When studying crystal lattices, I decided to learn which combinations of unit cells and lattice systems lead to the 14 Bravais lattices. I learnt that, for example, no centred triclinic unit cell exists.

Later while searching the Internet and Stack Exchange, I found an answer saying that an end-centred cubic cell can be transformed into a (smaller) tetragonal unit cell as shown in th picture below.

Cubic C and F vs tetragonal P and I

Then I thought on and realised that a face centred cubic lattice can be transformed into a (smaller) body-centred tetragonal unit cell, and into an even smaller one, too.

So, why are some centred unit cells (for hexagonal, triclinic and rhombohedral lattices) considered non-existent, while others (e.g. face-centred cubic) are considered to exist even though they, too, can be reduced?

  • 1
    $\begingroup$ By that logic we should remove all centered unit cells, not just FCC, because each of them can be reduced to some primitive non-centered cell. $\endgroup$ Commented Nov 17, 2015 at 12:47
  • $\begingroup$ I don't know whether it is correct to say"... each of them can be reduced to some primitive non-centered cell."Moreover my question is justified $\endgroup$
    – Sikander
    Commented Nov 17, 2015 at 12:55
  • $\begingroup$ The logic I gave for my question about FCC have earlier been given in this site ,quite unsatisfactorily, so I decided to impose this question again as it has always confused me as to why other type of unit cells are not included in bravais lattice for example end centred cubic unit cell. $\endgroup$
    – Sikander
    Commented Nov 17, 2015 at 13:17
  • $\begingroup$ I think your question is quite unclear. Please not that the 'usual' picture of an fcc lattice is drawn with a 'conventional' unit cell, consisting of multiple atoms, and drawn to highlight the actual cubic symmetry of that particular Bravais lattice. Something like the Wigner-Seitz unit cell, while an appropriate reduced unit cell containing only one atom, and still showing the full Bravais lattice symmetry, is just harder to look at and directly see the cubic nature. I believe that you have difficulties in understanding space groups vs Bravais lattices vs point groups vs conventional cells. $\endgroup$
    – Jon Custer
    Commented Nov 17, 2015 at 17:29
  • $\begingroup$ @Sikander Don’t worry, since an edit to a question on hold automatically puts said question into a reopen review queue. But even after your edit your question remains unclear to me. Are you asking why crystallography uses some centred unit cells even though they can be reduced to primitive ones? $\endgroup$
    – Jan
    Commented Nov 18, 2015 at 10:37

1 Answer 1


In an FCC lattice a=b=c while in a body centered tetragonal lattice a=b but c is unconstrained. The first example you gave should be that an end centered tetragonal is equivqlent to a primative tetragonal lattice. There is no end centered cubic lattice as that would violate the minimum cubic symmetry of 3 four fold symmetry axes. however the transition from FCC to a body centered tetragonal lattice would be displacive not transformational.


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