1 PHYSICAL REVIEW A VOLUME 58, NUMBER 6 DECEMBER 1998 Observation of the far wing of Lyman due to neutral atom and ion collisions in a laser-produced plasma J. F. Kielkopf Department of Physics, University of Louisville, Louisville, Kentucky N. F. Allard Observatoire de Paris-Meudon, Département Atomes et Molécules en Astrophysique, Meudon Principal Cedex, France and Centre National de la Recherche Scientifique, Institut d Astrophysique, 98 bis Boulevard Arago, Paris, France Received 4 June 1998 The time-resolved spectrum of a laser-produced plasma in H 2 exhibits a continuum that we attribute to radiative collisions of H atoms in a 2p state with ground-state neutral atoms and protons. This binary collision Lyman- 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 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- 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 atoms/cm 3, 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 H 2 and H 2. S X PACS number s : Jz, Qt, Tc, 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 cm 1 to account for the line wings and the asymptotic energy must be correct to within approximately cm 1 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 observed 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, H 2 and H 2 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 H 2 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 H 2 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 H 2, 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 m in diameter surrounding the focal point. As a consequence, a shock wave propagates outward and leaves behind a cooling mixture of neutral H, H 2, ions, and electrons to provide a source for studying radiative collisions. Simple /98/58 6 / /$15.00 PRA The American Physical Society
2 PRA 58 OBSERVATION OF THE FAR WING OF LYMAN 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- region. Time-resolved emission spectroscopy combined with models of the postshock gas allows us to identify features in the Lyman- 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 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 % purity H 2 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 H 2 O has strong structured continuum absorption in the Lyman- 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 MgF 2 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 MgF 2 windows of the photomultiplier and the monochromator absorb light with 1150 Å. The photocathode of the detector is not sensitive to 1800 Å. Separate calibrations of the detector quantum efficiency and grating reflectivity are combined to provide a system efficiency. The overall spectral response of the monochromator and detector is also checked by comparing the spectrum of H 2 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 m, 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 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 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- 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, 5800 Å, Å, and t 6 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 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
3 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 mj of laser energy is delivered to H 2 at an initial molecule density of cm 3. The electron density inferred from the H- linewidth is excess of cm 3 and the electron temperature inferred from the Balmer series line-tocontinuum ratio is of the order of 10 5 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 K and the electron density is cm 3. We also find that the H- 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 s ofits 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 s 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 2R 0 c s J 0 1/2 t c s 2 1 t 2 1/2, 1 FIG. 2. Shadowgraph image of the shocked plasma 1 s 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 s. where R 0 is a scale distance, c s is the speed of sound in the ambient gas, and constant parameters J 0 and 1 are determined 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 s after the the plasma-production laser pulse. FIG. 3. Shadowgraph measurement of the shock front. For times less than 2 s 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.
4 PRA 58 OBSERVATION OF THE FAR WING OF LYMAN DUE FIG. 4. Radial variation of temperature and the densities of H 2, H, and H or e at 20 ns. Lyman 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 H 2 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 H 2 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 H 2 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 H 2 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 H 2 into two atoms. The distribution of H atoms is shown in the three-dimensional representation of Fig. 6. The emission responsible for Lyman arises FIG. 5. Radial variation of temperature and the densities of H 2, H, and H or e at 50 ns. As in Fig. 4, Lyman 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 K in the ionized region to less than 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 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 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.
5 4420 J. F. KIELKOPF AND N. F. ALLARD PRA 58 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 cm 3 and the maximum ion density is about cm 3. 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 m 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 s. IV. OBSERVED SPECTRA FIG. 7. Lyman- wing near 1230 Å at three different times immediately after the laser shot. Lyman is reversed in the cooler outer parts of the plasma. Two prominent features at 1230 Å and 1240 Å attributed to interactions with H 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. 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- 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. Prior to 100 ns there is no evidence of emission from H 2, but it does appear weakly later. As we have seen from the models of the blast wave, compressed H 2 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 X 1 g ground state to the B 1 u or C 1 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- 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 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, 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 scattered 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. 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 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 H 2 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
6 PRA 58 OBSERVATION OF THE FAR WING OF LYMAN DUE FIG. 8. Lyman- line profile at 22 ns. The spectrum has been banned so that the signal F 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 K. and above Lyman the free-free and bound-free electron-ion continuum emission from an ionized hydrogenic plasma is proportional to 22 F exp kt hc and decreases exponentially with decreasing wavelength. The function log 10 F appears to have a linearly increasing base line corresponding to a plasma temperature of about 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 that highlights its line shape. At the very center, however, where Lyman 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 H 2 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 H 2, 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 H 2 excited states in a glow discharge 23. We have a database of 2 FIG. 9. Spectral image of the Lyman- 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 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 H 2 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 s have been summed. The neutral shell density is expected to be about atoms/cm 3 during that time. V. COMPARISON WITH THEORETICAL LINE SHAPES Structures in the Lyman- 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 per unit frequency interval F ( ) is proportional to the probability of a configuration that per-
7 4422 J. F. KIELKOPF AND N. F. ALLARD 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 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 T exp ng s, 5 FIG. 10. The 1600-Å region of the Lyman- 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 s after the laser pulse. turbs the system to make the energy difference between the upper and lower states give V i (R) V f (R) /h. Thus the photon emission rate is where the Fourier transform T is taken such that F ( ) is normalized to unity when integrated over all frequencies and is measured relative to the unperturbed line. When we assume that the perturber follows a rectilinear classical path trajectory at velocity v, we get 1 g s e,e D ee 2 ee e,e 0 2 d dx dee R 0 exp i 0 s dt2 e e R t F d N4 R 2 dr 3 d e e R s d ee R 0. 6 or, in terms of potentials, F N4 R 2 d V/h dr 1, 4 where V/h and V is the difference between upperand lower-state potentials. This leads to a satellite where V goes through an extremum. Five difference potentials for the allowed transitions of H 2 have extrema and are expected to exhibit associated excited atom-proton binary collision satellites. This is not the case for H 2, 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 H 2 and H 2 7. There are many such states and a tabulation of them is given in Ref. 5. For the calculations shown here, the H 2 potentials were taken from tabulations given by Sharp 25 and Wolniewicz and Dressler 26. The radiative dipole transition moments for H 2 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 H 2 the potentials calculated by Madsen and Peek 29 were used with dipole transition moments given given by Ramaker and Peek 30,31. The e and e label the energy surfaces on which the interacting atoms approach the initial and final atomic states of the transition as R. For Lyman there are several different energy surfaces that lead to a same asymptotic energy as R. The sum with weight factors ee 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 e e R E e R E e R /h, which become the isolated radiator frequency if 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 2 x vt 2 1/2, where is the impact parameter of the perturber trajectory and x is the position of the perturber along its trajectory. The term d ee R(s) given in Eq. 6 is d ee R s D ee R s exp i2 s, where D ee R(s) is the R-dependent dipole transition moment. The total line strength of the transition is the sum of its components e,e D ee 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 7 8 9
8 PRA 58 OBSERVATION OF THE FAR WING OF LYMAN DUE FIG. 11. Theoretical profiles of Lyman for three different combinations of densities of neutral atomic H and the same H ion density, computed with the unified line shape theory as described in the text. The temperature for the calculation is K, but the profiles are not very temperature sensitive. F is normalized such that its integral over the profile is unity when is given in cm 1. The far wing satellite at 1600 Å increases with N H,asdothe 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-H 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. 13. Comparison of the 1230-Å region with a theoretical profile for N H ions/cm 3 and N H atoms/cm 3. 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- 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 H density is as a low as cm 3. The theory does not include broadening by electrons, which is important close to the FIG. 12. Theoretical profiles of Lyman for three different combinations of densities of the H 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 K, but the profiles are not very temperature sensitive. F is normalized such that its integral over the profile is unity when is given in cm 1. The far wing satellite at 1400 Å increases with N H. FIG. 14. Comparison of the far Lyman- 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 atoms cm 3 and N H ions cm 3.
9 4424 J. F. KIELKOPF AND N. F. ALLARD PRA 58 TABLE I. Satellites on Lyman Å due to H-H collisions. Upper state Lower state Å Theory Å Observed R Å h 3 g i 3 g C 1 u C 1 u B 1 u b 3 u b 3 u X 1 g X 1 g X 1 g 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 H 2 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 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- wing. This experimental confirmation of the theory supports TABLE II. Satellites on Lyman Å due to H-H collisions. Upper state Lower state Å Theory Å Observed R Å 3d g 2p u d g 2p u s g 2p u p u 1s g f u 1s g p u 1s g d g 2p u 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 is normalized to unity for an integral of the line profile over in cm 1 and scaled for a neutral H density of atoms/cm 3. 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- spectra of white dwarf and 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- 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- 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. 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.
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