lecture3 pol mechnism

8/7/2019 Lecture3 Pol Mechnism

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Electronic polarization

343

V RT!

For calculating the effect of electronic polarization,

We consider an idealized atomwith perfect spherical

symmetry. I

t has a point likecharge + ze in the nucleus, andThe exact opposite charge zehomogeneously distributed inthe volume of the atom, whichis

with R= radius of the atom

Ques: Density of electrons?

Ques: Lorentz force = coulomb force =?


2/20

The density of electrons (Charge/Volume of atom) is then

In the presence of an electric field E;

Lorentz force: F1=-zeE acts on charges.

The positive charge in the nucleus and the center of thenegative charges from the electron "cloud" will thusexperience forces in different direction and will become

separated.We have the idealized situation shown in theimage in last slide.

33

4

z e

RV

T!


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The attractive force (between opposite charges)

F2 is given by

2 2

0

2

30

3 3

3 3

(charge of nucleus)(negative charge encolesd in sphere of radius d)

4

( )

4

negative charge encolesd in sphere of radius d

4 3 d= ze

4 3 R

Fd

ze

dR

zed

R

TI

TI

T

T

!

!

!

Q

* Q= charge density x volume

--

-


4/20

Equating F1 and F2 we have equilibrium distance dE as

Therefore the induced dipole moment is

As polarization is dipole moment per unit volume so

And Electronic Polarizability:

3

0

3

0

4 and

P4

P NR E

NR

E

TI

G T

I

!

!

3

04

Ezed R EQ TI! !

304

E

R Ed

ze

TI!

3

04

.

R

E

e

TIE

EQ

!

! Where e is defined byElectronic polarizability.Ques: What is the Dimension

of Polarizability?


5/20

P = NEeE = Io(Ir-1)E

This gives

Io(Ir-1) = NEe

Here N is the number of dipoles per unit volume.

There are a number of interesting points about the

last result:1. We have justified the "law" of a linear relationship

between E and P for the electronic polarizationmechanism (sometimes also called inducedpolarization).

2. We can easily extend the result to a mixture of different atoms: All we have to do is to sum over therelative densities of each kind of atom.


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3

0

3

0

4 and

= 4

P NR E

NRE

TI

TI

!

!

We can easily get an order of magnitude for .

Taking a typical density ofN

x 10

25

m

3

andR6 x 10 12 m, we obtain G8.14 x 105,

and r=1. 000 0814

Note:

1. The values ofmolecular dipole moments are usually

expressed in Debye units(3.3 x10-30 Cm). The Debye unit,

abbreviated as D, equals 10-18 electrostatic units (e.s.u.).

2. The permanent dipole moments of non-symmetricalmolecules generally lie between 0.5 and 5D.

3. In the case ofpolymers and biopolymers one can meet

much higher values ofdipole moments ~ hundreds or

even thousands of Debye units.


7/20

2.Ionic polarization

Consider a simple ioniccrystal, e.g. NaCl.

Each Na+ - Cl pair is a naturaldipole, no matter how you pair up two atoms.

The polarization of a givenvolume, however, is exactlyzero because for every dipolemoment there is a neighboring

one with exactly the samemagnitude, but opposite sign.

Note that the dipoles can notrotate; their direction is fixed.


8/20

In an electric field, theions feel forces in

opposite directions.

For a field acting asshown, the lattice distortsa little bit (hugelyexaggerated in thedrawing)

The Na+ ions moved a bitto the right, the Cl ionsto the left.

The dipole momentsbetween adjacent NaCl -

pairs in field direction arenow different and there isa net dipole moment ina finite volume now.

Ques: How large is d?


9/20

How large is d?

The force F1 increasing the distance is given by

F1 = q E

with q = net charge of the ion.

The restoring force F2: ford


10/20

Other relations

2

0

2

0

r

and

NqP .

Find out and

q EYd

E

Yd

Q

I

!

Exercise:


11/20

Summary: Ionic polarization

0

qEd

Yd!

2

02

0

r

and

q= .

ind out and

q E

Yd

E

Yd

Q

I

!

Net Displacement:

In the presence of E:

Dipole moment:

Polarization:

00

2

0 Yd

Nq

E

P

IIG !!

And r= (G+1)


12/20

Orientation Polarization In the case of Orientation polarization (sometime called Dipolar

polarization) we have a material with built-in dipoles that areindependent of each other, i.e. they can rotate freely- in sharp contrastto ionic polarization.

The prime example is liquid water, where every water molecule is a little dipole that can have any orientation

with respect to the other molecules.

Moreover, the orientation changes all the time because the molecules

moves! (O

rientation polarization for dielectric dipoles thus is pretty muchlimited to liquids)

It is like a snapshot (2D) with a very, very short exposure time. A fewnanoseconds later the same piece of water may look totally different indetail, but pretty much the same in general.

In a three-dimensional piece of water the blue and green circles would nothave to be in the same plane; but that is easy to imagine and is difficult to

draw


13/20

Orientation polarization: visualize it

in water Water molecules that form natural dipoles.

because the negatively charged oxygen atom and the two positivelycharged H - atoms have different centers of charge.

Each molecule carries a dipole moment which can be drawn as a vectorof constant length.

If we only draw a vector denoting the dipole moment, we get - in two

dimensions - a picture like this:

The sum of all dipole moments will be zero, if the dipoles arerandomly oriented.

If we now introduce a field E, the dipoles would have a tendency toturn into the field because that would lower their energy.


14/20

The dipoles are not sitting still, but moving around and rotating allthe time - because they contain thermalenergy and thus also someentropy.

The electrical field induces average orientation in field direction. Most of the time an individual dipole points in all kinds of directions.

The "real" picture (in the sense of a snapshot with a very short exposuretime) looks like this:

The orientation of all dipoles is just a little bit shifted so that anaverage orientation in field direction results. In the picture, the effectis even exaggerated!

With out FieldW

ithF

ield


15/20

Energy of a dipole in Electric field:

Consider a Polar moleculewhich caries a permanent

dipole moment (qd, like

water molecule) is placed in an

electric field E.

The Potential energy is givenby;

U= -.E=- Ecos()

E

+q(1)

(2)-q

d

cos( )EUkT kT

e e

Q U

!

In thermal equilibrium the relative number of molecules with the potentialenergy Uis proportional to (Boltzmann statistics)


16/20

Number ofdipoles in a solid angle:

Let n() be the number of dipoles per unit solid angle at

, we have

The number of dipoles in a solid angle d;

co s ( )

0( ) .

E

k Tn n eQ U

U !

0

cosexp( )2 sin

En d

kT

Q UT U U

Note:Where d;! r.dS/r3

Can be calculated

as follows:

********************************************************************************************


17/20

Average Dipole Moment The average dipole moment in an electric field is the net moment

of the assembly divided by the total numberof dipoles

Therefore,

00

00

exp( cos / )2 sin ( cos )

exp( cos / )2 sin

n E kT d

n E kT d

T

T

Q U T U Q U UQ

Q U T U U "!

cosE xk

Q U !where and E akQ !

L(a)L(a)is calledLangeven functionLangeven function

cos

0

cos

0

1cos sin

2 1

1sin

2[ ]1 1 1

cot ( ),[ ]

aE

xk

a

aE

xk

a

x x a a a

a

x a a a

a

e d e xdx

ae d e dx

xe e e eanha L a

a e e e a a

T Q U

T Q U

U U UQ

QU U

"! ! !

! ! ! !

Coth(a)


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....453

1

)(

3

!

aa

aaCoth

kT

Ea

where

aPP

aNN

a

so

vv

.

)(.

)(..

)(.

Q

QQ

QQ

!

!

!

!Where Ps is saturation polarization and

Po is orientation Polarization.

y=1/3a

0 1 2 4 5 6 7 80.0

0.2

0.4

0.6

0.8

1.0 y=1

y=L(a)

a

y

Langevin

In Fig. the angeven unction L(a)L(a) is

plotted against aa.

L(a)L(a) has a limiting value 11, which was to

be expected since this is the maximum o

coscosUU..

Case I: when a is very high : at Low Temp;

a>>1 L(a)=1, Po = Ps

Case II: when a is very ow: at high Temp;

a


19/20

TkT

Hence

ENP

and

kT

ENP

oo

voo

v

o

1

3

3

2

2

w!

!

!

EQ

E

E

QOrientation Polarization

Orientation Polarizability

OrDipolar Polarizability

Note: Derivation shows that tendency o the extended ield toalign the dipoles is contracted by thermal motion resulting in a

decreasing value o Po with increasing T


20/20

Total Polarizability of Polyatomic Gas:

P = Pe+ Pi + Po

P = NEe + NEi + NEo= 0 (r - 1) E

Hence

(r - 1) = =N./ 0

Where is called Total Polarizabilty

kie

3

2

QEEE !

Is knownas Langevin- DebyeEquation

lecture3 pol mechnism

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