National Research Center

“Kurchatov Institute”

Moscow, Russia

2-nd Tungsten meeting, Vienna, Oct.6-8,2014

Statistical Theory for Radiative-

Collisional Processes with Heavy Ions

in Plasmas

V.S. Lisitsa, A.V. Demura, M.B. Kadontsev, V.A. Shurygin

Numerical codes for atomic processes and radiative losses

calculations • Kinetic codes

• D.E. Post, R.V. Jensen, C.B. Tarter, W.H. Grasberger, W.A. Lokke, At. Data and Nucl. Data Tables 20, 397 (1977).

• V.M. Leonov, V.E. Zhogolev, Plasma Phys. Control. Fusion 47, 903 (2005).

• Complex codes: atomic structure +elementary processes +kinetics

• H.P. Summers, ADAS users manual, JET –IR 06 (Abingdon: JET Joint undertaking) (1994)

• D. Post, J. Abdallah, R. E. H. Clark, N. Putvinskaya, Physics of Plasmas 2, 2328 (1995)

• K. Asmussen, K.B. Fournier, J.M. Laming, J.F.Seely, R. Dux, W. Engelhardt, J.C. Fuchs, Asdex upgrade team, Nucl. Fusion 38, 967 (1998).

• R. Neu, R. Dux, A. Kallenbach, T. Putterich, M. Balden, J.C. Fuchs, A. Herrmann, C.F. Maggi, M. O’Mullane, R. Pugno, I. Radivojevic, V. Rohde, A.C.C. Sips, W. Suttrop, A. Whiteford, Asdex upgrade team, Nucl. Fusion 45, 209 (2005).

• T. Putterich, R. Neu, R. Dux, A.D. Whiteford, M.G. O’Mullane, H.P. Summers, Asdex upgrade team, Nucl. Fusion 50, 025012 (2010).

• J. Abdallah Jr, J. Colgan, R.E.H. Clark, C.J. Fontes, H.L Zhang, J. Phys. B 44, 075701 (2011)

Reasons for data discrepancy between

different codes

• There are no universal model of atomic processes for total temperature range : 1eV – 40 keV

• 1 keV < T < 40 keV – Coulomb-Born

• 100 eV < T < 1 keV- renormalized Born- scattering matrix

• 10 eV < T < 100 eV- distorted waves

• 1 eV < T < 10 eV - close coupling – number of states?

• Strong discrepancy in ionization balance description as well radiative losses

Statistical approach vs

detail calculations

• Analog: plasma oscillation – molecular

dynamics;

• Manipulation not by specific probabilities

(lines) but by envelopes (averaged over

configurations by statistical method);

• Statistical Atom = collection of classical

oscillators with local plasma frequencies

and oscillator strengths.

Statistical models

• Looking for scaling

•Atom as a system of classical oscillators

• Oscillator strength f_ij is expressed in terms of atomic electron density

•Local Plasma frequency method (LPF) – Brandt-Lundquist (1966) as a method for

atom response calculation on external actions: atomic transition frequencies are

equal to local plasma frequencies

Photoabsorption cross section s(w) is a functional from electron density

( ) 24iff n r r drp= × × ×

( ) ( ) ( )( ) ( )2 2 2 23 22 2

4( )

abs p

p

r r

n re ed r n r r r

d rmc mc

drw

ww

p ps w d w w p

w

=

æ öç ÷ç ÷

= - = × ×ç ÷ç ÷ç ÷è ø

ò

Local plasma frequency model (LPF) in Thomas-

Fermi statistical model

2

2 42

0

( , )

( , )3/

16 ( , )1

( , )

B L

ph

a

x qx

x qeu a u

Z c x q

x x q

ww

w

w

w w

ccw p

sw c

c

-¢æ ö

= = × × ×ç ÷è ø -

¢16

u a uZ c

where хw is determined by the solution of equation

3/41/2

2

( )128, /

9TF

a

xu x r r

Z x

ww w

w

cww p

æ öæ ö= = × =ç ÷ç ÷è ø è ø

c(хw , q) is the solution of TF equation for the ion with a nuclear charge

Z and ion charge qZ

3/2

1/20 0

0

( , ) 1, 0( , )

( , )) ,( , ) 0, ( , )

x q xx q

x q qx q x qx

x

cc

cc c

® ®ì = í ¢= = -ïî

E.Fermi equivalent photon (EPh)

method in electron excitation of

complex ions • The electric field of impact electron on the

ion is equal to the action of equivalent photon flux (Fermi equivalent photon method)

• Electron excitation = equivalent photon absorption (photoabsorption cross section)

• Photoabsorption cross section→

• Quasicontiuum spectra

Fermi EPh method for calculations of atomic excitation and

ionization processes

( ) 3/2

3/2

2

2

0

[ ( / )] 2

2 2 2( / )

Coulombeffua E

RyaZ s

e a T

Z ZdI Z Z Rydu e g s u

TRyn d

a

w ww

w w

¥- -

æ ö× ×ç ÷è ø

ì üï ï æ öé ù< >ï ï æ ö× × × × ×ç ÷ê úí ý ç ÷ç ÷æ ö× è øê úï ï ë ûè øç ÷ï ïè øî þ

ò2 2

du2 2

ò2 22 2

-

Radiative losses in statistical model =

absorption of EPh

2

/ 243

0

0

3/2

3/2

2

( , )

( , )3 1 2/ ( ) (2 ) ( )

16 6 ( , )1

( , )

2

2 2

Z

ssI Z Ry

s

abs e a

s

s s

effu

RyZ s

T

x qx

x qRyQ n a Ry Z ds s

T x q

x x q

Z Z Rydu e g s u

T

ccp

wp c

c

×

¥- -

æ ö× ×ç ÷è ø

¢æ ö= × × × × × × × × ×ç ÷ç ÷

è ø -¢

æ öé ùæ ö× × × × × ×ç ÷ê úç ÷ç ÷è øê úë ûè ø

ò

ò

Total losses need for the sum over ionization balance

/ 2

3 2

0 02

0

3/2

3/2

2

1 2/ ( ) (2 ) ( ) [ ( ) / ]

6

2

2 2

ZI Z Ry

abs e a photo

effu

RyZ s

T

c RyQ n a Ry Z ds s a

e T

Z Z Rydu e g s u

T

w sp

×

¥- -

æ ö× ×ç ÷è ø

æ ö= × × × × × × × ×ç ÷ç ÷

è ø

æ öé ùæ ö× × × × ×ç ÷ê úç ÷ç ÷è øê úë ûè ø

ò

ò

1 21 23 23 23 23 23 2æ öæ öæ ö1 21 2c Rc Rc R3 23 23 21 21 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 2

In the frame of LPF model

Ionization balance = Gaussian shape for W

0.00 0.05 0.10 0.15 0.2010

-5

10-4

10-3

10-2

10-1

100

4030201510Te=5eV

rela

tive a

bundance, arb

.un.

x = k / Z

0.0 0.1 0.2 0.3 0.4 0.510

-5

10-4

10-3

10-2

10-1

100

10003001003010

rela

tive a

bundance, arb

.un.

x = k / Z

Rates are taken from

K. Asmussen et al. Nuclear Fusion 1998, vol. 38, No 7,

p.967.

Comparison with numerical codes

Comparison with kinetic codes

100

101

102

103

104

105

1E-32

1E-31

1E-30

LPF

CA-LARGE

ADPAKZIMPUR

From Polevoy

EM

Radiative-Collisional processes

in Statistical Theory

• All atomic processes can be expressed in

terms of transition energies and oscillator

strengths determined by atomic plasma

parameters related to electron density

distribution inside atoms; it is the case for

DR expressed in terms of radiative and

autoionization decay rates connected with

the same parameters.

Electron Impact Ionization=

Photo-Ionization by Equivalent

Photon Fermi • The statistical theory takes into account in a

specific manner collective effects related

• to excitation-autoionization channels by

excitation of plasmons with frequencies

lager than ionization potential

Tests for W, Xe, Kr, Fe, Ar and U IMPACT ELECTRON IONIZATION RATES AND CROSS SECTIONS

W ionization cross sections W ionization cross sections

Xe ionization cross sections Kr ionization cross sections

Fe ionization cross sections Ar ionization cross sections

U ionization cross sections DR-rates:Burdgess+General

expression

( )( )

3/2 2

2

2exp exp

2

R A

DR

n R A

W W nE z

T T W W n n T

pa

æ öDæ ö æ ö= - ç ÷ç ÷ ç ÷ +è ø è ø è øå

DR rate for W-29+ DR rate for W-20+

Spectral distribution of

radiative losses –

quasicontinuum spectra of W-

MD

Spectral distribution of

photoabsorption cross

section=envelope of spectral lines

( ) ( )

( ) ( ) ( )( )

( ) ( )

( ) ( )( )

( )

3

2

22

4

4;

in n

n

p

p

ph p

g f

g n r r d r

e n rr

m

n rr r

c n r

ww w

w

w d w w

w d w w

pw

p ws w w w

= -

= -

=

= =¢

å

ò

Slater orbitals – more realistic case

• If plasma frequency is lager than its

maximum value coming from Slater

distribution – sharp cut of the spectra at

large frequencies (small wave lengths)

( ) ( ) ( )2 2

0 0 exp 2nn r n R r n r rk g= = -

0

2 , ;

;

ii iI I ionization potential

n normalization factor

outer shell parameter

g

k

= - -

- -

- - -

Quasicontinuum spectrum of W-22-W-27

ions near 50 A spectral range (compare

with modern modeling of LHD-spectra)

Peaks

correspond to

small shifts of

plasma

frequencies

for different

ions

5 6 70

1

2

3

4spectrum from PLT, T

e=750 eV

theory

(Te=700 eV, k=4.5, R=200)

inte

nsi

ty [

a.u]

wavelength [nm]

Quasicontinuun spectra from

Alcator-C Mod and ASDEX Upgrade

4.5 5.0 5.5

1

2

3

4

5

theory

(Wtot

, k = 4.5)

ASDEX Upgrade

Alcator-C Mod

spec

tral

rad

iance

[a.

u.]

wavelength [nm]

Conclusion

• 1. Statistical models for atomic processes: ionization cross sections and, rates DR rates are in good correspondence with detail calculations and experiments

• 2. Quasicontiuum spectra of heavy ions seems to be collective plasma effects of atomic shell – envelope of spectral lines arrays

• -3. Resume: the precision of statistical model for atomic processes and radiative losses of plasma with heavy ions is of the same magnitude as in detail calculations

Tungsten spectra recorded at

the LHD and comparison with

calculations

C S Harte1, C Suzuki2, T Kato2,

H A Sakaue2, D Kato2, K Sato2,

N Tamura2, S Sudo2, R

D’Arcy1, E Sokell1, J White1

and G O’Sullivan1

1 University College Dublin,

Belfield, Dublin 4, Ireland

2 National Institute for Fusion

Science, 322-6 Oroshi-cho, Toki

509-5292, Japan

J. PhysB,2010, 205004 (14pp)