Alkaline broadening in Stars

1.7 MB·English

by  A. De Kertanguy

ALKALINE LINES BROADENING IN STARS.
A.DEKERTANGUY
5
LERMA,
1
OBSERVATOIREDEPARIS-MEUDON
0 92195MEUDONFRANCE
2
[email protected]

n
a
J ABSTRACT. Givingnewinsightforlinebroadeningtheoryforatomswithmorestructure
thanhydrogeninmoststars.Usingsymbolicsoftwaretobuildprecisewavefunctionscor-
4
rectedforδs,δpquantumdefects.Theprofilesobtainedwiththatapproach,havepeculiar
1
trends,narrowerthanhydrogen,allquantumdefectsusedaretakenfromatomicdatabase

topbase.Illustrationofstrongereffectsofionsandelectronsonthealkalineprofiles,than
]
neutral-neutralcollisionmechanism.
h
keywords:Stars:fundamentalparameters-Atomicprocesses-Line:profiles.
p
-
n
e
g
. 1. INTRODUCTION
s
c Herewedefinewhatisneededforthetheory:Weintroducetheprofilefunctionnormal-
i
s ized: F(∆ω)requiringnormalization:
y
(cid:90) ∞
h (1) F(∆ω)d(∆ω)=1
p −∞
[
Theradiantpowerfunction,thatisthepoweremittedthougunitfrequencyis:

1 4ω4
v (2) P(∆ω)= F(∆ω)
3 3c3
8 WerecallthetworelationsdefiningtheFouriertransformsvariables: ω ands(timevari-
2 able)suchthatω.sisdimensionless.
5
(cid:90) ∞
0 Φ(s)= ei∆ω.sF(∆ω)d(∆ω)
2. −∞
1(cid:90) ∞
0 (3) I(∆ω)= Φ(s)ei∆ω.sds
5 π −∞
1
: 2. HYDROGENFACTS
v
Xi Recalling some facts on Hydrogen lines broadening: such as Balmer lines Hα, and
Hβ or Lyman lines : Lyα, and Lyβ h¯ω being the energy gap between levels defining
r 0
a a transition i→ f the detuning ∆ω giving rise to F(∆ω) for hydrogen lines is mainly
first order Stark effect: for instance exists for Hβ whose wavelength at the centre of the
n =4→n =2transitionis: λ =486.1nm.
f i 0
These Hydrogen lines are broad because of the degeneracy of levels defined by |nlm>
therecanexistsg=2×(2l+1)foranldefinedstatesuchas0≥l≤n−1foranidentified
HlinesuchasHβ n =4→n =2linewehaveN=4.n2×n2thatis512sublevelsimplied
f i f i
and128neglectingtheelectronspin. Goodliteratureexistsforsuchmatters,thatistaking
Date:February19,2015.
1
2A.DEKERTANGUYLERMA,[e-mail]
into account for perturbers (ions and electron) effects on the radiating atom. Feautrier
(1976).
3
(4) ∆E= n×(n −n )eF+O(F2)
1 2
2
(5) ∆E= h¯∆ω =h¯(ω −ω)
0
h¯2
(6) ∆E= h¯∆ω = (k2−k2)
2m i f
e
Thesetwoapproachesgivesrisetothecompletetheoryofbroadening:
1)whenthestaticfield|F|=Cstisconstantrelativelytodecayrate,infactafieldgenerated
bystaticheavychargesionsorprotons(relativelytotheelectron)). ThentheStarkprofile
hassuchdependence:
−5
(7) I(λ)(cid:39)|∆λ| 2
2)ThefieldF(t)variesquicklybeforetheatomrelaxes,
e2
(8) F(t)(cid:39)
r2(t)
(9) (cid:126)r(t)(cid:39)(cid:126)b+(cid:126)v.t
Thequantumtheoryofoneelectron+H(orradiatingatom)systemVanRegemorter(1972)
impliestherelationh¯∆ω = h¯2 (k2−k2)isthegoodanswertotakeintoaccountforlarge
2me i f
∆ω andreplacestheimpactparametersemi-classicalapproach.
3. ALKALINELINES
Dealing with atoms having a structure such as alkalines: Li, Ca, K, Mg, Na or atoms
such as He, and O. These atoms can be modelled using what is called since a long time
Born(1926)thequantumdefect. Theopticalelectron: theonethatgivesrisetotransition
(quantumjumps(α →β)suffersfromanadditionalpotential: thepolarizationpotential:
V (r)=−αD,α beingthedipolarstaticpolarizibilty. Thefulltheorybeginswithrecent
p 2r4 D
review work Schwerdtfeger (2006) and displays the development of an energy level as
ordersofthefieldstrengthF:(if(cid:126)F(t)=(cid:126)E )thisthewellknownStarkeffect.
0
dE 1d2E
E(F)=E + dF+ dF2+O(F3)
0 dF 2dF2
The polarization potential varying as r−4 is easily reckognized as the second order term
dF2. Let’usintroducethewaytodealwiththeV (r)potential.
p
(10) (cid:126)p= α .(cid:126)E
D
(11) dV = −(cid:126)p.d(cid:126)E
p
(12) dV = −α (cid:126)E.d(cid:126)E
p D
α
D
(13) V (r)= −
p 2r4
(14)
ALKALINELINESBROADENINGINSTARS. 3
3.1. Semi-classical expression. How to deal with: Let’us start with the semi-classical
formulafortheprofileF(ω)
(15) F(ω)= lim (cid:90) T2 dt.ei.ω.t|<Ψ(t)|(cid:126)D|Ψ (t)|2 1
i f
T →∞ −T 2πT
2
ThisequationisputforwardinVanRegemorter(1972)inhisreviewofspectrallinebroad-
ening. I adapt that question to a quite similar way: I need not have |Ψ(t)> as a time
i
dependantwavefunction,butthemodifyedradial|Ψ((cid:126)r)>withnotimedependence. Giv-
ingforalkalinespecies,withknownquantumdefects(therearedatafromTopbase):
(16) α = n l
i∗ i∗
(17) |Ψ ((cid:126)r)>= R ×Y (θ,φ)
α α α
(18) β = n l
f∗ f∗
(19) |Ψ ((cid:126)r)>= R ×Y (θ,φ)
β β β
orforHydrogenthewellknownbasicket|R (r)×Y (θ,φ)>
nl lm
(20) nl= n l
i∗ i∗
(21) |Ψ ((cid:126)r)>= R ×Y (θ,φ)
nl nl lm
(22) n(cid:48)l(cid:48)= n l
f f
(23) |Ψn(cid:48)l(cid:48)((cid:126)r)>= Rn(cid:48)l(cid:48)×Yl(cid:48)m(cid:48)(θ,φ)
Ourpurposeistodefinethemostefficientway,the|Ψ ((cid:126)r)>usingtruequantumdefects
α
leadington : theeffectivequantumnumber. Eachatomicspecieshasitspeculiarquan-
∗
tumdefect. ThesearenowavailablefromdatabasesuchasTopbase. Nowitisafactthat
the l kinetic momentum is defined by eigen value of the spherical harmonics for a pure
Coulombpotential,becomesl =l−δ thedegeneracyofthelevelsdisappears. Thequan-
∗ s
tumnumbersetis:α≡n =n−δ,l =l−δ.Thephysicaleffectproducedcanbeexplain
∗ l ∗ l
thisway:theopticalelectongettingawayfromtheclosedshellbeneathpolarizesthecore
shell, the higher the levels of the optical electron , the nearest to hydrogenic ”states” are
thetransitions.
L= 0≡S→δ
s
L= 1≡P→δ
p
δ ≡ δ ≥δ ≥δ ≥δ →0
l s p d f
(24)
4. TIMEDEPENDENTMETHOD
It is seen that one can change the semi-classical formula, into the way suggested by
Schiff(1968),withnolessgenerality.
Ψα((cid:126)r,t)=Ψα((cid:126)r,0)×eiEh¯αt. the same for the |β >| that is: Ψβ((cid:126)r,t)=Ψβ((cid:126)r,0)×eiEh¯βt .
Thetransitionprobabilityisproportionalto:
|<Ψα|V((cid:126)r)|Ψβ|>|2×ei.ωβα.t. FromthetextbookofL.I.Schiff,weusehisg(t)plateau
functiontotransformandtherelationT =1−S:
i (cid:90) ∞
(25) <β|(S−1)|α >=− <α|T|β > g(t)ei.ωβα.t
h¯ −∞
TheP(∆ω)functionisthenproportionalto:
w ∝|<α|T|β >|2thetransitionprobalibity.
αβ
4A.DEKERTANGUYLERMA,[e-mail]
Thefollowingoperatorsarethesame:
A(t)=|<Ψ ((cid:126)r,t)|(cid:126)D|Ψ ((cid:126)r,t)>|2and
α β
B(t)=|<Ψ ((cid:126)r)|(cid:126)D|Ψ ((cid:126)r>)|2×g(t,t )
α β 0
ALKALINELINESBROADENINGINSTARS. 5
5. TOWARDTHEMGPROFILES
ItisclearthatonecanobtainthewavefunctionsofthesealkalineelementssuchasMgI
neutralwithsomequantumdefects:
δ =1.52andδ =1.04andδ =0.56S=0(Singlet)
s p d
δ =1.63andδ =1.12andδ =0.17S=1(Triplet)seeRef. Born(1926),pp190
s p d
TheretheI istheKostelecky´ index,takingvaluessuchas1or2.Kostelecky´ (1984)
K
This is done to insure the positivity of quantum numbers n l . For a Mg3s→3p or a
∗ ∗
Mg3s→4p(S=0),thequantumnumbersarebelow:
(26) α = n =3−δ =1.48
i∗ s
(27) l = 0−δ +I
i∗ s K
(28) |Ψ ((cid:126)r)>= R (r)×Y (θ,φ)
α α α
(29) β = n =3−δ =1.96
f∗ p
(30) l = 1−δ
f∗ p
(31) |Ψ ((cid:126)r)>= R (r)×Y (θ,φ)
β β β
ThefulltheoryofthecalculationofwavefunctionsofsuchtypeisdoneinBates(1949),
itimpliessubtletransformationswiththetheoryoftheKostelecky´ index,relatedtosuper-
symmetry transformations Kostelecky´ (1984). The quantum analog to the classical Born
atomicmodel,withtheprecessionoftheellispse,isobtainedbyBates(1949). Thegood
quantumtheoryneedstoconsideramodificationoftheY (θ,φ)→Y (θ,φ)Itisuseful
lm l∗m∗
tocomparetwotransitionssuchas:Mg3p→4stripletquantumdefectstoconsiderwhose
wavelengthisλ =517.83nmwithanHydrogenlinesuchasH3p→4sλ =1875.11nm.
if if
Alldatatakenfrom NBS(1969).Theratiooftheintensitiesisgivenby:
|<αn ,l |r.cos(θ)|βn(cid:48),l(cid:48) >|2
(32) R= ∗ ∗ ∗ ∗
∑l |<n,l,m,|r.cos(θ)|n(cid:48),l±1,m>|2
m=−l
TheratioRcanbeconsideredastheratiooftheEinsteinA ofthetwolines,forsuchtwo
ik
lines: R=0.56×10−6
Formostionicspecies,whosesingleoropticalelectrongainshighdistances
<r >=n2a ≥20a , the quantum defects disappear, leading to simple hydrogenic be-
∗ 0 0
haviour,thewavefunctionsturntobetheseofHydrogen.
5.1. Towardtheglobaltheory. Weusetheupwarddefinitionforthelineshape:
(cid:90) ∞
(33) F(ω)= |<Ψ |(cid:126)D.f(t)|Ψ >|2dt
α β
−∞
(34) g(t)= f(t).f(t)
(cid:90) ∞
(35) F(ω)= |<Ψ |(cid:126)D|Ψ >|2 e−iωt.g(t)dt
α β
−∞
(36)
TheseFouriertransformsareeasilyperformedwith”Mathematica”,andIwriteherethe
g(t,t ),t beingthewidthofthe”plateau”inseconds.
0 0
(37) g(t,t )=e(−t+t0)H(t−t )+H(−t+t )+H(t−t )+e(t−t0)H(−t+t )
0 0 0 0 0
Hereisthepictureofthedistributionfunctiong(t,t ):
0
Itisrealllyinterestingtohavefastsymbolicsoftwaresuchas”Mathematica”,toperform
theFourierTransformofthefunctiong(t,t )assumingthatt ≥0togetF(ω,t ),andhave
0 0 0
6A.DEKERTANGUYLERMA,[e-mail]
negativesecondwaytogHtL Schifffunction
1.0
0.9
0.8
0.7
0.6
-10 -5 5 10
negativesecondwaytogHtL Schifffunction
1.0
0.9
0.8
0.7
0.6
-10 -5 5 10
FIGURE1. g(t,t0)functiont0=3st0=5s
itasagoodanalyticalfunction:
√
(38) F(ω,t )= I√1 2 (1−e−t0ω+2ωcos(ωt0))
0 πω 1+ω2
It is interesting to look at the behaviour of the profile function F(ω,t ) , this function
0
oscillatesbecauseofthephasescontainedintheexpresssion.
For a given ω the shorter is the t width the least the profile F(ω) oscillates. To obtain
0
the good former profile function we need to replace in the defined function F(ω,t ) the
0
variable ω by the variable ∆ω = ω−ω . Here are some pictures of the alkaline Mg
0
element:
Hereω isthefrequencyassociatedwiththecenterofthelineω = Eβ−Eα.
0 0 h¯
ALKALINELINESBROADENINGINSTARS. 7
TheparameterI =|<Ψ |(cid:126)D|Ψ >|2,shouldbetheintensityatthecenterofthelinethat
1 α β
isω =ω .ThatdefinitionofI shallbemodifiedtotakeintoaccountthepressureeffect.
0 1
6. THECUT-OFFRADIUSANDITSSIGNIFICANCEUPONTHEPROFILEF(∆ω,t0).
Asimpleandefficientwaytotakeintoaccountthepressureeffectwhoseorigincomes
−1
fromexternalparametersuchasdensityN andthusthecut-offisthefollowingL =N 3 .
A A A
Thiscut-offparameterL istheonetousewhentherearenocharges,andthesurrounding
A
of the emitters is mainly done of atomic species of density N . If the emitters of the
A
transitionα →β arepartofaplasma,withadegreeofionization: τ = Ne
Ne+NA
thegoodcut-offcouldbetheshorterofthethetwoStehle´ (1996)thatis:
(cid:114)
kT
e
(39) λD =
e 4πe2.N
e
(cid:114)
T
(40) λD = 6.9×108 e(nm,K,cm−3)
e
N
e
λD
e
(41) λD =
i (cid:113)
1+ <Z2.Te
<Z>.Ti
ρ = Min(λD(N ,T),L )
C e A
λDN( ) that is the Debye radius to compare the spatial extension L , because of the
ions A
distributionofthechargesintheplasma.
7. EFFECTOFTHECUT-OFFONTHEVALUEOBTAINEDFORTHEDIPOLARSQUARED
MATRIXELEMENTdαβ=|<α|(cid:126)D|β >|2
That simple explanation to take into account for pressure effects on radiating atoms
,partsofsuchmedia,suchasexistsinAstrophysics:-Staratmospheres,-Dustininterstellar
matter. -Molecular clouds. Finding the wayto define the upperlimit ρ for theemitting
C
atoms,andtobuildthewavefunctionswiththeirpeculiarquantumdefectsforeachspecies
(Li, Na, O ,Al , Ca ,Na ,Mg and even He) now easily reached with modern symbolic
calculationsoftware,IdefinetheretheaprobabilityfunctionP (x):
α,β
|(cid:82)x(T)<α|(cid:126)r|β >|2
(42) P (x,T)= 0
αβ |(cid:82)∞<α|(cid:126)r|β >|2
0
8. SOMERESULTSCOMINGFROMTHEPαβ(x,T)function
Thevariationdomainofthisfunctionisthe-halfrealaxis, strictlypositiveassuitable
foralength.
Wecanconsidersuchfunction:
(43) 0≤P (x(T)≤1
α,β
P (0)=0voidofparticle.
αβ
P (x)=0squeezedstateoftheopticalelectron.
αβ
P (∞)=1existenceoftheparticule.
αβ
8A.DEKERTANGUYLERMA,[e-mail]
9. ANALYTICALRESULTSFORF(∆ω,t0)TAKENFROMΦ(t).
Insuchmatterwedeltwith:twodistinctsproblemsaresolved:Thenonhydrogenicbe-
haviouroftheemittersorabsorbers(He,Li,O,Mg,Ca,Na,K)andtheirionizedspecies,is
modelledwiththequantumdefectsfromwhichexistalargelitteratureandevendatabases
suchasTopbase. Thecorrectbuildingofthewavefunctionsisasolvedproblem. deKer-
tanguy (1979 ). The second problem solved in this paper, is the good way to obtain the
profileF(∆ω)fromtheFouriertransformΦ(t),whosevalueincludestherestrictionofthe
wavefunctionthroughthecut-offparameter,ρ =x(T). Asamatteroffactanoscillation
C
effectundertheenvelopoftheF(∆ω)≤ Γ .
|∆ω2
Γ beingthelimitingimpactprofilefunctionforlightperturbersaselectrons. Therethe
∆ω2
transitionfrequencyω forans(δ )→mp(δ )transitionisgivenby: I =13.606eV and
0 s p H
e=|q|=1.6021910−19Cand∆E isgiveninatomicunitsua.
αβ
(44) ω = 2.48181014π×I ×∆E
0 H αβ
(45) ω = e×ω (Hz)
0 0
1 1 1
(46) ∆E = ( − )
αβ 2 n2 m2
∗ ∗
n = n−δ
∗ s
m = m−δ
∗ p
Setting∆ω =ω−ω .
0
Fixingthet parameterthatisthewidthofthesmoothfunctiong(t,t ).
0 0
I rewrite down the Fourier Transform with the modified constant I changed into a Γ
1 i→f
resulting from the shortening of the radial integration domain or ”incomplete” dipolar
squaredmatrixelement.
(cid:90) x
OpeQDFunction(x,ind,3,4)= (r2+ind)φ (r)φ (r)dr
n=3−δs n=4−δp
0
a istheBohrradius.
0
(47) I =n3×(OpeQDFunction(0.1,1,3,4)2)a3
1 ∗ 0
TheΓ parameterisallwaysdefinedbythechosentransitioni→ f isobtainedbysolv-
i→f
ing:
(48) g(t,t )×Γ =OpeQDFunction(t,1,n,n )
0 i→f i f
√
(49) F(∆ω,t0)= Γi√→πfω2×(1−e−t0ω1++2ωω2cos(ωt0))
This function F(∆ω,t ) can be integrated on the ∆ω variable from −∞→∞. Using the
0
Stehle´ (1996), as a reference a function such as F1(∆ω)= Γ can not be normalized
(∆ω)2
whenintegratedfrom−∞→∞whileafunctionsuchasF(∆ω,t )can.
0
Infact:
(cid:90) ∞
(50) F1((∆ω)d∆ω = 0
−∞
(cid:90) ∞ √
(51) F((∆ω,t0)d∆ω = Γi→fet0 2π
−∞
ALKALINELINESBROADENINGINSTARS. 9
10. CONCLUSIONS
Thisworkwhoseconclusionisanewinsighttocomplexatoms(morestructurethanHy-
drogen)linebroadening,liesontwodifferentpoints. First,thepressureeffectisincluded
inthecalculationoftheprobabilityfunctionP (x(T,N),wherex(N,T)isthecut-offpa-
αβ
rameter ρ to be used. The surrounding of the radiating atom will fix it. It is clear that
C
whenx(N,T)→∞theisolatedatomlineprofileisobtained,inbetweenthepressureeffect
istakenintoaccount!. Thesecondinterestingfactisincludedinthedefinitionofthewidth
t of the smoothed g(t,t ) distribution function: It can exhibit oscillations of the profile
0 0
F(∆ω,t ). The choice of the width t should be driven by a sound consideration of the
0 0
surrounding atoms or ions background. It seems plausible the densest is the medium of
thesurroundingplasmaorbathofneutralperturbers,theshorterthet shouldbe! Hereare
0
somesketchesinFig.4andFig.5oftheF(∆ω,t )fordifferentwidtht whoserangegoes
0 0
fromt =1µstot =20s
0 0
1A0.DEKERTANGUYLERMA,[e-mail]
11. ACKNOWLEDGEMENTS
TheauthorthanksB.AlbertofthecomputerdepartmentoftheLERMAforhistechnical
assistance.
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