A calibration-free model for laser-induced breakdown spectroscopy using non-gated detectors

Abstract

Calibration-free (CF) laser-induced breakdown spectroscopy (LIBS) is normally only applicable for gated detectors due to its dependence on the assumption of a steady-state plasma. However, most currently available LIBS systems are equipped with non-gated detectors such as charge-coupled device (CCD), which degrades the accuracy of CF method. In this paper, the reason for the less satisfactory quantification performance of CF for LIBS with non-gated detectors was clarified and a time-integration calibration-free (TICF) model was proposed for applications with non-gated detectors. It was based on an assumed temporal profile of plasma properties, including temperature and electron density, obtained from another pre-experiment. The line intensity at different time during the signal collection time window was estimated with self-absorption correction according to the temporal profile of the plasma properties. The proposed model was validated on titanium alloys and compared with traditional CF. The accuracy of elemental concentration measurement was improved significantly: the average relative error of aluminum and vanadium decreased from 6.07% and 22.34% to 2.01% and 1.92%, respectively. The quantification results showed that TICF method was able to extend the applicability of CF to LIBS with non-gated detectors.

Graphical abstract

Keywords

laser-induced breakdown spectroscopy / calibration-free / non-gated detector / self-absorption correction

1 Introduction

Laser-induced breakdown spectroscopy (LIBS) has been regarded as one of the most versatile elemental analysis techniques and has shown great potential in many applications [1-4], such as metals [5], coal [6], pollutants [7], and minerals [8]. Generally, quantitative LIBS analysis is performed by calibration models [9-14] and calibration-free (CF) models [15-18]. The CF model proposed by Ciucci et al. [17] avoids the use of a series of reference samples and thus improves the convenience of LIBS. However, the CF model relies on strict assumptions, including stoichiometric ablation, optical thinness (free of self-absorption), and spatial homogeneity and local thermodynamic equilibrium (LTE) of the plasma [19]. There is also an implicit assumption of a steady state plasma because the drastically changing plasma temperature and electron density are not considered in the CF model, since lumped constant temperature and electron density are applied. In general, emission with tens to thousands of nanoseconds of the earlier stage plasma is collected using gated signal detectors such as intensified charge-coupled device (ICCD) to meet the LTE condition and the nearly steady state assumption [19-21]. However, most currently available LIBS systems are equipped with non-gated detectors such as charge-coupled device (CCD) for their low cost and high stability, with an exposure time about 1 millisecond and do not satisfy the basic assumptions. Despite the possible violation of CF assumptions, some researchers attempted CF model with CCD detectors [22-26], but the quantification performance were not satisfactory compared with those with ICCD detectors. Grifoni et al. [27] proposed to obtain time-resolved spectra by the subtraction between two LIBS spectra at two different delays using a CCD detector. Although this method does not require any pre-calibration of the system and does not rely on theoretical assumptions or iterative procedures, the subtraction between two LIBS spectra actually increases the uncertainty of signal since essentially these two spectra were obtained from two different laser-induced plasmas, which is absolutely harmful to the final results. In addition, the line intensity estimated from subtraction may sometimes be very weak or even be negative, limiting its potential for CF analysis. The application of CF model for LIBS with CCD detectors is still a big challenge for LIBS.

Another problem which affects the quantification performance of CF is self-absorption [20, 28]. Self-absorption is an undesirable phenomenon in which plasma emission is re-absorbed by the plasma itself. It generally reduces the intensity and increases the width of an emission spectral line, resulting in inaccurate estimation of plasma temperature and elemental concentrations in CF-LIBS [29-33]. Various methods have been proposed to correct the self-absorption effect [34]. Bulajic et al. [35] used curve of growth (COG) for self-absorption correction, in which the effect of self-absorption on line profile was clarified and an iterative algorithm was proposed to calculate plasma temperature, electron number density, Gaussian broadening, Lorentzian broadening, and optical path length. Based on this method, Sherbini et al. [36] summarized the relationship between line broadening and self-absorption extent, and proposed a simplified iterative algorithm to correct self-absorption, but required at least one optically thin line with negligible self-absorption effect. Aragon et al. [37] applied a C-sigma method to compensate for the self-absorption effect and quantitatively analyzed elements in aluminum alloys. Moon et al. [38] proposed a duplicating mirror method to measure and correct self-absorption, but required an additional mirror and an optical shutter, and required to collect spectra with and without the mirror each time. Sun et al. [22] proposed a method that used one or several lines as internal reference, which were regarded free from self-absorption, for self-absorption correction (IRSAC), while the intensity of other lines was calculated by the reference line intensity and the plasma temperature was estimated based on theoretical equation. Donget al. [16] proposed the internal reference–external standard with the iteration correction (IRESIC) procedure based on IRSAC, in which one matrix-matched standard sample along with the genetic algorithm (GA) was utilized to estimate the accurate plasma temperature. Demidov et al. [39] proposed an improved Monte-Carlo (MC) method for standard-less analysis, where the concentrations were found by fitting model-generated synthetic spectra to experimental spectra. Zhu et al. [40] proposed an approach to overcome the self-absorption effect by utilizing molecular emission. Liet al. [41] proposed a new self-absorption correction method for CF-LIBS called blackbody radiation referenced self-absorption correction (BRR-SAC). It used an iterative algorithm to calculate the plasma temperature and the normally hard-to-obtain collection efficiency of the optical collection system by directly comparing the measured spectrum with the corresponding theoretical blackbody radiation for self-absorption correction.

However, none of these correction methods considered the influence of plasma evolution on the degree of self-absorption. Plasma properties such as temperature and electron density changes drastically during the exposure time of non-gated detectors [42-44]. The self-absorption also changes drastically due to the change of temperature and electron density [45], making the normal self-absorption correction methods not applicable for non-gated detectors. In this paper, a new CF model with self-absorption correction that can be used under non-gated conditions is proposed, namely time-integration calibration-free (TICF), and is applied for titanium alloy samples. The results show that TICF can effectively improve the measurement accuracy compared to traditional CF.

2 Methods

According to the theory of plasmas in LTE, the emission intensityε (W m⁻¹) of the line with the central wavelength of \lambda₀ (m) can be expressed as [46]

(1)\epsilon \left(\lambda \right)=F\cdot {\displaystyle \frac{hc}{4\pi {\lambda }_0}}gA\cdot {\displaystyle \frac{rNl}{U\left(T\right)}}\exp \left(-{\displaystyle \frac{E}{kT}}\right)V\left(\lambda \right),

and its wavelength-integrated emission intensity I(W) is

(2)I={\displaystyle \int \epsilon \left(\lambda \right)\mathrm{d}\lambda \text{=}F\cdot \frac{hc}{4\pi {\lambda }_0}gA\cdot \frac{rNl}{U\left(T\right)}\exp \left(-\frac{E}{kT}\right).}

where F (m²) is the coefficient of collection that takes into account the optical efficiency of the collection system as well as the region of the plasma in the viewing field of optical collection system [47], h (J·s) is the Planck constant, c (m·s⁻¹) is the speed of light in vacuum, and k (J·K⁻¹) is the Boltzmann constant. A (s⁻¹) is the transition probability, g (dimensionless) and E (J) are the degeneration and the energy of the upper level of the transition, respectively, and U (dimensionless) is the partition function of the emitting species. T (K) is the plasma temperature, N (m⁻³) is the elemental number density, l (m) is the plasma length, r (dimensionless)is the ionization factor of the emitting species, which represents the percentage of emitting species to the corresponding elemental species and can be obtained by the Saha-Boltzmann ionization equation, and V( \lambda) is the normalized spectral line profile. It should be noted that since N and l always appear at the same time, Nl (m⁻²) will be merged into one variable (columnar density) in the following discussion.

The representation of ln[I \lambda₀/gA] as a function of E, called a Boltzmann plot, leads to a straight line with a negative slope −1/( kT) and an intercept that characterizes the number of species, from which the temperature and elemental concentration can be obtained.

For non-gated LIBS, we treat the plasma as an evolving object with universal temporal profile for fixed experimental setup but spatially homogeneous. The expressions for plasma temperature, electron density, and number of species at moment t are

(3)T(t)=f_T(T_0,t),

(4)N_e(t)=f_{Ne}({N_e}_0,t),

(5)Nl(t)=f_{Nl}(N{l}_0,t).

where f_T, f_Ne and f_Nl represent the profile function of the change of plasma temperature, electron density, and particle number density with time, respectively. T₀, N_e0 and Nl₀ represent the parameters of the plasma at a certain delay time (the moment when we start to characterize the plasma evolution), which depends on and needs to be determined for each different sample. In this paper, a time-resolved pre-experiment with ICCD is used to determine the expression of f_T, f_Ne and f_Nl. Considering the temporal evolution, the expression of the intensity I in Equation 2 should be corrected to

(6)I=F\cdot {\displaystyle \frac{hc}{4\pi {\lambda }_0}}gA\cdot {\displaystyle \int {\displaystyle \frac{r(t)Nl(t)}{U\left(T(t)\right)}}\exp \left(-{\displaystyle \frac{E}{kT(t)}}\right)V\left(\lambda ,t\right)\text{d}t,}

The potential self-absorption effect of the plasma leads to the failure of the optical thinness assumption, which usually weakens the intensity of the spectral line and should be corrected. For spectrum collected under non-gated conditions, the plasma temperature T and electron number density N_e change along plasma evolution process, which leads to changes of self-absorption effect [48, 49]. A simple simulation of the self-absorption (SA) coefficient defined by Ref. [32] of Al(I) 394.4 nm changing with T and N_e is shown in Fig.1, which helps to understand the change of self-absorption effect under non-gated conditions. Significant changes in self-absorption coefficient can be observed in the typical ranges of T and N_e during the exposure time of non-gated detectors, which indicates the urgent need of self-absorption correction and the inapplicability of existing correction methods. Therefore, it is necessary to introduce and to improve the self-absorption correction for the CF-LIBS under non-gated conditions.

Self-absorption correction can be achieved by comparing the intensity of the plasma emission with the intensity of the blackbody radiation at the corresponding temperature. Our previous work [41] has elaborated on the principles of self-absorption correction and proposed a method called blackbody radiation referenced self-absorption correction (BRR-SAC) method. The basic idea of BRR-SAC is to imagine an ideal blackbody which has the same temperature as the plasma. By comparing the emission of the ideal blackbody and the plasma, the self-absorption effect can be evaluated and corrected. The plasma emission intensity with and without self-absorption effect can be described as a function of blackbody radiation intensity L_P( \lambda) and optical length τ( \lambda):

(7)L_P\left(\lambda \right)={\displaystyle \frac{2\pi h{c}^2}{{\lambda }^5}}{\displaystyle \frac{1}{{\mathrm{e}}^{hc/(\lambda kT)}-1}},

(8)\epsilon \left(\lambda \right)=F\cdot L_P\left(\lambda \right)\tau \left(\lambda \right),

(9){\epsilon }^{\ast }\left(\lambda \right)=F\cdot L_P\left(\lambda \right)\left[1-{\mathrm{e}}^{-\tau \left(\lambda \right)}\right],

where ε( \lambda) is the emission intensity without self-absorption and has the same meaning as in Eq. (1) while ε^*( \lambda) is the emission intensity which considers the self-absorption effect. The intensity I^{\ast } after considering the self-absorption effect can be obtained by integrating ε^*( \lambda) along time and wavelength:

(10)\begin{array}{rl}{I}^{\ast }=& F\iint L_P\left(\lambda ,t\right)\{1-\exp [-{\displaystyle \frac{1}{L_P\left(\lambda ,t\right)}}{\displaystyle \frac{hc}{4\pi {\lambda }_0}}gA\\ & \times {\displaystyle \frac{r(t)Nl(t)}{U\left(T(t)\right)}}\exp \left(-{\displaystyle \frac{E}{kT(t)}}\right)V\left(\lambda ,t\right)]\}\text{d}\lambda \text{d}t,\end{array}

In addition, the broadening of spectral line profile should be considered since it was affected by self-absorption. In this study, Stark broadening is considered as the main factor for the broadening of line profile, and the expression for V ( \lambda, t) is

(11)V(\lambda ,t)={\displaystyle \frac{1}{\pi \gamma (t)}}\left[{\displaystyle \frac{{\gamma }^2(t)}{{\left(\lambda -{\lambda }_0\right)}^2+{\gamma }^2(t)}}\right].

where \gamma is the scale parameter which specifies the half-width at half-maximum (HWHM) and can be expressed as a function of electron density N_e and Stark broadening coefficient α:

(12)\gamma (t)={\displaystyle \frac{1}{2}}\alpha N_e(t).

Since the Stark broadening coefficient α is not available or not accurate in many cases, it is set as an unknown variable in this study. Considering the strong positive correlation between α and the HWHM of a certain spectral line and the little effect of α on the wavelength-integrated intensity, α can be quickly adjusted to make the HWHM of the calculated line the same as the experimental line.

Based on the above derivation, the new CF procedure considering temporal evolution, namely TICF, is proposed and described as follows. Iterate the value of T₀, N_e0, and Nl₀, until the experimental spectral intensity is as close as possible to that described in Equation 10. Then, these parameters (T₀, Nl₀, and N_e0) can be substituted into Equation 6 to obtain the spectral intensity after self-absorption correction. The quantification results can be obtained by the relative number densities of different element byNl₀. The main procedure of TICF is shown in Fig.2.

3 Experimental

A standard LIBS setup was used in this experiment. A Q-switched Nd:YAG laser (Q-Smart 100, Quantel, France) was applied. The laser wavelength was 1064 nm and energy was set to 80 mJ/pulse, with a pulse duration of 6 ns and a repetition rate of 1 Hz. The spot size at the sample surface was about 0.2 mm in diameter. A spectrometer with non-gated CCD detectors (Avantes, Netherland) was used to acquire plasma spectra. The wavelength range of spectra was from 180 to 430 nm and the spectral resolution about 0.1 nm. Optical fiber and spectrometer were corrected by a standard light source (DH-3plus, Ocean Tec, USA). The sample was standard titanium alloys (Chinese National Standard No.02503~02507) with known concentrations, as shown in Tab.1.

To determine the form of the function f_T, f_Ne and f_Nl, a time-resolved pre-experiment with ICCD (LTB butterfly echelle spectrometer) was performed. Spectra of sample 1# were collected at delay times of 2, 3,…,14, 15, 17 and 20 μs with gate time of 0.1 μs. A short gate time was chosen to avoid strong changes in plasma temperature and electron number density during the measurement. Ten spectra were collected at each delay time and each spectrum is accumulated by 10 laser pulse signals to improve the signal-to-noise ratio.

For all samples, 10 non-gated spectra were collected at the delay time of 2 μs with the exposure time of 1 ms. Each spectrum was generated by a single pulse.

4 Result and discussion

Analytical lines selection. Due to the presence of titanium and vanadium element in the plasma is mainly in the form of ions, a number of titanium ion lines and vanadium ion lines can be found in the spectrum with high intensity. At the same time, several low-intensity titanium and aluminum atomic lines can also be observed. The selected lines for these four species are listed in Tab.2. The transition probability A_ki is an important parameter of the spectral line, which can be obtained by the NIST database.

Determination off_T, f_Neand f_Nl. For the spectrum of sample 1 with short gate, the temperature can be determined by Boltzmann plots. In order to eliminate the interference of self-absorption effects on these parameters, blackbody radiation referenced self-absorption correction method [41] was applied.

The electron density is obtained by the Saha−Bolzmann ionization equation:

(13)n_e={\displaystyle \frac{2{(2\pi m_{e}kT)}^{3/2}}{h^3}}{\displaystyle \frac{I_{mn}^I{A}_{ij}{g}_i^{II}}{I_{ij}^I{A}_{mn}{g}_m^I}}\exp \left(-{\displaystyle \frac{E_{ion}+E_i^{II}-E_m^I}{kT}}\right).

Here, m_e, k, T, andh are the mass of an electron, the Boltzmann constant, the temperature of the plasma, and the Planck’s constant, respectively. I_mn, A_mn, g_m^I andE_m^I are the observed intensity of the line transition from m-level to n-level, the Einstein coefficient of transition of transition probability for spontaneous transition, the degeneracy of the upper level, and the energy of the upper energy level of an atomic line. Similarly, I_ij, A_ij, g_i^II andE_i^II are parameters of an ionic line. E_ion is the 1^st ionization energy. Electron density was obtained by substituting data of titanium atom lines and titanium ion lines.

Fig.3 shows the profile of temperature T and the electron density N_e of sample 1# changing with delay time. According to the curve trend of T and N_e in Fig.3, we select power function to fit the curve. The results are as follows:

(14)T(t)=T_0\cdot t^{-0\text{.196}},

(15)N_e(t)={N_e}_0\cdot t^{-0\text{.296}}.

The curve of T and N_e for other samples was selected as the same form as formula 14 and 15 since the samples in Tab.1 have similar matrices, while T₀ and N_e0 are to be determined for each sample. The LTE condition of plasma was verified before quantitative analysis. The necessary condition of LTE can be expressed by McWhirter’s criterion [50]:

(16)n_e>1.4\times {10}^{14}{T}^{1/2}(\mathrm{\Delta }E{)}^3\text{c}{\text{m}}^{-3},

where ΔE (eV) is the highest energy difference between the upper and lower energy levels. According to the formula 16, the critical value of the electron density is 4×10¹⁶ cm⁻³, which shows that the plasma in this experiment meets the necessary condition of the local thermal equilibrium. It should be noted that McWhirter criterion is just a necessary condition for LTE, while two additional conditions (relaxation time and diffusion length) are needed to guarantee the LTE conditions [50]. Maybe the additional sufficient conditions of LTE (relaxation time and diffusion length) were difficult to be maintained at 20 microseconds. However, considering the fact that spectral line intensity decreased sharply with time and the main contribution of line intensity was the plasma at early delay which was easier to be at LTE condition [51], it should not be a big problem for CF, and the high accuracy of final CF result showed that this problem can be neglected in this work.

Nl actually represents the number of observed species, and is considered to be independent of time approximately:

(17)Nl\left(t\right)=N{l}_0.

Quantitative analysis. Here, the calibration-free method is used for quantitative analysis, that is, the elemental concentration is determined according to the elemental number density Nl₀. Since there are multiple lines, we need to adjust the value of T₀, Nl₀, and N_e0 to minimize the residual error Δ of the experimental line intensity I⁰ and the calculated line intensityI^*:

(18)\mathrm{\Delta }\text{=}{\displaystyle \sum {\ln }^2\left(\frac{I_i^{\ast }}{I_i^0}\right),}

where i=1,2,··· is the serial number of the line. The specific method of adjusting each parameter has been described in our previous work [41]. Due to space limitations, it will not be repeated here.

Tab.3 shows the results of calibration-free quantitative analysis. The result of traditional CF was selected as the baseline for comparison, where the non-gated spectra was directly used to build traditional CF model. It can be seen that the newly proposed method TICF can significantly improve the measurement accuracy. Since the line intensity of Fe, Si and C were undetectable and their concentrations were relatively low, the prediction of these elements was neglected. In the future, more samples and more different types of samples can be tested to verify the applicability of the TICF method. For a set of different samples with similar matrix, only one preliminary time-resolved experiment is necessary to get the evolution trend of plasma parameters.

5 Conclusion

A time-integration calibration-free (TICF) method that can be used under non-gated conditions is proposed for laser-induced breakdown spectroscopy (LIBS). In order to eliminate the effects of temporal inhomogeneity of plasma, the parameters of the plasma are considered as functions of time but spatially homogeneous. The spectrum under non-gated conditions is regarded as the integration of the plasma emission along time. The line intensity at each moment of the plasma is calculated according to the profile of plasma evolution, and the self-absorption correction is performed separately for the line intensity at each moment. The application of TICF on titanium alloys shows that the elemental concentration measurement accuracy was significantly improved compared with traditional calibration-free method. The proposed TICF method considers the temporal inhomogeneity of plasma parameters, which extends the applicability of calibration-free for LIBS with non-gated detectors. Spatial inhomogeneity effects should be considered in the future to improve the performance further.

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