The Molecular Tactics of a Crystal

Part 1

THE MOLECULAR TACTICS OF A CRYSTAL

LORD KELVIN

London HENRY FROWDE OXFORD UNIVERSITY PRESS WAREHOUSE AMEN CORNER, E.C.

New York MACMILLAN & CO., 66 FIFTH AVENUE

THE MOLECULAR TACTICS OF A CRYSTAL

LORD KELVIN, P.R.S.

PROFESSOR OF NATURAL PHILOSOPHY IN THE UNIVERSITY OF GLASGOW AND FELLOW OF PETERHOUSE, CAMBRIDGE

Being the Second ROBERT BOYLE LECTURE, delivered before the Oxford University Junior Scientific Club on Tuesday, May 16, 1893

WITH TWENTY ILLUSTRATIONS

Oxford AT THE CLARENDON PRESS 1894

Oxford PRINTED AT THE CLARENDON PRESS BY HORACE HART, PRINTER TO THE UNIVERSITY

ON THE MOLECULAR TACTICS OF A CRYSTAL

By LORD KELVIN, P.R.S.

§ 1. My subject this evening is not the physical properties of crystals, not even their dynamics; it is merely the geometry of the structure--the arrangement of the molecules in the constitution of a crystal. Every crystal is a homogeneous assemblage of small bodies or molecules. The converse proposition is scarcely true, unless in a very extended sense of the term crystal (§ 20 below). I can best explain a homogeneous assemblage of molecules by asking you to think of a homogeneous assemblage of people. To be homogeneous every person of the assemblage must be equal and similar to every other: they must be seated in rows or standing in rows in a perfectly similar manner. Each person, except those on the borders of the assemblage, must have a neighbour on one side and an equi-distant neighbour on the other: a neighbour on the left front and an equi-distant neighbour behind on the right, a neighbour on the right front and an equi-distant neighbour behind on the left. His two neighbours in front and his two neighbours behind are members of two rows equal and similar to the rows consisting of himself and his right-hand and left-hand neighbours, and their neighbours’ neighbours indefinitely to right and left. In particular cases the nearest of the front and rear neighbours may be right in front and right in rear; but we must not confine our attention to the rectangularly grouped assemblages thus constituted. Now let there be equal and similar assemblages on floors above and below that which we have been considering, and let there be any indefinitely great number of floors at equal distances from one another above and below. Think of any one person on any intermediate floor and of his nearest neighbours on the floors above and below. These three persons must be exactly in one line; this, in virtue of the homogeneousness of the assemblages on the three floors, will secure that every person on the intermediate floor is exactly in line with his nearest neighbours above and below. The same condition of alignment must be fulfilled by every three consecutive floors, and we thus have a homogeneous assemblage of people in three dimensions of space. In particular cases every person’s nearest neighbour in the floor above may be vertically over him, but we must not confine our attention to assemblages thus rectangularly grouped in vertical lines.

§ 2. Consider now any particular person C (Fig. 1) on any intermediate floor, D and D′ his nearest neighbours, E and E′ his next nearest neighbours all on his own floor. His next next nearest neighbours on that floor will be in the positions F and F′ in the diagram. Thus we see that each person C is surrounded by six persons, DD′, EE′ and FF′, being his nearest, his next nearest, and his next next nearest neighbours on his own floor. Excluding for simplicity the special cases of rectangular grouping, we see that the angles of the six equal and similar triangles CDE, CEF, &c., are all acute: and because the six triangles are equal and similar we see that the three pairs of mutually remote sides of the hexagon DEFD′E′F′ are equal and parallel.

§ 3. Let now A, A′, A″, &c., denote places of persons of the homogeneous assemblage on the floor immediately above, and B, B′, B″, &c. on the floor immediately below, the floor of C. In the diagram let a, a′, a″ be points in which the floor of CDE is cut by perpendiculars to it through A, A′, A″ of the floor above, and b, b′, b″ by perpendiculars from B, B′, B″ of the floor below. Of all the perpendiculars from the floors immediately above and below, just two, one from each, cut the area of the parallelogram CDEF: and they cut it in points similarly situated in respect to the oppositely oriented triangles into which it is divided by either of its diagonals. Hence if a lies in the triangle CDE, the other five triangles of the hexagon must be cut in the corresponding points, as shown in the diagram. Thus, if we think only of the floor of C and of the floor immediately above it, we have points A, A′, A″ vertically above a, a′, a″. Imagine now a triangular pyramid, or tetrahedron, standing on the base CDE and having A for vertex: we see that each of its sides ACD, ADE, AEC, is an acute angled triangle, because, as we have already seen, CDE is an acute angled triangle, and because the shortest of the three distances, CA, DA, EA, is (§ 2) greater than CE (though it may be either greater than or less than DE). Hence the tetrahedron CDEA has all its angles acute; not only the angles of its triangular faces, but the six angles between the planes of its four faces. This important theorem regarding homogeneous assemblages was given by Bravais, to whom we owe the whole doctrine of homogeneous assemblages in its most perfect simplicity and complete generality. Similarly we see that we have equal and similar tetrahedrons on the bases D′CF, E′F′C; and three other tetrahedrons below the floor of C, having the oppositely oriented triangles CD′E′, &c. for their bases and B, B′, B″ for their vertices. These three tetrahedrons are equal and heterochirally similar to the first three. The consideration of these acute angled tetrahedrons, is of fundamental importance in respect to the engineering of an elastic solid, or crystal, according to Boscovich. So also is the consideration of the cluster of thirteen points C and the six neighbours DEFD′E′F′ in the plane of the diagram, and the three neighbours AA′A″ on the floor above, and BB′B″ on the floor below.

§ 4. The case in which each of the four faces of each of the tetrahedrons of § 3 is an equilateral triangle is particularly interesting. An assemblage fulfilling this condition may conveniently be called an ‘equilateral homogeneous assemblage,’ or, for brevity, an ‘equilateral assemblage.’ In an equilateral assemblage C’s twelve neighbours are all equi-distant from it. I hold in my hand a cluster of thirteen little black balls, made up by taking one of them and placing the twelve others in contact with it (and therefore packed in the closest possible order), and fixing them all together by fish-glue. You see it looks, in size, colour, and shape, quite like a mulberry. The accompanying diagram shows a stereoscopic view of a similar cluster of balls painted white for the photograph.

§ 5. By adding ball after ball to such a cluster of thirteen, and always taking care to place each additional ball in some position in which it is properly in line with others, so as to make the whole assemblage homogeneous, we can exercise ourselves in a very interesting manner in the building up of any possible form of crystal of the class called ‘cubic’ by some writers and ‘octahedral’ by others. You see before you several examples. I advise any of you who wish to study crystallography to contract with a wood-turner, or a maker of beads for furniture tassels or for rosaries, for a thousand wooden balls of about half an inch diameter each. Holes through them will do no harm and may even be useful; but make sure that the balls are as nearly equal to one another, and each as nearly spherical, as possible.

§ 6. You see here before you a large model which I have made to illustrate a homogeneous assemblage of points, on a plan first given, I believe, by Mr. William Barlow (Nature, December 20 and 27, 1883). The roof of the model is a lattice-frame (Fig. 3) consisting of two sets of eight parallel wooden bars crossing one another, and kept together by pins through the middles of the crossings. As you see, I can alter it to make parallelograms of all degrees of obliquity till the bars touch, and again you see I can make them all squares.

§ 7. The joint pivots are (for cheapness of construction) of copper wire, each bent to make a hook below the lattice frame. On these sixty-four hooks are hung sixty-four fine cords, firmly stretched by little lead weights. Each of these cords (Fig. 4) bears eight short perforated wooden cylinders, which may be slipped up and down to any desired position. They are at present actually placed at distances consecutively each equal to the distance from joint to joint of the lattice frame.

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