New approach to obtain the correct chemical compositions by absorption correction using analytical transmission electron microscopy

Abstract

Analytical transmission electron microscopy (ATEM) is a powerful tool to obtain the chemical compositions of minerals at very small areas of minerals down to the nanometer scale. Most ATEM analytical systems are capable of performing quantitative calculations of chemical compositions by absorption correction. However, an appropriate procedure of obtaining the correct chemical compositions by absorption correction has not yet been established. In this study, we analyzed thin foils of garnet and olivine with known compositions and thicknesses using two different analytical systems to evaluate the certainty of the absorption correction based on the Cliff-Lorimer equation. The results show that the absorption correction using the real sample thicknesses at the analyzed spots did not yield the correct chemical compositions. The correct quantification data were obtained when using the sample thickness t corrected to minimize the value of the function SSR(t) (sum of squared residuals) = ∑_i[n_i⁰ − ∑_j n_ij(t)]², where n_i⁰ is the ideal atomic percentage of the i site, n_ij(t) is the atomic percentage of the j ion at the i site calculated by the software, and the sum holds all the ionic sites. Also, it was confirmed that the minimum value of the SSR obtained by calculating all existing elements as oxides (oxygen is not quantified) is in most cases smaller than that obtained by quantifying all elements, including oxygen, independently. This method of using the SSR without knowing the real thickness at the analyzed spot can be applied to the analytical results with absorption correction by any software. Therefore, this SSR has the potential of becoming a universal indicator to assess the results of quantitative chemical analyses by absorption correction in ATEM.

INTRODUCTION

Micro-analyses of materials down to the nanometer scale are essential for the study of Earth and planetary materials and the synthetic equivalents of those materials. For these purposes, transmission electron microscopy (TEM) and analytical transmission electron microscopy (ATEM) are very powerful tools. However, compared to the technical and theoretical developments of imaging and diffraction of the fine regions (e.g., Bradley and Dai, 2009; Kogure and Okunishi, 2010; Ishikawa et al., 2011; Midgley and Eggeman, 2015), developments in the quantitative chemical analysis of the fine regions are rather limited. The most widely used method for quantitative chemical analysis of the fine and thin Earth and planetary materials in ATEM is the Cliff-Lorimer equation (Cliff and Lorimer, 1975) coupled with the absorption correction (Goldstein et al., 1977), as discussed later. There is another method, the ζ-factor method, which considers the ionization cross sections and fluorescence yields of characteristic X-rays (Watanabe et al., 1996), but it is less common than the Cliff-Lorimer equation. However, for the Cliff-Lorimer equation, a method to obtain the correct quantitative chemical compositions of the fine areas has not yet been established. This is because many major software programs have problems with the absorption correction using the Cliff-Lorimer equation; how to choose the thickness to obtain the correct chemical composition and whether this thickness accurately reflects the real sample thickness at the analyzed spot. To overcome these hurdles in achieving reliable quantitative chemical analysis, we applied the Cliff-Lorimer equation coupled with the absorption correction, to the garnet and olivine samples with known compositions and thicknesses.

The Cliff-Lorimer equation is as follows:   

\begin{equation} C_{i}/C_{r} = k_{ir}(X_{i}/X_{r})\ (i = 1, 2,,,) \end{equation} (1),

where C_i and C_r are the weight percentages of the i and reference r elements, respectively; X_i and X_r are the intensities of the characteristic X-ray diffraction peaks of the i and r elements, respectively, and k_ir is called the k-factor. When all k_ir values are known, C_i values can be obtained by measuring the X_i values and using Eqs. (1) and (2).   

\begin{equation} \textstyle\sum C_{i} =100.0 \end{equation} (2).

Conversely, using a sample with a known chemical composition, the k-factors can be experimentally obtained via Eq. (1) by measuring the X-ray intensities of the sample. Figure 1 shows the k-factors k_OSi and k_MgSi plotted against the total X-ray counts of the pyrope garnet used in this study. The total X-ray counts serve as a proxy for the specimen’s thickness. k⁰_OSi and k⁰_MgSi are the k-factors of O and Mg, respectively, for zero-thickness.

Figure 1. Plots of k-factors of O and Mg for the pyrope garnet against the total X-ray counts.

In quantitative chemical analyses, the conventional method is often used when the k-factors of the respective elements of an unknown sample are derived from the calibration lines of the k-factors, as shown in Figure 1, of the reference sample with a known chemical composition. However, when the composition of the unknown sample differs to some extent from that of the reference sample, the slopes of the calibration lines of their k-factors could differ from each other. For these cases, the following theoretical k_ir values are used in the calculation of the quantitative chemical analysis:   

\begin{equation} k_{ir} = k_{ir}^{0}\frac{\mu_{i}^{m}}{\mu_{r}^{m}} {\cdot} \frac{[1 - \mathit{exp}(-\mu_{r}^{m}\rho t\mathit{cosec}\theta)]}{[1 - \mathit{exp}(-\mu_{i}^{m}\rho t\mathit{cosec}\theta)]} \end{equation} (3),

where $k_{ir}^{0}$ is the k-factor for zero-thickness; $\mu_{i}^{m}$ and $\mu_{r}^{m}$ are the mass absorption coefficients of the X-ray of elements i and r, respectively; ρ is the density of the sample; t is the thickness at the analyzed spot; and θ is the take-off angle of the X-ray detector (Goldstein et al., 1977). The value $k_{ir}^{0}$ is experimentally determined from the value for zero-thickness of the calibration line of the k-factor of the reference sample, as shown in Figure 1. However, the absorption correction is not widely used in the quantitative chemical analyses by ATEM because it is difficult to correctly estimate the absolute thickness of the sample at the analyzed spot, and it has also not been established whether the actual thickness will yield the correct chemical composition in the major software of the ATEM system, as described above.

There are two ways to calculate the quantitative chemical composition using the software associated with ATEM: one is to calculate oxygen as an independent element using the X-ray intensity of oxygen in the same way as the other existing elements in Eq. (1) (hereafter referred to as ‘Oxygen Independent mode’); and the other is to calculate all the existing elements except oxygen as oxides, without using the X-ray intensity of oxygen (hereafter referred to as ‘Oxide mode’). In the former case, the thickness used in the software is often selected so that the calculated atomic percentage of some key element becomes equal to that of the ideal chemical formula (in many cases Si or O is selected for silicate minerals), or the calculated atomic percentage of all the elements satisfies the electrical neutrality (Van Cappellen and Doukhan, 1994). In the latter case, the various thicknesses are inputted into the software until the calculated atomic percentage of some key element becomes equal to that of the ideal chemical formula because the electrical neutrality is already maintained. However, whether the estimated thickness gives an accurate chemical composition, and whether it actually coincides with the actual sample thickness at the analyzed spot have not been fully examined. In the present study, we examined these points by using garnet and olivine samples with known compositions and thicknesses.

EXPERIMENTAL METHODS

We examined the same garnet samples using different software A and B, which were attached to 200 kV ATEMs at different laboratories. The olivine sample was examined using software B. The garnet samples were a synthetic endmember grossular and a natural pyrope (Bohemia, Czech Republic). The olivine sample was a natural forsterite (San Carlos, Arizona, USA). The chemical composition of the synthetic grossular was determined by analyzing the homogeneous starting material glass for the synthesis of polycrystalline grossular, using a JEOL JSM-7000F field-emission scanning electron microscope (FESEM) equipped with an energy-dispersive spectrometer (EDS). The natural pyrope and forsterite were analyzed using a JEOL JXA-8200 electron probe microanalyzer (EPMA) with a wavelength-dispersive spectrometer (WDS). In both analyses, the chemical compositions were calculated using the ZAF method. These results are summarized in Table 1. For the weight percentage of pyrope in Table 1, the Fe content is shown as a component of FeO. This does not mean that this pyrope contains only Fe²⁺, but is just a way of expressing the result of the analysis. In Table 1, the cation numbers 1 of pyrope shows that the total number of cations at the Al site is less than 2, while the total number of cations at the Mg site is larger than 3. Actually, 16% of Fe in this pyrope is considered to be Fe³⁺, as discussed later in the Results. This introduction of 16% Fe³⁺ results in the proper cation numbers 2 of the pyrope (Table 1).

Table 1. Chemical compositions of samples analyzed using EPMA

Sample name		Grossular*
	wt%	Cation numbers
		(O =12)
Al2O3	23.02(0.14)	Al	2.00
SiO2	40.86(0.11)	Si	3.01
CaO	37.59(0.13)	Ca	2.97
Total	101.47(0.20)

^*Measured with the starting material glass. Average value of 10 points.

Sample name		Pyrope*
	wt%	Cation numbers 1				Cation numbers 2**
			(O = 12)				(O = 12)
Al2O3	21.36(0.19)	Al	1.80	$\bigg\}$	1.94	Al	1.79	$\Bigg\}$	2.01
TiO2	0.50(0.01)	Ti	0.03			Ti	0.03
Cr2O3	1.99(0.05)	Cr	0.11			Cr	0.11
						Fe3+	0.08
SiO2	41.94(0.22)	Si	2.99			Si	2.98
MgO	21.03(0.12)	Mg	2.24	$\Bigg\}$	3.10	Mg	2.23	$\Bigg\}$	3.01
CaO	4.37(0.06)	Ca	0.33			Ca	0.33
MnO	0.35(0.02)	Mn	0.02			Mn	0.02
FeO	8.49(0.13)	Fe2+	0.51			Fe2+	0.43
Total	100.04(0.43)

^*Average value of 9 points.

^**Fe³⁺ was estimated by SSR.

Sample name		Forsterite*
	wt%	Cation numbers
		(O = 4)
SiO2	40.81(0.11)	Si	1.00
Na2O	0.01(0.01)	Na	0.00
MgO	49.85(0.13)	Mg	1.82
Al2O3	0.01(0.01)	Al	0.00
CaO	0.07(0.01)	Ca	0.00
Cr2O3	0.02(0.02)	Cr	0.00
MnO	0.12(0.02)	Mn	0.00
FeO	8.68(0.07)	Fe	0.18
NiO	0.37(0.02)	Ni	0.01
Total	99.93(0.16)

^*Average value of 10 points.

The natural pyrope and natural forsterite were thinned by an Ar-ion beam at 2-4 kV (GATAN PIPS) to obtain the k-factors for zero-thickness. Experiments for determining the k-factors for zero-thickness of natural pyrope were performed with system A using the software A of 200 kV ATEM (JEOL JEM-2100F). The same experiments of natural pyrope and natural forsterite were performed with system B using the software B of another 200 kV ATEM (JEOL JEM-ARM200F) at the different laboratory. The k-factors for zero-thickness of pyrope were used for the calculations of both grossular and pyrope owing to their identical crystal structure. Regarding the foils used for the quantitative chemical analyses for system A, the synthetic grossular was thinned by an FIB (JEOL JEM-9310FIB) operated at an accelerating voltage of 30 kV. A Ga-ion beam was irradiated from the top to gradually thin the wall between two trenches, and the wall was finally thinned to a thickness of less than 100 nm. After the left and right ends of the foil were cut off, the foil was picked up using a glass needle and transferred onto a Cu grid. The Ar-ion thinned natural pyrope was used for the quantitative chemical analyses of system A. With the foils used for the quantitative chemical analyses of system B, all the grossular, pyrope, and forsterite foils were prepared using another FIB (Hitachi SMI4050). After the deposition of carbon protective layers, the minerals were thinned to a thickness of ∼ 2 µm and cut out using a Ga-ion beam at an accelerating voltage of 30 kV. The thin sections were mounted on Cu grids using a micromanipulator equipped with the FIB, and then ultrathinned to a thickness of less than 130 nm at a probe current of 40 pA in the finishing step.

The thicknesses of the thin foils at the analyzed spots were determined in the following manner. The thicknesses of the thin edge parts of the FIB foils of grossular, pyrope, and forsterite analyzed by system B were measured on the images of the FIB system (Fig. 2). The X-ray intensities of Si at the thin-edge parts of the FIB foils in Figure 2 were measured using the same condition as that of the analyzed spots. Then, the X-ray intensities of Si, I_si, and the thickness t at the thin-edge part and the analyzed spot in each FIB foil satisfy the following equation:   

\begin{equation} I_{\textit{Si}} = T \,[1 - \mathit{exp} \,(-\mu_{\textit{Si}}^{m} \rho t\mathit{cosec}\theta)] \end{equation} (4),

where $\mu_{\textit{Si}}^{m}$, ρ, t, θ are the same as Eq. (3), and T is determined so that I_si becomes the X-ray intensity of Si at the thin edge part when t is the thickness of the thin-edge part. Then, I_si and the thickness t at the analyzed spot on the FIB foil also satisfy Eq. (4). In this way, the thicknesses at the analyzed spots of grossular, pyrope, and forsterite were determined for system B. The X-ray intensity of Si at the part very close to the analyzed spot on the FIB foil of grossular in system A was also measured in system B. Therefore, the thickness at the analyzed spot on the grossular foil in system A could also be obtained in the same manner as described above. Although the pyrope foil in system A, which was thinned by Ar ion, was not measured in system B, the thickness at the analyzed spot in system A could be obtained using the following equation incorporating the ratio of the X-ray intensities of Si in the pyrope and grossular:   

\begin{equation} \frac{I_{\textit{Py}}}{I_{\textit{Gr}}} = \frac{I_{0\textit{Py}}}{I_{0\textit{Gr}}} {\cdot} \frac{\mu_{\textit{Gr}}^{m}\rho_{\textit{Gr}}}{\mu_{\textit{Py}}^{m}\rho_{\textit{Py}}} {\cdot} \frac{[1 - \mathit{exp}\,(-\mu_{\textit{Py}}^{m}\rho_{\textit{Py}}t_{\textit{Py}}\mathit{cosec}\theta)]}{[1 - \mathit{exp}\,(-\mu_{\textit{Gr}}^{m}\rho_{\textit{Gr}}t_{\textit{Gr}}\mathit{cosec}\theta)]} \end{equation} (5),

where I_Py and I_Gr are the X-ray intensities of Si in the pyrope and grossular, respectively, I_0Py and I_0Gr are the X-ray intensities of Si in the pyrope and grossular, respectively, generated by the unit length of the materials before absorption; $\mu_{\textit{Py}}^{m}$ and $\mu_{\textit{Gr}}^{m}$ are the mass absorption coefficients of Si in the pyrope and grossular, respectively; ρ_Py and ρ_Gr are the densities of the pyrope and grossular, respectively; and t_Py and t_Gr are the thicknesses of the pyrope and grossular, respectively, at the analyzed spots. In this equation, the value of I_0Py/I_0Gr before absorption is considered to be almost equal between systems A and B. The value of I_0Py/I_0Gr for system B is obtained from the X-ray intensities of Si in the pyrope and grossular with known thicknesses of system B in Eq. (5). Therefore, by introducing this value, the X-ray intensity of Si, and the sample thickness at the analyzed spot of the grossular in system A into Eq. (5), the relation of the X-ray intensity of Si and the thickness at the analyzed spot of pyrope in system A can be obtained. By plotting this relation, the thickness at the analyzed spot of pyrope of system A was obtained through the X-ray intensity of Si. All the thicknesses at the analyzed spots of grossular and pyrope in system A, and grossular, pyrope, and forsterite of system B were thus obtained and are summarized in Table 2.

Figure 2. Edge-on secondary electron images of the FIB sections of grossular (top), pyrope (middle), and forsterite (bottom). Close-up images of the left parts of the sections are in the insets.

Table 2. Thicknesses of the thin foil samples at the analyzed spots

System A		System B
Grossular	Pyrope	Grossular	Pyrope	Forsterte
77 ± 20 nm	68 ± 20 nm	88 ± 20 nm	112 ± 20 nm	113 ± 30 nm

Quantitative chemical analyses were performed using the different software A and B under two different ATEMs operated at 200 kV. Systems A and B use the silicon drift X-ray detector (SDD) with receiving areas of 60 mm² and 100 mm², respectively. In both systems, the X-ray signals were collected in the STEM mode. In system A, EDS spectra were obtained by scanning the circled area with a diameter of 100 nm with a spot size of 0.5 nm for 30 sec. The irradiated electron intensities were ∼ 50 pA on a small fluorescence screen without the sample. For system B, the spectra were collected by scanning the squared area of 100 × 100 nm² with the spot size of 0.1 nm for 30 sec. The irradiated electron intensities were ∼ 50 pA on the small fluorescence screen without the sample.

A preliminary examination of the outputs of the software of systems A and B showed that the output k_ir [= (C_i/C_r)·(X_r/X_i)] value was just equal to k_ir of Eq. (3) for system B and almost equal to k_ir of Eq. (3) (the difference was less than ∼ 0.5%) for system A. This means that Eq. (3) is correctly incorporated into the software of systems A and B. However, the input of the real thickness at the analyzed spot to the software of both systems A and B did not give the correct chemical composition, as shown later. This means that Eq. (3) may not be perfect, or the analyzer process to produce the X-ray net counts from the signal of the X-ray detector may have some problems, or both. However, the detailed processes to obtain the X-ray net counts from the signal of the X-ray detector are not known to us. Therefore, we searched another way to obtain the correct chemical compositions from the output of the software, rather than improve the software.

We here propose the use of the following SSR(t) (sum of squared residuals) function of the thickness t at the analyzed spot used in the software to obtain the chemical compositions:   

\begin{equation} \text{SSR}\ (t) = \textstyle\sum_{i}[n_{i}^{0} - \sum_{j}n_{ij}(t)]^{2} \end{equation} (6),

where n_i⁰ is the ideal atomic percentage of the i-site, and n_ij (t) is the atomic percentage of the j-th atom of the i site calculated by the absorption correction using the software of system A or B. For example, in the case of (Mg,Fe)₂SiO₄ = (Mg,Fe)_28.57Si_14.29O_57.14 (expressed in atomic percentage) forsterite, SSR(t) = [28.57 − n_Mg (t) − n_Fe (t)]² + [14.29 − n_Si (t)]² + [57.14 − n_O (t)]². The most reliable n_ij (t) values should be those that give the minimum of the SSR(t) function. This is the least squares method to obtain the most reliable atomic percentage of ions at their respective ionic sites. In the following, the calculated results of the SSR(t) for synthetic grossular and natural pyrope by systems A and B and natural forsterite by system B are presented for the cases where the absorption corrections were made in the Oxygen Independent and Oxide modes.

RESULTS

Fe³⁺ in natural pyrope derived from the EPMA data by SSR

Before applying the SSR method to the results of natural pyrope obtained with systems A and B, it was applied to the EPMA result of natural pyrope from Czech Republic to examine whether it can reproduce the presence of Fe³⁺ or not. In this case, the variable parameter of the SSR is Fe³⁺% in all Fe, and this Fe³⁺ is allocated to the octahedral site in pyrope. There is a report that about ∼ 8-12% of Fe in Bohemian pyrope is Fe³⁺ (Seifert and Vrána, 2005). In Figure 3, the SSR value of EPMA became a minimum (0.012), which is very low, when Fe³⁺ is 15.8% of Fe. This value is very close to the actual Fe³⁺ content of Bohemian pyrope. On the other hand, the smallest SSR values of natural pyrope obtained from the ATEM data of systems A and B, which are the smallest values among the various thicknesses, are not so small as that obtained by EPMA. This indicates that SSR is not very accurate at determining the Fe³⁺ content from the analyzed results by ATEM. When the Fe³⁺ content in all Fe is fixed to 15.8% in the ATEM data of the natural pyropes of systems A and B (Fig. 3), their SSR values increase to 0.49 (system A) and 0.38 (system B), respectively. Although the feasible Fe³⁺ content of pyrope is likely to be ∼ 15.8% of total Fe, we determined the Fe³⁺ content of pyrope by the SSR method using the ATEM data because quantitative analysis of very small areas can be only done by ATEM, which does not refer to the result of EPMA analysis.

Figure 3. Plots of the SSR (Fe³⁺%) against Fe³⁺% in Fe in pyrope, as obtained by EPMA, ATEM of system A, and ATEM of system B. The SSR (Fe³⁺%) values by ATEM of systems A and B are the minimum values among the various thicknesses.

Synthetic grossular

Figure 4 shows the calculated results of the SSR(t) versus thickness t of synthetic grossular based on the output of systems A and B. The Oxygen Independent SSR(t) was calculated based on the Oxygen Independent compositions, while that of the Oxide mode was calculated based on the Oxide compositions. In each SSR figure, the data points were approximated by a quadratic least-squares regression, and the minimum value of the SSR(t) was defined as the ‘minimum of the SSR(t) function’. For some data points of the SSR(t), the calculated atomic percentages of oxygens are also shown. Systems A and B in Figure 4 show that the minimum values of the curves of the SSR(t) for the Oxide mode are smaller than those of the Oxygen Independent mode, and are equal to or less than 0.05 for systems A and B. This means that the compositions estimated by the method by Van Cappellen, and Doukhan (1994) are not so accurate because the compositions estimated by their method are based on the calculations by the Oxygen Independent mode. The atomic percentage of oxygen at the minimum point of the SSR(t) for the Oxide mode is very close to the ideal value of 60.00 of garnet for systems A and B.

Figure 4. Plots of the SSR(t) against the thickness t for grossular garnet. Oxide means that all the ions except oxygen were calculated as oxides, and Oxygen Independent means that all the ions including oxygen were calculated as independent ions. At% denotes atomic%. SSR(t) = [15.00 − n_Ca(t)]² + [10.00 − n_Al(t)]² + [15.00 − n_Si(t)]² + [60.00 − n_O(t)]²

About the thicknesses t at the minimum value of the SSR for the Oxygen independent calculation, they are much larger than the real thicknesses for both systems A and B. However, the thickness t at the minimum SSR for the Oxide mode, ∼ 140 nm, for system A is larger than the real thickness, ∼ 77 nm, while ‘−1.4 nm’ for system B is much smaller than the real thickness, ∼ 88 nm. The compositions of the minimum SSR for the Oxide mode for systems A and B (Table 3) correspond well to the values of the end-member grossular, as seen for systems A and B. However, for the compositions of the minimum point of the SSR for the Oxygen Independent calculations, Al and Si are overestimated, and Ca is underestimated for both systems A and B.

Table 3. Chemical compositions of grossular garnet by the absorption correction using the thicknesses that give the minimum value of the SSR (t)

	EPMA	System A			System B
			O Independent	Oxide		O Independent	Oxide
			t = 308 nm	t = 140 nm		t = 172 nm	t = 0.0 nm*
Element	atomic%	k0iSi	atomic%	atomic%	k0iSi	atomic%	atomic%
O	60.04	0.997	59.99	59.99	1.076	59.90	59.97
Al	10.01	0.954	10.35	10.06	1.051	10.54	10.18
Si	15.08	1.000	15.31	14.95	1.000	15.29	14.85
Ca	14.86	1.102	14.35	15.00	1.204	14.27	15.00
Total	100.00		100.00	100.00		100.00	100.00

O Independent denotes Oxygen Independent.

^*SSR is minimum at t = −1.35. However, the software of the absorption correction accepts the thickness t as t = 0 or positive.

Natural pyrope

As shown by the cation numbers 2 of pyrope (Table 1) and described above, 16% of the total Fe is considered to be Fe³⁺ in the natural pyrope sample. However, we performed quantitative chemical analysis including Fe³⁺, using the SSR without using the constraint that the Fe³⁺content is 16% of total Fe. When we assume that this natural pyrope contains only Fe²⁺, the SSR patterns of pyrope for systems A and B (Fig. 5) show very different features from those of grossular. The minimum of the each SSR function for the Oxide mode is not as small as those for grossular. The minimum of the SSR function for system A exceeds ∼ 0.4, and for system B it becomes 1.0. Also, in systems A and B, the minimum SSRs for the Oxide mode are nearly identical with those for the Oxygen Independent mode. These large SSR values are reflected in the calculated compositions of pyrope in Table 4, where Fe is all treated as Fe²⁺. For system B (Table 4), the sum of the atomic percentages of the octahedral site (Al site) for the Oxide mode is 9.67, much lower than the ideal value of 10.00, while the sum of the atomic percentages of the dodecahedral site (Mg site) is 15.87, greatly exceeding the ideal value of 15.00. This indicates that the SSR value will be reduced when some of the ions are redistributed from the dodecahedral site to the octahedral site. Meanwhile, for the oxide mode in system A (Table 4), the sum of the atomic percentages of the octahedral site, 9.64, is much lower than the ideal value of 10.00, and the sum of the dodecahedral site, 14.77, is also lower than the ideal value of 15.00. Instead, the Si value, 15.45, fairly exceeds the ideal value of 15.00.

Figure 5. Plots of the SSR(t) against the thickness t for pyrope garnet. All Fe is assigned as Fe²⁺ at the dodecahedral site. SSR(t) = [15.00 − n_Mg(t) − n_Fe²⁺(t) − n_Ca(t) − n_Mn(t)]² + [10.00 − n_Al(t) − n_Cr(t) − n_Ti(t)]² + [15.00 − n_Si(t)]² + [60.00 − n_O(t)]²

Table 4. Chemical compositions of pyrope garnet by the absorption correction using the thicknesses that give the minimum value of the SSR (t)

	EPMA	System A					System B
			O Independent	Oxide				O Independent	Oxide
			t = 335 nm	t = 457 nm				t = 305 nm	t = 525 nm
Element	atomic%	k0iSi	atomic%	atomic%			k0iSi	atomic%	atomic%
O	60.00	0.997	60.08	60.15			1.076	60.14	59.79
Al	8.95	0.954	9.09	9.16	$\bigg\}$	9.64	1.051	8.84	9.09	$\bigg\}$	9.67
Ti	0.13	1.016	0.06	0.06			1.343	0.17	0.15
Cr	0.56	1.328	0.45	0.42			1.457	0.48	0.43
Fe3+	0.40
Si	14.91	1.000	15.47	15.45			1.000	14.61	14.67
Mg	11.14	0.945	11.15	11.30	$\Bigg\}$	14.77	0.998	11.76	12.32	$\Bigg\}$	15.87
Ca	1.67	1.102	1.36	1.29			1.204	1.62	1.46
Mn	0.11	1.118	0.07	0.06			1.562	0.20	0.17
Fe2+	2.13	1.684	2.28	2.12			1.536	2.18	1.92
Total	100.00		100.00	100.00				100.00	100.00

Here, all Fe is assigned as Fe²⁺ at the dodecahedral site.

We found that the minimum of the SSR function for the Oxide mode for system B reduced to 0.157 when 30% of the total Fe is assigned as Fe³⁺ and reallocated to the octahedral site, as shown in Figure 6. On the other hand, the minimum SSR for the Oxide mode in system A very slightly reduced to 0.40 when 5% Fe is assigned as Fe³⁺ and reallocated to the octahedral site, although the minimum SSR for the Oxygen Independent mode reduced to the slightly lower value of 0.37. Table 5 shows that for the Oxide mode for system B, the total sums of the atomic percentage of the octahedral and dodecahedral sites were fairly improved and slightly exceeded 10.00 and 15.00, respectively, while the total sum of Si is 14.70, a little smaller than 15.00. Meanwhile, for the Oxide mode in system A, where 5% Fe was assigned as Fe³⁺, although the total sums of the atomic percentages of the octahedral and dodecahedral sites are close to the ideal values, Si is 15.72, much larger than the ideal value of 15.00. These contrasting contents of Si and other elements between systems A and B originate from the differences of the k-factors between systems A and B. The k-factors at thickness 0, $k_{i\textit{si}}^{0}$, for the pyrope in system A (Table 5) are smaller than those for system B, except for Fe. These differences of the k-factors between systems A and B led to the decrease in the atomic percentages of ions other than Si and the increase in the atomic percentage of Si in system A compared with system B. When the slightly larger values are chosen as k-factors at thickness 0 in system A, the minimum SSR for the Oxide mode will further reduce to the lower value than that for the Oxygen Independent mode. Although the atomic percentage of Fe³⁺ was not correctly estimated by the SSR using the ATEM data for pyrope, the atomic percentages of other ions were fairly correctly estimated, particularly for system B, as the Fe content was not large.

Figure 6. Plots of the SSR(t) against the thickness t for pyrope garnet. In system A, 5% of Fe is assigned as Fe³⁺ at the octahedral site, while for system B, 30% of Fe is assigned as Fe³⁺ at the octahedral site. SSR(t) = [15.00 − n_Mg(t) − n_Fe²⁺(t) − n_Ca(t) − n_Mn(t)]² + [10.00 − n_Al(t) − n_Fe³⁺(t) − n_Cr(t) − n_Ti(t)]² + [15.00 − n_Si(t)]² + [60.00 − n_O(t)]²

Table 5. Chemical compositions of pyrope garnet by the absorption correction using the thicknesses that give the minimum value of the SSR (t)

	EPMA	System A					System B
			O Independent	Oxide				O Independent	Oxide
			t = 333 nm	t = 304 nm				t = 292 nm	t = 150 nm
Element	atomic%	k0iSi	atomic%	atomic%			k0iSi	atomic%	atomic%
O	60.00	0.997	60.16	59.46			1.076	60.00	59.94
Al	8.95	0.954	9.07	9.21	$\Bigg\}$	9.86	1.051	8.86	8.75	$\Bigg\}$	10.20
Ti	0.13	1.016	0.07	0.07			1.343	0.17	0.19
Cr	0.56	1.328	0.44	0.46			1.457	0.49	0.53
Fe3+	0.40		0.12	0.12				0.66	0.73
Si	14.91	1.000	15.44	15.72			1.000	14.67	14.70
Mg	11.14	0.945	11.12	11.26	$\Bigg\}$	14.98	0.998	11.78	11.49	$\Bigg\}$	15.17
Ca	1.67	1.102	1.36	1.41			1.204	1.64	1.77
Mn	0.11	1.118	0.07	0.07			1.562	0.20	0.22
Fe2+	2.13	1.684	2.16	2.24			1.536	1.54	1.69
Total	100.00		100.00	100.00				100.00	100.00

In system A, 5% of Fe is assigned as Fe³⁺ and allocated to the octahedral site, and for system B, 30% of Fe is assigned as Fe³⁺ and allocated to the octahedral site.

As shown in Figure 6, systems A and B showed a different tendency, that is, the thickness at the minimum of the SSR function for the Oxide mode in system A is much larger than the real thickness, while it is a little larger but close to the real thickness for system B.

Natural forsterite

Figure 7 shows the SSR of the natural forsterite olivine for system B. The SSR of forsterite for the Oxide mode has a very small minimum value of 0.015 in the very thin region. The relationship between the real thickness at the analyzed spot and the thickness that gives the minimum SSR for the Oxide mode is the same as that of the grossular for system B, that is, the thickness of the minimum SSR is much smaller than the real thickness. The chemical composition obtained from the minimum SSR for the Oxide mode (Table 6) is very similar to that obtained by EPMA.

Figure 7. Plots of the SSR(t) against the thickness t for forsterite olivine for system B. SSR(t) = [28.57 − n_Mg(t) − n_Fe²⁺(t)]² + [14.29 − n_Si(t)]² + [57.14 − n_O(t)]²

Table 6. Chemical compositions of forsterite olivine by the absorption correction using the thicknesses that give the minimum value of the SSR (t)

	EPMA	System B
			O Independent	Oxide
			t = 208 nm	t = 51 nm
Element	atomic%	k0iSi	atomic%	atomic%
O	57.12	1.110	57.18	57.11
Si	14.24	1.000	14.15	14.22
Mg	25.93	0.972	26.42	26.16	$\Big\}$	28.67
Fe2+	2.53	1.599	2.25	2.51
Na	0.01		100.00	100.00
Ca	0.02
Cr	0.01
Mn	0.04
Ni	0.10
Total	100.00

DISCUSSION

In the present study, the software used to calculate the chemical compositions by the absorption correction using ATEM did not give us the correct chemical compositions for the input of the real thickness at the analyzed spot with garnets and olivine. Therefore, we introduced SSR which uses the output chemical compositions for the input of the thicknesses to the software, and it was proved that the correct chemical composition is given when the SSR function becomes minimum. In the SSR calculation, the minimum SSR for the Oxide mode was almost always lower than that for the Oxygen Independent mode. This may reflect the fact that the X-ray intensity of oxygen is not as accurately measured as those of the other elements by the X-ray counter. Therefore, the widely used method by Van Cappellen and Doukhan (1994) is not appropriate to obtain the correct chemical compositions because it uses the absorption correction based on the Oxygen Independent mode.

The minimum value of the SSR appropriate for obtaining a reliable chemical composition may depend on the number of ions to determine their compositions and the number of different ionic sites. For cases where the number of different ions is less than approximately five or six, and the number of different ionic sites is equal to or less than four, the minimum SSR needed to obtain the correct chemical compositions will decrease to below ∼ 0.1, as observed for grossular garnet and forsterite olivine. When the number of different ions or different ionic sites increases, the minimum SSR producing the correct chemical composition will increase to above ∼ 0.1. From the present results, it will be possible to determine the site occupancies of different ions to some extent using SSR, although it depends on the species of ions. However, the determination of the valence state of ions by the SSR seems to be difficult at the present stage, as seen for the Fe³⁺ content in natural pyrope, for the following reason. In the case of site occupancies of different ions, the total amounts of respective ions are constrained by their X-ray intensities; however, in the case of ions with different valence states, these ions cannot be distinguished by an ATEM that is not equipped with an EELS spectrometer.

It was evident that there was a difference between the real thickness at the analyzed spot and the thickness of the minimum of the SSR function calculated based on the output of the software of ATEM. In the case of the Oxygen Independent mode, the thickness of the minimum SSR was always double or triple of the real thickness. However, in the case of the Oxide mode, the situation was different. For system A, the thicknesses of the minimum SSR were always larger than the real thicknesses, while for system B, they were roughly equal to or smaller than the real thicknesses. In addition, cases occurred where the minimum point of the SSR is at the point of the ‘negative thickness’, as seen in grossular garnet for system B. Two possibilities seem to be the cause of these differences. The first is that the present theory of the absorption collection may not be perfect. For example, Eq. (3) (Goldstein et al., 1977) assumes that the intensities of the characteristic X-rays generated by electron shooting are equal at different depths of the foil. However, the actual intensities of the generated X-rays would decrease with depth because the shooting electron beam decays during the penetration, even in a thin foil. This point is not considered in their model. The second possibility is that the analyzing process to transform the X-ray signals caught by the detector to the X-ray counts may not be properly functioning, as recent X-ray detectors are designed to catch X-rays in a wider area. When we consider that the deviation of the thickness at the minimum of the SSR function from the real thickness was opposite between systems A and B, the first possibility seems to be unlikely, while the second possibility seems more plausible. At present, we do not clearly understand the process of transforming the detected X-ray signals into the X-ray counts of respective elements. Therefore, this will be a future problem to be solved throughs the collaboration of users and the technical groups of manufactures of ATEM.

CONCLUDING REMARKS

A method to obtain the correct chemical compositions of silicate minerals by absorption correction using ATEM has not been established so far. To overcome this situation, we propose a new method for quantitative chemical analysis by applying the least-squares method to the results obtained from the EDS software of ATEM, instead of improving the software itself, the details of which are undisclosed. We introduced the function SSR(t) written in Eq. (6) to find the most likely thickness that gives us the correct chemical composition at the analyzed spot of the sample, based on the output of the EDS software. This approach to obtain the correct chemical compositions of silicate minerals by the absorption correction will be summarized as follows.

1. 1)    The value of the SSR function is a useful indicator to obtain the correct chemical compositions by the absorption correction from the output of the ATEM software without knowing the real thickness at the analyzed spot.
2. 2)    The minimum value of the SSR function determined by the Oxide calculation mode is lower than that by the Oxygen Independent calculation mode.
3. 3)    The minimum value of the SSR function that gives the correct chemical compositions depends on the number of different ions and the number of different ionic sites of the sample.
4. 4)    Site assignments of the different ionic species based on the SSR will be possible depending on the ionic species. However, site assignments of ionic species having different valence states seem difficult at the present stage.
5. 5)    The minimum value of the SSR(t) function and the thickness t for that for the Oxide mode can be easily obtained by calculating the SSR values of at least three arbitrarily chosen thicknesses and approximating them with a quadratic function of t. Therefore, the whole process to obtain the correct chemical composition can be easily incorporated into the software.
6. 6)    At the present stage, the thickness t at the analyzed spot that yields the minimum value of the SSR(t) does not coincide with the real thickness at the analyzed spot. Solving this issue will be the future work.
7. 7)    This SSR method can be applied to the results of the quantitative chemical analyses obtained using any ATEM method. Therefore, the minimum value of the SSR will become a universal indicator to assess the correctness of the results of quantitative chemical analyses using ATEM.

ACKNOWLEDGMENTS

We thank K. Shibuya and T. Suzuki for their discussions. Thanks are due to Yasuhiro Shibata, Jun-ichi Ando for EPMA analyses, and Koji Kawakami for FESEM-EDS analyses of the samples. We would like to thank the GRC, Ehime University, for using their equipments for the ATEM analysis data obtained in this study.

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