WO2021007726A1 - 由一张电子衍射花样重构晶体布拉菲格子的方法 - Google Patents

由一张电子衍射花样重构晶体布拉菲格子的方法 Download PDF

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WO2021007726A1
WO2021007726A1 PCT/CN2019/095872 CN2019095872W WO2021007726A1 WO 2021007726 A1 WO2021007726 A1 WO 2021007726A1 CN 2019095872 W CN2019095872 W CN 2019095872W WO 2021007726 A1 WO2021007726 A1 WO 2021007726A1
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lattice
diffraction
bravais
electron diffraction
crystal
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施洪龙
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中央民族大学
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01N—INVESTIGATING OR ANALYSING MATERIALS BY DETERMINING THEIR CHEMICAL OR PHYSICAL PROPERTIES
    • G01N23/00—Investigating or analysing materials by the use of wave or particle radiation, e.g. X-rays or neutrons, not covered by groups G01N3/00 – G01N17/00, G01N21/00 or G01N22/00
    • G01N23/20—Investigating or analysing materials by the use of wave or particle radiation, e.g. X-rays or neutrons, not covered by groups G01N3/00 – G01N17/00, G01N21/00 or G01N22/00 by using diffraction of the radiation by the materials, e.g. for investigating crystal structure; by using scattering of the radiation by the materials, e.g. for investigating non-crystalline materials; by using reflection of the radiation by the materials
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01N—INVESTIGATING OR ANALYSING MATERIALS BY DETERMINING THEIR CHEMICAL OR PHYSICAL PROPERTIES
    • G01N23/00—Investigating or analysing materials by the use of wave or particle radiation, e.g. X-rays or neutrons, not covered by groups G01N3/00 – G01N17/00, G01N21/00 or G01N22/00
    • G01N23/20—Investigating or analysing materials by the use of wave or particle radiation, e.g. X-rays or neutrons, not covered by groups G01N3/00 – G01N17/00, G01N21/00 or G01N22/00 by using diffraction of the radiation by the materials, e.g. for investigating crystal structure; by using scattering of the radiation by the materials, e.g. for investigating non-crystalline materials; by using reflection of the radiation by the materials
    • G01N23/207—Diffractometry using detectors, e.g. using a probe in a central position and one or more displaceable detectors in circumferential positions

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  • the invention relates to a method for reconstructing the Bravais lattice of a crystal by using an electron diffraction pattern, and belongs to the technical field of material microstructure characterization and crystal structure analysis.
  • Bravais lattices are necessary parameters for crystal structure analysis using X-ray diffraction and electron diffraction.
  • the measured diffraction peaks can be indexed by X-ray diffraction technology to obtain the Bravais lattice.
  • the advantage of the transmission electron microscope is that it can perform selected area electron diffraction or nano-beam diffraction on a single crystal while observing the microstructure of the sample under test in real time. Therefore, the transmission electron microscope has become an irreplaceable research tool for crystal structure analysis.
  • Electron diffraction is the projection of the three-dimensional reciprocal lattice of the crystal on the phosphor screen or CCD.
  • the recorded diffraction pattern is the two-dimensional reciprocal cross section of the crystal, which can visually display the information of the distance between the crystal planes and the angle between the crystal planes.
  • the traditional method of using electron diffraction to determine the Bravais lattice is to tilt the crystal around a low-index diffraction point, record a series ( ⁇ 3) of low-index axial electron diffraction patterns, and then perform geometric composition Method to derive the Braffi lattice.
  • This method has very high requirements for both the transmission electron microscope and the crystal to be tested: 1) The transmission electron microscope is required to have a larger objective lens pole shoe spacing for large-angle crystal tilt; 2) The crystal to be tested is required to have a larger The crystal grains have high symmetry, otherwise it is inconvenient for the tilting and geometric composition of the crystal; 3) The crystal needs to be strictly tilted to the low-index positive belt axis. Obviously, in the increasingly emerging research of nanomaterials, it is difficult to achieve small-grain, low-symmetry crystals using conventional transmission electron microscopes, especially high-resolution transmission electron microscopes (small objective lens pole shoes, tilt angle ⁇ 25°). Confirm the Braffi grid.
  • the present invention proposes a method for reconstructing the Bravais grid from only one electron diffraction pattern, which includes the following steps:
  • Step 1) Record the on-axis electron diffraction pattern of the crystal to be tested, and obtain an on-axis electron diffraction pattern containing high-order Laue diffraction;
  • Step 3 Measure the high-order Laue diffraction ring
  • Step 4) Measure high-order Laue diffraction points
  • Step 5 Reconstruction of three-dimensional reciprocal primitive cells
  • Step 6) Convert the reciprocal elementary cell obtained in step 5) into a positive lattice
  • the method includes the following steps:
  • Step 1' Record any of the on-axis electron diffraction patterns containing high-order Laue diffraction of the crystal to be tested;
  • Step 3' Measure the high-order Laue diffraction ring, and calculate the interlayer spacing CH of the reciprocal surface according to the following formula:
  • R is the radius of the high-order Laue diffraction ring
  • ⁇ is the wavelength of the electron beam
  • L is the camera length
  • Step 4' Measure the high-order Laue diffraction point: measure the position of the high-order Laue diffraction point H, and translate it to the two-dimensional primary cell in step 2') to obtain C 1 , C 1 is the three-dimensional inverted The projection point of the lattice point C of the initial elementary cell in the two-dimensional elementary cell;
  • Step 5') Reconstruction of the three-dimensional reciprocal primary cell OA, OB and OC constitute the three base vectors of the three-dimensional reciprocal primary cell; from the projection point C 1 to the adjacent edges OA and OB of the two-dimensional primary cell As a vertical line, the vertical feet are A 1 and B 1 respectively ; from the geometric relationship, it can be known that ⁇ OA 1 C, ⁇ OB 1 C, and ⁇ COC 1 are all right triangles, and thus:
  • OC 1 is the distance between the translated projection point C 1 and the origin
  • OA 1 and OB 1 are the distances from the two vertical feet to the origin
  • CC 1 is the layer distance CH of the reciprocal surface calculated in step 3');
  • Step 6' According to the reciprocal relationship, transform the reciprocal primitive cell obtained in step 5') into a positive lattice:
  • Step 7' Reduction processing: According to the positive lattice calculated in step 6'), the three shortest vectors that are not coplanar are calculated and defined as the basis vectors of the initial elementary cells of the positive lattice in order to satisfy the constraints of Niggli reduction ; The reduction process is as follows:
  • the grid point index u i, v i, w i preferably positive, negative or zero; Generally, -6 ⁇ u i, v i, w i ⁇ 6 may be sufficient for reductive; certain u i, Calculate within the range of v i and w i to obtain a list of ti ;
  • step S2) Find the three smallest t i values in the t i list calculated in step S1), which are defined as t 1 , t 2 and t 3 , provided that t 1 and t 2 are not collinear, that is, the vectors t 1 and The three exponents h, k, and l obtained by the cross product of t 2 are all non-zero; and three vectors with the smallest t i value are required, and t 3 ⁇ t 1 ⁇ t 2 cannot be zero, thus three The shortest vector that is not coplanar;
  • u 1 , v 1 , and w 1 are the grid index of t 1 calculated in step S1); u 2 , v 2 , and w 2 are the grid index of t 2 .
  • Step 8' Convert Niggli reduced cells to Bravais lattices: use the correspondence between 44 Nigg1i reduced cells and 14 Bravais lattices to pass the Niggli reduced cells obtained in step 7') 44 kinds of matrix conversions get 44 Bravais grids; each conversion matrix corresponds to a Bravais grid, which is called the target Bravais grid here; whether the Bravais grid calculated by checking meets the target Bravais grid The symmetry characteristics of the Fei lattice determine the type and lattice constant of the Bravais lattice.
  • the crystal to be tested in step 1') can be a known structure or an unknown structure
  • the crystal to be tested in step 1') can be bulk, powder, or single crystal, polycrystalline, microcrystalline or nanocrystalline;
  • a transmission electron microscope is used to record the on-axis electron diffraction pattern of the crystal to be tested;
  • the electron diffraction of the transmission electron microscope can be selected area electron diffraction, precession electron diffraction, nanobeam electron diffraction Diffraction, microbeam electron diffraction or convergent beam electron diffraction;
  • step 1' there is no restriction on the axis index or symmetry of the recorded axis electron diffraction pattern, and it can be any axis;
  • step 1' the recorded on-axis electron diffraction pattern is not required to meet strict positive-axis conditions.
  • the position of the high-order Laue diffraction point H is not limited, and it can be any position of the high-order Laue diffraction point H.
  • the symmetry characteristic of the target Bravais lattice has the following symmetry characteristics:
  • the present invention proposes a method for reconstructing a Bravais grid from a single electron diffraction pattern. Only a single electron diffraction pattern containing high-order Laue diffraction points is needed to achieve three-dimensional reconstruction, which can be used for phase identification of any crystal material, known or unknown
  • the reconstruction of the Bravais lattice of the crystal phase is especially suitable for situations where it is difficult to obtain multiple diffraction patterns.
  • the crystal grains are smaller than 300 nm, it may take up to tens of minutes to record an on-axis electron diffraction by tilting the crystal, and it is almost impossible to record multiple electron diffraction. Therefore, it is difficult to record multiple diffraction patterns for most microcrystals and nanocrystals.
  • the method of this application only needs to record an electron diffraction pattern containing high-order Laue diffraction points to achieve three-dimensional crystal reconstruction, which greatly reduces the amount of experiment and analysis. The workload and efficiency are significantly improved.
  • the method of the present invention has the following advantages: the method of the present application does not require a shaft index, does not need to be tilted to a strict positive belt shaft, and does not require complex crystal tilting to record multiple shaft electron diffraction. Greatly simplify the experimental operation of electron diffraction. Only by an axial electron diffraction pattern containing high-order Laue diffraction points, the lattice type and lattice constant of the measured crystal can be analyzed; in the actual electron microscope experiment and data analysis, the experiment and analysis can be greatly reduced The workload has significantly improved work efficiency.
  • the proposed analysis method is not affected by the crystal system and the level of symmetry, and is suitable for the determination of the Bravais lattice of any crystal system.
  • the method of the invention utilizes selected area electron diffraction, precession electron diffraction, microbeam electron diffraction, nanobeam electron diffraction or convergent beam electron diffraction on a transmission electron microscope to quickly analyze the Bravais lattice of the crystal to be measured.
  • Figure 1 is a schematic diagram of the principle of using a piece of electron diffraction to determine the Bravais lattice: (a) the geometric relationship between the high-order Laue diffraction ring and the interlayer spacing; (b) the reciprocal space reconstruction using high-order Laue diffraction points Principle diagram;
  • Fig. 2 shows the use of a single-crystal silicon on-axis electron diffraction to determine the Bravais lattice in Example 1: (a) single-crystal silicon’s on-axis electron diffraction pattern; (b) reciprocal space reconstruction;
  • Fig. 3 shows the Bravais lattice determined by the electron diffraction of a piece of titanium dioxide close to the axis in Example 2: (a) the axis electron diffraction pattern of titanium dioxide; (b) reciprocal space reconstruction.
  • the transmission spot is taken as the origin O of the two-dimensional primary cell, and the nearest neighboring diffraction points A and B are used as the adjacent edges to construct the two-dimensional primary cell, as shown in Figure 2b.
  • Measure OA, OB and ⁇ AOB, and the measurement results are listed in Table 1.
  • OC 1 is the distance between the translated projection point C 1 and the origin
  • OA 1 and OB 1 are the distances from the two vertical feet to the origin
  • CC 1 is the layer distance CH of the reciprocal surface calculated in step 3).
  • the resulting reduced cell parameters are:
  • the electron diffraction is not strictly belt-axis electron diffraction, and it deviates from the positive belt axis by about 0.11°.
  • OC 1 is the distance between the translated projection point C 1 and the origin
  • OA 1 and OB 1 are the distances from the two vertical feet to the origin
  • CC 1 is the layer distance CH of the reciprocal surface calculated in step 3).
  • the resulting reduced cell parameters are:

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Abstract

一种利用一张电子衍射花样重构晶体的布拉菲格子的方法，包括如下步骤：步骤1)：记录待测晶体的带轴电子衍射花样，得到一张含有高阶劳厄衍射的带轴电子衍射花样；步骤2)：测量二维初基胞；步骤3)：测量高阶劳厄衍射环；步骤4)：测量高阶劳厄衍射点；步骤5)：三维倒易初基胞的重构；步骤6)：将步骤5)中得到的倒易初基胞转化为正格子；步骤7)：约化处理；步骤8)：将Niggli约化胞转换为布拉菲格子。该方法仅由一张含有高阶劳厄衍射点的带轴电子衍射花样，即可分析出所测晶体的点阵类型及晶格常数；在实际的电镜实验和数据分析中，可大大减少实验和分析的工作量。

Description

由一张电子衍射花样重构晶体布拉菲格子的方法 技术领域
本发明涉及一种利用一张电子衍射花样重构晶体的布拉菲格子的方法，属于材料显微结构表征和晶体结构分析的技术领域。
背景技术
布拉菲格子的类型和大小是利用X射线衍射和电子衍射进行晶体结构解析的必要参数。对于结晶性较好、具有单一物相的粉末样品，利用X射线衍射技术对所测衍射峰进行指标化，可得到布拉菲格子。但在多物相或结晶较差的材料中，由于难以准确提取各物相的衍射峰，不能准确地确定各物相的布拉菲格子。
透射电子显微镜的优点是，在实时观察待测样品的显微结构的同时，可以对单个晶粒进行选区电子衍射或纳米束衍射，因此透射电子显微镜已成为晶体结构分析不可替代的研究工具。
电子衍射是晶体的三维倒易点阵在荧光屏或CCD上的投影。所记录的衍射花样是晶体的二维倒易截面，它能直观地展示晶面的晶面间距、晶面夹角的信息。在实验中，利用电子衍射确定布拉菲格子的传统方法是，绕某一低指数的衍射点倾转晶体、记录一系列(≥3张)低指数的带轴电子衍射花样，再经几何构图法推导出布拉菲格子。该方法对透射电子显微镜和待测晶体都有很高要求：1)要求透射电子显微镜具有较大的物镜极靴间距，以便进行大角度的晶体倾转；2)要求待测晶体具有较大的晶粒且具有高对称性，否则不便于晶体的倾转和几何构图；3)需要将晶体严格倾转到低指数正带轴。显然，在日益兴起的纳米材料研究中，利用常规的透射电子显微镜，尤其是高分辨透射电子 显微镜(小物镜极靴，倾转角<±25°)难以实现对小晶粒、低对称性的晶体进行布拉菲格子的确定。
所以，很有必要开发一种对样品要求低、实验操作简单的布拉菲格子的确定方法。
发明内容
为了改善上述技术问题，本发明提出一种仅由一张电子衍射花样重构布拉菲格子的方法，包括如下步骤：
步骤1)：记录待测晶体的带轴电子衍射花样，得到一张含有高阶劳厄衍射的带轴电子衍射花样；
步骤2)：测量二维初基胞；
步骤3)：测量高阶劳厄衍射环；
步骤4)：测量高阶劳厄衍射点；
步骤5)：三维倒易初基胞的重构；
步骤6)：将步骤5)中得到的倒易初基胞转化为正格子；
步骤7)：约化处理；
步骤8)：将Niggli约化胞转换为布拉菲格子。
根据本发明的实施方案，所述方法包括如下步骤：
步骤1’)记录待测晶体的任意一张含有高阶劳厄衍射的带轴电子衍射花样；
步骤2’)在零阶劳厄衍射上测量二维初基胞：以透射斑为二维初基胞的原点O，由最近邻的两个衍射点A和B为邻边形成的平行四边形作为二维初基胞，平行四边形中OA＝a *，OB＝b *，∠AOB＝γ *；
步骤3’)测量高阶劳厄衍射环，根据如下公式计算得到该倒易面的层间距CH：
CH＝R 2/(2λL 2)
其中，R为高阶劳厄衍射环的半径；λ为电子束的波长，L为相机长度；
步骤4’)测量高阶劳厄衍射点：测量高阶劳厄衍射点H的位置，并将其平移到步骤2’)中的二维初基胞中得到C 1，C 1即为三维倒易初基胞的格点C在所述二维初基胞内的投影点；
步骤5’)三维倒易初基胞的重构：由OA、OB和OC构成三维倒易初基胞的三个基矢；从投影点C 1向二维初基胞的邻边OA和OB作垂线，垂足分别为A 1和B 1；由几何关系可知ΔOA 1C、ΔOB 1C和ΔCOC 1均为直角三角形，由此可得：
a *＝OA
b *＝OB
Figure PCTCN2019095872-appb-000001
Figure PCTCN2019095872-appb-000002
Figure PCTCN2019095872-appb-000003
γ *＝∠AOB
其中，OC 1为平移后的投影点C 1到原点的间距，OA 1和OB 1为两垂足到原点的间距；CC 1为步骤3’)计算得出的倒易面的层间距CH；
步骤6’)根据倒易关系将步骤5’)中得到的倒易初基胞转化为正格子：
Figure PCTCN2019095872-appb-000004
Figure PCTCN2019095872-appb-000005
Figure PCTCN2019095872-appb-000006
Figure PCTCN2019095872-appb-000007
Figure PCTCN2019095872-appb-000008
Figure PCTCN2019095872-appb-000009
其中，
Figure PCTCN2019095872-appb-000010
Figure PCTCN2019095872-appb-000011
为倒易初基胞的体积。
步骤7’)约化处理：根据步骤6’)计算出的正格子，计算出三个不共面的最短矢量依次定义为正格子初基胞的基矢，使其满足Niggli约化的约束条件；约化过程如下：
S1)根据步骤6’)计算出的正格子计算出任意指数u i,v i,w i的格点到晶格原点的长度t i：
Figure PCTCN2019095872-appb-000012
其中，格点指数u i,v i,w i可取正数、负数或零；一般地，-6≤u i,v i,w i≤6可足以用于约化；在一定的u i,v i,w i范围内计算，得到t i列表；
S2)在步骤S1)所计算出的t i列表中找到三个最小的t i值，定义为t 1、t 2和t 3，条件是t 1和t 2不共线，即矢量t 1和t 2的叉乘积所得的三个指数h，k和l均不为零；且要求具有最小t i值的三个矢量，其t 3·t 1×t 2不能为零，由此得到三个不共面的最短矢量；
S3)计算三个矢量t 1、t 2和t 3间的夹角；矢量t 1和t 2的夹角可以采用如下公式计算：
Figure PCTCN2019095872-appb-000013
其中u 1，v 1，w 1为步骤S1)中计算t 1的格点指数；u 2，v 2，w 2为t 2的格点指数。
S4)由此得到约化胞：a′＝t 1，b′＝t 2，c ′＝t 3，
Figure PCTCN2019095872-appb-000014
Figure PCTCN2019095872-appb-000015
步骤8’)将Niggli约化胞转换为布拉菲格子：利用44种Nigg1i约化胞与14种布拉菲点阵之间的对应关系，将步骤7’)中得到的Niggli约化胞通过44 种矩阵转换得到44个布拉菲格子；每个转换矩阵都对应着一种布拉菲格子，在此称为目标布拉菲格子；通过检验计算出来的布拉菲格子是否满足目标布拉菲格子的对称性特征，确定布拉菲格子的类型和晶格常数。
根据本发明的实施方案，步骤1’)中的待测晶体可以是已知结构，也可以是未知结构；
根据本发明的实施方案，步骤1’)中的待测晶体可以是块材、粉末，也可以是单晶、多晶、微晶或纳米晶；
根据本发明的实施方案，步骤1’)中，采用透射电子显微镜记录待测晶体的带轴电子衍射花样；所述透射电子显微镜的电子衍射可以是选区电子衍射、旋进电子衍射、纳米束电子衍射、微束电子衍射或会聚束电子衍射；
根据本发明的实施方案，步骤1’)中，对所记录的带轴电子衍射花样没有带轴指数的限制，也没有对称性的限制，可以是任意带轴；
根据本发明的实施方案，步骤1’)中，不要求所记录的带轴电子衍射花样满足严格的正带轴条件。
根据本发明的实施方案，步骤2’)中，所述二维初基胞所围的面积最小且两邻边OA与OB的夹角∠AOB＝γ *≤90°；
根据本发明的实施方案，步骤4’)中，对所述高阶劳厄衍射点H的位置没有限制，可以为任意一高阶劳厄衍射点H的位置。
根据本发明的实施方案，步骤8’)中，目标布拉菲格子的对称性特征具有如下对称性特征：
Figure PCTCN2019095872-appb-000016
Figure PCTCN2019095872-appb-000017
有益效果
本发明提出由一张电子衍射花样重构布拉菲格子的方法，仅需一张含高阶劳厄衍射点的电子衍射花样实现三维重构，可用于任意晶体材料的物相识别、已知或未知晶体物相的布拉菲格子的重构，尤其适用于难以获得多张衍射花样的场合。一般地，当晶体晶粒小于300nm时，倾转晶体记录一张带轴电子衍射可能耗时长达数十分钟，几乎不可能记录多张电子衍射。所以大部分微晶、纳米晶都难以记录多张衍射花样，然而本申请的方法只需要记录一张含高阶劳厄衍射点的电子衍射花样就可以实现晶体三维重构，大大减少了实验和分析的工作量，效率显著提高。
综上，本发明的方法具有如下优点：本申请的方法没有带轴指数的要求，不需要倾转到严格的正带轴，也不需要复杂的晶体倾转以记录多张带轴电子衍射，大大简化了电子衍射的实验操作。仅由一张含有高阶劳厄衍射点的带轴电子衍射花样，即可分析出所测晶体的点阵类型及晶格常数；在实际的电镜实验和数据分析中，可大大减少实验和分析的工作量，明显提高工作效率。所提出的分析方法不受晶系、对称性高低的影响，适用于任何晶系的布拉菲格子的确定。
本发明的方法在透射电子显微镜上利用选区电子衍射、旋进电子衍射、微束电子衍射、纳米束电子衍射或会聚束电子衍射，可以快速分析出待测晶体的布拉菲格子。
附图说明
图1为利用一张电子衍射确定布拉菲格子的原理示意图：(a)高阶劳厄衍 射环与层间距的几何关系；(b)利用高阶劳厄衍射点进行倒易空间重构的原理示意图；
图2是实施例1利用单晶硅的一张带轴电子衍射确定布拉菲格子：(a)单晶硅的带轴电子衍射花样；(b)倒易空间重构；
图3是实施例2利用二氧化钛的一张靠近带轴的电子衍射确定布拉菲格子：(a)二氧化钛的带轴电子衍射花样；(b)倒易空间重构。
具体实施方式
下文将结合具体实施例对本发明的技术方案做更进一步的详细说明。应当理解，下列实施例仅为示例性地说明和解释本发明，而不应被解释为对本发明保护范围的限制。凡基于本发明上述内容所实现的技术均涵盖在本发明旨在保护的范围内。
除非另有说明，以下实施例中使用的原料均为市售商品，或者可以通过已知方法制备。
实施例1测量单晶硅的布拉菲格子
1)记录单晶硅的一张含高阶劳厄衍射的带轴选区电子衍射花样；图2a为JEOL JEM-2100型透射电子显微镜在200kV下记录的单晶硅的带轴电子衍射，相机长度L＝100mm。
2)在零阶劳厄衍射上测量二维初基胞
以透射斑为二维初基胞的原点O，以最近邻的衍射点A和B为邻边构建二维初基胞，如图2b所示。测量OA、OB和∠AOB，测量结果列于表1中。
3)测量高阶劳厄衍射环，根据公式CH＝R 2/(2λL 2)(其中R为高阶劳厄衍射环的半径，
Figure PCTCN2019095872-appb-000018
为200kV电子束的波长)得到该倒易面的层间距CH为
Figure PCTCN2019095872-appb-000019
4)测量高阶劳厄衍射点，并结合步骤3)得到的层间距CH重构出倒易初基胞，测量过程为：
测量图2a中的H点的坐标(也可以为任一高阶劳厄衍射点)，其坐标为(1114.5,351.438)；将H点平移到二维初基胞中，得到高阶劳厄衍射点的投影点C 1，其坐标为(728.25,417.766)。
5)三维倒易初基胞的重构
测量OA 1、OB 1、OC 1，以及垂线段A 1C 1和B 1C 1，测量结果见表1，利用几何关系可重构出倒易初基胞：
Figure PCTCN2019095872-appb-000020
Figure PCTCN2019095872-appb-000021
Figure PCTCN2019095872-appb-000022
γ *＝∠AOB＝60.53°。
其中，OC 1为平移后的投影点C 1到原点的间距，OA 1和OB 1为两垂足到原点的间距；CC 1为步骤3)计算得出的倒易面的层间距CH。
6)根据倒易关系，将步骤5)中得到的倒易初基胞转化为正格子：
Figure PCTCN2019095872-appb-000023
其中，
Figure PCTCN2019095872-appb-000024
Figure PCTCN2019095872-appb-000025
为倒易初基胞的体积。
7)约化处理：根据步骤6)计算出的正格子，计算出三个不共面的最短矢量：
a′＝-b 0
b′＝a 0-b 0
c′＝-a 0-2b 0-c 0
由此得到的约化胞参数为：
Figure PCTCN2019095872-appb-000026
α′＝59.90°，β′＝60.47°， γ′＝62.83°。
8)将Niggli约化胞转换为布拉菲格子：利用44种Nigg1i约化胞与14种布拉菲点阵之间的对应关系，将步骤7)中得到的Niggli约化胞通过44种矩阵转换为44个布拉菲格子；其中，所计算出来的格子
Figure PCTCN2019095872-appb-000027
Figure PCTCN2019095872-appb-000028
α＝90.33°，β＝91.46°，γ＝90.94°。在实验误差范围内满足立方晶系布拉菲格子的对称性要求，对应的点阵类型为cF，即面心立方结构。
9)将所计算的晶格常数利用零阶劳厄衍射点进行最小二乘法精修，所得晶格常数为：
Figure PCTCN2019095872-appb-000029
α＝β＝γ＝90°。该结果与单晶硅的晶体结构(PDF 77-2108，
Figure PCTCN2019095872-appb-000030
)高度吻合，因此认为上述分析方法是合理的。
表1单晶硅的电子衍射花样的测量结果
Figure PCTCN2019095872-appb-000031
实施例2测量二氧化钛的布拉菲格子
1)记录二氧化钛的一张含高阶劳厄衍射的带轴选区电子衍射花样；图3a为JEOL JEM-2100型透射电子显微镜在200kV下记录的二氧化钛的带轴电子衍射，相机长度L＝100mm。该电子衍射并非严格的带轴电子衍射，与正带轴偏离约0.11°。
2)在零阶劳厄衍射上测量二维初基胞
以透射斑为二维初基胞的原点O，以最近邻的衍射点A和B为邻边构建二维初基胞，如图3b所示。测量OA、OB和∠AOB，测量结果列于表2中。
3)测量高阶劳厄衍射环，根据公式CH＝R 2/(2λL 2)(其中R为高阶劳厄衍射环的半径，
Figure PCTCN2019095872-appb-000032
为200kV电子束的波长)得到该倒易面的层间距CH为
Figure PCTCN2019095872-appb-000033
4)测量高阶劳厄衍射点，并结合步骤3)得到的层间距重构出倒易初基胞
测量图3a中的H点的坐标(也可以为其他任一高阶劳厄衍射点)，其坐标为(592,109.25)；将H点平移到二维初基胞中，得到高阶劳厄衍射点的投影点C 1，其坐标为(707.859,427.844)。
5)三维倒易初基胞的重构
测量OA 1、OB 1、OC 1，以及垂线段A 1C 1和B 1C 1，测量结果如表2所示。利用几何关系可重构出倒易初基胞：
Figure PCTCN2019095872-appb-000034
Figure PCTCN2019095872-appb-000035
Figure PCTCN2019095872-appb-000036
γ *＝∠AOB＝85.19°。
其中，OC 1为平移后的投影点C 1到原点的间距，OA 1和OB 1为两垂足到原点的间距；CC 1为步骤3)计算得出的倒易面的层间距CH。
6)根据倒易关系，将步骤5)中得到的倒易初基胞转化为正格子：
Figure PCTCN2019095872-appb-000037
其中，
Figure PCTCN2019095872-appb-000038
Figure PCTCN2019095872-appb-000039
为倒易初基胞的体积。
7)约化处理：根据步骤6)计算出的正格子，计算出三个不共面的最短矢量：
a′＝a 0+b 0+c 0
b′＝a 0
c′＝-a 0+2b 0+c 0
由此得到的约化胞参数为：
Figure PCTCN2019095872-appb-000040
α′＝92.30°，β′＝90.23°，γ′＝94.37°。
8)将Niggli约化胞转换为布拉菲格子：利用44种Nigg1i约化胞与14种布拉菲点阵之间的对应关系，将步骤7)中得到的Niggli约化胞通过44种矩阵转换为44个布拉菲格子；其中，所计算出来的格子
Figure PCTCN2019095872-appb-000041
Figure PCTCN2019095872-appb-000042
α＝90.23°，β＝94.37°，γ＝92.30°在实验误差范围内满足四方晶系布拉菲格子的对称性要求，对应的点阵类型为tP，即四方相。
9)将所计算的晶格常数利用零阶劳厄衍射点进行最小二乘法精修，所得晶格常数为：
Figure PCTCN2019095872-appb-000043
α＝β＝γ＝90°。该结果与二氧化钛的金红石结构(PDF 75-1755,
Figure PCTCN2019095872-appb-000044
)高度吻合，因此认为上述分析方法也适用于带轴未严格倾转的情况。
表2 TiO 2电子衍射花样的测量结果
Figure PCTCN2019095872-appb-000045
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。

Claims (10)

  1. 一种仅由一张电子衍射花样重构布拉菲格子的方法，其特征在于，包括如下步骤：
    步骤1)：记录待测晶体的带轴电子衍射花样，得到一张含有高阶劳厄衍射的带轴电子衍射花样；
    步骤2)：测量二维初基胞；
    步骤3)：测量高阶劳厄衍射环；
    步骤4)：测量高阶劳厄衍射点；
    步骤5)：三维倒易初基胞的重构；
    步骤6)：将步骤5)中得到的倒易初基胞转化为正格子；
    步骤7)：约化处理；
    步骤8)：将Niggli约化胞转换为布拉菲格子。
  2. 根据权利要求1所述的方法，其特征在于，包括如下步骤：
    步骤1’)记录待测晶体的任意一张含有高阶劳厄衍射的带轴电子衍射花样；
    步骤2’)在零阶劳厄衍射上测量二维初基胞：以透射斑为二维初基胞的原点O，由最近邻的两个衍射点A和B为邻边形成的平行四边形作为二维初基胞，平行四边形中OA＝a *，OB＝b *，∠AOB＝γ *；
    步骤3’)测量高阶劳厄衍射环，根据如下公式计算得到该倒易面的层间距CH：
    CH＝R 2/(2λL 2)
    其中，R为高阶劳厄衍射环的半径；λ为电子束的波长，L为相机长度；
    步骤4’)测量高阶劳厄衍射点：测量高阶劳厄衍射点H的位置，并将其平移到步骤2’)中的二维初基胞中得到C 1，C 1即为三维倒易初基胞的格点C在所述二维初基胞内的投影点；
    步骤5’)三维倒易初基胞的重构：由OA、OB和OC构成三维倒易初基胞的三个基矢；从投影点C 1向二维初基胞的邻边OA和OB作垂线，垂足分别为A 1和B 1；由几何关系可知ΔOA 1C、ΔOB 1C和ΔCOC 1均为直角三角形，由此可得：
    a *＝OA
    b *＝OB
    Figure PCTCN2019095872-appb-100001
    Figure PCTCN2019095872-appb-100002
    Figure PCTCN2019095872-appb-100003
    γ *＝∠AOB
    其中，OC 1为平移后的投影点C 1到原点的间距，OA 1和OB 1为两垂足到原点的间距；CC 1为步骤3’)计算得出的倒易面的层间距CH；
    步骤6’)根据倒易关系将步骤5’)中得到的倒易初基胞转化为正格子：
    Figure PCTCN2019095872-appb-100004
    Figure PCTCN2019095872-appb-100005
    Figure PCTCN2019095872-appb-100006
    Figure PCTCN2019095872-appb-100007
    Figure PCTCN2019095872-appb-100008
    Figure PCTCN2019095872-appb-100009
    其中，
    Figure PCTCN2019095872-appb-100010
    Figure PCTCN2019095872-appb-100011
    为倒易初基胞的体积。
    步骤7’)约化处理：根据步骤6’)计算出的正格子，计算出三个不共面的最短矢量依次定义为正格子初基胞的基矢，使其满足Niggli约化的约束条件；
    步骤8’)将Niggli约化胞转换为布拉菲格子：利用44种Nigg1i约化胞与14种布拉菲点阵之间的对应关系，将步骤7’)中得到的Niggli约化胞通过44种矩阵转换得到44个布拉菲格子；每个转换矩阵都对应着一种布拉菲格子，在此称为目标布拉菲格子；通过检验计算出来的布拉菲格子是否满足目标布拉菲格子的对称性特征，确定布拉菲格子的类型和晶格常数。
  3. 根据权利要求1或2所述的方法，其特征在于，步骤1’)中的待测晶体可以是已知结构，也可以是未知结构；
    一般地，步骤1’)中的待测晶体可以是块材、粉末，也可以是单晶、多晶、微晶或纳米晶。
  4. 根据权利要求1-3任一项所述的方法，其特征在于，步骤1’)中，采用透射电子显微镜记录待测晶体的带轴电子衍射花样；所述透射电子显微镜的电子衍射可以是选区电子衍射、旋进电子衍射、纳米束电子衍射、微束电子衍射或会聚束电子衍射。
  5. 根据权利要求1-4任一项所述的方法，其特征在于，步骤1’)中，对所记录的带轴电子衍射花样没有带轴指数的限制，也没有对称性的限制，可以是任意带轴。
  6. 根据权利要求1-5任一项所述的方法，其特征在于，步骤1’)中，不要求所记录的带轴电子衍射花样满足严格的正带轴条件。
  7. 根据权利要求1-6任一项所述的方法，其特征在于，步骤2’)中，所述 二维初基胞所围的面积最小且两邻边OA与OB的夹角∠AOB＝γ *≤90°；
  8. 根据权利要求1-7任一项所述的方法，其特征在于，步骤4’)中，对所述高阶劳厄衍射点H的位置没有限制，其可以为任意一高阶劳厄衍射点H的位置。
  9. 根据权利要求1-8任一项所述的方法，其特征在于，步骤7’)中，约化过程如下：
    S1)根据步骤6’)计算出的正格子计算出任意指数u i,v i,w i的格点到晶格原点的长度t i：
    Figure PCTCN2019095872-appb-100012
    其中，格点指数u i,v i,w i可取正数、负数或零；一般地，-6≤u i,v i,w i≤6用于约化；在一定的u i,v i,w i范围内计算，得到t i列表；
    S2)在步骤S1)所计算出的t i列表中找到三个最小的t i值，定义为t 1、t 2和t 3，条件是t 1和t 2不共线，即矢量t 1和t 2的叉乘积所得的三个指数h，k和l均不为零；且要求具有最小t i值的三个矢量，其t 3·t 1×t 2不能为零，由此得到三个不共面的最短矢量；
    S3)计算三个矢量t 1、t 2和t 3间的夹角；矢量t 1和t 2的夹角可以采用如下公式计算：
    Figure PCTCN2019095872-appb-100013
    其中u 1，v 1，w 1为步骤S1)中计算t 1的格点指数；u 2，v 2，w 2为t 2的格点指数；
    S4)由此得到约化胞：a′＝t 1，b′＝t 2，c′＝t 3，
    Figure PCTCN2019095872-appb-100014
    Figure PCTCN2019095872-appb-100015
  10. 根据权利要求1-9任一项所述的方法，其特征在于，步骤8’)中，目标布拉菲格子的对称性特征具有如下对称性特征：
    Figure PCTCN2019095872-appb-100016
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HAN, MING: "A New Method to Reconstruct Reciprocal Primitive Cell From Selected Electron Diffraction Patterns", 浙江大学学报 (工学版 ) (JOURNAL OF ZHEJIANG UNIVERSITY (ENGINEERING SCIENCE)), vol. 33, no. 6, 25 November 1999 (1999-11-25), XP55775265, ISSN: 1008-973x *
LI, ZONGQUAN: "Principle and Method of Determining Crystal Structure Using Higher-Order Laue", PHYSICAL AND CHEMICAL TEST-PHYSICAL VOLUME, vol. 29, no. 5, 15 October 1993 (1993-10-15), pages 30 - 34, XP055775261 *

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