Progress in the description of Motional Stark Effect in fusion plasmas

1. Progress in the description of Motional Stark Effect in fusion plasmas O. Marchuk1, Yu. Ralchenko2,D.R. Schultz3, E. Delabie4,A.M. Urnov5, W. Biel1, R.K. Janev1andT. Schlummer1 1- Institute forClimateandEnergy Research, Forschungszentrum Jülich GmbH, 52425, Jülich, Germany 2- AtomicPhysics Division, NIST, Gaithersburg, MD 20886, USA 3- Physics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-6373 4- F.O.M. Institute for Plasma Physics "Rijnhuizen", P.O.Box 1207, 3430 BE Nieuwegein, Netherlands 5- P. N. Lebedev Institute of RAS, Leninskiipr 53, Moscow, 119991, Russia

2. Principlesofactiveplasmaspectroscopy H0 + {e,H+,Xz} →H∗ + {e,H+,Xz}→ ћω(1) H0 + Xz+1→ H++ X*z(nl) → ћω(2) H++ H0 →H∗ + H+ → ћω(3) beam-emission spectroscopy (BES) sourceofcharge-exchange diagnostic sourceof fast iondiagnosticandmainionratiomeasurements (ACX- activecharge-exchange, PCX- passive charge-exchange) • Plasma parameters: • Density = 1013…1014 cm-3 • Beam energy = 20 .. 200 keV/u • Temperature = 1..15 keV • Magneticfield = 1.. 5 T

3. Fields „observed“ bytheatom B Lorentz transformationforthefield: y y´ x x´ z z´ v (cgs) Beam atom • In therestframeoftheatomtheboundelectronexperiencestheinfluenceofthecrossedmagneticandelectricfields (x´y´z´) isthecoordinatesystem in therestframeof hydrogen atom (xyz) isthelaboratorycoordinate system • Example: B = 1 T, E = 100 keV/u → v = 4.4·108 cm/s → F = 44 kV/cm • Strong electricandmagneticfield in therestframeoftheatomisexperiencedbytheboundelectron • Externalfieldsareusuallyconsideredasperturbationappliedtothefield-freesolution

4. Beam emissionspectrameasuredat JET Hα(n=3 → n=2)Delabie E. et al. PPCF 52 125008 (2010) • 3 components in the beam (E/1, E/2, E/3) • Passive light fromtheedge • Emission of thermal H+and D+ • Coldcomponentsof CII Zeeman multiplet • Overlappedcomponentsof Stark effectspectra • Intensityof MSE multipletas a functionofobservation angle θ relative tothedirectionofelectricfield • Ratiosamongπ-(Δm=0)andσ- (Δm=±1)lineswithinthemultipletarewelldefinedandshouldbeconstant.

5. „Statistical“ descriptionfortheexcitedstates • The firsttheoreticalmodelsfortheexcitedstatesaresolvedonlyfortheprincipalquantumnumbern • Stark effect was ignoredcompletely, namelyit was observedin fusiona fewyearslater, LevintonFM Phys. Rev. Lett. 63 2060 (1989) • The populationofthefinestructurelevelswithinΔn=0 isassumedtobe proportional totheirstatisticalweights, Isler RC and Olson RE Phys. Rev. A 37 3399 (1988) • The crosssectionsused in these CRM (Δn>0) arebased on thereccomendeddata in sphericalstatesfromthecollisionaldatabases ALLADIN and ORNL, Summers HP 2004 The ADAS User Manual, version 2.6 http://adas.phys.strath.ac.uk • The dominant excitation/ionizationchannelsarecollisionswithions • For heavy particlescollisionsthedataof TDSE, AOCC, DW, EI, … areused

6. Currentstatusofstatisticalmodels (2011) • Formanyyearsthe different populationsofexcitedstatesof n=2 and n=3 werereportedfrom different models Eb = 40 keV/u Te=Ti= 2 keV Zeff= 1 Solid lineswithpoints – presentcalculations Dashedline - Hutchinson I PPCF 44 71 (2002) Delabie E. et al. PPCF 52 125008 (2010) • Thatisthefirst time thatthepopulationofexcitedstates (n-states) ofthe beam agreewithin 20% forthree different modelsin thedensityrangeof 1013-1014cm-3: • Key component: ionizationdatafrom n=2 and n=3 states

7. Linear Stark effectfortheexcitedstates • Hydrogen atomplaced in electricfieldexperiences Stark effect • Hamiltonianis diagonal in parabolicquantumnumbers • Sphericalsymmetryoftheatomisreplacedbythe axial symmetryaroundthedirectionofelectricfield. • The energyofthem –levelsis not degeneratedany more in thepresenceofelectricfield (multipletstructure) • The unitarytransformationbetweenthesphericalwavefunctions|nlm>andparabolicwavefunctions|nkm> exists. n k |m| 3 2 0 3 1 1 3 0 0 3-1 1 3-2 0 z „Good“ quantumnumbers: n=n1+n2+|m|+1, n1, n2 >0 (nkm) k=n1-n2 – electricquantumnumber m – z-projectionofmagneticmoment n=3 σ0 π4 2 1 0 2 0 1 2 -1 0 n=2 1/2 σ π

8. Calculationofthecrosssections in parabolicstates parabolicstatesnikimi θ nilimi– sphericalstates θ=π/2 for MSE • Calculationsincludetwotransformationsofwavefunctions • Rotation ofthecollisional (z‘) frame on the angle θtomatchzframeEdmonds A R 1957 Angular Momentum in Quantum Mechanics (Princeton, NJ: Princeton University Press) • Transformation betweenthesphericalandparabolicstates in the same frame zLandau L D and Lifshitz E M 1976 Quantum Mechanics: Non-Relativistic Theory

9. Calculationofthecrosssections in parabolicstates (2) • Transformation betweenthesphericalandparabolicstates 540 • Final expressionforthecrosssectioncanbewrittenas: • The coefficientsciandfinallythecrosssectiondepend on the angle betweenthefieldanddirectionoftheprojectile • This effect was observed in thel - mixingcollisionsof high Rydbergstates (Hickman A P 1983 Phys. Rev. A 28 111; de Prunele E 1985 Phys. Rev. A 31 3593) • Fortheexcitationfromthegroundstatetheexpressionsare still simple:

10. Influenceoftheorientation on thecrosssections. AOCC calculations. 210 201 • Energyisvaried in radial direction : 20…200 keV/u • Polar angle isthe angle betweenthefielddirectionandtheprojectile. (MSE – π/2) ? • Why do weneedthe angular dependenceif Cross sectiondepends on therelative velocitybetween beam andplasmaparticles. ITER Diagnostic beam: T=20 keV, E=100 keV/u relative velocity - F π/2 θ=π/2± π/6 The simple formulasfor rate coefficients beam-Maxwellianplasma do not workanymore. Beam direction

11. Populationsofexcitedlevels: pi = Ni/gi/N0 statisticalcasemeans Beam energy 50 keV/u Plasma density 3·1013 cm-3 Magneticfieldis 3 T Ionizationisexcluded Level-crossing + Ionizationbyelectricfield Index ofexcitedlevels

12. Influenceoftheorientation on the Stark multipletemission statisticalcalculations F θ v • The strongestdeviationtothestatisticalcaseisobservedfortheconditionsofMotional Stark effect • Increaseofπanddropofσcomponentsas a functionof angle θisobserved

13. Comparisonwith experimental JET data (1993) & (2010) • Points- experimental data E. Delabieet al. PPCF 52 125008 (2010)Error bars- experimental dataW. Mandl et al. PPCF 35 1373 (1993)Dashedline GA O. Marchuk et al. JPB 43011002 (2010) Solid linepresentresults , AOCC + GADashed-point linesdenotestatisticalvalues

14. Reductionofthe beam emission rate coefficients • Observation oflong-standing discrepancy on theorderof 20-30% betweenthemeasured (BES) andcalculateddensityof hydrogen beam in theplasmausingstatisticalmodels Measured beam emission Calculated beam attenuation • The non-statisticalsimulationsdemonstrate a reductionofthe beam emission rate relative tothestatisticalmodel on theorderof 15-30% atlowand intermediate density. E=100 keV/u T=3 keV

15. Summary • The completelym-resolvedmodel in parabolicstateupto n=10 was developed. In contrasttothesphericalstatestheparabolicstatesaretheeigenstatesofthe hydrogen beam in theplasma. The modeloperateswitharbitraryorientationbetweenthefieldandprojectiledirection. • The collisionalredistributionamongtheparabolicstates was takenintoaccount in CRM NOMAD • The systematicmeasurementson non-statisticalpopulationsofσ- andπ – componentswereexplainedusingthemodel in parabolicstates. An excellentagreementwith experimental datafrom JET was found. • The assumption on thestatisticaldistributionof m-resolvedstates in the hydrogen beam is not valid in theplasma.

17. Non-LTE model in parabolicstates • The experimental datafrom JET andotherdevicesclearlydemonstrate a needfor an m-resolvedcollisional-radiativemodel in parabolicstatesforcalculationof: • Line intensitiesfrom m-resolvedlevels • Populationsofexcitedstates • Beam-emission data • Beam attenuation in theplasma • The CRM modelshouldinclude: • Energylevels • Radiativetransitionprobabilities • Cross sectionsamongthe m-resolvedlevels

18. Statistical intensitiesof Stark effect σ Iij,a.u. • Are theobservedintensitiesof MSE components proportional totheirstatisticalweights ? π • E. Schrödinger, Ann d. Phys. 385(80), 437 (1926) σ1/σ0=0.353 • The experimentsat JET givetheanswer„NO“ σ1/σ0 λ(Hα) Displacementof Hαline W. Mandl et al. PPCF 35 1373(1993)

19. Ratio betweentheσtoπcomponents • In statisticalcasetheintensityoftheσandπcomponentsisthe same • The ratioisquite sensitive totheplasmadensitywiththesteepgradient in thedensityrangeof 1012-1014 cm-3 • The valuesforthecoronallimitaredeterminedthroughtheexcitationcrosssectionfromthegroundstate

20. Calculationofthecrosssections in parabolicstates (3) • „The knowledge on the m-resolveddata in sphericalstatesis not enough“ • Off-diagonal elements (coherence) wereobserved in quantumbeatsandpolarizationstudiesof Hαline (H++He) Densitymatrixfor n=2 states • In ordertoachievethe same qualityas in caseofstatisticalmodelsforthecross-sections in parabolicstatesweusedthefollowingmethods: • AOCC calculationsfortheexcitationto n=2 and n=3 states • Glauber approximationforexcitationbetween all otherstates. • Born approximation was usedonlytocontrolthecalculationsat high energies. * H+(E=210 keV/u) + Carbonfoil → H(n=2) Gaupp A. et al. Phys. Rev. Lett. 32268 (1973)

21. Calculationofthecrosssections in parabolicstates(4) s-p s-p blue - AOCC (presentresults) green - Glauber approximation (presentresults) dashed - Born approximation orange - CCC Schöller O et al. J. Phys B.: At. Mol. Opt. Phys. 19 2505 (1986) red - EA Rodriguez VD andMiraglia JE J. Phys. B: At. Mol. Opt. Phys. 1992 25 2037 black - AOCC (presentresults) blue - SAOCC Winter TG, Phys. Rev. A 2009 80 032701 green – Glauber approximation (presentresults) dashed - Born approximation orange - SAOCC Shakeshaft R Phys. Rev. A 1976 18 1930 red - EA Rodriguez VD andMiraglia JE J. Phys. B: At. Mol. Opt. Phys. 1992 25 2037 • Uptonowtherewereno urgent needsfors-pands-dcoherenceofexcitation

22. Exampleoftheinfluenceofionization on thepopulationofexcitedstates. ITER Heating beam *For n=5 thestatisticalassumption was assumed

23. Intensityof CXRS spectrallinesprovides: • Plasma temperature (Doppler broadeningofspectrallines) • Plasma rotation (Doppler shift) • Densityofimpurityions • (posterby T. Schlummer, thisconference) • In ordertoobtainthedensityofimpurityionsoneneedstoknowthedensityofexcitedstatesofthe beam: fisthecalibrationfunction Nz+1isthedensityofimpurity z+1 - isthedensityofthestateofthe beam (=1, 2, etc….)isthe CX rate coefficient cm3/s

24. PrincipleofLocalThermodynamicEqulibrium (LTE) for H-likeions (gl~2l+1) n, j1 Collisionalchannel • Collisionalchannelswithinthefinestructuredominateradiativedecaysfromthislevel • Heavy particlecollisions(H+,D+) areresponsiblefortheredistributionwithinthe same principalquantumnumber n, j2 Radiativechannel m, j3

25. Statistical modelsarebased on theatomicdata in sphericalrepresentation • The beam eigenstatesareclosetotheparabolicones ΔE Δn=0,F=0<< ΔE Δn=0,F≠0<< ΔE Δn>0(*) F≠0 F=0 ΔEΔn=0 n2l ΔEΔn=0 n2km ΔE Δn>0 ΔE Δn>0 n1km n1l at (*) • The statisticalresultsare still validifandonlyifthepopulationsofthereal states in theplasmaare proportional tothestatisticalweights

26. Contents • Principlesofactiveplasmaspectroscopy in fusion • Roleoftheexcitedstatesof hydrogen atoms in plasmadiagnostics • The needsforthe non-LTE models in the atom-plasma interaction • Collisionalatomicdatain different representations • Non-LTE collisional-radiativemodel NOMAD in parabolicstates • Comparisonwith experimental datafor Stark effect • Summary and Outlook