Local thermodynamic equilibrium modeling of ionization of impurities in argon inductively coupled plasma

Spectrochimica Acta Part B 65 (2010) 15–23

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Spectrochimica Acta Part B j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / s a b

Local thermodynamic equilibrium modeling of ionization of impurities in argon inductively coupled plasma Petras Serapinas ⁎, Julius Šalkauskas, Žilvinas Ežerinskis, Artūras Acus Institute of Theoretical Physics and Astronomy, Vilnius University, A. Goštauto 12, 01108 Vilnius, Lithuania

a r t i c l e

i n f o

Article history: Received 17 June 2009 Accepted 27 October 2009 Available online 2 November 2009 Keywords: Inductively coupled plasma Local thermodynamic equilibrium Mass spectrometry Ionization Elemental analysis Matrix effect

a b s t r a c t Essentially higher ionization degree of small concentrations of elements in inductively coupled plasma in comparison to the ionization of pure elements is emphasized. This conclusion is used to determine the relative dependence of the sensitivity of the inductively coupled plasma mass spectrometer on the atomic mass. The possibility of evaluation of the ionization temperature and electron density from mass spectrometric signals is proposed. Temperatures about 7000 K and 8000 K were obtained from the ionization ratio dependences on ionization potentials. Electron densities of the order of magnitude 1015 cm− 3, in excess to the local thermodynamic equilibrium values, follow from the application of the Saha equation to the measurement results and indicate the recombining character of the plasma in the mass spectrometer measurement region. Effects due to additional ionization from matrix were discussed. The effect is largest on minor abundant ionization state components. Matrix effect is restricted to some temperature interval, which depends on the whole matrix composition and the plasma state. The results show that the local thermodynamic equilibrium modeling, if adequately matching the sample composition, can be useful as a quantitative basis for both description of the plasma state and indication of the character of the nonequilibrium effects. © 2009 Elsevier B.V. All rights reserved.

1. Introduction Number of experimental studies of the analytical performance of the inductively coupled plasma (ICP) spectrometry constantly increases, especially in the recent years [1–14]. Robustness of the analytical techniques, possibilities to predict the matrix effects, accuracy of the analytical measurements are of permanent interest. In spite that analysis of different aspects of applicability of local thermodynamic equilibrium (LTE), partial-LTE, non-LTE approaches lasts from the early studies up to now [15–26] description of the plasma state in LTE related terms of the particle densities and temperatures remains the base for most models and analysis of the experimental data. In more detail, understanding of the degree of ionization as determined by temperature and ionization potentials is routine. Addition of easy to ionize elements is used for reduction of the ionization degree of analytes. However increased ionization degree as a consequence of decreasing the electron concentration, characteristic to modern instruments mainly designed and appropriate for analysis of low concentrations, is much less emphasized. In general, ionization in analytical LTE plasmas ought to be regarded as integrated result of Saha equations for each element or chemical compound in common with the balance equations and the ideal gas law. Especially at increased ionization degree quantitative effects of those relations

⁎ Corresponding author. E-mail address: serapinas@pfi.lt (P. Serapinas). 0584-8547/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.sab.2009.10.008

become comparable, the results can essentially differ from expectations from a single Saha equation only and are much less straightforward. Multi-element LTE study of ionization of the analytes and matrix in dependence on their densities and temperature, including matrix effect on the analytes (Sections 2 and 3), and test of the applicability of the model (Section 4) were the main goals of the present paper. On the other hand, accuracy and variety of the experimental studies increase. At the same time, in parallel to collision-radiative and similar balance models [16,20,27–31] often selections of elementary processes invoked for explanation of the experimental results are lacking corresponding quantitative arguments. Even if deviation from the equilibrium state is really important some part of the distribution functions can reach quasi equilibrium and the plasma can be analyzed in LTE terms. Character of deviations from the LTE usually can be detectable and formulated in the local (in energy or geometrical space) thermodynamic equilibrium terms also, for example: correspondence or difference between electron, excitation, ionization temperatures, correspondence or difference of the particle densities from the equilibrium ones, and so on. In such a manner one can expect that wider implementation of the LTE based quantitative data can aid understanding of the possible role of particular LTE or nonequilibrium phenomena, their influence to analytical applications. Search for means for testing applicability of the LTE model to the real ICP plasma was in between the goal of the study. Studies in the two directions mentioned above are strongly interconnected. Conclusion on almost total ionization of the low

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ionization potential analytes (ionization potentials about or less than 7 eV) was used to calibrate the mass spectrometric Element 2 measurement data. It enabled wider application of the experimental results, elaboration of additional means for plasma diagnostics and testing the accuracy of the model. Typical for recombination dominated plasma increment of the electron density (in comparison to the LTE results) was detected in the MS measurement region. In spite that because of such deviation from the equilibrium state the main conclusions on the character of the matrix effect as obtained from the approximate model ought to be valid quantitative results can be some different. Need for higher accuracy of the measurements, improvement of the model, close matching of the chemical composition of samples in calculations, integrated measurements of both matrix and analyte ionization and excitation states for more detailed analysis of the matrix effects is emphasized. 2. Theoretical Saha equation, in analogy to the law of active masses, is the main quantitative relationship between the densities of particles in the neighbour ionization states, i and i + 1, of the element and electrons. Because of the moderate temperatures characteristic to ICP, about or lower than 10 000 K, only atoms, the singly and double ionized ions are found at appreciable concentrations. Thus i takes values from 1 to 2 in relation to i + 1 equal to 2 and 3 (further i is used as an index to the physical quantity or as Latin numerals traditionally accepted for identification of the ionization stage in spectroscopy). Namely, for each element and every pair of the neighbour ionization stages, Ni

+ 1 Ne

Ni

=

ð2πme kTÞ3 = 2 2Qi + 1 ðTÞ −ðEi −ΔEi Þ = kT e : Qi ðTÞ h3

ð1Þ

Here T is temperature, me is the electron mass, h and k are Planck and Boltzman constants correspondingly, Q are the partition functions, E is the ionization potential of the lower ionization state, and ΔE is the lowering of the ionization energy in the plasma. Various ideas are used to estimate the ΔE [32,33], between those the Debye shielding approach: 3

1=2

ΔEi = 2ðzi + 1Þe ðπ =kTÞ


1 = 2 2 ∑ Ni zi i

ð2Þ

where e is the electronic charge, zi is the elementary charge of the i-th ion (zi = i− 1). Importance of temperature and ionization energies is evident from the equations above. As electrons are produced in any ionization step, the equations for each element or compound and all ionization stages must be regarded simultaneously. In addition, balance of the total number of particles, N, and dependence of the total particle number density on temperature must be included. Asymptotic approach to the true electron density value Ne can be used. Correspondence of the electron density to the total concentration of all ions and sums of the densities of ions to the given gas composition for each element should be applied as accuracy tests. If the values of the constants are introduced Saha equation can be rewritten in the form convenient for diagnostic purposes: ni

+ 1

ni

=

4:83⋅1015 T 3 = 2 −11605 ðE −ΔEi Þ e T i : Ne

ð3Þ

 Here ni = Ni Q is population of a single statistical ionization state. i Particle densities in Eq. (3) are expressed in cm− 3, T is in K and ionization energy is in eV. Both the index of the exponent and the coefficient before the exponent, number of the free electronic states attributed to one occupied state, can be determined from measure-

ments enabling simultaneous determination of the plasma electron temperature and density. For simplification let us take the Saha equation for atoms and the singly ionized ions of a single element in the form Ne ·N2 = N1 = K:

ð4Þ

At low temperatures Ne = N2 ≪ N1 ≈ N, with pffiffiffiffiffiffiffi N as the total concentration of analyte particles. Then N2 = NK . Dependence of the density of neutrals on the total concentration must be faster than linear. Tendencies to such square root ion analytical signal dependence on analyte concentrations or faster than linear atomic line calibration curves hardly have been observed in modern analytical spectrometry. In more general case, including higher temperatures (for single element, as before), N =N1 +N2, and N2/(N−N2) =K/Ne. The degree of ionization, N2/N =K/(Ne +K). It means that ionization is total, N2/N = 1, if Ne ≪K independent on the real values of the ionization potentials (chemical species) and temperatures. Addition of the elements with low ionization potentials should increase the total electron density and reduce the ionization degree of any individual component. In a similar way decrease of the absolute analyte (and, consequently, electron) concentration at constant temperature must increase the ionization degree. The mechanism of the dependence of the ionization degree on the absolute electron densities is obvious: at lower electron densities the frequency of collisions for recombination decreases faster than for ionization resulting in higher ionization degree. In modern analytical plasma sources plasma is created in foreign to the sample, usually inert gas, environment and only small quantities of the sample are introduced enabling to make almost no changes of the plasma state. Just the high ionization potential of the inert atmosphere, small sample quantities introduced and small concentrations of the analytes facilitate low electron densities, cumulatively increasing the ionization degree of the analyte and potential of the optical and mass spectrometric determination of ions from small quantities of a sample. Naturally, when sample throughput is to be increased, caution is needed on when the sample starts to change the plasma conditions. If it takes place, the absolute values of concentrations of all species must be specified for analysis of the data or comparison between experiments. Decisive role of the environment (relative to the analyte) in determining the electron density is the reason why the simplest form of the single Saha equation (Eq. (4)) at low temperatures hardly can be observed. As an example, comparison of the fraction of the singly ionized ions relative to the total number of particles of the element (degree of the first ionization, N2/N) as dependences on the ionization potential of the neutral atoms in gas mixture characteristic to argon ICP (Ar 97%, H 2%, O 1% and 10− 7% of the analyte element, electron density below 1015 cm− 3, top) and pure monoelemental substances (bottom, electron densities up to 1017 cm− 3 and higher) at atmospheric pressure and temperature 7000 K are presented in Fig. 1. Essential increase of ionization in ICP is evident, especially for elements with ionization potentials from 6 eV to about 8 eV. Some decrease of the first ionization of the elements with low ionization potentials (Ba, Sr) in ICP is because of the appreciable higher (secondary) ionization while at potentials about or higher than 8 eV the fraction of the neutral atoms increases. Dispersion of the data corresponding to different elements with high ionization potentials is due to the differences in partition functions. As some reference, data for a hypothetic “element” with partition functions equal to unity and high ionization potential of the first ion (negligible secondary ionization) and a varying ionization potential of the “atom” are included in Fig. 1 as a solid line. Naturally, maximum of the ion density in ICP (Fig. 2) for low ionization potential elements in comparison to the pure substance shifts to the lower temperatures. Decrease of the density of the singly

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Fig. 1. Degree of the first ionization at T = 7000 K as dependence on the ionization potential of the neutral atom for pure elements (○) and in argon gas (●). Argon plasma composition: Ar 97%, H 2%, O 1%, other elements 10− 7%. Solid line represents hypothetic elements with partition functions equal to 1 and negligible secondary ionization in argon gas.

ionized ions at higher temperatures takes place because of the law of the ideal gases (dotted line for Al II in Fig. 2) and, at low ionization potentials of the ions, due to successive ionization. As a consequence, essentially higher ionization of impurities in ICP plasma in comparison to the pure substances ought to take place the difference being dependent on the ionization degree of the surrounding gas. In pure He plasma, for example, at the same temperatures higher ionization (in comparison to Ar ICP) of the elements with ionization potentials about or exceeding 8 eV is expected. Densities of ions of elements with low (b7.5 eV) ionization potentials are largest at comparatively low temperatures, the differences being essentially due to the ideal gas law.

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the environment dependent density of electrons, i.e. N2/N1 =K(T)/Ne. In such a manner, at a given temperature matrix effect is an effect on the electron density. Quantitatively the effect on the ratio N2/N1 will be reciprocal to the effect on the electron density (or effects on Ne and N1/N2 are equal). As usually ionization potential of any added matrix is lower than that of argon increase of the electron density and, consequently, depletion of the densities of ions and increase of the density of the neutral atoms must take place. It is illustrated in Fig. 3 as effect of added sodium matrix on densities of atoms and ions of Cd and Al and in Fig. 4 as dependences of densities of the same species on relative water content in argon plasma. As mentioned above, effects on electron density and ionization ratios are reciprocal and the same for different elements (if the degree of successive ionization is small). Often analysts suppose that if the analytical signal of ions does not change the ionization degree is constant. Such a straightforward conclusion is an oversimplification. It is easy to understand that if the ionization degree is 99.9% decrease of the ion density by one percent results in increase of the atomic signal by more than a factor of ten. Because of the preservation of the total number of particles, the main effect is to the less abundant component, i.e. to the densities of atoms when the ionization degree is high and to the densities of ions if the neutral atoms prevail. In such a manner three questions are to be answered: what is the temperature interval in which an effect of the particular matrix must be expected, how large it is, and how the effect is distributed between atomic species and ions. As an example, carbon matrix effect, which is important for the analysis of organics containing samples, is discussed below. Naturally, as ionization degree is exponentially dependent on temperature, electron density is a sensitive function of this parameter.

3. LTE modeling of dependence of analytical signals on ICP sample composition Before taking attempts to account for the sample composition effects, or searching for the ways to diminish them in further analysis of the experimental data, more detailed LTE model study of those is presented in this section. Two aspects are regarded: dependence of the analytical signal on the general plasma composition, including atmospheric environment and sample components (matrix effects), and dependence of the analytical signal on the analyte concentration. As mentioned above, argon prevails in ICP plasma and ionization degree of an element can be regarded as determined by temperature and Fig. 3. Dependences of Al and Cd particle densities on Na particle fraction in sample gas at 7000 K and 5000 K.

Fig. 2. Dependence of the density of the singly ionized ions on temperature at argon ICP model conditions. Plasma composition: Ar 97%, H 2%, O 1%, other elements 10− 7%. Ionization potentials of the elements: Al 6.0 eV, Mg 7.6 eV, Si 8.1 eV, B 8.3 eV, Be 9.3 eV, As 9.8 eV.

Fig. 4. Illustration of independence of Al and Cd particle densities on water particle fraction in sample gas (T = 7000 K).

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An appreciable effect on electron density can be found at carbon particle fractions of about 0.001 or higher. Electron density increment (ratio of the density of electrons when the matrix is introduced to the corresponding density without matrix) in plasma gas consisting of Al (10− 9), Mg (10− 9), C(0.01 and 0.001), O (0.01), H (0.02) and Ar (up to 1 in total) is presented in Fig. 5. Essential dependence of the effect on temperature is evident. The effects on particle densities of atoms and ions are shown in Fig. 6a. Practically no effect is observed on densities of ions if the atoms are easy to ionize (Al, Mg) and on densities of atoms if the atom ionization potential is high. If in addition the contents of the easy to ionize elements is increased approaching the composition of samples of the practical interest, electron density increases and the matrix effect shifts to higher temperatures (from the peak at 4000 K in Fig. 6a to 6000 K in Fig. 6b). In spite that in general the dependences seem similar, the changes are essential. As in this temperature region the ionization degree of As, Se is appreciable, effect on densities of those atoms also increases while effect on densities of Al II and Mg II ions remains negligible. Effect on densities of Al and Mg atoms starts at temperatures about 5000 K but here absolute densities of those atoms are about two orders of magnitude lower (Fig. 6c) than at low temperatures (periphery of the plasma) where no matrix effect takes place. The effect on ions can be appreciable only if the corresponding atoms are hard to ionize (as As, Se). Thus the results show extremely composition and temperature dependent character of the matrix effects. Therefore any theoretical calculation must closely match the sample-plasma composition, measurements with space resolution would be of importance, or modeling ought to integrate space dependent changes. Due to higher electron density in complex plasma the matrix effect shifts to higher temperatures where relative effects on atomic and ion component densities can be essentially different then at lower temperatures. In general, magnitude of the effect, if shifted to higher temperatures, decreases. Concerning linearity of the calibration curve (self effect of the analyte) the conclusions formulated above are also valid. If the electron density is determined by the external to the analyte sources (inert gases, solvent and sample matrix) ratio N1/N0 does not depend on analyte concentration. Then the linear atomic and ion analytical signal dependences on concentration follow from the Saha equation. Because of the almost total ionization of the elements with the first ionization potentials about or lower 7.5 eV linear dependence of the analytical signal of the singly ionized ions on concentration can be expected up to high concentrations. Deviation from the linear dependence of the intensities of the atomic lines (and double ionized ions) of those elements could be expected (e.g., Fig. 7 for Ba) due to suppression of ionization because of the higher electron densities at large concentrations of the analyte. Explicitly for Ba, when electrons are mainly produced from ionization of the analyte atoms, N2 and Ne

Fig. 5. Temperature dependence of the electron density increment due to carbon content (carbon particle fractions 0.01 (♦) and 0.001 (■); Al (10− 9), Mg (10− 9), O (0.01), H (0.02) and Ar (remaining gas)).

Fig. 6. Carbon content (particle fraction 0.01) effect on atomic particle and ion densities at different temperatures: a) plasma composition Al (10− 9), Mg (10− 9), O (0.01), H (0.02), other elements (10− 9) and Ar (the remaining gas)); b) plasma composition Al (10− 5), Mg (10− 5), O (0.01), H (0.02), other elements (10− 9) and Ar (the remaining gas); c) calculated particle densities (plasma composition as in case b, without carbon matrix).

approach N and, as follows from Eq. (4), N1 approaches N2/K or N1 ∼ N2. At 6000 K (Fig. 7) transition to this region takes place at Ba fraction about 10− 4 while at 4000 K ionization degree of Ar is small and such dependence is characteristic to much lower Ba fractions. But when at high Ba fraction the ionization degree decreases appreciably, N1 becomes comparable to or larger than N2, the dependence on total concentration of analyte returns to linear (slope equal to 1 on logarithmic scales). Matrix effect of alkaline and rare earth elements in ICP can surpass the alkali elements effect because of the appreciable second ionization, providing two electrons from one atom. Effect on analytical signals of atoms and ions of the elements with high first ionization potential is more sensitive to the sample composition details and hardly can be formulated without an adequate calculation. The same concerns the relative role of depletion of the ionization potentials at high charged particle densities, the effect being opposite to the direct influence of the electron density.

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Fig. 8. Linear fit of the Element 2 response (normalized to 39K) dependence on atomic mass, y = 0.0225 M + 0.0302, R2 = 0.9731. Data for elements with ionization potentials between 7.5 eV and 8 eV and for elements with ionization potentials exceeding 8 eV are given for comparison only.

Fig. 7. Dependence of particle densities of Ba atoms and ions on Ba particle fraction.

4. Sensitivity of mass spectrometric measurements and ionization temperature 4.1. Experimental To test the approaches as considered above measurement results on a wide set of elements were regarded. Measurements were carried out by a double focusing sector field mass spectrometer Element2 (Thermo Finnigan MAT, Bremen, Germany). The main operating characteristics of the mass spectrometer and sample introduction system are presented in Table 1. Mainly stock ICP Multi-Element Standard Solution VI CertiPUR (Merck, Darmstadt, Germany) was used as a sample. Up to 2% of nitric acid (Suprapur®, 65.3%, Merck) was added to the diluent for stabilization of the subsamples. B, Na, Mg, Al, K, Ca, V, Cr, Mn, Fe, Co, Ni, Cu, Zn, Ga, As, Se, Rb, Sr, Ag, Cd, Sn, Ba, Tl, Pb, Bi, and U at mass fractions 10− 7–10− 9 g/g were measured as a general set. Simultaneously some other elements were added for some special tests. Ultra pure water prepared by NANOpure Barnstead/Thermolyne Co. (Dubuque, USA) water purification system was used for all sample dilutions. 4.2. Response of the instrument As follows from Figs. 1 and 2 degree of ionization of elements ought to be almost total at temperatures higher than 4000 K if the ionization potential of the neutral atom is less than 7 eV. If so, responses for those elements must be independent on the ionization conditions and represent the sensitivity of the instrument. The experimental responses of the mass spectrometer Element 2 at concentration of analytes 1 ng/g are presented in Fig. 8 as signal dependences on the atomic mass of the analyte. Isotopic composition

Table 1 Operating characteristics of the ICP mass spectrometer Element 2. Nebulizer, sample uptake rate Spray chamber Argon gas flow rates, l min− 1 Plasma Auxiliary Sample RF power, W Sample and skimmer cones Acquisition mode

Concentric, 0.7 ml/min (peristaltic pump) Cyclonic 14 0.75 1–1.4 1100 Nickel E-scan

of the elements was accounted for by division of the measured signals by atomic percentage of the isotope. Relative response scale is used in Fig. 8 and further for simplification of comparison of the different measurement sets and data for different elements. The data were normalized to the signal for 39K by division of all measured values by the response to this isotope (naturally, caution is needed to avoid spectral interference from 38Ar1H in the case, especially at lower sample gas rates). If the ionization potential is between 7.5 eV and 8 eV the response values are slightly lower while if the ionization potentials exceed 8 eV the responses are essentially less than the main population. Only the data for elements with ionization potentials less than 7.5 eV were used for instrument calibration. The dependence of the response of the low ionization potential elements with masses exceeding 23 Da is close to linear. Linear fit ought to be preferred in comparison to polynomial according to the residual test. Mean from two data sets is presented in Fig. 8. Standard error of the linear slope is about 4%. Coefficient at the linear term is about 0.02 for the scale accepted. The physical meaning of the slope is increase of the signal when the mass of the analyte increases by one atomic unit. It would be interesting to compare the slope values with those from the isotope ratio measurements. Naturally, in the method used the accuracy of the isotope ratio measurements was poor and no neighbour isotope response ratio dependence on mass could be determined because of the large measured ratio variations. Nevertheless the mean neighbour isotopes response ratio values, 1.025 ± 0.008 (k = 2) for the medium resolution data and 1.028 ± 0.015 for the high resolution, were close to the slope mentioned above. Time variation of the slopes was appreciable (from about 0.02 to 0.01) and caution is needed for practical applications. No difference between medium and high resolution data slopes in the same measurement sets was observed, only isotope ratio data population standard deviation for the high resolution was about twice larger than for the medium resolution. Some difference between low ionization potential elements can also be noticed in Fig. 8. The signals for alkali and alkaline earth elements at similar ionization potentials and masses are slightly larger than for the rest elements the effect possibly being due to sample atomization. 4.3. Ionization temperature and electron density The results on ionization state in ICP plasma and character of the matrix effects presented in Sections 2 and 3 are indications of some potentials of the LTE modeling in study of the problems of the analytical interest. However the results presented above and results from the model in general can be model dependent. Therefore elaboration of means for detection of the possible deviations from the model is of interest. If the plasma is not in the equilibrium state different measurement techniques based on the equilibrium concept

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ought to provide the results different from one to another, or from LTE model. Some means for acquisition of data on temperatures and particle densities in plasma from ICP MS measurements are proposed and discussed below. Discussion on possible deviation from the plasma equilibrium state will be presented also. We find the conclusion on almost total ionization of the low ionization potential atoms in ICP plasma, as presented in Section 2, natural and highly confident. In previous section it was applied to analysis of the MS response data and evaluation of the sensitivity function of the instrument. If the instrument sensitivity function discussed above is used to all measured data, input signal values for the measured elements can be restored. Relative values ought to be preferred because of the higher accuracy. Potassium can be selected as a good reference: because of the low ionization potential of the atom and high ionization potential of the first ion ionization degree of this element, N2/N, must be close to unity. Such relative signal values for different elements, normalized to the corresponding value for 39K, are plotted in Fig. 9 as dependences on the ionization potential of the atom. High ionization potential elements on the right of the figure are those beneath the total ionization response line in Fig. 8. Dependences of the sort as in Fig. 9 can be used to evaluate the temperature in the sampling region. Relative responses correspond to the ratio of the signal of ions to the total density of atomic particles of the element, the relative ionization degree, N2/N. In addition to dependence on the ionization potential the ionization degree, if much less than unity, depends on partition functions and (as in Fig. 9) is not a monotonic function of the potential. Corresponding LTE results are presented for comparison. It follows that the measured absolute ionization degrees of the elements with high ionization potentials are lower than LTE results at realistic temperatures (7000 K, 6000 K or even lower). Such estimation depends on the whole set of data needed for calculation, including accuracy of the electron density. It would be important to get the temperature values directly from experiment, or with minor application of the theoretical assumptions. Exponents of the Saha Eq. (3) are used as other estimates. Signal of the neutral atoms cannot be measured directly in mass spectrometry. If the second ionization is poor (it is true for the elements with high ionization potentials of the singly ionized ions), N = N2 + N1, and the ratio originating from the Saha equation, N2/N1, can be obtained as . N 2 N2 N  : = N1 1− N2 N

ð5Þ

provides direct estimate of the temperature independent on knowledge of the electron density. It is clear from Eq. (5) that accuracy of the values of the ratio N2/N1 will be poor if the ionization degree approaches 1. Because of this only data for the elements with ionization degree about 0.5 and lower (ionization potential of atoms exceeding 7.5 eV) were used for further analysis. Concerning partition functions, for example, at temperature 7000 K the ratio of the partition functions of the ions and corresponding atoms is between 2.2 and 0.2 mainly while in some cases the differences are even larger (e.g., Q2/Q1 = 19 for Cr and 0.12 for B). At high ionization degrees correlation between the ionization degree and ratio of the partition functions is negligible but ratios N2/N1or N3/N2 are proportional to the ratios of the partition functions. The values N2/N1 calculated from measured data divided by the calculated values of the partition functions, Q2/Q1, are plotted for Ag, Ni, Cu, Fe, Co, Zn, Se and As in Fig. 10. Index of the exponent (see Eq. (3)) corresponds to the ionization temperature about 7000 K. (Subsequent approximations enable to account for the proper temperature partition functions). Results of the corresponding dependence of the ratios of the number densities of the double and singly ionized ions (as ratios of the signals) on the ionization potential of the singly ionized ion are presented in Fig. 10 as rectangle marks. However, at high ionization potentials ratios are small and dispersion of the data is large, resulting in high uncertainty of the temperature value. Ionization temperatures from about 7300 K to 7700 K follow from the slope of the experimental results. Relative concentrations of elements were to be known to use for temperature measurements the ratios of the densities of the singly ionized ions and neutral atoms as the signals were to be normalized to the same scale. Measurements of the ratios of the signals of the double and singly ionized ions can be run on an unknown sample. Only the concentrations of the elements included must be enough to measure the comparatively small signals of the double ionized ions at sufficient accuracy. Experimental results for various isotopes of Ba, Sr, V, Mn and Pb are included in Fig. 10. The slopes of the asymptotic experimental and LTE calculated ratio dependences in Fig. 10 correspond within comparatively large uncertainty of about 15%. The same uncertainty (about 1000 K) is characteristic to the temperature values provided above. However the experimental ratio values are below the theoretical lines. Quantitatively it is evident from the equations accompanying each line in Fig. 10. Coefficients at the exponents of the experimental lines are 0.6 · 106 and 2 · 106 as compared to 3 · 106 and 107 at the LTE calculated lines of the same slopes (and temperatures). The same coefficient in Saha Eq. (3) is 4.83 · 1015 T3/2/Ne. It enables explicit calculation of the electron densities from the experimental values of

If the corresponding ratios are obtained for elements with different ionization potentials the index of the exponent of the dependence

Fig. 9. Dependences of the fraction of the singly ionized ions on the ionization potential of the atoms: measured (♦); LTE calculated: 6000 K (( ), Ne = 3.5 · 1013 cm− 3), 7000 K ((□), Ne = 3 · 1014 cm− 3), 7000 K at increased electron density ((Δ), Ne = 2 · 1015 cm− 3).

Fig. 10. Dependences of the ratios of populations of the neighbour ionization states on ionization potentials (populations of single statistical ionization states, ni = Ni/Qi, densities of ions divided by the corresponding partition functions, are used; see Eq. (3) also): experimental data for the ratios of populations of the states of the singly ionized ions and neutral atoms ((▲), n2/n1 = 2 · 106exp(− 1.7E1)); experimental data for the ratios of populations of the states of the double and singly ionized ions ((■), n3/n2 = 6 · 105exp(− 1.5E2)); LTE calculated dependences at 7000 K (dashed line, ni + 1/ni = 107exp(− 1.7E)) and 7700 K (dashed-dotted line, ni + 1/ni = 3 · 106exp(− 1.5E)).

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the coefficients and corresponding temperatures. Electron densities in the range from 1.4 · 1015 cm− 3 to 5 · 1015 cm− 3 were obtained from the experimental data while the theoretical values from 0.3 · 1015 cm− 3 at 7000 K to 1015 cm− 3 at temperature 7700 K follow from the LTE estimates indicating about five times increased electron densities as compared to the LTE values. The LTE results were obtained for plasma gas consisting of Ar (96.3%), H (2.5%) and O (1.3%). Effect of variation of the plasma composition up to Ar(70%), H(20%), O(10%) was tested resulting only in about 20% increase of the electron density (in correspondence to Fig. 3). Low concentrations of the test elements were used for measurements at minor possible input to electron density.

5. Discussion Conclusion on high ionization degree of small concentrations of elements in ICP plasma was used to determine the relative dependence of the sensitivity of the ICP mass spectrometer on atomic mass. Many reasons including phenomena in mass spectrometer optics, sampling system and plasma discharge determine this complicated instrument sensitivity dependence [4,6,7,34–39]. Close to linear signal dependence on mass for atomic masses M ≧ 23 Da was found for the equal mass fractions of elements in solutions. Dependence of the sensitivity to the equal particle numbers is parabolic. But for practical applications the linear dependences mentioned above can be useful as lower number of points is needed to get a linear dependence. Problem of the possible differences in response to the singly and double ionized ions remains. It is possible that reduction of the coefficient at the Saha exponent (Eq. (3), Fig. 10) for the ratios of the double to the singly ionized ions in some part is due to different sensitivity of the measurement system to the double and singly ionized ions. The measured data as analyzed in LTE approximation show similar LTE ionization conditions for wide set of different elements, including the singly and double ionized ions. Procedure for measurement of the electron density and electron density independent possibilities (compare two test element ionic to atomic line intensity ratio method [40]) to measure ionization temperatures from the LTE exponents for the first and second ionizations were proposed and applied. The measured temperature values are in acceptable correspondence with earlier results [18,20,21,26,28,41] while show some decrease to lower ionization stages. However the detected electron density exceeds the calculated LTE values by a factor of about five. The whole temperature and electron density picture indicates the recombining character of the plasma at the mass spectrometer sampling region with ionization temperature relaxation rate exceeding the recombination rate. At the same time ion sampling (mass spectrometry) and observation (optical spectrometry) geometry dependent character of the plasma state characteristics is evident. High ions originate only from the central plasma region and relative fraction of the lower ions can be increased due to radial temperature distribution. The same reason can explain some higher temperatures as detected from the ratios for the higher ions. From this point of view electron density measurement method originating from the ratio of number densities of the singly ionized ions and atoms but based on application of the data for singly ionized ions only (triangles in Fig. 10), while includes the same defect, physically is more transparent. If the electron density is really increased (it seems natural for recombining plasma flow) rates of electronic collisions also increase. It enables faster equilibration of populations of the excited states, particle, excitation and ionization temperatures. In a similar way, ionization potentials of most matrix elements are lower than the ionization potential of Ar, and matrix ought to decrease deviations from the equilibrium state. As a consequence, LTE approximation can be expected to be a useful starting point for quantitative description of

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the plasma and analysis of the possible role of particular nonequilibrium phenomena. It follows from LTE modeling, in some correspondence to experiments [11,42], that electron density dependent matrix effect starts only at some temperature which depends on both the plasma state and the ionization potential of the matrix elements. Dependence of the ionization temperature on the interval of energies involved in the measurements, as presented above, and electron densities characteristic to the recombining plasma is some indication that kinetic analysis ought to disclose the cross-over effect in more details. Especially if the limiting temperatures are low, the signal in the boundary region ought to be highly unstable due to sharp dependence of the ionization degree on temperature (Figs. 2, 5 and 6). Abrupt increase of fluctuations is expected not only in the peripheral regions of plasma but due to temperature gradients at aerosol particles on the microscopic scale also. If the matrix effect on ionization is essential it mainly manifests on the minor plasma component only. The effect can be quite different in magnitude to atomic and ion components the relative role being dependent on the ionization degree. Minor effects on the dominating ion component of each element are expected. For most elements in ICP that component is the singly ionized ions. This result is in good correspondence with the observed [14] carbon effects on atomic and ion spectral lines. At higher electron densities (as follows from Fig. 10) the magnitude of the effect would be some smaller while general character of the dependences ought to be preserved. However, if the optical spectrometry data are regarded, differences of the excitation potentials must be accounted for also — the low excitation energy atomic lines can be radiated from the low temperature regions where no electronic matrix effect is possible. LTE modeling, in addition, could be useful for selection of the internal standards, nevertheless role of the chemical effects and the differences in masses in mass spectrometric measurements [5,12,13,43–45] can be no less important. Due to detailed balance specific effects on population of the excited states or ionization are possible only if plasma is not in equilibrium. Energy interval dependent ionization temperatures and excess electron density are indication of importance of the kinetic effects. Determined densities of electrons are typical to recombining plasma and demonstrate non local character of the ionization balance and electron generation. A small but systematic difference in the ionization degree between alkali – alkaline earth and other low first ionization potential elements (Fig. 8) possibly could be explained by sample introduction effects but still no arguments can be presented to exclude the deviation from equilibrium as well. No element specific deviations from equilibrium ionization or specific deviations from the general population, for example, for Mg or Cu (characteristic elements for charge transfer from Ar [16,46–49]) or Tm (low second ionization potential [3]) were detected in this study. The highest accuracy measurements were not the goal of the present paper making preference to the wider material. On the other hand, populations of the particular excited levels could be more sensitive test than ionization. Possible dependence of the results on the set of elements selected is of interest. Mechanism of the deviation from equilibrium, if present, hardly can be determined from tests on the analyte only, as characteristic to many publications. Parallel information on the excitation–ionization state of the matrix and other main plasma components is equally important. As matrix usually only increases the electron density in general it ought to decrease deviations from the equilibrium state. Matrix effect on the ionization and excitation temperatures, especially in the case of high thermal capacity of the matrix components (lanthanides), ought to be regarded also. Concerning the plasma LTE modeling itself limited accuracy of the modeling is at hand (e.g., Figs. 9 and 10). Lower values of the ionization ratios, or coefficients at the Saha exponent, in comparison to LTE modeling (Eq. (3), Fig. 10) could be a cumulative effect of some

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reasons: really increased electron densities in the recombining plasma, charge dependent sensitivity of the instrument, different ion acquisition capabilities of the discharge and the measurement system. On the other hand, accuracy of calculations is also limited. Early publications [50–52] are good starting approach to evaluation of the partition functions, especially at lower temperatures. However values for other atoms and ions and, especially, accuracy of the values for higher temperatures ought to be increased accounting for more exhaustive information on high energy levels of atoms and ions and real depletion of the ionization potentials in the plasma. Decrease of the ionization potential due to Debye shielding is a physically clear effect. Effects of impurities, due to polyatomic and negative ions, large charges of the aerosol particles need more detailed analysis. Information on plasma dynamics, the temperature and particle density distributions in plasma ought to be included to get the integral values for comparison with experiment or application of the measurement data for plasma diagnostics. Importance of the close matching of the chemical composition of the sample and environment in models must be emphasized. Uncertainty due to the mentioned above effects especially increases if only one or two pairs of the ratios are used as a diagnostic tool.

6. Conclusions It follows from the results above that ICP is a plasma source essentially optimized for optical or mass spectrometric determination of low concentrations of analytes from measurements on ions. The inert gas atmosphere, low concentrations of analyte and matrix enable almost total ionization of most elements, low matrix effects and stability of the ion signals. Effects due to matrix ionization, extremely composition and temperature dependent, are most important on the minor plasma components and in the intermediate temperature region. The measured data as analyzed in LTE approximation show similar LTE ionization conditions for wide set of different elements. Electron density and electron density independent temperature measurement possibilities from mass spectrometric measurements were proposed and applied. The measured data indicate the recombining character of the plasma in the measurement region. LTE modeling if closely matching the sample composition was found essential for quantitative description of both the plasma state and character of the deviation from equilibrium. The approach used in the present paper can be a perspective for the quantitative analysis of the phenomena in plasmas of the complex samples.

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