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Chapter 4 crystal binding

Outline• Classification of solids based on configuration of valence electrons• Crystals of inert gases• Ionic crystals• Covalent crystals• Metallic crystals

2

In chapter 1: Classification of solid according to geometrical symmetry7 Crystal systems and 14 Bravais lattices.

Another classification scheme in this chapter: based on physical properties, more specifically configuration of valence electrons.

This classification reveals natures of crystal binding.

Cohesive energy (ground state energy) of solids will be discussed.

Distinction between metals and insulators

Qualitative observations:

Insulators: electronic distribution is more concentrated in the vicinity of ion coresMetals: electronic distribution is less concentrated in the vicinity of ion cores

More rigorous criterion to distinguish metals from insulators has to be described in the reciprocal space (band structure theory)

3

Calculated radial atomic wave function for neon [1s22s22p6]

Calculated radial atomic wave function for sodium [1s22s22p63s1]

Very small overlap of 2s and 2p orbitals

Essentially no overlap of 2s and 2p orbitals

Enormous overlap of 3s orbitals

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Classification of insulators

• Noble gas crystals (molecular crystals)• Ionic crystals • Covalent crystals

Crystals of noble (inert) gases

Examples: neon, argon, krypton, and xenon

• Completely filled outermost electronic shells (stable closed shell) ns2np6

• Distribution of electron charge in the free atom is spherically symmetric• All crystalize in monatomic fcc Bravais lattice

The solid is held together by very weak forces, which originates from slightly distorted electronic configuration. These forces are known as van der Waals forces

Origin of van der Waals forces: oscillating dipoles

If the charge distribution is rigidly spherically symmetric, there is no electric field outside a neutral atom, and there is no electrostatic interactions between atoms

nuclei

Electron cloud

A dipole is induced when the charge distribution is distorted from being spherical

nuclei

electron cloud

+ -

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The dipole will polarize a neighboring atom, because if induces a electric field proportional to

+ - + -

The charge distribution are oscillating around the symmetric spherical distribution and creating temporary fluctuating dipoles.

The fluctuating atoms induce dipole moments in each other, and the induced moments cause an attractive interaction between the atoms

+ - + - + - + - + -

+ - + - + - + - + -

+ - + - + - + - + -

+ - + - + - + - + -

3r

p

6

Consider two atoms (1 and 2) separated by distance r.

Estimate of van der Waals interaction:

The average charge distribution in a single inert gas atom is spherically symmetric. However, according to the

fluctuating dipole model, at any instant there may be a net dipole moment. Suppose the instantaneous dipole

moment of atom 1 is .1pr

The electric field induced by is , where r is the distance from atom 1. 1pr

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1~r

pE

rr

The electric field will induce a dipole moment in atom 2 proportional to the field:3

1

2 ~r

pEp

αα=

Dipole interaction energy: . This interaction lowers the total energy of the pair of dipoles.6

2

1

3

12 ~r

p

r

pp α

This interaction energy falls rapidly with distance, and therefore is a very weak. This explains the low melting and boiling points of the condensed inert gases.

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A quantum mechanical treatment of Van der Waals-London interaction

+ - + -

spring

x2x1

R

Assumptions: positive charges are stationary and separated by distance R. Negative charges with mass m oscillates around equilibrium.

2

2

2

2

2

1

2

102

1

2

1

2

1

2

1Cxp

mCxp

mH +++=

2

0ωmC =

Coulomb interaction2

2

1

2

21

22

1xR

e

xR

e

xxR

e

R

eH

−−

+−

−++=

Rxx <<21 ,

3

21

2

1

2

R

xxeH −≈

Diagonalize the Hamiltonian )(2

121 xxxs += )(

2

121 xxxa −=

)(2

11 as xxx += )(

2

12 as xxx −=

Similarly )(2

11 as ppp += )(

2

12 as ppp −=

8

Total Hamiltonian:

+++

−+= 2

3

222

3

22

)2

(2

1

2

1)

2(

2

1

2

1aass x

R

eCp

mx

R

eCp

mH

2/1

3

2

/)2

(

±= m

R

eCω

+−±≈ L

2

3

2

3

2

0 )2

(8

1)

2(

2

11

CR

e

CR

eωω

+

−

+= K

2

3

2

3

2

0

2

8

12

2

11

CR

e

CR

ea ωω

+

−

−= K

2

3

2

3

2

0

2

8

12

2

11

CR

e

CR

es ωω

( )

2

3

2

00

2

3

2

0

2

8

1

2

8

22

2

1

2

1

−=

+

−=

+=

CR

e

CR

e

U as

ωω

ω

ωω

hh

Kh

h

6

2

3

2

0

2

8

1

R

A

CR

eU −=

−=∆ ωh Interaction varies as the minus sixth power of the separation of the two

oscillators. This is known as van der Waals interaction, also known as London interaction.It is this weak attractive potential that hold s the inert gas solid together.

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Repulsive interaction:

Repulsive interaction between two atoms is a result of Pauli exclusion principle

Pauli exclusion principle: Two electrons can not have all their quantum numbers equal.

As the two atoms are brought together, their charge distributions gradually overlap. The Pauli exclusion principle prevents multiple occupancy, and electron distribution of atoms with closed shells can overlap only if accompanied by the partial promotion of electrons to unoccupied high energy states of the atoms. Thus the electron overlap increases the total energy of the system and gives a repulsive contribution to the interaction.

↑s1 ↓s1

H H He+

↓↑ ss 11

Total electron energy: -78.98 eV

H H He+

Total spin zero

↑s1 ↑s1 ↑↑ ss 21

Total electron energy: -59.38 eV

Total spin one

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The form of the repulsive potential energy was chosen to be:12

R

B

The precise form of this potential should not be taken too seriously. It is nothing more than a simple way of taking into account of the fact that the repulsive potential is stronger than the attraction at small separations.

The total potential energy of two atoms at separation R: (Lennard-Jones potential)

−

=

−=

612

612

4

)(

RR

R

A

R

BRU

σσε

where A≡64εσ B≡12

4εσ

Calculation of equilibrium lattice constants:

The cohesive energy of an inert gas crystal is given by summing the Lennard-Jones potential over all pair of atoms in the crystal

−

= ∑

ij ijij

totRR

U

612

'4

σσε

( )

−

= ∑∑

j ijj ij

totRpRp

NU

6

'

12

'4

2

1 σσε

R nearest neighbor distance

Rpij distance between atoms i and j

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For fcc structure

1213188.1212' ≈=∑ −

j

ijp 45392.146' =∑ −

j

ijp close to 12

There are 12 nearest neighbor sites in fcc structure. The interaction is short range. The nearest neighbors contribute most of the interaction energy of inert gas crystals.

For hcp structure

13229.1212' =∑ −

j

ijp 45489.146' =∑ −

j

ijp

Equilibrium value 0RR = given by 0=dR

dU tot

For fcc or hcp:

0=dR

dU tot

( )( ) ( )( ) 045.14613.121227

6

13

12

=

−−

RRN

σσε

σ09.10 =R

Cohesive energy

( ) ( )

−

=612

45.1413.122RR

NU tot

σσε σ09.1=R

( )( )εNURRtot 415.2

0

−==

12

Ionic Crystals:

General features:

• Ionic crystals are made up of positive and negative ions

• The ionic bond results from the electrostatic interaction of oppositively charged ions

• Electronic configuration of ions: closed electronic shell

• Charge distribution are spherically symmetric

• The charged ion spheres are considered “impenetrable” (Pauli exclusion principle)

LiF: Li 1s22s F 1s22s22p5

Li+ 1s2 F- 1s22s22p6

“The impenetrability is a consequence of the Pauli exclusion principle and the stable closed-shell electronic configuration of the ions. When two ions are brought so close together that their electronic charge distributions start to overlap, the exclusion principle requires that the excess charge introduced in the neighborhood of each ion by the other be accommodated in unoccupied levels. However, the electronic configuration of both positive and negative ions is of the stable closed-shell ns2np6 variety, which means that a large energy gap exists between thelowest unoccupied levels and the occupied ones. As a result it costs much energy to force the charge distribution to overlap; i.e., a strongly repulsive force exists between ions, whenever they are so close together that their electronic charge distribution interpenetrate.” (A&M p379)

Ionic crystals are similar to inert gas crystals in that the electronic charge distribution are highly localized in the neighborhood of the ion cores. But unlike inert gas crystals, the constituent microscopic units are not neutral atoms, but charged ions. In addition the electrostatic forces between ions are longer range interactions compared fluctuating dipole attractive potential ininert gas crystals.

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Electronic charge distribution of NaCl crystals: nearly spherical, but distorted but slightly distorted.

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Many ionic crystals are realized by alkali halides (I-VII ionic crystals)

The positive ion (cation) is one of the alkali metals (Li+, Na+, K+, Rb+, or Cs+).

The negative ion (anion) is one of the halogens (F-, Cl-, Br-, or I-)

They all crystallize under normal conditions in the sodium chloride structure except for CsCl, CsBr, and Cs I, which are most stable in the cesium chlorice structure.

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Why the basic entities in ionic crystals are ions instead of neutral atoms?

Consider RbBr

Br-: the additional electron has a binding energy (electron affinity) of 3.5 eV. Or in other words, 3.5 eV is released in forming the ion.

Rb+: It costs 4.2 eV to strip off the extra electron in order to form the ion.

It appears that a pair of isolated rubidium atom and bromine atom together would have an energy 0.7 eV lower than the corresponding ions.

However, when the pair of ions are brought together, the energy of the pair is lowered by their electrostatic interaction.

Assume a distance between Br- and Rb+ r=3.4 Å. A pair of ions at this distance will have an Coulomb energy of-4.2 eV, which more than compenstate for the 0.7 eV favoring the atoms over the ions at large separations.

Electrostatic or Madelung Energy

Interaction energy between ions i and j

( )

ij

r

ijr

qeU ij

2/ ±= − ρλ

Central field repulsive potentialλ and ρ: empirical parameters

Coulomb potential

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∑=j

iji UU '

In a crystal composed of N molecules or 2N ions

itot NUU = neglecting surface effects

Rpr ijij ≡ Where R is the nearest neighbor separation in the crystal

=ijU R

qe

R 2

−−

ρλ

R

q

pij

21

±

nearest neighbors

otherwise

−==

−

R

qezNNUU

R

itot

2αλ ρ

Where z is the number of nearest neighbors of any ion

( )≡

±≡∑

j ijp

'α Madelung constant

At equilibrium: 0=dR

dU tot 02

2

=+−=−

R

qNe

Nz

dR

dUN

Ri α

ρλ ρ

λραρ zqeRR

/22

0

0

=−

−−=

00

2

1RR

qNU tot

ρα

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Covalent crystals

A chemical bond formed by sharing a pair of electrons is called a covalent bond.

• It is a strong bond• Strong directional properties• Electrons forming the bond tend to partly localized in the region between the two toms joined by the bond•The spins of the two electrons in the bond are antiparallel.

Influence of Pauli principle

• Pauli principle gives a strong repulsive interaction between atoms with filled shells• If the shells are not filled, electron overlap can be accommodated without excitation of electrons to high energy states and the bond will be shorter

Electrons with like spins tend to repel each otherElectrons with unlike spins tend to attract each other

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Diamond

C: 1s22s22p2 Partially promote to 1s22s2p3

One electron in each of the four orbitals. Bond is formed when electrons with opposite spins from four nearest neighbors occupy these orbitals.

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Metals

• High electrical conductivity. A large number of electrons are free to move about.• Weak binding.• Interaction energy between the positively charged ions and the negatively charged electron gas play an essential role.