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

I looked up the value and I've found it to be

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

with two equal sides of the separation distance

then

Now, I'm having a hard time relating

Does anyone know what's going on?

*l*.I looked up the value and I've found it to be

*a*= [itex]\frac{4}{\sqrt{3}}[/itex]*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}[itex]\frac{1}{3}[/itex]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.*x*^{2}= 2*l*^{2}– 2*l*^{2}cosθthen

*x*= [itex]\frac{2}{\sqrt{3}}[/itex]*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*= 2*x*. But from the crystal structure, my mind tells me [itex]\sqrt{2}a[/itex] = 2*x*.Does anyone know what's going on?

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