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DOUBLE REFRACTION

THE optical phenomenon presented by many gem-stones is complicated by their property of splitting up a beam of light into two with, in general, differing characters. In this chapter we shall discuss the nature of double refraction, as it is termed, and methods for its detection. The phenomenon is not one that comes within the purview of everyday experience.

So long ago as 1669 a Danish physician, by name Bartholinus, noticed that a plate of the transparent mineral which at that time had recently been brought over from Iceland, and was therefore called “Iceland-spar,” possessed the remarkable property of giving a double image of objects close to it when viewed through it. Subsequent investigation has shown that much crystallized matter is doubly refractive, but in calcite—to use the scientific name for the species which includes Iceland-spar—alone among common minerals is the phenomenon so conspicuous as to be obvious to the unaided eye. The apparent separation of the pair of images given by a plate cut or cleaved in any direction depends upon its thickness. The large mass, upwards of two feet (60 cm.) in thickness, which is exhibited at the far end of the Mineral Gallery of the British Museum (Natural History), displays the separation to a degree that is probably unique.

Fig. 24.—Apparent doubling of the Edges of a Peridot when viewed through the Table-Facet.

Although none of the gem-stones can emulate calcite in this character, yet the double refraction of certain of them is large enough to be detected without much difficulty. In the case of faceted stones the opposite edges should be viewed through the table-facet, and any signs of doubling noted. The double refraction of sphene is so large, viz. 0·08, that the doubling of the edges is evident to the unaided eye. In peridot (Fig. 24), zircon (b), and epidote the apparent separation of the edges is easily discerned with the assistance of an ordinary lens. A keen eye can detect the phenomenon even in the case of such substances as quartz with small double refraction. It must, however, be remembered that in all such stones the refraction is single in certain directions, and the amount of double refraction varies therefore with the direction from nil to the maximum possessed by the stone. Experiment with a plate of Iceland-spar shows that the rays transmitted by it have properties differing from those of ordinary light. On superposing a second plate we notice that there are now two pairs of images, which are in general no longer of equal brightness, as was the case before. If the second plate be rotated with respect to the first, two images, one of each pair, disappear, and then the other two, the plate having turned through a right angle between the two positions of extinction; midway between these positions the images are all equally bright. This variation of intensity implies that each of the rays emerging from the first plate has acquired a one-sided character, or, as it is usually expressed, has become plane-polarized, or, shortly, polarized.

Fig. 25.—Wave-Motion.

Before the discovery of the phenomenon of double refraction the foundation of the modern theory of light had been laid by the genius of Huygens. According to this theory light is the result of a wave-motion (Fig. 25) in the ether, a medium that pervades the whole of space whether occupied by matter or not, and transmits the wave-motion at a rate varying with the matter with which it happens to coincide. Such a medium has been assumed because it explains satisfactorily all the phenomena of light, but it by no means follows that it has a concrete existence. Indeed, if it has, it is so tenuous as to be imperceptible to the most delicate experiments. The wave-motion is similar to that observed on the surface of still water when disturbed by a stone flung into it. The waves spread out from the source of disturbance; but, although the waves seem to advance, the actual particles of water merely move up and down, and have no motion at all in the direction in which the waves are moving. If we imagine similar motion to take place in any plane and not only the horizontal, we form some idea of the nature of ordinary light. But after passing through a plate of Iceland-spar, light no longer vibrates in all directions, but in each beam the vibrations are parallel to a particular plane, the two planes being at right angles. The exact relation of the direction of the vibrations to the plane of polarization is uncertain, although it undoubtedly lies in the plane containing the direction of the ray of light and the perpendicular to the plane of polarization. The waves for different colours differ in their length, i.e. in the distance, 2 bb (Fig. 25), from crest to crest, while the velocity, which remains the same for the same medium, is proportional to the wave-length. The intensity of the light varies as the square of the amplitude of the wave, i.e. the height, ab, of the crest from the mean level.

Various methods have been proposed for obtaining polarized light. Thus Seebeck found in 1813 that a plate of brown tourmaline cut parallel to the crystallographic axis and of sufficient thickness (cf. p. 11) transmits only one ray, the other being entirely absorbed within the plate. Another method was to employ a glass plate to reflect light at a certain critical angle. The most efficient method, and that in general use at the present day, is due to the invention of Nicol. A rhomb of Iceland-spar (Fig. 26), of suitable length, is sliced along the longer diagonal, dd, and the halves are cemented together by means of canada balsam. One ray, ioo, is totally reflected at the surface separating the mineral and the cement, and does not penetrate into the other half; while the other ray, iee, is transmitted with almost undiminished intensity. Such a rhomb is called a Nicol’s prism after its inventor, or briefly, a nicol.

Fig. 26.—Nicol’s Prism.

If one nicol be placed above another and their corresponding principal planes be at right angles no light is transmitted through the pair. In the polarizing microscope one such nicol, called the polarizer, is placed below the stage, and the other, called the analyser, is either inserted in the body of the microscope or placed above the eyepiece, and the pair are usually set in the crossed position so that the field of the microscope is dark. If a piece of glass or a fragment of some singly refractive substance be placed on the stage the field still remains dark; but in case of a doubly refractive stone the field is no longer dark except in certain positions of the stone. On rotation of the plate, or, if possible, of the nicols together, the field passes from darkness to maximum brightness four times in a complete revolution, the relative angular intervals between these positions being right angles. These positions of darkness are known as the positions of extinction, and the plate is said to extinguish in them. This test is exceedingly delicate and reveals the double refraction even when the greatest difference in the refractive indices is too small to be measured directly.

Doubly refractive substances are of two kinds: uniaxial, in which there is one direction of single refraction, and biaxial, in which there are two such directions. In the case of the former the direction of one, the ordinary ray, is precisely the same as if the refraction were single, but the refractive index of the other ray varies from that of the ordinary ray to a second limiting value, the extraordinary refractive index, which may be either greater or less. If the extraordinary is greater than the ordinary refractive index the double refraction is said to be positive; if less, to be negative. A biaxial substance is more complex. It possesses three principal directions, viz., the bisectrices of the directions of single refraction and the perpendicular to the plane containing them. The first two correspond to the greatest and least, and the last to the mean of the principal indices of refraction. If the acute bisectrix corresponds to the least refractive index, the double refraction is said to be positive, and if to the greatest, negative. The relation of the directions of single refraction, s, to the three principal directions, a, b, c, is illustrated in Fig. 27 for the case of topaz, a positive mineral. The refractive indices of the rays traversing one of the principal directions have the values corresponding to the other two. In the direction a we should measure the greatest and the mean of the principal refractive indices, in the direction b the greatest and the least, and in the direction c the mean and the least. The maximum amount of double refraction is therefore in the direction b.

Fig. 27.—Relation of the two

Directions of single Refraction to

the principal Optical Directions

in a Biaxial Crystal.

In the examination of a faceted stone, of the most usual shape, the simplest method is to lay the large facet, called the table, on a glass slip and view the stone through the small parallel facet, the culet. Should the latter not exist, it may frequently happen that owing to internal reflection no light emerges through the steeply inclined facets. This difficulty is easily overcome by immersing the stone in some highly refracting oil. A glass plate held by hand over the stone with a drop of the oil between it and the plate serves the purpose, and is perhaps a more convenient method. A stone which does not possess a pair of parallel facets should be viewed through any pair which are nearly parallel.

We have stated that a plate of glass has no effect on the field. Suppose, however, it were viewed when placed between the jaws of a tightened vice and thus thrown into a state of strain, it would then show double refraction, the amount of which would depend on the strain. Natural singly refractive substances frequently show phenomena of a similar kind. Thus diamond sometimes contains a drop of liquid carbonic acid, and the strain is revealed by the coloured rings surrounding the cavity which are seen when the stone is viewed between crossed nicols. Double refraction is also common in diamond even when there is no included matter to explain it, and is caused by the state of strain into which the mineral is thrown on release from the enormous pressure under which it was formed. Other minerals which display these so-called optical anomalies, such as fluor and garnet, are not really quite singly refractive at ordinary temperatures; each crystal is composed of several double refractive individuals. But all such phenomena cannot be confused with the characters of minerals which extinguish in the ordinary way, since the stone will extinguish in small patches and these will not be dark all at the same time; further, the double refraction is small, and on revolving the stone between crossed nicols the extinction is not sharp. Paste stones are sometimes in a state of strain, and display slight, but general, double refraction. Hence the existence of double refraction does not necessarily prove that the stone is real and not an imitation. Stones may be composed of two or more individuals which are related to each other by twinning, in which case each individual would in general extinguish separately. Such individuals would be larger and would extinguish more sharply than the patches of an anomalous stone.

Fig. 28.—Interference of Light.

An examination in convergent light is sometimes of service. An auxiliary lens is placed over the condenser so as to converge the light on to the stone. Light now traverses the stone in different directions; the more oblique the direction the greater the distance traversed in the stone. If it be doubly refractive, in any given direction there will be in general two rays with differing refractive indices and the resulting effect is akin to the well-known phenomenon of Newton’s rings, and is an instance of what is termed interference. It may be mentioned that the interference of light (Fig. 28) explains such common phenomena as the colours of a soap-bubble, the hues of tarnished steel, the tints of a layer of oil floating on water, and so on. Light, after diverging from the stone, comes to focus a little beneath the plane in which the image of the stone is formed. An auxiliary lens must, therefore, be inserted to bring the focal planes together, so that the interference picture may be viewed by means of the same eyepiece.

If a uniaxial crystal be examined along the crystallographic axis in convergent light an interference picture will be seen of the kind illustrated on Plate III. The arms of a black cross meet in the centre of the field, which is surrounded by a series of circular rings, coloured in white light. Rotation of the stone about the axis produces no change in the picture.

PLATE III

1. UNIAXIAL

2. UNIAXIAL

(Circular Polarization)

3. BIAXIAL

(Crossed Brushes)

4. BIAXIAL

(Hyperbolic Brushes)

INTERFERENCE FIGURES

A biaxial substance possesses two directions (the optic axes) along which a single beam is transmitted. If such a stone be examined along the line bisecting the acute angle between the optic axes (the acute bisectrix) an interference picture[4] will be seen which in particular positions of the stone with respect to the crossed nicols takes the forms illustrated on Plate III. As before, there is a series of rings which are coloured in white light; they, however, are no longer circles but consist of curves known as lemniscates, of which the figure of 8 is a special form. Instead of an unchangeable cross there are a pair of black “brushes” which in one position of the stone are hyperbolæ, and in that at right angles become a cross. On rotating the stone we find that the rings move with it and are unaltered in form, whereas the brushes revolve about two points, called the “eyes,” where the optic axes emerge. If the observation were made along the obtuse bisectrix the angle between the optic axes would probably be too large for the brushes to come into the field, and the rings might not be visible in white light, though they would appear in monochromatic light. In the case of a substance like sphene the figure is not so simple, because the positions of the optic axes vary greatly for the different colours and the result is exceedingly complex; in monochromatic light, however, the usual figure is visible.

It would probably not be possible in the case of a faceted stone to find a pair of faces perpendicular to the required direction. Nevertheless, so long as a portion of the figures described is in the field of view, the character of the double refraction, whether uniaxial or biaxial, may readily be determined.

There is yet another remarkable phenomenon which must not be passed over. Certain substances, of which quartz is a conspicuous example and in this respect unique among the gem-stones, possess the remarkable property of rotating the plane of polarization of a ray of light which is transmitted parallel to the optic axis. If a plate of quartz be cut at right angles to the axis and placed between crossed nicols in white light, the field will be coloured, the hue changing on rotation of one nicol with respect to the other. Examination in monochromatic light shows that the field will become dark after a certain rotation of the one nicol with respect to the other, the amount of which depends on the thickness of the plate. If the plate be viewed in convergent light, an interference picture is seen as illustrated on Plate III, which is similar to, and yet differs in some important particulars from the ordinary interference picture of a uniaxial stone. The cross does not penetrate beyond the innermost ring and the centre of the field is coloured in white light. If a stone shows such a picture, it may be safely assumed to be quartz. It is interesting to note that minerals which possess this property have a spiral arrangement of the constituent atoms.

It has already been remarked (p. 28) that if a faceted doubly refractive stone be rotated with one facet always in contact with the dense glass of the refractometer the pair of shadow-edges that are visible in the field move up or down the scale in general from or to maximum and minimum positions. The manner in which this movement takes place depends upon the character of the double refraction and the position of the facet under observation with regard to the optical symmetry of the stone. In the case of a uniaxial stone, if the facet be perpendicular to the crystallographic axis, i.e. the direction of single refraction, neither of the shadow-edges will move. If the facet be parallel to that direction, one shadow-edge will move up and coincide with the other, which remains invariable in position, and away from it to a second critical position; the latter gives the value of the extraordinary refractive index, and the invariable shadow-edge corresponds to the ordinary refractive index. This phenomenon is displayed by the table-facet of most tourmalines, because for reasons given above (p. 11) they are as a rule cut parallel to the crystallographic axis. In the case of facets in intermediate positions, the shadow-edge corresponding to the extraordinary refractive index moves, but not to coincidence with the invariable shadow-edge. The case of a biaxial stone is more complex. If the facet be perpendicular to one of the principal directions one shadow-edge remains invariable in position, corresponding to one of the principal refractive indices, whilst the other moves between the critical values corresponding to the remaining two of the principal refractive indices. In the interesting case in which the facet is parallel to the two directions of single refraction, the second shadow-edge moves across the one which is invariable in position. In intermediate positions of the facet both shadow-edges move, and give therefore critical values. Of the intermediate pair, i.e. the lower maximum and the higher minimum, one corresponds to the mean principal refractive index, and the other depends upon the relation of the facet to the optical symmetry. If it is desired to distinguish between them, observations must be made on a second facet; but for discriminative purposes such exactitude is unnecessary, since the least and the greatest refractive indices are all that are required.

The character of the refraction of gem-stones is given in Table V at the end of the book.