Marder M.P. Condensed Matter Physics

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Part III

MECHANICAL PROPERTIES

Condensed Matter

Physics,

Second Edition

by Michael P. Marder

11.

Cohesion of Solids

11.1 Introduction

The cohesive energy of a solid is the energy needed in order to rip a sample apart

into a gas of widely separated atoms. By

itself,

this energy does not have much

significance. It is not easy to measure experimentally, and it bears no relation to

the practical strengths of

solids;

practical strength is governed by resistance to flow

and fracture, physically quite distinct from cohesive energy.

The question that the cohesive energy makes it possible to address is how crys-

tals choose their equilibrium structure. Electronic structure calculations begin by

assuming that atomic positions are known, but in studying cohesive energy, one

asks which structure leads to the lowest energy and why. In the course of this

study, crystals divide roughly into five classes: molecular, hydrogen

bonded,

ionic,

covalent, and metallic. These classes blend into each other, but still represent con-

ceptual ideal types. The molecular crystals are composed from atoms whose shells

are filled, which hold tightly onto their electrons, and which bind together only be-

cause of small induced dipole moments. Hydrogen bonded solids involve hydrogen

atoms as part of the bonding process; this might seem too specialized to warrant

a separate category except that it includes much of biology. The ionic crystals are

composed of pairs of atoms, in which one member of the pair donates an electron

to the other, and vast numbers of such pairs are held together by dipole forces.

Figure 11.1(A) shows the valence charge density of NaCl; the visible charge sits

in a sphere around the chlorine, and charge density drops to very small values

between the atoms. The covalent crystals feature even more electrons wandering

away from their host atoms, and it is convenient to think of bonds between atoms,

because charge density is high along lines connecting near neighbors, as shown in

Figure

11.1

(B).

In metals, the conduction electrons are distributed quite uniformly

throughout the solid, and the net effect of the interaction between electrons and

nuclei is a hydrostatic pressure keeping the solid together. A plot of the conduction

electron density in a metal would be flat and almost featureless. As a final illustra-

tion of the connection between charge density and cohesion, Figure 11.2 presents

experimentally measured charge density for cuprous oxide. This metal oxide does

not easily fit classification. The charge density around the copper has the shape of

d orbitals, and leads to a phenomenon resembling covalent bonding between metal

ions.

295

Condensed Matter

Physics,

Second Edition

by Michael P. Marder

296

Chapter 11. Cohesion of Solids

Figure 11.1. (A) Valence charge density of NaCl along a (100) plane, computed by the

plane wave pseudopotential code of Stumpf and Scheffier (1994). The seven valence elec-

trons of

and one of Na are the only electrons depicted; all electrons sit on the chlorine, in

a nearly perfectly spherical configuration, so the sodium is invisible. The valence electrons

avoid the core of the atom, which is why their density drops to zero there. (B) Valence

charge density of Ge along a (100) plane. The fourfold coordination of the diamond struc-

ture is evident, as well as the large charge density lying along the lines between atoms,

characteristic of covalent bonding.

Figure 11.2. Experimentally measured charge density of cuprous oxide CU2O. The plot

shows the difference between the charge distribution in CU2O and the charge distribution

of free ions. The charge distribution around the copper atoms looks like d orbitals; the

presence of oxygen induces covalent bonding between metal ions. [Source: Zuo et al.

(1999),

p. 50.]

Introduction 297

Figure 11.3. Picture of the sodium chloride structure, indicating the sizes of the sodium

(small) and chlorine (large)

ions.

The chlorine ion is large because it

has

robbed

electron

from the sodium.

11.1.1 Radii of Atoms

In searching for ideas by which to explain how atoms assemble themselves, the

simplest is to assign each atom a radius, as if it were a hard sphere. An illustration

of NaCl, indicating the sizes of the two ions, appears in Figure 11.3. Such an

assignment would be without much consequence were it allowed to change every

time the atom formed a compound with a new neighbor, but in fact the apparent

radii of atoms remain roughly independent of their surroundings. This statement

must, unfortunately, be qualified a bit, because the apparent radii do depend upon

the class of crystal in which the atom is located.

Table 11.1 contains a summary of phenomenological rules that can be used

to estimate lattice constants if crystal structure is known; the origin of these rules

is discussed at length by Pearson (1972). Use of these rules should give lattice

constants within about 10% of experimental values, and they can serve as a quick

check on the results of either experiment, or theoretical calculation.

Example: Cristobalite. According to Wyckoff (1963-1971), quartz in the ß-

cristobalite form is cubic (a = 7.12 Â) and has a basis with eight silicon atoms and

sixteen oxygens, which in units of a/8 are at

Si:

(000) (440) (404) (044) (222) (266) (626) (662)

O: (111) (551) (515) (155) (177) (537) (573) (133)

(717) (357) (313) (753) (771) (331) (375) (735)

The nearest-neighbor distance for this structure is 1.54 Â. The silicon has four

neighboring oxygens, so Z = z = 4, while each oxygen has two neighboring sil-

icons,

and Z = 6, z = 2. According to Table 11.1, the covalent radius of silicon

is 1.17 Â, and that of oxygen is 0.74 -0.14 = 0.60 Â, which sum to 1.77 Â. The

discrepancy is more than 10%, so there are grounds for concern. Liu et al. (1993)

show that Wyckoff's structure is incorrect.

298

Chapter

11.

Cohesion of Solids

Table

11.1.

Effective radii

of the

elements

El.

Ag 1 +

Au 1 +

Cs 1 +

1.88

1.81

1.45

1.43

1.39

1.44

2.24

1.13

1.70

0.98

0.92

1.97

1.57

1.83

1.25

1.36

2.73

1.28

1.77

1.76

2.04

1.27

1.41

1.37

1.80

1.86

1.26

0.50

2.22

1.37

1.35

0.35

0.20

1.95

0.15

2.60

0.99

0.97

1.01

1.81

1.67

0.96

1.36

0.62

0.53

2.72

fli

1.53

1.25

1.21

1.52

1.98

0.89

1.52

0.80

1.14

0.77

1.74

1.49

1.65

0.99

2.35

1.35

1.59

1.57

1.85

0.72

1.27

1.22

1.61

El.

0.78

1.58

1.57

1.77

1.66

1.36

2.38

1.88

1.56

1.60

1.30

1.40

0.88

1.91

1.47

1.83

1.25

1.56

0.89

1.35

1.28

1.63

1.75

1.38

1.76

1.83

2.08

1.10

2.16

0.81

1.33

2.00

1.15

0.68

0.65

1.71

0.97

1.58

1.40

2.12

0.84

1.49

1.58

1.33

1.44

2.03

1.69

1.23

1.56

1.36

0.74

1.57

1.64

0.74

1.10

1.43

1.53

1.65

El.

1 +

1.39

1.58

2.55

1.38

1.35

1.34

1.27

1.59

1.64

1.40

1.32

1.80

1.62

2.15

1.47

1.78

1.60

1.80

1.46

1.72

1.75

1.56

1.35

1.41

1.80

1.94

1.39

1.60

Ä,

1.48 2.16

1.84 1.04

2.45

1.41

0.81

1.44

1.98 1.17

0.41

1.17

2.71

1.66

0.71

1.40

2.94

1.13 1.91

1.59

2.21

1.37

0.68

0.95

1.46

1.56

2.17

0.93

1.62

1.70

0.74

1.31

0.80

El.

Element;

Z =

Valency;

M -

Metallic radius;

I =

Ionic radius;

= Covalent radius.

The metallic radius

intended

for use in

close-packed structures with coordination number

12.

Ionic

and

covalent radii

may

need correction

additional factors.

The

ionic radius

intended

for

ions with coordination number near

6, and

should

multiplied

by the

factor

[z/6]

/("~

where

the

number

nearest neighbors,

and n is the

Born exponent

Table

11.2.

The

covalent radius gives best results

for

atoms with coordination number

4, and for

an atom with

valence electrons

and z

nearest neighbors

given

by R = R\

—0.13 In[Z/z]:

for

Z < 0, use

\Z\. Source: Tosi (1964)

and

Pearson (1972),

pp.

147-153.

Noble Gases

Table 11.2. Born exponent n

Ion Type He Ne Ar Kr Xe

(inert core) Cu

Born exponentn 5 7 9 10 12

Used in correction factor

[z/6]

/'"

-1

for ionic

radii. Source: Pearson (1972).

11.2 Noble Gases

The noble gases are characterized by a very weak attraction between the atoms.

The potential energy of a crystal conventionally takes the form of a two-body po-

tential

= ÌE^<;) (li.i)

with

σ\ΐ2'

Ç)-0

(11.2)

Here r^ is the distance between two atoms. The potential in Eq. (11.2) is called the

Lennard-Jones 6-12 potential and arises partly from basic calculations and partly

from phenomenological considerations. The term proportional to r

-6

accurately

represents the interaction at large distances between two molecules without a per-

manent dipole moment, the van der Waals interaction. It arises in the following

way.

The interaction between two dipoles is

Φ(τ) = \?\ ■p2-?,(p

-r){p2-r)]/r\ (H-3)

where r is the distance between the atoms, and r is a unit vector. Now it will be

protested that atoms without dipole moments do not have dipole moments. This

is true. However, quantum or thermal fluctuations continually induce tiny dipole

moments in each atom. The resulting electric field then polarizes the other atom,

producing a small dipole moment in it, of order r~

. The resulting interaction

is then of order r~

. The induced dipole moment is always of such a sign as to

lower the energy of the system; therefore, the interaction is always attractive. To

obtain such a result formally, see Problem 3. One places the quantum operator

for Coulomb interactions as a perturbation into the Hamiltonian for two widely

separated atoms and then finds a nonzero result in the second order of perturbation

theory. The final result is of the form

Φ—^ψ, (11.4)

where a, is the polarizability of atom i. It is only valid when the atoms are well-

separated. When they come close together, at distances comparable to the atomic

300

Chapter

11.

Cohesion of Solids

radii, the force between them becomes repulsive. For this reason, and with no

firmer foundation, one adds the term proportional to r

in Eq. (11.2).

In order to provide a firm check on the theory, one can measure the constants

e and σ in the gas phase and then use the results to make predictions about the

solid phase. The gas phase measurements use the following relation from the virial

expansion:

= - id? [1

—e~^

See

Landau andLifshitz(1980), pp. 231-232, (11.5)

2 J who use somewhat different notation.

The coefficient b

enters the virial expansion for the pressure P in the form

PV/k

T =

l-b

... (11.6)

so measurements of the equation of state at low densities can be used to find b

and hence fit σ and e (Table 11.3).

Table 11.3. Parameters σ and e for the noble gases

Noble Gas

e(eV)

σ(Α)

Source:

201.

He Ne

8.6· 10"

0.0031

2.56 2.74

Bernardes

0.0104

3.40

0.0104

3.65

(1958) and Hirschfelder et al

0.0200

3.98

. (1954),

The total energy of a crystal where atoms interact in this way is then

£/Ν

1*Σ

®'

-ψ

Taking the crystal to contain N atoms, and (\ 1.7)

adopting periodic boundary conditions so that

ϋφθ all atoms have identical neighborhoods.

Letting d be a nearest neighbor distance, one can rewrite the energy as

ε/*

^Σφ

-φ

di·«)

26[Α

φ'

-Α

φνίώΑ

/Ξ

Σ (|Y. (11.9)

Written in this way, one sees that the lattice sums can be carried out once and

for all for any particular crystal structure, so that the energy depends only upon

the nearest-neighbor separation. The sums converge rapidly because the powers

involved are so high. Except for helium, all the noble gases solidify into an fee

structure, and the appropriate lattice sums in this case are shown in Table 11.4.

One can now work out the nearest-neighbor spacing in equilibrium,

4> =

*(^)

1/6

(ii.io)

Ionic Crystals 301

Table 11.4.

Crystal

Al/2An

Lattice sums,

fee

14.4539

12.1319

8.6102

A(,

and Λ n for various crystal

bec

12.2537

9.1142

8.2373

structures

hep

14.4549

12.1323

8.6111

The sums are defined in Eq. (11.9). The final row is proportional to the energy of the

crystal. While fee and hep have noticeably lower energy than bec, the energies of the

two former structures are too close for these arguments to determine which will appear

experimentally; the rare gas solids choose fee.

Table 11.5. Properties of noble gases compared with predictions of Lennard-

Jones potentials

Noble Gas Ne AT Kr Xe

Experimental do (Ä)

do from Eq. (11.10) (Â)

Experimental £/N (eV/atom)

E/N

from Eq. (11.11)

Experimental B (dyne/cm

)

B from Eq. (11.13)

Results show agreement at roughly the 10% level. Source: Ashcroft and Mermin

(1976),

p. 401.

3.13

2.99

-0.02

-0.027

1.1 -10

1.81· 10

3.75

3.71

-0.08

-0.089

2.7 · 10

3.18· 10

3.99

3.98

-0.11

-0.120

3.5 · 10'°

3.46· 10

4.33

4.34

-0.17

-0.172

3.6 · 10

3.81

■

the cohesive energy

P I hi — -c

'2A

E/N

= -e—4-, (11.11)

and the bulk modulus B which is given by

B =

lzri

(11.12)

For an fee lattice, the volume per particle V/N is related to the nearest-neighbor

separation d by V/N = d

/\/2· Evaluating Eq. (11.12) gives

Comparisons of this simple theory with experiment are listed in Table 11.5.

11.3 Ionic Crystals

The simplest ionic crystals are the alkali halides, built by combining an element

from the first column of the periodic table with an element from the seventh col-

umn. The distinctive feature of these crystals is that the alkali metal (Li, Na, K,

302

Chapter 11. Cohesion of Solids

Rb,

or Cs) gives up an electron relatively easily so as to obtain a closed outer shell,

while similarly the halogen (F, Cl, Br, or I) has a strong affinity for acquiring an

extra electron to fill its outer shell. In fact, the benefits of closing the electronic

shell outweigh Coulomb repulsion, and isolated halogen atoms stably bind extra

electrons. Their positive electron affinities are listed in Table 11.6, along with the

first ionization potential of the alkalis. The affinity of the halogen is typically less

than the ionization potential of the alkali, but bringing the positively and negatively

charged ions close together gives enough Coulomb energy to make the resulting

structure stable. The Coulomb energy is easily estimated by treating the two ions

as possessing a single negative or positive charge and placing them at a distance of

about 3 Â apart; this gives a Coulomb energy of around —4 eV, which is enough

to overcome the difference between electron affinity and ionization potential. The

reason that the I-VII compounds are particularly simple, however, is that the struc-

ture stabilizes in such a way that the overlap of electronic charge from neighboring

ions is negligible, as shown in Figure 11.1(A).

Four particularly common lattice structures—the sodium chloride structure, the

cesium chloride structure, the zincblende structure, and the wurtzite structure—

were sketched in Section 2.3.

11.3.1 Ewald Sums

Actually carrying out the sum over positive and negative ions to calculate the en-

ergy of the ionic solids is technically tricky. If in the first unit cell the positive ion

(cation) is at the origin, and the negative one (anion) is at d, the potential energy

associated with a particular cation is

ΪΣ[ΓΗ]-Ϊ

Ϊ»»-*-Ί

(1U4)

and the energy of the crystal as a whole will be N/2 times this amount. The

dif-

ficulty is that the sum (11.14) converges very slowly, and in fact it can be done so

as to have any value whatsoever. Physically, different limiting values of the sum

correspond to crystals with differing amounts of surface charge, and the Coulomb

force is so long range that the surface charges are able to make the crystal energy as

Table 11.6. Electron affinities and ionization potential for the alkalis

and the hai ides

Atom Electron affinity (eV) Atom First ionization potential (eV)

Ö75

3.40

3.61

3.36

3.06

5.32

5.14

4.34

4.18

3.90

Source: Hotop and Lineberger (1975) and Emsley (1998).