# Collected Physical Papers/A Simple and Accurate Method of Determination of Index of Refraction for Light

VII

A SIMPLE AND ACCURATE METHOD OF DETERMINATION OF INDEX OF REFRACTION FOR LIGHT

The apparatus usually employed for the determination of the index of refraction is very elaborate and costly. Numerous adjustments have to be made, and a long time spent to secure accurate results. The following method for the determination of the optical index is a modification of the one which has for some time been employed by me in the determination of the index of electric refraction. The apparatus required is very simple, and I made a rough model of it at a trifling cost, and gave it to my pupils for trial. I was surprised to find how, even in their inexperienced hands, it gave results which would compare favourably with the various determinations of the indices hitherto made. The other advantages of this method are the quickness with which a determination can be made, and its adaptability to lecture demonstrations.

The method depends on the determination of the critical angle at which total reflection takes place. The principle of total reflection has also been employed by M. M. Terquem and Trannin, but the method described below is somewhat different, being much simpler, as no Collimator and observing Telescope are required. It has the additional advantage of being applicable to solids; and by a process of repetition, the value of the critical angle is obtained with a remarkably high degree of accuracy.

A beam of light is refracted from the given substance into air, and the angle of incidence gradually increased till total reflection just takes place. From the critical angle i thus determined, the index is found from the formula ${\displaystyle \mu ={\frac {1}{\sin i}}}$.

The necessary appliances for the determination are: (1) a hollow glass cylinder, which may be a beaker, for liquids (or a stoppered phial for volatile substances), or two semi-cylinders of the given solid; (2) a circular wooden table graduated into degrees and capable of rotation round a vertical axis. The table has at the centre a raised circular platform carrying an index. The platform can revolve round the common vertical axis independently of the table. When the platform is clamped down, the circle and the platform revolve together round a common axis.

The apparatus may be used for the following investigations:—

(1) Determination of the indices of solids and liquids.
(2) Variation of the index with the strength of different solutions.
(3) Variation of the index with temperature.
(4) Determination of the indices for the different rays, and of the dispersive power.

1. Determination of the index for liquids

A vertical, rectangular plate is suspended so as to divide the liquid in the glass cylinder into two halves. A narrow central and vertical slit is cut in the rectangular plate. Two thin microscope cover slips about one cm. square are superposed and cemented with an interposed air-film, and the piece is fixed against, say, the lower portion of the vertical slit. The liquid is, of course, continuous through the upper and uncovered portion of the slit; but, in the lower portion, the air-film separates the two semi-cylinders of liquid. Light passing through one of these semi-cylinders is thus incident on the second medium, the film of air, and when the angle of incidence is sufficiently large, the beam undergoes total reflection. In order that all the rays might undergo total reflection simultaneously, it is necessary that the incident beam should be parallel.

Adjustment for rendering the beam parallel.—The diagram represents a section of the liquid semi-cylinders: the one to the left we shall call P, that to the right Q.

Fig. 12. The source of light O and its image at O′

Suppose O to be the principal focus of P; light diverging from a source O, would emerge into the air-film as a parallel beam, and would be focussed by Q at an equal distance, O′ to the right of the air-film. This will be all the more exact if a narrow diaphragm cut off all but the central rays. To render the beam incident on the air-film parallel, it is therefore only necessary to bring gradually the source of light near the cylinder, till an image is formed at an equal distance on the other side, the distances being measured from the axis of the cylinder. It will be observed that the semi-cylinder P acts as a collimating lens, and Q as the object-glass of an observing telescope. It will also be seen that the value of the index may now be obtained with fair accuracy from the distance OO′ (which is equal to four times the focal distance of the cylinder) and the radius of the cylinder.

As the lens is cylindrical, the source of light may be a line instead of a point. A narrow slit may be illuminated by sodium light for monochromatic measurements; or the slit may be illuminated by sun-light. It is sometimes more convenient to use the incandescent filament of a glow lamp as the source of light.

Fig. 13. Four positions for Total Reflection.
Method of experiment.—The cylinder containing the liquid with the interposed air-film is placed on the platform, the axis of the cylinder passing through the axis of the graduated circle. If the cylinder is kept fixed, and the direction of incident light changed, it will be observed that there are four positions, A, B, C, D, for total reflection, the angles AOB or DOC being each equal to twice the critical angle.

The source of light may be kept fixed, and the angle of incidence varied by rotating the central platform on which the cylinder with the interposed film is placed, till total reflection just takes place. The image produced remains fixed, till at the critical angle it suddenly disappears from the screen; this is a great advantage, as the trouble entailed in following the deflected image in the prism-method is avoided. As the cylinder revolves round, total reflection will take place four times, twice

Fig. 14. The Spectrometer circle. There is a sliding tube with a slit at one end. The distance of the slit from the centre of the cylinder is adjusted till a sharp image is formed at an equal distance on the screen to the right. A vertical plate with a slit and air-film is suspended dividing the cylinder in two halves.

from the air-film surface of each semi-cylinder. The source of light and a screen being now fixed at the extremities of a diameter passing through the centre of the circle, the distance of the slit is adjusted for rendering the beam incident on the air-film parallel, in the manner already described. When the angle of incidence reaches the critical value, the image on the screen is suddenly extinguished. In the centre of the screen there is a circular aperture, behind which the eye itself may be placed to receive the light directly. When white light is used, as in the determination of the indices for the different rays, the image is cast on the slit of a small direct vision spectroscope, and the spectrum observed.

A portion of light may be allowed to pass through the upper part of the slit, where the liquid is continuous, and the corresponding portion of the image will never be extinguished. This portion of the image is used in lecture demonstrations as a marked line of reference. But in the lower portion of the image, light has to pass through the air-film, and the bright image suddenly disappears from the screen as soon as the angle of incidence exceeds the critical value.

During a complete rotation of the cylinder, two pairs of readings will be obtained for total reflection. In the first set, one-half of the semi-cylinder will act as the collimating lens, the other half acting as the focussing lens: in the next set (when the cylinder is rotated through 180°) the functions of the two halves are exchanged. From the mean of the differences obtained from the two sets of observations, the value of 2 i is obtained with great accuracy, provided the circle is accurately graduated, and the reading Vernier permits of small angular measurements being made.

The circle used in the simple form of apparatus being only divided into degrees, I could not expect to obtain from it, in the ordinary way, any very accurate results; but by using the following method, it was possible to obtain results which are highly accurate.

Fig. 15. The dotted lines represent the two positions of the air-film for total reflection.
The platform index is at first clamped down pointing to the zero of the scale, and the circle as a whole rotated till total reflection takes place. The circle is next clamped, and the platform carrying the cylinder unclamped and rotated in the direction of the lower arrow till light is again totally reflected, the index reading being now B. The difference of the two readings evidently gives twice the critical angle. The angle is repeated by clamping down the index and rotating the circle as a whole for total reflection in the direction of the upper arrow. The process may thus be repeated, the rotation of the platform in one direction being followed by the rotation of the circle in an opposite direction. There is thus produced at each operation a relative displacement of the index (in reference to the circle) through twice the critical angle. The value of i is by this means obtained with great accuracy, the errors due to eccentricity or defect in graduation being eliminated by repeating the angles n times where 2 ni is as near as possible equal to four right angles or any multiple of four right angles. If R be the difference between the first (zero) and the last readings, the critical angle ${\displaystyle i={\frac {R}{2n}}}$. It is evident that by merely increasing n, the value of i could be obtained with any degree of accuracy, even with circles graduated only into degrees.

The index is fixed at the beginning against the zero of scale. It is therefore only necessary to take one final reading, and count the number of repetitions. A complete determination could thus be made in the course of a few minutes.

The air-film, as has been said before, should be placed in the vertical plane dividing the cylinder into two halves. This is more easily accomplished by suspending a metallic plate on knife edges, from V grooves cut at the ends of a diameter of the upper end of the cylinder. The plate remains vertical under the action of gravity. A central slit is cut in this plate; on opposite sides of this central diaphragm are cemented, as previously stated, the two glass slips which contain the air-film.

One set of readings having been obtained with the semi-cylinder P turned towards the source of light, the cylinder is turned round through 180°, and observation repeated with Q occupying the previous position of P; it will be seen that by this procedure the air-film is also reversed. The mean value of the critical angle obtained from the two sets of observations, one "direct," the other "reverse," is thus free from any outstanding error.

Having explained the principle of measurement, I now proceed to give an account of the experiments, and compare the result obtained by this method with those obtained by previous observers. The values for carbon disulphide, absolute alcohol, glycerine have been determined, but the results obtained by different observers do not agree. The reason for this discrepancy is obvious; it is almost impossible to obtain two different specimens of these substances exactly alike. The only substance which can be obtained in a state of approximate purity is distilled water, but even here we have various contaminations by the absorption of different gases like ammonia and carbonic acid from the atmosphere; or glass itself may be dissolved in minute qualities. The values of the index for water obtained by different observers are therefore not very concordant. The following are the values of the index of water for the D ray:—

 Wollaston and Brewster, D ray (temp. not given)•          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          • 1·336
 Sir John Herschel (at density of 1)•          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          • 1·3336
The following are for the D ray at a temperature of 18°:—
 Terquem and Trannin•          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          • 1·3336
 Gladstone and Dale (reduced to 18°)•          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          • 1·3322

If we reject the first two values as somewhat indefinite and take the last two, we find a difference of more than 1 part in a thousand, Gladstone and Dale's result being the lower.

With C ray at the temperature 18°:—

 Fraunhofer•          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          • 1·3317
 Gladstone and Dale (reduced to 18°)•          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          • 1·3304

Here also the same difference is observed. Thus the minimum difference between the best observations is about 1 part in a thousand.

My determinations were made with ordinary distilled water, condensed by passing steam through a coil of copper immersed in cold water. The experiments were carried out with the two following apparatus:—

I. A large circle, roughly graduated into degrees, with a diameter of 45 cm.; the glass trough used is 25 cm. in diameter.
II. The circle was obtained from an old theodolite. The Vernier reads to 1′. The glass cylinder used has a diameter of 14·4 cm.

It must be remarked here that the cylinders at my disposal, though circular in section, were slightly conical in shape. I tried to obviate this defect by using a very small portion of the height of the cylinder for light to pass through, the length of the slit being reduced to about 5 mm.

Just before total reflection, a series of images of the slit are seen reflected by the air-film. When white light is used, and the image is observed through a direct vision spectroscope, interference bands are observed in the spectrum, which flit across it with the slightest rotation of the cylinder.

I give below the results obtained from determinations made on different days (temp. = 26°).

(1) The larger apparatus was used for this experiment (diameter of the cylinder = 25 cm.), and the angle repeated fifteen times.
(2) Experiment with the smaller apparatus (diameter of the cylinder = 14·4 cm.). Reading was only taken of two successive positions for total reflection with the air-film 'direct' and 'reverse.'
(3) The same as above but with a different air-film.
(1) Angle repeated fifteen times.
 Direct.⁠ ⁠Reverse. 1459° ⁠1460°

Mean value for ${\displaystyle \scriptstyle {i={\frac {1459\cdot 5}{2\times 15}}=48^{\circ }\ 39^{\prime }}}$

μ = 1·3321.

From (2) and (3), mean value of i obtained = 48° 40′.

μ = 1·3317.

The values of μ from the best determinations reduced to the temp. 26° are given below:—

 Gladstone and Dale•          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          • 1·3315
 Terquem and Trannin•          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          • 1·3329

The result obtained by me is thus seen to be practically the same as the above.

Having obtained the absolute value of the index of water for the D line and at the temperature of 26°, the following investigations were made to study the effect of different strengths of solution, of temperature, and of different rays in the variation of the index.

It must be remembered that we are concerned here in measurement of effect of small variations from the standard. The absolute value of the index under the changed condition is deduced from the value under standard condition and the small observed variation.

2. Variation of the index with the different strengths of solution

The apparatus is first adjusted for total reflection with distilled water. To the distilled water in the cylinder is added enough salt (e.g., sodium chloride) to make, say, a five per cent. solution. This produces an increase of refractive power, and the angle of incidence has to be decreased to reach the critical angle. The difference of the two readings subtracted from the critical angle for distilled water (obtained from a previous accurate determination) gives the critical angle for five per cent. solution, and hence the absolute index. It is to be noticed here that a large number of determinations can be made rapidly by merely adding requisite quantities of salt and taking the corresponding readings.

With a finely-graduated circle, provided with a Vernier, the differences of readings may easily be obtained. With a roughly-graduated scale, the angular differences may be determined by fixing a mirror over the upper portion of the slit, and measuring the small differences in the usual way, by observing a reflected scale through a telescope, or receiving the reflected spot on a scale. A curve of variation of the indices with the different strengths of solution may thus be easily obtained.

3. Effect of temperature

The index of a liquid is increased and the critical angle decreased with the lowering of temperature. Hot water is poured into the cylinder, and the cylinder rotated till the angle of incidence is just a little less than the critical angle for the particular temperature. The liquid is stirred at intervals, and a thermometer bulb placed in the same horizontal layer of the liquid through which light is passing. As the liquid slowly cools down, the critical angle is decreased, and at a certain temperature the image suddenly disappears. The corresponding temperature of the liquid is now observed. The angle is then decreased by a small known amount, and the new temperature for total reflection again observed. In this way the slight variation of the angle with the variation of temperature is found. The absolute value of the critical angle for a standard temperature is known from a previous experiment. Hence the indices for different temperatures may be easily deduced.

4. On the determination for different rays and the dispersive power

The value for the D line having been accurately determined by the method of repetition, the values for the other rays are found in the following way:—

Sunlight may be used for the experiment; the image falls on the slit of a direct vision spectroscope. A spectrum is thus formed, in which the well-known Fraunhofer's lines are present. As the cylinder is rotated, a shadow gradually moves along the spectrum, beginning with the most refrangible end. The shadow is made to coincide with the different absorption lines, and the differences between these readings and the "D" reading give the absolute critical angles for the different rays. In this way, the values of the indices for different rays are found, and hence the dispersive power. A greater degree of accuracy is obtained by repeating the difference.

The incandescent filament of a glow lamp may also be used as a source of light. The micrometer in the eye-piece of the spectroscope should have been previously calibrated.

The following determination was made to find the difference between D and F lines. This was repeated ten times, which gave a total difference of 2° 5′. The critical angle for the F ray is therefore less than the angle for D ray by 12·5′. Hence taking the mean critical angle for D to be 48° 39·5′.

μ for D=1·3319.
μ for F =1·3362.
 The difference between D and F, as found by me, is therefore about•          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          • 43 parts in 13,000 at 26°.
 Gladstone and Dale found a difference of•          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          • 42⁠„⁠„⁠„⁠at 15°.
 This difference will, however, be reduced to about•          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          •          • 41⁠„⁠„⁠„⁠at 26°.

Indices for solids.—The method for the determination of the indices for solids is precisely the same as that for liquids. The solids are cut in the form of two small semi-cylinders, a process which is not so difficult as the cutting of a prism.

Lecture Demonstration

A few words may now be said about the modification of the above for lecture demonstrations. As only a moderate degree of accuracy is required, the source of light (a slit illuminated with sunlight, or any other powerful source) is brought slightly nearer the cylinder, so that the image is cast on a distant screen.

Effect of temperature.—Half the height of the cylinder is filled with cold liquid a circular piece of mica being placed above. Hot liquid is now slowly and cautiously poured over mica. When the cylinder is rotated, the light from the lower half would be the first to undergo total reflection, and this totally-reflected image may be received on a second suitably-placed screen. Light would, however, be still transmitted through the hot portion of the liquid. There would thus be a transference of one-half of the image from one screen to the other. On further rotation the missing portion would join its other half on the second screen. A slight rotation in one direction or the other would produce corresponding transference of the images from one screen to the other.

Different indices for the different rays.—A spectrum is formed by allowing light which forms the image to pass through a carbon bisulphide prism. As the rotation of the cylinder is continued, different portions of the spectrum would be totally reflected in succession, and would appear on the second screen, the spectra on the two screens being complementary.

Conclusion.—My object has been to get a fairly accurate determination of the index and its variations with an improvised and inexpensive apparatus, which can easily be set up. Even with the simple apparatus described above, the quantitative results obtained were shown to possess great accuracy. The determinations were found to be capable of being made with considerable rapidity. Only one final reading was all that was necessary for the determination of the absolute index. With a finely graduated circle, a true cylinder, and a perfectly parallel air-film, there is no reason why the method described above should not give results possessing the highest degree of accuracy.

(From unpublished Paper, 1895.)