Extracting Physical Information from the Voigt Profile Using the Lambert W Function

Abstract

Spectral line shapes are a key ingredient of hot-plasma opacity calculations. Since resorting to elaborate line-shape models remains prohibitive for intensive opacity calculations involving ions in different excitation states, with L, M, etc., shells are populated, and Voigt profiles often represent a reliable alternative. The corresponding profiles result from the convolution of a Gaussian function (for Doppler and sometimes ionic Stark broadening) and a Lorentzian function, for radiative decay (sometimes referred to as “natural” width) and electron-impact broadening. However, their far-wing behavior is incorrect, which can lead to an overestimation of the opacity. The main goal of the present work was to determine the energy (or frequency) at which the Lorentz wings of a Voigt profile intersect with the underlying Gaussian part of the profile. It turns out that such an energy cut-off, which provides us information about the dominant line-broadening process in a given energy range, can be expressed in terms of the Lambert W function, which finds many applications in physics. We also review a number of representations of the Voigt profile, with an emphasis on the pseudo-Voigt decomposition, which lends itself particularly well to cut-off determination.

1. Introduction

The main recent improvements of calculations of photon absorption by plasma come mainly from a better treatment of the photo-excitation (bound-bound) spectrum, either with DLA (Detailed Line Accounting) or statistical methods. Theoretical photon absorption exhibits a strong dependence on spectral line-shape models. Unfortunately, it is impossible with the present codes to implement elaborate line-broadening mechanisms [1,2,3,4]. For this reason, the opacity codes employ simpler approximations. The usual procedure consists of performing the convolution of a Gaussian function, due to Doppler broadening, and a Lorentzian function, due to the natural lifetime plus electron impact widths. The absorption coefficient of a line (frequency ${\nu }_0$ and strength S) at the photon frequency $\nu$ reads as follows:

$${\kappa }_{\nu }(x,y)=\frac{S\sqrt{ln2}}{\sqrt{\pi }{\gamma }_G}K(x,y),$$

(1)

where $x=(\nu -{\nu }_0)\sqrt{ln2}/{\gamma }_G$, $y={\gamma }_L\sqrt{ln2}/{\gamma }_G$ (${\gamma }_L$ and ${\gamma }_G$ are the HWHM (half with at half maximum) values of the Lorentzian and Gaussian profiles, respectively), and

$$K(x,y)=\frac{y}{\pi }{\int }_{-\infty }^{\infty }\frac{e^{-t^2}}{{(x-t)}^2+y^2}dt.$$

(2)

The latter function is the so-called Voigt profile [5] and can be expressed as follows:

$$K(x,y)=\Re \left[w_1\left(z\right)\right],$$

(3)

where $\Re \left(z\right)$ denotes the real part of complex number z, and

$$\begin{array}{ccc}\hfill w_1\left(z\right)& =& e^{-z^2}\left(1+\frac{2i}{\sqrt{\pi }}{\int }_0^z{e}^{t^2}dt\right),\hfill \\ & & =e^{-z^2}[1+\mathrm{erf}\left(iz\right)]=e^{-z^2}[1-\mathrm{erf}(-iz)]=e^{-z^2}\mathrm{erfc}(-iz),\hfill \end{array}$$

(4)

is sometimes called the Faddeyeva–Terent’ev [6,7,8] function or complex error function [9]. Equation (4) is proven in Appendix A. Several methods have been proposed in order to calculate the Voigt function [10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26]. It can, for instance, be expressed as a series of confluent hypergeometric functions [27,28]. A good review of different families of methods was given by Schreier [29]. Some interesting relationships (particularly from the historical point of view) are recalled in Appendix B.

In hot plasma, near the line center, the Voigt profile is usually the convolution of a relatively broad Gaussian profile with a narrow Lorentzian profile, and so, it resembles the Gaussian profile. That is, at $\nu \approx {\nu }_0$, we can replace $e^{-t^2}$ in the integrand of the right-hand side of Equation (2) with $e^{-x^2}$ and integrate over the new variable $u=x-t$ so that

$$K(x,y)\approx \left(\frac{y}{\pi }\right)e^{-x^2}{\int }_{-\infty }^{\infty }\frac{du}{u^2+y^2}\approx e^{-x^2}.$$

(5)

The wings of the profile are governed by the Lorentzian function, which has much broader wings than the Gaussian function. This means that it is far away from the Doppler core where $x\gg 1$, so that (with $y\ll 1$, i.e., the full width at half maximum (FWHM) of the natural/collision broadening is much smaller than the FWHM of the Doppler broadening) replacing $x-t$ with x in the integrand of the right-hand side of Equation (2) results in the following:

$$K(x,y)\approx \frac{y}{\pi }{\int }_{-\infty }^{\infty }\frac{e^{-t^2}}{x^2}dt=\frac{y}{\pi }\frac{\sqrt{\pi }}{x^2}=\frac{y}{\sqrt{\pi }{x}^2}.$$

(6)

Due to the ionic microfield distribution, the quasi-static approximation applies for photon frequencies larger than the Weiskopf frequency and should decrease roughly as ${\nu }^{-5/2}$. The Voigt profile has incorrect far-wing behavior and, therefore, can significantly overestimate the photon absorption. The goal of the present work was to determine the energy at which the Lorentz wings of a Voigt profile intersect with the underlying Gaussian part of the profile. It was found that such a value can be expressed with the Lambert function.

In Section 2, we provide a non-exhaustive list of different families of representations of the Voigt function. The widely used pseudo-Voigt approximation is introduced in Section 3. The “competition” between the Lorentzian and Gaussian functions in the Voigt profile is studied in Section 4 with the help of the Lambert W function, and issues related to the far-wing behavior of the profile are discussed in Section 5.

2. Representations of the Voigt Function: A Non-Exhaustive Mini-Review

In this section, we provide a non-exhaustive list of representations of the Voigt function. A long time ago, Fettis [30] and Katriel [31] obtained the following (see also Ref. [32]):

$$\begin{array}{cc}\hfill K(x,y)=& exp(y^2-x^2)cos\left(2xy\right)-\frac{1}{\sqrt{\pi }}\left\{(y+ix){}_{1}F_1\left[1;\frac{3}{2};{(y+ix)}^2\right]\right.\hfill \\ \hfill & \left.+(y-ix){}_{1}F_1\left[1;\frac{3}{2};{(y-ix)}^2\right]\right\},\hfill \end{array}$$

(7)

where ${}_{1}F_1$ is the Kummer confluent hypergeometric function, and Yang proposed the following representations [33]:

$$\begin{array}{cc}\hfill K(x,y)=& \frac{1}{2\sqrt{\pi }}\left\{{(y-ix)}^{-1/2}exp\left[{(y-ix)}^2/2\right]W_{-1/4,-1/4}\left[{(y-ix)}^2\right]\right.\hfill \\ \hfill & \left.+{(y+ix)}^{-1/2}exp\left[{(y+ix)}^2/2\right]W_{-1/4,-1/4}\left[{(y+ix)}^2\right]\right\},\hfill \end{array}$$

(8)

where W represents the Whitaker function, and

$$K(x,y)=\frac{1}{2}\left\{exp\left[{(y-ix)}^2\right]erfc(y-ix)+exp\left[{(y+ix)}^2\right]erfc(y+ix)\right\}$$

(9)

and

$$K(x,y)=\frac{1}{\sqrt{2\pi }}\left\{exp\left[{(y-ix)}^2/2\right]D_{-1}\left[\sqrt{2}(y-ix)\right]+exp\left[{(y+ix)}^2/2\right]D_{-1}\left[\sqrt{2}(y+ix)\right]\right\},$$

(10)

where

$$D_{-1}\left(z\right)=e^{-z^2}{\int }_0^z{e}^{t^2}dt.$$

(11)

More recently, Di Rocco derived an expression as an infinite sum over Kummer confluent hypergeometric functions ${}_{1}F_1$ [34]:

$$K(x,y)=\sum _{n=0}^{\infty }{(-1)}^n\left\{\frac{1}{\Gamma (n+1)}{}_{1}F_1\left(n+\frac{1}{2},\frac{1}{2};y^2\right)-\frac{2y}{\Gamma \left(n+\frac{1}{2}\right)}{}_{1}F_1\left(n+1,\frac{3}{2};y^2\right)\right\},$$

(12)

while Zaghloul obtained [35]

$$\begin{array}{ccc}\hfill K(x,y)& =& \left[1-\mathrm{erf}\left(y\right)\right]exp\left[-x^2+y^2\right]cos\left(2xy\right)\hfill \\ & & +\frac{2}{\sqrt{\pi }}{\int }_0^{x}exp(-x^2+{\xi }^2)sin\left[2y(x-\xi )\right]d\xi \hfill \end{array}$$

(13)

and Limandri obtained [36]

$$\begin{array}{ccc}\hfill K(x,y)& =& exp(-x^2+y^2)\left(erfc\left(y\right)cos\left(2xy\right)\right)\hfill \\ & & +cos\left(2xy\right)\left\{erf\left(y\right)-\Re \left[erf(y+ix)\right]\right\}\hfill \\ & & \left.+sin\left(2xy\right)\Im \left[erf(y+ix)\right]\right),\hfill \end{array}$$

(14)

where $\Im \left(z\right)$ represents the imaginary part of complex number z. As mentioned in Ref. [25], He and Zhang claimed to have obtained an exact expression of the Voigt profile that is proportional to the product of an exponential and a cosine function. However, Pagnini and Saxena [25] pointed out that such a representation assumes negative values, which is not correct. Such an issue was also commented on by Zaghloul [37]. Pagnini and Saxena also proposed to express the Voigt function in terms of Fox functions [38,39,40]:

$$H_{pq}^{mn}\left[z|\begin{array}{c}(a_1,A_1),\cdots ,(a_p,A_p)\\ (b_1,B_1),\cdots ,(b_q,B_q)\end{array}\right]=\frac{1}{2\pi i}{\int }_{\mathcal{L}}h\left(s\right)z^{s}ds,$$

(15)

with

$${\displaystyle h\left(s\right)=\frac{{\prod }_{j=1}^m\Gamma \left(b_j-B_{j}s\right){\prod }_{j=1}^n\Gamma \left(1-a_j+A_{j}s\right)}{{\prod }_{j=m+1}^q\Gamma \left(1-b_j+B_{j}s\right){\prod }_{j=n+1}^n\Gamma \left(a_j-A_{j}s\right)}.}$$

(16)

The integration path $\mathcal{L}$ is the contour that separates the points $s=(b_j+k)/B_j$, $j=1,\cdots ,m$ and $k=0,1,\cdots$, which are the poles of $\Gamma (b_j-B_{j}s)$ (still for $j=1,\cdots ,m$) from the points $s=(a_j-k-1)/A_j$, $j=1,\cdots ,n$, which are the poles of $\Gamma (1-a_j+A_{j}s)$ ($j=1,\cdots ,n$). The more common Meijer functions ${\mathrm{G}}_{pq}^{mn}$ are special cases of Fox functions for which $A_j=B_k=1$, $j=1,\cdots ,p$, $k=1,\cdots ,q$, i.e.,

$$H_{pq}^{mn}\left[z|\begin{array}{c}(a_1,1),\cdots ,(a_p,1)\\ (b_1,1),\cdots ,(b_q,1)\end{array}\right]={\mathrm{G}}_{pq}^{mn}\left[z|\begin{array}{c}{a}_1,\cdots ,a_p\\ b_1,\cdots ,b_q\end{array}\right]$$

(17)

and the corresponding Voigt profile reads

$$V\left(x\right)=\frac{1}{2\pi {\omega }_G}\left\{H_{11}^{11}\left[\frac{2}{{\omega }_G}({\omega }_L+ix)|\begin{array}{c}(\frac{1}{2},\frac{1}{2})\\ (0,1)\end{array}\right]+H_{11}^{11}\left[\frac{2}{{\omega }_G}({\omega }_L-ix)|\begin{array}{c}(\frac{1}{2},\frac{1}{2})\\ (0,1)\end{array}\right]\right\}$$

(18)

or equivalently,

$$V\left(x\right)=\frac{1}{2\pi {\omega }_G}\left\{H_{11}^{11}\left[\frac{{\omega }_G}{2({\omega }_L+ix)}|\begin{array}{c}(1,1)\\ (\frac{1}{2},\frac{1}{2})\end{array}\right]+H_{11}^{11}\left[\frac{{\omega }_G}{2({\omega }_L-ix)}|\begin{array}{c}(0,1)\\ (\frac{1}{2},\frac{1}{2})\end{array}\right]\right\}.$$

(19)

Setting

$$Z=\frac{2({\omega }_L+ix)}{{\omega }_G},$$

(20)

and expressing the Voigt profile in the form

$$V\left(x\right)={\int }_{-\infty }^{\infty }L(x-\xi )G\left(\xi \right)d\xi$$

(21)

with the Gaussian function

$$G\left(x\right)=\frac{1}{\sqrt{\pi }{\omega }_G}exp\left[-{\left(\frac{x}{{\omega }_G}\right)}^2\right]$$

(22)

and the Lorentzian function

$$L\left(x\right)=\frac{{\omega }_L}{\pi }\frac{1}{x^2+{\omega }_L^2},$$

(23)

yield

$${\displaystyle V\left(x\right)=\frac{{\omega }_L}{{\omega }_G{\pi }^{3/2}}{\int }_{-\infty }^{\infty }\frac{exp\left[-{\left(\frac{\xi }{{\omega }_G}\right)}^2\right]}{{(x-\xi )}^2+{\omega }_L^2}d\xi ,}$$

(24)

and finally,

$$V\left(x\right)=\frac{1}{2{\pi }^{3/2}{\omega }_G}\left\{{\mathrm{G}}_{12}^{21}\left[\frac{Z^2}{4}|\begin{array}{c}\frac{1}{2}\\ (0,\frac{1}{2})\end{array}\right]+{\mathrm{G}}_{12}^{21}\left[\frac{{\overline{Z}}^2}{4}|\begin{array}{c}\frac{1}{2}\\ (0,\frac{1}{2})\end{array}\right]\right\}.$$

(25)

Throughout the manuscript, and especially in the present section, since we review different published approximations, we decided, in order to avoid the introduction of typographical errors, to keep the notations of the corresponding authors. This is also justified because the choices made by these authors are often of interest (in terms of simplicity, physical context, etc.). However, in every case, we try to indicate the correspondence between the different quantities. For instance, the parameters involved in $K(x,y)$ in Equation (2) and in $V\left(x\right)$ from Equation (21) are related by

$${\omega }_G=\frac{{\gamma }_G}{\sqrt{ln2}};{\omega }_L={\gamma }_L.$$

(26)

Figure 1 represents the Gaussian, Lorentzian, and Voigt functions. The Gaussian is such that $2{\sigma }^2=1$; and the Lorentzian, such that $\gamma =1$ (see Equations (28) and (29), respectively).

Finally, within a different approach, we believe that the Tepper-García function is worth mentioning. It consists of a combination of an exponential function and rational functions that approximate the line-broadening function $H(a,u)$ over a wide range of its parameters [41] (see Appendix C).

3. The Pseudo-Voigt Function

3.1. Main Principle

The pseudo-Voigt profile (or pseudo-Voigt function) is an approximation of the Voigt profile $V\left(x\right)$ using a linear combination of a Gaussian $G\left(x\right)$ and a Lorentzian $L\left(x\right)$ function instead of their convolution. The pseudo-Voigt function is often used for calculations of experimental spectral line shapes [42,43]. The mathematical definition of the normalized pseudo-Voigt profile is given by

$$V_p(x,f_V)=\eta L(x,{\gamma }_L)+(1-\eta )G(x,{\gamma }_G)$$

(27)

where

$$G(x;\sigma )\equiv \frac{e^{-x^2/\left(2{\sigma }^2\right)}}{\sigma \sqrt{2\pi }},$$

(28)

and $L(x;\gamma )$ is the centered Lorentzian profile:

$$L(x;\gamma )\equiv \frac{\gamma }{\pi (x^2+{\gamma }^2)}$$

(29)

with $0<\eta <1$, $\eta$ being a function of the HWHM parameters of the Voigt, Lorentzian, and Gaussian FWHM parameters (respectively, ${\gamma }_V$, ${\gamma }_L$, and ${\gamma }_G$). The HWHM of the Gaussian profile is

$${\gamma }_G=\sigma \sqrt{2ln\left(2\right)},$$

(30)

and the HWHM of the Lorentzian profile is

$${\gamma }_L=\gamma .$$

(31)

There are several possible choices for the $\eta$ parameter [44,45,46,47]. It turns out that a simple formula, accurate to 1%, is as follows [48,49]:

$$\eta =1.36603\left(\frac{{\gamma }_L}{{\gamma }_V}\right)-0.47719{\left(\frac{{\gamma }_L}{{\gamma }_V}\right)}^2+0.11116{\left(\frac{{\gamma }_L}{{\gamma }_V}\right)}^3,$$

(32)

where $\eta$ is a function of the Lorentz (${\gamma }_L$), Gaussian (${\gamma }_G$), and total (${\gamma }_V$) HWHM parameters, with the latter being described by

$${\gamma }_V={\left[{\gamma }_G^5+2.69269{\gamma }_G^4{\gamma }_L+2.42843{\gamma }_G^3{\gamma }_L^2+4.47163{\gamma }_G^2{\gamma }_L^3+0.07842{\gamma }_G{\gamma }_L^4+{\gamma }_L^5\right]}^{1/5}.$$

(33)

An approximate relation (accurate to within about 1.2%) between the widths of the Voigt, Gaussian, and Lorentzian profiles is as follows [50]:

$${\gamma }_{\mathrm{V}}\approx \frac{{\gamma }_{\mathrm{L}}}{2}+\sqrt{\frac{{\gamma }_{\mathrm{L}}^2}{4}+{\gamma }_{\mathrm{G}}^2}.$$

(34)

By construction, this expression is exact for a pure Gaussian or Lorentzian. A better approximation with an accuracy of 0.02% was given by Olivero et al. [51] (originally found by Kielkopf [52]):

$${\gamma }_{\mathrm{V}}\approx 0.5346{\gamma }_{\mathrm{L}}+\sqrt{0.2166{\gamma }_{\mathrm{L}}^2+{\gamma }_{\mathrm{G}}^2}.$$

(35)

Expression (35) is also exact for a pure Gaussian or Lorentzian.

3.2. Matveev’s Analytical Approximation of the Voigt Profile and the “Equivalent” Width of the Voigt Profile

Another approximation, resulting in the summation of a Gaussian and a Lorentzian, was proposed by Matveev [12]:

$${\kappa }_{\nu }(\eta ,\xi )=\frac{S}{\sqrt{\pi }{\gamma }_V}\left(\sqrt{ln2}(1-\xi )e^{-{\eta }^{2}ln2}+\frac{\xi }{\sqrt{\pi }}\frac{1}{1+{\eta }^2}\right),$$

(36)

where $\xi ={\gamma }_L/{\gamma }_V$, $\eta =(\nu -{\nu }_0)/{\gamma }_V$, and ${\gamma }_V$ is the equivalent width [51] of the Voigt profile, which can be calculated by

$${\gamma }_V=\frac{1}{2}\left({\gamma }_L+\sqrt{{\gamma }_L^2+4{\gamma }_G^2}\right)+0.05\left(1-\frac{2{\gamma }_L}{{\gamma }_L+\sqrt{{\gamma }_L^2+4{\gamma }_G^2}}\right).$$

(37)

The latter expression is exact in two limiting cases: $\xi =0$ and $\xi =1$ (pure Gaussian and Lorentzian). Note that the accuracy of Matveev’s approximation was improved by Dobrichev [53].

3.3. Equivalent Width According to He and Zhang

Using a Fourier transform and Taylor series, He and Zhang [54,55] obtained

$${\gamma }_V=\frac{{\gamma }_G}{\sqrt{1+8{\pi }^2(ln2){\left({\displaystyle \frac{{\gamma }_L}{{\gamma }_G}}\right)}^2}}$$

(38)

with

$$f(\nu -{\nu }_0)=\frac{f^{\prime }y}{\pi }{\int }_{-\infty }^{\infty }\frac{e^{-t^2}}{y^2+{(x-t)}^2}dt$$

(39)

with $y=ln2({\gamma }_L/{\gamma }_G)$, $x=(\nu -{\nu }_0)\sqrt{ln2}/{\gamma }_G$, and $f^{\prime }=\sqrt{ln2}/{\gamma }_G$, where ${\gamma }_L$ and ${\gamma }_G$ are the HWHM of the Lorentzian and Gaussian functions, respectively. Using the Fourier and inverse Fourier transforms, He and Zhang obtained an accurate Voigt profile as follows:

$$f\left(z\right)=\sqrt{\pi f^{\prime }\left[exp(4\pi y^2-z^2)cos\left(4\pi yz\right)+exp(4\pi y^2-x^2)cos\left(4\pi yx\right)\right]}$$

(40)

with

$$z=\left(\frac{\nu +{\nu }_0}{{\gamma }_G}\right)\sqrt{ln2}.$$

(41)

They also provided a relationship between the HWHM of the Voigt profile and the HWHM of the Gaussian and Lorentzian profiles [56]:

$$ln\left[cos\left(\frac{4\pi (ln2){\gamma }_L{\gamma }_V}{{\gamma }_G^2}\right)\right]=\frac{\left({\gamma }_V^2-{\gamma }_G^2\right)}{{\gamma }_G^2}ln2=ln\left[2^{\frac{\left({\gamma }_V^2-{\gamma }_G^2\right)}{{\gamma }_G^2}}\right].$$

(42)

Using the expansions

$$cosx=1-\frac{x^2}{2}+\frac{x^4}{4!}-\frac{x^6}{6!}+\cdots$$

(43)

as well as

$$2^x=1+xln2+\frac{x^2}{2}{(ln2)}^2+\frac{x^3}{3!}{(ln2)}^3+\cdots$$

(44)

we obtain the following, considering the first two terms only:

$$1-\frac{16{\pi }^2{(ln2)}^2{\gamma }_L^2{\gamma }_V^2}{2{\gamma }_G^2}=1+\frac{\left({\gamma }_V^2-{\gamma }_G^2\right)}{{\gamma }_G^2}ln2,$$

(45)

yielding

$${\left(\frac{{\gamma }_V}{{\gamma }_G}\right)}^2+8{\pi }^{2}ln2{\left(\frac{{\gamma }_L}{{\gamma }_G}\right)}^2=1.$$

(46)

It is important to mention that Wang et al. presented a simple approximation scheme to describe the half width of the Voigt profile as a function of the relative contributions of the Gaussian and Lorentzian broadening [57]. The proposed approximation scheme is highly accurate and provides an accuracy better than ${10}^{-17}$ for arbitrary ${\gamma }_L/{\gamma }_G$ ratios. In particular, the accuracy reaches an astonishing ${10}^{-34}$ (quadruple precision) in the domains $0\le {\gamma }_L/{\gamma }_G\le 0.2371$ and ${\gamma }_L/{\gamma }_G\ge 33.8786$.

4. Competition between Lorentzian and Gaussian Wings

4.1. Mathematical Formalism

In Hui’s algorithm [13], which computes the complex error function with rational approximation, the exponential decrease $e^{-x^2}$ is not well described for vanishing y. In order to avoid that, it was suggested by Karp [58] to replace Hui’s approximation for the real part of the complex error function with an ansatz for small y and $y/x^2<{10}^{-4}$. Using a Fourier transform expression [59,60],

$$K(x,y)=\frac{1}{\sqrt{\pi }}{\int }_0^{\infty }{e}^{-yu-\frac{u^2}{4}}cos\left(xu\right)du$$

(47)

and, for $y\ll 1$, the first-order approximation $e^{-yu}\approx 1-yu$, one finds that

$$\begin{array}{ccc}\hfill K(x,y)& \approx & \frac{1}{\sqrt{\pi }}{\int }_0^{\infty }{e}^{-\frac{u^2}{4}}cos\left(xu\right)du-\frac{y}{\sqrt{\pi }}{\int }_0^{\infty }u{e}^{-\frac{u^2}{4}}cos\left(xu\right)du\hfill \\ & =& \frac{2}{\sqrt{\pi }}{\int }_0^{\infty }{e}^{-t^2}cos\left(2xt\right)dt-\frac{2y}{\sqrt{\pi }}{\int }_0^{\infty }2t{e}^{-t^2}cos\left(2xt\right)dt,\hfill \end{array}$$

(48)

where we have made a change of variables $t=u/2$. One now has

$$\frac{2}{\sqrt{\pi }}{\int }_0^{\infty }{e}^{-t^2}cos\left(2xt\right)dt=e^{-x^2}$$

(49)

and

$$\begin{array}{ccc}\hfill {\int }_0^{\infty }2t{e}^{-t^2}cos\left(2xt\right)dt& =& 1-2x{\int }_0^{\infty }{e}^{-t^2}sin\left(2xt\right)dt\hfill \\ & =& 1-2x{e}^{-x^2}{\int }_0^x{e}^{u^2}du\equiv 1-2xF\left(x\right),\hfill \end{array}$$

(50)

which leads to the following equation. Note that the two terms on the right-hand side of Equation (51) are the first two terms of an expansion (see Appendix D).

$$K(x,y)\approx e^{-x^2}-\frac{2y}{\sqrt{\pi }}\left[1-2xF\left(x\right)\right]$$

(51)

Then, using the asymptotic expansion

$$1-2xF\left(x\right)=-\left[\frac{1}{2x^2}+\frac{1.3}{{\left(2x^2\right)}^2}+\frac{1.3.5}{{\left(2x^2\right)}^3}+\frac{1.3.5.7}{{\left(2x^2\right)}^4}+\cdots \right]$$

(52)

truncated at the first order in $x^2$, one obtains the following [10,29,60]:

$$K(x,y)\approx e^{-x^2}+\frac{y}{\sqrt{\pi }{x}^2}.$$

(53)

The Voigt profile corresponds to the same parameters. Figure 2 displays the forms $e^{-x^2}$, $1/\left(\sqrt{\pi }{x}^2\right)$ (see Equation (53)), and the same Voigt profile as in Figure 1. Figure 3 shows the functions $e^{-x^2}$, $\gamma /\left(\sqrt{\pi }{x}^2\right)$ (see Equation (53)) with $\gamma =0.2$, and the Voigt profile $K(x,0.2)$ (see Equation (2)). Compared to in Figure 2, the shape of the Lorentzian is “sharper” and more concentrated near the origin. In this case, the Voigt profile is dominated by the Lorentzian close to the center of gravity, but also for higher values of x. The Lorentzian wings are recovered much farther away.

One searches for the critical value $x_c$, for which the Lorentz wings intersect with the Gaussian, such that

$$e^{-x_c^2}=\frac{y}{\sqrt{\pi }{x}_c^2}.$$

(54)

which reads equivalently to

$$x_c^2{e}^{-x_c^2}=\frac{y}{\sqrt{\pi }}.$$

(55)

The solution of the latter equation can be written as

$$x_c=-\sqrt{W_{-1}\left(-\frac{y}{\sqrt{\pi }}\right)},$$

(56)

where $W_{-1}$ represents Lambert’s function [61,62]. This function appears in various fields of physics, such as Wien’s displacement law; the fringing fields of a capacitor, with the latter problem being representative of some problems solved using conformal transformations; the resolution of the Schrödinger equation in a generic radial grid; etc. [63,64,65,66,67,68], but also in mathematics, for instance, in the study of prime numbers [69]. The function (see Figure 4), named after Lambert, who was faced with a related problem [61], was described by Euler [70].

For each integer k there is one branch, denoted by $W_k\left(z\right)$, which is a complex-valued function of one complex argument. $W_0$ is known as the principal branch. When dealing with real numbers only, the two branches $W_0$ and $W_{-1}$ suffice: the equation $y{e}^y=x$ can be solved for y only if $x\ge -1/e$. We obtain $y=W_0\left(x\right)$ if $x\ge 0$, and the two values $y=W_0\left(x\right)$ and $y=W_{-1}\left(x\right)$, if $-1/e\le x<0$.

4.2. Application in the Case of the Pseudo-Voigt Function of Matveev

Using Matveev’s pseudo-Voigt function [12] (see Equation (36)), the value of parameter $\eta$ for which the Gaussian and the Lorentzian intersect with each other is obtained by solving

$$(1-\xi )e^{-{\eta }^{2}ln2}=\frac{\xi }{\sqrt{\pi ln2}}\frac{1}{1+{\eta }^2},$$

(57)

which can be written equivalently as

$$-(1+{\eta }^2)(ln2)e^{-(1+{\eta }^2)ln2}=-\frac{1}{2}\sqrt{\frac{ln2}{\pi }}\frac{\xi }{1-\xi }.$$

(58)

The solution of the latter equation is

$$\eta =\sqrt{-1-\frac{1}{ln2}{W}_{-1}\left[-\frac{1}{2}\sqrt{\frac{ln2}{\pi }}\frac{\xi }{1-\xi }\right]}.$$

(59)

4.3. Calculation of the Lambert W Function

Besides the theoretical advantages of providing an adequate analytical solution to our problem, another benefit of the Lambert W function is the availability of libraries in computer algebra systems, which allows for a convenient way to obtain values, expansions, plots, etc. We have the following expansion for the Lambert function W [71,72,73]:

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

(60)

where

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

(61)

with $S[k+m,k+1]$ being a Stirling number of the first kind [74,75], for which an explicit expression was provided by Karanicoloff [76]. Stirling numbers can be obtained with recursion relations. We suggest using Equation (60) truncated at the fourth order:

$$\begin{array}{ccc}\hfill W\left(x\right)& =& lnx-lnlnx+\frac{lnlnx}{lnx}+\frac{lnlnx(lnlnx-2)}{2{(lnx)}^2}\hfill \\ & & +\frac{lnlnx(2{(lnlnx)}^2-9lnlnx+6)}{6{ln}^{3}x}+O\left[{\left(\frac{lnlnx}{lnx}\right)}^5\right].\hfill \end{array}$$

(62)

The Lambert function satisfies many interesting properties. For instance, its derivative can be in the form

$$W^{\prime }\left(x\right)=\frac{W\left(x\right)}{x\left[1+W\left(x\right)\right]}$$

(63)

and, more generally ($n\ge 1$),

$$\frac{d^{n}W\left(e^x\right)}{d{x}^n}=\frac{q_n\left(W\left(e^x\right)\right)}{{\left[1+W\left(e^x\right)\right]}^{2n-1}}$$

(64)

where

$$q_n\left(w\right)=\sum _{k=0}^{n-1}{(-1)}^k⟪\begin{array}{c}n-1\\ k\end{array}⟫w^{k+1},$$

(65)

where $⟪\begin{array}{c}a\\ b\end{array}⟫$ represents second-order Eulerian numbers [77]. The anti-derivative of the Lambert W function reads

$$\int W\left(x\right)dx=x\left[W\left(x\right)-1+\frac{1}{W\left(x\right)}\right]$$

(66)

and one has

$$W\left(x\right)+W\left(y\right)=W\left(xy\left(\frac{1}{W\left(x\right)}+\frac{1}{W\left(y\right)}\right)\right).$$

(67)

5. Remarks on the Far Wings of the Voigt Profile

5.1. The Cut-Off of Iglesias et al.

It is well known by spectroscopists that the Voigt profile has incorrect far wing behavior and, therefore, can significantly overestimate the photon absorption. Iglesias et al. [78] proposed an extension of the Voigt profile that incorporates a far wing cut-off remaining computationally tractable. The second-order line width from electron collisions for a transition from level a to level b is proportional to the quantity $G_{ab}\left(\omega \right)$ [79,80,81,82] satisfying, under the assumptions of a Debye plasma and the dipole approximation for the electron-radiator interaction with minimum impact parameter cut-off $q_{ab}^{-1}$, the limits

$$G_{ab}(\omega \to 0)=\frac{1}{2}\left\{ln\left(1+{\lambda }_D^2{q}_{ab}^2\right)-\frac{{\lambda }_D^2{q}_{ab}^2}{1+{\lambda }_D^2{q}_{ab}^2}\right\}$$

(68)

and

$$G_{ab}(\omega \to \infty )=G_{ab}^{\infty }=\frac{1}{2}{E}_1\left[X_{ab}^2\right]\to \frac{e^{-X_{ab}^2}}{2X_{ab}^2}\mathrm{when}{X}_{ab}\to \infty$$

(69)

with

$$X_{ab}^2=\frac{m_e}{2q_{ab}^{2}T}{\left(\omega +|{\omega }_a-{\omega }_b|\right)}^2$$

(70)

and the exponential integral

$$E_1\left(x\right)={\int }_x^{\infty }\frac{e^{-t}}{t}dt.$$

(71)

The quantity ${\lambda }_D$ represents the electron Debye length, defined as ${\lambda }_D^2=T/\left(4\pi e^2{n}_e\right)$, where $n_e$ is the electron number density.

$$I_1\left(\omega \right)=Re\left[\mathcal{W}(\omega +ia,\omega )\right]$$

(72)

where $\mathcal{W}$ is the truncated error function (note that there is a typographical error in Equation (4.1.3) of Ref. [78]; $z-\omega$ should be replaced with $t-z$):

$$\mathcal{W}(z,\Omega )=\frac{1}{i\pi }{\int }_{\omega -\Omega }^{\omega +\Omega }dt\frac{e^{-t^2}}{t-z}$$

(73)

which can be evaluated with the sampling theorem [17]

$$\begin{array}{ccc}\hfill \mathcal{W}(z,\Omega )& =& \frac{h}{{\pi }^2}\Im \sum _{n=-N}^N\frac{e^{-t_n^2}}{z-t_n}\left\{\mathrm{Si}\left[\frac{\pi }{h}(\omega -t_n-\Omega )\right]-\mathrm{Si}\left[\frac{\pi }{h}(\omega +t_n+\Omega )\right]\right.\hfill \\ & & +\left\{\mathrm{Ci}\left[\frac{\pi }{h}(\Omega -ia)\right]-\mathrm{Ci}\left[\frac{\pi }{h}(-\Omega -ia)\right]sin\left[\frac{\pi }{h}(z-t_n)\right]\right\}\hfill \\ & & +\left\{\mathrm{Si}\left[\frac{\pi }{h}(\Omega +ia)\right]-\mathrm{Si}\left[\frac{\pi }{h}(-\Omega -ia)\right]cos\left[\frac{\pi }{h}(z-t_n)\right]\right\},\hfill \end{array}$$

(74)

where $\mathrm{Si}\left(z\right)$ represents the sine integral

$$\mathrm{Si}\left(z\right)={\int }_0^z\frac{sint}{t}dt$$

(75)

and $\mathrm{Ci}\left(z\right)$ represents cosine integral

$$\mathrm{Ci}\left(z\right)=-{\int }_0^z\frac{cost}{t}dt.$$

(76)

The decay parameter ${\tau }_{ab}$ is given by

$${\tau }_{ab}^2=\frac{m}{2q_{ab}^{2}T}$$

(77)

and $\Omega$ is obtained solving

$$\Omega =\frac{\sqrt{y}}{{\tau }_{ab}}+|{\omega }_b-{\omega }_a|$$

(78)

obtained as the solution of

$$G_{ab}\left(0\right)=G_{ab}^{\infty }(\omega =\Omega )$$

(79)

and y is the solution of

$$y{e}^y=\frac{1}{2G_{ab}\left(0\right)},$$

(80)

i.e.,

$$y=W\left(\frac{1}{2G_{ab}\left(0\right)}\right),$$

(81)

where W is the Lambert function.

5.2. Continued-Fraction Representation

The continued fraction for a function that has finite power moments of any order can be written in terms of the Laplace transform

$$\mathcal{F}\left(\omega \right)=\frac{1}{\pi }\Re \left\{{\mathcal{F}}^{\mathrm{Lap}}(s=-i\omega )\right\}.$$

(82)

For an even function, $\mathcal{F}\left(\omega \right)=\mathcal{F}(-\omega )$, the Laplace transform can be written as

$${\displaystyle {\mathcal{F}}^{\mathrm{Lap}}\left(s\right)=\frac{{\mu }_0}{s}-\frac{{\mu }_2}{s^3}+\frac{{\mu }_4}{s^5}+\cdots =\frac{b_0}{{\displaystyle s+\frac{b_1}{{\displaystyle s+\frac{b_2}{s+\cdots }}}}}}$$

(83)

The continued fraction coefficients $b_j$ are given in terms of the moments

$${\mu }_{2n}=2{\int }_0^{\infty }{\omega }^{2n}\mathcal{F}\left(\omega \right)d\omega$$

(84)

by a recursive relation [83,84]. The continued fraction representation can be expressed in the form

$$p\left(\omega \right)=\frac{1}{\pi }\Re \left\{\frac{1}{s+b_1{\psi }_1\left(s\right)}\right\}{|}_{s=-i\omega }$$

(85)

with

$${\psi }_n\left(s\right)=\frac{1}{s+b_{n+1}{\psi }_{n+1}\left(s\right)}$$

(86)

Inaccuracies can occur without an approximate expression for the termination function, ${\psi }_N\left(s\right)$. Unfortunately, the recursion relation for the $b_n$ is numerically unstable, due to the emphasis of the power moments on larger frequencies. Appropriately chosen polynomials, rather than power moments, yield more stable formulas [85]:

$${\pi }_{2n}=\frac{1}{{\left(2\lambda \right)}^{2n}}{\int }_{-\infty }^{\infty }d\omega H_{2n}\left(\lambda \omega \right)I_1\left(\omega \right)=\left(2n\right)!\sum _{k=0}^n\frac{{\mu }_{2k}}{\left(2k\right)!}{\left\{\frac{{\lambda }^2-1}{4{\lambda }^2}\right\}}^{n-k}$$

(87)

5.3. The Pearson Distribution: An Interesting Alternative?

In order to overcome the inadequacies of the Lorentzian, Gaussian, and Voigt functions, an alternative profile function, the Pearson VII function, is often used for peak fitting in, for example, X-ray diffraction [86]. The Pearson VII function is

$$I_P\left(\lambda \right)=I_0{\left[1+{\left\{\frac{2(\lambda -{\lambda }_0)\sqrt{2^{1/M}-1}}{{\Delta }_P}\right\}}^2\right]}^{-M},$$

(88)

where ${\Delta }_P$ is the FWHM, and M is known as the Pearson parameter. When M is equal to one, the distribution is identical to a Lorentzian, while as M becomes large, the distribution tends to a Gaussian. It is very much less computationally intensive than the Voigt function as it does not require numerical integration. To our knowledge, the Pearson VII distribution has not been used in the analysis of plasma spectra. This is probably because there is no direct relationship between the plasma properties and the Pearson VII fit parameters, ${\Delta }_P$ and M [87], whereas the widths of the Lorentzian and Gaussian distributions, which can be extracted from fits to the Voigt function, can readily be related to the properties of the plasma.

6. Conclusions and Future Plans

Hot-plasma opacity calculations are sensitive to spectral line shapes. Since elaborate line-shape codes can hardly be tractable for intensive opacity calculations involving billions of lines of multi-charged ions in many different excitation states, Voigt profiles are often used. However, their far-wing behavior, which can lead to an overestimation of the opacity, is a persistent issue of opacity calculations. The main goal of the present work was to show that the variable (energy, frequency, etc.) at which the Lorentz wings of a Voigt profile intersect with the underlying Gaussian part of the profile involves the Lambert W function, giving the solution x of an equation of the kind $x{e}^x=b$. Such a quantity is important, in the general case, to know the physical broadening mechanisms that are dominating in a given energy range of the profile, but it becomes particularly interesting in the framework of “pseudo-Voigt” approximations of the Voigt profile, consisting of representing the latter as a sum of a Gaussian and a Lorentzian function, with appropriate weights. We took advantage of the opportunity to review a number of representations of the Voigt profile and to point out their weaknesses, in particular, as terms of the far wings. The literature about the Voigt profile is very abundant, and we decided to choose some specific representations, involving special functions such as the hypergeometric ${}_{1}F_1$ of Fox functions, which, in our opinion, have not been completely exploited and may lead to the derivation of new useful approximations. It is important to attach importance to different formulations, because each of them has its own advantages. For instance, the pseudo-Voigt function may be considered rather limited if one is interested in a high numerical accuracy, but it remains very practical for the interpretation of atomic or molecular spectra, or in order to determine the abscissa at which the Gaussian and the Lorentzian intersect with each other. The high number of available relations for the hypergeometric functions should also benefit the computation of the Voigt profile, either in order to derive recurrence relations or to investigate asymptotic properties. Finally, it is important to keep in mind that properly accounting for the Stark effect requires the convolution of the Voigt profile with an electric-microfield distribution. It would be interesting to repeat the present study with a simple modeling of the latter, for instance, by resorting to the Holtsmark function, for which appropriate analytical formulations may be derived [88,89].