# Is the number of tetrahedral voids always equal to double the number of octahedral voids in any crystal?

I came accross the question whether the number of tetrahedal voids is always equal to twice the number of octahedral voids (in the case where both the voids are present). The key was given as the statement was true. Could anyone justify this?
I know that it is true for ccp and hcp but can this be generalized for all types of crystals?

• Well alpha polonium has a primative cubic structure. What types of voids would that crystal have? – MaxW Apr 16 '20 at 9:04
• @MaxW I asked that in the case when the crystal had both the voids would the condition be valid (I would edit the question) – An enthusiast Apr 16 '20 at 9:22
• Probably true for all closed packing lattices, not sure if can be generalized – Zenix Apr 16 '20 at 12:19
• @Zenix Then in the spinel structure tetrahedral voids and octahedral voids are occupied then in that case would there be any complexity in deciding the validity of this statement or is it's answer given is wrong – An enthusiast Apr 17 '20 at 3:57
• The title reads "Is the number of tetrahedral voids always equal to number of octahedral voids in any crystal?"; Didn't you mean double the number of octahedral voids? – Guru Vishnu Apr 24 '20 at 6:53

Note that, in case of tetrahedral voids, each sphere is in contact with three tetrahedral sites above, and three tetrahedral sites below, hence, there are two tetrahedral sites associated with each sphere. Thus,

Number of tetrahedral voids $$(N_{th}) = 2 \times$$ (Rank of unit cell)

Or alternatively, if you divide a FCC lattice into 8 octants, each octant will have a tetrahedral site at it's body centre.

For octahedral sites, each sphere is associated with one octahedral sites, thus,

Number of octahedral voids $$(N_o) =$$ Rank of unit cell

For FCC, besides the body center, there is one site at the center of each of the 12 edges. It is surrounded by 6 atoms, three belonging to the same cell unit cell, and the three belonging to two adjacent unit cells.

For clarity see:

For HCP,

• thanks for your answer but my question was was about generalizing the statement I know that it holds for CCP and HCP I think all these are associated with CCP and HCP – An enthusiast Apr 27 '20 at 6:58
• @Adithya I haven't personally encountered any source discussing octahedral and tetrahedral voids for end centred cubics or body centred cubics... – Zenix Apr 27 '20 at 12:43
• Generally tetrahedral and octahedral voids is formed in HCP and CCP structure, and don't form in other crystal structure like primitive cubic and BCC. – Manu Apr 27 '20 at 13:40