Marder M.P. Condensed Matter Physics

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Classical Potentials 313

£?

-C

(A)

1.2 1.4 1.6 1.

Wigner-Seitz radius, r^ (Â)

2.0°

(B)

Lattice constant a (Â)

Figure 11.6. (A) Comparison of cohesive energy of fee, bee, and hep structures for alu-

minum as a function of the Wigner-Seitz radius

rw-

The experimental value of

= 1.583

is indicated by the dotted line. The fee lattice has the lowest energy, but lies only a minute

amount below hep. The bec lattice has considerably higher energy. (B) Cohesive energy

of fee aluminum computed as a function of lattice constant a (squares), compared with

Eq.

( 11.50)

(solid line). Computations were performed with VASP of Kresse and Hafner

(1994) and Kresse and Furthmiiller (1996). The computations measure all energies rela-

tive to a rather arbitrary zero, so the fit to Eq. (11.50) is needed to determine the cohesive

energy, by extrapolating to widely separate atoms and calling the result the zero of energy.

For each element, define a scaled length

Πνο

(11.49)

where rwo is the radius of the Wigner-Seitz sphere of the metal in equilibrium, rw

is its radius for expanded or contracted crystals, and

is a dimensionless number

that characterizes the anharmonicity of each element. Upon choosing η appropri-

ately and letting So be the equilibrium cohesive energy of each element, a wide

variety of different elements have a cohesive energy described quantitatively by

the fitting function

E{r

) = E

* (-1 -a* -0.05^) . (11.50)

Table 11.11 records values of rwo and

for a number of metals. These data

can be used to determine pressure and bulk modulus over a wide range of lattice

parameters, and they are in good accord with experimental measurements of metals

under large pressures.

11.8 Classical Potentials

The molecular dynamics and Monte Carlo calculations described in Section 5.4

rely upon a functional giving the energy of a solid as a function of all the nuclear

positions. This functional exists; it is just the ground-state energy

ε = (Φ|5ί(Α! .../?*)

|Φ>

(11.51)

314 Chapter 11. Cohesion of Solids

Table

11.11.

Wigner-Seitz radius

rwo,

scaling parameter η, and cohesive energy £o

for selected elements

El.

rwo

(Â)

1.60

1.58

1.59

2.46

1.25

2.18

1.73

2.02

1.39

1.42

2.98

1.41

1.96

1.94

2.27

El.= Element

5.94

4.71

6.75

4.41

4.01

4.52

8.08

3.11

5.31

5.59

4.17

5.30

4.85

4.94

4.75

£o

(eV)

2.96

3.34

3.78

1.86

3.33

1.83

1.16

4.77

4.39

4.10

0.83

3.50

3.10

3.30

1.80

El.

rwo

(Â)

1.41

1.99

1.76

1.74

1.84

1.50

2.57

1.72

1.77

1.55

2.08

1.63

1.38

1.93

1.52

. Source: Rose et

al.

(1984).

■n

5.16

4.27

5.05

4.66

5.11

6.52

3.94

3.10

5.60

5.85

3.70

4.84

5.11

6.37

6.41

(eV)

4.29

4.14

3.87

6.35

2.60

6.93

0.94

1.65

1.53

6.81

1.13

7.47

4.44

2.04

3.94

El.

rwo

(Â)

1.53

2.75

1.52

1.48

1.68

1.62

1.99

1.62

1.90

1.49

1.56

1.99

1.54

1.77

6.47

4.18

6.15

6.04

4.88

4.92

4.12

4.76

5.74

4.81

5.69

4.23

3.94

7.16

4.48

£o

(eV)

5.85

0.86

8.10

6.62

4.64

8.09

5.93

4.86

1.87

5.30

8.66

4.39

1.60

1.35

6.32

of N nuclei, and their accompanying electrons, viewed as a function of ionic posi-

tions.

Of course, every time a nucleus moves, it is necessary to solve the electron

problem over again from scratch. The force on atom / is then

= -^i. (11.52)

dRi

When deviations of atoms from equilibrium are small, the subject can be developed

systematically and will occupy Chapter 13. For the purposes of molecular dynam-

ics,

small deviations are not sufficient, but it is impossible to evaluate Eq. (11.51)

exactly. One possibility is to guess that the energy takes the form of Eq. (11.1),

and to choose a function for φ that reproduces some desired feature of experiment

or calculation, such as ground-state crystal structure or melting temperature. Po-

tentials of the two-body form are often clearly inadequate. For example, there

does not exist any potential </>(r) for which the diamond structure is the ground

state,

which rules out study of silicon and germanium in addition to carbon. The

diamond structure can, however, be made the stable ground state of a classical

potential if one adds to the energy functional some terms that depend upon angles

between bonds. The angle

in Figure 11.7 is given by cos

-1

[(R\

— R2) ■ (R3

R2)]',

terms that are functions of angles depend simultaneously upon locations of three

atoms and thus are called three-body potentials. Another possibility is suggested

by the energy functionals of Section 11.4, which depend upon the overall density

Problems 315

of electrons; if terms of this sort are added to an energy functional, they depend

upon the total volume occupied by the atoms, which cannot be expressed in terms

either of pair potentials or three-body potentials.

Figure 11.7. Any term depending upon the

angle between two bonds must depend simul-

taneously upon

the

coordinates of three atoms

and therefore introduces triplet or three-body

terms into the energy.

By the time that potentials of this type have been employed in enormous com-

puter programs, it is easy to forget that the potentials themselves are not especially

well grounded, and represent a mix of functional forms suggested by calculation,

constraints provided by experiment, and calculational convenience. Classical po-

tentials often do not give reliable results for physical quantities they were not tuned

to get right. For example, a widely used potential for studying silicon was proposed

by Stillinger and Weber (1985), involving two- and three-body terms. Its parame-

ters are chosen to give the melting temperature of silicon correctly, but the potential

gets the density change at melting wrong by a factor of two and does not correctly

predict vibrational spectra or surface reconstructions. A particularly large number

of potentials has been proposed for silicon; it is difficult to know how to choose

between them. These and other matters related to the construction of classical

potentials are discussed by Carlsson (1990).

Problems

Sums for noble gases: Compute the sums Ag and

A12

to five decimal places

for an fee lattice, and verify the results in Table 11.4.

Zinc in copper: Copper is a monovalent fee nearly-free electron metal. Zinc

is an hep divalent metal. In small quantities, zinc mixes substitutionally with

copper, its main effect being to increase the electron concentration.

(a) Find the relationship between electron density n and the Fermi wave vector

kp within the nearly free electron approximation.

(b) Still working in the nearly free electron approximation, assume that the zinc-

copper mixture will undergo a phase transition to bec as soon as the Fermi

sphere first touches some point on the edge of the Brillouin zone. Show that

the transition occurs at 36% zinc. Remember that in an fee conventional unit

cell of lattice constant a there are four ions.

occurs when the Fermi sphere intersects the Brillouin zone of the bec lattice.

What is this second concentration of zinc?

316

Chapter 11. Cohesion of Solids

Quantum view of van der Waals force:

(a) Consider two widely separated hydrogen atoms. Let "K\ and

OÏ2

be the

Hamiltonians for the separate, unperturbed atoms, and let R be the distance

vector between them. Consider now the perturbation

3~C;

R \R\-R\

\R\

—R2Ì

(11.53)

where R\ and R

are the operators for the positions of the electrons at atoms

1 and 2 (there is no harm done in regarding the electrons as distinguishable,

because they are widely separated).

Show that at first order in perturbation theory, the effect of the perturbation is

exponentially small, because the ground state of the hydrogen atom is spheri-

cally symmetric, and the charge distributions of the two atoms hardly overlap.

(b) Argue that at second order in perturbation theory, all contributions to the

relevant integrals are negligible unless the electrons are very close to their

respective atoms:

|/?i|

<Λ and \R

-R\ «A. (11.54)

In this case, it is valid to expand Eq. (11.53) to leading (second) order in

the two small quantities appearing in Eq. (11.54). Show that as a result the

perturbation takes the form

3(R

-R)([R

-R]-R) Ri-[R

-R)

(11.55)

negative and varies as R

Ewald summation for metals:

(a) Show that when the electron density n is constant,Eq. (11.29) can be rewrit-

ten as

N-

5(0)

S(r)

(11.56)

where 5 is defined by Eq. (11.14).

(b) Show finally that if 5(0) is given by Eq. (11.21) then Eq. (11.56) can be

rewritten as

N-

5(0)

(11.57)

Ewald summation in one dimension: Suppose that one has a collection of

charged spheres containing charge of magnitude q\ . . . q^. Overall the

spheres are electrically neutral; YJi-\ qi = 0. Make many copies of these

References 317

spheres, and evenly space them along a line so that the sequence of charges qi

repeats periodically.

The energy per unit cell of such a collection of charges is

y^ y-v qiQj+l Π1 5XÌ

.->

/ j / j I -i · Sum over all positive and negative j φ 0. \ιι.~>ο)

]φ0 1=1 \

The goal of

this

problem is to use the technique of Ewald summation to rewrite

Eq. (11.58) in a form suitable for rapid numerical evaluation. It is useful to

define

= 2^

e m

11- Both / and k are

integers.

(11.59)

The final answer should have the form

N-Ì

I |2 N

\ "ô / v "

k=i

j^o ;

©)ί-^Σ^ΣΣ^. <>',o,

with

£I(JC)=

/°° — e~

. (11.61)

Find the functions

and G/^·.

6. Friedet model of J bands:

(a) Consider the tight-binding model, Eq. (8.67), and specialize to the case of

one dimension, with nearest-neighbor interactions. Write down the dispersion

relation for energy as a function of wave number, and from it find the density

of energy states £>(£).

(b) Friedel's model for electrons in a J band replaces the true three-dimensional

density of states D(E) with a function of the form

D(£)

~2^

Ö(£

+ W)Ö(W

£)

(1L62)

Estimate the relation between W and the coefficient of

the

hopping coefficient

t in Eq. (8.67) appropriate to an fee metal, assumed to have only nearest-

neighbor interactions, in three dimensions; set 2W equal to the bandwidth of

the tight-binding model.

ergy of the d band solids as a function of the number Z of electrons per atom

in the band.

318

Chapter 11. Cohesion of Solids

References

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Bernardes (1958), Theory of solid Ne, A, Kr, and Xe at 0°K, Physical Review, 112, 1534-1539.

A. E. Carlsson (1990), Beyond pair potentials in elemental transition metals and semiconductors,

Solid State Physics: Advances in Research and Applications, 43, 1-91.

Chemistry of Materials (1994), volume 6

M. L. Cohen and V. Heine (1970), The fitting of pseudopotentials to experimental data and their

subsequent application, Solid State Physics: Advances in Research and Applications, 24, 37-248.

A. H. Cottrell (1988), Introduction to the Modern Theory of Metals, The Institute of

Metals,

London.

G. Desiraju (1989), Crystal Engineering: The Design of Organic Solids, Elsevier, Amsterdam.

J. Emsley (1998), The Elements, 3rd ed., Clarendon Press, Oxford.

G. Grüner (1988), The dynamics of charge-density waves, Reviews of Modern Physics, 60, 1129-

1181.

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Physics: Advances in Research and Applications, 24, 250^4-63.

J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird (1954), Molecular Theory of Gases and Liquids,

John Wiley and Sons, New York.

H. Hotop and W. C. Lineberger (1975), Binding energies in the atomic negative ions, Journal of

Physical and Chemical Reference Data, 4, 539-576.

W. Hume-Rothery (1931), The Metallic State: Electrical Properties and Theories, Clarendon Press,

Oxford.

J. D. Joannopoulos and M. L. Cohen (1976), Theory of short-range order and disorder in tetrahedrally

bonded semiconductors, Solid State Physics: Advances in Research and Applications, 31,71-148.

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generalized gradient approximation, Physical Review B, 45,

293-13 297.

G. Kresse and J. Furthmiiller (1996), Efficiency of ab-initio total energy calculations for metals and

semiconductors using a plane-wave basis set, Computational Materials Science, 6, 15-50.

G. Kresse and J. Hafner (1994), Ab initia molecular-dynamics simulation of the liquid-metal-

amorphous-semiconductor transition in germanium, Physical Review B, 49, 14251-14269.

L. D. Landau and E. M. Lifshitz (1980), Statistical

Physics,

Part 1, 3rd ed., Pergamon Press, Oxford.

Liu, S. H. Garofalini, R. D. King-Smith, and D. Vanderbilt (1993), First-principles studies on

structural properties of /3-cristobalite, Physical Review Letters, 70, 2750-2753.

S. Nagasaka and

Kojima (1987), Interionic potentials based on the charge-transfer model for ionic

and partially ionic substances. I. Alkali halides, Journal of the Physical Society of Japan, 56,

408^114.

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

New York.

R. E. Peierls (1955), Quantum Theory of Solids, Clarendon Press, Oxford.

G. Pettifor (1987), A quantum-mechanical critique of the Miedema rules for alloy formation,

Solid State Physics: Advances in Research and Applications, 40, 43-92.

H. T. Philipsen and E. J. Baerends (1996), Cohesive energy of 3d transition metals: Density

functional theory atomic and bulk calculations, Physical Review B, 54, 5326-5333.

A. L. Roitburd (1978), Martensitic transformation as a typical phase transformation in solids, Solid

State Physics: Advances in Research and Applications, 33, 317-390.

J. H. Rose, J. R. Smith, R. Guinea, and J. Ferrante (1984), Universal features of the equation of state

of metals, Physical Review B, 29, 2963-2969.

H. Schlosser (1992), Cohesive energy-lattice constant and bulk modulus-lattice constant relation-

ships:

Alkali halides, Ag halides, Tl halides, Journal of Physics and Chemistry of Solids, 53,

855-856.

References

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H. Stillinger and T. A. Weber (1985), Computer simulation of local order in condensed phases of

silicon, Physical Review B, 31, 5262-5271.

R. Stumpf and M. Scheffler

(

1994),

Simultaneous calculation of the equilibrium electronic structure

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The code is available in source from the CPC program library.

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49-51.

12.

Elasticity

12.1 Introduction

The theory of elasticity describes the energy needed to deform a solid body, pro-

vided that the wavelength of the deformations is large in comparison with any

microstructure. The equations are largely derived through considerations of sym-

metry, depending in the end upon a small number of parameters that can be deter-

mined either from experiment or from atomic-level calculation.

12.2 Nonlinear Elasticity

Nonlinear elasticity describes reversible deformation of solid objects without as-

suming that the deformation is small. In order to describe deformation, one has to

decide that in a certain state a piece of material is not deformed. This corresponds

to the material sitting quietly without any external forces acting upon it, and in this

situation it is in the reference state. Describe locations of material points in this

reference state with the variable r. Now grab the solid, twist it, pull it, bend it.

Each material point originally located at ? is now at a new location s(r).

Before deformation After deformation

γ ?(?) = r + ü(r).

The theory of elasticity assumes that the new energy of the object depends

only on the change in distance of material points that were originally close to one

another. For example, one can think about point r and its nearby neighbor r + dr.

The original squared distance between these two points is dr

■

dr =

+ dr

+dr

After deformation, the new distance is

\s(r + dr)-s(r)\

^\—dr

+—dr

+ —dr

(12.2)

9? ds

aß

where g

aß

= -— · —. «and grange

over

(12.3)

Ora y, andz.

aß

The tensor g

ß is the metric tensor. The change in square distance caused by

deformation can be found from the

Lagrangian

strain

tensor

β ,

A |?(?+i/r)

—

?(r)|

—

\dr\ \ The factor of 1/2 is conventional. (12.4)

Σ \ ^

ß ~

δα

αβ dr

(12.5)

αβ αβ

321

Condensed Matter

Physics,

Second Edition

by Michael P. Marder

322

Chapter 12. Elasticity

When all distances between nearby points are unchanged, for example by moving

a body rigidly from one place to another or rotating it, the Lagrangian strain ten-

sor vanishes. Otherwise it keeps track how much material at each point has been

stretched in each direction.

12.2.1 Rubber Elasticity

Most solids are

stiff,

and deform irreversibly when stretched more than a fraction of

a percent, justifying some further simplifications of elastic theory that will occupy

succeeding sections. An exception to this general observation is rubber, which can

stretch to five or eight times its original length and then snap back to its original

form. Thus when using strain tensors to describe rubber, they appear in their full

nonlinear glory.

Vulcanized rubber is constituted from a densely intertwined network of poly-

mers that link together at occasional intervals, as shown in Figure 12.1. As de-

scribed in Section 5.5.8, polymer strands act like springs, and because the poly-

mers making up rubber are typically 350,000 units long and travel a few hundred

units between linking points, they can be extended a long way without undergoing

permanent damage.

One other experimental observation about rubber is crucial. All the low-energy

deformations keep the density of rubber fixed. Rubber can be sheared easily, and

it can be stretched, but if stretched in one direction it contracts in other directions

so as to keep the volume per molecule fixed. In this respect, rubber resembles an

incompressible fluid, such as water. This fact can be accounted for by modifying

the free energy of a polymer mixture given by Eq. (5.73), to read

T\j2 "4+Vßn

VC«

. . 1. (12.6)

Write the free energy in terms of the density n and volume V. The sum over j is taken

over all the N

polymers in the rubber. In Eq. (5.73) there is a term involving 1/DÎ

that is omitted here since the monomer density in rubber is so high that the argument

producing it is invalid.

The fact that rubber preserves density will follow if Bn and Cn

in Eq. (12.6) have

magnitude much larger than 1, and B is negative, so that the polymer mixture col-

lapses to a high density given by Eq. (5.77). Any deformations of the rubber that

involve changing the density n increase the energy much more than changes that

do not.

The basic experimental facts, then, are that the energy of rubber varies as the

square of how far it has been stretched. However, the energy is subject to the

constraint that the density of the rubber remain constant. In addition, since rubber

is isotropie, its energy should be invariant to changes of coordinate axes. Turning

now to the Lagrangian strain tensor Ε

β one asks how to construct a theory of

this sort. Suppose that the rubber is stretched uniformly by factors X

, X

, and

along the x, y, and z axes respectively. This stretch can be consistent with the