Emittance evolution of beams at the Fermilab
Booster and resonance crossing
S.Y. Lee, Indiana University
10/15/2009 UMER workshop
1.
2.
3.
4.
Motivation
Measurements and Modeling of emittance evolution of Fermilab Booster
Implications on the non-scaling FFAG accelerators
Possible measurements of scaling laws at UMER
1. X. Huang, S.Y. Lee, K.Y. Ng, and Y. Su, Emittance measurement and modeling for
the Fermilab Booster, PRSTAB 8, 014202 (2006)
2. S.Y. Lee, A Fundamental Limit of Non-scaling FFAG Accelerators, Phys. Rev. Lett.
97, 104801 (2006)
3. S.Y. Lee, G. Franchetti, I. Hofmann, F. Wang, and L. Yang, Emittance Growth
Mechanisms for the Space Charge Dominated Beams in the FFAG and Proton
Driver Rings, New J. Phys. 8, 291 (2006)
4. X. Pang, et al., Emittance Growth Scaling Laws, HB2008
Motivation
 What are mechanisms of emittance growth in synchrotrons? envelope instability
or half integer stopband; Montague resonances; linear coupling; nonlinear
resonances; integer resonances; ….
 Measurements? Can one measure emittance
turn-by-turn?
 FFAG → a favorable candidate for high power
proton drivers: Constant guide field → high rep
rate, in the kHz.
 Scaling design
 Non-scaling design.
 Beam quality can deteriorate when crossing
resonances.
 Tune-ramp rate: ~ -10-3 to -10-2 per turn
∆ν x , z
ν x , z ∆E
≈− 2
∆n
β E ∆n
Ionization beam profile monitors (IPM's) measure the beam profile by collecting
electrons from background gas ionization. Measurement accuracy will depend on
the number of electrons collected. The number of electrons collected depends on
pressure, bunch intensity, and data collection mode. The center-to-center distance
between microchannels is of the order of 1.5mm. Thus one can measure the
beam width to within 1.5-2.0 mm. For the IPM at Fermilab Booster, the space
charge correction has been shown to be important [J. Amundsen et al PRSTAB 6
102801 (2003)]
σ2
ε=
β
Vrf
Vrfsinφs
KE
Typical Injection scheme at the Fermilab
Booster: H- at 400 MeV from Linac is about
30 mA. Strip injection into the Booster gives
about 4.2×1011 protons per injection turn. At
400 MeV, the revolution frequency is
4.51×105 Hz. The Normal operation has 12
injection turns, and the total intensity is about
5×1012 ppp at 15 Hz rep-rate. In fact, when
the injection turn is larger than 12, the
Booster loss problem becomes severe.
500-rev
4000-rev
8000-rev
12000-rev
16000-rev
6t-inj
Space charge effects: Linac delivers about 30 mA beam current to the Fermilab
Booster, i.e. about 4.2×1011 particles in one injection turn.
t
ε = a0 + b1t + b2 ∫ K sc dt
0
Horizontal IPM measurements
σ x2 = β xε rms + D 2δ 2
= a + bt + ct 2 + Ae −αt cos[2π ( f1t + f 2t 2 ) + ϕ ]
δ = (∆p/p )rms
a + bt + ct 2 = β xε rms + D 2δ 2
f syn (t ) = f1 + 2 f 2t
2 A = D 2 (δ12 − δ 22 )
2 Α t∆E 1/ 2
γT
(
)
δ1 = 1/ 6
2
2
3 βτ ad Γ( 3 ) 3mc γ
ν s Αφδ
δ2 =
hηπδ1
Αφδ =
hω 0
Α t∆E
2
2
β γ T mc
1/ 3
τ ad
 πβ 2γ T4 mc 2 

=  2
 γω0 heV | cos φs | 
δ1
δ2
Summary:
1. dεz/dt ~ b+Ksc
2. The horizontal emittance and the off-momentum spread can
be separated by using different scaling law (energy
dependence).
3. The horizontal emittance is less affected by the space
charge force! Why?
4. The slow linear growth of the horizontal emittance is the
same as that of the vertical plane! εz increases linearly with t
at about 1 π-mm-mrad in 104 revolutions.
5. The post-transition horizontal bunch-width oscillation is
induced essentially by the longitudinal mis-match of the
bunch shape with space charge effect in the rf potential well.
Using the bunch shape mis-match, one can deduce the phase
space area.
An Overview of Booster’s lattice
19.76m
FD
FU
DU
DD
FD
L(6m)
S
beam
β x ,max = 33.7 m
FU
S(1.2m)
0.5m
β z ,max = 20.4m
• Superperiod=24
• In each period, four Combinedfunction magnets. Length 2.889 m.
Gradients are K1(F)=0.0542 m-2 and
K1(D)=-0.0577 m-2. The integrated
focusing strengths are 0.313 m-1 and
–0.333 m-1.
• Horizontal tune 6.7, vertical tune 6.8
• Extraction doglegs in L03 and L13
perturb the lattice functions
Dmax = 3.2m
γT
The Fermilab Booster
Circumference(m)
474.2
Inj/ext energy (GeV)
0.4/8.0
Cycling rate (Hz)
15
Hori/vert betatron tunes
6.7/6.8
Superperiod
24
Transition gamma
5.446
Harmonic number
84
Protons per pulse
5.0E12
Longitudinal 95% emittance (eV-s)
0.10 at injection
Transverse 95% emittance (mm-mr)
12π (normalized) at injection
Space charge Modeling algorithm:
We consider N-particle in Gaussian distribution and construct a model with
24 Superperiod FODO cells
γT
Obtained by ICA method
Space charge force is a local kick on every half cell (no change even when
more space charge kicks are added):
Space Charge Force
∆x '
∆z ' K sc ∧
−j
=
F sc ,
2
∆s
∆s
b
∧
− ( a + jb ) 2 + ( ar + j ) 2
2π
b
r
[ w(a + jb) − e
F sc = j
w(ar + j )],
r
(σ x2 − σ z2 )
Where,
w( Z ) = e − Z [1 +
2
2j
∫
π
Z
0
eζ dζ ].
2
But, for the round beam
∂V
x
Fx , sc = − sc = K sc 2 f x ( x 2 , z 2 )
∂x
σx
Fz , sc = −
∂Vsc
z
= K sc 2 f z ( x 2 , z 2 )
∂z
σz
1
fx (x , z ) =
1− r 2
2
2
r2
fz (x , z ) =
1− r 2
2
2
1
∫ dv e
− a 2 (1− v 2 ) − b 2 (
singularity when r=1
1
v2
r
1
∫
r
1
−1)
, where a =
dv − a 2 (1−v 2 ) −b 2 ( v 2 −1)
e
2
v
x
2(σ − σ )
2
x
2
z
,b =
2σ z2 + t
, v =
2
2
2σ x2 + t
2(σ x − σ z )
z
2
Define ε = 1 − r =
σ x −σ z
, then
σx
σx
fx (x , z ) =
σ x +σ z
2
2
x 2t (1 − εt / 2)
z 2t (1 − εt / 2)
∫0 dt exp[− σ x (σ x + σ z ) − σ x (σ x + σ z )(1 − εt ) 2 ]
1
Do Taylor' s expansion and use the integral :
1
f n ( w) = ∫ e
0
n
n!
wk
−w
t dt = n +1 (1 − e ∑
2), where
w
k = 0 k!
− wt n
x2
z2
w = p + q, p =
, q=
σ x (σ x + σ z )
σ x (σ x + σ z )
We obtain
σx
( p − 3q ) 2
p − 3q
2
{ f 0 ( w) + ε
fx (x , z ) =
f 2 ( w) + ε [−2qf 3 ( w) +
f 4 ( w)]
σ x +σ z
8
2
2
2
5q
( p − 3q ) 3
+ ε [−
f 4 ( w) − ( p − 3q )qf 5 ( w) +
f 6 ( w)]
2
48
5 pq − 23q 2
( p − 3q ) 2 q
( p − 3q ) 4
4
+ ε [−3qf 5 ( w) −
f 6 ( w) −
f 7 ( w) −
f 8 ( w)]
4
4
384
7
3 pq − 19q 2
5 p 2 q − 46 pq 2 + 93q 3
5
+ ε [− qf 6 ( w) −
f 7 ( w) −
f 8 ( w)]
2
2
16
( p − 3q ) 3 q
( p − 3q ) 5
f 9 ( w) +
f10 ( w)] + O (ε 6 )}.
−
24
3840
3
•
•
•
•
•
Sextupole nonlinearity on each half cell for nonlinearity in dipoles
Linear coupling,
Random quadrupoles with zero tune shifts
Random closed orbit error
Dynamical aperture of 80 by 50 pi-mm-mrad
b0 ( s )
1 b2 ( s ) 2
(x − z2 )
2 ρ
ρ
ρ
ρ
a0 ( s ) b1 ( s )
a1 ( s )
b2 ( s )
′
′
z + K z ( s) z = −
−
z+
x−
xz
x′′ + K x ( s ) x =
ρ
+
b1 ( s )
ρ
x+
a1 ( s )
ρ
z+
ρ
Random number generators are used to generate b0, a0, b1, and a1. The
quadrupole error is subject to a constraint with zero tune shift. The
integrated sextupole strengths are set to the systematic values: -0.0173 m-2
and -0.263 m-2 for focusing and defocusing dipoles respectively.
(For n>nt=9600) Gδ=2, Aδ=0.5, f=1/150, αδ=1/(15*150)
x
z
σz
σx
No sextupole
Summaries of FNAL Booster measurements:
1.
2.
3.
We are able to fit experimental measured IPM data (vertical and horizontal)
to deduce the betatron emittances of the beam. We note that the vertical
emittance increases rapidly in the initial stage of acceleration. The horizontal
emittance seems to be less affected by the space charge at the initial stage.
The bunch width oscillation after the transition crossing has been used to
deduce the bunch-mismatch factor, and the result is consistent with the rf
program in the Booster operation.
We use the rms space charge force model to carry out numerical simulations.
We are able to fit the observed data of emittance growth. The emittance
growth in the vertical plane has resulted mainly from the skew quadrupoles,
that induce a sum resonance at Qx+Qz=integer. This induces emittance
growth mainly in the vertical plane. The reason is that the Montague
resonance suppresses the growth of the horizontal emittance, and enhance
the vertical emittance by about 25%. The random dipole field error also
generates about 25% vertical emittance growth.
Systematic Resonances
 Sp Ch potential:
4th order
 Expansion:
1
2
2
2 1
{exp[
(
1
)
( 2 − 1)] − 1}dv
−
−
−
Vsc = K sc ∫
a
v
b
r 1− v2
v
2σ z2 + t
x
z
2
,b =
, v =
a=
2
2
2
2
2σ x2 + t
2(σ x − σ z )
2(σ x − σ z )
1
6th order
6th order Systematic Resonance
 In action-angle variables,
6νx= P
6νz= P
Can be factored out sp-ch dependent part of resonance strength, giving
dimensionless reduced strength gmnl:
lattice dependent
sp-ch dependent
Simulation Parameters
•
•
•
•
•
•
Proton injection rate: particles per turn:
4×1011
Harmonic number h= 84.
Circumference C=474.2m.
B = C /(h 2π σ s ) = 2
Bunching factor = 2
ε N ,rms = 8.33 ×10 −6 πm
Initial emittance:
Kinetic energy is kept constant at 1GeV during the tracking.
Sample Simulation






100 turns injection
systematic resonances:
6nx=P, 6nz=P, P=24
dnx,z/dn = -0.0004
Aperture=500 π mm-mrad
Emittance growth factor
(EGF):
Final emit./initial emit.
(ν x0 , ν z0 ) = (4.25,4.45)
⇓
(ν x , ν z ) = (3.69,3.89)
| g 60 P |= 0.00181
| g 06 P |= 0.00172
Turn 820
Turn 1270
Scaling Power Laws
 Tune ramp rate:
0.0004  0.0112
 Log-log plots
 Scaling Law
Critical tune ramp rate
Critical Tune Ramp Rate
| G06 P / 06 P | ∝ ∆ν sc , x / z
The number of turns that
the beam is under the influence
of the resonance :
∆n ≈ ∆ν sc / |
dν
|
dn
Can serve as a guideline for FFAG
design!
4th order Systematic Resonances
Conclusion:
1. We carried out systematic measurements of the emittance evolution at the
Fermilab Booster in order to model the effect of space charge force and other
effects on beams. We found that the random skew quadrupoles were
important on the emittance dilution. Stopband correction experiments would
be very important to confirm our results. Systematic measurements at other
accelerators would be very valuable!
2. Systematic Nonlinear space charge resonances can be important in high
intensity accelerators. For future neutron source design, one should try to
avoid the systematic nonlinear space charge resonance, if it is possible!
Otherwise, minimize space charge resonance strengths!
3. For the non-scaling FFAG, the nonlinear resonances induced by the space
charge potential can be the limiting factor. These resonances limit the phase
advance of each basic cell to within π/2 to π/3, and thus the momentum
acceptance is highly constrained.
4. We also find that the emittance growth factor for quadrupole and skewquadrupole errors obeys a simple scaling law.
Many thanks for your attention
 λ 2πg 2 
EGF = exp 

d
/
dn
ν


Comments after the workshop discussions
1. Using Quadrupole transfer function to measure the beam emittances
W. Guo and S.Y. Lee, Quadrupole-mode transfer function and the
nonlinear Mathieu instability, Phys. Rev. E65, 066505 (2002)
2. Using the measured betatron tune to determine the ring impedance
3. Once the barrier bucket is implemented, there is an opportunity to study
head-tail effects of high intensity beams.