The Theory of the Macroscopic Properties of Isotropic Dielectrics

BY B. K. P. SCAIFE

1 Introduction

A chemist is anxious to obtain as much information as possible about the molecular conformation, interaction, and dynamics of the substances which he studies. Because all materials are built up from combinations of electric charges, protons, and electrons, it is not surprising that certain molecular properties and motions give rise to macroscopic properties which determine the reaction of a particular material to the imposition of an electromagnetic field.

To relate the observed macroscopic phenomena to various molecular processes is, in general, a task of great difficulty. Such an undertaking requires a fully developed macroscopic theory to enable proper design and assessment of experimental procedures to be made. Furthermore, without an adequate theory of molecular processes and without a precise relationship between macroscopic and microscopic parameters, it will not be possible to obtain accurate information about events at a molecular level from experimental data.

The primary concern of this chapter is to provide an introduction to the theory of the macroscopic properties of dielectric bodies. Attention is restricted to isotropic materials and it is assumed that the external electric field is always sufficiently weak to allow any non-linear effects which might arise to be ignored.

The subject matter is such that it lends itself to mathematical description; nevertheless, the emphasis here will be on the physical meaning of the mathematical equations and we shall not be particularly concerned with matters of mathematical technique.

This introductory survey commences with a derivation of the macroscopic equation

D = ε0 E + P

based on Maxwell's equations for charges in free space. The concept of the relative permittivity for static fields, εg is introduced in Section 3. In the following section it is shown how this concept may be generali:ied for the case of harmonically varying applied fields.

A frequency (= ω/2π) dependent complex susceptibility,

x(ω) = x'(ω) -ix"(ω) = [ε(ω) -1] ε0,

is defined and the physical significance of x'(ω) and x"(ω) is discussed. It is also shown how x(ω) is related to the temporal behaviour of the dielectric polarization following the sudden application, or removal, of an electric field. Various forms of the Kramers-Kronig dispersion relations are introduced for x'(ω) and x"(ω) and for a number of functions of x(ω). The section closes with the definition of the frequency-dependent complex refractive index n(ω) = n(ω) – iκ(ω) and a discussion of its relation to ε(ω).

The final section discusses, in outline, various macroscopic manifestations of the underlying dynamic nature of a dielectric material. It is shown how certain macroscopically observable quantities are of direct significance on a microscopic scale.

2 Basic Considerations

The electric and magnetic fields set up in free space by a system of charges is described by Maxwell's equations, which read, in SI units, as follows:

curl E(r, t) = -[partial derivative],B(r, t) (1)

curl H(r, t) = [partial derivative]tD(r, t) + J(r, t) (2)

div D(r, t) = p(r, t) (3)

div B(r, t) = O (4)

B(r, t) = μo H(r, t) (5)

D(r, t) = εo E(r, t) (6)

The absolute permittivity and absolute permeability of free space are denoted by ε0 and μ0 respectively. The sources of the electromagnetic-field vectors D, E, B, and Hare the charge density, p(r, t), and the current density, due to charges in motion, J(r, t). The radius vector of a point with co-ordinates x, y, and z is denoted by r = ix + jy + kz, and the time is denoted by t. The use of the tilde, ~, on the field variables in equations (1H6) is to emphasize that these equations describe the electro-magnetic field in free space. Quantities without the tilde will have meanings to be defined below.

In view of equations (5) and (6) it would appear an unwarranted extravagance to use four variables when two would suffice. As we shall see presently, the distinction between D and E in free space, which stems from the SI system of units, will prove of considerable benefit when we come to setting up the equations for the potential gradient in polarizable matter.

For a great many applications in dielectrics, Maxwell's equations may be greatly simplified by neglecting the magnetic field on the ground that the motion of the charges is not sufficiently rapid to give rise to appreciable radiation. With this approximation equations (1) and (2) are uncoupled and we may use the following equations to describe the spatial variation of D and E:

curl E(r, t) = 0 (7)

div D(r, t) = p(r, t) (8)

D(r, t) = ε0 E(r, t) (9)

It is convenient and fruitful to regard Das the electric flux density and E as the force that would be exerted on a unit point charge.

The implications of equations (7) and (8) are (i) that E(r, t) may be described in terms of a scalar potential [??],(r, t), such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

and (ii) that the electric flux lines, whose density is D, begin on positive charges and end on negative charges. Notice that, for a system of charges in free space, at each and every point the electric flux density, D, is related to the electric field intensity, E, by the simple equation (9).

The charge system that we are concerned with is one in which the charges are the sub-atomic charged particles, electrons and protons, which make up dielectric materials. It is clear that the quantities D, E, and [??] must now be regarded as inaccessible to simple macroscopic measurements. Indeed we shall regard D, E, and [??] as variables at a microscopic level and our immediate task is to find a means of relating them to corresponding macroscopic quantities.

We must realize at once that for most purposes a 'small' distance – 'small' in a macroscopic sense – when expressed in terms of, say, the radius of a hydrogen atom appears to be very large on a microscopic or atomic scale. Thus when we speak of the macroscopic charge density 'at a point' we mean: the ratio of electric charge contained in a small (on a macroscopic scale) volume around the point in question divided by the volume. Expressed mathematically this last statement reads:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

in which the microscopic charge density, [??], is distinguished from the macroscopic charge density, p, by the tilde. In the rest of this chapter only microscopic quantities will carry the tilde. It is to be understood that the volume V, surrounding the point r and which appears in equation (11), is to be regarded as 'macroscopically small but microscopically large'. This method of deriving macroscopic variables from microscopic variables was introduced by Lorentz.

Equations (8) and (11) lead to the equation

div D(r, t) = p(r, t) (12)

which is a relation between macroscopic quantities.

An alternative definition of D, the macroscopic flux density, is embodied in the equation (see Figure 1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

in which the volume, Sh, over which the integration is taken is that of a thin disc of area S and thickness h. Both S and h are macroscopically small. Dn is the component of the macroscopic flux density normal to the disc and n is the unit normal pointing outwards from one of the faces of the disc.

However, it must not be concluded that every average of the kind used in equations (11) and (13) is related to a meaningful macroscopic quantity. A case in point is E: whereas it is perfectly possible to consider its volume average, as we did for [??], the result is of no relevance on a macroscopic level. In contrast to the Lorentzian approach, we choose to relate E, the microscopic negative potential gradient, to the macroscopic negative potential gradient which we shall denote by E. Thus we are led to the following expression for E:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

in which L is a distance at once macroscopically small and microscopically large. In view of equations (13) and (14) it is apparent that any relationship between the macroscopic quantities D and E must involve the charge configuration and dynamics of the system.

For simplicity we shall restrict our attention to those dielectric materials which are good insulators, that is materials for which transport of charge (over macroscopic distances) is practically impossible. In all such materials the constituent charges are strongly localized. This strong localization does not prevent the possibility of relative displacement between the constituent charges of opposite sign. Such displacements might arise from the imposition of an external potential gradient, or from the placing of charged particles in the midst of the material, or even from thermal agitation. Whatever their cause, these displacements manifest themselves as electrically polarized particles. It is important to note that electrical neutrality is preserved which means that, on a macroscopic scale, there is never an excess of charge of one sign.

The charge configuration on an atomic level can be described mathematically in terms of, so-called, multipole moments. As far as macroscopic effects are concerned, only the dipole moment is usually of importance. The dipole moment, m, of a group of charges e, (i = 1, ..., N) at points ri satisfies the equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

provided the total charge, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], is zero. In addition, and of supreme importance, is the fact that the dipole moment m is independent of the co-ordinate system used. This means that no matter how we set about calculating the dipole moment of a group of charges we shall always obtain the same result, provided only that the total charge vanishes. 4 As a result it is possible to describe the polarization of a material in two entirely equivalent ways, the one using macroscopic variables the other using microscopic variables.

Let us introduce a new vector quantity, denoted by P(r, t), called the polarization or the dipole moment per unit volume. The total dipole moment of a (macroscopically) small volume, V, is equal to the sum of the molecular dipole moments, Σimj, contained therein. Thus,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

With the aid of the polarization, P, we may describe the electrical state of a dielectric body. It is possible to calculate the electric potential, φ, in terms of P at points both inside and outside the body. We do this by adding (see Figure 2) the elementary contribution to φ from each polarized element of volume, dV, of the body. Treating each element of volume as a dipole of moment P dV and of negligible spatial extension, we may write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Poisson, in his celebrated memoir on the theory of magnetism, showed that, following integration by parts, this last equation could be written:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

The interpretation of this result is straightforward. It tells us that in calculating φ(r) we may do so by assuming a surface charge density, n·P, at the surfaces of the dielectric body and a volume charge density, of amount

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

The occurrence of the surface charge density is easy to understand since the surface will contain the ends of the polarized particles. The volume charge distribution is a much more subtle concept and it will be profitable to devote some attention to it.

In discussing this charge density, -div P, it is important to realize that equation (16) does not imply that isolated charges, of one sign or another, actually exist in the material. Remember that we set out with the understanding that, on a macroscopic level, electrical neutrality was preserved. The solution to the apparent paradox of electric charge appearing as a result of combining electrically neutral polarized particles can be obtained by considering the very simple system exhibited in Figure 3 in which are shown spatially compact but widely separated electric dipoles of varying magnitudes.

Suppose that the inter-dipole distance a is a macroscopically small quantity. The linear density of dipole moment at x = na is simply (mn/a) = p(x), say. The dipole moment mn merely specifies the product of charge and separation. It is possible to imagine a dipole of moment mn made up of two equal and opposite charges (mn/a) separated by a distance a. Applying the same reasoning to each dipole we find the net charge, midway between each dipole pair, to be (mn-1 - mn)/a = p(x -a) -p(x) = q(x), say. Because the distance a is small we may write, by Taylor's theorem,

q(x) = p(x) - a [partial derivative]αp(x) -p(x)

and hence the linear charge density, ρ(x), is given by

ρ(x) = -[partial derivative] αp(x) (18)

Thus, in this one-dimensional example at least, we have shown how the potential set up by a system of dipoles can be produced by a particular distribution of isolated charges. Equation (18) is the one-dimensional version of equation (17), ρp = -div P. It is well to remember that a mere spatial non-uniformity is not sufficient to give rise to a non-vanishing divergence of P. For example, the polarization surrounding a point charge, embedded in a dielectric medium, is divergence-free. Examples of the fact that div P does not always vanish will be found in Frohlich's monograph.

The fact that the potential distribution, in and around a polarized body, can be described in terms of a surface and a volume distribution of charge does not in any way contradict the fact that electrical neutrality, on a macroscopic scale, can be maintained at every point of the body.

Once the potential, φ(r, t) has been obtained we may calculate the value of the negative potential gradient, E(r, t), by means of equation (10).

We now turn to the problem of relating D, E, and P. Envisage a dielectric body of arbitrary shape which has a distribution of polarization, P, partly produced by a distribution of charges of density p and partly by particular charge configurations at a microscopic level resulting from thermal fluctuations. In any very small region of the body the polarization P will be sensibly spatially uniform. If we imagine a small surface drawn in the body perpendicular to Pit is clear that an electric flux density of amount P must pass through this surface. This may be seen by imagining the polarization to be frozen and the small surface turning into a thin disc-shaped fissure. P is only one contribution to the total flux density, D, at a particular point. The Poisson analysis, outlined earlier, shows that all polarization effects can be described by surface and volume distributions of charge. Once the magnitude and position of these equivalent charges have been determined the system is treated as a system of charges in free space without any dielectric material present. In the vicinity of the point of interest a negative potential gradient, E, is set up which is due to the imposed charges and the equivalent charges. This total negative potential gradient exists in free space and therefore we may associate with it a flux density ε0E.

The total electric flux density D is the sum of the two contributions P and ε0E, hence

D = ε0 E + P (19)

It is instructive to take the divergence of this equation; we find

div(D -P) = div ε0 E

From equation (12) div D is equal to p, the true, free, or external charge density. In view of equation (17), div P may be replaced by - ρp the polarization or bound charge density.

Consequently

div ε0 E = ρ + ρP (20)

which means that the flux density, ε0E, in the Poisson equivalent system, is determined by both the true and the bound charge densities.