making luminosity at the tevatron · 2015-03-03 · – imagine: particle circulating in ﬁeld, b;...

Making Luminosity at the Tevatron

Mike SyphersFermilab / Accelerator Division

Tuesday, July 13, 2010

Imagine two bunches passing through each other, each with N pariticles and of transverse cross sectional area A; the ``interaction cross section’’ for a collision is . If they pass through each other with frequency f, then the rate of particle interactions will be:

Σint

area, A

N particles

1, of N

Luminosity

R =�

Σint

A·N

�·N · f

=f N2

A· Σint

≡ L · Σint

Luminosity

Σint


Circular Accelerator/Collider

• Our model of accelerator:– Accelerating device + magnetic field to bring it back to accelerate again

• ex.: Tevatron: 48,000 rev/sec x 3600 sec/hr x 20 hr/store = …– Need for Stability of particle motion!!!

• Will discuss (a) acceleration and longitudinal motion, and (b) transverse motion – must show that it is valid to treat them independently


Acceleration and Longitudinal Motion• DC acceleration can only take us so far

– ~10 MeV, say, before electrostatic breakdown• What if circulate particles back through the DC

voltage again?– Net voltage along circuit! will be zero; must turn ! voltage on/off --> AC

• Use resonant cavities, with electric field in z direction, Ez = (V/g) sin2!frf t g = gap length– “radio frequency” cavities (10’s - 100’s MHz) typical


• Principal of phase stability– McMillan (U. California) and Veksler (Russia)– Imagine: particle circulating in field, B; along orbit, arrange particle to pass through a

cavity with field oscillating at frequency h x frev (h an integer); suppose particle arrives near time of zero-crossing -- net accel./decel. = eV sin("#t), V = max. voltage of cavity

– if arrives late, gets more accel; arrives early, gets less• thus, a restoring force --> energy oscillation

– in general, lower momentum particles take longer, arrive late gain extra momentum, etc. – next, slowly raise the strength of B; if raised adiabatically, the oscillations will continue

about the “synchronous” momentum, defined by p/e = B.R for constant R– particle momentum will oscillate about the ideal momentum associated with the guide

field strength...

• This is the principle behind the synchrotron, used in all major HEP accelerators today

The Synchrotron & Phase Focusing

“Synchrotron Oscillations”


Bunched BeamEx: Bunch by adiabatically raising voltage of RF cavities

eV sin(!"t)

(or, time)

E -

Eid

eal

A “Phase Space” plot


• Say ideal particle arrives at phase !s:• Particles arriving nearby in phase, and nearby in energy E = Es + #E

will oscillate about these ideal conditions

Synchrotrons...

• Adiabatic (on scale of energy oscillation period) increase of bend field generates stable phase space regions; particles oscillate, follow along– “bunched” beam; h = frf / frev = # of possible bunches

• Phase Space plot:

!

!E

Stable Phase Space regions

dEs

dt= f0eV sin!s !

dB

dt


Phase SpaceBuckets, Bunches, Batches, …

• Stable phase space region is called a bucket.– Boundary is the separatrix; only an approximation– $s = 0, ! -- particles outside bucket remain in the accelerator

! ! ! “DC beam”– Other values of $s -- particles outside bucket are lost

• DC beam from injection is lost upon acceleration

• Bunches of particles occupy buckets; but not all buckets need be occupied.

• Batches (or, bunch trains) are groupings of bunches formed in specific patterns, often from upstream accelerators


Acceleration Process


Stability and Synchrotron Tune• Combining the above difference equations, and taking the

differential limit leads us to the phase equation! ! ! ! ! ! ! ! where #$ = $ - $s.• For small angles we have simple harmonic motion, with

“frequency”

• The quantity %s is the synchrotron tune and is the number of synchrotron oscillations which occur per revolution. At 150 GeV in the Tevatron, for example, we get


• Groups of particles naturally form, accelerate together, using high-frequency (radio frequency) power systems

• Keep particle orbit constant --> adiabatically “ramp” magnetic field• Necessary magnetic field strength proportional to particle momentum:

Larger and Larger Synchrotrons...

• Final energy set by the maximum magnetic guide field strength• Iron saturates at B ~ 2 Tesla; superconducting magnets approach 10 T

• Tevatron: 4.4 T; LHC: 8 T; highest accel mag (lab): ~ 15 T

!mv2

R= evB !" R =

p/e

B

p/e !

!

10

3

T m

GeV/c

"

p


Transverse Motion• So far, can see process for creating bunches of particles and accelerating to

arbitrary energies; phase stability restores particles toward the ideal energy. However, not all particles (any??) travel along the ideal trajectory. Thus, need to also provide transverse restoring forces.

• Deflections in a Uniform magnetic field: ! Horizontal -- stable !! Vertical -- NOT! !

• Electric Forces too weak at high energies; need to use magnetic fields to restore trajectories.

• Reference coordinate system:• Assuming small angles,

B

xy

s


Weak Focusing vs. Strong Focusing• Original focusing schemes employed (now-called) Weak Focusing

-- use outwardly wedged-shape pole faces on magnets to produce horizontal field which varies with vertical displacement– As accelerator radii increased, so did their required apertures -- and hence

the required transverse real estate! (could stand inside some beam pipes!)• Strong Focusing -- alternate the wedge direction of pole faces on

successive magnets; makes field gradient vary from vertically focusing to vertically defocusing from magnet to magnet– Has a much stronger overall restoring force

• Eventually, realized could separate the focusing and bending functions -- uniform fields (dipoles) to bend the central trajectory, and quadrupole fields to provide a field gradient:

! First separated function synchrotron: Fermilab Main Ring N

N S

SB


Strong Focusing• Think of standard focusing scheme as alternating system

of focusing and defocusing lenses• Quadrupole will focus in one transverse plane, but defocus

in other; if alternate, can have net focusing in both– for equally space infinite set, net focusing requires -- F > L/2

– F = focal length, L = spacing

– FODO cells:

• Thin lens focal length:

Tevatron:

and L = 30 m

x(s)

sF


Separated Function• Until late 60’s, synchrotron magnets (wedge-shaped variety) both

focused and steered the particles in a circle. (“combined function”)• With Fermilab Main Ring (and CERN SpS), use “dipole” magnets to

steer, and use “quadrupole” magnets to focus• Quadrupole magnets, with alternating field gradients, “focus” particles

about the central trajectory -- act like lenses

Fermilab Logo


Example: FNAL Main Injector

Bending Magnets

Focusing Magnets


Superconducting Tevatron Magnet

• Outside is at room temperature;! inside is at 4oK!• Field is 4.4 Tesla @ ~4,000 A• Each magnet is ~20 ft long,! and weighs about 4 tons• ~1000 magnets in the Tevatron

B


Stability Criterion• System of lenses with appropriate separation and focal

lengths decouples the radius of the synchrotron from the transverse aperture -- strong focusing. Thus, can build synchrotrons of very large circumference.

• Do not want all “lenses” equally spaced, necessarily. However, if system of linear restoring forces, then is linear algebra problem:– For N linear (restoring force) elements once around,

! ! stable, if |Trace(M)| < 2.» Note: must analyze each plane separately, but for same lens system


Particle Trajectories

Analytical Description:

- Equation of Motion:

- Nearly simple harmonic; so, assume soln.:

0 5 10 15 20

-2-1

01

2

s

x

0 5 10 15 20

-2-1

01

2

s

x

(Hill’s Equation)

�K(s) =

e

p

∂By

∂x(s)

�

1 FODO “cell”

F FD


We see that an “amplitude function” exists, so taking and assuming , can show that

•

• and

In a “drift” region (no focusing fields), - beta function will be a parabola- if pass through a waist at s = 0, then,

So, optical properties of synchrotron (#) are now decoupled from particle properties (A, $) and accelerator can be designed in terms of optical functions; beam size will be proportional to #1/2

0 L 2L 3L 4L!4

!2

0

2

4

longitudinal position

x!mm"

Figure 2.10: Particles’ trajectories in a drift space. The beam looks like thisin a collision straight section of a collider. We adjust the beam so that its sizebecome minimum at the center of the physics detector.

32

Hill’s Equation and the “Beta Function”

�β~


Low-Beta “Squeeze”• A triplet of quadrupoles located on either side of

detector region provide the final focusing of beam• Triplet and other quadrupoles, located outside the

region, used to adjust beam size at the focus– quads outside do

most of work; like “changing the eyepiece” of a telescope to adjust magnification


Tune• Since! ! ! ! and

then the total phase advance around the circumference is given by

The tune, %, is the number of transverse “betatron oscillations” per revolution. The phase advance through one FODO cell in the Tevatron is given by

! ! !For Tevatron, L/2F = 0.6, and since there are about 100 cells, the

total tune is about 100 x (2 x 0.6)/2! ~ 20• Note: since betatron tune ~ 20, and synchrotron tune ~ 0.002, it is

(relatively) safe to consider these effects independently• “circular” accels --> resonance conditions; choose tunes carefully!


Emittance• Just as in longitudinal case, we look at the phase space trajectories, here

x-x’, in transverse space.• Viewed at one location, phase space trajectory of a particle is an ellipse:

x’

x Here &, ', ( are theCourant-Snyder parameters

While beta function changes along the circumference, the area of the phase space ellipse = !A2, and is independent of location!So, define emittance, ), of the beam as area of phase space ellipse containing some fraction (95%, say) of the particles (units = mm-mrad)

[det(M) = 1]


Emittance (cont’d)• Emittance of the particle distribution is thus a measure of beam

quality.– At any one location...

– note: if ' in m, ) in mm-mrad, then x will be in mm

• Variables x, x’ are not canonical variables; but x, px are; the area in x-px phase space is an adiabatic invariant; so, define a normalized emittance as

• The normalized emittance should not change as we make adiabatic changes to the system (accelerate, for instance). This implies that the beam size will shrink as p-1/2 during acceleration.

�x2�1/2(s) =�

�β(s)


Many other interesting effects to consider…• Momentum dispersion and chromatic effects (chromaticity)• Linear errors

– Steering errors and corrections; integer resonance– focusing errors and corrections; half-integer resonance

• Further field imperfections (nonlinearities)– Dynamic aperture, resonance conditions, …– Resonant extraction

• Coupling of degrees-of-freedom– Transverse linear coupling– Nonlinear coupling– Synchro-betatron coupling

• Emittance growth mechanisms• Synchrotron Radiation• Space charge interactions (mostly low-energies)• Beam-beam interactions

– Head-on collisions, long-range interactions

• Impedance / Wake Fields due to pipe, “cavities,” … – Coherent instabilities -- Head-tail, resistive wall, etc.

• RF Manipulations -- – coalescing, barrier buckets, cogging, slip stacking, …

• Emittance reduction -- stochastic cooling, electron cooling• …

Help is always ! welcome !!


• The Fermilab Accelerator System is made up of a ‘chain’ of accelerators, each delivering particles to the downstream accelerator.– Cockcroft-Walton style “pre-accelerator”

• 0 - 750 keV

– Linear Accelerator (Linac)• 0.75 MeV - 400 MeV

– Booster Synchrotron• 400 MeV - 8000 MeV

– Main Injector Synchrotron• 8 GeV - 150 GeV

– Tevatron Synchrotron• 150 GeV - 980 GeV

– Plus storage rings: – Antiproton accumulator (8 GeV)– Recycler (pbar storage) (8 GeV)

Fermilab’s Accelerators


The Tevatron• Commissioned in 1983

– Replaced 400 GeV “Main Ring” in the same tunnel (built ~1972)– 1st superconducting synchrotron– Circumference = 2! km (+/- 5 cm!) (~ 4 miles)– One round trip for a proton takes 21 µsec (48,000 revolutions/sec)

• Acceleration takes place with 8 RF cavities, total ~20 m. Rest of circumference is magnets, bringing particles back to the cavities!


Beam-Beam Mitigation• As particle beams “collide” (very few particles actually

“interact” each passage), the fields on one beam affect the particles in the other beam. This “beam-beam” force can be significant.

– on-coming beam can act as a “lens” on the particles, thus changing the focusing characteristics of the synchrotron, oscillation frequencies, etc.

• Beams are nowadays “separated” (if not in separate rings of magnets) by electrostatic fields so that the bunches interact only at the detectors

– “Pretzel” or “helical” orbits are common– Note, however, the “long-range” interactions can still affect performance– new “electron lenses” and current-carrying wires are being investigated which

can mitigate the effects of beam-beam interactions


Tevatron Beam Envelopes

Protons

antiprotons

Helical orbits through 4 standard arc cells

bunch length


Here, is form factor less than 1, caused by the longitudinal extent of the bunches

Back to Luminosity...We had, for equal, round, colliding beams...

If B bunches of each beam are made to collide, and the revolution frequency is f0, then f = Bf0

If the transverse extent of the beams are round and Gaussian, with variance , then the effective cross sectional area will give...

L =f0B N2

4πσ2· H

H

σ2

L =fN2

A

0 L 2L 3L 4L!4

!2

0

2

4

longitudinal position

x!mm"

Figure 2.10: Particles’ trajectories in a drift space. The beam looks like thisin a collision straight section of a collider. We adjust the beam so that its sizebecome minimum at the center of the physics detector.

32


For unequal beam emittances, bunch intensities ...

L =f0BN1N2

2π(σ∗12 + σ∗

22)· H

=3f0γBN1N2

β∗(�1 + �2)· H

Typically,

#* ~ 30 cm

%z ~ 50 cm

ϵ ~ 5-20 µm

H =√

π

�β∗

σz

�e(β∗/σz)2 [1− erf(β∗/σz)]

0.0 0.5 1.0 1.5 2.0

0.0

0.2

0.4

0.6

0.8

β∗/σz

H

σ∗ =

��β∗

6πγ

Unequal Bunch Intensities


Luminosity Evolution ...In the Tevatron we have more protons than antiprotons, which increases the luminosity shown previously. So, assume whenever a proton is lost, so is an antiproton.

Let N1(t) = N(t) + "N, and N2(t) = N(t); then,

From which:

BN = −L Σint n

L =f0BN1N2

4πσ∗2 · H =f0BN(N + ∆N)

4πσ∗2 · H

L(t) = L0∆N2e∆Nkt

(N01 e∆Nkt −N0

2 )2

k ≡ nL0Σ/BN01 N0

2 = nf0HΣ/4πσ∗2


... with Unequal Bunch Intensities

Integrating, we get

where ...

The time to integrate to a fraction f of I0:

I ≡� T

0L(t)dt =

BN02

nΣ·

e∆NkT − 1

e∆NkT − N02

N01

=⇒ I0, as t→∞

I0 ≡BN0

2

nΣ(given by beam of lesser intensity)

Tf =I0/L0

(1−N02 /N0

1 )ln

�1− fN0

2 /N01

1− f

�


Assume 36 bunches in each beam, N10 =250x109 (protons) and N20 =70x109 (pbars)... BN10 = 1013, BN20 = 252x1010

Use spot size of , and an hour glass factor H = 0.6; take an inelastic cross section of 60 mb

Suppose store stays in until 85% of I0 is reached, (for which luminosity reaches 10% of original value):

L0 ≈ 250× 1030/cm2/sec = 250 µb−1/sec = 0.9 pb−1/hr,I0 ≈ 21 pb−1/store,

I0.85 ≈ 18 pb−1/store,T0.85 ≈ 2 days

Put in some numbers ...

σ = 25 µm


Instantaneous (left) and integrated (right) luminosity vs. time through a ``perfect'' store, using parameters above. Here, the number of particles in one beam is ~30% that of the other beam.

0 10 20 30 40 50

050

100

150

200

250

time(hr)

Lum

inos

ity (/

mic

roba

rn/s

ec)

0 10 20 30 40 50

05

1015

20

time(hr)In

tegr

ated

Lum

inos

ity (/

pb)


Obviously not the whole story ...

Tevatron stores do not integrate to 18 pb-1 ...

In the above, particles are lost only due to collisions

Not the only source of particle loss!

Need to include effect of emittance growth and corresponding particle lifetime due to other causes, such as:

beam-gas scattering, RF noise, PS ripple, ...


Emittance Growth

Just before we ``initiate collisions,’’ the beam is scraped and the aperture is defined by the collimators

From then on, assume that single-particle emittance growth mechanisms drive particles transversely into the aperture

Assume an ``effective emittance’’ and an effective emittance growth rate

��


Diffusion Loss RatesIn absence of “collisions,” an equilibrium distribution would be reached; equilibrium lifetime develops

0 10 20 30 40 50

0.0

0.2

0.4

0.6

0.8

1.0

t (hr)

N(t)

/N(0

)

aperture at 3 timesinitial rms beam size Use this to describe “non-

lumi” particle loss

Assume loss mechanisms are same/similar for both beams, with equal equilibrium lifetimes in absence of collisions

τ =2a2

λ21d�x2�/dt

≈ 2�

�

λ1 = 2.405

details depend upon initial distribution


Differential Equationsfor bunch intensities

With this description in mind...

where,

Subtracting:

If let N2(t) = N(t):

N1 = −L · Σ · n/B − 1τ

N1 = −kN1N2 −1τ

N1

N2 = −L · Σ · n/B − 1τ

N2 = −kN1N2 −1τ

N2

k = L0Σn/(BN01 N0

2 ) = L0/(I0N01 )

N1(t)−N2(t) = (N01 −N0

2 )e−t/τ

N +�

1τ

+ k∆Ne−t/τ

�N + kN2 = 0


Curves, with diffusion

Curves with & = 10 hr, 25 hr, 50 hr, and infinity, using same parameter values as before. Note that for this set, indicative of recent Tevatron performance, I0/L0 ~22 hr.

0 10 20 30 40 50

050

100

150

200

250

time(hr)

Lum

inos

ity (/

mic

roba

rn/s

ec)

0 10 20 30 40 50

05

1015

20

time(hr)

Inte

grat

ed L

umin

osity

(/pb

)


Sources of emittance growth

Beam-gas scattering

Intra-beam and beam-beam effects

RF noise

Power supply noise

Orbital motion

...

(accounts for ~>0.5 ' mm-mr/hr)*

*V. A. Lebedev, L. Y. Nicolas† , A. V. Tollestrup , Residual Gas, Emittance Growth and Beam Lifetime in Tevatron at 150 GeV, Beams-doc-1155 (2004).


An Early Example: January 2007

black: data

red: model

assumes 6 equal stores, equally spaced, each using parameters that have been averaged over all 6 stores ! !!!

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

!!!!!!!!!!!!!!! !

0 50 100 150

010

2030

4050

time in week (hr)

Inte

grat

ed L

umin

osity

(/pb

)

Week beginning 1 Jan 2007

black dots: data

red line: model


Luminosity Optimization

For Tevatron, balance rate at which integrate luminosity against the rate at which we can produce antiprotons 100 200 300 400 500 600

30

40

50

60

70

Accumulated Antiprotons (e10) for next store

Int.

Lu

m p

er

we

ek (

1/p

b)

assumes accumulation rateof 2x1011 pbars/hr

0 5 10 15 20 25

010

020

030

040

0

Time (hrs)

B*N

2(t),

e10

Antip

roto

ns (e

10)

store

stash


Optimization ProcessKeeping stores in a long time, to build up pbars and get a larger initial luminosity, doesn’t necessarily help; eventually, not integrating much near the end of this time

If could have ``pbars on demand,’’ would fill frequently; shot set-up time becomes the dominant factor

For a given average pbar accumulation rate, there will be an optimum initial number of pbars which, if stores are reproducible, that would optimize the weekly integrated luminosity

We’ve been operating pretty close to these conditions; weekly numbers depend mostly upon downtimes, etc.


Tevatron Operation

Balancing the production rate of antiprotons to find optimum running conditions

two 7-day periods

0 50 100 150

020

40

60

80

Record Weeks

time in week (hr)

Inte

gra

ted

Lu

min

osity (

1/p

b)

9.5

6.5

# stores

Average Antiproton Production Rate

12/8/08 -- 18 e10/hr

1/1/07 -- 12 e10/hr


Tevatron Performance

Tevatron I design goal (1/µb/s) (Eng. run, 1980’s)

best achieved in Run I (25/µb/s) (5/92 - 2/96)

Final level reached in Run I (~100/pb) (5/92 - 2/96)

• Ecm = 1.96 TeV; operating up to 400 times original design luminosity

• Upgrades since 1986 -- – Linac upgrade; new Main Injector; Interaction Region magnets; improved magnet cooling; more bunches (6 -> 36); “Recycler” (antiproton storage); electron cooling; new stochastic

cooling systems; new Beam Position Monitoring systems, other diagnostics; much maintenance -- alignment, magnet fixes, etc.; much more ...


Further Schooling...US Particle Accelerator School:

- http://uspas.fnal.gov

- Twice yearly, January / June

CERN Accelerator School:

- http://cas.web.cern.ch

• Spring (specialized topics)

• autumn (intro/intermediate)

email: [email protected]


http://uspas.fnal.gov/


http://cas.web.cern.ch/


mailto:[email protected]

mailto:[email protected]

Some References• D. A. Edwards and M. J. Syphers, An Introduction to the Physics

of High Energy Accelerators, John Wiley & Sons (1993)• S. Y. Lee, Accelerator Physics, World Scientific (1999)• E. J. N. Wilson, An Introduction to Particle Accelerators, Oxford

University Press (2001)! and many others…

• Particle Accelerator Schools -- – USPAS: http://uspas.fnal.gov CERN CAS: http://cas.web.cern.ch

• Conference Proceedings --– Particle Accelerator Conference (2003, 2001, …)– European Particle Accelerator Conference (2004, 2002, …)

• Documents on luminosity, diffusion, ... :– Beams-doc’s ( http://beamdocs.fnal.gov/ ) : 3031, 2798, 2685, 1478, 1348, (MJS)– ... 1155, 2230, 2645, ... (many: search Luminosity)– PRST-AB 8:101001 (2005) Shiltsev, et al., ...






http://beamdocs.fnal.gov

http://beamdocs.fnal.gov