Mineral Oil Index of Refraction

Narumon Suwonjandee and Randy Johnson

March 23, 2000

Plots corrected July 6, 2001

We have used a "freshman physics" spectrometer to measure the index of refraction of two samples of mineral oil for six lines of the mercury spectrum. We have fit these measurements to a single resonance dense material model and find an indexo of refraction, n_D= 1.4627+/-.0008+/-.0004 with a dispersion (n_D-1)/(n_F-n_C) = 51.4 for the Witco oil and n_D= 1.4616+/-.0008+/-.0004 with a dispersion (n_D-1)/(n_F-n_C) = 51.3 for the Amoco oil. For the indices of refraction, the first error is the measurement error during measurement of the oil and the second is the calibration error for the angle of the prism.

Apparatus

We have used the above apparatus to measure the index of refraction of the Amoco and the Witco mineral oil as a fuction of wavelength. The apparatus consists of a atomic line source (we used helium and mercury for the prism calibration and just a mercury line source for the oil measurements), a spectrometer table which measured angles to a minute of arc, and an approximately equilateral plexiglass container (wall thicknesses of 1/8"). The triangular container was first filled with distilled water to calibrate the interior angle and then with the two samples of mineral oil. For each measurement, the container was rotated on the spectrometer table until the minimum deflection was observed. Under these conditions, the incoming angle with respect to the normal to the front face is equal to the outgoing angle with respect to the normal of the back face and the relationship between the total angular deflection (t), the interior angle of the prism (a), and the index of refraction (n) is given by:

Sin[(t+a)/2] = n Sin[a/2]

Calibration of Prism

The interior angle of the prism was calibrated by filling it with distilled water and measuring the resulting angle of deflection.

Table 1 gives the indices of refraction as a function of wave length.

Table 1
Water Indices of Refraction at 20 C

Wave Length (A)	Frequency (PHz)	Index	Calculated Index
7065	2.6680	1.3300	1.3301
5893	3.1986	1.3330	1.3328
5016	3.7579	1.3364	1.3361
4047	4.6577	1.3427	1.3428

The indices of refraction are fit to a "one resonance" model for dense media:

        n^2 - 1           a
        -----   = -------------
        n^2 + 2     (w_0^2 - w^2)

where a and w_0 are fit parameters. For the calculated values given in table 1, a=88.42 (PHz)^2 and w_0=20.98 PHz.

With distilled water in the prism, we measured the angular deviations given below in Table 2.

Table 2
Measured Angles with Distilled Water

Wave Length	Index	Angle	Measurement Sigma	Number of Measurements	Prism Angle
Mercury
5790	1.3331	23.083	.13	6	59.044
5461	1.3342	23.289	.10	6	59.272
4358	1.3402	23.578	.19	6	58.976
Helium
6563	1.3311	23.172	.08	3	59.489
4861	1.3369	23.256	.07	3	58.839
4340	1.3403	23.794	.05	3	59.352

The prism angle is calculated as:

alpha = 2 ArcTan[Sin[gamma/2]/(n-Cos[gamma/2])]

where alpha is the interior angle of the prism and gamma is the measured deflection angle.

From these measurements, we conclude that

The average error on a single measurement is +/- .13o; and
The interior angle of the prism is 59.14 +/- .04.

Oil Results

Measurements were made with six lines from mercury for both Amoco and Witco oil. The results of these measurements are summarized in tables 3 and 4. All of the indices have a correlated error of +/- .0004 due to the error in the measurement of the angle of the prism. This error is not included in the tables.

Table 3
Amoco Oil Measurements

Wave Length (A)	Frequency (EHz)	Measured Angle	Angle Sigma	Calculated Index	Index Error
6907	2.730	32.82	.04	1.4572	.0008
5790	3.257	33.22	.03	1.4621	.0008
5461	3.453	33.34	.04	1.4636	.0008
4916	3.836	33.78	.14	1.4689	.0008
4358	4.327	34.11	.07	1.4729	.0008
4047	4.660	34.38	.06	1.4762	.0008

Table 4
Witco Oil Measurements

Wave Length (A)	Frequency (EHz)	Measured Angle	Angle Sigma	Calculated Index	Index Error
6907	2.730	32.98	.12	1.4591	.0008
5790	3.257	33.29	.08	1.4629	.0008
5461	3.453	33.53	.17	1.4659	.0008
4916	3.836	33.76	.09	1.4687	.0008
4358	4.327	34.22	.08	1.4743	.0008
4047	4.660	34.52	.08	1.4779	.0008

These results are plotted in Figure 1 (click on the picture for a PostScript version) along with a single resonance fit to the data. The parameters of these fits are given in Table 5.

Figure 1

Table 5
Mineral Oil Index of Refraction Fit Parameters

Oil	a	w_0	n_D	Dispersive Index
Amoco	112.4	20.48	1.4616	51.3
Witco	114.9	20.68	1.4627	51.4