Fundamentals of the Physics of Solids / 07-The Structure of Crystals

7.2 Cubic Crystal Structures

213

two atoms. As listed in the table of Appendix B, a relatively large number of elements crystallize in this structure at room temperature, e.g., monovalent alkali metals (Li, Na, K, Rb, Cs), two heavy alkaline-earth metals (Ba, Ra), a transition metal (Cr), two forms (α and δ) of iron (Fe), and also Mo, Nb, Ta, V, and W. The latter one can be considered as the prototype, hence its traditional name: W structure (in the Strukturbericht notation: A2 structure).

This structure occurs more frequently than the simple cubic structure among the elements. This is because space filling is more e cient (but still not ideal) with an extra atom at the center of the cube. Each atom is surrounded by eight others in a cubic arrangement, the coordination number is thus 8. When the lattice is filled with one kind of spherical atom of radius r in such a way that atoms at the vertices touch those at the body centers, the space

	diagonal of the Bravais lattice is 4r – that is, the edge length of the cubic
	√											√		3	. Since each Bravais
	Bravais cell is a = 4r/		3, and so its volume is (4r/										3)
	cell contains two atoms, the packing fraction is
	4				4r		−3		√
	2			r3π √				=	3		π = 0.680 .				(7.2.8)
		3							8
						3

The W structure, which has a monatomic basis, possesses all rotation and

¯ 9

reflection symmetries of a cube; its space group is Im3m (Oh). When the basis consists of several atoms, there exist a number of structures for which the same space group is preserved. In La2O3 lanthanum atoms are located at vertices and body centers, and oxygen atoms at face and edge centers, however, on the average only half of the possible oxygen positions are occupied. The Pearson symbol for this arrangement is thus cI5. These positions or sites – face and edge centers – are of particular importance, as they are highly symmetric, surrounded in an octahedral arrangement by the atoms at the vertices and body centers of the Bravais lattice, as illustrated in Fig. 7.8(a). Besides these octahedral sites the arrangement has other high-symmetry positions: ( 14 12 0) and all the sites obtained from this through the symmetry transformations of the cube. These are called tetrahedral sites because of the tetrahedral arrangement of the nearest neighbors (Fig. 7.8(b)). Note that the A15 structure shown in Fig. 7.5(a) can also be regarded as an arrangement in which not only the vertices and centers of the Bravais lattice of a body-centered crystal are occupied but also half of the tetrahedral sites – in such a manner that the space group is P m3n.

There exist more complicated structures in which the symmetry of the space group Im3m is preserved, e.g., D81, D8f, and L22 structures, whose prototypes are Fe3Zn10, Ir3Ge7, and Sb2Tl7. In the first two cases the primitive cell contains two molecules and the Bravais cell four, while in the third case there are three molecules per primitive cell and six per Bravais cell. Their Pearson symbols are therefore cI52, cI40, and cI54.

In simpler structures some of the symmetries are broken because of the lower symmetry of the basis. Of particular interest is the MoAl12 structure,

214 7 The Structure of Crystals

a 3 2 a 2 a 2

(a)

a	3 2	5 4
	a
(b)

Fig. 7.8. Highly symmetric empty sites in a bcc lattice: (a) octahedral sites ( ); (b) tetrahedral sites ( )

illustrated in Fig. 7.9. Molybdenum atoms are located at the vertices and centers of the cubic Bravais cells. Each of them are surrounded by twelve aluminum atoms in an icosahedral arrangement. The space group is therefore

¯ 5

reduced to Im3 (Th ). Each Bravais cell contains two units, hence the Pearson symbol is cI26. The compounds MnAl12 and Al12W both crystallize in this structure; the latter is considered as the prototype.

Fig. 7.9. MoAl12 structure. For the sake of clarity, the twelve icosahedrally arranged Al atoms are drawn only around two sites, a vertex and a body center

7.2.3 Face-Centered Cubic Structures

Face-centered cubic (fcc) crystals are obtained by decorating an fcc Bravais lattice with identical groups of atoms. Disordered alloys (e.g., Cu3Au), in

7.2 Cubic Crystal Structures

215

which the two components are found at each lattice point with equal probability also belong here. Upon heating, the simple cubic structure of Cu3Au

(space group: P m3m) is transformed into a face-centered one at a critical temperature.

Primitive vectors point from the vertices of the Bravais cell into the face centers:

a1 =

a

(yˆ + zˆ) ,

a2 =

a

(xˆ + zˆ) ,

a3 =

a

(xˆ + yˆ) .

(7.2.9)

2

2

2

The face-centered cubic lattice can therefore be viewed as four interpenetrating simple cubic lattices, whose origins are displaced by the above vectors. The lattice is thus decomposed into four equivalent sublattices. The primitive cell spanned by the primitive vectors is rhombohedral, as shown in Fig. 7.10(a), so it does not possess the symmetries of a cube. This is why preference is often given to the conventional unit cell, which is four times bigger but more symmetric: the cubic Bravais cell with edge vectors

a = −a1 + a2 + a3 = axˆ ,
b = a1 − a2 + a3 = ayˆ ,	(7.2.10)
c = a1 + a2 − a3 = azˆ .

In what follows, we shall specify atomic positions in this Bravais cell.

a1

a2

(a)

a3

(b)

Fig. 7.10. Face-centered cubic crystals: (a) the Bravais cell with the primitive vectors; (b) the Wigner–Seitz cell

The symmetric Wigner–Seitz cell, obtained via Dirichlet’s construction, is shown in Fig. 7.10(b). For clarity, it is drawn into a cube that is displaced by half the space diagonal with respect to the Bravais cell – that is, lattice points are at the body and edge centers of the cube. Since each lattice point has 12 nearest neighbors, when the cube is cut by the perpendicular planes through the midpoints, a rhombic dodecahedron is obtained for the Wigner–Seitz cell. In the direction of the 12 adjacent lattice sites it is bordered by congruent rhombi. The Wigner–Seitz cell shows every symmetry of the cube. As we have seen, the Brillouin zone of a body-centered cubic crystal has the same shape.

216 7 The Structure of Crystals

The reciprocal lattice of an fcc lattice is determined in the usual way, by taking the inverse of matrix A constructed from the primitive vectors of the direct lattice:

Afcc = 2				1 0 1		,	Bfcc =	2a	−1	1	1	.	(7.2.11)
	a			0 1 1				π	1	1	1
										−1 −1
			1 1 0						1

The reciprocal-lattice vectors are then

b1 =	2π	(−1, 1, 1) =	2π	(−xˆ + yˆ + zˆ) ,
	a		a
b2 =	2π	(1, −1, 1) =	2π	(xˆ − yˆ + zˆ) ,	(7.2.12)
	a		a
b3 =	2π	(1, 1, −1) =	2π	(xˆ + yˆ − zˆ) .
	a		a

Comparison with (7.2.4) shows that the reciprocal of a face-centered cubic lattice is a body-centered cubic lattice with a lattice constant of 4π/a. Faceand body-centered cubic lattices are thus reciprocal to one another in the sense that the reciprocal lattice of an fcc lattice is a bcc lattice, and vice versa. This implies that the Brillouin zone of an fcc lattice is of the same shape as the Wigner–Seitz cell of a bcc lattice: a truncated octahedron. This is shown in Fig. 7.11, along with the special points of the Brillouin zone.

kz
4 a	b1
4 a

b2

ky

4 a

kx

kx

b3

kz
	L	U
			'
		Q
			ky
			X
	K	W

Fig. 7.11. The reciprocal lattice of the face-centered cubic lattice and its Brillouin zone with special points

The simplest crystal structure in a face-centered (F) cubic (c) lattice with a monatomic basis contains four atoms per Bravais cell, since only 1/8 of each atom at the vertices and 1/2 of each atom at the face centers belong to the cell. This arrangement is thus denoted by cF4. Each atom is surrounded by twelve identical ones as nearest neighbors, so the coordination number is 12. For spherical atoms with equal radii this is the highest possible number of

7.2 Cubic Crystal Structures

217

touching neighbors. When the radius of the spheres is r, it is related to the

√

edge length of the cube by 4r = 2a. Since the Bravais cell of volume

√ r	3	= √ r3	(7.2.13)
4		32

22

contains four atoms, the packing fraction for this crystal structure is

		4			32				−1	√
	4		r3π		√		r3		=	2		π = 0.740 .	(7.2.14)
		3								6
						2

This is higher than the values obtained for simple and body-centered cubic crystals. Indeed, spherical atoms fill the space optimally in this arrangement. This is why the face-centered cubic structure is also known as the cubic closepacked (ccp) structure.

Close packing is best illustrated by the atomic arrangement in the (111) atomic planes perpendicular to the space diagonal. As it can be seen in Fig. 7.12, within these planes atoms form a hexagonal lattice, which is the most e cient way of covering the plane with circles. In the next section we shall return to the question of how to stack such atomic layers to obtain an fcc or some other structure.

Fig. 7.12. Close packing of atoms in an fcc crystal, in the planes perpendicular to the space diagonal

As close packing is favored by the metallic bond, many metals crystallize in this structure. The prototype, after which this monatomic structure has been named, is copper. The space group of the Cu structure (also called A1

¯ 5

structure) is F m3m (Oh). Besides copper, several other elements crystallize

218 7 The Structure of Crystals

in this form: other noble metals, Ag and Au; two alkaline-earth metals, Ca, Sr; trivalent Al; several transition metals such as γ-Fe, α-Co, Ni, Ir, Pt, Rh, Pd, and Pb; some lanthanoids and actinoids, e.g. Ce, Pr, Th, Yb; and also a couple of noble gases in their low-temperature solid phase, Ar, Ne, Kr, and Xe.

In addition to the vertices and face centers of the cube, atoms have to be placed at other sites as well to obtain face-centered cubic structures with a polyatomic basis. These additional atoms tend to fill up the empty spaces among the atoms at the vertices and face centers. In face-centered cubic structures there are two typical sites that can be occupied.

One of them is the center of the cube, site 12 12 12 , and the equivalent positions at the midpoints of the edges, which can be reached by translations of the cube center through primitive vectors. These sites are surrounded by six lattice points in an octahedral geometry, therefore they are also called octahedral sites. Rotations and reflections that leave such a point invariant and transform the crystal into itself are just the elements of the point group Oh.

(a)

(b)

Fig. 7.13. Highly symmetric empty sites in an fcc lattice: (a) octahedral sites ( ); (b) tetrahedral sites ( )

Other typical sites are the centers of the octants of the cube: 1 1 1 and

4 4 4

equivalent points. When an atom is placed there, its four nearest neighbors in the fcc lattice form a regular tetrahedron. For this reason they are called tetrahedral sites. The local symmetry group at these sites is Td. The two types of site are shown in Fig. 7.13. At each site of the lattice the potential due to all other atoms (called the crystal-field potential) shows the symmetries of the point group of the site in question. As discussed in the previous chapter, this potential may give rise to crystal-field splitting. Since the main features of the splitting (apart from its magnitude) depend on the local symmetry, knowing it is important for understanding atomic energy spectra. Conversely, the type of splitting can be used to determine the local symmetry – and hence the position of the atoms.

There are eight tetrahedral and four octahedral sites in a Bravais cell. In the simplest fcc crystals with multiatomic bases, besides vertices and face centers these sites are occupied by atoms – either completely or partially. De-

7.2 Cubic Crystal Structures

219

pending on the occupancy of each site, various structures are possible. Because of their importance and the partial symmetry breaking occurring in them, diamond structure and the closely related sphalerite structure merit separate discussion. Below we shall present some other types in which symmetries of

¯	5
the space group F m3m (Oh) are fully preserved.

A common structure is the sodium chloride or rock-salt structure (in traditional notation: B1 structure). Sodium cations (Na+) are located at vertices

and face centers – at 000, 0 21				21 ,	21 0 21 , and 21	21 0, if the edge of the fcc Bravais
cell is chosen as unity –, and chlorine anions (Cl−) at the octahedral sites 21 00,
0 1 0, 00	1 , and	1 1 1 . This arrangement is illustrated in Fig. 7.14(a).
2	2	2 2 2
		Na	Cl			Ca

F

(a) (b)

Fig. 7.14. Face-centered cubic structures: (a) sodium chloride (NaCl, B1) structure; (b) fluorite (C1) structure

Sodium and chlorine ions are at alternate points of a simple cubic lattice, forming a three-dimensional checkerboard pattern. Each Na+ ion is surrounded by six Cl− ions and vice versa, the coordination number is thus 6. Cations and anions form two interpenetrating fcc sublattices that are displaced through half the space diagonal with respect to one another. From the viewpoint of symmetries, the overall structure is face-centered cubic, too. Since each Bravais cell contains four sodium chloride molecules, the Pearson symbol is cF8. The four times smaller primitive cell contains only two ions, Na+ at 000 and Cl− at 12 12 12 . These two ions make up the basis of the lattice. The halides of all alkali metals but Cs crystallize in the rock-salt structure

– and so do divalent salts, e.g., MgO, CaO, MgS, CaSe, and BaTe. We shall give the simple geometrical reason for this at the end of the chapter.

Fluorite structure or C1 structure – the prototype of which is fluorite (CaF2) – is obtained by placing atoms of the second type at the tetrahedral (rather than octahedral) sites. This structure is shown in Fig. 7.14(b). Calcium cations (Ca2+) are located at vertices and face centers, and fluorine anions (F−) at the centers of the octants. Translation of the basis – made up of a

Ca2+ ion at 000 and two F− ions at	1 1 1	and	3 3 3	– through the primitive
	4 4 4		4 4 4

220 7 The Structure of Crystals

vectors (7.2.9) generates the full crystal. The Bravais cell contains four CaF2 molecules, so the Pearson symbol is cF12. Each Ca2+ ion is surrounded by eight F− ions in a cubic arrangement, their coordination number is thus 8; F− ions, on the other hand, are surrounded by only four Ca2+ ions in a tetrahedral arrangement, so their coordination number is just 4. The oxides and sulfides of alkali metals (e.g., Li2O, Na2O, K2O, Na2S, K2S), as well as many other oxides and halides crystallize in this structure.

When both octahedral and tetrahedral sites are occupied by atoms of the same (second) type, a BiF3 or D03 structure – also known as AlFe3 structure – arises. This is shown in Fig. 7.15(a). Bismuth cations (Bi3+) are located at the vertices and face centers of the cube, and fluorine anions (F−) among them, at octahedral and tetrahedral sites. The Bravais cell contains four molecules, and so its Pearson symbol is cF16.

The octahedral and tetrahedral sites are all occupied – although by two di erent types of ion – in the L21 structure. The prototype for this structure is the Heusler alloy AlMnCu2, which is particularly noted for its magnetic properties.

Bi

F

U

(a)

(b)

B

Fig. 7.15. Face-centered cubic structures: (a) BiF3 structure (D03 structure), with 16 atoms per Bravais cell; (b) UB12 structure (D2f structure) with 52 atoms per Bravais cell. Only two of the four cuboctahedra are shown; each contains twelve boron atoms

Even more complicated is the D2f structure shown in Fig. 7.15(b), whose prototype is UB12. Among the uranium atoms at the vertices and face centers, octahedral sites are occupied not by single ions but groups of twelve boron atoms forming a cuboctahedron.4 Referred to the octahedral site (e.g., the center of the Bravais cell), the coordinates of the 12 boron atoms around it are 0(±61 )(±16 ); (±16 )0(±16 ), and (±16 )(±16 )0. With this choice the distance

4Also called a heptaparallelohedron or triangular gyrobicupola; a cube truncated at its vertices to its edge centers.

7.2 Cubic Crystal Structures

221

between atoms that belong to adjacent cuboctahedra is the same as the interatomic distance within a cuboctahedron. The Bravais cell contains four molecules, hence the Pearson symbol is cF52.

One may find face-centered cubic structures in which the basis contains a much higher number of atoms. The Bravais lattice of the NaZn13 or D23 structure contains 112 atoms. An even higher number of atoms is found in the basis of crystalline fullerite, formed by C60 molecules (shown in Fig. 2.8). In this allotropic form of carbon a C60 molecule is sitting at each point of an fcc lattice. With carbon atoms arranged in regular pentagonal and deformed hexagonal rings at the vertices of a truncated icosahedron, the molecule it-

¯¯ 5

self shows the symmetries of the icosahedral point group Ih (m35). If the orientation of the fivefold rotation axes of the C60 molecules is ordered, then

– because of the incompatibility of cubic and icosahedral symmetries – the entire crystal cannot show each symmetry of the cube. The space group of

¯ 6

fullerite is P a3 (Th ) in this low-temperature ordered phase. At higher temperatures, in the plastic-crystalline phase, fullerene molecules are free to rotate,

and thus full cubic symmetry is restored. The space group of the crystal is

¯ 5 then F m3m (Oh).

7.2.4 Diamond and Sphalerite Structures

Among fcc structures with a diatomic basis particularly important are those in which all vertices and face centers but only half of the tetrahedral sites are occupied. Depending on whether the atoms at the tetrahedral sites are identical with those at the vertices and face centers, the arrangement is either a diamond structure or a sphalerite (zincblende) structure. Their Strukturbericht designations are A4 and B3. In either case, the basis consists of two atoms, at 000 and 14 14 14 . The Bravais cell contains eight atoms, thus the Pearson symbol is cF8 for both of them. The two structures are shown in Fig. 7.16.

(a)

(b)

Fig. 7.16. Fcc crystal structures with a two-point basis: (a) diamond (A4) structure; (b) sphalerite (B3) structure

5Pentagons are bounded by hexagons, while hexagons are bounded by pentagons and hexagons alternately, corresponding to single and double bonds.

222 7 The Structure of Crystals

The figure clearly shows that in the diamond structure each atom is surrounded by four neighbors in a tetrahedral arrangement. The coordination number is therefore 4. This implies that space filling is much less e cient than in previously discussed structures. When this structure is built up of

touching spheres of radius r, the relation between the edge of the Bravais cell

√

and the radius is a 3/4 = 2r. Since each Bravais cell contains eight atoms, the packing fraction is

	4			8r			−3		√
8		r3π						=		3		π = 0.340 .	(7.2.15)
				√
	3								16
					3

This structure is realized in a particular form of carbon, diamond, as well as silicon, germanium, and gray tin (α-Sn). These elements are all in group 14 (IVA) of the periodic table where – as discussed in Chapter 4 – tetrahedral coordination is due to the covalent bonds created by the sp3 hybrid orbitals.

This structure can also be considered to consist of two interpenetrating face-centered cubic Bravais lattices, displaced by 14 a(xˆ + yˆ+ zˆ) relative to one another. The obtained diamond lattice is not a Bravais lattice. It should be emphasized: even though the crystal is made up of a single kind of atom, it is impossible to choose a primitive cell with a monatomic basis that can serve to generate the entire crystal.

Concerning translational symmetries, the diamond lattice has the same primitive vectors as a face-centered cubic lattice, thus its reciprocal lattice is the same as that of an fcc lattice. The Brillouin zones are also identical: truncated octahedra.

On the other hand, rotation and reflection symmetries of the cube are not fully preserved. Inversion symmetry is lost because of the atoms in the octants. Similarly, reflections σx, σy , σz , fourfold rotations C4x, C4y , C4z and twofold rotations around the face diagonals are no longer symmetries, either. In each case, however, the crystal can be brought into coincidence with itself by an additional translation along the space diagonal through one quarter of its

length (that is, by the vector 14 14 14 ).6 If instead of the origin we were to choose

the point 1 1 1 as the fixed point of the point-group operations, inversion would

8 8 8

remain a symmetry but invariance under other rotations would be broken – unless followed by a translation through one quarter of the space diagonal. Thus there are screw axes and glide planes among the symmetry elements of

¯

7

the diamond structure. The space group is F d3m (Oh).

When the two sublattices are made up of two di erent kinds of atom, as shown in Fig. 7.16(b), a sphalerite (B3) structure is obtained. In sphalerite – a polymorph of ZnS – the basis contains two ions, Zn2+ at 000 and S2− at

6This translation is neither along the screw axis nor in the glide plane. However, the same name may be used for convenience, as the contradiction with the definition of these symmetry operations is only apparent. See footnote on page 159. E.g., it

is readily seen that the diamond lattice has a “pure” fourfold screw axis parallel to the z-axis if it is chosen to go through the point 14 12 0.