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.

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   λ   is as follows: I λ k i = F C S A k i g k U S ( T ) e − ( E k k B T ) . (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 k i   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 ( T )   is the partition function at the temperature   T . Equation 1 should be transformed, and the logarithm of both sides should be considered: l n I λ k i A k i g k = − E k k B T + l n F C S U S ( T ) . (2)

Equation 2 can be rewritten in a linear form: y = m x + q S , (3) y = l n I λ k i A k i g k , (4) x = E k , (5) m = − 1 k B T , (6) q S = l n F C S U S ( T ) . (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 ( T )   is calculated as U S ( T ) = ∑ g k e − E k K B T . (8)

The value of   F   can be determined through normalization, which states that the sum of all species in the sample equals 1: ∑ C S = 1 F ∑ U S ( T ) e q S = 1 . (9)

Then, the concentration of each element in the sample can be determined as C S = 1 F U S ( T ) 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].

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 × 10 12 T 1 2 ( Δ E ) 3 , (11) where   Δ 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 ( λ ) = I 0 ( λ ) · E ( λ ) , (12) where   E ( λ )   is the relative efficiency;   I ( λ )   is the signal intensity output from the spectral detector; and   I 0 ( λ )   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].

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.

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.