Quantitative energy-dispersive X-ray fluorescence analysis for unknown samples using full-spectrum least-squares regression

Abstract

The full-spectrum least-squares (FSLS) method is introduced to perform quantitative energy-dispersive X-ray fluorescence analysis for unknown solid samples. Based on the conventional least-squares principle, this spectrum evaluation method is able to obtain the background-corrected and interference-free net peaks, which is significant for quantization analyses. A variety of analytical parameters and functions to describe the features of the fluorescence spectra of pure elements are used and established, such as the mass absorption coefficient, the $G_{i}$ factor, and fundamental fluorescence formulas. The FSLS iterative program was compiled in the C language. The content of each component should reach the convergence criterion at the end of the calculations. After a basic theory analysis and experimental preparation, 13 national standard soil samples were detected using a spectrometer to test the feasibility of using the algorithm. The results show that the calculated contents of Ti, Fe, Ni, Cu, and Zn have the same changing tendency as the corresponding standard content in the 13 reference samples. Accuracies of 0.35% and 14.03% are obtained, respectively, for Fe and Ti, whose standard concentrations are 8.82% and 0.578%, respectively. However, the calculated results of trace elements (only tens of μg/g) deviate from the standard values. This may be because of measurement accuracy and mutual effects between the elements.

1 Introduction

Energy-dispersive X-ray fluorescence (EDXRF) is an analytical method for determining the concentration of micro and major elements in different matrices. This technology is widely used in pharmaceutical analysis, antique authentication, and the exploration of mineral resources because of its rapid and nondestructive analysis process [1]. However, calibrating the efficiency of the instrument and matrix effect correction are the most realistically difficult problems in EDXRF. Many studies focusing on these two problems have been conducted. Since the 1950s, the mutual impact coefficient method and the fundamental parameter method (FPM) have been the main methods used in X-ray fluorescence analyses. The former consists in calibrating efficiency with standard samples to solve the problem of overlapping-peak peeling, but the X-ray fluorescence spectrometer has to be rescaled when there are large matrix differences in the sample [2]. The FPM is a widely used method based on physical parameters. It can effectively reduce the absorption and enhancement effects. The major advantage of the FPM is the minimum number of standard samples required for efficiency calibration [3,4,5]. However, most measurements based on the FPM are developed for wavelength-dispersive X-ray fluorescence (WDXRF) analysis [6,7,8,9]. By comparison, the ability to carry out in situ measurement works is one of the major advantages of EDXRF. In conventional EDXRF, elements with an atomic number below 19 are commonly difficult to analyze in the field. Thus, a “dark matrix” must exist in the sample. To correct for the variations caused by the matrix effect, the backscatter fundamental parameter method was proposed and developed [10, 11]. In addition, several methods were developed for EDXRF analysis, which can be referred to in a previous review [12]. In recent years, using an effective atomic number $Z_{\text{eff}}$ as a representative of the “dark matrix” has become the prevailing approach, and this parameter has also been studied in our previous works [13, 14].

Usually, the net characteristic X-ray intensity of each element is derived once the EDXRF spectrum is taken, and then, the element concentration is calculated via the FPM [15]. This independent quantitative method has proven to be workable. However, it remains a complex and time-consuming process that requires a high degree of experience and knowledge from the instrument user [16]. The full-spectrum least-squares (FSLS) method is a multivariate calibration method that is able to, based on the conventional least-squares principle, increase the selectivity of matrix components and provide the possibility of detecting samples as outliers. Theoretically, it can deal with considerable peak overlaps owing to its lower dependence on the shape of predefined peak lines [17]. However, this multivariate calibration method is rarely applied in EDXRF except for near-infrared and infrared spectroscopic analytical studies [16,17,18,19]. Therefore, this paper explores the feasibility of using FSLS regression for the quantitative EDXRF determination of the concentration of micro and major elements, such as titanium (Ti), iron (Fe), nickel (Ni), copper (Cu), and zinc (Zn), which are common components in solid samples.

2 Theoretical descriptions

In EDXRF, the basic equations that relate the measured X-ray fluorescence intensity and the corresponding element composition were derived by Sherman and other authors [16, 20, 21]. When an X-ray beam irradiates the surface of the sample, the absorption and scattering of the original irradiated beam will occur along the trajectory path. The attenuation of the original irradiated beam caused by absorption and scattering not only is proportional to the incident X-ray intensity but also depends on the thickness, density, and the number of encountered atoms per unit cross section of the absorber. Based on this theory, the absorption of the original irradiated beam can be divided into three processes: I, II, and III, as shown in Fig. 1.

Fig. 1

(Color online) Measurement system structure for EDXRF

Process I represents the attenuation of the original X-ray (whose wavelength is $\lambda$ and were produced by an X-ray tube) in the sample. Process II indicates the absorption of attenuated X-ray in the target element $i$ in an infinitesimal volume. $t$ is the incident depth of the X-ray along the surface normal. Element $i$, whose weight concentration is $C_{i}$, emits characteristic X-ray after being excited, the magnitude of which depends on the excitation factor $(E_{i} )$. $E_{i}$ is closely related to the jump ratio $(J_{i} )$ for the absorption edge, transition probability $(f_{i} )$, and fluorescence yield $(w_{i} )$, which can be expressed as follows [21]:

$$E_{i} = J_{i} f_{i} w_{i} .$$

(1)

These characteristic X-rays are radiated evenly in all directions and are then attenuated by being absorbed by matter in process III. Only the X-ray with a spatial solid angle of ${\varOmega \mathord{\left/ {\vphantom {\varOmega {4\pi }}} \right. \kern-0pt} {4\pi }}$ can enter the detector. Therefore, the primary fluorescence intensity $(S_{i,\lambda } )$ for a line of element $i$ can be expressed as follows [20]:

$$S_{{i,\lambda_{ } }} = ({\varOmega \mathord{\left/ {\vphantom {\varOmega {4\pi }}} \right. \kern-0pt} {4\pi }})\varepsilon_{i} \csc \varphi_{1} C_{i} E_{i} \mathop \smallint \limits_{{\lambda_{\hbox{min} } }}^{{\lambda_{{{\text{abs}},i}} }} \frac{{u _{i,\lambda } }}{{u_{s}^{{\prime }} + u_{s}^{{\prime \prime }} }}I_{\lambda } {\text{d}}\lambda ,$$

(2)

$$u_{\text{s}}^{{\prime }} = u_{{{\text{s,}}\lambda }} {\text{cosec}}\varphi_{1} ,$$

(3)

$$u_{\text{s}}^{{\prime \prime }} = u_{{{\text{s}},\lambda_{i} }} {\text{cosec}}\varphi_{2} .$$

(4)

Here, $\varphi_{1}$ is the angle between the central beam of the tube’s radiation and the surface of the sample. $\varphi_{2}$ is the takeoff angle of the measured radiation from the sample. $\varepsilon_{i}$ is the intrinsic detection efficiency of the characteristic X-ray of element $i_{ }$. $\mu_{i,\lambda }$ is the mass absorption coefficient of element $i_{ }$ for the original wavelength $\lambda_{ }$, and its unit is cm²/g. Similarly, $\mu_{\text{s},\lambda }$ is the mass absorption coefficient of the sample and $\mu_{{\text{s},\lambda_{i} }}$ is its mass absorption coefficient for the characteristic X-ray wavelength $\lambda_{i}$. It should be noted that the original X-ray spectrum produced by the X-ray tube includes the continuous spectrum and the characteristic spectrum [2]. The former is the main X-ray source that excites the sample. For any given experimental conditions, there is no exact analytical expression to obtain the incident spectrum. Thus, the continuous integral over $\lambda_{ }$ is divided into a number of intervals $\Delta \lambda_{k}$, and in this way, the spectral distribution of the original radiation can be obtained as follows:

$$\mathop \smallint \nolimits {}_{{\lambda_{\hbox{min} } }}^{{\lambda_{{{\text{edge}},i}} }} I_{\lambda } {\text{d}}\lambda \to \mathop \sum \nolimits {}_{{\lambda_{\hbox{min} } }}^{{\lambda_{{{\text{edge}},i}} }} I_{{\lambda_{k} }}\Delta \lambda_{k} = I_{0} \mathop \sum \nolimits {}_{{E_{{{\text{abs}},i}} }}^{{E_{\hbox{max} } }} \varphi (E).$$

Here, $\varphi (E)_{ }$ is the original X-ray spectral distribution, which can be simulated via Monte Carlo methods [22]. The simulation results for electrons (≤ 30.3 keV) bombarding rhodium (Rh) target atoms obtained with the MCNP program are shown in Fig. 2.

Fig. 2

Original X-ray spectral distribution obtained via Monte Carlo methods

In addition, $I_{0}$ is the total intensity of the original incident X-ray, which can be categorized, much like ${\varOmega \mathord{\left/ {\vphantom {\varOmega {4\pi }}} \right. \kern-0pt} {4\pi }}$, $\varepsilon_{i}$, and ${ \csc }\varphi_{1}$, into a factor related to the structure of the instrument. The synthetic effect of these four parameters is taken as an important factor, $G_{i}$, which can be expressed as follows [20, 21]:

$$G_{i} = _{ } ({\varOmega \mathord{\left/ {\vphantom {\varOmega {4\pi }}} \right. \kern-0pt} {4\pi }})\varepsilon_{i} \csc \varphi_{1} I_{0} .$$

(5)

Thus, Eq. (2) can be simplified as follows:

$$S_{i,E} = G_{i} C_{i} E_{i} \mathop \sum \nolimits {}_{{E_{{{\text{abs}},i}} }}^{{E_{\hbox{max} } }} \frac{{u _{i,E} }}{{u_{s}^{{\prime }} + u_{s}^{{\prime \prime }} }}\varphi (E),$$

(6)

where $S_{{i,E_{ } }}$, $u_{i,E}$, $u_{s,E}$, and $u_{{s,k_{\alpha } }}$ are equivalent to $S_{{i,\lambda_{ } }}$, $\mu_{i,\lambda }$, $\mu_{s,\lambda }$, and $\mu_{{s,\lambda_{i} }}$, respectively. For multi-element samples, when the excited wavelength of another interfering element $j$ (whose concentration is $C_{j}$) in the matrix is less than the wavelength of the absorption edge of element $i$, secondary fluorescence will happen. This process also includes processes I, II, and III. The secondary fluorescence formula can be expressed as follows [20, 23]:

$$S_{{ij,E_{ } }} = S_{{i,E_{ } }} e_{{ij,E_{ } }} C_{j} ,$$

(7)

where

$$e_{ij,E} = 0.5E_{j} u_{{i,E_{j} }} \left( {\frac{{u_{j,E} }}{{u_{i,E} }}} \right)\left[ {\frac{1}{{u_{s}^{{\prime }} }}\ln \left( {1 + \frac{{u_\text{s}^{{\prime }} }}{{u_{{\text{s},E_{j} }} }}} \right) + \frac{1}{{u_\text{s}^{{\prime \prime }} }}\ln \left( {1 + \frac{{u_\text{s}^{{\prime \prime }} }}{{u_{{\text{s},E_{j} }} }}} \right)} \right].$$

(8)

Here, $u_{{i,{{E}}_{j} }}$ and $u_{{s,E_{j} }}$ are the mass absorption coefficient of element $i_{ }$ and the mass absorption coefficient of the sample for the characteristic X-ray energy $E_{j}$ of element $j$, respectively. The percentage of the tertiary X-ray fluorescence in the total detected intensity does not exceed 4%, even in the most extreme cases, and can be ignored in the calculations. Therefore, the detected intensity $I_{{i,E_{ } }}$ of the fluorescence radiation of element $i_{ }$ in the sample can be expressed as the sum of the primary fluorescence intensity and the secondary fluorescence intensity [21].

$$I_{{i,E_{ } }} = S_{{i,E_{ } }} + S_{{ij,E_{ } }} .$$

(9)

Actually, in X-ray spectrum analyses, the count rate of one energy channel is not related only to the target element; it is also affected by other elements in the matrix. Therefore, the spectrum can be regarded as the sum of the contributions of a series of elements within the same matrix. Once the theoretical spectrum of a single element is obtained, the optimal response relationship between the content of that element and the detected X-ray spectrum can be established. Based on the fluorescence formulas, the calculated intensity is just a value equal to the total area of the detected X-ray peak for one element. Thus, the intensity at the peak position needs to be Gaussian-broadened [24] to the scale edge of the spectrometer, which is approximately 0–1023 channels. This can be described in the following procedure.

The Gaussian distribution prototype formula [25] is shown in the following.

$$f(x) = \frac{1}{{\sigma \sqrt {2\pi } }}e^{{\frac{{ - (x - \mu )^{2} }}{{2\sigma^{2} }}}} .$$

(10)

Here, $\mu$ is the distribution expectation, which is the characteristic peak position in EDXRF spectral analysis. $\sigma$ is the distribution’s standard deviation, which is indispensable in the Gaussian-broadening process. Experimentally, nine kinds of single-element samples, namely potassium (K), calcium (Ca), titanium (Ti), iron (Fe), nickel (Ni), zinc (Zn), strontium (Sr), zirconium (Zr), and molybdenum (Mo), were prepared, as given in Table 1 in the next section. The actual EDXRF spectra were Gaussian fitted with the OriginPro9 program, and then, the $\sigma$ of each element’s spectral line was obtained. The functional relationship between the characteristic energy and $\sigma$ is shown in Fig. 3.

Table 1 Component proportions of the 45 standard samples (the mass of each sample was 0.05 kg)

Fig. 3

(Color online) Relationship between $\sigma$ and the characteristic energy of different elements

Once the $\sigma$ of element $i$ is obtained via interpolation through Eq. (10), the characteristic X-ray intensity $I_{{i,E_{ } }}$ can be Gaussian-broadened into 1023 energy channels: $\left( {\begin{array}{*{20}c} {E_{i,0} ,} & {E_{i,1} ,} & {\begin{array}{*{20}c} {E_{i,2} ,} & \ldots & {E_{i,1023} } \\ \end{array} } \\ \end{array} } \right)^{\text{T}}.$ Unlike conventional quantitative EDXRF methods, the FSLS method directly relates element concentrations to the detected EDXRF spectrum. Therefore, it is able to establish the response relationship between the theoretical spectrum and the actual detected spectrum, which can be described in the following form:

$$\left( {\begin{array}{*{20}c} {E_{1,0} } & {E_{2,0} } & {\begin{array}{*{20}c} \ldots & {E_{i,0} } \\ \end{array} } \\ {E_{1,1} } & {E_{2,1} } & {\begin{array}{*{20}c} \ldots & {E_{i,1} } \\ \end{array} } \\ {\begin{array}{*{20}c} {E_{1,2} } \\ \vdots \\ {E_{1,1023} } \\ \end{array} } & {\begin{array}{*{20}c} {E_{2,2} } \\ \vdots \\ {E_{2,1023} } \\ \end{array} } & {\begin{array}{*{20}c} {\begin{array}{*{20}c} \ldots & {E_{i,2} } \\ \end{array} } \\ {\begin{array}{*{20}c} & \vdots \\ \end{array} } \\ {\begin{array}{*{20}c} \ldots & {E_{i,1023} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \right) \cdot \left( {\begin{array}{*{20}c} {C_{1} } \\ {C_{2} } \\ {\begin{array}{*{20}c} \vdots \\ {C_{i} } \\ \end{array} } \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {A_{1} } \\ {A_{2} } \\ {\begin{array}{*{20}c} {A_{3} } \\ {\begin{array}{*{20}c} \vdots \\ {A_{1023} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \right).$$

(11)

This equation can be simplified as follows:

$$P_{0} \cdot C_{x} = P.$$

(12)

Here, $P_{0}$ represents the theoretical pure-element matrix with a number of rows equal to the number of channels in the full spectrum and a number of columns equal to the number of pure elements of interest in the unknown sample. Matrix $C_{x}$ consists of the calculated contents of the corresponding elements, which are updated after each iteration. $P$ is the actual full spectrum measured with a portable EDXRF spectrometer. This instrument uses a Rh-anode X-ray tube with a tube voltage of 30 kV. A Si-PIN semiconductor detector with a resolution of 196 eV (at ⁵⁵Fe) was employed. The measuring time was 300 s for each sample. The overdetermined equations presented in Eq. (11) have a unique smallest least-squares solution, which can be referred to in previous literature [26, 27]. The determination of factor $G_{i}$ and each mass absorption coefficient will be described in detail in the next section.

3 Experiments

Determining factor $G_{i}$ and the mass absorption coefficients is the key issue in the entire algorithm. As shown in Eq. (5), factor $G_{i}$ can be seen as a structural constant for one specific element $i$ in a certain detector system. However, it is difficult to directly calculate it. Replacing the absolute X-ray intensity with the relative intensity is the most common method to eliminate some physical parameters, including factor $G_{i}$ [2, 28]. In this paper, a more innovative approach is proposed to calculate $G_{i}$ with standard single-element samples through Eq. (6) and then to establish a function of $G_{i}$ with the characteristic X-ray energy of different elements. Nine kinds of standard single-element (K, Ca, Ti, Fe, Ni, Zn, Sr, Zr, and Mo) samples with five different contents ($9 \times 5{ = }45$) were prepared in our laboratory. The component proportions of the 45 samples are listed in Table 1.

Here, SiO₂ was used as the matrix in the sample and H₃BO₃ was used as an adhesive for different contents. The $G_{i}$ profile versus the characteristic X-ray energies of K, Ca, Ti, Fe, Ni, Zn, Sr, Zr, and Mo is shown in Fig. 4.

Fig. 4

(Color online) Relationship between factor $G_{i}$ and the characteristic X-ray energies of different elements

As shown in Fig. 4, the fitted formula between factor $G_{i}$ and the characteristic X-ray energy $E$ is:

$$G_{i} = 2 \times 10^{6} E^{ - 1.393} .$$

(13)

The square of the correlation coefficient (R²) is 0.9721. Therefore, the $G_{i}$ factor of any element between K and Mo can be calculated via interpolation from Eq. (13). The mass absorption coefficient is another basic parameter that is closely related to the contents of unknown elements. The matrix information of the sample is the key to obtain the mass attenuation coefficient in EDXRF. In fact, the interaction of X-ray with compounds or mixtures along their trajectories can be equivalent to their interaction with single elements. The atomic number of this equivalent element is called the effective atomic number of this compound or mixture. Maynerod [29] first proposed a formula to calculate the effective atomic number $Z_{\text{eff}}$, which is given as follows:

$$Z_{\text{eff}} = \sqrt[m]{{\mathop \sum \nolimits f_{i} Z_{i}^{m} }}.$$

(14)

Here, $Z_{i}$ is the atomic number of element $i_{ }$ in the irradiated matter and $f_{i}$ is the electron percentage of element $i_{ }$ in the irradiated matter. Index “$m$” is 2.49. Zhang et al. [13] divided the elements in a multi-component sample into two groups, including the known one and the “dark matrix.” $XZ_{\text{eff}}$ is defined to represent the effective atomic number of the “dark matrix.” Thus, the calculation formula of $Z_{\text{eff}}$ for a multi-component sample can be expressed as follows:

$$Z_{\text{eff}}^{2.94} = \mathop \sum \nolimits C_{i} Z_{i}^{2.94} + (1 - \mathop \sum \nolimits C_{i} )XZ_{\text{eff}}^{2.94} ,$$

(15)

where $C_{i}$ is the concentration of element $i$. Figure 5 is the actual X-ray spectrum of sample No. 11, obtained with the EDXRF spectrometer. The blue line stands for the raw X-ray spectrum. After smoothing and deducing background scattering (red line), the net X-ray spectrum (black line) can be obtained.

Fig. 5

(Color online) Actual X-ray spectrum of sample No. 11 (for a test time of 300 s)

Rayleigh and Compton scattering peaks exist in the net EDXRF spectrum. Many researchers have concluded that the coherent-to-Compton scattering cross-sectional ratio depends only on the effective atomic number of composite materials [30]. Duvauchelle et al. [31] pointed out that a given $Z_{\text{eff}}$ should define a mixture or compound based on the intensity ratio of the Rayleigh-to-Compton scattering. İçelli et al. [32] successfully applied the coherent-to-Compton scattering cross-sectional ratio to obtain the $Z_{\text{eff}}$ of some elements ($26 \le Z \le 82$). Thus, in this paper, we define $R$ as the ratio of the coherence-to-Compton scattering net peak. As shown in Fig. 6, the functional relationship between $Z_{\text{eff}}$ and $R$ for different elements can be established as follows.

Fig. 6

(Color online) Relationship between $R$ and $Z_{\text{eff}}$ for different elements

The relationship between $Z_{\text{eff}}$ and $R$ is fitted with a quadratic equation, as shown in Eq. (16).

$$Z_{\text{eff}} = - 125.794R^{2} + 169.494R - 18.729.$$

(16)

The square of the correlation coefficient is 0.9608. The mass absorption coefficient for a multi-component sample is equal to the sum of the weighted mass absorption coefficient of each component, including the known contents and the “dark matrix.” This parameter can be derived [14] as follows:

$$\mu_{\text{S},E}^{ } = \mathop \sum \nolimits C_{i} \mu_{{i,{{E}}}} + (1 - \mathop \sum \nolimits C_{i} )\mu_{{XZ_{\text{eff}} , {{E}}}} .$$

(17)

Here, $C_{i}$ is the concentration of element $i_{ }$, which should be updated after each iteration of Eq. (11). $XZ_{\text{eff}}$ is the effective atomic number of the “dark matrix,” whose mass absorption coefficient is defined as $\mu_{{XZ_{\text{eff}} ,E}}$. Related parameters and formulas for calculating the mass absorption coefficients can be referred to in the cross-sectional manual [33]. $(1 - \mathop \sum \nolimits C_{i} )$ is the concentration proportion of the unknown “dark matrix.” It should be noted that each mass absorption coefficient will be updated and the $P_{0}$ matrix will be reconstituted after each iteration. A simplified flow diagram of the whole algorithm can be drawn as shown in Fig. 7.

Fig. 7

Simplified flow diagram of the whole algorithm

4 Results and discussion

In this paper, 13 national standard soil (Nss) samples were used to verify the feasibility of the FSLS algorithm. The iterations stop when the difference between the current and the former calculated contents is less than 0.005%. The results of the algorithm are given in Table 2.

Table 2 Calculated results and relative errors for elements Ti, Fe, Ni, Cu, and Zn, out of the 13 Nss samples

Ti, Fe, Ni, Cu, and Zn were the most common elements in the geological samples. From Table 2, Ti and Fe, whose concentration exceeded several thousands of ppm, were labeled using percentage symbols (%), whereas the unit used for Ni, Cu, and Zn was micrograms per gram (μg/g). It is clear that the calculated contents for high-concentration elements are close to the standard contents. In particular, the calculated results for Fe are in good agreement with the standard contents and the corresponding relative errors are comparatively lower than those of the low-concentration elements. To intuitively analyze the trend changes in the content of each element, the resulting data were plotted as column graphs, with the error bars [34] representing standard deviation, in Fig. 8. The red columns stand for the standard contents, and the shadow columns represent the calculated contents. The blue line is the relative error level.

Fig. 8

(Color online) Column charts showing calculated content and standard content for elements a Ti, b Fe, c Ni, d Cu, and e Zn

As shown in Fig. 8, the calculated content has the same variation tendency as the standard content among the 13 Nss samples, which reflects the feasibility of using the FSLS algorithm. There is a close relationship between the calculated content and the standard content except for Ni, whose relative error is larger than 100%. The analysis results are as follows.

• (1) For high-concentration elements, such as Fe (as shown in Fig. 8b), whose maximum standard content is 8.828%, the minimum relative error is 0.35% (Nss 5). However, the relative error increases with the decrease in its standard content. When the standard content is 2.462%, the relative error reaches 18.64% (Nss 2), which is acceptable in EDXRF analyses. This is mainly because Fe is the major element in the solid samples; its concentration exceeds 1.39%. The background scattering intensity of the original X-ray beam can be considered inversely proportional to the element contents. Therefore, comparatively low backscattered radiation contributes little to the characteristic peak area of Fe during the background-deducing step. In addition, the count rate of the detector is proportional to the content of Fe, as shown in Fig. 9a. Increasing the count rate would effectively reduce the statistical error. The calculation results of Ti, which is also a major element in the Nss samples, are similar to those of Fe. The differences between the red columns and the shadow columns for Ti in Fig. 8a exist for the same reason as for Fe.

  Fig. 9

  

  Linear relationship between the count rate of the detector and the standard content of a Fe, b Ti, and c Ni

• (2) Ni, Cu, and Zn are microelements in the 13 Nss samples because their average contents are just 33.91 μg/g, 62.85 μg/g, and 115.08 μg/g, respectively. Their actual characteristic X-ray peaks are probably affected by the high backscattered radiation. As shown in Fig. 8c–e, the shadow columns of the calculated contents for these elements are almost higher than the red columns of their standard contents. This may be due to the absorption enhancement effects of other trace elements, such as Ce, Ba, and Pb, which leads to the actual characteristic X-ray intensity line deviating from the true intensity line. Among the 13 Nss samples, the one with a high content of Ni, Cu, and Zn has a smaller relative error than the other samples. The differences found between them are significantly related to measurement accuracy.

• (3) However, for microelement Ni and as shown in Fig. 8c, the maximum relative error is 292.69%, which corresponds to 12 μg/g (Nss 3), and the minimum relative error is 76.49%, which corresponds to 64 μg/g (Nss 4). The average error is up to 149.80%, and the reason for this is not just related to the trace amount of Ni in the solid samples. Except for the absorption enhancement effect of the other microelements (which are neglected in this FSLS algorithm), the peak overlap effect caused by the elements (such as Co) adjacent to Ni is another main factor. Besides, the net characteristic X-ray intensity of Ni is heavily affected by the characteristic K_β rays of Fe, which are beyond the resolution capability of our detector. This leads to a strong increase in the weak peak area of the characteristic X-ray for Ni, and thus, the calculated content of Ni is far in excess of the standard value. The former can be used to calibrate the inter-element effects via empirical coefficient approaches and, in the subsequent step, increasing the number of the spectral variables in the P₀ matrix will, to some extent, reduce the mutual interference between elements. Furthermore, as shown in Fig. 9c, there is a nonlinear relationship between the content of the Ni element and the counting rate of the system. Therefore, statistical fluctuations during the spectral data acquisition will seriously influence the final calculated results. It is crucial to improve the accuracy of detection systems. In addition, a single residual matrix is needed to optimize the FSLS algorithm, which can effectively modify the mismatch between the algorithm model and the X-ray characteristic spectrum data caused by statistical fluctuations, backscattered radiation, etc.

5 Conclusion

This paper introduces an FSLS method to quantitatively perform EDXRF analysis for unknown solid samples. Compared with the conventional FPM approach, it is a multivariate calibration method that is able to increase component selectivity and provide the possibility of detecting a sample as an outlier. The $G_{i}$ factor and mass absorption coefficients are critical parameters to describe the features of the fluorescence spectra of pure elements. It is innovative to obtain the $G_{i}$ factor for different elements through a $G_{i} - E$ function. To take full account of the unknown components in the sample, $XZ_{\text{eff}}$ can be used to represent the effective atomic number of the “dark matrix.” The coherent-to-Compton scattering peak ratio is applied to obtain the $Z_{\text{eff}}$ of some of the elements. Based on fluorescence formulas, the calculated X-ray intensity should be Gaussian-broadened at the peak position to the scale edge (0–1023 channels) of the spectrometer. Subsequently, the response relationship between the theoretical spectral matrix and the actual X-ray spectrum can be established. The results show that using the FSLS method is feasible for light elements (≥ hundreds of μg/g) in unknown samples. However, for microelements, high backscattered radiation, measurement accuracy, and inter-elements effects are the main sources of relative errors. In particular, for Ni, the characteristic K_β rays of Fe affect its weak characteristic X-ray peak. In future researches, increasing the number of the spectral variables in the P₀ matrix and optimizing the FSLS algorithm with a residual matrix will, to some extent, modify the mismatch between the algorithm model and the X-ray data and reduce the mutual interference between elements.