Widely Used Hermann-Mauguin Notation, As Well As of The

J. Japan. Assoc. Min. Petr. Econ. Geol. 71, 299-307, 1976.

EXPRESSION OF SPACE GROUPS WITH NUMBERS OF TWO FIGURES AS THE KEYNOTE

YUKIO TAKANO Coll. Gen. Educ., University of Tokyo, Tokyo YOSHIHIKO SHIMAZAKI Geologica Survey of Japan, Tokyo

From 67 non-translation space groups, the authors separated 6 space groups having two each pairs of diffraction patterns . Classification numbers of two figures were given to the total 73 non-translation space groups, so as to express their crystal system, elements of symmetry and lattice type. The rest 157 space groups, derived from these non-translation space groups, were provided with letters (four in maximum) to be put before or after the classification numbers of the basic space groups, to express their screw axis and glide plane. With this new style of notation, 157 space groups can be expressed by not more than three letters inclusive of the basic numerals. Thus, almost all space groups are represented by the symbols that are two to four letters fewer than those of the existing Hermann-Mauguin notation. This new notation enables even the beginners to identify the crystal system and the point group to which the space group belongs.

widely used Hermann-Mauguin notation, INTRODUCTION as well as of the combination of elements It is a recent tendency that descriptive of symmetry in 230 space groups. Con data of chemical compounds are often sequently, looking at the symbol of a certain required to be accompanied by crystallogr space group, they will recognize no more aphic data. However, the reseachers who than the type of lattice, such as primitive, are not concerned with crystallography will body-centred, face-centred, etc., and whe find it quite troublesome to describe crystal ther it is of holohedry class of so-and-so class and space group of a chemical com crystal system. Most of crystallographers pound, not to speak of its chemical com may have to consult the International ponent, density, crystal system, unit cell, Tables for X-ray Crystallography (1952), in or the number Z of molecules contained order to know the orientation and combina therein. Also for the researchers who see tion of elements of symmetry for that space the description of crystal class or space group. group, it would be fairly difficult to discern The Schoenflies symbols and the which crystal class of what crystal system Hermann-Mauguin symbols are the most the crystal belongs to, if it was represented popular notations for expressing space only by the space group. Even the cryst groups. These two types of notations are allographers themselves generally have a often used together, as recommended by loose sense of the expression based on the the International Union of Crystallography.

(Manuscript received May 24, 1976) 300 Y. Takano and Y. Shimazaki

The Schoenflies notation, contrived in 1891, ainted themselves thoroughly with the was historical in that it introduced 230 space meaning of these symbols and the order of groups mathematically, by arranging point description, the Hermann-Mauguin nota groups to the respective lattice points of the tion is much more convenient than the space lattice and supplementing them with Schoenflies notation. Nevertheless, it translation. For all of the 32 point groups must be pointed out that few crystallogra the space group symbol has a numeral, rang phers would be able to answer promptly ing from the minimum 1 (C3h) to the that Ia3 belongs to the cubic system and maximum 28 (D2h), on the right shoulder. Ccc2 to the orthorhombic system, both as a Therefore, the point group to which the hemihedry class. Many crystallographers space group belongs can be indicated by might be perplexed when asked to tell the eliminating the numeral. In this respect, point groups for Ccc2 and Ccca. For the the Schoenflies notation is convenient. beginners it would take some time to com But, since the numerals are merely the prehend that mmm, 23 and 32 in the Her space group numbers that are mathemati mann-Mauguin notation are the expression cally systematized, they offer no informa of their belonging to the orthorhombic, tion on elements of symmetry of the space cubic and trigonal systems respectively. group, such as screw axis and glide plane. The authors, with their many years' In other words, so far as crystallography is experience of teaching crystallography, concerned, the Schoenflies symbols are not have been exploring a new method for informative on space groups, and so they describing space groups, a method which is are undesirable for educating beginners in easy to understand for the natural scientists crystallography. who are often forced to make crystallogra The Hermann-Maugiun notation, on the phic description in their reports, and is other hand, was formed in the present convenient for the crystallographers as century when analysis of crystal structure well. In the present paper the authors became popular. The notation makes it a report the results hitherto obtained and principle that the elements of symmetry for propose a new type of notation. primary description of point groups should be minimized. Elements of symmetry in NOTATION OF 67 LATTICE GROUPS space groups also are expressed in conformi ty with this principle. In the Hermann The authors thought that, in order to Mauguin notation of space groups, P, I, F make the space group symbols easily under and C denote primitive, body-centred and stood by natural scientists in general, such base-centred lattices respectively, 65, 43, symbols as P, I, F, C.... and m, c, b, a, 32 and 21 indicate screw axes, and c, b, a, n n, d.... should be avoided, because these symbols requirethe knowledgeof prescribed and d are glide planes. After the lattice rules for the usage. Also, giving considera indication, the symbols follow the prescribed tion to the occasion of putting the symbols order of description, beginning with the into the computer's memory, one of the data on the principal rotation axis, just authors decided to employ numbers of two like the symbols of point groups. There figuresas the keynote, like the case of classi fore, to the crystallographers who are acqu fication numbers for elements as proposed Expression of space groups with numbers of two figures as the keynote 301 previously (Takano, 1976). Since it is numbers) of one figure, 9 for the ones with practially impossible to identify all of 230 the mirror plane perpendicular and parallel space groups by numbers alone, the to the principal axis (holohedry class), 7 author gave classificationnumbers of two for the ones with the mirror plane parallel figures first to the basic 67 non-translation to the principal axis or rotoinversion axis, space groups. These 67 spacegroups among 5 for the ones with the mirror plane per 230 do not contain the glide planes repre pendicularly intersecting the principal axis, sented by c/2, b/2, n/2, etc., or screw axes 3 for the ones with 2-fold rotation axes represented by 1/2, 1/4, 1/6, etc., in the forming side axes, and 1 is used for the ones elements of symmetry denoted by the Her without the elements of symmetry except mann-Mauguin notation. As shown in center of symmetry. Table 1, the 67 non-translation space groups Other lattices than P-lattice are to be comprise32 for P-lattice, 15 for I-lattice, 8 expressed by even numbers, as mentioned for F-lattice, 6 for C-lattice, 5 for R-lattice before. So, I-lattice is represented by the and 1 for A-lattice.* number of P-lattice minus 1, F-lattice by Now, in numbering the lattices, the the number of P-lattice minus, 3. In mono authors made it a rule that P-lattice should clinic and trigonal systems where I-lattice have odd numbers and other lattices even is absent, C-lattice and R-lattice are expres numbers. With regard to ten figures, 0 is sed by the number of P-lattice minus 1. For assignedto the oblique system (triclinic and C-lattice in the orthorhombic system, the monoclinicsystems), 1 to the orthorhombic number of P-lattice minus 5 is given. For (rectangular) system with one 2-fold rota Amm2, number 10 is given. This way, all tion axis, and 2 to the orthorhombic system of 67 non-translation space groups are with three 2-fold rotation axes. Similarly provided with numbers of two figures. The 4, 3 and 6 are assigned to the ones with 4- authors propose to call these basic space fold, 3-fold and 6-fold rotation axes groups the lattice groups. These lattice respectively. In the case of the cubic groups divided by crystal system are as system, which consists of three 2-fold or follows: Triclinic 01, Monoclinic 02-07, 4-fold rotation axes, four 3-fold rotation Orthorhombic 10-29, Tetragonal 40-49, axes and six 2-fold rotation axes, the author Trigonal 30-37, Hexagonal 61-69, Cubic 50- used the sum of the numbers of the 59 (T, Th), 74-77 (Td) and 90-99 (0, Oh) components, that is, 5, 7 and 9 are the (Table 1), inclusive of the negative sign** numbers of ten figures to represent the used for roto-inversion axis, etc. cubic system (Smith and Kennard, 1974). Next, on the basis of expressing the SIX REVERSE LATTICE GROUPS elements of symmetry other than the prin By changing the rotation and the reflec cipal axis in P-lattice by numbers (odd tion into the screwing and the gliding for the

* In Bravais 14 space lattices A-lattice is included in C-lattice , but Amm2 of the orthorhombic system must be distinguished from Cmm2 in the orientation of elements of symmetry. and it corresponds to 4 space groups, C142V to C172V, among the 230 space groups. ** The , negative sign used in indices of crystal face is understood to mean the reverse as well as the minus. Therefore, such signs as fi, 4 and 1 expressing the roto-inversion axis and the center of symmetry are used as indicative of the reverse side. 302 Y. Takano and Y. Shimazaki

Table 1. 67 lattice groups and 6 reverse lattice groups.

above-mentioned basic 67 lattice groups of the hexagonal system (Iwasaki 1971). Of including Amm2, the entire 230 space groups these, the first two pairs are P-lattice and I- can be represented in the form of subgroups lattice belonging to the same crystal class, but (Buerger, 1963, Bloss, 1971). It must be the other four pairs are P-lattice of different noted here that, from the mathematical classes. These paired groups, as shown in standpoint, 5 point groups that account for Fig. 1, differ in the orientation of elements 6 lattice groups among them are regarded as of symmetry in space lattice, and cannot be respectively independent point groups, but superimposed with one another; in other in a crystallographical view these 6 lattice words, they are independent lattice groups. groups must be considered to be 5 separate Of course, they are different also in the crystal classes forming pairs, namely, P42 diffraction pattern and the manner of distri m(47) versus P4m2 and 142m(46) versus bution (Hermann, 1938). Among these 12 14m2 belonging to the tetragonal system; lattice groups, 6 groups on the fore side with P321(33) versus P312, P3m1(37) versus their classification numbers in parentheses P31m, and P3m1 (37) versus P31m of the are considered to be counted in the 67 trigonal system; and P62m(69) versus P6In2 lattice groups mentioned before. The other Expression of space groups with numbers of two figures as the keynote 303

Fig. 1. 6 pairs of front and reverse lattice groups.U3 304 Y. Takano and Y. Shimazaki

6 groups on the rear side can be regarded as remaining 157 space groups have more than the reverse lattices of the fore-side groups, one screw axis or glide plane. The transla so the authors distinguished them by putting tion is provided to one rotation axis or numbers of one figure with the negative plane of reflection of the basic 73 lattice sign. 2-fold axes are parallel to edges of groups. In order to express the elements crystal lattices in the front lattices, and the of translation, letters (four in maximum) axes exist in diagonal position in the are used for one space group. In describ reverse lattices. Thus, added with these ing the space group, the numbers of two reverse 6 groups*, the basic lattice groups figures representing the original lattice for the 230 space groups become 73 in total. group are put between these letters, as These 73 lattice groups corresponds to follows, symmorphic groups derived by E. V.Fed cSNand crow (Fedorow, 1896, Hilton, 1903). On Cs: the glide elements (a, b, c, n, d) of glide the other hand, the 32 point groups can plane perpendicular to the principal be also classified into five groups by the axis are put along with the screw axis combination of the elements of symmetry (one to five) of the principal axis. (Buerger, 1963), namely, Group 1 for a : the screw element (only one) of side axis primitive rotation, Group 2 for crossed intersecting the principal axis at right rotation, Group 3 for primitive reflection, angles, or the glide element of glide Group 4 for crossed reflection, and Group 5 for axial reflection, as shown in the left plane parallel to the principal axis. d: the screw element of the third rotation and right columns of Table 1. The numbers . axis, or the glide element of the third of lattice types, such as P-lattice and I lattice, belonging to the respective symmetry glide plane. The arrangement of the letters follows groups are also given in Table 1. In the table, A denotes the number of the Amm2 the style of the Hermann-Mauguin nota orthorhombic lattices, and D the number tion (Table 2), but the symbol m for the of the reverse lattices belonging to the reflection plane and the symbol for the trigonal, hexagonal and tetragonal systems. ordinary rotation axis are not put, leaving the place for the letters blank. Also the screw elements are not expressed by 6, NOTATION OF 230 SPACE GROUPS 41, 21, but just the numbers showing the As has been mentioned so far, 73 lattice elements, like 1, etc., are given.** groups, about one-third of the 230 space As a result, the letters P, I, F to show groups, can be expressed merely by combin the lattice types become unnecessary, and ing the numbers of two figures. The almost all space groups are expressed by the

* By dividing the lattice groups into the fore and rear lattices , one each in the tetragonal system and the hexagonal system and three in the trigonal system, the mathematical 32 point groups can become 37 crystallographic classes. As indicated in Fig. 1, when the side axis was once determined structurally, physical properties such as natural polarization and optical activity would become different depending on the orientation of the side axis and the diagonal position. Therefore, the direction of axis cannot be changed. ** In describing the screw axis , it is desired that the use of numbers on the lower left side, such as 56, 34, 12, will be allowed, in the same manner as 56, 4, 12. Expression of space groups with numbers of two figures as the keynote 305

Table 2. Notation of 230 space groups with numbers of two figures as keynote .

symbols two to four letters fewer than the and elements of symmetry other than the Hermann-Mauguin symbols that were principal axis, by paying attention to the thought to be most simple and easy. Of numbers of two figures. It might take time the 157 space groups that need letters, 84 before this new notation is generally adopted space groups require only one letter. When for description of actual crystals. The 73 space groups that require no letter at all authors looks for the scientists who would are added to these 84, the space groups that attempt to describe space groups by the can be expressed by not more than three parallel use of the Hermann-Mauguin nota letters amount to 157 in total. With this tion and this new notation, instead of the new style of notation, even inexperienced parallel use of the Hermann-Mauguin and students can easily get information on Schoenflies notations. crystal system, point group, lattice type 306 Y. Takano and Y. Shimazaki

ACKNOWLEDGEMENT REFERENCES The idea of this paper was conceived in Buerger, M. J. (1963). Elementary Crystallography, 1974 when one of the authors was studying John Wiley and Sons, Inc., New York. Bloss, F.D. (1971). Crystallography and Crystal at the Department of Crystallography, Chemistry, Holt, Rinehart and Winston, Inc., Birkbeck College, London University. The New York. authors extends their sincere thanks to Prof. Donnay, J. D. H. (1969). Symbolism of Rhombo hedral Space Groups in Miller axes. Acta H. Carlisle and Dr. A. L. Mackay of the Cryst., A25, 715-716. same Department, for their valuable Fedorow, E. (1896). Theorie der Kristallostructur. discussion on space group and other Z. Krist. 25, 113-224. Herman, C. (Editor), (1938). International Tables subjects. The authors are indebted also to for the Determination of Crystal Structures, I, Prof. Y. Takeuchi, Prof. R. Sadanaga, Prof. Tables on the Theory of Groups, G. Bell and S. Takagi and Dr. T. Nishida of the Sons, Ltd., London. Hilton, H. (1903). Mathematical Crystallography, University of Tokyo, for their kind help in Clarendon Press. Oxford. many ways. International Union of Crystallography (1952). International Tables for X-ray Crystallography, Symmetry Groups, Kynoch Press, Birming ham. Expression of space groups with numbers of two figures as the keynote 807

Iwasaki, H. (1971). Symmetry Relations , Classi New York. fication and Asymmetric Units of the Reflec Smith, G. and Kennard, C.H.L. (1974). Definition tion indices in X-ray Diffraction . Rept. of Cubic Crystal Class. Jour. Chem. Educ., Inst. Phy. Chem. Research Japan , 47, 61-80 51, 801. (in Japanese with English abstract). Takano, Y. (1976). The Endless Periodic Table. Shubnikov, A. V. and Koptisk, V. A. (1974). Kagaku, 46, 178-181 (in Japanese). Symmetry in Science and Art. Plenum Press ,

2桁の数字を根幹とする空間群の表現

高野幸雄・嶋崎吉彦

67の非並進空間群のうち,2組つつの対をなす回折斑点を有する6空間群を分離独立させた。これら73の非並進空間群全部にに封してその結晶系・対称操作および格子型を表示する2桁の分類番号を与えた。これら非並進空間群から導かれる残り157空間群に対しては,その根幹をなす非並進空間群の分類番号の前後に最高4字以内の添字を与えて,そのらせん要素および映進要素を表現した。この新方式によれば,前記の73空間群が2桁だけの数字で表現されるのはもちろんのこと,残り157空間群中84空間群は根幹の2字を含めて3字で表現され,ほとんどの空間群が現行のH. M.記号より2～4字少ない記号で記載できることになる。また,この表現方式は初心者にもその空間群の属する結晶系や点群の判別を容易にするものである。