Strong, Weak and Merging Lines in Atomic Spectra

Abstract

We present analytical estimates for the maximum line strength in a transition array, as well as for the numbers of strong and weak lines. For that purpose, two main assumptions are used as concerns the line strength distribution. The first one, due to Porter and Thomas, is more suitable for $J-J^{\prime }$ sets, where J is the total atomic angular momentum, and the second one, based on a decreasing-exponential modeling of the line-amplitude distribution, is more relevant for an entire transition array. We also review the different approximations of overlapping and blanketing (band model), insisting on the computation and properties of the Elsasser function. We compare different approximations of the Ladenburg–Reiche function giving the equivalent width of an ensemble of lines in a frequency bin and discuss the possibility of using statistical indicators, such as the Chernoff bound or the Gini coefficient (initially introduced in economics for the measurement of income inequality), in the statistical characterization of transition arrays.

1. Introduction

Knowing the number of lines in a transition array (ensemble of lines between two electronic configurations differing by an electric-dipole-allowed one-electron jump) is important for calculating the absorption and emission spectra of high-temperature plasmas, corresponding typically to $k_{B}T$ > 5 eV ($k_B$ being the Boltzmann constant). Of course, as can be seen in

Table 1. Exact number of lines calculated for several transition arrays in different coupling schemes: LS, jj, jℓ and IC (intermediate coupling). In the case of jℓ coupling, the two values correspond to the non-zero spin–orbit integral (specified inside brackets).

Transition Array	LS	jj	jℓ	IC
d4 − d3 p	637	859	1718 (p) − 1053 (d)	1718
d5 − d4 f	1903	2471	3286 (d) − 5470 (f)	5470
d5 − d4 p	1150	1572	3245 (p) − 1925 (d)	3245
d9 f − d8 f2	1331	1602	2780 (d) − 2727 (f)	3590
p3 d2 − p2 d3	6129	6915	13,611 (p) − 12,128 (d)	18,237
p3 − p2 d	46	56	71 (p) − 97 (d)	97

for different transition arrays, the number of lines depends on the angular-momentum coupling scheme; throughout this paper, we consider intermediate coupling. Indeed, when these lines are sufficiently numerous, it is reasonable to represent them by continuous, smooth structures, modeled statistically by simple functions, whose characteristics (mean value, variance) can be calculated exactly using Racah’s angular momentum-coupling techniques [1,2,3,4,5,6]. Although such approaches (UTA: unresolved transition arrays and SOSA: spin–orbit–split arrays) have been proven to be very powerful, it is important to be aware of more detailed properties of transition arrays, such as high-order moments of the distribution [7], value of the maximal strength, number of weak lines (i.e., which strength is smaller than a given small threshold), strong lines (i.e., which strength is higher than a high fraction of the maximal strength), etc.

Global properties of spectra, as well as regularities or trends, are important for checking atomic databases [8,9,10,11], as well as opacity tables [12,13]. They can help in detecting anomalies or opening the way to new discoveries.

In the field of quantitative analytical spectroscopy, the foundational experimental data generally comprise measurements of total emission or absorption (or transmission) coefficients as functions of photon wavelength or energy. While the resulting spectrum may appear straightforward in cases involving a single electronic transition—thus producing only one spectral line—such simplicity is the exception rather than the norm. More commonly, multiple transitions occur simultaneously, giving rise to complex spectra with overlapping lines. The accurate quantitative interpretation of extinction coefficients within these overlapping transmission spectra represents a significant analytical challenge, which has been the focus of considerable research efforts by Elsasser [14] and Benedict et al. [15], for instance. Except for the (ideal) case of an infinite series of equally spaced, identical lines, the quantitative effect of line overlap requires the use of approximation methods. In addition to the spacing between neighboring lines, another easily accessible parameter of a transmission spectrum is the area between the spectral line and its baseline. Investigations of the effect of line overlap on this particular parameter have been limited, but for the special case of a symmetric doublet of overlapping Lorentzian lines, Bykov and Hien found that the interference contribution to spectral area can be approximated using zero- and first-order Bessel functions of the second kind with an imaginary argument [16].

In Section 2, we propose different methods to predict the maximum line strength, based on two different assumptions for the distribution of the amplitudes: the Porter–Thomas law, valid for $J-J^{\prime }$ sets, and a decreasing exponential (or Poisson distribution) over a whole transition array. In Section 3, different estimates for the number of strong and weak lines are obtained, in the context of the two hypotheses mentioned above. In Section 4, we review the main historical models of line opacity, both in the non-overlapping and overlapping cases. In the same section, the equivalent width is discussed, together with the Ladenburg–Reiche function. Different approximations of the latter are compared. Promising statistical tools are briefly mentioned in the two appendices, in view of their potential application to the modeling of complex atomic spectra.

2. Estimation of the Maximal Line Strength

2.1. Porter–Thomas Distribution

Porter and Thomas have shown that the amplitudes of the lines between all the levels of two random matrices, i.e., two Gaussian Orthogonal Ensembles (GOE), are well described by a Gaussian distribution [17,18]. In the corresponding formalism, the number of lines whose amplitude lies between a and $a+da$ reads as follows:

$$P\left(a\right)={\displaystyle \frac{L}{\sqrt{2\pi v}}}exp\left(-{\displaystyle \frac{a^2}{2v}}\right).$$

Based on the strength being the square of the amplitude, $S=a^2$, we have the following:

$$P\left(S\right)\mathrm{d}S=P\left(a\right)\mathrm{d}a+P(-a)\mathrm{d}a=2P\left(a\right)\mathrm{d}a$$

(1)

yielding a distribution of the ${\chi }^2$ type [19,20]:

$$P\left(S\right)={\displaystyle \frac{L}{\sqrt{2\pi \langle S\rangle S}}}exp\left(-{\displaystyle \frac{S}{2\langle S\rangle }}\right),$$

(2)

where L and $\langle S\rangle$ represent the number of lines and the mean value of the line strength in a particular ($J,J^{\prime }$) set, respectively. A given ($J,J^{\prime }$) set represents the number of lines between a collection of levels with the same total angular momentum J to levels with the same total angular momentum $J^{\prime }$. Due to the electric-dipole selection rules, we have $J^{\prime }=J$ (with $J^{\prime }=J=0$ forbidden), $J^{\prime }=J+1$ or $J^{\prime }=J-1$. The variance of the amplitudes v is exactly the average strength $v=\langle S\rangle$ (total strength of the array divided by the number of lines).

Nuclear physicists have been pioneers in analyzing the statistical properties of microscopic systems composed of a large number of particles. Initially, they studied the energy levels of nuclei and introduced the concepts of random matrices and the Gaussian Orthogonal Ensemble (GOE). The fact that the amplitudes of the lines between all the levels of two random matrices obey a Gaussian distribution was first found to describe statistics of transition strengths between chaotic states in compound nuclei and has also proven its worth for the spectral statistics of heavy open-shell atoms [21,22,23,24]. By definition, all elements of a random matrix are independent and follow two Gaussian distributions. The variance of the distribution for the diagonal elements is twice that of the off-diagonal elements. It has been demonstrated that the eigenvalues of a high-order Hamiltonian matrix follow a Gaussian distribution, and that the distribution of their nearest-neighbor spacings corresponds to the derivative of a Gaussian function with respect to the line amplitudes. Porter and Thomas established that the transition amplitudes between all levels of two random matrices follow a Gaussian distribution. In both atomic and nuclear physics, it is tempting to consider that, within a given configuration, all energy levels corresponding to the same value of total angular momentum are eigenvalues of a random matrix. This is plausible since the total angular momentum J is a well-defined quantum number, as the Hamiltonian operator commutes with $\overrightarrow{J}$. As pointed out by Bauche [25], other exact quantum numbers are irrelevant in this context. Indeed, parity has a fixed value within each configuration, and the $(J,M_J)$ states are degenerate with respect to $M_J$ (projection of $\overrightarrow{J}$ along the z-axis) in the absence of external fields. The exponential term dominates in $P\left(S\right)$, except close to the origin (i.e., for very weak lines). This is due to the fact that Random Matrix Theory contains approximate symmetries, which are not sufficient to describe the vicinity of Russell–Saunders and $jj$ couplings [19,26]. In the model proposed by Wilson et al. [27], diagonal terms are calculated in a pure coupling using Cowan’s code [28], and off-diagonal elements are populated statistically beyond the GOE according to a bi-Gaussian distribution function where elements are correlated. They observed a huge number of off-diagonal elements of a small amplitude.

Starting from the Porter–Thomas assumption that the electric-dipolar line amplitude between two $(J,J^{\prime })$ sets (considered as Gaussian Orthogonal Ensembles) follows a Gaussian distribution, we can resort to the extreme-value theorem of Kendall and Stuart [29] to estimate the value of $S_{\max }$ such that, on average, strength values larger than $S_{\max }$ are greater than $1/2$. This constraint reads as follows:

$${\displaystyle \frac{L}{\sqrt{2\pi v}}}{\int }_{S_{\max }}^{\infty }{\displaystyle \frac{1}{\sqrt{S}}}exp\left(-{\displaystyle \frac{S}{2v}}\right)\mathrm{d}S=L\mathrm{erfc}\left(\sqrt{{\displaystyle \frac{S_{\max }}{2v}}}\right)={\displaystyle \frac{1}{2}},$$

(3)

where $\mathrm{erfc}=1-erf\left(z\right)$, with the usual error function

$$erf\left(x\right)={\displaystyle \frac{2}{\sqrt{\pi }}}{\int }_0^x{\mathrm{e}}^{-t^2}\mathrm{d}t.$$

Using the following asymptotic form of the complementary error function for large values of z:

$$\mathrm{erfc}\left(z\right)={\displaystyle \frac{2}{\sqrt{\pi }}}{\int }_z^{\infty }exp\left(-t^2\right)\mathrm{d}t\approx {\displaystyle \frac{1}{\sqrt{\pi }z}}exp\left(-z^2\right),$$

and setting $u=\sqrt{S_{\max }/\left(2v\right)}$, Equation (3) can be rewritten as follows:

$${\displaystyle \frac{2L}{\sqrt{\pi }}}=uexp\left(u^2\right),$$

(4)

or, in other words,

$$2u^{2}exp\left(2u^2\right)={\displaystyle \frac{8L^2}{\pi }},$$

whose solution can be expressed using the Lambert W function $2u^2=W\left[{\displaystyle \frac{8L^2}{\pi }}\right]$ [30,31,32] (see Figure 1). Finally,

$$S_{\max }=vW\left({\displaystyle \frac{8L^2}{\pi }}\right).$$

The Lambert W has the following expansion [33,34,35]:

$$W\left(x\right)=lnx-lnlnx+\sum _{k=0}^{\infty }\sum _{m=1}^{\infty }{c}_{km}{\displaystyle \frac{{(lnlnx)}^m}{{(lnx)}^{k+m}}},$$

(5)

where

$$c_{km}={\displaystyle \frac{{(-1)}^k}{m!}}{S}_{k+m}^{(k+1)},$$

$S_{k+m}^{(k+1)}$ is the Stirling number of the first kind [36,37], defined as follows:

$$x(x-1)\cdots (x-n+1)=\sum _{i=0}^n{S}_i^{\left(n\right)}{x}^j$$

and for which Karanicoloff proposed an interesting expression [38]. We suggest using Equation (5) truncated at the fourth order:

$$W\left(x\right)=lnx-lnlnx+{\displaystyle \frac{lnlnx}{lnx}}+O\left[{\left({\displaystyle \frac{lnlnx}{lnx}}\right)}^2\right],$$

since it provides a good compromise between simplicity and accuracy [39]. $S_{\max }$ can then be approximated as follows:

$$S_{\max }=v\left[ln\left({\displaystyle \frac{8L^2}{\pi }}\right)-lnln\left({\displaystyle \frac{8L^2}{\pi }}\right)+{\displaystyle \frac{lnln\left({\displaystyle \frac{8L^2}{\pi }}\right)}{ln\left({\displaystyle \frac{8L^2}{\pi }}\right)}}\right].$$

(6)

Bauche and Bauche–Arnoult obtained a different expression, making the additional assumption $u^2\approx ln\left(L\right)$ after taking the logarithm of both sides of Equation (4):

$$ln\left(2L\right)\approx {\displaystyle \frac{1}{2}}ln\left(\pi \right)+lnu+u^2,$$

(7)

and obtained the following [19]:

$$S_{\max }=v\left[2lnL-ln(lnL)-ln\left({\displaystyle \frac{\pi }{4}}\right)\right].$$

(8)

The relative difference between Formulas (6) and (8) for $S_{\max }/\langle S\rangle$ as a function of the total number of lines L is represented in Figure 2. It is always very small (below 2%) in the considered range of variation in the number of lines L.

2.2. Decreasing Exponential Distribution for the Line Amplitudes

The assumption made by Porter and Thomas is relevant inside $(J,J^{\prime })$ sets of lines of a transition array [20,25,40]. Bauche, Bauche-Arnoult, and Wyart studied the distribution of the intensities of the bound–bound transitions in an atomic transition array [25]. They found that the distribution of the line amplitudes is essentially a decreasing exponential function. However, there always appears an excess of weak lines. This excess is related to coupling effects, when the physical coupling is close to an extremal (pure) coupling, and to scars of symmetries, in the opposite case. The occurrence of several types of scars makes the whole intensity array very complex. If we are interested in the maximal strength over a whole transition array of lines, we can consider that the sum of several Gaussian distributions leads to a decreasing exponential distribution for the line amplitudes [41,42,43,44]:

$$P\left(a\right)={\displaystyle \frac{L\lambda }{2}}exp\left(-\lambda a\right),$$

where $\lambda$ is defined by $v=2/{\lambda }^2=\langle S\rangle =S_{\mathrm{tot}}/L$, $S_{\mathrm{tot}}$, representing the total strength of the transition array:

$$S_{\mathrm{tot}}=\sum _{\begin{array}{c}J\to J^{\prime }\\ J\in c,J^{\prime }\in c^{\prime }\end{array}}{S}_{J\to J^{\prime }},$$

where c and $c^{\prime }$ are the initial and final configurations of the transition array. A level should be denoted $\gamma J$, $\gamma$ representing the ensemble of quantum numbers (possibly) required to identify a level of total angular momentum J in a unique way. However, for simplicity, we label a level by a value of the total atomic angular momentum J. Using relation (1), we obtain the following:

$$P\left(S\right)={\displaystyle \frac{L\lambda }{2\sqrt{S}}}exp\left(-\lambda \sqrt{S}\right).$$

The theorem of extreme values [45,46,47] enables one to define the maximal strength of a transition array as follows:

$${\int }_{S_{\max }}^{\infty }P\left(S\right)\mathrm{d}S=Lexp\left(-\lambda \sqrt{S_{\max }}\right)={\displaystyle \frac{1}{2}},$$

leading to the simple form for $S_{\max }$:

$$S_{\max }={\displaystyle \frac{v}{2}}{\left[ln\left(2L\right)\right]}^2.$$

(9)

It is worth mentioning that Bauche-Arnoult et al. proposed an alternative approach [20]. Since it is easier to calculate integrals with finite bounds for an exponential function than for a Gaussian, the authors suggested to represent, at least locally, the Gaussian by an exponential. They performed “shaping” of the exponential by the right-half of a Gaussian distribution as follows:

$${\displaystyle \frac{2}{\sqrt{\pi }}}exp\left(-x^2\right)\mathrm{d}x=\eta exp(-\lambda \eta )\mathrm{d}\eta ,$$

thus defining the relation between the x and y variables. Since both sides of the latter equation are normalized to unity when integrated from 0 to ∞, they can be used to relate $x_{\max }$ and $y_{\max }$ by integrating both sides from $x_{\max }$ (resp. $y_{\max }$) to infinity. This provides the following equation:

$$\mathrm{erfc}\left(x_{\max }\right)=exp(-\eta y_{\max }).$$

Thus, one has the following:

$$S_{\max }={\displaystyle \frac{1}{{\eta }^2}}ln\left\{\mathrm{erfc}\left({\displaystyle \frac{1}{\sqrt{2}}}{\left[2ln\left(L\right)-ln(lnL)-ln\left({\displaystyle \frac{\pi }{4}}\right)\right]}^{1/2}\right)\right\}.$$

Of course, the result depends on the value of $\eta$ (Bauche et al. suggest to take $\eta =1$). Within that approach, it is possible to combine both assumptions, first the Gaussian one (since $y_{\max }$ is deduced from $x_{\max }$, stemming from the Gaussian assumption), and the exponential one (for the shape of the distribution). This might be interesting since both distributions can be viewed as limiting cases, the truth likely lying in between. However, the fact that the results depend on the (arbitrary) parameter $\lambda$ is clearly a weakness of the model.

The values of expressions (6) and (9) for $S_{\max }/\langle S\rangle$ as a function of the total number of lines L are compared in Figure 3. The maximal strength estimated using Formulas (6), (8) and (9) is compared to the exact value for two different transition arrays: Pd VII 4d⁴ − 4d³ 5p and Cd V 4d⁷ 5s − 4d⁷ 5p in

Table 2. Estimates of the maximum strength (calibrated so that the total strengths of the arrays are 2940 and 50,400 in Pd VII and Cd V, respectively [41]) for two different transition arrays: Pd VII 4d⁴ − 4d³ 5p and Cd V 4d⁷ 5s − 4d⁷ 5p. The radial integrals required for the diagonalization of the transition array Pd VII 4d⁴ − 4d³ 5p are given in Table 3.

	Pd VII	Cd V
	4d4 − 4d3 5p	4d7 5s − 4d7 5p
Total number of lines	1718	2082
Variance in the amplitudes v	1.7161	24.2064
Exact maximal strength $S_{\max }$	53	1050
Approximate $S_{\max }$ (Formula (6))	22.73	329.27
Approximate $S_{\max }$ (Formula (8))	22.53	326.55
Approximate $S_{\max }$ (Formula (9))	56.88	840.68

. (The Slater and spin-orbit integrals used for the diagonalization of Pd VII 4d⁴ − 4d³ 5p are given in

Table 3. Values of the parameters used for the diagonalization of the 4d⁴ − 4d³ 5p array in the Pd VII spectrum in cm⁻¹. $F^k$ and $G^k$ are direct and exchange Slater integrals, respectively, and $\zeta$ denotes a spin–orbit integral.

	4d4	4d3 5p
$F^2$(4d, 4d)	73,800	76,300
$F^4$(4d, 4d)	50,500	52,700
$F^2$(4d, 5p)		30,323
$G^1$(4d, 5p)		10,000
$G^3$(4d, 5p)		8020
${\zeta }_{4\mathrm{d}}$	2200	2390
${\zeta }_{5\mathrm{p}}$		5784

). We can see that Formula (9) gives the best agreement with the exact value, which is due to the fact that the decreasing exponential is more relevant than the Gaussian to describe the distribution of line amplitudes inside a whole transition array. The two transition arrays, computed using Cowan’s code [28], are displayed in Figure 4 and Figure 5, respectively.

2.3. Maximal Strength Associated with Learner’s Rule

In a seminal 1982 study, Learner measured a large number of line intensities in the atomic emission spectrum of neutral iron and uncovered a striking statistical regularity: a power-law relationship governing the density of spectral lines as a function of their intensity [48]. Specifically, he demonstrated that the logarithm of the number of lines, $N_n$, whose intensities fall within the interval $[2^n{I}_1,2^{n+1}{I}_1]$ (with n being an integer), decreases linearly with n. The reference intensity $I_1$ is chosen such that this linear law holds consistently over nine octaves ($1\le n\le 9$), based on an analysis of approximately 1500 spectral lines in the wavelength range from 290 nm to 550 nm. The number of lines is multiplied by ${10}^p$ when the size of the interval is multiplied by two. Learner’s observations imply the following ($S_1$ is the line strength corresponding to intensity $I_1$ in the interval $2^{n-1}{S}_1-2^n{S}_1$) [48,49,50]:

$${log}_{10}\left({\int }_{2^{n-1}{S}_1}^{2^n{S}_1}P\left(S\right)\mathrm{d}S\right)\approx q_0-p.n,$$

(10)

with $P\left(S\right)\approx k.S^{-\alpha }$ and $p>0$. Equation (10) can be put in the following form:

$$\begin{array}{ccc}\hfill {log}_{10}\left({\int }_{2^{n-1}{S}_1}^{2^n{S}_1}P\left(S\right)\mathrm{d}S\right)& \approx & {log}_{10}\left(k{\displaystyle \frac{{\left[S^{1-\alpha }\right]}_{2^{n-1}{S}_1}^{2^n{S}_1}}{1-\alpha }}\right)\hfill \\ & \approx & {log}_{10}\left({\displaystyle \frac{k{S}_1^{1-\alpha }\left(1-2^{\alpha -1}\right)}{(1-\alpha )}}\right)+(1-\alpha ){log}_{10}\left(2\right).n.\hfill \end{array}$$

(11)

Identifying Equation (11) with Equation (10), we obtain $q_0\equiv {log}_{10}\left({\displaystyle \frac{k{S}_1^{1-\alpha }\left(2^{1-\alpha }-1\right)}{(1-\alpha )}}\right)$ and $p\equiv -(1-\alpha ){log}_{10}\left(2\right)$. Since $p\approx -{\displaystyle \frac{1}{2}}{log}_{10}\left(2\right)$ (observation made by Learner), we find $\alpha ={\displaystyle \frac{3}{2}}$ and $P\left(S\right)\approx k.S^{-3/2}$, where

$$k\approx {\displaystyle \frac{(1-\alpha ){10}^{q_0}}{S_1^{1-\alpha }\left(1-2^{\alpha -1}\right)}}\approx {\displaystyle \frac{{10}^{q_0}\sqrt{S_1}}{2(\sqrt{2}-1)}}.$$

The extreme-value theorem yields the following:

$${\int }_{S_{\max }}^{\infty }P\left(S\right)\mathrm{d}S={\displaystyle \frac{2k}{\sqrt{S_{\max }}}}={\displaystyle \frac{1}{2}},$$

i.e.,

$$S_{\max }\approx 16k^2\approx {\displaystyle \frac{4.{10}^{2q_0}{S}_1}{{(\sqrt{2}-1)}^2}}.$$

3. On the Number of Strong and Weak Lines

Estimating the number of weak and strong lines can be useful in order to choose the best statistical representation for an array of lines and to derive hybrid models [51] such as the Mixed-UTA approach (see Section 4) below [52]. We can decide, following Bauche et al. [19], that strong lines are such that their strength is greater than a small value $S_0$ (e.g., a small fraction of the maximal strength $S_{\max }$ determined in the previous section).

3.1. Porter–Thomas Law

In the first approach (Porter–Thomas, Gaussian distribution of the amplitudes, $(J,J^{\prime })$ set), the number of weak lines reads, using Equation (2):

$$N_{\mathrm{weak}}={\int }_0^{S_0}{\displaystyle \frac{L}{\sqrt{2\pi \langle S\rangle S}}}exp\left(-{\displaystyle \frac{S}{2\langle S\rangle }}\right)\mathrm{d}S$$

or equivalently,

$$N_{\mathrm{weak}}={\int }_0^{S_0}{\displaystyle \frac{L}{\sqrt{2\pi vS}}}exp\left(-{\displaystyle \frac{S}{2v}}\right)\mathrm{d}S$$

giving the following based on the change in variables $S=x^2$:

$$N_{\mathrm{weak}}=L\sqrt{{\displaystyle \frac{2}{\pi v}}}{\int }_0^{\sqrt{S_0}}exp\left(-{\displaystyle \frac{x^2}{2v}}\right)\mathrm{d}x$$

and thus, setting $t=x/\sqrt{2v}=t/\sqrt{2\langle S\rangle }$:

$$N_{\mathrm{weak}}=L\mathrm{erf}\left(\sqrt{{\displaystyle \frac{S_0}{2\langle S\rangle }}}\right),$$

(12)

or equivalently:

$$N_{\mathrm{strong}}=L\left[1-\mathrm{erf}\left(\sqrt{{\displaystyle \frac{S_0}{2\langle S\rangle }}}\right)\right].$$

(13)

We suggest estimating the erf function through the following expansion:

$$erf\left(x\right)={\displaystyle \frac{2}{\sqrt{\pi }}}exp\left(-x^2\right)\left(x+{\displaystyle \frac{2}{3}}{x}^3+{\displaystyle \frac{4}{15}}{x}^5\right)+o\left(x^6{\mathrm{e}}^{-x^2}\right).$$

The latter formula is very accurate for the values of the argument encountered in the present work. It may be useful to have an expression in the form of a polynomial multiplied by a Gaussian (which is related to the moments of the Gaussian) for integration purposes, for instance. An even better compromise between simplicity and accuracy can be achieved via the following approximation [53]:

$$erf\left(x\right)\simeq 1-exp\left(-1.9x^{1.3}\right),$$

(14)

with a difference below 2.2 × 10⁻². This gives the following:

$$N_{\mathrm{weak}}=L\left\{1-exp\left[-1.9{\left({\displaystyle \frac{S_0}{2v}}\right)}^{0.65}\right]\right\},$$

(15)

or equivalently:

$$N_{\mathrm{strong}}=Lexp\left[-1.9{\left({\displaystyle \frac{S_0}{2v}}\right)}^{0.65}\right].$$

(16)

The approximation improves to yield a difference of less than 10⁻² for $x\ge 1$. Other useful methods for approximating the error function are recalled in Appendix A.

3.2. Decreasing Exponential Distribution of the Amplitudes

In the second approach (whole transition array, decreasing exponential of the amplitudes), one obtains the following:

$$N_{\mathrm{strong}}=L-N_{\mathrm{weak}}=Lexp\left[-\lambda \sqrt{S_0}\right].$$

We have seen that $v=2/{\lambda }^2$. On the other hand, $v=S_{\mathrm{tot}}/L$, where $S_{\mathrm{tot}}$ is the total strength of the transition array and L its number of lines. This yields the following:

$$\lambda =\sqrt{{\displaystyle \frac{2L}{S_{\mathrm{tot}}}}}$$

and thus, one obtains the following:

$$N_{\mathrm{strong}}=L-N_{\mathrm{weak}}=Lexp\left[-\sqrt{2L{\displaystyle \frac{S_0}{S_{\mathrm{tot}}}}}\right]=Lexp\left[-\sqrt{2{\displaystyle \frac{S_0}{\langle S\rangle }}}\right].$$

(17)

As can be checked in

Table 4. Number of lines with a strength greater than a fraction of the maximum strength (exact and estimated based on Formulas (13) and (17)) for transition array Pd VII 4d⁴ − 4d³ 5p (see Figure 4). The total number of lines is 1718.

$\mathit{ϵ}$	No. lines $\mathit{S}>\mathit{ϵ}{\mathit{S}}_{\mathbf{max}}$	No. lines $\mathit{S}>\mathit{ϵ}{\mathit{S}}_{\mathbf{max}}$	No. lines $\mathit{S}>\mathit{ϵ}{\mathit{S}}_{\mathbf{max}}$
	Exact	Formula (13)	Formula (17)
0.003	781	1460.33	1314.87
0.01	524	1253.99	1054.35
0.03	311	934.66	737.49
0.06	185	683.32	519.56
0.3	20	100.73	118.48
0.5	14	25.15	54.41
0.6	5	12.87	39.14
0.8	1	3.46	21.80

, the approximate Formulas (13) and (17) do not give accurate estimations of the numbers of lines whose strength is greater than a specific value. However, they both provide a general trend. What is more surprising in that case is that Formula (13) is closer to the exact values, although it is expected (Porter–Thomas) to be more reliable for $J-J^{\prime }$ sets than for a whole transition array. Actually, the disagreement can be easily understood here. Indeed, as can be see in Figure 4, the strongest line is “alone”, in the sense that its strength is much larger than the strength of the other lines. It is easily understandable, thus, that basing the estimator of the weak/strong lines on this singularity is not the most appropriate choice. It would be, in that case, much more relevant to consider the strong lines as lines having a strength larger than a fraction of the total strength of the array for instance, or having a strength larger than the average strength, etc.

3.3. Imposing a Number of Lines and Searching for the Threshold

In the computation of hot-plasma atomic spectra, one may be interested in imposing that the number of lines does not exceed a certain value. Thus, it may seem natural to keep only the strongest lines $N_{\mathrm{strong}}$. Equation (17) gives the following in the case of decreasing exponential variation in the amplitudes:

$$S_0={\displaystyle \frac{\langle S\rangle }{2}}{\left[ln\left({\displaystyle \frac{L}{N_{\mathrm{strong}}}}\right)\right]}^2.$$

(18)

However, in the case of the Porter–Thomas distribution, as can be seen from Equation (12), one needs to invert the $\mathrm{erf}$ function [54,55,56]. This can be described using a series expansion:

$${erf}^{-1}\left(z\right)=\sum _{k=0}^{\infty }{\displaystyle \frac{b_k}{2k+1}}{\left({\displaystyle \frac{\sqrt{\pi }}{2}}z\right)}^{2k+1}$$

where

$$c_k=\sum _{m=0}^{k-1}{\displaystyle \frac{b_m{b}_{k-1-m}}{(m+1)(2m+1)}}$$

and

$$\left\{b_0,b_1,b_2,b_3,\dots \right\}=\left\{1,1,{\displaystyle \frac{7}{6}},{\displaystyle \frac{127}{90}},\dots \right\}.$$

We obtain the following expansion:

$${erf}^{-1}\left(z\right)={\displaystyle \frac{1}{2}}\sqrt{\pi }\left(z+{\displaystyle \frac{\pi }{12}}{z}^3+{\displaystyle \frac{7{\pi }^2}{480}}{z}^5+{\displaystyle \frac{127{\pi }^3}{40320}}{z}^7+{\displaystyle \frac{4369{\pi }^4}{5806080}}{z}^9+{\displaystyle \frac{34807{\pi }^5}{182476800}}{z}^{11}+\cdots \right).$$

Using the simple and efficient formula (15), one obtains the following:

$$S_0=2\langle S\rangle {\left[{\displaystyle \frac{1}{1.9}}ln\left({\displaystyle \frac{L}{N_{\mathrm{strong}}}}\right)\right]}^{{\displaystyle \frac{20}{13}}}.$$

(19)

An approximation (14) of the $\mathrm{erf}$ function (more precisely, of $1-erf\left(x\right)$) is illustrated in Figure 6. A comparison between Formulas (18) and (19) is displayed in Figure 7 for the quantity $S_0/\langle S\rangle$.

4. Merging Lines Together: Band Models

The modeling of several lines, and in particular, their potential overlap has been a subject of investigation since the early times of spectroscopy. The models presented in this section have been widely used in the past, but mostly in the context of molecular spectroscopy, in particular by the National Aeronautics and Space Administration; for instance, for the determination of atmospheric temperature profiles from planetary limb radiance profiles [57], or the effects of transmission models in the rotational water-vapor band on radiance calculations and constituent inferences [58]. They have been applied to hot plasma as well, but to a lesser extent. They are interesting because they provide insight about the coalescence properties of spectra, which are directly related to the so-called “porosity” of transition arrays. This is a cornerstone of opacity theory, because the modeling of radiation transport requires, at least in the diffusion approximation, the knowledge of the Rosseland mean opacity:

$${\kappa }_R={\left({\int }_0^{i^{nfty}}{\displaystyle \frac{W_R\left(u\right)}{\kappa \left(u\right)}}\mathrm{d}u\right)}^{-1},$$

where $\kappa \left(u\right)$ is the spectral opacity and

$$W_R\left(u\right)={\displaystyle \frac{15}{4{\pi }^4}}{\displaystyle \frac{u^4{e}^{-u}}{{(1-e^{-u})}^2}}$$

is the Rosseland weighting function, proportional to the derivative of the Planck distribution with respect to the temperature. The Rosseland mean is thus a weighted harmonic mean of the spectral (monochromatic) opacity. Being able to quantify this “amount of merging” may be very useful in order to define coalescence criteria and thus correct the predictions of the statistical methods (UTA, …) using a coefficient accounting for the porosity. Such models are also useful to define a resulting average line full width at half maximum. However, these models are often dependent on strong assumptions. For instance, the Elsasser model [14] (see Section 4.2.1 below) assumes Lorentz line shape, equal spacing, and equal intensity for the lines.

The common assumptions of band models are as follows. We consider an infinite array of absorption lines of uniform statistical properties. Moreover, an interval of this array (containing several lines) is taken to have properties similar to those of an interval of the real interval under consideration. Finally, each interval is flawed by statistically similar intervals. This will not necessarily be true for a real band. In the following, we discuss two different cases: the non-overlapping model, in which all lines are modeled by a single profile, and the overlapping model, in which each line has its own profile.

4.1. Non-Overlapping Model

4.1.1. Lorentzian Shape

Let us define the absorptance as follows:

$$\overline{A}={\displaystyle \frac{1}{d}}{\int }_{-\infty }^{\infty }(1-e^{-\rho \ell {\kappa }_{\nu }})\mathrm{d}\nu ,$$

where d is the average spacing between lines, $\rho$ is the mass density and ℓ is the thickness of the sample ($\rho \ell$ is therefore the areal mass density). Assuming a Lorentzian shape for the opacity gives the following:

$${\kappa }_{\nu }=S{\displaystyle \frac{\alpha }{\pi }}{\displaystyle \frac{1}{{\alpha }^2+{\nu }^2}},$$

and we obtain the following for the absorptance:

$$\overline{A}=2\pi yu{e}^{-u}\left[J_0\left(iu\right)-i{J}_1\left(iu\right)\right]=2\pi y\mathcal{L}\left(u\right),$$

where $\mathcal{L}\left(u\right)$ is the Ladenburg–Reiche function [59,60,61] ($J_0$ and $J_1$ are Bessel functions of the first kind of orders 0 and 1, respectively [62]). We have introduced the following reduced variables:

$$x={\displaystyle \frac{\nu }{d}},y={\displaystyle \frac{\alpha }{d}}\mathrm{and}u={\displaystyle \frac{S\rho \ell }{2\pi \alpha }}.$$

For strong lines $u\gg 1$, $\mathcal{L}\left(u\right)\approx \sqrt{2u/\pi }$, and for weak lines, $L\left(u\right)\approx u$.

4.1.2. Gaussian (Doppler) Shape

In the case of the Gaussian assumption, we obtain the following for the opacity:

$$k_{\nu }={\displaystyle \frac{S}{\sqrt{\pi }\Delta {\nu }_D}}exp\left(-{\displaystyle \frac{{\nu }^2}{\Delta {\nu }_D^2}}\right)$$

with

$$x={\displaystyle \frac{\nu }{\Delta {\nu }_D}}\mathrm{and}w={\displaystyle \frac{S\rho \ell }{\sqrt{\pi }\Delta {\nu }_D}},$$

yielding the absorptance:

$$\overline{A}={\displaystyle \frac{\Delta {\nu }_D}{d}}{\int }_{-\infty }^{\infty }\left(1-exp\left[-w{e}^{-x^2}\right]\right)\mathrm{d}x.$$

For large values of w, we have the following:

$$\overline{A}={\displaystyle \frac{2\Delta {\nu }_D}{d}}\left[{(logw)}^{1/2}+{\displaystyle \frac{0.2886}{{(logw)}^{1/2}}}-{\displaystyle \frac{0.1335}{{(logw)}^{3/2}}}+{\displaystyle \frac{0.0070}{{(logw)}^{5/2}}}...\right].$$

Although these two non-overlapping models (together with the Lorentzian and Gaussian assumptions) are both based on a strong approximation, they were of great use to physicists in the early stages of opacity calculations [63].

4.2. Overlapping Models

The case of overlapping lines is of course more realistic and frequent [64] in the framework of the calculation of hot plasma absorption and emission coefficients.

4.2.1. The Elsasser Model

Within the Elsasser model [14], the absorption coefficient is modeled by a sum of Lorentzian lines:

$${\kappa }_{\nu }=\sum _{n=-\infty }^{\infty }{\displaystyle \frac{\alpha }{\pi }}{\displaystyle \frac{S}{{(\nu -nd)}^2+{\alpha }^2}}$$

yielding the absorptance coefficient:

$$\overline{A}=1-{\int }_{-{\displaystyle \frac{1}{2}}}^{{\displaystyle \frac{1}{2}}}exp\left[-2\pi uy{\displaystyle \frac{sinh\left(2\pi y\right)}{cos\left(2\pi y\right)-cos\left(2\pi x\right)}}\right]\mathrm{d}x=1-E(y,u),$$

where $E(y,u)$ is the so-called Elsasser function [14,65,66]:

$$E(y,u)={\displaystyle \frac{1}{\pi }}{\int }_0^{\pi }exp\left(-{\displaystyle \frac{uzsinh\left(z\right)}{cosh\left(z\right)-cos\left(y\right)}}\right)\mathrm{d}y.$$

The Elsasser function can be put in the following form [67]:

$$=1-{\int }_0^{uz}exp[-\mathrm{cotanh}\left(z\right).a]I_0\left({\displaystyle \frac{a}{sinhz}}\right)\mathrm{d}a,$$

where $I_0$ is the modified Bessel function of the first kind of order 0. The modified Bessel function of the first kind $I_n\left(z\right)$ can be defined by the contour integral:

$$I_n\left(z\right)={\displaystyle \frac{1}{2\pi i}}\oint e^{(z/2)(t+1/t)}{t}^{-n-1}\mathrm{d}t,$$

where the contour encloses the origin and is crossed in a counterclockwise direction [68]. $I_n\left(x\right)$ is related to $J_n\left(x\right)$ as follows:

$$I_n\left(x\right)=i^{-n}{J}_n\left(ix\right)=e^{-n\pi i/2}{J}_n\left(x{e}^{i\pi /2}\right).$$

When $y\to \infty$, $sinh\left(2\pi y\right)\to cosh\left(2\pi y\right)\to \infty$, this yields the following:

$$E(y,u)\approx e^{-2\pi yu}.$$

(20)

In the strong-line limit ($u\gg 1$), neglecting $y^2$ in the denominator of the Lorentz profile, one has $sinh\left(2\pi y\right)\approx 2\pi y$ and $cosh\left(2\pi y\right)$, leading to the following:

$$E(y,u)\approx 1-erf\left(\pi y\sqrt{2u}\right).$$

(21)

An approximation for moderately large y values can also be derived, noticing that

$${\displaystyle \frac{2\pi uysinh\left(2\pi y\right)}{cosh\left(2\pi y\right)-cos\left(2\pi x\right)}}\approx 2\pi yu\left[1+2e^{-2\pi y}cos\left(2\pi x\right)\right].$$

The corresponding form is as follows:

$$E(y,u)\approx I_0\left(4\pi yu{e}^{-2\pi y}\right)e^{-2\pi yu}.$$

(22)

The evaluation of the Elsasser function is discussed in [69]. Other tabulations have been given by [70,71,72,73]. Figure 8, Figure 9, Figure 10, Figure 11, Figure 12 and Figure 13 show the comparison between the exact calculation and different approximations of the Elsasser function as a function of y for different values of u, and as a function of u for different values of y. The three approximations are given by Equations (20), (21) and (22), respectively. The third one represents a good compromise between simplicity and range of reasonable accuracy.

4.2.2. The Mayer–Goody Model

Taking into account the line strength distribution $P\left(S\right)$, the mean absorptance reads as follows [63,74,75,76]:

$$\overline{A}=1-exp\left[-{\displaystyle \frac{1}{d}}{\displaystyle \frac{{\int }_{-\infty }^{\infty }{\int }_0^{\infty }P\left(S\right)(1-e^{-{\kappa }_{\nu }\rho \ell })\mathrm{d}S\mathrm{d}\nu }{{\int }_0^{\infty }P\left(S\right)\mathrm{d}S}}\right],$$

where different forms of $P\left(S\right)$ were introduced in Section 2. Two distributions were studied by Mayer and Goody.

The first one assumes that $P\left(S\right)={\displaystyle \frac{1}{\sigma }}{e}^{-{\displaystyle \frac{S}{\sigma }}}$. In this case, one has the following:

$$\overline{A}=1-exp\left[-{\displaystyle \frac{1}{d}}{\int }_{-\infty }^{\infty }{\displaystyle \frac{\rho \ell \sigma f(\nu ,\sigma )}{1+\rho \ell \sigma f(\nu ,\sigma )}}\mathrm{d}\nu \right]$$

with

$$f(\nu ,\sigma )={\displaystyle \frac{\alpha }{\pi }}{\displaystyle \frac{1}{{\nu }^2+{\alpha }^2}}$$

yielding

$$\overline{A}=1-exp\left[-{\displaystyle \frac{\rho \ell \sigma \alpha }{d{\left({\alpha }^2+\rho \ell \sigma \frac{\alpha }{\pi }\right)}^{1/2}}}\right].$$

The second model sets $P\left(S\right)={\displaystyle \frac{K}{S}}$, yielding the following for the absorptance:

$$\overline{A}=1-exp\left[-{\displaystyle \frac{2\pi K\alpha }{d}}\left\{e^{-u}{J}_0\left(iu\right)+2u{e}^{-u}\left[J_0\left(iu\right)+J_1\left(iu\right)\right]\right\}-1\right]$$

4.3. Equivalent Width of a Lorentz Line

The equivalent width W of a line is the width of a rectangular line with 100% absorption that has the same absorption as the actual line [77,78,79]. Using the definition of the absorptance of a line, we obtain the following:

$$W=\overline{A}={\displaystyle \frac{1}{d}}{\int }_{-\infty }^{\infty }(1-e^{-\rho \ell {\kappa }_{\nu }})\mathrm{d}\nu ,$$

which involves the Ladenburg–Reiche function (see the non-overlapping case discussed in Section 4.1.1) [59,60]:

$$\mathcal{L}\left(x\right)=x{e}^{-x}\left[J_0\left(ix\right)-i{J}_1\left(ix\right)\right],$$

where $J_n\left(x\right)$ is a Bessel function of the first kind. Since

$$J_n\left(ix\right)=i^n{I}_n\left(x\right),$$

where $I_n\left(x\right)$ is a modified Bessel function of the first kind, one has $\mathcal{L}\left(x\right)\equiv f\left(x\right)$, where

$$f\left(x\right)=x{e}^{-x}\left[I_0\left(x\right)+I_1\left(x\right)\right]$$

and different approximations were proposed. One has, for instance [80]:

$$f\left(x\right)\approx \sqrt{{\displaystyle \frac{2x}{\pi }}}\left[1-exp\left\{-{\left({\displaystyle \frac{\pi x}{2}}\right)}^{1/2}\right\}\right]$$

(23)

or Varanasi (quoted by Tien in Ref. [80]):

$$f\left(x\right)={\left[{\displaystyle \frac{2x}{\pi }}\left\{1-exp\left(-{\displaystyle \frac{\pi x}{2}}\right)\right\}\right]}^{1/2}.$$

(24)

The following approximation was also proposed (referred to as “Goldman 1” in Figure 14 and Figure 15) [81]:

$$f\left(x\right)\approx {\displaystyle \frac{x}{{\left({\displaystyle 1+\frac{\pi x}{2}}\right)}^{1/2}}}$$

(25)

derived from the function $F\left(x\right)=x/\sqrt{1+2x}$ by the transformation $\left(4\pi \right)F(x\pi /4)$. Finally, the last considered expression is as follows (referred to as “Goldman 2” in Figure 14 and Figure 15) [81]:

$$f\left(x\right)\approx {\displaystyle \frac{x}{{\left[1+{\left({\displaystyle \frac{\pi x}{2}}\right)}^{\alpha }\right]}^{\frac{1}{2\alpha }}}}\mathrm{with}\alpha =5/4.$$

(26)

A comparison of the accuracy of the four approximations, Equations (23)–(26), was already performed by Goldman [81] using the values obtained by Kaplan and Eggers [65] as exact values.

It turns out that the calculations were carried out for (approximation – exact function)/exact function. It is seen that the maximum error is about 17, 6, 8, and 1 percent for approximations (23), (24), (25) and (26), respectively. The largest errors in using Equations (23)–(25) occur for intermediate values of x, between x = 0.2 and $x=2$, while the error in using Equation (26) lies close to zero near x = 0.5. Approximation (26) is more accurate than the others and yet simple enough to manipulate mathematically. Approximation (25) is not as accurate as Equation (26), but it is convenient for a least squares fitting of a set of experimental data to the theoretical curve.

Martin, Olivares and Sotomayor [82] proposed the following approximant:

$$I_1\left(x\right)={\displaystyle \frac{xcosh\left(x\right)}{{(1+{\lambda }^2{x}^2)}^{3/4}}}{\displaystyle \frac{p_0+\sqrt{{\displaystyle \frac{2}{\pi }}}{\lambda }^{3/2}{q}_1{x}^2}{(1+q_1{x}^2)}}$$

with $\lambda =1/5$ and $p_0=1/2$ and

$$q_1={\displaystyle \frac{{\displaystyle \frac{3}{4}{\lambda }^2-\frac{3}{8}}}{\frac{4}{\sqrt{2\pi }}{\lambda }^{3/2}-1}}.$$

In terms of $\lambda$ only, one has the following:

$${\displaystyle I_1\left(x\right)=\frac{xcoshx}{{(1+{\lambda }^2{x}^2)}^{3/4}}\frac{{\displaystyle \frac{1}{2}+\sqrt{{\displaystyle \frac{2}{\pi }}}{\lambda }^{3/2}\left[{\displaystyle \frac{{\displaystyle \frac{3}{4}{\lambda }^2-\frac{3}{8}}}{{\displaystyle \frac{4}{\sqrt{2\pi }}{\lambda }^{3/2}-1}}}\right]x^2}}{1+\left[{\displaystyle \frac{{\displaystyle \frac{3}{4}{\lambda }^2-\frac{3}{8}}}{{\displaystyle \frac{4}{\sqrt{2\pi }}{\lambda }^{3/2}-1}}}\right]x^2},}$$

and with numerical values,

$$I_1\left(x\right)={\displaystyle \frac{xcosh\left(x\right)}{2{(1+0.04{\lambda }^2{x}^2)}^{3/4}}}{\displaystyle \frac{1+0.0574403x^2}{1+0.40244x^2}}.$$

For the integral $I_0\left(x\right)$ Olivares, Martin and Valero proposed the following approximation [83]:

$$I_0\left(x\right)={\displaystyle \frac{cosh\left(x\right)}{{(1+{\lambda }^2{x}^2)}^{1/4}}}{\displaystyle \frac{p_0+p_1{x}^2}{(1+q_1{x}^2)}}$$

with $\lambda =1/2$, $p_0=1$

$$p_1={\displaystyle \frac{{\displaystyle \frac{3}{4}{\lambda }^2-\frac{3}{8}}}{{\displaystyle \frac{4}{\sqrt{2\pi }}{\lambda }^{3/2}-1}}}.$$

and

$$q_1={\displaystyle \frac{{\displaystyle \frac{1}{4}-\frac{{\lambda }^2}{4}}}{1-\sqrt{{\displaystyle \frac{2}{\pi }}}{\lambda }^{1/2}}}.$$

In terms of $\lambda$ only, one has the following:

$${\displaystyle I_0\left(x\right)=\frac{coshx}{{(1+{\lambda }^2{x}^2)}^{1/4}}\frac{{\displaystyle 1+\sqrt{{\displaystyle \frac{2}{\pi }}}\left[{\displaystyle \frac{1-{\lambda }^2}{1-\sqrt{{\displaystyle \frac{2}{\pi }}}{\lambda }^{1/2}}}\right]\frac{{\lambda }^{1/2}{x}^2}{4}}}{{\displaystyle 1+\left[{\displaystyle \frac{1-{\lambda }^2}{1-\sqrt{{\displaystyle \frac{2}{\pi }}}{\lambda }^{1/2}}}\right]\frac{x^2}{4}}}}$$

and with numerical values,

$$I_0\left(x\right)={\displaystyle \frac{cosh\left(x\right)}{{(1+0.25x^2)}^{1/4}}}{\displaystyle \frac{1+0.24273x^2}{1+0.43023x^2}}.$$

Figure 14 and Figure 15 show the different approximations over two different ranges.

Figure 16 shows the relative difference between the exact value of the Ladenburg–Reiche function and the approximation obtained using the approximants of the Bessel functions $I_0$ and $I_1$ published by Olivares et al. [83] and Martin et al. [82], respectively. Figure 17 displays the same quantities as Figure 14 but over a wider range. Figure 18 represents the relative difference with respect to the exact value for different approximations of the Ladenburg–Reiche function. The Tien and Varanasi formulas are given by Equations (23) and (24), respectively, and the Goldman (1 and 2) formulas are given by Equations (25) and (26), respectively. The plotted quantity corresponds to (Exact – Approximation)/Exact and is expressed in %.

4.4. General Considerations About the Merging of Lines (Blanketing)

In 2007, Abdallah et al. introduced a novel approach to spectral line binning [84], leveraging the observation that many spectral lines often appear in close proximity—particularly in cases such as intermediate coupling in low-Z elements. When line centers are clustered and their individual line widths do not differ significantly, it becomes possible to sum their strengths into a single aggregate entity before generating the line profile. If the variation in line widths is modest, a weighted average can be used to assign a representative width to the resulting “super-line”. However, when line widths exhibit large disparities—such as the coexistence of both narrow and broad features—a more refined strategy is required. Abdallah et al.’s technique segments the bound–bound photon energy domain into very fine histogram bins, typically on the scale of one-tenth (or less) of the resolution step used in the final opacity grid. The method employs histogram arrays with sizes on the order of 10⁶, which is dramatically smaller than the number of individual spectral lines, which may reach hundreds of millions depending on physical conditions such as temperature and density. The entire spectral line database is first scanned to identify lines whose centers fall within each bin, and for each bin, the total line strength, a weighted average of the line widths, and a weighted average of the inverse widths are computed. These statistical measures help distinguish between bins dominated by broad or narrow lines. A second pass through the data separates the lines accordingly and aggregates them within their respective categories. Finally, for each populated bin, two distinct super-lines are constructed—one representing the broad lines, the other the narrow ones—mapped onto the final opacity calculation grid. Any bins with zero total line strength are discarded.

The mixed-UTA (MUTA) method [52] combines detailed and statistical treatments within the same transition array, as well as for a set of transition arrays used in the spectral or opacity calculations of light and mid-Z elements. By selecting the strongest lines within each transition array, the method delivers a spectral description with accuracy comparable to a fully detailed treatment, where all lines are explicitly included in the spectral calculation. The weaker lines are represented using a UTA-like functional form. As a result, the computational cost of this method is similar to that of the statistical UTA method. The approach was implemented in the Cowan atomic structure code and successfully applied to the calculation of spectra for both local thermodynamic equilibrium (LTE) and non-LTE plasmas of two mid-Z elements, xenon and iron. However, it is important to keep in mind that such an approach is difficult to use in intermediate coupling. Indeed, the MUTA approach involves gathering the weak lines (with the approach mentioned in Section 2) into a “global” structure, a kind of “pillow of lines”, and keeping only the strongest ones. This is possible close to pure LS coupling, when the UTA approach applies, or close to pure jj coupling, where the SOSA formalism is appropriate. In the latter case, the transition array shows two main relativistic sub-structures (see Figure 19) (the third one is very small and not visible). It is clear that in that case, using a single “pillow” (typically modeled by a Gaussian) would be deleterious, filling the gap between the two ensembles of SOSAs (i.e., the two relativistic subarrays). In that case, the approach may be extended, in the sense that we could attribute some of the weak lines to one of the subarrays, and the others to the second one. But in intermediate coupling, which is the most frequent case and the only way to account properly for configuration interactions, the applicability of the method may seem questionable. For instance, let us consider the transition array Br XI 3d⁴ 4f³ − 3d³ 4f⁴. In that case, two main relativistic substructures, corresponding to the relativistic subarrays 3d_3/2 − 4f_5/2 and 3d_5/2 4f_7/2 (the subarray 3d_5/2 − 4f_5/2 does exist, but it is too small to be clearly visible), are not clearly separated because the atomic number is not sufficiently high (spin–orbit splitting scales as $Z^4$).

Colgan et al. studied the spectra produced from a short-wavelength long-pulse (nanosecond) laser incident on Mg plasma [85]. A very complex experimental spectrum was analyzed in detail via comparison with large-scale atomic kinetics calculations using the recently developed mixed-UTA (MUTA) model. The authors found that the experimental spectrum exhibited lines from many inner-shell transitions from ions ranging from neutral Mg to Li-like Mg. In particular, lines from transitions such as 1s-3ℓ, 1s-4ℓ and from hollow atoms (in this context, hollow atoms refer to ions with an empty 1s subshell) were observed.

The resolved transition array (RTA) approach is a different kind of statistical method in which lines are picked up at random in a joint distribution of line energies and amplitudes. Unlike the UTA model, in which the main characteristics (mean energy variance) of the transition arrays are exact (calculated in the framework of the second-quantization formalism and Racah angular-momentum techniques), the RTA approach is approximate (the lines are not at their exact energy and do not have their exact strength) but is very useful in order to take into account the “porosity” of transition arrays when the detailed computations are not feasible and the usual statistical methods (UTA, SOSA) are not applicable (at low density, for instance). The RTA method was recently successfully applied in the context of kilonova emission [86].

Finally, let us finish this section with a rather surprising link between line merging in spectroscopy and number theory. Interference broadening coefficients describe the non-Lorentzian effect that arises as pressure causes lines to overlap. These coefficients, one for each line, are at moderate pressures related linearly to absorption and dispersion. In the course of a work on interference broadening coefficients [87,88], Eakin obtained the following expression:

$$\begin{array}{cc}\hfill & {\int }_{-\infty }^{\infty }〈1-exp\left(-\left[{\displaystyle \frac{1}{{(x-a)}^2+b^2}}+{\displaystyle \frac{1}{{(x+a)}^2+b^2}}\right]\right)\rangle \mathrm{d}x\hfill \\ \hfill & ={\displaystyle \frac{2\pi }{b}}\langle 1+\sum _{p=2}^{\infty }{\displaystyle \frac{{(-1)}^{p-1}\left[2(p-1)\right]!}{{\left(2b\right)}^{2p-2}p!{[(p-1)!]}^2}}\hfill \\ \hfill & \times \left\{1+\sum _{r=1}^{p-1}{\displaystyle \frac{p!\left[2(p-r-1)\right]!(2r-1)!(p-2)!b^{2r}}{{[(p-r-1)!]}^2(2p-3)!(p+2r)!{(a^2+b^2)}^r}}\right.\hfill \\ \hfill & \left.\times \left[\sum _{k=0}^{\lfloor {\displaystyle \frac{r-1}{2}}\rfloor }{\displaystyle \frac{{(-1)}^k(p+3r-2k-1)!}{(2r-k)!k!(r-2k-1)!}}\right]\right\}〉,\hfill \end{array}$$

(27)

giving, for completely overlapping lines, an identity for Mersenne numbers [89,90]:

$$2^n-1=\sum _{r=1}^n{\displaystyle \frac{(n+1)!\left[2\right(n-r\left)\right]!(2r-1)!(n-1)!}{{[(n-r)!]}^2(2n-1)!(n+2r+1)!}}\sum _{k=0}^{\lfloor {\displaystyle \frac{r-1}{2}}\rfloor }{\displaystyle \frac{{(-1)}^k(n+3r-2k)!}{(2r-k)!k!(r-2k-1)!}}.$$

Note that the left-hand side of Equation (27) is precisely the form for which Bykov and Hien [16] found an expression based on Bessel functions (see also Section 4.3).

5. Conclusions

We have presented analytical formulas in order to find the maximum line strength in a transition array. We have also given an approximation of the Ladenburg–Reiche formula for the equivalent width, and of the Elsasser function, which is a key ingredient of the overlapping model. We have also reviewed the different approximations of overlapping and blanketing, and we discussed the possibility of using a statistical indicator such as the Chernoff bound and the Gini coefficient (see Appendix B and Appendix C). The statistical properties of spectra are important for checking opacity databases [11,13]. We plan to investigate the merging of lines in the framework of other distributions, such as Holtzmark’s one [91]:

$${\displaystyle \varphi \left(u\right)=\frac{5sin\left(\frac{2\pi }{5}\right)}{4\pi w}\frac{1}{1+{\left({\displaystyle \frac{u}{w}}\right)}^{5/2}}.}$$

(28)

Stark broadening of hydrogen-like lines in plasma is predominantly ion quasi-static in character. Griem has proposed a generalized Holtsmark profile to describe the ion quasi-static profile of hydrogen-like lines [92]. The analytical expression in Equation (28) matches his function in the line wings and differs from it by no more than 10% over the line width. In practice, the central, ion quasi-static Stark structure is smoothed out by Doppler and electron impact broadening, and for many applications, the generalized Holtsmark and Equation (28) adequately describe the Stark profile of hydrogen-like lines.

In order to improve our understanding of the statistical properties of transition arrays, we plan to compute the moments of the Porter–Thomas distribution [93,94] and to compare them with values inferred from measured or computed complex atomic spectra. We believe that the strong connections with random matrix theory need to be developed further [95].