CN107589136A - The dual model approximating method and system of a kind of small angle X ray scattering - Google Patents
The dual model approximating method and system of a kind of small angle X ray scattering Download PDFInfo
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Abstract
The present invention proposes a kind of dual model approximating method of small angle X ray scattering, including：Obtaining step：Obtain the scattering strength experimental patterns of analyzed object；Modeling procedure：According to the feature construction dual model of scattering strength experimental patterns；Analyzing step：Adjust each adjustable parameter in the scattering strength calculation formula model of isotropic scatter and the scattering strength calculation formula model of orientation scattering object, so that the calculating collection of illustrative plates and the difference of the scattering strength experimental patterns that the scattering strength calculation formula model of isotropic scatter and the scattering strength calculation formula model of orientation scattering object obtain after being added are minimum, you can parse the parameter of each model.The invention also provides a kind of dual model of small angle X ray scattering is fitted system.The present invention provides more preferable data support to carry out the Non-Destructive Testing of effectively observation material mesoscopic mesostructure using small angle X ray scattering.
Description
Technical field
The invention belongs to small angle X ray scattering (SAXS) theoretical calculation field, more particularly to a kind of small angle X ray scattering
Dual model approximating method and system.
Background technology
Small angle X ray scattering (SAXS) is the effective laboratory facilities for probing into meso-scale structure, and its essence is in scattering object
The uneven caused scattering phenomenon of electron density in the range of one to hundreds of nanometers.Because SAXS scatterings have, sample preparation is simple, gas phase,
The measurable feature of liquid phase, solid phase is widely used in chemistry, chemical industry, material science, molecular biology, medicine and pharmacology, solidifying
In the multi-disciplinary researchs such as poly- state physics.Research object include there are various nanostructureds, as liquid crystal, liquid crystal state biomembrane it is each
Kind of phase change, surfactant associative structure, large biological molecule (protein, nucleic acid etc.), self-assembled supermolecular structure, micropore,
Colloidal sol fractal structure and interface structure, polymer solution, crystalline orientation polymer (industrial fiber and film), block ion from
Microstructure of polymers etc..
SAXS is micro- in fiber because it can have the features such as preferable particle statistics evenness with direct measurement body phase material
There is the unrivaled advantage of micro- Electronic Speculum in terms of the measurement for seeing pore space structure, so as to be obtained in the microtexture measurement of fiber
Attention and extensive use.Because micropore is in the presence of orientation in its system in the SAXS analyses of fibrous material, so as to scattered using two dimension
The method for penetrating spectrogram fitting is modeled analysis.The hole of existing stretching orientation in fiber is found in substantial amounts of experimental analysis,
Also there is non-oriented spherical hole, larger error be present from the scattering object progress digital simulation of single orientation.
The content of the invention
To solve the above problems, the present invention proposes the dual model approximating method and system of a kind of small angle X ray scattering, root
Modeled respectively according to the different scattering properties of scattering object, the digital simulation for SAXS two-dimensional scattering collection of illustrative plates provides more accurately data.
A kind of dual model approximating method of small angle X ray scattering proposed by the present invention, this method comprise the following steps：
Obtaining step：Obtain the scattering strength experimental patterns of analyzed object；
Modeling procedure：According to the feature construction dual model of scattering strength experimental patterns, it is specially：The analyzed object
Scattering system has isotropic scatter and is orientated the scattering object of scattering object both types, chooses isotropic scatter structure
The scattering strength calculation formula model of isotropic scatter is built, chooses the scattering strength of orientation scattering object structure orientation scattering object
Calculation formula model；Wherein, the scattering strength calculation formula model of isotropic scatter is spherical model, is orientated scattering object
Scattering strength calculation formula model is stick model；
Analyzing step：The scattering for adjusting the scattering strength calculation formula model and orientation scattering object of isotropic scatter is strong
Each adjustable parameter spent in calculation formula model so that the scattering strength calculation formula model and orientation of isotropic scatter dissipate
The calculating collection of illustrative plates and the difference of the scattering strength experimental patterns that the scattering strength calculation formula model of beam obtains after being added are minimum,
The parameter of each model can be parsed.
Further, in modeling procedure, the scattering strength calculation formula model and orientation of isotropic scatter scatter
The scattering strength calculation formula model of body is：
Wherein, q is vectorial for scattering, FnFor the scattered amplitude of n-th of scattering object, Fn’Scattering for the n-th ' individual scattering object is shaken
Width, N be scattering object quantity, rnFor the center of n-th of scattering object, rn'For the center of the n-th ' individual scattering object.
Further, in the modeling procedure, set the center of the scattering object as the origin of coordinates, then it is described respectively to same
Property scattering object and scattered amplitude F (q) calculation formula of the orientation scattering object be：
F (q)=∫vρdVe-iqr (2)
Wherein, q is scattering vector, and V is the volume of scattering object, and ρ is the electron density of scattering object, and r is any on scattering object
A little arrive the distance of the origin of coordinates.
Further, in modeling procedure, according to the scattered amplitude of scattering object and according to the Size Distribution of scattering object and
Distribution of orientations, to build above-mentioned scattering strength calculation formula model；
Constructed scattering strength calculation formula model is：
Wherein, C1For constant, R1、R2、R3For three and half axial lengths of scattering object,WithFor scattering object
The Size Distribution of three axles, h (ω) are to be distributed along axis of orientation zenith angle,For azimuthal distribution, ω is represented along axis of orientation day
Drift angle, ψ represent Scattering of Vector and qxThe angle of component direction,Represent azimuth；
For being orientated scattering object, in formula (8), R1、R2、R3Value differ, or wherein both are identical, the 3rd
Person is different from both；
For isotropic scatter, in formula (8), R1=R2=R3, h (ω) is constant, and azimuthal distribution is constant.
Further, wherein, for being orientated scattering object, describe the zenith angle using Von Mises distribution functions and be distributed,
Functional form is as follows：
Wherein I0For the bessel functions of the rank of the first kind 0 amendment, ω0It is true by variance parameter к for average value, the variance of distribution
It is fixed,It is defined as being uniformly distributed；In formula (10), ω0, κ be adjustable parameter.
Further, wherein, describe the axle of the orientation scattering object using log series model function and the isotropism dissipates
The radius of beam, shown in the form such as following formula (11) of the log series model function：
Wherein μ and σ is respectively the logarithmic average and logarithm standard deviation of logarithm normal distribution, its average value m and variance parameter υ
Tried to achieve by following formula：
M=exp (μ+σ2/2) (12)
υ=exp (2 μ+σ2)(expσ2-1) (13)
In formula (11), μ and σ are adjustable parameter.
A kind of dual model fitting system of small angle X ray scattering proposed by the present invention, the system include following modules：
Acquisition module：Obtain the scattering strength experimental patterns of analyzed object；
Modeling module：According to the feature construction dual model of scattering strength experimental patterns, it is specially：The analyzed object
Scattering system has isotropic scatter and is orientated the scattering object of scattering object both types, chooses isotropic scatter structure
The scattering strength calculation formula model of isotropic scatter is built, chooses the scattering strength of orientation scattering object structure orientation scattering object
Calculation formula model；Wherein, the scattering strength calculation formula model of isotropic scatter is spherical model, is orientated scattering object
Scattering strength calculation formula model is stick model；
Parsing module：The scattering for adjusting the scattering strength calculation formula model and orientation scattering object of isotropic scatter is strong
Each adjustable parameter spent in calculation formula model so that the scattering strength calculation formula model and orientation of isotropic scatter dissipate
The calculating collection of illustrative plates and the difference of the scattering strength experimental patterns that the scattering strength calculation formula model of beam obtains after being added are minimum,
The parameter of each model can be parsed.
Further, in modeling module, the scattering strength calculation formula model and orientation of isotropic scatter scatter
The scattering strength calculation formula model of body is：
Wherein, q is vectorial for scattering, FnFor the scattered amplitude of n-th of scattering object, Fn’Scattering for the n-th ' individual scattering object is shaken
Width, N be scattering object quantity, rnFor the center of n-th of scattering object, rn'For the center of the n-th ' individual scattering object.
Further, in the modeling module, set the center of the scattering object as the origin of coordinates, then it is described respectively to same
Property scattering object and scattered amplitude F (q) calculation formula of the orientation scattering object be：
F (q)=∫vρdVe-iqr (2)
Wherein, q is scattering vector, and V is the volume of scattering object, and ρ is the electron density of scattering object, and r is any on scattering object
A little arrive the distance of the origin of coordinates.
Further, in modeling module, according to the scattered amplitude of scattering object and according to the Size Distribution of scattering object and
Distribution of orientations, to build above-mentioned scattering strength calculation formula model；
Constructed scattering strength calculation formula model is：
Wherein, C1For constant, R1、R2、R3For three and half axial lengths of scattering object,WithFor scattering object
The Size Distribution of three axles, h (ω) are to be distributed along axis of orientation zenith angle,For azimuthal distribution, ω is represented along axis of orientation
Zenith angle, ψ represent Scattering of Vector and qxThe angle of component direction,Represent azimuth；
For being orientated scattering object, in formula (8), R1、R2、R3Value differ, or wherein both are identical, the 3rd
Person is different from both；
For isotropic scatter, in formula (8), R1=R2=R3, h (ω) is constant, and azimuthal distribution is constant.
Further, wherein, for being orientated scattering object, describe the zenith angle using Von Mises distribution functions and be distributed,
Functional form is as follows：
Wherein I0For the bessel functions of the rank of the first kind 0 amendment, ω0It is true by variance parameter к for average value, the variance of distribution
It is fixed,It is defined as being uniformly distributed；In formula (10), ω0, κ be adjustable parameter.
Further, wherein, describe the axle of the orientation scattering object using log series model function and the isotropism dissipates
The radius of beam, shown in the form such as following formula (11) of the log series model function：
Wherein μ and σ is respectively the logarithmic average and logarithm standard deviation of logarithm normal distribution, its average value m and variance parameter υ
Tried to achieve by following formula：
M=exp (μ+σ2/2) (12)
υ=exp (2 μ+σ2)(expσ2-1) (13)
In formula (11), μ and σ are adjustable parameter.
Beneficial effects of the present invention：The present invention is according to the meter for establishing spherical and bar-shaped dual model in fiber the characteristics of scattering object
Calculation method, the accurate hole information of fibrous inside is obtained, further expand the scope of application of model.
Brief description of the drawings
Fig. 1 is the flow chart of the dual model approximating method shown in the present invention；
Fig. 2 is the structured flowchart of the dual model fitting system shown in the present invention；
Fig. 3 is the phase difference schematic diagram of incident X-rays and scattered x-ray in the scattering object shown in the present invention；
Fig. 4 is the schematic diagram of orientation scattering object each angular relationship in a coordinate system shown in the present invention；
Fig. 5 is the schematic diagram of the two-dimentional dual model analysis method shown in the present invention；
Fig. 6 a and Fig. 6 b are the Von Mises function curves under the different parameters shown in the present invention；
Fig. 7 is logarithm normal distribution function curve under the different parameters shown in the present invention；
Fig. 8 is the small angle X ray scattering experimental patterns that the present invention is directed to；
Fig. 9 is the double model structure schematic diagram that the present invention proposes scattering object；
Figure 10 is the two-dimensional scattering collection of illustrative plates of the dual model digital simulation of the present invention.
Embodiment
For the object, technical solutions and advantages of the present invention are more clearly understood, below in conjunction with specific embodiment, and reference
Accompanying drawing, the present invention is described in more detail.But those skilled in the art know, the invention is not limited in accompanying drawing and following reality
Apply example.
As shown in figure 1, the present invention proposes a kind of small angle X ray scattering SAXS dual model approximating method, including it is following
Step：
Obtaining step：The scattering strength experimental patterns of analyzed object are obtained, as shown in Figure 8；
Modeling procedure：According to the feature construction dual model of scattering strength experimental patterns, it is specially：The analyzed object
Scattering system has isotropic scatter and is orientated the scattering object of scattering object both types, chooses isotropic scatter structure
The scattering strength calculation formula model of isotropic scatter is built, chooses the scattering strength of orientation scattering object structure orientation scattering object
Calculation formula model；Wherein, the scattering strength calculation formula model of isotropic scatter is spherical model, is orientated scattering object
Scattering strength calculation formula model is stick model, as shown in figure 9, the present invention is using spherical model and the dual model of stick model
To calculate two-dimensional scattering collection of illustrative plates, the scope of application of model is expanded, improves the accuracy of fitting；
Analyzing step：The scattering for adjusting the scattering strength calculation formula model and orientation scattering object of isotropic scatter is strong
Each adjustable parameter spent in calculation formula model so that the scattering strength calculation formula model and orientation of isotropic scatter dissipate
The calculating collection of illustrative plates and the difference of the scattering strength experimental patterns that the scattering strength calculation formula model of beam obtains after being added are minimum,
The parameter of each model can be parsed.
In modeling procedure, the present invention analyzes the scattering object that volume is V first, it is assumed that its electron density be ρ (r)=
ρ, the electron density outside scattering object is ρ (r)=0, then when beam of x-rays reaches the system, the electronics forced oscillation in scattering object
It is dynamic, produce secondary wave (scattered wave).The phase difference schematic diagram of its incident X-rays and scattered x-ray is as shown in Figure 3.
The scattered amplitude of scattering object can be written as：
F (q)=∫ ∫ ∫ dV ρ (r) eiqr (1)
It is abbreviated as：
F (q)=∫vρdVe-iqr (2)
Wherein, q is scattering vector, and V is the volume of scattering object, and ρ is the electron density of scattering object, and r is any on scattering object
The distance of the origin of coordinates is a little arrived, i is imaginary unit.
If there is N number of scattering object in a system, center is located at r respectively1, r2, r3... rN, electron density can be written as ρn
(r-rn) then the total electron density of system can be written as：
The scattered amplitude of system can be written as：
According to above-mentioned formula, the scattering strength I (q) of spherical model and stick model can be derived by modeling procedure
Formula be：
As n=n ',Above formula can be further written as：
The size and orientation of scattering object in system are introduced into when calculating scattering strength, wherein, define three axles of scattering object
Size Distribution isWithH (ω) is distributed as along axis of orientation zenith angle, azimuthal distribution isIts body
Product isBecause the interference between the scattering object in lax system is very weak, therefore the Section 2 in above formula (6)
It is zero.
Above formula (6) can be further simplified as：
Then the scattering strength of spherical model and stick model can be written as：
There is following relation in above formula between each angle：
γ represents scattering object and the angle of X-axis, and ω represents that along axis of orientation zenith angle 2 θ are incident X-rays and scattered x-ray
Angle, ψ represents Scattering of Vector and qxThe angle of component direction,Represent azimuth.Scattering strength meter for being orientated scattering object
Calculate formula model, i.e. stick model, R1、R2、R3Value differ, or wherein both are identical, the third party with both not
Together；For the scattering strength calculation formula model of isotropic scatter, i.e. spherical model, R1=R2=R3, h (ω) is constant,
Azimuthal distribution is constant.
It is assumed that orientation scattering object has rotational invariance, azimuthal distributionIt is defined as being uniformly distributed, is a constant.My god
Drift angle distribution h (ω) is defined as Von Mises distributions, and functional form is as follows：
Wherein I0For the bessel functions of the rank of the first kind 0 amendment, ω0For average value, the variance of distribution is determined by к.
Von Mises distributions are the most frequently used circular distributions, and span is generally [0,2 π], are suitable for describing scattering object
Distribution of orientations.As shown in figures 6 a and 6b, the Von Mises function curves under different parameters are described.
The axle R being orientated in modeling procedure in scattering object1、R2、R3Logarithm normal distribution f (R) is obeyed, isotropism dissipates
Axle in beam, that is, the radius R of spheroid also obey logarithm normal distribution f (R), and functional form is as follows：
Wherein μ and σ is respectively the logarithmic average and logarithm standard deviation of logarithm normal distribution, and average value m and variance v can lead to
Following formula is crossed to try to achieve：
M=exp (μ+σ2/2) (12)
υ=exp (2 μ+σ2)(expσ2-1) (13)
As shown in Figure 7, it is shown that logarithm normal distribution function curve under different parameters.
In analyzing step, each adjustable parameter value is given, you can calculate two-dimensional scattering collection of illustrative plates corresponding to each model, such as scheme
Shown in 10；The calculating collection of illustrative plates that described two scattering strength calculation formula are added to obtain and the scattering strength experimental patterns are carried out
Compare, by adjusting the parameter μ of size, σ and the parameter ω of direction of adjustment angle0, к it is strong to adjust the scattering of stick model
Angle value, the scattering strength value of spherical model is adjusted by adjusting parameter μ, the σ of size, so that stick model dissipates
Penetrate intensity level and both scattering strengths of obtained calculating collection of illustrative plates and experimental patterns error is added with the scattering strength value of spherical model
Minimum, as shown in Figure 5, you can parse the parameter of two models.Before experimental patterns and calculating collection of illustrative plates are compared, also need
Experimental patterns are modified, deduct the influence of laboratory apparatus and practical operation to experimental patterns.Experimental patterns are entered first
The various corrections of row, such as backscatter, dark current, beam shape correction, obtain high quality two dimension experiment scatter pattern.Pass through again
The method of two models of structure calculates two-dimensional scattering collection of illustrative plates, adjusts the parameter of each model so that experimental patterns and calculating collection of illustrative plates
Difference it is minimum, you can parse the parameter of each model, and pass through the validity that standard sample verifies model.
The invention also provides the dual model digital simulation system of small angle X ray scattering SAXS a kind of, as shown in Fig. 2 bag
Include following modules：
Acquisition module：The scattering strength experimental patterns of analyzed object are obtained, as shown in Figure 8；
Modeling module：According to the feature construction dual model of scattering strength experimental patterns, it is specially：The analyzed object
Scattering system has isotropic scatter and is orientated the scattering object of scattering object both types, chooses isotropic scatter structure
The scattering strength calculation formula model of isotropic scatter is built, chooses the scattering strength of orientation scattering object structure orientation scattering object
Calculation formula model；Wherein, the scattering strength calculation formula model of isotropic scatter is spherical model, is orientated scattering object
Scattering strength calculation formula model is stick model, as shown in figure 9, the present invention is using spherical model and the dual model of stick model
To calculate two-dimensional scattering collection of illustrative plates, the scope of application of model is expanded, improves the accuracy of fitting；
Parsing module：The scattering for adjusting the scattering strength calculation formula model and orientation scattering object of isotropic scatter is strong
Each adjustable parameter spent in calculation formula model so that the scattering strength calculation formula model and orientation of isotropic scatter dissipate
The calculating collection of illustrative plates and the difference of the scattering strength experimental patterns that the scattering strength calculation formula model of beam obtains after being added are minimum,
The parameter of each model can be parsed.
In modeling module, the present invention analyzes the scattering object that volume is V first, it is assumed that its electron density be ρ (r)=
ρ, the electron density outside scattering object is ρ (r)=0, then when beam of x-rays reaches the system, the electronics forced oscillation in scattering object
It is dynamic, produce secondary wave (scattered wave).The phase difference schematic diagram of its incident X-rays and scattered x-ray is as shown in Figure 3.
The scattered amplitude of scattering object can be written as：
F (q)=∫ ∫ ∫ dV ρ (r) eiqr (1)
It is abbreviated as：
F (q)=∫vρdVe-iqr (2)
Wherein, q is scattering vector, and V is the volume of scattering object, and ρ is the electron density of scattering object, and r is any on scattering object
The distance of the origin of coordinates is a little arrived, i is imaginary unit.
If there is N number of scattering object in a system, center is located at r respectively1, r2, r3... rN, electron density can be written as ρn
(r-rn) then the total electron density of system can be written as：
The scattered amplitude of system can be written as：
According to above-mentioned formula, the scattering strength I (q) of spherical model and stick model can be derived by modeling module
Formula be：
As n=n ',Above formula can be further written as：
The size and orientation of scattering object in system are introduced into when calculating scattering strength, wherein, define three axles of scattering object
Size Distribution isWithH (ω) is distributed as along axis of orientation zenith angle, azimuthal distribution isIts body
Product isBecause the interference between the scattering object in lax system is very weak, therefore the Section 2 in above formula (6)
It is zero.
Above formula (6) can be further simplified as：
Then the scattering strength of spherical model and stick model can be written as：
There is following relation in above formula between each angle：
γ represents scattering object and the angle of X-axis, and ω represents that along axis of orientation zenith angle 2 θ are incident X-rays and scattered x-ray
Angle, ψ represents Scattering of Vector and qxThe angle of component direction,Represent azimuth.Scattering strength meter for being orientated scattering object
Calculate formula model, i.e. stick model, R1、R2、R3Value differ, or wherein both are identical, the third party with both not
Together；For the scattering strength calculation formula model of isotropic scatter, i.e. spherical model, R1=R2=R3, h (ω) is constant,
Azimuthal distribution is constant.
It is assumed that orientation scattering object has rotational invariance, azimuthal distributionIt is defined as being uniformly distributed, is a constant.My god
Drift angle distribution h (ω) is defined as Von Mises distributions, and functional form is as follows：
Wherein I0For the bessel functions of the rank of the first kind 0 amendment, ω0For average value, the variance of distribution is determined by к.
Von Mises distributions are the most frequently used circular distributions, and span is generally [0,2 π], are suitable for describing scattering object
Distribution of orientations.As shown in figures 6 a and 6b, the Von Mises function curves under different parameters are described.
And the axle R in scattering object is orientated in modeling module1、R2、R3Logarithm normal distribution f (R) is obeyed, isotropism dissipates
Axle in beam, that is, the radius R of spheroid also obey logarithm normal distribution f (R), and functional form is as follows：
Wherein μ and σ is respectively the logarithmic average and logarithm standard deviation of logarithm normal distribution, and average value m and variance v can lead to
Following formula is crossed to try to achieve：
M=exp (μ+σ2/2) (12)
υ=exp (2 μ+σ2)(expσ2-1) (13)
As shown in Figure 7, it is shown that logarithm normal distribution function curve under different parameters.
In parsing module, each adjustable parameter value is given, you can calculate two-dimensional scattering collection of illustrative plates corresponding to each model, such as scheme
Shown in 10；The calculating collection of illustrative plates that described two scattering strength calculation formula are added to obtain and the scattering strength experimental patterns are carried out
Compare, by adjusting the parameter μ of size, σ and the parameter ω of direction of adjustment angle0, к it is strong to adjust the scattering of stick model
Angle value, the scattering strength value of spherical model is adjusted by adjusting parameter μ, the σ of size, so that stick model dissipates
Penetrate intensity level and both scattering strengths of obtained calculating collection of illustrative plates and experimental patterns error is added with the scattering strength value of spherical model
Minimum, as shown in Figure 5, you can parse the parameter of two models.Before experimental patterns and calculating collection of illustrative plates are compared, also need
Experimental patterns are modified, deduct the influence of laboratory apparatus and practical operation to experimental patterns.Experimental patterns are entered first
The various corrections of row, such as backscatter, dark current, beam shape correction, obtain high quality two dimension experiment scatter pattern.Pass through again
The method of two models of structure calculates two-dimensional scattering collection of illustrative plates, adjusts the parameter of each model so that experimental patterns and calculating collection of illustrative plates
Difference it is minimum, you can parse the parameter of each model, and pass through the validity that standard sample verifies model.
The embodiment of Von Mises function curves and Lognormal distribution curve under different parameters is given below.
Embodiment 1：
Arrange parameter ω0=-pi/2；κ=100, obtain the Von Mises function curves as corresponded to parameter in Fig. 6 a.
Embodiment 2：
Arrange parameter ω0=-π/4；κ=100, obtain the Von Mises function curves as corresponded to parameter in Fig. 6 a.
Embodiment 3：
Arrange parameter ω0=0；κ=100, obtain the Von Mises function curves as corresponded to parameter in Fig. 6 a.
Embodiment 4：
Arrange parameter ω0=π/4；κ=100, obtain the Von Mises function curves as corresponded to parameter in Fig. 6 a.
Embodiment 5：
Arrange parameter ω0=pi/2；κ=100, obtain the Von Mises function curves as corresponded to parameter in Fig. 6 a.
Embodiment 6：
Arrange parameter ω0=3 π/4；κ=100, obtain the Von Mises function curves as corresponded to parameter in Fig. 6 a.
Embodiment 7：
Arrange parameter ω0=0.95 π；κ=100, obtain the Von Mises function curves as corresponded to parameter in Fig. 6 a.
Embodiment 8：
Arrange parameter ω0=π；κ=100, obtain the Von Mises function curves as corresponded to parameter in Fig. 6 a.
Embodiment 9：
Arrange parameter ω0=0；κ=0, obtain the Von Mises function curves as corresponded to parameter in Fig. 6 b.
Embodiment 10：
Arrange parameter ω0=0；κ=4, obtain the Von Mises function curves as corresponded to parameter in Fig. 6 b.
Embodiment 11：
Arrange parameter ω0=0；κ=8, obtain the Von Mises function curves as corresponded to parameter in Fig. 6 b.
Embodiment 12：
Arrange parameter ω0=0；κ=12, obtain the Von Mises function curves as corresponded to parameter in Fig. 6 b.
Embodiment 13：
Arrange parameter ω0=0；κ=16, obtain the Von Mises function curves as corresponded to parameter in Fig. 6 b.
Embodiment 14：
Arrange parameter ω0=0；κ=20, obtain the Von Mises function curves as corresponded to parameter in Fig. 6 b.
Embodiment 15：
Arrange parameter ω0=0；κ=100, obtain the Von Mises function curves as corresponded to parameter in Fig. 6 b.
Embodiment 16：
Arrange parameter m=20；V=200, obtain the logarithm normal distribution function curve as corresponded to parameter in Fig. 7.
Embodiment 17：
Arrange parameter m=20；V=150, obtain the logarithm normal distribution function curve as corresponded to parameter in Fig. 7.
Embodiment 18：
Arrange parameter m=20；V=100, obtain the logarithm normal distribution function curve as corresponded to parameter in Fig. 7.
Embodiment 19：
Arrange parameter m=20；V=80, obtain the logarithm normal distribution function curve as corresponded to parameter in Fig. 7.
Embodiment 20：
Arrange parameter m=20；V=60, obtain the logarithm normal distribution function curve as corresponded to parameter in Fig. 7.
Embodiment 21：
Arrange parameter m=20；V=40, obtain the logarithm normal distribution function curve as corresponded to parameter in Fig. 7.
Embodiment 22：
Arrange parameter m=20；V=20, obtain the logarithm normal distribution function curve as corresponded to parameter in Fig. 7.
Embodiment 23：
Arrange parameter m=20；V=10, obtain the logarithm normal distribution function curve as corresponded to parameter in Fig. 7.
By setting different parameters, the zenith angle distribution and the normal distribution of major and minor axis of scattering object are adjusted so that pass through
What two models were added to obtain calculates the error minimum between collection of illustrative plates and experimental patterns, you can parses the parameter of each model.Root
It is modeled according to the feature of scattering strength experimental patterns so that the Porous Characteristic expression of fibrous material is more accurate.
More than, embodiments of the present invention are illustrated.But the present invention is not limited to above-mentioned embodiment.It is all
Within the spirit and principles in the present invention, any modification, equivalent substitution and improvements done etc., it should be included in the guarantor of the present invention
Within the scope of shield.
Claims (10)
1. the dual model approximating method of a kind of small angle X ray scattering, it is characterised in that this method comprises the following steps：
Obtaining step：Obtain the scattering strength experimental patterns of analyzed object；
Modeling procedure：According to the feature construction dual model of scattering strength experimental patterns, it is specially：The scattering of the analyzed object
System has isotropic scatter and is orientated the scattering object of scattering object both types, and it is each to choose isotropic scatter structure
To the scattering strength calculation formula model of same sex scattering object, the scattering strength for choosing orientation scattering object structure orientation scattering object calculates
Formula model；Wherein, the scattering strength calculation formula model of isotropic scatter is spherical model, is orientated the scattering of scattering object
Strength calculation formula model is stick model；
Analyzing step：Adjust the scattering strength calculation formula model of isotropic scatter and the scattering strength meter of orientation scattering object
Calculate each adjustable parameter in formula model so that the scattering strength calculation formula model and orientation scattering object of isotropic scatter
Scattering strength calculation formula model be added after obtained calculating collection of illustrative plates and the difference minimum of the scattering strength experimental patterns, you can
Parse the parameter of each model.
2. the method as described in claim 1, it is characterised in that in modeling procedure, the scattering strength of isotropic scatter
Calculation formula model and orientation scattering object scattering strength calculation formula model be：
Wherein, q is vectorial for scattering, FnFor the scattered amplitude of n-th of scattering object, Fn’For the scattered amplitude of the n-th ' individual scattering object, N is
The quantity of scattering object, rnFor the center of n-th of scattering object, rn'For the center of the n-th ' individual scattering object.
3. method as claimed in claim 2, it is characterised in that in the modeling procedure, set the center of the scattering object
For the origin of coordinates, then the scattered amplitude F (q) of the isotropic scatter and orientation scattering object calculation formula is：
F (q)=∫vρdVe-iqr (2)
Wherein, q is scattering vector, and V is the volume of scattering object, and ρ is the electron density of scattering object, and r is any point on scattering object
To the distance of the origin of coordinates.
4. method as claimed in claim 3, it is characterised in that in modeling procedure, according to the scattered amplitude of scattering object and
According to the Size Distribution and distribution of orientations of scattering object, to build above-mentioned scattering strength calculation formula model；
Constructed scattering strength calculation formula model is：
Wherein, C1For constant, R1、R2、R3For three and half axial lengths of scattering object, f (R1)、f(R2) and f (R3) it is three axles of scattering object
Size Distribution, h (ω) be along axis of orientation zenith angle be distributed,For azimuthal distribution, ω is represented along axis of orientation zenith angle, ψ
Represent Scattering of Vector and qxThe angle of component direction,Represent azimuth；
For being orientated scattering object, in formula (8), R1、R2、R3Value differ, or wherein both are identical, the third party with
Both is different；
For isotropic scatter, in formula (8), R1=R2=R3, h (ω) is constant, and azimuthal distribution is constant.
5. method as claimed in claim 4, it is characterised in that wherein, for being orientated scattering object, be distributed using Von Mises
Function describes the zenith angle distribution, and functional form is as follows：
<mrow>
<mi>h</mi>
<mrow>
<mo>(</mo>
<mi>&omega;</mi>
<mo>|</mo>
<msub>
<mi>&omega;</mi>
<mn>0</mn>
</msub>
<mo>,</mo>
<mi>&kappa;</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<msup>
<mi>e</mi>
<mrow>
<mi>&kappa;</mi>
<mi>c</mi>
<mi>o</mi>
<mi>s</mi>
<mrow>
<mo>(</mo>
<mi>&omega;</mi>
<mo>-</mo>
<msub>
<mi>&omega;</mi>
<mn>0</mn>
</msub>
<mo>)</mo>
</mrow>
</mrow>
</msup>
<mrow>
<mn>2</mn>
<msub>
<mi>&pi;I</mi>
<mn>0</mn>
</msub>
<mrow>
<mo>(</mo>
<mi>&kappa;</mi>
<mo>)</mo>
</mrow>
</mrow>
</mfrac>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>10</mn>
<mo>)</mo>
</mrow>
</mrow>
Wherein I0For the bessel functions of the rank of the first kind 0 amendment, ω0For average value, the variance of distribution is determined by variance parameter к,It is defined as being uniformly distributed；In formula (10), ω0, κ be adjustable parameter；
And/or
Wherein, the axle of the orientation scattering object and the radius of the isotropic scatter, institute are described using log series model function
Shown in the form such as following formula (11) for stating log series model function：
<mrow>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>R</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mi>R</mi>
<mi>&sigma;</mi>
<msqrt>
<mrow>
<mn>2</mn>
<mi>&pi;</mi>
</mrow>
</msqrt>
</mrow>
</mfrac>
<msup>
<mi>e</mi>
<mfrac>
<mrow>
<mo>-</mo>
<msup>
<mrow>
<mo>(</mo>
<mi>ln</mi>
<mi>R</mi>
<mo>-</mo>
<mi>&mu;</mi>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mrow>
<mn>2</mn>
<msup>
<mi>&sigma;</mi>
<mn>2</mn>
</msup>
</mrow>
</mfrac>
</msup>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>11</mn>
<mo>)</mo>
</mrow>
</mrow>
Wherein μ and σ is respectively the logarithmic average and logarithm standard deviation of logarithm normal distribution, and its average value m and variance parameter υ pass through
Following formula is tried to achieve：
M=exp (μ+σ2/2) (12)
υ=exp (2 μ+σ2)(expσ2-1) (13)
In formula (11), μ and σ are adjustable parameter.
6. the dual model fitting system of a kind of small angle X ray scattering, it is characterised in that the system includes following modules：
Acquisition module：Obtain the scattering strength experimental patterns of analyzed object；
Modeling module：According to the feature construction dual model of scattering strength experimental patterns, it is specially：The scattering of the analyzed object
System has isotropic scatter and is orientated the scattering object of scattering object both types, and it is each to choose isotropic scatter structure
To the scattering strength calculation formula model of same sex scattering object, the scattering strength for choosing orientation scattering object structure orientation scattering object calculates
Formula model；Wherein, the scattering strength calculation formula model of isotropic scatter is spherical model, is orientated the scattering of scattering object
Strength calculation formula model is stick model；
Parsing module：Adjust the scattering strength calculation formula model of isotropic scatter and the scattering strength meter of orientation scattering object
Calculate each adjustable parameter in formula model so that the scattering strength calculation formula model and orientation scattering object of isotropic scatter
Scattering strength calculation formula model be added after obtained calculating collection of illustrative plates and the difference minimum of the scattering strength experimental patterns, you can
Parse the parameter of each model.
7. system as claimed in claim 6, it is characterised in that in modeling module, the scattering strength of isotropic scatter
Calculation formula model and orientation scattering object scattering strength calculation formula model be：
Wherein, q is vectorial for scattering, FnFor the scattered amplitude of n-th of scattering object, Fn’For the scattered amplitude of the n-th ' individual scattering object, N is
The quantity of scattering object, rnFor the center of n-th of scattering object, rn’For the center of the n-th ' individual scattering object.
8. system as claimed in claim 7, it is characterised in that in the modeling module, set the center of the scattering object
For the origin of coordinates, then the scattered amplitude F (q) of the isotropic scatter and orientation scattering object calculation formula is：
F (q)=∫vρdVe-iqr (2)
Wherein, q is scattering vector, and V is the volume of scattering object, and ρ is the electron density of scattering object, and r is any point on scattering object
To the distance of the origin of coordinates.
9. system as claimed in claim 8, it is characterised in that in modeling module, according to the scattered amplitude of scattering object and
According to the Size Distribution and distribution of orientations of scattering object, to build above-mentioned scattering strength calculation formula model；
Constructed scattering strength calculation formula model is：
Wherein, C1For constant, R1、R2、R3For three and half axial lengths of scattering object, f (R1)、f(R2) and f (R3) it is three axles of scattering object
Size Distribution, h (ω) be along axis of orientation zenith angle be distributed,For azimuthal distribution, ω is represented along axis of orientation zenith angle,
ψ represents Scattering of Vector and qxThe angle of component direction,Represent azimuth；
For being orientated scattering object, in formula (8), R1、R2、R3Value differ, or wherein both are identical, the third party with
Both is different；
For isotropic scatter, in formula (8), R1=R2=R3, h (ω) is constant, and azimuthal distribution is constant.
10. system as claimed in claim 9, it is characterised in that wherein, for being orientated scattering object, be distributed using Von Mises
Function describes the zenith angle distribution, and functional form is as follows：
<mrow>
<mi>h</mi>
<mrow>
<mo>(</mo>
<mi>&omega;</mi>
<mo>|</mo>
<msub>
<mi>&omega;</mi>
<mn>0</mn>
</msub>
<mo>,</mo>
<mi>&kappa;</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<msup>
<mi>e</mi>
<mrow>
<mi>&kappa;</mi>
<mi>c</mi>
<mi>o</mi>
<mi>s</mi>
<mrow>
<mo>(</mo>
<mi>&omega;</mi>
<mo>-</mo>
<msub>
<mi>&omega;</mi>
<mn>0</mn>
</msub>
<mo>)</mo>
</mrow>
</mrow>
</msup>
<mrow>
<mn>2</mn>
<msub>
<mi>&pi;I</mi>
<mn>0</mn>
</msub>
<mrow>
<mo>(</mo>
<mi>&kappa;</mi>
<mo>)</mo>
</mrow>
</mrow>
</mfrac>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>10</mn>
<mo>)</mo>
</mrow>
</mrow>
Wherein I0For the bessel functions of the rank of the first kind 0 amendment, ω0For average value, the variance of distribution is determined by variance parameter к,It is defined as being uniformly distributed；In formula (10), ω0, κ be adjustable parameter；
And/or
Wherein, the axle of the orientation scattering object and the radius of the isotropic scatter, institute are described using log series model function
Shown in the form such as following formula (11) for stating log series model function：
<mrow>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>R</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mi>R</mi>
<mi>&sigma;</mi>
<msqrt>
<mrow>
<mn>2</mn>
<mi>&pi;</mi>
</mrow>
</msqrt>
</mrow>
</mfrac>
<msup>
<mi>e</mi>
<mfrac>
<mrow>
<mo>-</mo>
<msup>
<mrow>
<mo>(</mo>
<mi>ln</mi>
<mi>R</mi>
<mo>-</mo>
<mi>&mu;</mi>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mrow>
<mn>2</mn>
<msup>
<mi>&sigma;</mi>
<mn>2</mn>
</msup>
</mrow>
</mfrac>
</msup>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>11</mn>
<mo>)</mo>
</mrow>
</mrow>
Wherein μ and σ is respectively the logarithmic average and logarithm standard deviation of logarithm normal distribution, and its average value m and variance parameter υ pass through
Following formula is tried to achieve：
M=exp (μ+σ2/2) (12)
υ=exp (2 μ+σ2)(expσ2-1) (13)
In formula (11), μ and σ are adjustable parameter.
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| CN112037183A (en) * | 2020-08-12 | 2020-12-04 | 深圳大学 | 2D SAXS spectrum calculation method and device |
| CN113049615A (en) * | 2019-12-27 | 2021-06-29 | 株式会社理学 | Scatterometry analysis method, scatterometry analysis device, and information storage medium |
| CN113624168A (en) * | 2021-08-05 | 2021-11-09 | 中国科学院上海高等研究院 | Key size measuring system and method based on small-angle X-ray scattering technology |
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| CN112037183A (en) * | 2020-08-12 | 2020-12-04 | 深圳大学 | 2D SAXS spectrum calculation method and device |
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