Table Of Content1 Elements of Symmetry in Periodic Lattices, Quasicrystals
Walter Steurer
Institut fur Kristallographie und Mineralogie, Universitat Miinchen,
Miinchen, Federal Republic of Germany
List of Symbols and Abbreviations 3
1.1 Introduction 5
1.2 Symmetry of Crystals 8
1.2.1 Morphology 8
1.2.2 Crystallographic Axes 10
1.2.3 Crystal Faces - Miller Indices 12
1.2.4 Zones and Forms 14
1.2.5 Symmetry Elements 16
1.2.6 External Form and Internal Structure 20
1.3 Crystal-Lattice Symmetry 21
1.3.1 Crystal Patterns, Vector and Point Lattices 21
1.3.2 The 7 Crystal Systems 22
1.3.3 The 14 Bravais Lattices 22
1.3.4 The Reciprocal Lattice 23
1.3.5 Topological Properties of Lattices 27
1.3.6 Lattice Transformations: Axes, Indices and Coordinates 28
1.4 Crystallographic Point-Group Symmetry 28
1.4.1 Group-Theoretical Terminology 28
1.4.2 Symmetry Operations 30
1.4.3 The 32 Crystallographic Point Groups 31
1.5 Crystallographic Space-Group Symmetry 35
1.5.1 Symmetry Operations 35
1.5.2 The 230 Space Groups 39
1.5.3 Wyckoff Positions and Site Symmetries 41
1.5.4 Crystallographic Orbits and Lattice Complexes 44
1.5.5 Subgroups and Supergroups of Space Groups 45
1.5.6 Representation of Space Group Symmetry in the International Tables
for Crystallography 46
1.6 Quasicrystals 48
1.6.1 Morphology 48
1.6.2 Quasiperiodic Tilings 49
1.6.2.1 Fibonacci Chain 51
1.6.2.2 Penrose Tilings 52
1.6.3 Decoration of Tilings 53
1.6.4 Non-Crystallographic Point-Group Symmetry 54
Materials Science and Technology
Copyright © WILEY-VCH Verlag GmbH & Co KGaA. All rights reserved.
2 1 Elements of Symmetry in Periodic Lattices, Quasicrystals
1.6.5 JV-Dimensional Crystallography 55
1.6.5.1 Symmetry Operations 55
1.6.5.2 Space Groups for Some Icosahedral Structures 58
1.7 Acknowledgements 59
1.8 References 59
List of Symbols and Abbreviations 3
List of Symbols and Abbreviations
a, b, c Wyckoff positions
a, b, c, n, d glide planes with translation components a/2, b/2, c/2, (a + b)/2 ...,
a, m, o t, h c symbols for the crystal systems (families)
9 9
a, b, c or a Basis vectors of the direct lattice, i = 1, 2, 3
t
a, b c or a lengths of basis vectors of the direct lattice, cell parameters, i = 1, 2, 3
9 f
a Fibonacci numbers
n
Z) diagonal matrix
<4ki interplanar spacing
e unit element of a group
G metric tensor
G general group symbol
g , g elements of 6-D space group
E x
g h group elements
k9 k
g elements of G
ik
H reciprocal lattice vector
(hkl) Miller indices of a crystal face or components of the reciprocal lattice
vector H
{hkl} indices of a crystal form
(hkil) Bravais-Miller indices of a crystal face for hexagonal axes
{hkil} Bravais-Miller indices of a crystal form for hexagonal axes
k order of a group
K point group of order k
L, S long, short interval
m reflection (mirror) plane
M vector lattice on basis a, 6, c
m reflection plane perpendicular to x, y
x y
(m n o) Weiss indices of a crystal face
P transformation matrix
P A B, C I,F R symbols for the centering type of unit cells (Bravais lattice type)
9 9 9 9
r, r position vector and its length
R rotation matrix
R , R external, internal space component of the rotation matrix
E l
t translation vector
T translation group of infinite order
u, v, w integer coordinates of a lattice vector
[u v w] indices of a direction (zone axis)
V volume of the unit cell
V , V external, internal component of the n-D space
E x
x, y z or x coordinates of a point, i = 1, 2, 3
9 t
(xyz) vector components in direct space
z zone axis vector
4 1 Elements of Symmetry in Periodic Lattices, Quasicrystals
a, /?, y or a interaxial angles, cell parameters, i = 1, 2, 3
t
r (g^) matrix representation of the group element g
k
5 Kronecker symbol
tj
Q (r) electron density distribution
T golden mean
<p angles characterizing the unit face (111)
x>y>z
q> (N) Euler numbers
X character of a representation
i/^ angles between the normal of a face (mno) and the axes of the crystal
x y z
coordinate system
1, 2, 3 ... N AT-fold rotation axes
2 ...,N AT-fold screw axes
l9 m
T center of symmetry, inversion center
3, 4,... JV rotoinversion axes
* symbols referring to the reciprocal lattice, e.g., the basis vectors, are
marked with the index *
GDM generalized dual-grid method
HRTEM high-resolution transmission electron microscope
k klassengleich (characterization of subgroups of a spacegroup)
MI morphological importance
PBC periodic bond chain
SHG second-harmonic generation
t translationengleich (characterization of subgroups of a spacegroup)
1.1 Introduction
1.1 Introduction
The regular polyhedral shape of crystals
has long fascinated the observer by their
beauty and brightness and by the perfect
planarity of their faces (Fig. 1-1), which ex-
ceeds the proficiency of the work of arti-
sans. The beliefs of Babylonians and Egyp-
tians in the magical and healing powers of
minerals and gemstones has been passed
on to other civilizations, a revival taking
place for instance in the Middle Ages. In-
deed, the important learned man and
bishop "doctor universalis" Albertus Mag-
nus (1193-1280) dedicated a part of his
book De Mineralibus et Rebus Metallicis
Libri V, which appeared in 1276, to the
curative properties of crystals. The more
rational minds of antiquity dealt with the
problems of the formation and composi-
tion of minerals. This is reflected in
the application of the word %Q\)<j%(xX'koq,Figure 1-1. Rock-crystal (quartz, SiO). The single
2
which means something like "solidification crystals show plane faces and trigonal symmetry
by freezing" and had originally only been (from Hochleitner, 1981).
used for ice, to rock-crystal (quartz) during
the time of Platon (428-348 B.C.).
Following some of the ideas of antiquity with general validity, however, by Jean-
Georgius Agricola (1494-1555) was one of Baptiste Rome de Vlsle (1736-1790) as late
the learned men to overcome the mystical as 1783. He was able to verify this hypoth-
assumptions of medieval times. His books esis by angular measurements with a con-
are not only a collection of the empirical tact goniometer (Fig. 1-2), which was con-
mining knowledge about minerals of his structed in 1780 by his assistant Maurice
time but they contain many hypotheses Carangeot, thus opening the way to quan-
about crystal growth and properties. In the titative crystal morphology. In 1820
following centuries the external shape of William Hyde Wollaston (1766-1829)
crystals, their morphology, attracted more greatly increased the possible accuracy of
and more interest. Thus Niels Stensen measurement with his optical goniometer
(1636-1686) found from his crystallization (from about 1° to 1').
studies that a correlation exists between With the work of the Abbe Rene Just
crystal growth and form - and that the Haiiy (1743-1822) a new chapter of crys-
morphology of crystals was not accidental. tallography began. A structural way of
The law of constant angles between equiva- thinking received strong impetus from his
lent crystal faces was noticed by Stensen work relating the internal structure of crys-
1669 on the examples of quartz and he- tals to their external shapes. From the ob-
matite. It was formulated explicitly and servation that the fragments obtained by
1 Elements of Symmetry in Periodic Lattices, Quasicrystals
nal parameter coefficients which had im-
plicitly already been found by Haiiy in
1784. The more convenient way of index-
ing used today, based on the reciprocal val-
ues of the intercepts, was introduced by
William Hallowes Miller (1801-1880).
Franz Ernst Neumann (1798 -1895) found a
correlation between the morphology of a
crystal and the anisotropy of its physical
properties. His work on crystal physics was
continued by his former student Woldemar
Voigt (1850-1919).
Interest in crystal symmetry was ini-
tiated by the symmetric arrangement of
crystal faces. By way of analyzing the mor-
Figure 1-2. A historical contact goniometer for mea- phology of crystals, in 1830 Johann
suring the interfacial angles of crystals (from Haiiy, Friedrich Christian Hessel (1796-1876) or-
1801). dered them into 32 possible crystal classes
according to their symmetry. Using the
repeatedly cleaving a crystal preserved the concept of a mathematical point lattice,
initial crystal form, Haiiy derived primitive Auguste Bravais (1811-1863) deduced in
forms ("molecules integrantes") for the ba- 1848 the 14 possible 3-D space lattices in 7
sic building units of all crystals. His decres- groups which correspond to the 7 crystal
cency theory described the formation of dif- systems detected by Weiss. The symmetry
ferent crystal shapes from basic paral- of these point lattices (holohedries) was too
lelepipeds (Fig. 1-3). Haiiy's book Traite de high in many cases to explain the symme-
Miner alogie (Haiiy, 1801) soon became the try properties of the respective elastic ten-
standard work on crystallographic miner- sors determined experimentally. This con-
alogy of the nineteenth century. The mor- tradiction could be overcome by occupy-
phological school received new stimulus ing the lattice nodes with point complexes,
from Christian Samuel Weiss (1780-1856) so lowering the symmetry. After the pre-
who polemized against the atomistic basis liminary work of Leonhard Sohncke (1842-
of Haiiy in an appendix to his German 1897), who in 1879 detected 65 space
translation of the Traite de Mineralogie. groups (the subset containing symmetry
Weiss focused on the dynamic character of operations of the first kind only) using
matter and the dominating influence of the group theoretical tools, in 1891 all of the
external crystal form. He discovered the 230 possible space group symmetries were
vectorial nature of some physical proper- derived by Evgraf Stepanovic Fedorov
ties. From the symmetrical arrangement of (1853-1919) and Arthur Schonflies (1853-
sets of crystal faces he derived the existence 1929) independently. The atomistic struc-
of 2-, 3-, 4- and 6-fold zone axes. He then tural theory based on the space lattice con-
described the faces by their integer inter- cept was confirmed by the first X-ray
cepts with these axes to form three-dimen- diffraction experiment (Fig. 1-4) which was
sional coordinate systems. As a conse- suggested by Max von Laue (1879-1960)
quence, Weiss formulated the law of ratio- in 1912.
1.1 Introduction
(b) (d)
Figure 1-3. Haiiy's decrescency theory of crystal growth: crystals of the same chemical composition but with a
different habit are built from the same basic parallelepipeds ('molecules integrantes'). Schematic drawings of the
crystal forms and of their construction using cubic unit cells are shown: (a) and (b) rhomb-dodecahedron, (c) and
(d) pentagon-dodecahedron with inscribed cubes (from Hauy, 1801).
1 Elements of Symmetry in Periodic Lattices, Quasicrystals
#, ;
* igure 1-5. Monochromatic zero-layer X-ray pre-
cession photograph of the decagonal quasicrystal
Al Co Ni showing clearly non-crystallographic
70 15 15
tenfold rotational symmetry.
Figure 1-4. One of the very first X-ray photographs
taken by Laue, Friedrich and Knipping (1912). The
diffraction pattern of zinc blende (ZnS) reflects the 1.2 Symmetry of Crystals
fourfold rotation symmetry along one of the main
axes of this cubic crystal (from Laue, 1961).
1.2.1 Morphology
A very extensive and rich collection of
crystal drawings was edited by Victor
Mordechai Goldschmidt (1852-1933) in the
years 1913-1923. He ordered the pub-
The theory of crystal symmetry, i.e., of lished information about the morphology
symmetric transformations in 3-D space of natural crystals, systematically in a nine
under restrictions imposed by the existence volume atlas. Figure 1-6 shows one page of
of the periodic crystal lattice, appeared to Vol. VIII illustrating different natural crys-
be a rather closed part of crystallography tal forms of silver. Single crystals always
until 1984, when the sensational discovery have a convex polyhedral form, concave
of quasicrystals by Shechtman, Blech, Gra- parts indicate that two single crystals are
tias and Cahn occurred. The understand- grown together. If these two individuals
ing of well-ordered crystals yielding dif- can be transformed into each other by a
fraction patterns with non-crystallo- particular symmetry operation then we
graphic (icosahedral, decagonal, ...) sym- call the crystal twinned. A plane of inter-
metry, i.e., incompatible with a 3-D peri- growth, for instance, may correspond to a
odic translation lattice, has been a new reflection plane, and we say that this crys-
challenge for crystallography (Fig. 1-5). tal is twinned according to that particular
1.2 Symmetry of Crystals
Figure 1-6. One page of
the famous Atlas der Kry-
stallformen in nine volumes
edited by Goldschmidt
during the years 1913-1923.
Silver crystals with different
habits grown under differ-
ent conditions are shown
in schematical drawings
(from Goldschmidt, 1913 —
1923).
plane. Some of the drawings in Fig. 1-6 metry), transform a symmetrical object
show twinned crystals, e.g., the last one of into itself. The transformed object is not
row one, and the second and third ones in distinguishable from the untransformed
row three. The polyhedra characterizing one, its position in space and its shape co-
the shape of crystals grown under equilib- incide with the original ones.
rium conditions tend to show a particular Phenomenologically, crystals are de-
symmetry, i.e., they are invariant under fined as chemically homogenous materials
particular motions around symmetry ele- with anisotropic physical properties. The
ments centered in the crystal. These mo- most conspicuous manifestation of this an-
tions, which may be a rotation of the crys- isotropy is the formation of plane faces re-
tal around an axis or a reflection on a flecting the internal symmetry of the crys-
mirror plane or on a point (center of sym- tal. Whereas the crystal habit, defined by
10 1 Elements of Symmetry in Periodic Lattices, Quasicrystals
the relative sizes of the faces, may vary angles a, /?, y. Usually the ratio a':b':d is
widely between different crystal individu- given normalized to fe' = l. The disturbing
als of the same material due to different influence of the individual crystal habit
growth conditions, the interfacial angles may be eliminated by representing the
remain constant (law of constant angles). faces by their normal vectors. The com-
Examples of crystals with the same crystal monly used graphical method for the rep-
form of a cuboctahedron but with a differ- resentation of a crystal form is the stereo-
ent habit are shown in Fig. 1-6 (the second graphic projection of its pole figure. Figure
and third drawing in row two). The one 1-7 (a) shows how the face poles result from
polyhedron shows large hexahedron faces the intersection of the face normals with a
h and small octahedron faces o, for the circumscribed sphere having a common
other polyhedron the ratio is inverse. In center with the crystal. In the next step, the
both cases, however, the interfacial angles face poles of the northern hemisphere are
are the same as well as the directions of the connected with the south pole and those of
face normals. The interfacial angles can be the southern hemisphere with the north
measured by means of a contact goniome- pole [Fig. l-7(b)]. The stereographic pro-
ter or, more accurately, by means of an jection of the face poles is obtained by pro-
optical two-circle goniometer, as was de- jecting the poles along the connecting lines
veloped by Fedorov and Goldschmidt in upon the equatorial plane [Fig. l-7(c)]. The
1892, where a crystal is mounted on a stereographic projection of the face poles is
goniometer head so that the rotation axis independent of individual variations of the
of the goniometer is parallel to the edges relative dimensions of crystal faces, it
formed by the intersecting faces to be mea- shows the inherent symmetry of the crystal
sured. A collimated light beam falls on the form in an unbiased way. For the practical
crystal, and in each case, when a face is application of the stereographic projec-
rotated to a reflecting position a signal is tion, it is useful to perform the construction
seen in a telescope. The light beam, the on a Wulffs net, i.e., a stereographic projec-
normal to the reflecting face and the tele- tion of meridians and parallels with 2° divi-
scope have to be in a plane perpendicular sions (Fig. 1-8). The construction and eval-
to the rotation axis of the goniometer. uation is facilitated by the fact that interfa-
Hence, during one complete revolution of cial angles of a crystal form appear as true
the crystal the angles between the faces of angles in the projection whilst circles on
one zone can be measured. The angle be- the sphere appear as circles in the projec-
tween two face normals equals n minus the tion.
interfacial angle.
The complete set of interfacial angles al-
1.2.2 Crystallographic Axes
lows establishment of the eigensymmetry
(corresponding to the group of point sym- The symmetrical arrangement of faces
metry operations bringing the crystal into (zone-faces) bounding a crystal grown un-
self-coincidence) of the crystal. It is even der equilibrium conditions led Weiss to the
possible to derive a kind of morphological idea to refer all crystal faces to a 3-D coor-
unit cell which is characteristic for a mate- dinate system formed by three non-copla-
rial with a given chemical composition. It nar symmetry axes (zone-axes) (Fig. 1-9).
is represented by a parallelepiped with Usually they are given by the basis vectors
edge lengths a\ br, d in relative units and a, A, c with lengths a, b, c, coordinates x, y
9
