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C ollisional line broadening v ersus C ollisional depolarization : Similarities and differences

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versus

Collisionaldepolarization:

Similarities and differences

S. Sahal-Bréchot1 and V. Bommier21 Observatoire de Paris, LERMA CNRS UMR 8112, France2 Observatoire de Paris, LESIA CNRS UMR 8109, France

9th SCSLSA, Banja Koviljaca, Serbia, May 13-19, 2013

Atomicpolarization -1Whatisatomicpolarization?And whatiscollisionaldepolarization?

Whatis “atomicpolarization” ?

The Zeeman sublevels of the radiatingatom are not in LTE:

Different populations (diagonal elements of theatomicdensitymatrix) : N(αJM)≠N(αJM’)

Coherent superposition of states (off-diagonalelements of the densitymatrix)

Then the emitted line canbepolarized

Needs:Anisotropy (or dissymmetry) of excitation of the levels:

e.g.Directiveincident radiation

e.g.Directivecollisional excitation

Modification of the atomicpolarization:

Magneticfieldvector (Hanleeffect)

Anisotropicvelocityfields

Isotropic collisions restore LTE (collisionaldepolarization)

Multiple scattering (not opticallythinlines): depolarizingeffect

9th SCSLSA, Banja Koviljaca, Serbia, May 13-19, 2013

E electric vector of the radiation

< …> = average over a time interval large compared to the wave period

Radiation field propagating along z

- : Polarization matrix

Stokes

Parameters:

Short remind -1-Polarization matrix of radiation andStokes parameters9th SCSLSA, Banja Koviljaca, Serbia, May 13-19, 2013

MeasuredIntensity: analyser axis OX, angle αwithOx:

O

- Linearpolarizationdegreepl

- polarization direction α0(withinπ)

Circularpolarizationdegree

I-= right circular component = 〈E+(t) E*+ (t)〉

I+= leftcircular component = 〈E-(t) E*-(t)〉

Short remind -2-Physicalmeaning of the Stokes parameters9th SCSLSA, Banja Koviljaca, Serbia, May 13-19, 2013

Quantization axis :Symmetry axis (direction of incident radiation) a Zeeman LM states

- Photons σ+and σ-excite the sublevels
- M=1 and M=-1 are equallypopulated.
- M=0 is not populated
- The Zeeman sublevels are aligned
- Polarizationmatrix of the scattered radiation mirrorsthat of the atomicexcitedlevel
- Projection on the direction of observation (AZ) a Stokes Parameters of the observed radiation (linearlypolarizedalong AY)

Resonancescattering: quantum aspect

(normal Zeeman triplet)

vibratingdipoles

Natural incident radiation along Oz (unpolarized)

LinearlyPolarizedscatteredradiation along AZ

Linear polarization due to radiative scattering: basic quantum interpretationRight angle scattering

Y

x

z

O

A

y

Z

Polarizationdegree

withη= (I±-I0) /(I0+2I±)

N(LM) populations of the sublevels

9th SCSLSA, Banja Koviljaca, Serbia, May 13-19, 2013

Vocabulary:Alignment : populations of M and -M are equallinear polarizationOrientation: Imbalance of populations of Zeeman sublevels M and -M circular polarization

9th SCSLSA, Banja Koviljaca, Serbia, May 13-19, 2013

Maximum polarization degree

9th SCSLSA, Banja Koviljaca, Serbia, May 13-19, 2013

Line broadening and Atomicpolarization: briefsurvey of the theory

The density matrix ρof the whole systemis solution of the

Schrödinger equation

A is the atomicsubsystem

B is the bath of perturbers (P) and photons (R ) and are assumedindependent

HAgives the atomicwavefunctions (unperturbed) and the atomicenergies

The reduceddensity matrices ρA(t) andρB(t)are not described by an hamiltonian and thus are not solution of a Schrödinger equation

They are solution of a master equation

9th SCSLSA, Banja Koviljaca, Serbia, May 13-19, 2013

Collisional line broadening: short survey of the theory

Intensity (Baranger 1958abc)

ρisthe density matrix of the system:

atom (A)+ bath B (R photons, P perturbers)

dis the atomicdipole moment

T(s) the evolutionoperator of the system

- Twokeys approximations:
- No back reaction: ρ = ρA⊗ρB
- Impact approximation
- Mean duration of an interaction <<meanintervalbetweentwo interactions
- ρR(t) and ρP(t) are decoupled and their interactions with the atom are decoupled
- ⇒ The atom-perturber interactioniscomplete(no emission of photon
- duringthe time of interest). The collisional S-matrix willappear
- The calculation of the line profile becomes an application of
- the theory of collisions

9th SCSLSA, Banja Koviljaca, Serbia, May 13-19, 2013

Collisionalbroadening: short survey of the theoryIsolatedlines

No overlapof close levels(withΔl=1 for electron impacts) due to collisionallevel-widths

⇒The profile islorentzian

- ρAis the atomicdensity matrix
- Withoutpolarization: only diagonal elements:
- LTE: Boltzman factor,
- Out of LTE, itselements are solutions of the statisticalequilibriumequations (collisional radiative model
- Emissivity= Profile x Population of the upperlevel
- This iscomplete redistribution

9th SCSLSA, Banja Koviljaca, Serbia, May 13-19, 2013

Baranger’s formula for an isolated line i(αi Ji) – f(αf Jf)

TrP: trace over the perturbers, i.e. average over all perturbers =

The scatteringS matrix issymmetric and unitary

9th SCSLSA, Banja Koviljaca, Serbia, May 13-19, 2013

αiJi

αJ

α’ J’

αf Jf

α’ J

expression of the Baranger’s formula for the widthWith the T matrix: T=1-S, and usingT*T= 2 Re (T)

9th SCSLSA, Banja Koviljaca, Serbia, May 13-19, 2013

Atomicpolarization: briefsurvey of the theorycalculationof the atomicdensity matrix

- 1- HAgivesthe atomicwavefunctions (unperturbed)
- and the atomicenergies
- 2-Same key approximations as for collisional line broadening:
- • First key approximation: no back reaction
- ρ(t)= ρA(t) ⊗ ρB(t)
- • Second key approximation: the impact approximation
- Meanduration of an interaction <<meanintervalbetweentwo interactions
- ⇒ ρR(t) and ρP(t) are decoupled and their interactions with the atom are decoupled
- 4- The atom-perturber interactioniscompleteduring the time of interest (S-matrix appears)
- 5 Markov approximation: evolution of ρA(t) onlydependsρA(t0) on and not on hispasthistory
- 6Secularapproximation

9th SCSLSA, Banja Koviljaca, Serbia, May 13-19, 2013

Atomicpolarization: briefsurvey of the theory:calculation of the atomicdensity matrix

7-The radiation isweak:

Perturbation theorysufficient for atom radiation interaction

8-Consequence of Second order perturbation theory + Markov:

Transitions canonlybedonewith exact resonance in energy,

so:

Profile cannotbetakenintoaccount: δ-profile

Atomicpolarizationis a global information

9th SCSLSA, Banja Koviljaca, Serbia, May 13-19, 2013

Atomicpolarization: briefsurvey of the theory:processes to takeintoaccount in thecalculation of the atomicdensity matrix

- Atomic polarisation (master equation):
- Excitation by anisotropicprocessresponsible for the polarization (onlyalignment in astrophysics and thusonlylinearpolarization) : radiation or beam of particles (energeticelectrons, protons)
- Quantization axis in the direction of the preferrred excitation
- Excitation by radiative or isotropiccollisionalprocesses (decreasealignment)
- Breakdown of the cylindricalsymmetry: apparition of coherences in the master equation and thus modification of the degree and direction of polarization
- e.g. interaction with a magneticfield B: Hanleeffect. Quantization axis in the direction of B
- Depolarization and transfer of alignment by isotropic collisions
- Followed by deexcitation (radiative and collisional)
- If the (hyper)fine levels are separated (no overlap by the lifetime), the atomicpolarizationisdifferent for the different (hyper)fine lines

9th SCSLSA, Banja Koviljaca, Serbia, May 13-19, 2013

Atomicpolarization: briefsurvey of the theory:expression of the atomicdensity matrix and of the radiation polarization matrix

- At the stationary state ρA(t) = ρA
- ρAissolution of the “statisticalequilibriumequations”leading to populations (diagonal elements of ρA) and coherences (off-diagonal elements pf ρA)
- in the standard JMrepresentation
- and at the stationary state
- ρR(t) = ρR
- ⇒ The matricialtransferequationof the Stokes operators (I Q U V)

Case of a two-levelatomwithoutpolarization of the lowerlevel in the irreducibletensorial TKQrepresentation

Master equation for the atomicdensity matrix:

Population: K=0

Orientation: K=1

Alignment: K=2

Coherences: Q≠0

9th SCSLSA, Banja Koviljaca, Serbia, May 13-19, 2013

Examplesof expressions

Rates beween the Zeeman sublevels

(standard atomicbasis αJM )

Angularaverage: Gordeyevet al. (1969, 1971), Masnou-Seeuws & Roueff (1972), Omont(1977), Sahal-Bréchot (1974), Sahal-Bréchotet al. (A&A 2007)

9th SCSLSA, Banja Koviljaca, Serbia, May 13-19, 2013

Examples of expressions (cf. for instance Sahal-Bréchot et al. (A&A 2007)

Tkq basis (irreducibletensorialoperators)

- Master equation:off-diagonal elements

- Master equation:diagonalelements:
- Sum of contributions of
- elastic collisions (k-pole depolarization rates)
- inelastic collisions (relaxation rates,independent of k): lossterms

9th SCSLSA, Banja Koviljaca, Serbia, May 13-19, 2013

Collisions withneutralHydrogenbroadening and depolarization- Calculation of the S-matrix - the interaction potential

Semiclassical approximation issufficient :

classicalpath for the perturber

Perturbation expansion of the S-matrix is not valid:

Close-couplingnecessary:

First ordersemiclassicaldifferentialequations to solve

3. Long range expansion (Van der Waals) not valid

Typical impact parameters 10–20 a0

4. Integration over the impact parameter:

A lowercutoff has to beintroduced

9th SCSLSA, Banja Koviljaca, Serbia, May 13-19, 2013

Fromhydrogencollisionalbroadening to depolarization: the ABO method

- Nineties: Line broadening by collisions withneutralhydrogen: a new and powerfulapproximatemethod: the so-calledABO method
- O’Mara and Anstee, Barklem:(Anstee & O’Mara 1991, 1995, 1997; Barklem & O’Mara 1997, Barklem et al. 1998ab, 2000)
- Semiclassical close-couplingtheory
- Approximate interaction potential: time-independent second-order perturbation theorywithout exchange, Unsöld approximation, allowing the Lindholm-Foley average over m states to beremoved
- (Brueckner 1971; O’Mara 1976)

- 2000-2007: Extension of thismethod to line depolarization
- Derouich, Sahal-Bréchot & Barklem, A&A 2003, 2004ab, 2005ab, 2007
- Derouich, thesis 2004, Derouich 2006, Derouich & Barklem 2007
- Good resultswhencompared to quantum chemistrycalculations (20%)

9th SCSLSA, Banja Koviljaca, Serbia, May 13-19, 2013

Broadening and Depolarizationby isotropiccollisions

The fine structure (and a fortiori hyperfine structure) canbemostoftenneglectedduring the collision in astrophysical conditions (hightemperatures):

The spin has no time to rotateduring the collision time

⇒ if LS couplingisvalid, the fine structure components have the samewidth and shift, that of the multiplet.

For collisionaldepolarization rates the linearcombinations of the T-matrix elements are differentfromthose of the width

And expressions between J levels are required

Diagonal elementsof the S-matrix appearin the broadening formula and do not play a rolein depolarization

Numericalcalculations have to beperformedboth for broadening and for depolarization.

No analytical relation (evenapproximate) between the collisionalwidth and the disalignment, disorientation, alignmenttransfer, and orientation transfercollisional rates.

9th SCSLSA, Banja Koviljaca, Serbia, May 13-19, 2013

Beyond the impact and Markov approximation for atom-radiation interaction

Couplingof the atomicdensity matrix to the line profile and redistribution of radiation:

First work by Bommier A&A 1997: twolevelatomwithunpolarizedlowerlevel

Higherorders of the perturbation development of the atom-radiation interaction takenintoaccount:

⇒ twophotons caninteractwith the atomat the same time

(resonantscattering)

Beyond the Markov approximation:

⇒ the pastmemoryistakenintoaccount

⇒the lifetimes are no longer ignored

⇒Introduction of the line profile in the master equation and thus in the Stokes parameters of the observed line

9th SCSLSA, Banja Koviljaca, Serbia, May 13-19, 2013

Thankyou for your attention

9th SCSLSA, Banja Koviljaca, Serbia, May 13-19, 2013

Broadening and Depolarization by isotropiccollisionsexample of collisions withneutrals (Hydrogen)

N.B. remind : T-matrix issymmetric, <JM⎮T⎮J’M’>=<J’M’⎮T⎮JM>

and ⎮<JM⎮T⎮J’M’>⎮2Ang.Av= ⎮<J-M⎮T⎮J’-M’>⎮2Ang.Av

For collisions withneutrals The quenchingisnegligible.

Processes to takeintoaccount: elastic and fine structure inelasticcollisions

Example : normal Zeeman triplet: p-s transition without spin: n p 1Po1 – n’ s 1S0

Ji=1, Mi= 0±1 (upperlevel) Notation: <1M⎮T⎮1M’>= TMM’

Jf=0, Mf=0 (lowerlevel) <00⎮T⎮00>=t00

Disorientation D(1)∝⎮T01⎮2Ang.Av+2⎮T-1 +1⎮2Ang.Av

DisalignmentD(2) ∝ 3⎮T01⎮2Ang.Av

Particular case: one state case: t00=0 (does not workwit collisions with H)

width= elastic collision rate of the upperlevel:

⎮

2w∝(1/3) [ 4⎮T01⎮2Ang.Av+2⎮T-1 +1⎮2Ang.Av] + (1/3) [T-1 -1⎮2Ang.Av+[T1 1⎮2Ang.Av+[T00⎮2Ang.Av]

1 1

n’ p

1 0

1 -1

n s

0 0

9th SCSLSA, Banja Koviljaca, Serbia, May 13-19, 2013

Broadening and Depolarization by isotropiccollisionsexample of collisions withneutrals (Hydrogen) (following)

N.B. remind : T-matrix issymmetric, <JM⎮T⎮J’M’>=<J’M’⎮T⎮JM>

and ⎮<JM⎮T⎮J’M’>⎮2Ang.Av= ⎮<J-M⎮T⎮J’-M’>⎮2Ang.Av

For collisions withneutrals The quenchingisnegligible.

Processes to takeintoaccount: elastic and fine structure inelasticcollisions

Example : normal Zeeman triplet: p-s transition without spin: n p 1Po1 – n’ s 1S0

Ji=1, Mi= 0±1 (upperlevel) Notation: <1M⎮T⎮1M’>= TMM’

Jf=0, Mf=0 (lowerlevel) <00⎮T⎮00>=t00

Disorientation D(1)∝⎮T01⎮2Ang.Av+2⎮T-1 +1⎮2Ang.Av

DisalignmentD(2) ∝ 3⎮T01⎮2Ang.Av

General case: two state case: t00≠0 (case of collisions with H)

width= elastic collision rate of the upperlevel+elastic collision rate of the lowerrlevel+ interferenceterm

⎮

1 1

n’ p

1 0

1 -1

n s

0 0

9th SCSLSA, Banja Koviljaca, Serbia, May 13-19, 2013

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