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Fig.1: Total Internal Reflection in a Diamond.
Light reaching the facet at an angle larger than the critical angle will be reflected.

Brilliance is the degree of "brightness" of a gemstone, resulting from light reflected from and refracted off the crown and pavilion facets. Brilliance is an overall visual perception based on the physical laws of refraction and reflection.

When light hits the surface of a transparent optically denser medium (like a gemstone) it will be either reflected from the surface (bounced back) or partially refracted (bent and sent in another direction) inside the stone. When a light ray is refracted, it hits a pavilion facet and either is refracted out of the stone or reflected inside the stone depending on which angle the light ray approaches the facet (Fig.1, see its explanation below).

Every stone has an angle at which the light ray will either be refracted (sent along a different path) or reflected (bounced back). This angle is known as the critical angle and depends on the refractive index of the stone (how much the light ray is bent). The higher the refractive index, the smaller the critical angle.

When a light ray approaches a pavilion facet at an angle larger than the critical angle, it will be completely reflected inside the stone. The opposite dictates that light which falls at an angle smaller than the critical angle will be refracted outside the stone. This property is very important in the fashioning of a gemstone to create brilliance or "life". As every gemstone has its own critical angle, the design of the cut needs to be adjusted for the stone at hand.

Explanation of Fig.1.: In this example, a brilliant cut Diamond is shown, having a critical angle ("ca") of approximately 24°26' (24 degrees and 26 minutes). The path the light travels (the a blue line) is as follows:

  1. light reaches the crown of the stone and is refracted (bent) inside the stone, bending towards point number 2
  2. the light ray reaches a pavilion facet at an angle larger than the critical angle (ca), so it will be reflected (bounced) towards 3
  3. at point 3, the light ray again reaches the pavilion facet at an angle larger than the critical angle so will be reflected towards 4
  4. here, the light ray reaches the crown at an angle which is smaller than the ca, so it will be refracted out of the stone

Note: The critical angle is measured from an imaginary line named the normal (NO). This line is drawn at 90° to the surface of the facet at the point where the light reaches that facet.

In a well proportioned transparent stone, all light that enters the faceted stone through the crown will be trapped inside the stone for a while and then be refracted out of the stone through the crown. This behavior is known as Total Internal Reflection (often abbreviated as TIR) and it is the key ingredient in the design of a refractometer. It should be noted that this unique phenomenon only occurs on the boundaries of an optically denser medium (gemstone) and an optically rarer medium (such as air), when light travels inside the denser medium.

When a transparent stone is poorly cut (with either a too shallow or too deep pavilion), light will leave through the pavilion. Light bleeding through the pavilion facets causes a stone to appear either too light or too dark. Brilliance relies much on transparency, therefore stones with good transparency will show better brilliance when cut well.

Of course, in colored gemstones, other factors such as color zoning, pleochroism, etc also play a role in deciding how a stone needs to be cut for best optical performance.


<math>critical\ angle = \arcsin \left (\frac {1}{n}\right )\ \Rightarrow \ critical\ angle = \arcsin \left ( \frac {1}{2.417}\right )\ \approx\ 24^\circ26'</math>
Relationship between critical angle and index of refraction (click to enlarge)

The critical angle can be calculated as the inverse Sine of 1 divided by n (the refractive index) of a gemstone. For Diamond with n = 2.417, the calculation will be the inverse sine of 1/2.417 (where 1 is the refractive index of air).

Note: the actual formula is arcsin(n2 / n1), but as we gemologists usually are only concerned with the critical angle between air and the gem, n2 = 1.

From the critical angle formula, the relationship between the refractive index of a gemstone and the critical angle becomes apparent. The higher the refractive index, the smaller the critical angle.

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