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Derivation of lattice parameter of zinc blende crystal structure

  1. Mar 25, 2012 #1
    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 = [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.
    x2 = 2l2 – 2l2cosθ
    then
    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 = 2x. But from the crystal structure, my mind tells me [itex]\sqrt{2}a[/itex] = 2x.

    Does anyone know what's going on?

     
    Last edited: Mar 25, 2012
  2. jcsd
  3. Mar 26, 2012 #2
    Found my mistake! the bond angle is arcos(-⅓)!
     
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