Dynamics of Ions in anElectrostatic Ion Beam Trap

Daniel ZajfmanDept. of Particle Physics

Weizmann Institute of ScienceIsrael

andMax-Planck Institute for Nuclear Physics

Heidelberg, 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 OpticsOptical resonator

Stability condition for a symmetricresonator:

∞≤≤ f4L

Symmetricresonator

Optical resonator Particle resonator

Trapping of fast ion beams using electrostatic field

Photon optics - ion optics

L

M

V V

Ek, q

V>Ek/q

L=407 mm

Entrance mirror

Exit mirror

Fiel

d fr

ee r

egio

n

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 =τ

The lifetime of the beam is given by:

n: residual gas densityv: beam velocity: destruction cross sectionσ

Destruction cross section:Mainly multiple scatteringand electron capture (neutralization) from residual gas.

( )τt0eNN(t) −

=

Does it really works like an optical resonator?

∞≤≤ f4L

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 electrodeWnEk=4.2 keVAr+ (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 thesame velocities (∆v/v≈5x10-4).

• Not all the particles travel exactly the same path length per oscillation.

• The Coulomb repulsion force pushesthe particles apart.

After 1 ms (~350 oscillations) the packet of ions is as large as the ion trap

2220n ∆TnWW +=

Time evolution of the bunch width

∆T: Characteristic Dispersion Time

V1 V1

Characteristic dispersion time as a function of potential slope in themirrors.

∆T=0 ⇒ No more dispersion??

Steeper slope

Flatter slope

Wn

2220n ∆TnWW +=

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 ∆TnWW +=

Dispersion

No-dispersion

Asymptotic bunch length

Wn

n

Injection of a wider bunch:Critical (asymptotic) bunch size?

Bunc

h le

ngth

(µs)

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.rqq

Vij

jiij +=

Solve Newton equations of motion

<v>, ∆v

L

Higher densityStronger interaction

Ion-ion interaction:

interac

ting

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

2mpΗ 2121

2

22

1

21 −++++=

0U k)U( +∆=∆ 221 xx

1D Mean field model: a test ion in a homogeneously charged “sphere”:

03ερqk −= interaction strength

( negative k -> repulsive interaction)

for ∆x << L, the equations of motion are:

where X is the center of mass coordinate Exact analytic solutionalso exists.

Ion-sphereinteraction

xk(X)V∆xqp∆∆p/mx∆

∆−′′−≅

≅

What is the real Physics behind this “strange” behavior?

L

∆xV(X) q

Nqρ

Ion-trapinteraction

Sphere-trapinteraction

Ion-sphere interaction (inside the sphere)

E

x

∆xρ

21/r~r~

mapping matrix M:

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

1kT-T/m/mkT-1M

**2

Interaction strength

0

n

n ∆p∆x

M∆p∆x

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛

The mapping matrix produces a Poincarésection 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

TPη =and

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∆

x∆2Trace(M) <

For the repulsive Coulomb force: k < 0

0<−≡ m/ηm*

0

0

dPdT

TPη =Since

03ερqk −=

0

n

n ∆p∆x

M∆p∆x

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛

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

2pLm4T

Oscillation period in a 1D potential well:

L

m,pS=“slope”

2p2Lm

S4

dpdT

−=

⎪⎪⎩

⎪⎪⎨

⎧

<⇒>

>⇒<

0dpdT,

Lm2pS if

0dpdT,

Lm2pS 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

illat

ion

tim

e

v1<v2

Tim

e

∆p=Fc ∆t ∆p=Fc ∆t

<v>

The Coulomb Repulsive Force

221

c ∆zqqF =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()(

TTT

TTTMTrace

ωηηωω

ωηωω

+−

++

−=

mk /≡ωwhere|Trace(M)|<2

Stable exact condition

|Trace(M)|≥2 Unstable exact condition

Repu

lsiv

eA

ttra

ctiv

e

03ερqk −=

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 ∆m8mE

1L∆Tk

=

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/f∼1/300

4.2 keVAr+

∆f

Dispersive mode: dT/dp < 0

f (kHz)

Self-bunching mode: dT/dp > 0

<3 Hz

tmeas=300 ms

∆f/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