# Derivation of lattice parameter of zinc blende crystal structure

emmanuelpn
I need to derive the lattice parameter in terms of the Zn-S separation distance, l.

I looked up the value and I've found it to be
a = $\frac{4}{\sqrt{3}}$l

The way that I started my derivation was saying that each tetrahedron has a sulfide ion in the center, so then we can make a triangle from the center point, and two zinc adjacent atoms. This isosceles triangle will have an angle
θ = cos-1$\frac{1}{3}$
with two equal sides of the separation distance l, and an opposite side of the angle θ, let's call it x. Finding x is then easy using the law of cosines.
x2 = 2l2 – 2l2cosθ
then
x = $\frac{2}{\sqrt{3}}$l

Now, I'm having a hard time relating x to a. And the only way it seems to work out to get the answer I looked up is by saying a = 2x. But from the crystal structure, my mind tells me $\sqrt{2}a$ = 2x.

Does anyone know what's going on?

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