For the given 2-D parallelogram unit cell in hexagonal close packing, the packing fraction will be:
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Solution
Packing fraction 2-D :
From the figure, we can say, Total number of atom in an unit cell, Zeff=1
Packing faction =Area occupied by circlesArea of unit cell
Let us consider, Edge length =l
Thus, radius (r)=l2
From figure, it is clear that parallelogram has one complete circle.
Thus, Area of circle =πr2=π(l2)2 Area of parallelogram =2×Area of ΔABC
From area of triangle, we know Area of ΔABC=12BC×AD AD2=AB2−BD2 =l2−l24=3l24 ∴AD=√3(l2) ∴ Area of ΔABC=12(l×√3l2)=√3l24 ∴ Area of parallelogram =2× Area of ΔABC =2√3l24=√3l22 ∴ Packing fraction =πl24√3l22=π2√3=0.907