An Optimized Calibration Procedure for Determining Elemental Ratios

Article pubs.acs.org/ac

An Optimized Calibration Procedure for Determining Elemental Ratios Using Laser-Induced Breakdown Spectroscopy Jens Frydenvang,*,†,‡ Kjartan Münster Kinch,‡ Søren Husted,† and Morten Bo Madsen‡ †

Department of Plant and Environmental Sciences, Faculty of Science, University of Copenhagen, Copenhagen, Denmark The Niels Bohr Institute, Faculty of Science, University of Copenhagen, Copenhagen, Denmark

‡

ABSTRACT: A novel procedure to determine elemental ratios by laser-induced breakdown spectroscopy is presented. This procedure, which we name optimized calibration (OC-LIBS), is a hybrid between empirical methods like calibration curves or chemometrics and the theoretical calibration-free LIBS method (Ciucci, A.; Corsi, M.; Palleschi, V.; Rastelli, S.; Salvetti, A.; Tognoni, E. Appl. Spectrosc. 1999, 53, 960−964) and seeks to reduce the high sensitivity to matrix effects seen when using, e.g., calibration curves by employing the mathematical framework of the calibration-free LIBS method. The OC-LIBS procedure is calibrated using 22 different certified powdered reference samples, spanning numerous different rock types as well as a large variation in the elemental composition. Using the OC-LIBS procedure, the compositional ratios between Mg and the elements Al, Ca, Fe, K, Na, Si, and Ti are calculated using an absolute minimum of sample preparation. A correlation between the reference and calculated values of R2 ≥ 0.91 and a median relative prediction error ranging between 9.5% and 33% are achieved, despite this diverse set of samples and limited sample preparation. With total data processing times below 1 s, the OC-LIBS procedure allows for all the unique features of LIBS to be utilized, including the ability to provide very fast realtime in situ analyses. fitted to an often limited number of spectral lines applicable for Boltzmann plots.1 This exponential dependence leads to severe deterioration of the data quality for even slightly self-absorbed spectral lines. In the optimized calibration (OC-LIBS) procedure presented here, a better way to cope with the matrix effect problems seen in the empirical methods was sought as well as a way to alleviate the unfortunate exponential dependencies seen in the CF-LIBS approach. This was achieved using the fundamental mathematical framework of the CF-LIBS method in an empirical calibration using reference samples, thereby using the best from both types of methods.

C

ompared to other analytical methods that are capable of measuring the elemental composition of samples, the main advantage of laser-induced breakdown spectroscopy2,3 (LIBS) is its ability to make fast in situ measurements. Hence, the process of data acquisition and analysis should not require laborious sample preparation or other requirements that impede this unique advantage of LIBS. The different models that are currently used to interpret LIBS measurements can roughly be divided into two main groups: empirical- and theoretical methods. The group of empirical methods include the use of calibration curves2,4 as well as various chemometrical methods5−7 (e.g., principal component analysis). The theoretical approach is mainly defined by the CF-LIBS approach as described by Ciucci et al.1 but also includes plasma-modeling approaches like the one presented by Herrera et al.8 Calibration curves have been found to yield good results,4,9,10 but a very high sensitivity to matrix-effects is observed. This has in effect meant that calibration curves must be carefully matrixmatched to the unknown samples and that a substantial effort needs to be put into sample preparation to obtain reproducible surfaces for each measurement,11 thereby severely limiting the applicability of the method outside laboratories. The CF-LIBS1 approach is the method of choice today for unknown samples8 or samples where matrix-matched calibration samples cannot be acquired. The method does however face difficulties in obtaining reproducible results with sufficient accuracy.5 Improved accuracy and reproducibility have been reported when correcting for, e.g., self-absorbed spectral lines;12 but the CF-LIBS method suffers fundamentally from the exponential dependence of the predictions on values that are © 2012 American Chemical Society

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THEORETICAL FRAMEWORK The theoretical foundation of the OC-LIBS procedure is the same as that used in the CF-LIBS approach;1 hence, the area underneath a spectral line is described by

( ) E

Iki = CsFgk Aki

exp − k Tk B

Zs(T )

(1)

where Iki is the area under the spectral line originating from the electronic transition between the upper level k to the lower level i; Cs is the number density of the emitting species; F is an experimental parameter describing factors which affect the overall intensity of the spectrum; gk is the statistical weight of Received: September 13, 2012 Accepted: December 20, 2012 Published: December 20, 2012 1492

dx.doi.org/10.1021/ac3026652 | Anal. Chem. 2013, 85, 1492−1500

Analytical Chemistry

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the upper level of the electronic transition; Aki is the Einstein Acoeffcient for the electronic transition (k to i); Ek is the energy of the upper electronic level; kB is the Boltzmann constant; T is the temperature of the plasma; and Zs(T) is the partition function for the emitting species at the temperature T. Describing the spectral line by this equation requires the assumption that the plasma obtains local thermal equilibrium (LTE) and that the plasma is optically thin at the observed spectral line. Of the constants in eq 1, F describes any effects that impact the overall intensity of the spectrum, e.g., laser-matter coupling, and thus must be assumed to change from shot to shot due to the chaotic nature of this interaction.12,13 Like F, the temperature T must be assumed to vary from shot to shot, especially when extensive sample preparation is avoided. Also, if T changes from shot to shot, so will the partition function Zs(T). In contrast, Cs and the atomic constants gk, Ak, and Ek are the same for all consecutive shots on the same sample. Evidently, variations in F, T, and Zs(T) must be compensated in order to utilize eq 1 to determine the concentration of a given species. Rather than use Boltzmann-plots to calculate and compensate for temperature variations as is done in the CF-LIBS theory,1 it is noted11 that, taking the area ratio of two spectral lines from the same species, the following relation is obtained: τ=

⎛ kT ⎞ ⎛ 1 ⎞kT2 = k T1 exp⎜ 2 ⎟ = k T1 exp⎜ ⎟ ⎝T ⎠ Is 2 ⎝T ⎠

⇒

I = 1 κ ( T )τ δ , I2

1

g S A S2

k T2 =

2

(2)

ES2 − ES1 kB

τ being used to express this ratio between two spectral lines from the same species, and rewritable to an expression for the temperature of the plasma through: T = k T2(ln(τ ) − ln(k T1))−1

(3)

−1 1 ⎛1⎞ exp⎜ ⎟ = k T1 kT2 τ kT2 ⎝T ⎠

(4)

τ therefore gives a measure that covaries with the temperature of the plasma, which is the most troublesome factor in eq 1. Variation in F causes the overall intensity of the spectrum to vary, but temperature variations cause the ratios between different lines to vary, thereby changing the overall pattern of the spectrum from shot to shot. Because F is the same for all spectral lines, F can also be taken out of eq 1 by taking the ratio between two spectral lines from different species. ⎛ E − E2 ⎞ I2 C g A 2 Z1(T ) exp⎜ 1 = 2 2 ⎟ I1 C1 g1A1 Z 2(T ) ⎝ kBT ⎠ =

C2 ⎛ 1 ⎞k C 2 k C1(T ) exp⎜ ⎟ ⎝T ⎠ C1

(5)

where k C1(T ) =

g2A 2 Z1(T ) g1A1Z 2(T )

k C2 =

κ(T ) = k C1(T )k T1

−k C 2 k T2

δ=

k C2 k T2

where eq 4 is used to go from eq 5 to eq 6. Approximations. Even for typically reported gate times of around 1 μs,1,14,15 some temperature evolution must be expected within the time period a spectrum is recorded. This is especially true for the measurements presented here, where all light emitted after the initial delay is collected. Hence, using a single temperature and the associated values for the relevant partition functions to describe all spectral lines recorded in a spectrum is at best questionable. Rather than obtaining transition constants and the partition functions included in κ in eq 6, it is therefore more appropriate to introduce possible approximations initially. First of all, the temperature used to explain a given ratio in eq 6 is allowed to vary for different spectral lines, rather than being described by a single temperature determined by eq 3. Even though the final cumulative spectrum is recorded over a very large temperature variation, the various spectral lines are assumed to be emitted primarily at a narrow temperature interval as the temperature of the plasma drops. In practice, this change in temperature for different spectral lines is modeled by multiplying a constant (unique to every spectral line) to the temperature T in the above expressions. This amounts to redefining the constant kT2 to kT2′ = kT3·kT2, where kT3 is the extra constant that allows for different temperatures for different lines, and then using kT2′ in equation eq 6. Under the assumption that the temperature describing different lines can vary significantly, finding correct values for the partition function using NIST16 or similar databases makes little sense. Therefore, the κ(T)-expression in eq 6 is approximated by a linear function: κ(T) ≈ a · T + b with unknown a and b parameters. This linear expression is obviously no more than an approximation of the temperature dependence of κ(T), but for the sake of minimizing the number of variables compared to the number of samples available, this simple approximation is deemed most appropriate. Another significant, but necessary, assumption of the OCLIBS procedure compared to the CF-LIBS1 approach is that the entire elemental ratio is calculated using eq 6 and not just the concentration of the neutral or ionized species of that element. This is required because no prior knowledge can be obtained regarding how much of an element is ionized in each separate laser-induced plasma. Without this knowledge, it is not possible to determine the value of the defined constants by fitting to known reference values. This in effect means that the approximation of κ(T) in eq 6 with κ(T) ≈ a · T + b will also serve the purpose of approximating the temperature-dependent ratio of neutral and ionized species described by the Saha-equation.17,18 This approximation of, essentially, the Saha-equation with the same linear function used to approximate the ratio between the partition function of two different species, see eqs 5 and 6, is of course an even more crude approximation than only approximating κ in eq 6. As noted above, this is however deemed most appropriate considering the limited amount of

where gS A S1

−k C 2 k C 2 k T2 τ k T2

(6)

Is1

k T1 =

C1 I = 1 k C1(T )k T1 C2 I2

E1 − E2 kB 1493



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spectral lines are chosen to provide the best result for the type of samples used for the calibration, and thus cannot be assumed to be the optimal values for the constants or the best lines on a general basis. Considering how spectral lines will become selfabsorbed or too weak to detect depending on the concentration of an element, a specific spectral line will never be the optimal selection in all possible situations. Summary. On the basis of the above information, it is anticipated that this novel optimized calibration (OC-LIBS) procedure is less sensitive to changes in matrix-effects than are, e.g., calibration curves and chemometrical approaches due to the temperature corrections incorporated in the theory. This should provide an improved dynamic range for this approach compared to other calibration methods and hence the ability to use the same calibration for a more diverse set of samples. At the same time, the exponential relation between the calculated elemental composition and the intensity of a very limited number of spectral lines seen in the CF-LIBS approach has been removed, thereby reducing the sensitivity to lines becoming self-absorbed that the CF-LIBS approach suffer from. In an operational mode, the areas of only four lines, two for the elemental ratio and two for the τ-expression, have to be recorded to determine one elemental ratio. The choice of which four spectral lines to use is predetermined during the calibration, along with the optimal value of the four constants. This allows for fast processing of acquired spectra and hereby for LIBS to maintain its unique feature of potentially being used for realtime in situ measurements. Furthermore, the fact that only a limited, predetermined number of spectral-ranges have to be recorded allows for designing cheaper, custom spectrometers for specific applications.

samples analyzed presently. The more samples included in the subsequent calibration of this model, the more complex this κ(T) approximation can become without overfitting. OC-LIBS Equation. Applying the above assumptions to eq 6 gives the following equation. C1 I I = 1 κ(T )τ δ = 1 (aT + b)τ δ C2 I2 I2 =

I1 ⎡ a k T ′(ln(τ ) − ln(k T1))−1⎤⎦ + b τ δ I2 ⎣ 2

(

)

⎞ I ⎛ α = 1⎜ + γ ⎟τ δ I2 ⎝ ln(τ ) − β ⎠

(7)

where α = ak T2′

β = ln(k T1)

γ=b

with τ = (Is1/Is2) being the ratio between two lines from the same species. Equation 7 is the central equation in the OC-LIBS procedure. Using this equation, the ratio between the concentration of two elements may be found from the measured area of four spectral lines; one line for each of the two elements under consideration and two more lines from a third element used in the expression for the temperature dependent τ. Four unknown constants (α, β, γ, and δ) must be predetermined during calibration in order to calculate the ratio between two elements using this approach. Optimization of Constants and Line Selection. Each set of constants (α, β, γ, and δ) is unique to every possible set of four spectral lines but will be the same for all consecutive shots and for all samples. Hence, samples with known composition give (C1/C2), τ, and (I1/I2), the two latter from the acquired spectrum. Measurements of samples with known elemental composition thus allow for the four unknowns to be determined by requiring the calculated ratio between the content of two elements to be similar to the reference ratio. Choosing which of the many spectral lines found in a typical LIBS-spectrum to use for calculating the elemental composition is a nontrivial problem. General criteria such as high intensity, no self-absorption, no overlapping of lines, etc. can be implemented when evaluating whether a given spectral line will be suitable or not for calculating the elemental composition using eq 7. Whether or not a spectral line fulfills these requirements are, however, often difficult to validate.2,12,19 Considering the approximations made in eq 7, it is more appropriate to construct an algorithm that tries all possible combinations of ratios between spectral lines and τ, rather than choosing lines initially based on the general criteria described above. Optimizing the four constants for all possible line combinations means that the choice of which lines to use can be made from which line combinations provide the best prediction of the elemental ratio. This approach is further corroborated by noting the success of various chemometrical methods, where the choice of using the right latent variables is seen to provide promising results,6,20,21 despite the fact that the linear relationships assumed in most chemometrical methods should not be able to account for the nonlinear variations seen in eq 1. It is however important to note that this best-prediction based selection of which lines to use is a strictly empirical part of the OC-LIBS procedure. The set of fitted constants and

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EXPERIMENTAL METHOD The certified reference samples used for determining the four constants for each elemental ratio are presented in Table 1. Reference values for the samples can be found on the USGS22 and NIST23 webpages, respectively. Experimental Setup. Measurements were made using a Twins BSL 220 laser from Quantel, which consists of two Quantel CFR400 frequency doubled Nd:YAG lasers with stable resonators mounted on the same baseplate, each of which has an output energy of 220 mJ at 532 nm. The laser pulse was focused onto the samples using a 15 cm focal length lens giving a spot radius of ≈0.34 mm (calculated), and hence an irradiance of ≈9 GW/cm2 (maximum, calculated). The laser was operated at maximum power for all measurements as this was shown to provide the strongest spectra. A Lasertechnik Berlin (LTB) Aryelle 400 spectrometer utilizing a 2048 × 512 pixel Andor DV440 CCD-chip was used to record the spectra in the wavelength range of approximately 300−800 nm. Single-pulse measurements were conducted with a delay of 1 μs between the firing of the laser and recording of the spectrum. No gate time can be set, meaning that all light emitted after the initial delay was recorded. The light from the plasma was collected using a 1 in. i.d lens located 14.5 cm from the plasma, 27° off angle from the axis of the laser. The collected light was then focused into a fiberoptic cable and guided into the spectrometer. It was prioritized that the measurements made to calibrate this novel procedure involved an absolute minimum of sample preparation. Hence, the powdered reference sample was poured into a sample holder consisting of an approximately 0.5 cm deep and a 2 cm i.d. cylindrical depression in a slap of 1494



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Table 1. USGS and NIST Powdered Reference Materialsa name

type

AGV-1 AGV-2 BCR-2 BHVO-2 BIR-1 COQ-1 DNC-1 DTS-2b DGPM-1 GSP-1 GSP-2 NOD-A-1 NOD-P-1 PCC-1 QLO-1 SCo-1 SDC-1 SGR-1b W-2a (NIST) SRM2709 (NIST) SRM2710 (NIST) SRM2711

andesite andesite basalt basalt basalt carbonatite dolerite dunite diss. gold ore granodiorite granodiorite manganese nodule manganese nodule peridotite quartz latite shale-cody mica schist shale diabase San Joaquin soil - baseline trace elements Montana soil - highly elevated trace elements Montana soil - moderately elevated trace elements

measurements were seen to diverge significantly from others, but rather than increase the amount of sample preparation, these divergent measurements have to be dealt with in the data processing. After the three shots had been fired on the filled sample holder, the sample holder was repacked and the procedure repeated until at least 15 measurements had been conducted on each reference sample. No averaging was done over spectra from several measurements, so all spectra used for the subsequent data processing are the result of only one laser pulse. The measurements presented here were conducted on two different days a week apart. No systematic difference between the data collected on different dates was observed. Data Processing. All acquired spectra are dark corrected (dark spectrum recorded before each laser pulse), and any overlap between the different echelle orders are removed by the “Sophi” control software. Because of the number of lines seen in the recorded spectra, a calibration using all possible combinations of lines is not feasible. Instead, only all linecombinations originating from the elements included in the ratio of interest will be tested. These lines are the only ones which should be able to convey any direct information regarding the elemental content (cf. eq 1), making this a reasonable constraint. On the basis of the online NIST library of spectral lines,16 a program is coded in MATLAB24 to identify which spectral lines are present in the recorded spectra. This program checks whether or not a peak is present in a spectrum above a minimum intensity within a narrow range around a NIST library wavelength. If a peak is present, upper and lower linelimits are defined by the points where the intensity stops decreasing when moving outward from the maximum of the peak. By requiring the difference in intensity at these limits to not exceed a preset level, lines located either on the sloping tail of another line and lines with clear self-absorption are excluded. If one line ends up being registered as several different “librarylines”, the user is prompted for a decision. The area underneath each line is defined as the sum of intensities between the lower and upper wavelength limit, with the minimum intensity of the range subtracted from each value. It is this area which is used for the subsequent data analysis. This somewhat simple program is not error-proof, and misclassification of lines, the inclusion of slightly self-absorbed lines, and similar types of problems cannot be excluded as possible outcomes. As all possible combinations of lines classified as the elements of interest is checked, this is however not a problem. Misclassified or self-absorbed lines will simply provide an extra combination to test but will not exclude lines providing a good prediction of an elemental ratio. The results obtained on the basis of this program are thus fully valid, only it cannot be decided whether even better predictions could be obtained using lines which are presently misclassified as a different element. Similarly, it is not possible to try all possible combinations of τ either. Only potassium lines are therefore used for the τexpression described in eq 2; this being an element that i) is present in all the available samples in reasonable concentrations and ii) has a suitable number of detected spectral lines to allow for a substantial number of possible combinations, without making the calibration too computationally demanding. With a total of 17 detected potassium lines, this gives a total of 136 possible τ combinations that will be checked for each ratio of spectral lines.

a The samples AGV-1 and GSP-1 were not officially certified by the USGS at the time of the measurements, as the certification is time limited. The reference values found for these samples are however still deemed valid compared to the accuracy realistically achievable by LIBS.

composite material. The samples were not pressed, and excess material was simply scraped off to give a flat surface. On each sample holder, a total of three shots were made, this being the highest number of shots possible without getting overlapping craters. A picture of one of the samples after three shots is shown in Figure 1.

Figure 1. Picture of a filled sample holder after three single-shot measurements have been conducted. The tick-marks on the right are in millimeters.

The size of the craters created varied significantly for different samples; most likely due to different amounts of energy being absorbed by the sample from each laser pulse and differences in density, grain size, and texture of the samples. As this will also be the case when conducting in situ measurements, nothing was done to prevent this. No systematic dependence was observed on whether a spectrum originates from the first or last measurement on a sample holder. The residue spread from each shot therefore does not appear to cause any significant change in the sample surface compared to the initial lightly prepared surface. Some 1495



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Figure 2. Plots showing the reference ratio vs the median of the calculated ratios for each shot for each sample. The blue marks are the samples used for the calibration, and the red marks are validation samples. The horizontal error bars are calculated based on the uncertainty of the reference samples, and the vertical error bars are the median absolute deviation from the median value shown. The red line describes the desired 1:1 ratio.

Nod-A-1 and Nod-P-1 that contain a large fraction of Mn. The results obtained for the different ratios are shown in Figures 2 and 3. For each elemental ratio, three of the samples were selected to act as test samples. The test samples were chosen in order to represent the whole range of values covered by the total set of samples for each elemental ratio and are therefore not the same from one elemental ratio to another. The test samples were excluded prior to the calibration procedure for each ratio and thus acted as unknown samples for validation. For the results presented in Figure 2, all samples listed in Table 1 are included, whereas it proved necessary for the results presented in Figure 3 to exclude specific samples. These samples were excluded prior to processing based on reference compositional values, either because the content of one element is extreme compared to the other samples or because no reference value for a specific element is provided. Calculating all of the results reported in Figures 2 and 3 from LIBS spectra takes less than 40 s on a 2.2 GHz CPU or less than 0.2 s per spectrum. This new procedure is therefore seen to fully comply with the requirement of providing virtually instantaneous results. The simulated annealing routine used to optimize the value of the four constants was seen to give consistent but not entirely similar results when run for the same line-combination repeatedly. This is undoubtedly an artifact originating from the very complex LIBS-spectra,7 the fact that there are four unknowns to be fitted, and the random nature of how the simulated annealing routine works. For this reason, the complete optimization routine was run at least three times

For each elemental ratio treated, a simulated annealing optimization routine coded in MATLAB24 is used to determine the optimal value of the four constants by minimizing the difference between the ratio calculated using eq 7 and the reference elemental ratio. This routine is based on the “simulannealbnd”-function25 and optimizes the value of the four constants for all possible line-combinations for the given elemental ratio, i.e., all line combinations of the elements in the ratio, and all 136 possible potassium τ-combinations. This calibration routine is a substantial computational task and requires close to a week of computations on three 2.40 GHz cores of an Intel Xeon CPU to check all line combinations of Ti, which is the element with most lines detected. It is however only the calibration routine which is time-consuming. After the calibration is completed, the area of only four lines needs to be determined from a new spectrum to calculate a specific elemental ratio using eq 7, something which can be done almost instantly. When the calibration routine is completed, the optimal value of α,β,γ, and δ has been determined for all possible line combinations. The optimal line-combination to be used for unknown samples can then be chosen from any combination which is observed to provide the best prediction.

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RESULTS The results obtained for the ratios between the denominator Mg and the nominators Al, Ca, Fe, K, Na, Si, and Ti are presented. These eight elements together account for ≈99% of the cations in all of the samples treated, except for the samples 1496



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Figure 3. Plots showing the reference ratio vs the median of the calculated ratios for each shot for each sample. The blue marks are the samples used for the calibration, and the red marks are validation samples. The horizontal error bars are calculated based on the uncertainty of the reference samples, and the vertical error bars are the median absolute deviation from the median value shown. The red line describes the desired 1:1 ratio. No reference value was reported for the nominator in the ratios treated in parts b and d for the sample DGPM-1, and this sample is therefore not included in the data treatment. For the ratios treated in parts a and c, the reference ratio for the samples COQ-1 and DGPM-1, respectively, is abnormal compared to the elemental ratio for the remaining samples; these samples are therefore excluded in parts a and c, respectively.

quality of the predictions, while taking the risk of outliers into consideration. This test shows a rapid improvement over the first seven shots, where the R2 value grows from 0.51 to 0.88 and the median relative prediction error falls from 16.7% to 12.7%. A more slow improvement is seen from 7 shots to 15 shots, where the R2-value grows to 0.92 and the median relative prediction error falls to 11.8%. All seven ratios presented in Figures 2 and 3 show fairly good correlation coefficients with the calibration and test samples having a combined R2 ≥ 0.91; the lowest R2-values being obtained for the (Si/Mg) (0.91) and (Fe/Mg) (0.93) ratios. The median relative prediction error is seen to vary between 9.6% (Fe/Mg) and 33% (Si/Mg), with the top and bottom interestingly being the two ratios with the two lowest R2 values. As no outliers are seen in any of the results presented, the R2 value seems to provide the best description of the obtained results and was therefore the primary measure used for evaluating the obtained results for the different linecombinations.

for each elemental ratio. The selection of the best combination of lines and constants was then based on which provided the best prediction across all three runs and the lowest median absolute deviations from the median value reported in Figures 2 and 3. The use of the median value, rather than the mean value, of the results for the different shots on each sample is chosen in order to easily exclude outliers. For all the elements, one or more of the samples had extreme outliers for one of the shots. Rather than manually excluding these extremes, it was found more appropriate to use robust statistics to deal with these outliers automatically, and in order simplify the computations, the median value was chosen instead of other robust statistical methods. The median relative prediction error is presented, as it is noted that the mean relative prediction error is skewed upward for even very low absolute differences between calculated and reference values. This is due to the presence of a few very low reference ratios, which cause a division with a value close to zero when calculating the prediction error. The dependence of the median relative prediction error and the R2-value on the number of shots made on each sample was tested for the (Ca/Mg)-ratio (data not shown). The use of 1− 15 shots was simulated by randomly choosing the appropriate number of spectra from each sample and calculating the resulting predictions 50 times. Looking at the average value obtained over these 50 repetitions allows for estimating the

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DISCUSSION The fact that the Si-ratio gives the poorest result obtained (R2 = 0.91) poses a clear problem if the absolute cation-content of any of the elements is to be calculated, as this requires the relevant ratio to be normalized by the sum of all ratios. Because the Si-content is high in all samples, uncertainties in the calculated quantity of this element will have a major effect on 1497



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all other elements as well, a problem which is of course shared with associated methods of determining absolute content from relative values.1,9 However, being able to calculate the elemental ratio is perfectly suitable to solve a wide range of problems. When, e.g., looking for concentration of trace elements in soil samples26,27 or plant tissue,28,29 little extra knowledge is acquired by obtaining an absolute concentration compared to a ratio between a trace element and a somewhat constant major element. When classifying rock types, a primary purpose of the ChemCam LIBS instrument on the NASA Curiosity Mars rover,30−32 classes are typically defined by relations between different elements as well.7,33 For in-line industrial process control,34 where a large degree of a priori knowledge is available, these kinds of relative values will also often provide sufficient information. In this context, the R2 values achieved appear very promising compared to what is reported elsewhere,27,35,36 especially as the results presented here are obtained using a far less laborintensive sample preparation and on a diverse set of geological samples. As for the very important question on how robust this calibration is, it is very encouraging to note how well the testsamples are handled in all cases, indicating that the accurate results obtained are not a result of overfitting. Furthermore, the fact that the measurements were made on two different days and still provided the observed accuracy also underlines the robustness of the model. The values reported for the four constants can however not be used in any LIBS setup, even when treating the same samples as in this study. The fact that LTE allows for different temperatures in different parts of the plasma immediately leads to the requirement of an identical optical setup. Furthermore, the value of the four constants incorporates a number of parameters unique for the specific equipment used, e.g., the wavelength dependent quantum efficiency of the spectrometer. Hence, it is the OC-LIBS procedure as a whole which we propose and not the use of the specific values presented. Looking at the median relative prediction errors in Figures 2 and 3, some of these appear very large, especially considering it is the median of the absolute deviations of each measurement from the median value. These large deviations are however believed to be a truthful depiction of the large variance seen for these LIBS-measurements, these in turn being a result of a mix between the unpredictable nature of the plasma ignition, growth, and decay, the variances of the sample surface due to the minimum of sample preparation used, and potential inhomogeneities in the geological samples. Hence large variance is the price paid for keeping the sample preparation to an absolute minimum. The fact that a rapid improvement was seen for the R2 and median relative prediction error up to 7 shots also appear to indicate that this is the lowest number of shots advisable with the minimum sample-pretreatment used. Finding solutions with small deviations are however not something that is prioritized in the algorithms written, which simply minimize the misfit of the median values over all shots to the reference value. It is strongly considered to include some sort of preference toward these more stable results in the algorithms, as these are believed to provide more matrixinsensitive results; the optimal method of doing this has not been determined yet however. Furthermore, it is unexpected that lines from ionized species provide the best prediction for some elemental ratios, rather than lines from the neutral species (Table 2). This makes the κ-

Table 2. Table with Lines, Line-Limits, and Constants Used to Calculate the Elemental Ratios ratio

lines

range/nm

constants

(Al/Mg)

Al I @ 394.40 nm Mg I @ 518.36 nm K II @ 413.47 nm K I @ 693.88 nm

394.10−394.68 517.92−518.79 413.38−413.55 693.52−694.12

α = −3.898 β = −7.364 γ = 1.829 δ = −0.834

(Ca/Mg)

Ca I @ 646.26 nm Mg I @ 516.73 nm K I @ 475.39 nm K I @ 486.98 nm

646.83−646.44 516.35−516.97 475.26−475.58 486.93−487.07

α = −6.076 β = 3.719 γ = −0.416 δ = −0.279

(Fe/Mg)

Fe I @ 382.44 nm Mg II @ 448.11 nm K II @ 361.85 nm K I @ 766.49 nm

382.35−382.51 447.99−448.37 361.79−362.03 765.52−768.51

α = −1.409 β = −6.264 γ = −0.676 δ = −0.915

(K/Mg)

K II @ 361.85 nm Mg I @ 516.73 nm K II @ 413.47 nm K I @ 693.88 nm

361.79−362.03 516.35−516.97 413.38−413.55 693.52−694.12

α = 16.043 β = 19.183 γ = 1.091 δ = −1.658

(Na/Mg)

Na I @ 568.82 nm Mg I @ 518.36 nm K II @ 310.50 nm K I @ 404.72 nm

568.54−569.34 517.92−518.79 310.43−310.55 404.63−404.83

α = 2.610 β = 0.292 γ = 5.571 δ = −0.423

(Si/Mg)

Si II @ 385.60 nm Mg II @ 448.11 nm K II @ 426.34 nm K I @ 693.88 nm

385.46−385.74 447.99−448.37 426.23−426.39 693.52−694.12

α = −10.751 β = −6.240 γ = 2.748 δ = −2.345

(Ti/Mg)

Ti I @ 363.55 nm Mg I @ 516.73 nm K I @ 485.61 nm K I @ 486.98 nm

363.45−363.64 516.35−516.97 485.47−485.69 486.93−487.07

α = −39.675 β = −23.930 γ = 3.601 δ = −1.976

expression used more crucial, as the concentration of the more abundant neutral species has to be calculated from the concentration of ionized species and will presumably lead to a less stable calibration. The K II line at 361.85 nm, seen to be used twice in Table 2, appears particularly dubious, however. This line is not reported as a strong line in the NIST library,16 and it is therefore more likely that this line is the fairly strong Fe I line at 361.84 nm. The use of this line thus appears to be the result of a misclassification, and an apparent covariance between Fe and K in the samples, leading to the Fe-line providing the best description of the K-content, similar to what has been observed in other work.37 The results presented are still valid for these samples, but clearly the K-calibration would be severely compromised for samples without the same apparent covariance between K and Fe. Increasing the number of calibration samples, and the elemental variation (without covariance between elements) in these, is seen as the best way to alleviate the problem of using lines from different elements and will also further improve the overall robustness of the model. Furthermore, implementing more sophisticated techniques, like those presented by Amato et al.,38 for recognition of spectral lines should also lead to fewer misclassified lines. 1498



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(8) Herrera, K. K.; Tognoni, E.; Gornushkin, I. B.; Omenetto, N.; Smith, B. W.; Winefordner, J. D. J. Anal. At. Spectrom. 2009, 24, 426− 438. (9) Pandhija, S.; Rai, N. K.; Rai, A. K.; Thakur, S. N. Appl. Phys. B: Laser Opt. 2010, 98, 231. (10) Gornushkin, S. I.; Gornushkin, I. B.; Anzano, J. M.; Smith, B. W.; Winefordner, J. D. Appl. Spectrosc. 2002, 56, 433. (11) Tognoni, E.; Palleschi, V.; Corsi, M.; Christoforetti, G.; Omenetto, N.; Gornushkin, I.; Smith, B. W.; Winefordner, J. D. In Laser-Induced Breakdown Spectroscopy (LIBS): Fundamentals and Applications; Miziolek, A. W., Palleschi, V., Schechter, I., Eds.; Cambridge University Press: Cambridge, U.K., 2006; Chapter 3. (12) Praher, B.; Palleschi, V.; Viskup, R.; Heitz, J.; Pedarnig, J. D. Spectrochim. Acta, Part B: At. Spectrosc. 2010, 65, 671−679. (13) Capitelli, M.; Casavola, A.; Colonna, G.; de Giacomo, A. Spectrochim. Acta, Part B: At. Spectrosc. 2004, 59, 271−289. (14) Senesi, G. S.; Dell’Aglio, M.; Gaudiuso, R.; de Giacomo, A.; Zaccone, C.; de Pascale, O.; Miano, T. M.; Capitelli, M. Environ. Res. 2009, 109, 413−420. (15) Cremers, D. A.; Radziemski, L. J. In Laser-Induced Breakdown Spectroscopy (LIBS): Fundamentals and Applications; Miziolek, A. W., Palleschi, V., Schechter, I., Eds.; Cambridge University Press: Cambridge, U.K., 2006; Chapter 1. (16) NIST Spectra Database. National Institute of Standards and Technology, http://www.nist.gov/physlab/data/asd.cfm (17) Hahn, D. W.; Omenetto, N. Appl. Spectrosc. 2010, 64, 335−366. (18) Cristoforetti, G.; de Giacomo, A.; Dell’Aglio, M.; Legnaioli, S.; Tognoni, E.; Palleschi, V.; Omenetto, N. Spectrochim. Acta, Part B: At. Spectrosc. 2010, 65, 86−95. (19) Lazic, V.; Barbini, R.; Colao, F.; Fantoni, R.; Palucci, A. Spectrochim. Acta, Part B: At. Spectrosc. 2001, 56, 807−820. (20) Duchene, S.; Detalle, V.; Bruder, R.; Sirven, J. B. Curr. Anal. Chem. 2010, 6, 60−65. (21) Bousquet, B.; Sirven, J. B.; Canioni, L. Spectrochim. Acta, Part B: At. Spectrosc. 2007, 62, 1582−1589. (22) U.S. Geological Survey Geochemical Reference Materials and Certificates. United States Geological Survey, 2010; http://minerals.cr. usgs.gov/geo_chem_stand/powdered_RM.html (23) National Institute of Standards and Technology. NIST Standard reference materials, 2009; http://www.nist.gov/ts/msd/srm/ (24) The Mathworks, Inc. MatLab R2010a, http://www.mathworks. se/ (25) The Mathworks, Inc., Global Optimization Toolbox, http:// www.mathworks.com/help/pdf_doc/gads/gads_tb.pdf (26) Dell’Aglio, M.; Gaudiuso, R.; Senesi, G. S.; De Giacomo, A.; Zaccone, C.; Miano, T. M.; De Pascale, O. J. Environ. Monitor. 2011, 13, 1422−1426. (27) Essington, M. E.; Melnichenko, G. V.; Stewart, M. A.; Hull, R. A. Soil Sci. Soc. Am. J. 2009, 73, 1469−1478. (28) Galiová, M.; Kaiser, J.; Novotný, K.; Hartl, M.; Kizek, R.; Babula, P. Microsc. Res. Tech. 2011, 74, 845−852. (29) Batista Braga, J. W.; Trevizan, L. C.; Nunes, L. C.; Rufini, I. A.; Santos, D. J.; Krug, F. J. Spectrochim. Acta, Part B: At. Spectrosc. 2010, 65, 66−74. (30) Wiens, R. C.; et al. ChemCam Science Objectives for the Mars Science Laboratory (MSL) Rover. In 36th Annual Lunar and Planetary Science Conference, League City, TX, March 14−18, 2005; Presentation 1580. (31) Dyar, M. D.; Tucker, J.; Humphries, S.; Clegg, S. M.; Wiens, R. C.; Lane, M. D. Spectrochim. Acta, Part B: At. Spectrosc. 2010, 66, 39− 56. (32) Maurice, S.; et al. Space Sci. Rev. 2012, 170, 95−166. (33) Tucker, J.; Dyar, M. D.; Schaefer, M.; Clegg, S. Chem. Geol. 2010, 277, 137−148. (34) Noll, R.; Sturm, V.; Aydin, Ü .; Eilers, D.; Gehlen, C.; Höhne, M.; Lamott, A.; Makowe, J.; Vrenegor, J. Spectrochim. Acta, Part B: At. Spectrosc. 2008, 63, 1159−1166. (35) Barbafieri, M.; Pini, R.; Ciucci, A.; Tassi, E. Chem. Ecol. 2011, 27, 161−169.

It would be interesting to include nongeological samples as well in the calibration of this OC-LIBS procedure, as nongeological samples would allow for an enlargement of the elemental variation, without the potential geological correlations between the different elements. Whether or not this OCLIBS procedure is capable of handling these even larger matrix differences will be interesting to see.

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CONCLUSION With the choices and approximations described, it is deduced that the ratio between two elements can be described by the equation ⎞ C1 I ⎛ α = 1⎜ + γ ⎟τ δ C2 I2 ⎝ ln(τ ) − β ⎠

in which C1 and C2 are the concentrations of the two elements and I1 and I2 are the areas underneath two spectral lines originating from the same elements; τ is the area ratio of two spectral lines from a third element and is used for temperature corrections; and the constants α, β, γ, and δ are determined based on reference samples using a simulated annealing routine. Using this optimized calibration (OC-LIBS) procedure on 22 geological powdered reference samples originating from a large range of different rock types, correlations with R2 ≥ 0.91 are observed between the predicted and reference values for the elements Al, Ca, Fe, K, Na, Si, and Ti when rated to Mg. Considering that an absolute minimum of sample preparation was used prior to the measurements and negligible processing time is required to predict the elemental ratios of new samples, these are very encouraging results. This OC-LIBS procedure thus shows substantial promise for use in realtime in situ elemental analysis applications in which the LIBS technique shows its greatest advantages compared to other analytical methods.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors would like to thank the company Scandinavian Highlands Holding A/S for providing access to the LIBSequipment used in this article.

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REFERENCES

(1) Ciucci, A.; Corsi, M.; Palleschi, V.; Rastelli, S.; Salvetti, A.; Tognoni, E. Appl. Spectrosc. 1999, 53, 960−964. (2) Cremers, D. A.; Radziemski, L. J. Handbook of Laser-Induced Breakdown Spectroscopy; John Wiley & Sons, Ltd.: Chichester, U.K., 2006. (3) Miziolek, A. W., Palleschi, V., Schechter, I., Eds. Laser-Induced Breakdown Spectroscopy (LIBS): Fundamentals and Applications; Cambridge University Press: Cambridge, U.K., 2006. (4) Haider, A. F. M. Y.; Wahadoszamen, M.; Sadat, M. E.; Abedin, K. M.; Talukder, A. I. Opt. Laser Technol. 2010, 42, 969−974. (5) Sallé, B.; Lacour, J.; Mauchien, P.; Fichet, P.; Maurice, S.; Manhés, G. Spectrochim. Acta, Part B: At. Spectrosc. 2006, 61, 301−313. (6) Sirven, J.-B.; Sallé, B.; Mauchien, P.; Lacour, J.-L.; Maurice, S.; Manhès, G. J. Anal. At. Spectrom. 2007, 22, 1471. (7) Clegg, S. M.; Sklute, E.; Dyar, M. D.; Barefield, J. E.; Wiens, R. C. Spectrochim. Acta, Part B: At. Spectrosc. 2009, 64, 79−88. 1499



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An Optimized Calibration Procedure for Determining Elemental Ratios

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