William G. Gehman
Atomics International Canoga Pork, California
Standard Ionic Crystal Structures
Solid state chemistry involves the formation, motion, and interaction of defects within crystals. An appreciation of these factors involves, as a minimum requirement, a strong intuitive grasp of the structural relationships of the atoms and interstices of various crystal structures. Fortunately this minimum requirement can he readily acquired by limiting one's attention to the small number of idealized ionic crystal structures discussed in this paper. These structures are derived by use of the concept of the closest packing of spheres and approximate to varying degrees the actual structures possessed by a large proportion of real crystals. In addition this approach permits one to consider the remaining real crystal structures as being derived from the ideal ones by successively greater deviations from idealized closest packing which in the extreme cases lead to wholesale atomic rearrangements and distortions of the crystal lattices. The use of the concept of the closest packing of spheres as a key to the intuitive understanding of crystal structures was first suggested by Barlow ( I , 2) in 1883, some thirty years prior to the discovery of X-ray diffraction; since that time a number of workers have extended the concept (3-8). From this viewpoint crystals are considered to he built up by successively stacking together closest packed monolayers of atoms; the various standard crystal structures then differ from one another mainly in the way in which the interstices between the closest packed atoms are occupied by other atoms. The present approach d i e r s from the former ones in its emphasis on the information to be obtained by carefully considering the relationship between the atoms and interstices first in a single closest packed monolayer of atoms and then in a construct of two closest stacked monolayers called a double layer. The information so obtained proves to he s a c i e n t to derive all the remaining properties of infinite, three-dimensional constructs of closest packed spheres. I n the present paper a qualitative description of this information is presented and applied to the standard ionic crystal structures.' Cubic and Hexagonal Closest Packed Atom Lattices
Infinite, three-dimensional closest packings of spheres of equal size can be considered to built up by the successive stackmg together of two-dimensional closest packed monolayers of spheres. The stacking direction This work was carried out under the au~uspieesof the United States Atomic Energy Commission, Contract Number AT(11-1)-GEN-8, and the paper based upon a portion of North American Aviation -Special Report-6003. The paper was presented at the Pacific Southwest Regional Meeting of the American Chemical Society, Sau Diego State College, December, 1961.
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is parallel to a cube body diagonal or a (111) direction in cubic closest packing, ccp, and is parallel to the hexagonal rotation axis or [0001] direction in hexagonal closest packing, hcp. The perpendicular separation distance between midplanes of nearest neighbor monolayers is equal to d 8 / 3 r , , = 1.6330rC,, where r,, is the radius of the closest packed spheres. The spheres of immediately adjacent monolayers fit into three-dimensional hollows of the nearest neighbor monolayers and tangentially contact three spheres of these monolayers. Thus, individual spheres which contact six other spheres in their own monolayer also contact three spheres in each of the two immediately adjacent monolayers leading to the familiar twelve-fold coordination of closest packing. The spheres a t the top and bottom of finite, crystal-model representations of closest packed arrays obviously contact a total of only nine neighbors.
Figure 1.
The t w o possible types of closed pocked double layers.
Whenever a new monolayer is added to the topmost layer of a preexisting stack of spheres, a two fold choice must be exercised with respect to the manner in which the new monolayer will be added. This two-fold choice is illustrated in Figure 1 and can he stated as follows: The spheres of the new or upper monolayer (heavy circles in Figure 1) are placed either: choice a: exclusively over plane triangular interstices which "point" toward the bottom of the page and which are labeled B interstices in Figure 1, or choice b: exclusively over interstices which point. toward the top of the page and which are labeled C interstices in Figure 1.
Figure 1 actually illustrates a single pair of monolayers or a single double layer in which the first. or lower layer is represented by the light circles and the second or upper layer by the heavy circles. A single exercise of the two-fold choice, which is all that is necessary to construct a double layer, does not lead to physically distinct constructs, since either of the two forms of the double layer illustrated in Figure 1 can he converted into the other. For example, rotation by 60' about the the stackmg direction perpendicular to the monolayer 1 A detailed mathematical description, "Translation-Permution Operator for the Description of Crystal Structures. I. Ideal Closest Packing," has been submitted to Ada Cryatallographica.
midplanes interconverts the two forms as can be seen by simply rotating the page by 60". Nevertheless, when the two-fold choice is exercised a t least twice in succession as it must be to form a closest packed triple layer, it is thefundamental cause for the existence of the physically dist,inct cubic and hexagonal forms of closest packing. The first coordination shell of twelve nearest neighbors of each of these two well-known forms of closest packing is illustrated in an "exploded" fashion in Figure 2; the two structures in the figure are finite fragments of closest packed triple layers. Cubic closest packing, ccp, arises from a perfect repetition of the same two-fold choice made in forming the initiil double layer (spheres X-XI1 and spheres IV-IX), i.e., either choices a+a or b+b. Hexagonal closest packing, hcp, arises from a perfect alteration between the two possible choices, i.e., either choices a+b or b+a. Futhermore, when the two-fold choice is exercised many times in succession as it must be in forming closest packed multilayers, it is the fundamental cause for the possible existence of an infinite number of threedimensional closest packings. This infinity of additional possibilities (4) arises from sequences of this fundamental two-fold choice that are increasingly more complicated than the two simple sequential extremes found in ccp and hcp. Several other comparative features of ccp and hcp are illustrated in Figure 2. First, the dodecahedra1 coordination polyhedra inside the cages of nearest neighbor atoms are defined by the intersections of planes perpendicular to and bisecting the connecting lmes between the central shaded atom and its twelve nearest neighbors. Second, both the atom cages and the coordination polyhedra illustrate (1) the hcp mirror planes of symmetry which lie in the midplanes
+
+
+
(V, VI, 11) (VII, VIII, 111) (X, XI, XII) (IX (not shown), IV, I). For a first approach to the study of crystal structures one need only concentrate on ideal cubic closest packing, ccp, and ideal hexagonal closest packmg, hcp, since the large majority of the crystal structures based on the closest packing of spheres are derived from these two types of packing. However, other types of closest packmg do exist in crystals (3-8). I n addition, real crystals are usually deformed from perfect closest packing and also have stacking faults or errors in the sequence of twofold choices. But the study of these aspects of crystal structure can be profitably delayed until after the standard crystal structures are mastered. Interstice Lalfices
Of fundamental importance t o the effective study of crystal structure is the concept of L. Strock (9) that crystal structures are composed not only of lattices of atoms but also of lattices of interstices. The threedimensional interstices lie between the closest packed atoms. These interstices are important since they are the keys t o the understanding of the so-called "defect lattices" and of the crystal structures of metal alloys, ionic salts, and minerals. The three-dimensional interstices are of two types, which can be conveniently described in connection with the models illustrated in Section (a) of Figure 3 .
Figure 3 . Models of la1 distribution of cloest ~ o c k e datoms about tetrohedral and o d a h e d r a l interstices, and lbl distribution of tetrahedral ond octahedral interstices about o tingle closest packed atom in both cubic a n d hexagonal packing. Figure 2. Structure of t h e Rrrt shell dorest pocking.
of neighbors in cubic
and hexagonal
of the closest packed monolayers in Figure 2b, and (2) the ccp center(s) of symmetry located a t the shaded atom in Figure 2a and a t all the closest packed atom centers in an infinite three-dimensional ccp array. Third, in studies of the mechanical slip of crystals, hcp is found to have only a single set of closest packed glide planes whereas ccp has four such sets. Thus stacking closest packed monolayers in a ccp fashion automatically generates three other closest packed planes which taken together with the original closest packed plane, form sets of intersecting tetrahedra. The central portions of the faces of the two smallest, intersecting tetrahedra surroundme: the central shaded atom i n Figure 2a are represented by the two sets of triangular interstices (I, 11, 111) (IV, V, X) I , 1 1I (VIII, I X (not shown), XII) and
+
+
+
Octahedral interstices are enclosed by six closest packed atoms which lie a t the apexes of a regular octahedron and by eight plane, curved-triangular interstices that form the faces of the octahedron. Tetrahedral interstices are enclosed by four closest packed atoms and by four plane triangular interstices that form the faces of the tetrahedron. If the tetrahedron in Figure 3 is thought of as a three-dimensional arrowhead, the apex of the arrowhead points upward, i.e., in the positive stacking direction or positive z-direction; the tetrahedral interstice defined by such a tetrahedron of closest packed atoms is termed a positive tetrahedral interstice and is indicated by the symbol "(+)-tet" where "tet" is an abbreviation for tetrahedral interstice. A tetrahedral interstice defined by a tetrahedron pointing in the negative stacking direction is termed a negative tetrahedral interstice and is denoted as "(-)-tet." Volume
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The three-dimensional interstices are distributed in planes which are perpendicular to the stacking direction and which lie between the midplanes of immediately adjacent monolayers of closest packed atoms. These planes are illustrated in both top and side views in Figure 4 where the tetrahedral int,erstices are labeled by small plastir spheres and the octahedral interstices by larger cork spheres. The two sizes of spheres representing interstices in Figure 4a conform roughly to the tetrahedral radius ratio of 0.225 and to the octahedral radius ratio of 0.414. The vertical coordinate of the plane-of-centers of the interstices, expressed in fractions of the minimum interplanar distance parallel to the stacking direction, is an important characteristic
atoms, tetrahedral interstices, and octahedral interstices -,., nc-,_,.,. This respectively, and where n,., = nc+, atom-interstice ratio rule can be derived by considering only the space between the midplanes of two immediately adjacent monolayers as in the double layer illustrated in Figure 46, and then further restricting attention to a space filling prism within this space such as t,he hexagon outlined in Figure 4a. Since the rule holds a t the double layer stage it is independent of any macroscopic symmetry differences among the infinity of closest packing types since these symmetry differences only arise past the double layer stage. Thus the atominterstice ratio is constant for all closest packings and serves as a key guide to the stoichiometry of compounds with closest packed crystal structures. A clue to the nature of the interstice lattices in ccp and hcp is supplied by the orientations of the tetrahedral and octahedral interstices around the closest packed atoms in these two types of closest packing. These orientations are illustrated in Section (b) of Figure 3 and are also listed below:
+
~ p 8: tetrahedral interstices = simple cubez 6 octahedral interstices = octahedron hcp: 8 tetrahedral interstices = complex structure with C3,, symmetry,' and 6 octahedral interstices = triganel prism
Figure 40.
Closest packed double loye.;
top view.
of the different interstices. These fractional coordinates are: (+)-tet = '/&, oct = and (-)-tet = s / r . Due to a slight defect in the construction of the model the (+)-tet interstices in Figure 4b appear to he closer to the lower midplane than the correct value of one-quarter of the interplanar distance. I n all closest packed arrays the relative number density of closest packed atoms and three-dimensional interstices is constant and can be expressed by either of two equations:
where nap,n,,,, and no,,are the numbers of closest packed
Figure 4b.
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Particular attention should be paid to the distribution of the complete set of both positive and negative tetrahedral interstices around the closest packed atoms in ccp, a simple cubic arrangement, and the octahedral interstices in hcp, a trigonal prism arrangement. When these two types of units are separately combined together to form three-dimensional arrays, they give rise to the arrangements of tetrahedral interstices and octahedral interstices that can he seen by looking ahead to Figures8and 11, respectively. These two arrangements provide the "keys" to the visualization of closest packed structures that can be of great value, either when reading a journal article under such circumstances that textbooks and crystal models are unavailable, or when attending scientific meetings. These "keys" can be stated asfollows: (1) The simple cubic tetrahedral-interstice-lattice is the key to ccp ( 2 ) The simple hexagonal octahedral-interstice-lattice is the key to hcp
For example, Figure 5 illustrates that the tetrahedralinterstice-lattice in ccp forms a three-dimensional checkerboard of simple cubes, half of which are occupied by closest packed atoms and the other half being occupied by octahedral interstices. Figure 10 illustrates that the octahedral-interstice-lattice in hcp forms a peculiar type of simple hexagonal, three-dimensional checkerboard of trigonal prisms, half of which are occupied by closest packed atoms, with the other half being
* Note that these 8 interstices are actually split into two sets of 4(+)-tet and 4(-)-tet interstices, each set defining the apexes of a. regular tetrahedron, however it is more valuahle to ignore the distinction between the two sets in the case of the particular point being presently considered. C8b symmetry has a threefold rotation axis perpendicular to a mirror plane of reflection symmetry. The complex structure can he viewed as either two intersecting, twinned regular tetrahedra or as a. compressed trigonal prism with a triangular prism on each triangular face.
occupied by pairs of tetrahedral interstices that are oriented parallel to the z-axis. The Approach of A. F. Wells The use of the concept of the closest packing of spheres, as a key t o the intuitive understanding of crystal structures, is best illustrated in the case of ionic crystals. Using a single theme throughout his whole text, Wells (3) is able to bring together a myriad of crystal structures into a unified picture. This theme has two parts. First, the closest packed ions which occupy the bulk of the space in the crystal are picked out. Second, the distribution of the remaining ions over the interstice lattices of the closest packed structure i s determined. In one sense, the larger, closest packed ions "determine" the crystal structure, in that (a) Their structure will be reflected in the macroscopic form of the crystal, and (b) Their structure determines the symmetry of the interstice lattices. and therefore limita the symmetry of thelattices of the smaller ions.
However, it should be emphasized that the role of the smaller ions (usually the cations) is not entirely passive, but may well be the dominant factor in determining whether the anion structure of strongly ionic crystals is ccp or hcp. The importance of the smaller ions arises from the differing symmetries of the lattices of the interstices in ccp and hcp. This difference can, in certain cases, violate Pauling's Rules (3) and lead to extra repulsive forces being present in the hcp structure that are not present in the ccp structure, and which are not compensated by any additional attractive forces. Thus, in these cases, the potential energy, and hence the thermodynamic free energy, of the hcp structure will be higher than that of the ccp ~ t r u c t u r e . ~ Standard Crystal Structures of Type Mpb The standard ionic crystal structures are defined as those structures which (1) contain only two types of ion, (2) possess either a cubic or hexagonal closest packed structure of one of the ions (usually the anions), and (3) have the smaller ions (usually the cations) distributed over the planes of tetrahedral and octahedral 4 A paper entitled "Instability of Hexagonal Analogues of Certain Cubic Crystalsu is in preparation.
Figure 5. The LiaW rtrudure (note that in Figures 5-9, the positive z-axis extends from the iower left front corner to the upper right rear corner of the cubic slrudurel.
interstices in such a manner that all the planes of a given type are either completely orcupied or completely empty (but not partially occupied). I n the case of the standard crystal structures the first part of the above two-part theme is carried out by simply limiting one's attention to either ccp or hcp arrays of the anions, X. Then in carrying out the second part of the theme, d i e tributing the small cations, M, throughout the octahedral and tetrahedral interstice lattices, two parametws are available: (1) the nature of the interstice occupied, i.e., either small tetrahedral interstices (radius ratio = p = 0.225) or larger octahedral interstices ( p = 0.414), and (2) the number of the interstices that are occupied. The first parameter depends, to a large extent, upon the radius ratio, but by no means completely on this factor; the electronic configuration of the cation, and the relative polarizabilities of tl-e anions and cations, markedly affect the situation. For example, a classic counterexample is found in the mineral spinel, MgA1204in which t h e larger magnesium ions (Pauling radius = 0.65 A) occupy tetrahedral interstices and the smaller aluminum ions (Pauling radius = 0.50 A) are in octahedral interstices. The second parameter is determined by the requirement that the crystal, as a whole, he electrically neutral. Applying these two parameters to the 16 comhinations of anion and cation charges ranging from *1 through *4, and selecting relatively simple distributions of the cations over the interstice lattices, leads to the different types of possible crystal structures listed in Table Examples are presented for as many different types as are presently known and listed in the third edition of Wells' text (3); it should he noted that in some cases a number of examples are known other than the single example listed and that several of the examples listed exist in more than one crystal form. Not all of the charge combinations are possible in closest packed crystals, and not all the possible cases can be realized without such severe distortions occurring that they destroy any simple crystal symmetry. For example, as frequently noted by Wells ( 3 , cations with charges of +3 and +4 have such high ionic potentialsV, where V = (cation charge)/(cation radius)-that they severely distort the electron density of the surrounding anions and tend to form covalent bonds. On the other hand, anions with charges of -3 and -4 are highly polarizable, strongly tend to form covalent bonds, and tend not to exist as simple ions at all. Deprnding upon the p:~rtirulnrromhinnrions of atom&,r l ~ e dirt.rtiona1 rhnrwrrri.;ti~.sof ~ v a l m bond. t 11~111 rither to a molecular crystal structure due to the formation of neutral molecules (e.g., SiF4) or to complex structures. These complex structures include examples with finite complex ions (such as AIF,-), infinite chains (as in the silicates), or infinite, three-dimensional structures (such as AIFa). Thus, the structures to the lower left of the Note that the entries in Table 1 are the standard names for the types of structure without regard to the actual ionic charges involved. For example, any ccp structure with a l : l ratio of cations to anions in which all the octahedral interstices are occupied, is called an NaCl type of structure; examples of compounds with this structure and varying charges are: f2, MgO; +3, LaN; *4, ZrC (although the last two are more likely of the nature of intermetallic compounds than ionic crystals, since ions of such extreme charges are unlikely, except in gaseous discharges). Volume 40, Number 2, February 1963
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t,ahle (below the lower heavy line) and to the upper right of the table (ahove t.he upper heavy line) in Table 1 are usually not rcaliaed in any simple form. The structures between the two heavy lines contain all the standard ionic crystal slructzrres. However, a few of the entries lying between tke two heavy lines in Tahle 1 are exceptions to the definition of a standard crystal structure. For example, the common a-Also8structure (corundum) is not a standard structure since the layers of octahedral interstices are only two-thirds occupied and neither completely filled nor completely empty. A more complicated exception occurs in the cases of Allsea and Ga:Ss where the fraction of tetrahedral interstices occupied is incommensurahle with the eightfold coordination of tetrahedral sites aro~rnda closest packed atom. In these structures the Group I11 atoms are considered to he randomly distributed over either the set of positive or negative tetrahedral interstices and to occupy each site of the given set with a two-thirds occupation-prohability. However, this assumption of a random occupation implies (1) that. the system is what is termed an order-disorder syst.em ( l o ) , (2) that the temperature at which the crystal structure was determined was ahove the order-disorder transition temperature since the disordered state was observed, and (3) that at low enough temperatures, i.e., below the order-disorder transition temperature, an ordered state would exist in which the Group 111 atoms would occupy fixed positions around the closest packed Group I1 atoms and lead to a relat,ivelycomplicated over-all crystal structure. The Standard CCP Crystal Structures
With the preceding information as background, the chief aim of this paper can be presented quickly; that aim is the development of an "intuitive grasp" of the Table 1. Typical Compound
I
Halide
I
The standard cryrtol structures
oxide
Legend: uct An octahedral interstice tet A tetrahedral interstire Z The crp ZnS structure, sinchlende or sphalerite W Thc hcp ZnS strur:t,ure, wurtzite X Combined u6t.h '‘0 " indicat,es t h a t the stated fraction is ineommensurnbjH with the total numher of interstices oi the given t,ype (i.e., 8 tet per cp atom and 6 0,-t per el, atom) ? The given type of compound is possible, but not highly ~
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whole field of crystal structure, which can be gained hy an appreciation of the derivation and interrelationships of the standard crystal structures. The approach starts with closest packed arrays of anions which have all t,heir octahedral and tetrahedral interstices occupied hg cations, and proceeds by systematically emptying out the cation sites. The ccp case will first be considered, and then the hcp case uill be separately considered. The starting point of the standard structures based on ccp anion arrays is illustrated, in an "exploded" fashion, in Figure 5 . The large spheres are the ccp anions, the intermediate sized spheres are the cations in octahedral interstices, and the small spheres are cations in tetrahedral interstices. It should he immediately noted that the least negative charge that the anions can have, in this structure, is -3; since, even xith rat,ions of the least positive charge (+1) there will be three cations per anion, due to the fundamental ratio of 2 tetrahedral interstices to 1 octahedral interstire per 1 closest packed atom. (1) The LisBi Structure. The structure illustrat,ed in Figure 5 is found for materials such as Li,Bi, and is called the Li,Bi structure. The material Li3Bi is more properly termed an intermet,allic compound than an ionic crystal, since the distortion of the strurture that would be expected with a highly polarizable-3 anion is absent. ( 2 ) T h e NaCl Structure. The Li,Bi structure is a relatively uncommon type of standard crystal structure. However, the grandfather of all crystal st,ructureanalysis, the NaCl stmcture, is readily derived from the L i a i structure by simply emptying out all the tetrahedral interstices, to give the result illustrated in Figure G. (3) T h e CdCL Structure. Then emptying out m e half of the remaining octahedral sites, by removing
I
Nitride
Carb>de
likely without severe distortion; and i t has not heen reported in ;my simple stnlcturr, tr, date ? * Compounds of this type are unlikrly, due tu excessive eoulomhic repulsion, in the hrp rase XX Comprunds of this t > v c art. impc,ssihle, due to electrical non-neutrality
NOTEADDEDIN PROOF: I n f4,-1 structures the entries " I / $ oct ? ?" should rend "(1/4 oct) X X", and in the + I , - 3 structure the entry under hcp shanld read "?I".
Figure 6.
The NoCl structure.
Figuri7.
alternutc (111) planes of cations, results in the CdCL strurture illustrated in Figure 7.O A simple rationale of the emptying-out process can he given in terms of the ionir rharges: since divalent cadmium has twice the charge of sodium, only half as many cadmium ions are required to neutralize the uninegative chloride ions; this rat,ionale is, of course.. im~licit in the derivation of . Table 1. (4) Th,e I&O Structure. If, instead of the tetrahedral sit,cs, all the oct,ahedral interstices are emptied, thc Li,Ri structure is convert,ed to the Li20 structure, illustrated in Fignre 8, which clearly exhibits the key t,etrahedral interstice latt,ice of the ccp structure. ( 5 ) The ZnS (Sphalerite) Structure. Finally, if one half of the tetrahedral interdices of t,he LipO struct,ure are systematirally emptied out, the cubic zinc 6 This structure is named the cadmium chloride structure, although CdCI* itself is distorted along a < I l l > axis to form a rhombohedra1 structure derived from a cube that has been compressed apprmirnately 3% along the body diagonal. If, instead of alternate planes of uctahedral sites, every plane had one-half its catitms removed, each closest packed anion would have three eatirm neighbors forming a right angle instead of an equilnt~rwltriangle, as in CdCl*, and the resulting charge asymmetry causes the hypothetical structure to (undergo a hypothet,icnl phase change and) form the rut,ile structure.
Figure 9.
The Irincblendel Zn.5
structure.
Figure 8.
TheICdClr structure.
The Li?O rtructure.
sulfide st,ructure of sphalerite is formed (Figure 9). In the figure the (+)-tet positions were all emptied out. The Standard HCP Crystal Structures
(1) The Analogue of the IJirBi Structure. The hcp analogues of these struct,ures can he considered to be derived either by: (a) Shifting the rrp anion lattice, in each case, to a hcp strurture and moving the cations to the appropriate interstices, or ( b ) Just shifting the ccp Li3Bistructure to its formally possible hcp structure, and then systematically emptying out the ration sites. Either way, the result is t,he same; hut, for convenience, we shall use the second approach. The hcp analogue of I&Bi is illustrated in Figure 10. (Sote in Figures 10, 13, and 14, that. atoms in tetrahedral interstices t,hat are behind t,he ep atoms, and mould be completely hidden in the figure, have heen indicated by heing lightly "ghosted-in.") This hcp analogue of a ccp structure does not exist in highly ionic crystals, since there are pairs of cations in hoth tetrahedral and octahedral interstices t,hat are much closer t,ogether than in the cubic case. This leads to electrostatic repulsive forces that render the struct,ure unstahle(1I ) . (2) The NiAs Structure. The hcp analogue of NaCl, formed by empt,ying out all the tetrahedral
Figure 10. The hexagon01 ~ n ~ l o g uofethe cubic LhBi structure.
Figure 1 1 .
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The NiAs structure.
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Figure 12.
The Cdh rtructure.
Figure 13. The h e x a g o n a l onologue of t h e cubic Li10 structure.
interstices of the starting hcp structure, is the NiAs structure. It is also not found in highly ionic crystals due to excess coulombic repulsion forces. However, it is observed in (1) such polarizable compounds as FeS, CoS, etc., and (2) in the compound NiAs which is compressed along the co axis to such an extent that the Ni atoms exchange electrons to form what might be termed one-dimensional metal lattices in a matrix of As atoms. Thus, it is only in the case of very polariaable salts, intermetallic compounds, and alloys that the hcp analogues of the ccp structures are expected wit,h the exceptions of the ZnS and CdL structures (see following sections). The N i structure, illustrated in Figure 11, clearly exhibits the key octahedral interstice lattice of hcp structures. (3) The Crll? Structure. Cont,inuing on from the N i s case, and emptying out one-half of the remaining octahedral sites by removing alternate (111) planes, leads to the CdIz structure (Figure 12) which is the hcp analogue of CdCL Unlike CdCI2,Cd12 closely approximates ideal closest packing and exhibits a co/ao ratio of 1.61, compared to the ideal value of 1.633. (4) The Analogue of the Li20 Structure. The hcp analogue of Li20 is formed by emptying out all t,he octahedral sites of the hcp starting structure; and, like the parent structure, is also not observed in ionic crystals (Figure 13). ( 5 ) The ZnS (Wurtzite) Structure. Finally, emptying out one half of the tetrahedral interstices of the hcp analogue of LizO leads to the hcp zinc sulfide structure of the mineral, wnrtzite, which is considered a standard crystal structure (Figure 14). I n this figure the (+)-tet interstices were emptied out. Deviations from ldeol Closest Packing
The discussion, so far, has illustrated the great usefulness of t,he concept of closest packed spheres in t,ying together a great number of diverse crystal struct,ures in a qualitative fashion. However, in a quantitative sense almost all real crystals deviate to some extent from ideal closest packing. I n fact, only the 27 face centered cubic elemental metals are truly cubic closest packed, hut this circumstance is actually "by definition." Of the 24 nominally hexagonally closest packed metals, three have co/m ratios in large excess of t,he 60
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Figure 14.
The ZnS (wurtzitel structure.
ideal value of 1.6330, five closely approximate the ideal value of the ratio with values of 1.633 +0.04 but none of these exactly equals 1,6330, and the remaining 16 all have values of the ratio that are markedly less than the ideal value. In the rase of the ionic crystals, no example of ideal closest packing is known simply because no cation-anion pair has a rigid-sphere radius ratio of the exact value for either tetrahedral or oct,ahedral interstices. For example, using Panling radii, the closest fit to the tet,rahedral radius ratio of 0.225 is fonnd in BeO, where (r~,++/ro-) = 0.31/1.40 = 0.221. No example closely fitting the octahedral radius rat,io of 0.414 is known; two of the closest are found in the cases of LiCl and MgO which straddle t,he ideal value: (rLi+/rCL-)= 0.60/1.81 = 0.331 and (rMg++/rO-)= 0.65/1.40 = 0.164. Nevert,heless the majority of crystal structures can he considered to approximate to a greater or lesser extent the ideal state exemplified by the ideal closest packing of spheres. Even the silicates can he viewed as consisting of very closely packed triple and quadruple layers of oxide ions and +2, +3, or +4 cations, wit,h the triple and quadruple oxide layers being "spaced" by the presence of relatively large alkali or alkaline earth cations. Literature Cited (1) BARLOW, W., h'atuw, 29, 186, 205, 404 (1883). (2) BARI.OW,W., A N D POPE,W. J., J. Clzem. Soc., 91, 113% 1214(1907). (3) WELLS,A. F., "Structural Inorganic Chemistry," 3rd ed., Clarendon Press, Oxford, 1962. (4) PAULING,L., T h e Nature of the Chen~ical Band," 3rd ed., Cornell Universit,y Press, Ithaca, F. Y., 1960, pp. 4n4-41 . . ...R..
BELOY, N. V., Compl. Rend. (Ihklady) ?cad. Sci. UKSS,23, 17(tli4 (103!)), (in English). Bnmr, G. B., L'Introductionto Crystal Chemistry," AECtr-4138. Ofice of Technical Services, Department of Comm&ce, Washington, 1960 (translated f;om il publication of the Moscow University Publishing House, 1954), pp. 132-146. L., "Intr~dueti~n to Salids," XlcGrarr-Hill, S e w AZAROFF, York, 1960, pp. 55-68. STILLWELL, C. W., J. CHEM.EDTC.,10, 590, 66i (1933); 13, 415, 469, 521, 566 (1937); 14, 34, 131 (1938). L. W., Z. I
