MEASUREMENT OF REFRACTIVE INDICES
The methods available for the measurement of refractive indices are of
two kinds, and make use of two different principles. The first, which
is based upon the very simple relation found in the last chapter to
subsist at total-reflection, can be used with ease and celerity, and is
best suited for discriminative purposes; but it is restricted in its
application. The second, which depends on the measurement of the angle
between two facets and the minimum deviation experienced by a ray of
light when traversing a prism formed by them, is more involved, entails
the use of more elaborate apparatus, and takes considerable time, but
it is less restricted in its application.
(1) THE METHOD OF TOTAL-REFLECTION
We see from equation _b_ (p. 18), connecting the angle of
total-reflection with the refractive indices of the adjacent media,
that, if the denser medium be constant, the indices of all less dense
media may be easily determined from a measurement of the corresponding
critical angle. In all refractometers the constant medium is a glass
with a high refractive index. Some instruments have rotatory parts, by
means of which this angle is actually measured. Such instruments give
very good results, but suffer from the disadvantages of being neither
portable nor convenient to handle, and of not giving a result without
some computation.
[Illustration: FIG. 16.—Refractometer (actual size).]
For use in the identification of cut stones, a refractometer with a
fixed scale, such as that (Fig. 16) devised by the author, is far
more convenient. In order to facilitate the observations, a totally
reflecting prism has been inserted between the two lenses of the
eyepiece. The eyepiece may be adjusted to suit the individual eyesight;
but for observers with exceptionally long sight an adapter is provided,
which permits the eyepiece being drawn out to the requisite extent.
The refractometer must be held in the manner illustrated in Fig.
17, so that the light from a window or other source of illumination
enters the instrument by the lenticular opening underneath. Good,
even illumination of the field may also very simply be secured by
reflecting light into the instrument from a sheet of white paper laid
on a table. On looking down the eyepiece we see a scale (Fig. 18), the
eyepiece being, if necessary, focused until the divisions of the scale
are clearly and distinctly seen. Suppose, for experiment, we smear a
little vaseline or similar fatty substance on the plane surface of the
dense glass, which just projects beyond the level of the brass plate
embracing it. The field of view is now no longer uniformly illuminated,
but is divided into two parts (Fig. 19): a dark portion above, which
terminates in a curved edge, apparently green in colour, and a bright
portion underneath, which is composed of totally reflected light. If
we tilt the instrument downwards so that light enters the instrument
from above through the vaseline we find that the portions of the field
are reversed, the dark portion being underneath and terminated by
a red edge. It is possible so to arrange the illumination that the
two portions are evenly lighted, and the common edge becomes almost
invisible. It is therefore essential for obtaining satisfactory results
that the plate and the dense glass be shielded from the light by the
disengaged hand. The shadow-edge is curved, and is, indeed, an arc of a
circle, because spherical surfaces are used in the optical arrangements
of the refractometer; by the substitution of cylindrical surfaces it
becomes straight, but sufficient advantage is not secured thereby to
compensate for the greatly increased complexity of the construction.
The shadow-edge is coloured, because the relative dispersion,
_n{v}_ − _n{r}_
——————————————— (_n{v}_ and _n{r}_ being the refractive indices for
_n_
the extreme violet and red rays respectively), of the vaseline differs
from that of the dense glass. The dispersion of the glass is very
high, and exceeds that of any stone for which it can be used. Certain
oils have, however, nearly the same relative dispersion, and the edges
corresponding to them are consequently almost colourless. A careful eye
will perceive that the coloured shadow-edge is in reality a spectrum,
of which the violet end lies in the dark portion of the field and the
red edge merges into the bright portion. The yellow colour of a sodium
flame, which, as has already been stated, is selected as the standard
for the measurement of refractive indices, lies between the green and
the red, and the part of the spectrum to be noted is at the bottom of
the green, and practically, therefore, at the bottom of the shadow,
because the yellow and red are almost lost in the brightness of the
lower portion of the field. If a sodium flame be used as the source of
illumination, the shadow-edge becomes a sharply defined line. The scale
is so graduated and arranged that the reading where this line crosses
the scale gives the corresponding refractive index, the reading, since
the line is curved, being taken in the middle of the field on the
right-hand side of the scale. The refractometer therefore gives at
once, without any intermediate calculation, a value of the refractive
index to the second place of decimals, and a skilled observer
may, by estimating the tenths of the intervals between successive
divisions, arrive at the third place; to facilitate this estimation
the semi-divisions beyond 1·650 have been inserted. The range extends
nearly to 1·800; for any substance with a higher refractive index the
field is dark as far as the limit at the bottom.
[Illustration: FIG. 17.—Method of Using the Refractometer.]
[Illustration: FIG. 18.—Scale of the Refractometer.]
[Illustration: FIG. 19.—Shadow-edge given by a singly refractive
Substance.]
A fat, or a liquid, wets the glass, _i.e._ comes into intimate contact
with it, but if a solid substance be tested in the same way, a film of
air would intervene and entirely prevent an observation. To displace
it, a drop of some liquid which is more highly refractive than the
substance under test must first be applied to the plane surface
of the dense glass. The most convenient liquid for the purpose is
methylene iodide, CH_{2}I_{2}, which, when pure, has at ordinary room
temperatures a refractive index of 1·742. It is almost colourless when
fresh, but turns reddish brown on exposure to light. If desired, it
may be cleared in the manner described below (p. 66), but the film
of liquid actually used is so thin that this precaution is scarcely
necessary. If we test a piece of ordinary glass—one of the slips used
by microscopists for covering thin sections is very convenient for
the purpose—first applying a drop of methylene iodide to the plane
surface of the dense glass of the refractometer (Fig. 20), we notice a
coloured shadow-edge corresponding to the glass-slip at about 1·530 and
another, almost colourless, at 1·742, which corresponds to the liquid.
If the solid substance which is tested is more highly refractive than
methylene iodide, only the latter of the shadow-edges is visible, and
we must utilize some more refractive liquid. We can, however, raise
the refractive index of methylene iodide by dissolving sulphur[2] in
it; the refractive index of the saturated liquid lies well beyond
1·800 and the shadow-edge corresponding to it, therefore, does not
come within the range of the refractometer. The pure and the saturated
liquids can be procured with the instrument, the bottles containing
them being japanned on the outside to exclude light and fitted with
dipping-stoppers, by means of which a drop of the liquid required is
easily transferred to the surface of the glass of the instrument.
So long as the liquid is more highly refractive than the stone, or
whatever may be the substance under examination, its precise refractive
index is of no consequence. The facet used in the test must be flat,
and must be pressed firmly on the instrument, so that it is truly
parallel to the plane surface of the dense glass; for good results,
moreover, it must be bright.
[Illustration: FIG. 20.—Faceted Stone in Position on the Refractometer.]
[Illustration: FIG. 21.—Shadow-edges given by a doubly refractive
substance.]
We have so far assumed that the substance which we are testing is
simple and gives a single shadow-edge; but, as may be seen from Table
V, many of the gem-stones are doubly refractive, and such will,
in general, show in the field of the refractometer two distinct
shadow-edges more or less widely separated. Suppose, for example, we
study the effect produced by a peridot, which displays the phenomenon
to a marked degree. If we revolve the stone so that the facet under
observation remains parallel to the plane surface of the dense glass
of the refractometer and in contact with it, we notice that both the
shadow-edges in general move up or down the scale. In particular cases,
depending upon the relation of the position of the facet selected to
the crystalline symmetry, one or both of them may remain fixed, or
one may even move across the other. But whatever facet of the stone
be used for the test, and however variable be the movements of the
shadow-edges, the highest and lowest readings obtainable remain the
same; they are the principal indices of refraction, such as are stated
in Table III at the end of the book, and their difference measures
the maximum amount of double refraction possessed by the stone. The
procedure is therefore simplicity itself; we have merely to revolve
the stone on the instrument, usually through not more than a right
angle, and note the greatest and least readings. It will be noticed
that the shadow-edges cross the scale symmetrically in the critical and
skewwise in intermediate positions. Fig. 21 represents the effect when
the facet is such as to give simultaneously the two readings required.
The shadow-edges _a_ and _b_, which are coloured in white light,
correspond to the least and greatest respectively of the principal
refractive indices, while the third shadow-edge, which is very faint,
corresponds to the liquid used—methylene iodide. It is possible, as we
shall see in a later chapter, to learn from the motion, if any, of the
shadow-edges something as to the character of the double refraction.
Since, however, each shadow-edge is spectral in white light, they will
not be distinctly separate unless the double refraction exceeds the
relative dispersion. Topaz, for instance, appears in white light to
yield only a single shadow-edge, and may thus easily be distinguished
from tourmaline, in which the double refraction is large enough for
the separation of the two shadow-edges to be clearly discerned. In
sodium light, however, no difficulty is experienced in distinguishing
both the shadow-edges given by substances with small amount of double
refraction, such as chrysoberyl, quartz, and topaz, and a skilled
observer may detect the separation in the extreme instances of apatite,
idocrase, and beryl. The shadow-edge corresponding to the greater
refractive index is always less distinct, because it lies in the bright
portion of the field. If the stone or its facet be small, it must
be moved on the plane surface of the dense glass until the greatest
possible distinctness is imparted to the edge or edges. If it be moved
towards the observer from the further end, a misty shadow appears to
move down the scale until the correct position is reached, when the
edges spring into view.
Any facet of a stone may be utilized so long as it is flat, but the
table-facet is the most convenient, because it is usually the largest,
and it is available even when the stone is mounted. That the stone need
not be removed from its setting is one of the great advantages of this
method. The smaller the stone the more difficult it is to manipulate;
caution especially must be exercised that it be not tilted, not only
because the shadow-edge would be shifted from its true position and an
erroneous value of the refractive index obtained, but also because a
corner or edge of the stone would inevitably scratch the glass of the
instrument, which is far softer than the hard gem-stones. Methylene
iodide will in time attack and stain the glass, and must therefore be
wiped off the instrument immediately after use.
(2) THE METHOD OF MINIMUM DEVIATION
If the stone be too highly refractive for a measurement of its
refractive index to be possible with the refractometer just described,
and it is desired to determine this constant, recourse must be had
to the prismatic method, for which purpose an instrument known as
a goniometer[3] is required. Two angles must be measured; one the
interior angle included between a suitable pair of facets, and the
other the minimum amount of the deviation produced by the pair upon a
beam of light traversing them.
[Illustration: FIG. 22.—Path at Minimum Deviation of a Ray traversing a
Prism formed of two Facets of a Cut Stone.]
Fig. 22 represents a section of a step-cut stone perpendicular to a
series of facets with parallel edges; _t_ is the table, and _a, b, c_,
are facets on the culet side. The path of light traversing the prism
formed by the pair of facets, _t_ and _b_, is indicated. Suppose that
_A_ is the interior angle of the prism, _i_ the angle of incidence of
light at the first facet and _i´_ the angle of emergence at the second
facet, and _r_ and _r´_ the angles inside the stone at the two facets
respectively. Then at the first facet light has been bent through an
angle _i - r_, and again at the second facet through an angle _i´ -
r´_; the angle of deviation, _D_, is therefore given by
_D = i + i´ - (r + r´)_.
We have further that
_r + r´ = A_,
whence it follows that
_A + D = i + i´._
If the stone be mounted on the goniometer and adjusted so that the
edge of the prism is parallel to the axis of rotation of the instrument
and if light from the collimator fall upon the table-facet and the
telescope be turned to the proper position to receive the emergent
beam, a spectral image of the object-slit, or in the case of a doubly
refractive stone in general, two spectral images, will be seen in
white light; in the light of a sodium flame the images will be sharp
and distinct. Suppose that we rotate the stone in the direction of
diminishing deviation and simultaneously the telescope so as to retain
an image in the field of view, we find that the image moves up to and
then away from a certain position, at which, therefore, the deviation
is a minimum. The image moves in the same direction from this position
whichever way the stone be rotated. The question then arises what are
the angles of incidence and refraction under these special conditions.
It is clear that a path of light is reversible; that is to say, if a
beam of light traverses the prism from the facet _t_ to the facet _b_
it can take precisely the same path from the facet _b_ to the facet
_t_. Hence we should be led to expect that, since experiment teaches us
that there is only one position of minimum deviation corresponding to
the same pair of facets, the angles at the two facets must be equal,
_i.e._ _i = i´_, and _r = r´_. It is, indeed, not difficult to prove by
either geometrical or analytical methods that such is the case.
_A_ _A_ + _D_
Therefore at minimum deviation _r_ = ——— and _i_ = —————————, and,
2 2
since sin _i_ = _n_ sin _r_, where _n_ is the refractive index of
the stone, we have the simple relation—
_A_ + _D_
sin —————————
2
_n_ = —————————————
_A_
sin ———
2
This relation is strictly true only when the direction of minimum
deviation is one of crystalline symmetry in the stone, and holds
therefore in general for all singly refractive stones, and for the
ordinary ray of a uniaxial stone; but the values thus obtained even in
the case of biaxial stones are approximate enough for discriminative
purposes. If then the stone be singly refractive, the result is the
index required; if it be uniaxial, one value is the ordinary index
and the other image gives a value lying between the ordinary and the
extraordinary indices; if it be biaxial, the values given by the two
images may lie anywhere between the greatest and the least refractive
indices. The angle _A_ must not be too large; otherwise the light will
not emerge at the second facet, but will be totally reflected inside
the stone: on the other hand, it must not be too small, because any
error in its determination would then seriously affect the accuracy of
the value derived for the refractive index. Although the monochromatic
light of a sodium flame is essential for precise work, a sufficiently
approximate value for discriminative purposes is obtained by noting the
position of the yellow portion of the spectral image given in white
light.
In the case of a stone such as that depicted in Fig. 22 images are
given by other pairs of facets, for instance _ta_ and _tc_, unless the
angle included by the former is too large. There might therefore be
some doubt, to which pair some particular image corresponded; but no
confusion can arise if the following procedure be adopted.
[Illustration: FIG. 23.—Course of Observations in the Method of Minimum
Deviation.]
The table, or some easily recognizable facet, is selected as the facet
at which light enters the stone. The telescope is first placed in the
position in which it is directly opposite the collimator (_T_{0}_ in
Fig. 23), and clamped. The scale is turned until it reads exactly
zero, 0° or 360°, and clamped. The telescope is released and revolved
in the direction of increasing readings of the scale to the position
of minimum deviation, _T_. The reading of the scale gives at once the
angle of minimum deviation, _D_. The holder carrying the stone is now
clamped to the scale, and the telescope is turned to the position,
_T_{1}_,in which the image given by reflection from the table facet is
in the centre of the field of view; the reading of the scale is taken.
The telescope is clamped, and the scale is released and rotated until
it reads the angle already found for _D_. If no mistake has been made,
the reflected image from the second facet is now in the field of view.
It will probably not be quite central, as theoretically it should be,
because the stone may not have been originally quite in the position
of minimum deviation, a comparatively large rotation of the stone
producing no apparent change in the position of the refracted image at
minimum deviation, and further, because, as has already been stated,
the method is not strictly true for biaxial stones. The difference in
readings, however, should not exceed 2°. The reading, _S_, of the scale
is now taken, and it together with 180° subtracted from the reading for
the first facet, and the value of _A_, the interior angle between the
two facets, obtained.
Let us take an example.
Reading _T_ (= _D_) 40° 41´ Reading T_{1} 261° 35´
less 180° 180 0
———————
81 35
Reading _S_ 41 30
———————
½_D_ 20 20½ _A_ 40 5
½_A_ 20 2½ ½_A_ 20 2½
———————
½(_A_ + _D_) 40 23
Log sin 40° 23´ 9.81151
Log sin 20 2½ 9.53492
———————
Log _n_ 0.27659
_n_ = 1.8906.
The readings _S_ and _T_ are very nearly the same, and therefore we may
be sure that no mistake has been made in the selection of the facets.
In place of logarithm-tables we may make use of the diagram on Plate
II. The radial lines correspond to the angles of minimum deviation and
the skew lines to the prism angles, and the distance along the radial
lines gives the refractive index. We run our eye along the line for the
observed angle of minimum deviation and note where it meets the curve
for the observed prism angle; the refractive index corresponding to the
point of intersection is at once read off.
This method has several obvious disadvantages: it requires the use
of an expensive and elaborate instrument, an observation takes
considerable time, and the values of the principal refractive indices
cannot in general be immediately determined.
Table III at the end of the book gives the refractive indices of the
gem-stones.
[Illustration: _PLATE II_
REFRACTIVE INDEX DIAGRAM]