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Unformatted text preview: 148 Chapter 3 / Crystalline State Table 3.4 All Possible Triplet Combinations of Crystallographic Rotation Axes Onefold Twofold Threefold Fourfold Sixfold Onefold 111
112
113
114
116
Twofold 212 222
213 223
214 224
216 226
Threefold 313 323 33 3
314 324 334
316 326 336
Fourfold 414 424 434 444
416 426 436 446
Sixfold 616 626 636 646 666 3.4, consider the possible combination of three twofold axes, 222. Each of the
rotation angles oz, B, and y is 77, so sin(a/2) = sin(B/2) = sin(y/2) = 1, and
cos(a/2) = cos(B/2) = cos(y/2) = 0. Thus, Eqs. 3.26 show u = U = w = 77/2.
Hence, one crystallographic axial combination is three mutually perpendicular two—
fold axes. This combination is plainly evident in the rectangular parallelepiped. Each
face has twofold rotational symmetry, the faces are mutually orthogonal, and the twofold axes intersect at the center of the body.
All axial combinations that yield values of the cosine function outside the range l to l are not possible. For instance, the combination 3 6 6 with a — 27/3,
B: y: 7r/3hascosw=3. The combination 3 3 3 has the solution it = v = w = 0; this combination thus is
equivalent to a single threefold axis and hence is not considered as a distinct axial
combination (i.e., l 1 3 is the single threefold axis). The only combinations for which there are valid nontrivial solutions to Eqs. 3.26
are 2 2 2, 2 2 3, 2 24, 2 2 6, 2 3 3, and 2 3 4. Table 3.5 lists the rotation angles and
interaxial angles given by Eqs. 3.26 for each of these cases. The resulting six spatial
arrangements of axes are illustrated with perspective sketches in Figure 3.56. Each of the combinations 2 2 2, 2 2 3, 2 2 4, and 2 2 6 involves two twofold axes
forming a plane, with the third axis (2, 3, 4, or 6) perpendicular to the other two
(see Figure 3.56). Because of their twofold axes, these combinations form the basis
of what are known as the dihedral point groups, to be further discussed below. The axial combinations 2 3 3 and 2 3 4 involve interaxial angles that are important
angles in the geometry of a cube (see Figure 3.56). Thus these combinations com—
prise the basis of the isometric point groups, discussed in greater depth below. Table 3.5 Arrangements of Crystallographic Rotation Axes Producing Allowed
Axial Combinations Axial
Combination oz ﬂ 3/ ZAOB = w ABOC = u AAOC = v 2 2 2 180° 180° 180° 90° 90° 90° 2 2 3 180° 180° 120° 60° 90° 90° 2 2 4 180° 180° 90° 45° 90° 90° 2 2 6 180° 180° 60° 30° 90° 90° 2 3 3 180° 120° 120° 54°44’ 70°32’ 54°44’ 2 3 4 180° 120° 90° 35°16’ 54°44' 45° 60°
90°
222 223
45° 30”
224 226 233 234 Figure 3.56 Spatial arrangements for the six permissible combinations of three rotational symmetry axes in crystals (after Buerger, 1978, p. 43). 149 150 Chapter 3 / Crystalline State So far, we have allowed for the possibility of just three rotation axes intersecting
at a point. However, it is possible for more than three axes to pass through a point.
For instance, if we allow the threefold rotation axis in the combination 2 2 3 illus
trated in Figure 3.56 to operate, an additional twofold axis, coplanar with the ones
initially present, must also be present. Similarly, all of the rotation axes in each of
the remaining illustrations in Figure 3.56 can operate, in order to generate the com—
plete set of rotation axes for each basic axial combination of three axes. Each of the
sets so generated is completely self—consistent; that is, there is a unique and ﬁnite
set of axes that can be generated from the initial combination of three axes. The
complete set of rotational symmetry axes for each of the six permissible crystallo
graphic axial combinations is illustrated in Figure 3.57. 224 226 233 234 Figure 3.57 Spatial arrangements for the six permissible combinations of rota—
tional symmetry axes in crystals, after allowing all rotational repetitions (after Buerger, 1978, p. 44). 3.2 The Crystallography of Three Dimensions 151 Inspection of Figure 3.57 for the combination 23 3 indicates that the complete
set of axes shown includes three twofold axes and four threefold axes intersecting
at a point. The lack of a permissible combination 33 3 among those said to be
possible may be noted. However, if a nontrivial combination 3 3 3 were permitted
by Eqs. 3.26, then threefold axes would have to be the only (nontrivial) rotational
symmetry axes intersecting the point. So, when several threefold axes pass through
a point, they do so in combination with other rotational symmetry axes: twofold
axes in the case of the combination 2 3 3, and two and fourfold axes in the case of
the combination 2 3 4. It is also possible to have mutually consistent combinations of nonparallel rota
tional symmetry axes that do not intersect at a point. Careful examination of this
possibility reveals that it necessitates the introduction of translational symmetry in
the form of screw symmetry axes (e.g., see Burger, 1978, p. 229 ff). The angles
between screw axes meeting at a point, or between screw axes and pure rotation
axes, must be the same as derived above for the six permissible crystallographic axial combinations. 3.2.4 The ThirtyTwo Crystallographic Point Groups The 10 crystallographic plane point groups were derived in Section 3.1.4. By deﬁ—
nition, point group symmetries do not involve translational symmetry. In two di
mensions, the set of possible symmetry operations is {rotations, translations, reﬂec
tions, glides}. Rotations and reﬂections are the only members of this set not involving
translations. The 10 crystallographic plane point group symmetries were developed
from the various ways in which rotational and reﬂectional symmetry can be present
at a point, either alone or in combination. Another way to consider the development of the crystallographic point groups is
to start with the possible pure rotational symmetries and then add other symmetries
as extenders in a systematic effort to derive additional distinct point group symme
tries. In two dimensions, the ﬁve allowed (monaxial) rotational symmetries can be
augmented by adding reﬁectional symmetry as the extender. Section 31.4 used this
approach to explore the possibilities for new twodimensional point group symme
tries. The same general procedure can be followed in three dimensions, but the
additional spatial degree of freedom allows a greater ﬂexibility, for the following reasons: 0 Rotational symmetries at a point in three dimensions can involve the intersec
tion of several rotational symmetry axes, as is obvious from the arrangement
of three orthogonal fourfold rotational symmetry axes about the center of a
cube (derived above in Section 3.2.3). This leads to additional point groups
that only involve rotational symmetry. 0 In three dimensions, rotational symmetry at a point can be augmented by in
version symmetry in addition to reflectional symmetry. 0 In three dimensions, there are often a variety of symmetrically distinct 0rien~
rations in which reﬂectional symmetry can combine with rotations (recall that 152 Chapter 3 / Crystalline State in two dimensions, mirrors were only added “vertically,” as described in Sec
tion 3.1.4). The procedure for developing the threedimensional point groups involves sys—
tematic combination of mirror planes and inversion centers to the crystallographi
cally permissible rotational symmetries, the complete set of which incorporates the
monaxial symmetries l, 2, 3, 4, and 6 plus the axial combinations 222, 223,
2 24, 2 2 6, 2 3 3, and 23 4. An additional possibility requiring discussion is that
other point groups could arise if the rotational symmetry axes become improper
axes, that is, to allow rotoreﬁections and rotoinversions. We digress brieﬂy and
demonstrate that all but one of the improper rotation axes (4) has an equivalent
combination which involves rotation and either reﬂection or inversion. Decomposition of Improper Rotation Axes With only one exception, the improper crystallographic rotation axes can be “de
composed” (i.e., reexpressed) as a combination, either of a proper rotation axis and
a perpendicular mirror plane or of a proper rotation axis and an inversion center.
We make a careful distinction here between a combination, which involves simul
taneous operation of the two types of symmetry, and the compound symmetry op
erations such as rotoinversion and rotoreﬂection, which are twostep symmetry op
erations (see Sections 3.1.2 and 3.2.1, respectively). The combination of an n—fold
rotation axis and a perpendicular mirror plane is denoted in international symbols
by n/m; the combination of an n—fold axis and an inversion center can be represented
by n ' i. Consideration of the patterns produced by the rotoinversion axes 7i shown
in Figure 3.35 gives the following relationships for the decomposed crystallographic
rotoinversion axes: 0 1: This is equivalent to the combination of a proper onefold axis and an inver
sion center, that is, T = 1 ' i. 0 2: This is equivalent to the combination of a proper onefold rotation axis and
a perpendicular mirror plane; that is, 2 = l/m. O 3: This is equivalent _to the combination of a proper threefold rotation axis and an inversion center, 3 = 3  i.
0 4: This improper axis cannot be decomposed into a simpler combination. Com parison with Figure 3.37 shows that 4 = 4.
0 6: This is equivalent to the combination of a proper threefold rotation axis and a perpendicular mirror plane; 6 = 3/m. Similarly, consideration of Figure 3.37 gives the following results for the decom
position of the crystallographic rotoreﬂection axes ﬁ: 0 i: This is equivalent to the combination of a onefold proper rotation axis and
a perpendicular mirror plane, denoted l/m; that is, l = l/m. 0 2: This is equivalent to the combination of a onefold proper rotation axis and
an inversion center; 2 = l ' i. 3.2 The Crystallography of Three Dimensions 153 ' 3: This is equivalent to the combination of a threefold proper rotation axis and
a perpendicular mirror plane; 3 = 3/m. ¢ 4: This improper axis cannot be decomposed into a simpler combination. ' 5: This is equivalent to the combination of a threefold proper rotation axis and
an inversion center, (5 = 3 ~ i. Thus, with the exception of the improper axis 4 = 4, all improper crystallographic
rotation axes can be reexpressed as combinations of proper rotation axes with per—
pendicular mirror planes or inversion centers. Therefore, by considering inversion
centers and perpendicular mirror planes as extenders, the possibilities for point group
symmetries can be explored without otherwise invoking improper axes, so long as the case of 4 = 4 is also considered. Derivation of Point Groups by Adding Extenders to Permissible Axial Combinations In Section 3.1.4, we considered the combination of rotation axes with parallel mirror
planes (i.e., the rotation axis was a line in the plane of the mirror, called a “vertical”
mirror plane). There may be more than one way to introduce such mirror planes as
extenders, depending upon their orientation relative to other rotational symmetry
axes which may be present. It is also necessary to consider the possibility of mirror
planes perpendicular to axes of rotational symmetry. Thus, we shall include below
both “diagonal” and “horizontal” mirror planes as extenders. Rather than presenting a complete derivation of the 32 crystallographic point
groups, we present below a select group of representative cases for discussion and
illustration. A complete table of the 32 crystallographic point groups is presented as
Figure 3.58. The columns in the table are labeled for the 12 basic axial combinations
of rotational symmetries: l, 2, 3, 4, 6, 222, 3 22, 42 2, 622, 2 3 3, 234, and 4.
The rows in the table relate to the addition of the extenders, which are, respectively,
horizontal mirror plane, vertical mirror plane, diagonal mirror plane, and inversion
center. Whenever the extender produces a new point group symmetry, there is an
entry in the table. A very important companion ﬁgure to Figure 3.58 is the key to entries for each
point group, provided as Figure 3.59. This key provides an explanation for the way
information is presented in Figure 3.58. Each entry provides a stereographic illus—
tration of how an asymmetric object gets repeated by the symmetry operations of
the point group, a stereographic projection of the symmetry elements present in the
point group, and the international and Schoenﬂies symbols for the point group. Several examples of the introduction of extenders to the axial symmetries will
now be considered in order to provide some insight into the methods by which the
point groups are systematically derived. We recommend careful study of each ex—
ample and the depiction of the relevant point group in Figure 3.58. Example I : Combination of a onefold axis and a horizontal mirror plane. Because
1 is the trivial rotational symmetry, the mirror plane introduced is the only additional
symmetry element. Thus, this combination is simply point symmetry m (= C,), which
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69 156 3.2 The Crystallography of Three Dimensions 157 International symbol,
abbreviated form Stereographic projection
of symmetry elements Projection of pattern produced
by placing asymmetric object
in general position Proper axes:
O A O 0 —_
2 3 4 6 Mirror plane Asymmetric object
0 above plane of paper Enantlomorph of
Q asymmetric object
above plane of paper Asymmetric object
' below plane of paper Enantiomorph of
2:3“: asymmetric object
' below plane of paper Eclipsed asymmetric
object and its enantlo (B morph, above and below
plane of paper lm ro er axes:
D P O A (Q) @ Inversion center:
g 7i '6 center of symmetry Figure 3.59 Key to Figure 3.58. Example 2: Combination of a onefola’ axis and a vertical mirror plane. As in
Example 1, the mirror plane introduced is the only additional symmetry element.
This symmetry is only different from Example 1 by orientation and therefore is
equivalent, yielding point group m (=C5). Example 3: Combination of a onefola' axis and an inversion center. Because 1 is
the trivial rotational symmetry, the inversion center introduced is the only additional
symmetry element. Thus, this combination yields the point symmetry l (=Ci). Example 4: Combination of a twofold axis and a horizontal mirror plane. Be—
cause, by deﬁnition, the horizontal mirror plane is perpendicular to the twofold axis,
the point group symmetry is 2/m (=C2h). Note from the projected pattern for point
group 2/m in Figure 3.58 that this symmetry combination contains an inversion
center. Example 5: Combination of a twofold axis and a vertical mirror plane. By def—
inition the vertical mirror plane contains the twofold axis, so the result of the appli—
cation of Eq. 3.7 may be recalled—that the combination implies a second mirror
plane perpendicular to the ﬁrst and also containing the twofold axis. This point group
symmetry will be recognized as the familiar 2mm (=C2u) 158 Chapter 3 / Crystalline State Example 6: Combination of a twofold axis and an inversion center. This com
bination involves the simultaneous presence of a twofold axis and an inversion cen
ter. Inspection shows that this combination requires the presence of a mirror plane
perpendicular to the twofold axis passing through the inversion center. This com—
bination is thus equivalent to the 2/m point group derived in Example 4. Example 7: Combination of the 2 2 2 axial combination and a horizontal mirror
plane. By deﬁnition the horizontal mirror plane is perpendicular to one of the twofold
axes. Because this means that both of the other twofold axes lie in the mirror plane,
Eq. 3.7 can be applied to determine that two additional vertical mirror planes which
pass through the vertical twofold axis must exist along with one of the horizontal
twofold axes. The point group symmetry of this combination is 2/m 2/m 2/m [the
common abbreviated form in international notation is mmm (=D2h)]. Consideration
of Figure 3.58 shows that this combination contains an inversion center. Example 8: Combination of the 222 axial combination and vertical mirror
planes. Equation 3.7 dictates that two orthogonal mirror planes will intersect along
the vertical twofold axis. Because they are introduced into the combination 2 2 2,
there are two distinct orientations of the mirror planes to consider: the “vertical”
orientation, in which the mirror planes contain pairs of twofold axes, and the “di—
agonal” orientation, in which they contain only one twofold axis, and bisect the other
two (see Figure 3.58). The vertical orientation simply rederives the point group
m mm. The diagonal orientation gives rise to a new point group symmetry called
42 m (=D2d). Note that what was originally the vertical twofold axis becomes a
4 axis when the diagonal mirrors are introduced. Example 9: Combination of the 2 3 4 axial combination and a horizontal mirror
plane. This combination gives rise to the highest crystallographic point group sym—
metry, that of a cube, which in international notation of the International Tables is
called 4/m 3 2/m and is commonly abbreviated as m 3m (=0h).11 Schoenﬂies Notation for the Crystallographic Point Groups
The general form for the notation is AM, where these three characters are from the sets 0 A: {C, D, T, O, S}.
0 n: {__, 2, 3, 4, 6}. Note: The “_” indicates that the symbol may be absent.
0 x: {_, s, i, h, U, d}. The main symbol A is explained as follows: 0 C: “cyclic,” refers to monaxial groups, with the exception of the special case 4.
0 D: “dihedral,” the case of twofold axes at right angles to a third axis. 0 T: “tetrahedral,” derived from the axial combination 2 3 3. “The abbreviated form m3 m was formerly written m 3 m. Both notations are in common use. 3.2 The Crystallography of Three Dimensions 159 ' 0:“octahedral,” derived from the_axial combination 2 3 4.
 S: used only for the special case 4. The subscript n designates the repetition interval of the highest symmetry rotation
axis. It is omitted when A = T or A = 0 or for the special cases when x = s or
x = z. The subscript x is the only subscript for the n = 1 cases, in which x = i for T
(inVersion) and x = s for m (recall that Spiegel is German for “mirror”). The other
possibilities, x = h, x = v, and x = d, indicate the presence of horizontal, vertical,
and diagonal mirror planes, respectively. If none of these is present, the subscript is
omitted. The possibility of noncrystallographic axial combinations allowed by Eqs.
(3.26a—c) requires special mention. Recall that point symmetries of objects lacking
translational symmetry do have meaning and that rotational symmetries of such
objects are not restricted to the set {1, 2, 3, 4, 6}. In fact, Eqs. (3.26a—c) can be
used to investigate noncrystallographic axial combinations and predict interaxial
angles for such combinations. Perhaps the most familiar allowed noncrystallographic
combination is 2 3 5, which is the basis of the pattern on a soccer ball and also the
point symmetry of the “Buckyball” molecule, C60. Table 3.6 provides additional information about the 32 crystallographic p...
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