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

C ollisional line broadening v ersus C ollisional depolarization : Similarities and differences. S . Sahal- Bréchot 1 and V. Bommier 2 1 Observatoire de Paris, LERMA CNRS UMR 8112, France 2 Observatoire de Paris, LESIA CNRS UMR 8109, France.

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

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  1. Collisionalline broadening 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

  2. 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

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

  4. Intensity of the radiation: I 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 parameters 9th SCSLSA, Banja Koviljaca, Serbia, May 13-19, 2013

  5. 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 interpretation Right 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

  6. Vocabulary:Alignment : populations of M and -M are equallinear polarizationOrientation: Imbalance of populations of Zeeman sublevels M and -M circular polarization 9th SCSLSA, Banja Koviljaca, Serbia, May 13-19, 2013

  7. Maximum polarization degree 9th SCSLSA, Banja Koviljaca, Serbia, May 13-19, 2013

  8. 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

  9. 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

  10. 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

  11. 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

  12. αJ αiJi αJ α’ J’ αf Jf α’ J expression of the Baranger’s formula for the width With the T matrix: T=1-S, and usingT*T= 2 Re (T) 9th SCSLSA, Banja Koviljaca, Serbia, May 13-19, 2013

  13. 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

  14. 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

  15. 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

  16. 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

  17. 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

  18. 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

  19. 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

  20. 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

  21. 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

  22. 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

  23. Thankyou for your attention 9th SCSLSA, Banja Koviljaca, Serbia, May 13-19, 2013

  24. 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

  25. 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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