CN100545644C - A kind of semi-automatic X-ray crystal plane index calibration and lattice constant Calculation Method - Google Patents

Abstract

The present invention relates to X ray crystal face index calibration method and lattice constant computing method.According to LaNi ₅The base hydrogen storage material has and LaNi ₅The ultimate principle of identical peak shape has been worked out the semi-automatic calibrating procedure of computing machine and has been realized crystal face calibration on X-ray spectrum, utilizes this crystal face calibration result to adopt least square method to work out lattice constant accurate Calculation program simultaneously, adopts the present invention can carry out LaNi ₅The crystal face index calibration of base hydrogen storage material X-ray diffraction spectrum and the accurate Calculation of lattice constant.

Description

A kind of semi-automatic X-ray crystal plane index calibration and lattice constant Calculation Method

Technical field

The present invention relates to X ray crystal face index calibration method and lattice constant computing method, a kind of LaNi of being used for is provided especially ₅The semi-automatic X-ray crystal plane index calibration and the lattice constant Calculation Method based on MATLAB of base hydrogen storage material.

Background technology

Lattice constant is one of crystalline material important physical parameter, and accurately measuring lattice constant, to help to study the bonded energy and the key of this material strong; The component of theory of computation density, anisotropic thermal expansion coefficient and compressibility coefficient, solid solution and solid solubility, macroscopical unrelieved stress size; Determine the phase boundary of phase solubility curve and phasor; The research phase transition process; The relation of analysis of material lattice constant and various physical properties.At LaNi ₅Base hydrogen storage material, scientists often need to determine that its suction puts before and after the hydrogen intensity of variation of lattice constant and conclude to inhale and put the intensity of material self strain of bearing in the hydrogen process, thereby further infer the service life and the resistance to chalking energy of this hydrogen storage material.The lattice constant of simple substance or compound crystal changes very little with temperature, pressure, component etc., generally 10 ^-3～10 ^-5The nm order of magnitude, thereby must accurately measure.The main experimental methods of accurately measuring lattice constant is the polycrystalline diffraction approach, main calculation methods ills extrapolation method, and least square method, line is to method, and line is to least square method.And must carry out crystal face index calibration to the polycrystalline X-ray spectrum of sample before the accurate Calculation lattice constant to set up the one-to-one relationship between the indices of crystal plane and the angle of diffraction.At present, scholars generally believe that versatility and the accuracy of least square method when calculating lattice constant of Ke Heng proposition is the highest.The least square method data volume is huge, is suitable for Computing.This computation process relates to matrix operation, adopts traditional computer language such as C, C++ or Fortran to realize then relatively difficulty, and easily makes mistakes.The present invention adopts the MATLAB language to realize at LaNi ₅The semi-automatic X-ray crystal plane index calibration and the lattice constant least square accurate Calculation of base hydrogen storage material.

Summary of the invention

The object of the present invention is to provide a kind of LaNi of being used for ₅The semi-automatic X-ray crystal plane index calibration and the lattice constant Calculation Method based on MATLAB of base hydrogen storage material.

Technical scheme of the present invention is:

A kind of LaNi that is used for ₅The semi-automatic X-ray crystal plane index calibration method and the lattice constant computing method based on MATLAB of base hydrogen storage material comprise the steps:

1) 20～30um dusting cover is also crossed in sample powder process, on X-ray diffractometer, used CuK α radiation then Sample is carried out continuous sweep, sweep velocity: 0.12～2 second/step (sec./step), scanning step: in 0.02～0.04 degree/step (deg./step), the scope of 2 θ: 20-90 degree (deg.) obtained the X-ray diffraction spectrum of sample.

2) data file with the LaNi5-CuKa-standard-xrd by name of file in crystal parameter and the index calibrating file imports in the work space (workspace) of MATLAB, and the user only needs the alternative Sample_xrd data with existing of the diffraction data of sample X-ray diffraction spectrum is got final product.Sample_xrd is that a columns is two array, and first classifies angle of diffraction as, and second classifies diffracted intensity as.

3) in the command window (command window) of MATLAB, key in cal_cell_para (Sample_xrd then, lumda, standxrd_JCPDS, LaNi5xrd_reference) a personal-machine interactive window can be occurred on the screen of order back, two X-ray diffraction spectrums and Red Cross LaNi can be shown in the window ₅The normal place at X ray peak (data from the PDF card of " JCPDS (JCPDS) "), and one can be followed the cross that mouse moves and select the peak calibration tool.That wherein, demonstration is blue is the LaNi that is used for user's reference ₅X-ray diffraction spectrum, show black for needing indexing and calculating the sample X-ray diffraction spectrum of lattice constant.

4) user passes through with reference to LaNi ₅The X-ray diffraction spectrum indices of crystal plane of coming calibration sample, and finally calculate the lattice constant of sample by the one-to-one relationship of the indices of crystal plane and angle of diffraction.

5) whole calibrating and computation process are finished by following program, and whole procedure comprises 3 functions: cal_cell_para function, cohenh function, indexing function; Wherein the indexing function is used for the X-ray diffraction spectrum of calibration sample, and input variable is the angle of diffraction and the corresponding diffracted intensity of the X-ray diffraction spectrum of sample, and LaNi ₅The angle of diffraction of standard X-ray diffraction peak and the corresponding indices of crystal plane, output variable are the angle of diffraction and the corresponding indices of crystal plane of sample; The cohenh function is used for the lattice constant of calculation sample, and input variable is the angle of diffraction of sample and the wavelength of the corresponding indices of crystal plane and diffracted ray, and output variable is the lattice constant a of sample ₀, c ₀, V ₀, used mathematic calculation is a least square method.

The used canonical matrix of coefficients of program cohenh function that least square method is calculated lattice constant is:

$\left\{\begin{array}{c}\mathrm{\Σ}{\mathrm{\α}}_i{\sin }^2{\mathrm{\θ}}_i=\mathrm{A\Σ}{\mathrm{\α}}_i^2+\mathrm{C\Σ}{\mathrm{\α}}_i{\mathrm{\γ}}_i+\mathrm{D\Σ}{\mathrm{\α}}_i{\mathrm{\δ}}_i\\ \mathrm{\Σ}{\mathrm{\γ}}_i{\sin }^2{\mathrm{\θ}}_i=\mathrm{A\Σ}{\mathrm{\α}}_i{\mathrm{\γ}}_i+\mathrm{C\Σ}{\mathrm{\γ}}_i^2+\mathrm{D\Σ}{\mathrm{\γ}}_i{\mathrm{\δ}}_i\\ \mathrm{\Σ}{\mathrm{\δ}}_i{\sin }^2{\mathrm{\θ}}_i=\mathrm{A\Σ}{\mathrm{\α}}_i{\mathrm{\δ}}_i+\mathrm{C\Σ}{\mathrm{\γ}}_i{\mathrm{\δ}}_i+\mathrm{D\Σ}{\mathrm{\δ}}_i^2\end{array}\right.$

Wherein $A=\frac{{\mathrm{\&lambda;}}^2}{{{3a}_0}^2},$ α＝h ²+hk+k ²， $C=\frac{{\mathrm{\&lambda;}}^2}{{{4\mathrm{<figure-callout\; id="0"\; label="c"\; filenames="C20051004713100211.png"\; state="\{\{state\}\}">c</figure-callout>}}_0}^2},$ γ＝1 ²， $\mathrm{\&delta;}={\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="C20051004713100202.png"\; state="\{\{state\}\}">sin</figure-callout>}}^{2}2\mathrm{\&theta;}(\frac{1}{\sin \mathrm{\&theta;}}+\frac{1}{\mathrm{\&theta;}}),$ λ is the X ray wavelength, and θ is an angle of diffraction, and h, k, l are the indices of crystal plane, a ₀, c ₀Be lattice constant.

Described cal_cell_para function flow process is as follows:

At first program brings into operation, and imports the X-ray spectrum data of sample by the operator, and computing machine is made LaNi automatically ₅The X ray of alloy supplies the operator in the timing signal reference, from the X-ray spectrum extracting data of sample go out intensity and refraction angle data respectively assignment give angle of diffraction thet2a, the strong intensity in peak, call the indexing function then and finish the demarcation of the indices of crystal plane, calling the cohenh function utilizes above-mentioned calibration result to finish the calculating of lattice constant, result of calculation is shown on the screen, EOP (end of program).

Described indexing function flow process is as follows:

At first program brings into operation, input sample X-ray spectrum angle of diffraction and peak strong data thet2a, intensity and LaNi ₅Standard diffraction data standxrd carries out normalized to the X ray intensity of sample, then from LaNi ₅Extract indices of crystal plane ordered series of numbers and angle of diffraction ordered series of numbers difference assignment among the standard diffraction data standxrd and give h_standard, k_standard, l_standard, thet2a_standard, set up a displacement ordered series of numbers p who is used to store the empty array thet2a_peak of sample diffraction peak angle of diffraction and is used for determining the indices of crystal plane then, to n and but initialize, wherein n is the subscript that is used for determining the ordered series of numbers numerical digit then, and but is used to characterize keyboard by the key assignments of building; Judge the button value of keyboard, obtain the location point and the key assignments of mouse if not 27 (are esc key corresponding to the button on the keyboard), be stored in xi respectively, yi is among the but, by computing machine at xi, search obtains sample peak position angle of diffraction near the yi, is shown in the relevant position, judges key assignments then, if key assignments is 1 (corresponding to the left button of mouse), with the left side arest neighbors LaNi at sample X ray peak ₅The indices of crystal plane of standard peak position correspondence are the indices of crystal plane at the selected peak of sample, deposit in the computing machine, if key assignments is 97 (corresponding to the button on the keyboard a) with the left side time neighbour LaNi at sample X ray peak ₅The indices of crystal plane of standard peak position correspondence are the indices of crystal plane at the selected peak of sample, deposit in the computing machine, if key assignments is the arest neighbors LaNis of 32 (corresponding to the button spacebar on the keyboard) with sample X ray peak ₅The indices of crystal plane of standard peak position correspondence are the indices of crystal plane at the selected peak of sample, deposit in the computing machine, if key assignments is the right side arest neighbors LaNis of 3 (corresponding to right mouse buttons) with sample X ray peak ₅The indices of crystal plane of standard peak position correspondence are the indices of crystal plane at the selected peak of sample, deposit in the computing machine, if key assignments any button that is other, with the right side time neighbour LaNi at sample X ray peak ₅The indices of crystal plane of standard peak position correspondence are the indices of crystal plane at the selected peak of sample, deposit in the computing machine, repeat the above-mentioned peak process of selecting, press esc key up to the operator, finish to demarcate storing process, show calibration result in window, the form storage output nominal data with array supplies other routine call, EOP (end of program) simultaneously.

Described cohenh function flow process is as follows:

At first program brings into operation, the indices of crystal plane ordered series of numbers h of the X-ray spectrum of input sample, k, l and corresponding angle of diffraction ordered series of numbers thet2a, and X ray wavelength lumda, by computing machine respectively to theta, alpha, gama, deta assignment (theta, alpha, gama, these are intermediate variable for deta, just help the expression of mathematical derivation and programming process, do not have what physical significance), set up the matrix of coefficients a of canonical equation group then, b, to matrix of coefficients carry out a left side remove operation (left side remove operation be in the MATLAB programming language at the specific operational form of matrix, its objective is to obtain reliably separating of linear equations group Ax=b) x=a b ' (b ' represent the transposition of matrix b), obtain an ordered series of numbers, the preceding two number assignment that extract ordered series of numbers are given A, and C is according to formula $A=\frac{{\mathrm{\&lambda;}}^2}{{{3a}_0}^2},$ $C=\frac{{\mathrm{\&lambda;}}^2}{{{4\mathrm{<figure-callout\; id="0"\; label="c"\; filenames="C20051004713100211.png"\; state="\{\{state\}\}">c</figure-callout>}}_0}^2},$ ${\mathrm{<figure-callout\; id="0"\; label="V"\; filenames="C20051004713100211.png"\; state="\{\{state\}\}">V</figure-callout>}}_0={a_0}^2\&CenterDot;{\mathrm{<figure-callout\; id="0"\; label="c"\; filenames="C20051004713100211.png"\; state="\{\{state\}\}">c</figure-callout>}}_0\&CenterDot;\sin \left(\frac{\mathrm{\&pi;}}{3}\right),$ Calculate and obtain lattice constant a ₀, c ₀, V ₀, EOP (end of program).

The invention has the beneficial effects as follows:

The present invention adopts the recognized standard LaNi ₅X ray diffracting data is demarcated LaNi ₅The X-ray diffraction spectrum of base hydrogen storage material sample, and foundation is demarcated the indices of crystal plane of acquisition and the lattice constant that corresponding angle of diffraction employing least square method is calculated the acquisition sample.The method can obviously improve the user from LaNi ₅Obtain the speed and the precision of lattice constant data in the base hydrogen storage material X-ray diffraction spectrum.Help the novel LaNi of user's R and D ₅The base hydrogen storage material.

Description of drawings

Fig. 1 imports to the middle software interface figure of work space (workspace) of MATLAB for the data file of the inventive method LaNi5-CuKa-standard-xrd.

Fig. 2 a-c is the MATLAB program flow diagram.

Embodiment

The invention provides a kind of LaNi of being used for ₅The semi-automatic X-ray crystal plane index calibration method and the lattice constant computing method based on MATLAB of base hydrogen storage material are according to LaNi ₅The base hydrogen storage material has and LaNi ₅The ultimate principle of identical peak shape has been worked out the semi-automatic calibrating procedure of computing machine and has been realized crystal face calibration on X-ray spectrum, utilizes this crystal face calibration result to adopt least square method to work out lattice constant accurate Calculation program simultaneously, comprises the steps:

1) 20～30um dusting cover is also crossed in sample powder process, on X-ray diffractometer, used CuK α radiation (wavelength then

) sample is carried out continuous sweep, sweep velocity: in 0.12～2 (second/step), scanning step: in 0.02～0.04 (degree/step), the scope of 2 θ: 20-90 (degree) obtains the X-ray diffraction spectrum of sample.

2) derive at LaNi according to the least square method mathematical principle ₅The canonical equation group that is used for lattice constant calculating of base hydrogen storage material:

$\left\{\begin{array}{c}\mathrm{\&Sigma;}{\mathrm{\&alpha;}}_i{\sin }^2{\mathrm{\&theta;}}_i=\mathrm{A\&Sigma;}{\mathrm{\&alpha;}}_i^2+\mathrm{C\&Sigma;}{\mathrm{\&alpha;}}_i{\mathrm{\&gamma;}}_i+\mathrm{D\&Sigma;}{\mathrm{\&alpha;}}_i{\mathrm{\&delta;}}_i\\ \mathrm{\&Sigma;}{\mathrm{\&gamma;}}_i{\sin }^2{\mathrm{\&theta;}}_i=\mathrm{A\&Sigma;}{\mathrm{\&alpha;}}_i{\mathrm{\&gamma;}}_i+\mathrm{C\&Sigma;}{\mathrm{\&gamma;}}_i^2+\mathrm{D\&Sigma;}{\mathrm{\&gamma;}}_i{\mathrm{\&delta;}}_i\\ \mathrm{\&Sigma;}{\mathrm{\&delta;}}_i{\sin }^2{\mathrm{\&theta;}}_i=\mathrm{A\&Sigma;}{\mathrm{\&alpha;}}_i{\mathrm{\&delta;}}_i+\mathrm{C\&Sigma;}{\mathrm{\&gamma;}}_i{\mathrm{\&delta;}}_i+\mathrm{D\&Sigma;}{\mathrm{\&delta;}}_i^2\end{array}\right.$

Wherein

$A=\frac{{\mathrm{\&lambda;}}^2}{{{3a}_0}^2},$ α＝h ²+hk+k ²， $C=\frac{{\mathrm{\&lambda;}}^2}{{{4\mathrm{<figure-callout\; id="0"\; label="c"\; filenames="C20051004713100211.png"\; state="\{\{state\}\}">c</figure-callout>}}_0}^2},$ γ＝1 ²， $\mathrm{\&delta;}={\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="C20051004713100202.png"\; state="\{\{state\}\}">sin</figure-callout>}}^{2}2\mathrm{\&theta;}(\frac{1}{\sin \mathrm{\&theta;}}+\frac{1}{\mathrm{\&theta;}}),$ λ is the X ray wavelength, and θ is an angle of diffraction, and h, k, l are the indices of crystal plane, a ₀, c ₀Be lattice constant.

3) according to this system of equations establishment MATLAB function cohenh, its input variable is the angle of diffraction of sample and the wavelength of the corresponding indices of crystal plane and diffracted ray, and output variable is the lattice constant a of sample ₀, c ₀, V ₀

4) because LaNi ₅The base hydrogen storage material has and LaNi ₅Similar cell configuration is so show LaNi on the X-ray spectrum ₅The base hydrogen storage material has and LaNi ₅Identical peak shape, just change has taken place in its peak position, and according to this principle establishment MATLAB function indexing, its input variable is the angle of diffraction and the corresponding diffracted intensity of the X-ray diffraction spectrum of sample, and LaNi ₅The angle of diffraction of standard X-ray diffraction peak and the corresponding indices of crystal plane, output variable are the angle of diffraction and the corresponding indices of crystal plane of sample.

5) establishment MATLAB function cal_cell_para finishes the distribution of experimental data, show sample and LaNi ₅Alloy X-ray spectrum and LaNi ₅The standard peak position distributes, and calls indexing function and cohenh function respectively, finish demarcate and calculate after with data presentation in window.The indexing function is used for the X-ray diffraction spectrum of calibration sample, and input variable is the angle of diffraction and the corresponding diffracted intensity of the X-ray diffraction spectrum of sample, and LaNi ₅The angle of diffraction of standard X-ray diffraction peak and the corresponding indices of crystal plane, output variable are the angle of diffraction and the corresponding indices of crystal plane of sample; The cohenh function is used for the lattice constant of calculation sample, and input variable is the angle of diffraction of sample and the wavelength of the corresponding indices of crystal plane and diffracted ray, and output variable is the lattice constant a of sample ₀, c ₀, V ₀, used mathematic calculation is a least square method.

Just set up the mathematical derivation process of canonical equation group and the program of crystal face index calibration below respectively and realize that thought is described in detail.

The purpose of least square method is to obtain the optimum value of measured data, the condition of optimum value be the quadratic sum of difference of it and measured value for minimum, that is: ∑ (optimum value-measured value) ²=minimum value

LaNi ₅And LaNi ₅The structure cell of base hydrogen storage material all belongs to hexagonal system structure, must set up Accuracy Error function correspondingly before setting up the canonical equation group of hexagonal system.

The expression of error function:

The foundation of measuring lattice parameter with X-ray diffraction method is the position of diffracted ray, i.e. 2 θ angles.On the basis of exponentiate, can calculate lattice parameter at diffraction pattern by Bragg equation and interplanar distance formula.According to Bragg equation:

${\sin }^2\mathrm{\&theta;}=\frac{{\mathrm{\&lambda;}}^2}{{4d}^2}---\left(1\right)$

Take the logarithm in both sides:

$\mathrm{In}{\sin }^2\mathrm{\&theta;}=\mathrm{In}\frac{{\mathrm{\&lambda;}}^2}{4}-2\mathrm{Ind}---\left(2\right)$

Both sides differential (for avoiding obscuring the differential sign Δ with interplanar distance d):

$\frac{{\mathrm{\&Delta;}\sin }^2\mathrm{\&theta;}}{{\sin }^2\mathrm{\&theta;}}=2(\frac{\mathrm{\&Delta;\&lambda;}}{\mathrm{\&lambda;}}-\frac{\mathrm{\&Delta;d}}{d})---\left(3\right)$

Thereby obtain:

$\frac{\mathrm{\&Delta;d}}{d}=\frac{\mathrm{\&Delta;\&lambda;}}{\mathrm{\&lambda;}}-\cot \mathrm{\&theta;}\&CenterDot;\mathrm{\&Delta;\&theta;}---\left(4\right)$

As seen the error of d value is mainly derived from angle of diffraction error delta θ and wavelength error Δ λ, and the precision of X ray wavelength reaches

And this error of any measurement is all equated, thus ignore, i.e. Δ λ=0, so:

$\frac{{\mathrm{\&Delta;}\sin }^2\mathrm{\&theta;}}{{\sin }^2\mathrm{\&theta;}}=-2\frac{\mathrm{\&Delta;d}}{d}---\left(5\right)$

For the relative error of interplanar distance, Nelson and Riley and Taylor and Sinelair etc. have studied the sample of different absorbent properties, draw more strict error expression:

$\frac{\mathrm{\&Delta;d}}{d}=k\&CenterDot;{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="C20051004713100202.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2\mathrm{\&theta;}(\frac{1}{\sin \mathrm{\&theta;}}+\frac{1}{\mathrm{\&theta;}})---\left(6\right)$

(5) formula substitution (6) formula is got (making D=-(k/2)):

$\mathrm{\&Delta;}{\sin }^2\mathrm{\&theta;}=-\frac{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="C20051004713100202.png"\; state="\{\{state\}\}">k</figure-callout>}}{2}{\sin }^{2}2\mathrm{\&theta;}(\frac{1}{\sin \mathrm{\&theta;}}+\frac{1}{\mathrm{\&theta;}})=D{\sin }^{2}2\mathrm{\&theta;}(\frac{1}{\sin \mathrm{\&theta;}}+\frac{1}{\mathrm{\&theta;}})---\left(7\right)$

Again because

Δ sin ²θ=sin ²θ (optimum value) with certain systematic error-

Sin ²θ ₀(actual value), according to the diffraction equation of hexagonal system:

${\sin }^2{\mathrm{\&theta;}}_0=\frac{{\mathrm{\&lambda;}}^2}{{{3\mathrm{<figure-callout\; id="0"\; label="a"\; filenames="C20051004713100211.png"\; state="\{\{state\}\}">a</figure-callout>}}_0}^2}({\mathrm{<figure-callout\; id="2"\; label="h"\; filenames="C20051004713100202.png"\; state="\{\{state\}\}">h</figure-callout>}}^2+\mathrm{hk}+k^2)+\frac{{\mathrm{\&lambda;}}^2}{{{4\mathrm{<figure-callout\; id="0"\; label="c"\; filenames="C20051004713100211.png"\; state="\{\{state\}\}">c</figure-callout>}}_0}^2}\&CenterDot;1^2---\left(8\right)$

So get by (7) formula and (8) formula:

$+D{\sin }^{2}2\mathrm{\&theta;}(\frac{1}{\sin \mathrm{\&theta;}}+\frac{1}{\mathrm{\&theta;}})---\left(9\right)$

Here it is hexagonal system sin ²The expression formula of θ error function.

Determining of canonical equation group:

At sin ²On the basis of θ error function, utilize least square method to determine the target equation.

Sin ²θ (optimum value)=A α+C γ+D δ (10)

In the formula $A=\frac{{\mathrm{\&lambda;}}^2}{{{3a}_0}^2},$ α＝h ²+hk+k ²， $C=\frac{{\mathrm{\&lambda;}}^2}{{{4\mathrm{<figure-callout\; id="0"\; label="c"\; filenames="C20051004713100211.png"\; state="\{\{state\}\}">c</figure-callout>}}_0}^2},$ γ＝1 ²， $\mathrm{\&delta;}={\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="C20051004713100202.png"\; state="\{\{state\}\}">sin</figure-callout>}}^{2}2\mathrm{\&theta;}(\frac{1}{\sin \mathrm{\&theta;}}+\frac{1}{\mathrm{\&theta;}}).$

For the diffraction peak of the n in the diffraction spectra, can obtain their sin respectively ²θ, alpha, gamma, δ substitution (10) formula can obtain n such equation, and the factor alpha and the addition of each equation be multiply by unknown number A get:

∑α _isin ²θ _i＝A∑α _i ²+C∑α _iγ _i+D∑α _iδ _i (11)

The coefficient gamma and the addition of each equation be multiply by C get:

∑γ _isin ²θ _i＝A∑α _iγ _i+C∑γ _i ²+D∑γ _iδ _i (12)

The coefficient δ and the addition of each equation be multiply by D get:

∑δ _isin ²θ _i＝A∑α _iδ _i+C∑γ _iδ _i+D∑δ _i ² (13)

Corresponding canonical equation group is:

$\left\{\begin{array}{c}\mathrm{\&Sigma;}{\mathrm{\&alpha;}}_i{\sin }^2{\mathrm{\&theta;}}_i=\mathrm{A\&Sigma;}{\mathrm{\&alpha;}}_i^2+\mathrm{C\&Sigma;}{\mathrm{\&alpha;}}_i{\mathrm{\&gamma;}}_i+\mathrm{D\&Sigma;}{\mathrm{\&alpha;}}_i{\mathrm{\&delta;}}_i\\ \mathrm{\&Sigma;}{\mathrm{\&gamma;}}_i{\sin }^2{\mathrm{\&theta;}}_i=A\mathrm{\&Sigma;}{\mathrm{\&alpha;}}_i{\mathrm{\&gamma;}}_i+\mathrm{C\&Sigma;}{\mathrm{\&gamma;}}_i^2+\mathrm{D\&Sigma;}{\mathrm{\&gamma;}}_i{\mathrm{\&delta;}}_i\\ \mathrm{\&Sigma;}{\mathrm{\&delta;}}_i{\sin }^2{\mathrm{\&theta;}}_i=\mathrm{A\&Sigma;}{\mathrm{\&alpha;}}_i{\mathrm{\&delta;}}_i+\mathrm{C\&Sigma;}{\mathrm{\&gamma;}}_i{\mathrm{\&delta;}}_i+\mathrm{D\&Sigma;}{\mathrm{\&delta;}}_i^2\end{array}\right.---\left(14\right)$

Solve the canonical equation group and obtain A and C, and then obtain the lattice parameter a of hexagonal cells ₀And c ₀

The program of index calibrating realizes thought:

Because LaNi ₅The base hydrogen storage material has and LaNi ₅Similar cell configuration is so show LaNi on the X-ray spectrum ₅The base hydrogen storage material has and LaNi ₅Identical peak shape, just change has taken place in its peak position.So, at first by making LaNi ₅Standard xrd peak value figure and have a same crystal formation LaNi ₅The xrd spectrum of base hydrogen storage material. according to LaNi ₅Xrd spectrum and standard xrd peak value figure user can must determine the peak position of sample xrd spectrum and the corresponding relation of the indices of crystal plane very soon.The fine notion that must embody man-machine collaboration of this method: determine that by computing machine the peak position of sample xrd spectrum judged the crystal face and the LaNi of peak position correspondence then by the people ₅The similarity relation of corresponding crystal face among the standard xrd peak value figure, thus finally determine peak position and indices of crystal plane one-to-one relationship by computing machine.

MATLAB program flow diagram shown in Fig. 2 a-c, whole procedure comprise 3 function cal_cell_para functions, cohenh function, indexing function; Wherein the indexing function is used for the X-ray diffraction spectrum of calibration sample, and input variable is the angle of diffraction and the corresponding diffracted intensity of the X-ray diffraction spectrum of sample, and LaNi ₅The angle of diffraction of standard X-ray diffraction peak and the corresponding indices of crystal plane, output variable are the angle of diffraction and the corresponding indices of crystal plane of sample; The cohenh function is used for the lattice constant of calculation sample, and input variable is the angle of diffraction of sample and the wavelength of the corresponding indices of crystal plane and diffracted ray, and output variable is the lattice constant a of sample ₀, c ₀, V ₀, below be the MATLAB program of whole calibrating and computation process; The cal_cell_para function is used to finish the distribution of experimental data, show sample and LaNi ₅Alloy X-ray spectrum and LaNi ₅The standard peak position distributes, and calls indexing function and cohenh function respectively, finish demarcate and calculate after with data presentation in window.

This subprogram is used for the indices of crystal plane of calibration sample X-ray diffraction spectrum:

function[out1，out2，out3，out4]＝indexing(thet2a，intensity，standxrd)

%thet2a, intensity-are respectively the 2theta of sample X-ray diffraction spectrum, intensity

%standxrd is a LaNi5 standard diffraction data

%LaNi5 standard diffraction data derives from the PDF card of " JCPDS (JCPDS) "

% obtains the 2theta value and the corresponding indices of crystal plane at required peak

％copyright@alex12cheng，Cheng?Honghui

％---------------

％load?the?standard-xrd?of?LaNi5

％get?the?standard?thet2a?and?h，k，l

intensity＝intensity./max(intensity).*100；％normalizing

thet2a_standard＝standxrd(:，1)；

h_standard＝standxrd(:，3)；

k_standard＝standxrd(:，4)；

l_standard＝standxrd(:，5)；

％plot?the?peak?of?the?standxrd?and?the?requested?xrd

plot(standxrd(:，1)，standxrd(:，2)，′R+′，thet2a，intensity，′black-′)；

set(gcf，′menubar′，′none′，′Position′，[139?1024?698])；％make?the?interface?window?fully?open

set(gca，′position′，[0.03，0.03，0.96，0.96])％make?the?axis?more?big

axis([20，90，0，102])；

hold?on

thet2a_peak＝[]；

p＝[]；％permutation?vector

n＝0；

but＝1；％when?push?the?ESC?key，the?choosing?peak?manipulation?terminate

while?but～＝27％?ESC?key

[xi，yi，but]＝ginput(1)；

if?but～＝27

％To?improve?the?accurancy，the?peak?will?be?acquired?nearthe

％location?of?cursor

upi＝xi+0.2；

downi＝xi-0.2；

Irange＝find(thet2a＞downi&thet2a＜upi)；

[int_peaki，Ipeak]＝max(intensity(Irange))；

thet2a_peaki＝thet2a(Irange(Ipeak))；

plot(thet2a_peaki，int_peaki，′m^′)；

n＝n+1；

thet2a_peak(n，1)＝thet2a_peaki；

ifbut＝＝1％click?the?left?button?of?mouse?if?the?standard?peak?is?close

％to?the?left?side?of?the?sample?xrd?peak

Ip＝find(thet2a_standard＞(thet2a_peaki-5)&thet2a_standard＜thet2a_peaki)；

p(n)＝Ip(end)；

hi＝h_standard(Ip(end))；

ki＝k_standard(Ip(end))；

li＝l_standard(Ip(end))；

elseif?but＝＝97％clicka?key?on?keyboard?if?the?standard?peak?is?not

％so?close?to?the?left?side?of?the?sample?xrd?peak

Ip＝find(thet2a_standard＞(thet2a_peaki-5)&thet2a_standard＜thet2a_peaki)；

p(n)＝Ip(end-1)；

hi＝h_standard(Ip(end-1))；

ki＝k_standard(Ip(end-1))；

li＝l_standard(Ip(end-1))；

elseifbut＝＝32％click?spacebar?if?the?manipulator?can?not?figure?out?the?location

％relation?between?the?standardxrd?peak?and?the?sample?xrd?peak

Ip＝find(thet2a_standard＞(thet2a_peaki-0.5)&thet2a_standard＜(thet2a_peaki+0.5))；

[C，I]＝min(abs(thet2a_standard(Ip)-thet2a_peaki))；

p(n)＝Ip(I)；

hi＝h_standard(Ip(I))；

ki＝k_standard(Ip(I))；

li＝l_standard(Ip(I))；

elseifbut＝＝3％?click?right?button?of?mouse?if?the?standard?peak?is?close?to?the

％right?side?of?the?sample?xrd?peak

Ip＝find(thet2a_standard＞thet2a_peaki&thet2a_standard＜(thet2a_peaki+5))；

p(n)＝Ip(1)；

hi＝h_standard(Ip(1))；

ki＝k_standard(Ip(1))；

li＝l_standard(Ip(1))；

elseifbut＝＝100％?click?d?key?on?keyboard?if?the?standard?peak?is?not?so

％close?to?the?right?side?of?the?sample?xrd?peak

Ip＝find(thet2a_standard＞thet2a_peaki&thet2a_standard＜(thet2a_peaki+5))；

p(n)＝Ip(2)；

hi＝h_standard(Ip(2))；

ki＝k_standard(Ip(2))；

li＝l_standard(Ip(2))；

end

string＝[num2str(thet2a_peaki)，′(′，num2str(hi)，″，num2str(ki)，″，num2str(li)，′)′]；

text(thet2a_peaki，int_peaki，string，...

′VerticalAlignment′，′bottom′，′HorizontalAlignment′，′center′)；

％display?the?result?of?indexing

end

end

hold?off

h＝h_standard(p)；

k＝k_standard(p)；

l＝l_standard(p)；

hkl＝[h，k，l]；

thet2a_peakhkl＝[thet2a_peak，h，k，l]；

if?nargout＝＝1

out1＝thet2a_peakhkl；

elseifnargout＝＝2

out1＝thet2a_peak；

out2＝hkl；

elseifnargout＝＝4

out1＝thet2a_peak；

out2＝h；

out3＝k；

out4＝l；

end

The lattice constant that this subprogram is used for sample calculates:

function[a0，c0，V0]＝cohenh(h，k，l，thet2a，lumda)

％[a0，c0，V0]＝cohenh(h，k，l，thet2a，lumda)

%a0, c0, the lattice parameter of V0 for obtaining

%cohenh is a function name

%h, k, l are the indices of crystal plane

%thet2a is corresponding angle of diffraction, and unit is degree

%lumda is the X ray wavelength, and unit is a dust

theta＝thet2a./2；

alpha＝h.^2+h.*k+k.^2；

gama＝1.^2；

deta＝sin(thet2a.*pi./180).^2.*(sin(theta.*pi./180)+theta.*pi./180)./(theta.*pi./1?80)./sin(theta.*pi./180)；

a(1，1)＝sum(alpha.^2)；

a(1，2)＝sum(alpha.*gama)；

a(1，3)＝sum(alpha.*deta)；

a(2，2)＝sum(gama.^2)；

a(2，3)＝sum(gama.*deta)；

a(3，3)＝sum(deta.^2)；

for?i＝2:3

forj＝1:2

ifi～＝j

a(i，j)＝a(j.i)；

end

end

end

b(1)＝sum(alpha.*sin(theta.*pi./180).^2)；

b(2)＝sum(gama.*sin(theta.*pi./180).^2)；

b(3)＝sum(deta.*sin(theta.*pi./180).^2)；

x＝a\b′；

A＝x(1)；

C＝x(2)；

a0＝sqrt(lumda^2./3./A)；

c0＝sqrt(lumda^2./4./C)；

V0＝a0^2.*c0.*sin(pi/3)；

This subprogram is finished the distribution of experimental data, show sample and LaNi ₅Alloy X-ray spectrum and LaNi ₅The standard peak position distributes, and calls indexing function and cohenh function respectively, finish demarcate and calculate after with data presentation in window:

function?cal_cell_para(Sample?xrd，lumda，standxrd_JCPDS，LaNi5xrd_reference)

plot(LaNi5xrd_reference(:，1)，LaNi5xrd_reference(:，2)，′b-′)；

Text (60,40, ' blueness is the LaNi5-x alpha spectrum, ' color ', ' blue ', ' fontsize ', 15);

holdon

thet2a＝Sample_xrd(:，1)；

intensity＝Sample_xrd(:，2)；

[thet2a，h，k，l]＝indexing(thet2a，intensity，standxrd_JCPDS)；

[a0，c0，V0]＝cohenh(h，k，l，thet2a，lumda)

s＝sprintf(′a0＝％0.4f\nc0＝％0.4f\nV0＝％0.4f′，a0，c0，V0)；

text(60，80，s，′color′，′red′，′fontsize′，15)；

With purity (at%) is La 99.3, and Ni 99.9, and the raw material of Al99.7 is pressed 34: 61.05: 4.95 counterweights of design mix.The material for preparing is placed water cooled copper mould, carry out melting in the electric arc furnaces under the argon gas atmosphere protection.With alloy turn-over remelting five times, carry out electromagnetic agitation simultaneously in the fusion process, to guarantee the homogeneity of alloy.After the melting alloy pig is encapsulated in the vitreosil pipe vacuum tightness＜10 ^-2A puts into heat-treatment furnace and is warming up to 1100 ℃ with stove, and being incubated quenched after 8 hours makes sample.Ground sample is crossed 20～30um dusting cover, use Japan (Rigaku) D/max-2500pc X-ray diffractometer of science and CuK α radiation (

) powder is carried out continuous sweep, sweep velocity: 0.12 (second/step), scanning step: 0.04 (degree/step), the scope of 2 θ: 20-90 (degree).Whole experiment obtains 1750 data points altogether.The data file of the LaNi5-CuKa-standard-xrd by name of file in crystal parameter and the index calibrating file is imported in the work space (workspace) of MATLAB, then the alternative Sample_xrd data with existing of the diffraction data of sample X-ray diffraction spectrum is got final product (as shown in Figure 1).(standxrd JCPDS LaNi5xrd_reference) a personal-machine interactive window can occur on the screen of order back for Sample_xrd, lumda to key in cal_cell_para then in the command window (command window) of MATLAB.Carry out crystal face index calibration and lattice constant and calculate, obtain LaNi _4.25Al _0.75The lattice constant of alloy

(V ₀Be unit cell volume).

Claims (5)

1, a kind of semi-automatic X-ray crystal plane index calibration and lattice constant Calculation Method is characterized in that comprising the steps:

1) with LaNi ₅The sample powder process of base hydrogen storage material is also crossed 20～30um dusting cover, on X-ray diffractometer, uses CuK α radiation, wavelength then Sample is carried out continuous sweep, sweep velocity: 0.12～2 second/step, scanning step: in 0.02～0.04 degree/step, the scope of 2 θ: the 20-90 degree obtained the X-ray diffraction spectrum of sample;

2) data file with crystal parameter and index calibrating imports in the work space of MATLAB software, the diffraction data of sample X-ray diffraction spectrum is substituted the Sample_xrd data with existing, Sample_xrd is that a columns is two array, and first classifies angle of diffraction as, and second classifies diffracted intensity as;

3) in the command window of MATLAB, key in order then and make appearance one personal-machine interactive window on the screen;

4) user passes through with reference to LaNi ₅The X-ray diffraction spectrum indices of crystal plane of coming calibration sample, and finally calculate the lattice constant of sample by the one-to-one relationship of the indices of crystal plane and angle of diffraction;

Whole calibrating and computation process are finished by following program, and whole procedure comprises 3 functions: cal_cell_para function, cohenh function, indexing function; Wherein the cal_cell_para function is finished the distribution of experimental data, show sample and LaNi ₅Alloy X-ray spectrum and LaNi ₅The standard peak position distributes, and calls indexing function and cohenh function respectively, finish demarcate and calculate after with data presentation in window; The indexing function is used for the X-ray diffraction spectrum of calibration sample, and input variable is the angle of diffraction and the corresponding diffracted intensity of the X-ray diffraction spectrum of sample, and LaNi ₅The angle of diffraction of standard X-ray diffraction peak and the corresponding indices of crystal plane, output variable are the angle of diffraction and the corresponding indices of crystal plane of sample; The cohenh function is used for the lattice constant of calculation sample, and input variable is the angle of diffraction of sample and the wavelength of the corresponding indices of crystal plane and diffracted ray, and output variable is the lattice constant a of sample ₀, c ₀, V ₀, used mathematic calculation is a least square method.

2, according to the described method of claim 1, it is characterized in that: the used canonical matrix of coefficients of program cohenh function that least square method is calculated lattice constant is:

$\left\{\begin{array}{c}\mathrm{\&Sigma;}{\mathrm{\&alpha;}}_i{\sin }^2{\mathrm{\&theta;}}_i=\mathrm{A\&Sigma;}{{\mathrm{\&alpha;}}_i}^2+\mathrm{C\&Sigma;}{\mathrm{\&alpha;}}_i{\mathrm{\&gamma;}}_i+\mathrm{D\&Sigma;}{\mathrm{\&alpha;}}_i{\mathrm{\&delta;}}_i\\ \mathrm{\&Sigma;}{\mathrm{\&gamma;}}_i{\sin }^2{\mathrm{\&theta;}}_i=\mathrm{A\&Sigma;}{\mathrm{\&alpha;}}_i{\mathrm{\&gamma;}}_i+\mathrm{C\&Sigma;}{{\mathrm{\&gamma;}}_i}^2+\mathrm{D\&Sigma;}{\mathrm{\&gamma;}}_i{\mathrm{\&delta;}}_i\\ \mathrm{\&Sigma;}{\mathrm{\&delta;}}_i{\sin }^2{\mathrm{\&theta;}}_i=\mathrm{A\&Sigma;}{\mathrm{\&alpha;}}_i{\mathrm{\&delta;}}_i+\mathrm{C\&Sigma;}{\mathrm{\&gamma;}}_i{\mathrm{\&delta;}}_i+\mathrm{D\&Sigma;}{{\mathrm{\&delta;}}_i}^2\end{array}\right.$

Wherein $A=\frac{{\mathrm{\&lambda;}}^2}{3{a_0}^2},$ α＝h ²+hk+k ²， $C=\frac{{\mathrm{\&lambda;}}^2}{4{c_0}^2},$ γ＝l ²， $\mathrm{\&delta;}={\sin }^{2}2\mathrm{\&theta;}(\frac{1}{\sin \mathrm{\&theta;}}+\frac{1}{\mathrm{\&theta;}}),$ D=-(k/2), λ are the X ray wavelength, and θ is an angle of diffraction, and h, k, l are the indices of crystal plane, a ₀, c ₀Be lattice constant.

3,, it is characterized in that described cal_cell_para function flow process is as follows according to the described method of claim 1:

At first program brings into operation, and imports the X-ray spectrum data of sample by the operator, and computing machine is made LaNi automatically ₅The X ray of alloy supplies the operator in the timing signal reference, from the X-ray spectrum extracting data of sample go out intensity and refraction angle data respectively assignment give angle of diffraction thet2a, the strong intensity in peak, call the indexing function then and finish the demarcation of the indices of crystal plane, calling the cohenh function utilizes above-mentioned calibration result to finish the calculating of lattice constant, result of calculation is shown on the screen, EOP (end of program).

4,, it is characterized in that described indexing function flow process is as follows according to the described method of claim 1:

At first program brings into operation, input sample X-ray spectrum angle of diffraction and peak strong data thet2a, intensity and LaNi ₅Standard diffraction data standxrd carries out normalized to the X ray intensity of sample, then from LaNi ₅Extract indices of crystal plane ordered series of numbers and angle of diffraction ordered series of numbers difference assignment among the standard diffraction data standxrd and give h_standard, k_standard, l_standard, thet2a_standard, set up a displacement ordered series of numbers p who is used to store the empty array thet2a_peak of sample diffraction peak angle of diffraction and is used for determining the indices of crystal plane then, to n and but initialize, wherein n is the subscript that is used for determining the ordered series of numbers numerical digit then, and but is used to characterize keyboard by the key assignments of building; Judge the button value of keyboard, if not 27 location point and the key assignments that obtain mouse, be stored in xi respectively, yi is among the but, by computing machine at xi, search obtains sample peak position angle of diffraction near the yi, is shown in the relevant position, judges key assignments then, if key assignments is 1, with the left side arest neighbors LaNi at sample X ray peak ₅The indices of crystal plane of standard peak position correspondence are the indices of crystal plane at the selected peak of sample, deposit in the computing machine, if key assignments is the 97 left side time neighbour LaNi with sample X ray peak ₅The indices of crystal plane of standard peak position correspondence are the indices of crystal plane at the selected peak of sample, deposit in the computing machine, if key assignments is the 32 arest neighbors LaNi with sample X ray peak ₅The indices of crystal plane of standard peak position correspondence are the indices of crystal plane at the selected peak of sample, deposit in the computing machine, if key assignments is the 3 right side arest neighbors LaNi with sample X ray peak ₅The indices of crystal plane of standard peak position correspondence are the indices of crystal plane at the selected peak of sample, deposit in the computing machine, if key assignments any button that is other, with the right side time neighbour LaNi at sample X ray peak ₅The indices of crystal plane of standard peak position correspondence are the indices of crystal plane at the selected peak of sample, deposit in the computing machine, repeat the above-mentioned peak process of selecting, press esc key up to the operator, finish to demarcate storing process, show calibration result in window, the form storage output nominal data with array supplies other routine call, EOP (end of program) simultaneously.

5,, it is characterized in that described cohenh function flow process is as follows according to the described method of claim 1:

At first program brings into operation, indices of crystal plane ordered series of numbers h, k, l and the corresponding angle of diffraction ordered series of numbers thet2a of the X-ray spectrum of input sample, and X ray wavelength lumda, by computing machine respectively to theta, alpha, gama, the deta assignment, set up the matrix of coefficients a of canonical equation group then, b, to matrix of coefficients carry out a left side remove operation x=a b ', obtain an ordered series of numbers, the preceding two number assignment that extract ordered series of numbers are given A, and C is according to formula $A=\frac{{\mathrm{\&lambda;}}^2}{3{a_0}^2},$ $C=\frac{{\mathrm{\&lambda;}}^2}{4{c_0}^2},$ $V_0={a_0}^2\&CenterDot;c_0\&CenterDot;\sin \left(\frac{\mathrm{\&pi;}}{3}\right),$ Calculate and obtain lattice constant a ₀, c ₀, V ₀, EOP (end of program).

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