1 I. INTRODUCTION

Atomic populations are of fundamental importance in a variety of areas in both pure and applied science. Examples include the equation of state in thermodynamics; absorption, emission, and scattering processes in matter; lasing; radiation transport; radiative cooling and energy loss; diagnostic and spectroscopic methods that employ the radiative properties of matter; astrophysics and planetary science; the physics of radiation sources; and fusion science and technology.^1,2

In a plasma, several charge states usually exist simultaneously, and therefore the total radiation emission arises from excited states of different ionic charges. This indicates that not only are the populations of excited states and their excitation mechanisms relevant but so too is the ionization balance. In hot plasmas, while the excited-state populations are essentially driven by electron collisional excitation, the ionization balance depends strongly on ionization and recombination processes. As shown by Burgess³ in the context of an analysis of solar emission, electron collisional ionization and radiative recombination alone could not account for observations, and dielectronic recombination was proposed to explain a low level of ionization. In considering radiation balance, it is therefore important to take account of dielectronic recombination. Similarly, the concept of local thermodynamic equilibrium (LTE) depends not only on radiative and collisional rates (as in the traditional description), but on autoionization rates too.

Dielectronic recombination (DR) is the capture of an electron with simultaneous excitation of an atomic core and subsequent radiative stabilization of the core. Within the framework of a simplified model, DR can be viewed as a sequence involving dielectronic capture of a continuum electron into the state nl and subsequent radiative stabilization. The first step is the capture of an electron into the state nl with simultaneous excitation of the atomic core from state α₀ to state α:After dielectronic capture, autoionization or radiative stabilization can take place:The radiative stabilization involves the core and the captured electron (spectator electron):

Here, A^+(Z−1)** denotes a doubly excited state, A^+(Z−1)* a singly excited state, and γ the ground state of the ion in charge state Z − 1, i.e., A^+(Z−1)(γ) = A^+(Z−1)(ground). The radiation emission of the core [the relation (1.3)] is known as dielectronic satellite emission and has properties that are of great importance for plasma temperature diagnostics, as revealed in the pioneering work by Gabriel.⁴ The study of dielectronic satellite emission has become a major aspect of the characterization of a variety of complex phenomena in nonequilibrium plasmas: determination of electron density by angular-momentum-changing collisions^2,5–7 and by the Stark effect;^8,9 characterization and measurement of hot electrons and classification of related instabilities;^10–12 determination of Auger electron heating phenomena in X-ray free-electron laser (XFEL) interaction with solid matter;¹³ radiation field analysis via hollow ion X-ray emission;^14,15 collisional phenomena induced by laser-produced plasma jets;^16,17 impurity transport and charge-exchange phenomena with the neutral background in magnetic fusion plasmas;^18,19 relaxation phenomena in fluctuating plasmas;²⁰ disappearance of resonance line emission followed by accumulation of dielectronic satellite emission;^21–24 ionization potential depression analysis via two-dimensional maps of hollow-ion X-ray emission in XFEL–solid matter interaction.²⁵ It should be noted here that for diagnostic purposes, radiation transport of radiative transitions originating from autoionizing states should be avoided, because re-emission of the photons is considerably reduced owing to the high autoionization rates.¹⁶

Comparison of the relations (1.1) and (1.4) shows that effective recombination occurs because an ion A^+Z is finally transformed into an ion A^+(Z−1). As was demonstrated in Refs. 3 and 26, DR can even be the most important recombination process. As a rule, the DR rate is high for ions with a complex core that exhibits transitions between levels without any change in principal quantum number, i.e., transitions with Δn = 0, such as 2 s→2p transitions in lithium-like and more complex ions. It is obvious that, particularly for ions with a complex core, the number of angular momentum coupling possibilities for the whole series of captured electrons is enormous. Therefore, the corresponding atomic structure calculations that combine all the single contributions (1.1)–(1.4) to give the total DR of each ion are numerically prohibitive, and this has manifested itself in a continuing controversy regarding the calculation of the ionic fractions.^27–29

Moreover, the total DR rate is not an issue of atomic structure calculations alone, but also requires consideration of populations that are strongly out of LTE and consideration of the plasma electric microfield. Therefore, the theoretical determination of the total DR in a nonequilibrium plasma cannot be done within the framework of atomic structure calculations alone.

The present paper is therefore devoted to a critical analysis of methods for the determination of the total DR rate and to develop a new framework to address the challenges faced. In this respect, a rather surprising element is discovered that can be illustrated via the above-mentioned Δn = 0 transitions, e.g., 2s → 2p transitions in lithium-like and more complex ions. The transition energy ΔE = ℏω_core for Δn = 0 and Z ≫ 1 is of the order of ZRy, while the ionization energy is of the order of Z²Ry ≫ ΔE. The 2s → 2p transition is the main channel for both the autoionization and radiative decay. Since the energy E of the incident recombining electron is in all cases smaller than the excitation energy, this implies the following inequality [using Ry=12me(cα)2 and E=12meve2]:where η is the Coulomb parameter (note that the standard Coulomb parameter determines the possibility of a quasiclassical consideration). The condition (1.5) indicates the validity of the quasiclassical regime in which, for example, the spectrum from the quantum theory of bremsstrahlung³⁰ coincides with the classical spectrum.³¹ Thus, an important part of the overall DR processes is described well by a quasiclassical approach. This opens up new ways to treat important phenomena occurring in plasmas other than purely quantum mechanical atomic structure calculations (e.g., the multiconfiguration Dirac–Fock method).

On the other hand, DR related to Δn = 1 transitions may require quantum mechanical approaches. A typical example is provided by the core transitions 1s → 2p in H-like, He-like, etc. ions. Here, the 1s → 2p transition is the main radiation channel, while the autoionization channel depends strongly on the state nl of the captured electron: for large quantum numbers n, we encounter autoionization according to 2pnl → 2s + e, while for small quantum numbers, the autoionization channel 2pnl → 1s + e is dominant. For the case 1s → 2p, the transition energy ΔE = ℏω_core is of the order of Z²Ry and thus of the same order as the ionization energy. Therefore, the Coulomb parameter is about η ≈ 1 and quantum mechanical calculations might be required.

The paper considers the most recent developments that allow a unique description of the relevant phenomena in DR in nonequilibrium plasmas. For a review of standard methods in the theory of DR, the reader is referred to Refs. 26 and 32–34. In Sec. II, general DR formulas are derived in terms of atomic structure and collisional–radiative decay probabilities. In Sec. III, quantum mechanical multichannel methods are developed and then in Sec. IV, excited-state coupling driven by electron collisional excitation is considered. In Sec. V, the impact of angular-momentum-changing collisions on DR is considered and visualized with a simple analytical approach. Plasma electric field effects are discussed in Sec. VI, where the impacts of ionization potential depression and of symmetry perturbations of autoionizing matrix elements are described. Finally, Sec. VII develops a local plasma frequency statistical approach to calculate DR rates for very complex ions.

2 II. AUTOIONIZATION, DIELECTRONIC CAPTURE, AND DIELECTRONIC RECOMBINATION

Let us recall the essence of the DR process. An incident electron with energy E excites an ion core with excitation energy ΔE = ℏω_core. In this case, if the energy E is smaller than ΔE, the electron is finally captured by the ion into a state with energy Ef≈−RyZeff2/nf2 (Z_eff is the effective charge of the excited ion core and n_f is the principal quantum number of the captured electron) obeying the conditionThis capture results in a doubly excited state of the ion; i.e., the ion core electron is excited with energy ΔE, while the captured electron occupies a highly excited level of the ion. This state of the ion can decay in two possible ways:(i)

by relaxation of the ion core electron into the initial ground state with simultaneous ejection of the captured electron from the ion: this process is known as autoionization [compare with the relation (1.2)];

For illustration, Fig. 1 shows the relevant energy–level diagram for the He-like 2l2l′ autoionizing levels from which the so-called Ly_α satellites originate. The energy of the 2l2l′ levels is given approximately by E2l2l′≈2Zeff2Ry/4=Zeff2Ry/2 (in the H-like approximation), which is about half of the ionization energy of the H-like ground state Zn2Ry (where Z_n is the nuclear charge). The series limit of the autoionizing levels 2lnl′ is the first excited state 2l. Radiative decay (dielectronic satellite emission) from the 2l2l′ levels populates the singly excited levels 1s2l^1,3L, from which further radiative decays proceed (e.g., the resonance line W = 1s2p¹P₁ − 1s^{2 1}S₀ and the intercombination line Y = 1s2p³P₁ − 1s^{2 1}S₀) that finally populate the ground state 1s^{2 1}S₀.

Fig. 1. Energy-level diagram of the He-like autoionizing levels 2l2l′ and their associated radiative decays, so-called Ly_α satellites. After radiative decay, the singly excited states 1s2l^1,3L are formed, from which further radiative decay proceeds (e.g., the resonance and intercombination lines W and Y, respectively). Also indicated are the Li-like autoionizing levels 1s2l2l′.

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The chain of processes of dielectronic capture (1s + e → 2l2l′), radiative decay to singly excited levels (2l2l′ − 1s2l + hν), and radiative decay to the ground state (1s2l^1,3L → 1s^{2 1}S₀ + hν′) is called dielectronic recombination (the DR channel) because an effective recombination has taken place from the H-like ground state 1s²S_1/2 to the He-like ground state 1s^{2 1}S₀.

Thus, the DR process as well as the photorecombination (PR) process result in the capture of an incident electron and its simultaneous photon emission. The difference is that the photon is emitted by the ion core electron in the DR process rather than by the incident electron as in the PR process. Note that the relationship between the PR and DR processes is analogous to that between standard bremsstrahlung and polarization bremsstrahlung.³⁵

The DR rate can be calculated from the autoionization rate of a given atomic state with the help of the principle of detailed balance. The first step is the application of the principle of detailed balance to dielectronic capture, i.e.,where njZ is the atomic population of the autoionizing state, ΓjkZ,Z+1 is the autoionization rate from state j to a state k with population nkZ+1, and 〈DC〉_kj is the dielectronic capture rate from state k to the upper state j. To solve Eq. (2.2) for the dielectronic capture rate, we need to specify the populations. For this purpose, we consider a system in thermodynamic equilibrium. In this case, the populations njZ and nkZ+1 are linked via the Saha–Boltzmann equation, because states j and k belong to different ionic states Z and Z + 1, respectively, i.e.,where gjZ and gkZ+1 are the statistical weights of states j and k, n_e is the electron density, m_e is the electron mass, and T_e is the electron temperature. The energy difference ΔEkjZ+1,Z is related to the so-called dielectronic capture energy EkjDC by (see also Fig. 1)where EkjDC is the energy of the Auger electron if the autoionizing state j decays via autoionization to state k. Combining Eqs. (2.2)–(2.4), we find the general expression for the dielectronic capture rate:or, in convenient units (with ΓjkZ,Z+1 in s⁻¹, and EkjDC and T_e in eV),[note that Eq. (2.6) assumes a Maxwellian electron energy distribution function with temperature T_e]. If Pj,grZ is the probability that the autoionizing state j of charge state Z decays to the ground state of the same charge state, then the quantity Pj,grZDCkj is called the dielectronic recombination rate coefficient (with units cm³ s⁻¹) into state k via the intermediate state j:

Note that it is important to underline that the assumption of thermodynamic equilibrium to eliminate the populations of Eq. (2.2) with the help of Eq. (2.3) is only a convenient method to apply the principle of detailed balance, but does not mean that relations derived with the help of the detailed balance relations are only valid under the assumption of thermodynamic equilibrium. In fact, the dielectronic capture rate [Eq. (2.6)] can also be derived from purely quantum mechanical relations (micro-reversibility), providing a dielectronic capture rate that is also valid for an arbitrary electron energy distribution function F(E):²⁰or, in convenient units (with ΓjkZ,Z+1 in s⁻¹, and F(E), EkjDC, and T_e in eV),

3 III. TOTAL RATES OF DIELECTRONIC RECOMBINATION: LOW-DENSITY APPROXIMATION

3.1 A. General considerations

In general, the probability Pj,grZ from Eq. (2.7) is a function of density and temperature, i.e.,Pj,grZ=Pj,grZ(ne,Te).(3.1)The probability function (3.1) has to be determined from numerical calculations of a multilevel multi-charge-state atomic population kinetics that explicitly involves all necessary autoionizing states as “active levels” (“active” here means that the populations of the autoionizing levels are calculated in a similar way to the ground and singly excited states in the population kinetics). If collisions are negligible compared with spontaneous radiative decay rates as well as with autoionization rates, then the probability Pj,grZ can be approximated by the so-called satellite branching factors BjiZ:³Pj,grZ→∑iBjiZ=∑iAjiZ∑lAjlZ+∑kΓjkZ,Z+1.(3.2)That is to say, Eq. (3.2) is the low-density approximation of the probability Pj,grZ. In this case, the DR rate is given by the following approximate expression:DRkjZ+1,Z≈∑iBjiZ.DCkjZ+1,Z=∑iAjiZ∑lAjlZ+∑kΓjkZ,Z+1DCkjZ+1,Z.(3.3)With the help of Eq. (2.5), Eq. (3.3) can be written as follows:DRkjZ+1,Z≈12gkZ+12πℏ2me3/2exp(−EkjDC/kTe)(kTe)3/2∑igjZΓjkZ,Z+1AjiZ∑lAjlZ+∑kΓjkZ,Z+1.(3.4)The term in braces {⋯} is the so-called dielectronic satellite intensity factorQk,jiZ+1,Z=gjZΓjkZ,Z+1AjiZ∑lAjlZ+∑kΓjkZ,Z+1,(3.5)which has been calculated by the pioneering work in Refs. 36 and 37 with unprecedented precision via the multiconfiguration Z-expansion method. Therefore, under the assumptions made in Eq. (3.2), the DR rate is given by the sum of the dielectronic satellite intensity factors. One can see that the essential quantities that appear in Eqs. (3.4) and (3.5) to calculate the DR rate in the low-density approximation are the dielectronic capture energy, statistical weights, and radiative and autoionizing decay rates. These quantities can nowadays routinely be generated from atomic structure calculations.

However, as one can see, even for the simplest configurations 2lnl′, the numerical calculations are rather cumbersome because very large quantum numbers nl′ have to be involved to achieve convergence for the DR rates. For large quantum numbers, however, convergence is difficult to achieve in purely quantum numerical atomic structure calculations. Moreover, to obtain the total DR rate from H-like to He-like ions, one needs to invoke all possible intermediate states j = 3lnl′, 4lnl′, 5lnl′, ….

One can easily understand that for more complex configurations, the number of autoionizing states that need to be involved rapidly becomes numerically prohibitive for purely quantum mechanical numerical calculations. For practical applications, it is therefore mandatory to invoke additional methods, such as the Burgess approach,^3,32 Vainshtein’s simplified quantum mechanical dielectronic recombination model,^26,38 a quasiclassical approach,^31,39 or a statistical approach.^40,41

3.2 B. The Burgess approximation

To illuminate the essence of the various approximations currently employed, we consider first the most general expression for the total dielectronic recombination rate DRtotZ+1,Z. For the total rate, all DR rates DRkjZ+1,Z=Pj,grZDCkjZ+1,Z have to be summed with respect to the initial state k and also with respect to the intermediate states j, i.e.,DRtotZ+1,Z=∑k∑jDRkjZ+1,Z=∑k∑jPj,grZDCkjZ+1,Z.(3.6)Because the probability Pj,grZ is a function of density and temperature [see Eq. (3.1)], it is very difficult to obtain general and closed formulas for the DR rate coefficient. Only in the low-density approximation, where Eq. (3.3) holds, can general formulas for the DR rate coefficients be obtained.

One of the most widely used general approximate empirical formulas in the framework of the approximation (3.2) is the so-called Burgess formula,³ in which it is assumed that the nl spectator electron is not interacting with the core and can be treated in the hydrogenic approximation and that the capture cross section averaged over the resonances can be obtained with the aid of the correspondence principle by extrapolating below threshold the partial cross section for the core excitation α₀ → α:DRkjZ+1,Z:=DZ+1,Z(α0→α,nl).(3.7)For the total DR rate, we haveDRtotZ+1,Z:=DZ+1,Z=∑α0∑α∑n∑l=0n−1DZ+1,Z(α0→α,nl).(3.8)For the simplest example of autoionizing states 2l2l′ outlined in Fig. 1, α₀ = 1s and α₁ = 2p, i.e., the transition α₀ → α corresponds to the Ly_α transition in H-like ions. For these configurations, DR into the ground state is the most important transition; i.e., there exists a single state k = α₀ = 1s. Therefore, α₀ coincides with the atomic ground state, and the sum over α₀ can be suppressed:DZ+1,Z≈∑α∑n∑l=0n−1DZ+1,Z(α0→α,nl).(3.9)The DR rate coefficient D^Z+1,Z(α₀ → α, nl) can then be expressed via the following analytical empirical expression:³DZ+1,Z(α0→α,nl)=4.8×10−11fα0αBdβ3/2e−βχd(cm3/s),(3.10)withβ=(z+1)2RykTe,(3.11)χd=χ1+0.015z3(z+1)2,(3.12)χ=ΔE(α0→α)(z+1)2Ry.(3.13)Here, z is the so-called spectroscopic symbol of the doubly excited ion after recombination, z = Z_n − N_bound + 1, where N_bound is the number of bound electrons and Z_ne is the nuclear charge. If the first resonance transition is a Δn = 0 transition, then the branching factor B_d is given by the following fitting formula (the so-called Burgess–Mertz formula):³²Bd=zχz2+13.41/211+0.105(z+1)χ+0.015(z+1)2χ2.(3.14)For Δn ≠ 0 the fitting function for the branching factor B_d is different:³²Bd=zχz2+13.41/20.51+0.210(z+1)χ+0.030(z+1)2χ2.(3.15)Let us recall the meaning of the branching factor B_d: after dielectronic capture, a doubly excited state is formed that can decay via autoionization or radiative decay. For DR, only the radiative decays contribute finally to recombination, since autoionization returns the autoionizing state to the original state.

According to Eq. (3.10), α₀ is the ground state, and therefore f_α0α is the electric dipole absorption oscillator strength for the resonance transition α₀ → α with transition energy ΔE(α₀ → α) in eV. As the absorption oscillator strength decreases rapidly with increasing principal quantum number of the upper level, it is usually sufficient to consider only the first two α-terms in the sum in Eq. (3.9), and we finally obtain the desired expression for the total DR coefficient:DZ+1,Z≈DZ+1,Z(α0→α1)+DZ+1,Z(α0→α2),(3.16)withDZ+1,Z(α0→α):=∑n∑l=0n−1DZ+1,Z(α0→α,nl).(3.17)Let us consider DR into neutral helium as an example (note that a single * indicates a singly excited state, while ** corresponds to a multiply excited state):He1+(1s)+e→He0+**(nln′l′)→He0+(1s2).(3.18)For this example, α₀ = 1s, α₁ = 2p, α₂ = 3p, …. Therefore, fα0α1 corresponds to the electric dipole absorption oscillator strength of the resonance line, namely, the H-like Ly_α line of singly ionized helium, while fα0α2 corresponds to the Ly_β line. The oscillator strengths are f_1s→2p = 0.4164 and f_1s→3p = 0.079 14, and their transition energies are ΔE(1s → 2p) = 40.81 eV and ΔE(1s → 3p) = 48.37 eV. The spectroscopic symbol is z = 1 and Δn ≠ 0 [therefore, Eq. (3.15) applies]. As one can see, higher-n oscillator strengths make almost negligible contributions to the total DR rate (note that this has to be distinguished from the fact that high-n spectator electrons can make quite important contributions). From Eqs. (3.11)–(3.13), we obtain β = 4Ry/kT_e, B_d(1s → 2p) = 0.0825, B_d(1s → 3p) = 0.0846, χ_d(1s → 2p) = 0.747, and χ_d(1s → 3p) = 0.886. For the rate coefficients at kT_e = Ry (β = 4), we obtain DHe1+,He0 +(1s→2p)=1.65×10−12cm3/s, and DHe1+,He0 +(1s→3p)=3.2×10−13cm3/s. This confirms that the leading terms for DR are indeed given by Eq. (3.16).

3.3 C. Quantum mechanical multichannel approach

Comparison of the results from Eqs. (3.10)–(3.15) with more precise quantum mechanical calculations carried out with Vainshtein’s ATOM code^42,43 show that the χ_d values are in quite good agreement, whereas the B_d values differ strongly. For the resonance transitions 1s → 2p and 1s → 3p, the B_d values obtained from Vainshtein’s simplified quantum mechanical multi-channel (QMMC) approach^38,44 for the above example of helium are significantly different, namely, B_d,ref(1s → 2p) = 0.155, B_d,ref(1s → 3p) = 0.0144, χ_d,ref(1s → 2p) = 0.744, and χ_d,ref(1s → 3p) = 0.888, giving DrefHe1+,He0 +(1s→2p)=3.10×10−12cm3/s and DrefHe1+,He0 +(1s→3p)=5.46×10−14cm3/s, i.e., the Burgess formula underestimates the B_d value for α₀ → α = 1s → 2p by a factor of 2 and overestimates the B_d value for α₀ → α = 1s → 3p by a factor of 6. These are general observations: the precision of Eqs. (3.10)–(3.15) is very difficult to estimate: it might be about a factor of 2 for the strongest resonance transition, but it might also deviate by orders of magnitude.

Understanding the large overestimate of the B_d value for the transition 1s → 3p is of particular importance and is related to the fact that the Burgess formulas take into account only one autoionizing channel. For example, the 3lnl′-configurations (which are related to the transition α₀ → α₂ = 1s → 3p in the above example) autoionize not only to the ground state but to excited states too:3lnl′→1s+eAuger2l+eAuger.(3.19)Complex numerical multiconfiguration Hartree–Fock calculations show^23,45 that the autoionization rates to the excited states “2l” are even more important than to the ground state “1s.” This reduces considerably the branching factor for DR [the B_d factor in Eq. (3.10)]. In fact, as one can see from Eq. (3.15), very similar branching factors are given for the transitions α₀ → α₁ = 1s → 2p and α₀ → α₂ = 1s → 3p because only one autoionizing channel is taken into consideration.

It is very important for the practical application of DR rates in the modeling of ionization balance to explore in more detail the influence of the various multiple channels for Auger and radiative decay. Below, we perform QMMC calculations for DR in the simplified Vainshtein approach,^38,42–44 and we fit the numerical results to an analytical expression in order to facilitate the application and allow direct comparison with the Burgess formulas:DZ+1,Z(α0→α,nl)=10−8×m2l0+1Bdβ3/2e−βχd(cm3/s),(3.20)β=Z2RykTe,(3.21)where Ry = 13.6057 eV, kT_e is the electron temperature in eV, m is the number of equivalent electrons of state α₀, Z is the charge of the ion where the core transition α₀ → α takes place (e.g., for the 2lnl′ autoionizing states of He-like argon, the core transition is the 1s → 2p transition in H-like argon, Z = 18), and l₀ is the corresponding orbital momentum of state α₀. The physical meaning of the parameter χ_d is related to the fact that all contributions from the configurations with different spectator electrons nl have to be summed up to give the total DR rate with different energies [see Eq. (2.1)]. The parameter χ_d provides a fit to the numerical results to replace the sum of different energies in the best manner through an average energy parameter χ_dβ. Finally, the total sum is replaced by an average amplitude B_d to provide a simple analytical expression without the need for summation.

As a demonstration, let us consider the various mechanisms via a study of the DR related to the core transition 2s–4p. For example, the numerical calculations in the single-channel approximation for Be atoms give Bd(1- channel)(2s−3p)=3.4×10−5, whereas Bd(1- channel)(2s−4p)=1.6×10−5; i.e., 2s–4p transitions are only reduced by a factor of about two compared with 2s–3p transitions. Numerical calculations including multichannel decay provide an entirely different picture, with Bd(6- channel)(2s−3p)=2.0×10−6, but Bd(6-channel)(2s−4p)=3.5×10−7; i.e., the QMMC numerical calculations indicate that higher-order DR rates are strongly suppressed. This is a general observation that multichannel decay can reduce DR considerably and can even lead to a quite different interpretation of its importance.

Table I shows the numerical calculation for the DR B_d factors for single- and multichannel decay into Li-like ions for different orders and elements in comparison with the standard Burgess formula. One observes that the Burgess formula is in reasonable agreement with the numerical results for single-channel decay, although it differs by up to a factor of 3 in some cases. However, comparison with the numerical calculations using the QMMC approach reveals extremely large overestimates of DR obtained by the Burgess formula. In particular for light elements, the overestimation can be by as much as one or two orders of magnitude: for example, for the DR related to the autoionizing states 1s²4lnl′ of Be, we have Bd(multichannel)(2s−4p)=3.47×10−7 and Bd(Burgess)(2s−4p)=1.10×10−5; i.e., the Burgess formula overestimates B_d by more than a factor of 30. It is therefore not recommended to calculate higher-order contributions to DR via the Burgess approach.

In Tables II–IV, we present the numerical results of the QMMC approach for H-, He-, and Li-like ions, which are the most important for K-shell spectroscopy. For ease of application, we have fitted all results to the simple analytical formula (3.20).

Table II presents the results of a numerical calculation of the total DR rate into H-like ions for the core transitions 1s–2p and 1s–3p for all elements from He (Z = 2) up to Mo (Z = 42) and the corresponding fitting parameters according to Eq. (3.20). It can be seen that for low-Z elements, the DR related to the core transition 1s–2p is dominant, whereas for large Z, the relative contribution of the DR with the core transition 1s–3p increases. The Burgess formula provides amplitudes B_d that are about a factor of 3 smaller than the present numerical calculations.

For the 3lnl′ states, the Burgess formula considerably overestimates the DR rate because it does not take into account multichannel radiative and Auger decays. This is of particular importance for low-Z elements. For example, for C, the single-channel approximation gives B_d = 6.75 × 10⁻⁵, whereas the four-channel approximation gives B_d = 6.32 × 10⁻⁶, i.e., a reduction by a factor of 10. Multichannel decay is much less important for higher Z: for example, for Fe, B_d = 5.13 × 10⁻⁶, whereas the four-channel approximation gives B_d = 2.60 × 10⁻⁶.

Table III presents numerical results for the total DR rate into He-like ions for the core transitions 1s–2p and 1s–3p for all elements from He (Z = 2) up to Mo (Z = 42) and the corresponding fitting parameters according to Eq. (3.20). It can be seen that for low-Z elements, the DR related to the core transition 1s–2p is dominant, whereas for large Z, the relative contribution of the DR with the core transition 1s–3p increases. The Burgess formula gives amplitudes B_d that are about a factor of 3 smaller than the present numerical calculations. For the 1s3lnl′ states, the Burgess formula considerably overestimates the DR rate because it does not take into account multichannel radiative and Auger decays. This is of particular importance for low-Z elements. For example, for C, the single-channel approximation gives B_d = 6.76 × 10⁻⁵, whereas the four-channel approximation gives B_d = 2.98 × 10⁻⁶, i.e., a reduction by a factor of 20. Multichannel decay is much less important for higher Z: for example, for Fe, B_d = 5.34 × 10⁻⁶, whereas the four-channel approximation gives B_d = 2.60 × 10⁻⁶.

Table IV presents numerical results for DR into Li-like ions related to the core transition 2s–2p, i.e., a Δn = 0 transition. Therefore, the fitting parameter χ_d is rather small and the associated exponential factor for DR does not vary much. In addition, the configurations 1s²2lnl′ are only autoionizing for rather high principal quantum numbers. This is quite different to the DR related to the core transition 2s–3p, where the states are autoionizing for rather low quantum numbers nl and the temperature dependence is very different owing to an order-of-magnitude difference in the χ_d factor.

Unlike DR into H- and He-like ions (Tables II and III), the DR related to the n = 3 core transition is very important compared with the 2s–2p related recombination. For this reason, the temperature dependence of the total recombination rate (which is the sum of the DR rates related to the 2s–2p, 2s–3p, …, core transitions) is complex and differs qualitatively from the rates of DR into H- and He-like ions, which are dominated by a single exponential factor.

4 IV. EXCITED-STATE COUPLING OF DIELECTRONIC RECOMBINATION IN DENSE PLASMAS

Table V shows the DR rates related to the excited states 1s²2p of Li-like ions. It can be seen from a comparison of the numerical data in Tables IV and V that the contribution from the excited states is even more important than that from the ground state. For example, for Be, B_d(2s − 3p) = 1.97 × 10⁻⁶, whereas B_d(2p − 3d) = 1.78 × 10⁻⁴ and B_d(2p − 4d) = 1.88 × 10⁻⁵. This means that the excited-state contribution is up to two orders of magnitude larger than the ground-state contribution. Therefore, even for rather moderate densities with small populations of the excited states, their contribution to DR can be important.

Particular important cases are encountered if the first excited states are related to Δn = 0 radiative transitions. Because these transition probabilities are orders of magnitude lower than those for Δn > 0 transitions, Boltzmann populations with respect to the ground state are already achieved for rather low electron densities. For example, for Be, at densities of about 10¹⁵ cm⁻³, the population of the excited state 1s²2p is more important than that of the ground state 1s²2s.⁴⁶ Therefore, all excited states of beryllium (e.g., for tokamaks at typical divertor densities) make larger contributions than the ground state.

The excited-state contribution could even be important at very low densities if the excited states are metastable states. Therefore, for heavy ions, where we encounter excited states that are close to ground states, related either by a dipole-allowed radiative transition or by multipole transitions, DR is extremely complex even at rather low densities. This is the main reason why, to date, ionic balance calculations of heavy elements differ strongly from one method to another and why DR remains an active field of research and of considerable interest for a number of applications (nuclear fusion, astrophysics, radiation sources, spectroscopic diagnostics, etc.).

In conclusion, the excited-state contribution is driven by atomic kinetics that can have a much greater impact than any other complicated effects related to ground-state contributions. This points again to the great practical importance of approximate methods, including the quasiclassical approach, that provide the possibility of obtaining numerical data even for large quantum numbers (which can be quite important for DR).

We underline that the excited-state contributions to DR up to high quantum numbers for the corresponding core transitions may exceed the ground-state contribution by many orders of magnitude. It is for this reason that it is essential to include excited-state contributions as much as possible, even if these are based on atomic structure calculations of limited precision, rather than attempting to improve via sophisticated atomic structure calculations the simplest core-transition-related DR rates while ignoring higher-order and excited-state contributions.

5 V. ANGULAR-MOMENTUM-CHANGING COLLISIONS

A strict consideration of angular-momentum-changing collisions requires a very extended atomic-level system that includes all details of the autoionizing states in order to treat properly the collisional redistribution of populations. We restrict ourselves here to a discussion of principles with the help of the most frequently employed formula for DR and proceed from dielectronic capture from channel k and with radiative transition j → i [see also Eqs. (3.3)–(3.5)]:Let us now consider a simple illustrative example, namely the Ly_α dielectronic 2l2l′ satellites of He-like ions and depict two levels, one that has very large autoionization rate and one that has a negligible one. For the first case, we consider the level j′=2p21D2, k=1s2S1/2 and the radiative transition j′=2p21D2→i′=1s2p1P1. Atomic structure calculations for carbon (Z_n = 6) deliver³⁷

For the second case, we consider the autoionizing configuration j=2p23P1, k=1s2S1/2 and the radiative transition j=2p23P1→i=1s2p3P2. Atomic structure calculations (again for Z_n = 6) deliverfrom which it follows that Qk,jiZ+1,Z=0.

Assuming a two-level system where only dielectronic capture and angular momentum changing collisions (characterized by the rate coefficient C_j′j) contribute, the atomic populations njZ and nj′Z are given bywherewith q = j, j′. In the absence of collisions, Eqs. (5.1)–(5.4) becomewhere the superscript “(0)” indicates the low-density case. As can be seen from Eq. (5.5), the low-density dielectronic intensity satellite factor [Eq. (3.5)] is reproduced from Eq. (5.2) if the angular-momentum-changing collisions are negligible, i.e., if

Note that for very closely spaced levels, ion–ion collisions might also be of importance.

To understand the effect of angular-momentum-changing collisions on the total DR rate, we need to consider the sum for the two levels, i.e.,where the subscript “coll” for the single DR rates DRcollk,jiZ+1,Z and DRcollk,j′i′Z+1,Z indicates that these rates include collisional processes. This has to be distinguished from Eq. (5.1), which is a low-density approximation. It is of principal interest to understand the change in DR due to collisions with reference to the low-density case, i.e., we consider the ratioThe collisional DR rates cannot be determined from relations like Eq. (5.5) but need to be determined directly from the populations, i.e.,because the product of the level population with the radiative decay is the rate at which the excited state decays to the ground state, which is equivalent to DR [note that the usual branching ratios that appear in formulas like Eq. (5.1) are already included via the equilibrium population] if the right-hand sides of Eqs. (5.2) and (5.3) are driven by dielectronic capture and angular-momentum-changing collisions between the autoionizing levels under consideration. Combining Eqs. (5.7) and (5.8), we obtaini.e.,Because EkjDC≈Ekj′DC, we have for the population ratio in the low-density case (for the example given above)since ΓjkZ,Z+1≪Γj′kZ,Z+1 and ΓjkZ,Z+1≪∑lAjlZ (see the above example). Therefore, population is essentially transferred by angular-momentum-changing collisions from level j′ to level j, but not vice versa. Let us now consider the above example with the populations given by Eqs. (5.2) and (5.3):andEquations (5.2), (5.3), and (5.8) indicate that for autoionizing levels with very large autoionization rates, the populations are close to the low-density case. Equation (5.4) therefore corresponds to the low-density case [Eq. (5.5)], i.e.,Substituting Eqs. (5.11), (5.12), and .(5.14) into Eq. (5.9), we obtainBecause g_j′C_j′j ≈ g_jC_jj′ for closely spaced levels, Eq. (5.15) takes the formIf ∑lAjlZ≈neCjj′, then the term in parentheses in Eq. (5.16) is about one-half, and the relation indicates that the total DR rate is enhanced (i.e., DRcolltotZ+1,Z/DRtotZ+1,Z>1) owing to angular-momentum-changing collisions. This can be understood in a transparent qualitative picture: for the level j′ with high autoionization rate, the dielectronic capture is high and, owing to the large autoionization rate, the branching factor for radiative deexcitation is small. If, however, a certain fraction of population is collisionally transferred to another level before autoionization and radiative decay disintegrate the upper level j′, then level j is effectively populated by collisions from j′ → j (because the population of level j is small since dielectronic capture is insignificant owing to a small autoionization rate). The transferred population, however, has a very favorable branching factor for level j compared with level j′. In the above example,for level j, whereasfor level j′. Therefore, the transferred population is more effectively transferred to the ground state to finally contribute to DR.

The impact of angular-momentum-changing collisions on the total DR rate is difficult to observe in dense plasmas, because the impact of such collisions is only indirect, namely, via the change in ionization balance, where other recombination processes (e.g., three-body recombination and radiative recombination) also come into play. However, angular-momentum-changing collisions can be directly observed via the collisional induced change in the spectral distribution of the corresponding dielectronic satellite transitions.

Figure 2 demonstrates this effect via He-like Ly_α transitions and their associated satellite transitions 2lnl′ → 1s2l + hν. Numerical calculations of the spectral distribution have been carried out using the MARIA suite of codes,^10,47–49 taking into account an extended level structure, with LSJ-split levels of different ionization stages for ground, single, and multiple excited states being simultaneously included. Strong density effects are indicated by red arrows in Fig. 2. Not only do the 2l2l′ satellites show strong density effects near λ ≈ 0.853 nm, but so do the 2l3l′ satellites near λ ≈ 0.847 nm. The density sensitivity of the 2l3l′ satellites starts at lower densities, because the collisional rates between the 2l3l′ configurations are in general larger than those for the 2l2l′ configurations [the collisional rates C(2lnl′–2lnl″) increase with principal quantum number n, the corresponding radiative rates A(2lnl″–1snl″) are almost independent of n, and the autoionization rates Γ(2lnl″–1s) decrease with n]. Also indicated in Fig. 2 is the so-called “blue satellite” emission located on the blue wing of the Ly_α resonance line. These satellite transitions exhibit negative screening⁷ that is due to the strong effect of angular-momentum coupling (F states). [Note that the term “negative screening” arises from use of the Bohr formulato match the actual energy E via a screening constant σ: in cases where the match can only be obtained for effective charges Z_eff = Z_n − σ, the screening constant is negative; see also Ref. 12.] As can be seen from Fig. 2, angular-momentum-changing collisions have little effect on blue satellites (because their autoionization rates are quite large).

Fig. 2. MARIA simulations of dielectronic satellite emission near Ly_α of H-like Mg ions for different values of the electron density at kT_e = 100 eV. The red arrows indicate the rises in intensity of particular satellite transitions with increasing density. Satellites indicated in blue have effective negative screening due to strong angular-momentum coupling effects.

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In dense-plasma spectroscopy, the perturbation of the spectral dielectronic satellite distribution due to angular-momentum-changing collisions is employed for density diagnostics.^5–7 Note that although the satellite intensity shows variations with temperature too, density determination via dielectronic satellite transitions is essentially not a two-parameter problem: the temperature determination relies on the intensities of satellite transitions (with high autoionization rates) relative to the resonance line, while the density determination is based on the relative intensities of dielectronic satellites among the transitions themselves (with low and high autoionization rates). The density diagnostic is usually rather independent of the electron temperature, because all cross sections of angular-momentum-changing collisions are typically in the Born limit (the energy differences are typically much smaller than the electron temperature).

6 VI. ELECTRIC FIELD EFFECTS

6.1 A. Ionization potential depression of spectator electron orbitals

Atomic population kinetics of gases and plasmas has been applied very successfully to the study of low-density environments, where atoms and ions are essentially field-free. As the density increases, however, the free-atom model breaks down, resulting in a perturbation of the atomic energy levels. This perturbation manifests itself essentially in a broadening and a shift. Such perturbations can be observed in high-resolution spectroscopic experiments via analysis of the line broadening, the line shift, and the disappearance of the line emission corresponding to the ionization potential depression (IPD) of the upper level. The IPD is of great interest for applications in thermodynamics and also for the understanding of various radiative properties, such as emission, absorption, and scattering (for further reading on current developments in IPD research with spectroscopic precision, the interested reader is referred to Refs. 50 and 51 and references therein).

The fundamental origin of the perturbation of energy levels is the plasma electric microfield, which in turn also limits the number of bound states. Electric field ionization starts at a critical field strength F_crit given by^39,52Fcrit≈6.8×108Zeff3nF4[V/cm],(6.1)where Z_eff is the effective ion charge and n_F is the principal quantum number at which field ionization starts. To estimate the limited number of quantum states that effectively take part in the recombination process, we identify the critical field strength F_crit with the mean field strength, which can be expressed in terms of the Holtsmark normal mean field strength F₀ (Z_P is the perturber charge and N_P is the perturber density),^52aFmean≈3.41F0=3.4×4π4152/3Ryea0ZPNP2/3≈1.3×10−6ZPNP2/3[cm−3][V/cm],(6.2)and we identify the principal quantum number n_F with the maximum principal quantum number of the spectator electron nl of the autoionizing configuration, i.e.,nF≈nmaxspectator≈4.8×103Zeff3/4ZP1/4NP1/6(cm−3).(6.3)nmaxspectator limits the contribution of high-n spectator electrons according to Eq. (3.17) [and therefore effectively limits the B_d values in Eqs. (3.14), (3.15), and .(3.20)]. Equation (6.3) is useful to understand the various effects that limit DR (apart from the electric field effect, kinetic effects, for example, may likewise limit the effective rate coefficient; see below). Note, however, that Eq. (6.3) itself provides only a rough estimate of the principal quantum number of the spectator electrons due to the electric field effect, since (a) it neglects the fact that usually up to five perturber charge states are present in a dense plasma, (b) the field strength distribution is only taken into account via the mean field value, and (c) charged-particle correlations that influence the field strength distribution itself are ignored.

Effective DR is not only limited by the maximum principal quantum number nmaxspectator from Eq. (6.3), but also by collisional disintegration of the autoionizing level: in fact, in a dense plasma, not only does the high density of charged particles result in a high microfield, but the high density also implies a significant rate of collisional processes. For autoionizing states, we need to consider two stages: a first stage involving collisional processes associated with core relaxation and a second stage following core relaxation. Concerning the first stage, we need to compare the radiative stabilization rate of the autoionizing configuration (which is the origin of effective recombination) with the collisional ionization rate of the spectator electron (which is the origin of collisional disintegration of the autoionizing level). Therefore, effective DR requiresArad(α→α0)≫neIn,(6.4)where A_rad(α → α₀) is the radiative stabilization rate from Eqs. (3.10) and (3.20) (note that this rate is distinctly different from the radiative transition rate of the spectator electron itself), n_e is the electron density, and I_n is the electron collisional ionization rate coefficient of the spectator electron with principal quantum number n. Note that for autoionizing states, the stabilization rate is almost independent of the high-n spectator electron, which is distinctly different from the scaling law for singly excited states (approximately A ∝ 1/n³). If the radiative stabilization is associated with a change in principal quantum number, then the H-like approximation can be used to estimate A_rad(α → α₀):Aradα→α0,Δn>0≈1.57×1010Zeff4nα0nα3(nα2−nα02)[s−1],(6.5)where Z_eff is the effective ion charge. If A_rad(α → α₀) is associated with a transition without change in principal quantum number, i.e., Δn = 0, then numerical Hartree–Fock calculations including angular-momentum coupling are required to obtain reasonable estimates of the transition probability. Numerical calculations indicate that for Li-like and Be-like ions, transition probabilities and oscillator strengths can be roughly estimated from the expressionsArad(1s22p2P→1s22s2S)≈1.2×108(Zn−3)(s−1),f(1s22s2S→1s22p2P)≈1.2/(Zn−2),Arad(1s22s2p1P→1s22s21S)≈8×108(Zn−3)[s−1],f(1s22s21S→1s22s2p1P)≈4.5/(Zn−2).The ionization rate of the spectator electron can be estimated from the following expression (note that there are many competing formulas available in the literature,²⁶ but for the present estimates, the simple Lotz-like formula appears to provide a good balance between complexity and precision):In≈6×10−8RyEn3/2βne−βn⁡ln1+0.562+1.4βnβn(1+1.4βn)[cm3/s],(6.6)where Ry = 13.6056 eV andβn=EnkTe.(6.7)E_n is the ionization potential of the spectator electron, which can be approximated byEn≈(Zeff−1)2Ryn2.(6.8)

It is important to point out that the condition (6.4) is necessary for DR to be effective, but it is not sufficient. After radiative stabilization α → α₀ of the core, we need to radiatively stabilize the spectator electron nl to the ground state “gr” (if it disappears in the continuum, no recombination is encountered), i.e., we must haveArad(nl→gr)≫neIn.(6.9)For spectator electrons, Δn = 0 transitions are not relevant and an expression similar to Eq. (6.5) describes all cases of practical interest:Arad(nl→gr,Δn>0)≈1.57×1010(Zeff−1)4ngrn3(n2−ngr2)[s−1].(6.10)Whether condition (6.4) or (6.9) is more stringent depends on the type of core transition. Let us consider as an example a magnetically confined deuterium fusion plasma containing Li-like iron impurities. Assuming Z_P = 2, N_P = 10¹³ cm⁻³, n_e = 10¹⁴ cm⁻³, and kT_e = 10 keV, we obtain from Eq. (6.3) with Z_eff = 24 a maximum principal quantum number of spectator electrons of about nmaxspectator≈300. For Li-like iron, DR is associated with the core transition α₀ → α = 1s²2s²S → 1s²2p²P with A_rad(1s²2p²P → 1s²2s²S) ≈ 3.5 × 10⁹ s⁻¹ and f(1s²2s²S → 1s²2p²P) ≈ 0.066. As the relevant principal quantum numbers are of the order of 100, we haveβn=EnkTe=(24−1)2×13.6104n2≈10−4,and we can approximate the β_n-dependent terms in Eq. (6.6) roughly by 0.1 and solve Eq. (6.4) for the critical principal quantum number of the spectator electron ncritspectator [note that the critical principal quantum number is obtained from Eq. (6.4) with equality sign]: ncritspectator≈(7×1021/ne)1/3≈400.

Let us now investigate the condition (6.9). For Li-like iron, n_gr = 2 and Eq. (6.10) provides roughly A_rad(nl → gr, Δn > 0) ≈ 2 × 10¹⁵/n⁵ [s⁻¹]. Taking into account the n dependence of A_rad(nl → gr, Δn > 0), we can solve Eq. (6.9) for the critical principal quantum number ncritspectator [note that the critical principal quantum number is obtained from Eq. (6.10) with equality sign] of the spectator electron: ncritspectator≈(4×1023/ne)1/8≈16. Therefore, in the above example, the condition (6.10) is much more stringent than conditions (6.3) and (6.4).

Let us now consider the above example for parameters typical of inertial fusion plasmas. We assume that iron is employed as a diagnostic tracer element in a compressed dense plasma. Assuming Z_P = 2, N_P = 10²³ cm⁻³, n_e = 10²⁵ cm⁻³, and kT_e = 10 keV, we obtain from Eq. (6.3) with Z_eff = 24 a maximum principal quantum number of the spectator electron of about nmaxspectator≈6. As before, α₀ → α = 1s²2s²S → 1s²2p²P with A_rad(1s²2p²P → 1s²2s²S) ≈ 3.5 × 10⁹ s⁻¹ and f(1s²2s²S → 1s²2p²P) ≈ 0.066. As the relevant principal quantum numbers are of the order of 10, we haveβn=EnkTe=(24−1)2×13.6104n2≈10−2,and we can approximate the β_n-dependent terms in Eq. (6.6) roughly by 0.4 and solve Eq. (6.4) for the critical principal quantum number of the spectator electron ncritspectator [note that the critical principal quantum number is obtained from Eq. (6.4) with equality sign]: ncritspectator≈(2×1021/ne)1/3<1. Therefore, DR associated with the Li-like core transition α₀ → α = 1s²2s²S → 1s²2p²P is ineffective. The essential physical reason is related to the low transition probability for the Δn = 0 core transition. Therefore, total recombination rates are effectively related to core transitions involving the K shell because, in this case, the radiative decay rates are very large owing to the strong Z scaling [see Eq. (6.5)]: assuming Z_eff = 24 as before, but taking n_α0 = 1 and n_α = 2 in Eq. (6.5), we have A_rad(α → α₀, Δn > 0) ≈ 1.1 × 10¹⁴ s⁻¹, i.e., a value five orders of magnitude higher compared with the Δn = 0 core transition discussed above. In this case, K-electron involvement in a core transition allows a few principal quantum numbers of the order of 1 to survive and to contribute effectively to DR.

What is the conclusion from the above estimates? They indicate that it might not be appropriate to adopt a purely atomic structure point of view to obtain convergence of the sum of DR rates over a large range of principal quantum numbers of the spectator electron nl in Eq. (3.17): owing to microfields and collisional–radiative competition, large principal quantum numbers might not effectively contribute to the total DR rate. Therefore, total DR rates calculated with rather approximate methods (quasiclassical methods or Vainshtein’s simplified QMMC method) but taking into account the plasma microfield and collisional processes might be more accurate than purely adopting sophisticated atomic structure calculations.

In view of these results, we now address the influence of the plasma microfield itself on the autoionization rates.

6.2 B. Perturbed autoionization rates

The influence of the electric field on the autoionization and corresponding DR rates was first studied in Refs. 53–55 in the context of the simplest atomic system of He-like autoionizing states 2l2l′. It was realized that forbidden autoionizing processes (i.e., forbidden in the LS-coupling scheme) become allowed by electric field mixing of autoionizing bound-state wavefunctions. The allowed autoionization width is given by the first-order transition rateΓ(d→c)=2πℏdVc2δ(Ed−Ec),(6.11)where V is the electrostatic interaction. Because V is a scalar operator, the autoionization vanishes unless there are available adjacent continuum states c with the same angular momentum and parity as those of the discrete levels d.³² Because of the absence of even-parity P states below the second ionization threshold, the 2p²³P-state of He-like ions is metastable against autoionization decay. In the presence of perturbing electric fields, however, autoionization of the state a = 2p²³P may occur by a second-order process involving a field-induced transition to the nearby autoionizing state d = 2s2p³P. In a quasistatic ion field, the field-induced autoionization rate is given byΓ(a→c)=2πℏ|∑daQ⃗⋅E⃗ddVc(Ea−Ed)+iℏ(Γd+Ad)/2|2δ(Ea−Ec),(6.12)where Q⃗ is the electric dipole moment operator, and Γ_d and A_d are respectively the autoionization and radiative widths of the state d. Therefore, the first-order contribution from the field-induced transition decays directly into the nonresonant continuum c = 1sεp³P.

It should be noted that for practical applications, not only field-induced transitions have to be considered, but intermediate coupling, configuration, and magnetic interactions too. In particular for highly charged ions, these “non-electric-field effects” may make a considerable contribution to the forbidden autoionization width, as is demonstrated by the results in Table VI, which have been calculated using the FAC code.⁵⁶ In addition, the Breit interaction^56a induces an autoionization rate for the 2p²³P₁ state.

Table VI also illustrates the general effect that if the nuclear charge increases, then the autoionization widths are more evenly distributed over the levels. Therefore, electric field effects are best studied for low-Z elements. The table also demonstrates that autoionization rates are strongly dependent on LSJ quantum numbers: therefore, simple summations over l quantum numbers might be a quite inappropriate way to simplify complex atomic structures in kinetics.⁴⁸

From the relationship between the corresponding capture and autoionization rates, it follows that the electric field can induce DR through normally inaccessible high-angular-momentum states that have large statistical weights.^54,55 In fact, in a plasma, the angular momentum l is no longer a good quantum number, because the presence of an electric field destroys the spherical symmetry. However, the projection L_Z of L⃗, which generates the magnetic quantum number m, defined with respect to the direction of the electric field, remains a good quantum number. For nonzero quantum numbers m, this results in a twofold degeneracy of the outer electrons in addition to the twofold degeneracy due to spin. The appropriate transformation of the field-free substates l has the formnλm=∑l=mn−1nlmnlmnλm,(6.13)where the electric quantum number λ, which replaces l in the presence of the electric field, can take integer values in the range λ = 0, …, n − |m| − 1. Calculations^54,55,57,58 demonstrate that the dependence of the autoionization rates on the quantum number λ is rather smooth, in contrast to the field-free case, where the autoionization rates decrease rapidly with quantum number l. For this reason, dielectronic capture increases in the presence of an electric field, because it is proportional to the autoionization rate and the statistical weight: DCk,jZ+1,Z∝gjZΓjkZ,Z+1. Because the DR rate is proportional to dielectronic capture rate [see Eq. (2.7)], this results in a considerable increase in the total DR rate. For example, for the autoionizing states 1s²2pnl in Be-like Fe²²⁺, an approximately threefold increase in the DR rate was found even for densities as low as 10¹⁴ cm⁻³.⁵⁴ This dramatic increase at rather low densities is connected in particular with the fact that for the 1s²2pnl configuration, the resonance spontaneous transition probability 2s–2p is not very large and high-n states have autoionization rates larger than radiative decay rates for n quantum numbers up to about 100. Consequently, high-n states contribute considerably to the DR rate. As high-n states are likewise strongly affected by rather small electric fields, a considerable impact on the total recombination rate is encountered even for rather low plasma densities (being of importance for typical densities of the solar corona or magnetic fusion plasmas).

Interaction with an electric field makes atomic structure calculations extremely complex, and it is difficult to derive general conclusions. However, it has been demonstrated^57,58 that the quasiclassical approach combined with a transformation to parabolic quantum numbers [Eq. (6.13)] gives results that are in surprisingly good agreement with those of extremely complex numerical calculations.⁵⁹ Moreover, the quasiclassical approach combined with the transformation to parabolic quantum numbers^59a,59b enables the derivation of a closed-form expression for the autoionization rate in an electric field:Γ(n,λ,m)=∫lminlmaxP(nl;λm)Γ(nl)dl,(6.14)withlmin2=12{[(n−1)2+m2−λ2]−[(n−1)2+m2−λ2]2−4(n−1)2m2}(6.15)andlmax2=12{[(n−1)2+m2−λ2]+[(n−1)2+m2−λ2]2−4(n−1)2m2},(6.16)where Γ(nl) is the standard autoionization rate in spherical polar coordinates (which is independent of m owing to spherical symmetry) and P(nl; λm) is a joint probability (with normalization equal to unity) for the appearance of spherical (nl) and parabolic (nλm) quantum numbers that can be expressed in terms of Clebsch–Gordan coefficients. For large quantum numbers and the condition m < l « n (the quasiclassical limit of Clebsch–Gordan coefficients that is of practical interest), the joint probability can be approximated by⁵⁸P(nl;λm)≈1π2l(l2−lmin2)(lmax2−l2).(6.17)Substituting quasiclassical values for the autoionization rate^{31,32,36,38,42}Γ(nl) into Eq. (6.14) and using Eq. (6.13), we obtain an autoionization rate in parabolic quantum numbers expressed in terms of universal functions (t = l/l_eff, leff=(3Z2/ω)1/3):Γ(n,λ,m)=fijπn3I(tmin,tmax),(6.18)I(tmin,tmax)≈2lmax3Z2ω2/3Ylmin(ω/3Z2)1/3,(6.19)Y(x)≈0.284⁡exp(−2x3),(6.20)where f_ij is the oscillator strength of the core transition with charge Z (e.g., the oscillator strength corresponding to the transition 1s–2p in H-like Al for the He-like 2lnl′ satellites, Z = 13). The expressions (6.14)–(6.20) demonstrate similarly a broad distribution over the electric quantum number λ that finally results in an increase of the DR rate.