The Paul Trap Simulator Experiment:The Paul Trap Simulator Experiment: Studying Transverse Beam Dynamics in a
Compact Laboratory Experiment*Compact Laboratory Experiment
Erik P. GilsonP i t Pl Ph i L b tPrinceton Plasma Physics Laboratory
March 15th 2012March 15th, 2012
Presented at:Presented at:Accelerator Seminar
Thomas Jefferson National Accelerator Facility
*This work is supported by the U.S. Department of Energy.1
Thanks!
Summer StudentsAnkit AroraPatrick BradleySteve Crowell
Graduate StudentsMoses ChungMikhail DorfWei Liu
PPPL StaffAndy CarpeRonald C. DavidsonPhilip Efthimion
Andrew GodbehereNate Thomas
Adam SefkowHua WangWei-Hua Zhou
Igor KaganovichRichard MajeskiHong QinEdward Startsev
2
Plasma Science and Technology at PPPL
PPPL vision statement
3
PPPL vision statementEnabling a world powered by safe, clean and plentiful fusion energy while leading discoveries in plasma science and technology.
PTSX Simulates Nonlinear Beam Dynamicsin Magnetic Alternating-Gradient Systems
• Purpose: PTSX simulates, in a compact experiment, the transverse nonlinear dynamics ofintense beam propagation over large distances through magnetic alternating-gradient transportsystems.y
• Applications: Accelerator systems for high energy and nuclear physics applications, heavy ionfusion, spallation neutron sources, and high energy density physics.
4
The Goal of PTSX is to Study Key Issues in the Physics of Intense Beams
Issues:
•Beam mismatch and envelope instabilities;
•Collective wave excitations;
•Chaotic particle dynamics and production of halo particles;
•Mechanisms for emittance growth;
•Compression techniques; and
•Effects of distribution function on stability propertiesproperties.
Today: transverse beam compression, the effects of random lattice noise, and transverse beam modes
5
transverse beam modes.
Paul Traps
R Blümel et al Phys Rev A 40 808 (1989)
Paul W., Steinwedel H. (1953)."Ein neues Massenspektrometer ohne Magnetfeld",R Zeitschrift für Naturforschung A 8 (7): 448-450
( )tfV π2cos
R. Blümel, et al., Phys Rev A, 40 808 (1989)
Nobel Prize: 1989
( )tfV π2cos0
( )2V f( ) ( ) ( )0 2 2 22 20 0
cos 22
2trapV f t
x y zr z
πφ = + −
+x
x x⎛ ⎞ ⎛ ⎞
q = 0.908
[ ]2
2 2 cos(2 ) 02
x xd y q yd
z zτ
τ
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟+ =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟−⎝ ⎠ ⎝ ⎠
( ) ( )0
2 2 20 0
42 2
eVq ftm f r z
τ ππ
= =+ 6
A Hyperbolic Potential in Any Pair of Variables Will Do – r-z or x-y
)2cos()2/sin()(4),,(2
1
0 θππ
φ ll
ll
l⎟⎟⎠
⎞⎜⎜⎝
⎛= ∑
∞
= wap r
rtVtyx-V0(t)
2 21( , , ) ( )( )2b ap q
e x y t t x yφ κ= −+V0(t) +V0(t)
02
8 ( )( )
bq
b w
e V ttm r
κπ
=
( ) sin( )V t V tω=
-V0(t)
0 0 max( ) sin( )V t V tω=
The ponderomotive force can be written as where and
ξπ
ωf rm
Ve
wb
bq 2
max 08=22
02 2p
p
EF
εωω
= − ∇uur ur 2p b qF m rω= −
ur r 12 2
ξπ
=
The ponderomotive force… …can be written as… …where… …and
7
Analogy Between AG System and Paul Trap
))((21),,( 22 yxtmtyxe qap −′= κφ
( ) ( ) ( )yxqfocq xyzB eexB ˆˆ +′=( ) ( ) ( )yxqfoc yxz eexF ˆˆ −−= κ
Quadrupolar Focusing
20
)(8)(
wq rm
teVtπ
κ =′
( ) ( ) ( )yxqfoc
2
)()(
cmzBZe
z qq βγκ
′=
[ ]Ze [ ]),,(),,(22 syxAsyxcmZe
zβφβγψ −=
∫ ′′=⎟⎟⎞
⎜⎜⎛ ∂
+∂ fydxdKπψ 2
22
Self‐Forces usual φself(x,y,t)
Field Equations Poisson’s Equation
( ) ( ) ⎫⎧ ∂⎟⎞⎜⎛ ∂∂⎞⎛ ∂∂∂∂ ψψ
∫−=⎟⎟⎠
⎜⎜⎝ ∂
+∂ b
fydxdNyx
ψ22 Field Equations Poisson’s Equation
( ) ( ) 0=⎭⎬⎫
⎩⎨⎧
′∂∂
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+−−′∂
∂⎟⎠⎞
⎜⎝⎛
∂∂
+−∂∂′+
∂∂′+
∂∂
bqq fyyys
xxxs
yy
xx
sψκψκVlasov Equation
The resulting ponderomotive force is a radial linear restoring force with characteristic frequency ωq.
ξπ
ωf rm
eV
wq 2
max 08= 22
1π
ξ = 3423 ηη
ξ−
=for a sinusoidal waveform V(t).
for a periodic step function waveform V(t) with fill factor η.
maxvq
v fσ
ωσ
Smooth Focusing Equilibria are Parameterized by the Normalized Intensity s
( ) ( ) ( )⎥⎥⎦
⎤
⎢⎢⎣
⎡ +−=
kTrqrm
nrns
q
22
exp022 φω
In thermal equilibrium,
( )0
1 s qn rrr r r
φε
∂ ∂=
∂ ∂Poisson’s equation becomes a nonlinear equation for φ
s
that must be solved numerically.
s = 0 0 9 Solutions are
s ν/ν0
s 0…0.9
s = 0.99
0 9999 s = 12pω
characterized by the normalized intensity parameter s.
0.1 0.950.2 0.900.3 0.840.4 0.770.5 0.71
PTSX-accessible
s = 0.9999 s = 1 12 2
Paul trap electrodesPTSX Apparatus
1 22′The PTSX collector disk is a 5 mm diameter copper disk, held at
ion source Collector
))((21),,( 22 yxttyxe qap −′= κφ
ppground, that is mounted to a linear motion feedthrough and movesalong a null of the time‐dependent oscillating potential ±V0(t).
20
)(8)(
wq rm
teVtπ
κ =′ ξπ
ωf rm
eV
wq 2
max 08=
Plasma length 2 m Wall voltage 140 V
Wall radius 10 cm End electrode voltage 20 V
Plasma radius ~ 1 cm Frequency 60 kHz
Cesium ion mass 133 amu Pressure 5x10-10 Torr
Ion source grid voltages < 10 V Trapping time 100 ms10
Electrodes, Ion Source, and Collector
Broad flexibility in applying V(t) to electrodes with arbitrary function
t
Increasing source current creates plasmas with intense space-charge.
generator.g
1.25 in
Large dynamic range using sensitive electrometer.
115 mm
Measures average Q(r).
Plasmas are Manipulated Using anInject-Hold-Dump Cycle
f = 75 kHzT = 13.3 μsVs = 3 V 1.7 ms < 300 ms > 5 ms > 1 ms
12
vz = 2 m/ms
Figures: M. Chung, Ph.D. Thesis, Princeton University (2008).
Transverse Dynamics are the Same – Including Self-Field Effects
•Long coasting beamsIf…
•Beam radius
Instability of Single Particle Orbits – An Early Experiment
• Experiment - stream Cs+ions from source to collector without axial trapping of the plasma.
fπω
• V0(t) = V0 max sin (2π f t)• V0 max = 387.5 V• f 90 kHz
Electrode parameters:
fq 2ω = • f = 90 kHz
Ion source parameters:• Vaccel = -183.3 V• Vdecel = -5.0 V
Mismatch Between Ion Source and Focusing Lattice Creates Halo Particles
“Si l ti ” iText box.
“Simulation” is a 3D WARP simulation that includes injection Streaming- jfrom the ion source.
gmode experiment
Qualitatively similar to C K Allen et al Phys Rev Lett 89
• s = ωp2/2ωq2= 0.6.
• ν/ν0 = 0.63Qualitatively similar to C. K. Allen, et al., Phys. Rev. Lett. 89(2002) 214802 on the Los Alamos low-energy demonstration accelerator (LEDA).
• V0 max = 235 V f = 75 kHz σv = 49o
WARP Simulations Reveal the Evolution of the Halo Particles in PTSX
Oscillations can be seen at both fand ωq near z = 0.
Downstream, the transverse profile relaxes to a core plus a broad,
1/f
pdiffuse halo.
2π/ωq
• s = ωp2/2ωq2= 0.6.
• ν/ν0 = 0 63Go back to s ~ 0.2. ν/ν0 0.63• V0 max = 235 V
f = 75 kHz σv = 49o
Oscillations From Residual Ion Source Mismatch Damp Away in PTSXThe injected plasma is still mismatched because a circular cross-section ion source is coupling to an oscillating quadrupole transport system.
Over 12 ms, the oscillations in the on-axis plasma density damp away…
… and leave a plasma that is
kT = 0.12eV2/2 2
nearly Gaussian.
nearly the thermal temperature of the ion source.
• s = ωp2/2ωq2= 0.2.
• ν/ν0 = 0.88• V = 150 V• V0 max = 150 V
f = 60 kHz σv = 49o
Radial Profiles of Trapped Plasmas are Gaussian – Consistent with Thermal Equilibrium
•Ib = 5 nA
•V0 = 235 V
• thold = 1 ms
• σv = 49o
The charge Q(r) is collected through the 1 cm aperture is averaged over 2000 plasma shots.
•f = 75 kHz • ωq = 6.5 × 104 s-1
( ) ( )plasmaaperturelre
rQrn 2π=
The density is calculated from
2 plasmaaperture
2.02 2
2
==q
psω
ω
• n(r ) = 1 4 × 105 cm-3
18
• n(r0) = 1.4 × 10 cm
• Rb = 1.4 cm
• s = 0.2
PTSX Simulates Equivalent PropagationDistances of 7.5 km
• At f = 75 kHz a lifetime
• s = ωp2/2ωq2= 0.18.
• At f = 75 kHz, a lifetime of 100 ms corresponds to 7,500 lattice periods.
• If S is 1 m the PTSX• V0 = 235 V f = 75 kHz σv = 49o
• If S is 1 m, the PTSX simulation experiment would correspond to a 7.5 km beamline.
19
Phys. Rev. Lett. 92, 155002 (2004).
Transverse Bunch Compression by Increasing ωq
oq
NqkTRmπε
ω4
22
22 +=
If line density N is constant and kT doesn’t change too much, then increasing ωq decreases R, and the bunch i dis compressed.
ξω eV max08= ξπ
ωf rm w
q 2=
Either1 ) i i (i i i fi ld h)1.) increasing V0 max (increasing magnetic field strength) or 2.) decreasing f (increasing the magnet spacing)increases ωq
20
Adiabatic Amplitude Increases Transversely Compress the Bunch
20% increase in V0 max 90% increase in V0 max
σv = 63o σv = 111o
0 9Instantaneous
0 93Adiabatic
0 63Baseline
0 83 R = 0.79 cmkT = 0.16 eVs = 0.18Δ = 10%
R = 0.93 cmkT = 0.58 eVs = 0.08Δ = 140%
R = 0.63 cmkT = 0.26 eVs = 0.10Δ = 10%R kT
R = 0.83 cmkT = 0.12 eVs = 0.20
21
Δε = 10% Δε = 140%Δε = 10%• s = ωp2/2ωq2 = 0.20 • ν/ν0 = 0.88 • V0 max = 150 V • f = 60 kHz • σv = 49o
ε ~R√ kT
Nucl. Inst. and Meth. in Phys. Res. A 577, 117 (2007).
Less Than Four Lattice Periods Adiabatically Compress the Bunch
σv = 111o
σv = 63o
σv 111σv = 81o
22• s = ωp2/2ωq2 = 0.20 • ν/ν0 = 0.88 • V0 max = 150 V • f = 60 kHz • σv = 49o
Phys. Rev. ST Accel. Beams 10, 064202 (2007).
2D WARP PIC Simulations Corroborate Adiabatic Transitions in Only Four Lattice Periods
InstantaneousInstantaneous Change.
Change gOver Four Lattice Periods.
23
Nucl. Inst. and Meth. in Phys. Res. A 577, 117 (2007).
Peak Density Scales Linearly with ωq
2~22
kTR
kTRm qε
ω Constant emittance
.~
~
2 constR
kTR
qω
ε
⇓
emittance
Constant
.~)0( 2 constNRn =
)0(
energy
qn ω~)0(
ξeV max08 ξπ
ωf rm w
q 2max 0=
Phys. Rev. ST Accel. Beams 10, 064202 (2007).24
Increasing ωq adiabatically by decreasing f
300 1.5
( ) ( )VV φi
( ) ( )tVtV φsinmax 0=
hase
(rad
)
150
200
250
plitu
de (a
rb)
0.0
0.5
1.0
πφ2
)(t&
( ) ( )tVtV φsinmax 0=( )tφ
0.0 0.2 0.4 0.6 0.8 1.0
ph
0
50
100
0.0 0.2 0.4 0.6 0.8 1.0
Am
-1.5
-1.0
-0.5
200
Time (ms)
100
1 0
1.5
Time (ms)
( )⎥⎦
⎤⎢⎣
⎡+
−−−+= 1
2tanh
22)( 21
τπφ ttt
fftft fif
( ) ( )tVtV φsinmax0=( )tφ
phas
e (r
ad)
50
100
150
kHz
-50
0
50
Am
plitu
de (a
rb)
-0.5
0.0
0.5
1.0
φ )(t&
( ) ( )a0( )tφ
Time (ms)
0.0 0.2 0.4 0.6 0.8 1.00
50
Time (ms)
0.0 0.2 0.4 0.6 0.8 1.0-150
-100
Time (ms)
0.0 0.2 0.4 0.6 0.8 1.0-1.5
-1.0π2
25
Adiabatically Decreasing f Compresses the Bunch
ξeV08• s = ωp2/2ωq2
33% d i f
ξπ
ωf rm
eV
wq 2
max08=p q
= 0.2.
• ν/ν0 = 0.88• V0 max = 150 V
f = 60 kHz 33% decrease in ff = 60 kHz σv = 49o
Good agreement with KV-equivalent beam envelope solutions.
Phys. Rev. ST Accel. Beams 10, 124201 (2007).26
Transverse Confinement is Lost When Single-Particle Orbits are Unstable
( )⎥⎦
⎤⎢⎣
⎡+
−−−+= 1
2tanh
22)( 21
τπφ ttt
fftft fif
πφ2
)(t& tt
πφ2
)(
2
1ff
qv ∝=
ωσ
τ = τccMeasured τc (dots)Set σv max = 180o and solve for τc (line)
60 kH
τcf 0
f0 = 60 kHzf0 = 60 kHz
τ c f
f1 (kHz)
Phys. Rev. ST Accel. Beams 10, 124201 (2007).27
Good Agreement Between Data and KV-Equivalent Beam Envelope Solutions
f 0
τc
τf0 = 0
f = 19 9f0 = 60 kHz
τc τf0 = 19.9
τ = τc
τf0 = 260
Phys. Rev. ST Accel. Beams 10, 124201 (2007).28
The Effects of Noise on Beam Propagation
Trapping/ e (V
)
InjectionTrapping/ Relaxing
Noise Application
Relax/ Dumping
Volta
ge
0 5 10 15 20 25 30 35 40 45Time (ms)
Vary the amplitude of each half-period by
Δmax = 1.5% maximum noise amplitude
an amount chosen from a uniform distribution. 155 V
max p
Vn = 150(1+δn) Volts is the applied waveform
|δn| < Δmax are the random amplitude perturbations
29
amplitude for half-period n 145 V
Noise Drives a Continuous Increase in RMS Radius
bb
r
b
r
bb
NrdrrnrR
rdrrnNw
w
/2)(
2)(
2
0
π
π
∫
∫
=
=
1
1 4
1.6
WARP simulation
bbb)(
0∫
Experiment for 1% uniform noise
0.1
C)
1% error Baseline 10 msec 20 msec 30 msec 1.0
1.2
1.4
Rb/R
b0
WARP simulation
0.01
Cha
rge
(pC
1.4
1.60 5 10 15 20 25 30
0.8
Experiment
1E-4
1E-3
C
0 8
1.0
1.2
Rb/R
b0
30
0 1 2 3 4 5 6 7 8 91E-4
Radius2 (cm2)
0 5 10 15 20 25 300.8
Time (msec)
Phys. Rev. ST Accel. Beams 12, 054203 (2009).
Noise Drives Continuous Emittance Growth
• Continuous emittance growth ~ linear with the noise duration
Experiments WARP 2D PIC Simulations
o
b
Bbq
ibi
b
i
qNTkRmTRTR
πεω
εε
42,
2
22 +==⊥
⊥
⊥
2.5
3.0
Amplitude of uniform noise 0.5 % 1.0 % 1 5 %
2.5
3.0
Amplitude of uniform noise 0.5 % 1.0 % 1 5 %
Experiments WARP 2D PIC Simulations
1 5
2.0
1.5 %
ε / ε
i
1 5
2.0
ε / ε
i
1.5 %
0 5 10 15 20 25 30
1.0
1.5ε
0 5 10 15 20 25 30
1.0
1.5ε
31
0 5 10 15 20 25 30
Duration of noise (msec)0 5 10 15 20 25 30
Duration of noise (msec)
1800 lattice periodsPhys. Rev. ST Accel. Beams 12, 054203 (2009).
Noise Drives the Continuous Development of a Non-Thermal Tail Distribution
10-2
10-1
100
(pC
)
0.5% Noise 0 ms 10 ms 20 ms 30 ms
0 4 8 12 1610-4
10-3
100Q
(r)
10-2
10-1
(r) (p
C)
1.0% Noise 0 ms 10 ms 20 ms 30 ms
A straight line in the log of Q(r) versus r2 plot indicates that the radial profile is a Gaussian function of r
1000 4 8 12 16
10-4
10-3
1.5% Noise
Q(
10-3
10-2
10-1 0 ms
10 ms 20 ms 30 ms
Q(r)
(pC
)
320 4 8 12 1610-4
r2 (cm2)Phys. Rev. ST Accel. Beams 12, 054203 (2009).
Colored Noise with Finite Autocorrelation Time is Less Detrimental Than White Noise
0 5
0.6(b)(a)
0.4
0.5
rge
(pC
)
0.2
0.3
axis
cha
Noise duration = 1 ms Noise amplitude = 1%
0.0
0.1On- Colored noise (τac = 2.5T)
Colored noise (τac = 0.5T) White noise
No noise Colored noise (τac = 2.5T) White noise
0 1 2 3 4 50.0
0 5 10 15 20
Noise amplitude (%) Noise duration (ms)
33Phys. Rev. ST Accel. Beams 12, 054203 (2009).
Colored Noise Can Enhance Halo FormationDuring Beam Mismatch
10020 msec duration
10020 msec duration
Experiments WARP 2D PIC Simulations
10-1
pC)
No mismatch + No noise Mismatch + No noise Mismatch + 1% colored noise (τac= 5T)
10-1
pC)
No mismatch + No noise Mismatch + No noise Mismatch + 1% colored noise (τac= 5T)
10-3
10-2
Cha
rge
(p
10-3
10-2
Cha
rge
(p0 4 8 12 16
10-4
2 2
0 4 8 12 1610-4
2 2Radius2 (cm2) Radius2 (cm2)Beam mismatch is induced by instantaneously increasing the voltageamplitude by 1.5 times, and switching back to the originalvalue after one focusing period
34
value after one focusing period.
Phys. Rev. ST Accel. Beams 12, 054203 (2009).C. L. Bohn and I.V. Sideris, Phys. Rev. Lett. 91, 264801 (2003).
Studies of Beam Modes Begins with Simple Expressions for Two Particular Modes
2/11 ⎞⎛Breathing mode:
2112 ⎟
⎠⎞
⎜⎝⎛ −= sqB
)ωωUsing KV distribution and Quadrupole mode:
2/1
4312 ⎟
⎠⎞
⎜⎝⎛ −= sqQ
)ωωsmooth focusing approximation.
Quadrupole mode:
2
2
2ˆ
q
psω
ω= kHzf
qq 08.16~2
22
πω
=q
kHzfs
6023.0~
0 =kHzf BB 13.15~2
ωπ
ω=
0
kHzf QQ 63.14~2πω
=deg6.48~vσ 35
An Initially Hollow Beam Changes from Hollow to Peaked and Back Again as it Streams From
Source to Collector
120
Source to Collector
80
100
Cur
rent
(pA
)
40
60
20
40
Radius (cm)
-5 -4 -3 -2 -1 0 1 2 3 4 50
ωqttransit = 1.5π36
An ℓ = 1 Dipole Mode Can Be Excited By Masking the Ion Source
2
3
Off-axis Mask 1/26Off-axis Mask 1/27
ion
(cm
)
0
1
Pos
it
-2
-1
ω t
1π 2π 3π 4π 5π 6π 7π 8π-3
-2
ωq ttransit
PTSX operated in “streaming” mode.
ttransit decreased by increasing Injection voltage from 3 V to 30 V.
ωq decreased by raising f from 60 kHz to 90 kHz
V0 is varied to vary ωq
ωqttransit = 1.5π37
Collective Mode Diagnostics Were Installed But Were Not Sensitive to the Modes and Degraded
ConfinementConfinement
38
The Modes Can Be Excited Using Externally Applied Perturbations Near the Mode Frequency
• Sum-of-sines is applied to arbitrary function generator
where f1 is near the mode frequency
)2sin()2sin()( 100 tfVtfVtV πδπ +=)
1 q y
• Typical Operating Parameters
VV 140~0) )%5.0(7.0~ 0VVV
)δ
KHzf 60~0 varying~1fKHzf 600 varying1f
kHzfq 083.16~2
39
The Modes Can ALSO Be Excited Using the Beat Frequency Between f0 and f1
• Sum-of-sines is applied to arbitrary function generator
where f1 is near f0 ± fmode
)2sin()2sin()( 100 tfVtfVtV πδπ +=)
1 0 mode
• Typical Operating Parameters
VV 140~0)
)%5.0(7.0~ 0VVV)
δ
KHzf 60~0 varying~1f
kHzfq 083.16~2 mst onperturbati 30=
40
Beat-Method Frequency Scans With Different Initial Amounts of Space-Charge Attempt to Find
the Space-Charge DependenceChange of on-axis density under different perturbation amplitudes
1 0
the Space-Charge Dependence
0.8
1.0 30ms, 1.0% perturbation
rge/
cycl
e (p
C)
0 4
0.6
increasess
Cha
r
0.2
0.4
initial s=0.44 initial s=0.32initial s=0.23initial s=0.15
Frequency (KHz)
15.0 15.5 16.0 16.5 17.0 17.5 18.00.0
initial s=0.09
qf2
41
Beat-Method with Larger Amplitude Perturbations
Change of on-axis density under different perturbation amplitudes
1.2
1.4
30ms, 1.5% perturbation
cycl
e (p
C)
0.8
1.0≈
≈
≈
≈
Cha
rge/
c
0.4
0.6
initial s=0.44initial s=0.32
increasess≈ ≈
≈ ≈
15.0 15.5 16.0 16.5 17.0 17.5 18.00.0
0.2initial s=0.23initial s=0.15initial s=0.09
qf2
≈ ≈
Frequency (KHz)q
42
The Measured Frequencies Are Larger Than Those Computed in the Simple Model
Collecive Mode Frequency VS Normalized IntensityK
Hz) 16
18
higher peaklower peak2fq*(1-0.5s)
1/2
e Fr
eque
ncy
(K
14
2fq*(1-0.75s)1/2
Col
lect
ive
Mod
e
10
12
Normalzied Intensity s
0.0 0.2 0.4 0.6 0.8 1.0
C
8
Normalzied Intensity s
43
The Linear Drive is Stronger But Does Not Exhibit a Clear Dependence on Space-Charge
Change of on-axis density under different perturbation amplitudes
30ms, 0.5% perturbation1.82.0
cle
(pC
)
1.2
1.4
1.6
initial s=0.44initial s=0.32initial s=0.23initial s=0.15
≈ ≈
≈ ≈increasess
Cha
rge/
cyc
0.6
0.8
1.0 initial s=0.092fq
≈ ≈
≈ ≈
8 10 12 14 16 18 200.0
0.2
0.4≈ ≈
Frequency (KHz)
44
Observed Mode Frequencies are Larger Than Those in Warp or the Simple Model
Mode Frequency vs Normalized Density s
z)
16 experiment30ms, 0.5% perturbation
simulation1ms, 0.5% perturbationfre
quen
cy (k
Hz
12
14
higher peak experiment , p
mod
e f
10
higher peak experimentlower peak experimenthigher peak simulationlower peak simulation2w_q(1-0.5s)^0.52w_q(1-0.75s)^0.5
normalized density s
0.0 0.2 0.4 0.6 0.8 1.08
45
Using a Second Arbitrary Function Generator to Break the Quadrupole Symmetry
( ) ( )∫=π
θθ2
0
cos41 nVAn
h ( )
( )∑ ⎟⎟⎠
⎞⎜⎜⎝
⎛=
n
n
wn nr
rCr θθφ cos),(
‐1At t = 0, with V0(0) = 1(1, ‐1, 1, ‐1), thenA2 = 1A6 = 0.333
If a dipole is applied as (1, 0, ‐1, 0), thenA1 = 0.5A3 = 0.167A5 = 0.111
‐1
0
11
‐1
1‐1
A perturbation (1+δ, ‐1, 1, ‐1), can be decomposed as(1+δ/4, ‐1‐δ/4, 1+δ/4, ‐1‐δ/4) + (δ/2, 0, ‐δ/2, 0) + (δ/4, δ/4, δ/4, δ/4) and then
A0 = δπ/8
0
‐1A1 = δ/4A2 = 1+δ/4A3 = δ/12A5 = δ/20A6 = 1/3 + δ/12
1+δ1A6 1/3 δ/12 The higher‐order terms are less
significant because the contribution of each term is proportional to (r/rw)nAn.‐1 46
ℓ= 1 Perturbations Excite Dipole Modes Near the Dipole Mode Frequency
0.4
0.5
0.4
0.5
axis
Cha
rge
(pC
)
0.2
0.3
axis
Cha
rge
(pC
)
0.2
0.3
0.10%0.30%0.50%
On-
a
0 0
0.1
On-
a
0 0
0.1
0.70%
As expected, the ℓ= 1 dipole mode frequency does not depend on the amount of space-charge.
Perturbation Frequency (kHz)
7.8 8.0 8.2 8.4 8.6 8.8 9.00.0
Perturbation Frequency (kHz)
7.4 7.6 7.8 8.0 8.2 8.4 8.6 8.80.0
p , p q y p p g
Stronger perturbations lead to an unexplained shift in the peak frequency.
47
Dipole Gaussian White Noise Leads to Radius Temperature and, Thus, Emittance Growth
1 )
4x107
5x107
sity 0.20
0.25
0.30
Line
Cha
rge
(m-
2x107
3x107
0.30%0.10%
0.50% 0.70%
Nor
mal
ized
Inte
ns
0.10
0.15
0.10%0.30%0.50%
Perturbation Time (ms)
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40
107%
Perturbation Time (ms)
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.00
0.05 0.70%
2.0
2.5
)0.8
1.0
0.10%0.30%0.50%0.70%
Rad
ius
(cm
)
1.0
1.5
0 10%
Tem
pera
ture
(eV
)
0.4
0.6
2
Perturbation Time (ms)
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.0
0.50.10%0.30%0.50%0.70%
Perturbation Time (ms)
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.0
0.22
2 2 24
bq b B
o
N qm R k Tωπε⊥
= +48
2 5
As Expected the Dipole Noise has a Larger Effect Than the Quadrupole Noise
amet
ers
1.5
2.0
2.5
Line ChargeRadiusNormalized IntensitykTEmittance
Nor
mal
ized
Par
a
0.5
1.0
2 0
2.5
Line ChargeRadiusN li d I i
2.5
Line Charge
Noise Amplitude (%)
0.0 0.1 0.2 0.3 0.4 0.50.0
aliz
ed P
aram
eter
s
1.0
1.5
2.0 Normalized IntensitykTEmittance
ized
Par
amet
ers
1 0
1.5
2.0 RadiusNormalized IntensitykTEmittance
0 0 0 1 0 2 0 3 0 4 0 5
Nor
ma
0.0
0.5
0 0 0 1 0 2 0 3 0 4 0 5
Nor
mal
0.0
0.5
1.0
Noise Amplitude (%)
0.0 0.1 0.2 0.3 0.4 0.5
Noise Amplitude (%)
0.0 0.1 0.2 0.3 0.4 0.5
Using the arbitrary function generators, the dipole noise and the quadrupole noise can be considered separately.
49
The Right Mode Structure and Mode Frequency Excite Dipole and Quadrupole Modes
0.6
(pC
)
0.4
0.5O
n-ax
is C
harg
e
0.2
0.3
Dipole
O
0 0
0.1
pQuadrupole
Perturbation Frequency (kHz)
6 7 8 9 10 11 12 13 14 15 16 17 180.0
Dipole perturbations do not strongly excite the quadrupole mode, an quadrupole perturbations no not strongly excite the dipole mode.
50
Understanding the Statistical Nature of Noise Applied to One Electrode
Variation With New Waveform Generated for Each Shot30 ms (1786 period) Noise Duration
Noise on One Electrode
0.4
0.5C
harg
e (p
C)
0.3
On-
axis
0.1
0.2
Noise Amplitude (%)
0.00 0.25 0.50 0.75 1.00 1.25 1.500.0
51Focus on 0.50% noise amplitude
Time Series and Histogram of On-axis Charge Measurements for 200 Sets of Random Numbers
Time Series for 0.5% Noise0.5
Histogram30
rge
(pC
)
0.3
0.4
20
25
On-
Axis
Cha
r
0 1
0.2
Cou
nt
10
15
Trial Number
0 20 40 60 80 100 120 140 160 180 200 2200.0
0.1
On-Axis Charge (pC)
0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.28 0.32 0.36 0.40 0.44 0.480
5
Trial Number g (p )
As before, this is a predominantly the result of the dipole perturbation.
52
Fix the Waveform and Manipulate the Spectrum to See How the Noise Acts By Coupling to the
ModesModesBefore After
First 15 of 1000 periods First 15 of 1000 periodsM i lManipulate
Example with 10% noise
53First 100 kHz of the FFT of the Waveform
Noise Applied to One Electrode Damages the Beam Through Its Interaction with the ℓ = 1 Dipole
ModeModeOne Set of 0.5% Noise Applied to One Electrode
A Notch Filter Removes Frequencies from Zero to f kHzOne Set of 0.5% Noise Applied to One Electrode
A Notch Filter Removes One Frequency Component
pC)
0.4
0.5
pC)
0.4
0.5
On-
Axi
s C
harg
e (p
0.2
0.3
On-
Axi
s C
harg
e (p
0.2
0.3
0 2 4 6 8 10 12 14 16 18 20
O
0.0
0.1
8.0 8.2 8.4 8.6 8.8 9.0O
0.0
0.1
f (kHz) Frequency (kHz)
54
PTSX is a Compact Experiment for Studying the Propagation of Beams Over Large Distances
• PTSX is a versatile research facility in which to simulate collectiveprocesses and the transverse dynamics of intense charged particle beampropagation over large distances through an alternating-gradientpropagation over large distances through an alternating gradientmagnetic quadrupole focusing system using a compact laboratory Paultrap.
• PTSX explores important beam physics issues such as:p p p y
•Beam mismatch and envelope instabilities;
•Collective wave excitations;
•Chaotic particle dynamics and production of halo particles;
•Mechanisms for emittance growth;
•Compression techniques; and•Compression techniques; and
•Effects of distribution function on stability properties.
55