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Unformatted text preview: 3.1 The Crystallography of Two Dimensions 107 special position. Only objects whose point symmetry equals or exceeds the symmetry
at the special position can occupy the special position. International and Schoenﬂies Symbols TWO different systems for designating crystal symmetries are‘ in common use: the
HermannMauguin symbols and the Schoenﬂies symbols. While both are used in
the International Tables for Crystallography, HermannMauguin symbols, com—
monly known as international symbols, predominate. The international symbols are
more generally used by materials scientists, and the Schoenﬂies symbols by chemists
and physicists. In both notations, Arabic numerals are used to designate axes of
rotational symmetry, with the numeral corresponding to the value of n of the axis
(e. g., a repetition angle of 277/3 is designated by the numeral 3 to indicate a three
fold axis of rotational symmetry). In the international notation, plane point group symmetries lacking intersecting
mirror lines are designated by the appropriate numeral: 1, 2, 3, 4, or 6. The com
bination of the axis 1 with a single mirror line is designated as point symmetry m.
The other combinations with mirror lines are designated as 2mm, 3m, 4mm, and 6mm
in international notation. Regarding the notation for the plane point groups containing mirror lines, it
should be noted that a double m (mm) appears in the point groups 2mm, 4mm, and
6mm, whereas in the point groups In and 3m, it does not. The double m is required
when the mirror lines are interleaved, that is, when the angular interval between the
mirrors is half that of the rotational symmetry axis of the point group. Thus, in plane
point group 4mm, the angle between mirrors is 77/4. In plane point group 3m, all of
the mirrors can be generated simply through the operation of the axis A2,” on the
original mirror line; the new mirrors identiﬁed by Eq. 3.7 superimpose on the ones
generated by Aha alone. The Schoenﬂies notation uses a capital letter from the set {C, D, T, 0, S} and
one or more subscripts. All of the plane point groups are designated by the capital
letter C, for “cyclic” (because of the cyclic rotation about a single axis perpendicular
to the plane of the pattern).5 All of the “C” plane point groups involve a single
rotation axis and are thus said to be “monaxial” groups. The subscripts for the plane
point groups can include integers 1, 2, 3, 4, and 6 to indicate the value of n of the
axis. When no mirror lines intersect the rotation axis, the plane point group sym—
metrics are designated respectively as C ,, C2, C3, C4, and C6. For the combinations
of two, three, four—, and sixfold axes and mirror lines, the symbols Czu, C3,, C4,”
and C6,, are used, respectively. The subscript v indicates a “vertical” mirror plane;
that is, in three dimensions, a vertical mirror plane intersects the rotation axis along
its length (this reduces to a mirror line perpendicular to the rotation axis in a two
dimensional pattern). The symbol C5 is used for the case of intersection of a onefold
axis with a mirror line (the s abbreviates Spiegel, German for “mirror”). Figure 3.18 5Note that there are additional point groups identiﬁed with a C that are relevant to three—dimensional
symmetries. These are considered in Section 3.2.4. 108 Chapter 3 / Crystalline State includes international and Schoenﬂies symbols for each of the 10 plane point group
symmetries. 3.1.5 The Five Distinct Plane Lattices Now it is possible to make a systematic study of rotational symmetry axes and/or
mirror lines combining with translations in a selfconsistent manner. In the process,
we will describe the ﬁve plane lattice types and ﬁnd how the rotational symmetry
axes and mirror lines can become distributed within the unit cell in certain cases. A ﬁnite set of ﬁve plane lattices arises from the possible combinations of the
crystallographic plane point group symmetries with translational symmetry in two
dimensions. Different lattice types arise because the presence of special symmetry
(i.e., rotation axes or mirror lines) usually places restrictions on the basis vectors t1
and t2 and the interaxial angle 3/. For the most general case, when there are no
restrictions, t] and t2 form a parallelogram cell which results in a parallelogram
lattice. Consideration of the parallelogram lattice illustrated in Figure 3.3 will reveal
that this lattice contains twofold rotation axes. First, remember combining an axis of rotational symmetry with a translation,
covered in Section 3.1.3. Figure 3.19 illustrates a rotation axis A“ and the translation
t. The line 1 is drawn at an angle 05/2 to the line perpendicular to t, such that the
rotation axis An moves the point P1 and the line 1 to the point P2 and the line 2,
respectively. The translational symmetry operation t moves the point P2 and the line
2 to the point P3 and the line 3, respectively. Consideration of Figure 3.19 shows
that the combination of symmetry operations Au and t is thus consistent with a
rotation axis Ba located at the intersection of lines 1 and 3. The rotation axis at B is
through the same angle a and in the same direction (counterclockwise in Figure
3.19) as for the axis Ad. The location of the axis B is related geometrically to a and
ltl. The general conclusion of this result can be stated as follows (Buerger, 1978, p. 72): A rotation about an axis A through an angle at, followed by a translation t
perpendicular to the axis, is equivalent to a rotation through the same angle Figure 3.19 Combination A“  t is equivalent to
rotation axis Ba (after Buerger, 1978, p. 71). 3.1 The Crystallography of Two Dimensions 109 a, in the same sense, but about an axis B situated on the perpendicular bisector
ofAA ’ and at a distance d = (ltl/Z) cot(a/2)fr0mAA ’. Symbolically, this combination of symmetry operations is represented by
Al  t = Ba (3.8) An n—fold rotation axis A“ is an axis about which rotations of oz (= 277m), 2a, . . . ,
na are all symmetry operations. Thus, in order fully to apply Eq. 3.8 to an axis Au,
an allowance must be made for each of these 21 rotations. We shall demonstrate, for
example, that when considering the combination Am  t, Eq. 3.8 will dictate that
additional fourfold and twofold axes must be present in the primitive cell. Plane Lattice Nets Arising from Crystallographic Rotation Axes and Translations The set of rotational symmetry axes consistent with lattice translation in two dimen
sions results in restrictions on t1, t2, and 'y and yields three of the ﬁve distinct two
dimensional lattice nets. In order to identify all of the rotational symmetry axes
contained within the primitive cell, Eq. 3.8 must be applied to each of the translations
t1, t2, and t1 + t2 because the end points of these three vectors plus the origin deﬁne
the vertices of the cell. All of these cases are presented in detail by Buerger (1978,
pp. 72—78). Here, to illustrate the basic process of derivation, we consider the com
binations involving two— and fourfold axes. The following treatment ignores the
application of Eq. 3.8 to the onefold axis A2,” as the ﬁnding of other onefold axes
at various points is a trivial result. Consider the combination A,r  t. Figure 3.20 shows the axis A, the vector t, and
the axis A’ produced from A by t. Equation 3.8 implies the existence of an axis BW
located along the perpendicular bisector (shown solid in Figure 3.20) of the lineAA’.
The distance of B7, from the line AA' is proportional to cot(a/2) = cot(7T/2) = 0.
Because arbitrary basis vectors t1 and t2 deﬁne a parallelogram lattice containing a
twofold axis at the origin, a twofold axis must also exist halfway along t1. Similar
reasoning results in twofold axes existing halfway along t2 and halfway along
t] + t2. Figure 3.21 shows the resulting primitive cell and the locations of all rotation
axes within the cell. Note that the twofold axis at the origin puts no restrictions on
allowed vectors for t2, thus resulting in a primitive cell that is in the general case a Figure 3.20 Combination A1,  t gives rise to twofold
axis 8,, at the point halfway along line AA’. 110 Chapter 3 / Crystalline State Figure 3.21 Primitive cell obtained by combination A17
and perpendicular translations. In addition to twofold
axes at cell comers, other twofold axes appear midway
along cell edges and at center of cell. parallelogram and which repeats on a parallelogram lattice. (The lenticular symbols
appear in four different orientations in Figure 3.21 to indicate that the axes are
crystallographically nonequivalent; that is, the environment around each of these
four regions in the cell is generally different. Similar convention is used in several
of the ﬁgures that follow.) Now consider the combination of a fourfold axis and t. The fourfold axis contains
the set of rotations A7,,2, Am Aha, and A2". Because of the fourfold axis, the vectors
defining the primitive cell, t1 and t2, are perpendicular and of equal magnitude,
resulting in a square lattice. Considering ﬁrst the combination AW  t1, Eq. 3.8 gives
the result that a fourfold axis must exist along the perpendicular bisector of AA’, at
a distance (AA’IZ) cot(1r/4) = AA’IZ. A moment’s reﬂection will show that this point
is in the center of the primitive cell, shown as Bl in Figure 3.22. The combinations
Am;  t2 and Ava  (t1 + t2) do not yield additional results. Next, because the rotation
A7, is a subgroup of the fourfold axis, we must consider the combination A1T  t2,
which was treated above and illustrated in Figure 3.20. This combination produces
a new twofold axis halfway along AA’, shown as B2 in Figure 3.22. Similar reasoning
can be used to derive the remaining twofold axes shown in Figure 3.22. Plane lattices consistent with threefold axes are similarly derived with the aid of
Eq. 3.8. The resulting cell forms a 120° rhombus lattice. Thus, threefold symmetry
forces a unique value to 7 and also requires ltll = ltzl. When sixfold axes are
combined with lattice translations, the resulting cell is also a 120° rhombus, but, in
addition to the sixfold axes, there arise three and twofold axes in the cell, due to
the fact that the rotations A273 and A7, are a subgroup of the rotation A215. The
resulting cells, and the locations of rotational symmetry axes within the cells, are illustrated in Figure 3.23. Figure 3.22 Primitive cell obtained by combination Am; and per
pendicular translations. In addition to fourfold axes at cell corners,
another fourfold axis exists at center of the cell and twofold axes appear midway along cell edges. 3.1 The Crystallography of Two Dimensions 111 (a) (b)
Figure 3.23 Primitive cells obtained by combination (a) A273 and (b) A",3 with
perpendicular translations. Both combinations give rise to a 120° rhombus lattice.
Different orientations of symbols indicate symmetry elements are at nonequivalent positions in unit cell. To summarize results so far for the combinations of each of the rotational sym
metries l, 2, 3, 4, and 6 with translations, in the absence of mirror symmetry (the
cases discussed immediately above), there are only three twodimensional arrange—
ments of lattice points (called lattice nets): the parallelogram, the 120° rhombus, and the square. Example Problem 6
Superpose the symmetry elements shown in Figure 3.22 on the periodic pattern shown in Figure EP6a so as to outline a primitive cell in the pattern. Convince
yourself that all of the four— and twofold axes shown in Figure 3.22 are present in the pattern. %
% %%% 8%
%%
&% &%%%
ﬁﬁ‘ﬁ
Elﬁﬁgé as;
%% %%%E%% as (a) (b)
Figure EP6 (a) Pattern containing two and fourfold rotational symmetry
axes. (b) Superposition of symmetry elements within single conventional unit
cell of pattern in (a). 112 Chapter 3 / Crystalline State Solution
Figure 3.22 is superposed on the pattern in Figure EP6b. Note the two nonequivalent locations of fourfold axes: One set is centered on the small repeated motif, and the
other is centered midway between four neighboring motifs. Either location can serve as a suitable origin for the primitive cell of this pattern. Lattice Nets Arising from Mirror Lines and Translations An additional derivation is required when mirrors are present and the highest rota
tional symmetry is a one or twofold rotation axis. This leads to two additional lattice
nets. These two lattices can be developed by ﬁrst considering the one—dimensional
combination of a translation vector tl with a line of reﬂection symmetry m, as shown
in Figure 3.24:1.6 We show in the following section (see Eq. 3.9) that this combi
nation results in a periodic array of lines of mirror symmetry with spacing equal to
half the magnitude of the lattice translation, as shown in Figure 3.24b. The dots in
Figures 3.24a and b represent lattice points; note that the spacing of lattice points is
twice the spacing of the lines of mirror symmetry. Figure 3.24c shows a periodic
pattern containing the symmetry elements of Figure 3.24b. Note that while the pat
tern in Figure 3.240 has onedimensional periodicity (a single translation vector t1),
the repeated object (an isosceles triangle) is two dimensional. Adding a second lattice translation t2 to the pattern of Figure 3.24%) will extend
the repetition of the pattern from one dimension to two. The goal is to incorporate
reﬂectional symmetry into the twodimensional pattern; thus, the points of mirror
symmetry in the onedimensional repetition must extend to lines of reﬂectional sym '" m P 1.1 E1: 5
:E t —*—'" P (a) (b) (c)
Figure 3.24 (a) Combination of lattice translation t1 and
perpendicular line of reﬂectional symmetry. (b) Combi
nation introduces lines of mirror symmetry spaced at half
the magnitude of the translational symmetry vector.
(c) Portion of periodic pattern that conforms to arrange
ment of symmetry elements illustrated in (b). °ln order that the resulting pattern have onedimensional periodicity, it is necessary that the lattice trans
lation be perpendicular to the line of reﬂectional symmetry. A mirror m oriented with respect tot by
some angle other than 90° leads to the two—dimensional centered—rectangular lattice. 3.1 The Crystallography of Two Dimensions 113 metry in two dimensions. In order for the additional lines of reﬂectional symmetry
to be present, t2 must be constrained in one of two ways: Either it must be perpen—
dicular to t2, or its component parallel to t1 must be equal to t1/2. Both cases are
illustrated in Figure 3.25. The ﬁrst is illustrated in Figure 3.25a, with the arrangement
of symmetry elements and a set of lattice points shown on the left and a periodic
pattern containing this arrangement on the right. The resulting lattice net is the
rectangular lattice and the conventional unit cell is a primitive rectangular cell as
overlaid on the right in Figure 3.25a. The second case is presented analogously in
Figure 3.25b. Because of the constraint on t2, it becomes possible to deﬁne a rec—
tangular cell that is a double cell, resulting in the centered rectangular lattice, out
lined in Figure 3.25b in the upper right. This is the conventional unit cell for this
arrangement of symmetry elements (i.e., it conforms to the terminology in the In
ternational Tables for Crystallography). An alternative choice of unit cell for the
centered rectangular lattice is also illustrated in Figure 3.25b in the lower right. In t T
 
4m.» 3 5 5 55 55
V v V—
V V v v V v (a) v v v v
I
l
l
 5533§S§§ l l l Figure 3.25 (a) Development of rectangular lattice by adding second
translation to pattern of Figure 3.241). In this case, mirror symmetry is
preserved by taking t2 perpendicular to t1. Arrangement of symmetry
elements shown on left, and a pattern with this symmetry shown on
right with conventional unit cell outlined. (b) Development of centered
rectangular lattice. Here, t; is constrained to fall on a mirror line mid
way between lattice points, as shown on left. Conventional cell for this
symmetry arrangement is a double cell, as shown in upper right, and
lattice net is known as centered rectangular lattice. Primitive cell shown
at lower right is an alternative unit cell known as a diamond cell or a
rhombus cell. 114 Chapter 3 / Crystalline State this case, the basis vectors are of equal length and the cell is a primitive cell, known
as the diamond lattice or alternatively as the rhombus lattice. (Note that there is no
constraint on the interaxial angle 3/ of the rhombus) A thorough analysis of the addition of lines of mirror symmetry to the other
lattices we have derived previously (square and 120° rhombus) shows that no ad
ditional lattice geometries arise. We have therefore developed the complete set of
ﬁve types of plane lattices: parallelogram, 120° rhombus, square, rectangle, and cen
tered rectangle. The information obtained concerning the plane lattice types is sum
marized in Table 3.1, which speciﬁes, given the highest point symmetry present in
the pattern, the least specialized lattice type consistent with that point symmetry.
(Higher symmetry lattices may be present but are not required by symmetry.) The
ﬁve plane lattices are illustrated in Figure 3.26, along with the conventional choices
for basis vectors and their positions relative to special symmetry elements that are
present. Note that the 120° rhombus lattice associated with rotational symmetries 3 and 6
is a specialized centered rectangular lattice. The 120° interaxial angle is consistent
with rotational symmetries 3 and 6, both of which are higher than the symmetries 2
and m associated with the centered rectangular lattice. When either symmetry 3 or
6 is present, the conventional unit cell is chosen to be the 120° rhombus, rather than
the centered rectangle, to emphasize the presence of higher (rotational) symmetry. 3.1.6 Plane Groups So far, we have derived the ten plane point groups and the ﬁve associated plane
lattices. In addition to deriving the shapes of the plane lattices, we have demonstrated
that axes of rotational symmetry are often present within the unit cell and their
locations are known. The ﬁnal task in deﬁning the set of crystallographically distinct
and consistent ways to arrange symmetry elements in a plane is to allow systemat—
ically for reﬂectional symmetry and glide (translation—reﬂection) symmetry in the
arrangements of symmetry elements that have been derived so far. The end result
of this exercise will be the determination of the 17 plane groups: the complete set Table 3.1 The Five Plane Lattice Types Consistent
with the Ten Crystallographic Plane Point Groups __________________._.________————————— Highest Point Group Symmetry Present Lattice Type"‘
1, 2 Parallelogram
m, 2mmb Rectangle
m, 2mmb Centered rectangle
4, 4mm Square
3, 3m, 6, 6mm 120° rhombus _______________________.__.————
“A plane lattice with higher symmetry is possible, but not required.
bCrystallographically distinct arrangements result when the centering translation
1/2, 1/2 is added to a pattern with m or 2mm point group symmetry. 3.1 The Crystallography of Two Dimensions 115 (t1+t2)/2 Figure 3.26 Five plane lattices—parallelogram, square, 120° rhombus, rectangle,
and centered rectangle—conventional unit cells for each and locations of rotational
symmetry axes within unit cells. Lines of mirror symmetry not shown. 116 Chapter 3 / Crystalline State of distinctly different ways of arranging symmetry elements in a two—dimensional
crystal. Any pattern that has two‘dimensional translational symmetry (e.g., a two
dimensional crystal, ﬂoor tiles, patterned textiles) will have its symmetry elements
arranged in one of these 17 plane groups. For twodimensional crystal structures,
the complete description of such a pattern can then be given simply by designating
the appropriate symbol for the plane group and specifying the contents (e.g., atom
types and coordinates) of the unit cell. Addition of Reﬂectional Symmetry to Plane Lattices The ad...
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 Fall '10
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