ibs - a view from the tevatronibs – a view from tevatron, v. lebedev, electron-ion workshop,...
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IBS – A view from Tevatron, V. Lebedev, Electron-Ion workshop, Jefferson Lab, March 15-17, 2004 1
IBS - A View from the Tevatron
Talk outline 1. Introduction 2. Parametric model of the Luminosity evolution 3. Interplay of Diffusion and Beam-beam effects
Conclusions
Electron-Ion Collider Workshop
March 15-17, 2004 Jefferson Lab
Valeri Lebedev FNAL
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IBS – A view from Tevatron, V. Lebedev, Electron-Ion workshop, Jefferson Lab, March 15-17, 2004 2
1. Introduction IBS in the Tevatron complex ♦ Electron cooler (4.3 MeV, energy recuperation, part. loss from collec. ~10-5) Ø Single scattering introduces tails in longitudinal distr. ⇒ particle loss
from the collector of electrons ♦ Accumulator Ø Transverse IBS depends strongly on lattice functions Ø Two lattices
• Stacking lattice – optimized for antiproton stacking • Shot lattice – optimized for final cooling
♦ Recycler Ø Longitudinal bunch compression to decrease IBS
• Longitudinal and transverse temperatures are expected be equal for cooled beam
♦ Tevatron Ø IBS major effect leading to the luminosity decay
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IBS – A view from Tevatron, V. Lebedev, Electron-Ion workshop, Jefferson Lab, March 15-17, 2004 3
IBS physics ♦ Standard theory (Bjorken-Mtingwa, Piwinsky) Ø Gaussian distributions in all degrees of freedom Ø Not self-consistent Ø Single scattering is neglected Ø Formulas are quite complicated but still not sufficiently accurate in many
practical application ♦ In many practical cases including colliders the long. temperature is much
smaller than the trans. one in the beam frame Ø Significant simplification in theory Ø Single and multiple scattering can be treated together by comparatively
simple integro-differential equation Ø As a rule in data analysis the IBS has to be considered together with
other mechanisms affecting evolution of the distribution function • Gas scattering • Noises (RF noise, EMI, quad motion, etc.) • Beam-beam effects • Electron or stochastic cooling
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IBS – A view from Tevatron, V. Lebedev, Electron-Ion workshop, Jefferson Lab, March 15-17, 2004 4
Peak luminosity during Run II
Ø Steady growth for three years of Run II Ø Peak luminosity is close to Run IIA (no electron cooling) design of 8⋅1031 cm-2s-1 Ø There is potential for further luminosity growth (factor ~ 1.5) before electron
cooling is available Ø Final design peak luminosity is 30⋅1031 cm-2s-1 (4 –5 times of present best)
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IBS – A view from Tevatron, V. Lebedev, Electron-Ion workshop, Jefferson Lab, March 15-17, 2004 5
2. Parametric Model of Luminosity Evolution in Tevatron The model takes into account the major beam heating and particle loss mechanisms
• Phenomena taken into account ⇒ Interaction with residual gas
♦ Emit. growth and particle loss due to E-M and nuclear scattering ⇒ Particle interaction in IPs (proportional to the luminosity)
♦ Emit. growth and particle loss due to E-M and nuclear scattering ⇒ IBS
♦ Energy spread and emittance growth due to multiple scattering ♦ Particle loss due to single scattering (Touschek effect)
⇒ Longitudinal dynamics ♦ Nonlinearity and finite size of potential well ♦ Bunch lengthening due to RF noise and IBS ♦ Particle loss from the bucket due to single IBS (Touschek effect) and
due to heating longitudinal degree of freedom (multiple IBS and RF noise)
♦ Absence of tails after acceleration
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IBS – A view from Tevatron, V. Lebedev, Electron-Ion workshop, Jefferson Lab, March 15-17, 2004 6
• Phenomena presently ignored in the model
⇒ Beam-beam effects ♦ Important for first few hours into the store
⇒ Non-linearity of the lattice ♦ Plays no role in during a store
⇒ Diffusion amplification by coherent effects including strong-strong beam-beam effects
♦ We have no evidence that it makes any effect in Tevatron • Thus, the model presents the best-case scenario
⇒ It describes comparatively well our present stores
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IBS – A view from Tevatron, V. Lebedev, Electron-Ion workshop, Jefferson Lab, March 15-17, 2004 7
Beam Evolution in Longitudinal Degree of Freedom ♦ Longitudinal acceptance grows from
4 to 10 eV s during acceleration Ø Absence of tails after
acceleration ♦ Interplay of single and multiple
scattering ♦ The model based on a solution
of integro-differential equation which describes both single and multiple IBS
( )∫∞
′−′′=∂∂
0
d),(),(),( ItIftIfIIWtf
Here the kernel is (A. Burov, Private communications, 2003)
−≤′′−
′+
′
+≥′−′
+
′−
′=′
,,21
,,21
)(),(~ 2
0
EEEEE
I
EEEEE
I
EELD
EEWC
δω
δωωωω
0 1 2 30
1
2
3
RF phase [rad]
Numerical simulation of the
longitudinal bunch profile evolution during store.
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IBS – A view from Tevatron, V. Lebedev, Electron-Ion workshop, Jefferson Lab, March 15-17, 2004 8
Measurements of beam distribution in the longitudinal phase space
A. Tollestrup , Feb. 2004.
♦ Distribution functions are computed from signals of the resistive wall monitor (SBD)
♦ Entire bucket of 4.2 eV⋅s is filled at injection
♦ There is no long. emittance growth during acceleration
♦ Entire proton bucket of 10.7 eV⋅s is filled at the end of store
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IBS – A view from Tevatron, V. Lebedev, Electron-Ion workshop, Jefferson Lab, March 15-17, 2004 9
Parametric model of luminosity evolution ♦ Compromise between simplicity of the model and accuracy of the description
Ø Finite accuracy of the measurements
♦ System of eight ordinary differential equations
−−−
++
++−−−
++
++
=
−
−
bppLascata
totalpa
gasayIBSayBBay
gasaxIBSaxBBax
bppLpscatp
totalpp
gaspyIBSpyBBpy
gaspxIBSpxBBpx
a
pa
ay
ax
p
pp
py
px
nLdtdNN
dtd
dtddtddtd
dtddtddtdnLdtdNN
dtd
dtddtddtd
dtddtddtd
N
N
dtd
στ
σ
εεε
εεεστ
σ
εεε
εεε
σεε
σεε
2
2
22
2
2
1
2
1
2
2
2
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IBS – A view from Tevatron, V. Lebedev, Electron-Ion workshop, Jefferson Lab, March 15-17, 2004 10
Luminosity Evolution
0 5 10 15 20 250
1 .1031
2 .1031
3 .1031
Time, hour
Lum
inos
ity, c
m^-
2 s^
-1
0 5 10 15 20 250
10
20
30
40
Time, hour
Lum
inos
ity li
fetim
e, h
our
B0D0
D0 Model
Model
B0
Good regular store !!! ( Store 2138, Jan.5 2003 ) ♦ Three free parameters are used in the model Ø Residual gas pressure - P=1⋅10-9 Torr of N2 equivalent Ø Spectral density of RF noise- ( ) Hz/rad105 211−⋅≈sf fPφ (70µrad in ∆f=100Hz) Ø X-Y coupling - κ = 0.4 (strong coupling due to beam-beam effects)
♦ Their values are not very critical for the luminosity prediction but important for detailed comparison
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IBS – A view from Tevatron, V. Lebedev, Electron-Ion workshop, Jefferson Lab, March 15-17, 2004 11
Bunch population per bunch (Store 2138 )
0 5 10 15 20 251.4 .1010
1.6 .1010
1.8 .1010
2 .1010
2.2 .1010
2.4 .1010
Time, hour
Num
ber o
f pba
rs/b
unch
0 5 10 15 20 251.6 .1011
1.7 .1011
1.8 .1011
1.9 .1011
Time, hour
Num
ber o
f pro
tons
/bun
ch
Pbars Protons
♦ At the store beginning Ø Pbar loss is large due to large initial luminosity Ø Proton loss is small due to short bunch length/absence of tails and,
consequently, low longitudinal loss ♦ Model describes well the particle loss during the store
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IBS – A view from Tevatron, V. Lebedev, Electron-Ion workshop, Jefferson Lab, March 15-17, 2004 12
0 5 10 15 20 2510
15
20
25
30
35
40
Time, hour
Em
ittan
ce, m
m m
rad
0 5 10 15 20 2510
15
20
25
30
35
40
Time, hour
Em
ittan
ce, m
m m
rad
εpy εay
εax εpx
0 5 10 15 20 2550
60
70
80
Time, hour
Rm
s bu
nch
leng
th, c
m proton
pbar
Emittances and bunch lengths on time for Store 2138 ♦ Beam-beam effects ? Ø Vertical pbar emit. grows
faster at the store beginning Ø Pbar bunch length does not
grow at the second half of
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IBS – A view from Tevatron, V. Lebedev, Electron-Ion workshop, Jefferson Lab, March 15-17, 2004 13
Time [hour] Time [hour]
0 5 10 15 20 250
5 .104
1 .105
1.5 .105Proton loss per bunch
Time, hour
1/s
0 5 10 15 20 250
5 .104
1 .105
1.5 .105Pbar loss per bunch
Time, hour
1/s
Gas
Longitudinal
Luminosity
Luminosity
Gas
Longitudinal
Particle loss computed for different loss mechanisms for Store 2138.
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IBS – A view from Tevatron, V. Lebedev, Electron-Ion workshop, Jefferson Lab, March 15-17, 2004 14
Comparison of Model Predictions to Store 2328 (Mar. 20 2003) The store is strongly affected by the beam-beam effects !!!
0 5 10 15 201.5 .10
10
2 .1010
2.5 .1010
Time, hour
Num
ber o
f pba
rs/b
unch
0 5 10 15 201.4 .10
11
1.6 .1011
1.8 .1011
2 .1011
2.2 .1011
Time, hour
Num
ber o
f pro
tons
/bun
ch
♦ At the store beginning both proton and pbar bunch intensities decay faster
than the model predictions Ø Incorrect tune or too long bunch or both
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IBS – A view from Tevatron, V. Lebedev, Electron-Ion workshop, Jefferson Lab, March 15-17, 2004 15
0 5 10 15 2010
15
20
25
30
35
Time, hour
Em
ittan
ce, m
m m
rad
0 5 10 15 2010
15
20
25
30
35
Time, hour
Em
ittan
ce, m
m m
rad
εpy εay
εpxεax
0 5 10 15 2050
60
70
80
Time, hour
Rm
s bu
nch
leng
th, c
m proton
pbar
Emittances and bunch lengths on time for Store 2328 ♦ Beam-beam effects ? Ø Proton bunch length and hor. emit.
grow slower at the store beginning • Protons with large synchrotron
amplitudes are lost due to beam-beam effects
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IBS – A view from Tevatron, V. Lebedev, Electron-Ion workshop, Jefferson Lab, March 15-17, 2004 16
0 5 10 15 200
1 .1031
2 .1031
3 .1031
4 .1031
Time, hour
Lum
inos
ity, c
m^-
2 s^
-1
0 5 10 15 200
10
20
30
Time, hour
Lum
inos
ity li
fetim
e, h
our
B0 D0
D0 Model
Model
B0
♦ Luminosity decays faster than the model predictions but the difference is
small
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IBS – A view from Tevatron, V. Lebedev, Electron-Ion workshop, Jefferson Lab, March 15-17, 2004 17
Computed beam-beam linear tune shifts for Store 2328
0 5 10 15 200
0.005
0.01
0.015
0.02
Time [hour]0 5 10 15 20
0
0.005
0.01
0.015
0.02
Time [hour]
ξx
ξy
ξx ξy
Pbars Protons
♦ We already close to the design linear tune shift of 0.02 for pbar beam (~80%) Ø and only about 20% for proton beam
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IBS – A view from Tevatron, V. Lebedev, Electron-Ion workshop, Jefferson Lab, March 15-17, 2004 18
3. Interplay of Diffusion and Beam-Beam effects ♦ 1 store ~ 4⋅109 turns – too much for any computer in visible future ♦ Conclusion following from parametric model study: for correctly tuned
collider at present intensities the beam-beam effects and machine nonlinearity do not produce harmful effects on the beam dynamics while beams are in collisions
♦ We can not accept any significant worsening of the lifetime if we want to maximize the luminosity integral
♦ Theory should be build as perturbation theory to the diffusion model
♦ Diffusion amplification by resonances Ø Motion inside resonance island is fast
comparing to the beam lifetime • 100-10,000 turns depending on ξ and
the resonance order Ø Flattening distribution over resonance
0 1 2 3 40
1
2
3
4ν 0 0.578=
ξ 1 0.01=
ξ 2 0.01=
σs 0=
Phase trajectories in vicinity of 12-th order resonance νx=7/12; two Tevatron IPs but zero length of counter-rotating bunch, and zero synchrotron motion amplitude
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IBS – A view from Tevatron, V. Lebedev, Electron-Ion workshop, Jefferson Lab, March 15-17, 2004 19
0 2 4 60
0.2
0.4
0.6
2.1 3.15
ν=0.325, ξ=0.02, ∆p/p=0, σs
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IBS – A view from Tevatron, V. Lebedev, Electron-Ion workshop, Jefferson Lab, March 15-17, 2004 20
Long Range collisions
δp = 0 δp = 1.25⋅10-4
Swing of the normalized transverse amplitudes on the 5th order resonances and their synchrotron satellites at synchrotron amplitude δp = 0 (left) and δp = 1.25⋅10-4 (right), lattice chromaticity is zero, νx = 20.585, νy = 20.575. Courtesy of Yu. Alexahin
2νx+3ν
3νx+2ν
4νx+νy
5νx
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IBS – A view from Tevatron, V. Lebedev, Electron-Ion workshop, Jefferson Lab, March 15-17, 2004 21
Beam-beam simulations with Lifetrack (D. Shatilov, BINP, Novosibirsk) ♦ Features Ø Weak-strong code, symplectic Ø Linear maps between IPs Ø Gaussian distribution in the strong beam Ø Measured coupling for both beams is taken into account Ø Build-in chromaticities of the tunes and the beta-functions Ø Presently, use ~50% of PC farm of 64 Pentium processors Ø In near future we plan to increase computing power by factor of four
♦ Preliminary results Ø Still testing the code and physics behind it Ø ~(2-5)⋅106 turns are required to get to the right scaling Ø Strong effect of optics distortions on the beam-beam
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IBS – A view from Tevatron, V. Lebedev, Electron-Ion workshop, Jefferson Lab, March 15-17, 2004 22
Conclusions 1. During last three years we made significant progress in understanding how
IBS affects the Tevatron complex operation 2. It includes
a. Mitigation of IBS in Accumulator with dual lattice operation b. Quantitative understanding of Tevatron luminosity evolution c. Building credible scenario of recycler operation
3. We are still far away to understand all details of the interplay between the beam-beam effects and IBS
• Significant progress has been achieved during last year
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IBS – A view from Tevatron, V. Lebedev, Electron-Ion workshop, Jefferson Lab, March 15-17, 2004 23
Backup slide Interaction with Residual Gas Beam lifetime
( ) ∑∑ +
+
+=−
iii
my
y
mx
x
iiii
pscat cnZZn
crβσ
ε
β
εβ
βγ
πτ 1
232
21
where: m701 ,, ≈= ∫ dsC yxyx ββ εmx, εmy – acceptances are chosen to be 62⋅20 mm mrad
♦ Average vacuum is adjusted to match the beam lifetime and the emittance growth rate for small intensity beam, P=1⋅10-9 Torr of N2 equivalent Ø Coulomb scattering (~6000 hour) Ø Nuclear absorption (~400 hour) Ø Total gas scattering lifetime (~380 hour) Ø Gas composition used in the simulations
Gas H2 CO N2 C2H2 CH4 CO2 Ar Pressure [nTorr] 1.05 0.18 0.09 0.075 0.015 0.09 0.15
Emittance growth time due to gas scattering
( ) yxi
Ciiipyx
iLZZn
cr
dt
d,32
2, 1
2β
βγ
πε
+= ∑ ⇒ mrad/hourmm2.0≈≈ dt
d
dtd yx εε
• Beam based measurements of vacuum were carried out in July 2003. Analysis will follow.
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IBS – A view from Tevatron, V. Lebedev, Electron-Ion workshop, Jefferson Lab, March 15-17, 2004 24
Backup slide Scattering in IP ♦ Nuclear interaction Ø Main mechanism for loss of antiprotons Ø pp − cross-section ~ 69 mbarn
• Inelastic – 60 mbarn • Elastic – 15 mbarn
§ 40% scatters within the beam (3σ) ♦ Electromagnetic scattering Ø Emittance growth
( )( )ayaxpypxbbpyx fNLr
dt
d
εεεεβγ
ε
++=
320
2, 4
Ø dε /dt ≈ 0.01 mm mrad and is negligible in comparison with gas scattering
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IBS – A view from Tevatron, V. Lebedev, Electron-Ion workshop, Jefferson Lab, March 15-17, 2004 25
Backup slide Intrabeam Scattering ♦ Pancake distribution function
allows one to use simple IBS formulas
♦ Integration over Tevatron lattice was carried out and results were compared to the smooth lattice approximation Ø Comparison yielded
coincidence within 10% ♦ Therefore the smooth lattice
approximation has been used for IBS to simplify the model
♦ The following corrections has been taken into account Ø Bunch length correction due to non-linearity of longitudinal focusing Ø Average dispersion and dispersion invariant, Ax, were calculated using
lattice functions Backup slide
0 500 1000 1500 2000 2500 30000.1
1
10
100
s [m]
Ang
ular
spr
ead
[ura
d]
σp
γ106⋅
0 500 1000 1500 2000 2500 30000
0.5
1
1.5
2
s [m]
Bea
m s
ize
[mm
]
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IBS – A view from Tevatron, V. Lebedev, Electron-Ion workshop, Jefferson Lab, March 15-17, 2004 26
Intrabeam Scattering (Continue)
( ) ( )223332
||42
||2||
,
241
yxsyxiip
yxCi
cm
LNe
p
p
dtd
dtd
θθσσσβγ
θθθ
+
Ξ=
≡ ,
( )s
xx
dt
dA
dtd
2||1
θκ
ε−= ,
s
xy
dt
dA
dt
d 2||θκε
=
where
( )2
22
2222
|| 055.02ln
21,
+−
−
++≈Ξ
yxyx
xyyx
yxπ
, 2
||2θβεσ Dyxx += , yyy βεσ = , xxx βεθ = and yyy βεθ =
( )
sx
xxx
DDDA
βαβ 22 +′+
=
κ – coupling coefficient (measurements yield that presently κ ~ 0.4) Ax = 19.7 cm, βx=βy = 48.5 m, D = 2.84 m
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IBS – A view from Tevatron, V. Lebedev, Electron-Ion workshop, Jefferson Lab, March 15-17, 2004 27
Backup slide Beam Evolution in Longitudinal Degree of Freedom ♦ Diffusion mechanisms Ø IBS
• Multiple and single scattering Ø RF noise
• Phase noise • Amplitude noise
♦ Diffusion differently depends on action for all three mechanisms ♦ The first iteration of the model solved diffusion equation in a sinusoidal
potential well under constant diffusion, ,)( DID =
∂∂
∂∂
=∂∂
If
dIdEI
ID
tf
/
where
( )φcos12
22
−Ω+= sp
E , ∫= φπ pdI 21
Ø Equation is solved numerically for initial distribution f(I) = δ(I) Ø The boundary condition f(I) = 0 at the RF bucket boundary is used
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IBS – A view from Tevatron, V. Lebedev, Electron-Ion workshop, Jefferson Lab, March 15-17, 2004 28
Backup slide Beam Evolution in Longitudinal Degree of Freedom (continue)
0 0.5 1 1.5 20
0.2
0.4
0.6
0.8
12D distrib. function as function of E
0 1 2 3 4 50
0.2
0.4
0.6
0.8
12D distrib. function as function of I
In
0 1 2 30
0.2
0.4
0.6
0.8
1Distrib. function over bunch length
0 0.5 1 1.5 20
0.2
0.4
0.6
0.8
1Distrib. function over momentum
f(I(E)) f(I) fφ(φ) fp (p) Distribution functions as functions of the beam energy, action, longitudinal coordinate and the particle momentum deviation
where: 1)(max
0
=∫I
dIIf , ( ) ( )∫−
=)(
)(
max
max
),(φ
φφ φφ
p
p
dppIff , ( ) ( )∫−
=)(
)(
max
max
),(p
pp dpIfpf
φ
φ
φφ .
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IBS – A view from Tevatron, V. Lebedev, Electron-Ion workshop, Jefferson Lab, March 15-17, 2004 29
Backup slide Beam Evolution in Longitudinal Degree of Freedom (continue)
0 0.5 1 1.5 20
0.5
1
D t
D t⋅σφ
σp
0 2 4 6 80
0.5
1
D t
exp 1.35− D⋅ t⋅( )
♦ Asymptotic behavior Ø Shape of distribution function does not depend on time Ø Exponential decay of beam intensity
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IBS – A view from Tevatron, V. Lebedev, Electron-Ion workshop, Jefferson Lab, March 15-17, 2004 30
Backup slide Beam Evolution in Longitudinal Degree of Freedom (continue) To find compromise between completeness and simplicity of the model the following approximate relations were deduced from the numerical solution:
∆+
∆+Γ≈ ∆∆∆
3
/
2
// /
2
61
/
2
41
1sep
pp
sep
ppppss PPPP
σσσσ
( )( )
( ) ( )
+
Γ
+≈ ∆
RFIBS
pp
RF
s
sRF
s
dt
d
dt
d
dtdN
N
22/
2
77
7 2
265.1
2425.21 φσσ
λπ
πσλ
πσ
( ) ( ) ( )
Γ
+
∆−≈ ∆∆∆
RFs
RF
IBS
pp
sep
pp
total
pp
dt
d
dt
d
PPdt
d 222/5
/2
/
2/765.0
21 φ
σ
πλσσσ
where ( ) ( )sRFMs q πνλγα 2/1 2−=Γ is the parameter of longitudinal focusing
-
IBS – A view from Tevatron, V. Lebedev, Electron-Ion workshop, Jefferson Lab, March 15-17, 2004 31
Backup slide Beam Evolution in Longitudinal Degree of Freedom (continue) The bunch lengthening due to RF phase noise ♦ At small amplitude the bunch lengthening due to RF phase and amplitude noise
is determined by its spectral density at synchrotron frequency, ( )
( ) ( )
Ω+ΩΩ= sAss
RF
PPdt
d2
21 22
2
φφφ σπ
σ ,
where the spectral density of RF phase noise is normalized as
( ) ( )∫ ∫∞
∞−
∞
∞−
== ωωδ
ωωδφ φ dPA
AdP A
RF
RFRF 2
22 ,
♦ Spectral density and bunch lengthening measurement are in decent agreement, and they yield that
( ) ( ) Hz/rad10542 211−⋅≈Ω=Ω ssf PP φφ ππ ( )
hour/mrad2200 22
≈RF
dtd φσ
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IBS – A view from Tevatron, V. Lebedev, Electron-Ion workshop, Jefferson Lab, March 15-17, 2004 32
Backup slide Beam-beam effects ♦ Beam-beam effects are important at all stages Ø Injection Ø Acceleration Ø Squeeze Ø Collision
♦ Two types of the beam-beam effects Ø Head-on
• Run IB proton bunch population of ~2.7⋅1011 proton/bunch was set by the head-on collisions
• We aim to achieve the same number of protons per bunch § Linear beam-beam tune shift ξ ≈ 0.01 for each of two interaction points
Ø Long range • Much stronger than for Run IB • Additional tune spread within one bunch § ∆ν ≈ 5⋅10-3
• Tune spread between bunches (Np=2.7⋅1011) § At injection: ∆νx ≈ 5⋅10-3, ∆νy ≈ 2.5⋅10-3 § At flat top: ∆νx ≈ ∆νy ≈ 8⋅10-3
-
IBS – A view from Tevatron, V. Lebedev, Electron-Ion workshop, Jefferson Lab, March 15-17, 2004 33
Backup slide Beam-beam effects (continue) ♦ Tunes are between 5-th and 7-th, and on
12-th order resonance Ø 5-th and 7-th order are excited by long
large and lattice nonlinearity Ø 12-th order are excited by head-on
♦ Long range interactions make different tune shifts for different bunches Ø It can and must be mitigated
♦ Distance between 5-th and 7-th order resonances is 0.0285 Ø Pbars from Protons
• Head-on – 2⋅0.01=0.02 • Long range within a bunch – 0.005 • Bunch to bunch difference – 0.007 Ø Protons experience only half of this due
to smaller pbar intensity
0.580 0.585 0.590 0.595 0.600 0.605
0.570
0.575
0.580
0.585
0.590
0.595
0.580 0.585 0.590 0.595 0.600 0.605
0.570
0.575
0.580
0.585
0.590
0.595
Footprint of pbar bunch #6 in the tune diagram with νx=0.580, νy=0.580 (green dot) and nominal beam parameters. Dots show small amplitude tunes for other bunches. Footprint lines go in 2σ and 22.5 deg in the space of actions (angle or (ax2 + ay2)1/2 =const on a line). Courtesy of Yu. Alexahin