Gem-Stones and Their Distinctive Characters

by George Frederick Herbert Smith

We have further that

r + r´ = A,

whence it follows that

A + D = i + i´.

If the stone be mounted on the goniometer and adjusted so that the edge of the prism is parallel to the axis of rotation of the instrument and if light from the collimator fall upon the table-facet and the telescope be turned to the proper position to receive the emergent beam, a spectral image of the object-slit, or in the case of a doubly refractive stone in general, two spectral images, will be seen in white light; in the light of a sodium flame the images will be sharp and distinct. Suppose that we rotate the stone in the direction of diminishing deviation and simultaneously the telescope so as to retain an image in the field of view, we find that the image moves up to and then away from a certain position, at which, therefore, the deviation is a minimum. The image moves in the same direction from this position whichever way the stone be rotated. The question then arises what are the angles of incidence and refraction under these special conditions. It is clear that a path of light is reversible; that is to say, if a beam of light traverses the prism from the facet t to the facet b it can take precisely the same path from the facet b to the facet t. Hence we should be led to expect that, since experiment teaches us that there is only one position of minimum deviation corresponding to the same pair of facets, the angles at the two facets must be equal, i.e. i = i´, and r = r´. It is, indeed, not difficult to prove by either geometrical or analytical methods that such is the case.

Therefore at minimum deviation r = ^A/₂ and i = ^{A + D}/₂ and, since sin i = n sin r, where n is the refractive index of the stone, we have the simple relation—

n = ^{sin A + D/₂} / _{sin ^A/2}

This relation is strictly true only when the direction of minimum deviation is one of crystalline symmetry in the stone, and holds therefore in general for all singly refractive stones, and for the ordinary ray of a uniaxial stone; but the values thus obtained even in the case of biaxial stones are approximate enough for discriminative purposes. If then the stone be singly refractive, the result is the index required; if it be uniaxial, one value is the ordinary index and the other image gives a value lying between the ordinary and the extraordinary indices; if it be biaxial, the values given by the two images may lie anywhere between the greatest and the least refractive indices. The angle A must not be too large; otherwise the light will not emerge at the second facet, but will be totally reflected inside the stone: on the other hand, it must not be too small, because any error in its determination would then seriously affect the accuracy of the value derived for the refractive index. Although the monochromatic light of a sodium flame is essential for precise work, a sufficiently approximate value for discriminative purposes is obtained by noting the position of the yellow portion of the spectral image given in white light.

In the case of a stone such as that depicted in Fig. 22 images are given by other pairs of facets, for instance ta and tc, unless the angle included by the former is too large. There might therefore be some doubt, to which pair some particular image corresponded; but no confusion can arise if the following procedure be adopted.