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. Re iew o he 1s Spec al Line Shapes in Plasmas code compa ison wo kshop.
High Ene gy Densi y Phys. 2013,9, 528–534.
6. Gigosos, M.A.; Gonzalez, M.A. Commen on “A s udy o ion-dynamics and co ela ion e ec s o
spec al line b oadening in plasmas: K-shell lines”. J. Quan . Spec osc. Radia . T ans . 2007,
105, 533–535.
7. Alexiou, S. S a k b oadening o hyd ogen lines in dense plasmas: Analysis o ecen expe imen s.
Phys. Re . E S a . Nonlin. So Ma e Phys. 2005,71, 066403, doi:10.1103/PhysRe E.71.066403.
8. Alexiou, S. Implemen a ion o he F equency Sepa a ion Technique in gene al lineshape codes.
High Ene gy Densi y Phys. 2013,9, 375–384.
9. S ambulchik, E.; Ma on, Y. A s udy o ion-dynamics and co ela ion e ec s o spec al line
b oadening in plasma: K-shell lines. J. Quan . Spec osc. Radia . T ans . 2006,99, 730–749.
10. S ambulchik, E.; Alexiou, S.; G iem, H.R.; Kepple, P.C. S a k b oadening o high p incipal
quan um numbe hyd ogen Balme lines in low-densi y labo a o y plasmas. Phys. Re . E S a .
Nonlin. So Ma e Phys. 2007,75, 016401, doi:10.1103/PhysRe E.75.016401.
11. Gomez, T.A.; Mancini R.C.; Mon gome y M.H.; Winge D.E. Whi e dwa line shape heo y
including ion dynamics and asymme ies. 2014, in p epa a ion.
12. Lo enzen, S. Compa a i e S udy on Ion-Dynamics o B oadening o Lyman Lines in Dense
hyd ogen Plasmas. Con ib. Plasma Phys. 2013,53, 368–374.
13. S ambulchik, E.; Ma on, Y. Quasicon iguous equency- luc ua ion model o calcula ion o
hyd ogen and hyd ogen-like S a k-b oadened line shapes in plasmas. Phys. Re . E S a . Nonlin.
So Ma e Phys. 2013,87, 053108, doi:10.1103/PhysRe E.87.053108.
14. Iglesias, C.A.; Vijay, S. Robus algo i hm o compu ing quasi-s a ic S a k b oadening o spec al
lines. High Ene gy Densi y Phys. 2010,6, 399–405.
15. Wol z, L.A.; Hoope , C.F., J . Calcula ion o spec al line p o iles o mul ielec on emi e s in
plasmas. Phys. Re . A 1988,38, 4766–4771.
16. Mancini, R.C.; Kilc ease, D.P.; Wol z, L.A.; Hoope , C.F. Calcula ional Aspec s o he S a k Line
B oadening o Mul ielec on Ions in Plasmas. Compu . Phys. Commun. 1991,63, 314–322.
A oms 2014,2317
17. Calis i, A.; Khel aoui, F.; S amm, R.; Talin, B.; Lee, R.W. Model o he line shapes o complex
ions in ho and dense plasmas. Phys. Re . A 1990,42, 5433–5440.
18. Alexiou, S.; Poqué usse, A. S anda d line b oadening impac heo y o hyd ogen including
pene a ing collisions. Phys. Re . E S a . Nonlin. So Ma e Phys. 2005,72, 046404, doi:10.1103/
PhysRe E.72.046404.
19. Rosa o, J.; Capes, H.; S amm, R. In luence o co ela ed collisions on S a k-b oadened lines in
plasmas. Phys. Re . E S a . Nonlin. So Ma e Phys. 2012,86, 046407, doi:10.1103/
PhysRe E.86.046407.
20. Boe cke , D.B.; Iglesias, C.A.; Du y, J.W. Radia i e and anspo p ope ies o ions in s ongly
coupled plasmas. Phys. Re . A 1987,36, 2254–2264.
21. Talin, B.; Calis i, A.; Godbe , L.; S amm, R.; Lee, R.W.; Klein, L. F equency- luc ua ion model
o line shape calcula ions in plasma spec oscopy. Phys. Re . A 1995,51, 1918–1928.
22. Hol sma k, J. Übe die Ve b ei e ung on Spek allinien. Ann. Phys. 1919,58, 577–630.
23. Iglesias, C.A.; Boe cke , D.B.; Iglesias, C.A. Elec ic ield dis ibu ions in s ongly coupled
plasmas. Phys. Re . A 1985,31, 1681–1686.
24. Gigosos, M.A.; Gonzalez, M.A.; Ca deñoso, V. Compu e simula ed Balme -alpha, -be a and
-gamma S a k line p o iles o non-equilib ium plasmas diagnos ics. Spec ochim. Ac a Pa B
2003,58, 1489–1504.
25. Gigosos, M.A.; Ca deñoso, V. New plasma diagnos ics able o hyd ogen S a k b oadening
including ion dynamics. J. Phys. B-A . Mol. Op . Phys. 1996,29, 4795–4838.
26. Hege eld , G.; Kes ing, V. Collision- ime simula ion echnique o p essu e-b oadened spec al
lines wi h applica ions o Ly-α.Phys. Re . A 1988,37, 1488–1496.
27. Filon, L.N.G. On a quad a u e o mula o igonome ic in eg als. P oc. R. Soc. Edinb. 1928,
49, 38–47.
28. P ennig, H. On he Time-E olu ion Ope a o in he Semiclassical Theo y o S a k-B oadening o
hyd ogen Lines. J. Quan . Spec osc. Radia . T ans . 1972,12, 821–837.
29. Lisi sa, V.S.; Sholin, G.V. Exac Solu ion o he P oblem o he B oadening o he hyd ogen Spec al
Lines in he One-Elec on Theo y. So . J. Exp. Theo . Phys. 1972,34, 484–489.
30. Gigosos, M.A.; González, M.A. Calcula ions o he pola iza ion spec um by wo-pho on
abso p ion in he hyd ogen Lyman-αline. Phys. Re . E S a . Nonlin. So Ma e Phys. 1998,
58, 4950–4959, doi:10.1103/PhysRe E.58.4950.
31. Dju o i´
c, S.; ´
Ci i¨
san, M.; Demu a, A.V.; Demchenko, G.V.; Nikoli´
c, D.; Gigosos, M.A.;
Gonzalez, M.Á. Measu emen s o HβS a k cen al asymme y and i s analysis h ough s anda d
heo y and compu e simula ions. Phys. Re . E S a . Nonlin. So Ma e Phys. 2009,79, 046402,
doi:10.1103/PhysRe E.79.046402.
32. F isch, U.; B issaud, A. Theo y o S a k b oadening—I soluble scala model as a es . J. Quan .
Spec osc. Radia . T ans . 1971,11, 1753–1766.
33. F isch, U.; B issaud, A. Theo y o S a k b oadening—II exac line p o ile wi h model mic o ield.
J. Quan . Spec osc. Radia . T ans . 1971,11, 1767–1783.
34. Gün e , S.; Hi zschke, L.; Röpke, G. Hyd ogen spec al lines wi h he inclusion o dense-plasma
e ec s. Phys. Re . A 1991,44, 6834–6844.
A oms 2014,2318
35. Lo enzen, S.; Oma , B.; Zammi , M.C.; Fu sa, D.V.; B ay, I. Plasma p essu e b oadening o
ew-elec on emi e s including s ong elec on collisions wi hin a quan um-s a is ical heo y. Phys.
Re . E S a . Nonlin. So Ma e Phys. 2014,89, 023106, doi:10.1103/PhysRe E.89.023106.
36. Du y, J.W. Spec al Lines Shapes; Wende, B., Ed.; De G uy e : New Yo k, NY, USA, 1981.
37. S ambulchik, E.; Ma on, Y. S a k e ec o high-n hyd ogen-like ansi ions: Quasi-con iguous
app oxima ion. J. Phys. B-A . Mol. Op . Phys. 2008,41, 095703, doi:10.1088/0953-4075/
41/9/095703.
38. Mossé, C.; Calis i, A.; S amm, R.; Talin, B.; Lee, R.W.; Klein, L. Redis ibu ion o esonance
adia ion in ho and dense plasmas Phys. Re . A 1999,60, 1005–1014.
39. Iglesias, C. E icien algo i hms o s ochas ic S a k-p o ile calcula ions. High Ene gy
Densi y Phys. 2013,9, 209–221.
40. Rosa o, J.; Rei e , D.; Ko o , V.; Ma ande , Y.; Capes, H.; Godbe -Mou e , L.; Koubi i, M.;
S amm, R. P og ess on adia i e ans e modelling in op ically hick di e o plasmas.
Con ib. Plasma Phys. 2010,50, 398–403.
41. Rosa o, J.; Capes, H.; S amm, R. Di e gence o he S a k collision ope a o a la ge impac
pa ame e s in plasma spec oscopy models. Phys. Re . E S a . Nonlin. So Ma e Phys. 2013,88,
035101:1–035101:3.
42. Rosa o, J.; Capes, H.; S amm, R. Ideal Coulomb plasma app oxima ion in line shape models:
P oblema ic issues. A oms 2014,2(2), 277–298.
43. Lisi sa, V. P i a e communica ion.
44. Calis i, A.; Demu a, A.V.; Gigosos, M.A.; Gonzà ˛alez-He e o, D.; Iglesias, C.A.; Lisi sa, V.S.;
S ambulchik, E. In luence o Mic o ield Di ec ionali y on Line Shapes. A oms 2014,2, 259–276.
45. Chung, H.-K.; Lee, R.W. Applica ion o NLTE popula ion kine ics. High Ene gy Densi y Phys.
2009,5, 1–14.
46. Woolsey, N.C.; Hammel, B.A.; Keane, C.J.; As aw, A.; Back, C.A.; Mo eno, J.C.; Nash, J.K.;
Calis i, A.; Mossé, C.; S amm, R.; e al. E olu ion o elec on empe a u e and elec on densi y
in indi ec ly d i en sphe ical implosions. Phys. Re . E S a . Nonlin. So Ma e Phys. 1996,56,
2314, doi:10.1103/PhysRe E.56.2314.
47. Haynes, D.A.; Ga be , D.T.; Hoope , C.F.; Mancini, R.C.; Lee, Y.T.; B adley, D.K.; Dele ez, J.;
Eps ein, R.; Jaanimagi, P.A. E ec s o ion dynamics and opaci y on S a k-b oadened a gon line
p o iles. Phys. Re . E S a . Nonlin. So Ma e Phys. 1996,53, 1042–1050.
48. Mancini, R.C.; Iglesias, C.A.; Fe i, A.; Calis i, S.; Flo ido, R. The e ec s o imp o ed sa elli e
line shapes on he a gon Heβspec al ea u e. High Ene gy Densi y Phys. 2013,9, 731–736.
c
2014 by he au ho s; licensee MDPI, Basel, Swi ze land. This a icle is an open access a icle
dis ibu ed unde he e ms and condi ions o he C ea i e Commons A ibu ion license
(h p://c ea i ecommons.o g/licenses/by/3.0/).