lecture+6+mak crystalbinding

7/30/2019 Lecture+6+MAK CrystalBinding

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PHY 3201 FIZIK KEADAAN PEPEJAL

CHAPTER 3

CRYSTAL BINDING


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CRYSTAL BINDING

What holds a crystal together?

The attractive electrostatic interaction between the negatives

charges of electrons and the positive charges of the nuclei is

entirely responsible for the cohesion of the solids.

Magnetic forces have only a weak effect on the cohesion.

Gravitational forces are weak and negligible.


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The energy of the crystal is lower than that of thefree atoms by an amount equal to the energy

required to pull the crystal apart into a set of freeatoms. This is called the binding (cohesive)energy of the crystal.

NaCl is more stable than a collection of free Na

and Cl. Ge crystal is more stable than a collection of

free Ge.

Cl Na NaCl


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Cohesive energy

Cohesive energy of a crystal

The energy that must be added to the

crystal to separate it into neutral free atoms

at rest at infinite separation

There is a correlation between cohesive

energy and

-melting temperature

-bulk modulii


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Melting points


7/25PHY 3201 FIZIK KEADAAN PEPEJAL

Bulk modulii


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Cohesive energy

The observed differences between the forms of solids are

caused by difference in the distribution of the outermostelectrons and the ion cores.

In any situation, the net attractive force between the twoatoms are given by

natt rAU

As the atoms get closer, the will experience a repulsive

force given by

mrep r

BU


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This typical curve has aminimum at equilibrium

distance R0 R > R0 ;

the potential increasesgradually, approaching0 as R

the force is attractive

R < R0;

the potential increasesvery rapidly,

approaching at smallseparation.

the force is repulsive

R

r R

V(R)

0 R0

Repulsive

Attractive


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Cohesive energy

The repulsive interaction between theatoms arise generally from the electrostaticrepulsion of overlapping charge distributionand Paulis exclusion principle.

Pauli exclusion principleWhen electronic wavefunction of two atomsoverlap to a large degree, some of the electrons

must move to higher energy levels to avoidbeing in the same quantum state as the otherelectrons


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The Pauli principle prevents multiple

occupancy, and electron distributions of

atoms with closed shells can overlap only if

accompanied by the partial promotion of

electrons to unoccupied high energy statesof the atoms. Thus the electron overlap

increases the total energy of the system

and gives repulsive contribution to theinteraction


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H

1s

H

1s

He

1s 1s

Total spin zero

(a) Total

electron

energy:

-78.98 eV

H

1s

H

1s

He

1s 2s

Total spin one

(a) Total

electron

energy:

-59.38 eV

Two hydogen atom are pushed together until the protons are

almost in contact. Only the energy of the electron are taken

in the observation of the atomic He. The repulsive coulomb

energy of the two protons have been omitted.


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The net potential energy between two interacting atoms is the

sum of both the attracting and repulsive interaction betweenthe two atoms

mnrepatt r

B

r

AUUU

11

mn r

mB

r

nA

dr

dUF

The forces between these atoms is given by


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Potential energy and

force between twoatoms as a function of

interatomic distance r.

The opposite forces

are in equilibrium at

r=r0. The distance r0

is called the

equilibrium

separation


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Van der Waals-London

Interaction

First, we consider crystals of inert gases.The electron distribution in such crystals isvery close to that in free atoms. The noblegases such as neon (Ne), argon (Ar),

krypton (Kr) and xenon (Xe) arecharacterized by filled electron shells and aspherical distribution of electronic clouds inthe free atoms. In the crystal the inert gas

atoms pack together within the cubic fccstructure.


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Interaction

What holds atoms in an inert gas crystal

together? Consider two inert gas atoms

(1 and 2) separated by distance R. The

average charge distribution in a single

atom is spherically symmetric, whichimplies that the average dipole moment of

atom 1 is zero.


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However, at any moment of time there

may be a non-zero dipole moment

caused by fluctuations of the electronic

charge distribution.

Symmetrical

distribution

of electron charge

Asymmetrical

Distribution

(Changes with time)


Interaction

The dipoles can be

formed as a result

of unbalanced

distribution of

electrons in

asymettricalmolecules. This is

caused by the

instantaneous

location of a few

more electrons on

one side of the

nucleus than on

the other.

Therefore atoms or molecules

containing dipoles are attracted to

each other by electrostatic forces.


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Dipoles are created when positive and negative

charge centers exist.

-q

Dipole moment= =q.d

q= Electric charge

d = separation distance

+q

d

Van der Waals-

London Interaction


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Interaction According to electrostatics this dipole moment produces an

electric field, which induces a dipole moment on atom 2. The dipole moments of the two atoms interact with each

other. The energy is therefore reduced due to this interaction.

It is these forces which hold atoms together in inert gas solids


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Interaction

So we see that the coupling between the two

dipoles, one caused by a fluctuation, and theother induced by the electric field produced by

the first one, results in the attractive force,

which is called the Van der Waals force. The

energy of the interaction is

6R

A

U


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InteractionAt small separations a very strong repulsive forces cause

by the overlap of the inner electronic shells start todominate. This repulsive interaction can be fitted quitewell by the potential of the form B/R12, where B is apositive constant. Thus the total potential energy of twoatoms at separation Rcan be represented as

where 46A and 412B . This potential is known asLennard-Jones potential.

The force between 2 atoms is given by -dU/dR

612

4RR

U


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Lennard-Jones potential


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Equilibrium Lattice Constant

The cohesive energy of an inert gas crystal is given by

summing the Leonard-Jones potential over all pairs ofatom in the crystal. If there are N atoms in the crystal, the

total potential energy is

wherepijRis the distance between atom iand atomj,

expressed in terms of the nearest neighbour distance R.

The factor of occurs with the N to make sure we dontcount pairs


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If we take Utotas the total energy of the crystal, the

equilibrium value Ro is obtained by

The observed values of R0/ for the inert gas elements

are


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The cohesive energy of inert gas crystals at absolute zero

and at zero pressure can be obtained by substituting theabove equations

lecture+6+mak crystalbinding

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