THE VALIDITY OF THE LAW OF RATIONAL INDICES, AND THE ANALOGY BETWEEN THE FUNDAMENTAL LAWS OF CHEMISTRY AND

CRYSTALLOGRAPHY.

BY AUSTIN F. ROGERS.

(Read March 1, 1912.)

Some fundamental law of nature governs the position of the faces of a crystal and limits in number the faces which occur on the crystals of any one substance. Crystal faces are designated by intercepts on coördinate axes, which are chosen so as to yield simple relations. Now it is found that the intercepts of the various crystal faces of a given substance, on each coördinate axis taken separately,

FIG. 1. The coördinate axes of a crystal.

usually bear a simple ratio to each other such as 1:0, 1:2, 1:3, 2:3, 3: I, etc. A selected face chosen because of its prominence is taken as a standard and the other faces are expressed in terms of it. The selected face is called the unit face, as its intercepts on the three axes establish a unit which, in general, is different for each axis, as represented in Fig. 1. The intercepts of the unit face which are, in

general, irrational constitute the axial ratios which are constants for each crystallized substance. For convenience in calculation the reciprocal ratios of the intercepts are used. These reciprocals are called indices or Miller indices, as Miller, an English crystallographer, was the first to make extensive use of this method. The indices are usually simple numbers such as (110), (210), (130), (211), (321), (441), etc., the unit face being (111).

If we examine the statements concerning the rationality of the indices of crystal faces in text-books and treatises we find a difference of opinion as to the exact definition of the law. Some authors insist that the indices are small whole numbers, while others simply state the fact that the indices are whole numbers, usually, but not necessarily small. One crystallographer, Viola,1 goes so far as to doubt the validity of the law of rational indices. Another investigator, G. H. F. Smith,2 believes that the law of simple rational indices is valid except in one particular instance, that of calaverite from Cripple Creek, Colorado. But, as he shows, by assuming several interpenetrant space-lattices it may be valid even in this case.

Thus there are three possibilities to consider: (1) The indices are always small rational numbers. (2) The indices are rational numbers, but not necessarily small. (3) The indices are not always rational and the law has no meaning. This subject is such a fundamental one in both theoretical and practical crystallography that it seems advisable to enquire into the history and status of the law. Such is the object of this paper.

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The credit of the discovery of the rationality of the indices is due to Haüy, professor of the humanities in the University of Paris, who developed it from his theory of crystal structure based upon cleavage observations. Haüy believed that crystals are composed of minute cleavage fragments which he called molécules intégrantes. Primary faces, according to his view, are due to the association of the molécules in parallel position, while secondary faces are due to the omission of molécules on the exterior of the crystal in step-like

1

1 Zeitschrift für Krystallographie und Mineralogie, Vol. 34, pp. 353-388

(1901).

2

3.66

Mineralogical Magazine, Vol. 13, p. 122 (1902).

Essai d'une Theorie sur la Structure des Crystaux." Paris, 1784.

arrangement. According to Haüy the omission is usually of one, two or three, rarely of four or five rows of molécules. Fig. 2 shows the production of an (110) face in this manner. If the cubes were very minute the (110) face would appear to be smooth. This epoch-making discovery laid the foundation of crystallography as an exact science and entitles Haüy to the title "father of crystallography." With some modification it has been the guiding principle in crystallography since that time and should not be abandoned unless the evidence is clearly against it.

Some authors express the fundamental law of crystallography as the law of simple mathematical ratio. Thus Williams says: “Experience has shown that only those planes occur on any crystal whose axial intercepts are either infinite or small even multiples of unity." Tutton also says: "The indices of any and every face on a crystal are three small numbers." Small in these quotations is usually interpreted as not more than six. Faces with indices larger than six, according to this view, are accidental and are usually relegated to the list of uncertain forms. There is a tendency to consider forms.

"Elements of Crystallography," 3rd ed., p. 26 (1901). Similar statements are also made in the text-books of Bayley, Brush-Penfield, Moses and Parsons, Patton, and Van Horn.

566

Crystallography and Practical Crystal Measurement,” p. 70, 1911.

with indices at all complex as doubtful even when the measurements indicate the form.

The law of simple mathematical ratio is untenable. There are

hundreds of measurements to prove this statement. The accompanying tabulated list gives faces with complex indices for a number of minerals, which list could be greatly extended if space permitted.

Palache and Wood, American Journal of Science, Vol. 18, p. 355, 1904.

These are selected because of the good agreement between the measured and calculated angles. Outside of its position in certain zones the only proof of a face lies in this agreement. Ordinarily an agreement as close as ten to thirty minutes of arc is sufficient to establish a face. For the common form-rich minerals, such as orthoclase, tourmaline, fluorite, magnetite, pyrite, barite, anglesite, calcite, aragonite, cerussite, stibnite, hematite, etc., it is certain that some of the faces have complex indices. To be convinced of this fact let one look over the list of forms of the above mentioned minerals in Goldschmidt's "Krystallographische Winkeltabellen." For calcite one half of the forms (162 out of 325) have indices greater than 10. The law of simple mathematical ratio is hardly compatible

with this fact.

Many crystals have what are called vicinal faces. These are faces with very high indices which replace faces with very simple indices. Thus apparent cubic crystals of fluorite from the north of England are in reality bounded by faces of a tetrahexahedron with the symbol (32.1.0). Here each cube face is replaced by a very low four-faced pyramid. Vicinal faces are often regarded as accidental or in some way irregular and are usually excluded from the law of rational indices as they are of course inconsistent with the law of simple mathematical ratio. As they lie in prominent zones and as their arrangement conforms to the symmetry of the crystal on which they occur, they can hardly be excluded from the list of faces, though their origin is not clearly understood. The only possible argument for excluding them is that the exact indices of such faces can not always be determined, for the agreement between measured and calculated angles must be exceptionally good to establish the face. Miers found that on alum very flat trisoctahedral faces replace the octahedral faces. In one case the measurements indicated the symbol (251-251-250). As Miers says, this form can not be regarded as established. It may be some other form with a little different

'For recent additions to these lists see Whitlock, School of Mines Quarterly, Vol. 31, p. 320; Vol. 32, p. 51 (1910).

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Philosophical Transactions of the Royal Society, A, Vol. 202, pp. 459523 (1903).