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

7.6 Relationship Between Structure and Bonding

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overlap is usually much weaker between layers than within them. This gives rise to highly anisotropic physical properties. For example, in crystals built up of HMTTF and TCNQ (or similar pairs of) molecules electric conductivity is much larger in one direction than in the other two. La2−xSrxCuO4 and other crystals with CuO2 planes are much better conductors in the plane than in the perpendicular direction. This property may in itself be crucial for some applications. It is also of great importance at the phenomenological level: as we shall see, fundamentally new phenomena may be observed in materials in which the system of electrons is practically oneor two-dimensional. This explains why the study of chain-like and layered structures has become one of the hottest research topics in solid-state physics.

7.6 Relationship Between Structure and Bonding

We saw in Section 4.4 that unlike other types of bonds, covalent bonds are highly directional. This has a strong influence on the structures occurring for each particular type of chemical bonding.

7.6.1 The Structure of Covalently Bonded Solids

As mentioned in Subsection 4.4.7 and shown in Fig. 4.11, sp3 hybrid orbitals can create bonds in the directions of the four vertices of a tetrahedron. This is the underlying reason for the tetrahedral coordination of atoms in the diamond structure. Tetrahedral arrangement of nearest neighbors is often observed in two-component covalent compounds, too. The commonest are the sphalerite and wurtzite structures. In such tetrahedrally bonded covalent crystals a covalent radius can be determined for each atom from the requirement that the sum of covalent radii should give the bond length (also called bond distance) – i.e., the distance between the atoms in the covalent bond. The covalent radius derived from the bond length may also be introduced for covalently bonded materials with other structures, however, the lengths are di erent for atoms linked by single and double covalent bonds. Covalent radii – with the above uncertainty – are listed in Table 7.3 for some elements.

If besides the s-electron only two p-electrons (px and py) participate in bonding, then the following sp2 hybrid wavefunctions arise:

1

ψ1 = √ 3

1

ψ2 = √ 3

1

ψ3 = √ 3

φ2s +		/		3				φ2px ,
		2
		1								1			(7.6.1)
φ2s − √						φ2px + √						φ2py ,
			6								2
	−	1							−	1
		6								2
φ2s		√				φ2px				√		φ2py .

As a consequence of the form of s and p wavefunctions shown in Fig. 4.10, these hybrid states give high electron densities in three directions of the (x, y)

234 7 The Structure of Crystals

Table 7.3. Covalent radii in Å for elements that participate in covalent bonds

	B	C	N	O
	0.81	0.77	0.70	0.66
	Al	Si	P	S
	1.25	1.17	1.10	1.04
Zn	Ga	Ge	As	Se
1.25	1.25	1.22	1.21	1.17
Cd	In	Sn	Sb	Te
1.41	1.50	1.40	1.41	1.37
Hg	Tl	Pb	Bi	Po
1.44	1.55	1.54	1.46	1.46

plane, at 120◦ degrees to each other. This is illustrated in Fig. 7.25(a). These hybrid bonding orbitals give rise to the two-dimensional honeycomb-like network shown in Fig. 7.25(b).

+
-
-	+
-
+
(a)	(b)

Fig. 7.25. (a) Spatial distribution of electrons in states characterized by sp2 hybrid wavefunctions. (b) Two-dimensional network arising from these bonds

This is the case for an allotrope of carbon, graphite, where only two 2pelectrons hybridize with one 2s-electron. In addition to the three electrons that participate in saturated covalent bonds, a fourth electron makes a nonlocalized and unsaturated π bond with the three neighbors. This electron is responsible for the finite conductivity of graphite. The spatial structure of graphite – the prototype of A9 structure – is shown in Fig. 7.23(a): covalently bonded hexagonal planes are held together by weak van der Waals forces. The same planar network is found in trivalent As.

Besides s- and p-states, one or more 3d-states may also participate in the formation of hybrid bonding orbitals. Considering the form of d-states, illustrated in Fig. 6.1, it can be demonstrated that in a dsp2 hybrid the wavefunctions

states are combined with φdz2

7.6 Relationship Between Structure and Bonding

235

		1
	ψ1,2 =		φs ± √2φpx			+ φdx2−y2
		2
		1
	ψ3,4 =		φs ± √2φpy			− φdx2−y2
		2

,

(7.6.2)

can create bonds along the ±x and ±y directions of the (x, y) plane, in a tetragonal geometry.

In the d2sp3 hybrid φs, φpx , φpy , and φpz and φdx2 −y2 to give

1		1				1
ψ1,2 = √		φs ± √		φpz	+	√					φdz2 ,
	6		2							3
1		1				1						1
ψ3,4 = √		φs ± √		φpx	+				φdx2−y2			−	√				φdz2 ,	(7.6.3)
								2
	6		2												12
1		1				1						1
ψ5,6 = √		φs ± √		φpy	−				φdx2−y2			−	√				φdz2 .
							2
	6		2											12

In states associated with these functions electrons create bonds of octahedral configuration.

In covalently bonded solids each atom (molecule) is usually surrounded by the same number of nearest neighbors as the covalent bonds it can form. This is why the hexagonal and tetrahedral structures illustrated in Figs. 7.25 and 7.16 occur frequently. When an atom can form only two covalent bonds, a chain-like structure arises. An example for this, γ-selenium is shown in Fig. 7.24(a). The two covalent bonds do not lie along a straight line, which is why atoms are located along a spiral.

The strong directionality of bonds is preserved even when the covalently bonded elements make up a disordered, amorphous structure instead of a regular crystal. The bond lengths and angles are close to their values in a crystalline material, therefore these amorphous systems exhibit short-range order on atomic scales. On larger scales the order may disappear, as shown in Fig. 10.1.

7.6.2 Structures with Nondirectional Bonds

Unlike covalently bonded solids, the constituents of molecular crystals, ionic crystals, and metals with nondirectional van der Waals, ionic, or metallic bonds tend to arrange themselves as closely packed as possible. In the foregoing we have seen that the most e ective filling of space is o ered by facecentered cubic and hexagonal close-packed lattices; this is why most metals crystallize in one of these structures. Similar close-packed arrangements are found in molecular crystals, too.

The situation is similar for ionic crystals, although the di erent size of cations and anions plays an important role there. Although electron density decreases continuously with the distance from the ion core, considering the ions as rigid spheres of finite ionic radius ri seems to give a fair approximation.

236 7 The Structure of Crystals

Because of the tendency to reach the energy minimum, the spheres will try to fill space in the most closely packed arrangement. One would expect that the nearest-neighbor distance is the sum of the ionic radii of an anion and a cation. As we shall see, for geometrical reasons this cannot be the case for arbitrary ionic radii.

Ionic radii cannot be determined unambiguously. Relying on the assumption that atoms are close-packed, several attempts have been made to estimate ionic radii from the lattice constant of the ionic crystal and the separation d between neighboring cations and anions using

d = rc + ra ,

(7.6.4)

where rc is the radius of the cation and ra is the radius of the anion. V. M. Goldschmidt (1926) chose the radii of O2− and F− ions as reference, assuming the values: ri(O2−) = 1.32 Å and ri(F−) = 1.33 Å. One year later (1927) L. C. Pauling worked out a system in which the values ri(O2−) = 1.40 Å and ri(F−) = 1.36 Å were chosen. To illustrate the variations of ionic radii in the periods and groups of the periodic table, the Pauling ionic radii of some singly and doubly charged ions are listed in Table 7.4.

Table 7.4. Pauling ionic radii (in Å) of some singly and doubly charged ions

Li+	Be2+			O2−	F−
0.60	0.31			1.40	1.36
Na+	Mg2+			S2−	Cl−
0.95	0.65			1.84	1.81
K+	Ca2+	Cu+	Zn2+	Se2−	Br−
1.33	0.99	0.96	0.74	1.98	1.95
Rb+	Sr2+	Ag+	Cd2+	Te2−	I−
1.48	1.13	1.26	0.97	2.21	2.16
Cs+	Ba2+	Au+	Hg2+
1.69	1.35	1.37	1.10

In reality, ionic radii may depend on the number of nearest neighbors (the coordination number). Data listed in Table 7.5 clearly show that when the radius of the same ion is determined from the lattice constants of di erent types of crystals, highly disparate values are obtained.

According to Table 4.4, cesium chloride has the largest Madelung constant among simple structures with a diatomic basis. Therefore when ions are considered as classical objects with a spherical charge distribution whose energy can be determined in terms of the energies of point charges, this structure should be more stable than sodium chloride, sphalerite, or wurtzite structures. Nevertheless the latter occur naturally quite frequently. To understand this, finite ionic radii are taken into account through the assumption that in

7.6 Relationship Between Structure and Bonding

237

Table 7.5. Ionic radius (in Å) in environments with di erent coordination numbers, estimated from lattice constants

	Coordination	Al3+	Mn2+	Fe2+	Fe3+	Cu+	Zn2+	O2−
	number
	4	0.39	0.66	0.63	0.49	0.60	0.60
	6	0.54	0.83	0.61	0.55	0.77	0.74	1.40
	8		0.96	0.92	0.78		0.90	1.42

the energetically most favorable situation cations and anions are as close together as possible, i.e., in contact with each other inside the crystal. This is illustrated in Fig. 7.26, where the arrangement of touching ions is shown in the (110) plane of a CsCl structure for various values of the ionic radius ratio.

		√			√			√
ra	<	3 + 1	ra	=	3 + 1	ra	>	3 + 1
r		2	r		2	r		2
c			c			c

Fig. 7.26. Touching rigid spherical ions in the (110) plane of the cesium chloride structure, for various values of the anion–cation radius ratio

Taking the radius of the anions fixed and gradually reducing the radius of cations, anions and√cations will be in contact along the space diagonal provided 2(ra + rc) = 3a and 2ra ≤ a both hold, that is, as long as

		√		+ 1
ra	≤		3		= 1.366 .	(7.6.5)
rc		2

Using the Pauling ionic radii, the ratio ra/rc for CsCl is 1.07, the above condition is thus met.

However, for Na+ and Cl− ions the same ratio is found to be 1.91. In a CsCl structure these anions and cations could not touch one another. In such cases, NaCl structure is energetically more favorable: although its Madelung constant is slightly smaller, it allows the two ions to be in contact. As illustrated in Fig. 7.27, cations and anions are in direct contact – and so sodium

238	7 The Structure of Crystals
chloride structure is stable – as long as 2(ra + rc) = a and 4ra ≤							√		a, that is
								2
		ra		√
			≤	2 + 1 = 2.415 .			(7.6.6)
		rc

When the radius ratio exceeds this value, tetrahedrally coordinated sphaleritetype structure may occur. Similar considerations lead to the stability condition

ra		√
	≤ 2 +	6 = 4.444			(7.6.7)
rc

in this case.

a

rc		rc		a	rc
ra		ra			ra
ra	√	ra	√		ra	√
rc	< 2 + 1	rc	= 2 + 1		rc	> 2 + 1

Fig. 7.27. Touching rigid spherical ions in the (110) plane of the sodium chloride structure, for various values of the anion–cation radius ratio

In the stable structure cations are visibly surrounded by eight anions when 1 < ra/rc < 1.366, by six anions when 1.366 < ra/rc < 2.415, and by just four anions when 2.415 < ra/rc < 4.444. When the ratio of the ionic radii exceeds this value a trigonally coordinated planar√configuration becomes stable. For even higher values of this ratio, ra/rc > 2 3 + 3 = 6.464 linear coordination becomes stable.

Compared to the 1920s, we now have a much more profound understanding of the spatial distribution of electrons in ionic crystals. High-resolution di raction measurements permit us to determine the density distribution of electrons. This is shown in Fig. 7.28 for rock salt.

The electron density reconstructed from measurements varies continuously, and it is not really spherically symmetric either – nevertheless it is reasonable to choose the minimum along the axis joining the two ions as the border between them. The “physical” ionic radius riph determined this way is listed in Table 7.6 for various ions at sites with coordination number six. “Physical” and Pauling ionic radii are seen to di er by as much as 0.1–0.2 Å, and should therefore be considered as just indications of the size of ions.

7.6 Relationship Between Structure and Bonding

239

Fig. 7.28. Electron density in the (100) plane passing through atomic centers in a NaCl crystal, determined via the inverse Fourier transform of di raction patterns [R. Brill, Solid State Physics 20, 1 (1967)]

Table 7.6. “Physical” ionic radii (in Å) of some singly and doubly charged ions

Li+	Be2+			O2−	F−
0.94	0.59			1.26	1.16
Na+	Mg2+			S2−	Cl−
1.17	0.86			1.70	1.64
K+	Ca2+	Cu+	Zn2+	Se2−	Br−
1.49	1.14	0.91	0.88	1.84	1.80
Rb+	Sr2+	Ag+	Cd2+	Te2−	I−
1.63	1.32	1.29	1.09	2.07	2.05
Cs+	Ba2+	Au+	Hg2+
1.86	1.49	1.51	1.16

Further Reading

1.J. L. C. Daams, P. Villars, and J. H. N. van Vucht, Atlas of Crystal Structure Types for Intermetallic Phases, ASM International, Materials Park, OH (1991).

2.Materials Science and Technology, A Comprehensive Treatment, Edited by R. W. Cahn, P. Haasen, E. J. Kramer, Vol. 1. Structure of Solids, Volume Editor: V. Gerold, VCH Publishers Inc., New York (1993).

3.W. B. Pearson, The Crystal Chemistry and Physics of Metals and Alloys, Wiley-Interscience, New York (1972).

240 7 The Structure of Crystals

4.B. K. Vainshtein, V. M. Fridkin, and V. L. Indenbom, Modern Crystallography, Volume 2; Structure of Crystals, Third edition, Springer-Verlag, Berlin (2000).

5.P. Villars and L. D. Calvert, Pearson’s Handbook of Crystallographic Data for Intermetallic Phases, Second edition, ASM International, Materials Park, Ohio (1991).

6.R. W. G. Wycko , Crystal Structures, Second Edition, John Wiley & Sons Inc., New York (1963).