small θ "MAX = $n ==> 2 760 4#" 2 * ! c p ' " $ + 0 ( % A [MKS] Gas ( 2 % ! P[Torr] ! c M n0 ) Z Inc Ze & !T
US Particle Accelerator School Inelastic scattering lifetimes
✺ Beam-gas bremsstrahlung: if EA is the energy acceptance
153.14 1 $ Brem[hours] # " ln(!EA E0 ) P[nTorr]
✺ Inelastic excitation: For an average ßn
2 E 0[GeV ] $A[µm] "Gas[ hours] #10.25 P[nTorr] %n [m ]
!
US Particle Accelerator School 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
US Particle Accelerator School Touschek effect: Intra-beam Coulomb scattering
✺ Coulomb scattering between beam particles can transfer transverse 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 the relative motion of the particles is non-relativistic ➙ Then boost the result to the lab frame
1 1 N 1 1 N 1 # beam # beam 3 2 3 ˆ "Tousch. $ % x% y% S (&pA p0 ) $ Abeam% S VRF
US Particle Accelerator School ! Transverse quantum lifetime
✺ At a fixed s, transverse particle motion is purely sinusoidal
x = a " sin # t + $ T = x,y T n ( " n ) ✺ 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 particle envelope beyond acceptance in some azimuthal point ➙ The particle is lost
US Particle Accelerator School Quantum lifetime was first estimated by Bruck & Sands
" 2 T exp A2 2 2 T x, y # QT ! # DT 2 ( T " T ) = AT Transverse quantum lifetime
% ( 2 2 " E where "T = #T$T + ' DT * T = x,y & E 0 ) transverse damping time + DT , " # " exp $E 2 2% 2 QL DL ( A E ) ! Longitudinal quantum lifetime ✺ Quantum lifetime varies very strongly with the ratio ! between acceptance & rms size. Values for this ratio ≥6 are usually required
US Particle Accelerator School Lifetime summary
US Particle Accelerator School In colliders the beam-beam collisions also deplete the beams
This gives the luminosity lifetime
US Particle Accelerator School LEP-3 Life gets hard very fast
William A. Barletta Director, US Particle Accelerator School Dept. of Physics, MIT Economics Faculty, University of Ljubljana
US Particle Accelerator School Physics of a Higgs Factory
✺ Dominant decay reaction is e+ + e- => H =>W + Z
2 ✺ MW+MZ = 125 + 91.2 GeV/c ==> set our CM energy a little higher: ~240 GeV
✺ Higgs production cross section ~ 220 fb (2.2 x 10-37 cm2)
✺ Peak L = 1034 cm-1 s-1 = < L> ~ 1033 cm-1 s-1 ✺ ~30 fb-1 / year ==> 6600 Higgs / year ✺ Total cross-section at ~ 100 pb•(100GeV/E)2
We don’t have any choice about these numbers
US Particle Accelerator School Tune shift limited luminosity of the collider
2 & ) & ) N c" 1 Nri EI 1 Nri Pbeam L = * = 2 ( * + = 2 ( * + i = e, p 4#$n% SB erimic 4#$n ' % * erimic 4#$n ' % * Linear or Circular Tune shift
! Or in practical units for electrons # " & # 1 cm&# E &# I & L 2.17 1034 1 x Q = o % + ( y% * (% (% ( $ " y ' $ ) '$ 1 GeV'$ 1 A'
# & " x At the tune shift limit %1 + (Q y ) 0.1 $ " y ' ! 33# 1 cm&# E &# I & L = 2.17 o10 % * (% (% ( ! $ " '$ 1 GeV'$ 1 A'
US Particle Accelerator School
! We can only choose I(A) and ß*(cm)
✺ For the LHC tunnel with fdipole = 2/3, ρ ~ 3000 m ✺ Remember that # p & # 1& # 1 T& "(m) = 3.34 % ( % ( % ( $ 1 GeV/c' $ q' $ B '
✺ Therefore, Bmax = 0.134 T ✺ Each beam! particle will lose to synchrotron radiation E 4 (GeV) U (keV) = 88.46 o "(m) or 6.2 GeV per turn I << 0.1 A ! beam
US Particle Accelerator School Ibeam = 7.5 mA ==> ~100 MW of radiation
$ ' ✺ Then 33 1 cm L " 2.6 o10 & * ) % # (
✺ Therefore to meet the luminosity goal * * 1/2 ! <ß xß y> ~ 0.1 cm ✺ Is this possible? Recall that is the depth of focus at the IP
The “hourglass effect” lowers L ==> ß* ~ ß* σz
US Particle Accelerator School Bunch length is determined by Vrf
✺ The analysis of longitudinal dynamics gives
3 c# " p c p & ' " p " = C = 0 0 C S 2 ˆ $sync p0 2%q h f 0 V cos((S ) p0 -5 where αc = (ΔL/L) / (Δp/p) must be ~ 10 for electrons to remain in the beam pipe ! ✺ To know bunch length we need to know Δp/p ~ ΔE/E ✺ For electrons to a good approximation
"E # E beam < E photon > and
E[GeV ]3 # [keV ] = 2.218 = 0.665! E[GeV ]2 ! B[T ] c "[m] ! US Particle Accelerator School For our Higgs factory εcrit = 1.27 MeV
✺ Therefore "E $ p # # 0.0033 E p ✺ 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 ) # "E & qVmax % ( = (2cos+s + (2+s , ))sin+s $ E ' )h* E max c sync ✺ The greater the over-voltage, the shorter the bunch c# " c3 p & # " ! " = C p = 0 0 C p S 2 ˆ $ p0 2%q h f0 Vmax cos('S ) p0
US Particle Accelerator School
! 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
* * " i = #i $i for i = x,y ✺ Therefore, we must estimate the natural emittance which is determined by the synchrotron radiation ΔE/E !
US Particle Accelerator School The minimum horizontal emittance for an achromatic transport
& 2 ) $13 % min "x,min = 3.84 #10 ( + F meters ' Jx * 3 $13 2 & - ) , 3.84 #10 % ( achromat + meters ' 4 15 *
εy ~ 0.01 εx
! US Particle Accelerator School Because αc is so small, we cannot achieve the minimum emittance
✺ For estimation purposes we will choose 20 εmin as the mean of the x & y emittances ✺ For the LHC tunnel a maximum practical ipole length is 15 m ➙ A triple bend achromat ~ 80 meters long ==> θ = 2.67x10-2
<ε> ~ 7.6 nm-rad ==> σtransverse = 2.8 µm
How many particles are in the bunch? Or how many bunches are in the ring?
US Particle Accelerator School We already assumed that the luminosity is at the tune-shift limit ✺ We have
2 & ) & ) N c" 1 Nri EI 1 Nri Pbeam L = * = 2 ( * + = 2 ( * + i = e, p 4#$n% SB erimic 4#$n ' % * erimic 4#$n ' % * Linear or Circular Tune shift
! Nr 4"#$ ✺ Or Q = e % N = Q 4"#$ re
11 ✺ So, Ne ~ 8 x 10 per bunch !
✺ Ibeam = 7.5 mA ==> there are only 5 bunches in the ring
US Particle Accelerator School 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?
US Particle Accelerator School 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 synchrotron radiation (beamstrahlung) E[GeV ]3 # [keV ] = 2.218 = 0.665! E[GeV ]2 ! B[T ] c "[m]
✺ For LEP-3 Ecrit = 35 GeV ! (There are quantum corrections) The rf-bucket cannot contain such a big ΔE/E Beamstrahlung limits beam lifetime & energy resolution of events
US Particle Accelerator School There are other problems
✺ Remember the Compton scattering of photons up shifts the energy by 4 γ2
2 E=γmc λout
λin ✺ 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
US Particle Accelerator School