The three points P 1 = ( a / 2 ) ( 0, - 1, - 1 ), P 2 = ( a / 2 ) ( - 1, 0, - 1 ), and P 3 = ( a / 2 ) ( - 1, - 1, 0 ), describe a regular triangle (of edge a 2 / 2) orthogonal to the body diagonal unit vector n = ( 1 / 3 ) ( 1, 1, 1 ) and belong to the plane A, distant a / 3 from the origin. To see this, we remember that in the fcc structure the atom at the origin has 12 neighbors in the positions ( a / 2 ) ( 0, ± 1, ± 1 ) and cyclic permutations. These closed-packed hexagonal planes are arranged orthogonally to the body diagonal of the conventional fcc cube. where C is rotated by 2 π / 6 with respect to the B plane. The ideal closed-packed hexagonal structure occurs in the case atoms behave as hard sphere touching each other.įinally it is worth to mention that another ideal close-packing arrangement is provided by the fcc structure the sequence of hexagonal planes is in this case ABC, ABC, etc. This ratio is only approximately verified in actual hcp crystals. The ratio between 2 a / 3and a 2 / 2equals the ideal value 8 / 3. The six points given by ( a/2)(0, 1, −1) and ( a/2)(0, −1, 1) and cyclic permutations, describe a regular hexagon (orthogonal to n) around the atom at the origin, and belong to plane B finally, the three points given by ( a/2)(0, 1, 1) and cyclic permutations, belong to plane C. The three points P 1= ( a/2)(0, −1, −1), P 2=( a/2)(−1, 0, −1) and P 3=( a/2)(−1, −1, 0), describe a regular triangle (of edge a 2 / 2orthogonal to the body diagonal versor n = ( 1 / 3 ) ( 1, 1, 1 )and belong to the plane A, distant a / 3from the origin. To see this, we remember that in the fcc structure the atom at the origin has twelve neighbours in the positions ( a/2)(0, ☑, ☑) and cyclic permutations. These closed packed hexagonal planes are arranged orthogonally to the body diagonal of the conventional fcc cube. where C is rotated by 2π/6 with respect to the B plane. Several metals have the hcp structure, among them Be ( a = 2.29 Å c = 3.58 Å c/ a = 1.56), Cd ( a = 2.98 Å c = 5.62 Å c/ a = 1.88), Mg ( a = 3.21 Å c = 5.21 Å c/ a = 1.62), Ti ( a = 2.95 Å c = 4.69 Å c/ a = 1.59), Zn ( a = 2.66 Å c = 4.95 Å c/ a = 1.86).įinally it is worth to mention that another close-packing arrangement is provided by the fcc structure the sequence of hexagonal planes is in this case ABC, ABC, etc.

Perfect hexagon measurements