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Of the two rays produced by the double refraction of biaxial crystals, neither can be strictly denominated the ordinary one, since neither of them is refracted according to the ordinary law of single refraction. Both of them then are extraordinary rays, since they are refracted according to the laws of extraordinary refraction. Another peculiarity of biaxial crystals is that the position of the optic axes is not constant, but varies in the same crystal, according to the colour of the intromitted ray, and the temperature of the crystal. Thus a violet ray is separated into two pencils when incident in the same direction in which a red one is refracted singly. Sir John Herschel, to whom we are indebted for this discovery, found that the inclination of the resultant axes, in Rochelle salt, is for violet light 56°, and for red light 76°, but in the case of nitre, the inclination of the axes for violet light is greater than for red light, and Dr. Brewster discovered that glauberite has two axes for red light inclined about 5°, and only one axis for violet light. The changes produced on the inclinations of these axes by heat, I shall hereafter have occasion to notice.

In conclusion, then, crystals considered with respect to their singly or doubly refractive properties may be thus arranged:

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2. Form of Crystals.-A remarkable connexion exists between the optical properties and the geometrical forms of crystals; and to this I have now to beg your attention.

A crystal, like every other solid, possesses length, breadth, and thickness; and the measures of these are three imaginary lines which pass through the centre of the crystal, and are termed the axes. They may be denominated crystallographical or geometrical axes, to distinguish them from the optic axes with which they do not always coincide. Rose defines them to be "certain lines which pass through the centre of the crystal, and around which the faces are symmetrically disposed."

In some forms all these axes are equal in length, as in the cube; and in such cases it is said, that the axes are similar or alike. Such crystals are termed equiaxed. But in a very large proportion of cases the axes are not all equal, and these crystals are said to be unequiaxed. Now it is a remarkable circumstance, that the equiaxed crystals are single refractors, while the unequiaxed are double refractors. This is the first fact demonstrative of the connexion between the forms and the optical properties of crystals.

Of the unequiaxed crystals some have two, others three kinds of axes. If, for example, the length and the breadth of a crystal be alike, but the thickness different, the axes are of two kinds. Such crystals are usually said to have two dissimilar axes, but I shall term them di-unequiaxed. Other unequiaxed crystals have all their axes unequal; in other words, their length, their breadth, and their thickness are all unequal. Such crystals are generally said to have three dissimilar axes, but I shall call them tri-unequiaxed. Now, it is most remarkable that the di-unequiaxed crystals are double refractors, with one axis of [no] double refraction, while the tri-unequiaxed are double refractors with two axes of [no] double refraction. Here is another curious fact, illustrative of the relation which exists between the shape and optical properties of crystals.

Modern crystallographers arrange crystals in six groups, called systems. The equiaxed crystals constitute one system, called the cubic, octohedral or tessular system. The di-unequiaxed crystals comprehend two systems; one termed the square prismatic or pyramidal system, the other called the rhombohedric or rhombohedral system. The tri-unequiaxed crystals include three systems: one denominated the right rhombic or rectangular prismatic system; a second termed the oblique rhombic or rectangular prismatic system; and a third, called the doubly oblique prismatic system. The following table will, perhaps, render these statements more intelligible:

CLASS 1.

GEOMETRICAL CLASSIFICATION OF CRYSTALS.

Equiaxed crystals ......

(single refractors).

CLASS 2. Unequiaxedcrystals (double refractors)

......

....

Systems.

1. Cubic or Octohedral.

Order 1. Di-unequiaxed (one axis (2. Square Prismatic.
of [no] double refraction).. 3. Rhombohedric.
4. Right Rhombic Prismatic.
Order 2. Tri-unequiaxed (two axes
5. Oblique Rhombic Prismatic.
of [no] double refraction)..
6. Doubly Oblique Prismatic.

I shall not at present enter into any further details respecting the geometrical peculiarities of each of these systems, as the subject will be more appropriately considered presently.

3. Expansibility.-Between the particles of matter there exist two classes of forces, the one attractive, the other repulsive. By the first, particles are approximated and united to form masses; by the second, they are separated to greater or less distances. Hence attraction and repulsion are antagonizing forces.

Caloric or heat is a repulsive force. It augments the distance between particles and thereby weakens their attractive force; for molecular attraction rapidly diminishes as the distance between the particles increases. Hence solids and fluids, when heated, expand or dilate:

But the force of attraction which exists between the particles

of different bodies (solids and liquids) varies considerably: in some being much greater than in others. Hence, the same amount of heat gives rise to a very different degree of expansion in different bodies. In other words, each solid or liquid has an expansion peculiar to itself, owing to the greater or less attractive force which exists between the molecules.

Some crystals, when heated, expand equally in all directions, and such I shall accordingly denominate equiexpanding. Now it is obvious that in these the existence of equally attractive forces in all directions must be inferred; and it is a curious and striking confirmation of this inference that crystals, which suffer equal expansion in all directions, are singly refracting and equiaxed.

A very large number of crystals, however, dilate, when heated, unequally in different directions; and such may be conveniently denominated unequiexpanding. In them expansion in one direction is accompanied in some, if not in all cases, with contraction in another direction; and it is, therefore, obvious, that the force of attraction between their particles must be unequal in different directions, the attractive or cohesive force being least in that direction in which the expansion is the greatest. Crystals of this class are doubly refracting and unequiaxed.

The essential difference in shape between an equiexpanding and an unequiexpanding crystal is, that the first can be inscribed within a sphere, the second cannot. We may rudely illustrate this in the lecture-room, by diagram, substituting planes for solids, by inscribing a square, or an equilateral triangle in a circle (fig. 23, A and B). The first will represent the face of a cube, the second that of the regular tetrahedon. Now, it will be perceived that the circumference of the circle passes through all the angular points of the figure about which it is described. All these forms are equiexpanding.

FIG. 23.

C

D

E

00000

The regular six-sided prism expands unequally in some directions, but equally in others. If now we describe a circle around the terminal faces, it will be perceived that it passes through all the angular points of this face (fig. 23, C), and in all directions, in this plane, the crystal expands equally. The rhombohedron cannot be inscribed within the sphere, because its axes are unequal. If, for example, we attempt to describe a circle around the

rhombic face of Iceland spar (fig. 23, D), it will be found that while the obtuse angles (a a') are contained within the circle, the acute ones (b) project beyond it. Now, under the influence of heat, this face expands in the direction of the shortest axis, but contracts in that of the longest axis, by which the rhomb approaches to the square, the obtuse angles becoming more acute, the acute ones more obtuse (fig. 23, E).

These illustrations will serve to give some general notions of the relations which exist between the forms and expansibilities of crystals.

The di-unequiaxed crystals—that is, the doubly-refracting crystals, which have only one axis of [no] double refractionexpand equally in the direction of the equal crystallographical axes, but differently in that of the remaining one; and we may, therefore, denominate them di-unequiexpanding crystals. If, for example, a rhombohedron of Iceland spar be subjected to heat, it expands in the direction of its shortest axis, but contracts in all directions perpendicular to this, and in an intermediate direction it neither dilates nor contracts. Thus, according to Mitscherlich and Dulong, when heated from 32° to 212° Fahr, it actually expands, in the direction of the shorter axis, 0.00286, and contracts in a direction perpendicular to this 0.00056; so that its apparent or relative expansion in this axis is 0.00342 (that is 0.00286+0.00056). Now a necessary consequence of this unequal expansion is an alteration in the angles of the crystal: the obtuse ones become more acute, the acute ones more obtuse. In other words, the rhombohedron approximates to the cube; and in proportion to this change of form is the diminution of doubly refracting energy. Mitscherlich had conjectured that the latter effect would take place, and Rudberg has verified the conjecture. The last mentioned philosopher found, that while the ordinary refraction of Iceland spar underwent little or no change, the extraordinary refraction was considerably diminished by an augmentation of temperature.

FIG. 24.

Crystal of
Selenite.

The tri-unequiaxed crystals expand when heated, unequally in the direction of all their axes, and, therefore, they may be denominated triunequiexpanding crystals. When the temperature of selenite is augmented, the inclinations of all its faces suffer changes. Thus according to Mitscherlich by heating it from 32° to 212°, the inclination of the faces cc' was altered 10'50", that of the faces a a' 8'25", and that of the edges b b' only 7′26′′.

I have already explained what is meant by the terms positive or attractive, and repulsive or negative axes. They refer to optical differences in crystals, for which we find no corresponding

geometric or crystallographical differences. Now there have been observed, in the effects of heat on crystals, differences analogous to the optical ones just referred to. Thus, in crystals with a repulsive or negative axis, as Iceland spar, expansion is greatest in the direction of the shortest axis, showing that the molecular attraction in this direction is the weakest; whereas in positive or attractive crystals, as selenite, heat produces less dilatation in a direction parallel to the axis than in a direction perpendicular to it.

"The inclination of the optic axes, in biaxial crystals," says Mr. Lloyd, "is a simple function of the elasticities of the vibrating medium in the direction of three rectangular axes, and the plane of the optic axes is that of the greatest and least elasticities. If, then, these three principal elasticities be altered by heat in different proportions, the inclination of the axes will likewise vary; and if, in the course of this change, the difference between the greatest elasticity and the mean, or between the mean and the least, should vanish and afterwards change sign, the two axes will collapse into one, and finally open out in a plane perpendicular to their former plane. All these variations have been actually observed. Professor Mitscherlich found, that in sulphate of lime the angle between the axes (which is about 60° at the ordinary temperature) diminishes on the application of heat; that, as the temperature increases, these axes approach until they unite; and that, on a still further augmentation of heat, they again separate, and open out in a perpendicular plane. The primitive form of the crystal undergoes a corresponding change, the dilatation being greater in one direction than in another at right angles to it. Sir David Brewster has observed an analogous and even yet more remarkable property in glauberite. At the freezing temperature, this crystal has two axes for all the rays of the spectrum, the inclination of the axes being greatest in red light and least in violet. As the temperature rises, the two axes approach, and those of different colours unite in succession; and at the ordinary temperature of the atmosphere, the crystal possesses the singular property of being uniaxial for violet light, and biaxial for red. When the heat is further increased, the axes which have united open out in order, and in a plane at right angles to that in which they formerly lay, and at a temperature much below that of boiling water, the planes of the axes for all colours are perpendicular to their first position." The inclination of the optic axes in topaz, on the other hand, augments with the increase of temperature, and the variation M. Marx has observed, is much greater in the coloured than in the colourless varieties of this mineral+."

*

* Edin. Trans., vol. xi. ; and Phil. Mag., 3d series, vol. i., 417. † Jahrb. der Chemie, vol. ix.

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