Gem-Stones and Their Distinctive Characters

CHAPTER IV

MEASUREMENT OF REFRACTIVE INDICES

THE methods available for the measurement of refractive indices are of two kinds, and make use of two different principles. The first, which is based upon the very simple relation found in the last chapter to subsist at total-reflection, can be used with ease and celerity, and is best suited for discriminative purposes; but it is restricted in its application. The second, which depends on the measurement of the angle between two facets and the minimum deviation experienced by a ray of light when traversing a prism formed by them, is more involved, entails the use of more elaborate apparatus, and takes considerable time, but it is less restricted in its application.

(1) The Method of Total-Reflection

We see from equation b (p. 18), connecting the angle of total-reflection with the refractive indices of the adjacent media, that, if the denser medium be constant, the indices of all less dense media may be easily determined from a measurement of the corresponding critical angle. In all refractometers the constant medium is a glass with a high refractive index. Some instruments have rotatory parts, by means of which this angle is actually measured. Such instruments give very good results, but suffer from the disadvantages of being neither portable nor convenient to handle, and of not giving a result without some computation.

Fig. 16.—Refractometer (actual size).

For use in the identification of cut stones, a refractometer with a fixed scale, such as that (Fig. 16) devised by the author, is far more convenient. In order to facilitate the observations, a totally reflecting prism has been inserted between the two lenses of the eyepiece. The eyepiece may be adjusted to suit the individual eyesight; but for observers with exceptionally long sight an adapter is provided, which permits the eyepiece being drawn out to the requisite extent. The refractometer must be held in the manner illustrated in Fig. 17, so that the light from a window or other source of illumination enters the instrument by the lenticular opening underneath. Good, even illumination of the field may also very simply be secured by reflecting light into the instrument from a sheet of white paper laid on a table. On looking down the eyepiece we see a scale (Fig. 18), the eyepiece being, if necessary, focused until the divisions of the scale are clearly and distinctly seen. Suppose, for experiment, we smear a little vaseline or similar fatty substance on the plane surface of the dense glass, which just projects beyond the level of the brass plate embracing it. The field of view is now no longer uniformly illuminated, but is divided into two parts (Fig. 19): a dark portion above, which terminates in a curved edge, apparently green in colour, and a bright portion underneath, which is composed of totally reflected light. If we tilt the instrument downwards so that light enters the instrument from above through the vaseline we find that the portions of the field are reversed, the dark portion being underneath and terminated by a red edge. It is possible so to arrange the illumination that the two portions are evenly lighted, and the common edge becomes almost invisible. It is therefore essential for obtaining satisfactory results that the plate and the dense glass be shielded from the light by the disengaged hand. The shadow-edge is curved, and is, indeed, an arc of a circle, because spherical surfaces are used in the optical arrangements of the refractometer; by the substitution of cylindrical surfaces it becomes straight, but sufficient advantage is not secured thereby to compensate for the greatly increased complexity of the construction. The shadow-edge is coloured, because the relative dispersion, ^{n_v - n_r} /_n (n_v and n_r being the refractive indices for the extreme violet and red rays respectively), of the vaseline differs from that of the dense glass. The dispersion of the glass is very high, and exceeds that of any stone for which it can be used. Certain oils have, however, nearly the same relative dispersion, and the edges corresponding to them are consequently almost colourless. A careful eye will perceive that the coloured shadow-edge is in reality a spectrum, of which the violet end lies in the dark portion of the field and the red edge merges into the bright portion. The yellow colour of a sodium flame, which, as has already been stated, is selected as the standard for the measurement of refractive indices, lies between the green and the red, and the part of the spectrum to be noted is at the bottom of the green, and practically, therefore, at the bottom of the shadow, because the yellow and red are almost lost in the brightness of the lower portion of the field. If a sodium flame be used as the source of illumination, the shadow-edge becomes a sharply defined line. The scale is so graduated and arranged that the reading where this line crosses the scale gives the corresponding refractive index, the reading, since the line is curved, being taken in the middle of the field on the right-hand side of the scale. The refractometer therefore gives at once, without any intermediate calculation, a value of the refractive index to the second place of decimals, and a skilled observer may, by estimating the tenths of the intervals between successive divisions, arrive at the third place; to facilitate this estimation the semi-divisions beyond 1·650 have been inserted. The range extends nearly to 1·800; for any substance with a higher refractive index the field is dark as far as the limit at the bottom.

Fig. 17.—Method of Using the Refractometer.

Fig. 18.—Scale of the Refractometer.

Fig. 19.—Shadow-edge given by a singly refractive Substance.

A fat, or a liquid, wets the glass, i.e. comes into intimate contact with it, but if a solid substance be tested in the same way, a film of air would intervene and entirely prevent an observation. To displace it, a drop of some liquid which is more highly refractive than the substance under test must first be applied to the plane surface of the dense glass. The most convenient liquid for the purpose is methylene iodide, CH₂I₂, which, when pure, has at ordinary room temperatures a refractive index of 1·742. It is almost colourless when fresh, but turns reddish brown on exposure to light. If desired, it may be cleared in the manner described below (p. 66), but the film of liquid actually used is so thin that this precaution is scarcely necessary. If we test a piece of ordinary glass—one of the slips used by microscopists for covering thin sections is very convenient for the purpose—first applying a drop of methylene iodide to the plane surface of the dense glass of the refractometer (Fig. 20), we notice a coloured shadow-edge corresponding to the glass-slip at about 1·530 and another, almost colourless, at 1·742, which corresponds to the liquid. If the solid substance which is tested is more highly refractive than methylene iodide, only the latter of the shadow-edges is visible, and we must utilize some more refractive liquid. We can, however, raise the refractive index of methylene iodide by dissolving sulphur[2] in it; the refractive index of the saturated liquid lies well beyond 1·800 and the shadow-edge corresponding to it, therefore, does not come within the range of the refractometer. The pure and the saturated liquids can be procured with the instrument, the bottles containing them being japanned on the outside to exclude light and fitted with dipping-stoppers, by means of which a drop of the liquid required is easily transferred to the surface of the glass of the instrument. So long as the liquid is more highly refractive than the stone, or whatever may be the substance under examination, its precise refractive index is of no consequence. The facet used in the test must be flat, and must be pressed firmly on the instrument, so that it is truly parallel to the plane surface of the dense glass; for good results, moreover, it must be bright.

Fig. 20.—Faceted Stone in Position on the Refractometer.

Fig. 21.—Shadow-edges given by a doubly refractive substance.

We have so far assumed that the substance which we are testing is simple and gives a single shadow-edge; but, as may be seen from Table V, many of the gem-stones are doubly refractive, and such will, in general, show in the field of the refractometer two distinct shadow-edges more or less widely separated. Suppose, for example, we study the effect produced by a peridot, which displays the phenomenon to a marked degree. If we revolve the stone so that the facet under observation remains parallel to the plane surface of the dense glass of the refractometer and in contact with it, we notice that both the shadow-edges in general move up or down the scale. In particular cases, depending upon the relation of the position of the facet selected to the crystalline symmetry, one or both of them may remain fixed, or one may even move across the other. But whatever facet of the stone be used for the test, and however variable be the movements of the shadow-edges, the highest and lowest readings obtainable remain the same; they are the principal indices of refraction, such as are stated in Table III at the end of the book, and their difference measures the maximum amount of double refraction possessed by the stone. The procedure is therefore simplicity itself; we have merely to revolve the stone on the instrument, usually through not more than a right angle, and note the greatest and least readings. It will be noticed that the shadow-edges cross the scale symmetrically in the critical and skewwise in intermediate positions. Fig. 21 represents the effect when the facet is such as to give simultaneously the two readings required. The shadow-edges a and b, which are coloured in white light, correspond to the least and greatest respectively of the principal refractive indices, while the third shadow-edge, which is very faint, corresponds to the liquid used—methylene iodide. It is possible, as we shall see in a later chapter, to learn from the motion, if any, of the shadow-edges something as to the character of the double refraction. Since, however, each shadow-edge is spectral in white light, they will not be distinctly separate unless the double refraction exceeds the relative dispersion. Topaz, for instance, appears in white light to yield only a single shadow-edge, and may thus easily be distinguished from tourmaline, in which the double refraction is large enough for the separation of the two shadow-edges to be clearly discerned. In sodium light, however, no difficulty is experienced in distinguishing both the shadow-edges given by substances with small amount of double refraction, such as chrysoberyl, quartz, and topaz, and a skilled observer may detect the separation in the extreme instances of apatite, idocrase, and beryl. The shadow-edge corresponding to the greater refractive index is always less distinct, because it lies in the bright portion of the field. If the stone or its facet be small, it must be moved on the plane surface of the dense glass until the greatest possible distinctness is imparted to the edge or edges. If it be moved towards the observer from the further end, a misty shadow appears to move down the scale until the correct position is reached, when the edges spring into view.

Any facet of a stone may be utilized so long as it is flat, but the table-facet is the most convenient, because it is usually the largest, and it is available even when the stone is mounted. That the stone need not be removed from its setting is one of the great advantages of this method. The smaller the stone the more difficult it is to manipulate; caution especially must be exercised that it be not tilted, not only because the shadow-edge would be shifted from its true position and an erroneous value of the refractive index obtained, but also because a corner or edge of the stone would inevitably scratch the glass of the instrument, which is far softer than the hard gem-stones. Methylene iodide will in time attack and stain the glass, and must therefore be wiped off the instrument immediately after use.

(2) The Method of Minimum Deviation

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

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