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Lecture10.pptx
1. 02/07/2014 PHY 712 Spring 2014 -- Lecture 10 1
PHY 712 Electrodynamics
10-10:50 AM MWF Olin 107
Plan for Lecture 10:
Complete reading of Chapter 4
A. Microscopic macroscopic
polarizability
B. Clausius-Mossotti equation
C. Electrostic energy in dielectric
media
3. 02/07/2014 PHY 712 Spring 2014 -- Lecture 10 3
r
p
p
r
p
r
r
E
r
p
r
r
r
r
p
p
3
5
2
0
3
0
3
3
4
3
4
1
:
dipole
single
from
field
tic
Electrosta
4
1
:
dipole
single
from
potential
tic
Electrosta
:
of
extent
the
outside
For
'
'
'
:
moment
Dipole
r
r
r
r
)
ρ(
r
d
Focus on dipolar fields:
4. 02/07/2014 PHY 712 Spring 2014 -- Lecture 10 4
Microscopic origin of dipole moments
Polarizable isotropic atoms/molecules
Charge anisotropic molecules
Polarizable isotropic atoms/molecules
+q
x
E
k=mw0
2 0
:
m
equilibriu
At
2
0
2
0
w
w
m
q
m
q
E
x
x
E
5. 02/07/2014 PHY 712 Spring 2014 -- Lecture 10 5
E
+q
x
k=mw0
2
E
E
x
E
x
x
E
mol
m
q
q
p
m
q
m
q
w
w
w
0
2
0
2
2
0
2
0
:
moment
dipole
Induced
0
:
m
equilibriu
At
Polarizable isotropic atoms/molecules – continued:
6. 02/07/2014 PHY 712 Spring 2014 -- Lecture 10 6
Alignment of molecules with permanent dipoles p0:
E
p0
E
E
E
p
mol
kT
pE
kT
E
p
mol
kT
p
e
d
e
p
d
0
2
0
/
cos
/
cos
0
3
1
cos
:
on
distributi
Boltzmann
a
assuming
estimated
field,
electric
an
in
moment
average
its
dipole
rotating
freely
a
For
0
7. 02/07/2014 PHY 712 Spring 2014 -- Lecture 10 7
Field due to collection of induced dipoles
1
7
3
2
5
4
6
9
8
0
0
)
(
0
i
tot
i
j
ext
j
i
site
ext
i
i
tot
E
E
E
E
E
E
E
E
r
p
p
r
p
r
r
E 3
5
2
0 3
4
3
4
1
:
dipole
single
from
field
tic
Electrosta
r
r
8. 02/07/2014 PHY 712 Spring 2014 -- Lecture 10 8
Field due to collection of induced dipoles -- continued
1
7
3
2
5
4
6
9
8
P
E
p
E
E
E
p
r
p
r
r
E
E
r
p
p
r
p
r
r
E
0
0
)
(
5
2
0
)
(
3
5
2
0
3
1
3
1
1
3
4
1
3
4
3
4
1
tot
tot
i
site
ext
i
j
j
j
i
site
ext
i
i
i
i
tot
V
r
r
r
r
0
0
)
(
0
i
tot
i
j
ext
j
i
site
ext
i
i
tot
E
E
E
E
E
E
E
E
9. 02/07/2014 PHY 712 Spring 2014 -- Lecture 10 9
Field due to collection of induced dipoles -- continued
1
7
3
2
5
4
6
9
8
tot
e
mol
tot
mol
tot
mol
site
mol
tot
i
site
V
V
V
V
E
E
P
P
E
p
P
E
p
P
E
E
0
0
0
0
0
0
)
(
1
3
1
1
3
1
2
/
1
/
3
3
1 0
0
V
V
V
mol
mol
mol
e
Claussius-Mossotti equation
10. 02/07/2014 PHY 712 Spring 2014 -- Lecture 10 10
Example of the Clausius-Mossotti equation –
Pentane (C5H12) at various densities
Density (g/cm3) Mol/m3 e/e0 3V*(e/e0-1)/(e/e0+2)
0.613 5.12536E+27 1.82 1.25646E-28
0.701 5.86114E+27 1.96 1.24084E-28
0.796 6.65544E+27 2.12 1.22536E-28
0.865 7.23236E+27 2.24 1.2131E-28
0.907 7.58353E+27 2.33 1.2151E-28
11. 02/07/2014 PHY 712 Spring 2014 -- Lecture 10 11
Re-examination of electrostatic energy in dielectric media
)
(
)
(
2
1
)
(
)
(
2
1
0
)
(
)
(
2
1
)
(
)
(
2
1
)
(
2
1
)
(
:
field
nt
displaceme
of
In terms
)
(
)
(
2
1
3
3
3
3
3
3
r
E
r
D
r
E
r
D
r
r
D
r
r
D
r
D
r
D
r
r
r
d
W
r
d
r
d
r
d
r
d
W
r
d
W
mono
mono
12. 02/07/2014 PHY 712 Spring 2014 -- Lecture 10 12
Comment on the “Modern Theory of Polarization”
Some references:
R. D.King-Smith and D. Vanderbilt, Phys.
Rev. B 47, 1651 (1993)
R. Resta, Rev. Mod. Physics 66, 699 (1994)
R, Resta, J. Phys. Condens. Matter 23,
123201 (2010)
P
D
E
D
P
E
0
0
:
equations
Basic
mono
bound
mono
bound
tot
Note: In general P is highly dependent on the boundary values;
often it is more convenient/meaningful to calculate DP.
13. 02/07/2014 PHY 712 Spring 2014 -- Lecture 10 13
Comment on the “Modern Theory of Polarization”
-- continued
D
D
D
D
D
n
n
n
crystal
electronic
electronic
nuclear
bound
w
w
V
e
bound
bound
0
0 r
P
P
