It has already been shown that crystals are optically divisible into two classes characterised respectively by single and by double refraction. Singly refractive crystals belong exclusively to the system of highest symmetry, the cubic. They afford obviously only one index of refraction, which is generally symbolised by the Greek letter μ, the value of this constant being the same for all directions throughout the crystal. Crystals of the other six systems of symmetry are all doubly refractive. Those of the trigonal, tetragonal, and hexagonal systems have been shown in the last chapter to possess two refractive indices, a maximum and a minimum, one represented by ε corresponding to light vibrating parallel to the singular axis of the system, the trigonal, tetragonal, or hexagonal axis of symmetry, and another signified by ω corresponding to light vibrations perpendicular to that axis. For the properties are identical in all directions around this axis, which is thus the optic axis as well as the predominating crystallographic one. Such crystals are consequently known as “uniaxial.” When ε is the larger refractive index the crystal is positive, while if ω be the maximum the crystal is said to be negative. It has been shown in the last chapter that quartz belongs to the positive category, while calcite is negative. Along the one direction of the optic axis these uniaxial crystals behave like singly refractive crystals do in all directions.

Crystals of the rhombic, monoclinic, and triclinic systems of symmetry have also a minimum refractive index, symbolised by α, and a maximum index indicated by γ, corresponding to light vibrating parallel to two directions at right angles to each other; the third direction perpendicular to both these and normal to their plane does not afford an index of refraction equal to either of these, however, as in the case of a uniaxial crystal, but one of an intermediate value, for which the second letter β of the Greek alphabet is reserved. Whether this value β is nearer to the minimum α or to the maximum γ determines the conventional optical sign of the crystal, whether positive or negative. In the case of the rhombic system the three rectangular directions in question are identical with the three rectangular crystallographic axes. In the monoclinic system the single symmetry axis normal to the unique plane of symmetry is identical in direction with either the α, β, or γ optical direction, but in the triclinic system there are no coincidences between the crystal axes and those of the optical ellipsoid. Along none of these axial directions of the optical ellipsoid which can be imagined to express graphically the refractive index—an ellipsoid known as the optical “indicatrix,” and which has been shown by Fletcher to be a more convenient mode of expressing the optical characters of a crystal than the vibration-velocity ellipsoid of Fresnel—do the optical properties resemble those of a uniaxial crystal along the optic axis, or of a cubic singly refractive crystal, the crystal being doubly refractive along all three axes.

But it is a remarkable fact, nevertheless, that there are two directions in such a crystal along which the latter is apparently singly refractive, and these two directions are known as the “optic axes,” and the crystals of the three systems of lower symmetry are consequently said to be “biaxial.” These two singular directions are symmetrical to two of the three rectangular axes of the ellipsoid, those corresponding to the extreme indices α and γ, in the plane containing which two axes they lie, and they are perpendicular to the third β. For if we draw the ellipse of which the minimum and maximum axes are represented in length by α and γ, there will obviously be four symmetrical positions on the curve where a line drawn to the centre of the ellipse would be equal to the intermediate value β. If we join opposite pairs of these four points by diameters (lines passing through the centre of the ellipse) we have two directions each of which, together with the perpendicular direction of the β axis, lies on a circular section of the ellipsoid, for all radii from the centre lying in each of these sections are alike equal to β. Consequently, light transmitted along the two directions in the crystal normal (perpendicular) to these two circular sections will suffer no apparent double refraction, the refractive index being the same, namely β, and the velocity of vibration equal in all directions in the crystal parallel to the two circular sections. Hence, we have two directions in biaxial crystals in which the optical properties are similar to those of uniaxial crystals along their singular optic axis. But the optical properties along the two optic axes of a biaxial crystal are advisedly stated to be “similar” to, and not “identical” with those along the optic axis of a uniaxial crystal; for although they are identical to all ordinary experimental tests, they are not quite so when we come to ultimate details, which, however, are beyond the purview of this book, but an account of which will be found in the author’s “Crystallography and Practical Crystal Measurement” (Macmillan & Co., 1911).

With these prefatory theoretical remarks, which are necessary in order that the experiments now to be described should be understood, we may proceed to consider a graduated series of experimental demonstrations which it is hoped will render clear some of the more important features of crystal structure which have been dealt with in previous chapters. Our principal agent will be polarised light, that is, light which has been reduced to vibration in a single plane by means of the well-known Nicol’s prism. This latter is a rhomb of calcite which has been cut in two parts along a specific diagonal direction, and the two parts of which have been re-cemented together with Canada balsam, in such a manner that one of the two rays, known as the “ordinary” and which corresponds to the ω refractive index, into which the doubly refracting calcite crystal divides the ordinary light which it receives from the lantern or other source of light, is totally reflected at the layer of balsam, while the other ray, known as the “extraordinary” and corresponding to a refractive index of intermediate value between ω and ε, and composed of vibrations at right angles to those of the totally reflected ray, is alone transmitted, as a ray of plane polarised light.

We employ a pair of such Nicol prisms (a very large pair being shown in Fig. 71), together with a convenient system of lenses for focussing either the object-crystal or the phenomena displayed by it, as a “polariscope,” which is the most powerful weapon of optical research on crystals which has ever been invented. When the two prisms are arranged so that the vibration planes of the polarised light which they would singly transmit are parallel, we speak of them as “parallel Nicols,” and light is transmitted unimpeded through the pair thus placed in succession; but when one of them is rotated the light diminishes, until when the vibration planes are at right angles no light escapes at all if the Nicols are properly constructed, there being produced what is known as the “dark field” of the “crossed Nicols.” For the plane polarised light reaching the analyser from the polariser cannot get through the former, its plane of possible light vibration being perpendicular to that of the already polarised beam.

The phenomena exhibited by crystals in polarised light are of two kinds, namely, those observed when a parallel (cylindrical) beam of fight is passed through the crystal, and those exhibited when a converging (conical) beam of fight is employed and concentrated on the crystal, the centre of which should occupy the apex of the cone. The disposition of apparatus in the former case of parallel light will be described in the next chapter and illustrated in Fig. 79. The arrangement for convergent light, as employed for projections on the screen, has already been referred to in connection with the Mitscherlich experiment with gypsum, and illustrated in Fig. 51 (page 92). The arrangement is shown again here for convenience, in Fig. 71. The parts of the apparatus are briefly as follows: (1) the electric lantern with self-adjusting Brockie-Pell or Oliver arc lamp and a 4½ or 5–inch set of condensers; (2) the water cell; (3) the polarising Nicol with a parallelising concave lens at its divided-circle end; (4) a condensing lens; (5) the convergent system of three lenses closely mounted in succession; (6) the crystal; (7) the collecting system of three lenses equal and similar to the convergent system; (8) the field lens; (9) the projection lens; and (10) the analysing Nicol. The ten parts are separately mounted in the author’s apparatus, which confers greater freedom in experimenting and more power of varying the conditions; the converging and collecting lens systems, however, are mounted in a separately adjustable manner on a common standard, which carries in the centre complete goniometrical adjustments for the crystal.

When we place on the stage of the polariscope, the Nicols being crossed, a plate of a uniaxial crystal cut perpendicularly to the optic axis, and subsequently a similar plate of a biaxial crystal cut perpendicularly to that axis of the optical ellipsoid, either α or γ, which is the bisectrix of the acute angle between the two optic axes, and use the system of lenses which converges the light rays received from the polarising Nicol prism on the crystal, as shown in Fig. 71, we observe in the two cases quite different and very beautiful interference phenomena, which at once distinguish a uniaxial from a biaxial crystal. The two appearances are illustrated in Plate XIV., by Figs. 72, 73, and 74, which are reproductions of the author’s direct photographs. Fig. 72 shows the interference figure afforded by uniaxial calcite, which is the same for all positions of the crystal plate when rotated in its own plane by the rotation of the stage. Figs. 73 and 74 represent the interference figures given by biaxial aragonite, the orthorhombic form of carbonate of lime, calcite and aragonite being the two forms of this substance, which has been shown in Chapter VII. to be dimorphous. The effect shown in Fig. 73 is afforded when the line joining the two optic axes is parallel to the plane of vibration of either of the crossed Nicols, and the interference figure represented in Fig. 74 is given when the stage and crystal (or the two Nicols simultaneously) are rotated 45°.

The uniaxial calcite figure (Fig. 72) consists of circular spectrum-coloured rings resembling the well-known Newton’s rings, but with a dark cross, fairly sharp near the centre but shading off towards the margin of the field, marking the directions of the vibration planes of the Nicols.

The biaxial aragonite figures (Figs. 73 and 74) show two series of rings surrounding the two optic axes and thus locating the positions of their emergence, equidistant from the centre of the field, where the bisectrix emerges. They are not circular, but are curves known as lemniscates, which are complete rings nearest to the two optic axes, but soon pass into figure-of-eight loops, and eventually into ellipse-like lemniscates enveloping both optic axes, and more and more approaching circles in their curvature as the margin of the field is approached. Moreover, when the direction of the fine joining the two optic axes is parallel to the vibration plane of either of the Nicols, as was the case when Fig. 73 was produced and photographed, a black rectangular cross is seen, one bar, which is much the sharper one, passing through the optic axes and the other lying between them at right angles to the first bar, the centre of the cross being in the middle of the field.

On rotating the crystal plate in its own plane, while no change occurs with the calcite, the aragonite figure changes as regards the black cross, which breaks up into hyperbolic curves currently spoken of as “brushes,” until when the plate has been rotated 45° the appearance is that shown in Fig. 74, the eye being supposed to have followed the rotation. Or, keeping the eye still, the effect shown in Fig. 74 is equally produced by the simultaneous rotation of both Nicols for 45°. The vertices of the hyperbolæ now mark the positions of the optic axes, and the angle between them is the apparent angle of the optic axes as seen in air, which is considerably different from the true angle between the optic axes within the crystal, owing to the very different refraction of light in air and in the crystal substance.

Now some crystals exhibit a very different optic axial angle at different temperatures, and one of the most beautiful experiments which have ever been performed is the Mitscherlich experiment with gypsum, which has already been described in Chapter VII. in connection with the work of Mitscherlich, and illustrated in Plate XII., Figs. 52 to 55. Other substances, on the other hand, show a marked change of optic axial angle as the wave-length of the light is changed, and such a case has already been described in Chapter VIII. and illustrated in Plate XIII., Fig. 58. The figure afforded by such a substance in ordinary white light is, however, a complicated one, quite different from the normal one of Fig. 73 afforded by aragonite, as will be clear on reference to the interference figure shown at f in Fig. 58, which represents the figure given by ethyl triphenyl pyrrholone in white light.

In order to understand such biaxial interference figures thoroughly, they should be studied in monochromatic light, when one obtains a clear and sharp figure consisting of black curves as well as the cross or brushes, and very sharp vertices to the brushes when the crystal is arranged as in Fig. 74. The optic axial angle can then be measured for each important wave-length of light in turn, and the variation for wave-length followed throughout the whole spectrum. For this purpose it is very convenient to have a source of monochromatic light of any or every wave-length always at hand, and the author some years ago devised a spectroscopic monochromatic illuminator,^[17] for use with any observing instrument, and which is particularly convenient for use with the polariscopical goniometer which is employed in practice for the measurement of optic axial angles. It is shown, along with the latter instrument, in Fig. 75. The spectroscope has a single but very large prism of heavy but colourless flint glass, and the spectrum produced—the electric lantern being the source of light, its rays being concentrated on the slit—is filtered through a second slit at the other end of the spectroscope, where the detachable eyepiece is situated when the instrument is used as an ordinary spectroscope, and for the calibration (with the Fraunhofer solar lines) of the circle on which the prism is mounted. The escaping narrow slit of monochromatic light includes only the 250th part of the spectrum, so is monochromatic in a high sense of the word. It impinges on a little ground glass diffuser carried in a very short tube in front of this exit slit, and the optic axial angle polariscope is brought up almost into contact with the ground glass, and is thus supplied with an even field of pure monochromatic light. With this apparatus it is easy to observe the exact crossing wave-length in all cases of crossed-axial-plane dispersion such as that illustrated in Fig. 58; for the reading of the graduated circle on which the prism is mounted, and which is rotated in order to cause monochromatic light of the different wave-lengths in turn to stream through the exit slit, affords the exact wave-length with the aid of the calibration curve once for all prepared. This calibration of the graduations is readily carried out by using sunlight, and determining the readings corresponding to the adjustment of the principal Fraunhofer lines in the middle of the exit slit.

Having thus rendered clear the nature of ordinary interference figures afforded by crystals of the two types, uniaxial and biaxial, in convergent polarised light, we may pass on to see what happens when we take a number of plates of quartz of different thicknesses, cut perpendicularly to the optic axis in all cases, instead of a plate of calcite. We will examine first a fine pair of hexagonal quartz plates so cut, each 1 millimetre thick exactly, and about 2 inches in diameter. One was cut from a right-handed hexagonal prism, and the other from a similar left-handed one.

Employing the lantern projection polariscope shown in Fig. 71, arranged for convergent light just as for the Mitscherlich experiment, and with the Nicols crossed, we will now see what happens when each of these plates in turn is placed at the focus of the light rays, between the two convergent systems of lenses. On the screen we observe in each case a somewhat similar interference figure to that given by calcite, a black cross and rainbow coloured circular rings, the smallest ring, however, being very large relatively to the innermost ring given by calcite, and the other rings being also further separated from each other. Moreover, the black cross appears broadened out, this spreading of both rings and cross being due to the thinness of the plate combined with the low double refraction of quartz. Further, the right-handed and left-handed plates both afford apparently identical figures. In order to obtain a sharp figure like that of calcite we require to add a fourth lens, kept in reserve for such cases, to each of the two similar convergent lens systems, one on each side of the crystal plate, in order to increase the convergence of the light rays. The figure then obtained with one of the two plates is reproduced in Fig. 76, Plate XV.

Let us now observe, however, what occurs when a thicker plate of quartz is used. Taking one of 7.5 mm. thickness, and placing it in the focus of the converging rays, after removing the two extra lenses, we see on the screen quite a different effect, an attempt to reproduce which photographically in black and white is made in Fig. 77 on the same Plate XV. The rings are closer together (using the same degree of convergence), and the innermost is smaller; moreover, within it all signs of the central part of the black cross have disappeared, and instead a brilliant violet colour is shown, which alters to bright red of the first order spectrum with the least rotation of the analysing Nicol in one direction from its crossed position with respect to the polarising Nicol, while if the rotation be in the opposite direction the deep blue of Newton’s second order is produced. The arms of the cross, however, appear towards the margin of the field. The violet colour shown for the exact position of crossing of the Nicols is the tint of passage between the first and second orders of Newton’s spectra, and this illumination of the central part of the interference figure is obviously the effect of the optical activity of quartz, for the tint is the same as is produced with the plate in ordinary parallel plane polarised fight, and is, in fact, due to the central axial rays of the convergent cone being practically parallel.

On rotating the analysing Nicol for a few more degrees to the right we observe that the innermost ring widens out and that the red passes into orange and yellow, the quartz plate being a right-handed one. But when a similar plate cut from a left-handed quartz crystal is used instead, the inner ring closes up somewhat for the same rotation of the analyser, moving inwards instead of outwards, and the blue colour given with the first slight rotation passes into green and yellow as the rotation is continued. Moreover, the circular character of the rings is altered, and so much so that when the rotation has proceeded as far as 45° the shape of the rings has changed almost to a square. These alterations in the interference figure are characteristic of the two varieties of quartz crystals. A useful rule to remember is, that for a right-handed crystal rotation of the analyser to the right causes the colours to appear in the order of their refrangibility, namely, the least refrangible red first, then orange, yellow, green, blue and violet in their order; while for a left-handed crystal the converse is true when the direction of rotation of the analyser is the same, that is, to the right, clockwise; obviously also the colours appear in the opposite order when the rotation of the analyser is to the left.

It will now prove of interest to examine the effects produced by two plates of opposite varieties of quartz of half this thickness, namely, 3.75 mm. The phenomena are very similar to those just described, but the rings are a little wider, and the larger area within the innermost ring is now filled with yellow light instead of violet, when the analyser is exactly crossed to the polariser. It passes into a bright green when the analyser is rotated slightly on one side, and into orange when the Nicol is rotated in the reverse direction. But the most interesting thing of all is to observe what occurs when these two plates of 3.75 mm. thickness, one of right-handed quartz and the other of left-handed, are superposed and placed in contact together as one plate, of double the thickness, 7.5 mm., at the convergent focus. A beautiful spiral figure is produced on the screen, composed of the celebrated “Airy’s spirals” as if the black cross were being reproduced in the central part, but with each of its bars distorted into the shape of the letter S, as shown in Fig. 78 at the foot of Plate XV. The contrary effects of the two opposing rotations are thus extraordinarily indicated visually in the interference figure afforded by the composite plate.

Now, it is of great practical interest that certain quartz crystals are found in nature which show Airy’s spirals directly, on cutting a plate 7.5 mm. thick or thereabouts, perpendicular to the optic axis. For instance, one in the author’s collection of quartzes, a single plate of an apparently homogeneous and perfectly limpid crystal, shows the spirals exceedingly well and clearly defined. As a matter of fact, it is a twin, a right and a left-handed crystal being twinned together with an invisible plane of composition, which is only revealed on examining the crystal in polarised light, as will be demonstrated in the next chapter by the use of parallel polarised light. The fact of such a plate of quartz affording Airy’s spirals in convergent polarised light is, however, of itself an excellent proof of the twinning of two crystal individuals of the opposite varieties.

Now the very shape of these spiral figures suggests screw action of the molecular structure of the crystals on the waves of light passing through them, and moreover, of the action of two screws of opposite directions of winding, one clockwise and the other anti-clockwise, thus remarkably confirming the supposition that the point-systems of the structure of the right and left-handed varieties of quartz are of a helical nature and respectively of opposite modes of winding.

Another experiment, devised by Reusch, which still further enhances the probability that this supposition as to the structure of quartz crystals is correct, may next be introduced. A thin film of biaxial mica has been cut into twenty-four narrow strips, which have been laid over each other at angles of 60°, so that a screw-shaped pile has been formed of the central overlapping parts, consisting of four complete rotations; that is, there are four repetitions of the “pitch” of the screw, each composed of six films. On placing this composite plate of mica at the convergent focus of the lantern polariscope, so that the overhanging ends of any four identically superposed strips occupy the focus, the ordinary biaxial interference figure of mica—two sets of rings and hyperbolic brushes, very much like Fig. 52, Plate XII.—is observed on the screen. But when the plate is moved so that the central part comes into the focus, where all the twenty-four films overlap in their six different orientations 60° apart, and so that all the light rays have to traverse the whole helical pile of the twenty-four films, a uniaxial figure exactly like that of quartz is produced, namely, one composed of circular rings, with a black cross only visible, however, at the marginal part, and with the inner ring filled with brightly coloured light. Moreover, on slightly rotating the analysing Nicol the innermost ring moves outwards or inwards and the colour changes to blue or red, according to the direction in which the helix had been wound, in exact accordance with the rule stated above for quartz.

If now a second such helical pile of mica films, but one for which the opposite manner of winding has been adopted, anti-clockwise if the first had been clockwise, be examined at the convergent focus, precisely the same appearance will be observed with crossed Nicols, but the opposite changes will occur on rotating the analyser. Finally, to complete the interesting proof of the helical nature of quartz crystals, when these two oppositely wound composite mica plates are superposed—each being marked carefully to indicate the direction of the helix and the proper mode of superposition in order to effect precise oppositeness of arrangement, mirror-image symmetry, in fact, about the plane of contact—and placed in the convergent beam near its focus, there is at once seen on the screen a magnificent display of Airy’s spirals, as perfect as those afforded by the fine natural twin last experimented with. Hence, there can be no doubt whatever that the remarkable optical behaviour of quartz is due to its point-system being of a helical nature, a right or a left-handed screw structure being apparently produced in nature with equal facility. The circumstances of environment during the formation of the crystal probably determine which variety shall be produced, and when the nature of the environment becomes changed during the operation of formation either twins are produced of the two varieties, or separate individual crystals.

This may well conclude our experiments in convergent polarised light, which—including the beautiful Mitscherlich experiment described in Chapter VII., of exhibiting the crossing of the optic axial plane in the case of gypsum, and the production of all the types of interference figures in succession, as the crystal becomes warmed by the heat rays accompanying the beam of convergent light—will have introduced the reader to a typical series of such experiments, and such as were actually exhibited by the author to the British Association at Winnipeg. We may pass, therefore, in the next chapter to the consideration of an equally interesting series in which a parallel beam of polarised light will be used, which will still further elucidate the internal structure in the especially instructive case of quartz crystals, and that of crystals in general.