Shih-Ming Ha Westinghouse Research Loborotories and Bodie E. Douglas University of Pittsburgh
Pittsburgh, Pennsylvania 15213
Structures of the Elements in the PTOT System
M o s t inoreanic crvstal structures can be regarded ns based upon a ciose-packing scheme. I n the case of salts, ions of one type are in close-packed array (P positions) with other ions in octahedral (0) and/or tetrahedral (T) holes. A simple system of classification and notation was proposed (1) to include over 1000 compounds. This scheme is based upon the sequence of layers, . . .PTOTP. . ., which is common to all close-packed structures, although some layers can be vacant or only partially filled. The NaCl structure, 6P0, can be regarded as made up of a cubic close-packed (ccp) array of C1- ions (P positions) with Na+ ions filling all of the octahedral holes. There are six layers in the repeating sequence, PAOPBOPcO. Wurtzite, ZnS, has a hexagonal close-packed (hcp) array of 8 2 - ions with Zn2+ ions filling all of the T holes in one T layer (there are two T layers between adjacent P layers), or 4PT (PATPBT). I n fluorite, CaFZ, both T layers are filled by F- ions in a ccp arrangement described as 9PTT. This notation is useful for describing the essential chemical features of a structure a n d in showing relationships among many structures. I t was not intended to provide the more detailed information of interest to the crystallographer. The PTOT system is easier to learn and remember, for those not working with crystal structures frequently, than the approach where the features of each structure must be treated independently. It was shown (2) that by focusing upon the essential features of close-packed arrangements, other than the space-filling efficiency, one can include the simple cubic structure, 3P0, where the same atoms fill the P and 0 sites in an NaCl type structure. Similarly, the body-centered cubic (bcc) structure can be described as a ccp array with the same atoms also filling all T and 0 sites, or 3.2PTOT. Thus the same simple notation can be used to describe structures not strict,ly close-packed in terms of space-filling efficiency. Close-packed structures are particularly common for metals and noble gases. Some unusual structures are encountered, however, for elements which involve specific bonding interactions. It will be shown that most of these structures can be described clearly in terms of further extensions of the broad view of close packing.
Structurer of the p Elements Close-packed structures are expected for elements unless specific bonding forces are operative. Since the p elements of the same family have the same number of p electrons, the structural trends should depend upon the relative importance of these bonding forces or the degree of metallic character. Families differ in the number of p electrons and hence in the types of specificbondinginteractions. There are 31 elements in the ' ' p block" of the period table, including the six noble gases. Most of them are nonmetals or metalloids. Gene~allyspeaking, elements of this block with more than three and fewer than eight electrons in the outer electron shell have predominantly directional covalent bonds, and therefore crystallize in such a way as to satisfy the covalent, requirements.
The Noble Gases: He, Ne, Ar, Kr, Xe, Rn-3P cubic, 4M
(ccp),
All six noble gases have the 3P (ccp) crystal structure with 4 atoms per unit cell (4M), except that He has two additional allotropic structures which are also close-packed: 2P (hcp) and 3.2PTOT (bcc). The noble gas atoms are spherically symmetrical without specific bonding interactions, so that the 3P structure would be expected. These three structures are shown in Figures 1,2, and 3. Halogas: Ck,Err, I-3Pr (ccp), orthorhombic, 4Al All halogen elements occur in the solid state as covalent diatomic molecules, X2. Since there are only nondirectional van der Waals' interactions between molecules, their arrangement basically follows the ccp pattern. This pattern, however, is much distorted (Fig. 4) because of the dumbbell shaped molecules (3). The designation 3P' indicates a distorted 3P structure with an orthorhombic rather than a cubic unit cell.
Crystal Struclures of the Elements Most elements are metals which, with few exceptions, have normal close-packed structures. The border between metals and nonmetals is not distinct, and it is more instructive to consider by periodic groups the p elements, i.e., the six families to the right in the long form of the periodic table. 74
/ Journal of Chemical Education
Figure 1. The 3P structure lsubis close-pocked, ccp, or face-centered cubic). The cubic unit cell is outlined by double lines.
Figure 2. The 2P structure (heiogond dare-pocked, hcp). agonal vnit cell ir outlined b y double liner
9
-.
P~
The hex-
- T-
I
Figure 5.
The 3P kcp) structure of On.
Figure 6.
Tho 3P' structure of the molecular cry+d of Sa
6 Layers (unit cell1
Figure 3. The 3.2PTOT structure (body-centered cubic, bcd. vnit cell is outlined b y doubleliner
Tho cubic
rhombic form is not a close-packed array, and the structure is not known for the monoclinic form. The stable, metallic form of selenium (6) basically belongs to the ccp system, with deviation from a normal ccp structure due to the shift of some packing layers from their normally occupied relative positions (A, B, or C). This tendency is shown in Figure 7 where the normal relative positions are indicated as broken circles. This shift gives each atom two nearest neighbors forming infinite helical chains. The Se-Se distance within any packing layer is much lgnger (4.36 A) than between neighboring layers (2.32 A), as indicated in the figure. Selenium has also two other unstable polymorphs, the so-called a and P forms. A s in the case of sulfur, these two forms are made up of cyclic Ses molecules. Figure 4.
The 3P' structure of CIS(elongated csp).
A pro/ection down the
a. ad,.
Chalcogas: 0 ~ 3 Pcubic, , 4M Se-3P', rhombohedral, 1M S e 3 P 1 , hexagonal, 3M T e 3 P 1 ,hexagonal, 3M Po-3P0, simple cubic, 1M The elements of this family form polyatomic molecules from O2 (multiple bonded), to rings of sulfur atoms in Se or Ss, to infinite chains. There are three solid modifications of Oz which have not been completely characterized. I n the gamma form the 0 1 molecules are in a ccp pattern (4). Unlike the halogen crystals, molecules are oriented so that cubic symmetry the 0% of the unit cell is retained (Fig. 5). The S6 rings can be viewed as flattened octahedra (6) which are ananged in an essentia'y ccP array, 3P' wig. 6 ) . There are two structures for Ss: the
Figure 7. me j v structure or tno newat >e cryrrm. w j *roleenon on a PI. pe,pendi,.~a,to the. ..is. (b) clinographic projection.
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The one form of tellurium is isostructural with metallic Se. The Te-Te distances are 4.47 within each layer and 2.86 A between layers (7,8). The last member of this family, polonium, is perhaps the only element having, as its P form, a simple cubic structure (9) which is denoted 3PO. It can be viewed as the 6PO (NaC1) structure with each site occupied by Po. The uniform Po-Po distances indicate less covalent character than the other members of the family, and, in fact, polonium is a metallic conductor (10). The other allotropic modification of polonium is the low temperature alpha form which can he regarded as a minor distortion of t,he beta form tending towards the structure of selenium (3P'). Nitrogen Family:
Nz-3P, low temperature form, cubic, 4111 N2-2P, high temperature form, hexagonal, 2M P-2ID, orthorhombic, 8M As, Sb, and Bi-6(PO)', rhombohedral, 2M.
For nitrogen, the triple-bonded Nz molecules are arranged and oriented to form normal ccp (3P) or hcp (2P) structures. Its 3P structure is similar to that of 0% (Fig. 5). The structure of black phosphorus, the most stable form of the element, is an unusual layer structure in which each atom forms strong bonds to its three nearest neighbors (11). Each packing layer is actually a double layer consisting of two distorted close-packed layers separated by approximately the P-P bond distance. One of these layers is slightly displaced relative to the other when viewed along the packing
direction (perpendicular to the packing layers). This unusual double layer, denoted as a ID layer and read as a double D layer, is encountered only for black phosphorus and gallium. The packing pattern of these I D layers is in an hcp (. . .ABAB.. . or actually . . .AA'BBfAA'BB'. . .) arrangement. This special packing structure is shown in Figure 8. It has an orthorhombic unit cell occupied by 8 atoms. A different type of double layer structure for graphite will be discussed later. Arsenic, antimony, and bismuth all have the same type of structure in which each atom is covalently bonded to three others to form a puckered sheet which consists really of two normal packing layers (18). Therefore, their structures can be viewed 8s simple cubic-packed (scp) with different degrees of distortion. The regular scp arrangement is 3P0, as in the case of polonium, but for these three elements, the distance from an atom in any basal packing layer to its nearest neighboring atoms in the 0 layer above is different from that below. Therefore, their structures are designated 6(PO)'. A typical 6(PO)' structure is basically a cubic packing pattern. Both the rhombohedra1 unit cell (6 packing layers, 2M) and the deformed simple cubic unit cell (3P0, 3 packing layers, 1M) are marked by heavy lines in Figure 9. A comparison of the crys~ellographic data of these structures with those calculated from the corresponding normal scp structure (3P0, but 6P0 is used for comparison) is given in Table 1, which should be considered in conjunction with Figure 9.
the rhombohedra1 unit cell I2 atoms). The distorted simple cubic cell is shown above the rhombohedron. Thick circles are 0 (octahedral) layers.
Table I.
Comparison of the 6 P 0 Structure with that of As, Sb, and Bi Diatanee between neighboring atoms
A Rhombohedra1 unit cell (8 packing Layers. 2M) a d ) a
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Journol of Chemical Education
Hexagonal unit cell ( packing 6 iayers. 6M)--
~ ath
c(A)
~ { a
within basal plane
B a'-A
to up plane
(A)
C
A/B
to down plane
A/C
C-B
I n the column "rhombohedral unit cell," the angle ar for normal 6 P 0 is 60'. The deviation from this angle is least for Bi (Z044') and greatest for As (6'50'). The unit cell parameter a = 4 for 6 P 0 is taken arbitrarily for convenience of comparison. In the column "distance between neighboring atoms," three types of atomic distance-within the basal plane; between basal and the "up" plane; and between basal and the "down" plane-are listed. The differences or ratios in the table are significant in comparison with those of the normal 6P0 structure. It can be seen that, overall, the structure of Bi is closest to the 6P0 structure and As deviates most. The same conclusion is obtained in comparing the c/a values in the hexagonal cell column. These comparisons indicate greatest covalent character for arsenic, or, conversely, most metallic character for bismuth. The bonding characteristics and general properties of antimony are between those of As and Bi. Carbon Family:
(?(diamond), Si,Ge, Sn-GPT,
cubic,
RM --.A
C(diamond)-4PT, hexagonal, 4M C(graphite)-2IP, hexagonal, 4M, and 3IP, rhombohedral, 21LI Pb-3P, cubic, 4M Since the elements in this family are quadrivalent, the possibility arises of a structure which owes its coherence in three dimensions to covalent bonds alone. The first four elements in this family, as shown above, all have the 6PT cubic structure using their sp3 hybrid orbitals and the whole crystal constitutes a giant molecule as shown in Figw e 10.
Figure 11. graphite.
The ZIP (double P layer1 structure of hexagono1 close-packed
scheme. This plane, a normal double packing layer denoted as I P (double P layer), is entirely different from the double layer, ID, of black phosphorus. Since there are three relative packing positions, there are also only three possible combinations for the double layers: AB, AC, and BC. Graphite has two different structures. The familiar hexagonal form, which has an elongated hexagonal unit cell, is shown in Figure 11. For hexagonal graphite one position (designated C) is common to all double layers, but, of course, the spacing between layers is greater than within layers. The layers are . . .P,,PBo PAC.. . or . . .IPAIPaIPA. . . and the designation is ZIP for the bcp double layer structure. The other form of graphite, the so-called rhombohedral form, is shown in Figure 12. Adjacent layers
Figure 10. The 6PT Wudure (cubic diamond). The packing (PI loyerr ore in o ccp panern. The cubic unit cell is outlined b y double liner.
Carbon has two crystalline forms-diamond and graphite. Diamond is a typical 6PT ccp structure with carbon atoms filling the P sites and all T sites of one T layer. Its interatomic distance (1.64 A) is characteristic of the C-C single bond. The carhon atoms in graphite lie in layers and each layer has a special hexagonal network. The atomic positions in such a layer were first determined by A. W. Hull (15) who proposed that graphite consists of puckered layers. Other investigators (14, 15) later showed that the atoms in such a layer are coplanar within their limits of error. In this coplanar layer of graphite each atom has only three nearest neighbors, not six as in a real close-packed layer. However, if each of the -graphite layers be imagined to consist of two closepacked layers forced into One plane, graphite can be regarded as a special case of the general close packing
~i~~~~12. it..
The 31P (double P layer) structure of cubic close-pocked graph-
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have one position in common, . . .PABPBCPCA.. .. From the proposed concept, this form is assigned as cubic close-packed graphite with the symbol 3IP. Here also the big gap between layers (16) gives a ratio of atomic distance within the layers to that between two neighboring layers of 2.4 (3.40:1.42), compared to a ratio of unity in the normal close-packed 3P strnoture. Therefore, although the graphite layers follow exactly the cubic packing sequence, PABPBCPC*, the unit cell is not cubic but rhombohedra1 with two atoms per unit cell or hexagonal with six a t o m per unit cell. Recently, due to advances in high pressure technology, both cubic (6PT) and hexagonal (4PT) diamond have been synthesized from graphite (17-20). The close packing nature of graphite structures described above provides a possible model for those transformations. One can imagine that when very high pressure is applied, cubic graphite could close the big gaps between layers causing each double layer to split into two normal close packing layers. In the final diamond structure one of these (equivalent) layers would be a P layer and one a T layer. The conversion of rhombohedral graphite into cubic diamond can be . . -t . . .PATBrepresented as . . .IPABIPBcIPcAIPAB. P,T,PcTAPAT~.. . for 31P(graphite)-6PT(diamond). For hexagonal graphite (21P), in order to produce tetrahedral coordination, it is necessary that, after the splitting of the double layers, some of the simple close-packed layers must slide to change their relative positions. The structure of lead gives no evidence of covalent bonding. It is a cubic close-packed (ccp) arrangement which is characteristic of a true metal. Boron Family:
BIZ-3Pf, rhombohedral, 1M Al-3P, cubic, 4M Ga-2ID, orthorhombic, 8M In-3P1, tetragonal, 4M T1-2P, hexagonal, 2M, and 3.2PTOT, cubic, 2M.
Boron is the only nonmetallic element in this family. Of the several molecular structures, only one, the red crystal with a rhombohedral unit cell, is described here. The atomic arrangement in this colorful crystal can be considered as a nearly perfect ccp arrangement (3P) of B12icosahedra, as shown in Figure 13. This structure is possible because of the nearly spherical B12units ($1). Aluminum has the ccp structure (3P) which is characteristic of a true metal. Gallium, as in the case of black phosphorus, has a hexagonal close-packed structure of puckered double layers, 2ID. Indium has a distorted cubic structure (22, 25). This structure can be considered either as deformed body-centered cubic, 3.2(PTOT)', with 2 atoms in a tetragonal unit cell, or as pseudo face-centered cubic, 3P'. Thallium has both the hcp, 2P, and the body-centered cubic 3.2PTOT structures (24). Crysfal Sfrucfures o f the Mefollic Elements
The structures of the elements are shown in Table 2. Only the s, d, and j elements have not been discussed. All of these, except hydrogen with the 2P structure, are metals which, with few exceptions, have one of 78 / journal o f Chemical Educafion
Figure 13.
The 3P' ttructuremntoining BIS icomhedro.
three typical structures in the close packing scheme, ccp, hcp, or bcc (Figs. 1, 2, and 3). This result is expected in the absence of discrete bonds. However, the cyclic occurrence of the three close-packed structures in the periodic table, as shown in Table 2, is an indication that the preference for one of the three structures by any element is related to its particular bonding characteristics. Theories of the Metallic State. There are two basic approaches to the theory of metals, using valence bond theory or molecular orbital theory. In the band theory the energy states are analogous to molecular orbitals extending over the whole crystal. This approach has been particularly successful in explaining the electrical and magnetic properties of metals. The various valence bond approaches differ in the relative importance of s, p, d, and hybrid orbitals in bonding. The latter approaches are perhaps more useful for periodic structural correlations than the band theory. Pauling (25, 26) used the number of s and d bonding electrons per atom to correlate with structures of transition metals and concluded that the higher this number the more stable the structure will be. Hume-Rothery ($7) gave a critical review of the difficulties involved in Pauling's approach. A concept of bond hybrids ($8) has been used to differentiate the three closepacked structures encountered for metals. The corresponding hybrids for each structure were postulated to match the necessary geometrical symmetry of that particular structure. From these hybrids, then, the weight of d electrons per orbital was calculated for these three structures as follows Cvstal Structure
Hybridization
Weight of d elect~onsper orbital
Zener (29) and Cornish (SO) pointed out an important basic difference between ccp and bcc structures. In
.. ...
." . . * .,;.I""..i"
".#"
*,
":..'."-.."....- .
Elctkti
' T a b l e 2:' Cbse-Pocked .Cr+tdl,Stru&r&s'oP
2P: 3P: PTOT: 3PO: 68):
1 1 F; 1 lpToT! ::1 PTOT
23 Y
24 Cr
25 Mn
26 Fo
2P
2P PTOT
PTOT
PlOT
PTOT 3p
PTOT 3P
39 Y
40 Zr
41 Nb
42 Ho
43 Tc
Wt Ru
PTOT
PTOT
ZP
2P
27 t o
2P 3P 45 Rh
28 N , 29 Cu
3p
3p
46 Pd 47 Rg
1 1 / 1 I 1 1 1 w
75
76
o
3P
r 77 1r
71 Lu
72 H f
73 l a
74
2P
2P PTOT
PTOT
PTOT
2P
2P
?P
3P
3P PTOT
PTOT
PTOT
PTOT
PTM
2P 3P 4P
RE
'r;"H-;l
rm: Double D Layer ' : Mans Distorted
22 TI
2
".:"
R: Double P Layer
21 Sc
PTOT
PTOT
hcp ccp bcc scp scp IPseudo1
,
the cop structure, every atom has nearest neighbors which are nearest neighbors of each other. This is not true for the bcc structure in which there are two sets of atoms-two-fold identity. In the bcc (3.2PTOT) structure, atoms in positions P and 0 are equivalent and those in the two T layers are equivalent. Metals with high net electron spin should have a tendency to crystallize in a structure which permits nearest neighbors to have antiparallel spin. The bcc structure fulfills this requirement. The net spin reaches its maximum value when a subshell is half-filled. The alkali metals, Li to Cs, the refractory metals of the Ti, V, and Cr groups, and europium (Eu) have halffilled s, d, and f shells, respectively. These metals crystallize in the hcc structure. On the other hand, when the sub-shells are empty or completely filled and the net spin is zero, the ccp structure is preferred. For intermediate values of net spin, they prefer the hcp structure. This is an appealing approach, but neutron diffraction results were not consistent with the predict,ions (51). Finally, the so-called Engel-Brewer theory (33-57) has attracted much attention during the last decade. This is mainly because this theory can be used in predictions of multi-component phase diagrams (%), although there are still many inconsistencies in its applications (59). In general, the Engel-Brewer view that the stabilit,y of a metallic structure depends on the interplay of the promotion energy and the number of bonding electrons is preferable to the usual accounts of Pauling's ideas which consider only the latter. The Engel-Brewer theory states that the d electrons play
3P
3P
78 P f 79 Au
3P
3P
an important role in determining structure indirectly, but the bonding is due to s and p electrons only. A correlation between structures and the number of sand p electrons per atom is shown as follows Crystal Structwe
s and p eledrons per atom
CCP
2.5-3.0 1.7-2.1
