Lineshape calculations

Evgeny Stambulchik

Faculty of Physics, Weizmann Institute of Science, Rehovot 7610001, Israel

Joint ICTP-IAEA School on Atomic Processes in PlasmasFebruary 27 – March 3, 2017

Trieste, Italy

Introduction

Spectroscopy is a unique tool for nonintrusive plasmadiagnostics.

Light→ discrete spectrum (energy/wavelength and intensity)→ line width and shift→ line shape

plasma→ electron temperature→ density or iontemperature/velocity→ + electromagnetic fields

Line broadening also affects the radiation transfer and, hence,the level populations in non-optically-thin plasmas.

What is lineshape? Probability to emit (or absorb) photon of agiven energy/frequency/... Usually, normalized to unity:∫

L(ω)dω = 1

One talks about emission or absorption lineshapes, respectively.Often (but not always!) they are the same.

Entities and units

Frequently used entities and units.

frequency ν [Hz]

angular frequency ω = 2πν [rad/s]

wavenumber σ ≡ ν = ν/c [cm−1], [R∞](1 R∞ ≈ 109,737 cm−1)

energy E = hω [eV], [Ry](1 Ry ≈ 13.606 eV)

wavelength λ = 1/σ [Å], [nm](1 nm = 10 Å)

Units: good and bad

X eV ≈ 8065.5 X cm−1 ≈ 2.4180× 1014 X Hz ≈1.5193× 1015 X rad/s ≈ 12,398 X−1 Å

Line-broadening effects are usually much smaller than theunperturbed values, δX� X. Then:

δX eV ≈ 8065.5 δX cm−1 ≈ 2.4180× 1014 δX Hz ≈1.5193× 1015 δX rad/s ≈ 12,398 δX/X2 Å

Expressions involving λ’s are unnecessarily complex and ugly.

Atomic physics is all about energy, not wavelength! Hamiltonian,Lagrangian—all have units of energy.

Below, I will often use word “energy” for ω (h ≡ 1; atomic units).

Atomic physics phenomena

Motion of radiator (Doppler effect)

Magnetic field (Zeeman effect)

Electric field (Stark effect)

Doppler effect

“Static” Doppler shift:

ω0 → ω = ω0 +vqc

ω0 ; ω−ω0 =vqc

ω0 .

vq — projection of the radiator velocity toward the observer. Forbrevity, I omit the q subscript below.

In a plasma, the particles move with different velocities. Hence,

L(ω)dω = P(vω)dv

P(vω) — probability to find radiator moving at such a vω that theDoppler-shifted photon energy is ω.

L(ω) = P(vω)

[dω

dv

∣∣∣∣v=vω

]−1

Thermal Doppler broadening

Thermal motion→ Maxwellian distribution of v:

Pv(v)dv =

√M

2πkTexp

(−Mv2

2kT

)dv ,

L(ω) =1√

2πσDexp

[− (ω−ω0)

2

2σ2D

].

This is a Gaussian with the standard deviation σD =√

kTMc2 ω0.

FWHM =√

8 ln 2 σD ≈ 7.72× 10−5ω0

√T(eV)

M(a.m.u.)

Peak× FWHM = 2

√ln 2π≈ 0.94

Gaussian profiles

-10 -5 0 5 100

0.2

0.4

0.6

0.8σ = 1σ = 2σ = 0.5

Quasistatic/statistical broadening

Average over an ensemble of radiators (atoms/molecules) is called“statistical” or “quasistatic” (QS) broadening.

Thermal Doppler broadening is an example of such a broadening.Recall:

L(ω) = P(vω)

[dω

dv

∣∣∣∣v=vω

]−1

.

More generally,

L(ω) = P(αω)

[dω

dα

∣∣∣∣α=αω

]−1

.

Here, α is a property of the radiator or its surrounding causingenergy shift (for example, electric field due to neighbour plasmaparticles); when α = αω the radiator emits/absorbs photons at ωinstead of ω0.

Microfield distributions

Neglecting interactions between particles, the probability to findnormalized electric field β ≡ F/F0 is given by [Holtsmark, 1919]

H(β) =2π

β∫ ∞

0x sin(βx) exp(−x3/2)dx.

F0 = 2π

(4

15

)2/3

ZpeN2/3p

is the Holtsmark normal field.β� 1⇒ H(β) ∝ β2

β� 1⇒ H(β) ∝ β−5/20 2 4 6 8 10

β0

0.1

0.2

0.3

0.4

Note 1: similar distribution for gravitating masses[Chandrasekhar and von Neumann, 1942].Note 2: plasma coupling alters the distribution.

Linear & quadratic Stark effect

Consider a two-level system.

Perturbation due to the electricfield~F = (0, 0, F):

V = −~d~F = ezF =

(0 ez12F

ez12F 0

)

H = H0 + V =

(E0

1 ez12Fez12F E0

2

)0 0.5 1 1.5 2

F (∆/ez12)

E1

E2

Ener

gy

∆

|H− E| = 0⇒ E1,2 = (E01 + E0

2)/2±√(∆/2)2 + (ez12F)2

ez12F� ∆⇒ E1,2 ≈ E01,2 ± (ez12F)2/∆⇒ quadratic effect

ez12F� ∆⇒ E1,2 ≈ ± ez12F⇒ linear effect

Stark effect of Ly-α

-5 0 5Wavelength (arb. units)

0

0.1

0.2

0.3

0.4

0.5

0.6

Inte

nsity

(arb

. uni

ts)

StaticQuasistatic

Time-dependent perturbation

Power spectrum of a physical quantity evolving with time f (t) isgiven via its Fourier transform:

I(ω) ∝ |F (f )|2 =

∣∣∣∣∫ +∞

−∞f (t)e−iωt dt

∣∣∣∣2Dipole radiation:Electric field~F in an EM wave is proportional to the dipole moment~d of radiator:

~F(t) ∝~d(t)

(For classical and quantal harmonic oscillators alike; for the latter,~d→~dif , for transition between states i and f .)

I(ω) ∝∣∣∣~dω

∣∣∣2 , with ~dω ≡∫ +∞

−∞~d(t) e−iωt dt

Example: Natural broadening

Radiative decay |i〉 → |f 〉, characteristic time = Aif (Einsteincoefficient). Decay rate is Γ ≡ A.

|Ψi(t)|2 ∝ e−Γt; Ψi(t) ∝ eiωit−Γt/2, t ≥ 0 .

If |f 〉 is stable (GS), ωi = w0, 〈i| d |f 〉 ∝ Ψi(t) ∝ eiω0t−Γt/2η(t)(η(t) - Heaviside unit step function). Then

dω ∝ F[eiω0t−Γt/2η(t)

]=

1i(ω−ω0) + Γ/2

,

I(ω) ∝ |dω|2 ∝1

(ω−ω0)2 + (Γ/2)2 .

Area-normalized lineshape is a Lorentzian:

L(ω) =1π

Γ/2(ω−ω0)2 + (Γ/2)2 .

Lorentzian profiles

-10 -5 0 5 100

0.1

0.2

0.3

0.4

0.5

0.6 γ = 1γ = 2γ = 0.5

Voigt profiles

Convolution of a Lorentzian and a Gaussian:

V(x; γ, σ) ≡ L(x; γ) ∗ G(x; σ) =∫ ∞

−∞L(x′; γ)G(x− x′; σ)dx′

With a ∼ 1% accuracy,

δωV ≈[(

δωL2

)2

+ δω2G

]1/2

+δωL

2

Voigt profiles

-10 -5 0 5 100

0.1

0.2

0.3

0.4

0.5

σ = 1, γ = 0σ = 0, γ = 1σ = 1, γ = 1

The wings of Voigt are determined by the Lorentzian contribution.

Formal theory of line broadening I

I(ω) ∝∣∣∣~dω

∣∣∣2 , with ~dω ≡∫ +∞

−∞~d(t) e−iωt dt

Using the cross-correlation theorem F (f )F (g) = (2π)−1F (f ∗ g):∣∣∣~dω

∣∣∣2 =1

2π

∫ +∞

−∞C(t) e−iωt dt ,

where C(t) is the dipole auto-correlation function:

C(t) ≡∫ +∞

−∞~d∗(τ) ·~d(τ + t) dτ ,

or (using C(t) = C(−t))

I(ω) ∝1π

Re∫ +∞

0C(t) e−iωt dt

Formal theory of line broadening II

Now we need to average over an ensemble of radiators:

I(ω) ∝1π

Re{∫ +∞

0C(t) e−iωt dt

}av

=1π

Re∫ +∞

0{C(t)}av e−iωt dt

However, C(t) is already “averaged” by time, recall

C(t) ≡∫ +∞

−∞~d∗(τ) ·~d(τ + t) dτ '∑

τ

~d∗(τ) ·~d(τ + t)

From the ergodicity argument (time average = ensemble average),it suffices to keep only one term in the infinite sum, e.g. at τ = 0:

{C(t)}av '{~d∗(0) ·~d(t)

}av

I(ω) ∝1π

Re∫ +∞

0

{~d∗(0) ·~d(t)

}av

e−iωt dt

Formal theory of line broadening III

Generalization for multiple components of transition(s) and levelpopulations (via density matrix ρ):

C(t) = Tr[{

~D†(0) · ~D(t)ρ}

av

]Note 1: one often sees in the literature

C(t) = Tr[~D†(0) · ~D(t)ρ

]This expression, strictly speaking, is only valid in context of furtheraveraging over an ergodic ensemble!

Note 2: C(t) is sometimes called auto-correlation function of thelight amplitude.

Formal theory of line broadening IV

C(t) = Tr[{

~D†(0) · ~D(t)ρ}

av

]

It looks easy, but ~D(t),{}av, and even ρ are,in principle, results ofcomplex N-bodyplasma dynamicswhere there is no strictseparation between theradiators and the“bath”.

Broadening of isolated lines

According to [Baranger, 1958], a u→ ` transition assumesLorentzian shape with FWHM defined by

w = Ne

∫ ∞

0vF(v)dv

(∑

u′,uσuu′(v) + ∑

`′,`

σ``′(v) + |fu(v)− f`(v)|2)

,

where F(v) is the (Maxwellian) electron velocity distribution, σik(v)is impact cross section from i to k, and fk(v) is elastic scatteringamplitude.

w ≡ win + wel

The inelastic part:

win ≡ Ne

∫ ∞

0vF(v)dv

(∑

u′,uσuu′(v) + ∑

`′,`

σ``′(v)

)or

win = ∑u′,u〈σuu′Nev〉v + ∑

`′,`

〈σ``′Nev〉v

Broadening of isolated lines (cont.)

The derivation assumes the broadenings of Stark-coupled levelsdo not overlap; hence the name “isolated” lines.

Typically (except for near-threshold energies), wel < (or �) win.

w ≈ win = ∑u′,u〈σuu′Nev〉v + ∑

`′,`

〈σ``′Nev〉v

Compare to the natural broadening:

w = ∑relevant phenomena

(time of life of |u〉)−1 +(time of life of |`〉)−1

Generalizing to other impact mechanisms (ionization,recombination of any kind),

w ≈ ∑all mechanisms

depop. rate of |u〉 + ∑all mechanisms

depop. rate of |`〉

“Standard theory” of line broadening

In the “standard theory” (ST), the ions are usually given a[quasi]static role, while electrons are dynamic (impact):

I(ω) =1π

Re Tr∫ ∞

0dFiW(Fi){d†[iω− iHs(Fi) + φe(Fi)]

−1d}av

Alternatively, if applicable, ions may be treated in the impactapproximation, like electrons (φe → φe + φi):

I(ω) =1π

Re Tr{∆d[iω− iH0 + φe + φi]−1}av

Intermediate cases?

Computer simulations

The closest to ab initio calculations; since[Stamm and Voslamber, 1979].

The shape of a spectral line is calculated in three steps:

The perturbing fields are simulated using the Particle FieldGenerator (PFG), by calculating the motion of a finite numberof interacting electrons and ions (of a few types).

Using this field as a perturbation, the emitter oscillatingfunction is calculated by the Schrödinger Solver (SS).

The power spectrum of the emitter oscillating function isevaluated using the Fast Fourier Transformation (FFT)method, giving the spectral line profile.

Computer simulations (cont.)

(Particle Field Generator)

PFG

SS(Schrödinger Solver)

FFT(Fast Fourier Transform)

N-body simulation

Line-shape calculation

Û(t) → <D→(t)>

+ External fields

Computer simulations (cont.)

The Hamiltonian of the atomic system:

H = H◦ + V(t).

The perturbation V(t) is due to the plasma electric field (simulatedby the PFG) and external electric and magnetic fields. We solvethe Schrödinger equation

idΨ(t)/dt = HΨ(t)

using the time-development operator U in the interactionrepresentation:

idU(t)/dt = V(t)U(t).

Computer simulations (cont.)

The evolution of the dipole operator D(t) is then obtained:

~D(t) = U(t)†~D(0)U(t).

The Fourier transform of the dipole operator ~D(ω) is further usedto calculate the line spectrum:

Iλ(ω) ∝ ∑i

∑f

ω4fi|~eλ · 〈~Dfi(ω)〉|2.

The angle brackets denote an averaging over several runs of thecode (which corresponds to the averaging over an ensemble ofemitters).

Computer simulations are accurate, but very time consuming.

Quasi-contiguous (QC) approximation

-10 -8 -6 -4 -2 0 2 4 6 8 10q

Inte

nsity

-qQC qQC

πσ

Static Stark effect of Ly9

Intensities of the π and σ components form two parabolae, which,on average, can be substituted with a simple rectangular shape.

Quasi-contiguous (QC) approximation

Therefore:

In(ω) =

{I(0)n

2αnF/h for |hω| ≤ αnF0 for |hω| > αnF,

where I(0)n is the total line intensity, and αn is the linear-Stark-effectcoefficient:

αn =32(n2 − 1)

ea0

Z.

Generalization for n′ > 1:

αnn′ =32(n2 − n′2)

ea0

Z.

QC approximation: quasistatic shape

Convolution with a microfield distribution W(F):

I(ω) = I(0)nn′

∫ ∞

hω/αn

W(F)dF2αnn′F/h

≡ I(0)nn′Lqs(ω),

or, with the reduced field strength β = F/F0 and detuningω = ω/∆0,

Lqs(ω) =12

∫ ∞

β

H(β)

βdβ,

whereH(β) = W(F)/F0

and

∆0 =αnn′F0

h.

QC approximation: quasistatic shape

In ideal plasma:

Lqs(ω) = S(ω),

where the S function is defined as

S(ω) =1π

∫ ∞

0cos(ωx) exp(−x3/2)dx.

[Stambulchik and Maron, 2008]; also corrections due to moderateplasma coupling.

QC approximation: quasistatic shape

0 1 2 3 4 5ω

0

0.1

0.2

0.3

0.4

0.5

ω1/20

Ly9Ly10

S(ω)

Quasistatic shapes of Ly9 and Ly10(the central component of Ly10 is not shown)

(only “blue” half of the symmetric profiles are shown).

QC-FFM I

Frequency-fluctuation model (FFM) [Calisti et al., 2010] accountsfor field fluctuations with the typical field frequency

wdyn =〈v〉〈r〉 =

√kTm∗p

(4πNp

3

)1/3

.

Instead of the quasistatic lineshape Lqs(ω), the dynamic one is

L(ν; ω) =1π

ReJ(ν; ω)

1− νJ(ν; ω),

where

J(ν; ω) =∫ Lqs(ω′)dω′

ν + i(ω− ω′),

ν ∼ wdyn/∆0.

QC-FFM II

Applying FFM to quasi-contiguous lineshape, for idealone-component plasma one gets:

J(ν; ω) =∫ ∞

0dτ exp (−τ3/2 − i(ω− iν)τ) .

[Stambulchik and Maron, 2013]; straightforward extension formulti-component non-ideal plasmas.

QC-FFM :: High-n series & continuum lowering

The approach (closely following [Griem, 1997]):

Calculate bound-bound (BB) n` → nu shapes for a series ofnu until FWHM exceeds |Enu − Enu+1| (the “Inglis-Teller”reasoning).

(Optional) continue for a few more nu with the same width.

Assume the free-bound (FB) edge at the Enu+1 energy.

Convolve the FB continuum with the last (n` → nu) BBlineshape.

Sum up.

No ionization potential depression assumed.

QC-FFM :: High-n series & continuum lowering (cont.)

0.465 0.47 0.475 0.48 0.485 0.49 0.495Energy (hartree)

0

2

4

6

8

Inte

nsity

(arb

. uni

ts)

E∞

BBFBTotalSimU with all n ≤ 10

H I Lyman series, ne = 1017 cm-3, T = 1 eV(without the Boltzmann factor)

n = 4

5

6 7 8 9 10

SimU: Hamiltonian with 385 fully interacting states.SimU CPU time: > 1 month. QC-FFM: ∼ 1 sec (∼ ×3, 000, 000).

QC-FFM :: High-n series & continuum lowering example

[Wiese et al., 1972]

4000 4500 5000λ (Å)

1e-19

1e-18

1e-17

1e-16

Inte

nsity

(arb

. uni

ts)

Ne = 9.3×1016

Ne = 5.7×1016

Ne = 2.9×1016

Ne = 1.8×1016

∼ 1 s CPU time

Data sources and tools

Stark-B database (isolated lines):http://stark-b.obspm.fr/

[Griem, 1974] appendices (isolated lines)

Plasma Formulary Interactive (H-like and Rydberg):http://plasma-gate.weizmann.ac.il/pf

NIST Atomic Spectral Line Broadening BibliographicDatabase http://physics.nist.gov/cgi-bin/ASBib1/LineBroadBib.cgi

Notes on lineshape accuracy

Lineshape calculations are complex; reliably assessing theoreticaluncertantities can be more difficult than the calculationsthemselves. Claimed accuracy should be taken with a grain of salt.Some rules of thumb:

Accuracy is not mentioned at all — assume factor two orworse;

Claimed 20%–40% — must be discussed and explained, atleast shortly;

Claimed 10%–20% — there must be a section devoted toaccuracy estimates;

Claimed 3%–10% — accuracy is the subject of the study;

When using published data, pay attention what is meant by the“width”—it could be FWHM or HWHM!

Bibliography I

Books and reviews

Griem, H. R. (1974).Spectral Line Broadening by Plasmas.Academic Press, New York.

Kunze, H.-J. (2009).Introduction to Plasma Spectroscopy, volume 56 of SpringerSeries on Atomic, Optical, and Plasma Physics.Springer, Berlin, Heidelberg.

Gigosos, M. A. (2014).J. Phys. D: Appl. Phys., 47(34):343001.

Stambulchik, E. and Maron, Y. (2010).High Energy Density Phys., 6(1):9–14.

Bibliography II

References

Holtsmark, J. (1919).Ann. Phys. (Leipzig), 58:577–630.

Chandrasekhar, S. and von Neumann, J. (1942).The Astrophysical Journal, 95:489.

Baranger, M. (1958).Phys. Rev., 112:855–865.

Wiese, W. L., Kelleher, D. E., and Paquette, D. R. (1972).Phys. Rev. A, 6(3):1132–1153.

Stamm, R. and Voslamber, D. (1979).J. Quant. Spectr. Rad. Transfer, 22:599–609.

Bibliography III

Stambulchik, E. and Maron, Y. (2008).J. Phys. B: At. Mol. Opt. Phys., 41(9):095703.

Calisti, A., Mossé, C., Ferri, S., Talin, B., Rosmej, F., Bureyeva,L. A., and Lisitsa, V. S. (2010).Phys. Rev. E, 81(1):016406.

Stambulchik, E. and Maron, Y. (2013).Phys. Rev. E, 87(5):053108.

Griem, H. R. (1997).Principles of Plasma Spectroscopy.Cambridge University Press, Cambridge, England.