Crystallographic Symmetry Point Group Notation G. L. Breneman Eastern Washington University. Cheney, WA 99004 The vast majority of our molecular structure information comes from X-ray diffraction structure determinations of solid crystals. Many new structures are published every month in various iournals such as Acta Crvstallo~raohica and Inorganic ~hkmistry.The description-of these structures deoends verv heavilv on understandine crvstal svmmetry and its notation. ~ h i s " ~ a pwill e r briefly i e k r i h e ~"rystallographic symmetry point group notation and present a flow chart for the systematic determination of the correct point group. Sharma (1,2) and Ladd (3) have recently discussed some of the differences between the Schoenflies system for point group notation used by spectroscopists and-the ~ e m a n n Mauguin system used by crystallographers. The two systems are not just a difference in notation hut differ in three other aspects also. First, the set of symmetry operations used to form a group are not identical: crystallographers use a rotary-inversion axis while spectroscopists use a rotary-reflection axis. Second. the crvstalloeranhers' - . svstem uses onlv rotation axes allowed in crystals with their three-dimensional repeating structures (one., two., three., four., and sixfold) leading to 32 point groups. These two differences are well described hy Sharma. The third difference is that the Hermann-Mauguin notation is easily expandable to include not only point svmmetrv but also the various types of translationai symm&y found in the regular repeating structure of crystals. T h y e , along with the different types of crystal lattices, lead to 230 space groups (4).
referring to a single different or several different directions. The directions referred to depend on the crystal system to which the point group belongs and is described by Sands (5). The crystal system basically is determined by the type of rotation or rotary-inversion axis present. Rotation axes are represented by a number, for example, 6 for a sixfold rotation axis. Rotary-inversion axes are repre: sented by a number with a bar on top of it, for example, 3 ("three bar") is a threefold rotary-inversion axis. Mirrors are represented by the letter m. An inversion center is just a onefold rotary-inversion axis, or 1. An example is 4lmmm (D4h). The 4 represents a fourfold rotation axis (along z in this crystal class). The l m is part of the first component of the symboland refers to amirror plane perpendicular to that same direction. The second m refers to mirror planes perpendicular to both the x and y axes while the third m refers to mirrors perpendicular to the bisectors of the x and y axes. Of course these symmetry elements will necessitate others being present such as twofold axes perpendicular to the fourfold and an inversion center where the axes and mirrors intersect. Sands (6) lists all 32 point groups with both the Hermann-Mauguin and Schoenflies notations along with their corresoondine cwstal classes. A number of flow charts have been presented in the past for determining the proper Schoenflies point group notation (7-9). The figure presents a flow chart to help determine the proper Hermann-Mauguin point group symbol once the &&metry elements of asystem have been analyzed.
Hermann-Mauguln Notatlon The Schoenflies notation uses certain symbols to represent a certain combination of svmmetrv elements. for example, Cn, representsa threefold rotation axis with three mirror planes ~arallelt , ~this n~tntionaxis. The Hermann-Maueuin hotation uses symbols for the individual symmetry elements and uses these as components to build up the point group symbol with the position of each component of the symbol
1. Sharma, B. D. J. Chom. Edue. 1982.59.557. 2. Sharma.B.0. J. Chem.Educ. 1983.60462, 3. h d d , M. F.C. J. them E ~ U C 1986.sj. . 4 . Henry, N. F. M.: Lonsd.de, K., Eds. Int~rnorionolTobloa for X - m y Crrarollography; Kynmh: Birmingham. England, 1969: Vol. 1. 5. Sands, D. E. Introduction to Crysfollogmphy;Benjamin: New York, 1969: pp &55. 6. Sands, 0. E. Introdvetion to Crysto1logrophy:Benjsmin: New York, 1969: p 51. 7. Krub8aek.A. J. J. C h e m E d u r 1975.52.366. 8. Noeele. J. H. J . C h m . Edue. 1976.53. 190. 9. ~ a & r , J . P. J. Chsm. Educ. 1918.56 81
216
Journal of Chemical Education
636.
start
system
2's and/or m's
system
Hermann-Mauguin symmetry ndation flow chart. (11 reads "parallel to.")
Volume 64
Number 3
March 1967
217
