Application of the Levenberg-Marquardt Algorithm to X-Ray Diffraction Quantitative Phase Analysis
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摘要: 常规的X射线衍射物相定量分析方法有各自的优点，但这些方法往往也存在一些不足.在实际应用中迫切需要一种简便、高效的普适性多物相无标样定量分析方法.选择麦夸特算法、粒子群算法、遗传算法、差分进化算法这4种迭代搜索领域的经典算法构建了基于非线性模型参数估计方法的4模冗余系统，以19个配制的4相样品中各相“前三强线”的积分强度之和作为计算的原始数据，通过Matlab软件进行了含量计算.理论分析及试验结果表明，运用麦夸特算法进行定量分析具有更小的计算复杂度、更快的收敛速度及更好的全局搜索能力，各相含量的计算值与配比值的绝对误差在5%以内的约占总计算量的83%.为了验证该算法，计算了昭苏黄土剖面82个混合样品及青海湖二郎剑钻孔359个混合样品中刚玉的含量，刚玉含量的配比值与计算值的相关性分别达到0.83和0.63，刚玉含量的误差超过5%的分别占总计算量的4.88%和9.75%.基于麦夸特算法的定量分析方法在批量化处理多物相定量中具有效率高、可操作性强、准确度高等优点.Abstract: The conventional X-ray diffraction quantitative phase analysis methods are over-reliant on pure standard substances, working curve and K value. If the phases are more than 3, the fitting results are not good by Rietveld method. In addition, the versatility of quantitative methods with large calculation and fussy operation also need to be expanded. A new non-standard quantitative phase analysis method based on nonlinear model parameters estimation method of 4 modular redundant systems that consist of Levenberg-Marquardt, Particle Swarm Optimization, Genetic Algorithm and Differential Evolution is proposed. Taking the content of 4 phases in 19 mixture powder as the original data, performing the whole process of computing in the Matlab environment, the experimental results show that the Levenberg-Marquardt algorithm is an effective tool with smaller computing complexity, faster convergence speed and better global searching capability and other advantages. It is no need to add reference phase to the samples, which overcomes the problems that all the samples must be determined more than one time, and the method with no need for K value which enlarges the applications and enhances the accuracy of the X-ray diffraction method for quantitative phase analysis of the mixture samples. Replacing the conventional specific single spectrum line intensity or intensity rations by the sum of the integrated intensity of the top three peaks can improve the precision of the X-ray diffraction quantitative phase analysis. With this method, the content of Corundum in 82 samples of Zhaosu section in Ili basin and 359 samples in ELJ drilling core of ICDP in Qinghai Lake are computed. The correlation coefficient of the match ratio and the calculated value of Corundum in Zhaosu section and ELJ drilling core have reached 0.83 and 0.63. Practice has proved that it is a feasible, effective, rapid and correct technique of quantitative analysis of minerals, and the stability is satisfactory. It can be used for quantifying the phases in more than 9-phase materials.
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图 1 非线性模型参数估计方法的4模冗余系统示意
Fig. 1. The process of 4 modular redundant systems based on nonlinear model parameters estimation method
图 2 4种矿物的X射线衍射谱与标准卡片中衍射谱的对比
Fig. 2. Comparison of 4 kinds of minerals' X-ray diffraction patterns with standard card
图 3 4个混合样品的X射线衍射谱
Fig. 3. X-ray diffraction patterns of 4 samples in 19 prepared samples
图 4 各物相含量的计算值与配比值之间的绝对误差
Fig. 4. The absolute error between matched value and computed results of 4 kinds minerals in 19 samples
图 5 各样品总质量的配比值与计算值的对比
Fig. 5. Comparison between the matched value and the calculated value of 19 samples' total mass
图 6 刚玉的计算值与配比值的相关分析
Fig. 6. Relationship of calculated value and matched value of Corundum in ZSP section and Erlangjian drilling
表 1 4模冗余系统的计算结果
Table 1. The computed results of the 4 modular redundant system
项目 Q1 Q2 Q3 Q4 收敛指标 1E-10 1E-10 1E-10 1E-10 参数设置 - 轮盘选择法 k=2 α=0.7 α=0.7 C1=2.05 β=0.85 β=0.85 C2=2.05 均方差 1.521E-02 1.521E-02 1.521E-02 2.658E-02 残差平方和 2.775E-03 2.775E-03 2.775E-03 8.479E-03 时间开销(s) 0.47 0.78 0.219 0.188 系数($\hat \theta $) 3.693E-04 6.931E-04 3.693E-04 4.996E-04 4.771 9E-05 4.772 0E-05 4.771 9E-05 4.226 4E-05 3.978 0E-05 3.977 89E-05 3.978 0E-05 5.262 1E-05 5.208 2E-05 5.207 7E-05 5.208 2E-05 2.843 8E-07 
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