Transcript polarization

MATERIAL SCIENCE
2012-2013
What modules will be taught?
1.
2.
3.
4.
5.
6.
Dielectric materials
Magnetic materials
Polymers and Ceramics
Super-conductivity
Optical Materials
Thermoelectric materials
***Notable exception, we will not teach
Semiconductor materials******
Books used for this course
1)
2)
3)
4)
A Text book of solid state physics by S O Pillai
Material Science and engineering an Introduction by
W D Callister Jr
Science of Engineering Materials by Srivastava and
Srinivasan
Elements of Material Science and Enginnering by
Lawrence H Van Vlack
BASICS OF DIELECTRIC
MATERIALS
Basic Questions
• What is a dielectric?
• Why there is any electrical effect if the
insulators do not conduct electricity?
• Why should a field induce a dipole moment
in an atom if the atom is not a conducting
sphere?
What is a dielectric?
1.
A non conducting or insulating substance that can
sustain an electric field but does not conduct electric
current.
2.
Or an electrical insulator, a material with a low
(compared
with
that
of
a
metal)
electrical conductivity.
3.
Most generally, a dielectric is an insulator, a
substance that is highly resistant to flow of electric
current.
Layers of such substances are commonly inserted
into capacitors to improve their performance, and the
term
dielectric refers specifically to this
application.
4.
Can insulator be affected by electric
field?
• Faraday’s experimental observation: Capacitance of a
capacitor is increased when an insulator is placed between
the plates, the capacitance is increased by a factor  if the
insulator completely fills the space between the plates.
•  depends only on the nature of the insulating material and
is called dielectric constant
Q: What should be the dielectric constant of vacuum?
Consider a parallel plate capacitor; then
Capacitance
Charge
C
0 A
d
Q  CV .
Where
A= Area of plates,
d = Plate separation
C= Capacitance
Q= Charge on plate
V= Voltage difference
Now if we put a piece of insulating material like glass between the
plates, we find that the capacitance increases. That means, the
voltage decreases for the same charge.
But voltage difference is the integral of the electric filed across the
capacitor, therefore we conclude that, electric field is reduced
even though the charges on the plates remain unchanged.
Why should a field induce a dipole moment in an atom if
the atom is not a conducting sphere?
• Consider a single atom.
• For spherically symmetric system; center of
gravity of negative charges (electron cloud)
coincides exactly with the location of the nucleus.
Atom is unpolarised.
• If we now apply an electrical field, the centers of
charges (+ve and –ve) will be separated. The
electron cloud will be pulled in the direction of the
positive pole of the field, the nucleus to the
negative one. The atom will be polarized.
The effect of electrical field is that:
1. It induces electric dipoles in an unpolarized material
2. and tries to align them in the field direction.
3. The total effect of an electrical field on a dielectric material
is called the polarization of the material.
Review of some basic formula
1. Electric dipole:
2. Dipole Moment:


p  qr
3. Torque on the dipole
exerted by an E-field
 
  pxE    pE sin 
 
V   p.E   pE cos 
4. Potential energy of dipole
in an E-field
  0, V   pE
   ,V  pE
DKR-JIITN-2010-MS
5. Polarization: Defined as dipole moment per unit volume.
If the number of dipoles per unit volume is N, and if each has
moment p then polarization is given as (assuming that all the
dipoles lie in the same direction)


P  Np
Example: Suppose there are 3.34x1028 molecules per unit
volume of water each having dipole moment 6x10-30 C-m.
Solution: If all dipoles are oriented parallel to each other
then ‘Polarization’
P = 3.34x1028 x 6x10-30
= 0.2004 C/m2
DKR-JIITN-2010-MS
6. ELECTRIC FLUX DENSITY (Dielectric displacement) AND
POLARIZATION
E0



Electric fluxdensity  D   0 E0  P
Where, D 
q
= Electric flux density*
A
And P is called Polarization
*According to Gauss law,

 
Q
E.dA  encl
0
Polarization results in a reduction of the field inside the
dielectric medium.


 
Further, D  E   E  P
0

 


  0 r E   0 E  P   0 E( r 1)  P


P
Where,  r 
 ( r  1)    
0
0E
Here  is known as electric susceptibility and r is known as
relative dielectric constant of the medium.
POLARIZABILITY
Polarization of a medium is produced by field therefore, it is
reasonable to assume that,


p  E
Here ‘’ is known as polarizability of the
molecule representing dipole moment per
unit applied electric field


The polarization can now be written as, P  NE
Thus,

 


D   0 E  P   0 E  NE

N 
 D   0 (1 
)E


But, D   0 r E
0

N 
  0 r E   0 (1 
)E
N
  r  1
0
0
In all above expressions, N can be expressed in terms of
density , molar mass M of the material and Avogadro's
number NA as
N 
N A
M
Thus dielectric constant can be written as:
N
r  1

0
N A
r  1 (
)
 0M
However, experiments show that though above equations
hold good in gases but not for liquids and solids i.e. in the
condensed physical systems.
Section II : Summary
• Dipoles in solid dielectrics;
Polarization.
• Polarization is dipole
moment per unit volume:
• A relation between E & P:
• Connection between the
Polarization P and the
Electrical Displacement D
• Polarizability
N
  r  1
0
p= q·E
P = p· N V
P  0 E

 
D  0E  P
or D = ε0 (+1) E
p =E  =P / N E
V
Experiments show that last equation hold good in gases but not
for liquids and solids i.e. in the condensed physical systems. So
we need some corrections in the formulae for solids.
Local Field and Clausius - Mosotti
Equation
• Here, we are looking the effect of external field
on atoms and molecules in a solid or liquid
system.
• What an atom "sees" as local electrical field or
the local field Eloc or EL to be the field felt by
one particle (mostly an atom) of the material.
• we may express EL as a superposition of the
external field E0 and some field Emat introduced
by the surrounding material of an atom.
• EL = E0 + Emat
CLAUSIUS MOSOTTI RELATION
LOCAL FIELD


Eloc  E0  E1  E2  E 3
E0 = External field
E1 = Field due to polarization charges
lying on the surface of the sample.


E0

E1

E2 = Field due to polarization charges
lying on the surface of Lorentz sphere.
E3 = Field due to other dipoles lying within
the Lorentz sphere.
Lorentz
sphere

E2
Central dipole
Calculation of various fields:
Depolarizing field E1:
E1  
P
0
This field depends on the geometrical shape of the external
surface. Above equation is for a simple case of an infinite slab.
Field for a standard geometry is given as
NP
E1  
0
Here N is known as depolarizing factor. The values of N for
other regular shapes are given below:
Shape
Sphere
Thin slab
Thin slab
Cylinder
Cylinder
Axis
any
normal
in plane
Longitudinal
Transverse
N
1/3
1
0
0
½
Calculation of E2:
Surface area dA of the sphere lying
between  and +d is given as
dA  2r 2 sin d
Charge on the surface dA would be
dq  P cos  (2r 2 sin d )
Field due to this charge at the
centre of the sphere would be
dE

dE 
dq
40 r 2
Field in the direction of applied field would be
dE2  dE cos  
dq cos 
40 r 2
Field due to charges on the entire cavity thus would be,

E2   dE2
0

dq cos 

2
4

r
0
0


0
P 2r 2 sin  cos 2 d
40 r 2
 E2 
P
3 0
Calculation of E3:
The field due to other dipoles in the cavity may be calculated
by using the equation
 
2 
1 3( p.r )r  r p
E
40
r5
The result depends on crystal structure of the solid under
consideration. However for highly symmetrical structure like
cubic it sum up to zero. Thus
E3  0
(In other structure E3 may not vanish and it should be included
in the equation).
Thus Eloc would be


Eloc  E0  E1  E2  E 3

 Eloc

P


P
 E0  
 0 3 0


2P
 E0 
3 0

 P
E
3 0
Eloc = EL= Lorentz field, E is known as Maxwell field.
Now the polarization would be given as


P  NEL



 P
N P

N

E

 N ( E 
)
3 0
3 0


N
 P (1 
)  N E
3 0




N E
N
Again P (1 
)  N E  P 
N
3 0
1
3 0
Now
1


 
D   0 r E   0 E  P



NE
  0 r E   0 E 
N
1
3 0
 r  1 N

 r  2 3 0
CLAUSIUS MOSOTTI RELATION
N
  r  1
N
 0 (1 
)
3 0
N /  0
( r  1) 
N
(1 
)
3 0
3
( r  2) 
N
(1 
)
3 0
Reconsider CLAUSIUS MOSOTTI relation,
 r  1 N

 r  2 3 0
Molar mass
 r 1 M
N M
(
) (
)
r  2 
3 0 
Since,
Therefore ,
NM

Density
 NA
 r 1 M
N A
(
)

r  2 
3 0
 M
MOLAR POLARIZABILITY
EXAMPLE:
An elemental dielectric material has r = 12 and it contains
5x1028 atoms/m3. Calculate its electronic polarizability
assuming Lorentz field.
SOLUTION:
Using CLAUSIUS MOSOTTI relation,
 r  1 N

 r  2 3 0
12  1
5 10 28

12  2 3  8.85 10 12
   4.17 1020 Fm2
Section III : Polarization
Mechanisms
1. Types of Dielectrics
• Polar
• Non Polar
2. Electronic polarization:
3. Ionic polarization:
4. Orientation (Dipolar) polarization :
• interface polarization
Types of Dielectrics:
• Polar dielectrics:
– Materials having permanent dipole moments
– Net dipole moment– Not zero
– Many natural molecules are examples of
systems with a finite electric dipole moment
(permanent dipole moment), since in most
types of molecules the centers of gravity of the
positive and negative charge distributions do
not coincide.
– Ex. Water
Dipole moment of
water molecule.
• Non Polar dielectrics:
• Net dipole moment – zero, (in the absence of E)
• centers of gravity of the positive and negative charge
distributions coincide with each other.
• Ex. O2, N2 and Nobel gases
Polarization Mechanisms
• Dielectric Polarization is nothing but the displacement of
charged particles under the action of the electric field to
which they are subjected.
• Therefore this displacement of the electric charges results
in the formation of electric dipole moment in atoms, ions or
molecules of the material.
– There are essentially
mechanisms:
three
basic
kinds
of
polarization
1. Electronic polarization: also called atomic
polarization. An electric field will always displace the
center of charge of the electrons with respect to the
nucleus and thus induce a dipole moment. e.g
noble
gases.
Polarization Mechanism
2. Ionic polarization: In this case a (solid) material must
have some ionic character. It then automatically has
internal dipoles, but net dipole moment is zero. The
external field then induces net dipoles by slightly
displacing the ions from their rest position. Ex. simple
ionic crystals like NaCl.
3. Orientational polarization: Some time called
“Dipolar polarization”; Here the (usually liquid or
gaseous) material must have natural dipoles which can
rotate freely. In thermal equilibrium, the dipoles will be
randomly oriented and thus carry no net polarization.
The external field aligns these dipoles to some extent
and thus induces a polarization of the material. Ex. is
water, i.e. H2O in its liquid form.
NOTE:
• Some or all of these mechanisms may act simultaneously.
• Atomic polarization is always present in any material and thus
becomes superimposed on whatever other mechanism there might be.
• All three mechanisms are essential for basic consideration and
calculations.
**************************************************************************************
However interface polarization is also found in materials:
Surfaces, grain boundaries, interface boundaries may be charged, i.e.
they contain dipoles which may become oriented to some degree in an
external field and thus contribute to the polarization of the material.
– There is simply no general way to calculate the charges on
interfaces nor their contribution to the total polarization of a
material. Interface polarization is therefore often omitted from the
discussion of dielectric properties.
SOURCES OF POLARIZABILITY
1. Electronic Polarizability
E 0
2. Ionic Polarizability
E 0
3. Dipolar or orientational
Polarizability
E 0
1. ELECTRONIC POLARIZATION:
Volume of the atom is,
4
V   R3
3
Where, R = Radius of spherically
symmetric atom
E 0
If z be the atomic number then
charge/ volume of atom would be
3 ze
 
4 R 3

In presence of field E
Force on the charges


F1  ZeE
This leads to the separation of
charges.
Coulomb force between separated charge would be

F2  Ze X Field produced by displaced charges on nucleus
Ze X Charge enclosed in the sphere of radius d

4o d 2
Ze
4 3
Z 2e 2 d
Ze
4 3 3Ze

 d   

 d 
2
2
3
4o d
3
40 R 3
4o d
3
4R
In the equilibrium position, the two forces , F1 and F2 are
equal, thus
Z 2e2d
ZeE 
40 R 3
Zed
E
40 R 3
40 R 3 E
d 
Ze
This is equilibrium separation between charges, which
is proportional to the field.
Now the induced electric dipole moment would be
40 R 3 E
pe  Zed  Ze(
)
Ze
 pe  40 R 3 E
pe  40 R 3 E
But according to the definition of polarizability,
pe   e E
Comparing,
 e  40 R 3
(e =
electronic
polarizability)
Thus, Electronic Polarization can be given as



Pe   0 ( r  1) E  N e E
  r 1   
N e
0
Where N is number of atoms/ m3.
2. IONIC POLARIZATION:


pi   i E
Polarization Mechanisms
• Dielectric Polarization is nothing but the displacement of
charged particles under the action of the electric field to
which they are subjected.
• Therefore this displacement of the electric charges results
in the formation of electric dipole moment in atoms, ions or
molecules of the material.
– There are essentially
mechanisms:
three
basic
kinds
of
polarization
1. Electronic polarization: also called atomic
polarization. An electric field will always displace the
center of charge of the electrons with respect to the
nucleus and thus induce a dipole moment. e.g
noble
gases.
Polarization Mechanism
2. Ionic polarization: In this case a (solid) material must
have some ionic character. It then automatically has
internal dipoles, but net dipole moment is zero. The
external field then induces net dipoles by slightly
displacing the ions from their rest position. Ex. simple
ionic crystals like NaCl.
3. Orientational polarization: Some time called
“Dipolar polarization”; Here the (usually liquid or
gaseous) material must have natural dipoles which can
rotate freely. In thermal equilibrium, the dipoles will be
randomly oriented and thus carry no net polarization.
The external field aligns these dipoles to some extent
and thus induces a polarization of the material. Ex. is
water, i.e. H2O in its liquid form.
Torque=p X E
3. DIPOLAR POLARIZATION
Without field
with field
Consider a molecule which carries a permanent dipole
moment p (like water molecule) is placed in an electric
field. The potential energy of the dipole would be:
 
U   p.E   pE cos 
The electric field will apply a torque which will rotate the
dipole in the direction of the EF. The energy of the dipole
is minimum when it is aligned with the field (-pE) and it is
maximum when it is antiparallel to the EF(+pE)According
to Boltzmann distribution, no. of molecules with energy U
at equilibrium temperature T would be:
n n 0 e
U
kT
n

e
n0
pE cos
kT
Let n(θ) be the number of dipoles per unit solid angle at θ,
we have
n( )  n0 e
pE cos 
kT
The number of dipoles in a solid angle dW
n0 e
pE cos 
kT
Note: Here dW is
calculated as follows:
dW
 n0 e
pE cos 
kT
2 sin d
Dipole moment of dipoles making angle  with the field
(along x-axis) is
p x  p cos 
Therefore, the dipole moment along the field within angle dW
 n0 e
pE cos 
kT
2 sin d ( p cos  )
Now, average dipole moment (Total dipole moment divided
by total no. of dipoles) can be written as

n e
pE cos 
kT
0
p
2 sin  ( p cos  )d
0

n e
0
0
pE cos 
kT
2 sin d

n e
pE cos 
kT
0
p
2 sin  ( p cos  )d

0

n e

pE cos 
kT
0
2 sin d
p

p
0

e
sin  cos d
pE cos
kT
sin d
0
0
Let
e
pE cos
kT
pE cos 
x
kT
Therefore, a cos  x
and
and,
pE
a
kT
 a sin d  dx
a a
Limits
Substituting all above, the integral becomes,
a
p
1

p
a
x
e
 xdx
a
a
x
e
 dx
a
p 1 [ xe x  e x ] aa
e a  e a 1

 a

x a
a
p a
a
[e ]  a
e e
p ea  ea 1
 a

a
p e e
a
(Langevin Function)
Or,
p
1
 coth( a) 
 L (a )
p
a
From the above equation,
a
p  pL(a )
L (a )
Thus polarization would be,
Po  NpL(a)  Ps L(a)
Ps = Saturation polarization
a
pE
kT
Po  Ps L(a)
Where,
pE
a
kT
CASE 1: When a is very high (at
low temperature) i.e. a >> 1,
L(a) = 1
Po = Ps
CASE 2: When a is very low (at
high temperature) i.e. a <<1
1 a a3
1 a
coth( a)   
 ...  
a 3 45
a 3
1 1 a 1
 L(a)  coth( a)    
a a 3 a
a
 L(a ) 
3
a
 Po  Ps
3
Np 2 E
 Po 
3kT
Np 2 E
Orientation polarization  Po 
3kT
And in terms of orientation polarizability
P0  N o E
p2
 o 
3kT
1
 o 
T
Thus orientation polarizability is inversely proportional to T.
Total polarization
P  Pe  Pi  Po  N e E  N i E  N o E
P   0 ( r  1) E  N ( e   i ) E  N o E
 0 ( r  1)  N ( e   i )  N o
Np 2
  0 ( r  1)  N ( e   i ) 
3kT
In general, therefore, we may write total polarizability as
  e  i  o
or
p2
   ei 
3kT
temperature
independent
or
p2
  e  i 
3kT
Substituting  into ClausiusMosotti relation, we have
 r 1 M N A
p2
M  (
) 
( ei 
)
 r  2  3 0
3kT
Polar substances
N A ei
3 0
N A p2
slope 
9 0 k
Non-polar substances
Clausius-Mosotti relation (in terms of refractive inex) may
alternatively be written as
 r  1 n 2  1 N Lorentz-Lorenz Relation


2
 r  2 n  2 3 0
If the material consists of different types of molecules then
Clausius-Mosotti relation may be written as
 r 1 1

N i i

 r  2 3 0
Where Ni is the no. of molecules per unit volume and i is
polarizability of ith kind of molecule.
Section III : Polarization
Mechanisms
1. Types of Dielectrics
• Polar
• Non Polar
2. Electronic polarization:
3. Ionic polarization:
4. Orientation (Dipolar) polarization :
• interface polarization
Types of Dielectrics:
• Polar dielectrics:
– Materials having permanent dipole moments
– Net dipole moment– Not zero
– Many natural molecules are examples of
systems with a finite electric dipole moment
(permanent dipole moment), since in most
types of molecules the centers of gravity of the
positive and negative charge distributions do
not coincide.
– Ex. Water
Dipole moment of
water molecule.
• Non Polar dielectrics:
• Net dipole moment – zero, (in the absence of E)
• centers of gravity of the positive and negative charge
distributions coincide with each other.
• Ex. O2, N2 and Nobel gases
SOURCES OF POLARIZABILITY
1. Electronic Polarizability
E 0
2. Ionic Polarizability
E 0
3. Dipolar or orientational
Polarizability
E 0
Orientational polarization: Some time called “Dipolar polarization”;
Here the (usually liquid or gaseous) material must have natural dipoles
which can rotate freely. In thermal equilibrium, the dipoles will be
randomly oriented and thus carry no net polarization. The external field
aligns these dipoles to some extent and thus induces a polarization of the
material. Ex. is water, i.e. H2O in its liquid form.
1. Molecules that have permanent dipole moment show orientational
polarization.
2. One atom is always more electronegative than the other.
Eg in H Cl, Cl is more electronegative than H so the shared electrons
shift more towards Cl atom. So a HCl molecule has a net dipole moment
with dipole directed from Cl to H. These are AB type molecules.
3. Molecules like A-A have no permanent dipole moment. Eg H2 , O2
4. Let us now look at AB2 type molecules. They can have net dipole
moment(each bond has a dipole moment) if the molecule has no centre of
symmetry. In general a molecule ABCD… has dipole moment if it has
no centre of symmetry. Important thing to remember if that each
bond is an individual dipole and total dipole moment of the molecule
is the vector sum of all the dipoles due to all the bonds.
Dipole moment=0
Dipole moment nonzero