The Paul Trap Simulator Experiment:The Paul Trap Simulator … · 2012. 3. 15. · Paul trap electrodes PTSX Apparatus 1 ′ 2 2 The PTSX collector disk is a 5 mm diameter copper

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.


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)


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


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 %


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


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


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%


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


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%


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


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

Documents

The Paul Trap Simulator Experiment:The Paul Trap Simulator … · 2012. 3. 15. · Paul trap electrodes PTSX Apparatus 1 ′ 2 2 The PTSX collector disk is a 5 mm diameter copper