The Refractive Index of a Solid

1. The Refractive Index of a Solid An unusual application of spectroscopy

2. Background The goal of this experiment is to obtain the optical dispersion of a solid using a reference liquid of known optical dispersion at various, but specific, temperatures. The distortion polarizability and molar refraction of a solid will be calculated as a function of the wavelength of light. Determination of the optical properties of solids is not only of scientific interest but has a wide range of applicability from telecommunications to the design of optical components.

3. The refractive index of a solid is defined as the ratio of the speed of light of a specific wavelength in a vacuum to its speed in the solid. The change of the refractive index with wavelength is called the dispersion. In a solid, the refractive index may also be dependent on the direction the light takes through the material. It is relatively easy to measure the refractive index of a fluid and many instruments exist for that purpose. It is significantly more difficult to measure the refractive index for a solid. Regardless of the medium, a light beam entering a medium of higher refractive index (the more optically “dense” medium) will bend toward the normal to the surface. This is described by Snell’s Law.

4. Snell’s Law is given by:

5. Most materials of interest to the chemist are dielectrics, substances that do not conduct electrical charge. When an electrical potential is applied to them the electrons in the medium shift toward the positive electrode and positive charges shift toward the negative electrode. Thus, the applied electric field induces a dipole in the medium. The dipole moment per unit volume is called the polarization, P. It can be seen this motion of charges will decrease the size of the electric field in the medium from that in the vacuum. Electric Field in vacuum: Eo = 4πσ (σ – surf. chg. density) Electric Field in medium: E = 4πσ/ε Division is by a constant, the dielectric constant (ε), so the magnitude of the field in the medium is decreased.

6. The value of ε is specific to the material. Since the polarization partially cancels the electric field in the medium, the field may be represented as: E = 4π(σ – P) or 4πP = (ε – 1)E. The polarization can be related to the dielectric constant of the material and the external (vacuum) electric field by: The number of molecules of molecular weight, M, in the unit volume that contribute to the polarization is Noρ/M whereρ is the density. If the average dipole induced in each molecule is μ then the dipole in a cm3 of material is μNoρ/M = P.

7. The polarization can now be abandoned for an expression in terms of the microscopic quantity, μ: And since μ = α Eo the Clausius-Mossotti equation is derived: Where it is explicitly noted that both the polarizability and the dielectric constant depend upon the frequency of the field.

8. At optical frequencies, it can be shown that ε(v) = n(v)2 so that: R is the molar refraction which is characteristic of a material. However, it is found that different functional groups and types of bonds may be assigned partial molar refractions that are approximately additive and from them the molar refraction of a material can be estimated. Finally, the polarizability can be obtained from quantum mechanics:

9. The determination of the refractive index of the solid is based on the fact that a solid suspended in a liquid cannot be seen when its refractive index equal to that of the liquid. When the refractive indices of the two media differ, then light is reflected from the interface between the two. This reflected light Appears as an absorption from the beam of light traveling directly through the sample, i.e. it looks like an absorption. The refractive index of a liquid is temperature dependent as well as frequency dependent:

10. The refractive index of a solid is only weakly dependent on the temperature. The dispersion of a solid can be expressed by the Cauchy equation: Where A, B and C are specific to the material. The specular reflection that occurs when two media have different refractive indices is expressed by the Fresnel equation:

11. The light beam traveling through the suspension of solid particles will undergo a large number ( N) of reflections that cause the beam to be of diminished intensity and, therefore, decreased transmittance: T = ( 1 – R)N When the refractive indices of the liquid and solid match then T ~ 1. The refractive index of the liquid can be made to match that of the solid by varying the temperature. Thus, by measuring the transmission spectrum at several temperatures, the optical dispersion of the solid can be obtained.

12. Procedure • Allow cell to be used to equilbrate to 25 C while the sample is being prepared. • Fill a cuvette with pure toluene, the index matching liquid. • Samples will be of KCl, KSO4 or KClO3 or other related salts. • Grind samples separately to very fine powder. • Allow sample to equilibrate to 25 C.

13. Sprinkle a small amount of the solid onto the liquid in the cuvette but only so that it remains suspended without much sinking. • Measure the transmission spectrum from 350nm to 700nm with a UV-Vis spectrophotometer at ~ 1 nm resolution. • Measure subsequent transmission spectra at by increasing by 5 C degrees until the last measurement is made at 70 C. Measure the tranmission of a blank at each temperature.

14. Sprinkle more material at each temperature to assure material is always in suspension. • It is vital that the sample is equilibrated at each temperature. • At conclusion clean cuvettes and dispose of sample as instructed.

15. Data Analysis • Obtain the temperature dependence of the refractive index of toluene from the literature. • Fit the data with an equation that gives the refractive index of toluene as a function of temperature and light wavelength. • Calculate the toluene dispersion curves from 350nm to 700nm for each temperature.

16. Mark the wavelength on the dispersion curve where the corresponding transmission spectrum had its peak transmittance (both curves for the same temperature). • Connect the marks to obtain the dispersion curve of your solid over the measured wavelength region. • Obtain the Cauchy relations for each of the samples. • Plot the wavelength dependence of the polarizability and the molar refraction for your samples.