The indexing problem

In powder diffraction, the very first step in solving the crystal structure, i.e. finding the true unit cell, may present considerable difficulties because the experimental data are a one-dimensional projection of the three-dimensional reciprocal lattice recorded as a function of a single independent 

Regardless of the nature of the diffraction experiment, finding the unit cell in a conforming lattice is a matter of selecting the smallest parallelepiped in reciprocal space, which completely describes the array of the experimentally registered Bragg peaks. Obviously, the selection of both the lattice and the unit cell should be consistent with crystallographic conventions (see section 1.12, Chapter 1), which impose certain constraints on the relationships between unit cell symmetry and dimensions. 

This process is commonly known as indexing of diffraction data and in three dimensions it usually has a unique and easy solution when both the lengths and directions of reciprocal vectors, d A , are available. On the contrary, when only the lengths, / /, / of the vectors in the reciprocal space are known, the task may become extremely complicated, especially if there is no additional information about the crystal structure other than the array of numbers representing the observed Vdhu [= d hkl values. 

The difficulty and reliability of indexing are closely related to the absolute accuracy of the array of d Hki values, i.e. to the absolute accuracy with which positions of Bragg reflections have been determined. For 

As established earlier (see section 2.8, Chapter 2), the interplanar distances, d, are related to both the unit cell dimensions and Miller indices of the families of crystallographic planes by means of a well defined function, which in general form can be written as follows  

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Attention is drawn to a special feature of this Volume—inclusion of a generalized Cumulative Index to Volumes 1-10. It is believed that use of this Cumulative Index, in conjunction with the detailed Cumulative Subject Index to Volumes 1-5 (in Volume 6) and the detailed, individual Subject Indexes to Volumes 6-10, will afford the reader speedy access to sources of detailed information in the various Volumes. This solution to the indexing problem was chosen for economic reasons. 

Pantelides [27, 28], in his work with Sargent, defined the index in a manner that exposes its potential to cause problems in initialization as well as in the integration error. They too showed that the index problem can be eliminated by differentiation. Noting that only some of the equations need to be differentiated, they use a method based on the structural properties of the equations to discover these equations. They cite several examples in which the index problem is almost certain to occur in setting up and solving dynamic simulation models, e.g., calculations of flash dynamics and problems in which the trajectory of a state variable is specified. 

Finally, one should be concerned about the index problems for mixed sets of PDEs and algebraic equations. How many problems have been solved where some derivable independent equations have been missed ... 

It is the custom of CA to publish an Introduction to its Subject Indexes. The indexes are built to stand on their own feet. The introduction is not essential to the ready and effective use of the CA Subject Index. Nevertheless, for the best results in the use of any index the user must meet the index maker part way in understanding the indexing problems and nomenclature in particular. Use of the information in our Subject Index Introductions is recommended to the searcher who is doing more than incidental searching. 

A consideration of the indexing problems associated with the completed abstract throws considerable light on the philosophy of abstracting itself. 

We conclude this section with a simple notion it is impossible to solve the crystal structure of a material using an incorrect unit cell. Thus, proper indexing of the experimental powder diffraction pattern is of utmost importance, and in this chapter we shall consider various strategies leading to the solution of the indexing problem and how to find the most precise unit cell dimensions. 

Solving the indexing problem becomes a matter of identifying the differences that result in whole numbers when divided by a common divisor and c, respectively). The expected whole numbers are shown in Table 5.14 through Table 5.16 for several small h, k and /. It only makes sense to consider these small values because successful indexing is critically dependent on the availability of low Bragg angle peaks, which usually have small values of indices. 

The indexing problem is essentially a puzzle (. . . ). It would be quite an easy puzzle if errors of measurement did not exist. 

The most common and widely used indexing programs are ITO, TREOR " and DICVOL91. All three classic programs are present in the indexing Crysfire suite. Their approach to the indexing problem is different and will be briefly described. 

The indexing problem is usually solved in a few minutes if (a) the symmetry is not lower than monoclinic (b) the cell volume is less than 2000 A (c) the cell parameters are less than 20 A. More computing time is required for triclinic symmetry indeed the main drawback of the McMaille approach is the high request of computing time in the case of low crystal symmetry. 

The integration of a DAE system can be performed by transformation in an ODE system. It is worthy to note that this operation might be confronted with the index problem. Index is the minimum number order of differentiation needed to transform a DAE system into a set of first-order ODEs. Problems of index one can be solved by means of standard differentiation methods. When the index is higher than one then the DAE system needs a special treatment. Modem codes have capabilities for automatic detection of index higher-than-one, diagnose the problem and suggest modifications. 

To illustrate the index problem let s consider the following simple DAE system ... 

Even if one solves the indexing problem and then proceeds with the analysis by an evaluation of measured reflection intensities, one cannot expect to achieve an accuracy in the crystal structure data which would be comparable to those of low molar mass compounds. This is not only a result of the lack of single crystals, but represents also a principal property In small crystallites, as they are found in partially crystalline polymers, lattice constants can be affected by their size. In many cases crystallites are not only limited in chain direction by the finite thickness of the crystalline lamellae but also laterally since polymer crystallites are often composed of mosaic blocks. Existence of these blocks is indicated in electron microscopic investigations on... 

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