Electrodynamics Lecture Notes Chapter 4 Discussion

PHY 712 Electrodynamics 10-10:50 AM MWF Olin 103 Notes for Lecture 11 -- Complete reading Chapter 4 (Sec. 4.5-4.7 in JDJ) A. Microscopic dielectric functions and susceptibility macroscopic polarizability and B. Clausius-Mossotti equation C. Electrostatic energy in dielectric media D. Comment on modern theory of polarization 02/07/2025 PHY 712 Spring 2025 -- Lecture 11 1

02/07/2025 PHY 712 Spring 2025 -- Lecture 11 2

Note 1 r d that for : r d r r d = r ( P ) a d + 1 d q = 0 02/07/2025 PHY 712 Spring 2025 -- Lecture 11 3

Review -- Focus on dipolar fields: p Dipole moment : 3 p ' ' r r ' d r ( ) ( ) r For outside the extent of Electrostatic potential from single dipole: 1 4 Electrostatic field from single dipole: = E r : r p r ( ) r = 3 r 0 ( ) 2 3 r p r p r 1 4 ( ) ( ) r 3 p 5 4 3 r 0 (Assuming that p is located at the origin.) 02/07/2025 PHY 712 Spring 2025 -- Lecture 11 4

Now consider a distribution of monopoles and dipoles -- ( ) r Monopole electric charge density : mono ( ) r ( ) 3 r r q mono i i i ( ) P r Electric polarization due to collection of dipoles: ( ) ( ) 3 P r p r r i i i It will be convenient to define some new electrostatic fields -- ( ) ( ) ( ) ( ) + = D r E r P r Define Displacement field: 0 ( ) r D r Macroscopic form of Gauss's law: mono 02/07/2025 PHY 712 Spring 2025 -- Lecture 11 5

Electrostatic potential for a single monopole charge and a single dipole : 1 4 Electrostatic potential for collections of monopole charges and dipoles : i ( ) ( ) o i i i = r r q p p r q r ( ) r = + 3 r 0 q i p ( ) ( ) 3 3 r r r P r p r r q mon i i i ( ) ' r ( ) ( ' P r r ) r r r ' ' 1 ( ) r mono r = + 3 3 ' ' d r d r 3 4 r ' 0 ( ) ( ' P r ) ( ) ' r r r P r ' ' ' 1 ( ) P r = 3 3 3 Note: ' ' ' ' ' d r d r d r 3 r r r ' ' 02/07/2025 PHY 712 Spring 2025 -- Lecture 11 6

Coarse grain representation of macroscopic distribution of dipoles -- continued: Electrostatic potential for collections of monopole charges and dipoles : i q i p ( ) ' r ( ) ' r r P r ' ' ' 1 ( ) r = mono r 3 3 ' ' d r d r r 4 0 1 ( ) ( ) r ( ) ( ) r ( ) = = 2 E r P r mono 0 ( ) ( ) ( ) D r ( ) r ( ) ( ) + = E r P r 0 mono ( ) ( ) + = E r P r Defi ne Displacement field: 0 ( ) r D r Macroscopic form of Gauss's law: mono 02/07/2025 PHY 712 Spring 2025 -- Lecture 11 7

More relationships due to the properties of materials -- Many materials are polarizable and produce a polarization field in the presence of an electric field with a proportionality constant Note: In practice, some dielectric functions may be dependent on time and/or space. : e ( ) ( ) = ( ) + = P r E r 0 e ( ) ) e ( ) ( ) ( ) E r ric function of the material ( ) = + D r E r P r E r 1 0 + 0 e ( 1 represents the dielect 0 Boundary value problems in dielectric materials ( ) ( ) ( ) 0 and At a surface between two dielectrics, in terms of surface normal : continuous and = r D r = r For 0 mono ( ) ( ) = = = D r E r D r E r 0 r ( ) ( ) = r E r continuous 02/07/2025 PHY 712 Spring 2025 -- Lecture 11 8

Boundary value problems in the presence of dielectrics example: 1 E1 2 = For 2 2 E2 1 Specifically, 2 = = D1 2 0 D2 1 0 ( ) r ( ) ( ) = = = D r E r For 0: 0 and 0 mono r At a surface between two dielectrics, in terms of surface normal : continuous and = r D r ( ) ( ) = r E r continuous = = D D E E 1 2 1 D 1 2 2 n n n n For isotropic dielectrics: D = = E E 1 2 t t 1 2 t t 1 2 02/07/2025 PHY 712 Spring 2025 -- Lecture 11 9

More details = = D E D E 1 1 1 2 2 2 2 1 = For 2 2 E1 1 E2 Specifically, 2 = = 2 0 D1 D2 1 0 = = A E E 0 d d = = D A D 0 dV d = = E E D D 1 2 t t 1 2 n n 02/07/2025 PHY 712 Spring 2025 -- Lecture 11 10

Boundary value problems in the presence of dielectrics example: E0 E a z ( ) r r ( ) r r ( ) r r ( ) r For = a = D E 0 and 0 = = At : r a 0 r ( ) ( ) r ( ) ( ) r = D r r r ( ) ( ) r = = D For a 0 02/07/2025 PHY 712 Spring 2025 -- Lecture 11 11

Boundary value problems in the presence of dielectrics example -- continued: ( ) r ( ) r r r r = = At : r a ( ) r ( ) = l 0 = l cos A r P l l ( ) r ( ) 0 = C ( ) r ( ) = l = + l cos l B r P ( ) r l l + 1 l r = For cos r E r 0 0 = Solution -- only B E = = 1 contributes l 1 0 + 3 / 1 = 3 0 A E C a E 1 0 1 0 + 2 / 2 / 0 0 02/07/2025 PHY 712 Spring 2025 -- Lecture 11 12

Boundary value problems in the presence of dielectrics example -- continued: 3 E + ( ) r = cos r 0 2 / 0 2 3 / + 1 a ( ) r = 0 cos r E 0 2 / r 0 (r = = ) ) = = 102 1 r/a 02/07/2025 PHY 712 Spring 2025 -- Lecture 11 13

Microscopic origin of dipole moments Polarizable atoms/molecules Anisotropic charged molecules aligned in random directions Polarizable systems E=0 E k=m 2 -q +q E = 2 0 E x At equilibrium: 0 q m q m x = x 02/07/2025 PHY 712 Spring 2025 -- Lecture 11 14 2 0

Polarizable atoms/molecules continued: k=m 2 -q +q E x At equilibrium: q E = 2 0 E x 0 m q = x Molecular susceptibility for this model: 2 0 m Induced dipole moment: 2 q = 2 q = = mol x E E p q 2 0 m 0 mol 2 0 m 0 02/07/2025 PHY 712 Spring 2025 -- Lecture 11 15

For a neutral atom E=0 E 02/07/2025 PHY 712 Spring 2025 -- Lecture 11 16

Alignment of molecules with permanent dipoles p0: E p0 For a freely rotating dipole its average moment in an electric field, estimated assuming a Boltzmann distribution: 2 0 0 3 kT Molecular susceptibility for this model: cos / p E kT cos d p e 2 0 0 1 3 p kT p E kT 0 = p E for 1 0 mol cos / p E kT d e 0 2 0 1 p 1 3 p p E E mol mol mol kT 0 02/07/2025 PHY 712 Spring 2025 -- Lecture 11 17

Now consider a superposition of dipoles in an electric field Eext Edipole Eext Edipole Eext Edipole 02/07/2025 PHY 712 Spring 2025 -- Lecture 11 18

Field due to collection of induced dipoles = + 0 E ( ) r E ( ) r E ( ) r tot i ext i = + 0 E ( ) r E ( ) r E ( ) r 1 ( ) site i j ext 2 j i 7 = 0 E ( ) r E ( ) r 3 tot i 6 5 Eext 4 9 8 Electrostatic field from single dipole: = ( ) ( ) ( r r ) 2 r r p r r r r p 3 1 4 ( ) ( ) i i i i i 0 i 3 E r p r r i i 5 4 3 0 PHY 712 Spring 2025 -- Lecture 11 i 02/07/2025 19

Field due to collection of induced dipoles -- continued = + 0 E ( ) r E ( ) r E ( ) r 1 tot i ext 2 i 7 3 = + 0 E ( ) r E ( ) r E ( ) r 6 ( ) site i j ext 5 j i 4 = 0 9 E ( ) r E ( ) r tot i 8 Eext ( ) ( ) ( r r ) 2 r r p r r r r p 3 1 4 ( ) ( ) ( ) r i i i i i = + 3 E r p r r E i i ext 5 4 3 tot i 0 i ( ) ( ) ( ) 2 r r p r r r r p 3 1 ( ) ( ) ( ) r ( ) ( ) j j j j j = + = 0 i E r E E r E r ext 5 4 ( ) site i tot ( ) site i r r j i 0 j Averaging: 1 1 3 V 1 = + = + E E p E P 1 V ( ) site i tot t o t 3 3 d r 0 0 V 02/07/2025 PHY 712 Spring 2025 -- Lecture 11 20

Field due to collection of induced dipoles -- continued ( ) ( ) ( ) ( ) = 0 i E r E r E r ( ) site i tot ( ) site i ( ) ( ) ( r r ) 2 r r p r r r r p 3 1 4 ( ) ( ) i i i i i = 3 E r p r r i i 5 4 3 tot 0 i ( ) ( ) ( r r ) 2 r r p r r r r p 3 1 4 ( ) i i i i i = 3 E E p r r ( ) s ite i tot i i 5 4 3 0 i 1 1 3 V 1 = + = + E E p E P ( ) site i tot tot 3 0 0 02/07/2025 PHY 712 Spring 2025 -- Lecture 11 21

Field due to collection of induced dipoles -- continued 1 = + E E P ( ) site i tot 3 1 2 0 7 = p E 3 0 mol site 6 5 1 V 1 = = + P p E P 0 mol 4 9 tot 3 V 8 Eext 0 E tot = = P E 0 mol 0 e tot V mol V 1 Claussius-Mossotti equation 3 mol V + / / 1 2 = = = 1 3 0 V e mol 1 mol V 0 0 3 02/07/2025 PHY 712 Spring 2025 -- Lecture 11 22

02/07/2025 PHY 712 Spring 2025 -- Lecture 11 23

Example of the Clausius-Mossotti equation Pentane (C5H12) at various densities / 0 3V*( / 0-1)/( / 0+2) Density (g/cm3) Mol/m3 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 mol = 1.2 x 10-28 m3= 0.12 nm3 02/07/2025 PHY 712 Spring 2025 -- Lecture 11 24

Re-examination of electrostatic energy in dielectric media 1 2 = ( ) ( ) r 3 r W d r mono In terms of displacement field: ( ) 1 2 = D r mono 1 2 1 2 ( ) = ( ) = ( ) ( ) D r 3 3 3 D r r ( ) D r ( ) r W d r d r d r 1 = + 3 ( ) D r E r 0 2 ( ) d r 1 2 = 3 ( ) D r E r ( ) W d r 02/07/2025 PHY 712 Spring 2025 -- Lecture 11 25

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) N. A. Spaldin, J. Solid State Chem. 195, 2 (2012) 0 Basic equations : = = + E = tot bound mono P bound = D mono = + E D P 0 Note: In general P is highly dependent on the boundary values; often it is more convenient/meaningful to calculate P. 02/07/2025 PHY 712 Spring 2025 -- Lecture 11 26

Comment on the Modern Theory of Polarization -- continued = bound = nuclear + electronic P bound bound e n = electronic P 0r w w 0 n n crystal V Note: The concept of the polarization of a periodic solid is not unique: 02/07/2025 PHY 712 Spring 2025 -- Lecture 11 27

P example 02/07/2025 PHY 712 Spring 2025 -- Lecture 11 28

P example -- linear visualization Effects on the electronic distribution Na Cl 02/07/2025 PHY 712 Spring 2025 -- Lecture 11 29

Summary of electrostatics Chapter 1-4 in Jackson By statics we mean that all properties are constant in time Fiel d equations in differential form: = E r r ( ) = ( ) 0 = = + = D r ( ) where ( ) r D r E r P r E r ( ) ( ) E r ( ) ( ) ( ) 0 mono 0 Electrostatic potential: r ( ) = = E r ( ) r ( ) r ( ) 0 02/07/2025 PHY 712 Spring 2025 -- Lecture 11 30