# The number of populated electronic configurations in a hot dense plasma

Menahem Krief<EMAIL_ADDRESS>Racah Institute of Physics, The
Hebrew University, 9190401 Jerusalem, Israel

###### Abstract

In hot dense plasmas of intermediate or high-Z elements in the state of local
thermodynamic equilibrium, the number of electronic configurations
contributing to key macroscopic quantities such as the spectral opacity and
equation of state, can be enormous. In this work we present systematic methods
for the analysis of the number of relativistic electronic configurations in a
plasma. While the combinatoric number of configurations can be huge even for
mid-Z elements, the number of configurations which have non negligible
population is much lower and depends strongly and non-trivially on temperature
and density. We discuss two useful methods for the estimation of the number of
populated configurations: (i) using an exact calculation of the total
combinatoric number of configurations within superconfigurations in a
converged super-transition-array (STA) calculation, and (ii) by using an
estimate for the multidimensional width of the probability distribution for
electronic population over bound shells, which is binomial if electron
exchange and correlation effects are neglected. These methods are analyzed,
and the mechanism which leads to the huge number of populated configurations
is discussed in detail. Comprehensive average atom finite temperature density
functional theory (DFT) calculations are performed in a wide range of
temperature and density for several low, mid and high Z plasmas. The effects
of temperature and density on the number of populated configurations are
discussed and explained.

##  I Introduction

The calculation of radiative transport properties and equations state from
first principles are of key importance in the modeling of a wide variety of
high energy density plasmas, which exist both in stellar interiors [1, 2, 3,
4, 5, 6, 7, 8, 9, 10, 11] and in terrestrial laboratories, such as Z-pinch and
high-power laser facilities [12, 13, 14, 15, 16, 17, 18, 19, 20]. These
macroscopic quantities entail a very sophisticated interplay between plasma
physics and atomic physics.

In the calculation of spectral opacities, the bound-bound photoabsorption
spectra results from all radiative transitions between all levels from all
pairs of electronic configurations. In a hot dense plasma, the number of lines
between each pair of configurations may be enormous ([21, 22]) and statistical
methods must be used. The Unresolved-Transition-Array (UTA) method ([23, 24,
25, 26]) treats all levels between each pair of configurations statistically,
using analytic expressions for the energy moments of the transition array. In
many cases, the number of configurations may also be extremely large and a
such detailed-configuration-accounting (DCA) calculations are intractable as
well. Similarly, in the calculation of equations of state, a huge number of
electronic configurations should be taken into account, for an accurate
calculation of the total partition function.

In this work we discuss in detail the number of electronic configurations in a
hot dense plasma. The combinatoric number of possible configurations as well
as the number of configurations which have a non negligible population are
examined. The combinatoric number of configurations is calculated by using
exact recursive relations [27]. Two methods for the estimation of the number
of populated configurations are considered and compared. The calculations are
performed in a very wide range of plasma temperature (100eV-$10$keV) and
density ($10^{-3}-10^{3}\text{g/cm}^{3}$) for various low, mid and high Z
elements, using finite temperature average atom DFT calculations as well as
super-transition-array (STA) calculations, employing the opacity code STAR
[28, 29]. The effects of temperature and density on the number of populated
configurations are discussed and explained.

A relativistic electronic configuration $C$ is defined by a set of occupation
numbers $\left\\{q_{s}\right\\}$ on relativistic $s=(nlj)$ orbital shells,
which are full solutions of the Dirac equation. In the statistical
configuration approximation (neglecting the atomic structure within
configurations [30, 31]), the occupation of a configuration $C$ is given by
the Boltzmann distribution:

$P_{C}=\frac{g_{C}e^{-\left(E_{C}-\mu Q_{C}\right)/k_{B}T}}{U_{tot}},$ (1)

where $Q_{C}=\sum_{s}q_{s}$ is the number of bound electrons in $C$, $\mu$ is
the chemical potential, the statistical weight of $C$ is:

$g_{C}=\prod_{s}\binom{g_{s}}{q_{s}},$ (2)

where orbital degeneracy is $g_{s}=2j_{s}+1$, the total partition function is:

$U_{tot}=\sum_{C}g_{C}e^{-\left(E_{C}-\mu Q_{C}\right)/k_{B}T},$ (3)

and the the configuration average energy is:

$E_{C}=\sum_{s}q_{s}I_{s}+\frac{1}{2}\sum_{r,s}q_{r}(q_{s}-\delta_{rs})H_{rs},$
(4)

where, the residual energy is:

$I_{s}=\epsilon_{s}-\left\langle
s\left|\frac{Z}{r}+V\left(r\right)\right|s\right\rangle,$ (5)

where $\epsilon_{s}$ is the orbital energy, $V\left(r\right)$ is the self-
consistent mean field central atomic potential, and $H_{rs}$ is the average
interaction energy of the two-electron configuration $rs$, given in terms of
relativistic direct and exchange Slater integrals (explicit expression can be
found in Refs. [32, 33, 26]).

##  II Combinatoric number of configurations

First we demonstrate how to calculate the combinatoric number of
configurations with $Q$ electrons, which are distributed over a set of shells
$A=\left\\{s_{1}s_{2}...s_{N}\right\\}$, defined by:

$\displaystyle\mathcal{N}_{Q}^{A}$ $\displaystyle=$
$\displaystyle\sum_{\begin{subarray}{c}\\{q_{s}\\}_{s=1}^{N}\\\ \text{with
}\sum_{s=1}^{N}q_{s}=Q\end{subarray}}1,$

where we have expressed the constraint that only configurations
$\\{q_{s}\\}_{s=1}^{N}$ with a total number of $Q$ electrons are summed. This
sum can be expressed in terms of partials sums over the population of the
$N$th shell:

$\mathcal{N}_{Q}^{A}=\sum_{q_{N}=0}^{\text{min}(g_{N},Q)}\sum_{\begin{subarray}{c}\\{q_{s}\\}_{s=1}^{N-1}\\\
\text{with }\sum_{s=1}^{N-1}q_{s}=Q-q_{N}\end{subarray}}1.$ (6)

Since the inner sum is actually $\mathcal{N}_{Q-q_{N}}^{A/\\{s_{N}\\}}$, where
$A/\\{s_{N}\\}$ denotes the set $A$ excluding the shell $s_{N}$, we get the
recursive relation:

$\displaystyle\mathcal{N}_{Q}^{A}$ $\displaystyle=$
$\displaystyle\sum_{q_{N}=0}^{\text{min}(g_{N},Q)}\mathcal{N}_{Q-q_{N}}^{A/\\{s_{N}\\}}.$
(7)

We note that this recursion relation is performed simultaneously over the
numbers of electrons $Q$ and the number of shells $N$, and should obey the
initial condition $\mathcal{N}_{Q}^{A}=\delta_{Q,0}$ for an empty shell group
$A$. The derivation above, which results from a simple combinatoric argument,
can be proved by employing the powerful and general method of generating
functions, as was done in detail Ref. [27], which deals with stable algorithms
for the calculation of canonical partition functions, which were further
generalized and improved in Refs. [34, 35]. We note that in a recent work
[36], a novel approach, which leads to new recurrence and analytic relations
as well as a novel statistical modeling of the combinatoric number of
configurations is developed in detail.

The total combinatoric number of configurations for an element with an atomic
number $Z$, is given by summing the number of configurations over all
ionization levels:

$\mathcal{N}_{C}^{\text{combin}}=\sum_{Q=0}^{Z}\mathcal{N}_{Q}^{A},$ (8)

where $A$ is the set of all shells from which configurations are constructed.
We note that, in general, the number of bound shells is a function of the
atomic number, temperature and density, which determines the self consistent
central potential - for example, a higher Z element has a larger number of
bound shells, due to the higher nucleus charge and a higher density plasma may
have a smaller number of shells due to an increased pressure ionization effect
(bound states dissolving into the continuum [37, 38, 39, 40, 41, 33, 42, 43,
44, 45]).

One can also define the number of configurations taking into account only
charge states with probability larger than $p$:

$\mathcal{N}_{C}^{\text{combin}}\left(p\right)=\sum_{Q,\
P_{Q}>p}\mathcal{N}_{Q}^{A},$ (9)

where the charge state distribution is given by the sum of probabilities of
all configurations with a total charge $Q$:

$P_{Q}=\sum_{C,\ Q_{C}=Q}P_{C}.$ (10)

We note that eq. 9 may depend strongly on the parameter $p$, and may give a
severe overestimation for the number of populated configurations, since each
charge state $Q$ can correspond to a huge number of configurations, some of
which have an extremely low probability. In addition, an exact computation of
the ion charge distribution 10 (via eq. 1) is in many cases intractable, due
the huge number of configurations that need to be taken into account. However,
many approximated methods for the calculation of the charge state distribution
in a plasma exist (for example, via the well known Saha equations) and can be
used “externally” in the calculation of eq. 9. In this work we will use a more
accurate charge state distribution which is obtained from an STA calculation
(see below).

##  III Number configurations within superconfigurations

In the STA method [46, 47, 48, 49, 33, 50, 28, 51, 52, 53], a large number of
configurations $C$ are grouped into super-configurations (SCs), commonly
denoted by $\Xi=\Pi_{\sigma}\sigma^{Q_{\sigma}}$, which are defined as sets of
configurations which have $Q_{\sigma}$ electrons in each ’supershell’
$\sigma$, which is a group of shells. The total occupation of a super
configuration is naturally:

$P_{\Xi}=\sum_{C\in\Xi}P_{C},$ (11)

and combinatoric number of configurations within a superconfiguration is:

$\mathcal{N}_{C}\left(\Xi\right)=\sum_{C\in\Xi}1.$ (12)

As was already noted in Ref. [27], $\mathcal{N}_{\Xi}$ can be evaluated
exactly, using the recursive relation 7. Since electron occupation numbers in
different supershells are independent, eq. 7 can be used for each supershell,
to give the combinatoric number of configurations withing a
superconfiguration:

$\mathcal{N}_{C}\left(\Xi\right)=\prod\limits_{\sigma}\mathcal{N}_{Q_{\sigma}}^{\sigma},$
(13)

where $\mathcal{N}_{Q_{\sigma}}^{\sigma}$ is calculated by applying the
recursion relation 7 for each supershell.

A simple estimate for the number of populated configurations, is given by the
number of configurations within all populated superconfigurations with a
probability larger than $p$:

$\mathcal{N}_{C}^{\text{in SCs}}\left(p\right)=\sum_{\begin{subarray}{c}\Xi\\\
P_{\Xi}>p\end{subarray}}\mathcal{N}_{C}\left(\Xi\right).$ (14)

where we have expressed the constraint that only superconfigurations with a
non-negligible occupation are included.

We note that the combinatoric number of superconfigurations, for a given set
of supershells, can also be calculated using the recursion relation 7, by
treating the supershells $\sigma$ as shells - defining
$A=\left\\{\sigma_{1}...\sigma_{N}\right\\}$, with total degeneracies
$g_{\sigma}=\sum_{s\in\sigma}g_{s}$ and summing over all ionization levels.

##  IV Estimation of the number of populated configuration

Figure 1: (Color online) A schematic description of the population of electrons over shells according to the Fermi-Dirac distribution (red thick line). The occupation of electronic shells which have energies nearby the Fermi-Dirac step (which is located at $\epsilon_{s}\approx\mu$ and has a width of the order of $k_{B}T$), fluctuate and result in wide range of possibilities to distribute electrons over the magnetic quantum numbers, while shells which are far from the Fermi-Dirac step are either filled or empty. The fluctuating occupation numbers may give rise the a huge number of populated electronic configurations.  supershell $\sigma$ |  $Q_{\sigma}$ range | degeneracy $g_{\sigma}$
---|---|---
(1s) | [2,2] | 2
(2s) | [0,2] | 2
$(2\text{p}_{-})$ | [0,2] | 2
$(2\text{p}_{+})$ | [0,4] | 4
$(3\text{s}...5\text{f}_{-})$ | [0,7] | 74
$(5\text{f}_{+}...8\text{i}_{+})$ | [0,7] | 294
Table 1: The converged relativistic supershell structure (left column) for Iron at $T=182\text{eV}$, $\rho=0.13\text{g/cm}^{3}$. The range for the number of electrons in each supershell (middle column), chosen such that the superconfiguration occupation is larger than $10^{-7}$, as well as the total supershell degeneracy (right column), are also given. The total number of superconfigurations is $\mathcal{N}_{\Xi}=2880$.  supershell $\sigma$ |  $Q_{\sigma}$ range | degeneracy $g_{\sigma}$
---|---|---
(1s) | [2,2] | 2
(2s) | [2,2] | 2
$(2\text{p}_{-})$ | [2,2] | 2
$(2\text{p}_{+})$ | [4,4] | 4
$(3\text{s}3\text{p}_{-}3\text{p}_{+})$ | [7,8] | 8
$(3\text{d}_{-}3\text{d}_{+})$ | [9,10] | 10
$(4\text{s}...4\text{f}_{+})$ | [3,27] | 32
$(5\text{s}...7\text{d}_{+})$ | [0,12] | 140
$(7\text{f}_{-}...8\text{k}_{+})$ | [0,7] | 208
Table 2: Same as table 1, for Gold at $T=200\text{eV}$,
$\rho=0.1\text{g/cm}^{3}$. The total number of superconfigurations is
$\mathcal{N}_{\Xi}=10400$.

Figure 2: (Color online) The Fermi-Dirac distribution (red solid lines, as
illustrated in Fig. 1) and the occupation fluctuation $\delta q_{s}^{2}/g_{s}$
(black dashed lines, see eq. 18)) as a function of orbital energy, for Iron at
$T=182\text{eV}$, $\rho=0.13\text{g/cm}^{3}$ (upper figure) and Gold at
$T=200\text{eV}$, $\rho=0.1\text{g/cm}^{3}$ (lower figure). The relativistic
Dirac bound orbitals are listed and represented as vertical dashed thin lines
at their appropriate bound energies. The resulting chemical potential is given
in the title.

Figure 3: (Color online) The estimate given in eq. 19 for the number of
populated configurations, as a function of the parameter $\alpha$ (red solid
lines), for Iron at $T=182\text{eV}$, $\rho=0.13\text{g/cm}^{3}$ (upper
figure) and Gold at $T=200\text{eV}$, $\rho=0.1\text{g/cm}^{3}$ (lower
figure). Also shown are the combinatoric number of configurations over all
ionization levels (eq. 8, black dashed lines) and over all ionization levels
with probabilities larger than $10^{-2},10^{-4},10^{-5}$ and $10^{-6}$
respectively (eq. 9, magenta lines), as well as the number of configurations
within populated superconfigurations (eq. 14), with probabilities larger than
$10^{-6},10^{-7}$ and $10^{-8}$, respectively (eq. 14, blue lines).

Figure 4: (Color online) The charge state distribution calculated using an STA
model (red dashed lines, left axis) and the combinatoric number of
configurations for each charge state (eq. 8, blue lines, on the right axis),
for Iron at $T=182\text{eV}$, $\rho=0.13\text{g/cm}^{3}$ (upper figure) and
Gold at $T=200\text{eV}$, $\rho=0.1\text{g/cm}^{3}$ (lower figure).

In this section we discuss a very simple way to estimate the number of
populated configurations, without using a state of the art sophisticated STA
method, as was suggested in the previous section. This method was presented in
the seminal book [54] and was used subsequently in Refs. [28, 29]. In this
method, only the bound shells and chemical potential are needed \- so that the
estimate for the number of populated configurations can be obtained, for
example, by solving the Dirac equation in a Thomas-Fermi potential [55] or in
a more advanced average atom model potential [37, 38, 39, 41, 40, 33, 42, 43,
44, 45].

Let us estimate the number of populated configurations. Neglecting the
electron-electron interaction effects beyond the self consistent field model,
the configuration average energy given in eq. 4 can be approximated as a first
order polynomial of the occupation numbers:

$E_{C}\approx\sum_{s}q_{s}\epsilon_{s}.$ (15)

Under this approximation, the total partition function is:

$U_{tot}=\prod_{s}\left(1+e^{-(\epsilon_{s}-\mu)/k_{B}T}\right)^{g_{s}},$ (16)

and the configuration probability 1 becomes a simple multivariate binomial
distribution of the occupation numbers $\left\\{q_{s}\right\\}$, given by:

$\displaystyle P\left(\left\\{q_{s}\right\\}\right)\equiv P_{C}$
$\displaystyle=$
$\displaystyle\prod_{s}\binom{g_{s}}{q_{s}}n_{s}^{q_{s}}\left(1-n_{s}\right){}^{g_{s}-q_{s}},$
(17)

where $n_{s}=1/\left(e^{(\epsilon_{s}-\mu)/k_{B}T}+1\right)$ is the Fermi-
Dirac distribution. The variance of the population of each shell is given by:

$\displaystyle\delta q_{s}$
$\displaystyle\equiv\sqrt{\left\langle\left(q_{s}-\left\langle
q_{s}\right\rangle\right){}^{2}\right\rangle}$
$\displaystyle=\sqrt{g_{s}n_{s}\left(1-n_{s}\right)}.$ (18)

As illustrated in Figs. 1-2, it is evident that the occupation of shells whose
energies are near the Fermi-Dirac step, which is located around
$-k_{B}T\lesssim\epsilon_{s}-\mu\lesssim k_{B}T$ fluctuate, while the other
shells are either filled or empty. These fluctuating shells may have a wide
range of possibilities to distribute electrons over the magnetic quantum
numbers $0\leq m_{s}\leq g_{s}$. The fluctuating occupation numbers may give
rise the a huge number of populated configurations, which increases
exponentially with the number of fluctuating shells.

The number of populated configurations can be estimated as the number of
possibilities to put electrons in each shell, within a few standard deviations
$\delta q_{s}$ around the average occupation $\left\langle
q_{s}\right\rangle=g_{s}n_{s}$ of the multivariate distribution 17. The number
of possible occupation numbers for each shell is estimated as
$\alpha\times\delta q_{s}$, where $\alpha/2$ is the number of standard
deviations. Therefore, the number of populated configurations can be estimated
by:

$\displaystyle\mathcal{N}_{C}^{\text{approx}}$ $\displaystyle=$
$\displaystyle\prod_{s}\left(\alpha\delta q_{s}+1\right),$ (19)

which is simply the multidimensional “width” of the multivariate binomial
distribution (17).

We note that the result may depend on the somewhat arbitrary value chosen for
$\alpha$, but it can be expected that a reasonable value should be in the
range $2\lesssim\alpha/2\lesssim 4$ \- corresponding to a range of two to four
standard deviations for the occupation of each shell. In order to demonstrate
this, $\mathcal{N}_{C}^{\text{approx}}$ was calculated as a function of
$\alpha$ in the range $1\leq\alpha\leq 10$ and compared with the number of
configurations within populated superconfigurations (eq. 14) of a converged
STA calculation. Two cases are considered: (1) Iron (Z=26) at typical
conditions of the recent Sandia Z experiments [12, 15], with temperature
$T=182\text{eV}$ and density $\rho=0.13\text{g/cm}^{3}$ and (2) Gold (Z=79)
with temperature $T=200\text{eV}$ and density $\rho=0.1\text{g/cm}^{3}$. The
calculations were performed using the relativistic average-atom model
implemented in the STA code STAR [31, 26, 3, 28, 29, 51, 43]. The number of
bound shells was limited to a principle atomic number of $n_{\text{max}}=8$
(which corresponds to 64 relativistic orbitals), since highly excited bound
orbitals can be accounted for by using the method detailed in Ref. [56], and
therefore need not be accounted in the estimation of the number of populated
configurations that are used in the calculation of spectral opacities.
Illustration of the bound orbitals and the Fermi-Dirac step for the two cases
is given Fig. 2. The supershell structure for each case are given in tables
1-2. In Fig. 3 we present the number of populated configurations as a function
of $\alpha$, in comparison to the combinatoric number of configurations over
all ionization levels (eq. 8) and over all ionization levels with occupation
probabilities larger than $10^{-2},10^{-4},10^{-5}$ and $10^{-6}$ (eq. 9) as
well as the number of configurations within populated superconfigurations (eq.
14), with occupation probabilities larger than $10^{-6},10^{-7}$ and
$10^{-8}$. These results are also given explicitly in Table 3. The strong
dependence on $\alpha$ is evident, and it seems, as expected, that a good
choice is $\alpha=6$, which corresponds to $3$ standard deviations for the
occupation of each shell.

It is evident that the probability thresholds affect the resulting number of
configurations by about 1-2 orders of magnitude, which is a reasonable
accuracy for the number of configurations, which can be of the order of
$10^{15}-10^{40}$. It is evident that for the Iron case, the results for
$\mathcal{N}_{C}^{\text{combin}}\left(p\right)$ and $\mathcal{N}_{C}^{\text{in
SCs}}\left(p\right)$ agree to within 1-2 orders of magnitude. However, it is
evident that for the gold case, the
$\mathcal{N}_{C}^{\text{combin}}\left(p\right)$ values give a severe
overestimation (by about ten orders of magnitude) compared to
$\mathcal{N}_{C}^{\text{in SCs}}\left(p\right)$, which represents the correct
estimate for the number of configurations that need to be taken into account
in opacity calculations. As mentioned in section II, this is to be expected
since each charge state $Q$ can correspond to a huge number of configurations,
some of which have extremely low probabilities. This is more likely to happen
for a high Z element, for which the supershell structure (see table 2) gives
rise to only a small fraction of all possible configurations for some charge
states. In this way $\mathcal{N}_{C}^{\text{in SCs}}$ only accounts for
configurations with non-negligible probabilities while
$\mathcal{N}_{C}^{\text{combin}}\left(p\right)$ accounts for all
configurations for non-negligible charge states, without taking into account
the confrontational structure (which determines the configuration probability
in eq. 1).

| $\begin{subarray}{c}\text{Fe, }T=182\text{eV},\rho=0.13\text{g/cm}^{3}\\\ \\\ \end{subarray}$ | $\begin{subarray}{c}\text{Au, }T=200\text{eV, $\rho$=0.1$\text{g/cm}^{3}$}\\\ \\\ \end{subarray}$
---|---|---
$\mathcal{N}_{C}^{\text{combin}}$ | $9.78\times 10^{21}$ | $4.64\times 10^{39}$
$\mathcal{N}_{C}^{\text{combin}}\left(10^{-2}\right)$ | $2.08\times 10^{13}$ | $2.71\times 10^{31}$
$\mathcal{N}_{C}^{\text{combin}}\left(10^{-4}\right)$ | $6.34\times 10^{14}$ | $2.6\times 10^{32}$
$\mathcal{N}_{C}^{\text{combin}}\left(10^{-5}\right)$ | $3.2\times 10^{15}$ | $5.39\times 10^{32}$
$\mathcal{N}_{C}^{\text{combin}}\left(10^{-6}\right)$ | $1.53\times 10^{16}$ | $1.1\times 10^{33}$
$\mathcal{N}_{C}^{\text{in SCs}}\left(10^{-6}\right)$ | $2.71\times 10^{13}$ | $9.01\times 10^{19}$
$\mathcal{N}_{C}^{\text{in SCs}}\left(10^{-7}\right)$ | $1.22\times 10^{15}$ | $3.35\times 10^{21}$
$\mathcal{N}_{C}^{\text{in SCs}}\left(10^{-8}\right)$ | $2.26\times 10^{16}$ | $1.1\times 10^{22}$
$\mathcal{N}_{C}^{\text{approx}}$$\left(\alpha=6\right)$ | $1.25\times 10^{16}$ | $1.63\times 10^{20}$
Table 3: Various values for the number of configurations for the two cases
shown in Fig. 3.

Fig. 4 shows the charge state distributions, which were obtained from
converged STA calculations, together with the combinatoric number of
configurations as a function of the number of bound electrons (eq. 7), for the
Iron and Gold cases. The exponential growth for the combinatoric number of
configurations for large values of bound electrons is evident, as expected. We
note that the number of configurations in eq. 9 is obtained by summing the
number of configurations per charge state in Fig. 4 in the overlapping range
with the charge state distribution.

Finally, we note that it is possible to overcome the binomial approximation 17
using the correlated probability formalism. In Ref. [57] electron-electron
interactions are taken into account in the screening constant model which is
applicable for small interactions. The resulting correlation coefficients are
calculated to second order in the interaction energy. As noted by Perrot and
Blenski in Ref. [58], this method is complicated to implement in practice and
performs poorly for low temperatures. Perrot and Blenski introduced in Ref.
[58] a simple method, which overcomes these difficulties by replacing the
binomial distribution with its correlated Gaussian continuous limit, which is
applicable for orbital shells with large degeneracy and which are not close to
being full or empty. As was shown in Ref. [59], this approximation is accurate
to less than 1%. Hence, by using this Gaussian approximation in order to
calculate the population variance of shells with a large degeneracy and which
are not close to being full or empty, and the binomial approximation for the
remaining shells (as was done above), can give a better approximation for the
number of configurations, which takes into account electron-electron
interactions. However, this is beyond the scope of this manuscript, and will
be performed in a future work.

##  V Results

Figure 5: (Color online) Various color plots for Silicone (Z=14), as a
function of temperature and density (left to right, top to bottom): the number
of bound shells $N_{\text{shells}}$, the combinatoric number of relativistic
configurations $\mathcal{N}_{C}^{\text{combin}}$ over all ionization levels
(eq. 8) and over all ionization levels with probability larger than $10^{-5}$
(eq. 9), the number of superconfigurations in a converged STA calculation
$\mathcal{N}_{\Xi}$, the number of configurations within superconfigurations
$\mathcal{N}_{C}^{\text{in SCs}}$ (eq. 14 with $p=10^{-7}$), approximated
number of populated relativistic $\mathcal{N}_{C}^{\text{approx}}$ and non-
relativistic $\mathcal{N}_{C,\text{NR}}^{\text{approx}}$ configurations, the
average ionization $\overline{Z}$ and the normalized chemical potential
$\mu/k_{B}T$.

Figure 6: (Color online) Same as Fig. 5, for Iron (Z=26).

Figure 7: (Color online) Same as Fig. 5, for Xenon (Z=54).

Figure 8: (Color online) Same as Fig. 5, for Gold (Z=79). Figure 9: (Color
online) The Fermi-Dirac distribution (solid lines) and the occupation
fluctuation $\delta q_{s}^{2}/g_{s}$ (dashed lines, see eq. 18) as a function
of orbital energy, for Iron at $\rho=0.1\text{g/cm}^{3}$, and different
temperatures in the range $10^{2}-10^{4}\text{eV}$ (lines are arranged from
right to left.  Figure 10: (Color online) The Fermi-Dirac distribution (solid
lines) and the occupation fluctuation $\delta q_{s}^{2}/g_{s}$ (dashed lines,
see eq. 18) as a function of orbital energy, for Iron at $T=182\text{eV}$, and
different densities in the range $10^{-3}-10^{3}\text{g/cm}^{3}$ (lines are
arranged from left to right).

Detailed finite temperature density functional theory (DFT) calculations in
the spherical average atom approximation [37, 38, 40, 39, 41, 33, 42, 43, 44,
45] followed by STA calculations using the atomic code STAR [28, 29], were
performed over a wide range of plasma temperatures: 100eV-10keV and densities:
$10^{-3}-10^{3}\text{g/cm}^{3}$, for the following low, mid and high Z
elements - Silicone (Z=14), Iron (Z=26), Xenon (Z=54) and Gold (Z=79). The
results are shown in Figs 5-8, and include the average ionization, the
chemical potential, the number of bound shells, the combinatoric number of
configurations over all ionization levels (eq. 8) and over all ionization
levels with probabilities larger than $10^{-5}$ (eq. 9, the charge
distribution was obtained from the STA calculations), the number of populated
superconfigurations in a the converged STA calculation, the number of
configurations within these superconfigurations and the approximated number of
relativistic and nonrelativistic populated configurations. First, it is
evident that, as expected, the number of configurations grows exponentially
with the atomic number. It is seen that even for mid $Z$ plasmas, the number
of populated configurations can be huge (larger than $10^{13}$) in a wide
range of temperature and density, and a detailed configuration accounting
(DCA) calculation may be extremely costly, while a full detailed-line-
accounting (DLA) calculation is probably impossible. For a higher Z plasma
such as Xenon and Gold, there is a wide range of temperature and density with
a populated number of configurations larger than $10^{20}$, which is
completely intractable for a DCA calculation, and highlights the need for an
STA [46, 47, 48, 33, 50, 28, 51, 52] or Average-Atom [60, 61, 62] method for
opacity calculations.

A comparison of the number of configurations within superconfigurations
$\mathcal{N}_{C}^{\text{in SCs}}$ and approximated number of populated
relativistic configurations $\mathcal{N}_{C}^{\text{approx}}$, shows a good
qualitative and even quantitative agreement - which proves that
$\mathcal{N}_{C}^{\text{approx}}$ is a very good simple estimate for the
number of populated configurations. In addition, the plots for the number of
superconfigurations $\mathcal{N}_{\Xi}$ in a converged STA calculation in
comparison the the number of populated configurations highlights the strength
of the STA method - a computationally tractable number of superconfigurations
(in the range of $10^{3}-10^{5}$), which may contain a huge number of
configurations (more than $10^{25}$ in some cases) - yields a converged
opacity calculation, which would have been completely prohibitive in a
configuration based DCA calculation.

It is seen that the $\mu/k_{B}T$ contours are approximately straight lines
(when the density and temperatures are plotted on a log scale), a fact which
agrees with the ideal gas result:

$\mu_{\text{ideal}}=k_{B}T\ln\left(\Lambda^{3}\bar{n}\right),$ (20)

where $\bar{n}$ is the number density and
$\Lambda=\left(2\pi\hbar^{2}/mk_{B}T\right)^{\frac{1}{2}}$ is the thermal
wavelength, so that in the thermodynamic range studied here, the chemical
potential is a decreasing function of temperature and an increasing function
of density, as shown in Figs 9-10.

It is also evident from Figs 5-8, that the combinatoric number of
configurations over all ionization levels correlates perfectly with the number
of bound shells that exists in the atomic potential. This is to be expected
since this number of configurations depends on temperature and density only
through the number of existing bound shells - and not on their properties
(i.e. bound energies, wave functions etc). As was explained in the previous
section, since we are concerned in the number of configurations which should
be taken into account in opacity calculations, the number of bound shells is
limited here to $64$ \- which results in a sharp front in the plots for the
combinatoric number of configurations. It is also evident that
$\mathcal{N}_{C}^{\text{combin}}\left(p=10^{-5}\right)$ has a better agreement
with $\mathcal{N}_{C}^{\text{in SCs}}$ and $\mathcal{N}_{C}^{\text{approx}}$
for the lower Z elements and for cases with a smaller number of bound
orbitals. As was discussed in the previous section, this is to be expected,
since as apposed to the superconfiguration accounting approach, the charge
probability distribution does not contain information about the configuration
structure and as a result, some charge states may contain a huge amount of
very low probability configurations, which are taken into account in eq. 9.

Next, we discuss the temperature and density behavior for the populated number
of configurations. As expected, it is seen in Figs 5-8 that the number of
populated configurations has a maximum as a function of temperature and
density. For low temperatures, most shells have energies $\epsilon_{s}<\mu$
and are therefore “frozen” - either full or empty (see Fig. 1), while for
moderate temperatures (which are different for each element and density) the
Fermi-Dirac distribution has a shape of a step function, but with a finite
width which allow large fluctuations for the occupation numbers of shells with
energies nearby the step (see Fig. 9). For higher temperatures, on the one
hand, more shells are ionized \- which results in a decrease in the number of
populated configurations, and on the other hand, the number of bound shells
can be larger, due to a wider spatial extent of the atomic central potential -
which reduces the effect of pressure ionization, as seen in the plots of the
number of bound shells. The latter effect results in the slight tilted maxima
for the number of populated configurations as a function of temperature and
density, seen in Figs 5-8. In addition, it is evident that for low densities
the number of populated configurations is small due to the decrease in the
chemical potential (see Fig. 10), which reduces the number of fluctuating
shells, while for very high densities most shells are pressure-ionized (as
seen in the plots of the number of bound shells) and those which are not are
occupied and frozen - which leads again to a small number of populated
configurations.

##  VI Summary

Two useful methods for the estimation of the number of populated
configurations in a hot dense plasma were studied. In the first method, an
exact calculation of the total combinatoric number of configurations within
superconfigurations in a converged super-transition-array (STA) calculation
was used. In the second method, electron exchange and correlation effects are
neglected, leading to a multivariate binomial distribution for the electronic
occupation numbers, whose multidimensional width is an approximation for the
number of populated configurations. The mechanism which leads to the huge
number of populated configurations \- namely, the fluctuations of electronic
occupation numbers of bound shells nearby the Fermi-Dirac step, is
demonstrated and discussed in detail. Comprehensive average atom finite
temperature DFT calculations are performed in a wide range of temperature and
density for several low, mid and high Z plasmas, showing a good agreement
between these two methods. In addition, the temperature and density dependence
is discussed and explained.

The second method, which is much more simple than the first, only the bound
shells and chemical potential are needed - so that the estimate for the number
of populated configurations can be obtained, for example, by solving the Dirac
equation in a Thomas-Fermi potential or in a more advanced average atom model
potential. This simple estimate can be very useful in order to asses the
computational ability to perform configuration based, or even line based
opacity and equation of state calculations.

###### Acknowledgements.

We thank the anonymous referees for useful suggestions and comments, and in
particular, for suggesting the examination of the number of configurations
with respect to probability thresholds and a discussion of possible
improvements of the binomial approximation.

## References

  * [1] Jørgen Christensen-Dalsgaard, Maria Pia Di Mauro, Günter Houdek, and Frank Pijpers.  On the opacity change required to compensate for the revised solar composition.  Astronomy & Astrophysics, 494(1):205–208, 2009.
  * [2] FL Villante and AM Serenelli.  A quantitative analysis of the solar composition problem.  Physics Procedia, 61:366–375, 2015.
  * [3] M. Krief, A. Feigel, and D. Gazit.  Line broadening and the solar opacity problem.  The Astrophysical Journal, 824(2):98, 2016.
  * [4] Claudio Mendoza.  Computation of atomic astrophysical opacities.  Atoms, 6(2):28, 2018.
  * [5] Jean-Christophe Pain, Franck Gilleron, and Maxime Comet.  Detailed opacity calculations for astrophysical applications.  Atoms, 5(2):22, 2017.
  * [6] CJ Fontes, CL Fryer, AL Hungerford, P Hakel, J Colgan, DP Kilcrease, and ME Sherrill.  Relativistic opacities for astrophysical applications.  High Energy Density Physics, 16:53–59, 2015.
  * [7] CJ Fontes, HL Zhang, J Abdallah Jr, REH Clark, DP Kilcrease, J Colgan, RT Cunningham, P Hakel, NH Magee, and ME Sherrill.  The Los Alamos suite of relativistic atomic physics codes.  Journal of Physics B: Atomic, Molecular and Optical Physics, 48(14):144014, 2015.
  * [8] J. Colgan, D. P. Kilcrease, N. H. Magee, M. E. Sherrill, J. Abdallah Jr., P. Hakel, C. J. Fontes, J. A. Guzik, and K. A. Mussack.  A new generation of los alamos opacity tables.  The Astrophysical Journal, 817(2):116, 2016.
  * [9] Carlos A Iglesias.  Iron-group opacities for b stars.  Monthly Notices of the Royal Astronomical Society, 450(1):2–9, 2015\.
  * [10] Carlos A Iglesias and Forrest J Rogers.  Updated OPAL opacities.  The Astrophysical Journal, 464:943, 1996.
  * [11] Nigel R Badnell, MA Bautista, K Butler, F Delahaye, C Mendoza, Patrick Palmeri, CJ Zeippen, and MJ Seaton.  Updated opacities from the opacity project.  Monthly Notices of the Royal Astronomical Society, 360(2):458–464, 2005.
  * [12] JE Bailey, T Nagayama, GP Loisel, GA Rochau, C Blancard, J Colgan, Ph Cosse, G Faussurier, CJ Fontes, F Gilleron, et al.  A higher-than-predicted measurement of iron opacity at solar interior temperatures.  Nature, 517(7532):56–59, 2015.
  * [13] T Nagayama, JE Bailey, G Loisel, GA Rochau, JJ MacFarlane, and I Golovkin.  Calibrated simulations of z opacity experiments that reproduce the experimentally measured plasma conditions.  Physical Review E, 93(2):023202, 2016.
  * [14] T Nagayama, JE Bailey, G Loisel, GA Rochau, SB Hansen, C Blancard, P Cosse, G Faussurier, F Gilleron, J-C Pain, et al.  Model uncertainties of local-thermodynamic-equilibrium k-shell spectroscopy.  High Energy Density Physics, 2016.
  * [15] T Nagayama, JE Bailey, GP Loisel, GS Dunham, GA Rochau, C Blancard, J Colgan, Ph Cossé, G Faussurier, CJ Fontes, et al.  Systematic study of l-shell opacity at stellar interior temperatures.  Physical review letters, 122(23):235001, 2019.
  * [16] CA Back, JD Bauer, OL Landen, RE Turner, BF Lasinski, JH Hammer, MD Rosen, LJ Suter, and WH Hsing.  Detailed measurements of a diffusive supersonic wave in a radiatively heated foam.  Physical review letters, 84(2):274, 2000.
  * [17] CL Fryer, E Dodd, W Even, CJ Fontes, C Greeff, A Hungerford, J Kline, K Mussack, I Tregillis, JB Workman, et al.  Uncertainties in radiation flow experiments.  High Energy Density Physics, 18:45–54, 2016.
  * [18] Alastair S Moore, Thomas M Guymer, John Morton, Benjamin Williams, John L Kline, Nicholas Bazin, Christopher Bentley, Shelly Allan, Katie Brent, Andrew J Comley, et al.  Characterization of supersonic radiation diffusion waves.  Journal of Quantitative Spectroscopy and Radiative Transfer, 159:19–28, 2015.
  * [19] Noël Fleurot, Claude Cavailler, and JL Bourgade.  The laser megajoule (lmj) project dedicated to inertial confinement fusion: Development and construction status.  Fusion Engineering and Design, 74(1):147–154, 2005.
  * [20] C Decker, RE Turner, OL Landen, LJ Suter, P Amendt, HN Kornblum, BA Hammel, TJ Murphy, J Wallace, ND Delamater, et al.  Hohlraum radiation drive measurements on the omega laser.  Physical review letters, 79(8):1491, 1997.
  * [21] HA Scott and SB Hansen.  Advances in NLTE modeling for integrated simulations.  High Energy Density Physics, 6(1):39–47, 2010.
  * [22] F Gilleron and J-C Pain.  Efficient methods for calculating the number of states, levels and lines in atomic configurations.  High Energy Density Physics, 5(4):320–327, 2009.
  * [23] Steven A Moszkowski.  On the energy distribution of terms and line arrays in atomic spectra.  Progress of Theoretical Physics, 28(1):1–23, 1962.
  * [24] C Bauche-Arnoult, J Bauche, and M Klapisch.  Variance of the distributions of energy levels and of the transition arrays in atomic spectra.  Physical Review A, 20(6):2424, 1979.
  * [25] J Bauche, C Bauche-Arnoult, and M Klapisch.  Transition arrays in the spectra of ionized atoms.  Adv. At. Mol. Phys, 23:131–195, 1988.
  * [26] Menahem Krief and Alexander Feigel.  Variance and shift of transition arrays for electric and magnetic multipole transitions.  High Energy Density Physics, 17, Part B:254 – 262, 2015.
  * [27] Franck Gilleron and Jean-Christophe Pain.  Stable method for the calculation of partition functions in the superconfiguration approach.  Physical Review E, 69(5):056117, 2004.
  * [28] M Krief, A Feigel, and D Gazit.  Solar opacity calculations using the super-transition-array method.  The Astrophysical Journal, 821(1):45, 2016.
  * [29] Menahem Krief, Alexander Feigel, and Doron Gazit.  A new implementation of the sta method for the calculation of opacities of local thermodynamic equilibrium plasmas.  Atoms, 6(3):35, 2018.
  * [30] F Gilleron, J-C Pain, Q Porcherot, J Bauche, and C Bauche-Arnoult.  Corrections to statistical modeling of spectra for plasmas at moderate or low temperatures.  High Energy Density Physics, 7(4):277–284, 2011.
  * [31] Menahem Krief and Alexander Feigel.  The effect of first order superconfiguration energies on the opacity of hot dense matter.  High Energy Density Physics, 15:59–66, 2015.
  * [32] A Bar-Shalom, J Oreg, and WH Goldstein.  Effect of configuration widths on the spectra of local thermodynamic equilibrium plasmas.  Physical Review E, 51(5):4882, 1995.
  * [33] AA Ovechkin, PA Loboda, VG Novikov, AS Grushin, and AD Solomyannaya.  RESEOS–a model of thermodynamic and optical properties of hot and warm dense matter.  High Energy Density Physics, 13:20–33, 2014.
  * [34] Brian G Wilson, Franck Gilleron, and Jean-Christophe Pain.  Further stable methods for the calculation of partition functions in the superconfiguration approach.  Physical Review E, 76(3):032103, 2007.
  * [35] Jean-Christophe Pain, Franck Gilleron, and Brian G Wilson.  Optimized recursion relation for the computation of partition functions in the superconfiguration approach.  High Energy Density Physics, 37:100891, 2020.
  * [36] Jean-Christophe Pain and Michel Poirier.  Analytical and numerical expressions for the number of atomic configurations contained in a supershell.  Journal of Physics B: Atomic, Molecular and Optical Physics, 53(11):115002, 2020.
  * [37] David A Liberman.  Self-consistent field model for condensed matter.  Physical Review B, 20(12):4981, 1979.
  * [38] Balazs F. Rozsnyai.  Relativistic Hartree-Fock-Slater calculations for arbitrary temperature and matter density.  Phys. Rev. A, 5:1137–1149, Mar 1972.
  * [39] Thomas Blenski and Kenichi Ishikawa.  Pressure ionization in the spherical ion-cell model of dense plasmas and a pressure formula in the relativistic pauli approximation.  Physical Review E, 51(5):4869, 1995.
  * [40] B Wilson, V Sonnad, P Sterne, and W Isaacs.  Purgatorio - a new implementation of the inferno algorithm.  Journal of Quantitative Spectroscopy and Radiative Transfer, 99(1-3):658–679, 2006.
  * [41] VG Novikov and AA Ovechkin.  Calculations of the equation of state by the Liberman model.  Mathematical Models and Computer Simulations, 3(3):290–298, 2011\.
  * [42] AA Ovechkin, PA Loboda, and AL Falkov.  Plasma opacity calculations using the starrett and saumon average-atom model with ion correlations.  High Energy Density Physics, 30:29–40, 2019.
  * [43] Menahem Krief, Yair Kurzweil, Alexander Feigel, and Doron Gazit.  The effect of ionic correlations on radiative properties in the solar interior and terrestrial experiments.  The Astrophysical Journal, 856(2):135, 2018.
  * [44] Nathanael Matthew Gill and Charles Edward Starrett.  Tartarus: A relativistic green’s function quantum average atom code.  High Energy Density Physics, 24:33–38, 2017.
  * [45] Charles E Starrett, Nathanael Matthew Gill, Travis Sjostrom, and Carl William Greeff.  Wide ranging equation of state with tartarus: A hybrid green’s function/orbital based average atom code.  Computer Physics Communications, 235:50–62, 2019.
  * [46] A Bar-Shalom, J Oreg, WH Goldstein, D Shvarts, and A Zigler.  Super-transition-arrays: A model for the spectral analysis of hot, dense plasma.  Physical Review A, 40(6):3183, 1989.
  * [47] T Blenski, A Grimaldi, and F Perrot.  A superconfiguration code based on the local density approximation.  Journal of Quantitative Spectroscopy and Radiative Transfer, 65(1):91–100, 2000.
  * [48] G Hazak and Y Kurzweil.  A Configurationally-Resolved-Super-Transition-Arrays method for calculation of the spectral absorption coefficient in hot plasmas.  High Energy Density Physics, 8(3):290–297, 2012.
  * [49] Y Kurzweil and G Hazak.  Summation of the spectra of all partially resolved transition arrays in a supertransition array.  Physical Review E, 94(5):053210, 2016.
  * [50] Brian G Wilson, Carlos A Iglesias, and Mau H Chen.  Partially resolved super transition array method.  High Energy Density Physics, 14:67–73, 2015.
  * [51] M Krief and D Gazit.  Star: A new sta code for the calculation of solar opacities.  In Astronomical Society of the Pacific Conference Series, volume 515, page 63, 2018.
  * [52] Jacques Bauche, Claire Bauche-Arnoult, and Olivier Peyrusse.  Atomic Properties in Hot Plasmas: From Levels to Superconfigurations.  Springer, 2015.
  * [53] Jean-Christophe Pain.  Super transition arrays: A tool for studying spectral properties of hot plasmas.  Plasma, 4(1):42–64, 2021.
  * [54] Arnold F Nikiforov, Vladimir G Novikov, and Vasili B Uvarov.  Quantum-Statistical Models of Hot Dense Matter: Methods for Computation Opacity and Equation of State, volume 37.  Springer Science & Business Media, 2006.
  * [55] Richard Phillips Feynman, Nicholas Metropolis, and Edward Teller.  Equations of state of elements based on the generalized Fermi-Thomas theory.  Physical Review, 75(10):1561, 1949.
  * [56] Jean-Christophe Pain and Franck Gilleron.  Accounting for highly excited states in detailed opacity calculations.  High Energy Density Physics, 15:30–42, 2015.
  * [57] Joseph Matthew Green.  The statistical mechanics of the interdependent electrons in the screening constant model of the many-electron-atom.  Journal of Quantitative Spectroscopy and Radiative Transfer, 4(5):639–662, 1964.
  * [58] F Perrot and T Blenski.  Electronic structure and statistical mechanics of ionic configurations in hot plasmas.  Le Journal de Physique IV, 10(PR5):Pr5–473, 2000.
  * [59] Brian G Wilson.  Evaluating orbital occupation number correlations in high-temperature plasmas.  Journal of Quantitative Spectroscopy and Radiative Transfer, 49(3):241–258, 1993.
  * [60] D Shalitin, J Stein, and Akiva Ron.  Level and line broadening for Thomas-Fermi atoms at finite temperature.  Physical Review A, 29(5):2789, 1984.
  * [61] J Stein, D Shalitin, and Akiva Ron.  Average-atom models of line broadening in hot dense plasmas.  Physical Review A, 31(1):446, 1985.
  * [62] Balazs F Rozsnyai.  Solar opacities.  Journal of Quantitative Spectroscopy and Radiative Transfer, 71(2):655–663, 2001.