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Dynamics of Ions in anElectrostatic Ion Beam Trap
Daniel ZajfmanDept. of Particle Physics
Weizmann Institute of ScienceIsraeland
Max-Planck Institute for Nuclear PhysicsHeidelberg, Germany
•Oded Heber•Henrik Pedersen ( MPI)•Michael Rappaport•Adi Diner•Daniel Strasser•Yinon Rudich•Irit Sagi•Sven Ring•Yoni Toker•Peter Witte (MPI)•Nissan Altstein•Daniel Savin (NY)
Charles Coulomb (1736-1806)
The most common traps: The Penning and Paul trap
Penning trapDC electric + DC magnetic fields
Paul trapDC + RF electric fields
Ion trapping and the Earnshaw theorem: No trapping in DC electric fields
A new class of ion trapping devices: The Electrostatic Linear Ion Beam Trap
Physical Principle: Photon Optics and Ion Optics
are Equivalent
Photons can be Trapped in an Optical Resonator
Ions can be Trapped in anElectrical Resonator?
R R
L
V1 V2
V1<V2
V V
Ek, q
V>Ek/q
Photon Optics
Optical resonator
Stability condition for a symmetricresonator:
f4
L
Symmetricresonator
Optical resonator Particle resonator
Trapping of fast ion beams using electrostatic field
Photon optics - ion opticsPhoton optics - ion optics
L
M
V V
Ek, q
V>Ek/q
L=407 mm
Entrance mirror
Exit mirror
Field
fre
e r
eg
ion
Phys. Rev. A, 55, 1577 (1997).
V1
V4
V2V3
Vz
V1
V4
V2V3
Vz
Field free region
Trapping ion beams at keV energies
• No magnetic fields• No RF fields• No mass limit• Large field free region• Simple to operate• Directionality• External ion source• Easy beam detection
Why is this trap different from the other traps?
Detector (MCP)Ek
Neutrals
Beam lifetime
σnv1
The lifetime of the beam is given by:
n: residual gas densityv: beam velocity : destruction cross section
Destruction cross section:Mainly multiple scatteringand electron capture (neutralization) from residual gas.
t0eNN(t)
Does it really works like an optical resonator?
f4
L
f
Vz (varies the focal length)Left mirror of the trap
Step 1: Calculate the focal length as a function of Vz
Number of trapped particles asa function of Vz.
Step 2: Measure the number of stored particles as a function of Vz
Step 3: Transform the Vz scale to a focal length scale
f4L
Physics with a Linear Electrostatic Ion Beam Trap
• Cluster dynamics• Ion beam – time dependent laser spectroscopy• Laser cooling• Stochastic cooling• Metastable states • Radiative cooling• Electron-ion collisions• Trapping dynamics
Ek, m, q
W0
Pickup electrodeWn
Ek=4.2 keV Ar+ (m=40)
2Wn
280 nsT
2930 ns(f=340 kHz)
Induced signal on the pickup electrode.
Digital oscilloscope
Time evolution of the bunch length
The bunch length increases because:
• Not all the particles have exactly the same velocities (v/v5x10-4).
• Not all the particles travel exactly the same path length per oscillation.
• The Coulomb repulsion force pushes the particles apart.
After 1 ms (~350 oscillations) the packet of ions is as large as the ion trap
2220n ΔTnWW
Time evolution of the bunch width
ΔT: Characteristic Dispersion Time
V1 V1
Characteristic dispersion time as a function of potential slope in themirrors.
ΔT=0 No more dispersion??
Steeper slope
Flatter slope
Wn
2220n ΔTnWW
How fast does the bunch spread?
T=15 msT=5 msT=1 ms
T=30 ms T=50 ms T=90 ms
“Coherent motion?”
Expected
Observation:No time dependence!
Shouldn’t the Coulomb repulsionspread the particles?What happened to the initialvelocity distribution?
2220n ΔTnWW
Dispersion
No-dispersion
Asymptotic bunch length
Wn
n
Injection of a wider bunch:Critical (asymptotic) bunch size?
Bunch
length
(s
)
Oscillation number0 1 2 3 X 104
0
0.5
1
1.5
Self-bunching?
Injection of a “wide” bunch
Asymptotic bunch length
n
Q1: What keeps the charged particles together?Q2: Why is “self bunching” occurring for certain slopes of the potential? Q3: Nice effect. What can you do with it?
There are only two forces working on the particles: The electrostatic field from the mirrors and
the repulsive Coulomb force between the particles.
It is the Repulsive Coulomb forces that keeps the ions together.
+ -
(Charles Coulomb is probably rolling over in his grave)
Simple classical system: Trajectory simulation for a 1D system.
W0
const.r
qqV
ij
jiij
Solve Newton equations of motion
<v>, v
L
Higher densityStronger interaction
Ion-ion interaction:
interacti
ng
non-interacting
Stiff mirrors
“Bound”!
interacting
non-interacting
Soft mirrors
Trajectory simulation for the real (2D) system.
Trajectories in the real field of the ion trap
Without Coulomb interaction With Repulsive Coulomb interaction
E1>E2
)xqU(x)qV(x)NqV(x2mp
2mp
Η 21212
22
1
21
0U k)U( 221 xx
1D Mean field model: a test ion in a homogeneously charged “sphere”:
03ερq
k interaction strength( negative k -> repulsive interaction)
for Δx << L, the equations of motion are:
where X is the center of mass coordinate Exact analytic solution also exists.
Ion-sphereinteraction
xk(X)VΔxqpΔ
Δp/mxΔ
What is the real Physics behind this “strange” behavior?
L
Δx
V(X) qNq
ρIon-trap
interaction
Sphere-trapinteraction
Ion-sphere interaction (inside the sphere)
E
x
Δxρ
21/r~r~
mapping matrix M:
1kT-
T/m/mkT-1M
**2
Interaction strength
0
n
nΔp
ΔxM
Δp
Δx
The mapping matrix produces a Poincaré section of the relative motion as it passes through the center of the trap: x
p
Self-bunching:stable elliptic motion in phase space
T: half-oscillation time
m/ηm* 0
0dPdT
TP
ηand
Solving the equations of motion using 2D mapping
Phys. Rev. Lett., 89, 283204 (2002)
Stability and Confinement conditions for n half-oscillations in the trap:
4/ * mkT 0 2
Stability condition in periodic systems:
0dPdT
0
p
x2Trace(M)
For the repulsive Coulomb force: k < 0
0 m/ηm*
0
0
dPdT
T
PηSince
03ερq
k
0
n
nΔp
ΔxM
Δp
Δx
Self bunching occurs only for negative effective mass, m*
The system is stable (self-bunched) if the fastest particles have the longest oscillation time!
English:
Sp
2pLm
4T
Oscillation period in a 1D potential well:
L
m,pS=“slope”
2p2Lm
S4
dpdT
0dpdT
,Lm2p
S if
0dpdT
,Lm2p
S if
2
2
Synchronization occurs only if dT/dp>0
? 0dPdT
0
“Weak” slope yields to self-bunching!
Physics 001
What is the kinematical criterion dT/dP > 0?
Ion velocity
Osc
illati
on t
ime
v1<v2
Tim
e
p=Fc t p=Fc t
<v>
The Coulomb Repulsive Force
221
c Δzqq
F slow fast
dT/dv>0
Is dT/dP>0 (or dT/dE>0) a valid condition in the “real” trap?
Negative mass instabilityregion
dT/dE is calculated on the optical axis of the trap, by solving the equations of motion of a single ion in the realistic potential of the trap.
Impulse approx. works
for repulsive interaction
(k < 0)
Exact solution for any periodic system
22 )()1(
)cos()cos(4
)1(
)sin()cos()(
T
TT
T
TTMTrace
mk /where |Trace(M)|<2
Stable exact condition
|Trace(M)|≥2 Unstable exact
condition
Repuls
ive
Att
ract
ive
03ε
ρqk
4η/mkT- 0 2
Q1: What is the difference between a steep and a shallow slope? Q2: What keeps the charged particles together?Q3: Nice effect. What can you do with it?
High resolution mass spectrometry
Example: Time of flight mass spectrometry
laser
Ek,m,q Time of flight:k2E
mLT
L Δm8mE
1LΔT
k
The time difference between twoneighboring masses increases linearlywith the time-of-flight distance.
Target(sample) Detector
The Fourier Time of Flight Mass Spectrometer
MALDIIon Source
Camera
Laser
Ion trapMCPdetector
Lifetime of gold ions in the MS trap
Excellent vacuum – long lifetime!
Fourier Transform of the Pick-up Signal
.
Resolution: 1.3 kHz, f/f1/300
4.2 keVAr+
f
Dispersive mode: dT/dp < 0
f (kHz)
Self-bunching mode: dT/dp > 0
<3 Hz
tmeas=300 ms
Δf/f< 8.8 10-6
Application to mass spectrometry: Injection of more than one mass
FFT
m<mEk
132Xe+, 131Xe+
“Real” mass spectrometry: If two neighboring masses are injected, will they “stick” together because of the Coulomb repulsion?
Mass spectrum of polyethylene glycol H(C2H4O)nH2ONa+
H(C2H4O)nH2OK+
Even more complicated:
Combined Ion trap and Electron Target
Future outlook:• Complete theoretical model, including critical density and bunch size• Peak coalescence• Can this really be used as a mass spectrometer?• Study of “mode” locking• Transverse “mode” measurement• Stochastic cooling• Transverse resistive cooling• Trap geometry• Atomic and Molecular Physics
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