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The packing fractionin fcc and hcp struc...

The packing fractionin fcc and hcp structure is same. Moreover, this is the highest packing fraction of all the possible unit cells with same atoms. Can you explain this ?

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The arrangement of atoms in hcp is of ABABAB … type and in case of fcc is of ABCABCABC … type. Both in hcp and fcc, the layers are formed of same atoms in similar arrangements but the only difference is the placement of layers on top of each other. Hence, both have same packing fraction. In both the arrangements, the number of atom of same size present around the atom is maximum, i.e. 6. So, the packing fraction is highest.
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packing fraction of a unit cell is drfined as the fraction of the total volume of the unit cell occupied by the atom(s). P.E=("Volume of the atoms(s) present in a unit cell")/("Volume of unit cell")=(Zxx(4)/(3)pir^(3))/(a^(3)) and % of empty space = 100- P.F.xx100 where Z= effective number of stoms in s cube . r= radius of a an atoms a = edge lenght of the cube Packing fraction in face centered cubic unit cell is :