crystal systems

Crystal Systems and General Chemistry B. D. Sharma California State University, Los Angeles, Los Angeles, CA 90032 and Los Angeles Pierce College, Woodland Hills, CA 91371 Crystal chemistry is a subject of great importance (1,2) and, with good reasons. some asvects of this tooic are included in chemistry. The of thi; subject invariably includes either a table or eraohics of c m t a l svswms alone with axial length and interaxial angle relatfouships. In the a&ence of svmmetrv considerations. the relationshiv of axial leneths and interaxial angles for each crystal system can he ;is- leading. We present the definitions of each crystal system from the point of minimum symmetry inherent in each crystal system relating the packing of the chemical motif in the three-dimensional array and the consequence of the same toward the interaxial angles and axial length ratios. For detailed and precise discussion from a crystallographic point of view the reader is referred to texts on crystallography and physical chemistry (3,4,5). Tricllnlc System The symmetry of a triclinic system is either just onefold rotation axis or just center of symmetw. These svmmetrv elements place no restrictionson~either<he interaxial angles or axial lengths. 'l'hus, the unit cell for a triclinic system can have any value of interaxial angles and any ratio of axial lengths, as long as a parallelepiped is defined. This, obviously, includes a unit cell with a = b = c and a = 0 = y = 90°, which defines a cube. Just because we have defined a cube does not mean that we have a cubic system. In general, though, the unit cell of real crystals belonging to triclinic system is such that there is no specific relationship between axial lengths and between interaxial angles, a consequence of the nature of the chemical motif packed in the three-dimensional array exhibiting either just onefold rotation axis or just center of symmetry. Monocllnlc Svstem This system requires either just one mirror plane or just a single twofold rotation axis. Either of these svmmetrv ron- ditions demand one axis normal to the otheitwo, thus re- stricting two interaxial angles to a value of exactly 90°. However, there is no restriction as to the ratio of axial lengths. Once again a unit cell with a = b = c and a = 0 = y = 90° is fully compatible with a monoclinic system. Orlhorhombic System Any packing of a chemical motif in a three-dimensional array that exhibits just three mutually perpendicular twofold rotation axes as a minimum symmetry belongs to the ortho- rhombic system. By the very befinitidn of th6 orthorhomhic systrm it is clear that the three interaxial angles of the unit parallelepiped must he exactly 90°, hut there is no restriction upon the axial length ratios. Therefore, aunit cell with a = b = c and a = 0 = y = 90" is an acceptable example of an or- thorhombic system as long as the symmetry of the packing conforms to the minimum of three mutually perpendicular twofold rotation axes. Tetragonal Syslem This system requires the presence of a single fourfold rotation axis as a minimum symmetry. Under this condition all three interaxial angles are 90" and the two axes in the plane perpendicular to the fourfold axis equal in length, hut the axis coinciding with the fourfold axis can have any length. This does not exclude the unit cell with a = b = c and a = B= y = 90" from the tetragonal system. Rhombohedral System Four body diagonals of the parallelepiped exhihiting threefold rotation axis characteristics define a rhombohedral system. No other symmetry is necessary. The intersection of the body-diagonals may he a t any angle. This results in the interaxial angles to he equal to each other and the axial lengths to be equal to each other. Thus, the rhombohedra1 system demands a unit cell with a = b = c and a = 0 = y. Hexagonal System The three-dimensional array of chemical motif exhibiting just single sixfold rotation axis results in a hexagonal system. The angle between axes perpendicular to sixfold rotation axis is necessarily 120°, and these two axes are equal in length. However, the axis exhibiting sixfold rotation axis character can be of any length. Note that a unit cell with a = b = c and y = 120' and a = = 90° is also compatible with triclinic or monoclinic systems, if symmetry criteria are not used. A Correct Relatfonshlp of lnteraxlal Angles and Axial Length Ratlos for Seven Crystal Systems Crystal unit Cell System Axial Interaxial Angles* Lenglhs a:b:y a:b:c Triclinic NOResbictions, other than the No Restrictions requirement of forming a Parallelepiped Monoclinic a = y = 90'; Pany value less No Restrictions than 180' and greater than 0 ' Onhorhombic a = b = y = 90' NOResnictions Tetragonal a=b=r=goa a = b: c any value Rhombohedra1 a = 8 = y a=b=c Hexagonal y=120°:a=8=900 a= b cany value Cubic a=b=r=goO a=b=c f ~ngle a is oppohne a: angle 6 is appmile b: angle y is oppooite c: For monociinic System b axis is the symmetry axis: For tetragonal and hexagonal oystem c axis is me symmetry axis. 742 Journal of Chemical Education

crystal systems

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