PHYSICAL REVIEW A
VOLUME 58, NUMBER 6
DECEMBER 1998
Observation of the far wing of Lyman a due to neutral atom and ion collisions
in a laser-produced plasma
J. F. Kielkopf
Department of Physics, University of Louisville, Louisville, Kentucky 40292
N. F. Allard
Observatoire de Paris-Meudon, Département Atomes et Molécules en Astrophysique, 92195 Meudon Principal Cedex, France
and Centre National de la Recherche Scientifique, Institut d’Astrophysique, 98 bis Boulevard Arago, 75014 Paris, France
~Received 4 June 1998!
The time-resolved spectrum of a laser-produced plasma in H2 exhibits a continuum that we attribute to
radiative collisions of H atoms in a 2 p state with ground-state neutral atoms and protons. This binary collision
Lyman-a line wing observed from 1150 Å to 1700 Å is emitted by the shock dissociated gas that surrounds the
focal region. The observed temperature, electron density, and shock front speed confirm models of the shock
that predict that Lyman a arises from a shell of atomic hydrogen in which the neutral atom and ion densities
are sufficient to create the observed line broadening. The experimental far wing spectrum agrees with unified
theory calculations of the Lyman-a line, which allow for the dependence of the radiative dipole moment on
internuclear separation during a radiative collision. Broad neutral atom effects become evident at 1180 Å, 1260
Å, and 1600 Å when the density of neutral perturbers is of the order of 1020 atoms/cm3, while features at 1230
Å, 1240 Å, and 1400 Å appear when the plasma is highly ionized. These satellites result from free-free
transitions of a H atom during collisions with other neutral atoms and protons and are correlated with potential
curves and radiative dipole moments of H2 and H21 . @S1050-2947~98!02811-X#
PACS number~s!: 32.70.Jz, 52.25.Qt, 52.35.Tc, 52.50.Jm
I. INTRODUCTION
The shape of an atomic spectral line emitted by an excited
free atom in a gas is determined by its interaction with perturbing neutral atoms, ions, and electrons. When the gas density is sufficiently high, the effect of collisions on the line
shape dominates Doppler broadening due to the random velocities of the emitting atoms and natural broadening due to
the finite lifetime of the excited state. Great progress has
been made in recent years as experimental and theoretical
methods for studying the effects of collisions on spectral
lines have developed. For the case of broadening by neutral
perturbers, in the limit of binary collisions, the unified theory
of the line wing and core provides a semiquantitative explanation of the spectra observed for alkali-metal atoms and a
few other select cases perturbed by noble gases @1,2#.
In this work, one limiting factor is our lack of detailed
knowledge of the interaction potentials for excited states of
the radiator and the perturbing atom. In applications to spectral line shapes, the potentials must be known to approximately 1 cm21 to account for the line wings and the
asymptotic energy must be correct to within approximately
0.001 cm21 to predict the core shift and width properly. Excited atom-atom interactions, even the simpler cases of
alkali-metal–noble-gas collisions, have a multitude of interacting potentials that strongly mix states. This leads to potentials that exhibit shallow maxima or minima at long range
and radiative transition probabilities that depend on the separation of the emitting atom from its neighbors. Indeed, one
motivation for studying spectral line shapes in such systems
is to use them as a tool to measure the interaction potentials
and to detect collision-induced transitions. The line-shape
theory, however, has been tested only by comparing ob1050-2947/98/58~6!/4416~10!/$15.00
PRA 58
served profiles with those predicted using a parametrized potential, that is, a potential adjusted to achieve agreement between experiment and theory. This procedure offers insight
into the interactions, but it is not necessarily unique and it
does not test the line-shape theory thoroughly.
In the case of the hydrogen molecule and its ion, the
potentials and the transition moments for the low-lying states
are now well established theoretically with ab initio calculations and confirmed experimentally with measurements of
bound-bound transitions. Therefore, from the theoretical perspective, H2 and H21 are ideal systems with which to explore
spectral line shapes. Unfortunately, the experiments are not
as ideal because, unlike spectroscopy of alkali-metal–raregas systems, static cell experiments are not possible since H2
remains bound at temperatures below 3000 K. Although a
low-current electrical discharge will dissociate the molecule
~in a Wood’s tube the dominant species is the H atom!, the
density in the positive column is too low for line wing measurements. While atmospheric pressure arcs produce a high
degree of dissociation and a partially ionized plasma, they
are subject to impurities and intense molecular emission. Our
approach is to use a laser-produced plasma in a pure gaseous
H2 target to generate a dense atomic gas for line profile measurements.
When a 1064-nm 6-ns Nd:YAG laser pulse ~where YAG
denotes yttrium aluminum garnet! with an energy of the order of 300 mJ is focused into a cell containing H2, additional
self-focusing of the beam causes most of its energy to be
delivered suddenly to a cylinder about 0.5 cm long and 50
mm in diameter surrounding the focal point. As a consequence, a shock wave propagates outward and leaves behind
a cooling mixture of neutral H, H2, ions, and electrons to
provide a source for studying radiative collisions. Simple
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© 1998 The American Physical Society
PRA 58
OBSERVATION OF THE FAR WING OF LYMAN a DUE . . .
models of the shock excitation, dissociation, and ionization
are consistent with space- and time-resolved diagnostic spectroscopy of this source @3,4#. Of particular interest to us is
the extraction of the spectral line profile in the vacuum ultraviolet Lyman-a region. Time-resolved emission spectroscopy combined with models of the postshock gas allows us
to identify features in the Lyman-a profile that are the result
of radiative collisions of an excited H atom and an unexcited
atom or ion perturber @4#. We describe the experiments here
and compare the observed spectra with unified theory Lyman
a calculations @5–7#.
II. EXPERIMENT
The plasmas were made inside a 60-mm-i.d. stainlesssteel six-way cross. Fluorescence-free ultraviolet grade fused
silica windows along one axis permit a laser beam to be
focused at the center of the cross and then exit to a beam
dump. An 80-mm-focal-length fused silica lens was located
about 1 cm outside the entrance window, positioned so that
the focal point of the lens for the laser light was centered in
the cross. The target region was pumped by a liquid-nitrogen
trapped diffusion pump. Vacuum integrity and the presence
of residual gases were monitored with a quadrupole mass
spectrometer. The clean out-gassed cell was filled with
99.995% purity H2 and the cell pressures were measured
with a capacitance manometer. Once filled, the cell was used
for spectroscopy in a static mode, but the pressure was continuously monitored. We observe a slight rise in pressure ~2
Torr out of 800 Torr! due to heating of the filling gas by the
laser, but otherwise the cell pressures are stable over a data
run. Nevertheless, we repumped and refilled frequently to
minimize the effect of outgassing, particularly since H2O has
strong structured continuum absorption in the Lyman-a region.
Light from a Continuum Surelite-II Nd:YAG laser was
directed into this system with steering mirrors. In the experiments described here, the fundamental output from the laser
was used: 600 mJ, 1064 nm, Gaussian profile, and 5-ns pulse
duration at 10 Hz. Other experiments with the second ~300
mJ at 532 nm! and third ~110 mJ at 355 nm! harmonics yield
results similar to those described here, but use of the
1064-nm wavelength keeps the intense plasma-forming light
out of the sensitivity range of the spectroscopic detectors and
makes it easier to distinguish the visible and ultraviolet emission of the newly formed plasma from the scattered incident
laser light.
The spectrum of the plasma was observed at 90° to the
plasma-forming laser beam. In the vacuum ultraviolet the
emitted light passed along one axis of the cross, through a
MgF2 window, through an entrance slit, and into an evacuated 0.2-m Acton VM502 monochromator. This instrument
has a 1200-groove/mm holographic grating with an iridium
coating for a wide, nominally flat, spectral reflectivity. The
dispersed spectrum is imaged by the grating onto an exit slit
and the light then continues into an EMI CsI solarblind photomultiplier with a 2 ns risetime and 3.4 ns full pulse width.
The MgF2 windows of the photomultiplier and the monochromator absorb light with l,1150 Å. The photocathode
of the detector is not sensitive to l.1800 Å. Separate calibrations of the detector quantum efficiency and grating re-
4417
flectivity are combined to provide a system efficiency. The
overall spectral response of the monochromator and detector
is also checked by comparing the spectrum of H2 observed
from a positive column discharge with a theoretical spectrum, as described later. The dispersion of this grating on the
exit slit is 40 Å/mm and with matched entrance and exit slits
of 100 mm, the nominal instrumental linewidth is 4 Å.
For the detection of weak emission at low light levels, the
photomultiplier is connected to a fast Stanford Instruments
SR240 fast preamp and a SR430 multichannel scaler. The
scaler discriminates against noise pulses and then counts
photons and adds the total to 5-ns bins starting 45 ns after the
arrival of a trigger pulse, derived from a photodiode that
monitors the Nd:YAG laser output. The scaler is controlled
by a Pentium-based computer running Linux, which repetitively sums data from many laser pulses ~typically 200–2000
at 10 Hz!, transfers the sum to its memory, and steps the
vacuum monochromator in 2-Å increments. A twodimensional database is built in this way, which can span
1150–1800 Å. These data can be viewed as a spectral image
in which the intensity is proportional to the number of detected photons and the position is determined on one axis by
the wavelength and on the other by the delay from the initiation of the plasma. The data may be projected along the
time axis to obtain lifetimes at specific wavelengths or along
the wavelength axis to obtain spectra at specific times.
In the Lyman-a region, the first 100 ns of emission from
the plasma is far too intense for photon counting since the
counter saturates, or does not respond at all, for count rates
above about one photon every 5 ns. To take advantage of the
high initial flux we use an Stanford SR250 analog boxcar
detector typically with a 10-ns gate. An 3.5-ns analog delay
line permits observation of the rising edge of the plasma
emission. In this mode the system makes an exponential
moving average over 30 laser pulses of the total charge from
the photomultiplier anode. It produces an analog signal proportional to the number of photons detected, which is digitized and stored by the control computer system. Only one
time bin may be measured in each cycle of this process.
A narrow-band tunable dye laser, pumped by another synchronized Nd:YAG laser, provides a probe pulse ~2 mJ, l
55800 Å, Dl50.001 Å, and Dt56 ns) for schlieren and
shadowgraph imaging. The dye laser beam is spatially filtered and collimated to make a plane wave front with a diameter of about 2 cm, which passes through the plasma at
90° to the axis. In shadowgraph imaging the probe laser light
simply passes directly to a screen about 2 m from the cell
without any intervening lenses or masks. This makes a
‘‘bright field’’ image that is recorded with a low-light level
peltier-cooled charge coupled device camera and is analyzed
to derive gas density behind the shock front @8–10#. The
system may also record ‘‘dark field’’ schlieren by inserting a
lens and mask to block undeviated light in the optical path to
the screen.
III. DIAGNOSTICS AND MODELS
OF THE LASER-PRODUCED PLASMA
The densities behind the shock front derived from these
shadowgraph images and the blast wave theory described
below are sufficiently high that the requirements for local
4418
J. F. KIELKOPF AND N. F. ALLARD
PRA 58
FIG. 1. Schlieren measurement of the shock front, compared to
a cylindrical blast wave model from Eq. ~1!.
thermodynamic equilibrium ~LTE! are met and spectroscopic
diagnostics to determine the temperature and the electron
density are straightforward @3#. In a typical experiment 300–
600 mJ of laser energy is delivered to H2 at an initial molecule density of 231019 cm23. The electron density inferred
from the H-b linewidth is excess of 231018 cm23 and the
electron temperature inferred from the Balmer series line-tocontinuum ratio is of the order of 105 K at the initiation of
optical emission from the plasma. Both electron temperature
and density decrease rapidly and approximately 50 ns after
the laser fires the temperature is 20 000 K and the electron
density is 1018 cm23. We also find that the H-b emission
arises in a cylindrical shell expanding symmetrically about
the the laser beam axis. This shell has a radius of approximately 0.5 mm after 100 ns, with initial expansion speeds of
approximately 20 km/s. The appearance of a hot fast moving
shell of atomic H suggests that there is a blast wave originating from the focal region.
Figure 1 shows measurements of the radius of the shock
front from schlieren images during the first 1.6 ms of its
expansion. Shadowgraph images such as the one shown in
Fig. 2 allow measurements at later times and distinguish the
front and the hot gas remaining behind it. Figure 3 shows
how the shock front propagates until it gets beyond the
chamber window. We also distinguish between the expanding shock and a nearly static postshock bubble of hot gas.
Since the initial velocity of the shock wave is greater than
Mach 10, it compresses, heats, and dissociates the ambient
molecular gas, and leaves behind hot atoms, ions, and electrons. The strength of the shock decreases as it moves outward and after about 2 ms it separates from the heated gas.
The spectral observations apply to the strong shock prior to
this separation. The radius of the shock front is in itself an
indicator of the conditions in the gas when it is compared to
models of blast wave propagation @11,12#.
For a cylindrical blast wave, after time t the shock front
expands to radius R given by
R ~ t ! 5 ~ 2R 0 c s J 21/2
t2c 2s l 1 t 2 ! 1/2,
0
~1!
where R 0 is a scale distance, c s is the speed of sound in the
ambient gas, and constant parameters J 0 and l 1 are deter-
FIG. 2. Shadowgraph image of the shocked plasma 1 ms after
the laser pulse. The box is 2 cm on a side. The bright ring marks the
edge of the shock front and the outer boundary of the hot atomic H.
The initially cylindrical plasma region has become spheroidal at
this time and will be nearly spherical at 5 ms.
mined numerically @13,14#. The curve plotted in Figs. 1 and
3 demonstrates that a cylindrical blast wave offers an effective simple model of the observed shock propagation. The
scaling of the shock expansion is determined by the initial
gas density and the laser energy per unit length in the focal
region. The value used is chosen to give the best overall
agreement between the model and the observed expansion,
but it is within 10% of the experimentally observed ratio of
the laser energy absorbed to the plasma length. Additional
details of the formation of the shock and the late time behavior of the postshock gas are described elsewhere @15,16#.
Here we are concerned primarily with the characteristics of
the hot plasma and surrounding dense neutral gas from 20 ns
to 2 ms after the the plasma-production laser pulse.
FIG. 3. Shadowgraph measurement of the shock front. For times
less than 2 ms the front marks the boundary between the hot plasma
and the ambient gas, but for later times a static bubble of hot gas
remains and the front propagates out of the field of view.
PRA 58
OBSERVATION OF THE FAR WING OF LYMAN a DUE . . .
FIG. 4. Radial variation of temperature and the densities of H2,
H, and H1 or e 2 at 20 ns. Lyman a arises from the atomic H shell.
The solution of the self-similar one-dimensional shock
wave propagation given by Sakurai @13,14# does not take
into account dissociation of the ambient gas directly. The
theory yields analytical expressions for the radius of the
shock, and the density and temperature in the heated gas
behind the front. Because the densities are high, the gas
comes into LTE quickly in comparison to the motion of the
front. Consequently, in the first approximation we use thermal equilibrium to calculate the dissociation of H2 and Saha
ionization equilibrium to calculate the excitation and ionization of the atoms behind the moving front. This method
yields the density, temperature, and composition of the
shock-heated gas as a function of time in a laser-produced
plasma. It has been used successfully to model the spectra of
laser produced plasmas in other cases @17#. A more complete
model for dissociation, excitation, and ionization with shock
propagation in a real gas has been developed by Steiner and
Gretler @18#. Preliminary results from an application of their
methods to H2 indicate that the behavior of the shock radius
is similar in both cases, although the real gas is somewhat
cooler and denser behind the shock than in the idealized
model we use here @19#.
In the plasma model calculations of Figs. 4 and 5 we
assume an initial pressure of 800 Torr of H2 at 300 K into
which the laser deposits 600 mJ in a 0.5-cm-long channel to
start the expansion. A cylindrical bubble of hot ionized gas is
formed behind the shock with a radius that expands rapidly
after the initial laser pulse. Once the expansion is under way
and the excitation pulse is off the density is very low close to
the axis where the temperature high. Farther from the axis
and just behind the front the gas is compressed and the density is greater than ambient. With time the outward flow
excavates the cavity. Neutral atoms in the shock heated volume lie on a ridge in the r-t plane, which is defined on its
outer edge by the dissociation of H2 and on its inner edge by
the ionization of the atoms. The atomic density in this ridge
exceeds the ambient molecular density immediately after the
laser pulse because of the compression and also because of
the dissociation of diatomic H2 into two atoms. The distribution of H atoms is shown in the three-dimensional representation of Fig. 6. The emission responsible for Lyman a arises
4419
FIG. 5. Radial variation of temperature and the densities of H2,
H, and H1 or e 2 at 50 ns. As in Fig. 4, Lyman a arises from the
atomic H shell. The expansion is nearly self-similar as the peak in
the neutral density distribution moves outward and decreases.
in this outwardly moving shell. Inside it there is a similar
ridge in the electron density, where the outer edge is defined
by the ionization of the atoms and the inner edge by the
density gradient of postshock gas.
The relevant densities for radiative collisions involving
excited atoms, neutral-ground state atoms, ions, and electrons can be found for any time by taking slices through the
three-dimensional representations of the model. Sections at
20 and 50 ns are plotted in Figs. 4 and 5. At these times the
ions and neutral atoms are localized in thin overlapping
shells about 0.05 cm from the axis where the temperature
decreases from about 30 000 K in the ionized region to less
than 10 000 K in the neutral atom shell. The outward expansion evident in comparison of these two examples is accompanied by a decrease in the density of the neutral and ion
regions.
The predicted atom density is so high that the shell is
optically thick for Lyman a at line center and consequently
will require a solution of radiative transfer to fully model.
The theoretical line profiles described later predict that the
line wing is optically thin for all times when l.1300 Å.
The observed emission in the wing is an integration along
FIG. 6. Density of atomic H versus time and distance. The white
line marks the boundary of the shock wave.
J. F. KIELKOPF AND N. F. ALLARD
4420
the line of sight including the high-density atom and ion
shells, but weighted by the number of emitters within each
volume element. The average conditions for the formation of
the line wing will be in the densest parts of the neutral shell
where the ionization tends to be low. According to the blast
wave models, in the period 20 ns after the plasma is formed
the maximum neutral atom density is 231020 cm23 and the
maximum ion density is about 1019 cm23.
Recent work on self-focusing in laser-produced plasmas
suggest this scene for the onset of the shock @20,21,15#. The
leading edge of the laser pulse arrives close to the vacuum
focal point, but through multiphoton processes it creates a
temporary charge-density gradient that leads to ionization
defocusing. The defocusing causes the plasma formation to
begin back toward the entering beam, but as more energy is
deposited into the plasma, the ions also move off the axis.
Thermal channel formation creates an index of refraction
gradient that focuses the incoming beam and delivers its energy several millimeters along the axis. In the 6 ns that
elapse during the pulse, the shock wave has traveled more
than 100 mm outward and the plasma behind it may be fully
stripped. From this time onward the expansion is selfsimilar, but the shock intensity decreases. Except for the
scaling on both axes, the distributions shown in Fig. 4 and 5
appear to hold for times up to and beyond 1 ms.
IV. OBSERVED SPECTRA
The earliest spectra that we observe reveal an intense continuum with contributions from free-free and free-bound
electron-ion collisions in the plasma. In the visible region,
this continuum extends throughout the Balmer series lines,
which are broadened by electron-atom collisions to the degree that even H-b is hardly distinguishable as a line. In the
vacuum ultraviolet, the free-free bremsstrahlung continuum
for frequencies above the Balmer limit extends down to Lyman a. Prior to 100 ns there is no evidence of emission from
H2, but it does appear weakly later. As we have seen from
the models of the blast wave, compressed H2 should be
present in a thin layer just outside the atomic region, but the
temperature there is too low to excite the molecule from its
1 1
1
X 1S 1
g ground state to the B S u or C P u electronic states
from which it radiates in the vacuum ultraviolet.
In these early phases the degree of ionization is very high
and there are large contributions from free-free and boundfree electron-ion collisions, while the neutral atomic emission that appears from the ionized region is subject to high
collision rates with electrons and ions. The cooler neutral
atomic shell is optically thick at the center of the the
Lyman-a line, leading to a redistribution of the emission into
the far line wing where the gas is optically thin. At 20 ns the
full width of the self-reversal is approximately 20 Å.
The spectrum of the Lyman a region from 2 to 22 ns after
the peak of the emission in the vacuum ultraviolet is shown
in Fig. 7, corrected for instrumental response and transformed to be proportional to F n , photons per frequency interval. The self-reversal and the continuum to the red are the
most obvious features here. While the reversal makes it impossible to see the center of the line or to normalize the
observed profiles on an absolute scale, it is essential to the
detection of the far wing. Without it, the instrumental scat-
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FIG. 7. Lyman-a wing near 1230 Å at three different times
immediately after the laser shot. Lyman a is reversed in the cooler
outer parts of the plasma. Two prominent features at 1230 Å and
1240 Å attributed to interactions with H1 are indicated. Interactions
with neutral H appear as broad contributions to the wing at 1180 Å
and 1260 Å in the initial spectrum. Times are given relative to the
time of maximum emission in the optically thin far line wing.
tered light from the center of the line would be comparable
in strength to the line wing. Self-reversal provides a built-in
filter that removes the line core. In a very rapid sequence, the
reversal decreases and the line wing is revealed closer and
closer to the center of Lyman a. At the time the data in Fig.
7 were recorded, features at 1230 Å and 1240 Å made a
distinct appearance. These structures are ubiquitous in the
prompt emission from the plasma and in the interval from 0
to 50 ns we see them emerge and then fade back into the
wing. The disappearance of the 1230-Å and 1240-Å features
is coordinated with the disappearance of the bremsstrahlung
continuum, suggesting that they are due to ion-atom collisions. Other broad features apparent in Fig. 7, one with a
maximum at approximately 1180 Å and the other around
1260 Å, coincide with predicted satellites due to neutral H.
Their appearance is consistent with the hypothesis that Lyman a is emitted from the dense atomic H shell produced by
the shock front and excited optically by radiation from the
hotter ionized inner zone.
In the same time frame, a weaker feature at 1400 Å appears out of the falling electron-ion continuum. Figure 8
shows this region about 22 ns after the laser pulse. This too
is apparently correlated with the ion-atom collision rate.
There is a small increase in intensity at 1400 Å, with a very
noticeable drop to longer wavelengths. As noted, in this early
time there is no evidence of emission due to bound-bound
transitions of the H2 molecule. Another broader feature at
1600 Å appears in this spectrum, but is influenced by the
underlying electron-ion continuum that extends upward to
the Balmer series limit. The intensity of the apparently
atomic emission in the 1600-Å region grows relative to the
1400-Å feature with increasing time.
The strength of the bremsstrahlung continuum is a problem for the detection of weak structures such as these. As
noted earlier, this continuum extends back to the Balmer
series limit at 3645 Å. In the region below the Balmer limit
OBSERVATION OF THE FAR WING OF LYMAN a DUE . . .
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FIG. 8. Lyman-a line profile at 22 ns. The spectrum has been
banned so that the signal F n is given per frequency interval. Satellites near 1400 Å and 1600 Å are apparent. At longer wavelengths
the free-free and bound-free continuum due to electron-ion collisions boosts the observed line wing. The model shown is for 10 000
K.
and above Lyman a the free-free and bound-free electron-ion
continuum emission from an ionized hydrogenic plasma is
proportional to @22#
S
F n ~ l ! ;exp 2
hc
lkT
D
~2!
and decreases exponentially with decreasing wavelength.
The function log10 Fn appears to have a linearly increasing
base line corresponding to a plasma temperature of about
18 000 K as shown in Fig. 8. If we assume that this is due
entirely to electron-ion contributions, it can be removed as
shown in Fig. 14 to reveal weak but distinct satellites.
After the first 100 ns, the emission rate is low enough to
use photon counting and we can record a two-dimensional
map of the intensity as a function of time and wavelength
efficiently. The spectral image is shown in Fig. 9. When the
count rate is too high there is no detection at all. This results
in a contour close to Lyman a that highlights its line shape.
At the very center, however, where Lyman a is reversed, the
flux drops below the saturation threshold and a narrow line
reappears in the spectral image. Apart from that unusual aspect of interpreting the image, it reveals a broad prominent
feature at 1600 Å. In part this is due to bound-bound transitions of H2 and weak individual lines or blends are readily
distinguished on close inspection. Between those lines there
is still a continuum that appears to be about 80% of the total
emission in that region.
It would take very high spectral resolution to extract the
continuum precisely by resolving fully the individual boundbound B-X band lines, but we can remove the line spectrum
approximately with a fitting algorithm based on the known
energy levels and transition probabilities for H2, given that at
these times the electron-ion continuum may be neglected because the ion density is low. The basic procedure has been
described in earlier work to measure populations of H2 excited states in a glow discharge @23#. We have a database of
4421
FIG. 9. Spectral image of the Lyman-a region. The darkness of
the image is proportional to the number of detected photons at the
time and wavelength corresponding to the coordinates of the point
in the image. The flux near Lyman a is too high to count, which is
why it is apparently contoured by the dark band. The feature at
1600 Å is mostly the neutral 1600-Å satellite.
energy levels for the B and X states of H2 based on experimental measurements and we have theoretical values for the
oscillator strengths and Höln-London factors for the transition arrays. We calculate all possible allowed transitions in
the vacuum ultraviolet, assign a line shape to represent the
instrumental response, and simulate the spectrum with a sum
of all possible lines. Each contribution is weighted according
to the population of its initial state. Since each initial state
contributes linearly to the final spectrum, we can perform a
least-squares decomposition of the observed spectrum to determine the excited-state populations. In this case, where
there is also an unknown continuum present in addition to
the line spectrum, we make a first estimate of the continuum
by spline fitting to minimum intensity points between obvious lines. The fitting process provides the parameters of the
line spectrum contribution, defining the population of the B
states. Once these are known the bound-bound spectrum is
calculated and subtracted from the observed spectrum to obtain an improved estimate of the free-free neutral atom continuum. The resulting spectrum for the 1600-Å region is
shown in Fig. 10, where data from 800 ns to 2 ms have been
summed. The neutral shell density is expected to be about
1019 atoms/cm3 during that time.
V. COMPARISON WITH THEORETICAL LINE SHAPES
Structures in the Lyman-a line wing have been been identified with free-free transitions that take place during binary
close collisions of the radiating H atom and a perturbing
atom or ion @24,5#. A region in which the intensity goes
through a shoulder or even an increase with increasing separation from the line center is termed a ‘‘satellite’’ in the
line-shape literature. In the simple picture of a static distribution of perturbing atoms, the probability of the emission of
a photon of frequency n per unit frequency interval F n ( n ) is
proportional to the probability of a configuration that per-
J. F. KIELKOPF AND N. F. ALLARD
4422
PRA 58
The complete profile depends not only on the potentials
and collision statistics as suggested by Eq. ~4!, but on collision dynamics and the radiative dipole moment as a function
of interatomic separation. A unified theory has been developed to calculate neutral atom line broadening and complete
details of the derivation of the theory and explicit calculations for Lyman a are given elsewhere @6,7,32#. In this approach, the profile is given as the Fourier transform of an
autocorrelation function which is written as
F n ~ D n ! 5T$ exp@ ng ~ s !# % ,
~5!
where the Fourier transform T is taken such that F n (D n ) is
normalized to unity when integrated over all frequencies and
Dn is measured relative to the unperturbed line.
When we assume that the perturber follows a rectilinear
classical path trajectory at velocity v , we get
FIG. 10. The 1600-Å region of the Lyman-a wing from the
image in Fig. 9, showing the extracted continuum ~—! and the fit
with molecular contributions. The experimental spectrum ~—! and
the fit ~-•-•-! are not distinguishable on this scale. The spectrum is
integrated from 800 ns to 2 ms after the laser pulse.
turbs the system to make the energy difference between the
upper and lower states give n 5 @ V i (R)2V f (R) # /h. Thus the
photon emission rate is
F n ~ n ! d n 5N4 p R 2 dR
~3!
or, in terms of potentials,
U
F n ~ n ! 5N4 p R 2
d ~ DV/h !
dR
U
21
,
~4!
where n 5DV/h and DV is the difference between upperand lower-state potentials. This leads to a satellite where DV
goes through an extremum.
Five difference potentials for the allowed transitions of
H21 have extrema and are expected to exhibit associated
excited atom-proton binary collision satellites. This is not the
case for H2, but there are still several satellites expected
from free neutral atom collisions. Tables I and II list the
predicted satellites, the radius of the interatomic separation
of the potential extremum, and upper- and lower-state identifications. Several of these features are seen in these experiments and the measured wavelengths are also given in the
tables.
Although methods of calculating the complete profiles
have been known for some time, only recently comprehensive calculations have been made that include perturbations
by both neutral atoms and ions and all possible quasimolecular states of H2 and H21 @7#. There are many such states and
a tabulation of them is given in Ref. @5#. For the calculations
shown here, the H2 potentials were taken from tabulations
given by Sharp @25# and Wolniewicz and Dressler @26#. The
radiative dipole transition moments for H2 were taken from
tabulations of Dressler and Wolniewicz @27# for the singlet
states and from preliminary ab initio results of Spielfieldel
@28# for the triplet states. For H21 the potentials calculated
by Madsen and Peek @29# were used with dipole transition
moments given given by Ramaker and Peek @30,31#.
g~ s !5
1
( e,e 8 u D ee 8 u 2
3
E
1`
0
( p ee 8
e,e 8
2p r dr
F SE
3 exp i
s
0
E
1`
2`
dx d ee 8 „R ~ 0 ! …
dt2 pn e 8 e „R ~ t ! …
G
D
3d e 8 e „R ~ s ! …2d ee 8 „R ~ 0 ! … .
~6!
The e and e 8 label the energy surfaces on which the interacting atoms approach the initial and final atomic states of
the transition as R→`. For Lyman a there are several different energy surfaces that lead to a same asymptotic energy
as R→`. The sum with weight factors p ee 8 accounts for this
degeneracy. The asymptotic initial- and final-state energies
are E `i and E `f , such that E e (R)→E `i as R→`. We then
have R-dependent frequencies
n e 8 e ~ R ! [ @ E 8e ~ R ! 2E e ~ R !# /h,
~7!
which become the isolated radiator frequency n i f when perturbers are far from the radiator. The radiative dipole transition moment of each component of the line depends on R
and changes during the collision. At time t from the point of
closest approach for a rectilinear classical path
R ~ t ! 5 @ r 2 1 ~ x1 v t ! 2 # 1/2,
~8!
where r is the impact parameter of the perturber trajectory
and x is the position of the perturber along its trajectory. The
term d ee 8 „R(s)… given in Eq. ~6! is
d ee 8 „R ~ s ! …5D ee 8 „R ~ s ! …exp~ 2i2 pn a s ! ,
~9!
where D ee 8 „R(s)… is the R-dependent dipole transition moment. The total line strength of the transition is the sum of its
components ( e,e 8 u D ee 8 u 2 .
The evaluation of Eqs. ~5! and ~6! has been done for
temperatures and densities that are expected in the laserplasma source on the basis of the blast wave analysis such as
shown in Fig. 5. The profile dependence on neutral density is
shown in Fig. 11 and on ion density in Fig. 12. There are two
PRA 58
OBSERVATION OF THE FAR WING OF LYMAN a DUE . . .
FIG. 11. Theoretical profiles of Lyman a for three different
combinations of densities of neutral atomic H and the same H1 ion
density, computed with the unified line shape theory as described in
the text. The temperature for the calculation is 10 000 K, but the
profiles are not very temperature sensitive. F n is normalized such
that its integral over the profile is unity when n is given in cm21.
The far wing satellite at 1600 Å increases with N H , as do the
‘‘shoulders’’ at 1180 Å and 1260 Å closer to the line center.
very distinct satellites predicted by the theory, one due to
ions at 1400 Å and the other due to neutrals at 1600 Å. There
are also shoulders on the line core at 1180 Å and 1260 Å due
to the neutral satellites. The other satellites blend into the
profile.
In Fig. 13 we compare the profile of the close 1230-Å and
1240-Å satellites due to H-H1 interactions with the experimental profile. The 1230-Å satellite is very intense because it
occurs at R'11 Å and the interaction volume on the righthand side of Eq. ~4! is therefore large. When the conditions
FIG. 12. Theoretical profiles of Lyman a for three different
combinations of densities of the H1 ion and the same neutral H
density, computed with the unified line shape theory as described in
the text. The temperature for the calculation is 10 000 K, but the
profiles are not very temperature sensitive. F n is normalized such
that its integral over the profile is unity when n is given in cm21.
The far wing satellite at 1400 Å increases with N H 1 .
4423
FIG. 13. Comparison of the 1230-Å region with a theoretical
profile for N H 1 51019 ions/cm3 and N H 5231020 atoms/cm3. Satellite regions due to neutral atoms at approximately 1180 Å and to
ions at approximately 1230 Å appear at the observed strengths.
are optimum the satellite is a very distinct feature in the
Lyman-a wing and its intensity in the theoretical profile relative to the intensity of the 1400-Å satellite is in approximate
agreement with the observations.
The 1400-Å and 1600-Å satellites are compared in Fig. 14
with the experimental wing profile. The ion and neutral densities determine the relative contributions of the 1400-Å and
1600-Å satellites to the complete theoretical profile. The experimental profile is scaled to match the 1600-Å satellite and
consequently the line core and 1400-Å satellite appear to be
relatively weaker than the theory predicts. However, these
different spectral regions may originate under different
physical conditions in the source. In the theoretical profiles,
the 1400-Å satellite is difficult to distinguish when the H1
density is as a low as 1018 cm23. The theory does not include
broadening by electrons, which is important close to the
FIG. 14. Comparison of the far Lyman-a wing with theoretical
models. The experimental data ~—! from Fig. 8 are shown here
with the electron-ion bremsstrahlung contribution removed. It is
compared with a theoretical profile for N H 5231020 atoms cm23
and N H 1 51019 ions cm23.
4424
PRA 58
J. F. KIELKOPF AND N. F. ALLARD
TABLE I. Satellites on Lyman a ~1215.7 Å! due to H-H collisions.
Upper state
h 3s 1
g
i 3p g
C 1P u
C 1P u
B 1S 1
u
Lower state
b
b
X
X
X
s1
u
s1
u
1 1
Sg
1 1
Sg
1 1
Sg
3
3
l ~Å!
Theory
1172
1179
1214.1
1267
1623
l ~Å!
Observed
1180
1260
1600
R ~Å!
2.7
2.4
4.6
2.1
2.1
core, but much less than neutral broadening in the extreme
wing at these densities.
The values for neutral and ion densities used here were
taken from the blast wave model and the resulting profile
compares well with the experiment. The satellite at 1600 Å is
due to collisions of excited atom with a neutral perturbing
atom and the potential extremum responsible for it is at 2.1
Å ~see Table I!. The 1400-Å satellite is due to collisions with
a proton, but the extremum of the H21 potential curve responsible for this satellite is at 4.5 Å ~see Table II!. In order
for the 1600-Å satellite to appear stronger than the one at
1400 Å the neutral perturber density must be much higher
than the ion density. The shock models establish that the
density ratio of neutral atoms to ions is about 200:1 when
these spectra were recorded.
Figure 15 compares a line-shape calculation for the
1600-Å region observed in the time period when ion densities are low and charged-particle collisions produce a negligible effect on the profile. The theoretical profile matches the
observed shape of the 1600-Å satellite well. This is an important point because the line-shape theory predicts that this
satellite is enhanced by an increase in the radiative dipole
moment in the region of internuclear separation that contributes to this satellite.
VI. CONCLUSIONS
The laboratory observations of a laser-produced plasma
confirm that satellites appear on Lyman a due to collisions
with neutral atoms and protons. The satellites due to ions
were predicted by Stewart, Peek, and Cooper @33# and those
due neutral atoms were predicted by Sando, Doyle, and Dalgarno @34# in their early theoretical work on the Lyman-a
wing. This experimental confirmation of the theory supports
TABLE II. Satellites on Lyman a ~1215.7 Å! due to H-H1
collisions.
Upper state
Lower state
l ~Å!
Theory
3d p g
3d s g
2s s g
2ppu
4 f su
2ppu
3d s g
2p s u
2p s u
2p s u
1s s g
1s s g
1s s g
2p s u
966
1020
1107
1215.2
1234
1241
1405
l ~Å!
Observed
R ~Å!
1230
1240
1400
1.6
1.3
2.7
13.6
11.0
4.7
4.5
FIG. 15. Comparison of the 1600-Å region from Fig. 10 with
theoretical models with and without variable D(R). The line wing
is optically thin in this region. F n is normalized to unity for an
integral of the line profile over n in cm21 and scaled for a neutral H
density of 1019 atoms/cm3. The observed shape of the satellite is in
good agreement with the variable D(R) theory; the constant D
theory underestimates the strength of the satellite. Both theories
predict an oscillatory structure between the satellite and the line.
the identification of these features in the Lyman-a spectra of
white dwarf and l Bootis stars as seen with the International
Ultraviolet Explorer and Hubble Space Telescope @35#. Blast
wave models of the plasma are consistent with the observed
strength of ion and neutral satellite features in the Lyman-a
wing. For stellar atmospheres the line-shape models have
proved useful in diagnostic measurements of ion and neutral
densities and thereby temperature. It now appears that the
same is true for laboratory plasmas and that a comparison of
the Lyman-a wing with line-shape models is a tool for determining neutral and proton densities in a hydrogenic
plasma.
These experiments also confirm that the variation of the
radiative dipole moment is an important factor in determining the far wing emission of Lyman a. Most previous work
on far wing broadening has made the simplifying assumption
that D(R) is a constant, but based on this work it appears
that may not be valid. We have noted that when D(R) differs
significantly from its asymptotic value at an R close to the
region forming a satellite, the strength of the wing may be
enhanced ~or diminished! considerably @7#. Certainly when
satellites are used as density diagnostics for plasmas, this is a
factor that needs to be considered.
ACKNOWLEDGMENTS
The work at the University of Louisville was supported
by a grant from the U.S. Department of Energy, Division of
Chemical Sciences, Office of Basic Energy Sciences, Office
of Energy Research. The contributions of Frank Tomkins
from Argonne National Laboratory are acknowledged with
gratitude.
PRA 58
OBSERVATION OF THE FAR WING OF LYMAN a DUE . . .
@1# N. Allard and J. Kielkopf, Rev. Mod. Phys. 54, 1103 ~1982!,
reprint 208.
@2# J. Kielkopf, J. Phys. B 16, 3149 ~1983!.
@3# J. Kielkopf, Phys. Rev. E 52, 2013 ~1995!.
@4# J. F. Kielkopf and N. F. Allard, Astrophys. J. 450, L75 ~1995!.
@5# N. F. Allard, D. Koester, N. Feautrier, and A. Spielfiedel, Astron. Astrophys., Suppl. Ser. 108, 417 ~1994!.
@6# N. F. Allard, J. F. Kielkopf, and N. Feautrier, Astron. Astrophys. 330, 782 ~1998!.
@7# N. F. Allard et al., Astron. Astrophys. 335, 1124 ~1998!.
@8# F. Huang, Master’s thesis, University of Louisville, 1994 ~unpublished!.
@9# S. Huang, Master’s thesis, University of Louisville, 1997 ~unpublished!.
@10# J. Xu, Master’s thesis, University of Louisville, 1997 ~unpublished!.
@11# Y. B. Zel’dovich and Y. P. Raiser, Physics of Shock Waves
and High-Temperature Hydrodynamic Phenomena ~Academic,
New York, 1967!, Vol. I.
@12# Y. B. Zel’dovich and Y. P. Raiser, Physics of Shock Waves
and High-Temperature Hydrodynamic Phenomena ~Academic,
New York, 1967!, Vol. II.
@13# A. Sakurai, J. Phys. Soc. Jpn. 8, 662 ~1953!.
@14# A. Sakurai, J. Phys. Soc. Jpn. 9, 256 ~1954!.
@15# J. Kielkopf ~unpublished!.
@16# J. Kielkopf ~unpublished!.
@17# P. Shah, R. L. Armstrong, and L. J. Radziemski, J. Appl. Phys.
65, 2946 ~1989!.
@18# H. Steiner and W. Gretler, Phys. Fluids 6, 2154 ~1994!.
4425
@19# T. Tsuchiya, Master’s thesis, University of Louisville, 1998
~unpublished!.
@20# A. J. Mackinnon, in Atomic Processes in Plasmas, Proceedings
of the 11th Topical Conference, edited by E. Oks and M. S.
Pindzola ~American Institute of Physics, New York, in press!.
@21# R. Fedosejevs, X. F. Wang, and G. D. Tsakiris, Phys. Rev. E
56, 4615 ~1997!.
@22# A. P. Thorne, Spectrophysics ~Chapman and Hall, London,
1988!, pp. 366–368.
@23# J. Kielkopf, K. Myneni, and F. Tomkins, J. Phys. B 23, 251
~1990!.
@24# N. Allard and J. Kielkopf, Astron. Astrophys. 242, 133 ~1991!.
@25# T. E. Sharp, At. Data 2, 119 ~1971!.
@26# L. Wolniewicz and K. Dressler, J. Chem. Phys. 88, 3861
~1988!.
@27# K. Dressler and L. Wolniewicz, J. Chem. Phys. 82, 4720
~1985!.
@28# A. Spiefiedel ~unpublished!.
@29# M. M. Madsen and J. M. Peek, At. Data 2, 171 ~1971!.
@30# D. E. Ramaker and J. M. Peek, J. Phys. B 5, 2175 ~1972!.
@31# D. E. Ramaker and J. M. Peek, At. Data 5, 167 ~1973!.
@32# N. F. Allard, A. Royer, J. F. Kielkopf, and N. Feautrier ~unpublished!.
@33# J. C. Stewart, J. M. Peek, and J. Cooper, Astrophys. J. 179,
983 ~1973!.
@34# K. Sando, R. O. Doyle, and A. Dalgarno, Astrophys. J. 157,
L143 ~1969!.
@35# D. Koester and G. Vauclair, in White Dwarfs, edited by J.
Isern, M. Hernanz, and E. Garcia-Berro ~Kluwer, Dordrecht,
1997!, p. 429.