Research Article | Open Access
A Model of Calculating Radiative Opacities of Hot Dense Plasmas Based on the Density-Functional Theory
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 temperature 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.
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  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 .
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 Section 2, 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 nuclei of nuclear charge and electrons in a volume . The system is in thermal equilibrium with temperature (in energy unit).
The Hamiltonian of the system is as follows (hereafter, we use atomic units): where , are nuclear mass, momentum of a nucleus, respectively, and and 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: where , are principle quantum number and angular momentum of electron and is the effective potential at finite temperature including the exchange-correlation potential: where is the continuum electron density, and is the bound electron density of the average ion, The chemical potential of electron subsystem is determined by ensuring the charge neutrality of the system where is the ion density of the plasma. The function in (3) is the charge density, composed of three parts (Figure 1): (i) the charge density 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 , as follows: The function is the radial distribution function for the average ion.
The radial distribution function is obtained by hypernetted chain (HNC) approximation  as follows: where functions and are the direct correlation function, and these functions must satisfy the next Ornstein-Zernike’s relation: The function in (8), the effective interatomic potential, is the electrostatic interaction between two average ions separated distance , and is approximately calculated as follows: where the polarization of continuum electrons is neglected.
3. A Modeling of Absorption Cross Section: TDDFT
In our previous theory  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 Section 2.
The relaxation time 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 . 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 ).
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 ), 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 with integer occupation numbers , there are many terms with different energies in general; however, we assume that the average energy of an electron configuration of an ion  can be calculated as follows: where and are the exchange potential and the bound electron density, respectively; denotes a set of occupation numbers , and and are the bound electron energy and radial wave function of orbital obtained by the Kohn-Sham equation which is obtained by replacing the potential in (2) by the following potential:
The probability of existence for the ion (nuclear charge ) with an electronic configuration of integer occupation numbers is given as the change in the total energy of the ion. The probability of existence for the ion with an electron configuration is where is a normalization factor, is the angular momentum, and is the maximum occupation number of the orbital .
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 . If the incident photon energy (in atomic units) is in the vicinity of the difference between two orbital energies and (), that is, with , one of the electrons can attain an energy above the ionization energy via the following successive excitation processes: consequently, the electron can be ejected. Another process can coincide with this process; that is, one of the electrons can directly absorb the photon energy : 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 and the do not satisfy the condition , the channel mixing cannot occur in our calculation, but a direct bound-bound transition can occur if the photon energy satisfies the resonance condition . 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:
In previous theory , 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 magnitude directed along the -axis. The interaction between this field and the electrons of an ion is , where is the bound electron density, and The induced charge density in the presence of the external field is calculated by means of the single particle response function [22–24] as follows: where is the self-consistent field produced by the photoresponse of the electrons. The self-consistent field is determined by the following relations: where is the frequency-dependent induced potential In the above equation, the 2nd term in the integrand is a functional derivative of the exchange potential . From (13), the functional derivative of exchange potential is in proportion to
When a photon energy is in the vicinity of the energy deference of optically allowed two- bound states and , namely, , the response function in (21) is given as follows: where , is the unit step function, and , are given as In (25), summations for are over only occupied orbitals, as distinct from for finite temperature where summation is over all occupied and unoccupied orbitals . In the second term on the right-hand side of (25), the double summation runs over all bound states on the condition , and the prime denotes the omission of two bound states and .
In (27), the factor is a infinitesimal positive quantity. The factor in this term is real value and given as follows:
The in (25) is the radial wave function of a photoionized electron of positive energy and angular momentum , 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: where the term of in above sum corresponds to the first term of (25) and the factors , and the function are given as follows: The other terms () in the sum equation (29) correspond to the nonvanishing terms on the second term of (25), and the factors , and the function are determined by a combination of two bound states as follows: Here, we assume that the number of all nonvanishing terms in the second term of (25) is .
In similar manner, numbering , for all nonvanishing terms on the third term of (25), we rewrite it as follows: where the factors , and the function are, respectively,
Then, (25) can be expressed as
Substituting (34) into (21), we obtain the radial part of the induced electron density as follows: where complex coefficients and are given as follows: From (22), (23), and (35), the self-consistent field satisfies The function is given by
The self-consistent field is obtained by simultaneously solving (36) and (37). Substituting (37) into (36), we obtain the following relation: where the factor is and is the matrix given as follows: The factor in the above matrix elements is given as
The determinant of the matrix can be expanded in terms of its cofactors as follows: where and are the determinants obtained from the cofactors of . becomes (the infinitesimal positive quantity in (or , (27)) put 0 in previous theory ) where , and and , . The value of (45) may be negative because of the dependence of , but if it has a positive value, a positive solution of the relation in (44) is considered as a resonance energy position of Fano profile, and this position is approximately given by
The complex coefficients and satisfy the following relations the same as (43):
then, we obtain the following: and the others are obtained in the same way as .
When , for in (49), we obtain the bound-free photoabsorption corss section as follows The photoabsorption cross section equation (50) is the superposition of the “Fano profile” and “Lorentz profile” .
3.2.3. Bound-Bound Absorption
When the photon energy is in the vicinity of the difference of two orbital energies and , but satisfies the condition for any other orbital energies, the matrices and in (43) become real values, namely, , ; the cofactors and become real values and autoionization resonance does not occur, though an ordinary bound-bound transition may occur. In this case, the second term of the right-hand side of (49) becomes , but the limit of the first term as is a non-zero value. For calculating the first term in (49), we may expand the in (44) into power-series, but neglecting terms in high powers of as follows: Then in (49) becomes Therefore the 1st term of the right-hand side in (49) is rewritten as follows: as a result, the photoabsorption cross section for bound-bound is expressed as From (53), we consider the quantity as the natural line width of the transition .
The is the positive solution of the relation , 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 superimposing the photoabsorption cross section for an ion with a configuration , 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 temperature eV and density 0.01 g/cm3 are obtained by means of FTDFT as mentioned in Section 2. The chemical potential of the electron subsystem and the charge state (both are calculated by (6)) for average ion of this Al plasma are eV and , 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 are also shown, and these were all smaller than the ion-sphere radius 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 , but such orbitals are not discard for calculating the bound electron density (see, (5)).
The radial distribution function obtained by (8) is shown in Figure 2. The coupling constant of the plasma, which is defined in this paper as , is 1.2, so this Al plasma may be regarded as a strongly coupled plasma.
For calculation the photoabsorption cross section by (55) using DCA, the probability distributions (see, (16)) must be estimated for congurations as many as possible, and the normalization factor is the sum of those distributions in order to satisfy the conservation of probability; that is, 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 . 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: (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.
The population fraction of ions with charge state is calculated using the probability as follows:
Figure 3 shows the charge state distribution of ions obtained for this Al plasma. In this Al plasma, there was no ion of charge state (neutral atom) and population fraction of is about . The average of charge state is 4.07952, about 1.5 percent less than obtained by FTDFT (see, (5)).