Photoabsorption and photoionization of the 1s² ground state of He-like ions near 2s2p ¹P₁ autoionizing resonance

Photoionization cross-sections are important to analyse spectra emitted by laboratory or astrophysical plasmas. Many spectroscopic data have already been collected in atomic banks. In particular, the Opacity Project bank [1] contains data derived from close-coupling wavefunctions computed by the non-relativistic R-matrix code [2]. The most important feature, of the close-coupling method, is to give a precise description of collisional resonances, due to autoionization states interacting with continuum, the so-called interference effect or Fano profile [3]. Unfortunately, for highly ionized ions, the resonance lifetime becomes strongly affected by spontaneous emission: there is a competition between autoionization and radiative emission. To include these supplementary effects in the close-coupling formalism, two different approaches have been proposed in the literature called radiative damping theories. The first one, introduced by Davies and Seaton [4], below DS, is based on the traditional time-dependent perturbation theory (e.g. 'Quantum Mechanics' by Bransden and Joachim [5]): close-coupling wavefunctions are not affected by radiation. While, for the second one, proposed by Robicheaux et al, below RGPB [6], an optical imaginary potential is used to insert radiative effects directly in the R-matrix code.

The inverse process to photoionization is photo-recombination. In hot plasmas, recombination is usually more important than photoionization, cold electrons being more abundant than energetic photons. It is traditional to separate photo-recombination data in two non-interfering parts: (i) radiative recombination, corresponding only to the continuum contribution; (ii) dielectronic recombination, corresponding to the autoionizing resonance contribution. To obtain dielectronic recombination data, the most used formalism is time-independent perturbation theory, which includes two perturbations: electron–electron interaction describing autoionization, and radiative interaction describing spontaneous emission. The zero-order Hamiltonian considers autoionizing states as ordinary bound states. Corresponding wavefunctions can be computed by some atomic structure codes. In particular, for He- and Li-like ions, recombination data are essential to study the so-called satellite lines, mainly due to dielectronic recombination [7–9].

In the case of photoionization, only a few initial bound states are important, e.g. ground and metastable levels. While in the case of photo-recombination, the final bound states can be numerous, e.g. large n Rydberg levels. Bell and Seaton [10] proposed a theory to deal with such a situation. It combines DS theory to quantum defect theory [11]. The scattering matrix used in collision theory is transformed to include radiative damping, and is therefore no longer unitary. This method has a further interest, it can be extended easily to include radiative damping for electron excitation processes involving large n [12]. In particular, this method has been used for He-like excitations [13].

The original R-matrix code was non-relativistic [2]. It was well adapted to the first degrees of ionization, where autoionization is important but radiative effects are negligible. Recent versions of the R-matrix code [14] include relativistic effects and can extend the possibilities of the close-coupling wavefunction approach to highly ionized atoms. Autoionization is still very important, but radiative effects become comparable: damping effects should therefore be included, either by DS or RGPB theories. DS theory is apparently more difficult to use: it requires the computation of principal value (PV) integrals containing poles, which might be solved by analytical fitting of computed data [15–17]. RGPB theory turns off the difficulty, but all close-coupling computations have to be restarted from scratch. In this paper, we show that the PV problem is not unsolvable numerically, as long as data have been computed on a fine grid. We also describe a very simple approximation, satisfying the unitarity condition of DS theory, which gives approximate results, but still a lot better than original undamped data. Such approximation does not require any PV integral calculations. It can easily be applied to already published data computed without damping, e.g. [1, 18].

Davies and Seaton [4] assume that they can generalize the scattering matrix S_scatt, used in electron collision theory, to a 'electron–photon' S matrix, by introducing 'photon channels': the number of incoming particles, either electrons or photons, must be balanced by the number of outcoming particles, i.e. the unitarity of the S matrix. The DS theory cannot be used for bound states because no electron enters or exits, but it can be used for autoionizing levels.

The DS electron–photon S matrix contains four sub-matrices: S_{p − p}, S_{p − e}, S_{e − p}, S_{e − e}:

$$\begin{equation} {\bf S} = \left(\begin{array}{@{}cc@{}}\bf S_{p-p} {\bf S}_{p-e} \\ {\bf S}_{e-p} {\bf S}_{e-e} \end{array} \right) , \end{equation} \tag{ 2.1 }$$

where S_{p − p} and S_{e − e} are squares, and S_{p − e} and S_{e − p} are rectangular matrices. S_{e − p} and S_{p − p} are related to photon absorption followed either by electron or photon emissions. From [15], below STB, equations (2), (3), (6), (7) and using equation (A.8) from the appendix, we obtain

$$\begin{eqnarray} &&\begin{array}{l} \dsty{\bf S}_{p-p} = ({1}-{\bf L}_{p-p})({1}+{\bf L}_{p-p})^{-1}= {\bf S}_{p-p}^{t} \\\ms \dsty{\bf S}_{p-e} = -2{\rm i}\pi ({1}+{\bf L}_{p-p})^{-1} {\bf D}_{e-p}^{\dag } = -{\bf S}_{e-p}^{t} \\\ms \dsty{\bf S}_{e-p} = 2{\rm i}\pi {\bf D}_{e-p}^*({1}+{\bf L}_{p-p})^{-1} = -{\bf S}_{p-e}^{t} \\\ms \dsty{\bf S}_{e-e} = [ {\bf S}_{{\rm scatt}} + 2\pi ^2 {\bf D}_{e-p}^* ({1}+{\bf L}_{p-p})^{-1} {\bf D}_{e-p}^{\dag } ] = {\bf S}_{e-e}^{t}, \end{array} \end{eqnarray} \tag{ 2.2 }$$

where

$$\begin{equation} {\bf L}_{p-p}(E)= -{\rm i}\pi \lim _{\tau \rightarrow 0} \int {{\bf D}_{e-p}^{\dag }(E^{\prime }) {\bf D}_{e-p}(E^{\prime }) \over E^{\prime }-E-{\rm i}\tau }\, {\rm d}E^{\prime } . \end{equation} \tag{ 2.3 }$$

In the appendix, D_{e − p} is defined, (A.6). In particular, equation (A.9) shows that D†_{e − p} D_{e − p} is a real and symmetric matrix. L_{p − p} is then complex and symmetric, i.e. L_{p − p} = L^t_{p − p}. In (2.3), the L_{p − p} matrix can be separated into real and imaginary parts:

$$\begin{eqnarray} &&\fl {\bf L}_{p-p}(E) = {\bf L}^{{\mathcal {\rm R}}{\rm e}}_{p-p}(E)+ {\rm i} {\bf L}^{{\rm \mathcal{I}m}}_{p-p}(E):\nonumber\\&&\ms \fl {\rm where}\quad {\bf L}^{{\rm \mathcal{R}e}}_{p-p}(E) = \pi ^2 {\bf D}_{e-p}^{\dag }(E) {\bf D}_{e-p}(E); \nonumber \\&&\ms \fl {\rm and}\quad {\bf L}^{{\rm \mathcal{I}m}}_{p-p}(E)=-\pi \left[ {\rm PV}{\displaystyle \int {{\bf D}_{e-p}^{\dag }(E^{\prime }) {\bf D}_{e-p}(E^{\prime }) \over E^{\prime } -E}} {\rm d}E^{\prime } \right]\!. \end{eqnarray} \tag{ 2.4 }$$

The Cauchy principal value integral (PV) can be written as follows:

$$\begin{eqnarray} &&\fl {\bf L}^{{\rm \mathcal{I}m}}_{p-p}(E)\nonumber\\ &&\fl {=}{-}\,\pi\! \lim _{R\rightarrow \infty } \int _{E-R}^{E+R} {{\bf D}_{e-p}^{\dag }(E^{\prime }) {\bf D}_{e-p}(E^{\prime })-{\bf D}_{e-p}^{\dag }(E) {\bf D}_{e-p}(E) \over E^{\prime } -E} {\rm d}E^{\prime }\!. \nonumber\\ \end{eqnarray} \tag{ 2.5 }$$

Due to the fact that the integrand is now continuous, the integral can be carried out numerically. From former equations, DS has proved that the electron–photon S matrix is unitary:

$$\begin{equation} {\bf S}^{\dag } {\bf S} ={\bf S} {\bf S}^{\dag }={1} . \end{equation} \tag{ 2.6 }$$

When D_{e − p} is small, i.e. L_{p − p} ≃ 0, the damping effect becomes negligible. One may consider the 'first-order' (i.e. undamped) approximation S⁽¹⁾, from equation (2.2):

$$\begin{equation} \begin{array}{l} \dsty {\bf S}^{(1)}_{p-p}={1}_{p-p},\quad {\bf S}^{(1)}_{p-e}=-2{\rm i}\pi {\bf D}_{e-p}^{\dag },\\\ms \dsty {\bf S}^{(1)}_{e-p}= 2{\rm i}\pi {\bf D}_{e-p}^*,\quad {\bf S}^{(1)}_{e-e}={\bf S}_{{\rm scatt}}. \end{array} \end{equation} \tag{ 2.7 }$$

S⁽¹⁾ is not a unitary matrix, while the scattering matrix, S_scatt, is. To obtain unitarity (2.6), the simplest method is to replace, in (2.2), L_{p − p} by its real part:

$$\begin{equation} {\bf L}^{({\rm ZD})}_{p-p}(E)=\pi ^2 {\bf D}_{e-p}^{\dag }(E) {\bf D}_{e-p}(E) , \end{equation} \tag{ 2.8 }$$

i.e. to neglect ${\bf L}^{{\rm \mathcal{I}m}}_{p-p}$, Cauchy principal value integrals, (2.5). In the next sections, it will be called the ZD approximation.

The following He-like ions are considered: Fe^{24 +}, Ca^{18 +} and Si^{14 +}. For each of them, two photoionization cross-sections (1s² and 1s2s ¹S₀ states) have been computed by Mendoza [19], using the Breit–Pauli R-matrix code (without damping effects) [14], available at http://amdpp.phys.strath.ac.uk/tamoc/codes/serial/asy-damp/.

The 2s2p ¹P₁ resonance interacts only with a one-electron channel, 1s + _1sp, where _1s is the energy of the photoionized electron and p represents, l = 1. In the same website, a code can also be found to compute photoionization cross-sections with damping effects, using RGPB damping theory. In section 3.1, we consider only photoionization of 1s2s ¹S₀, i.e. the one-photon channel. In section 3.2, we consider both photoionization processes, i.e. two-photon channels.

The original 'undamped' photoionization cross-sections for 1s² and 1s2s ¹S₀ (Q₁(E^R) and Q₂(E^R)) are given in Megabarn (1 Mb= 10⁻¹⁸ cm²), and energy E^R is in Rydberg (Ryd) (E^R = 2E = 2_1s). They can be converted to area atomic units a²₀ (a₀ = 0.529 18 × 10⁻⁸ cm). For simplicity, j = 1 corresponds to 1s² and j = 2 to 1s2s. Photoionization of 1s² and 1s2s ¹S₀ (J' = 0, even) gives only J = 1, odd states, according to the electric dipole selection rule. Using STB (19), we obtain

$$\begin{eqnarray} &&\fl {\displaystyle {2\alpha ^2\omega _j^2\over \pi } {2J^{\prime }+1\over 2J+1}} Q_j^{(a.u)}(E^R) = \big|\big(S^{(1)}_{e-p}\big)_{1 j}(E)\big|^2\nonumber\\ &&\hphantom{\fl {\displaystyle {2\alpha ^2\omega _j^2\over \pi } {2J^{\prime }+1\over 2J+1}} Q_j^{(u.a)}(E^R)}=4\pi ^2 ({\bf D}_{e-p}^{\dag }(E) {\bf D}_{e-p}(E))_{j j} , \end{eqnarray} \tag{ 3.1 }$$

where α ≃ 1/137 is the fine-structure constant, ω_j are photon energies in atomic unit (au): for Fe^{24 +}, ω₁ = _1s + 324.84 (au) and ω₂ = _1s + 79.5 (au). From (2.4)

$$\begin{equation} \big({\bf L}^{{\rm \mathcal{R}e}}_{p-p}\big)_{j j}= \case{1} {4} \big|\big(S^{(1)}_{e-p}\big)_{1 j}\big|^2 . \end{equation} \tag{ 3.2 }$$

The reactance matrix K has only one term, K₁₁, a real number (see the appendix). Defining δ such that K₁₁ = tan δ, it gives, (A.8):

$$\begin{eqnarray*} &&\fl ({\bf D}_{e-p})_j = 2{\rm i} {\big({\bf D}^{(K)}_{e-p}\big)_j \over 1+ {\rm i} \tan \delta } \nonumber\\ &&\hphantom{\fl ({\bf D}_{e-p})_j}= 2{\rm i} \exp (-{\rm i} \delta ) \cos \delta \big({\bf D}^{(K)}_{e-p}\big)_j \quad (j=1,2), \end{eqnarray*}$$

i.e.

$$\begin{equation} ({\bf D}_{e-p})_j = d_j \exp \big({-}{\rm i} \big(\delta - \case{1}{2}\pi \big)\big) , \end{equation} \tag{ 3.3 }$$

where d_j is a real (positive or negative) number. Therefore, from (2.4)

$$\begin{equation} \big({\bf L}^{{\rm \mathcal{R}e}}_{p-p}\big)_{i j} = \pi ^2 d_i d_j . \end{equation} \tag{ 3.4 }$$

Using (3.2), we obtain d²₁, d²₂. For a given energy E, we may choose any sign for d₁, d₂, e.g. both positive, only by changing the sign of 1s or 2s radial functions. For other energies, d₁ and d₂ being continuous and differentiable functions of E, their signs are defined (see figure 6(a)).

3.1. One-electron channel and one-photon channel

Here, as Sakimoto et al [15], we consider 1s2s ¹S₀ photoionization for Fe^{24 +}, i.e. the one-photon channel only. It corresponds to j = 2, defined before. For simplicity, we use the following notations : ${\bf L}^{{\rm \mathcal{R}e}}={\bf L}^{{\rm \mathcal{R}e}}_{p-p}$ and ${\bf L}^{{\rm \mathcal{I}m}}= {\bf L}^{{\rm \mathcal{I}m}}_{p-p}$.

From the undamped photoionization cross-section, we obtain directly ${\bf L}^{{\rm \mathcal{R}e}}_{2 2}$. Using (2.2),

$$\begin{eqnarray} &&\fl \big|\big({\bf S}^{(1)}_{e-p}\big)_{1 2}\big|^2= 4 a^2 \quad {\rm and } \quad |({\bf S}_{e-p})_{1 2}|^2= {4 a^2\over (1+ a^2)^2 +b^2} , \nonumber\\ \end{eqnarray} \tag{ 3.5 }$$

where

$$\begin{equation*} a^2={\bf L}^{{\rm \mathcal{R}e}}_{2 2}=\pi ^2 d^2_2 \qquad {\rm and} \qquad b={\bf L}^{{\rm \mathcal{I}m}}_{2 2}. \end{equation*}$$

Since a² and b² are positive numbers, we obtain

$$\begin{equation} \big|\big({\bf S}_{e-p}\big)_{1 2}\big|^2 \le \big|\big({\bf S}^{(1)}_{e-p} \big)_{1 2}\big|^2, \end{equation} \tag{ 3.6 }$$

which means for the one-photon channel, a 'damped' |S_{e − p}(E)|² is always smaller than the corresponding 'undamped' |S⁽¹⁾_{e − p}|².

As pointed out in [15], the function ${\bf L}^{{\rm \mathcal{R}e}}_{2 2}$, shown in figure 1, has almost a Lorentzian profile. It is possible by fitting ${\bf L}^{{\rm \mathcal{R}e}}_{2 2}$ to an analytical expression to obtain easily $\big({\bf L}^{{\rm \mathcal{I}m}}_{2 2}\big)^2$. From (2.3) we obtain, using a residue theorem,

$$\begin{equation} {\bf L}^{{\rm \mathcal{I}m}}_{2 2}= {\bf L}^{{\rm \mathcal{R}e}}_{2 2} {2 \big(E^R-E^R_a\big)\over \Gamma _a} , \end{equation} \tag{ 3.7 }$$

where E_a = 351.295 27 Ryd and Γ_a = 0.0094 Ryd. ${\bf L}^{{\rm \mathcal{I}m}}_{2 2}$ can be computed directly from (2.5). In figure 2, we present both computed (solid line) and analytical (dashed line) results. One can see that they are almost identical.

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We define a damping factor for 1s2s photoionization as follows: $\mathcal{D}(1{\rm s}2{\rm s})=(1+ a^2)^2 +b^2$. In figure 3(a), we plot $\mathcal{D}(1{\rm s}2{\rm s})$ (solid line), compared to ZD approximation $\mathcal{D}^{{\rm ZD}}(1{\rm s}2{\rm s})=(1+ a^2)^2$ (dashed line), obtained by neglecting b². At the top, there is no difference between both curves, but differences appear in the wings. To see them better, in figure 3(b), we plot the ratio (1 + a²)²/(1 + a²)² + b². The value of ratio is 1, for E^R = E_a, and it tends to 1, for the smallest and larger values of E^R. In figure 3(b) there are two minima. They appear in figure 4 as two maxima. The separation between two minima is ΔE^R = 0.015 Ryd. In figure 4, we plot undamped |(S⁽¹⁾_{e − p})₁₂|² and damped |(S_{e − p})₁₂|², as well as approximate damped |(S^(ZD)_{e − p})₁₂|² defined as

$$\begin{eqnarray} &&\fl \big|\big({\bf S}^{({\rm ZD})}_{e-p}\big)_{1 2}\big|^2= {4 a^2\over (1+ a^2)^2};\nonumber\\ &&\fl \big|\big({\bf S}_{e-p}\big)_{1 2}\big|^2 \le \big|\big({\bf S}^{({\rm ZD})}_{e-p}\big)_{1 2}\big|^2 \le \big|\big({\bf S}^{(1)}_{e-p}\big)_{1 2}\big|^2 . \end{eqnarray} \tag{ 3.8 }$$

The two maxima are separated by ΔE^R, which requires a photon resolution of ΔE^R/(2ℏω₂) = 3.0× 10⁻⁵. In the case when ion Fe^{24 +} is in a plasma, we must include Doppler broadening: for an ion temperature of 1 keV, ΔE^R/(2ℏω₂) = 3.2× 10⁻⁴: the two maxima are mixed as a broaden single peak. To better estimate differences in figure 4, we compare the following numerical integrals, I, I⁽¹⁾ and I^(ZD). They are defined as follows:

$$\begin{eqnarray} &&I^{(1)}={\int }_{351.15}^{351.45} {\rm d}E^R \big|\big(S^{(1)}_{e-p}\big)_{1 2}\big|^2=0.0807, \nonumber \\ &&I={\int }_{351.15}^{351.45} {\rm d}E^R |(S_{e-p})_{1 2}|^2=0.0326,\nonumber\\ &&I^{({\rm ZD})}={\int }_{351.15}^{351.45} {\rm d}E^R \big|\big(S^{({\rm ZD})}_{e-p}\big)_{1 2}\big|^2=0.0359 \end{eqnarray} \tag{ 3.9 }$$

We see that I⁽¹⁾/I = 2.47 and I⁽¹⁾/I^(ZD)=2.25, i.e. ZD approximation includes 9% less damping than the damping expected by DS theory.

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3.2. One-electron channel and two-photon channels

We now consider both photoionizations of 1s² and 1s2s ¹S₀, i.e. two-photon channels. From (2.2), we obtain

$$\begin{eqnarray} &&\fl (({\bf S}_{e-p})_{1 1} ({\bf S}_{e-p})_{1 2})=2{\rm i}\pi (({\bf D}^*_{e-p})_{1 1} ({\bf D}^*_{e-p})_{1 2}) \nonumber\\ \quad\qquad\times\left(\begin{array}{@{}cc@{}}1+{\bf L}_{1 1} & {\bf L}_{1 2} \\ {\bf L}_{1 2} & 1+{\bf L}_{2 2} \end{array}\right)^{-1} \nonumber\\ &&\fl =2{\rm i}\pi (({\bf D}^*_{e-p})_{1 1} ({\bf D}^*_{e-p})_{1 2}) {\displaystyle {1\over (1+{\bf L}_{1 1})(1+{\bf L}_{2 2})- {\bf L}_{1 2}^2}} \nonumber\\ \quad\qquad\times\left(\begin{array}{@{}cc@{}}1+{\bf L}_{2 2} &-{\bf L}_{1 2} \\ -{\bf L}_{1 2} & 1+{\bf L}_{1 1}\end{array}\right) . \end{eqnarray} \tag{ 3.10 }$$

${\bf L}^{{\rm \mathcal{R}e}}_{2 2}$ and ${\bf L}^{{\rm \mathcal{I}m}}_{2 2}$ have already been presented in figures 1 and 2. ${\bf L}^{{\rm \mathcal{R}e}}_{1 1}$ and ${\bf L}^{{\rm \mathcal{I}m}}_{1 1}$ curves are plotted in figure 5. For comparison with ${\bf L}^{{\rm \mathcal{R}e}}_{1 1}$ curve (solid line), we also plot the continuum contribution curve (dotted line), represented as a straight line between E^R = 351.15 and 351.45 Ryd. By subtracting this contribution from ${\bf L}^{{\rm \mathcal{R}e}}_{1 1}$, one can estimate the positive and negative contributions of the resonance to photoionization (see later). In figure 6, ${\bf L}^{{\rm \mathcal{R}e}}_{1 2}$ and ${\bf L}^{{\rm \mathcal{I}m}}_{1 2}$ are plotted. Here, ${\bf L}^{{\rm \mathcal{R}e}}_{1 2}$ changes sign, because d₁ has changed sign (${\bf L}^{{\rm \mathcal{R}e}}_{1 2}=\pi ^2\,d_1 d_2$). The maxima of $|{\bf L}^{{\rm \mathcal{R}e}}_{2 2}|$ and $|{\bf L}^{{\rm \mathcal{I}m}}_{2 2}|$ are about 1 (figures 1 and 2); the maxima of $|{\bf L}^{{\rm \mathcal{R}e}}_{1 1}|$ and $|{\bf L}^{{\rm \mathcal{I}m}}_{1 1}|$ are about 5.0× 10⁻⁴ (figure 5); and the maxima of $|{\bf L}^{{\rm \mathcal{R}e}}_{1 2}|$ and $|{\bf L}^{{\rm \mathcal{I}m}}_{1 2}|$ are about 2.0× 10⁻² (figure 6). We can use these orders of magnitude to derive the following formula:

$$\begin{eqnarray} &&|({\bf S}_{e-p})_{1 1}|^2= 4 {{\bf L}^{{\rm \mathcal{R}e}}_{1 1}+\pi ^2\big(d_1 {\bf L}^{{\rm \mathcal{I}m}}_{2 2} -d_2 {\bf L}^{{\rm \mathcal{I}m}}_{1 2}\big)^2 \over \big(1+{\bf L}^{{\rm \mathcal{R}e}}_{2 2}\big)^2 +\big({\bf L}^{{\rm \mathcal{I}m}}_{2 2}\big)^2 }\nonumber\\ &&\hphantom{|({\bf S}_{e-p})_{1 1}|^2}=4 {c^2+f^2 \over (1+a^2)^2+b^2}, \end{eqnarray} \tag{ 3.11 }$$

where

$$\begin{eqnarray} &&\fl c^2={\bf L}^{{\rm \mathcal{R}e}}_{1 1}=\pi ^2 d_1^2\quad {\rm and}\quad f^2=\pi ^2\big(d_1 {\bf L}^{{\rm \mathcal{I}m}}_{2 2} -d_2 {\bf L}^{{\rm \mathcal{I}m}}_{1 2}\big)^2. \nonumber\\ \end{eqnarray} \tag{ 3.12 }$$

In figure 7(a), f² is plotted versus energy E^R. We fit it into 0.000 21× a², and the agreement is excellent. One can rewrite (3.12)

$$\begin{equation} |({\bf S}_{e-p})_{1 1}|^2=4 {c^2 \over (1+a^2)^2+b^2}+ 4 {f^2 \over (1+a^2)^2+b^2} . \end{equation} \tag{ 3.13 }$$

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It is similar to two independent contributions to 1s² photoionization: (i) c² contains the direct effect giving the famous Fano profile, and this direct process is independent from 2s2p resonance decaying to 1s2s; (ii) f² = 0.000 21 × a², which contains the indirect dependance from 2s2p resonance decaying to 1s2s. Both processes have the same damping factor, i.e. $\mathcal{D}(1{\rm s}2{\rm s})$. In figure 7(b), two ratios are plotted: c²/(c² + f²) and f²/(c² + f²). They cross for a ratio = 0.5, the corresponding width is ΔE^R = 0.013 Ryd, which is smaller than the Doppler width, which is ten times larger. In figure 8, undamped (RPGB) data |(S⁽¹⁾_{e − p})₁₁|², DS damped results |(S_{e − p})₁₁|² calculated in this paper and RPGB damped data |(S^(RGPB)_{e − p})₁₁|² are plotted. We represent RPGB data as stars to distinguish them from DS data (solid line). Also in figure 8, the continuum contribution, |S_c|², and the ZD approximation, |(S^(ZD)_{e − p})₁₁|², are shown. The ZD approximation consists of neglecting b² and f²:

$$\begin{equation} \big|\big({\bf S}^{({\rm ZD})}_{e-p}\big)_{1 1}\big|^2=4 {c^2 \over (1+a^2)^2} . \end{equation} \tag{ 3.14 }$$

We can compare quantitatively, the importance of the damping process by subtraction of the continuum contribution,|S_c|² (a straight line, which is tangent to |(S⁽¹⁾_{e − p})₁₁|² at E = 351.15 and 351.45 Ryd), see figure 5. This difference, positive or negative, can be separated into two parts which are integrated separately, but for positive integrand:

$$\begin{eqnarray*} &&\fl I^{(1)}_+= {\int }_{351.15}^{351.45} {\rm d}E^R \big(\big|\big(S^{(1)}_{e-p}\big)_{1 2}\big|^2-|S_c|^2\big)^{(\ge 0)}=2.78(-5)\\&&\ms \fl I^{(1)}_-= {\int }_{351.15}^{351.45} {\rm d}E^R \big(|S_c|^2-\big|\big(S^{(1)}_{e-p}\big)_{1 2}\big|^2\big)^{(\ge 0)}=2.08(-5); \\&&\ms \fl I_+={\int }_{351.15}^{351.45} {\rm d}E^R \big(\big|\big(S_{e-p}\big)_{1 2}\big|^2-|S_c|^2\big)^{(\ge 0)}=1.34(-5)\\&&\ms \fl I_-= {\int }_{351.15}^{351.45} {\rm d}E^R \big(|S_c|^2-\big|\big(S_{e-p}\big)_{1 2}\big|^2\big)^{(\ge 0)}=2.14(-5);\\&&\ms \fl I^{({\rm ZD})}_+={\int }_{351.15}^{351.45} {\rm d}E^R \big(\big|\big(S^{({\rm ZD})}_{e-p}\big)_{1 2}\big|^2-|S_c|^2\big)^{(\ge 0)}=1.37(-5)\hspace{170.71652pt}\\&&\ms \fl I^{({\rm ZD})}_-= {\int }_{351.15}^{351.45} {\rm d}E^R \big(|S_c|^2-\big|\big(S^{({\rm ZD})}_{e-p}\big)_{1 2}\big|^2\big)^{(\ge 0)}=2.48(-5), \end{eqnarray*}$$

where, for example, 2.48( − 5) = 2.48× 10⁻⁵. We see that the integral of the undamped positive part, I⁽¹⁾₊, is strongly decreased by damping, I⁽¹⁾₊/I₊ = 2.07, whereas the integral of the undamped negative part, I⁽¹⁾₋, is slightly increased by damping, I⁽¹⁾₋/I₋ = 0.97. The present approximation (ZD) reproduces well the damping of the positive part I⁽¹⁾₊/I^(ZD)₊ = 2.02, but overestimates the increase of the negative part I⁽¹⁾₋/I^(ZD)₋ = 0.84. We also see that undamped (I⁽¹⁾₊ − I⁽¹⁾₋) > 0, whereas both (I₊ − I₋) < 0 and (I^(ZD)₊ − I^(ZD)₋) < 0. It means that 1s² photoionization has been partly destroyed by 2s2p ¹P₁ resonance, because it decays more to 1s2s than to (1s + _1sp).

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3.3. Comparison results of RGPB theory and ZD approximation, for S^{14 +}, Ca^{18 +}

As damping effects increase with Z, it is interesting to study other atomic elements of helium isoelectronic sequence. More particularly, it is well known that radiative probabilities depend on Z⁴, while the autoionization probability stays almost constant. For Fe^{24 +}, radiative damping decreases the photoionization of 1s2s by a factor of 2.5. We expect these decrease effects to be smaller for Ca^{18 +} and even less for Si^{14 +}. For these two He-like ions, we have carried out a similar study to those in subsections 3.1 and 3.2. As for Fe^{24 +}, we find that DS calculations and RGPB data are almost identical. Then, in figures 9 and 10, we compare only RGPB data with ZD data. For Ca (Z = 20) and S (Z = 14), we see only a single peak for 1s2s photoionization. Indeed, the two peaks appear only for |(S⁽¹⁾_{e − p})₂₂|² > 4, i.e. Z ⩾ 24. The difference between RGPB (solid line) and ZD (dashed line) is very small in comparison with undamped results (dash–dotted line). Therefore, most damping effects are taken into account by ZD approximation for photoionization of 1s2s and 1s². As for Fe^{24 +}, in the case of 1s² photoionization, we can compare the integrals I₊, I₋, I⁽¹⁾₊ and I⁽¹⁾₋, after subtraction of the continuum contribution (dotted line). By opposition to Fe^{24 +}, the 2s2p resonance gives a positive contribution, but strongly decreased, compared to undamped ones: Si^{14 +}, (I⁽¹⁾₊ − I⁽¹⁾₋)/(I₊ − I₋) = 1.96; Ca^{18 +}, (I⁽¹⁾₊ − I⁽¹⁾₋)/(I₊ − I₋) = 4.54. We also obtain for Si^{14 +}, I⁽¹⁾₊/I₊ = 1.21, I⁽¹⁾₋/I₋ = 0.97; and for Ca^{18 +}, I⁽¹⁾₊/I₊ = 1.42, I⁽¹⁾₋/I₋ = 0.97. The value I⁽¹⁾₋/I₋ = 0.97 apparently does not change with Z, which means the accuracy is not good enough to be able to see any variation of this quantity.

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Photoabsorption of 1s² corresponds to the sum of two processes: (i) photoionization (1s² + ℏω₁ → 1s + _1sp), |(S_{e − p})₁₁|²; (ii) photon transfer (1s² + ℏω₁ → 1s2s + ℏω₂), |(S_{p − p})₂₁|².

The first matrix element (i) was already presented, see (3.11). The second one (ii) is a matrix element of S_{p − p}, defined in (2.2):

$$\begin{eqnarray} \fl {\bf S}_{p-p} =\left(\begin{array}{@{}cc@{}}1-{\bf L}_{1 1} & -{\bf L}_{1 2} \\ -{\bf L}_{1 2} & 1-{\bf L}_{2 2} \end{array}\right) {\displaystyle {1\over (1+{\bf L}_{1 1})(1+{\bf L}_{2 2})- {\bf L}_{1 2}^2}} \nonumber\\\ms \qquad\quad\times\left(\begin{array}{@{}cc@{}}1+{\bf L}_{2 2} & -{\bf L}_{1 2} \\ -{\bf L}_{1 2} & 1+{\bf L}_{1 1} \end{array}\right). \end{eqnarray} \tag{ 4.1 }$$

Using orders of magnitude, already discussed for (3.11), we obtain

$$\begin{eqnarray} &&\fl ({\bf S}_{p-p})_{2 1} = {\displaystyle {-2 \big({\bf L}^{{\rm \mathcal{R}e}}_{1 2} + {\rm i} {\bf L}^{{\rm \mathcal{I}m}}_{1 2}\big)\over \big(1+{\bf L}^{{\rm \mathcal{R}e}}_{2 2}\big)+ {\rm i} {\bf L}^{{\rm \mathcal{I}m}}_{2 2}}}; \nonumber \\ &&\fl |({\bf S}_{p-p})_{2 1}|^2= 4 {\displaystyle {g^2+h^2\over (1+a^2)^2 + b^2}}\nonumber\\ &&\fl {\rm where }\quad g^2= \big({\bf L}^{{\rm \mathcal{R}e}}_{1 2}\big)^2=a^2 c^2 {\rm \quad and\quad } h^2=\big({\bf L}^{{\rm \mathcal{I}m}}_{1 2}\big)^2 \end{eqnarray} \tag{ 4.2 }$$

and the ZD expression is

$$\begin{equation*} \big|\big({\bf S}^{({\rm ZD}\big)}_{p-p})_{2 1}\big|^2= 4 {\displaystyle {a^2 c^2\over (1+a^2)^2}}. \end{equation*}$$

Of course, the undamped value is |(S⁽¹⁾_{p − p})₂₁|² = 0. In figures 11, 12 and 13, results (i) of photon transfer: ω₁ → ω₂ and results (ii) of photoabsorption by S^{16 +}, Ca^{20 +} and Fe^{24 +} are presented. The continuum contribution |S_c|² (dotted lines) is given for comparison purpose. For photon transfer, we see that the ZD approximation includes almost 40% of the DS results.

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In this paper, we try to formulate clearly Davies and Seaton's (DS) radiative damping theory. We applied accurately DS theory to photoionization computations of He-like ions, (Z = 16, 20, 26), near 2s2p ¹P₁ autoionizing resonance. For Z = 26, we found the same results as those computed using Robicheaux et al damping theory (RGPB). We also present a new approximation (ZD) of DS theory, requiring almost no calculations. The analysis of numerical results shows that ZD approximation has accuracy better than 90% for 1s2s photoionization and slightly less for 1s². We also compare results obtained by ZD with RGPB damped data for (Z = 16, 20). Finally, we extend DS theory to the calculation of photoabsorption data.

The authors thank Dr Claudio Mendoza for helpful discussions and computing undamped and damped photoionization data obtained by RGPB theory.

In the close-coupling approximation, the (N + 1)-electron wavefunctions, ψ_γ(JE), are projected on N-electron target wavefunctions, coupled to the angular momenta of the projectile electron, $\phi _{\gamma ^{\prime \prime }}$. The projection coefficients are the projectile radial functions. They are solutions of n (coupled) radial equations with real coefficients, with n being the number of channels (called electron channels in this paper), γ''. There are n independent solutions of the total (N + 1)-electron system, γ = 1, ..., n. The total energy is conserved: $E=E_{\gamma ^{\prime \prime }}+k_{\gamma ^{\prime \prime }}^2/2$. Solutions of these radial equations can be chosen as either real or complex. The solutions, ψ^(K)_γ, with real radial functions, satisfy the following asymptotic expression:

$$\begin{eqnarray} &&\fl \displaystyle \psi ^{(K)}_{\gamma }(E)= \mathcal{A}\sum _{\gamma ^{\prime \prime }}\phi _{\gamma ^{\prime \prime }} F^{(K)}_{\gamma ^{\prime \prime } \gamma }(r),\nonumber\\ &&\fl \displaystyle F^{(K)}_{\gamma ^{\prime \prime } \gamma }(r)\mathop {\sim }_{r\rightarrow \infty } {(2\pi k_{\gamma ^{\prime \prime }})^{-1/2}\over r} \lbrace \sin (k_{\gamma ^{\prime \prime }}r+\cdots )\delta (\gamma ^{\prime \prime },\gamma ) \nonumber\\ &&\qquad \quad +\cos (k_{\gamma ^{\prime \prime }}r+\cdots ) K(\gamma ^{\prime \prime },\gamma )\rbrace \end{eqnarray} \tag{ A.1 }$$

where $\bf K$ is called the reactance matrix, (n × n). It is real and symmetric: $\bf K= \bf K^t$. $\mathcal{A}$ is the antisymmetrization operator. There are two possible choices for complex radial functions: either the boundary condition corresponds to recombination (incoming electron in channel γ):

$$\begin{eqnarray} &&\fl \displaystyle \psi ^{(+)}_{\gamma }(E) = \mathcal{A}\sum _{\gamma ^{\prime \prime }}\phi _{\gamma ^{\prime \prime }} F^{(+)}_{\gamma ^{\prime \prime } \gamma }(r)\nonumber \\ &&\fl \displaystyle F^{(+)}_{\gamma ^{\prime \prime } \gamma }(r)\mathop {\sim }_{r\rightarrow \infty } {(2\pi k_{\gamma ^{\prime \prime }})^{-1/2}\over r} \big\lbrace \exp (-{\rm i}k_{\gamma ^{\prime \prime }}r+\cdots )\delta (\gamma ^{\prime \prime },\gamma ) \nonumber\\ &&\qquad \quad -\exp (+{\rm i}k_{\gamma ^{\prime \prime }}r+\cdots ) S_{{\rm scatt}}(\gamma ^{\prime \prime },\gamma )\big\rbrace \end{eqnarray} \tag{ A.2 }$$

or the boundary condition corresponds to photoionization (outgoing electron in channel γ):

$$\begin{eqnarray} &&\fl \displaystyle \psi ^{(-)}_{\gamma }(E) = \mathcal{A}\sum _{\gamma ^{\prime \prime }}\phi _{\gamma ^{\prime \prime }} F^{(-)}_{\gamma ^{\prime \prime } \gamma }(r) \nonumber \\ &&\fl \displaystyle F^{(-)}_{\gamma ^{\prime \prime } \gamma }(r)\mathop {\sim }_{r\rightarrow \infty } {(2\pi k_{\gamma ^{\prime \prime }})^{-1/2}\over r} \lbrace \exp (+{\rm i}k_{\gamma ^{\prime \prime }}r+\cdots )\delta (\gamma ^{\prime \prime },\gamma ) \nonumber\\ &&\qquad \quad -\exp (-{\rm i}k_{\gamma ^{\prime \prime }}r+\cdots ) S_{{\rm scatt}}^{\dag }(\gamma ^{\prime \prime },\gamma )\rbrace \end{eqnarray} \tag{ A.3 }$$

where S_scatt is the scattering matrix. It is unitary, symmetric and related to K:

$$\begin{eqnarray} &&\fl {\bf S}_{{\rm scatt}}=({1}+{\rm i}{\bf K}) ({1}-{\rm i}{\bf K})^{-1} ; \quad {\bf S}_{{\rm scatt}} {\bf S}_{{\rm scatt}}^{\dag }= {\bf S}_{{\rm scatt}}^{\dag }{\bf S}_{{\rm scatt}}=1 ;\nonumber\\ &&\qquad \quad {\bf S}_{{\rm scatt}}={\bf S}^t_{{\rm scatt}} . \end{eqnarray} \tag{ A.4 }$$

Relations between the solutions Ψ^(K), Ψ^{( + )}, Ψ^{( − )} (row matrices) are

$$\begin{eqnarray} &&\fl \begin{array}{l} \Psi ^{(+)}= -2{\rm i} \Psi ^{(K)} ({1}-{\rm i}{\bf K})^{-1} ; \quad {\bf F}^{(-)}=({\bf F}^{(+)})^* , \\\ms \Psi ^{(-)}= +2{\rm i} \Psi ^{(K)} ({1}+{\rm i}{\bf K})^{-1} ; \quad \Psi ^{(-)}=-\Psi ^{(+)} {\bf S}_{{\rm scatt}}^{\dag } . \end{array} \end{eqnarray} \tag{ A.5 }$$

According to STB (4), the definition of D_{e − p} (matrix element) is

$$\begin{eqnarray} &&\fl D_{e-p} (J \gamma , J^{\prime } \gamma ^{\prime }; E) = {1 \over \sqrt{2J+1}}\left[{2\omega ^3_{\gamma ^{\prime }} \alpha ^3\over 3\pi }\right]^{1/2}\nonumber\\ &&\qquad \quad\times \big(\Psi ^{(+)}_{\gamma } J E||{\bf R}|| \Psi ^{({\rm bound})}_{\gamma ^{\prime }} J^{\prime } \big) , \end{eqnarray} \tag{ A.6 }$$

where $\omega _{\gamma ^{\prime }}$ is the photon energy (atomic unit) and α is the fine-structure constant. D_{e − p}(Jγ, J'γ'; E) is a complex number only because the radial functions of Ψ^{( + )}_γ are complex (phase factor choice). Using instead Ψ^(K)_γ, we obtain now a real quantity:

$$\begin{eqnarray} &&\fl D^{(K)}_{e-p} (J \gamma , J^{\prime } \gamma ^{\prime }; E) = {1 \over \sqrt{2J+1}}\left[{2\omega ^3_{\gamma ^{\prime }} \alpha ^3\over 3\pi }\right]^{1/2}\nonumber\\ &&\qquad \quad\times \big(\Psi ^{(K)}_{\gamma } J E||{\bf R}|| \Psi ^{({\rm bound})}_{\gamma ^{\prime }} J^{\prime } \big) . \end{eqnarray} \tag{ A.7 }$$

Inserting (A.5) in (A.6) and (A.7), we have

$$\begin{equation} {\bf D}_{e-p}=2{\rm i} ({1}+{\rm i}{\bf K})^{-1} {\bf D}^{(K)}_{e-p} ; \quad {\bf D}^*_{e-p}=-{\bf S}_{{\rm scatt}}{\bf D}_{e-p} \end{equation} \tag{ A.8 }$$

and

$$\begin{equation} {\bf D}^{\dag }_{e-p}{\bf D}_{e-p}=4 \big({\bf D}^{(K)}_{e-p}\big)^t ({1}+{\bf K}^2)^{-1} {\bf D}^{(K)}_{e-p} , \end{equation} \tag{ A.9 }$$

which is a real and symmetric matrix.