Polarizability: A Microscopic View
F = qE
(1)
CHARGE
DISTRIBUTION
t0
t1
+
t2
Induced Polarization
+
INDUCED
POLARIZATION
Electric Field
Polarization = m = aE
(2)
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Application of an Oscillating Electric Field such as Light
APPLIED FIELD
INDUCED POLARIZATION
• Application of an oscillating electric field will induce an oscillating polarization in
a material.
• For linear polarization, this electric field will have the same frequency as the
applied electric field, although its phase may be shifted (not shown).
• This induced electric field is, itself, light and in the absence of scattering will
propagate through the material in the same direction as the light beam that
created it.
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Mechanisms of Polarization
+
+
-
-
ELECTRONIC
-
_
_
_
_ _
_
+ + +
+
+
+
+
+ -
VIBRATIONAL
+ -
+ -
d+
+ -
M
+ -
With E Field
-
d-
-
O
+
+ -
M
No E Field
+
O
+
_ + _ _
_
+ _
+
+
_
+ _
+
+ -
The oscillating electric field of light affects all charges in the optical medium,
not only the electron. In ionic materials, the ions move relative to one another.
Dipolar and ionic motions involve nuclear reorientation.
ROTATIONAL
IONIC MOTION
For materials that contain electric dipoles, such as water molecules, the
dipoles themselves stretch or reorient in the applied field.
P
q
+F
2a
q
E
–F
t
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E
Anisotropic Nature of Polarizability
E
E
m
CH3
perpendicular
polarization
+
H3 C
Applied field
Induced field
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+-
+ - + - + -
CH3
+-
H3 C
CH3
+-
H3 C
m
Tensorial Nature of Polarization
Polarizability is a tensor quantity as shown below.
induced polarization
applied field
axx axy a xz
ayx ayy a yz
azx azy a zz
Each entry of the tensor is a component of the polarizability.
Deforming
Force
x'
x
Deforming
Force
z
z'
y'
y
The balloon diagram illustrates this point crudely. If a balloon is stretched in
one direction (z) then the dimension of the balloon will change in all three directions.
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Polarizability: A Macroscopic View
In bulk materials, the linear polarization is given by:
Pi(w) =
S cij(w)
i,j
Ej(w)
(4)
where cij(w) is the linear susceptibility of an ensemble of molecules
The total electric field (the "displaced" field, D) within the material becomes:
D = E + 4pP = (1 + 4pc)E
(5)
Since P = cE (Equation (4)), 4pcE is the internal electric field created by the
induced displacement (polarization) of charges
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The Dielectric Constant
The dielectric constant e(w) and the refractive index n(w) are two common
bulk parameters that characterize the susceptibility of a material.
The dielectric constant in a given direction is defined as the ratio of the displaced
internal field to the applied field (e = D/E) in that direction.
eij(w) = 1 + 4pcij(w) .
(6)
The frequency dependence of the dielectric constant provides insight into the
mechanism of charge polarization.
RADIO
MICROWAVE
VISIBLE
a e + a i + ad
a e+ a i
e
Frequency
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ae
The Index of Refraction
The ratio of the speed of light in a vacuum, c, to the speed of light in a material, v, is
called the index of refraction (n):
n = c/v.
(7)
At optical frequencies the dielectric constant equals the square of the
refractive index:
e∞(w)
= n2(w).
(8)
Consequently, we can relate the refractive index to the bulk linear (first-order)
susceptibility:
n2(w) = 1+ 4pc(w).
(9)
Index of refraction depends therefore on chemical structure.
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