A Brief Review of Calibration-Free Laser-Induced Breakdown Spectroscopy

The calibration-free laser-induced breakdown spectroscopy (CF-LIBS) technique requires no reference samples of the same matrix to establish the calibration curve, not affected by the matrix effect. In recent years, the CF-LIBS technology has greatly progressed, and the accuracy of quantitative analysis has gradually improved. The purpose of this review was to introduce the CF-LIBS fundamental and modified algorithms. The Boltzmann plot method, Saha–Boltzmann plot method, and column density Saha–Boltzmann plot (CD-SB) method were discussed. Moreover, as a critical factor in CF-LIBS, the self-absorption effect and its influence on CF-LIBS were also introduced. CF-LIBS has been applied in a variety of fields, such as environmental protection, explorations of space, cultural heritage preservation, and geological survey, which were also described in this review.

1 Introduction

The types and compositions of elements in materials have an impact on their properties, either directly or indirectly. It is crucial to make elemental analyses evaluate material performance. Conventional methods include atomic absorption spectrometry (AAS), inductively coupled plasma mass spectrometry (ICP-MS), inductively coupled plasma emission spectrometry (ICP-AES), X-ray fluorescence spectrometry (XRF), tunable diode laser absorption spectroscopy, quartz-enhanced photoacoustic spectroscopy, quartz-tuning-fork enhanced photothermal spectroscopy (QEPTS), and dual-comb absorption spectroscopy [1–3]. Due to their complicated operation and time-consuming process, these methods are usually used in laboratories. In recent years, scientists have been looking for and have developed new analytical assays with rapid response, easy operation, and high reliability.

Laser-induced breakdown spectroscopy (LIBS) is a new promising atomic spectrometry, more versatile than traditional methods [4]. LIBS is also often referred to as laser-induced plasma spectroscopy (LIPS) or laser spark spectroscopy. As the excitation source in LIBS, a pulsed laser beam is focused onto the sample surface by using a focusing lens. Through multiphoton ionization, atoms, ions, and molecules in the laser focus focal area absorb the laser energy and generate initial free electrons. With the inverse bremsstrahlung effect, the free electrons are accelerated by the electromagnetic field of the laser beam and then collide with particles in the ambient gas and sample materials to produce more free electrons. The newly created free electrons are also accelerated by the electric field, resulting in an electron avalanche ionization (EAI) process throughout the laser pulse duration [5]. During a breakdown phenomenon, plasma is generated on the sample surface. The surface species can be quantitatively deduced by analyzing the plasma emission spectrum [6]. LIBS has become an attractive and popular technique in the field of chemical analysis due to its unique advantages, such as its application to liquids [7, 8], gases [9, 10], and solids [11, 12], no sample pretreatment, simultaneous detection of multiple elements, and noncontact remote detection in many fields, including laser cleaning [13, 14], environmental protection [15, 16], space exploration [17, 18], and cultural heritage preservation [19, 20].

Generally, a series of certified samples of similar matrices are required for the quantitative analysis to establish the calibration curves in LIBS, called the referenced calibration method (RCM). However, it is extremely difficult or even impossible to obtain similar referenced samples in many cases, such as soil, mining, and biological tissues, where the matrix effect is hardly avoided [21, 22]. Furthermore, the RCM requires consistency in the experimental conditions, such as laser power density, temperature, and humidity. The limitations to the reference sample have hindered the development of LIBS.

Ciucci was the first to propose a determination method without referenced samples: the calibration-free laser-induced breakdown spectroscopy (CF-LIBS) [23]. The elemental concentration information is determined by describing the physical states of the laser-induced plasmas through mathematical models. There is no need for referenced samples or calibration curves, and matrix effects can be effectively avoided. CF-LIBS has piqued researchers’ interest since its introduction in 1999, although analytical accuracy is less satisfactory than the RCM.

In the past few decades, experimental modification and physical algorithmic improvements in CF-LIBS have been made by scholars all around the world. Analytical accuracy keeps increasing. To overview the development and the state-of-the-art CF-LIBS, this review included three parts: the essential assumptions and the basic mathematical model, a modified model combined with the Saha-Eggert equation, and the self-absorption and its effect on CF-LIBS. The purpose of this review was to give LIBS researchers some inspiration to promote the exploration.

2 Fundamental Algorithm

The basic assumptions of CF-LIBS include (1) chemometric ablation, in which elemental composition and content in plasmas are the same as in samples [23, 24]; (2) local thermal equilibrium (LTE), ensuring the particles are in the excited energy level, following the Boltzmann distribution [25]; (3) optical thinness, meaning that the self-absorption in the selected spectral line can be ignored for calculation; and (4) elemental information wholeness, observed spectra including all the species of elements [26]. Based on the aforementioned assumptions, the spectral intensity at wavelength $\lambda$ is as follows:

$I_{\lambda }^{ki}=F{C}_S{A}_{ki}\frac{g_k}{U_S\left(T\right)}{e}^{-\left(\frac{E_k}{k_{B}T}\right)}.(1)$

$F$ is the experimental parameter involving the receiving system optical efficiency and plasma number density; $C_S$ is the concentration of the emitting species $s$; $A_{ki}$ is the spontaneous transition probability; $g_{k }$ and $E_k$ are the statistical weight and energy of the upper level $k$; $k_B$ is the Boltzmann constant; $U_S\left(T\right)$ is the partition function at the temperature $T$. Equation 1 should be transformed, and the logarithm of both sides should be considered:

$ln\frac{I_{\lambda }^{ki}}{A_{ki}{g}_k}=-\frac{E_k}{k_{B}T}+ln\frac{F{C}_S}{U_S\left(T\right)}.(2)$

Equation 2 can be rewritten in a linear form:

$y=mx+q_S,(3)$

$y=ln\frac{I_{\lambda }^{ki}}{A_{ki}{g}_k},(4)$

$x=E_k,(5)$

$m=-\frac{1}{k_{B}T},(6)$

$q_S=ln\frac{F{C}_S}{U_S\left(T\right)}.(7)$

A relationship of $E_k$ and $y$ can be linearly fitted, called the Boltzmann plot. The linear plot can be drawn by each type of atom and ion. The plasma temperature and concentration of species s can be deduced by the line slope $m$ and intercept $q_S$, according to Eqs 6, 7, respectively.

The partition function $U_S\left(T\right)$ is calculated as

$U_S\left(T\right)={\displaystyle \sum g_k{e}^{-\raisebox{1ex}{$E_k$}\!\left/ \!\raisebox{-1ex}{$K_{B}T$}\right.}}.(8)$

The value of $F$ can be determined through normalization, which states that the sum of all species in the sample equals 1:

${\displaystyle \sum C_S}=\frac{1}{F}{\displaystyle \sum U_S\left(T\right)e^{q_S}=1}.(9)$

Then, the concentration of each element in the sample can be determined as

$C_S=\frac{1}{F}{U}_S\left(T\right)e^{q_S}.(10)$

Ideally, an accurate elemental determination requires two conditions [27–29]: (1) the fitting linear curve of the same species has high linearity ($R^2$ close to 1) and (2) the fitting lines of various species are nearly parallel (shown in Figure 1 [30]). However, the analytical accuracy is generally influenced by five factors: (I) the measured spectral intensities are inaccurate; (II) the Boltzmann plot established for atomic lines generally yields a lower plasma temperature than for ionic lines; (III) the transition species are close to but not in LTE conditions (due to ionization/recombination reactions through electronic impacts); (IV) plasma in LIBS is optically thick and thermally inhomogeneous, and the temperature in the plasma center is much higher than that at the plasma periphery; and (V) Equation 1 is ideal and cannot accurately describe the plasma [19, 31].

FIGURE 1

FIGURE 1. Typical Boltzmann plots for estimating the plasma temperature. Emission lines from singly ionized Cu and Ni are used for obtaining the temperature. Reproduced with permission from [30], ©2016 Cambridge core.

The laser-induced plasma is transient and inhomogeneous. It only approximately meets the conditions for LTE within the appropriate temporal and spatial window [19]. Deviation from LTE conditions will badly influence the analytical accuracy. The McWhirter criterion is the most commonly used criterion for verifying LTE, especially because in plasma with the presence of high-density particles, the collisional transitions dominate the radiative transitions between all states. It is a necessary but not sufficient condition for LTE because it only applies to homogeneous and static plasmas [32]. The McWhirter criterion can be used as

$N_e>1.6\times {10}^{12}{T}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\left(\mathrm{\Delta }E\right)}^3,(11)$

where $\Delta E$ represents the maximum adjacent energy level gap.

The obtained spectral intensity is proportional to the relative efficiency: the light emitted by the plasma is coupled to the spectrometer, where the detector converts the optical signal into an electrical signal. Because the transmission and conversion efficiency of the optical system and the spectrometer are wavelength-dependent, the spectral intensity obtained directly must be corrected:

$I\left(\lambda \right)=I_0\left(\lambda \right)·E\left(\lambda \right),(12)$

where $E\left(\lambda \right)$ is the relative efficiency; $I\left(\lambda \right)$ is the signal intensity output from the spectral detector; and $I_0\left(\lambda \right)$ is the spectral intensity emitted from the plasma. The spectral response can vary significantly in different spectral regions, so spectral correction must be performed across a wide range of wavelengths [31, 33]. A deuterium-halogen tungsten lamp, a combination of deuterium/halogen broadband source, a mercury lamp, and diffusely scattered pulsed laser light sources are general calibration light sources [34].

The Boltzmann plot was applied by researchers in various fields. Fahad et al. quantified the composition of the limestone, as shown in Figure 2; the results of the CF-LIBS technique compared to scanning electron microscopy combined with energy-dispersive X-ray spectroscopy and electron probe microanalysis are in good agreement [35]. Pandhiji et al. used the CF-LIBS method to quantify the elements in the coral skeleton, except for Sn(Certified data: 10 ppm and CF-LIBS data: 6 ppm); all of the values were in general agreement with the verified values. The results were somewhat different compared to ICP-MS, and the reason for this disagreement may be that the CF-LIBS data were related to the surface of the coral, while the ICP-MS data were related to its overall mass [36]. In the following year, they determined toxic heavy metals (Cd, Co, Pb, Zn, Cr, etc.) in soil samples from four industrial areas by the calibration curves, CF-LIBS, and ICP-OES methods. The results showed that the limits of detection (LOD) for Cd and Zn in soil were 0.2 and 1.0 ppm, respectively, and the ICP-OES method was in good agreement with the CF-LIBS method [37]. Similarly, Kumar et al. created a Boltzmann plot for different elements (Ca, Fe, and Pb) in the sludge to determine the concentrations of toxic elements Cr and Pb [38]. Agrawal et al. used CF-LIBS as a quality control tool to monitor the composition of various mineral elements in food additives. The quantitative analysis results were consistent with those on the additive label. The presence of new (not on the label) elements and non-detected elements may be related to errors in the food additive and was not reported by the manufacturer due to low concentrations [39]. Yang et al. used CF-LIBS to quantify the H/D concentration ratio in titanium alloys, and the relative error of the H/D concentration ratio was only 1.33% when a Boltzmann plot was used to calculate the plasma temperature (choosing the spectral line of titanium) due to the scarcity of H and D elemental spectral lines, as well as their susceptibility to interference from titanium spectral lines [40].The quantitative analysis of the composition of elements in different karats of gold by Ahmed et al. showed that the Au content increases from 75.9% to 92.7%, with increasing gold purity (18–22 K), and the corresponding Cu content decreases from 17% to 5.7% [41]. Hamad et al. analyzed the composition of pressed cement samples, and the calculated elemental concentrations agreed with the XRF results with a maximum relative percentage error of 5% [42].

FIGURE 2

FIGURE 2. Relative elemental abundance of the limestone sample at the fundamental harmonic (1064 nm) of the Nd:YAG laser. Reproduced with permission from [35], ©2018 Institute Of Physics.

Under optimized experimental conditions, CF-LIBS allows more accurate elemental content analysis, with results comparable to those of XRF, ICP-MS, ICP-OES, AAS, and RCM in LIBS. Moreover, a series of standard samples are not required, making it more cost-effective and less time-consuming. When referenced samples are unavailable for establishing calibration curves, CF-LIBS is the only choice for the quantitative analysis using LIBS.

3 Modified Algorithm

3.1 Saha–Boltzmann Plot

Species in the same ionized state generally do not have enough spectral lines for representing the whole energy level and poor calculation accuracy of the plasma electron temperature. Yalcin et al. proposed a method of introducing the Saha–Eggert equation with the Boltzmann plot (Saha–Boltzmann plot method) in 1999 [43] and studied the effects of environmental conditions and laser energy on the plasma temperature. This method indeed improved the accuracy and reliability of temperature measurements.

The Saha–Boltzmann plot method must satisfy the condition of LTE, as well as the Boltzmann plot method. The Saha–Eggert equation describes the totality of neutral and singly ionic states of the same elements under the LTE condition:

$\frac{N^{II}}{N^I}=\frac{1}{N_e}\frac{2U^{II}\left(T\right)}{U^I\left(T\right)}\text{exp}(-\frac{E_{ion}}{k_{B}T})\frac{{\left(2\pi m_e{k}_{B}T\right)}^{\frac{3}{2}}}{h^3},(13)$

where $N^I$ and $N^{II}$ represent the number density of species in the atomic and single ionic states of the same element, respectively; $m_e$ is the electron mass; $E_{ion}$ is the first ionization energy.

$N$ is proportional to $C_S$, $C_S=bN$. Equation 1 can be modified as

$I_{\lambda }^{ki}=F{C}_S{A}_{ki}\frac{g_k}{U_S\left(T\right)}{e}^{-\frac{E_k}{k_{B}T}}=F_{N}N{A}_{ki}\frac{g_k}{U_S\left(T\right)}{e}^{-\frac{E_k}{k_{B}T}}.(14)$

Combining Equations 13, 14, the intensity of the ionic line can be rewritten as

$I_{jh}^{II}=F_N{N}^I{A}_{jh}^{II}{g}_j^{II}\frac{\exp \left(-\frac{E_j^{II}}{k_{B}T}\right)}{U^I\left(T\right)}\left(\frac{2{\left(2\pi m_e{k}_{B}T\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{N_e{h}^3}\right)\text{exp}(-\frac{E_{ion}}{k_{B}T}).(15)$

Rearranging the aforementioned equation and taking the logarithm of both sides, we obtained the following equation:

$y^{\ast }=m·x^{\ast }+q,(16)$

where $y^{\ast }=\ln \left(\frac{I_{jh}}{A_{jh}{g}_j}\right)-\text{ln}\left(\frac{2{\left(2\pi m_e{k}_{B}T\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{N_e{h}^3}\right)$, $x^{\ast }=E_{ion}+E_j$, and $m=-\frac{1}{k_{B}T}$, $q=\text{ln}\left(\frac{F_N{N}^I}{U_S\left(T\right)}\right)$. To improve the accuracy and precision of plasma temperature, several emission lines that cover a wide range of upper energies were used (as shown in Figure 3, Ref. [44]). To some extent, this method mitigates the effect of line strength decline due to self-absorption [45]. This method is becoming increasingly popular among researchers in a variety of studies. Zhang et al. assessed the Ca/Na and Mg/Na ratios in human biological tissues and compared them to the ICP-OES method, revealing that the relative errors in hair and nails were less than 10% (the specific comparison graph, as shown in Figure 4), demonstrating the analytical accuracy [45]. Veis et al. quantified the H/D ratio in Be/W mixture coatings [46]. Pribula et al. studied the composition of tungsten-based samples with protective carbon layers using the W III spectral line to obtain more accurate results (quantitative results significantly influenced by the self-absorption effect of single ionized atoms) [47]. Alicia et al. explored the quantitative analysis of ps-CF and ns-CF LIBS for tungsten-based model materials (WCu) and found that the high linearity of the Saha–Boltzmann plot using the ps state resulted in a more accurate estimation of plasma temperature [48]. Horáková et al. measured the composition of acid pitchstone and found that the results agreed well with those of the electron microprobe analysis (EMPA) [49]. Wang et al. studied the emission spectra of Codonopsis pilosula to determine the elemental contents of Mg and Ca and compared them with liquid cathode glow discharge-atomic emission spectrometry (LCGD-AES) and inductively coupled plasma mass spectrometry (ICP-MS) [50].

FIGURE 3

FIGURE 3. Saha–Boltzmann plot for silicon and aluminum. Reprinted from [44], Copyright (2013), with permission from Elsevier.

FIGURE 4

FIGURE 4. Comparison of ICP-OES and LIBS results in hair and nails. Reprinted from [45], Copyright (2021), with permission from Elsevier.

3.2 Columnar Density Saha–Boltzmann Plot

The methods based on the Boltzmann plot and the Saha–Boltzmann plot both have significant limitations: (I) optical thinness is needed; (II) the plasma electron temperature cannot be accurately deduced by the slope of the Boltzmann or Saha–Boltzmann plot when only a small number of spectral lines for elements in the same ionization state can be observed.

The column density Saha–Boltzmann (CD-SB) method can effectively overcome the aforementioned limitations [51], where the column density of the ground state can be directly calculated. Furthermore, the presence of self-absorption in the resonance lines ensures long-term atomic evolution [52]. This method proposed by Cristoforetti and Tognoni opened up a new avenue for accurate plasma temperature calculation (the columnar density Saha–Boltzmann plot, shown in Figure 5 [51]).

FIGURE 5

FIGURE 5. Modified Saha–Boltzmann plot built by using the columnar density of self-absorbed lines from Al and Mg species. Reprinted from [51], Copyright (2013), with permission from Elsevier.

Similar to the conventional CF-LIBS method, the plasma is assumed to be spatially homogeneous over the measured time interval in the CD-SB plot method. Equation 13 can be rewritten according to [51]:

$\frac{n_i^{II}l}{g_i^{II}}=\frac{2{\left(2\pi m_e{k}_{B}T\right)}^{\frac{3}{2}}}{N_e{h}^3}\frac{n^{I}l}{U^I\left(T\right)}{e}^{-\frac{E_i^{II}+E_{ion-}{\text{ΔE}}_{ion}}{k_{B}T}}.(17)$

This equation describes the equilibrium population of different ionization stages in terms of the number density of the lower level of an ionic transition ($n_i^{II}l$). $E_i^{II}$ is the energy of the ionic transition at the lower energy level, and $\text{Δ}{E}_{ion}$ is the reduced ionization energy due to the plasma environment, which is 1–2 orders of magnitude lower than the sum of $\left( E_i^{II}+E_{ion}\right)$ and is generally negligible. The columnar density $n^{I}l$ can be determined as the following:

$n^{I}l=1770\frac{\mathrm{\Delta }{\lambda }_0}{f{\lambda }_0^2}k\left({\lambda }_0\right)l\times {10}^{17},(18)$

where ${\lambda }_0$ and $\text{Δ}{\lambda }_0$ are in $\dot{A}$ units; the value of $k\left({\lambda }_0\right)l$ can be evaluated by the self-absorption coefficient, and f is the line oscillator strength.

Similarly, the Boltzmann Equation can be rewritten as

$\frac{n_i^{I}l}{g_i^I}=\frac{n^{I}l}{U^I\left(T\right)}{e}^{-\frac{E_i^I}{k_{B}T}},(19)$

where $E_i^I$ is the energy of the atomic transition at the lower energy level $i$.

Combining Equations 17, 19, the column density Saha–Boltzmann equation can take the following format:

$y^{\ast }=m{x}^{\ast }+q,(20)$

where $m=-\frac{1}{k_{B}T}$ and $q=ln\frac{n^{I}l}{U^I\left(T\right)}$; for atomic lines, $x^{\ast }=E_i^I$ and $y^{\ast }=ln\frac{n_i^{I}l}{g_i^I}$; for ionic lines, $x^{\ast }=E_i^{II}+E_{ion}$ and $y^{\ast }=ln\frac{n_i^{II}l}{g_i^{II}}-ln\frac{2{\left(2\pi m_e{k}_{B}T\right)}^{\frac{3}{2}}}{N_e{h}^3}$. The plasma temperature is deduced by the slope of the linear fitting curves in the CD-SB plot method. The $y^{\ast }$ coordinate is calculated based on the column density of the atomic and ionic lines, rather than the intensity of spectral lines; the $x^{\ast }$ coordinate indicates the lower (instead of higher in the Boltzmann plot method) energy level value.

Since its introduction in 2013, this method has piqued the interest of many researchers because there is no need to search for optically thin spectral lines, calibrate the detection system, or correct self-absorption (instead, using self-absorbed lines to establish the CD-SB plot directly). Safi et al. determined the electron temperature of plasmas in aluminum alloys, which shows that the CD-SB plot is more suitable for plasma temperature determination, especially in the later stages of plasma evolution [52]. Hu et al. utilized the CD-SB plot in conjunction with the standard reference line method to determine the elemental composition of aluminum-bronze and aluminum alloy samples, demonstrating that this method outperformed the traditional CF-LIBS method in terms of precision and accuracy [53]. As shown in Figure 6, the CD-SB method combined with the standard reference line improves the results of quantitative elemental analysis compared to the traditional CF-LIBS method. Overall, this method not only improves accuracy compared to the traditional CF-LIBS method but also eliminates the need for complex self-absorption correction procedures.

FIGURE 6

FIGURE 6. AEs of aluminum-bronze alloy (A) and aluminum alloy (B) were calculated by classical CF-LIBS and CF-LIBS with CD-SRL. Reprinted from [53], Copyright (2021), with permission from Elsevier.

4 Self-Absorption Correction

4.1 Effect of Self-Absorption on CF-LIBS

The plasma is optically thin under ideal LIBS conditions, where the light emitted from the plasma is free from self-absorption. The intensities of spectral lines and elemental concentrations have a linear relationship. However, according to the classical radiation theory of spontaneous radiation and stimulated absorption, self-absorption is bound to exist, especially at higher elemental contents, corresponding to optically thick plasmas. The light emitted from the plasma center would be absorbed by atoms and ions at the plasma periphery. The self-absorption effect of the emission spectral lines increases the full width at half-maximum (FWHM), reduces the intensity, and even produces severe self-reversal phenomena. As a result, the optical information emitted from the plasma is distorted, far away from the original relationship with elemental contents. The complexity of laser-matter interactions, the inhomogeneity of the plasma, and the transient nature of plasma evolution make self-absorption a very complex phenomenon. The principle of the self-absorption process in plasma, including self-absorption and self-reversal, is shown in Figure 7 [54, 55]), influenced by laser energy [56, 57], delay time [56–58], ambient gas [57, 59, 60], gas pressure [60], geometrical optical configuration [61, 62], and other methods [55, 63–65].

FIGURE 7

FIGURE 7. Self-absorption process in the plasma (Refs. [54, 55]). Reproduced with permission from [54], ©2019 Institute Of Physics; Reproduced with permission from [55], ©2015 Optical Society of America.

CF-LIBS was based on optically thin plasmas, without self-absorption. The electron temperature of the plasma was evaluated using the Boltzmann or Saha–Boltzmann plot, and the content of the elements in the sample was determined. Actually, inevitable self-absorption reduces the spectral intensity, resulting in an unrealistic higher value of the calculated plasma electron temperature, while the calculated intercept is lower.

4.2 Self-Absorbing Correction Improves the Accuracy of CF-LIBS

4.2.1 Curve of Growth

The curve of growth is a self-absorption correction model that can be applied to CF-LIBS to calculate plasma-related parameters in an iterative form based on the corrected experimental intensities. Gornushkin et al. first used the COG method for elemental analysis in stainless steel [66], establishing a Boltzmann plot for the iron ion lines with different laser energies, and the results showed that the higher the temperature, the higher was the excitation of the higher energy states. Bulajic applied the COG model to CF-LIBS and used it to correct for self-absorption, elucidating the effect of self-absorption on the line profile [67]. The self-absorption was corrected by the plasma electron temperature, electron number density, Gaussian broadening, Lorentzian broadening, and optical path length using the COG method. Three different steel and ternary alloy samples were used to validate the COG model. The precious alloy Au917 was used to create a Boltzmann plot without self-absorption correction and with the COG method after self-absorption correction, as shown in Figures 8, 9. It illustrates the findings of its quantitative analysis. The results demonstrated that the COG model could be applied with CF-LIBS, and the quantitative analysis results after self-absorption correction are very close to the certified values. Based on Bulajic’s method, Praher et al. investigated the relationship between line broadening and self-absorption and proposed a simplified model [68]. Alfarraj et al. used the COG model, number density N, and absorption path length l to calculate the optical depth and self-absorption of strontium and aluminum lines under various conditions of different laser energies, gate delay time, and gate width time [69]. The COG method has been demonstrated to effectively correct self-absorption to improve the LIBS analysis performance. Nevertheless, this method and its variants have high algorithmic complexity, limiting practical applications.

FIGURE 8

FIGURE 8. Boltzmann plot for precious alloy Au917 (without self-absorption corrections) and the Boltzmann plot for precious alloy Au917 (with self-absorption corrections). Reprinted from [67], Copyright (2002), with permission from Elsevier.

FIGURE 9

FIGURE 9. Quantitative analysis for precious alloys. Reprinted from [67], Copyright (2002), with permission from Elsevier.

4.2.2 Self-Absorption Coefficient

The so-called self-absorption coefficient method is to select an optically thin line (or $H_{\alpha }$ line) as an internal reference line (or theoretical FWHM) for self-absorption correction. Sun et al. proposed an internal reference for the self-absorption correction (IRSAC) method [27]. Several lines with ignorable self-absorption were selected as references to correct other lines with self-absorption based on the initial temperature and the intensity of the reference line. Finally, the optimal plasma temperature was determined by an iterative procedure until the convergence of the correlation coefficients on the Boltzmann plot. The Boltzmann plot of the aluminum alloy before and after correction is shown in Figure 10 (see Ref. [27] for Fe-Cr alloy and Fe-Cr-Ni alloy). Similarly, Shakeel et al. applied the CF-LIBS method to Al-Si alloys, optimized the experimental conditions, removed background signals, and corrected for self-absorption with an internal reference line [70]. It is worth noting that the effectiveness of an optically thin plasma can be verified by comparing the intensity of two observed lines of the same element in the same state and transition energy level with the intensity calculated from the known atomic parameters [71, 72]. Based on this, Ahmed et al. constructed a Boltzmann plot with optically thin lines and compared it to the IRSAC method [73]. Dong et al. proposed an internal reference-external standard with the iteration correction (IRESIC) procedure based on the IRSAC approach, which requires a standard sample to estimate the plasma temperature using a genetic algorithm [74]. Furthermore, the internal reference line can be chosen manually or programmatically based on the emission coefficient [75], and temperature estimation can be optimized using a particle swarm algorithm [76]. However, the method of IRSAC still has some limitations: (I) the choice of the internal reference line has a significant impact on the final result, while it is not always possible to select the spectral line with the self-absorption coefficient $f_{\lambda }^b=1$; (II) a spectral line with almost no self-absorption was chosen as the internal reference line for each element. After the last iteration, the Boltzmann plot may reveal that the fitted lines for various elements are not parallel. Eventually, setting the initial temperature of the element with the highest temperature estimated as the mean value of the temperatures determined by all elements may not be the best choice.

FIGURE 10

FIGURE 10. (A) Initial Boltzmann plot derived from the raw line intensity of the aluminum alloy sample. (B) Boltzmann plot corrected by the IRSAC for the aluminum alloy sample. Reprinted from [27], Copyright (2019), with permission from Elsevier.

The self-absorption coefficient can be also expressed as [77]

$SA={\left(\frac{\text{Δ}\lambda }{\text{Δ}{\lambda }_0}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\alpha $}\right.},(21)$

where $\alpha =-0.54$; $\text{Δ}\lambda$ is the FWHM of the actual measured spectral line, and the Stark broadening can be separated t by the deconvolution method. The deconvolution method, however, is excessively time-consuming and can be approximated in the computation by assessing the actual measured width minus the Gauss instrumental broadening $\text{Δ}\lambda \approx \Delta {\lambda }_{actual}-\text{Δ}{\lambda }_{Gauss}^2/\Delta {\lambda }_{actual}$; $\text{Δ}{\lambda }_0$ is the line FWHM, generally calculated by the $H_{\alpha }$ line. Using the $H_{\alpha }$ line for electron density measurement has the distinct advantage of providing a result that is not affected by self-absorption. Furthermore, there is also no need to scrounge around for electronic collision parameters [78, 79]. The specific formula for calculating the electron number density using the $H_{\alpha }$ is as follows:

$N_e\left(H_{\alpha }\right)=8.02\times {10}^{12}{\left(\frac{\text{Δ}{\lambda }_{1/2}}{{\alpha }_{1/2}}\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.},(22)$

where ${\alpha }_{1/2}$ is the half-width of the reduced Stark profiles and is a weak function of electron density and temperature, whose value can be found in [80]. Mansour obtained a more accurate electron temperature by analyzing the electron density ratio of the observation line to the optically thin $H_{\alpha }$ line, corrected for the self-absorption effect of the aluminum atomic line [81]. Similarly, Iqbal et al. compared the effect of self-absorption correction on the emission intensity of spectral lines using the internal reference line and density methods, respectively [82]. Based on Sun’s method, Yang et al. proposed a modified method [83]: the spectral intensity was first corrected using the IRSAC method, and second the self-absorption effect was calculated, according to the electron number density and theoretical broadening.

4.2.3 Microwave-Assisted Excitation and Geometrical optical Configuration

The mechanism of microwave-assisted excitation is similar to the LIBS method of stimulated absorption, where the ground-state atoms in the plasma absorb microwave energy coupled to near-field radiation and transition to a higher energy level state. By adjusting the position of the sample relative to the microwave radiator, sharper peaks and better profiles were observed [84], and the schematic diagram of the device is shown in Figure 11. Over a wide spectral range, the microwave-assisted approach can reduce multiple elemental self-absorption in LIBS. In addition, some exceptional geometries of optical systems can reduce the effects of self-absorption to a certain extent. In unusual parallel laser irradiation, the sample is ablated by a shockwave generated from the air breakdown plasma formed near the sample surface [61]; a dual pulse system with an orthogonal configuration of pre-ablation (the first pulse laser used to generate air breakdown plasma; the second laser is propagated perpendicular to the sample for sample ablation) and reheating models (the first laser is focused perpendicularly to the sample surface for sample plasma generation; the second laser propagated parallel to the sample for plasma heating) [85, 86]; the dual pulse system in collinear configuration [87, 88].

FIGURE 11

FIGURE 11. Schematic presentation of two different microwave radiator locations related to the laser plasma and the sample in measurements: the two needles of a microwave radiator located about 2 mm above the sample surface and (A) 0.5 mm horizontally away from the ablation spot; (B) a 5-mm horizontal pass away from the ablation spot. Reprinted with permission from [84], ©The Optical Society.

Although few people have studied the microwave-assisted and geometrical optical configuration in CF-LIBS, it gives us some inspiration to utilize the aforementioned methods in CF-LIBS.

5 Conclusion

Taking the matrix as part of the analysis interest, CF-LIBS can effectively avoid the matrix effect. It is based on basic assumptions of chemometric ablation, local thermal equilibrium, and optical thinness to describe the spectral intensity by mathematical models. The plasma electron temperature and elemental ratio are obtained by the slope and intercept of the Boltzmann plot. After normalization, the concentration of each element can be obtained. Generally speaking, the higher the linearity of the fitted lines for individual elements (same ionized state) and the more parallel the fitted lines for different elements, the higher will be the accuracy of the calculated results. The accuracy of the plasma electron temperature calculated by the Boltzmann plot method is low when only a small number of spectral lines of species in the same ionized state can be observed, or the corresponding energy level distribution range is small. The method of the Saha–Boltzmann plot was a modified method for solving this problem. CD-SB is another modified method, which can directly use atomic and ionic lines in the ground state. According to classical radiation theory, self-absorption exists. The self-absorption will inevitably affect the calculation of the plasma temperature and CF-LIBS accuracy. The methods for mitigating self-absorption are required, including the COG method, the self-absorption coefficient method, and the microwave-assisted and geometrical optical configuration methods. In recent years, CF-LIBS attracts increasing attention in a variety of fields, such as environmental protection, explorations of space, cultural heritage preservation, and geological survey.

Author Contributions

NZ completed the manuscript, TO acquired the right to the pictures, and MW, ZL, CL, and YQ completed the collection of manuscripts. JL and HY revised the manuscript and provided funding support. NZ and QZ provided supervision and funding support. All the authors discussed the structure of the manuscript and commented on the manuscript.

Funding

This work was supported by the Key-Area Research and Development Program of Guangdong Province (2020B090922006), the National Natural Science Foundation of China (62005081 and 62105105), the Guangdong Basic and Applied Basic Research Foundation (2021A1515011932, 2020A1515110985, and 2019A1515111120), the Science and Technology Program of Guangzhou (202002030165), the Featured Innovation Project of Guangdong Education Department (2019KTSCX034), the Young Scholar Foundation of South China Normal University (19KJ13), and the Special Funds for the Cultivation of Guangdong College Students’ Scientific and Technological Innovation (“Climbing Program” Special Funds) (pdjh2020b0153). Key R & D plan of Guangdong Province (2020B090924001), Natural Science Foundation of Top Talent of SZTU (2020103).

Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s Note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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