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PHYS2012 EMP10_03 DIELECTRICS – MICROSCOPIC VIEW DIELECTRIC MATERIALS The term dielectric comes from the Greek dia + electric, where dia means through, thus dielectric materials are those in which a steady electric field can be set up without causing an appreciable current. Matter is usually neutral with an equal numbers of negative and positive charges. In dielectric materials, these charges are not free to move far under the influence of an applied external electric field, as are conduction electrons in a metal conductor. However, the forces due to an external field do cause small relative displacements (on an atomic scale) of the charges of each sign. The extent of such displacements depends upon the tightness with which the charges are held fixed. Also, polar molecules rotate in the external electric field try to align themselves with the external electric field. This displacement of the charges and rotation of molecules resulting from an applied external electric field is called polarization P of the material. The dielectric constant r is a measure of the extent of the polarization. The parameter that directly relates the polarization of the material to the electric field is called the electric susceptibility e where e r 1 P e 0 E We can start with a very crude model to explain the behaviour of dielectric materials. We assume a continuum of two uniform charge distributions of opposite signs. In the absence of an applied electric field, the positive and negative charge distributions are exactly superimposed. When an external electric field is applied, the positive distribution is displaced in the direction of the external electric field and the negative distribution is displaced in the opposite direction. This results in a cancellation of the charges in the interior of the dielectric material, bound charge density [C.m-3] b 0 and results in induced bound (polarized) surface charges +qb and -qb at the end surfaces of the dielectric material. The bound surface charge density b [C.m-2] is dq b b dA In this case, the bound surface charge density determines the polarization where n̂ is a unit normal vector pointing away from the dielectric. b P nˆ On the atomic level, there is another possible point of view. In the dielectric material each atom or molecule is distorted to produce an electric dipole and a dielectric material can be though of as consisting of large number of electric dipoles. We can attribute the effect of polarization of the material to be the sum of all the fields of all the dipoles. Separation of charge electric dipole polarization emp10_03.doc 4-May-17 3.1 A dipole consists of two equal and opposite charges +q and –q separated by a vector distance d pE q d Electric dipole moment p = pE = q d d -q +q pe points from negative to positive Induced dipole moment – helium atom E +2e -e Zero electric field – helium atom symmetric zero dipole moment -e -e +2e A -e d B effectively charge +2e at A and -2e at B dipole moment p = 2ed p emp10_03.doc 4-May-17 3.2 Potential and electric field from an electric dipole V ( P) 1 q q q r2 r1 4 r1 r2 4 r 1r2 Er E r d V ( P) P q d cos d2 4 r 2 cos2 4 r2 r + (d/2)cos q d cos p cos p r 2 2 4 r 4 r 4 r 3 r1 r – (d/2)cos r r extends from the centre of the dipole to the point P (d/2)cos The radial and tangential components of the field at point P are V 2 p cos r 4 r 3 1 V p sin E r 4 r 3 E E rˆ E ˆ -q Er r d +q along the axis of the dipole along the right bisector of the dipole = 0 E = 0 = /2 Er = 0 Electric field approaches zero much more quickly than a point charge. ??? Why ? ELECTRIC DIPOLE PLOT– MATLAB Why is difficult to plot the potential in a plane passing through the axis of the dipole? Potential: Electric Dipole 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 emp10_03.doc 4-May-17 3.3 % electric_dipole.m % Ian Cooper School of Physics,University of Sydney close all clear all clc % emconstants ------------------------------------------------------c = 3.00e-8; % speed of light e = 1.602e-19; % elementary charge eps0 = 8.85e-12; % permittivity of free space NA = 6.02e23; % Avogadro constant me = 9.11e-31; % electron rest mass mp = 1.673e-27; % proton rest mass mn = 1.675e-27; % neutron rest mass h = 6.626e-34; % Planck's constant kB = 1.38e-23; % Boltzmann's constant kC = 8.988e9; % Coulomb constant mu0 = 4*pi*1e-7; % permeability of free space amu = 1.66e-27; % atomic mass unit % Setup ------------------------------------------------------------q = e; % dipole charge d = 1.6795e-018; % dipole separation distance q1 = q; q2 = -q; % separated charges kc = 1/(4*pi*eps0); % constant in Coulomb's Law x1 = d/2; x2 = -d/2; % position of dipole y1 = 0; y2 = 0; scale = 1.25; % plotting region xmax = scale * d; ymax = xmax; xmin = -xmax; ymin = -ymax; % plane above diople num = 100; x = linspace(xmin,xmax,num); y = x; [xx yy] = meshgrid(x,y); r1 = sqrt((xx-x1).^2 + (yy-y1).^2); % distance from charges % to test point to calc. potential r2 = sqrt((xx-x2).^2 + (yy-y2).^2); V1 = kc .* q1 ./ (r1); % potential from each charge V2 = kc .* q2 ./ (r2); Vtot = V1 + V2; Vmax = max(max(Vtot)); sat = 0.5; % saturate the potential Vtot(Vtot > sat*Vmax) = sat * Vmax; % potential near a charge % is extremely large Vtot(Vtot < -0.5*Vmax) = -sat * Vmax; Vtot = Vtot/(max(max(Vtot))); figure(2); % [3D] plot surf(xx/d,yy/d,Vtot,'FaceColor','interp',... 'EdgeColor','none',... 'FaceLighting','phong') daspect([1 1 1]) axis tight; view(-45,20) camlight left; colormap(jet) grid off; axis off colorbar title('Potential: Electric Dipole') emp10_03.doc 4-May-17 3.4 POLARIZATION The quantity of real interest is not an individual dipole moment but the electric dipole moment per unit volume. In a region of uniform polarization, the polarization is then Pn p where p is the induced atomic dipole moment and n is the number of electric dipoles per unit volume. The word polarization has two meanings: a qualitative one referring to any relative displacements of positive and negative charge and the quantitative one, giving the resulting vector dipole moment per unit volume, P . The lines of P connect bound charges (negative to positive). The polarization describes the extent to which permanent or induced dipoles become aligned. The polarization gives rise to a surface bound charge density b bound and a volume bound charge density b bound . Consider the polarization of the dielectric between the plates of a charged parallel plate capacitor. dA + + + + + + + + + d For a cylinder of the dielectric of cross-sectional area dA extending from one plate to the other +f Throughout the body of the dielectric, the charges on adjacent ends of the polar molecules neutralize one another. At both the top and bottom of the dielectric the charges do not neutralize each other bound surface charges b . -f -b +b - - - - - - - - - electric dipole moment pE = q d dpE ( b dA)d P dpE dpE ( b dA)d d dA d dA d P b P b nˆ where n̂ is the normal outward pointing unit vector. Thus, the polarization equals the magnitude of the bound (induced) charge per unit area on the surface of the dielectric material. Also, the polarization can be obtained through the relationship b P (no proof) where b is the volume density of the bound charges. emp10_03.doc 4-May-17 3.5 Homogenous dielectric – unif ormly polarized Eexternal n̂ b P + + + + + + + Dielectric is neutral - Edep Edielectric - b 0 n̂ - - P qb qb b b - b P cos - P b nˆ The electrical f ield is reduced in the dielectric material How is the macroscopic measureable quantity, the dielectric constant related to quantities at an atomic level? Conductors Contain charges that are free to move and in the presence of an electric field, redistribute themselves on the surface of the conductor so that the electric field is zero in the interior. Dielectrics Induced dipoles (electronic) – in an external electric field, positive charge (nucleus) and negative charges (electron cloud) pushed in opposite directions induced electric dipole moment. + - + - Electric Field Polar molecules eg H2O, N2O – neutral but a lopsided charge distribution – one side excess positive and the other excess negative charge permanent electric dipole moment – zero electric field, random orientation of molecules in gases and liquids net electric dipole moment is zero. In an external electric field – dipoles experience a torque to orientate them with the electric field – thermal agitation of the molecules opposes the alignment align is not perfect. Ionic contribution to electric dipole moment – in a molecule some of the atoms have an excess positive or negative charge resulting from the ionic nature of the bond: in an electric field, the + ions and – ions are shifted in opposite directions. Net electric dipole moment = (induced dipoles - electronic + permanent dipoles – orientation + ionic dipoles - ionic) p pe po pi emp10_03.doc p pE 4-May-17 3.6 Net polarization = (electronic polarization + orientation polarization + ionic polarization) P n p Pe Po Pi The net polarization is related to a surface charge density bound and the density of bound charges bound bound b P nˆ n̂ is the normal pointing out of the volume Px Py Pz three dimensional variation of the polarization x y z bound b P If the polarization depends on time, then we may expect that the effect is similar to that of a current P polarization current density J b t and needs to be added to possible currents associated with free charges. Consider a medium when an applied electric field is turned on. As a consequence the atoms or molecules form small dipoles where none existed before – the alignment of the molecules constitutes a current. DEPOLARIZATION FACTOR When a dielectric material is placed in an electric field, the induced polarization charge always acts to decrease the average electric field within the dielectric from its value it had before the dielectric was inserted. In general, the polarized charge produces a non-uniform electric field, so the original electric field is modified differently at different regions in the dielectric material. However, we will only consider the application of a uniform external electric field Eext and dielectrics in which the polarization is also uniform. The induced field in the dielectric is called the depolarizing electric field Edep . Dielectrics which have an ellipsoidal shape satisfy this criteria (uniform polarization within the dielectric when an external electric field is applied). In cases where the shape approximates an ellipsoidal shape, we can than refer to the average electric field within the dielectric Edielectric Edielectric Eext Edep The depolarizing electric field depends upon a geometric factor L and the magnitude of the polarization P 1 Edep L P 0 The geometrical factor L is called the depolarizing factor and can take values from 0 to 1, depending on the shape of the dielectric. emp10_03.doc 4-May-17 3.7 Flat plate with its plane perpendicular to the external electric field (~ flat broad ellipsoid): similar to a parallel plate capacitor, the average electric field in the dielectric is reduced by the factor r. The average electric field within the flat plate dielectric is E plate Eext Edep Eext r E plate Edep E plate Eext ( r 1) E plate e E plate P 0 e E plate Edep 1 0 P L 1 E plate Eext 1 0 P This flat shape gives the maximum value for the depolarizing electric field for a given polarization of the material. Long thin rod with its axis parallel to the external electric field (~ long thin ellipsoid): if the rod is long and thin enough, the induced charges at the ends are small then the depolarizing electric field is essentially zero Edep 0 and the depolarizing factor has its smallest value, L 0. The average electric field in the rod (ignoring the ends) is Erod Eext Sphere: the value for the depolarizing factor for a sphere is L = 1/3. The surface charge density is given by b P cos where is measured with respect to the direction of the electric field and the polarization. The depolarizing electric field Edep and the average electric field within in the sphere Esphere are 1 1 Edep P Esphere Eext P 3 0 3 0 Eext - + Edep - - - Edep 0 - + + Thin long rod L = 0 Zero polarization - Edep 0 + Erod Eext Flat plate L = 1 Max polarization Edep 1 0 1 0 Edep - + + + + Sphere L = 1/3 Concentration of charges At surf ace given by b P cos 1 P 3 0 Esphere Eext P + + - Edep P E plate Eext - + 1 P 3 0 The external electric field can arise because of a distribution of free charges. Historically, a new vector was introduced, the electric displacement D such that D = f where f is the 1 DP surface charge density that gives the external electrical field. Hence, Edielectric 0 emp10_03.doc 4-May-17 3.8 RESPONSE OF A MOLECULE TO AN ELECTRIC FIELD The electric susceptibility e r 1 tells us about the polarizability of the atoms in matter. From a macroscopic view (only consider cases in which the electric field and polarization are uniform), the polarization P depends upon the electric field within the dielectric Edielectric P e 0 Edielectric r 1 0 Edielectric However, the macroscopic or average electric field Edielectric is not a satisfactory measure of the local electric field Eloc producing the polarization of each atom. We can assume the electric dipole moment p and polarization P are proportional to the local electric field Eloc experienced by the molecule. Taking into account the three contributions leading to the polarization of atoms or molecules p (e o i ) Eloc Eloc Electric dipole moment P n p n ( e o i ) Eloc n Eloc Polarization atomic polarizability e electronic or molecular polarizability o orientation polarizability i ionic polarizability We can now relate the macroscopic quantities - the electric susceptibility e and the dielectric constant r to a property of the molecules, called the atomic polarizability . The atomic polarizability relates to the ease in which electric dipoles moments can be formed giving rise to the polarization of the material and hence to the dielectric constant of the material. E073 What is the local (inner) field Eloc that acts upon an individual molecule within the dielectric? Eloc – the local (inner) field – electric field acting upon a molecule within a dielectric Dielectric – not continuous – composed of molecules S Consider dielectric between the plates of a parallel plate capacitor, where the average electric field with in the dielectric is E = Edielectric The local electric field at the point O consists of 4 parts: +f -b O +b r -f emp10_03.doc 4-May-17 3.9 1 Field at O due only to the charged plates E1 f 0 2 Polarization of the charges on the surface of the dielectric P E2 b 0 0 3 Polarization of charges on the surface of S which would be formed if the spherical section of the dielectric was removed P E3 see following proof 3 0 4 Polarization from the polar molecules within the spherical section, E4 P E4 K 4 K4 some constant, usually, E4 can’t be calculated exactly. 0 Hence, the local electric field Eloc at O is Eloc E1 E2 E3 E4 Eloc Eloc f P P P K4 0 0 3 0 0 DP 0 Eloc E P P K4 3 0 0 P P K4 3 0 0 Eloc E K P 0 electric field inside dielectric E DP 0 K is some positive constant. This equation gives the electric field Eloc that acts upon a single molecule of the dielectric. For dielectrics with E4 0, K4 0 and K = 1/3, the total local electric field at O is Eloc E P 3 0 This is a useful result, as this equation is applicable to cubic crystals, dilute solutions and gases. emp10_03.doc 4-May-17 3.10 Calculation of E3 E3 P 3 0 Assume a spherical section is removed from the dielectric. E3 is found by summing the contributions to the field of all ring elements of polarization charge on the surface S. Width of ring r d Radius of ring r sin + + + r - - r sin Area of the shaded ring between and + d E surface S 2 r sin rd d - Pcos P The charge density s on the spherical surface is given by the component of the polarization normal to S where is measured w.r.t. E s P nˆ P cos Surface area of the ring element is dS 2 r sin r d The charge on the surface element dS that lies between and + d is dq P cos 2 r sin r d By symmetry, all components of the electric field that are not normal to the capacitor plates cancel. Only the electric field due to elements of charge is in the vertical direction contribute. + + + E electric f ield at O due to charge dq e E0 cos Hence, the electric field due to an element of charge dqe (Coulomb’s Law) iis 1 dqe E0 cos cos 4 0 r 2 emp10_03.doc - 4-May-17 - E0 - element of charge dq e 3.11 The electric field due to the ring with charge dq is dE3 dq 1 4 0 r 2 cos 1 P cos 2 r sin rd 4 0 r 2 cos P 20 cos2 sin d The resultant field at the centre of the sphere is obtained by integrating over = 0 E3 E3 E147 P 20 0 cos 2 sin d P P cos3 1 1 0 (2)(3) 0 (2)(3) 0 P 3 0 The local electric field is greater than the electric field within the dielectric because of the contribution of E3. E896 emp10_03.doc 4-May-17 3.12 NONPOLAR DIELECTRICS Molecules without any intrinsic dipole moment will acquire an induced dipole moment in an external electric field, and so such molecules have a dielectric constant. The electrons shift position slightly inside their molecules and do so very quickly, in around 10-15 s, so that temperature and frequency have little effect. Monatomic gases (nonpolar) We will consider the rare gases such as helium and argon because of the simple theoretical model that can be used, although for most practical purposes it is not very useful. Simple model of a single atom (gives results that are correct to an order of magnitude) Positive nucleus +Ze and electrons -Ze Atomic nucleus: diameter ~ 10-10 m nuclear diameter ~ 10-15 m Nucleus point charge and electron cloud of charge –Ze distributed homogeneously throughout a sphere of radius a 10-10 m When the atom of radius a placed into external electric field Eext (Eext =Eloc) nucleus and electron cloud move in opposite directions to create an induced electric dipole equilibrium established with the nucleus shifted slightly relative to the centre of the electron cloud by a distance d. +Ze +Ze d a a d << a E The nucleus will experience a force in the direction of the electric field FE FE = Ze Eext and an opposing force Fec due to the electric field of the negative charge of the electron cloud which is assumed to acts as a point charge at the centre of the cloud The electric field Eec experienced by the nucleus at a distance d from the centre of the negatively charged electron cloud is determined by the application of Gauss’s Law Z e (d 3 / a 3 ) Eec 4 d 2 0 2 Z e2 d Fec Ze Eec 4 0 a 3 emp10_03.doc where the charge enclosed is (-Ze)(d3/a3) 4-May-17 3.13 Fec FE 4 0 a3 d Eext Ze The displacement distance d is proportional to the external electric field Eext. For the single atom Eloc = Eext, the molecular (electronic) polarizability of a monatomic gas is e pe Eloc Eext ( Ze) d ( Ze) 4 0 a 3 Eext Ze e 4 0 a3 The electronic (molecular) polarizability is proportional to the volume of the electron cloud ( a3) the larger the atom, the greater the charge separation and the greater the induced dipole moment: bigger the atom the larger r 10-40 F.m2 He 0.18 Ne 0.35 A 1.43 Kr 2.18 Xe 3.54 Now, we consider a rare gas containing n molecules.m-3 in an electric field E. We can neglect any interactions between the induced dipoles in the atoms (good approximation for a gas). Microscopic view: the polarization of the gas P is P n pe n E Eloc = E Macroscopic view: the polarization of the gas is P e 0 E r 1 0 E Edielectric = E Therefore we can relate the microscopic molecular polarizability with the macroscopic dielectric constant r n r 1 1 4 n a 3 0 We have obtained a relationship between the measurable quantity r and the microscopic quantities and a. How good is our simple model? Helium gas: temperature, T = 293 K and pressure, pg = 1 atm = 1.013105 Pa Dielectric constant r = 1.0000684 1/ 3 1 pg V N k B T n= N / V = 2.510 atoms.m a r 4 n Correct order of magnitude !!! our simple model not too bad 25 -3 6 1011 m We can estimate the relative shift d between the nucleus and the centre of the electron cloud E ~ 105 V.m-1 a ~ 10-10 m Z ~ 2 d ~ 10-17 m d is very small – very slight perturbing influence of the applied electric field on the atom emp10_03.doc 4-May-17 3.14 Number density n The number density for a gas is obtained from the ideal gas equation pg N kB T n kB T n V kB T o For a gas at atmospheric pressure and 20 C, the number density n is pg V N k B T pg pg = 1 atm = 1.013105 Pa T = 20 oC = 293 K kB = 1.381023 J.K-1 n = 2.51025 molecules.m-3 For a solid or liquid (density , molecular mass m, number of molecules N, mass of sample msample) the number density n is obtained as follows msample n V Nm V M NA m m M NA n N V n M NA NA M Avogadro’s number NA = 6.021023 molecules.mol-1 Molar mass M (in kilograms) For copper = 8.93103 kg.m-3 M = 63.5 g = 63.510-3 kg n = 8.51028 atoms.m-3 Note: the number density of solids is much greater than that of gases. E041 emp10_03.doc 4-May-17 3.15 Gases, dilute solutions and simple solids (nonpolar) Gases, dilute solutions, solids - one kind of atom eg diamond, phosphorus (cubic crystals), No permanent dipole moments or ions Polarization due to relative displacement of electron clouds and nuclei Local electric field same for all atoms K = 1/3 (no polar molecules E4 = 0 K4 = 0) = e P P r 1 0 E Edielectric = E 3 0 Combining these three equations gives the Clausius-Mossotti relationships P n pe n Eloc Eloc E 3 0 r 1 n r 2 r 1 n r 2 3 0 The distance between atoms in a solid is affected only slightly by temperature and therefore, n, , K and r are in a first approximation independent of the temperature. For a solid, a typical value for the number density is n ~ 51028 m-3. The dielectric constant for three solids with a diamond structure are: r (C) = 5.68 r (Si) = 12 r (Ge) = 16 The dielectric constant for the gases are very close to 1 eg r(H2) = 1.000132. Why is the dielectric constant for a solid much greater than for a gas? If r very close to 1 r+2 3 r 1 E293 E888 emp10_03.doc n 0 the same equation as for monatomic gases E039 4-May-17 3.16 POLAR DIELECTRICS Consider the dielectric material containing n molecules.m-3. Assume that each molecule has a permanent electric dipole moment p. Water is a typical polar liquid. Its constituent molecules have a permanent dipole moment. The polarization is due to the electronic polarization Pe (nucleus shifted slightly relative to the centre of the electron cloud) and the ionic polarization Pi (ionic nature of bond between atoms) and the orientation polarization Po (rotation and alignment of the polar molecules in the external electric field). Pe and Pi are essentially independent of the temperature but Po is very temperature dependent. At a temperature T and zero external electric field, the molecules will be randomly oriented zero polarization. When there is an external electric field, the molecules will try to align with the field. Each polar molecule can be considered to be a simple dipole. The force on the dipole provides the torque to rotate the molecule so that they will be in the lowest energy state where they are parallel to the field. If there were no thermal motion, all dipoles would line up along the external field direction. +Q F p d F -Q E The electric force on the dipole produces a couple and the torque acting to rotate the dipole about its centre is d d F sin F sin Q E d sin p E sin p E 2 2 Set the potential energy U( ) of the dipole to zero when = 90o. The potential energy of the dipole for an arbitrary angle is then given by 90o + E p E sin d p E cos p E - + U ( ) + U=- p E Lowest energy state emp10_03.doc - =0 U=0 4-May-17 = 90o - U=+ p E highest energy state = 180o 3.17 The dipole has the lowest potential energy when the dipole is parallel to the electric field and the highest energy when anti-parallel to the field small angles are preferred over larger ones. And if they were no thermal motion, all dipoles would line along the direction of the external electric field. The greater the temperature, the greater the thermal motion reduced alignment of the dipoles with the field. +pE U 0 -p E π/2 0 π The orientation polarization Po is given by the Langevin function (1905) pE 1 Po n p coth pE k T B k B T pE kB T 1 Po Po n p T 1 Complete alignment – this does not occur in gases pE kB T Po np pE Po n p 3 kB T Most practical case: 1 Po p 2 Po T 1 slope = 1/3 0 pE/kT 10 The total polarization of a polyatomic gas is given by where Eloc = E p2 P Pe Pi Po n e i E 3 kB T Macro view P e 0 E r 1 0 E r is related to the molecular properties by n p2 e i 3 kB T 0 r 1 How well does this prediction agree with experiment? emp10_03.doc 4-May-17 3.18 If dielectric constant r plotted against 1/T straight line measurement of p Slope n p2 / 3 kB Intercept n (e + i) measurement of (e + i) r - 1 intercept n 0 slope n p2 3 kB e i 1/T Dipole moments of gases in debye units (3.3310-30 C.m) NO 0.1 CO 0.11 HCl HBr 0.79 HI 0.38 NO2 CO2 0 CH4 0 H2O H2 0 A 0 NH3 1.04 0.4 1.84 1.4 Dielectric constant measurements have played an important part in determining molecular structure: CO2 has zero resultant dipole moment, whereas each CO bond does have a nonzero dipole moment O=C=O. H2O molecule must have a triangular structure. At ordinary temperatures and electric fields, the average dipole moment and hence polarization are greatly reduced by the thermal agitation – the molecules point in every way with only a slight alignment with the external electric field. In the low electric field approximation pE pE 1 Po n p kB T 3 kB T k T is the thermal energy and p E is the energy of the dipole molecule when aligned with the effective electric field. The polarization is reduced by the factor (p E / 3 kB T) and the greater the temperature the greater the reduction in the polarization and the polarization increases with increasing electric field strength. emp10_03.doc 4-May-17 3.19 The dielectric constant of water is 80 and the molecule dipole moment of a water molecule is 6.210-30 C.m. Find the electric field required to maximize the polarization of water. Water: M = 1810-3 kg = 103 kg.m-3 p = 6.210-30 C.m NA = 6.0210-23 mol-1 To maximize the polarization, all the water molecules need to be aligned. The polarization and electric field are given by P P e 0 E r 1 0 E E r 1 0 N n A 3.34 1028 molecule.m3 Number density M P n p = 0.20 C.m-2 Polarization Electric field E P 0.20 V.m -1 3.0 108 V.m -1 r 1 0 80 1 8.854 1012 This means that the water molecules are not all aligned up until the electric field reaches 300 MV.m-1, which is enough to break down water and produce an electric arc. E582 E039 E305 E041 emp10_03.doc E073 E147 E293 4-May-17 E305 E582 E888 E896 3.20