We can let a-0-jv in(4.81) to obtain the secondary current in a plasma with E(r, o)+ (a-ju)(-ju)2-c2) ute×E(r,o)+ r From this we find the dyadic permittivity E 1+ o[(o-jv)2-a21 (a-j)o(-j)2-a2l wcue Assuming that Bo is aligned with the z-axis we can use(4.85)to find the components of the dyadic permittivity matrix: o[(a-j)2-c2) e()=6(1--6 (u-j) (4.91) We see that [E] is not hermitian when v #0. We expect this since the plasma is lossy when collisions occur. However, we can decompose E] as a sum of two matrices [e]= [8 where [2] and [a] are hermitian [141. The details are left as an exercise. We also note that, as in the case of the lossless plasma, the permittivity dyadic obeys the symmetry ondition E( ( Bo)= E(Bo) 4.6.3 Simple models of dielectrics We define an isotropic dielectric material (also called an insulator) as one that obeys e macroscopic frequency-domain constitutive relationship D(r, o)=E(r, E(r, o) Since the polarization vector P was defined in Chapter 2 as P(r, t)=D(r, t)-EOE(r, t) an isotropic dielectric can also be described through P(r,)=((r,)-∈0)E(r,o)=元e(r,ω)∈0E(r,a) 2001 by CRC Press LLC
We can let ω → ω − jν in (4.81) to obtain the secondary current in a plasma with collisions: J˜s (r,ω) = jω − 0ω2 p(ω − jν) ω[(ω − jν)2 − ω2 c ] E˜(r,ω)+ + j 0ω2 p(ω − jν) ω(ω − jν)[(ω − jν)2 − ω2 c )] ωc × E˜(r,ω) + + ωc (ω − jν)2 0ω2 p(ω − jν) ω[(ω − jν)2 − ω2 c ] ωc · E˜(r,ω) . From this we find the dyadic permittivity ˜¯ c (ω) = 0 − 0ω2 p(ω − jν) ω[(ω − jν)2 − ω2 c ] ¯ I + j 0ω2 p ω[(ω − jν)2 − ω2 c )] ω¯ c + + 1 (ω − jν) 0ω2 p ω[(ω − jν)2 − ω2 c ] ωcωc. Assuming that B0 is aligned with the z-axis we can use (4.85) to find the components of the dyadic permittivity matrix: ˜ c xx (ω) = ˜ c yy (ω) = 0 1 − ω2 p(ω − jν) ω[(ω − jν)2 − ω2 c ] , (4.88) ˜ c xy (ω) = −˜ c yx (ω) = − j0 ω2 pωc ω[(ω − jν)2 − ω2 c )] , (4.89) ˜ c zz(ω) = 0 1 − ω2 p ω(ω − jν) , (4.90) and ˜ c zx = ˜ c xz = ˜ c zy = ˜ c yz = 0. (4.91) We see that [˜ c] is not hermitian when ν = 0. We expect this since the plasma is lossy when collisions occur. However, we can decompose [˜¯ c ] as a sum of two matrices: [˜¯ c ] = [˜¯] + [σ˜¯ ] jω , where [˜¯] and [σ˜¯ ] are hermitian [141]. The details are left as an exercise. We also note that, as in the case of the lossless plasma, the permittivity dyadic obeys the symmetry condition ˜ c i j(B0) = ˜ c ji(−B0). 4.6.3 Simple models of dielectrics We define an isotropic dielectric material (also called an insulator ) as one that obeys the macroscopic frequency-domain constitutive relationship D˜ (r,ω) = (˜ r,ω)E˜(r, ω). Since the polarization vector P was defined in Chapter 2 as P(r, t) = D(r, t) − 0E(r, t), an isotropic dielectric can also be described through P˜(r,ω) = ((˜ r,ω) − 0)E˜(r,ω) = χ˜e(r, ω)0E˜(r,ω)
where Xe is the dielectric susceptibility. In this section we shall model a homogeneous electric consisting of a single, uniform material type. We found in Chapter 3 that for a dielectric material immersed in a static electric field he polarization vector P can be viewed as a volume density of dipole moments. We choose to retain this view as the fundamental link between microscopic dipole moments and the macroscopic polarization vector. Within the framework of our model we thus describe the polarization through the expression P(r, t) (4.92) Here p i is the dipole moment of the ith elementary microscopic constituent, and we form the macroscopic density function as in 8 1.3.1 We may also write(4.92)as P(r,t)= △V」NB Pi|=N(r, t)p(r, t) here NB is the number of constituent particles within AV. We identify p(r, t) as the average dipole moment within AV, and N(r,t)s N as the dipole moment number density. In this model a dielectric material does not require higher-order multipole moments to describe its behavior. Since we are only interested in homogeneous materials in this section we shall assume that the number density constant: N(r, t)= N To understand how dipole moments arise, we choose to adopt the simple idea that mat- ter consists of atomic particles, each of which has a positively charged nucleus surrounded by a negatively charged electron cloud. Isolated, these particles have no net charge and no net electric dipole moment. However, there are several ways in which individual par ticles, or aggregates of particles, may take on a dipole moment. When exposed to an external electric field the electron cloud of an individual atom may be displaced, resulting in an induced dipole moment which gives rise to electronic polarization. When groups of atoms form a molecule, the individual electron clouds may combine to form an asym- metric structure having a permanent dipole moment. In some materials these molecules are randomly distributed and no net dipole moment results. However, upon application of an external field the torque acting on the molecules may tend to align them, creating an induced dipole moment and orientation, or dipole, polarization. In other materials, the asymmetric structure of the molecules may be weak until an external field causes the displacement of atoms within each molecule, resulting in an induced dipole moment causing atomic, or molecular, polarization. If a material maintains a permanent polar ization without the application of an external field, it is called an electret(and is thus similar in behavior to a permanently magnetized magnet) To describe the constitutive relations, we must establish a link between P(now describ- able in microscopic terms)and E. We do this by postulating that the average constituent 2001 by CRC Press LLC
where χ˜e is the dielectric susceptibility. In this section we shall model a homogeneous dielectric consisting of a single, uniform material type. We found in Chapter 3 that for a dielectric material immersed in a static electric field, the polarization vector P can be viewed as a volume density of dipole moments. We choose to retain this view as the fundamental link between microscopic dipole moments and the macroscopic polarization vector. Within the framework of our model we thus describe the polarization through the expression P(r, t) = 1 V r−ri(t)∈B pi . (4.92) Here pi is the dipole moment of the ith elementary microscopic constituent, and we form the macroscopic density function as in § 1.3.1. We may also write (4.92) as P(r, t) = NB V 1 NB NB i=1 pi = N(r, t)p(r, t) (4.93) where NB is the number of constituent particles within V. We identify p(r, t) = 1 NB NB i=1 pi(r, t) as the average dipole moment within V, and N(r, t) = NB V as the dipole moment number density. In this model a dielectric material does not require higher-order multipole moments to describe its behavior. Since we are only interested in homogeneous materials in this section we shall assume that the number density is constant: N(r, t) = N. To understand how dipole moments arise, we choose to adopt the simple idea that matter consists of atomic particles, each of which has a positively charged nucleus surrounded by a negatively charged electron cloud. Isolated, these particles have no net charge and no net electric dipole moment. However, there are several ways in which individual particles, or aggregates of particles, may take on a dipole moment. When exposed to an external electric field the electron cloud of an individual atom may be displaced, resulting in an induced dipole moment which gives rise to electronic polarization. When groups of atoms form a molecule, the individual electron clouds may combine to form an asymmetric structure having a permanent dipole moment. In some materials these molecules are randomly distributed and no net dipole moment results. However, upon application of an external field the torque acting on the molecules may tend to align them, creating an induced dipole moment and orientation, or dipole, polarization. In other materials, the asymmetric structure of the molecules may be weak until an external field causes the displacement of atoms within each molecule, resulting in an induced dipole moment causing atomic, or molecular, polarization. If a material maintains a permanent polarization without the application of an external field, it is called an electret (and is thus similar in behavior to a permanently magnetized magnet). To describe the constitutive relations, we must establish a link between P (now describable in microscopic terms) and E. We do this by postulating that the average constituent
dipole moment is proportional to the local electric field strength e CE where a is called the polarizability of the elementary constituent. Each of the polarization effects listed above may have its own polarizability: ae for electronic polarization, aa for atomic polarization, and ad for dipole polarization. The total polarizability is merely the sum a=ae aa +ad o In a rarefied gas the particles are so far apart that their interaction can be neglected Here the localized field E is the same as the applied field E. In liquids and solids where particles are tightly packed, e depends on the manner in which the material is polarized and may differ from E. We therefore proceed to determine a relationship between E nd P The Clausius-Mosotti equation. We seek the local field at an observation point within a polarized material. Let us first assume that the fields are static. We surround the observation point with an artificial spherical surface of radius a and write the field at the observation point as a superposition of the fielde applied, the field eg of the polarized molecules external to the sphere, and the field E, of the polarized molecules within the sphere. We take a large enough that we may describe the molecules outside the sphere in terms of the macroscopic dipole moment density P, but small enough to assume that P is uniform over the surface of the sphere. We also assume that the major contribution to E2 comes from the dipoles nearest the observation point. We then approximate E2 using the electrostatic potential produced by the equivalent polarization surface charge on the phere pps=n P(where n points toward the center of the sphere). Placing the origin of coordinates at the observation point and orienting the z-axis with the polarization P so that P= PoZ, we find that f P=-cos 0 and thus the electrostatic potential at any point r within the sphere is merely d(r)= Po cos′ This integral has been computed in8 3.2.7 with the result given by (3.103)Hence 3eo r cos 8=_Po and therefore En Note that this is uniform and independent of a The assumption that the localized field varies spatially as the electrostatic field, even when P may depend on frequency, is quite good. In Chapter 5 we will find that for a frequency-dependent source(or, equivalently, a time-varying source), the fields very near the source have a spatial dependence nearly identical to that of the electrostatic case We now have the seemingly more difficult task of determining the field Es produced by the dipoles within the sphere. This would seem difficult since the field produced by dipoles near the observation point should be highly-dependent on the particular dipole arrangement. As mentioned above, there are various mechanisms for polarization, and the distribution of charge near any particular point depends on the molecular arrange. ment. However, Lorentz showed [115 that for crystalline solids with cubical symmetry, 2001 by CRC Press LLC
dipole moment is proportional to the local electric field strength E : p = αE , (4.94) where α is called the polarizability of the elementary constituent. Each of the polarization effects listed above may have its own polarizability: αe for electronic polarization, αa for atomic polarization, and αd for dipole polarization. The total polarizability is merely the sum α = αe + αa + αd . In a rarefied gas the particles are so far apart that their interaction can be neglected. Here the localized field E is the same as the applied field E. In liquids and solids where particles are tightly packed, E depends on the manner in which the material is polarized and may differ from E. We therefore proceed to determine a relationship between E and P. The Clausius–Mosotti equation. We seek the local field at an observation point within a polarized material. Let us first assume that the fields are static. We surround the observation point with an artificial spherical surface of radius a and write the field at the observation point as a superposition of the field E applied, the field E2 of the polarized molecules external to the sphere, and the field E3 of the polarized molecules within the sphere. We take a large enough that we may describe the molecules outside the sphere in terms of the macroscopic dipole moment density P, but small enough to assume that P is uniform over the surface of the sphere. We also assume that the major contribution to E2 comes from the dipoles nearest the observation point. We then approximate E2 using the electrostatic potential produced by the equivalent polarization surface charge on the sphere ρPs = nˆ · P (where nˆ points toward the center of the sphere). Placing the origin of coordinates at the observation point and orienting the z-axis with the polarization P so that P = P0zˆ, we find that nˆ · P = − cos θ and thus the electrostatic potential at any point r within the sphere is merely (r) = − 1 4π0 S P0 cos θ |r − r | d S . This integral has been computed in § 3.2.7 with the result given by (3.103) Hence (r) = − P0 30 r cos θ = − P0 30 z and therefore E2 = P 30 . (4.95) Note that this is uniform and independent of a. The assumption that the localized field varies spatially as the electrostatic field, even when P may depend on frequency, is quite good. In Chapter 5 we will find that for a frequency-dependent source (or, equivalently, a time-varying source), the fields very near the source have a spatial dependence nearly identical to that of the electrostatic case. We now have the seemingly more difficult task of determining the field E3 produced by the dipoles within the sphere. This would seem difficult since the field produced by dipoles near the observation point should be highly-dependent on the particular dipole arrangement. As mentioned above, there are various mechanisms for polarization, and the distribution of charge near any particular point depends on the molecular arrangement. However, Lorentz showed [115] that for crystalline solids with cubical symmetry
or for a randomly-structured gas, the contribution from dipoles within the sphere is zero Indeed. it is convenient and reasonable to assume that for most dielectrics the effects of the dipoles immediately surrounding the observation point cancel so that e3 =0. This was first suggested by O.F. Mosotti in 1850 52 With e2 approximated as(4.95 )and E, assumed to be zero, we have the value of the exulting local field P(r) E(r)=E(r)+ (4.96) This is called the Mosotti field. Substituting the Mosotti field into(4.94) and using P=Np, we obtain P(r) P(r)=NaE(r)=Nae(r)+ Solving for P we obtain 3E0-No E(r)=Eoe(r) So the electric susceptibility of a dielectric may be expressed as N Using xe =Er-I we can rewrite(4.97)as ∈=∈0∈r=∈0 3+2Na/∈0 (4.98) which we can arrange to a=ae+ aa+ad= This has been named the Clausius-Mosotti formula, after O F Mosotti who proposed it in 1850 and R. Clausius who proposed it independently in 1879. When written in terms of the index of refraction n(where n2=6r), it is also known as the Lorentz-Lorenz formula, after H. Lorentz and L. Lorenz who proposed it independently for optical materials in 1880. The Clausius-Mosotti formula allows us to determine the dielectric constant fro simple gases(with pressures up to 1000 atmospheres) but becomes less reliable for liqi p the polarizability and number density of a material. It is reasonably accurate for cert: and solids, especially for those with large dielectric constants The response of the microscopic structure of matter to an applied field is not instanta- neous. When exposed to a rapidly oscillating sinusoidal field, the induced dipole moments may lag in time. This results in a loss mechanism that can be described macroscopically by a complex permittivity. We can modify the Clausius-Mosotti formula by assuming that both the relative permittivity and polarizability are complex numbers, but this will not model the dependence of these parameters on frequency. Instead we shall (in later paragraphs)model the time response of the dipole moments to the applied field An interesting application of the Clausius-Mosotti formula is to determine the permit- vity of a mixture of dielectrics with different permittivities. Consider the simple case in which many small spheres of permittivity E2, radius a, and volume V are embedded 2001 by CRC Press LLC
or for a randomly-structured gas, the contribution from dipoles within the sphere is zero. Indeed, it is convenient and reasonable to assume that for most dielectrics the effects of the dipoles immediately surrounding the observation point cancel so that E3 = 0. This was first suggested by O.F. Mosotti in 1850 [52]. With E2 approximated as (4.95) and E3 assumed to be zero, we have the value of the resulting local field: E (r) = E(r) + P(r) 30 . (4.96) This is called the Mosotti field. Substituting the Mosotti field into (4.94) and using P = Np, we obtain P(r) = NαE (r) = Nα E(r) + P(r) 30 . Solving for P we obtain P(r) = 30Nα 30 − Nα E(r) = χe0E(r). So the electric susceptibility of a dielectric may be expressed as χe = 3Nα 30 − Nα . (4.97) Using χe = r − 1 we can rewrite (4.97) as = 0r = 0 3 + 2Nα/0 3 − Nα/0 , (4.98) which we can arrange to obtain α = αe + αa + αd = 30 N r − 1 r + 2 . This has been named the Clausius–Mosotti formula, after O.F. Mosotti who proposed it in 1850 and R. Clausius who proposed it independently in 1879. When written in terms of the index of refraction n (where n2 = r), it is also known as the Lorentz–Lorenz formula, after H. Lorentz and L. Lorenz who proposed it independently for optical materials in 1880. The Clausius–Mosotti formula allows us to determine the dielectric constant from the polarizability and number density of a material. It is reasonably accurate for certain simple gases (with pressures up to 1000 atmospheres) but becomes less reliable for liquids and solids, especially for those with large dielectric constants. The response of the microscopic structure of matter to an applied field is not instantaneous. When exposed to a rapidly oscillating sinusoidal field, the induced dipole moments may lag in time. This results in a loss mechanism that can be described macroscopically by a complex permittivity. We can modify the Clausius–Mosotti formula by assuming that both the relative permittivity and polarizability are complex numbers, but this will not model the dependence of these parameters on frequency. Instead we shall (in later paragraphs) model the time response of the dipole moments to the applied field. An interesting application of the Clausius–Mosotti formula is to determine the permittivity of a mixture of dielectrics with different permittivities. Consider the simple case in which many small spheres of permittivity 2, radius a, and volume V are embedded
within a dielectric matrix of permittivity E1. If we assume that a is much smaller than the wavelength of the electromagnetic field, and that the spheres are sparsely distribute within the matrix, then we may ignore any mutual interaction between the spheres. Since the expression for the permittivity of a uniform dielectric given by(4.98) describes the effect produced by dipoles in free space, we can use the Clausius-Mosotti formula to define an effective permittivity Ee for a material consisting of spheres in a background electric by replacing Eo with EI to obtain ee=∈1 (4.99) In this expression a is the polarizability of a single dielectric sphere embedded in the background dielectric, and N is the number density of dielectric spheres. To find a we use the static field solution for a dielectric sphere immersed in a field(8 3.2.10) Remembering that p=ae and that for a uniform region of volume V we have p=VP, e can make the replacements Eo→∈and∈→∈2in(3.117) to get ∈2+2∈ Defining f=Nv as the fractional volume occupied by the spheres, we can substitute (4.100)into(4.99) to find that 1+2fy here ∈2-∈1 ∈2+2∈1 This is known as the Mazwell-Garnett miring formula. Rearranging we obtain which is known as the Rayleigh miring formula.. As expected,∈e→∈lasf→0.Even though as f l the formula also reduces to ee =E2, our initial assumption that f< (sparsely distributed spheres) is violated and the result is inaccurate for non-spherical nhomogeneities90. For a discussion of more accurate mixing formulas, see Ishimaru [90]or Sihvola [175 The dispersion formula of classical physics. We may determine the frequency de- pendence of the permittivity by modeling the time response of induced dipole moments This was done by H. lorentz using the simple atomic model we introduced earlier. Con- sider what happens when a molecule consisting of heavy particles(nuclei) surrounded by clouds of electrons is exposed to a time-harmonic electromagnetic wave. Using the same arguments we made when we studied the interactions of fields with a plasma in $4.6.1 we assume that each electron experiences a Lorentz force Fe= We neglect the magnetic component of the force for nonrelativistic charge velocities, and ignore the mo- tion of the much heavier nuclei in favor of studying the motion of the electron cloud However, several important distinctions exist between the behavior of charges within a plasma and those within a solid or liquid material. Because of the surrounding polarized matter, any molecule responds to the local field E instead of the applied field E. Also, as the electron cloud is displaced by the Lorentz force, the attraction from the positive 2001 by CRC Press LLC
within a dielectric matrix of permittivity 1. If we assume that a is much smaller than the wavelength of the electromagnetic field, and that the spheres are sparsely distributed within the matrix, then we may ignore any mutual interaction between the spheres. Since the expression for the permittivity of a uniform dielectric given by (4.98) describes the effect produced by dipoles in free space, we can use the Clausius–Mosotti formula to define an effective permittivity e for a material consisting of spheres in a background dielectric by replacing 0 with 1 to obtain e = 1 3 + 2Nα/1 3 − Nα/1 . (4.99) In this expression α is the polarizability of a single dielectric sphere embedded in the background dielectric, and N is the number density of dielectric spheres. To find α we use the static field solution for a dielectric sphere immersed in a field (§ 3.2.10). Remembering that p = αE and that for a uniform region of volume V we have p = VP, we can make the replacements 0 → 1 and → 2 in (3.117) to get α = 31V 2 − 1 2 + 21 . (4.100) Defining f = N V as the fractional volume occupied by the spheres, we can substitute (4.100) into (4.99) to find that e = 1 1 + 2 f y 1 − f y where y = 2 − 1 2 + 21 . This is known as the Maxwell–Garnett mixing formula. Rearranging we obtain e − 1 e + 21 = f 2 − 1 2 + 21 , which is known as the Rayleigh mixing formula. As expected, e → 1 as f → 0. Even though as f → 1 the formula also reduces to e = 2, our initial assumption that f 1 (sparsely distributed spheres) is violated and the result is inaccurate for non-spherical inhomogeneities [90]. For a discussion of more accurate mixing formulas, see Ishimaru [90] or Sihvola [175]. The dispersion formula of classical physics. We may determine the frequency dependence of the permittivity by modeling the time response of induced dipole moments. This was done by H. Lorentz using the simple atomic model we introduced earlier. Consider what happens when a molecule consisting of heavy particles (nuclei) surrounded by clouds of electrons is exposed to a time-harmonic electromagnetic wave. Using the same arguments we made when we studied the interactions of fields with a plasma in § 4.6.1, we assume that each electron experiences a Lorentz force Fe = −qeE . We neglect the magnetic component of the force for nonrelativistic charge velocities, and ignore the motion of the much heavier nuclei in favor of studying the motion of the electron cloud. However, several important distinctions exist between the behavior of charges within a plasma and those within a solid or liquid material. Because of the surrounding polarized matter, any molecule responds to the local field E instead of the applied field E. Also, as the electron cloud is displaced by the Lorentz force, the attraction from the positive