A data selection method for matrix effects and uncertainty reduction for laser-induced breakdown spectroscopy

1.   Introduction

Laser-induced breakdown spectroscopy (LIBS) is regarded as a future 'superstar' method for elementary analytical chemistry due to its advantages of simple sample preparation, microdamage and in situ/online real-time monitoring [1]. LIBS has been widely used in many scenarios, such as coal analysis [2–4], food quality monitoring [5, 6], biological tissues [7], the nuclear industry [8, 9] and measurement of rare earth elements [10, 11]. However, the quantitative performance of LIBS is severely limited by high signal uncertainty and matrix effects. These problems hinder the potential of LIBS for use in large-scale industrial applications. Signal uncertainty is caused by variations in plasma properties and is reflected by the inconsistent signals of different pulses of the same sample [12]. Matrix effects are due to chemical and physical differences between samples, such as sample composition, melting point, moisture content [13] and surface roughness [9, 14]. For a specific element with the same concentration in different samples, such differences can lead to changes in the spectral intensity of that element. Due to high signal uncertainty and matrix effects, the relationship between analyte concentration and spectral intensity tends to be complex and nonlinear, which poses a serious challenge to the quantitative performance of LIBS [15].

Many studies have attempted to optimize the system, modulate the plasma and process the spectra to reduce signal uncertainty [16, 17]. Some popular methods have included spatial confinement [18, 19], the use of gas mixtures [20] and beam shaping [21]. Basically, the initial aim of system optimization is to obtain highly repeatable signals by optimizing experimental parameters, such as energy, delay time and light collection. However, the reported results suggest that system optimization may not be as important in reducing signal uncertainty [17]. Plasma modulation reduces signal uncertainty by limiting the jitter of the plasma, providing a suitable environment for plasma expansion or avoiding partial overheating of the plasma. Nevertheless, plasma modulation requires an additional experimental setup, which increases the complexity of the LIBS system. Normalization is a simple and common data processing method to reduce signal uncertainty. It uses reference signals such as acoustic waves [22], ablation mass/volume [23], an internal reference line [24] and total/segmental spectral areas to reduce fluctuations in spectra by indirectly compensating for variation in laser energy. However, devices such as an intensified charge-coupled device camera, microphone and a scanning electron microscope (SEM) are needed to obtain reference signals. Sample preparation methods such as binder addition [25, 26] and surface assistance [27, 28] are practical ways to reduce matrix effects. These provide more controllable and stable physical states or more similar chemical components, but the process is often time consuming.

Notably, since signal uncertainty and matrix effects are reflected by the plasma parameters, direct compensation of the plasma parameters can effectively reduce both signal uncertainty and matrix effects [29–33]. For example, Wang et al considered the difference between actual and standard plasma parameters, including plasma temperature, electron density and total number density. The actual intensity can then be converted to the intensity in the standard state according to the basic LIBS formula [30, 31]. This method was tested on brass alloys, and the results show that the root-mean-square error of prediction (RMSEP) and the average relative standard deviation (RSD) decreased from 3.28% to 2.72% and from 8.61% to 5.29%, respectively.

In this work, a data selection method based on plasma temperature matching (DSPTM) was proposed to reduce signal uncertainty and matrix effects. Firstly, the intensity simulation of the LIBS spectra and sensitivity analysis of the three plasma parameters identified plasma temperature as the most sensitive parameter, which was then considered as the main impacting factor both between and within samples. Secondly, spectra with smaller plasma temperature differences were selected for quantification. As the total laser energy is fixed, if spectra with a similar plasma temperature are selected, their ablation process will be more similar and fluctuation of the total number density will decrease relatively, so signal uncertainty is reduced; at the same time, matrix effects are reduced as the plasma temperature difference between samples is reduced by choosing spectra with similar plasma temperatures. This method is very convenient, easy to implement and can directly correct differences in plasma temperature between samples. The proposed method was used to quantify zinc content in brass alloy, and the results demonstrate that DSPTM can be a viable approach to reduce signal uncertainty and matrix effects.

2.   Methods and materials

2.1   Theory and method

Although the effects of signal uncertainty and matrix effects on spectral intensity and LIBS quantification performance have rarely been studied, it is clear that both matrix effects and signal uncertainty affect LIBS signal by changing the parameters, including plasma temperature, electron density and total number density at the signal-collecting temporal-spatial window [12]. The difference is that, for a single sample, parameter differences are ultimately reflected as uncertainty and for different samples they are reflected as matrix effects and signal uncertainty. Besides, signal uncertainty is mainly contributed by the fluctuation of total number density [34], while plasma temperature difference plays an important role in matrix effects. Spectral intensity is determined by the plasma temperature, electron density and total number density of the measured element according to basic theory and can be expressed as follows when local thermal equilibrium (LTE) assumption is qualified [12, 35]:

where F is constant associated with light collection, n is total atomic or ionic number density of the element of interest and $\alpha$ is ionization degree. In our experiment, only atomic and single particles were considered: for atomic particles $n=n_{\mathrm{s}}(1-\alpha )$ and for ionic particles $n=n_{\mathrm{s}}\alpha ,$ $n_{\mathrm{s}}$ is the total number density of the element of interest, $E_i$ is the upper energy and $g_i,$ $U(T),$ $A_{ij},$ $k,$ h and $E_{\mathrm{ion}}$ are statistical weight, partition function, transition probability, Boltzmann constant, Planck constant and ionization potential, respectively. m_e, n_e and T are electron mass, electron number density and plasma temperature, respectively. △E is the negligible ionization potential lowering factor.

Based on the formulae mentioned above, simulations of atomic and ionic lines under the assumption of constant total species number density were conducted. Cu II 202.548 nm and Cu I 216.509 nm were used in our simulation and the results are shown in figure 1. As we can see: (1) for the atomic line, as plasma temperature increases, the intensity increases first and then decreases, and as the electron density increases, the intensity increases; (2) for the ionic line, an increase in plasma temperature or electron density results in an increase in spectral intensity; (3) for both atomic and ionic lines, the intensity is significantly affected by plasma temperature.

Figure  1.  Intensity simulation of (a) the atomic line at 216.509 nm and (b) the ionic line at 202.548 nm.

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Further, the results for the influence of plasma temperature and electron density on intensity during different stages of evolution are listed in table 1. The results of the sensitivity analysis show that: (1) for the three evolutionary stages of the atomic spectrum, a relative change in plasma temperature of 5% has a maximum effect of 35.7% and a minimum effect of 5.2% on the relative amplitude of the intensity, while a relative change in electron density of 5% has a maximum effect of 4.3% and a minimum effect of 0.1% on the relative amplitude of the intensity; (2) for the three evolutionary stages of the ion spectrum, a relative change in plasma temperature of 5% has a maximum effect of 77.5% and a minimum effect of 36.7% on the relative magnitude of the intensity, while a relative change in electron density of 5% has a maximum effect of 4.8% and a minimum effect of 0.4% on the relative magnitude of the intensity; (3) compared with plasma temperature, variation of electron density has little influence on the spectral intensity; (4) for the atomic line, there exists an insensitive area where variations in plasma temperature and electron density have little effect on the intensity; (5) for both atomic and ionic lines, plasma temperature variation has the greatest influence on the plasma intensity compared with electron density.

Table  1.  Sensitivity analysis of atomic and ionic spectra in different evolution stages.

Ionic △I/I	$T=0.6\mathrm{eV}$	$T=0.8\mathrm{eV}$	$T=1.0\mathrm{eV}$
	$n_{\mathrm{e}}=1.06\times {10}^{17}{\mathrm{cm}}^{-3}$	$n_{\mathrm{e}}=1.06\times {10}^{17}{\mathrm{cm}}^{-3}$	$n_{\mathrm{e}}=1.06\times {10}^{17}{\mathrm{cm}}^{-3}$
$\mathrm{△}\mathrm{T/T}=5\%$	77.5%	57.7%	36.7%
$\mathrm{△}{n}_{\mathrm{e}}/n_{\mathrm{e}}$ = 5%	4.8%	2.4%	0.4%
Atomic △I/I	$T=0.6\mathrm{eV}$	$T=0.8\mathrm{eV}$	$T=1.0\mathrm{eV}$
	$n_{\mathrm{e}}=1.06\times {10}^{17}{\mathrm{cm}}^{-3}$	$n_{\mathrm{e}}=1.06\times {10}^{17}{\mathrm{cm}}^{-3}$	$n_{\mathrm{e}}=1.06\times {10}^{17}{\mathrm{cm}}^{-3}$
$\mathrm{△}\mathrm{T/T}=5\%$	35.7%	5.2%	22.6%
$\mathrm{△}{n}_{\mathrm{e}}{n}_{\mathrm{e}}$ = 5%	0.1%	2.5%	4.3%

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As shown above, besides the total number density, which cannot be measured directly, plasma temperature is a more sensitive parameter than electron density, which directly affects the distribution of particle number for each energy level through the Saha–Boltzmann equation and reflects the state of the plasma. Such results indicate that temperature is an essential indicator for spectral intensity. If similar temperatures of different plasmas induced by laser were selected in their evolution stages, the term f$\left(T,n_{\mathrm{e}}\right)$ of all samples would be much more similar and thus there would be a better linear relationship between intensity and sample concentration, so matrix effects get reduced. Besides, as the total laser energy is fixed, if spectra with similar plasma temperatures are selected, their ablation process will be more similar and the fluctuation of the total number density will decrease relatively, so signal uncertainty is reduced. As both signal uncertainty and matrix effects are reduced, the accuracy and precision of quantitative LIBS analysis are improved.

In practical experimental operation, in order to control plasma temperature differences between samples, a plasma temperature matching parameter Q was used. In this method, spectra with plasma temperatures between $T\times (1-Q)$ and $T\times (1+Q)$were selected for quantification. For example, when the value of Q was set to 1%, plasma temperature was 10 000 K and spectra with plasma temperatures between 9900 K and 10 100 K were used for quantification. By traversing between the highest plasma temperature and the lowest plasma temperature, the plasma temperature interval corresponding to the largest number of spectra for a single sample was used for quantification in order to eliminate random noise from instrument and environmental factors.

2.2   Experiment and materials

Integrated LIBS equipment (ChemReveal, TSI, USA) was used to collect spectra in this experiment; more information about the system is given in [36]. In our experiment, laser energies and delay time were optimized to 84 mJ and 1.05 μs by repeatability (RSD) and signal-to-noise ratio (SNR). For each sample, a laser pulse was applied on a fresh surface and a total of 45 pulses were collected. Eleven ZBY series of brass alloy samples with certified reference concentration were used for quantitative analysis; their mass concentrations of the major elements (Cu, Zn) are listed in table 2. The test sample contained copper and zinc as the main elements, with small amounts of iron and lead, ranging from 56.62% to 95.9% copper and from 4.02% to 41.76% zinc. Standard leave-one-out cross validation (LOOCV) was used to verify the effectiveness of this method. To prevent model extrapolation, the highest and lowest concentrations were excluded from the validation.

Table  2.  Element concentration (%) of brass alloy samples.

Number	Sample label	Cu (%)	Zn (%)	Number	Sample label	Cu (%)	Zn (%)
1	ZBY901	73	23.99	7	ZBY907	59.55	34.92
2	ZBY902	64.43	33.45	8	ZBY922	61.88	37.53
3	ZBY903	60.28	38.79	9	ZBY924	80.9	18.75
4	ZBY904	59.14	38.85	10	ZBY925	85.06	14.79
5	ZBY905	58.07	39.59	11	ZBY927	95.9	4.02
6	ZBY906	56.62	41.76

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3.   Results and discussion

In the theory section, we proves that plasma temperature matching can simultaneously reduce matrix effects and uncertainty through simulation and sensitivity analysis of atomic and ionic lines. Here, the feasibility of our method in experiments was verified by determining the zinc concentration in brass alloy.

3.1   Plasma spectroscopy and plasma temperature calculation

As figure 2 shows, a typical spectrum of sample ZBY901 was acquired in the spectral range of 300–650 nm. Copper lines labeled in the spectrogram are used to calculate plasma temperature and zinc lines are used for the quantitative model. The two strongest zinc lines, Zn I 472.52 nm and Zn I 481.05 nm, become saturated and thus are discarded.

Figure  2.  Spectra of ZBY901 in the spectral range of 300–650 nm.

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The population distribution between different energy levels is determined by Boltzmann's law when the LTE assumption is qualified. Then plasma temperature can be calculated from the Boltzmann plot with spectral lines from the same ionization state of the same element:

The plasma temperature of the species can be thus obtained from the slope, –1/kT. In our experiment, the plasma temperature is calculated from the copper atomic lines at 453.07, 515.32, 570.03 and 578.21 nm because they are free from spectral line interference and are not easily self-absorbed. The corresponding atomic spectral data for these lines were taken from the NIST database and are shown in table 3. Figure 3 shows a typical Boltzmann plot at time delay t = 1.05 μs with a good coefficient of determination R² = 0.975; plasma temperature can be calculated from the slope, and for this plot it is 8491 K.

Table  3.  Spectral information used for calculation of plasma temperature.

λ (nm)	Ei (eV)	Ej (eV)	Aij (107s-1)	gi	gj
453.07	6.55	3.82	0.84	2	4
515.32	6.19	3.78	6.0	4	2
570.03	3.81	1.64	0.024	4	4
578.21	3.78	1.64	0.165	2	4

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Figure  3.  Boltzmann plot at time delay t = 1.05 μs.

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3.2   Improvement of precision and accuracy

Precision and accuracy are two key issues that stand in the way of large-scale industrial use of LIBS [37, 38]. In the following two subsections, the effects of the proposed method on improvement in accuracy and precision are discussed.

3.2.1   Uncertainty reduction

Figure 4 shows the results for uncertainty reduction for Zn I 330.29 nm in 11 samples. As shown in the figure, after plasma temperature matching, the uncertainty of most samples was reduced, with the mean RSD of 11 samples decreasing from 20.5% to 14.0%. More results for reduction in spectral uncertainty are shown in table 4. As we can see, the uncertainty of spectral intensity for almost all samples was reduced after plasma temperature matching. The mean RSD of Zn 330.29 nm, Zn 334.59 nm, Cu 249.21 nm and Cu 578.21 nm decreased from 20.5%, 18.2%, 14.0% and 17.7% to 14.0%, 13.3%, 10.7% and 12.1%, respectively. Further, we can find that different spectral lines correspond to different samples when the signal uncertainty is reduced the most. For Zn 300.29, Zn 334.59, Cu 249.21 and Cu 578.21 nm, samples 11, 5, 7 and 5 showed the greatest reduction in signal uncertainty from 42.0%, 19.1%, 20.8% and 19.7% to 25.4%, 8.2%, 13.8% and 10.5%, respectively.

Figure  4.  Uncertainty reduction for 11 samples (Zn I 330 nm).

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Table  4.  Uncertainty reduction of different spectral lines in 11 samples.

Lines	Method	1	2	3	4	5	6	7	8	9	10	11	Ave
Zn 330.29	/	17.1	17.9	17.1	19.6	20.3	17.4	24.4	13.2	21.0	15.1	42.0	20.5
	DSPTM	13.5	14.7	15.3	10.4	9.2	8.0	14.1	16.8	16.8	9.9	25.4	14.0
Zn 334.59	/	16.9	17.4	14.2	17.8	19.1	16.8	22.1	12.4	20.7	12.2	30.8	18.2
	DSPTM	11.3	13.8	12.9	9.8	8.2	8.8	13.2	15.2	19.4	9.1	25.0	13.3
Cu 249.21	/	12.6	13.7	12.2	14.5	14.6	14.0	20.8	12.7	11.8	11.1	15.8	14.0
	DSPTM	11.5	9.7	10.7	13.5	8.4	10.1	13.8	11.8	6.9	9.5	11.3	10.7
Cu 578.21	/	17.3	18.0	19.3	15.8	19.7	15.8	19.9	17.6	16.6	17.3	17.8	17.7
	DSPTM	11.2	11.0	15.3	10.0	10.5	10.8	17.3	15.2	11.4	8.8	11.6	12.1
Notes: DSPTM is plasma temperature matching, Ave is the mean RSD of 11 samples of the selected spectral line. '/' indicates original data without plasma temperature matching.

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3.2.2   Accuracy improvement

A univariate model is a classical calibration model. A univariate model has only one variable and is the basis of the use of LIBS for quantification. However, univariate models rely heavily on selected analysis lines and cannot fully utilize all the spectral information. Multivariate analysis methods such as multiple linear regression (MLR) are often used for quantification of sample concentration. MLR seeks the maximum correlation between multiple spectral lines and sample concentration. In this work, both a univariate model and MLR were used to verify the performance of this method. The spectral line at 330.28 nm was selected as the analytical line of zinc for the univariate model and zinc lines at 334.59, 330.29, 468.04 and 636.35 nm were used for MLR analysis.

Figure 5 shows that plasma temperature difference between samples was noticeably reduced after plasma temperature matching; more specifically, the standard derivation between samples decreased from 452 K to 24 K and the mean RSD decreased from 5.00% to 0.26%. The maximum plasma temperature difference between samples decreased from 3500 K to 90 K. The above results show that the differences in the term f(T, n_e) become smaller after plasma temperature matching and thus matrix effects get reduced. Besides, as total energy is fixed, the ablation process of spectra with a similar plasma temperature will be more similar, so signal uncertainty is improved.

Figure  5.  Sample plasma temperature at 1050 ns (a) and sample plasma temperature after temperature matching (b).

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In this work, three evaluation parameters were used to verify the accuracy and precision of the proposed method; they are shown in table 5. R² is the determination coefficient, RMSEP is the root-mean-square error of cross validation, RSD is the mean relative standard deviation of the predicted concentration of samples. UNI and DSPTM-UNI are the univariate model and univariate model after plasma temperature matching, respectively. MLR and DSPTM-MLR are the multiple linear regression model and multiple linear regression model after plasma temperature matching. Figures 6 and 7 show that: (1) for the univariate model, the values of RMSEP and R² are improved from 3.30% and 0.864 to 1.06% and 0.986, respectively, and the uncertainty of measurement also is reduced from 18.8% to 13.5%; (2) for MLR, the values of RMSEP and R² are improved from 3.22% and 0.871 to 1.07% and 0.986, respectively. The accuracy prediction performance of MLR is better than that of the univariate model, but the robustness becomes worse than that of the univariate model; (3) both univariate and multivariate models have better prediction performance after plasma temperature matching.

Table  5.  Comparison of results for Zn cross validation.

Model	R2	RMSEP (%)	RSD (%)
Univariate	0.864	3.30	18.8
DSPTM-univariate	0.986	1.06	13.5
MLR	0.871	3.22	26.2
DSPTM-MLR	0.986	1.07	17.4

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Figure  6.  Univariate model results: (a) at 1050 ns and (b) after plasma temperature matching (Q = 1%).

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Figure  7.  MLR results: (a) at 1050 ns and (b) after plasma temperature matching (Q = 1%).

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It is worth noting that even if the plasma temperature consistency of all samples is guaranteed, there is still no perfect linear relationship between spectral line intensity and sample concentration, indicating that other factors such as total number density and electron density also need to be considered.

3.3   Influence of the plasma temperature matching interval

Q = 1% is considered in the above discussion of accuracy and precision improvement. That means that for a plasma temperature of 10 000 K, spectra with a plasma temperature between 9900 and 10 100 K were selected for quantification. In this section, the influence of different plasma temperature intervals on quantification is investigated by changing the Q value. The results of the univariate model and MLR are shown in tables 6 and 7.

Table  6.  Univariate results of different Q values at 1050 ns.

Model	R2	RMSEP (%)	RSD (%)	Number of matched spectra
Q = 1%	0.986	1.06	13.5	7
Q = 2%	0.947	2.06	14.0	13
Q = 3%	0.915	2.61	15.4	19
Q = 5%	0.926	2.44	16.6	27
Q = 10%	0.894	2.91	18.2	41
Q = 100%	0.864	3.30	18.8	45

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Table  7.  MLR results for different Q values at 1050 ns.

Model	R2	RMSEP (%)	RSD (%)
Q = 1%	0.986	1.07	17.4
Q = 2%	0.968	1.59	21.9
Q = 3%	0.930	2.37	25.2
Q = 5%	0.903	2.79	21.4
Q = 10%	0.882	3.08	28.2
Q = 100%	0.870	3.23	26.4

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As the results show, for univariate models and MLR, the quantification performance decreased with increase in the Q value overall. More specifically, when the Q value increased from 1% to 100% (indicating that the plasma temperature difference between samples gradually increased), R², RMSEP and RSD decreased from 0.986, 1.06% and 13.5% to 0.864, 3.30% and 18.8%, respectively, for univariate models and from 0.986, 1.07% and 17.4% to 0.87, 3.23% and 26.4%, respectively, for MLR.

The number of matched spectra increased from 7 to 45 with Q value increase from 1% to 100%, in our experiment, at least 7 spectra are needed in order to avoid random noise. These results prove that reducing the plasma temperature difference between samples can effectively improve both accuracy and precision of LIBS quantification.

4.   Conclusion

In this work, a data selection method based on plasma temperature matching was proposed to reduce signal uncertainty and matrix effects, namely DSPTM. Firstly, an intensity simulation of the LIBS spectra and sensitivity analysis of the three plasma parameters identified plasma temperature as the most sensitive parameter, which was then considered as the main factor affecting both intra- and intersample differences. Secondly, spectra with smaller plasma temperature differences were selected for quantification. As the total laser energy is fixed, if spectra with a similar plasma temperature are selected, their ablation process will be more similar and the fluctuation of the total number density will decrease relatively, so signal uncertainty is reduced. At the same time, matrix effects are reduced as the plasma temperature difference between samples is reduced by the choice of spectra with similar plasma temperatures. This method is very convenient, easy to implement and can directly correct plasma temperature differences between samples. The proposed method was used to quantify the zinc content in brass alloy, and the results demonstrate that DSPTM can be a viable approach for reducing signal uncertainty and matrix effects. More specifically, the RMSEP of univariate and MLR models decreased from 3.30% and 3.22% to 1.06% and 1.07%, respectively, while the RSD of univariate and MLR models decreased from 18.8% and 26.2% to 13.5% and 17.4%, respectively. The maximum plasma temperature difference between samples decreased from 3500 to 90 K. This method is very convenient, easy to implement and can directly correct plasma temperature differences between samples. In future practical industrial applications, it may be possible to develop a real-time closed-loop control system that can calculate the plasma temperature, so that a sufficient number of sample spectra with the same plasma temperature can be obtained in less time, thus improving the accuracy and precision of LIBS.