CN107589136B - A kind of the dual model approximating method and system of small angle X ray scattering - Google Patents
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Abstract
本发明提出了一种小角X射线散射的双模型拟合方法，包括：获取步骤：获得被分析对象的散射强度实验图谱；建模步骤：根据散射强度实验图谱的特征构建双模型；解析步骤：调整各向同性散射体的散射强度计算公式模型和取向散射体的散射强度计算公式模型中的各可调参数，使得各向同性散射体的散射强度计算公式模型和取向散射体的散射强度计算公式模型相加后得到的计算图谱与所述散射强度实验图谱之差最小，即可解析出各模型的参数。本发明还提出了一种小角X射线散射的双模型拟合系统。本发明为利用小角X射线散射进行有效观测材料介观尺度结构的无损检测提供了更好的数据支持。
The present invention proposes a dual-model fitting method for small-angle X-ray scattering, comprising: an acquisition step: obtaining an experimental spectrum of scattering intensity of an analyzed object; a modeling step: constructing a dual model according to the characteristics of the experimental spectrum of scattering intensity; and an analysis step: Adjust the adjustable parameters in the calculation formula model of the scattering intensity of the isotropic scatterer and the calculation formula of the scattering intensity of the oriented scatterer, so that the calculation formula of the scattering intensity of the isotropic scatterer and the calculation formula of the scattering intensity of the oriented scatterer The difference between the calculated spectrum obtained after model addition and the scattering intensity experimental spectrum is the smallest, and the parameters of each model can be analyzed. The invention also proposes a double-model fitting system for small-angle X-ray scattering. The invention provides better data support for effectively observing the non-destructive testing of the mesoscale structure of materials by using small-angle X-ray scattering.
Description
技术领域technical field
本发明属于小角X射线散射(SAXS)理论计算领域，尤其涉及一种小角X射线散射的双模型拟合方法及系统。The invention belongs to the field of small-angle X-ray scattering (SAXS) theoretical calculations, in particular to a double-model fitting method and system for small-angle X-ray scattering.
背景技术Background technique
小角X射线散射(SAXS)是探究介观尺度结构的有效实验手段，其本质是散射体内一至数百纳米范围内电子密度不均匀引起的散射现象。由于SAXS散射具有制样简单，气相、液相、固相均可测量的特点被广泛的应用于化学、化工、材料科学、分子生物学、医药学、凝聚态物理等多学科的研究中。研究对象包括具有各种纳米结构，如液晶、液晶态生物膜的各种相变化、表面活性剂缔合结构、生物大分子(蛋白质、核酸等)、自组装超分子结构、微孔、溶胶分形结构和界面层结构、聚合物溶液、结晶取向聚合物(工业纤维和薄膜)、嵌段离子离聚物的微观结构等。Small-angle X-ray scattering (SAXS) is an effective experimental method to explore mesoscopic structures, and its essence is the scattering phenomenon caused by the uneven electron density in the range of one to hundreds of nanometers in the scattering body. Because SAXS scattering has the characteristics of simple sample preparation and measurement of gas phase, liquid phase and solid phase, it is widely used in multidisciplinary research such as chemistry, chemical engineering, material science, molecular biology, medicine, and condensed matter physics. Research objects include various nanostructures, such as liquid crystals, various phase changes of liquid crystal biofilms, surfactant association structures, biomacromolecules (proteins, nucleic acids, etc.), self-assembled supramolecular structures, micropores, sol fractal Structural and interfacial layer structures, polymer solutions, crystallographically oriented polymers (industrial fibers and films), microstructure of block ionomers, etc.
SAXS因其可以直接测量体相材料，有较好的粒子统计平均性等特点，在纤维内微观孔洞结构的测量方面具有显微电镜无可比拟的优势，从而在纤维的微观结构测量中得到重视和广泛应用。在纤维材料的SAXS分析中由于其体系内微孔存在取向，从而采用二维散射谱图拟合的方法进行建模分析。在大量的实验分析中发现纤维中既有拉伸取向的孔洞，也有未取向的球状孔洞，选用单个取向的散射体进行计算拟合存在较大的误差。Because SAXS can directly measure the bulk material, it has the characteristics of good statistical average of particles, etc., and has incomparable advantages in the measurement of the microscopic hole structure in the fiber, so it has been paid attention to in the measurement of the microscopic structure of the fiber. and widely used. In the SAXS analysis of fiber materials, due to the orientation of micropores in the system, the two-dimensional scattering spectrum fitting method is used for modeling analysis. In a large number of experimental analysis, it is found that there are both stretch-oriented holes and unoriented spherical holes in the fiber, and there is a large error in the calculation and fitting of a single-oriented scatterer.
发明内容Contents of the invention
为解决上述问题，本发明提出了一种小角X射线散射的双模型拟合方法及系统，根据散射体的不同散射特性分别建模，为SAXS二维散射图谱的计算拟合提供更精准的数据。In order to solve the above problems, the present invention proposes a dual-model fitting method and system for small-angle X-ray scattering, which can be modeled separately according to different scattering characteristics of scatterers, and provide more accurate data for the calculation and fitting of SAXS two-dimensional scattering patterns .
本发明提出的一种小角X射线散射的双模型拟合方法，该方法包括下列步骤：The double model fitting method of a kind of small-angle X-ray scattering that the present invention proposes, this method comprises the following steps:
获取步骤：获得被分析对象的散射强度实验图谱；Obtaining step: obtaining the experimental spectrum of the scattering intensity of the analyzed object;
建模步骤：根据散射强度实验图谱的特征构建双模型，具体为：所述被分析对象的散射体系具有各向同性散射体和取向散射体这两种类型的散射体，选取各向同性散射体构建各向同性散射体的散射强度计算公式模型，选取取向散射体构建取向散射体的散射强度计算公式模型；其中，各向同性散射体的散射强度计算公式模型为球状模型，取向散射体的散射强度计算公式模型为棒状模型；Modeling step: construct a dual model according to the characteristics of the scattering intensity experimental spectrum, specifically: the scattering system of the analyzed object has two types of scatterers: isotropic scatterers and oriented scatterers, and the isotropic scatterers are selected Construct the calculation formula model of the scattering intensity of the isotropic scatterer, and select the oriented scatterer to construct the calculation formula model of the oriented scatterer; among them, the calculation formula model of the scattering intensity of the isotropic scatterer is a spherical model, and the The strength calculation formula model is a rod model;
解析步骤：调整各向同性散射体的散射强度计算公式模型和取向散射体的散射强度计算公式模型中的各可调参数，使得各向同性散射体的散射强度计算公式模型和取向散射体的散射强度计算公式模型相加后得到的计算图谱与所述散射强度实验图谱之差最小，即可解析出各模型的参数。Analysis step: adjust the adjustable parameters in the calculation formula model of the scattering intensity of the isotropic scatterer and the calculation formula model of the scattering intensity of the oriented scatterer, so that the calculation formula model of the scattering intensity of the isotropic scatterer and the scattering of the oriented scatterer The difference between the calculated spectrum obtained after adding the intensity calculation formula model and the scattering intensity experimental spectrum is the smallest, and the parameters of each model can be analyzed.
进一步地，在建模步骤中，各向同性散射体的散射强度计算公式模型和取向散射体的散射强度计算公式模型均为：Further, in the modeling step, the calculation formula model of the scattering intensity of the isotropic scatterer and the calculation formula model of the scattering intensity of the oriented scatterer are both:
其中，q为散射向量，Fn为第n个散射体的散射振幅，Fn’为第n’个散射体的散射振幅，N为散射体的数量，rn为第n个散射体的中心，rn'为第n'个散射体的中心。where, q is the scattering vector, F n is the scattering amplitude of the nth scatterer, F n' is the scattering amplitude of the n'th scatterer, N is the number of scatterers, r n is the center of the nth scatterer , r n' is the center of the n'th scatterer.
进一步地，在所述建模步骤中，设定所述散射体的中心为坐标原点，则所述各向同性散射体和所述取向散射体的散射振幅F(q)的计算公式为：Further, in the modeling step, the center of the scatterer is set as the coordinate origin, then the calculation formula of the scattering amplitude F(q) of the isotropic scatterer and the oriented scatterer is:
F(q)＝∫vρdVe-iqr (2)F(q)＝∫ v ρdVe -iqr (2)
其中，q为散射向量，V为散射体的体积，ρ为散射体的电子密度，r为散射体上任意一点到坐标原点的距离。Among them, q is the scattering vector, V is the volume of the scatterer, ρ is the electron density of the scatterer, and r is the distance from any point on the scatterer to the coordinate origin.
进一步地，在建模步骤中，根据散射体的散射振幅以及根据散射体的尺寸分布和取向分布，来构建上述散射强度计算公式模型；Further, in the modeling step, the above-mentioned scattering intensity calculation formula model is constructed according to the scattering amplitude of the scatterer and the size distribution and orientation distribution of the scatterer;
所构建的散射强度计算公式模型为：The constructed formula model for calculating the scattering intensity is:
其中，C1为常数，R1、R2、R3为散射体的三个半轴长，和为散射体三个轴的尺寸分布，h(ω)为沿取向轴天顶角分布，为方位角分布，ω表示沿取向轴天顶角，ψ表示散射矢量与qx分量方向的夹角，表示方位角；Among them, C 1 is a constant, R 1 , R 2 , R 3 are the three semi-axis lengths of the scatterer, and is the size distribution of the three axes of the scatterer, h(ω) is the zenith angle distribution along the orientation axis, is the azimuth angle distribution, ω represents the zenith angle along the orientation axis, ψ represents the angle between the scattering vector and the direction of the q x component, Indicates the azimuth;
对于取向散射体，公式(8)中，R1、R2、R3的取值均不相同，或者其中两者相同，第三者与这两者不同；For oriented scatterers, in formula (8), the values of R 1 , R 2 , and R 3 are all different, or two of them are the same, and the third one is different from these two;
对于各向同性散射体，公式(8)中，R1＝R2＝R3，h(ω)为常数，方位角分布为常数。For an isotropic scatterer, in formula (8), R 1 =R 2 =R 3 , h(ω) is a constant, and the azimuth distribution is a constant.
进一步地，其中，对于取向散射体，利用Von Mises分布函数描述所述天顶角分布，函数形式如下：Further, wherein, for an oriented scatterer, the Von Mises distribution function is used to describe the zenith angle distribution, and the function form is as follows:
其中I0为第一类0阶修正的bessel函数，ω0为平均值，分布的方差由方差参数к确定，定义为均匀分布；在公式(10)中，ω0、κ为可调参数。Among them, I 0 is the Bessel function of the first kind of 0-order correction, ω 0 is the average value, and the variance of the distribution is determined by the variance parameter к, is defined as a uniform distribution; in formula (10), ω 0 and κ are adjustable parameters.
进一步地，其中，利用对数分布函数描述所述取向散射体的轴和所述各向同性散射体的半径，所述对数分布函数的形式如下式(11)所示：Further, wherein the axis of the oriented scatterer and the radius of the isotropic scatterer are described by a logarithmic distribution function, the form of the logarithmic distribution function is shown in the following formula (11):
其中μ和σ分别为对数正态分布的对数均值和对数标准差，其平均值m和方差参数υ通过下式求得：Among them, μ and σ are the logarithmic mean and logarithmic standard deviation of the lognormal distribution respectively, and its mean value m and variance parameter υ are obtained by the following formula:
m＝exp(μ+σ2/2) (12)m=exp(μ+ σ2 /2) (12)
υ＝exp(2μ+σ2)(expσ2-1) (13)υ＝exp(2μ+σ 2 )(expσ 2 -1) (13)
在公式(11)中，μ和σ为可调参数。In formula (11), μ and σ are adjustable parameters.
本发明提出的一种小角X射线散射的双模型拟合系统，该系统包括下列模块：The dual model fitting system of a kind of small-angle X-ray scattering that the present invention proposes, this system comprises following modules:
获取模块：获得被分析对象的散射强度实验图谱；Acquisition module: obtain the experimental spectrum of the scattering intensity of the analyzed object;
建模模块：根据散射强度实验图谱的特征构建双模型，具体为：所述被分析对象的散射体系具有各向同性散射体和取向散射体这两种类型的散射体，选取各向同性散射体构建各向同性散射体的散射强度计算公式模型，选取取向散射体构建取向散射体的散射强度计算公式模型；其中，各向同性散射体的散射强度计算公式模型为球状模型，取向散射体的散射强度计算公式模型为棒状模型；Modeling module: Construct a dual model according to the characteristics of the scattering intensity experimental spectrum, specifically: the scattering system of the analyzed object has two types of scatterers: isotropic scatterers and oriented scatterers, and the isotropic scatterers are selected Construct the calculation formula model of the scattering intensity of the isotropic scatterer, and select the oriented scatterer to construct the calculation formula model of the oriented scatterer; among them, the calculation formula model of the scattering intensity of the isotropic scatterer is a spherical model, and the The strength calculation formula model is a rod model;
解析模块：调整各向同性散射体的散射强度计算公式模型和取向散射体的散射强度计算公式模型中的各可调参数，使得各向同性散射体的散射强度计算公式模型和取向散射体的散射强度计算公式模型相加后得到的计算图谱与所述散射强度实验图谱之差最小，即可解析出各模型的参数。Analysis module: adjust the adjustable parameters in the calculation formula model of the scattering intensity of the isotropic scatterer and the calculation formula model of the scattering intensity of the oriented scatterer, so that the calculation formula model of the scattering intensity of the isotropic scatterer and the scattering of the oriented scatterer The difference between the calculated spectrum obtained after adding the intensity calculation formula model and the scattering intensity experimental spectrum is the smallest, and the parameters of each model can be analyzed.
进一步地，在建模模块中，各向同性散射体的散射强度计算公式模型和取向散射体的散射强度计算公式模型均为：Further, in the modeling module, the calculation formula model of the scattering intensity of the isotropic scatterer and the calculation formula model of the scattering intensity of the oriented scatterer are both:
其中，q为散射向量，Fn为第n个散射体的散射振幅，Fn’为第n’个散射体的散射振幅，N为散射体的数量，rn为第n个散射体的中心，rn'为第n'个散射体的中心。where, q is the scattering vector, F n is the scattering amplitude of the nth scatterer, F n' is the scattering amplitude of the n'th scatterer, N is the number of scatterers, r n is the center of the nth scatterer , r n' is the center of the n'th scatterer.
进一步地，在所述建模模块中，设定所述散射体的中心为坐标原点，则所述各向同性散射体和所述取向散射体的散射振幅F(q)的计算公式为：Further, in the modeling module, the center of the scatterer is set as the coordinate origin, then the calculation formula of the scattering amplitude F(q) of the isotropic scatterer and the oriented scatterer is:
F(q)＝∫vρdVe-iqr (2)F(q)＝∫ v ρdVe -iqr (2)
其中，q为散射向量，V为散射体的体积，ρ为散射体的电子密度，r为散射体上任意一点到坐标原点的距离。Among them, q is the scattering vector, V is the volume of the scatterer, ρ is the electron density of the scatterer, and r is the distance from any point on the scatterer to the coordinate origin.
进一步地，在建模模块中，根据散射体的散射振幅以及根据散射体的尺寸分布和取向分布，来构建上述散射强度计算公式模型；Further, in the modeling module, the above-mentioned scattering intensity calculation formula model is constructed according to the scattering amplitude of the scatterer and the size distribution and orientation distribution of the scatterer;
所构建的散射强度计算公式模型为：The constructed formula model for calculating the scattering intensity is:
其中，C1为常数，R1、R2、R3为散射体的三个半轴长，和为散射体三个轴的的尺寸分布，h(ω)为沿取向轴天顶角分布，为方位角分布，ω表示沿取向轴天顶角，ψ表示散射矢量与qx分量方向的夹角，表示方位角；Among them, C 1 is a constant, R 1 , R 2 , R 3 are the three semi-axis lengths of the scatterer, and is the size distribution of the three axes of the scatterer, h(ω) is the zenith angle distribution along the orientation axis, is the azimuth angle distribution, ω represents the zenith angle along the orientation axis, ψ represents the angle between the scattering vector and the direction of the q x component, Indicates the azimuth;
对于取向散射体，公式(8)中，R1、R2、R3的取值均不相同，或者其中两者相同，第三者与这两者不同；For oriented scatterers, in formula (8), the values of R 1 , R 2 , and R 3 are all different, or two of them are the same, and the third one is different from these two;
对于各向同性散射体，公式(8)中，R1＝R2＝R3，h(ω)为常数，方位角分布为常数。For an isotropic scatterer, in formula (8), R 1 =R 2 =R 3 , h(ω) is a constant, and the azimuth distribution is a constant.
进一步地，其中，对于取向散射体，利用Von Mises分布函数描述所述天顶角分布，函数形式如下：Further, wherein, for an oriented scatterer, the Von Mises distribution function is used to describe the zenith angle distribution, and the function form is as follows:
其中I0为第一类0阶修正的bessel函数，ω0为平均值，分布的方差由方差参数к确定，定义为均匀分布；公式(10)中，ω0、κ为可调参数。Among them, I 0 is the Bessel function of the first kind of 0-order correction, ω 0 is the average value, and the variance of the distribution is determined by the variance parameter к, is defined as a uniform distribution; in formula (10), ω 0 and κ are adjustable parameters.
进一步地，其中，利用对数分布函数描述所述取向散射体的轴和所述各向同性散射体的半径，所述对数分布函数的形式如下式(11)所示：Further, wherein the axis of the oriented scatterer and the radius of the isotropic scatterer are described by a logarithmic distribution function, the form of the logarithmic distribution function is shown in the following formula (11):
其中μ和σ分别为对数正态分布的对数均值和对数标准差，其平均值m和方差参数υ通过下式求得：Among them, μ and σ are the logarithmic mean and logarithmic standard deviation of the lognormal distribution respectively, and its mean value m and variance parameter υ are obtained by the following formula:
m＝exp(μ+σ2/2) (12)m=exp(μ+ σ2 /2) (12)
υ＝exp(2μ+σ2)(expσ2-1) (13)υ＝exp(2μ+σ 2 )(expσ 2 -1) (13)
公式(11)中，μ和σ为可调参数。In formula (11), μ and σ are adjustable parameters.
本发明的有益效果：本发明根据纤维内散射体的特点建立球状和棒状双模型的计算方法，得到纤维内部准确的孔洞信息，进一步扩大了模型的适用范围。Beneficial effects of the present invention: the present invention establishes a spherical and rod-shaped double model calculation method according to the characteristics of the scattering body in the fiber, obtains accurate hole information inside the fiber, and further expands the scope of application of the model.
附图说明Description of drawings
图1为本发明所示的双模型拟合方法的流程图；Fig. 1 is the flow chart of double model fitting method shown in the present invention;
图2为本发明所示的双模型拟合系统的结构框图；Fig. 2 is the block diagram of the dual model fitting system shown in the present invention;
图3为本发明所示的散射体内入射X射线与散射X射线的相位差示意图；Fig. 3 is a schematic diagram of the phase difference between incident X-rays and scattered X-rays in the scatterer shown in the present invention;
图4为本发明所示的取向散射体在坐标系中各角度关系的示意图；Fig. 4 is the schematic diagram of the relationship of each angle in the coordinate system of the orientation scatterer shown in the present invention;
图5为本发明所示的二维双模型分析方法的示意图；Fig. 5 is the schematic diagram of the two-dimensional dual-model analysis method shown in the present invention;
图6a和图6b为本发明所示的不同参数下的Von Mises函数曲线；Fig. 6 a and Fig. 6 b are the Von Mises function curve under the different parameters shown in the present invention;
图7为本发明所示的不同参数下对数正态分布函数曲线；Fig. 7 is the logarithmic normal distribution function curve under different parameters shown in the present invention;
图8为本发明针对的小角X射线散射实验图谱；Fig. 8 is the small-angle X-ray scattering experimental atlas of the present invention;
图9为本发明提出散射体的双模型结构示意图；Fig. 9 is a schematic diagram of a dual-model structure of a scatterer proposed by the present invention;
图10为本发明的双模型计算拟合的二维散射图谱。Fig. 10 is the two-dimensional scattering spectrum fitted by the double model calculation of the present invention.
具体实施方式Detailed ways
为使本发明的目的、技术方案和优点更加清楚明白，以下结合具体实施例，并参照附图，对本发明进一步详细说明。但本领域技术人员知晓，本发明并不局限于附图和以下实施例。In order to make the object, technical solution and advantages of the present invention clearer, the present invention will be described in further detail below in conjunction with specific embodiments and with reference to the accompanying drawings. However, those skilled in the art know that the present invention is not limited to the drawings and the following embodiments.
如图1所示，本发明提出了一种小角X射线散射SAXS的双模型拟合方法，包括下列步骤：As shown in Figure 1, the present invention proposes a kind of double model fitting method of small-angle X-ray scattering SAXS, comprises the following steps:
获取步骤：获得被分析对象的散射强度实验图谱，如图8所示；Obtaining step: obtaining the experimental spectrum of the scattering intensity of the analyzed object, as shown in Figure 8;
建模步骤：根据散射强度实验图谱的特征构建双模型，具体为：所述被分析对象的散射体系具有各向同性散射体和取向散射体这两种类型的散射体，选取各向同性散射体构建各向同性散射体的散射强度计算公式模型，选取取向散射体构建取向散射体的散射强度计算公式模型；其中，各向同性散射体的散射强度计算公式模型为球状模型，取向散射体的散射强度计算公式模型为棒状模型，如图9所示，本发明采用球状模型和棒状模型的双模型来计算二维散射图谱，扩大了模型的适用范围，提高了拟合的准确性；Modeling step: construct a dual model according to the characteristics of the scattering intensity experimental spectrum, specifically: the scattering system of the analyzed object has two types of scatterers: isotropic scatterers and oriented scatterers, and the isotropic scatterers are selected Construct the calculation formula model of the scattering intensity of the isotropic scatterer, and select the oriented scatterer to construct the calculation formula model of the oriented scatterer; among them, the calculation formula model of the scattering intensity of the isotropic scatterer is a spherical model, and the The intensity calculation formula model is a rod model, as shown in Figure 9, the present invention uses the double model of spherical model and rod model to calculate the two-dimensional scattering spectrum, which expands the scope of application of the model and improves the accuracy of fitting;
解析步骤：调整各向同性散射体的散射强度计算公式模型和取向散射体的散射强度计算公式模型中的各可调参数，使得各向同性散射体的散射强度计算公式模型和取向散射体的散射强度计算公式模型相加后得到的计算图谱与所述散射强度实验图谱之差最小，即可解析出各模型的参数。Analysis step: adjust the adjustable parameters in the calculation formula model of the scattering intensity of the isotropic scatterer and the calculation formula model of the scattering intensity of the oriented scatterer, so that the calculation formula model of the scattering intensity of the isotropic scatterer and the scattering of the oriented scatterer The difference between the calculated spectrum obtained after adding the intensity calculation formula model and the scattering intensity experimental spectrum is the smallest, and the parameters of each model can be analyzed.
在建模步骤中，本发明首先分析一个体积为V的散射体，假设其电子密度为ρ(r)＝ρ，散射体外的的电子密度为ρ(r)＝0，则一束X射线达到该体系时，散射体内的电子受迫振动，产生次生波(散射波)。其入射X射线与散射X射线的相位差示意图如图3所示。In the modeling step, the present invention first analyzes a scatterer whose volume is V, assuming that its electron density is ρ(r)=ρ, and the electron density outside the scatterer is ρ(r)=0, then a beam of X-rays reaches In this system, the electrons in the scattering body are forced to vibrate, generating secondary waves (scattering waves). The schematic diagram of the phase difference between the incident X-ray and the scattered X-ray is shown in FIG. 3 .
散射体的散射振幅可写为：The scattering amplitude of a scatterer can be written as:
F(q)＝∫∫∫dVρ(r)eiqr (1)F(q)＝∫∫∫dVρ(r)e iqr (1)
简写为：Abbreviated as:
F(q)＝∫vρdVe-iqr (2)F(q)＝∫ v ρdVe -iqr (2)
其中，q为散射向量，V为散射体的体积，ρ为散射体的电子密度，r为散射体上任意一点到坐标原点的距离，i为虚数单位。Among them, q is the scattering vector, V is the volume of the scatterer, ρ is the electron density of the scatterer, r is the distance from any point on the scatterer to the coordinate origin, and i is the imaginary unit.
如果一个体系中有N个散射体，中心分别位于r1，r2，r3，...rN，电子密度可写为ρn(r-rn)则体系总的电子密度可写为：If there are N scatterers in a system, the centers are located at r 1 , r 2 , r 3 ,...r N , and the electron density can be written as ρ n (rr n ), then the total electron density of the system can be written as:
体系的散射振幅可写为：The scattering amplitude of the system can be written as:
根据上述公式，在建模步骤中能够推导得到球状模型和棒状模型的散射强度I(q)的公式为：According to the above formula, the formula for the scattering intensity I(q) of the spherical model and the rod model can be derived in the modeling step as:
当n＝n′时，上式可进一步写为：When n=n', The above formula can be further written as:
在计算散射强度时引入体系中散射体的尺寸和取向，其中，定义散射体三个轴的尺寸分布为和沿取向轴天顶角分布为h(ω)，方位角分布为其体积为由于在稀松体系中散射体之间的干涉非常弱，因此上式(6)中的第二项为零。When calculating the scattering intensity, the size and orientation of the scatterers in the system are introduced, where the size distribution of the three axes of the scatterers is defined as and The zenith angle distribution along the orientation axis is h(ω), and the azimuth angle distribution is Its volume is Since the interference between scatterers is very weak in a loose system, the second term in the above formula (6) is zero.
上式(6)可以进一步简化为：The above formula (6) can be further simplified as:
则球状模型和棒状模型的散射强度可写为：Then the scattering intensity of spherical model and rod model can be written as:
上式中各角度之间有如下关系：The relationship between the angles in the above formula is as follows:
γ表示散射体与X轴的夹角，ω表示沿取向轴天顶角，2θ为入射X射线和散射X射线的夹角，ψ表示散射矢量与qx分量方向的夹角，表示方位角。对于取向散射体的散射强度计算公式模型，即棒状模型，R1、R2、R3的取值均不相同，或者其中两者相同，第三者与这两者不同；对于各向同性散射体的散射强度计算公式模型，即球状模型，R1＝R2＝R3，h(ω)为常数，方位角分布为常数。γ represents the angle between the scatterer and the X-axis, ω represents the zenith angle along the orientation axis, 2θ represents the angle between the incident X-ray and the scattered X-ray, and ψ represents the angle between the scattering vector and the direction of the q x component, Indicates the azimuth. For the calculation formula model of the scattering intensity of oriented scatterers, that is, the rod model, the values of R 1 , R 2 , and R 3 are all different, or two of them are the same, and the third is different from the two; for isotropic scattering The calculation formula model of the scattering intensity of the body is a spherical model, R 1 =R 2 =R 3 , h(ω) is a constant, and the azimuth angle distribution is a constant.
假定取向散射体具有旋转不变性，方位角分布定义为均匀分布，为一常数。天顶角分布h(ω)定义为Von Mises分布，函数形式如下：Assuming that the orientation scatterers are invariant to rotation, the azimuthal distribution Defined as a uniform distribution, it is a constant. The zenith angle distribution h(ω) is defined as the Von Mises distribution, and the function form is as follows:
其中I0为第一类0阶修正的bessel函数，ω0为平均值，分布的方差由к确定。Among them, I 0 is the Bessel function of the first kind of 0th order correction, ω 0 is the average value, and the variance of the distribution is determined by к.
Von Mises分布是最常用的圆形分布，取值范围一般为[0，2π]，适合于描述散射体的取向分布。如图6a和图6b所示，描述了不同参数下的Von Mises函数曲线。The Von Mises distribution is the most commonly used circular distribution, and its value range is generally [0, 2π], which is suitable for describing the orientation distribution of scatterers. As shown in Figure 6a and Figure 6b, the Von Mises function curves under different parameters are described.
在建模步骤中取向散射体中的轴R1、R2、R3均服从对数正态分布f(R)，各向同性散射体中的轴，也就是球体的半径R也服从对数正态分布f(R)，函数形式如下：In the modeling step, the axes R 1 , R 2 , and R 3 in the oriented scatterer all obey the lognormal distribution f(R), and the axes in the isotropic scatterer, that is, the radius R of the sphere also obey the logarithm Normal distribution f(R), the function form is as follows:
其中μ和σ分别为对数正态分布的对数均值和对数标准差，平均值m和方差v可以通过下式求得：Among them, μ and σ are the logarithmic mean and logarithmic standard deviation of the lognormal distribution, respectively, and the mean m and variance v can be obtained by the following formula:
m＝exp(μ+σ2/2) (12)m=exp(μ+ σ2 /2) (12)
υ＝exp(2μ+σ2)(expσ2-1) (13)υ＝exp(2μ+σ 2 )(expσ 2 -1) (13)
如图7所示，显示了不同参数下对数正态分布函数曲线。As shown in Figure 7, the lognormal distribution function curves under different parameters are shown.
在解析步骤中，给定各可调参数值，即可计算出各模型对应的二维散射图谱，如图10所示；将所述两个散射强度计算公式相加得到的计算图谱与所述散射强度实验图谱进行比较，通过调整尺寸大小的参数μ、σ和调整取向角度的参数ω0、к来调整棒状模型的散射强度值，通过调整尺寸大小的参数μ、σ来调整球状模型的散射强度值，从而使得棒状模型的散射强度值和球状模型的散射强度值相加得到的计算图谱和实验图谱的散射强度二者误差最小，如图5所示，即可解析出两个模型的参数。在实验图谱和计算图谱进行比较之前，还需要对实验图谱进行修正，扣除实验仪器和实际操作对实验图谱的影响。首先对实验图谱进行各种校正，如背景散射、暗电流、光束形状等校正，得到高质量二维实验散射图谱。再通过构建两个模型的方法计算出二维散射图谱，调整各模型的参数，使得实验图谱和计算图谱之差最小，即可解析出各模型的参数，并通过标准样品验证模型的有效性。In the analysis step, given each adjustable parameter value, the two-dimensional scattering spectrum corresponding to each model can be calculated, as shown in Figure 10; the calculation spectrum obtained by adding the two scattering intensity calculation formulas and the above Scattering intensity experimental maps are compared, the scattering intensity value of the rod model is adjusted by adjusting the size parameters μ, σ and the orientation angle parameters ω 0 , к, and the spherical model is adjusted by adjusting the size parameters μ, σ Intensity value, so that the error of the calculated spectrum and the scattering intensity of the experimental spectrum obtained by adding the scattering intensity value of the rod model and the scattering intensity value of the spherical model is the smallest, as shown in Figure 5, the parameters of the two models can be analyzed . Before comparing the experimental spectrum with the calculated spectrum, the experimental spectrum needs to be corrected to deduct the influence of the experimental instrument and actual operation on the experimental spectrum. First, various corrections are made to the experimental spectrum, such as background scattering, dark current, beam shape, etc., to obtain a high-quality two-dimensional experimental scattering spectrum. Then calculate the two-dimensional scattering spectrum by constructing two models, adjust the parameters of each model to minimize the difference between the experimental spectrum and the calculated spectrum, then analyze the parameters of each model, and verify the validity of the model through standard samples.
本发明还提出了一种小角X射线散射SAXS的双模型计算拟合系统，如图2所示，包括下列模块：The present invention also proposes a dual-model calculation and fitting system for small-angle X-ray scattering SAXS, as shown in Figure 2, including the following modules:
获取模块：获得被分析对象的散射强度实验图谱，如图8所示；Obtaining module: obtaining the experimental spectrum of the scattering intensity of the analyzed object, as shown in Figure 8;
建模模块：根据散射强度实验图谱的特征构建双模型，具体为：所述被分析对象的散射体系具有各向同性散射体和取向散射体这两种类型的散射体，选取各向同性散射体构建各向同性散射体的散射强度计算公式模型，选取取向散射体构建取向散射体的散射强度计算公式模型；其中，各向同性散射体的散射强度计算公式模型为球状模型，取向散射体的散射强度计算公式模型为棒状模型，如图9所示，本发明采用球状模型和棒状模型的双模型来计算二维散射图谱，扩大了模型的适用范围，提高了拟合的准确性；Modeling module: Construct a dual model according to the characteristics of the scattering intensity experimental spectrum, specifically: the scattering system of the analyzed object has two types of scatterers: isotropic scatterers and oriented scatterers, and the isotropic scatterers are selected Construct the calculation formula model of the scattering intensity of the isotropic scatterer, and select the oriented scatterer to construct the calculation formula model of the oriented scatterer; among them, the calculation formula model of the scattering intensity of the isotropic scatterer is a spherical model, and the The intensity calculation formula model is a rod model, as shown in Figure 9, the present invention uses the double model of spherical model and rod model to calculate the two-dimensional scattering spectrum, which expands the scope of application of the model and improves the accuracy of fitting;
解析模块：调整各向同性散射体的散射强度计算公式模型和取向散射体的散射强度计算公式模型中的各可调参数，使得各向同性散射体的散射强度计算公式模型和取向散射体的散射强度计算公式模型相加后得到的计算图谱与所述散射强度实验图谱之差最小，即可解析出各模型的参数。Analysis module: adjust the adjustable parameters in the calculation formula model of the scattering intensity of the isotropic scatterer and the calculation formula model of the scattering intensity of the oriented scatterer, so that the calculation formula model of the scattering intensity of the isotropic scatterer and the scattering of the oriented scatterer The difference between the calculated spectrum obtained after adding the intensity calculation formula model and the scattering intensity experimental spectrum is the smallest, and the parameters of each model can be analyzed.
在建模模块中，本发明首先分析一个体积为V的散射体，假设其电子密度为ρ(r)＝ρ，散射体外的的电子密度为ρ(r)＝0，则一束X射线达到该体系时，散射体内的电子受迫振动，产生次生波(散射波)。其入射X射线与散射X射线的相位差示意图如图3所示。In the modeling module, the present invention first analyzes a scatterer whose volume is V, assuming that its electron density is ρ(r)=ρ, and the electron density outside the scatterer is ρ(r)=0, then a beam of X-rays reaches In this system, the electrons in the scattering body are forced to vibrate, generating secondary waves (scattering waves). The schematic diagram of the phase difference between the incident X-ray and the scattered X-ray is shown in FIG. 3 .
散射体的散射振幅可写为：The scattering amplitude of a scatterer can be written as:
F(q)＝∫∫∫dVρ(r)eiqr (1)F(q)＝∫∫∫dVρ(r)e iqr (1)
简写为：Abbreviated as:
F(q)＝∫vρdVe-iqr (2)F(q)＝∫ v ρdVe -iqr (2)
其中，q为散射向量，V为散射体的体积，ρ为散射体的电子密度，r为散射体上任意一点到坐标原点的距离，i为虚数单位。Among them, q is the scattering vector, V is the volume of the scatterer, ρ is the electron density of the scatterer, r is the distance from any point on the scatterer to the coordinate origin, and i is the imaginary unit.
如果一个体系中有N个散射体，中心分别位于r1，r2，r3，...rN，电子密度可写为ρn(r-rn)则体系总的电子密度可写为：If there are N scatterers in a system, the centers are located at r 1 , r 2 , r 3 ,...r N , and the electron density can be written as ρ n (rr n ), then the total electron density of the system can be written as:
体系的散射振幅可写为：The scattering amplitude of the system can be written as:
根据上述公式，在建模模块中能够推导得到球状模型和棒状模型的散射强度I(q)的公式为：According to the above formula, the formula that can be derived to obtain the scattering intensity I(q) of the spherical model and the rod model in the modeling module is:
当n＝n′时，上式可进一步写为：When n=n', The above formula can be further written as:
在计算散射强度时引入体系中散射体的尺寸和取向，其中，定义散射体三个轴的尺寸分布为和沿取向轴天顶角分布为h(ω)，方位角分布为其体积为由于在稀松体系中散射体之间的干涉非常弱，因此上式(6)中的第二项为零。When calculating the scattering intensity, the size and orientation of the scatterers in the system are introduced, where the size distribution of the three axes of the scatterers is defined as and The zenith angle distribution along the orientation axis is h(ω), and the azimuth angle distribution is Its volume is Since the interference between scatterers is very weak in a loose system, the second term in the above formula (6) is zero.
上式(6)可以进一步简化为：The above formula (6) can be further simplified as:
则球状模型和棒状模型的散射强度可写为：Then the scattering intensity of spherical model and rod model can be written as:
上式中各角度之间有如下关系：The relationship between the angles in the above formula is as follows:
γ表示散射体与X轴的夹角，ω表示沿取向轴天顶角，2θ为入射X射线和散射X射线的夹角，ψ表示散射矢量与qx分量方向的夹角，表示方位角。对于取向散射体的散射强度计算公式模型，即棒状模型，R1、R2、R3的取值均不相同，或者其中两者相同，第三者与这两者不同；对于各向同性散射体的散射强度计算公式模型，即球状模型，R1＝R2＝R3，h(ω)为常数，方位角分布为常数。γ represents the angle between the scatterer and the X-axis, ω represents the zenith angle along the orientation axis, 2θ represents the angle between the incident X-ray and the scattered X-ray, and ψ represents the angle between the scattering vector and the direction of the q x component, Indicates the azimuth. For the calculation formula model of the scattering intensity of oriented scatterers, that is, the rod model, the values of R 1 , R 2 , and R 3 are all different, or two of them are the same, and the third is different from the two; for isotropic scattering The calculation formula model of the scattering intensity of the body is a spherical model, R 1 =R 2 =R 3 , h(ω) is a constant, and the azimuth angle distribution is a constant.
假定取向散射体具有旋转不变性，方位角分布定义为均匀分布，为一常数。天顶角分布h(ω)定义为Von Mises分布，函数形式如下：Assuming that the orientation scatterers are invariant to rotation, the azimuthal distribution Defined as a uniform distribution, it is a constant. The zenith angle distribution h(ω) is defined as the Von Mises distribution, and the function form is as follows:
其中I0为第一类0阶修正的bessel函数，ω0为平均值，分布的方差由к确定。Among them, I 0 is the Bessel function of the first kind of 0th order correction, ω 0 is the average value, and the variance of the distribution is determined by к.
Von Mises分布是最常用的圆形分布，取值范围一般为[0，2π]，适合于描述散射体的取向分布。如图6a和图6b所示，描述了不同参数下的Von Mises函数曲线。The Von Mises distribution is the most commonly used circular distribution, and its value range is generally [0, 2π], which is suitable for describing the orientation distribution of scatterers. As shown in Figure 6a and Figure 6b, the Von Mises function curves under different parameters are described.
而建模模块中取向散射体中的轴R1、R2、R3均服从对数正态分布f(R)，各向同性散射体中的轴，也就是球体的半径R也服从对数正态分布f(R)，函数形式如下：In the modeling module, the axes R 1 , R 2 , and R 3 in the oriented scatterer all obey the logarithmic normal distribution f(R), and the axes in the isotropic scatterer, that is, the radius R of the sphere also obey the logarithm Normal distribution f(R), the function form is as follows:
其中μ和σ分别为对数正态分布的对数均值和对数标准差，平均值m和方差v可以通过下式求得：Among them, μ and σ are the logarithmic mean and logarithmic standard deviation of the lognormal distribution, respectively, and the mean m and variance v can be obtained by the following formula:
m＝exp(μ+σ2/2) (12)m=exp(μ+ σ2 /2) (12)
υ＝exp(2μ+σ2)(expσ2-1) (13)υ＝exp(2μ+σ 2 )(expσ 2 -1) (13)
如图7所示，显示了不同参数下对数正态分布函数曲线。As shown in Figure 7, the lognormal distribution function curves under different parameters are shown.
在解析模块中，给定各可调参数值，即可计算出各模型对应的二维散射图谱，如图10所示；将所述两个散射强度计算公式相加得到的计算图谱与所述散射强度实验图谱进行比较，通过调整尺寸大小的参数μ、σ和调整取向角度的参数ω0、к来调整棒状模型的散射强度值，通过调整尺寸大小的参数μ、σ来调整球状模型的散射强度值，从而使得棒状模型的散射强度值和球状模型的散射强度值相加得到的计算图谱和实验图谱的散射强度二者误差最小，如图5所示，即可解析出两个模型的参数。在实验图谱和计算图谱进行比较之前，还需要对实验图谱进行修正，扣除实验仪器和实际操作对实验图谱的影响。首先对实验图谱进行各种校正，如背景散射、暗电流、光束形状等校正，得到高质量二维实验散射图谱。再通过构建两个模型的方法计算出二维散射图谱，调整各模型的参数，使得实验图谱和计算图谱之差最小，即可解析出各模型的参数，并通过标准样品验证模型的有效性。In the analysis module, given each adjustable parameter value, the two-dimensional scattering spectrum corresponding to each model can be calculated, as shown in Figure 10; the calculation spectrum obtained by adding the two scattering intensity calculation formulas and the above Scattering intensity experimental maps are compared, the scattering intensity value of the rod model is adjusted by adjusting the size parameters μ, σ and the orientation angle parameters ω 0 , к, and the spherical model is adjusted by adjusting the size parameters μ, σ Intensity value, so that the error of the calculated spectrum and the scattering intensity of the experimental spectrum obtained by adding the scattering intensity value of the rod model and the scattering intensity value of the spherical model is the smallest, as shown in Figure 5, the parameters of the two models can be analyzed . Before comparing the experimental spectrum with the calculated spectrum, the experimental spectrum needs to be corrected to deduct the influence of the experimental instrument and actual operation on the experimental spectrum. First, various corrections are made to the experimental spectrum, such as background scattering, dark current, beam shape, etc., to obtain a high-quality two-dimensional experimental scattering spectrum. Then calculate the two-dimensional scattering spectrum by constructing two models, adjust the parameters of each model to minimize the difference between the experimental spectrum and the calculated spectrum, then analyze the parameters of each model, and verify the validity of the model through standard samples.
下面给出不同参数下Von Mises函数曲线和对数正态分布曲线的实施例。Examples of Von Mises function curves and lognormal distribution curves under different parameters are given below.
实施例1：Example 1:
设置参数ω0＝-π/2；κ＝100，得到如图6a中对应参数的Von Mises函数曲线。Set the parameters ω 0 =-π/2; κ=100 to obtain the Von Mises function curve of the corresponding parameters as shown in Fig. 6a.
实施例2：Example 2:
设置参数ω0＝-π/4；κ＝100，得到如图6a中对应参数的Von Mises函数曲线。Set the parameters ω 0 =-π/4; κ=100 to obtain the Von Mises function curve of the corresponding parameters as shown in Fig. 6a.
实施例3：Example 3:
设置参数ω0＝0；κ＝100，得到如图6a中对应参数的Von Mises函数曲线。Set the parameters ω 0 =0; κ=100 to obtain the Von Mises function curve of the corresponding parameters as shown in Fig. 6a.
实施例4：Example 4:
设置参数ω0＝π/4；κ＝100，得到如图6a中对应参数的Von Mises函数曲线。Set the parameters ω 0 =π/4; κ=100 to obtain the Von Mises function curve of the corresponding parameters as shown in Fig. 6a.
实施例5：Example 5:
设置参数ω0＝π/2；κ＝100，得到如图6a中对应参数的Von Mises函数曲线。Set the parameters ω 0 =π/2; κ=100 to obtain the Von Mises function curve of the corresponding parameters as shown in Fig. 6a.
实施例6：Embodiment 6:
设置参数ω0＝3π/4；κ＝100，得到如图6a中对应参数的Von Mises函数曲线。Set the parameters ω 0 =3π/4; κ=100 to obtain the Von Mises function curve of the corresponding parameters as shown in Fig. 6a.
实施例7：Embodiment 7:
设置参数ω0＝0.95π；κ＝100，得到如图6a中对应参数的Von Mises函数曲线。Set the parameters ω 0 =0.95π; κ=100 to obtain the Von Mises function curve of the corresponding parameters as shown in Fig. 6a.
实施例8：Embodiment 8:
设置参数ω0＝π；κ＝100，得到如图6a中对应参数的Von Mises函数曲线。Set the parameters ω 0 =π; κ=100 to obtain the Von Mises function curve of the corresponding parameters as shown in Fig. 6a.
实施例9：Embodiment 9:
设置参数ω0＝0；κ＝0，得到如图6b中对应参数的Von Mises函数曲线。Set the parameters ω 0 =0; κ=0 to obtain the Von Mises function curve of the corresponding parameters as shown in Fig. 6b.
实施例10：Example 10:
设置参数ω0＝0；κ＝4，得到如图6b中对应参数的Von Mises函数曲线。Set the parameters ω 0 =0; κ=4 to obtain the Von Mises function curve of the corresponding parameters as shown in Fig. 6b.
实施例11：Example 11:
设置参数ω0＝0；κ＝8，得到如图6b中对应参数的Von Mises函数曲线。Set the parameters ω 0 =0; κ=8 to obtain the Von Mises function curve of the corresponding parameters as shown in Fig. 6b.
实施例12：Example 12:
设置参数ω0＝0；κ＝12，得到如图6b中对应参数的Von Mises函数曲线。Set the parameters ω 0 =0; κ=12 to obtain the Von Mises function curve of the corresponding parameters as shown in Fig. 6b.
实施例13：Example 13:
设置参数ω0＝0；κ＝16，得到如图6b中对应参数的Von Mises函数曲线。Set the parameters ω 0 =0; κ=16 to obtain the Von Mises function curve of the corresponding parameters as shown in Fig. 6b.
实施例14：Example 14:
设置参数ω0＝0；κ＝20，得到如图6b中对应参数的Von Mises函数曲线。Set the parameters ω 0 =0; κ=20 to obtain the Von Mises function curve of the corresponding parameters as shown in Fig. 6b.
实施例15：Example 15:
设置参数ω0＝0；κ＝100，得到如图6b中对应参数的Von Mises函数曲线。Set the parameters ω 0 =0; κ=100 to obtain the Von Mises function curve of the corresponding parameters as shown in Fig. 6b.
实施例16：Example 16:
设置参数m＝20；v＝200，得到如图7中对应参数的对数正态分布函数曲线。Set the parameters m=20; v=200 to obtain the lognormal distribution function curve of the corresponding parameters as shown in Fig. 7 .
实施例17：Example 17:
设置参数m＝20；v＝150，得到如图7中对应参数的对数正态分布函数曲线。Set the parameters m=20; v=150 to obtain the lognormal distribution function curve of the corresponding parameters as shown in Fig. 7 .
实施例18：Example 18:
设置参数m＝20；v＝100，得到如图7中对应参数的对数正态分布函数曲线。Set the parameters m=20; v=100 to obtain the lognormal distribution function curve of the corresponding parameters as shown in Fig. 7 .
实施例19：Example 19:
设置参数m＝20；v＝80，得到如图7中对应参数的对数正态分布函数曲线。Set the parameters m=20; v=80 to obtain the lognormal distribution function curve of the corresponding parameters as shown in Fig. 7 .
实施例20：Example 20:
设置参数m＝20；v＝60，得到如图7中对应参数的对数正态分布函数曲线。Set the parameters m=20; v=60 to obtain the lognormal distribution function curve of the corresponding parameters as shown in Fig. 7 .
实施例21：Example 21:
设置参数m＝20；v＝40，得到如图7中对应参数的对数正态分布函数曲线。Set the parameters m=20; v=40 to obtain the lognormal distribution function curve of the corresponding parameters as shown in Fig. 7 .
实施例22：Example 22:
设置参数m＝20；v＝20，得到如图7中对应参数的对数正态分布函数曲线。Set the parameters m=20; v=20 to obtain the lognormal distribution function curve of the corresponding parameters as shown in Fig. 7 .
实施例23：Example 23:
设置参数m＝20；v＝10，得到如图7中对应参数的对数正态分布函数曲线。Set the parameters m=20; v=10 to obtain the lognormal distribution function curve of the corresponding parameters as shown in Fig. 7 .
通过设置不同参数，调整散射体的天顶角分布以及长短轴的正态分布，使得通过两个模型相加得到的计算图谱与实验图谱之间的误差最小，即可解析出各模型的参数。根据散射强度实验图谱的特征进行建模，使得纤维材料的孔洞特征表达更加准确。By setting different parameters and adjusting the zenith angle distribution of the scatterers and the normal distribution of the long and short axes, the error between the calculated spectrum and the experimental spectrum obtained by adding the two models is minimized, and the parameters of each model can be analyzed. Modeling is carried out according to the characteristics of the scattering intensity experimental spectrum, so that the expression of the hole characteristics of the fiber material is more accurate.
以上，对本发明的实施方式进行了说明。但是，本发明不限定于上述实施方式。凡在本发明的精神和原则之内，所做的任何修改、等同替换、改进等，均应包含在本发明的保护范围之内。The embodiments of the present invention have been described above. However, the present invention is not limited to the above-mentioned embodiments. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principles of the present invention shall be included within the protection scope of the present invention.
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