As an example, the etched figures on the faces of a hexagonal prism and
the basal plane are illustrated in figs. 92-94 for three of the several
symmetry-classes of the hexagonal system. The classes chosen are those
in which nepheline, calcite and beryl (emerald) crystallize, and these
minerals often have the simple form of crystal represented in the
figures. In nepheline (fig. 92) the only element of symmetry is a hexad
axis; the etched figures on the prism are therefore unsymmetrical,
though similar on all the faces; the hexagonal markings on the basal
plane have none of their edges parallel to the edges of the face;
further the crystals being hemimorphic, the etched figures on the basal
planes at the two ends will be different in character. The facial
development of crystals of nepheline give no indication of this type of
symmetry, and the mineral has been referred to this class solely on the
evidence afforded by the etched figures. In calcite there is a triad
axis of symmetry parallel to the prism edges, three dyad axes each
perpendicular to a pair of prism edges and three planes of symmetry
perpendicular to the prism faces; the etched figures shown in fig. 93
will be seen to conform to all these elements of symmetry. There being
in calcite also a centre of symmetry, the equilateral triangles on the
basal plane at the lower end of the crystal will be the same in form as
those at the top, but they will occupy a reversed position. In beryl,
which crystallizes in the holosymmetric class of the hexagonal system,
the etched figures (fig. 94) display the fullest possible degree of
symmetry; those on the prism faces are all similar and are each
symmetrical with respect to two lines, and the hexagonal markings on the
basal planes at both ends of the crystal are symmetrically placed with
respect to six lines. A detailed account of the etched figures of
crystals is given by H. Baumhauer, _Die Resultate der Atzmethode in der
krystallographischen Forschung_ (Leipzig, 1894).
(b) _Optical Properties._
The complex optical characters of crystals are not only of considerable
interest theoretically, but are of the greatest practical importance. In
the absence of external crystalline form, as with a faceted gem-stone,
or with the minerals constituting a rock (thin, transparent sections of
which are examined in the polarizing microscope), the mineral species
may often be readily identified by the determination of some of the
optical characters.
According to their action on transmitted plane-polarized light (see
POLARIZATION OF LIGHT) all crystals may be referred to one or other of
the five groups enumerated below. These groups correspond with the six
systems of crystallization (in the second group two systems being
included together). The several symmetry-classes of each system are
optically the same, except in the rare cases of substances which are
circularly polarizing.
(1) Optically isotropic crystals--corresponding with the cubic system.
(2) Optically uniaxial crystals--corresponding with the tetragonal and
hexagonal systems.
(3) Optically biaxial crystals in which the three principal optical
directions coincide with the three crystallographic axes--corresponding
with the orthorhombic system.
(4) Optically biaxial crystals in which only one of the three principal
optical directions coincides with a crystallographic axis--corresponding
with the monoclinic system.
(5) Optically biaxial crystals in which there is no fixed and definite
relation between the optical and crystallographic
directions--corresponding with the anorthic system.
_Optically Isotropic Crystals._--These belong to the cubic system, and
like all other optically isotropic (from [Greek: isos], like, and
[Greek: tropos], character) bodies have only one index of refraction for
light of each colour. They have no action on polarized light (except in
crystals which are circularly polarizing); and when examined in the
polariscope or polarizing microscope they remain dark between crossed
nicols, and cannot therefore be distinguished optically from amorphous
substances, such as glass and opal.
_Optically Uniaxial Crystals._--These belong to the tetragonal and
hexagonal (including rhombohedral) systems, and between crystals of
these systems there is no optical distinction. Such crystals are
anisotropic or doubly refracting (see REFRACTION: _Double_); but for
light travelling through them in a certain, single direction they are
singly refracting. This direction, which is called the optic axis, is
the same for light of all colours and at all temperatures; it coincides
in direction with the principal crystallographic axis, which in
tetragonal crystals is a tetrad (or dyad) axis of symmetry, and in the
hexagonal system a triad or hexad axis.
For light of each colour there are two indices of refraction; namely,
the ordinary index ([omega]) corresponding with the ordinary ray, which
vibrates perpendicular to the optic axis; and the extraordinary index
([epsilon]) corresponding with the extraordinary ray, which vibrates
parallel to the optic axis. If the ordinary index of refraction be
greater than the extraordinary index, the crystal is said to be
optically negative, whilst if less the crystal is optically positive.
The difference between the two indices is a measure of the strength of
the double refraction or birefringence. Thus in calcite, for sodium (D)
light, [omega] = 1.6585 and [epsilon] = 1.4863; hence this substance is
optically negative with a relatively high double refraction of [omega] -
[epsilon] = 0.1722. In quartz [omega] = 1.5442, [epsilon] = 1.5533 and
[epsilon] - [omega] = 0.0091; this mineral is therefore optically
positive with low double refraction. The indices of refraction vary, not
only for light of different colours, but also slightly with the
temperature.
The optical characters of uniaxial crystals are symmetrical not only
with respect to the full number of planes and axes of symmetry of
tetragonal and hexagonal crystals, but also with respect to all vertical
planes, i.e. all planes containing the optic axis. A surface expressing
the optical relations of such crystals is thus an ellipsoid of
revolution about the optic axis. (In cubic crystals the corresponding
surface is a sphere.) In the "optical indicatrix" (L. Fletcher, _The
Optical Indicatrix and the Transmission of Light in Crystals_, London,
1892), the length of the principal axis, or axis of rotation, is
proportional to the index of refraction, (i.e. inversely proportional to
the velocity) of the extraordinary rays, which vibrate along this axis
and are transmitted in directions perpendicular thereto; the equatorial
diameters are proportional to the index of refraction of the ordinary
rays, which vibrate perpendicular to the optic axis. For positive
uniaxial crystals the indicatrix is thus a prolate spheroid
(egg-shaped), and for negative crystals an oblate spheroid
(orange-shaped).
In "Fresnel's ellipsoid" the axis of rotation is proportional to the
velocity of the extraordinary ray, and the equatorial diameters
proportional to the velocity of the ordinary ray; it is therefore an
oblate spheroid for positive crystals, and a prolate spheroid for
negative crystals. The "ray-surface," or "wave-surface," which
represents the distances traversed by the rays during a given interval
of time in various directions from a point of origin within the crystal,
consists in uniaxial crystals of two sheets; namely, a sphere,
corresponding to the ordinary rays, and an ellipsoid of revolution,
corresponding to the extraordinary rays. The difference in form of the
ray-surface for positive and negative crystals is shown in figs. 95 and
96.
[Illustration: FIG. 95.--Section of the Ray-Surface of a Positive
Uniaxial Crystal.]
[Illustration: FIG. 96.--Section of the Ray-Surface of a Negative
Uniaxial Crystal.]
When a uniaxial crystal is examined in a polariscope or polarizing
microscope between crossed nicols (i.e. with the principal planes of the
polarizer or analyser at right angles, and so producing a dark field of
view) its behaviour differs according to the direction in which the
light travels through the crystal, to the position of the crystal with
respect to the principal planes of the nicols, and further, whether
convergent or parallel polarized light be employed. A tetragonal or
hexagonal crystal viewed, in parallel light, through the basal plane,
i.e. along the principal axis, will remain dark as it is rotated between
crossed nicols, and will thus not differ in its behaviour from a cubic
crystal or other isotropic body. If, however, the crystal be viewed in
any other direction, for example, through a prism face, it will, except
in certain positions, have an action on the polarized light. A
plane-polarized ray entering the crystal will be resolved into two
polarized rays with the directions of vibration parallel to the
vibration-directions in the crystal. These two rays on leaving the
crystal will be combined again in the analyser, and a portion of the
light transmitted through the instrument; the crystal will then show up
brightly against the dark field. Further, owing to interference of these
two rays in the analyser, the light will be brilliantly coloured,
especially if the crystal be thin, or if a thin section of a crystal be
examined. The particular colour seen will depend on the strength of the
double refraction, the orientation of the crystal or section, and upon
its thickness. If now, the crystal be rotated with the stage of the
microscope, the nicols remaining fixed in position, the light
transmitted through the instrument will vary in intensity, and in
certain positions will be cut out altogether. The latter happens when
the vibration-directions of the crystal are parallel to the
vibration-directions of the nicols (these being indicated by cross-wires
in the microscope). The crystal, now being dark, is said to be in
position of extinction; and as it is turned through a complete rotation
of 360 deg. it will extinguish four times. If a prism face be viewed
through, it will be seen that, when the crystal is in a position of
extinction, the cross-wires of the microscope are parallel to the edges
of the prism: the crystal is then said to give "straight extinction."
In convergent light, between crossed nicols, a very different phenomenon
is to be observed when a uniaxial crystal, or section of such a crystal,
is placed with its optic axis coincident with the axis of the
microscope. The rays of light, being convergent, do not travel in the
direction of the optic axis and are therefore doubly refracted in the
crystal; in the analyser the vibrations will be reduced to the same
plane and there will be interference of the two sets of rays. The result
is an "interference figure" (fig. 97), which consists of a number of
brilliantly coloured concentric rings, each showing the colours of the
spectrum of white light; intersecting the rings is a black cross, the
arms of which are parallel to the principal planes of the nicols. If
monochromatic light be used instead of white light, the rings will be
alternately light and dark. The number and distance apart of the rings
depend on the strength of the double refraction and on the thickness of
the crystal. By observing the effect produced on such a uniaxial
interference figure when a "quarter undulation (or wave-length)
mica-plate" is superposed on the crystal, it may be at once decided
whether the crystal is optically positive or negative. Such a simple
test may, for example, be applied for distinguishing certain faceted
gem-stones: thus zircon and phenacite are optically positive, whilst
corundum (ruby and sapphire) and beryl (emerald) are optically negative.
[Illustration: FIG. 97.--Interference Figure of a Uniaxial Crystal.]
_Optically Biaxial Crystals._--In these crystals there are three
principal indices of refraction, denoted by [alpha], [beta] and [gamma];
of these [gamma] is the greatest and [alpha] the least ([gamma] > [beta]
> [alpha]). The three principal vibration-directions, corresponding to
these indices, are at right angles to each other, and are the directions
of the three rectangular axes of the optical indicatrix. The indicatrix
(fig. 98) is an ellipsoid with the lengths of its axes proportional to
the refractive indices; OC = [gamma], OB = [beta], OA = [alpha], where
OC > OB > OA. The figure is symmetrical with respect to the principal
planes OAB, OAC, OBC.
In Fresnel's ellipsoid the three rectangular axes are proportional to
1/[alpha], 1/[beta], and 1/[gamma], and are usually denoted by a, b and
c respectively, where a > b > c: these have often been called "axes of
optical elasticity," a term now generally discarded.
[Illustration: FIG. 98.--Optical Indicatrix of a Biaxial Crystal.]
[Illustration: FIG. 99.--Ray-Surface of a Biaxial Crystal.]
The ray-surface (represented in fig. 99 by its sections in the three
principal planes) is derived from the indicatrix in the following
manner. A ray of light entering the crystal and travelling in the
direction OA is resolved into polarized rays vibrating parallel to OB
and OC, and therefore propagated with the velocities 1/[beta] and
1/[gamma] respectively: distances Ob and Oc (fig. 99) proportional to
these velocities are marked off in the direction OA. Similarly, rays
travelling along OC have the velocities 1/[alpha] and 1/[beta], and
those along OB the velocities 1/[alpha] and 1/[gamma]. In the two
directions Op1 and Op2 (fig. 98), perpendicular to the two circular
sections P1P1 and P2P2 of the indicatrix, the two rays will be
transmitted with the same velocity 1/[beta]. These two directions are
called the optic axes ("primary optic axis"), though they have not all
the properties which are associated with the optic axis of a uniaxial
crystal. They have very nearly the same direction as the lines Os1 and
Os2 in fig. 99, which are distinguished as the "secondary optic axes."
In most crystals the primary and secondary optic axes are inclined to
each other at not more than a few minutes, so that for practical
purposes there is no distinction between them.
The angle between Op1 and Op2 is called the "optic axial angle"; and the
plane OAC in which they lie is called the "optic axial plane." The
angles between the optic axes are bisected by the vibration-directions
OA and OC; the one which bisects the acute angle being called the
"acute bisectrix" or "first mean line," and the other the "obtuse
bisectrix" or "second mean line." When the acute bisectrix coincides
with the greatest axis OC of the indicatrix, i.e. the
vibration-direction corresponding with the refractive index [gamma] (as
in figs. 98 and 99), the crystal is described as being optically
positive; and when the acute bisectrix coincides with OA, the
vibration-direction for the index [alpha], the crystal is negative. The
distinction between positive and negative biaxial crystals thus depends
on the relative magnitude of the three principal indices of refraction;
in positive crystals [beta] is nearer to [alpha] than to [gamma], whilst
in negative crystals the reverse is the case. Thus in topaz, which is
optically positive, the refractive indices for sodium light are [alpha]
= 1.6120, [beta] = 1.6150, [gamma] = 1.6224; and for orthoclase which is
optically negative, [alpha] = 1.5190, [beta] = 1.5237, [gamma] = 1.5260.
The difference [gamma] - [alpha] represents the strength of the double
refraction.
Since the refractive indices vary both with the colour of the light and
with the temperature, there will be for each colour and temperature
slight differences in the form of both the indicatrix and the
ray-surface: consequently there will be variations in the positions of
the optic axes and in the size of the optic axial angle. This phenomenon
is known as the "dispersion of the optic axes." When the axial angle is
greater for red light than for blue the character of the dispersion is
expressed by [rho] > [upsilon], and when less by [rho] < [upsilon]. In
some crystals, e.g. brookite, the optic axes for red light and for blue
light may be, at certain temperatures, in planes at right angles.
[Illustration: FIG. 100.]
[Illustration: FIG. 101.