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Chapter 14. Linear Dielectric Properties. Area under curve. Relative dielectric constant …. Very important. It is a measure of how much charge a solid can store relative to vacuum. In general. D =    P. where D is the displacement (C/m 2 ) E is the applied electric

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Linear dielectric properties

Chapter 14

Linear Dielectric Properties



Linear dielectric properties

Relative dielectric constant …. Very important

It is a measure of how much charge a solid

can store relative to vacuum


Linear dielectric properties

In general

D = P

where D is the

displacement (C/m2)

E is the applied electric

field, (V/m) and P is the

polarization (C/m2)

of your material.

In vacuum, P = 0

and D = 



Linear dielectric properties

Something happens in the

solid that allows the parallel

plate capacitors to store more

charge.

That something is called

polarization

P = qd

Thus understanding polarization is the key to

understanding dielectric properties.


Linear dielectric properties

Electronic polarization

Pe

Ionic polarization

Pi

Dipolar or orientation

polarization

Pdip

Convince yourself that when E is applied the center -ve charge

is no longer coincident with the center of +ve charge.


Linear dielectric properties

N = number of diploes/m3

polarizability, which is an ionic/

atomic property.

Only valid for dilute gases or when Eapplied = Elocal

Only valid for cubic symmetry

but used in many situations


Most important equation in this chapter
Most Important Equation in This Chapter

  • Why?

  • Because it is link between micro and macro…

  • Always recall that it is an approximate expression and if it agrees with experiment it is because you were born under a lucky star

  • and your mother loves you ....


Linear dielectric properties

4 Fundamental Polarization Mechanisms in Solids

Electronic Polarization

Ionic Polarization

Dipolar Polarization ( linear)

is also referred to as orientational.

Space charge - occurs at electrodes and is very

important in electrochemistry… Not discussed in this

class…


The effect of frequency
The Effect of Frequency

Purple = V

Red = q

Blue = I

When charges

are in perfect

sync with Eapp

you have a

perfect dielectric,

with no losses.

If charges are in

phase what is

the current doing?

Hint:

I = dq/dt

In a perfect

dielectric the current

?? the voltage by ??


Ideal vs real dielectric
Ideal vs. Real Dielectric

  • Charge in phase with voltage; since i = dQ/dt, then current, Ichg, is 90° out of phase. This current is called a displacive current and does NOT lead to energy loss.

  • In a real dielectric, there is energy loss.

  • To take these into account:

G = 1/R =conductance

Power dissipation, W/m3


A digression on i
A digression on i

  • It is a shorthand notation that - in the context of dielectric and optical properties you have two components..

  • a brilliant tool to solve DE and describe various physical phenomena..

  • When you see i, then your first thought should be: there must be some form of energy loss somewhere in this system…

  • The last jewel:


Vectorial sum
Vectorial Sum

Some charges are in

phase with V and result

in a charging current - but

no loss.

Others are 90° out of

phase and lead t energy

dissipation.

Itot = Ichg + I loss

Tan = I loss/Ichg


Measuring dielectric properties
Measuring Dielectric Properties

If  is 90°

then you have

a perfect

dielectric

with no

losses.

If  is 0°, then

you have a

perfect

conductor and

no capacitance.

You use a something called a lock-in analyzer ..


How can you measure k
How can you measure k’’

  • In principle, one way would be to simply measure the temperature change in the system…

  • If k” is zero there is no energy loss in the system and thus no heat increase.


Electronic polarization
Electronic Polarization

Assumptions:

i) Applied field = local field….

ii) The electrons are collectively attached

by a spring to the nucleus, with a natural frequency of

vibration of o and a spring constant = So.

iii) Recall:


Electronic polarization1
Electronic Polarization

Newton’s Law

F = ma

Zi = atomic number of atom/ion

f is a friction factor, and is thus related to k’’


Resonance a thing of beauty
Resonance; a thing of beauty

Any examples from real life?


Linear dielectric properties

Region 1

DC limit, viz.  = 0 and

Region 2

Near resonance:  =  and ke increases dramatically.

and would go to infinity, if there was no friction, i.e. if f =0

Region 3

 <<  then charges cannot follow the field and

drops out. Then ke goes to 1.



What determines k e
What determines, ke’

Radius of atom


Linear dielectric properties

Very important result… Electronic polarizability is

Proportional to the volume of an atom or ion.

1- Size

2 - Charge

3 - Presence

of d-electrons

which are

less

shielding.


Linear dielectric properties

Quantum mechanics tells us:

Other simplification that local E = Applied E, does not

change the physics, but only resonant frequency.


Ionic polarization
Ionic Polarization nucleus attached with a spring…. Life is more complicated.


Linear dielectric properties

In DC limit nucleus attached with a spring…. Life is more complicated. = 0 and k”= 0 then:

Nion = number of ion pairs/m3


What determines k ion
What determines k nucleus attached with a spring…. Life is more complicated.ion

What is

r0???


Linear dielectric properties

r nucleus attached with a spring…. Life is more complicated.o


Linear dielectric properties

- nucleus attached with a spring…. Life is more complicated.

+

P

ABO3

Dipolar Polarization

Teams teams… teams ..


Dipolar polarization
Dipolar Polarization nucleus attached with a spring…. Life is more complicated.

Dipole moment = q


Linear dielectric properties

In English: What determines k’ nucleus attached with a spring…. Life is more complicated.dip ??

charge on the dipole is a big one

density of dipoles is another

jump distance… another big one

and finally, T ……. and here

comes Mr. Entropy!


Debye model
Debye Model nucleus attached with a spring…. Life is more complicated.

  • At high frequency,

  • As freq goes to 0,


Debye equations
Debye Equations nucleus attached with a spring…. Life is more complicated.

For DC what

is k’dip??

How about at

very high 

For DC what

is k’’dip??

How about at

very high 

For DC what is ??

How about at very high 


Relaxation
Relaxation nucleus attached with a spring…. Life is more complicated.


Temperature dependence
Temperature nucleus attached with a spring…. Life is more complicated.Dependence


Linear dielectric properties

Total polarization nucleus attached with a spring…. Life is more complicated.

P = Pe + Pi + Po

With

increasing

frequency

you tend to

lose the

various

mechanisms

in order

shown.


Dielectric loss
Dielectric Loss nucleus attached with a spring…. Life is more complicated.


Slight digression what is p v for dc conditions does anybody recognize the expression
Slight digression: nucleus attached with a spring…. Life is more complicated.What is Pv for DC conditions?Does anybody recognize theexpression?


Dielectric breakdown
Dielectric Breakdown nucleus attached with a spring…. Life is more complicated.

Intrinsic: That’s when the

electrons go ballistic or

postal. :-)

Thermal Breakdown:

Lossy dielectric, leads to

T increase - leads to

more current - leads to more

Heat - lead to more current

leads to death of capacitor.


Worked example

Worked Example nucleus attached with a spring…. Life is more complicated.

Note # of ion pairs is also equal to # of each ion individually

Consider CsCl: Lattice parameter = 0.412 nm

Cs+: e = 3.35x10-40 Fm2 Cl-: e = 3.4x10-40 Fm2

Mean ionic polarizability per ion pair = 6x10-40 Fm2

Estimate the dielectric constant of CsCl at low and optical frequencies.

_______________________________________________________

If you solve for k’ you get 7.56.


Insulators and capacitors
Insulators and Capacitors nucleus attached with a spring…. Life is more complicated.

  • For capacitor functions:

    • k’ should be maximum of minimum??

    • k” should be low or high??

  • For insulator functions:

    • k’ should be maximum of minimum??

    • k” should be low or high??


How about at optical frequencies
How about at optical frequencies? nucleus attached with a spring…. Life is more complicated.

Solving for k’e gives: 2.71.

Experimental values are: 7.2 and 2.62, respectively.

Moral of the story: If you want to use CsCl as a window

what is k’??

How about if you want to use it as a capacitor, then what?


Linear dielectric properties

MP of compound nucleus attached with a spring…. Life is more complicated.

Effect of dipolar polarization on k’.


Linear dielectric properties

Capacitors and Insulators nucleus attached with a spring…. Life is more complicated.

Material Dielectric constant

Vacuum 1 (by definition)

Air 1.00054

Polyethylene 2.25

Paper 3.5

PTFE (Teflon(TM)) 2.1

Polystyrene 2.4-2.7

Pyrex glass 4.7

Rubber 7

Silicon 11.68

Methanol 30

Water (20°) 80.4

Barium titanate 1200


Summary of linear dielectrics
Summary of Linear Dielectrics nucleus attached with a spring…. Life is more complicated.

  • Polarization - separation of charge - is key

  • Electronic polarization = rapid, T indep., low k’ and occurs in ALL solids, liquids and gases.

  • Ionic polarization = rapid, T indep.- low and need ions!

  • Dipolar polarization: T dep. - intermediate k’; requires permanent dipole!

  • ke and kion examples of resonance; kdip e.g. of relaxation: difference is in restoring force.

  • Charges in phase with voltage give rise to k’. Analogy with elastic loading

  • Charges 90° out of phase with voltage give rise k” and dissipation of energy.


From here on optional
From here on optional… nucleus attached with a spring…. Life is more complicated.

  • Have fun..


Linear dielectric properties

Effect of Frequency nucleus attached with a spring…. Life is more complicated.

With

increasing

frequency

you tend to

lose the

various

mechanisms

in order

shown.

Total polarization

P = Pe + Pi + Po


Ferroelectric ceramics
Ferroelectric Ceramics nucleus attached with a spring…. Life is more complicated.

Ferroelectric Ceramics are dipolar below Curie TC = 120ºC

  • cooled below Tc in strong electric field - make material with strong dipole moment

Fig. 18.35, Callister 7e.


Linear dielectric properties

As BaTiO nucleus attached with a spring…. Life is more complicated.3 is cooled below 120 °Cit goes from cubic to tetragonalspontaneously.The tetragonal unit cellhas a permanent dipole moment that is huge,which is why most capacitors today aremade from BaTiO3or related compounds.


Piezoelectric materials
Piezoelectric Materials nucleus attached with a spring…. Life is more complicated.

Piezoelectricity – application of pressure produces current

at rest

compression induces voltage

applied voltage induces expansion

Application of voltage produces dimensional changes


Summary ii
Summary II nucleus attached with a spring…. Life is more complicated.

• Solids that do not conduct electricity are called

dielectric solids.

The relative dielectric constant, ’ is a measure of the

charge storing capacity of a material relative to vacuum.

The 3 most common polarization mechanisms are:

Electronic: occur in all solids

Ionic: Only occurs in ionically bonded solids

Orientation: Where permanent dipoles orient

in the applied electric field.

The first two are rather small, the third one is the most

important and includes ferroelectric solids.

Piezoelectric solids can convert mechanical energy

to electrical energy and vice versa.