Ion dynamics effect on stark-broadened line shapes: A cross-comparison of various models

A oms 2014,2, 299-318; doi:10.3390/a oms2030299 OPEN ACCESS a oms ISSN 2218-2004 www.mdpi.com/jou nal/a oms A icle Ion Dynamics E ec on S a k-B oadened Line Shapes: A C oss-Compa ison o Va ious Models Sand ine Fe i 1,*, Anne e Calis i 1, Ca oline Mossé 1, Joël Rosa o 1, Be na d Talin 1, Spi os Alexiou 2, Ma co A. Gigosos 3, Manuel A. González 3, Diego González-He e o 3, Na i idad La a 3, Thomas Gomez 4, Ca los Iglesias 5, Sonja Lo enzen 6, Robe o C. Mancini 7 and E geny S ambulchik 8 1Aix-Ma seille Uni e si é, CNRS, PIIM UMR7345, 13397 Ma seille, F ance; E-Mails: anne e.calis i@uni -amu. (A.C.); ca oline.mosse@uni -amu. (C.M.); joel. osa o@uni -amu. (J.R.); [email p o ec ed] (B.T.) 2TETY, Uni e si y o C e e, 71409 He aklion, TK 2208, G eece; E-Mail: [email p o ec ed] (S.A.) 3Depa men de Óp ica y Física Applicada, Uni e sidad de Valladolid, Valladolid 47071, Spain; E-Mails: [email p o ec ed]a.es (M.A.G.); [email p o ec ed]a.es (M.A.G.); die[email p o ec ed] (D.G.-H.); [email p o ec ed]a.es (N.L.) 4Depa men o As onomy, Uni e si y o Texas, Aus in, TX 78731, USA; E-Mail: [email p o ec ed]xas.edu (T.G.) 5LLNL, Li e mo e, CA 94550, USA; E-Mail: [email p o ec ed] (C.I.) 6Ins i u ü Physik, Uni e si ä Ros ock, D-18051 Ros ock, Ge many; E-Mail: [email p o ec ed] (S.L.) 7Physics Dep ., Uni e si y o Ne ada, Reno, NV 89557, USA; E-Mail: cman@un .edu (R.C.M.) 8Facul y o Physics, Weizmann Ins i u e o Science, Reho o 7610001, Is ael; E-Mail: E geny[email p o ec ed] (E.S.) *Au ho o whom co espondence should be add essed; E-Mail: sand ine. e i@uni -amu. ; Tel.: +33-49128-8623. Recei ed: 30 Ap il 2014; in e ised o m: 10 June 2014 / Accep ed: 16 June 2014 / Published: 4 July 2014 Abs ac : Modeling he S a k b oadening o spec al lines in plasmas is a complex p oblem. The p oblem has a long his o y, since i plays a c ucial ole in he in e p e a ion o he obse ed spec al lines in labo a o ies and as ophysical plasmas. One di icul y is he cha ac e iza ion o he emi e ’s en i onmen . Al hough se e al models ha e been p oposed o e he yea s, he e ha e been no sys ema ic s udies o he esul s, un il now. He e, calcula ions om s ochas ic models and nume ical simula ions a e compa ed o he A oms 2014,2300 Lyman-αand -βlines in neu al hyd ogen. Also discussed a e esul s om he Helium-α and -βlines o A XVII. Keywo ds: S a k b oadening; line shapes; plasmas; nume ical simula ions; models 1. In oduc ion Line shape analysis is one o he mos impo an ools o plasma diagnos ics, as i p o ides in o ma ion on he unde lying physical p ocesses in ol ed in he line o ma ion. Wi h he inc easing numbe o applica ions in di e en a eas o plasma physics, he modeling o line b oadening om neu al o cha ged emi e s has been in pe pe ual de elopmen and emains a keys one in plasma spec oscopy [1]. In he o ma ion o a line shape, S a k b oadening is he mos compu a ionally challenging con ibu ion, since he main di icul y is o p ope ly cha ac e ize he emi e en i onmen . I in ol es a complex combina ion o a omic physics, s a is ical mechanics and de ailed plasma physics [2]. In pa icula , i is well known ha he quasi-s a ic ion app oxima ion can lead o disc epancies wi h expe imen al da a nea he line cen e . This happens whene e he elec ic mic o ields p oduced a he emi e by he su ounding ions luc ua e du ing he in e se hal -wid h a hal -maximum (HWHM) ime scale. The i s a emp s o accoun o ion dynamics in heo e ical models we e done in he 1970s, ollowed by expe imen al p oo (see he his o ic in oduc ion in [3] and he e e ences he ein). Since hen, se e al models based on s ochas ic o collisional app oaches ha e been de eloped, oge he wi h nume ical simula ions ([4] and he e e ences he ein). Necessa ily, hei limi o applicabili y, accu acy and, hus, esul s di e om one ano he , and up o now, no sys ema ic compa ison ha e exis ed [5]. The pu pose he e is o p esen c oss-compa isons o di e en models ha accoun o he ion dynamics e ec . The line shape o malism is b ie ly ecalled in Sec ion 2, which se es o in oduce no a ion. The speci ics o he a ious models and nume ical simula ions a e also p esen ed in his sec ion. We e iew he simula ions Eule –Rod igues (ER)-simula ion [6], HSTRK [7], HSTRK_ equency sepa a ion echnique (FST) [8], SimU [9,10], Xenomo ph [11] and he models Quan ST.MMM (MMM—model mic o ield me hod) [12], quasicon iguous (QC)- equency luc ua ion model (FFM) [13], mul i-elec on line-shape (MELS) [14], mul i-elec on adia o line-shape (MERL) [15,16], PPP [17], ST-PST [18] and UTPP [19] ha ha e been used o he p esen pu pose. The ion dynamics e ec on he hyd ogen Lyman-αand -βlines is discussed in Sec ion 3.1, demons a ing he di icul y o such modeling e en o hese well-known lines. In Sec ion 3.2, esul s on helium-α and -βlines o A XVII p oduced by he wo s ochas ic models (Boe ke –Iglesias–Du y (BID) [20] and FFM [4,21]) a e discussed wi h he help o he nume ical simula ion (SimU). The eliabili y o such calcula ions is o in e es in he diagnos ics o ine ial con inemen usion co e plasma condi ions. Conclusions a e gi en in Sec ion 4. A oms 2014,2301 2. Theo y, Models and Simula ions We ecall ha he line shape is gi en by: I(ω) = 1 πRe ∫∞ 0 d eiω C( )(1) whe e C( )is he au oco ela ion unc ion o he adia o dipole ope a o d, which can be exp essed in Liou ille space as: C( ) =≪d†|U( )|dρ0≫(2) whe e he double b a and ke ec o s a e de ined as usual in Liou ille space. He e, ρ0is he densi y ope a o o he emi e only a he he modynamical equilib ium and U( ) = {Ul( )}l∈Fis he ba h a e aged e olu ion ope a o o he emi e . lbelongs o a measu able unc ional space, {F}, which p o ides a s a is ical me hod o he calcula ion o a e age quan i ies. The main p oblem is o de e mine U( ). One has hus: • o ind he ime e olu ion o Ul( ) o a gi en mic o ield con igu a ion, which means sol ing he ollowing equa ion: dUl( ) d =−i[L0−d·Fl( )] Ul( ), Ul(0) = 1 (3) whe e L0 ep esen s he Liou illian o he unpe u bed adia o and d·Fl( ) ep esen s he S a k e ec ha connec s he dipole ope a o d o he mic o ield c ea ed by su ounding cha ged pa icles Fl(including ions and elec ons), •and o a e age i o e a s a is ical ensemble o he mic o ields { }l∈F. In i s gene al o m, he p oblem canno be ea ed analy ically. Ne e heless, U( )can be ob ained by nume ical simula ion in eg a ing Equa ion (3) on simula ed sampling o mic o ield his o ies. Usually, such a calcula ion is spli in o wo independen s eps [3]. Fi s , he plasma pa icle ajec o ies a e ob ained by a nume ical solu ion o New on’s equa ions o mo ion o an al e na i e me hod. Knowing he ajec o y o each pa icle, he elec ic ields a he emi e a e e alua ed and s o ed o be used in he second s ep. Then, he line shape simula ion ollows: a s ep-by-s ep in eg a ion o Equa ion (3) is pe o med using hese ield his o ies. The e olu ion ope a o o he emi e is calcula ed, and he whole p ocedu e is epea ed se e al imes in o de o a e age o e a ep esen a i e sample se o independen pe u bing ield his o ies { 1, 2... N}. As a esul , C( )is gi en by: C( ) = 1 N N ∑ i=1 Ci( )(4) and he line shape is ob ained by a Fou ie ans o m o C( ). Al hough all line shape simula ions a e based on he same scheme, we will see in he nex sec ion ha hey can di e sligh ly depending on he de ails o he models. Al e na i ely, e icien analy ical models based on undamen al assump ions and app oxima ions ha e been de eloped [1]. In he s anda d heo y (ST), he line shape calcula ion is based on he sepa a ion be ween he ions and he elec ons due o he adically di e en dynamical p ope ies o he mic o ields A oms 2014,2302 hey c ea e. Indeed, he ypical luc ua ion a e o he elec ic ield c ea ed by pe u be species pwi h a eloci y ela i e o he cen e o mass pand a densi y npis de ined by: νp= p/dp(5) whe e dp= (3/4πnp)1/3is a ypical in e pa icle dis ance. Assuming equal empe a u e o ions and elec ons and plasma neu ali y, one has [3]: νe νi ∼(µi µe)1/2 Z1/3 i.(6) Thus, he pe u ba ion due o he elec ons (wi h educed mass µe) is nea ly wo o de s o magni ude as e han ha o ions (wi h educed mass µiand cha ge Zi). This allows o ea ing he elec ons and he ions in a di e en way. The as elec ons a e assumed o pe u b he emi e by means o collisions, ea ed in he impac app oxima ion, and he slow ions a e assumed o be quasi-s a ic. This esul s in a quan um-emi e sys em pe u ba ion ope a o l=−d·Fi,l +iϕe, con aining a non-He mi ian homogeneous elec on-impac b oadening con ibu ion ϕeand he ion mic o ield in e ac ion −d·Fi,l, which has o be nume ically a e aged wi h a s a ic- ield p obabili y dis ibu ion Q(Fi), o because o iso opy, wi h dW(Fi)=4πF2 iQ(Fi)dFi. The la e can be calcula ed nume ically in he ideal gas limi o pe u bing ions [22] o using mo e sophis ica ed models ha accoun o ion co ela ions [23]. Using he se o abo e assump ions, he quasi-s a ic line shape is w i en as: Is(ω) = −1 πIm ≪d†|∫dFiQ(Fi)Gs(ω, Fi)|dρ0≫(7) in which he esol en ope a o is gi en by: Gs(ω, Fi) = (ω−L0+d·Fi−iϕe)−1.(8) Al hough he elec ons a e o en well desc ibed wi hin he impac app oxima ion, a quasi-s a ic ea men o he ions can lead o la ge e o s o plasma condi ions, such as he ion mic o ields luc ua e du ing he in e se HWHM ime scale. In he nex sec ion, we b ie ly e iew he simula ions and he models ha ha e been de eloped o accoun o he ion dynamics e ec and ha ha e been used o he p esen c oss-compa isons. 2.1. The Nume ical Simula ions The esul s om ou nume ical simula ion codes based on di e en models ha e been submi ed. They di e ei he in he way hey model he mo ions o he plasma pa icles o in he p ocedu e o he in eg a ion o he Sch ödinge equa ion. In he ER-simula ion, he simula ed plasma is an elec ically neu al ensemble o s a is ically independen cha ged pa icles made o Niions and Neelec ons mo ing along s aigh line ajec o ies wi hin a sphe ical olume. An emi e is assumed o be placed a he cen e o such a box. The empo al e olu ion o he whole sys em is measu ed along a disc e e ime axis om ze o o a de ini e numbe o imes o a ixed inc emen . E e y empo al s a e is gi en by he se o alues o he posi ions and eloci ies o he pa icles in he sys em. A e e y ime s ep, he elec ic ield p oduced by ions and A oms 2014,2303 elec ons is calcula ed using Coulomb’s law o a Debye-sc eened ield. This elec ic ield is an inpu o he Sch ödinge equa ion ha compu es he emi e ime e olu ion ope a o . Fo hyd ogen and when he no-quenching app oxima ion is conside ed, he a om s a e is desc ibed wi h he Eule –Rod igues pa ame e s [24]. The HSTRK and HSTRK_FST codes also use he Gigosos–Ca deñoso app oach [25]. Bo h codes ely on he Hege eld–Kes ing–Seidel me hod o collision- ime s a is ics [26] and compu e C( ). Depending on he app op ia e op ion, HSTRK can do an elec on only, ion only o join simula ion, bu one can also do combina ions, e.g., elec on simula ion and quasi-s a ic ions o impac elec ons and ion simula ion. Fo he Fou ie ans o m, i a long- ime exponen ial beha io is de ec ed o imes > τ, hen he con ibu ion o he Fou ie ans o m o he (τ, ∞) egion is compu ed analy ically using he de ec ed exponen ial decay and added o ∫τ 0d C( )eıω .τis de e mined ia s a -up uns, e.g., a un wi h a small numbe o con igu a ions is done o ob ain a ough idea o he HWHM and τis adjus ed o co e a leas a numbe o in e se HWHMs. The in eg al is done by Filon’s ule [27]. HSTRK_FST implemen s he equency sepa a ion echnique, which i s iden i ies he “impac ” phase space o ion pe u be s (e.g., impac pa ame e s and eloci ies), which p oduce a wid h much less (in hese uns, “much less” was 10- imes less) han he ield luc ua ion equency. This mean : HWHM(Ω) = 0.1 Ω (9) whe e he HWHM is compu ed by including all ion pe u be s wi h impac pa ame e ρand eloci y > Ωρ. Hence, he calcula ion is essen ially he same, excep ha only slow ions < Ωρa e included in he simula ion. The C( )ob ained om he simula ion o hese slow ions is hen mul iplied by e−HW HM(Ω) , and he Fou ie ans o m is aken as in HSTRK. The use o a pu e exponen ial o m o he apidly luc ua ing (impac ) pa is a consequence o using he comple e collision assump ion o sol ing he impac pa [28,29] and esul s in a C( ) ha is no co ec o e y sho imes. This is mani es ed in he ( a ) wing beha io o he HSTRK_FST p o iles and can be emedied by using he incomple e collision o mulas o he abo e-ci ed analy ical solu ions. SimU is a combina ion o wo codes: a molecula dynamics (MD) simula ion o a iable complexi y and a sol e o he e olu ion o an a omic sys em wi h he MD ield his o y used as a ( ime-dependen ) pe u ba ion. A echnical di e ence om o he nume ical simula ion me hods is he way he spec um is calcula ed. Ins ead o employing he dipole au oco ela ion unc ion ia Equa ion (1), SimU calcula es he Fou ie ans o m o he dipole ma ix:  d(ω) = ∫∞ 0 d e−iω  d( )(10) and hen uses i di ec ly ins ead o C( ): Iλ(ω)∝1 2π∑ i ρi∑ ω4 i|eλ· ⟨ d i(ω)⟩|2(11) whe e eλis he ligh pola iza ion di ec ion and each ini ial s a e iis assigned a popula ion ac o ρi. Simila ly o o he me hods, his p ocedu e is epea ed many imes and a e aged (c . Equa ion (4)). The ecen ly de eloped code, Xenomo ph, is based on he models o Gigosos and González [30], whe e a s aigh line assump ion is made. A gene al Sch ödinge sol e desc ibed in [31] is used o A oms 2014,2304 ob ain he eigen alues En( )and eigen ec o s |n( )⟩a e e y ime s ep o he simula ion. The emi e ime e olu ion ope a o is hen e alua ed: Ul( + ∆ ) = {∑ n e−iEn( )∆ /¯hn( )⟩⟨n( )|}Ul( )(12) and is used o ob ain he dipole ma ix. The Fou ie ans o m o he la e is compu ed o ob ain he line shape unc ion, as is done in SimU (c . Equa ions (10) and (11)). 2.2. The Models The main di icul y in in oducing he ion dynamics in he S a k line shape calcula ions is o de elop a model ha p o ides a su icien ly accu a e solu ion o he e olu ion Equa ion (2) assuming an idealized s ochas ic p ocess ha conse es he s a is ical p ope ies o he “ eal” in e ac ion be ween he mic o ields and he adia ing a om. A success ul model de eloped o neu al emi e s— he model mic o ield me hod (MMM), due o B issaud and F isch [32,33]—in ol es s ochas ic ields ha a e cons an in a gi en ime in e al and suddenly jump om one alue o he nex one a andom imes. The ampli udes o he ield sequences a e de e mined in o de o be consis en wi h he s a ic p ope ies o he mic o ield, i.e., he s a ic- ield p obabili y dis ibu ion Q(F). The jumping equency ν(F)has o be chosen p ope ly in o de o ep oduce he dynamics p ope ies o he mic o ields ep esen ed by hei au oco ela ion unc ion <F( )·F(0) >. In Quan S .MMM, MMM ( o ions) is combined wi h a quan um-s a is ical app oach o calcula e p essu e b oadening due o plasma elec ons. The pe u ba ion by elec ons is conside ed o second o de in he po en ial [34,35]. MELS and MERL a e based upon he BID model. The la e de i es om he MMM, bu i s o mula ion is based on s a is ical mechanics [36] and p o ides a uni ied desc ip ion o adia i e and anspo p ope ies o cha ged emi e s [20]. The s ochas ic line shape is w i en as: Id(ω) = −1 πIm ≪d†|∫dFQ(Fi)GBID(ω, Fi) 1 + iν(ω)∫dFQ(Fi)GBID(ω, Fi)|dρ0≫(13) in which he esol en is gi en by: GBID(ω, Fi) = (ω−L0+d·Fi−iν(ω))−1(14) The jumping equency ν(ω)is chosen as: ν(ω) = ν0 1 + iωτ .(15) whe e he wo pa ame e s ν0and τa e de ined in his model by he low- and high- equency limi s o he momen um au oco ela ion unc ion. He e, τis assumed o be null. Ano he app oach is he equency luc ua ion model (FFM), on which he PPP code and, ecen ly, he QC-FFM code ely. The la e is a hyb id model using he quasi-con iguous app oxima ion [37] o H-like ansi ions and he FFM o modeling he mic o ield dynamics e ec . The FFM elies on a di e en idealiza ion o he s ochas ic p ocess han MMM and BID. He e, he quan um sys em pe u bed A oms 2014,2305 by a ime-dependen mic o ield beha es like a se o ield-d essed wo-le el ansi ions (SDT) subjec o a collision- ype mixing p ocess. Mo e p ecisely, he luc ua ion mechanism o hese SDT obeys a s a iona y Ma ko p ocess de ined by he ins an aneous p obabili y o s a es pj=aj/∑kak(ajbeing he in ensi y o he SDT, j) and he ansi ion a es be ween hese s a es Wk,j =−Γjδk,j +Wk,j, whe e Γk,j =νδi,j and Wk,j =νpj. The ypical luc ua ion a e νFFM o he elec ic ield, gi en by Equa ion (5), is used. Wo king in he Liou ille space o he d essed wo-le el adia o s, he line shape is w i en as [38]: Id(ω) = 1 πRe ∑ j,k i≪Dk|GFFM(ω)|Djpj≫(16) wi h he esol en : GFFM(ω) = (ω−Lω+iW)−1(17) whe e Lωis he Liou ille ope a o in ol ing he ansi ion ene gies o he SDT (ωi) and Dia e he ma ix elemen s o he dipole momen o he SDT. Due o he pa icula o m o he ma ix o ansi ion a es W, he dynamic line shape is w i en as [4]: Id(ω) = ∑kak πRe ∑k pk ν+i(ω−ωk) 1−ν∑k pk ν+i(ω−ωk) (18) Despi e he ac ha he wo s ochas ic models lead o di e en unc ional o ms, i ollows ha bo h BID and FFM eco e he s a ic limi o νBID = 0 in Equa ion (14) and o νFF M = 0 in Equa ion (18). In he opposi e limi , bo h models eco e he as luc ua ion limi (ν→ ∞) ha should app oxima e he “no ions” p o ile. Howe e , BID eco e s he impac limi in he line cen e whene e νis la ge, while he FFM does no (see [39] o a mo e de ailed discussion). We no e ha QC-FFM uses he FFM app oxima ion o ions and elec ons alike. Fo he la e , co ec ly app oaching he impac app oxima ion in he as luc ua ion limi becomes especially impo an . To his end, a modi ica ion o he e ec i e luc ua ion a e was in oduced: ˜ν=ν+ν2 ν0 (19) whe e ν0is an empi ically ob ained cons an ( o de ails, see [13]). Two o he models based on he collisional app oach ha e been used, oo. The ST-PST model is based on he s anda d heo y wi h a numbe o op ions. Speci ically, apa om he pu e ST esul s, ST-PST can (and by de aul does) also compu e he esul s o ST wi h pene a ing collisions co ec ly accoun ed o analy ically [18]. In addi ion, an FST-FFM calcula ion is also done [8]: i s , an Ωis de e mined, exac ly as desc ibed abo e o HSTRK_FST. Nex , he FFM is applied o he ield ha excludes he as , impac pa . Las , he wo p o iles a e con ol ed. As a esul , he impac limi is co ec ly buil in and eco e ed, hence ex ending he FFM alidi y wi hou sac i icing i s speed. No e, howe e , ha wi h he cu en FST implemen a ion, which uses he comple ed collision assump ion o he impac phase space, he a wings a e no accu a e, as al eady discussed. The UTPP code is de o ed o he calcula ion o hyd ogen line shapes in egimes whe e he impac app oxima ion o ions is easonably accu a e. Such a egime is a ained o lines wi h a low p incipal A oms 2014,2306 quan um numbe in magne ic usion expe imen s in he absence o Dopple b oadening (Dopple - ee line shape models we e equi ed o adia ion anspo simula ions, e.g., [40]). In UTPP, a line shape is calcula ed using he ollowing o mula: I(ω) = 1 πRe ≪d†|1 s+iL0+K(s)|dρ0≫(20) whe e s=−iω and K(s)is a collision ope a o calcula ed in a amewo k simila o ha used in he Voslambe uni ied heo y (Bogoliubo -Bo n-G een-Ki kwood-Y on (BBGKY) hie a chy), bu he e adap ed o ions [19]. The main ad ance wi h espec o he uni ied heo y is ha he collision ope a o accoun s o he ini e li e ime o he a om du ing each collision; his li e ime yields an e ec i e ange o he ac ion o he mic o ield o he o de o /¯γ, whe e ¯γis a ypical ma ix elemen o he collision ope a o (see he discussion in [41]). This model (and i s adap a ion o elec ons) does no lead o a di e gen collision ope a o i he Debye leng h is assumed in ini e, which is in con as o s anda d hyd ogen models (see [42]); his makes i sui able o he p esen ed cases, p o ided he pe u bing species unde conside a ion is s ongly dynamic. 3. Compa isons and Discussion To es he accu acy o he di e en nume ical codes based ei he on s ochas ic and collisional models o nume ical simula ions, calcula ions o s anda dized case p oblems we e ca ied ou and analyzed [5]. A p eselec ed se o ansi ions on a g id o elec on densi ies (ne) and empe a u es (T=Te=Ti) ha e been p oposed, and o each case, he a omic and plasma models ha e been speci ied. In his way, a ious con ibu ions ha can a ec he S a k b oadened line shape, such as he in luence o pa icle co ela ions on elec ic mic o ields, he e ec s o ex e nal ields, he high-n me ging wi h con inuum o he sa elli e b oadening, ha e been in es iga ed. Fo he p esen pu pose, we will only ocus on cases whe e he ion dynamics e ec was s udied. 3.1. Hyd ogen Lyman-αand Lyman-βLines The ollowing examples conside he hyd ogen Lyman-αand Lyman-βlines in an ideal plasma consis ing o p o ons o elec on densi ies ne= 1017 −1019 cm−3and empe a u es T= 1 −100 eV. These cases a e no necessa ily p ac ical, bu pe mi basic compa isons o assess he in luence o ion dynamics on he line p o iles. He e, only pu e ionic linea S a k e ec is conside ed (∆n= 0 in e ac ions a e igno ed) and he ine s uc u e is no aken in o accoun . The concep o ideal plasma means ha unsc eened pa icles mo ing along s aigh pa h ajec o ies a e conside ed in he nume ical simula ions, and he Hol sma k s a ic- ield dis ibu ion unc ion [22] is used in he models. An o e all compa ison o he esul s is p esen ed in Figu e 1. Fo each subcase (de e mined by a combina ion o (ne, T)) and o each code, a ios be ween he ull-wid h a hal -maximum (FWHM) and an a e age o FWHM o all submi ed esul s ha e been e alua ed [5]: Ri=FWHM <FWHM >.(21) A oms 2014,2307 The g aph is di ided in wo egions: he le side co esponds o esul s o he Lyman-αline and he igh side o he Lyman-βline. Each egion is di ided in o h ee sub- egions ha co espond o he h ee densi ies chosen. Finally, in each sub- egion, each se o esul s co esponds o he empe a u es, T= 1,10,100 eV, espec i ely. Fo he Lyman-αcase, he esul s p esen a la ge dispe sion, de ia ing om he a e age by mo e han a ac o o i e in each di ec ion. In con as , he sca e o he Lyman-β shows a a he good ag eemen be ween he codes. In ac , hese wo lines p esen a comple ely di e en beha io conce ning he ion dynamics e ec . Figu e 1. O e all compa ison o he wo kshop esul s o he ion dynamics e ec on Lyman-αand -βhyd ogen lines. Fo each subcase, i.e., di e en pai s o (ne, T), he sca e o a ios be ween he di e en esul s and an a e age alue is plo ed. The di e en symbols co espond o: (black do ) SimU; ( ed squa e) UTPP; (blue iangle) PPP; (blue as e isk) Xenomo ph; (cyan open iangle) HSTRK; (cyan iangle) HSTRK_FST; ( ed diamond) ER-simula ion; (g een ci cle) Quan ST.MMM; (black c oss) QC-FFM. 5 6 7 8 0.1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 10 FWHM/<FWHM> n e (cm -3 ) 10 17 cm -3 H Lyman- β H Lyman- α 10 19 cm -3 10 18 cm -3 10 17 cm -3 10 18 cm -3 10 19 cm -3 T=1eV T=10eV T=100eV 3.1.1. The Lyman-αLine The s a ic S a k e ec o he Lyman-αline (as all he ∆n=n−n′= 1 lines, whe e nand n′a e he p incipal quan um numbe o he uppe and lowe s a es, espec i ely) ea u es a s ong unshi ed componen ha is highly sensi i e o he ion dynamics e ec . Thus, e en hough he Lyman-αline is he simples case om he a omic s uc u e poin o iew, i p esen s a non- i ial S a k-b oadening beha io . In Figu e 2, only esul s om he nume ical simula ions a e plo ed o he sake o cla i y. One sees ha in he ange o 1 o 100 eV, he simula ions ei he p edic ha he wid h inc eases when he plasma empe a u e inc eases ( o he ixed densi y ne= 1019 cm−3, hey p esen a empe a u e dependency as ∼T1/3) o p edic ha he wid h is mos ly insensi i e o he empe a u e’s ise ( o he ixed densi y ne= 1017 cm−3). Conce ning he dependence on he plasma densi y, he wid h, which is mainly due o he wid h o he cen al componen o T= 1 eV, inc eases as n1/3 e. Fo T= 100 eV, he esul s show a n2/3 edependence, co esponding o he quasi-s a ic beha io o he la e al componen s [2]. We men ion, howe e , ha he cu o o he Coulomb in e ac ion a a ini e box size may no accu a ely ep oduce an ideal plasma [42]. A oms 2014,2314 Figu e 10. The He-αline, he s ong componen o T= 1 keV and ne= 5 ×1023 cm−3: SimU (black ci cles); FFM wi h ν= 3 eV (solid blue); ν= 5.62 eV (solid ed); and ν= 8 eV (solid black). 0.001 0.01 0.1 no malized in ensi y 3160315031403130312031103100 ene gy (eV) Finally, he He-βline is p esen ed in Figu e 11. A he chosen plasma condi ions, as he S a k spli ing o he He-βquasi-s a ic line shape is g ea e han he luc ua ion a e and he elec on wid h is la ge , he ion dynamics e ec is less p onounced han on he He-α. Figu e 11 shows SimU, BID and FFM in a he good ag eemen ela i e o he disc epancies o hei quasi-s a ic p o iles. The measu e o he dynamics- o-s a ic ela i e dep h is de ined by: Dd−s=Idyn(ω0)−Is a (ω0) Idyn(ω0)(23) The e is a ai ly good ag eemen be ween he BID and he FFM (see Table 2). Figu e 11. The He-βline o T= 1 keV and (a)ne= 5×1023 cm−3; (b)ne= 2×1024 cm−3. S a ic ions: MERL ( ed do ), PPP (blue do ); SimU (black do ); BID (solid ed); FFM (solid blue). 50x10 -3 40 30 20 10 0 no malized in ensi y 37203700368036603640 ene gy(eV) MERL QS BID PPP QS FFM SimU a) 20x10 -3 15 10 5 0 no malized in ensi y 376037403720370036803660364036203600 ene gy(eV) MERL QS BID PPP QS FFM SimU b) A oms 2014,2315 Table 2. Dynamics- o-s a ic ela i e dip (%) measu ed on he a gon He-βline o T= 1 keV. Models BID FFM Ne= 5 ×1023 cm−358 57 Ne= 1 ×1024 cm−350 51 Ne= 2 ×1024 cm−347 48 4. Conclusions Line shape calcula ions om di e en nume ical codes ha accoun o he ion dynamics e ec we e p esen ed. To es he accu acy o he di e en codes, s anda dized case p oblems ha e been chosen and a sys ema ic c oss-compa ison has been done. Resul s om ou nume ical simula ions based on di e en algo i hms and se en models elying on ei he s ochas ic o collisional p ocesses, ha e been hen submi ed. Su p isingly, he esul s ob ained on he hyd ogen Lyman-αline in an ideal OCP plasma consis ing o p o ons p esen s a la ge dispe sion. While he nume ical simula ions show a ela i ely good ag eemen be ween each o he , he FFM and MMM models sys ema ically display a weake wid h han he a e aged esul s. This can be explained by an incomple e desc ip ion o he ion dynamics e ec on he cen al componen o his line. The de ailed s udy on he in luence o he mic o ields di ec ionali y in he line shape p esen ed in his olume, [44] o o he me hods discussed he e can help imp o e he modeling o lines wi h unshi ed componen s. The o e es ima e o he UTPP code based on a collisional app oach is explained by an incomple e desc ip ion o ion s a ic e ec s. The esul s ob ained on he H Lyman-βline p esen a be e ag eemen be ween all codes. Conce ning he ion dynamics e ec on he a gon He-αand -βlines, BID and FFM show a di e en beha io ha has been a ibu ed, up o now, o nume ical inaccu acies, due o he e y weak alue o he line in ensi ies. The ecen ly de eloped nume ical simula ion code, SimU, could no help disc imina e be ween he wo models, bu highligh ed ano he p oblem: i seems ha di e en alues o luc ua ion a es ha e o be used o ep oduce di e en po ions o he simula ed p o ile. As bo h he linea and quad a ic S a k e ec , which a e linked o he weak and s ong alues o mic o ields, espec i ely, a e in ol ed in p oducing he shape o his line, one can wonde i a equency- (o ield-) dependen luc ua ion a e is needed o gi e a be e desc ip ion o ion dynamics on his line. Acknowledgmen s The au ho s would like o acknowledge he In e na ional A omic Ene gy Agency B.J. B aams and H.-K. Chung o he o ganiza ional and inancial suppo o his wo kshop. Au ho Con ibu ions The p esen wo k is based on codes de eloped by all au ho s, who also pa icipa ed in all aspec s o his wo k. A oms 2014,2316 Con lic s o In e es The au ho s decla e no con lic s o in e es . Re e ences 1. G iem, H.R. P inciples o Plasma Spec oscopy; Camb idge Uni e si y P ess: Camb idge, UK, 1997. 2. G iem, H.R. Spec al Line B oadening by Plasmas; Academic P ess: New Yo k, NY, USA, 1974; ISBN:0-12-302850-7. 3. S ambulchik, E.; Ma on, Y. Plasma line b oadening and compu e simula ions: A mini- e iew. High Ene gy Densi y Phys. 2010,6, 9–14. 4. Calis i, A.; Mossé, C.; Fe i, S.; Talin, B.; Rosmej, F.; Bu eye a, L.A.; Lisi sa, V.S. Dynamic S a k b oadening as he Dicke na owing e ec . Phys. Re . E S a . Nonlin. So Ma e Phys. 2010,81, 016406, doi:10.1103/PhysRe E.81.016406. 5. S ambulchik, E. 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