Patterson, Bailey - Solid State Physics Introduction to theory

4 1 Crystal Binding and Structure

	d1		d2
		R
–e	+e	+e	–e

Fig. 1.2. Simple model for the van der Waals forces

The total energy of the vibrating dipoles may be written

E =	1		(P		2 + P2 )				+	1 k(d 2				+ d 2 ) +							e2
	2M			1			2			2	1				2			4πε0 (R + d1 + d2 )				(1.1)
									e
+		e2				−				2			−				e2				,
		4πε	0	R			4πε	0	(R + d )						4πε	0	(R + d		2	)
											1

where ε0 is the permittivity of free space. In (1.1) and throughout this book for the most part, mks units are used (see Appendix A). Assuming that R >> d and using

	1	1−η +η2 ,
	1+η

if |η| << 1, we find a simplified form for (1.1):

	1	(P2	+ P2 ) +	1	k(d 2	+ d 2 ) +	2e2d d
E							1 2	.
	2M	1	2	2	1	2	4πε0R3

(1.2)

(1.3)

If there were no coupling term, (1.3) would just be the energy of two independent oscillators each with frequency (in radians per second)

ω0 = k M .

(1.4)

The coupling splits this single frequency into two frequencies that are slightly displaced (or alternatively, the coupling acts as a perturbation that removes a twofold degeneracy).

By defining new coordinates (making a normal coordinate transformation) it is easily possible to find these two frequencies. We define

Y+	= 1	(d1 + d 2 ),	Y− = 1 (d1 −d 2 ),
	2			2	(1.5)
P = 1				1
		(P + P ),	P =		(P − P ).
+		1 2	−	2	1 2
	2

By use of this transformation, the energy of the two oscillators can be written

1

2

e

2

2

1

2

e

2

2

E

P

k

+

k

+

+

Y

P

+

−

Y

. (1.6)

2M

2

4πε

2M

2

4πε

+

0

R3

+

−

0

R3

−

1.1 Classification of Solids by Binding Forces (B)	5

Note that (1.6) is just the energy of two uncoupled harmonic oscillators with frequencies ω+ and ω− given by

ω± =	1		e2	(1.7)
		k ±	.
	M		2πε0R3

The lowest possible quantum-mechanical energy of this system is the zero-point energy given by

	E		(ω+ +ω−) ,	(1.8)
		2

where is Planck’s constant divided by 2π.

A more instructive form for the ground-state energy is obtained by making an assumption that brings a little more physics into the model. The elastic restoring force should be of the same order of magnitude as the Coulomb forces so that

4πεe2R2 kdi .

0

This expression can be cast into the form

	e2		R	k .
	4πε0R3		di
It has already been assumed that			R >> di so that the above implies

e2/4πε0R3 << k. Combining this last inequality with (1.7), making an obvious expansion of the square root, and combining the result with (1.8), one readily finds for the approximate ground-state energy

E ω0 (1−C / R6 ) ,						(1.9)
where
C =		e4			.
	32π	2	k	2 2
				ε0

From (1.9), the additional energy due to coupling is approximately −Cћω0/R6. The negative sign tells us that the two dipoles attract each other. The R−6 tells us that the attractive force (proportional to the gradient of energy) is an inverse seventh power force. This is a short-range force. Note that without the quantummechanical zero-point energy (which one can think of as arising from the uncertainty principle) there would be no binding (at least in this simple model).

6 1 Crystal Binding and Structure

While this model gives one a useful picture of the van der Waals forces, it is only qualitative because for real solids:

1.More than one dimension must be considered,

2.The binding of electrons is not a harmonic oscillator binding, and

3.The approximation R >> d (or its analog) is not well satisfied.

4.In addition, due to overlap of the core wave functions and the Pauli principle there is a repulsive force (often modeled with an R–12 potential). The totality of R–12 linearly combined with the –R–6 attraction is called a Lennard–Jones potential.

1.1.2 Ionic Crystals and Born–Mayer Theory (B)

Examples of ionic crystals are sodium chloride (NaCl) and lithium fluoride (LiF). Ionic crystals also consist of chemically saturated units (the ions that form their basic units are in rare-gas configurations). The ionic bond is due mostly to Coulomb attractions, but there must be a repulsive contribution to prevent the lattice from collapsing. The Coulomb attraction is easily understood from an electrontransfer point of view. For example, we view LiF as composed of Li+(1s2) and F−(1s22s22p6), using the usual notation for configuration of electrons. It requires about one electron volt of energy to transfer the electron, but this energy is more than compensated by the energy produced by the Coulomb attraction of the charged ions. In general, alkali and halogen atoms bind as singly charged ions. The core repulsion between the ions is due to an overlapping of electron clouds (as constrained by the Pauli principle).

Since the Coulomb forces of attraction are strong, long-range, nearly two-body, central forces, ionic crystals are characterized by close packing and rather tight binding. These crystals also show good ionic conductivity at high temperatures, good cleavage, and strong infrared absorption.

A good description of both the attractive and repulsive aspects of the ionic bond is provided by the semi-empirical theory due to Born and Mayer. To describe this theory, we will need a picture of an ionic crystal such as NaCl. NaCllike crystals are composed of stacked planes, similar to the plane in Fig. 1.3. The theory below will be valid only for ionic crystals that have the same structure as NaCl.

Let N be the number of positive or negative ions. Let rij (a symbol in boldface type means a vector quantity) be the vector connecting ions i and j so that |rij| is the distance between ions i and j. Let Eij be (+1) if the i and j ions have the same signs and (–1) if the i and j ions have opposite signs. With this notation the potential energy of ion i is given by

	Ui = ∑all j(≠i) Eij	e	2	,	(1.10)
		4πε0	| rij |

1.1 Classification of Solids by Binding Forces (B)	7

.

.

.

- + -

. . . + - + . . .

-+ -

.

.

.

Fig. 1.3. NaCl-like ionic crystals

where e is, of course, the magnitude of the charge on any ion. For the whole crystal, the total potential energy is U = NUi. If N1, N2 and N3 are integers, and a is the distance between adjacent positive and negative ions, then (1.10) can be written as

	U	i	= ∑	′		(−)N1 +N2 +N3	e2	.	(1.11)
				(N1	,N2	,N3 )	4πε0a
						N12 + N22 + N32

In (1.11), the term N1 = 0, N2 = 0, and N3 = 0 is omitted (this is what the prime on the sum means). If we assume that the lattice is almost infinite, the Ni in (1.11) can be summed over an infinite range. The result for the total Coulomb potential energy is

U = −N			M NaCl e2			,		(1.12)
				4πε0a
where
M NaCl = −∑N′∞,N				(−)N1 +N2 +N3
		,N		=−∞			+ N 2	(1.13)
1	2		3	N	2			+ N 2
				1			2	3

is called the Madelung constant for a NaCl-type lattice. Evaluation of (1.13) yields MNaCl = 1.7476. The value for M depends only on geometrical arrangements. The series for M given by (1.13) is very slowly converging. Special techniques are usually used to obtain good results [46].

As already mentioned, the stability of the lattice requires a repulsive potential, and hence a repulsive potential energy. Quantum mechanics suggests (basically from the Pauli principle) that the form of this repulsive potential energy between ions i and j is

R
			| rij |
Uij	= Xij exp	−			,	(1.14)
			R
			ij

where Xij and Rij depend, as indicated, on the pair of ions labeled by i and j. “Common sense” suggests that the repulsion be of short-range. In fact, one usually

8 1 Crystal Binding and Structure

assumes that only nearest-neighbor repulsive interactions need be considered. There are six nearest neighbors for each ion, so that the total repulsive potential energy is

U R = 6NX exp(−a / R) .

(1.15)

This usually amounts to only about 10% of the magnitude of the total cohesive energy. In (1.15), Xij and Rij are assumed to be the same for all six interactions (and equal to the X and R). That this should be so is easily seen by symmetry.

Combining the above, we have for the total potential energy for the lattice

		M	NaCl	e2
U = N	−					+ 6NX exp	−
		4πε0a

. (1.16)

R

The cohesive energy for free ions equals U plus the kinetic energy of the ions in the solid. However, the magnitude of the kinetic energy of the ions (especially at low temperature) is much smaller than U, and so we simply use U in our computations of the cohesive energy. Even if we refer U to zero temperature, there would be, however, a small correction due to zero-point motion. In addition, we have neglected a very weak attraction due to the van der Waals forces.

Equation (1.16) shows that the Born–Mayer theory is a two-parameter theory. Certain thermodynamic considerations are needed to see how to feed in the results of experiment.

The combined first and second laws for reversible processes is

TdS = dU + p dV ,

(1.17)

where S is the entropy, U is the internal energy, p is the pressure, V is the volume, and T is the temperature. We want to derive an expression for the isothermal compressibility k that is defined by

1		∂p
	= −		.	(1.18)
kV
		∂V
			T

The isothermal compressibility is not very sensitive to temperature, so we will evaluate k for T = 0. Combining (1.17) and (1.18) at T = 0, we obtain

	1			∂2U
			=			.	(1.19)
				∂V 2
kV
		T =0			T =0

There is one more relationship between R, X, and experiment. At the equilibrium spacing a = A (determined by experiment using X-rays), there must be no net force on an ion so that

	∂U	= 0 .	(1.20)
	∂a a= A

1.1 Classification of Solids by Binding Forces (B)	9

Thus, a measurement of the compressibility and the lattice constant serves to fix the two parameters R and X. When we know R and X, it is possible to give a theoretical value for the cohesive energy per molecule (U/N). This quantity can also be independently measured by the Born–Haber cycle [46].3 Comparing these two quantities gives a measure of the accuracy of the Born–Mayer theory. Table 1.1 shows that the Born–Mayer theory gives a good estimate of the cohesive energy. (For some types of complex solid-state calculations, an accuracy of 10 to 20% can be achieved.)

Table 1.1. Cohesive energy in kcal mole–1

Solid	Born–Mayer theory	Experiment
LiCl	196.3	201.5
NaCl	182.0	184.7
NaBr	172.7	175.9
NaI	159.3	166.3

Adapted from Born M and Huang K, Dynamical Theory of Crystal Lattices, selected parts of Table 9 (p.26) Clarendon Press, Oxford, 1954. By permission of Oxford University Press.

1.1.3 Metals and Wigner–Seitz Theory (B)

Examples of metals are sodium (Na) and copper (Cu). A metal such as Na is viewed as being composed of positive ion cores (Na+) immersed in a “sea” of free conduction electrons that come from the removal of the 3s electron from atomic Na. Metallic binding can be partly understood within the context of the Wigner-Seitz theory. In a full treatment, it would be necessary to confront the problem of electrons in a periodic lattice. (A discussion of the Wigner–Seitz theory will be deferred until Chap. 3.) One reason for the binding is the lowering of the kinetic energy of the “free” electrons relative to their kinetic energy in the atomic 3s state [41]. In a metallic crystal, the valence electrons are free (within the constraints of the Pauli principle) to wander throughout the crystal, causing them to have a smoother wave function and hence less 2ψ. Generally speaking this spreading of the electrons wave function also allows the electrons to make better use of the attractive potential. Lowering of the kinetic and/or potential

3The Born–Haber cycle starts with (say) NaCl solid. Let U be the energy needed to break

this up into Na+ gas and Cl– gas. Suppose it takes EF units of energy to go from Cl– gas to Cl gas plus electrons, and EI units of energy are gained in going from Na+ gas plus electrons to Na gas. The Na gas gives up heat of sublimation energy S in going to Na solid,

and the Cl gas gives up heat of dissociation D in going to Cl2 gas. Finally, let the Na solid and Cl2 gas go back to NaCl solid in its original state with a resultant energy W. We are back where we started and so the energies must add to zero: U – EI + EF – S – D – W = 0. This equation can be used to determine U from other experimental quantities.

10 1 Crystal Binding and Structure

energy implies binding. However, the electron–electron Coulomb repulsions cannot be neglected (see, e.g., Sect. 3.1.4), and the whole subject of binding in metals is not on so good a quantitative basis as it is in crystals involving the interactions of atoms or molecules which do not have free electrons. One reason why the metallic crystal is prevented from collapsing is the kinetic energy of the electrons. Compressing the solid causes the wave functions of the electrons to “wiggle” more and hence raises their kinetic energy.

A very simple picture4 suffices to give part of the idea of metallic binding. The ground-state energy of an electron of mass M in a box of volume V is [19]

	E =	2π 2	V −2 / 3 .
		2M

Thus the energy of N electrons in N separate boxes is

	EA = N	2π 2	V −2 / 3 .	(1.21)
		2M

The energy of N electrons in a box of volume NV is (neglecting electron–electron interaction that would tend to increase the energy)

	EM = N	2π 2	V −2 / 3N −2 / 3 .	(1.22)
		2M

Therefore EM/EA = N –2/3 << 1 for large N and hence the total energy is lowered considerably by letting the electrons spread out. This model of binding is, of course, not adequate for a real metal, since the model neglects not only electron– electron interactions but also the potential energy of interaction between electrons and ions and between ions and other ions. It also ignores the fact that electrons fill up states by satisfying the Pauli principle. That is, they fill up in increasing energy. But it does clearly show how the energy can be lowered by allowing the electronic wave functions to spread out.

In modern times, considerable progress has been made in understanding the cohesion of metals by the density functional method, see Chap. 3. We mention in particular, Daw [1.6].

Due to the important role of the free electrons in binding, metals are good electrical and thermal conductors. They have moderate to fairly strong binding. We do not think of the binding forces in metals as being two-body, central, or shortrange.

4A much more sophisticated approach to the binding of metals is contained in the pedagogical article by Tran and Perdew [1.26]. This article shows how exchange and correlation effects are important and discusses modern density functional methods (see Chap. 3).

1.1 Classification of Solids by Binding Forces (B)	11

1.1.4 Valence Crystals and Heitler–London Theory (B)

An example of a valence crystal is carbon in diamond form. One can think of the whole valence crystal as being a huge chemically saturated molecule. As in the case of metals, it is not possible to understand completely the binding of valence crystals without considerable quantum-mechanical calculations, and even then the results are likely to be only qualitative. The quantum-mechanical considerations (Heitler–London theory) will be deferred until Chap. 3.

Some insight into covalent bonds (also called homopolar bonds) of valence crystals can be gained by considering them as being caused by sharing electrons between atoms with unfilled shells. Sharing of electrons can lower the energy because the electrons can get into lower energy states without violating the Pauli principle. In carbon, each atom has four electrons that participate in the valence bond. These are the electrons in the 2s2p shell, which has eight available states.5 The idea of the valence bond in carbon is (very schematically) indicated in Fig. 1.4. In this figure each line symbolizes an electron bond. The idea that the eight 2s2p states participate in the valence bond is related to the fact that we have drawn each carbon atom with eight bonds.

C C C

C C C

C C C

Fig. 1.4. The valence bond of diamond

Valence crystals are characterized by hardness, poor cleavage, strong bonds, poor electronic conductivity, and poor ionic conductivity. The forces in covalent bonds can be thought of as short-range, two-body, but not central forces. The covalent bond is very directional, and the crystals tend to be loosely packed.

5More accurately, one thinks of the electron states as being combinations formed from s and p states to form sp3 hybrids. A very simple discussion of this process as well as the details of other types of bonds is given by Moffatt et al [1.17].

12 1 Crystal Binding and Structure

Molecular crystals are bound by the van der Waals forces caused by fluctuating dipoles in each molecule. A “snap-shot” of the fluctuations.

Example: argon

+	+	+
ion	ion	ion
+	+	+
ion	ion	ion

+-

ion ion

-+

ion ion

Ionic crystals are bound by ionic forces as described by the Born–Mayer theory. Example: NaCl

++

ion ion

++

ion ion

Metallic crystalline binding is described by quantum-mechanical means. One simple theory which does this is the Wigner–Seitz theory.

Example: sodium

Valence crystalline binding is described by quantum-mechanical means. One simple theory that does this is the Heitler– London theory.

Example: carbon in diamond form

Fig. 1.5. Schematic view of the four major types of crystal bonds. All binding is due to the Coulomb forces and quantum mechanics is needed for a complete description, but some idea of the binding of molecular and ionic crystals can be given without quantum mechanics. The density of electrons is indicated by the shading. Note that the outer atomic electrons are progressively smeared out as one goes from an ionic crystal to a valence crystal to a metal

1.1.5 Comment on Hydrogen-Bonded Crystals (B)

Many authors prefer to add a fifth classification of crystal bonding: hydrogenbonded crystals [1.18]. The hydrogen bond is a bond between two atoms due to the presence of a hydrogen atom between them. Its main characteristics are caused by the small size of the proton of the hydrogen atom, the ease with which the electron of the hydrogen atom can be removed, and the mobility of the proton.

The presence of the hydrogen bond results in the possibility of high dielectric constant, and some hydrogen-bonded crystals become ferroelectric. A typical example of a crystal in which hydrogen bonds are important is ice. One generally thinks of hydrogen-bonded crystals as having fairly weak bonds. Since the hydrogen atom often loses its electron to one of the atoms in the hydrogen-bonded molecule, the hydrogen bond is considered to be largely ionic in character. For this reason we have not made a separate classification for hydrogen-bonded

1.2 Group Theory and Crystallography 13

crystals. Of course, other types of bonding may be important in the total binding together of a crystal with hydrogen bonds. Figure 1.5 schematically reviews the four major types of crystal bonds.

1.2 Group Theory and Crystallography

We start crystallography by giving a short history [1.14].

1.In 1669 Steno gave the law of constancy of angle between like crystal faces. This of course was a key idea needed to postulate there was some underlying microscopic symmetry inherent in crystals.

2.In 1784 Abbe Hauy proposed the idea of unit cells.

3.In 1826 Naumann originated the idea of 7 crystal systems.

4.In 1830 Hessel said there were 32 crystal classes because only 32 point groups were consistent with the idea of translational symmetry.

5.In 1845 Bravais noted there were only 14 distinct lattices, now called Bravais lattices, which were consistent with the 32 point groups.

6.By 1894 several groups had enumerated the 230 space groups consistent with only 230 distinct kinds of crystalline symmetry.

7.By 1912 von Laue started X-ray experiments that could delineate the space groups.

8.In 1936 Seitz started deriving the irreducible representations of the space groups.

9.In 1984 Shectmann, Steinhardt et al found quasi-crystals, substances that were neither crystalline nor glassy but nevertheless ordered in a quasi periodic way.

The symmetries of crystals determine many of their properties as well as simplify many calculations. To discuss the symmetry properties of solids, one needs an appropriate formalism. The most concise formalism for this is group theory. Group theory can actually provide deep insight into the classification by quantum numbers of quantum-mechanical states. However, we shall be interested at this stage in crystal symmetry. This means (among other things) that finite groups will be of interest, and this is a simplification. We will not use group theory to discuss crystal symmetry in this Section. However, it is convenient to introduce some group-theory notation in order to use the crystal symmetry operations as examples of groups and to help in organizing in one’s mind the various sorts of symmetries that are presented to us by crystals. We will use some of the concepts (presented here) in parts of the chapter on magnetism (Chap. 7) and also in a derivation of Bloch’s theorem in Appendix C.