Mm 1W18 072 if

"^^^"^ Space Groups and Lattice Complexes; —

— Werner FISHER

Mineralogisches Institut der Philipps-Universitat

D-3550 Marburg/Lahn, German Federal Republic

Hans BURZLAFF

Lehrstuhl fiir Kristallographie der

Friedrich-Alexander-Universitat Erlangen-Niirnberg

D-8520 Erlangen, German Federal Republic

Erwin HELLNER

Fachbereich Geowissenschaften der Philipps-Universitat

D-3550 Marburg/Lahn, German Federal Republic

J. D. H. DONNAY

Department de Geologic, Universite'de Montreal

CP 6128 Montreal 101, Quebec, Canada

Formerly, The Johns Hopkins University, Baltimore, Maryland

Prepared under the auspices of the Commission on International Tables,

International Union of Crystallography, and the Office of Standard

Reference Data, National Bureau of Standards.

U.S. DEPARTMENT OF COMMERCE, Frederick B. Dent, Secreiory

NATIONAL BUREAU OF STANDARDS, Richard W. Roberts, Dnecior

Issued May 1973

Library of Congress Catalog Number: 73-600099

National Bureau of Standards Monograph 134

Nat. Bur. Stand. (U.S.), Monogr. 134, 184 pages (May 1973)

CODEN: NBSMA6

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402

(Order by SD Catalog No. C13,44:134). Price: $4.10, domestic postpaid; $3.75, GPO Bookstore

Stock Number 0303-01143

Abstract

The lattice complex is to the space group what the site set is to

the point group - an assemblage of symmetry- related equivalent points.

The symbolism introduced by Carl Hermann has been revised and extended.

A total of 402 lattice complexes are derived from 67 Weissenberg com-

plexes. The Tables list site sets and lattice complexes in standard

and alternate representations. They answer the following questions:

What are the co-ordinates of the points in a given lattice complex?

In which space groups can a given lattice complex occur? What are the

lattice complexes that can occur in a given space group? The higher

the symmetry of the crystal structures is, the more useful the lattice-

complex approach should be on the road to the ultimate goal of their

classification.

Keywords: Crystallography; crystal point groups ; crystal structure;

lattice complexes; site sets; space groups.

Acknowledgments

We wish to acknowledge support received from the Philipps-

Universitat, The Johns Hopkins University, the National Bureau of

Standards and the International Union of Crystallography.

Special thanks go to our students and assistants, both in

Marburg and in Baltimore, for their enthusiastic participation in

the construction and the checking of the tables.

The authors

III

T 0

THE

MEMORY

0 F

CARL HEINRICH HERMANN

(1898 - 1961)

Dr. Phil, of the University of Gottingen,Professor of Crystallography at the University

Marburg-on-the-Lahn, Germany

CONTENTS

Page

TEXT

Introduction 1

Part I. SITE SETS 3

1. Generalities 3

1 .1 Definitions 3

1 .2 Rules of Spl itting 5

1 .3 Special Positions 6

2. Description of Site Sets 6

3. Occurrence's of Site Sets 9

4. Subgroup Relations 9

5. Site Sets and Their Occurrences in Oriented Subgroups 9

5.1 Cubic Degradation 9

5.2 Hexagonal Degradation 10

Part II. LATTICE COMPLEXES 11

6. Generalities 11

6.1 Definitions 11

6.2 The Lattice-Complex Symbol 12

6.21 Invariant Lattice Complexes 12

6.22 Variant Lattice Complexes 16

6.3 Rules for Selecting the Generating Lattice Complex 16

6.4 Distribution Symmetry 17

7. Weissenberg Complexes as Generating Complexes 18

7.1 The Weissenberg Complexes 18

7.2 Tables of Weissenberg Complexes 19

7.3 Figures of Weissenberg Complexes 19

8. List of the Lattice Complexes and Their Occurrences 20

9. List of Space Groups with Lattice-Complex Representations 23

Appendix. Subgroup Relations 23

References 26

FIGURES

1-47 Site Sets in Stereographic Projection 29

48 Subgroup Relations in Point Groups 3449-53 Cubic Lattice Complexes 3554-57 Hexagonal Lattice Complexes 4058-60 Tetragonal Lattice Complexes 4461-63 Orthorhombic Lattice Complexes 4764-65 Monoclinic Lattice Complexes 50

TABLES

1 Rules of Splitting 54Site Sets:

2 Cubic 55

3 Hexagonal and Rhombohedral 56

4 Rhombohedral (in R co-ordinates) 58

5 Tetragonal 59

6 Orthorhombic 607 Monoclinic , 61

8 Anorthic (Triclinic) 61

9-15 Site Sets in Positions in Point Groups 6216 Cubic Degradation 64

17 Hexagonal Degradation 66

Weissenberg Complexes18 Cubic 69

19 Hexagonal and Rhombohedral 72

20 Rhombohedral (in R co-ordinates) 75

21 Tetragonal 76

22 Orthorhombic 79

23 Monoclinic 8224 Anorthic (Triclinic) 83

V

Occurrence of Lattice Complexes

25 CubicInvariant 85Univariant 88

Bi variant 92

Tri variant 93

27 Hexagonal and RhombohedralInvariant 95Univariant 98Bi variant 101

Trivariant 10328 Tetragonal

Invari ant 105

Univariant 109

Bivariant 114

Trivariant 117

29 OrthorhombicInvariant 119

Univariant 121

Bivariant 124Trivariant 127

30 Monocl inicInvariant 129

Univariant 129

Bivariant 130Trivariant 130

31 Anorthic (Triclinic) 131

Space Groups with Lattice-Complex Representations32 Cubic 133

33 Hexagonal and Rhombohedral 13834 Rhombohedral (in R co-ordinates) 14835 Tetragonal 14936 Orthorhombic 1 58

37 Monocl inic 16538 Anorthic (Triclinic) 167

Appendix

43 Subgroup Relations 169

Index 175

Errata 177

VI

SPACE GROUPS AND LATTICE COMPLEXES

Werner Fischer, Hans Burzlaff, Erwin Hellner, and J.D.H. Donnay

INTRODUCTION

A lattice is defined, sensu striata, as the

assemblage of all the points that are the termini

of the vectors

t.{uvw) = ua + uIj + wc,

where a, b, c are three non-coplanar vectors and

u, V, w are integers that can take all possible

values, positive, negative or zero. The lattice

points are called nodes so they can be distinguish-

ed from other points in space. A lattice is the

geometrical expression of a translation group: if

a node is taken as origin, any other node can be

obtained from it by means of a translation that is

one of the vectors t.{uvw)

.

As defined above, a lattice would always be

primitive. In the case of symmetrical lattices, a

multiple cell can be used to define the lattice;

examples: cl, cubic body-centered; cF, cubic face-

centered; etc. Centered lattice modes are useful

in that they display the lattice symmetry, but they

are no more than alternate descriptions of lattices

that could be defined by primitive cells. Whatever

description is chosen, the lattice nodes are the

same, and, therefore, the translation group is also

the same.

After 1912 the word lattice was given various

physical meanings, in addition to its original geo-

metrical meaning, in the description of crystal

structures. A structure in which all the atoms were

of one kind, situated at the nodes of a lattice, could

be described as a "lattice" of such atoms. For example

the "lattice" of Cu atoms was used to designate the

crystal structure of copper metal. Atoms of one kind

were referred to as a "lattice" even in cases where

they did not occupy the nodes of any lattice; as the

carbon atoms in diamond, for instance, which -- to

this day! -- are described as forming the "diamond

lattice". In a crystal structure composed of more

than one kind of atoms, CsCl for example, as many

"atomic lattices" had to be distinguished as there

were different kinds of atoms, whence "the cesium

lattice" and "the chlorine lattice". The CsCl

structure was thus described by means of "two inter-

penetrating lattices" instead of "two atoms repeated

by a primitive cubic lattice". We realize nowadays

how unfortunate this concept of interpenetration was.

for there are as many interpentrating lattices as one

cares to pick points in the cell, whether these points

are occupied by atoms or not. As the crystal struc-

tures that were worked out increased in complexity, the

word lattice came to be used for any set of equi-

valent atoms in a structure, whether the atoms were

equivalent by lattice translations or by other sym-

metry operations in the space group. Even when the

atomic sets did occupy the nodes of lattices (s.s.),

different "lattices" were found to occur in the same

structure; as in fluorite, for instance, where the

"calcium lattice" is cF (cubic face-centered) with

cell edge a, whereas the "fluorine lattice", meaning

the "lattice" of fluorine atoms, is cP (cubic primi-

tive) with cell edge a/2. Confusion reached a climax

when the word "lattice" was given another additional

meaning to designate the collection of all the sets

of atomic centers in the crystal structure, in other

words, the crystal structure itself!

Paul Niggli (1919) introduced the term lattice

complex to designate a set of crystal! bgraphically

equivalent atoms in a crystal structure. Examples:

the Na atoms in NaCl , the C atoms in diamond, the

Cs atoms in CsCl, etc. Consider the carbon atoms

in diamond. They can be described in two ways:

either eight "interpenetrating carbon lattices" or

a cell content of 8 carbon atoms repeated by one

lattice. 1 Niggli visualized the former, as was

common at that time among German-speaking crystallo-

graphers. Not only are the carbon atoms of any one

of the eight "lattices" equivalent (both crystallo-

graphically and chemically) by lattice translations,

but the "carbon lattices" themselves are equivalent,

either by lattice translation or by some other space-

group- symmetry operation. The set of "atomic latt-

ices" could thus be called a complex of equivalent

lattices or a lattice complex for short. By the

same token. a crystal form, being a set of equivalent

faces, could be dubbed a face complex.

Niggli discusses the analogy between the crystal

forms of crystal morphology and the lattice complexes

^ Note that this lattice is taken as primitive (cP)

;

with a face-centered lattice (cF) , two atoms ar'e re-

peated by the lattice.

1

of crystal structure, pointing out that this analogy

is not complete. He also shows the importance of

lattice complexes^ in three fields: (1) the des-

cription of structures; (2) the crystal -structure

determination itself, in which structure factors

^{.hkl) should be computed by summing over lattice

complexes rather than individual atoms; (3) the des-

cription of space groups.

Niggli broaches the subject over and over

again: in the several editions of his Lehrbuch der

Mineralogie (1920, 1924, 1941) and in his Grundbe-

griffe (1928). Although he attempts to systematize

the nomenclature, he does not have any easy symbolism

for lattice complexes.

Carl Hermann recognized the importance of the

lattice complexes and devised a method for symbol-

izing invariant and uni variant lattice complexes, which

appeared in the 1935 edition of the international

Tables (see Chapters VI and VII). Such a complex was

designated by tne symbol of the space group of higiiest

symmetry that contains tne lattice complex as a point

position'^ of highest symmetry; the space-group symbol

being followed by the conventional Wyckoff letter of

this point position. If more than one such position

is available, the one that includes the origin (000)

among its sites is the one to be chosen. This scheme

is applicable to all lattice complexes, including the

tr I vari ant ones

.

For practical applications of lattice complexes

in 2 and 3 dimensions, see Burzlaff, Fischer and

Hellner (1968 and 1969); Fischer (1968); Smimova

(1966) ; Smimova and Poteshnova (1966) ; Hellner

(1965); Loeb (1967 and 1970).

^ Point position or, more simply, position is used

as a collective noun to designate a set of equivalent

points (either in a point group or in a space group)

.

It corresponds to the German PunktZage and the French

position. The term point position was used in the

English text of the 1935 edition of the Intematicmal

Tables; it has priority over the periphrase "set of

positions" used in the 1952 edition.

In 1953, in a paper given at the Erlangen meeting

of the Crystallography Section of the German Mineral

-

ogical Society, Hermann outlined a symbolism for uni-

variant lattice complexes, based on the splitting of

the points of an invariant lattice complex in a

number of directions depending on the available de-

grees of freedom. Hermann published this talk in

1960, adding new designations for the bivariant latt-

ice complexes, the space-group symbol being slightly

altered by the underlining of the appropriate m.

P. Niggli 's original concept of lattice complex

was somewhat modified by Hermann (1935) and by J.D.H.

Donnay, E. Hellner and A. Niggli (1966). This last

publication is a report on a Symposium held at the

University of Kiel, February 17-28, 1964, at which

the symbolism of lattice complexes was discussed by

A. de Vries and P. Th. Beurskens (University of

Utrecht), R. Allmann, H. Burzlaff, W. Fischer, U.

Wattenberg, the late Gudrun Weiss, and the three

rapporteurs

.

The fundamental feature of a lattice complex;

which was pointed out by Hermann in 1953, is that its

points either are themselves invariant or are dis.-

tributed in symmetrical clusters around the (in-

variant) points of an invariant complex. Each in-

variant point in its surroundings possesses its own

point symmetry, so that the problem largely reduces

to symbolizing sets of equivalent points in a point

group. Part I, accordingly, deals with site sets

in point groups. If we draw a sphere around a point

taken as the origin and let the symmetry elements of

a point group go througii the origin, symmet ry- re 1 a ted

points will lie on the surface of the sphere. The

poles of the faces of a crystal form constitute a

familiar example of a set of equivalent points. The

co-ordinates of one face pole are of the form ha*, kb*

,

la*, with h, k, I integers; they are the co-ordinates

of a node in the reciprocal lattice. In this work the

co-ordinates will be taken in direct space: x = u'a,

y = v'b, z = w'a, where u' , v' , w' need not be

integers

.

Part II applies the results of part I to lattice

complexes.

7

PART I. SITE SETS

1. GENERALITIES

1.1 Symbolism and definitions

The lattice, the infinite assemblage of points

that geometrically expresses the triple periodicity of

a crystal, is the basis for Bravais' definition of the

crystal systems. It possesses, at any one of its

nodes, a point symmetry which is one of the 32 crystal

point groups. There exist seven such lattice sym-

metries, to which Bravais' seven crystal systems cor-

respond. The point group of a lattice, together with

the subgroups that constitute its merohedries, are all

the point groups that are possible as crystal sym-

metries in the corresponding crystal system. When

the crystal symmetry is the same as the lattice sym-

metry, it is called holohedral synmetry or holohedry.

Examples: 32/m is the rhombohedral -lattice symmetry;

in the rhombohedral system it is the holohedry; 3m,

32, 5 and 3 are its merohedries; any substance with

a rhombohedral lattice can only have, as crystal sym-

metry, one of these five trigonal symmetries. Tiie

hexagonal-lattice symmetry, on the other hand, is

5/m 2/m 2/m; it has eleven merohedries (down to its

ogdohedry 3); any substance with a hexagonal lattice

can have, as crystal symmetry, any one of these

twelve point groups, which include the five trigonal

symmetries

.

The Hermann-Hauguin symbolism rests on the funda-

mental concept of the symmetry directions in the lat-

tice. A symmetry direction is defined as a symmetry

axis of proper or improper rotation; in other words, a

rotation axis^ or the normal to a mirror. Equivalent

symmetry directions that pass through the node taken

as the origin of a lattice, are symbolized in a col-

umn of the Hermann-Mauguin symbol: an Arabic numeral

indicates a symmetry axis along the direction under

consideration, the letter m represents a mirror

perpendicular to it. The number of columns in the

Hermann-Mauguin symbol is the number of kinds of equi-

An alternating axis (of rotatory inversion or

rotatory reflection) is a symmetry direction since it

contains a rotation axis as a subgroup. Examples: 4

contains 2; 6 and 3 both include 3. Note that 2 is

the normal to a mirror.

valent symmetry directions. Example: in the hexa-

gonal system, the three columns of the symbol refer

to: (1) the principal direction (or primary axis);

(2) the lateral axes of the 1st kind {Neben axes or

secondary axes); and (3) the lateral axes of the 2nd

kind {zwisahen axes or tertiary axes). The most sym-

metrical point group is the holohedry, 6/m 2/m 2/m or

^/mmm for short. The symmetry directions of the lat-

tice are vJiiat Carl Hermann called t\\e. Bliokriohtungen~

the directions in which you look to see the symmetry.

Note that, in the five trigonal classes, the

Hermann-Mauguin symbol can easily be made to distin-

guish between the case of a rhombohedral lattice and

that of an hexagonal one (Donnay, 1969): two-column

symbols (Table 11) include 3l and 31; three-column

symbols (Table 10) are explicited as 311 and 311. In

every case the figure 1 stands for symmetry directions

of the lattice that are not symmetry directions for the

crystal as a whole.

A orystallographia site set (here called simply

site set for short) is defined as a set of N equi-

valent points that are symmetry related in at least

one of the 32 crystal lographic point groups and can,

therefore, occupy a position^ in this point group. A

site set will be called an H-pointer (after the German

H-Punktner or H-Punkter) .. Every site of the N-pointer

is invariant under certain symmetry operations, which

constitute a subgroup of order N of the crystal lo-

graphic point group. This subgroup is called point sym-

metry of the site or site symmetry of the point jgroup.

A site thus lies on one, or more than one, synmetry

element, except in the trivial case where the site sym-

metry reduces to identity. (Symmetry elements are

mirrors, axes and the center.) The actual symmetry

elements in the various sites of a site set differ from

site to site: although equivalent mirrors and equi-

valent axes are found in every site, they are in

general differently oriented from one site to another.

Only one point can have the center as an element of

the site symmetry -- it is the fixed point in the point

group, taken as the origin 000. P. Niggli calls it

Hauptsymmetriepunkt (HSP), which we propose to trans-

late by highest symmetry point.

^ See Introduction, last footnote.

3

Examples. A site set is provided by the corners

of the cube. The eight points have the following co-

ordinates in the usual Cartesian system, with the

origin in the center:

They constitute a set of equivalent sites in any one

of three point groups: m3m (of order 48), 432 (of

order 24) and m3 (of order 24), with site symmetry

3m (of order 6) in mZm and 3 (of order 3) in both

432 and m3. In all three point groups, each site thus

lies on a 3-fold rotation axis, which coincides with

the line of intersection of three mirrors in the holo-

hedral point group m3m. In every case the site sym-

metry is a subgroup of index 8 of the appropriate

point group.

Another example is the assemblage of the 48 points

whose co-ordinates are obtained by replacing xxx, in

the first example, by each of the six complete permut-

ations of syz. These points are equivalent sites in

point group nQm, and their site symmetry is 1, the

trivial subgroup of index 48.

In a point group, as a limiting case resulting

from a specialization of site co-ordinates, a position

may, without change in multiplicity or site symmetry,

simulate a position that actually exists, with a dif-

ferent site symmetry, in another point group within the

same crystal system. The site set that is realized in

the latter position is called the limiting site set.

Example. In point group 23, the co-ordinates may

be made to obey the conditions y = z = the

general position ixyz; then simulates the special

position :xxO: of point groups n^m and 432. In this

limiting case the site set in -.xxQ: is the limiting

site set that is normally realized in the other point

groups; it will be designated by the same site-set

symbol, which, however, will be enclosed between

parentheses. ^

The site set realized in the general position

will be called general site set; that realized in any

special position, special site set. This terminology

is analogous to one that is applied to crystal forms

In Table 9 (point group 23) the parentheses around

12xx recall the fact that 12xx occurs as a limiting

site set of 6z2xy. The multiplicity and the site

symmetry of position :a:j/z : , where 6z2xy is realized,

remain the same in column :xa;Q : , where (12xx) occurs.

in morphological crystal lography.^

A limiting site set can occur only in a special

position in certain point groups, but not ever spe-

cial position in a given group can receive a limiting

site set (see Tables 9-15).

In the Tables the site symmetry is given in its

oriented form. In an oriented symbol dots stand for

Bliakriahtungen that are not used. In the hexagonal

crystal system, for instance site synmetry m may

appear as m.. , .m. , or . .m, according as the mirror

is the primary mirror, one of the secondary, or one

of the tertiary mirrors. In the rhombohedral crystal

system, them of the oriented symbol must always api-

pear in the second column of the two-column symbol

{.m).

The oriented symbol is easily recognized as the

Hermann-Mauguin symbol of the site synmetry, simply

by omitting the dots. By taking the dots into account,

the elements of symmetry of the site are identified

among those of the point group. Examples. In the

tetragonal system, 2mm. indicates that the 2 is a

subgroup of the 4, the two mirrors are the axial

mirrors (represented by a single letter in the

second column of the tetragonal symbol) and the dot

expresses the fact that the diagonal mirrors of the

tetragonal symmetry are not part of this site sym-

metry. Note that 2mm. is the tetragonal way of

writing the orthornombic mmZ; in both symbols the

2-fold axis is directed vertically, that is, parallel

with z.

Systems of co-ordinate axes.- The co-ordinate

axes used in the present Tables are those of Inter-

national Tables (1952). In the monoclinic crystal

system, we follow the 1951 Stockholm convention (i

unique). In the orthorhombic crystal system, the

three columns of a symmetry symbol refer to a, b, a,

in this order. In the tetragonal crystal system, the

smallest cell is used, so that provision must be made

^ Note, however, a different usage in Int. Tab.

(1935) , where the crystal forms are listed on pp.

49-63: here the term "special form" is used in all

limiting cases, while "general form" applies to all

other forms. We follow the older conventional usage:

general form {hkl} , a f9nn whose faces are oblique to

every element of symmetry; any other form is a special

form.

4

for the two settings 42m and 4m2; the co-ordinate axes

being taken along the cell edges. In the hexagonal

crystal system, the smallest cell is used and desig-

nated P (formerly called C) ; as in the tetragonal

system, provision must be made for two possible set-

tings, 5m2 and 62m. The rhombohedral crystal system

is treated twice: once in rhombohedral co-ordinates,

the co-ordinate axes being taken along the lower edges

of the primitive cell, so that they look like the

three legs of an inverted stool, and with the axis

pointing towards the observer; and again in hexagonal

co-ordinates, with a triple cell resulting from the

so-called "R centering". The relations between the

two systems of co-ordinates are shown in International

Tables (1952, vol. 1, p. 20, Fig. (a)), where the

orientation of the rhombohedral axes with respect to

the hexagonal ones is called obverse, positive or

direct. The additional nodes are placed at 2/3, 1/3,

1/3 and 1/3, 2/3, 2/3, so that the face (lOTl) slopes

toward the observer.

All our systems of co-ordinates are right-handed.

1.2 Rules of Splitting.

If the point at the fixed origin is made to move

away from it, the symmetry operations that are acting

on it will generate N equivalent points. This is what

we mean when we say that the point at the origin split

spl its into the N points. If the point at the origin

is made to stay on an element of symmetry as it moves

away from the origin, partial splitting takes place,

and further splitting will be necessary to reach com-

plete splitting, when the site syimetry of the final

points is reduced to identity.

Splitting of points is the basis of our symbolism

of site sets. It will be carried out according to the

following rules (see Table 1).

First Splitting.- Two cases are distinguished

according as the point group does or does not possess

at least one homopolar axis; in the second case the

point group possesses a single polar axis (also called

heteropolar axis).

A. At least one homopolar axis. In this case

there exists only a single point, taken as the origin,

that is invariant under all the symmetry operations of

the point group.

A(a). In point groups T, 4 and 3, complete

splitting is immediate; the first splitting yields,

respectively, the 2, 4 and 6 sites of the general

position

.

/'i(b). In 2/m, 4/m, 3/m(=6) and 6/m, as well as

in settings 4m2, 42m, 3ml, 3lm and 3m (which is treat-

ed in rhombohedral co-ordinates), the splitting point

is made to stay in a mirror, which is either normal

to, or contains, tiie principal axis.

A(c). In tne remaining point groups (Table 1,

column 1), the point is made to move in the plane that

passes through the origin and is perpendicular to a

primary symmetry axis; in addition the moving point

must retain the highest possible site symmetry, that

is to say, it must move along a secondary or tertiary

symmetry axis, or along another primary symmetry axis

if available (along x or xx in noncubic groups, along

2 in cubic groups)

.

Tnis moving point reaches a site that is desig-

nated by its literal co-ordinates; symmetry related

points move symmetrically and complete the assemblage

of equivalent sites produced by the first splitting.

B. A single polar axis. When there exists more

than one invariant point, the fixed origin of the

point group is not completely determined by the sym-

metry. One invariant point is arbitrarily chosen as

origin. The first "splitting" then consists in let-

ting the point move in the invariant space: in point

group 1, the moving pdint can reach any point in

three-dimensional space; in point group m, any point

in the mirror; in any other hemimorphic point group,

any point on the polar symmetry axis.

In both cases A and B, the first splitting is

described by the number of equivalent points, follow-

ed by the variable co-ordinates of one of the points.

Examples: 2xy2, 4xz, 2xz, 4x, 6x; Ixyz, Ixz,

ly, Iz, Ixxx (Table 1). Note that, in the cubic

system, in order to bring out analogies with the

tetragonal system, the co-ordinate chosen to symbol-

ize the first splitting is 2, and the resulting six-

pointer ^ is 6z.

Second Splitting.- If, after the first splitting,

the site symmetry is not the identity, the site set

that was produced by this splitting becomes a gener-

ating site set, in that its sites will serve as start-

ing points- for further splitting.

In crystal morphology the cube, or hexahedron, is

the crystal form that corresponds to the six-pointer

5z . The Dana System of Mineralogy symbolizes the

form as {001} in preference to {100} , which is the

symbol used by many other authors. Similar problems

call for similar solutions!

5

A. If the site symmetry contains a single element

(m, 2, 3, 4 or 6), only one more splitting, by means

of this element, is needed to bring the site synmetry

down to 1

.

B, In every remaining case, the site symmetry

consists of n mirrors intersecting in an n-fold axis.

If the mirrors are of one kind, the second splitting

takes place in_ these mirrors (example: 3xx in 3ml,

but 3x in 31m). If the mirrors are of two kinds, the

point is made to move in the mirror that contains the

next co-ordinate axis, in the cyclic sequence (exam-

ple: splitting 2x follows splitting Iz in mmZ) , In

case B, a third splitting is necessary to reduce the

site symmetry to 1

.

Third Splitting.- When the generating point still

lies in a mirror after the second splitting, the third

splitting takes it off the mirror, by making use of

the third co-ordinate. Although the last splitting is

taking place astride the mirror, the two resulting

points need not lie on a line perpendicular to the

mirror. This V-shaped splitting simplifies the use

of the third co-ordinate. The site symmetry is now

reduced to 1 (Table 1).

The final site-set symbol, after complete split-

ting, is given (column 9, Table 1) for each of the 32

crystal lographic point groups and for both settings

wherever two settings are possible. The rhombohedral

co-ordinates (Miller axes) are provided for, as well

as the hexagonal ones (Bravais axes), in trigonal

point groups.

1.3 Special Positions

In the process of deriving the site-set symbols

for the general positions in the crystal! ographic

point groups, a number of special positions have been

encountered as intermediate steps. Additional special

positions occur in certain point groups.

Some of the additional special positions possess

one degree of freedom and, in them, the site symmetry

reduces to a hemimorphic subgroup of the point group.

The symbols of the corresponding site set are given

here, each one followed by the oriented symbols^ of

its possible site symmetries, enclosed between paren-

theses: in the cubic point groups, 8xxx (.3m, .3.),

4xxx (.3m, .3.), 12xx (m.m2, ..2); in the hexagonal

point groups, 2z (6mm, 3.m, 3m., 6.., 3..), 6xx (mm2,

..2)i in the tetragonal point groups, 2z {^mm, Zrm. ,

See Sn. 1.1.

2.rm, 4.., 2..); 4xx {m.mZ, ..2); in the rhombohedral

point groups, 2xxx (3^, 3.), 6xx (.2). Note that the

hemimorphic symmetries, 2, 4, 3, 6, mmZ, 4mm, 3m, 6mm,

are given their oriented symbols in the above list.

Other special positions have two degrees of free-

dom and, in them, the site symmetry goes down to m,

either ..m in 6z4xx (cubic), lz4xx, 4xx2z (tetra-

gonal), 6x2z and 3x2z (hexagonal); or .m. in 4x2z

(tetragonal), lz6xx, 6xx2z and 3xx2z (hexagonal).

Note that the letters z, x, y that occur in a

site-set symbol express successive degrees of freedom,

as opposed to co-ordinates of points. Note also,

however, that points on the Zwischen axes are sym-

bolized 3xx, where the letter x indicates that the

first degree of freedom is being used and xx indi-

cates that the second co-ordinate is negative and has

the same numerical value as the first one.

2. DESCRIPTION OF SITE SETS

The site sets occur in the 32 crystallographic

point groups as assemblages of equivalent points.

These symmetry- re la ted points lie on a sphere around

the origin. If they are considered to be the poles

of crystal faces, the assemblage is a representation,

in spherical projection, of a arystal form. The lines

radiating from the origin to the face poles would

thus be face normals. If, on the other hand, the sym-

metry-related points on the sphere are considered to

be the vertices of a polyhedron, the assemblage be-

comes a representation of the co-ordination around the

origin. In the polyhedron that idealizes a crystal

form, the faces are equivalent, and the polyhedron is

called isohedral; in the co-ordination polyhedron the

vertices are equivalent, and the polyhedron is called

isogonal (referring to the solid angles at the ver-

tices). The two polyhedra are thus dual of each other.

It follows that we have two ways of describing

co-ordination of sites around the origin: (1) the

co-ordination polyhedron of the chemists; (2) the

dual polyhedron, whose face normals are the equivalent

elements.

The site sets are shown in stereographic projec-

tions (Figs. 1-47); they are classified on the basis

of their raltiplicity. A site set of multiplicity N

is called an W-pointer. It is easy to see how the

crystal -form polyhedron gives rise to the co-ordina-

tion polyhedron and to coin a name or provide a des-

cription for the latter.

6

2.1 The One-pointer.- The only one-pointer is the

lOINT (Fig. 1), corresponding to the MONOHEDRON (or

pedion)

.

2.2 The Two-pointers.- The two two-pointers are the

LINE SEGMENTS, one of w.iicii is col linear witli the

origin (Fig. 2) and corresponds to the pinacoid (or

paral lelehedron) , while the other is not collinear with

the origin (Fig. 3) and corresponds to the DIHEDRON^

.

2.3 The Three-pointers.- The two three-pointers are

the TRIGONS (or equilateral triangles), one coplanar

with the origin (Fig. 4), the other not (Fig. 5).

Thei-r duals are, respectively, the trigonal prism and

t(ie TRIGONAL PYRAMID.

2.4 The Four-pointers.- There exist seven four-

pointers: two RECTANGLES, one coplanar with the origin

(Fig. 6) (dual, RHOMBIC PRISM), the other not (Fig. 7)

(dila^ , RHOMBIC PYRAMID); two TETRAGONS (or squares),

likewise coplanar with 0 (Fig. 8) or not (Fig. 9)

(duals: TETRAGONAL PRISM and TETRAGONAL PYRAMID, res-

pectively); and three tetrahedra, which have the same

names as their duals: RHOMBIC TETRAHEDRON (or dis-

phenoid) (Fig. ^0) , TETRAGONAL TETRAHEDRON (ordis-

phenoid) (Fig. 11), and REGULAR TETRAHEDRON {Ft g. 12).

2.5 The Six-pointers.- Eight six-pointers comprise:

two TRUNCATED TRIGONS, coplanar with 0 (Fig. 13) (dual,

DITRIGONAL PRISM) or not (Fig. 14) (dual, DITRIGONAL

pyramid); two HEXAGONS, coplanar with 0 (Fig. 15) or

not (Fig. 16), corresponding to the HEXAGONAL PRISM

and HEXAGONAL PYRAMID, respectively; the OCTAHEDRON

(Fig. 17), whose dual is tne CUBE {or hexahedron); the

TRIGONAL ANTIPRISM {V-ig. 18), whose dual is the RHOM-

BOHEDRON (in crystal -form language, a trigonal anti-

prism is described as the combination of a rhombo-

nedron with the pinacoid); and two more co-ordination

polyhedra that are best described as combinations of

forms: TRIGONAL PRISM WITH PINACOID (Fig. 19) and

TRIGONAL TRAPEZOHEDRON WITH PINACOID (Fig. 20), whose

duals are the TRIGONAL DIPYRAMID and TRIGONAL TRAPE-

ZOHEDRON, respectively.

2.6 The Eight-pointers.- The seven eight-pointers

The single term dihedron replaces the two terms

used by Groth, dome and sphenoid. A crystal form

should have the same name regardless of the face sym-

metry (Boldyrev, 1925)

.

I

and their duals are as follows: TRUNCATED TETRAGONS,

coplanar with 0 (Fig. 21) or not (Fig. 22) [DITETRA-

GONAL PRISM and DITETRAGONAL PYRAMID) ; three TETRA-

GONAL site sets that are described by combinations of

TETRAHEDRON (Fig. 2Z) , PRISM (Fig. 24), and TRAPEZO-

HEDRON (Fig. 25), each with the PINACOID, correspond-

ing to TETRAGONAL SCALENOHEDRON , TETRAGONAL DIPYRAMID

and TETRAGONAL TRAPEZOHEDRON, respectively; t\\e RECT-

ANGULAR PARALLELEPIPED (Fig. Zi,) [RHOMBIC DIPYRAMID);

the CUBE [octahedron) (Fig. 27).

2.7 The Twelve-pointers.- Ten twelve-pointers, six

of which belong to the hexagonal system and four to

the cubic system, have the following designations.

The names of the duals are enclosed in parentheses.

truncated hexagon, coplanar with 0 (Fig. 28) [dihexa-

GONAL prism) or not coplanar with 0 (Fig. 29) [dihexa-

GONAL pyramid); PINACOID WITH RHOMBOHEDRON (Fig. 30)

[hexagonal SCALENOHEDRON) ; TRUNCATED TRIGONAL PRISM

WITH PINACOID (Fig. 31) [DITRIGONAL DIPYRAMID); HEXA-

GONAL PRISM WITH PINACOID (Fig. 32) [HEXAGONAL DI-

PYRAMID); HEXAGONAL TRAPEZOHEDRON WITH PINACOID (Fig.

33) [HEXAGONAL TRAPEZOHEDRON); TRUNCATED TETRAHEDRON

(Fig. 34) [TRIGON-TRITETRAHEDRON)^ ; CUBE WITH TWO

TETRAHEDRA (Fig. 35) [TETRAGON-TRITETRAHEDRON)'^ ;

CUBOCTAHEDRON (Fig. 36) [RHOMB-DODECAHEDRON) ; PENTAGON-

TRITETRAHEDRON with two TETRAHEDRA (Fig. 37) [PENTAGON-

TRITETRAHEDRON) ; DIHEXAHEDRON WITH OCTAHEDRON (Fig. 38)

[DIHEXAHEDRON or PENTAGON- DODECAHEDRON)

.

2.8 The Sixteen-pointer .- The only si xteen-pointer

is a combination of the TRUNCATED TETRAGONAL PRISM

WITH THE PINACOID (Fig. 39), whose dual is the DI-

TETRAGONAL DIPYRAMID.

2.9 The 24-pointers.- There exist six 24-pointers.

They are listed here with the dual polyhedron shown

between parentheses: TRUNCATED HEXAGONAL PRISM WITH

PINACOID (Fig. 40) [DIHEXAGONAL DIPYRAMID); TRUNCATED

CUBE (Fig. 41) [TRIGON-TRIOCTAHEDRON)^ ; RHOMBICUBOC-

TAHEDRON (Fig. 42) [TETRAGON-TRIOCTAHEDRON)^; CUBE

WITH TWO TETRAHEDRA (Fig. 43) [HEXATETRAHEDRON) \

TRUNCATED OCTAHEDRON (Fig. 44) [TETRAHEXAHEDRON) ;

CUBE WITH OCTAHEDRON AND DIHEUHEDRON (Fig. 45)

[DIDODECAHEDRON); SNUB CUBE (Fig. ^6) [PENTAGON-TRI-

OCTAHEDRON) .

The two polyhedra belong to the same site set

6z2xx (Table 2) .

^ Two two polyhedra belong to the same site set

6z4xx (Table 2) .

7

2.10 The 48-pointer.- The only 48-pointer is the

TRUNCATED CUBOCTAHEDRON (Fig. 47), whose dual is the

HEUOCTAHEDRON.

Note that, in the above nomenclature, a truncated

polygon is symmetrically truncated on the angles, a

truncated polyhedron (tetrahedron, cube, octahedron)

is likewise symmetrically truncated on the corners,

but a prism, being an open form with no vertices, is

truncated on its edges. The rhombicuboctahedron re-

sults from the combination of a cube with octahedron

and rnomb-dodecahedron. The snub cube is a cube modi-

fied by octahedron and pentagon-trioctahedron. The

truncated cuboctahedron shows the twelve rhomb-dode-

cahedral faces truncating the twelve corners of the

cuboctahedron.

The site sets that occur in the 32 crystallo-

graphic point groups are listed in Tables 2 to 8,

limiting cases not included. Under a subheading con-

sisting of the names of the co-ordination polyhedron

and dual crystal form, the symbol of the site set is

followed by the co-ordinates of the equivalent sites,

expressed in a condensed manner (see below), and the

symbols of the positions where the site set occurs.

The position is given by the point-group symbol and

the position letter between parentheses. This letter

is taken from Carl Hermann's description of the point

groups (so-called crystal classes)^.

The multiplicity of each site set can be obtained

immediately by making the product of the numerals that

indicate splitting in the site-set symbol. Example:

6z2x has multiplicity 12.' No explicit mention of the

multiplicity is made in the Tables. A given site set

may occur in different positions, in the same point

group or in different point groups, and can be de-

scribed by different symbols, which will be found

listed under one another, together with the co-ordi-

nates of the sites and the positions of occurrence.

When a site-set symbol corresponds to two names, these

are listed under each other in the subheading. Alter-

nate orientations of point groups (4m2, 42m, for in-

stance) have been taken into consideration. The five

trigonal point groups are listed twice: once in rhom-

bohedral co-ordinates (Miller axes) under rhombohedral

system and again in hexagonal co-ordinates (Bravais

axes) under hexagonal system.

The above presentation is largely based on the

Report of the Kiel Symposium on Co-ordination Poly-

Chapter III in Int. Tab. (1935)

.

hedra^ , especially as regards the nomenclature of the

polyhedra and the occurrence of the site sets In point

groups. For additional data, such as topological and

metrical site-set symbols, and limiting cases, this

Report should be consulted.

The following remarks will help in the under-

standing of the tabular presentation.

Condensed description of the site co-ordinates.

-

Instead of listing the co-ordinates of all the equi-

valent sites in the position, it may be advantageous

to use a scheme of condensation. One triplet of co-

ordinates specifies one of the sites of the position,

various signs explain how the other sites are derived

from the first.

The Greek letter kappa (<) stands for the initial

of kyklos, circle, and indicates the three cyclic per-

mutations of the preceding triplet or of two (or more)

triplets enclosed between parentheses.

The Greek letter pi {-n) indicates the six com-

plete permutations of the preceding triplet.

The Greek letter tau (t) is used to signify that

the first two co-ordinates must be interchanged.

In the hexagonal system when three co-ordinates

are used (a-j, 6.2, c axes) (in contradistinction to

Weber's four co-ordinates), the Greek letter iota (i)

indicates the following operations: {xys)\ = xyz;

y, x-y , z; y-x, x, z. (They correspond to the cyclic

permutations of the first three co-ordinates in a four-

co-ordinate symbol in which the third co-ordinate is

zero: xyO; Oxy = 0-y, x-y, y-y; yOx = y-x, 0-x, x-x.

The ± sign, which should be read as "plus and

minus", is the expression of centrosymmetry; it

doubles the number of equivalent sites. The ± sign

may precede all other statements or a single letter

(as in ±x, y , z) .

Triplets of signs (example: +--) can be followed

by a Greek letter, with the same obvious meaning.

Example: the last site set in Table 2 is the 48-

pointer; the 48 triplets of co-ordinates that would be

necessary to give all the co-ordinates explicitly can

be condensed into the statement

+ (+++, +— k) {xyzTi),

which is easily deciphered as follows: take the six

complete permutations of the triplet xyz, first with

the signs +++ and then with the three cyclic per-

mutations of +--; double the number of points, from

24 to 48, by changing all the signs. Another way of

J.D.H. Donnay, E. Hellner and A. Niggli (1964).

8

condensing would be ±{x, y , ±2; x, y, ±z)-n, which is

just as easily interpreted. We consider such state-

ments helpful in evoking the picture of the assemblage

of points. Stimulating and interpretative, as well as

space-saving, they are preferable to the bland catalog,

xyz yzx zxy yxz xzy zyx

xyz yzx zxy yxz xzy zyx

xyz yzx zxy yxz xzy zyx

xyz yzx zxy yxz xzy zyx

xyz yzx zxy yxz xzy zyx

xyz yzx zxy yxz xzy zyx

xyz yzx zxy yxz xzy zyx

xyz yzx zxy yxz xzy zyx.

which dulls the senses and discourages visualization.

In the description of hexagonal site sets, and of

rhombohedral ones in hexagonal co-ordinates, we re-

commend the elegant Weber four-index symbol xj/js for

the co-ordinates of points. All four Bravais axes

must be used: the third co-ordinate, j, which is made

equal to the negative sum of the first two, is not

superfluous: it cannot be left out at will, as can

the i index in a face symbol {hkil) . The advantage of

these symbols (Weber, 1922; Donnay, 1947a, b) is that

permutations of the first three co-ordinates are made

available to express the symmetry of the point assem-

blage. In fact, the parallelism between the presen-

tation of the co-ordinates oq^jz of equivalent points

and that of the indices hkil of equivalent faces is

so perfect that the two symbols are interchangeable.

This property is the consequence of the mathematical

symmetry of the equation of zone control in four-index

symbolism,

hx + ky + ij + Iz = 0,

which expresses the condition for the face {hkil) to

be contained in the zone [an/Jz].

For calculations, however, symbols with three co-

ordinates are more convenient. For this reason they

are used in Table 3. The transformation from four-

index to three-index symbols is inmediate; it consists

in subtracting the third co-ordinate j from each of

the first three, so that xyjz becomes x~ j , y- j ,

(0.) 2. Example: 5T42 gives 5-4, T-4, 2 = 932. Con-

versely, transforming from three to four co-ordinates:

932 gives 9-(9+3)/3, 3-(9+3)/3, -(9+3)/3. 2 = 5142.

It may not be superfluous to stress that any co-

ordinate can be assigned negative or positive values.

It follows that the single point of the site set Iz,

which may have a negative as well as a positive value

for 2 in the co-ordinate triplet OOz, corresponds to

two crystal forms in morphology: the upper monohedron

{001} and tne lower monohedron {001}, This is the

reason why two positions are listed in such a case.

Example: 4mm{B.) and 4mm(b) (Table 5).

3. OCCURRENCES OF SITE SETS

Site sets occur in positions in point groups. For

each crystal system (Tables 9-15), we give a double

entry tabulation: each row stands for one point group,

each column for a type of co-ordinate triplet. At the

intersection of row and column, the symbol of the site

set^ appears over the corresponding site symmetry of

the occupied position. This site symmetry is sym-

bolized in the format of the point-group symbol, that

is to say that it is oriented with respect to the

symmetry directions of the crystal lattice (see

Sn. 1.1).

4. SUBGROUP RELATIONS

The relations between groups and subgroups among

the 32 crystal lographic point groups have been pre-

sented, in the form of a line diagram, by Carl

Hermann in the 1935 edition of the International

Tables.

This diagram is reproduced here as Fig. 48.

5. SITE SETS AND THEIR OCCURRENCES

IN ORIENTED SUBGROUPS

5.1 Cubic Degradation (Table 16).-

The top lines show the site sets and the positions

in the cubic point groups. Going down in the first

column, we list the various subgroups in order of de-

creasing symmetry. The tetragonal point groups, for

instance, are listed in the first column in their

cubic orientation; they are repeated in the second

column in their standard tetragonal notation.

Further down the Table, the trigonal point groups

are likewise shown twice: as subgroups of the cubic

groups, in cubic orientation, in the first column and

as standard trigonal point groups, in the third

column.

The orthorhombic point groups appear three times:

in the first column, as oriented subgroups of the

cubic symmetries; in the second column, as oriented

' Or the symbol of the limiting site set, which is

written between parentheses where it appears as a

limiting case (see Sn. 1.1) .

tetragonal subgroups; and, in the fourth column, as

orthorhombic point groups in their own orientation.

The same scheme is carried out all the way down to

the triclinic point groups.

5.2 Hexagonal Degradation (Table 17).-

The subgroups of the hexagonal holohedry are listed,

including the orthorhombic, monoclinic, and triclinic

point groups.

10

PART II. LATTICE COMPLEXES

6. GENERALITIES

6.1 Definitions .-

A configuration means a set of n specified^ con-

gruent primitive lattices, whose nodes are symmetry-

related sites in at least one of the 230 space groups.

Every one of the n lattices may be represented by one

node that is a site in the conventionally chosen cell.

Every 'site in the configuration is invariant under

certain symmetry operations, whicn constitute a sub-

group of the point group to which the space group be-

longs. The subgroup, which may be the trivial sub-

group 1, is called point symmetry of the site or site

symmetry. A site may happen to lie on an element of

symmetry whose operation involves a translation com-

ponent (a screw axis or a glide plane of symmetry),

but such an element cannot be part of the site sym-

metry, which by definition is a point group.

In a given space group, two configurations are

said to occupy one and the same point position (or

position for short) if, to every site in the one con-

figuration, there corresponds, in the other config-

uration, a site that lies on the very same elements of

site symmetry.

The condition for two positions, in the same

space group or in different space groups, to belong to

the same lattioe complex (or complex for short) is

that, to every configuration of the first position,

there corresponds, in the second position, a congruent

or enantiomorphous , configuration -- and conversely.

As a consequence of this definition, all the positions

in which a lattice complex can be realized occur in

one and the same crystal system.

^

The positions that belong to the same lattice

^ Specified in the sense that the co-ordinates of

the nodes have specified numerical values, in any con-

venient system of absolute co-ordinates

.

^ The two lattice complexes tP and cP are distinct

complexes because tP (class of all the sets of points

that occupy the nodes of a tetragonal primitive lat-

tice) contains, in addition to all the configurations

of cP , configurations' that are not included in cP

,

i.e. configurations of points that do not occupy the

nodes of a cubic primitive lattice.

complex, either in a given space group or in different

space groups, may be differently oriented and their

site symmetries may also be different. Such positions

constitute different representations of the lattice

complex.

One of the representations of a lattice complex,

in tiie space group where it shows the highest site

symmetry, is cnosen as the standard representation.

Our choice follows, in general, that already made by

Carl Hermann (Int. Tab., 1935). The multiplicity of

a lattice complex is the number of its equivalent

points comprised within the conventional cell of its

standard representation.

General lattice complexes are tri variant; special

lattice complexes can be bi variant, uni variant or in-

variant. If a position contains all the configurations

of a second position, but the second position does not

contain all the configurations of the first, then the

lattice complex realized in the second position will

be a limiting lattioe complex (LLC) of the lattice com-

plex of the first position. The subset of configura-

tions (in the first position) that coincide with the

set of configurations of the limiting lattice complex

will simulate this complex as a limiting case.

Example. In space group 23 the lattice complex that

is realized in position (j) -.xyz'. possesses a subset

of configurations that coincides with the set of con-

figurations of another complex, which is realized in

positions (i) ixxQ: of space groups Pmim and P432.

In space group ?23, position (j) , for y = x, z = 0,

simulates position (i) of Pm2>m and p432, while retain-

ing its own multiplicity and site symmetry.

The lattice complex of only one special position,

namely, position 48(/'):a:0^: in laZd is not a limiting

lattice complex: it cannot occur as limiting case in

any position of any space group. All other special

lattice complexes are limiting lattice complexes.

The space group in which a lattice complex is

defined in its standard (or normal) representation is

called the characteristic space group of the lattice

complex. Other representations for which the site

symmetry is equal can only occur in the same charac-

teristic space group; other representations, with

lower site symmetry, may be found in another space

group or in several other space groups, as well as in

11

501-563 0 - 73 -2

the characteristic space group; nonstandard represen-

tations win be referred to as alternate represen-

tations.

6.2 The Lattice-Complex Symbol .-

6.21 Invariant Lattice Complexes.- An invariant

lattice complex, that is to say, a complex that can

occupy a parameterless position in a space group, is

designated by a short symbol, which consists in a

single letter when referring to the standard represen-

tation. Example: in the anorthic (triclinic) system

(Table 25), P is the symbol of an invariant lattice

complex in standard representation, which occurs in

space group P\ as position (a), with site symmetry 1

and multiplicity 1. PI is its characteristic space

group.

The same lattice complex occurs in an alternate

representation in P^[a), with site symmetry 1. In

this space group the complex occupies a general

position -.xyz: and is symbolized P[xyz]. The co-

ordinates are placed between brackets to indicate

that they represent another origin for the complex

which remains invariant in spite of this choice of

origin.

In the monoclinic system (Table 26) P and C are

invariant lattice complexes in standard orientations.

Among the lattice-complex letters, we find all the

letters that are used to characterize lattice center-

ings, since the nodes of any lattice, according to

definition, are the equivalent points of a lattice

complex.

Some of the symbols listed (Table A) for invar-

iant lattice complexes are topological symbols, which

means that they can be used in more than one crystal

system. The symbol must be expanded in order to

specify the system in which the complex is to be con-

sidered. This is achieved simply by using the ini-

tial of the name of the system as a prefix to the

lattice-complex symbol. Examples: cP, hP, rP, tP,

oP, mP, aP. Note that t stands for tetragonal, r

for rhombohedral , a for anorthia (which is used in-

stead of triclinia in order to avoid conflict between

initials). The system need not be specified, of

course, unless there is danger of confusion.

12

CNj iroI— IfOCM ICO

r— iroO CM ICO

CM 1— in I— ICMI— ICMI— ICM

roicoLTJIOO

1— inI— ICMI— ICM

I— iror-iroCM iro

r-^icoO0i<i-I— ICM

r-.icor^ico

•—KX)inicoLOIOO

r— i<=i-

i~ICMr^ioo

r-ICM

roi**

CM ICOCM iroI— iro

•—t<:j-I— ICMIT) ICO

roiCOCO ICO

I— KMr-ICMI— ICM

I— ICM.— ICMI— ICM

r-ICO

COw

I— 1^r-l'::^-

1— ICMr— ICMr— ICM

r— 1-:^

CMiroiro

13

Distinction between a lattice complex and its re-

presentations.- We may distinguish between the con-

cept of the lattice complex and the various represen-

tations that the complex can have in one space group

or in different space groups. The standard represen-

tation will be designated by the symbol of the lat-

tice complex itself. The symbols of all the invariant

lattice complexes in their standard representations

are listed alphabetically in Table A (column 1).

Examples of alternate representations will be found in

column 3. The co-ordinates of the equivalent points

in the conventional cell (column 2) and the multi-

plicity m (column 4) complete the Table.

The symbols used for the invariant lattice com-

plexes are essentially a typographical simplification

of those that were proposed by Carl Hermann (1960).

The new symbols were agreed upon at the Kiel Symposium

(Donnay et al., 1966). They can be remembered without

too much difficulty, as each letter chosen recalls

some structural feature of the complex. When two re-

presentations of a lattice complex of multipl icity w

combine to form a new complex with multiplicity 2m,

the complex letter is usually^ retained but an asterisk

follows it in superscript. Example: J* (pronounce:

J star). If the process is repeated, the complex of

multiplicity 4m still has the same letter, this time

with a double asterisk. Whenever a complex possesses

two enantiomorphous representations, the standard re-

presentation is symbolized by a letter preceded by a

pi us sign in superscript. Example: Q. The represen-

tation chosen as standard is the one that contains

left-handed screw axes. In this convention, the sign

placed in superscript is that of the optical activity.

Alternate representations can be enantiomorphous

or congruent to the standard orientation. The minus

representation of a pi us complex will ca.rry the minus

sign in superscript in front of the letter. A con-

gruent representation that is obtained from the stand-

ard representation by a translation will retain the

same letter, but it will be preceded by the shift

vector expressed by its three components in fractions

of the cell edges. Examples: illj is a shifted re-in 13'^

presentation of J; ^^^F and p^F are shifted represen-

tations of F. All other congruent representations

will have their symbolic letter of the standard repre-

sentation preceded by a prime sign. Examples: 'R is

Exairples of exceptions: F + --- F is not de-

signated F* , it is called P2 instead (see below);

P + P is I, not P*.

obtained from R by a 180° rotation around the a axis

(in hexagonal co-ordinates); 'S is derived from S by

a screw rotation of power 1 or 3 around a 4i axis. It

is true that 'S can also result from S by an inversion

through the origin (or the center of the cell), but

this inversion can always be combined with another

operation of the 2nd kind that is a symmetry operation

of the complex to produce an operation of the 1st kind.

To obtain a minus representation from the plus repre-

sentation, on the other hand, the only operation

available is one of the 2nd kind, the inversion

through the origin, for instance.

Other alternate representations can be obtained

by changing the scale in certain directions (not nec-

essarily all directions), in such a way that the crys-

tal system is preserved. The transformation matrix

that expresses the cell edges a', 1j', c' in terms of

the edges a, o, c, of the complex in standard repre-

sentation is given in Tables 25-31. Where no matrix

is listed, unit matrix is to be understood.

Examples: P^^ (Table 26) is a representation of P

with b = b'/2, so that two standard representations of

the complex occupy the volume of the cell. The trans-

formation matrix^ that gives the cell edges from the

edges of the complex is 100/020/001.

A small letter in subscript thus indicates that

the corresponding edge of the complex is to be multi-

plied by 2 in order to give the length of the cell of

the lattice in which the complex is represented. Two

small letters in subscript, as in P^^, mean that both

a and b are to be doubled to give the dimensions of

the cell. If all three edges are to be doubled, the

subscript used is 2. P2, for instance (Table 31), is

a representation in which the cell volume contains 8

times the P complex in standard representation. The

transformation matrix "complex to cell" is 200/020/002.

Examples of representations that result from a

rotation combined with a contraction or an expansion

of a mesh are found in the tetragonal system (Table

28) and in the hexagonal system (Table 29-30). Con-

sider a tetragonal F cell, with its 4 nodes. It has

twice the volume of an I cell, which is oriented at

45° to it. The standard representation of the lattice

complex is I, with multiplicity 2. The alternate re-

presentation, in the F cell, has multiplicity 4. The

F cell is obtained from the cell of the standard com-

plex by transformation 110/110/001. It is rather in-

felicitous to describe this transformation as a 45°

See Sn. 8.1, footnote.

14

counterclockwise rotation of the mesh accompanied by

an expansion of the lengths in the ratio 1 to •11.

While it is true that, if the o length is kept con-

stant, this peculiar operation will transform the

small cell into the large one, it is also true that

the same operation will not turn an I cell into an F

cell! The symbols proposed by Carl Hermann make use

of the letters v and V (from the Latin vertere, to

turn) to indicate the 45° "turn" in the tetragonal

system and the 30° "turn" in the hexagonal system,

respectively. We did not feel it would be justified

to modify this symbolism, misleading though it might

be.

In the above example, the I complex in the F re-

presentation could have been designated 1^. Likewise

a P complex in the C representation could have been

called P . (The familiar lattice letters F and C have

been retained for the sake of convenience, even though

the unity of the symbolism is thereby impaired.)

Consider now a D complex in a tetragonal cell,

that is, a tD complex. (In the cubic system, cD is

exemplified by the diamond structure.) The tetragonal

cell in question will be an F cell, which we will want

to describe in the conventional I setting. The trans-

formation matrix ^^^/^^^/OOl expresses the 45° clock-

wise rotation with shrinkage of the cell. The repre-

sentation of the D complex in the small cell will have

multiplicity 4, instead of 8, and the complex symbol

will be ^D.

A mnemonic rule may be found useful: to go to a

larger cell, the co-ordinate axes a-] and are ro-

tated counterclockwise, that is, to the right, and the

subscript u is placed to the right of the letter; to

go to a smaller cell, the co-ordinate axes a-^ and

are rotated clockwise, that is, to the left, and the

superscript v is placed to the left of the letter.

The same rule applies to the hexagonal case, where the

capital letter V is used for the 30° turn.

In the hexagonal system (Table 29-30) one addi-

tional symbol is needed whenever the o axis of the c

complex in standard orientation is to be multiplied by

3 to give the e' length of the cell. P^ indicates

this transformation. indicates the doubling of o.c ^

It follows that P(,^ stands for a multiplication of c

by 3x2 or 6.

Relations between invariant lattice complexes.

-

Using the abbreviation P' to mean iUp. F" to mean111 333444 ^» ' ^° ""^^^ 444 F, and similar abbreviations

F', J', D', etc., we can write the following equations:

P + P' = I (note tnat I is preferred to P*)

,

J + J' = J*,

F + F' = P^ (note that F* is not used).

P + J = F, P' + J'

I + J* =

F + F" = D, F' + F'

hP + G = hPy = H, hP + N

= F',

= D',

= hP.^1^2-

Explanation of the symbolic letters used for in-

variant lattice complexes.- The letters that also

describe the variously centered cells of the Bravais

lattices are, of course, self-explanatory. The other

letters recall some structural feature of the complex,

as follows:

D Diamond structure.

E Hexagonal close packing {Italian: esagonale).

The I'lg atoms in the ilg structure occupy the equi-

valent points of the E lattice complex.

G Graphite structural layer. The B atoms in AlBg

occupy the G complex.

J Defined in standard representation as 3 points in

the midpoints of the cell faces. Consider the

six half-points within the cell and connect oppo-

site half-points: the resulting figure has the

shape of the American toy called "jack". (The

center point of the jack is the center of the

cell, ^^^.) The atoms of oxygen in perovskite

occupy the J complex.

M Complex that results from the repetition of the

equi points of N by a rhombohedral lattice, in a

hexagonal cell. The Ni atoms in Ni2Pt2S2 occupy

the M complex.

N Net. Three points per mesh, extending into a

two-dimensional 4-connected net, are repeated by

the hexagonal lattice. The Co atoms in CoSi

illustrate the N complex.

Q Quartz. The Si atoms in high-quartz occupy the

complex.

S This complex has site symmetry 4 and contains

tetragonal tetrahedra that have 4 synmetry,

in Schoenflies notation, formerly called spheno-

hedral . The Th atoms in ThjP^ illustrate the S

complex.

T Tetrahedra sharing all four corners. The Cu

atoms in MgCu2 ^^'-^P'^ ^ ^ complex, and so do the

oxygen atoms in Cristobal ite.

This complex has site symmetry 222 and contains

rhombic tetrahedra that have symmetry 222, or V

in Schoenflies notation. The Ca atoms in garnet

occupy the V* complex (V* = V + "V).

15

W Non-intersecting {German: windschief), mutually

perpendicular rows of equivalent points. The Cr

atoms in Cr^Si occupy the W complex.

In the invariant complexes + ^Y' = """Y* and

"Y + "y' = "Y*, three equipoints of Y surround

one equipoint of Y', and viae versa, so as to

form the letter "Y". The Li atoms in LiFej-Oo

occupy the Y' complex. The Au atoms in Ag2AuTe2

occupy *Y*. The K atoms in high-temperature

leucite, K(AlSi20g), occupy Y**.

As further examples, J* is illustrated by the Pt

atoms in Pt^O^, S* is occupied by Si in garnet

Ca2Al2(SiO^)2. and W* accommodates the B atoms in

Zn40(B02)g.

6.22 Variant Lattice Complexes.- We have seen

in Part I that the splitting of the point at the fixed

origin of a point group gives rise to a site set,

which is an assemblage of equivalent points around the

origin. Likewise the points of an invariant lattice

complex may be made to split into as many assemblages

of equivalent points, the sum of which constitute a

variant complex. Examples: Starting from the in-

variant lattice complex P, in the cubic system (Table

31), splitting along the principal symmetry directions

of the site symmetry gives 6z (see Table 1, last line)

and the result is the univariant lattice complex P5z.

If we take I, instead of P, as the generating complex,

the resulting univariant complex will be I6z, which is

composed of two octahedra of equivalent points, one

around each of the two points of I.

The direction in which splitting is to take place

is expressed by using, once, twice, or tnree times,

tne letter that designates the variable, as was done

for site sets in point groups. Examples: P12xx is a

univariant complex composed of points that lie on the

bisectrices of the interaxial angles and thus are the

apices of a cuboctahedron. ISxxx is also univariant,

but the length x is plotted three times in succession,

parallel to the three unit vectors a-j

, a2 and a^, so

that the resultant lies along the body diagonal of the

cell and, within the cell, the complex consists of two

cubes, one around each of the two points of I, the

generating complex. In the orthorhombic system (Table

27), F2y2z is made up of four site sets 2y2z, one .

around each of the four points of the generating com-

plex F; it is a bivariant complex, because y and z

have different independent numerical values. In the

same system, a simple trivariant complex is P2x2y2z,

the symbol of which is self-explanatory.

It may happen that any one of several lattice

complexes could be used as generating complex to pro-

duce, by splitting, a given variant complex. Our first

task is to set up rules, according to which a unique

choice of generating complex can be made in all cases.

6.3 Rules for selecting the generating lattice com-

plex (GLC).-

The problem may be stated as follows: Given a

variant lattice complex, select a generating lattice

complex, that is, either an invariant lattice complex

(ILC) or a limiting lattice complex (LLC), whose

points will by splitting produce the given variant

complex. (We note that, if the characteristic space

group of the variant lattice complex is different from

that of the generating lattice complex, the generating

lattice complex can be an ILC or a LLC.)

If more than one generating complex is available,

consider the site symmetries of the several possible

generating complexes, taken as positions in the space

group, and use as many of the following rules as need-

ed.

Rule 1: If the site symmetries are of different

orders, take the highest order (n in Table B).

TABLE B

Possible site symmetries for an invariant latticecomplex listed according to decreasing Laue sym-metry for each order n of the point group and each

number F of degrees of freedom

n F=0 F=l F=2 F=3

48 4/m 3 2/m

24 432, 43m; 2/m 3; 6/m 2/m 2/m

16 Mm 2/m 2/m

12 23; 622, 62w; 6/m ; 3 2/m 6mm

8 422, 42m; 4/m; 2/m 2/m 2/m 4mm — —6 6; 32; 3 6; 3m — —4 4; 222; 2/m 4;mm2 — —3 3 — —2 T 2 m —1

Example: In space group PmSf?!, position 6(e) xOO

is the characteristic position for a univariant lat-

tice complex. This LC may be described from two dif-|

ferent generating complexes, both of which are invari-

ant lattice complexes: P in 1(a) with site symmetry

mjn and UL in 2{d) with site symmetry 4/mmm. The

16

order of the group of site symmetry is 48 in 1(a) and

16 in 3(d). P must be chosen as generating complex,

and the uni variant lattice complex will be written P6z.

If the site symmetries have the same order, it

may be useful to consider synonyms, that is to say, to

provide, for the variant LC, alternate descriptions

based on different generating complexes that belong to

the same crystal system (see Tables 25-31). Two cases

can occur: the site symmetries are different point

groups or they are the same point group. The follow-

ing rul-es indicate which of the synonyms must be se-

lected in order to get as short a symbol as possible.

Rule 2: If the site symmetries are different

point groups of the same order, take the smallest num-

ber of degrees of freedom.

Example: To describe the tri variant LC that is

realized in the general position 8(e) in Pbam, we

choose site symmetry T in 4(a), 000, with no degree of

freedom, in preference to 2.. in 4(c), a-, 1/4,0,

which has one degree of freedom, and . .m in 4(ci)

,

x,y,]/i, which has two.

Rule 3: If the site symmetries are different

point groups of the same order and rule 2 does not

apply, take tne highest Laue symmetry (see Table B and

note that 6/m arbitrarily takes precedence over 3 2/m)

.

Example: For 16(n) in PA/mao, site symmetry 422

in 2(a) is chosen in preference to 4/m in 2{b) because

it has a higher Laue symmetry.

Rule 4: If the site symmetries are different

point groups of the same order and rule 3 does not

apply, take the group of proper rotations.

Example: For 16(n) in P'i/nbm, site symmetry 422

in 2(a) takes precedence over 42m in 2(e).

Rule 5: If the site symmetries are one and the

same point group in different orientations, take the

coordinate axes of the LC parallel to those of the

space group, rather than parallel to the diagonal

directions

.

Example: For 8(p) in P4222, site symmetry 222

oriented 222. in 2(a) and 2(c) is preferred to 222

oriented 2.22 in 2(e); the Neben axes are taken

rather than the Zwisohen axes.

Rule 6: If the site symmetries are one and the

same point group in different orientations and rule 5

does not apply, the x direction takes precedence over

the y direction, wnich takes precedence over the z

direction.

Example: For 8(d) in -r2i2i2^, site symmetry 2

is available in three orientations; 2.. in 4(a) is

chosen rather than .2. in 4(Z)) or ..2 in 4(c).

Rule 7: If the site symmetries are one and the

same point group in only one orientation, give two or

more GLC symbols, writing first the GLC obtained by

setting x = y = z = Q in the co-ordinates of the

given variant position. (If there are more than tv;o

generating complexes, list all but the first one in

al phabeti c order.

)

Example: In the example given for rule 5, there

were two positions with site symmetry 222., namely

2(a) with the ILC and 2(c) with 0^01. The sym-

bol ..2 P^I2x2yz of the complex of tne general

position 8(p) in P4222, therefore, makes use of two

generating complexes, of which P^ comes first.

In Taoles 18-24, which relate the lattice com-

plexes to the Weissenberg complexes, symbols are given

for all synonyms. Synonyms that violate rules 2-6 are

written in full and placed between brackets. If the

same site set can be used after any of the generating

complexes, tne synonym is given in abbreviated nota-

tion. Example: S*[V*]2x instead of S*2x[V*2x].

Remark.- The shift vector that is placed in

front of the LC symbol to designate a shifted represen-

tation of the LC will always refer to the first GLC

when there is more than one in the symbol.

6.4 Distribution symmetry.

-

We have designated (Sn. 1.1) the fixed point of a

point group by the term high symmetry point (HSP).

The generation of a variant lattice complex may be

visualized as consisting of the following steps: (1)

the splitting, into a set of equivalent points, of the

HSP of the point group that belongs to the space group

under consideration; (2) the subdivision of this set

into a number of subsets to be distributed in the cell

so that each one becomes associated with one (or more

than one) point of the GLC. The subsets may have more

than one orientation.

The symmetry operation that carries a subset from

near the HSP to its assigned location near a point of

the GLC is represented by the symbol of a symmetry ele-

ment of the space group.' Such a symbol will stand for

the first power of the symmetry operation if the sym-

metry element is of order 2; if the order is greater

than 2, the power to be used will be explicitly stated

17

(if no power is stated, all the powers are meant to be

used). The symbol of the symmetry element will be

oriented by means of dots that stand for the several

columns of the Hermann-Ilaugui n symbol. It will be

written in front of the symbol of the GLC. Enough

symmetry operations must be stated to specify the

orientations of all the subsets. Together they con-

stitute the distribution symmetry, which is not a

group.

The following rules have been applied in express-

ing the distribution symmetries.

Rule 1 : All the symmetry elements used to ex-

press the distribution symmetry should pass between

the equivalent points of the GLC.

Rule 2: The number of syirmetry elements used

must be kept to a minimum.

Rule 3: The number of translation components

must be kept to a minimum.

Rule 4: Symmetry planes are to be preferred to

axes; axes to center; among axes, 4 to 4.

Rule 5: In order of preference the directions

are x, y, z.

Rule 6: Whenever possible, splitting is done

from the standard representation of the GLC, except

when the chirality of an enantiomorphous GLC is lost

in the resulting variant lattice complex.

Example. In j4^32 the sites of 24(/)2.. can be

obtained oy splitting from the standard representation

or from the alternate representation "V. {Either

one will do.) In such a case, instead of using both

representations and writing .3. V"V2z, we omit the

sign and write .3.V2z. (In so doing, we follow a

rule proposed by Carl Hermann.) In the same space

group, however, we write4i

. . Y*3xx and 42-. Y*3xx

for the two uni variant lattice complexes realized in

24(?) and 24 {h) , each of which retains the chirality

of its GLC: "Y* and "^Y*, respectively.

^

Rule 7: In a C centered space group, it is not

allowed to bring 000 to coincidence witn ool by an

1 It would be superfluous to repeat the sign in

front of the univariant complex symbol and write:

~{4^. .~Y*3xx) and (4^ . .*Y*3xx) .

operation of the distribution symmetry; the latter

must bring 000 onto instead. Likewise in an

I centered space group, the distribution symmetry

should bring 000 onto ^^O' "ot onto OOg.

Rule 7 is necessary to avoid ambiguity whenever

C^ is the GLC. Without this rule, two different com-

plexes would get the same symbol, namely: ..mC^2x2yz

for both Caamim) and ibam {k) . This situation arises

from the fact that C may be constructed as C + Ooic1

C /L

or as I + 00^1.

The complete lattice-complex symbol may contain

up to four parts: (1) the generating -lattice -complex

symbol, which consists of a letter for an invariant

lattice complex (possibly modified by a subscript or

a superscript), followed by (2) the site set that

indicates the splitting at each point of the GLC, and

preceded by (3) the distribution symmetry. To avoid

ambiguity it may be necessary to specify the crystal

system; this is done by prefixing (4) the initial

letter of the name of the system, in lower case, and

hyphenating it to the symbol. In addition, in case of

a representation symbol, parts (3), (1), (2) may be

enclosed in a parenthesis modified by a subscript or

a superscript, and the symbol so-far described may be

preceded by a shift vector.

7. WEISSENBERG COMPLEXES AS GENERATING COMPLEXES

7.1 The Weissenberg complexes.

-

There exist lattice complexes, 67 in all, for

which the multiplicity does not decrease for any spec-

ial values of the co-ordinates. Such a lattice com-

plex can simulate an invariant lattice complex as a

limiting complex . These 67 complexes were first re-

cognized by Weissenberg, who gave them the name Haupt-

gitter. We propose to call them the Weissenberg com-

plexes.

The Weissenberg complexes may be regarded as a

basic set of generating complexes. By splitting, they

lead to 402 lattice complexes. The Weissenberg com-

plexes include all the invariant lattice complexes and

also some univariant, bivariant and trivariant ones.

All of them, except the trivariant ones, could be used

as generating complexes; according to our rules, how-

ever, a few of them are not needed. The reason for

this is that we give synonyms of lattice-complex sym-

bols only when the site symmetry is of the same multi-

plicity for all the possible generating lattice com-

plexes; if we were to give all possible synonyms, all

the Weissenberg complexes would be necessary.

18

7.2 Tables of Weissenberg complexes.

-

The Tables of Weissenberg complexes (Tables 18-

24) give the following information, under each crystal

system.

Each Weissenberg complex appears as a subheading,

in which its symbol is followed by its characteristic

space group, data pertaining to the position it oc-

cupies in its standard representation (multiplicity,

Wyckoff letter and site symmetry) and the co-ordin-

ates of the equivalent points.

Under this subheading comes the list of all the

lattice complexes that are derived by splitting. Each

one is given in its standard representation only, its

symbol being followed by that of the characteristic

space group with Wyckoff position letter, and the co-

ordinates of the equivalent sites.

It may happen that the co-ordinates of the sites

of the position given in the second column are not

those listed in the third column but are shifted by

a constant vector. ^ In such a case, the Wyckoff

letter is marked by a prime ('). Example: the sixth

lattice complex listed under P in Table 18 is

4..P23XX, I432(i'); the co-ordinates given in the In-

ternational Tables (1952) under -^432 (i) are those of

the shifted representation 4..P23xx.

The co-ordinates are presented in condensed form.

Moreover, the several origins that are provided by a

centered lattice mode will be indicated by means of

the conventional lattice letter: i" + (xxx; ...) in-

stead of (000; 22^) + (xxx; ...).

7.3 Figures of Weissenberg complexes.

-

7.31 Cubic System.- The a co-ordinates are ex-

pressed in eighths of the cell edge.

The 16 invariant lattice complexes are figured

in standard orientation (Fig. 49). They are repeated,

with examples of their representations, on Figs. 50-51.

The equations that relate the complexes and their re-

presentations can be read off these figures.

There are five uni variant cubic lattice complexes

that are Weissenberg complexes (Figs. 52-53). They

are drawn in standard representation, each in its char-

acteristic space group. The full range of values of

the parameter that is permitted by the one degree of

freedom cannot be shown graphically, but arrows in-

dicate the range between the two or three limiting

cases that are used as generating complexes. Example:

1 The shift vector is not given in Tables 18-24;

it is specified in Tables 25-31 and 32-38.

2i2^..FYlxxx in P2i3. In this example, position (a)

of P2i3 has co-ordinates xxx\ [^Oj + {xxx)'\Tt. For

special values of the co-ordinates, the position sim-

ulates two invariant complexes, F and Y, each one in

four different representations, as follows: for

a; = 0, F; for x = 1/8, \; for x = 1/4, F; for

X = 3/8, 'Y; for x = 1/2, 555 F; for x = 5/8,

Y; for X = 3/4, ^ F; for x = 7/8, Y. Accord-

ing to our rules (Sn. 6.3), we use F as the first gen-

erating complex and Y, which stands for the four re-

presentations of Y, as the second one. The distri-

bution symmetry 2.-^2-^.. is found by applying the rules

(Sn. 6.4). Note that the figure illustrates only the

standard representations of F and ^Y; this is done to

avoid cluttering up the drawing.

7.32 Hexagonal and rhombohedral systems, in

hexagonal description.- The z co-ordinates are ex-

pressed in twelfths of the a cell edge.

There are 7 invariant complexes: P, G, E, N,

in the hexagonal system; R and M in the rhombohedral

system (Fig. 54)

.

Examples of non-shifted representations are il-

lustrated (Fig. 55)

.

There are 5 univariant and 2 bivariant complexes,

in these two systems, that are Weissenberg complexes

(Figs. 56-57).

In case of a bivariant complex, the range of

variation is indicated by hachured triangles. Note

that these triangles do not give the complete 2-di-

mensional range. The height of each triangle can be

read at its three corners. Any site inside one of

the hachured triangles will have corresponding sites

in all the others.

7.33 Tetragonal system.- The z co-ordinates are

expressed in eighths of the a cell edge.

The tetragonal invariant lattice complexes are:

P (alternate representation C = P^), I (alternate re-

presentation F = ly), D (which occurs only as ^D) , T

(which occurs only as ^T). Both and ^T are used as

generating complexes. Another representation of P is

C^. These tetragonal lattice complexes and the alter-

nate representation F can be found among the figures

of the cubic complexes (Figs. 49-51): only the a/a

ratio must be taken f 1. Additional representations

C, C^, ^D, and ^T do not appear on the cubic drawings

and are presented in Fig. 58

<

There are eight variant tetragonal Weissenberg

complexes: 7 univariant and one bivariant (Figs. 59-

19

60). The symbol is that of the complex in its stand-

ard orientation; the drawing figures the complex in

the representation realized in the position of the

space group, and the necessary shifts are indicated in

the legend of the signs used to show the equivalent

points of the generating complexes. The shift of the

first generating complex is thus specified and the

symbol of the figured representation can be obtained

by prefixing this shift to the given Weissenberg-com-

plex symbol

.

7.34 Orthorhombic system.- The z co-ordinates

are expressed in eighths of the a cell edge.

The orthorhombic invariant lattice complexes are:

P, I, F, D, T, which result from easily visualized de-

formations of the cubic lattice complexes that are de-

signated by the same letters (Fig. 49), and C, which

results from the orthorhombic deformation of the tet-

ragonal C (Fig. 58), itself a representation of the

tetragonal lattice complex P.

Alternate representations of orthorhombic in-

variant lattice complexes are as follows: V^^, the

orthorhombic analog of the cubic ?2 (Fi9- 50); C^,

tne orthorhombic analog of the tetragonal (Fig. 58);

and additional representations P^, P^, P^^^, P^^,

P,., 1.. I.. A, B, A, , A . B, (Fig. 61).ao a b c b a b

There are 5 univariant orthorhombic Weissenberg

complexes, 2 bivariant arid one trivariant (Figs. 62-

63). Two of them need special explanations.

The bivariant complex 1.2, B, A,FI Ixz, in Pnma(c)I D a a

is illustrated in shifted representation by means of

two orthogonal projections; the aa projection showing

the partial range of site location by means of ha-

chured 2-dimensional fields.

The drawing of the trivariant complex, which is

realized in P2-^2-^Z-^{a) , shows partial 3-dimensional

range of site location by means of irregular tetra-

hedra.

7.35 Monoclinic system.- The y co-ordinates are

expressed in eighths of the b cell edge.

There are two monoclinic invariant complexes, P

and C in standard representations (Fig. 64). Examples

of alternate representations, P^, P^, P^^ and A, I,

C , F, for P and C, respectively, are also illustrated

because they are used in the text (Tables 23, 30 and

37).

There are three variant monoclinic Weissenberg

complexes: two univariant and one bivariant (Fig. 65).

7.35 Anorthic system.- The anorthic system has

only one Weissenberg complex. It is the anorthic P,

which results from anorthic deformation of any other

P.

8. LIST OF THE LATTICE COMPLEXESAND THEIR OCCURRENCES

8.1 Arrangement of the Tables.-

The crystal systems are listed in order of de-

creasing symmetry (Tables 25-31).

The tables can have from 2 to 5 columns. The

first column lists the lattice complexes, each one

followed by its alternate representations. If needed

a footnote lor a column gives the transformation ma-

trix^ from the cell of the standard representation to

the cell of the space group in which the occurrences

of the alternate representation are being considered.

Unit transformations are not explicitly given.

The next column gives the multiplicity of the

lattice complex, in each of its representations, re-

ferred to the cell of the space group. Example: in

the tetragonal system (Table 28), the multiplicity is

1 for P, 2 for P and C, 4 for P , and C .

c ab c

The next column gives the site symmetry, ori-

ented in the symbolism of the relevant crystal system.

Example: in the orthorhombic system (Table 29), the

site symmetry of P^i^ is 222 in Cmma{a) and Crma{b)

,

2/m.. in Cmma{o) and Cmma(d) , .2/m. in Crma{e) and

Crmaif), ..2/m in Cmrm{e) and Cmmm{f) , and T in

Pbanie) , Pban{f) , Pmmn{a) and Prmn{d)

.

The last column lists all the occurrences of

every lattice complex, in its standard and alternate

representations. The first space group mentioned is

the characteristic space group and the first Wyckoff

letter is that of the position which defines the stand-

ard representation. All representations that are

shifted with respect to the standard one have the

shift specified after the Wyckoff letter by its three

fractional components.

The matrices are presented in the linear form,

long ago proposed by T.V. Barker, in which the three

rows are written on one line, separated by oblique

fraction bars. Example: letting a', S' , c' be the

vectors that define the conventional cell of the space

group and a, S, c those that define the cell of the

lattice complex in standard representation, the trans-

formation

a = a, b = b, c = 2c - a

will be represented by the matrix 100/010/102.

20

Each crystal system has its table subdivided into

four parts, which gather, respectively, the invariant,

univariant, bivariant and trivariant complexes. The

only exception is the anorthic system, in which only

one invariant and one trivariant complex occur.

8.2 Remark.

-

In the tetragonal system Carl Hermann defined

and in the smaller cell, whereas we define U and T

in tne larger cell. This difference in outlook

accounts for the fact that, in Hermann's tables, the

transformation matrix for these two complexes is the

unit matrix, whereas it is yo/^^O/OOl in our own

tabulation.

9. LIST OF SPACE GROUPSWITH LATTICE-COMPLEX REPRESENTATIONS

The crystal systems are here listed in order of

decreasing symmetry (Tables 32-38).

Each system is subdivided into as many parts as

there are point groups in it. Under each point group,

the space groups are taken up in the sequence of in-

creasing superscripts in the Schoenflies symbols.

For every space group, we give, for each position,

the lattice complex in the appropriate representation.

The position data appear under the lattice-complex

symbol. In order to save space, when two consecutive

positions representing the same lattice complex have

the same multiplicity and the same oriented site sym-

metry, the multiplicity is written in front of the

first Wyckoff letter and the site symmetry, behind the

second. This typographical presentation retains the

rule, followed by the International Tables, that the

multiplicity precedes the Wyckoff letter and the site

symmetry follows it. The site symmetry is oriented by

means of dots, which stand for the unused symmetry

directions of the lattice in the crystal system being

considered. This scheme is particularly useful to

symbolize point groups that belong to lower crystal

systems: it helps bring out subgroup relations.

For the sake of brevity, only the standard lat-

tice-complex symbol is given - no synonym, no abbre-

viated notation of synonyms. The shift vector, of

course, cannot be dispensed with. The positions are

listed in the alphabetical order of the Wyckoff let-

ters, with a few exceptions. In Pm3, for instance, it

is clear that position {h) must inmediately follow

position (e), since the two positions accommodate the

same lattice complex P6z, (e) in its standard repre-

sentation and [h) in a shifted representation.

The position that serves to define a lattice com-

plex in its characteristic space group is enclosed in

a frame. It is the first occurrence of the lattice

complex in the characteristic space group. Our choice

is, in most cases, the same as that made by Carl Her-

mann. Note that, when the LC occurs in several re-

presentations, the standard one is chosen. Note also

that tnere are cases where the LC is defined by means

of a shifted representation. Example: in Paom, pos-

ition 8(r)l. This is a consequence of the choice of

origin. In the example given, if the origin were to

be chosen at 222 instead of 2/m, the standard re-

presentation would be obtained.

In any centrosymmetric space group, the origin

is chosen at a center, unless otherwise stated.

9.1 Remarks.

-

Even though our method of expressing the orien-

tation of the site symmetry enables us to distinguish

between 2.. and ..2 in the cubic system, some

ambiguity persists in each case. Examples: In P4232,

the oriented site-symmetry symbol is 2.. for both

I6z, in 12(;!), and .3.W2x, in 12(i). In 1432, both

I12xx, in 24(/2), and . .P^3y.x, in 24(i), have ..2

as site symmetry. The differences between the 2-fold

axes in the two positions in each space group stem

from their relations to the origin.

In the hexagonal system, space group P3^^2, the

co-ordinates of the sites of position 3ia) were given

as follows, in the 1935 edition of the International

Tables:

X, z, 0; X, Zx, 1/3; 2x, x, 2/3.

In the 1952 edition, the z co-ordinates were increased

by 1/3, so that the sites are now written:

X, X, 1/3; X, 2x, 2/3; 2x, x, 0;

and agree with the co-ordinates given for position

3(a) in space group P3i21 in the 1935 edition:

X, 0, 1/3; 0, X, 2/3; x, x, 0.

In the tetragonal system, space group PMmaa,

note that P2z occurs in 4(^) as (P2z) . This should

not be written Pj,2z: the subscript must modify the

whole LC, which is P2z, defined by position l{g) in

its characteristic space group P/^/mmm.

In PMmno, position 8(^)..2, the LC is written

0^^( .b.C2xx)^. Note that the distribution symmetry

comes from the characteristic space group PMmbm. The

subscript c, being attached to the parenthesis, re-

calls the fact that the LC inside the parenthesis was

defined in another space group; in other words, that

the space group in which it occurs here is not its

21

cnaracteri sti c space group.

When several lattice complexes can be chosen as

generating lattice complexes, we do not explicitly

symbolize those that would be designated by the same

letter. Example: In P4/nce, 4(e)4.. is described

as oioC . .2CIlz)_. (The complex has been defined in

1

FMnrm, position 2(e)4mm, as O2O..2CIIZ.) If we set

2=0, 1/8 and 1/4, successively, in the co-ordinates:

0^2; ^Oi; 0,^,^+2; ^,0,^-2, we obtain three possible

generating complexes: ^^1^,0^. This fact is not ex-

plicitly expressed in our symbolism, not even by the

synonyms of the standard symbol of the LC.

In hemimorphic space group P^^ma, positions

4(e). m. and 8(/)l correspond to the LC symbols

..2P^I2x[z] and . .2P^I2x2y[z]. In both of these

the distribution symmetry ..2 comes from P^„/rma,

position 8(q)m. ., which defines the lattice complex

..2 P^I2x2y. Note that the [z] operates on the whole

complex, which can be shifted by any arbitrary 2 value,

and that we have 2=0 in the characteristic space

group. It follows that the 2-fold axis (..2) must

move with the complex.

In the orthorhombic system, we keep the holohedral

space group in the same setting as in the 2nd edition

of the International Tables (1952), but for hemimorphic

space groups, we select a setting that retains the

same systematic omissions and, therefore, shows the

subgroup relations. Example: Space group 02^^^ is

described in setting Prma. Space group Z^^^ is des-

cribed in setting PmaZ in the International Tables,

we keep this setting to facilitate comparisons, but we

also give an alternate setting, Pm2a.

22

APPENDIX

SUBGROUP RELATIONS(TABLES 39-43)

The consanguinity between space groups is deter-

mined by their subgroup relations. The description

of the positions by means of lattice complexes pro-

vides a method to follow the effects of a certain sub-

group relation on all the positions of a space group.

A complete discussion of the questions raised by

subgroup relations for all the space groups would be

an enormous task. We shall, therefore, limit our-

selves to a presentation of a few examples to show

what types of relations are to be expected and the

kind of results that can be obtained.

In a first example (Fig. Al), we consider the

cubic asymmorphic^ space groups and shpw subgroup re-

lations connection F2-^3 to its minimal supergroups,

the latter to their minimal supergroups, and so on, up

to the common symmorphic supergroup Im3m. In a second

example (Fig. A2), the filiation goes from Fddd,

through its minimal supergroups and the supergroups of

the latter, to the common supergroup P>i2m. In pre-

paring these examples, we were able to take advantage

of a pre-publication copy of an article on subgroup

relations by Neubuser and Wondratschek (Priv. Comm.,

1958), which was of great help to us. In the figures,

a solid line connects a group to a subgroup of index

2, while a dashed line goes to a subgroup of index 4

(Fig. Al) or of index 3 (Fig. A2). Along the left

margin, the scale indicates the multiples of the low-

est order, 12, of group PZ-jS. A space-group symbol

enclosed between parentheses and followed by a sub-

script 2 indicates that a block of eight cells, two in

each of the three co-ordinate directions, is being

L space group is called symmorphia if it contains

a position whose site symmetry is the same as the

point group isomorphic with the space group; hemisym-

morphia , if it is of the second kind and contains a

position whose site symmetry is the highest proper sub-

group of the first kind of the isomorphic point group;

asymmorphia, if neither of the above. Examples: Pm3m

is symmorphic, Pw3n is hemisymmorphic , Pm3n is asym-

morphic. Site symmetry m2m is available in Pm3m,

432 in Pn3n, but neither m3w nor 432 is site symmetry

in PrnJn.

considered.

If we compare corresponding positions in a space

group and in one of its subgroups, we see that one of

three situations may obtain:

1. The site symmetry of the position remains un-

changed in the subgroup, but the multiplicity is de-

creased.

2. The multiplicity remains unchained, then it

is the site symmetry that is decreased.

3. Both site symmetry and multiplicity can be

decreased at the same time; this may happen when the

subgroup is of index 4 or index greater than 4.

These three rules apply without any modification

to the description of positions by means of lattice

complexes, whenever the positions have no degree of

freedom. Examples: the relation between 1^ and P^,

which occur as 16(a) in Ia3d and 8(a) in Ia3 respec-

tively, is an example of Rule 1; the complex I, which

occurs as 2(a) in Im3m and as 2(a) in Pm3n, illustrate

Rule 2; and the relation between in (Pw3n)2(e) and

V* and S*, which occur in Ic3d as 24(c) and 24 {d)

,

respectively, is explained by Rule 3 (Table 40).

For positions with degrees of freedom, on the

other hand, the naming of the corresponding complex

depends not only on the site symmetry, but also on

the behavior of the generating (invariant) complex.

Rule 1 can be subdivided into two cases:

la. The site symmetry of the invariant position

remains unchanged; then the invariant complex splits

into complexes of lesser multiplicity. Example:

4.. Y**2xxx in Za3d and 22.. Y*2xxx in J4i32 (Table

39).

lb. The site symmetry of the invariant position

does not remain unchanged; then both the splitting

number and the invariant complex cnange. Example:

4.. Y**2xxx in Ja3d and 4.. l2Y**lxxx in J?3J (Table

39).

Rule 2 gives rise to three cases.

2a. The position and the invariant generating

complex suffer a decrease in site symmetry without any

change in the naming of the generating complex; then

the variant lattice complex remains the same. Example

I5z in Im3m, ]2{e)4m.m, remains I5z in Pm3n, 12(/)ot7)2.

(Table 40).

2b. In the naming of the position, the invariant

23

generating complex remains unchanged, but the point

complex changes; then some distribution symmetry may

have to be specified. Example: I12xx in Im3m{h) and

..2 I6z2x in Pm3n{k) (Table 40).

2c. The change in site symmetry requires a com-

pletely different generating complex. Example:

4a.. Y**3xx in Ia3d{g) and 22.. P26xyz in Ia3(e)

(Table 40).

Relationships are shown between asymmorphic cubic

body-centered space groups, which have the same cell:

Ia3d, Il3d, 14^32, Id3, i"2^3 (Table 39). The series

Jm3m, Pm3n, Ia3d, Ia3 , Pa3, P2^3 is taken from Fig.

Al and the subgroup relations are explained in Table

40. Here a set of points in the supergroup, Ia3d or

(Im3m)2, is followed in all the subgroups until the

general position is reached in a subgroup. In Table

41 the positions in space group Pa3 are related to

those in the minimal supergroups Ia3 and Fm3. In

Table 42 two space groups, f4^32 and J4^32, have a

common subgroup P^^^^i the solid lines show where the

special positions of f4^32 and I4i32 are to be found

in the subgroup. Finally, Table 43 shows the relation-

ships between positions in different crystal systems,

illustrated by the space groups Fd3m, Fd3, I^-^/amd

and Fddd.

24

Fig. Al, Subgroup relations in cubic asymmorphic space groups;

'^3m 43m 432 m3 23

i-(Jw3m)

Fig. A2. Subgroup relations in space groups of different systems.

25

REFERENCES

Belov, N.V., N.N. Neronova, J.D.H. Donnay and G.

Donnay (1960) Tables of magnetic space groups.

Acta Cryst . 13 , 1085.

Belov, N.V., N.N. Neronova, J.D.H. Donnay and G.

Donnay (1962) Tables of magnetic space groups.

II. Special positions. Jour. Phys. Soc. Japan

J7(B-II), 332-335.

Boldyrev, A.K. (1925) Die vom Fedorow-Institut ange-

nommene kri stall ographische Nomenklatur. Z. Krist .

62, 145-150.

Burzlaff, H., W. Fischer and E. Hellner (1968) Die

Gitterkomplexe der Ebenengruppen . Acta Cryst .

A24, 57-67.

Burzlaff, H., W. Fischer and E. Hellner (1969) Bemer-

kungen zu "Die Gitterkomplexe der Ebenengruppen"

(Acta Cryst. A24, 57, 1968) und "Kreispackungs-

bedingungen in der Ebene" (Acta Cryst. A24, 67,

1968). Acta Cryst. A25, 710-711

.

Donnay, J.D.H. (1947a) Hexagonal four-index symbols.

Amer. Mineral . 32, 52-58.

Donnay, J.D.H. (1947b) The superabundant index in the

hexagonal Bravais symbol. Amer. Mineral . 32,

477-478.

Donnay, J.D.H. (1969) Symbolism of rhombohedral space

groups in Miller axes. Acta Zryst .A25, 715-716.

Donnay, J.D.H., E. Hellner and A. Niggli (1964)

Coordination polyhedra. Z. Krist . 120, 364-374.

Donnay, J.D.H., E. Hellner and A. Niggli (1966)

Symbolism for lattice complexes revised by a Kiel

Symposium. Z. Krist. 12S, 255-262.

Fischer, W. (1968) Kreispackungsbedingungen in der

Ebene. Acta Cryst . A24, 67-81.

Hellner, E. (1965) Descriptive symbols for crystal

-

structure types and homeotypes based on lattice

complexes. Acta Cryst . 3g, 703-712.

Hermann, C. (1935) Gitterkomplexe. Int. Tab, zur

Bestimmung von Kristal Istrukturen , vol. l,

chapters VI and VII.

Hermann, C. (1953) Paper presented at the Deutsche

Mineralogische Gesellschaft, Sektion fiir Kristall-

kunde, Erlangen meeting.

Hermann, C. (1960) Zur Nomenklatur der Gitterkomplexe.

Z. Krist ., 113, 142-154.

Internationale Tabellen zur Bestimmung von Kristall-

strukturen (1935) 2 vols. Berlin, Borntraeger.

International Tables for X-ray Crystallography, vol.il

(1952, 1965, 1969). Birmingham, England, The

Kynoch Press.

Loeb, A.L. (1967a) Crystal algebra of cubic lattice

complexes. Tech. Rept . 130 , Ledgemont Labora-

tory, Lexington, Mass.

Loeb, A.L. (1957b) Domains, interstices and lattice

complexes: derivatives from the b.c.c. lattice.

Tech. Rept . 133, Ledgemont Laboratory, Lexington,

Mass.

Loeb, A.L. (1970) A systemative survey of cubic

crystal structures. J. Solid State Chem . 1,

237-267.

Niggli, P. (1919) Geometrische Kristallographie des

Diskontinuums. Leipzig, Borntrager.

Niggli, P. (1920) Lehrbuch der Mineralogie. Berlin,

Borntraeger. (1st ed., pp. 125 et ss.); 2nd ed.,

1924, pp. 168 et ss.; 3rd ed. , 1941, IPP. 275 et ss.

Niggli. P. (1928) Kri stall ographische und struktur-

theoretische Grundbegriffe. Leipzig, Akad.

Verlagsges.

Smirnova, N.L. (1966) An analytical method of repre-

sentation of inorganic crystal structures. Acta

Cryst . 21, A33.

Smirnova, N.L., and L.I. Poteshnova (1966) Vestnik

Moskovskogo Univ.. Ser. Geol . No. 6.

Weber. L. (1922) Das viergl iedrige Zonensymbol des

hexagonalen Systems. Z. Krist . 57, 200-203.

26

FIGURES

501-563 0 - 73 -3

Figs. 1- 47. Site sets in stereographic projection.Name of site set as co-ordination polyhedron and of itsdual as crystal form.

Fig. 1. Point. Monohedron.2. Line segment (through 0). Pinacoid.3. Line segment (off 0) . Dihedron.4. Trigon (through 0) . Trigonal prism.5. Trigon (off 0). Trigonal pyramid.5. Rectangle (through 0). Rhombic prism.7. Rectangle (off 0). Rhombic pyramid.8. Tetragon (through 0) . Tetragonal prism.9. Tetragon (off 0). Tetragonal pyramid.

10. Rhombic tetrahedron. Rhombic tetrahedron.11. Tetragonal tetrahedron. Tetragonal tetrahedron.12. Regular tetrahedron. Regular tetrahedron.13. Truncated trigon (through 0). Ditrigonal prism.14. Truncated trigon (off 0) . Ditrigonal pyramid.15. Hexagon (through 0). Hexagonal prism.16. Hexagon (off 0) . Hexagonal pyramid.

28

29

Figs. 1-47. Site sets in stereographic projection.Name of site set as co-ordination polyhedron andof its dual as crystal form.

Fig. 17. Octahedron. C\±ie.

18. Trigonal antiprism. Rhombohedron

.

19. Trigonal prism & pinacoid. Trigonal dipyramid.20. Trigonal trapezohedron & pinacoid. Trigonal trapezohedron.21. Truncated tetragon (through 0) . Ditetragonal prism.22. Truncated tetragon (off 0) . Ditetragonal pyramid.23. Tetragonal tetrahedron & pinacoid. Tetragonal scalenohedron.24. Tetragonal prism & pinacoid. Tetragonal dipyramid.25. Tetragonal trapezohedron & pinacoid. Tetragonal trapezohedron.26. Rectangular parallelepiped. Rhombic dipyramid.27. Cube. Octahedron.28. Truncated hexagon (through 0) . Dihexagonal prism.29. Truncated hexagon (off 0) . Dihexagonal pyramid.30. Pinacoid & rhombohedron. Hexagonal scalenohedron.31. Truncated trigonal prism & pinacoid. Ditrigonal dipyramid.32. Hexagonal prism & pinacoid. Hexagonal dipyramid.

30

31.

Figs. 1-47. Site sets in stereographic projection.Name of site sets as co-ordination polyhedron andof its dual as crystal form.

Fig. 33. Hexagonal trapezohedron & pinacoid. Hexagonal trapezohedron.34. Truncated tetrahedron. Trigon-tritetrahedron.35. Cube & 2 tetrahedra. Tetragon-tritetrahedron.35. Cuboctahedron. Rhomb-dodecahedron.37. Pentagon-tritetrahedron & 2 tetrahedra.

Pentagon-tritetrahedron

.

38. Dihexahedron & octahedron. Dihexahedron.39. Truncated tetragonal prism & pinacoid. Ditetragonal dipyramid.40. Truncated hexagonal prism & pinacoid. Dihexagonal dipyramid.41. Truncated cube. Trigon-trioctahedron.42. Rhombi cuboctahedron. Tetragon-trioctahedron.43. Cube & 2 tetrahedra. Hexatetrahedron.44. Truncated octahedron. Tetrahexahedron.45. Cube & octahedron & dihexahedron. Didodecahedron.46. Snub cube. Pentagon-trioctahedron.47. Truncated ciiboctahedron. Hexaoctahedron.

o

32

33

Order of

Fig. 48. The point group a-

line is a subgroup of that <

triple) lines mean that theupper one in two (or three)Dashed lines show subgroups

(After C. Hermann, Int.

; the lower end of a connectingit the upper end. Double (or

lower group is subgroup of thenon-equivalent orientations,of index > 2

.

Tab., vol. I, p. 49, 1935.)

34

r-

0- -0

0-

0-

I

4-

0

IT

4

D

4^26-n4

•0

•0

26— 0-

0

0-

h^^^—

I

26

0— 26

ho-6

0-H42

- H\—

4- -c)- 4

i\-

—1

—

1

1-

1

—1

—

- 1-

1

-3--f-

1

-5-H-

1

-7--H-t-

-3- - 1--7- -5-

H--h-5-

—

K

-7--+-

- 1- -3-—

U

-7-1

-5-1

-+--3-

1

- 1 -

1

T

TV 6^37-

4-| [-4— 0-1-4

15

ho2

-37-

37 15'

-04-

04- ()-

\- 04- 4

—1

—

1 -—1

—

3

-h-

1- _ e

- 1-

_l-3 -

1

—1—

'

-1 -1

-5-

-H

1

-7-1

—h-

1

-3--h-H

-7-

-h- 3--\-

- 1-

H--5--4-

-3--\-

-7-—\—

- 5- - 1-

-5-1

- 1-

1

—\—-3-

1

-7-1

15

oH ho

•37

4-^0—4-1

4-hO— 4-| |-4

6L

-1 5-

0^42

•15

637

V

The 16 cubic invariant lattice complexes in standardorientation. Heights in eighths of cell edge.

35

0-

0-

0-

0-

0

-0

-0 04

04

-0 04

2 2 2"

04-

26

26-

04

04

26

26-

04-

04-

04-

04 0-

04

04 0-

04'

04

04

0

04

04

04

-0

•0

o ob c. b

o - r>cc co

0

J- ±± p4 4 4^2

•O-

0-J.J_J_ p2 2 2'

•0

0-

0-

4 4 4 r

J

•4-

0

— cf-

-C)-- -4

1

-0

-6-4-

-0

D

Fig. 50.

0-

0

3 1 3. p4 4 4 r

0

0-111

I

0-

0

i.± j_r)2 2 2"^

-0

1

-15-1

-37-1

-15-1

-37-

-37-

-hH--15- -37- -15-

-15- -37- -15- -37-

-37-1

H--15-

1

-37-1

+-15-

1

±±1 p8 8 8 "2

26-04-26

26-

26

26

•26

26-

04-26 04

26-

04-^26-^04-26 04-

26-

-0 04

J-

04-

-0-

04

J"

1

-37-1

-15-1

-37-1

-15-

-15- -37- -15- -37-

-h-37- -15- -37- -1 5-

-15-1

-37-1

-15-1

-f--37-

1

13. 3 IT

8 8 8 '2

26t04t26^04^26

26- 04

04

26

04

26-^04-^26-^04-^26

04-

04

-26

04

a. a. a.

26

0246

--04--

46

0246

04 26-]-2|-|-26-^02_[.26

02 4-04-LQ2

26^°i^26^b/f ^2646

26-

0246

46

02'46

04-'

46

04-

0246

26

0246

.0246

Cubic invariant lattice complexes and representations,Heights in eighths of cell edge.

36

-26

•26

W

-0—

r2

ho6

6

0-h42_L

26

i -

v

4-r26

-0

-0-

4-1- 26-

w

H26

H

26 26

5I I I

7-"T-6

h4-

-0-

oH

V

15- •37-

1-37 -15

|_4—0^4 0-|

-15- •37-

I

-1-1

-3^1

-5-r

-

-7--\--3-

-\-^-1- -7-

—1—-h--5-

-H-5-

1-h-7- -1- -3-

-7-1

-H-5-

1

-3-1

-h-1-

1

T

-0- 4-

37-

|_0— 4+ 0— 4-| [-4

—

15

ho2

6-+

•15

OH

37-H

4-|-0

637- 15^-1

—

r

-5--4—

1

-7-—1

—

-1 — 1

-3-

-7- -5- -3- -1-

-1 --+--3- -5-

—1—-7-

1

-3-1

-1-1

-7-1

-5-1

2 2 2'

4q

0

|-04

26

I-+26

1357

-26

.1 35

04 -[-04 --04-j

,1 357

26

26

-^--|-26

.1 357

-26

26|-04--04-|-04— 04-|

26 H—I—|- 26-1—I—1-26I I 13 I I I 1 3_1_J

57Wc

57'

-1

-1

-7-

3

7 5

-3_i_

Fig. 51.

5-

3-

-7

3-

2 2 2 '

-3-

-5-

5+7--+

1

-7_l_

-7-

Y

-I— 5 —7

3 7— 5

-5 1 3

1 5-

•3-

-1-

-7

Y

-5-

-7-

2 2 2 ^

04T26T04-r26-r04:

,

I ] I T I

T

I

26404--26--04--26

26--04--26

04--26--04--26

04-L 26 -L04-L26 -L 04

04—26

04

Cubic invariant lattice complexes and representations(continued). Heights in eighths of cell edge.

37

®

•©(§)-

D©

I2j3(^7) 2^2^.. F| Y Ixxx

OP,

12^3 (Z>)

os

2^3.SVlz® UP® DOWN

Fig. 52. Cubic univariant Weissenberg complexes in standard

representation and characteristic space group.

Heights in eighths of cell edge. (See Section 7.31,

p. 19.)

38

Fig. 53. Cubic univariant Weissenberg complexes in standard

representation and characteristic space group.Heights in eighths of cell edge. (See Section 7.31,

p. 19.) (continued).

39

E

0N

M

40

0,6 0,4,8 0,4,8

41

P6|22/2';3,2.Pc,EcQ;iXX

P<6jaJ 3,2,..Pc.,Ec^Qclxy

Cc

PZm\fd) .2.GE1Z

OG E

P32i2/'^7; 32..PcQlxx P322I r^7; s^. . p/q Ix

Fig. 56,

OOfP, 00|+Q

Hexagonal univariant and bivariant Weissenbergcomplexes. Heights in twelfths of c. (See

Section 7.32, p. 19.)

42

PZz{a) 32..R,"^Qlxy

A 3 3^n:

Fig. 57. One more hexagonal bivariant Weissenberg complex.Heights in twelfths of a. (See Section 7.32, p. 19.)

501-563 O - 73 - 4

43

44

©

©®[SI-

P4/nmm ic) ..2CI1Z

OO^OC DO^^I

I4/acd ie)

[7][5H (D [3][5}- ®

(3)®-

4..I,P^,lx

1—J a. a. a ir4 4 8 "ca

<2>-

(p-

I42d(^)

mm-

0)

4..nf^lx

O VT n ILL F

(D©-

ID [2]

®©

P4322(d7) 4,.. P T Ix3 CC C

©©

2 4 xc

Fig. 59. Four tetragonal univariant Weissenberg complexes,Heights in eighths of o. (See Section 7.33.)

45

..22^TC,,lx

'-'4 4 8 ^CC

<3>

Fig. 60. Three more univariant tetragonal Weissenberg complexesand the only bivariant one. Heights in eighths of a.

(See Section 7.33.)

46

0 0 0 04-

04

04

04

04-

04-

04-

04-Pbc

•04 04-

04

04 04-

ac

04 0^

04 0

-04 0- 0—

0-

0

0 04-

04-

26

-04

•04

-4-

A

-0 0 0 0—4 0— 4—

0

0 0—4— 0— 4—0

0-

Aa

0 0-

0 4

0 0-

0-

•0-

Bb

Fig. 61. Representations of orthorhombic invariant Weissenbergcomplexes. Heights in eighths of o. For the invariantcomplexes themselves, look up cubic P, I, F, D, T (Fig. 16)

and tetragonal representation C (Fig. 58), which are theirtopological analogs.

47

CO)® 2

6®C0)

©(pd?

Pmma (e) .2. Pa Biz

O^OOPa DiO^B

©@ Klh {6^< C6X2

©(2)

-©©[2

Pmmn /"^y

OC

Zj.. CIlz

00^1

®© >{I]- 6 he C6X2

ZxwzxT\(c) 2j..CcFly

OOOjC O^jF

Cmma /^p^y 2.. PabFlz

H]®©

Imma (e)

© O^OBb

©@-{2}

.2. B^AJz PbcmiW 2.T Pbc Ab CeF Uy

Fig. 62. Five \inivariant and one bivariant orthorhombic

Weissenberg complexes. Heights in eighths of o.

48

(3)

lev,

(3

(0

o<GO

C\J

o£cCL

A

O

<

O

CDOOO

u•H Q)

0)

s0 "

—

0 •

A-t

4->

M0 0

+J

m•rH

> 0)

HM cP •rH

>i CO

rH -P

0 CPrH0)

-P

•

CO

rO 0)

4J 0)

CfC &-H s

0u •

>•H^! !h •

Q)

CD

M0 0) 0g CO •H

W +J

(D •H u<U CD

O in

•H

49

/ / / /J 4 0 4 0 4

I I I I0 4

Fig. 54. The two monoclinic invariant Weissenberg complexes,in standard and alternate representations. Heightsin eighths of h.

50

X <

P2/C fej cF^Aly

X <

C^/cfeJ T CcFlyOooic^ OO^iF

Fig. 65.

pz^/rr^reJ 2^ f^ACIlxz

oo^op^ doHa Oiioc A^^^I

The two univariant and the only bivariant monoclinic

Weissenberg complexes. Heights in exghths of b.

51

TABLES

i

Table 1. Rules of Splitting

Point group:

homopolar single, 1st Site 2nd Site 3rd Site Point-complexprimary polar Split. Symm. Split. Symm. Split. Symm. symbol

axis axis

1

1 Ixyz 1 Ixyz2xyz 1 Immediate 2xyz

4 4xyz 1 4xyz3 6xyz 1 6xyz *

m Ixz m 2y 1 lxz2y2/m 2xz m 2y 1 2xz2yMm 4xy m.

.

2z 1 4xy2zZ/m ** 3xy rn.

,

2z 1 3xy2z6/m 6xy m.

.

2z 1 Stay in m. 6xy2z<4 m 2 4xz .m. 2y 1 then out. 4xz2y'4 2 m 4xxz . .m 2y 1 4xxz2y(3 m 1 6xxz .m. 2y 1 6xxz2y3 1 m 6xz . .m 2y 1 6xz2y

i 3 m 6xxz .m 2y 1 6xxz2y

2 2 2xz 1 ly2xz4 Iz 4.. 4xy 1 lz4xy

J- i IZ J.

.

0 V Y/jxy 1 izoxy1 Ixxx 3. 3yz 1 lxxx3yz

6 Iz 6.. 6xy 1 lz6xy2 2 2 2x 2.. 2yz 1 Stay on axis, 2x2yz4 2 2 4x .2. 2yz 1 4x2yz

(3 2 1 3x .2. 2yz 1 then off. 3x2yz3 1 2 3xx . .2 2yz 1 3xx2yz, ***

3xx2yz(3 2 3xx .2 2yz 1

6 2 2 6x .2. 2yz 1 6x2yz2 3 6z 2.. 2xy 1 6z2xy4 3 2 6z 4.. 4xy 1 6z4xy

2 m m Ix 2mm 2y . .m 2z 1 Ix2y2zm 2 m ly m2m 2z m.

.

2x 1 Iy2z2xm m 2 Iz mm2 2x ,m. 2y 1 Iz2x2y4 m m Iz 4mm 4x .m. 2y 1 Iz4x2ym 1 Iz 3m. 3XX .m. 2y 1 jlz3xx2y

3 1 m Iz 3.m 3x . .m 2y 1 Iz3x2y3 m Ixxx 3m 3y .m 2z 1 Ixxx3y2z6 m m Iz 6mm 6x . .m 2y 1 Iz6x2y

m m m 2x 2mm 2y . .m 2z 1 2x2y2z4/m m m 4x m2m. 2y m.

.

2z 1 4x2y2z1 3/m 2 m 3x m2m 2y m.

.

2z 1 3x2y2zt 3/m m 2 3XX mm2 2y m.

.

2z 1 3xx2y2z6/m m m 6x m2m 2y m.

.

2z 1 6x2y2zmm3 6z 2mm. .

.

2x m.

.

2y 1 6z2x2y4 3m 6z 2..Tmi 2xx . .m 2y 1 6z2xx2ym 3 m 6z 4x m.

.

2y 1 6z4x2yStay on axis; off axis. in m; out of m.

*: The symbol is the same in hexagonal and rhombohedral axes.

**'3/m = 6

*** The same symbol serves for 312 in the hexagonal system and for 32 in rhombohedral

co-ordinates. The preceding symbol is used both for 321 in the hexagonal system and

for 32 in hexagonal co-ordinates.

54

Table 2. Site Sets^ Found in the Cubic System

Co-ordination polyhedron. Dual crystal form. Site-set symbol . Co-ordinates. Occurrences.

Tetrahedron. Tetrahedron .

4xxx: xxx; xxx k 43m(a); 43m(b); 23(a); 23(b).

Octahedron. Cube .

6z: ±(00z) K m3m(a); 43m(c) ; 432(a); m3(a); 23(c).

Cube. Octahedron .

8xxx: ±(xxx; xxxk) m3m(b); 432(b); m3(b).

Cuboctahedron. Rhomb-dodecahedron .

12xx: ±(x x,0) m3m(c); 432(c).

Di hexahedron + octahedron. Pi hexahedron (or pentagon-dodecahedron )

.

6z2x: ±(x,0,±z) k m3(c)

Truncated tetrahedron. Tri gon-tri tetrahedron . (For

Cube + 2 tetrahedra. Tetragon-tri tetrahedron . (For

6z2xx: (xxz; xxz)k ; xxz it 43m(d)

.

Pentagon-Tri tetrahedron + 2 tetrahedra. Pentagon-Tri tetrahedron .

6z2xy: (xyz; xyz; xyi; xyz)K 23(d).

Truncated octahedron. Tetrahexahedron

.

6z4x: ±(x,0,±z) ir m3m(d)

.

Truncated cube. Tri gon-tri octahedron

.

(For Jz|<|xRhombi cuboctahedron. Tetragon-tri octahedron . (For

. .)

x|<|z| .)

6z4xx: ±(x,x,ifcz k; xxz it) m3m(e).

Snub cube. Pentagon-trioctahedron .

6z4xy: (xyz; xyz; xyz; xyz; yxz; yxz; yxz; yxz)K 432(d),

Cube + octahedron + di hexahedron. Didodecahedron .

6z2x2y: ±(x,y,±z; x,y,+z) k m3(d).

Cube + 2 tetrahedra. Hexatetrahedron

.

6z2xx2y: (xyz; xyz; xyz; xyz) it 43m(e).

Truncated cuboctahedron. Hexoctahedron .

6z4x2y: ±(x,y,±z; x,y,±z) tt m3m(f).

Limiting cases not included.

55

Table 3. Site Sets^ Found in the Hexagonal and Rhombohedral Systems(in hexagonal co-ordinates)

Co-ordination polyhedron. Dual crystal form . Si te-set symbol . Co-ordinates. Occurrences.

Point. Monohedron ,

Iz: 00 z

Line segment (col linear with origin)

2z: 0,0, ±z

6mm(a)

;

3ml (a)

;

31m(a)

;

Paral 1 el ehedron .

6/mmm(a)

;

6/mCa)

;

321(a)

Trigon coplanar with origin. Trigonal prism .

xxO3XX3x: xOO'

Trigon. Trigonal pyramid .

Iz3xx:lz3xlz3xy:

xxzxOz'xyzj

6m2Cb);62mCb);

3ml (c).

31m(c).

311 (c).

5mm(b)

3ml Cb)

31m(b)

6m2(a)

;

6(a);312(a);

6m2(c).321(c);

Truncated trigon coplanar with origin. Pi trigonal prism .

3xx2y:

3x2y:

Truncated trigon.

1 z3xx2y:Iz3x2y:

(xyO, yxO)(xyO^)^

'

Di trigonal pyramid .

(xyz, yxz)

(xyO J ,

^

5m2(e).

S2mCe)

.

3ml (d).

31m(d).

Hexagon coplanar with origin. Hexagonal prism .

6x16xx:

6xy:

(±x,0.0)±(xxO )

^

±(xyO')

Hexagon. Hexagonal pyramid .

Iz6xx:

lz6x:

lz6xy:

(xxZt-)^

(±x,a,z}^(xyz, xyz)^

6/mmm(b)

;

6/mmmCc)

;

6/mCb).

6mmCc)

,

6mm (d)

.

6(c).

622Cb);

622 Cc);

Trigonal prism + pinacoid. Trigonal dipyramid .

3xx2z:3x2z:3xy2z

:

(x,x,±z)

Cx.O,±z)^Cx,y,±z)^

6m2(d).62m (d).

5Cc).

6(a); 6(b);

311(a); 3lUb),

62m(a); 522(a);3ml (a); 31m(a);511 (a).

62mCc); 321(b).

3ml (b)

31m(b),

^ Limiting cases not included.

56

Table 3 (concluded)

Trigonal trapezohedron + pinacoid. Trigonal trapezohedron .

3x2yz: (xyz; ^xz)^ 321(d).

3xx2yz: (xyz; yxz)^ 312(d).

Trigonal anti prism. Rhotnbohedron .

5xxz: ±(xxz) 5ml (c).

6xz: ±(xOz)^ 31m(c).6xyz: ±(xyz)J 511(b).

Truncated hexagon coplanar with origin. Dihexagonal prism.

6x2y: ±(xyO^) ^ 6/mmm(f).

Truncated hexagon. Dihexagonal pyramid .

Iz6x2y: CCxyz; xyz)^)^ 6mm(e).

Pinacoid + rhombohedron . Hexagonal scalenohedron .

6xxz2y: ±(xyz; yxz) 3ml(d).6xz2y: ±(xyz^)

^SlmCd).

Truncated trigonal prism + pinacoid. Pi trigonal di pyramid .

3xx2y2z: (x,y,±z; y,x,±z) Sm2(f).3x2y2z: (x,z,±z^)^ ^ 62m(f )

.

Hexagonal prism + pinacoid. Hexagonal di pyramid .

6x2z: (±x,0,±z) 6/mmmCd).6xx2z: (xjXjiz^-)^ 6/mmmCe).5xy2z: +Cx,y,±z)| 6/mCc).

Hexagonal trapezohedron + pinacoid. Hexagonal trapezohedron .

6x2yz: ( xyz, xyz,yxz,yxz )^ 622(d).

Truncated hexagonal prism + pinacoid. Dihexagonal di pyramid .

6x2y2z: ±(x,y,±z^)^ 6/mmm(g).

57

Table 4. Site Sets'^ Found in the Rhombohedral System

(in rhombohedral co-ordinates)

Co-ordination polyhedron. Qua! crystal form . Si te-set symbol . Co-ordinates. Occurrences.

Point, iionohedron ( pedion )

.

Ixxx: XXX 3m(a); 3m(b); 31(a); 31(b).

Line segment (col linear vn'th origin). Paral lelehedron ( pinacoid )

.

2xxx: ±(xxx) 3m(a); 32(a); 31(a).

Trigon coplanar with origin. Trigonal prism .

3xx: xxO K 32(b); 32(c).

Trigon. Trigonal pyramid .

Ixxx3z: xxz < 3m(c).Ixxx3yz: xyz k 31(c) .

Hexagon coplanar with origin. Hexagonal prism .

6xx: ±(xxO k) 5m(b).

Truncated trigon. Pi trigonal pyramid .

Ixxx3y2z: xyz tt 3m(d).

Trigonal trapezohedron + pinacoid. Trigonal trapezohedron .

3xx2yz: (xyz, xzy)K 32(d).

Trigonal anti prism. Rhombohedron .

6xxz: ±(xxz k) 3m(c).6xyz: ±(xyz <) 31(b).

Pinacoid + rhombohedron. Hexagonal scalenohedron .

5xxz2y: ±(xyz v) 3m(d).

Limiting cases not included.

58

Table 5. Site Sets^ in the Tetragonal System

Co-ordination polyhedron. Dual crystal form . Site-set symbol. Co-ordinates. Occurrences

Point. Monohedron ( pedion )

.

Iz: OOz 4mm(a); 4mm(b); 4(a); 4(b).

Line segment (collinear with origin). Parallelehedron ( pinacoid )

.

2z: ±(00z) 4/mmm(a); 42m(a); 4m2(a); 422(a); 4/m(a); 4(a).

Tetragon (square) coplanar with origin. Tetragonal prism .

4x: +(xOO)t 4/mmm(b); 42m(b); 422(b).4xx: ±(x,±x,0) 4/mmm(c); 4m2(b); 422(c).4xy: ±(xyO,yxO) 4/m(b).

Tetragon. Tetragonal pyramid .

Iz4x: +x,0,z; 0,±x,z. 4mm(c).Iz4xx: ±x,±x,z. 4mm(d).Iz4xy: (+++, —+) (xyz ,yxz) . 4(c).

Tetragonal tetrahedron. Tetragonal tetrahedron .

4xz: ±x,0,z; 0,±x^z. 4m2(c).4xxz: xxz; xxz; xxz^ _ _ 42m(c).4xyz: xyz, xyz, yxz, yxz 4(b).

Truncated tetragon coplanar with origin. Pi tetragonal prism .

4x2y: ±(xyO; yxO)^ 4/mmm(f).

Tetragonal tetrahedron + pinacoid. Tetragonal scalenohedron .

4xz2y: ±x,±y,z; ±y,±x,z^ _ 4m2(d).4xxz2y: (xyz; xyz; yxz; yxi)^ 42m(d).

Truncated tetragon. Pi tetragonal pyramid .

Iz4x2y: ±x,±y,z; ±y,+x,z. 4mm(e).

Tetragonal prism + pinacoid. Tetragonal di pyramid .

4x2z: ±(x,0,±z)t- _ 4/mmm(d).4xx2z: ±(x,x,±z; x,x,±z) 4/mmm(e)

.

4xy2z: ±(x,y,±z; y,x,±z) 4/m(c).

Tetragonal trapezohedron + pinacoid. Tetragonal trapezohedron .

4x2yz: xyz; xyz; xyz; xyz; yxz; yxz; yxz; yxz 422(d).

Truncated tetragonal prism + pinacoid. Pi tetragonal di pyramid .

4x2y2z: (x,y, z; y,x,±z)^ 4/mmm(g).

^ Limiting cases not included.

59

501-563 O - 73 - 5

Table 6. Site Sets^ Found in the Orthorhombic System

Co-ordination polyhedron. Dual crystal form . Site-set symbol. Co-ordinates. Occurrences.

Point. Monohedron ( pedion )

,

Iz: OOz. mm2(a), mm2(b)

Line segment (col linear with origin). Parallelehedron ( pinacoid )

.

2x:

2y:

2z:

±xOO.±OyO.

±00z.

mmm(a)

;

mmm(b)

;

mmm(c)

;

222(a),

222(b),

222(c),

Line segment. Dihedron .

Iz2y:

lz2x:0, ±y, z

±x, 0, z

mm2 ( c

)

mm2(d)

,

Rectangle coplanar with origin. Rhombic prism .

2y2z:2x2z:

2x2y:

±(0, y, ±z)

±(±x, 0, z)

±(x, ±y, 0)

mmm(d)

.

mmm(e)

.

mmm(f )

.

Rectangle. Rhombic pyramid .

Iz2x2y: ±x, y, z; +x, y, z mm2(e)

Rhombic tetrahedron. Rhombic tetrahedron .

2x2yz: xyz; xyz; xyz; xyz 222(d),

Rectangular parallelepiped. Rhombic di pyramid .

2x2y2z: ±(x, y, ±z; x, y, ±z) mmm(g)

,

Limiting cases not included.

60

Table 7. Site Sets^ Found in the Monoclinic System

Co-ordination polyhedron. Dual crystal form . Site-set symbol . Co-ordinates. Occurrences,

Point. Monohedron ( pedion ).

ly: OyO 2(a), 2(b).

Ixz: xOz m(a).

Line segment (collinear with origin). Parallelehedron (pinacoid) .

2y: ±(OyO) 2/m(a).2xz: +(xOz) 2/m(b).

Line segment. Dihedron .

Iy2xz: xyz, x^z 2(c).Ixz2y: xyz, xyz m(b).

Rectangle coplanar with origin. Rhombic prism .

2xz2y: ±(x, ±y, z) 2/m(c).

Table 8. Site Sets^ Pound in the Anorthic (Triclinic) System

Co-ordination polyhedron. Uual crystal form . Site-set symbol. Co-ordinates. Occurrences,

Point. Monohedron ( pedion ).

Ixyz: xyz 1(a).

Line segment (collinear with origin). Paral 1 elehedron ( pinacoid )

.

2xyz: ±(xyz) T(a).

^ Limiting cases not included.

61

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Weissenberg Complexes

Table 18: Cubic system

P Pm3m l(a)mZm 000

P4-XXX t<icm[ e) tXjCCCCJ CCCOt^\^

rbz 'PmZvn (e) Z

{

UUZ

)

K

reXXX Pm3m(g) ± (XXXI XXXK )

P12xx -i- //VI -t-/v* /~i )± ( Xj ±iCj UJ K

22. .P 2xxx laZ(c) I± (xxx);{l0ll± (xxx)}k

4. .P^3xx 1432 d') {Jt (xxO) ; J+ (xxO; xOx; Oxx) } k

. r 2 "XX J-fflOffl [ U J

Jr DZ^XX ir^ Of 11 ( L J ( XXZ y XXZ J XXZ 7T

r DZZX Pw 'K f n ) Z ( C/j xS / K

r DZHX XVm 'Km \ Z ( ZS / TT

r DZM-XX tTnoTii ( 777/ — ( *Cj ins Kj XXZ TT /

OP C rrr OVzr^bZ/X nmoO I 7^/

P6z2xy P23(j) (xyz;xyz;xyz;xyz) k

P6z2xx2y P43m(j) (xyz;xyz;xyz;xyz) tt

PSzIxy r a::^2; xyz; xyz; yxz; yxz; yxz; yxz) k

P6z2x2y Pm3(l) ± 2/J ±2; Xj y^±z)KP6z'4x2y PmZmin) ±(x,yj ±z;x,y,±z)T^

22. . P 6xyz Ia3(e) {J± (xyz) ; \0\I± (xyz;yzx;zxy ) }k

nP 6z2xy F43o (h) {F-\-(xyz;xyz;xyz;xyz) ;\\ \F-\- (yxz; yxz; yxz; yxz j } k

m. .P 6z4xy Fm3c (j') {F+(xyz; xyz; xyz; xyz; yxz; yxz; yxz; yxz) ;

{ . .2P 6z2x2y} 5 2 \F-\- (xyz; 3[yz;xyz; xyz; yxz; yxz; yxz; yxz )}<

1 Im3m 2(a)m3m 000;

I4xxx I43m(a) I+(xxx;xxxk)I6z Im3m(e) I±(00z)k1 8XXX Im3m(f) I±(xxx;xxxk)112XX Im3m(h) I± (x, ±X, 0) K

. .2I4XXX Pn3m(e) XXXy XXX }^ J 2 2 2^ ^XXXy XXx^

)

I6z2xx I43m(g) !+{(xxz; xxz ) k; xxz tt }

I6z2x Im3 (g) J± (x^O^±z)kI6z4x Im3m(j) I±(x, 0, ±z)-n

IBz^xx Im3m(k) I± (Xj Xj±z k; xxz tt )

. .2I6z2x Pm3n(k) ±{x,0,±z;ir2+(±z^0,x)}K

. . 2I6z2xx Pn3m(k) {xxs^ xxz y xxz y xxZy '2 2 2*^ (xxzy xxZy xxZy xxz ^ }"

K

I6z2xy 123(f) I+(xyz;3:yz;xyz;xyz) k

I6z2xx2y Il3m(h) 1+ (xyz;xyz;xyz;o!yz)-nI6zHxy I432(j) 1+ (xyz; xyz; xyz; xyz; yxz; yxz; yxz; yxz) k

I6z2x2y Im3(h) I±(x,y3±z;x,yy±z)KI6z4x2y Im3m(l) I± (x^y, ±z;Xjyj ±z)t\

. . cI6z2xy P43n(i) {xyz; xyz; xyz; xyz; Hi+ (yxz;yxz;yxz;yxz) }k

. . 2I5z2xy P4 32(m) {xyz;xyz;xyz;xyz; \\\+(yxz;yxz;yxz;yxz) }k

n. . I5z2>Qr Pni(h) {xyz;xyz;xyz;xyz; lll+(ocyz;xyz;xyz;xyz) }k

n. . I6z4xy Pn3n(i) {xyz; xyz; xyz; xyz; yxz; yxz; yxz; yxz;

\\\+(xyz; xyz; xyz; xyz; yxz; yxz; yxz; yxz) } k

69

..2I6z2x2y PmZn(l) tfajji/j ±3;a;j^, ±z; H2+ TyjiCj ±2;yja:j ±2^}k

..2I6z2xx2y PnZm(l) {xyzjx^Sjxyz;xyss^ii+(xysjxyz;c!ysixyz)}'nd.21^6z2-xy Fd3o(h) F-^-{xyz;xyz;5yz;5yz; 111+ (yxz;yxz;yxz;yxz) ;

5 H+ Cyxz; yxz; yxz; yxz) ; 111+ (xyz; xyz; a^z; xyz) } k

Pm3m Z(c)4/mm.m \0\k

.3.J2X Tm3(f) ±(\+x,0^\)k

.3.J4X Pm3m(h) ±(l+x^0^i)T\

F FmSm 4(a)m3m 000;\0\k

F4xxx F43m(e) F+(xxx;o^xk)F6z Fm2m(e) F±(00z)kF8xxx FmZm(f) Ft (o!X)!:x;xxxk)

F12XX Fm3m(h) F± (x^±x^O)kbe. . F2xxx Pa3(a) ±(xxx) ;

{

101±(xxx)}k.FSxx P4^32(k') {xxO; 2(95+ rxxO; xOx; Oxx) } k

F6xx Pn3m(i'

)

± {xxO; rxxO; xOx; Oxx) } k

42, . F^Sxx Fd3o(g') F+ {xxO; lOl+ (xxO; xOx; Oxx) ; H 2+ TxxO)

;

Ti\l+(xxO; xOx; Oxx) } <

F5z2xx F43m(h) F+{(xxz; XXz ) k; xxz tt }

F6z2x Fm3(h) F+ (x^O, ±z)kF6z4x Fm3m(j) Ft (x^O^tz)TT

F6z4xx Fm3m(k) Fi(XjXj+s k; xxz tt )

F6z2xyF6z2xx2yF6z4xyF6z2x2yF5z4x2y

be. . F6xyz

F23(h) F+ (xyz; xyz; xyz; xyz)

K

F43m(i) F+(xyz;xyz;xyz;xyz)T\F432 (j) F+ (xyz; xyz; xyz; xyz; yxz; yxz;yxz; yxz)KFm3(i) Fi(Xjyj+s;Xjyj+z)KFm3m(l) F±(Xjyj+z;Xjyj+z)-nPa3(d) t{xyz;\0\+(xyz;zxy;yzx)}K

P4232 4 (a). 32 111 . 25 3.8 8 8.9 8 8 8 I'

Y2xxx'Y3xx

42.. Y3xx2yz

P4 32(c)P4p2(d)

P4^32(e)

8 8 8 — (XXX) ; { 8 8 st (XXX } K

{ 5H+ CxxO) ;l\l+(xxO; xOx; Oxx) } k

{ 5 5 5 + ro^jz; yxz );lll+ (xyz; zxy; yzx; yxz; zyx; xzy) } k

Pm3n 6(c)lm. 2 t(OU)<

.3.W2Z Pm3n(g){.3. J*2z}

.3.W4XX Pm3n(j)

t(0^l^l+z)K

J* Im3m 6(h)4/rm.m I+(\0\)k

.3.J*2x Im3(e) I±(1+x^0,1)k

.3.J*4x Im3m(g) It(\+x^O^\)-n

70

D Fd3m 8 (a) 43m F+(000;HV

D6z FdZm(f) F+ (0, 0, ±z; 1^ l±z) k

..2DM-XXX FdZm(e) F-^{xxx;xxxk: )}

..2D6z2xx FdZm(g) F+{xxz; + (xxz: )}k

..2D6z2xy F4^22(h) F+{xyz;xyz;xyz;ayz; Hl+(yxz;yxz;yxz;yxz)}Kd..D6z2xy Fdi(g) F+{3^jz;xyz;5yz;syz; l%l-\r(xyz;xyz;xyz;xyz)}K..2D6z2xx2y FdZmd) F+{xyz;xyz;3yz;xyz; lll+(xyz;xyz;xyz;xyz)]T^

"^Y* m^32 8(a). 32 I+(lll;iU^)

22..Y*2xxx 14 32(e) I+[Ul±(xxx); {IU±(xxx)}k]42.. Y*3xx I4^32(h) I+{lU+(xxO); iU+(xxO;xOx;Oxx)}K

22 . . Y*3xx2yz 14 ^32(i) I+{lll-\-(xyz;yxz) ; 111+ (xyz; zxy; yzx; yxz; zyx; xzy ) }k

W* ImSm 12(d)4m.2 I±(OH)<

S Ii+3d 12(a)4.. I+(OU;OU)k

.3.S2Z I43d(d) I+(0,1,1+z;0,1,\±z)k

. 3dS4xyz I43d(e) I+{Oli+ (xyz;xyz;yxz;yxz) ; OII+ (yxz;yxz;xyz;xyz) }k

"^V 14^32 12(0)2.22 I+(OH;OU)^

.3.V2Z I4^32(f) I+(0,l,l±z;0,i,l±z)K

T Fd3m 16(o).3m F+(U\;UU)

22..T3XX F4 32(g) F+{lU^(xxO) ;IU+(xxO;xOx;Oxx)}k4..T6XX FdZm(h) F+-{lll±(xxO) ; UI±(xxO;xOx;Oxx)]k

Y** Ia3d 16(b). 32 ItdlUiU^)

4..Y**2xxx Ia3d(e) I±[lH±(xxx) ; {Ui + (xxx) }k]

{4. . I 22xxx}4a..Y**3xx laZd(g) I+{UU(xxO) ;IU+(xxO;xOx;Oxx)}k

4a. .Y**3xx2yz Ia3d(h) I±{\\1+ (o^jz;yxz) ; l^l-\r(xyz;zxy;yzx;yxz;zyx;xzy)}K{4.2l26xyz}

S* laSd 24(d)4.. I±(OU;OU)k

.3.S*2z Ia3d(f) I±(0,l,l±z;0,l,l±z)K{.3.V*2z}

71

V* Ia3d 24(c) 2. 22 I±(OU;Oli)>i

2^2^..FYlxxx P2^3 4 (a). 3. XXX; {\0\-i-(XXX ) } K

2^2^. .FYlxxx3yz P2^3(b) {xyz; \0\+(xyz;yzx;zxy ) }k

2^2^. .P^Y^lxxx 12^3 8(a). 3. J+ [xxx; { (xxx) } k)

2^2^. .P^Y^lxxxSyz I2^3(c) I+{xyz; \0\+ (xyz;yzx; zxy) } K

2^3,SVlz 12^3 12(b)2.

.

5j 8+2j 5 J s~z) K

I43d 16(c). 3. X+ { 2C3CCCy 1+ u 4 "f" (cCCCCC ) ^

{ lOl+(xxx) ; lH+(xxx) }k]

. 3.J2S*V*lx Ia3 24(d) 2.. I±(l+x^O, I; l+x, ^^V <

Table 19 : Hexagonal System

(and rhombohedral system in hexagonal description)

P P5/mmm l(a)6/mmm 000

P2z P6/rrmn(e) 0^0, ±z

P3xx P6m2(j) xxO;Xj2XjO; 2x,Xj,0

P3x P62m(f) xOO;OxO;xxOP6x P6/mmm(j) ±(xOO;OxO;xxO)P6xx PB/mmmd) ±(xxOjXj2x,0; 2XjXjO)

. . 2P 3X P6 /mom (g ' ) xOO; 0x0; xxO; 00\+(xOO; 0x0; xxO

)

. 2 . P^E3xx P6 /mmc (h ' )xxO; x^ 2x, 0; 2x, x, 0; 001+ (xxO; 2x, 0; 2x^ x, 0)

3 . .P^+Q2x P6 22(g) ±(xOO) ;00\±(xxO) ;00\±(OxO)32..P^+Q2xx P6^22(i') ±(xxO) ;00\±(2x,x,0);00\±(x,2x,0)

P3xy P6(3) xyO;y,x-y,0;y-x^x^OP3xx2y Pdm2(l) xyO;yjX-y3 0;y-XjXjO;yxO;y-Xjy,0;x,x-y,0P3xx2z P6m2(n)P3x2z PB2m(i) x,0,±z;0,x,±z;x,x,±zP3x2y P62m(j) (xy0;y,x-y,0;y-x^x,0) t

P6xy P6/m(j) ±(xyO;y,x-y,0;y-x,x,0)P5xz P31m(k) ±(xOz;Oxz;xxz)P6xxz P3ml(i) ±(xxz;Xj2XjZ; 2XjXjZ)P6x2z P6/mmi(n) ±(XjOj±z;0^x^±z;XjX^±z)P6xx2z P6/mmm(o) ±(x,x,±z;x,2x,±z;2x,Xj±z)

72

P6x2y. .2P 3xy.2.P'^E3xy

2 ..?^E3yy3^..P^+Q2xy72.P 6xy. .2P'^3x2y

. .2P^3x2z{m. .P6xz}

.2.P^E3xx2y

.2.P E3xx2zc

{m. . P6xxz}

P6/mrm(p)P6c2(k')P62c(h')P6 /m(h')

Pefmccd)P6 /mcm(j')P6ymam(k')

P6yrmo(j')

P6 /mmo(k')

±(xyOjiijX-y^O;y-x,x,0)TxyO; z/j x-y, Oj y-x, x, 0; 001+ (yxO; y-Xj y, 0; x, x-y, 0)

xyO; y^ x-y, 0; y-x, x, 0; 00\+ (yxO; x-y, y, 0; x, y-x, 0)

xyO; y^ x-y, 0; y-Xj x, 0; 00\+ (xyO; y, y-x, 0; x-y,x, 0)

± (xyO) ;00i+ (y-x, x, 0) ; 00i± (y, x-y, 0)+ (xyO;y, x-y, 0; y-x, x,0);00\± (yxO; x-y, y, 0; x, y-x, 0)

{xyO; y, x-y, 0; y-x, x, 0; 00i+ (xyO; y-x, y, 0; x, x-y, 0)}t

X, 0, ±z;0,x, +z;x,x, ±2;00l+(x, 0, ±z;0, x, ±z;x,x, +z)

xyO; y,jc-y, 0; y-x, x, 0; yxO; y-x, y, 0; x,jc-y,J);

001+ (xyO; y, y-x, 0; x-y, x, 0; yxO; x-y, y, 0; x, y-x, 0)

X, X, ±z; X, 2x, ±z; 2x, x, ±z; 00\+ (x, x, ±z; x, 2x, ±z;

2x, X, ±z

)

P3xy2zP3xx2yzP3x2yzP6xyzP3xx2y2z

P3x2y2zP6x2yz

P6xy2zP6xz2yP6xxz2yP6x2y2z

. .2P 3xy2z{m. .P 3xx2yz}

.2.P E3xy2z{m..P 3x2yz}

32..P^'^Q2x2yz

. .2P 3x2yz{?2.P E3xx2yz}

2^. .P E3xy2z{m. . P^6xyz}

. .cP E3xx2yz{?.2P 6xyz}

.n.P 3x2yz{?2.P 6xyz}

m. . P 5x2yz{?2.P 6xy2z}

c

. 2P 3x2y2z{m. .Pj,6xz2y}

P6(l)PS12(l)P321 (g)

PZjg)P6m2(o)

P62m(l)P622(n)

P6/m(l)PZlmd)PZml (j)

P6_/mnm(r)

P6c2(l')

P62a(i')

P6^22 (k)

P6,22 (i)

Pe^md')

P21a(i')

PZcl(g')

P6/maa (m'

)

PB^mcmd')

.2.P^E3x^2y2z PB^rmod'){m. . P^6xxz2y}

x,y,±z;y,x-y,±z;y-x,x,jLzxyz; y, x-y, z; y-x, x, z; yxzj y-x, y, zj x, x-y, z

xyz; y, x-y, z; y-x, x, zj yxz; x-y, y, z; x, y-x, z

+ (xyz;y,xry,z;y-x,x,zjX, y, ±z; y, xry, +z; y-x, x, ±z; y, x, ±z; y-x, y, ±z;

x,3^y,±z^(X, y,_±z; y, s^y, ±z;y-x,_x, ±z ) x

xyzj y, x-y, zj y-x, x, zj xyzj y, y-x, z; x-y, x, z;

yxz; x-y, y,j.; x, y-x, z; yxz; y-x, y, z; x, x-y, z

±(x,y, ±z; y, x-y, ±z; y-x, x, ±z )

±(xyz;y,x-y,z;y-x,x,z)T± (xyz; y, x-y, z; y-x, x, z; yxz; x-y, y, z; x, y-x, z)

±(x,y,±z;y,x-y,±z;y-x,x,±z)TX, y, ±z; y, x-y, ±z; y-x, x, ±z; 00\+ (y, x, ±z; y-x, y, ±z;

X, x-y, ±z)

X, y, ±z; y, x-y, ±z; y-x, x, ±z; 00\+ (y, x, ±z; x-y, y, +z;

x,y-x,±z)xyz; xyz;x-y,y, z_;y-x,y, z_; 001+ (y-x, x^, z; x-y,x,z_;_X, y-x, z; X, 3:^y, z) ;00\+(y, x-y, z;y, y-x, z; yxz, yxz

)

xyz; y, x-y, z; y-x, x, z; yxz; x-y^y, z; x, y-x, z;

00\+ (xyz; y, y-x, z; x-y, x, z; yxz; y-x, y, z; x, x-y, z)

^, y> ±3; y, xry, ±z; y-x, x, ±z; 00\+ (x, y, ±z; y, y-x, ±z;

x-y,x, ±z)

xyz; y, x-y, z; y-x, x, z; yxz; y-x, y, z; x, x-y, z;

00\ + (yxz; x^y, y, z;x, y-x,z;xyz;y, y-x, z; x-y, x, z

)

xyz; y, x-y, z; y-x, x, z; yxz; x-y_,y, z; x, y-x, z;

00\+(yxz; y-x, y, z;x, x-y, z; xyz, y, y-x, z; x-y, x,j

)

xyz; y, x-y, z; y-x, x,_z; xyz; y, y-x, z; x-y, x, z; yxz;^

x-yJ 2; X, y-x, ZJ yxz; y-x, y, z; x, x-y, z; 001+ (xyz;

y, x-y, z; y-x, x, z; xyz; y, y-x, z; x-y, x, z; yxz; x-y, y, z;

X, y-x, z; yxz; y-x, y, z; x, x-y, z )

(X, y, ±z; y, x-y, ±z; y-x, x, +z) x; 00\+ (x, y, ±z; y, y-x, ±z;

x-y,x,±z)T

^1 y> ±2; y, x-y, +z; y-x, x,jLz; y, x, ±zj y-x, y, ±z; x, x-y, ±z;

001+ (y^ X, ±z; x-y, y, ±z; x, y-x, ±z; x, y, ±z; y, y-x, ±z;

x-yjX, ±z)

P6/mmm 2(o)6m2 ±(\\0)

G2z P6/iwrm(h)

73

P6 /mineo

2(o)6m2 - ( 3 3 U/

E2z P6^/mma(f)

R3Tn 3 fa J 3/77.

R2zR3xxR3xR6x

.n. 'R 3xc

RSxyRBxxzR3xx2y

.n. 'R^3xy

000;±(m)

R3m(c)RZm(h')R32 (d)

RZm(f)R3o (e')

R3(b')RZm(h)R3m(a')R3o(b')

R+(00z)R+ (xxO; X, 2Xi 0; 2x, x^O)R+ (xOO; 0x0; xxO)R± (xOO; 0x0; xxO)

R+{xOO; 0x0; xxO; 00\\ (xOO; 0x0; xxO)

}

R+ (xyO; x-y^ 0; y-x, x, 0)

R± (xxz; X, 2x, z; 2x, x,z)R+ (xyO; y, x-y, 0; y-x, x, 0; yxO; y-x, y, 0; x, x-y, 0)

R+ { xyO; y, x-y, 0; y-x, x, 0; 001+ (yxO; y-x,y, 0;x,x-y, 0)

}

R3x2yz

R6xyzR6xxz2y

,n. 'R 3x2yz{ .5. 'R 6xyz}

R32(f)

Rl(f)R3m(i)

R3o(f')

R+ (xyz;y, x-y, z; y-x, x, z; yxz; x-y, y, z;

X, y-x, z

)

R± (xyz; y, x-y, z; y-x, x,z)

R+ (xyz;y, x^y, z; y-x, x, z; yxz; x-y, y, z;

X, y-x, z

)

R+{xyzjJi, x-y, z; y-x, x, z; yxz;^x-y, y, z; x, y-x, z;

001+ (yxz; y-x, y, z; x, x-y, z; xyz; y, y-x, z;x-y,x,z)

}

N P5/mmm 3(f)mrm \00;0\0;\\0

N2z P6/rmm(i) \,0,±z;0, \,±z;\,\,±z

3(c) 222 \00;0\l;\\\

Q2z P6^22 (f) 2,0, ±z; 0, 2, \-z; \, 2 J \-z

R3m 9(e).2/m. R+(\00;010;\\0)

,2.GElz P3ml 2(d)3.m ±(\lz)

P3212 3(a')..2

.P^'''Qlxx2yz P3^12(a') xyz; yxz; 00\+(y, x-y, z; x, x-y, z)

;

00\+(y-x, X, z; y-x, y,z)

32-. P^ Qlx P3221 3(a').2. xOO; 0x%;xx\

32-.?^ Qlx2yz P3^21(c') xyz; x-y, y;_z; 001+ (y, x-y, z; yxz )

;

00\+ (y-x, X, z; x, y-x, z)

74

3,.2P^"""Q Ix P6,22

1 Cc c 16(a) .2. xOO; xxl; Oxi; ocOj; xxl; Oxi

3 .2P "^Q lx2yz P6 22(a){3^2.g^^E^+Qlxx2yz}

xyz; x-y, y, z; 001+ (x-y , x, s; x, x-y^z); 00\+(y>x-y, z;yxz) ; 001+ (xyz;y-Xjy, z) ;00l+(y-x, x,

Xi y-x^z) ; 00\+(y,y-x, z;yxz)

3,2.P- e/q Ixx P6.221 Cc C c 1

6 Cb ) » • 2 ooooO^ 2cCj 6 J 2cc^ 3 j cccc 2^

23(j^ cc^ 3 J cc^ 2cc^ 6

3^ . . Pj^"'"Qlxy P32 3(a')l xyO;y,x-y,\;y-x,x^\

''1 6(a')l xyO; x-y^ X, I; y, x-y^ \; xyl;— 2 5

2/— Xj 3 z/j y^x^ 6

Table 2o: Rhombohedral complexes in rhombohedral description

p P3m 1 (a) 3m 000

P2xxxP3yP3xxP6xx

.nI3xx

P3m(a)P3m(b')P32 (d)

P3m(f)P3o(e ')

± (xxx)00ZKxxOk± (XXO)

K

{xxO; 1^^+(xxO)}k

P3yzP6xxzP3y2z

.nI3yz

P31(b')P3m(h)P3m(c')P3c(b')

OyzK± (xxz ) K

OyzT\

{Oyz;\\\+(Ozy)]K

P3xx2yzP6xyzP6xxz2y

.nI3xx2yz{.2I6xyz}

P32 (f)P31 (f)PZm(i)

P3c(f')

(xyz; xzy) k

±(xyz)K±(xyz)-n

{xyz;xzy; \l\+(xzy;xyz) }k

J P3in 3(e). 2/m '^0\k

501-563 0-73-6

75

Table 21: Tetragonal System

P4/mmm 1 (a)4/mmm 000

.2

P2zPHxxPlx

.b.C2xx

. .2P I2x

.2.P^2xx

.b.C*^2xx

P4XXZP4xzP4xyP4x2yP4xx2zP4x2z.P I2xy,P%xy

. . 2c5xxz

. .mC4xy

{ .b.C2xx2y}.b.C2xx2z. . mCUxz

{. .2CIlz4x}m. . P 4-xz

{?.2P I2x2z}..2P I2x§y.2.P^2xx2ym. .P i+xxz

{?2.P 2xx2z}.b2C 2xy^, .mC 4-xy

{?2.C 2xx2y}.b.C 4xxz

{?b.C 2xx2z}c

P4xyzP4xxz2yPHxz2yP4x2yzP4xy2zP4x2y2z

.2.Pc4x7z

. .mC^xyz

{. 2i.CIlz2xx2y}..2Pcifxyz

{4^. .P^2xx2yz}. .2C'4xyz

{.b.C2xx2yz}.2 .C2xx2yz

{..2CIlz4xy}

,P I2x2yz}c '

P4/rnmm(g)

P4/mmm(j)P4/mmm(l)P4/mbm(g')P4^/mmo(Q)P4 /mom(i)I47mom(h'

)

P42m(n)P4m2(j)P4/m(j)P4/rnmm(p)

P4/mmm(r>)

P4/rrmn(s)

P4Mj)P47moo(m)P4/nbm(m'

)

P4/rribm(i)

P4/mbm(k')P4/nim('C)

P4^/rmc(o'

)

P4^/rmc(q)P4^/mam(n)P4^/mam(o ')

P4jmbo(h)I4/mam(k)

±(00z)± (Xj ±Xj 0)

± (xOO) T

±(3:x0);\\0±(xx0)+ (x00;0x\)±(xxO;xx\)I± (xxO;xx\)

xxz;xxz;xxzTtXj Oj z; 0, ±Xj z

±(xyO;yxO)± (xyO; yxO) t

± (Xf Xj ±z; Xj ±2 )

±(x^0, +z)t±(xyO;yx\)±(xyO;yxO;xyl;_yxl)xxz; xxz; xxz t; i\0+(xxz; xxz; xxz x )

± (xyO; yxO); \ \0± (xyO; yxO)

±(XjXj±z);\ \0± (x^ X, ±z

)

iXj Oj z; ixCj z; ii0+(0j ±Xj z; ±Xj 0,z)

iXj z; 0^ +Xj z; 001+ (±Xj 0^ z; 0, tXj z)

±{xyO;xyO;yx\;yx\)±(xyO;xyl)Txxz; xxz; xxzt; 001+ (xxz; xxz; xxzx)

±(xyO) ;\\0±(xyO) ;OOl+(yxO) ;\\\±(yxO)I±(xy 0; yxO; xy\;yx\)

I4/mcm(l' ) 1+ {xxz; xxz; xxzt; \\0+ (xxzt; xxz; xxz) }

P4(h)P42m(o)P4m2('l)

P422(p)P4/m(l)P4/mmm (u)

P42a(n)

P42fi(f)

Plo2(3)

Plb2(i)

P42^2(g)

xyz; xyz; yxz; yxz(xyz; xyz; yxz; yxz

)

x

x^ ±y^ z; x^ ±y^ z;_y^ ±x, z; y^ ±x^ z

xyz;xyz;xyz; xyz;yxz;yxz; yxz; yxz±(Xjy^ ±z;y,x,±z)tfXjj/j ±z;y^x,±z)Txyz; xyz; yxz; yxz; 00\+ (xyz; xyz; yxz; yxz

)

xyz; xyz; yxz; yxz; 5 lO+ (yxz; yxz; xyz; xyz )

xyz; xyz; yxz; yxz; 00\+ (yxz; yxz; xyz; xyz )

xyz; xyz; yxz; yxz; \\0+(yxz; yxz; xyz; xyz

)

xyz; xyz; yxz; yxz; 2 10+ (xyz; xyz; yxz;yxz)

76

..2P^I2x2yz P4^22(p){.2.P^2xx2yz}

m. .P^i+xyz P4jm(k').? 2xy2z}

lCi4xyz P4/n(g){ICIlz^xy}

m, .P^4x2yz P4/moo(n'

)

{.2.P 4xy2z}..mC4x2yz P4/nbm(n)

{

.

.2C4xxz2y}..mC4xy2z P4/mbm(l)

{ .b.C2xx2y2z}..TnC4xz2y P4/nrm(k)

{..2CIlz4x2y}. .c2C 4xyz P4/noc(g)

{?2^cC 2xx2yz}{1.2(c5lz) i+xy}

m. .P^M-xz2y

{. .2P I2x2y2z}m. .P 4xxz2y

{?2.P 2xx2y2z}.22C i4xyz

{?b2C 2x2yz}

{ .2cC^2xx2yz}m.2C 4-xyz

{?b2C 2xy2z}{nib.C^2xx2yz}

. .m2C Hxyz{?2 mC 2xx2yz}{4. rF2xxz2y}{.2^2(CIlz)^2xx2y}

. . 2C Hxyz I4c2(i)2xx2yz}

.b.C 4-x2yz I4/mcm(m'

)

{?b.C 4xxz2y}{.b.c'^4xy2z}

{.b.c"^2xx2y2z}c '

P4^/mmc(r'

)

P4^/mcm(p'

)

P4^/nho(k)

P4^/mbc(i ')

P4^/nam(j)

xyz; xyz; xyz; xyz; 00\+ (yxz; yxz;yxz; yxz

)

xyz; xyz; yxz; yxz; 001+ (xyz; xyz; yxz; yxz

)

xyz; xyz;yxz; yxz; \ lO+ (xyz; xyz; yxz; yxz)

xyz; xyz; xyz; xyz; yxz; yxz; yxz; yxz;001+ (xyz; xyz; xyz; xyz; yxz; yxz; yxz ; yxz

)

xyz; xyz; xyz; xyz; yxz; yxz; yxz; yxz;

5 iO+ (yxz; yxz; yxz; yxz; xyz; xyz; xyz; xyz)±(x,y, +z;y^x, +z); \\0±(y,x, ±z;x,y, ±z)

X, +y_, z; X, ±y, z; y, ±x, z; y, +x, z; l\0+(±y,x, z;±y,x,^zj±x,y^,zj±x,y,z)

xyz; o:yz;yxz; yxz; \ \0+ (xyz; xyz; yxz; yxz )

;

00\+(yxz;yxz;xyz;xyz) ; m+ (yxz; yxz; xyz; xyz)

X, ±y, z; X, ±y, z;y^,

+x^ z; y, ±x, z; 00\+ (x, ±y , z;

x,±y,zjy,±x_,z;y,±x,z)(xyz;xyz;yxz;yxz)T;00\+(xxjz;'xyz; yxz;yxz)x

xyz; xyz; yxz; yxz; \ \0+ (yxz; yxz; xyz; xyz )

;

00\ + (xyz; xyz; yxz; yxz ) ;\\\+(yxz; yxz; xyz; xyz)

oryz; xyz; yxz; yxz; \ \0+(yxz; yxz; xyz; xyz)

;

001+ (xyz; xyz; yxz; yxz ) ; ^l^+ (yxz; yxz; xyz; xyz)

xyz; xyz; yxz; yxz; 2 5O+ (yxz; yxz; xyz; xyz);00\+ (yxz; yxz; xyz; xyz) ; H 2+ (xyz; xyz; yxz; yxz

)

I+{xyz;xyz;yxz;yxz; \\0+ (xyz; xyz; yxz; yxz)

}

I+{xyz; xyz; xyz; xyz; yxz; yxz; yxz; yxz;001+ (xyz; xyz; xyz; xyz; yxz; yxz; yxz;yxz) )

m/mmm 2(a)4/mrim 000;\\\

I2z I4/mmm (e) I±(00z)I4xx I4/mrm(h) I± (x, ±x^ 0)

mx I4/mmm(i) It (xOO) T

. n. I2xx P4 /rmm(f) ±(xxO); l\\±(xxO)

.2.F 2xxc

I4yacd(f') I+{+ (xxO); \ \0+ (xxO); \0l± (xxO); 0\ l± (xxO) }

IHxz I4m2(i) 1+ ( ±x, 0, z; 0, ±x, z )

IHxxz I42m(i) 1+ (xxz; xxz; xxz x )

mxy I4/m(h) I± (xyO;yxO)I4x2y I4/rnrm(l) I+(xyO;yxO)TI4xx2z I4/rrmn (m) It (cCj Xj tz; X, Xj tz )

I4x2z I4/mrm(n) It (x, 0,tz)i:. .2I4xy P4/mnc(h) '± (xyO; yxO) ;\\\t (yxO;xijO). . 2I4XXZ P4ynnm (m) 32CCZ ^ CCCCS J CCCCS T ^ 222"^ (cCCCZ j OCXS j CCCCS T )

.n. I2xx2y P4^rmmd) t(xyO) T;V2\±(xyO)T

77

.n.I2xx2z

. . cli+xz

5. . F2xxz{.2 2(CIlz) 2xx}

.bdF 2xy^

c ^

IM-xyz

I'4xz2y

I4xxz2yIi+x2yz

I4xy2zI4x2y2z

. . cI4-xyz

..2I4xyz

{ .n. I2xx2yz},2^. I2xx2yzn . . I4xyz. . cI4x2yz

. ,2mxy2z

. ,2I4xxz2y•n. I2xx2y2z, . cI4xz2y

. 22F 4xyz{?b2.F^2xx2yz}

P4ymnm(j )

P4yrmc (g)

P4ynom(i'

)

I4^od(b')

14(g)I4m2 (j)

I42m(j)1422 (k)

I4/m(i)I4/mmm(o)P4_2 o(e)P4n2(i)

P4 2 2(g)P4jn(g)P4/nnc(k)

P4/rma(i)P4 /nnm(n)P4yrmm(k)P4ynmo (h)

I4yaad(g)

±(x,X3 ±z); lll±(x,x, ±z)

+x, (9j z; ±Xj z;\\\+(Oj tx^ z; ±Xj 0,z)± (xxz);ilO±(XXz );lOl+(xxz);0l\+ (xxz

)

1+ { ± (xyO) ;l\0± (xyO) ;lOl± (yxO) ;0\l± (yxO) }

1+ (xyz; xyz; yxz; yxz)-ft (X, +V J z ; X, ±y,z;y,±x,z;y3±x,z)J+ (xyz; xyz; yxz; yxz ) x

J+ (xyz; xyz; xyz; xyz; yxz; yxz; yxz; yxz)I± (x,y, ±z;y,x, ±z)

I± ( y, ±z; y,x,±z)Txyz; xyz; yxz; yxz; l\\+(yxz; yxz; xyz;xyz)xyz; xyz; yxz; yxz; \\\+ (yxz; yxz; xyz; xyz

)

xyz; xyz; yxz; yxz; 555+ (xyz;xyz;yxz;yxz)xyz; xyz; yxz; yxz; \\\+(xyz; xyz; yxz; yxz

)

xyz; xyz; xyz; xyz; yxz; yxz; yxz; yxz;

5H+ (yxz; yxz; yxz; yxz; xyz; xyz; xyz; xyz

)

± (x, y^J-z; y,jc, ±z)j\\l±(y, x, ±z; x, y, ±z)

(xyz; xyz; yxz; yxz) i;\\\\(yxz; yxz; xyz; xyz ) x

±(x,y,±zJ\;\\\±(x,y,±zjTx^ ±y, z; X, +y, z; y^ +Xj, z; y, +x, z; 111+ (y, z;

y,±x,^zjx,±y^,zjx,±y,z)

xyz; xyz; yxz; yxz; \ (yxz; yxz; xyz; xyz )

;

\0l+ (xyz; xyz; yxz; yxz) ; Oi 5+ ('yxz; yxz; xyz; xyz)

D F4^/ddm 8 (a) 42m F+(000;UI)

14^/amd 4(a)4m2 I+(000;OU)

VD2z

^D4xx. . d"^D2x

.2,^D2xx

I4yamd(e)I4^/amd(g)14 md(h')14y2(d)

I+(0, 0, ±z;0^l^l±z)I+{±(x,±x^O);OU± (x, ±x, 0) }

I+{+(xOO);0\l±(OxO)}1+ { ± (xxO) ;Oll± (xxO) }

1 V4^. . D2x71 V.2. DHxz. .d^D2x2y

14 (b')

14 /amd(h)I4^r"d(G')

I+{±(xyO);0\l±(yxO)]1+ { ±x, 0, z; 0, ±x, z; 0\ 1+ ( ±sc, 0, z; 0, ^x^z)}1+ {

±

(x, ±y, 0);Oll±(±y, x, 0) }

.2.'^D4xyz

.2."^D2xx2yz

a. . D4xyz.2.^D4xz2y

I42d(e)14 22(g)

I4\/a(f)I4yamd(i)

1+ {xyz; xyz; yxz; yxz; 0\ 1+ (xyz; xyz; yxz; yxz)

}

I+{xyz; xyz; yxz; yxz; 0H+ Txyz;xyz;yxz; yxz) }

I+{xyz; xyz; yxz; yxz; 011+ (xyz; xyz; yxz; yxz ) }

I+{x, ±y, z; X, ±y, z; y^ tx^ z; y^ ±Xj z; 0H+ (x, ±y, z;

X, ±2/ J z;y^ ±x, z; y, ±x^z)}

T F4^/ddm

14^/amd

16(c).. 2/m

8(c).2/m.

Pj. / 111. 313)CTl 88 8j 888'!/

1+ rO^gj 0^^; ^Oj; ^Oj)

. .2^T2x I4yamd(f) 1+ (iXj Uj 8 J iXj 5j 8j 5j —Xj 8j iXj s)

78

. .2CIlz P4/nmm 2(o' )4rm 00z;\\z

.2^.CIlz2xx P42^m(e') lO+ (XXZT

)

..2CIlz'+xx P4/nrm(j') xxz;xxz;xxzt;\10+(xxzt;xxz;xxz)

4^. .P I Ix P4„223 cc c 3

4(a' ).2. xOO; 0x1; xOl; 0x1

4-. .P I lx2yz P4^22(d')^ {5^.?P ^Dlxx2yz}

^xyz; xyz; 00i+(yxz; yxz ) ;00\+(xyz; xyz ) ;

001+ (yxz; yxz)

4^..P ^Dlxx PU^223 cc 3

4^. . I ^Dlxx3 c

P432^2 4 (a)..

2

lU+(xx0)

. I^^Dlxx2yz P4^2^2(b) xyz; yxz; 5H+ Tyxz; xyz );00\+ (xyz; yxz )

;

\\l+(yxz;xyz)

4..^TF Ixc

I42d 8(d). 2. -Z^+ (X u Q y X 83' U'^' 8 J 8 )

. . 22^TC Ixcc

14^22 8(f). 2. X+ (X L, Q y X Q y iiX Q y nX 8 )

^••Vc2l^ m^/acd 16(e').2. 1+ (xOO; x\ \; xO\; x\0;1 1.3 3, 1.3 1, 1.

I 3_ 11

^g.-P^^ Dl^lxy P4g 4(a')l xyO;yxl;xy\;yxl

Table 22: Orthorhombic System

P Pmmm l(a)mmm 000

P2x Pmmm(j) ±(x00)

P2y2z Pmmm(u) ±(0^y^±z)2..P 2>Q^ Pccm(q) +(xyO) ;±(3:y\) .

m. .P^2xz Prma(i) +(xOz);\00±(xOz){?2.P^Blz2x}

.m.Pgj^2yz Crma(m) C±(Oyz;\yz){2..P Flz2y}

79

P2x2yzP2x2y2z

.mP 2x2yz{5. .P 2xy2z}

m. .P,2xz2y{72. P Blz2x2y}

,22P 2xyz{c?2P Fly2xz}{12.(f^Blz) 2xy}

2.inP 2xyz^

{.§jin(P Clx) 2yz}{2.1P^ A^C ^^lxy2z}

be Td cP^l-Sx^yz{ .m.P 2yz2x}{m. .P 2xz2y}{2. .P^Flz2x2y}P22xyr{:22(P Fix) 2yz}{2.2(P^^Fly)^2xz}{22.(P^^Flz)^2xy}

ab c

m

22

P222(u) xyz;xyz;xyz;xyz

Pcom (r' ) xyz;xyz; xyz; xyz; 00\+ (xyz; xyz; xyz; xyz)

Pmma(l) ±(x^±y^z); iOO+(x^±y,z)

Poaa (f) ±(xyz);lOO±(xyz) ;00\±(xyz);lO\±(xyz)

Pbcm(e) ±(xyz) ;OlO±(xyz);00\ + (xyz) ;Ol\+(xiyz)

Crma(o') C+{xyz;xyz;xyz;xyz; lOO+(xyz;xyz;3yz;xyz) }

Iboa(f) 1+ { ± (xyz ) ; loot (xyz); 0\0± (xyz );00\±(xyz)}

Cmmm 2(a)mm 000; l\0

C2xC2z

Cnvm(g)Cmmm(k)

C± (xOO)

C±(00z)

C2y2zC2x2y

.2.B2yzb. .C2xy.n.C 2yz

{§. .C Fly2z}n. .C F2xyb. .C^2xy.2.B^2yz

{.2.B^A lz2y}b a -'

C2x2yzC2x2y2z

b. .C2x2yz2.2A 2xyz

{?/n. .B A lz2xy}{. .2/aB^F5x2yz}

.2.B2yz2xb. .C2xy2zb2,C^2xyz

.maB^2xyz{1.2,B A FI lxz2y}

.n.C 2yz2x ^ ^

n. .C^2x2yz{n. .C F2xy2z}

b. ."C 2x2yz{%. .C 2xy2z}

Cmrm(n)Cmmm(p)Pmna(h)Pbam(g)Cmam (f)

Ccom (I)

Ibam (j)

Imma(h)

C222(l)Cmrm(v)Pban(m)Pnna (e)

Prma(i)Pbam(i)Pbcn (d)

Pnma (d)

Cmcm(h)Cccm (m'

)

C±(0,y,±z)C± (xj, ±y, 0)

±(0yz);\0\±(0yz)±(xyO);UO±(xyO)C+{±(0yz);00l±(0yz)}

C±(xyO;xy\)I± (xyO;xy\)I+{±(0yz);0l0±(0yz)}

C+ (xyz;xyz;xyz;3yyz)C±(x,y,±z;x,y_^_+z)

xyz;xyz;xyz;xyz;WO-¥ (xyz; xyz; xyz; xyz)

±(xyz);0\\±(xiiz) ; hOO+ (xiiz J ; U ii/^ys;

±(±x,y,z);lO\±(±x,y^z)±(x,y, ±z);\\0±(x:,y,±z)

±(xyz); \\0±(xyz);00\±(xyz) ; \\\±(xyz)

±(xyz); lO\±(xyz);0\0±(xyz);\ll±(xyz)

C+{±(±x,y,_z)_;iJl±(+x,y,z)_}C+{xyz;xyz;xyz;xyz; m+(xyz; xyz; xyz; xyz) }

Ibam(k^ ) 1+ {xyz; xyz; xyz; xyz; 1^0+ (xyz; xyz; xyz; xyz)}

80

,2.B 2yz2x{2. .A 2xz2y}{.2.B^A Iz2x2y}

b a

ImmaCj) I+{±(±x,y,z);lO\±(±x^y,z)]

Immm 2(a)mmm 000;\\\

I2x

I2y2zn. . I2xy

I2x2yzI2x2y2z

n. . I2x2yzn. • I2xy2z

Irrmn(e)

Imrm(t)Pnnm(g)

1222 (k)

Immm(o)Pnnn (m)

Pnnm(h)

I± (xOO)

l±(0,y,±z)±(xyO);\\\±(xyO)

J+ (xyz; xyz; xyz; xyz

)

I±(x,y,±z;x,y,±z)xyz; xyz; xyz; xyz; 1^1+ (xyz; xyz; xyz; xyz )

±(x,y, ±z) ;\\\ + (x^y,±z)

Fmmm 4(a)mrrm 000; OUk

F2x Fmmm(g)

F2y2z Fmrm(m).2.F2yz Cmaa(f)

F2x2yz F222(k)F2x2y2z Fmmm(p)

c.2F2xyz Pcon(e)

{12-L.(CIlz) 2xy}bc.F2xyz ^ Pbca(c).2.F2yz2x Cmca(g)c..F2x2yz Caaa(i)

F± (xOO)

F±(0,y,±z)C+{±(0yz);\0\±(0yz)}

F+ (o^yz; xyz; xyz; xyz

)

F± CxjZ/j ±z;x,y^_+z)

±(xyz);0\\±(xyz); \Ol+(xyz); l\0±(xyz)

±(xyz) ;0\l+(xyz); lOl±(xyz) ; HOtfxyz)C+{±(±x,y,_z)_;lOH(±x,y^z)}C+{xyz;xyz;xyz;xyz; \0\+(xyz;xyz;xyz;xyz) }

D2x

d. .D2xy

d, ,D2x2yz

Fddd 8(a)222 F+(000;lU)

Fddd (e) Ft {

±

(xOO) ;lll± (xOO) }

Fdd2(b') F^{±(xyO);Ul+(xyO)}

Fddd(h) F+ {xyz; xyz; xyz; xyz; Till+(xyz;xyz;o!yz;xyz) }

Fddd 16(c)l Vj. fill • 111^ )TTl 88 8j 888''^/

.2.P Blz Pmma3.

2(e' )rm2 OOz; jOz

.2.P^Blz2y

.2.P^Blx2yz{2. .P^Aly2xz}

Pmma(k '

)

P222^(e)

o,±y,z; I:, ±y:,z

xyz; xyz; 00\+ (xyz; xyz )

81

.2^. .CIlz Pmmn 2(a)mm2 00z;\\z

2^. .CIlz2y Pmmn(e) 0,^yyz;\,\±y,z

2 . .CIlz2xy

2^, .CIlz2x2yP2^2 2(c)PmmnJcf

)

xyz;xys; ^lOi-(xyz;xyz)

—y J '^'j •A'j J *^ J 2 2*-^ ' I —i7 J "J ^ +7

2, . .C Fly1 c

Cmcm 4(c')m2m C+(OyO;Oyl)

2...C Fly2x1 c

Cmom(g'

)

C+(±X:,y,0;±x,y,\)

.2 .C Flx2yz(5 C Flv2xz)

1 c

C222^(o) C+{xyz;xyz; l^l+(xyz;xyz)}

2. .P ^Flzab

Cmma 4(g')rrm2 C+(00z;0\z)

.2.B^A Izb a

Imma 4(e')rm2 I+(00z;0\z)

. .2C^Bjjlx2yz 12 2 2Jd') I+{xyz;xyz; \\0+ (xyz;xyz)}{2.. A C ly2xz}

{.2.BfA'^lz2xy}D a

Pbcm 4(d')..m xyO;xy\;x, \^y,0;Xy 1-77 12 y 3 2

1.2,B, A FI Ixz1 b a a

Pnma 4(c').m. XOZ J 2 S J 2 "^Xy 0^ 3l—y. 1 1

2 't'j 2J 2 +2

2, 2,. FA B.C I LI Ixyz11 a b c a D c ^ P2,2,2,111 4(a)l xyz^ \~X3y 3 \+zj \+x,

Xj i+y } 2~z3;

Table 23: Monoclinic System

P2/m 1(a) 2/m

P2y P2/m(i) ±(OyO)

P2xz P2/m(m) ±(xOz)

P2xz2ymPj32xyz

{2^Pj^ACIlxz2y}2P 2xyz{cP Aly2xz}

P2/m(o)

P2^/m(f)

P2/a(g)

±(X3 ty^z)±(xyz);OlO±(xyz)

±(xyz) ;00l±(xyz)

82

C C2/m 2(a)2/m 000;\\0

C2y C2/m(g) C±(OyO)

C2xz C2/m(i) C±(xOz)

C2xz2y C2/m(j) C±(x,±y,z)cA2xyz P2 /c(e) ±(xyz) ;Ol^±(xyz)

2 CcF2xyz C27o(f) C+{±(xyz) ;0\\±(xyz)}{lC^Fly2xz}

cP Alyc

P2/c 2(e')2 OyO;Oy\

IC Flyc ^ C2/c 4(e ')2 C+(OyO;Oyl)

2,P^ACIlxz1 D

P2^/m 2(e')m xOz;x\z

Table 24: Anorthic System

P PI 000

P2xyz Pl(i) ±(xyz)

83

Occurrence of Lattice Complexes

Table 25

Lattice complex Multi- Site Occurrence

:

in all its plicity symmetry space group, Wyckoff letter.

representations (oriented) shift from standard representation

Cubic, invariant

11

1 mZm Pm3m(a), (b)\\\

42m P43m(a)j (b)lH

432 P432(a) , (b)\\\

m3. Pm3 (a), (b)\\\

23. P23 (a)j, (b)\l\

( )P2 8 43m Fm2m(c)lll

432 Fm2a(a)

m3. Fm3c(b)

23. F43c(a), (b)Hl

F432(c)lU-T-l >-? / 1111Fm3 [G)ll\

. 2m-r- n / » 1 1 1

Im3m(o) lii

. 32 Pm3n(e) HI

1432(a) 5U

.3. Pn3n (a)

Im3 (a)lH

Ia2 (a)^ (b)lll

1

1

2 m2m Im2m(a)

43m Pn3m(a)

Td ^n )

432 'Pn3n(a)

1432(a)

m3. Pm3n (a)

Im3 (a)

23. P43n(a)

P4^32(a)

Pn3 (a)

123 (a)

(^)200/020/002. From standard representation to cell.

85

(^) I2 23.

• 3.

Fd3o(a)

0

(^) J2

3

24

4/rm.m

42.

m

42.2

mrm.

.

222.

.

4m. 2

4/m..

4.

.

m.rm

2. 22

2/m.

,

Pm3m(a), (d)\\\

P43m(o), (d)U{

P432(a), (d)Ui

Pm3 (c)Jd)lV2

P23 (c),(d)m

Fm3c(c)lU

Fm3o(d)

F43o(g), (d)lU

Fm3m(d)

F4 32(d)

FmS (aj

4

32

m3m

43m

432

m3.

23.

. 3m

.32

.3.

.32

. 0.

Fm3m(a), (b)\ll

F43m(a), (b)Ul, (c)Uh (dJUi

F432(a), (b)\\l

Fm3 (a)Jb)V2i

F23 (a), (b)n\, (o)Hh (d)mPn3m(h)Hh (o)Ui

P4^32(b)lU, (c)lU

Pn3 (b)Hh(<2)Ui

Pa3 (a),(b)\\\

Fd3o(b)lH

CCiOa ( 8 8 8

) Y

4

4

.32

. 32

P4^32(d), (b)Ul

P4 32(a) (b)^^

6 4m. 2

4.

.

2. 22

Pm3n(c), (d)\\\

P~43n(e)\\\, (d)

P4 32(e) (f)\\\j: ^ ^0 Ct \ t:^ / y \ J y 2 2 2

6 4/mm.m

42. m

Im3m(b)

Pn3m(d)

(^)200/020/002. (^) 100/010/001. .

86

I43m(h)

42. 2 Pn3n (b)

1432(b)

mmm.

.

Pm3n (b)

ImS (b)

222. . P43n(b)

P4^32(d)6

Pn3 (d)

123 (b)

48 4. . Fd3c(d)

0 8 43m Fd3m(a), (b)in

22. F4^32(a), (b)\l\

Fd3 (a),(b)lH

8 .32 14 ^32 (a)

~Y# 8 .32 I4^32(b)

12 4m. 2 Im3m (d)

4.

.

Pn3n(d)

I43m(d)

2.22 Pn2m(f)

1432(d)

El 12 4. . I43d(a)

's 12 4. . I43d(b)

12 2.22 14 ^32 (a)

(^) ~V 12 O 0 0 T/l 7 9 )14 jOii ( CLJ

B 16 . 3m Fd3m(c), (d)V2\

.32 F4^32(c), (d)\i\

.3. Fd3 (c),(d)\\\

16 .32 Ia3d(b)

(^)200/020/002, () 100/0 10/001.

87

24 4. . Ia3d(d)

24 2. 22 IaZd(a)

Cubic, univariant

P4xxx

(^) (PHxxx )2

4

32

. 3m

.3.

.3.

P43m(e)

P23(e)

Fl3c(e)

2^2^. .FYlxxx 4 . 3. P2^3(a)

P6z

(^) (P6z)2

6

48

4m. m

4. .

mm2.

.

2m mm

2. .

4.

.

mm2.

.

2. mm

2..

Pm3m(e), (fJli'z

P432(e), (f)\\\

Pm3 (e)Jh)\\l

P43m(f) . (a)iH

P23 (f),(i)l\l

Fm3c(f)lU

Fm3c(e)

Fm3m(g)Hi

Fl3a(f), (g)lH

F432(i)lll

Fm3 (g)lH

6 mm2.

.

2..

Pm3 (f),(g)\\\

P23 (g),(h)m

.3.J2x

P8xxx

(^) (P Sxxx)^

8

64

. 3m

.3.

.3.

Pm3m(g)

P432(g)

Pm3 (i)

Fm3o(g)

. . 2mxxx 8 . 3m

. 3.

Pn3m(e)

P4^32(g)

(^)200/020/002.

88

(^) (..2I4xxx)2 54 . Z.

Pn3 (e)

FdScie)

I4xxx 8 . 2m

.3.

I43m(o)

P43n(e)

123 (a)

8

0CI

.3.

. o.

P4^32(a).'''Y2XXX

r) '4^..~Y2xxx

be. . F2xxx 8 . 3. Pa3 (a)

Qo•2

. c. T9 7.f/-r)J.CI \ a J2^2^. .P2Y^flxxx

. 3. J4x 12 Tnm2.

.

2..

Pm3m(h)

P43m(h)

P432(n)

P12xx

(^) (I=12xx)2

12

96

m, m2

..2

..2

Pm3m(i), fjiHi

P432('C)^

Fm3o(h)iH

.3.W2Z 12 mm2.

.

2..

Pm3n(g),(h)\V2

P43n(g)\Ui (h)

P4^2{%) ,{3)1^^

I6z 12 4m.m

4..

rm2.

.

2. mm

2..

Im3m(e)

Pn3n(e)

1432(e)

PmZn (f)

Im3 (d)

Pn3m(g)

I43m(e)

P43n(f)

P4^32(h)

Pn3 (f)

(^)200/020/002. (^) 100/010/001.

89

9

12 S rdj

14^. .F3xx 22 . . 2 1P4 ^32 (k) Ti'ii'ii* ( 1^ ) Ti

6

[4 /Y3xi

&)

<

"Y3xx

22

i2

. .2

. . 2

P4^32(d)

P4^32(d)1

.3.J^2x 22 m?772. .

2.

.

Im3 (e)

PnZ (g)

160 \ &

)

2^3. SVlz 12 2. . I2^3(b)1

1 8XXX 16 . Sm

. 3.

ImSm (f)

Pn3n (f)

PmSnd)

1432(f)

Im3 (f)

FM-xxx 16 . 3m

m tJ »

F43m(e)

F23 (e)

lb TA 7/7

)

4. . I^Yff-jflxxx

16 . 3. I4^32(e)22. . Y*2xxx

J. 0 Tn3 (r

)

22. .P22XXX

. 3. WHxx 24 . . 2 PrnZyi ( 1

)

Lilt KJ ' *^ ^ fj /

5. . F6xx 24 . . 2 Pn3m(i)Hh (j)iU

F6z 24 4m. m

4. .

mm2. .

2. mm

Fm3m(e)

F432(e)

Fm3 (e)

-T-i n t-> / r% \ / 1111F43m(f), (g)Hl

(^)200/020/002. (^) 100/010/001,

90

2. . F22 (f)Jg)Ul

. 3

.

K 24 mm2. .

2. .

Im2m(g)

Pn2n(g)

Pn2m(h)

I42m(f)

1422(g)

1 12XX 24 m. m2

..2

Im2m(h)

Pn2n (h)

1422(h)

.3.S2Z 24 2. . I42d(d)

24 ..2 I422(i)lll4. .P^Sxx

.3.V2Z 24 2. . 14^22 (f)

4^. ."'"Y*3xx 24

24

' 2

. . 2

I4^22(h)

I4^22(g)( ) 4^. . Y*-3xx

24 2. . Ia2 (d).3.J2S^fV*lx

F8xxx 22 . 2m

. 2.

Fm2m(f)

F4 22(f)

Fm2 (f)

22 . 2m

. 2.

Fd2m(e)

F4^22(e)

Fd2 (e)

. .2D4XXX

22 . 2. Ia2d(e)4. . Y**2xxx

F12XX 48 m.m2

..2

Fm2m(h), (i)l\\

F422(g), (h)\\\

(^) 100/010/001

91

501-563 0 - 73 -7

D6z 48 2. mm

2.

.

Fd3m(f)

F4^32(f)

FdZ (f)

4. .P^exx 48 ..2 ImZmd) III

. 3.S^2z 48 2.

.

Ia3d(f)

48 ..2 Ia3d(g)^a. . Yjf*3xx

22. .T3xx 48 . . 2 F4^32(g)

4. .T5xx 96 ..2 Fd3m(h)

42, .F^Sxx 96 ..2 Fd3o(g)U\

Cubic, bivariant

P6z2xx 12 . . m P43m(i)

P6z2x 12 m.

.

Pm3 (3)Jk)l\\

P6z4x 24 m.

.

Pm3m(k),

P6z4xx 24 . .m Pm3m (m)

24 m.

.

Pm3n (k). .2I6z2x

24 . .m Pn3m(k). .2I6z2xx

I6z2xx 24 . , m I43m(g)

I6z2x 24 m.

.

Im3 (g)

92

I6z4x 48 m. . ImZm (j )

48 . .m Im2m(k)I5z4xx

48 . . m F42m(h)F6z2xx

F5z2x 48 m.

.

Fm3 (h)

F6z4x 96 m.

.

FmZm (j)

F6z4xx 96 . . m Fm2m(k)

. .2P26Z2X 96 m.

.

FmZad)

. .2D5z2xx 96 . . m FdZm(g)

Cubic, tri

The 12-pointers

:

P6z2xy P23 (3)

P2^3(b)2^2^. . FYlxxx3yz

The 2M--pointers

:

P6z2xx2y P43m(j)

. . cI6z2xy P43n(i)

P6z4xy P432(k)

. .2I6z2xy P4^32(m)

4 . . Y3xx2yz

—h

( ) H . . Y3xx2yz

P4^32(e)

P4^32(e)

(^)ioo/oio/ooi.

iant (site symmetry 1)

P6z2x2y Pm3 (I)

Pn3 (h)n. . I6z2xy

Pa3 (d)be. . F5xyz

I6z2xy 123 (f)

12^3(0)2^2^. .P2Y*#lxxx3yz

The M-S-pointers

:

P5z4x2y Pm3m(n)

n. . I6zM-xy Pn3n(i)

. .2I6z2x2y Pm3n(l)

93

. .2I6z2xx2y

I5z2xx2y

, 3dS4xyz

I5z4xy

22. . YJf3xx2yz

I5z2x2y

22. .P^Bxyz

F6z2xy

I5z4x2y

43. . Y*)f3xx2yz

The 96-pointers:

F6z2xx2y

,nP25z2xy

F6zM-xy

. .2D6z2xy

F6z2x2y

d. .D6z2xy

PnZm(l)

I43m(h)

I4Sd(e)

I432(j)

Im3 (h)

Ia3 (e)

F23 (h)

Im3m(l)

Ia3d(h)

F43m(i)

F43a(h)

F432(q)

F4^32 (h)

Fm3 (i)

Fd3 (g)

,2D6z2xx2y Fd3m(i)

d.2l26z2xy Fd3o(h)

The 192-pointers

;

F5z4x2y Fm3m(l)

m. . P^Bz^jQ^ Fm3o

94

Occurrence of Lattice Complexes

Table 26-27

Lattice complex

in all its

representations

Mill ?

plicity

oixe

symmetry

(oriented)

uccurrence

.

space group, Wyckoff letter,

shift from standard representation

Hexagonal and rhombohedral , invariant

0

P[z]

(^)

6/rrmm

6m2

62m

622

6/m.

.

6..

3m.

3.m

22.

2.2

2..

6mm

6.

.

2m.

2.m

2.

.

6m2

62m

622

6/m.

.

6.

.

3m.

'2.m

P6/mmm(a)y (b)OOl

P6m2(a), (b)OOl, (c)\\0, (d)\n,

(e)\\0,(f)n\

P62m(a), (h)00\

P622(a)j, (b)00\

P6/m(a), (b)OOl

P6 (a), (b)OOl, (c) \\0, (d) Wz,

(e)nO,(f)m

P2ml(a), (b)00\

P21m(a)^ (b)OOl

P221(a)^ (b)00\

P212(a), (b)OOl, {e}\\0, {d)\\\,

P2 (a), (b)00\

P6mm(a)

P6 (a)

P2ml(a), (b)ilO, (a)\\0

P21m(a)

P2 (a)Jb)\\0,(c)%\0

P6jmmc(b)00l

P6^/mom(a)00l

P6/moQ(a)00l

P6/mao(b)

P6a2(b)00l, (d)\U, (f)mP62c(b)00l

P6^/m(a)00l

P6^/mmo(a)

P6^/mcm(h)

( ) 100/010/002. From standard representation to cell.

95

32. P62c(a)

P6^22(a)

P3cl(a)00l

Z.2 P6c2(a), (c)HO, (e)ViO

P6^22(b)00i

PZlc(a)00i

3.. P6^/m(b)

PZcl(b)

PZlc(b)

2 6. . P6cc (a)

3m. P6^o(a)

Z.m P6^cm(a)

3.

.

P6^ (a)

PZal(a), (b)\iO, (c)no

PZlc(a)

(^) Z 222 P6^22(a), (b)OOl

P6^22(a), (b)OOl

(^) P^W Z 2. . P6^ (a)

0I. J2 6m2 P6/mrm(o), (d)OOl

6.. P62m(a), (d)OOl

P6/m (a), (d)OOl

Z. 2 P622(o), (d)OOl

PZlm(c), (d)00\

G[z] 3m. P6mm(b)

3.

.

P6 (b)

PZlm(b)

(^) Gc

4 6.

.

P6/mco(d)

P6^/mcm(o)00l

Z.2 P6/mco(c)00l

P6^/mcm(d)

(^) G [z] 4 Z. . P6ca(b)

irO 2 ^ '

1

1

HI 2 6m2 P6jmmo(o), (d)00\

(^)100/010/002, (^)100/010/003.

96

6.

.

P62o(c), (d)00\

P6jm(c), (d)00\3

3. 2 P6^22(o), (d)00\0

P31c(c), (d)00\

E [z] 2 3m. P6^mo(h)

3. . P6^ (b)

P31a(b)

B 3 3m. R3m (a), (b)00\

32. R32 (a), (b)00\

3. . R3 (a), (b)OOl

R[z] 3 3m. R3m (a)

3. . R3 (a)

'Rc

6 32. R3o (a) 00

1

3. . R3c (b)

(^) 6 3.

.

R3a (a)

i_j3 mmrn P6/mm(f), (g)00\

222 P622(f), (g)OOl

2/m. . P6/m(f), (g)00\

.2/m. P3ml (e), (f)00\

..2/m P31m(f), (g)00\

1 P3 (e),(f)00h

N fz] 3 2mm P6mm(c)

2.

.

P6 (c)

Nc

6 222 P6/mco (f)OOl

2/m. . P6/mca(g)

. 2/m. P6 ^/mma(g)6

. . 2/m P6Jmom{f)6

1 P6jm(g)6

P3cl (e)

P31o(g)

(^) NJz] 6 2. . P6co(c)

3 222 P6^22(a), {d)00\2

+Q[zJ 3 2. . P6^ (b)

(^) -Q 3 222 P6^22(c), (d)00\

(^)100/010/002. ('^)010/110/002. ('^)100/010/001.

97

(^) 3 2. . PS, (b)

[71 9 . 2/m. H3m (d)00\, (e)

1 R3 (d)OOl, (e)

'Mc

18 1 R3g (d)

Hexagonal and rhombohedral , univariant

P2z

( ) (P2z),

r) (P2z),

6mm P6/mrm(e)

6.. P622(e)

P6/m(e)

3m. P6m2(g)Jh)\lO, (i)\\0

P3ml (a)

3.m P62m(e)

P31m(e)

3.. P6 (g),(h)\\0. ii)\\0

P321(c)

P312(g), (h)\iO, (i)l\0

P3 (c)

6. . P6/moc(e)

3m. P6^/mma(e)

3.m P6^/mcm(e)

3.

.

P6c2(g),(h)\\0, ii)\\0

P62o(e)

P6^22(e)

P6^/m(e)

P3al (c)

P31o(e)

2. . P6^22(e)

P6^22(e)

.2.GE1Z 3m.

3..

P3ml (d)

P321 (d)

P3 (d)

(^) 100/010/002. (^) 100/010/003. ('')010/110/002 . C'^) 100/010/001

98

(^) (.2.GElz)c

4 3.

.

P3al (d)

P3xx 3

3

au

rm2

..2

. m.

9

P6m2(j), (k)OOl

P312(j), (k)OOi

P3ml (d)

( -i

)

P3xx£z}

(^) (P3xx)c

3

3

0

m2m

.2.

. . m

9

P62m(f), (g)00\

P321(e), (f)00\

P31m(c)

P3x

P3x[z]

(^) (P3x)c

32..P(."^Qlxx 3

3

..2

. . 2

P3^12(a)00i, (b)OOl

P3^12(a)00\, (b)OOl(d) 3^..P^ Qlxx

32..Pc^Qlx 3 .2.

9• ^ •

P3^21(a)00i, (b)OOi

P3 21(a)00\. (h)OOii^) 3^..P^"q1x

4

8

3m.

3. .

3. -

P6/mmm(h)

P62m(h)

P622(h)

P6/m(h)

P31m(h)

P6/mao (h)

Jrb ^/mom{ nj

G2z

(^) (G2z)2

4 3m.

3.

.

P6^/mmc (f)

P62o(f)

P6^22(f)

P6^/m(f)

P?7^ Z' -P)ro 1(J { J J

E2z

6

12

3m.

3. .

3. .

R3m (o)

R32 (a)

R3 (c)

R3o (c)

R2z

•(R2z)c

(^)100/010/002. (^)010/110/002. ( ^) 100/010/001.

99

N2z

( ) (N2z) 12

2rm

2.

.

P6/mrnm(i)

P622(i)

P6/m(i)

P6/mco(i)

P5x

P6x [z]

(^) (P6x)

6

12

m2m

.2.

. .m

P6/rrmn(j), (k)00\

P622(j), (k)OOl

Pdml(g), (h)OOl

P6rm(d)

P6/mcc (3)001

P6^/mmc(i)

P6xx

P6xx [z]

(^) (P6xx)

6

12

rm2

..2

. m.

..2

P6/mrm(l), (m)00\

P622(l), (m)OOl

PZlmd), (c)00\

P6mm(e)

P6/moc(k)00l

P6^/mcm(i)

2P 3xc

,2? 3x[z]c

m2m

. 2.

. , m

P6 /mom(g)00i

P6^22(g)

P'Zcl(f)00l

P6^om(c)

,2.P E3xx

c

,2.P E3xx [z]c

mm2

..2

. m.

P6 Jmino(h)00l

P6^22(h)00l

P31o(h)00l

P6^mc(o)

(d) 32.2P^^-Q^lx

2.

2.

P6 222(a)

P6^22(a)

..2

..2

P6^22(b)00'^

P6^22(b)OOh.

Q2z

(d) Q2z

6

6

2.

2.

P6^22(f)

P6^22(f)

(^)100/010/002. (d)100/010/001.

100

32..P^'^Q2x

(d) 3^..P^-(32x

6

6

.2.

. 2.

P6^22(g), (h)OOl

P6^22(g)j (h)00\

32 - .Pp''"Q2xx

(d) 3^..P^ (32XX

6

6

..2

. . 2

P6^22(i)00\, (j)OOi

P6^22(zJ00ij (3)001

R3xx 9 , m. R3m (b)

R3x 9 .2. R32 (d),(e)00\

R6x 18 .2. R3m (f),(g)00\

.n. 'R 3xc

18 .2. R3c (e)OOl

Hexagonal and rhombohedral, bivariant

P3xy

P3xy fz]

3

3

m,

.

1

P6 (0), (k)OOl

P3 (d)

32..P^'^Qlxy [z]

(d)3^. .P^"Qlxy[z]

3

3

1

1

P32 (a)

P3^ (a)

P3xx2y

P3}<x2y fz]

6

6

m. .

1

P6m2(l), (m)OOl

P3ml (e)

P3xx2z 6 . m. P6m2(n)

6

6

m.

,

1

P'Bo2(k)00l

P3g1 (d)

. . 2P 3xyc

. .2P3xy [z]

P3x2z 6 . , m P62m(i)

P3x2y

P3 K2y [z]

6

6

m. .

1

P62m(o)Jk)00\

P31m(d)

(^)100/010/001.

101

.2.P^E3xy

.2.p^e;3xy [z]

6

6

m. .

1

P62c(h)00l

PZlc(c)

P6xy

P6xy {zl

6

6

m. .

1

P6/m(j), (k)00\

P6 (d)

6

6

m.

.

1

P6^/m(h)00l

P6 ^ (a)

2, . .P ESxy1 c

2^ . .P E3xy{zl1 c

P6xz 6 . . m P31m(k)

6 . m. PZml (i)P6xxz

6

i)6

1

1

P6^ (a)

P6 ^ (a)

32..P^^Q2x3

(d) 3^..P^-

r tz)

Q2xy{z]

6

6

1

1

P6^ (a)

P6^ (c)

R3xy iz] 9 1 R3 (h)

P6x2z 12 . . m P6/mmm(n)

P6xx2z 12 . m. P6/mmm(o)

P6x2y

P6 x2y[z]

12

12

m. .

1

P6/rmm(p), (q)OOl

P6mm(f)

. 2.P^6xy

.2.P 6xy{z}c

12

12

m.

.

1

P6/moc(l)

P6co(d)

. .2P^3x2y

. .2P 3-

c<2y{z]

12

12

m.

.

1

P6^/mam(j)00l

. .2P 3x2zc

12 . . m P6^/mGm(k)00l

(^)100/010/001.

102

.2.P E3xx2yc

.2.P E3xc

x2y(zl

12

12

m. .

1

P6^/mma(j)00i

P6^mo(d)

.2.P E3xx2zc

12 . m. P6 Jrmo(k)00lo

R6xxz 18 . m. R3m (h)

R3xx2y [z) 18 1 RZm (a)

.n. ' R 3xy [z]c

18 1 R3c (b)

Hexagonal and rhombohedral , trivariant (site symmetry 1)

The 6-pointers:

P3xy2z P6 (I)

P3xx2yz P312(l)

P3x2yz P321 (g)

P3^12(c)OOi

P3^12(o)00\

32..P^^Qlxx2yz

C^) 3^. .P^-Qlxx2yz

32..P^'^Qlx2yz

{^) 3^. .P^-Qlx2yz

P3^21(c)OOi

P3^21(c)00\

P6xyz P3 (g)

The 12-pointers:

P3xx2y2z P6m2 (o)

..2P^3xy2z P6g2(1)001

P3x2y2z P62m(l)

(d)lOO/010/001.

.2.P^E3xy2z P62Q(i)00l

P6x2yz P622(n)

3r2Pcc\^^2yz P6^22(o)

P6^22(o)(d) 3^.2P_ "Q lx2yz

z LC c

32- .P^^Q2x2yz P6^22(k)

P6^22(k)(d) 3^..P^-Q2x2yz

P6^22(i). .2P 3x2yzc

P6xy2z P6/m(l)

P6^/m(i)00l2, . .P E3xy2z1 c

P6xz2y P31m(l)

P31a(i)00l. .cP^E3xx2yz

103

P6xxz2y P2ml (j)

cP?,nl (n)OOT-

The 18-pointers:

R3x2yz R32 (f)

R6xyz R3 (f)

The 24--pointers

:

P6x2y2z P6/mrnm(r)

m. . P 6x2yzc

P6/mca (m)OOl

. .2P 3x2y2zc

P6^/mam(l)00l

P6jmmc(l)00lO

.2.P^E3xx2y2z

The 36-pointers:

R6xxz2y RZm (i)

•n. ' R 3x2yzc

RZo (f)OOi

104

Occurrence of Lattice Complexes

Table 28

Lattice complex Multi- Site Occurrence

:

in all its plicity symmetry space group, Wyckhoff letter.

representations (oriented) shift from standard representation

Tetragonal , invariant

0 1 4/mmm P4/mrm(a), (b)00\, (c)\\0, (d)\\l

42m P42m(a), (b)\H, (c)00\, (d) \\0

4m2 P4m2(a), (h)l\0, (c)Uh (d)00\

422 P422(a), (b)00\, {c)\\0, (d)\\\

4/m.

.

P4M(a), (b)OOi, (o)\\0, (d)\V2

4. . P4 (a)Jh)00\,(o)\\OJd)\\\

P{z} 1 4mm P4rm(a), (b)llO

4. . P4 (a),(b)\\0

(^) Pc

2 42m P4^/mom(a)00l, (b)\\l

4m2 P4^/mmc(e)00l, (f)\\l

422 P4/mcc(a)00l, (c)\\l

4/m. . P4/maa(b), (dJhlO

4. . P42a(e), (f)\\0

P4c2(c), (d)\lO

P4^/m(e)00l,(f)\\l

mmm. P4^/nmo(a), (b)V20

m. mm P4^/mom(b), (d)\\0

222. P42c(a)00l, (c)\\l

P4^22(a), (b)\\0

2. 22 P4G2(a)00l, (b)\\l

P4^22(e)00l, (f)\U

2/m. . P4^/m(a), (b)V20

2 4. . P4ac(a), (b)\\0

2mm. P4^mc(a), (b)llO

2. mm P4^am(a), (b)\\0

2.

.

P4^ (a), (b)UO

&) c 2 42m P4/nbm(c)0\0, (d)0\\

4m2 P4/nxm(a), (b)00\

(^)100/010/002. () 110/110/001. From standard representation to cell.

105

&) C(2)

ab

i') C

422 P4/nbm(a), (b)OOl

4/m. . P4/mbm(a), (b)OOl

4. . P42jm(a), (b)00\

P4b2(a)j, (b)00\

P4/n(a), (b)00\

mm. P4/mmm(e)0\\j (f)0\0

m. mm P4/mbm(o)0\l, (d)0\0

222. P42m(e)lOO, (f)\0\

P422(e)\OOj, (f)\0\

2. 22 P4b2(o)OiO, (d)0\\

P42^2(a)j (b)00\

2/m. . P4/m(e)0\0, (f)Oll

4. . P4bm(a)

2mm. P4mm(o)\00

2. mm P4bm(b)\00

2. . P4 (a)0\0

. . 2/m P4/nbm(e)U0, (f)Ul

P4/nmm(d)llO, (e)Ul

1 P4/n(d)U0, (e)ll\

42m I4/mcm(b)0\l

4m2 I4/mmm(d)0ll

422 I4/mcm(a)00l

4/m.

.

I4/mcm(o)

4. . P4/nnc(d)l0l

P4/noc (b)

P4^/nbo(d)

P4^/mba(b)00l

P4^/mnm(d)0U

P4^/nam(b)

I4a2(b), (o)0U

I42m(d)0\l

I4/m(d)0U

mmm. I4/mmm(a)OlO

m. mm I4/mam(d)0\0

222. P4/mco(e)0\l

P4/nnc(c)\00

( )110/110/001. (^)200/020/001. () 110/110/002.

106

P4^/mcm(e)0\l

P4^/nho(a)0U^ (b)OOl

P4^/nnm(a)OlO

I42m(o)0\0

I422(o)0\0

2.22 P4/mna(d)0U

P4/nac(a)00l

P4^/nho(o)0\0

P4^/nnm(d)0\l

P4^/rribc(d)0U

P4^/ncm(a)00l

I4G2(a)00l,(d)0\0

1422 (d)OU

2/m. . P4/moc(f)0\0

P4/mnc(c)OlO

P4^/mom(f)0\0

P4^/rrbo(a), (c)0\0

P4^/wnm(o)0\0

I4/m(c)OiO

4.. I4om(a)

2rnm. I4rm(b)OlO

2. mm I4cm(b)\00

2.. P4^om(o)0\0

P4^nm(b)0\0

P4oc(q)010

P4no(b)0\0

P4^bo(a), (b)OlO

14 (b)0\0

..2/m I4/mrm(f)lU

I4/mom(e) HI1 P4/nnc(f)Ul

P4/naa(d)H0

P4jnbo(e)lU

P4^/rmo(e)lll

I4/m(f)\ll

(^)110/110/002. (®)200/020/002.

107*

501-563 O - 73 - 8

I[z]

F

(^) F

( ) F fz)c

(^) I2 16

4/mmm

42m

4m2

422

4/m. .

mmm.

m. mm

222.

2.22

2/m. .

4mm

4. .

2mm.

2. mm

2. .

. . 2/m

1

4. .

2. 22

2. .

1

I4/mmm(a), (h)00\

P4^/nnm(a), (b)OOl

I42m(a), (h)00\

P4^/nma(a), (b)00\

I4m2(a), (h)OOl, (c)0\l, (d)OU

P4/nnc(a), (b)00\

1422(a), (b)OOl

P4/mnG(a), (b)00\

I4/m(a), (b)00\

P42^c(a), (b)OOl

P4n2(a), (b)00\

P4^/n(a), (b)OOl

14 (a),(b)00l,(c)0\l,(d)0\l

P4^/nma(c)0\0, (d)0\\

P4^/mnm(a), (b)00\

P42c(b)\0l, (d)0\l

P4^22(o)0\0, (d)On

P4n2(c)0U, (d)Ol3

' 2 u

P4^2^2(a), (b)00\

P4^/m(c)0\0, (d)OH

I4mm(a)

P4nc (a)

14 (a)

P4^mc(G)0\0

P4 ^nm(a)

P4^ (c)OlO

P4^/nnm(e)lll, (f)

P4^/ncm(a)Uh (d)Ui

P4^/n(o)lU, (d)mI4^/aad(b)

I4^/acd(a)00l

I4^od(a)

I4^/aad(c)0U

135

(^)110/110/001. (^)110/110/002. ( )200/020/002.

108

[3 8 42m F4 Jddrn-1

(^) 4 4m2 I4^/amd(a), (b)00\

4.

.

I42d(a), (b)00\

14^/a (a), (b)00\

2.22 14 ^22 (a), (b)00\

4 2mm. I4^md(a)

2. . 14^ (a)

16 . . 2/m F4^/ddm^

8 . 2/m. I4^/amd(c), (d)00\

1 I4ja(o), (d)OOl1

'

Tetragonal, univariant

2 4mm P4/mmm(g), (h)UOP2z

4.

.

P422(g), (h)^{0

P4/m(g), (h)llO

2mm. P4m2(e), (f)l\0

2. mm P42m(g), (h)llO

2. . P4 (e)Jf)\lO

(^) (P2z)n

4 4. . P4/mco(g), (h)UO

2mm. P4^/mmc(g), (h)UO

2. mm P4^/mom(g), (h)\\0

2.. P42o(k), (l)\\0

P4c2(g),(h)\\0

P4^22(g), (h)\lO

P4^/m(g),(h)nO

C2z 4 4.

.

P4/nbm(g)

P4/mbm(e)

2mm. P4/mmm(i)OlO

P4/nmm(f)

2. mm P4/nbm(h)OlO

P4/rribm(f)0\0

Unconventional setting. Cp. next entry.

(^)100/010/002. (b)110/110 001. (f )lfo/Ho/ooi.

109

2.. P42m(m)OiO

P42^m(d)

P4b2(e), (f)OlO

P422(i)0\0

P42^2(d)

P4/m(i)OlO•

P4/n(f)

r) (C2z )c

8 4. . I4/mGm(f)

2mm. I4/mmm(g)0\0

2. mm I4/mam(g)0i0

2.

.

P4/mca (i)0\0

P4/nno(g)\00

P4/mno(f)0\0

P4/nao(e)

P4^/mcm(k)OlO

P4^/nba(f)OlO, (g)

P4^/nnm(h)0\0

P4^/rribc(e), (f)0\0

P4^/mnm(h)OlO

P4^/ncm(f)

I4c2(f), (g)0\0

I42m(h)0\0

I422(f)0\0

I4/m(g)0\0

. .2CI1Z 2 4mm P4/rmn(c)0\0

4.. ^42^2(0)010

P4/n(G)0\0

2mm, P4m2(g)0l0

2. mm P42^m(c)0\0

2.. P4 (g)0\0

(^) (..2CIlz)^ 4 4. . P4/noc {o)0\0

2mm. P4^/nmo{d)0\0

2. mm P4^/ncm(e)0\0

2.

.

V42^(d)0\0

.

(^)110/110/002. (^) 100/010/002.

110

P4a2 (i)OlO

M^2^2(d)0\0

P4^/n(e)0\0

P4xx

Pi|xx{z]

(^) (P4xx)

(^) C4x

{^) (CUx) 16

m. 2m

..2

. .m

..2

.2.

.2.

P4/mrm(j)^ (k)00\

P4m2(h), (i)OOl

P422(j), (k)00\

P4rm(d)

P4/moc(3 )00l

P4^/rmo(n)00l

P4/nbm(k), (l)00\

I4/mem(j)00i

P4x

Pi+x[z}

(^) (PHx)

r) C4xx

( ) (C4xx) 16

m2m.

.2.

. m.

.2.

..2

P4/mmm(l), (m)OOl, (n)l\0, (o)\\\

P42m(i), (k)00\, (l)\\0

P422(l), (m)\\h (n)00\, (oJHO

P4rm(e), (f)\\0

P4/mco(k)00l, (DWlP4^/mcm{l)00l, (m)

P4/nbm(i), (j)OOl

P4/nrm(g), (h)00\

I4/rrmm(k)0ll

I4/mam('L)00l

1112 2 U

.b.C2xx

.b.C2xx{z)

(^) (.b.C2xx)

m. 2m

..2

. .m

..2

P4/mbm(g)0\0, ('h)0\\

P4b2(g)0\0, (h)Ol\

P42^2(e), (f)00\

P4hm(e)0\0

P4/mnc(g)0\l

P4^/mbc(g)0 1 ^

2 t

.2P I2xc

m2m.

.2.

P4^/mme(j), (k)Uh (l)00\, MWOP42c(g)OOl, (h)\\l, (iJVii, (j)OOl

P4^22(j), (k)^U, (l)00\, (m)\\0

( )100/010/002. ( )110/110/001, ( )110/110/002,

111

..2P I2x{z). c

( ) .2.C F2xxc

.m. P4^mo(d), (e)liO

P4^/nnm(k)0U, (l)0\

P4jncm(g)00l, (h)OOl

32 14

,2.P 2xxc

.2.P 2xx(z). c

r) ..2C 2xc

m. 2m

..2

. . m

.2.

P4^/mom(i), (j)OOh

P4c2(e)00l, (f)OOl

P4^22(n)00h, (o)OOl

P4^cm(d)

P4^/nba(h)00l, (i)OOi

.n. I2xx

,n. I2xx{z)

m. 2m

..2

. . m

P4^/mnm(f), (g)\\i

P4n2(f)m, (g)OU

P4^2^2(e), (f)00\

P4^nm(o)

I2z

{^) (F2z) 16

4mm

4.

.

2mm.

2. mm

2.

I4/mmm(e)

P4/nnG (e)

P4/mna(e)

1422(e)

I4/m(e)

P4^/mmo(i)OlO

P4^/nmc(c)

I4m2(e), (f)OU

P4^/nnm (g)

P4^/mnm(e)

I42m(e)

P42c(m)0U

P42^a(o)

P4n2(e), (h)0\l

P4^22(i)0\0

P4^2^2(o)

P4^/m(i)0\0

P4^/n(f)

14 (e), (f)OH

I4^/acd(d)

(^)110/110/001. (^)110/110/002.

112

i+^..P I Ix3 cc c

(g) 4, . .? I1 cc

^Ix

4

4

.2.

. 2.

P4^22(a)00l, (h)\U

P4^22(a)00i, (h)\\i

H^..? "^Dlxx3 cc

1 ccDlxx

4

4

..2

. . 2

P4^22(c)00l

P4 ^22(o)00i

H^. .1 ^Dlxx

(g) ""d1 c

Lxx

4

4

..2

..2

P4^2^2(a)

P4^2^2(a)

mxx

I4xx{z}

8

8

m. 2m

..2

. . m

I4/mrm(h)

P4/nnc(k)

P4^/nma(f)

I4m2(g), (h)OU

1422(g)

I4mm(G)

lUx1

mx(z}

8

8

m2m.

.2.

. m.

I4/mrm(i), (3)\lO

P4/nnc(i), (j)OOl

P4^/nnm(i), (j)00\

I42m(f), (g)00\

1422(h), (i)00\

I4rm(d)

[z),

8

•

8

m. 2m

..2

• .m

I4/mom(h)0\0

P4/noc(f)00l

P4^/nhe(o)0\0

I4o2(e)00l, (h)OlO

1422(3 )0H

I4Gm(c)\00

.b.C 2xxc

.b.C 2xxc

VD2z 8 2mm.

2..

I4^/amd(e)

I42d(o)

I4^22(c)

I4^/a(e)

U.7tF Ixc

8 .2. I42d(d)

(^)100/010/001.

113

. .d D2x [z) 8 . m. I4^md(h)

.2.^D2xx

(^) .2.^ D2xx

8

8

..2

..2

I4^22(d)

I4^22(e)

..22^TC Ixcc

8 . 2. I4^22(f)

. .2^T2x 16 . 2. I4^/amd(f)

VD4xx lb . . 2 14 ^/amaig)

1 0 0 1^ j/CZOCX{ e J Ui;-§

.2.F 2xxc

1 0 ? T4 /nr>rl ( F)nn^

Tetragonal, bivariant

PUxxz 4 . .m P42m(n)

.2^.CIlz2xx 4 . . m P42^m(e)0\0

P4xz 4 .m. P4m2(j), (kJllO

PHxy

Pi+xy (z)

4

4

m.

.

1

P4/m(c), (k)OOl

P4 (d)

U""-. .P^I2xy

51. .P^I'2xy (zj

4

4

m. .

1

P4/m(j)

P4^ (d)

4 ..P ^DI l>Qr|{z)o CC C 1

(g) 4, . .P "^DI Ixy(z)1 cc c

4

4

1

1

P4^ (a)

P4^ (a)

P4x2y

Pt+x2y{z}

8

8

m. .

1

P4/rrmn(p), (q)00\

P4rm(g)

(^)100/010/001.

114

P4xx2z 8 . .m P4/mmm(v)

P4x2z 8 . m. t'i/mnmi s J , (r/ 22^/

.2.P Hxyc

.2.P ^4xy (z)

8

QO

m. .

1

P4/mcc(m)

Vdnn ( rl

)

i tr 0 C- I H, /

. . 2C^xxz 00 Pd /vihm (m )C)-n

. .mC+xy

4xy [z)

8

8

m. .

1

P4/mbm(i), (j)OOl

P4bm (d)

8 . . m t4/TnDTn { KJU2U.b.C2xx2z

. .2I4xy

. .21'4xy [z]

8

8

m. .

1

P4/mnc(h)

P4nc(a)

. . mCM-xzna . m. r'i/ YlTWl ( i-y

8 . .m P4/nrm(g)0\0. .2CIlz4xx

m. .P Hxzc

aU m Ilia P4 /mmc (0)00 1 Td J H

E

8

QO

m.

.

1

P4^/minc(q)

VA mr> ( -P )

2 ^

. .2P I2x2yc

. .2P I2x2yfzlc

8

s

m.

.

2

P4 /mom(n)

P4 om(e)

.2.P^2xx2y

.2.P^2xx2y iz]

m. . P M-xxzc

Qo ITI P4 /mem (o)OOt

. .2mxxz QO ^/ riruit \ III/

.b2C 2xyc -'

.b2C 2xy{z}c

8

8

m. .

1

P4^/rrbo(h)

t4^OOKQ)

.n. I2xx2y

.n.I2xx2y{z}

8

8

m.

.

1

P4^/rmm('L)

P4^nm(d)

115

.n. I2xx2z 8 . . m P4^/mnm(j)

. . cIM-xz 8 .m. P4^/nmc(g)

4. . F2xxz 8 . .m P4^/ncm(i)lll

14-xz 8 .m. I4m2(i)

8 . .m I42m(i)I4xxz

I4xy

] 4xy(z)

8

8

m. .

1

I4/m(h)

14 (a)

8 1 14^ (b)1 V

4^ . . D2x]J {z]

mx2y

x2y[z)

16

16

m. .

1

I4/rrmn(l)

I4rm (e

)

16 . . m I4/imw(m)I4xx2z

I'+x2z 16 . m. I4/mrrm(n)

. ,mC 4xyc

. .mCc

16

16

m.

.

1

I4/mam(k)

I4cm(d)

.b.C 4xxzc

16 . . m I4/mcm(l)0H

. 2.

^D4xz 16 . m. I4^/amd(h)

. .d^D2x2y (z) 16 1 I4^md(o)

.bdF 2xyc

(z) 1 0 7 14 aS (b

)

_L ^ ^ OLi \ U J

116

Tetragonal, trivariant (site symmetry 1)

The M--pointers:

PM-xyz

The 8-pointers;

P4xxz2y

,2.P 4xyzc

, mC4xyz

, cI4>Q?'z

P4xz2y

, 2P 4xyzc

, 2C4xyz

,2I4xyz

Pi+x2yz

>2^.C2xx2yz

4^..P I lx2yz3 cc c ^

P4 (h)

(g) 4, . .? I lx2yz1 cc c ^

4„. . I ^Dlxx2yz3 c i_

(g) 4^. .1^ Dlxx2yz

,2P I2x2yzc ^

, 2^.I2xx2yz

P42m(o)

P42c(n)

P42^m(f)

P42^a(e)

P4m2(l)

P4g2 (j)

P4b2(i)

P4n2(i)

P422(p)

P42^2(g)

P4 ^2 2(d) 00

1

P4^22(d)OOl

P4^2^2(b)

P4^2^2(b)

P4^22(p)

P4^2^2(g)

(^)100/010/001.

P4xy2z

m. . P 4xyzc

lC4xyz

n . . I4xyz

I4xyz

The" 16-pointers:

P4x2y2z

m. .P^4x2yz

.mC4x2yz

, cI4x2yz

,mC4xy2z

.2I4xy2z

. .mC4xz2y

. .c2C 4xyzc

m. .P 4xz2yc

m. . P 4xxz2vc

, 22C 4xyzc

P4/m(l)

P4^/m(k)00l

P4/n(g)

P4/n(g)

14 (g)

P4/mmm(u)

P4/mac (n)OOl

P4/nhm (n)

P4/nna(k)

P4/rribm(l)

P4/mna(i)

P4/nrm(k)

P4/nca (g)

P4^/mmc(r)00l

P4^/mGm(p)00i

P4^/nbc(k)

117

. .2I4xxz2y P4^/nnm(n)

Tn.2C 4-xyzc

P4^/mbG(i)00l

P4^/rmm(k).n. I2xx2y2z

. . cmxz2y P4^/nma(h)

. .in2C 4xyzc

P4^/ncm(j)

I4xz2y I4m2 (j)

I4o2(i). .2C 4xyzc

mxxz2y I42m(j)

I42d(e).2.^D4xyz

mx2yz 1422 (k)

I4^22(g).2.^D2xx2yz

mjQ?2z I4/m(i)

Va. . DHxyz I4^/a(f)

The 32-pointers

:

I4x2y2z I4/nmn(o)

.b.C^4x2yz I4/mam(m)00l

.2."^D4xz2y I4^/amd(i)

.22F^4xyz I4^/aod(g)

118

Occurrence of Lattice Complexes

Table 29

Lattice Transfor- Multi- Site Occurrence

:

complex mation from plicity symmetry space group, Wyckoff letter.

in all its standard (oriented) shift from standard representation

represen~ representa-

tations tion to

cell

Orthorhombic, invariant

0 1 mrrm Pmmm(a) , (h)\00, {o)00\, {d)\0\.

(e) 0\0, {f)\\0, (g)0\\, (h)\\\

222 P222(a) , (h)\00, (o)0\0, (d)00\.

(e) \\0, {f)\0\, {q)0\\, {h)\\\

P{z} 1 TWl2 Pmm2 (a) . (h)0\0, (o)iOO, (d)liO

pa

200/010/001 2 . 2/m. Pmma(a. , {h)0\0, (a)OO^j (d)OU

P (z)a

200/010/001 2 . . 2 Pma2(a) . (b)OW

Pc

100/010/002 2 222 Poom (e

,

OOL (f)'20L (q)0\l, (h)\\l

. . 2/m Pccm (a) (h)\\0, (o)0\0, {d)\00

P (z)c

100/010/002 2 . . 2 Poo2(aj , (h}0\0, {c)\00, (d)\\0

pab

200/020/001 4 222 CmmcL (a. 100, (b ) lOi

2/m.

.

CmmcL (o J , (d)OOi

. 2/m. Cmma (e jlln f-p)lllU J U 4 2

. . 2/m Cxwrvmie > 77 7T Ly A I / / 77 77 ^U * J ' 4 4 Z

1 Phcoi (e /lln (-P)lll

Prnmioi lln (A)lll

P , (z}ab

200/020/001 4 ..2 Cmm2(o/ llo

Pac

200/010/002 4 1 Paca (a/ , (b)OlO

^bc100/020/002 4 1 Pbom(aj , (b)lOO

Pbcf^> 100/020/002 4 ..2 Abm2(ay , (b)lOO

^2 200/020/002 8 222 Fmrm(fj 111U U 14

2/m.

.

Fmmm(Oy

.2/m. Fmmm(dy

. . 2/m Fmmm(e,

1 Cmom(d,

Cmoa ic,)

Coca (a,

119

200/020/002

Immm(k) HiIbam(e) HIIbca(a), (b)lU

Frm2(h)U0

0

Cfz]

A[z}

B

C

001/100/010

010/001/100

100/010/002

C (z]c

A [z)a

100/010/002

002/100/010

002/100/010

010/002/100

rrrnn

222

. . 2/m

rm2

. . 2

rm2

..2

2/m. .

222

2/m. .

. . 2/m

1

..2

. 2/m.

1

..2

1

2/m. .

Cmnm(a), (b)\00, (c)\0\, (d)00\

Phan(a), (b)\00, (c)\0\, (d)00\

C222(a), (b)0\0, (c)\0l, (d)OOl

Pbcm(a), (b)00\, (c)OlO, (d)Oll

Cmm2(a), (b)0\0

Pba2(a), (b)0\0

Amm2(a), (b)\00

Pnc2(a), (b)\00

Pmna(a), (b)\00, (c)\\0, (d)0\0

Caom(a)00l, (b)OU

rbam(a)00l, (b)\Ol

Cmcm(a), (b)0\0

Cccm(c), (d)0\0

Ibcm(c), (d)\00

Pbcn(a), (b)0\0

Ccc2(a), (b)0\0

Iba2(a), (b)OlO

Imma(c) Hij (d) Hi

Pnna(a), (b)00\

Ama2(a)

Ima2 (a)

Pnma(a), (b)00\

Imma(a), (b)00\

I[z]

rrrnn

222

. . 2/m

mm2

..2

Immm(a), (bJOll, (c)U0, (d) \0\

Pnnn(a), (b)\00, (c)00\, (d)0\0

1222(a), (b)lOO, (c)00\, (d)0\0

Pnnm(a), (b)00\, (c)0\0, (d)0\\

Imm2(a), (b)0\0

Pnn2(a), (b)OlO

120

E 4 mrm Fmmm(a), (h)00\

222 Coca(a), (b)00\

F222(a), (b)00\, (a)lU, (d)lH

2/m. . Cmca(a), (h)00\

. . 2/m Ccom(e)llO, (f)llO

1 Pnnn(e)lU, (f)Ui

Pccn(a), (b)OOh

Pbaa(a), (b)OOl

F(z) 4 mm2 Fmm2(a)

..2 Coo2(c)U0

Aba2 {a)

0 8 222 Fddd(a), (b)OOl

DlzJ 8 . . 2 Fdd2 (a)

1 0 i raaaloj ^ (aj 222

Orthorhombic , univariant

P2x 2 2mm Pmmm(i), (j)OOi, (k)0\0, (DOW2. . P222(i), (g)00\, (k)OlO, {l)0\\

P2x{z} 2 . m. Pmm2(e)^ (f)0\0

P2y 001/100/010 2 m2m Pmmn(m), (n)00\, (o)\00, (p)^Oh

. 2. P222(m), (n)00\, {o)\00, {p)\0\

P2y[z] 001/100/010 2 m. . Pmm2(g), (h)\00

P2z 010/001/100 2 mm2 Pmmm(q), (t)0\0, (s)\00, (t)\\0

. . 2 P222(q), M\00, (s)0\0, {t)\\0

(P2x)c

100/010/002 4 2. . Pcom(i)00l, (3)011

(P2y)d

002/100/010 4 .2. Pmma(g), (h)OOi

(P2y)c

001/100/020 4 .2. Pccm(k)00l, (l)\Ol

(P2z)c

010/001/200 4 . . 2 PcQm(m), (n)\lO, (o)0\0, (p)lOO

200/020/001 8 2. . Cmma(h), (i)00\

002/200/010 8 .2. Cmma(3)lOO, (k) 10\

(P2-)ah 020/002/100 8 . . 2 Cmmm(m) HOCmma(l)lOO

121

(P2x),

(P2y).

(P2z),

200/020/002

002/200/020

020/002/200

16

16

16

2.

.2

Fmmm(l) III

FmrmCk) HIFmrrm(j ) HI

,2.P Biza.2,P Blx

c

2. .P Alyc

2. .P^Alyfz]

..2P Cly[z]3.

(.2.P Biz)a c

(. .2P^Clx)b c

(2. .P Aly)c a

(2. .P Aly) ,

c ab

001/010/100

010/001/100

010/001/100

100/001/010

100/010/002

001/100/020

020/001/100

020/002/100

2

2

2

2

2

4

4

mm2

2.

.

.2.

m.

.

m, .

..2

2.

.

. 2.

.2.

Prma(e)l00^ (fJllO

P222^(a), (b)0\0

P222^(c)00l, (d)lOl

Pmc2^(a), (b)lOO

Pma2 (a) lOO

Pcca(d) 100, (e)l\0

Fbom(c)OlO

Prrma(g)OlO

Cmca(e) HI

2^. ,CIlz

. .2^BIly

(2. . .CIlz)1 c

010/001/100

100/010/002

2

4

mm2

..2

m. .

..2

Pmrm(a), (h)0\0

P2^2^2(a), (b)OlO

Prm2^ (a)

Paan(a)UOj, (d)HO

C2x

C2x(z)

C2y

C2y(z)

B2x

A2y(z)

(C2x)

(C2y)

(B2x)j

(A2y).

010/100/001

010/100/001

100/001/010

001/100/010

100/010/002

010/100/002

100/002/010

002/100/010

4

4

4

2rm

2.

.

. m.

m2m

.2.

m. .

.

2. .

m. .

2. .

2.

.

.2.

Cmmm(g), (h)OOl

Phan(g), (h)OOl

C222(e), (f)OOl

Crm2 (d)

Cmrm(i), (o)00\

Pban(i), (j)OOl

0222 (g), (h)00\

Crm2(e)

Prma(e), (f)0\0

Amm2(d), (e)\00

Cmcm (e)

Coom(g)00l

Ibam(f)00l

Caom(h)00l

Tbam(g)00l

Irma (f)

Imma (g) 111U 14 U

122

C2z

A2xlz]

(C2z)

001/100/010

100/010/002

rm2

..2

. m.

CmrmCk), (l)0\0

Pban (k), (l)0\0

Pbam(e), (f)0\0

0222 (i), (j)OhO

Arm2 (a)

Ccom(i), (j)OlO

Ibam(h), (i)0\0

2, . .C Fly1 c

2,..C Fly(zl1 c

.2,.C Fix1 c

. .2.B, Fix± D

..2,A Fly{z)la

010/100/001

010/001/100

001/010/100

m2m

.2.

m.

.

2. .

2. .

m. .

Cmcm(c)00l

Phcn(c)00l

0222 ^(b) 00

1

Cmc2^(a)

0222 ^(a)

2nna(d) III

Ama2 (b)lOO

2. .P ^Flzah

(2,

.2P^^Fly

,2P^ Flxlz]DC.P , Flz)ab c

(.2.P, Fix)DC a

(..2P Fly),ac b

010/001/100

001/010/100

100/010/002

002/100/010

010/002/100

mm2 CTma(g)OlO

..2 0222 (k)UO

.2. Tcca(o)00l

.m. Abm2 (c)OlO

. . 2 Coca (h) HOIbca(e)OlO

2. . Ibca(c)00l

.2. Iboa(d) loo

I2x

I2x{zJ

I2y

I2y{zl

I2z

001/100/010

001/100/010

010/001/100

2rm Irrmn(e), (f)0\0

2. . Pnnn(g), (h)00\

1222(e), (f)00\

. m. Imm2 (c)

m2m Immm(g), (h)00\

.2. Pnnn(i), (j) lOO

1222(g), (h)\00

m. . Irm2 (d)

rm2 Imrmd), (j)lOO

..2 Pnnn(k), (l)0\0

'Pnnm(e), (f)0\0

1222 (i), (c)0\0

123

501-563 0 - 73 -9

.2.B^A Izb a

4 rm2 Irma(e)0l0

..2 Pnna(c)lOl

12.2^2 Jc)OU111. . 2C Ix

c b001/100/010 4 2. . 12^2,2 Ja)lOl111

2. .A Ca

lyc ^ 010/001/100 4 .2. 12 ^2^2^(b) HO

2. .A C ly(zla c

010/001/100 4 . 2. Ima2 (b ) lOO

F2x 8 2rm Fmmm(g)

2.. Cmca (d)

Ccaa (e)

F222(e), (o)HlF2x[z) 8 .m. Frm2 (d)

F2y 001/100/010 8 m2m Fmmm(h) .

. 2. Caca(f)

F222(f), (i)lll

F2y[z} 001/100/010 m.

.

Fnm2(c)

F2z 010/001/100 rm2 Fmrm(i)

..2 Coom(k)UO

Ccaa (g)

F222(a) . (h)iii

D2x 16 2.

.

Fddd(e)

D2y 001/100/010 16 .2. Fddd(f)

D2z 010/001/100 16 . . 2 Fddd(g)

Orthorhombic, bivariant

4 m.

.

Prrmn(u),(v)\00P2y2z

P2x2z 001/100/010 4 . m. Pmmm(w)j (x)0\0

P2x2y 010/001/100 4 . . m Vrnmiy), (z)00\

P2x2y(z) 010/001/100 4 1 I>rm2(i)

4

4

. , m

1

Pcom(q)

PGG2(e)

2. .P 2xyc

2. .Pc2xy{zl

124

m. .P 2xzam. .P. 2xy{z) 100/001/010

4

4

.m.

1

Prrma(i), (j)O^O

Pma2 (d)

010/001/100

4

4

m.

.

1

Prma(k) lOO

Pmc2^(c)

.2.P^Blz2y

2. .P^Aly2x(z}

.2.B2yz

2. .a:lxy{z] 010/001/100

4

4

m.

.

1

Pmna(h)

Pnc2 (c)

b. .C2xy

h..c:Ixyiz]

4

4

. . m

1

Pbam(g), (h)00\

Pba2(a)

010/100/001

4

4

. .m

1

Pbcm(d)00l

Poa2^(a).21P B C Flxylzlac a c

n. . I2xy

n. .i:;^x/lzl

4

4

. .m

1

PWLm(g)

Pnn2(o)

010/100/001

010/001/100

4

4

4

m.

.

.m.

1

Pmrm(e)

Pmrm(f)

Pim2^(b)

2^. .CIlz2y

.2^.CIlz2x

. .2^BIly2xIz)

1.2,B, A FI Ixz1 b a a

100/001/010

4

4

.171.

1

Pnma(o)OlO

Pna2^(a)12,. C A FI1 c a a

Ixy (z)

C2y2z

C2

A5

x2z

x2yiz)

010/100/001

001/010/100

8

8

8

m.

.

.m.

1

Cmmm(n)

Cmmm(o)

Arm2 (f)

C2x2y

C2 x2y{zl

8

8

. .m

1

Cmmm(p)^ (q)00\

Crm2 (f)

.n.C 2yzc

. .nA ^2xy(z) 001/010/100

8

8

m. .

1

Cmom(f)

Ama2(a)

2,..C Fly2x1 c

2...C Fly2x{zli c

8

8

. . m

1

Cmcm(g)00l

Cmo2 ^ (b)

125

,2.F2yz

.2.F2xy[zl 001/010/100

m. .

1

Cmca (f)

Aba2(b)

n. .C^F2xy

n. .C^F2xy{z)

. m Ccom (t)

Coa2 (d)

.m.P ^2yzabm. .P ^2xz

ab.m.P^ 2xy[z]

be

010/100/001

001/010/100

m. .

. m.

1

Cmma (m)

Cmma(n)OlO

Ahm2 (d)

I2y2z

I2x2z

I2x2y

I2x2y(z)

001/100/010

010/001/100

010/001/100

m. .

. m.

. .m

1

Immm (Z)

Immm(m)

Irmrm(n)

Imm2 (e)

b. .C 2xyc

b. .C 2xy[zlc

, m Ibam(j)

Iba2(o)

,2.B^2yz

2. .A 2xza

2. .A 2xy(z]a

m.

010/100/001

010/001/100

. m.

Irma(h)

Irmad)

Ima2 (c)

F2y2z

F2x2z

F2x2y

F2x2y[zl

001/100/010

010/001/100

010/001/100

16

16

16

16

m.

, m.

, m

Fmmm(m)

Frrmm(n)

Frrmn(o)

Fmm2 (e)

d, . D2xy {zl 16 Fdd2(h)

126

Orthorhombic , trivariant (site symmetry 1)

The 4-pointers:

P2x2yz P222 (u)

P222^(e).2,P Blx2yzc

P2^2^2(a)2^. .CIlz2xy

P2^2^2^(a)2,2, .FA B^C I LI Ixyz11 abcabc

The 8-pointers:

P2x2y2z Prrmn(a)

Pnnn (m)n. . I2x2yz

Paom(r)00l. .mP^2x2yz

b. .C2x2yz Pban (m)

Prma(l)m. .P 2xz2ya ^

2.2A 2xyza ^ Pnna(e)

.2.B2yz2x Pmna (i)

Paca(f).22P 2xyz3.C

b. .C2xy2z Pbam(i)

c.2F2xyz Pccn(e)

2.mP^ 2xyzbe

Pbom(e)

n. . I2xy2zj

Pnnm (h)

2^. .CIlz2x2y Pmmn(g)

b2.C 2xyzc

Phcn (d)

be. F2xyz Phoa(o)

. maBj^2xyz Pnma (d)

C2x2yz C222(l)

C222^(c).2^,C^Flx2yz

I2x2yz 1222 (k)

I2^2^2^(d)lOl. .2C B^lx2yzc b

The 15-pointers:

C2x2y2z Cmmm(r)

.n.C 2yz2xc

Cmcm(h)

.2.F2yz2x Cmca (g)

n. .C^2x2yz Ccom(m)00l

Crrma(o) lOOm. .P ^2x2y2ab

c. .F2x2yz Coca ('I)

I2x2y2z Immm(o)

b. .C 2x2yzc

Ibam(k)00l

22.P22xyz Ibca(f)

. 2.B^2yz2xb

Imma(j)

F2x2yz b ZZ2

(

kJ

The 32-pointers:

F2x2y2z Fmmm(p)

Fddd(h)d. .D2x2yz

128

Table 30

Lattice complex Multi- Site Occurrence

:

in all its plicity symmetry space group, Wyckoff letter.

representations (oriented J shift from standard representation

Monoclinic

,

invariant

0 1 2/m P2/m(a), (b)OlO, (a)00\, (d)\00.

(e)HO, (f)0\\,(g)m,(h)m

P{xzl 1 m Pm (a),(b)OlO

P(y} 1 2 P2 (a), (h)00\, (a) iOO, (d) \0l

(^) 2 1 F2^/m(a), (h)\00, (o)00\, (d)\0\

(^Pc

2 1 F27o(a), (b)UO, (o)0\0, (d) \00

^ ^ ^ab4 7J. Li Cj/ IIL \ i:^ J 4UL-'_, \ J J 1442

2 2/m C2/m(a), (h)0\0, (c)OOl, (d)0\\

C[xzl 2 m Cm (a)

Clyl 2 2 C2 (a), (b)00\

(^) A 2 1 P2^/c(a), (b)lOO, (c)00\, (d)lOl

4 1 C2/c(a), (b)0\0

^ e ^

( ) F 4 1 C2/c(c)llO^ (d)U2

Monoclinic, univariant

P2y

P2y{xz]

2

2

2

1

P2/m(i), (j)iOO, (k)OOl, (l)lOl

Pm (c)

2

2

2

1

P2/a(e)00l, (f)\Ol

Po (a)

CP^Aly

CP^AlyCxz)

C2y

C2y{xzl

4

4

2

1

C2/m(g), (h)00\

Cm (b)

(^)100/020/001. (^)100/010/002. (^)200/020/001. (^)001/010/100. ( ® ) 100/010/102

.

From standard representation to cell.

129

IC Flyc

IC Fly[xz]c

C2/a(e)00l

Cc (a)

Monoclinic, bivariant

2

2

m

1

P2/m(m), (n)OlO

P2 (e)

P2xz

P2xz(y]

2

2

m

1

P2^/m(e)OlO

P2^ (a)

2,P^ACIlxz1 b

2,P^ACIlxz(y]lb

C2xz 4

4

m

1

C2/m(i)

C2 (a)C2xz[y]

Monoclinic, trivariant

P2xz2y 4 1 P2/m(o)

TiiP^2xyz 4 1 P2^/m(f)

2P 2xyzc

4 1 P2/o(g)

cA2xyz 4 1 P2^/c(e)

C2xz2y 8 1 C2/m(J)

2,C F2xyz1 c

8 1 C2/c(f)

130

Table 31

Lattice complex

in all its

Multi-

plicity

Site

symmetry

Occurrence

:

space group, Wyckoff letter,

qIh T "Fi~ "F"pnm c5i~3'nria"Prl 'PPr>'PP=5PTii~;^"t" "T on

Anorthic (triclinic) , invariant

0Pixyz]

1

1

1

1

PI ra;, rbjool, rc;oio, rdj^oo^

reJHo, (f)ioh (g)o\h (h)mPI (a)

Anorthic (triclinic), trivariant

P2xyz 2 1 PI (i)

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Subgroup Relations

Tables 39-43

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173

INDEX

Anorthic system, 12

Asymmorphic, 23

Axis

alternating --, 3

co-ordinate 4

Neben --, 3

Zwischen --, 3

Bl ickrichtungen , 3

Complex, 1

lattice -- (LC), 11, 85-131

Weissenberg complex, 18, 69-83

Configuration, 11

Co-ordinates

condensed description of, 8

Crystal

- class, 8

- form, 1 , 6

- symmetry, 3

Degradation

cubic --, 9, 64

hexagonal --, 10, 66

Distribution symmetry, 17

Dome (=dihedron), 7

Face complex, 1

Form, crystal form, 1 , 4

General --, 4

special --, 4

Hauptgitter, 18

Hemisymmorphic, 23

High symmetry point (HSP), 3, 17

Isogonal polyhedron, 6

Lattice, 1

- complex, 1 , 11

- symmetry, 3

Lattice complex (L.C.)

general --, 11

special --, 11

limiting --, (L.L.C.), 11

invariant --, (I. L.C), 12, 35,

40, 44, 47, 50

variant --, 16, 38, 42, 48, 51

generating -- (G.L.C.), 16

Limiting site set, 4

Multiplicity

- of a lattice complex,' 11

- of a site set, 8

Nodes, 1

17 5

Parallelohedron (=pinacoid) , 7

N-pointer, 3, 6

N-punkter

N- punktner, 3

Position, 2, 11

point --, 2, 11

Prime

prime sign on Wyckoff letter, 19

Punktlage, 2

References, 26

Representation

- of a lattice complex, 11

standard --, 11

alternate --, 12

shifted --,14

congruent --, 14

enantiomorphous --, 14

Rhombicuboctahedron , 8

Shift vector, 14

Site

- set, 3, 6, 29, 55

- symmetry, 3, 11, 16, 62

general site set, 4

special site set, 4

generating site set, 5

Snub cube, 8

Space groups, 133-167

characteristic -- of a L.C.,

11, 12

Sphenoid (=dihedron), 7

Splitting, 5, 54

partial --, 5

complete --, 5

Subgroup relations

- in point groups, 9, 34

- in space groups, 23, 169-173

Symbol

s

Hermann-Mauguin --, 3

oriented --, 4

Symmetry

- direction, 3

- element, 3

- operation, 17

Symmorphic, 23

Synonyms of G.L.C. , 17

Transformation of co-ordinates,

4 - and 3 - index symbols, 9

Triclinic (=anorthic) system, 12

Truncations, 8

Weber four-index symbols, 9

176

ERRATA

p. 17, Sn. 6.4, line 4: in front of space group insert characteristic

c9p. 133, Im3m(0, ), second line of entry, in 6(c). 3m:

replace 6 with 8

3p. 138, P6„/mcm(D ), position 4(e)3.m is correctly given. [Note erratum

in I. T. (1965, p. 303); instead of 4(e)6 read 4(3)3m]

p. 150', P4/nmm(D, ,^) , first line of entry, above (h)..2:

replace OiO with OOi

gp. 150, P4/ncc(D^^ ), second line of entry, after 4(c):

replace 4.. with 4..

2p. 154, P4bm(C, ), first line of entry, above 4(c)..m:

replace OiO.b.CZxx with 0i0.b.C2xx[z]

p. 155, P422^2(D^ ), second line of entry:

replace 4(c) with 4(c)2..replace (d)2.. with 4(d)2..

4p. 158, Pban(D2j^ ), under Pban add choice of origin : 0 in 222

p. 165, C2/c(C2^ ), second line of entry:replace (b) with (b)l

replace (c) with 4(c)

Remark.- Some of the conventional position letters used by Wyckoff (1930) haveoccasionally been interchanged in Vol. I of the I.T. (1952 and reprintings) . Theoriginal Wyckoff letters have been used in the present lattice-complex tables.Agreement with the I.T. can be achieved by means of the following changes.

p. 105, under P, at P , 2, 42m, P42/mcm:replace (a) with (b)

replace (b) with (d)

[Note that (b) should have read (c)]

p. 105, under P, at P , 2, m.mm, P4„/mcm:C 2.

replace (b) with (a)

replace (d) with (c)

p. 108, under I, at F^, 8, 4.., 14^/acd:replace (b) with (a)

on the next line, at 2.22, 14, /acd:

replace (a) with (b)

p. 150, P42/mcm(D^^"'"^) : interchange (a) and (b); likewise (c) and (d)

20p. 151, l4^/acd(D^^ ): interchange (a) and (b)

p. 152, P42c(D2^^): replace (g)(h)(i)(j) with (j)(i)(h)(g)

Wyckoff, R.W.G. (1930). The analytical expression of the results of thetheory of space groups. Second edition. "Carnegie Institution of Washington.

177

FORM NBS-114A (1-71)

U.S. DEPT. OF COMM.BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO.

NBS MN-134

2. Gov't AccessionNo.

3. Recipient's Accession No.

4. TITLE AND SUBTITLE

Space Groups and Lattice Complexes

5. Publication Date

May 1973

6. Performing Organization Code

7. AUTHOR(S)W. Fischer, H. Burzlaff, E. Hellner and J.D.H. Donnay

8. Performing Organization

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDSDEPARTMENT OF COMMERCEWASHINGTON, D.C. 20234

10. Project/Task/ Work Unit No.

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12. Sponsoring Organization Name and Address

Same as No. 9.

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Final

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15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significantbibliography or literature survey, mention it here.)

The lattice complex is to the space group what the site set is to the point group—-an assemblage of symmetry-related equivalent points. The symbolism introduced byCarl Hermann has been revised and extended. A total of 4Q2 lattice complexes arederived from 67 Weissenberg complexes. The Tables list site sets and latticecomplexes in standard and alternate representations. They answer the followingquestions: What are the co-ordinates of the points in a given lattice complex?In which space groups can a given lattice complex occur? Wfiat are the latticecomplexes that can occur in a given space group? The higher the symmetry of thecrystal structures is, the more useful the lattice-complex approach should be onthe road to the ultimate goal of their classification.

17. KEY WORDS (Alphabetical order, separated by semicolons) Crystallography;' crystal point grOupS ;

crystal structure; lattice complexes; site sets; space groups.

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