introduction to qualitative plasma spectroscopy · yr, ictp-iaea workshop, 2012 2 a few textbooks...

1 YR, ICTP-IAEA Workshop, 2012

Introduction to

Qualitative Plasma Spectroscopy

Yuri Ralchenko

National Institute of Standards and Technology

Gaithersburg, MD, USA

IAEA-ICTP Workshop on Fusion Plasma Modelling using Atomic and Molecular Data, Jan 23-27 2012


A Few Textbooks on Plasma Spectroscopy

● H.R. Griem

– Plasma Spectroscopy (1964)

– Principles of Plasma Spectroscopy (1997)

● H.-J. Kunze

– Introduction to Plasma Spectroscopy (2009)

● D. Salzmann

● Atomic Physics in Hot Plasmas (1998)

● T. Fujimoto

– Plasma Spectroscopy (2004)

● R.D. Cowan

– Theory of Atomic Structure and Spectra (1981)


Diagnostics on ITER

© 2011, ITER Organization

The system will comprise about 50 individual measuring systems

drawn from the full range of modern plasma diagnostic techniques,

including lasers, X-rays, neutron cameras, impurity monitors,

particle spectrometres, radiation bolometers, pressure and gas

analysis, and optical fibers.

http://www.iter.org/disclaimer

http://www.iter.org/disclaimer


Life of a photon

plasma

Plasma spectroscopy tries to understand what the number of photons (or energy)

registered at a specific energy/wavelength can tell us about plasma properties

(density, temperature, motion, fields...)


Units

● Energy

– 1 Ry = 13.61 eV = 109 737 cm-1 (ionization energy of H)

– 1 eV = 8065.5447 cm-1

● Temperature

– 1 eV = 11604 K

● Length

– a0 = 5.29 10-9 cm = 0.529 Å (radius of H atom)

● Area (cross section)

– a02 = 8.8 10-17 cm2 (area of H atom)

http://physics.nist.gov/cuu/Units/


Three major sources of photons

A. Free-free transitions (bremsstrahlung)

– Az+ + e → Az+ + e + hν

B. Free-bound transitions (radiative recombination)

– Az+ + e → A(z-1)+ + hν

C. Bound-bound transitions

– Ajz+ → Ai

z+ + hν

Ener

gy

Continuum

Bound states

A

B

C

Ionization energy


Bremsstrahlung (free-free)

● Calculation is straightforward for Maxwellian electrons off

bare nuclei of Z:

● Total power loss

● Multicomponent plasma:

+

E1

E2

,1

33

322

2/1

2

3

0e

ffT

hc

e

eZ

ff TGeT

RyZNN

Ryace

3

2/1

2451051.4cmsr

WNN

Ry

TZ ez

eff

Dominant at longer wavelengths

zi

i

zi

zi

i

zi

zi

i

zi

e

eff

ff

eff

ff

Nz

NzNz

NzHz

,

,

2

,

21;

Maximum emission at eVT

nm

e

620max


Bound-bound

● Bound-bound transitions between atomic states

e-

hν

A. How many ions in the plasma

are in the upper state?

B. What is the probability for an ion

to emit a photon during electron jump? C. Is the energy of the recorded photon

different from that of the emitted one?

These questions are directly related to the environment properties

j

i

Spectral line intensity

Iij=Nj Aij h ij


Spectral Line Intensity

Einstein coefficient or

transition probability (s-1) Upper state density (cm-3)

Photon energy (erg) Energy emitted due to a specific

transition from a unit volume

per unit time

(almost) purely

atomic parameters strongly depends

on plasma conditions ijijjij EANI

Question 1: which physical processes affect populations of atomic states?


Atomic Processes

,...,,ˆ,...',',' cbacba if

initial state final state interaction operator

Most of related physics is inside this matrix element


Hydrogen and H-like ions

Hydrogen atom H-like ion

Radius:

Energy: 1 R

y=

13.6

1 e

V

0

2

~ aZ

nan

2

2

n

RyZEn


Energy structure of an ion

Electrons are grouped into shells nl

(K n=1, L n=2, M n=3,...)

producing configurations (or even superconfigurations)

~Z2

~Z ~Z4

terms levels

electron-nucleus

interaction

electron-electron

interaction

spin-orbit

(relativistic effects)

Every state is defined by a set of

quantum numbers which are mostly

approximate

The only exact discrete

quantum numbers:

total angular momentum J

parity=(-1) l (odd or even)

Z is the spectroscopic charge, Z=z+1


Types of coupling

● LS coupling: electron-electron » spin-orbit

– light elements

● jj coupling: spin-orbit » electron-electron

– heavy elements

● Intermediate coupling: neither is stronger

● Other types of couplings exist

SLJssSllL

...,..., 2121

......,, 21222111 jjJsljslj


Standard notations

● Ar I = Ar0+, Ar II = Ar+, Ar III = Ar2+,…

● Z = z+1 = Zn-N+1: spectroscopic charge

● Configurations: (ground: lowest)

● Terms

– LS coupling (low- to mid-Z elements) 2S+1LJ, 2S+1Lo

J

– jj coupling (heavy elements) ( j1, j

2 )J

– other couplings (LK, jK, L1L

2,...)

● Pauli principle for equivalent (n1l1=n2l2) electrons!

● Seniority etc.

● Examples:

– 1s22s22p2 1D2; 3p4(3P)3d2(3F) 5D

0

– 3d94s (5/2,1/2)3; 2p54f 2[5/2]

3; 3s23p(2Po)4f G 2[7/2]3 (LS1 or LK coupling)

– 3d2: only 1S, 3P, 1D, 3F, 1G are allowed

...21

2211

wwlnln

http://www.nist.gov/pml/pubs/atspec/index.cfm


State mixing

expansion coefficients

He-like Na9+: 1s2p 3P1 = 0.999 3P + 0.032 1P

He-like Fe24+: 1s2p 3P1 = 0.960 3P + 0.281 1P

He-like Mo40+: 1s2p 3P1 = 0.874 3P + 0.486 1P

s-o coupling increases with Z

change of coupling scheme

...,...,,,...,,,...,,,...,, 321 cbacbacbacba


From LS to jj: 1s2p in He-like ions

1P1


Radiative transitions

● Classical rate of loss of energy: dE/dt ~ |a|2, and decay rate ~ |r|2

● Quantum treatment:

– eikr = 1+ikr+… 1 (electric dipole or E1 = allowed)

– Velocity form:

– Length form:

● Spectroscopic charge: Z = ion charge + 1

– e.g. Ar III = Ar2+

– Isoelectronic sequence: different ZN, same number of electrons

● Li I, Be II, B III, C IV,…

i

rki

f ea

if

if r

must be equal for an exact wavefunction (good test)


S, f, and A

● Line strength

– Symmetric w/r to initial-final

● Oscillator strength (absorption)

– gj fij = gi fji (gj = 2Jj+1); dimensionless

– Typical values for strong lines: ~ 0.1-1

● Transition probability (or Einstein coefficient)

– Typical values for neutrals: ~ 108 s-1

j

i

ji

j

iij feVE

g

gA

27103.4

ijji SjriS2

SRy

E

gf

i

ji3

1

jiiijjijij BgBgBc

hA ,

22

3


Selection rules and Z-scaling

● n ≠ 0

– r Z-1 S Z-2

– E Z2 f Z0

– A Z4

● n = 0

– r Z-1 S Z-2

– E Z f Z-1

– A Z

Fundamental law: parity and J do not change

Before: After:

jj JP

phiphi JJPP

Exact selection rules: Pj = - Pi

| J| ≤ 1, 0 0 (Jj+Ji 1)

Pph = -1 Jph(E1) = 1

Approximate selection rules (for LS coupling): S = 0, | L| ≤ 1, 0 0

Intercombination transitions: S ≠ 0 2s2 1S0 – 2s2p 3P1


Some useful info

● “Left” is stronger than “right”

– f( l=-1) > f( l=+1)

– He I

● f(1s2p 1P1 – 1s3s 1S0) = 0.049

● f(1s2p 1P1 – 1s3d 1D2) = 0.71

● Level grouping

– Average over initial states

– Sum over final states

– Example: from levels to terms

– Any physical parameter

s p

d

p

j

i

A

B

i

j

j

j

ijj

BAg

g


Principal quantum number n

● n-dependence for f

● n-dependence for A

● Total radiative rate from a specific n

3

2

12

3

2

5

1

3

2

2

2

1

21

1

245.033

41

1111

33

32

nnnf

nnnf

nnnnnnf

2/9

410106.1

n

ZnAZ

3

2

12

1

nnnA


Forbidden transitions (high multipoles)

● QED: En, Mn (n=1, 2,…)

● E1/M1 dipole, E2/M2

quadrupole, E3/M3

octupole,…

● Selection rules

– Pj Pi

● +1 for M1, E2, M3,…

● -1 for E1, M2, E3,…

– Jph(En/Mn) = n

● M3 and E3 were measured!

● Magnetic dipole (M1)

– Stronger within the same

configuration/term

– A Z6 or stronger

– Same parity, | J| ≤ 1, Jj+Ji 1

● Electric quadrupole (E2)

– Stronger between

configurations/terms

– A Z6 or stronger

– Same parity, | J| ≤ 2, Jj+Ji 2


Scaling in Ne-like ions

Jönsson et al, 2011


Aurora borealis

January 24, 2012


Forbidden transitions: auroras

O I 2p4

3P

1D

1S E2: 5577 Å M1: 6300 Å M1: 6364 Å

Wavelength Transition A(s-1)

2958 1S0-3P2 E2: 2.42(-4)

2972 1S0-3P1 M1: 7.54(-2)

5577 1S0-1D2 E2: 1.26(+0)

6300 1D2-3P2 M1: 5.63(-3)

6300 1D2-3P2 E2: 2.11(-5)

6364 1D2-3P1 M1: 1.82(-3)

6364 1D2-3P1 E2: 3.39(-6)

6392 1D2-3P0 E2: 8.60(-7)


j

i

t ~ 1/A

Why are the forbidden lines sensitive to density?


Forbidden transitions: highly-charged W

W52+

W52+

W52+

W51+

W53+

W53+

W51+

Practically all strong lines are due to M1 transitions within 3dn ground configurations

EBIT spectra: EB = 5500 eV


Atomic Structure Methods and Codes

● Coulomb approximation (Bates-Darmgaard)

● Single-configuration Hartree-Fock (self-consistent field)

– Cowan’s code, online interfaces available

● Model potential (including relativistic)

– HULLAC, FAC, AUTOSTRUCTURE

● Multiconfiguration HF (http://nlte.nist.gov/MCHF)

● Multiconfiguration Dirac-Fock (MCDF)

– GRASP2K (http://nlte.nist.gov/MCHF)

– Desclaux’s code

● Various perturbation theory methods

● B-splines

http://plasma-gate.weizmann.ac.il/directories/free-software/


Atomic Structure & Spectra Databases

● Extensive list

– http://plasma-gate.weizmann.ac.il/directories/databases/

● Evaluated and recommended data

– NIST Atomic Spectra Database http://physics.nist.gov/asd

● Level energies, ionization potentials, spectral lines, transition probabilities

● Other data collections

– VALD (Sweden)

– SPECTR-W3 (Russia)

– CAMDB (China)

– CHIANTI (USA/UK/…)

– Kurucz databases (USA)

– GENIE (IAEA)

– …


Autoionization

1s2 1s2l 2l2l’

IP ~ ¾ IP

Energy of 2l2l’

∆E

∆E

A** → A+ + e


Selection rules

● Examples of AI states: 1s2s2, 1s22pnl (high n)

● Same old rule: before = after

● A** A* + l

– Exact: Pj = Pi; J = 0

– Approximate (LS coupling): S = 0, L = 0

● 2p2 3P 1s + p: parity/L violation!

– BUT: (2p2 3P2 ) = (2p2 3P2 ) + (2p2 1D2 ) + …

– and (2p2 3P0 ) = (2p2 3P0 ) + (2p2 1S0 ) + …

– YET: Aa(2p2 3P1) 0


Autoionization + DR

● Typical values Aa ~ 1013-1014 s-1 and are (almost) independent of Z

● Inverse process: dielectronic capture (DC)

– 1s2 + e → 1s2l3l

● Main decay channels for an AI state

– A** → A+ + e (autoionization)

● DR = DC + RS (1964, Burgess)

● Unlike collisional exc/ion,

DC is a resonant process

A* + hν (radiative stabilization) ∆E

∆E


Collision (excitation)

e

ΔE

ΔE

.

. .

Continuum

Atom

Main binary quantity: cross section (E) [cm2]

Effective area for a particular process

Process rate in plasmas:

dEEfENNsR

E

E

vv][max

min

1

dEfE2

,,

rate coefficient


Basic Parameters

● Typical values for atomic cross sections

– a0 ~ 5·10-9 cm a

02 ~ 10-16 cm2

● Cross sections are probabilities

– Classically:

● Collision strength Ω (dimensionless, on the order of unity):

– Ratio of cross section to the de Broglie wavelength squared

– Symmetric w/r to initial and final states

dEEPEE0

,,2,

EEg

RyaE ij

j

ij

2

0

ρ is the impact parameter


Types of transitions

● Optically(dipole)-allowed

– P·P' = -1 (different parity)

– |∆l| = 1

– ∆S = 0

– (E ∞) ~ ln(E)/E

● Optically(dipole)-forbidden

– ∆S = 0

– (E ∞) ~ 1/E

● Spin-forbidden

– ∆S ≠ 0

– (E ∞) ~ 1/E3

Examples in He I:

1s2 1S 1s2p 1P

1s2p 3P 1s4d 3D

1s2s 1S 1s3s 1S

1s2s 3S 1s4d 3D

1s2 1S 1s2p 3P

1s2p 3P 1s4d 1D


Order of cross sections

● General order

– optically allowed > optically forbidden > spin forbidden

● OA: long-distance, similar to E1 radiative transitions

● The larger Δl, the smaller cross section

Li-like ions: 2s – 2p excitation

Excitation cross sections for ions are NOT zero at the threshold

+ e-


From cross sections to rates

dEeEEmT

dEEfE TE

E

Maxw

E

E

/

2/1

3

8vv

max

min

For (E) = A/E: T

e T

E

v

Rate coefficients for an arbitrary energy distribution function

Often only threshold is important:

dET

EE

TT

ee

e exp1

eji

ie

eji TRyg

a

mTTR

2

0

2/1

8

Effective collision strength:


van Regemorter-Seaton formula

● Optically-allowed excitations Gaunt factor

oscillator

strength

X≡E/ΔEij ij

ij

ij fX

Xg

E

RyaE

2

2

03

8

2

2

14 )ln(1051.6ln

2

3: cmf

X

X

eVEEXXgX ij

“Recommended” Gaunt factors:

XX

Xng

XXX

Xng

ln18.0

2

3,0

ln08.03.0

33.0,02

Atoms:

2ln2

3,22.0,0

ln2

36.0

17.0

11,0

XforXXXng

XnZ

Xng

Ions:


Scaling of Excitations

2

ij

ijE

fE

• n-scaling • n=1

• f~n, E~n-3, ~ n7, ~ n4

• Into high n • f~n-3, E~n0, ~ 1/n3

• Z-scaling • n=0

• f~Z-1, E~Z, ~ 1/Z3, < v> ~ 1/Z2

• n=1 • f~Z0, E~Z2, ~ 1/Z4, < v> ~ 1/Z3


Direct and Exchange

e

ΔE

ΔE

.

. .

e

ΔE

ΔE

.

. .

Direct channel Exchange channel

Not for ΔS≠0 For all cases


Two cases

Ne IX 1s2 1S → 1s2p 1P 1s2 1S → 1s2p 3P

Ne VII 2s2 1S → 2s2p 1P 2s2 1S → 2s2p 3P


Direct and Exchange (cont’d)

1s2

1s2p

2s2

2s2p

Ne IX

Ne VII

He-like ion Be-like ion


Resonances in excitation

e e e

Direct excitation Intermediate states (non-resonance contr.)

Intermediate AI states


Experiment vs theory

Zatsarinny and Bartschat, 2010


Collisional Methods and Codes

● Plane-wave Born

● Coulomb-Born (better for highly-charged ions)

● Distorted-wave method

● Close-coupling methods

– Convergent CC (CCC)

– R-matrix (with pseudostates)

– B-splines

– TDCC

– …

● Relativistic versions available: RCCC, DARC, DBSR…


Ionization cross sections

Z

Lotz formula:

X

X

Z

na

E

IE

I

RyaEn n

n

ion

ln76.2

/ln76.2,

4

42

0

22

0

Same theoretical methods as for excitation: Born, Coulomb-Born, DW, CC, CCC, RMPS…


Excitation-Autoionization

3s23p63d104s2 Xe24+: Pindzola et al, 2011 3p63d Ti3+: van Zoest et al, 2004

IP

When EA is important: • few electrons on the outermost shell • Mid-Z multielectron ions • …but less important for higher Z (rad!)

EA in ionization cross sections is not required for detailed modeling with AI states!


3-Body Recombination

AZ+ + e- ↔ A(Z+1)+ + e- + e-

2/9

21

4

21

2

21

Tvv

nvv

Nvv e


Radiative Recombination

AZ+ + e A(Z-1) + h h = E + I

E

I

Semiclassical Kramers cross section: 2

0

3

5

4

33

64a

IE

Ry

n

ZEKr

Quantummechanical cross section: EGEE bf

nKrph

Cross section Z-scaling:

22

1

ZZ

h


Dielectronic Recombination

A++e A** A++e DC AI

A*+h

Example: Δn=0 for Fe XX 2s22p3 2s22p3 4S3/2 + e 2s2p4(4P5/2)nl 2s22p3 4S3/2 + e 2s2p4(4P5/2)nl 2s22p3 4S3/2 + e 2s2p4(4P5/2)nl

RS

Savin et al, 2004

n ≥ 7


Dielectronic Recombination

A++e A** A++e DC AI

A*+h

Examples: 1s2 + e 1s2pnl 1s2nl + h 1s22s2 + e 1s22s2pnl 1s22s2nl + h 1s22s2 + e 1s2s22pnl 1s22s2nl + h

RS

1s

2s 2p n=0: high n important, Major difficulty: n ∞

n>0: high n not as important

1s2 + e 1s2pnl 1s2nl + h

1s22s + e 1s22pnl 1s22snl + h 1s22s + e 1s23pnl 1s22snl + h 1s22s + e 1s2s2pnl 1s22snl + h


Is DR important?..

RR and DR for O VII (Kunze, 2009)

Answer: YES Burgess (1964) was the first to show importance of DR for solar corona ionization balance

W ions: Te = 9000 eV, Ne = 1014 cm-3

(NLTE-7 Workshop, unpublished)

There exist a number of recommended formulas for rate coefficients of variable quality; Z < 30 (Burgess-Merts-Magee-Cowan, Badnell et al, Mazzotta et al, Hahn, Gu,…)

i e

ii

e

DRT

EC

Texp

12/3


Heavy-particle collisions

In thermal plasmas electrons are almost always more important for excitations than heavy particles Exception: closely-spaced levels (e.g., 2s and 2p in H-like ions) Neutral beams: E ~ 100 keV heavy particle collisions are of highest importance

Charge exchange H + Az+ H+ + A(z-1)+(n) • Very large cross sections > 10-15 cm2

• High excited states populated: n ~ Z0.77

• High l values are preferentially populated

Ionization cross sections


Databases for Collisions

● IAEA (most diverse)

● NIFS (most extensive)

● TIPbase (iron group elements)

● CCC Database (limited but good)

● CAMDB

● …

http://plasma-gate.weizmann.ac.il/directories/databases/


Thermodynamic equilibrium

● Principle of detailed balance

– each direct process is balanced by the inverse

● excitation ↔ deexcitation

● ionization ↔ 3-body recombination

● photoionization ↔ photorecombination

● autoionization ↔ dielectronic capture

● radiative decay (spontaneous+stimulated) ↔ photoexcitation



● Four “systems”: photons, electrons, atoms and ions

● Same temperature Tr = Te

● We know the equilibrium distributions for each of them

– Photons: Planck

– Electrons: Maxwell

– Populations within atoms/ions: Boltzmann

– Populations between atoms/ions: Saha


Ener

gy

Continuum

Bound states

Maxwell

Saha

Boltzmann

Ionization energy


Boltzmann: eT

EE

g

g

N

N 21

2

1

2

1 exp


Planck and Maxwell

● Planck distribution ● Maxwell distribution

dET

EE

TdEEf

ee

M exp2 2/1

2/32/11

12/22

3

TEech

EEB


Saha Distribution

Z Z+1

AZ (+ e) ↔A

Z+1 + e (+ e)

ionization 3-body recombination

autoionization dielectronic capture

i

T

EE

iZZ

T

I

e

e

Z

Z

Z

Z

e

i

e

Z

egg

eNh

mT

g

g

N

N

0

,

2/3

2

1

1 122

Which ion is the most abundant?

11

Z

Z

N

N10~1

e

Z

T

I


Local Thermodynamic Equilibrium

● (Almost) never complete TE: photons decouple

easily

● LTE = Saha + Boltzmann + Maxwell

● Griem’s criterion for Boltzmann: collisional rates >

10*radiative rates

● Saha criterion for low Te:

72/13

01

143 104.1 ZeVTeVEcmN ee

62/12/5143 101 ZeVTeVIcmN eze

H I (2 eV): 2×1017 cm-3

C V (80 eV): 2×1022 cm-3

H I (2 eV): 1017 cm-3

C V (80 eV): 3×1021 cm-3


LTE Line Intensities

● No atomic data (only energies and statweights) are

needed to calculate populations

● Intensity ratio eT

EE

AEg

AEg

AEN

AEN

I

I 21

222

111

222

111

2

1 exp

EgA

Iln

Aragon et al, J Phys B 44, 055002 (2011)

Boltzmann plot


Excitation Deexcitation

Ni

Nj Principle of detailed balance: jiejijei vNNvNN

In thermodynamic equilibrium:

T

E

g

g

N

N ij

i

j

i

jexp

T

Evgvg

ij

jijiji exp

''''22

'

2/1

2/1

0

2/1

2/1

dEeEEm

EegdEeEE

m

Eg T

E

jiT

E

jT

E

ij

E

i

Substitution: E E+

dEeEEgdEeEEEg T

E

jijT

E

ijiji

00

Must be valid for any T, therefore: EEgEEEEg dxc

ijj

exc

iji

Only Maxwell is needed here!


Other direct inverse

● A + e A+ + e + e (ionization and 3-body recomb.)

● A + h A+ + e (photoionization and

photorecombination)

T

Ig

mTg z

brziz expvv2

2v 3211

2/3

2

zprzpiz IEhEgh

Emchg ,

2122

2

Milne formula

Conclusion: only direct cross sections are sufficient


Saha-LTE conclusions

● Collisions >> radiative processes

– Saha between ions

– Boltzmann within ions

● Since collisions decrease with Z and radiative

processes increase with Z, higher densities are

needed for higher ions to reach Saha/LTE conditions


Coronal Equilibrium

Luc Viatour / www.Lucnix.be

Small electron densities!

C:/Documents and Settings/yralchen/Desktop/ICTP/snatch.flv.dv


Line Intensities under CE

Ng

N1

Rexc Arad

Balance equation:

ERNEANI

A

NN

A

RNN

ANRN

excgrad

rad

eg

rad

excg

radexcg

1

1

1

v

I Ne

Line intensity does NOT depend on Arad!

small populations!

If more than one radiative transition:

Ng

Nj

jk

kj

ij

jgegijijjij

ji

ij

jgeg

ji

ij

excg

j

ji

ijjexcg

A

ANNAENI

A

NN

A

RNN

ANRN

v

v

Also cascades may be important

303~ N

e

Z ZT

IMost abundant ion:


Ionization Balance in CE

Z Z+1

Electron-impact ionization: Ne

Photorecombination and DR: Ne

DReRRe

ione

Z

Z

vNvN

vN

N

N 1

Independent of Ne!


Ionization Balance in a General Case

Z Z+1

Electron-impact ionization: Ne

Photorecombination and DR: Ne 3-body recombination: Ne

2

Z

Z

N

N 1lg

lgNe

Corona

Ne0 Saha

Ne-1

Ionization from excited states


From Corona to PLTE

Co

llision

al stron

ger Rad

iati

ve s

tro

nge

r

PLTE

Corona

1/n4.5 n4

Griem limit: 17/1

17/2

7.0

140~ e

e

TN

Zn


Time-Dependent Corona

Ng

N1

Rexc Arad

Initial condition: N1(t=0) = 0

tA

rad

excg

radexcg

radeA

RNtN

AtNRNdt

tdN

11

11

Solution:

Linear regime:

tRNtN excg1

Characteristic time depends only on Arad:

rad

radeqA

tt1


General 2-Level Case

Ng

N1

Rexc Arad Rdxc

Balance equation:

dxce

rad

TE

graddxc

exc

g

raddxcexcg

vN

AW

W

e

g

g

AR

R

N

N

ANRNRN

,1

/

11

11

Time-dependent case:

tARR

raddxcexc

exc

raddxcexcexc

g

raddxcexcg

raddxcexceARR

RNtN

ARRtNRNdt

tdN

NNN

AtNRtNRtNdt

tdN

1

~

~

~

1

11

1

111

Again at small t: tRNtN excg1

But: raddxcexc

eqARR

1


He-like lines and satellites

O.Marchuk et al, J Phys B 40, 4403 (2007)


Energy levels in He-like Ar

● Ground state: 1s2 1S0

● Two subsystems of terms

– Singlets 1snl 1L, J=l (example 1s3d 1D2)

– Triplets 1snl 3L, J=l-1,l,l+1 (example 1s2p 3P0,1,2)

● Radiative transitions within each subsystem are

strong, between systems depend on Z


He-like Ar Levels and Lines

1s2 1S0

1s2s 1S0

1s2p 1P1

1s2s 3S1

1s2p 3P0

1s2p 3P1

1s2p 3P2

W

X

Y Z

E1

2E1

Line He0 Ar16+ Fe25+ Kr34+

W 1.8(9) 1.1(14) 4.6(14) 1.5(15)

Y 1.8(2) 1.8(12) 4.4(13) 3.9(14)

X 3.3(-1) 3.1(8) 6.5(9) 9.3(10)

Z 1.3(-4) 4.8(6) 2.1(8) 5.8(9)


Z-scaling of A’s

● W[E1]: A(1s2 1S0 – 1s2p 1P1) Z4

● Y[E1]: A(1s2 1S0 – 1s2p 3P1)

– Z10 for low Z

– Z8 for large Z

– Z4 for very large Z

● X[M2]: A(1s2 1S0 – 1s2p 3P2) Z8

● Z[M1]: A(1s2 1S0 – 1s2s 3S1) Z10


n=2 populations


Ar XVII Line Ratios


1s2lnl satellites

• 1l2l2l’ • 1s2s2: 2S1/2

• 1s2s2p: • 1s2s2p(1P) 2P1/2,3/2

• 1s2s2p(3P) 2P1/2,3/2; 4P1/2,3/2,5/2

• 1s2p2

• 1s2p2(1D) 2D3/2,5/2

• 1s2p2(3P) 2P1/2,3/2; 4P1/2,3/2,5/2

• 1s2p2(1S) 2S1/2

• 1s2lnl’ • Closer and closer to W • Only 1s2l3l can be reliably resolved • Contribute to W line profile


Temperature diagnostics with DS

Ionization limit of 1snl

Ionization limit

of 1s2nl

Li-like

He-like

1s2

1s2p

Ionization limit of 1s2l Excitation rate for 1s2p ~

DC rate for 1s2l2l ~

2/1T

e T

EW

1s2p2l 1s2p3l

2/3T

e T

Es

Reminder: for (low-density) coronal conditions line intensity = population influx Therefore:

TT

T

E

I

I

W

s 1~

exp

Independent of ionization balance since the initial state is the same!


Temperature dependence: Ly satellites

H-like Ne X 100 eV

130 eV

160 eV

Ly

1s2l-2l2l

1s1/2-2p1/2

1s1/2-2p3/2

1snl-2l nl, n=2,3,4,…


Density diagnostics with DS

Ne X Ly and satellites 1snl-2pnl 1019

1021

3×1020

1020

Ly

B

A C A. 1s2s 3S1 – 2s2p 3P0,1,2

B. 1s2p 3P0,1,2 – 2p2 3P0,1,2

C. 1s2p 1P1 – 2p2 1D2 (J satellite)

Collisions Dielectronic Capture

Radiative Decay

1s 2S

1s2p 3P

2s2p 3P 2p2 3P