Download Lecture Notes 12: Microscopic Theory of Dielectrics, Clausius-Mossotti Eqn, Langevin and DeBye Eqns; Ferro-, Piezo- and Pyro-Electric Materials

UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 12
Prof. Steven Errede
LECTURE NOTES 12
ELEMENTARY MICROSCOPIC THEORY OF DIELCTRICS
Consider a Class-A/linear dielectric material consisting of non-polar molecules (i.e. having no
permanent electric dipole moments) in gas, liquid or solid form, immersed in an external electric
field Eext ( r ) .
-
each molecule will have an induced electric dipole moment of pmol (Coulomb-meters)
each molecule is assumed (here, for simplicity) to be spherical in shape.
In a real Class-A/linear non-polar dielectric, e.g. at room temperature, at the microscopic
level, due to the thermal energy associated with the material making up the dielectric, from one
instant in time to the next at any given point r inside the dielectric, random fluctuations of
significant size can/do occur in the electric field at that point. If one simultaneously monitors a
group of neighboring, microscopically nearby points in space at a particular instant in time, there
are also significant, random fluctuations about an average value of the electric field in this
region.
Thus, what we need to know is the time-and-space averaged local electric field as seen by a
single molecule in the dielectric at the point r - call this electric field Eloc ( r ) . Note that this
time-and-space averaged electric field does not include a contribution from the electric field
associated with the induced dipole moment of the molecule in question at the point r !!!
Conceptually, in order to determine Eloc ( r ) , we imagine that we (momentarily) “freeze”
the thermal motion of all molecules in the dielectric at a given instant in time – thus obtaining a
“snapshot” of the microscopic configuration of the dielectric at that instant in time. We then
imagine that we remove the molecule in question at the point r , keeping all other molecules
“frozen” in their positions and orientations at that instant in time. We then calculate the spaceaveraged electric field intensity inside the (assumed) spherical cavity that was occupied by the
molecule in question at the point r . We then momentarily replace the molecule, allow time to
progress forward to some other instant in time, where we again freeze the motion of all
molecules at this new instant in time, remove the same molecule, and repeat the space-averaged
electric field calculation at the point r . We repeat this procedure many, many times until we
obtain a time-and-space-averaged value of the local electric field at the point r , ≡ Eloc ( r ) .
Thus, we also expect the time-and-space averaged local electric field at the point r ,
Eloc ( r ) to be larger than the macroscopic electric field E ( r ) at the same point, precisely
because Eloc ( r ) does not include the (time-and-space-averaged) electric field of the molecule
under scrutiny at the point r , E ( r )
mol
dipole
which points in the direction opposite to that of the
applied external electric field, Eext ( r ) as shown in the figure below:
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2008. All Rights Reserved.
1
UIUC Physics 435 EM Fields & Sources I
Eext
Fall Semester, 2007
Lecture Notes 12
Prof. Steven Errede
ΔV
= Eo xˆ = Vo d
E
−Q
pmol
E
+ x̂
+Q
mol
dipole
+σ free
−σ free
In P435 Lecture Notes 10 (p. 1-6) {again, see also Griffiths problem 3.41(a-c)}, we learned that
the average electric field intensity within a sphere of radius R containing an arbitrary charge
distribution, which we identify as equal to the macroscopic, microscopically time and spaceaveraged electric field associated with the induced molecular electric dipole moment pmol ( r ) of
the molecule in question at the point r is given by:
E (r )
mol
dipole
=−
pmol ( r )
where R = radius of a single spherical molecule
4πε o R 3
Define: nmol ≡ # molecules per unit volume = molecular number density (#/m3).
Then the macroscopic (i.e. microscopically space-and-time-averaged) electric dipole moment per
unit volume (a.k.a. electric polarization) is:
Ρ ( r ) = nmol pmol ( r ) and thus: pmol ( r ) = Ρ ( r ) nmol
Then:
E (r )
mol
dipole
=−
pmol ( r )
Ρ (r )
Ρ (r )
Ρ (r )
=−
=−
=−
3
3
4πε o R
4πε o nmol R
3ε o ( vmol ∗ nmol )
3ε o
Where: vmol = 4π R 3 3 = spherical volume occupied by a single spherical molecule, and
note also that: ( vmol ∗ nmol ) = 1.
Thus:
2
E (r )
mol
dipole
=−
Ρ(r )
3ε o
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2008. All Rights Reserved.
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 12
Prof. Steven Errede
Thus it can be seen that the macroscopic E -field at the point r , E ( r ) is the linear superposition
of a.) the space-and-time-averaged local electric field Eloc ( r ) at that point, plus b.) the spaceand-time averaged electric field associated with the induced electric dipole moment pmol ( r ) of
the molecule in question at the point r , E ( r )
mol
dipole
E ( r ) = Eloc ( r ) + E ( r )
But
E (r )
mol
dipole
=−
, i.e.:
mol
dipole
Ρ (r )
Ρ (r )
(for spherical molecules), and thus: E ( r ) = Eloc ( r ) −
.
3ε o
3ε o
Turning this around, the space-and-time-averaged local electric field is thus:
Eloc ( r ) = E ( r ) − E ( r )
mol
dipole
= E (r ) +
Ρ (r )
3ε o
Thus, this last relation explicitly shows that the magnitude of the space-and-time averaged local
electric field Eloc ( r ) is larger than the macroscopic electric field E ( r ) , since the electric
polarization Ρ ( r ) (and also pmol ( r ) ) point in the same direction as the macroscopic electric field
E ( r ) . Note that Eloc ( r ) also points in the same direction as E ( r ) (and thus Ρ ( r ) and pmol ( r ) ),
whereas the space-and-time-averaged electric field due to the induced dipole moment of the
molecule in question at the point r , E ( r ) mol points in the opposite direction to E ( r ) ,
dipole
Eloc ( r ) , Ρ ( r ) and pmol ( r ) .
Note also that in general:
Eloc ( r ) = E ( r ) + b
Ρ (r )
εo
where b = 1/3 for spherical molecules.
For molecules with shapes other than a sphere, the constant b = some number ~ Ο (1) .
In a linear/Class-A dielectric, the molecular charge separation is proportional to the local
electric field, i.e. pmol ( r ) = qd mol and Eloc ( r ) are linearly related to each other by:
pmol ( r ) = α mol Eloc ( r )
where α mol ≡ molecular electric polarizability (SI Units: Coulombs2/Newton/meter).
But: Ρ ( r ) = nmol pmol ( r ) Thus: Ρ ( r ) = nmol pmol ( r ) = nmolα mol Eloc ( r )
But: Eloc ( r ) = E ( r ) +
⎛
Ρ (r )
Ρ (r ) ⎞
Thus: Ρ ( r ) = nmolα mol ⎜⎜ E ( r ) +
⎟
3ε o
3ε o ⎟⎠
⎝
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2008. All Rights Reserved.
3
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 12
Prof. Steven Errede
Then solving for the electric polarization, Ρ ( r ) in terms of the macroscopic electric field, E ( r ) :
⎛
Ρ (r ) ⎞
Ρ (r )
Ρ ( r ) = nmolα mol ⎜⎜ E ( r ) +
⎟⎟ = nmolα mol E ( r ) + nmolα mol
3ε o ⎠
3ε o
⎝
⎛ nmolα mol ⎞
⎜1 −
⎟ Ρ ( r ) = nmolα mol E ( r )
3ε o ⎠
⎝
Thus: Ρ ( r ) =
nmolα mol
⎛ nmolα mol
⎜1 −
3ε o
⎝
⎞
⎟
⎠
E (r )
But: Ρ ( r ) = ε o χ e E ( r ) for a linear/Class-A non-polar dielectric.
Solving for χ e :
nmolα mol
E ( r ) = ε o χe E ( r )
⎛ nmolα mol ⎞
⎜1 −
⎟
3ε o ⎠
⎝
nmolα mol
= ε o χe
⎛ nmolα mol ⎞
⎜1 −
⎟
3ε o ⎠
⎝
⎛ nmolα mol ⎞
⎜
⎟
εo ⎠
⎝
Thus: χ e =
= electric susceptibility of a linear/Class-A non-polar dielectric
⎛ nmolα mol ⎞
(assuming spherically-shaped molecules).
⎜1 −
⎟
3ε o ⎠
⎝
For gases (e.g. @STP), note that ( nmolα mol ε o )
Ο (10−3 )
1 , since nmol =
ρ
mmol
N A where:
nmol = number density of molecules (# / m3)
ρ = mass density (kg / m3)
NA = Avogadro’s # (6.022 × 1023 molecules / mole)
mmol = molecular weight (# kg / mole)
Thus for non-polar gases @ STP, we can safely neglect the ( nmolα mol 3ε o ) term in the
denominator of the expression for the electric susceptibility, approximating it (quite well) as:
( nmolα mol ε o )
nmolα mol
because ( nmolα mol ε o ) 1
χ egas =
1 − ( nmolα mol 3ε o )
εo
And:
4
K egas = 1 + χ egas = 1 +
( nmolα mol ε o )
1 − ( nmolα mol 3ε o )
1+
nmolα mol
εo
because
( nmolα mol ε o )
1
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2008. All Rights Reserved.
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 12
Prof. Steven Errede
The table below compares theory vs. experiment for electric susceptibility χ egas and dielectric
constant K egas = 1 + χ egas for a few simple non-polar gases:
ρgas
Gas
(gms/cc)
0.00339
0.00143
0.00489
0.00178
CS2
O2
CCl4
A
mmol
(gms/mole)
76.0
32.0
153.8
39.9
nmol
αmol
(#/m3) (C2/{N/m})
2.7×1025 9.55×10−40
2.7×1025 1.72×10−40
1.9×1025 1.39×10−40
2.7×1025 1.80×10−40
χ egas ( pred )
K egas ( pred )
K egas (expt )
0.0029
0.0005
0.0030
0.0055
1.0029
1.0005
1.0030
1.0055
1.0029
1.0005
1.0030
1.0055
It can be seen that for these simple non-polar gases, the theory predictions obtained from this
simple microscopic model of a linear/Class-A non-polar dielectric vs. experimental
measurements of the electric susceptibility χ egas and dielectric constant K egas = 1 + χ egas are in
excellent agreement with each other.
For liquids (and solids), ( nmolα mol ε o ) Ο (1) and thus this term cannot be neglected in the
formula for the electric susceptibility.
The table below compares theory vs. experiment for electric susceptibility χ eliquid and dielectric
constant K eliquid = 1 + χ eliquid for the liquid-form versions of the entries in the above table. The
results for the liquid-form are obtained using the results from the gas-form, scaled by the ratio of
volume mass densities.
Liquid
CS2
O2
CCl4
A
ρliquid
ρliquid
(gms/cc) ρgas
1.29
1.19
1.59
1.44
381
832
325
810
nmol
(#/m3)
1.0×1028
2.2×1028
6.2×1028
2.2×1028
nmolα mol ε o
χ eliquid ( pred )
K eliquid ( pred )
K eliquid (expt )
1.11
0.435
0.977
0.441
1.76
0.51
1.45
0.52
2.76
1.51
2.45
1.52
2.64
1.51
2.24
1.54
Here the agreement of theory prediction vs. experiment is quite good.
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2008. All Rights Reserved.
5
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 12
Prof. Steven Errede
THE CLAUSIUS-MOSSOTTI EQUATION
We have obtained an expression for the local electric field Eloc ( r ) inside a linear/Class-A nonpolar dielectric:
Eloc ( r ) = E ( r ) +
Ρ(r )
3ε o
We have also obtained an expression for the electric polarization associated with this dielectric:
⎛
Ρ (r ) ⎞
Ρ ( r ) = nmolα mol ⎜⎜ E ( r ) +
⎟
3ε o ⎟⎠
⎝
However, inside a linear/Class-A non-polar dielectric the electric displacement D ( r ) ,
the electric field E ( r ) and electric polarization Ρ ( r ) are related to each other by:
D (r ) = E (r ) + Ρ (r )
Thus: Ρ ( r ) = D ( r ) − ε o E ( r ) but inside a linear/Class-A non-polar dielectric the electric
displacement D ( r ) and the electric field E ( r ) are also related to each other by: D ( r ) = ε E ( r ) .
Thus: Ρ ( r ) = ( ε − ε o ) E ( r ) = ε o ( K e − 1) E ( r ) where K e ≡ ε ε o = 1 + χ e
Thus, inside a linear/Class-A non-polar dielectric the electric polarization Ρ ( r ) and
the electric field E ( r ) are also related to each other by: Ρ ( r ) = ε o χ e E ( r )
⎛
Ρ (r ) ⎞
Then since: Ρ ( r ) = ε o ( K e − 1) E ( r ) = ε o χ e E ( r ) and: Ρ ( r ) = nmolα mol ⎜⎜ E ( r ) +
⎟
3ε o ⎟⎠
⎝
⎛
ε o ( K e − 1) E ( r ) ⎞
⎛
Ρ(r ) ⎞
Thus: ε o ( K e − 1) E ( r ) = nmolα mol ⎜⎜ E ( r ) +
⎟
⎟⎟ = nmolα mol ⎜⎜ E ( r ) +
⎟
3ε o ⎠
3εo
⎝
⎝
⎠
⎛ ( K − 1) ⎞
⎛ 3 + ( K e − 1) ⎞
( Ke + 2)
∴ ε o ( K e − 1) = nmolα mol ⎜1 + e
⎟ = nmolα mol ⎜
⎟ = nmolα mol
3 ⎠
3
3
⎝
⎝
⎠
⎛ K − 1 ⎞ ⎛ nmolα mol ⎞
⎛ χ e ⎞ ⎛ nmolα mol ⎞
or: ⎜ e
⎟=⎜
⎟ or equivalently: ⎜
⎟=⎜
⎟
⎝ K e + 2 ⎠ ⎝ 3ε o ⎠
⎝ χ e + 3 ⎠ ⎝ 3ε o ⎠
6
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2008. All Rights Reserved.
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 12
Prof. Steven Errede
If the Class-A/linear non-polar dielectric is a compound dielectric composed of N different types
of (spherical) non-polar molecules, then we obtain the so-called Clausius-Mossotti Equation:
⎛ Ke − 1 ⎞ 1
⎜
⎟=
⎝ K e + 2 ⎠ 3ε o
N
∑n
mol
i
α imol ⇐ Clausius-Mossotti Equation
i =1
The Clausius-Mossotti Equation relates the dielectric constant K e = ( ε ε o ) = (1 + χ e ) for a
Class-A/linear non-polar dielectric material (n.b. assumed to have spherically-shaped molecular
electric dipoles) to the mass density ρ (kg/m3) of the Class-A dielectric material and the
molecular electric polarizability, α mol .
If nmol =
ρ
mmol
N A then for a single molecular species the Clausius-Mossotti equation becomes:
⎛ K e − 1 ⎞ ⎛ nmolα mol ⎞
⎛ K e − 1 ⎞ ρ N Aα mol
1 ⎛ K e − 1 ⎞ ⎛ N Aα mol
or:
⎜
⎟=⎜
⎟ then: ⎜
⎟=
⎜
⎟=⎜
ρ ⎝ K e + 2 ⎠ ⎝ 3ε o
⎝ K e + 2 ⎠ ⎝ 3ε o ⎠
⎝ K e + 2 ⎠ 3ε o mmol
⎞ 1
⎟
⎠ mmol
⎛N ⎞
We define: Ρ molar = molar electric polarization ≡ ⎜ A ⎟ α mol
⎝ 3ε o ⎠
Then:
1 ⎛ K e − 1 ⎞ Ρ molar
⎜
⎟=
ρ ⎝ K e + 2 ⎠ mmol
n.b. The RHS of this equation (and thus also the LHS) is independent of the mass density ρ !!!
{This is true for a wide variety of gases and also non-polar liquids (approximately ~ 50)}
⎛ K − 1 ⎞ nmolα mol
For the single molecular species the Clausius-Mossotti Equation: ⎜ e
⎟=
3ε o
⎝ Ke + 2 ⎠
Note that if nmol →
3ε o
α mol
⎛α
then the RHS of C-M Equation nmol ⎜ mol
⎝ 3ε o
⎞
⎛ 3ε o ⎞⎛ α mol ⎞
⎟→ ⎜
⎟⎜
⎟ →1
⎠
⎝ α mol ⎠⎝ 3ε o ⎠
⎛ K −1 ⎞
⎛ 3ε ⎞
Thus, when nmol → ⎜ o ⎟ , then ⎜ e
⎟ → 1, i.e. K e → ∞ !!!
⎝ Ke + 2 ⎠
⎝ α mol ⎠
Physically, this corresponds to (spherical) molecules that become infinitely polarizable!!
For gases and liquids, the molecular electric polarizabilities α mol are relatively small,
⎛ 3ε ⎞
so nmol → ⎜ o ⎟ never actually happens for gases and/or liquids.
⎝ α mol ⎠
For crystalline solids, this simple model of molecular polarization is simply too crude – it doesn’t
work / doesn’t agree well with crystal data…
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2008. All Rights Reserved.
7
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 12
Prof. Steven Errede
Polar Dielectrics and the Langevín Equation
Polar dielectrics (e.g. water, calcite, quartz, . . .) have permanent molecular electric dipole
moments. Molecules consisting of two (or more) dissimilar atoms can exhibit permanent electric
dipole moments. Diatomic molecules such as O2, N2, etc. with identical atoms cannot, due to
symmetry considerations – there is no asymmetric way to arrange charge with identical atoms!!!
The potential energy of a single molecule with a permanent electric dipole moment pmol ( r )
inside a polar dielectric is:
W1 = P.E.1 = − pmol ( r )i Eloc ( r )
(n.b. W1 = P.E.1 is a minimum when pmol
Eloc )
θ
pmol
Eloc
The Langevín Equation
In a gaseous or liquid dielectric, the thermal energy of the medium (due to it being at finite
temperature) causes collisions between the molecules, which tend to destroy / randomize any net
alignment of the polar molecules with the local field Eloc (i.e. thermal energy / thermal
agitation depolarizes the macroscopic alignment of such molecules). However, the local electric
field Eloc exerts a restoring force (via a torque) on the electric dipoles between collisions.
Thus a partial net macroscopic alignment, or electric polarization will exist, i.e. a net Ρ (electric
dipole moment per unit volume) will exist, macroscopically.
For no applied external electric field (i.e. Eext = 0 ) the dipoles are oriented at random.
If there are nmol molecules per unit volume, then for random orientation of dipoles, the fraction
dnmol nmol of polar molecules within angles θ and (θ + dθ ) {see figure above} is just:
d cosθ
dnmol d Ω 2π d cos θ 2π sin θ dθ 1
=
=
=
= sin θ dθ
nmol
Ω
4π
4π
2
where nmol = # polar molecules per unit volume and dnmol = # polar molecules per unit volume
within θ and (θ + dθ ) . The factor of 2π arises from azimuthal (ϕ ) symmetry – the dot product
pmol i Eloc only cares about the (polar) angle θ between pmol and Eloc . Note that random
orientations are flat probability distributions in (ϕ , cos θ ) with ϕ ranging over 0 ≤ ϕ ≤ 2π and
cos θ ranging over −1 ≤ cos θ ≤ 1 (i.e. where the polar angle variable θ ranges over 0 ≤ θ ≤ π ).
8
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2008. All Rights Reserved.
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 12
Prof. Steven Errede
If the permanent dipoles are then subjected to an externally-applied electric field Eext and are
also in thermal equilibrium with each other, then number dnmol of molecules per unit volume
possessing a particular potential energy W is given by the Boltzmann distribution law:
dnmol = C e−W kBT where k B =Boltzmann’s Constant = 1.381 × 10−23 J/K
constant
T = Absolute Temperature (Kelvin degrees)
For a single polar molecule in a gas of identical particles: W1 = − pmol i Eloc = − pmol Eloc cos θ
The number of polar electric dipoles / unit volume within θ and (θ + dθ ) is:
dnmol = Ce
(
+ pmol i Eloc
= Ce
u cosθ
k BT
)
+ p
sin θ dθ = Ce ( mol
Eloc cosθ k BT )
sin θ dθ
sin θ dθ where u ≡ pmol Eloc k BT
The constant C is a normalization constant such that the total # of molecules / unit volume nmol is
given by:
π
nmol = C ∫ eu cosθ sin θ dθ
0
Now the polar molecules whose permanent electric dipole moments lie within θ and (θ + dθ )
possess a total electric dipole moment per unit volume in the direction of the local electric field
of:
eu cosθ cos θ sin θ dθ
d Ρ = dnmol pmol cos θ = nmol pmol π
u cosθ
∫ e sin θ dθ
0
Thus the NET electric dipole moment per unit volume (= electric polarization) P is given by:
Ρ = ∫ d Ρ = nmol pmol cos θ = nmol pmol
∫
π
0
eu cosθ cos θ sin θ dθ
∫
π
0
We define: t ≡ u cos θ
⎛p
E ⎞
thus: t = ⎜ mol loc ⎟ cos θ = u cos θ
⎝ k BT
⎠
+u
n p
n p
Then: Ρ = mol mol t = mol mol
u
u
∫
∫
−u
+u
−u
Or:
eu cosθ sin θ dθ
et tdt
et dt
=
t
t
nmol pmol ⎡⎣te − e ⎤⎦
u
et +− uu
⎡
⎛ k BT
⎛p
E ⎞
Ρ = nmol pmol ⎢ coth ⎜ mol loc ⎟ − ⎜⎜
⎝ k BT
⎠
⎝ pmol Eloc
⎣⎢
+u
−u
⎞⎤
⎟⎟ ⎥ ⇐ Langevín Equation
⎠ ⎦⎥
At room temperature (T ~ 300K) the <thermal energy> is:
k BT ∼ 4 × 10−21 Joules ( ∼
1
40
eV ( electron − Volt ) )
(1 eV = 1.602×10−19 Joules)
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2008. All Rights Reserved.
9
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 12
Prof. Steven Errede
Typical permanent/polar molecular electric dipole moments are on the order of pmol ∼ Ο (10−31 )
Coulomb-meters, and for a typical value of Eloc ∼ 107 V / m , then:
⇒ u (T ) =
pmol Eloc
k BT
2 × 10−3
(
1) @ T = Troom ~ 300 K
Now since u (T ) is quite small in this situation, we can expand the Langevín Equation in a
Taylor series expansion, retaining terms up to order u 3 :
⎤
1 u u 3 2u 5
1 ⎡ u 2 u 4 2u 6
coth ( u ) = + − +
+ .... = ⎢ − +
+ ....⎥
u 3 45 945
u ⎣ 3 45 945
⎦
2 + u2
2u ⎡⎣1 + ( u 2 6 ) ⎤⎦
Then:
1⎤
⎡
Ρ = nmol pmol ⎢ coth u − ⎥
u⎦
⎣
Or:
Ρ (r )
Thus: Ρ ( r )
⎡
2 + u2
1 ⎥⎤
⎢
nmol pmol
−
⎢ 2u ⎡1 + ( u 2 6 ) ⎤ u ⎥
⎦
⎣ ⎣
⎦
⎡(2 + u2 ) ⎛ u2 ⎞ 1 ⎤
nmol pmol ⎢
⎜1 − ⎟ − ⎥
6 ⎠ u⎥
⎢⎣ 2u ⎝
⎦
2
nmol pmol u ( r ) nmol pmol
=
Eloc ( r )
3
3k BT
2
nmol pmol
Eloc ( r ) for T ~ 300K {i.e. pmol Eloc
3k BT
⇒ When pmol Eloc
k BT }
k BT the electric polarization Ρ ( r ) in a polar dielectric is linearly
proportional to the local electric field Eloc ( r ) .
Now for a linear/Class-A polar dielectric we also have the relation: Ρ ( r ) = ε o χ e E ( r )
But: Eloc ( r ) = E ( r ) +
Ρ (r )
Ρ (r )
or: E ( r ) = Eloc ( r ) −
for spherical polar molecules.
3ε o
3ε o
⎛
Ρ (r ) ⎞
Thus: Ρ ( r ) = ε o χ e E ( r ) = ε o χ e ⎜⎜ Eloc ( r ) −
⎟
3ε o ⎟⎠
⎝
Or:
⎛ K −1 ⎞
ε o χe
Eloc ( r ) = 3ε o ⎜ e
Ρ (r ) =
⎟ Eloc ( r ) then using: Ρ ( r )
⎛ χe ⎞
⎝ Ke + 2 ⎠
⎜1 + ⎟
3 ⎠
⎝
n p2
We see that: mol mol Eloc ( r )
3k BT
⎛ K −1 ⎞
or: ⎜ e
⎟
⎝ Ke + 2 ⎠
10
2
nmol pmol
Eloc ( r )
3k BT
⎛ K −1 ⎞
3ε o ⎜ e
⎟ Eloc ( r )
⎝ Ke + 2 ⎠
2
nmol pmol
when pmol Eloc
9ε o k BT
k BT , i.e. when T ~ 300K .
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2008. All Rights Reserved.
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 12
Prof. Steven Errede
We can solve the above relation for the dielectric constant K e and electric susceptibility
χ e = K e − 1 associated with a polar dielectric. Defining: a ≡
(i.e. pmol Eloc
2
nmol pmol
, then for T
9ε o k BT
300 K
k BT ):
2
⎛ 2nmol pmol
+
1
⎜
9ε o k BT
⎛ 1 + 2a ⎞ ⎜
=
Ke = ⎜
⎟ ⎜
2
nmol pmol
⎝ 1− a ⎠
−
1
⎜
9ε o k BT
⎝
⇒ When pmol Eloc
2
⎞
⎛ nmol pmol
⎟
⎜
⎟ ≥ 1 and: χ e = K e − 1 = ⎛⎜ 1 + 2a ⎞⎟ − 1 = ⎛⎜ 3a ⎞⎟ = ⎜ 3ε o k BT
2
⎟
⎝ 1− a ⎠
⎝ 1 − a ⎠ ⎜ 1 − nmol pmol
⎟
⎜
9ε o k BT
⎠
⎝
⎞
⎟
⎟≥0
⎟
⎟
⎠
k BT , dielectric constant K e and electric susceptibility χ e of a polar
dielectric are inversely proportional to the absolute temperature T. Note that non-polar dielectrics
have no such temperature dependence! For polar gases, where χ e ~ 0 and K e ~ 1 , note that:
χ egas ≈
2
nmol pmol
2n p 2
≥ 0 and K egas = 1 + χ egas ≈ 1 + mol mol ≥ 1
3ε o k BT
9ε o k BT
p
E
1⎤
Ρ
⎡
A plot of the Langevín function f L ( u ) = ⎢ coth ( u ) − ⎥ =
versus u = mol loc :
u ⎦ nmol pmol
k BT
⎣
Slope @ u = ∞ : m = 0
fL (u → ∞ ) = 1
1⎤
⎡
f L ( u ) = ⎢coth ( u ) − ⎥ vs. u
u⎦
⎣
Initial slope @ u = 0: minit = 1/3
fL (u )
fL (u = 0) = 0
0 1 2 3 4 5
10
u = pmol Eloc k BT
The DeBye Equation for Polar Dielectrics
Consider a real polar dielectric – i.e. one in which both permanent and induced molecular
electric dipoles are taken into account. Then by the principle of linear superposition:
ΡTOT ( r ) = Ρinduced ( r ) + Ρ polar ( r )
Now: Ρinduced ( r ) = nmol pmol
induced
(r )
Induced electric dipole
moment per unit volume
nmol = number density of molecular dipoles (#/m3)
pmol
induced
( r ) = induced electric dipole moment at point
r
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2008. All Rights Reserved.
11
UIUC Physics 435 EM Fields & Sources I
But:
pmol
induced
( r ) = α mol
Fall Semester, 2007
Lecture Notes 12
⇒ ∴ Ρinduced ( r ) = nmol pmol
Eloc ( r )
induced
molecular
polarizability
Prof. Steven Errede
( r ) = nmolα mol
Eloc ( r )
And (from the Langevín Equation, for u << 1):
Ρ polar ( r )
2
nmol pmol
permanent
3k BT
Eloc ( r )
for T ~ 300K ( i.e. pmol Eloc
∴ Ρ = ΡTOT ( r ) = Ρ induced ( r ) + Ρ polar ( r ) = nmolα mol Eloc ( r ) +
But:
Eloc ( r ) = E ( r ) − Emolecular ( r ) = E ( r ) +
dipoles
k BT )
2
nmol pmol
permanent
3k BT
Eloc ( r )
Ρ (r )
3ε o
n.b. This is an important (but simplifying) assumption here - because it implicitly assumes
b = 1/ 3 spherical-shaped molecules…
However, for linear/Class-A dielectrics we also have: Ρ ( r ) = ε o χ e E ( r ) = ε o ( K e − 1) E ( r )
Then: Eloc ( r ) = E ( r ) +
ε o ( K e − 1) ⎞
Ρ (r ) ⎛
⎛ ( K − 1) ⎞
⎛ K +2⎞
= ⎜1 +
E (r ) = ⎜ e
⎟ E ( r ) = ⎜1 + e
⎟
⎟ E (r )
⎟
3ε o ⎜⎝
3
3
3 εo
⎝
⎠
⎝
⎠
⎠
Thus we obtain an important relationship between the space-and-time-averaged local electric
field Eloc ( r ) and the macroscopic electric field, E ( r ) :
⎛ 3 ⎞
or: E ( r ) = ⎜
⎟ Eloc ( r )
⎝ Ke + 2 ⎠
Again, it can be seen from this relation that Eloc ( r ) is parallel to E ( r ) but because K e ≥ 1 we
⎛ K +2⎞
Eloc ( r ) = ⎜ e
⎟ E (r )
⎝ 3 ⎠
also see that Eloc ( r ) > E ( r ) .
From the above relation(s) for the electric polarization, we see that:
2
nmol pmol
⎛ K −1 ⎞
permanent
Ρ = nmolα mol Eloc ( r ) +
Eloc ( r ) = ε o ( K e − 1) E ( r ) = 3ε o ⎜ e
⎟ Eloc ( r )
3k BT
2
+
K
⎝ e
⎠
2
⎛
pmol
⎛ K − 1 ⎞ ⎛ n mol ⎞ ⎜
permanent
Or: ⎜ e
⎟=⎜
⎟ ⎜ α mol +
3k BT
⎝ K e + 2 ⎠ ⎝ 3ε o ⎠ ⎜
⎝
12
⎞
⎟
⎟ ⇐ DeBye Equation for Class-A polar dielectrics
⎟
⎠
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2008. All Rights Reserved.
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 12
Prof. Steven Errede
Compare this to:
⎛ K e − 1 ⎞ ⎛ n mol ⎞
⎜
⎟=⎜
⎟ α mol ⇐ Clausius-Mossotti Equation for Class-A non-polar dielectrics
⎝ K e + 2 ⎠ ⎝ 3ε o ⎠
Thus we see that the DeBye Equation essentially is the C-M Equation, just with an additional
term on the RHS, depending on 1/T due to the contribution arising from the polar molecules, as
seen from the Langevín equation for polar molecules at finite temperature.
⎛m ⎞
Let us now multiply both sides of the DeBye equation by: ⎜ mol ⎟
molecular mass (kg/mole)
⎝ ρ ⎠
Recall that:
( m ) = mρ
nmol = # density of molecules #
3
NA
NA = Avogadro’s number 6.022 × 1023 molecules/mole
mol
Rearranging the above relation for polar dielectrics, we obtain the molar polarization as :
Ρ molar ≡
2
⎞
mmol ⎛ K e − 1 ⎞ ⎛ N A ⎞ ⎛
pmol
α
=
+
⎜
⎟ ⎜
⎟ ⎜ mol
⎟ ⇐ DeBye Equation for polar molecules
ρ ⎝ K e + 2 ⎠ ⎝ 3ε o ⎠ ⎝
3k BT ⎠
Again, note that the RHS of this equation is independent of mass density (hence so is the LHS).
Note also that this equation is that for a straight line, i.e. y ( x ) = mx + b with x = 1/ T , and
2
⎛ N A pmol
⎞
⎛ N Aα mol ⎞
mmol ⎛ K e (T ) − 1 ⎞
⎜⎜
⎟⎟ , slope m = ⎜
⎟ and intercept b = ⎜
⎟
ρ (T ) ⎝ K e (T ) + 2 ⎠
⎝ 9ε o k B ⎠
⎝ 3ε o ⎠
as shown in the figure below:
y ( x ) = Ρ molar (1/ T ) ≡
⎛ mmol
⎜
⎝ ρ
⎞ ⎛ Ke − 1 ⎞
⎟
⎟⎜
⎠ ⎝ Ke + 2 ⎠
2
N A pmol
slope =
9ε o k B
N Aα mol
3ε o
0
1/T
2
pmol
m
=
Note also that the ratio of slope/intercept, many things cancel, except:
b 3k Bα mol
In real life, it is only possible to reliably measure α mol and pmol for gases, or for dilute solutions
of polar molecules in non-polar solvents.
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2008. All Rights Reserved.
13
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 12
Prof. Steven Errede
Ferro-Electricity:
⎛ K − 1 ⎞ nmolα mol
In a certain crystalline solids, the condition ⎜ e
≈ 1 is satisfied, i.e. K e → ∞ and
⎟=
3ε o
⎝ Ke + 2 ⎠
χ e = ( K e − 1) → ∞ ! In this situation, such materials exhibit permanent polarization
(e.g. permanently polarized materials, such as electrets).
Thus,
nmolα mol
≈ 1 can be taken as a necessary condition for permanent polarization to occur.
3ε o
The Making of a “Ferro-Electret”:
Suppose the (absolute) temperature T is very high (i.e. let T → ∞). No permanent polarization
Ρ permanent can exist, because the thermal energies ∼ k BT are >> pmol Eloc , randomizing the (net)
dipoles
polarization orientation. However, ∃ (there exists) a temperature Tc (known as the Curie
Temperature) at (or below) which a ferro-electric material spontaneously generates a net
permanent electric polarization Ρ (this is precisely what happens e.g. for ferro-magnets,
spontaneously developing a permanent magnetization Μ (= magnetic dipole moment per unit
volume). This spontaneous electrically-polarized state is relatively stable and can exist for a
very long time!!!
Examples of Ferro-Electric Materials:
Barium Titanate (BaTiO3), Curie Temperature Tc 120oC
Lead Zirconate (PbZrO3)
Potassium Tantalum Niobiate (KTaNbO3) has Ke = 34,000 @ 0oC!!!
Lead Titanate
(PbTiO3)
Certain (organic) polymers such as Kynar ® Film = Polyvinylidene Flouride (a.k.a. PVDF)
Examples of Ferro-Magnetic Materials:
Iron (Tc ~ 770oC)
AlNiCo Alloys
Samarium-Cobalt
Other rare earth magnets using Neodymium, e.g. Neodymium-Boron…
Thermal Stability of Kynar ® (PVDF) Film
% Retention
of Electric
Polarization Ρ
T = 22oC
T = 60oC
T = 80oC
100
80
60
40
20
2
100
1000
# Days
A polarized ferro-electric material is stable against a reversed external electric field, provided it
is not too large. The electric polarization Ρ of such a material, exhibits hysteresis (= “to lag
behind”)
14
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2008. All Rights Reserved.
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
“Typical” hysteresis curve of Ρ vs. Eloc
Lecture Notes 12
Prof. Steven Errede
for a ferro-electric material:
Ρ
Points a & c are where
Eloc = 0 but Ρ ≠ 0
a
b
ϑ
c
d
Eloc
Points b and d are where
Eloc is (just) large enough
to permanently reverse/flip
the polarization Ρ
Compare the above curve for a ferro-electric material with that for a class-A dielectric
(where Ρ = ε o χ e E is valid):
Ρ
slope = ε o χ e
E
For ferro-electric materials, we see that no simple relation between Ρ and Ε exists – it is in fact
(highly) non-linear and depends on previous history of sample of material!
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2008. All Rights Reserved.
15
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 12
Prof. Steven Errede
Piezo-electricity
Piezo-electricity (= “pressure” electricity) is the capability (or ability) of certain crystalline
materials to change their dimensions when subjected to an externally-applied electric field; or
conversely, the ability to produce electrical signals (i.e. voltages) when mechanically deformed!
Examples of Piezo-Electric materials:
Quartz crystals (discovered by Pierre & Jacques Curie in 1880’s)
Rochelle Salts
Tourmeline
Kynar ® Piezo-Electric Film (PVDF film)
Some Barium Salts (e.g. Barium Titanate – BaTiO3)
Human & other mammal bones!!! (crucial for remodeling of bones, esp. after breaking one –
evolution at work!!!)
S (r ) = d E (r )
E (r ) = g x (r )
Resultant stress
From applied
Piezo strain
constant of
Piezo-Electric
voltage constant
Eext field
material
((V/m)/(N/m2))
(meters/meter)
((m/m)/(Volt/m))
stress (N/m2)
Piezo-Electricity arises from the electrical polarization produced by mechanical strains in
crystals (belonging to certain classes of symmetries). The polarization Ρ is proportional to
strain. If a crystal is centro-symmetric, it cannot be Piezo-Electric.
The existence of a polar axis in a crystal gives rise to an inherent, spontaneous polarization Ρ .
Pyro-Electricity
Pyro-Electricity (= “heat” electricity) is the ability (or capability) of certain crystalline materials
to produce electrical signals (i.e. voltages) when exposed to changes in temperature (i.e. changes
in internal thermal energy). This phenomenon is due to electric polarization Ρ induced by
thermal energy absorption in certain crystals. The amount of polarization ΔΡ is proportional to
the change in the thermal energy ΔU thermal and hence is proportional to the change in the
temperature ΔT .
Examples of Pyro-Electric materials:
Kynar ® (PVDF) film
Tri-glycerine sulfate (TGS)
Lead-zirconate-titanate (PZT)
16
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2008. All Rights Reserved.
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 12
Prof. Steven Errede
ΔQB = (bound) charge produced by a change in Temperature ΔT
ΔQB =
charge
(coulombs)
ρ
ΔT
pyro-electric
coefficient
( coul 2 o )
m / K
A
Temp cross-sectional
change
area
For Kynar ® film, ρ ≈ 170 nC ( cm 2 K )
Nano Coulombs = 10−9 Coulombs
The Relationship Between the Index of Refraction n and the Dielectric Constant Ke
In free space / vacuum, the speed of propagation of electromagnetic radiation (real photons)
is c = 3 x 108 m/s, which is related to the macroscopic parameters of the vacuum - ε o and μo by:
1
c=
ε o μo
ε o = 8.85 ×10−12 Farads/m = electric permittivity of free space
μo = 4π ×10−7 Henrys/m = magnetic permeability of free space
ε o and μo are macroscopic electric and magnetic properties (respectively) of the (physical) QED
vacuum {QED = Quantum Electro-Dynamics} At the microscopic level, the (physical) QED
vacuum consists of electrically-charged, virtual fermion-antifermion pairs flitting in/out of
existence, as allowed by the Heisenberg uncertainty principle, ΔE Δt ≤ .
The vacuum is relativistically Lorentz invariant (i.e. no absolute origin exists), thus ε o and μo
cannot/do not/must not have any frequency dependence - i.e. ε o ≠ ε o ( f ) and μo ≠ μo ( f )
because any frequency dependence of the vacuum is forbidden by Lorentz invariance.
In physical matter, electromagnetic waves propagate through physical matter in a manner
analogous to that of EM waves propagating through the QED vacuum. One major difference:
physical matter is made up of composite atoms-bound states of electrons and nuclei – thus, ∃
resonances at certain frequencies of EM radiation – whenever:
Eγ = hf = energy of (real) photon
atom
ΔΕ atom
− Ε natom
n ,m = Ε m
As a consequence of this, in matter, electromagnetic radiation travels at speed v ≤ c :
For non-conductors – e.g. dielectric media:
v( f ) =
Frequency dependence
of v is e.g. responsible
for rainbows!!!
1
ε ( f )μ( f )
⇐ Frequency dependent!!!
Because of/due to frequency-dependence of:
ε =ε( f )
μ = μ( f )
= electric permittivity of medium
= magnetic permeability of medium
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2008. All Rights Reserved.
17
UIUC Physics 435 EM Fields & Sources I
1
Compare: v ( f ) =
But v ( f ) =
ε ( f )μ( f )
Fall Semester, 2007
Lecture Notes 12
(matter)
c=
vs:
1
ε o μo
Prof. Steven Errede
(vacuum)
ε ( f )μ( f )
c
c
=
in a dielectric material, i.e. n ( f ) =
= index of refraction
ε o μo
v( f )
n( f )
Many dielectric materials have no magnetic properties – i.e. their μ ( f ) = μo to a high degree.
ε(f)
= K e ( f ) since (for linear dielectrics)
εo
Thus for many dielectric materials n ( f ) =
Ke ( f ) ≡
ε(f)
= 1 + χe ( f )
εo
In general: v ( f ) =
Km ≡
1
ε ( f )μ( f )
c
Ke ( f ) Km ( f )
where (for linear magnetic materials)
μ( f )
= 1 + χ m ( f ) . In dielectric materials that are non-magnetic, K m 1 because
μo
μ ( f ) μo , then v ( f )
18
=
1
ε ( f ) μo
=
c
Ke ( f )
.
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2008. All Rights Reserved.