02/08/2013 PHY 712 Spring 2013 -- Lecture 10 1
PHY 712 Electrodynamics11-11:50 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
02/08/2013 PHY 712 Spring 2013 -- Lecture 10 2
02/08/2013 PHY 712 Spring 2013 -- Lecture 10 3
rpprprrE
rpr
r
rrp
p
35
2
0
30
3
343
41
:dipolesinglefromfieldticElectrosta4
1:dipolesinglefrompotentialticElectrosta
:ofextent theoutsideFor
'''
:moment Dipole
rr
r
r
)ρ(rd
Focus on dipolar fields:
02/08/2013 PHY 712 Spring 2013 -- Lecture 10 4
Microscopic origin of dipole moments Polarizable isotropic atoms/molecules Charge anisotropic molecules
Polarizable isotropic atoms/molecules
+q
x
E
k=mw02 0
:mequilibriuAt
20
20
w
w
mq
mqEx
xE
02/08/2013 PHY 712 Spring 2013 -- Lecture 10 5
E+q
xk=mw0
2
EEx
Ex
xE
molmqq p
mq
mq
w
w
w
020
2
20
20
:momentdipoleInduced
0:mequilibriuAt
Polarizable isotropic atoms/molecules – continued:
02/08/2013 PHY 712 Spring 2013 -- Lecture 10 6
Alignment of molecules with permanent dipoles p0:
Ep0
EE
Ep
mol
kTpE
kTEp
mol
kTp
ed
epd
0
20
/cos
/cos0
31
cos
:ondistributiBoltzmann aassumingestimatedfield,electrican in moment averageitsdipolerotatingfreely aFor
0
02/08/2013 PHY 712 Spring 2013 -- Lecture 10 7
Field due to collection of induced dipoles
17
32
5
4
6
98
00)(
0
itotij
jisite
iitot
EEEE
EE
rpprprrE 3
5
2
0 343
41
:dipolesinglefromfieldticElectrosta
r
r
02/08/2013 PHY 712 Spring 2013 -- Lecture 10 8
Field due to collection of induced dipoles -- continued
173
2
54
6
98
00)(
0
itotij
jisite
iitot
EEEE
EE
PEpEE
prprrE
rpprprrE
00)(
5
2
0)(
35
2
0
31
311
34
1
343
41
tottotisite
ij
jjisite
ii
iitot
V
rr
rr
02/08/2013 PHY 712 Spring 2013 -- Lecture 10 9
Field due to collection of induced dipoles -- continued
173
2
54
6
98
totemol
totmol
totmol
sitemol
totisite
VV
VV
EE
P
PEpP
Ep
PEE
00
0
0
0
0)(
1
311
31
2/1/3
31 0
0
V
V
Vmol
mol
mol
e
Claussius-Mossotti equation
02/08/2013 PHY 712 Spring 2013 -- 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
02/08/2013 PHY 712 Spring 2013 -- Lecture 10 11
Re-examination of electrostatic energy in dielectric media
)()(21
)()(210
)()(21)()(
21)(
21
)(:fieldnt displacemeofIn terms
)()(21
3
3
333
3
rErD
rErD
rrDrrDrD
rD
rr
rdW
rd
rdrdrdW
rdW
mono
mono
02/08/2013 PHY 712 Spring 2013 -- 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)
PDEDPE
0
0
:equationsBasic
mono
bound
monoboundtot
Note: In general P is highly dependent on the boundary values; often it is more convenient/meaningful to calculate DP.
02/08/2013 PHY 712 Spring 2013 -- Lecture 10 13
Comment on the “Modern Theory of Polarization” -- continued
D
DDDD
nnn
crystal
electronic
electronicnuclearbound
wwVe
boundbound
00 rP
P