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.
[Illustration: 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.
[Illustration: 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.
[Illustration: 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_.
[Illustration: 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.
[Illustration: 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.
[Illustration: _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.