Hahn Th. (ed.) International tables for crystallography. Vol. A. Space-group symmetry
Подождите немного. Документ загружается.
9.3. Further properties of lattices
BY B. GRUBER
9.3.1. Further kinds of reduced cells
In Section 9.2.2, a ‘reduced basis’ of a lattice is defined which
permits a unique representation of this lattice. It was introduced into
crystallography by Niggli (1928) and incorporated into Interna-
tional Tables for X-ray Crystallography (1969), Vol. I. Originating
from algebra (Eisenstein, 1851), a reduced basis is defined in a
rather complicated manner [conditions (9.2.2.2a) to (9.2.2.5f )in
Section 9.2.2] and lacks any geometrical meaning. A cell spanned
by a reduced basis is called the Niggli cell.
However, unique primitive cells may be introduced also in other
ways that – unlike the Niggli cell* – have significant geometrical
features based mainly on extremal principles (Gruber, 1989). We
shall describe some of them below.
If a (primitive) cell of the lattice L fulfils the condition
a b c min
on the set of all primitive cells of L, we call it a Buerger cell. This
cell need not be unique with regard to its shape in the lattice. There
exist lattices with 1, 2, 3, 4 and 5 (but not more) Buerger cells
differing in shape. The uniqueness can be achieved by various
additional conditions. In this way, we can arrive at the following
four reduced cells:
(i) the Buerger cell with minimum surface;†
(ii) the Buerger cell with maximum surface;
(iii) the Buerger cell with minimum deviation;‡
(iv) the Buerger cell with maximum deviation.
Equivalent definitions can be obtained by replacing the term
‘surface’ in (i) and (ii) by the expression
sin sin sin
or
sin sin sin ,
and by replacing the ‘deviation’ in (iii) and (iv) by
jcos jjcos jjcos j
or
jcos cos cos j:
A Buerger cell can agree with more than one of the definitions
i, ii, iii, iv: 9:3:
1
:1
For example, if a lattice has only one Buerger cell, then this cell
agrees with all the definitions in (9.3.1.1). However, there exist also
Buerger cells that are in agreement with none of them. Thus, the
definitions (9.3.1.1) do not imply a partition of Buerger cells into
classes.
It appears that case (iv) coincides with the Niggli cell. This is
important because this cell can now be defined by a simple
geometrical property instead of a complicated system of conditions.
Further reduced cells can be obtained by applying the definitions
(9.3.1.1) to the reciprocal lattice. Then, to a Buerger cell in the
reciprocal lattice, there corresponds a primitive cell with absolute
minimum surface§ in the direct lattice.
The reduced cells according to the definitions (9.3.1.1) can be
recognized by means of a table and found in the lattice by means of
algorithms. Detailed mutual relationships between them have been
ascertained.
9.3.2. Topological characteristic of lattice characters
In his thorough analysis of lattice characters, de Wolff (1988)
remarks that so far they have not been defined as clearly as the
Bravais types and that an exact general definition does not exist.
Gruber (1992) tried to base such a definition on topological
concepts.
The crucial notion is the decomposition of a set M of points of the
n-dimensional Euclidean space E
n
into equivalence classes called
components of the set M. They can be defined as follows: Two
points X,Y of the set M belong to the same component if they can be
connected by a continuous path which lies entirely in the set M (Fig.
9.3.2.1). This partition of the set M into components is unique and is
determined solely by the set M.
Now let us return to lattices. To any lattice L there is attached a
point in E
5
called the Niggli point of L. It is the point
a a
c c
,
b b
c c
,
2b c
c c
,
2a c
c c
,
2a b
c c
9:3:2:1
provided that the vectors a, b, c describe the Niggli cell of L and
fulfil the conditions (9.2.2.2a) to (9.2.2.5f ) of Section 9.2.2. If L is a
set of lattices then the set of Niggli points of all lattices of L is
called the Niggli image of L .
Thus we can speak about the Niggli image of a Bravais type T .
This Niggli image is a part of E
5
and so can be partitioned into
components. This division of Niggli points induces back a division
of lattices of the Bravais type T . It turns out that this division is
identical with the division of T into lattice characters as introduced
in Section 9.2.5. This fact, used conversely, can be considered an
exact definition of the lattice characters: Two lattices of Bravais
type T are said to be of the same lattice character if their Niggli
points lie in the same component of the Niggli image of T .
We can, of course, also speak about Niggli images of particular
lattice characters. According to their definition, these images are
connected sets. However, much more can be stated about them:
these sets are even convex (Fig. 9.3.2.2). This means that any two
points of the Niggli image of a lattice character can be connected by
a straight segment lying totally in this Niggli image. From this
property, it follows that the lattice characters may be defined also in
the following equivalent way:
Fig. 9.3.2.1. A set M E
2
consisting of three components.
*
See, however, later parts of this section.
{
Meaning that this cell has the smallest surface of all Buerger cells of the lattice.
{
The deviation of a cell is the number j90
jj90
jj90
j.
}
This cell need not be a Buerger cell.
756
International Tables for Crystallography (2006). Vol. A, Chapter 9.3, pp. 756–760.
Copyright © 2006 International Union of Crystallography
We say that two lattices of the same Bravais type belong to the
same lattice character if one of them can be deformed into the other
in such a way that the Niggli point of the deformed lattice moves
linearly from the initial to the final position while the Bravais type
of the lattice remains unchanged.
Unlike convexity, nothing can be said whether the Niggli images
of lattice characters are open sets (with regard to their dimension) or
not. Both cases occur.
The lattice character of a lattice L can also be recognized [instead
of by means of Table 9.2.5.1 or by Tables 1 and 3 in Gruber (1992)]
by perpendicular projection of the c vector onto the ab plane
provided the vectors a, b, c describe the Niggli cell of L and fulfil
the conditions (9.2.2.2a) to (9.2.2.5f ) in Section 9.2.2 (de Wolff &
Gruber, 1991). See also Figs. 9.2.4.1 to 9.2.4.5.
9.3.3. A finer division of lattices
The 44 lattice characters form a subdivision of the 14 Bravais types.
There is another commonly known subdivision of the Bravais types,
namely the 24 Delaunay sorts (symmetrische Sorten) (Delaunay,
1933; International Tables for X-ray Crystallo graphy , 1952, Vol. I;
cf. Section 9.1.8). However, both divisions, being based on quite
different principles, are incompatible: the 44 lattice characters do
not form a subdivision of the 24 Delaunay sorts.
A natural problem arises to construct a division of lattices which
would be a subdivision of both the lattice characters and the
Delaunay sorts. However, we do not admit a purely mechanical
intersection of both these divisions; we insist that their common
subdivision be crystallographically meaningful.
Such a division was proposed recently (Gruber, 1997a). It uses
the fact that the Niggli points of all lattices lie in two five-
dimensional polyhedra, say
and
. The underlying idea,
originating from H. Wondratschek, is based on the distribution of
Niggli points among the vertices, edges, faces, three- and four-
dimensional hyperfaces, and the interior of
and
. This leads
to a natural division of Niggli points and further to a division of
lattices. This division has 67 classes, but is not suitable for
crystallography because it does not constitute a subdivision of the
Bravais types.
A modification of the idea is necessary. It consists of representing
a lattice L by several points (instead of by one Niggli point) and the
addition of two minor conditions. One of them concerns the
diagonals of the Niggli cell and the other the bases of L which
describe the Niggli cell.
Though these conditions are of little importance in themselves,
they lead to a very useful notion, viz the division of all lattices into
127 classes which is a subdivision of both the lattice characters and
the Delaunay sorts. The equivalence classes of this division are
called genera. They form, in a certain sense, building blocks of both
lattice characters and Delaunay sorts and show their mutual
relationship.
The distribution of genera along the Bravais types is the
following (the number of genera is given in parentheses): cP(1),
cI(1), cF(1), tP(2), tI(5), oP(1), oC(8), oI(7), oF(3), hP(3), hR(4),
mP(5), mC(43), aP(43). Thus, genera seem to be especially suitable
for a finer classification of lattices of low symmetry.
The genus of a given lattice L can be determined – provided that
the Niggli point of L is known – by means of a table containing
explicit descriptions of all genera. These descriptions are formed by
open linear systems of inequalities. Consequently, the ranges of
conventional parameters of genera are open unlike those concerning
the lattice characters.
Genera are denoted by symbols derived from the geometrical
shape of
and
. They can be visualized in the three-
dimensional cross sections of these bodies. This gives a fairly
good illustration of the relationships between genera.
However, the most important feature of genera seems to be the
fact that lattices of the same genus agree in a surprisingly great
number of crystallographically significant properties, such as the
number of Buerger cells, the densest directions and planes, the
symmetry of these planes etc. Even the formulae for the
conventional cells are the same. The genus appears to be a
remarkably strong bond between lattices.
9.3.4. Conventional cells
Conventional cells are dealt with in Chapter 9.1. They are illustrated
in Fig. 9.1.7.1 and described in Table 9.1.7.2. This description,
however, is not exhaustive enough for determining the Bravais type.
In mathematical terms, the conditions in Table 9.1.7.2 are necessary
but not sufficient. For example, the C-centred cell with
a 6, b 8, c 5, cos 7=15, 90
9:3:4:1
has the typical shape of a conventional cell of an mC lattice. But the
lattice generated by the C-centred cell (9.3.4.1) is actually hR with
the conventional rhombohedral basis vectors
c, a b=2, a b=2:
It is a natural goal to establish a system of conditions for the
conventional cells which would be not only necessary but also
sufficient. This is done in Table 9.3.4.1. In order to make the
conditions as simple as possible, the usual mC description of the
monoclinic centred lattices is replaced by the mI description. The
relation between the two descriptions is simple:
a
I
c
C
, b
I
b
C
, c
I
a
C
c
C
:
The exact meaning of Table 9.3.4.1 is as follows: Suppose that a
Bravais type different from aP is given and that its symbol appears
in column 1 in the ith entry of Table 9.3.4.1. Then a lattice L is of
this Bravais type if and only if there exists a cell (a, b, c)inL such
that
(i) the centring of (a, b, c) agrees with the centring mode given in
column 2 in the ith entry, and
(ii) the parameters of the cell (a, b, c) fulfil the conditions listed
in column 3 in the ith entry of Table 9.3.4.1.
9.3.5. Conventional characters
Lattice characters were defined in Section 9.3.2 by dividing the
Niggli image of a certain Bravais type T into components. Doing
Fig. 9.3.2.2. A convex set in E
2
.
757
9.3. FURTHER PROPERTIES OF LATTICES
the same – instead of with the Niggli points – with the parameters of
conventional cells* of lattices of the Bravais type T we obtain a
division of the range† of these parameters into components. This
leads to a further division of lattices of the Bravais type T into
equivalence classes. We call these classes – in analogy to the Niggli
characters – conventional characters. There are 22 of them.
Two lattices of the same Bravais type belong to the same
conventional character if and only if one lattice can be deformed
into the other in such a way that the conventional parameters of the
deformed lattice change continuously from the initial to the final
position without change of the Bravais type. The word ‘con-
tinuously’ cannot be replaced by the stronger term ‘linearly’
because the range of conventional parameters of the monoclinic
centred lattices is not convex.
Conventional characters form a superdivision of the lattice
characters. Therefore, no special notation of conventional char-
acters need be invented: we write them simply as sets of lattice
characters which constitute the conventional character. Denoting
the lattice characters by integral numbers from 1 to 44 (according to
the convention in Section 9.2.5), we obtain for the conventional
characters symbols like f8, 19, 42g or f7g.
Conventional characters are described in Table 9.3.5.1.
9.3.6. Sublattices
A sublattice L
0
of an n-dimensional lattice L is a proper subset of L
which itself is a lattice of the same dimension as L. A sublattice L
0
of
Table 9.3.4.1. Conventional cells
Bravais
type
Centring
mode of the
cell (a, b, c) Conditions
cP P
a b c,
90
cI I
a b c,
90
cF F
a b c,
90
tP P
a b 6 c,
90
tI I
c=
2
p
6 a b 6 c,*
90
oP P
a < b < c, y
90
oI I
a < b < c,
90
oF F
a < b < c,
90
oC C
a < b 6 a
3
p
, z
90
hP P
a b,
90
, 120
hR P
a b c,
,
6 60
, 6 90
, 6 !x
mP P
2c cos <a < c, {
90
<
mI I
c cos <a < c,**
90
<, 9:3:4:2
but not a
2
b
2
c
2
,
a
2
ac cos b
2
, yy 9:3:4:3
nor a
2
b
2
c
2
,
b
2
ac cos a
2
, zz 9:3:4:4
nor c
2
3b
2
9a
2
,
c 3a cos , xx 9:3:4:5
nor a
2
3b
2
9c
2
,
a 3c cos {{ 9:3:4:6
Note: All remaining cases are covered by Bravais type aP.
* For a c=
2
p
, the lattice is cF with conventional basis vectors c, a b, a b.
† The labelling of the basis vectors according to their length is the reason for
unconventional Hermann–Mauguin symbols: for example, the Hermann–Mauguin
symbol Pmna may be changed to Pncm, Pbmn, Pman, Pcnm or Pnmb. Analogous
facts apply to the oI, oC, oF, mP and mI Bravais types.
‡ For b a
3
p
, the lattice is hP with conventional vectors a, b a=2, c.
§ ! arccos1=3109
28
0
16
00
. For 60
, the lattice is cF with conventional
vectors a b c, a b c, a b c; for !, the lattice is cI with
conventional vectors a b, a c, b c.
{ This means that a, c are shortest non-coplanar lattice vectors in their plane.
** This means that a, c are shortest non-coplanar lattice vectors in their plane on
condition that the cell (a, b, c) is body-centred.
†† If (9.3.4.2) and (9.3.4.3) hold, the lattice is hR with conventional vectors
a, a b c=2, a b c=2, making the rhombohedral angle smaller than 60
.
‡‡ If (9.3.4.2) and (9.3.4.4) hold, the lattice is hR with conventional vectors
a, a b c =2, a b c=2, making the rhombohedral angle between 60 and
90
.
§§ If (9.3.4.2) and (9.3.4.5) hold, the lattice is hR with conventional vectors
a, a b c=2, a b c=2, making the rhombohedral angle between 90
and !.
{{ If (9.3.4.2) and (9.3.4.6) hold, the lattice is hR with conventional vectors
c, a b c=2, a b c=2, making the rhombohedral angle greater than !.
Table 9.3.5.1. Conventional characters
Bravais type Conditions Conventional character
cP {3}
cI {5}
cF {1}
tP a < c {11}
c < a {21}
tI a < c=
2
p
{15}
c=
2
p
< a < c {7}
c < a {6, 18}
oP {32}
oI {8, 19, 42}
oF {16, 26}
oC b < a
3
p
{13, 23}
a
3
p
< b {36, 38, 40}
hP {12, 22}
hR* <60
{9}
60
<<90
{2}
90
<<!† {4}
!< {24}
mP {33, 34, 35}
mC {10, 14, 17, 20, 25, 27, 28,
29, 30, 37, 39, 41, 43}
aP <90
{31}
90
{44}
* The angle refers to the rhombohedral description of the hR lattices.
† ! arccos1=3109
28
0
16
00
.
*
For aP lattices, these parameters are derived from the Niggli point [see (9.3.2.1)].
{
This range is a subset of E
k
, where k 6.
758
9. CRYSTAL LATTICES
L causes a decomposition of the set L into, say, i mutually congruent
sublattices, L
0
itself being one of them (Fig. 9.3.6.1). The number i
is called the index of the sublattice L
0
and indicates how many times
L
0
is ‘diluted’ with respect to L.
Sublattices are defined in a natural way in those lattices that have
centred conventional cells, being generated by the vertices of these
cells (‘decentring’). They are primitive and belong to the same
crystal family as the given lattice. Thus, in the cI, cF, tI, oI, oF, oC,
mC and hR * lattices, we can meet sublattices of indices 2, 4, 2, 2, 4,
2, 2 and 3, respectively.
Theoretically (though hardly in crystallographic practice), the
Bravais type of centred lattices can also be determined by testing all
their sublattices with the suspected index and finding in any of these
sublattices the Niggli cell.
All sublattices of index i of an n-dimensional lattice L can
be constructed by a procedure suggested by Cassels (1971). If
a
1
, ..., a
n
is a primitive basis of the lattice L then primitive bases
a
0
1
, ..., a
0
n
of all sublattices of index i of the lattice L can be found
by the relations
a
0
1
, ..., a
0
n
a
1
, ..., a
n
R
T
,
where the matrix R r
ij
fulfils
0 r
ij
for 1 i < j n,
0 r
ij
< r
jj
for 1 j < i n,
r
11
...r
nn
i :
9:3:6:1
The number D
n
,
i
of these matrices is equal to the number of
decompositions of an n-dimensional lattice L into sublattices of
index i. To determine this number, it is not necessary to construct
explicitly the matrices fulfilling (9.3.6.1). The following formulae
(Gruber, 1997b) can be used:
(i) If i p
q
, where p > 1 is a prime number, then
D
n
,
i
p
n
1
p 1
p
n1
1
p
2
1
p
n2
1
p
3
1
...:
|{z}
q times
(ii) If i p
q
1
1
...p
q
m
m
( p
1
, ..., p
m
mutually different prime
numbers, m > 1), we deal with any factor p
q
j
j
j 1, ..., m
according to point (i) and multiply all these numbers to obtain the
number D
n
,
i
.
For example, for n 3andi = 2, 3, 4 and 6, we obtain for D
n
,
i
the values 7, 13, 35 and 91, respectively.
In all considerations so far, the symmetry of the lattice L was
irrelevant. We took L simply as a set of points and its sublattices as
its subsets. (Thus, for illustrating sublattices, the ‘triclinic’ lattices
are most apt; cf. ‘derivative lattices’ in Chapter 13.2.)
However, this is not exactly the crystallographic point of view. If,
for example, the mesh of the lattice L in Fig. 9.3.6.1 were a square,
the sublattices in cases (a)and(b) would have the same symmetry
(though being different subsets of L) and therefore would be
considered by crystallographers as one case only. The number D
n
,
i
would be reduced. From this aspect, the problem is treated in
Chapter 13.1 in group-theoretical terms which are more suitable for
this purpose than the set-theory language used here.
Fig. 9.3.6.1. Three possible decompositions of a two-dimensional lattice L
into sublattices of index 2.
*
When choosing their hexagonal description.
759
9.3. FURTHER PROPERTIES OF LATTICES
References
9.1
Aza´roff, L. V. & Buerger, M. J. (1958). The powder method in X-ray
crystallography. New York: McGraw-Hill.
Bravais, A. (1866). Etudes cristallographiques. Paris.
Burckhardt, J. (1966). Die Bewegungsgruppen der Kristallographie,
2nd ed., pp. 82–89. Basel: Birkha¨user.
Burzlaff, H. & Zimmermann, H. (1977). Symmetrielehre. Part I of
the series Kristallographie, pp. 96–135. Stuttgart: Thieme.
Burzlaff, H. & Zimmermann, H. (1993). Kristallsymmetrie–Kristall-
struktur, pp. 90–105. Erlangen: R. Merkel.
Delaunay, B. N. (1933). Neuere Darstellung der geometrischen
Kristallographie. Z. Kristallogr. 84, 109–149.
Donnay, J. D. H. (1943). Rules for the conventional orientation of
crystals. Am. Mineral. 28, 313–328, 470.
Donnay, J. D. H. & Ondik, H. M. (1973). Editors. Crystal data, Vol.
2, 3rd ed. Introduction, p. 2. Washington: National Bureau of
Standards.
International Tables for X-ray Crystallography (1952). Vol. I, edited
by N. F. M. Henry & K. Lonsdale. Birmingham: Kynoch Press.
[Abbreviated as IT (1952).]
International Tables for X-ray Crystallography (1969). Vol. I, 3rd
ed., edited by N. F. M. Henry & K. Lonsdale. Birmingham:
Kynoch Press. [Abbreviated as IT (1969).]
Minkowski, H. (1897). Allgemeine Lehrsa
¨
tze u
¨
ber die konvexen
Polyeder. In Gesammelte Abhandlungen. Leipzig, 1911.
[Reprinted: Chelsea, New York (1967).]
Neubu
¨
ser, J., Wondratschek, H. & Bu
¨
low, R. (1971). On crystal-
lography in higher dimensions. I. General definitions. Acta Cryst.
A27, 517–520.
Patterson, A. L. & Love, W. E. (1957). Remarks on the Delaunay
reduction. Acta Cryst. 10, 111–116.
Schwarzenberger, R. L. E. (1980). N-dimensional crystallography.
San Francisco: Pitman.
Selling, E. (1874). U
¨
ber bina
¨
re und terna
¨
re quadratische Formen.
Crelles J. Reine Angew. Math. 77 ,143ff.
Trotter, J. (1975). Editor. Structure reports. 60-year structure index
1913–1973. A. Metals and inorganic compounds. Utrecht: Bohn,
Scheltema & Holkema. [Now available from Kluwer Academic
Publishers, Dordrecht, The Netherlands.]
Wolff, P. M. de, Belov, N. V., Bertaut, E. F., Buerger, M. J., Donnay,
J. D. H., Fischer, W., Hahn, Th., Koptsik, V. A., Mackay, A. L.,
Wondratschek, H., Wilson, A. J. C. & Abrahams, S. C. (1985).
Nomenclature for crystal families, Bravais-lattice types and
arithmetic classes. Report of the International Union of Crystal-
lography Ad hoc Committee on the Nomenclature of Symmetry.
Acta Cryst. A
41, 278–280.
Wondratschek, H., Bu
¨
low, R. & Neubu
¨
ser, J. (1971). On crystal-
lography in higher dimensions. III. Results in R4. Acta Cryst.
A27, 523–535.
9.2
Buerger, M. J. (1957). Reduced cells. Z. Kristallogr. 109, 42–60.
Buerger, M. J. (1960). Note on reduced cells. Z. Kristallogr. 113,
52–56.
Delaunay, B. N. (1933). Neuere Darstellung der geometrischen
Kristallographie. Z. Kristallogr. 84, 109–149.
Delaunay, B. N., Galiulin, R. V., Dolbilin, N. P., Zalgaller, V. A. &
Stogrin, K. I. (1973). On three successive minima of a three-
dimensional lattice (in Russian). Dokl. Akad. Nauk SSSR, 209,
309–313.
Eisenstein, G. (1851). Tabelle der reducierten positiven quadra-
tischen Formen nebst den Resultaten neuer Forschungen u
¨
ber
diese Formen, insbesondere Beru
¨
cksichtigung auf ihre tabellar-
ische Berechung. J. Math. (Crelle), 41, 141–190.
Gruber, B. (1973). The relationship between reduced cells in a
general Bravais lattice. Acta Cryst. A29, 433–440.
Gruber, B. (1978). Comments on Bravais lattices. Research report,
Charles University, Prague, Czech Republic. Details can be
obtained from the author on request.
Gruber, B. (1989). Reduced cells based on extremal principles. Acta
Cryst. A45, 123–131.
International Tables for X-ray Crystallography (1969). Vol. I, 3rd
ed., edited by N. F. M. Henry & K. Lonsdale. Birmingham:
Kynoch Press. [Abbreviated as IT (1969).]
Kr
ˇiv
y
´
, I. & Gruber, B. (1976). A unified algorithm for determining
the reduced (Niggli) cell. Acta Cryst. A32, 297–298.
Mighell, A. D. (1976). The reduced cell: its use in the identification
of crystalline materials. J. Appl. Cryst. 9, 491–498.
Mighell, A. D. & Rodgers, J. R. (1980). Lattice symmetry
determination. Acta Cryst. A36, 321–326.
Mighell, A. D., Santoro, A. & Donnay, J. D. H. (1975). Addenda to
International Tables for X-ray Crystallography. Acta Cryst. B31,
2942.
Niggli, P. (1928). Kristallographische und strukturtheoretische
Grundbegriffe. Handbuch der Experimentalphysik, Vol. 7, Part
1. Leipzig: Akademische Verlagsgesellschaft.
Santoro, A., Mighell, A. D. & Rodgers, J. R. (1980). The
determination of the relationship between derivative lattices.
Acta Cryst. A36, 796–800.
Wolff, P. M. de, Belov, N. V., Bertaut, E. F., Buerger, M. J., Donnay,
J. D. H., Fischer, W., Hahn, Th., Koptsik, V. A., Mackay, A. L.,
Wondratschek, H., Wilson, A. J. C. & Abrahams, S. C. (1985).
Nomenclature for crystal families, Bravais-lattice types and
arithmetic classes. Report of the International Union of Crystal-
lography Ad hoc Committee on the Nomenclature of Symmetry.
Acta Cryst. A41, 278–280.
9.3
Cassels, J. W. S. (1971). An introduction to the geometry of numbers,
p. 13. Berlin: Springer.
Delaunay, B. N. (1933). Neuere Darstellung der geometrischen
Kristallographie. Z. Kristallogr. 84, 109–149.
Eisenstein, G. (1851). Tabelle der reducierten positiven quadra-
tischen Formen nebst den Resultaten neuer Forschungen u
¨
ber
diese Formen, insbesondere Beru
¨
cksichtigung auf ihre tabellar-
ische Berechung. J. Math. (Crelle), 41, 141–190.
Gruber, B. (1989). Reduced cells based on extremal principles. Acta
Cryst. A45, 123–131.
Gruber, B. (1992). Topological approach to the Niggli lattice
characters. Acta Cryst. A48, 461–470.
Gruber, B. (1997a). Classification of lattices: a new step. Acta Cryst.
A53, 505–521.
Gruber, B. (1997b). Alternative formulae for the number of
sublattices. Acta Cryst. A53, 807–808.
International Tables for X-ray Crystallography (1952). Vol. I, edited
by N. F. M. Henry & K. Lonsdale, pp. 530–535. Birmingham:
Kynoch Press.
International Tables for X-ray Crystallography (1969). Vol. I, 3rd
ed., edited by N. F. M. Henry & K. Lonsdale, pp. 530–535.
Birmingham: Kynoch Press.
Niggli, P. (1928). Kristallographische und strukturtheoretische
Grundbegriffe. Handbuch der Experimentalphysik, Vol. 7, Part
1. Leipzig: Akademische Verlagsgesellschaft.
Wolff, P. M. de (1988). Definition of Niggli’s lattice characters.
Comput. Math. Appl. 16, 487–492.
Wolff, P. M. de & Gruber, B. (1991). Niggli lattice characters:
definition and graphical representation. Acta Cryst. A47, 29–36.
760
9. CRYSTAL LATTICES
references
10.1. Crystallographic and noncrystallographic point groups
BY TH. HAHN AND H. KLAPPER
10.1.1. Introduction and definitions
A point group* is a group of symmetry operations all of which leave
at least one point unmoved. Thus, all operations containing
translations are excluded. Point groups can be subdivided into
crystallographic and noncrystallographic point groups. A crystal-
lographic point group is a point group that maps a point lattice onto
itself. Consequently, rotations and rotoinversions are restricted to
the well known crystallograph ic cases 1, 2, 3, 4, 6 and
1, 2
m,
3, 4, 6(cf. Chapter 1.3); matrices for these symmetry operations
are listed in Tables 11.2.2.1 and 11.2.2.2. No such restrictions apply
to the noncrystallographic point groups.
The numbers of the crystallographic point groups are finite: 2 for
one dimension, 10 for two dimensions and 32 for three dimensions.
The numbers of noncrystallographic point groups for dimensions
n 2 are infinite. The two- and three-dimensional crystallographic
point groups and their crystal systems are summarized in Tables
10.1.1.1 and 10.1.1.2. They are described in detail in Section 10.1.2.
The two one-dimensional point groups are discussed in Section
2.2.17. The noncrystallographic point groups are treated in Section
10.1.4.
Crystallographic point groups occur:
(i) in vector space as symmetries of the external shapes of
crystals, i.e. of the set of vectors normal to the crystal faces
(morphological symmetry);
(ii) in point space as site symmetries of points in lattices or in
crystal structures and as symmetries of atomic groups and
coordination polyhedra.
General point groups, i.e. crystallographic and noncrystallographic
point groups, occur as:
(iii) symmetries of (rigid) molecules (molecular symmetry):
(iv) symmetries of physical properties of crystals (e.g. tensor
symmetries); here noncrystallographic point groups with axes of
order infinity are of particular importance, as in the symmetries of
spheres or rotation ellipsoids:
(v) approximate symmetries of the local environment of a point
in a crystal structure, i.e. as local site symmetries. Examples are
sphere-like atoms or ions in crystals, as well as icosahedral atomic
groups. These noncrystallographic symmetries, however, are only
approximate, even for the close neighbourhood of a site.
A (geometric) crystal class is the set of all crystals having the
same point-group symmetry. The word ‘class’, therefore, denotes a
classificatory pigeon-hole and should not be used as synonymous
with the point group of a particular crystal. The symbol of a crystal
class is that of the common point group. (For geometric and
arithmetic crystal classes of space groups, see Sections 8.2.3 and
8.2.4.)
Of particular importance for the structure determination of
crystals are the 11 centrosymmetric crystallographic point groups,
because they describe the possible symmetries of the diffraction
record of a crystal:
1; 2=m; mmm;4=m;4=mmm; 3; 3m;6=m;
6=mmm; m
3; m3m. This is due to Friedel’s rule, which states that,
provided anomalous dispersion is neglected, the diffraction record
of any crystal is centrosymmetric, even if the crystal is
noncentrosymmetric. The symmetry of the diffraction record
determines the Laue class of the crystal; this is further explained
in Part 3. For a given crystal, its Laue class is obtained if a
symmetry centre is added to its point group, as shown in Table
10.2.1.1.
In two dimensions, six ‘centrosymmetric’ crystallographic point
groups and hence six two-dimensional Laue classes exist: 2; 2mm;
4; 4mm;6;6mm. These point groups are, for instance, the only
possible symmetries of zero-layer X-ray photographs.
Among the centrosymmetric crystallographic point groups in
three dimensions, the lattice point groups (holohedral point groups,
holohedries) are of special importance because they constitute the
seven possible point symmetries of lattices, i.e. the site symmetries
of their nodes. In three dimensions, the seven holohedries are:
1;
2=m; mmm;4=mmm;
3m;6=mmm; m3m. Note that 3m is the point
symmetry of the rhombohedral lattice and 6=mmm the point
symmetry of the hexagonal lattice; both occur in the hexagonal
crystal family (cf. Chapter 2.1). Point groups that are, within a
crystal family, subgroups of a holohedry are called merohedries;
they are called specifically hemihedries for subgroups of index 2,
tetartohedries for index 4 and ogdohedries for index 8.
In two dimensions, four holohedries exist: 2; 2mm;4mm;6mm.
Note that the hexagonal crystal family in two dimensions contains
only one lattice type, with point symmetry 6mm.
Another classification of the crystallographic point groups is that
into isomorphism classes. Here all those point groups that have the
same kind of group table appear in one class. These isomorphism
classes are also known under the name of abstract point groups.
There are 18 abstract crystallographic point groups in three
dimensions: the point groups in each of the following lines are
isomorphous and belong to the same abstract group:
Order 1 : 1 Order 8 : 422, 4mm,
42m
2 :
1, 2, m 12 : 6=m
3 : 312:
3m, 622, 6mm, 62m
4 : 2=m, 222, mm212: 23
4 : 4,
416: 4=mmm
6 :
3, 6, 624: 6=mmm
6 : 32, 3m 24 : m
3
8 : mmm 24 : 432,
43m
8 : 4=m 48 : m
3m:
In two dimensions, the ten crystallographic point groups form nine
abstract groups; the groups 2 and m are isomorphous and belong to
the same abstract group, the remaining eight point groups
correspond to one abstract group each.
Table 10.1.1.1. The ten two-dimensional crystallographic point
groups, arranged according to crystal system
Dashed lines separate point groups with different Laue classes within one
crystal system.
Crystal system
General
symbol
Oblique (top)
Rectangular (bottom) Square Hexagonal
n 12436
nmm m 2mm 4mm 3m 6mm
*
For reasons of simplicity, in this chapter the same term ‘point group’ is used for a
‘particular point group’ and a ‘type of point group’. For space groups, this
distinction is explained in Section 8.2.2. For a different use of the term ‘point group’
see Section 8.1.6.
762
International Tables for Crystallography (2006). Vol. A, Chapter 10.1, pp. 762–803.
Copyright © 2006 International Union of Crystallography
10.1.2. Crystallographic point groups
10.1.2.1. Description of point groups
In crystallography, point groups usually are described
(i) by means of their Hermann–Mauguin or Schoenflies symbols;
(ii) by means of their stereographic projections;
(iii) by means of the matrix representations of their symmetry
operations, frequently listed in the form of Miller indices (hkl)of
the equivalent general crystal faces;
(iv) by means of drawings of actual crystals, natural or synthetic.
Descriptions (i) through (iii) are given in this section, whereas for
crystal drawings and actual photographs reference is made to
textbooks of crystallography and mineralogy; this also applies to the
construction and the properties of the stereographic projection.
In Tables 10.1.2.1 and 10.1.2.2, the two- and three-dimensional
crystallographic point groups are listed and described. The tables
are arranged according to crystal systems and Laue classes. Within
each crystal system and Laue class, the sequence of the point groups
corresponds to that in the space-group tables of this volume: pure
rotation groups are followed by groups containing reflections,
rotoinversions and inversions. The holohedral point group is always
given last.
In Tables 10.1.2.1 and 10.1.2.2, some point groups are described
in two or three versions, in order to bring out the relations to the
corresponding space groups (cf. Section 2.2.3):
(a) The three monoclinic point groups 2, m and 2=m are given
with two settings, one with ‘unique axis b’ and one with ‘unique
axis c’.
(b) The two point groups
42m and 6m2 are described for two
orientations with respect to the crystal axes, as
42m and 4m2 and as
6m2 and 62m.
(c) The five trigonal point groups 3,
3, 32, 3m and 3m are treated
with two axial systems, ‘hexagonal axes’ and ‘rho mbohedral axes’.
(d) The hexagonal-axes description of the three trigonal point
groups 32, 3m and
3m is given for two orientations, as 321 and 312,
as 3m1 and 31m, and as
3m1 and 31m; this applies also to the
two-dimensional point group 3m.
The presentation of the point groups is similar to that of the space
groups in Part 7. The headline contains the short Hermann–
Mauguin and the Schoenflies symbols. The full Hermann–Mauguin
symbol, if different, is given below the short symbol. No
Schoenflies symbols exist for two-dimensional groups. For an
explanation of the symbols see Section 2.2.4 and Chapter 12.1.
Next to the headline, a pair of stereographic projections is given.
The diagram on the left displays a general crystal or point form, that
on the right shows the ‘framework of symmetry elements’. Except
as noted below, the c axis is always normal to the plane of the figure,
the a axis points down the page and the b axis runs horizontally
from left to right. For the five trigonal point groups, the c axis is
normal to the page only for the description with ‘hexagonal axes’; if
described with ‘rhombohedral axes’, the direction [111] is normal
and the positive a axis slopes towards the observer. The
conventional coordinate systems used for the various crystal
systems are listed in Table 2.1.2.1 and illustrated in Figs. 2.2.6.1
to 2.2.6.10.
In the right-hand projection, the graphical symbols of the
symmetry elements are the same as those used in the space-group
diagrams; they are listed in Chapter 1.4. Note that the symbol of a
symmetry centre, a small circle, is also used for a face-pole in the
left-hand diagram. Mirror planes are indicated by heavy solid lines
or circles; thin lines are used for the projection circle, for symmetry
axes in the plane and for some special zones in the cubic system.
In the left-hand projection, the projection circle and the
coordinate axes are indicated by thin solid lines, as are again
some special zones in the cubic system. The dots and circles in this
projection can be interpreted in two ways.
(i) As general face poles, where they represent general crystal
faces which form a polyhedron , the ‘general crystal form’ (face
form) fhklg of the point group (see below). In two dimensions,
edges, edge poles, edge forms and polygons take the place of faces,
face poles, crystal forms (face forms) and polyhedra in three
dimensions.
Face poles marked as dots lie above the projection plane and
represent faces which intersect the positive c axis* (positive Miller
index l), those marked as circles lie below the projection plane
(negative Miller index l). In two dimensions, edge poles always lie
on the pole circle.
(ii) As general points (centres of atoms) that span a polyhedron
or polygon, the ‘general crystallographic point form’ x, y, z. This
interpretation is of interest in the study of coordination polyhedra,
atomic groups and molecular shapes. The polyhedron or polygon of
a point form is dual to the polyhedron of the corresponding crystal
form.†
The general, special and limiting crystal forms and point forms
constitute the main part of the table for each point group. The
theoretical background is given below under Crystal and point
forms; the explanation of the listed data is to be found under
Description of crystal and point forms.
Table 10.1.1.2. The 32 three-dimensional crystallographic point groups, arranged according to crystal system (cf. Chapter 2.1)
Full Hermann–Mauguin (left) and Schoenflies symbols (right). Dashed lines separate point groups with different Laue classes within one crystal system.
General
symbol
Crystal system
Triclinic
Monoclinic (top)
Orthorhombic (bottom) Tetragonal Trigonal Hexagonal Cubic
n 1 C
1
2 C
2
4 C
4
3 C
3
6 C
6
23 T
n 1 C
i
m 2 C
s
4 S
4
3 C
3i
6 3=mC
3h
––
n=m 2=mC
2h
4=mC
4h
––6=mC
6h
2=m3 T
h
n22 222 D
2
422 D
4
32 D
3
622 D
6
432 O
nmm mm2 C
2v
4mm C
4v
3mC
3v
6mm C
6v
––
n2m ––42mD
2d
32=mD
3d
62mD
3h
43mT
d
n=m 2=m 2=m 2=m 2=m 2=mD
2h
4=m 2=m 2=mD
4h
––6=m 2=m 2=mD
6h
4=m 32=mO
h
*
This does not apply to ‘rhombohedral axes’: here the positive directions of all
three axes slope upwards from the plane of the paper: cf. Fig. 2.2.6.9.
{
Dual polyhedra have the same number of edges, but the numbers of faces and
vertices are interchanged; cf. textbooks of geometry.
763
10.1. CRYSTALLOGRAPHIC AND NONCRYSTALLOGRAPHIC POINT GROUPS
The last entry for each point group contains the Symmetry of
special projections, i.e. the plane point gr oup that is obtained if the
three-dimensional point group is projected along a symmetry
direction. The special projection directions are the same as for the
space groups; they are listed in Section 2.2.14. The relations
between the axes of the three-dimensional point group and those of
its two-dimensional projections can easily be derived with the help
of the stereographic projection. No projection symmetries are listed
for the two-dimensional point groups.
Note that the symmetry of a projection along a certain direction
may be higher than the symmetry of the crystal face normal to that
direction. For example, in point group
1 all faces have face
symmetry 1, whereas projections along any direction have
symmetry 2; in point group 422, the faces (001) and 00
1 have
face symmetry 4, whereas the projection along [001] has sym metry
4mm.
10.1.2.2. Crystal and point forms
For a point group P a crystal form is a set of all symmetrically
equivalent faces; a point form is a set of all symmetrically
equivalent points. Crystal and point forms in point groups
correspond to ‘crystallographic orbits’ in space groups; cf. Section
8.3.2.
Two kinds of crystal and point forms with respect to P can be
distinguished. They are defined as follows:
(i) General form:Aface is called ‘general’ if only the identity
operation transforms the face onto itself. Each complete set of
symmetrically equivalent ‘general faces’ is a general crystal form.
The multiplicity of a general form, i.e. the number of its faces, is the
order of P. In the stereographic projection, the poles of general
faces do not lie on symmetry elements of P.
A point is called ‘general’ if its site symmetry, i.e. the group of
symmetry operations that transforms this point onto itself, is 1. A
general point form is a complete set of symmetrically equivalent
‘general points’.
(ii) Special form:Aface is called ‘special’ if it is transformed
onto itself by at least one symmetry operation of P, in addition to
the identity. Each complete set of symmetrically equivalent ‘special
faces’ is called a special crystal form. The face symmetry of a
special face is the group of symmetry operations that transforms this
face onto itself; it is a subgroup of P. The multiplicity of a special
crystal form is the multiplicity of the general form divided by the
order of the face-symmetry group. In the stereographic projection,
the poles of special faces lie on symmetry elements of P. The Miller
indices of a special crystal form obey restrictions like fhk0g,
fhhlg, f100g.
A point is called ‘special’ if its site symmetry is higher than 1. A
special point form is a complete set of symmetrically equivalent
‘special points’. The multiplicity of a special point form is the
multiplicity of the general form divided by the order of the site-
symmetry group. It is thus the same as that of the corresponding
special crystal form. The coordinates of the points of a special point
form obey restrictions, like x, y,0;x, x, z
; x, 0, 0. The point 0, 0, 0 is
not considered to be a point form.
In two dimensions, point groups 1, 2, 3, 4 and 6 and, in three
dimensions, point groups 1 and
1 have no special crystal and point
forms.
General and special crystal and point forms can be represented by
their sets of equivalent Miller indices fhkl g and point coordinates
x, y, z. Each set of these ‘triplets’ stands for infinitely many crystal
forms or point forms which are obtained by independent variation of
the values and signs of the Miller indices h, k, l or the point
coordinates x, y, z.
It should be noted that for crystal forms, owing to the well known
‘law of rational indices’, the indices h, k, l must be integers; no such
restrictions apply to the coordinates x, y, z, which can be rational or
irrational numbers.
Example
In point group 4, the general crystal form fhklg stands for the set
of all possible tetragonal pyramids, pointing either upwards or
downwards, depending on the sign of l; similarly, the general
point form x, y, z includes all possible squares, lying either above
or below the origin, depending on the sign of z. For the limiting
cases l 0orz 0, see below.
In order to survey the infinite number of possible forms of a point
group, they are classified into Wyckoff positions of crystal and point
forms, for short Wyckoff positions. This name has been chosen in
analogy to the Wyckoff positions of space groups; cf. Sections
2.2.11 and 8.3.2. In point groups, the term ‘position’ can be
visualized as the position of the face poles and points in the
stereographic projection. Each ‘Wyckoff position’ is labelled by a
Wyckoff letter.
Definition
A ‘Wyckoff position of crystal and point forms’ consists of all
those crystal forms (point forms) of a point group P for which the
face poles (points) are positioned on the same set of conjugate
symmetry elements of P; i.e. for each face (point) of one form
there is one face (point) of every other form of the same
‘Wyckoff position’ that has exactly the same face (site)
symmetry.
Each point group contains one ‘general Wyckoff position’
comprising all
general crystal and point forms. In addition, up to
two ‘special Wyckoff positions’ may occur in two dimensions and
up to six in three dimensions. They are characterized by the
different sets of conjugate face and site symmetries and correspond
to the seven positions of a pole in the interior, on the three edges,
and at the three vertices of the so-called ‘characteristic triangle’ of
the stereographic projection.
Examples
(1) All tetragonal pyramids fhklg and tetragonal prisms fhk0g in
point group 4 have face symmetry 1 and belong to the same
general ‘Wyckoff position’ 4b, with Wyckoff letter b.
(2) All tetragon al pyramids and tetragonal prisms in point group
4mm belong to two special ‘Wyckoff positions’, depending on
the orientation of their face-symmetry groups m with respect to
the crystal axes: For the ‘oriented face symmetry’ .m., the forms
fh0lgand f100gbelong to Wyckoff position 4c; for the oriented
face symmetry ..m, the forms fhhlg and f110g belong to
Wyckoff position 4b. The face symmetries .m. and ..m are not
conjugate in point group 4mm. For the analogous ‘oriented site
symmetries’ in space groups, see Section 2.2.12.
It is instructive to subdivide the crystal forms (point forms) of
one Wyckoff position further, into characteristic and nonchar-
acteristic forms. For this, one has to consider two symmetries that
are connected with each crystal (point) form:
(i) the point group P by which a form is generated (generating
point group), i.e. the point group in which it occurs;
(ii) the full symmetry (inherent symmetry) of a fo rm (considered
as a polyhedron by itself), here called eigensymmetry C . The
eigensymmetry point group C is either the generating point group
itself or a supergroup of it.
Examples
(1) Each tetragonal pyramid fhklgl 6 0
of Wyckoff position 4
b
in point group 4 has generating symmetry 4 and eigensymmetry
764
10. POINT GROUPS AND CRYSTAL CLASSES
4mm; each tetragonal prism fhk0g of the same Wyckoff
position has generating symmetry 4 again, but eigensymmetry
4=mmm.
(2) A cube f100g may have generating symmetry 23, m
3, 432, 43m
or m
3m, but its eigensymmetry is always m3m.
The eigensymmetries and the generating symmetries of the 47
crystal forms (point forms) are listed in Table 10.1.2.3. With the
help of this table, one can find the various point groups in which a
given crystal form (point form) occurs, as well as the face (site)
symmetries that it exhibits in these point groups; for experimental
methods see Sections 10.2.2 and 10.2.3.
With the help of the two groups P and C, each crystal or point
form occurring in a particular point group can be assigned to one of
the following two categories:
(i) characteristic form, if eigensymmetry C and generating
symmetry P are the same;
(ii) noncharacteristic form, if C is a proper supergroup of P.
The importance of this classification will be apparent from the
following examples.
Examples
(1) A pedion and a pinacoid are noncharacteristic forms in all
crystallographic point groups in which they occur:
(2) all other crystal or point forms occur as characteristic forms in
their eigensymmetry group C;
(3) a tetragonal pyramid is noncharacteristic in point group 4 and
characteristic in 4mm;
(4) a hexagonal prism can occur in nine point groups (12 Wyckoff
positions) as a noncharacteristic form; in 6=mmm, it occurs in
two Wyckoff positions as a characteristic form.
The general forms of the 13 point groups with no, or only one,
symmetry direction (‘monoaxial groups’) 1, 2, 3, 4, 6,
1, m, 3, 4, 6
3=m,2=m,4=m,6=m are always noncharacteristic, i.e. their eigen-
symmetries are enhanced in comparison with the generating point
groups. The general positions of the other 19 point groups always
contain characteristic crystal forms that may be used to determine
the point group of a crystal uniquely (cf. Section 10.2.2).*
So far, we have considered the occurrence of one crystal or point
form in different point groups and different Wyckoff positions. We
now turn to the occurrence of different kinds of crystal or point
forms in one and the same Wyckoff position of a particular point
group.
In a Wyckoff position, crystal forms (point forms) of different
eigensymmetries may occur; the crystal forms (point forms) with
the lowest eigensymmetry (which is always well defined) are called
basic forms (German: Grundformen) of that Wyckoff position. The
crystal and point forms of higher eigensymmetry are called limiting
forms (German: Grenzformen)(cf. Table 10.1.2.3). These forms are
always noncharacteristic.
Limiting forms† occur for certain restricted values of the Miller
indices or point coordinates. They always have the same multi-
plicity and oriented face (site) symmetry as the corresponding basic
forms because they belong to the same Wyckoff position. The
enhanced eigensymmetry of a limiting form may or may not be
accompanied by a change in the topology‡ of its polyhedra,
compared with that of a basic form. In every case, however, the
name of a limiting form is different from that of a basic form.
The face poles (or points) of a limiting form lie on symmetry
elements of a supergroup of the point group that are not symmetry
elements of the point group itself. There may be several such
supergroups.
Examples
(1) In point group 4, the (noncharacteristic) crystal forms
fhklgl 6 0 (tetragonal pyramids) of eigensymmetry 4mm
are basic forms of the general Wyckoff position 4b, whereas the
forms fhk0g (tetragonal prisms) of higher eigensymmetry
4=mmm are ‘limiting general forms’. The face poles of forms
fhk0g lie on the horizontal mirror plane of the supergroup 4=m.
(2) In point group 4mm, the (characteristic) special crystal forms
fh
0lg with eigensymmetry 4mm are ‘basic forms’ of the special
Wyckoff position 4c, whereas f100g with eigensymmetry
4=mmm is a ‘limiting special form’. The face poles of f100g
are located on the intersections of the vertical mirror planes of
the point group 4mm with the horizontal mirror plane of the
supergroup 4=mmm, i.e. on twofold axes of 4=mmm.
Whereas basic and limiting forms belonging to one ‘Wyckoff
position’ are always clearly distinguished, closer inspection shows
that a Wyckoff position may contain different ‘types’ of limiting
forms. We need, therefore, a further criterion to classify the limiting
forms of one Wyckoff position into types: A type of limiting form of
a Wyckoff position consists of all those limiting forms for which the
face poles (points) are located on the same set of additional
conjugate symmetry elements of the holohedral point group (for the
trigonal point groups, the hexagonal holohedry 6=mmm has to be
taken). Different types of limiting forms may have the same
eigensymmetry and the same topology, as shown by the examples
below. The occurrence of two topologically different polyhedra as
two ‘realizations’ of one type of limiting form in point groups 23,
m
3 and 432 is explained below in Section 10.1.2.4, Notes on crystal
and point forms, item (viii).
Examples
(1) In point group 32, the limiting general crystal forms are of four
types:
(i) ditrigonal prisms, eigensymmetry
62m (face poles on
horizontal mirror plane of holohedry 6=mmm);
(ii) trigonal dipyramids, eigensymmetry
62m (face poles on
one kind of vertical mirror plane);
(iii) rhombohedra, eigensymmetry
3m (face poles on second
kind of vertical mirror plane);
(iv) hexagonal prisms, eigensymmetry 6=mmm (face poles on
horizontal twofold axes).
Types (i) and (ii) have the same eigensymmetry but different
topologies; types (i) and (iv) have the same topology but
different eigensymmetries; type (iii) differs from the other three
types in both eigensymmetry and topology.
(2) In point group 222, the face poles of the three types of general
limiting forms, rhombic prisms, are located on the three (non-
equivalent) symmetry planes of the holohedry mmm. Geome-
trically, the axes of the prisms are directed along the three non-
equivalent orthorhombic symmetry directions. The three types
*
For a survey of these relations, as well as of the ‘limiting forms’, it is helpful to
consider the (seven) normalizers of the crystallographic point groups in the group of
all rotations and reflections (orthogonal group, sphere group); normalizers of the
crystallographic and noncrystallographic point groups are listed in Tables 15.4.1.1
and 15.4.1.2.
{
The treatment of ‘limiting forms’ in the literature is quite ambiguous. In some
textbooks, limiting forms are omitted or treated as special forms in their own right:
other authors define only limiting general forms and consider limiting special forms
as if they were new special forms. For additional reading, see P. Niggli (1941, pp.
80–98).
{
The topology of a polyhedron is determined by the numbers of its vertices, edges
and faces, by the number of vertices of each face and by the number of faces
meeting in each vertex.
765
10.1. CRYSTALLOGRAPHIC AND NONCRYSTALLOGRAPHIC POINT GROUPS
of limiting forms have the same eigensymmetry and the same
topology but different orientations.
Similar cases occur in point groups 422 and 622 (cf. Table
10.1.2.3, footnote *).
Not considered in this volume are limiting forms of another kind,
namely those that require either special metrical conditions for the
axial ratios or irrational indices or coordinates (which always can be
closely approximated by rational values). For instance, a rhombic
disphenoid can, for special axial ratios, appear as a tetragonal or
even as a cubic tetrahedron; similarly, a rhombohedron can
degenerate to a cube. For special irrational indices, a ditetragonal
prism changes to a (noncrystallographic) octagonal prism, a
dihexagonal pyramid to a dodecagonal pyramid or a crystal-
lographic pentagon-dodecahedron to a regular pentagon-dodecahe-
dron. These kinds of limiting forms are listed by A. Niggli (1963).
In conclusion, each general or special Wyckoff position always
contains one set of basic crystal (point) forms. In addition, it may
contain one or more sets of limiting forms of different types. As a
rule,† each type comprises polyhedra of the same eigensymmetry
and topology and, hence, of the same name, for instance
‘ditetragonal pyrami d’. The name of the basic general forms is
often used to designate the corresponding crystal class, for instance
‘ditetragonal-pyramidal class’; some of these names are listed in
Table 10.1.2.4.
10.1.2.3. Description of crystal and point forms
The main part of each point-group table describes the general and
special crystal and point forms of that point group, in a manner
analogous to the positions in a space group. The general Wyckoff
position is given at the top, followed downwards by the special
Wyckoff positions with decreasing multiplicity. Within each
Wyckoff position, the first block refers to the basic forms, the
subsequent blocks list the various types of limiting form, if any.
The columns, from left to right, contain the following data
(further details are to be found below in Section 10.1.2.4, Notes on
crystal and point forms):
Column 1: Multiplicity of the ‘Wyckoff position’, i.e. the number
of equivalent faces and points of a crystal or point form.
Column 2: Wyckoff letter. Each general or special ‘Wyckoff
position’ is designated by a ‘Wyckoff letter’, analogous to the
Wyckoff letter of a position in a space group ( cf. Section 2.2.11).
Column 3: Face symmetry or site symmetry, given in the form of
an ‘oriented point-group symbol’, analogous to the oriented site-
symmetry symbols of space groups (cf. Section 2.2.12). The face
symmetry is also the symmetry of etch pits, striations and other face
markings. For the two-dimensional point groups, this column
contains the edge symmetry, which can be either 1 or m.
Column 4: Name of crystal form. If more than one name is in
common use, several are listed. The names of the limiting forms are
also given. The crystal forms, their names, eigensymmetries and
occurrence in the point groups are summarized in Table 10.1.2.3,
which may be useful for determinative purposes, as explained in
Sections 10.2.2 and 10.2.3. There are 47 different types of crystal
form. Frequently, 48 are quoted because ‘sphenoid’ and ‘dome’ are
considered as two different forms. It is customary, however, to
regard them as the same form, with the name ‘dihedron’.
Name of point form (printed in italics). There exists no general
convention on the names of the point forms. Here, only one name is
given, which does not always agree with that of other authors. The
names of the point forms are also contained in Table 10.1.2.3. Note
that the same point form, ‘line segment’, corresponds to both
sphenoid and dome. The letter in parentheses after each name of a
point form is explained below.
Column 5: Miller indices (hkl) for the symmetrically equivalent
faces (edges) of a crystal form. In the trigonal and hexagonal crystal
systems, when referring to hexagonal axes, Bravais–Miller indices
(hkil) are used, with h k i 0.
Coordinates x, y, z for the symmetrically equivalent points of a
point form are not listed explicitly because they can be obtained
from data in this volume as follows: after the name of the point
form, a letter is given in parentheses. This is the Wyckoff letter of
the corresponding position in the symmorphic P space group that
belongs to the point group under consideration. The coordinate
triplets of this (general or special) position apply to the point form
of the point group.
The triplets of Miller indices (hkl) and point coordinates x, y , z
are arranged in such a way as to show analogous sequences; they are
both based on the same set of generators, as described in Sections
2.2.10 and 8.3.5. For all point groups, except those referred to a
hexagonal coordinate system, the correspondence between the (
hkl)
and the x, y, z triplets is immediately obvious.‡
The sets of symmetrically equivalent crystal faces also represent
the sets of equivalent reciprocal-lattice points, as well as the sets of
equivalent X-ray (neutron) reflections.
Examples
(1) In point group
4, the general crystal form 4b is listed as
hkl
hklkhlkhl: the corresponding general position 4h
of the symmorphic space group P
4 reads x, y, z; x, y, z; y, x, z;
y, x, z.
(2) In point group 3, the general crystal form 3b is listed as (hkil)
(ihkl)(kihl) with i h k; the corresponding general
position 3d of the symmorphic space group P3 reads x, y, z;
y, x y, z; x y, x, z .
(3) The Miller indices of the cubic point groups are arranged in one,
two or four blocks of 3 4 entries. The first block (upper left)
belongs to point group 23. The second block (upper right)
belongs to the diagonal twofold axes in 432 and m
3m or to the
diagonal mirror plane in
43m. In point groups m3 and m3m,the
lower one or two blocks are derived from the upper blocks by
application of the inversion.
10.1.2.4. Notes on crystal and point forms
(i) As mentioned in Section 10.1.2.2, each set of Miller indices of
a given point group represents infinitely many face forms with the
same name. Exceptions occur for the following cases.
Some special crystal forms occur with only one representative.
Examples are the pinacoid f001g, the hexagonal prism f10
10g and
the cube f100g. The Miller indices of these forms consist of fixed
numbers and signs and contain no variables.
In a few noncentrosymmetric point groups, a special crystal form
is realized by two representatives: they are related by a centre of
symmetry that is not part of the point-group symmetry. These cases
are
(a) the two pedions (001) and 00
1;
{
For the exceptions in the cubic crystal system cf. Section 10.1.2.4, Notes on
crystal and point forms, item (viii)
{
The matrices of corresponding triplets
~
h
~
k
~
l and
~
x,
~
y,~z, i.e. of triplets generated by
the same symmetry operation from (hkl) and x, y, z, are inverse to each other,
provided the x, y, z and
~
x,
~
y,~z are regarded as columns and the (hkl) and
~
h
~
k
~
l as
rows: this is due to the contravariant and covariant nature of the point coordinates
and Miller indices, respectively. Note that for orthogonal matrices the inverse
matrix equals the transposed matrix; in crystallography, this applies to all coordinate
systems (including the rhombohedral one), except for the hexagonal system. The
matrices for the symmetry operations occurring in the crystallographic point groups
are listed in Tables 11.2.2.1 and 11.2.2.2.
766
10. POINT GROUPS AND CRYSTAL CLASSES