The packing fractions of hexagonal close packed unit cell and face centred cubic unit cell come out to be same when calculated geometrically.

Is it just a coincidence? Differently put, could it be predicted that the packing fractions of HCP and FCC will be same without calculating them geometrically?

  • $\begingroup$ Look at them hexagonal layers. They are the same in both structures. Now stack one layer on top of the other. This is also done in exactly the same way in both structures. How could the volumes be any different? $\endgroup$ – Ivan Neretin Mar 6 '19 at 8:00
  • $\begingroup$ @IvanNeretin Can you explain more? From what I understand, HCP has AB type packing, and FCC has ABC type packing, and hence are different. Or are they ultimately the same thing? $\endgroup$ – Anubhab Das Mar 6 '19 at 10:16
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    $\begingroup$ True, they are different, but in a way that's of no importance to us when we consider density. Look again at the layers: they are identical. Look at the way one layer is placed on top of the other, and consider the distances between layers: they are identical. What else do we need? Nothing. $\endgroup$ – Ivan Neretin Mar 6 '19 at 10:23
  • $\begingroup$ The C layer is just the A layer placed differently and causes no difference in the density. Is that right? $\endgroup$ – Anubhab Das Mar 6 '19 at 10:46
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    $\begingroup$ Yes, but that's not quite my point. Take just two adjacent layers out of the structure. Look at them as hard as you can. What are they? Maybe A and B? Or B and C? Or C and A? Where are they from, HCP or FCC? You won't be able to tell. There is simply no way of knowing. Any pair is precisely identical to any other. The distance between the layers is the same, and hence so is the packing density. $\endgroup$ – Ivan Neretin Mar 6 '19 at 11:32

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