The finite-temperature random-phase approximation for the spectroscopic properties of dense plasmas

High Energy Density Physics 3 (2007) 65e75 www.elsevier.com/locate/hedp

The finite-temperature random-phase approximation for the spectroscopic properties of dense plasmas J. Colgan a,*, C.J. Fontes b, G. Csanak a,c, L.A. Collins a b

a Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA Applied Physics Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA c Department of Physics, University of Nevada, Reno, NV 89557, USA

Available online 7 February 2007

Abstract The finite-temperature random-phase-approximation (FTRPA) is used to calculate selected spectral properties for 1P excitations of dense neon plasmas. We present FTRPA calculations that include coupled-channel effects. The linear algebraic method is used to solve the resulting set of coupled-channel equations and, as a test of our method, oscillator strengths are calculated at zero-temperature and density and compared with previous calculations. Coupled-channel calculations of the oscillator strength for neon at various temperatures and densities are given; the inclusion of these effects is shown to cause significant differences in the oscillator strength when compared with single-channel and with average-atom calculations. Some trends as a function of temperature and density are also discussed. Ó 2007 Elsevier B.V. All rights reserved.

1. Introduction Radiative properties of hot dense plasmas, such as those that occur in inertial confinement fusion devices, in astrophysical environments, and produced by high-power lasers, remain a subject of much current interest [1]. One of the earliest models applied to these cases was the ‘average-atom’ (AA) model, introduced by Rozsnyai [2] who performed a series of calculations for the radiative properties of hot dense plasmas [3e5]. The AA model assumes that the quantum-mechanical states of the plasma can be described by that of a selected ion and plasma-electrons in its immediate surroundings, all within a ‘neutrality sphere’ (the immersed-ion model). The effect of the other ions and plasma-electrons can be taken into account by a WignereSeitz type of boundary condition at the edge of the neutrality sphere (the ion-sphere model) or by an extra potential added to that of the central nucleus (ion-correlation model) [6]. For the electrons within the ion-sphere, an independent

* Corresponding author. Atomic and Optical Theory, Los Alamos National Laboratory, T-4 MS B283, Los Alamos, NM 87545, USA. Tel.: þ1 505 6650291; fax: þ1 505 6671931. E-mail address: [email protected] (J. Colgan). 1574-1818/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.hedp.2007.02.002

electron description was adopted, the various one-electron states were assumed to be occupied according to the FermieDirac distribution (for a given temperature and chemical potential), and the HartreeeFockeSlater potential was assumed in a self-consistent calculation to be used in a oneelectron Dirac equation. Subsequently the AA model was obtained as the implementation of density-functional theory [7,8] for the ‘immersed-ion’ model [9]. The next development was the application of adiabatic timedependent density-functional theory (ATDDFT) for these problems [10e12].1 The ATDDFT used was a generalization for the finite-temperature, immersed-ion model of the ATDDFT used by Zangwill and Soven [13] and Stott and Zaremba [14] for ground-state atoms at zero temperature. The ATDDFT is an intuitive scheme for the description of atomic electrons in a self-consistent manner via the KohneSham equation in the presence of a given time-dependent external potential. The ATDDFT has given good results for response properties of atoms and molecules in the T ¼ 0 temperature case (for a review see, e.g. Mahan and Subbaswamy [15]). However, the potential that is implied for the excited electron in ATDDFT has the 1

In Refs. [10,11] ATDDFT is referred to as RPA.

66

J. Colgan et al. / High Energy Density Physics 3 (2007) 65e75

asymptotic behavior of Q/r, where Q is the charge of the ion, instead of the correct (Q þ 1)/r (see p. 68 of Ref. [15]), and special care is needed to remove the self-interaction to obtain the correct potential (see Section 3.5 of Ref. [15]). In order to address this issue in the present work, we adopted the finite-temperature random-phase approximation (FTRPA) for the calculation of spectral properties of a neon plasma. The FTRPA was formulated as a matrix eigen-value problem by Mermin [16] and Vautherin and Vinh Mau [17] using temperature Green’s functions [18,19]. The same eigen-value equations were also obtained by des Cloizeaux [20] using a density operator technique, and who gave a very complete formulation of the problem, including the normalization conditions. FTRPA is the approximation scheme for the linear response function of a fermion system which is obtained when the time-dependent HartreeeFock (TDHF) approximation is used for the description of the fermion system at finite-temperatures in the presence of an arbitrary time-dependent external potential. The TDHF approximation in turn assumes that the two-particle density matrix factorizes in terms of the one-electron densities (with due consideration of the anti-symmetry requirement) in the presence of the external field. The FTRPA has been widely used in nuclear physics mainly for the description of giant resonances in hot nuclei [21,22], but also for the inclusion of the effect of collective states on thermodynamic functions [23,24]. The application of the FTRPA for the description of spectroscopic properties in finite-temperature plasmas was suggested by Csanak and Kilcrease [25], who also reformulated the fundamental equations of the FTRPA as coupled, partial integro-differential equations. Individual channels are defined by the removal of a particle from a (partially) occupied orbital. Thus independent particle ‘excited states’ can be created by the removal of particles from different occupied orbitals and placing them in different ‘excited’ orbitals. In the FTRPA these ‘individually excited states’ (associated with different channels) interact through channel-coupling, forming a ‘collective state’. Subsequently Csanak and Meneses [26], assuming an ion-sphere or ion-correlation model [25], performed angular-momentum and spin analysis on the coupled-channel form of the FTRPA equations obtaining a coupled system of integro-differential equations. They also introduced the approximation that assumes that the coupling in that integro-differential equation system can be neglected (in a zeroth order approximation), obtaining thereby a system of uncoupled integro-differential equations, which they called the single-channel, single-component random-phase approximation (SCRPA). They then used the SCRPA to obtain numerical results for the spectroscopic properties of He in dense, hot plasmas [26]. Csanak and Daughton [27] also used the SCRPA to obtain numerical results, using both ion-sphere and ion-correlation models, for the spectroscopic properties of He and Li in dense, hot plasmas. The purpose of this paper is to introduce channel-coupling and perform FTRPA calculations using its coupled-channel form for the description of spectroscopic properties of neon plasmas. The numerical method used is the linear algebraic method developed by Collins et al. [28] for the problem of electron-atom (ion) or molecule scattering

and which was adapted here for handling coupled-channel Eigen-value problems. In Section 2 we introduce the ion-sphere/ion-correlation model that will lend itself to a simple angular-momentum and spin analysis [29] of the FTRPA equations. As most of the relevant equations of FTRPA have previously been published, we give a complete development of the FTRPA equations in Appendix A. In Section 2, we also exhibit the operational equations for neon plasmas and introduce the numerical methods [28] used in this work. In Section 3 we present our numerical results for both zero-temperature and finite-temperature systems, and discuss their physical meaning. Finally, in Section 4, we draw some conclusions.

2. Theoretical background In this work (just as in Refs. [25e27]), we shall use the ionsphere and/or ion-correlation models. These models assume that there is a central nucleus surrounded by electrons. The ion-sphere model assumes that the system lies within a sphere centered at that nucleus, and boundary conditions on the surface of the sphere represent the plasma residing outside of the sphere (electrons and ions). The electrons inside the sphere are in thermal equilibrium with those outside. In the case of the ion-correlation model [30], we assume that electrons are located in a large box. Inside the box there is one central ion and additional ions correlated to it. Their correlated motion results in a potential that is added to the potential created by the central ion. The total charge of electrons is compensated by the charge of the central ion plus the total charge of the additional ions that are distributed inside the large box. In both models the electrons are assumed to be in thermal equilibrium, and their motion is determined by one central potential. Since the development of FTRPA has been given in several previous publications [25e27], we provide a complete description of the relevant equations in Appendix A. The coupled-channel FTRPA equations in differential form are given in Eqs. (A.19eA.21). These equations are coupled in two ways, in the n0 l0 and l indices. The channel indices n0 l0 represent the principal and orbital angular momentum quantum numbers of the average-atom orbitals occupied with fractional occupation numbers from which an electron is removed and placed into an ‘excited orbital’. The component index l represents an angular momentum component in the expansion of the excited orbital; every excited orbital (in principle) can have l ¼ 0, 1, 2, . components. The previous work of Csanak and Meneses [26] and Csanak and Daughton [27] made the ‘single-channel, single-component’ approximation, where it was assumed that coupling was negligible and the most important terms in the sums on the right-hand-side of Eq. (A.19) are the n0 l0 ¼ njlj and l ¼ li terms. These assumptions may be adequate for the helium and lithium systems considered in [26,27], but are unlikely to be good approximations for the neon systems considered here. We also note that the terms on the right-hand-side of Eq. (A.19) are independent of M and so the solutions PLM ðn0 l0 Þnl ðrÞ will also be independent of M.

J. Colgan et al. / High Energy Density Physics 3 (2007) 65e75

The diagonal parts of the first two summations on the righthand-side of Eq. (A.19) (i.e. the terms for which n0 l0 ¼ njlj and l ¼ li) can be interpreted as representing the exchange and direct contributions of the Pnj lj orbital to the HF potential SHF li , respectively. Moving these two terms to the left-handside of Eq. (A.19) results in a subtraction that creates exactly the proper potential for the excited electron denoted as nj lj SlNn , with N designating the initial number of electrons, rei calling that nnj lj represents the (fractional) occupation number of the initial orbital to be removed. This subtraction is a generalization of the zero-temperature case where the excited electron is calculated in a VN1 potential (see, e.g. [33]). The off-diagonal terms contained within the first two summations on the right-hand-side of Eq. (A.19) (for which n0 l0 s njlj and/or l s li) represent the coupling between the various channel and component functions. The last term in Eq. (A.19) can be interpreted as an orthogonality term that controls the overlap of the PLM ðnj lj Þnli ðrÞ orbital with those Pni li ðrÞ orbitals that are incorporated in that sum. In order to be able to use orbitals and energies from the average-atom code APATHY of Daughton and Snell [34], we introduce on the left-hand-side of Eq. (A.19) the local-exchange approximation, which results in replacement of the SHF li potential by the appropriate average-atom potential SAA li . (The same approximation was used by Csanak and Meneses [26] for single-channel calculations.) The major step in this replacement is the substitution of the exact exchange SHF li by the local-exchange SAA li . We note that the subtraction discussed in the previous paragraph is now no longer exact in the local-exchange approximation. However, this choice is unavoidable since we are unaware of any average-atom formalism which treats the exchange in a non-local manner. We also point out that the subtraction of the diagonal direct part of the potential still creates exactly the proper direct potential for the excited electron. In this work, we use the linear algebraic method [28] to solve this set of coupled differential equations. This method has been used with much success in electron-atom and electronmolecule scattering [35] and has been adapted here for handling coupled-channel eigen-value problems. A detailed description of the computational techniques applied in the linear algebraic method has already been given [35], and here we give a brief summary of the method in Appendix B. 2.1. Normalization The general form for normalization of the FTRPA orbitals was given by Csanak and Meneses [26] in the form: X 1  nni li  nnj lj XnnLM X nLM ¼ 1 ð1Þ i li ;nj lj ni li ;nj lj ni li ;nj lj

where XnnLM was defined as: i li ;nj lj ZN ¼ drPni li ðrÞPLM XnnLM i li ;nj lj ðn l Þnl ðrÞ: j j

0

i

In a practical calculation the number of channel indices (njlj) and component indices (li) are limited (in the cases considered here, they are two and three, respectively) and thus the sum in Eq. (2) over (njlj) is finite. For a given eigen-value unLM, the corresponding set of unnormalized solutions, PLM ðnj lj Þnli ðrÞ, are substituted into the left-hand-side of Eq. (2) and the summation is calculated. pffiffiffiffi If we denote this result by C, then functions of the form ð1= CÞPLM ðnj lj Þnli ðrÞ will give the properly normalized set of solutions of the coupled integro-differential equation system. In the T ¼ 0 temperature case the nni li and nnj lj occupation numbers are 0 and 1, respectively. Thus for T ¼ 0 we obtain the normalization condition: X

XnnLM XnLM ¼ i li ;nj lj ni li ;nj lj

ni li ;nj lj

X XZ nj l j

Z

dr 0 Pni li ðrÞPni li ðr 0 Þ

dr

ni l i

LM 0  PLM ðnj lj Þnli ðrÞPðnj lj Þnli ðr Þ XZ LM ¼ drPLM ðn l Þnl ðrÞPðn l Þnl ðrÞ j j

nj l j l i



XZ

Z dr

nj l j l i

i

j j

i

dr 0 PLM ðn l Þnl ðrÞ j j

i

0 0  PLM ðn l Þnl ðr Þrli ðr; r Þ ¼ 1 j j

i

ð3Þ

where rli ðr; r 0 Þ is the lith component of the ground-state density matrix, defined as: rli ðr; r 0 Þ ¼

X

Pni li ðrÞPni li ðr 0 Þ

ð4Þ

ni

where ni runs through the principal quantum numbers associated with li angular-momentum ground-state orbitals. The normalization procedure for arbitrary T is the same as for T ¼ 0 except that of course, in the case of arbitrary T, the occupation numbers nni li and nnj lj generally become some fraction between 0 and 1.

2.2. Oscillator strength We also calculate optical oscillator strengths for a neon plasma as this is a convenient quantity for examining the effects of coupling between the channels. At zero-temperature and density, comparisons can be made with previous studies designed to examine bound-state properties of isolated atoms. We do note, however, that our theory can easily be extended to examine other spectral quantities of ions in dense plasmas. We start from the definition of the dipole matrix element given by Csanak and Meneses [26]: Dn ¼

ð2Þ

67

Z

dr!1

Z

ds1 z1 cn ðr!1 s1 ; ! r 1 s1 Þ

ð5Þ

with cn as defined in Eq. (A.11), and z1 ¼ r1 cos q1. By performing the integrals over the coupled spin and

J. Colgan et al. / High Energy Density Physics 3 (2007) 65e75

68

spherical harmonics defined in Eqs. (A.12) and (A.13), we obtain: X 1 l l0 2ð2l þ 1Þð2l0 þ 1Þ1=2 DnLM ¼ 3 0 0 0 l;l0  ð  1Þ

ll0

ZN dL1 dM0

dr1 r1

X

L¼1;M¼0 Pðn ðr1 ÞPn0 l0 ðr1 Þ; 0 l0 Þnl

1 P 1  e  u HlAA 2p 3 P Pð2pÞ3s ðrÞ i ¼0 8 <

1 ¼ n2p  : 2p

ð6Þ



n0

0

1 2p

ZN

ZN

1P

dr 0 y1 ðr; r 0 ÞPð2sÞ3p ðr 0 ÞP2s ðr 0 ÞP2p ðrÞ

0 1P

dr 0 y1 ðr; r 0 ÞPð2pÞ3s ðr 0 ÞP2p ðr 0 ÞP2p ðrÞ

0

pffiffiffi ZN 1P 2 dr 0 y1 ðr; r 0 ÞPð2pÞ3d ðr 0 ÞP2p ðr 0 ÞP2p ðrÞ þ 2p

1

where we restrict ourselves to dipole transitions from the S ground state to singlet states, i.e. to 1P states. We define the optical oscillator strength (OS) as: OS ¼ 2unLM jDnLM j

2

0

ð7Þ

þ

where unLM is the transition energy in atomic units from the ground state of a neon atom. We note that, when reduced to the single-channel form, this definition is in agreement with the oscillator strength defined by Csanak and Meneses [26].

1 4p

ZN

1P

dr 0 y0 ðr; r 0 ÞP2p ðr 0 ÞP2p ðr 0 ÞPð2pÞ3s ðrÞ

0

þ

1 4p

ZN

1P

dr 0 y1 ðr; r 0 ÞP2p ðr 0 ÞP2s ðr 0 ÞPð2sÞ3p ðrÞ

0

9 pffiffiffi ZN = 1 2 P  dr 0 y2 ðr; r 0 ÞP2p ðr 0 ÞP2p ðr 0 ÞPð2pÞ3d ðrÞ ; ; 4p

2.3. The FTRPA equations for neon

ð9Þ

0

In this section we write out the explicit form of the FTRPA Eq. (A.19) for the case of the 2l ‘hole’ to 3l0 ‘particle’ dipoleallowed transitions in neon. Therefore, we consider Eq. (A.19) only with two coupled-channels (2s, 2p), which are excited via a dipole transition (L ¼ 1) to any one of the 3s, 3p or 3d component states. Explicitly stating the coupled equations for this example serves to inform the subsequent discussion and to illustrate how the angular algebra simplifies considerably, resulting in a compact form. With these considerations, the coupled equations take the form:

and

1 P 1P P HlAA  e  u 2p 3 ð2pÞ3d i ¼2 8 < pffiffi2ffi ZN 1P ¼ n2p þ dr 0 y1 ðr; r 0 ÞPð2sÞ3p ðr 0 ÞP2s ðr 0 ÞP2p ðrÞ : 2p 0

pffiffiffi ZN 1P 2 dr 0 y1 ðr; r 0 ÞPð2pÞ3s ðr 0 ÞP2p ðr 0 ÞP2p ðrÞ þ 2p 0



1 P 1 HlAA  e  u 2s 3 P Pð2sÞ3p ðrÞ i ¼1 8 <

1 ¼ n2s  : 2p ZN

ZN



1 p

ZN

1P

dr 0 y1 ðr; r 0 ÞPð2pÞ3d ðr 0 ÞP2p ðr 0 ÞP2p ðrÞ

0 1P

1 þ 4p

dr 0 y1 ðr; r 0 ÞPð2sÞ3p ðr 0 ÞP2s ðr 0 ÞP2s ðrÞ

0

ZN

1P

dr 0 y0 ðr; r 0 ÞP2p ðr 0 ÞP2p ðr 0 ÞPð2pÞ3d ðrÞ

0

dr 0 y1 ðr; r 0 ÞPð2pÞ3s ðr 0 ÞP2p ðr 0 ÞP2s ðrÞ

pffiffiffi ZN 1P 2 dr 0 y2 ðr; r 0 ÞP2p ðr 0 ÞP2p ðr 0 ÞPð2pÞ3s ðrÞ  4p

pffiffiffi ZN 1P 2 dr 0 y1 ðr; r 0 ÞPð2pÞ3d ðr 0 ÞP2p ðr 0 ÞP2s ðrÞ þ 2p

pffiffiffi ZN 1P 2 dr 0 y1 ðr; r 0 ÞP2p ðr 0 ÞP2s ðr 0 ÞPð2sÞ3p ðrÞ  4p

1  2p

1P

0

0

0

1 þ 4p þ

1 4p

ZN 0 ZN

0 1P

dr 0 y0 ðr; r 0 ÞP2s ðr 0 ÞP2s ðr 0 ÞPð2sÞ3p ðrÞ

þ

0 1P

dr 0 y1 ðr; r 0 ÞP2s ðr 0 ÞP2p ðr 0 ÞPð2pÞ3s ðrÞ

0

9 pffiffiffi ZN = 1 2 P dr 0 y1 ðr; r 0 ÞP2s ðr 0 ÞP2p ðr 0 ÞPð2pÞ3d ðrÞ ;  ; 4p 0

1 4p

ZN

ð8Þ

1P

9 =

dr 0 y2 ðr; r 0 ÞP2p ðr 0 ÞP2p ðr 0 ÞPð2pÞ3d ðrÞ : ;

ð10Þ

In these equations n2s and n2p are the values of the Fermie Dirac distribution functions (see Eq. (A.4)) associated with a given ei for the 2s and 2p orbitals, respectively, for the temperature T and chemical potential m selected. In the zero-temperature case both of these coefficients are equal to 1, and in the finite-temperature case these coefficients are fractional

J. Colgan et al. / High Energy Density Physics 3 (2007) 65e75

values between 0 and 1. We note that these three coupled equations have three ‘sets’ of solutions, each one corresponding to a 1P excited state case where the p, s, or d component function dominates, and where each set has a different eigen-value or transition energy u3 1 P. We have chosen here to denote the solutions corresponding to the first (n ¼ 1) excitation with a ‘3’, since these solutions are analogous to the results obtained from a traditional configuration-interaction approach that produces the usual ‘3l’ excited levels after diagonalization of the system Hamiltonian. In Eqs. (8e10), we have omitted the ‘orthogonality term’, since this term was not included in the zero-temperature calculations in the following section. This orthogonality term is included, however, in the finite-temperature and density calculations discussed in Section 3.2. 3. Results 3.1. Zero-temperature and density comparisons As a check on our method and computer programs, we first solved the FTRPA equations for various neutral atoms at zero temperature and density. In the zero-temperature limit, our equations become identical to TammeDancoff-approximation equations [36]. These calculations, presented in Table 1, were made to compare with previous RPA calculations made for an isolated atom by Amusia [33]. To be consistent with our finitetemperature and density calculations made using the averageatom approach, we also employ a local HartreeeFockeSlater (HFS) potential in these zero-temperature and density calculations. This potential was calculated from atomic orbitals generated using the CATS program [37,38]. The previous calculations of Amusia [33] used a non-local HartreeeFock (HF) exchange potential, which can be considerably more accurate, especially for neutral atoms. The comparison of our FTRPA calculations at zero-temperature and density with the RPA calculations of Amusia [33], presented in Table 1, show some differences in the oscillator strengths for the

Table 1 Oscillator strengths for various noble gas atoms at zero-temperature and density

He 1s-2p 1s-3p 1s-4p Ne 2p-3s 2p-3d Ar 3p-4s 3p-3d

OSRPA/HF

OSFTRPA/HFS

OSCATS/HF

OSCATS/HFS

0.252 0.070 0.029

0.394 0.052 0.019

0.315 0.096 0.005

0.383 0.128 0.007

0.163 0.021

0.143 0.062

0.127 0.012

0.153 0.018

0.298 0.167

0.232 0.838

0.240 0.193

0.265 1.970

We compare the calculations of Amusia [33] (OSRPA/HF), made using a Hartreee Fock approach, with the current FTRPA calculations (OSFTRPA/HFS) which were made using a HartreeeFockeSlater potential. For comparison we also show calculations using the CATS code [37,38] using both a HartreeeFock (OSCATS/HF) and HartreeeFockeSlater (OSCATS/HFS) approach.

69

various transitions in He, Ne, and Ar, which are likely due to the use of these different potentials. We remark that for the lowest energy transitions, the agreement between the current calculations and those of Amusia is fairly good, but for the higher transitions the agreement worsens. To further illustrate how the HF and HFS exchange terms can produce quite different oscillator strengths, we also show in Table 1 two sets of calculations made using the CATS program [37], in which the only difference between the two sets of calculations was the use of a HF versus a HFS exchange potential. The CATS program [37], which is an adaptation of Cowan’s atomic structure programs [38], was used to calculate HartreeeFock orbitals, energy levels, and oscillator strengths. Configuration-interaction effects between the various excited configurations in the calculation were included in the usual way, by diagonalizing the system Hamiltonian. The large differences in the oscillator strengths obtained from these two calculations indicate that a plausible reason for the difference between our FTRPA calculations and the RPA calculations of Amusia [33] are due to the different exchange potentials employed. Further work is necessary to formulate a non-local HF exchange approach which can be used for both finite-temperature and zero-temperature calculations. 3.2. Finite-temperature and density results We now turn to an exploration of the solution of the FTRPA equations for systems at finite-temperatures and densities. Here, we examine, the effects of channel and component coupling in neon plasmas (Z ¼ 10) at various temperatures and densities. As previously discussed, the average-atom code APATHY [34] was used to generate average-atom orbitals and potentials for six different density/temperature pairs. We note that, when these average-atom calculations are extended to compute the equation-of-state, they produce total plasma pressures in reasonable agreement with the SESAME database [39] produced at Los Alamos National Laboratory. The APATHY code solves the KohneSham equation for both the bound and continuum wave functions to construct the electron density within the average-atom model, where the solutions for both bound and continuum equations were obtained fully quantum-mechanically. These equations are solved self-consistently, with the correlation between the average-atoms computed using hypernetted chain theory [34]. The average ionization values, as well as the chemical potential for each case, which are obtained from the self-consistent solution of these equations, are listed in Table 2, along with the various plasma parameters, occupation numbers, and energies of the first six average-atom orbitals. The temperature/density region chosen for this set of calculations is such that the average number of electrons present in the average atom is between 4 and 7. We present oscillator strengths for the solutions of the FTRPA equations for 1P excited states where the 3p, 3s, and 3d components are dominant, for a neon plasma at various densities and temperatures, in Tables 3e5. We compare the single-channel FTRPA (SC) results with the coupled-channel

J. Colgan et al. / High Energy Density Physics 3 (2007) 65e75

70

Table 2 Average-atom parameters from the APATHY code at various temperatures and densities for a neon plasma T ¼ 20.0 eV

T ¼ 25.0 eV w

E (au)

w

E (au)

r ¼ 0.01 (g/cc)

Z* ¼ 3.83 m ¼ 6.16 G ¼ 1.14

1s 2s 2p 3s 3p 3d

2.00 1.22 2.16 0.03 0.06 0.08

35.03 4.85 4.10 1.38 1.18 0.96

Z* ¼ 4.48 m ¼ 6.33 G ¼ 1.25

1s 2s 2p 3s 3p 3d

2.00 0.91 1.67 0.02 0.06 0.08

36.20 5.65 4.95 1.76 1.57 1.37

r ¼ 0.05 (g/cc)

Z* ¼ 3.20 m ¼ 4.73 G ¼ 1.36

1s 2s 2p 3s 3p 3d

2.00 1.36 2.54 0.06 0.13 0.16

33.97 4.03 3.25 0.88 0.69 0.46

Z* ¼ 3.80 m ¼ 4.89 G ¼ 1.53

1s 2s 2p 3s 3p 3d

2.00 1.09 2.09 0.05 0.12 0.16

34.90 4.66 3.92 1.15 0.95 0.72

r ¼ 0.10 (g/cc)

Z* ¼ 2.95 m ¼ 4.11 G ¼ 1.46

1s 2s 2p 3s 3p 3d

2.00 1.41 2.71 0.08 0.18 0.23

33.52 3.67 2.88 0.66 0.48 0.25

Z* ¼ 3.65 m ¼ 4.23 G ¼ 1.78

1s 2s 2p 3s 3p 3d

2.00 1.17 2.28 0.07 0.17 0.23

34.32 4.20 3.44 0.87 0.67 0.44

Z* is the average ionization, m is the chemical potential, and G is the plasma coupling parameter. We also list the energies, E, and occupation numbers, w, of the first six average-atom orbitals.

FTRPA results (CC). The FTRPA excited-state absolute energies (E3l) are also listed, and were found by adding the transition energy u3 1 P to the appropriate average-atom ‘hole’ orbital energy, E2l. The orthogonality term, as defined in Eq. (A.22), was included in all of these calculations. This term, as well as the exchange terms in the FTRPA equations, was observed to have a significant effect on the excited-state energies (E3l) and on the amplitudes of orbitals, and a corresponding effect on the resulting oscillator strengths. This behavior is unlike that observed in the single-channel FTRPA calculations presented previously for helium [27], where the exchange and orthogonality terms made little difference. The reasons for this difference are evident: the previous calculations for helium were in a region where the atoms were almost fully stripped and so there were few (or no) significantly occupied average-atom orbitals which could increase the

magnitude of the exchange and orthogonality terms. In these previous calculations, the average atoms were almost completely in the 1s state. On the other hand, our current calculations are performed for neon, where the coupling effects may be expected to be significant. While the effects of coupling on the transition energies is observed to be minimal, the effect on the oscillator strengths are rather significant. For the solution where the 3p component dominates, we see that the coupled-channel calculations result in oscillator strengths which are around 15e30% lower than the single-channel calculations over the range of temperatures and densities presented. For the solution in which the 3s component dominates, a similar trend is evident, with the coupled-channel OS now around 40e50% lower than those for the single-channel calculations. This trend is continued for the solution in which the 3d component dominates,

Table 3 Excited-state energies, E3p, transition energies, u, and oscillator strengths for the 2s-3p transition to a 1P level in a neon plasma at various temperatures (T ) and densities (r) T ¼ 20.0 eV

r (g/cc)

T ¼ 25.0 eV

E3p

u

OS

E3p

u

OS

0.01

APATHY SC CC

1.180 1.518 1.515

3.672 3.335 3.338

0.323 0.373 0.235

1.570 1.839 1.837

4.083 3.813 3.815

0.400 0.431 0.281

0.05

APATHY SC CC

0.690 1.029 1.028

3.337 2.996 2.997

0.253 0.314 0.285

0.950 1.247 1.245

3.707 3.413 3.415

0.324 0.365 0.250

0.10

APATHY SC CC

0.480 0.817 0.815

3.183 2.849 2.851

0.218 0.286 0.176

0.670 0.972 0.970

3.525 3.227 3.229

0.283 0.330 0.265

We compare calculations made in the single-channel FTRPA approximation (SC) with coupled-channel FTRPA calculations (CC), and with calculations made using the ‘excited’ orbitals obtained directly from the average-atom (APATHY) calculations. All energies are given in atomic units.

J. Colgan et al. / High Energy Density Physics 3 (2007) 65e75

71

Table 4 Excited-state energies, E3s, transition energies, u, and oscillator strengths for the 2p-3s transition to a 1P level in a neon plasma at various temperatures (T ) and densities (r) T ¼ 20.0 eV

r (g/cc)

T ¼ 25.0 eV

E3s

u

OS

E3s

u

OS

0.01

APATHY SC CC

1.380 1.571 1.571

2.723 2.529 2.529

0.166 0.202 0.127

1.760 1.924 1.924

3.181 3.021 3.021

0.159 0.197 0.092

0.05

APATHY SC CC

0.880 1.085 1.085

2.371 2.163 2.163

0.166 0.196 0.140

1.150 1.329 1.329

2.769 2.588 2.588

0.161 0.194 0.110

0.10

APATHY SC CC

0.660 0.874 0.874

2.213 2.005 2.005

0.163 0.191 0.144

0.870 1.051 1.051

2.576 2.389 2.389

0.160 0.189 0.116

We compare calculations made in the single-channel FTRPA approximation (SC) with coupled-channel FTRPA calculations (CC), and with calculations made using the ‘excited’ orbitals obtained directly from the average-atom (APATHY) calculations. All energies are given in atomic units.

where the channel-coupling results in an oscillator strength that is around 30% lower. We also list the transition energies and oscillator strengths that are obtained if one uses the average-atom ‘excited’ orbitals, P3l, to compute the oscillator strength. For all cases, the energies of the 3l excited states, P(2l )3l0 , computed from the FTRPA equations are quite different from the energies of the average-atom ‘excited’ 3l orbitals. We recall that the FTRPA excited states are computed in the proper SNnnj lj potential for the excited electron (see Section 2), whereas the average-atom ‘excited’ orbitals are computed in the unphysical SN potential. Consequently, the FTRPA excited states are more tightly bound than the average-atom ‘excited’ orbitals. This difference results in transition energies for the averageatom calculation which are typically higher than the FTRPA transition energies by around 5e10%. Interestingly, the oscillator strengths computed from the average-atom ‘excited’ orbitals are quite near those calculated from the SC FTRPA approach, for several cases, where the average-atom oscillator strength falls between the SC and CC FTRPA oscillator strength calculations. We regard this agreement as somewhat fortuitous due to the unphysical nature of the average-atom ‘excited’ orbitals, as previously discussed. We also note that

for the solution in which the 3d component dominates, which results in a large oscillator strength, the average-atom approach over-estimates the oscillator strength by around a factor of two. We also note some trends as a function of density and temperature. In Fig. 1, we show the oscillator strengths at a temperature of 25 eV, and as a function of density. We compare the oscillator strengths calculated from the single-channel FTRPA equations with those from the full coupled-channel FTRPA equations, and with oscillator strengths computed using the AA ‘excited’ orbitals. The oscillator strengths from the AA calculations fall between those from the SC and CC FTRPA calculations for the transitions where the 3p and 3s components dominate (Fig. 1a and b). The CC calculations differ from the SC calculations by between 20 and 60% for these cases. For the transition where the 3d component dominates, the AA calculations are significantly higher than the SC and CC FTRPA calculations. The SC and CC calculations differ by around 50% over the density range considered. As the density is increased for a given temperature, the excited-state energy increases due to pressure ionization. The resulting transition energy between the two bound states also decreases as the density is increased; this behavior is a result of the bound states becoming squeezed closer together at

Table 5 Excited-state energies, E3d, transition energies, u, and oscillator strengths for the 2p-3d transition to a 1P level in a neon plasma at various temperatures (T ) and densities (r) T ¼ 20.0 eV

r (g/cc)

T ¼ 25.0 eV

E3d

u

OS

E3d

u

OS

0.01

APATHY SC CC

0.960 1.113 1.114

3.144 2.987 2.988

2.490 1.455 0.971

1.370 1.494 1.494

3.578 3.450 3.450

2.981 1.280 0.862

0.05

APATHY SC CC

0.460 0.620 0.620

2.792 2.628 2.628

1.906 1.301 0.917

0.720 0.868 0.868

3.192 3.049 3.049

2.452 1.277 0.902

0.10

APATHY SC CC

0.250 0.415 0.415

2.625 2.464 2.464

1.551 1.174 0.844

0.440 0.584 0.584

3.001 2.856 2.856

2.113 1.204 0.867

We compare calculations made in the single-channel FTRPA approximation (SC) with coupled-channel FTRPA calculations (CC), and with calculations made using the ‘excited’ orbitals obtained directly from the average-atom (APATHY) calculations. All energies are given in atomic units.

J. Colgan et al. / High Energy Density Physics 3 (2007) 65e75

72

0.4

(a)

AA SC CC

2s-3p

0.2

OS

0 0.001

0.2

(b)

0.01

0.1

1

0.01

0.1

1

0.1

1

2p-3s

0.1 0 0.001 4 3

(c)

2p-3d

2 1 0 0.001

0.01

density (g/cc) Fig. 1. Oscillator strengths for neon 1P excitations as a function of density, at a temperature of 25 eV, using our coupled-channel approach. We show again results for three transitions; (a) the case where the 3p component is dominant, (b) the case where the 3s component is dominant, and (c) the case where the 3d component is dominant. We compare AA results (dashed lines) with single-channel FTRPA results (dot-dashed lines) and coupled-channel results (solid lines).

higher densities. As the density is further increased, the excited state will eventually transition to the continuum, and the oscillator strength will then drop to zero. In Fig. 2, we show the oscillator strengths as a function of temperature, and at a constant density of 0.01 g/cc. We again compare the oscillator strengths calculated from the singlechannel FTRPA equations with those from the full coupledchannel FTRPA equations. As the temperature is increased, 0.8 0.6

(a)

2s-3p

AA SC CC

0.4 0.2 0 16 0.4

OS

0.3

20

(b)

24

28

32

36

40

24

28

32

36

40

24

28

32

36

40

2p-3s

0.2 0.1 0 16 4 3

20

(c)

2p-3d

2 1 0 16

20

temperature (eV) Fig. 2. Oscillator strengths for neon 1P excitations as a function of temperature, at a density of 0.01 g/cc, using our coupled-channel approach. We show results for three transitions; (a) the case where the 3p component is dominant, (b) the case where the 3s component is dominant, and (c) the case where the 3d component is dominant. We compare AA results (dashed lines) with single-channel FTRPA results (dot-dashed lines) and coupled-channel results (solid lines).

the excited-state orbitals become more tightly bound. This behavior is probably due to the increase in the number of free electrons as the temperature is increased, so that the remaining bound electrons feel more of the attractive nuclear potential. The transition energy also increases with temperature since the core (ground state) electrons will be more tightly bound than the excited electrons. Again, the oscillator strengths for all possible 3l solutions display different trends. The oscillator strength for the dominant 3p component transition generally increases with temperature, for the SC, CC, and AA solutions. The oscillator strengths corresponding to the solutions in which the 3s component is dominant and in which the 3d component is dominant decrease with increasing temperature, for all these calculations, except for the AA, 3d results, where the trend is strongly increasing. For the solution in which the 3s component is dominant, the difference between the SC and CC results increases as the temperature is raised, until the difference at the highest temperature considered (35 eV) is almost 70%. These differences further underscore the importance of including the channel-coupling in the calculations. We also remark that several numerical issues prevent us from extending these current calculations to lower temperatures at this time. As previously noted [34], the average-atom code APATHY can give inconsistent results at low temperature, due to complications arising from the use of an effective interaction potential in the treatment of the ions in the average-atom model. Discontinuities in the ion radial distribution function may occur [34] and further work is required to correct this anomaly. We also found that at lower temperatures, the energies of the FTRPA solutions in which the 3p and 3s components are labelled dominant become almost degenerate, which requires a more refined numerical analysis. Finally, at quite low temperatures, some of the excited states forming part of the coupled-channel solution can be pushed into the continuum by pressure ionization. Since the resonance treatment of a bound state pushed into the continuum is not treated correctly in our current average-atom model, discontinuous transitions can also be expected in this region. All of these aspects will be the subject of further work in this area. Finally, we recall that the single-channel transition energies are very similar to the coupled-channel transition energies, for all temperature and density cases, although the oscillator strengths can be quite different, due to differences in the component functions. This behavior implies that coupled-channel effects on any spectra will be limited to the magnitude of spectral lines, rather than to their position. Presently, there is no other theoretical or experimental work with which to compare these new coupled-channel finite-temperature and density calculations. We look forward to comparing our results with any spectroscopic properties of plasmas under investigation by other groups as they become available. 4. Conclusions In this paper, we have expanded previous work [25e27] on the finite-temperature random-phase-approximation (FTRPA) by including coupled-channel effects in solving the FTRPA

J. Colgan et al. / High Energy Density Physics 3 (2007) 65e75

equations. We have used the linear-algebraic method [28] for the solution of the coupled equations, which relies upon the average-atom code APATHY [34] to generate the temperatureand density-dependent average-atom orbitals. Our method has been tested by comparing results at zero density and temperature with independent calculations and we have presented, for the first time, coupled-channel oscillator strengths for neon plasmas at various temperatures and densities. The effect of coupling on these quantities was observed to be quite significant (30e50%). We intend to develop this work in a number of directions. First, we aim to calculate more spectroscopic quantities so that comparisons may be made with previous work, and the effects of coupling on these spectroscopic plasma properties may be investigated more fully. We intend to explore the possibility of using a fully non-local HartreeeFock exchange potential in the average-atom formalism instead of the localexchange approximation used currently. Also, we aim to formulate the FTRPA equations in the canonical ensemble, so that plasma properties may be computed for each ion stage within a plasma. This approach will be of prime importance as it will allow direct comparison with other kinetics modeling calculations which use detailed atomic data obtained without temperature/density effects included from the outset. It is expected that, particularly at high densities, the temperature and density effects on the atomic orbitals may be significant. Work on these extensions is well underway.

Acknowledgments We thank W. Daughton for invaluable advice in the running of the APATHY program and D. Kilcrease for useful discussions. This work was performed under the auspices of the US Department of Energy through the Los Alamos National Laboratory.

73

the potential originating from the other ions, rHF(x, x0 ) is the HF density matrix of the electrons, which can be written in the form: X rHF ðx; x0 Þ ¼ ni ji ðxÞji ðx0 Þ ðA:2Þ i

where ji(x) are the solutions of the HF equations,  Z2 HHF ji ðxÞh  V2 þ SHF ji ðxÞ ¼ ei ji ðxÞ 2m

ðA:3Þ

with ji referring to the associated eigen-value and, ni ¼

1 1 þ ebðei mÞ

ðA:4Þ

is the FermieDirac distribution function with b ¼ 1/kT, where k is the Boltzmann constant, T is the absolute temperature and m is the chemical potential. Relativistic effects and spineorbit coupling are also neglected. We hope to include both of these effects in our future work. The aim of the FTRPA is to calculate the linear response function, which is obtained in the form, FTRPA

L



x1 ; x2 ; x10 ; z



    X cn x1 ; x10 cn x2 ¼ un  z n

ðA:5Þ

where cn(x) h cn(x, x). In Eq. (A.5) un can be interpreted as the ‘excitation energy’ of the finite-temperature system and cn(x) as the transition density from the initial (finite-temperature) state to the nth excited state of the system. If cn(x, x0 ) is expanded on the basis of the HF eigenfunctions, ji(x), cn ðx; x0 Þ ¼

X

cnij ji ðxÞjj ðx0 Þ

ðA:6Þ

i;j

then the FTRPA eigen-value equation is obtained in the form: Appendix A



The starting point for the description of the FTRPA is the finite-temperature (or thermal) [31] HartreeeFock (FTHF) approximation [32]. The HartreeeFock (HF) potential is defined by the formula: Z SHF ðx; x0 Þ ¼  dðx  x0 Þ þ Vext ðrÞdðx  x0 Þ r Z 0 0 r ! r ÞrHF ðx1 ; x1 Þdx1 þ dðx  x Þ Vð! 0  Vð! r ! r ÞrHF ðx0 ; xÞ

 X n  ei  ej cnij  ni  nj ckl ½hiljyjjki  hiljyjkji ¼ un cnij k;l

ðA:7Þ where hiljyjkji ¼

Z

Z dx

dx0 ji ðxÞjj ðx0 Þ

0  Vð! r ! r Þjk ðxÞjl ðx0 Þ:

ðA:1Þ

where xhð! r ; sÞ refers to both the spatial ð! r Þ and spin (s) coordinates of the electrons, integration over x1 signifies integration over ! r 1 in coordinate space and summation over s1 in 0 0 spin space, rhj! r j, Vð! r ! r Þ ¼ e2 =j! r ! r j is the Coulomb interaction potential, Z is the nuclear charge, Vext(r) is

ðA:8Þ

Introducing the function: cnj ðxÞ ¼

X

cnij ji ðxÞ

ðA:9Þ

then permits the reformulation of the FTRPA eigen-value problem given by Eq. (A.7) as a coupled-channel system in the form,

J. Colgan et al. / High Energy Density Physics 3 (2007) 65e75

74



 HHF  ej  un cnj ðxÞ ¼ nj

XZ l

 nj

dx1 jj ðx1 ÞVð! r ! r 1 Þjl ðx1 Þcnl ðxÞ

XZ

dx1 cnl ðx1 ÞVð! r ! r 1 Þjl ðx1 Þjj ðxÞ

l

þ ni

XZ

Z dx1

l

 ni

Z2 d2 Z2 li ðli þ 1Þ þ þ SHF ðA:17Þ li : 2r 2 2m dr 2 LM 0 The Xn;l;l 0 ðr; r Þ functions defined by Eq. (A.11) can be expanded in the form, ¼ HlHF i

XZ

Z dx1

LM 0 rr 0 Xn;l;l 0 ðr; r Þ ¼

HF

dx2 r ðx1 ; xÞjj ðx2 Þ

 Vð! r 1! r 2 Þjl ðx2 Þcnl ðx1 Þ:

ðA:10Þ

Under the assumptions of the ion-cell or ion-correlation model, the electrons move in a spherically symmetric potential. Csanak and Meneses [26] assumed that cn(x, x0 ) can be factorized in the form 0 0 cn ðx; x0 Þhcn ð! r s; ! r s0 Þ/cnLM ð! r ;! r ÞzSn MSn ðs; s0 Þ

h

LM 0 LM Xn;l;l r ;br 0 ÞzSn MSn ðs; s0 Þ 0 ðr; r ÞYll0 ðb

 enj lj  unLM PLM HlHF ðnj lj Þnli ðrÞ i

¼ nnj lj

ðA:11Þ

 

where ð  1Þ

1=2þm0s



ms ;m0s

: l;l0 ;l

X

l l ;ll0

Ali j

ðA:12Þ

m;m0

with Ylm ðbr Þ a spherical harmonic as defined by Edmonds [29]. In this paper, we shall be interested in dipole transitions; therefore, we shall only search for singlet excited states, restricting our consideration to Sn ¼ 0, MSn ¼ 0 states. Since in the ion-sphere or ion-correlation model, the external potential in which the electrons move is spherically symmetric, the HF orbitals factorize in the form:

and the HF potential can be expanded as: X 0 SHF ðx; x0 Þ ¼ SHF r ;br 0 Þds;s0 : l ðr; r ÞPl ðb

ðA:14Þ

N XZ n0

0

dr 0 yl ðr; r 0 ÞPnj lj ðr 0 ÞPn0 l0 ðr 0 ÞPLM ðn0 l0 Þnl ðrÞ

0

nni li Pni li ðrÞInnli lii ;nj lj

9 = ;

ðA:19Þ

l l ;ll0

1=2 1 ð2li þ1Þð2lj þ1Þð2lþ1Þð2l0 þ1Þ 4p   0  l li l l l lj L li lj Lþl ð1Þ ð2lþ1Þ ; 000 000 l l0 l

Ali j ¼

l l ;ll0

Bli j

¼

ðA:20Þ

1=2 1 ð2li þ 1Þð2lj þ 1Þð2l þ 1Þð2l0 þ 1Þ 4p    li lj L l l0 L 2 dlL ; 0 0 0 0 0 0

ðA:21Þ

and Innli lii ;nj lj

¼

X l;l0 ;l

XZ

ZN

N

l l ;ll0 Bli j

n0

dr

0

0 0  PLM ðn0 l0 Þnl ðr ÞPn0 l0 ðr Þ þ

dr 0 Pni li ðrÞPnj lj ðrÞyl ðr; r 0 Þ

0

X l;l0 ;l

XZ

ZN

N

l l ;ll0 Ali j

n0

0  Pnj lj ðr 0 Þyl ðr; r 0 ÞPLM ðn0 l0 Þnl ðrÞPn0 l0 ðr Þ:

0

dr

dr 0 Pni li ðrÞ

0

ðA:22Þ

ðA:15Þ

l

The Pni li ðrÞ radial functions are eigen-functions of the HlHF i operator such that, HlHF Pni li ðrÞ ¼ eni li Pni li ðrÞ i

n0

0 0 dr 0 yl ðr; r 0 ÞPLM ðn0 l0 Þnl ðr ÞPn0 l0 ðr ÞPnj lj ðrÞ

where

is a coupled spin function with cms ðsÞ referring to the Pauli spin function, and X r ;br 0 Þ ¼ ðlm; l0 m0 jll0 LMÞYlm ðbr ÞYl0 m0 ðbr 0 Þ ðA:13Þ YllLM 0 ðb

Pn l ðrÞ ji ðxÞhji ð! r ; sÞ ¼ i i Yli mi ðbr Þcmsi ðsÞ; r

X

XZ

N

l l ;ll0

Bli j

ni

 1 1 1 1  m0s ; ms

Sn MSn 2 2 22

 cm0s ðsÞcms ðs0 Þ

where

8
l;l0 ;l

X

ðA:18Þ

When the angular momentum and spin factorizations given by Eqs. (A.11eA.15) are introduced into Eq. (A.10) and the definitions given by Eqs. (A.17) and (A.18) used, we obtain the following coupled system of integro-differential equations for the PLM ðn0 l0 Þnl ðrÞ functions,

l;l0

zSn MSn ðs; s0 Þ ¼

0 PLM ðn0 l0 Þnl ðrÞPn0 l0 ðr Þ:

n0

dx2 rHF ðx1 ; xÞVð! r 1! r 2 Þjl ðx2 Þcnl ðx2 Þ

l

X

X

ðA:16Þ

The quantity vl(r1, r2) is the multipole component in the expansion of the Coulomb potential given by: yl ðr1 ; r2 Þ ¼

4p rlþ1

with r< ¼ min(r1, r2) and r> ¼ max(r1, r2).

ðA:23Þ

J. Colgan et al. / High Energy Density Physics 3 (2007) 65e75

Appendix B

References

We re-write Eq. (A.19) in the form: 

1 d2 li ðli þ 1Þ AA þ þ Sli  enj lj  unLM PLM  ðnj lj Þnli ðrÞ 2 dr 2 2r 2 X  LM ¼ ðrÞ PLM VSLM ðrÞ þ Wex ðn0 l0 Þnl ðrÞ n0 l0 ;l

 LLM ðrÞPLM ðnj lj Þnli ðrÞ;

ðB:1Þ

where, in this equation, we have now used atomic units, VLM S (r) represents the static part of the potential on the righthand-side of Eq. (A.19) given by: N X l l ;ll0 Z i j LM VS ðrÞ ¼ nnj lj Al dr 0 yl ðr; r 0 ÞPnj lj ðr 0 ÞPn0 l0 ðr 0 Þ; ðB:2Þ l

0

WexLM(r) is an operator that represents the exchange part of the potential on the right-hand-side of Eq. (A.19), given by: N X li lj ;ll0 Z LM LM Wex ðrÞPðn0 l0 Þnl ðrÞ ¼ nnj lj Bl dr 0 yl ðr; r 0 Þ l

0

0 0  PLM ðn0 l0 Þnl ðr ÞPn0 l0 ðr ÞPnj lj ðrÞ;

ðB:3Þ

and finally an orthogonality operator can be defined according to: X LLM ðrÞPLM nni li Pni li ðrÞInnli lii ;nj lj : ðB:4Þ ðnj lj Þnli ðrÞ ¼ ni

This set of coupled differential equations is then converted into coupled integral equations by writing: XZ

N

ðrÞ PLM ðnj lj Þnli

¼G

1

ð nj l j Þ

ðrÞ þ 2 nl i

n0 l0 ;l

h Gðn0 l0 Þnl ðr; r 0 Þ SAA li þ VS ðrÞ

0

i 0 0 þWex ðrÞ þ LðrÞ PLM ðn0 l0 Þnl ðr Þdr

ðB:5Þ

where, ( 0

75

Gðn0 l0 Þnl ðr; r Þ ¼

G1ðn0 l0 Þnl ðrÞG2ðn0 l0 Þnl ðr 0 Þ; r > r 0 G1ðn0 l0 Þnl ðr 0 ÞG2ðn0 l0 Þnl ðrÞ; r < r 0

ðB:6Þ

and G1ðn0 l0 Þnl ðrÞ ¼ kn rfl ðkn rÞ G2ðn0 l0 Þnl ðrÞ ¼ rhl ðkn rÞ:

ðB:7Þ

Here fl(knr) and hl(knr) are regular and irregular Coulomb functions, respectively, with kn the channel electron momenta. The linear algebraic approach proceeds by introducing a quadrature to the integrals and a discrete mesh to the functions and solving the equations on a numerical grid [35]. By further re-writing these equations in matrix form, standard numerical methods can be applied to solve the set of equations in an efficient manner.

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