Analysis of rocks by CSigma laser-induced breakdown spectroscopy with fused glass sample preparation

Abstract

We report a validation experiment of the Cσ-LIBS method, proposed recently by our group for quantitative laser-induced breakdown spectroscopy. This method is applied to determine eight elements (Si, Al, Ca, Mg, Fe, Ti, Mn, and V) in rocks prepared as fused glass samples, having a wide range of certified concentrations. The characterization stage of the method is performed using only two standard fused glass samples prepared from mixtures of pure compounds (SiO₂, Al₂O₃, Fe₂O₃ and CaCO₃). The average precision obtained is 3.4% for concentrations higher than 0.1 wt%. This validation suggests that the accuracy and precision of this technique, if combined with fused glass sample preparation under favourable conditions, may become as good as those produced by certification laboratories.

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1 Introduction

Among the efforts to improve the performance of laser-induced breakdown spectroscopy (LIBS) for quantitative analysis, an increasing interest exists in methods in which the conventional calibration procedure with a wide set of standard samples is avoided or at least simplified. These methods take advantage of the broad information about atomic data available in databases, compilations and research articles to switch from univariate calibration to procedures that use several spectral lines of each element of interest. The starting point of all these analytical procedures is the physical equations governing the spectral emission of plasmas in local thermodynamic equilibrium. The first approach proposed within such methods was the calibration-free procedure (CF-LIBS), introduced in 1999 by Ciucci et al.¹ This approach is based on the determination of the temperature of the plasma from Boltzmann plots constructed including spectral lines of all the elements of the sample. Concentrations are obtained from the intercept of the Boltzmann plots, after applying a closure relation. Since its publication, CF-LIBS has been applied to analyse many types of samples, as described in a review² and in a recent article³ published by the group that proposed the method. Also, improvements of CF-LIBS to account for self-absorption⁴ and plasma inhomogeneity⁵ have been introduced. A different group of methods for calibration-free LIBS were proposed by Gornushkin et al.,⁶ Yaroshchyk et al.,⁷ D'Angelo et al.,⁸ and Beldjilali et al.,⁹ based on the comparison of measured spectra to theoretical spectral radiance computed for a plasma in local thermodynamic equilibrium. These methods differed in the model of the plasma, and the algorithms used to determine the parameters such as temperature and elemental concentrations, which were based on fitting procedures^7–9 or Monte Carlo optimization.⁶ Besides calibration-free, other efforts for calibration in LIBS are based on different chemometric methods, as discussed in the review article by Hahn and Omenetto.¹⁰

In recent years, some approaches for quantitative LIBS have relaxed the complete absence of calibration by including a previous measurement with a reduced number of reference standard samples, with the aim of increasing the trueness in the analysis.³ Gaudiuso et al.¹¹ proposed the so-called calibration-free inverse method (CF-IM), in which the plasma temperature, a key parameter of all these methods, is determined as the one providing the best agreement between the composition determined from the original calibration-free algorithm and the known composition of the certified standard. Cavalcanti et al.¹² presented a variation of the original CF-LIBS approach, called the one-point calibration (OPC) method, based on the empirical determination of the product of an experimental and spectroscopic factor, whose knowledge is often imprecise or lacking, from the spectrum measured for one standard of known composition.

Our group has also recently proposed an approach for quantitative analysis by LIBS that may be applied using a few standard samples.^13,14 This method takes its name (Cσ-LIBS) from the so-called Cσ graphs, generalized curves of growth that allow including several lines of different elements in the same plot.¹⁵ In Cσ-LIBS, conventional calibration for each element using a wide set of standards is replaced with a characterization procedure based on Cσ graphs, performed using usually one or two samples of known composition. Once the characteristic plasma parameters have been determined, the elemental concentrations of unknown samples are obtained by an iterative procedure supported also in a Cσ-graph representation. The first validation of the Cσ-LIBS method was carried out using fused glass samples prepared from certified slags.¹³ A satisfactory accuracy was obtained, especially for relative concentrations, for which the average relative error was 3.2%, whereas the error for absolute concentrations was 9.2%. In recent work, Grifoni et al.³ have compared three methods for quantitative LIBS using spectra acquired on bronze samples, obtaining an average percent error for Cσ-LIBS results of 19%. This accuracy may be considered reasonable taking into account the high difficulty of the direct analysis of samples, as opposed to their dilution in fused glass, due to the possible failure of the homogeneous model of the plasma and the higher complexity of the spectra. In fact, fused glass sample preparation, introduced in LIBS by Pease,¹⁶ has important advantages to check calibration models using samples from many different sources. These advantages are mainly due to the improved line-to-background of the spectra obtained with those samples, whose borate matrix contributes with only a few spectral lines to plasma emission.¹⁷ Moreover, the reduced concentrations of the elements of interest in fused glass samples result in low self-absorption. Although this method for developing standards eliminates the readiness of direct analysis, it is valuable for the initial validation of new approaches for quantitative analysis.

In this work, we apply the Cσ-LIBS method to analyse certified rocks prepared as fused glass disks. Quantitative analysis of rocks by LIBS has attracted great attention in recent years, due to its relevant applications in environmental science and geology. Also, the interest in rock analysis has been enhanced by the contribution of the LIBS ChemCam instrument on board the Curiosity rover on its Mars mission.¹⁸ Different approaches for quantitative LIBS have been applied to rock analysis, including conventional calibration,^16,19,20 chemometric methods^16,20,21 and CF-LIBS.¹⁹ Our aim in the present work is to improve and evaluate the analytical performance of the Cσ-LIBS method, including fused glass sample preparation, in the analysis of rocks, where the range of concentrations of the elements determined is much wider than in our first validation performed with certified slags.¹³

2 Experimental

The LIBS system is the same as that used previously,¹⁷ so it is only described briefly. For plasma generation, a Nd:YAG laser (1064 nm, 4.5 ns, 60 mJ per pulse, 20 Hz) is focused by a lens of 126 mm focal length at right angles to the sample, placed in air at atmospheric pressure. A system consisting of a pair of flat and concave mirrors collects the emission from the plasma at a small angle from the laser beam, and forms a 1 : 1 image on the entrance slit of a spectrometer (Czerny-Turner, 0.5 m focal length, 3600 lines per mm grating), equipped with an intensified charge-coupled device (1200 × 256 effective pixels). We have measured the spectral efficiency of the system using deuterium and tungsten calibration standards of spectral irradiance. The sample rotates at 100 rpm during the measurements, in which the emission from 100 laser shots is accumulated.

The samples analysed are four certified rocks; two of them (basalt and granodiorite) have been supplied by the United States Geological Survey (USGS) and the other two (dolomite and bauxite) by the Bureau of Analysed Samples (BAS). Table 1 shows the certified concentrations (wt%) of the eight elements analysed in this work. As can be seen, the four rocks selected for this study present a wide range of concentrations for these elements, which include the main constituents of the rocks with the exceptions of K and Na, whose emission lines are out of the spectral range of our system. In addition, we have prepared for characterization two fused glass samples from mixtures of compounds (SiO₂, Al₂O₃, Fe₂O₃, and CaCO₃ from Alfa Aesar) in powder form, having purities of at least 99.99%. Both the certified reference materials and the pure compounds were dried at 120 °C for 24 hours prior to their preparation as fused glass disks by borate fusion. The flux used consisted of a mixture of lithium tetraborate and lithium borate of 50/50 composition and a total mass of 6 g. The mass fraction of the dissolved material was 5 wt% in all cases. To assess that the purity of the flux is enough to avoid contamination in the analysis of minor elements, we have compared spectra obtained from the fused glass samples prepared from rocks to those measured for a sample prepared by fusion of the flux itself. The compared spectra are shown in Fig. 1, where we have indicated the analytical lines for elements present in rocks at concentrations lower than 0.05 wt%. As can be seen, for V in basalt and Mn in granodiorite, the analytical lines are not observed in the flux spectra, which allows us to deduce that contamination of these elements is negligible in our experiment. In the case of Ca and Mg in bauxite, the analytical lines are observed with low intensity in the flux spectra, indicating the presence of traces of these elements. From the line intensity ratios between both spectra, we have estimated threshold values for the contents in the flux of 1.9 ppm for Ca and 0.17 ppm for Mg that, taking into account the dilution in the fused samples, correspond to concentrations of 38 and 3.4 ppm, respectively, in the bauxite sample. Comparing these values to the certified contents of 360 ppm for Ca and 120 ppm for Mg, we deduce that contamination has a reduced effect on the analytical results obtained for these elements. The fluxer (Claisse Fluxy) allows simultaneous fusion of three samples in Pt–Au crucibles and molds. The general fusion program has been used, comprising six heating stages with increasing values of gas flow and rotation speed of the crucible, and a total operation time of nearly 2 minutes.

Table 1 Concentrations (wt%) of the elements analysed in certified rocks. The values in italics correspond to concentrations too small to be detected by our system

	Basalt^a	Granodiorite^b	Dolomite^c	Bauxite^d
a Original reference: USGS BHVO-2. b Original reference: USGS GSP-2. c Original reference: BAS 782-1. d Original reference: BAS 395.
Si	23.3	31.1	0.124	0.580
Al	7.16	7.88	0.055	27.7
Ca	8.17	1.50	21.68	0.036
Mg	4.36	0.58	12.84	0.012
Fe	8.63	3.43	0.314	11.4
Ti	1.63	0.40	0.0025	1.16
Mn	0.1290	0.0320	0.063	—
V	0.0317	0.0052	—	—

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Fig. 1 Spectra obtained from the fused glass samples prepared from rocks, showing the analytical lines for the elements present in contents in the rocks lower than 0.05 wt%, compared to the same spectra obtained from a sample prepared by fusion of the flux.

3 Results and discussion

The Cσ-LIBS procedure comprises two separate stages, described in the next subsections: a characterization procedure using standard samples and the analysis of samples with unknown composition.

3.1 Characterization

The Cσ-LIBS method relies on characterization of the plasma and the LIBS system by a set of parameters, namely βA (instrumental factor β times transverse area of the plasma A), ηNl (concentration unit factor η times columnar density Nl), T (temperature), and N_e (electron density).

3.1.1 Cσ graphs obtained with standard samples. The parameters βA, ηNl and T are determined by fitting experimental Cσ graphs obtained using reference standard samples with known composition. Characterization is performed separately for neutral atom and ion emissions. Indeed, the different distribution of the two ionization states in the plasma results in different apparent parameters for neutral atoms and ions, even if emission lines from both ionization states are measured in the same time window. Nonetheless, in our case the parameters for both ionization states are different, as we have used two different time windows to detect neutral atom (3.0 μs delay, 1.0 μs width) and ion (1.6 μs delay, 0.4 μs width) emissions. The criteria used to select the time windows were high line-to-background ratio for each ionization state, delay not too short for ions to avoid excessive line broadening caused by a high electron density, and delay not too large for neutral atoms to cause deviations from local thermodynamic equilibrium due to low electron density. For neutral atoms, the standard is a fused glass sample containing a mixture of SiO₂, Al₂O₃ and Fe₂O₃, whereas the ion emission has been characterized by means of a sample prepared from a mixture of Fe₂O₃ and CaCO₃.

To accomplish characterization, we determined first the electron density of the plasma from the Stark broadening of the H_α line.¹⁷ The electron density was the same within the error for the two standard samples. The resulting N_e values were (0.82 ± 0.01) × 10¹⁷ cm⁻³ and (1.78 ± 0.02) × 10¹⁷ cm⁻³ for neutral atoms and ions, respectively. Then, experimental Cσ graphs, defined as plots with ordinate I/(L_PΔλ_L) and abscissa Cσ_l, are constructed, using experimental and atomic data for selected lines of the elements of interest. In the ordinate of Cσ graphs, I is the measured line intensity, L_P is the Planck radiance of a blackbody calculated at the plasma temperature and at the central wavelength of the transition, and Δλ_L is the Lorentzian width, obtained by multiplying the Stark width by the electron density. In the abscissa, C is the concentration in the sample expressed in mol g⁻¹ and σ_l(T,N_e) is the line cross-section, a key parameter defined as^13,15

where the quantities k_t(T) and r_i(T,N_e) are obtained from plasma parameters and the atomic data (level energies and degeneracies, oscillator strength, and central wavelength) of the transition involved. The reason for expressing the concentration in mol g⁻¹ is that the equations of Cσ-LIBS use the number density of the elements in the plasma, which is assumed to be proportional to the concentration of the element in the sample. Therefore, this concentration must be the atomic (or mole) fraction. To provide analytical results, we have transformed the concentrations obtained into wt%.

The lines used in the present work for characterization and analysis, together with their atomic data are listed in Table 2. The line cross-sections of the lines are also provided in the last column for typical values of T and N_e. Next, we describe the criteria used to select neutral atom or ion lines for a given element. (1) For some elements, at the typical temperature of the plasma, ion lines in the near ultraviolet to visible region are too weak, having small k_t values. For example, at 10 000 K, k_t values for Si II and Al II lines in the spectral range 200–700 nm are lower than 0.1 × 10⁻²⁰ m² Å. Consequently, these lines are not observed or they show a very low line-to-background ratio, so neutral lines are used for these elements. (2) As described in our former work,¹³ too intense lines exceed the model limit of the method. As the plasma is strongly ionized, ion lines most commonly exceed the model limit than neutral ones for the same concentration. Therefore, for elements with high concentrations, neutral lines are generally preferred, unless weak ion lines are available. In the case of Mg, for example, the intense ion lines are avoided and Mg I lines are used except for the bauxite sample, which has a small Mg content. (3) For some elements, especially transition metals, both neutral and ion lines are available that do not exceed the model limit; in this case, ion lines are usually preferred, due to their higher overall line-to-background ratio. Fig. 2 shows the experimental Cσ graphs obtained for characterization of neutral atom and ion emissions using the selected spectral lines.

Table 2 Spectral lines used to construct the Cσ graphs, with their atomic data and typical σ_l values

	λ (Å)	E i (eV)	E k (eV)	g i	g k	A ki (10^8 s^−1)	f	Acc.	w (Å)	Acc.	σ l (10^−20 m^2)
a References: Fe I, Fe II;^22 Si I;^23 Al I;^24 Ca I, Mg I, Mg II, Ti II, V II;^25 Ca II;^25,26 Mn II.^27 b Stark widths at electron density Ne = 10^17 cm^−3. References: Fe I;^28 Si I;^29 Al I;^30 Ca I;^31 Fe II;^17,32,33 Ca II;^34 Mn II;^35 Mg II;^36,37 Ti II.^38 When the uncertainty is not displayed in the table, the experimental Stark width is not available, and the average of known Stark widths for lines of the same multiplet has been used. An uncertainty M means that the data provided come from a rough measurement performed in our laboratory. When the Stark width is not provided in the table, a value of 0.1 Å has been used. c Calculated for T = 10 000 K and Ne = 10^17 cm^−3. d The Mg II lines at 2797.930 Å and 2797.998 Å have been grouped, and the resulting data are indicated.
Fe I	3631.463	0.96	4.37	7	9	0.517	0.131	A	—	—	8.73
	3647.843	0.91	4.31	9	11	0.291	0.0711	A	—	—	6.46
	3734.864	0.86	4.18	11	11	0.901	0.189	A	0.08	—	23.4
	3749.485	0.91	4.22	9	9	0.763	0.161	A	0.08	—	15.4
	3763.789	0.99	4.28	5	5	0.544	0.116	A	0.08	20	5.70
	3765.539	3.24	6.53	13	15	0.951	0.233	B+	0.07	20	2.20
	3767.192	1.01	4.30	3	3	0.639	0.136	A	0.08	20	3.92
Si I	2207.978	0.00	5.61	1	3	0.262	0.0575	B	0.065	34	11.4
	2210.892	0.01	5.62	3	5	0.346	0.0423	B	0.061	38	26.6
	2211.745	0.01	5.61	3	3	0.181	0.0133	B	0.063	—	8.11
	2216.669	0.03	5.62	5	7	0.454	0.0469	B	0.063	—	46.9
	2218.057	0.03	5.62	5	5	0.109	0.00805	B	0.063	—	8.06
	2506.897	0.01	4.95	3	5	0.547	0.0859	B	0.141	29	30.0
	2516.112	0.03	4.95	5	5	1.68	0.159	B	0.117	31	110
	2524.108	0.01	4.92	3	1	2.22	0.0708	B	0.104	33	34.0
	2528.508	0.03	4.93	5	3	0.904	0.052	B	0.107	35	39.7
Al I	2652.475	0.00	4.67	2	2	0.142	0.015	12	0.78	M	0.337
	2660.386	0.01	4.67	4	2	0.284	0.0151	11	0.78	M	0.671
	3082.153	0.00	4.02	2	4	0.59	0.168	12	0.45	B	11.3
	3092.710	0.01	4.02	4	6	0.71	0.153	2	0.49	B	18.6
	3944.006	0.00	3.14	2	2	0.47	0.11	10	0.37	B	14.4
	3961.520	0.01	3.14	4	2	0.99	0.117	6	0.38	B	29.7
Ca I	4283.011	1.89	4.78	3	5	0.434	0.199	C+	0.15	M	7.07
	4289.367	1.88	4.77	1	3	0.600	0.500	C+	0.15	M	5.99
	4298.988	1.89	4.77	3	3	0.466	0.129	C+	0.15	M	4.62
	4302.528	1.90	4.78	5	5	1.360	0.378	C+	0.15	M	22.3
	4318.652	1.90	4.77	5	3	0.740	0.120	C+	0.155	D	4.59
Mg I	2776.690	2.71	7.18	3	5	1.320	0.254	B+	—	—	14.4
	2778.271	2.71	7.17	1	3	1.820	0.632	C+	—	—	12.0
	2779.834	2.72	7.18	8	8	3.010	0.356	B	—	—	53.5
	2781.416	2.71	7.17	3	1	5.430	0.210	C+	—	—	11.9
	2782.971	2.72	7.17	5	3	2.140	0.149	B+	—	—	14.1
Fe II	2582.584	1.08	5.88	4	4	0.88	0.088	B	0.051	15	14.8
	2585.876	0.00	4.79	10	8	0.894	0.0717	B+	0.0411	15	131
	2591.543	1.04	5.82	6	6	0.572	0.0576	B	0.0523	—	14.9
	2592.785	4.08	8.86	14	16	2.74	0.316	C	0.045	15	6.53
	2598.370	0.05	4.82	8	6	1.43	0.108	B+	0.0391	—	158
	2599.396	0.00	4.77	10	10	2.35	0.239	B+	0.045	15	402
	2607.088	0.08	4.84	6	4	1.73	0.117	B	0.0394	14	123
	2611.874	0.05	4.79	8	8	1.2	0.122	B+	0.0368	14	192
	2613.825	0.11	4.85	4	2	2.12	0.109	B+	0.0384	14	76.7
	2617.618	0.08	4.82	6	6	0.488	0.0501	B	0.038	15	55.1
	2621.670	0.12	4.85	2	2	0.56	0.0577	B+	0.0391	—	19.7
	2711.842	3.15	7.72	12	14	0.436	0.056	C	0.045	—	3.17
	2714.413	0.99	5.55	8	6	0.57	0.0473	B	0.0541	—	18.4
	2716.218	3.42	7.99	6	6	1.15	0.127	C	0.045	—	2.63
	2724.884	1.04	5.59	6	6	0.0958	0.0107	C+	0.0524	—	3.04
	2730.734	1.08	5.62	4	4	0.279	0.0313	B	0.05	14	5.99
	2743.197	1.10	5.62	2	4	1.97	0.445	B+	0.0515	14	40.7
	2753.288	3.27	7.77	10	12	1.89	0.258	C	0.0532	14	9.28
Ca II	3158.869	3.12	7.05	2	4	2.83	0.847	10	0.53	15	20.1
	3179.331	3.15	7.05	4	6	3.44	0.782	10	0.49	15	39.4
	3706.024	3.12	6.47	2	2	0.837	0.172	10	0.66	15	4.47
	3736.902	3.15	6.47	4	2	1.57	0.164	10	0.67	15	8.27
	3933.663	0.00	3.15	2	4	1.47	0.682	C	0.17	15	2890
	3968.469	0.00	3.12	2	2	1.40	0.033	C	0.16	15	1510
Mn II	2593.724	0.00	4.78	7	7	2.77	0.2795	4.7	0.14	16	509
	2605.684	0.00	4.76	7	5	2.72	0.1979	4.8	0.148	16	344
	2933.055	1.17	5.40	5	3	2.01	0.1556	5	0.163	16	56.7
	2939.308	1.17	5.39	5	5	1.95	0.2527	4.6	0.122	16	124
	2949.205	1.17	5.38	5	7	1.94	0.3544	4.6	0.16	16	133
Mg II	2790.777	4.42	8.86	2	4	4.01	0.937	A	0.162	B+	20.5
	2795.528	0.00	4.43	2	4	2.60	0.608	A+	0.087	B+	4210
	2797.998^d	4.43	8.86	8	10	3.19	0.469	A	0.144	B+	45.8
	2802.705	0.00	4.42	2	2	2.57	0.303	A+	0.0945	B+	1940
Ti II	3341.875	0.57	4.28	6	8	1.68	0.375	B+	0.0887	15	126
	3361.212	0.03	3.72	8	10	1.58	0.335	C	0.0702	15	287
	3372.793	0.01	3.69	6	8	1.41	0.321	B+	0.0724	15	212
	3380.277	0.05	3.72	10	10	0.137	0.0235	C+	0.075	—	24.8
	3383.758	0.00	3.66	4	6	1.39	0.358	B+	0.0688	15	160
V II	2893.311	0.37	4.65	9	7	1.20	0.120	B	—	—	59.8
	2908.808	0.39	4.65	11	9	1.60	0.170	B	—	—	102
	2924.636	0.37	4.61	9	9	1.20	0.150	B	—	—	76.4

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Fig. 2 Experimental Cσ graphs for characterization of neutral atom (a) and ion (b) emissions. The samples were prepared from mixtures of SiO₂, Al₂O₃ and Fe₂O₃ for neutral atoms, and Fe₂O₃ and CaCO₃ for ions.

3.1.2 Determination of the characteristic parameters. To obtain the plots of Fig. 1, one needs to first determine the temperature, to introduce it in the line cross-section σ_l. This in turn requires calculating the ordinate values of the Cσ graph, so that a fitting of experimental and theoretical data may be performed. For a detailed description of the theoretical framework of the Cσ-LIBS method, the reader is referred to our previous articles.^13,15 Here, it is enough to recall that a single-region integration of the radiative transfer equation leads to the following expression for the line intensity

The optical depth is factorized as

τ(λ) = ηNlCk_tr_iV(λ), (3)

where V(λ) is the normalized line shape, described by a Voigt profile. In summary, the intensity for a particular line and concentration value is calculated from the parameters βA, Nl, T, and N_e and the atomic data by numerical integration of eqn (2). Once we are able to calculate the ordinate of the Cσ graph, the fitting of the experimental data provides the characteristic parameters. The curves resulting from this fitting are shown as solid lines in Fig. 2. In the case of neutral atoms, the parameter values obtained are ηNl = (53 ± 10) × 10²⁰ g mol⁻¹ m⁻² and T = 9200 ± 200 K. The value of βA is not relevant, as it depends on the instrumental factor β of the particular system. For ions, the Cσ graph is almost linear, which prevents obtaining accurate separate values for the parameters βA and ηNl. In this case, the fitting provides the product of these two parameters, in addition to the temperature, T = 12 500 ± 200 K.

3.2 Analysis results

The knowledge of the characteristic parameters of the plasma and the LIBS system allows the analysis of unknown samples. The procedure¹³ starts from the measurement of spectra and construction of initial experimental Cσ graphs for neutral atoms and ions including selected spectral lines and using an arbitrary value of concentration for all elements. To perform the analysis, an iterative process is carried out by varying the concentrations and obtaining the value of χ² of the experimental data with respect to the curve of the LIBS system, calculated from the plasma parameters. The convergence of this process leads to the absolute concentrations of the elements and to the final Cσ graphs, as shown in Fig. 3 for neutral atoms and in Fig. 4 for ions. From the comparison of the plots of the four rocks in each figure, we notice the high difference of the concentrations among the samples, particularly between dolomite and bauxite in respect of the similar composition of basalt and granodiorite.

Fig. 3 Cσ graphs of neutral atom emission lines, providing the elemental concentrations for basalt (a), granodiorite (b), dolomite (c) and bauxite (d) samples.

Fig. 4 Cσ graphs for ion emission lines, providing the elemental concentrations for basalt (a), granodiorite (b), dolomite (c) and bauxite (d) samples.

The analytical results obtained are presented in Table 3. For the four rocks prepared as fused glass samples, the elemental concentrations resulting from Cσ-LIBS analysis are compared to certified values. The last column of the table shows the relative difference between Cσ-LIBS and certified values. To summarize, the difference is 13% on average for concentrations lower than 0.1 wt%, and 3.4% for higher concentrations. It is worth noting that, taking into account the dilution at 5 wt% of the rock powder in the borate solvent, the real concentration in the fused glass samples is in the range 6.0–31 μg g⁻¹ for the elements with concentrations in the rock lower than 0.1 wt%. For these low concentrations in the fused glass sample, the signal-to-noise ratio of the analytical lines becomes poor, which explains the higher error of the determined concentrations. Overall, the precision is improved with respect to that obtained in our previous validation with slag fused glass samples.¹³

Table 3 Concentrations determined by Cσ-LIBS of rocks prepared as fused glass samples, compared to certified concentrations

Sample	Element	Certified (wt%)	σ lab (%)	Cσ-LIBS (wt%)	Diff.^b (%)
a Standard deviation of concentrations provided by different laboratories. b Relative difference between Cσ-LIBS and certified concentration.
Basalt	Si	23.3	1.3	22.7	2.4
	Al	7.16	1.1	7.04	1.7
	Ca	8.17	1.5	7.73	5.4
	Mg	4.36	1.5	4.37	0.2
	Fe	8.63	1.6	8.59	0.5
	Ti	1.63	1.2	1.59	2.5
	Mn	0.1290	3.1	0.1295	0.4
	V	0.0317	3.4	0.0308	2.8
Granodiorite	Si	31.1	1.3	30.4	2.3
	Al	7.88	1.4	6.96	12
	Ca	1.50	2.7	1.52	1.2
	Mg	0.58	3.4	0.558	3.8
	Fe	3.43	3.2	3.41	0.5
	Ti	0.40	2.5	0.401	0.3
	Mn	0.0320	6.2	0.0381	19
Dolomite	Si	0.124	5.0	0.131	5.6
	Al	0.055	8.2	0.067	21
	Ca	21.68	0.4	22.30	2.8
	Mg	12.84	1.3	12.09	5.9
	Fe	0.314	3.0	0.295	6.1
	Mn	0.063	5.5	0.060	5.4
Bauxite	Si	0.580	3.2	0.565	2.6
	Al	27.73	3.8	25.98	6.2
	Ca	0.036	20	0.028	23
	Mg	0.012	50	0.011	7.2
	Fe	11.40	0.6	11.62	1.9
	Ti	1.157	3.6	1.085	6.5

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To estimate the limit of detection in Cσ-LIBS, we first obtain the quantity (Cσ_l)_L = 3s/b, where s is the standard deviation of the ordinate of the graph for low Cσ_l values corresponding to data of low concentration C, and b is the slope of the linear limit, given by the product of parameters b = βA × Nl. Then, (Cσ_l)_L is divided by the line cross-section σ_l to obtain the detection limit C_L. We have determined in this way the detection limit for the element with lower concentration in each rock, with the following results: 4.0 μg g⁻¹ for V in the basalt fused sample, 1.5 μg g⁻¹ for Mn in that prepared from granodiorite, 1.4 μg g⁻¹ for Mn in that prepared from dolomite, and 0.6 μg g⁻¹ for Mg in the sample prepared from bauxite. It is worth mentioning that these limits of detection correspond to the analysis of the fused glass samples, where the rocks are diluted at 5 wt%, and therefore they are higher, in the 10–100 ppm range, in the rock samples.

It is interesting to compare the differences of Cσ-LIBS results relative to certified values with the data provided by the laboratories participating in the certification of the reference samples. To this aim, Table 3 includes the column with heading σ_lab, meaning the relative standard deviation of concentrations provided by different laboratories, taken from the certification sheet. As can be seen, the difference between Cσ-LIBS and certified values is not far from σ_lab, which means that the accuracy of the technique obtained under the conditions of the present work is comparable to that of the methods used by the laboratories, such as XRF, ICP-MS, etc. However, it should be noted that the results of the present validation have been obtained under favourable conditions using fused glass samples that, as mentioned before, are particularly suitable for initial validation of calibration methods, specifically for the CSigma-LIBS approach. The four main factors that contribute to improved analytical performance when using fused glass sample preparation are (1) the accurate measurement of line intensities, resulting from the high line-to-background of spectra obtained with these samples (2) the high sample homogeneity, (3) the absence or limited influence of matrix effects due to the common borate matrix, and (4) the low elemental concentration range resulting from dilution in the borate solvent. All these factors contribute to the accurate results collected in Table 3, which therefore may not be extrapolated to possible applications of the method involving other types of samples. The latter factor has particular relevance in Cσ-LIBS, as low elemental concentrations help to overcome its limitations to describe plasma inhomogeneity. Indeed, as described in detail in our previous work,¹³ the method is based on a so-called double homogeneous plasma model. This simple model is only an approximation of the true complex distribution of physical parameters in the plasma. Therefore, the model is expected to fail in describing the emission of elements with high ionization energies, such as H, C, N and O. Also, we have checked that it fails to describe Cσ data for intense lines or high concentrations. Therefore, a further development of the method including a more realistic model of the plasma will be necessary to apply it to samples with higher elemental contents or in cases where the mentioned elements have to be determined.

4 Conclusions

The Cσ-LIBS method for quantitative laser-induced breakdown spectroscopy has been applied to analyse certified rocks prepared as fused glass samples. The characterization stage prior to analysis is performed using only two standard samples prepared from mixtures of pure compounds. The four rocks selected for this experiment have a wide range of concentrations for the eight elements analysed. The average precision obtained for absolute concentrations is 3.4% for concentrations in the rock higher than 0.1 wt%. The comparison of the precision obtained with the standard deviation of values provided by certification laboratories shows that the accuracy of Cσ-LIBS, when combined with fused glass sample preparation under favourable conditions, is comparable to that of the methods used in the certification analysis. It is clear that this type of sample preparation removes the versatility and speed of LIBS analysis. Moreover, the dilution of the sample of interest prevents the determination of trace (ppm level) elements. However, as a counterpart, various problems that limit the accuracy of the analysis of non-prepared samples are mitigated by fused glass preparation, which turns out to be very convenient to perform the initial validations of the method. In particular, the main limitation of Cσ-LIBS in its present stage, related to the failure of the simple plasma model used, is overcome due to the low elemental concentrations present in the diluted samples. Further work is necessary to extend the applicability of the method to other types of samples, in particular to direct analysis by LIBS, without sample preparation. However, we consider that this new validation of the Cσ-LIBS method is a step forward towards the acceptance of laser-induced breakdown spectroscopy as one of the established techniques for reliable quantitative analysis. Our next goal is to extend the applicability of the method to direct analysis of alloy samples.

Acknowledgements

This work has been supported by the project FIS2014-54285-P of the Spanish Ministerio de Economía y Competitividad.

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