CHAPTER VIII.

RINGS AND BRUSHES PRODUCED BY CRYSTAL

PLATES.

THE phenomena hitherto discussed are those produced by a beam of parallel rays; it remains to consider those due to systems of convergent or divergent rays. It was seen in the former case that the effect produced by a plate of crystal upon polarised light depended partly upon the direction of the ray with reference to the natural structure (or more particularly to the optic axis or axes) of the crystal, and partly to the thickness of crystal traversed by the ray. And it is clear that if a system of convergent or divergent rays fall upon a plate of crystal of uniform thickness, the rays will strike the plate at various angles, and traverse various thicknesses of the crystal. Hence it follows that the effects produced will not be, as in the case of parallel rays, fields of uniform tint, but figures of various form and tint.

It was explained above that in Iceland spar there is a particular direction, viz. that of the line joining the two opposite obtuse angles of the natural crystal, in which there is no double refraction, and in which all rays travel with the same velocity. This direction

(that is to say, this line and all lines parallel to it) bears the name of the optic axis. There are many other crystals having the same property in one and only one direction-in other words, having a single optic axis. There is, moreover, another class of crystals having two such axes. Crystals of the first class, or uni-axal crystals, are again divided into two groups, viz. positive, in which the extraordinary ray is more refracted than the ordinary, and negative, in which the ordinary ray is the more refracted. It will be remembered that the ray which travels slowest is the most refracted. Among the former may be mentioned

Every crystal belongs to one or other of six systems or types of symmetry. These are conveniently distinguished by the various kinds of axes to which the faces are referred for the purpose of geometrical comparison.

(1.) The regular system, which is based upon a system of three equal rectangular axes. Any form derived from this will be perfectly symmetrical with reference to the three axes, and will present no distinguishing feature in relation to any of them. Crystals belonging to this system have no optic axis nor any doubly refracting property. The cube and regular octahedron are examples of forms in this system.

(2.) The tetragonal system, based upon a system. of three rectangular axes, whereof two are equal, but the third greater or less than the other two. Crystals belonging to this system have one optic axis coinciding with the last-mentioned crystallographic axis. The square-based prism is a form of this system.

(3.) The hexagonal system, referred by some crystallographers to three equal axes lying in one plane, inclined at 60° to one another, and a fourth axis at right angles to the other three. Crystals of this system have one optic axis coinciding with the fourth axis above mentioned. The rhombohedron (the six faces of which are equal rhombs) is a form of this system. Iceland spar, or calcite, has cleavages parallel to the faces of a rhombohedron, the obtuse plane angles of which are 101° 5′. Some crystallographers refer the crystals of this system to axes parallel to the edges of a rhombohedron; these axes being three in number obliquely but equally inclined to each other, and of equal lengths.

(4.) The orthorhombic system, having three rectangular but unequal axes.

(5.) The clinorhombic system, which differs from the rhombic in this, that while one of the three axes is perpendicular to the other two, these two are oblique to one another.

(6.) The anorthic system, in which all the axes are oblique.

All crystals belonging to the last three systems have two optic axes. In the rhombic system they lie in a plane containing two of the three crystallographic axes; in the monoclinic they lie either in the plane containing the oblique axes or in a plane, or for different colours in planes, at right angles thereto. In the triclinic no assignable relation between the optic and the crystallographic axes has been determined.

It was shown above that the retardation due to any doubly refracting crystal, and consequently the colour produced by it, is dependent on the thickness; and that with a crystal of constantly increasing thickness the colours go through a complete cycle, and then begin again. Suppose, then, a divergent beam to fall perpendicularly upon a uni-axal crystal plate cut at right angles to the optic axis; the central rays will fall perpendicularly to the surface; but the rays which form conical shells about that central ray will fall obliquely. The rays forming each shell will fall with the same degree of obliquity on different sides of the central ray, those forming the outer shells having greater obliquity than the inner. Now, the more obliquely any ray falls upon the surface the greater will be the thickness of the crystal which it traverses;

and this will still be the case even though it suffers refraction, or bending towards the perpendicular, on entering the crystal. Each incident cone of rays will consequently still form a cone when refracted within the crystal, although less divergent than at incidence, in its passage through the plate; and the successive refracted cones will be more and more oblique, as were the incident cones, but in a less degree, as we pass from the more central to the more external members of the assemblage forming the beam of light.

Now, not only will the actual thickness of crystal traversed by the ray, but so also will the double refraction, increase with the obliquity or divergence of direction from that of the axis; and the effect of the former is in fact small compared with that of the latter in producing retardation.

Let A B C D (Fig. 21) represent the crystal plate, OP the direction of the optic axis and of the central ray, On, On' those of any two other rays. The ray O P will not be divided; but on will be separated by the double refraction of the plate into two, n s, n r, the one ordinary, the other extraordinary; and these will emerge parallel to one another, and may be represented by the lines st, rv. Similarly, the effect of double refraction on on' may be represented by n' s, n'v', st, v. Suppose now that the process were reversed, and that two monochromatic rays, one ordinary, the other extraordinary, reach the plate at s and ✔ in the directions ts, v r respectively; these would meet at n and travel together to O. Suppose, further, that the difference in length of s n and r n is equal to one wave length; then, since one of them is an ordinary