Impact Ionization and Excitation of Multicharged Ions in ...euvlsymposium.lbl.gov/pdf/2005/poster/2-SO-43 Zakharov_Keldiysh I… · 4th International Extreme Ultra Violet Lithography

4th International Extreme Ultra Violet Lithography (EUVL) Symposium, 07-09 November 2005, San Diego, California

Impact Ionization and Excitation of Multicharged Ions in DPP & LPP

V.S. Zakharov 1*, V.G. Novikov 1, S.V. Zakharov 2*

1 Keldysh Institute of Applied Mathematics RAS, 125047 Moscow, Russia2 EPPRA sas, 91961 Courtaboeuf, France

* also RRC Kurchatov Institute, 123182 Moscow, Russia

[email protected]


Correct calculations of ionization & excitation cross-sections and rate coefficients are one of the critical issues in simulations of ionization state and EUV emission of DPP and LPP sources. Indirect impact ionization processes, for instance impact excitation and subsequent autoionization, give a significant contribution to the total ionization cross-section, up to 70% of it. Cross-sections of inelastic electron-ion interaction, the ionization and excitation rate coefficients are examined on the basis of Hartree-Fock-Slater quantum-statistical model using the method of distorted waves (DW) for calculations. Sn+8÷+12,Xe+5÷+11 and Ar+3÷+18 ions are considered in details. Calculation results of total ionization cross-sections are compared to available experimental data. In dense laser plasmas, the density effects exert essential influence on impact processes with respect to calculations for free ions due to variation of the ion potential, energy levels and other effects. Qualitative differences take place in cross-section values of excitation and ionization depending on plasma density. As DW calculation shows, the difference in cross-section value may reach 50% with plasma density variation in the range of 10-6 – 10-1 g/cm3. Discharge and laser plasmas have a non-equilibrium distribution of electrons with presence of fast (high energy) particles. The fast electrons produce significant changes for the ionization balance. DW calculations for argon plasma show that adding of 10keV electrons completely changes the impact rate coefficients especially below 100eV plasma temperature.The ionization and excitation rates for multi-charged ion plasma of Sn and Xe, producing a high emission in EUV, obtained with DW method exceed considerably the values calculated with well known and usually used formulas.

Abstract

1


ii ir

ai

�ωωωω

dc

drex dex

Ionization

Excitation

Recombination

Deexcitation

Dielectroniccapture and dielectronic

recombination

Autoionization

Multicharged ionsSn(+4÷+13), Xe(+8÷+13)

Electron - ion impact processes

2


In the model of average ion, the system of level-kinetics equations by the states for the average population is the following:

ν

ννN

νννν

νν LNSgN

dtdN −��

�

��

�−= 1

( ) ( ) ( )dcphrirdexemexabs ZNNS ννννµ

µνµνµνµ

µνµνµν ααααααα ++++++= ��><

0

( ) ( ) aiphiiiexabsdexem

g

N

g

NL ννν

νµνµνµ

µ

µ

νµνµνµ

µ

µν ααααααα ++++

��

�

�

��

�

�−++

��

�

�

��

�

�−= ��

><

11

Here is the total rate of processes in 1/sec, leading to an increase of electron population in the state . , is the total rate of processes, leading to a decrease of electron population in this state. The state means a single-electron state with indexes : , where is principal quantum number, . is orbital quantum number, is quantum number of full moment of electron (accordingly ).

νSν νL

nl j

,

,

Level-kinetics equations

nlj=ν''' jln=µ

3


Method of distorted waves

)(',;, 00000000012

)12)(12)(12)(12( 0000 κλεελκ

κκκ

λλnlln

d Rllllll

P ×��

��

�×��

�

��

�×

+++++=

Here is the energy of colliding electron, are quantum moments of colliding electron before and after impact. Direct and exchange radial integrals , may be introduced through 3jm and 6jWigner coefficients as follows:

[ ]� +=κ

κκαα σσσ ),(''),(' 00,0llllex

� ��

��

��

�

��

�

+=

� ��

��

��−�

�

��

�

+=

λλ κκκκ

λλ κκκκκκ

πε

πσ

πε

πσ

, ''

, ''

0

0

22

0

200

2

0

200

2)12(216),(''

2)12(216),('

e

dd

Pl

all

PPPl

all

Cross-sections of impact processes may be obtained from single-electron approach and perturbation theory. For the transition in the ion from state to state (with electron excitation )

α 0αnlln →00

Here and are single-electron cross-sections, at that includes direct andinterference parts, is the exchange part:

),(' 0 llκσ),('' 0 llκσ

),(' 0 llκσ),('' 0 llκσ

,

,

ε λλ ,0 dPκeP κκ '

I.I. Sobelm

an, L.A. V

aishtein, E.A

. Yukov, E

xcitation of atoms and broadening of spectral lines. S

pringer, 1980. 4a



The rate coefficient of the impact process may be calculated from the cross-section in that way

)'(',;,

'

0000

000

00

'000

'000

'

)12)(12)(12)(12)(12(

κλεελ

κκ

λκλκλκλκ

λλκ

nlln

e

Rl

lll

llP

×�

��

×��

��

�×��

�

��

�×

×+++++=

.

( ) ( ) ( ) ( ) ( ) −= +>

<� � ''''''''' 1,;, drdrrRrRrr

rRrRR δγκ

κ

βακ

δγβα

Slater’s integrals include radial parts of single-electron wave functions of discrete and continuous spectra.

( ) ( ) ( ) ( )rRrRrRrR δγβα ,,,

,)()(2000

εεεσεσα ααεααεαα dFnvnE�∞

∆

>=<=

where v is the velocity of the free colliding electron, n�

is the electron density, �E is the threshold or transition energy, F(�) is the function of distribution of free electrons by energy, e.g. Maxwell function. Integrating is carried out by the energy of the free electron before impact.

4b



Similarly, the expressions for the ionization cross-section are obtained

( ),''

''

''

0

0

0ε

εεσ

σεε

αααα d

d

di

�−

=

here is the energy of ionization.iεDifferential cross-section of ionization may be written in the form of the sum of direct and

interference, and exchange parts:

( ) ( )[ ]

( ) ( )

( ) ( ) '.'2122

16*,''

,''2122

16*,'

,*,''*,'

2

''*,,

2

0

200

''*,,

2

0

200

00

0

0

0

επε

πλσ

επε

πλσ

λσλσσ

κκκλλλ

κ

κκκ

κκλλλ

κ

κκκ

αα

dPl

ald

dPPPl

ald

ldldd

e

edd

��

��

��

��

�

+=

��

��

� −��

��

�

+=

+=

ΣΣ

ΣΣ

Σ

Where is a quantum moment of the ionized electron, is the energy of the ionized electron.Direct and exchange radial integrals may be calculated the same way as for excitation. The threshold energy in this approach is equal the energy of ionization .

*λ ''εiε 4c


Cro

ss-s

ectio

n, ππ ππ

a 02

Cro

ss-s

ectio

n, ππ ππ

a 02

Electron energy, eVElectron energy, eV

Effective cross-section of 4d-4f excitation in Xe XI by distorted wave calculation with a various number of terms in sum by ( = 10, 20, 30, 40, 50).

maxκ κ maxκ

Effective cross-section of 4d-4f excitation in Sn X by distorted wave calculation (DW�), Born’sapproach (Born) and Van Regemorter’s formula(Regemorter).

Excitation cross-section

5


Temperature, eVTemperature, eV

Rat

e, 1

/sec

Rat

e, 1

/sec

Rate coefficients of 4d-5p impact excitation process in Xe XI by distorted wave calculation (DWC), Van Regemorter’s formula(Regemorter) and Born’s approach (Born).

Rate coefficients of 4d shell impact ionization process in Xe X by distorted wave calculation (DWC), Born’s approach (Born), Lotz’s (Lotz) and Thomson’s (Thomson) formulas.

Excitation & ionization rate coefficients

6


Ionization cross-section

Impact ionization calculation of cross-section in Xe VII gives lower values than experimental data. The reason of such mismatch is a significant contribution of indirect processes of ionization like the collision excitation of ion into autoionization state and follow-up autoionization.When the outer shell is incompletely populated, like in Xe VIII – Xe XIII, the processes of excitation with further autoionization may significantly increase the value of ionization cross-section. The major contribution is brought by the transition 4d-4f with the threshold energy in the range of 87-108 eV. Also excitations from 4d to 5d and 5f shells (and 5p shell in Xe VI) are important.

Cro

ss-s

ectio

n, ππ ππ

a 02

Electron energy, eV

Ionization cross-sections in Xe VII with the excitation-autoionization process accounting (Total DWC) and without it (Direct DWC) by distorted wave calculations in comparison with experimental data (Experimental).

7a


Ionization cross-section

If the excitation of the ion to autoionization state is weak, i.e. the cross-section of such process is small, total cross-section of the ionization process is fully determined by the process of direct ionization. Similar case takes place in Xe IX. Since in the ground state 4d shell is completely populated, the direct ionization brings the main contribution into the ionization cross-section (ionization threshold is about 170 eV). Excitation processes into autoionization states in this case is improbable.

Cro

ss-s

ectio

n, ππ ππ

a 02

Electron energy, eV

Ionization cross-section in Xe IX by distorted wave calculation (DWC) in comparison with experimental data (Experiment).

7b


Analytical approximations

From a number of approximations for the calculation of the rate coefficients of ionization and excitation in ions we are mark out the following approximation formulas:

for the rate coefficient computation of impact ionization processes (at 1/sec):

/91

2

0

10 ,

,

iie

e A

N a ea

ZN N

A

ε θ θαθ

ρ

−−=+

=

here is the temperature (a.u.), is the ionization energy of the level, and areparameters, is the average ion charge, � is the density in g/cm3, is Avogadroconstant, is the atomic weight;

θ iε1a 2a

0ZAN

A

Impact ionization rate coefficients in Sn XI at density10-5 g/cm3. Distorted wave calculation (DW�), approximation formula (Approximation), Born’sapproach (Born) and Lotz’s formula (Lotz).

Temperature, eV

Rat

e, 1

/sec

8a


where d1, d2, d3 are parameters.

for the computation of the rate coefficient of the impact excitation:

( )9 /1

2

/ 110 ,ex

eN c ec

ε θ ε θ θα

θ− −∆ ∆ +

=+

here is the energy of transition,c1 and c2 are parameters.

In some cases for the better corresponding to numerical calculations it is necessary to introduce an additional parameter:

,103

2/1/

19 2

dedN d

eex

+= −−

θθα θ

ε∆

Impact excitation rate coefficients of transition 4d-4f in Sn XI at density 10-5 g/cm3. Distorted wave calculation (DW�) and approximations.

Rat

e, 1

/sec

Temperature, eV

Analytical approximations

8b


Influence of density effects

Radius, a.u.

r V

(r)

In the computation of the ionization and excitation cross-sections, the single-electron wave functions calculated in self-consisted HFS model are used. The electron density and the potential depend on the plasma density due to variation of the radius of atomic cell. This leads to the shift of the energy levels, to the change of the wave functions and other effects. Thus, the cross-sections of the impact processes depend on the plasma density. Due to the different number of the discrete and continuous wave functions and their combinations in the cross-section calculations it leads to the different behaviors of the ionization and excitation with density. Potential function r V(r) in elementary cell at various

densities (where r is the radius).

9a


Electron energy, eV Radius, a.u.

Radial parts of numerical wave functions of continuous spectra in Sn X ( =7 �.�., l =3) at various densities.

Effective excitation cross-section of 4d-4f transition in Sn X by distorted wave calculation at various densities.

ε

Cro

ss-s

ectio

n, ππ ππ

a 02


R(r

)

9b


Cro

ss-s

ectio

n, ππ ππ

a 02

Electron energy, eV

Effective ionization cross-section of 4d level in SnX by distorted wave calculation at various densities.

Effective excitation cross-section of 4d-5p transition in Sn X by distorted wave calculation at various densities.

Electron energy, eV


Cro

ss-s

ectio

n, ππ ππ

a 02

9c


Fast electrons calculations

Cro

ss-s

ectio

n, ππ ππ

a 02

Rat

e, 1

/sec

Ionization cross-sections of 2p level in Ar XI at density 6x10-6 g/cm3 by distorted wave calculation (DWC) in comparison with Thomson’s formula (Thomson).

Electron energy, eV Temperature, eV

Impact ionization rate coefficients of 2p level in ArXI at density 6x10-6 g/cm3. Distorted wave calculation (DW�), Thomson’s formula (Thomson) with Maxwell and Fast electron’s (nonMaxwell) distributions of electrons and Lotz’s formula (Lotz).

10aWe introduce Fast electron’s distribution like a sum of 90% of Maxwell distribution of free electrons by the energy and 10% of Maxwell distribution of free electrons at the 10 keV temperature.


Fast electrons calculations

Cro

ss-s

ectio

n, ππ ππ

a 02

Rat

e, 1

/sec

Electron energy, eV Temperature, eV

Excitation cross-sections of 2p-3s transition in ArXI at density 6x10-6 g/cm3 by distorted wave calculation (DWC) and by Van Regemorter’s formula (Van Regemorter).

Impact excitation rate coefficients of 2p-3s transition in Ar XI at density 6x10-6 g/cm3. Distorted wave calculation (DW�), Van Regemorter’sformula (Regemorter) with Maxwell and Fast electron’s (nonMaxwell) distributions of electrons and Regemorter’s rate formula (Regemorter).

10b


Near future work directions

11

Building of sets of the approximation parameters for Xe and Sn ions for the impact processes including density dependence.

Extended research of the density effects in the multicharged ion plasma.

Non LTE calculations with non Maxwell electron distributions for Xe, Sn and Ar ion plasmas.

Documents

Impact Ionization and Excitation of Multicharged Ions in ...euvlsymposium.lbl.gov/pdf/2005/poster/2-SO-43 Zakharov_Keldiysh I… · 4th International Extreme Ultra Violet Lithography