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Hcp
1. Ref. p.4581 Appendix
Fig. 8. First Brillouin zone of the fee lattice. Symmetry )
points and symmetry lines are indicated (BSW notation
[36Bo]). The following coordinates refer to the Cartesian
system k,, k,,, k,, (seeFig. 7) and are given in units of 2x/a.
Coordinates of points:
I-: (000); x: (010); L: (l/2; l/2; l/2); w: (l/2 10)
K: (3/4 3/4 0); u: (l/4 1 l/4);
along lines:
A: [OOQ withO<[<l; C: [CYO]with 0<5<3/4
forT--%KandO<[<lforTAKAX’;
A: [c[[] with 0<5<1/2;
(The coordinates are those in the drawn BZ or in a neigh-
bouring repeated zone, as commonly usedin the literature).
-
Distances: l-X=2x/a; TKX =1/2 2x/a; TL=fix/a;
3 Hexagonal closepacked (hcp, A3) metals: Be, (C), Cd, a-Co, Dy, Er, Gd, Hf,
Ho, Lu, Mg, OS,Re, Ru, SC,Tb, Tc, Ti, Tl, Tm, Y, Zn, a-Zr
Fig. 10. The primitive unit cell of the hexagonal lattice (full
characterized by lattice parameters a and c*). The size of lines) with its basisvectors ti (i = 1,2,3, bold arrows) relative
the atoms (circles) is drawn arbitrarily. Representative to the conventional (Bravais) unit cell (dashedwith full lines).
positions of the first to eighth neighbours of atom “0.” are The basis consists of two atoms (hatched circles): one “0.”
indicated (the order of successionmay change if the ratio at (000)and a second one “1.” at ((a/l/j, 0 c/2) (see Fig. 9).
of c/a is not ideal). Their Cartesian coordinates (xyz) are
also given.
t, = (fi a/Z -a/2, O),
tz = (0, a, 01,
*) (The ideal ratio of c/a = 1.633).
1,= (0, 0, c).
(Components are given in the Cartesian system x, y. z).
Volume of the conventional (Bravais) unit cell:
62,= (3/2)fi a2c.
Volume of the primitive unit cell: Q= G&/3.
Fig. 11. Reciprocal lattice (hexagonal, full lines), reciprocal )
basis vectors gj (j =l, 2,3, bold arrows) and first Brillouin
zone (dashedlines) of the hcp lattice. k,, k,, k, indicate the
Cartesian coordinate system in reciprocal spaceparallel to
the x, y, z system in real space(seeFig. 10).The following
components of the gj are referred to this system:
gl=(4@a, 0, O),*)
g,=(Wjba, Wa, O),
g,=@, 0, 297/C).
Volume of the reciprocal hexagonal unit cell:
0,=(6/~@(2x)~/a~c.
Volume of the first Brillouin zone: 62,,=8,/3.
*) Seecaption of Fig. 3.
449
![Ref. p.4581 Appendix
Fig. 8. First Brillouin zone of the fee lattice. Symmetry )
points and symmetry lines are indicated (BSW notation
[36Bo]). The following coordinates refer to the Cartesian
system k,, k,,, k,, (seeFig. 7) and are given in units of 2x/a.
Coordinates of points:
I-: (000); x: (010); L: (l/2; l/2; l/2); w: (l/2 10)
K: (3/4 3/4 0); u: (l/4 1 l/4);
along lines:
A: [OOQ withO<[<l; C: [CYO]with 0<5<3/4
forT--%KandO<[<lforTAKAX’;
A: [c[[] with 0<5<1/2;
(The coordinates are those in the drawn BZ or in a neigh-
bouring repeated zone, as commonly usedin the literature).
-
Distances: l-X=2x/a; TKX =1/2 2x/a; TL=fix/a;
3 Hexagonal closepacked (hcp, A3) metals: Be, (C), Cd, a-Co, Dy, Er, Gd, Hf,
Ho, Lu, Mg, OS,Re, Ru, SC,Tb, Tc, Ti, Tl, Tm, Y, Zn, a-Zr
Fig. 10. The primitive unit cell of the hexagonal lattice (full
characterized by lattice parameters a and c*). The size of lines) with its basisvectors ti (i = 1,2,3, bold arrows) relative
the atoms (circles) is drawn arbitrarily. Representative to the conventional (Bravais) unit cell (dashedwith full lines).
positions of the first to eighth neighbours of atom “0.” are The basis consists of two atoms (hatched circles): one “0.”
indicated (the order of successionmay change if the ratio at (000)and a second one “1.” at ((a/l/j, 0 c/2) (see Fig. 9).
of c/a is not ideal). Their Cartesian coordinates (xyz) are
also given.
t, = (fi a/Z -a/2, O),
tz = (0, a, 01,
*) (The ideal ratio of c/a = 1.633).
1,= (0, 0, c).
(Components are given in the Cartesian system x, y. z).
Volume of the conventional (Bravais) unit cell:
62,= (3/2)fi a2c.
Volume of the primitive unit cell: Q= G&/3.
Fig. 11. Reciprocal lattice (hexagonal, full lines), reciprocal )
basis vectors gj (j =l, 2,3, bold arrows) and first Brillouin
zone (dashedlines) of the hcp lattice. k,, k,, k, indicate the
Cartesian coordinate system in reciprocal spaceparallel to
the x, y, z system in real space(seeFig. 10).The following
components of the gj are referred to this system:
gl=(4@a, 0, O),*)
g,=(Wjba, Wa, O),
g,=@, 0, 297/C).
Volume of the reciprocal hexagonal unit cell:
0,=(6/~@(2x)~/a~c.
Volume of the first Brillouin zone: 62,,=8,/3.
*) Seecaption of Fig. 3.
449](https://image.slidesharecdn.com/hcp-200914161203/85/Hcp-1-320.jpg)
