chapter 3. crystal binding

7/28/2019 Chapter 3. Crystal Binding

1/38

Chapter 3Crystal Binding

Phys 175A

Dr. Ray Kwok

SJSU


2/38

4 basis categories


3/38

Molecular Bonds Introduction

To understand the crystal binding, one shouldunderstand how molecules bind together

The bonding mechanisms in a molecule arefundamentally due to electric forces

The forces are related to a potential energyfunction

A stable molecule would be expected at aconfiguration for which the potential energyfunction has its minimum value


4/38

Features of Molecular Bonds

The force between atoms is repulsive at very small

separation distances This repulsion is partially electrostatic and partially due to

the exclusion principle

Due to the exclusion principle, some electrons in overlappingshells are forced into higher energy states

The energy of the system increases as if a repulsive forceexisted between the atoms

The force between the atoms is attractive at largerdistances (e.g. due to shifted charge distribution,

induced dipole-dipole interaction)


5/38

Potential Energy Function

The potential energy for a system of twoatoms can be expressed in the form

r is the internuclear separation distance

m and n are small integers (usually)

A is associated with the attractive force

B is associated with the repulsive force

( )n m

A BU r

r r

= +


6/38

Graph U(x)

At large separations, theslope of the curve is positive

Corresponds to a net

attractive force (F = dU/dr)

At the equilibrium separationdistance, the attractive andrepulsive forces just balance

At this point the potential

energy is a minimum

The slope is zero (F=0)


7/38

Molecular Bonds Types

Simplified models of molecular bonding

include Ionic

Covalent

van der Waals

Hydrogen


8/38

Ionic Bonding

Ionic bonding occurs when two atomscombine in such a way that one or moreouter electrons are transferred from one

atom to the other Ionic bonds are fundamentally caused by

the Coulomb attraction between oppositely

charged ions (e.g. Na+ Cl)


9/38


10/38

Ionic Bonding, NaCl Example

The graph

shows the totalenergy of themolecule vs the

internucleardistance

The minimumenergy is at the

equilibriumseparationdistance

Binding energy = 4.2 eV


11/38

Ionic Bonding,final

The energy of the molecule is lower than

the energy of the system of two neutralatoms

It is said that it is energetically favorablefor the molecule to form

The system of two atoms can reduce its energy

by transferring energy out of the system andforming a molecule


12/38


13/38

Covalentbonds

Bonding can occurwithout outrightremoval or addition

of an electron. Inthese types ofbonds, the

connection occursthrough orbitaloverlap.


14/38

Wave Function Two Atoms Far Apart

Each atom has a wavefunction (1s1)

There is little overlapbetween the wave

functions of the two atomswhen they are far awayfrom each other

1 3

1( ) or a

s

o

r e

a

=


15/38

Wave Function Molecule

The two atoms are

brought closetogether

The wave functions

overlap and form thecompound waveshown

The probabilityamplitude is largerbetween the atomsthan on either side


16/38

Covalent Bonding, Final

The probability is higher that the electrons

associated with the atoms will be located betweenthem

This can be modeled as if there were a fixed

negative charge between the atoms, exertingattractive Coulomb forces on both nuclei

The result is an overall attractive force between the

atoms, resulting in the covalent bond


17/38

Van der Waals Bonding

Two neutral molecules are attracted to each

other by weak electrostatic forces called vander Waals forces (typically 0.1 eV)Atoms that do not form ionic or covalent bonds are

also attracted to each other by van der Waalsforces

The van der Waals force is due to the fact

that the molecule has a charge distributionwith positive and negative centers at differentpositions in the molecule


18/38

Van der Waals Bonding, cont.

As a result of this charge distribution, the

molecule may act as an electric dipoleBecause of the dipole electric fields, two

molecules can interact such that there is anattractive force between them

Remember, this occurs even though themolecules are electrically neutral

e.g. Liquid nitrogen molecules N2


19/38

Types of Van der Waals Forces

Dipole-dipole force

An interaction between two molecules eachhaving a permanent electric dipole moment

Dipole-induced dipole forceA polar molecule having a permanent dipole

moment induces a dipole moment in a nonpolarmolecule


20/38

Types of Van der Waals Forces, cont.

Dispersion force An attractive force occurs between two nonpolar molecules

The interaction results from the fact that, although theaverage dipole moment of a nonpolar molecule is zero, the

average of the square of the dipole moment is nonzerobecause of charge fluctuations

The two nonpolar molecules tend to have dipole momentsthat are correlated in time so as to produce van der Waals

forces


21/38

Hydrogen Bonding

In addition to covalent bonds, a hydrogen

atom in a molecule can also form ahydrogen bond (weak 0.5 eV)

Using water (H2O) as an exampleThere are two covalent bonds in the molecule

The electrons from the hydrogen atoms are more

likely to be found near the oxygen atom than thehydrogen atoms


22/38

Hydrogen Bonding H2O cont.

This leaves essentially bare protons at thepositions of the hydrogen atoms

The negative end of another molecule cancome very close to the proton

This bond is strong enough to form a solid

crystalline structure


23/38

Hydrogen Bonding, Final

The hydrogen bond isrelatively weak comparedwith other electrical bonds

Hydrogen bonding is acritical mechanism for thelinking of biological

molecules and polymers DNA is an example


24/38

Bonding in Solids

Bonds in solids can be of thefollowing types

Ionic

Covalent

Metallic


25/38

Ionic Bonds in Solids

The dominant interaction between ions isthrough the Coulomb force

Many crystals are formed by ionic

bonding (I-VII, II-VI) e.g. NaCl Ions are closed electronic shells. [e.g.

LiF, Li (1s22s) becomes Li+ (1s2), F

(1s2

2s2

2p5

) becomes F

(1s2

2s2

2p6

)]


26/38

Electrostatic Energy

The net effect of all the interactions is anegative electric potential energy

is a dimensionless number known as the

Madelung constantThe value of depends only on the crystalline

structure of the solid

2

attractive e

eU

k

r=


27/38

Total Energy in a Crystalline Solid

As the constituent ions of a crystal arebrought close together, a repulsive forceexists

The potential energy term B/rm accounts forthis repulsive force

This repulsive force is a result of electrostaticforces and the exclusion principle


28/38


29/38

Ionic Bonds, NaCl Example

The crystalline structure is shown (a)

Each positive sodium ion is surrounded by six negative chlorine ions (b)

Each chlorine ion is surrounded by six sodium ions (c)


30/38

Na-Cl

Na + 5.14 eV Na+ + e-

(ionization energy = 5.14 eV)

Cl + e- Cl

+ 3.61 eV

(electron affinity = 3.61 eV)

Na+ + Cl

NaCl + 7.9 eV

(cohesive energy = 7.9 eV)

i.e. the energy per molecule of NaCl is(7.9 5.1 + 3.6) = 6.4 eV lower thanthe energy of separated neutralatoms.


31/38

More properties of Ionic Crystals

They form relatively stable, hardcrystals

They are poor electrical conductors

They contain no free electrons (filled shells)Each electron is bound tightly to one of the

ions

They have high melting points


32/38

Properties of Solids with Covalent Bonds

Properties includeUsually very hard

Due to the large atomic cohesive energies

High bond energiesHigh melting points

Good electrical conductors


33/38

More about Covalent Bonds

Share electrons (usually 2 e-)

Directional (along orbital that share e-)

Electrons tend to localize between atoms

Prefer anti-parallel spins (Pauli)Act as glue to atoms

No clear cut range to be ionic or covalent


34/38

Cohesive Energies for Some

Covalent Solids


35/38

Covalent Bond Example

Diamond

Each carbon atom in a diamond crystal is covalentlybonded to four other carbon atoms

This forms a tetrahedral structure


36/38

Metallic Solids

Metallic bonds are generally weaker thanionic or covalent bonds

The outer electrons in the atoms of a metal

are relatively free to move through thematerial (high mobility)

The number of such mobile electrons in a

metal is large (high carrier density)High electrical conductivity (& thermal)


37/38

Metallic Solids, cont.

The metallic structure

can be viewed as asea or gas of nearlyfree electronssurrounding a lattice of

positive ions The bonding

mechanism is theattractive force betweenthe entire collection ofpositive ions and theelectron gas


38/38

Properties of Metallic Solids

Light interacts strongly with the free

electrons in metalsVisible light is absorbed and re-emitted quite

close to the surface (reflective)

This accounts for the shiny nature of metalsurfaces (screening, plasma frequency)

High electrical conductivity

More discussions on metal later.

chapter 3. crystal binding

Documents