2. THEORY AND DETAILS OF CALCULATION

The mix of interaction physics depends on a range of parameters now experimentally accessible. These include laser intensity, wavelength, spatial and temporal coherence, pulse duration and shape, and the nature of the plasma. It is not the aim of this work to give the experimental details of dedicated experiments for fundamental studies in atomic physics. In this article, we carry out low-order relativistic perturbation-theory calculations for DR rate coefficients associated with the 2s–2p excitation in Al¹⁰⁺. Resonance structures arise from the 38 levels of the Al⁹⁺ 1s²2p8l configurations that straddle the Al⁹⁺ 1s²2s² → Al¹⁰⁺ 1s²2s ionization threshold. This paper's main concern is with ensuring the completeness of negative-energy resonance effects in dielectronic data for plasma modeling and not with all the consequential modeling of populations and ion distributions in plasma.

The various applications of laser plasmas, each has their own requirements for the coupling. The inertial confinement fusion, for example, requires that inverse bremsstrahlung be the dominant absorption process. This preferred absorption mechanism is simply described as collisional heating. In inverse bremsstrahlung, the coherent energy of the oscillating electrons in the laser field is thermalized by elastic scattering with the plasma ion. In an electromagnetic wave, electrons acquire a coherent energy oscillation. This coherent energy is converted into random thermal energy at the rate at which electrons are scattered by ions. By conservation of energy, this heating represents a damping of the laser energy. Inverse bremsstrahlung leads to bulk heating of the plasma and can be very efficient. Assuming the plasma created on Al target by a single 1 ns, temporally squared shaped pulse of 0.53 µm, with intensity on target of 1.6 × 10¹⁴ W/cm², we used the FLYCHK code (Chung et al., Reference Chung, Chen, Morgan, Ralchenko and Lee2005) in “runfile mode” at NIST (http://nlte.nist.gov/FLY) to execute time-dependent collisional-radiative calculation. The Al layer was considered tamped on both sides with a thin layer of plastic. The driver is an undiluted radiation field emitted from the rear side of a thin, 1000 Å Au foil. The intensity on the target is considered of 1.6 × 10¹⁴ W/cm² over an area that is 1500 µm square. The solution of time dependent coupled state equations including radiative transfer gives not only the temporal evolution of the population but also the temporal evolution of the mean ionic charge state into this plasma. Figure 1 presents this result as output from the code. Once the population distributions on the excited states have been determined, the code calculates the temporal evolution of the plasma parameters. Figure 2 and Figure 3 presents, respectively, the calculated time-dependence of the electron density and temperature into the plasma. The effect of excitation-autoionization and dielectronic channels on ionization balance is taken into account. Autoionizing doubly-excited states are included as Δn = 1 DR channels, which are critical in ionization balance for low Z atoms. The absorption efficiency is a function of the damping rate of the laser energy times the plasma density scale length. The dependence of the plasma absorption efficiency, the most basic signature of the coupling physics, as function of Iλ² for lasers with no coherence control, is an important indicator of the absorption mechanism. For Iλ² values on the order of 10¹⁴ W μm²/cm² the inverse bremsstrahlung becomes the dominant absorption mechanism with small component of collective processes. In region dominated by inverse bremsstrahlung, the bulk electrons are heated to several keV. Inverse bremsstrahlung is efficient when the absorption length, the group velocity of the light divided by the damping rate is less than the plasma dimension. The radiation transport equation is written in terms of specific intensity. The specific intensity of radiation I(r, n, ν, t) at position r in direction n, with frequency ν is defined as the amount of energy δɛ that is carried by radiation of frequency interval (ν, dν), across the surface element dS into a solid angle dΩ in a time interval dt:

(1)$${\rm \delta} {\rm \varepsilon} = I\lpar r\comma \; n\comma \; {\rm \nu} \comma \; t\rpar n \quad dSd\Omega d{\rm \nu} dt.$$

The atomic data, continuum lowering and radiation trapping effects (or opacity effects) are included in the FLYCHK code. The emissivity and opacity are determined for all transitions, both bound-bound and bound-free by recourse to the populations in the file generated by FLYCHK. The emissivity for a bound-bound transition between the upper level “u” and the lower level “l” is given by:

(2)$${\rm \varepsilon} _{\rm \nu} = N_{\rm \nu} A_{ul} h{\rm \nu} _{ul} /4 {\rm \pi} {\rm \varphi} \lpar {\rm \nu} \rpar \quad \hbox{\lpar ergs/cm}^3/\hbox{Hz}/\Omega\rpar \comma \;$$

where the population density of the upper level u is N _u and ν _ul is the frequency of the transition. The opacity of the same transition is given by:

(3)$${\rm \kappa} _{\rm \nu} = N_l \lpar 1 - N_u g_l /N_l g_u \rpar {\rm \pi} e^2 /mcf_{ul} {\rm \varphi} \lpar {\rm \nu} \rpar \quad \lpar {\rm cm}^{-1}\rpar .$$

A _ul and f _lu are the spontaneous emission rate and oscillator strength of the radiative transition, respectively. The line profile function ϕ(ν) that determines the frequency dependence of the emissivity and opacity is considered with a Voigt function. We performed this computation work and results are given in Figures 4 and 5, respectively. Figure 4 presents the simulated bound-bound emissivity as function of photon energy. Figure 5 presents the bound-free opacity as function of photon energy, as output from the code. These figures correspond to t = 5 × 10⁻¹¹ s, T _e = 45 eV, N _e = 2.7 × 10¹⁹ cm⁻³ parameters in the FLYCHK code. The post-processor FLYSPEC synthesizes the emission and absorption spectra using the population distributions from the FLYCHK output file.

Fig. 1.

The time evolution of the mean ion charge in laser-produced aluminum plasma. The plasma is formed at the interaction of a single 1ns, temporally squared shaped laser pulse of 0.53 µm, with intensity on target of 1.6 × 10¹⁴ W/cm².

Fig. 2.

The time evolution of the electron density (in cm⁻³ units) into the plasma.

Fig. 3.

The time evolution of the electron temperature (in eV units) as output from the code.

Fig. 4.

(Color online) Bound-bound opacity (in cm⁻¹) as function of photon energy (in eV units), corresponding to t = 5 × 10⁻¹¹ s, T _e = 45 eV, N _e = 2.7 × 10¹⁹ cm⁻³.

Fig. 5.

Bound-free opacity (in cm⁻¹) as function of photon energy (in eV), as output from the code. Results correspond to the following parameters: t = 5 × 10⁻¹¹ s, T _e = 45 eV, N _e = 2.7 × 10¹⁹ cm⁻³.

The second example presented herein is the laser produced plasma with Li-like Al ions, i.e., the recombining laser-produced plasma. Amplification of the X-ray-ultraviolet radiation on Li-like Al ion 3d–5f (λ = 10.57 nm), 3d–4f (λ = 15.46 nm), and 4f–5g (λ = 33.47 nm) transitions has been experimentally demonstrated 20 years ago (Carillon et al., Reference Carillon, Edwards, Grande, Henshaw, Jaegle, Jamelot, Key, Kiehn, Klisnick, Lewis, O'neill, Pert, Ramsden, Regan, Rose, Smith and Willi1990). The plasma has been generated by irradiating cylindrically an Al fiber of 7 µm diameter by a laser pulse of following characteristics: intensity 10¹³ W/cm², wavelength 0.53 µm. Gaussian temporal shape of 120 ps duration (full width at half maximum) with maximum at 70 ps. Some discrepancies have been observed between the measured and calculated values of the gain coefficient. The complete simulation (time- and space-dependent density and temperature being taken into account), has then been carried out with, and without, DR (Stancalie et al., Reference Stancalie, Sureau, Klisnick, Moller, Gouennou, Berete and Rose1995). Following this precursory work, DR doesn't modify significantly the temporal profiles and appears only as a lowering of amplitude of about 8% for each transition. Depending on the plasma density and temperature, there are different paths for Al¹⁰⁺ ion to temporarily capture an electron into a high Rydberg state. Owing to the electron temperature during the recombination phase (≤200 eV), the most probably ways involve the autoionizing levels 1s²2pnl ^1,3L'_J' and 1s²3pnl ^1,3L'_J' (with initial ion 1s²2s) and 1s²3dnl ^1,3L'_J' (with initial ion 1s²2p). For each way characterized by a core transition, i-j, the population density of the initial level decays according to the relation dn _i/dt = −n _in _eK _d (i − j) where n _e is the free-electron density and K _d(i-j) the total rate coefficient (sum over all possible intermediate and final states) of the considered way. During the time interval dt the total DR of the considered species is then: $\sum\limits_{i\comma j} {n_i } n_e K_d \lpar i - j\rpar $. If the populations are expressed in terms of the Saha-Boltzmann factors (Summers et al., Reference Summers, Dickson, O'mullane, Badnell, Whiteford, Brooks, Lang, Loch and Griffin2006), then they show strong overpopulation of the high n-shells at low density due to DR. As the electron density is increased, the DR influence becomes less due to ionization of the high n-shell populations with all the Saha-Boltzmann factors tending to 1. Complete detailed calculation of a K _d coefficient requires a lot of intermediate data, and is therefore not easily manageable in a simulation calculating, for each time step, the populations of many levels in a number of plasma cells. For positive resonance energies, the values of K _d (cm³s⁻¹) as a function of electron temperature (kT in eV) has shown a first maximum (near 10 eV) due to the core transitions 2s–2p while the other two recombination ways (core transitions 2s–3p and 2p–3d) are dominant above 50 eV and maximum at 150 eV. We reproduce in Figure 6 the total DR rate of Li-like into Be-like Al ions in this recombining plasma regime (Stancalie et al., Reference Stancalie, Sureau, Klisnick, Moller, Gouennou, Berete and Rose1995). The only positive-energy resonance states have been included into the calculation.

Fig. 6.

The total DR rate coefficients (cm³s⁻¹) in X-ray laser produced plasma with Li-like Al ions (Stancalie et al., Reference Stancalie, Sureau, Klisnick, Moller, Gouennou, Berete and Rose1995).

The electron scattering with Li-like Al has been subject of our previous works (Stancalie Reference Stancalie2000; Stancalie & Pais, Reference Stancalie and Pais2006). The detailed analysis of the radiative damping effect on DR for this particular ionic species has also been reported (Stancalie, Reference Stancalie2005). In order to investigate the negative-energy resonance states contribution to DR of Li-like and Be-like Al ions, in the present work we use the low-order relativistic perturbation theory to calculate the DR in Al¹⁰⁺ above and below the first ionization threshold. The Oxford code, the GRASP multiconfiguration Dirac-Fock (MCDF) code (Dyall et al., Reference Dyall, Grant, Johnson, Parpia and Plummer1989), was used to obtain the wave functions and energies of the Li-like Al target. The first autoionizing state in Al⁹⁺ corresponds to n = 9 (Stancalie, Reference Stancalie2005). We are interested in evaluating atomic data as level energies, radiative transition probabilities, line-strengths, and Auger rates, for transitions between the n = 8 levels. To this end, configurations such that 1s²2s², 1s²2p², 1s²2s8s, 1s²2s8d, 1s²2s8f, 1s²2s8g, 1s²2p8s, 1s²2p8p, 1s²2p8f, and 1s²2p8g have been included in the average level (EAL) calculation, while 1s², 2s² and 2p² shells were kept full. These jj-coupled configuration state functions result in a set of 74 levels that are limited to the following symmetry-parity combinations J ^π = 0^±, 1^±, 2^±, 3^±, 4^±, 5⁺. We have used the extended average level MCDF-EAL option where in the Hamiltonian matrix we minimize a weighted trace (proportional to 2 J +1). Additional relativistic corrections arising from the Breit interaction and quantum electrodynamics are also included. For determining the radiative and autoionizing rates the Dirac atomic R-matrix Code was used (Norrington & Grant, Reference Norrington and Grant1987, and references herein). To stabilize the order of the levels (odd or even), an initial analysis on each parity has been done. For the even parity group, the reference set contains (1s²2s², 1s²2p², 1s²2s8s, 1s²2s8d, 1s²2s8g, 1s²2p8p, and 1s²2p8f), and for the odd parity group the reference set contains {1s²2s8p, 1s²2p8s, 1s²2p8d}. Once the order has been stabilized, we refer to Δn = 0, n = 8 transitions. There is no level energy data in the Atomic Structure Database of the National Institute for Standards and Technology (http://www.nist.gov). To check the accuracy of the resonance positions we used the Opacity Table (1995).

The relativistic approach based on the Dirac equation has the advantage that all the relativistic effects are included not only for the Eigen energies but, most importantly, in the radial wave functions. In the present calculation, we included Breit interaction in the bound-bound part of the continuum Hamiltonian. The effect is to modify the thresholds and move the resonance structures.

The Dirac Hamiltonian for the (N + 1)-electron system, in atomic units, is written as:

(4)$$H^{N + {1}} = \sum\limits_{i = {1}}^{N + {1}} { - ic\, {\rm \alpha} } \cdot \nabla _i + \lpar {\rm \beta} - {1}\rpar c^{2} - \displaystyle{Z \over {r_i }} + \sum\limits_{\,j = i + {1}}^{N + {1}} \displaystyle{{1} \over {\left\vert {r_j - r_i } \right\vert }}\comma \;$$

where i and j index the individual electrons, Z is the charge of an infinitely heavy point nucleus, the electron mass has been subtracted, and α and β are the usual Dirac matrices constructed from Pauli spin and unit matrices. The first three one-electron terms in the Hamiltonian are a momentum term, mass term, and the electron-nucleus Coulomb attraction. The final two-electron term is the Coulomb electron-electron repulsion.

In the isolated-resonance and independent process approximation, the DR cross section for a given N electron initial level i through an intermediate (N + 1) electron intermediate level j is given by (Pindzola et al., Reference Pindzola, Loch and Robicheaux2011, and references herein):

(5)$${\rm \sigma} _j \lpar {\rm \varepsilon} \rpar = \displaystyle{{{\rm \pi} ^{2} } \over {\left\vert {E_{\,ji} } \right\vert }}\displaystyle{{g_j } \over {{2{\rm g}}_i }}\displaystyle{{\displaystyle{{\Gamma _j } \over {{2{\rm \pi} }}}} \over {\left({{\rm \varepsilon} - E_{\,ji} } \right)^{2} + \displaystyle{{\Gamma _j^{2} } \over {4}}}}\sum\limits_k {\Gamma _A } \lpar j \to k\rpar B_j\comma \;$$

where E _j is the energy and g _j is the statistical weight of the (N + 1) electron doubly excited level, E _i is the energy and g _i is the statistical weight of the N electron initial target level, and E _ij = E _j − E _i. The branching ratio for radiative stabilization is given by:

(6)$$B_j = \displaystyle{{\sum\limits_n \Gamma _R \lpar \,j \to n\rpar } \over {\sum\limits_k \Gamma _A \lpar \,j \to k\rpar + \sum\limits_n \Gamma _R \lpar \,j \to n\rpar }}\comma \;$$

where the radiative Γ_R and autoionizing Γ_A rates are evaluated using relativistic calculations. When a single state is embedded in a continuum or Rydberg series, the short range S matrix has the property (Robicheaux et al., Reference Robicheaux, Gorczyca and Pindzola1995):

(7)$${{1} - \left\vert {S_{{11}} \lpar E\rpar } \right\vert ^{2} = \displaystyle{{\Gamma _R \Gamma _A } \over {\lpar E - E_r \rpar ^{2} + \left[{\left({\Gamma _R + \Gamma _A } \right)/{2}} \right]^{2} }}}\comma \;$$

where E _r is the energy of the compact state (which can be positive or negative). Γ_A is related to the CI coupling between the state and the channel. This form does not depend on whether E _r is positive or negative. If the resonance is at positive energy, the rate coefficient for DR from a thermal distribution is calculated as:

(8)$$K_d = \left({\displaystyle{{{2{\rm \pi} }} \over T}} \right)^{{3}/{2}} \displaystyle{{{2{\rm J}}_r + {1}} \over {{2}\lpar {2{\rm J}}_c + {1}\rpar }}\displaystyle{{\Gamma _R \Gamma _A } \over {\Gamma _R + \Gamma _A }}e^{ - E_r /T}$$

where T is the temperature (T≫Γ_R + Γ_A), J _r is the angular momentum of the resonance, and J _c is the angular momentum of the core. If the resonance is at negative energy, the probability to be in an n manifold with total angular momentum J _r and core electrons with angular momentum J _c is (Robicheaux et al., Reference Robicheaux, Loch, Pindzola and Balance2010):

(9)$$P\lpar n\comma \; J_r \rpar = \left({\displaystyle{{{2{\rm \pi} }} \over T}} \right)^{{3}/{2}} \displaystyle{{{2{\rm J}}_r + {1}} \over {{2}\lpar {2{\rm J}}_c + {1}\rpar }}{\exp}\lpar \!- E_n /T\rpar {\rm \zeta} \lpar E_n \rpar \comma \;$$

with E _n = −Z ²/(2(n ²) is the energy of the n manifold and ζ (E _n) is the Saha-Boltzmann deviation factor (Summers et al., Reference Summers, Dickson, O'mullane, Badnell, Whiteford, Brooks, Lang, Loch and Griffin2006) depending on plasma temperature and density.

Relativistic calculation has been done to output energy levels for 1s²2s8l and 1s²2p8l levels in Al⁹⁺. For the even-parity resonance levels the radiative rates were determined for the decay to the multiconfiguration levels involving the odd-parity configurations: 1s²2s8p and 1s²2s8f, while the autoionization rates were determined for the decay of the even-parity resonance levels to the single-configuration levels involving the even-parity configurations: 1s²2sɛs, 1s²2sɛd, and 1s²2sɛg, ɛ ≤ 8. For the odd-parity resonance levels, radiative rates were determined for the decay to the multiconfiguration levels involving the even-parity configurations 1s²2p², 1s²2s8s, and 1s²2s8d, while the autoionization rates were determined for the decay of the odd-parity resonance levels to the single-configuration levels involving the odd-parity configurations: 1s²2s8p, 1s²2p8d, and 1s²2s8f levels. Results are shown in Table 1. To check the accuracy of the resonance positions we compared our calculated energies with those reported by the Opacity team, and a very good agreement has been observed. In Table 2, we present these data. The resonance-level energies are calculated relative to the ionization threshold ²S^e_.Figure 7 presents the temperature-independent contribution of the Al⁹⁺ states to DR rate coefficient as a function of energy, relative to the ionization threshold. Resonances attributed to 1s²2p8l configurations straddle the threshold, with a strong resonance feature found at −0.02 eV. We recall here that for positive resonance energies (Fig. 6), the values of K _d (cm³s⁻¹) as a function of electron temperature (kT in eV) has shown a first maximum (near 10 eV) due to the core transitions 2s–2p while the other two recombination ways (core transitions 2s–3p and 2p–3d) are dominant above 50 eV and maximum at 150 eV. The effect of including the negative-energy resonance-level into the rate coefficient calculation is shown in Figure 8. In this figure the electron temperature has been considered 1 eV.

Fig. 7.

The temperature-independent contribution of the Al⁹⁺ states to DR rate coefficient, Γ ≡ (2J_r + 1)/[2(2J_c + 1)][Γ_A Γ_R/(Γ_A + Γ_R)], as function of energy, E, relative to the ionization threshold. The energies were obtained from DARC code shifting the core energies to the NIST values.

Fig. 8.

The DR rate coefficient (in cm³s⁻¹) for Al¹⁰⁺: a relativistic perturbation theory calculation at 1eV.

Table 1.

Al⁹⁺ resonance-level energies, in atomic units (au), the radiative transition rates, Γ_R, and the autoionization transition rates, Γ_A. a ^−b read s a × 10^b

Table 2.

Comparison between the resonance –level energies (this work), in Rydberg units (Ryd.), and those reported by the Opacity team calculation. a ^−b reads a × 10^b