The type of interference figure exhibited by a biaxial crystal in
convergent polarized light between crossed nicols is represented in
figs. 100 and 101. The crystal must be viewed along the acute bisectrix,
and for this purpose it is often necessary to cut a plate from the
crystal perpendicular to this direction: sometimes, however, as in mica
and topaz, a cleavage flake will be perpendicular to the acute
bisectrix. When seen in white light, there are around each optic axis a
series of brilliantly coloured ovals, which at the centre join to form
an 8-shaped loop, whilst further from the centre the curvature of the
rings is approximately that of lemniscates. In the position shown in
fig. 100 the vibration-directions in the crystal are parallel to those
of the nicols, and the figure is intersected by two black bands or
"brushes" forming a cross. When, however, the crystal is rotated with
the stage of the microscope the cross breaks up into the two branches of
a hyperbola, and when the vibration-directions of the crystal are
inclined at 45 deg. to those of the nicols the figure is that shown in
fig. 101. The points of emergence of the optic axes are at the middle of
the hyperbolic brushes when the crystal is in the diagonal position: the
size of the optic axial angle can therefore be directly measured with
considerable accuracy.
In orthorhombic crystals the three principal vibration-directions
coincide with the three crystallographic axes, and have therefore fixed
positions in the crystal, which are the same for light of all colours
and at all temperatures. The optical orientation of an orthorhombic
crystal is completely defined by stating to which crystallographic
planes the optic axial plane and the acute bisectrix are respectively
parallel and perpendicular. Examined in parallel light between crossed
nicols, such a crystal extinguishes parallel to the crystallographic
axes, which are often parallel to the edges of a face or section; there
is thus usually "straight extinction." The interference figure seen in
convergent polarized light is symmetrical about two lines at right
angles.
In monoclinic crystals only one vibration-direction has a fixed position
within the crystal, being parallel to the ortho-axis (i.e. perpendicular
to the plane of symmetry or the plane (010)). The other two
vibration-directions lie in the plane (010), but they may vary in
position for light of different colours and at different temperatures.
In addition to dispersion of the optic axes there may thus, in crystals
of this system, be also "dispersion of the bisectrices." The latter may
be of one or other of three kinds, according to which of the three
vibration-directions coincides with the ortho-axis of the crystal. When
the acute bisectrix is fixed in position, the optic axial planes for
different colours may be crossed, and the interference figure will then
be symmetrical with respect to a point only ("crossed dispersion"). When
the obtuse bisectrix is fixed, the axial planes may be inclined to one
another, and the interference figure is symmetrical only about a line
which is perpendicular to the axial planes ("horizontal dispersion").
Finally, when the vibration-direction corresponding to the refractive
index [beta], or the "third mean line," has a fixed position, the optic
axial plane lies in the plane (010), but the acute bisectrix may vary in
position in this plane; the interference figure will then be symmetrical
only about a line joining the optic axes ("inclined dispersion").
Examples of substances exhibiting these three kinds of dispersion are
borax, orthoclase and gypsum respectively. In orthoclase and gypsum,
however, the optic axial angle gradually diminishes as the crystals are
heated, and after passing through a uniaxial position they open out in a
plane at right angles to the one they previously occupied; the character
of the dispersion thus becomes reversed in the two examples quoted. When
examined in parallel light between crossed nicols monoclinic crystals
will give straight extinction only in faces and sections which are
perpendicular to the plane of symmetry (or the plane (010)); in all
other faces and sections the extinction-directions will be inclined to
the edges of the crystal. The angles between these directions and edges
are readily measured, and, being dependent on the optical orientation of
the crystal, they are often characteristic constants of the substance
(see, e.g., PLAGIOCLASE).
In anorthic crystals there is no relation between the optical and
crystallographic directions, and the exact determination of the optical
orientation is often a matter of considerable difficulty. The character
of the dispersion of the bisectrices and optic axes is still more
complex than in monoclinic crystals, and the interference figures are
devoid of symmetry.
_Absorption of Light in Crystals: Pleochroism._--In crystals other than
those of the cubic system, rays of light with different
vibration-directions will, as a rule, be differently absorbed; and the
polarized rays on emerging from the crystal may be of different
intensities and (if the observation be made in white light and the
crystal is coloured) differently coloured. Thus, in tourmaline the
ordinary ray, which vibrates perpendicular to the principal axis, is
almost completely absorbed, whilst the extraordinary ray is allowed to
pass through the crystal. A plate of tourmaline cut parallel to the
principal axis may therefore be used for producing a beam of polarized
light, and two such plates placed in crossed position form the polarizer
or analyser of "tourmaline tongs," with the aid of which the
interference figures of crystals may be simply shown. Uniaxial
(tetragonal and hexagonal) crystals when showing perceptible differences
in colour for the ordinary and extraordinary rays are said to be
"dichroic." In biaxial (orthorhombic, monoclinic and anorthic) crystals,
rays vibrating along each of the three principal vibration-directions
may be differently absorbed, and, in coloured crystals, differently
coloured; such crystals are therefore said to be "trichroic" or in
general "pleochroic" (from [Greek: pleon], more, and [Greek: chroa],
colour). The directions of maximum absorption in biaxial crystals have,
however, no necessary relation with the axes of the indicatrix, unless
these have fixed crystallographic directions, as in the orthorhombic
system and the ortho-axis in the monoclinic. In epidote it has been
shown that the two directions of maximum absorption which lie in the
plane of symmetry are not even at right angles.
[Illustration: FIG. 102.--Dichroscope.]
The pleochroism of some crystals is so strong that when they are viewed
through in different directions they exhibit marked differences in
colour. Thus a crystal of the mineral iolite (called also dichroite
because of its strong pleochroism) will be seen to be dark blue, pale
blue or pale yellow according to which of three perpendicular directions
it is viewed. The "face colours" seen directly in this way result,
however, from the mixture of two "axial colours" belonging to rays
vibrating in two directions. In order to see the axial colours
separately the crystal must be examined with a dichroscope, or in a
polarizing microscope from which the analyser has been removed. The
dichroscope, or dichroiscope (fig. 102), consists of a cleavage
rhombohedron of calcite (Iceland-spar) p, on the ends of which glass
prisms w are cemented: the lens l is focused on a small square aperture
o in the tube of the instrument. The eye of the observer placed at e
will see two images of the square aperture, and if a pleochroic crystal
be placed in front of this aperture the two images will be differently
coloured. On rotating this crystal with respect to the instrument the
maximum difference in the colours will be obtained when the
vibration-directions in the crystal coincide with those in the calcite.
Such a simple instrument is especially useful for the examination of
faceted gem-stones, even when they are mounted in their settings. A
single glance suffices to distinguish between a ruby and a
"spinel-ruby," since the former is dichroic and the latter isotropic and
therefore not dichroic.
The characteristic absorption bands in the spectrum of white light which
has been transmitted through certain crystals, particularly those of
salts of the cerium metals, will, of course, be different according to
the direction of vibration of the rays.
_Circular Polarization in Crystals._--Like the solutions of certain
optically active organic substances, such as sugar and tartaric acid,
some optically isotropic and uniaxial crystals possess the property of
rotating the plane of polarization of a beam of light. In uniaxial
(tetragonal and hexagonal) crystals it is only for light transmitted in
the direction of the optic axis that there is rotatory action, but in
isotropic (cubic) crystals all directions are the same in this respect.
Examples of circularly polarizing cubic crystals are sodium chlorate,
sodium bromate, and sodium uranyl acetate; amongst tetragonal crystals
are strychnine sulphate and guanidine carbonate; amongst rhombohedral
are quartz (q.v.) and cinnabar (q.v.) (these being the only two mineral
substances in which the phenomenon has been observed), dithionates of
potassium, lead, calcium and strontium, and sodium periodate; and
amongst hexagonal crystals is potassium lithium sulphate. Crystals of
all these substances belong to one or other of the several
symmetry-classes in which there are neither planes nor centre of
symmetry, but only axes of symmetry. They crystallize in two
complementary hemihedral forms, which are respectively right-handed and
left-handed, i.e. enantiomorphous forms. Some other substances which
crystallize in enantiomorphous forms are, however, only "optically
active" when in solution (e.g. sugar and tartaric acid); and there are
many other substances presenting this peculiarity of crystalline form
which are not circularly polarizing either when crystallized or when in
solution. Further, in the examples quoted above, the rotatory power is
lost when the crystals are dissolved (except in the case of strychnine
sulphate, which is only feebly active in solution). The rotatory power
is thus due to different causes in the two cases, in the one depending
on a spiral arrangement of the crystal particles, and in the other on
the structure of the molecules themselves.
The circular polarization of crystals may be imitated by a pile of mica
plates, each plate being turned through a small angle on the one below,
thus giving a spiral arrangement to the pile.
_"Optical Anomalies" of Crystals._--When, in 1818, Sir David Brewster
established the important relations existing between the optical
properties of crystals and their external form, he at the same time
noticed many apparent exceptions. For example, he observed that crystals
of leucite and boracite, which are cubic in external form, are always
doubly refracting and optically biaxial, but with a complex internal
structure; and that cubic crystals of garnet and analcite sometimes
exhibit the same phenomena. Also some tetragonal and hexagonal crystals,
e.g. apophyllite, vesuvianite, beryl, &c., which should normally be
optically uniaxial, sometimes consist of several biaxial portions
arranged in sectors or in a quite irregular manner. Such exceptions to
the general rule have given rise to much discussion. They have often
been considered to be due to internal strains in the crystals, set up as
a result of cooling or by earth pressures, since similar phenomena are
observed in chilled and compressed glasses and in dried gelatine. In
many cases, however, as shown by E. Mallard, in 1876, the higher degree
of symmetry exhibited by the external form of the crystals is the result
of mimetic twinning, as in the pseudo-cubic crystals of leucite (q.v.)
and boracite (q.v.). In other instances, substances not usually regarded
as cubic, e.g. the monoclinic phillipsite (q.v.), may by repeated
twinning give rise to pseudo-cubic forms. In some cases it is probable
that the substance originally crystallized in one modification at a
higher temperature, and when the temperature fell it became transformed
into a dimorphous modification, though still preserving the external
form of the original crystal (see BORACITE). A summary of the literature
is given by R. Brauns, _Die optischen Anomalien der Krystalle_ (Leipzig,
1891).
(c) _Thermal Properties._
[Illustration: FIG. 103.--Conductivity of Heat in Quartz.]
The thermal properties of crystals present certain points in common with
the optical properties. Heat rays are transmitted and doubly refracted
like light rays; and surfaces expressing the conductivity and dilatation
in different directions possess the same degree of symmetry and are
related in the same way to the crystallographic axes as the ellipsoids
expressing the optical relations. That crystals conduct heat at
different rates in different directions is well illustrated by the
following experiment. Two plates (fig. 103) cut from a crystal of
quartz, one parallel to the principal axis and the other perpendicular
to it, are coated with a thin layer of wax, and a hot wire is applied to
a point on the surface. On the transverse section the wax will be melted
in a circle, and on the longitudinal section (or on the natural prism
faces) in an ellipse. The isothermal surface in a uniaxial crystal is
therefore a spheroid; in cubic crystals it is a sphere; and in biaxial
crystals an ellipsoid, the three axes of which coincide, in orthorhombic
crystals, with the crystallographic axes.
With change of temperature cubic crystals expand equally in all
directions, and the angles between the faces are the same at all
temperatures. In uniaxial crystals there are two principal coefficients
of expansion; the one measured in the direction of the principal axis
may be either greater or less than that measured in directions
perpendicular to this axis. A sphere cut from a uniaxial crystal at one
temperature will be a spheroid at another temperature. In biaxial
crystals there are different coefficients of expansion along three
rectangular axes, and a sphere at one temperature will be an ellipsoid
at another. A result of this is that for all crystals, except those
belonging to the cubic system, the angles between the faces will vary,
though only slightly, with changes of temperature. E. Mitscherlich found
that the rhombohedral angle of calcite decreases 8' 37" as the crystal
is raised in temperature from 0 deg. to 100 deg. C.
As already mentioned, the optical properties of crystals vary
considerably with the temperature. Such characters as specific heat and
melting-point, which do not vary with the direction, are the same in
crystals as in amorphous substances.
(d) _Magnetic and Electrical Properties._
Crystals, like other bodies, are either paramagnetic or diamagnetic,
i.e. they are either attracted or repelled by the pole of a magnet. In
crystals other than those belonging to the cubic system, however, the
relative strength of the induced magnetization is different in different
directions within the mass. A sphere cut from a tetragonal or hexagonal
(uniaxial) crystal will if freely suspended in a magnetic field (between
the poles of a strong electro-magnet) take up a position such that the
principal axis of the crystal is either parallel or perpendicular to the
lines of force, or to a line joining the two poles of the magnet. Which
of these two directions is taken by the axis depends on whether the
crystal is paramagnetic or diamagnetic, and on whether the principal
axis is the direction of maximum or minimum magnetization. The surface
expressing the magnetic character in different directions is in uniaxial
crystals a spheroid; in cubic crystals it is a sphere. In orthorhombic,
monoclinic and anorthic crystals there are three principal axes of
magnetic induction, and the surface is an ellipsoid, which is related to
the symmetry of the crystal in the same way as the ellipsoids expressing
the thermal and optical properties.
Similarly, the dielectric constants of a non-conducting crystal may be
expressed by a sphere, spheroid or ellipsoid. A sphere cut from a
crystal will when suspended in an electro-magnetic field set itself so
that the axis of maximum induction is parallel to the lines of force.
The electrical conductivity of crystals also varies with the direction,
and bears the same relation to the symmetry as the thermal conductivity.
In a rhombohedral crystal of haematite the electrical conductivity along
the principal axis is only half as great as in directions perpendicular
to this axis; whilst in a crystal of bismuth, which is also
rhombohedral, the conductivities along and perpendicular to the axis are
as 1.6 : 1.
Conducting crystals are thermo-electric: when placed against another
conducting substance and the contact heated there will be a flow of
electricity from one body to the other if the circuit be closed. The
thermo-electric force depends not only on the nature of the substance,
but also on the direction within the crystal, and may in general be
expressed by an ellipsoid. A remarkable case is, however, presented by
minerals of the pyrites group: some crystals of pyrites are more
strongly thermo-electrically positive than antimony, and others more
negative than bismuth, so that the two when placed together give a
stronger thermo-electric couple than do antimony and bismuth. In the
thermo-electrically positive crystals of pyrites the faces of the
pentagonal dodecahedron are striated parallel to the cubic edges, whilst
in the rarer negative crystals the faces are striated perpendicular to
these edges. Sometimes both sets of striae are present on the same face,
and the corresponding areas are then thermo-electrically positive and
negative.
The most interesting relation between the symmetry of crystals and their
electrical properties is that presented by the pyro-electrical phenomena
of certain crystals. This is a phenomenon which may be readily observed,
and one which often aids in the determination of the symmetry of
crystals. It is exhibited by crystals in which there is no centre of
symmetry, and the axes of symmetry are uniterminal or polar in
character, being associated with different faces on the crystal at their
two ends. When a non-conducting crystal possessing this hemimorphic type
of symmetry is subjected to changes of temperature a charge of positive
electricity will be developed on the faces in the region of one end of
the uniterminal axis, whilst the faces at the opposite end will be
negatively charged. With rising temperature the pole which becomes
positively charged is called the "analogous pole," and that negatively
charged the "antilogous pole": with falling temperature the charges are
reversed. The phenomenon was first observed in crystals of tourmaline,
the principal axis of which is a uniterminal triad axis of symmetry. In
crystals of quartz there are three uniterminal dyad axes of symmetry
perpendicular to the principal triad axis (which is here similar at its
two ends): the dyad axes emerge at the edges of the hexagonal prism,
alternate edges of which become positively and negatively charged on
change of temperature. In boracite there are four uniterminal triad
axes, and the faces of the two tetrahedra perpendicular to them will
bear opposite charges. Other examples of pyro-electric crystals are the
orthorhombic mineral hemimorphite (called also, for this reason,
"electric calamine") and the monoclinic tartaric acid and cane-sugar,
each of which possesses a uniterminal dyad axis of symmetry. In some
exceptional cases, e.g. axinite, prehnite, &c., there is no apparent
relation between the distribution of the pyro-electric charges and the
symmetry of the crystals.
The distribution of the electric charges may be made visible by the
following simple method, which may be applied even with minute crystals
observed under the microscope. A finely powdered mixture of red-lead and
sulphur is dusted through a sieve over the cooling crystal. In passing
through the sieve the particles of red-lead and sulphur become
electrified by mutual friction, the former positively and the latter
negatively. The red-lead is therefore attracted to the negatively
charged parts of the crystal and the sulphur to those positively
charged, and the distribution of the charges over the whole crystal
becomes mapped out in the two colours red and yellow.
Since, when a crystal changes in temperature, it also expands or
contracts, a similar distribution of "piezo-electric" (from [Greek:
piezein], to press) charges are developed when a crystal is subjected to
changes of pressure in the direction of a uniterminal axis of symmetry.
Thus increasing pressure along the principal axis of a tourmaline
crystal produces the same electric charges as decreasing temperature.