REFLECTION, REFRACTION, AND DISPERSION
It is obvious that, since a stone suitable for ornamental use must
appeal to the eye, its most important characters are those which depend
upon light; indeed, the whole art of the lapidary consists in shaping
it in such a way as to show these qualities to the best advantage. To
understand why certain forms are given to a cut stone, it is essential
for us to ascertain what becomes of the light which falls upon the
surface of the stone; further, we shall find that the action of a
stone upon light is of very great help in distinguishing the different
species of gem-stones. The phenomena displayed by light which impinges
upon the surface separating two media[1] are very similar in character,
whatever be the nature of the media.
Ordinary experience with a plane mirror tells us that, when light is
returned, or reflected, as it is usually termed, from a plane or flat
surface, there is no alteration in the size of objects viewed in this
way, but that the right and the left hands are interchanged: our right
hand becomes the left hand in our reflection in the mirror. We notice,
further, that our reflection is apparently just as far distant from the
mirror on the farther side as we are on this side. In Fig. 13 _MM´_
is a section of the mirror, and _O´_ is the image of the hand _O_ as
seen in the mirror. Light from _O_ reaches the eye _E_ by way of _m_,
but it appears to come from _O´_. Since _OO´_ is perpendicular to the
mirror, and _O_ and _O´_ lie at equal distances from it, it follows
from elementary geometry that the angle _i´_, which the reflected ray
makes with _mn_, the normal to the mirror, is equal to _i_, the angle
which the incident ray makes with the same direction.
[Illustration: FIG. 13.—Reflection at a Plane Mirror.]
Again, everyday experience tells us that the case is less simple when
light actually crosses the bounding surface and passes into the other
medium. Thus, if we look down into a bath filled with water, the
bottom of the bath appears to have been raised up, and a stick plunged
into the water seems to be bent just at the surface, except in the
particular case when it is perfectly upright. Since the stick itself
has not been bent, light evidently suffers some change in direction as
it passes into the water or emerges therefrom. The passage of light
from one medium to another was studied by Snell in the seventeenth
century, and he enunciated the following laws:—
1. The refracted ray lies in the plane containing the incident ray and
the normal to the plane surface separating the two media.
It will be noticed that the reflected ray obeys this law also.
2. The angle _r_, which the refracted ray makes with the normal, is
related to the angle _i_, which the incident ray makes with the same
direction, by the equation
_n_ sin _i_ = _n´_ sin _r_, (_a_)
where _n_ and _n´_ are constants for the two media which are known as
the indices of refraction, or the refractive indices.
This simple trigonometrical relation may be expressed in geometrical
language. Suppose we cut a plane section through the two media at right
angles to the bounding plane, which then appears as a straight line,
_SOS´_ (Fig. 14), and suppose that _IO_ represents the direction of
the incident ray; then Snell’s first law tells us that the refracted
ray _OR_ will also lie in this plane. Draw the normal _NON´_, and with
centre _O_ and any radius describe a circle intersecting the incident
and refracted rays in the points _a_ and _b_ respectively; let drop
perpendiculars _ac_ and _bd_ on to the normal _NON´_. Then we have
_n.ac = n´.bd_, whence we see that if _n_ be greater than _n´_, _ac_
is less than _bd_, and therefore when light passes from one medium
into another which is less optically dense, in its passage across the
boundary it is bent, or refracted, away from the normal.
[Illustration: FIG. 14.—Refraction across a Plane Surface.]
We see, then, that when light falls on the boundary of two different
media, some is reflected in the first and some is refracted into the
second medium. The relative amounts of light reflected and refracted
depend on the angle of incidence and the refractive indices of the
media. We shall return to this point when we come to consider the
lustre of stones.
We will proceed to consider the course of rays at different angles of
incidence when light passes out from a medium into one less dense—for
instance, from water into air. Corresponding to light with a small
angle of incidence such as _I_{1}O_ (Fig. 15), some of it is reflected
in the direction _OI´_{1}_ and the remainder is refracted out in the
direction _OR_{1}_. Similarly, for the ray _I_{2}O_ some is reflected
along _OI´_{2}_ and some refracted along _OR_{2}_. Since, in the case
we have taken, the angle of refraction is greater than the angle of
incidence, the refracted ray corresponding to some incident, ray
_I_{c}O_ will graze the bounding surface, and corresponding to a
ray beyond it, such as _I_{3}O_, which has a still greater angle of
incidence, there is no refracted ray, and all the light is wholly
or totally reflected within the dense medium. The critical angle
_I_{c}ON_, which is called the angle of total-reflection, is very
simply related to the refractive indices of the two media; for, since
_r_ is now a right angle, sin _r_ = 1, and equation (_a_) becomes
_n_ sin _i_ = _n´_ (_b_)
Hence, if the angle of total-reflection is measured and one of the
indices is known, the other can easily be calculated.
[Illustration: FIG. 15.—Total-Reflection.]
The phenomenon of total-reflection may be appreciated if we hold a
glass of water above our head, and view the light of a lamp on a table
reflected from the under surface of the water. This reflection is
incomparably more brilliant than that given by the upper surface.
The refractive index of air is taken as unity; strictly, it is that of
a vacuum, but the difference is too small to be appreciated even in
very delicate work. Every substance has different indices for light
of different colour, and it is customary to take as the standard the
yellow light of a sodium flame. This happens to be the colour to
which our eyes are most sensitive, and a flame of this kind is easily
prepared by volatilizing a little bicarbonate of soda in the flame of
a bunsen burner. A survey of Table III at the end of the book shows
clearly how valuable a measurement of the refractive index is for
determining the species to which a cut stone belongs. The values found
for different specimens of the species do in cases vary considerably
owing to the great latitude possible in the chemical constitution
due to the isomorphous replacement of one element by another. Some
variation in the index may even occur in different directions within
the same stone; it results from the remarkable property of splitting up
a beam of light into two beams, which is possessed by many crystallized
substances. This forms the subject of a later chapter.
Upon the fact that the refractive index of a substance varies for
light of different colours depends such familiar phenomena as the
splendour of the rainbow and the ‘fire’ of the diamond. When white
light is refracted into a stone it no longer remains white, but is
split up into a spectrum. Except in certain anomalous substances the
refractive index increases progressively as the wave-length of the
light decreases, and consequently a normal spectrum is violet at one
end and passes through green and yellow to red at the other end. The
width of the spectrum, which may be measured by the difference between
the refractive indices for the extreme red and violet rays, also
varies, though on the whole it increases with the refractive index. It
is the dispersion, as this difference is termed, that determines the
‘fire’—a character of the utmost importance in colourless transparent
stones, which, but for it, would be lacking in interest. Diamond excels
all colourless stones in this respect, although it is closely followed
by zircon, the colour of which has been driven off by heating; it is,
however, surpassed by two coloured species: sphene, which is seldom
seen in jewellery, and demantoid, the green garnet from the Urals,
which often passes under the misnomer ‘olivine.’ The dispersion of the
more prominent species for the _B_ and _G_ lines of the solar spectrum
is given in Table IV at the end of the book.
We will now proceed to discuss methods that may be used for the
measurement of the refractive indices of cut stones.