Fig. 26.
Fig. 27.
If the optical character is positive the line joining these dark spots is perpendicular to the direction c of the mica plate, see Fig. 26.
If negative the line joining the dark spots coincides with the direction c of the mica plate, see Fig. 27.
The (+) and (−) character is easily determined by remembering that the line, joining the dark spots, makes the + and − sign respectively with the direction c of the mica plate. The direction c of the mica plate (represented in the figures by an arrow) is of course not seen, but its position must be borne in mind when making this test. This test can be made with either monochromatic or white light.
If the mica plate does not give satisfactory results, which will be the case when the double refraction of the crystal to be tested is very weak or when the section is very thin, use a selenite plate, cut the proper thickness to give the red color of the first order.
This plate must be introduced with its vibration direction a (previously determined) making an angle of 45° with the planes of vibration of the nicols. Instead of the dark spots being seen there will appear two blue and two red quadrants. The diagonally opposite quadrants being of the same color.
In determining the (+) and (−) character consider the blue quadrants as the equivalent of the dark spots in the preceding case. This test must be made with white light.[72]
Biaxial Interference Figures.
Fig. 28.
Fig. 29.
(a) Sections perpendicular to an optic axis exhibit the interference figures shown in Figs. 28 and 29, the curves being nearly circular and a straight black bar bisecting these curves, whenever the trace of the plane of the optic axes coincides with the vibration direction of either nicol. As the stage, carrying the section, is rotated the bar changes into one arm of a hyperbola and back again into a bar. This arm or bar will rotate in the opposite direction to the motion of the stage.
As previously stated sections of biaxial crystals, perpendicular to an optic axis, do not remain dark during rotation of the stage between crossed nicols in parallel light. On the contrary these sections remain uniformly illuminated.[73]
(b) Sections perpendicular to the acute bisectrix (see p. 5), exhibit interference figures like those shown in Figs. 30 and 31.
Fig. 30.
Fig. 31.
Fig. 30 shows the appearance of the interference figure when the plane of the optic axes is parallel to the plane of vibration of either nicol, and Fig. 31 shows the appearance when this plane is inclined 45° to the planes of vibration of the nicols.
As the stage, carrying the section, is rotated the dark cross seems to dissolve into two branches of a hyperbola, which again unite to form a cross.
In sections perpendicular to a bisectrix, with a large axial angle, the figure will appear, during a rotation of 90° (in the direction of the hands of a watch), as in Fig. 32, top row. When the section is somewhat oblique to an “optic axis,” the figure appears as in middle row; and when still more oblique, as in bottom row.
The black centres[74] of the small ellipses and the black hyperbolic curves mark the points of emergence of the optic axes, and therefore indicate approximately the size of the axial angle, 2E.
Sections in other positions, relative to the optic axes, give interference figures less definite in appearance than those just described; and the same conditions affect the appearance of all figures as in the case of uniaxial crystals. Very thin sections, of weak double refraction, may only show indistinct dark crosses or hyperbolic curves, without any ellipses.
The section perpendicular to the acute bisectrix, which gives the most characteristic interference figure, cannot generally be recognized except by an examination in convergent light. It is never, however, the section giving the maximum interference color. The interference colors in different sections, normal to the optical elements, grade downward in the following order: (1) optic normal, (2) obtuse bisectrix, (3) acute bisectrix, (4) optic axis. Sometimes cleavage may furnish a clue as to the best section to test as in Topaz and Mica, where the acute bisectrix is normal to the cleavage.
Fig. 32.—Biaxial Interference Figures (from Reinisch). Top row: Almost perpendicular to bisectrix, large axial angle. Middle row: Somewhat oblique to an “optic axis.” Bottom row: More oblique to an “optic axis.”
It must be remembered that this uncertainty, in the choice of sections for testing, does not exist in uniaxial crystals; where the best sections are indicated by the fact that they remain dark or nearly so during complete rotation between crossed nicols.
The uniaxial or biaxial character of a mineral section, which only shows an indistinct bar, may be determined as follows: A bar (one arm of the cross) of a uniaxial interference figure moves in the same direction as the rotating stage, and always remains straight, while the biaxial bar rotates in the opposite direction to the stage and becomes curved.
Optical Character, Positive or Negative. When the axial angle is very small, so that the interference figure approaches that of a uniaxial crystal, the methods used for testing uniaxial figures are employed.
When, however, the axial angle is large, the following method can be used:
After having obtained an interference figure, from a section as nearly at right angles to the acute bisectrix[75] as possible, the stage is rotated until the plane of the optic axes (the trace of which on the plane of the section is the line joining the points of emergence of the two optic axes) makes an angle of 45° with the planes of vibration of the crossed nicols or the cross-wires in the eye-piece.
A quartz wedge[76] is now pushed in between the mineral section and the analyzer,[77] so that its axis ć = c (previously determined and marked on the wedge) is either at right angles or parallel to the plane of the optic axes of the mineral section.
The optical character of the mineral is positive when the ellipses, surrounding the points of emergence of the two optic axes on the convex sides of the hyperbola, appear to expand or open out towards the centre when the quartz wedge is pushed in with its axis parallel to the plane of the optic axes.
The optical character is negative when the ellipses appear to expand or open out when the wedge is pushed in with its axis at right angles to the plane of the optic axes.
As the ellipses expand they move from the points of emergence of the optic axes towards the centre of the interference figure, and finally open into lemniscates which move outward from the plane of the optic axes.
Even when the section is very thin and the double refraction very weak, only the black hyperbolas without ellipses being seen, the test can be made; and colored ellipses will appear, after the pushing in of the quartz wedge, which will act in the same way as the ellipses of the interference figure.
In a section at right angles to the obtuse bisectric these results are all reversed.
When the section is perpendicular to one optic axis, rotate the section until the plane of its optic axes is 45° to the planes of vibration of the nicols. The interference figure will now have the appearance as shown in Fig. 29, the hyperbola being convex towards the acute bisectrix. Insert the ¼ undulation mica plate, so that its direction c is parallel to the plane of the optic axes. If the optical character is positive the hyperbola will move towards the acute bisectrix and if negative away from it. When the gypsum plate is used the blue color will appear on the convex side for (+) and on the concave side for (−) minerals.
Determination of the Axial Angle.[78] This can be approximately determined with a petrographical microscope, if equipped with a micrometer eye-piece. Have the axial plane of the crystal section in the diagonal position, Fig. 31; and measure the distance d from the centre to either hyperbola with a micrometer (or average the distance to both). Then sin E = d/C, in which C is a constant for the same combination of lenses and is obtained by using a crystal section (mica cleavage) of known axial angle. For example, in a mica with 2E = 91° 50′ and d = 41.5 divisions on the micrometer scale, C = d/sin E = 57.78 for that special combination of lenses. The true axial angle can be obtained from the equation sin V = d/βC.
Optical Distinctions between Orthorhombic, Monoclinic, and Triclinic Crystal Sections (perpendicular to acute and obtuse bisectrices). The interference figures are always symmetrical in shape and distribution of color to the planes and axes of symmetry of the crystal system; hence are most symmetrical in the orthorhombic, less so in the monoclinic and still less so in the triclinic system.
Orthorhombic crystals show the figures always in two of the pinacoids and in white light the color distribution will be symmetrical to the trace of the axial plane and the line through the centre at right angles to this trace and also to the central point.
Monoclinic crystals show the figures in the clino pinacoid or in sections at right angles to this. In white light the color distribution is never symmetrical to two lines, but is symmetrical either to the trace of the axial plane (inclined dispersion[79]), or to the line through the centre at right angles to this trace (horizontal dispersion), or to the central point (crossed dispersion).
Triclinic crystals show in white light figures with distribution of color unsymmetrical to any line or point.
In white light the “color fringes” of the hyperbola are due to the “dispersion”[79] of the optic axes and bisectrices. That is, for each color (for light of each wave-length) there is a particular interference figure; the overlapping of these superposed figures producing the color fringes.
When the axial angle is larger for red light than for violet, the dispersion is said to be ρ > ν and the interference figure, in the position of Fig. 31, will show the hyperbolic curves fringed with red towards the centre (inside). In general the color with the larger axial angle is nearer the centre of the field. This is due to the extinguishing of light of each color at the axial points, the resulting colors at these points being produced by white light minus the absorbed color. When the dispersion is ν > ρ the reverse distribution of color fringes will take place.
By measuring the axial angle in red and blue light, this dispersion of the optic axes can also be obtained.
Resumé of the Uses of Parallel and Convergent Polarized Light.
Parallel light is used to detect pleochroism, to distinguish between isotropic and anisotropic substances, to study interference colors, to determine the strength of the double refraction, to locate directions of vibration, to measure extinction angles, to find the directions of vibration of the faster and slower rays, to determine the relative value of the indices of refraction of the two rays, and to investigate the crystal structure in general.
Convergent light is used to distinguish between uniaxial and biaxial crystals, to determine whether a section that appears to be isotropic is really so or only perpendicular to an optic axis and to determine the optical character, grade of symmetry (system), axial angle and dispersion.