Download
1 / 28
E N D
The first step in analysing unknown powder pattern is often an attempt to find a unit cell that explains all observed lines in the spectrum. You do not need additional crystallographic data, although if it exists it makes for faster and more reliable results. The material to be analysed must be single phased and the experimental material must be very accurate.
Indexing programs use only the positional information of the pattern and try to find a set of lattice constants (a,b,c,a,b,g) and individual Miller indices(hkl) for each line. The form of equations to solve is complicated for the general case (triclinic) in direct space but is straightforward in reciprocal space. In the latter the set of equations is：
Q = h2A + k2B + l2C+ hkD + hlE + klF where the Q-values are easily derived from the diffraction angle Q. This set has to be solved for the unknowns, A, B, C, D, E, F, which are in a simple way related to the lattice constants. Finding the proper values for the lattice parameters so that every observed d-spacing satifies a particular combination of Miller indices is the goal of indexing. It is not easy even for the cubic system, but it is very difficult for the triclinic system.
There are two general approaches to indexing, the exhaustive and the analytical approach. Both of these approaches require very accurate d-spacing data. The smaller the errors, the easier it is to test solutions because there are often missing data points due to intensity extinctions related to the symmetry or the structural arrangement or due to lack of resolution of the d-spacing themselves. The earliest approaches were of the exhaustive type and were done by graphical fitting or numerical table fitting.
Indexing Programs • The methods currently implemented are shown bold. They are selected through the item Indexing in the main menu. • Program Author Type • ITO Visser analytical • TREOR Werner exhaustive • POWDER • DICVOL • CUBIC
The programs use a set of common parameters, e.g. the wavelength and a method specific set. After you have clicked on a program with the left mouse button indexing is immediately started with the active parameter set. Depending on the problem and computer type the program run can take from seconds to many minutes. All solutions are computed and stored internally. The "best" one is displayed at the top of the screen, the next to best at the bottom. For an overview of all solutions select Solutions in the main menu.
Miller Indices Triplet of integer numbers uniquely assigned to a Bragg reflection. The notation is usually in the form of (hkl). Formally spoken, Miller Indices represent coordinates of lattice points in reciprocal space.
Reciprocal Space Direct space is composed of unit cells and its contents, whereas reciprocal space is a lattice whose lattice points are Bragg reflections. Direct and reciprocal space are tightly coupled.
Lattice constants A set of maximally six floating numbers representing the unit cell. As crystal symmetry grows the number of lattice constants needed to describe the metrics of a unit cell reduce. In the cubic system there is only one constant.
Q-value The Q-value is one of several possible forms to represent the positional information of peaks in powder pattern. They are defined as: Q = 1/d2= ( 4 sin2Q)/l2 Q can be represented as a quadratic form of the (hkl)'s. Q = h2A + k2B + l2C + hk D + hl E + kl F, where A, B, C, D, E, F are related to the lattice constants. In ITO each Q-value is multiplied by 10000.
d-Spacing The d-spacing is one of several possible forms to represent the positional information of peaks in powder pattern. Other forms are the angle Q, or 2Q, or Q-values. Win_DIFFRAC assumes input in the form of d-spacings. They are internally converted, e.g. ITO works only with Q's.
TREOR This program uses the exhaustive approach. It is widely used for indexing of powder pattern throughout the world and is believed to be one of the most powerful engines for solving high and medium symmetries. It is useful as well for low symmetry at the cost of rather high time consumption.
We recommend that you start with a diffraction pattern whose result is well known to you. If you are already familiar with the TREOR program you may decide to straightforward indexing , using the default parameters. In case you are a novice user you should swiftly go through the parameters and change settings only when they are obviously wrong. With the results obtained you may then turn to the more sophisticated parameters, e.g. Select Baseline Set. The parameters are described in detail under general and TREOR-specific parameters.
In contrast to ITO a multitude of non-systematic extinctions among the first lines may not appreciably affect the power of trial-and-error methods. However, 'powder indexing is not like structure analysis, which works well on good data, and will usually get by on poor data given a little more time and attention. Powder indexing works beautifully on good data, but with poor data it will usually not work at all.' (op.cit Ref.[4]).
The standard procedure using TREOR is to start with the high symmetries: cubic, tetragonal, hexagonal and orthorhombic in one run. Next the monoclinic symmetry should be tried. More than one run may be necessary, successively increasing the number of baseline sets, the cell volume and cell edge. If the formula weight and density are known, they should be used. The CPU-time needed will then usually be strongly reduced. Unfortunately they are mostly not well known and therefore not often usable. If all previous tests have failed the triclinic test remains for which in general we recommend ITO.
Analytical Approach The crystal is assumed to be triclinic. Solutions are approached in three steps. In the first, one tries to find reasonable zones from the diffraction data. Each pair of lines, together with the origin forms a crystallographic zone. A zone is relevant to the solution finding algorithm if some other diffraction lines lie on it, if not it is discarded. In the second step the best zones are then tried in combinations to find complete lattices. This method is known as the ITO method. Once a lattice is found that satisfies the experimental d-spacing data, the geometry of the lattice is examined for possible higher metric symmetry. This is the final step.
衍射分析应用 物质对射线的衍射产生了衍射花样或衍射谱，对于给定的单晶试样，其衍射花样与入射线的相对取向及晶体结构有关；对于给定的多晶体也有特定的衍射花样。衍射花样具有三要素：衍射线（或衍射斑）的位置、强度和线型。测定衍射花样三要素在不同状态下的变化，是衍射分析应用的基础。
衍射分析应用 • 基于衍射位置的应用 ⑴ 点阵参数的精确测定，膨胀系数的测定； ⑵ 第一类（即宏观残余）应力的测定； ⑶ 由点阵参数测定相平衡图中的相界； ⑷ 晶体取向的测定； ⑸ 固溶体类型的测定，固溶体组分的测定； ⑹ 多晶材料中层错几率的测定； ⑺ 点缺陷引起的Bragg峰的漂移。
衍射分析应用 • 基于衍射强度测量的应用 （1）物相的定量分析，结晶度的测定 (2) 平衡相图的相界的测定； (3) 第三类应力的测定； (4) 有序固溶体长程有序度的测定； (5) 多晶体材料中晶粒择优取向的极图、反极图和三维取向分布的测定； (6) 薄膜厚度的测定。
衍射分析应用 • 基于衍射线型分析的应用 (1) 多晶材料中位错密度的测定，层错能的测定，晶体缺陷的研究； (2) 第二类（微观残余）应力的测定； (3) 晶粒大小和微应变的测定；
衍射分析应用 • 基于衍射位置和强度的测定 (1) 物相的定性分析 (2) 相消失法测定相平衡图中的相界; (3) 晶体(相)结构,磁结构,表面结构,界面结构的研究
衍射分析应用 • 同时基于衍射位置、强度和线型的Rietveld多晶结构测定 需输入原子参数（晶胞中各原子的坐标、占位几率和湿度因子）、点阵参数、波长、偏正因子、吸收系数、择优取向参数等。
衍射分析应用 • Rietveld profile refinement • What is the Rietveld method? • A comprehensive, computer based method fro the analysis of X-ray and neutron powder diffraction patterns by least squares fitting of a calculated diffraction pattern to the observed data. • What is special for Rietveld method? • This method does not use integrated powder diffraction intensities,but employs directly the profile intensities obtained from step-scanning measurements of the powder diagram.
衍射分析应用 • Rietveld profile refinement • How useful of Rietveld method? • The Rietveld method has been developed into a valuable method for structure analyses of nearly all class of crystalline materials (Young 1995) by a number endeavours.It has been extensively used to investigate the structure and phase problems. A lot of useful information can be retrieved by Rietveld method.
对于大多数固溶体，其点阵参数随溶质原子的浓度呈近似线对于大多数固溶体，其点阵参数随溶质原子的浓度呈近似线 性关系，即服从费伽(Vegard)定律： 式中，aA和aB分别表示固溶体组元A和B的点陈参数。因此，测得含量为x的B原子的因溶体的点阵参数工ax，用上式即求得固溶体的组分。 实验表明，固溶体中点阵参数随溶质原子的浓度变化有不少呈非线性关系，在此情况下应先测得点阵参数与溶质原子浓度的关系曲线。 实际应用中，将精确测得的点阵参数与已知数据比较即可求得固溶体的组分。
