X: The Classijkation of Crystals KAROL J. MYSELS University of Southern California, Los Angeles, California
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classification of crystals into the several systems such as cubic, tetragonal, orthorhombic, etc., is generally based in textbooks2 on a consideration of crystal axes, particularly their relative lengths and direction, as shown in the table. This approach gives correct assignments in a great majority of cases hut occasionally leads to an error. What is more serious, it obscures the real basis of the classification, which is that of the symmetry of the crystal in all its properties and not only in its axes. The subject of crystal symmetry is correctly treated in books on crystallography and X-ray diffraction3 but is rather lengthy and involved. We can indicate only some basic conclusions here. A crystal, as is well known, is a regular repeating arrangement of atoms, molecules or ions. The repeating unit, or unit cell, must correspond to the symmetry of this structure hut still can be chosen in an infinite number of ways. This is like the distance between trees in an orchard which can have a large number of values depending on the direction in which it is measured. Among these infinite ways, a simple one, directly related to the crystal shape, is chosen as the conventional unit cell and determines the axes of the crystal with respect to both angle and length. The shape of the unit cell generally reflects all the symmetry properties of the structure, but occasionally, due to accidental compensations, this shape may have much higher symmetry. To use a somewhat exaggerated illustration: a jack-in-the-box does not have cubic symmetry even when enclosed in a perfectly symmetrical box. The symmetry of the structure of a crystal may he defined very precisely in terms of rotations, inversions, and translations which superimpose the strncture upon itself. There are 230 possible combinations of these operations corresponding to the 230 so-called space groups. These in turn fall into the systems listed in the table.
' Suggestions and material suitable for this column are eagerly soneht and will be scknowledeed. Thev should be sent with as knawledged as guest authors. Since the purpose of this column is to prevent the spread and continuation of errors and not the evaluation of individual texts, The error the source of the errors discussed dl not be must occur in a t least two independent standard books to be presented. "ee, for example, PHILLIPS,F. C., "An Introduction to Crystdlography," Longmans, Green and Co., New York, 1946, or BUERGER, M. J., "X-Ray Crystallography," John Wley & Sons, Inc., New York, 1942.
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The strnctural symmetry of a crystal manifests itself in all its directional properties. Crystal habit (the relative development of different possible faces), refractive indices, coefficients of thermal expansion, pyroelectricity, piezoelectricity, and above all the single crystal X-ray diffraction pattern are among the properties useful in determining experimentally the symmetry properties of the crystal. The relative lengths and angles of the crystal axes can often he ohtained from a study of only its outer shape. Although there are not many, cases do exist in which a classification can he in error if it is based only on a consideration of the axes. Thus, a unit cell with a z b ~ and c a = @ = y=90° does not necessarily mean that the crystal has orthorhomhic symmetry; in the tabulation of Donnay and Nowacki4there are over fifty monoclinic substances listed having unit cells of the above shape. Among these are cobalt tungstate, potassium iodate, calcium zirconate. The same source lists twenty orthorhombic (pseudotetragonal) substances having two equal axes, e.g., elemental gallium or potassium ferricyanide, and ten tetragonal (pseudocubic) substances having all three axes equal such as silver mercuric iodide, and analcite [NaAl(SiOa)zHzO]. The erroneous assignment in these examples depends on the numerical values of the axes. It depends therefore also on the accuracy with which the measurements are made. I n the case of gallium, which is orthorhombic with a/b = 1.0002, measurements had to be Erroneous Classification of Crystal Systems Cubic Tetragonal Orthorhambic Monoclinic Triclinic Hexagonal
a a a a a a
=b=c = b # c # b Z c # b # c Z b # c =b#e # =
either a = b = c or
a = b # c
,
. . , , I .
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Trigonal
a = # = r =goo a=#=Y=goO a=fl=r=90" B#a=y=90° af##v#9O0 a=y=90m
B=120
a=@=u#90D (rhombohedral) a=r=9O0 (sometimes included under hexagonal)
made with high precision to show that the unit cell was not tetragonal in shape. Since the thermal ex~ansioncoefficient follows the StrnctUral Symmetry o f t h e crystal it will differ along two axes which are only "accidentally" equal. Hence the equality can he exact only a t one temperature. For the same reason, however, the "accidental" equality can become absolutely exact at such a temperature, and 'DONNAY,J . D. H., AND W. NOWACKI,"Crystal Data," Geological Sooiety of America, Memoir 60, New York, 1954.
JOURNAL OF CHEMICAL EDUCATION
the measurement of the axes leads to an erroneous assignment no matter how accurately it is made. On the other hand, if the stmctural symmetry of a crystal places it in a given system, then the relative lengths of the axes and their angles not only have the exact values required but keep these in spite of changes in temperature and pressure. Thus while consideration of the lengths and angles of crystal axes forms a simple basis for the classification of crystals, the student should be made aware that these are not the most fundamental consideration. They
VOLUME 34, NO. 1, JANUARY, 1957
are only one of the manifestations (and not an infallible one) of the basic symmetry of the crystal. ACKNOWLEDGMENT
Professor R. F. Kruth of the University of Arkansas suggested a discussion of this topic and helped in presenting it. Professor J. Donohue of the University of Southern California Chemistry Department has kindly reviewed the manuscript and contributed many improvements.
