Some Simple AX and AX2 Structures A. F. Wells The University of Connecticut, Department of Chemistry and Institute of Materials Science. Storrs, CT 06268
Descriptions of a number of simple crystal structures of compounds AX and AX2 have been introduced into elementary chemistry textbooks over the past few decades, and usuallv a unit cell is illustrated. Since i t is important to understand precisely what is meant by the statement that a particular compound has the sodium chloride (or other) structure we eiamine here three uf the simplest of these structures, the s d i u m chloride, rutile, and fluorite structures. In any crystallinr compound AX, there are necessarily contacts (bonds) between an A (X) atom and its nearest X (A) neiyhhors, hut for a complt~teand correcr descriptiun of the structure we also need to know whether therr arr A -A contacts and/or X - ?( contacts. The unit cell dimensions and .... ~. ~- atomic coordinates alone do not furnish this information. There are three essentially different ways of illustrating the structure of a compound AX,, namely, (1) by showing the positions of the atomic centers in a unit cell as small circles of arbitrary radii connected where appropriate by lines between nearest neighbors; (2) by representing the atoms (ions) as spheres having the correct relative radii to show which atoms are in contact with neinhborina o n e s s u c h a diamam, or a section through the structure showing the contacts in a selected plane. mav be descrihed as a "packing diagram;" or (3) as a a y s t e k 0: linked coordinati& poljhedra. Corresponding- to these three kinds of illustration of a crystal structure there are three types of model, "ball-and-spoke," sphere-packing, and polyhedral model. Since the reader may he less familiar with the third typeof illustrationor model we shall describe it briefly. The group of X atoms surrounding an A atom in a crystalline compound AX, is the coordinationgroup. In both NaF (sodioum chloride structure) and MgFz (rutile structure) this is a group of six F- ions arranged around each cation at the vertices of a remlar odahedron. i.e.. the coordination number (c.n.) of A is 6.in both these crystals the X:A ratio is less than 6, the value for an isolated octahedral group, and therefore each F- ion must belong to a number of cation coordination groups. This number, the c.n. of X, is 6 in NaF and 3 in MgF2, i.e., these are structures of 6%and 6:3 coordination, respectivelv. More eenerallv, if a is the c.n. of A and x the c.n. of X in a crystalline compound A,X, then am must be equal to m, because these two products represent alternative ways of
Figure 1. Unit cell of me sodium chloride structure. 830
Journal of Chemical Education
counting the number of A-X bonds; we are assuming that the c.n. has the value a for all A atoms and the value x for all X atoms. The structures of these compounds, therefore, may he represented as aggregates of octahedra joined together, by having vertices, edges, or faces in common, in such a way that each X atom belongs to x octahedra. This is clearly the least reali~tictype of representation of a crystnl structure, for not only are the inetril amms at the centers of the polyhedra omitted but also all the A-X bonds are omitted. The edges of the polyhedra represent only van der H'aals bunds between ?( atomsof ration courdination groups. On the other h:md, the type of coordination of the cation is immediately evident, as also is the c.n. of theanim (the number of . vdvhedra mretine . a t a vertex). Moreover, i t seems that many students can visualize and remember certain structures more easily as polyhedral models than as ball-and-spoke or sphere-packing models. The Sodlum Chloride Structure
This structure, also referred to as the halite or rock-salt structure, was described and illustrated in 1883 by Barlow and confirmed by X-ray diffraction some thirty years later. I t is the structure of more than two hundred compounds AX of all chemical tvnes. raneine from most of the alkali halides and the oxides i d k f i l e s i f magnesium, the alkaline earths, and some transition metals, to interstitial carbides and nitrides of transition metals; evidently it is not characteristic only of simple ionic compounds such as the alkali halides. I t is the only AX structure in which there is regular octahedral coordination of both A and X atoms, and it is stable for atoms (ions) A and X with a large range of relative sizes (radii, rA and r ~ )In. this structure (Fig. 1)the arrangement of the A atoms is the same as that of the^ atoms. The two sets of atoms are related by a simple translation; other structures with this nrooertv are the cubic and hexaeonal ZnS structures and the bsdl st;ucture. Therefore, in tce "anti-NaCl" structure, in which the ~ositionsof cations and anions are interchaneed. ~ X are unchanged in contrast with thehu: the c.n.'s d f and orite and "anti-fluorite" structures, which are discussed later. We shall deal with two features of this structure: There is no simple relation between rA and rx for compoundswith this structure,and (h) The mode of packing of the atoms (the numbers and types of atoms which they touch) is quite differentin different compounds with this structure,or inother words, thereis nouniquespherepacking model or diagram. Therefore. we must be careful how we discuss two essentiallv geometrical topics which are also introduced into many introductorv texts. namelv. .. the radius-ratio concept and the closest df equal spheres. I t is tempting to introduce the radius-ratio concept because. of its geom&i& simplicity and because the ratio df the radii of cations and anions is certainly related to their coordination numbers in many essentially ionic crystals (fluorides and oxides). However, the concept is more valuable in complex than in simple ionic structuris-with the notable exception of fluorides and oxides AXz-and it certainly has little relevance to the structures of the simplest of all ionic crystals, the fluorides and monoxides of the more electropositive elements: (a)
c.n. of M
4
MF MO
6
LiF. NaF MgO. CaO
None
BeO. ZnO
8
I
I KF. RbF. CsF
f
SrO.
None None
BaO
In each series the radius of the cation increases from left to right, and the dotted line marks the point a t which we would expect a change to a structure of higher c.n. if the only operative factor were the relative sizes of the ions. We must also be particularly careful to make clear the distinction between the statement that atoms are in the oositions of closest packing and the statement that atoms i r e most closely packed. The relative positions of either the A or the X atoms are those of cubic closest packing (ccp), hut this does not mean that either the A or the X atoms are most closely packed in any particular compound with the sodium chloride structure. There are clearly two extreme configurations of the structure: ~ A atoms closest packed, X in octahedral interstices ( r >> rx), or X atoms closest packed, A in octahedral interstices (rx >> r ~ )and , an indefinite number of intermediate configurations. The feature common to all forms of this structure is the regular octahedral coordination of A by 6X and of X by 6A, but as regards the numbers and types of atom with which each A or X atom is in contact they range between the two extremes (a) and (c) of Table 1 and Figure 2. Between them we may recognize a special case where A and X are equal in size. The term "sodium chloride structure," therefore, covers a range of configurations. If we regard the atoms (ions) as "hard" spheres, the relative radii of A and X are those in Table 1. The value for the configuration (a) is the reciprocal of that for (c), which is (&-I). The radius ratios for the alkali halides with this structure range from a value less than 0.4 for LiI through the value 1.0 for KF to nearly 1.3 for CsF, though there is not general agreement as to the precise values of ionic radii or, for that matter, to the validity of the picture of all these halides as ionic crystals. Comparison of the X-X distances with twice the appropriate ionic radii shows that only the Li compounds are close to the configuration (c); the values for the fluorides are F
-F
LiF
NaF
KF
RbF
CsF
2.84
3.26
3.77
3.98
4.25A
which should be compared with 2.7A (twice the radius of F-), the value to be expected for cp anions. For the monoxides of Mg and the alkaline earths, the picture is somewhat similar to that for the alkali halides. The configuration (c) is a reasonable representation of the struc-
(a) (b) Fgm 3. Models rewesewingthe configurations (=)and (b)of me sodium chlwke SlWCtWe.
ture of MgO, in which 0-0 is 2.98A. Comparison with 2.8A, twice the radius of 0 2 - , shows that an 02-ion is nearly in contact with its 12 anion neighbors. The structure of BaO is close to the configuration (h), for the radius of Ba2+ (1.36A) is similar to that of 02-(1.40A), while SrO and CaO are intermediate cases, as shoyn by the ionic radii and the 0-0 distances: MgPf ionic radius Configuration
0-0
0.72
c
Table 1.
lnteratornlc Contacts In the Sodlurn Chlorlde Structure (b)
(a)
Contacts A X r6:ry ,. "
Examples Figure 2
12 A
6X 6A 2.4
6X 6A 1.0
TIC. ZrC (4
KF, BaO (bl
(c) 6X 6A 12X 0.414
LiF. MgO (cl
1.16
1.36A
2.98
(bl 3.40
3.65
3.90A.
The Rutlle and Fluorite Structures
In contrast to AX compounds the ionic fluorides MF2 provide an excellent illustration of the change in c.n. with increasing size of M2+ except for the fact that 8-coordination persists beyond the point where it would he expected to change to a higher value; it remains 8 even in BaF2, in which the ions have the same size. The failure to change to structures of higher coordination (for example, 10:5 or 126 coordination) is apparently due to the fact that such structures cannot be constructed; that is, the problem is one of geometry rather than chemistry.
rr Shucture
b
1.00
Ba2+
@I
4
BeFl
a
Sr2+
Closely approximating to the other extreme configuration (a) are the itructures of a numher ot interstitial compounds (Tic, TiN, ZrC, and ZrN) in which the metal atoms are in contact or nearly so; in the titanium compounds the Ti-Ti distance is approximately 3A as in the metal. The configurations (a) and (b) are illustrated as models in Figure 3; (c) would be represented by a model similar to (a) if the larger spheres correspond to the anions.
c.n. of A
Figure 2. Horizonlal sections of the sodium chloride struchne showing achral contacls between atoms (ions).
Ca2+
6
8
MgFl
CaFTBaF2
0.35
0.72
1.00-1.36~
Silica-like
Rutile
Fluorite
Metal dioxides show a clear-cut division into structures of 6and 8-coordination. The smaller ions M4+ (of Ti. V. Cr. and Mn) form the 6-coordinated rutile structure and the larger ones (of Hf. Po. 4f and Sf elements) the 8-coordinated fluorite structure. The RuMe Structure This structure of 6 3 coordination is the most important octahedral :ID structure for ionic compounds AX? (fluorides and oxides). Rv a 3D structure, we mean one in which the octahedral coordination groups are linked together so as to form an arrangement extending indefinitely in three dimensions, as opposed to a layer (2D) or chain (ID) structure. Many dichlorides, dihromides, diiodides, and disulfides (and some dihydroxides) crystallize with one or both of the octahedral Cd12and CdC12layer structures. Volume 59 Number 8 August 1982
631
Figure 4. Unit m i l of me rutile structure.
The rutile structure is not easily visualized from a unit cell diagram (Fig. 4) or from the corresponding ball-and-spoke model. The way in which the octahedral AX6 groups are joined together is seen more easily from the model of Figure 5(a). Each octahedron shares two opposite edges to form chains which are then joined into a 3D structure by sharing vertices. I t is not possible to have a t the same time regular octahedra and the most symmetrical coordination group around X, that is. three A atoms conlanar with X and situated at the comers of an equilateral triangle (all angles A-X-A equal to 120'). Regular octahedra imply that the bond angles at X are one of 90Dand two of 135". Two difluorides, CrF2 and CuF2, have a distorted form of this structure due to the symmetry of their d shells (d4and d9), and afew transition metal dioxides (those of V. Nb. Mo, W. Tc, and Re) have a less svmmetrical variant of the st'ruct&edue to interactions betwken alternate pairs t these exce~tionsthe of -~ metal atoms in the chains. A ~ a rfrom structures of all difluorides &d dioxides with the rutile structure approximate closely to the "ideal" model. In all these crystals the cation is appreciably smaller than the anion. When the radius of M2+ (in M F 3 or M4+ (in MOs) increases beyond a certain value, ionic difluorides and dioxides adopt the Bcoordinated fluorite structure. I t is, therefore, not necessary to consider cunfigurati~msof this structure for lager M ions as we did for the sodium chloride structure and as we shall do for the fluorite structure. The arrangement of the anions in the rutile structure approximates to hexagonal closest packing, rather more closely in CaCh, SrC12, and CaBr:! than in the fluorides with this structure. The (idealized) structure may, therefore, be illustrated as a cp model (Fig. 5(h)) which is shown in the same orientation as that of Figure 5(a) with the planes of the cp layers vertical. In the anti-rutile structure the metal atoms are arranged in closest packing with the nonmetal atoms in octahedral interstices: one of the rare examdes is Ti2N. We noted that the two extremeconfigu~tionsof the sodium chloride structure, with either A or X atoms (ions) in contact, are geometrically similar because the spatial arrangement (and therefore the c.n.) of the A atoms is the same as that uf the X atoms. On the other hand. therearediffermt numbers of A-A and X-X contacts in thd two extreme configurations of an AX2 structure, as we shall see for the fluorite structure. We do not describe here the other configuration of the rutile structure, in which there would be A-A as opposed to X-X contacts, since no examples of this structure are known. I t may interest the reader to visualize the structure which results from increasing the size of the A atoms until A-A contacts occur. ~
~~
~~
The Fluorite Structure Before dealing in detail with this structure we remind the reader that the cubical coordination of A is not the preferred arrangement of 8 neighbors either in finite molecules or complex ions or in 3D structures. The best packing is achieved 632
Journal of Chemical Education
Figure 5. (a) me rutile structure built from octahedral AX, coordination groups. m e small spheres represent X atoms (ions). (b) The rutile structure represented as a CP array of X ions (large transparent spheres) with metal ions in one half
F l g ~ r e6. Unll cell of the fluorite snucture. m e atoms shownas shaded circles are m o p e of the packing doagrams of F i g m 7.
if coordination polyhedra have triangular faces, and of the following: 8-cwrdination polyhedmn Triangulated dodecahedron (bisdisphenoid) Square antiprism Cube
Faces All triangular Triangular and square All square
the first two are preferred in complex ionic crystals, in most 8-coordinated molecules and ions, and in AX4 structures of 8:2 coordination (ThLZrF4, and ZrCl4). Cubical cwrdination occurs in CaFz because it is not possible to construct a 3D AX2 structure with antiprismatic or trigonal dodecahedra1 coordination of A and tetrahedral coordination of X. In AX4 structures the geometrical restrictions are far less severe than in an AX:, structure because an X atom is common tn only two 8-coordination polyhedra as opposed to four in an ASs structure. I'l'his areument a ~ o l i e with s still more force to an AX structure such& the ce&m chloride structure, in which each X must be common to eight AX* coordination groups.) The unit cell diagram (Fie. 6) shows that in a com~ound with the fluorite str&ture.~has 8 nearest neighbors ar;anged at the vertices of n cuhe while X is in contact with 4 A atoms arranged tetrahedrally. In order to show all the contacts in various configuratiohs of this structure we select the atoms shown as shaded circles which lie in a diagonal vertical plane. The extreme configurations correspond to contacts between A or contacts between X atoms (ions) as set out in Table 2 and illustrated in Figure 7 (a) and (c). In all configurations of this structure the centeU, of the A atoms are in the positions of cubic closest packing, and the X atoms are in the positions of the tetrahedral interstices. However, the A atoms are most closely packed (that is, in contact with 12 other A atoms) only in the configuration (a) which is, therefore, correctly described as a ccp array of A atoms in which X atoms occupy all the
Table 2. lnleratomlc Contacts In the FluorHe and Antllluorlte Structures Fluorite (AXd sbucture (a) (b) Contacts
A X
12A
8X 4A 4.44
r~:rx Example Figures 7 and 8
YH2
(a)
8X 4A 1.0 BaFn (b)
(cl
'
8X 4A BX 0.73 CaF2 (c)
Antitluaite (A2X) structure la)
Contacts
rr:rx Example
A X
0.225
4x 8A 1.0
L1;Te
Kg0
4 x 8A
12 X
Ic)
Ihl
4X 8A
6A
1.37 Be&
(c) Figure 8. Models cwesponding to the packing diagrams of Figure 7.
C Figue 7. Atomic mntacts in Uw wee canfigrations (a), (b). and (c) of UW Ruxite structure as set out in Table 2.
tetrahedral interstices. This implies, of course, that X is much smaller than A. This configuration represents the structure of YH2, in which a Y atom has, in addition to its 8 H neighbors, 12 Y neighbors a t the same distance (3.65 A) as in metallic yttrium. In the other extreme configuration, (c), the X atoms are in contact. They are situated a t the points of a primitive cubic lattice, and each is in contact with 6 others, in contrast t o the 12 A with which an A atom is in contact in (a). Thus, this form
of the structure could be described as a CsCl structure from 'which one half of the cations have been removed, leaving the remaining A ions in alternate cubical holes along each cubic axis. The positions of the A ions are such as to form tetrahedral groups around each X ion. In CaF2 the distance between Fand its 6 nearest F-neighbors (2.7A) is equal to twice the radius of this ion, showinn that this confirnation represents the structure of CaF2. he description oca ccp arrangement of Caz+ ions with F- ions in tetrahedral interstices is clearly unrealistic since F- is larger than Ca2+ (radius 1.OA). The radius of Ba2+ is the same as that of F-; the structure of BaFz is accordingly represented by Figure 7(h); that of SrFz lies between the configurations (b) and (c). In the anti-fluorite structure the positions of cations and anions are interchanged, and again we have two extreme configurations. The nearest approach to the cp structure (a) is found for the combination of the smallest alkali metal and the largest chalcogen. In LizTe each T e atom has 12 T e neighbors a t 4.6 A, a distance close to twice the radius of Te2(4.4A). In the alkali metal oxides the 02-ions w e not most clmely parked; rumpart, 0-0 in LijO.:i.3 A,nnd in Na20,J.g A, w ~ t ht w i c e thc radius of O1' 12.8 A). Their structures are intermediate between the configurations (a) and (b). An example of the other extreme configuration, (c), is Be&, in which Be has 6 Be neighbors a t 2.2 A, which is the distance between a Be atom and its nearest neighbors in the metal. The configurations of the fluorite structure shown as packing diagrams in Figure 7 are illustrated as actual packings of spheres of appropriate radii in Figure 8.
Volume 59 Number 8 August 1982
633
