Digitized by the Internet Archive in 2008 with funding from IVIicrosoft Corporation http://www.archive.org/details/analyticalexpresOOwyckrich THE ANALYTICAL EXPRESSION OF THE RESULTS OF THE THEORY OF SPACE-GROUPS -*.--'iar- BT RALPH W. G. WYCKOFF PUBUSHED BY THE CARNEGIE INSTITUTION OF WASHING^N Washington, October, 1922 V'^ CARNEGIE INSTITUTION OF WASHINGTON Publication No. 318 THE TECHNICAL PRESS WASHINGTON, D. C. PREFACE. With the development of such methods of stud5dng the arrangement of the atoms in crystals as are furnished by the phenomena of the diffraction of X-rays, the geometrical theory of space groups becomes of the utmost importance. Until recently the work published upon this theory has been directed primarily toward the preparation of a statement of all the different kinds of symmetry (groupings of elements of symmetry) which are crystal- lographically possible. This statement, to be complete, must necessarily give all of the possible ways of arranging points in space which, by their arrangement, will express crystallographic symmetry. In its most general form such an analytical expression of the results of this theory was given by Schoenflies.* Before it is applicable to the study of the structures of crystals, however, modifica- tions in this original representation are necessary. First, there must be selected such a portion of the grouping that in its calculated effects upon X-rays it can be taken as typical of the entire arrangement. It is thus necessary to state a space group in terms of the equivalent positions which lie within a unit cell rather than by giving, as Schoenflies does, the equiva- lent positions about one point of the lattice underlying the grouping. This rather obvious modification has of course been made by those who have used the space groups as a guide in studying crystals. The second modifi- cation, or rather amplification, is not so readily made. The X-ray experi- ments which have already been carried out show that the number of particles (atoms) contained in the unit cell is commonly smaller than the number of most generally placed equivalent points of the space group having the sym- metry of the crystal. The special arrangements of the equivalent points (upon axes, planes, and other elements of symmetry), whereby the number of most generally placed equivalent positions is reduced, are thus of great importance and it becomes essential to be able to state all of them in any particular case. Nigglif has already given the simpler of these special cases. For some time the writer has been engaged in working out all of them and the following tables are an expression of the results of these computations. It was the original intention simply to state these results and to outline the method whereby they were obtained. The writer is firmly convinced, however, that sure and definite progress in this relatively new field of crystal structure can be realized only by making the fullest possible use of the added information which the theory of space-groups furnishes; and since any dis- cussion of this theory is almost completely absent from work published in English, it has seemed worth while to add a brief introduction in order to give such details of the space groups and of their development as seem to furnish sufficient background for the appreciation of the importance of the theory in this new field of physical science. * Krystallsysteme und Erystallstructur (Leipzig, 1891). t Geometrische Krystallographie des Discontinuums (Leipzig, 1919). Ill 3005 25 IV PREFACE. At present a Imowledge of the method of derivation is not required by the crystal analyst or by the person primarily interested in using the results of such X-ray studies. Those interested in the theory as a geometrical problem will of course find the development thoroughly given by Schoenflies. In the only publication available to Enghsh readers Hilton* has summarized, in excellent form, the work of Schoenflies, introducing at the same time some of the methods of representation employed by Federov. A thorough under- standing of the manner of developing the theory, however, is best attained from a study of the original work of Schoenflies. The discussion in the pres- ent book is intended for those who wish only to get a sufficient idea of the nature of the results of the theory of space-groups so that these results can be intelligently used. For the substance of this discussion the writer's obligation to Schoenflies is obvious; the work of Hilton has also been used with entire freedom. In the book to which reference has already been made Niggli has given the positions within the unit cell (of each space-group) of all of its elements of symmetry. This information, while of no aid in the actual determination of the structures of crystals, may prove useful in the attempt to derive from these structures additional information, such as that bearing upon the internal symmetries of their constituent atoms. In comparing the partial analytical expression given by Niggli with his results based directly upon those of Schoenflies, the writer found that particularly in the case of the tetragonal space-groups there were many differences, owing chiefly to the choice of different points as the origin of coordinates. Because of the possible usefulness of the additional data relating to the positions of elements of symmetry that are furnished by Niggli, it has seemed desirable, in spite of some loss of logicality, to recalculate these groups so that they would accord with those already published. Similar differences exist in orthorhombic and monoclinic groups; the changes necessary to reconcile the two descriptions are in these cases sufficiently obvious, however, that it has seemed worth while only to indicate in some more or less illustrative instances the nature of the translations necessary to bring about a general coincidence. The writer wishes to express his gratitude to Dr. S. Nishikawa for the advice and criticism given him when in 1917 he began to familiarize himself with the theory of space-groups. Geophysical Laboratory, March, 1921. * Mathematical Crystallography (Oxford, 1903). CONTENTS. Paob Chapter I. Historical Introduction 1-3 Chapter II. Nature of the Space-Groups 4-38 Elements of Symmetry 4 Point-Groups 6 Analytical Representation of the Point-Groups: TricUnic System 11 Monochnic System 12 Orthorhombic System 13 Tetragonal System 14 Cubic System 16 Hexagonal System 18 Space Lattices 22 Space-Groups 24 An Outline of the Derivation of the Space-Groups: Triclinic System 26 Monochnic System 27 Orthorhombic System 27 Tetragonal System 30 Cubic System 33 Hexagonal System 35 Chapter III. The Apphcation of the Theory of Space-Groups to Crystals 39-46 Units of Structure 39 Space-Groups and Crystals 42 Special Cases of the Space-Groups 44 The Treatment of Calcite as a Typical Case 45 Chapter IV. The Complete Analytical Expression of the Space-Groups 47-180 Trichnic System: A. Hemihedry, Ci 48 B. Holohedry, Cj 48 Monochnic System: A. Hemihedry, Cg 49 B. Hemimorphy, C2 49 C. Holohedry, C| 49 Orthorhombic System: A. Hemimorphy, C2 52 B. Hemihedry, V 56 C. Holohedry, V*' 59 Tetragonal System: A. Tetartohedry of the Second Sort, S4 73 B. Hemihedry of the Second Sort, V*^ 73 C. Tetartohedry, C4 79 D. Paramorphic Hemihedry, C4 80 E. Hemimorphic Hemihedry, C4 83 F. Enantiomorphic Hemihedry, D4 86 G. Holohedry, D^ 89 V V I CONTENTS. Cubic Sj'^stem: The Special Cases of the Cubic System 103 A. Tetartohedry, T 121 B. Paramorphic Hemihedry, T*^ 123 C. Hemimorphic Hemihedry, T 128 D. Enantiomorphic Hemihedry, O 132 E. Holohedry, O^ 138 Hexagonal System — Rhombohedral Division: A. Tetartohedry, C3 151 B. Hexagonal Tetartohedry of the Second Sort, C3 151 C. Hemimorphic Hemihedry, C3 152 D. Enantiomorphic Hemihedry, D3 153 E. Holohedry, D^ 155 Hexagonal System — Hexagonal Division: A. Trigonal Paramorphic Hemihedry, C3 157 B. Hemihedry with a Three-fold Axis, D'3* 158 C. Hexagonal Tetartohedry, Cq 160 D. Hemimorphic Hemihedry, Cg 161 E. Paramorphic Hemihedry, Cg 162 F. Enantiomorphic Hemihedry, Dg 163 G. Holohedry, Dg 166 Summarizing Tables 170 TABLES. Table 1. A Comparison of Some Current Systems of Point-Group (Crystal Class) Nomenclature 10 Table 2. The Unit Cells of Each of the 14 Space Lattices 42 Tables Summarizing the Numbers op Special Cases op Space-Groups Having THE Symmetry op Each op the Systems op Crystal Symmetry: Table 3. Triclinic System 170 Table 4. Monochnic System 170 Table 5. Orthorhombic System 171 Table 6. Tetragonal System 174 Table 7. Cubic System 176 Table 8. Hexagonal System — Rhombohedral Division 178 Table 9. Hexagonal System — Hexagonal Division 179 THE ANALYTICAL EXPRESSION OF THE RESULTS OF THE THEORY OF SPACE-GROUPS By Ralph W. G. Wyckoff vn CHAPTER I. HISTORICAL INTRODUCTION.* The investigation of the structure of crystals involves the study both of the substance from which the crystals are made and of the way in which this material is arranged in space. Until very recently, practically all of the information bearing upon the first of these points has arisen from the realiza- tion of the probable physical reality of the chemical atom. How these atoms are associated together in crystals and whether the chemical mole- cule, or some other aggregate of atoms, has the significance in solids which it possesses in gases and Hquids are questions which have been answered only by conjecture and inference. The development in the other direction, however, presenting a problem which in its most general statement is inde- pendent of current hypotheses concerning the nature of the material from which crystals are built, has been capable on the other hand of a far-reaching and apparently satisfactory growth. In the days when an atomic structure of matter was a crude working hypothesis without any basis in experimentally determined fact, we find Robert Hookef reproducing the forms of alum by properly pihng up "a company of bullets and some few other very simple bodies," very much as we represent the structure of a crystal on the basis of X-ray measurements. It was the phenomenon of regular cleavage, however, that suppHed the evidence upon which early hypotheses of the regular arrangement of the material of crystals were based. For instance, Westfeld| considered calcite as built up of tiny rhombohedrons; and Bergman,§ basing his behefs partly on the observation of Gahn that a skalenohedron of calcite yields a rhombo- hedron on cleaving, developed what might be called the first geometrical theory of crystal structure. For just as the crystals of calcite could be considered as an aggregate of minute rhombohedrons placed parallel to one another, so garnet or pyrite or other crystals can be developed similarly from certain fundamental forms. These ideas seem to be essentially the same as those held by Hauy.^ He, also, considered cleavage as the guiding factor. The cleavage units, his molecules iniegrantes, were either tetrahedra, triangular prisms, or parallelopipeda, and he showed how crystals with vari- ously developed faces could be represented by the aggregation of these units. These ideas of Hauy were built around the law of rational indices, though they were fundamentally independent of it. Many objections to the details of the hypothesis of Hauy arose, as indeed they must arise against any theory based primarily upon cleavage. Not only does the existence of the many * Most of the material for this introduction is given by L. Sohncke, Entwickelung einer Theorie der Krystallstruktur (Leipzig, 1879). It is given in English and brought up to date in a report of the Brit. Assoc. 297-337. 1901. t Micrographia (London, 1665), p. 85. t Mineralogische Abhandlungen, Stuck I. 1767. § Nov. Acta. Reg. Soc Se. Upsal. 1773, i; Opusc. (Upsala) 1780, ii. q Essai de Cristallographie (Paris) 1772; etc. r^;.^". HISTORICAL INTRODUCTION. crystals which show no cleavage necessitate many supplementary hypotheses, but the observed cleavage of such substances as fluorite (with octahedral cleavage) is not readily accounted for by any kind of close-fitting units. Simultaneously with the extension of the belief in the atomic nature of substances, and perhaps because of this belief, emphasis came to be shifted from the shape of the crystal units to the relative positions of their centers of gravity as centers of some sort of crystal molecules. Thus there evolved from these different speculations the basis for a suitable geometrical study in the definite conception of a crystal as composed of units of undefined shape repeated in some regular fashion throughout space. In such a regular pattern for repeating the crystal unit we have a space lattice. All of the symmetrical networks of points which can have crystallo- graphic symmetry were found geometrically by Frankenheim. * Some years later this was done more accurately and rigidly by Bravais.f As a result of his work, Bravais looked upon a crystal as built up by placing units of a suitable symmetry all in the same orientation at the points of one of these symmetrical networks. Thus the unit of a cubic crystal might have cubic or even tetrahedral symmetry, but it could not, for instance, have monoclinic or hexagonal symmetry. As a matter of fact, Bravais thought of his units as groups of atoms forming some sort of a crystal molecule, though such a view is not a necessary part of the geometrical development. In this theory of Bravais, in which a crystal is composed of aggregates of atoms repeated regularly and indefinitely through space, is to be found the beginning of an adequate treatment of the possible groupings of matter in crystalline bodies. The objections to Bravais' theory, however, are many and obvious. In the first place, all of the space lattices have the complete symmetry of some one of the crystal systems, so that, in order to account for the lower degrees of symmetry, it was necessary for him to ascribe the degradation in such cases to the shape of the crystal units, or molecules, without at the same time being able satisfactorily to treat these units. Again this theory implies a distinct restriction, and one which had not been proved necessary, that all of the crystal molecules must have the same orientation throughout the crystal. In the course of a general study of the theory of groups of movements JordanI gave a perfectly general method for defining all of the possible ways of regularly repeating an identical grouping of points indefinitely throughout space. By combining this treatment of Jordan with the principle (laid down by Wiener) that regularity in the arrangement of indentical atoms is attained when "every atom has the other atoms arranged about it in the same fashion," Sohncke§ eventually deduced all of the typical ways of regu- larly repeating identical groupings of atoms throughout space so that the * Die Lehre von der Cohasion (Breslau, 1835). t Journ. de I'ficole Polytech. (Paris) XIX, 127. 1850; XX, 102. 1851. t Annali di matematica pura ed applicata (2) 2, 167, 215, 322. 1869. § L. Sohncke, op. cit. HISTOEICAL INTRODUCTION. 3 total assemblage will possess erystallographic symmetry.* This method of treatment in attacking the problem of the arrangement of the points within what was the crystal unit or molecule of Bravais brings the problem towards its final solution. None of the systems of Sohncke can be made to account in an entirely satisfactory manner for the enantiomorphic (mirror-image) characteristics of many crystals. SchoenfUesf was led to consider that every point of an assemblage must have all of the other points ranged about it in a "hke fash- ion," where "likeness" may refer either to an identical arrangement or to a mirror-image similarity. Starting from this basis, he obtained the 230 space groups which represent all of the possible typical ways of arranging (symmetry-less) points in space so that the grouping will possess the sym- metry of one of the thirty-two crystal classes. The same derivation of the space groups was accomplished independently by Federov| and by Barlow, but at present the work of Schoenflies is the most useful because it is pre- sented in a form that is of immediate application. With the aid of this final theory of space groups the different degrees of symmetry exhibited by crystals can at last be traced back definitely and precisely to the arrangement of the atoms in the crystals (without postulating any characteristics of sym- metry for them) . Besides indicating the elements of symmetry which are characteristic of each of the 230 typical ways of arranging points in space, Schoenflies gives, in general terms, the coordinates of the points in each of these groupings which are equivalent to one another. The discovery of the diffraction of X-rays and the consequent develop- ment of the physical methods for studying the structure of crystals have made this analytical expression of the results of the theory of space groups of the utmost importance. It is the purpose of the present work to give these results a detailed expression, thereby putting them into a form in which they will be immediately useful as an aid to the study of the arrangement of the atoms in crystals. X-ray experimentation thus far carried out shows that the special cases which result when equivalent points (the atoms in crystals) lie in some element or elements of symmetry, such as axes or planes, are the ones which are physically most important. As a consequence the prepara- tion of this detailed expression, in so far as it introduces material which is not outlined in the work of Schoenflies, has made necessary the working out of all of these special cases for all of the space-groups. * At first Sohncke seems to have been inclined to view all of the points of a point system as regular and all of one kind. When the insufficiency of this theory was emphasized he postulated the presence of a few different kinds of points (which can be made to correspond with different kinds of atoms). The partial grouping composed of the points of any one kind is homogeneous; at the same time the different groupings all have the axes and the other elements of symmetry in common. t A. Schoenflies. Krystallsysteme u. Krystallstruktur (Leipzig, 1891). t E. Federov. Z. Kryst. 24, 209. 1895; W. Barlow. Z. Kryst. 23, 1. 1894. Federov's work appeared, in Russian, before that of either of the other two. CHAPTER II. NATURE OF THE SPACE - GROUPS. ELEMENTS OF SYMMETRY. Axes of symmetry. — An axis of rotation of a figure* is a line about which the figure can be rigidly turned. The angle of the rotation is the angle between the final and initial positions of a plane which contains the axis of rotation. A figure is said to possess an axis of symmetry when rotation through a definite angle about an axis of rotation will cause the figure to assume the same point-for-point configuration that it originally possessed. The angle of the rotation about an axis which is required to bring about this coincidence is called the angle of the axis of symmetry. Every figure has an infinite number of 27r axes of symmetry; that is, a complete rotation of 360" about any line through a body will cause it to assume its original configuration. The operation of such a 27r (one-fold) axis is called the iden- tical operation of symmetry (or simply the identity). If a rotation of 180° is sufficient to effect a coincidence, the axis of rotation is a 180", or two-fold axis of symmetry; more generally, an n-fold axis of symmetry is one for which 27r a rotation of angle — brings about coincidence. One-, two-, three-, four- and n six-fold axes are found in crystals (and in figures possessing crystallographic symmetry). (Figure 1.) \ \ I r 60\ / /I / >a*^ Fig. 1. The crystallographically significant rotational axes of symmetry. PZane of symmetry. — In figure 2 the line POP' is perpet dicular to the plane ABCD. If then PO equals OP' in length, the point P' stands in a mirror- image relation to the point P. If a plane can le passed through a figure so that every point of the figure upon one side of this plane has a corresponding * By a figure is meant any sort of a collection of points, lines, planes, and 4 so on. ELEMENTS OF SYMMETRY. 5 point in a mirror-image position upon the other side of the plane, the plane is a plane of symmetry. Center of sijmmeiry. — A point of a figure is a center of symmetry if a line drawn from any point of the figure to it and extended an equal distance beyond will encounter a point corresponding to the arbitrarily chosen point. (Figure 3.) Fig. 2. Fig. 3. O is the center of symmetry of a figure in which P and Pi are corresponding points. Screw-axes of symmetry. — A figure is said to experience a translation when every point of the figure is moved by the same amount in the same direction. A rotation about an axis accompanied by a translation along the axis of rotation is called, a rotary translation. This screw-motion must be defined 6 THE NATURE OF THE POINT-GROUPS. both by the angle of the rotation and by the amount of the translation. The axis of the rotation (and the line of the translation) is called a screw- axis. If such a rotary translation will bring the points of a figure into co- a- :> Fig. 4. The crystallographically significant screw axes of symmetry. incidence, the axis of the motion is a screw-axis of symmetry. In a figure having crystallographic symmetry these screw-axes may be one-, two-, three-, four- or six-fold. (Figure 4.) Glide planes of symmetry. — If a figure can be brought into point-for-point co- incidence by a reflection in a plane combined with a" translation of a defi- nite length and direction in the plane, the plane is called a glide plane of symmetry. In this case the transla- tion-reflection must be defined both by the position of the plane and by the length and direction of the translation. (Figure 5.) POINT-GROUPS. The thirty-two ways of suitably com- bining these planes, axes, and centers of sjnnmetry give the elements of sym- metry which are characteristic of the 32 classes of crystallographic symme- try. Each one of these combinations of symmetry elements is a point-group. Thus, a point-group may be defined by stating either the elements or operations* of symmetry which characterize it. * By an operation of sjonmetry is meant any movement which will bring about a point-for- point coincidence. For instance, a six-fold axis of rotation possesses six operations of symmetry Fig. 5. Pi is a glide reflection of P in the plane shown in the figure. THE NATURE OF THE POINT-GROUPS. 7 The elements of symmetry characteristic of each of the 32 point-groups will now be given. The cycUc groups have only one axis of symmetry. They may be written symbolically as where n may be either 1, 2, 3, 4 or 6. A will be taken as the symbol of a rotation so that the term within the braces is to be considered as defining a rotation of angle — . n 27r Dieder-groups have one principal axis of symmetry of angle — and n two- fold axes in a plane at right angles to the principal axis. where U (Umklappung) will be used to represent the two-fold rotation of the secondary axes. The value of n may be 1, 2, 3, 4 or 6. The group Di is clearly identical with C2, however. The positions of the axes of the other groups are shown in figure 6. The group for which n = 2 furnishes the spe- cial case of three two-fold axes at right angles to one another; this group is more commonly known as the vierer-group and is designated as V. Fig. 6. The tetrahedral group (symbol = T) has 3 two-fold axes at right angles to one another (like the vierer-group) and 4 three-fold axes so placed that if the two-fold axes are taken to bisect the sides of a circumscribed tetrahedron, the 4 three-fold axes will each one pass through the point of intersection of the two-fold axes and through one of the corners of the tetrahedron (figure 7). The octahedral group (symbol = 0) has 3 four-fold, 4 three-fold, and 6 two-fold axes arranged in the same manner as are the altitudes, the body- diagonals, and the face-diagonals of a cube (figure 8). The groups which have so far been considered require only simple rota- tion axes for their expression; they are commonly called groups of the first sort. Those that now follow are groups of the second sort. 8 THE NATURE OF THE POINT-GROUPS. Cyclicgroups of the second sort possess one screw-axis of symmetry =■ = {<!)}■ where the symbol in brackets may be taken as a rotary translation of angle — . The value of n may be 1, 2, 3, 4 or 6. When n = 1 the rotary translation is clearly equivalent to a reflection in a plane at right angles to the axis of rotation. Thus, in figure 9a the rotary translation of angle 27r will bring the point P to the position Pi; this operation is, however, equivalent to a reflection in the plane through normal to pp', where Op is equal to one- half of the length r of the translation component of the rotary translation. When n = 2, the resulting rotary translation Op is equivalent to an inversion through the point of figure 9b. These two groups may thus be written Ci={A(27r)) = {S} and C2= {A(7r)} = {1} where S (Spiegelung) stands for a reflection and I for an inversion. In a similar fashion it will be seen that when n==4, this group is identical with one obtained by combining a rotation A( 2 ) with a reflection S^ in a hori- zontal plane of symmetry. Thus C. = Ki)} = {A(i>Bfs.. Other groups of the second sort can be obtained by combining a principal axis of rotation with a plane or with a center of symmetry. Three types of such groups having but one axis of symmetry are possible: (1) when the plane of symmetry is normal to the axis of symmetry (a horizontal reflecting plane), (2) when the plane of symmetry contains the axis of symmetry (a vertical THE NATURE OF THE POINT-GROUPS. 9 reflecting plane) and (3) when the new element of symmetry is a center of symmetry. These three types may then be written (1) CS= {Cn, Sh} (2) Cl= {Cn, Sv} (3) C^= {Cn, 1} It can be shown that if n is odd, all three of these types are possible. When, however, n is even, the number of different groups for any value of n is but two. The groups of this sort that are thus possible are the following: When n = l. — The group Ci is clearly the same as the group Cl; further- more it is identical with the group Ci. Similarly the group Ci is identical with the group Cg. Fig. 8. When n = 2, 4 or 6.— CS = CL, so that groups of the types CS = CL and Cl are possible. When n = 3. — Point-groups of all three types are possible. Some new groups arise by combining the axes of a group of the type Dn with a reflection plane. The plane of symmetry may Ue in the horizontal position (normal to the principal axis of symmetry) ; if it lies in the vertical position new groups will be obtained only when the plane bisects the angle between secondary axes (a diagonal plane). It can furthermore be shown that in the latter case groups of crystallographic significance will be obtained only when n = 2 and when n=3. Thus, when n = 2, 3, 4, or 6, we have the new groups D|-V^, D5 = V^ Dg, DS, The groups T* and T** arise from the tetrahedral group, T, by combining the axes of T with a horizontal and with a diagonal-vertical reflecting plane, respectively. One new group, O, can be produced from the octahedral group O. All of the 32 groups have now been defined. On the basis of their total symmetry these 32 point-groups can be placed in 6 (or 7) systems, the systems of crystallographic symmetry.* This classification of the point-groups is given in Table 1, together with the names of the classes of crystal symmetry (according to Schoenflies, Dana, and Groth) corresponding to each. * A basis for this classification will become evident when the point-groups are discussed sep- arately and given an analytical expression. 10 NOMENCLATURE OF THE POINT-GROUPS. Table 1. Symbol. Class of symmetry. S) a o 1 o d of symmetry and | of equivalent j points. 1 SCHOENFUBB. Dana. GftOTH. I. Triclinic system: 1. Ci Hemihedry Asymmetric Asymmetric pedial 1 2. C,«Sj=Ci II. Holohedry Monoclinic system. Normal Pinacoidal 2 3. Ci-C^-Cg Hemihedry Clinohedral Domatio 2 4. C« Hemimorphic hemihedry Hemimorphic Monoclinic sphenoidal 2 5.CJ III. Holohedry Orthorhombic system. Normal Monoclinic prismatic 4 6.0- Hemimorphic hemihedry Hemimorphic Rhombic pyramidal 4 7. D8=V Enantiomorphic hemihedry Sphenoidal Rhombic bispbenoidal 4 8. dJ-V** Holohedry Normal Rhombic bipyramidal 8 IV. Tetragonal system. ^ 9. S4=C4 Tetartohedry of second sort Tetartohedral Tetragonal bispbenoidal 4 MO. v'^-dJ Hemihedry of second sort Sphenoidal Tetragonal scalenohedral 8 ^ 11. C« Tetartohedry Pyramidal hemimorphic Tetragonal pyramidal 4 \ 12. Cj Paramorphic hemihedry Pyramidal Tetragonal bipyramidal 8 13. Cj Hemimorphic hemihedry Hemimorphic Ditetragonal pyramidal 8 ^ 14. D4 Enantiomorphic hemihedry Trapezohedral Tetragonal trapezohedral 8 \ 16. dJ V. Holohedry Cubic system. Normal Diteti agonal bipyramidal 16 16. T Tetartohedry Tetartohedral Tetrahedral pentagonal dode- cahedral 12 17. T^ Paramorphic hemihedry Pyritohedral Diacisdodecahedral 24 18. T'^ Hemimorphic hemihedry Tetrahedral Hexacistetrahedral 24 19. Enantiomorphic hemihedry Plagihedral Pentagonalicositetrahedral 24 20. O'* VI. Holohedry Hexagonal system. Normal Rhombohedral Division Hexacisoctahedral 48 21. C. Tetartohedry 24. Trigonal pyramidal 3 22. C\ Hexagonal tetartohedry of second sort Trirhombohedral Rhombohedral 6 23. C^ Hemimorphic hemihedry Ditrigonal pyramidal Ditrigonal pyramidal 6 24. ^r^ 25. Df Enantiomorphic hemihedry Trapezohedral Trigonal trapezohedral 6 Holohedry Rhombohedral Ditrigonal scalenohedral 12 26. Cj Hexaponal Division Trigonal paramorphic hem -23. Trigonal bipyramidal 6 _h hedry 27. D^ Trigonal holohedry Trigonotype Ditrigonal bipyramidal 12 28. C« Tetartohedry Pyramidal hemimorphic Hexagonal pyramidal 6 29. C, Paramorphic hemihedry Pyramidal Hexagonal bipyramidal 12 80. C^ Hemimorphic hemihedry Hemimorphic Dihexagonal pyramidal 12 31. D. Enantimorphio hemihedry Trapezohedral Hexagonal trapezohedral 12 82. dJ Holohedry Normal Dihexagonal bipyramidal 24 Note. — It may be remarked that the numbers of the first column have no particular sig- ni^p^ce and do not refer to any of the current systems of designating symmetry classes. THE TRICLINIC POINT-GROUPS. 11 The analytical expression of the point-groups. — On the basis of the defini- tions of the 32 point-groups, the operations of symmetry (footnote on page 6) that characterize each of them can be immediately written. Furthermore, it is evident that if any point, x, y, z, is subjected to each of the operations of a point-group, a group of equivalent points will result whose symmetry is Fig. 9. that of the point-group; thus its analytical expression is obtained. The operations of S3anmetry which are characteristic of each of the point-groups will now be stated and through them analytical representations given to each of these groups. TRICLINIC SYSTEM. Point-group Ci. — This group has but one element of symmetry, the identical operation (symbol =1). Since the identity brings any point x, y, z into coincidence with itself, any single point, xyz, serves as an analytical rep- resentation of this group. The coordinate axes to which these coordinates refer obviously can be any three Unes in space, of unequal unit lengths and making unequal angles with one another. Such axes will be called the tri- cUnic axes of reference. They are equally serviceable for the point-group, Ci, which follows. Pointr-group d. — The operations of S5rmmetry characteristic of this group are the identity (obviously an operation of every group) and an inversion 12 THE MONOCLINIC POINT-GROUPS. (symbol = I). Since an inversion through the origin of coordinates changes the signs of all three coordinates (figure 3) the operations of symmetry and the coordinates of equivalent points of this point-group are Operations of symmetry: 1, I. Coordinates of equivalent points: xyz; xyz. MONOCLINIC SYSTEM. In their analytical expressions all of the point-groups having the symmetry of this system can be referred to a system of axes, two of which (the X- and Y-axes) make any angles with one another; the third axis (the Z-axis) is normal to the plane of these other two. The Z-axis consequently is taken to coincide with the principal two-fold axis, where such exists. Point-group Cg. — The single operation of symmetry (besides the identity) of this group is a reflection to be taken in the horizontal (XY-) plane. Since such a reflection (symbol = Sii) changes the sign of the z-coordinate (figure 10), the operations and equivalent and equivalent points of this group are Operations of symmetry: 1, S^. Coordinates of equivalent points: xyz; xyz. Fig. 10. Fig. 11. Point-group C2. — A two-fold rotation about an axis (the Z-axis) normal to the plane of the other two axes of coordinates, changes the signs of these two coordinates (figure 11). Consequently the point-group C2 can be ex- pressed as Operations of symmetry: 1, A(7r). Coordinates of equivalent points: xyz; xyz. Point-group C|.— Since this group is developed by mirroring C2 in a hori- zontal (XY-) plane of symmetry, it is to be expressed as follows: Operations of symmetry: 1, A(ir), Sh, A(7r)Sh. The operation whose symbol is A(ir)Si„ the product of A(7r) and S^, is to THE ORTHORHOMBIC POINT-GROUPS. 13 be understood as a two-fold rotation followed by a reflection in the hori- zontal plane.* Coordinates of equivalent points, xyz; xyz; xyz; xyz. ORTHORHOMBIC SYSTEM. The orthorhombic axes of reference are three mutually perpendicular axes of unequal unit lengths. Point-group Cl- — Since reflection in a plane containing two of the axes of reference and normal to the third changes the sign of the coordinate value for the third (confer C2), this point-group may be expressed as Operations of sjmimetry: 1, A(7r), Sy, SvA(7r). Sy is a reflection in a vertical plane (taken through Y and Z). Coordinates of equivalent points: xyz; xyz; xyz; xyz. Fig. 12. Point-group V. — The rotations about the three mutually perpendicular two-fold axes will be designated as U, V, and W (figure 12). Operations of sjTnmetry : 1, U, V, w. Coordinates of equivalent points : xyz; xyz; xyz; xyz. Point-group V. — ^As usual, the XY-plane is taken as the horizontal mir- roring plane. Operations of symmetry: 1, U, V, W, S„ USb, vs„ ws,. Coordinates of equivalent points: xyz; xyz; xyz; xyz; xyz; xyz; xyz; xyz. * The order of combining the operations in such a product is immaterial. It could equally well have been called a reflection followed by a two-fold rotation. 14 THE TETRAGONAL POINT-GROUPS. TETRAGONAL SYSTEM. The three tetragonal axes of reference, mutually perpendicular to one another, are two (the X- and the Y-axes) of equal unit length. Point-group 04 = 84. — Operations of symmetry: 1, S,a(^), A(7r)*, Coordinates of equivalent points : S.A - (!) xyz; yxz; xyz; yxz. Point-group C4. — As we have just seen, the rotation of angle - about the Z-axis interchanges the X and Y coordinates and leaves the new X-coordi- nates reversed in sign (figure 13). -y yx \y Operations of symmetry: 1, A^|\ A(7r), Fig 13. AHf (W Coordinates of equivalent points : xyz; yxz; xyz; yxz. Point-group V. — The diagonal reflecting plane contains the Z-axis and bisects the angle between the X- and Y-axis. Reflection in such a plane (Sd) interchanges the X- and Y-coordinates (figure 14). Operations of symmetry. 1, U, V, W, Sd, USd, VSd, WSd. Coordinates of equivalent points. : xyz; xyz; xyz; xyz; yx^; yxz; yxz; yxz. * This arises from the observation that two reflections in the same plane nullify one another. THE TETRAGONAL POINT-GROUPS. 15 Point-group C4. — Operations of symmetry: 1, a(^^, A(7r), A^y), S„ S^A(^y S,A(x), S,a(^^ This may be more conveniently written as Cl= {C4, Sn}, signifying that the operations of C4 are those of C4 plus the reflections of these operations in the horizontal plane.* Fig. 14. Coordinates of equivalent points : xyz; yxz; xyz; yxz; xyz; yxz; xyz; yxz. These coordinates illustrate the fact, of which use will commonly be made in the work which follows, that a two-fold rotation about an axis combined with a reflection in a plane normal to the axis is equivalent to an inversion (figure 15). Thus C4={C4, 1} is an alternative expression of the point- group C4. In this latter case the coordinates of equivalent points would be written in the following order. yxz; xyz; yxz; xyz; yxz; xyz; yxz. xyz; Point-group C4. — Operations of symmetry : C4 = { C4, Sy } . Sv is again a mirroring in the vertical, YZ-plane. Coordinates of equivalent points: xyz; yxz; xyz; yxz; xyz; yxz; xyz; yxz. Point-group D4. — The four two-fold axes lying in the XY-plane coincide with the X- and Y-axes and bisect the angles between them (figure 6). The * In the future this abbreviated representation will be used when no ambiguities are thereby introduced. 16 TETRAGONAL AND CUBIC POINT-GROUPS. operations of symmetry of D4 may consequently be obtained by appljdng the operations of one of the two-fold axes (the one coinciding with the X- axis will be employed) to those of C4. Operations of sjonmetry: D4={C4,U}. Coordinates of equivalent points: xyz; yxz; xyz; yxz; xyz; yxz; xyz; yx2. If other two-fold axes were used, the order of the last four coordinate values would be changed. Fig. 15. Point-group D4. — Operations of symmetry: DS={D4,S,} = {D4,I}. Coordinates of equivalent points: xyz; yxz; xyz; yxz; xyz; yxz; xyz; yKz; xyz; yxz; xyz; yxz; xyz; yxz; xyz; yxz. CUBIC SYSTEM. The cubic axes of reference are three mutually perpendicular axes with units all of the same length. Point-group T. — Operations of symmetry 1, AHf U, Ai V, A21 (t)' A^f)' ^{i} w, A.( CUBIC POINT-GROUPS, 17 A, Ai, A2, As are rotations about the trigonal axes of a, ai, as, at of figure 7. Coordinates of equivalent points: xyz; xyz; xyz; xyz; zxy; zxy; zxy; zxy; yzx; yzx; yzx; yzx; xyz; xyz; xyz; xyz; zxy; zxy; zxy; zxy; yzx; yzx; yzx; yzx. Point Group T^.— Operations of symmetry : T»'={T,S,} = {T,I}. In writing the coordinates of equivalent points the second of the representa- tions will be used. Coordinates of equivalent points: The 12 coordinate positions of T, and II. Point-group T"*. — The diagonal mirroring plane is taken to bisect the angle between the X- and Y-axes. Operations of symmetry: T«={T,S,}. Coordinates of equivalent points: The 12 coordinate positions of T, and III Point-group O U, yxz; yxz; yxz; yxz; xzy; xzy; xzy; xzy; zyx; zyx; zyx; zyx. of symmetry: The 12 < )perati( /t> k ^ / '37r\ U., B (2^ ) M .^> /7r> L / 'Sir\ V„ 6,(2^ )' »■( j)' W2, b/|^ ), B,( '3x\ .2/ Wx Rotations about the various axes of figure 8 are represented by the corre- sponding capital letters. The four-fold axes b, bi and bs have the positions of u, V and w of figure 7. Coordinates of equivalent positions: The 12 coordinate positions of T, and IV. yxz; yxz; yxz; yxz; xzy; xzy; xzy; xzy; zyx; zyx; zyx; zyx.* * The order of writing these coordinates has been changed about to make it conform with its later uses. 18 CUBIC AND HEXAGONAL POINT-GROUPS. Point-group 0^. — Operations of symmetry: 0^={0,S,} = {0,I}. Coordinates of equivalent positions: The 48 coordinate positions of I, II, III, and IV. HEXAGONAL SYSTEM. Rhombohedral Division. The point-groups of this division will be described in terms of two kinds of axes of reference. The rhombohedral axes (1), all of the same unit length and making equal angles with one another, are arranged symmetrically about the three-fold axis (figure 16). Two of the hexagonal set of axes (2) are of equal unit lengths and make an angle of 120° with one another (figure 17) ; the third, the Z-axis, is of a different unit length and is normal to the plane of the X-and Y-axes. This second set is thus a special case of the monoclinic axes; the cubic axes, on the other hand, are a special case of the rhombohedral (1) axes. Coordinates according to the rhombohedral axes are given below under I, according to the hexagonal axes under II. Fig. 16. Point-group C3. — Operations of symmetry: 1, a(|), a(|). Coordinates of equivalent points: I. xyz; zxy; yzx. II. xyz; y-x, x, z; y, x-y, z.* Point-group C3. — Operations of symmetry: 03={C3,I}. ♦ A reference to figure 17 will show the source of these coordinate values. HEXAGONAL POINT-GROUPS. 19 zxy; x-y, X, z; fzx. y, y-x, z. Coordinates of equivalent positions : I. xyz; zxy; yzx; xy2; II. xyz; y-x, X, z; y,x-y, z; xyz; Point-group C3. Operations of symmetry: Cl= {C3, Sy}. This vertical reflecting plane can have two possible positions, one containing both the X- and the Z-axes (hexagonal axes II), the other containing the Z-axis and a line in the XY-plane which makes an angle of 30° with the X-axis (see figure 26). The reflection in a plane occupying the first of these two positions will be designated as Sa, the reflection in the other plane by Sg. Coordinates of equivalent points : Fig. 17. xyz; zxy; yzx; I. yxz; II. xzy; zyx. WhenCl={C3, Sa}: xyz; y-x, x, z; y, x-y, z; x-y, y, z; yxz; X, y-x, z. WhenCl={Ca,S3}: xyz; y-x, x, z; y, x-y, z; y-x, y, z; yxz; x, x-y, z. Point-group D3. — Operations of symmetry: D3= {C3, U}. The two-fold axis of rotation may lie either in the X-axis or in a line in the XY plane which makes an angle of 30° with the X-axis. A rotation about the first-named axis will be called Ua, about the second, Ug. There may thus be two different sets of coordinates of equivalent points for the point-group D3 corresponding to the two sets already defined for C3. 20 HEXAGONAL POINT-GROUPS. Coordinates of equivalent points: I. xyz; zxy; yzx; yxz; xzy; zyx. II. WhenD3={C3, Ua}: xyz; y-x, X, z; y, x-y, z; x-y, y, z; yxz; x, y-x, 2. WhenD3-{C3,U«}: xyz; y-x, X, z; y, x-y, z; y-x, y, z; yxz; x, x-y, z. Point-group D3. — It can be shown that this point-group arises from the combination of D3 with an inversion. Just as there are two ways of ex- pressing D3 in terms of hexagonal axes of reference (depending upon the position of the two-fold axis) so there must be two ways of expressing Df. Operations of symmetry: Di={D3,Sd} = {D3,I}. Coordinates of equivalent points : I. xyz; zxy; yzx; yxz; xzy; zyx; xyz; zxy; yzx; yxz; xzy; zyx. II. When the operation of the two-fold axis is Ua: xyz; y-x, x, z; y, x-y, z; x-y, y, z; yxz; x, y-x, z. xyz; x-y, X, z; y, y-x, z; y-x, y, z; yxz; x, x-y, z. When the operation of the two-fold axis is Us : xyz; y-x, X, z; y, x-y, z; y-x, y, z; yxz; x, x-y, z; xyz; x-y, x, z; y, y-x, z; x-y, y, z; yxz; x, y-x, z. Hexagonal Division The point-groups of this division of the hexagonal system will be expressed only in terms of the hexagonal axes. Point-group C3. — Operations of symmetry: C§={C3,SU. Coordinates of equivalent points : xyz; y-x, X, z; y, x-y, z; xyz; y-x, x, z; y, x-y, z. Point-group D3. — Operations of symmetry : D§={D3. SJ. HEXAGONAL POINT-GROUPS. 21 Just as there are two ways of expressing D3, so there will be two ways of stating D3. Coordinates of equivalent points: When the two-fold axis has the position of the X-axis (Ua) : xyz; y-x, x, z; y, x-y, z; x-y, y, z; yxz; x, y-x, 2; xyz; y-x, x, z; y, x-y, z; x-y, y, z; yxz; x, y-x, z. When the two-fold axis makes an angle of 30° with the X-axis (Us) : xyz; y-x, x, z; y, x-y, z; y-x, y, z; yxz; x, x-y, z; xyz; y-x, x, z; y, x-y, z; y-x, y, z; yxz; x, x-y, z. Point-group Ce- — The operations of this group can be written as those arising from the opera- tion of a 60° axis of symmetry. Taken thus the operations of Cg are: Operations of symmetry : 1. a(|), a(|), aw, A(f), a(|). Coordinates of equivalent points: xyz; y, y-x, z; y-x, x, z; xyz; y, x-y, z; x-y, x, z; Point-group CI- — Operations of symmetry : C^={Ca,S,}. Coordinates of equivalent points: xyz; y, y-x, z; y-x, x, z; xyz; y, x-y, z; x-y, X, z; xyz; y, y-x, z; y-x, X, z; xyz; y, x-y, z; x-y, X, z. Point-group CI- — Operations of symmetry: Q={C6,Sv}.* Coordinates of equivalent points: xyz; y, y-x, z; y-x, x, z; xyz; y, x-y, z; x-y, x, z; X, y-x, z; y-x, y, z; yxz; x, x-y, z; x-y, y, z; yxz. Point-group Df — Operations of symmetry: D«={Ce,U}. U is a rotation of 180° about axes in the XY-plane, one of which coincides with the X-axis. Coordinates of equivalent points: xyz; y, y-x, z; y-x, x, z; xyz; y, x-y, z; x-y, x, z; X, y-x, z; y-x, y, z; yxz; x, x-y, z; x-y, y, z; yxz. * This group is of course equally the result of operating upon Cg by a two-fold axis coincident with the Z-axis. That is, C^={C3,Uil. 22 HEXAGONAL POINT-GROUPS — SPACE LATTICES. Point-group Dq. — Operations of symmetry: Dg={D6,S.} = {De,I}. Coordinates of equivalent points. xyz; y, y-x, z; y-x, x, z; xyz; y, x-y, z; x-y, x, z; X, y-x, z; y-x, y, z; yxz; x, x-y, z; x-y, y, z; yxz. Xyz; y, x-y, z; x-y, x, z; xyz; y, yx, z; y-x, x, z; X, x-y, z; x-y, y, z; yxz; x, y-x, z; y-x, y, z; yxz. SPACE LATTICES. A series of parallel planes such that the distance between any two consecu- tive planes of the series is constant is called a set of planes. The sum total of the points of intersection of any three sets of planes is called a regular space lattice. z Fia. 18. A sjrmmetrical lattice. The intersection points of this figure are points of the lattice. If some point of a lattice (0 of figure 18) is taken as the origin of coordi- nates, the neighboring points of the lattice are given by the translations ±2t„ ±2Ty, ±2r, along the X, Y and Z axes; and in general any point of che lattice is given by the composite translation Ti = d: 2mT j =t 2nTy ± 2pT, where m, n and p are any integers or zero. The three translations, 2tx, 2Ty, 2t„ giving neighboring points of the lattice, are called the primitive trans- THE 14 SPACE LATTICES. 23 lations. It is customary to define a lattice by stating its primitive trans- lations with respect to the axes of reference.* This definition is sufficient since the primitive translations of a lattice can be considered as those transla- tions which will yield all of the points of the lattice by their continued appUcation, first to a point of the lattice chosen as origin, and to the new points continually derived from this and succeeding applications. It can be shown that but fourteen symmetrical lattices are possible; each of them has the complete symmetry of one of the seven systems of crystallo- graphic symmetry (counting the rhombohedral division of the hexagonal system as a separate system). The primitive translations of these 14 space-lattices, identical with the lattices of Bravais, are as follows. The axes of reference are the same as those used for the point-groups of corresponding symmetry. Primitive translations. 2r,;2ry;2T.. 2r,;2Ty;2r,. 2r,; 2ry; 2t,. '"x) ■'■yJ Tx> "TyJ 2t,. •^"^X) ''yj ^z'> "^7) ''z* ''yj "^ty ''«' ''x> ''x> '''y 2Tx; 2Ty] 2r^', Tjj, Ty, T,. 2r,;2Ty;2r,. Tij Ty'y ^x) "'''yl 2t,. "^yy "^zj "^zy '^xy "^xy "^y 2tx; 2Ty; 27^; t,, Ty, r,. Symbol. Tridinic system. 1. Ttr Monoclinic system. 2. r„. 3. r.' Orthorhombic system. 4. To 5a. To' (a) b. To' (b) 6. To" 7. To'" Tetragonal system. 8a. Ft (a) b. rt(b) 9a. r/ (a) b. Ft' (b) Cubic system. 10. r. 11. Tc' 12. Tc" Hexagonal system. 13. Trb 14. Tu 2T,;2Ty;2T,. '^yy '^z'y ^2> '^x'y'^xi "^y 2t^; 2Ty; 2t,; t,, Ty, t,. 2tx ; 2Ty ; 2tj . (Rhombohedral Axes) 2tx; 2Ty; 2t^. (Hexagonal Axes) ♦ Different groups of primitive translations for a single lattice are possible by taking the unit directions differently. We shall have use for the primitive translations just defined and for no others. t By Ty, Tz is meant a translation Ty along the Y-axis followed by one of length Tz along the Z-axis. The translation Ty, — Tz is similar except that Tz is here taken in the — z direction. These are written by Schoenflies as Ty + Tz and Ty — tz respectively. 24 THE NATURE OF SPACE-GROUPS. Lattices 13 and 14 belong to the rhombohedral division; lattice 14 has the complete symmetry of the hexagonal division of the hexagonal system. Lattices 2, 4, 8a, 10, 13 and 14 are all special cases of lattice 1, in which the lengths of the units along the axes or the angles between the axes have par- ticular values. The lattices having the symmetry of the tetragonal and of ^he cubic system can be looked upon as special cases of the orthorhombic space lattices; in this process of speciaUzation, for lattices of tetragonal sym- metry, if the axes are taken after the manner of lattice 4, 8a is obtained, if according to 5, 8b results. The two forms of 8 are, however, identical. In a similar fashion 9a and 9b arise from 6 and 7. SPACE-GROUPS. In giving analytical representations to each of the 32 point-groups the different ways have been expressed in which points can group themselves about a central position so that the aggregate of points will by their arrange- ment exhibit crystallographic symmetry. We are not, however, primarily Fig. 19. The point-group Cl- The points P, P/, P//, P/// are the four equivalent points of this point-group. interested in such an aggregate of points about a single position in space but rather in the indefinite extension in all directions of such a symmetrical grouping of points. In order to accomplish this, it is necessary to distribute point-groups (or perhaps other suitably symmetrical groupings of points), properly oriented according to some regular pattern which repeats itself indefinitely in all directions. Such a regular pattern must be one of the 14 space lattices. The indefinitely extended symmetrical arrangement of points all equivalent to one another, which is obtained by placing such groups of A TYPICAL SPACE-GROUP, Ca. 25 equivalent points with their centers at the points of one of the regular space lattices, is a space-group.* For the sake of illustration the very simple space-group which is obtained by placing the point-group Cg, the holohedry of the monoclinic system, (figure 19) at the points of the monocUnic space lattice Tm will be considered, f A portion from this space-group is shown in figure 20. The four equivalent Fig. 20. A portion from the monoclinic space-group C^u- points P, P/, P// and P/// (and the two-fold axis of symmetry and the plane normal to it) of C2 repeat themselves about each of the points O, A, B of the first monoclinic lattice Fm. Taking O as the origin, then the coordin- ates of the points of the group about A, the point of the lattice obtained by the primitive translation 2tx, are x-f-2Tx, y, z; 2t^-x, y, z; 2r^-x, y, z; x-F2t„ y, z. In a similar way the coordinates of the equivalent points about the other neighboring points of the lattice and in general about any point of the lattice * The view which one takes of a space-group will depend largely upon his interests. For instance, the crystallographer will in all probability consider a point-group as a particular aggre- gation of elements of symmetry arranged in some definite fashion. The space-groups will then, first and above all, describe to him the way in which these elements of symmetry can be dis- tributed throughout a crystal. On the other hand, the physicist or chemist who is accustomed to think of a crystal essentially as an orderly arrangement of atoms or molecular groupings of atoms will probably incline to the more analytical view of point-groups and space-groups as aggre- gates of equivalent points which are potential positions for the atoms in crystals. Because we are interested here in discussing only those phases of the theory of space-gioups which are of immediate use to the physical study of the structures of crystals, the characteristics of symmetry possessed by the various space-groups will receive only such treatment as is required for the building up of an analytical expression of the results of the theory. t Figure 18 will illustrate Fm if X and Y have any unit lengths and make any angle with one another, and if Z is norma! to the plane XY. 26 THE TRICLINIC SPACE-GROUPS. are given by one of the following sets which, taken together, completely define this space-group: x±2mT,, y±2nTy, z±2pr,; ±2mTx— X, ±2nTy— y. z±2pT,; ±2mTx — X, ±2nTy— y, ±2pT, — z; x±2mTx, y±2nTy, ±2prj — z; where, as before, m, n and p can be any integers or zero. Some of the space-groups are obtained by thus placing point-groups at the points of the lattice of corresponding symmetry; the rest of the 230 typical ways of arranging points so that the assemblage will exhibit crystallographic symmetry may be obtained by placing, at the points of these lattices, groups of points analogous to the point-groups, and derived from them, of such a nature that the symmetry of the aggregate is that of one of the point-groups themselves. It is obvious that a space-group is completely defined (analytically) when the coordinates of the equivalent points ranged about one point of the lattice (the points of a point-group or of a "modified point-group") and the primitive translations of the lattice are given; for, as we have just seen in the case of the monochnic space-gi"Oup, with this information it is always possible to reconstruct the space-group. AN OUTLINE OF THE DERIVATION OF THE SPACE-GROUPS. The nature of each of the space-groups will be apparent from the following tabular outline. Under each class of symmetry a brief discussion of the development of the space-groups exhibiting its symmetry will be given. This will be followed by a statement under three headings of (1) the symbol of the space-group, (2) an abbreviated indication of its particular derivation, and (3) the fundamental lattice underlying it. TRICLINIC SYSTEM. Hemihedry. — The single space-group of this class is obtained by placing the single equiva- lent point of the point-group Ci at the points of the lattice Ftf. 1. {C}=a,r,r}.* T^ Holohedry. — The single space-group having this symmetry is obrained by placing the equivalent points of Q at the points of the lattice Ta. 2. c| = {c.,r,,}. r„ * The space-group symbol is a simple adaptation of the symbols used for the point-groupa. The letters to be found in exponent position in the symbols for point-groups are reduced to the subscript position. The different space-groups isomorphous with a particular point-group are distinguished by numbers in the exponent position. Thus Cj^j^ is the hfth space-group (isomor- phoua with the point-group C^) that is defined. 3. cl = 4. c^= 5. c^= 6. ct= THE MONOCLINIC SPACE-GROUPS. 27 MONOCLIJIC SYSTEM. Hemihedry. — The space-groups having this symmetry can be developed by combining the space-group C| when it has the speciahzed form of either Tm or Fm, with a ghding reflection in a plane which is taken as that of the X- and Y- axes.* |r„„S,(r)). F„ 1 Fna » ^h ) • Fm Hemimorphic hemihedry. — Since the point-group C2 is obtained by combining Ci with a two-fold axis, the space-groups isomorphous with C2 can be obtained by combining the lattices Fn, and F^' with screw axes of symmetry. The translation com- ponents of these screw-axes are either zero or half a primitive translation in the direction of the Z-axis. 7. C^={F„,A(7r)}. r^ 8. Ci = {F„, A(7r, T.)}. F„, 9. C^ = { F„', ACtt) I = { F„.', A(7r, Tr) } . F„.' Holohedry. — The space-groups isomorphous with Cj can be obtained by multiplying (= combining) space-groups isomorphous with C2 with the operation of a glide plane of symmetry. Since a rotation of 180° combined with a reflec- tion in a plane at right angles to the axis of rotation is equivalent to an inversion, these space-groups result also from multiplying the groups iso- morphous with C2 by an inversion. 10. C2h= {C2, Sh}. F, 11. C2h — 1 C2, feh } . F, 12. C2h= (C2, Shj. F, 13. C2l,= {CLS,(r)}. F, 14. ci^{ci,s^(t)}. F, 15. CA={Cf,S,(r)}. F, ORTHORHOMBIC SYSTEM. Hemimorphic hemihedry. — The intersections with the XY-plane of the axes of space-groups C™ (the space-groups having the symmetry of C2) when the angle between the axes has the special value of 90°, is given by the points A, B, C, D, Ai . . . . of figure 21. The space-groups isomorphous with Cg can be developed j^y * A glide plane the translation component of which is zero is of course a simple reflecting plane, r, a primitive translation in the XY-plane, may then be chosen as either t^ or xy. 28 ORTHORHOMBIC SPACE-GROUPS. multiplying groups ismorophous with Ca by a vertical glide plane of sym- metry, that is, one parallel to or containing the Z-axis. The various possible positions of the intersections of these planes with the XY-plane are shown by c, o-m, etc. of figure 21a and c^, a^', etc. of figure 21b. zzy A d B A, 6 m. 6 nix ^1 C d' di D ^ ^3 1 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. ZTX Fig. 21. r* - p2 _ ^2v — C4 2v = 5 ^2v — |CLS} = {CLSi(rJi |CLS(r,)} 1C2, S(tJ} = {C2, Sm [CLS(rJ}. ^2v *^2v p9 _ V-'2v — = {C2, S(Tx + tJ}. = {CiS(r,+r,)}. = {C2, Sni(Tx)}. 1C2, Sni(Tx)}. {C2, Sni(Tx+Tz)}. {C2, Sd} = {C2, Sflij {C2, Sd}. C2v= {C2, Sdfrj,)}. C'2v ~ { ^2> ^ ) • C2v={C2, S(tJ}. S(rJ}. C^' = {C V^2V ^2v' {C2^s}. {C2, Sin[§(l =+rJ]}. ''2> ^2V ^22 -'2V 3 ^•^^' 2> SdCrJ}. = {C2, Sd(Tj}. KO' To To To To To To To To To To ro'(a) ro'(a) ro'(a) r«'(b) ro'(b) ro'(b) ro'(b) To" rin « rin rtn erA: Enantiomorphic hemihedry. — Definition. — If a certain portion of the operations of a group when taken alone themselves form a group, they define a sub-group. The space-groups isomorphous with the point group V are best described by giving the sub-groups whose axes are parallel to the X, Y- and Z- axes of the lattice (and of the coordinates). ORTHORHOMBIC SPACE-GROUPS. 29 38. Y'={Cl,Cl,Cl}. To 39. Y'={Cl,Cl,Cl}. To 40. Y'^{Cl,Ci,Cl\. To 41. Y'=={Cl,ClCl\. To 42. Y^={Cl,ClCl}. ro'(a) 43. V«={CLCLC^}. ro'(a) 44. Y'=={ClCl,Cl\. To" 45. V«=lC^,C,^q}. To'" 46. Y'={ClCl,Cl\. To'"* Holohedry. — The space-groups isomorphous with V^ can bo obtained by combining groups isomorphous with V with a horizontal ghding reflection. It is more simple, however, to consider them as developed by combining certain groups V™ with inversions. The locations of these points of inversion will be clear from a reference to figure 22. 47. Vi={VMl. To 48. V^={VM^}. To 49. V^={VM.}. To 50. V^={VM,1. To 51. V^={V^I}. To 52. V«={VM„}. To 53. V^={VMJ. To 54. V^={VM,}. To 55. V«={VM}. To 56. V^.°={VM^}. To 57. Vy = {V',I,}. To 58. V^h'={VMw}. To 59. V\f={VM,}. To 60. V\,^={VM,}. To 61. Y'^^{Y\\\. To 62. V^,«={VM,}. To 63. V»J={VM}. ro'(a) 64. V^.'={VMJ. ro'(a) 65. V^,«={V«,I}. ro'(a) 66. Vl''={V«,I„,}. ro'(a) 67. VV={V«, IJ. ro'(a) 68. Vi^ = {V«,I,}. ro'(a) 69. Vi^^lV, I}. To" 70. C1*={V,I^}. To" 71. V=if={V«,I}. To'" 72. Vl«={VM„}. To'" 73. VY={VSI}. To'" 74. Vl«={V«,I,}. To'" * These two last space-groups differ in the manner of distribution of their axes. For the former the axis of rotation lies in the line AD, for the latter in the line BC of Figure 21. 30 TETRAGONAL SPACE-GROUPS. TETRAGONAL SYSTEM. Tetartohedry of the second sort. — The groups Sf can be obgained by combining groups isomorphous with Cj with a rotary-reflection (a rotation combined with a reflection) having the same axis as the group C^. 75. s1={CLa}. r, 76. SI={C|,A}. r/ Hemihedry of the second sort. — The space-groups isomorphous with V* can be obtained by multiplying groups isomorphous with V by the operation of a diagonal vertical glide plane of symmetry. A reflection in the plane WMGA of figure 22 will be called (Td, one in the parallel plane through F, 0-^,. ^ , \ y/ / Ay j>Tr y X-H- -H ^ -y 1 / rx- / / U ^ir / yo '--7% / y y / / y yy . r Fig. 22. 77. V^={VSS,}. Te 78. V,^={VSS,(rJ} r» 79. Va'={V^S,}. r» 80. V^={V3,S,(rJ}. r. 81. v^={V6,s,}. r* 82. V«={VSS,(r.)}. r. 83. v^=|v«,s..(^^)} r, 84. V«=|ve,S^(^^^+r.)| T, 85. V,»={V7,S,}. r/ 86. ¥^^={^,8,(7,)}. v: 87. vy = {V8,s,)}. r/ 88. Vl,='={V»,S,(Tr)}.* r/ \2' 2' 2j' ' ~\ 2'2'2/''~V2' 2'2j'' ~V2'2' 2/ TETRAGONAL SPACE-GROUPS. 31 Tetartohedry. — The space-groups C™ can be derived by arranging screw-axes of s)nmmetry according to the two tetragonal lattices. 89. Cl = 90. 91. 92. 93. 94. i={A(0r.} cl={A(-;,|'),r.}. Cl={A(||-),r.}. cf={AQ,r.'}. cS={A(|f),r.'} Paramorphic hemihedry. — The groups C^ are most readily obtained by inverting groups isomorphous with C4 either through a point lying in a four-fold axis or midway of a line joining two four-fold axes. This second inversion will be represented by Ii. Tt . Ft . r, Ft' Hemimorphic hemihedry. — The groups C^ are obtained by multiplying groups C" by vertical gliding reflections. The positions of these reflecting planes are shown in figure 23. 95. CA = {C1,I}. 96. ^4li — 1^4) !}• 97. CA = {Cl,Ix} 98. CA={CU,} 99. C4h= {C4, I}. 100. C4h= {C4, II} 101. C4V= {C4, Sg}. Ft 102. C4v= {C4, So}. Ft 103. C4v= {C4, Sg}. Ft 104. C/v={Ct.Se}. Ft 105. C/v={Cl,S,(rJ}. Ft 106. C4^={CLSe(r.)}. Ft 107. C4;={Ct,S3(rJ}. Ft 108. C4v= {C4, Se(Tj)}. Ft 109. C4v= {C4, Sg}. Tt' 110. Clv°={Cps(rJ}. Tt' 111. C4v= {C4, Sc}. Tt' 112. Cl^={aSe(rJ}. Tt' 32 TETRAGONAL SPACE-GROUPS. Enanliomorphic hemihedry. — Since the point-group D4 results from the multiplication of C4 by a two-fold axis lying in the plane normal to the four-fold axis of C4, the groups D^n arise by multiplying certain of the groups C tn by two-fold axes lying in the XY-plane. The positions of these axes are shown in figure 23, if the Hnes AB and C1C2 define the axes Ug and Uc respectively. Fig. 23. 113. Dl={Cl,UJ. Ft 114. Dl={Cl,Ue}. Ft 115. T>l={Cl,V,\. Ft 116. T>t ={ClVJ. Ft 117. D|={CtUJ. Ft 118. D^={Cf,Ue|. Ft 119. Dl={Ct,UJ. Ft 120. D|={CtUe}. Ft 121. DS={C|, UJ. Tt' 122. D^4°={C«, U.}. Tt' Holohedry- - The space-groups D^ may be deiived by combining groups of D^ with an inversion. If the axes striking the XY-plane in A, Ai, etc. (figure 23) are called o and those meeting the plane in points corresponding to B are called b. then the points of inversion are located (1) at the intersection of a with an axis parallel to U„ (2) midway between two such points of intersection, (3) on an axis parallel to Ub, midway between a and b or (4) half of the way between a and b and half way between axes parallel to Ug. The inversions through these four points will be denoted by I, I', Ii and Ii'. These four inversions TETRAGONAL AND CUBIC SPACE-GROUPS. 33 are equivalent to inversions I, I, G, and M of figure 22. Igj and Ija about the four points A, W, 123. DA = Dl,Il Ft 124. DA = Dl,I. !. Tt 125. DA = DL IJ Tt 126. DA = Dl,I„, }• Tt 127. Dil}. Ft 128. DA = DLlw 1- Tt 129. d;,= Bl, I J Tt 130. ^4h — DMn, |. Tt 131. n^ — Dtl} Ft 132. -nio_ J^4h — Dtlw !• Tt 133. n^^- ^4b — D|,IJ • Tt 134. J-'4h — D|, In, }• Tt 135. ^4h — Dl,I}. Ft 136. ^4h — Dt I. [. Tt 137. ^4h — D^ I,} . Tt 138. ^4h — DMn. }• Tt 139. n^^ — D4M}. Tt' 140. r)i8_ J-'4h — D^ Iw !• r/ 141. -ni9_ ^4h — D^", I« I. r/ 142. ^4h — D^.°. I„ J. r/ CUBIC SYSTEM. Tetartohedry. — The space-groups isomorphous with T can be obtained by combining certain groups V™ with the operation of a three-fold rotation axis. Except in the case of the group derived from V, when it must be AA', the position of this three-fold axis can be that of any diagonal of figure 22. This rotation of angle / will be represented by A. 143. 144. 145. 146. 147. T»={VS A}. T2={V^, A}. T3={V8, A}. T4={V^ A}. Ts={V«, A}. Paramorphic hemihedry. — Since the point-group Tjj can be derived from the point-group T by com- bining it with an inversion (as well as with the operation of a horizontal plane of symmetry), the groups isomorphous with T^ can be obtained from the groups T™ by combining them with an inversion. This center of symmetry 34 CUBIC SPACE-GROUPS. lies either at a corner of the cube of figure 22 (A) or at M. sions will be called I and 1^ respectively. These two inver- 148. 149. 150. 151. 152. 153. 154. Tl = T^ = T^ = T^ = n= ^7— ( TM}. T2,I}. T3,I}. TSI}. T^={T^I}. Hemimorphic hemihedry. — The groups isomorphous with T^ can be derived by combining groups T™ with a gliding reflection in a diagonal plane. This plane can be taken as WMGA of figure 22. / / / A. / > / Hi Y / / z. ) y- / 4 } / / 1/ / Fig. 24. Fig. 25. 155. Tl={TSS,}. To 156. T^={T2,Sd}. To' 157. T^={T3,S,}. To" 158. T^={TSS,(r)}. To 159. T,^={TSS,(r)}. r ' *■ c 160. T^={T^S,(r)|. r " ^ c Enantiomorphic hemihedry. — The groups 0™ result from combining groups T™ with the operation of a two-fold rotation axis. This axis may be taken parallel to UK of figure 22. If it passes through the point M of figure 22 the rotation will be denoted by Urn, (2) if it has a parallel position through the point A by U, (3) if it hes in the line bisecting AM (see figure 24) by Ui, or (4) if it bisects MA' by Us. CUBIC AND HEXAGONAL SPACE-GROUPS. 35 161. 0^={TSU}. Te 162. 02={TSU„}. Te 163. 03={'P, U}. r/ 164. 0*={T2, U„}. Te' 165. 0«={T3, U}. Tc" 166. 06={T*, Ui}. Tc 167. 0'={T*, U2I. Te 168. 08={T5, U}. To" Holohedry. — Since the point-group O^ results from O by the operation of a center of symmetry, as well as of a horizontal reflecting plane, the groups O" iso- raorphous with O'* can be obtained by combining groups 0™ with an inversion. These centers may be at A, A', M or M' of figure 25; the corresponding inver- sions will be called I, I', Im, Im'. 169. Oi = {OSI}. Tc 170. Og={OSI„,}. Tc 171. 0^={0^I| To 172. 0^ = {0M^}. r„ 173. 0^={0' I}. To' 174. 0«={03,I'}. To' 175. 0^ = {0M^}. To' 176. Og={OM:„}. To' 177. 0« = {0M}. To" 178. 0^t"-{0«,I}. To" HEXAGONAL SYSTEM. RHOMBOHEDRAL DIVISION. Tetartohedry. — The space-groups isomorphous with C3 can be obtained by combining the lattices Th and Trh with a three-fold screw axis. The translation com- ponent of this screw-motion is to be taken along the Z-axis. 179. c^=|A(|'),r,|. r„ 180. C| = |A^y,y),rb|. r^. 181. c^ = {A(|,|'),r,}. r, 182. Ct = ^A(~yT,^ Trt 36 HEXAGONAL SPACE-GROUPS. Paramorphic hemihedry. — The two space-groups Cs™ can be obtained by combining groups C™ with an inversion (I). 183. C3\ = {CLI}. r^ 184. Cl = {Ct,l}. Tr, Hemimorphic hemihedry. — The vertical reflecting plane will contain the vertical (Z) axis and either (1) the X-axis — of the point and isomorphous space-group — (A A' of figure 26), or (2) a line (AB of figure 26) which Ues in the XY-plane and makes an angle of 60" with the X-axis. In the first case the reflection will be desig- nated So, in the second S.. ^x Fig. 26. 185. C3V={ 0^,83}. T^ 186. C3l = {cLsj. r, 187. C3l = {cLS3(r,)}. r, 188. C3t={CLS.(r,)}. r, 189. C3^ = {c*.Sa|. r,^ 190. C3^ = {c^.s,(tJ}. r,, Enantiomorphic hemihedry. — The space-groups D" result from operating upon groups C™ with a two- fold axis which has the position either of AA' of figure 26, (11^), or of AB, (Ue). 191. D^={CLUJ. 192. DI = {C^,UJ. 193. T>l = {Cl,VJ. 194. D^={Ci, UJ. 195. D|={C3^UJ. 196. D!={C3^UJ. 197. D^={C3^U3}. HEXAGONAL SPACE-GROUPS. 37 Holohedry. — The groups D^ are most easily obtained by combining groups of D™ with an inversion. This point of inversion will lie either at the intersection of a three-fold and a two-fold axis, (I), or midway between two such intersec- tions (I'). 198. 199. 200. 201. 202. 203. D3^, = {DJ,I}. D/, = {D LI) D3t. = {DLr) Trh HEXAGONAL DIVISION. Trigonal paramorphic hemihedry. — The single space-group isomorphous with C3 is obtained by reflecting Cj in a horizontal plane. 204. C/.= {CLS,}. r. Trigonal holohedry. — The groups D3™ arise by reflecting groups D™ in a horizontal plane which either contains the two-fold axes, (Sn), or lies midway between them, (So,). 205. D3'k={DLSh] 206. D3l.= {D^S„, 207. D3l={DLS,l 208. D3l.= {DLS^ Hexagonal ietartohedry. — The space-groups isomorphous with Cg result from combining a six-fold screw-axis with the hexagonal lattice. 209. CJ = M3>4- 210. ci= K3't)4 211. ci= r\3' Ty' ^T 212. ct- K3.|').4- 213. cl= Ka-T').-}- 214. c?= Ki n). 4. 38 HEXAGONAL SPACE-GROUPS. Hemimorphic hemihedry. — The groups C^ are obtained by combining groups C" with the operation of a vertical reflecting plane which passes through either the line AA' or the line AB of figure 26. The reflection in the plane through AA' will be desig- nated as Sft. 215. C6V={cS,sj. r, 216. Cel={a,S,(rJ}. T, 217. Ce^={C«,SJ. r^ 218. Ce*.= {C«,S,(r.)}. T, Pammorphic hemihedry, — The space-groups isomorphous with Cg can be obtained by reflecting groups C ™ in a horizontal plane. 219. C6\,= {CJ,S,}. r^ 220. c=L={c«,s^}. r, Enantiomorphic hemihedry. — The space-groups D™ are most simply derived by combining groups C" with the operation of a two-fold axis which coincides with the X-axis of coor- dinates of the point and isomorphous space-groups (AA' of figure 26). This two-fold rotation will be represented by U». 221. DJ={CJ,UJ. r^ 222. D^ = {q,Ua}. r,. 223. D^ = {C^,Ua}. r^ 224. D^={C*, UJ. r^ 225. Di={C6^Ua}. r^ ' 226. D«={C^,UJ. r^ Holohedry. — The groups Dg™ result by combining groups D^ with an inversion which lies in the six-fold axis either at its intersections with the two-fold axes (I) or at points midway between such intersections, (I'). 227. DeV'iDj,!}. r». 228. De1.= {Dj,I'}. T^ 229. D/. = {D«,I}. Th 230. DA = {D«.I'}. r. CHAPTER III. THE APPLICATION OF THE THEORY OF SPACE- GROUPS TO CRYSTALS.* UNITS OF STRUCTURE. A space lattice has been definedf as the sum total of the points of inter- section of any three sets of planes. These sets of planes partition the space into units of structure, all of the same size and shape. Such a unit is OABDEGFC of figure 18. There will thus be a unit corresponding to each of the 14 lattices; points of the lattice will be found at each of the corners of the unit prisms and in some cases other points of the lattice will lie in the center of the unit or at the centers of faces (as examples, To'" and To"). If the lattice is a monocUnic lattice, the unit will be some sort of a monoclinic prism; if the lattice is cubic, the unit will be a cube, and so on. / Fia. 27. The unit cell derived from Ttr. The edges of this unit are of unequal lengths and make unequal angles with one another. Just as a simple lattice can be divided intd unit prisms by three sets of planes parallel to the axes of coordinates, so any space grouping of points, built upon some lattice, can be similarly divided. The fourteen units of structure characteristic of the fourteen space lattices are shown in figures 27 to 34. The number of the points of the lattice to be associated with a unit prism can be readily told. For instance, in the case of the simple cubic lattice, Fc, this number is one since each of the eight points of the lattice located at the eight corners of the cube is shared by the seven other cubes meeting at this point and there are no other lattice points contained in or touch- ing the unit. For the same reason the unit cube of a space grouping having this lattice fundamental to it will have a single group of equivalent points (the n points about a single point of the lattice) associated with it; each of the 8 corner-points of the lattice will contribute to the cube one eighth, and each a different eighth, of the equivalent points ranged about it, ♦P. Niggli, op. cit.; Ralph W. G. Wyckoff, Am. J. Sci. 1, 127. 1921. t See p. 22. 39 40 UNIT OF STRUCTURE FOR SPACE-GROUP CL A consideration of the unit of the space-group aheady discussed in detail, Cah, will make this more clear. The unit prism, OAFCGBDE of figure 20 (see also figure 18), contains four equivalent points M, M', M", and M'", the coordinates of which are M(xyz), M'(2rx — x, 2ry — y, z), M"(x, y, 2rz — z) and M'"(2rx— X, 2ry— y, 2rz — z). Since, however, the arrangement about every point of the lattice is the same as that about every other, it follows that corresponding points of the groups about neighboring points of the lattice are entirely similar. It is, then, so far as the expression of the relative posi- tions of equivalent points is concerned, permissible to consider 2rx— x= — x, — y = 2ry — y, and — z = 2r^ — z.* The coordinates of the four equivalent positions of the unit of structure of the space-group CgV are thus: xyz; -X, -y, z; x, y, -z; -x, -y, -z or, as it will hereafter be written: xyz; xyz; xyz; xyz. The number of points of the lattice to be associated with the units of each of the other lattices can be similarly obtained and from this the coordinates which can be taken as typical of the posi- tions of equivalent points within the unit of any space-group can be written down. The treatment of a slightly more com- pUcated space-group will outhne the necessary procedure. For this purpose we will take the space-group C2I1 obtained by placing the point-group C2 at the points of the second monocHnic lattice Fm (figure 29). The unit prism of this lattice proves to be a monoclinic prism with additional points of the lattice at the centers of two of its faces. The eight points of the lattice that are located at the corners of the prism serve, as with the space-group C2h, to place within it the equivalent points of one group (in this instance, by definition, a point-group). One half of the points about each of the two points of the lattice at the diagonals of faces (and opposite halves) he within the unit prism so that these two points of the lattice together contrive to place within the unit a second group of equivalent points. If O of figure 29 is taken as * This simplification is geometrically justified (1) since the unit prism that has been chosen has no particular physical significance but serves rather as a unit that is conveniently visualized and (2) because the coordinates adopted actually define a group of equivalent points which re- peated along and parallel to the axes of coordinates will build up the entire assemblage. It is, moreover, justified analytically as an expression of the points associated with the unit prism itself (if one prefers to think of this unit) because as applied to the study of the structure of crystals, these coordinates define the interference effects to be expected from atoms placed at these positions; this definition involves sine and cosine terms within which 2tx, 2Ty, and 2tz in 2tx— x, etc., dis- appear. Fig. 28. If OX j^OY 5^ YZ and ZY is normal to the plane YOX but ZYOXt^ 90°, this is the unit of TrnJ if the three edges are mu- tually perpendicular and (1) 0X?^0Y?^YZ, the unit corre- sponds to To, (2) if OX=OY?^YZ it corresponds to rt(a) or (3) if OX = 0Y = YZ the unit is that of r.. THE UNIT OF STRUCTURE FOR SPACE-GROUP CI- 41 the origin, the centers of the second group of equivalent points will be for the half of the equivalent points at P(0, Ty, Tz) and for the other half at the oppo- site point P'(2tx, Ty, Tz). Keeping in mind the analogous case of C2h (figure 20) the actual coordinates of the equivalent points within this unit are:* xyz ; 2rx-x, 2ry-y, z ; x, y, 2tz-z ; 2rx-x, 2ry-y, 2t,-z; X, y+Ty, z+Tz; 2rx-x, Ty-y, z+r^; x, y+Xy, Tz-z; 2t^-x, Ty-y, Tz— Z. Just as was done for the space-group Cjh these coordinates can be reduced to : xyz ; xyz ; xyz ; xyz ; x, y+Ty, z+Tzi 5c, Ty-y, z+Tz; x, y+Ty, Tz-z; x, Ty-y, Tz-z. It will be observed that this process is equivalent to placing a group of equiva- lent points (in this case a point-group) at the origin and at one other point (0, Ty, Tz). Fig. 29. If YZ J_ plane YOX and ZY?^ YO 5^ OX and (1) if Z YOX 5^ 90°, the unit corresponds to Tm', (2) if Z YOX =90°, it corresponds to To' (b). Fig. 30. This unit is a rectangular parallelopiped; if YZp^YO^^OX it corresponds to To' (a), if^ZY 5^Y0=0X to rt(b). The positions of the equivalent points within a unit for each of the space- gi'oups can be expressed in the same way as the coordinates of the character- istic groups of equivalent points placed at typical points of the lattice. f The typical point or points of the lattice corresponding to a particular unit are in all cases the origin, as well as sometimes the center of the unit or, as in this latter instance, Cah, the center of a side or the centers of several sides. The extension of this same line of thought to the rest of the 14 lattices will show the number of groups of equivalent points to be associated with the unit. Thus the coordinates of typical points of the lattice which serve as centers of these groups are those of Table 2. * This is true if x is less than tx, y than Ty and z than tz. A slight and obvious modification which would yield final and reduced values the same as these, would define the points within this unit prism if one or all of x, y and 2 exceed tx, ry or t,. t The general case of each space-group (Chapter IV) in which there are three vaiiable par^ meters is obtained by placing the characteristic group of equivalent points at the typical points of the underlying lattice. 42 THE 14 UNITS OF STRUCTURE. Table 2. Number of Lattice. associated Coordinates of lattice points. typical points. Triclinic System. 1. Ttr 1 (COO). Fig. 27. MoNocLiNic System. 2. r„, 1 (COO). Fig. 28. 3. r^' 2 O(CCO); P(0, ry,r.). " Fig. 29. Orthorhombic System 4. To 1 (COO). Fig. 28. 5a. To' (a) 2 O(OOO); Pi(T.,ry,0). Fig. 30. b. To' (b) 2 O(CCC); P(0,Ty,r.). Fig. 29. 6. To" 4 O(CCC);P(0,ry,r.); Pi(Tx,Ty,C); P2(r.,0,r.). Fig. 31. 7. To"' 2 O(CCC); Pa (tx, Ty, r.). Fig. 32. Tetragonal System. 8a. Ft (a) 1 (CCO). Fig. 28. b. Ft (b) 2 O(CCC); P, {T.,Ty,0). Fig. 30. 9a. Ft' (a) 4 O(CC0);P(0,r„T.); Pi(rx,ry,0); P2 (r., 0, r.). Fig. 31. b. T,' (b) 2 O(OGO); P3 (tx, Ty, T.). Fig. 32. Cubic System. 10. Tc 1 (COO). Fig. 28. 11. r/ 4 O(OOO); P(0,Ty,rO; Pi(rx,ry,0); P2 (r., 0, r,). Fig. 31. 12. To" 2 O(000);P3(tx, Ty, r.). Fig. 32. Hexagonal System.* 13. Trh 1 (CCO). Fig. 33. 14. Th 1 (CCO). Fig. 34, SPACE-GROUPS AND CRYSTALS. Every crystal, considered as a regular arrangement of atoms in space, must possess the symmetry of some one of the 230 space-groups. The theory of space-groups, then, supplies a method with the aid of which it should be possible to represent all of the ways in which the atoms of a crystal can be arranged in space. If an atom of a crystal occupies such a position that it corresponds with the coordinate position xyz of an equivalent point of the space-group having the symmetry of the crystal, then symmetry demands that exactly similar atoms shall be found at positions corresponding to those of * The unit cell for Fh can also Tbe taken as a base-centered rhombic prism, the lengths of whose sides stand in the ratio of a : b : c = \/3 : 1 : c. Niggli (op. cit.) has, worked out upon this basis the analytical expression for all of the groups having Th as the fundamental lattice. Such a unit is useful when it is desired to compare an hexagonal crystal with one exhibiting rhombic, tetragonal or cubic symmetry. SPACE-GROUPS AND CRYSTALS. 43 each of the other equivalent points of the space-group. Most crystals are built of atoms of more than one sort. As a consequence if we find the atoms of kind A occupying the positions of equivalent point xyz and the other points equivalent to it, the atoms of B will be found at some other positions developed from x' y' z', and so on. The atoms of a crystal may thus be thought of as occupying the positions of a sort of composite space-group developed by superimposing several sets of equivalent positions upon the same set of axes (and other elements of sym- metry). The atom.s of a crystal, as a result, must be arranged in groups with centers at the points of one of the space lattices. Such a group of atoms has Fig. 31. = 0X A rectangular parallelopiped. If YOt^OX it corresponds to To", if YO=OX to rt'(a), or if YZ = YO to Tc'. Fig. 32. A rectangular parallelopiped. If YZ9^Y0 9^0X it corresponds to To'", if YZ?^YO=OX to rt'(b), or if YZ = YO=OXtorc". been called a crystal molecule. In this sense the crystal molecule is a purely geometrical conception and except under special conditions would not be thought of as possessing any physical significance. It is possible, of course, to think of a crystal as divided, in the same way that a space-group can be divided, into a large number of unit prisms by sets of planes passing parallel to the three planes each of which contains two of the axes of coordinates. Measurements of the X-ray spectrum from the face of a crystal together with a knowledge of the density of the crystal can be made to yield the nmnber of chemical molecules that are to be associated with this unit of structure.* If a compound were of the type AB, where A is one kind of atom and B another, and if the atoms of A occupy the most general equivalent positions one of which is xyz, then there will be as many chemical molecules of AB associated with the unit prism as there are equiva- * The factor actually determined is n^/m, where n is the "order" of the reflection spectium and m is the number of chemical molecules associated with the unit prism. The value of n cannot, however, in general be determined so that to may usually have one of two or perhaps three values. 44 SPECIAL CASES OF SPACE-GROUPS. lent points in the unit. This number may under certain conditions be rela- tively great. For instance, in the case of the space-groups having the sym- metry of the holohcdry of the cubic system, the number of equivalent points of the point-group 0**, and of the other groups of points associated with a single point of the lattice, is 48. If then the fundamental lattice of a holo- hedral cubic space-group is the simple cubic lattice Fc and the compound crystalhzes with this symmetry (as sodium chloride does, for instance), 48 (if all of the A atoms are alike and all of the B atoms are also alike, and more if they are not alike) chemical molecules of AB must be placed within the unit cell; if the lattice were, on the other hand, the face-centered lattice Fc with four points of the lattice associated with the unit, this number of molecules of AB must be at least 192. Fig. 33. If the three edges meeting at O are of equal legths and make equal, angles with one another, this unit cor- responds to Tth. Fig. 34. If ZO i plane YOX and ZYOX = 120°, a rhombic prism two of the sides of whose base are XO and OY and of height OZ serves as the unit for Th. SPECIAL CASES. If, however, the values of x, y and z which express the positions of the atoms of A and B are such that the atoms lie upon some element of symmetry, two or more of the equivalent positions coincide and this number of molecules to be placed within the unit cell will be reduced. For instance if a point were to lie upon a plane of symmetry, it would of course be identical with its mirror image; or if it stood in a three-fold or four-fold axis of symmetry, three or four of the equivalent points would occupy the same position. In the space- group C21J (figure 20) if z is equal to t«, that is, to one half of the height of the unit prism, then the four equivalent points of the unit would occupy two positions (M coincides with M" and M' with M'") or if x is equal to r,, and y to Ty, the four points will have two equivalent positions (M will coincide with M' and M'' with M'") . Ifx=y = z=0 then the four points will all unite THE TYPICAL CASE OF CALCITE. 45 at the origin and there will be but one equivalent position within the unit; the same is true if x = rx, y=ry and z=Tj.. The results of all of the X-ray experimentation which has thus far been carried out seem to point to the fact that this number of chemical molecules to be contained within a unit cell is in all probabmty very much less than the number of most generally placed equivalent positions. As a consequence the determination of these special cases of the space-groups becomes of the utmost importance to the person interested in the structure of crystals. A discussion of calcite, which has already been treated in detail by this procedure,* will serve to indicate the need for these special cases of the space- groups. The X-ray measurements show that almost certainly two chemical molecules of calcium carbonate are to be associated with a unit rhombohedron. Calcite crystallizes with a symmetry which is that of the point-group D3. Two space-groups isomorphous with D3, namely Dg^ and Dgd, have Trh as the fundamental lattice. Since two chemical molecules of calcium carbonate are to be associated with the unit rhombohedron, two calcium atoms, two carbon atoms and six oxygen atoms must be placed within it. These two calcium atoms may conceivably be alike or they may be different one from the other; the same is true for the two carbon atoms; and the oxygen atoms may be for instance (1) all ahlce, (2) all different, (3) four ahke and two different, (4) two sets of three hke atoms or (5) three sets of two Uke atoms. Copjdng from page 157 it is seen that all of the potential atomic positions consistent with the space groups Ds^ and Y>^^ are Space-Group Dgdi Oitie equivalent position: (a) 0. (b) H i. Two equivalent positions: (c) uuu; tiuu. Three equivalent positions: (d) OOi; OiO; ^00. (e) OH; HO; |0i &ix equivalent positions: (f) utiO; uOu; Ouu; uuO; uOu; Ouu. (g) uu|; u|u; ^uti; uu^; u^ti; |uu. (h) uuv; uvu; vuu; uuv; uvti; viiu. Twelve equivalent positions : (i) xyz; yzx; zxy; yxz; xzy; zyx; xyz; yzx; zxy; yxz; xzy; zyx. Space-Group Dg^: Two equivalent positions : (a) 000; Hi (b) Hi; f f f . Four equivalent positions : (c) uuu; tiuu; ^-u, |-u, |-u; u-j-|, u-{-i u-|-i * Ralph W. G. Wyckoff, Am. J. Sci. 50, 317. 1920. 46 THE TYPICAL CASE OF CALCITE. Six equivalent positions : {A\ 111. 111. 113. 131. 111. Ill Vu; 444> 4441 444> 444> 444? 444' (e) uuO; uOu; Ouu; |-u, u+^, |; u+|, |, |-u; 2> 2 U, 11+2- TweZi'g equivalent positions: (f) xyz; yzx; zxy; yxz; xzy; zyx; 2 X, 2 Y) 2~2;; 2- y, 2 ^^ 2 ^j 2 z, 2 x, 2 yj y+i x+i z+i; x+i z+i y+^; z+i y+i x+|. The attempt to write down on the basis of these coordinate positions the different arrangements of the atoms in calcite that are possible in the light of its symmetry immediately eliminates many of the possibihties just discussed. For instance it is clear that in neither case are there enough special cases of one equivalent position so that the two calcium atoms can be different and the two carbon atoms also different. The same fact shows that possibility (2) for the arrangement of the oxygen atoms may also be omitted from con- sideration; it can be similarly shown that there are in neither space-group sufficient special cases so that four of the oxygen atoms can be alike and two dift'erent. All of the possible ways for the atoms of calcite to be arranged can then be written as:* Arrangements arising from Dg^: (a) Ca = u u u ; u u u. C = Ui U] ui ; til til ti2. O = U2ti2 0; ti2 0u2; Ou2ti2; 112 U2O; U2OU2; 0ii2U2. (b) Ca and C as in (a). O = U2 112 1; 112 1 U2; § U2 112; ti2 U2 1; U2 ^ 112; I U2 U2 (c) Ca and C as in (a). = U2 U2 v; U2 V U2; V U2 U2; tia ti2 v; 112 v 112; v ti2 ti2. (d) Ca and C as in (a). 0=U2U2U2; ti2ti2ti2. U3U3U3; tiaiistis. U4U4U4; ii4 ti4 ti4. (e) Ca and C as in (a). = 00^; 10; 1 0. OH; HO; |0i Arrangements arising from DaV (f) Ca = iH; HI or 000; Hi C=000; H^ or iii; f f f . 0—133. 331. 111. 111. 311. Ill ~444; 444> 444> 444; 4 4 4> 4 44- (g) Ca and C as in (f). = uuO; tiOu; Ouu; ^-u, u-Hi ^; u+^, |, i-u; 2> 2~U, U+2. In this same manner all of the ways of arranging the atoms in any crystal can be written down from a knowledge of the number of molecules to be associated with the unit cell (as furnished by the X-ray spectrum measure- ments) and from a consideration of the special cases of the different space- groups possessing the symmetry of the crystal. * These arrangements, giving as we have seen the positions of the atoms within a unit cell which by simple translations along the axes of refeience will locate all of the atoms in the ciystal are in a form which is immediately usable for testing them by fuither X-ray measurements. CHAPTER IV. THE COMPLETE ANALYTICAL EXPRESSION OF THE SPACE-GROUPS. Niggli has already recorded many of the simpler cases for the various space- groups. For some time the present writer has been engaged in working out analytically all of the special cases of the space-groups. The tables which follow are the results of these computations. They purport to give the coordi- nates of the most generally placed equivalent points and all of the special cases of these equivalent points contained within the unit of structure of each of the 230 space-groups. The analytical determination of the special cases can be quite simply carried out by equating the coordinates of one point xyz with those of each of the other equivalent positions within the unit cell. This will yield a series of special cases (if any exist) which can be further speciaHzed by applying this same process to the coordinates of these special positions. The continued use of this procedure will eventually yield all of the special cases for a space- group.* By way of illustration the special cases of the space-group C2h (page 49) will be deduced. The positions of the most generally placed equivalent points in the unit cell of this space-group are xyz; xyz; xyz; xyz. Equivalent point xyz will have the same position as equivalent point xyz when (1) x=x, y = y, z = z; that is, when x=0 or | (Xa), y = or -| (Xb) and z=w(Xc) where w is any fractional part of c. The lengths a, b, c are unit lengths along the X-, Y- and Z-axes. It will have the same position as the point xyz when (2) x = x, y = y, z = z; that is, when x=u(Xa), y = v(Xb), z = or ^(Xc); u and v are any fractional parts of a and b, respectively. The points xyz and xyz will coincide in position when (3) x = x, y = y, z = z; that is, when x = or |(Xa), y = or |(Xb), z = Oor|(Xc). The special cases of this space-group then arise from using these values for X, y and z. They are From (1) : (a) when x=0, y=0, and z = w;t then OOw; w. * The algebra of this process differs in certain details from the more ordinary kind. For in- stance there arises from our previous definitions the fact that = 1=2= Further- more x=x=0 or I, and X = K— X = i or J, and more generally x = l/n — x = "^" , where n = l, 2, 3, fin this example and in all of the tables which follow only the fractional parts of the unit lengths along the diffeient coordinate axes will be stated. If for any reason absolute distances of points are desired, it is of course necessary to multiply the coordinate values given in these tables by the proper values of a, b and c. 47 48 THE TRICLINIC SPACE-GROUPS c} AND cj. (b) whenx = 0, y = |, z=w; then 0|w; | w. (c) whenx = |, y = 0, z = w; then |0w; ^ w. (d) whenx = f, y = h z = w; then H w; | i w. From (2) : (e) whenx = u, y = v, z = 0; then uvO; tivO. (f) whenx = u, y = v, z = §; then uv|; u v |. From (3) : (g) whenx = y = z=0; then 0. (h) whenx = ^, y = z = 0; then ^0 0. (i) whenx = z = 0, y = ^-; then 0^0. (j) whenx = y = 0, z = ^; then 0|. (k) whenx=0, yandz = |; then 0| 1 1 2- (1) when x and z = |, y = 0; then ^ |. (m) when x and y = ^, z = 0; then ^ | 0. (n) when x, y and z = | ; then § 1 1. We must now speciaHze by the same procedure each of the special cases (a) to (f). Inspection, however, shows that in the present instance this will lead to no new special positions. All of the special cases of the space-group C2h are then defined by (a) to (n). The other space-groups can all be specialized in the same fashion. These special positions for each space-group are given in the tables which follow. TRICLINIC SYSTEM. A. HEMIHEDRY. Space-Group C}. One equivalent position: (a) xyz. B. HOLOHEDRY. Space-Group C}. One equivalent position: (a) 0. (e) H 0. (b)OOi (f) |0|. (c) 0^0. (g)OH. (d)iOO. (h)HI. Two equivalent positions: (i) xyz; xyz. THE MONOCLINIC SPACE-GROUPS Cg-Czh. 49 MONOCLINIC SYSTEM. A. HEMIHEDRY. Space-Group Cl. One equivalent position: (a) u V 0. (b) u V |. Two equivalent positions: (c) xyz; xyz. Space-Group Cl. Two equivalent positions: (a) xyz; x+i y, z. Space-Group Cf. Two equivalent positions : (a) u V 0; u, v-f ^, ^. Four equivalent positions : (b) xyz; xyz; x, y-j-f, z+|; x, y+i |-z. Space-Group Cg. Four equivalent positions: (a) xyz; x-|-i y, z; x, y+|, z-f^; x-M, y+|, |-z, B. HEMIMORPHY. Space-Group Cl. One equivalent position: (a) u. (b) I u. (c) 1 u. (d) H u. Two equivalent positions : (e) xyz; xyz. Space-Group Cg. Two equivalent positions: (a) xyz; x, y, z-f ^ Space-Group Cl. Two equivalent positions: (a) OOu; 0, I, u-M. (b)^Ou; i |, u-Ff. Four equivalent positions: (e) xyz; xyz; x, y-f-i z-{-^; x, ^-y, z+^. C. HOLOHEDRY. Space-Group €2^. One equivalent position: (a) 0. (e) OH- (b)OOi (f) hO-l (c)iOO. (g)HO. (d)OiO. (h)Hi 50 THE MONOCLINIC SPACE-GROUPS C2h-C2l, Space-Group C2h {continued). Two equivalent positions: (i) OOu; OOu. (1) Hu; HQ. (j) 0|u; OiQ. (m) u vO; uvO. (k) |0u; |0u. (n) uv|; U V |. Four equivalent positions : (o) xyz; xyz; xyz; xyz. Space-Group Cgh- Two equivalent positions: (a) OOi; oof. (d)Hi; 113 2 2 4- (b)0H; OH. (e) uvO; u v|. (c) hOh |0f. Four equivalent positions : (f) xyz; X, y, z+|; xyz; x, y, ^-z. These coordinate positions can be simplified by transferring the origin to the point { i/ ) of this first set. They then become : Two equivalent positions: (a) 0; OOi. (d)HO; Hi (b) 0|0; OH. (e) uvi; u vf. (c) |0 0; ^Oi Four equivalent positions: (f) xyz; X, y, z+l; X, y, |-z; xyz. Space-Group C2I. Two equivalent positions : (a) 0; n i i u 2 2. (c) 10 0; Ill 222. (b)OOi; 0^0. (d) HO; hoh Four equivalent positions: (e) OH; nil. 3 U 4 4 , U 4 f; Oil. (f) 244; 13 1. 13 2 4 4) 2 4 3 . 113 4) 244. (g) OOu; OOti; 0, h u+A; 0, 1 1 _i, 2)2 l^l. (h) iOu; iOu; h h, u+l; h i Hu. (i) uvO; uvO; u, v+i 1; u, 2 V, 2" Eight equivalent positions: (J) xyz; xyz; x, y-fi z-|-|; x, ^-y, z+i; xyz; xyz; X, ^- -y, Hz; X, y+h h-z. Space-Group C2l». Two equivalent positions: (a) iOO; f 0. (d)HO ; HO. (b) i i i; IH; (e) OOu ; |0u. (c) iO|; fO|. (f) Hu ; 0|u. THE MONOCLINIC SPACE-GROUPS CztrCzh- 51 Space-Group C2h {continued). Four equivalent positions : (g) xyz; xyz; x+i y, z; ^-x, y, z. These coordinate positions can be simplified by transferring the origin to the point ( ;c ) of this first set. They then become : Two equivalent positions : (a) 0; |0 0. (b) II; III. (c) 001; lOi (d)0|0; (e) fOu; (f) flu; 110. f Ou. A i fi 4 2 U. Four equivalent positions: (g) xyz; l-x, y, z; x+l, y, z; xyz. SpACE-Group C2h. Two equivalent positions : (a) iOi; fOf. /'K^i 1 1 1 • 3 13 \") 4 2 4> 4 2 4- (c) iOf; (d)ilf; 3 n 1 4^4' 3 11 4 2 4- Four equivalent positions: (e) xyz; x, y, z+|; x+|, y, z; |-x, y, |-z. These coordinate positions can be simplified by transferring the oiigin to the point l-^, -^ ) of this first set. They then become : Two equivalent positions : (a) 0; 1 0|. (c) 0|; |0 0; (b)0|0; III. (d) Oil; 1 1 0. Four equivalent positions : (e) xyz; |-x, y, z+|; x+|, y, |-z; xyz. Space-Group CgV Four equivalent positions : (a) iOO; fOO; 111. 4 2 2> III. (b)iO|; fO|; 1 1 n- 4 2 »-'> f|0. (c) Hi; 3 3 1. 3 13. Ill 4 4 4; 4 4 4> 444. (d)fff; 111. 111. 111 4 4 4; 4 4 4> 444. (e) OOu; |0u; 0, 1, u+l; 1, 1, 1 .. _. _. t-u. Eight equivalent positions: (f) xyz; xyz; x+|, y, z; |-x, y, z; X, y+l, z+^; X, |-y, z+l; x+i y+i |-z; 2~X, 2~y> 2~Z. A change of origin to the point ( ;^ ) of this set of coordinates would simplify (a) and (b). 62 THE ORTHORHOMBIC SPACE-GROUPS Cav-Ca^y ORTHORHOMBIC SYSTEM. A. HEMIMORPHY. Space-Group CgV. One equivalent position: (a) OOu. (c) iOu. (b) I u. (d) H u. Two equivalent positions : (e) uOv; tiOv. (g) Ouv; u v. (f) u|v; u|v. (h) |uv; ^ u v. Four equivalent positions : (i) xyz; xyz; xyz; xyz. Space-Group Ciy. Two equivalent positions : (a) uOv; u, 0, v-f-|. Four equivalent positions : (c) xyz; X, y, z+|; xyz; Space-Group Cav- Two equivalent positions: (a) OOu; 0, 0, u+|. (b) O^u; 0, h u+i Four equivalent positions : (e) xyz; xyz; x, y, z+^; x, y, z+|. Space-Group Cgtr. Two equivalent positions : (a) OOu; ^u. (b) Hu; 0|u. (c) iuv; fuv. Four equivalent positions: (d) xyz; xyz; x-|-|, y, z; ^-x, y, z. Space-Group Cgy. Four equivalent positions: (a) xyz; x, y, z-f-|; x+i, y, z; ^-x, y, z-\-^. Space-Group CaV Two equivalent positions: (a) OOu; i 0, u+|. (b)Hu; 0, i u+|. Four equivalent positions: (c) xyz; xyz; x+i y, z-\-^; |-x, y, z-|-i (b) u|v; u, i v+i X, ; y, z+i (c) iOu; h 0, u-l-i (d) Hu; 1 2> 1 2"> u+|. THE ORTKOREOMBIC SPACE-GROUPS Cay-Cay. 53 Space-Group Cay. Two equivalent positions: (a) |uv; f, u, v+§. Four equivalent positions: (b) xyz; x, y, z+|; x+i y, z+l; |-x, y, z. A slight simplification can be effected by transferring the origin of coordi- nates to -~ of this first set. They then become : Two equivalent positions : (a) Ouv; I, u, v+|. Four equivalent positions: (b) xyz; |-x, y, z+|; x+i y, z+i; xyz. Space-Group CgV Two equivalent positions: (a) OOu; Hu. (b) ^Ou; 1 u. Four equivalent positions: (c) xyz; xyz; x-|-|, |-y, z; ^-x, y+|, z. Space-Group Cay. Fowr equivalent positions: (a) xyz; x, y, z+|; x-fi |-y, z; |-x, y-^i, z-{-i Space-Group Cgy. Tioo equivalent positions: (a) OOu; i i u-hi (b)Oiu; I, 0, u-f^. Fowr equivalent positions : (c) xyz; xyz; x+i |-y, z+i; ^-x, y+|, z+|. Space-Group CH. Two equivalent positions: (a) OOu; Hu. (b) ^Ou; § u. Four equivalent positions : (c) Hu; f fu; Hu; f iu. (d)uOv; uOv; u-}-|, |, v; ^-u, |, v. (e) Ouv; Ouv; ^, u+^, v; i, ^-u, v. iJzg/i< equivalent positions: (f) xyz; xyz; xyz; xyz; x+i y+iz; l-x, §-y, z; x-|-i ^-y, z; |-x, y-j-|, z. 54 THE ORTHORHOMBIC SPACE-GROUPS cH-cl Space-Group Cl^. Four equivalent positions: (a) uOv; fi, 0, v+|; u-I-^ §, v; |-u, i v+ 1 2- Eight equivalent positions: (b) xyz; x, y, z+l; xyz; x, y, z+i; x+i y+l, z; ^-x, |-y, z+l; x+|, |-y, z; i-x, y+i z+|. Space-Group Cgv- /^owr equivalent positions : (a) OOu; Hu; 0, 0, u+i; i i u+i. (b)Oiu; lOu; 0, i, u+A; i 0, u+i. (c) 4 4 u; 4 4 u; 4, 4, u+fJ 4> 4> u-|-2. Efy/i/ equivalent positions : (d) xyz; xyz; x, y, z+|; x, y, z+^; x+^y+iz; l-x, ^-y, z; x+|, |-y, z+|; i-x, y+i z+|. Space-Group Cgv- Two equivalent positions : (a) OOu; 0, i u+|. (b) H u; |, 0, u+i Four equivalent positions: (c) uOv; uOv; u, |, v-j-|; u, I, v+|. (d)Ouv; Ouv; 0, u+|, v+i; 0, |-u, v+|. (e) |uv; |uv; i u-M, v+|; i |-u, v+|. ^^gf/if equivalent positions: (f) xyz; xyz; xyz; xyz; X, y+l, z+^; X, ^-y, z-i-l; x, |-y, z+§; x, y-|-|, z+|. Space-Group Czv- Four equivalent positions : (a) OOu; O^u; 0, 0, u4-|; 0, i u-}-|. (b)Hu; |0u; hh^+i; I, 0, u-f-|. (c) uiv; uf v; u, f, v-|-|; u, i, v-f-|. Eight equivalent positions: (d) xyz; syz; x, y, z+|; x, y, z-f-J; X, y+iz+l; X, ^-y, z4-i; x, |-y, z; x, y+l, z. Space-Group Cay. Four equivalent positions : (a) OOu; |0u; 0, |, u+i; h h u+i (b) iuv; fuv; i u-l-^ v+^; f, ^-u, v+i THE ORTHORHOMBIC SPACE-GROUPS cl^-cly. 55 Space-Group C2V (continued). Eight equivalent positions : (c) xyz; xyz; x+|, y, z; ^-x, y, z; X, y+^z-l-^; X, ^-y, zH-l; x+|, |-y, z-hl; i-x, y-hi z-hi. Space-Group CH. Four equivalent positions : (a) OOu; iiu; |, 0, u+i; 0, i, u-j-i Eight equivalent positions: (b) xyz; xyz; x+|, y, z+i; ^-x, y, z+|; X, y+iz+^; X, i-y, z+l; x+i ^-y, z; |-x, y+^, z. Space-Group CH. Four equivalent positions: (a) OOu; Hu; i 0, u-h|; 0, i u-|-i Eight equivalent positions: (b) 4 4U; 4 4U; T) T> U"r2; 4> 4j u~r2J 13„. 3i„. 13 ,,4.1. 3 1 n_l_l 4 4 U, 4 4 U, 4, 4, UT^2; 4j 4) '^T^2' (c) uOv; uOv; u+|, i v; |-u, I, v; u+i, 0, v+§; |-u, 0, v-(-i; u, ^, v-H|; u, |, v-|-|. (d) Ouv; Ouv; i u-hl, v; i |-u, v; iu, v-hl; ^, u, v-h^; 0, u-M, v-|-|; 0, ^-u, v+|. Sixteen equivalent positions : (e) xyz; xyz; xyz; xyz; x+2) y~r2> z; 2~x, ^— y, z; x-|-2, 2"~y> z; 2~x, yi-2> z; x+iy, z-M; ^-x, y, z+l; x+i y, z-|-^; |-x, y, z-|-|; X, y+iz-f-|; X, i-y, z-l-i; x, f-y, z-l-|; x, y-|-|, z-hi Space-Group C2V. Eight equivalent positions: (a) OOu; Hu; i i u+i; f, f, u-hi; iO,u+i; 0,iu-F^; f, i u-^f; i f, u-|-|. Sixteen equivalent positions: (b) xyz; xyz; x-|-i J-y, z+|; i-x, y-Hi, z-f-i; x-Fiy-F^, z; ^-x, |-y, z; x+f, f-y, z-Hi; f-x, y-f-i z-i-l; x+iy, z-1-^; l-x, y, z-l-l; x-f-f, f-y, z-f-f; 4~x, y-rj, Z-I-4; X. y+i z-l-l; X, |-y, z+|; x-f-^ f-y, z-f-f ; i-x, y+i z+i 56 THE ORTHORHOMBIC SPACE-GROUPS cly-Y^. Space-Group Cly, Two equivalent positions: (a)OOu; i i u+i (b)|Ou; 0, i u+|. Four equivalent positions: (c) uOv; uOv; u+^, ^, v-f-^; §-u, ^, v4-^. (d) Ouv; Otiv; f, u+^, v+|; I, §-u, v+^. Eight equivalent positions: (e) xyz; xyz; xyz; xyz; x+2, y"r^) z+2j 2~x, 2~y> z-i-2> x+2> 2~y> Z+2> 2-x, y+i z-hi Space-Group C"- Fowr equivalent positions: (a) OOu; Uu; 0, 0, u4-^; i i u+|. (b)0|u; §0u; 0, i u+l; i 0, u-h|. J5JzgfJ^f equivalent positions: (c) xyz; xyz; x, y, z-}-|; x, y, z-f-^; x+iy+2, z+l; ^-x, ^-y, z-Hi; x+i |-y, z; i-x,y+^, z. Space-Group C|v. Four equivalent positions: (a) OOu; |0u; i i u-f-^; 0, i u-}-|; (b)iuv; |uv; i u-M, v4-|; i, §-u, v-|-|. ^fgf^f equivalent positions: (c) xyz; xyz; x+^, y, z; |-x, y, z; x+iy+izH-^; ^-x, ^-y, z+|; x, ^-y, z+^; X, y+l, z-h^ B. HEMIHEDRY, Space-Group V^. One equivalent position : (a) 0. (d)OOi. (g) OH- (b)iOO. (e)HO. (h)iH. (c) 0^0. (f) hOh. Two equivalent positions: (i) uOO; uOO. (m) OuO; OuO. (q) u; OOu (j) uO^; uOi (n) Ou^; Ou^ (r) |0u; |0u (k) uH; ti^O. (o) ^uO; ^uO. (s) OH; 0|u (1) uH; QH. (P)|ui; |ui (t) Hu; Hu Four equivalent positions: (u) xyz; xyz; xyz; xyz. THE ORTHORHOMBIC SPACE-GROUPS V^-V^. 57 Space-Group V^. Two equivalent positions: (a) uOO; uOi (c) Oui; u f . (b)uH; u*0. (d) inl; i- u f. Four equivalent positions: (e) xyz; xyz; x, y, |-z; x, y, z+i Space-Group V. Two equivalent positions: (a) OOu; Hu- (b) 0|u; |Oti. Four equivalent positions: (c) xyz; x-fl, |-y, z; |-x, y-}-^, z; xyz. Space-Group V*. Four equivalent positions: (a) xyz; x+|, |-y, z; x, y-|-i |-z; ^-x, y, z-|-|. Space-Group V*. Four equivalent positions: (a)uOO; uO^; u+i i 0; ^-u, i i (b) Oul; Ouf; |, u+i i; |, ^-u, f. ^tfif/if equivalent positions: (c) xyz; xyz; x, y, |-z; x, y, z-{-|; x+iy+l, z; x-l-il-y, z; ^-x, y+i ^-z; |-x, l-y, z+^. Space-Group V. Two equivalent positions: (a) 0; HO. (c) OH; |oi (b) ^00; 0^0. (d) HI; OOi Four equivalent positions: (e) uOO; uOO; u+i i 0; |-u, i 0. (f) uH; QH;' u+i o, |; |-u, o, i (g)OuO; OuO; i u+i 0; i i-u, 0. (h) |u|; |u*; 0, u+i |; 0, |-u, i (i) OOu; OOu; Hu; H u. (j) 0|u; Oiu; ^Ou; |0u. (k) i^u; 44U; 44U; 44 u. Eight equivalent positions: (1) xyz; xyz; xyz; xyz; x+iy+iz; x+il-y, z; |-x, y+i z; ^-x, |-y, z. 58 THE ORTHORHOMBIC SPACB-GR0UP3 V^-V\ Space-Group V^. Four equivalent positions: (a) 000; HO; ^01; OH. (b) iOO; OiO; OOi; Hi (c) ill. 4 4 4; 13 3. 4 4 4> fH; Hi (d) HI; Hi; IH; IH. Eight equivalent positions: (e) uOO; uOO; u+i i 0; l-u, i 0; uH; QH ; u+i 0, 1; ^-u, 0, -|. (f) OuO; OuO i u+i 0; i l-u, 0; *u|; n^ ; 0, u+i 1; 0, l-u, i (g) OOu; OOu; i 0, u+^; -I, 0, ^-u; Hu; Hu; 0, h u+l; 0, i i-u. (h) Hu; ffu; 3 3 l_i,. 1 1 l_,v 4; 4> 2 '^J 4j 4) 2 "> ifu; fiQ; 3 1 n-4-i- i ^ ii4-i 4> 4) U-t-2, 4> 4> "^2' (i) |ui; 3 ,, 3 4 U 4, 3 1_,, 3. 1 1_,, 1. 4> 2 "> 4> 4> 2 "> 4> if, 3 . 4 U 4, ftii ; i u+i \; \, u+i f. Q) ,,11. U 4 4, uff 1_11 3 3. 1_,, 1 1. 2 "j 4> 4> 2 "> 4; 4> f, 1 3 U 4 4 uH ; u+l. f, i; u+i i i Sixteen equivalent pos itions : (k) xyz; xyz; xyz; xyz; x+i y+iz ; x+i^-y, z; ^-x, y+i z; ^-x, ^- -V, 7; x+i y, z+l ; x+iy, |-z; ^-x, y, ^-z; ^-x, y, z+i X, y+ iz+l ; X, |-y, l-z; X, y+i |-z; X, ^-y, z+i Space-Group V*. Ttoo equivalent positions : (a) 000; Hi (b) iOO; OH- Four equivalent positions : (c) 001; HO. (d) |0|; 0^0. (e) uOO; (f) uO|; (g) OuO; (h)Ou|; (i) OOu; (J) 0|u; uOO QO^ OuO Oui OOu 0|u u+i i i; l-u, i i u+i i 0; i-u, i 0. i u-l-i i; i ^-u, i i u+i 0; i Ku, 0. 2> 5> U-|-2; 2> 2> ^~U. i 0, u-hl; i 0, i-u. £?tgi/if equivalent positions : (k) xyz; x+i y+i z+l; xyz; x+i i •y, ^-z; xyz; xyz; l-x, y+i |-z; ^-x, ^-y, z-i-i 59 Space-Group V*. Four equivalent positions: (a) uOi; l-u, 0, f; uH; u+i h f. (b) iuO; f, l-u, 0; iQi; f, u+i i (c) Oiu; 0, i l-u; IH; h I u+|. jE^i'gf/if equivalent positions: (d) xyz; x, y, |-z; |-x, y, z; x, |-y, z; x+5, y+iz+l; x+i |-y, z; x, y+i |-z; |-x, y, z+|; C. HOLOHEDRY. Space-Group V^. One equivalent position : (a) 0. (d)Hi (k) OH. (b) iOO. (e) 0|0. (h) hhh (c) oo|. (f) HO. Two equivalent positions : (i) uOO; uOO. (m)OuO; OuO. (q) OOu; OOu (j) uO|; uOi (n) Oui; Oui (r)OH; OH (k) u|0; u*0. (o) HO; HO. (s) |0u; |0u (I) uH; QH. (p)Hi; huh (t) 1 1 u; IH Four equivalent positions: (u) Ouv ; Ouv; Ouv Ouv. (v) H V , H v; Hv Hv. (w) u V uOv; uOv uOv. (x) u|v , ufv; uH uH* (y) u V uvO; uvO uvO. (z) uv| uv|; uv| uvi Eight equivalent positions: (a) xyz; xyz; xyz; xyz; xyz; xyz; xyz; xyz. Space-Group V^. Two equivalent positions: (a) 000; Hi (c) 06| HO. (b)HO; OH. (d)HI 0|0. Four equivalent positions: (e) Hi IH; IH; Hi (f) HI, fii; HI; IH. (g) uOO tiOO; l-u, i 1; u-Fi i i. (h) uO| Q0|; i-u, §, 0; u+i I 0. (i) OuO OuO; i i-u, 1; i u+i i (j) Oui Ou|; i i-u, 0; i u-hi 0. (k) u OOti; i i I-u; i i u+i (1) OH. OH; i 0, I-u; i 0, u+i 60 THE ORTHOREOMBIC SPACE-GROUPS yI-Y^ Space-Group Yl {continued). Eight equivalent positions: (m)xyz; ^— X, xyz; ■y, l-z; xyz; xyz; 2-x, y+i z-\-\; x-M, l-y, z-f-L' x+2> y"i~2> 2~2;. Space-Group V'. Two equivalent positions: (a) OOi; (b)Hi; (c) OH; (d)^Oi; oof. ill 2 2 4- nil U 2 4* 2 '-' 4- Four equivalent positions (i) uOO tiOO; (j) uH U 2 2> (k) OuO OuO; (1) ^u| h^h (m) u OOu; (n) H u Hu; (o) i u OH; (p) ^Ou ^Ou; (q) uv| ; uvf; (e) 0, OOi (f) 100 |0i (g) OH, 0|0. C! • (h)HI, HO. Lb. uOi uO|. uiO u^O. Ou| Ou|. ^uO iuO. 0, 0, l-u; 0, 0, u-Kl 1 1 2> 2> 1 „ . 1 ^~U, 2, i u+l 0, i l-u; 0, i u-f-l i 0, l-u; i 0, u+l uvf ; tivi ^tg^^ equivalent positions: (r) xyz; xyz; X y, |-z; X, y, z+^; Space-Group Yt. xyz; xyz; X, y, z-M; X, y, \-z. Two equivalent positions : (a) 0; ^^0 (b) 10 0; 2 2 OiO. (c) OH; (d)HI; |oi 00 i Four equivalent positions: HO 111 4 I 5 (e) (f) (g) uOO (h)uH OuO |u| (i) (j) (k) u (1) 0|u 1 1 n. 4 4 ^> 13 1. I.l f > uOO; U 2 2, OuO; 2^2, OOti; 0|u; fiO; ffO. ff|. h 0; 0, h u, 0; 0, l-u, I; fil; l-u, l-u, 1 1 2) 2 u+l, I, 0. u+l, 0, |. I, u+l, 0. 0, u+l, i 1 1 fS . ill, 2 ^ U, 2 2"- lOu. |0u; Eight equivalent positions : (m)xyz; xyz; xyz; xyz; l-x, l-y, z; l-x, y+l, z; x+|, |-y, z; x+|, y+|, z. THE ORTHORHOMBIC SPACE-GROUPS Y^-vl. 61 Space-Group Vh. Two equivalent positions: (a) 0; OOi (d)HI; HO. (b)|00; |0i (e) Oui; Ouf. (c) OH; 0|0. (f) Iu|; 1 ,-; 3 ^U 4. Four equivalent positions: (g) uOO; uOO; uO|; uOi (h) uH; uH; u-^O; u|0. (i) Ouv; Ouv; 0, u, l-v; 0, ti, v+l (j) §uv; |uv; i u, h-y; h u, v+i (k;)uv|; uvf; uv| ; uvf. jEzgf/i^ equivalent positions: (1) xyz; xyz; x, y, §-z; x, y, z+|. xyz; xyz; x, y, z+i; x, y, |-z. Space-Group Vh. Four equivalent positions : (q\ iin- iin- 111- 331 yaj 44U, 44U, 442> 44^' /^Mlll. 131. 3in. 33n \"J 442> 442> 44'-'> 44'-'' (c) uOO; uO|; |-u, i 0; u+i i i (d) Oui; Ouf; i i-u, f; |, u+l, i Etgf/i< equivalent positions: (e) xyz; xyz; x, y, |-z; x, y, z+|; l-x, l-y, z; ^-x, y+l, z; x+|, ^-y, z+i; x+5, y+i ^-z. A slight simplification of the two uniquely defined positions [(a) and (b)] can be effected if the origin of coordinates is changed to the point ( "o "2^ ) ^^ this first set. Space-Group V^. Two equivalent positions : (a) iOO; fOi (c) i H; f i 0. (b) foo; ioi. (d) fH; HO. Four equivalent positions: (e) uOO; u0|; ^-u, 0, 0; u+i 0, ^ (f) uH; tii-O; l-u, I, 1; u+i §, 0. (g) Oui; Ouf; Uf; |ui (h) iuv; itiv; f, u, |-v; f, u, v+i ^tgf/i< equivalent positions : (i) xyz; xyz; x, y, |-z; x, y, z+^; l-x, y, z; §-x, y, z; x+|, y, z+^; x-f-^, y, l-z. 62 THE ORTHORHOMBIC SPACE-GROUPS vl-vL Space-Group Yl (continued). By shifting the origin of coordinates to the point ( "o ) of this first set, these positions become; Two equivalent positions: (a) 000; AOi (b) iOO; OOi (c) OH; HO. (d)Hi; oio. Four equivalent positions : (e) uOO; uOO; ^-u, (f) uH; uH; l-u, (g) iui; iuf; fuf; (h) Ouv; Ouv; |, u, 0, 1; u+i 0, h h 0. u+i, i 0. 3 ,, 1 4 U 4. l-v; h u, v-h^ £'ig'/i< equivalent positions: (i) xyz; xyz; ^-x, y, ^-z; |-x, y, z-f|; xyz; xyz; x+|, y, z+§; x+l, y, |-z. Space-Group V^. Four equivalent positions: (a) OiO; (b) i i I; (c) uOO; (d)Oui; (e) |ui; OfO; i 3. 1- 2 4 2; uO^; Ouf; iQf; i 1- ^ 4 2 > HO; ulO; 0, l-u, f; "4 2- 1 3 n 2 4^' 11 i i U 2 2- 1 2> i — 11 ^• 2 "> 4> 0, u+i, i Eight equivalent positions: (f) xyz; xyz; X, y+i z; y, i -z; z+l; X, y, z+^; X, y+i ^-z. X, t-y, z; x, yi-t, z; x, f-y, The unique cases can be simplified by transferring the origin to the point ( o^ )• Space-Group Vu- Two equivalent positions: (a) 0; HO. (b)00|; Hi Fow equivalent positions: (c) 0|0; (d)OH; (e) OOu (f) Oiu (g) uvO (h)uvf OOu; o§u; uvO; uv^; 1 1 f, . 2 2 "> ? f u. ^Ou. |0u; u+i ^-v, 0; u+i i_ u, v-fi 0. •V, I; |-u, v-f-l, Eight equivalent positions: (i) xyz; xyz; x+i ^-y, z; |-x, y+i z; xyz; |-x, y+i z; x+i I i_ y. z; xyz. THE ORTHORHOMBIC SPACE-GROUPS Vt-V". 63 Space-Group Vh". Four equivalent positions: /'o^lll• 313. 133. Ill V<^y 444; 444} 444> 444- (V,\ 11^- 111. 111. Ill \") 444; 444; 444; 44 4* (c) OOu; Hu; hh, |-u; 0, 0, u+i (d)0|u; |0u; i 0, |-u; 0, i u+i ^ipAi equivalent positions: (e) xyz; x+i |-y, z; |-x, y+i z; xyz; i-x, |-y, ^-z; X, y, z+|; x, y, z+l; |-x, |-y, z+l. By shifting the origin to f ^' o^' o' ) ^^^ uniquely placed arrangements can be sUghtly simplified. Space-Group V". Four equivalent positions: (a) OiO; HO; HO; OfO. rMnii- 111- 131. c\ii. \") ^42; 242; 242; "42' (c) OOu; Hu; 0|u; |0u. (d) iuv; f uv; f, |-u, v; \, u+|, v. Eight equivalent positions: (e) xyz; x+i h-y, z; |-x, y+i z; xyz; X, h-y, z; i-X; y, z; x+|, y, z; x, y+l, z. The unique cases can be simplified by placing the origin at the point ( 2^ )• Space-Group V". Two equivalent positions: (a) OOi; Hi (c) OH; iof. (b)00|; Hi. (d)0H; loi Four equivalent positions : (e) g-C^«; i-|u; i i u-|-^; 0, 0, Hu. (f) 6|u; iOu; i 0, u+|; 0, \, \-m. (g) uvi; uvi; u+i Hv, f; |-u, v+i f. £'tgf/i< equivalent positions: (h) xyz; x+l, |-y, z; Hx, y+i z; xyz; X, y; ^-z; l-x, y+i z-f-l; x+i HY; z+^; x, y, ^-z. The unique cases can be simpUfied by changing the origin to ( 2 )• Space-Group Vh^ Two equivalent positions: (a) OOu; Hu- (b) 0|u; ^Oa. 64 THE ORTHORHOMBIC SPACE-GROUPS Vfa-Vh. Space-Group Vh (continued). Four equivalent positions: (c) i i 0; 3 1 n- 4 4 U, 4 4 U. (d)iH; 3 11. 4 4 2; 13 1. 3 3 1 4 4 2> 4 4 2- (e) Ouv; Ouv; h, l-u, v; h, u+i v. (f) uOv; uOv; ^-u, 1, v; u+i 1, v. Eight equivalent positions : (g) xyz; x+i |-y, z; ^-x, y+^, z; xyz; l-x, |-y, z; xyz; xyz; x+i y+i z. The unique cases can be simplified by changing the origin to ( ^' o^ j. Space-Group V^h • Four equivalent positions: (rt\ ini- ^ii- 113. ani V.ct; 4'-'4> 424; 424> 4'-'4' Ch'i^n^' 111- 311. 10^ \'->J 4'-'4> 424; 424; 4^4' (c) OOu; Hii; i 0, |-u; 0, i u-|-i Eight equivalent positions : (d) xyz; x+i §-y, z; ^-x, y+f, z; xyz; l-x, y, §-z; X, y+i z+l; x, |-y, z+l; x+i y, |-z. By changing the origin to the point f -^' ^ j the unique cases are simplified. Space-Group V^^ Four equivalent positions: (a) 000; HO; OH; §0i (b) HI; i; i 0; 010. Eight equivalent positions : (c) xyz; x+i |-y, z; x, y-{-^, |-z; |-x, y, z+i xyz; l-x, y+l, z; x, |-y, z-|-§; x-|-i y, |-z. Space-Group Y'^. Four equivalent positions : (a) Xlf). 3 in. 111. Ill \aj 44 W, 44 U, 442; 44 2- CM ill- 114' 3. 3 (). lln W>' 442; 44l; 44*^; 44'-'' (c) Ouv; h l-u, v; 0, u+i |-v; i u, v-}-^ ^z'gfAf equivalent positions: (d) xyz; x+i |-y, z; x, y+|, |-z; ^-x, y, z+^; l-x, l-y, z; xyz; x+|, y, z+|; x, y+|, ^-z. The unique cases are simplified when the origin is changed to ( ^ (f ?)• THE OETKOREOMBIC SPACE-GROUPS V^^V^,«. 65 Space-Group Vh^. Four equivalent positions : (a) 0; 1; HO; Hi (b) |00; iOi; 0|0; OH- (c) Ou|; Ouf; I u+i h Eight equivalent positions: 1 i_„ 2> 2 "> ill. 4 4 2; /'rl^lln- 3. If). 311. 3 3 n. 3 1 A. 4 4 U, 4 4 "> (e) uOO; uOO; uO^; uO|; 3 3 1. 4 4 2> 111 4 4 2- 1 ,, 1 1 2~U, 7j, 5 "T^2> 2> 1 u+i i 0; l-u, i, (f) Ouv; 0, u, ^-v; i u+i v; i u+i Ouv; 0, u, v-hh; h, i ■u, v; f, ^-u, i-v; v+i (g) uvi; uvf; u+i v+^, i; u+f, |-v, f; uvi; i-u, l-v, f; |-u, v+i |. uvf; Sixteen equivalent positions : (h) xyz; xyz; x, y, ^-z; x, y, z+|; xyz; xyz; x, y, z+|; x, y, |-z; x+l, y+i, z; x+i |-y, z; ^-x, y+i §-z; ^-x, i-y, z+^; 2-x, i-y, z; ^-x, y+l, z; x+i §-y, z+|; x+i, y+i l-z. Space-Group Vh^. jPowr equivalent positions : (a) iOO; 10^; HO; (b) fOO; 10 h IhO; Eight equivalent positions : 111 4 2 2- 3 11 4 2 2- (c) OH HO (d) u uH (e) Oui Ouf (f) i u V iu V OfO; OH; Ofl; iin- lai. Ill 24 "> 242> 24 2' uOi; u-iO; Uf; Hu, 0, 0; u+i 0, ^; i— 11 i A- ii4-i 1 n i l-u, f; 0, uH, i; 2, u+l, i; 0, l-u, f. i u, i-v; f, u+i v; f, u+i |-v; i % v-|-§; I, §-u, v; i|-u, v+i |u Sixteen equivalent positions: xyz; (g) xyz; x+i y+l, z; x+l, Hy, ^ X, y, f-z; X, y, z+i; X, y, z; ^-x, y, z; x+i y, z+^; x+i y, ^- z: l-x, y+i Hz; ^-x, Hy, z+l; X, Hy, z; X, y+l, z; X, Hy, z-H; x, y-f-i f •z. 66 THE ORTHORKOMBIC SPACE-GROUPS V Space-Group V". Two equivalent positions: (a) 0; HO. (c) OH |0|. (b) 10 0; 0|0. (d)IH 00|. Four equivalent positions : (e) HO; HO; 3 1 n. 3 3 n 4 4 U, 4 4 U. (f) HI, 13 1. 4 4 2) 3 11. 3 3 1 4 4 5> 4 4 2- (g) uOO uOO, u+l, 1, 0; 1- -u, 1, 0. (h)uH fill. U 2 2 J u+l, 0, 1; 1- -u, 0, |. (i) OuO OuO; h u+l, 0; 1, l-u, 0. (i) huh, 1 fi 1 . 2^2, 0, u+l, 1; 0, l-u, |. (k) OOu, OOu; 2 1 U; 2 2 u. (1) 0|u, 0|u; |0u; |0u. Eight equivalent positions: (m)Hu Hu, Hu; Hu; IfQ fiu, 13,,. 1 1 fi 4 4 U, 4 4 U. (n) Ouv ; Ouv ; 1, u+l, v; 1, u+l, v; Ouv ; Ouv ; 1, l-u, v; 1, l-u, V. (o) uOv ; uOv ; u-f 1, 1, v; l_n 1 2 "j 2> v; uOv ; uOv ; u+l, 1, v; 1 n 1 2— U, 2> V. (p) u v ; uvO ; u+l, v+l, 0; l-u, 1- -V, 0; uvO ; uvO ; u+l, l-v, 0; l-u, V- fl, 0. (q) u V 1 ; u V ^ ; u+i v+l, 1; 1 n 1 2— U, 2" -V, 1; uv| ; u v| u+l, l-v, 1; l-u, V- fl, 1 2' Sixteen equiva ilent pos itions (r) xyz; xyz; xyz; xyz; xyz; xyz; xyz; xyz x+l, y+l, z ; x+i l-y, z; l-x, y+ 1, z ; l-x, l-y, z; ^ — X, l-y, z ; l-x, y+l, z; v-4-i i — A-r2, 2 y, z 1 X + 2, y+l, z. Spa ;e-Group V1°. Four equivale Qt positi ons: (a) 00 00^; HO; HI. (b)|00 |0|; 0^0; OH. (c) OOi OOf; IH; Hi (d)OH OH; |0i; |0i (e) iH HI; fH; Hi (0 Hf, 111. 4 4 4; 111. Ill 4 4 4 > 4 4 4* £Jifif/i< equivalent positions: (g) uOO uOO (h) u OuO uO|; u+l, I, UU 2, Oui; Ou|; l-u, I, I, u+l, I, l-u, 1 1 . 2, ?> 1 1 U + l, l-u, I, I, u+l, I; I, l-u, |. THE ORTHORHOMBIC SPACE-GROUPS Vh^-v". 67 Space-Group Y^^ {continued). (i) OOu; Hu OOti; Hu (j) 0|u; lOu O^u; —- (k) iiu; 3 3,,. (1) uvi; uv 2 u u 1 3 ,-, 4 4 U 3 1 ,-, 4 4 U i-r 3 uv|; ti vf 0, 0, u+l; 0, 0, i-u; 0, i u+l; VJ, 2j 2 '^J 1 1 l—u- 4» 4> 2 "^J 3 3 1_,,. 4> 4) 2 "^J 1 — 11 l_v !• 2 U) 2 V, 4, Sixteen equivalent positions : (m) xyz; xyz; 2> 2> 2 "^' h 0, u+l; i 0, i-u. 1 3 1,4.1. 4) 4> "T^2> 3 1 ii_L.l 4) 4> U-t-2. u+l, i-v, f; l-u, v+i f. xyz; xyz; X, y, t-z; X, y, z+l; X, y, z+^; X, y, |-z; x+i y+i, z; x+i ^-y, z; ^-x, y+|, z; i-x, |-y, z; 2~x, 2~yj 2~z; 2— X, y+^'j z+2 j x+j, 2~y> z-|-2> x+i, y+i i-z. Space-Group Vh^ jPowr equivalent positions: (a) 0; JOO; HO; o\o. (b)OOi; 5 1; 111. nil 2 2 2» U 2 2- (c) iOO; fOO; fH; HO. (d)HI; 3 11. 4 2 2; 101; ioi (e) OiO; OfO; HO; HO. (f ) 2 4 2 5 13 1. 2 4 2» n a 1- nil ^ 4 2» ^4 2- (g) i-iu, 13,-.. 4 4 U, Hu; Hu. Eigr/i< equivalent posit ions: (h) u ulO Hu, h 0; u+i 0, 0; tiOO uiO u+l, i 0; Hu, 0, 0. (i) uH uOi Hu, 0, \; u+l, 1, h aH uOi u+2> 0, 1; Hu, ^, 1 2- (j) OuO HO i Hu, 0; 0, u+i 0; OuO HO i u+i 0; 0, Hu, 0. (k) hnh Ou^ 0, Hu, \', i u+i 1 . \^\ Ou^ 0, u-hi i; 1, Hu, 1 2' (1) OOu 2-Ou OH; Hu; OOu Hu OH; H u. (m) J u V f U V ; i u-l-i v; i u+i v; J U V 1 U V ; i Hu, v; \, Hu, v. (n) u i V uf V ; u-l-i f, v; u-l-i \, v; ti J V uf V ; Hu, i v; Hu, i V. Sixteen equiva lent po£ jitions: (o) xyz; xyz; xyz; xyz; i-x, y, z; Hx, y, z; x+i y, z; x+i y, z; x+i y+i z ; x+iHy, z; Hx, y+i z; Hx, Hy, z; x,|- ■y, z; X, y+l, z; x, H y, z; X, y+i z. 68 THE ORTHORHOMBIC SPACE-GROUPS Vh^-Vh^ Space-Group Y^. Four equivalent positions : (a) 0; i (b) iOO; - i- 2 ^ 2 > 1 1 0- 2 2 "> OOi; 0|0; "2 2- 111 2 2^' Eight equivalent positions: (c) |0i 3 11 4 2 4 (d) H (e) 13 1 2 4 4 (f) uOO uOO OuO OuO (g) OOu OOti 1 (h) _ u 3 3 „ 4 4" 1- 3 n 3. ^ 4 J 4^4) 1 3. 2 4 > 3 3 4 4 J 1 3 . 4 4> Oil; Oil: 1 3. 2 4> 1 3 4 4> 3 3. 4 4; u+i 0, i; 2~U, 0, 2) 0, u+i i; '-' 4> 1 1 2 4- 3 1 . 4 4 > 1 1 4 4- 1 2> 1 2? 0, 0, 0, -u, i; u+l; 1 3 t-u; - - IT 13 3 1 ri • 3 1 4 4 U, 4, 4, i u+l, 0; 2> 2~u, 0; I, 0, u+l; 2^> 0, 2 ^5 l-u; 1_„. 3 3 2 U, 4, 4, 1, 1 1. U 2 2 > fl i i U 2 2* 1 „ 1. 2 U 2 ) i f] i 2 U 2- 1 1 „. 2 2 "^^ Hu. u+l; u+i Sixteen equivalent positions : (i) xyz; xyz; xyz; xyz; l-x, y, |-z; l-x, y, z+l; x+i y, z+|; x+i, y, |-z; x+iy+l, z; x+l, |-y, z; |-x, y+|, z; f-x, |-y, z; X, l-y, |-z; X, y+i z+^; x, §-y, z+|; x, y+i ^-z. Space-Group Vh^ /^owr equivalent positions: (a) 00 1 1 n. 2 2 ^j 1 n 1 2^2) OH. (b)^OO 0|0; OOi, Hi t equivalent positions: (c) OH 13. ^ 4 4j n 1 3 U 4 4 Ofi; Hi 113. 2 4 4) 13 3 2 4 4 Hi. (d) \oi 1 n a- 4 '-' 4» fOf fOi; 3 11 4 2 4 3 13. 4 2 4) iH iH. (e) HO ifO; fiO; HO; 111 4 4 2 H^; ii^; iH. (f) Hi 13 3. 4 4 4) 3 13. 4 4 4) Hi; 3 3 1 4 4 4 fii; ifi. 111 444. (g) uOO u+i 0, ^; u+i i 0; u uOO l-u, 0, 1; l-u, h, 0; u (h) u 0, u+i h; h, u+i 0; ^ OtiO 0, h- -u, h; i Hu, 0; ^ (i) OOu 0, i n+h i 0, u+^; 1 2 OOu 0) i h-n; i 0, H -u; ^ u ^; 1 2. i. 2) u|. ^u; ^u. TEE CRTEORKOMBIC SPACE-GROUPS Vh^-vl*. 69 Space-Group V^^ (continued). Sixteen equivalent positions: (J) (k) (1) (m) (n) (o) i i 11 4 4 U 13,,. 4 4 U, i h u+l; 1 4> 3 4j u+L- IfQ 3 1 n • 4 4 U, 3 3 1 4> 4> 2~ -u; 3 4) 1 4) 1 2- -u; flu 3 1,,. 4 4 U, 3 3 ,i_J_l. 4> 4) Ui-2, i 1 47 u+l; HQ; 13.-,. 4 4 U, 111 4> 4> 2 -u; 1 4> 3 47 ^- ■u. 1 n 1 4 U 4, 1 n 1 . 4 U 4, i u+i 1 . 4 > 1 47 u- f|7 f; ftif, 3 ,-, 1 • 4 U 4, i l-u, 3. 4> 3 4) 1 2 ■ -u, 1 . 47 fuf, 3 ,, 1 . 4 U 4 , 3 ,,_Ll 4, U-1-2, 3 . 4> 3 4> u- H, 1 . 47 1 Ti 1 4 U 4, iQf; 1 1_,] 4> 2 "> 1 . 4 7 1 47 1 2 " -u, 3 4- 11 1 1 U 4 4, uH; u+i h 1. 4» u ■f^ 1 7 47 3. 47 f, 3 3 U 4 4, f, 3 1 . U 4 4, 1 — 11 ^ 2 ") 4> 3 . 47 1 2 -u 3 7 47 1 . 4 7 ,,3 3 U 4 4, uH; u+i I, 3 . 4) u H 3 7 47 1 . 4> vi 1 1 U 4 4) GH; l-u, i, 1 . 4> 1 2 -u 1 7 47 3 4- Ouv 0, u+i, v+^; 1 2> u+i v; 1 27 u, v+§; Ouv ; 0, 1- -u, l-v; 1 2> 1 2 u, v; 1 27 U7 l-v; Ouv 0, u-Hi ^-v; i u+i v; 1 2> u, i-v; Ouv 0, 1- ■u, v+l; 1 2j 1 2~ U7 v; h U7 v-l-i uOv ; u+i 0, v+l; u 4-i i v; U7 1 27 v+^; uOv u+i 0, l-v; u H, 1 27 v; U7 h l-v; uOv ; l-u, 0, l-v; ^ -u, h v; U7 1 27 l-v; uOv ; l-u, 0, v+^; 1 2 -u, h v; u, 1, v-h|. uvO ; u+i v-}-i 0; u 4-i V, h; u, v+i 1; uvO ; u+i i-v, 0; u ■fi V, 1. 2> u, 1. 2 -V, 1; uvO ; ^-u, v+i 0; 1 1- -u, V, h U7 V H, 1; u vO ; l-u, l-v, 0; f -u, V, 1 . 27 Q, 1 2" -V, |. Thirty-two equivalent positions: (p) xyz; xyz; xyz; xyz; xyz; xyz; xyz; xyz; X+I7 y+i z; x-}-^, |-y, z; l-x, ^-y, z; l-x, y+l, z; x+l, y, z+l; x-l-l, y, |-z; l-x, y, |-z; l-x, y, z-\-i; X, y+l, z+l; X, l-y, l-z; X, |-y, |-z; X, y-fi z+l; l-x, y+i, z; x+27 2~y7 z; 2— X, y, ^ — z; f-X7 l-y, z; x+l, y+i z; l-x, y, z-f-§; X+I7 y, z+l; x+i y, §-z; X, y+i l-z; X, i-y, z+i* x> l-y,' z+l; X, y+i i-z. Space-Group V h*. Eight equivalent positions: (a) 0; HO; 10^; 0|i ill. 4447 13 1. 4 4 47 111. 4 4 4 7 (b) §0 0; 0^0; OOi; ^ 1 1 !• 4 4 4; 13 1, 4 4 2; 111. 4 4 4; 2 2 ; 113 444. 111. 2 2 2; 1 3 3. 444. 70 24 ,.25 THE ORTEOEEOMBIC SPACE-GROUPS Vh-Vh. Space-Group Vh* (continued). Sixteen equivalent positions: 111 S 5 8 13 3 8 8 8 13 7 8 8 8 5 15 8 8 8 5 5 5 8 8 8 5 7 7 8 8 « 17 3 8 ? 8 111 8 8 8 111- 8 8 8 7 111. 8 8 8> 3 5 7. 8 8 8) 111. 8 8 8 > 5 1 3. 8 8 8 > 7 5 1. 8 8 8 > 113. 8 8 8> 1 5 1. 8 8 8) 7 17. ¥88) 111. 8 8 8) 15 5. 8 8 8) 3 15. 8 8 8) 3 5 3. 8 8 8) 115. 8 8 8) 111. 8 8 8) 7 3 1. ¥ 8 8) 111. 8 8 ¥) 5 5 1. 8 8 8) 1 5 1. 8 8 8) 13 5 8 8 8* 3 3 5. 8 8 8) 114. 8 8 8) 111. 8 8 8) 111 8 8 8- (c) (d) (e) (f) (g) Thirty-two equivalent positions: (h) xyz; xyz; xyz; xyz; i-x, i-y, i-z; i-x, y-\-h z+i; x-f-i, l-y, z+i; x+i y+i i-z; x+l, y+l, z; x-\-i,l-y,z; |-x, y-f-^, z; ^-x, ^-y, z; l-x, f-y, i-z; l-x, y-M, z+i; x+f, f-y, z-f-i; x+f, y+f, i-z; x+l) y) z-f-l; x-l-iy, ^-z; i-x, y, ^-z; J-x, y, z+|; f-x, i-y, f-z; f-x, y+i, z-|-f; x-|-f, i-y, z-h|; x+l, y+i f-z; X, y+5, z-}-^; X, |-y, |-z; x, y+i ^-z; x, |-y, z-f-|; i-x, f-y, f-z; J-x, y-l-f, z-f-f; x-hi f-y, z+f; x-ff, y+f, f-z. uOO u- ■f-l, 0, 1; u+i h i; u+i i, f; uOO 1. 2 -u, 0, 1; 1 4" -11 i i- ") 4) 4) 3 4 " -U) i f; uH uH) i 0; u+i i 1; ii-J_l 1 1. •^ r4) 4) 4) uH ^ -u, i 0; i_ 4 -u, i 1; 1_ 4 -u, i h OuO 0, u+i i; 1 4) u+i) f; 1 4) u+l, f; OuO 0, l-u, 1; i) i-u, i; f) f-u, f; hul i u+i 0; 1 4) u+i, f; 1 4) u+f, i; HI 1 2) l-u, 0; 1 4) i-u, f; 3 4) l-u, h OOu 0, h u+l; 1 4) i u+i; 1 4) I u+f; OOu 0, i l-u; 1) i i-u; i) f) f-u; Hu i 0, u+l; f, 1 iij-i- 4) Ui-4, f) 1, u+f; Hu , i 0, l-u; 3 4) 3 1 ,,. 5) 4— U, f, f, f-u. Space-Group V^^. Two equivalent positions: (a) 0; Hi (c) 01; HO. (b)HO; OH. (d)| 01 0|0. Four equivalent positions: (e) uOO; uOO; u- H, hh 1- -u, i i (f) uO|; tiOl; U + l) h 0; 1- -u, 1, 0. (g) OuO; OuO; i u+l) 1; 1 2) l-u, |. (h)Ou|; onh; i u+i 0; 1) l-u, 0. (i) OOu; OOVl; i h u+l; i 1, l-u. a) OJu; OH; i 0, u+J; 1, 0, l-u. THE ORTHORHOMBIC SPACE-GROUPS Y\'-y'l 71 Space-Group Vh^ (continued). Eight equivalent positions: (k)Hi; 3 3 3, 4 4 4> 13 3. 4 4 4> 3 11. 4 4 4; (1) Ouv ; Ouv; Ou V ; Ouv; (m) u V ; uOv; uOv , uOv; (n) u V ; uvO; uvO ; uvO; 3 13. 331. 4 4 47 4 4 4) 13 1. 113 4 4 4j 4 4 4- i u+i v+l; i u+l, i-y; h, Hu, l-v; h l-u, v-H. u+l, i v+l; u+i i i-v; Hu, h, l-v; Hu, i v+|. u+i v+^, 1; ii_l_l 1 lr !• Ui"2> 2~V, 5, 1 11 1 -jr 1 • 2~U, 2 V, 2; f-u, v+i i Sixteen equivalent positions : (o) xyz; xyz; xyz; xyz; xyz; xyz; xyz; xyz; x+l, y+i z-H; x+i |-y, |-z; |-x, y+i |-z; 2~x, 2~y> z+fj 2~x, 2~y> 2~z; 2'~x, y+2) z-f-a; x+2, 2~y> z+g-; x+i y+i l-z. Space-Group V^". T^'owr equivalent positions : (a) 0; 0^; HI; HO (b) iOO; 1 n i- 2 " 2> OH; 0|0 (c) OOi; oof; HI; Hi (d)OH; OH; lOf; |oi £^?gf/i< equivalent positions: (e) HO, 1 3 A. 4 4 *J) 3 3 1. 111. 4 4 2) 4 4 2) (f) uOO uOA; uOO uO|; (g) OuO Ou|; OuO Ou|; (h) u Hu; OOu HQ; (i) 0|u |0u; 0|u |0u; (J) uvi ; uvf; Gvi ; uvf; 3 1 n- 3 3 n- 4 4 U, J 4 U, 13 1. Ill 7 4 2) 442' u+i h 0; i_i, 1 n- 2 '^) 2) ") i u-fi 0; i Hu, 0; i i u-H; 1 1 1_1T 2) ^> 5 ") I, 0, u+l; i 0, Hu; u+l, Hv, I; Hu, v+l, i; 11-4.1 1 1. U^2) 2) 2) i— 11 i i 2 ") 2) 2- 1 „_Ll 1. 1) U-f-2) ?) i Hu, i 0, 0, u+i; 0, 0, Hu. 0, i u+l; 0, i Hu. u+i v-l-i f ; Hu, Hv, f. Sixteen equivalent positions : (k) xyz; xyz; xyz; xyz; X, y, Hz; x, y, z+l; x, y, z+|; x+i y+i z-\-^; x-i-i Hy, Hz; Hx, Hy, z; y, t-z; 2~x, y-r^, 2~z; 2~x, ^— y, z-|-f; Hx, y+l, z; x+i Hy) z; x-fi y-f|, z. 72 Space-Group \^h • Eight equivalent positions: (a) 000 • 001; |00; 0^0; 111 2 2 2 HO; OH; ioh (b) 1 H Hi; 113. 3 11. 4 4 4> 4 4 4; 3 3 3 3 13. 3 3 1. 13 3 4 4 4) 4 4 4> 4 4 4> 4 4 4- (c) uOi ; uH; u+i 0, i; u+i i f; iiOf ; QH; Hu, 0, f; Hu, h h (d)iuO Uh h u+i 0; 1, u+i i; f uO luh; i Hu, 0; i Hu, i (e) 0|u Hu; 0, h u+i; i f, u+i; Ofu Hu; 0, i Hu; i i Hu. Sixteen equivalent positions: (^) xyz, X, y, i- -z; Hx, y, z; x, Hy, z; xyz; X, y, z+|; x+i y, z; x, y+i z; x+l, y+i z+l; x+i, Hy, z; x, y+i Hz; 2~x, y, z+5; Hx, Hy, Hz; Hx, y+l, z; x, Hy, z+i; x+i y, Hz. Space-Group Vf . Fowr equivalent positions: (a) OOi; HI; OH; HI (b)Hi; OOf; HI; OH (c) HO; HI; HO; IH (d)HO; 111. 4 4 2, HO; fH (e) 0|u; ^, 4, Hu; 1 3 , 2, 4, ^ Ezfirj'if equivalent positions: (f) uO| uOi (g) Iu0 f uO (h) Ouv |u v (i) ujv 11 i i- "2 4, u+i 0, f; u-hl I f; uH; Hu, 0, I; Hu, i f. iul; i, Hu, 0; J, u-H, |; f ul; f, Hu, 0; i u-f-i i 0, u, Hv; i Q, v-fl; h u-H, v-H; 0, Hu, v; I, Hu, ^; 0, u-hi Hv. u, i Hv; u-H, h v; u-l-i f, v-f-^; v; fi 4 i — u, J, 2 Hu, i v; Hu, f, v-F^. Sixteen equivalent positions: (j) xyz; X, y, Hz; Hx, y, z; x, Hy, z; Hx, Hy, z; xyz; X, Hy, z; x+i y, z; x+i y+iz-H; x+iHy, z; x, y+i Hz; Hx, y, z+|; X, y, Hz; Hx, y+i z+^; x-|-|, y, z+|; x, y-f-l, Hz. The unique cases can be simplified by transferring the origin to the point ©"■ this first set of axes. TEE TETRAGONAL SPACE-GROUPS sl-Vfl. 78 TETRAGONAL SYSTEM.^ A. TETARTOHEDRY OF THE SECOND SORT. Space-Group S\. One equivalent position : (a) 0. (b) 00^ (c) i |0. Two equivalent positions: (e) OOu; OOu. (f) Hu; Ha. (g) 0|u; ion. Four equivalent positions: (h) xyz; yxz; xyz; yxz. Space-Group Si Two equivalent positions: (a) 0; Hi (b)OOi; HO. (c) (d) OH; lof OH (d) i 1 i 3 3?. Four equivalent positions : (e) OOu; OOu; |, i u-fi; §, h i-u. (f) 0|u; Hu; h 0, u+h; 0, i |-u. Eight equivalent positions: (g) xyz; yxz; xyz; yxz; x+5, y+i z-f-i; i-y, x-H, |-z; l-x, Hy, z+i; y+i 5-x, h-z- B. HEMIHEDRY OF THE SECOND SORT. Space-Group VJ. One equivalent position : (a) 0. (b) Hi (c) 0^ (d) h h 0. Two equivalent positions: (e) |0 0; OH- (g) OOu; OOu. (f) OH; Hi (h)Hu; HQ. Four equivalent positions: (i) uOO ; uOO; OuO OQO. (j) uH ; uH; Hi Hi (k) u ^ , QO^; Ou^ Otii (1) uH uH; HO HO. (m) H ; OH; Hu Hu. (n) u u V, u u v; uu V utiv. ^ Of the tetragonal space-groups those marked with an asterisk will be found to have co- ordinates differing from the definitions previously given. These differences, which arise from changes of origin, have been introduced to bring about agreement with the descriptions of Niggli (op. cit.). 74 THE TETRAGONAL SPACE-GROUPS Vi-Vfl. Space-Group VJ (continued). Eight equivalent positions: (o) xyz; xyz; xyz; xyz; yxz; yxz; yxz; yxz. Space-Group Vj. Two equivalent positions: (a) 000; OOi (d) 0^0; §0i (b)nO; OH- (e) OOi; OOf. (c) IH; HO. (f) Hi; Hi Four equivalent positions : (g) uOO; uOO; Ou§; Oui (h) uH; uH; Iu0; ^uO. (i) u|0; u^O; i-u|; ^ui (j) uO^; uO^; OuO; OuO. (k) OOu; OOu; 0, 0, u+|; 0, 0, l-u. (1) Hu; Hu; i i u-h^; ^, i ^-u. (m)0§u; OH; i 0, u+i; i 0, i-u. J&tfl'Af equivalent positions: (n) xyz; xyz; xyz; xyz; y, X, z-f-l; y, X, l-z; y, x, |-z; y, X, z+i Space-Group Va. Two equivalent positions: (a) 000; HO. (c) § u; iOa. (b)00|; Hi Four equivalent positions: (d) OOu; OOu; Hu; H^- (e) u, |-u, v; |-u, u, v; u, u+J, v; u-}-§, u, v. Eight equivalent positions : (f) xyz; yxz; xyz; yxz; i-x, y-Fi, z; l-y, ^-x, z; x-hi ^-y, z; y+i x-^^, z. Space-Group V^. Ti^o equivalent positions: (a) 000; Hi (b) 00|; HO. Four equivalent positions: (c) OOu; OOu; i i ^-u; i i uH-i. (d)OH; §0u; i 0, ^-u; 0, i u+i J5igf/i< equivalent positions : (e) xyz; yxz; xyz; yxz; l-x, y+i §-z; ^-y, ^-x, z-f-|; x-|-|, §-y, ^-z; y+i x+i z+i THE TETRAGONAL SPACE-GROUPS yI-yI. 75 Space-Group V^. Two equivalent positions: (a) 0; HO. (b) i 0; A 0. Four equivalent positions: (c) OH; loh (d) hhh ooi (e) OOu; OOu 1 1 ,T . 11,, (f) 0|u; OiQ 1 u; J u. (g) Hu; ifu ; flu; Hu. t equivalent positions : (h) uOO; OuO i u+i 0; u+i i 0; uOO; OuO h l-u, 0; i-u, f, 0. (i) uH; |u| 0, u+i i; u+i 0, i; ,T 1 1 . 1 Vi 1 U 2 2; 2^2 0, l-u, i; l-u, 0, |. (j) uuv; tiuv, ^-u, u+l, v; u+i u+i v; uuv; uuv l-u, §-u, v; u+i i-u, V. (k) u, u+^, v; u, u+^, v; u+|, u, v; u+^, u, v; u, ^-u, v; u, |-u, v; |-u, u, v; |-u, u, v. Sixteen equivalent positions: (1) xyz; xyz; xyz; xyz; yxz; yxz; yxz; yxz; x+l, y-l-iz; x+l, |-y, z; |-x, y+|, z; |-x, ^-y, z; y+i x+l, z; i-y, x-|-|, z; y+i |-x, z; |-y, §-x, z. Space-Group V^. Foitr equivalent posit] ons: (a) 0, OOi; HO; H 1 2- (b) i 0, OH; OH; H 1- (c) OOi, OOf; 111. 11 2 2 4; 11 3 4- (d)OH, OH; HI; H i. Eight equivalent positions : (e) uOO Ou^ u+i i 1 > 2} u+l, L- uOO Ou^ Hu, i 1 2> i — 11 i 2 ^} 2' (f) uH hnO u+i, 0, i 0, u+l, 0; Qii HO Hu, 0, 1 0, Hu, 0. (g) OOu Hu 1 1 n-Ll 2> 2j U-t-2 0, 0, u+i; OOti HQ h h Hu , 0, 0, Hu. (h) A u Hu i 0, u+^ 0, i u+l; OH Hti i> 0, Hu 0, 1, Hu. (i) Hu iH 1 3 1_„ 4) 4> 2 U . 1 f 4> i u+i; flu HQ i h Hu 1, i u+i 76 THE TETRAGONAL SPACE-GROUPS ¥^-¥4. Space-Group V® (continued). Sixteen equivalent positions: (j) xyz; xyz; xyz; xyz; y, X, z+l; y, X, |-z; y, x, ^-z; y, x, z+|; x+i y+i z; x+l, l-y, z; l-x, y+i z; ^-x, |-y, z; y+i x+i z+i; l-y, x+^, ^-z; y+|, ^-x, |-z; 2~y> 1~X, Z+2. Space-Group "V^* Four equivalent positions : (a) 0; iOO; -HO; 0^0. (b)HI; OH; OOi; |0i. (c) HO; ifO; HO; HO. (d)H^; 13 1. 3 3 1. 3 11 4 4 2; 4 4 2; 4 4 2* Eight equivalent positions: (e) OOu; OOu; |0u; §0U; r Hu; Hu; O^u; Olu. (f) Hu; Hu; Hu; Hu; » HQ; Hu; HQ; Hu. (g) iuO; ufO; f u 0; u K 3; l-u, i 0; 1 'i+lj i) . 0; i u+i 0; i 1 2 -u, 0. (h)iu|; uH; fu|; U H; l-u, 3 1. 4j 2) u+l, hh i u+i i; 1 4j i-u, i Sixteen equivalent positions: (i) xyz; yxz; xyz; yxz; ^-x, y, z; l-y, X, z; x+i y, z > y+i X, z; x+i y+i z; y+i, l-x, z ; l-x, l-y , z ; i-y, x-Fi z; X, y+i z; y, 1- X, z; X, h-y, z > y, x+i z. Space-Group V^* Four equivalent positions: (a) 0; i 1; HO; OH. (b) HI; 0§0; 00|; 10 0. (c) Hi; HI; fH; IH- (d)iH; HI; HI; Hi £^ifif/i< equivalent positions : (e) OOu; OOu; i 0, l-u; i 0, u+l; Hu; 1 1 ii . ^ 2 u, 0, i l-u; 0, i u+|. (f) iui; uHi 1 ftH; u H; l-u, 3 3 . 4» 4> u+i if; i u-Fi i; 1 4, l-u, i (g) HI; uH; fuf; u H; l-u, 3 1. 4) 4 > u+i i i i u-f-i 1; 1 4> l-u, i (h)i|u; HQ; Hu; f iu; hh l-u; i, 1, u+l; 1 3 i_„. 4> 4j 2 "j f, i u-hi THE TETRAGONAL SPACE-GROUPS Vfl-Va. 77 Space-Group V^ {continued). Sixteen equivalent positions: (i) xyz; yxz; xyz; yxz; |-x, y, |-z; |-y, x+i y+i z; y+i X, y+i -2--z; y, 1- X, Z + 2 j 2 X, z; ■X, z+i; x+i y, l-z; y+i x, z+|; l-x, i-y, z; i-y, x+iz; X, §-y, l-z; y, x+iz+|. Space-Grou? Vd. Four equivalent positions : (a) 0; HO; hOh (b) ^0 0; 0§0; 0^; rp^lll. 111. 111. \.W444) 444> 444> (d)iil; Hi; fH; OH. HI. 3 3 1 444. Iff. Eight equivalent positions: (e) OOu; Hu; I, 0, u+|; 0, |, u+A; I, 0, 1— u; 0, I, |— u. Ill 11. 11 n_Ll. 4; 4> 2 "> 4> 4> ""r^> OOu; Hu; |, 0, ?— u; 0, |, |-u. (f) Hu; Hu 33,,. 31,-., 3 3 l_n. 3 1 ,,_Ll 4 4 U, 4 4 U, 4, 4, 2 "> 4> 4> u rj. ; u 1 2 1. 2 1 ; u 1 h 1 > 2 u 1 . 2 J ; 1 u 1; ; u 1 4 3 . 4> J u 3 4 1. 4> ; i u 1; ; f u 1 . 4 ; ; u + 1 2> ; u + h 1 > 2 — u, ; 1 - u, u+l, i 0; u+i 0, ^; l-u, i 0; l-u, 0, 1; i u-H, 0; 0, u+i 1; i i-u, 0; 0, l-u, i i i-u, 1 . 4> u+i i f; l_n i 2 "j 4> 1 . 4; u+i i i; l-u, i f; 1, u+i f; 3 1,, 4) 2~U, 3 . 4; i u+i i. u+i v; U ■H, u, v+l; u, u+i v+i- 5-u, v; U -H, u, l-v; u, §-u, l-v; u+i v; 1 2 ■ -u, u, |-v; u, u+i l-v; l-u, v; h -u, u, v+i; u, i-u, v+i Sixteen equivalent positions: (g) uOO uOO OuO OtiO (h) iui uH ufi 1 11 1 4 u 4 (i) u u v UU V uuv uti V Thirty-two equivalent positions: (J) xyz; xyz; xyz; xyz; yxz; yxz; yxz; yxz; x+iy+iz; x+il-y, z; |-x, y+i z; y+i x+i z; |-y, x+iz; y+i f-x, z; x+iy, z+i- x+i y, A- z; |-x, y, |-z; y+2, X, z+2; 2~y> X, 2~z; y+2; X, 2~z; X, y+iz+l; X, l-yH-z; x, y+i |-z; y, x+i z+i y, x+i \-z; y, |-x, |-z; Space-Group V^". Eight equivalent positions: (a) 0; 01; HO; hh\. iOi- iOO; OH; 0|0. 2~x, 2~y> z; 2~y> 2~"X, z; 5^~x, y, z+2; §-y, X, z+l; X, h-Y, z+i- y, §-x, z+i 78 THE TETRAGONAL SPACE-GROUPS Vd°-Vd. Space-Group V^^ (continued). (b)0 0|; OOf; H rl 1 1 2 2 3. 4; -|0f; |0i; oi 3 4 Oi h (c) iiO; 1 3 n. 4 4 ^> 3 1 4 4 3 3 4 4 3; 111. 4 4 2; 3 11. 4 4 2; 1 3 4 4 1 H 1 2- (d)HI; 13 3. 4 4 4) 3 1 4 4 3 4) 3 3 4 "4 1. 4; 113. 4 4 47 3 11. 4 4 4 J 1 3 4 4 1 4 3 3 4 4 f. ?en equivalent positions : (e) uOO; Oiil; 0, u+l, 0, u+l, i 0; uOO; Ou|; 0, 1 2" -u, ¥-u, i 0; 11 i i- "22) ^uO; 1 2) u+i 1 u+f, 0, ^; nil. U 2 2> iQO; 1 2j 1 2 " -u, i i-u, 0, i (f) OOu; O^u; 0, 1 2> u-M 0, 0, u+§; OOu; 0|u; 0, 1 2> i-u 0, 0, l-u; Hu; iOu; h 0, u-^i 1 1 2) 2j u+^; 1 1 f, . 2 2 "> |0u; 1 2) 0, i-u i i i--u. (g) Hu; Hu; 1 4> 1 4> u-hl . 1 3 4; 47 u+l; iiti; 13,-,. 4 4 U, 1 4> 1 4> i-u 3 1 4} 4} l-u; 3 3,,. 4 4 U, 3 1,,. 4 4 U, 3 4; 3 4> u-hl !' !' u+^; ffti; 3 1 fi . 4 4 U, 3 4) 3 4; l-u 4> 4; i-u. (h) iui; n 1 1 • U 4 4, 1 4> 1 2 ■ -u, i u+i i h 1 f] 3 . 4 U 4, 11 i ^• U 4 4, 1 4; u+l, f l-u, 1 3. 4> 4; 3 fi 1 . 4 U 4, 1, 3 1. U 4 4, 3 4) u+i i l-u, 3 1. 4j 4> 3 „ 3, 4 U 4, f, 3 3 . U 4 4, 3 4; 1 2 " -11 s. U, 4 u+i f, I. Thirty-two equivalent positions : (i) xyz; xyz; xyz; xyz; y, X, z+^; y, x, ^-z; y, x, |-z; y, x, z-|-^; x+i y+i 2; x-f^, i-y, z; ^-x, y-fi z; |-x, |-y, z; y+2> X-1-2) Z+2> 2~y> X-}-2, 2~Z; y-r2> 2~X, 2~^'> 2~y> 2~x, z-f-j; x+i y, 55-f-l; x+l, y, l-z y-l-i X, z; |-y, x, z; 2 X, y, i y+i X, z; i-x, y, z-l-l; l-y, X, z; X, y+i z-l-l; X, |-y, |-z; x, y+|, |-z; x, |-y, z+^; y, x-l-l, z; y, x-f-|, z; y, ^-x, z; y, § — x, z. Space-Group V^.* Two equivalent positions : (a) 0; H i Fowr equivalent positions: (e) 100; OH; (d)OH-; OH; (e) OOu; OOu; OH; (b)00§; HO. iOf. 1 2) u-H; 1 2} Eight equivalent positions: (f) uOO; OuO; u+i i |; i u+|, 1; uOO; OaO; ^— u, §, 1; 1, |-u, §. THE TETRAGONAL SPACE-GROUPS ¥^-€4. 79 (g) uO| (h) I u 0|u (i) u u V tiu V Space-Group Vd^ (continued). Oui; u+i i, 0; ^ u-f-i 0; Ou|; A-u, I, 0; i |-u, 0. |0u; 0, h u+l; I, 0, u+i; |0u; 0, i l-u; i 0, i— u. uuv; u+i, u+i v+l; u+i ^-u, |-v; uuv; ^-u, l-u, v+l; |-u, u+|, |-v. Sixteen equivalent positions: (J) xyz; xyz; xyz; xyz; yxz; yxz; yxz; yxz; x+i y+i z+i; x+i |-y, ^-z; |-x, y+|, ^-z; §-x, i-y, z+^; y+i x+i z+l; |-y, x+|, |-z; y+|, ^-x, |-z; 5-y, l-x, z+i Space-Group V^ *• Four equivalent positions: (a) 000; iOi; HI; OH- (b)00|; |0|; HO; OH- i equivalent positions (c) OOu; OOu; i 0, i-u; i 0, u+i; i i u-hl; 1, i l-u; 0, |, f-u; 0, i u+i (d)|u|; uH; fui; ail; i u+i, f; u+l, i f; I l-u, f; l-u, f, f. Sixteen equivalent positions: (e) xyz; yxz; xyz; yxz; l-x, y, i-z; l-y, x, z+J; x-hi y, i-z; y+i x, z+i; x+i y+i z-l-l; y-t-i |-x, |-z; |-x, |-y, z+|; l-y, x+i I-z; X, y+l, f-z; y, |-x, z+f ; x, |-y, |-z; y, x-M, z+f. C. TETARTOHEDRY. Space-Group CJ. One equivalent position : (a) OOu. (b) Hu. Ttyo equivalent positions: (c) 0|u; |0u. Four equivalent positions: (d) xyz; yxz; xyz; yxz. Space-Group Ci Four equivalent positions: (a) xyz; y, x, z+l; x, y, z+|; y, x, z-|-f. 80 TEE TETRAGONAL SPACE-GROUPS C4-C^. Space-Group Ct Two equivalent positions: (a) OOu; 0, 0, u+i (c) O^u; I 0, u+i (b) H u; i I, u+i Four equivalent positions : (d) xyz; y, x, z+|; xyz; y, x, z+^. Space-Group Ct Four equivalent positions : (a) xyz; y, x, z+|; x, y, z+^; y, x, z+i. Space-Group C|. Two equivalent positions: (a) OOu; i i, u+i Four equivalent positions: (b) Olu; iOu; i 0, u+^; 0, i u-^i Eight equivalent positions: (c) xyz; yxz; xyz; yxz; x+2> y+2> z+2; 2~y> x+2, z+2; 2~x, 2~y) z4-2j y"r2> 2~X, z+2' Space-Group C^. /^owr equivalent positions : (a) OOu; 0, i u+i; |, 0, u+|; i i u+i ^zgi/if equivalent positions: (b) xyz; y, ^-x, z+|; xyz; |-y, x, z+f; x+2> y"F2> z+2; y+2> X, z+j; 2~x, 2~y> z+2; y, x+i z+i. D. PARAMORPHIC HEMIHEDRY. Space-Group 04^,. One equivalent position: (a) 000. (b) 00|. (c) HO. (d) Hi Two equivalent positions: (e) 0|0; ^0 0. (g) OOu; OOu. (f) OH; ^0|. (h) Hu; HQ. Four equivalent positions: (i) O^u; iOu; 0|u; ^Ou. (j) uvO; vuO; uvO; v u 0. (k) uv|; vu|; uv|; vu^. THE TETRAGONAL SPACE-GROUPS cA-Cil,. 81 Space-Group C4h (continued). Eight equivalent positions : (1) xyz; yxz; xyz; yxz; xyz; yxz; xyz; yxz. Space-Group C^. Two equivalent positions: (a) 0; i. (d) OH; ^0 0. (b)HO; HI. (e) OOi; f . (c) 0^0; |0i (f) Hi; Hi Four equivalent positions : (g) OOu; OOu; 0, 0, u-}-|; 0, 0, |-u. (h) 2 2 U; 2 2 U; 2] 2) U"l 2; 2) 2> 2~U. (i) Olu; Oiu; i 0, u4-i; i 0, ^-u. (j) uvO; uvO; vu|; vuf. E'zgf/if equivalent positions : (k) xyz; y, x, z-hi; xyz; y, x, z-Hi; xyz; y, x, Hz; xyz; y, x, ^-z. Space-Group CA. Two equivalent positions: (a) 010; 100. (b)0i-3; 1 i. (c)OOu; H u. Four equivalent positions : (d) HO; fiO; HO; ffO. fa\ ill.lli»131.331 \pj 442> 442> i 4 2 ) 442* (f) O^u; |0u; |0u; H- iJzgf^i equivalent positions: (g) xyz; yxz; xyz; yxz; Hx, Hy, z; y+i i-x, z; x+|, y-l-i z; Hy, x+|, z. Space-Gi cup C/h. Tiwo equivalent positions: (a) OH; |of. (b) OH; ioi Four equivalent positions : /'p^ll^J• 311. 33n. 131 V W 44^> 442; 44'-'> 442- \M) 442> 44"-'> 442> 44'-'' (e) OOu; Hu; i i |-u; 0, 0, u+i (f) O^u; iOu; i 0, u-hi; 0, |, Hu. Eight equivalent positions: (g) xyz; y, x, z-H; xyz; y, x, z-|-§; ^-x, Hy, z; y+i i-x, Hz; x-Fi y+|, z; Hy, x+l, i_. 82 THE TETEAGONAL SPACE-GROUPS C^-C^- Space-Group C^. Two equivalent positions: (a) 0; -HI. (b) 1 ; HO. Four equivalent positions: (c) 0|0; |0 0; i 1, OH. (d)OH; Hi; OH; lOf. (e) OOu; OOu; i i u+l; h i l-u. Eight equivalent positions: (f) iii; fii; ffi; ill; 113, 133. 1 11. 4 4 3> 4 4 2> 4 4 4> fH. (g) 0|u; |0u; i 0, u+l; 0, 1, u+l; Olu; |0u: i 0, l-u; 0, i l-u. (h) uvO; vuO; |— v, u+l, 1; u+i y-\-h h uvO; vuO; v+l, l-u, 1; l-u, l-v, |. Sixteen equivalent positions: (i) xyz; yxz; xyz; yxz; xyz; yxz; xyz; yxz; x+i y+l, z+l; l-y, x-hi z+l; |-x, |-y, z+f; y+i l-x, z+l; l-x, §-y, |-z; y+l, |-x, |-z; x+|, y+l, |-z; l-y, x-}-|, \-z. Space-Group C^.* ♦ Four equivalent positions: (a\ nil. nl'. 115. 133 (\<\ n35. nl3. 111. 137 ? 4 8- Eight equivalent positions: (c) 000; iil; |0|; Hi; o|0; Hi; IH; Hi (d)OOA; iii; ^00; HO; n 1 1. 111. 4 4 4> 9 3 3.. 4 4 4; 13 1. 4 ¥?» 3 11 744. (e) Oiu; i i, u+f; i f, u+|; Of u; h, f, i-u; i i, 1 i_ t-u; 0, f, u+i; 0, i, f-u. Sixteen equivalent positions: (f) xyz; y-f-i, i-x, z+f; \-x, y, z-j-^; X, y+i z; y-f-i f-x, i i_ z: I-x, I x+l, y+l, z-l-l; y+f, f-x, z-Hi; x, x+l, y, l-z; y+f, i-x, f-z; xyz; i-y, x-f-f, z-f-i; -y, l-z; i-y, x-l-i, f-z ; l-y, z; f-y, x-F- i, z+f; f-y, x+f, i-z. THE TETRAGONAL SPACE-GROUPS civ-Ci^v- 83 E. HEMIMORPHIC HEMIHEDRY. Space-Group C4V One equivalent position : (a) u. (b) H u. Two equivalent positions: (c) 0|u; iOu. Four equivalent positions: (d)uuv; uuv; uuv; uQv. (e) uOv; Ouv; uOv; Otiv. (f) u^v; |uv; u ^ v; ^uv. Eight equivalent positions: (g) xyz; yxz; xyz; yxz; yxz; xyz; yxz; xyz. Space-Group C4V. Two equivalent positions : (a) OOu; Hu- (b) O^u; |0u. Four equivalent positions: (c) u, |-u, v; u-M, u, v; u, u-M, v; |-u, u, v. Eight equivalent positions: (d) xyz; yxz; xyz; yxz; y+l, x+l, z; x-f-i^-y, z; ^-y, ^-x, z; ^-x, y+i z. Space-Group Ct,. Two equivalent positions : (a) OOu; 0, 0, u-hi (b) H u; i i u-j-^ Four equivalent positions: (c) Oiu; iOu; i 0, u+l; 0, i u-|-i (d) uuv; uuv; u, u, v-f-^; u, u, v-|-|. Eight equivalent positions: (e) xyz; y, x, z+|; xyz; y, x, z-\-\; yxz; X, y, z-fl; yxz; x, y, z-f-§. Space-Group C4V.* Two equivalent positions: (a) OOu; h h u+|. Four equivalent positions: (b)0|u; 0, i u+i; ^Ou; i 0, u-^. (c) uuv; u+l, |-u, v-f|; uuv; ^-u, u-M, v-|-^. 84 THE TETRAGONAL SPACE-GROUPS C4V-C4V. Space-Group C^y {continued). Eight equivalent positions : (d) xyz; y+i |-x, z+|; xyz; |-y, x+|, z+|; l-x, y+i z+^; yxz; x-Hl, |-y, z4-^; yxz. Space-Group C/v Two equivalent positions : (a) OOu; 0, 0, u+i (b)Hu; i §, u+i Fowr equivalent positions : (c) O^u; iOu; i 0, u+|; 0, |, u-^. ^tgf/i< equivalent positions: (d) xyz; yxz; xyz; yxz; y, X, z+§; X, y, z+|; y, x, z+i; x, y, z+|. Space-Group C/y. Two equivalent positions: (a) OOu; i I, u+i i^owr equivalent positions: (b)0|u; iOu; i 0, u+|; 0, i u+i ^tg/if equivalent positions : (c) xyz; yxz; xyz; yxz; y I 2> X+2) Z-t-j-; X+2) 2~y> ^12) 2~y) 2~X, Z+2J 2~x, y-|-2> z+2. Space-Group C/y. Tw/'o equivalent positions : (a) OOu; 0, 0, u+|. (b)Hu; i i u+i (c) Oiu; i 0, u-hi Four equivalent positions : (d)uOv; uOv; 0, u, v+l; 0, u, v-hi (e) u^v; u|v; i u, v-f-^; i u, v-\-^. Eight equivalent positions: (f) xyz; y, x, z+^; xyz; y, x, z-f-^; y, X, z+l; xyz; y, x, z-|-^; xyz. Space-Group C^y. Four equivalent positions : (a) OOu; Hu; i i u-^; 0, 0, u-hi (b)0|u; ^Ou; i 0, u+f; 0, i u-l-|. Eight equivalent positions: (c) xyz; y, x, z-}-|; xyz; y, x, z+i; y+i x+l, z+l; x4-|, |-y, z; ^-y, |-x, z-|-|; ^-x, y+i, z. THE TETKAGONAL SPACE-GROUPS Ciy-Cli- 85 Space-Group C'^. Two equivalent positions: (a) OOu; i, i u+i Four equivalent positions : (b)Oiu; lOu; i 0, u+^; 0, i u+i Eight equivalent positions : (c) u u V uti V (d) u V tiOv uuv; u+i ^-u, v+^; u+i u+|, v+^; uuv; ^-u, u+l, v4-^; i-u, |-u, v-|-^. Ouv; I, u+l, v+^; \\-\-\, i v+|; Ouv; i ^-u, v+^; |-u, i v+^. Sixteen equivalent positions: (e) xyz; yxz; xyz; yxz; yxz; xyz; yxz; xyz; x+i y+i z+^; |-y, x+|, z+^; |-x, ^-y, z+^; y+i l-x, z-l-l; y+i x+i z+^; x+i l-y, z4-^; |-y, ^-x, z-h|; l-x, y-l-i z+\. Space-Group Cl". Four equivalent positions: (a) OOu; Hu; \, h, n+h 0, 0, u-Hi (b)Oiu; iOu; i 0, u-f-i; 0, i u+i ^tgr/i< equivalent positions: (c) u, u+i v; ^-u, u, v; % u+^, v+|; u-f-|, u, v+^; u, ^-u, v; \i-\-\, u, v; u, ^-u, v+^; ^-u, u, v-}-^. Sixteen equivalent positions: (d) xyz; yxz; xyz; yxz; y, X, z-f^; x, y, z-F-|; y, x, z+\; x, y, z-\-\', x+l, y+l, z-f^; \-y, x-f-|, z-fl; |-x, ^-y, z-j-|; y+i l-x, z-M; y+i x-fi z; x-l-i |-y, z; |-y, |-x, z; |-x, y+i z. Space-Group Civ-* i^ow?' equivalent positions: (a) OOu; 0, i, u-hi; i 0, u+|; i i u+i Eight equivalent positions: (b) Ouv; u, i v-f-J; i u+|, v-f-^; |-u, 0, v-f-f; Ouv; % i v+i; i |-u, v-|-|; u-|-i 0, v+|. 86 THE TETRAGONAL SPACE-GROUPS C\;-T>1. Space-Group Cl* (continued). Sixteen equivalent positions: (c) xyz; y, ^-x, z+i; xyz; |-y, x, z+f; xyz; y, |-x, z+J; xyz; y+i x, z+f; x+i y+l, z+§; y+i X, z+f; |-x, §— y, z+|; 2~x, y+2> z+^; ^• Space-Group CH* Eight equivalent positions: ■y, X, z+f; y, x+i z+^; x+i |-y, z+l; y, x+i z+i (a) OOu; 0, i u+i; |, 0, u+f; 0, 0, u+i; Hu; I, 0, u+i; 0, I, u+f; i i, u+i Sixteen equivalent positions : (b) xyz; y, |-x, z+|; xyz; §-y, x, z+f; X, y, z+l; y, l-x, z+f; x, y, z+§; y+|, x, z+f; x+l, y+l, z+l; y+l, x, z+f; §-x, f-y, z+|; y, x+i z+f; l-x, y+i z; i-y, X, z+f; x+i §-y, z; y, x+|, z+f. F. ENANTIOMORPHIC HEMIHEDRY. Space-Group D4. One equivalent position : (a) 0. (c) H .0. (b) |. (d)H h Two equivalent positions : (e) OiO; §0 0. (g) OOu; OOu (f) OH; 1 n i 2 2- (h)Hu; HG Four equivalent positions: (i) Olu; §0u; 0|u §0u. (j) uuO; uuO; utiO ; uuO. (k) uu§; uu|; uu| uu|. (1) uOO; OuO; uOO OuO. (m)uH; §u|; uH Hi (n) uO§; Ou§; uO§ oa|. (0) uiO; |uO; u|0; |uO. Eight equivalent positions: (P) xyz; yxz; xyz; yj 4z; yxz; xyz; yxz; x] ^z. Space-Group D^* Two equivalent positions: (a) 0; H 0. (b) 001; Hi (c) 0§u; §0u. THE TETRAGONAL SPACE-GROUPS DJ-D^. 87 Space-Group D4 {continued). Four equivalent positions: 1 1 r, . 2 2 u, i i 11 2 2 Li* (d) OOu; OOu; (e) uuO; uuO; u+^, ^—\i, 0; |— u, u+|, 0. (f) uu|; ua|; u+i |-u, |; §-u, u+i f. ^tgr/i^ equivalent positions: (g) xyz; y+i |-x, z; ^-x, y+l, z; yxz; Space-Group Df.* Four equivalent positions: (a) OuO; uOf; Ou|; u i (b) |ui; uH; IQO; u H- xyz; l-y, x+i z; x+i l-y, z; yxz. (c) uuf; uu|; uu|; uuf. Eight equivalent positions: (d) xyz; y, x, z+f; x, y, z+|; y, x, z+i; xyz; y, x, i-z; x, y, |-z; y, x Space-Group Dt* l-z. Four equivalent positions: (a) uuO; uu^; u+l, ^-u, f; |-u, u-|-|, J. ^ig^i equivalent positions: (b) xyz; y+^, ^-x, z+f; ^-x, y+i i-z; y, X, Space-Group Dt* Two equivalent positions: X, 5^, z+l; x+i h-z -y, x+l, z+i; ■y, f-z; yxz. (a) 0; ooi. (d)OH HO. (b)HO; Hi (e) 00 f; oof. (c) 0^0; ^oi (f) Hf; Hf. /^owr equivale Qt positions: (g) OOu OOu; 0, 0, u-f-i; 0, 0, Hu. (h) Hu HQ; i i u-}-^; i i 5-u. (i) 0|u OH; 1, 0, u+l; 1, 0, l-u. (j) uOO uOO; Ou|; Oui (k)uH uH; HO; 1 u 0. (1) uOi uO§; OuO; OuO. (m) u 1 uiO; H^; Hi (n) u u i ; uuf; uuf; uuf. (0) uuf ; uuf; uuf; uuf. Eight equivale nt positi ons: (P) xyz; y, X, z+l; xyz; y, x, z+i xyz; y, X, 1- -z; xyz; y, x, Hz. 88 THE TETRAGONAL SPACE-GROUPS D4-D4. Space-Group Dt* Two equivalent positions; (a) 000; HI- (b) 00^; HO. Four equivalent positions: (c) OOu; OOu; h h, |-u; h i u+h (d)Oiu; |0u; i 0, |-u; 0, i u+i. (e) uuO; uuO; u+|, |-u, |; ^— u, u+|, | (f) uu|; uu|; u+i |-u, 0; ^-u, u+|, 0. Eight equivalent positions: (g) xyz; y+^, |-x, z+|; xyz; ^-y, x+i, z+|; §-x, y+i i-z; yxz; x+i l-y, |-z; yxz. Space-Group D4.* Four equivalent positions: (a) OuO; uOi; Ou|; uOf. (b) |ui; uH; HO; u H- (c) uu|; uuf; uuf; u u |. Eight equivalent positions: (d) xyz; y, x, z+i; x, y, z+|; y, x, z-f-f; xyz; y, x, f-z; x, y, |-z; y, x, |-z. Space-Group Df.* Four equivalent positions: (a) uuO; util; u-H, |-u, i; |-u, u-f-^ f. Eight equivalent positions : (b) xyz; y-f-^ |-x, z-f-|; x, y, z4-|; |-y, x+|, z-f-|; yxz; l-x, y-f-j, |-z; y, x, |-z; x-j-^ |-y, i-z. Space-Group D4. Ti^o equivalent positions: (a) 0; HI- (b) J; HO. Four equivalent positions: (c) 0^0; 1 0; HI; OH- (d) OH; hoh HI; OH- (e) OOu; OOu; i i u-|-|; |, I, |-u. Eight equivalent positions : (f) OH; OH; 0, h u+l; 0, 1, Hu; Hu; HQ; i 0, u-fl; 1, 0, Hu. (g) uuO; uuO; u-f-i u-l-l, 1; Hu, u+l, 1; utiO; utiO; Hu, Hu, 1; u+i Hu, I. THE TETRAGONAL SPACE-GROUP D4-D4. 89 Space-Group D4 {continued), (h) uOO; OuO; tiOO; 1 . 5> OuO; Oui; It u+^, 1 1 . I' i u+i 1 . i-u, 1 1 . 2> i i-u, 1 I- u-hi i, 0; i u+i 0; l-u, i 0; 1 1? l-u, 0. (i) (j) u, u+i, I; u, i-u, J; u+i u, f ; i-u, u, |; ^-u, u, 1 . U + ^ u, i-u, f ; ii, u-Hi f. Sixteen equivalent positions: (k) xyz; yxz; xyz; yxz; yxz; xyz; yxz; xyz; x+l y+i z-|-§; |-y, x+|, z+|; y+2> x-f-2, 2~z; x-|-2> 2~y> 2~z; Space-Group D\°.* i^our equivalent positions: (a) 000; OH; iOf; Hi (b) 00^; OH; iOi; HO. Etgf/i< equivalent positions: l-y, z+^; y+i Hx, z-l-^; l-y, i-x, l-z; 2~x, y-|-^, 2~z. (c) (d) (e) (f) OOu 0, OOu, 0, uuO ; u, uuO ; u, uuO » a, uuO u, uH, h GH 3 4? u+i u+l, Hu, u-hi 1 2) ^> 4 U 1 1 ¥ 1 4 1 4 a 8 3 8 h 0, u+f; 1 1 2> 2> Hu; io, l-u; hh u+i l-u, l-u, 1 . l-u, u, 1; u-hi u+i 1; u+i u, f. u-f-i l-u, ^; u+l, u, 1; Hu, u+l, 1 . 2) l-u, ti, f. l-u, 1 5. 4) 8) iu|; u+i 3 5. 4; 87 |a|. Sixteen equivalent positions: (g) xyz; 2 X, 4 Z, y, Hx, z+l; 2~y> ^~x, 2~ x+i y+i z+l; y+i x, z+f; X, y+i i-z; yxz; xyz; Hy, x, z+|; x+i y, l-z; yxz; l-x, i-y, z+l; y, x+i z+i; X, |-y, i-z; y+i x+i Hz. G. HOLOHEDRY. Space-Group DA One equivalent position: (a) 0. (c) (b)OOi (d) Two equivalent positions: (e) OH; io|. (f) 0^0; ^00. Ill (g) OOu; (h)Hu; OOu. 1 1 vi 90 THE TETRAGONAL SPACE-GROUPS B^-D^n. Space-Group D^ {continued). Four equivalent positions: (i) OH; |0u; Hu, OH. (j) uuO uuO, uuO uuO. (k) uui uu| uu| ; Qui (1) uOO OuO; OuO uOO. (m) u 1 Ou|; Ou| uOi (n) uH, iuO; |uO u|0. (o) uH, |u|; huh uH. Eight equivale nt posit ions: (p) uvO vuQ vuO ; uvO; vuO uvO uvO ; vuO. (q) u V 1 vu| vu| ; uv^; vu| ; \uuv ; uuv u V ^ ; vui (r) u u V ; uuv ; uuv; uuv ; uuv ; uuv. (s) Ouv tiOv uOv ; Ouv; uOv Ouv Ouv ; uOv. (t) |uv uiv u| V ; Hv; U^ V iuv |uv ; u|v. Sixteen equivalent positions : (u) xyz; yxz; xyz; yxz; yxz; xyz; yxz; xyz; xyz; yxz; xyz; yxz; yxz; 5cyz; yxz; xyz. Space-Group D^. iiO; (c) f f (d)Hi; hoh 111 ^ 2 !• 113 ^2 4' Two equivalent positions: (a) 0; 0|. (b) OOi; oof. Four equivalent positions: (e) OH; ^00; OH; (f) OH; Hi; HI; oH- (g) OOu; OOu; 0, 0, Hu; 0, 0, u+i (h) Hu; Hu; i i l-u; i i u+i Etgf/if equivalent positions : (i) OH, Hu; OH, HQ; (J) uuO ; UU 0; uuO ; QuO; (k) uOO, OuO; uOO; OuO; (1) uH; H§; QH, HI; (m) ^ V i, vui; uvi, vui; 0, i u+l; h, 0, u-f-l; 0, i Hu; i, 0, Hu. uu^l uQ§; Qui; Qui uOi; Ou|; QOi; OQi uH; HO; uH; HO. vuf; uvf; vQf; Qvf. THE TETRAGONAL SPACE-GROUPS Biir^*^' 91 Space-Group D^ {continued). Sixteen equivalent positions : (n) xyz; yxz; xyz; yxz; yxz; xyz; yxz; xyz; X, y, l-z; y, x, ^-z; x, y, l-z; y, X, i-z; y, X, z+l; X, y, z+l; y, x, z+i; X, y, z+i :e-Group Dtt. Two equivalent positions : (a) 000; HO. (c) 0|0; 100. (b) 001; 111. (d) OH; 101. Four equivalent positions: (e) HO; HO; i f 0; ff 0. (f) iH; fil; iH; HI- (g) OOu; OOu; Hu; H u. (h) 0|u; |0u; ^Ou; 0|u. Eight equivalent positions: (i) uuO; uuO; u-H, u-h^, 0; u+l, i— u, 0: utiO; tiuO; §— u, ^— u, 0; |— u, u+^, 0. (j) uu|; uu^; u-f-^, u+i |; u+i |-u, |; uu|; uu|; l-u, l-u, 1; l-u, u+i i (k) uOO; OuO; u+i h 0; i u+i 0; uOO; OuO; ^-u, i 0; i ^-u, 0. (1) uH; |u|; u+i 0, h 0, u+i i; tiH; |u|; l-u, 0, 1; 0, l-u, i (m)u, ^- ■u, v; u+i u i, v; 1- u, u, v; u, u i+i v; l-u, u, v; u, i-u i, v; u, u+i v; u+t ^, Q, V. Sixteen equivalent positions: (n) xyz; yxz; xyz ; yxz; ~ yxz; xyz; yxz ; xyz; ^-x, ?-y, z; y+h §-x, z; x+i y+i z; 1 -y, x-hi z; l-y, i-x, z; ^-x, y+i z; y+h x+i z; x+i 5-y, z. Space-Group D^- Two equivalent positions: (a) 0; Hi (b) OOi ; HO. Four equivalent positions: (c) 10 0; 0|0; OH; ^Oi (d)^Oi; OH; OH; lOf. (e) OOu; OOu; hh i-u; 1, h u+1. Eight equivalent positions : (f) Hi; Uh iH; IH; iH; fH; Hf; Hi 92 THE TETRAGONAL SPACE-GROUPS dA-d/i,. Space-Group D4 (continued). (g) ^Ou, O^u; ^Oii, OiQ; (h) uuO ; uuO; uuO ; uu 0; (i) uOO OuO; uOO OuO; (3) uOl Ou^; uO| Oui; 0, i u+i; 1, 0, u+l; 0, i ^-u; h 0, i-u. u+i l-u, 1 . M + h u+l, i; l-u, u+i 1; l-u, 2~U, 2- 1 „_Ll 1- u+i 1 1. 2> 2> i i-u, ^; ^-u, 1 1 2') 2' i u+l, 0; u+i i 0; i l-u, 0; l-u, h 0. Sixteen equivalent positions: (k) xyz; yxz; xyz; yxz; yxz; xyz; yxz; xyz; 'l~x, ^— y, 2~2:; y+2> 2~x, g — z; x+^, y+2j 2"~z; 2~y? x+2, 2~z; ¥-y, ¥-x, z+l; |-x, y-Hi z+l; y+|, x+l, z+|; X+2} 2~y} 2+2' Space-Group D^. ^0. (c) OH; loi (d) iOO; 0^0. Two equivalent positions : (a) 0; 2 (b) 2 ^ ¥ i Four equivalent positions: (e) OOu; OOu; Hu; Hu. (f) Oiu; 0|u; ^Ou; ^ u. (g) u, u-l-i 0; |-u; u, 0; u+i u, 0; (h) u, u+i i; |-u, u, I; u+i u, |; Eigf/if equivalent positions: u, i—u, 0. (i) uvO; vuO; v+i u+|, 0; u+|, ^-v, 0; uvO; vuO; 5— v, |— u, 0; i— u, v+|, 0. (j) uv^; vu|; v+i u+i |; u+i |-v, |; uv|; vu|; |-v, ^-u, |; ^-u, v-Hi |. (k) u, ^-u, v; u-f-i u, v; ^-u, u, v; u, u+|, v; vi, \i-\-i, v; l-u, ti, v; u+i u, v; u, |-u, v. Sixteen equivalent positions: (1) xyz; yxz; xyz; yxz; y+ix-|-|, z; x-l-i, i-y, z; ^-y, i-x, z; |-x, y+i, z; xyz; yxz; xyz; yxz; h-y, |-x, z; i-x, y-\-^, z; y-|-|, x+h z; x+|, |-y, z. Space-Group D^.* Two equivalent positions: <n^ 000; Hi (b) 00^: HO THE TETRAGONAL SPACE-GROUPS D^-D^. 93 Space-Group D^ (continued). Four equivalent positions: (c) OJO; iOO; |0|; OH- (d)OH; hoh OH; iof. (e) OOu; OOu; i I, u+|; i i ^-u. EtgfTi^ equivalent positions : (f) O^u; |0u; i 0, u+|; 0, i u+l; 0|u; ^Ou; |, 0, ^-u; 0, §, |— u. (g) u, u+i i; u+i u, i; u, |-u, i; |-u, u, |; u, u+i f; u+l, u, I; 0, |-u, f; ^-u, u, f. (h) uvO; vQO; v+|, u+i ^; u+i §-v, |; uvO; vuO; |-v, ^-u, |; ^-u, v+i ^. Sixteen equivalent positions : (i) xyz; yxz; xyz; yxz; xyz; yxz; xyz; yxz; l-x, y+i z+^; ^-y, |-x, z+l; x+|, |-y, z+^; y+i x+l, z+l; l-x, y+i ^-z; |-y, |-x, f-z; x+i |-y, |-z; y+2> ^+2, 2~Z. Space-Group D/h.* Ttyo equivalent positions: (a) 0; HO. (b)OOi; Hi Four equivalent positions : (d)iiO; fiO; ffO; (c) 0|u; lOu. i 3 n 4 4 U. 111. 3.31. 131 442> 44^> 44 ¥• (e) Hi (f) OOu; OOu; Hu; Hu. ^z'gr/if equivalent positions : (g) uu 0; uti uu 0; u uO (h) uu|; uu| uu|; uu| (i) uOv; Ouv uOv; Ouv u+i Hu, 0; u+i, u+l, 0; Hu, u+i 0; Hu, Hu, 0. u+i Hu, i U+I, u+i I; 5~U, U+2, 2J 2~U, 2~U, 2- U+i i v; i u+i v; Hu, i v; i-u, V. (J) u, u+i v; u, Hu, v; u, Hu, v; u, u+i v; u+i u, v; u+i u, v; Hu, u, v; Hu, u, v. Sixteen equivalent positions: (k) xyz; y+i Hx, z; xyz; Hy, x+i z; x+i y+i z; yxz; Hx, Hy, z; yxz; xyz; Hy, Hx, z; xyz; y+i x+i z; Hx, y+i z; yxz; x+i Hy, z; yxz. 94 THE TETRAGONAL SPACE-GROUPS D^-D^. Space-Group D^.* Four equivalent positions : (a) 0; HO; H*; 0^ (b)OOi; Hi; OOf; Hi (c) Oiu; iOu; i 0, |-u; 0, i u-^|. Eight equivalent positions : (A\ XXX- 311. 111. 111. \p.) 444, 444; i 4 4k ) 444; 313. 113. 133. 313 4 4 4; 4 4 4; 4 4 4; 4 4 4- (e) OOu; Hu; h, h, u+|; OOu; Hu; |, h, |-u; (f) uuO; uu|; u+|, ^-u, 0; uuO; uu|; |-u, u+|, 0; 0, 0, u+§; 0, 0, l-u. u+l, u-t-l, l-u, |-u, Sixteen equivalent positions: (g) xyz; y+l, |-x, z; • x-f-iy+i l-z; y, X, X, y, Z+2; 2~Yf 2~X, |-x, y+l, z; yxz; Space-Group DiV* l-z; 1. 2 ; 1 2' xyz; l-y, x-|-|, z; l_-u- l_tr 1 2 X, t-y;t-z; y, X, f-z; X, y, z+l; y+l, x+i z+l; x+i |-y, z; yxz. According to the previous definitions (page 33), this space group is D, 10 and the following one is D4h, NiggU's descriptions. Two equivalent positions : (a) 0; OOi (b) H 0; H I- (c) 0|0; |0|. Four equivalent positions: The two are here interchanged to conform with (d) (e) (f) OH; 100. OOi; 111. ^ ^ 4; 00 i 113 ar 5 4- (g) (h) (i) (J) (k) (1) uO| (m) u i OOu i i 11 2 2 U 0|u uOO 11 i i U 2 2 OOu; 1 1 f, . 2 2"; 0|u; uOO; "2 2; OtiO; 2^2, 0, 0, u-M; 1 2; 2; '^r 2 ) h, 0, I— u; Ou|; luO; u. 0, 0, 1 2; 0, u+i 1 1 1—11 2; 2; 2 *^- 1 ^; Ou^. u w 2; u|0; OuO. ill i 2 U 2. Eight equivalent positions : (n) uui uuf (o) Ouv Ouv (p) |uv |uv (q) uvO tivO uu n 3. 4; Qui; uuf; uuj; utif; Qui u, 0, v-l-l; OQv; u,. 0, w+h u, 0, | — v; OQv; Q, 0, I— v. u, I, v-l-l; |Qv; Q, |, v-|-|; u, i |-v; |Qv; u, i |-v vQ|; uvO; vu|; vQ|; uvO; vu|. THE TETRAGONAL SPACE-GROUPS D^-D^i. 95 Space-Group D^ (continued). Sixteen equivalent positions: (r) xyz; y, X, z+l; xyz; y, x, z+i; xyz; y, X, 1- -z; xyz; y, x, 2 z; xyz; y, X, z+l; xyz; y, x, z+l; xyz; y, X, 1- -z; xyz; y, x, Hz. Space-Group DlS * According to the previous definitions this group is D4b. Two equivalent positions: (a) 000 0|. (c) HI; HO. (b)OOi OOf. (d)Hi, Hi i^owr equivalent positions: (e) 0|0; |oi; 100; OH. (f) OH, lOf; OH; |0i (g) OOu OOu; 0, 0, u+i; 0, 0, Hu. (h)Hu Hu; h h u+^; i 1 1 11 2> 2~U. (i) uui uuf; uuf; uuf. (J) uuf ; uu?; uuf; uuf.' Eight equivale nt positions: (k) A u iOu; i 0, u+l; 0, i u-hl; 0|u §0u; i 0, l-u; 0, i Hu. (1) uOO, Ou^; uO|; OuO; tiOO Ou|; uO|; OuO. (m) u H |uO; u^O; |u|; uH |uO; u^O; |ui (n) u V i vuf; uvf; vuf; f uvf vui; uvf; vuf. (o) u u v ; uuv; u, u, v+l; u, u, v+i; ti u V ; utiv; u, u, l-v; u, u, Hv. Sixteen equiva lent positions: (p) xyz; y X, z+l; xyz; y, X, z+l; X, y, |-z; yxz; x, y, ^-z ; yxz; . X, y, z+^; yxz; x, y, z+| ; yxz; xyz; y , X, Hz; xyz; y, X, Hz. Space-Group Dli,* The space-groups D|J and D4h are interchanged to conform with the descrip- tions of Niggli. Four equivalent positions : (a) 0; 0^; HO; (b) OH; HI; (c) oof; OOf; (d)0H; Hf; HO; 113. 2 1 4> 1 n 1 • Hi OH. Ill !■ 2 4' n 1 3 0^4. 96 THE TETEAGONAL SPACE-GROUPS ^4h Space-Group D^ (continued). Eight equivalent positions: (e) (f) (g) (h) (i) (J) 1 1 n- 4 4 ^J 13 1. 4 4 ?> 111 4 4 2) 1 3 n. 4 4 ^> OOu Hu; OOu Hu; 0§u , ^Ou; O^ti , hOu; uOO Ou|; uOO , Ou^; uH |uO; an |uO; UU J ; uuf; tiu 1 ; uuf; 3 3 1. 4 4 2) 3 1 n- 3 1 1. 4 4 '-'> 4 4 2 J 3 1 n 4 4 U. 2> U+2I 1 l—ii- 2> 2 ^> 0, u+l; 0, ^-u; ^+2) 2) i — 11 i- 2 "> 2> 0, u+l, 0; 0, l-u, 0; u+l, u+l, f ; 1 n 1 11 3 . 2— U, I — U, 4, 0, 0, u+i; 0, 0, l-u. 0, i u+l; 0, i ^-11. u+l, I, 0; l-u, i 0. u+i 0, ^; l-u, 0, i l-u, u+i i. Sixteen equivalent positions: (k) xyz; y, x, z+|; xyz; y, x, z+^; x+2> y I 2> z; y+2> 2~^) 2~z; 2~x, 2~y> 2j 2~yj X"r2) 2~z; i-x, y+l, z; |-y, |-x, z-1-^; x+|, |-y, z; y+i x+i z+l; xyz; y, x, |-z; xyz; y, x, |-z. Space-Group D]^.* This group is DH of the previous definitions. Two equivalent positions: (a) 0; Hi (b) 1; HO. Four equivalent positions: (c) 0^0; OH; hoh |oo. (d)OH; OH; Hf; ^oi (e) Hi; HI; IH; Hi (f) IH; iH; HI; Hi (g) OOu; OOu; i i u+§; i i Hu. igf/i^ equivalent positions: (h)Oiu; Hu; i 0, u-hi; 0, i u+|; OH; ^Ou; i 0, Hu; 0, i f-u. (i) uOO; OuO; i u+i §; u-HH, 1; uOO; OuO; i Hu, I; Hu, h h (j) uO|; OuA; |, u-hi 0; u-h|, i 0; uOi; Ou|; i ^-u, 0; |-u, |, 0. (k) u, u-l-i \; u, Hu, i u, l-u, i % u+i f; u+i u, i; u-hi u, f; Hu, u, i; ^ -u, u, i (1) u, u-fi 1; u, Hu, i; u, l-u, f; Q, u+i 1. 4> u+i u, f; u-H, u, \; Hu, u, f ; 1- -u, u, i (m)uuv; u-l-i Hu, v-H; uuv; uuv; \i-\-\, U-I-5, Hv; uuv; l-u, u+l, v-H; ^-u, Hu, Hv. THE TETRAGONAL SPACE-GROUPS Dlu-Dlh. 97 Space-Group T)^ (continued). Sixteen equivalent positions : (n) xyz; y+i |-x, z+|; xyz; J-y, x+|, z+|; x+i y+i l-z; yxz; ^-x, ^-y, ^-z; yxz; ^-x, y+i z+J; yxz; x+i ^-y, z+|; yxz; xyz; |-y, |-x, |-z; xyz; y+i x+|, |-z. Space-Group D]^.* Space-groups D]^ and D|^ also are interchanged, i^owr equivalent positions : (a) 0; (b)OOi; (c) 0|0; (d)OH; OOf; 2 ^ 2 > 2 '-' 4 > 1 1 n- A 2 2"? 2 1 i 1- ^ 2 4> 100; OH; ill 2 2 2' 113 2 2 4- n i A t» 2 2- i n A 2^4' Eight equivalent positions: (e) OOu; Hu; OOti; (f) Oiu; 1 1 f] . 2 2 ^J iOu; |0u; 1 27 1 2> 2> 1 2} ^\2} l-u; u+l; 1 0, 0, u+i; 0, 0, i-u. 0, i u-h§; 0, h i-u. 0, 0, t-u; (g) u, u-l-i J; u-}-^, u, f; u, ^-u, I; u, u4-|, f; u-l-i u, i; ti, |-u, f; (h) uvO; uvO; |— u, v-|-|, 0; u+l, t-u, u, i-u, u, -V, 0; 3 . 4> 1 4- vu|; vu§; |-v, |-u, |; v-fi u-f-|, |. Sixteen equivalent positions: (i) xyz; y, x, z+|; xyz; y, x, z-\-^; xyz; y, x, §-z; xyz; y, x, |-z; l-x, y+^, z; ^-y, ^-x, z+^; x+i |-y, z; y-l-i x-fl, z-f-^; ^-x, y-Fi z; i-y, |-x, |-z; x+i |-y, z; y I 2"> X-pz, 2 2' Space-Group D^.* Two equivalent positions : (a) 0; 111 2 2 2- (b)OOi; HO. Four equivalent positions : (c) (d) (e) (f) (g) 0|0 n A i U 2 4 OOu uuO uu| OH; n 1 3 . OOu; uu 0; uu^; hO^ ^, lOO. 1 n ^- A n A 2 " 4> 2 " 4- 1 1 n_l_l- l-u, u-f-i ^; l-u, u+i 0; 1 2> i-u. u+i l-u, u+i l-u, Ez'gi/ii equivalent positions : (h)O^u on (i) uvO tivO (j) uuv U U V |0u; *0u; vuO; vuO; uuv; uuv: I, 0, u+h h 0, l-u; v+l, i-n, I; i_v ii4-A A. 2 V, U-t-2, 2j u+i §-u, l-v; ^-u, u-M, v-Hi; 0, h u+l; 0, i J-u. u+l, l-v, I; l-u, v+i i u+i l-u, v-H; |-u, u+^, l-\. 98 THE TETRAGONAL SPACE-GROUPS Dli-D^t. Space-Group D]^ (continued). Sixteen equivalent positions: (k) xyz; y+i, |-x, z+|; xyz; xyz; y+i |-x, |-z; xyz; l-x, y+i z+l; yxz; x+i, |-x, y-}-|, |-z; yxz; x-HI, Space-Group D1^.* Space-groups D]^ and Dj^ are here interchanged Two equivalent positions: i-y, x+l, z+l; y, 1 2 1-y, i-y, x+l, z+l; 2 ^> 2 ~z; yxz; yxz. (b) 01; (a) 000; Hi i^owr equivalent positions: (c) OOu; OOu; h, h §-u; h h u+i (d) 0|u; |0u; |, 0, A-u; 0, i u+i ^tgr/i< equivalent positions : (e) 111 4 4 4; 3 13. 4 4 4; 3 3 1. 13 3. 4 4 4; 4 4 4; 3 3 3. 4 4 4; 13 1. 4 4 4; 113. 3 11 4 4 4; 4 4 4- (f) uuO ; ti u 0; u+l, J-u, 1; l-u, u+l, 1; utiO ; u u 0; u+i u+l, 1; l-u, l-u, i (g) Ouv ; Ouv; u+i i v+l; i-u, 1, v-f-l; uOv ; uOv; h u+l, l-v; i, 1- -u, l-v. Sixteen equivalent positions : (h) xyz; y-f-^ |-x, z+|; xyz; |-y, x-|-|, z+f; x+i y+h l-z; yxz; f-x, ^-y, |-z; yxz; xyz; |-y, |-x, z+|; xyz; y+|, x-|-|, z+|; |-x, y-M, i-z; yxz; x+|, |-y, |-z; yxz. Space-Group D1^* Four equivalent positions: (a) 000; IH; 001; (b) ('p^ 1 1 n- 111 V.W 4 4 ^> 4 4 2 fiO; |0u; 113. 5 2 4; 3 11. 4 4^; OOi; OOf; HO; f 2 "• 111 ^2 4" 13 1 4 4^- 111. 4 4 7; (d) (e) Oiu; A 3 1 . 4 4 2; 1 3 n 4 4"' 1 n 1 — 11- 2, U, 2 U, 0, h u+i. E^'gf/ii equivalent positions : (f) OOu Hu; OOu Hu; (g) uuO uu|; uuO Qui; (h) uu| uuO; utii ; QuO; h h u+i; 0, 0, u-f-i; i> h |— u; 0, 0, |— u. u-M, l-u, I; u+i u+i 0; l-u, l-u, 0. u+i u-f-i I; l-u, u+l, I; u+l, l-u, 0; l-u, u+l, 0; l-u, l-u, |. (i) u, u+l, v; u+l, u, v; u, -u, v+l; u, l-u, v; u, u+|, v+|; u+l u, |-v; |-u, u, v: |-u, u, l-v. THE TETRAGONAL SPACE-GROUPS DiS-DlJ. 99 Space-Group D]^-{contimied). Sixteen equivalent positions: (j) xyz; y+l, |-x, z-l-|; " '-hi X, y, z+l; l-x, y+l, x-hi y+i z; y, x ^-z; A— 7' 2 ^j xyz; l-y, x-h|, z+|; '-X, l-y, z; y, x, z; X, y, z+l; y+|, x+^, z; -z; yxz; x+i, |-y, |-z; yxz; i-y, •J X, Space-Group DIJ. Two equivalent positions: (a) 0; Hi Fowr equivalent positions : (c) 0^0; ^0 0; ^Of; (d) OH; |oi; OH; (e) OOu; OOu; Eight equivalent positions: (b) OOi; HO. 1 1 2; 2) OH. *oi i — IT i 2 l^; 2j U + |. (i) 4 4*; 3 3 3. 4 4 4; (g) O^u; O^u; (h) uuO; uti 0; uOO; tiOO; (i) (J) 2 > 1. 2 ) 13 1. 4 4 4 7 111. 4 4 4> ^Ou; |0u; uu 0; tiuO; OuO; OuO; Oui; Ou^; 3 11. 4 4 4> 13 3. 4 4 4> 3. 1 1 . 4 4 4; 113 4 4 4- 2> h 0, u+^; h> 0, 2~u; i u+i h, h-u, 0, h u+l; 0, i ^-u. 1 ,, 1 „ 1 ^— u, ^— U, 5. u+i 1 1 . 2} 1> 1 2> u+l, I, 0; ^-u, h 0. Sixteen equivalent positions: % |-u, i; "-U, u, I; 0, i •u, f ; f ■u, ti, i; (k) u, u-H, i; |-u, u, i; u+i u, f; u, ^-u, f ; § u+i u, f; u, u-l-i f; u, u+i i; u+l, u, i; (1) uvO; vuO; uvO; vtiO; vuO; uvO; vuO; tivG; u+iv+l, ^; §-v, u+il; v+iu+l, i; u+l, l-v, i; (m)uuv; uuv; uuv; uuv; uuv; uuv; uuv; uuv; u+i l-u, v+l; u+i u+i v-Hl; Hu, u+i l-v; u+i u+i 2-v; u+i u, i; u, u+l, f ; ^-u, u, f; u, |-u, h u, I- 1 2 1 — v i- 2 V, 2 V, I; v+iHu, 1; u, I; |-u, v+i |. |-u, u+l, v+^; |-u, l-u, v-l-l; u+i l-u, ^-v; l-u, §-u, i-v. (n)Ouv; uOv; Ouv; uOv; uOv; OQv; uOv; Ouv; i u+iv-f-i; l-u, iv-l-l; i^-u, v-|-^; u+i i v-l-^- u+l, i §-v; i Hu, l-v; ^-u, i |-v; |, u+i -J-v. 100 THE TETRAGONAL SPACE-GROUPS dII~J)1 Space-Group D^l-icontinued). Thirty-two equivalent positions: (o) xyz; yxz; xyz; yxz; yxz; xyz; yxz; xyz; xyz; yxz; xyz; yxz; yxz; xyz; yxz; xyz; x+i, y+f, z+l; l-y, x-hl, z+|; ^-x, |-y, z+l; y+h, l-x, z+§; y+i x+i, ^-z; x-f-|, i-y, |-z; i-y, |-x, |-z; 5~x, y+2, 2~2; l-x, l-y, i-z; y+l, ^-x, l-z; x+^, y+f, §-z; i-y, x+l, ^-z; l-y, ^-x, z+l; l-x, y+l, z+l; y+l, x-|-|, z+|; x+i l-y, z+|. Space-Group D^. Four equivalent positions: (a) 0; (b)OiO; (c) OOi; (d)OH; 001; 100; 111. 2 2 2; n 1 1 • 1 i 3 . 2 2 4; 2 vJ 4, 2 *-* 4> t tw- in A 2 " 2' 4 i i 2 2 4' nil <J 2 4' Eight equivalent positions: /p^ 1 in- 3 in- iin- iio- Vc^ 4 4"; 4 4^; 4 4^; 4 4 ^> (f) 1 3 1. 4 4 2; 1 3. 1 • 4 4 2^; OOu; OOu; (g) 0|u; 0|u; (h) u, u+l, i; u+i u, f; 1 l^l- 2 2 l-l; 1 1 f, • 2 2"; AOu; |0u; 111. 4 4^; 1 1} 1 ¥; 1 ^; 1 ^; 3 11 4 4 2' ^; ^+2; 1 2; 2 U; 0, u+l; 0, l-u; t-u, u, u, i-u, 1 . 4; 1- 4; 0, 0, u+l; 0, 0, i-u. 0, I, u-f-l; 0; h i-u. U; |-u, 1; u-M, u, |-u, u, f; u, u-M, Sixteen equivalent positions: (i) uuO; uu| uuO; uu| utiO; uu| uu 0; uu I (j) uOO; u0| OuO; Ou| uOO; uO| OuO; Oui (k) uvi; vuf vu uv| u vi; vti f vui; " u vf 1 1—11 2; 2 " (1) u, u+l, v; u+i u, v; u, -2--U, i-v; ^-u, u, v+l; u+l, u+l, |-u, u+l, l-u, l-u, u+l, |-u, u+l, i i u+i ^-u, I, 0; 0; hi -I -7 'V \ u-l-l, u, |-v; U; u-Hi v-f-f ; v+i u+i h u+i l-v, 1. 4; ^-v, l-u, 1- 4; l-u. v+l, i; i-u, u, v; u, |- -u, v; u-i-l, u+h h l-u, u+l, I; 1 11 l_n !• 2 ^; 2 "; 2^» u+l, i-u, |. n-4-1 1 1- U-r2; 2; 2; 1 n _L1 1 • 2; U-j-2; ^, 1 11 1 1- ¥ — U, J, 2, I; | — U, |. u+l, v+l, f; l-v, u+l, f; l-u, l-v, f ; v+i l-u, f. u, l-u, v; u+l, u, v; l-u, % v; u, u+l, v; u, u+l, l-v; l-u, u, |-v; u+l, u, v+l; u, |-u, v+|. THE TETRAGONAL SPACE-GROUPS Dl^-bljj.' , : ; : " lOJ: Space-Group D^ (continued). Thirty-two equivalent positions: (m)xyz; yxz; xyz; yxz; yxz; xyz; yxz; xyz; X, y, i-z; y, x, J-z; x, y, ^-z; y, x, |-z; y, X, z-\-i; X, y, z+i; y, x, z+|; x, y, z+i; x+i y+i z+l; |-y, x+l, z+i; |-x, ^-y, z-f-^; y I 2) 'a X, z-i-2> y+i x+l, |-z; x+l, ^-y, |-z; ^-y, |-x, ^-z; 2 X, y+2j 2 z; |-x, |-y, z; y+i, |-x, z; x+|, y+i z; i-y, x+|, z; |-y, l-x, z; ^-x, y+l, z; y+i x+i z; x+i |-y, z. Space-Group T)]^* Four equivalent positions : (a) 000; OH; iOf; Hi (b) OOi; OH; ^01; HO. Eight equivalent positions: lis. 313. 135. 3 f\ 7 5^48>458>^48>4'-'8' /JN 111. inl. 131. 111. \yj ^48; 4"8> ^48; 428> ni5. 3.ni« nil- 3.11 ^48> 4"8> '-'48> 42 8' (e) OOu; 0, i u+i; i i u+|; i 0, u-hf ; OOu; 0, i i-u; i i, Hu; i 0, |-u. Sixteen equivalent positions : (i) U4 §; 4> 2~u, g; u 4 §; T^f; n4_l 11. 3 i_,, 3. i — n 3 5. 3,, 7. ^^2> 4j 8> 4> ? "j 8> 2 l^) 4> 8> 4 *^ 8) r, 11- 3 ,i_l_l 3. nil. 3ril. l_n 15. 1 ii_Ll 3. n_Ll 3 1. 1 f; 7 ^ ") 4) 8} 4> U-t-2> 8j U-t-2> 4) 8> 4 U ||. (g) uuO; u, Hu, i; uuO; ^-u, u, f; u+i u, f ; u-l-l, Hu, I; Hu, ti, |; u, u+|, i; uuO; u, i-u, I; uuO; u, u+i |; 2— U, U-i-2> 2; 2~U, 2~U, 2; U+2> U, 4; U+2> U+2, 2- (h) Ouv; u, i, v+l; Otiv; Hu, 0, v+f; i u, f-v; u+f, i Hv; I, u, |-v; uOv; iu+iv+l; u+i 0, v+l; i Hu, v-M; u, i v+i; 0, u-M, i-v; u v; 0, i-u, |-v;Hu, |, |- v. Thirty-two equivalent positions: (i) xyz; y, Hx, z-hi; xyz; |-y, x, z+|; X+-2-, y, f-z; y+i J-x, §-z; Hx, y, |-z; yxz; xyz; y, |-x, z+J; xyz; y+i x, z+f; §-x, y, |-z; Hy. Hx, i-z; x+|, y, f-z; yxz; x+i y+i z+i y+i x, z+f; Hx, Hy, z+i y, x+i z+l; X, y+i i-z; yxz; x, Hy, l-z; Hy, x+i Hz; Hx, y+i z+i- Hy, x, z+f; x+i Hy, z+i- y, x+i z+i; X, y+i i-z; yxz; x, Hy, J-z; y+i x+i ^-z. •102 THE TETRAGONAL SPACE-GROUP D^. Space-Group DlS-* Eight equivalent positions: (a) 000; OH; uuf; (b)OOi; OH; 0|0; 100; 1 1 n. 111 Hf; Hi Sixteen equivalent positions: 2 4 5) 'J 4 8> (c) Oj I; i^ I; ? 4 8 7 f Oi H (d) u OOu Hu Hu (e) (f) 11 i i U 4 8 uH uff uuO u4-^ 3 15. 4 ^ 8> 3 11. 4 2 8; HI; h 3 3 . 1 8> 0, h 0, i 0, i 0, i u+i; u+f; 3_„. 4 ", -u; 2-u, f; l-u, I; U, 4> 8; 4 '^ 8> fOf; iOf. h 0, u+i; i 0, u-hf; i 0, f-u; i 0, i-u; f, 3 1 . U 4 8> 1 2; 1 2> uH; 1 2 u; u: 0, 0, 0, 0, u-f-i u, t; u. u-l-i I; U~f"2j 4> fj u, l-u, u+i ^-u, 0; 1 2 ~ 3 ¥> 1 4; 1 . 4; 1 . 8> 3 . 8> 1. 8> uuO; -u, u, 4 U 8, 3 „ 1. 4 U 8> 4 U¥, 1 f! 3 ?; u-}-i v uu^; Thirty-two equivalent positions: n 1- u> 4> u, §-u, i; u, f; •u, I; 5-u, u, I; u u i; u, u-f-l, J; ^-u, u+^, 0. (g) xyz; y, l-x, z+i; x+i y, i-z; yH-i, ^-x, z; X, y, z+i; y, l-x, z+l; ^-x, y, f-z; ^-y, |-x, ^-z; x-f-|, y, f-z; yxz; x+i y+i z+l; y+i, X, z+f; ^-x, |-y, z+|; xyz; i-y, x, z+f; i-x, y, i-z; y, x, §-z; X, y, z+l; y+l, X, z+i; y, x+l, z+J; X, y+i f-z; y, x, |-z; x, |-y, f-z; |-y, x+|, z; ^-x, y+l, z; l-y, X, z+i; x+|, |-y, z; y, x+|, z+f; X, y+l, i-z; yxz; x, |-y, f-z; y+|, x+|, ^-z. SPECIAL CASES OF THE CUBIC SPACE-GROUPS. 103 CUBIC SYSTEM. THE SPECIAL CASES OF THE CUBIC SPACE-GROUPS. ONE Equivalent Position. (la) 0. (lb) Hi TWO Equivalent Positions. (2a) 0; Hi THREE Equivalent Positions. (3a) h \ 0; |0i; OH- (3b) 10 0; OH; 0|. FOUR Equivalent Positions. (4a) uuu; uuu; tiuu; utiu. (4b) 0; HO; |0|; OH. (4c) \\\, iOO; OH; ooi (4d)Hi, HI; 3 13. 331 4 4 4; 4 4 4' (4e) Iff; 3 11. 4 4 4; HI; Hi (4f ) uuu ; u+i ^-u, u; u, u+i ^-u; |-u, u, u+i (4g) HI; IH; HI 3 7 5 8 8 8- (4h)H|; 17 3. 8 8 8 > III 13 1 8 8 8- (4i) IH, HI; fH Iff. (4j) Hi Iff; 111 8 8 5 HI. SIX Equivalent Positions. (6a) uOO ; OuO OOu; (6e) 0^0; OOi; HO; uOO OuO ooti. HI; HO; OH. (6b) iuO ; 0|u uO|; (6f) OH; HI; HO; HO ; OH tiOi OH; fO^; HO. (6c) Ou^ ; |0u uH; (6g)HI; HO; OH; Ou| ; Hu uH. HI; fH; OH. (6d)|u| ; Hu uH; |Gi ; HQ QH . EIGHT Equivalent Positions. (8a) uuu; uuu; u+|, u+|, u+|; |-u, u+|, |-u; uuti; uuu; u+l, |— u, |— u; |— u, |— u, u+l- (8b) uuu; |-u, u, ti; u+|, u+|, u+|; u, u+|, |-u; u, u, |-u; u, |-u, u; u+|, |-u, u; |-u, u, u+|, (8c) uuu; uuti; tiuu; Qtiu; tititi; uuu; utiu; uuti. (8d) uuu; tiuti; utiti; titiu; (8e) 1 1 1; Hu, l-u, §-u; u+l, l-u, u+l; |-u, u+l, u+l; u+l, u+l, |-u. 13 3. 4 4 4; 13. 1 1 1. 4 4; 4 4 7; 3 13. 4 4 4; 3 3 1. 4 7 4; 4 4*; 4 t f • 104 SPECIAL CASES OF THE CUBIC SPACE-GROUPS. EIGHT Equivalent Positions. — Continued. (8f) OH; Hi; HO; 0; 13 3. 4 4 4; 3 13. 4 4 4) 3 3 1. 4 4 4; iih (8g) 10 0; 0|0; 001; 111. 2 5^, 3 11. 4 4 4> 13 1. 4 4 4> HI; 3 3 3 4 4 4- (8h) uuu; u+i l-u, u ; ti, u+- J; Hu; 1 -u, u, u+l; uuti; ^-u, u+l, u ; u, l-u, u+J; u +i u, i-u. (8i) 0; HO; 1 n 1 • OH; IH; OOi; OiO; |0 0. (8j) uuu; u+i, l-u, u ; u, u+- h i-u; i -u, u, u+j; l-u, i-u, i — u; u- ■M, l-u, uH; f- -u, u+l, u+f; u+i u+i l-u (8k) uuu; u+i l-u, u ; U; U + - 11 ,,. 1 2; ^~U, 2 -u, u, u+l; f-u, 3_„ 3 4 ^) 4: -u; u+i i-u, u+f; \- -u, u+f, u+f; u+f; u+f, f-u (81) HI; HI; IH; III; Iff; IH; HI; III. (8m) Iff; 3 5 1. 8 8 8; 13 5. 8 8 8; 5 13. 8 8 8; 8 8 8; HI; 5 11. 8 8 8; 17 5 8 8 8- TWELVE Equivalent Positions. (12a) u ; uOO u+l, hh \- -u, i 1; OuO ; OuO 1 ,,_Ll 1. 1 2; ""r2; 5; 2; 1 1, 1 . 2— U, ^, OOu ; OOti ii u+l; i 1; l-U. (12b) u^O ; u|0 u+i 0, 1; ^ -u, 0, 1; Ou^ ; Ou^, 1; u+l, 0; i l-u, 0; |0u ; iou, 0; i u+l; 0, 1, l-u. (12c) uOi ; uH, u+l, 13. 1 ^; f; 2" -u, 0, f ; iuO ; HI; f, u4 -I;i; f; Hu, 0; Oiu ; HQ; 1 3 2; 4; u+l; 0, f. l-u. (12d) u V ; Ouv ; Ouv; Ouv; vOu ; vOu ; vOu; vOu; uvO ; uvO ; uvO; uvO. (12e) |uv ; 5UV |uv; |uv; v^u ; v§u viu; v^ti; U V2 ; uv| uv|; u v|. (12f) uOi uOi; u|0; ti|0; |uO |uO; Ou|; Ou|; O^u 0|u; Hu; |0u. (12g) uuv ; uuv ; tiu V utiv: vuu ; vuu ; vuu vuu; u vu ; uvu ; uvti uvu. (i2h)ni; |0f OH; OH; iiO; HO. HO; HO: OH; Ofi iO|; foi SPECIAL CASES OF THE CUBIC SPACE-GROUPS. 105 TWELVE Equivalent Positions. — Continued. (12i) uOi , uO^; u+l, 0, h |-u, 0, 1; |uO ; ^uO; i u+ h 0; h l-u, 0; 0|u ; OH; 0, i u+^; 0, 1, l-u. (12j) u^O ; ti^; u+l, h 0; l-u, 1, 0; Oui ; Oui; 0, u+ 1,1; 0, l-u, 1; ^Ou ; \0u; i 0, u+i; 1, 0, l-u. (12k) fOi; iOf; Hi; fH; ifO; HO; HI; Hf; OH; OH; HI; 13 7. 5 4 8, (121) 13 3. 2 4 8; HI; OH; Ofl; 3 13. 8 2 4; Hi; fof; f|0; . 3 3 1. HI; HO; |0f. 4 8?; • (12m) uiiO ; uuO; uuO; tiuO Ouu ; Ouu; Otiu; Ouu uOu ; uOu; uOu; uOu (12n) uu^ ; uu ^; ii u 1; uu| |uu ; 1 uu; 1 uti; |uu u^u ; u|u; u|u; u|u (12o) u, h- -u, i; u, u+i f; u, l-u, f; U; u+i 1; h u, ^-u; f, u, u+l; f, u, l-u; f, u, u+l; l-u, i, u; u+i i u; l-u, f, u; u+l, f, u. (12p) u, h- -u, f ; u, u+l, 1 . 4; U; l-u, f; U; u+i, f; f , u, |-u; i, u, u+l; f, u, l-u; f, u, u+l; ^-u, f, u; u+i i, u; l-u, f, u; u+l, f, u. (12q) i-u, u, 1; f-u, 1- -u, 1; u+f, u+l, f; u+f, ti, f ; i i- -u, u; 1, f-u, l-u; i u+f, u+i; 1; u+i, u; u, i i-u; l-u, 1, f-u; u+l, f; u+f; Q; i U + i. (12r) l-u, u, f; f-u, 1- -u, 1; u+f, u+l, 1; U+f, Q, 1; if- -u, u; 1, f-u, l-u; i u+f, u+l; 1, u+l, u; 11 3 U; ¥; f-u; l-u, f, f-u; u+l, i u+f; u, 1, u+f. (128) |0i IH; fOf; ill. 4 8 2, iio, fH; ffO: Hf; OH; HI; ofi; HI. SIXTEEN Equivalent Positions. (16a) uuu ; u+l. u+l, u; u+l. u, u+l; U; U + l, U+l; uuu ; u+l, ^-u, u; u+l. % l-u; U; l-U; l-u; tiuu ; l-u. u+l, u; l-u. u. l-u; u, U + l; l-u; uuu ; l-u, |-u, u; |-u, u, u+l; u, l-u. u+|. (16b) Hi; ill; HI; IH; SSI. 8 8 8; HI; 3 5 7. ¥ 8 ¥, 5 15. 155. 8 8 8; 888, 5 7 3. 113. 8 8 8; 8 8 8; ¥ 8 ¥; ¥ ¥ s; fit; 1 1 1- 106 SPECIAL CASES OF THE CUBIC SPACE-GROUPS. SIXTEEN Equivalent Positions. — Continued. (16c) (16d) (16e) 13 7 8 8 8 15 1 8 8 8 111 8 8 8 15 7 8 8 8 uuu uuti uuu uuu uuu uuu 111. 8 8 8) ill. 8 8 8 7 1 1 1- 8 8 8 > 1 1 5.- 8 8 8> 111. 8 8 8; 1 1 1. 8 8 8) 111. 8 8 8> 111. 8 8 8) 3 5 3. 8 8 8) 5 5 5. 8 8 8) 111 8 8 8- uuu; uuu; uuu; uuu; 2 u, 2 U) 2 u; |-u, u+i u+l; u+i l-u, u+l; u+l; (16f) (16g) u+^, u+l, u+^; u+i l-u, |-u; |-u, u+l, ^-u: ^-u, |-u, u+l; u+l, u+l, |-u. u, u, ^-u; ^-u, u, u; u, ^-u, u; u, u, u+^; u+l, u, u; u, u+^, u; u+i u+l, u+l; u+l, ^-u, u; u, u+|, |-u; ^-u, u, u-i u-|. u-|; ^-u, u+l, u; u, |-u, u+|; U + 5) u, |-u. uuu; u, u, | — u; | — u, u, Q; u, | — u, u; u+i, u+i u+i; i-u, u+i f-u; u+i, f-u, i-u; f-u, i-u, u+i; u+i u+l, u+l; u+l, |-u, u; u, u+|, |-u; l-u, u, u+l; u+f, u+l, u+f; f-u, u+f, i-u; u+f, f-u, f-u; i-u, f-u, u+f. uuu; u, u, |— u; |— u, u, u; u, | — u, u; i-u, i-u, i-u; u+l, l-u, u+f; f-u, u+f, u+f; u+f, u+f, f-u; u+l, u+l, u+l; u+l, |-u, u; u, u+|, |-u; l-u, u, u+l; f-u, f-u, f-u; u+f, f-u, u+f; f-u, u+f, u+f; u+f, u+f, f-u. (16h) (16i) 000; l|0; 1 . 10^ "22, 1 1 1- 8 8 8) 7 7 7. 8 8 8) 3 3 3. 8 8 8) 5 5 5. 8 8 8) 1 1 1- 4 4 4) 1 1 1- 4 4 4) 113. 4 4 4, 111. 4 4 4, 1 1 1- 8 8 8, 115. 8 8 8) 111. 8 8 8) 1 1 1- 8 8 8) 3 1 3. 4 4 4, 1 1 1- 4 4 4) 1 1 !• 4 4 4) 111. 4 4 4) 1 1 1- 8 8 8) 111. 8 8 8) 3 5 1. 8 8 8) 113. 8 8 8, 2^2) 001; 0|0; 10 0. 111. 8 8 8) 111. 8 8 8> 5 13. 8^8, 111 8 8 8* TWENTY-FOUR Equivalent Positions. (24a) uOO aoo OuO OtiO GOu GOu u+l, I, 0; u+l, 0, I; u||; 2~u, 2) 0; 2~u, 0, 2) ^2^^; I, u+l, 0; |u|; 0, u+|, |; I, l-u, 0; |u|; 0, |-u, |; ||u; I, 0, u+l; 0, |, u+|; l|u: I, 0, l-u; 0, I, l-u. SPECIAL CASES OF THE CUBIC SPACE-GROUPS. 107 TWENTY-FOUR Equivalent Positions. — Continued. (24b) (24c) (24d) (24e) (24f) (24g) (24h) i ill 4 4 U 3 3 „ 4 4 U 11 i i U 4 4 Oi i U 4 4 i n i 4 U 4 13 1 ? 4 4 4 ? 2 ^ i 4 4 4? Ouv Otiv Ouv Ouv vOu vOu uOi iuO Oiu uOf f tiO Ofu; uO| ^uO 0|u uO| no on UU V V uu u vu uu V vuu u vu ioo Oi 00 fO 1 4 Of oof r, 3 1 u 4 4 „ 1 3 U 4 4 4 U 4 3 ,-, 1 4 U 4 1 Tl a 4 U 4 3 „ 3 4 U 4 1 3 n nil U 4 4 3 1 2 1 4 3 4 01 3 3 4 4 H 3 1 4 ? vOu vOu uvO uvO uvO uvO aH uu 11 1 3 U^ 4 3 n 1 jU 2 ? 4 U u|0 Ou^ iOu u§0 Qui |0u ti u V vuu u vu uti V vuu u vti 13 1 2 4? 1 i i 4 2? i i 1 2 2 4 111 ? 4 2 111 T ? ? 3. 4) 3 4; 1 4; ?-u, ?-u, „_l_l 1 3. U-ri, 4) 4> 3 n_L.i 1. 4> U-t-2> 4> 11 n !• 4 ) ? ".' 4 ; 3 1 n. 3 3 rj. 4 4 U, I 4 "> 4> 3 4> 1 4> 1 3 4; ¥> nil U 4 4 1 n 1 4 U 4 111 2 4 7 111 4 2 4 111 4 4? 1 2> 1 ?> 1 ?> 1 ?> 1 1- " 4 4> 1 n 1. 4 ^ 4> 113. 111. 4 ? 4> Hi u+i v+i l-u, ^-v u+i ^-v l-u, v+l v+?, ?, u+l 1 17- 1 1 n ^-u, 0. f; I l-u, 0; 0, i i-u; 1—11 1- 2 ll> 4> u+i f; ?-u; ?-u; u+i 0, i; i u+i 0; 0, h n+h u+i h 0; 0, u+i ^; i 0, u+l; §— u, i 0; 0, ? — u, ^; h, 0, ^-u; u+i u+i v+i; v4-?, u+i u-l-^; u+i v+i u+i; u+?, ?-u, |-v; l-v, u+i |-u; l-u, l-v, u+i; 1 IT 1 nJ-l- v4-l 1 1 — 11- VT^2> 2» 2 U, u+?, v+i I; l-u, i-v, ^; ?-u, v+i i u+?, ?, h i u+i ?-u, i ii 1 i_n 4> 2 U, ?, i ^-u. u+i 0, I; i u+i 0; 0, h u+i l-u, 0, I i ^-u, 0; 0, h h-u. |-u, u+l, ^-v; l-v, l-u, u+l; u+?, ?-v, ^-u; ^-u, l-u, v+l; v+?, ?-u, l-u; ^-u, v+i l-u. Hi; 3 1 n- 4 ?^; 13 0- ? 4 U, HO; HO; OH; \0h n 3 1 . u 4 ?> 3 n 1 • 4 'J?; |0i; OH; OH. 108 SPECIAL CASES OF THE CUBIC SPACE-GROUPS. TWENTY-FOUR Equivalent Positions. — Continued. (24i) (24j) (24k) (241) (24m) uOi; ^-u, 0, f ; uH; i f-u, 0; HO; f, l-u, 4 ^ 2 J l-u, 0. i; Oiu; 0, I i-n 1 1 ,1 • 2 4 U, 0, h l-u; i u+i I f, i-u, 1 ,,_1_1 1 3. i u+l, 0, u+i 1 i i-u, i 1 i u+l, 1; u+f, 0, f ; i i, u+l i h i-u, 1, i u+§; 0, 1, u+|. uuO; uu 0; u+l, 1- -u, 1; ^-u, 1 „ 1 . 2— U, ^, Outi; Ouu; ^, u+l, l-u; i ^- -u, i-u; liOu; u u; 5-u, i u+l; l-u, 11 „. 2) ^ — U, uuO; uu 0; u+l, u+l, ^; ^-u, u+i f; Ouu; Ouu; ^, u+^, u+i; i i- -u, u+l; uOu; uO u; u+i h u+^-; u+l, i l-u. u, l-u, i ; u, u+i f ; u, ^-u, f ; u, u+l, f; i u, ^-u , h u, u+f ; i u, ^-u ; i u, u+l; 5-u, i, u u+i h u ; ^-u, f, u ; u+l, 1, u; u+i u, f 2-u, ti, i ; u+l, u, 1 ; l-u. u, f; u, f, u+i ti, i l-u ; u, i u+^ ; u, f, l-u; i u+i u i ^-u, ti , i, u+i u ; 1, l-u. u. uOi; U+2> ?> 4) l-u, 0, 1; QH; iuO; i u+l, 1; f, ^-u, 0; iai; 0|u; i i u+l; 0, f, i-u; HG; h i-u, 0; 3 3 1, 1 . 4> 4~U, 2j h u+f, i; i u+i 0; i-u, 0, I; f-u, i 1, u+i i i, u+i 0, f ; 0, i i-u; i i l-u; i i u+f; 0, i u+i u, u+i, 1 u, i-u, 1 ; u, l-u, 1 ; u, u+i f ; i u, u+i i u, i-u ; i u, l-u ; f, u, u+l; u+i i u i-u, i u ; f-u, i u ; u+i i ti; u+i i-u , 1; i-u, u +h h u+^ \, u+i f; l-u. i u+i i- -u; i l-u , u+J; 1, u +1, u+l; i l-u, l-u; i-\i, I, u+l; u+i I, l-u; u+i i u+|; |— u, f, |— u. (24n) u, i-u, I; ti, u+i |; u, u+i f ; i u, u+l; i u, u+l; u+i i u; u+i i u; u+l, u+i I; l-u, l-u, I; u+|, |-u, f; l-u, u+i I; u; i u+l, f-u; u. l-u, i; i u, l-u; 1- -u, i u; ti, l-u, f; i ti, l-u; l-u, i u; I, u+l, u+l; i l-u, i u+i I, u+l; l-u, i I- i l-u, u+l; l-u, f, u+l; u+i f, |-u. SPECIAL CASES OF THE CUBIC SPACE-GROUPS. 109 TWENTY-FOUR Equivalent Positions. — Continued. (24o) Ouv; Ouv vOu; V u u vO; u V uOv; uOv Ovu; Ovu vuO; vu (24p) |uv; u|v; |u V uv| u^ V h vu vuf; ?u^ (24q) uu v; uu v vuu; vuu u vu; ti vu tiuv; uuv vuti; vuu tivu; uvti (24r) Ouv; Ouv vOu; vOu uvO; tivO h ^-v, ^-u (24s) u, ^-u, I h u, |~u i-u, i, u u, u+i f i u, u-f-^ u+i f, u (24t) u, l-u, i i u, ^-u ^-u, i u Ouv vOu uvO uOv Ovu vuO iuv v^ u uv^ u| V I vu vu^ uuv vuu u vu uuv vuu u vu Ouv vOu uvO Ouv vOu uvO uOv Ovu vuO |u V v^u ti v^ U§ V ^ vu vu^ uti V vuu uvti uuv vuu U V u Ouv vOii iivO l-u, u, I h l-u, u h, v+i u+^ v+i u+i ^ i u, u+^ u+i i u u, l-u, i i ti, ^-u i-u, i, ti u, u+i f i u, u+l u+i i u u+l, u, I u, I, u+l i u+l, u l-u, i v+^; u+l, ^, l-v; i v+i |-u; i l-v, u+^; v+il-u, I; i-v, u+i |. ti, ^-u, f i ti, ^-u ^-u, i u u, u+i i i, u, u+l u+i i, u ti, i-u, f i ti, ^-u f, ti f, 1 1 11, 4, 2 1 1. 4, 2 ti, u+i i; i u, u+i; u+i i a; u, l-u, f; i u, l-u; i-u, i u. ti, u+i i; h ti, u+l; u+i i ti; u+i u, i; u, i u+§; i, u+i ti. ^-u, u, f; ■u; ■u, ti; (24u) uuv; uuv; tiuv; titiv; vuu; vuu; vtiu; vuti; uvu; tivu; uvti; tivQ; i-u, ^-u, §-v; l-u, u+l, v+l; u+l, |-u, v+J; u+i, u+^, l-v; ^-v, l-u, J-u v+i ^-u, u+^; v+i u+^, ^-u; ^-v, u+i u+^; •u, l-v, ^-u; u+i v+l, ^-u; ^-u, v+|, u+^; u+i l-v, u+i. ^-^ no SPECIAL CASES OF THE CUBIC SPACE-GROUPS. TWENTY-FOUR Equivalent Positions. — Continued. (24v) lOi; HI , fO|; 7 11. 8^4; i|0; H^ HO; iH; OH; HI > ^ 4 8> 117. ^4 8, |0|; IH 113. > 8 2 4, fOi; f|0; HI , HI; ifO; OH; 113 ? 4 8 IH; OH. (24w) fOi; 113 8 4 4 iOf; 5 11. 8^4, HO; 3 7 1 4 8^ HO; 15 1. 4 8 5, OH; HI OH; 115. 5 ¥ 8, fOl; Hi 3 13. 8^4; |0i; HO; HI 4 8 ^> HO; OH; Hi 13 3. 1 4: 8> OH. THIRTY-TWO Equivalent Positions. (32a) u u u uuti tiuu utiu uuu uuu utiu uuu (32b) uuu utiu tiuti tiuu i-u, i u+l u+l l-u l-u l-u l-u u+l u+l u+l u+l l-u l-u 1 2 l-u, u; I u+l, u; u+l, u, u+l |-u, u; u+l, u, |-u u+l, ti; |-u, u, |-u u, u; |-u, u, u+l u, u, |-u u+l, u; |-u, u, u+l |-u, u; u+l, u, u+l u+l, ti; u+l, u, |-u u+l, u; u+l, u, u+l |-u, u; u+l, ti, |-u u+l, ti; |-u, u, |-u |— u, u; |-u, ti, u+l -u; l-u, f-u, l-u; u. u+l, u+l; u. l-u. l-u; u, u+l. l-u; u. l-u. u+l; u, l-u. l-u; u, u+l, u+l; u, l-u. u+l; u. u+l. l-u. u, u+l. u+l; u. l-u. l-u; Q, u+l. l-u; u. l-u. u+l; t-u, t-u, f-u; 4 — u, 4— u, 4— u; i-u, u+i, u+i; f-u, u+f; u+i; f-u, u+i, u+f ; l-u, u+f, u+f; u+l, l-u, u+l; u+f, f-u, u+l; u+f, |-u, u+f; u+l, f-u, u+f; u+l, u+l, l-u; u+f, u+f, l-u; u+f, u+|, f-u; u+l, u+f, f-u. (32c) uuu; u+l, u+|, u; u+|, u, u+|; u, u+|, u+|; uuu; u+l, |-u, ii; u+|, u, |-u; u, |-u, |-u; uuti; |-u, u+l, u; |-u, u, |-u; ti, u+|, |-u; uuu; |-u, |-u, u; |-u, u, u+|; u, |-u, u+|; u+l, u+l, u+l; u, u, u+l; u, u+|, u; u+|, u, u; |-u, u+l, l-u; u, u, l-u; u, u+|, ti; |-u, u, u; u+l, l-u, l-u; u, ti, |-u; u, |-u, u; u+|, u, u; |-u, |-u, u+l; ti, ti, u+l; u, l-u, u; I u, u, u. SPECIAL CASES OF THE CUBIC SPACE-GROUPS. Ill THIRTY-TWO Equivalent Positions. — Continued. (32d) i I i (32e) If 8 8 ) 1 3. 8 S} 1 1- 8 8 ) 3 3.. 8 8 > 5 7. 8 8 ) 3. 1- 8 8 } 1 5 . 8 8 7 1 1 • 8 8> 3 3. 8 8> 3 1. 8 8 7 5 5. 8 8 > 1 5. 8 8 ; 3 5. 8 8 f 1 5. 8 8> 5 3. 8 8 ; 5 7. 8 8 7 17 7 8 8 8 113 8 8 8 7 5 7 8 8 8 3 3 5 8 8 8 7 5 3 8 8 8 1 k 1 8 8 8 3. 1 5 8 8 8 111 8 8 8 111 8 8 8 111 8 8 8 5 7 5 8 8 8 115 8 8 8 7 5 5 8 8 8 115 8 8 8 5 5 1 8 8 8 111 8 8 8 (32f) uuu; u, u, 4 »J> 4 uuu; u+i 111 8 8 8 111 8 8 8 17 5 8 8 8 5 5 i 8 8 8 5 1 1 8 8 8 111 8 8 8 13 5 8 8 8 15 5 8 8 8 111 8 8 8 3 7 7 8 ? 8 111 8 8 8 15 1 8 8 8 111 8 8 8 5 1 1 8 8 8 111 8 8 8 5 1 1 8 8 8 l-u; -u; 111 8 8 8 15 5 8 8 8 113 8 8 ¥ 5 3. 7 8 8 ¥ 111 8 8 8 111 8 8 8 111 8 8 8 111 8 8 8 111 8 8 8 113, 8 8 8 111 8 8 8 111 8 8 8 111 8 8 8 111 8 8 8 111 8 8 8 5 11 8 8 8 u, u, u-hl; u+i, u+i; u+i u+^, U+-2-; 1 4 -u, 1 4 -u, 3 4 -u; 1 2 -u, 1 2 -u, 1 -u; u +i u +i u _i_i. T^4> u, u, u; u, t — u, u; u+i i-u, u+f; i-u, u+f, u+i; u+f, u+i i-u; u-f-l, ti, u; u, u+i u; i-u, u+i f-u; u+i f-u, i-u; f-u, l-u, u+l; u+l, |-u, u; u, u4-i ^-u; l-u, u, u+l; u+f, f-u, u+i; f-u, u+i, u+f; u+i u+f, f-u; |-u, u+l, u; u, |-u, u+^; u+l, u, |-u; f-u, u+f, i-u; u+f, i-u, f-u; i-u, f-u, u+f. FORTY-EIGHT Equivalent Positions. (48a) 1 1 u iiu 1 3 4 4 3 1 f, I lu 4 4 U II i i U 4 4 fl i ^ U 4 4 fl ^ i U 4 4 n 1 1 U 4 4 i 11 i 4 U 4 1 f, 1 4 U 4 1 y, 3 4 U 4 3 „ 1 4 U 4 3 4 1 4 u 1 4 3 4 u 1 4 1 4 ti u 1 4 1 4 u 3 4 1 4 u 1 4 3 4 fl i i U 4 4 1 f, 1 4 U 4 1 1, 1 4 U 4 1 1, 1 4 U 4 i fl 1 4 U 4 ,,_|-1 1 i- l^ I 2, 4, 4> 1 — ll 1 1- 2 Uj 4j 4> l_n 1 1- 2 LI, 4, 4> „J_1 1 1. l-l I 2j 4, 4> 1 ii-l-l 1- 4, ^^ I 2j 4, 1 l_n 1- 4, 2 ^,4, 1 1_,] 1- 4 > 2 <-l> 4 » 1 u_l_l 1- 4, Ui-2, 4, 1_11 1 1- 2 ", 4, 4, ,l_Ll 1 1. UT^2, 4, 47 n_|_l 1 1- UT^27 4» 4> l_il 1 1- 2 lA; 4 7 4 > 1 l — i] 1- 47 2 1^7 4 7 f7 U + l, h u+l. 47 1. 47 i— 11 i- 2 U, 4, u+^; ^-u; i-u; u+^; 5-u; 1 1 n-i-l- 47 47 ^^2f 1 3 ,i4.1. 47 47 l^ I 27 3 1 1 11 47 47 5 — U. 1 1 47 47 1 1 47 47 i 1 47 47 1 1 47 47 1 1 47 47 112 SPECIAL CASES OF THE CUBIC SPACE-GROUPS. FORTY-EIGHT Equivalent Positions.— Continued. (48b) (48c) Ouv Ouv Ouv Ouv vOu vOu vOu vOu uvO uvO uvO uvO uOO uOO OuO OuO OOu OOu l-u 1 i) 1 i-u, v; , v; i u+l, v; h 2-u, v; ^-v, I, u; v+i i u; u+i v+i ^-u, |-v, u-Hl, i-v, i-u, v+i u+l, i 0; l-u, i 0; h u+i 0; i l-u, 0; 11,-1. 2 2 U, i. 1 1 • 4,) if 1. 4> i u, v+l; 1 f, 1 _ h u, l-v; v; i u, v-f-l; v+i, 0, u+i 0, u-hl, v-l-^; 0, i— u. ^— v: ■2> l-v, 0, ^-u i-v, 0, u-Hi v+h 0, i-u u+l, V, i l-u, V, § u+i V, ^ l-u, V, ^ u+i 0, I u, 0, I 1 2 1 n 1 2 U^ 1 ii 1 • u+i i-v; V, i i-u; u, v+i i; u, h V, i; ti, v-|-§, It u|i Q^ 4) f-u, 3 1 i> h 0, u-f-l; 3_„ 1 4 "^J 4> 3 0, 0, 0, 0, 2 '^J 2 ) 3 . 4> 3 3.. 7 4; 4> -U 1 ii_|_i 1- 4> "14) 4> ii_l_3 3. 1. ■n_l_3. 1 3. n_I_l 3 3. U-r4> 4> 4) U-t-4, 4, 4, U-t-4, 4, 4, 1 3_,, 1. 3 l_,i 3. ' - 4> 4 ^i if 4) 4 "^j 4j 3 „ J^3 1 . 3 ,,_ll 3 . " I 4j 3 n_|_3 1 4> U-^4, 4, 3 4j 3 1 4) 4> 3 4» 4 U, 1 3_,, 3. 4j 4 *^i 4j 1 ii_]_3 3.. J) ^"Ti, 4) 1 3 3_,,. 4j 4» 4 "> (48d) 4j 4) 4 U> 4) 4> 4 U> 4j 4) 4 ^j 4j 4» 4 ^J 1 1 -ii-Ll- 3 3 ,,_Ll. 3 1 ,,_J_3.. 1 3 -, I 3 4> 4> U-hj, 4, 1, U-I-4, 4, t, U-f-f, 4, :j, Ui-:f. uuv; utiv; tiuv; tiuv; vuu; vuti; vuu; vuu" uvu; uvu; uvu; uvti u+l, u+t, v; u-Hl, ^-u, V v+i, u+5, u; |-v, u+§, u u+l, v-l-iu; ^-u, i-v, u (48e) u+l, u, v+l v+l, u, u-hl U+5, V, u-|-§ u, u-hi, v-i-| V, u+l, u+^ u, v-h^, u-f-^ uOO OuO; OOu; uOO; OuO; OOu: nJ_i 1 -1- |-v, u, l-u ^-u, v,u-}-i u, l-u, l-v V, u-hl, ^-u u,^-v, u+l §-u, u+iv; ^-u, ^-u,v 4 — \T i — 11 n* v-4-i i — 11 ii _ V, |-u, u; v-f-i, |-u, u u+l, f-v, u; ^-u, v-M, ii |-u, u, |-v; l-u, u, v-l-l |-v, ti, u+i; v+l, u, |-u ' u; l-u, V, l-u V, ^-u, l-u u-Fi V, ^ 11,-,. 2 2 *i> uO^; Oui; . -, cr, ^-u, \ \, 0, l-u; 0, i ^-u u^O; u+i 0, 0, u-f-l, 0; HO; SPECIAL CASES OF THE CUBIC SPACE-GROUPS. 113 FORTY-EIGHT Equivalent Positions.— Confmucd. 1, h u+l; 0, 0, u+i; 0|u; ^Ou; 2 > 2 ^J 2 > (48f) 2) 5> 2"~u; 0, 0, 2 (48g) (48h) utiO Ouu uOu uuO Ouu uOu uuO Ouu uOu uuO Ouu uOti uu^ |uu u|u uu| |uu u^u uu^ |uu u|u uui^ |uu u^u u-l-i i-u, 0; i u+i u l-u, I, u u+i u+i I, u-l-i u u+i I, u u, l-u 1 2 i |-u, u ^-u, i, u |-u, u-l-l i, §-u, u u+^, ^-u 0, u+l, u §— u, 0, u u-fl, u-f-| 0, u+l, u u+i 0, u |-u, ^-v 0, ^-u, u |-u, 0, u |-u, u-M 0, ^-u, u u+^, 0, u 0, i-u, 0; iuO; tiiO; l-u, 0, 0; 0|u; |0u. 2> u, ^— u; 0, u+§, l-u; 0; l-u, 0, u+i; u, i u+^; u+l, u, ^; u, u+i ^; I, u, u+l; 0, u-l-^, u+l; u+l, 0, u-\-i; u, i u+§; 0; 0; 1 . 2 > 1. 2 > 1. 2 > 1 . 2 > 2— U, U, ^, u, ^-u, ^; I, u, i-u] 0, ^-u, |-u; i-u, 0, |-u; u, i ^-u; ^-u, u, I; u, u+l, I; 0, ^-u, u+§; u, f-u, 0; 1 ii_i_i 1 11- 1) u-f-^, 5 — u, u, t-u, t; i u, i-u; 4-u, i u; u+i f-u, i; u, u+f, I; I, u, u+f; u+f, h u; u+l, u+i I; i u, u+§; u+i 0, ^-u; u+i Q, 0; 0, u, |-u; i-u, i u+l; u, 0, u+i; u+l, u, 0; u, u+l, 0; 0, u, u+l; I, u+l, u+l; u+i i u+l; u, 0, u+^; i-u, u, 0; u, ^-u, 0; 0, u, ^-u; i l-u, ^-u; ^-u, h i-u; u, 0, |-u; ^-u, u, 0; u, u+^, 0; 0, u, u+l; i ^-u, u+l; u+i i l-u; u, 0, l-u. u, u+f, 1; i u, u+f; u+f, h u; ^-u, u+i, i; 8> ^~u, u+4; u+i I, u; u+i i-u, t; u+l, u+f, I; ^-u, f-u, f ; |-u, u+f, f ; i u+l, f-u; f, u+iu+f; u+f, f, u; f-u, f, u; u, i-u, I; I, u, f-u; f-u, I, u; 2 U, 4 U, 8, ii-u, f-u; l-u, I, ti; i u, f-u; f, u, u+f; f, u, f-u; f, u, u+f; f-u, iu+^; u+f, I, u+l; f-u, |,|-u; u+f,i|-u; u, f -u, I; u, u+f, I; u, f-u, t; u, u+f, f; iu+if-u; I, u+iu+f; |, f •u; i l-u, u+f; (48i) f-u, I, u+l; u+f, I, u+l; f-u, f, |-u; u+f, f, |-u. fOO; f|0; fO|; ii f 0; Of 0; OfO: IfO; HI; 4 2^, 4 " 2 , i 3 n- i 1 !• 2 4 "> 2 ¥ 2 I 4 1 1; 3 1 !• 4 2 2, n 3 1 . '-'4 2. 114 SPECIAL CASES OF THE CUBIC SPACE-GROUPS. FORTY-EIGHT Equivalent Positions.— Con^wwed. OOi; 111. 2 2 4; HI; OH; OOf; 113. 2 2 4; Hi; OH; Hi; 3 11. 4 2 4; 3 n 3. 113. 4 '-' 4; 4 2 4; Hi; 3 1 • 4^4; 3 13. 1 n 3 . 4 2 4; 4 " 4 ; 'J 4 4; 13 1. 2 4 4; 113. n 3 3 . 2 4 4; 'J 4 4; 111. 2 4 4; nil. "J 4 4; All- 13 3. '-'4 4; 2 4 4; i|0; 3 3 f). 4 4^; 3 11. 13 1. 4 4 2; 4 4?; 111. 4 4 2; 3 3 1. 4 4 2; 3 1 n. 1 3 n 4 4 U, 4 4 U. (48j) Ouv Ouv; i u+i v+^; i u+i Hv; vOu vOu; v+i i u+l; l-v, i u+l; uvO uvO; u+l, v+i 1; u+i l-v, ^; uOv aOv; 1 nil ,,• 2""U, 2; 2~V, ^-u, i v+^; Ovu Ovu; i l-v, ^-u; i v+i |-u; vuO vuO; i-y, Hu, 1; v-l-i 1 — 11 i- V-r2; 2 ^) 2; Ouv Ouv; i l-u, Hv; 1 1 n ir-Ll- 2, 2— U, V-|-2, vOu vOu; i-v, i l-u; v+l, i |-u; uvO uvO; Hu, Hv, 1; |-u, v+l, ^; uOv uOv; u+i i v+i; u+i i i-y; Ovu Ovu; i v+i u+^; i, l-v, u+l; vuO vuO; v+i u+i 1; ^-v, u+i, 1; (48k) UU V ; u u V ; u+i u+i v+l; i-u, u+l, v+^; vuu ; vuu ; v+i u+i u+i; v+l, Hu, u+l; u vu ; u V u ; u+i v+i u+l; u+i v+l, Hu; u ti V ; u u V ; Hu, Hu, l-v; l-u, u+i Hv; u vu ; u V ti ; Hu, l-v, l-u; u+i l-v, |-u; vuu ; vtiu ; i-v, ^-u, l-u; l-v, Hu, u+l; UU V ; UU V ; u+i l-u, Hv; Hu; l-u, v+l; vuu ; vuu ; i-v, u+i l-u; v+i Hu, l-u; U V u ; ti vti ; l-u, |-v, u+l; |-u, v+i |-u; uti V ; uu V ; u+i Hu, v+l; u+i u+i |-v; u vu ; u V u ; ^-u, v+i u+^; u+l, |-v, u+l; vuti ; vuu ; v+l, u+l, ^-u; |-v, u+l, u+|. (481) u, u- j_i 1 • r2; 4; u, ^-u, i; ti, u+i 3. f, 1_,, 1. 4 ; "; 2 i^; 4 ; i u, u+^; f; u, ^-u; i u, u+l; i u, ^-u; u+i h u; ^-u, f, u; u+i i u; l-u, i, a; u+i u, i; ^-u, u, 1; u+l, ti, f; l-u, ti, i; u, i u+i; u, i ^-u; u, f, u+^; u, I, |-u; iu^ hi u; f, i-u, u; f, u+i u; i i-u, u; u, ^- -u, f; u, u+l, i; u, l-u. i; u, u+l, 1; f , u, l-u; i, ti, u+l; i, u, 1- -u; i u, u+l; ^-u, f; ti; u+i i ti; ^-u, i, u; u+l, f, u; ^-u, U; f; u+i ti, i; |-u, u, i; u+l, u, f; u, f, l-u; u, i, u+l; u, i, ^- -u; u, f, u+l; f, ^ -u, u; i u+i ti; i ^-u, u; f, u+l, u; SPECIAL CASES OF THE CUBIC SPACE-GlROtJPS. Il5 FORTY-EIGHT Equivalent Positions.— Con^mwed. (48m) uO|; l,_Li 1 3, l-u, 0, f; ,-,11. "2 4; iuO; i u+l, i; i l-u, 0; |u|; Oiu; i i u+l; 0, i l-u; §ia; i i-u, 0; 3 3_,, 1. 4> 4 ^> 2 } i u+i 1; i u+i i-u, 0, h 3_„ 1 3. 4 ^) 2) 4} u+i h i; u+i 0, f 0, h i-u; 1 3 3_,,. 2) 4> 4 l^; 1 1 11-4-3. 2j 4> LI 14? 0, i u+i tiOf; i — 11 A i- 2 "> 2> 4; „ 1 3. U 2 4; u+i 0, i |Q0; i l-u, 1; fu|; i u+i Ofti; i i ^-u; Hu; 0, i U+I h u+i, 1; 1, u+f, 0; i f-u, 0; 3 1 „ 1 4 J 4— U, ^ u+i, i h u+i 0, f; f-u, 0, I; i — 11 i ^ 4 ^} 2> i i i u+i; 0, f, u+l; 0, i f-u; 1 3 1_,, 2> 4> 4 U (48n) u, u+i i; % i-u, i; u, f-u, f; u, U+i f i, u, u+J; i u, i-u; i u, f-u; i u, U+f u+i i u; i-u, i u; f-u, i u; u+i i a u+i i-u, I; l-u, u+i I; u+i u+i f ; |-u, f-u, f ; 8, u+2, 4— u; 8, 2— u, u+j; ■§, u+^, u+4; g, 2— u, 4— u; 4 — u, 8) u+2; u+4, 8, 2— u; u+4, 8> u+2; 4— u, g, 2— u; u, f-u, I; u, u+i f; u+i f-u, i; u, u+f, |; i u, f-u; i u, u+i i u+i f-u; i u, u+f; f-u, I, u; u+i f, u; f-u, i u+|; u+i |, u; l-u, i-u, I; |-u, u+f, i; u, i-u, f ; u+i u+i f; i|-u, j-u; i, l-u, u+f; i u, f-u; i u+i u+f ; f-u, i i-u; u+ii|-u; f-u, i u; u+f , i u+i SIXTY-FOUR Equivalent Positions. (64a) uuu; uuti; uuu; tiuu; tiuu; tiuu; uuu; uuu; |-u, |-u, |-u; |-u, u+i u+i u+i i-u, u+i u+i u+i i-u; u+i u+i u+i u+i l-u, i-u; l-u, u+i |-u; l-u, |-u, u+i u+i u+i u; u+i|-u, u; ^-u, u+iu; |-u, |-u, u; |-u, |-u, u; |-u, u+iu; u+i|-u, u; u+i u+i u; u, u, |-u; u, u, u+i u, u, u+i u, u, |— u; u, u, u+i u, u, |-u; u, u, i-u; u, u, u+i u+i u, u+i u+iti, |-u; |-u, u, |-u; |-u, u, u+i |-u, u, i--u; |-u, u, u+i u+i u, u+i u+i u, |-u; u, |-u, u; u, u+i u; u, |-u, u; u, u+i u; u, u+i u; u, |-u, u; u, u+i u; u, |-u, u; u, u+i u+i u, |-u, §-u; u, u+i|-u; u, |-u, u+i u, l-u, |-u; u, u+i u+i u, l-u, u+i u, u+i|-u; |-u, u, u; |-u, u, u; u+i u, u; u+i u, u; u+i u, u; u+i u, u; |-u, u, u; |-u, u, u. 116 SPECIAL CASES OF THE CUBIC SPACE-GROUPS. SIXTY-FOUR Equivalent Positions. — Continued. (64b) uuu; uuu; uuti; utiu; l-u, i-u, i-u; u+i J-u, u+i; |-u, u+J, u+i; u+i, u+l, J-u f-u, f-u, f-u; l-u, u+f, u+f; u+|, f-u, u+f; u+f, u+f, f-u u+i, u+l, u+l; |-u, u+l, |-u; u+^, |-u, ^-u; ^-u, ^-u, u+i u+^, u+l, u; u+l, §— u, u; |-u, u+§, u; |-u, |-u, u f-u, f-u, i-u; u+f, f-u, u+i; f-u, u+f, u+f; u+f, u+f, f-u f-u, f-u, f-u; f-u, u+f, u+f; u+f, f-u, u+f; u+f, u+f, f-u u, u, u+l; u, u, ^-u; u, u, |-u; u, u, u+|; u+l, u, u+l; u+l, u, ^-u; |-u, u, ^-u; |-u, u, u+| f-u, f-u, f-u; u+f, f-u, u+f; f-u, u+f, u+f u+f, u+f, f-u f-u, f-u, f-u; f-u, u+f, u+f; u+f, f-u, u+f; u+f, u+f, f-u u, u+l, u; u, u+l, u; u, §-u, u; u, |-u, u; u, u+^, u+l; u, ^-u, |-u; u, u+|, |-u; u, ^-u, u+| f-u, f-u, f-u; u+f, f-u, u+f; f-u, u+f, u+f u+f, u+f, f-u f-u, f-u, f-u; f-u, u+f, u+f; u+f, f-u, u+f; u+f, u+f, f-u u+l, u, u; |-u, u, u; u+|, u, u; |— u, u, u. NINETY-SIX Equivalent Positions. (96a) Ouv Ouv Ouv Ouv; vOu vOu vOu vOu; uvO uvO uvO uvO; uOv uOv uOv uOv; Ovu Ovu Ovu Ovu; vuO vuO vuO vuO; h u+i v; v+i I, u; 1 1 —11 1) 2 U v; l-v, i u; h u+l, v; 2-v, I, u; h, l-u, v; v+l, I, u; u+l, v+l, 0; l-u, |-v, 0; u+|, |-v, 0; |-u, v+|, 0; |-u, I, v; u+l, I, v; |-u, |, v; u+|, 0, v; I, |-v, H; I, v+l, u; |, v+|, u; |, |-v, u; |-v, l-u, 0; v+l, u+l, 0; v+|, |-u, 0; |-v, u+|, 0; I, u, v+l; I, u, |-v; |, u, |-v; |, u, v+|; v+l, 0, u+l; l-v, 0, |-u; |-v, 0, u+|; v+|, 0, |-u; u+l, V, I; l-u, V, I; u+|, v, |; |-u, v, |; l-u, 0, |-v; u+l, 0, v+l; |-u, 0, v+|; u+|, 0, |-v; I, V, u+l; I, V, l-u; |, v, u+|; I, V, l-u; SPECIAL CASES OF THE CUBIC SPACE-GROUPS. 117 NINETY-SIX Equivalent 'Posttio^s.— Continued. ^-v, u, I; v+l, u, i; 0, u+iv-1-^; 0, l-u, |-v; V, h u+l; V, i ^-u; u, v-l-l, ^; ti, i ^-v; 0, ^-v, ^-u V, l-u, I; (96b) uuv; utiv vuu; vuu uvu; tivu utiv; uuv uvu; uvu vuu; vuu u+i u+^, V v+iu-l-iu u-h^, v-l-^, u ^-u, ^-u, V l-u, ^-v,u l-v, i-u,u u+l, u, v+l v+l, u, u+l u-}-|, V, u+l i-u, u, |-v 1-u, V, l-u ^-v, u, |-u; v+^, u, |-u u, u+iv+^; u, i-u, i-v v,u-H|, u+l; V, u+l, i-u u, v+^, u+^; u, |-v, u+l u, ^-u, i-v; u, ^-u, v+l u, ^-v, |— u; Q, v+^, u+^ ^^, ^-u, i-u; V, u+ii-u (96c) uuO; u+i u, i; Ouu; ^, u, u-f-|; uOu; u+^, 0, u+l; u, i-v, i; u, i v+l; 0, v+i u+l; V, u+i I; utiv vtiii ti V ti uuv uvu vuu u+t, t-u, V i-v, u+iti ^-u, |-v,u u+i |-u, V |-u, v+iu v+iu+iti u+i u, i-v |-v, u, i-u ^-u, V, U+I u+i u, v+l |-U, V, U+I v+i u, I; l-v, u, I; 0, u+i|-v; 0, l-u, v+l; V, i u+i V, i l-u; u, l-v, i ti, v+i i- u, i v+i u, i |-v; 0, v+i |-u; 0, l-v, u+i V, l-u, i V, u+i i uuv V tiu u V ti tiu V u V ti vtiu atiO; |-u, u, i 1 . l-u, 0, 1) u, ^- 1 2 tiOti; ti ti; l-u, l-u i l-u, l-u l-u, i l-u u+i u+i I u+i i U+I i u+i U+I l-u; u; ti, |-u, 0; 0, l-u, ti; u|u; u, u+i 0; u|u; 0, u+i u; |-u, u+iv; |-u l-v, |-u,u u+i |-v,ti l-u, u+iv u+i v+i u v+i|-u, u l-u, u, l-v |-v, U, U+I u+i V, l-u l-u, U, V+I u+i V, l-u V+I, ti, U+I u, u+i l-v V. |-u, U + I u, l-v, l-u u, u+i V+I u, v+i l-u V, l-u, U+I uuO Ouu uOu uuO Ouu uOu |— u, U+I, I i l-u, U+I u+i I, |— u u+i l-u, I i u+i l-u |— u, i U+I 1 ) 2 U, V V+I, l-u, u l-u, v+i u u+i u+i V u+i |-v,u l-v, u+i u l-u, U, V+I v+i ti, l-u l-u, V, l-u u+i u, |-v U+I, V, U+I l-v, U, U+I '-' i-u,v + | 1 U, 2 1 V, t-u, f-u u, V+I, l-u u, U+I, l-v u, l-v, U+I V, u + i U + I u+i ti, I; I, u, l-u; l-u, 0, U+I; l-u, u, I; I, ti, U+I; u+i 0, l-u; ti, u+i 0: 0, |-u, u; u|u; u, |-u, 0; 0, u+i ti; u|u; 118 SPECIAL CASES OF THE CUBIC SPACE-GROUPS. NINETY-SIX Equivalent Positions. — Continued. M+h u-l-l, 0; u, u+i I; u+|, |-u, 0; u, i-u, |; i u+^, u; 0, u+i i-u; l-u, i, u; u, J, u+l; l-u, u+l, 0; u, u+l, ^; i i-u, u; 0, ^-u, u+l; uu§; |-u, u, 0; 0, u, u+§; |uu; u, 0, |-u; u-}-|, 0, u; i u+i u; 0, u+l, u+l; u+i i u; u, i u+l; |-u, l-u, 0; u, |-u, i; ^-u, u, 0; 2 u> 2> u; 2» 2~u, u; 0, u, ^-u; u, 0, |-u; l-u, 0, u; u+i u, 0; u, 0, u+l; u+l, 0, u; 0, u, u+l; ^uu; (96d) uvi; iuv; 2 "? 2 V, 4 1 vuf; ufv; f vQ; uvf; f uv; u iv; 1 4 vu; uvf; f uv; vfu; vui; ujv; ?vu; uvi; iuv; viu; i-v, i l-u v+i u+i I u+i i v+i I, v+i u+i l-u, v+i f f, l-u, v+l vf u; v+i f, l-u vui; |-v, u+i i u+i i ^-v i |-v, u+i i u+i l-v i-v, f, u+l v+i i-u, i ^-u, i v+l i, v+i ^-u u+i v+i i i u+l. v+l v+i i, U+I vuf; l-v, i-u, I ufv; l-u, f, l-v f vu; f, |-v, l-u (96e) uuv; utiv; uuv; u ti v; vuu; vuu; vuu; vuu; uvu; uvu; uvti; uvti; l-u, i-u, f-v; u+f, f-u, v+i; f-u, u+f, v+|; u+i u+f, f-v; f-u, i-v, l-u; f-u, v+i u+i; u+f, v+i f-u; u+i i-v, u+i utii' u+i u, 0; 0, u, i- ■u, |uu; u, 0, u+i |-u, 0, u. i u+i V ; i u, l-v; v+i f, u ; V, i U+I; i-u, i V ; u, f, V+I; i l-v, u h V, l-u; i u+i V ; i u, V+I; l-v, i u ; V, f, U+I; l-u, i V u, i ^-v; i v+i u h V, l-u; 1 1 n 4> 2 U, V u+i V, f ; v+i i u l-v, u, i u+i f, V u+i V, i; f, l-v, u v+i u, f ; 3 1_„ 4j 2 ^} V l-u, V, f; h-y, I u l-v. u, f; u+i i V l-u, V, f; i v+i u v+i u, f; f, u, i- •V u, l-v, f; ■Tr 3 1 V, i, 2- -u V, u+i I; u, i, V+I u, v+i f; i V, U+I V, u+i f ; i u, v+i u, l-v, f; V, i 1- ■u, V, l-u, f; n 3 1 U, "4, 1- ■V, u, v+i f ; f, V, U+I; u, l-u, f. SPECIAL CASES OF THE CUBIC SPACE-GROUPS. 119 NINETY-SIX Equivalent Positions. — Continued. i-v, l-u, i-u; v+l, u+i i-u; v+i, i-u, u+i; l-Y, u+i u+i; u+l, u+l, v; u+i|-u, v; ^-u, u+|, v; |-u, |-u, v; v+l, u+iu; l-v, u-f-|, u; ^-v, ^-u, u; v+|, ^-u, ti; u+iv+iu; l-u, l-v, u; u+i|-v, u; ^-u, v+iu; f-u, f-u, i-v; u+f, f-u, v+i; f-u, u+f. v+|; u+f, u+f, T-v; f-u, f-v, i-u; f-u, v+f, u+i; u+f, v+f, i-u; u+f, f-v, u+i; f-v, f-u, i-u; v+f, u+f, l-u; v+f, f-u, u+i; f-v, u+f, u+i; u+l, u, v+l; u+iu, ^-v; |-u, u, |-v; ^-u, ti, v+|; v+iu, u+l; l-v, u, ^-u; |-v, u, u+^; v+i ti, i-u; u+^, V, u+l; ^-u, V, u+^; u+|, v, ^-u; ^-u, v, ^-u; f-u, i-u, f-v; u+f, i-u, v+f; f-u, u+i, v+f; u+f, u+i f-v; f-u, l-v, f-u; f-u, v+i u+f; u+f, v+i f-u; u+f, i-v, u+f; f-v, i-u, f-u; v+f, u+f, f-u; v+f, i-u, u+f; f-v, u+f, u+f; u, u+iv+§; u, ^-u, ^-v; u, u+i|-v; u, ^-u, v+^; V, u+l, u+l; V, u+l, |-u; v, ^-u, u+^; v. ^-u, |-u; u, v+iu+l; u, l-v, u+^; u, |-v, ^-u; u, v+i |-u; f-u, f-u, f-v; u+f, f-u, v+f; f-u, u+f, v+f; u+f, u+f, f-v; f-u, f-v, f-u; f-u, v+f, u+f; u+f, v+f, f-u; u+f, f-v, u+f; f-v, f-u, f-u; v+f, u+f, f-u; v+f, f-u, u+f; f-v, u+f, u+f. (96f) u, f-u, i; u, u+f, I; u, f-u, |; u, u+f, i; h u, f-u; I, u, u+f; |, u, f-u; i u, u+f; f-u, i, u; u+f, I, u; f-u, i u; u+f, i u; u+i f-u, i; u+i u+f, I; |-u, f-u, |; |-u, u+f, i; i u+i f-u; I, u+i u+f; f, ^-u, f-u; f, i-u,u+f; f-u, f, u; u+f, I, u; f-u, f, u; u+f, |, u; u+i f-u, f; u+i u+f, f; ^-u, f-u, |; ^-u, u+f, f; f, u, f-u; f, u, u+f; f, u, f-u; f, u, u+f; f-u, i u+i u+f, I, u+i f-u, i ^-u; u+f, i|-u; u, f-u, f; u, u+f, t; u, f-u, f; u. u+f, |; i u+i f-u; I, u+i u+f; |, |-u, f-u; i|-u, u+f; f-u, f, u+i u+f, I, u+i- f-u, I, l-u; u+f, f, ^-u; f-u, u, i; u+f, u, I; f-u, u, |; u+f, u, |; u, i f-u; u, I, u+f; u, i f-u; u. f. u+f; 120 SPECIAL CASES OF THE CUBIC SPACE-GROUPS. NINETY-SIX Equivalent Positions. — Continued. h i-u, u; f-u, u+i I; U+2> 8> 4 Uj h u+f, u; u+i I, u+f ; h i-u, u; l-u, ^-u, I; l-u, f, i-u; I, u+f, u; l-u, I, u+f; 1, l-u, u; i u+f, u; i f-u, u; i u+f, u; l-u, u+il; u+f, u+l, f ; f-u, ^-u. f; u+f, i-u,|; u, f, f-u; u, 1, u+f; u, i f-u; % I u+f; 1, f-u, u+l; 1, u+f, u+i; h f-u, | — u; iu+f,|-u; f-u, u, f ; u+f, u, 1; f-u, u, 1; u+f, u, 1; u+i 1, f-u; u+i 1, u+f; i-u, 1, f- u; l-u, iu+f; f, f-u, u+l; f, u+f, u+l; 3 1 „ 1 ¥, 4— U, 2 — u; i u+f,|-u. (96g) uOO; u+l, h 0; u+i 0, 1; u 2 2 , tiOO; ^-u, 1, 0; ^— u, 0, ^; uH; OuO; h u+i. 0; ^u|; 0, u+l, A; OuO; i ^-u, 0; Hi; 0, i-u, 1; OOu; Hu; i 0, u+l; 0, 1, u+l; OOu; Hu; 2, 0, i-u; 0, i, h-u; h f-u, f ; f, f-u, f; 3 1_,, 3. 4> 4 ", 4, h f-u, f; f, u+f, f ; 3 1,4.3 1. 4> '^14, 4, 3 „4_1 3. 4> "T^4, 4, h u+f, f ; 4 U> 4, 4> 3_,, 3 1. 4 11, 4, 4> 3 „ 1 3 . 4 — U, 4j 4, i_ii 3 3. 4 ^) 4> 4> U+f, f, f ; il_J_3 3 1. ^lif 4, 4, U+f, f, f ; U+f, f, f ; i, h i-u; f, i f-u; 3 1 3_„. 4> 4> 4 >^, 1 3 3_,,. 4> 4> 4 ", i i u+f; f, f, u+f; f, h U+f; h f, U+f; f-u, f, f; f-u, f, f; f-u, f, f; f-u, f, f; u+f, f, f ; u+f, f, f ; u+f, f, f; u+f, f, f; f, f-u, f; h f-u, f; f, f-u, f; f, f-u, f; f, u+f, f ; h u+f, f ; f, u+f, f; i u+f, f ; 3 3 3_,,. 4> 4, 4 "> f, f, f-u; h f, f-u; 3 1 i_„. 4> 4> 4 ", f, f, U+f; 1 1 ii_L3. 4> 4, U-r4, f, i u+f; f, f, U+f; h u+i, 1; Oui; 0, u+^, 0; iuO; 2) 3 U, 2J OQA; 0, i-u, 0; itiO; u+l, h ^; uOJ; uiO; u+l, 0, 0; l-u, i i; uO|; QH; l-u, 0, 0; h h, u+l; 0, 0, u+l; O^u; |0u; i i l-u; 0, 0, i-u; 0|u; |0u; (96h) u, f-u, J; u, u+f, 1; u, f-u, 1; u, u+f, i; i u, f-u; 1, u, u+f; 1, u, f-u; i u, u+f; f-u, i u; u+f, h u; f-u, 1, u; u+f, i u; u+if-u, i; u+i u+f, 1; i-u, f-u, 7 . 8j l-u, u+f, 1; 1, u+l, f-u; f,u+|, u+f; ii-u,f- u; f,|-u,u+f: f-u, f, u; u+f, 1, u; f-u, f, u, u+f, 1, u; u+i f-u,|; u+i u+f, 1; i-u, f-u. 1; |-u,u+f, t; i u, f-u; f, u, u+f; i u, f-u, i u, u+f; f-u, |,u+^; u+f, |,u+|; f-u, 1,1- u; u+f, il-u; THE CUBIC SPACE-GROUPS T -T.^ 121 NINETY-SIX Equivalent Positions. — Continued. u, l-u, f; u, u+i, I; u, f-u, |; u, u+i, f; iu+if-u; I, u+iu+i; |, |-u, f-u; ii-u,u+i; i-u, |,u-f-|; u+f, f,u+i; i-u, f, §-u; u+f, f,^-u; f-u, u+l, I; u+iu+l, I; f-u, |-u, f; u+i,|-u, |; u+ii f-u; u+it,u+i; |-u, |,f-u; |-u,f, u+i; I, f-u, u+l; t,u+iu-f-|; f, f-u, |-u; f,u+i,^-u; f-u, u, I; u+f, u, f; f-u, u, f; u+f, u, f; u, i f-u; u, I, u+f; u, |, f-u; u, i u+f; if-u, u+l; I, u+f, u+l; |, f-u, |-u; i,u+f,|-u; f-u, u, I; u+f, u, I; f-u, u, |; u+f, u, 1; u+iif-u; u+i|,u+f; ^-u, |, f-u; |-u, iu+f; t, f-u, u; I, u+f, u; f, f-u, u; f, u+f, u; f-u, u+l, I; u+f, u+il; f-u, |-u, |; u+f, |-u, |; u, i f-u; u, f, u+f; u, f, f-u; u, |, u+f; i f-u, u; I, u+f, u; |, f-u, u; i, u+f, u; A. TETARTOHEDRY. Space-Group T*. One equivalent position : (a) la. (b) lb. Three equivalent positions: (c) 3a. (d) 3b. Four equivalent positions: (e) 4a. Six equivalent positions: (f) 6a. (h) 6c. (g) 6b. (i) 6d. Twelve equivalent positions: (j) xyz; xyz; xyz; xyz; zxy; zxy; zxy; zxy; yzx; yzx; yzx; yzx. Space-Group 'P. Four equivalent positions: (a) 4b. (c) 4d. (b) 4c. (d) 4e. Sixteen equivalent positions: (e) 16a. Twenty-four equivalent positions: (f) 24a. (g) 24b. 122 THE CUBIC SPACE-GROUPS T^-T*. xyz; xyz; zxy; zxy; yzx; yzx; x+i h-y, z; l-x, y+i z l-x, |-y, z; l-z, x+iy; h-z, l-x, y z+i l-x, y; h-y, ^-z, x; y+i 5-z, X i-y, z+i x; x+l, y, i-z; l-x, y, l-z ^-x, y, z+l; l-z, X, l-y; ^-z, X, y+i z+i x, l-y; i-y, z, x+i; y+i z, l-x h-y, z, i-x; X, h-y, l-z; X, y+i i-z X, |-y, z+l; z, x+i l-y; z, i-x, y+i z, 2"~x, 2~yj y, ^-z, x+i; y, i-z, ^-x y, z+i §-x. Space-Group T^ (continued). Forty-eight equivalent positions: (h) xyz; xyz; zxy; zxy; yzx; yzx; x+i y+i z; z+i x+iy; y+i z+i x; x+i y, z+i z+l, X, y+i y+i z, x+^; X, y+l, z+l; z, x+i, y+i y, z+i x+li Space-Group T'. Two equivalent positions: (a) 2a. Six equivalent positions : (b) 6e. Eight equivalent positions: (c) 8a. Twelve equivalent positions : (d) 12a. (e) 12b. Twenty-four equivalent positions : (f) xyz; xyz; xyz; zxy; zxy; zxy; yzx; yzx; yzx; x+l, y+l, z-}-^; x+i l-y, |-z; J-x, y-\-^, \-z; ^-x, ^-y, z-Fl; z+i x-l-l, y-t-l; \-z, x-hi ^-y; §-z, |-x, y+|; z+i, l-x, l-y; y+i z-j-i x+^; i-y, ^-z, x+l; y-f-^, J-z, |-x; i-y, z+l, ^-x. Space-Group T*. Four equivalent positions: (a) 4f. Twelve equivalent positions: (b) xyz; \-\-\, ^-y, z; x, y-\rh, |-z; ^-x, y, z+^; zxy; z, x+^, |-y; |-z, x, y+^; z-F^ \-x, y; yzx; l-y, z, x+^; y-f-J, ^-z, x; y, z-f-^, ^-x. xyz; zxy; yzx; THE CUBIC SPACE-GROUPS T^-l^. 123 Space-Group T*. Eight equivalent positions: (a) 8b. Twelve equivalent positions : (b) 12c. Twenty-four equivalent positions: (c) xyz; X, y, |-z; |-x, y, z; x, |-y, z; zxy; l-z, X, y; z, ^-x, y; z, x, |-y; yzx; y, §-z, x; y, z, |-x; |-y, z, x; x+i y+i z+^; x+i |-y, z; x, y+i |-z; i-x, y, z+^; z+i x+i y+^; z, x+i ^-y; |-z, x, y+§; z+2> 5"~x, y; y+i z+i x+^; ^-y, z, x+^; y+^, ^-z, x; y, z+i ^-x. B. PARAMORPHIC HEMIHEDRY. Space-Group T^. One equivalent position: (a) la. (b) lb. Three equivalent positions: (c) 3a. (d) 3b. Six equivalent positions : (e) 6a. (g) 6c. (f) 6b. (h) 6d. ^tgf/i< equivalent positions: (i) 8c. Twelve equivalent positions: (j) 12d. (k) 12e. Twenty-four equivalent positions: (1) xyz; xyz; xyz; xyz; zxy; zxy; zxy; zxy; yzx; yzx; yzx; yzx; xyz; xyz; xyz; xyz; zxy; zxy; zxy; zxy; yzx; yzx; yzx; yzx. Space-Group T^. Two equivalent positions: (a) 2a. Four equivalent positions: (b) 4d. (c) 4e. 124 THE CUBIC SPACE-GROUPS T^-T^. Space Group Ti (continued). Six equivalent positions: (d) 6e. Eight equivalent positions: (e) 8d. Twelve equivalent positions: (f) 12a. (g) 12b. Twenty-four equivalent positions: (h) xyz; xyz; xyz; zxy; zxy; zxy; yzx; yzx; yzx; xyz; zxy; yzx; 2 X, 2 Yf 2 Zj 2 2, f X, 2 -x, y+l, z+^; x+l, i-y, z+|; X+2> y+2> 2 ■y; z+l, l-x, y+l; z+i x;+i |-y; •z; ■y, 2-z, 2 ^) 2— Z, X+2, y+2i y+i z+i l-x; ^-y, z+i x+§; yi2) 2~Z, x+2. Space-Group T^ Four equivalent positions: (a) 4b. (b) 4c. Eight equivalent positions: (c) 8e. Twenty-four equivalent positions: (d) 24c. (e) 24a. Thirty-two equivalent positions : (f) 32a. Forty-eight equivalent positions: (g) 48a. (h) 48b. Ninety-six equivalent positions: (i) xyz; zxy; yzx; xyz; zxy; yzx; xyz; zxy; yzx; xyz; zxy; yzx; x+i y+i z; z+i x+i y; y+h z+i x; j-x, i-y, z; i-z, i-x, y; l-y, h-z, x; xyz; zxy; yzx; xyz; zxy; yzx; x+l, h- xyz; zxy; yzx; xyz; zxy; yzx; •y, z l-z, x+iy 2~y> 2~z, X l-x, y+i z z+i ^-x, y y+i z+i X 2-x, y+l, z i-z, ^-x, y y+i?-z, X x+l, |-y, z z+i x+i y h-y, z+l, X J-x, l-y, z; z+i^-x, y; |-y, z+i x; x+J, y+l, z; l-z, x+i y; y+l, l-z, x; THE CUBIC SPACE-GROUPS I?-tJ. 125 Space-Group T^ (continued). x+i y, z+l; x+i y, l-z ^-x, y, l-z l-x, y, z+l z+i X, y+l; l-z, X, ^-y l-z, x, y+l z+i X, l-y y+i z, x+i; i-y, z, x+l y+i z, §-x l-y, z, l-x ^-x, y, i-z; l-x, y, z+l x+i y, z+i x+i y, l-z ^-z, X, ^-y; z+l, X, y+l z+l, X, l-y l-z, X, y+l l-y, z, l-x; y+i z, l-x i-y, z, x+l y+l, z, x+l X, y+i z+l; X, i-y, ^-z X, y+i l-z X, l-y, z+l z, x+l, y+l; z, x+i i-y , z, i-x, y+l z, l-x, l-y y, z+i x+l; y, i-z, x+l y, i-z, l-x y, z+l, l-x X, ^-y, ^-z; X, y+i z+l X, h-y, z+l X, y+l, l-z z, l-x, i-y; z, ^-x, y+^ z, x+l, l-y z, x+l, y+l y, l-z, ^-x; y, z+i ^-x y, z+i x+l y, l-z, x+l Space-Group T^. Eight equivalent positions: (a) 8f. (b) 8g. Sixteen equivalent positions: (c) 16b. (d) 16c. Thirty-two equivalent positions: (e) 32b. Forty-eight equivalent positions: (f) 48c. Ninety-six equivalent positions: (g) xyz; xyz; xyz; zxy; zxy; zxy; yzx; yzx; yzx; x+l, y+l, z z+l, x+l, y y+l, z+l, X x+l, y, z+l z+l, X, y+l y+l, z, x+l X, y+l, z+l z, X+l, y+l y, z+l, x+l xyz zxy yzx 4 X, 4 y, 4 x+|. l-y, z l-z, x+l, y l-y, l-z, X x+l, y, l-z l-z, X, |-y l-y, z, x+l X, |-y, |-z z, x+l, l-y y, l-z, x+l z; l-x, y+i 4~z, 4— X, 4— yj z+4, 4 X, i-y, i-z, i-x; y+i z+|, l-x, f-y, i-z; l-x, y+l, |-x, y+l, z; l-x, l-y, z; 2 — z, 2— X, y; z+2, 2 — X, yj y+l, l-z, x; l-y, z+l, x; |-x, y, l-z; l-x, y, z+l; l-z, X, y+l; z+l, X, |-y; y+l, z, l-x; l-y, z, |-x; X, y+l, l-z; X, l-y, z+|; z, 2— X, y+2; z, 2^- X, ^— y; y, l-z, l-x; y, z+l, l-x; z+i; x+i l-y, z+l; x+i y+i j-z; y+l; z+l, x+l, l-y; l-z, x+l, y+l; l-x; l-y, z+l, x+l; y+l, l-z, x+l; z+l; x+l, l-y, z+l; x+l, y+|. l-z; 126 THE CUBIC SPACE-GROUPS T^-Th. Space-Group T^ {continued). 4 z, 4 X, 4 y 4~y, 4 z, 4— X 4 X, 4 y? 4 z 4 z, 4 X, 4 y 4~y, 4 z, 4— X i-x, |-y, i-z 4~~z, 4— X, 4— y 4 y, 4 z, 4 X Space-Group T^. Two equivalent positions: (a) 2a. Six equivalent positions : (b) 6e. Eight equivalent positions : (c) 8e. Twelve equivalent positions : (d) 12a. (e) 12b. Sixteen equivalent positions : (f) 16d. Twenty-four equivalent positions ; (g) 24d. Forty-eight equivalent positions : z+i f-x, y+i z+i x+f, i-y; f-z, x-\-l y-hl; y+i z+i i-x, f-y, z-f-f, x+i; y+i f-z, x+i; 4 X, y+i z+f x+i i-y, z+f; x+i y+i f-z; Z + i l-x, y+f z+i x+i f-y; f-z, x+i y+f; y+i z+i f-x f-y, z+i x+f; y+i f-z, x+f; i-x, y+i Z + I x+i f-y, z+f; x+i y+i f-z; z+i l-x, y+f z+i x+f, f-y; l-z, x+i y+f; y+i z+i f-x ; l-y, z+i x+f; y+i f-z, x+i xyz; xyz; xyz; xyz; zxy; zxy; zxy; zxy; yzx; yzx; yzx; yzx; xyz; xyz; xyz; xyz; zxy; zxy; zxy; zxy; yzx; yzx; yzx; yzx; x+i y+i Z+I; x+i i-y, l-z; z+i x+i y+i; |-z, x+i ^-y; 2~x, y+^, 2 — z; ?-x, i-y, Z+I; l-z, |-x, y+l; z+i | — X, 2~yi THE CUBIC SPACE-GROUPS T^-TI,. 127 Space-Group T^ {continued). y+i z+i x+l; |-y, |-z, x+|; 2~x, 2~y> 2~z; -J— X, y+^, z+2J 2~Z, 2"~X, 2~y> 2 + 2, 2~X, y+2J i~y> 2~z, 2~x; y+2> z+2, 2~x; Space-Group TJ. Fowr equivalent positions: (a) 4b. (b) 4c. Eight equivalent positions : (c) 8h. Twenty-four equivalent positions: y+2> 2~z, 5— x; h-y, z+i §-x; x+i |-y, z+l; x+i y+h l-z; z+i x+i l-y; 2~Z, X+2> y4"2J l-y, z+i x+2; y"r2> I~Z, X-|-2. (d) xyz; x+i |-y, z zxy; z, x+i ^-y yzx; |-y, z, x-|-^ xyz; l-x, y+i z zxy; z, |-x, y+^ yzx; y-hi z, ^-x X, y+2j 2~z; 2"~Z, X, y+2; y+2) 2~z, x; X, 2""y> z+2> z+2> X, 5— y; 2~y> z+2, x; 2~x, y, z+2; zt"?> 2 X, y; y> z+2> 2~"X; x+i y, ^-z; l-z, x+i y; y, l-z, x+^ Space-Group TJ. ^*^/ii equivalent positions: (a) 8i. (b) 8e. Sixteen equivalent positions: (c) 16e. Twenty-four equivalent positions: (d) 24e. Forty-eight equivalent positions : (e) xyz; x, y, ^-z; |-x, y, z; x, |-y, z; zxy; |-z, X, y; z, ^-x, y; z, x, |-y; yzx; y, ^-z, x; y, z, |-x; ^-y, z, x; xyz; X, y, z+|; x+|, y, z; x, y+i z; zxy; z+l, X, y; z, x+|, y; z, x, y+§; yzx; y, z+i x; y, z, x+^; y+|, z, x; x+i y+h z+^; x+i l-y, z; x, y+|, |-z; |-x. y, z-M; z+i x-f-l, y+l; z, x+i l-y; ^-z, x, y+|; z+i, |-x, y; y+i z+i x+l; |-y, z, x+|; y-H|, ^-z, x; y, z-fi |-x; 128 THE CUBIC SPACE-GROUPS tJ-T^. Space-Group T^ {continued), i~x, 2~y> 2~z; 2— X, y+27 z; x, 2~y> z-j-^; x+i y, ^-z; i-z, |-x, |-y; z, J-x, y+|; z+i x, ^-y; ^-z, x+l, y; h-y, i-z, |-x; y+i z, |-x; ^-y, z+J, x; y, i-z, x+i C. HEMIMORPHIC HEMIHEDRY. Space-Group T^. One equivalent position : (a) la. (b) lb. Three equivalent positions: (c) 3a. (d) 3b. Four equivalent positions: (e) 4a. Six equivalent positions : (f) 6a. (g) 6d. Twelve equivalent positions: (h) 12f. (i) 12g. Twenty-four equivalent positions: (j) xyz; zxy; yzx; yxz; xzy; zyx; Space-Group T^. Four equivalent positions: (a) 4b. (c) 4d. (b) 4c. (d) 4e. Sixteen equivalent positions: (e) 16a. Twenty-four equivalent positions: (f) 24a. (g) 24b. Forty-eight equivalent positions: (h) 48d. xyz; xyz; xyz; zxy; zxy; zxy; yzx; yzx; yzx; yxz; yxz; yxz; xzy; xzy; xzy; zyx; zyx; zyx; THE CUBIC SPACE-GROUPS T^-tS. 129 Space-Group T^ (continued). Ninety-six equivalent positions: (i) xyz; xyz; xyz; xyz zxy; zxy; zxy; zxy yzx; yzx; yzx; yzx yxz; yxz; yxz; yxz xzy; xzy; xzy; xzy zyx; zyx; zjrx; zyx x+i y+i z; x+i h-y, z; l-x, y+i z l-x, l-y, z z+i x+i y; ^-z, x+i y; l-z, l-x, y z+i l-x, y y+i z+l, x; l-y, 1-z, x; y+i l-z, X l-y, z+i X y+i x+i z; i-y, x+i z; y+i l-x, z ; l-y, l-x, z x+i z+l, y; x+i l-z, y; l-x, l-z, y ; l-x, z+i y z+i y+i, x; l-z, l-y, x; l-z, y+i X ; z+i l-y, X x+i y, z+l; x+iy, ^-z; l-x, y, l-z l-x, y, Z+I z+i,x, y+l; i-z, X, l-y; l-z, X, y+l z+i X, l-y y+i z, x+i; l-y, z. x+i; y+i z, l-x ; l-y, z, l-x y+i X, z+^; i-y, X, i-z; y+i X, l-z l-y, X, Z+I x+i z, y+l; x+i z, i-y; l-x, z, y+l l-x, z, l-y z+i y, x+^; l-z, y, X+I; l-z, y, l-x z+i y, l-x X, y+i z+l; X, i-y, l-z; X, y+i l-z X, l-y, Z+I z, x+i y+l; z, x+i l-y; z, l-x, y+l z, l-x, l-y y, z+i x+^; y, l-z, x+i y, l-z, l-x y, z+i l-x y, x+i z-\-i; y, x+i i-z; y, l-x, l-z y. l-x, Z+I X, z+l, y+^; X, l-z, i-y; X, l-z, y+l, X, z+i l-y z, y+i x+l; z, l-y, X+I; z, y+i l-x z, l-y, l-x Space-Group T|. Two equivalent positions: (a) 2a. Six equivalent positions: (b) 6e. Eight equivalent positions: (c) 8a. Twelve equivalent positions: (d) 12h. (e) 12a. Twenty-four equivalent positions: (f) 24f. (g) 24g. Forty-eight equivalent positions: xyz; xyz; xyz; xyz; zxy; zxy; zxy; zxy; yzx; yzx; yzx; yzx; yxz; yxz; yxz; yxz; xzy; xzy; xzy; xzy; zyx; ^; zyx; zyx; 130 THE CUBIC SPACE-GROUPS T?-!^. Space-Group T^ {continued). x+i y+l, z+l z+i x+l, y+l y+i z+i x+l y+i, x+l, z+l x+i z+l, y+l z+i y+i x+i x+2) ^~yj 2~z; ^— z, x+2, 2~~yi i-y, l-z, x+l; 2~y> x+2, 2~~z; x+i |-z, J-y; §-z, l-y, x+l; Space-Group T\. Two equivalent positions: (a) 2a. Six equivalent positions: (b) 6e. (d) 6g. (c) 6f. Eight equivalent positions: (e) 8a. Twelve equivalent positions : (f) 12a. (h) 12j. (g) 12i. Twenty-four equivalent positions : (i) xyz; xyz; xyz; xyz; ixy; zxy; zxy; zxy; yzx; yzx; yzx; yzx; y+l, x+l, z+\; i-y, x-f-|, |-z; x+i z+i y+l; x+i l-z, |-y; z+i y+i x+l; l-z, |-y, x+|; Space-Group T^. Eight equivalent positions: (a) 8i. (b) Be. Twenty-four equivalent positions: (c) 24c. (d) 24h. Thirty-two equivalent positions : (e) 32c. 2~x, y+2j 2~z; 2~x, 2~y> Z+2I 2 z, 2 X, y+2J Z+2> 2~X, 2~yj y'T2> 2~z, 2"~x; 2"~y> z+2, ^~x; yi2> 2~x, 2~z; ^~y, 2~x, z+2; 2~"X, ^ — z, y+2j ^ — X, z+2, 2~y; §-z, y+i l-x; z+2> 2~y> 2~~x; y+i ¥-x, l-z; 5~yj 5~x, z+2; 2~x, ^ — z, y+^; |-x, z+l, |-y; i~z, y+2) ?~~x; z+l, l-y, l-x. THE CUBIC SPACE-GROUPS T^-I^. 131 Space-Group T* (continued). Forty-eight equivalent positions: (f) 48e. (g) 48a. Ninety -six equivalent positions: (h) xyz; xyz; xyz; xyz; zxy; zxy; zxy; zxy; yzx; yzx; yzx; yzx; y+i x+i z+i; |-y, x+^, |-z; y+i ^-x, §-z; h-y, i-x, z+^; x+i z+l, y+l; x+i ^-z, |-y; ^-x, |-z, y+l; 2~X, Z+2> 2~y> z+i y+i x+^; ^-z, |-y, x+|; |-z, y+i ^-x; z+i h-y, ^-x; x+i, y+iz; x+i|-y, z; |-x, y+i z; ^-x, |-y, z; z+l, x+l, y; l-z, x+l, y; 5-z, |-x, y; z+i ^-x, y; y+i z+i x; |-y, i-z, x; y+i |-z, x; |-y, z+i x; y, X, z-h^; y, X, ^-z; y, X, l-z; y, X, z+^; X, z, y+l; X, z, |-y; X, z, y+l; X, z, l-y; z, y, x+^; z, y, x+l; z, y, i — V 2 ■''■> z, y, 5-x; x+l, y, z+^; x+iy |-z; |-x, y, ^-z; |-x, y, z+|; z+ix, y-l-^; ^-z, x, |-y; *-z, x, y+l; z+|, x |-y; y+i, z, x+l; i-y, z, x+l; y+|, z, ^-x; |-y, z, ^-x; y, x+i z; y, x+l, z; y, |-x, z; y, |-x, z; X, z+i y; X, l-z, y; x, |-z, y; x, z+|, y; z, y+l, x; z, |-y, x; z, y+i x; z, |-y, x; X, y+iz+l; X, |-y, |-z; x, y+i ^-z; x, J-y, z+|; z, x+iy+l; z, x+l, §-y; z, |-x, y+|; z, |-x, |-y; y, z+l, x+l; y, l-z, x+l; y, |-z, |-x; y, z+|, |-x; y+l, X, z; |-y, x, z; y+|, x, z; |-y, x, z; x+l, z, y; x+l, z, y; |-x, z, y; |-x, z, y; z+i y, x; |-z, y, x; |-z, y, x; z+|, y, x. Space-Group T^. Twelve equivalent positions : (a) 12k. (b) 121. Sixteen equivalent positions : (c) 16f. Twenty-four equivalent positions : (d) 24i. Forty-eight equivalent positions : (e) xyz; x, y, |-z; |-x, y, z; x, |-y, z; zxy; |-z, X, y; z, |-x, y; z, x, |-y; yzx; y, |-z, x; y, z, |-x; |-y, z, x; 132 THE CUBIC SPACE-GROUPS Tfl-O^ Space-Group T' {continued). y+i x+l, z+i; i-y, x+\, f-z; y+i f-x, |-z; 4~y> 4~X, Z + jI x+i z+l, y+l; x+i f-z, \-y; f-x, f-z, y+f; 4— X, z+j, 4~yj z+i, y+i x+f; f-z, f-y, x+f; f-z, y+f, f-x; z+4> 4~y> 4~x; x+J,y+|,z+|; x+l, ^-y, z; x, y+|, |-z; l-x, y, z+^; z, x+l, ^-y; i-z, X, y+|; z+2) 2~x, y; |-y, z, x+l; y+i, |-z, x; y, z+l, ^-x; l-y, x-hf, f-z; y+f, f-x, f-z; 4~y> 4~x, z-f-i) x+f, f-z, f-y; f-x, f-z, y+f; 4~x, z+4, 4— y; z+i x+l, y+l y+i z+i x+l y+i x+f, z+f x+f, z+f, y+f z+f, y+f, x+f ■z, 4, y, x+4; 4— z, y+4, 4~x; z+f, f-y,f-x. D. ENANTIOMORPHIC HEMIHEDRY. Space-Group O*. One equivalent position: (a) la. (b) lb. Three equivalent positions: (c) 3a. (d) 3b. Six equivalent positions: (e) 6a. (g) 6c. (f) 6b. (h) 6d. Eight equivalent positions: (i) 8c. Twelve equivalent positions: G) 12m. (k) 12n. Twenty-four equivalent positions: (1) xyz; xyz; xyz; zxy; zxy; zxy; yzx; yzx; yzx; yxz; yxz; yxz; xzy; xzy; xzy; zyx; zyx; zyx; xyz; zxy; yzx; yxz; xzy; zyx. THE CUBIC SPACE-GROUPS O^-O'. 133 Space-Group O^. Two equivalent positions: (a) 2a. Four equivalent positions: (b) 4d. (c) 4e. Six equivalent positions: (d) 6e. (f) 6g. (e) 6f. Eight equivalent positions: (g) 8d. Twelve equivalent positions: (h) 12a. (k) 12o. (i) 12i. (1) 12p. a) 12j. Twenty-four equivalent positions: (m) xyz; xyz; xyz; xyz: zxy; zxy; zjcy; zxy; yzx; yzx; yzx; yzx; h-y> i-x, i-z; y+i |-x, z+|; ^-y, x+J, z+|; y+i x+i i-z; ^-x, i-z, ^-y; i-x, z+i y+|; x+|, z+|, ^-y; x+l, i-z, y+l; i-z, ^-y, l-x; z-hi y+h ^-x; z+|, ^-y, x+J; ^-z, y+l, x+^. Space-Group 0'. Four equivalent positions: (a) 4b. (b) 4c. Eight equivalent positions: (c) 8e. Twenty-four equivalent positions : (d) 24c. (e) 24a. Thirty-two equivalent positions: (f) 32a. Forty-eight equivalent positions : (g) 48f. (i) 48a. (h) 48g. 134 THE CUBIC SPACE-GROUPS 0^-0*. Space-Group 0^ {continued). Ninety-six equivalent positions: (j) xyz; xyz; xyz; xyz zxy; zxy; zxy; zxy yzx; yzx; yzx; yzx yxz; yxz; yxz; yxz xzy; xzy; xzy; xzy zyx; zyx; zyx; zyx x+i y+i z x+l, i-y, z i-x, y+i z l-x, |-y, z; z+i x+l, y ; ^-z, x+i y A-z, i-x, y r z+l, l-x, y; y+i z+i X i-y, ^-z, X y+i 5-z, X l-y, z+l, x; ^-y, l-x, z y+i l-x, z l-y, x+i z y+l, x+l, z; i-x, l-z, y l-x, z+i y x+l, z+i y x+l, l-z, y; l-z, ^-y, X , z+i y+l, X z+i l-y, X l-z, y+l, x; x+i y, z+^ x+i y, i-Z; l-x, y, l-z l-x, y, z+l; z+i X, y-l-^ i-z, X, ^-y l-z, X, y+l z+l, X, l-y; y+i z, x+^ ^-y, z, x+l y+i z, |-x l-y, z, l-x; ^-y, X, l-z y+l, X, z+l h-Y, X, z+l y+l, X, l-z; i-x, z, l-y l-x, z, y+^; x+i z, |-y x+l, z, y+l; |-z, y, ^-x z+i y, l-x z+l, y, x+l l-z, y, x+l; X, y+l, z+l X, ^-y, l-z, X, y+l, l-z X, l-y, z+l; z, xH-l, y+l z, x+i |-y z, l-x, y+l z, l-x, l-y; y, z+i x+l y, i-z, x+h y, l-z, l-x y, z+l, l-x; y, l-x, l-z y, ^-x, z+l y, x+l, z+l y, x+l, l-z; X, i-z, l-y X, z+i y+l, X, z+l, |-y, X, l-z, y+l; z, ^-y, i-x z, y+i i-x, z. l-y, x+l z. y+l, x+|. Space-Group 0^. Eight equivalent positions: (a) 8f. (b) 8g. Sixteen equivalent positions: (c) 16b. (d) 16c. Thirty-two equivalent positions: (e) 32b. Forty-eight equivalent positions: (f) 48c. (g) 48h. Ninety-six equivalent positions: (h) xyz; xyz; xyz; zxy; zxy; zjiy; yzx; yzx; yzx; xyz; zxy; yzx; i-y, i- X, i-z; y+i \- •X, i-z, i-y; i-x, z+l. z+i; l-y, x+i, z-\-\) y+i, x+i i-z; y+i; x+i, z+i i-y; x+i i-z, y+J; THE CUBIC SPACE-GROUPS 0*-0^. 135 Space-Group O* (continued). — z. y, l-x; z+i y+i ^-x; z+i i-y, x-\-i; i-z, y-l-i, x+i; x+iy+iz; x-l-i^-y, z; |-x, y+i z; |-x, |-y, z; z+ix-|-|, y; i-z, x-j-^ y; |-z, ^-x, y; z+i |-x, y; y+iz+l, x; ^-y, i-z, x; y+i |-z, x; |-y, z+^, x; f-y, f-x, i-z; y+l f-x, z+i; f-y, x+f, z+i; y+i x+i i-z; f-x, f-z, i-y; f-x, z+f, y+i; x+f, z+f, f-y; x+f, f-z, y+i; f-z, f-y, i-x; z+f, y+f, i-x; z+f, f-y, x+i; ? — z, y+4, x+j; x+iy, z+l; x+iy, i-z; 2 ~ X, y, 2 — z ; •X, y, z+^; z+ix, y+^; |-z, x, |-y; ^-z, x, y+^; z+i x, |-y; y+iz, x+l; ^-y, z, x+^; y+i z, |-x; ^-y, z. §-x; f-y, ¥-x, f-z; y+f, i-x, z+f; f-y, x+f, z+f; y+f, x+f, f-z; f-x, f-z, f-y; f-x, z+f, y+f; x+f, z+f, f-y; x+f, l-z, y+f; f-z, f-y, f-x; z+f, y+f, f-x; z+f, f-y, x+f; f-z, y+f, x+f; X, y+iz+^; x, ^-y, ^-z; x, y+|, ^-z; x, i-y, z+|; z, x+iy+l; z, x+i|-y; z, ^-x, y+|; z, |-x, |-y; y, z+ix+^; y, |-z, x+l; y, ^-z, |-x; y, z+i |-x; -y, f-x, f-z; y+f, f-x, z+f; f-y, x+f, z+f; y+i x+f, f-z; ■X, f-z, f-y; i_ X, z+f, y+f; x+f, z+f, f-y; x+f, f-z, y+f; -z, f-y, f-x; z+f, y+f, f-x; z+f, f-y, x+f; f-z, y+f, x+f Space-Group 0^. Two equivalent positions: (a) 2a. Six equivalent positions: (b) 6e. Eight equivalent positions: (c) 8e. Twelve equivalent positions: (d) 12h. (f) 12b. (e) 12a. Sixteen equivalent positions: (g) 16d. 136 THE CUBIC SPACE-GEOUPS 0*-0'*. Space-Group 0^ (continued). Twenty-four equivalent positions: (h) 24j. (i) 24k. Forty-eight equivalent positions: xyz; xyz; xyz; xyz; zxy; zxy; zxy; zxy; yzx; yzx; yzx; yzx; yxz; yxz; yxz; yxz; xzy; xzy; xzy; xzy; zyx; zyx; zyx; zyx; x+i y+h z+§; x+i ^-y, i-z; z+i x+i y+J; 5-z, x+i |-y; ^— X, y+^, ^ — z; i-x, h-y, z+^; 2~z, ^— X, y+a; z+i l-x, ^-y; y+i z+i x+l; l-y, l-z, x+l; y+J, |-z, |-x; i-y, z+i ^-x; l-y, l-x, |-z; y+i §-x, z+l; |-y, x+|, z+i; y+i x+i |-z; i-x, i-z, |-y; l-x, z+i y+|; x+|, z+|, ^-y; x+l, i-z, y+l; i-z, |-y, |-x; z+i y+l, ^-x; z+i |-y, x-f|; l-z, y+5, x+i. Space-Group 0". Four equivalent positions : (a) 4g. (b) 4h. Etgf/i< equivalent positions: (c) 8j. Twelve equivalent positions: (d) 12q. Twenty-four equivalent positions: (e) xyz; x-fl, |-y, z; x, y+i |-z; |-x, y, z-|-|; zxy; z, x-ff, §-y z, X, y+l; z+i i-x, y; yzx; ^-y, z, x-j-|; y-f-i |-z, x; y, z+|, §-x; i-y, i-x, i-z; y+i |-x, z+i; f-y, x+i z+f; y+J, x-l-f, f-z; i-x, i-z, J-y; l-x, z-fi y+|; x+i z+f, |-y; x+f, f-z, y+i; l-z, f-y, f-x; z+f, y+f, f-x; z+f, f-y, x+f; f-z, y+f, x+f. It is evident that a suitable transformation would simplify the two unique cases. THE CUBIC SPACE-GROUPS O'^-O*. I'M Space-Group O^. Four equivalent positions: (a) 4i. (b) 4j. Eight equivalent positions: (c) 8k. Twelve equivalent positions: (d) 12r. Twenty-four equivalent positions: (e) xyz; x+i ^-y, z; x, y-j-|, |-z; |-x, y, z+|; zxy; z, x+i ^-y; ^-z, x, y+|; z+|, |-x, y; yzx; |-y, z, x+|; y+i ^-z, x; y, z+i ^-x; l-y, f-x, l-z; y+i, i-x, z+f; i-y, x+f, z+i; y+4, x-f-j, 4 — z; l-x, f-z, l-y; i-x, z+f, y+i; x+f, z+i, |-y; x+i i-z, y+l; l-z, l-y, l-x; z+f, y+i i-x; z+i l-y, z+f; i-z, y+f, x+i Space-Group O*. Eight equivalent positions : (a) 81. (b) 8m. Twelve equivalent positions: (c) 12s. (d) 121. Sixteen equivalent positions : (e) 16g. Twenty-four equivalent positions : (f) 241. (g) 24m. (h) 24n. Forty-eight equivalent positions: (i) xyz; x, y, i-z; ^-x, y, z; x, ^-y, z; zxy; i-z, X, y; z, ^-x, y; z, x, ^-y; yzx; y, |-z, x; y, z, |-x; ^-y, z, x; ¥-y, i-x, i-z; y+i i-x, z+f; f-y, x+f, z+i; y"!"*, x+4, 4 — z; i-x, i-z, l-y; i-x, z+f, y+i; x+f, z+f, f-y; x+f, f-z, y+f; f-z, f-y, f-x; z+f, y+f, f-x; z+f, f-y, x+f; f-z, y+l, x+f; x+i y+l, z+l; x+i l-y, z; x, y+i |-z; l-x, y, z+l; 138 THE CUBIC SPACE-GROUPS 0*-0i. Space-Group O* (continued). z+i, x+i y+l; z, x+l, ^-y; |-z, x, y+^; z+l, l-x, y; y+i z+i 5C+I; l-y, z, x+|; y+i l-z, x; y, z+i l-x; f-y, l-x, f-z; y+l, f-x, z+i; f-y, x+i z+f; y"r4) x-|-4, 4 — z; f-x, f-z, l-y; f-x, z+i y+f; x+i z+f, f-y; x+f, f-z, y+f; l-z, f-y, f-x; z+f, y+f, f-x; z+f, f-y, x+f; f-z, y+f, x+f. E. HOLOHEDRY. Space-Group OJ. One equivalent position: (a) la. (b) lb. Three equivalent positions: (c) 3a. (d) 3b. Six equivalent positions: (e) 6a. (f) 6d. Eight equivalent positions: (g) 8c. Twelve equivalent positions: (h) 12f. (j) 12n. (i) 12m. Twenty-four equivalent positions: (k) 24o. (m) 24q. a) 24p. Forty-eight equivalent positions: (n) xyz; xyz; xyz: xyz; zxy; zxy; zxy; zxy; yzx; yzx; yzx; yzx; yxz; yxz; yxz; yxz; xzy; xzy; xzy; xzy; zyx; zyx; zyx; zyx; xyz; xyz; xyz; xyz; zxy; zxy; zxy; zxy; yzx; yzx; yzx; yzx; yxz; yxz; yxz; yxz; xzy; xzy; xzy; xzy; zyx; zyx; gyx; zyx; THE CUBIC SPACE-GROUPS 0,-0^. 139 Space Group 01. Two equivalent positions : (a) 2a. <Sta; equivalent positions: (b) 6e. Eight equivalent positions: (c) 8e. Twelve equivalent positions : (d) 12h. (e) 12a. Sixteen equivalent positions: (f) 16d. Twenty-four equivalent positions : (g) 24f. (h)24j. Forty-eight equivalent positions: (i) xyz; xyz; xyz; xyz; zxy; zxy; zxy; zxy; yzx; yzx; yzx; yzx; yxz; yxz; yxz; yxz; xzy; xzy; xzy; xzy; z^x; zyx; zyx; zyx; i-x, ^-y, |-z; l-x, y+i z+^; x+i i-y, z+l; x+i y+h i-z; \-z, i-x, |-y; z+§, |-x, y+i; z+i x+|, ^-y; ^— z, x+2> y+ij 2~y> 2~z, 2~x; y+f, z+2, 2~x; 2~y> z+2, x+2> y~r2j 2 z, X+2J y+i x+i z+i; \-y, x+l, §-z; y+i |-x, \-z; l~y, 2~X, z+2; x+i z+i y+^; x+l, §-z, ^-y; §-x, |-z, y+^; l-x, z+l, |-y; z+i y+i x+l; l-z, |-y, x+^; |-z, y+|, |-x; z+i 5-y; i-x. Space-Group 0^. Two equivalent positions: (a) 2a. Six equivalent positions: (b) 6e. (c) 6f. (d) 6g. Eight equivalent positions : (e) 8e. 140 THE CUBIC SPACE-GROUPS 0^-0^. Space-Group 0^ (continued). Twelve equivalent positions: (f) 12a. (g) 12i. (h) 12j. Sixteen equivalent positions: (i) 16d. Twenty-four equivalent positions: (j) 24s. (k) 24r. Forty-eight equivalent positions : (1) xyz; xyz; xyz; xyz; zxy; zxy; zxy; zxy; yzx; yzx; yzx; yzx; ^-y, l-x, l-z; y+i, i-x, z+i; ^-y, x+i z+|; y~r2> ^+2, 2~z; l-x, ^-z, l-y; l-x, z+l, y+|; x+i z+|, |-y; X+2J 2~z, y+2j h-z, l-y, l-x; z+i y+i, i-x; z+i |-y, x+|; 2~z, y+2, X+2J xyz; xyz; xyz; xyz; zxy; zxy; zxy; zxy; ygX; yzx; yzx; yzx; y+i, x+l, z+l; l-y, x+|, |-z; y+i |-x, i-z; l-y, l-x, z+l; x+l, z+l, y+l; x+l, l-z, |-y; |-x, |-z, y+|; 2— X, z+2, ^~y; z+l, y+l, x+l; l-z, l-y, x+l; |-z, y+|, |-x; z+l, l-y, l-x. Space-Group 0^. Two equivalent positions: (a) 2a. Four equivalent positions: (b) 4d. (c) 4e. Six equivalent positions : (d) 6e. Eight equivalent positions : (e) 8d. Twelve equivalent positions : (f) 12h. (g) 12a. Twenty-four equivalent positions: (h) 24f. (i) 24t. (j) 24u. THE CUBIC SPACE-GROUPS Oh-Oh. 141 Space-Group O^ (continued). Forty-eight equivalent positions: (k) xyz; xyz; xyz; xyz; zxy; zxy; zxy; zxy; yzx; yzx; yzx; yzx; a—y? 2~x, 2~z; y-r2) 2~x, z+2 A-x, l-z, |-y; ^-x, z+l, y+| i-z, ^-y, i-x; z+i y+i ^-x ^-x, ^-y, i-z; |-x, y+i z+^ 2"~Z, 2~X, 2~y> ^12) 2~X, y+2 |-y, ^-z, ^-x; y+i z+^, |-x yxz; yxz; yxz; yxz; xzy; xzy; xzy; xzy; zyx; zyx; zyx; zyx. Space-Group 0^. Four equivalent positions: (a) 4b. (b) 4c. ^zfif/i^ equivalent positions: (c) 8e. Twenty-four equivalent positions: (d) 24c. (e) 24a. Thirty-two equivalent positions: (f) 32a. Forty-eight equivalent positions: (g) 48a. (h) 48f. (i) 48g. Ninety-six equivalent positions : (j) 96a. (k) 96b. One hundred ninety-two equivalent positions: (1) xyz; xyz; xyz; xyz; zxy; zxy; zxy; zxy; yzx; yzx; yzx; yzx; yxz; yxz; yxz; yxz; xzy; xzy; xzy; xzy; zyx; zyx; zyx; zyx; xyz; xyz; xyz; xyz; zxy; zxy; zxy; zxy; 2~y< xH-2, Z+2I y+i x-l-l, |-z; x+2) Z+2, ^— y; X+2> 2~z, y+2> z+2, 2~yj X+2J l-z, y+i x+l; x+i 5-y, z+l; x+l, y+i l-z; z+2, X+2, 2~yj 2 — Z, X+2, y+2 • 2~y> z+2, x+2; y+2, |-z, x+^; 142 THE CUBIC SPACE-GROUPS O^-OS. Space-Group 0^ (continued). yzx; yzx; yzx; yzx; yxz; yxz; yxz; yxz; xzy; xzy; xzy; xzy; zyx; zyx; zyx; zyx; x+l, y+i z; x+i ^-y, z; l-x, y+l, z; 2 X, 2 y, z z+i x+i y; 2~z, x-i-2> y> l-z, l-x, y; z+l, l-x, y y+i z+i X, h-y, 2-z, X, y+l, l-z, X, l-y, z+l, X 2~y> 2~x, z; y+h l-x, z; l-y, x+l, z, y+l, x+l, z l-x, l-z, y; 2~x, z-|-2, y, x+l, z+l, y, x+l, l-z, y f-z, |-y, x; z+l, y+i X, Z+2J 2~y> X l-z, y+l, X l-x, |-y, z, ^-x, y+i z x+l, |-y, z x+l, y+l, z l-z, l-x, y; z+2) 2 X, y; z+l, x+l, y; l-z, X+l, y l-y, i-z, X, y+l, z+ix; l-y, z+l, X y+l, l-z, X y+i x+i z l-y, x+l, z y+l, l-x, z l-y, l-x, z x+l, z+i y x+i l-z, y l-x, l-z, y l-x, z+l, y z+i y+i X l-z, ^-y, X l-z, y+l, X z+l, l-y, X x+i y, z+i x+i y, l-z l-x, y, l-z l-x, y, z+l z+i X, y+J l-z, X, l-y l-z, X, y+l z+l, X, l-y y+l, z, x+^ l-y, z, x+^ y+l, z, l-x , l-y, z, l-x ^-y, X, i-z ; y+i, X, z+i , l-y, X, z+l ; y+l, X, |-z ^-x, z, ^-y ; |-x, z, y+l ; x+l, z, l-y ; x+l, z, y+l h-2, y, 2-x ; z+l, y, l-x ; z+l, y, x+l ; l-z, y, x+l l-x, y, i-z , |-x, y, z+l x+l, y, z+l , x+l, y, l-z i-z, X, l-y ; z+l, X, y+l ; z+l, X, l-y ; l-z, X, y+l l-y, z, ¥-x ; y+i z, i-x ; l-y, z, x+l ; y+l, z, 3^+1 y+i X, z+l ; l-y, X, |-z ; y+l, X, l-z ; l-y, X, z+l x+i z, y+l ; x+i z, |-y ; |-x, z, y+l ; l-x, z, l-y z+l, y, x+l ; l-z, y, x+l ; l-z, y, l-x ; l-z, y, l-x X, y+l, z+l ; X, l-y, l-z ; X, y+l, l-z ; X, l-y, z+l z, x+i y+l ; z, x+i |-y ; z, l-x, y+l ; z, l-x, l-y y, z+l, x+l , y, l-z, x+l ; y, l-z, l-x ; y, z+l, l-x y, l-x, l-z ; y, l-x, z+i ; y, x+l, z+l ; y, x+l, l-z X, i-z, i-y ; X, z+i y+l ; X, z+l, l-y ; X, l-z, y+l z, l-y, §-x ; z, y+i l-x ; z, l-y, x+l ; z, y+l, x+l X, h-J, h-z ; X, y+l, z+l ; X, l-y, z+l ; X, y+l, l-z z, ^-x, |-y ; z, l-x, y+l ; z, x+l, l-y ; z, x+l, y+l y, l-z, l-x ; y, z+l, l-x ; y, z+l, x+l ; y, l-z, x+l y, x+l, z+l ; y, x+l, |-z ; y, l-x, l-z ; y, l-x, z+l X, z+i y+l ; X, l-z, |-y ; X, l-z, y+l ; X, z+l, l-y z, y+l, x+l ; z, l-y, x+l ; z, y+l, l-x • z, l-y, l-x Space-Group 0^. Eight equivalent positic ns: (a) 8i. (b) 8e. Twenty-four equivalent positions : (c) 24c. (d ) 24h. THE CUBIC SPACE-GROUP O^. 143 Space-Group 0° (continued). Forty-eight equivalent positions: (e) 48a. (f) 48e. Sixty-four equivalent positions : (g) 64a. Ninety-six equivalent positions: (h) 96c. (i) 96d. One hundred ninety-two equivalent positions: (J) xyz; xyz; xyz; xyz; zxy; zxy; zxy; zxy; yzx; yzx; yzx; yzx; yxz; yxz; yxz; yxz; xzy; xzy; xzy; xzy; zyx; zyx; zyx; zyx; l-x, l-y, h- -z; h- -X, y+i z+l; x+l, ^-y, z+^; x+i y+i ^-z; 2 z, 2 X, 2 y; z-t-2, 2 X, y+2; Z+2, X+2> 2~y; l-z, x+i y+i; l-y, ^-z, l-x; y+l, z+i §-x; ^-y, z+i x+|; y+i l-z, x+^; y+i x+i z+l; |-y, x+|, |-z; y+f, |-x, |-z; l-y, i-x, z+§; x+i z+i y+J; x+i ^-z, |-y; |-x, ^-z, y-(-^; l-x, z+i l-y; z+i y+i x-l-^; ^-z, |-y, x+^; ^-z, y+l, A-x; z+i ^-y, 5-x; x+i y+i z; x+i ^-y, z; |-x, y-|-|, z; f-x, |-y, z; z+i x+l, y; |-z. x+i y; ^-z, ^-x, y; z+|, |-x, y; y+i z+i x; |-y, |-z, x; y+i, |-z, x; |-y, z+^, x; l-y, ^-x, z; y+^, |-x, z; |-y, x+i, z; y+|, x+^, z; |-x, l-z, y; i-x, z+i y; x+i ^+^, y; x+i |-z, y; ^-z, l-y, x; z+i y+i x; z+|, i-y, x; ^-z, y+i x; X, y, ^-z; X, y, z+^; x, y, z+^; x, y, |-z; z, X, ^-y; z, X, y+|; z, x, |-y; z, x, y+|; y, z, l-x; y, z, |-x; y, z, x+^; y, z, x+|; y, X, z+§; y, x, ^-z; y, x, ^-z; y, x, z+|; X, z, y+l; x, z, |-y; x, z, y+|; x, z, |-y; z, y, x-f-l; z, y, x-l-i; z, y, |-x; z, y, |-x; x+i y, z+l; x+i y, ^-z; ^-x, y, |-z; ^-x, y, z+^; z+i X, y+l; i-z, X, ^-y; |-z, x, y+|; z+|, x, ^-y; y+iz, x+^; ^-y, z, x+l; y+|, z, ^-x; |-y, z, |-x; h-y, X, h-z; y+i x, z+l; |-y, x, z+|; y+i x, ^-z; ^-x, z, ^-y; l-x, z, y+|; x+§, z, ^-y; x+§, z, y+^; ^-z, y, |-x; z+i y, |-x; z-f-^ y, x-F§; ^-z, y, x+|; 144 THE CUBIC SPACE-GROUPS 0^-0^. Space-Group 0^ (continued). X, l-y, z; X, y+l, z; X, I-y, z; X, y+l. z; z, l-x, y; z, §-x, y; z, x+l, y; z, x+l, y; y, ^-z, x; y, z+§, x; y, z+l, x; y, l-z, x; y, x+l, z; y, x+l, z; y, l-x, z; y, l-x. z; X, z + l, y; X, i-z, y; X, l-z, y; X, z+l, y; z, y+i x; z, I-y, x; z, y+l, x; z, I-y, x; X, y+i z+l; X, h-y, l-z; X, y+l, l-z; X, I-y, z+l Z, x+2, y+l; z, x+i I-y; z, |-x, y+l; Z, 2 X, I-y y, z+i x+l; y? 2 Z, X+2i y, l-z, |-x; y, z+l, l-x y, ¥-x, l-z; y, l-x, z+l; y, x+l, z+l; y, x+l, l-z X, ^-z, i-y; X, z+l, y+l; X, z+l, I-y; X, l-z, y+l z, i-y, 1 — V 2 X, z, y+l, l-x; z, I-y, x+l; z, y+l, x+l i-x, y, z; l-x, y, z; x+l, y, z; x+l, y, z; ^-z, X, y; z+l, X, y; z+l, X, y; l-z, x. y; i-y, z, x; y+l, z, x; I-y, z, x; y+l, z. x; y+l, X, z; i-y, X, z; y+l, X, z; I-y, X, z; x+i z, y; x+l, z, y; l-x, z, y; l-x, z, y; z+i y, x; l-z, y, x; l-z, y, x; z+l, y. X. Space-Group 0^. Eight equivalent positions: (a) 8f. (b) 8g. Sixteen equivalent positions : (c) 16b. (d) 16c. Thirty-two equivalent positions : (e) 32b. Forty-eight equivalent positions : (f) 48c. Ninety-six equivalent positions : (g) 96e. (h) 96f. One hundred ninety-two equivalent positions: (i) xyz; xyz; xyz; xyz; zxy; zxy; zxy; zxy; yzx; yzx; yzx; yzx; -y, i_. :-z; y+ 1 i_' A X, A Z, t-z, t-y. y; — x. X, z+i; i-y, x+i z+i; y+4, x+j, 4 — z; l-x, z+i y+i; x+i z+i, i-y; x+i l-z, y+i; i-x; z+i, y+i, i-x; z+i, i-y, x+i; i-z, y+i, x+i; i-z; i-x, y+i, z+i; x+i, i-y, z+i; x+i, y+i, i-z; 4 THE CUBIC SPACE-GROUP 0^. 145 Space-Group O^ {continued). 1—7 i — V i — 4 ^) 4 ^) i 4 Jj 4 ^> 4 yxz; yxz; xzy; xzy; zyx; zyx; x+i y+i z; z+i x+i y; y+i z+l, x; 4~y> 4~X, j- y; z+i, i-x, x; y+i z+i y+l; z+i x+l, |-y; l-z, x+i y+i; 4~x; 4~y, z+4, x+j; y+i ?-z, x+l; yxz; yxz; xzy; xzy; zyx; zyx; x+i |-y, z; l-x, y+^z; i-x, |-y, z; ^-z, x+i y; i-y, h-z, x; ■z; y+i f-x, 3_v 1 4 *> 4 •z, t-y 1—7 1 — V i 4 ^> 4 J> 4 i-x 4 X, 4 y> 4 — z 3_7 4 Z, ■X, 4 -y 4 y> 4 z, 4 X l-z, I-x, y; z+l, I-x, y; y+i l-z, x; |-y, z+l, x; z+4j 4— y> x+4, z+4; y+i x+l, z; x+l, z+l, y; z+l, y+l, x; x+l, y, z+l; z+l, X, y+l; y+l, z, x+l; f-y y+i x+f, t-z; y+i; x+f, z+f, i-y; x+f, f-z, y+i; 4 — x; z+4, 4— y, x+j; 4~z, y+4, x+j; f-x, y+f, z+i; x+f, f-y, z+i; x+4, y+4, 4 — z; y+i; z+f, x+f, i-y; 4 — z, x+4, y+y; 4— x; 4— y, z+4, x+4; y+i f-z, x+i; 4 X, z+4, z+i y+i Z+4J 4— X, y+i z+i 1_Y 1. 4 A, 4 l-y, x+l, z x+l, l-z, y ^-z, |-y, x: x+l, y, l-z 2 — z, X, ^— y |-y, z, x+l z; y+i i-x. 4— X, 4 — z, 4 y ■z, t-y. 1_Y i — 4 A, 4 y, f — z 1—7 1_Y 4 ^> 4 X, y> 4 Z, 4 4 X, Z+4, z+i y+i f-x, y+i, z + 4, 4— X, f-x; y+i z+i y+l, X, z+l x+l, z, y+l z+l, y, x+l X, y+l, z+l z, x+l, y+l l-y, X, l-z; x+l, z, |-y; 2— z, y, x+2; X, l-y, l-z; z, x+l, l-y; y+i I-x, z; l-y, I-x, z; I-x, l-z, y; I-x, z+l, y; l-z, y+l, x; z+l, l-y, x; I-x, y, l-z; I-x, y, z+|; l-z, X, y+l; z+l, X, l-y; y+i z, |-x; l-y, z, I-x; z+f; f-y, x+i z+f; y+4j x+4, 4 — z; y+f; x+i z+i f-y; x+i i-z, y+f; f-x; z+i i-y, x+f; 4 — z, y+4, x+j; z+f; x+i f-y, z+f; x+i y+i f-z; y+f; z+i x+i f-y; f-z, x+i y+f; f-x; f-y, z+i x+f; y+i i-z, x+f; y+i X, l-z; l-y, x, z+|; I-x, z, y+l; I-x, z, |-y; 2 — z, y, ^— x; z+2, y, 2— x; X, y+i l-z; X, l-y, z+|; z, 2-x, y+l; z, |-x, l-y; 146 THE CUBIC SPACE-GROUPS 0^-0^. Space-Group Ol {continued). y, z+i x+l; y, ^-z, x+|; y, |-z, ^-x; y, z+|, ^-x; 4~y> 4~x, 4 — z; y+i, 4— X, z+4; 4 — y, x-f-j, z+4; y+jj x~|-4, 4~z; i"~x, 4— z, 4— y; 4~x, Z-I-4, y4-4; x+j, z+4, 4— y; x+4) 4 2, y+4i l-z, f-y, f-x; z+i, y+f, |-x; z+|, f-y, x+f; l-z, y+i x+l; l-x, l-y, l-z; i-x, y-f-f, z+i; x+i f-y, z+f; ^+4> y+4j 4~z; j-z, f-x, f-y; z+i f-x, y+f; z+|, x+f, f-y; 4 — z, x+4, y+i; f-y, f-z, f-x; y+f, z+f, f-x; f-y, z+f, x+f; y+4> 4""^, x+4; y, x+l, z+l; y, x+|, ^-z; y, |-x, |-z; y, i-x, z+|; X, z+2, y+2; X, 2~z, ^— y; x, 2~z, y+2l X, z+^, 2~y5 z, y+i x+l; z, |-y, x+^; z, y+|, l-x; z, |-y, |-x. Space-Group O*. Sixteen equivalent positions: (a) 16h. Thirty-two equivalent positions : (b) 32d. (c) 32e. Forty-eight equivalent positions : (d) 48i. Sixty-four equivalent positions: (e) 64b. Ninety-six equivalent positions: (f) 96g. (g) 96h. One hundred ninety-two equivalent positions: (h) xyz; xyz; xyz; xyz; zxy; zxy; zxy; zxy; yzx; yzx; yzx; yzx; f-y, f-x, f-z; y+f, f-x, z+f; f-y, x+f, z+f; y+4, x+4, 4 — z; f-x, f-z, f-y; f-x, z+f, y+f; x+f, z+f, f-y; x+f, f-z, y+f; f-z, f-y, f-x; z+f, y+f, f-x; z+f, f-y, x+f; f-z, y+f, x+f; f-x, f-y, f-z; f-x, y+f, z+f; x+f, f-y, z+f; x+f, y+f, f-z; f-z, f-x, f-y; z+f, f-x, y+f; z+f, x+f, f-y; f-z, x+f, y+f; THE CUBIC SPACE-GROUP OJ. 147 Space-Group 0' {contirvaed). l-y, l-z, l-x; y+l, z+f, |-x; |-y, z+f, x+f; y+i l-z, x+l; y+i x+i z+^; |-y, x+i \-z; y+i ^-x, |-z; l-y, l-x, z+l; x+i z+l, y+l; x+l, l-z, |-y; |-x, |-z, y+|; 2~~x, z+2, 2~yj z+i y+l, x+l; l-z, |-y, x+|; |-z, y+|, |-x; z+i, ^ — y> i" — x; x+l, y+l, z; x+l, ^-y, z; |-x, y+|, z; |-x, |-y, z; z+l, x+l, y; l-z, x+iy; |-z, |-x, y; z+|, ^-x, y; y+l, z+l, x; l-y, l-z, x; y+|, |-z, x; ^-y, z+i x; f-y, l-x, i-z; y+f, l-x, z+i; f-y, x+f, z+i; y+4, x+4, J — z; l-x, l-z, i-y; f-x, z+l, y+i; x+f, z+|, i-y; x+4, 4 — z, y+4j 4 — z, 4— y, 4 — x; z+4, y+4, 4— x; z+j, 4— y, x+j; 4— z, y+4, x+jj i-x, i-y, i-z; i-x, y+i z+|; x+i, f-y, z+|; x+i, y+i l-z; i-z, i-x, l-y; z+i i-x, y+|; z+i x+i f-y; 4 — z, x+j, y+4; i-y, i-z, l-x; y+i z+i f-x; i-y, z+i x+|; y+i i-z, x+l; y, X, z+l; y, x, |-z; y, x, |-z; y, x, z+|; X, z, y+l; X, z, |-y; x, z, y+|; x, z, |-y; z, y, x+l; z, y, x+l; z, y, ^-x; z, y, |-x; x+l, y, z+l; x+l, y, i-z; |-x, y, |-z; |-x, y, z+|; z+l, X, y+l; l-z, X, |-y; f-z, x, y+|; z+|, x, ^-y; y+i z, x+l; l-y, z, x+|; y+i z, |-x; |-y, z, |-x; l-y, i-x, l-z; y+f, i-x, z+|; f-y, x+i z+f; y+i x+i f-z; f-x, i-z, f-y; f-x, z+i y+f; x+i z+i f-y; x+i i-z, y+f; f-z, i-y, f-x; z+i y+i f-x; z+i i-y, x+f; f-z, y+i x+f; i-x, f-y, i-z; i-x, y+i z+f; x+i, f-y, z+f; x+i y+i i-z; i-z, f-x, i-y; z+i f-x, y+f; z+i x+i i-y; \-z, x+i y+i; i-y, l-z, i-x; y+i z+i i-x; i-y, z+i x+i; y+i f-z, x+i; y, x+l, z; y, x+l, z; y, ^-x, z; y, ^-x, z; X, z+i y; x, l-z, y; x, |-z, y; x, z+|, y; z, y+l, x; z, l-y, x; z, y+i x; z, |-y, x; X, y+l, z+i; X, l-y, l-z; x, y+|, |-z; x, |-y, z+|; 148 THE CUBIC SPACE-GROUPS 0^-02- Space-Group 0^ (continued). z, x+iy+l; z, x+i^-y; z, |-x, y+l; z, |-x, |-y; y, z+i x+l; y, f-z, x+^; y, |-z, ^-x; y, z+i ^-x; i-y, f-x, f-z; y+i f-x, z+f; i-y, x+f, z+f; y+h x+i l-z; l-x, f-z, l-y; l-x, z+l, y+f; x+i, z+f, f-y; x+i f-z, y+f; l-z, f-y, f-x; z+l, y+f, f-x; z+|, f-y, x+f; i-z, y+i x+f; f-x, i-y, i-z; f-x, y+i, z+|; x+f, |-y, z+f; x+i y+i f-z; f-z, f-x, f-y; z+f, f-x, y+f; z+f, x+f, f-y; ■J— z, x+4, y+j; l-y, i-z, f-x; y+f, z+f, f-x; f-y, z+f, x+f; yij, 4 — z, x+j; y+l, X, z; l-y, x, z; y+|, x, z; ^-y, x, z; x+i z, y; x+l, z, y; ^-x, z, y; |-x, z, y; z+i y, x; i-z, y, x; f-z, y, x; z+f, y, x. Space-Group O^. Two equivalent positions: (a) 2a. Six equivalent positions : (b) 6e. Eight equivalent positions: (c) 8e. Twelve equivalent positions: (d) 12h. (e) 12a. Sixteen equivalent positions: (f) 16d. Twenty-four equivalent positions : (g) 24f. (h) 24j. Forty-eight equivalent positions: (i) 481. (j) 48j. (k) 48k. Ninety-six equivalent positions: (1) xyz; xyz; xyz; xyz; zxy; zxy; zxy; zxy; yzx; yzx; yzx; yzx; yxz; yxz; yxz; yxz; xzy; xzy; xzy; xzy; • THE CUBIC SPACE-GROUPS 0^- Ufa. 149 Space-Group Oh (continued). zyx; zyx; zyx; zyx; xyz; xyz; xyz; xyz; zxy; zxy; zxy; zxy; yzx; yzx; yzx; yzx; yxz; yxz; yxz; yxz; xzy; xzy; xzy; xzy; zyx; zyx; zyx; zyx; x+l, y+i z+§ z+i, x+i y+l y+l, z+i x+l 2~y> 2~x, ^— z |-x, §-z, l-y 2 — z, 5— y, 2~x l-x, |-y, i-z l-z, i-x, §-y l-y, ^-z, ^-x y-hh x+l, z+i x+i z+i y+l z+i y+i x+l Space-Group 0^*. Sixteen equivalent positions: (a) 16h. (b) 16i. Twenty-four equivalent positions : (c) 24v. (d) 24w. Thirty-two equivalent positions : (e) 32f. Forty-eight equivalent positions : (f) 48m. (g) 48n. X+2J 2~y> 2~Z 2~z, x+2, 2~y h-y, §-z, x-M y+2> 2~x, z4-2 ^-x, z+l, y+l z+i y+h ¥-x l-x, y+i z+l z+i ^-x, y-{-^ y+2> Z+2, 2~X 2~y> X4-2j ?~z x+i l-z, l-y 2~Z, 2~y) X-f-2 2~x, y+2) 2~z; 2"~x, 2~y) z-j-2; 2~Z, 2'~X, y+2; Z+2> 2~X, 2~yj y I 2) 2~z, 2~x; 2~y> z-f-2> 2~x; 2~y> x+2> z+j-; y+i x+l, l-z; ^+h z+i |-y; x+2) 2~Z, y+2; z+2> ^~y) x+2"; 2~z, y+i, x+2; x+i, 2~y) z+2; x+i y+i |-z; Z+2> X+2, 2~y; ^-z, 3C+5, y+^; 2~y> z+2, x+2; yr2, 2~z, x+2; y~r2> 2^~x, 2~z; h-y, l-x, z+^; ^-x, i-z, y+l; i-x, z+i ^-y; ^-z. y+i i-x; z+i ^-y, i-x. 150 THE CUBIC SPACE-GROUP 0^. Space-Group 0^° (continued). Ninety-six equivalent positions: (h) xyz; x, y, ^-z; |-x, y, z; x, |-y, z; zxy; i-z, X, y; z, ^-x, y; z, x, ^-y; yzx; y, |-z, x; y, z, |-x; ^-y, z, x; j-y> l-x, i-z; y+i, i-x, z+f; J-y, x+f, z+i; y"r4> x+4, j — z', l-x, l-z, i-y; i-x, z+f, y+i; x+f, z+i f-y; x+i, i-z, y+f ; i-z, i-y, i-x; z+f, y+f, f-x; z+f, f-y, x+f; f-z, y+f, x+f; xyz; X, y, z+|; x+i y, z; x, y+|, z; zxy; z+i X, y; z, x+i y; z, x, y+|; yzx; y, z+|, x; y, z, x+|; y+|, z, x; y+f, x+f, z+f; f-y, x+f, f-z; y+f, f-x, f-z; f-y, f-x, z+f; x+f, z+f, y+f; x+f, f-z, f-y; f-x, f-z, y+f; 4~x, z+j, 4— y; z+f, y+f, x+f; f-z, f-y, x+f; f-z, y+f, f-x; z+4, 4— y, 4— x; x+l, y+i z+l; x+i |-y, z; x, y+i |-z; i-x, y, z+l; z+i x+i y+i; z, x+i |-y; |-z, x, y+|; z+i, l-x, y; y+i z+§, x+l; |-y, z, x+l; y+i |-z, x; y, z+2, 2— x; f-y, f-x, f-z; y+f, f-x, z+f; f-y, x+f, z+f; y+f, x+f, f-z; f-x, f-z, f-y; f-x, z+f, y+f; x+f, z+f, f-y; x+f, f-z, y+f; f-z, f-y, f-x; z+f, y+f, f-x; z+f, f-y, x+f; f-z, y+f, x+f; l-x, |-y, |-z; l-x, y+l, z; x, |-y, z+|; x+l, y, |-z; l-z, l-x, l-y; z, l-x, y+|; z+|, x, |-y; |-z, x+l, y; l-y, l-z, l-x; y+l, z, l-x; |-y, z+|, x; y, |-z, x+l; y+f, x+f, z+f; f-y, x+f, f-z; y+f, f-x, f-z; f-y, f-x, z+f; x+f, z+f, y+f; x+f, f-z, f-y; f-x, f-z, y+f; f-x, z+f, f-y; z+f, y+f, x+f; f-z, f-y, x+f; f-z, y+f, f-x; z+f, f-y, f-x. THE HEXAGONAL SPACE-GROUPS C3-C3I. 151 HEXAGONAL SYSTEM. RHOMBOHEDRAL DIVISION. A. TETARTOHEDRY. Space-Group CJ. — (Hexagonal Axes.) One equivalent position: (a)OOu. (b)Hu. (c)Hii. Three equivalent positions: (d) xyz; y-x, x, z; y, x-y, z. Space-Group C3. — (Hexagonal Axes.) Three equivalent positions: (a) xyz; y-x, x, z-f-f; y, x-y, z+f, Space-Group C|. — (Hexagonal Axes.) Three equivalent positions: (a) xyz; y-x, x, z+f; y, x-y, z+f Space-Group C3. — (Rhombohedral Axes.) One equivalent position: (a) uuu. Three equivalent positions: (b) xyz; zxy; yzx. B. HEXAGONAL TETARTOHEDRY OF THE SECOND SORT. Space-Group C3,. — (Hexagonal Axes.) One equivalent position : (a) 000. (b) 00|. Two equivalent positions: (c) OOu; GOii. (d) Hu; f i u- Three equivalent positions: (e) iH; OH; iOl (f) HO; O^O; HO. Six equivalent positions: (g) xyz; y-x, x, z; y, x-y, z; xyz; x-y, x, z; y, y-x, z. Space-Group C|,. — (Rhombohedral Axes.) One equivalent position: (a) 000. (b)Hi 152 THE HEXAGONAL SPACE-GROUPS C3I-C3V. Space-Group Ca^i (continued). Two equivalent positions: (c) uuu; tiiiii. Three equivalent positions: (d)OO^; ^00; 0^0. (e) H 0; OH; iO^ Six equivalent positions : (f) xyz; zxy; yzx; xyz; zxy; yzx. C. HEMIMORPHIC HEMIHEDRY. Space-Group Cay. — (Hexagonal Axes.) One equivalent position: (a)OOu. (b) Hu. (c)f|u. Three equivalent positions: (d) utiv; 2u, u, v; u, 2u, v. Six equivalent positions : (e) xyz; y-x, x, z; y, x-y, z; yxz; X, x-y, z; y-x y z. Space-Group Cay. — (Hexagonal Axes.) One equivalent position : (a) OOu. Two equivalent positions : (b) Hu; Hu. Three equivalent positions: (c) uuv; Ouv; ti v. Six equivalent positions: (d) xyz; y-x, x, z; y, x-y, z; yxz; x, y-x, z; x-y, y, z. Space-Group CaV — (Hexagonal Axes.) Two equivalent positions: (a) OOu; 0, 0, u-Fi (c) H u; f , I u-f-f (b) Hu; i f, n-\-h Six equivalent positions: (d) xyz; y-x, x, z; y, x-y, z; X, x-y, z+^; y, x, z+i; y-x, y, z-H. THE HEXAGONAL SPACE-GROUPS Csy-Ds- 153 Space-Group C*^. — (Hexagonal Axes.) Two equivalent positions : (a)OOu; 0, 0, u+i (b)Hu; f, i u+i. Six equivalent positions : (c) xyz; y-x, x, z; y, x-y, z; y, X, z+i; X, y-x, z+l; x-y, y, z+|. Space-Group Cat- — (Rhombohedral Axes.) One equivalent position: (a) u u u. Three equivalent positions : (b) uu v; vuu; u vu. Six equivalent positions: (c) xyz; zxy; yzx; xzy; Z3rx; yxz. Space-Group Cay. — (Rhombohedral Axes.) Two equivalent positions: (a) uuu; u+l, u-M, u-|-^. Six equivalent positions: (b) xyz; zxy; yzx; x+i z+i y-|-|; z-M, y-hi x+^; y-f-i x-Hi z+|. D. ENANTIOMORPHIC HEMIHEDRY. Space-Group D3. — (Hexagonal Axes.) One equivalent position : (a) 000. (c) HO. (e) f | 0. (b)OOi (d)Hi (f) IH. Two equivalent positions: (g) OOu; OOu. (i) Hu; H u. (h)Hu; HQ. TAree equivalent positions : (j) uuO; 2u, u, 0; u, 2u, 0. (k) ua^; 2u, u, I; u, 2u, ^. <Sfa; equivalent positions: (1) xyz; y-x, x, z; y, x-y, z; x, x-y, z; yxz; y-x, y, z. 154 THE HEXAGONAL SPACE-GROUPS D3-D3. Space-Group D3. — (Hexagonal Axes.) One equivalent position: (a) 000. (b) OOi Two equivalent positions: (c) OOu; OOu. (d) Hu; IH- Three equivalent positions: (e) uuO; OuO; tiOO. (f) uu|; Ou|; ti §. Six equivalent positions: (g) xyz; y-x, x, z; y, x-y, z; yxz; X, y-x, z; x-y, y, z. Space-Group D|. — (Hexagonal Axes.) Three equivalent positions: (a) utii; 2u, u, f; u, 2u, 0. (b) uuf; 2u, u, I; u, 2u, ^. Six equivalent positions: (c) xyz; y-x, x, z-f^; y, x-y, z-ff; y-x, y, z; y, x, f-z; x, x-y, f-z. Space-Group D3. — (Hexagonal Axes.) Three equivalent positions: (a) uOO; uui; Ouf. (b) uO^; Quf; Ouf. Six equivalent positions: (c) xyz; y-x, x, z+^; y, x-y, z+f; x-y, y, z; y, X, |-z; x, y-x, ^-z. Space-Group D3. — (Hexagonal Axes.) Three equivalent positions: (a) uu|; 2u, u, |; u, 2u, i. (b) utif; 2u, u, ^; u, 2u, 0. Six equivalent positions: (c) xyz; y-x, x, z+f; y, x-y, z-}-^; y-x, y, z; y, x, f-z; x, x-y, f-z. Space-Group D*. — (Hexagonal Axes.) Three equivalent positions: (a) uOO; Out; utif. (b)uO^; Ouf; Qui Six equivalent positions: (c) xyz; y-x, x, z+f; y, x-y, z+l; x-y, y, z; y, x, ^-z; x, y-x, f-z. THE HEXAGONAL SPACE-GROUPS Dj-dIj. 155 Space-Gboup D3. — (Rhombohedral Axes.) One equivalent position : (a) 0. (b) H h Two equivalent positions: (c) u u u; ti u u. Three equivalent positions : (d) uuO; tiOu; Ouu. (e) uu^; u|u; ^uti. Six equivalent positions: (f) xyz; yzx; zxy; yxz; xzy; zyx. E. HOLOHEDRY. Space-Group Dl^. — (Hexagonal Axes.) One equivalent position: (a) 0. (b) OOi Two equivalent positions: (c) HO; fiO. (e) OOu; u. /JN 13 1.211 \^) 3 5 2 J 3 3 2- Three equivalent positions: (f) HO; o|0; ^00. (g) Hi; OH; ^0^. Four equivalent positions: /U^ 12,1. 1 2 ,-, . 21,,. 2 1 fi Wl^U, ^gU, 33U, 3lU. Six equivalent positions: (i) uiiO; 2u, ti, 0; u, 2u, 0; uuO; 2u, u, 0; u, 2u, 0. (j) uui; 2u, ti, ^; u, 2u, i; uu|; 2u, u, ^; ti, 2u, ^ (k) uuv; Otiv; u v; uOv; uuv; Ouv. Twelve equivalent positions: (1) xyz; y-x, x, z; y, x-y, z; x, x-y, z; y.xz; y-x, y, z; xyz; x-y, x, z; y, y-x, z; x, y-x, z; yxz; x-y, y, z. Space-Group Dgj. — (Hexagonal Axes.) Two equivalent positions : (a) 000; OOi (c) HO; f H- (b)OOi; OOf. (d)Hi; fio. Four equivalent positions : (e) OOu; OOti; 0, 0, ^-u; 0, 0, u+i (f) Hu; Hti; f, h l-u; f, i u-M. y-x, X, z; y, X- -y, z; yxz; y-x, y, z; x-y, X, l-z; y, y- -X, i-z y, X, z+^; x-y, y, z+l 156 THE HEXAGONAL SPACE-GROUPS D^-Dm- Space-Group D^ (continued). Six equivalent positions : (a\ 111' nil- ini- in3' 113. nis (h) uuO; 2u, u, 0; u, 2u, 0; uu|; 2u, u, ^; Q, 2u, §. Twelve equivalent positions: (i) xyz; X, x-y, z; X, y, i-z; X, y-x, z+l; Space-Group D'a. — (Hexagonal Axes.) One equivalent position : (a) 0. (b) i Two equivalent positions: (c) OOu; OOti. (d) Hu; Hu. Three equivalent positions : (e) HO; OiO; ^00. (f) Hi; OH; ioi Six equivalent positions : (g) uuO (h) uu| (i) u ti V Twelve equivalent positions: (j) xyz; y-x, x, z; y, x-y, z; X, y-x, z; yxz; x-y, y, z; xyz; x-y, x, z; y, y-x, z; X, x-y, z; yxz; y-x, y, z. Space-Group T>za- — (Hexagonal Axes.) Two equivalent positions: (a) 0; 0|. (b) 0^; f . Four equivalent positions: (c) OOu; OOQ; 0, 0, ^-u; 0, 0, u-hi (d)Hu; fiti; i i ^-u; i I, u-f-i Six equivalent positions: (e) OH; |0i; Hi; oH; iof; Hi (f) uuO; OuO; uOO; uu^; Ou|; uOi Twelve equivalent positions : (g) xyz; y-x, x, z; y, x-y, z; X, y-x, z; yxz; x-y, y, z; X, y, ^-z; x-y, x, ^-z; y, y-x, ^-z; X, x-y, z+i; y, x, z-F^; y-x, y, z-f^. OuO; tiOO; uuO; OuO; u 0. Ou|; uO^; uu^; Ou^; u 1. 2u, u, v; u, 2u, v; uuv; 2u, u, v; u, 2u, v. THE HEXAGONAL SPACE-GROUPS 0^-0^. 157 Space-Group Di^. — (Rhombohedral Axes.) One equivalent position: (a) 0. (b) H i Two equivalent positions: (c) uuu; uuti. Three equivalent positions: (d)OO^; 0|0; ^00. (e) H 0; |0i; OH- Six equivalent positions: (f) utiO; uOu; Outi; uuO; uOu; Otiu. (g) uu|; u|u; ^uu; tiu^; u|u; ^uu. (h) uuv; uvu; vuu; tiuv; uvti; vuu. Twelve equivalent positions: (i) xyz; yzx; zxy; yxz; xzy; zyx; xyz; yzx; zxy; yxz; xzy; zyx. Space-Group DgV — (Rhombohedral Axes.) Two equivalent positions: (a) 000; Hi (b)iii; f f f . Four equivalent positions : (c) uuu; uuu; |-u, |-u, |-u; u+|, u+i, u+^. Six equivalent positions : fr\\ 133. 331. 113. 13,1. 3.11. 113 \MJ 44 7> 444) 444) 444; 444) 44 f* (e) uuO; uOu; Outi; l-u, u-f-i h; u+l, I, |-u; i, l-u, u-|-|. Twelve equivalent positions: (f) xyz; yzx; zxy; yxz; xzy; zyx; i-x, h-y, |-z; h-y, l-z, 5-x; §-z, |-x, ^-y; y+i x-l-i z+l; x-hi z-f-i y-f-^; z-|-^, y-H, x-H. HEXAGONAL DIVISION. A. TRIGONAL PARAMORPHIC HEMIHEDRY. Space-Group Cgh. — (Hexagonal Axes.) One equivalent position: (a) 000. (c) HO. (e) UO. (b)00|. (d)Hi (f) IH. Two equivalent positions: (g) OOu; OOu. (i) \ liu; Hu. (h)Hu; HQ. 158 THE HEXAGONAL SPACE-GROUPS Cgh-D^. Space-Group C^^ {continued). Three equivalent positions: (j) uvO; V— u, u, 0; v, u— v, 0. (k) uv^; v-u, u, I; v, u-v, |. Six equivalent positions : (1) xyz; y-x, x, z; y, x-y, z; xyz; y-x, x, z; y, x-y, z. B. HEMIHEDRY WITH A THREE-FOLD AXIS. {Trigonal Holohedry.) Space-Group DaV. — (Hexagonal Axes.) One equivalent position: (a) 0. (c) H 0. (b)ooi. (d)Hi (e) f 1 0. Two equivalent positions: (g) OOu; OOu. (i) (h) Hu; Hii. fiu; fiu. Three equivalent positions: (j) uuO; 2u, u, 0; u, 2u, 0. (k) uu|; 2u, u, ^; u, 2u, ^. Six equivalent positions: (1) uvO; v— u, u, 0; v, u— v, 0; u, u— V, 0; vuO; v— u, v, 0. (m)uv^; v-u, u, i; v, u-v, |; u, u-v, ^; vu^; v-u, v, ^. (n) uuv; 2u, u, v; u, 2u, v; uuv; 2u, u, v; u, 2u, v. Twelve equivalent positions: (o) xyz; y-x, x, z; y, x-y, z; X, x-y, z; yxz; y-x, y, z; xyz; y-x, x, z; y, x-y, z; X, x-y, z; yxz; y-x, y, z. Space-Group Da^u- — (Hexagonal Axes.) Two equivalent positions : (a) 0; ooi (d)iH; Hi (b)OOi; OOf. (e) fiO; IH. (c) HO; Hi (f) fH; fH. /^owr equivalent positions: (g) OOu; OOu; 0, 0, ^-u; 0, 0, u+i (h)Hu; Hu; i f, l-u; i I, u-fi (i) Hu; IH; f, i ^-u; f, i u-fi THE HEXAGONAL SPACE-GROUPS D^-Dgt- 159 Space-Group D^ (continued). Six equivalent positions : (j) uuO; 2u, u, 0; u, 2u, 0; uu|; 2u, u, ^; u, 2u, ^. (k) uvi; v-u, u, i; v, u-v, i; u, u-v, I; vuf; v-u, v, f. Twelve equivalent positions: (1) xyz; y-x, x, z; y, x-y, z; X, x-y, z; yxz; y-x, y, z; X, y, h-z; y-x, x, ^-z; y, x-y, ^-z; X, x-y, z+^; y, x, z+|; y-x, y, z-f-i Space-Group Dsh. — (Hexagonal Axes.) One equivalent position : (a) 0. (b) 1. Two equivalent positions : (c) HO; fiO. (e) OOu; u. K^J 3 1 2 > 3 3 2- Three equivalent positions : (f) uuO; OuO; uOO. (g) uu|; Ou|; uO^ Four equivalent positions : (V,\ 12,,. 2 1 fi . 12,-,. 21,, W 3 I U, 3 3 U, 3 3 U, 3 3 U. Six equivalent positions: (i) u u V (j) uvO vuO (k) uvi vu| Ouv; uOv; tiOv; uuv; Ou^. v-u, u, 0; V, u— V, 0; u, v-u, 0; u-v, V, 0. v-u, u, I; V, u-v, I; u, v-u, I; u-v, V, |. Twelve equivalent positions: (1) xyz; y-x, x, z; y, x-y, z; X, y-x, z; yxz; x-y, y, z; xyz; y-x, x, z; y, x-y, z; X, y-x, z; yxz; x-y, y, z. Space-Group Dg^. — (Hexagonal Axes.) Two equivalent positions: (a) 000; OOi (c) Hi; IH. (b)ooi; oof. (d) HI; Hi. Four equivalent positions: (e) OOu; OOti; 0, 0, |-u; 0, 0, u-|-|. (0 Hu; Hu; i i l-u; |, i u-h|. 160 THE HEXAGONAL SPACE-GROUPS Da^-Ce. Space-Group Da^^ (continued). Six equivalent positions: (g) uuO; OtiO; uOO; uu|; Ou|; u 0^ (h) uv|; v-u, u, i; v, u-v, i; vuf; u, v-u, f; u-v, v, f. Twelve equivalent positions : xyz; y-x, X, z; y, x-y. z; X, y-x, z; yxz; x-y, y, z; X, y, i-z; y-x, X, h-z; y, x-y. 5-z; X, y-x, z+^; y, X, z+^; x-y, y, z+i C. HEXAGONAL TETARTOHEDRY. Space-Group CJ. — (Hexagonal Axes.) One equivalent position : (a) OOu. Two equivalent positions: (b)i!u; flu. Three equivalent positions: (c) Hu; 0|u; fOu. Six equivalent positions : (d) xyz; y-x x, z; y, x-y, z; xyz; x-y, x, z; y, y-x, z. Space-Group Ce- — (Hexagonal Axes.) Six equivalent positions : (a) xyz; y-x, x, z-j-^; y, x-y, z-|-f; X, y, z+l; x-y, X, z+f; y, y-x, z+i Space-Group Ce- — (Hexagonal Axes.) Six equivalent positions: (a) xyz; y-x, x, z+f; y, x-y, z+f; X, y, z+f; x-y, X, z+f; y, y-x, z+f. Space-Group Ce. — (Hexagonal Axes.) Three equivalent positions: (a) OOu; 0, 0, u+f; 0, 0, u+f. (b) ffu; 0, f, u+f; f, 0, u+f. Six equivalent positions: (c) xyz; y-x, x, z+f; y, x-y, z+f; xyz; x-y, x, z+f; y, y-x, z+f. THE HEXAGONAL SPACE-GROUPS Cfl-Coy. 161 Space-Group C*. — (Hexagonal Axes.) Three equivalent positions: (a) OOu; 0, 0, u+h 0, 0, u+f. (b) Hu; 0, h u+h h 0, u+|. Six equivalent positions: (c) xyz; y-x, x, z+|; y, x-y, z+f; xyz; x-y, x, z-\-\; y, y-x, z+f. Space-Group Q%. — (Hexagonal Axes.) Two equivalent positions : (a) OOu; 0, 0, u+|. (b) H u; f i u+i. Six equivalent positions : (c) xyz; y-x, x, z; y, x-y, z; X, y, z+l; x-y, x, z+|; y, y-x, z+|. D. HEMIMORPHIC HEMIHEDRY. Space-Group Cjy. — (Hexagonal Axes.) One equivalent position: (a) OOu. Two equivalent positions: (b)Hu; fiu. Three equivalent positions: (c) Hu; OJu; iOu. Six equivalent positions : (d) uuv; Ouv; uOv; utiv; Ouv; uOv. (e) utiv; 2u, % v; u, 2u, v; uuv; 2u, u, v; u, 2u, v. Twelve equivalent positions: (f) xyz; y-x, x, z; y, x-y, z; xyz; x-y, x, z; y, y-x, z; X, y-x, z; yxz; x-y, y, z; X, x-y, z; yxz; y-x, y, z. Space-Group Cgy. — (Hexagonal Axes.) Two equivalent positions: (a) OOu; 0, 0, u-|-^. Four equivalent positions: (b)Hu; f^u; f, i u+^; i f, u+i 162 THE HEXAGONAL SPACE-GROUPS Cey-Cfi. Space-Group Cev (continued). Six equivalent positions: (c) Hu; 0|u; ^Ou; h h u+l; 0, i u+l; i 0, u+i Twelve equivalent positions: (d) xyz; y-x, x, z; y, x-y, z; xyz; x-y, X, z; y, y-x, z; x, y-x, z+l; y, x, z+J; x-y, y, z+^; X, x-y, z+^; y, x, z+|; y-x, y, z+i Space-Group Cev- — (Hexagonal Axes.) Two equivalent positions : (a) OOu; 0, 0, u+l. Four equivalent positions : (d) 3 3 U; 3, 3, U-t-2j 3 3 U> t> 3? ^"r^* Six equivalent positions: (c) uuv; Otiv; uOv; u, u, v+l; 0, u, v+l; u, 0, v+|. Twelve equivalent positions: (d) xyz; y-x, x, z; y, x-y, z; X, y, z+l; x-y, X, z+|; y, y-x, z+^; X, y-x, z; yxz; x-y, y, z; X, x-y, z+l; y, x, z+l; y-x, y, z+^. Space-Group CeV — (Hexagonal Axes.) Two equivalent positions : (a) OOu; 0, 0, u+|. (b)Hu; I, i u+i Six equivalent positions: (c) utiv; 2u, u, v; u, 2u, v; u, u, v-h^; 2u, u, v+?; ti, 2u, v+§. Twelve equivalent positions: (d) xyz; y-x, x, z; y, x-y, z; X, y, z+l; x-y, X, z+?; y, y-x, z+l; X, y-x, z+J; y, x, z-M; x-y, y, z+|; X, x-y, z; yxz; y-x, y, z. E. PARAMORPHIC HEMIHEDRY. Space-Group CeV — (Hexagonal Axes.) One equivalent position : (a) 0. (b) i THE HEXAGONAL SPACE-GROUPS Ceh-Dj. 163 Space-Group Ceh (continued). Two equivalent positions : (c) HO; UO. (e) OOu; OOu. K^J 3 3 2 > 3 3 2' Three equivalent positions: (f) HO; o|0; 100. (g) HI; OH; Hi Four equivalent positions: /'V.^ 12,,. 21,,. 1 2 ,T . 2 1 ,T W33U, 3 3 U, 33U, ^lU. Six equivalent positions : iOu. (i) Hu; Oiu; |0u Hu; 0|u (j) uvO; v-u, u, 0; V, u-v, 0; uvO; u-v, u, 0; V, v-u. 0. (k)uv|; v-u, u, 1; V, u-v. 1; Q^l; u-v, u, h; V, v-u, 1 2' Twelve equivalent positions: (1) xyz; y-x, x, z; y, x-y, z; xyz; x-y, x, z; y, y-x, z; xyz; y-x, x, z; y, x-y, z; xyz; x-y, x, z; y, y-x, z. Space-Group Cei. — (Hexagonal Axes.) Two equivalent positions: (a) 0; 00 i (b) 0|; oof. (c) HO; fH (d)HI; HO Four equivalent positions: (e) OOu; OOu; 0, 0, |-u; 0, 0, u-f-i /f\ 12,,. 12,-;. 2 1 1 ,, . 2 1 ,, I 1 W 3 3U, 35U, 5, ^, jr — U, 3, 3, Ui-2. Six equivalent positions: ('c^lil- n4i' 401- 111. oil. ini (h) uvO; v-u, u, 0; v, u-v, 0; uv|; u-v, u, I; v, v-u, i Twelve equivalent positions: (i) xyz; y-x, x, z; y, x-y, z; X, y, z-f-|; x-y, X, z^-§; y, y-x, z+|; xyz; y-x, x, z; y, x-y, z; X, y, l-z; x-y, x, |-z; y, y-x, |-z. F. ENANTIOMORPHIC HEMIHEDRY. Space-Group DJ. — (Hexagonal Axes.) One equivalent position: (a) 0. (b) h 164 THE HEXAGONAL SPACE-GROUPS dJ-d|. Space-Group DJ {continued). Two equivalent positions: (c) HO; UO. (e) OOu; OOu. f/\\ 12 1. 2 11 \MJ ■532; 332' Three equivalent positions: (f) HO; oio; ioo. (g) Hi; OH; hoh Four equivalent positions: \M) 3 3^; 3 3 U; 3 3 U; 3 3 U. Six equivalent positions: (i) Hu (J) uuO (k) uu| (1) utiO (m) u u I 0|u; iOu; Hti; OH; |0u. OuO; uOO; uuO; OuO; u 0. Ou|; GO I; uu|; Ou|; u i. 2u, u, 0; u, 2u, 0; uuO; 2u, u, 0; u, 2u, 0. 2u, u, I; u, 2u, I; uu|; 2u, u, ^; u, 2u, |. Twelve equivalent positions : (n) xyz; y-x, x, z; y, x-y, z; xyz; x-y, x, z; y, y-x, z; X, y-x, z; yxz; x-y, y, z; X, x-y, z; yxz; y-x, y, z. Space-Group T>1. — (Hexagonal Axes.) Six equivalent positions: (a) OuO; uO|; tiuf; Ou|; uOf; uu^. (b) uura; 2u, u, f; u, 2u, Y2; ^^¥2', 2u, u, i; u, 2u, rz- Twelve equivalent positions: (c) xyz; y-x, x, z+i; y, x-y, z+|; X, y, z+i; x-y, x, z+|; y, y-x, z+i; X, y-x, z; x-y, y, f-z; y, x, |-z; X, x-y, l-z; y-x, y, i-z; y, x, f-z. Space-Group 1l>1. — (Hexagonal Axes.) Six equivalent positions: (a) OuO; uOf; uu^; Ou|; uO^; uuf. (b)uura; 2u, u, f ; u, 2u, i^; uufa; 2u, u, i; u, 2u, Twelve equivalent positions: (c) xyz; y-x, x, z-|-f; y, x-y, z+^; X, y, z+i; x-y, X, z-f-^; y, y-x, z-M; X, y-x, z; x-y, y, ^-z; y, x, f-z; X, x-y, i-z; y-x, y, f-z; y, x, ^-z. --- y 12- THE HEXAGONAL SPACE-GROUPS I>t-T>t 165 Space-Group Dg. — (Hexagonal Axes.) Three equivalent positions: (a) 0; oof; 00 i (b) 00^; OOi; OOf. Six equivalent positions: (c) ^ ^ 3 ; 0^0; nil. 1 n 2 (e) OOu OOu (f) Hu iOu (g) uui (h) uuf (i) uu| (J) uuf 0, 0, u+f ; 0, 0, u+i; 0, 0, l-u; 0, 0, f-u. 0, h u+f; i 0, u+f; 0, 2} 3~Ui ^> ?> a ^• OuO; uOf; Oui; uOf; 2u, ti, 0; u, 2u, u, I; u, Twelve equivalent positions: (k) xyz; y-x, x, xyz; x-y, x, X, y-x, z; x-y, y, 2 3 uuf; uuf; 2u, 2u, OuO; Ouf; uuf; uuf; uO uO 2u, 2u, 0; 1 . u, 2u, u, 2u, z+f; z+f; 3 ^7 X, X-y, z; y-x, y, t-z; Space-Group Dg. — (Hexagonal Axes.) Three equivalent positions: (a) 0; OOf; OOf. (b) OOf; y, x-y, z+f; y, y-x, z+f; y, X, f-z; y, X, f-z. /n\ 111. n 1 1. \^J "5 2 6) " ^ 3 > (d) Iff; OfO; i n 1 2^3' Six equivalent positions: (e) OOu OOu (f) ffu fOu (g) uuf (h) uuf (i) uuf 0, 0, u+f; 0, 0, u+f; 0, 0, f-u; 0, 0, f-u. 0, f, u+f; f, 0, u+f; 0, f, f-u; f, f, f-u. Ouf; uOf; uuf; Ouf; uOf. OuO; uOf; uuf; OuO; u f . 2u, Q, f; u, 2u, f; uuf; 2u, u, f; u, 2u, f. 2u, u, 0; u, 2u, f; uuf; 2u, u, 0; u. 2u, f. (J) uuf Twelve equivalent positions: (k) xyz; y-x, x, z+f; y, x-y, z+f; xyz; x-y, x, z+f; y, y-x, z+f; X, y-x, z; x-y, y, f-z; y, x, f-z; X, x-y, z; y-x, y, f-z; y, x, f-z. Space-Group DJ. — (Hexagonal Axes.) Two equivalent positions: (a) 0; (b)OOf; (c) (d) 12 1. 2 13 3 3 4> 3 3 4- 12 3.. 3 3 4; 2 11 3 ^ ?• 166 THE HEXAGONAL SPACE-GROUPS De-Dgh. Space-Group Dg (continued). Four equivalent positions: (e) OOu; OOu; 0, 0, u+f; 0, 0, |-u. ^.1/ 3 3 "> 3 3 ^) 3) 3j l^ r2> 3> 3> 2 '^• Sto; equivalent positions : (g) uuO; OuO; uOO; uu^; Ou^; uOi (h) uui; 2u, u, i; u, 2u, i; uuf; 2u, u, f; u, 2u, i Twelve equivalent positions: (i) xyz; y-x, x, z; y, x-y, z; X, y, z+l; x-y, x, z+|; y, y-x, z+|; X, y-x, z; x-y, y, z; yxz; X, x-y, |-z; y-x, y, ^-z; y, x, §-z. G. HOLOHEDRY. Space-Group Dgh. — (Hexagonal Axes.) One equivalent position : (a) 0. (b) 1. Two equivalent positions: (c) HO; (d)P^ HO. IH. (e) OOu; OOu. 1 1> Three equivalent positions: (f) HO; 0|0; fOO. Four equivalent positions: (g) 111. ^ "2 It OH; 1 i 2 '-' 2- (h) 3 3 U, 3 3 U, I 3 U, ^ ^ U. Six equivalent positions: (i) Hu; Oiu; §0u; Hu; (j) uuO; OiiO; uOO; tiuO; (k) uu|; Otii; uO|; au^; (1) uQO; 2u, u, 0; u, 2u, 0; (m)uu^; 2u, u, ^; u, 2u, |; Twelve equivalent positions : OH; iOu. OuO; uOO. Oui; uOi auO; 2u, u, 0; u, 2u, Qui; 2u, u, i; u, 2u, i. (n) u u V uu V (o) uuv uu V (p) u V uvO Ouv; uOv; tiuv; Ouv; uOv; Ouv; tiOv; uuv; Ouv; uOv. 2u, u, v; u, 2u, v; uuv; 2u, u, v; u, 2u, v; v; uuv; 2u, u, v; u, 2u, v. V, u-v, 0; V, v-u, 0; vuO; vuO. 2u, ti, v; u, 2u V— u, u, u— V, u, Q, V— u, 0; u— V, V, u, u— V, 0; V— u, V, THE HEXAGONAL SPACE-GROUPS Dei-Doh- 167 Space-Group Dgh (continued). v-u, u, I; V, u-v, i; u-v, u, I; V, v-u, ^; u, v-u, I; u-v, V, h; vu|; u, u-v, ^; v-u, V, ^; vu|. (q) uv^; Twenty-four equivalent positions: (r) xyz; xyz; X, y-x, z; X, x-y, z; xyz; xyz; X, y-x, z: y-x, X, z; y, x-y, z; x-y, X, z; y, y-x, z; x-y, y, z; yxz; y-x, y, z; yxz; y-x, X, z; y, x-y, z; x-y, X, z; y, y-x, z; x-y, y, z; yxz; X, x-y, z; y-x, y, z; yxz. Space-Group Deh- — (Hexagonal Axes.) Two equivalent positions: (a) 000; OOi (b) OOJ; OOf. Four equivalent positions: HI. Hi; HI; Hi (e) OOu; OOQ; 0, 0, |-u; 0, 0, u+i /Six equivalent positions: (c) H Oj f I Oj 3 3 5? (d) i ! i; 1 1 . 113. 1 1 J) 11) 1 3 . 1 4, (f) HO; 0^0; 100; | (g) Hi; OH; Hi; Etgf^f equivalent positions: (h) Hu; fiu; f, i u+i; i f, u+i; Hu; HQ; !, i I-u; i !, i-u. Twelve equivalent positions: (i) Hu; Oiu; iOu; Hii; Oiu; iOO; i I, i-u; 0, i, i-u; i 0, i-u; i i u+i; 0, i, u+i; i, 0, u+i. OuO; tiOO; uuO; OuO; uOO; Oui; uOi; uui; Oui; uOi. 2u, u, 0; u, 2u, 0; tiuO; 2u, u, 0; u, 2u, 0; 2u, u, i; u, 2u, i; uui; 2u, u, i; ti, 2u, i. v-u, u, i; V, u-v, i; u-v, V, i; V, v-u, i; Q, v-u, I; u-v, V, f; vui; u, u-v, f; v-u, v, |. (j) uuO uui (k) uuO uui (1) u v| u vi vuf 168 THE HEXAGONAL SPACE-GROUPS Da^-De^ . ■z; Space-Group Del {continued). Twenty-four equivalent positions: (m) xyz; xyz; X, y-x, z; X, x-y, z; X, y, |-z; X, y, |-z; X, y-x, z+l; X, x-y, z+i; Space-Group Dgh. — (Hexagonal Axes.) Two equivalent positions: (a) 0; OOi (b) OOJ; f . Four equivalent positions : (c) IfO; (d)Hi; (e) OOu; OOii; 0, 0, |-u; 0, 0, u+|. Six equivalent positions: y-x, x, z; x-y, x, z; x-y, y, z; y-x, y, z; y-x, X, i- x-y, X, ^-z; x-y, y, z+i; y-x, y, z+l; y, x-y, z; y, y-x, z; yxz; yxz; y, x-y, §-z; y, y-x, h-z; y, X, z-fl; y, X, z+^. 111. 3 3 2, 2 13. 3 3 4, 2 1 f). 3 3*-', 2 11. 3 3 4, 111 3 3 2- ill 3 3 4' /f\ 111. nil- inl' 111- 04^' i-Ol W 15 4, "¥4, 2^4, ¥¥4, "5 4, t " 4- (g) uuO; OuO; uOO; Qui; Ou^; uO|. Etgf^f equivalent positions: (h) Hu; Hu; f , i u+l; i i u-|-|; i ^ U, 53U, S, ^, 5 — U, 3, S, ^ U. Twelve equivalent positions: (i) uu i uuf (J) uvO u v| vuO vG| (k) UU V UUV 2u, u, i; u, 2u, i; uuf; 2u, u, f; 2u, u, f; u, 2u, f; uui; 2u, u, i; V— u, u, 0; V, u— V, 0; V, u-v, i; u-v, V, 0; v-u, V, J. uOv; QOv; u-v, u, i; u, V— u, 0; u, u-v, i; Oflv; Oflv; % u, v-l-i; 0, u, v+^; u, 0, v-f-^; G, G, i-v; 0, u, i-v; u, 0, |-v. Twenty-four equivalent positions: (1) xyz; X, y, z-M; X, y-x, z; X, x-y, i-z; xyz; X, y, i-z; X, y-x, z; X, x-y, z-l-l; y-x, X, z; y, x-y, z; x-y, X, z+i; y, y-x, z+i; x-y, y, z; y-x, y, i-z; y-x, X, z; x-y, X, i-z; x-y, y, z; y-x, y, z+l; yxz; y, X, i-z; y, x-y, z; y, y-x, i-z; yxz; y, X, z-\-^. G, 2u, f ; G, 2G. I THE HEXAGONAL SPACE-GROUP DeV • 169 Space-Group D6*h. — (Hexagonal Axes.) Two equivalent positions : (a) 0; 00 i (b) OOi; OOf. Four equivalent positions : V^/ 3 3 4 ) 3 3 4- (d) I 12 3. 3 3 4; 111 3 3 4- (e) OOu; OOti; 0, 0, ^-u; 0, 0, u+f. (0 H 2 1 n_|_l • 2 1,-,. 3j 3> " r2> 3 3"? 12 1 3j 3> ^ -u. Six equivalent positions : (g) HO; 010; ^00; H^; OH; hoh (h) uui; 2u, u, i; u, 2u, i; uuf; 2u, u, |; u, 2u, f. Twelve equivalent positions : (i) uuO uu^ (j) uvi ti v| vuf V ti J OuO; uOO; uuf; Ou|; uO|; Oti^; uO^; uuO; OuO; uOO. v-u, u, t 3 . V, u-v, I; u-v, u, t; V, v-u, i; u, V u, f; u-v, V, f; u, u-v, i; v-u, V, i. (k) uuv; 2u, u, v; u, 2u, v; uuv; 2u, u, v; u, 2u, v; u, ti, ^-v; 2u, u, §-v; u, 2u, |-v; u, u, v+l; 2u, u, v+h; % 2u, v+^. Twenty-four equivalent positions : (I) xyz; X, y, z+^; X, y-x, z; X, x-y, \-z] X, y, §-z; xyz; y-x, x, z; y, x-y, z; x-y, X, z-\-\; y, y-x, z-f-^; yxz; x-y, y, z; y-x, y, h- y-x x-y, x, z; z; X, 2~z; y, X, t-z; y, x-y, \-z] Y, y-x, z; X, y-x, z+i; x-y, y, z-|-|; y, x, z-H; X, x-y, z; y-x, y, z; yxz. 170 tables: triclinic and monoclinic systems. TABLES. The following tables provide a summary of the number and kinds of the different arrangements to be obtained from each of the space-groups. In these tabulations the symbol 1 (0), for instance, signifies one arrangement (a special case) having no variable parameters; similarly the symbol 3 (2) would mean three arrangements having two variable parameters each. Table 3.— TRICLINIC SYSTEM. Space-Group. Number of equivalent positions. 1 2 A. Hemihedry: Ci 1(3) 8(0) 1(3) B. Holohedry: ci Table 4.— MONOCLINIC SYSTEM. Space-Group. Number of equivalent positions. 1 2 4 8 A. Hemihedry: CI 2(2) 4(1) 8(0) 1(3) 1(3) 1(2) 1(3) 1(3) 2(1) 4(1); 2 (2) 4 (0); 1 (2) 4(0) 4(0); 2(1) 4(0) 1(3) 1(3) 1(3) 1(3) 1(3) f2(0) 2(1) ll(2) 1(3) 1(3) 4 (0); 1 (1) CI c^ ct B. Hemimorphic hemihedry: d c^ c; C. Holohedry: CL 1 (3) CL Cjh CSh l'(S) tables: orthorhombic system. 171 .2 ■s •s J s i C5 CO . . . . 00 c? o "* cv? CO CO CO CO cv? J^ CO c^ CO "-^ ?? 3 S ""' ""* d d •^'-^'~' 1— li— li-Hi— IrHi-HrHi-Hi-Hj— 1 ^ i— 1 CO CO ^^ t— 1 i-H • CI C<J ^ w o B N rH PQ •41 E-I ■^ c c 1 c ei C a J > 5 > I ^ '''.'.'.'.'.'.'....■..'.'.'.'.'.'. '. <» ■ ■ ■. "c^a i; ^ .^ ''''.'.'.'.'.'.'.'.'.'.'.'.'.'.'.'. '. :. J • • : : : : -.:::;: r«ft .a ■ ..... j|00000U00000Q0Q0U0000U0 172 tables: orthorhombic system. «*5 O o S3 "3 a 00 -«*< CO CO tH CO CO 1-1 1-4 O r-l rH COCOCOCOCOCOCOCOCOCO CO cocococo COCOCOCOi— ^i-hOi— ii— • ,_(_<r-(rHC^t^Tt<C>CO 'I (N 1-1 (M !-< (N --I IM 1-1^ ■«0 1— I CO CO (N t-l CO (N "® O* ^ O' Ch" o' Ch* o' Ch' o' ,'^^ '^ '~', Ch o'o'<^ (MO0C<J(M(NCO(N<N(N (M(M(>J . O . O, rH 0,30, (M -^ 00 -ti O 00 QO tables: orthorhombic system. 173 fsi [SSw PO • 1-H i-H ^^.-. (M ^ ^^ s ^-^ CO CC CO CC '^ '^ CO ■v ^^"^ ^•"•^ vO M M ", ^ CO CO i-H T-l ^ ^ ^ ^ ^g^ CO <M 1— 1 rH r-4 .1 ^_^ ^ ^^ ^^ ^^ ^,— , , ^ ^,_^ a (M C^???^— 1 C^ »-i <N » +J N*-^ >*-^ t3 a 00 s 'S5 ^C^O'^ -^■*.-i(N->*C03-C0 sE 'S'll'^. 1— ( I— 1 i-H (M r-l (M '-' ^■^i^o'o''^ O -,-H -* — b3 c O p2 o3 ,-( lO iQ IM '^ T-H (N (N _> :^ '3 ^— . \ <^* ^-"^ .'■"V ^-^s o* (N >-H T-H 1-^ T-H 0) ^ >— » ■S—^ N..^ **— ^ *© l-H T-H o ; "=° S •"' o o :5 ■ ■» • ^H c» Tj< ■^ . • *^ CO J O ^ 2 (M g-O 3-0^ (N • «C ■>* ■ s O a (N (N c^ ■* s s o „ ^ ,— ^ w M 5 , P5 ■<*< -^ o w H S ? 1 >-^ H .J n £5 C u c *. C s i g 1. a: i j^ flO » ^ C4 <0 -W lA <e -■-■ 1 3 •-« - a -.j3«j:o»jc»xc-ij3«jaNj a « J a Mjsr<d > > > > > > > > > > > > > 174 tables: tetragonal system. Table 6.— TETRAGONAL SYSTEM. Space-Group. Number of equivalent positions. 16 32 A . Tetartohedry of the second Sort: S] SJ 4(0) B. Hemihedry of the i>econd Sort: V^ 4(0) Yl Yl yi VI. yi v^ Vi» vr 3(1) 4(0) 2(0); 2(1) 6(0) 2 (0); 1 (1) 2(0) 4(0) C. Tetartohedry: CI CI CI c\ c\ c: 2(1) D. Paramorphic hemihedry: Cih Cih Cib G4h 4(0) 2(0) 1(1) 3(1) l'(l) 2(0); 2(1) 6(0) 2 (0); 1 (1) 2(0) 2(0) 1(3) 2(1) (1); 1 7(1) (1); 1 2(1) 3(1) 4(0) 4(0) 4(0) 4(0) 2(0); 1(1) 2(0) 1(3) 1(3) 1(3) 1(3) 1(3) (1); 2 5(1) 4(1) 4(1) 2(1) 4(0) (1); 1 2(1) (3) (3) (3) (3) (1) (1) 1(1); 2(2) 3(1); 1(2) 2(0); 1(1) 2(0); 2(1) 2(0); 1(1) 2(0) (2) (2) 1(3) 1(3) 1(3) 1(3) (1); 1 4(1) 1(3) 1(3) (2) (3) (3) (3) (3) 1(3) 1(3) tables: tetragonal system. Table 6.— TETRAGONAL SYSTEM (Continued). 175 Space-Group. E. HemiTnorphic hemihedry: a a. V^4v. CIO 4v pll C12 4v Number of equivalent positions. 2(1) F. Enantiomorphic hemihedry: Dl D^ DJ DJ DJ d: D^ DJ d: Dl" G. Holohedry: Dl,. DL DL DJh ■L'4h DJ, dIh D8 4h D» 4b Dl A^4h. ■L'4h- ■L'4h- D18 4h- ■'-'4h- ^4h- 4(0) 4(0) 1(1) 2(1) 2(1) 1(1) 2(1) 1(1) 3(1) 1(1) 2(0); 2(1) 2 (0); 1 (1) 6(0) 2(0) 2(0) (0); 2 4(0) 4(0) 2(0) 4(0) 2(0) (0); 1 6(0) 4(0) 2'(0) 2(0) 2(0) 2(0) (1) (1) 3(2) 1(2) (1); 1 (1); 1 1(1) 1(1) 2(2) 2(1) 1(1) 2(1) 1(1) 7 3 3 1 9 4 3 1 2(0) 2 7 (0) (0) (0) 4 (0) (0) (0) 7 (0) 4 (0) 4 (0) 2 (0) (0) 4 2 1) 1) 1) 1) 1) 1) 1) 1) 1(1) 0) 1) 2(1) 2(1) 1(1) 1) 1(1) 1(1) 1(1) 1) (1) (1) (1) (1) 1(1) 0) 0) 1(3) (3) (3) (3) (3) (3) (3) (3) (2) (2) (2) (1) 1(3) 1(3) (3) (3) (3) (3) (3) (3) (1) (1) 5(2) (1); 1 (1); 1 (0);4 3(2) 2(1); 2(1); 1(0); 1(1); 3(1); 1(0); 5 (1); 3(1); 1(1); (2) (2) (1) (2) (2) (1) (2) (2) (1) (2) (2) (2) (0); 1(1); 1(2) 3(1); 1(2) 1(0); 4(1) 1(0); 3(1) 2(0); 1(1) 2(0) 16 1(3) 1(3) 1(3) 1(3) (3) (3) (3) (3) (3) (3) (3) (3) (3) (3) (3) 1(3) 1(3) 1(3) 1(3) 1(3) 1(3) 1 (3) 1(1); 3 (2) 2(1); 2(2) 2 (1); 1 (2) 1(0); 3(1) 32 1(3) 1(3) 1(3) 1(3) 176 tables: cubic system. On 0\ . CC . .00 s 00 i-H CO .00 CO O I— I a o S O <M 00 VO . 1— I CO .CO • Ol T-H -1—1 CO . 1— I CO w ■ 2- O CO CO . (N 00 2-C 1— ( 1-H O O O i— I o ?— I 1— I T— I CI 1— ( I— I C<J O o CO I— I £2- "^ ^^ .-H (N ■^ '-I (N 1-H i-H 1— I (M (M • 1-1 CO I— I Ttl O o o o ex I 03 CO ^^ ^H^HHH 11 %^ • c>» '^ J3 e^ j3 <• X *«■ j3 ><» JS «o o: 1^ .C a gHHHHHHH •8 HHHHEhH cq tables: cubic system. 177 1— t . . . . CO CO CO CO . . • I-H 1-H I-H I-H ss s : :gg; : : : : '.'.'.'. C^^^^CO CO • 1— * 1— ( 1-H I-H s (1)1 (1)1 s 00 ; ; ^^cc ; ; n^ ^-V .^V ^-V ^-V ^— V ..— V-V ^-V C^ ^ COCOCOCOi-Hi-Hi— lO r-< ■ CO M 1—1 • 1— 1 i-Hi-Hl-Hi-HCO(Ml— li-HiZn^C^ 1 a 12; 1-H CN| ! ! i-H ^H i i ! • • • • 3 • 3§ • 3 <*i • 1— t 1— ( ■ I-H • I-H C^ -I-H T— ( <M (M i-i o CO CO ""^ ! -— t CO CO 1-H I-H T— ( (3 • C^ 1—1 I-H CO I-H S3TlTlTls '■ •3b CO oa Ch" ^ S "^J • • (N c^ i-l C^ 1-H o O i 2(0) 1(1) 1(1) • 33 • • • §§32 • 1-H I-H . (M ^ ^ Cs^ 3 T-H 1-H 1-H t-4 33 • •^.33s C^ lO • 3 '-H 1-H (N ' — ^ 1-H ' — ^ I-H - • • • I-H • 1— ' 1— < .... CO S'^ s ■ ■ ■ ■ s ■ R 1-H I-H 1-H 1-H 00 ^ciro'o^o'333 3§§3§^^ iS ! 9 ^Hi-H^HM^H^Hi-HOfl ,-i ^ ,-, r-, -^ C^ C<i -I-H • T «o 32^ • • 2 • ^O 0,0 . . . . O, . M 1^ CO • • I-H • • • <N 1-H CO I-H • • • • r-l • i ^ : So : : So : : : SS : : : ; : • <N (M • • (N IM • • • • (N (N »*) S :::::: ; (M S :;::.::: : M :".■.; C4 : S : : o : : : • SSS •■■•§• • rH • • —1 • • • fH i-H 1— 1 -I-H »-« s :::::: S C 1 U ) i i ! D. Enantiomorphic hemihedry: o» C 666666 -2--. t?^ "c oc oooooo 1^ 178 tables: hexagonal system. Table 8.— HEXAGONAL SYSTEM. I. Rhombohedral Division Space-Group. A. Tetartohelry: CI C^ CI Ci B. Paramorphic hemihedry: CJi Csi C. Hemimorphic hemihedry: CI Cjv Csv Cjv C.v D. Enantiomorphic hemihedry: Dl D^ Df D^ T>1 D'a DI E. Holohedry: DL DL Dir...... DJa DL, Number of equivalent positions. 3(1) 1(1) 2(0) 2(0) 3(1) 1(1) 1(1) 6(0) 2(0) 2(0) 2(0) 2(0) 2(0) 2(1) 1(1) 1(3) 1(3) 1(3) 1(3) 2(0) 2(0) 1(2) 1(1) 1(2) 3(1) 2(1) 1(2) 1(1) 3(1) 2(1) 2(1) 2(1) 2(1) 2(1) 2(1) 2(1) 1(1) 2(1) 2(0); 1(1) 2(0) 4(0) 2(1) 2(0) 2(0) 1(1) 2(0) 2(0) 1(1) 2(1) 2 0) I'd) (3) (3) (3) (3) (3) (3) (3) (3) (3) (3) (3) (3) (3) (3) (3) 12 2(1); 1(0); 2(1); 1(0); 2(1); 1(0); 1(2) 1(1) 1(2) 1(1) 1(2) 1(1) 1(3) 1(3) 1(3) 1(3) 1(3) 1(3) tables: hexagonal system. 179 o O si tJ S -< o ^ o o ^ o » < K u 1—4 T 1 05 » »:3 n < H a B 3 00 CO CO CC CO CO ■^co "^ CO CO CO CO CO CO r-4 O o O . o CC ec • (M • •55 -SAg^ "e ?> s -« b S <» o <» §j s -« ^ = Oj-s; ^i 1 ») m -5 m •» S . Tn para hemi ■ £ -^0 QQ Q ft ; ; ; ; ; ; s « l*-2 : : : : : : 6 05 180 tables: hexagonal system. • CO CO CO CO 1-H 1-H 1-H I-H c^^c^c^ <VI CO fC fC M~ CO cv? CO CO CO CO CO CO ^»-H (M (M 1-H t— 1 »— 1 F- i-H »— t T}^^^^ CO r-H I-H 00 . 1-H 1-H • I-H 1-H I S^ I-H I-H d m'^m"?^ ~ IM 1- Ss"^ '-' a o (N "-I r-H r- - I-H r- lO (N (N «0 «0 (M ^ c^ o o I-H r-i > V o ■^ i 1-H 1— 1 s= I-H ! ! ! ! 1-H 1-H 1-H s • T— 1 .—1 rH <N rH • • • • (N (M (N »o ■—I o s • -ss • o ; : : 1-H (N (N • ■ M< M< • Cv| . . . I-H 1-H 1-H N 1— 1 1— 1 »— 1 i-H Tj • 71 • • -s "^ o Sg 1-H 1— 1 I— 1 04 g . . . .^ (M *H y~-> o ■• t ; ; ; : : g : : : 1-H (N ■<5y '■'■* N (N • • • 6 1 CO 5 lie °c °c °c ^ 1 -1 'c 1 -1 : : : : : : IQQQQQQ : . t . 'o 1 R^raW TO U^K^l^^^„ BORROWED lOAN DEPT. Ranewed books are subjac. ,o immediate recall. LD 21A-50w-12.'60 (BC.22l8l0)476B .General Library Universuy of California Berkeley ^& M^t^l^