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UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 12 Prof. Steven Errede LECTURE NOTES 12 ELEMENTARY MICROSCOPIC THEORY OF DIELCTRICS Consider a Class-A/linear dielectric material consisting of non-polar molecules (i.e. having no permanent electric dipole moments) in gas, liquid or solid form, immersed in an external electric field Eext ( r ) . - each molecule will have an induced electric dipole moment of pmol (Coulomb-meters) each molecule is assumed (here, for simplicity) to be spherical in shape. In a real Class-A/linear non-polar dielectric, e.g. at room temperature, at the microscopic level, due to the thermal energy associated with the material making up the dielectric, from one instant in time to the next at any given point r inside the dielectric, random fluctuations of significant size can/do occur in the electric field at that point. If one simultaneously monitors a group of neighboring, microscopically nearby points in space at a particular instant in time, there are also significant, random fluctuations about an average value of the electric field in this region. Thus, what we need to know is the time-and-space averaged local electric field as seen by a single molecule in the dielectric at the point r - call this electric field Eloc ( r ) . Note that this time-and-space averaged electric field does not include a contribution from the electric field associated with the induced dipole moment of the molecule in question at the point r !!! Conceptually, in order to determine Eloc ( r ) , we imagine that we (momentarily) “freeze” the thermal motion of all molecules in the dielectric at a given instant in time – thus obtaining a “snapshot” of the microscopic configuration of the dielectric at that instant in time. We then imagine that we remove the molecule in question at the point r , keeping all other molecules “frozen” in their positions and orientations at that instant in time. We then calculate the spaceaveraged electric field intensity inside the (assumed) spherical cavity that was occupied by the molecule in question at the point r . We then momentarily replace the molecule, allow time to progress forward to some other instant in time, where we again freeze the motion of all molecules at this new instant in time, remove the same molecule, and repeat the space-averaged electric field calculation at the point r . We repeat this procedure many, many times until we obtain a time-and-space-averaged value of the local electric field at the point r , ≡ Eloc ( r ) . Thus, we also expect the time-and-space averaged local electric field at the point r , Eloc ( r ) to be larger than the macroscopic electric field E ( r ) at the same point, precisely because Eloc ( r ) does not include the (time-and-space-averaged) electric field of the molecule under scrutiny at the point r , E ( r ) mol dipole which points in the direction opposite to that of the applied external electric field, Eext ( r ) as shown in the figure below: © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2008. All Rights Reserved. 1 UIUC Physics 435 EM Fields & Sources I Eext Fall Semester, 2007 Lecture Notes 12 Prof. Steven Errede ΔV = Eo xˆ = Vo d E −Q pmol E + x̂ +Q mol dipole +σ free −σ free In P435 Lecture Notes 10 (p. 1-6) {again, see also Griffiths problem 3.41(a-c)}, we learned that the average electric field intensity within a sphere of radius R containing an arbitrary charge distribution, which we identify as equal to the macroscopic, microscopically time and spaceaveraged electric field associated with the induced molecular electric dipole moment pmol ( r ) of the molecule in question at the point r is given by: E (r ) mol dipole =− pmol ( r ) where R = radius of a single spherical molecule 4πε o R 3 Define: nmol ≡ # molecules per unit volume = molecular number density (#/m3). Then the macroscopic (i.e. microscopically space-and-time-averaged) electric dipole moment per unit volume (a.k.a. electric polarization) is: Ρ ( r ) = nmol pmol ( r ) and thus: pmol ( r ) = Ρ ( r ) nmol Then: E (r ) mol dipole =− pmol ( r ) Ρ (r ) Ρ (r ) Ρ (r ) =− =− =− 3 3 4πε o R 4πε o nmol R 3ε o ( vmol ∗ nmol ) 3ε o Where: vmol = 4π R 3 3 = spherical volume occupied by a single spherical molecule, and note also that: ( vmol ∗ nmol ) = 1. Thus: 2 E (r ) mol dipole =− Ρ(r ) 3ε o © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2008. All Rights Reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 12 Prof. Steven Errede Thus it can be seen that the macroscopic E -field at the point r , E ( r ) is the linear superposition of a.) the space-and-time-averaged local electric field Eloc ( r ) at that point, plus b.) the spaceand-time averaged electric field associated with the induced electric dipole moment pmol ( r ) of the molecule in question at the point r , E ( r ) mol dipole E ( r ) = Eloc ( r ) + E ( r ) But E (r ) mol dipole =− , i.e.: mol dipole Ρ (r ) Ρ (r ) (for spherical molecules), and thus: E ( r ) = Eloc ( r ) − . 3ε o 3ε o Turning this around, the space-and-time-averaged local electric field is thus: Eloc ( r ) = E ( r ) − E ( r ) mol dipole = E (r ) + Ρ (r ) 3ε o Thus, this last relation explicitly shows that the magnitude of the space-and-time averaged local electric field Eloc ( r ) is larger than the macroscopic electric field E ( r ) , since the electric polarization Ρ ( r ) (and also pmol ( r ) ) point in the same direction as the macroscopic electric field E ( r ) . Note that Eloc ( r ) also points in the same direction as E ( r ) (and thus Ρ ( r ) and pmol ( r ) ), whereas the space-and-time-averaged electric field due to the induced dipole moment of the molecule in question at the point r , E ( r ) mol points in the opposite direction to E ( r ) , dipole Eloc ( r ) , Ρ ( r ) and pmol ( r ) . Note also that in general: Eloc ( r ) = E ( r ) + b Ρ (r ) εo where b = 1/3 for spherical molecules. For molecules with shapes other than a sphere, the constant b = some number ~ Ο (1) . In a linear/Class-A dielectric, the molecular charge separation is proportional to the local electric field, i.e. pmol ( r ) = qd mol and Eloc ( r ) are linearly related to each other by: pmol ( r ) = α mol Eloc ( r ) where α mol ≡ molecular electric polarizability (SI Units: Coulombs2/Newton/meter). But: Ρ ( r ) = nmol pmol ( r ) Thus: Ρ ( r ) = nmol pmol ( r ) = nmolα mol Eloc ( r ) But: Eloc ( r ) = E ( r ) + ⎛ Ρ (r ) Ρ (r ) ⎞ Thus: Ρ ( r ) = nmolα mol ⎜⎜ E ( r ) + ⎟ 3ε o 3ε o ⎟⎠ ⎝ © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2008. All Rights Reserved. 3 UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 12 Prof. Steven Errede Then solving for the electric polarization, Ρ ( r ) in terms of the macroscopic electric field, E ( r ) : ⎛ Ρ (r ) ⎞ Ρ (r ) Ρ ( r ) = nmolα mol ⎜⎜ E ( r ) + ⎟⎟ = nmolα mol E ( r ) + nmolα mol 3ε o ⎠ 3ε o ⎝ ⎛ nmolα mol ⎞ ⎜1 − ⎟ Ρ ( r ) = nmolα mol E ( r ) 3ε o ⎠ ⎝ Thus: Ρ ( r ) = nmolα mol ⎛ nmolα mol ⎜1 − 3ε o ⎝ ⎞ ⎟ ⎠ E (r ) But: Ρ ( r ) = ε o χ e E ( r ) for a linear/Class-A non-polar dielectric. Solving for χ e : nmolα mol E ( r ) = ε o χe E ( r ) ⎛ nmolα mol ⎞ ⎜1 − ⎟ 3ε o ⎠ ⎝ nmolα mol = ε o χe ⎛ nmolα mol ⎞ ⎜1 − ⎟ 3ε o ⎠ ⎝ ⎛ nmolα mol ⎞ ⎜ ⎟ εo ⎠ ⎝ Thus: χ e = = electric susceptibility of a linear/Class-A non-polar dielectric ⎛ nmolα mol ⎞ (assuming spherically-shaped molecules). ⎜1 − ⎟ 3ε o ⎠ ⎝ For gases (e.g. @STP), note that ( nmolα mol ε o ) Ο (10−3 ) 1 , since nmol = ρ mmol N A where: nmol = number density of molecules (# / m3) ρ = mass density (kg / m3) NA = Avogadro’s # (6.022 × 1023 molecules / mole) mmol = molecular weight (# kg / mole) Thus for non-polar gases @ STP, we can safely neglect the ( nmolα mol 3ε o ) term in the denominator of the expression for the electric susceptibility, approximating it (quite well) as: ( nmolα mol ε o ) nmolα mol because ( nmolα mol ε o ) 1 χ egas = 1 − ( nmolα mol 3ε o ) εo And: 4 K egas = 1 + χ egas = 1 + ( nmolα mol ε o ) 1 − ( nmolα mol 3ε o ) 1+ nmolα mol εo because ( nmolα mol ε o ) 1 © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2008. All Rights Reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 12 Prof. Steven Errede The table below compares theory vs. experiment for electric susceptibility χ egas and dielectric constant K egas = 1 + χ egas for a few simple non-polar gases: ρgas Gas (gms/cc) 0.00339 0.00143 0.00489 0.00178 CS2 O2 CCl4 A mmol (gms/mole) 76.0 32.0 153.8 39.9 nmol αmol (#/m3) (C2/{N/m}) 2.7×1025 9.55×10−40 2.7×1025 1.72×10−40 1.9×1025 1.39×10−40 2.7×1025 1.80×10−40 χ egas ( pred ) K egas ( pred ) K egas (expt ) 0.0029 0.0005 0.0030 0.0055 1.0029 1.0005 1.0030 1.0055 1.0029 1.0005 1.0030 1.0055 It can be seen that for these simple non-polar gases, the theory predictions obtained from this simple microscopic model of a linear/Class-A non-polar dielectric vs. experimental measurements of the electric susceptibility χ egas and dielectric constant K egas = 1 + χ egas are in excellent agreement with each other. For liquids (and solids), ( nmolα mol ε o ) Ο (1) and thus this term cannot be neglected in the formula for the electric susceptibility. The table below compares theory vs. experiment for electric susceptibility χ eliquid and dielectric constant K eliquid = 1 + χ eliquid for the liquid-form versions of the entries in the above table. The results for the liquid-form are obtained using the results from the gas-form, scaled by the ratio of volume mass densities. Liquid CS2 O2 CCl4 A ρliquid ρliquid (gms/cc) ρgas 1.29 1.19 1.59 1.44 381 832 325 810 nmol (#/m3) 1.0×1028 2.2×1028 6.2×1028 2.2×1028 nmolα mol ε o χ eliquid ( pred ) K eliquid ( pred ) K eliquid (expt ) 1.11 0.435 0.977 0.441 1.76 0.51 1.45 0.52 2.76 1.51 2.45 1.52 2.64 1.51 2.24 1.54 Here the agreement of theory prediction vs. experiment is quite good. © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2008. All Rights Reserved. 5 UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 12 Prof. Steven Errede THE CLAUSIUS-MOSSOTTI EQUATION We have obtained an expression for the local electric field Eloc ( r ) inside a linear/Class-A nonpolar dielectric: Eloc ( r ) = E ( r ) + Ρ(r ) 3ε o We have also obtained an expression for the electric polarization associated with this dielectric: ⎛ Ρ (r ) ⎞ Ρ ( r ) = nmolα mol ⎜⎜ E ( r ) + ⎟ 3ε o ⎟⎠ ⎝ However, inside a linear/Class-A non-polar dielectric the electric displacement D ( r ) , the electric field E ( r ) and electric polarization Ρ ( r ) are related to each other by: D (r ) = E (r ) + Ρ (r ) Thus: Ρ ( r ) = D ( r ) − ε o E ( r ) but inside a linear/Class-A non-polar dielectric the electric displacement D ( r ) and the electric field E ( r ) are also related to each other by: D ( r ) = ε E ( r ) . Thus: Ρ ( r ) = ( ε − ε o ) E ( r ) = ε o ( K e − 1) E ( r ) where K e ≡ ε ε o = 1 + χ e Thus, inside a linear/Class-A non-polar dielectric the electric polarization Ρ ( r ) and the electric field E ( r ) are also related to each other by: Ρ ( r ) = ε o χ e E ( r ) ⎛ Ρ (r ) ⎞ Then since: Ρ ( r ) = ε o ( K e − 1) E ( r ) = ε o χ e E ( r ) and: Ρ ( r ) = nmolα mol ⎜⎜ E ( r ) + ⎟ 3ε o ⎟⎠ ⎝ ⎛ ε o ( K e − 1) E ( r ) ⎞ ⎛ Ρ(r ) ⎞ Thus: ε o ( K e − 1) E ( r ) = nmolα mol ⎜⎜ E ( r ) + ⎟ ⎟⎟ = nmolα mol ⎜⎜ E ( r ) + ⎟ 3ε o ⎠ 3εo ⎝ ⎝ ⎠ ⎛ ( K − 1) ⎞ ⎛ 3 + ( K e − 1) ⎞ ( Ke + 2) ∴ ε o ( K e − 1) = nmolα mol ⎜1 + e ⎟ = nmolα mol ⎜ ⎟ = nmolα mol 3 ⎠ 3 3 ⎝ ⎝ ⎠ ⎛ K − 1 ⎞ ⎛ nmolα mol ⎞ ⎛ χ e ⎞ ⎛ nmolα mol ⎞ or: ⎜ e ⎟=⎜ ⎟ or equivalently: ⎜ ⎟=⎜ ⎟ ⎝ K e + 2 ⎠ ⎝ 3ε o ⎠ ⎝ χ e + 3 ⎠ ⎝ 3ε o ⎠ 6 © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2008. All Rights Reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 12 Prof. Steven Errede If the Class-A/linear non-polar dielectric is a compound dielectric composed of N different types of (spherical) non-polar molecules, then we obtain the so-called Clausius-Mossotti Equation: ⎛ Ke − 1 ⎞ 1 ⎜ ⎟= ⎝ K e + 2 ⎠ 3ε o N ∑n mol i α imol ⇐ Clausius-Mossotti Equation i =1 The Clausius-Mossotti Equation relates the dielectric constant K e = ( ε ε o ) = (1 + χ e ) for a Class-A/linear non-polar dielectric material (n.b. assumed to have spherically-shaped molecular electric dipoles) to the mass density ρ (kg/m3) of the Class-A dielectric material and the molecular electric polarizability, α mol . If nmol = ρ mmol N A then for a single molecular species the Clausius-Mossotti equation becomes: ⎛ K e − 1 ⎞ ⎛ nmolα mol ⎞ ⎛ K e − 1 ⎞ ρ N Aα mol 1 ⎛ K e − 1 ⎞ ⎛ N Aα mol or: ⎜ ⎟=⎜ ⎟ then: ⎜ ⎟= ⎜ ⎟=⎜ ρ ⎝ K e + 2 ⎠ ⎝ 3ε o ⎝ K e + 2 ⎠ ⎝ 3ε o ⎠ ⎝ K e + 2 ⎠ 3ε o mmol ⎞ 1 ⎟ ⎠ mmol ⎛N ⎞ We define: Ρ molar = molar electric polarization ≡ ⎜ A ⎟ α mol ⎝ 3ε o ⎠ Then: 1 ⎛ K e − 1 ⎞ Ρ molar ⎜ ⎟= ρ ⎝ K e + 2 ⎠ mmol n.b. The RHS of this equation (and thus also the LHS) is independent of the mass density ρ !!! {This is true for a wide variety of gases and also non-polar liquids (approximately ~ 50)} ⎛ K − 1 ⎞ nmolα mol For the single molecular species the Clausius-Mossotti Equation: ⎜ e ⎟= 3ε o ⎝ Ke + 2 ⎠ Note that if nmol → 3ε o α mol ⎛α then the RHS of C-M Equation nmol ⎜ mol ⎝ 3ε o ⎞ ⎛ 3ε o ⎞⎛ α mol ⎞ ⎟→ ⎜ ⎟⎜ ⎟ →1 ⎠ ⎝ α mol ⎠⎝ 3ε o ⎠ ⎛ K −1 ⎞ ⎛ 3ε ⎞ Thus, when nmol → ⎜ o ⎟ , then ⎜ e ⎟ → 1, i.e. K e → ∞ !!! ⎝ Ke + 2 ⎠ ⎝ α mol ⎠ Physically, this corresponds to (spherical) molecules that become infinitely polarizable!! For gases and liquids, the molecular electric polarizabilities α mol are relatively small, ⎛ 3ε ⎞ so nmol → ⎜ o ⎟ never actually happens for gases and/or liquids. ⎝ α mol ⎠ For crystalline solids, this simple model of molecular polarization is simply too crude – it doesn’t work / doesn’t agree well with crystal data… © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2008. All Rights Reserved. 7 UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 12 Prof. Steven Errede Polar Dielectrics and the Langevín Equation Polar dielectrics (e.g. water, calcite, quartz, . . .) have permanent molecular electric dipole moments. Molecules consisting of two (or more) dissimilar atoms can exhibit permanent electric dipole moments. Diatomic molecules such as O2, N2, etc. with identical atoms cannot, due to symmetry considerations – there is no asymmetric way to arrange charge with identical atoms!!! The potential energy of a single molecule with a permanent electric dipole moment pmol ( r ) inside a polar dielectric is: W1 = P.E.1 = − pmol ( r )i Eloc ( r ) (n.b. W1 = P.E.1 is a minimum when pmol Eloc ) θ pmol Eloc The Langevín Equation In a gaseous or liquid dielectric, the thermal energy of the medium (due to it being at finite temperature) causes collisions between the molecules, which tend to destroy / randomize any net alignment of the polar molecules with the local field Eloc (i.e. thermal energy / thermal agitation depolarizes the macroscopic alignment of such molecules). However, the local electric field Eloc exerts a restoring force (via a torque) on the electric dipoles between collisions. Thus a partial net macroscopic alignment, or electric polarization will exist, i.e. a net Ρ (electric dipole moment per unit volume) will exist, macroscopically. For no applied external electric field (i.e. Eext = 0 ) the dipoles are oriented at random. If there are nmol molecules per unit volume, then for random orientation of dipoles, the fraction dnmol nmol of polar molecules within angles θ and (θ + dθ ) {see figure above} is just: d cosθ dnmol d Ω 2π d cos θ 2π sin θ dθ 1 = = = = sin θ dθ nmol Ω 4π 4π 2 where nmol = # polar molecules per unit volume and dnmol = # polar molecules per unit volume within θ and (θ + dθ ) . The factor of 2π arises from azimuthal (ϕ ) symmetry – the dot product pmol i Eloc only cares about the (polar) angle θ between pmol and Eloc . Note that random orientations are flat probability distributions in (ϕ , cos θ ) with ϕ ranging over 0 ≤ ϕ ≤ 2π and cos θ ranging over −1 ≤ cos θ ≤ 1 (i.e. where the polar angle variable θ ranges over 0 ≤ θ ≤ π ). 8 © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2008. All Rights Reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 12 Prof. Steven Errede If the permanent dipoles are then subjected to an externally-applied electric field Eext and are also in thermal equilibrium with each other, then number dnmol of molecules per unit volume possessing a particular potential energy W is given by the Boltzmann distribution law: dnmol = C e−W kBT where k B =Boltzmann’s Constant = 1.381 × 10−23 J/K constant T = Absolute Temperature (Kelvin degrees) For a single polar molecule in a gas of identical particles: W1 = − pmol i Eloc = − pmol Eloc cos θ The number of polar electric dipoles / unit volume within θ and (θ + dθ ) is: dnmol = Ce ( + pmol i Eloc = Ce u cosθ k BT ) + p sin θ dθ = Ce ( mol Eloc cosθ k BT ) sin θ dθ sin θ dθ where u ≡ pmol Eloc k BT The constant C is a normalization constant such that the total # of molecules / unit volume nmol is given by: π nmol = C ∫ eu cosθ sin θ dθ 0 Now the polar molecules whose permanent electric dipole moments lie within θ and (θ + dθ ) possess a total electric dipole moment per unit volume in the direction of the local electric field of: eu cosθ cos θ sin θ dθ d Ρ = dnmol pmol cos θ = nmol pmol π u cosθ ∫ e sin θ dθ 0 Thus the NET electric dipole moment per unit volume (= electric polarization) P is given by: Ρ = ∫ d Ρ = nmol pmol cos θ = nmol pmol ∫ π 0 eu cosθ cos θ sin θ dθ ∫ π 0 We define: t ≡ u cos θ ⎛p E ⎞ thus: t = ⎜ mol loc ⎟ cos θ = u cos θ ⎝ k BT ⎠ +u n p n p Then: Ρ = mol mol t = mol mol u u ∫ ∫ −u +u −u Or: eu cosθ sin θ dθ et tdt et dt = t t nmol pmol ⎡⎣te − e ⎤⎦ u et +− uu ⎡ ⎛ k BT ⎛p E ⎞ Ρ = nmol pmol ⎢ coth ⎜ mol loc ⎟ − ⎜⎜ ⎝ k BT ⎠ ⎝ pmol Eloc ⎣⎢ +u −u ⎞⎤ ⎟⎟ ⎥ ⇐ Langevín Equation ⎠ ⎦⎥ At room temperature (T ~ 300K) the <thermal energy> is: k BT ∼ 4 × 10−21 Joules ( ∼ 1 40 eV ( electron − Volt ) ) (1 eV = 1.602×10−19 Joules) © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2008. All Rights Reserved. 9 UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 12 Prof. Steven Errede Typical permanent/polar molecular electric dipole moments are on the order of pmol ∼ Ο (10−31 ) Coulomb-meters, and for a typical value of Eloc ∼ 107 V / m , then: ⇒ u (T ) = pmol Eloc k BT 2 × 10−3 ( 1) @ T = Troom ~ 300 K Now since u (T ) is quite small in this situation, we can expand the Langevín Equation in a Taylor series expansion, retaining terms up to order u 3 : ⎤ 1 u u 3 2u 5 1 ⎡ u 2 u 4 2u 6 coth ( u ) = + − + + .... = ⎢ − + + ....⎥ u 3 45 945 u ⎣ 3 45 945 ⎦ 2 + u2 2u ⎡⎣1 + ( u 2 6 ) ⎤⎦ Then: 1⎤ ⎡ Ρ = nmol pmol ⎢ coth u − ⎥ u⎦ ⎣ Or: Ρ (r ) Thus: Ρ ( r ) ⎡ 2 + u2 1 ⎥⎤ ⎢ nmol pmol − ⎢ 2u ⎡1 + ( u 2 6 ) ⎤ u ⎥ ⎦ ⎣ ⎣ ⎦ ⎡(2 + u2 ) ⎛ u2 ⎞ 1 ⎤ nmol pmol ⎢ ⎜1 − ⎟ − ⎥ 6 ⎠ u⎥ ⎢⎣ 2u ⎝ ⎦ 2 nmol pmol u ( r ) nmol pmol = Eloc ( r ) 3 3k BT 2 nmol pmol Eloc ( r ) for T ~ 300K {i.e. pmol Eloc 3k BT ⇒ When pmol Eloc k BT } k BT the electric polarization Ρ ( r ) in a polar dielectric is linearly proportional to the local electric field Eloc ( r ) . Now for a linear/Class-A polar dielectric we also have the relation: Ρ ( r ) = ε o χ e E ( r ) But: Eloc ( r ) = E ( r ) + Ρ (r ) Ρ (r ) or: E ( r ) = Eloc ( r ) − for spherical polar molecules. 3ε o 3ε o ⎛ Ρ (r ) ⎞ Thus: Ρ ( r ) = ε o χ e E ( r ) = ε o χ e ⎜⎜ Eloc ( r ) − ⎟ 3ε o ⎟⎠ ⎝ Or: ⎛ K −1 ⎞ ε o χe Eloc ( r ) = 3ε o ⎜ e Ρ (r ) = ⎟ Eloc ( r ) then using: Ρ ( r ) ⎛ χe ⎞ ⎝ Ke + 2 ⎠ ⎜1 + ⎟ 3 ⎠ ⎝ n p2 We see that: mol mol Eloc ( r ) 3k BT ⎛ K −1 ⎞ or: ⎜ e ⎟ ⎝ Ke + 2 ⎠ 10 2 nmol pmol Eloc ( r ) 3k BT ⎛ K −1 ⎞ 3ε o ⎜ e ⎟ Eloc ( r ) ⎝ Ke + 2 ⎠ 2 nmol pmol when pmol Eloc 9ε o k BT k BT , i.e. when T ~ 300K . © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2008. All Rights Reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 12 Prof. Steven Errede We can solve the above relation for the dielectric constant K e and electric susceptibility χ e = K e − 1 associated with a polar dielectric. Defining: a ≡ (i.e. pmol Eloc 2 nmol pmol , then for T 9ε o k BT 300 K k BT ): 2 ⎛ 2nmol pmol + 1 ⎜ 9ε o k BT ⎛ 1 + 2a ⎞ ⎜ = Ke = ⎜ ⎟ ⎜ 2 nmol pmol ⎝ 1− a ⎠ − 1 ⎜ 9ε o k BT ⎝ ⇒ When pmol Eloc 2 ⎞ ⎛ nmol pmol ⎟ ⎜ ⎟ ≥ 1 and: χ e = K e − 1 = ⎛⎜ 1 + 2a ⎞⎟ − 1 = ⎛⎜ 3a ⎞⎟ = ⎜ 3ε o k BT 2 ⎟ ⎝ 1− a ⎠ ⎝ 1 − a ⎠ ⎜ 1 − nmol pmol ⎟ ⎜ 9ε o k BT ⎠ ⎝ ⎞ ⎟ ⎟≥0 ⎟ ⎟ ⎠ k BT , dielectric constant K e and electric susceptibility χ e of a polar dielectric are inversely proportional to the absolute temperature T. Note that non-polar dielectrics have no such temperature dependence! For polar gases, where χ e ~ 0 and K e ~ 1 , note that: χ egas ≈ 2 nmol pmol 2n p 2 ≥ 0 and K egas = 1 + χ egas ≈ 1 + mol mol ≥ 1 3ε o k BT 9ε o k BT p E 1⎤ Ρ ⎡ A plot of the Langevín function f L ( u ) = ⎢ coth ( u ) − ⎥ = versus u = mol loc : u ⎦ nmol pmol k BT ⎣ Slope @ u = ∞ : m = 0 fL (u → ∞ ) = 1 1⎤ ⎡ f L ( u ) = ⎢coth ( u ) − ⎥ vs. u u⎦ ⎣ Initial slope @ u = 0: minit = 1/3 fL (u ) fL (u = 0) = 0 0 1 2 3 4 5 10 u = pmol Eloc k BT The DeBye Equation for Polar Dielectrics Consider a real polar dielectric – i.e. one in which both permanent and induced molecular electric dipoles are taken into account. Then by the principle of linear superposition: ΡTOT ( r ) = Ρinduced ( r ) + Ρ polar ( r ) Now: Ρinduced ( r ) = nmol pmol induced (r ) Induced electric dipole moment per unit volume nmol = number density of molecular dipoles (#/m3) pmol induced ( r ) = induced electric dipole moment at point r © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2008. All Rights Reserved. 11 UIUC Physics 435 EM Fields & Sources I But: pmol induced ( r ) = α mol Fall Semester, 2007 Lecture Notes 12 ⇒ ∴ Ρinduced ( r ) = nmol pmol Eloc ( r ) induced molecular polarizability Prof. Steven Errede ( r ) = nmolα mol Eloc ( r ) And (from the Langevín Equation, for u << 1): Ρ polar ( r ) 2 nmol pmol permanent 3k BT Eloc ( r ) for T ~ 300K ( i.e. pmol Eloc ∴ Ρ = ΡTOT ( r ) = Ρ induced ( r ) + Ρ polar ( r ) = nmolα mol Eloc ( r ) + But: Eloc ( r ) = E ( r ) − Emolecular ( r ) = E ( r ) + dipoles k BT ) 2 nmol pmol permanent 3k BT Eloc ( r ) Ρ (r ) 3ε o n.b. This is an important (but simplifying) assumption here - because it implicitly assumes b = 1/ 3 spherical-shaped molecules… However, for linear/Class-A dielectrics we also have: Ρ ( r ) = ε o χ e E ( r ) = ε o ( K e − 1) E ( r ) Then: Eloc ( r ) = E ( r ) + ε o ( K e − 1) ⎞ Ρ (r ) ⎛ ⎛ ( K − 1) ⎞ ⎛ K +2⎞ = ⎜1 + E (r ) = ⎜ e ⎟ E ( r ) = ⎜1 + e ⎟ ⎟ E (r ) ⎟ 3ε o ⎜⎝ 3 3 3 εo ⎝ ⎠ ⎝ ⎠ ⎠ Thus we obtain an important relationship between the space-and-time-averaged local electric field Eloc ( r ) and the macroscopic electric field, E ( r ) : ⎛ 3 ⎞ or: E ( r ) = ⎜ ⎟ Eloc ( r ) ⎝ Ke + 2 ⎠ Again, it can be seen from this relation that Eloc ( r ) is parallel to E ( r ) but because K e ≥ 1 we ⎛ K +2⎞ Eloc ( r ) = ⎜ e ⎟ E (r ) ⎝ 3 ⎠ also see that Eloc ( r ) > E ( r ) . From the above relation(s) for the electric polarization, we see that: 2 nmol pmol ⎛ K −1 ⎞ permanent Ρ = nmolα mol Eloc ( r ) + Eloc ( r ) = ε o ( K e − 1) E ( r ) = 3ε o ⎜ e ⎟ Eloc ( r ) 3k BT 2 + K ⎝ e ⎠ 2 ⎛ pmol ⎛ K − 1 ⎞ ⎛ n mol ⎞ ⎜ permanent Or: ⎜ e ⎟=⎜ ⎟ ⎜ α mol + 3k BT ⎝ K e + 2 ⎠ ⎝ 3ε o ⎠ ⎜ ⎝ 12 ⎞ ⎟ ⎟ ⇐ DeBye Equation for Class-A polar dielectrics ⎟ ⎠ © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2008. All Rights Reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 12 Prof. Steven Errede Compare this to: ⎛ K e − 1 ⎞ ⎛ n mol ⎞ ⎜ ⎟=⎜ ⎟ α mol ⇐ Clausius-Mossotti Equation for Class-A non-polar dielectrics ⎝ K e + 2 ⎠ ⎝ 3ε o ⎠ Thus we see that the DeBye Equation essentially is the C-M Equation, just with an additional term on the RHS, depending on 1/T due to the contribution arising from the polar molecules, as seen from the Langevín equation for polar molecules at finite temperature. ⎛m ⎞ Let us now multiply both sides of the DeBye equation by: ⎜ mol ⎟ molecular mass (kg/mole) ⎝ ρ ⎠ Recall that: ( m ) = mρ nmol = # density of molecules # 3 NA NA = Avogadro’s number 6.022 × 1023 molecules/mole mol Rearranging the above relation for polar dielectrics, we obtain the molar polarization as : Ρ molar ≡ 2 ⎞ mmol ⎛ K e − 1 ⎞ ⎛ N A ⎞ ⎛ pmol α = + ⎜ ⎟ ⎜ ⎟ ⎜ mol ⎟ ⇐ DeBye Equation for polar molecules ρ ⎝ K e + 2 ⎠ ⎝ 3ε o ⎠ ⎝ 3k BT ⎠ Again, note that the RHS of this equation is independent of mass density (hence so is the LHS). Note also that this equation is that for a straight line, i.e. y ( x ) = mx + b with x = 1/ T , and 2 ⎛ N A pmol ⎞ ⎛ N Aα mol ⎞ mmol ⎛ K e (T ) − 1 ⎞ ⎜⎜ ⎟⎟ , slope m = ⎜ ⎟ and intercept b = ⎜ ⎟ ρ (T ) ⎝ K e (T ) + 2 ⎠ ⎝ 9ε o k B ⎠ ⎝ 3ε o ⎠ as shown in the figure below: y ( x ) = Ρ molar (1/ T ) ≡ ⎛ mmol ⎜ ⎝ ρ ⎞ ⎛ Ke − 1 ⎞ ⎟ ⎟⎜ ⎠ ⎝ Ke + 2 ⎠ 2 N A pmol slope = 9ε o k B N Aα mol 3ε o 0 1/T 2 pmol m = Note also that the ratio of slope/intercept, many things cancel, except: b 3k Bα mol In real life, it is only possible to reliably measure α mol and pmol for gases, or for dilute solutions of polar molecules in non-polar solvents. © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2008. All Rights Reserved. 13 UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 12 Prof. Steven Errede Ferro-Electricity: ⎛ K − 1 ⎞ nmolα mol In a certain crystalline solids, the condition ⎜ e ≈ 1 is satisfied, i.e. K e → ∞ and ⎟= 3ε o ⎝ Ke + 2 ⎠ χ e = ( K e − 1) → ∞ ! In this situation, such materials exhibit permanent polarization (e.g. permanently polarized materials, such as electrets). Thus, nmolα mol ≈ 1 can be taken as a necessary condition for permanent polarization to occur. 3ε o The Making of a “Ferro-Electret”: Suppose the (absolute) temperature T is very high (i.e. let T → ∞). No permanent polarization Ρ permanent can exist, because the thermal energies ∼ k BT are >> pmol Eloc , randomizing the (net) dipoles polarization orientation. However, ∃ (there exists) a temperature Tc (known as the Curie Temperature) at (or below) which a ferro-electric material spontaneously generates a net permanent electric polarization Ρ (this is precisely what happens e.g. for ferro-magnets, spontaneously developing a permanent magnetization Μ (= magnetic dipole moment per unit volume). This spontaneous electrically-polarized state is relatively stable and can exist for a very long time!!! Examples of Ferro-Electric Materials: Barium Titanate (BaTiO3), Curie Temperature Tc 120oC Lead Zirconate (PbZrO3) Potassium Tantalum Niobiate (KTaNbO3) has Ke = 34,000 @ 0oC!!! Lead Titanate (PbTiO3) Certain (organic) polymers such as Kynar ® Film = Polyvinylidene Flouride (a.k.a. PVDF) Examples of Ferro-Magnetic Materials: Iron (Tc ~ 770oC) AlNiCo Alloys Samarium-Cobalt Other rare earth magnets using Neodymium, e.g. Neodymium-Boron… Thermal Stability of Kynar ® (PVDF) Film % Retention of Electric Polarization Ρ T = 22oC T = 60oC T = 80oC 100 80 60 40 20 2 100 1000 # Days A polarized ferro-electric material is stable against a reversed external electric field, provided it is not too large. The electric polarization Ρ of such a material, exhibits hysteresis (= “to lag behind”) 14 © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2008. All Rights Reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 “Typical” hysteresis curve of Ρ vs. Eloc Lecture Notes 12 Prof. Steven Errede for a ferro-electric material: Ρ Points a & c are where Eloc = 0 but Ρ ≠ 0 a b ϑ c d Eloc Points b and d are where Eloc is (just) large enough to permanently reverse/flip the polarization Ρ Compare the above curve for a ferro-electric material with that for a class-A dielectric (where Ρ = ε o χ e E is valid): Ρ slope = ε o χ e E For ferro-electric materials, we see that no simple relation between Ρ and Ε exists – it is in fact (highly) non-linear and depends on previous history of sample of material! © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2008. All Rights Reserved. 15 UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 12 Prof. Steven Errede Piezo-electricity Piezo-electricity (= “pressure” electricity) is the capability (or ability) of certain crystalline materials to change their dimensions when subjected to an externally-applied electric field; or conversely, the ability to produce electrical signals (i.e. voltages) when mechanically deformed! Examples of Piezo-Electric materials: Quartz crystals (discovered by Pierre & Jacques Curie in 1880’s) Rochelle Salts Tourmeline Kynar ® Piezo-Electric Film (PVDF film) Some Barium Salts (e.g. Barium Titanate – BaTiO3) Human & other mammal bones!!! (crucial for remodeling of bones, esp. after breaking one – evolution at work!!!) S (r ) = d E (r ) E (r ) = g x (r ) Resultant stress From applied Piezo strain constant of Piezo-Electric voltage constant Eext field material ((V/m)/(N/m2)) (meters/meter) ((m/m)/(Volt/m)) stress (N/m2) Piezo-Electricity arises from the electrical polarization produced by mechanical strains in crystals (belonging to certain classes of symmetries). The polarization Ρ is proportional to strain. If a crystal is centro-symmetric, it cannot be Piezo-Electric. The existence of a polar axis in a crystal gives rise to an inherent, spontaneous polarization Ρ . Pyro-Electricity Pyro-Electricity (= “heat” electricity) is the ability (or capability) of certain crystalline materials to produce electrical signals (i.e. voltages) when exposed to changes in temperature (i.e. changes in internal thermal energy). This phenomenon is due to electric polarization Ρ induced by thermal energy absorption in certain crystals. The amount of polarization ΔΡ is proportional to the change in the thermal energy ΔU thermal and hence is proportional to the change in the temperature ΔT . Examples of Pyro-Electric materials: Kynar ® (PVDF) film Tri-glycerine sulfate (TGS) Lead-zirconate-titanate (PZT) 16 © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2008. All Rights Reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 12 Prof. Steven Errede ΔQB = (bound) charge produced by a change in Temperature ΔT ΔQB = charge (coulombs) ρ ΔT pyro-electric coefficient ( coul 2 o ) m / K A Temp cross-sectional change area For Kynar ® film, ρ ≈ 170 nC ( cm 2 K ) Nano Coulombs = 10−9 Coulombs The Relationship Between the Index of Refraction n and the Dielectric Constant Ke In free space / vacuum, the speed of propagation of electromagnetic radiation (real photons) is c = 3 x 108 m/s, which is related to the macroscopic parameters of the vacuum - ε o and μo by: 1 c= ε o μo ε o = 8.85 ×10−12 Farads/m = electric permittivity of free space μo = 4π ×10−7 Henrys/m = magnetic permeability of free space ε o and μo are macroscopic electric and magnetic properties (respectively) of the (physical) QED vacuum {QED = Quantum Electro-Dynamics} At the microscopic level, the (physical) QED vacuum consists of electrically-charged, virtual fermion-antifermion pairs flitting in/out of existence, as allowed by the Heisenberg uncertainty principle, ΔE Δt ≤ . The vacuum is relativistically Lorentz invariant (i.e. no absolute origin exists), thus ε o and μo cannot/do not/must not have any frequency dependence - i.e. ε o ≠ ε o ( f ) and μo ≠ μo ( f ) because any frequency dependence of the vacuum is forbidden by Lorentz invariance. In physical matter, electromagnetic waves propagate through physical matter in a manner analogous to that of EM waves propagating through the QED vacuum. One major difference: physical matter is made up of composite atoms-bound states of electrons and nuclei – thus, ∃ resonances at certain frequencies of EM radiation – whenever: Eγ = hf = energy of (real) photon atom ΔΕ atom − Ε natom n ,m = Ε m As a consequence of this, in matter, electromagnetic radiation travels at speed v ≤ c : For non-conductors – e.g. dielectric media: v( f ) = Frequency dependence of v is e.g. responsible for rainbows!!! 1 ε ( f )μ( f ) ⇐ Frequency dependent!!! Because of/due to frequency-dependence of: ε =ε( f ) μ = μ( f ) = electric permittivity of medium = magnetic permeability of medium © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2008. All Rights Reserved. 17 UIUC Physics 435 EM Fields & Sources I 1 Compare: v ( f ) = But v ( f ) = ε ( f )μ( f ) Fall Semester, 2007 Lecture Notes 12 (matter) c= vs: 1 ε o μo Prof. Steven Errede (vacuum) ε ( f )μ( f ) c c = in a dielectric material, i.e. n ( f ) = = index of refraction ε o μo v( f ) n( f ) Many dielectric materials have no magnetic properties – i.e. their μ ( f ) = μo to a high degree. ε(f) = K e ( f ) since (for linear dielectrics) εo Thus for many dielectric materials n ( f ) = Ke ( f ) ≡ ε(f) = 1 + χe ( f ) εo In general: v ( f ) = Km ≡ 1 ε ( f )μ( f ) c Ke ( f ) Km ( f ) where (for linear magnetic materials) μ( f ) = 1 + χ m ( f ) . In dielectric materials that are non-magnetic, K m 1 because μo μ ( f ) μo , then v ( f ) 18 = 1 ε ( f ) μo = c Ke ( f ) . © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2008. All Rights Reserved.