# Light Refraction and Reflection in Diamonds

Diamonds and other precious gems sparkle because of the way they reflect, refract, and disperse light. When light hits a polished and cut diamond in air, some of the light bounces off (reflection). But some light bends at a new angle inside the diamond (refraction), bounces off the surfaces inside the diamond (internal reflection), and then bends at another angle to exit the diamond (more refraction). The pattern of reflection and refraction depends on a property tof the diamond called the refractive index, and just as importantly the cut of the diamond. Certain cuts, such as the round brilliant, make the most of a diamond's refractive properties. The branch of physics that deals with the behavior of light is called optics. Here we explain some fundamental equations in optics to show how light behaves with diamonds.

## Refraction

Refraction is when light passes through one medium to another and bends at the interface of the two media. How much the light bends depends on the refractive index of each medium. When light passes from a low refractive index medium to a higher one, the light bends closer to the normal or perpendicular. However, when light passes from a high refractive index medium to a lower one, the light bends away from the perpendicular. The precise relation among the angles and refractive indices is is given by Snell's Law:

n_{1 }* sin(θ_{1}) = n_{2 }* sin(θ_{2})

where n_{1} is the refractive index of Medium 1, θ_{1} is the angle of the incident ray of light with respect the the perpendicular, n_{2} is the refractive index of Medium 2, and θ_{2} is the angle of the refracted light with respect to the perpendicular.

This is illustrated in the diagram on the right, where the incident angle is denoted by α and the angle of refraction is denoted by β. If for example, α = 50°, n_{1} = 1.08, and n_{2} = 2.19, then we can solve for angle β using Snell's Law above. The equation is

1.08*sin(50) = 2.19*sin(β)

(1.08/2.19)*sin(50) = sin(β)

arcsin[(1.08/2.19)*sin(50)] = β

22.2° ≈ β

The refraction index of air is 1 and the refraction index of diamond is about 2.42, one of the highest values for clear solids. Higher density materials tend to have higher refractive indices. For comparison, the R.I. of ice is 1.31 and the R.I. of liquid water is 1.33. The refractive index of diamond's cheaper competitor, cubic zirconia, is about 2.17. Pyrex glass has a refractive index of 1.47.

When light at an incident angle of "i" degrees passes through air and into a diamond, we can again use Snell's Law to compute the refracted angle "r" degrees:

1*sin(i) = 2.42*sin(r)

sin(i)/2.42 = sin(r)

r = arcsin(sin(i)/2.42)

The graph below shows r as a function of i, for i between 0 and 90 degrees. Notice that the maximum value of r is 24.4 degrees. This is called the critical angle for diamond refraction.

The critical angle is important because if the incident ray of light originates inside the diamond, it can only pass through to the air is if the incident angle is less than 24.4 degrees. If the incident angle originating inside the diamond is greater than 24.4 degrees, then when it hits the interface it will reflect back inside the diamond. This is illustrated below.

The phenomenon of refraction and reflection inside the diamond, light bouncing around inside and finally being transmitted back out through the surfaces of the diamond, is what gives diamonds and other gems their sparkle and fire. How the light bends and reflects depends greatly on the angles of the diamond cut, which leads to the next topic.

## The Ideal Diamond Cut

The ideal diamond cut is one such that when light enters from the crown, it is refracted and reflected back out through the crown for the most part. With a good diamond cut, you will have more internal reflection so that the light bounces back out through the table and the facets surrounding the table. with a poor diamond cut, too much light will refract and reflect out through the pavilion, which is hidden in the jewelry setting.

The cut that is ideal for a particular diamond depends on its color and clarity, which can affect the refractive index. Also, what is an ideal cut for a diamond is not necessarily ideal for a cubic zirconia gem, since the materials have different refractive indices. Below is a diagram that shows how light behaves in a diamond with an ideal shape, versus diamonds that are too steep or too shallow.

In the ideal diamond shape, light hits the sides of the pavilion at angles greater than the critical angle, thus the light reflects back inside rather than refracting out. But in the sub-ideal cuts, some light hits the sides at angles less than the critical angle, so the light escapes through the pavilion. Besides internal reflection, there is also external reflection, which also causes a diamond to sparkle.

## External Reflection

Some incident light bounces back on the surface of the diamond rather than refracting inside. This is called reflectivity. The proportion of light that bounces back depends on the refractive indices of the two media. If the refractive indices of Mediums 1 and 2 are n_{1} and n_{2} respectively, then when light passes from Medium 1 to Medium 2, the proportion that reflects back is given by the expression

[(n_{2 }- n_{1})/(n_{2 }+ n_{1})]^2

As you can see from the algebraic form of the formula, it doesn't matter which refraction index is higher or whether you switch the order of Medium 1 and Medium 2. Squaring something always returns a positive value. Using n_{1} = 1 for air and n_{2} = 2.42 for diamond, we have

Reflectivity

= [(n_{2}-n_{1})/(n_{2}+n_{1})]^2 * 100%

= [1.42/3.42]^2 * 100%

= 17.24%

In other words, 17.24% of light reflects back, while the remaining 86.76% is transmitted through the diamond where it refracts and internally reflects. And some of that internal light is also subject to reflectivity before it refracts out of the diamond and into the air, producing even more of the sparkle and fire that make diamonds so beautiful.

In comparison, Pyrex glass has a refractive index of 1.47, so when light hits it from the air, the amount that is reflected back is

[(1.47-1)/(1.47+1)]^2 * 100%

= [0.47/2.47]^2 * 100%

= 3.62%.

In reality, glass seems more reflective than that. For example, on a sunny day a glass window act more like a mirror. This is because reflectivity also depends on the polarization of light.