3.3 Lattice
39
Fig. 3.5 Primitive
translations of the fcc
lattice. These vectors
connect the origin with the
face-center points. The
primitive unit cell is the
rhombohedron spanned by
these vectors. The primitive
translations a , b and c are
given in (3.2). The angle
between the vectors is 60 ◦
a'
b'
c'
a
Fig. 3.6 Primitive
translations of the bcc
lattice. These vectors
connect the origin with the
lattice points in the cube
centers. The primitive unit
cell is the rhombohedron
spanned by these vectors.
The primitive translations
a , b and c are given in
(3.3). The angle between
the vectors is ≈ 70.5 ◦
a'
b'
a
c'
3.3.2.2 Hexagonally Close Packed Structure (hcp)
The 2D hexagonal Bravais lattice fills a plane with spheres (or circles) with maximum filling factor.
There are two ways to fill space with spheres and highest filling factor. One is the fcc lattice, the other
is the hexagonally close packed (hcp) structure.
1 Both fcc and hcp have a filling factor of 74%.
For the hcp, we start with a hexagonally arranged layer of spheres (A), see Fig. 3.7. Each sphere has
six next-neighbor spheres. This could, e.g., be a plane in the fcc perpendicular to the body diagonal.
the next plane B is put on top in such a way that each new sphere touches three spheres of the previous
layer. The third plane can now be added in two different ways: If the spheres of the third layer are
vertically on top of the spheres of layer A, a plane A’ identical to A has been created that is shifted
from A along the stacking direction (normally called the c-axis) by
c hcp =
8/3 a ≈ 1.633 a .
(3.4)
The vectors spanning the primitive unit cell are
a 1 =
a
2
1, −
√
3, 0
, a 2 =
a
2
1,
√
3, 0
, a 3 = c (0, 0, 1) .
(3.5)
1 The hcp structure is not a Bravais lattice since the individual lattice sites are not completely equivalent.
39
Fig. 3.5 Primitive
translations of the fcc
lattice. These vectors
connect the origin with the
face-center points. The
primitive unit cell is the
rhombohedron spanned by
these vectors. The primitive
translations a , b and c are
given in (3.2). The angle
between the vectors is 60 ◦
a'
b'
c'
a
Fig. 3.6 Primitive
translations of the bcc
lattice. These vectors
connect the origin with the
lattice points in the cube
centers. The primitive unit
cell is the rhombohedron
spanned by these vectors.
The primitive translations
a , b and c are given in
(3.3). The angle between
the vectors is ≈ 70.5 ◦
a'
b'
a
c'
3.3.2.2 Hexagonally Close Packed Structure (hcp)
The 2D hexagonal Bravais lattice fills a plane with spheres (or circles) with maximum filling factor.
There are two ways to fill space with spheres and highest filling factor. One is the fcc lattice, the other
is the hexagonally close packed (hcp) structure.
1 Both fcc and hcp have a filling factor of 74%.
For the hcp, we start with a hexagonally arranged layer of spheres (A), see Fig. 3.7. Each sphere has
six next-neighbor spheres. This could, e.g., be a plane in the fcc perpendicular to the body diagonal.
the next plane B is put on top in such a way that each new sphere touches three spheres of the previous
layer. The third plane can now be added in two different ways: If the spheres of the third layer are
vertically on top of the spheres of layer A, a plane A’ identical to A has been created that is shifted
from A along the stacking direction (normally called the c-axis) by
c hcp =
8/3 a ≈ 1.633 a .
(3.4)
The vectors spanning the primitive unit cell are
a 1 =
a
2
1, −
√
3, 0
, a 2 =
a
2
1,
√
3, 0
, a 3 = c (0, 0, 1) .
(3.5)
1 The hcp structure is not a Bravais lattice since the individual lattice sites are not completely equivalent.