A developed method of qu­anti­tative phase analysis (QPA) to solve the effect of preferred orientation in phosphate materials

Chen, H.; Wang, C.; Li, J.

doi:10.1107/S2053229625003237

research papers

STRUCTURAL
CHEMISTRY

ISSN: 2053-2296

Volume 81| Part 5| May 2025| Pages 282-287

https://doi.org/10.1107/S2053229625003237

A developed method of quantitative phase analysis (QPA) to solve the effect of preferred orientation in phosphate materials

He Chen,^a Chunjian Wang ^a,^b ^* and Jingmin Li ^c

^aFaculty of Material Science and Engineering, Kunming University of Science and Technology, Kunming 650093, People's Republic of China, ^bAnalytic & Testing Research Center of Yunnan, Kunming University of Science and Technology, Kunming 650093, People's Republic of China, and ^cYunnan Ledford Technology Co. Ltd, Kunming 650093, People's Republic of China
^*Correspondence e-mail: wangchunjian@kust.edu.cn

Edited by M. Rosales-Hoz, Cinvestav, Mexico (Received 12 January 2025; accepted 9 April 2025; online 16 April 2025)

Quantitative phase analysis (QPA) by X-ray diffraction is widely used in materials, minerals, metallurgy, etc. But when preferred orientation exists in samples, QPA by single peaks in diffraction patterns will be seriously affected and becomes less exact, for example, for phosphate materials. As an alternative, whole pattern methods (especially the Rietveld method) can be utilized positively and the effect of preferred orientation can be solved mathematically. But application of the Rietveld method generally takes a lot of time, not only in high-accuracy pattern acquisition, but also in continuously refining many parameters for multiple iterative computation, which is not applicable to situations where rapid or automatic QPA is required, such as industrial production, customs inspection, and so on. In this article, a new mathematical method was developed and discussed, and was then reasonably simplified for convenient operation. The simplified method was tested and examined using the N₂H₉PO₄ phase, which can produce preferred orientation easily. The results indicated that the QPA deviation is reduced from about 33% using the single-peak method to less than 1% using the new simplified method. Use of the new method and its simplified version is recommended when preferred orientation exists and rapid or automatic QPA is required.

Keywords: X-ray diffraction; XRD; quantitative phase analysis; QPA; phosphate; preferred orientation; reference intensity ratio; RIR; Rietveld; crystal structure.

1. Introduction

X-ray diffraction (XRD) has become one of the most commonly used characterizing technologies in scientific research and industrial production in recent decades (Rabiei et al., 2020 ; Fattouh et al., 2023 ; Surdu & Győrgy, 2023 ; Santos et al., 2023 ; Ali et al., 2022 ; Narayanaperumal et al., 2024 ; Zhang et al., 2022b ). Quantitative phase analysis (QPA) by XRD plays an important role in determining the components of materials (Cabrera-German et al., 2017 ; Popović, 2020 ; Arita-Merino et al., 2020 ). But when the orientation of grains in materials is distributed regularly (named preferred orientation) (Viganò & Ludwig, 2020 ; Ulusoy, 2023 ; Behera et al., 2021 ), the peak intensities in the XRD patterns can change and QPA by the single-peak method traditionally become inexact (de Matos et al., 2022 ). Many methods have been developed to reduce or eliminate the effect of preferred orientation on QPA (Akinbodunse et al., 2024 ; Zhang et al., 2022a ; Lyza et al., 2023 ). For example, by grinding the powder into finer particles and subjecting the samples to high-speed rotation when conducting an XRD test, the number of irradiated grains can be increased (Li et al., 2022 ; Fu et al., 2023 ).

The whole-pattern methods of QPA (especially the Rietveld method) are well known to reduce the effect of preferred orientation, in which all the peaks of the XRD patterns are utilized for analysis (Paiva et al., 2019 ; Rowles, 2021 ; Nakamura et al., 2024 ). But it takes much time to gather fully detailed and precise XRD patterns before analysis using the Rietveld method (Saville et al., 2021 ). In addition, when executing the Rietveld method, much time is still needed to carefully adjust and refine many parameters, such as profile parameters, microstructure parameters, and so on, in order to acquire better iteration results (Spiliopoulou et al., 2020 ; Savage et al., 2023 ). Due to its limited testing and analysis efficiency, the Rietveld method is not suitable for rapid, automatic and standardized QPA in scenarios such as industrial production and customs inspection (van Ravenhorst et al., 2021 ).

In this artcicle, mixtures with different weight ratios of diammonium phosphate (N₂H₉PO₄), easily producing preferred orientation, and tetrasodium pyrophosphate (Na₄P₂O₇) were prepared and tested by XRD, and then the effect of preferred orientation was thoroughly examined. A new mathematical method was developed and is discussed according to the multiplicity of crystal planes, and was reasonably simplified for convenient operation. The QPA deviation using the new simplified method was smaller than 1%, while the deviation was about 33% using the reference intensity ratio (RIR) method (a widely used single-peak method). The new simplified method can be used easily and results obtained rapidly because it is not necessary to adjust many parameters.

2. Experimental and results

The experimental samples with different weight ratios were prepared by mixing N₂H₉PO₄ powder with preferred orientation and Na₄P₂O₇ powder. The XRD test was carried out using an EMPYREAN X-ray diffraction instrument produced by Malvern Panalytical. XRD parameters: Cu Kα, Pixcel 1D detector, BBHD module filter, voltage 40 kV, current 40 mA, step length 0.02626° and 75 s per step (testing about 20 min). Data analysis was conducted using JADE standard software (https://www.icdd.com/mdi-jade/) with Pseudo-Voigt Function fitting applied. The QPA results obtained via the Rietveld method are shown in Table 1, demonstrating that the sample was mixed uniformly.

Table 1
N₂H₉PO₄–Na₄P₂O₇ weight ratio and QPA deviation by the Rietveld method

N₂H₉PO₄–Na₄P₂O₇, mixtures with preferred orientation
Weight ratio	9:1	7:3	1:1	3:7	1:9
N₂H₉PO₄ wt%	89.70	69.80	50.10	29.60	9.70
N₂H₉PO₄ deviation (%)	−0.30	−0.20	+0.10	−0.40	−0.30

N₂H₉PO₄ grains consistently exhibit regular distribution, which can be attributed to the tendency of certain N₂H₉PO₄ grains to gather at specific positions, thereby leading to a consistent orientation, as shown in Fig. 1(c) (left). This phenomenon results in a sudden increase in the diffraction peak intensity of the corresponding crystal plane, deviating significantly from the theoretical Bragg intensity, as demonstrated in Fig. 1(a). Conversely, Na₄P₂O₇ grains exhibit a random orientation and irregular distribution, as illustrated in Fig. 1(c) (right), resulting in no preferred orientation for the Na₄P₂O₇ phase.

Figure 1
(a) XRD patterns of N₂H₉PO₄ powder, (b) XRD patterns of Na₄P₂O₇ powder and (c) grain orientation distribution.

Measuring the diffraction peak intensity of N₂H₉PO₄ and Na₄P₂O₇ before calculating the percentage content for each phase is essential. It was observed that for the N₂H₉PO₄ phase, all Bragg lines were closely matched with actual measured intensities. However, one diffraction peak exhibited a significant intensity increase, as shown in Fig. 2(b), indicating a regular distribution of grains. For further QPA, a mixture of N₂H₉PO₄ and Na₄P₂O₇ in a 9:1 weight ratio was used as an example. The eight strongest peaks were fitted, as depicted in Fig. 2(c), and the measured intensity of each diffraction peak was obtained.

Figure 2
(a) XRD data with different weight ratios and phase analysis, (b) the change of the preferred orientation of N₂H₉PO₄ and (c) the eight strongest peaks of N₂H₉PO₄–Na₄P₂O₇ mixtures with a 9:1 weight ratio fitted using PVE to obtain peak shape parameters.

The primary XRD quantitative principle should obey the fundamental equations relating diffraction intensity (I_a) to concentration (W_a). The RIR method with a powder specimen is:

$[W_a = {{I_a}\over{{\rm RIR}_a}} \times \left( \sum _{i=1}^{n} {{I_i}\over{{\rm RIR}_i}} \right)^{-1} \eqno(1)]$

where W_a is the percentage by weight (wt%) of the phase in the mixture, RIR_a is the reference intensity ratio and I_a is the diffraction intensity (integral area). Subsequently, the RIR values of N₂H₉PO₄ and Na₄P₂O₇ were chosen as 0.74 and 1.07, respectively, and the QPA results obtained by the RIR method are shown in Table 2.

Table 2
QPA results of the RIR method for N₂H₉PO₄–Na₄P₂O₇ mixtures with different weight ratios

N₂H₉PO₄–Na₄P₂O₇, artificial mixtures
Weight ratio	9:1	7:3	1:1	3:7	1:9
N₂H₉PO₄ wt%	97.48	90.74	83.44	37.39	7.45
N₂H₉PO₄ deviation (%)	+7.48	+20.74	+33.44	+7.39	−2.25

Table 2 presents the percentage content of the N₂H₉PO₄ phase as determined by the RIR method. Notably, the QPA deviation with this method was significant. The maximum deviation reached 33.44% and more than half of the mixtures showed very large deviations. This high deviation is mainly due to the regular distribution of N₂H₉PO₄ grains, which leads to a distortion of the strongest peak intensity and results in inaccurate quantitative analysis (QPA) results.

3. Mathematical model and derivation

In powder diffraction, to ensure that the diffraction peaks of all the crystal planes appear in the diffraction pattern and that each diffraction peak intensity is as pronounced as possible, it is generally required during measurement that the grains in the sample exhibit a completely random spatial orientation. Assuming that the orientation of the grains in the sample is entirely random, the number of each crystal plane (assuming no equivalent crystal plane) participating in the diffraction can be regarded as consistent at this time, and the number is assumed to be ρ. When considering equivalent crystal planes, the diffraction signals generated by multiple equivalent crystal planes will overlap at a diffraction position, resulting in the signal of the diffraction position superimposed multiple times, and this overlap number is expressed as m_i, where m_i represents the multiplicity of crystal planes. The number of crystal planes corresponding to each measured diffraction peak is then ρ·m_i.

In conventional symmetric diffraction geometry, all the diffracting crystal planes are parallel to the sample surface, meaning that each grain – assuming that all the crystal planes can participate in diffraction – has only one crystal plane participating in diffraction. It is assumed that the total number of crystal planes participating in the diffraction is N, which can be expressed as follows:

$[ N = \rho \cdot \sum _{i=1}^{n} m_i \eqno(2)]$

where N also represents the total number of grains participating in diffraction. For a specific sample under specific testing conditions (where the number of grains within the X-ray beam irradiation area remains constant, ensuring that the number of grains remains unchanged), N can be regarded as a constant.

When grains are completely randomly oriented, diffraction peak intensity is influenced by factors such as multiplicity, crystal structure, diffraction geometry, environmental temperature, material absorption and phase content. If these factors remain stable, variations in peak intensity can be primarily attributed to preferred orientation. When the grains in a sample are no longer completely randomly oriented, some grain orientations are consistent at this time (called preferred orientation), the number of crystal planes participating in the diffraction of the corresponding orientation will increase accordingly and the intensity of the diffraction peak of the crystal plane will increase accordingly. Since the total number of grains remains unchanged, the number of corresponding other crystal planes (which are still randomly oriented) will decrease accordingly, leading to a reduction in the intensity of their corresponding diffraction peaks. This is the principle whereby the preferred orientation affects the diffraction peak intensity.

Based on this principle, as shown in Fig. 3, assuming that all grains are oriented uniformly and that each grain provides the same crystal plane to participate in the diffraction, all diffracted signals from these planes overlap at a single diffraction position, forming a single diffraction peak. The total number of participating planes corresponding to this orientation is

$[ \rho \cdot \sum _{i=1}^{n} m_i]$

Figure 3
$[\sum _{i=1}^{n} m_i]$ mechanism.

The diffraction peak intensity of this plane is the total intensity from all the crystal planes, denoted as I_i^t. Therefore, the diffraction intensity generated at this position by each grain is:

$[ I_i = {{I_i^t}\over{\rho \cdot \sum _{i=1}^{n} m_i}} \eqno(3)]$

where I_i^t is the intensity of the i-th diffraction line of the preferred material. Since the diffraction intensity is related to the multiplicity of the crystal planes, the diffraction intensity corresponding to each crystal plane is:

$[ I_i ^{'}= {{I_i^t}\over{\rho \cdot \sum _{i=1}^{n} m_i}}\cdot \rho \cdot m_i \eqno(4)]$

The diffraction intensity of any other k-th crystal plane is then:

$[ I_k = {{I_i^t}\over{\rho \cdot \sum _{i=1}^{n} m_i}}\cdot \rho \cdot m_i \cdot {{I_k^{\rm PDF}}\over{I_i^{\rm PDF}}} \eqno(5)]$

where

$[ {{I_k^{\rm PDF}}\over{I_i^{\rm PDF}}}]$

is the intensity ratio of two diffraction lines in a non-preferred oriented state; it can be calculated by theory or checked by the PDF (Powder Diffraction File) entry (i = 1, 2, …, n). The corrected total intensity is then:

$[ I_k^{'} = \sum \left( I_i^t \cdot {{I_k^{\rm PDF}}\over{I_i^{\rm PDF}}} \cdot {{m_i}\over {{\sum _{i=1}^{n} m_i}}} \right) \eqno(6)]$

where $[I_k^{'}]$ is the corrected total intensity in a preferred material. Equation (6) provides a mathematical model to calculate the corrected intensity of a specific crystal plane in the preferred material. This method only needs to know the diffraction intensity of each crystal plane to determine its corrected intensity under the assumption of uniform orientation. The principle is the same as that originally established by Gullberg & Lagneborg (1966 ) for quantifying the phase content in preferred materials. In addition, we define an RIR value for each diffraction peak, such that:

$[ {{I_k^{\rm PDF}}\over{R_k}} = {{I_i^{\rm PDF}}\over{R_i}} \eqno(7)]$

The corresponding RIR value for each diffraction peak is then:

$[ {{I_k^{\rm PDF}}\over{I_i^{\rm PDF}}} = {{R_k}\over{R_i}} \eqno(8)]$

Equation (6) can be simplified as follows:

$[ {{I_k^{'}}\over{R_k^{'}}} = \sum {{m_i}\over{\sum _{i=1}^{n}m_i}}\cdot {{I_i^t}\over{R_i}} \eqno(9)]$

Equation (9) represents the corrected $[ {{I_k^{'}}\over{R_k^{'}}}]$ , which can be substituted into the RIR method to calculate the phase content.

4. Discussion

In order to verify the accuracy of Equation (9), the percentage content is shown in Table 3.

Table 3
QPA deviation results obtained by Equation (9)

N₂H₉PO₄–Na₄P₂O₇, artificial mixtures
Weight ratio	9:1	7:3	1:1	3:7	1:9
N₂H₉PO₄ wt%	90.48	70.33	50.04	30.35	10.71
N₂H₉PO₄ deviation (%)	+0.48	+0.33	+0.04	+0.35	+0.71

As shown in Table 3, the QPA results calculated using Equation (9) exhibit deviations of less than 1%. This approach effectively corrects for preferred orientation and its deviation is also controlled within a reasonable range. However, applying Equation (9) requires knowledge of the multiplicity factor for each crystal plane. Unfortunately, many PDF entries lack complete multiplicity factor data. In cases where the crystal structure information is absent, indexing and calibrating the multiplicity factors for each diffraction peak become necessary, complicating the application of the method.

To simplify this method, it was assumed that all crystal plane multiplicity factors are equal (m₁ = m₂ = m₃ = … m_i), deriving the simplified method. However, in practice, the m values vary significantly. Applying this simplified method (where all m values are considered equal) is expected to introduce substantial deviations. Since each diffraction peak corresponds to a specific m value, but a single phase typically contains multiple diffraction lines, the sum $[ \sum _{i=1}^{n} m_i]$ in Equation (9) is generally large. When a sufficient number of diffraction peaks are considered, the result should approximate $[1 \over n]$ . Taking NaCl as an example, this behaviour is illustrated in Fig. 4.

Figure 4
Calculation results by $[ {{m_i}\over{\sum _{i=1}^{n} m_i}}]$ and $[1 \over n]$ , respectively.

The values of m_i were chosen as 4 and 2, respectively, and the plane multiplicity factors were calculated by progressively increasing the number of diffraction peaks. It is found that as the number of diffraction peaks increases, the plane multiplicity factors tended to be consistent, as shown in Fig. 4. When

$[ {{m_i}\over{\sum _{i=1}^{n} m_i}}]$

is regarded as $[1 \over n]$ , the curve trend is similar to that calculated using

$[ {{m_i}\over{\sum _{i=1}^{n} m_i}},]$

which indicates that the same result can still be obtained by substituting $[1 \over n]$ for

$[ {{m_i}\over{\sum _{i=1}^{n} m_i}}]$

and Equation (9) can be simplified to the following Equation (10):

$[ {{I_k^{'}}\over{R_k^{'}}} = {{1}\over{n}} \sum _{i=1}^{n} {{I_i^t}\over{R_i}} \eqno(10)]$

Taking N₂H₉PO₄ as an example, the quantitative results calculated using Equation (10) are shown in Fig. 5. It is worth noting that as the number of diffraction peaks increases, the QPA deviation decreases gradually. And when using eight diffraction peaks for the calculation, the deviation of the simplified method is controlled within 1%. This fully demonstrates that the simplified method, which treats the plane multiplicity factor as $[1 \over n]$ , can achieve comparable accuracy. Moreover, this method eliminates the need to determine and calculate plane multiplicity factors, significantly simplifying and accelerating the process.

Figure 5
QPA deviation results obtained using Equation (10)

5. Conclusions

(1) To solve the influence of preferred orientation on QPA, a new mathematical method has been derived and simplified rationally; see Equation (10) above.

(2) The simplified method was verified experimentally using mixtures of N₂H₉PO₄ with preferred orientation and Na₄P₂O₇ powder. The results confirmed that when the number of diffraction peaks involved in the calculation is sufficient (more than eight diffraction peaks), the QPA deviation can be controlled within 1%.

(3) This method does not require many structural parameters to be adjusted and is particularly suitable for fast automated results, but it requires the use of many diffraction peaks and is therefore not suitable for cases where preferred orientation is so severe that diffraction peaks are missing.

Acknowledgements

The authors thank the Kunming University of Science and Technology Analysis and Testing Research Center for assistance in facility support. This work was supported by the Science Research Foundation of Yunnan Education Bureau.

Conflict of interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Funding information

Funding for this research was provided by: Science Research Foundation of Yunnan Education Bureau (grant No. 2023J0128).

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