A oms 2014,2, 259-276; doi:10.3390/a oms2020259
OPEN ACCESS
a oms
ISSN 2218-2004
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A icle
In luence o Mic o ield Di ec ionali y on Line Shapes
Anne e Calis i 1,*, Alexande V. Demu a 2, Ma co A. Gigosos 3, Diego González-He e o 3,
Ca los A. Iglesias 4, Vale y S. Lisi sa 2and E geny S ambulchik 5
1Aix Ma seille Uni e si é, CNRS, PIIM UMR 7345, 13397 Ma seille, F ance
2Na ional Resea ch Cen e “Ku cha o Ins i u e”, Moscow 123182, Russia;
E-Mails: [email p o ec ed] (A.D.); lisi sa@yandex. u (V.S.L.)
3Depa amen o de Óp ica, 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 (D.G.-H.)
4Law ence Li e mo e Na ional Labo a o ies, P.O. Box 808, Li e mo e, CA 94550, USA;
E-Mail: [email p o ec ed] (C.A.I.)
5Facul 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: anne e.calis i@uni -amu. ;
Tel.: +33-4-912-827-19.
Recei ed: 15 Ap il 2014; in e ised o m: 2 June 2014 / Accep ed: 5 June 2014 /
Published: 19 June 2014
Abs ac : In he amewo k o he Spec al Line Shapes in Plasmas Code Compa ison
Wo kshop (SLSP), la ge disc epancies appea ed be ween he di e en app oaches o accoun
o ion mo ion e ec s in spec al line shape calcula ions. Fo a be e unde s anding o hese
e ec s, in he second edi ion o he SLSP in Augus , 2013, wo cases we e dedica ed o he
s udy o he ionic ield di ec ionali y on line shapes. In his pape , he e ec s o he di ec ion
and magni ude luc ua ions a e sepa a ely analyzed. The e ec s o wo a ian s o elec ic
ield models, (i) a pu e o a ing ield wi h cons an magni ude and (ii) a ime-dependen
magni ude ield in a gi en di ec ion, oge he wi h he e ec s o he ime-dependen ionic
ield on shapes o he He II Lyman-αand -βlines o di e en densi ies and empe a u es,
a e discussed.
Keywo ds: spec al line shapes in plasmas; ion dynamics e ec s; ion mic o ield luc ua ions
A oms 2014,2260
1. In oduc ion
The e ec o ionic ield luc ua ions on spec al line shapes o hyd ogen and hyd ogen-like emi e s
has been s udied o a long ime by di e en g oups. In he 1970s, he obse ed de ia ions be ween
expe imen s and heo ies we e a ibu ed o ion mo ion. A he same ime, he i s a emp s o include
ion mo ion e ec s in heo ies appea ed [1–5], and he expe imen al p oo on hyd ogen had been ob ained
nea ly concomi an ly by Wiese and co-wo ke s [6,7]. The i s N-body molecula dynamics simula ions
appea ed in he he la e 1970s [8].
Nowadays, la ge di e ences s ill appea be ween he a ious app oaches o ake in o accoun he
e ec o ion dynamics in he line shape calcula ions. Fo a be e unde s anding o he o igin o hese
di e ences highligh ed du ing he SLSP con e ence in 2012 [9], a s udy o he ionic ield di ec ionali y
on line shapes has been p oposed a he second edi ion o he SLSP wo kshop in 2013 [10].
In his pape , we epo he s udy o he e ec s o he di ec ion and magni ude luc ua ions o he ionic
ield analyzed sepa a ely. To his end, “ o a ional” and “ ib a ional” mic o ields ha e been de ined as:
~
F o ( ) = F0
~
F( )
|~
F( )|(1)
and:
~
F ib( ) = ~nz|~
F( )|(2)
wi h F0 he Hol sma k ield (F0= 2.603Zen2/3wi h Z he ionic cha ge numbe , e he elemen a y
cha ge in s a coulomb, n he ion numbe densi y in cm−3) and ~
F( ) he ield c ea ed a he emi e
by he su ounding ion cha ges in e ac ing h ough an elec on-Debye-sc eened Coulomb po en ial.
The conside ed plasmas consis o an impu i y (o a small pe cen age) o hyd ogenic helium in a bulk o
p o ons. All o he cha ged pa icles in e ac oge he h ough an e ec i e Yukawa po en ial o accoun
o he in luence o elec ons on he ionic s uc u e.
The e ec s o hese wo a ian s o elec ic ield models oge he wi h he e ec s o ~
F( )on spec al
line shapes a e discussed o he He II Lyman-αand -βlines o di e en densi ies and empe a u es.
The spec al p o iles o hese wo lines p esen e y di e en beha io s when he luc ua ing elec ic
ields a e aken in o accoun due o he exis ence o absence o an unshi ed cen al componen . They a e
good candida es o s udies compa ing he di e en app oaches, such as simula ion modeling [11–14] o
kine ics models [15–19], de eloped o accoun o ionic ield luc ua ions.
2. Spec al Line Shape Calcula ions
The spec al line shape is ela ed o he Fou ie ans o m o he adia o dipole ope a o co ela ion
unc ion, Cdd( ) =<~
d( )·~
d(0) >, by:
I(ω) = <e1
πZ∞
0
d eiω <~
d( )·~
d(0) >(3)
The co ela ion unc ion, Cdd( ), o he adia o dipole ope a o , ~
d, can be w i en in Liou ille space
as [20,21]:
Cdd( ) =~
d†|{Ul( )}ba h|~
dρ0(4)
A oms 2014,2261
whe e he double b a and ke ec o s a e de ined as usual in Liou ille space, ρ0is he equilib ium densi y
ma ix and {Ul( )}ba h is he ba h-a e aged e olu ion ope a o o he emi e . Ul( )is a solu ion o he
ollowing s ochas ic Liou ille equa ion (SLE):
dUl( )
d =−iLl.Ul( )(5)
wi h he condi ion U(0) = I, he iden i y ope a o . Lldesigna es he Liou illian o he adia o in
he ba h. We ha e Ll=L0+l( ), whe e L0is he Liou illian o he ee adia o and l( )a andom
pe u ba ion o he he mal ba h ( he plasma). The mos di icul pa o a line-b oadening p oblem is o
iden i y co ec ly he en i onmen o he emi e , l( ). In pa icula , accoun ing o he luc ua ions o
elec ic ields p oduced a emi e s, by mo ing elec ons and ions, has been o cons an in e es o bo h
he expe imen al and heo e ical poin s o iew since he 1960s [22]. In he s anda d heo y [23], due o
hei g ea di e ence o mass, ions and elec ons a e ea ed in di e en ways, leading o:
Ll( ) = L0−~
d·~
Fl( )−iΦ(6)
whe e ~
Fl( )is he elec ic ield p oduced by su ounding ions in a gi en con igu a ion land Φis he
elec onic collisional ope a o .
In his pape , in o de o enhance he ionic ield luc ua ion e ec s on spec al line shapes, he
elec ic ields p oduced a emi e s by mo ing elec ons is neglec ed. Thus, Ll( )is educed o:
Ll( ) = L0−~
d·~
Fl( ). Addi ionally, calcula ions o he spec al p o ile ha e been pe o med assuming
h ee a ian s o he luc ua ing elec ic ield, ~
Fl( ):
•The ime-dependen ield c ea ed by he p o ons in e ac ing h ough a Debye-sc eened
Coulomb po en ial;
•A pu e o a ing ield ollowing Equa ion (1);
•A pu e ib a ing ield ollowing Equa ion (2).
2.1. The Di e en Codes and App oaches
A s aigh o wa d way o ake in o accoun he luc ua ions o he elec ic ield a he emi e is
he nume ical simula ion, which sol es he Sch ödinge equa ion desc ibing he ime e olu ion o he
emi e wa e unc ions in he ime-dependen ield o su ounding cha ges p oduced by molecula
dynamics (MD) o an al e na i e echnique and hen a e age o e a s a is ically ep esen a i e numbe o
con igu a ions o ob ain he inal esul . In he ollowing, he MD simula ion esul s will be conside ed
as benchma ks.
Fou ypes o simula ion codes a e in ol ed in he p esen s udy:
•The SimU code [11]: The pe u bing ields a e simula ed by he pa icle ield gene a o , whe e
he mo ion o a ini e numbe o plasma pa icles (elec ons and ions) is calcula ed assuming
ha classical ajec o ies a e alid. Then, using his ield as a pe u ba ion, he adia o dipole
oscilla ing unc ion is calcula ed by he Sch ödinge sol e . Finally, using he as Fou ie
ans o ma ion (FFT) me hod, he powe spec um o he adia o dipole oscilla ing unc ion is
e alua ed, gi ing he spec al line shape. The esul s o epea ed uns o his p ocedu e a e hen
A oms 2014,2262
a e aged o ob ain a smoo h spec um. Al hough, in p inciple, he pa icle ield gene a o may
accoun o in e ac ions be ween all pa icles, o he cases p esen ed in his s udy, pe u bing
p o ons we e modeled as educed-mass Debye quasipa icles in e ac ing only wi h he s a iona y
adia o ia he Debye po en ial.
•The BinGo code [12]: This code uses s anda d classical MD simula ion o compu e he pe u bing
ields. In his wo k, he plasma model consis s o classical poin ions in e ac ing oge he h ough
a Coulombic po en ial sc eened by elec ons and localized in a cubic box wi h pe iodic bounda y
condi ions. New on’s equa ions o pa icle mo ion a e in eg a ed by using a eloci y-Ve le
algo i hm using a ime-s ep consis en wi h ene gy conse a ion. The simula ed ime-depending
ield his o ies a e used in a s ep-by-s ep in eg a ion o he Sch ödinge equa ion o ob ain Ul( )
and, hus, Cdd,l( )in he Liou ille space. An a e age o e a se o his o ies is necessa y o e alua e
Cdd( ). Again, he spec al line shape is ob ained using FFT.
•The Eule –Rod igues (ER)-simula ion code [13]: The plasma model o he simula ion o
ime-dependen ield his o ies consis s o an emi e a es in he cen e o a sphe ical olume
and se in a ba h o s a is ically independen cha ged quasi-pa icles mo ing along s aigh line
ajec o ies. A einjec ion echnique ensu es s a is ical homogenei y and s abili y. The simula ed
elec ic ield his o ies a e used in a sol e o he e olu ion o he a omic sys em. Fo hyd ogen,
i he SO (4) symme y is alid, Eule –Rod igues (ER) pa ame e s a e used; o he wise he
diagonaliza ion p ocess is done using Jacobi’s me hod.
•The DM-simula ion code [14]: This code uses he same sol e as he ER-simula ion code, bu he
ime-dependen ield his o ies a e simula ed using he MD simula ion echnique in o de o accoun
o he pa icle in e ac ions.
E en hough hese nume ical simula ion echniques ha e, success ully, been used as model labo a o y
expe imen s o compa e wi h line shapes esul ing om o he me hod calcula ions o expe imen s, hey
can be imp ac ical when he ele an a omic s uc u e becomes oo complex. Se e al app oaches and
models ha e been de eloped o o e come his di icul y [22] and implemen ed in nume ical codes. Th ee
di e en models (o codes) ha e con ibu ed o he p esen s udy:
•The mul i-elec on line shape (MELS) code [24]: The “s anda d” heo y (quasi-s a ic ions and
impac elec ons) and he Boe ke -Iglesias-Du y (BID) model [15] o accoun o ion dynamics
e ec s. The mic o ield dis ibu ion is om he adjus able-pa ame e exponen ial app oxima ion
(APEX) model [25,26].
•The PPP code [27]: The S a k b oadening is aken in o accoun in he amewo k o he s anda d
heo y by using he s a ic ion app oxima ion and an impac app oxima ion o he elec ons o
including he e ec s o ionic pe u be dynamics by using he luc ua ion equency model [16,17].
The mic o ield dis ibu ion unc ions equi ed a e calcula ed using he APEX model o an ex e nal
MD simula ion code.
•Quasicon iguous (QC)- equency luc ua ion model (FFM) [18,19]: The s a ic line p o ile o a
ixed ield is ep esen ed by a ec angula shape, which is hen in eg a ed o e he mic o ield
dis ibu ion wi h he dynamic p ope ies o mic o ields accoun ed o ia FFM.
A oms 2014,2263
2.2. Plasma Cha ac e is ics
We conside a plasma con aining an impu i y (o a small pe cen age) o He II in p o ons a wo
empe a u es, (T= 1 and T= 10 eV) and wo ion numbe densi ies (n= 1018 and n= 1019 cm−3).
The plasma condi ions a e summa ized in Table 1 oge he wi h some dimensionless pa ame e s use ul
o cha ac e izing he plasma, such as Γ, he coupling plasma pa ame e , α, he a io o he in e pa icle
dis ance o he sc eening leng h and ypical equency alues, such as he plasma ion equency, ωpi,
and an es ima e o he cha ac e is ic equency o he pe u bing ield, νdyn. Wi h he in e pa icle
dis ance and he sc eening leng h gi en by 0= (3/4πn)1/3and λD= kBT
4πne2, espec i ely, we ha e
Γ = e2/( 0kT),α= 0
λD
,ωpi = 4πne2
m,mbeing he p o on mass and νdyn = / 0, whe e is he
he mal eloci y in he adia o ’s e e ence ame.
Table 1. Plasma cha ac e is ics.
Ne(cm−3)T(eV) Γ α ωpi( ad/s) νdyn( ad/s)
1018 1 0.23 0.83 1.32 ×1012 1.77 ×1012
1018 10 0.02 0.26 −5.57 ×1012
1019 1 0.50 1.22 4.16 ×1012 3.80 ×1012
1019 10 0.05 0.39 −1.20 ×1013
Figu e 1a illus a es he empo al a ia ion o he ionic ield a he emi e o n= 1018 cm−3and
T= 1 eV. In he same igu e, he g aph in he subwindow shows he ield au oco ela ion unc ion,
CFF ( )whose decay gi es an es ima e o he ime scale o he ield luc ua ions. Figu e 1b shows he
co esponding o a ional z-componen ( ull line) and he ib a ional ields (dash-do ).
Figu e 1. (a) Tempo al a ia ion o he h ee componen s and co ela ion unc ion
(subwindow) o he pe u bing ionic ield; (b) empo al a ia ion o he z-componen o he
o a ional ield ( ull line) and o he ib a ional ield (dash-do line) a he emi e . The plasma
condi ions a e n= 1018 cm−3and T= 1 eV o bo h (a) and (b).
15
10
5
0
-5
F( )/F
0
14x10
-12
1210
86420
(s)
F
z, o
( )
F
ib
( )
(a)
(b)
40
20
6x10
-11
543210
(s)
-40
-20
-10
-5
0
5
10
F
x,y,z
( )/F
0
(a)
1.0
0.5
0.0
C( )/C(0)
10
-14
10
-13
10
-12
10
-11
(s)
The cha ac e is ic ime scale o he ield luc ua ions has o be compa ed o he adia o ime o in e es ,
which is usually de e mined by he in e se o he HWHM (hal wid h a hal maximum) due o all

A oms 2014,2264
b oadening mechanisms o he conside ed line p o ile. This ime depends on he plasma condi ions
and on he a omic da a o he conside ed line and can be conside ed as he physical ime o in e es
cha ac e izing he esponse ime o a plasma measu emen de ice.
I he adia o ime o in e es is small compa ed o he ime scale o he ield luc ua ions
(HWHM−1ν−1
dyn, o example), he ion ield a ies li le o e he adia o ime o in e es and
may be conside ed as a s a ic ield well cha ac e ized by a s a ic ield dis ibu ion unc ion (quasi-s a ic
app oxima ion). As soon as he wo cha ac e is ic imes a e o he same o de o magni ude, i becomes
necessa y o accoun o ion mo ion on he line p o ile.
The ollowing esul s ha e been ob ained o ex eme condi ions ha allow o a be e unde s anding
o he di e ences obse ed o he a ious codes and me hods. Two di e en lines ha e been chosen: he
hyd ogen-like helium Lyman-αand -βlines. The spec al line shapes o hese wo lines calcula ed
in he amewo k o he quasi-s a ic app oxima ion a e plo ed in Figu e 2and compa ed o he
esul s accoun ing o ion mo ion ob ained wi h a simula ion code. The wid h o he Lyman-αline
(n= 2 −n0= 1) is e y sensi i e o he ion mo ion e ec due o he unshi ed cen al componen ,
whe eas he wid h o he Lyman-βline (n= 3 −n0= 1), which is b oade han he Lyman-alpha,
due o a la ges uppe p incipal quan um numbe and does no ha e unshi ed cen al componen , is
less a ec ed.
Figu e 2. Spec al line shapes o he Lyman-α(a) and Lyman-βline (b) o hyd ogen-like
Helium calcula ed in a quasi-s a ic (dash line) and in a luc ua ing ( ull line) ionic elec ic
ield p oduced by p o ons a n= 1019 cm−3and T= 10 eV.
5x10
-15
4
3
2
1
0
No malized in ensi y
-5x10
14
0x10
14
5x10
14
∆ω
( d.s
-1
)
(b)
3.0x10
-14
2.5
2.0
1.5
1.0
0.5
0.0
No malized in ensi y
2x10
14
1x10
14
0x10
14
-1x10
14
-2x10
14
∆ω
( d.s
-1
)
(a)
In he ollowing, we will sepa a ely analyze how changing he di ec ion and magni ude o he ionic
elec ic ield −→
F( )will in luence he line shape. Compa isons be ween he di e en codes and me hods
a e also pe o med.
3. Resul s
3.1. Gene ali ies
Simula ion esul s o he e ec s o he magni ude and di ec ion luc ua ions o he elec ic ield on
he Lyman−αline a e plo ed in Figu e 3a,b, espec i ely, o n= 1019 cm−3. I can be seen ha he
ib a ion o he elec ic ield wi h a ixed di ec ion (Figu e 3a) does no a ec he cen al componen ,
A oms 2014,2265
whe eas each la e al componen ends o me ge a ound i s g a i y cen e . As expec ed, he wings o he
line a e well ep oduced by he s a ic p o ile ( he ele an imes o in e es in he line wings a e small
compa ed o he cha ac e is ic imes o he ield luc ua ions). Inc easing he empe a u e inc eases he
luc ua ion a e, bu as he s a ic line wid h also depends on empe a u e (due o he Debye-sc eened
po en ial), he a io be ween hese wo alues does no change signi ican ly, and he e ec s o ib a ions
on he p o ile a e e y simila . Conce ning he pu e o a ing ield case (Figu e 3b), he si ua ion is
comple ely di e en . As he ield has a ixed magni ude, F0, he s a ic p o ile is composed o h ee
un-b oadened S a k componen s, one cen al unshi ed componen and wo la e al ones (in Figu e 3b,
he componen s ha e been b oadened a i icially o allow o plo ing). When he luc ua ions o he ield
a e aken in o accoun , he ield magni ude is unchanged, only i s di ec ion luc ua es. He e, an inc ease
o empe a u e esul s in an inc ease o dynamics e ec s. All he S a k componen s a e a ec ed by he
ield di ec ion changes by being b oadened and me ged o he cen e o g a i y.
Figu e 3. Spec al line shapes o he Lyman-αline in a pu e ib a ing ield (a) and a pu e
o a ing ield (b) compa ed o he co esponding s a ic p o iles, a n= 1019 cm−3and wo
empe a u es, T= 1 and T= 10 eV. The dynamic case esul s ha e been ob ained wi h he
BinGo simula ion code.
20
15
10
5
0
40.9040.8540.8040.75
Ene gy (eV)
(a)
S a ic case
1 eV
10 eV
Dynamic case
1 eV
10 eV
140
120
100
80
60
40
20
0
40.8440.8240.80
Ene gy (eV)
T=1eV
s a ic case
T=10eV
(b)
In Figu e 4, he Lyman−βline p o iles calcula ed by nume ical simula ions ha e been plo ed o he
same condi ions as p e iously. He e, again, he ib a ions o he elec ic ield a ec he “blue” and “ ed”
la e al componen s by me ging hem o hei espec i e cen e o g a i y. As has been al eady men ioned,
he dynamics e ec s a e less impo an in his case, because he s a ic line-wid h is b oade han in he
p e ious case. In Figu e 4b, he e ec s o he ield di ec ion changes on he line p o ile a e shown o
he wo di e en empe a u es and compa ed o he s a ic p o ile. He e, he e is no unshi ed componen ,
and he e ec s o luc ua ions a e o b oaden he S a k componen s, illing p og essi ely he dip be ween
he wo se s o S a k componen s.
To ha e a be e idea o he pu e dynamics e ec , esul s co esponding o a ious educed masses
be ween adia o and pe u be s (µ0,4µ0and 16µ0) ha e been plo ed in Figu e 5 o he Lyman-αline
a n=1018 cm−3and T= 10 eV. He e, µ0= 0.8is he educed mass in he He-H+cen e o mass ame.
I can be seen on he ull p o ile (Figu e 5a) ha o he la ges alue o he educed mass, he dynamics
p o ile s ill p esen s a s uc u e in he wings, and when he dynamics e ec s inc ease (when he educed
A oms 2014,2266
mass dec eases), he line b oadening inc eases and any s uc u e disappea s. I he educed mass we e
smalle , we would see a na owing o he p o ile. Fo he o a ional p o ile (Figu e 5b), his na owing
is al eady seen, o he educed mass equals µ0. The ull p o ile is a complex combina ion o bo h
o a ional and ib a ional e ec s. The shape o he Ly-αand -βline cen e is de e mined mainly by he
e ec s ela ed o he o a ion o he elec ic ield. The e o e, p ope conside a ion o hese e ec s in he
models seems o be one o he keys o he ion dynamics issue.
Figu e 4. Spec al line shapes o he Lyman-βline in a pu e ib a ing ield (a) and a pu e
o a ing ield (b) compa ed o he co esponding s a ic p o iles, a n= 1 ×1019 cm−3and
wo empe a u es, T= 1 and T= 10 eV. The dynamic case esul s ha e been ob ained wi h
he BinGo simula ion code.
16
14
12
10
8
6
4
2
0
48.648.548.448.348.2
Ene gy (eV)
T=1eV
Dynamic case
S a ic case
T=10eV
Dynamic case
S a ic case
(a)
40
30
20
10
0
48.4448.4048.3648.32
Ene gy (eV)
s a ic case
T=1eV
T=10eV
(b)
Figu e 5. Reduced-mass e ec on he Lyman-αline shape a n= 1018 cm−3and T= 10 eV
in he h ee models o ield ( ull (a); o a ion (b) and ib a ion (c)). The colo codes black,
ed and blue co espond o µ0,4µ0and 16µ0, espec i ely. The calcula ions ha e been done
wi h he SimU simula ion code.
6
8
10
2
4
6
8
100
2
4
No malized in ensi y
1.0x10
-2
0.80.60.40.20.0
Ene gy
(a)
1
2
4
6
10
2
4
6
100
2
4
No malized in ensi y
8x10
-3
6420
Ene gy
(b)
3
4
5
6
10
2
3
4
5
6
100
No malized in ensi y
8x10
-3
6420
Ene gy
(c)
3.2. Code Compa isons
In his wo k, bo h MELS and PPP ha e been modi ied o accoun o ib a ion only.
The MELS code uses a s ochas ic mic o ield desc ip ion o he plasma o he ion dynamics e ec .
The model ob ains an exac solu ion o he line p o ile assuming an idealized s ochas ic p ocess. I is
possible o cons ain he model by p ese ing known impo an p ope ies o he ac ual p oblem by
adjus ing ee pa ame e s in he heo y. Fo he p esen calcula ions, he ee pa ame e , he luc ua ion
A oms 2014,2267
a e, was chosen o sa is y he ion impac limi , which, in p inciple, is equi alen o ep oducing he
sho ime limi o he momen um au o co ela ion unc ion [24]. The o a ional ield luc ua ions we e
compu ed by pe o ming he in eg a ions o e he elec ic ield di ec ions as be o e [24], bu a a single
alue o he ield magni ude. The ib a ional luc ua ions we e compu ed by es ic ing he angle a e age
o a single ield di ec ion in he z-axis.
In he PPP code, he ion dynamics e ec s a e ea ed by means o he equency luc ua ion model
(FFM) [16,17]. The FFM is based on he p emise ha a quan um sys em pe u bed by an elec ic
mic o ield beha es like a se o ield-d essed wo-le el ansi ions, he S a k d essed ansi ions. I he
mic o ield is ime a ying, he ansi ions a e subjec o a collision- ype mixing p ocess—a Ma ko
p ocess—induced by he ield luc ua ions. In p ac ice, he FFM line-shape is he esul o in ensi y
exchanges be ween di e en spec al domains o he s a ic line-shape wi h an exchange a e gi en by
νdyn (Table 1). Owing o he ac ha a pu e ib a ional ield will mix nei he he cen al componen
wi h he la e al componen s no he la e al componen s oge he , he code has been adap ed in o de o
mix only he S a k componen s ha mus be mixed. Fo he o a ional model, he s a ic p o ile has been
ob ained by se ing he elec ic ield o F0~nz, and he usual “ion-dynamics” models, FFM, ha e been
applied. I is he mixing o he s a ic S a k componen s all oge he ha mimics he di ec ion changes
o he elec ic ield. The chosen mixing p ocess is a Ma ko p ocess, sugges ing ha he cause o he
change in s a es is so iolen , ha in i s inal s a es, he sys em has no memo y o i s ini ial s a e.
3.2.1. Full Cases
In his example, he line p o iles a e calcula ed o he ime-dependen ield c ea ed by he p o ons
in e ac ing h ough an elec on-sc eened Coulomb (i.e., Debye–Yukawa) po en ial. As a i s obse a ion,
he h ee codes, BinGo, DM-simula ion and SimU, gi e always e y simila esul s (c . Figu e 6).
The BinGo and DM-simula ion codes a e simila and accoun o all o he in e ac ions be ween
pa icles, whe eas he SimU code simula es elec on Debye-sc eened quasi-pa icles, which in e ac
only wi h he cha ged emi e . A ew mino di e ences appea in he cen e o he lines, wi hin he
unce ain ies o he di e en me hods. Fo ins ance, looking a he Lyman−αline (Figu e 6a), he
de ia ion maximum appea s o n= 1019 cm−3and T= 1 eV and ep esen s abou 6% on he peak alue.
The nume ical simula ions, BinGo, DM-simula ion and SimU, will hen be conside ed as equi alen
labo a o y nume ical expe imen s, and excep o he line-wid h compa isons, esul s co esponding o
hese h ee simula ions will be plo ed only once, o cla i y.
The esul s ob ained o bo h lines, wi h he di e en codes, ha e been compa ed and plo ed in
Figu e 7 o n= 1019 cm−3and T= 1 eV. I can be seen ha he lines calcula ed by he ER-simula ion
code a e oo b oad. In his code, he ields measu ed a he emi e a e sc eened by elec ons, bu no
pa icle in e ac ions a e aken in o accoun ( he pa icles mo e on s aigh -pa h ajec o ies); hus, he
s a ic ield dis ibu ion unc ion is shi ed owa d he la ge ields. This leads o b oade s a ic p o iles
and o changing he a io be ween s a ic and dynamic e ec s. The S a k p o ile is, in his case, less
a ec ed by ion dynamics e ec s. This beha io is mo e p onounced a low empe a u e when he plasma
coupling pa ame e inc eases.
A oms 2014,2274
equency-dependen luc ua ion a e o a di usion model on he basis o wha has been done o Dopple
p o iles [30] could be es ed.
Acknowledgmen s
The o ganiza ional and inancial suppo om he In e na ional A omic Ene gy Agency is
highly app ecia ed.
Au ho Con ibu ions
The p esen a icle is based on compa isons o a ious codes and models de eloped by he di e en
au ho s. All o hem ha e been in ol ed in all aspec s o he p esen wo k.
Appendix A: Field Co ela ion Func ion in he BID Model
Conside a sys em o ions wi h elec ic cha ge Ze and mass min e ac ing h ough a sc eened Coulomb
po en ial wi h numbe densi y nand T he empe a u e in ene gy uni s. The ans o m o he ield
au o-co ela ion unc ion a an ion o his sys em is app oxima ed in he BID model by [15]:
e
CFF (ω) = Z∞
0
d eiω <−→
F·−→
F( )>≈3mT
Z2e2
ωλ(ω)
ω+iλ(ω)(A1)
The unc ion λ(ω)is gi en by:
−iλ(ω) = Z2e2
3mT Z∞
0
dP()2
ω+iν =ω2
pγ
3ν
1
ω+iν (A2)
whe e νis he ee pa ame e in he BID o mula ion he e assumed a cons an , P()is he p obabili y
dis ibu ion o he ield magni ude a an ion, ωpis he ion plasma equency and [25,26,29]:
< F2>=γ < F2>OCP = 4πnTγ (A3)
de ines he co ec ion γ, o he ield dis ibu ion second momen compa ed o ha o he sys em o ions
in e ac ing h ough a pu e a he han a sc eened Coulomb po en ial. Subs i u ion o hese esul s in o
he in e se ans o m yields:
CFF ( ) =<−→
F·−→
F( )>=< F2>nω+e−ω+ −ω−e−ω−
ω+−ω−o(A4)
whe e:
ω±=ν
2"1± 1−4γω2
p
3ν2#(A5)
No e ha : Z∞
0
d CFF ( ) = 0 (A6)
sa is ying an exac p ope y [15].

A oms 2014,2275
Con lic s o In e es
The au ho s decla e no con lic s o in e es .
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c
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