Law of rational intercepts

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. 

Figure 9-8. Inclined edges from cleavage units and illustration for the law of rational intercepts.

The famous French crystallographer, Ren Just Hatty, enun-ciator in 1784 of the Law of Rational Intercepts, believed that substances of identical crystal form must haVe the same chemical composition as well as the same constitution. This we now know to be absolutely true. This rule must not be confused with Mitscherlich s Law of Isomorphism which, of course, is not rigidly true — only approximately so. 

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. 

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. 

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... 

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. 

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