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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.
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