Quantitative analysis by laser-induced breakdown spectroscopy based on generalized curves of growth

Spectrochimica Acta Part B 110 (2015) 124–133

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Quantitative analysis by laser-induced breakdown spectroscopy based on generalized curves of growth C. Aragón ⁎, J.A. Aguilera Departamento de Física, Universidad Pública de Navarra, Campus de Arrosadía, E-31006 Pamplona, Spain Institute for Advanced Materials (INAMAT), Public University of Navarre, Campus de Arrosadía, E-31006 Pamplona, Spain

a r t i c l e

i n f o

Article history: Received 13 February 2015 Accepted 13 June 2015 Available online 19 June 2015 Keywords: Laser-induced breakdown spectroscopy LIBS Csigma graph Quantitative analysis Standardless analysis

a b s t r a c t A method for quantitative elemental analysis by laser-induced breakdown spectroscopy (LIBS) is proposed. The method (Cσ-LIBS) is based on Cσ graphs, generalized curves of growth which allow including several lines of various elements at different concentrations. A so-called homogeneous double (HD) model of the laserinduced plasma is used, defined by an integration over a single-region of the radiative transfer equation, combined with a separated treatment for neutral atoms (z = 0) and singly-charged ions (z = 1) in Cσ graphs and characteristic parameters. The procedure includes a criterion, based on a model limit, for eliminating data which, due to a high line intensity or concentration, are not well described by the HD model. An initial procedure provides a set of parameters (βA)z, (ηNl)z, T z and N ze (z = 0, 1) which characterize the plasma and the LIBS system. After characterization, two different analytical procedures, resulting in relative and absolute concentrations, may be applied. To test the method, fused glass samples prepared from certified slags and pure compounds are analyzed. We determine concentrations of Ca, Mn, Mg, V, Ti, Si and Al relative to Fe in three samples prepared from slags, and absolute concentrations of Fe, Ca and Mn in three samples prepared from Fe2O3, CaCO3 and Mn2O3. The accuracy obtained is 3.2% on the average for relative concentrations and 9.2% for absolute concentrations. © 2015 Elsevier B.V. All rights reserved.

1. Introduction The interest in laser-induced breakdown spectroscopy (LIBS), both from a fundamental and applied perspective, has increased over the last years. The number and variety of applications of LIBS to elemental analysis have grown rapidly, driven by technological improvements. However, as pointed out in a recent review by Hahn and Omenetto [1], quantitative aspects remain a major limitation of the technique, as opposed to its general success in the qualitative ones. In its original formulation, quantitative analysis by LIBS is based on a calibration curve of signal (generally spectral line intensity) versus concentration of the element to be detected [2]. An accurate calibration process requires a relatively wide set of matrix-matched standards, which are not available in many applications. Hence, an effort has been made in the last years to develop standardless methods, capable of providing quantitative results without the use of standard reference materials or, at least, such that standards need to be used only infrequently [1]. The methods proposed for standardless LIBS (also known as calibration-free methods) share two hypotheses, one related to the ablation process and the other to the plasma formed. The first hypothesis is that the stoichiometry of the plasma is the same as that of the sample. ⁎ Corresponding author. Fax: +34 948169565. E-mail address: [email protected] (C. Aragón).

http://dx.doi.org/10.1016/j.sab.2015.06.010 0584-8547/© 2015 Elsevier B.V. All rights reserved.

The second one is that local thermodynamic equilibrium (LTE) is satisfied in the radiating laser-induced plasma, where collisions with electrons are assumed to be the dominant process. The existence of LTE allows using the Boltzmann distribution for excited atomic levels and the equation of radiative transfer to account for self-absorption of the spectral lines. The first proposal of a standardless approach for LIBS was the calibration-free method introduced by Ciucci et al. [3]. The method was based in the construction of Boltzmann plots for the different species in the plasma. The plasma temperature is obtained from the slope of the plots, while the intercept value is exploited to determine the relative number density of the individual species. The original approach for calibration-free LIBS introduced in Ref. [3] included two additional hypothesis: that the plasma can be described as a homogeneous source and the optically thin condition for the spectral lines used. The latter hypothesis is not fulfilled generally in laser-induced plasmas, in which spectral lines often experience a significant degree of self-absorption. A procedure for correcting self-absorption in calibration-free LIBS was developed in [4], leading to an improvement in the precision of the method of approximately one order of magnitude. In a recent article [5], the proposers of calibration-free LIBS contributed a critical review of the method, which included an overview of its numerous applications, the variants of the procedure developed later, the possible difficulties due to the plasma non-ideality and other relevant experimental issues. They concluded that the method is suited

C. Aragón, J.A. Aguilera / Spectrochimica Acta Part B 110 (2015) 124–133

for the global characterization of the matrix of an unknown sample, while it can only provide a qualitative estimation of trace elements. The optically thin condition was removed in subsequent standardless methods for LIBS. In these approaches, the equation of radiative transfer is used to calculate a spectrum, which is compared to the measured spectrum to deduce the relative elemental concentrations in the sample. The approaches in this group differ mainly in the model used to describe the laser-induced plasma, and also in the calculation algorithms, where the characteristic plasma parameters, in particular the temperature, play a major role. Yaroshchyk et al. [6] proposed a method in which temperature and relative elemental concentrations are determined from the iterative routine of spectra comparison. For the majority of elements investigated in bauxites, mineral samples and laboratory air, agreement within 25% was achieved between estimated and certified values. A similar approach with a different calculation algorithm was used by Beldjilali et al. [7] to evaluate the minor element concentrations in potatoes. Plasma temperature, electron density and the relative concentrations are adjusted to obtain the best agreement between measured and computed spectra. The estimated mass fractions for the detected elements were in good agreement with published data obtained using standard techniques for analysis of aliments. In a recent work [8], the same group has performed elemental analysis of glass, including oxygen, and determined all the main constituents of the samples, achieving a measurement precision of about 10%. Also, in [9], they have demonstrated the method on carbon-based materials from the inner wall of a fusion reactor. In [6–8], a model of uniform plasma is considered, whereas a plasma composed of two different uniform zones is used in [9] to account for the observed temperature gradient. Thus, setting a suitable plasma model is probably the most relevant and difficult issue in standardless methods. Gornushkin et al. [10] and Shabanov et al. [11] have developed more elaborated radiation dynamic models intended for use in calibration-free LIBS, where concentrations of species are inferred from plasma spectra. The experimental verification of the model of laser-induced plasma expanding into vacuum [12] provided agreements within 10–70% between determined and certified concentrations [13]. Our aim in this work is developing an approach for quantitative LIBS taking the curve-of-growth methodology as a starting point. The analytical calibration functions in conventional LIBS are in fact curves of growth, in which the concentration of the element plotted in the abscissa is proportional to the optical depth [14]. The curve-of-growth methodology was first applied to LIBS by Gornushkin et al. [15]. Later, our group has made contributions to the subject [16,17], in which the problem of plasma inhomogeneity is present. Specifically, in [17], we demonstrated that two different apparent temperatures are necessary to describe the curves of growth of neutral atom and singly-charged ion lines measured at a given instant of plasma evolution. This result was in agreement with that deduced previously [18] from the local and spatially-integrated characterization of a laser-induced plasma: as a result of plasma inhomogeneity, different apparent temperatures are obtained in spatially-integrated measurements from the Boltzmann plots of neutral atoms and ions. In a recent work [19], we have introduced a generalization of curves of growth called Cσ graphs, which allow including several lines of various elements at different concentrations. In agreement with our previous results [17,18], different Cσ graphs corresponding to distinct apparent parameters need to be constructed for neutral atoms and ions. This separated treatment for the two main ionization states defines a plasma model which overcomes the root problem of the unrealistic homogeneous model. Then, the complex inhomogeneous character of the laser-induced plasma is addressed using a model limit. In [19], Cσ graphs are proposed as a tool for characterization of laser-induced plasmas. In the present work, we present a method for quantitative analysis by LIBS, based on the use of Cσ graphs (Cσ-LIBS). Two ways of applying the method, resulting in relative and absolute concentrations, are described. To test the method, fused glass samples prepared from slags and from pure

125

compounds are analyzed. A patent application on the method has been registered by the Public University of Navarre [20]. 2. Method The theoretical framework leading to Cσ graphs is described in detail in our previous work [19]. Here, the main definitions and equations are recalled for clarity. 2.1. Line cross section To construct a Cσ graph, a line cross section σl(T,Ne) is calculated for each spectral line, starting from the atomic data of the transition, the temperature T and the electron density Ne. The steps leading to the key definition of line cross section start from the transfer of radiation through a homogeneous plasma source that, taking into account the absorption and stimulated emission processes, may be described using one of the following expressions for the optical depth τ(λ)[21,22]: 0

τðλÞ ¼ k ðλÞl ¼ σ 0 ðλÞNzα l:

ð1Þ

In Eq. (1), k′(λ) is the effective absorption coefficient, l is the length of the plasma along the line-of-sight, σ ′(λ) is the effective absorption cross section and N zα is the numerical density of emitters (element α, ionization state z). In our approach, we consider the averages of τ(λ) and k′(λ) over the line shape, defining a line absorption coefficient as kl ¼

1 ΔλL

Z

0

line

k ðλÞ dλ;

ð2Þ

where ΔλL is the Lorentzian width. A similar definition is made of a line optical depth τl. Under LTE, the Saha equation provides the ratio S10(T,Ne) between the density of singly-charged ions N1α and that of neutral atoms N 0α [23] S10 ðT; Ne Þ ¼

N1α N0α

¼

2U 1α N e U 0α



mkT 2πℏ2

3=2 exp −

! E0∞ −ΔE0∞ ; kT

ð3Þ

where Uzα(T) is the partition function of the emitting species, m is the electron mass, k is Boltzmann's constant, h is Planck's constant (ℏ = h/2π), E0∞ is the ionization energy and ΔE0∞ is the correction thereof due to interactions in the plasma. Assuming that the laser-induced plasma is formed only by neutral atoms and singly-charged ions, an ionization factor ri(T,Ne) is defined for the two ionization states as follows ri ¼

ri ¼

1 1 þ S10 S10 1 þ S10

for neutral atoms;

ð4aÞ

for singly‐charged ions:

ð4bÞ

The ionization factor allows relating the density of an emitting species N zα with the total density of the element Nα = N 0α + N1α by the expression Nzα ¼ r i Nα :

ð5Þ

Using the ionization factor, a new absorption cross section σ⁎(λ) may be defined by σ  ðλÞ ¼

0

k ðλÞ ¼ r i σ 0 ðλÞ Nα

ð6Þ

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C. Aragón, J.A. Aguilera / Spectrochimica Acta Part B 110 (2015) 124–133

whose integration over the line shape provides the line cross section σl ¼

1 ΔλL

Z line

σ  ðλÞ dλ ¼ r i

1 ΔλL

Z line

σ 0 ðλÞ dλ:

ð7Þ

The final step allowing the calculation of the line cross section is the following expression for the effective absorption cross section, valid for a LTE plasma [23] σ 0 ðλÞ ¼ kt V ðλÞ;

ð9Þ

where V(λ) is the normalized line shape, described by a Voigt profile. The coefficient kt(T), dependent on the atomic data and the temperature, is deduced making use of the Boltzmann distribution for the population of excited levels, with the result [17] Ek −Ei  e2 λ20 g e−kT  1−e− kT ; f i 4ε0 mc2 U zα ðT Þ

known. In this case, Cat may be calculated for all elements, so atomic fraction is a suitable concentration unit for Cσ graphs. However, the present work deals with analysis of certified samples whose elemental concentrations are provided as mass fractions, and no concentration values are provided for some elements. Consequently, atomic fractions may not be calculated, so we have used the mole to gram fraction in the sample (equivalent to molality in a solution) as the concentration unit in Cσ graphs. Expressed in mol g−1, C is related to Cat by a proportionality factor η = Cat/C (g mol−1), so that Eq. (13) may be written as Nα ¼ CηN

ð14Þ

Substituting Nα from Eq. (14) into Eq. (5) and then N zα from Eq. (5) and σ ' (λ) from Eq. (9) into Eq. (1), we get the following expression for the optical depth

Ei

kt ¼

ð10Þ

where e is the elementary charge, λ0 is the central wavelength of the transition, ε0 is the permittivity of free space, c is the speed of light in vacuum, f is the transition oscillator strength, gi is the degeneracy of the lower energy level and Ei and Ek are the energies of the lower and upper energy levels respectively. Substituting Eq. (9) into Eq. (7), the line cross section is obtained as σ l ¼ kt r i

1 : ΔλL

ð11Þ

Assuming that the main broadening mechanism in the laser-induced plasma is the Stark effect, the Lorentzian width may be deduced from the electron density if the Stark width of the line is known. Thus, according to Eq. (11), the line cross section results from the product of three factors which may be calculated for a given temperature and an electron density, provided that the atomic data, including the Stark width, are known. In Table 1, the lines selected for this experiment are listed together with their atomic data. Calculated values of kt and σl are also listed for typical values of T and Ne. As the laser-induced plasma has a high ionization ratio, we have selected ionic lines when possible. However, Si II and Al II lines are relatively weak and were not observed in the spectra, so Si I and Al I lines, together with Fe I lines, have been used for these elements. 2.2. Cσ graphs The integration of the equation of radiative transfer for a homogeneous plasma leads to the following expression for the line intensity measured in counts and corrected by the system spectral efficiency [17] Z I ¼ βALP

 line

 1−e−τðλÞ dλ

ð12Þ

where A is the transverse area of the region of the of plasma whose emission is detected, β is the instrumental factor of the system (counts W−1 sr) and LP = LP(λ0,T) is the Planck radiance of a blackbody, considered constant in the integration over the line profile and calculated at the plasma temperature and at the central wavelength of the transition. Assuming that the stoichiometry of the sample is preserved in the formation of the plasma, the number density of element α is related to the total number density N by Nα ¼ C at N

ð13Þ

where Cat is the concentration of the element in the sample expressed as atomic (or mole) fraction. In our former work [19], Cσ graphs were obtained for characterization purposes starting from home-made samples prepared from pure compounds, whose composition was completely

τðλÞ ¼ ηNl C kt r i V ðλ Þ

ð15Þ

Numerical integration of Eq. (12), using Eq. (15) for the optical depth, allows calculating a curve of growth defined as a plot of the line intensity versus concentration, as done in [15–17]. These conventional curves of growth, which are also the calibration curves in LIBS, depend strongly on the damping ratio a, which makes them very different for lines of varying Lorentzian widths. In our former work [19], we introduced Cσ graphs as follows: in the functional dependence of I/LP versus τ(λ), we divide both quantities by ΔλL to obtain a new dependence I ¼ f ðCσ l Þ LP ΔλL

ð16Þ

The Cσ graph is a plot with abscissas and ordinates defined by Eq. (16). Experimental Cσ graphs are obtained measuring the line intensities for given concentrations and obtaining LP, ΔλL and σl from the atomic data and estimations of temperature and electron density. Also, making use of Eqs. (12) and (15), theoretical Cσ graphs or curves may be calculated starting from the parameters βA, ηNl, T, and Ne. In [19], it was shown that, when lines of different widths are included in a calculated Cσ graph, the data follow a smooth behavior for low to moderate values of the abscissa and Lorentzian widths higher than 0.01 Å for a typical Doppler width of 0.03 Å (damping ratios a N 1). Moreover, the Cσ graph has a common linear limit as Cσl → 0, with slope given by the product βA × ηNl. As an example, Fig. 1 shows a theoretical Cσ graph including all the ionic lines listed in Table 1, calculated using typical values of the parameters and concentrations in our experiment. The linear limit of the graph is also displayed in the figure. As can be seen, in spite of the very different widths of the lines included, the data follow a smooth behavior. As a consequence, in experimental Cσ graphs, taking into account the errors, the data for different Lorentzian widths will form a single curve within a good approximation. Cσ graphs may be considered as a generalization of curves of growth allowing the inclusion of several lines of different elements. In [19], Cσ graphs are proposed as a tool for characterization of laserinduced plasmas. In a recent work [25], they have been used for measurement of transition probabilities. In the present work, Cσ graphs are used for two purposes: a previous step of characterization of the plasma and the LIBS system using a sample with known concentration and a later stage of analysis, in which the concentration C is used as a fitting parameter. Two different analytical procedures, providing relative and absolute concentrations, are described in Section 4. It is worth stressing that, when the second procedure is applied, it is the use of Cσ graphs which allows performing separated stages of characterization and analysis. The characterization stage, which replaces calibration in conventional LIBS, is a distinctive feature of our approach over previous methods proposed for calibration-free or standardless analysis [3,4, 6–9].

C. Aragón, J.A. Aguilera / Spectrochimica Acta Part B 110 (2015) 124–133

127

Table 1 Spectral lines used to construct the Cσ graphs, with their atomic data and typical kt and σl values. λ

Fe II

Ca II

Mn II

Mg II

Ti II

V II

Fe I

Si I

Al I

a

Ei

Ek

(Å)

(eV)

(eV)

2395.626 2404.887 2410.520 2411.069 2413.311 2430.079 2432.262 2439.302 2444.516 2585.876 2587.945 2591.543 2592.785 2598.370 2599.396 2611.874 2617.618 2103.235 2112.757 3158.869 3179.331 3181.275 3706.024 3736.902 2576.105 2593.724 2605.684 2933.055 2939.308 2949.205 2790.777 2797.998e 2936.510 2928.633 3341.875 3361.212 3372.793 3380.277 3383.758 2893.311 2908.808 2924.006 2924.636 3734.864 3749.485 3758.233 3763.789 3765.539 3767.192 3795.002 3797.515 3805.342 3815.840 3820.425 3821.178 3827.822 3834.222 2210.892 2216.669 2506.897 2516.112 2524.108 2528.508 3082.153 3944.006 3961.520

0.05 0.08 0.11 0.12 0.12 2.83 2.84 3.15 2.58 0.00 4.15 1.04 4.08 0.05 0.00 0.05 0.08 3.12 3.15 3.12 3.15 3.15 3.12 3.15 0.00 0.00 0.00 1.17 1.17 1.17 4.42 4.43 4.43 4.42 0.57 0.03 0.01 0.05 0.00 0.37 0.39 0.39 0.37 0.86 0.91 0.96 0.99 3.24 1.01 0.99 3.24 3.30 1.48 0.86 3.27 1.56 0.96 0.01 0.03 0.01 0.03 0.01 0.03 0.00 0.00 0.01

5.22 5.24 5.25 5.26 5.26 7.93 7.94 8.23 7.65 4.79 8.94 5.82 8.86 4.82 4.77 4.79 4.82 9.02 9.02 7.05 7.05 7.05 6.47 6.47 4.81 4.78 4.76 5.40 5.39 5.38 8.86 8.86 8.65 8.65 4.28 3.72 3.69 3.72 3.66 4.65 4.65 4.63 4.61 4.18 4.22 4.26 4.28 6.53 4.30 4.26 6.50 6.56 4.73 4.10 6.51 4.80 4.19 5.62 5.62 4.95 4.95 4.92 4.93 4.02 3.14 3.14

gi

gk

fa

Aaki 8

(10 s 8 6 4 2 2 8 6 12 6 10 8 6 14 8 10 8 6 2 4 2 4 4 2 4 7 7 7 5 5 5 2 8 4 2 6 8 6 10 4 9 11 11 9 11 9 7 5 13 3 5 13 9 9 11 11 7 7 3 5 3 5 3 5 2 2 4

10 8 6 2 4 10 8 14 8 8 10 6 16 6 10 8 6 4 6 4 6 4 2 2 9 7 5 3 5 7 4 10 2 2 8 10 8 10 6 7 9 11 9 11 9 7 5 15 3 7 13 11 7 9 13 5 5 5 7 5 5 1 3 4 2 2

2.59 1.96 1.55 2.37 1.02 1.91 1.57 2.25 2.78 0.894 1.69 0.572 2.74 1.43 2.35 1.20 0.488 0.421 0.593 2.83 3.44 0.71 0.837 1.57 2.82 2.77 2.72 2.01 1.95 1.94 4.01 3.19 2.30 1.15 1.68 1.58 1.41 0.137 1.39E 1.20 1.60 1.70 1.20 0.901 0.763 0.634 0.544 0.951 0.639 0.115 0.457 0.860 1.12 0.667 0.554 1.05 0.452 0.346 0.454 0.547 1.68 2.22 0.904 0.59 0.47 0.99

Acc.

−1

)

wb

Acc.

(Å) 0.279 0.227 0.203 0.207 0.179 0.211 0.186 0.234 0.332 0.0717 0.213 0.0576 0.316 0.108 0.239 0.122 0.0501 0.0559 0.0596 0.847 0.782 0.11 0.172 0.164 0.361 0.279 0.198 0.156 0.253 0.354 0.937 0.469 0.149 0.148 0.375 0.335 0.321 0.0235 0.358 0.120 0.170 0.220 0.150 0.189 0.161 0.134 0.116 0.233 0.136 0.0347 0.0989 0.228 0.190 0.120 0.143 0.165 0.0713 0.0423 0.0469 0.0859 0.159 0.0708 0.052 0.17 0.11 0.12

B+ B+ B+ B+ B+ C+ C+ C C B+ C B C B+ B+ B+ B 15 15 10 10 15 10 10 1.4 4.7 4.8 5 4.6 4.6 A A A A B+ C B+ C+ B+ B B B B A A A A B+ A A B+ B+ A A B+ A A B B B B B B 12 10 6

0.040 0.043 0.0431 0.041 0.039 0.044 0.043 0.039 0.046 0.0411 – 0.047 0.045 0.040 0.045 0.0368 0.038 0.67 0.73 0.53 0.49 0.51 0.66 0.67 0.144 0.14 0.148 0.163 0.122 0.16 0.162 0.144 0.3 0.29 – – – – – – – – – 0.08 0.08 0.08 0.08 0.07 0.08 0.08 – 0.07 0.11 0.09 0.07 0.14 0.09 0.078 0.075 0.145 0.117 0.104 0.107 4.5 3.7 3.8

k tc (10

– 15 15 15 15 15 15 15 15 15 – 15 15 – 15 14 15 15 15 15 15 – 15 15 – 16 16 16 16 16 B+ B+ B+ B+ – – – – – – – – – – – – 20 20 20 – – – 20 20 – 20 – 29 – 28 31 33 35 B B B

σld −20

2

m Å)

16.0 9.45 5.50 2.76 2.39 0.494 0.321 0.568 0.785 6.32 0.121 0.915 0.346 7.28 21.3 8.30 2.45 0.327 0.682 11.1 20.1 2.83 3.07 5.75 102 79.9 57.1 10.4 16.9 23.9 3.72 7.38 1.29 0.646 13.5 30.6 22.6 2.65 17.1 6.41 10.9 14.3 8.19 15.6 10.25 6.34 3.79 1.46 2.60 1.15 0.630 0.936 6.46 10.3 0.753 4.03 3.50 4.79 8.70 12.5 38.0 10.4 12.5 39.9 42.1 88.8

(10−20 m2) 338 187 108 57.2 52.1 9.54 6.35 12.4 14.5 131 1.03 16.5 6.53 157 402 192 55.1 0.470 0.900 20.1 39.4 5.34 4.47 8.27 630 509 344 56.7 124 133 20.5 45.8 3.84 1.99 126 287 212 24.8 160 59.8 102 133 76.4 29.3 19.3 11.9 7.12 3.14 4.90 2.16 0.948 2.01 8.83 17.3 1.62 4.33 5.86 20.8 39.4 29.2 110 34.0 39.7 1.12 1.44 2.96

Data from Refs. [24] (Fe II, Fe I), [25] (Ca II), [26] (Mn II), [27] (Mg II, Ti II), [28] (Si I), and [29] (V II). Stark widths at electron density Ne = 1017 cm−3 from Refs. [30–32] (Fe II), [33] (Ca II), [34] (Mn II), [35,36] (Mg II), [16,17] (Fe I), [37] (Si I), and [38] (Al I). 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. 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. d Calculated for T = 10,000 K and Ne = 1017 cm−3. e The Mg II lines at 2797.930 Å and 2797.998 Å have been grouped, and the resulting data are indicated. b

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2.3. Plasma model and model limit As mentioned in the Introduction, the definition of a suitable model to describe the inhomogeneous laser-induced plasma is probably the most difficult issue in the methods that pursue quantitative or standardless LIBS. The most direct consequence of plasma inhomogeneity, resulting from the different spatial distribution the ionization states, is the need of two different apparent temperatures to fit Boltzmann plots [18] and curves of growth [17] of neutral atoms and singlycharged ions measured with spatial integration at a certain time window. In our former work [19], this idea was extended to the rest of plasma parameters, specifically the columnar density Nl, when they are obtained by fitting of experimental Cσ graphs measured with spatial integration. Thus, within this approach, two different Cσ graphs have to be constructed for neutrals (z = 0) and ions (z = 1). If Eqs. (12) and (15) are used to calculate the fitting function, each Cσ graph leads to four parameters, named (βA)z, (ηNl)z, Tz and N ze. We will refer to the plasma model described by this set of parameters as a homogeneous double (HD) model, were the term “homogeneous” refers to the single-region integration of the equation of radiative transfer leading to Eq. (12), and the term “double” represents the separated treatment of neutral atoms and singly-charged ions. It is worth noting that the description of the inhomogeneous plasma by this model is an approximation, which is limited to elements whose ionization potentials are not very different. The experiments described in Section 4, together with previous results reported in [19], have shown that the HD model works well for elements with ionization energies from 5.99 eV (Al I) to 8.15 eV (Si I). However, for materials that contain also elements with strongly different ionization energies, such as H, C N and O, the characterization of atoms and ions by two temperatures will probably not hold, and a different model will be necessary. With respect to the electron density, previous experiments [39,40] have shown that laser-induced plasmas often have a central region of nearly constant electron density. Moreover, Cσ graphs show a smooth dependence on the Lorentzian width for low to moderate values of the abscissa. As a consequence, a single value of electron density may be considered in the HD model when applied to a given time window. However, as we use different time windows for detection of neutral atom and singlycharged ion lines, the four parameters are different in our experiment for the two ionization states, and we will refer to them as (βA)z, (ηNl)z, Tz and N ze. Of course, the simple HD model is far from the true inhomogeneous distribution of physical parameters in the laser-induced plasma.

Fig. 1. Theoretical Cσ graph constructed for singly-charged ion lines. The values of the plasma parameters used in the calculation are T = 14,000 K, Ne = 1017 cm−3, (ηNl)1 = 40 × 1020 g mol−1 m−2, βA = 103 counts W−1 m2 sr, and the concentrations are CFe = 0.01 × 10−2 mol g−1, CCa = 0.04 × 10−2 mol g−1, CMn = 0.003 × 10−2 mol g−1, CMg = 0.003 × 10−2 mol g−1, CV = 0.0004 × 10−2 mol g−1, and CTi = 0.0002 × 10−2 mol g−1.

Therefore, the model is not able to describe data corresponding to intense lines or high concentrations, for which radiation is emitted from wider regions of the plasma where gradients of the parameters exist. To account for this problem, a limit of validity may be established [19]. With this aim, we have looked for a parameter which increases with the columnar density, the concentration in the sample and the line intensity. Thus, for each of the data in a Cσ graph the following quantity is evaluated ðτl ΔλL Þz ¼ ηNl C kt r i

ð17Þ

The parameter (τlΔλL)z has been found to be useful to predict the failure of the HD model. A procedure, described in Section 4, is used to eliminate from the Cσ graph the data for which (τlΔλL)z exceeds the model limit previously established. The use of a quantitative model limit that allows discarding data in the analytical procedure is also a distinctive feature of our method from previous approaches [3,4,6–9]. Following the idea expressed previously, it should be stressed again that, for elements with strongly different ionization energies, a different value of the model limit or an additional criterion might be required. 3. Experiment The experimental setup is the same used in previous works [30,32], so it is only described briefly here. Laser-induced plasmas are generated by a Nd:YAG laser (wavelength 1064 nm, pulse energy 60 mJ, pulse width 4.5 ns), focused at right angles to the sample surface, placed in air at atmospheric pressure. The focusing lens has a focal length of 126 mm and the lens-to-sample distance is 122 mm. The emission from the plasma is collected at a small angle with the laser beam direction by a system of plane and concave mirrors, which forms a 1:1 image onto the entrance slit of a spectrometer (Czerny Turner, focal length 0.75 m, gratings of 3600 and 1200 lines mm−1). The detector is an intensified charge-coupled device (ICCD) with 1200 × 256 effective pixels. The spectral efficiency of the system has been measured using calibrated tungsten and deuterium lamps. The time window used for detection is 1.6–2.0 μs for singly-charged ion lines and 2.5–3.5 for neutral atom lines. The spectra are measured accumulating the emission from 100 laser shots while the sample rotates at 100 rev min− 1. All the experimental data correspond to the average of five measurements at different positions in the sample, the error bar in the figures representing the standard deviation of the average. For the test of the Cσ-LIBS method reported here, fused glass disks have been chosen as suitable samples, based on results of previous works. In a study with geologic materials by Pease [41] it was shown that the approach of fused glass sample preparation produces better quantitative results in LIBS than does the more commonly used method of pressed powder pellets. In later works [32,33] our group has also used fused glass samples for measurement of Stark widths by LIBS, showing that spectra with greater line-to-background ratios are obtained compared to those measured with alloys. Thus, the samples have been prepared by borate fusion using a fluxer and Pt–Au crucibles and molds. Two types of samples have been prepared. On one side, three slag certified reference materials (Institut de Recherches de la Sidérurgie Française, IRSID, distributed by Bureau of Analysed Samples, BAS) have been dissolved to form fused glass disks named C1, C2, and C3. The certified composition of the slags is shown in Table 2. The mass percentage of slag powder dissolved to obtain the fused glass samples has been 5% in all the analytical experiments. On the other side, three home-made standards S1, S2, and S3 have been prepared by borate fusion of pure compounds (Fe2O3, CaCO3 and Mn2O3). The content of compounds has been selected to obtain concentrations of Fe, Ca and Mn similar to those of the fused glass samples of certified slags.

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eliminating the data with higher (τlΔλL)z. The Cσ graph for ion lines obtained using sample C1, which contains 5% of slag, is shown in Fig. 3a. As can be seen, the data for the more intense Ca II and Mn II lines, indicated in the figure, are deviated from the curve formed by the rest of data. This deviation is due to the failure of the HD model, so these data are eliminated from the Cσ graph, i.e., they are excluded in the fitting performed for characterization, although they have been included in the figure to illustrate their behavior. The fact that the observed deviation of the intense lines is mainly due to the failure of the model and not to errors in the transition probabilities may be checked from the inspection of the Cσ graph of Fig. 3b, measured with a sample containing 0.6% of the same slag. As can be seen, the data for the three lines appear in this case in the same curve as the rest of data, in agreement with their lower (τlΔλL)1 value corresponding to a lower concentration [see Eq. (17)]. We notice also that the Cσ graph of Fig. 3b is linear, due to the negligible self-absorption which takes place at lower concentration. The values of the parameters resulting from this characterization procedure applied to the data of Fig. 3a are (ηNl)1 = (44 ± 6) × 1020 g mol−1 m−2 and T1 = 13,400 ± 200 K. The model limit is estimated as (τlΔλL)1lim = 0.05 Å. The value of the (βA)1 parameter is not relevant, as it depends on the instrumental factor β of our own system. In fact, the characterization procedure includes the determination of physical parameters of the plasma such as the columnar density, the temperature, and the electron density, as well as the parameter βA, which characterizes the experimental arrangement of the LIBS system. As described in [19], the linear limit of a Cσ graph leads, in a logarithmic plot, to a multi-element Boltzmann plot such as that shown in Fig. 4, obtained using the same data of Fig. 3b. The temperature resulting from the slope of the linear fitting of this plot is the same deduced from the Cσ graph within the experimental errors. As explained in Section 2, the parameters (βA)1, (ηNl)1 and T1, together with N1e , describe the HD model for singly-charged ions. A similar characterization has been performed for neutral atoms. However, self-absorption is not

Table 2 Elemental concentrations (wt.%) in certified slags used in the preparation of fused glass samples C1, C2 and C3.

Fe Ca Mn Mg V Ti Si Al P S a b c

C1a

C2b

C3c

11.92 36.88 1.48 0.88 0.46 0.152 2.59 0.42 7.67 0.127

14.87 34.96 1.59 1.12 0.514 0.205 3.1 0.326 7.07 0.092

17.89 32.97 4.6 1.82 0.288 0.302 5.48 0.477 0.110 0.288

Original reference: ECRM 806-1. Original reference: ECRM 805-1. Original reference: ECRM 804-1.

4. Results and discussion 4.1. Characterization of the plasma and the LIBS system. Model limit An initial characterization has been performed for the laser-induced plasma generated from sample C1. A home-made computer program based on least-squares fitting is used to obtain the line intensities and widths from the measured spectra. The electron density of the laser-induced plasma has been measured from the Stark broadening of the Hα line by the same procedure described in previous works [30–33]. The electron density obtained at the time window 1.6–2.0 μs used for detection of ionic lines is (1.8 ± 0.2) × 1017 cm− 3. A flux diagram of the algorithm used for characterization is shown in Fig. 2. The main steps are the fitting of the Cσ graph, which provides the parameters (βA)z, (ηNl)z and Tz, and the determination of the model limit (τlΔλL)zlim from the convergence of χ2 in an iterated process of

Estimation of

Estimation of

Selection of elements and spectral lines

Measurement of line intensities

Calculation of initial values of

Determination of parameters , and by fitting of C graphs

,

,

Calculation of

Elimination of datum with higher

no

129

Convergence of 2 yes Determination of

Fig. 2. Flux diagram of the procedure used for characterization of the plasma and the LIBS system.

130

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a

b

4.2. Determination of relative concentrations Once the characterization process based on the HD model has been performed, the relative concentrations of elements are determined. To this aim, the graphs shown in Figs. 5 (for singly-charged ions) and 6 (for neutral atoms) have been obtained, using the parameters resulting from characterization. These graphs correspond to sample C1, and similar graphs have been used to analyze samples C2 and C3. All the elements of the slags with certified concentrations have been included except P and S, whose intense emission lines are outside of the spectral range of our system. As described previously, the measured data that exceed the model limit have been eliminated. In the graphs of Figs. 5a and 6a, an arbitrary initial concentration is used, which causes that the data for different elements appear in different curves. Then, the elemental concentrations are varied until convergence of χ2 takes place, in a process whose final result is shown in Figs. 5b and 6b. As can be seen, at the end of the iteration process, once the relative concentrations have been obtained, the data for all elements form approximately a single curve. The values obtained for the relative concentrations, taking Fe as the reference element, are shown in Table 3, where they are compared to those obtained from the certified values in the slags. The relative difference of determined and certified values is also shown in the table. The average of these relative differences for all elements and samples is 3.2%. It is worth stressing that accurate results are obtained for all elements, including those whose concentration in the sample is very

a Fig. 3. (a) Experimental Cσ graph measured using a fused glass sample prepared from 5% of certified slag. The solid line is the fitting to a Cσ curve, performed eliminating the data for the three lines indicated, which exceed the model limit. The dashed line is the linear limit of this curve. Wavelength is expressed in Å. (b) A similar graph for a sample containing 0.6% of slag.

noticeable for neutral atom lines in our experiment, even for the sample with a 5% of slag. As a consequence, the Cσ graph for neutral atom lines is linear, so the parameters (βA)0 and (ηNl)0, as well as the model limit (τlΔλL)0lim are undetermined. In this case, only the slope (βA)0 × (ηNl)0 and the temperature T0 = 9200 ± 200 K can be determined from the fitting. The electron density, determined as described previously, is N 0e = (1.0 ± 0.1) × 1017 cm−3 at the time window 2.5–3.5 μs.

b

Fig. 4. Muti-element Boltzmann plot obtained from the same spectrum used to construct the Cσ graph of Fig. 3b. The temperature resulting from the slope of the linear fitting is indicated.

Fig. 5. Cσ graph for singly-charged ions, showing the procedure for determination of relative elemental concentrations. Sample C1 prepared from certified slag has been used in the measurement. (a) Initial graph obtained using an arbitrary concentration. (b) Final graph resulting from the iterative process of varying the concentrations. The inset shows a zoom where the data for V II and Ti II can be better seen.

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131

4.3. Determination of absolute concentrations

a

To investigate the determination of absolute concentrations, we have used the three home-made standard samples S1, S2, and S3, with Fe, Ca and Mn concentrations similar to those of the fused glass samples prepared from slags. The application of Cσ graphs to determine absolute concentrations requires performing previously the characterization procedure, as described in Subsection 4.1. In this case, suitable ionic lines of the elements of interest are observed in the spectra, so only Cσ graphs for ions are used in the characterization and analysis procedures. The Cσ graph for characterization has been constructed from Fe II lines using a fourth standard sample prepared from a single compound (Fe2O3), with a Fe concentration of 0.015 × 10−2 mol g−1. The fitting of the experimental Cσ graph provides the parameters (βA)1, (ηNl)1 and T1. From these parameters, together with Ne, the Cσ curve shown as a solid line in Fig. 7a is obtained. This curve characterizes the LIBS analytical system, allowing the determination of the absolute elemental concentrations. To this aim, a Cσ graph, shown in Fig. 7a, is constructed for the elements of interest using an arbitrary initial value of concentration, as done in the previous subsection. The convergence of χ2 in the iteration process of variation of concentrations leads to the graph shown in Fig. 7b, where the data for the different elements have been shifted to the reference Cσ curve of the system. The concentrations obtained at the end of the iteration process are the absolute concentrations of the elements. The results for the absolute elemental concentrations determined using the Cσ-LIBS method are shown in Table 4, compared to the

b

a Fig. 6. Cσ graph for neutral atoms, showing the procedure for determination of relative elemental concentrations. Sample C1 prepared from certified slag has been used in the measurement. (a) Initial graph obtained using an arbitrary concentration. (b) Final graph resulting from the iterative process of varying the concentrations. The inset shows a zoom where the data for Al I can be better seen.

small. For example the data for Ti II (inset of Fig. 5b) and Al I (inset of Fig. 6b) correspond to concentrations in the samples in the ranges 72– 140 ppm for Ti and 150–230 ppm for Al (concentrations converted to mass fraction). These results show the high accuracy that can be obtained with Cσ-LIBS. This approach for quantitative analysis is advantageous with respect to conventional calibration as, from the knowledge of the parameters obtained using a single sample, the relative concentrations for all the elements are determined from only one or two Cσ graphs (depending on the need to use one or both ionization states). Moreover, the method may be also applied to determine absolute concentrations, as described in the next subsection.

b

Table 3 Elemental concentration, relative to Fe concentration, in fused glass samples prepared from slags. Results obtained by Cσ-LIBS, compared to certified values. C1

Ca Mn Mg V Ti Si Al a b

C2 a

Cσ-LIBS

Cert.

4.05 0.124 0.184 0.0406 0.0146 0.443 0.0712

4.31 0.126 0.170 0.0423 0.0149 0.432 0.0730

Diff. (%) 6.1 2.0 8.6 4.0 2.0 2.5 2.4

b

C3 a

Cσ-LIBS

Cert.

3.00 0.108 0.167 0.0389 0.0163 0.404 0.0430

3.28 0.109 0.173 0.0379 0.0161 0.415 0.0454

Diff. (%) 8.4 1.0 3.7 2.8 1.5 2.5 5.1

b

Cσ-LIBS

Cert.a

Diff.b (%)

2.64 0.2615 0.245 0.0178 0.0202 0.6087 0.0536

2.57 0.2614 0.234 0.0176 0.0197 0.6091 0.0552

2.9 0.1 4.7 0.9 2.8 0.1 2.9

Concentration relative to Fe obtained from certified elemental concentration. Relative difference.

Fig. 7. Cσ graph for singly-charged ions, showing the procedure for determination of absolute elemental concentrations. Sample S1 prepared from pure compounds has been used in the measurement. (a) Initial graph obtained using an arbitrary concentration. The solid line is the Cσ curve obtained by the characterization procedure (b) Final graph resulting from the convergence of the iterative process of varying the concentrations. The inset shows a zoom where the data for Ca II can be better seen.

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Table 4 Absolute elemental concentration in fused glass samples prepared from Fe2O3, CaCO3 and Mn2O3. Results obtained by Cσ-LIBS, compared to nominal values. S1

Fe Ca Mn a b c

S2

S3

Cσ-LIBSa

Nom.b

Diff.c (%)

Cσ-LIBSa

Nom.b

Diff.b,c (%)

Cσ-LIBSa

Nom.b

Diff.c (%)

0.0139 0.0454 0.00143

0.0127 0.0416 0.00140

9.3 9.2 2.2

0.0109 0.0483 0.0019

0.0101 0.0439 0.0017

7.9 10.2 10.7

0.0166 0.0465 0.0042

0.0152 0.0392 0.0040

9.3 18.6 5.6

Determined elemental concentration (10−2 mol/g). Nominal elemental concentration (10−2 mol/g). Relative difference.

nominal values. The relative difference between determined and nominal values, shown in the table, has in this case an averaged value of 9.2%. It is worth noting that, unlike the method used for relative concentrations, the procedure for determination of absolute concentrations is affected by matrix effects which may take place in the formation of the plasma. This is due to the fact that the Cσ curve used as a reference in this procedure is obtained from the characteristic parameters of the plasma, which may vary depending on the sample used. The higher relative error that we have obtained for absolute concentrations compared to the relative values is possibly due to a matrix effect in the generation of the plasma from the fused glass samples. Although the matrix of all the samples is the borate mixture used as solvent, a slight matrix effect may still be taking place, leading to a variation of the plasma parameters depending on the dissolved material. An indication of the existence of matrix effects is the slightly different value of the product βA × Nl obtained for fused glass samples prepared from different slags or from Fe2O3. Actually, Cσ graphs are particularly suited to investigate matrix effects through plasma characterization, but this subject is beyond the scope of this work. 5. Conclusions We have described and tested an approach for quantitative analysis by LIBS based on Cσ graphs, which have been introduced in a recent work for plasma characterization purposes. Cσ graphs play in our method a role similar to that of calibration curves in conventional quantitative LIBS. The advantage of such graphs is that they allow including several elements in the same plot, avoiding having to calibrate each element separately. The method comprises a procedure for characterization, which replaces conventional calibration, plus two alternative procedures for analysis, resulting in relative and absolute concentrations respectively. Contrary to calibration-free or standardless methods proposed so far, in which plasma parameters and concentrations in the sample are handled within a single process, in the present approach characterization may be performed as a previous step, providing the parameters and Cσ curves which allow the later stage of analysis. Thus, the use of a closure equation including the whole set of elements in the sample is not required to determine absolute concentrations. Instead, the previous characterization step using a sample with known concentration is performed, which means that, applied in this way, our approach may not be strictly considered as a calibration-free or standardless method. Nevertheless, the procedure proposed for determination of relative concentrations is a calibration-free method that may also provide absolute concentrations if a closure relation is applied. The characterization and analysis stages of the method are performed using a homogeneous double model of the plasma, in which different parameters describe line emissions from neutral atoms and singly-charged ions, while the equation of radiative transfer is integrated over a single homogeneous region. This model allows overcoming the difficulties of the conventional homogeneous plasma model, maintaining a high simplicity. As the homogeneous double model is only a convenient simplified approximation of the complex distribution of parameters in the plasma, a criterion based on a model limit is used to discard data described wrongly by the model.

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