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Unformatted text preview: 3.2 The Crystallography of Three Dimensions 127 Figure 3.33 An inversion center is created
between right and left hands when they are
positioned as illustrated. Rotoinversion We previously discussed proper rotation axes in Section 3.1.3. A proper axis pro
duces a sequence of objects all of the same handedness. A series of objects alternating
in handedness is produced through improper rotation axes by combining rotation
with either reﬂection or inversion symmetry operations. In threedimensional pat—
terns, rotation axes can combine with an inversion center to create an improper
rotoinversion axis. Additionally, an axis of rotational symmetry can combine with
a reﬂection plane to produce an improper rotoreﬂection axis (see below). A rotoinversion axis is a compound symmetry operation that combines a rotation
axis with an inversion center located at a point 0 on the axis. The sequence of
operations in an nfold rotoinversion axis which produces a series of objects in space
related by rotoinversion is as follows: Rotate the object by 27/11, invert the object
through 0, and then draw the inverted image of the object; rotate the inverted object
by 27r/n and invert through 0; and draw the image of the object (which is now of
the same handedness as the original starting object). Continue until the object is
repeated at its original location with its original handedness. The object superposes
after n rotation intervals if n is even; otherwise it repeats after 2n intervals. An
overbar is used to indicate a rotoinversion axis (e.g., 8). Figure 3.34 illustrates the operation of the fourfold rotoinversion axis Z. It also
illustrates a view looking down the axis, in the form of a two—dimensional projection.
Note that the symmetry operation Z does not have the proper rotation 4 as a subgroup;
it does have the proper rotation 2 as a subgroup, however. The rotoinversion axis
T is equivalent to the inversion operation, i. Similar projections illustrating repetitions by the crystallographic axes T, E, 3, Z,
and 5 are presented in Figure 3.35. These should be compared with the patterns
produced by the operation of proper rotation axes, shown in Figure 3.11. The ro
toinversion axes T, 3, Z, and 6 have special symbols to distinguish them from the
corresponding proper rotation axes; these are shown in Figure 3.35 (there is no
special symbol for the axis 5, as it is equivalent to a horizontal mirror plane, i.e., a
mirror perpendicular to the 5 axis). A good example of rotoinversion symmetry is provided by the ethane molecule,
CZHG, in the low—energy, staggered conformation, illustrated in Figure 1.6. This 128 Chapter 3 / Crystalline State (a) (b)
Figure 3.34 Operation of fourfold rotoinversion axis: (a) perspective
sketch; (b) projection along axis. Open and ﬁlled triangles are of opposite
handedness. Note the open square with solid lens shape inside, which
symbolizes the 1 symmetry element. Figure 3.35 Patterns produced by crystallographic rotoin—
version axes. Large circles indicate midplane, e.g., the
plane of the page. Filled triangles lie above the midplane,
and open triangles lie below it. Where ﬁlled and open
triangles superpose, they are drawn gray. Open and ﬁlled
triangles are of opposite handedness. Note set of symbols
at centers of each illustration, which indicate types of
rotoinversion axes. 3.2 The Crystallography of Three Dimensions 129 conformational state of ethane has a 3 axis coaxial with the C—C bond, with the
inversion center at the bond’s midpoint. Figure l.6 should be compared with the illustration of the 3 axis in Figure 3.35. Rotoreﬂection A rotoreﬂection axis is a compound symmetry operation, and its operation is de
scribed below. An nfold rotation axis is perpendicular to a reﬂection (mirror) plane
m and passes through m at an origin 0. The object to be repeated is rotated by
277/n, then reﬂected by m, then drawn. The process is repeated until the object is
reproduced in its original location and with its original handedness. If n is even, the
object will be superposed in its original location after 11 steps (total rotation 277).
If n is odd, the object will not superpose at its original location until a rotation of
477' (2n steps) has been completed. Rotoreﬂection axes are denoted by an integer
with a tilde above it (e.g., 3). Figure 3.36 illustrates the operation of the threefold
rotoreﬂection axis 3. It also illustrates a view looking down the axis in the form of
a two— dimensional projection4. Similar projections that illustrate repetitions by the crystallographic axes 1,2, 3,4,and 6 are presented in Figure 3. 37. These should be
compared with Figures 3 11 and 3. 35 Comparison of Figures 3.11, 3.35, and 3. 37 will show that the set of rotoreﬂection
axes are equivalent to the set of rotoinversion axes For instance, 1 = 2. These and
other aspects of equivalencies of improper rotational symmetry axes with simpler
symmetry combinations, are discussedby Buerger (1978, pp. 27—30), and will be
considered below in Section 3.2.4. Because rotoreﬂectional symmetry can generally
be expressed by an equivalent combination of a proper rotation axis and a perpen—
dicular mirror plane, there are no special symbols for rotoreﬂection axes, unlike
those introduced above for rotoinversion symmetry (the axis 4 = 4 is the only special case). 53 (a) (b)
Figure 3.36 Operation of threefold rotoreﬂection axis: (a) perspec
tive sketch; (b) projection along axis. Because triangles are super
posed, they are drawn gray. 130 Chapter 3 / Crystalline State 993
<35? Figure 3.37 Patterns produced by crystallographic ro
toreﬂection axes. Large circles indicate the mirror mid—
plane, e.g., the plane of the page. Where ﬁlled and open
triangles superpose, they are drawn gray. Filled triangles
lie above the midplane, and open triangles lie below it. Screw Axes
A screw axis is a compound symmetry operation involving a proper rotational sym
metry axis and a translation. Recall that in Sections 3.1.3 and 3.1.5 we considered
this type of combination in two dimensions and hence necessarily constrained the
translation to lie perpendicular to the rotational symmetry axis. We now consider
combinations of rotation axes and translations that lie parallel to the rotation axis.
Figure 3.38 demonstrates the combination of a rotation axis A0, and a parallel
translation 1'; an object (in this case, a group of atoms) located at P is repeated at Figure 3.38 Combination of rotation axis
Am and parallel translation 1'. The ﬁrst two
repetitions of the object at P are shown at Q
and R. 3.2 The Crystallography of Three Dimensions 131 Q, then R, and so on. Depending on the sequence of the rotation and the translation,
this can be expressed by either of the paths PVQ or PUQ; that is, the component
operations of rotation and translation are commutable. Such a combination can be
considered as a screw motion. This can be represented symbolically by Aa'T = A,, (3J5) where the symbol A,” represents a screw axis; a is called the rotation component
of the screw axis and '7 is called the translation component of the screw axis. When
screw symmetry is applied to an object, a helical arrangement of otherwise identical
copies of the object results. Helical symmetry is quite common, and many of the
threedimensional crystallographic symmetry groups possess one or more screw axes.
The allowed rotational components of crystallographic screw axes are the same as for pure rotation axes, namely 277, 77, 277/3, 77/2, and 77/3. Clearly, the translational
component of the screw axis cannot be arbitrary. After n rotational repetitions, the
total rotation amount must be 277. This will produce a total translation n7, which
must be an integer multiple m of the lattice translation TH parallel to the screw axis. Thus, 711' = MT”, 01'
¢=En (mm
n Therefore the only values of m that need to be considered in Eq. 3.16 are those
between 0 and n — I (see Buerger, 1978, p. 203). Permissible values of n and Fr for
each of the possible crystallographic screw axes, along with the conventional des
ignation of the various screw axes, are presented in Table 3.3. In international no
tation, a screw axis is represented by an integer n representing the symmetry of the
rotation, followed by a subscript denoting the value of m in Eq. 3.16 (e.g., a 63 screw
axis has n = 6 and m = 3). Figure 3.39 illustrates the process by which patterns showing the crystallograph
ically allowed types of fourfold screw axes can be generated. Figure 3.39a illustrates
a 41 screw axis. Beginning with the representation of an object at the top of the axis,
a clockwise rotation of 77/2 is performed, followed by a shift '7 parallel to the axis.
The process is repeated three times, until the total shift is TH = 41', and the object
is returned to its original orientation, which coincides precisely with the requirement
that an identical object and orientation exist at all multiple distances of T”. Note that
this 41 screw axis produces a helical arrangement of objects, with a righthand
orientation of the screw motion. Figure 3.3% illustrates the process by which a pattern with 4; symmetry is gen—
erated. By deﬁnition, for a 42 screw axis, the translation parallel to the axis is TH/2.
The motif is repeatedly rotated (clockwise as viewed from above) by 77/2 and trans
lated Til/2 downward. Successive positions are indicated as l, 2, 3, 4. Because 132 Chapter 3 / Crystalline State Table 3.3 Allowed Crystallographic Screw Axes ____—_______—__..__————————————————_—— Proper
Rotation The Eleven Permissible
n Components Axes Crystallographic Screw Axes
l a O (or 277)
T O (or TH)
Designation l
'2 a 72' 7T
Designation 2 21
3 a §7T §7T %7T
Designation 3 31 32
4 a £77 %7r %7 in
7 0 iTH ﬁTil iTu
Designation 4 41 42 43
6 at %’2T §7r §7T 317r §7r §7T
T 0 éTll éTH 3TH 3TH 3TH
Designation 6 61 62 63 64 65 _______________—_____________.____——————————— Source: Buerger, 1978, p. 204, m = 2 for this screw axis, the full rotation of 360° occurs over a distance of 2TH
But the translational symmetry operation TH applies to the pattern as a whole, and
thus it operates on objects 2, 3, and 4, as shown in the lower Figure 3.391? [new
objects can be generated in the pattern by applying the fundamental lattice transla
tions to any object(s) generated by a symmetry operation]. Inspection of Figure 3.3919
reveals that the 42 screw axis is neutral, in that there is no sense of rotation of the
object; in fact, a 42 screw axis also contains a twofold axis. Figure 3.390 illustrates the process by which a pattern with 43 symmetry is gen—
erated. Here the translation parallel to the axis is iTH. Successive positions are again
indicated 1, 2, 3, 4. Since m = 3 for this screw axis, the full rotation of 360° is
accomplished over a distance of 3TH. Again, translational symmetry is applied to
place equivalent objects within the basic interval (0, T“). This results in a pattern
similar to the 4, screw axis, except that the direction of rotation of successive objects now forms a lefthanded screw.
The sense of handedness of the series of objects produced by a screw axis (not
of a single object, which may or may not possess handedness and, in either case, 3.2 The Crystallography of Three Dimensions 133 T” = 4T 2T” = 4T 3T”: 41' 41 42 4 3 (a) (b) (C)
Figure 3.39 Repetitions of an object (here a radial spoke normal to screw axis)
by operation of 41, 42, and 43 screw axes. Bottom ﬁgures are the ﬁnal patterns after
applying TH to all objects generated by screw symmetry. Sense of rotation is clock
wise (CW) when viewed from above. Note compressed vertical scale of ﬁgures ([7)
and (c) relative to (a) [i.e., TH is progressively shorter on going from (a) to (6)]. will have its handedness preserved by the screw symmetry operation) is related to
the relative values of m and 11, according to the relations '1' < %TH righthand screw axis
7 = éT” neutral screw axis 1' > %T” left—hand screw axis The repetitions of an object produced by each of the crystallographic screw axes
are presented in Figure 3.40. In addition, at the top of each axis in Figure 3.40 the
conventional symbol for the axis is shown. Note that the screw axes are represented
by extending the conventional symbol for a pure rotation axis such that it appears
to have “spokes” and these spokes have a sense of rotational direction that relates
to the pattern of the repetition. 134 Chapter 3 / Crystalline State Proper
rotation
axes Screw axes of 2 21
‘
i
3 31 32
4 41 42 43
6 61 62 63 64 65 Figure 3.40 Repetitions of an object by operation of screw axes. Object is indi
cated by an open circle; larger circles are oriented toward reader, i.e., on reader’s side of the rotation axis (Buerger, 1978, p. 205). Polymer molecules often display helical symmetry. For example, polyethylene in
its orthorhombic crystal form has a 21 screw axis parallel to the chain axis. Figure
3.41 shows an array of unit cells with the c axis (parallel to the molecular Chain
axes) oriented approximately along the viewing direction. The 2i screw axes centered
along the molecular axes should be apparent from the staggered orientations of
adjacent —CH2——— units. The three crystalline forms8 of isotactic polypropylene all 3.2 The Crystallography of Three Dimensions 135 Figure 3.41 Crystal structure of polyethylene has 21 screw
axes coaxial with molecular chain axes. In this View, portion
of polyethylene crystal is shown with its orthorhombic unit
outlined. Screw axes are viewed approximately end—on, par
allel to the c axis of the unit cell. The international symbol
for space group of this crystal is P 21/n 2,/a 2,/m. exhibit a threefold screw axis, again parallel to the chain axis. Another familiar
material containing screw axes is graphite, which has a primitive hexagonal Bravais
lattice (see Section 3.2.5), the base of the unit cell forming what is termed the basal
plane, with the “c axis” normal to the basal plane. Both 63 and 21 screw axes are
present in the structure, oriented parallel to the c axis. The same screw axes are also
present in the “hexagonal close—packed” structure (e.g., in Mg). The double—helix
structure of DNA is perhaps the most well—known of all molecules with screw symmetry. 3.2.2 Techniques for ThreeDimensional Spatial Relationships Specifying points, lines, and planes in three dimensions is straightforward. However,
some tools are necessary for geometric analysis in three dimensions, and for this
purpose, we introduce some concepts in spherical trigonometry. Visualization of
these structures becomes more challenging, especially as twodimensional ﬁgures
often must be used to represent three—dimensional objects; we later describe how
stereographic projection fulﬁlls this need. Rational Intercept Plane: Miller Indices In Section 3.1.1, we introduced a notation for specifying points and lines in two
dimensional lattices in relation to a coordinate system determined by the lattice
translations t1 and t2. In threedimensional lattices, not only is there a third coor»
dinate to contend with, but also there is often a need to specify twodimensional
planes and, in particular, the orientation of families of parallel planes. The orienta—
tions of planes are speciﬁed by means of Miller indices. A plane that intersects a set of lattice points is called a rational plane. The equa
tion of a plane in three dimensions can be written in terms of its intercepts as follows.
Let the basis vectors of the lattice be t1, t2, and t3 and the corresponding lattice
constants be a, b, and c. A coordinate system with axes x, y, and z parallel to t1, t2,
and t3 is deﬁned. The unit distances along x, y, and z are taken to be the lattice 8A substance with more than one crystalline form is said to be polymorphic. 136 Chapter 3 / Crystalline State constants a, b, and c, respectively. A plane can then be described in terms of its
intercepts P, Q, and R along the axes x, y, and z, respectively, as x y z — + — + — = l 3.17
P Q R ( )
The coefﬁcients of x, y, and z in Eq. 3.17 specify the orientation of a plane. For
convenience, it is customary to specify these coefﬁcients as integers by multiplying
both sides of Eq. 3.17 by PQR, and deﬁning h E QR, k E RP, and l —=— PQ which
leads to hx + ky + lz = PQR (3.18) Equation 3.18 is the equation of the rational intercept plane, that is, the plane
with intercepts P, Q, and R (proportional to llh, 1/k, and 1/], respectively) along the
axes x, y, and z. Planes are commonly speciﬁed by writing the numbers h, k, and l
in parentheses [i.e., (hkl)]. In this form, (/1 kl) are called the Miller indices of the
plane. In the Miller index notation, a negative number is indicated by an overbar.
Thus, for the planes with h = —3, k = 2, and l = 5, the Miller indices would be
written (3 2 5). Common multiplicative factors among the coefﬁcients h, k, and [are
usually divided out when reporting Miller indices. It is therefore preferable to write
(2 l 1) rather than (6 3 3). If a plane is parallel to an axis, its intercept with that axis
is at inﬁnity; therefore, the corresponding Miller index for that axis is zero. When a unit cell is chosen to have basis vectors that are among the smallest of
the lattice translations for the crystal structure, small values of the Miller indices 11,
k, and 1 correspond to the more widely spaced planes in the crystal and to planes in
which the number of lattice points per unit area on the plane is high. Such planes
tend to make up the external faces of naturally occurring faceted crystals and to play
a dominant role in plastic deformation of crystals by the motion of dislocations. The deﬁnition of Miller indices does not rely on a Cartesian coordinate system.
The axes x, y, and z may make arbitrary angles with each other, and the lattice
constants along each axis can be different. For example, Figure 3.42 ShOWS a plane
with intercepts at A, B, and C along x, y, and z, where A = 3, B = 9, and C = 6.
Thus, Eq. 3.17 becomes (3.19) mm
\o‘<
cum
II
as and Eq. 3.18 is 54x + 18y + 272: = 162 (3.20) As it is conventional to divide Miller indices by any common factors, the Miller
indices of the plane ABC in Figure 3.42 are (6 2 3). As all three planes shown in 3.2 The Crystallography of Three Dimensions 137 Z Figure 3.42 A stack of parallel rational planes be
C longing to the {6 2 3} family. >y Figure 3.42 are parallel, they have the same Miller indices (6 2 3). Figure 3.43 pro
vides additional examples by illustrating the Miller indices of several planes. A family of planes refers to the set of planes in a crystal which all have identical
atomic arrangements but lie in different orientations (i.e., the planes of a family are
crystallographically equivalent). A family of planes is denoted by braces, for ex
ample, the hkl family of planes is denoted {hkl}. In a cubic crystal, the {1 l 1}
family of planes is comprised of planes in four different orientations: (1 l 1),
(1 1T), (1 T 1), and (1 TT). It is important to note that unless a crystal possesses
special symmetries, atomic conﬁgurations on planes with permuted Miller indices
will be different; for instance, in a crystal with unspecialized lattice constants a, b,
c, a, [3, and y, the (h kl) plane will be geometrically different from the (h l k) plane.
In such cases, the (h kl) and (h l k) planes are not of the same family. In Section 3.1.1, we presented the method of designating points and lines in two
dimensional lattices. The same approach applies in three—dimensional lattices. In
particular, lines can be represented by vectors, for example, I' = ”t1 + Ut2 + Wt3 (3.21) and the direction of the line is speciﬁed by [u v w], where the numbers u, v, and w
are by convention cleared of fractions and reduced by dividing out any common {10 X
Figure 3.43 Threedimensional lattice with several ra
tional planes indicated by Miller indices. 138 Chapter 3 / Crystalline State factors. Families of crystallographically equivalent directions are gi...
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