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Coordination number counts the touching neighbours around one atom, and it is fixed by the lattice: six for simple cubic, eight for body-centred cubic, twelve for both close-packed structures and only four for diamond. Since neighbours touch, the centre-to-centre distance is twice the atomic radius regardless of structure - 256 pm for copper's 128 pm radius.
Nearest neighbour distance
d = 2r for touching spheres
Structure relations
SC: CN 6; BCC: CN 8; FCC/HCP: CN 12; diamond: CN 4
Both are close packings of spheres - six neighbours in the plane, three above and three below. They differ only in stacking sequence.
Because bonding is directional. Four covalent bonds at tetrahedral angles cannot be packed densely, leaving only 34% occupancy.