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Calculation of the effects of velocity gradients and opacity on line transfer in laser-produced plasmas
PeQamoa
.I. QWII. Spectrosc. Radial. Transfer Vol. 52, No. 5, pp. 531-544, 1994 Elpcvier Science Ltd. Printed in Great Britain
0022-4073(94)ooo70-0
CALCULATION OF THE EFFECTS OF VELOCITY GRADIENTS AND OPACITY ON LINE TRANSFER IN LASER-PRODUCED PLASMAS A. DJAOUI,~$S. J. RosE,~§ and J. S. WARK~ TRutherford Appleton Laboratory, Chilton, Didcot OX1 1 OQX, $Department of Physics and Space Science, University of Birmingham, Birmingham B15 2TT, and YDepartment of Physics, Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, U.K. (Received 18 February 1994)
Abstract-A method that allows the simulation of K-shell resonance line emission in low density as well as high density laser-produced plasmas is described. The effect of resonance line trapping on ionic populations is taken into account using escape probabilities. A multifrequency line transfer algorithm that allows for plasma flow velocity gradients, and includes a choice of Doppler, Voigt, or a convolution of Doppler and Stark profiles, is described. The application of the code is illustrated by a set of calculations for the Ly-cc (AIXIII ls-2~7) line in an aluminium plasma created with a nanosecond, 0.53 pm laser beam and for the Ly-p (AlXIII ls-3~) and He-/l (AlXII Is*-1x3~) lines in a plasma created with a 12 psec, 0.268 pm laser beam. The effect of velocity gradients is found to be of critical importance in determining the line shapes and intensities.
1. INTRODUCTION
An understanding of radiative energy transfer in astrophysical and laser-produced plasmas is important for the prediction of plasma behaviour and for the interpretation of spectroscopic diagnostics. iv2Whilst the calculation of the emission of radiation from a static optically thin plasma is in general a complex task, inclusion of the effects of temperature, density, and velocity gradients as well as optical depth is a more challenging problem. In a previous study,3 the effects of temperature and density gradients on the diagnostic use of spectral features from a spherical plasma were shown to be important. Velocity gradients were not relevant to that study and were not included. In this paper we investigate plasmas produced by direct laser irradiation where the inclusion of velocity gradients is found to be critical in determining the intensity and shape of emitted spectral lines. An important aspect of laser-produced plasmas is the departure of the ionic excitation and ionisation from a steady state. When the characteristic hydrodynamic timescales are of the order of, or smaller than, the timescales associated with atomic processes, it is necessary to solve the time-dependent rate equations in conjunction with the hydrodynamics. A description of a plasma which involves a self-consistent solution of the equations of plasma hydrodynamics, atomic physics of excitation/ionization, and radiative transfer is generally more complex than is necessary for the modelling of laser-produced plasmas. Most laser-produced plasmas are of small size (a few hundred microns) and as a result are optically thin to many lines in the spectrum as well as continuum radiation (free-free and bound-free). The coupling between radiation and the ionic populations then reduces to the calculation of the transfer of a few optically thick lines. This problem was first solved for the case of a uniform plasma4 by considering the effect of photon trapping for a line with a Doppler, Voigt, or Lorentz profile. $To whom all correspondence
should be addressed. 531
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Previous investigations have used various models in an attempt to solve the coupled hydrodynamic, radiative transfer, and atomic physics problem. 5-7The emphasis in these studies was on plasma opacity and no account of macroscopic Doppler shifts2 was included. For an isolated line, the effect of a large velocity gradient is to inhibit reabsorption beyond a small distance from the point of emission of a photon. a Many previous workers (starting with Sobolev) used escape probabilities to calculate line transfer in the presence of large velocity gradients.‘-I4 More recently models for the calculation of x-ray line emission taking into account macroscopic Doppler shifts, as well as the coupling between radiation and ionic populations, have been developed.‘s-‘7 The emphasis in these studies is on detailed radiative transfer and the coupling with ionic populations. These models however still need to be extended to deal with Stark broadened lines. Recent experiments at the Rutherford Appleton Laboratory have made high resolution measurements of hydrogen-like resonance line emission from a nanosecond laser produced aluminium plasma as a function of angle’* with respect to the normal to the target, in a quasi-1D planar situation. Typically, in these plasmas, the velocity gradients in the normal direction to the target surface exceed lo9 set-’ and result in a bulk Doppler shift of about 12 mA, which is not only larger than the Doppler width (N 3 mA) but also the Al 1~,,~-2p,~~and Al 1~,,~-2p~,~separation (_ 5 mA). Both components of the doublet should therefore be considered and their interaction taken into account. Another series of experiments also recently conducted at the Rutherford Appleton Laboratory used time-resolved K-shell emission from a short pulse laser-produced aluminium plasma to infer average densities in excess of 10z3crne3 from Stark broadening of the lines.” These plasmas are characterized by larger line widths (-50 mA) and also larger velocity gradients, exceeding 10” set’. Understanding the effect of velocity, as well as temperature and density gradients, on line transfer is also critical for x-ray diagnostics based on Stark broadened lines, which have been the subject of many experimental investigations in the context of inertial confinement fusion studies20-22and the production of dense plasmas using short pulse lasers.23 In work reported in this paper, a radiation hydrodynamics model which incorporates time-dependent atomic physics, opacity, and velocity gradient effects is presented. A multifrequency radiative transfer algorithm which includes a choice of Doppler, Voigt, or a convolution of Doppler and Stark profiles’4 is used for the calculation of detailed hydrogen or helium-like resonance line profiles. This paper is organized as follows. The hydrodynamics and ionic population calculation is described in Sec. 2, while Sec. 3 covers the line transfer algorithm. Ly-cc (1.7-2,~) emission as a function of angle in the nanosecond experimentI is analysed in Sec. 4, and as an example of Stark broadened line transfer, the He-p (ls*-1.~3~) and Ly-/3 (1.~3~) emission from the picosecond experiment” is treated in Sec. 5. Section 6 contains a discussion of the results. 2. HYDRODYNAMICS
AND
IONIZATION
KINETICS
CALCULATION
The numerical model used to simulate the experiments is based on the 1D Lagrangian hydrodynamics code MEDUSA.25 The model follows the excitation and ionization in each Lagrangian cell by using the time-dependent non local thermodynamic equilibrium (non-LTE) average-atom model.26 Energies involved in atomic processes are included in the free electron energy balance equation which is iterated with the rate equations to ensure self consistency. The plasma is assumed to be optically thin to continuum radiation and to most line radiation. The effect of reabsorption in optically thick resonance lines on state populations is accounted for only approximately using escape probabilities as multiplicative factors to the spontaneous radiative rates in the average atom model. In the presence of large velocity gradients significant absorption occurs only over a distance for which the Doppler shift is less than the line width. This distance is referred to as the Doppler-decoupling distance.8 The escape probability for the case of a large velocity gradient is therefore larger than in the static case. This mechanism is an important contributor to the successful operation of recombination-pumped X.U.V.lasers *’ because reabsorption of resonance line photons pumps the lower laser level, reducing the population inversion. In the presence of large velocity gradients, escape probabilities are in good agreement with detailed line transfer calculations.‘R,29
Effect of velocity gradients and opacity on line transfer
533
For a static plasma, the probability that a photon escapes before being absorbed is determined by integrating over the emission line profile and over all directions.
P,=
&
s
dCI dv@(v)e-‘(KR) s
(1)
where n is the angle of propagation, v the frequency, @ the normalized line profile (assumed to be the same in emission and absorption), and t the optical thickness over the distance from the point of emission to the plasma boundary. Assuming a Doppler line profile this reduces to C=$
s
dxexp[-x2-r,,exp(-x2)] _mm
(2)
where r. is the optical thickness at line centre. Equation (2) can also be used for a plasma with velocity gradients, provided r. is calculated over the Doppler-decoupling distance’ which is usually much smaller than the plasma dimension. For a resonance transition from level 1 to n, it is given approximately by (3) where m, e are the electron mass and electric charge,f,, is the oscillator strength and vln frequency of the transition, N, is the number density of the ground state ions, and dV/dR the velocity gradient. 3.
LINE
TRANSFER
CALCULATION
Given the temperatures, densities, velocities, and ionic populations from the calculation described in the previous section, the emitted spectrum of a selected number of lines can be calculated. This line transfer calculation post-processes the output from the hydrodynamic/ionization simulation. The fact that the absorption profiles of ions that absorb are Doppler-shiftedzJ relative to the emission profile of the emitting ions because of the velocity difference between them, is taken into account. In this section we concentrate on the aluminium Ly-a doublet in order to illustrate the model. The hydrogenic ground state Is,,~, the 2~,,~, and the 2p,,, states will be denoted by g, a, and b respectively. In planar geometry, the intensity of photons emitted from Lagrangian cell m whose width is Axm, through an angle 8 with respect to the laser beam direction and having frequency v in the frame of reference of cell m, is Im(v) = 1
Sgm,k(v){ 1 - exp[-pmrc;_k(v)Axm/lcos
0 I]}.
k = a,b
The first term (k = a) represents photons emitted in the l~,,~--+2p,,, line (g --+ a) (line centre 1726.7 eV) and the second term (k = b) from the Is,,, + 2p,,, line (g + 6) (line centre 1728.0 eV). As can be seen from Eq. (4) within the cell in which the photons originate, the radiation transfer does not allow interaction between the two lines. This is a reasonable approximation within a Lagrangian cell in which all the material is moving with the same velocity. The opacity in cell m from the g --+ k transition (k = a or 6) is
PmKgm,kb) =
$ fg+kNy [ 1-$-+,,,v)
where fg_k is the oscillator strength (0.1387 for g -+ a and 0.2775 for g -+ 6). The number density of state g and k in Lagrangian cell m and their degeneracies are denoted by N ,“, N ‘/, gn, and gk respectively. @r-k(v) is the frequency normalized line profile for cell m. This is either a Doppler profile where the width is calculated using the ion temperature predicted by MEDUSA for Lagrangian cell m, a Voigt profile where radiative and collisional depopulation rates of the levels are calculated from the average atom rates, or a convolution of the Doppler profile with a Stark
A. DJAOUIet al
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profile” appropriate to the ion and electron densities and temperatures predicted by MEDUSA in cell m. The source function for the two components S,,, (k = a or 6) is
.
(6)
The number density of state a and b in Lagrangian cell m are calculated from the time-dependent shell populations in the average-atom model using the method described in Ref. 26. This method predicts the total density in the ground state (Is& and in the n = 2 states (2s,,*, 2~.+, and 2pvz). The population in 2~4,~(NY) and 2p3,2 (Nr) is then calculated by assuming n = 2 states are populated according to their degeneracies. Although stimulated emission is not important for the cases considered in the next two sections, because N;g,/N;g,<< 1, (k = a or b), it is included in Eqs. (5) and (6) for consistency. The photons from cell m are transported to the edge of the plasma through successive cells 1using the recurrence relation Z”‘(v) = I”‘(‘- ‘)(v)exp[ -p’rc’(v’“‘)Ax’/Jcos 0 I]
(7)
with
,&‘(vm’)
=
1 k=a.b
+&_kN'
g[
1-
(9)
$]@;-k@“‘,
mc
where Zm’(v) is the intensity of photons originating in cell m that have emerged from cell 1 and the opacity ~‘(v’“‘)includes the Doppler shift originating from the difference in velocity (V’ - V”) between the cell in which the photons were born (m) and that in which absorption is being considered (I). The intensity of photons which are born in cell m that emerges from the last cell must then be corrected for the Doppler shift between cell m and N and is observed I:!_, (vobserver) the observer ) = Z”N(v) I”observer (Vobserver
7.160
7.165
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7.175
7.160
7.165
7.190
7.195
7.200
Fig. 1. The measured Al Ly-a emission for various observation angles (in degrees) with respect to the laser beam.
Effect of velocity
535
gradientsand opacity on line transfer
where (11) In deriving Eqs. (8) and (11) a Galilean transformation which is valid for a non relativistic fluid flow, is used. The solution of the above equations proceeds as follows: for every frequency in the laboratory frame, Eq. (11) is used to find the corresponding frequency in the frame of reference of the emitting cell m, where the emission (Eqs. 4, 5, and 6) and transport to the observer (Eqs. 7, 8, 9, and 10) is performed. To compare with experimental measurements, the contribution from all cells m are added. The photon flux per unit area, per unit frequency, per unit solid angle is then obtained by division by the photon energy (hvobseWer). 4. Ly-a
EMISSION
FROM NANOSECOND
LASER-PRODUCED
PLASMA
As a first application of this model, we present calculations of Ly-cr emission from a nanosecond laser-produced aluminium plasma. ‘* In the experiment a single beam containing 80J of 0.53 pm
Distance [pm]
Distance [pm] Fig. 2. Plasma conditions as a function of distance, at the time of peak emission in the nanosecond laser-produced plasma. (a) Electron density, temperature, and ion temperature profiles. (b) Velocity profile, ground state, and first excited state hydrogen-like ion density profiles.
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light in a 1.2 nsec full width at half maximum (FWHM) pulse was incident normally onto a 10 pm thick, 500 pm diameter aluminium foil. The focal spot diameter on target was varied between about 100 and 500 pm giving incident intensity from 5 x lOI Wcm-* to 10” Wcm-*. Time integrated emission as a function of observation angle was recorded and is shown in Fig. 1 for which the absorbed intensity was about lOI Wcm- *. For 150 and 180” data (i.e., observing the emission through the target towards the laser beam) the target was 1 pm Al coated on 10 pm plastic (CH,). For these data, the reduction in intensity due to the x-rays passing through the cold, unablated portion of the target has been removed. In order to explain the experimental results, a series of hydrodynamic and line transfer calculations are presented in this section. An intensity of lOI Wcm-* is used for these calculations. Figure 2(a) shows the plasma conditions at the time of the peak of the laser pulse which corresponds to the peak of emission according to our calculation. In Fig. 2(b) the density of hydrogen-like ions in the n = 2 excited state shows that the source function peaks at a point near the critical surface where the electron density [Fig. 2(a)] is N 5 x lo*’ cm-3 and the temperature -600 eV. The density of n = 1 ground state shows that the absorbing plasma between the point of peak emission and the plasma boundary is characterized by large velocity (greater than lO’sec-‘), density and temperature gradients. There is also a significant difference between ion and electron temperature indicating that one temperature hydrodynamic models can significantly overestimate the ion temperature and hence the Doppler width. Figure 3 shows the peak optical depth of the 1~,,,-2p,~, component in the frame of the emission cell, as seen by an escaping photon emitted at different positions in the plasma. Curve A is obtained assuming a static plasma, while curve B includes the effect of velocity gradients. Doppler-decoupling results in a reduction of opacity by at least a factor of 4. Line interaction is non-negligible due to some overlap of the components, but also because high energy photons from the 1~-2~~~~ component from the region of peak emission appear red-shifted to the outer expanding plasma and are absorbed there by the 1~~,~-2p~,~ component. The absorption by the 1~,,~-2p’,~is included in curve B but not in curve C. The effect of using a Voigt or a convolution of Doppler and Stark profiles is negligible in this Ly-a case; most of the emission comes from plasma at densities sufficiently low that the line profile is, to a good approximation, Doppler. We next investigate the effect of trapping in the hydrodynamic/excitation/ionization calculation on line emission. The effect of including trapping in the calculation of ionic population on the emitted Ly-a line profile is shown in Fig. 4. The strong blue shift of the line (see Fig. 1) is correctly predicted whether trapping is included or not. The line intensity is, however, strongly dependent on the amount of trapping which transfers ground state populations to the n = 2 excited state 40
I
I
I
I
350
400
I
35 -
300
Distance [pm] Fig. 3. Optical thickness of a photon born at the peak of the 1~,,~-2p3,~Al Ly-cl component as a function of position in the plasma. Curve A is for a static plasma while curve B includes the effect of bulk Doppler shifts resulting from plasma flow. Curve C does not include the contribution from 1~,,~-2p,,~contribution while curve B does.
Effect of velocity gradients and opacity on line transfer 1400
537
1'1'1'1'1'1'1"
1200 t
P,
No trapping (x10) 800 600 -
I 7.150
I
I
I
I
I
I
,
I
I
It
I
It
7.155 7.160 7.165 7.170 7.175 7.180
I
I
I
7.185 7.190
Fig. 4. Al Ly-a front emission at 30” with respect to the laser beam as calculated from ion densities obtained with the use of escape probabilities from Eq. (2) and with no trapping (i.e., escape probability = I). The dotted curve has been multiplied by the factors shown in parentheses. The arrows on the wavelength axis indicate the position of the peaks of the doublet for a static plasma.
populations and results in a higher intensity. Thus absolute line intensity measurements can provide valuable information on the radiative transfer problem. The effect of opacity and velocity gradients can be studied using the radiative transfer algorithm. Curve A in Fig. 5 is the result of a calculation where the plasma is assumed static (all flow velocities set to zero) and optically thin [IC’(V~‘) in Eq. (7) set to zero]. The line shows two peaks corresponding to the 1~~,~-2p~,~ and 1~,,~-2p,,~components. Curve B assumes an optically thin plasma, but includes the effect of the velocity field. The effect of velocity gradients is to blue shift and smear out the whole spectrum, since the emitting plasma is moving towards the observer. Curve C assumes a static plasma, but includes opacity. In addition to a reduction in intensity, the line shape is completely
7.150 7.155 7.160 7.165 7.170 7.175 7.180 7.185 7.190
Fig. 5. AI Ly-a profile as calculated by the line transfer algorithm at 30” with respect to the laser beam. Curve A assumes a static and optically thin [K’(v’“‘)in Eq. (7) set to zero] plasma. Curve B assumes an optically thin plasma but includes velocity effects. Curve C assumes a static plasma but includes absorption. Curve D includes the effect of velocity fields as well as absorption. The last two curves have been multiplied by the factors shown in parentheses. The arrows on the wavelength axis indicate the position of the peaks of the doublet for a static plasma.
A. DJAOUIet al
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ii: 250 0 7.15
7.16
7.17
7.16
7.19
7.20
7.21
Fig. 6. Al Ly-cl emission at 30, 60, 150, and 180” with respect to the laser beam. The front emission is blue-shifted while the back emission is red-shifted with respect to the position of the peaks of the doublet in a static plasma, as indicated by the arrows on the wavelength axis. The absorption by the cold material is not included in the 150 and 180” data.
different and shows a reversal at the 1~,,~-2p~,~and 1~,,~-2p,,~at the peak positions in the opacity. Curve D includes both opacity and velocity effects. Its shape is intermediate between curve A and B and bears no resemblance to curve C while its intensity is more nearly that of curve C. This shows that the line shape is mainly determined by the velocity profile while the intensity is dominated by level populations which depend strongly on the enhanced probability of escape as a result of Doppler-decoupling. This conclusion is also in agreement with recombination x-ray laser studies, that show that without the Doppler-decoupling mechanism on the resonance lines, trapping would reduce the predicted gain, and is entirely inconsistent with that observed experimentally.27~29 The dependence of emission on angle with respect to target normal is shown in Fig. 6. The relative intensities and shift at 30 and 60” agree well with the experimental measurements (see 1400 1200 1000 600 600 400 200 0 7.165 7.170 7.175 7.180 7.165 7.190 7.195 7.200 7.205 7.210
Fig. 7. Al Ly-a profile as calculated by the line transfer algorithm at 150” with respect to the laser beam. Curve A includes the effect of velocity fields as well as absorption. Curve B assumes an optically thin [~‘(v’“‘)in Eq. (7) set to zero] plasma but includes velocity effects. Curve C assumes a static plasma but includes absorption. Curve B was divided by the factor shown in parentheses. The arrows on the wavelength axis indicate the position of the peaks of the doublet for a static plasma.
Effect of velocity gradients and opacity on line transfer
539
1250
f
1000
% A
750
:I50
7.155
7.160
7.165
7.170
7.175
7.180
7.185
7.190
Fig. 8. Al Ly-u front emission at 30 and 60” with respect to the normal to the laser beam. Full curves are calculated using the algorithm of Eqs. (7). (S), and (9), which calculate the absorption in cell I by using the material densities and temperatures for cell 1. For the dotted curves the absorption in cell 1is calculated by using the material densities and temperatures from cell m where the photons originate. The arrows on the wavelength axis indicate the position of the peaks of the doublet for a static plasma.
Fig. 1). The back emission at 150 and 180” does not include the absorption by the cold material. The calculated line shape for the back emission can be better understood in Fig. 7. If the plasma is assumed static and absorption is included (curve C), a similar reversed profile to the one obtained in Fig. 5 is obtained. The effect of the velocity profile without absorption (curve B) results in a red shift and a smoothing of the shape. Including absorption as well as the velocity field results in the profile shown by curve A which shows a peak on the red wing of line. Such a peak results from the fact that photons emitted in the expanding plasma are red-shifted away from the peak of absorption in the relatively stationary absorbing plasma near the critical surface. A central assumption in the use of escape factors in a plasma with a large velocity gradients is that the plasma properties do not change appreciably over the Doppler-decoupling distance. ,,,,,,,1,1,1,1,1,1,
A-n 3000 T H % ,200o 0 Ll 2 B 1000 0 iE 0 7.140 7.145 7.150 7.155 7.160 7.165 7.170 7.175 7.180 7.185 7.190 h [AI Fig. 9. Al Ly-a front emission at 30” with respect to the normal to the laser beam. Curve D corresponds to an intensity of 2 x IO”, C to 10I4, B to 2 x 1014,and A to 5 x lOI Wcmm2. B, C, and D have been multiplied by the factors shown in parentheses, The arrows on the wavelength axis indicate the position of the peaks of the doublet for a static plasma.
A. DJAOUIet al
540
7.165
Fig. 10. Al Ly-cc emission
7.170
7.175
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7.195
for various observation angles (in degrees) with respect calculated for an absorbed intensity of lOI W/cm*.
7.200
to the laser beam as
We investigate this approximation by recalculating the emission using not the algorithm of Eqs. (7), (8), and (9), which calculates the absorption in cell I by using the material densities and temperatures from cell 1, but by using the material densities and temperatures from cell m where the photons originate. Figure 8 shows that this only slightly alters the predicted profiles proving that the assumption inherent in the escape probability is valid. This is also in agreement with the previous work on the use of escape probabilities” which showed good agreement between escape probabilities and detailed line transfer calculations in laser produced plasmas. We now look at the effect of varying the incident laser intensity on emission. It is seen in Fig. 9 that higher laser intensities result in an increase in line intensity as a result of the larger volume of plasma created. There is also a small increase in the blue component as a result of the higher velocity gradients. In order to compare with the experimental results of Fig. 1, we show in Fig. 10 a calculation which uses an absorbed intensity of lOI W/cm*. There is good agreement in line shape as well as line intensity. This shows that line transfer in laser-produced plasmas having large velocity gradients can be calculated using the simple approach described in this paper, where trapping of radiation is taken into account by use of escape probabilities and velocity gradients accounted for by taking into account the Doppler shift between the emission and absorption points in the plasma. 5. LINE
EMISSION
FROM
A
PICOSECOND
LASER-PRODUCED
PLASMA
For high laser intensities, when the pulse length is of the order or smaller than the hydrodynamic time scale of expansion, high density, high temperature plasmas are produced. In a series of experiments,” a 35 prepulse free, high brightness, Raman amplified KrF laser (0.268 pm wavelength) was used to irradiate solid aluminium targets with a 12 psec (FWHM) pulse at intensities of 6 x 10” Wcm-*. Time resolved K-shell emission was recorded and showed Stark widths of the order 55 mA for the He-/I and 40 mA for the Ly-jJ line widths at early times, which narrowed as the plasma expanded in a time of about 40 psec to a value (N 20 mA) dominated by instrumental broadening. Average electron densities of greater than lO23cme3 were inferred from the Stark widths of the He-/I and Ly-8 lines at the early times. In this section, we present simulations of the time-dependent emission from such a plasma. Although trapping for Stark profiles has already been considered for resonance lines of hydrogenic iond0v3 and was found to be more important than in the case of Voigt profiles, this is not implemented in our model at this moment. The ionic populations are therefore calculated without any trapping. In view of the analysis in the previous
Effect of velocity
gradientsand opacity on line transfer
541
10199 25
30
35
40°
35
40
Distance [pm] (b)
25
30
Distance [pm] Fig. 11. Plasma conditions as a function of distance, at the time of peak emission in the picosecond laser-produced plasma. (a) Electron density, temperature, and ion temperature profiles. (b) Velocity profile, ground state, and second excited state helium-like ion density profiles.
section the effect of trapping is not expected to affect the line profiles or widths which are used for density determination. Average temperatures are however deduced from the ratio of the He-P to the Ly-8, and although trapping has a direct effect on the intensities, the ratio of the intensities should be less sensitive to it. The quantitative extent to which the above diagnostics are affected by trapping will be investigated in separate work. Figure 1l(a) shows the simulated electron and ion temperatures, and electron density at the time of peak emission (10 psec after peak of laser beam). Figure 1l(b) shows the velocity profile as well as the helium-like ground state and n = 3 excited state ion densities. The He-/? emission zone is characterized by higher densities than in the nanosecond situation, and larger velocity gradients (greater than lO”sec-‘), although the temperatures are similar. The time-dependent He-p and Ly-B emission at a fixed angle of 45” with respect to the laser beam, as calculated by the line transfer algorithm is shown in Figs. 12 and 13 respectively. In both cases, the emission peaks at around 10 psec after the peak of the laser. Strong Stark broadening near the peak of emission results in line widths of around 55 rnA for the He-/? and 35 rnA for the Ly-p line, in fairly good agreement with the experimental values. I9The line widths decrease rapidly as a result
A. DJAOUI et al
542
Al Hep
2.5~10"
I 6.56
6.56
6.60
6.62
6.64
6.66
6.68
6.70
6.72
Fig. 12. Time resolved Al He-p front emission at 45” with respect to the laser beam at different respect to the peak of the laser.
times with
of plasma expansion and lower electron densities to less than 20 mA, 30 psec after the peak of the laser beam. These calculations do not include any instrumental broadening (about 20 mA) which dominates the line profile at later times as the plasma expands. The effect of the velocity gradients on line emission is shown in Fig. 14. Only the Ly-p time integrated spectrum is shown. Similar results were obtained for the time-resolved profile, as well as for the He-j? line. The calculation for a static plasma shows a symmetric line profile. This is strongly blue-shifted when the velocity field is included in the calculation. Although the Stark line widths are much larger than the line widths in the nanosecond experiment, the velocity gradients and the corresponding bulk Doppler shifts are also much higher. The blue shift in the position of the peak is about 10 mA compared to the 4 mA in the nanosecond case (see curves A and B in Fig. 5).
6.00
Fig. 13. Time resolved
6.02
6.04
6.06
6.08
Al Ly-/3 front emission at 45” with respect to the laser beam, with respect to the peak of the laser.
6.10
at different
times
Effect of velocity gradients and opacity on line transfer
6.00
I
I
I
I
6.02
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6.66
6.06
543
6.10
~[AI Fig. 14. Time integrated (up to 140psec) Al Ly-B front emission at 45” with respect to the laser beam as calculated for a static plasma (dotted curve) and by taking into account the velocity field (full curve).
6. CONCLUSION
We have developed a two stage 1D planar model for calculating line transfer in laser-produced plasmas where temperature, density, and velocity gradients, as well as the effect of optically thick line reabsorption on level populations, are taken into account. In the first stage, the hydrodynamics and the ionic populations are solved in a self-consistent manner. The atomic physics of excitation and ionization is calculated using a time-dependent non-LTE average atom model, where resonance photon trapping in hydrogen and helium-like ions is treated approximately using escape probabilities. A detailed line transfer algorithm, which takes the output from the hydrodynamic/ionization calculation, is used in the second stage to calculate line profiles for hydrogen and helium-like resonance lines in aluminium. A choice of a Doppler, Voigt, or a convolution of Doppler and Stark profiles is available. In its present form this is already a powerful tool for interpreting and understanding radiative transfer in laser-produced plasma. Simulation of time integrated Ly-a emission as a function of angle in a long pulse (nanosecond) experiment show that velocity gradients are critical in determining the line shape. In the case of a short pulse (picosecond) experiment, the Stark broadened line widths and their narrowing with time as the plasma expands are well reproduced. It is also seen that even for the relatively broad emission from high densities, the velocity gradients are important in determining line shapes. Thus spectroscopic methods which are based on fitting line shapes from a static, constant density and temperature model can be in error if plasma flow velocities are not taken into account. By extending this model to include trapping for Stark broadened lines it will be possible to address the effect of trapping on level populations and on line ratios, as used for the determination of average temperatures, in high density a well as low density plasmas. REFERENCES 1. D. Mihalas, Stellar Atmospheres, Freeman, San Francisco, CA (1978). 2. D. Mihalas and B. W. Mihalas, Foundation of Radiation Hydrodynamics, OUP, Oxford (1984). 3. R. W. Lee, JQSRT 27, 87 (1982). 4. E. H. Avrett and D. G. Hummer, Mon. Not. R. ast. Sot. 130, 295 (1965). 5. D. G. Colombant, K. G. Whitney, D. A. Tidman, N. K. Winsor, and J. Davis, Physics Fluids 18, 1687 (1975). 6. D. Duston, R. W. Clark, J. Davis, and J. P. Apruzese, Phys. Rev. A 27, 1441 (1983). 7. D. Duston, R. W. Clark, and J. Davis, Phys. Rev. A 31, 3220 (1985). 8. G. J. Pert, J. Phys. B 9, 3301 (1976). 9. V. V. Sobolev, Moving Envelopes of Stars, Harvard University Press, Cambridge, MA (1960).
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10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31.
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J. I. Castor, Mon. Not. R. astr. Sot. 149, 111 (1970). L. B. Lucy, Aswophys. J. 163, 95 (1971). G. B. Rybicki and D. G. Hummer, Astrophys. J. 219, 654 (1978). D. G. Hummer and D. G. Rybicki, Astrophys. J. 254, 767 (1982). G. B. Rybicki and D. G. Hummer, Astrophys. J. 274, 380 (1983). D. Mostacci, L. M. Montierty, J. P. Dinguirard, and R. L. Morse, Physics Fluids B 1, 2106 (1989). D. C. Eder and H. A. Scott, JQSRT 45, 189 (1991). 0. Peyrusse, Physics Fluids B 4, 2007 (1992). J. S. Wark, A. Djaoui, S. J. Rose, H. He, 0. Renner, T. Missalla, and E. Foerster, Whys. Rev. Lett. 72, 1826 (1994). D. Riley, L. A. Gizzi, F. Y. Khattak, A. J. Mackinon, S. M. Viana, and 0. Willi, Phys. Rev. Lett. 69, 3739 (1992). J. D. Kilkenny, R. W. Lee, M. H. Key, and J. G. Lunney, Phys. Rev. A 22, 2746 (1980). C. F. Hooper Jr, D. P. Kilcrease, R. C. Mancini, L. A. Woltz, D. K. Bradley, P. A. Jaanimagi, and M. C. Richardson, Phys. Rev. Lett. 63, 267 (1989). B. A. Hammel, C. J. Keane, M. D. Cable, D. R. Kania, J. D. Kilkenny, R. W. Lee, and R. Pasha, Phys. Rev. Lett. 70, 1263 (1993). C. Y. Chien, J. S. Coe, G. Mourou, J. C. Kieffer, M. Chaker, Y. Beaudouin, 0. Peyrusse, and D. Gilles, Opt. Left. 18, 1535 (1993). R. W. Lee, JQSRT 40, 561 (1988). J. P. Christiansen, D. E. T. F. Ashby, and K. V. Roberts, Comput. Phys. Commun. 7, 271 (1974). A. Djaoui and S. J. Rose, J. Phys. B 25, 2745 (1992). G. J. Pert and S. J. Rose, Appl. Phys. B 50, 307 (1990). Y. T. Lee, R. A. London, and G. B. Zimmermann, Phys. Fluids B 2, 2731 (1990). D. C. Eder, H. A. Scott, S. Maxon, and R. A. London, Appl. Opt. 31, 4962 (1992). J. C. Weisheit, JQSRT 22, 585 (1979). R. C. Mancini, R. F. Joyce, and C. F. Hooper Jr, J. Phys. B 20, 2975 (1987).
.I. QWII. Spectrosc. Radial. Transfer Vol. 52, No. 5, pp. 531-544, 1994 Elpcvier Science Ltd. Printed in Great Britain
0022-4073(94)ooo70-0
CALCULATION OF THE EFFECTS OF VELOCITY GRADIENTS AND OPACITY ON LINE TRANSFER IN LASER-PRODUCED PLASMAS A. DJAOUI,~$S. J. RosE,~§ and J. S. WARK~ TRutherford Appleton Laboratory, Chilton, Didcot OX1 1 OQX, $Department of Physics and Space Science, University of Birmingham, Birmingham B15 2TT, and YDepartment of Physics, Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, U.K. (Received 18 February 1994)
Abstract-A method that allows the simulation of K-shell resonance line emission in low density as well as high density laser-produced plasmas is described. The effect of resonance line trapping on ionic populations is taken into account using escape probabilities. A multifrequency line transfer algorithm that allows for plasma flow velocity gradients, and includes a choice of Doppler, Voigt, or a convolution of Doppler and Stark profiles, is described. The application of the code is illustrated by a set of calculations for the Ly-cc (AIXIII ls-2~7) line in an aluminium plasma created with a nanosecond, 0.53 pm laser beam and for the Ly-p (AlXIII ls-3~) and He-/l (AlXII Is*-1x3~) lines in a plasma created with a 12 psec, 0.268 pm laser beam. The effect of velocity gradients is found to be of critical importance in determining the line shapes and intensities.
1. INTRODUCTION
An understanding of radiative energy transfer in astrophysical and laser-produced plasmas is important for the prediction of plasma behaviour and for the interpretation of spectroscopic diagnostics. iv2Whilst the calculation of the emission of radiation from a static optically thin plasma is in general a complex task, inclusion of the effects of temperature, density, and velocity gradients as well as optical depth is a more challenging problem. In a previous study,3 the effects of temperature and density gradients on the diagnostic use of spectral features from a spherical plasma were shown to be important. Velocity gradients were not relevant to that study and were not included. In this paper we investigate plasmas produced by direct laser irradiation where the inclusion of velocity gradients is found to be critical in determining the intensity and shape of emitted spectral lines. An important aspect of laser-produced plasmas is the departure of the ionic excitation and ionisation from a steady state. When the characteristic hydrodynamic timescales are of the order of, or smaller than, the timescales associated with atomic processes, it is necessary to solve the time-dependent rate equations in conjunction with the hydrodynamics. A description of a plasma which involves a self-consistent solution of the equations of plasma hydrodynamics, atomic physics of excitation/ionization, and radiative transfer is generally more complex than is necessary for the modelling of laser-produced plasmas. Most laser-produced plasmas are of small size (a few hundred microns) and as a result are optically thin to many lines in the spectrum as well as continuum radiation (free-free and bound-free). The coupling between radiation and the ionic populations then reduces to the calculation of the transfer of a few optically thick lines. This problem was first solved for the case of a uniform plasma4 by considering the effect of photon trapping for a line with a Doppler, Voigt, or Lorentz profile. $To whom all correspondence
should be addressed. 531
A. DIAOUIet al
532
Previous investigations have used various models in an attempt to solve the coupled hydrodynamic, radiative transfer, and atomic physics problem. 5-7The emphasis in these studies was on plasma opacity and no account of macroscopic Doppler shifts2 was included. For an isolated line, the effect of a large velocity gradient is to inhibit reabsorption beyond a small distance from the point of emission of a photon. a Many previous workers (starting with Sobolev) used escape probabilities to calculate line transfer in the presence of large velocity gradients.‘-I4 More recently models for the calculation of x-ray line emission taking into account macroscopic Doppler shifts, as well as the coupling between radiation and ionic populations, have been developed.‘s-‘7 The emphasis in these studies is on detailed radiative transfer and the coupling with ionic populations. These models however still need to be extended to deal with Stark broadened lines. Recent experiments at the Rutherford Appleton Laboratory have made high resolution measurements of hydrogen-like resonance line emission from a nanosecond laser produced aluminium plasma as a function of angle’* with respect to the normal to the target, in a quasi-1D planar situation. Typically, in these plasmas, the velocity gradients in the normal direction to the target surface exceed lo9 set-’ and result in a bulk Doppler shift of about 12 mA, which is not only larger than the Doppler width (N 3 mA) but also the Al 1~,,~-2p,~~and Al 1~,,~-2p~,~separation (_ 5 mA). Both components of the doublet should therefore be considered and their interaction taken into account. Another series of experiments also recently conducted at the Rutherford Appleton Laboratory used time-resolved K-shell emission from a short pulse laser-produced aluminium plasma to infer average densities in excess of 10z3crne3 from Stark broadening of the lines.” These plasmas are characterized by larger line widths (-50 mA) and also larger velocity gradients, exceeding 10” set’. Understanding the effect of velocity, as well as temperature and density gradients, on line transfer is also critical for x-ray diagnostics based on Stark broadened lines, which have been the subject of many experimental investigations in the context of inertial confinement fusion studies20-22and the production of dense plasmas using short pulse lasers.23 In work reported in this paper, a radiation hydrodynamics model which incorporates time-dependent atomic physics, opacity, and velocity gradient effects is presented. A multifrequency radiative transfer algorithm which includes a choice of Doppler, Voigt, or a convolution of Doppler and Stark profiles’4 is used for the calculation of detailed hydrogen or helium-like resonance line profiles. This paper is organized as follows. The hydrodynamics and ionic population calculation is described in Sec. 2, while Sec. 3 covers the line transfer algorithm. Ly-cc (1.7-2,~) emission as a function of angle in the nanosecond experimentI is analysed in Sec. 4, and as an example of Stark broadened line transfer, the He-p (ls*-1.~3~) and Ly-/3 (1.~3~) emission from the picosecond experiment” is treated in Sec. 5. Section 6 contains a discussion of the results. 2. HYDRODYNAMICS
AND
IONIZATION
KINETICS
CALCULATION
The numerical model used to simulate the experiments is based on the 1D Lagrangian hydrodynamics code MEDUSA.25 The model follows the excitation and ionization in each Lagrangian cell by using the time-dependent non local thermodynamic equilibrium (non-LTE) average-atom model.26 Energies involved in atomic processes are included in the free electron energy balance equation which is iterated with the rate equations to ensure self consistency. The plasma is assumed to be optically thin to continuum radiation and to most line radiation. The effect of reabsorption in optically thick resonance lines on state populations is accounted for only approximately using escape probabilities as multiplicative factors to the spontaneous radiative rates in the average atom model. In the presence of large velocity gradients significant absorption occurs only over a distance for which the Doppler shift is less than the line width. This distance is referred to as the Doppler-decoupling distance.8 The escape probability for the case of a large velocity gradient is therefore larger than in the static case. This mechanism is an important contributor to the successful operation of recombination-pumped X.U.V.lasers *’ because reabsorption of resonance line photons pumps the lower laser level, reducing the population inversion. In the presence of large velocity gradients, escape probabilities are in good agreement with detailed line transfer calculations.‘R,29
Effect of velocity gradients and opacity on line transfer
533
For a static plasma, the probability that a photon escapes before being absorbed is determined by integrating over the emission line profile and over all directions.
P,=
&
s
dCI dv@(v)e-‘(KR) s
(1)
where n is the angle of propagation, v the frequency, @ the normalized line profile (assumed to be the same in emission and absorption), and t the optical thickness over the distance from the point of emission to the plasma boundary. Assuming a Doppler line profile this reduces to C=$
s
dxexp[-x2-r,,exp(-x2)] _mm
(2)
where r. is the optical thickness at line centre. Equation (2) can also be used for a plasma with velocity gradients, provided r. is calculated over the Doppler-decoupling distance’ which is usually much smaller than the plasma dimension. For a resonance transition from level 1 to n, it is given approximately by (3) where m, e are the electron mass and electric charge,f,, is the oscillator strength and vln frequency of the transition, N, is the number density of the ground state ions, and dV/dR the velocity gradient. 3.
LINE
TRANSFER
CALCULATION
Given the temperatures, densities, velocities, and ionic populations from the calculation described in the previous section, the emitted spectrum of a selected number of lines can be calculated. This line transfer calculation post-processes the output from the hydrodynamic/ionization simulation. The fact that the absorption profiles of ions that absorb are Doppler-shiftedzJ relative to the emission profile of the emitting ions because of the velocity difference between them, is taken into account. In this section we concentrate on the aluminium Ly-a doublet in order to illustrate the model. The hydrogenic ground state Is,,~, the 2~,,~, and the 2p,,, states will be denoted by g, a, and b respectively. In planar geometry, the intensity of photons emitted from Lagrangian cell m whose width is Axm, through an angle 8 with respect to the laser beam direction and having frequency v in the frame of reference of cell m, is Im(v) = 1
Sgm,k(v){ 1 - exp[-pmrc;_k(v)Axm/lcos
0 I]}.
k = a,b
The first term (k = a) represents photons emitted in the l~,,~--+2p,,, line (g --+ a) (line centre 1726.7 eV) and the second term (k = b) from the Is,,, + 2p,,, line (g + 6) (line centre 1728.0 eV). As can be seen from Eq. (4) within the cell in which the photons originate, the radiation transfer does not allow interaction between the two lines. This is a reasonable approximation within a Lagrangian cell in which all the material is moving with the same velocity. The opacity in cell m from the g --+ k transition (k = a or 6) is
PmKgm,kb) =
$ fg+kNy [ 1-$-+,,,v)
where fg_k is the oscillator strength (0.1387 for g -+ a and 0.2775 for g -+ 6). The number density of state g and k in Lagrangian cell m and their degeneracies are denoted by N ,“, N ‘/, gn, and gk respectively. @r-k(v) is the frequency normalized line profile for cell m. This is either a Doppler profile where the width is calculated using the ion temperature predicted by MEDUSA for Lagrangian cell m, a Voigt profile where radiative and collisional depopulation rates of the levels are calculated from the average atom rates, or a convolution of the Doppler profile with a Stark
A. DJAOUIet al
534
profile” appropriate to the ion and electron densities and temperatures predicted by MEDUSA in cell m. The source function for the two components S,,, (k = a or 6) is
.
(6)
The number density of state a and b in Lagrangian cell m are calculated from the time-dependent shell populations in the average-atom model using the method described in Ref. 26. This method predicts the total density in the ground state (Is& and in the n = 2 states (2s,,*, 2~.+, and 2pvz). The population in 2~4,~(NY) and 2p3,2 (Nr) is then calculated by assuming n = 2 states are populated according to their degeneracies. Although stimulated emission is not important for the cases considered in the next two sections, because N;g,/N;g,<< 1, (k = a or b), it is included in Eqs. (5) and (6) for consistency. The photons from cell m are transported to the edge of the plasma through successive cells 1using the recurrence relation Z”‘(v) = I”‘(‘- ‘)(v)exp[ -p’rc’(v’“‘)Ax’/Jcos 0 I]
(7)
with
,&‘(vm’)
=
1 k=a.b
+&_kN'
g[
1-
(9)
$]@;-k@“‘,
mc
where Zm’(v) is the intensity of photons originating in cell m that have emerged from cell 1 and the opacity ~‘(v’“‘)includes the Doppler shift originating from the difference in velocity (V’ - V”) between the cell in which the photons were born (m) and that in which absorption is being considered (I). The intensity of photons which are born in cell m that emerges from the last cell must then be corrected for the Doppler shift between cell m and N and is observed I:!_, (vobserver) the observer ) = Z”N(v) I”observer (Vobserver
7.160
7.165
7.170
7.175
7.160
7.165
7.190
7.195
7.200
Fig. 1. The measured Al Ly-a emission for various observation angles (in degrees) with respect to the laser beam.
Effect of velocity
535
gradientsand opacity on line transfer
where (11) In deriving Eqs. (8) and (11) a Galilean transformation which is valid for a non relativistic fluid flow, is used. The solution of the above equations proceeds as follows: for every frequency in the laboratory frame, Eq. (11) is used to find the corresponding frequency in the frame of reference of the emitting cell m, where the emission (Eqs. 4, 5, and 6) and transport to the observer (Eqs. 7, 8, 9, and 10) is performed. To compare with experimental measurements, the contribution from all cells m are added. The photon flux per unit area, per unit frequency, per unit solid angle is then obtained by division by the photon energy (hvobseWer). 4. Ly-a
EMISSION
FROM NANOSECOND
LASER-PRODUCED
PLASMA
As a first application of this model, we present calculations of Ly-cr emission from a nanosecond laser-produced aluminium plasma. ‘* In the experiment a single beam containing 80J of 0.53 pm
Distance [pm]
Distance [pm] Fig. 2. Plasma conditions as a function of distance, at the time of peak emission in the nanosecond laser-produced plasma. (a) Electron density, temperature, and ion temperature profiles. (b) Velocity profile, ground state, and first excited state hydrogen-like ion density profiles.
A. DJAOUIet al
536
light in a 1.2 nsec full width at half maximum (FWHM) pulse was incident normally onto a 10 pm thick, 500 pm diameter aluminium foil. The focal spot diameter on target was varied between about 100 and 500 pm giving incident intensity from 5 x lOI Wcm-* to 10” Wcm-*. Time integrated emission as a function of observation angle was recorded and is shown in Fig. 1 for which the absorbed intensity was about lOI Wcm- *. For 150 and 180” data (i.e., observing the emission through the target towards the laser beam) the target was 1 pm Al coated on 10 pm plastic (CH,). For these data, the reduction in intensity due to the x-rays passing through the cold, unablated portion of the target has been removed. In order to explain the experimental results, a series of hydrodynamic and line transfer calculations are presented in this section. An intensity of lOI Wcm-* is used for these calculations. Figure 2(a) shows the plasma conditions at the time of the peak of the laser pulse which corresponds to the peak of emission according to our calculation. In Fig. 2(b) the density of hydrogen-like ions in the n = 2 excited state shows that the source function peaks at a point near the critical surface where the electron density [Fig. 2(a)] is N 5 x lo*’ cm-3 and the temperature -600 eV. The density of n = 1 ground state shows that the absorbing plasma between the point of peak emission and the plasma boundary is characterized by large velocity (greater than lO’sec-‘), density and temperature gradients. There is also a significant difference between ion and electron temperature indicating that one temperature hydrodynamic models can significantly overestimate the ion temperature and hence the Doppler width. Figure 3 shows the peak optical depth of the 1~,,,-2p,~, component in the frame of the emission cell, as seen by an escaping photon emitted at different positions in the plasma. Curve A is obtained assuming a static plasma, while curve B includes the effect of velocity gradients. Doppler-decoupling results in a reduction of opacity by at least a factor of 4. Line interaction is non-negligible due to some overlap of the components, but also because high energy photons from the 1~-2~~~~ component from the region of peak emission appear red-shifted to the outer expanding plasma and are absorbed there by the 1~~,~-2p~,~ component. The absorption by the 1~,,~-2p’,~is included in curve B but not in curve C. The effect of using a Voigt or a convolution of Doppler and Stark profiles is negligible in this Ly-a case; most of the emission comes from plasma at densities sufficiently low that the line profile is, to a good approximation, Doppler. We next investigate the effect of trapping in the hydrodynamic/excitation/ionization calculation on line emission. The effect of including trapping in the calculation of ionic population on the emitted Ly-a line profile is shown in Fig. 4. The strong blue shift of the line (see Fig. 1) is correctly predicted whether trapping is included or not. The line intensity is, however, strongly dependent on the amount of trapping which transfers ground state populations to the n = 2 excited state 40
I
I
I
I
350
400
I
35 -
300
Distance [pm] Fig. 3. Optical thickness of a photon born at the peak of the 1~,,~-2p3,~Al Ly-cl component as a function of position in the plasma. Curve A is for a static plasma while curve B includes the effect of bulk Doppler shifts resulting from plasma flow. Curve C does not include the contribution from 1~,,~-2p,,~contribution while curve B does.
Effect of velocity gradients and opacity on line transfer 1400
537
1'1'1'1'1'1'1"
1200 t
P,
No trapping (x10) 800 600 -
I 7.150
I
I
I
I
I
I
,
I
I
It
I
It
7.155 7.160 7.165 7.170 7.175 7.180
I
I
I
7.185 7.190
Fig. 4. Al Ly-a front emission at 30” with respect to the laser beam as calculated from ion densities obtained with the use of escape probabilities from Eq. (2) and with no trapping (i.e., escape probability = I). The dotted curve has been multiplied by the factors shown in parentheses. The arrows on the wavelength axis indicate the position of the peaks of the doublet for a static plasma.
populations and results in a higher intensity. Thus absolute line intensity measurements can provide valuable information on the radiative transfer problem. The effect of opacity and velocity gradients can be studied using the radiative transfer algorithm. Curve A in Fig. 5 is the result of a calculation where the plasma is assumed static (all flow velocities set to zero) and optically thin [IC’(V~‘) in Eq. (7) set to zero]. The line shows two peaks corresponding to the 1~~,~-2p~,~ and 1~,,~-2p,,~components. Curve B assumes an optically thin plasma, but includes the effect of the velocity field. The effect of velocity gradients is to blue shift and smear out the whole spectrum, since the emitting plasma is moving towards the observer. Curve C assumes a static plasma, but includes opacity. In addition to a reduction in intensity, the line shape is completely
7.150 7.155 7.160 7.165 7.170 7.175 7.180 7.185 7.190
Fig. 5. AI Ly-a profile as calculated by the line transfer algorithm at 30” with respect to the laser beam. Curve A assumes a static and optically thin [K’(v’“‘)in Eq. (7) set to zero] plasma. Curve B assumes an optically thin plasma but includes velocity effects. Curve C assumes a static plasma but includes absorption. Curve D includes the effect of velocity fields as well as absorption. The last two curves have been multiplied by the factors shown in parentheses. The arrows on the wavelength axis indicate the position of the peaks of the doublet for a static plasma.
A. DJAOUIet al
538
ii: 250 0 7.15
7.16
7.17
7.16
7.19
7.20
7.21
Fig. 6. Al Ly-cl emission at 30, 60, 150, and 180” with respect to the laser beam. The front emission is blue-shifted while the back emission is red-shifted with respect to the position of the peaks of the doublet in a static plasma, as indicated by the arrows on the wavelength axis. The absorption by the cold material is not included in the 150 and 180” data.
different and shows a reversal at the 1~,,~-2p~,~and 1~,,~-2p,,~at the peak positions in the opacity. Curve D includes both opacity and velocity effects. Its shape is intermediate between curve A and B and bears no resemblance to curve C while its intensity is more nearly that of curve C. This shows that the line shape is mainly determined by the velocity profile while the intensity is dominated by level populations which depend strongly on the enhanced probability of escape as a result of Doppler-decoupling. This conclusion is also in agreement with recombination x-ray laser studies, that show that without the Doppler-decoupling mechanism on the resonance lines, trapping would reduce the predicted gain, and is entirely inconsistent with that observed experimentally.27~29 The dependence of emission on angle with respect to target normal is shown in Fig. 6. The relative intensities and shift at 30 and 60” agree well with the experimental measurements (see 1400 1200 1000 600 600 400 200 0 7.165 7.170 7.175 7.180 7.165 7.190 7.195 7.200 7.205 7.210
Fig. 7. Al Ly-a profile as calculated by the line transfer algorithm at 150” with respect to the laser beam. Curve A includes the effect of velocity fields as well as absorption. Curve B assumes an optically thin [~‘(v’“‘)in Eq. (7) set to zero] plasma but includes velocity effects. Curve C assumes a static plasma but includes absorption. Curve B was divided by the factor shown in parentheses. The arrows on the wavelength axis indicate the position of the peaks of the doublet for a static plasma.
Effect of velocity gradients and opacity on line transfer
539
1250
f
1000
% A
750
:I50
7.155
7.160
7.165
7.170
7.175
7.180
7.185
7.190
Fig. 8. Al Ly-u front emission at 30 and 60” with respect to the normal to the laser beam. Full curves are calculated using the algorithm of Eqs. (7). (S), and (9), which calculate the absorption in cell I by using the material densities and temperatures for cell 1. For the dotted curves the absorption in cell 1is calculated by using the material densities and temperatures from cell m where the photons originate. The arrows on the wavelength axis indicate the position of the peaks of the doublet for a static plasma.
Fig. 1). The back emission at 150 and 180” does not include the absorption by the cold material. The calculated line shape for the back emission can be better understood in Fig. 7. If the plasma is assumed static and absorption is included (curve C), a similar reversed profile to the one obtained in Fig. 5 is obtained. The effect of the velocity profile without absorption (curve B) results in a red shift and a smoothing of the shape. Including absorption as well as the velocity field results in the profile shown by curve A which shows a peak on the red wing of line. Such a peak results from the fact that photons emitted in the expanding plasma are red-shifted away from the peak of absorption in the relatively stationary absorbing plasma near the critical surface. A central assumption in the use of escape factors in a plasma with a large velocity gradients is that the plasma properties do not change appreciably over the Doppler-decoupling distance. ,,,,,,,1,1,1,1,1,1,
A-n 3000 T H % ,200o 0 Ll 2 B 1000 0 iE 0 7.140 7.145 7.150 7.155 7.160 7.165 7.170 7.175 7.180 7.185 7.190 h [AI Fig. 9. Al Ly-a front emission at 30” with respect to the normal to the laser beam. Curve D corresponds to an intensity of 2 x IO”, C to 10I4, B to 2 x 1014,and A to 5 x lOI Wcmm2. B, C, and D have been multiplied by the factors shown in parentheses, The arrows on the wavelength axis indicate the position of the peaks of the doublet for a static plasma.
A. DJAOUIet al
540
7.165
Fig. 10. Al Ly-cc emission
7.170
7.175
7.180
7.185
7.190
7.195
for various observation angles (in degrees) with respect calculated for an absorbed intensity of lOI W/cm*.
7.200
to the laser beam as
We investigate this approximation by recalculating the emission using not the algorithm of Eqs. (7), (8), and (9), which calculates the absorption in cell I by using the material densities and temperatures from cell 1, but by using the material densities and temperatures from cell m where the photons originate. Figure 8 shows that this only slightly alters the predicted profiles proving that the assumption inherent in the escape probability is valid. This is also in agreement with the previous work on the use of escape probabilities” which showed good agreement between escape probabilities and detailed line transfer calculations in laser produced plasmas. We now look at the effect of varying the incident laser intensity on emission. It is seen in Fig. 9 that higher laser intensities result in an increase in line intensity as a result of the larger volume of plasma created. There is also a small increase in the blue component as a result of the higher velocity gradients. In order to compare with the experimental results of Fig. 1, we show in Fig. 10 a calculation which uses an absorbed intensity of lOI W/cm*. There is good agreement in line shape as well as line intensity. This shows that line transfer in laser-produced plasmas having large velocity gradients can be calculated using the simple approach described in this paper, where trapping of radiation is taken into account by use of escape probabilities and velocity gradients accounted for by taking into account the Doppler shift between the emission and absorption points in the plasma. 5. LINE
EMISSION
FROM
A
PICOSECOND
LASER-PRODUCED
PLASMA
For high laser intensities, when the pulse length is of the order or smaller than the hydrodynamic time scale of expansion, high density, high temperature plasmas are produced. In a series of experiments,” a 35 prepulse free, high brightness, Raman amplified KrF laser (0.268 pm wavelength) was used to irradiate solid aluminium targets with a 12 psec (FWHM) pulse at intensities of 6 x 10” Wcm-*. Time resolved K-shell emission was recorded and showed Stark widths of the order 55 mA for the He-/I and 40 mA for the Ly-jJ line widths at early times, which narrowed as the plasma expanded in a time of about 40 psec to a value (N 20 mA) dominated by instrumental broadening. Average electron densities of greater than lO23cme3 were inferred from the Stark widths of the He-/I and Ly-8 lines at the early times. In this section, we present simulations of the time-dependent emission from such a plasma. Although trapping for Stark profiles has already been considered for resonance lines of hydrogenic iond0v3 and was found to be more important than in the case of Voigt profiles, this is not implemented in our model at this moment. The ionic populations are therefore calculated without any trapping. In view of the analysis in the previous
Effect of velocity
gradientsand opacity on line transfer
541
10199 25
30
35
40°
35
40
Distance [pm] (b)
25
30
Distance [pm] Fig. 11. Plasma conditions as a function of distance, at the time of peak emission in the picosecond laser-produced plasma. (a) Electron density, temperature, and ion temperature profiles. (b) Velocity profile, ground state, and second excited state helium-like ion density profiles.
section the effect of trapping is not expected to affect the line profiles or widths which are used for density determination. Average temperatures are however deduced from the ratio of the He-P to the Ly-8, and although trapping has a direct effect on the intensities, the ratio of the intensities should be less sensitive to it. The quantitative extent to which the above diagnostics are affected by trapping will be investigated in separate work. Figure 1l(a) shows the simulated electron and ion temperatures, and electron density at the time of peak emission (10 psec after peak of laser beam). Figure 1l(b) shows the velocity profile as well as the helium-like ground state and n = 3 excited state ion densities. The He-/? emission zone is characterized by higher densities than in the nanosecond situation, and larger velocity gradients (greater than lO”sec-‘), although the temperatures are similar. The time-dependent He-p and Ly-B emission at a fixed angle of 45” with respect to the laser beam, as calculated by the line transfer algorithm is shown in Figs. 12 and 13 respectively. In both cases, the emission peaks at around 10 psec after the peak of the laser. Strong Stark broadening near the peak of emission results in line widths of around 55 rnA for the He-/? and 35 rnA for the Ly-p line, in fairly good agreement with the experimental values. I9The line widths decrease rapidly as a result
A. DJAOUI et al
542
Al Hep
2.5~10"
I 6.56
6.56
6.60
6.62
6.64
6.66
6.68
6.70
6.72
Fig. 12. Time resolved Al He-p front emission at 45” with respect to the laser beam at different respect to the peak of the laser.
times with
of plasma expansion and lower electron densities to less than 20 mA, 30 psec after the peak of the laser beam. These calculations do not include any instrumental broadening (about 20 mA) which dominates the line profile at later times as the plasma expands. The effect of the velocity gradients on line emission is shown in Fig. 14. Only the Ly-p time integrated spectrum is shown. Similar results were obtained for the time-resolved profile, as well as for the He-j? line. The calculation for a static plasma shows a symmetric line profile. This is strongly blue-shifted when the velocity field is included in the calculation. Although the Stark line widths are much larger than the line widths in the nanosecond experiment, the velocity gradients and the corresponding bulk Doppler shifts are also much higher. The blue shift in the position of the peak is about 10 mA compared to the 4 mA in the nanosecond case (see curves A and B in Fig. 5).
6.00
Fig. 13. Time resolved
6.02
6.04
6.06
6.08
Al Ly-/3 front emission at 45” with respect to the laser beam, with respect to the peak of the laser.
6.10
at different
times
Effect of velocity gradients and opacity on line transfer
6.00
I
I
I
I
6.02
6.64
6.66
6.06
543
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~[AI Fig. 14. Time integrated (up to 140psec) Al Ly-B front emission at 45” with respect to the laser beam as calculated for a static plasma (dotted curve) and by taking into account the velocity field (full curve).
6. CONCLUSION
We have developed a two stage 1D planar model for calculating line transfer in laser-produced plasmas where temperature, density, and velocity gradients, as well as the effect of optically thick line reabsorption on level populations, are taken into account. In the first stage, the hydrodynamics and the ionic populations are solved in a self-consistent manner. The atomic physics of excitation and ionization is calculated using a time-dependent non-LTE average atom model, where resonance photon trapping in hydrogen and helium-like ions is treated approximately using escape probabilities. A detailed line transfer algorithm, which takes the output from the hydrodynamic/ionization calculation, is used in the second stage to calculate line profiles for hydrogen and helium-like resonance lines in aluminium. A choice of a Doppler, Voigt, or a convolution of Doppler and Stark profiles is available. In its present form this is already a powerful tool for interpreting and understanding radiative transfer in laser-produced plasma. Simulation of time integrated Ly-a emission as a function of angle in a long pulse (nanosecond) experiment show that velocity gradients are critical in determining the line shape. In the case of a short pulse (picosecond) experiment, the Stark broadened line widths and their narrowing with time as the plasma expands are well reproduced. It is also seen that even for the relatively broad emission from high densities, the velocity gradients are important in determining line shapes. Thus spectroscopic methods which are based on fitting line shapes from a static, constant density and temperature model can be in error if plasma flow velocities are not taken into account. By extending this model to include trapping for Stark broadened lines it will be possible to address the effect of trapping on level populations and on line ratios, as used for the determination of average temperatures, in high density a well as low density plasmas. REFERENCES 1. D. Mihalas, Stellar Atmospheres, Freeman, San Francisco, CA (1978). 2. D. Mihalas and B. W. Mihalas, Foundation of Radiation Hydrodynamics, OUP, Oxford (1984). 3. R. W. Lee, JQSRT 27, 87 (1982). 4. E. H. Avrett and D. G. Hummer, Mon. Not. R. ast. Sot. 130, 295 (1965). 5. D. G. Colombant, K. G. Whitney, D. A. Tidman, N. K. Winsor, and J. Davis, Physics Fluids 18, 1687 (1975). 6. D. Duston, R. W. Clark, J. Davis, and J. P. Apruzese, Phys. Rev. A 27, 1441 (1983). 7. D. Duston, R. W. Clark, and J. Davis, Phys. Rev. A 31, 3220 (1985). 8. G. J. Pert, J. Phys. B 9, 3301 (1976). 9. V. V. Sobolev, Moving Envelopes of Stars, Harvard University Press, Cambridge, MA (1960).
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