A New Method for Determination of Self-Absorption ... - SAGE Journals

A New Method for Determination of Self-Absorption Coefficients of Emission Lines in Laser-Induced Breakdown Spectroscopy Experiments ´ N´ CTOR O. DI ROCCO, HUGO M. SOBRAL, MAYO VILLAGRA FAUSTO O. BREDICE,* HE MUNIZ, and VINCENZO PALLESCHI Centro de Investigaciones Opticas, P.O.Box 124 C. P.1900 La Plata, Argentina (F.O.B.); Instituto de Fı´sica Arroyo Seco (IFAS), Universidad Nacional del Centro, Pinto 399 Tandil, Argentina (H.O.D.R.); Laboratorio de Fotofı´sica, Centro de Ciencias Aplicadas y Desarrollo Tecnolo´gico, Universidad Nacional Autońoma de Me´xico, Apartado Postal 70-186, Me´xico D.F. 04510, Me´xico (H.M.S., M.V.-M.); and Applied Laser Spectroscopy Laboratory, IPCF-CNR, Area della Ricerca di Pisa, Via G. Moruzzi 1, 56124 Pisa, Italia (V.P.)

In this paper we present a new method for determining the self-absorption coefficients of emission lines in laser-induced breakdown spectroscopy (LIBS) experiments. With respect to other methods already present in the literature, the proposed approach has the advantage of not requiring, providing some conditions are fulfilled, any knowledge of the plasma parameters such as temperature and electron number density and of the emission line spectral coefficients such as transition probability. An example of the application of the approach is given for emission lines measured at different delay times after laser ablation of a silver target. Index Headings: Self-absorption; Optical spectroscopy; Laser-produced plasmas; Laser-induced breakdown spectroscopy; LIBS; Atomic parameters.

INTRODUCTION The problem of determining line shape and intensity of a given spectral line emitted by a plasma has been widely discussed in recent literature.1–10 It was shown by El-Sherbini et al.7 that the self-absorption coefficient can be defined as the ratio of the actual intensity of the emission line at its maximum over the value np0(k0) obtained by extrapolating the curve of growth in the optically thin regime of the same emitter’s number density of the actual measurement, i.e.: 1 ekl ð1Þ SA ¼ kl where k is the absorption coefficient, which depends, among other things, exponentially on the ratio between the energy of the lower level of the transition and the plasma temperature (times the Boltzmann constant). In Ref. 7 it was demonstrated that the integral intensity of the self-absorbed emission line I over the non-self-absorbed one I0 scales as: I ¼ SAb ð2Þ I0 with b ¼ 0.46, while the actual full width at half-maximum (FWHM) of the line Dk over the non-self-absorbed one, Dk0, scales as: Dk ¼ SA1b ð3Þ Dk0 The latter relationship was used in Ref. 7 as a way of determining the SA coefficient by measuring the linewidth Received 4 September 2009; accepted 4 December 2009. * Author to whom correspondence should be sent. E-mail: faustob@ciop. unlp.edu.ar.

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from the LIBS spectrum and calculating Dk0 from the knowledge of the Stark broadening of the line. For that purpose, the electron number density and temperature must be measured and the Stark coefficient at the same temperature must be known (or calculated according to Refs. 11 and 12). This method, although quick and powerful, has some drawbacks that in many cases may limit its application. First of all, the measurement of the width of the emission line of interest is not trivial, in most cases, because of the unavoidable presence of the instrumental broadening, due to the finite spectral resolution of the spectrometer. When the line broadening due to self-absorption is comparable with the instrumental width, the errors in calculation of the selfabsorption coefficient may grow exponentially. Moreover, for the determination of the non-self-absorbed linewidth Dk0, the knowledge of the electron number density and of the Stark coefficient(s) at the plasma temperature must be available. We will demonstrate that, under the same hypotheses used in Ref. 7, the self-absorption coefficient of a given line may be obtained without having recourse to any knowledge of the plasma parameters such as temperature and electron number density and of the emission line spectral coefficients such as transition probability. Moreover, the method presented may be applied even when the instrumental broadening is of the same order of magnitude (or larger) than the measured emission line FWHM.

PRINCIPLES OF THE METHOD The method illustrated here is based on the time variation of spectral line intensities in laser-induced plasmas.9 According to the well-known Boltzmann plot method, each emission line of a given species may be represented as a point in a twodimensional space defined by the upper-level energy Ek (xaxis) and the parameter ln(I/gkAki) (y-axis). In the local thermal equilibrium approximation and in the absence of self-absorption, the points corresponding to the emission lines of the same species would arrange themselves on a straight line in the Boltzmann plot whose slope is m ¼1/ KT, where T is the plasma temperature and K is the Boltzmann constant. The evaluation of the self-absorption coefficient of a given line seems straightforward. In fact, the measured integral intensity of the line must be corrected, using its self-absorption coefficient, for bringing back the point on its theoretical position on the Boltzmann plot. In other words:

0003-7028/10/6403-0320$2.00/0 Ó 2010 Society for Applied Spectroscopy

APPLIED SPECTROSCOPY

Ek2 ¼

1 1 1 1 Ek1 1 1 1 Tb Ta Tb Ta ¼ K Tb Ta KðEk2 Ek1 Þ ð5Þ

In the limit case of Ek1 ¼ 0, we would have ln[I01(Ta)/I01(Tb)] ¼ 0 and thus, considering a line with upper energy Ek2 ¼ Ek: I0 ðTa Þ ln 1 1 1 I0 ðTb Þ ¼ ð6Þ K Tb Ta Ek

FIG. 1. Linear plot of the logarithm of the intensity ratios vs. Ek. The slope of the curve is tan(a), defined in Eq. 5.

"

# IðSAÞb Ek N0 þ ln ln ¼ gk Aki KT UðTÞ

ð4Þ

In principle, Eq. 4 should also contain an intercept, which would take into account the changes in N0 with temperature (due, for example, to temperature-dependent chemical reactions) and in the partition functions. As long as these effects are negligible, the Boltzmann equation can be safely written as in Eq. 4. According to this approach, the parameters of the best fitting linear function on the Boltzmann plot should be determined independently (fitting the intensities of non-self-absorbed lines, for example); the logarithm term at the right-hand side (RHS) of the equation, in particular, may be affected by large errors since it corresponds to the extrapolation at zero energy of the best fitting line on the Boltzmann plot. Moreover, for calculating the SA coefficient the transition probability Aki must be known with reasonable accuracy, since the indetermination on Aki would reflect proportionally on the indetermination on the measured SAb. In many cases, the Aki of the transitions are known with accuracies not better than 20% or even worse (it is not unusual to find transition probabilities known with uncertainties higher than 50%); therefore, the direct evaluation of SA from the Boltzmann plot may be plagued by very large errors. For overcoming these problems, we propose considering the logarithm ln[I(Ta)/I(Tb)] of the ratio between the integral intensities of the same emission line at two different temperatures Ta and Tb. This logarithm can be plotted as a function of Ek, and again in the same approximation of local thermal equilibrium and in the absence of self-absorption the points corresponding to different emission lines of the same species would align along a straight line forming an angle a with the x-axis (see Fig. 1). The slope of this line, tan(a), can be expressed as: I02 ðTa Þ I01 ðTa Þ ln ln I02 ðTb Þ I01 ðTb Þ tanðaÞ ¼ ðEk2 Ek1 Þ

i.e., this straight line has zero intercept. If an emission line is self-absorbed, the logarithm of the ratio of the measured intensities must be corrected taking into account the self-absorption coefficients at the temperatures considered: ( ) IðTa Þ ðSAÞðTb Þ b ln IðTb Þ ðSAÞðTa Þ 1 1 1 ¼ ð7Þ K Tb Ta Ek The absorption coefficient of the plasma depends on the difference of population between the higher and lower level of the transition. If the temperature is high enough, these populations tend to equilibrate and absorption is compensated by stimulated emission. In practice, the absorption coefficient of the plasma depends exponentially on the ratio between the energy of the lower level of the transition and the plasma temperature (times the Boltzmann constant). If the temperature Ta is high enough, the plasma becomes optically thin for this transition (i.e., kl 1) and thus [SA(Ta)]b ; 1. Therefore, if the approximation of the optically thin plasma holds, at T ¼ Ta, for the transition considered, we have: IðTa Þ ½ðSAÞðTb Þb ln 1 1 1 IðTb Þ ’ ð8Þ K Tb Ta Ek from which the calculation of SA is straightforward. Since the energies of the levels of the transition are known, it is quite simple to assess whether the typical plasma temperatures corresponding to the early stages of the plasma expansion are compatible with an optically thin plasma or not. We would like to stress that the only action required for validating Eq. 8 is the check that the approximation kl 1 holds for the temperature Ta; this check is, of course, much simpler than the actual measurement of the self-absorption coefficient. Equation 8 is similar to Eq. 4 but it does not require the knowledge of the Aki of the transition nor the extrapolation of the best fitting line on the Boltzmann plot at zero energy; therefore, the only information needed would be the measured intensities of the spectral line at two different temperatures and the knowledge of Ta and Tb. These latter temperatures can be measured using non-self-absorbed lines of the same species or, making further use of the approximation of local thermal equilibrium, even using non-self-absorbed lines emitted by other species present in the plasma. However, if at least one non-self-absorbed line of the same species is available, Eq. 8 can be further simplified using the relation


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TABLE I. Spectroscopic parameters of the lines studied in this paper. Wavelength (nm)

Aki (s1) 3 108

Ek (eV)

Ei (eV)

gk

gi

1.38 1.22 5.26 3 102 5.86 3 101 7.88 3 101

3.778 3.664 6.434 6.044 6.046

0 0 3.664 3.664 3.778

2 2 2 2 4

4 2 2 4 6

328.068 338.289 447.604 520.908 546.550

ln

IR ðTa Þ 1 1 1 IR ðTb Þ ¼ K Tb Ta EkR

ð9Þ

where IR(Ta) and IR(Tb) are the integrated intensities of this ‘‘reference’’ line at the temperatures Ta and Tb and EkR is the energy of the upper level of the corresponding transition. Substituting Eq. 9 in Eq. 8 gives: IðTa Þ IR ðTa Þ ½ðSAÞðTb Þb ln ln IðTb Þ IR ðTb Þ ’ ð10Þ Ek EkR from which we obtain: SAðTb Þ ¼

IðTb Þ IðTa Þ

b1

IR ðTa Þ IR ðTb Þ

bEEk

kR

ð11Þ

Thus, for determining the self-absorption coefficients of the lines of interest without knowing specifically the two temperatures Ta and Tb at which the two intensities I(Ta) and I(Tb) are measured, we need to measure the intensities IR(Ta) and IR(Tb) of a non-self-absorbed reference line of the same species. The actual validity of the assumption of the optically thin plasma at the wavelength of this reference line can be checked by applying one of the many methods proposed in the literature,8,10–12 or even by direct measurement using the mirror approach recently proposed by Moon et al., in Ref. 14, which, in its simplest implementation, does not require any assumption. However, we would like to stress again that the only information needed for validating Eq. 11 is just the assessment that the approximation kl 1 holds for the line considered at the two temperatures Ta and Tb, not a precise measurement of the self-absorption coefficient.

APPLICATION The method discussed in the previous section was applied to the determination of the temporal evolution of the selfabsorption coefficients of the neutral silver emission lines Ag I at 328.068, 338.289, 447.604, and 546.550 nm (using the Ag I line at 520.908 nm as reference). The spectroscopic parameters of the lines studied are reported in Table I. The plasma was obtained by focusing on the samples the beam of a pulsed Nd:YAG laser (Surelite I from Continuum), operating at k ¼ 1.06 lm. The laser delivers pulses of 25 mJ at a 3 Hz repetition rate, with a 7 ns pulse FWHM. The timeresolved spectra of a pure (99.9%) silver target were acquired using a 50 cm monochromator (Spectra Pro 500i from Roper Scientific, diffraction grating of 1200 lines/mm), with a slit width of 10 lm. A 20 cm focal lens was used to form the image of the plasma onto the entrance slit of the spectrometer.

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FIG. 2. K tan(a) as a function of the delay time for the spectral lines considered. (Open diamonds) Ag I line at 328.068 nm; (solid inverted triangles) Ag I line at 338.289 nm; (open triangles) Ag I line at 447.604 nm; (solid circles) Ag I line at 520.908 nm; and (solid squares) Ag I line at 546.550 nm.

In typical laser-induced plasmas, the temporal variations of electron temperature and density are very steep; therefore, large temporal acquisition windows cannot be used, since in this case very different values of these parameters would be integrated on the detector, and the resulting intensities of the emission lines would be affected by this integration. 13 In our experimental configuration, the plasma spectral emission was acquired by an intensified charge-coupled device (ICCD) camera (PIMAX:1024 UV from Princeton Instruments) with a fast 20 ns gate, starting from 500 ns up to 10000 ns after the arrival of the laser pulse, with a step of 500 ns. In order to improve the signal-to-noise ratio, fifty measurements, taken in the same experimental conditions, were averaged. The integrated intensities of the lines of interest were obtained by fitting the line shape with a Voigt profile.7 Figure 2 shows the time dependence of the parameter K*tan(a) calculated for the lines of interest and the ‘‘reference’’ line as a function of the time delay. All the points are calculated taking Ta as the temperature corresponding to the delay td ¼ 500 ns and Tb as the one at the time delay considered (the actual value of the temperatures Ta and Tb is not needed for the calculation); according to Eq. 6 (or Eq. 9) for all the non-self-absorbed lines, this parameter should be the same and equal to [(1/Tb) (1/Ta)]. Deviations from this behavior indicate the presence of self-absorption, whose amount can be calculated using Eq. 11, if a non-self-absorbed line is available. The absorption coefficient of the plasma for a given transition at the peak wavelength k0 is given by k¼

1 Ek k40 Aki gk Ei N0 e KT eKT Dk 8pc UðTÞ

ð12Þ

where N0 is the emitter’s number density of the emitting species, Aki is the transition probability between the two levels Ek and Ei, gk is the degeneracy of the upper level, U(T) is the partition function of the species at temperature T, and Dk is the FWHM of the transition, assuming a Lorentzian line shape.11 According to Eq. 12, among the lines considered, the best

induced breakdown spectroscopy experiments. With respect to a similar method recently proposed in the literature by one of the authors, providing some conditions are fulfilled, the present approach does not require any knowledge of the plasma parameters and of the emission line spectral coefficients. Since only the measurement of line intensities is involved, the present approach can be applied in experiments performed using low spectral resolution spectrometers and is, therefore, particularly interesting for development of low-cost LIBS instrumentation for quantitative analysis of materials. ACKNOWLEDGMENTS This work was supported by the National Autonomous University of Me´xico GAPAUNAM and the National Council of Science and Technology of Me´xico CONACyT. The authors also want to thank the Comisioń de Investigaciones Cientı´ficas de la Provincia de Buenos Aires and the Consejo Nacional de Investigaciones Cientı´ficas y Tećnicas, CONICET where two of the authors (FB) and (HODR) work as researchers, respectively. FIG. 3. SA(t)/SA(t0) as a function of the delay time for the spectral lines considered. (Solid inverted triangles) Ag I line at 328.068 nm; (open triangles) Ag I line at 338.289 nm; (open circles) Ag I line at 447.604 nm; and (solid squares) Ag I line at 546.550 nm.

choice for the non-self-absorbed reference line would thus be the line at 520.908 nm. As an order of magnitude, considering typical plasma values of N0 ffi 1018, T ffi 1 eV, Dk ffi 0.1 nm, and l ffi 1 mm, the kl coefficient for the Ag I line considered results on the order of 0.05, i.e., 1 as required for the correct application of the method proposed. The self-absorption coefficients of the four Ag I lines at 328.068, 338.289, 447.604, and 546.550 nm, calculated using Eq. 11, are reported in Fig. 3. The Ag I lines at 328.068 and 338.29 nm have a self-absorption coefficients that goes very quickly to zero (corresponding to very high self-absorption) as the time delay increases and the plasma temperature, correspondingly, decreases, as expected for a resonant line (Ei ¼ 0). On the contrary, the self-absorption coefficient of the Ag I lines at 447.604 and 546.550 nm (non-resonant lines) decreases more slowly with the temperature decrease. However, for the 546.550 nm line, the self-absorption coefficient is not negligible (SA ; 0.6/0.7) even at relatively short time delays (less than 1 ls).

CONCLUSION We have presented a new method for calculating the selfabsorption coefficient of spectral lines obtained from laser-

1. I. B. Gornushkin, J. M. Anzano, L. A. King, B. W. Smith, N. Omenetto, and J. D. Winefordner, Spectrochim. Acta Part B 54, 491 (1999). 2. C. Aragoń, J. Bengoechea, and J. A. Aguilera, Spectrochim. Acta, Part B 56, 619 (2001). 3. H. Amamou, A. Bois, B. Ferhat, R. Redon, B. Rossetto, and M. Ripert, J. Quant. Spectrosc. Radiat. Transfer 77, 365 (2003). 4. I. B. Gornushkin, C. L. Stevenson, B. W. Smith, N. Omenetto, and J.D. Winefordner, Spectrochim. Acta, Part B 56, 1769 (2001). 5. J. A. Aguilera, J. Bengoechea, and C. Aragoń, Spectrochim. Acta, Part B 58, 221 (2003). 6. D. Bulajic, M. Corsi, G. Cristoforetti, S. Legnaioli, V. Palleschi, A. Salvetti, and E. Tognoni, Spectrochim. Acta, Part B 57, 339 (2002). 7. A. M. El Sherbini, Th. M. El Sherbini, H. Hegazy, G. Cristoforetti, S. Legnaioli, V. Palleschi, L. Pardini, A. Salvetti, and E. Tognoni, Spectrochim. Acta, Part B 60, 1573 (2005). 8. H. Amamou, A. Bois, B. Ferhat, R. Redon, B. Rossetto, and P. Matheron, J. Quant. Spectrosc. Radiat. Transfer 75, 747 (2002). 9. V. Bulatov and I. Schechter, ‘‘Plasma Morphology’’, in Laser-Induced Breakdown Spectroscopy (LIBS) Fundamentals and Applications, A. Miziolek, V. Palleschi, and I. Schechter, Eds. (Cambridge University Press, New York, 2006). 10. F. Bredice, F. O. Borges, H. Sobral, M. Villagran-Muniz, H. O. Di Rocco, G. Cristoforetti, S. Legnaioli, V. Palleschi, L. Pardini, A. Salvetti, and E. Tognoni, Spectrochim. Acta, Part B 61, 1294 (2006). 11. A. El Sherbini, Th. El Sherbini, H. Hegazy, G. Cristoforetti, S. Legnaioli, L. Pardini, V. Palleschi, A. Salvetti, and E. Tognoni, Spectrosc. Lett. 40, 643 (2007). 12. F. Bredice, F. O. Borges, H. Sobral, M. Villagran-Muniz, H. O. Di Rocco, G. Cristoforetti, S. Legnaioli, V. Palleschi, A. Salvetti, and E. Tognoni, Spectrochim. Acta, Part B 62, 1237 (2007). 13. E. Tognoni, V. Palleschi, M. Corsi, and G. Cristoforetti, Spectrochim. Acta, Part B 57, 7 (2002). 14. H.-Y. Moon, K. K. Herrera, N. Omenetto, B. W. Smith, and J. D. Winefordner, Spectrochim. Acta, Part B 64, 702 (2009).


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