Shih-Ming Ho Westinghouse Research Laboratories Pittsburgh, Pennsylvania 15235 and Bodie E. Douglas University of Pittsburgh Pittsburgh, Pennsylvonia 15213
I
I 1
A System of Notation and Classification for Typical close-Packed Structures --
Studies of crystal chemistry have attracted much greater interest during the last decade than ever before. This is the result mainly of the increasingly severe requirements for materials in every phase of technology and scientific research. The number of publications on crystal chemistry continues to increase. In order to understand much of the discussion of crystal structures in the current literature, it is necessary that one remember the features of hundreds of three dimensional structures and the relationships among these strnctures. Only specialists can be expected to have the necessary information at their fingertips. Structures are commonly described in these terms: "it has a NaCl structure," "its structure is either of the wurtzite or zinc blende type," or "the disordered spinel type structures." Such descriptions require not only that the reader or listener remember the details of structures by name, but also that he make comparisons which require one to be able to visualize subtle structural changes. Most of the common inorganic crystal stmctures are closely related, one to another, based upon a broad close packing concept as discussed in a previous paper (1). Consequently, it is necessary that one be familiar with the features of close-packed structures. A simple system of notation is proposed in this paper to describe the essential features of a structure in terms of the broad close-packing concept. This proposed notation makes it unnecessary to remember many individual structures by name. I t shows how a strncture is related to the general close-packing scheme and thereby simplifies comparisons among structures. The time presently allocated to teaching several isolated structures could be used for presenting the concepts of close packing nnd thereby the basis for describing many hundreds of inorganic structures. A Broad View of Close Packing
The fundamental characteristics of both ideal close packing, in which each sphere touches its 12 neighbors, and the proposed broad concept, which includes cases in which the packing atoms no longer touch, have previously been described (1). Figure 1is a summary of the main features of the Close-packed nature of most crystals. It is a section along the packing direction in a plane perpendicular to the packing layer planes. The two packing atom layers (P) shown in Figure 1 1 Tho distance hotweon two spheres or two atoms or two sites is tho distance between their centers. 2 Tho T+ sites are those for which the tetrahedra point in the packing direction, while the T- sites are those for which the tetrahedra point in the opposite direction.
P Layer
T- Layer
0 Layer T+ Layer P Layer
P: Packing Atom
0: Octahedral Site 1: Tetrahedra Site A , 0.C: The Three Relative Packing Positions Figure 1. Construction of the boric close-pocked unit-the orrongement between two pocking .tom layerr for any close-packed struchlre.
constitute the basic packing unit for any close-packed structure. Between these two layers, which are sepa-. rated by the distance d, the "holes" or sites form three more layers: an octahedral site layer (0)at 1/2 d and two tetrahedral site layers (T), one a t 1/4 d and one a t 3/4 d. The distance between any two neighboring sites in any one of the three layers is exactly equal to the distance between two neighboring atoms1 in the packing atom layers. Thus, there are actually four packing layers within one close-packed unit in the sequence PTl - - , OT - - - - P.? The two most important basic close packing structures are PATOTPeTOT and PATOTPeTOTPcTOT in which A, B, and C are the three relative positions in close packing. The former is the conventional hexagonal close packing (hcp) structure as shown in Figure 2. The latter is the cubic close packing (ccp) structure as shown in Figure 3. All types of crystal structures discussed in this paper are derived from these two basic models by the following variations of these close packing arrays 1) The atoms occupying the four basic packing layers (P, T+, 0 , T-) in the packing units could he all alike, d l different, or any intermediate combination. Thus they might be alike in two pairs, e g., atoms in P and 0 layers are the same; three alike, e.g., atoms in T+, T-,and 0 layers are the same; or other less likely combinations 2) One or more of the four basic pecking layers could be vacant, as in the structure of NaC1. Since both tetrahedral site layers T+ and T- are vzcmt, this packing sequence is PO. 3) Any of the four basic packing layers could he partially filled in s. symmetrical pattern, such as in the rutile structure, in which only half of the positions in every octahedral site layer are occupied. Its paekmg sequence is therefore POliz. 4) Any one of the four basic packing layers could contain more than one kind of atom, the different kinds being symmetrically distributed in the packing layer. I n the CaTiOa structure, for instance, the packing atom layer contain8 Ca and 0 in a ratio of 1 3 , and the Ti atoms occupy 1/4 of the octahedral positions. The packing sequence in this case is POlx. Volume
46, Number 4, April 1969
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positions in two T layers, forming a thin triangular sandwich with the packing sphere in the center a t a B position. The remaining two tctrahedral sites arc located one above and one bclow the packing sphere (B positions). The six octahedral sites are all located a t C positions forming a thick triangular sandwich with the packing sphere in the center as shown in Figure 4. In ccp (-ARC-), however, both the eight tetrahedral sites and the six octahedral sites are symmetrically arranged around the packing sphere. The eight tetrahedral sites, which are located in four consecutive tetrahedral site Pac ing Sequence layers, form a cube, and the Figure 2. The basic hexogonol close pocking (hcpl rtructure IBPTOT structure). P, 1, 0: pocking layers. A, B, C: six octahedral sites form a the three relative pocking positions regular octahedron, as shown in Figure 5 . In actual close-packed structures the distance The relative arrangement -hetween the octahedral between any two neighboring packing atoms is usually sites and the tetrahedral sites in hcp is also entirely larger than in the idealized case. In other words, the different from that in ccp. In hcp, each octahedral site packing atoms do not touch each other in the actual is surrounded by six tetrahedral sites from two tetrastructures because they are pushed apart by the atoms hedral site layers in a much deformed octahedral in octahedral and/or tetrahedral sites or because of pattern. Each tetrahedral site, on the other hand, has specific bonding interactions. Consequently, the relaonly three nearest octahedral site neighbors forming a trigonal pyramid, as in the structure of NH3. In ccp tionship between the relative sizes of the packing atoms the octahedral sites are equivalent to the packing atom and the atoms in sites is less important for the actual positions, so every octahedral site is located in the structures than for the ideal case. Their relative discenter of a cube formed by eight tetrahedral sites tances and positions are, however, still essentially the contributed from four consecutive tetrahedral site same as under the ideal closc-packed conditions. layers. Every tetrahedral site has only four octahedral Generally, the P-P and 0 - 0 distances are all equal, and the P-T, and 0 - T distances are equal. The P - 0 dissite neighbors arranged tetrahedrally about it. These relationships are shown in Figures 4 and 5 . tance is shorter than that for P-P or 0-0 and the P-T and 0-T distances are the shortest of all, as shown in Figure 1. The solid lines in the figure are located'in the plane of the paper. The broken lines are out .of the plane of the paper but are projected on it. The three angles marked in the figure indicate the positions of the packing atoms, the octahedral site atoms, and the tetrahedral site atoms relative to the plane of the packing atom layer. In either hexagonal close packing (hcp) or cubic close packing (ccp), each packing sphere has eight tetrahedral sites as nearest neighbors and six octahedral sites as second nearest neighbors. The arrangements of these sites about the wackinn are entirely ..swhere . different ill these tlvo cases. Figure 3. The bosic cubic close pocking kcp) structure (cubic PTOT structure). This is the body-centered cubic (bccl structure when all the pocking layers (P, T, and 0 ) ore filled by one kind of otom. Also it is the CsCl six tetra- ~twcture In hcp when P and 0 layers are filled by one kind of otom and both T layer. b y the other. P, T, 0; packing hedral sites arc located a t A layerr. A, B, C: the three relative packing positions.
t(
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Journal o f Chemical Education
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of the four basic close packing layers, in the order PTOTP, between any two packing atom layers (P) is always obeyed. Any of the layers can be vacant in a particular structure, hut the positions of the vacant layers or vacant sites will still he there. For instance, the PO structure means that only P layers and O layers are occupied by atoms and that all the T layers are vacant in the basic packing unit PTOT. Symbols T+ and T- are used only when the distinction is necessary; thus, (AB) ( PTOT) PT+PT-, but P T or PTT. 4 A When any of the four basic packing layers is partially occupied in a symmetrical Figure 4. Arrangement of the eight tetrohedrol neighbors (broken-line circler) ond rix octahedral neighbon pattern, a fractional number (broken-line triangles) of one hcp packing otom (4. P, T, 0: packing loyen. A. 8, C: the three relative packing poritionr indicating is occupancy appears as a subscript, as in the previously described structures POljz and POll4. Any structure type has a numerical index to indicate the total number of packing layers in its unit cell. For example, 4PT means that this structure has four packing layers, PTPT, in its unit cell. I n addition, this numerical index also shows indirectly the basic packing sequence of the packing atom layers (P). Two of the four layers for 4PT are packing atom layers (P). The two P layers must be in the AB sequence, corresponding to hexagonal close packing. I n (ABC) the case of GPOl/4,three of the t Packing Sequence six layers are P layers. Since this is the repeating unit, the Figure 5. Arrangement of the eight tetrahedral neighbors lo cube) and six octahedral neighbors (shaded, on P layers must be in the ABC octahedron) of one ccp pocking atom 1x1. P, T, 0:pocking layers. A, 8, C: the three relative pocking positions. sequence, corresponding to cubic close packing. For 12Notation PTOT there are three P Recently, the significance of the close packing nature layers in the repeating unit, corresponding to ABC or of crystals has attracted tremendous attention of many cubic close packing, PATOTPBTOTPcTOT. The same results are obtained by dividing the index by the investigators in different fields (2-9). Most of these investigators have tried to correlate the relationships number of distinct packing layers to give two (AB) for among differentstructure types using this packing treathop or three (ABC) for ccp. Thus, in the 4 P 0 strucment. This important approach has not, however, been ture, for instance, four divided by two (P 0 ) gives used systematically in nomenclature and classification. two. This result means the P layers are in an hcp In this paper the authors propose a system of classifica(-AB-) sequence; a hexagonal unit cell is expected. tion based upon the notation used in describing the I n case of the index six divided by two giving characteristics of close packing. This system should three means that the P layers in this structure are in a simplify the understanding and further studies of huuccp (-ABC-) packing sequence, commonly giving a dreds of complicated three-dimenstional crystal struccubic unit cell. I n case of 12PTOT, there are four tures. layers in one repeating packing unit; the index 12 The packing atom layer, the tetrahedral site layer, divided by four gives three. The quotient indicates and the octahedral site layer are indicated by the letthat in this structure the P layers are also in a ccp ters P, T, and 0, respectively. The definite sequence (-ABC-) sequence.
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46, Number 4, April 1969
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In structures in which one layer contains more than one kind of atom or is not completely filled, the number of packing cycles in a unit cell is sometimes neither two (Ibr hcp) nor three (for ccp) but some multiple of two or three. The higher indices, indicating the number of layers in the unit cells, are given as 2. n or 3. n in order to show the basic packing pattern.of the Players. For example, the chalcopyrite (CuFeSz) structure corresponds to that of zinc hlende (ZnS, 6PT) except that Zn is replaced by Cu and Fe in the T sites. The Cu and Zn atoms are in an ordered array (2 Cu and 2 Zn attached to each S) so that there are six, not three, packing cycles or 12 packing layers in the unit cell (PATPBTPCTPATPBTPCT). The structure is designated as 3.4PT to emphasize the ccp pattern. Similarly, the structure of SuL is designated 3. 6PTtlsTlis instead of 18PTlisTllsin order to make it clear that Table 1.
Structure Type 2P 3P 4P 6P 4PO 4POm 4POln 4POm 2.4PO1m 2.6POzn 6PO
. ..
6POm 6PanOtir 3.4POm 12POc1~P0m
PPO (Layer Structures)
PTOT
+
Classification
Most crystal structures of the elements and inorganic compounds can be classified into well-defined systems based upon the two basic close-packed structures, hcp and ccp, as shown in Figures 5 and 3. Only eight of the main systems are described in this paper. Although there are within each system many different structure types, only those types for which there are known
Classification of Close-Packing Structures
Basic Packing (P layers only) (-AB-) (-ABC) (-ABAC) (-ABCACB-) hcp (412 = 2) ~ C P
hcp hop hop h c ~ ccp (612
...
=
3)
CCP COP
CCP CCP
4PT
hcp (412 = 2)
4PTain
hcp
2.6PTm 2.6PTzm
~ C P ~ C P
6PT 6PTw 6PTzn
ccp (6/2 = 3) COP CcP
3.4PT 3.4PTsx 3.4PT1n 3.4P8/,T
CCP CCP CCP COP
2.6PTtiaTm 9PTT 9PT~mTliz
~ C P
9PTmTlla
CCP
3.6PT314Tm
CCP
3.6PTrnT1m 2.312PPO 2.3/2PPOlir 2.9/2PP02is 9PPO 9PPOm 9PP0r~ 8PT1a0~izTm
CCP hcp hcp hcp
...
PTT
the arrangement of the packing atom layers (P) is ccp. The role of each element in the structure is usually obvious from the atom ratio and the notation; e.g., for SnL, with PTllsTllspacking, Sn must he in the T sites. When this is not the case, e.g., CaTiOa, the roles of the atoms can be indicated in parentheses, e.g., CaTiOI (Ca 0 :P).
.. .
ccp (9/3 = 3) CCP
cep (913 = 3)
CCP ~ C P
hcp (8/4 = 2)
CCP 3.2PTOT CCP 3.2PTOT 12PTOT cep (1214 = 3) POPTOT cep (1816 = 3) 18P0&PT,i4O1nTlia PPOPO 1OPPOd'Ou~ hcp (1015 = 2) ~ . ~ / ~ P P O I , ~ P O I I C~' V a The first number in the parentheses is the reference number, the is the page number. b M = Molecule.
210 / Journal o f Chemical Education
Unit Cell
Example
References
Mg (hcp structure) ... ... Ni (ccp structure) Ls. .. . CoaV (12) NiAs (nicolite) (11, 61) Ti02 (rutile) (3, 77, 125) CsNiCls (10, n,425) UCls (10, 11, 205) MnO(OH) (11, 124) n-Altoa (2, 128); Fig. 6 NaCl Fig. 7 FeSz (pyrite) (11, 66) CaTiOs Fig. 8 ReOs (3, 67) Ti02 (anatrtse) (2, 126) CU~CI(OH)~ (11, 42; 3, 394) (ntacamite) Hexa., 2M ZnS (wurtzite) Fig. 9 Orth., 2M CusAs& (enargite) (11, 60) (13) Hexa.,l/2 M AlaZnS, (Al Zn: T ) Random Hexa., 12M 8-ZnCI, (10, I, 311) Aexs., 6M n-Ga& (10, 11, 22) (high temp, form) Cubic, 4M ZnS (blende) Cubic, 1M InzCdSe* ii,'533) Cuhic,4/3M y G a & (low temp. (14) form, random) Tetra., 4M CuFeSz (11, 55) Tetra., 2M 8-AgzHgIn (3, 533) Tetra., 4M ar-ZnClz (3, 119) Tetra., 2M Cu3ShS3 (3, 532) (tetrabedrite) Mono., ZM Al&h (3, 352; 10, II, 57) Cubic, 4M CaFz Fig. 10 Tetra., 4M PtS (Planar Pt) (3, m ) (Pt: P layers) Tetra., ZM PbO (Pyramld Pb) (3, 122) (Pb: P laycrs) Cuhic, 16M Mn& (3, 465) (Mn:P layers) (2, 69) Cubic, 8M SnIl (3, 350) Hexa., 1M CdI. Fig. 11 Hexa., 1M KlGeFe (3, 376; 10, 111, 349) Rhom., 2M BiIa (10, II, 46) Hoxa., 3M CdCI. (3, 349; 10, I, 267) Hexs., 6M CrCla (10, 11, 54) K.PtCls (3, 376; 3, 67) Cubic, 4M Orth., 4M MgsS~04 Fig. 12; (forsterite) (11, 174) Cubic, 2M Cr (bee structure) Fig. 3 Cubic, 1M CsCl Fig. 3 Cubic, 4M BiLia Fig. 3 Cubic, 8M MgAIxO. (spinel) Fig. 13 Hexa., 2M CssTlzCIg (distorted) (3, 376) Hexa., 1M CsrAszC1. (distorted) (3, 376) roman numeral is the volume number, and the Arabic numeral Hexa., 2Mb Cubic, 4M Hexa., 4M Hexa., 6M Hexa., 2M Tetra., 2M Hexa., 2M Hexa., 3M Mono., 8M Rhom., 2M Cubic, 4M Cubic, 4M Cubic, 1M Cubic, 1M Tetra., 4M Orth., 4M
+
examples are listed. Many other theoretically possible types in each system can be derived by varying either t l ~ rornpositiu~ r or the grornrrry of t h e peking layers; such extensions will be rrserwd for futurc studies. The eight systems, structure types for each system, unit cell descriptions, and specific examples are summarized in Table 1. Gcnrrally, an hcp structure has a hexagonal unit cell and a ccp structure has a cubic unit cell. This is always tme for simple compounds of fnlly occupied packing layers. When one or more layers are not fully occupied or occupied by more than one kind of atom, however, the structure usually has a crystal system other than hexagonal or cubic, as shown in Table 1. For instance, ZnS (murzite) and CusAsSa 110th have the 4PT structure type. The former has a simple hexagonal unit cell containing 2 ZnS. The latter, however, has 6 Cu, 2 As, and 8 S in an orthorhombic unit cell because of the ordered occupancy of the T sites by both Cu and As. P System
The P system includes the typical close-packed nt,ruct,urmmith only the packing atom layers (P) occupied; all the octahedral site layers (0)and tetrahedral site Iayors (T) are vacant. This systcm is fnlly described i n , most sonrces dealing with r:ryst,al structures, About 50 elements and many intermetallic compounds, especially those of the XY3 composition, have strnctures belonging to thiv system. The structure type 2P corresponds to hcp; the typc 3P corresponds to ccp. The types 4P, 6P, etc., correspond to more complex packingarrangements (see Tnhle 1).
oxides because of its high melting temperature and chemical inertness. The most important of the several polymorphic structures for alumina, the hexagonal form, c-AIzOZ,also belongs to the PO system; its notation is 2.6POzI2. A detailed drawing indicating its rhombohedra1 unit cell formed by 12 packing layers is shown in Figure 6; the packing atom layers (P) are in an hcp pattern. The "NaC1 structure," which is encountered for more t,han 200 known compounds is the most familiar of the ccp types in the PO system; it is classified here as GPO. Its cubic unit cell, which contains four molecules, is shown in Figure 7. Another very interesting type in this system is the "perovskite structure" (CaTi03), with calcium and oxygen atoms in the packing atom layers and Ti atoms occupying only 1/4 of the octahedral sites in each layer, i.e., only those forming Ti06 units. The basic ccp nature of the P layers and the cubic unit cell are shown in Figure 8. The structure of ReOs is closely related to' the 6PO114 (perovskite) structure. The ReOa structure is classified as 6P31101,4,with each packine laver onlv three-auarters filled by oxygen atoms. f h e packing atom positions
R hombohedral Unit Cell 2.6 Layers
PO Sysfem
The PO system has only P and 0 layers occupied mith all T laycrs vacant. It is the most common system, including more than 500 kno\vti compounds, most of which are ionic. The most simple hcp type in the PO system is the conventional "NiAs stmcture," classified as 4P0, in which all octahedral sites are filled and the parking atom lavers (P) are in an hrp arrangement. Many other similar types are obtained by partially filling the octahedral sites: ex.. for CsNiClr, 4P01/6 for UCle, and 2.4p0,,2 for MnO(OH), (A1z03) is One the most intensively studied
I
( A B ) (PO
4
2'3
1
0 layerr occupy the some relative positions lcl. I31 There ore 1 2 pocking layen IP and 01 in the rhombohedra1 unit cell. (4) There ore 2 molecules 12A12031 in the unit cell. 151 The relative positions in planes perpendicular
to the plane of paper are indicated by numbers in the circles in t h e upper part of the figure. The layers ore from the highest I l l to t h e lowest 191. Small circle: A1 atom; large circle; 0 otom; broke I vacant; lined circle; otom in the unit cell.
Volume 46, Number 4, April 1969
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of ALBra. The ccp types arc t,he most common structures in this system; of these 9PTT (Fig. 10) is the most import,ant. I t is the so-called "fluorite" structure (CaF,), with more than 100 known examples. Some other t,ypes helonging to this system arc listed in Tahle 1.
c
P
j
Layers Iunit cell)
PPO System
The PPO system represents layer type structures resulting from the vacancy of alternate octahedral site layers. The type 2.3/2 PPO is the hcp CdI2 structure and the type 9PPO is the ccp CdC12 structure: a total of about 77 known examples exist for both structures. These structures are related to the 4 P 0 (NiAs) and (ABC) (PtO) GPO (NaCI) structures, rePacking Sequence spectively, with alternate octa12) There are hedral Figure 7. The 6 P 0 structure I N K 1 rtrudurel. (11 The cubic vnit cell is outlined by dark line.. site (cation) layers six packing loyerr (P and 0 ) in the unit cell. (31 There is a ccp pattern of the P loyerr land of the 0 layers) varant. The 2,3/2 PPO structure is shown in Figure 11 as an examvle of the laver strucoccupied by Ca atoms in CaTiOs are vacant and Ti tures. Four more types in this system derived from atoms are replaced by Re atoms. these two basic layer structures are listed in Table 1. The 9PP0114type for IGPtCls is an interesting structure PT System in which the atoms of K and Cl form the P layers, and Many covalent compounds and most "tetrahedral P t atoms occupy one-quarter of every other O layer. structures" (4) belong to the P T system, in which only If,however, the PtCls2- octahedron is treated as a unit, every other tetrahedral site layer is occupied. There then K2PtC16can be described as having a 9PTT strucare two very common structure types in this system. These types are the hcp murtaite structure, which is 4PT (Fig. 9), and t:he ccp zinc hlende structure, mhich is WT. The latter structure with both P and T layers occupied by carbon atoms is thc cubic diamond structure. Some (b) ot,her structure types in this 6 Layers systcm arc list,ed in Table 1.
( u n i t cell)
PTT System
In the PTT system both tetrahedral site layers (T+ and T-) in the basic packing unit are filled, but the octahedral layer is vacant. I n hcp types such as GPTT the distance bet,meen the two neighboring tetrahedral layers is too close to be stable (see Fig. 2). However, other hcp types in mhich the T layers are only partially occupied should he good hesago~ial st,ructures for hinary compounds; e.g., OPTI/CTI/C, mhich describes the stmrt,urc 212
/
0
P 0
P
(a) 6 Layers (unit cell)
(ABC) (PO,)
t
t"
packing seq;ence Figure 8. The 6POl,r rtructure ICoTiOs, perovskite rtructure). (11 Cubic unit cell: within the dark lines ( 0 ) or the cube formed b y 8 light atoms in the center (bl. (2) There are six packing loyen in the vnit cell. (3) % of the 0 sites ore filled by Ti otomr (only thore surrounded octahedrolly b y oxygen atoms). ( 4 ) There is a ccp pattern of the P byers (and of the 0 loyerrl.
Journal o f Chemical Education
4 Layers (unit cell) T
P
packing sequence The 4PT rtrvcture (ZnS, wurtzite structure). ( I ) Hexagonal unit cell within the dork lines. 131 There is a hcp pattern of the P layers land of the T ore four pocking layer. in the vnit
Figure 9.
(2) There
layers).
cell.
ture (fluorite type), with the PtC1, octahedra forming, the packing layers and the I
