40
3 Crystals
Fig. 3.7 Structure of the a
hcp and b fcc lattice. For
hcp the stacking (along the
c-axis) is ABABAB. . ., for
fcc (along the body
diagonal) it is
ABCABCABC. . .
(a)
(b)
The hcp stacking order is ABABAB. . . for hcp, the coordination number is 12. In the fcc structure,
the third layer is put on the thus far unfilled positions and forms a new layer C. Only the forth layer is
again identical to A and is shifted by
c fcc =
√
6 a ≈ 2.45 a .
(3.6)
The fcc stacking order is ABCABCABC. . .
In the hexagonal plane of the fcc lattice (which will later be called a {111} plane) the distance
between lattice points is a = a 0 /
√
2, where a 0 is the cubic lattice constant. Thus c =
√
3 a 0 , just what
is expected for the body diagonal.
For real materials with hexagonal lattice the ratio c/a deviates from the ideal value given in (3.4).
Helium comes very close to the ideal value, for Mg it is 1.623, for Zn 1.861. Many hcp metals exhibit
a phase transition to fcc at higher temperatures.
3.3.3 Unit Cell
The choice of the vectors a i making up the lattice is not unique (Fig. 3.1). The volume that is enclosed
in the parallelepiped spanned by the vectors a 1 , a 2 and a 3 is called the elementary cell. A primitive
elementary cell is an elementary cell with the smallest possible volume (Fig. 3.1b). In each primitive
elementary cell there is exactly one lattice point. The coordination number is the number of nextneighbor lattice points. A primitive cubic (pc) lattice, e.g. has a coordination number of 6.
The typically chosen primitive elementary cell is the Wigner–Seitz (WS) cell that reflects the symmetry of the lattice best. The Wigner–Seitz cell around a lattice point R 0 contains all points that are
closer to this lattice point than to any other lattice point. Since all points fulfill such a condition for
some lattice point R i , the Wigner–Seitz cells fill the volume completely. The boundary of the Wigner–
Seitz cell is made up by points that have the same distance to R 0 and some other lattice point(s). The
3 Crystals
Fig. 3.7 Structure of the a
hcp and b fcc lattice. For
hcp the stacking (along the
c-axis) is ABABAB. . ., for
fcc (along the body
diagonal) it is
ABCABCABC. . .
(a)
(b)
The hcp stacking order is ABABAB. . . for hcp, the coordination number is 12. In the fcc structure,
the third layer is put on the thus far unfilled positions and forms a new layer C. Only the forth layer is
again identical to A and is shifted by
c fcc =
√
6 a ≈ 2.45 a .
(3.6)
The fcc stacking order is ABCABCABC. . .
In the hexagonal plane of the fcc lattice (which will later be called a {111} plane) the distance
between lattice points is a = a 0 /
√
2, where a 0 is the cubic lattice constant. Thus c =
√
3 a 0 , just what
is expected for the body diagonal.
For real materials with hexagonal lattice the ratio c/a deviates from the ideal value given in (3.4).
Helium comes very close to the ideal value, for Mg it is 1.623, for Zn 1.861. Many hcp metals exhibit
a phase transition to fcc at higher temperatures.
3.3.3 Unit Cell
The choice of the vectors a i making up the lattice is not unique (Fig. 3.1). The volume that is enclosed
in the parallelepiped spanned by the vectors a 1 , a 2 and a 3 is called the elementary cell. A primitive
elementary cell is an elementary cell with the smallest possible volume (Fig. 3.1b). In each primitive
elementary cell there is exactly one lattice point. The coordination number is the number of nextneighbor lattice points. A primitive cubic (pc) lattice, e.g. has a coordination number of 6.
The typically chosen primitive elementary cell is the Wigner–Seitz (WS) cell that reflects the symmetry of the lattice best. The Wigner–Seitz cell around a lattice point R 0 contains all points that are
closer to this lattice point than to any other lattice point. Since all points fulfill such a condition for
some lattice point R i , the Wigner–Seitz cells fill the volume completely. The boundary of the Wigner–
Seitz cell is made up by points that have the same distance to R 0 and some other lattice point(s). The