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?

 

[emmanuelpn]

Found my mistake! the bond angle is arcos(-⅓)!