A. F. Wells Unlvexsity of Connecticut stws. 06268
Some Structural Principles for Introductory Chemistry
More than 60 vears have elaosed since the determination of the first cry& structure ( N ~ C I1913) , and nearly a century since the puhlication of Barlow's work on the packing of spheres which led him to suggest a number of simple arrangements of equal spheres and of spheres of different sizes.' I t was shown subsequently that these sphere packings included the most imoortant structures adopted by metals and by compounds AX.-1n spite of these considerable time intervals it is still unfortunately true that the solid state is virtually ignored in the teaching of chemistry. The lack of knowledge of structure in the solid state did not hinder the development of an elaborate stereochemistry of organic compounds because (1)most solid organic compounds consist of the same finite molecules that exist in solution or in the vapor state, and (2) considerable developments were possible based on two early suggestions regarding the spatial arrangement of honds from carbon atoms in aliphatic and aromatic compounds, suggestions that were later proved to be qualitatively correct by structural studies. Nevertheless the organic chemist now relies almost exclusivelv on crvstal structure determinations for precise information about the structures of complex molecules. Also, certain topics in organic chemistry call for a knowledge of the spatial relations between one molecule and another, for example, a solid state reaction such as the photcdimerization of astilbene to form a substituted cyclobu&e. A more subtle matter is the relation between molecular structure and optical activity, with which we deal in another arti~le.~ The exclusion of the results derived from the studv of the solid state has had more far-reaching effects in inorganic chemistrv. for not onlv are some 80%of the elements solids under oriinary condicons but so also are the majority of inorganiccompounds. Diagrams of the unit cells of a few of the simplest structures are now included in many elementary texts: their effectiveness would he enhanced if thev were preceded by some discussion of patterns that repeat in one, two. or three dimensions. There are few sirns that the maioritv . . of teachers of chemistry regard the stru&res ofsolidsasan internal Dart of inorganic chemistrv. This implies a recornition thatth; structures-of crystals containing infinite arrays of atoms are the logical result of the same processes which are of atoms-in fact responsible for the formation of finite that the formation of asolid reaction product is synonymous with the growth of a crystal. It requires that we consider how atoms forming 2,3,4, or some other number of honds can join together to fonn not only finite groups but also chains, layers, or three-dimensional frameworks, and further that the same principles apply to groups of atoms such as triangular BOs or tetrahedral Si04 groups etc. which can be joined by sharing anorooriate .. . numbers of vertices (here 0 atoms). This is what we mean by regarding the structures of crystalline compounds as an internal uart of inorzanic chemistrv: it imolies more than simply ad&nginformatioi here and there aboui the structures of individual elements or compounds. If diagrams illustrating simple crystal structures are not oreceded hv an adeauate introduction to reoeatine oatterns it is difficul; for the kader of an i n t r t d u m j teat ieconcile the chemical formula of a compound with the numbers of Barlaw, W., Nature, 29,186,205,404 (1883). Wells, A. F., to be published.
atoms (ions) of various kinds shown in the illustration of the unit cell of thestructure. For example, the unit wll of the NaCl structure appears to contain 13CI- m d 14 Na7 ions (or 14 CIand 13 Na' ions) rather than four of earh, and the usual cell of CaF2 shows 8 F- ions hut 14 instead of the expected .I Caz* ions. It is therefore necessary to give'rules' for counting the numhers of atoms in a unit cell allowing for the fact that their centers may lie within the cell, on faces or edges, or at theorigin. The reason for counting atoms in unit cells is not usuallp explained in elementary texts. Certainly it is necessary to know the number (Z) of atoms in the unit ceU of a crystalline element or the numher of formula-units in the unit cell of a compound if we are concerned with the relation between the density, cell volume, and absolute masses of atoms. Otherwise i t would seem that the value of Z is of interest only if it is necessary to check thc chemical formula against a model or diagram of the crystal structure of a simple ompound. Assuming that we wish todo this it is nreferahle codetennine the numhers of nearest neighbors of each kind of atom (coordination numhers). these nearest neiehhors beine atoms (ions) of the other kind: Representing thecoordinatik numbers as suoerscrints. AOX'.. the number of bonds A-X (or X-A) in'any l~rge'vdk& of the crystal containing M formulaweights may he counted in two ways, either as M(ma) or as M(nx), from which i t follows that am = xn. The essential features of a structure are not understood anv better bv counting atoms in unit cells, for Z has no chemical or structural significance, in contrast to the values of a and x which are the actual numbers of honds formed by an A or X atom. In order to explain why the value of Z is irrelevant we deal first with the question of choice of unit cell, and then suggest ways of describing simple structures. These will not include descrintions in terms of internenetratine lattices. which intraduce an entirely unnecessary complication and like the values of Z have no olace in the teachine of elementarv chemistry. I t is true hithe pattern of Na+ ions in NaCl is t h l same as that of the CI- ions. the nositions of one set of ions heing related tn thoseof the dthekhy a translation ofone half of the edge of the cubic unit cell. However. the descriotion of the struciure as two interpenetrating latLces is not b n ~ yincorrect (because a lattice is a framework of lines and points. not of atoms) but it adds nothing to our understandini of the structure. which is one in which each kind of ion is surrounded octahedrally by six ofthe other kind. Similar ohjectionsapply to the descriotion of the fluorite structure in terms of interpenetrating 'lattices.' The only structures which should be described in terms of interpenetrating systems of atoms are those in which some fraction (usually one half) of all the atoms in the crystal are bonded mgeth~rto form a framework within which the normal chemical hondineof theatoms issatisfied. The crystal then consists of two (or more) completely i n t ~ r nenetratine (usuallv identical) frameworks which are held together b; bonds of a kind different from those operating within the frameworks. A simole examole. in fact the first example of a structure of this type to he determined, is'the structure of CunO, the mineral cuprite. Unlt Cells In Repeating Patterns The subject of unit cells in crystals is conveniently and loeicallv " annroached bv considerine in turn patterns that repeat periodically in one, two, or three dimensions. The repeat
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Volume 54, Number 5, May 1977 / 273
Tabie 1. Structures Having A and X Atoms With the Same Highly Symmetrical Arrangement of Nearest Neighbors Coordination number
Coordination
Sfrucfure
zinc blende (rmalerite) sodium Chloride Cesium chloride
4 6 8
regular tetrahedral regular octahedral cubic
NDC
Table 2. Three Simplest + Thrae-Dimensional Structures for AX, Figure 4. Alternative "hombohedral" unit cells in lhe NaCl structure: (a) Z = 1. (b) Z = 2. Figure 4 is reprcdueed from Wells. A. F., "Structural lnwganlc Chemlsby," 4th. Ed, Clarendon Press, Oxford, 1875 (Fig. 6.3, p. 197).
I
center of symmetry
Figure 5. (lefl) "Staggered arrangement of bonds from adjacent atoms in lhe (cublc) dlamond structure. Fgue 6. (righl)The fluaiie (CaF2)sbuchre. The smaller circles represem m l ions. Figure 6 is reprcduced from Weiis. A. F.. "The ThirdOimenslon inchemistry." Clarendon Press, Oxford. 1956 (Fig. 79(a), p. 98).
which they pack together in the different polymorphs (allotropes). Again we see that the unit cell of the crystallographer is not of primary importance to the chemist. As in a twodimensional nattern so also in a three-dimensional structure a centeredcd~lis not the smallest one that may be chosen. Fieure 3 shows alternative cells in hodv-centered cuhic and face-centered cuhic structures, in which the normal cuhic cells have Z = 2 and 4. resnectivelv. Crystalline N ~ C has I cuhic symmetry and theconventional unit cell is a cube (all-face-centered)containina four NaCI. Alternatively the structure may be referred to any of the following (among other) cells ~~~~~~~~~~
~
hody-centered"tetragonal" hody-centered"rhombohedral" primitive "rhombohedral" (In crystallography the terms tetragonal and rhombohedral define the s3mmetrr of a crystal, and its structure, which in the presentcase is cubic. we use the terms "tetragonal" and "rhomhohedral" here to mean that the unit cells are dimensionally tetragonal or rhombohedral, that is, they have the shapes of a rectangular prism (a = b # c ) and a rhombohedron, respectively.) The cells listed above are the same as for cubic closest packing, of which two have been illustrated in Figure 3 ( ( b )and (c)); the two "rhomhohedral" cells are illustrated for NaCl in Figure 4. In view of this arbitrariness in choice of unit cells and the resulting different ways of describing the same structure perhaps we should question the wisdom of introducine crvstalloera~hicterms unless we are prepared to delve fuzheiinto c&s~allography. The structure of (cubic) diamond also is based on a facecentered cubic lattice, and there is accordingly the same choice of unit cells a9 for NaCl or a cuhic closest-packed metal. Here the conventional cuhic cell contains eight carhon atoms and the simpler cells four or two atoms. The reason why this
Structure Crirtobalite Rutile Fluorite
C~~rdinafion numbers Coordination A X of A 4 6
8
Tetrahedral Octahedral Cubic
2 3 4
Examples BeF, MgF2 CEF,
Si0, TiOl 210,
structure cannot he referred to a cell containing only one carhon atom is as follows, The smallest unit from which this structure can he built allowing only translations is a pair of atoms the honds from which are in the "staggered" relation (Fig. 5 ) , or in other words there is a center of symmetry hetween each pair of bonded atoms. The atom A cannot reproduce B simply by a translation, and therefore the basic structural unit is the pair of atoms. (This relation between the bonds from anvpair of adjacent atoms is an essential feature of the structu;e-and, togkther with the regular tetrahedral bond aranaement, is the only information required to build a model ofthe cubic diamond structure.) Dercrlptlons of Slmple Structures For compounds AX very few highly symmetrical structures are geometrically possible, and three in which both the A and X atoms have the same highly symmetrical arrangement of nearest neighbors (of the other kind) are shown in Table 1. The zinc blende structure is most simply described as derived from the (cuhic) diamond structure by replacing C atoms alternately by Zn and S atoms. An alternative description of a compound AX with this structure as a cubic closest packing of X atoms with A atoms occupying one half of the tetrahedral interstices is valid for some compounds in which each X atom is in contact with 12 other X atoms, and is intelligible to students who have studied models based on closest-packed spheres. The sodium chloride structure is adopted by more than 200 compounds of a variety of chemical types; it is not restricted to ionic compounds, as the following examples show: MgS, hut also US, MgO, hut also UO, VO, VC, VN, UN, SnP, etc. This structure is stable for ionic compounds AX in which the ratio of the radii of A and X varies over a wide range, and other factors favoring its widespread adoption are the stability of covalent or metallic character octahedral hon& with and the fact already mentioned that there are few other highly symmetrical structures possible for compounds AX. The relative positions of the Na+ or of the C1- ions are those of cubic rlose~tpacking, hut the description as a c.c.p. array of CI- ions with the Na- ions in all the octahedral interstices is not nhvsicallv realistic since the C1- ions are not actually in - contact with one another; the "closest-packed'' description is more nearly valid for LiCI. The CsCl structure is relatively unimportant, for it is adopted only by a few cesium salts (CsC1, CsBr, and CsI a t ordinary temperatures, CsCN, and CsSH) and by some intermetallic compounds. The cuhic coordination found in this ytructure is abnormal in that the best packing is achieved if coordination polyhedra have triangular faces. and of the fullowing
.
d
8-coordination polyhedron triangulated dodecahedron (bisdisphenoid)
Faces
all triangular
Volume 54, Number 5, May 1977 / 275
square antiprism cube
triangular and square all square
the first two are preferred in complex ionic crystals and in most molecules and c o m ~ l e xions. However. three-dimensional structures with thkse more satisfact& types of 8coordination for hoth kinds of ion are geometrically impossihle, and the CsCl structure is found for a few salts containing large, highly polarizable, ions. If we wish to include AX2 structures it would seem logical to set out the three s i m ~ l e sthree-dimensional t structures shown in Table 2. In th&e structures there is the same sequence of types of coordination of the A ions as in the AX structures previously listed, and as in the CsCl structure the abnormal cubic coordination of the cation in CaF? is apparentlv due to the imnossihilitvof buildinea three-dimensional strukure with oneLofthe p;eferred types of coordination to which we have referred. The essential features of the fluorite structure are the 8-coordination (cubic) of the cations and the 4-coordination (tetrahedral) of the anions. In CaFz the F- ions are in contact (each with six others) so that the structure may he described as a s i m ~ l cubic e ~ a c k i n of e these ions with Ca2+ ions occupying alternate cubicboles along any axial direction.
276 1 Jourml of Chemical Education
The ~ositionsof the cations are such as to form a regular tetrahehral arrangement of cations around each anion. Figure 6 shows that the structure can he rezarded as derived from the CsCl structure by removing one half of the cations and replacing them by ions carrying twice the charge. Inspection of a diagram or a model of the CaFz structure shows that the CaZ+ ions are arranged in the same way as the Na+ (or C1-) ions in NaCl or the metal atoms in a cubic closest packed metal, and that the F- ions are in the positions of (all) the tetrahedral interstices. While this description is justified if we are systematizing structures derived from the various types of closest packin& of spheres it is unrealistic to describe the structure of CaFz in this way because (1) the Ca2+ions are not in contact with one another, and (2) the description implies that the F- ions are smaller than the Ca2+ ions, whereas the reverse is true (radii, Ca2+, 1.00.4, F-, 1.33.4). The "closest-packed" description would be nearer the truth for Liz0 (radii, Li+, 0.74.4, 02-, 1.40.4) which has the same structure but with the positions of cations and anions interchanged (anti-fluorite structure). The author would like to thank Dr. E. Kostiner and a referee for a number of constructive criticisms of the original manuscript.
