Usage of Snell’s Law
by Will Moats

Instead of relying on the usual tables to obtain recommended cutting angles, I decided to use Snell’s Law to better determine the critical angle for the new Nigerian tourmaline rough I recently acquired. Paul Hlava revealed during the March meeting of the New Mexico Faceters Guild that he determined this particular tourmaline species to be liddicoatite, a calcium tourmaline. I was not able to locate the specific refractive indices (RI) for liddicoatite in the references I have at home, but I was able to find refractive indices for “calcium tourmaline” in my old optical mineralogy book.

Remember that tourmaline is optically uniaxial and has both a minimum and a maximum refractive index. The refractive indices listed for “calcium tourmaline” are 1.641 (along the A axis) and 1.621 (along the C axis). These refractive index values are somewhat less than those reported for the various varieties of elbaite tourmaline.

For the special case involving the usage of Snell’s Law, where a ray of light enters a gem from the air, the critical angle (CA) is found by the relationship:

1/RI = sin(CA)

Using Snell’s Law for the Nigerian tourmalines that I have, the critical angle for stones with the tables cut perpendicular to the C axis is 38.1 degrees. Also with Snell’s Law, the critical angle for the stones with the tables cut perpendicular to the A axis is 37.5 degrees.

The round pink tourmalines that I showed during the Guild meeting in May were cut with the table perpendicular to the C axis at 38.5 degrees, at a slightly higher angle than what I calculated for the critical angle. When viewed through the table, these stones showed no “fish-eye” effect.

I suggested during the meeting that persons in the Guild who have the necessary equipment could measure the refractive index specifically for this Nigerian tourmaline. I look forward to knowing their results.

{Editor’s comment: For more information regarding optical properties of gemstones, Snell’s Law, refractive indices, and critical angles, please check The Gemstone and Mineral Data Book by John Sinkankas, page 293.}