Rational indices, law

Rational Indices, Law of A rational number is an integer or the quotient of two integers. The Law of Rational Indices states that all of the faces of a crystal may be described, with respect to their intercepts on three noncolinear axes, by three small whole numbers. 

In 1784, Haiiy formulated the Law of Rational Indices, which states that all faces of a crystal can be described by Miller indices (hid), and for those faces that commonly occur, h, k, and l are all small integers. The eight faces of an octahedron are (111), (111), (ill), (111), (111), (111), (III), and (III). The form symbol that represents this set of eight faces is 111. The form symbol for the six faces of a cube is 100. Some examples in the cubic system are shown in Figs. 9.1.3. and 9.1.4. 

Therefore, in compliance with the Law of Rational Indices, only n-axes with n = 1,2,3,4 and 6 are allowed in crystals. The occurrence of the inversion center means that the rotation-inversion axes I, 2(= m), 3, 4 and 6 are also possible. 

It has been recognized already in the earliest stages in the history of crystallography [8] that the most important characteristic of the outer symmetry of the crystals is not really the form itself but rather two phenomena expressed by two rules. One is the constancy of the angles made by the crystal faces. The other is the law of rational intercepts or the law of rational indices. 

Usually the crystal faces are described by the reciprocals of the multiples of the standard intercepts, hence the name the law of rational indices. In Figure 9-8 three lines are adopted as axes which may also be directions of the crystal edges. A reference face ABC makes intercepts a, b, c on these axes. Another face of the crystal, e.g., DEC, can be defined by intercepts alh, blk, dl. Here h, k, l are simple rational numbers or zero. They are called Miller indices. The intercept is infinite if a face is parallel to an axis, and horkorl will be zero. For orthogonal axes the indices of the faces of a cube are (100), (010), and (001). The indices of the face DEC in Figure 9-8 are (231). 

It is important to describe each crystal face in a numerical way if data on different crystals or from different laboratories are to be compared. The method used to describe crystal faces is derived from the Law of Rational Indices, proposed by Haiiy and Arnould Carangeot. This Law states that each face of a crystal may be described, by reference to its intercepts on three noncollinear axes, by three small whole numbers (that is, by three rational indices)/ From this law, William Whewell introduced a specific way of designating crystal faces by such indices, and William Hallowes Miller popularized it. The integers that characterize crystal faces are called Miller indices h, k, and 1. When this method is used to describe crystal faces, it is rare to find h, k, or / larger than 6, even in crystals with complicated shapes. An example of the buildup of unit cells to give crystals with different faces is shown in Figure 2.11. 

An important fact about crystal faces was known long before there was any knowledge of crystal interiors. It is expressed as the law of rational indices, which states that the indices of naturally developed crystal faces are always composed of small whole numbers, rarely exceeding 3 or 4. Thus, faces of the form 100, 111, iTOO, 210, etc., are observed but not such faces as 510, 719, etc. We know today that planes of low indices have the largest density of lattice points, and it is a law of crystal growth that such planes develop at the expense of planes with high indices and few lattice points. 

Hatiy enunciated the laws governing crystal symmetry, and paved the way for his later discovery of the law of rational indices, which, in 1801, he substantiated by a comprehensive survey of the mineral kingdom. Thus by the first year of the nineteenth century the fundamental laws of morphological crystallography had been established. 

Molecular forms bring concepts back developed during the nineteenth century in which crystal growth forms were systematically investigated. At that time, atoms were not considered to be real but only as a way of expressing chemical laws [4], and lattice periodicity an hypothesis compatible with the empirical law of rational indices with no consequence for the physical nature of crystals [5]. The present molecular situation is reversed one knows that there are atoms and where the atoms are. A molecular lattice allows an interpretation of the molecular morphology, also expressible in terms of rational indices, but without a theoretical basis, even if one can speak of molecular crystallography [6]. 

Usually, the crystal faces are described by the reciprocals of the multiples of the standard intercepts, hence the name the law of rational indices. In Figure 9-8 three lines are adopted as axes which may also be directions of the... 

Nineteenth-century crystallography may be considered to be the mathematical branch of mineralogy. It is based on two empirical laws, the law of constancy of angle and the law of rational indices. These laws will be presented in the following pages after a discussion of some mathematical principles fundamental to crystallography, non-unitary coordinate systems and reciprocal coordinates. 

Cleavage of crystals, in particular of calcite CaC03, and the law of rational indices generated the idea of the periodicity of crystal structures and the theory of translational lattices ... 

The analogy between the equations representing the edges and faces of a crystal on the one hand, with lattice lines and lattice planes on the other is the foundation of the theory of the periodic nature of crystal structures. This interpretation of the law of rational indices was formulated by the French abbe Auguste Bravais (1811-1863) as follows ... 

Law of rational indices The lengths of intercepts of different crystal faces on any crystallographic axis are in ratios of small integers. Rhodes G (1999) Crystallography made crystal clear a guide for users of macromolecular models. Elsevier Science and Technology Books, New York. 

Nikolaus Steno, as early as 1669. formulated the law of constant angles, which states that two like edges of crystals of the same substance always make equal angles with each other. In 1823. Franz Ernst Neumann discovered the law of simple rational indices. Planes drawn through faces of a cry.stal intersect three coordinate axes, constructed parallel to crystal edges, at minima lengths a. b. c that are in a constant ratio to one another. The other faces of the crystal can be... 

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The activity coefficient fz is sometimes called the rational activity coefficient, since it gives the most direct indication of the deviation from the ideal behavior required by Raoult s law. It is, however, not often used in connection with measurements on solutions of electrolytes, and so the coefficients/c and/, which are commonly employed, are described as the practical activity coefficients. The coefficient / , from which the... 

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