A full spectrum fitting method of powder X-ray diffraction pattern based on statistical modeling and expectation maximization algorithm

Abstract

The present invention discloses a whole pattern fitting method for polycrystalline powder X-ray diffraction patterns based on statistical modeling: Whole Pattern Fitting of Powder XRD by Expectation Maximum algorithm (WPEM for short). It can be used for whole pattern fitting of polycrystalline powder X-ray diffraction patterns, and extract crystal structure information such as peak position, peak width, peak shape, integrated diffraction intensity, etc. of each diffraction peak. It realizes functions such as "statistical background calculation", "precise determination of component lattice constant", and "quantitative determination of component volume fraction". The technology of the present invention can reduce or eliminate the noise of asymmetric peak shape formed by diffraction geometric factors, and can perform full pattern decomposition and fitting of X-ray diffraction patterns of mixed polycrystalline systems, and accurately and quantitatively determine the lattice constants, peak shapes and volume fractions of each crystal system. Compared with mainstream commercial fitting software such as Fullprof and TOPAS, it shows superior fitting accuracy.

Description

Technical Field

[0001] The invention relates to the field of material structure characterization and X-ray diffraction technology, and in particular to a powder X-ray diffraction pattern expectation maximization algorithm full spectrum line fitting method based on statistical modeling. Background Art

[0002] Powder X-ray diffraction (XRD) plays a vital role in characterizing the atomic-scale geometric structure of materials. Although Debye and Scherrer proposed the powder diffraction method for the determination of crystal structure as early as 1916, it was not until the introduction of the Rietveld full spectrum fitting method in 1967 that the problem of overlapping peaks in the measured diffraction spectrum was basically solved, enabling it to develop rapidly and be so widely used today. The Rietveld method fits the entire spectrum based on the diffraction intensity theory and the least squares method, and will eventually converge to a local optimal solution. Therefore, all kinds of mainstream commercial fitting software currently developed based on the Rietveld method still face the convergence and matrix singularity problems that may be caused by the use of the least squares method.

[0003] The WPEM method interprets the full spectrum fitting process as a probability density estimation problem and solves it through the expectation maximization algorithm. WPEM uses profile parameters to constrain the theoretical intensity error to obtain a more accurate description of the Bragg peak intensity. A statistical method is used to distinguish the diffraction intensity of the background, that is, the background distribution function is obtained by quantitative analysis in the reciprocal space through fast Fourier transform and Savitzky-Golay (SG) filter. WPEM is good at determining complex crystal structures and decoupling severely overlapping Bragg peaks from mixed X-ray or mixed crystal system polycrystalline powder diffraction spectra. Compared with other widely used refinement methods such as Fullprof and TOPAS, this method has fewer model parameters and higher fitting accuracy, and has obvious advantages in terms of effectiveness and goodness of fit. Summary of the invention

[0004] The technical problem to be solved by the present invention is to extract crystal information from the diffraction data of complex polycrystalline mixed powders. It is used to perform full spectrum peak shape fitting on the polycrystalline diffraction powder X-ray diffraction pattern, and extract crystal structure information such as the peak position, peak width, peak shape, and integrated diffraction intensity of each diffraction peak.

[0005] Technical solution: The present invention adopts the following technical solution to solve the above technical problems: a powder X-ray diffraction pattern expectation maximization algorithm full spectrum fitting method based on statistical modeling, the method comprises the following steps:

[0006] Step 1: Measure the polycrystalline powder sample by X-ray polycrystalline powder diffractometer to obtain experimental diffraction data, and calculate and remove the background noise contained in the experimental measurement data by Fourier transform and SG filtering method;

[0007] Step 2: Use the mixed probability density model in the statistical method to fit the diffraction data after removing the background noise, and use the expectation maximization algorithm and the Bragg formula alternately iterative method to solve the established fitting model;

[0008] Step 3: Calculate the fitting intensity of each compound mixed in the powder from the solved fitting model, and calculate the volume proportion of each compound in combination with the X-ray diffraction intensity theory.

[0009] Furthermore, the specific method of step 1 is as follows:

[0010] Step 1.1) A set of discrete experimental diffraction data (2θ j ,I(2θ j )), contains n diffraction angle values ​​and the corresponding diffraction intensity value I(2θ j ), j = 1, ..., n, with the diffraction angle 2θ as the independent variable and the intensity value I as the dependent variable, a continuous function I = I(2θ) is defined in the real number domain. When 2θ = 2θ j , I=I(2θ j ), the function I=I(2θ j ) performs fast Fourier transform and decomposes it into the superposition of n wave functions Each wave function F f (2θ) is a signal component, which is a complex function under the expansion of Euler formula, f = 1, .., n. In the reciprocal space, the DC signal component with zero imaginary part and the diffraction wave with amplitude less than the preset value due to Auger electrons and fluorescence effects during diffraction are filtered out, and the first n / 3 wave functions with the largest modulus length are retained;

[0011] The residual wave function is re-superimposed by inverse Fourier transform and converted to real space, and the signal-to-noise ratio of the diffraction function is further reduced by SG filtering technology to obtain the filtered diffraction function I filter (2θ), the experimental diffraction angle 2θ j Substitute into function I filter (2θ) to obtain the filtered diffraction data (2θ j ,I filter (2θ j ));

[0012] Step 1.2) The de-noised diffraction data (2θ j ,I filter (2θ j)) is divided into P intervals. For the first P-1 intervals, the one with the lowest diffraction intensity is selected in each interval. points, of which If it is not divisible, round down and the number selected in the last interval is background points, and get a total of K background points (2θ k ,I filter (2θ k )), where K is a constant, and the recommended value is one tenth of the total number of diffraction angles n, k = 1, ..., K, K < n;

[0013] Step 1.3) By The background intensity distribution I is calculated from the discrete background points in bac (2θ), background intensity distribution I bac (2θ) is derived from the background mean function and background variance function Composition, including three different calculation modes:

[0014] (1) Constant mode: This mode assumes the background variance function is a constant function ∈, with background mean function By polynomial fitting, Among them, C 0 , C 1 , C 2 , C 3 is the constant to be fitted, the background variance function is a constant function ∈, where

[0015] (2) Polynomial mode: This mode assumes the background variance function is the function of the diffraction angle 2θ, the background mean function By polynomial fitting, Among them, C 0 , C 1 , C 2 , C 3 is the constant to be fitted, the background variance function Based on polynomial fitting, Among them, B 0 , B 1 , B 2 , B 3 is the constant to be fitted;

[0016] (3) Multivariate Gaussian model: This model uses the random process assumption that the set The background intensity I of the background points included in bac (2θk ) and the background strength of other nK points not included The Gaussian random process assumption is satisfied, that is, the vector:

[0017] Y 1 =(I bac (2θ 1 ),I bac (2θ 2 ),…,I bac (2θ k ))

[0018] And the vector:

[0019] The joint distribution of satisfies the multivariate mixed Gaussian distribution, k=1,…,K,K<n,h1,…,nK, and the mean vector of the multivariate mixed Gaussian distribution is The diagonal elements of the skew variance matrix are and Solved by Gaussian process regression method;

[0020] Step 1.4) Set the experimental diffraction angle 2θ j Substitute the background mean function Get the angle 2θ j The background strength The original experimental measurement intensity I(2θ j ) minus the background mean function Obtain background-free diffraction data (2θ j ,I nobac (2θ j )),in,

[0021] Furthermore, the specific method of step 2 is as follows:

[0022] Step 3.1) Remove the background from the diffraction data nobac (2θ j ), modeled as m mixed probability models, each probability model is fitted using a PV function, that is, j=1,…,n,i=1,…,m;where w i is the weight of the i-th PV function, m is the total number of mixed models, is the i-th PV function at the diffraction angle 2θ j The value of I fit (2θ j ) is m PV functions at a diffraction angle of 2θ j The weighted sum of the values ​​at is defined as follows:

[0023]

[0024] Among them, μ i is the peak value of the i-th PV function, γ i is half of the full width at half maximum of the i-th PV function, is the variance of the Gaussian component in the i-th PV function, π is a constant, Δ i is the weight of the Lorentz component in the i-th PV function. The weights of the m probability models and their parameters w are iteratively solved by the expectation maximization algorithm and the Bragg formula. i , Δ i , μ i , γ i , Make I nobac (2θ j ) and I fit (2θ j ) is the smallest residual, i=1,…,m;

[0025] Step 3.2) According to the characteristics of the diffraction pattern, the initial values ​​of the model parameters are given Update the parameters according to the direction of expected maximization. If the value of the model parameter in step t is in, The parameters w are i , Δ i , μ i , γ i , The value in the t-th iteration is the t-th step parameter and the diffraction angle 2θ j Substitute into formula (2)(3)(4) to calculate the three intermediate variables of step t

[0026]

[0027]

[0028]

[0029] Where, j = 1, 2, ..., n; i = 1, 2, ..., m;

[0030] The intermediate variable Substitute into formula (5)(6)(7)(8)(9) to obtain the parameter value of step t+1

[0031]

[0032]

[0033]

[0034]

[0035]

[0036] Among them, I area is the integrated diffraction intensity value, according to the formula Calculated;

[0037] Step 3.2) Update the parameters in step t+1 according to Bragg's diffraction law According to Bragg's diffraction law, the relationship between the crystal structure and the diffraction peak position is:

[0038]

[0039] in, is the Prague peak position, d i (HKL) is the interplanar spacing of the crystal in the direction of the crystal plane with a diffraction index of HKL. For a given diffraction index HKL, the calculation formula for the interplanar spacing is:

[0040]

[0041] Among them, the function Determined by the geometric symmetry of the measured crystal, a, b, and c are the unit cell basis vectors The modulus of α, β, and γ are vectors The angle between them, λ is the wavelength of the X-ray used in the experiment, and the multivariable equation shown in formula (12) is solved by the nonlinear least squares method to calculate a, b, c, α, β, γ:

[0042]

[0043] Substitute the obtained solutions a, b, c, α, β, γ into the function Calculate the interplanar spacing d i (HKL), the interplanar spacing d i (HKL) into the formula Get updated Prague Peak location use replace Get the updated model parameters in step t+1, namely: i=1,…,m;

[0044] Step 3.3) Set the value of the model parameter at step t+1 Substitute into formula (1) to calculate the fitted diffraction intensity I under this set of parameters: fit(2θ j ) and the residual at this time j=1,…,n, repeat steps 3.1)-3.3) to obtain the value of the parameter in step t+2 and the corresponding residual;

[0045] Step 3.4) Iterates step 3.1)-step 3.3) continuously until I nobac (2θ j ) and I fit (2θ j ) stops iterating and outputs the parameter value in the last iteration: i=1,…,m.

[0046] Furthermore, the specific method of step 3 is as follows:

[0047] Step 4.1) The final model parameter value is i=1,…,m, and put it into formula (1) to get the fitted diffraction intensity value I fit (2θ j ) and the fitted diffraction intensity function I fit (2θ), j = 1,…,n;

[0048] Step 4.2) Substitute into function I fit (2θ) Calculate the fitting peak intensity values ​​of m diffraction peaks Get the peak intensity data of the diffraction peak i=1,…,m;

[0049] Step 4.3) Peak intensity data of all diffraction peaks according to The values ​​are arranged from large to small, and the largest diffraction peak intensity is selected and its corresponding diffraction angle And calculate the factors in, is the structure factor, P is the multiplicity factor, V 0 is the unit cell volume;

[0050] Step 4.4) The volume fraction of the compound mixed in the polycrystalline powder is calculated by formula (13):

[0051]

[0052] Wherein, V is the volume fraction of the compound in the polycrystalline powder, and C is a constant.

[0053] Beneficial effects: Compared with the prior art, the present invention adopts the above technical solution and has the following technical effects:

[0054] A: This method analyzes the powder diffraction spectrum based on statistical viewpoints and introduces the expectation maximization algorithm optimization. Based on this, it is convenient to introduce mature statistical sampling algorithms such as the Monte Carlo-Markov algorithm on the basis of this method model to simplify and accelerate the calculation of more complex peak shape functions.

[0055] B: Compared with the Rietveld method solved using the least squares method, it has better fitting accuracy.

[0056] C: It can handle the refinement of more complex mixed powder diffraction patterns. Type 1: Quantitative determination of mixed compounds with very similar crystal structures; Type 2: Quantitative determination of complex mixed compounds with five or more components. BRIEF DESCRIPTION OF THE DRAWINGS

[0057] Figure 1 is a flow chart of the method;

[0058] Figure 2 It is the iterative process of solving the parameters of the “Expectation Maximization-Bragg Formula” in this method;

[0059] Figure 3 It is the result of this method before and after filtering the diffraction function by fast Fourier transform;

[0060] Figure 4 are the calculation results of three different calculation modes of background distribution in this method;

[0061] Figure 5 This is the fitting result of this method for the powder X-ray diffraction data of three different types of compounds. DETAILED DESCRIPTION

[0062] The technical solution of the present invention is further described in detail below in conjunction with the accompanying drawings:

[0063] like Figure 1 The calculation process of this method is shown in Figure 1. The polycrystalline powder sample is tested by an X-ray polycrystalline powder diffractometer, and the background noise contained in the experimental measurement data is calculated and removed based on the experimental diffraction data obtained. The diffraction data with background noise removed is fitted by a mixed probability model, and the parameters in the model are solved by the expectation maximization algorithm and the Bragg formula alternating iteration method.

[0064] like Figure 2The iterative process of parameter solution in this method is shown. The optimization process is divided into two mutually nested parts, namely the full spectrum expectation maximization iteration process and the Bragg convergence process of each crystal system. In the expectation maximization iteration process, the full spectrum simultaneously processes the X-ray diffraction patterns of the polycrystalline system, and each iteration gives the position and shape parameters of each peak based on expectation maximization. The peak position updated in the expectation maximization process is matched one by one with the currently calculated Bragg peak position according to the principle of proximity, and assigned to different crystal systems. Then, the crystal parameters are iterated according to the Bragg law and the lattice structure and diffraction index corresponding to each crystal system, and the peak position is adjusted. The adjusted peak position will be used as the value of a new round of expectation maximization iteration, and the full spectrum expectation maximization fitting will be performed. The full spectrum expectation maximization iteration process and the Bragg convergence process of each crystal system are performed alternately until convergence.

[0065] like Figure 3 The filtering result of the fast Fourier transform is shown in Figure 1. The experimental spectrum measured by the polycrystalline powder diffractometer is the result of the superposition of diffraction on the crystal planes in all directions, such as Figure 3 (a) As shown. Through fast Fourier transform, the observed diffraction function is decomposed into wave functions before superposition and sorted according to their respective frequency sizes, as shown in Figure 3 (b). The diffraction waves with zero imaginary parts and the diffraction waves with amplitudes less than the preset value due to Auger electrons and fluorescence effects during diffraction are filtered out, and the first n / 3 wave functions with the largest modulus are retained, as shown in 3(c). The remaining wave functions are re-superimposed by inverse Fourier transform and converted to real space to obtain the filtered diffraction spectrum, as shown in 3(d).

[0066] like Figure 4 The background mean function curves and corresponding one-time standard deviation confidence intervals obtained under three different calculation modes for background distribution are shown in the figure. The background mean function calculated by the constant mode and the polynomial mode changes smoothly, and its one-time standard deviation confidence interval is narrow, as shown in Figure 1. Figure 4 (a), 4(b). The background mean function obtained by the mixed Gaussian mode changes significantly, and the one-time standard deviation confidence interval is large in the interval of non-background points, indicating that the noise variance near the diffraction peak is large, such as Figure 4 (c).

[0067] like Figure 5 The figure shows the calculation results of this method for polycrystalline powder diffraction data. Figure 5 (a) is the method for compound Tb 2 BCO 5 The fitting of the crystal powder diffraction data accurately calculated the lattice constant of the compound. The accuracy of the full spectrum fitting is the profile factor Rp = 3.738%, and the weighted profile factor Rwp = 6.887. Figure 5 (b) This method is used for the compound PbSO 4The fitting of the crystal powder diffraction data accurately calculated the lattice constant of the compound. The accuracy of the full spectrum fitting is the profile factor Rp = 2.595%, and the weighted profile factor Rwp = 7.118%. Compared with the mainstream commercial fitting methods such as Fullprof and TOPAS, this method shows superior fitting accuracy, that is, lower Rp and Rwp values, as shown in Table 1. Figure 5 (c) is the fitting of this method with the diffraction data of the compound crystal powder with the atomic ratio of Ti-15Nb. The compound contains three different phases: β phase, α phase and αprim phase. This method successfully distinguishes and determines the lattice parameters and peak position values ​​of the three phases. It has extremely high fitting accuracy: Rp = 3.234%, Rwp = 5.425%. As shown in Table 1, the fitting accuracy of this method is compared with mainstream commercial fitting methods such as Fullprof, TOPAS, etc.

[0068]

[0069] Table 1

[0070] The above descriptions are only some embodiments of the present invention. It should be pointed out that, for ordinary technicians in this technical field, several improvements and modifications can be made without departing from the principles of the present invention. These improvements and modifications should also be regarded as the scope of protection of the present invention.

Claims

1. A powder X-ray diffraction pattern expectation maximization algorithm full spectrum fitting method based on statistical modeling, characterized in that: The method comprises the following steps: Step 1: Measure the polycrystalline powder sample by X-ray polycrystalline powder diffractometer to obtain experimental diffraction data, and calculate and remove the background noise contained in the experimental measurement data by Fourier transform and SG filtering method; Step 2: Use the mixed probability density model in the statistical method to fit the diffraction data after removing the background noise, and use the expectation maximization algorithm and the Bragg formula alternately iterative method to solve the established fitting model; Step 3: Calculate the fitting intensity of each compound mixed in the powder from the solved fitting model, and calculate the volume proportion of each compound in combination with the X-ray diffraction intensity theory; In step 2, the solution process is divided into two mutually nested parts, namely, the full spectrum expectation maximization iterative process and the Bragg convergence process of each crystal system; In the expectation maximization iteration process, the full spectrum simultaneously processes the X-ray diffraction patterns of multiple crystal systems. Each iteration gives the position and shape parameters of each peak based on expectation maximization. The peak positions updated in the expectation maximization process are matched one by one with the currently calculated Bragg peak positions according to the principle of proximity and assigned to different crystal systems. Then, the crystal parameters are iterated according to the Bragg law and the lattice structure and diffraction index corresponding to each crystal system to adjust the peak position. The adjusted peak position will be used as the value of a new round of expectation maximization iteration, and the full spectrum expectation maximization fitting will be performed. The full spectrum expectation maximization iteration process and the Bragg convergence process of each crystal system are performed alternately until convergence.

2. The powder X-ray diffraction pattern expectation maximization algorithm full spectrum fitting method based on statistical modeling according to claim 1, characterized in that: The specific method of step 1 is as follows: Step 1.1) A set of discrete experimental diffraction data (2θ j ,I(2θ j )), contains n diffraction angle values ​​and the corresponding diffraction intensity value I(2θ j ), j = 1, ..., n, with the diffraction angle 2θ as the independent variable and the intensity value I as the dependent variable, a continuous function I = I(2θ) is defined in the real number domain. When 2θ = 2θ j , I=I(2θ j ), the function I=I(2θ j ) performs fast Fourier transform and decomposes it into the superposition of n wave functions Each wave function F f (2θ) is a signal component, which is a complex function under the expansion of Euler formula, f = 1, .., n. In the reciprocal space, the DC signal component with zero imaginary part and the diffraction wave with amplitude less than the preset value due to Auger electrons and fluorescence effects during diffraction are filtered out, and the first n / 3 wave functions with the largest modulus length are retained; The residual wave function is re-superimposed by inverse Fourier transform and converted to real space, and the signal-to-noise ratio of the diffraction function is further reduced by SG filtering technology to obtain the filtered diffraction function I filter (2θ), the experimental diffraction angle 2θ j Substitute into function I filter (2θ) to obtain the filtered diffraction data (2θ j ,I filter (2θ j )); Step 1.2) The de-noised diffraction data (2θ j ,I filter (2θ j )) is divided into P intervals. For the first P-1 intervals, the one with the lowest diffraction intensity is selected in each interval. points, of which If it is not divisible, round down and the number selected in the last interval is background points, and get a total of K background points (2θ k ,I filter (2θ k ), where K is a constant, and the recommended value is one tenth of the total number of diffraction angles n, k = 1, ..., K, K <n; Step 1.3) By The background intensity distribution I is calculated from the discrete background points in bac (2θ), background intensity distribution I bac (2θ) is derived from the background mean function and background variance function Composition, including three different calculation modes: (1) Constant mode: This mode assumes the background variance function is a constant function ∈, with background mean function By polynomial fitting, Among them, C0, C1, C2, C3 are constants to be fitted, and the background variance function is a constant function ∈, where (2) Polynomial mode: This mode assumes the background variance function is the function of the diffraction angle 2θ, the background mean function By polynomial fitting, Among them, C0, C1, C2, C3 are constants to be fitted, and the background variance function Based on polynomial fitting, Among them, B0, B1, B2, B3 are constants to be fitted; (3) Multivariate Gaussian model: This model uses the random process assumption that the set The background intensity I of the background points included in bac (2θ k ) and the background strength of other nK points not included The Gaussian random process assumption is satisfied, that is, the vector: Y1=(I bac (2θ1),I bac (2θ2),…,I bac (2θ k )) And the vector: The joint distribution satisfies a multivariate mixture Gaussian distribution, \(k = 1,\ldots,K\), \(K < n\), \(h = 1,\ldots,n - K\), and the mean vector of this multivariate mixture Gaussian distribution is The diagonal elements of the covariance matrix are and solved by the Gaussian process regression method; Step 1.4) Set the experimental diffraction angle 2θ j Substitute the background mean function Get the angle 2θ j The background strength The original experimental measurement intensity I(2θ j ) minus the background mean function Obtain background-free diffraction data (2θ j ,I nobac (2θ j )),in, 3. The powder X-ray diffraction pattern expectation maximization algorithm full spectrum fitting method based on statistical modeling according to claim 1, characterized in that: The specific method of step 2 is as follows: Step 3.1) Remove the background from the diffraction data nobac (2θ j ), modeled as m mixed probability models, each probability model is fitted using a PV function, that is, Among them, w i is the weight of the i-th PV function, m is the total number of mixed models, is the i-th PV function at the diffraction angle 2θ j The value of I fit )2θ j ) is m PV functions at a diffraction angle of 2θ j The weighted sum of the values ​​at is defined as follows: Among them, μ i is the peak value of the i-th PV function, γ i is half of the full width at half maximum of the i-th PV function, is the variance of the Gaussian component in the i-th PV function, π is a constant, Δ i is the weight of the Lorentz component in the i-th PV function. The weights of the m probability models and their parameters w are iteratively solved by the expectation maximization algorithm and the Bragg formula. i , Δ i , μ i , γ i , Make I nobac (2θ j ) and I fit (2θ j ) is the smallest residual, i=1,…,m; Step 3.2) According to the characteristics of the diffraction pattern, the initial values ​​of the model parameters are given Update the parameters according to the direction of expected maximization. If the value of the model parameter in step t is in, The parameters w are i , Δ i , μ i , γ i , The value in the t-th iteration is the t-th step parameter and the diffraction angle 2θ j Substitute into formula (2)(3)(4) to calculate the three intermediate variables of step t Where, j = 1, 2, ..., n; i = 1, 2, ..., m; The intermediate variable Substitute into formula (5)(6)(7)(8)(9) to obtain the parameter value of step t+1 Among them, I area is the integrated diffraction intensity value, according to the formula Calculated; Step 3.2) Update the parameters in step t+1 according to Bragg's diffraction law According to Bragg's diffraction law, the relationship between the crystal structure and the diffraction peak position is: in, is the Prague peak position, d i (HKL) is the interplanar spacing of the crystal in the direction of the crystal plane with a diffraction index of HKL. For a given diffraction index HKL, the calculation formula for the interplanar spacing is: Among them, the function Determined by the geometric symmetry of the measured crystal, a, b, and c are the unit cell basis vectors The modulus of α, β, and γ are vectors The angle between them, λ is the wavelength of the X-ray used in the experiment, and the multivariable equation shown in formula (12) is solved by the nonlinear least squares method to calculate a, b, c, α, β, γ: Substitute the obtained solutions a, b, c, α, β, γ into the function Calculate the interplanar spacing d i (HKL), the interplanar spacing d i (HKL) into the formula Get updated Prague Peak location use replace Get the updated model parameters in step t+1, namely: Step 3.3) Set the value of the model parameter at step t+1 Substitute into formula (1) to calculate the fitted diffraction intensity I under this set of parameters: fut (2θ j ) and the residual at this time Repeat steps 3.1) to 3.3) to obtain the value of the parameter in step t+2 and the corresponding residual; Step 3.4) Iterates step 3.1)-step 3.3) continuously until I nobac (2θ j ) and I fit (2θ j ) stops iterating and outputs the parameter value in the last iteration:

4. The powder X-ray diffraction pattern expectation maximization algorithm full spectrum fitting method based on statistical modeling according to claim 1, characterized in that ,The specific method of step 3 is as follows: Step 4.1) The final model parameter value is Substitute it into formula (1) to obtain the fitted diffraction intensity value I fit (2θ j ) and the fitted diffraction intensity function I fit (2θ), j = 1,…,n; Step 4.2) Substitute into function I fit (2θ) Calculate the fitting peak intensity values ​​of m diffraction peaks Get the peak intensity data of the diffraction peak Step 4.3) Peak intensity data of all diffraction peaks according to The values ​​are arranged from large to small, and the largest diffraction peak intensity is selected and its corresponding diffraction angle And calculate the factors in, is the structure factor, P is the multiplicity factor, and V0 is the unit cell volume; Step 4.4) The volume fraction of the compound mixed in the polycrystalline powder is calculated by formula (13): Wherein, V is the volume fraction of the compound in the polycrystalline powder, and C is a constant.

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