Research: Science & Education
The Language of Lattices and Cells M. F. C. Ladd Department of Chemistry, University of Surrey, UK In a language, a given situation may be described by more than one word, or a particular word may be used to define more than one entity; this state of affairs occurs also with lattices and cells. The terms lattice and unit cell are well known in solid-state chemistry and physics. Nevertheless, there are a number of instances where their use is not entirely consistent (1, 2), and this situation may create a learning difficulty. This paper addresses topics associated with three-dimensional figures and concepts, the nomenclature used currently to describe them in different contexts, and attempts to formulate designations that both apply to and respect tradition in the typical example situations used. Lattice A lattice may be defined as a regular, infinite arrangement of points in space for which any point has the same environment as every other point. This description applies equally well in one-, two-, and three-dimensional spaces, and the numbers of lattices are distributed in these spaces as indicated in the box. Since we are concerned with the solid state, our interest centers on the Bravais, or space, lattices. Frankenheim (3) derived 15 lattices, but Bravais (4– 6) showed that two of them referred to one and the same monoclinic C unit cell; for his sin, DimenNumber of Lattice Frankenheim is ususionality Lattices Descriptor ally ignored in this context. 1 1 Row Each Bravais 2 5 Net lattice may be 3 14 Bravais specified by three noncoplanar vectors a, b, and c, parallel to reference axes x, y, and z, respectively. The axes form a right-handed set, and the interaxial angles α, β and γ refer to b^c, c^a and a^b, respectively. From any point as an origin, any other point in the lattice has a vector distance ruvw from the origin given by r uvw = Ua + Vb +Wc
that the stacking of nets a,b at another given spacing c leads to a Bravais lattice. We note that the framework shown in the figure is not a part of the lattice; the lattice is the set of mathematical points, the lines being drawn to assist us in appreciating the lattice geometry. The symmetry at each point of the lattice in Figure 1 is 1¯ (i). When a lattice possesses symmetry higher than 1¯ at each point, the vectors a, b, and c may be selected in a specialized manner. For example, if a twofold symmetry axis passes through each point in the lattice, the symmetry at each point is then 2/m (C2h): the symmetry 2 at each point combined with the symmetry 1¯ present in the lattice results in symmetry 2/m at each point. The vectors a, b, and c may be selected such that a ≠ b ≠ c, with α = γ = 90°, and β ≠ 90° or 120°. In principle, the unique angle could be α, β or γ; the choice of β is a crystallographic convention. It is possible to choose other axial vectors for this lattice, such that no interaxial angle is equal to 90° or 120°. In such a choice, however, the symmetry of the lattice would be revealed in the axial relationships. For example, if the lattice referred to a conventional monoclinic C-centered unit cell (a,b,c) is referred instead to a primitive P unit cell (a9,b9,c9) under the transformation a9 = a/2 – b/2;
b9 = a/2 + b/2;
c9 = c
(2)
then, we find the relationships a9 = b9 = (a2 + b2)1/2 / 2
(3)
γ = cos{1 {(a 2 – b2) / (a2 + b2)}
(4)
and
The symmetry of a lattice is invariant under choice of unit cell. Thus, it is pertinent to consider how the particular vectors a, b, and c may be selected for each lattice.
(1)
where the integers U, V, and W are the coordinates of the lattice point. Since U, V, and W may be positive, negative, or zero, it follows that all lattices are centrosymmetric. The most general lattice is represented in Figure 1, in which a ≠ b ≠ c, and α ≠ β ≠ γ ≠ 90° or 120°; these numerical values would imply lattice rotational symmetry 4 (C4 ), 2 (C2 ), or 6 (C 6), 3 (C3 ), respectively. The symmetry symbols employed here are those of Hermann (7, 8) and Mauguin (9), followed by those of Schönflies (10) (in parentheses) the first time a given symbol is introduced. The ≠ sign should be read as “not constrained by symmetry to equal” rather that just “not equal to”. This difference may not be important in studying the lattice itself, but it becomes meaningful in a structure, where chemical entities are distributed regularly and symmetrically with respect to the lattice points. From Figure 1, we can see that the aligning of rows of spacing a at another given spacing b builds up a net, and
z c
a x
b y
Figure 1. Rows (a) aligned to form oblique nets (a ≠ b, γ ≠ 90° or 120°), and the nets (a,b) then stacked regularly to form a triclinic Bravais lattice (a ≠ b ≠ c , α ≠ β ≠ γ ≠ 90° or 120°); the symmetry at each point in the lattice is 1¯ (i ).
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Research: Science & Education Unit Cell We use the term unit cell here to refer to the conventional crystallographic parallelepiped, based on three noncoplanar vectors, a, b, and c; the face-to-face stacking– sharing of these parallelepipeds builds up the entire lattice. Table 1 sets out the Bravais lattices in the seven crystal systems, together with the conventional unit cells and their axial relationships. An important feature of the conventional unit cells is that each cell, itself, clearly displays the symmetry of its lattice. Thus, for example, the three cubic lattices are represented by unit cells (P, I, and F), which have the same symmetry at each lattice point, m3m (Oh). The conventional unit cells are, in fact, the parallelepipeds of smallest volume in which a, b, and c are aligned with the highest symmetry directions of the lattice. The volume V of a unit cell (a,b,c) is given by V=a·b×c
(5)
which, from standard manipulation, may be written as V = abc (1 – cos2α – cos2β – cos2γ + 2cosα cosβ cosγ)1/2 (6)
and the number of lattice points associated with the unit cells of volume V is as follows: P 1
C 2
I 2
F 4
[2/3,1/3 ,z] and [1/3,2 /3 ,z], a lattice is defined that is trigonal but has a triply-primitive unit cell Rhex, with unique lattice points at 0,0,0 and ±(2/3 ,1 /3,1/3) in the unit cell. We may note in passing that a hexagonal array of points (honeycomb) or a hexagonal prismatic array, with or without (pseudo) bodycentering, does not constitute a lattice. The triply primitive unit cell Rhex may be transformed to a primitive rhombohedral unit cell R. There are two ways of setting the rhombohedron with respect to the triply primitive hexagonal unit cell (11), obverse and reverse; that listed in Table 1, the obverse setting, is the standard. It is not desirable in all solid-state studies to use centered unit cells, so we consider next how primitive cells may be defined for each of the Bravais lattices. Translation Unit Cells First, we note that all the Bravais lattices may be represented by true unit cells that are primitive. In seven cases, they are the P (and R) unit cells of Table 1. For the others, transformations may be set up in such a way that the transformed (primed) vectors form a right-handed set and obey the condition that (a92 + b92 + c92) is a minimum. Many metals crystallize with cubic structures, in either I or F unit cells. They may be transformed to primitive cells, as follows:
R 1
Cubic I:
The unit cells C (A), I, and F are centered unit cells, and the unit cells contain unique lattice points in addition to that at the origin, 0,0,0, as follows. The A centered unit cell is equivalent to C, but is needed with certain orthorhombic space groups: P C I F R
0,0,0 0,0,0; 0,0,0; 0,0,0; 0,0,0
1/ ,1 / ,0 2 2 1/ ,1 / ,1/ 2 2 2 0,1 /2,1/2;
(A 0,0,0; 1/
1 2,0, /2 ;
0,1 /2 ,1/2)
1 / ,1/ ,0 2 2
The trigonal system calls for special mention. A twodimensional unit cell with a = b and γ = 120° is compatible with both threefold and sixfold symmetry. Hence, the hexagonal Bravais lattice (P unit cell) may be used for crystals in the trigonal system, with 3 along the line [0,0,z]. However, when threefold axes are present also along the lines
Cubic F
a9 = { a / 2 + b / 2 + c / 2 b9 = a / 2 – b / 2 + c / 2 c9 = a / 2 + b / 2 – c / 2
(7)
a9 = b / 2 + c / 2 b9 = c / 2 + a / 2 c9 = a / 2 + b / 2
(8)
In each case the primitive cell is a rhombohedron, with an angle α of cos {1 ({1/3) or 60° for I or F, respectively. If thought ¯ of in isolation, the symmetry of the rhombohedral cell is 3m. Of course, the symmetry of the lattice that the cell repre¯ is a subgroup of m3m. Figure 2 illussents is still m3m; 3m trates the transformation (shown in eq 8) for a cubic F unit cell. The cubic symmetry is implicit in the specialized value for α; it is the angle between vectors, say, r110 and r101, the directions normal to faces on a rhombic dodecahedron for the cubic F unit cell. For the cubic I unit cell, it is the angle between, say, r111 and r1¯ 11 ¯ , the tetrahedral angle. Similar standard transformations (11) of unit cells can be carried out for the other five conventional centered unit
Table 1. Crystal Systems, Bravais Lattices, and Conventional Unit Cells System
462
Symbols of conventional unit cells
Axial relationships in the conventional unit cell
Symmetry at each lattice point
Triclinic
P
a≠b≠c α ≠ β ≠ γ ≠ 90° or 120°
Monoclinic
P, C
a≠b≠c α = γ = 90°; β ≠ 90° or 120°
2 / m (C2h )
Orthorhombic
P, C, I, F
a≠b≠c α = β = γ = 90°
mmm (D2h)
Tetragonal
P, I
a=b≠c α = β = γ = 90°
4 m mm D 4h
Cubic
P, I, F
a=b=c α = β = γ = 90°
m3m (Oh )
Hexagonal
P
a=b≠c α = β = 90°; γ = 120°
6 m mm D 6h
Trigonal
R
a=b=c α = β = γ ≠ 90°, < 120°
3¯ m (D3d )
Journal of Chemical Education • Vol. 74 No. 4 April 1997
1¯ (Ci )
Research: Science & Education
Figure 2. Transformation of an all-face-centered conventional unit cell F (a, b, c) to a rhombohedral primitive (translation) unit cell R (a9, b9, c9) both in the same cubic lattice of m 3m (O h) symmetry. Each set of axial vectors is a right-handed triplet. In isolation, the rhombohedral unit cell shows symmetry 3¯ m (D3d), a subgroup of m 3m , at each point, whereas the conventional cubic unit cell displays the full lattice symmetry.
b
a
Figure 3. Oblique net (a ≠ b, γ ≠ 90° or 120°). The Voronoi polygons are obtained by drawing vectors from one point to the nearest (six) points, and then bisecting each such vector perpendicularly; the closed figure so obtained is the Voronoi polygon. For any one polygon only six points are needed in forming it; the bisectors of vectors to points further away would not lie within the particular defined polygon.
cells listed in Table 1. It is suggested that the totality of these cells may be distinguished by the term translation unit cells. The important factor is that they will generate the fourteen Bravais lattices by face-to-face stacking–sharing, although they are not all conventional crystallographic unit cells.
so that a* is parallel to the direction b9 × c9, or ({ i + j + k). In a similar manner,
Reciprocal Lattice
By analogy with 7, 11–13 represent a primitive translation cell in a lattice that may also be represented by a conventional I unit cell, which was to be proved. In a like manner, it may be shown that an F unit cell reciprocates to an I unit cell. Other centered unit cells reciprocate without a change in their mode of centering.
The reciprocal lattice is another lattice, and may be defined by three noncoplanar vectors a*, b*, and c* of a unit cell in reciprocal space. The reciprocal lattice has the same symmetry as the corresponding direct space lattice, and the reciprocal unit cell vectors are given by
b* = (1 / a) (i – j + k)
(12)
c* = (1 / a) (i + j – k)
(13)
and
Wigner-Seitz cells
a* = K b × c ; b* = K c × a ; c* = K a × b (9) a⋅b×c a⋅b×c a⋅b×c where a, b, and c are vectors of a translation unit cell in real space and the denominator is, in each case, the volume of the translation unit cell. The constant K may be given the value of unity, frequently in theoretical discussions of reciprocal space, or of λ, generally in experimental diffraction studies, or of 2π in work with Brillouin zones and related aspects of the solid state. Thus, Brillouin zones are defined in reciprocal space, usually called k-space. This space is like the crystallographic reciprocal space, but whereas crystallographic studies focus on the properties of the integer reciprocal lattice points hkl, k-space is concerned with the whole of the reciprocal space volume under consideration. We show next that an F unit cell reciprocates to an I unit cell. Let a conventional F unit cell have vectors a, b, and c, where a = b = c; then a = ai;
b = aj;
c = ak
(10)
where i, j, and k are unit vectors along a, b, and c, respectively. From eqs 8 and 9 with K = 1, we have by standard manipulation, remembering that any vector (cross) product p × q is equal to { q × p, a2/4 k + i × i + j a* = b′′× c′′ = = 1/a { i + j + k a′′⋅ b × c′′ a3/8 j + k ⋅ { i + j + k
(11)
Consider a general two-dimensional lattice, symmetry 2 at each lattice point, and, from any point as an origin, let lines be drawn to the lattice points nearest to it. These lines are then bisected perpendicularly, and the bisectors produced until they meet. The smallest area thus enclosed is a Voronoi polygon (12) (also known as a Dirichlet domain), and is a space-filling structure. Figure 3 shows the Voronoi polygons from this construction for a number of adjacent points in the two-dimensional lattice. The Voronoi construction may be applied to any system of points: for example, it can be used to define parish boundaries (13); here, however, we are concerned in using it in the context of the regular arrays of points comprising lattices. We may extend this construction to a three-dimensional lattice, whereupon the bisecting elements become planes. The extensions of the planes to intersection defines a closed figure which is a Voronoi polyhedron. Where the lattice is a Bravais lattice, the Voronoi polyhedron is also known as a Wigner–Seitz cell (14) (Fig. 4). It is evident that the planes of largest area (longest lines in two dimensions) are those that are closest to the central, generating point. The Bravais lattices give rise directly to twenty-four Wigner–Seitz cells, constructed in this manner. For example, in the tetragonal system, an I unit cell in which c > a has a Wigner–Seitz cell that is different in shape from that with c < a (Fig. 5a,b). It is for this sort of reason that these Wigner–Seitz cells are more numerous than the Bravais lattices from which they are derived. In the three instances where the conventional unit cell is primitive and rectangular (cubic P, tetragonal P, and orthorhombic P), the
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Research: Science & Education Figure 4. Wigner– Seitz cell obtained from a body-centered cubic unit cell, side a, and its environs. Its shape is a combination of the cube and octahedron forms: the y eight octahedral x faces are obtained from the vectors (length a√–3/2) from the central lattice point to those at the corners of the cube, and the six cube faces are the planes bisecting the vectors (length a) from the central lattice point to the central points of six adjacent cubes in the lattice. Thus, the octahedral faces are of larger area than the cube faces, and no other bisecting planes that could be drawn would intersect the closed polyhedron. z
Wigner–Seitz cells are of the same shape as the conventional unit cell. When applied in reciprocal space (k-space), the Wigner–Seitz cells correspond to the first Brillouin zones. The Wigner–Seitz cells are primitive, with one lattice point at the center of the cell; they are also space-filling, and generate the corresponding Bravais lattices by stacking-sharing. Except for the three special cases mentioned above, the Wigner–Seitz cells do not have a set of unique translation vectors, emanating from each corner. It is suggested that the term Wigner–Seitz cell, rather than Wigner–Seitz unit cell, be used in this context. Federov and Plato Solids
z y x
Figure 5. Wigner–Seitz cells for conventional body-centered I tetragonal unit cells. (a) Axial ratio c / a > 1; the cell shows a combination of tetragonal prism and tetragonal bipyramid forms. (b) c / a < 1; the cell shows a combination of pinacoid, tetragonal prism, and tetragonal bipyramid forms.
follows that m (2n – 4) / n < 360 / 90
Conventional unit cell
cube rhombic dodecahedron cube + octahedron hexagonal prism elongated rhombic dodecahedron
(15)
which simplifies to 1 / m + 1 / n > 1 /2
(16)
Descartes showed (17) that the discrepancy of inequality (16) is equal to 1 / e for special cases, such as the Plato solids. Since m and n must each be greater than 2, the only values of m and n that satisfy inequality 16 are the following: m,n 3,3 3,4 3,5 4,3 5,3
Federov (15) showed that there are only five polyhedra, each of which can be packed in the same orientation to fill space; they are illustrated in Figure 6. They are also Wigner–Seitz cells (derived from conventional unit cells), as follows: Wigner-Seitz cell
(b)
(a)
e 6 12 30 12 30
f 4 8 20 6 12
c 4 6 12 8 20
Solid tetrahedron octahedron icosahedron cube (a Federov solid) pentagonal dodecahedron
We see that for m ≠ n, the Plato solids are related in pairs by interchange of f and c. The solids in each pair have ¯ similar symmetries: m3m for the octahedron and cube; 53m for the pentagonal dodecahedron and the icosahedron. Further treatments of the properties of polyhedra can be found in the excellent discussions by Coxeter (17, 18) and references therein, and in Vainshtein (19).
cubic P cubic I cubic F hexagonal Rhex tetragonal I
The numbers of faces f, edges e, and corners c of a polyhedron are related by the equation c+f=e+2
(14)
as deduced by Euler (16), but sometimes called the Descartes–Euler equation. Much earlier, Plato (ca. 400 B.C.E.) had shown that there are only five regular solids possible (i.e., solids in which the faces are identical, and with equal edges); one of the Plato solids is also a Federov solid. The following argument may be used to determine the Plato solids. Consider a polygon with n sides, and let m such polygons form a corner of a polyhedron. Since the internal angle of a poly-n-gon is {90(2n – 4) / n}° and the sum of the angles formed by the faces at any corner must be less than 360°, it
464
(a)
(b)
(c)
(d)
(e)
Figure 6. The five Federov solids: (a) cube, (b) rhombic dodecahedron, (c) cube + octahedron, (d) hexagonal prism, (e) elongated rhombic dodecahedron.
Journal of Chemical Education • Vol. 74 No. 4 April 1997
Research: Science & Education Literature Cited 1. Kettle, S. F. A.; Norrby, L. J. J. Chem. Educ. 1993, 70, 959–963. 2. Kettle, S. F. A.; Norrby, L. J. J. Chem. Educ. 1994, 71, 1003–1006. 3. Frankenheim, M. L. System der Krystalle; Grass, Barth KG: Breslau, 1842. 4. Bravais, A. Note to C. R. Acad. Sci. 1848, followed by “Mémoire sur les systèms formés par des points distribués réguliérment sur un plan ou dans l’espace”. J. l’École Polytech. 1850, Cahier 33, 19, 1–128. 5. Bravais, A. On the systems formed by points regularly distributed on a plane or in space; Shaler, A. J. Trans.; Crystallographic Society of America, Memoir Number 1; The Book Concern: Hancock, MI, 1949. (Translation of ref 4.) 6. Bravais, A. “Études Crystallographiques”; J. l’École Polytech. 1850, 19, 127; 20, 102, 197.
7. Hermann, C. Z. Kristallogr. 1928, 68, 257–287; 1929, 69, 226–270, 533. 8. Hermann, C. Z. Kristallogr. 1931, 76, 559–561. 9. Mauguin, Ch. Z. Kristallogr. 1931, 76, 542–558. 10. Schönflies, A. Krystallsysteme une Krystallstruktur. Leipsig, 1891. 11. International Tables for Crystallography; Hahn, O. Ed.; Reidel: Dordrecht, 1983; Vol. A. 12. Voronoi, G. J. Reine Angew. Math. 1908, 134, 198–289. 13. Loeb, A. L. Space Structures; Addison-Wesley; Reading, PA, 1976; p 112. 14. Wigner, E; Seitz, F. Phys. Rev. 1933, 43, 804. 15. Federov, E. S. Symmetry of Crystals; American Crystallographic Association Monograph 7; ACA: , 1971. 16. Euler, L. Akad. nauk Leningrad 1752, 4, 109–160. 17. Coxeter, H. S. M. Introduction to Geometry; Wiley: New York, 1963. 18. Coxeter, H. S. M. Regular Polytopes; Dover: New York, 1973. 19. Vainshtein, B. K. Modern Crystallography I; Springer: Berlin, 1981.
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