# A Model of Calculating Radiative Opacities of Hot Dense Plasmas Based on the Density-Functional Theory

Authors: Shuji Kiyokawa
Journal: Journal of Atomic and Molecular Physics
 (2014)

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## Abstract

We determine the radiative opacity of plasmas in a local thermal equilibrium (LTE) by time-dependent density-functional theory (TDDFT) including autoionization resonances, where the photoabsorption cross section is calculated for an ion embedded in the plasma using the detailed configuration accounting (DCA) method. The abundance of ion with integer occupation numbers is determined by means of the finite temperature density-functional theory (FTDFT). For an Al plasma of temperatureT=20 eV and density 0.01 g/cm3, we show the opacity and the photoabsorption cross section of b-f and b-b transitions with Doppler and Stark width, and also show a result that the Planck and Rosseland mean opacities are 28,348 cm2/g and 4,279 cm2/g, respectively.


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## Body

## 1. Introduction

For investigation of hot dense plasmas, the density-functional theory has been used to calculate their atomic properties and has provided reliable data such as electronic structure, equation of state (EOS), and opacity [1–7]. Particularly, the study of radiative properties of inertial confinement plasmas, interior of stars, and so on is important and theoretically great interest for the reason that the thermal properties and the electronic ones of plasmas are closely correlated with each other.The most popular model for the hot dense plasmas is the average atom (AA) model [5, 8–11], and it has been employed vigorously to study the opacity [12] and so on. However, as an actual LTE plasma is composed of various ions in different excited states and charge states; the spectral structure of LTE plasma is very complex because of the enormous number of transition lines. The method of the supertransition array (STA) [7, 12–14] has been used to analyze such a complex line spectrum of an ion in a LTE plasma.For the dense plasmas, autoionization is an intrinsically crucial atomic process and is important for treatments of plasma opacity, but the autoionization and the ion-ion pair distribution function are not treated in calculations of the opacities by STA. One of methods of calculating the autoionization in the dense plasmas is the time-dependent density-functional theory which is treated the autoionization resonance as the dynamical linear response of electronic system.To calculate the opacity of plasmas, we have considered the time-dependent density-functional theory (TDDFT) to treat the photoabsorption cross section of plasmas, where the autoionization process is included without using any other code [15].In this method, LTE plasmas are treated by finite temperature density-functional theory (FTDFT) [16, 17] and all the calculations are carried out within the framework of density-functional theory (DFT). The method is fast and stable for the numerical calculation of autoionization resonance. The resonance energy point obtained by our method is found to be equal to the difference of two related orbital energies shifted by an amount due to the relaxation effects of electrons in the time-dependent external electric field.In Section2, we review the formulation of LTE plasmas by FTDFT [16, 17] in brief. In Section 3, a model of the calculation of the photoabsorption cross section of LTE plasmas is shown by TDDFT, where ions in the plasmas are “real ion” with integer numbers of bound electrons. The opacity of an Al plasma is shown in Section 4, and we compare our results with other experimental and theoretical results. We conclude with a short summary in Section 5.


## 2. A Self-Consistent Model of Plasmas: FTDFT

Int this section, we review the finite temperature density functional theory for plasmas (FTDFT) [15, 18–20], briefly. We consider a plasma containing Nn nuclei of nuclear charge Za and NnZa electrons in a volume Vsys. The system is in thermal equilibrium with temperature T (in energy unit).The Hamiltonian of the system is as follows (hereafter, we use atomic units):(1)Hsys=∑j=1NnPj22M+∑i>j(Za)2|Ri-Rj|+He,He=∑j(-12∂2∂rj2)+∑i>j1|ri-rj|-∑i,jZa|Ri-rj|,
where M, P are nuclear mass, momentum of a nucleus, respectively, and R and r denote the positions of nucleus and electron, respectively. We assumed here that the nuclear motion can be treated classically, while the electron subsystem obeys the quantum mechanics. We also assumed the plasma consists of average ions and uniform continuum electrons, where the electronic structure of the average ion is determined by the Kohn-Sham equation as follows:
(2)1r2ddr(r2dRnldr)+2(εnl-veff(r)-l(l+1)2r2)Rnl=0Rnl(∞)=0,
where n, l are principle quantum number and angular momentum of electron and veff(r) is the effective potential at finite temperature including the exchange-correlation potential:(3)veff(r)=-Zar+∫ρb¯(r′)-Qs(r′)|r-r′|dr′‍+vxc[ρb¯(r)+ρc]-vxc[ρc],
where ρc is the continuum electron density,
(4)ρc=2π2∫0∞‍εeβ(ε-μe)+1dε,
and ρb¯(r) is the bound electron density of the average ion,(5)ρb¯(r)=14π∑nl‍2(2l+1)exp⁡[β(εnl-μe)]+1Rnl(r)2.
The chemical potential μe of electron subsystem is determined by ensuring the charge neutrality of the system(6)Za=∑nl‍2(2l+1)exp⁡[β(εnl-μe)]+1+ρcρion,
where ρion is the ion density of the plasma. The function Qs(r) in (3) is the charge density, composed of three parts (Figure 1): (i) the charge density Zaρiongii(r) of the other nucleus crowded around the ion located at the origin of the coordinate system, (ii) the bound electrons density of these nucleus, and (iii) the uniform background electron charge density -ρc, as follows:
(7)Qs(r)=Zaρiongii(r)-ρion∫ρb¯(r-r′)gii(r′)dr′-ρc.
The function gii(r) is the radial distribution function for the average ion.Figure 1
The image of the charge densityQs(r) of (7). In this figure, the black circle • shows a nucleus located at the origin of the coordinate system. “Ion distribution” means the sum of the first and second terms of the right-hand side in (7).The radial distribution functiongii(r) is obtained by hypernetted chain (HNC) approximation [21] as follows:
(8)log⁡[gii(r)]=-βϕ(r)+ρion∫C(|r-r′|)Gii(r′)dr′,
where functions Gii(r)≡gii-1 and C(r) are the direct correlation function, and these functions must satisfy the next Ornstein-Zernike’s relation:
(9)Gii(r)=C(r)+ρion∫C(|r-r′|)Gii(r′)dr′.
The function ϕ(r) in (8), the effective interatomic potential, is the electrostatic interaction between two average ions separated distance R, and is approximately calculated as follows:(10)ϕ(r)=Za2r-2∫‍Zaρb¯(r′)|r-r′|dr′+∬ρb¯(r1)ρb¯(r2)|r+r1-r2|dr1dr2,
where the polarization of continuum electrons is neglected.


## 3. A Modeling of Absorption Cross Section: TDDFT

In our previous theory [15] of the photoabsorption cross section for plasmas by means of time-dependent density functional theory (TDDFT) [22–25], we employed the average ion model of calculating the electronic structure of ions in the plasma, where all ions have the same electronic structure. In this section, we extend our previous formula for the photoabsorption cross section to apply in the case of the photoabsorption by real ions in the plasma.


### 3.1. Probability of Existence of an Ion

When plasmas are in LTE, electronic state of ion in the plasma will be in every possible state, that is, the number of bound electrons in an orbital of the ion fluctuates around the average value. The electronic configuration of average atom model is a virtual image obtained by a time average for these deviated states. In our presented model, these average values of occupation are calculated by means of FTDFT as mentioned in Section2.The relaxation timeτ of this fluctuation will estimate roughly by the mean time between two-electron or ion-electron collisions. For the spatial distribution of surrounding an ion (given by (8)), the relaxation time of its fluctuation will roughly be equal to the mean collision time between ions, and it is very large compared to the electron collision time. On the other hand, the photoabsorption occurs in very short time compared with that fluctuation time τ. Therefore, electronic state of ions going to absorb photon is in one of deviated states from the average atom, and this ion will be in electric field caused by deviation of spatial distribution of ions. Then, we must prepare a deviated electronic state, that is, a electronic configuration with integer occupation for an initial state of ion, but we here assume that the ion spatial distribution is given by (8) without fluctuation, and other ions around an ion at origin are replaced with average atoms obtained by FTDFT (so-called the constant density method [12]).In the calculation of the opacity of LTE plasmas, it is necessary to know the probability distribution of excited electronic state of an ion with integer occupation, where the plasma effects (the ionization potential depression (IPD [26]), pressure ionization, etc.) must be correctly taken into account. To consider this probability distribution, we calculate the energy of an ion with integer occupation numbers of bound electrons embedded in the plasma, where the electrostatic potential surrounding that ion is determined by continuum electrons, the spatial distribution of neighboring ions, and their electronic structure. Therefore, the IPD is included in calculation of opacity.For a given electron configuration(11)(1s)q1(2s)q2(2p)q3(3s)q4(3p)q5(3d)q6⋯
with integer occupation numbers (q1,q2,q3,q4,…), there are many terms with different energies in general; however, we assume that the average energy of an electron configuration of an ion [27] can be calculated as follows:
(12)E{q}=∑i‍qiεi-12∬‍ρ{q}(r)ρ{q}(r′)|r-r′|drdr′-34∫‍ρ{q}(r)vx[ρ{q}(r)]dr,
where vx[ρ{q}(r)] and ρ{q}(r) are the exchange potential and the bound electron density, respectively;
(13)vx[ρ{q}(r)]=-(3πρ{q}(r))1/3,(14)ρ{q}(r)=14π∑i‍qiRi(r)2,{q} denotes a set of occupation numbers (q1,q2,q3,…), and εi and Ri are the bound electron energy and radial wave function of orbital i obtained by the Kohn-Sham equation which is obtained by replacing the potential veff(r) in (2) by the following potential:(15)V{q}(r)=-Zar+∫‍ρ{q}(r′)-Qs(r′)|r-r′|dr′+vx[ρ{q}(r)].The probabilityP{q} of existence for the ion (nuclear charge Za) with an electronic configuration of integer occupation numbers {q} is given as the change in the total energy of the ion. The probability P{q} of existence for the ion with an electron configuration {q} is
(16)P{q}=1ZGion∏i(giqi)exp⁡[-β(E{q}-μe∑i‍qi)],
where ZGion is a normalization factor, li is the angular momentum, and gi=2(2li+1) is the maximum occupation number of the orbital i [15].


### 3.2. Photoabsorption Cross Section

We now discuss the photoresponse of an ion in plasmas to a frequency-dependent external field. The electronic configuration of this ion is assumed to be(1s)q12sq2(2p)q3(3s)q4⋯. If the incident photon energy ω (in atomic units) is in the vicinity of the difference between two orbital energies εnili and εnflf (|li-lf|=1), that is, ω~εnflf-εnili with εnflf-εnili>|εnflf|, one of the (nflf) electrons can attain an energy above the ionization energy via the following successive excitation processes:
(17)(nili)qi(nflf)qf⟶(nili)qi-1(nflf)qf+1⟶(nili)qi(nflf)qf-1(εl)1;
consequently, the (nflf) electron can be ejected. Another process can coincide with this process; that is, one of the (nflf) electrons can directly absorb the photon energy ω:
(18)(nili)qi(nflf)qf⟶(nili)qi(nflf)qf-1(εl)1.
These two processes compete with each other; namely, a channel mixing occurs, and consequently, the Fano profile appears on the photoabsorption cross section. If the energies of the εnili and the εnflf do not satisfy the condition εnflf-εnili>|εnflf|, the channel mixing cannot occur in our calculation, but a direct bound-bound transition
(19)(nili)qi(nflf)qf⟶(nili)qi-1(nflf)qf+1
can occur if the photon energy satisfies the resonance condition ħω=εnflf-εnili. To investigate the above processes, we study how the electrons respond to a frequency-dependent external field.


#### 3.2.1. Photoabsorption Cross Section of an Ion:σ(ω:{q})

In previous theory [15], we considered only bound-free contribution to the calculation of the photoabsorption cross section, but here we extend the theory to include the bound-bound absorption.We assume that the frequency-dependent external electric field is of magnitudeE0 directed along the z-axis. The interaction between this field and the electrons of an ion is ∫‍ρb(r)ϕext(r,ω)eiωtdr, where ρb(r) is the bound electron density, and
(20)ϕext(r,ω)=E04π3rY10(r^)≡E0ϕωext(r)Y10(r^).
The induced charge density δρω(r) in the presence of the external field ϕωext(r) is calculated by means of the single particle response function χω0(r,r′) [22–24] as follows:
(21)δρω(r)=∫0∞χω0(r,r′)ϕωscf(r′)r′2dr′,
where ϕωscf(r) is the self-consistent field produced by the photoresponse of the electrons. The self-consistent field ϕωscf(r) is determined by the following relations:
(22)ϕωscf(r)=ϕωext(r)+ϕωind(r),
where ϕωind(r) is the frequency-dependent induced potential
(23)ϕωind(r)=∫0∞(4π3r<r>2+δvx[ρ{q}(r)]δρ|ρ(r′)δ(r-r′)r′2)×δρω(r′)r′2dr′≡∫0∞K(r,r′)δρω(r′)dr′.‍
In the above equation, the 2nd term in the integrand is a functional derivative of the exchange potential vx[ρ(r)]. From (13), the functional derivative of exchange potential vx[ρ(r)] is in proportion to ρ(r)-2/3(24)∫δvx[ρ]δρ|ρ{q}(r′)δ(r-r′)r′2δρω(r′)r′2dr′=vx[ρ{q}(r)]3ρ{q}(r)δρω(r).When a photon energyω is in the vicinity of the energy deference of optically allowed two- bound states nili and nflf, namely, ω~εf-εi≡Δε, the response function χω0(r,r′) in (21) is given as follows:
(25)χω0(r,r′)≃2fi(1-ff)ClilfηifRnili(r)Rnflf(r)×Rnflf(r′)Rnili(r′)+∑s<s′′2fs(1-fs′)Clsls′ηs,s′Rs(r)Rs′(r)1-11fs′×Rs′(r′)Rs(r′)-iπ∑s∑l=ls±12fs(1-f(εs+ω))Clsl×Rs(r)R(εs+ω)l(r)R(εs+ω)l(r′)×Rs(r′)θ(εs+ω),
where fs=qs/[2(2ls+1)], θ(εs+ω) is the unit step function, and Cll′, ηnl,n′l′ are given as
(26)Cll′={l+l′+18πfor|l-l′|=10for|l-l′|≠1,(27)ηif=1ω-Δε+iδ-1ω+Δε+iδ,Δε=εf-εi.
In (25), summations for s are over only occupied orbitals, as distinct from χω0(r,r′) for finite temperature where summation is over all occupied and unoccupied orbitals [15]. In the second term on the right-hand side of (25), the double summation runs over all bound states on the condition εs<εs′, and the prime denotes the omission of two bound states (nili) and (nflf).In (27), the factor δ is a infinitesimal positive quantity. The factor ηs,s′ in this term is real value and given as follows:(28)ηs,s′=1εs-εs′+ω-1εs′-εs+ω.TheREl(r) in (25) is the radial wave function of a photoionized electron of positive energy E and angular momentum l, which is obtained by solving the Kohn-Sham equation.


#### 3.2.2. Bound-Free Absorption

We rewrite the first term and the sum in the second term of (25) to a single sum as follows:
(29)∑k=0,1,2,…kmax⁡ckηkUk(r)Uk(r′),
where the term of k=0 in above sum corresponds to the first term of (25) and the factors c0, η0 and the function U0(r) are given as follows:
(30)c0=2fi(1-ff)Clilf,η0=ηif,U0(r)=Rnili(r)Rnflf(r).The other terms (k=1,2,…) in the sum equation (29) correspond to the nonvanishing terms on the second term of (25), and the factors ck, ηk and the function Uk(r) are determined by a combination of two bound states as follows:(31)ck=2fs(1-fs′)Clsls′,ηk=ηs,s′,Uk(r)=Rs(r)Rs′(r).
Here, we assume that the number of all nonvanishing terms in the second term of (25) is kmax⁡.In similar manner, numberingkmax⁡+1, kmax⁡+2,… for all nonvanishing terms on the third term of (25), we rewrite it as follows:
(32)∑k>kmax⁡ckηkUk(r)Uk(r′),
where the factors ck, ηk and the function Uk(r) are, respectively,
(33)ck=2fs(1-f(εs+ω))Clslθ(εs+ω),ηk=-iπ,Uk(r)=Rs(r)R(εs+ω)l(r).Then, (25) can be expressed as
(34)χω0(r,r′)=∑k=0ckηkUk(r)Uk(r′).Substituting (34) into (21), we obtain the radial part of the induced electron density δρω(r) as follows:
(35)δρω(r)=c0U0(r)a0+∑k=1ckηkUk(r)ak,
where complex coefficients a0 and ak are given as follows:
(36)a0=η0∫0∞‍U0(r)ϕωscf(r)r2drak=∫0∞‍Uk(r)ϕωscf(r)r2dr,k>0.
From (22), (23), and (35), the self-consistent field ϕωscf(r) satisfies
(37)ϕωscf(r)=ϕωext(r)+c0B0(r)a0+∑k=1ckηkBk(r)ak.
The function Bk(r) is given by
(38)Bk(r)=∫0∞K(r,r′)Uk(r′)r′2dr′.The self-consistent fieldϕωscf(r) is obtained by simultaneously solving (36) and (37). Substituting (37) into (36), we obtain the following relation:
(39)V^(a0a1a2⋮)=(d0d1d2⋮),
where the factor dk is
(40)dk=∫0∞‍Uk(r)ϕωext(r)r2dr,
and V^ is the matrix given as follows:
(41)(1η0-V00-η1V01-η2V02-η3V03⋯-V101-η1V11-η2V12-η3V13⋯-V20-η1V211-η2V22-η3V23⋯-V30-η1V31-η2V321-η3V33⋯⋮⋮⋮⋮⋱).
The factor Vkk′ in the above matrix elements is given as
(42)Vkk′=ck′∫0∞Uk(r)Bk′(r)r2dr.The determinantD of the matrix V^ can be expanded in terms of its cofactors as follows:
(43)D=(1η0-V00)A+B,
where A and B are the determinants obtained from the cofactors of D. |D|2 becomes
(44)|D|2=|A|2[(ω2-εr(2)(ω)2Δε)2+(λ+ωΔεδ)2],
(the infinitesimal positive quantity δ in η0 (or ηif, (27)) put 0 in previous theory [15]) where Δε=εnflf-εnili, and
(45)εr(2)(ω)≡Δε2+2Δε(V00-γ)+δ2,
and λ≡Im⁡(B/A), γ≡Re(B/A). The value εr(2)(ω) of (45) may be negative because of the ω dependence of γ, but if it has a positive value, a positive solution of the relation ω2-εr(2)(ω)=0 in (44) is considered as a resonance energy position ωr of Fano profile, and this position is approximately given by(46)ωr≃lim⁡δ→0εr(2)(ω)=Δε2+2Δε(V00-γ).The complex coefficientsa0 and ak satisfy the following relations the same as (43):
(47)Da0=(1η0-V00)A+B′;Dak=(1η0-V00)Pk+Qk,then, we obtain the following:(48)a0=1D|d0-η1V01-η2V02⋯d11-η1V11-η2V12⋯d2-η1V211-η2V22⋯⋮⋮⋮⋱|≡G0D,a1=1D|1η0-V00d0-η2V02⋯-V10d1-η2V12⋯-V20d21-η2V22⋯⋮⋮⋮⋱|≡G1D,
and the others are obtained in the same way as a1.The photoabsorption cross section is calculated as follows [22, 24]:
(49)σ(ω:{q})=-4παω∬ϕωscf*(r′)[Im⁡χω0(r,r′)]-4παkωπαω×ϕωscf(r)r′2r2dr′dr,=4παω(c0|G0|2|D|2ωΔεδ+π∑k>kmax⁡ck|Gk|2|D|2),
where α is the fine structure constant.Whenλ≠0, for δ→0 in (49), we obtain the bound-free photoabsorption corss section as follows
(50)σ(ω:{q})=4π2αω∑k>kmax⁡ck|Gk|2|D|2.
The photoabsorption cross section equation (50) is the superposition of the “Fano profile” and “Lorentz profile” [15].


#### 3.2.3. Bound-Bound Absorption

When the photon energy is in the vicinity of the difference of two orbital energiesεnili and εnflf, but satisfies the condition ω<|εi| for any other orbital energies, the matrices A and B in (43) become real values, namely, λ=0, ck=0(k>kmax⁡); the cofactors A and B become real values and autoionization resonance does not occur, though an ordinary bound-bound transition nili→nflf may occur. In this case, the second term of the right-hand side of (49) becomes 0, but the limit of the first term as δ→0 is a non-zero value. For calculating the first term in (49), we may expand the ω2-εr(2)(ω) in (44) into power-series, but neglecting terms in high powers of (ω-ωr) as follows:
(51)ω2-εr(2)(ω)≃ωr2-εr(2)(ωr)+[2ωr-dεr(2)(ωr)dω](ω-ωr)=(2ωr+2Δεdγdω)(ω-ωr)≃2ωr(ω-ωr).
Then |D|2 in (49) becomes
(52)|D|2=|A|2[(ωrΔε)2(ω-ωr)2+(ωΔεδ)2].
Therefore the 1st term of the right-hand side in (49) is rewritten as follows:
(53)c0|G0|2|D|2ωΔεδ=c0|G0|21|A|2Δεωr(ω/ωr)δ[(ω-ωr)2+((ω/ωr)δ)2];
as a result, the photoabsorption cross section for bound-bound is expressed as
(54)lim⁡δ→0σ(ω:{q})=4παωc0|G0|2lim⁡δ→0(ω/Δε)δ|D|2≃4π2αc0Δε|G0|2|A|2δ(ω-ωr).
From (53), we consider the quantity 2δ as the natural line width of the transition nflf→nili.Theωr is the positive solution of the relation ω2-εr(2)(ω)=0, and it is the energy point of the resonance absorption of this bound–bound transition. As mentioned above, this energy point is shifted from the difference of the energies of two orbitals involved in the photoabsorption, as for the case of autoionization resonance.The photoabsorption cross section for a plasma is expressed as(55)σ(ω)=∑qP{q}σ(ω:{q}),
superimposing the photoabsorption cross section σ(ω:{q}) for an ion with a configuration q=(q1,q2,q3,…), where summation is over electronic configurations ready beforehand (detailed configuration accounting (DCA) method).


## 3.1. Probability of Existence of an Ion

When plasmas are in LTE, electronic state of ion in the plasma will be in every possible state, that is, the number of bound electrons in an orbital of the ion fluctuates around the average value. The electronic configuration of average atom model is a virtual image obtained by a time average for these deviated states. In our presented model, these average values of occupation are calculated by means of FTDFT as mentioned in Section2.The relaxation timeτ of this fluctuation will estimate roughly by the mean time between two-electron or ion-electron collisions. For the spatial distribution of surrounding an ion (given by (8)), the relaxation time of its fluctuation will roughly be equal to the mean collision time between ions, and it is very large compared to the electron collision time. On the other hand, the photoabsorption occurs in very short time compared with that fluctuation time τ. Therefore, electronic state of ions going to absorb photon is in one of deviated states from the average atom, and this ion will be in electric field caused by deviation of spatial distribution of ions. Then, we must prepare a deviated electronic state, that is, a electronic configuration with integer occupation for an initial state of ion, but we here assume that the ion spatial distribution is given by (8) without fluctuation, and other ions around an ion at origin are replaced with average atoms obtained by FTDFT (so-called the constant density method [12]).In the calculation of the opacity of LTE plasmas, it is necessary to know the probability distribution of excited electronic state of an ion with integer occupation, where the plasma effects (the ionization potential depression (IPD [26]), pressure ionization, etc.) must be correctly taken into account. To consider this probability distribution, we calculate the energy of an ion with integer occupation numbers of bound electrons embedded in the plasma, where the electrostatic potential surrounding that ion is determined by continuum electrons, the spatial distribution of neighboring ions, and their electronic structure. Therefore, the IPD is included in calculation of opacity.For a given electron configuration(11)(1s)q1(2s)q2(2p)q3(3s)q4(3p)q5(3d)q6⋯
with integer occupation numbers (q1,q2,q3,q4,…), there are many terms with different energies in general; however, we assume that the average energy of an electron configuration of an ion [27] can be calculated as follows:
(12)E{q}=∑i‍qiεi-12∬‍ρ{q}(r)ρ{q}(r′)|r-r′|drdr′-34∫‍ρ{q}(r)vx[ρ{q}(r)]dr,
where vx[ρ{q}(r)] and ρ{q}(r) are the exchange potential and the bound electron density, respectively;
(13)vx[ρ{q}(r)]=-(3πρ{q}(r))1/3,(14)ρ{q}(r)=14π∑i‍qiRi(r)2,{q} denotes a set of occupation numbers (q1,q2,q3,…), and εi and Ri are the bound electron energy and radial wave function of orbital i obtained by the Kohn-Sham equation which is obtained by replacing the potential veff(r) in (2) by the following potential:(15)V{q}(r)=-Zar+∫‍ρ{q}(r′)-Qs(r′)|r-r′|dr′+vx[ρ{q}(r)].The probabilityP{q} of existence for the ion (nuclear charge Za) with an electronic configuration of integer occupation numbers {q} is given as the change in the total energy of the ion. The probability P{q} of existence for the ion with an electron configuration {q} is
(16)P{q}=1ZGion∏i(giqi)exp⁡[-β(E{q}-μe∑i‍qi)],
where ZGion is a normalization factor, li is the angular momentum, and gi=2(2li+1) is the maximum occupation number of the orbital i [15].


## 3.2. Photoabsorption Cross Section

We now discuss the photoresponse of an ion in plasmas to a frequency-dependent external field. The electronic configuration of this ion is assumed to be(1s)q12sq2(2p)q3(3s)q4⋯. If the incident photon energy ω (in atomic units) is in the vicinity of the difference between two orbital energies εnili and εnflf (|li-lf|=1), that is, ω~εnflf-εnili with εnflf-εnili>|εnflf|, one of the (nflf) electrons can attain an energy above the ionization energy via the following successive excitation processes:
(17)(nili)qi(nflf)qf⟶(nili)qi-1(nflf)qf+1⟶(nili)qi(nflf)qf-1(εl)1;
consequently, the (nflf) electron can be ejected. Another process can coincide with this process; that is, one of the (nflf) electrons can directly absorb the photon energy ω:
(18)(nili)qi(nflf)qf⟶(nili)qi(nflf)qf-1(εl)1.
These two processes compete with each other; namely, a channel mixing occurs, and consequently, the Fano profile appears on the photoabsorption cross section. If the energies of the εnili and the εnflf do not satisfy the condition εnflf-εnili>|εnflf|, the channel mixing cannot occur in our calculation, but a direct bound-bound transition
(19)(nili)qi(nflf)qf⟶(nili)qi-1(nflf)qf+1
can occur if the photon energy satisfies the resonance condition ħω=εnflf-εnili. To investigate the above processes, we study how the electrons respond to a frequency-dependent external field.


### 3.2.1. Photoabsorption Cross Section of an Ion:σ(ω:{q})

In previous theory [15], we considered only bound-free contribution to the calculation of the photoabsorption cross section, but here we extend the theory to include the bound-bound absorption.We assume that the frequency-dependent external electric field is of magnitudeE0 directed along the z-axis. The interaction between this field and the electrons of an ion is ∫‍ρb(r)ϕext(r,ω)eiωtdr, where ρb(r) is the bound electron density, and
(20)ϕext(r,ω)=E04π3rY10(r^)≡E0ϕωext(r)Y10(r^).
The induced charge density δρω(r) in the presence of the external field ϕωext(r) is calculated by means of the single particle response function χω0(r,r′) [22–24] as follows:
(21)δρω(r)=∫0∞χω0(r,r′)ϕωscf(r′)r′2dr′,
where ϕωscf(r) is the self-consistent field produced by the photoresponse of the electrons. The self-consistent field ϕωscf(r) is determined by the following relations:
(22)ϕωscf(r)=ϕωext(r)+ϕωind(r),
where ϕωind(r) is the frequency-dependent induced potential
(23)ϕωind(r)=∫0∞(4π3r<r>2+δvx[ρ{q}(r)]δρ|ρ(r′)δ(r-r′)r′2)×δρω(r′)r′2dr′≡∫0∞K(r,r′)δρω(r′)dr′.‍
In the above equation, the 2nd term in the integrand is a functional derivative of the exchange potential vx[ρ(r)]. From (13), the functional derivative of exchange potential vx[ρ(r)] is in proportion to ρ(r)-2/3(24)∫δvx[ρ]δρ|ρ{q}(r′)δ(r-r′)r′2δρω(r′)r′2dr′=vx[ρ{q}(r)]3ρ{q}(r)δρω(r).When a photon energyω is in the vicinity of the energy deference of optically allowed two- bound states nili and nflf, namely, ω~εf-εi≡Δε, the response function χω0(r,r′) in (21) is given as follows:
(25)χω0(r,r′)≃2fi(1-ff)ClilfηifRnili(r)Rnflf(r)×Rnflf(r′)Rnili(r′)+∑s<s′′2fs(1-fs′)Clsls′ηs,s′Rs(r)Rs′(r)1-11fs′×Rs′(r′)Rs(r′)-iπ∑s∑l=ls±12fs(1-f(εs+ω))Clsl×Rs(r)R(εs+ω)l(r)R(εs+ω)l(r′)×Rs(r′)θ(εs+ω),
where fs=qs/[2(2ls+1)], θ(εs+ω) is the unit step function, and Cll′, ηnl,n′l′ are given as
(26)Cll′={l+l′+18πfor|l-l′|=10for|l-l′|≠1,(27)ηif=1ω-Δε+iδ-1ω+Δε+iδ,Δε=εf-εi.
In (25), summations for s are over only occupied orbitals, as distinct from χω0(r,r′) for finite temperature where summation is over all occupied and unoccupied orbitals [15]. In the second term on the right-hand side of (25), the double summation runs over all bound states on the condition εs<εs′, and the prime denotes the omission of two bound states (nili) and (nflf).In (27), the factor δ is a infinitesimal positive quantity. The factor ηs,s′ in this term is real value and given as follows:(28)ηs,s′=1εs-εs′+ω-1εs′-εs+ω.TheREl(r) in (25) is the radial wave function of a photoionized electron of positive energy E and angular momentum l, which is obtained by solving the Kohn-Sham equation.


### 3.2.2. Bound-Free Absorption

We rewrite the first term and the sum in the second term of (25) to a single sum as follows:
(29)∑k=0,1,2,…kmax⁡ckηkUk(r)Uk(r′),
where the term of k=0 in above sum corresponds to the first term of (25) and the factors c0, η0 and the function U0(r) are given as follows:
(30)c0=2fi(1-ff)Clilf,η0=ηif,U0(r)=Rnili(r)Rnflf(r).The other terms (k=1,2,…) in the sum equation (29) correspond to the nonvanishing terms on the second term of (25), and the factors ck, ηk and the function Uk(r) are determined by a combination of two bound states as follows:(31)ck=2fs(1-fs′)Clsls′,ηk=ηs,s′,Uk(r)=Rs(r)Rs′(r).
Here, we assume that the number of all nonvanishing terms in the second term of (25) is kmax⁡.In similar manner, numberingkmax⁡+1, kmax⁡+2,… for all nonvanishing terms on the third term of (25), we rewrite it as follows:
(32)∑k>kmax⁡ckηkUk(r)Uk(r′),
where the factors ck, ηk and the function Uk(r) are, respectively,
(33)ck=2fs(1-f(εs+ω))Clslθ(εs+ω),ηk=-iπ,Uk(r)=Rs(r)R(εs+ω)l(r).Then, (25) can be expressed as
(34)χω0(r,r′)=∑k=0ckηkUk(r)Uk(r′).Substituting (34) into (21), we obtain the radial part of the induced electron density δρω(r) as follows:
(35)δρω(r)=c0U0(r)a0+∑k=1ckηkUk(r)ak,
where complex coefficients a0 and ak are given as follows:
(36)a0=η0∫0∞‍U0(r)ϕωscf(r)r2drak=∫0∞‍Uk(r)ϕωscf(r)r2dr,k>0.
From (22), (23), and (35), the self-consistent field ϕωscf(r) satisfies
(37)ϕωscf(r)=ϕωext(r)+c0B0(r)a0+∑k=1ckηkBk(r)ak.
The function Bk(r) is given by
(38)Bk(r)=∫0∞K(r,r′)Uk(r′)r′2dr′.The self-consistent fieldϕωscf(r) is obtained by simultaneously solving (36) and (37). Substituting (37) into (36), we obtain the following relation:
(39)V^(a0a1a2⋮)=(d0d1d2⋮),
where the factor dk is
(40)dk=∫0∞‍Uk(r)ϕωext(r)r2dr,
and V^ is the matrix given as follows:
(41)(1η0-V00-η1V01-η2V02-η3V03⋯-V101-η1V11-η2V12-η3V13⋯-V20-η1V211-η2V22-η3V23⋯-V30-η1V31-η2V321-η3V33⋯⋮⋮⋮⋮⋱).
The factor Vkk′ in the above matrix elements is given as
(42)Vkk′=ck′∫0∞Uk(r)Bk′(r)r2dr.The determinantD of the matrix V^ can be expanded in terms of its cofactors as follows:
(43)D=(1η0-V00)A+B,
where A and B are the determinants obtained from the cofactors of D. |D|2 becomes
(44)|D|2=|A|2[(ω2-εr(2)(ω)2Δε)2+(λ+ωΔεδ)2],
(the infinitesimal positive quantity δ in η0 (or ηif, (27)) put 0 in previous theory [15]) where Δε=εnflf-εnili, and
(45)εr(2)(ω)≡Δε2+2Δε(V00-γ)+δ2,
and λ≡Im⁡(B/A), γ≡Re(B/A). The value εr(2)(ω) of (45) may be negative because of the ω dependence of γ, but if it has a positive value, a positive solution of the relation ω2-εr(2)(ω)=0 in (44) is considered as a resonance energy position ωr of Fano profile, and this position is approximately given by(46)ωr≃lim⁡δ→0εr(2)(ω)=Δε2+2Δε(V00-γ).The complex coefficientsa0 and ak satisfy the following relations the same as (43):
(47)Da0=(1η0-V00)A+B′;Dak=(1η0-V00)Pk+Qk,then, we obtain the following:(48)a0=1D|d0-η1V01-η2V02⋯d11-η1V11-η2V12⋯d2-η1V211-η2V22⋯⋮⋮⋮⋱|≡G0D,a1=1D|1η0-V00d0-η2V02⋯-V10d1-η2V12⋯-V20d21-η2V22⋯⋮⋮⋮⋱|≡G1D,
and the others are obtained in the same way as a1.The photoabsorption cross section is calculated as follows [22, 24]:
(49)σ(ω:{q})=-4παω∬ϕωscf*(r′)[Im⁡χω0(r,r′)]-4παkωπαω×ϕωscf(r)r′2r2dr′dr,=4παω(c0|G0|2|D|2ωΔεδ+π∑k>kmax⁡ck|Gk|2|D|2),
where α is the fine structure constant.Whenλ≠0, for δ→0 in (49), we obtain the bound-free photoabsorption corss section as follows
(50)σ(ω:{q})=4π2αω∑k>kmax⁡ck|Gk|2|D|2.
The photoabsorption cross section equation (50) is the superposition of the “Fano profile” and “Lorentz profile” [15].


### 3.2.3. Bound-Bound Absorption

When the photon energy is in the vicinity of the difference of two orbital energiesεnili and εnflf, but satisfies the condition ω<|εi| for any other orbital energies, the matrices A and B in (43) become real values, namely, λ=0, ck=0(k>kmax⁡); the cofactors A and B become real values and autoionization resonance does not occur, though an ordinary bound-bound transition nili→nflf may occur. In this case, the second term of the right-hand side of (49) becomes 0, but the limit of the first term as δ→0 is a non-zero value. For calculating the first term in (49), we may expand the ω2-εr(2)(ω) in (44) into power-series, but neglecting terms in high powers of (ω-ωr) as follows:
(51)ω2-εr(2)(ω)≃ωr2-εr(2)(ωr)+[2ωr-dεr(2)(ωr)dω](ω-ωr)=(2ωr+2Δεdγdω)(ω-ωr)≃2ωr(ω-ωr).
Then |D|2 in (49) becomes
(52)|D|2=|A|2[(ωrΔε)2(ω-ωr)2+(ωΔεδ)2].
Therefore the 1st term of the right-hand side in (49) is rewritten as follows:
(53)c0|G0|2|D|2ωΔεδ=c0|G0|21|A|2Δεωr(ω/ωr)δ[(ω-ωr)2+((ω/ωr)δ)2];
as a result, the photoabsorption cross section for bound-bound is expressed as
(54)lim⁡δ→0σ(ω:{q})=4παωc0|G0|2lim⁡δ→0(ω/Δε)δ|D|2≃4π2αc0Δε|G0|2|A|2δ(ω-ωr).
From (53), we consider the quantity 2δ as the natural line width of the transition nflf→nili.Theωr is the positive solution of the relation ω2-εr(2)(ω)=0, and it is the energy point of the resonance absorption of this bound–bound transition. As mentioned above, this energy point is shifted from the difference of the energies of two orbitals involved in the photoabsorption, as for the case of autoionization resonance.The photoabsorption cross section for a plasma is expressed as(55)σ(ω)=∑qP{q}σ(ω:{q}),
superimposing the photoabsorption cross section σ(ω:{q}) for an ion with a configuration q=(q1,q2,q3,…), where summation is over electronic configurations ready beforehand (detailed configuration accounting (DCA) method).


## 3.2.1. Photoabsorption Cross Section of an Ion:σ(ω:{q})

In previous theory [15], we considered only bound-free contribution to the calculation of the photoabsorption cross section, but here we extend the theory to include the bound-bound absorption.We assume that the frequency-dependent external electric field is of magnitudeE0 directed along the z-axis. The interaction between this field and the electrons of an ion is ∫‍ρb(r)ϕext(r,ω)eiωtdr, where ρb(r) is the bound electron density, and
(20)ϕext(r,ω)=E04π3rY10(r^)≡E0ϕωext(r)Y10(r^).
The induced charge density δρω(r) in the presence of the external field ϕωext(r) is calculated by means of the single particle response function χω0(r,r′) [22–24] as follows:
(21)δρω(r)=∫0∞χω0(r,r′)ϕωscf(r′)r′2dr′,
where ϕωscf(r) is the self-consistent field produced by the photoresponse of the electrons. The self-consistent field ϕωscf(r) is determined by the following relations:
(22)ϕωscf(r)=ϕωext(r)+ϕωind(r),
where ϕωind(r) is the frequency-dependent induced potential
(23)ϕωind(r)=∫0∞(4π3r<r>2+δvx[ρ{q}(r)]δρ|ρ(r′)δ(r-r′)r′2)×δρω(r′)r′2dr′≡∫0∞K(r,r′)δρω(r′)dr′.‍
In the above equation, the 2nd term in the integrand is a functional derivative of the exchange potential vx[ρ(r)]. From (13), the functional derivative of exchange potential vx[ρ(r)] is in proportion to ρ(r)-2/3(24)∫δvx[ρ]δρ|ρ{q}(r′)δ(r-r′)r′2δρω(r′)r′2dr′=vx[ρ{q}(r)]3ρ{q}(r)δρω(r).When a photon energyω is in the vicinity of the energy deference of optically allowed two- bound states nili and nflf, namely, ω~εf-εi≡Δε, the response function χω0(r,r′) in (21) is given as follows:
(25)χω0(r,r′)≃2fi(1-ff)ClilfηifRnili(r)Rnflf(r)×Rnflf(r′)Rnili(r′)+∑s<s′′2fs(1-fs′)Clsls′ηs,s′Rs(r)Rs′(r)1-11fs′×Rs′(r′)Rs(r′)-iπ∑s∑l=ls±12fs(1-f(εs+ω))Clsl×Rs(r)R(εs+ω)l(r)R(εs+ω)l(r′)×Rs(r′)θ(εs+ω),
where fs=qs/[2(2ls+1)], θ(εs+ω) is the unit step function, and Cll′, ηnl,n′l′ are given as
(26)Cll′={l+l′+18πfor|l-l′|=10for|l-l′|≠1,(27)ηif=1ω-Δε+iδ-1ω+Δε+iδ,Δε=εf-εi.
In (25), summations for s are over only occupied orbitals, as distinct from χω0(r,r′) for finite temperature where summation is over all occupied and unoccupied orbitals [15]. In the second term on the right-hand side of (25), the double summation runs over all bound states on the condition εs<εs′, and the prime denotes the omission of two bound states (nili) and (nflf).In (27), the factor δ is a infinitesimal positive quantity. The factor ηs,s′ in this term is real value and given as follows:(28)ηs,s′=1εs-εs′+ω-1εs′-εs+ω.TheREl(r) in (25) is the radial wave function of a photoionized electron of positive energy E and angular momentum l, which is obtained by solving the Kohn-Sham equation.


## 3.2.2. Bound-Free Absorption

We rewrite the first term and the sum in the second term of (25) to a single sum as follows:
(29)∑k=0,1,2,…kmax⁡ckηkUk(r)Uk(r′),
where the term of k=0 in above sum corresponds to the first term of (25) and the factors c0, η0 and the function U0(r) are given as follows:
(30)c0=2fi(1-ff)Clilf,η0=ηif,U0(r)=Rnili(r)Rnflf(r).The other terms (k=1,2,…) in the sum equation (29) correspond to the nonvanishing terms on the second term of (25), and the factors ck, ηk and the function Uk(r) are determined by a combination of two bound states as follows:(31)ck=2fs(1-fs′)Clsls′,ηk=ηs,s′,Uk(r)=Rs(r)Rs′(r).
Here, we assume that the number of all nonvanishing terms in the second term of (25) is kmax⁡.In similar manner, numberingkmax⁡+1, kmax⁡+2,… for all nonvanishing terms on the third term of (25), we rewrite it as follows:
(32)∑k>kmax⁡ckηkUk(r)Uk(r′),
where the factors ck, ηk and the function Uk(r) are, respectively,
(33)ck=2fs(1-f(εs+ω))Clslθ(εs+ω),ηk=-iπ,Uk(r)=Rs(r)R(εs+ω)l(r).Then, (25) can be expressed as
(34)χω0(r,r′)=∑k=0ckηkUk(r)Uk(r′).Substituting (34) into (21), we obtain the radial part of the induced electron density δρω(r) as follows:
(35)δρω(r)=c0U0(r)a0+∑k=1ckηkUk(r)ak,
where complex coefficients a0 and ak are given as follows:
(36)a0=η0∫0∞‍U0(r)ϕωscf(r)r2drak=∫0∞‍Uk(r)ϕωscf(r)r2dr,k>0.
From (22), (23), and (35), the self-consistent field ϕωscf(r) satisfies
(37)ϕωscf(r)=ϕωext(r)+c0B0(r)a0+∑k=1ckηkBk(r)ak.
The function Bk(r) is given by
(38)Bk(r)=∫0∞K(r,r′)Uk(r′)r′2dr′.The self-consistent fieldϕωscf(r) is obtained by simultaneously solving (36) and (37). Substituting (37) into (36), we obtain the following relation:
(39)V^(a0a1a2⋮)=(d0d1d2⋮),
where the factor dk is
(40)dk=∫0∞‍Uk(r)ϕωext(r)r2dr,
and V^ is the matrix given as follows:
(41)(1η0-V00-η1V01-η2V02-η3V03⋯-V101-η1V11-η2V12-η3V13⋯-V20-η1V211-η2V22-η3V23⋯-V30-η1V31-η2V321-η3V33⋯⋮⋮⋮⋮⋱).
The factor Vkk′ in the above matrix elements is given as
(42)Vkk′=ck′∫0∞Uk(r)Bk′(r)r2dr.The determinantD of the matrix V^ can be expanded in terms of its cofactors as follows:
(43)D=(1η0-V00)A+B,
where A and B are the determinants obtained from the cofactors of D. |D|2 becomes
(44)|D|2=|A|2[(ω2-εr(2)(ω)2Δε)2+(λ+ωΔεδ)2],
(the infinitesimal positive quantity δ in η0 (or ηif, (27)) put 0 in previous theory [15]) where Δε=εnflf-εnili, and
(45)εr(2)(ω)≡Δε2+2Δε(V00-γ)+δ2,
and λ≡Im⁡(B/A), γ≡Re(B/A). The value εr(2)(ω) of (45) may be negative because of the ω dependence of γ, but if it has a positive value, a positive solution of the relation ω2-εr(2)(ω)=0 in (44) is considered as a resonance energy position ωr of Fano profile, and this position is approximately given by(46)ωr≃lim⁡δ→0εr(2)(ω)=Δε2+2Δε(V00-γ).The complex coefficientsa0 and ak satisfy the following relations the same as (43):
(47)Da0=(1η0-V00)A+B′;Dak=(1η0-V00)Pk+Qk,then, we obtain the following:(48)a0=1D|d0-η1V01-η2V02⋯d11-η1V11-η2V12⋯d2-η1V211-η2V22⋯⋮⋮⋮⋱|≡G0D,a1=1D|1η0-V00d0-η2V02⋯-V10d1-η2V12⋯-V20d21-η2V22⋯⋮⋮⋮⋱|≡G1D,
and the others are obtained in the same way as a1.The photoabsorption cross section is calculated as follows [22, 24]:
(49)σ(ω:{q})=-4παω∬ϕωscf*(r′)[Im⁡χω0(r,r′)]-4παkωπαω×ϕωscf(r)r′2r2dr′dr,=4παω(c0|G0|2|D|2ωΔεδ+π∑k>kmax⁡ck|Gk|2|D|2),
where α is the fine structure constant.Whenλ≠0, for δ→0 in (49), we obtain the bound-free photoabsorption corss section as follows
(50)σ(ω:{q})=4π2αω∑k>kmax⁡ck|Gk|2|D|2.
The photoabsorption cross section equation (50) is the superposition of the “Fano profile” and “Lorentz profile” [15].


## 3.2.3. Bound-Bound Absorption

When the photon energy is in the vicinity of the difference of two orbital energiesεnili and εnflf, but satisfies the condition ω<|εi| for any other orbital energies, the matrices A and B in (43) become real values, namely, λ=0, ck=0(k>kmax⁡); the cofactors A and B become real values and autoionization resonance does not occur, though an ordinary bound-bound transition nili→nflf may occur. In this case, the second term of the right-hand side of (49) becomes 0, but the limit of the first term as δ→0 is a non-zero value. For calculating the first term in (49), we may expand the ω2-εr(2)(ω) in (44) into power-series, but neglecting terms in high powers of (ω-ωr) as follows:
(51)ω2-εr(2)(ω)≃ωr2-εr(2)(ωr)+[2ωr-dεr(2)(ωr)dω](ω-ωr)=(2ωr+2Δεdγdω)(ω-ωr)≃2ωr(ω-ωr).
Then |D|2 in (49) becomes
(52)|D|2=|A|2[(ωrΔε)2(ω-ωr)2+(ωΔεδ)2].
Therefore the 1st term of the right-hand side in (49) is rewritten as follows:
(53)c0|G0|2|D|2ωΔεδ=c0|G0|21|A|2Δεωr(ω/ωr)δ[(ω-ωr)2+((ω/ωr)δ)2];
as a result, the photoabsorption cross section for bound-bound is expressed as
(54)lim⁡δ→0σ(ω:{q})=4παωc0|G0|2lim⁡δ→0(ω/Δε)δ|D|2≃4π2αc0Δε|G0|2|A|2δ(ω-ωr).
From (53), we consider the quantity 2δ as the natural line width of the transition nflf→nili.Theωr is the positive solution of the relation ω2-εr(2)(ω)=0, and it is the energy point of the resonance absorption of this bound–bound transition. As mentioned above, this energy point is shifted from the difference of the energies of two orbitals involved in the photoabsorption, as for the case of autoionization resonance.The photoabsorption cross section for a plasma is expressed as(55)σ(ω)=∑qP{q}σ(ω:{q}),
superimposing the photoabsorption cross section σ(ω:{q}) for an ion with a configuration q=(q1,q2,q3,…), where summation is over electronic configurations ready beforehand (detailed configuration accounting (DCA) method).


## 4. Results and Discussion

For the purpose of demonstrating the usefulness of our model, we show opacities including autoionization resonances of Al plasma. At first, some physical properties of an Al plasma of temperatureT=20eV and density 0.01 g/cm3 are obtained by means of FTDFT as mentioned in Section 2. The chemical potential μe of the electron subsystem and the charge state Z*(=Za-∫‍ρ¯bdr) (both are calculated by (6)) for average ion of this Al plasma are μe=-127.4928557eV and Z*=4.1399487, respectively, and these values are in good agreement with other typical theoretical results. Orbital energies of average atom obtained by (2) are shown in Table 1. In this table, mean orbital radii 〈r〉 are also shown, and these were all smaller than the ion-sphere radius aion though wave functions are solved under the boundary condition as shown in (2). For the case of comparatively high temperature, some mean orbital radii of bound electrons become larger than the aion, but such orbitals are not discard for calculating the bound electron density (see, (5)).Table 1
Orbital energies and mean orbital radii〈r〉 of average atom obtained by FTDFT for Al plasma of temperature T=20 eV and density 0.01 g/cm3. aion is the ion-sphere radius, in this case, aion=19.3563 in a.u.
Orbital
Occupation
Energy [eV]
〈
r
〉
/
a
ion
1s
1.0000000
-
1593.2399450
0.0062490
2s
0.9471505
-
185.2130378
0.0307858
2p
0.7507994
-
149.5504598
0.0284118
3s
0.0170534
-
46.4087164
0.0893144
3p
0.0112252
-
37.9267667
0.0944112
3d
0.0065214
-
26.9702796
0.0984221
4s
0.0040830
-
17.5564467
0.1868533
4p
0.0035343
-
14.6586288
0.2001951
4d
0.0029352
-
10.9321117
0.2172469
4f
0.0027064
-
9.3044024
0.2041051
5s
0.0024418
-
7.2412225
0.3263015
5p
0.0022922
-
5.9737677
0.3498977
5d
0.0021112
-
4.3251798
0.3842115
5f
0.0020295
-
3.5337236
0.3796523
5g
0.0019966
-
3.2065049
0.3361391
6s
0.0019489
-
2.7221846
0.5264264
6p
0.0018911
-
2.1186225
0.5678503
6d
0.0018186
-
1.3359917
0.6364388
6f
0.0017813
-
0.9203438
0.6548017
6g
0.0017597
-
0.6762144
0.6174240
7s
0.0017625
-
0.7081767
0.8533281The radial distribution functiongii(r) obtained by (8) is shown in Figure 2. The coupling constant Γ of the plasma, which is defined in this paper as Γ=(Z*)2/(aionT), is 1.2, so this Al plasma may be regarded as a strongly coupled plasma.Figure 2
The radial distribution functiongii(r) of Al plasma (0.01 g/cm3, 20 eV) obtained by HNC (see (8) and (9)). The coupling constant Γ is 1.2.For calculation the photoabsorption cross section by (55) using DCA, the probability distributions P{q} (see, (16)) must be estimated for congurations q as many as possible, and the normalization factor ZGion is the sum of those distributions in order to satisfy the conservation of probability; that is,
(56)ZGion=∑{q}‍P{q}.
The summation is over all configurations used in (55), where the number of terms is usually huge even if we limit the maximum number of orbitals [13]. However, here we carried out the summation of (55) on two assumptions.(I)
The maximum principle quantum number is 8; namely, the electron configuration is described as follows:(57)(1s)q1(2s)q2(2p)q3(3s)q4(3p)q5(3d)q6⋯(8l=7)q36.(II)
The number of excited electrons from grand state is 3 or less.For an Al ion, the total number of electron configurations on these assumptions is 703,236.The total energy of an ion with the above assumed electron configuration cannot always be obtained because the ion is not isolated but is immersed in the plasma. Hence, it is regarded than an ion with such an electron configuration is unstable in the plasma and we omit this configuration from the summation of (55).The number of stable electron configurations for this Al plasma was 288,737; the conservation of probability equation (16) was established by these 288,737 electron configurations. Table 2 shows the ten greatest probabilities obtained.Table 2
Probability distribution of electronic configuration{q} of ion in the Al plasma (0.01 g/cm3, 20 eV) calculated by (22). Charge state Q means Q=Za-∑i=136‍qi. The top 10 probabilities are shown in descending order starting from the highest probability.
Configuration
{
q
}
P
{
q
}
Charge stateQ
1
s
2
2
s
2
2
p
5
(2, 2, 5, 0, 0, 0, 0,…)
0.297933
4
1
s
2
2
s
2
2
p
4
(2, 2, 4, 0, 0, 0, 0,…)
0.201386
5
1
s
2
2
s
2
2
p
6
(2, 2, 6, 0, 0, 0, 0,…)
0.030392
3
1
s
2
2
s
1
2
p
5
(2, 1, 5, 0, 0, 0, 0,…)
0.027722
5
1
s
2
2
s
2
2
p
4
3
p
1
(2, 2, 4, 0, 1, 0, 0,…)
0.022315
4
1
s
2
2
s
2
2
p
4
3
d
1
(2, 2, 4, 0, 0, 1, 0,…)
0.021047
4
1
s
2
2
s
1
2
p
6
(2, 1, 6, 0, 0, 0, 0,…)
0.015942
4
1
s
2
2
s
2
2
p
5
3
d
1
(2, 2, 5, 0, 0, 1, 0,…)
0.014986
3
1
s
2
2
s
2
2
p
5
3
p
1
(2, 2, 5, 0, 1, 0, 0,…)
0.014871
3
1
s
2
2
s
2
2
p
5
5
f
1
(2, 2, 5, 0, 0, 0, 1,…)
0.011916
3The population fractionWQ of ions with charge state Q(=Za-∑i=136‍qi) is calculated using the probability P{q} as follows:
(58)WQ=∑∑qi=Za-Qq1,q2,q3,…P{q}.Figure3 shows the charge state distribution of ions obtained for this Al plasma. In this Al plasma, there was no ion of charge state Q=0 (neutral atom) and population fraction of W1(Q=1) is about 1.54×10-5 [1]. The average of charge state 〈Q〉 is 4.07952, about 1.5 percent less than Z*(=Za-∫‍ρ¯bdr) obtained by FTDFT (see, (5)).Figure 3
Charge state distributionWQ of (58) for the Al plasma (ρion = 0.01 g/cm3, T = 20 eV). The average of charge state 〈Q〉 is 4.07952, although FTDFT in Section 2 gives this value as 4.1399487.The photoabsorption cross section (55) was calculated by selecting the top 500 probable electron configurations (from the total 288,737 configurations). In Figure 4, we show the opacity obtained by (50) and (54) for the Al plasma. In this result, the Doppler width was not considered and the profile of (54) was taken to be rectangle profile of width 4×10-4 eV. Figure 5 shows the absorption cross section for the photon energy range 35–45 eV. There are large two peaks in the vicinity of 38 eV; these are superposition of some Fano proles with respect to the transition 2p→3s as shown in Figure 4, we show the two Fano profiles by black lines in Figure 5, which are most contributed to make that structure.Figure 4
Red solid line: opacity equations (55) obtained by (50) and (54) for the Al plasma (ρion = 0.01 g/cm3, T= 20 eV). Blue solid line: opacity included contribution of autoionization resonance only (50). Both of the red and blue lines don't include Doppler width ans Stark width. 2p→3s absorption lines of Al+2 and Al+3 are indicated in the figure.Figure 5
Opacity equations (55) included contribution of autoionization resonance only (50) for the Al plasma (ρion = 0.01 g/cm3, T = 20 eV). It is the same as the opacity shown by red line in Figure 4, but photon energy range is 35 eV–45 eV. Two black lines are Fano profiles; these mostly contributed the photoabsorption cross section with respect to the transition 2p→3s of Al+3.Figure6 shows the opacity obtained by (50) including the Stark width (electron impact width:FWHM=0.4 eV [1]). Resonance points (see, (46)) and transitions of some Fano profiles (a–g in Figure 6) are shown in Table 3. In this table, two transitions for each profiles compete with each other as mentioned in Section 3.2. For example, profile “a” is identified as channel mixing of two processes b-b (2s→2p) and b-f (3d→free) for an initial state (1s22s22p43d1) of Al4+. In this case, b-f transition from 2p cannot occur because the incident photon energy is smaller than the threshold energy of 2d→free.Table 3
Assignment of lines a–g in Figure6. Electronic configurations and two transitions with respect to autoionization resonance are shown for each line.
Symbol
Initial state
Transition
a
Al
4
+
:
1s
2
2s
2
2p
4
3d
1
2
s
→
2p, 3d
→
free
b
Al
3
+
:
1s
2
2s
2
2
p
5
3
[spd
]
1
2
p
→
3s, 3p
→
free
c
Al
3
+
:
1
s
2
2
s
2
2p
5
4
[
spf
]
1
2
p
→
3s, 4f
→
free
d
Al
3
+
:
1s
2
2s
2
2p
5
3
[
pd
]
1
2
p
→
3d, 3d
→
free
e
Al
3
+
:
1s
2
2s
2
2p
5
[
4d4f5d5f
]
1
2
p
→
3d, 4, 5d
→
free
f
Al
4
+
:
1s
2
2s
2
2p
4
3
[
pd
]
1
2
p
→
3d, 3d
→
free
g
Al
4
+
:
1s
2
2s
2
2p
4
3
[
pd
]
1
2
p
→
3d
,
3d
→
freeFigure 6
Opacity of Al plasma (ρion = 0.01 g/cm3, T = 20 eV) obtained by (55) without b-b resonance (see, (54)) but included Stark width. Spiking profiles are all Fano profiles, and symbols a–g indicate comparatively large Fano profiles.Figure7 shows the opacity of ordinary b-b resonance by calculated by (54) including the Stark width (electron impact width). Symbols a–k in this figure show some major resonances; these resonance points and transition processes are shown in Table 4.Table 4
Assignment of lines a–k in Figure7. Electronic configurations and transitions are shown for each line.
Symbol
Initial state
Transition
a
Al
4
+
:
1s
2
2s
2
2p
5
2
s
→
2
p
b
Al
3
+
:
1s
2
2s
2
2p
6
2
p
→
3
s
c
Al
3
+
:
1s
2
2s
2
2p
6
2
p
→
3
d
d
Al
4
+
:
1s
2
2s
2
2p
5
2
p
→
3
s
e
Al
4
+
:
1s
2
2s
2
2p
5
2
p
→
3
d
f
Al
5
+
:
1s
2
2s
2
2p
4
2
p
→
3
s
g
Al
4
+
:
1s
2
2s
2
2p
5
2
p
→
4
d
h
Al
5
+
:
1s
2
2s
2
2p
4
2
p
→
3
d
i
Al
5
+
:
1s
2
2s
2
2p
4
2
p
→
4
d
j
Al
5
+
:
1s
2
2s
2
2p
4
2
s
→
3
p
k
Al
5
+
:
1s
2
2s
2
2p
4
2
p
→
5
dFigure 7
Opacity for b-b resonance of Al plasma (ρion = 0.01 g/cm3, T = 20 eV) obtained by (55) with (54), not including autoionization resonance. Spiking profiles are all b-b resonance profiles with Stark width.The opacity of Al plasma under consideration is the sum of two opacities equations (50) and (54), or the sum of Figure 6 and Figure 7; then its result is shown in Figure 8. The blue solid line in Figure 8 shows the same opacity but without the contribution from (54), showing several narrow autoionization resonance profiles and the dominant autoionization above 100 eV.Figure 8
Opacity of Al plasma (ρion = 0.01 g/cm3, T = 20 eV) summed up b-f (Figure 6) and b-b (Figure 7) results.The transmissionF(ω)=e-ρκ(ω)L of the Al plasma is shown in Figure 9, where the path length L is chosen to be L = 0.003 cm [1] and the Stark width is considered by convolution of the obtained photoabsorption cross section and a Gaussian type profile. In the transmission, we considered Kramers cross section for free-free absorption, but we have not included the scattering cross section.Figure 9
Transmission spectra obtained for Al plasma (ρion = 0.01 g/cm3, T = 20 eV). The Doppler width is considered by convolution of the cross section obtained and a Gaussian profile.


## 5. Conclusion

We propose a new model of calculating radiative opacity of hot dense plasmas in LTE using the detailed configuration accounting (DCA) including not only bb and bf contributions but also autoionization contributions. Electronic structures for ion with integer number of bound electrons, embedded in the plasma, are calculated by Kohn-Sham equation, and estimate the probability distribution of energies of such ions in the plasmas. The electrostatic potential surrounding an ion is determined by means of FTDFT [16]. The photoabsorption cross section of an ion with integer occupations is obtained by the previously derived formula by means of TDDFT [15], where Doppler and Stark width are both included.In conclusion, we have calculated the opacity and transmission of an Al plasma (T = 20 eV, 0.01 g/cm3) in the energy region 0–300 eV, but not considered the width caused of the unresolved transition array (UTA). The results are in good agreement with the experimental results of Winhart et al. [28] and theoretical results of Zeng et al. [1], in spite of the small number (500) of electron configurations used to calculate the photoabsorption. The Planck mean opacity and Rosseland mean opacity obtained were 28,348 cm2/g and 4,279 cm2/g, respectively. These mean opacities were calculated from the photoabsorption cross section obtained considering the Doppler width. The Rosseland mean opacity is very sensitive to the width of the line profile, so calculating this mean opacity is necessary to account for the autoionization resonance profile and other spectral widths. It is important to include the UTA width in our model so that width may be the most dominant in hot dense plasmas, and it is a subject for future analysis.



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