physics of and with electron beam ion traps · 2018. 11. 14. · electron-ion interactions number...

Physics of and with Electron Beam Ion Traps and Sources

EBIS/TFred Currell

Centre for Plasma PhysicsQueen’s University, Belfast

Overview

• Trapping the ions• Electron ion interactions to build the

charge balance• Energetics of the ions in the trap• Controlling the EBIST to study physics of

electron - ioninteractions– Electron impact ionisation– Dielectronic Recombination

• Summary2

Everything should be made as simple as possible, but not one bit simpler.

Albert Einstein, (attributed)

Laplace’s equation

Poisson’s equation

Why are Highly Charged Ions Interesting?

H He+ U91

+

Binding energy

Orbital velocity of the 1s electron

Relativistic

0 20 40 60 80 100Z0

5.0�107

1.0�108

1.5�108

2.0�108

2.5�108

3.0�108

3.5�108v�m�s�

Quantum Electro Dynamics (QED)

In QED, the electromagnetic field is quantized.QED coupling constant is ve/c

QED calculations look like series (Zα)n

Quantum Electro Dynamical

0 20 40 60 80 100Z0.0

0.2

0.4

0.6

0.8

1.0ZΑn

Orbital radius of the 1s electron

Nuclear Size Affected

20 40 60 80Z

1.�10�9

2.�10�9

3.�10�9

4.�10�9

5.�10�9

r�m�

viii THE PHYSICS OF MULTIPLY AND HIGHLY CHARGED IONS

Land of knowledge

Sea of ignorance

Ionic Charge

Atomic Number

092

92

1 Neutral

matter

SSJ

Figure I.1. Parametersation of the world of atomic physics

number) and ionic charge (the difference between the number of electrons and

protons). The claim at the beginning of the preface is that most of chemistry and

a large portion of atomic physics is concerned with 276 distinct species. These

276 species are contained in the thin sliver to the left hand side of this figure

labelled ‘neutral matter’ and two equally thin slivers either side corresponding

to singly charged positive and negative ions. The world of multiply and highly

charged ions is the remainder of this figure. When considering the overlap

between atomic and nuclear physics (for example, hyperfine interactions) one

has to introduce an additional dimension to account for isotope effects related

to variation in the nuclear structure. We will not consider this dimension here.

The figure above is based on a somewhat artistic reproduction of a figure

which has appeared elsewhere in several guises and a very loose interpretation

of what constitutes sufficient knowledge (i.e. the array of transitions elec-

trons bound in the system can undergo has been identified). A less artistic

version of this figure can be found in "On The Structure and Spectra of Mul-

tiply Charged Ions" by I. Martinson and I. Kink, in Trapping Highly Charged

Ions: Fundamentals and Applications, ed. J. Gillaspy, Nova Scientifc (2001).

Notwithstanding the looseness of the interpretation some trends are readily ap-

parent. Most of our knowledge lies to the left hand side. Not surprisingly, these

multiply charged ions are relatively easy to produce. Consequently they are

the most amenable to experimental study. For the more highly charged ions,

Astrophysics

Fusion, Xe, Mo, W ...

QED tests, high Z , few e

Nuclear effects

World of Atomic Physics

Electron Beam Ion Trap (EBIT)

The Electron Beam Ion Trap

V(z

)

ZB

(z)

EnergyDynamics

Ion SpatialDistributions

Charge ChangingReactions

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overlap factors a nd wo rk

done expand ing ion cl oud

temp erature determ

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overlap factors scaleelectron-ion interactions

number of ions normalisesdistributions and effects ion space-charge compensation

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See F Currell + G Fussmann “Physics of Electron Beam Ion Traps and Sources”, IEEE Trans. Plas. Sci. 33 1763-1777 (2005) + refs therein

Typical Time Scales•Radial motion of the ions 10nsec

•Axial motion of the ions 100nsec

•Ion-ion collisions 0.1 - 1msec

•Cross-field diffusion 1 - 10msec

•Charge changing 10msec - 10sec

•Axial escape 100msec - hours

EnergyDynamics



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EnergyDynamics



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‘state’ variablesaxial densitytemperature

index i runs over all charge states of all species in the trap.This is not a unique choice but it provides a computationally convenient description

EnergyDynamics



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The trapping mechanisms for the ions

• Treat electrostatic terms as separable in r and z– good approximation for EBITs, even better for

longer EBISes• Ignore effects of B-field

– OK for ion clouds as crossing B costs no energy

• Include effects of ion charge in a self-consistent way based on Poisson’s equation 23

Trapping Potentials

Exercise 1

Treating it non-relativistically, work out the charge per unit length in a (cylindrical) electron beam of current I and energy E

Do the same for a relativistic beam

Some basic electrostatics from Hyperphysics

Gauss' Law, Integral FormThe area integral of the electric field over any closed surface is equal to the net charge enclosed in the surface divided by the permittivity of space. Gauss' law is a form of one of Maxwell's equations, the four fundamental equations for electricity and magnetism. Gauss' law permits the evaluation of the electric field in many practical situations by forming a symmetric Gaussian surface surrounding a charge distribution and evaluating the electric flux through that surface.

Exercise 2 (from a QUB 1st year tutorial question)

Often fundamental physics investigations involve the interaction of charged particle beams (protons, electrons etc.). In this question you will calculate the electric field due to such a beam. i) Consider an infinitely long, cylindrical, uniform charged particle beam of radius R, with a charge per unit length of Q. What is the charge in a cylinder of unit length and radius r lying along the axis (consider both cases r < R and r > R)?Hint uniform means the charge density is the same everywhere inside the beam. ii) Identify the relevant symmetries and hence determine the direction of the electric field everywhere, due to the charge in the beam. iii) Apply Gauss’s law to work out an expression for the magnitude electrostatic field everywhere (remember two cases r < R and r > R).

Exercise 3

Take the expressions from the last exercise and from them derive an expression for the potential due to the beam we considered in exercise 1. In doing this define your zero of potential at r=0

Radial Trapping Potential

typically about 10 V

Axial Trapping Potential• Treat as a square

well (reasonable approximation)

• Depth of the potential determined by applied voltages

• Needs a small correction for the space-charge of the electron beam This term comes from the requirement

that our radial potential matches the boundary condition at the drift tubes. In the exercises we set the potential on-axis to be zero.

Magnetic effects• Clearly the magnetic field effects the motion of the ions. • However, when collisions are considered, the magnetic field

doesn’t affect the spatial (equilibrium) distribution of the ions– Moving across the lines of B costs no energy– A more sophisticated treatment requires solution of the

Vlassov equation – There is a class of experiments where the e-beam is

turned off -> like a penning trap (already discussed by Jose)

31

[see F. J. Currell “The physics of electron beam ions traps, section 5” in Trapping highly charged ions: fundamentals and applications, ed. John Gillaspy, Nova Science Publishers, ISBN 1-56072-725-X. (1999), G. Fussmann, C. Biedermann, and R. Radtke,“EBIT: An Electron Beam Source for the Production and Confinement of Highly Ionized Atoms” in: ”Advanced Technologies Based on Wave and Beam Generated Plasmas”, Eds. H. Schluter and A. Shivarova, Dordrecht: Kluwer Academic Publishers, 429 (1999)]

Trap terms, a summary

32

RadialAxial

square well, applied voltages plus small correction

Magneticnegligible when ion cloud distributions are considered

Trap terms, a summary

33

Axial

A real ‘static’ trap for allpositive charge states

EnergyDynamics



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Typical Time Scales•Radial motion of the ions 10nsec

•Axial motion of the ions 100nsec

•Ion-ion collisions 0.1 - 1msec

•Cross-field diffusion 1 - 10msec

•Charge changing 10msec - 10sec

•Axial escape 100msec - hours

Suggests that maybe each charge state should have its own temperature but that it is meaningful to ascribe a single temperature for all degrees of freedom for a given charge state

Density Distribution

re

Using the Vlasov equation it is possible to show that due to the effect of ion-ion collisions, the magnetic field has no effect on the ion density distribution at equilibrium. It is determined by the electrostatic potential alone.

Some integrals over the spatial distribution

38

No. of ions in the trap, of species i

No. of ions in the electron beam, of species i

The ratio of the two is called the overlap factor. It acts to scale electron - ion interactions

Poisson’s equation for radial spatial distributions

figure from the Lu paper

40

X. Lu and F.J.Currelll PHYSICAL REVIEW SPECIAL TOPICS - ACCELERATORS AND BEAMS 12, 014401 (2009)

Electron beam: energy, 2.65 keV; radius, 35 um; current 68 mA. The positive charged ions here used were Si ;axial density of 2.5 E+9/cm ; the temperature is 81 eV.

14+

3

How important is the ion space charge?

• Depends on the machine set up. • Compensation for the charge of the

electron beam can range from negligible to close to 100%

• small compensation: shallow axial trap, axial ion escape dominates

• near 100%: deep axial trap, radial escape dominates

41

EnergyDynamics



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Ion Creation + Trapping•Use a high current, high energy electron beam to

• stepwise ionize neutrals/low charged ions • provide radial trapping

•Use a series of drift tubes to provide axial trapping

Overview

• In principle many charge states can be present in both the ground state and excited states

• ionisation moves ions up the charge staircase

• radiative recombination and charge exchange move ions down the charge staircase

• escape allows ions to jump right off• system is fed by source neutral or

lowly charged ions

45

Electrons in a strong fieldA simple picture

Potential due to the nucleus

Kinetic energy of electron

Momentum of electron

Free electron

Bound electronn=1

n=2

n=3

n=4

Ee

Eb

Bound MotionLike planets round the sun

n=1

n=2

n=3

n=4

Free MotionRutherford scattering

n=1

n=2

n=3

n=4

Some Electromagnetic effects

Classically, an accelerated charged particle emits electromagnetic waves.

Quantum mechanically, this results in transitions accompanied by emission of photons.

Transitions between bound states

n=1

n=2

n=3

n=4

hν

Due to high charge multipole contributions are significant

Radiative Recombination

n=1

n=2

n=3

n=4

Ee

Eb

hν

N.B. ion charge reduces by one

Electron Impact Ionisation

n=1

n=2

n=3n=4

Ee

N.B. ion charge increases by one

Charge Exchange

N.B. ion charge reduces by one, or possibly more (multiple electron transfer)

Excited states

• Because of the high charge states we are concerned with, lots of forbidden transitions have a high transition rate

• As a result there are almost no metastable states in highly charged ions– there is a rare exception in Be-like ions

• Hence we can consider all of the ions to be in their ground electronic state

• They might be in excited hyperfine states but that doesnʼt affect the ionisation dynamics

54

Overview• We can model the system by

considering the populations of each charge state

• Can consider ions as being in the ground electronic state

• Ions are not always in the electron beam

• Take this into account using ‘overlap factors’

55

Coupled set of rate equations

56

or diagrammatically like this (includes multiple charge exchange)

ionisation

radiativerecombination

chargeexchange

escape

EnergyDynamics



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Energy mechanisms• Electron-ion collisions

– Electron beam heating• Ion-ion collisions

– Sharing of energy among the ions

– Cross-field diffusion into spatial equilibrium distribution

• Axial escape– Preferential loss of low-

charge/ high-energy ions

– Depletes energy faster than population

p pzr

Electron beam heating• Mostly due to many long range collisions

– Well known in plasma physics (e.g. Spitzer)• For an ion inside the electron beam, the

heating rate is Coulomb logarithm

Ion-ion collisions• Equipartition time for ions of species i interacting

with species j

For similar charge states q-4

This is strongest for highly charged ions

Axial escape• High energy tails of the velocity

distribution escape– Axial component so 1-D

• They are replenished by velocity diffusion caused by collisions

Smaller but cooler

Initial axial velocity

distribution

Ion measurement system

Acceleration Lens

DischargeIon Source (MEVVA)

Ionsfrom the EBIT

Analysing Magnet

Secondary Electron Multiplier(movable)

ElectrostaticBender

QuadrupoleLens

Deceleration Lens

Einzel Lens1

Einzel Lens2

Einzel Lens3

movableAperturefor Beam Profiling

Deflector 2

Deflector 1

Position SensitiveDetector(movable)

To Ion-Surface Experiment

Raw Data

Warning, peak heights can not simply be interpreted as giving the charge balance dynamics without proper consideration of the iontemperatures as the extraction efficiency is temperature dependent

Energy Dynamics•Compare escape rate to number trapped for two or more charge states

•Temperature dynamics

•Ion cloud distributions from Maxwellian

Surprise! kT(t=0) in not 0eV

Evidence for ionisation heating

Density Distribution

re

Due to the effect of ion-ion collisions, the magnetic field has no effect on the ion density distribution. It is determined by the electrostatic potential alone.

Reconstruction

Work done expanding the ion cloud

Work done expanding the ion cloudUndergrad thermo dynamics, C = no. degrees of freedom/2v

5 degrees of freedom1 associated with axial motion (free particle/gas atom)2 for each direction perpendicular to the electron beamc.f. ion in a crystal

3 degrees of freedom1 associated with each direction x,y,zfree particle/gas atom

2nd order phase transition of the electron beam + ion-cloud system

TypicalEBIT

TypicalEBIS

Controllableparameter

Axial escape ratesBoth the number and the temperature decrease exponentially but the time constants are different

Av. Energy of trapped ions Av. Energy of

escaping ions

Pulsed Evaporative Cooling (PEC)

• Gradually lower the trap barrier– Successively cooler ions, using evapouration

• Model the process including– Evaporative cooling equations– Electron beam heating equations– Electron-Ion overlap equations

Set of coupled differential equations which provide a description of PEC

Simulation of PEC

Time through dump [msec]

Rel

ativ

e el

ectro

n-io

n in

tera

ctio

n ra

te

Ba46+, Ee=10keV, Ie=36mA

DumpX-ray emission from Ba46+

Ee=10keV Ie=36mA

Trap refill

Measurement of PEC

Cooling by PEC

Time through dump [msec]

Mea

n fr

ee p

ath

[m]

•Progressive lowering of the trap leads to

•cooling of the ions

• a collapse of the ion cloud

•increased electron-ion interaction rate

•decreased mean free path

Rel

ativ

e el

ectro

n-io

n in

tera

ctio

n ra

tekT

[eV

]

Collapse of the Ion Cloud,‘phase-transition’-like behaviour

EBIST programming• Quasi-equilibrium: set d(anything)/dt = 0

– electron impact ionisation, closed shell dielectronic recombination

• ‘Impulse’: fast changes, assume charge balance and temperatures are constant– open shell dielectronic recombination

• Normalise cross sections/rates using other (well known) rates

78

EnergyDynamics



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Beam energy Ee

Beam current Ie

Substanceinjected (Z)

Trap depth Vt

viii THE PHYSICS OF MULTIPLY AND HIGHLY CHARGED IONS

Land of knowledge

Sea of ignorance

Ionic Charge

Atomic Number

092

92

1 Neutral

matter

SSJ

Figure I.1. Parametersation of the world of atomic physics

number) and ionic charge (the difference between the number of electrons and

protons). The claim at the beginning of the preface is that most of chemistry and

a large portion of atomic physics is concerned with 276 distinct species. These

276 species are contained in the thin sliver to the left hand side of this figure

labelled ‘neutral matter’ and two equally thin slivers either side corresponding

to singly charged positive and negative ions. The world of multiply and highly

charged ions is the remainder of this figure. When considering the overlap

between atomic and nuclear physics (for example, hyperfine interactions) one

has to introduce an additional dimension to account for isotope effects related

to variation in the nuclear structure. We will not consider this dimension here.

The figure above is based on a somewhat artistic reproduction of a figure

which has appeared elsewhere in several guises and a very loose interpretation

of what constitutes sufficient knowledge (i.e. the array of transitions elec-

trons bound in the system can undergo has been identified). A less artistic

version of this figure can be found in "On The Structure and Spectra of Mul-

tiply Charged Ions" by I. Martinson and I. Kink, in Trapping Highly Charged

Ions: Fundamentals and Applications, ed. J. Gillaspy, Nova Scientifc (2001).

Notwithstanding the looseness of the interpretation some trends are readily ap-

parent. Most of our knowledge lies to the left hand side. Not surprisingly, these

multiply charged ions are relatively easy to produce. Consequently they are

the most amenable to experimental study. For the more highly charged ions,

Astrophysics

Fusion, Xe, Mo, W ...

QED tests, high Z , few e

Nuclear effects

World of Atomic Physics

atom

ic n

umbe

r (Z

)

charge (Q)

Fundamentals and Applications

Astrophysics

Fusion, W, Mo, Xe, ...

QED tests, high Z , few

eNuclear effects

H-like Electron Impact Ionisation

At equilibrium, H-like ionisation rate must equal the bare recombination rate,

provided we correct for charge exchange and assume effects of overlap factor and

escape cancel

Electron Impact Ionisation Measurements

H-like Bare

……

Mo data65keV 175mA electron beam

H-like

bare

1-2s

2-3s

18-19s

19-20s

H-like

Bare

Intensity variation with timeMo 64.4keV 170mA

Equilibrium 7-16s

Measurement of the Electron Impact Ionization Cross Section

measuretheory

theory

The Electron Impact Ionization Cross Section of H-like Fe

IRON (Z=26)

B. O'Rourke B et al. J.Phys.B: 34 4002 (2001)

The Electron Impact IonizationCross Section of H-like Mo

MOLYBDENUM (Z=42)

R. Marrs et al Phys Rev A 56 1338 (1997) H. Watanabe et al. J.Phys.B: 35 (2002)

Comparison to theory

89

compilation of all measurements Z>20

Electron Energy/Ionisation Potential

O’Rourke,BE; Watanabe,H; Currell,FJ, “Electron Impact Ionisation of Hydrogen-like Ions” 2003, in “The Physics of Multiply and Highly Charged Ions: Vol.1 Sources, Applications and Fundamental Processes”, Ed. F.J. Currell Kluwer Academic Publishers, ISBN 1-4020-1565-8

Bare Mo time evolutionMo+ and re varied

lines (from slowest to fastest onset) correspond to a beam radius of 66 , 62, 58, 52 and 48 µm respectively

B = 4.0T, Ee= 64.4 keV, Ie = 170mA, Vt = 50V

X.Lu and F Currell, Phys Rev special topics, accelerators and beams, 12 014401 (2009)

H-like Mo time evolutionparameters as per previous

lines (from slowest to fastest onset) correspond to a beam radius of 66 , 62, 58, 52 and 48 µm respectively

B = 4.0T, Ee= 64.4 keV, Ie = 170mA, Vt = 50V


Discrepancy, calls some of the lower charge state ionisation cross-sections into question

Dielectronic recombination,He-like ions

Change beam energy quickly enough: ‘frozen’ charge balance. Correlate detected photons with

beam energy.

Dielectronic Recombination (DR)

• Resonant version of radiative recombination

• continuum electron trapped while another electron in the target is excited

• cross-section/rate can be orders of magnitude greater than for RR

• system stabilises by emission of one or more photons

• Photon(s) emitted, ion charge state decreased by one unit

Dielectronic Recombination (DR)

A q++ e- → [ ]A (q-1)+ ∗∗ → A(q-1)+ n ⋅ hν+

one or more photons: energy gives transition infodown-charged ion: gives total cross-section info

Resonance manifold denoted in inverse Auger notatione.g. KLM = 1s-1 2l 3l

Dielectronic Recombination

Experiment Layout

DR via the x-rays

KLL

KLMKLN

KMM KMN

RadiativeRecombination

He-like system

Scaling law: H. Watanabe et al.J. Phys. B 34 5095-5102 (2001)

Ti data point: B.O'Rourke et al.J.Phys B: 37 2343-2353 (2004)

€

S = 1m1Z 2 + m2Z−2

Fit to

hydrogen and helium like Ti

B. O'Rourke, et al Physical Review A, (2008), 77, 6, 062709

2l2

1s2l

1s2

KLL cascade

KLM cascade

2D Plot of Ti Data

Observe Li-like Ions at Early Times

Use Reject

Run charge dynamics modelonly Ti+ abundance fitted


Run charge dynamics modelonly Ti+ abundance fitted but DR now included


Dielectronic recombination of open L-shell ions

Look at energy dependence of ratios of neighbouring charge states (nq-1/nq), when we hit a DR resonance there

should be a maximum. Infer from extracted ions.

Ion measurement system

Acceleration Lens

DischargeIon Source (MEVVA)

Ionsfrom the EBIT

Analysing Magnet

Secondary Electron Multiplier(movable)

ElectrostaticBender

QuadrupoleLens

Deceleration Lens

Einzel Lens1

Einzel Lens2

Einzel Lens3

movableAperturefor Beam Profiling

Deflector 2

Deflector 1

Position SensitiveDetector(movable)

To Ion-Surface Experiment

Dielectronic recombination of open L-shell ions

Look at energy dependence of ratios of neighbouring charge states (nq-1/nq), when we hit a DR resonance there

should be a maximum. Infer from extracted ions.

Data analysis

Extracted ion energy dependence

Ratios of neighbouring charge states

Unknown, slowly varying with Ee

DR by change in ratio at eqm as a function of energy?

measured, rapidly varying with Ee

measured, rapidly varying with Ee

Corrected for charge exchange/escape

Add to He-like scaling data

Kavanagh,A.P et al, Phys. Rev. A, 81 022712 (2010)

€

S = 1m1Z 2 + m2Z−2

Fit to

and generate similar scaling laws for Li- and Be-like

€

S =1

m1Z2 + m2Z

−2Fit to

Kavanagh,A.P et al, Phys. Rev. A, 81 022712 (2010)

Li-like ions Z-dependence

N. Nakamura et alPRL 100, 7, 073203 (2008)

Z-dependence of GBI in Li-like ions

Be-like Bi line shape

N. Nakamura et al PRA 80, 014503 (2009)

Summary

EnergyDynamics



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We have looked at a model in terms ofelectron/ion interactions

(atomic physics, energetics etc.)

Summary

EnergyDynamics



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This model can provide a reasonable account of the observed evolution of the

charge states in EBIST machines...

... and also provide a framework for the design of experiments

which probe these same electron-ion interactions

Electron impact ionisation Dielectronic Recombination

EnergyDynamics



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Summary

Thank you for your time and attention

physics of and with electron beam ion traps · 2018. 11. 14. · electron-ion interactions number...

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