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or,in other words, in which no double refraction exists or is evident. These directions are called the optic axes or the axes of double refraction. I have already stated that the phrase axes of so double refraction would be more intelligible. These axes may be regarded as positions of equilibrium where certain forces, which exist within the crystal and act in opposition, balance each other. In crystals of certain forms they coincide with the geometrical or crystallographical axes,whereas in crystals of other shapes they do not; but to these points I shall again have to beg your attention.
If we consider doubly refracting crystals in regard to the number of their optic axes we may divide them into two orders; one including those that possess only one axis, and another comprehending such as have two axes. The first are called uniaxial, the second biaxial crystals. As this distinction is connected with other remarkable optical peculiarities, as well as with the geometric and thermotic properties of crystals, it will be necessary to notice it a little more in detail.
a. Uniaxial Crystals.—Those crystals which have only one axis of [no] double refraction, and which, in consequence, are termed uniaxial crystals, or crystals with one optic axis, belong to the square prismatic or rhombohedric systems. In them the geometric or crystallographic axis is coincident with the optical one; that is, the line or direction in the crystal, around which the figure is symmetrically disposed, or about which every thing occurs in a similar manner on all sides, is coincident with the optic axis, or the axis around which the optical phenomena are the same in all directions. You must not, however, suppose that the axis is a single line; for there must be as many axes as there may be lines parallel to each other, so that the word is merely synonymous with a fixed direction.
In all other directions but the one called the optic axis, these crystals doubly refract; and of the two rays thus produced, one follows the ordinary laws of simple refraction, and is accordingly called the ordinary ray, while the other, being subject to an extraordinary law, is denominated the extraordinary ray.
These two rays advance with unequal degrees of velocity; the one suffering greater retardation than the other. When the ordinary ray advances more rapidly than the extraordinary one, the crystal is said to have a negative or repulsive axis of [no] double refraction; but when the ordinary ray advances less rapidly, the crystal is said to possess a positive or attractive axis In other words, when the extraordinary ray is refracted towards the axis, the crystal is said to have a positive axis; but when the ray is refracted from the axis, the crystal is said to have a negative axis. These terms are not very expressive of the property they are intended to represent. Biot used the terms
attractive and repulsive to designate the attractive or repulsive forces which he supposed to emanate from the axes cf crystals. For it is obvious that if the extraordinary ray be most retarded, it will be refracted from the axis, that is, it will appear to be repelled by a force emanating from the axis; whereas, if it be the least retarded, it will be refracted towards the axis, or will appear to be attracted by a force emanating from the axis. Now it was to obviate the hypothesis which these terms involve, that Brewster substituted the words positive and negative for the terms attractive and repulsive, merely meaning to denote by them the opposition, but not the nature, of the forces. Table of Uniaxial Crystals.
Kegatire (—) or repulsive Crystals
Positive (+) or attractive Crystals.
In uniaxial crystals the position of the optic axis is constant, whatever be the colour of the light; whereas in biaxial crystals this is not the case, as I shall presently show.
b. Of Biaxial Crystals.—A very large number of crystals, including all which belong to the right rhombic prismatic, oblique prismatic, and doubly oblique systems, have two axes of double refraction, which are more or less inclined to each other. Such crystals are, in consequence, denominated biaxial crystals, or crystals with two optic axes. In them there is no single line or axis around which the figure is symmetrical, as in uniaxial crystals; and the optic axes do not always, or even frequently, coincide with any fixed line in the crystals. Now this fact has led Dr. Brewster to believe that the optic axes are not the real axes of the crystals, but only the resultants of the real, or polarising, axes, or lines, in which the opposite actions of the two real axes compensate each other. Hence he terms them the resultant axes, or axes of no polarization, or of compensation.
The following is a list of a few biaxial crystals; and for a more extensive one I must refer my auditors to Dr. Brewster's works:
Table of Biaxial Crystals.
Character of Principal Inclination of Resultant
Glauberite , Negative 2° or 3°
Nitrate of Potash Negative 5° 2C
Carbonate of Lead Negative 10° 35'
Arragonite Negative 18" 18'
Borax Positive 28" 42'
Sugar Negative 50°
Selenite Positive 60°
Bochelle Salt Positive 80°
• The principal axis is, according to Dr. Brewster, the middle point '^'ween the two nearest poles of no polarization.—Phil Tans., 1818.
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 refractedaccordingtothe 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:
OPTICAL CLASSIFICATION OF CRYSTALS.
nmihlv refracHne^rvstals i order '• Uniaxial., l ,th ( a. Repulsive (negative) or Doubly redacting crystals .. J 0rder % Biaxial . . } citner £ b Attractive (positivc)
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:
GEOMETRICAL CLASSIFICATION OF CRYSTALS.
Class 1. 1
Equiaxed crystals >, 1. Cubic or Octohedral.
(Order 1. Di-unequiaxed (one axis (2. Square 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.
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