PRINCIPLES DETERMINING T H E ARRANGEMENT OF ATOMS AND IONS I N CRYSTALS* BY M. L. HUGGINS
So many types of crystal structure have been worked out and so much is known concerning the structures of the atoms of which they are composed that it is now possible to determine to a large extent what are the principles determining the arrangement assumed. Such principles are usefc' in arriving at the structures of other substances. They are also of aid in studies of interatomic forces, and so of molecular structures. Although some of the ideas to be expressed in this paper originated with the author, many are due to Lewis,' the B r a g g ~ , ~Goldschmidt,4 ,~ CuyJ5 PaulingB and others. The first principle to be mentioned is that the influence of an atom on other atoms decreases very rapidly with distance, so that the effect on all but those atoms immediately adjacent to the given atom is almost negligible. Second, atoms of the same kind crystallizing in the same environment, tend to be surrounded similarly. As a result of this, all like atoms or ions are almost invariably surrounded in a similar fashion throughout a given crystal, except of course in such cases as crystals of organic molecules where the surroundings of the atoms were in part determined long before the crystallization. Combining these two principles, one concludes that the arrangement in a crystal should repeat itself a t rather short intervals-that the unit cell should contain relatively few atoms or ions or molecules-and this is known to be the case. The third principle is that of neutrality. Neglecting the surfaces, which might possess slight excesses of electrical charges, the whole crystal (and hence the unit cell) must be electrically neutral. If a crystal is built up of ions, this determines for simple salts or limits for complex salts the relative numbers of the different kinds of ions, and so the chemical formula. Another closely related principle may be stated as follows: Negatively charged ions tend to be surrounded by positively charged ions, and vice-versa. No exceptions to this rule are knowm. Both this and the preceding rule are of course derivable from potential theory or from a consideration of the fact that during crystallization an ion would always be drawn toward an ion or group of ions of opposite sign. One can generalize still further from potential theory and conclude that if a crystal contains anions of more than one kind, those having the greatest negative charge would tend to be in the positions of greatest positive potential,
* Presented a t the Pacific Coast Intersectional Meeting of the American Chemical Society, Eugene, Oregon, June 20, 1930. Contribution from the Chemistry Department of Stanford University.
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FIG.I A layer of spheres in a “close-packed” assemblage. The sphere centers in the next layer are over points marked “B.” In the “hexagonal close-packed arrangement” the spheres in the third layer are directly over those in the fist, giving the sequence ABABAB. . . . In the “cubic close-packed arrangement” the sequence is ABCABCABC. . . .
FIG.2 The unit cube of the CaCl structure.
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FIG.3 The unit cube of the CaF2 structure.
FIG.4 Illustrating by a two-dimensional analogy, t h e pririciple of relxtive ionic size. This arrangement of ionic centers would he stable for ions of 8pproxitnately the same size (as shown on the left) but unstable for ions differing greatly in size :as o n the right).
those haring 2 smaller charge would tend to be in a porition of less pwitive potential (although still surrounded by positive ions. 1‘ This is equivalent, to Pauling’s “electrostatic valence That scinething else is involved than potential theory, as applied t o point charges, is evident from t,he fact that the corresponding relationship for cations does not hold. The next principle t o he mentiontiti is that of “close-packing”.2 It can readily be shown that the mcst st,nble arrangements for atoms with spherical force-fields are the same as the arraiigernents one mould get by packing spheres of equal size as closely together as ;;oreit)le,-arrangement~ in which each atom or sphere is surrounded at equal distances by twe!ve others ( f i g . I j .
A R R b S G E J I E S T S OF ATOMS AND I O S S 15 CRTST.ILS
FIG.5 The unit cube of the KaCl structure. The arrangement of like atoms is the “face-centered cubic” or “cubic close-packed” arrangement. Each atom is surrounded symmetrically by six of the other kind.
FIG.6 The unit cube of the ZnS structure. Th distrihu?ion of the valence electron pairs is indicated bv the small circles. Each atom is surrounded symmetrically by four of the other kind. The arrangement of iike atoms, considered by themselves, is the same as in the XaCl structure.
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The most symmetrical of these arrangements-that of “cubic close-packing” is the one found for solid argon. The close-packing principle is important also for crystals composed of ions. If the ions are of approximately equal size the closest packing consistent with the rule that each ion is to be surrounded by ions of opposite charge, will be that in which each is surrounded by as many as possible ions of the other kind. If the compound is of the type AB, with equal numbers of A and B ions, this number is eight and the structure is that known as the cesium chloride structure (Fig. 2). I n an AB2 compound there can a t most be eight
FIQ.7 Illustrating in two dimensions the distribution of oxygen atoms around Si and AI in silicates.
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FIG.8 The probable distribution in s ace of the atoms in PCls, with the corresponding electron formug.
B ions around each A ion and four A ions around each B ion-aa in the calcium fluoride structure (Fig. 3 ) . If the ions are of quite different size4Jz6* arrangements such as these would place like ions too close together as compared with the distances between unlike ions. Closer packing and greater stability is obtained then if the like ions, considered by themselves, form a close-packed lattice. One can roughly approximate the facts by considering the ions as spheres, and assuming that any structure with the spheres of like ions in contact is unstable (Fig. 4). The cesium chloride structure then becomes less stable than the sodium chloride structure (Fig. 5 ) and that in turn less stable than the zinc sulfide
ARRANGEMENTS OF ATOMS AND IONS IN CRYSTALS
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The probable arrangement of atoms in SFs, with the corresponding electron formula. (This arrangement is the eame aa that Sound for SiFT in crystals.) 00
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Probable atomic distribution in ICls, with the corresponding electron formula.
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The arran ement of the atoms in the Is- and IC12- ions, wit% the corresponding electron formula.
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structure (Fig. 6) as the ratio of the two radii departs further from one. In both the sodium chloride and the zinc sulfide distribut,ions each ion is equidistant from twelve others of the same sign while in the cesium chloride st,ructure each is equidistant from only eight other like ions. Similar considerations oft.en determine the number of atoms or ions of opposite sign around a given atom or ion in more complex structures. If the central atom is relatively large compared with the outer atoms, the co-ordination number will be large (say 6 or 8) unless other factors interfere, while if the central atom is relatively small the coordination number must be small, usually four. Thus silicon in silicates seems to be surrounded invariably by four oxygen atoms, while aluminum, somewhat larger, always has six oxygen atoms around it3 (Fig. 7). The substitution of fluorine for oxygen, however, increases the coordination number of silicon to six, due partly to the smaller size of the fluorine and partly to the decreased repulsion between the negative atoms.
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FIG.1 2 The arrangement of the atoms in the carbonate and nitrate ions in crystals, with the corresponding electron formula.
Another factor in this and some other cases is the mutual repulsion between the electron pairs of the valence shell of the central atom. If this valence shell has a relatively small radius, as is usually the case for electronegative atoms, only four pairs can surround the kernel without making them too close to each other. However, with more electropositive atoms the valence shells are relatively large and six or more valence pairs can be accommodated. The determining factor then seems to be the repulsion between the surrounding atoms rather than between the elect,ron pairs. Examples of electronegative atoms probably having 5 or 6 pair valence shells are found in phosphorous pentachloride (Fig. 8), sulfur hexafluoride (Fig. 9) , iodine trichloride (Fig I O ) , t>hetri-iodide and dichloroiodide ions (Fig. 1 1 ) and the carbonate and nitrate ions (Fig. 12). Atoms of electronegative atoms frequently complete their valence shells by sharing electron pairs with other electronegative atoms.' From the chemical formula and the number of valence electrons each atom furnishes one can readily calculate the number of shared pairs necessary to give each negative atom its complete shell. Thus in iodine crystals, with seven valence electrons per atom, each atom shares one pair; in selenium and tellurium and probably
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also sulfur, each atom shares two pairs; in phosphorous, arsenic, antimony and bismuth each atom shares three pairs; and in carbon and silicon each atom shares four (Fig. 13). I n the last case the angles between the bonds are those between the lines joining the center of a regular tetrahedron with it,s corners. If only two or three of the four pairs around an atomic kernel serve as bonds, this angle tends to be somewhat larger, due t o the repulsion between t’he bonded atoms. As another example may be mentioned “arsenic trioxide”, so-called. Each arsenic atom furnishes five valence electrons and each oxygen six. To complete the valence shells each arsenic shares three pairs with oxygens and each oxygen two pairs with arsenic atoms. This determines the empirical
FIG.13 Illustrating the formation of (A) diatomic molecuies as in I?, ,B) string molecules as in Se, (C)layer molecules as in As, and (D) whole crystal molecules as in the diamond, as the result of the tendency of electronegative atoms to complete their valence shells.
formula As203. The angles between bonds in a molecule of this complexity however would be far from those of a regular tetrahedron and like atoms would be drawn too close together for great stability, SO a molecule of the formula Xs406, having th? structure shown in Fig. 14 is formed.’ For this molecule the objections mentioned do not hold. The principles thus far mentioned do not in every case determine the orientation of the valence shells of the electronegative atoms or ions with respect to the surrounding electropositive atoms or ions. In some cases the valence electron pairs must be on or near the centerlines joining these atoms, and it would seem reasonable to assume that this would be the case wherever it is structurally possible. In a crystal of the zinc sulfide type for instance, n e should expect the sulfur valence t’etrahedra to be pointed toward the zinc kernels, giving a tetrahedron also around the latter (Fig. 6). I n ferrous disulfide or pyrite, Fe&, one of the four valence pairs around each sulfur kernel is shared with another sulfur; the others we should expect to be pointing
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FIG.14 The arrangement of atoms and of valence electron pairs (small circles) in a molecule of Asr08, as found in crystals of “arsenic trioxide.” Large dots denote As, large circles 0 atoms.
FIG.I j Showing the arrangement of pairs of valence electrons (small circles) and iron kernels (large dots) around the sulfur kernels (large circles) in a crystal of FeS2, pyrite. The structure is like that of NaCl (Fi . 5 ) m t h Na and C1 replaced by Fe and S, groups, respectivefy.
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toward the three neighboring doubly charged iron kernels (Fig. j) giving an octahedron of six pairs around each of the latter. One more example will be given. Mercuric iodide consists8 of layers in which each mercury is tetrahedrally surrounded by four iodines, while each iodine has two mercury atoms adjacent roughly a t two corners of a tetrahedron. Orientation of the valence tetrahedra around the iodine kernels toward the mercury atoms gives a similar but larger tetrahedron around each mercury kernel (Fig. 16). Structures such as this cannot be attributed solely to the attractions between spherical ions. There is apparently also a definite attraction between the kernels of the metal atoms and the valence pairs, resulting usually in tetrahedrsl or octahedral or cubic valence shells around these metal kernels, depending on the size relationships and structural limitations. These structural limitations are largely a matter of pure geometry. It frequently happens that all of the tendencies I have mentioned cannot be completely satisfied. Either one of the weaker tendencies remains unsatisfied, as in sodium chloride, where it is impossible for the chlorine valence pairs to be on the atomic centerlines-there being six centerlines and only four pairs -or a compromise is reached, as in corundum, A 1 2 0 3 . Each aluminum wants FIG.16 to be surrounded by six oxygens at Showing, in plan and elevation, the corners of a regular octahedron, and arrangement of atomic kernels and valence each oxygen wants to have four alumi- electron pairs in a layer of the HgIg structure. Large dots denote Hg, large circles num atoms around it a t corners of a I atomic centers. regular tetrahedron, but both of these tendencies cannot be satisfied. So actually each aluminum is surrounded by oxygens at corners of a very much distorted octahedron and each oxygen by aluminums at corners of a very much distorted tetrahedron. Another principle of which considerable use has been made is that of the approximate constancy of size of an atom in similar structures. What is meant by “size” depends on one’s definition or method of calculation, but in spite of differences of opinion in this regard, the idea has proved a most useful one. One can also generalize regarding the relative sizes of different atomsthe effect of increasing the kernel charge or the number of shells in the kernel, ebo.-and the differences in size of the same kind of atom in different structures but tiilip will not permit of more discussion of this now.
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In presenting these principles no attempt has been made at completeness, nor have they been given in what will probably be their simplest or most useful forms. However if this paper will help to promote a better understanding of the reasons for the various types of crystal structures observed, my object will have been fulfilled. The better these reasons are understood the easier it will be t o work out new arrangements and the better we will understand atomic structures, molecular structures and the explanation of properties in terms of these structures.
References
* 3
Lewis: “Valence and the Structure of Atoms and Molecules”. (1923). W. H. and W. L. Bragg: “X-Rays and Crystal Structure”, Chapter XI1 (1924). W.L. Bragg: Trans. Faraday SOC.,25, 291 (1929). Goldschmidt: Ber., 60, 1263 (1927);2. techn. Physik, 8,2jI (1927). Cuy: J. Am. Chem. Sot.? 49, 201 (1927). Pauling: J. Am. Chem. Soc., 49,765 (1927);51, 2868 (1929);2 . Krist., 67,377 (1928). Bozorth: J. Am. Chem. SOC.,45, 1621 (1923). Huggins and Magill: J. Am. Chem. SOC.,49, 2357 (1927).
Stanford Cniz3ersity.
