Physica Scripta. Vol. T65,183-187, 1996
Paschen Lines of Hydrogen and He+ Ion
C. Stehlk
UPR 176 du CNRS, DARC,Observatoire de Paris, F-92195Meudon Cedes France
Received October 11,1995; Accepted November 29,1995
Abstract
New theoretical results concerning the Stark broadened profiles of the
Paschen lines of hydrogen and He' ion are presented. Effects due to the
ionic motions are taken into account. Anal~dexpressions are proposed
for the halfwidths at low densities. Comparisons with available experimental data are performed.
1. Introduction
Due to their large broadening, the lines of hydrogen and
hydrogenic ions are used since a long time to determine the
electron density in laboratory plasmas and also the gravity
or effective temperature in stellar atmospheres. Astrophysical applications require a correct knowledge of the line
centers and of the line wings under a wide range of plasma
conditions, although the fine details of the intrinsic line
shapes may be partly masked by the radiative transfer. As
different lines are formed in the different layers of the atmosphere, they may be used as a local diagnostics of the atmospheric plasma conditions.
Several theoretical approaches have been devoted to this
study, starting with the well known static theories. In these
approaches, the ionic plasma electric microfield, which is
responsible for the broadening, is assumed to be time independent on time scales which are relevant for the line
broadening. This electric microfield is the sum of elementary
fields due to the plasma ions and is characterized by the
distribution function P Q . It is responsible for an energy
splitting of the eigenenergies of the radiating atom (i.e. the
radiator). The resulting line intensity is then obtained after
summing these elementary field dependent intensities
between Stark eigenstates and averaging over the field distribution function. This static approach was used for the
ions whereas a collisional description in terms of damping
rates y was usually taken for the electronic contribution.
These damping rates are frequency independent in the first
theories [l] which are appropriated for the line centers. The
Unified Theories used a frequency dependent damping rate
y(Ao) which allowed to calculate in an unified manner the
line centers and the line wings [2-41. A further progress was
the inclusion of the ion dynamic effects in the line shapes.
This means that the time variations of the ionic microfield
are no more neglected. In comparison with theoretical line
shapes obtained with static ions, these ionic motion effects
tend to broaden the lines like Lya, Hu . . . and fill the dips
of other lines like Lyp, HP . . . This effect has been proved
experimentally [SI and different theoretical approaches tried
to describe this effect, like the Molecular Dynamics Simulation [SI, the Relaxation Theory [7], the Kinetic Theory [SI,
the Line Mixing Theory [SI and Microfield Method [lo,
111. All these approaches have their own advantages and
disavantages, but among them only the last one is appropriate for astrophysical purposes. Hence it has a wide range of
applications, is rather economic in computer time, allows to
obtain the line wings and the line centers. We already used
it in hydrogen opacity calculations [13] and for the tabulation of the Lyman and Balmer lines of hydrogen [14].
Astrophysical applications require intensive line shape
tabulations. This large work could be greatly simpified by
the use of analytical expressions. We present in Section 2 an
analytical expression for the halfwidth of all the lines of one
electron systems (hydrogen and hydrogenic ions) which has
been presented elsewhere for the general case [lS] and will
be illustrated here for the Paschen lines. This expression is
only valid in the impact or collisional limit which is the
regime where the ionic motions are the most important.
Section 3 presents new results obtained within the Model
Microfield Theory for the Paschen lines of hydrogen. These
lines lie in the Infrared Spectral range and have been seldom
studied experimentally and theoretically. This is not the case
for the Paschen a line of He' which is in the visible and for
which we can compare our data with several experimental
ones.
2. Collisional regime
Let us consider a radiating one electron system. Neglecting
the fine structure effects, the degenerated eigenstates are
denoted by I nlm). The potential of interaction between the
radiator and all the plasma charges (electrons and ions) is
the sum of all the elementary coulombic interactions. It may
be developed in multipolar expansion, assuming that the
perturbing particles remain far from the radiator. The
monopole contribution has no incidence on the internal
structure of the radiator and thus does not contribute to the
line profile. One retains thus only the dipolar contribution
which is
v(t)= -dE,on(t)
- dEe~ec(t)
(1)
In this expression d is the electronic dipole of the radiator
Eelccare the total electric microfields due to the
and EiOn,
ions and to the electrons, with a typical time scale for fluctuations, t, equal to the inverse of the ionic and electronic
plasma frequencies up,ion, up,elcc, defined as
w
~N
/F
4zNee'
,
~
~5.64 ~ 104&(rds
~
N
*
s-', ~ m - ~ ) ,
where Ne is the electron density per unit of volume, me and
mi, the electron and ion masses.
Physica Scripta T65
184
C. Stehlt
Due to the different time scales the electron and the ion
contributions are usually described independently from each
other.
The line shape expression may be written as
L ( o ) = N Re
yif,i,f.
= 2nN,
I m / ( u ) u du [ b db
& U) - 6ii'
dte'*'
i , i',
relative motions are characterized by the impact parameter
b and the relative velocity D(u, Q,) and one has
(10)
Where N, is the density of perturbers of type p (electrons
or ions). In the line wings, assuming that the impact limit is
still valid, one has
f,f'
O M * / (Tf,f(t,
0)dfi)plO)
f f'}Ang. AV.
(3)
where p f ) is the population of the radiator in state i, the
brackets denote the average over all the plasma configu7
rations at time 0. The summation is performed over the L(0)= II AwZ
radiator states and the constant N insures the normalization
with
of L(o),i.e.
x
(Ti*(t,
(4)
This equivalent halfwith 7 allows to obtain the validity
The time evolution operator T(t, 0) of the radiator perturbed by the plasma is a solution of the usual Schrodinger conditions of the collisional limit,
equation
t(o)> t, or 7 < up.
(12)
This condition is always satisfied for the electrons for
0).
(5) which a collisional treatment is thus justified, as it was done
also in the former theories (see introduction). It is also the
In the Liouville 'pace, the eq* (3) can be
case for the ionic contribution at low densities [30].
written as
The equivalent halfwidth can be calculated using a perturbational expansion of S(b, U) in the potential of interL(o)= N Re
dt eicot(U if , ,(t,O))Dip i f pip)
(6) action. If one also uses the no quenching approximation (no
if,i'f' 0
Stark mixing of states with different principal quantum
where U i f , i , f t ( T , 0) = Ti.(t, O)qfp(t, 0) and D i , f , , i f = dits. numbers), one obtains for 7 the following analytical expression
dfi
The typical time scale for the line broadening t(o)is, for
angular frequencies o close to the line center value coo, 7, = 8.62 10-6N, Zi
knntf(b,,, , b,)
approximately equal to the inverse of the halfwidth l / A c ~ ~ , ~ .
The comparison between t(o)and the plasma time of interx (rad s-', ~ m - K),
~ ,
est allows to choose the adequate theoretical description of
the problem. In the line wings, putting Ao = 0 - oo,this
(13)
time of interest is shorter, due to the properties of the
Fourier transform. Thus one has
Ln (bi,Jbz)
f(b,,,, b,) -0.78
t(o)= Min (Am;,;, I Am I-').
(7) for hydrogen,
wm,
~~
J
it
9
E
+
At low densities the bath-radiator interactions history is a
succession of short independent events (collisions).As long
as we are concerned with the line centers, the impact
approximation is valid, which allows to describe the line
shape in terms of collisional damping rates y. We refer the
reader to the text books for the details of this derivation
[12]. We recall only the basic assumption which is that it is
possible to find some time scale At large compared to the
collisional duration t, and such that the cumulative effect of
the plasma-radiator interactions is negligible. One obtains
for the Liouville time evolution operator (U@, 0)) the following equation
for ionic radiators.
In these equations m, is the reduced mass of the colliding
pair, Z , and N, are the perturber charge and density, T the
temperature, a = Z,Z,e2/(2kT) = 1.67 10-3Z,ZT/T (cm,
K), Z, is the net charge of the one electron ion, and b,, =
A, ,where A, is the Debye length defined by
8
,IDe= 6.9 - (cm, K, ~ m - ~ )
for electrons,
and the line shape is simply
L(w)= N Re
[Lo- if~ylif,',,~,D i r f p , i f pi!).
(9)
if,i'f'
The matrix elements of the damping rate operator y, are
expressed in terms of the usual scattering operators S. In the
semi-classical collisional approach, the trajectories for the
Physica Scripta T65
(14)
for ions of charge Z , .
The positive sign in the last line of the previous equation
is for the case where the perturbers are electrons, and the
negative sign for ionic perturbers.
Paschen Lines of Hydrogen and He' Ion
I (units of l / a )
€ ' """'
'
""""
'
""""
'
""""
'
'
""""
"7
]
\+
Aa
f
lo-'o 1
1o'2
'
'
,.,'.('
IO"
P.l
'
",,.,'.I
1 oo
'
'
,.,,','
'
IO'
, , , ' ' ' I
1o2
'
' ' . " , ( '
10'
'
\ " '
1
Paschen a line in Fig. 1 where are shown the variations of
the line intensity versus detuning in reduced units. The line
center is well described by a Lorentzian shape up to a
detuning smaller than the ionic plasma frequency (in wavelength units). At higher detunings Aw = o - w o the impact
approximation breaks down because the radiative time of
interest decreases (7) and now is smaller than the duration
of a collision w;' . At very large detunings, this time of
interest is so short that all the perturbers can be supposed
as frozen on this scale. One recovers the static wing limit in
IC,,,,,I Aw I - 5/2 reported also on the figure.
This static limit may be also reached at high densities for
exceeds the
the whole profile, when the halfwidth Aw
inverse of the plasma time of interest.
1 o4
Act
Fig.
185
3. Model microfield method
I , Line shape of the Paschen U line of hydrogen perturbed by protons
In between these two extreme limits, a wide density range
exists where the effects of ionic motions are not negligible.
Up to now, no analytical theory is able to treat this general
problem due to the Wicult many body interactions. As was
pointed out in the introduction, the Model Microfield
Method [lo, 111 is well adapted to the study of the lines in
the general case.
This Model Microfield Method is based upon a modeling
The constant k,,,,, depends on the transition and may be of the statistics of the time fluctuations of the plasma microexpressed in terms of dipolar line strengths SnZ,nit, Snl,nzt, field. The field is supposed to be constant on time intervals.
The jumping times follow a Poisson statistics with a density
n,ztj, and so on [ l 5 ] . The numerical results for knntwere
fitted by a polynomial expansion in n and n'. The following v(E). In this model the Fourier transform of the generalized
Liouville operator U(t, 0) > (eq. (8))is given by
expression
and electrons. The plasma temperature is 104 K and the electron density is
N e = 3.17* 1012cm-3.The detUning is given in reduced wavelengths units
a defined by ha = A@)/Fo (ues - cgs). The normalized intensity is thus in
l/a units. The normal Holstmark field is defined by F, = e ( 3 / ( 4 ~ N J )=
~/~
0.27ues - cgs. The Doppler broadening is not included. The detuning corresponding to the ionic plasma frequency is equal to 0.32 (a units). The
and the impact limit (0.1461Aa)-2) are
static wing limit (0.1021Aal-5/2)
also reported
<w4>= {uEcw + {vK4-4)
x
{VI - v~UE(Q)}-'{VUE(~~)}.
(17)
reproduces the numerical results for n, n' varying between 1
and 50 with a relative deviation of
The preceding
expression was derived analytically for the Lyman lines
(obvious) and for the Balmer lines [16]. This allows to
aErm that the preceding expression is the exact analytical
one. This is one of the main points of this paper.
The preceding expression includes the broadening of the
upper n and lower n' states, as well as their (negative) interference contribution. Interference effects are very important
for the n, n + 1 transitions. This effect and the contribution
of the lower state to the broadening may be neglected as
long as n is much larger than n'.
For the Paschen lines one obtains
In this expression U&l), for example, is the Laplace transform at 2.! = w + iv(E) of the time evolution operator U&,
0) for a fixed value E of the microfield and { 1 denotes the
average over the field distribution function P(E).
The relevant quantities of the method are the field distribution functions P(E)which are in the present case taken
from the tables of Hooper [31, 321 and the frequency jump
v(E). This frequency jump is a free parameter. It is chosen to
reproduce the time variations of the field autocorrelation
function CEE(t) = (E(O)E(t)).Noting that the perturbational
expansion of the impact damping rates obtained previously
can be also rewriten in terms of the integral over the time of
this quantity, it is now understandable that this choice for
'1
the frequency jump allows to recover the impact limit at low
J
[3n4 - 57n2 2161.
k densities [ 1 7 ] . In fact it is an interpolating formalism
3n - 2(Z, - 1)2
between the impact and the static limits. This point will be
Looking at the general expression of the line shape in the illustrated in the next sections. The Model Microfield
impact regime (eq. (9)), there are no reasons to assume that Method can be applied to both the electronic and ionic conthe line has a Lorentzian shape. Hence for example, in the tributions.
are absent, one obtains a
case where coherence terms yV,
superposition of Lorentzian shapes which, of course, is not 3.1. Hydrogen lines
of Lorentzian type, except in the line wings. Nevertheless we Using the Model Microfield Method we calculated the lines
already proved, that, due to the structures of the relaxation of the Paschen series (3-n transitions). The electronic contrioperator and of the generalized dipole operator D, the line bution was calculated also with the Model Microfield
shape of Lyman and Balmer lines are pure Lorentzians. We Method. This first calculation allowed to introduce a frebelieve that this rule is still valid for higher series. Hence, quency depending electronic damping rate y(w) which is
this behaviour is verified in all our numerical calculations at considered in its tensorial form for the Paschen a and p
low densities in the frame of the Model Microfield theory, lines and as an isotropic quantity (i.e. replacing y(w) by T(o)
which will be presented below. An example is shown for the defined like in expression 11) for transitions 3-7 up to 3-9.
+
itf,
Physica Scripta T65
186
C. Stehlt
For higher lines the contribution of the lower state (n = 3)
to the broadening was neglected. This last assumption is
often done for numerical convenience. It is justified for large
n values [lS]. For n = 7 the approximation of no lower
state interaction overestimates the broadening by a factor of
lo%, in the line intensity. These calculations have been performed for plasma-temperatures conditions corresponding
to the stellar enveloppes (T = 104-10sK, Ne =
1014-10’7~ m - ~ In
) . all these calculations perturbers are
assumed to be protons. The corresponding tables complement the previous tables of the Lyman and the Balmer
series. They are implemented at the Centre de Donnks
Fig. 2. Halfwidth value (HWHM) of the Paschen a of hydrogen us. elec- Astronomiques de Strasbourg (CDS) and are accessible by
tronic density N e . Units are Angstroms and cm-j. The calculation does
not include the Doppler broadening (A& = 0.67A). The temperature is anonymous ftp (130.79.128.5).
The variations of the halfwidth of the Paschen U line
10 000 K and the perturbing ions are protons. The impact and static limits
are also plotted.
(A = 1.87 pm) are plotted in Fig. 2 for different electron den-
q,p)
’
0-
0
1 o2
P
L
O
3
P”
:
/ / O x
/
.
.
10’
results towards these limits at low and high densities respectively.
The Fig. 3 shows the variations of the halfwidth value for
the Paschen p line (A = 1.28 pm) us. the electron density.
Calculations are performed for different temperatures and
assuming protons as ionic perturbers. Doppler effect is
included. Theoretical results are relatively insensitive to the
ionic temperature. Experimental results are also reported
[26-281. They are in very good agreement with the theoretical calculations.
N,(c~’~)
The Model Microfield Method was initially developed for
the broadening of neutral radiators. We have extended this
method to the broadening of ionic lines. These calculations
were restricted to the line centers. The electronic damping
operator y (which is supposed to be frequency independent)
was calculated using the perturbational semi-classical collisional treatment mentioned in the previous section. The
details of the theoretical expansion and its justification are
given in [18]. We report here results concerning the
Paschen U line of He’ (A = 4689A). In Fig. 4 the variations
of the halfwidth value are plotted us. the electron density for
different types of ionic perturbers (He’ and H’) and for
- +’
different temperatures. Doppler effect is included. The static
limit is also reported for comparison. In this static calculculation the ions are supposed to be static whereas the elec10’
trons are treated within the Model Microfield Method. This
static limit is in good agreement with the theoretical result
of Greene [29] using static ions and the unified semiclassical theory for the electrons. The results obtained
within the Model Microfield Method indicate the large
effect of ionic motions in the broadening. Various experimental results are also reported [19-251. They are in very
1 oo
10l6
io”
1
good agreement except at high densities ( N e loi8~ m - ~ )
Fig. 4. Halfwidth value 0
of the Paschen a line of He+ us. elec- where the theoretical and experimental halfwidths disagree
tronic density N e . Units are hgstroms and
The calculations include by a factor of two.
Fig.3. Halfwidth value (HWHM) of the Paschen fl line of hydrogen us.
electron density N e at different temperatures ( l O W K , 20000K, 4OOOOK,
80000K). Units are hgstroms and
The calculation includes the
Doppler broadening. Perturbing ions are protons. The corresponding
curves are almost superposed. Experimental results: x [2q; 0 [2q; 0
C281.
t
-
the Doppler broadening. They have been performed for different temperatures and different perturbing ions: full line 10’K and He+;
10’ K and H+ ;
4 * 104 K and H+ ; - - - - 4 l e K and He+. The
dotted line gives the results obtained with static ions (lo5K), which cross
the theoretical result of Greene [29] (static ions, 0).Experimental results
are also reported: V [19];
[20];
[24]; 0 [22]; 0 [23];
[25]; x
c211.
-e-*
--a-
+
Physica Scripta T65
4. Conclusion
New results for the Paschen lines of hydrogen and helium
have been presented. We have derived a new expression for
Paschen Lines of Hydrogen and H e f Ion
187
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1767 (1971).
12. Griem, H. R., "Spectral Line Broadening by Plasmas" (Academic
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13. Stehlt,C. and Jacquemot, S.,Astron. Astrophys. 271,348 (1993).
14. Stehlt, C., Astron. Astrophys. Suppl. Ser. 104,509 (1994).
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16. Stehlb, C. and Feautrier, N., J. Phys. B17,1477 (1984).
17. Stehlt, C., J. de Physique C1,121 (1991).
18. Stehlt, C., Astron. Astrophys 292,699 (1994).
19. Solt~isch,H. and Kusch, H. J., Z.Naturforsch. 34a,310 (1979).
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the halfwidth value in the impact regime. The ability of the
Model Microfield Method to reproduce the impact and
static limits has been proved again. The present theoretical
results are in very good agreement with the experimental
ones. They illustrate the influence of the ion motions on the
line center. These new data for hydrogen lines can be used
for plasma diagnostics in the infrared.
Physica Scripta T65