sin C = 1/n. Why does diamond sparkle? n = 2.42.

Total Internal Reflection — Critical Angle of Diamond

Question

The refractive index of diamond is 2.42. Calculate the critical angle for total internal reflection at the diamond-air interface.

$\sin C = \frac{1}{n}$

Find $C$ and explain why diamond sparkles.

Solution — Step by Step

Write Snell's Law at the critical angle

At the critical angle $C$ , the refracted ray travels along the interface — meaning the angle of refraction is exactly $90°$ . Applying Snell's Law at this boundary:

$n_1 \sin C = n_2 \sin 90°$

Since $\sin 90° = 1$ and $n_2 = 1$ (air), this reduces to:

$\sin C = \frac{n_2}{n_1} = \frac{1}{n}$

Substitute the refractive index of diamond

We have $n = 2.42$ for diamond.

$\sin C = \frac{1}{2.42} = 0.4132$

Find the critical angle

$C = \sin^{-1}(0.4132)$

$\boxed{C \approx 24.4°}$

This is a remarkably small critical angle — one of the smallest of any gem material.

Why This Works

When light travels from a denser medium (diamond, $n = 2.42$ ) into a rarer medium (air, $n = 1$ ), it bends away from the normal. As the angle of incidence increases, the refracted ray bends further and further. At the critical angle, it bends so far that it grazes the surface at $90°$ .

Any angle beyond $C$ — and there's no refracted ray at all. The light has nowhere to go but back into the diamond. This is total internal reflection.

Diamond's $n = 2.42$ is exceptionally high, which pushes $C$ down to just $24.4°$ . Gem cutters know this and shape diamonds with many faces (facets) angled so that light entering from the top hits those faces at angles greater than $24.4°$ . The light bounces internally several times before shooting back out the top — intense, concentrated, brilliant. That's the sparkle.

💡 Expert Tip

For any TIR problem, just remember: $\sin C = \frac{1}{n}$ holds only when the second medium is air (or vacuum). If both media are non-air, the formula is $\sin C = \frac{n_2}{n_1}$ where $n_1 > n_2$ .

Alternative Method

We can also verify this using the full Snell's Law form without assuming the rarer medium is air — useful when the examiner changes the setup (say, diamond-water interface):

$\sin C = \frac{n_{\text{rarer}}}{n_{\text{denser}}}$

For diamond-water ( $n_{\text{water}} = 1.33$ ):

$\sin C = \frac{1.33}{2.42} = 0.549 \implies C \approx 33.3°$

Notice $C$ is larger now — the contrast between the two media is smaller, so you need a steeper angle before TIR kicks in. Diamond underwater sparkles less. Jewellers prefer air gaps for this exact reason.

Common Mistake

⚠️ Common Mistake

Students often write $\sin C = n$ instead of $\sin C = \frac{1}{n}$ .

This comes from confusing which medium the light is travelling from. Remember: the light is going from dense → rare (diamond → air). The denser medium's $n$ goes in the denominator. If you get $\sin C > 1$ after substituting, you've flipped the fraction — catch it immediately, since $\sin$ of any angle can never exceed 1.

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

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