Tratamiento de la equivalencia en el ámbito de derecho de asilo

We know already that the conventional cell for the trigonal system (a rhombohedron) can be constructed inside a hexagonal prism (see Fig. 31).

In such a construction, the sixfold symmetry axis of the hexagonal prism

Table 5 Experimental lattice parameters for arsenic, antimony, bismuth, and mercury. All

this elements crystallize in trigonal crystal structures. The axes a1

, a2
, and a3
define a

rhombohedral P unit cell that in the case of As, Sb, and Bi contains two atoms. The atoms are placed with respect to a lattice point like it is shown in Fig. 54b. In the case of Hg the basis is composed of one atom.

Element Lattice parameters r a (Å), α_r Number of atoms in a rhombohedral P unit cell

Coordinates of the basis atoms given in terms of

vector (a
₁+a
₂+a
₃) As 4.13 54 10 r r a α = ′ = ° 2 x = ±0.226 Sb 4.51 57 6 r r a α = ′ = ° 2 x = ±0.233 Bi 4.75 57 14 r r a α = ′ = ° 2 x = ±0.237 Hg (5 K) 2.99 70 45 r r a α = ′ = ° 1 x = 0

becomes a threefold symmetry axis of

that a trigonal lattice is just a centered hexagonal additional lattice points in a trigonal lattice

lattice, reduces the sixfold hexagonal prism axis to shown in Fig. 55. We can see in

that are inside the hexagonal prism define two equilateral triangles i

orthogonal to the sixfold hexagonal prism symmetry axis. The axis is crossing these planes

distribution of the trigonal lattice points prism, reduces the sixfold

a trigonal lattice. The basis vectors

rhombohedrally centered hexagonal unit cell for a trigonal lattice is called a triple hexagonal unit cell

hexagonal cell for a trigonal lattice and the symbol of a trigonal lattice is just

In Fig. 56a, we show the projections of the triple hexagonal cell

shows the projections

inside the hexagonal prism from Fig. 55. origin of the triple hexagonal unit cell The coordinate of each point

Figure 55 Primitive rhombohedral and a

lattice may be considered as

hexagonal cell, reduce the sixfold symmetry ax symmetry axis.

threefold symmetry axis of the rhombohedron. We will see now trigonal lattice is just a centered hexagonal one. The presence of lattice points in a trigonal lattice, with respect to the hexagonal reduces the sixfold hexagonal prism axis to a threefold one, what is shown in Fig. 55. We can see in this figure that the trigonal lattice points

he hexagonal prism define two equilateral triangles i

orthogonal to the sixfold hexagonal prism symmetry axis. The axis is crossing these planes at the geometric centers of the triangles. Just such distribution of the trigonal lattice points, which are inside the hexagonal reduces the sixfold axis of a hexagonal lattice to the threefold axis of a trigonal lattice. The basis vectors ah

, b_h
, ch
in Fig. 55 define a rhombohedrally centered hexagonal unit cell for a trigonal lattice. This cell

triple hexagonal unit cell and contains three lattice points. A triple

hexagonal cell for a trigonal lattice is also called a triple hexagonal and the symbol of a trigonal lattice is just hR.

we show the projections of the centering points of the triple hexagonal cell R from Fig. 55 on the cell base. Whereas, Fig. 56b

the projections on the prism base of 6 trigonal lattice points

the hexagonal prism from Fig. 55. In this figure O represents the origin of the triple hexagonal unit cell, defined by basis vectors ah

, The coordinate of each point in the ch

axis is shown next to the lattice point mitive rhombohedral and a R centered hexagonal unit cells. A primitive trigonal

considered as a R centered hexagonal lattice. The centering points, reduce the sixfold symmetry axis of the hexagonal prism to a

rhombohedron. We will see now . The presence of with respect to the hexagonal threefold one, what is that the trigonal lattice points he hexagonal prism define two equilateral triangles in planes orthogonal to the sixfold hexagonal prism symmetry axis. The axis is the geometric centers of the triangles. Just such are inside the hexagonal axis of a hexagonal lattice to the threefold axis of define a . This cell contains three lattice points. A triple called a triple hexagonal cell R

points of the Fig. 56b that are represents the , b_h
, ch
. lattice point A primitive trigonal within the is of the hexagonal prism to a threefold

projection and is expressed

base of the hexagonal prism in consideration translated vector (a_h+b )_h

. For changes from O to

lattice to a translation vector we obtain the same trigonal lattice.

Figure 56 (a) Projections of the centering points

on the base of the cell. The coordinates of these points axes ah
, b_h
, ch
. (b) P

base. The coordinates of these points are given in terms of the base translated by a translation vector

Figure 57 The reverse

rhombohedral cell.

expressed in units of c. In Fig. 56b, there is also shown the base of the hexagonal prism in consideration translated by a translation

For this case, the origin of the triple hexagonal unit cell O′ . We can see in Fig. 56b that by translating a trigonal lattice to a translation vector (a_h+b )_h

of the R centered hexagonal lattice we obtain the same trigonal lattice.

rojections of the centering points of the triple hexagonal cell R from Fig. 55 . The coordinates of these points are given in terms of the hexagona Projections of the 6 points that are inside the hexagonal prism on its base. The coordinates of these points are given in terms of the c
_h axis. The hexagonal prism

translation vector

₍

a_h+b_h

₎

is also shown.

reverse setting of a triple hexagonal cell in relation to the primitive there is also shown the a translation origin of the triple hexagonal unit cell translating a trigonal hexagonal lattice, from Fig. 55 ven in terms of the hexagonal the hexagonal prism on its The hexagonal prism

The setting of the triple hexagonal unit cell in relation to the primitive rhombohedral unit cell is not unique.

of the triple hexago

cell and in Fig. 56a, as we know, the plane orthogonal to

ch

for the hexagonal cell

reverse setting of a triple hexagonal unit cell in relation to the primitive

rhombohedral cell. The positions of the depend on the setting

Figure 58 shows two triple hexagonal cells

in obverse setting in relation to the primitive rhombohedral cell, while the cell from Fig. 58b is in

Both figures show the positions of the three lattice points within the hexagonal unit cell

points expressed in terms of the axes (of course, the vectors

We have learned here that it is possible to describe a trigonal lattice in terms of the hexagonal axes. M

a R centered hexagonal lattice. Moreover, it is more convenient to see this lattice as a R centered hexagonal one since the hexagonal axes are easier to visualize. The relations between the basis vectors that define a rhombohedral cell and the ones that define a triple hexagonal cell

Figure 58 Positions of the three points within the triple hexagonal

setting and (b) in reverse setting in relation to the primitive rhombohedral unit cell. The coordinates are expressed in units of

The setting of the triple hexagonal unit cell in relation to the primitive rhombohedral unit cell is not unique. In Fig. 55 is shown the obverse setting triple hexagonal unit cell with respect to the primitive rhombohedral

, as we know, is displayed the projection of this cell on the plane orthogonal to the ch

axis. If we propose the basis vectors

hexagonal cell in the way done in Fig. 57, then we obtain the of a triple hexagonal unit cell in relation to the primitive rhombohedral cell. The positions of the centering points in a hexagonal cell depend on the setting in consideration.

58 shows two triple hexagonal cells R. The cell from Fig. 58a in relation to the primitive rhombohedral cell, while the is in reverse setting with respect to the rhombohedral cell. oth figures show the positions of the three lattice points within the hexagonal unit cell R. We can observe that the coordinates of the centering

in terms of the axes ah

and b_h

are in each case different

, the vectors ah

and b_h

are also different).

We have learned here that it is possible to describe a trigonal lattice in terms of the hexagonal axes. More strictly speaking, a trigonal lattice is just centered hexagonal lattice. Moreover, it is more convenient to see this centered hexagonal one since the hexagonal axes are easier to visualize. The relations between the basis vectors that define a rhombohedral cell and the ones that define a triple hexagonal cell R are

Positions of the three points within the triple hexagonal R unit cell (a) in obverse in reverse setting in relation to the primitive rhombohedral unit cell. The coordinates are expressed in units of a and c.

The setting of the triple hexagonal unit cell in relation to the primitive

obverse setting

to the primitive rhombohedral the projection of this cell onto we propose the basis vectors ah

, b_h

, then we obtain the of a triple hexagonal unit cell in relation to the primitive points in a hexagonal cell . The cell from Fig. 58a is in relation to the primitive rhombohedral cell, while the to the rhombohedral cell. oth figures show the positions of the three lattice points within the centering different We have learned here that it is possible to describe a trigonal lattice in ore strictly speaking, a trigonal lattice is just centered hexagonal lattice. Moreover, it is more convenient to see this centered hexagonal one since the hexagonal axes are easier to visualize. The relations between the basis vectors that define a rhombohedral

in obverse in reverse setting in relation to the primitive rhombohedral unit cell. The

Figure 59 Three types of

cells shown in the figure is defined by basis vectors combinations. Inside the hexagonal prism there is a rhombohedral vectors ar
, b_r
, and cr
vectors a
_c , bc
, and c
_c

. All three unit cells have the same origin O.

Three types of unit cells of the bcc lattice. Each of the three triple hexagonal cells shown in the figure is defined by basis vectors ah

, b_h
, and ch
or their linear combinations. Inside the hexagonal prism there is a rhombohedral P unit cell defined by basis

cr

. Besides that, there is a cubic I cell of the bcc lattice defined by c

. All three unit cells have the same origin O.

e. Each of the three triple hexagonal R or their linear unit cell defined by basis lattice defined by

a_h=a_r −b , b_{r h}=b_r−c , c_{r h}=a_r + b_r+c_r

,

in the case of the obverse setting and

a_h=c_r−b , b_{r h}=a_r −c , c_{r h}=a_r+ b_r+c_r

,

in the case of the reverse setting.

In document Bases metodológicas para la traducción jurídica: traductología, derecho comparado y lexicografía bilingüe (página 54-63)