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If the stone be too highly refractive for a measurement of its refractive index to be possible with the refractometer just described, and it is desired to determine this constant, recourse must be had to the prismatic method, for which purpose an instrument known as a goniometer[3] is required. Two angles must be measured; one the interior angle included between a suitable pair of facets, and the other the minimum amount of the deviation produced by the pair upon a beam of light traversing them.

Fig. 22.—Path at Minimum Deviation of a Ray

traversing a Prism formed of two Facets of a

Cut Stone.

Fig. 22 represents a section of a step-cut stone perpendicular to a series of facets with parallel edges; t is the table, and a, b, c, are facets on the culet side. The path of light traversing the prism formed by the pair of facets, t and b, is indicated. Suppose that A is the interior angle of the prism, i the angle of incidence of light at the first facet and i´ the angle of emergence at the second facet, and r and r´ the angles inside the stone at the two facets respectively. Then at the first facet light has been bent through an angle i - r, and again at the second facet through an angle i´ - r´; the angle of deviation, D, is therefore given by

D = i + i´ - (r + r´).

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.

Fig. 23.—Course of Observations in the Method of Minimum Deviation.

The table, or some easily recognizable facet, is selected as the facet at which light enters the stone. The telescope is first placed in the position in which it is directly opposite the collimator (T0 in Fig. 23), and clamped. The scale is turned until it reads exactly zero, 0° or 360°, and clamped. The telescope is released and revolved in the direction of increasing readings of the scale to the position of minimum deviation, T. The reading of the scale gives at once the angle of minimum deviation, D. The holder carrying the stone is now clamped to the scale, and the telescope is turned to the position, T1,in which the image given by reflection from the table facet is in the centre of the field of view; the reading of the scale is taken. The telescope is clamped, and the scale is released and rotated until it reads the angle already found for D. If no mistake has been made, the reflected image from the second facet is now in the field of view. It will probably not be quite central, as theoretically it should be, because the stone may not have been originally quite in the position of minimum deviation, a comparatively large rotation of the stone producing no apparent change in the position of the refracted image at minimum deviation, and further, because, as has already been stated, the method is not strictly true for biaxial stones. The difference in readings, however, should not exceed 2°. The reading, S, of the scale is now taken, and it together with 180° subtracted from the reading for the first facet, and the value of A, the interior angle between the two facets, obtained.

Let us take an example.

Reading T (= D)	40°	41´	Reading T1	261°	35´
			less 180°	180	0
				———————
				81	35
			Reading S	41	30
				———————
½D	20	20½	A	40	5
½A	20	2½	½A	20	2½
	———————
½(A + D)	40	23
Log sin	40°	23´	9.81151
Log sin	20	2½	9.53492
		————
Log n			0.27659
	n = 1.8906.

The readings S and T are very nearly the same, and therefore we may be sure that no mistake has been made in the selection of the facets.

In place of logarithm-tables we may make use of the diagram on Plate II. The radial lines correspond to the angles of minimum deviation and the skew lines to the prism angles, and the distance along the radial lines gives the refractive index. We run our eye along the line for the observed angle of minimum deviation and note where it meets the curve for the observed prism angle; the refractive index corresponding to the point of intersection is at once read off.

This method has several obvious disadvantages: it requires the use of an expensive and elaborate instrument, an observation takes considerable time, and the values of the principal refractive indices cannot in general be immediately determined.

Table III at the end of the book gives the refractive indices of the gem-stones.

PLATE II

REFRACTIVE INDEX DIAGRAM