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)(

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2

0

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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