Dipole moment per unit volume

Molecules do not consist of rigid arrays of point charges, and on application of an external electrostatic field the electrons and protons will rearrange themselves until the interaction energy is a minimum. In classical electrostatics, where we deal with macroscopic samples, the phenomenon is referred to as the induced polarization. I dealt with this in Chapter 15, when we discussed the Onsager model of solvation. The nuclei and the electrons will tend to move in opposite directions when a field is applied, and so the electric dipole moment will change. Again, in classical electrostatics we study the induced dipole moment per unit volume. 

It is important to distinguish various components of the solvent polarization which play different roles in the process of electron transfer. Most generally, the total polarization of the solvent (i.e., the dipole moment per unit volume) P,(r) at a spatial point of the solvent r may consist of four components ... 

In condensed phases, it is more convenient to consider the dipole moment per unit volume or polarization (P)... 

Dielectric polarization is the polarized condition in a dielectric resulting from an applied AC or DC field. The polarizability is the electric dipole moment per unit volume induced by an applied field or unit effective intensity. The molar polarizability is a measure of the polarizability per molar volume thus it is related to the polarizability of the individual molecules or polymer repeat unit. 

Light-matter interactions can be described via an induced polarization, i.e., the induced dipole moment per unit volume. Ultrafast laser pulses, which are used in laser scanning microscopes, have high enough intensity to induce a nonlinear polarization in various materials. For intense optical electric field E, the polarization vector P can be expanded in the power series (Boyd 1992)... 

The interaction between permanent or induced electric dipoles with an applied electric field is called polarization, which is the induced dipole moment per unit volume. Polarization causes positive charge to accumulate on the bottom surface next to the... 

Let us first consider the dielectric constant of a gas. We assume that the molecules are far enough apart for them to contribute independently to the polarization and that the electric field E induces an electric dipole moment aE in each molecule. The quantity a is called the electric polarizability of the molecule. The number of moles per unit volume of gas is the density p divided by the tnolecular weight M, and the number of molecules in unit volume is this ratio multiplied by Avogadro s number A. Hence the polarization of the gas (the induced dipole moment per unit volume) is given by the following equation ... 

A vector quantity representing the dipole moment per unit volume of a dielectric medium. In rationalized units, the electric induction in a dielectric is given by D = cE, which can be written... 

A polarized material can be regarded as made up of elementary dipolar prisms, the end faces of which carry surface charge densities of +(7P and —av as shown in Fig. 2.26. The dipole moment per unit volume of material is termed the polarization P and can vary from region to region. From Fig. 2.26 the magnitudes of the vectors are given by... 

Material Media and Their Reaction to External Fields. In a material medium, a charge distribution can induce some charge separations, or dipoles, which help to minimize the total energy. Similarly, an external magnetic field will induce some magnetic dipoles in the medium to counteract this field. To handle these effects, an electric polarization (or electrical dipole moment per unit volume) P and a magnetization (or magnetic dipole moment per unit volume) M are defined. If the medium is linear and isotropic, these two new vectors P and M are proportional to E and to H, respectively ... 

With the only exception of thiourea, all aliphatic compounds are characterized by positive dl/dE values.54 Within the limits in which we can exclude an appreciable reorientation of the aliphatic molecules with a change in the applied potential E, the increase of l with an increase in E is to be ascribed to a gradual reorientation of the adsorbed water molecules. Water molecules have a particularly high dipole moment per unit volume, and are known to be readily orientable under the influence of an electric field. A progressive shift of E towards more positive values causes to pass gradually from negative to positive values. Consequently, the contribution to aM required to keep E constant upon removing v adsorbed water molecules by one... 

If M is the magnetization (magnetic dipole moment per unit volume, analogous to the dielectric polarization P) induced by the field H, the volume magnetic susceptibility x is defined by the equation... 

Here Pind is the induced dipole moment per unit volume, and X X and X are the first-, second-, and third-order electric susceptibilities of the sample, where the order refers to the power of electric field (not of X). Equation (1) represents the macroscopic (bulk) form for the polarization in terms of single molecule properties, the induced dipole moment per molecule is written as... 

The constant of proportionality a is called the polarisability of the molecule. For isotropic molecules, which include the monatomic molecules of the inert gases, the average direction of the induced molecular dipoles is in the direction of the applied field. (Since the local field is, as we shall see later, proportional to the overall applied field, we have m = const.E.) The total dipole moment per unit volume developed in this way, the polarisation P, is then related to the number of molecules per unit volume Nq . 

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, contained therein. Thus,... 

An axially symmetric molecule is characterized by its linear polarizability in the principal axes a x and a y = a" and a" = af/. It is a good approximation to assume that its second- and third-order polarizability tensors each have only one component and respectively, which is parallel to the z principal axis of the molecule. For linear and nonlinear optical processes, the macroscopic polarization is defined as the dipole moment per unit volume, and it is obtained by the linear sum of the molecular poiarizabilities averaged over the statistical orientational distribution function G(Q). This is done by projecting the optical fields on the molecular axis the obtained dipole is projected on the laboratory axes and orientational averaging is performed. The components of the linear and nonlinear macroscopic polarizabilies are then given by ... 

An important group of piezoelectric ceramics are solid solutions of PbZrOj and PbTiOs represented as Pb(Zr, Ti)03 (and commonly referred to as PZT). At high temperature these compounds have the cubic perovskite structure (Fig. 1). In the ferroelectric phase, which is stable at room temperature, the lattice is distorted, and the asymmetry of the positive and negative ions results in a net dipole moment. Spontaneous polarization is the dipole moment per unit volume. 

Several important relationships are relevant to the properties of a dielectric. The vector P defined for a dielectric is the net dipole moment per unit volume. When it is combined with the electrical field, one obtains the definition of the electric displacement D. Thus,... 

The polarization of a solid dielectric medium, P, is defined as the dipole moment per unit volume averaged over the volume of a crystal cell. A component of P can be expanded as a function of the electric field E ... 

Figure 26.1(c) shows a cylindrical element of the dielectric with its axis in the direction of the polarizing field. If the area of each face of the cylinder is A, then the charges on the faces are -h(7p A and —dp A. These charges are separated by the length of the cylinder a, so that the cylinder has a dipole moment equal to (dp A)a. Since the volume of the cylinder is aA, the dipole moment per unit volume is... 

Therefore the polarization dp in addition to being the charge density on the surface is also equal to the dipole moment per unit volume of the dielectric. 

The dipole moment per unit volume of the dielectric is made up of contributions from all the molecules in the unit volume. If N is the number of molecules per unit volume, and m is the average dipole moment per molecule induced by the field, then the dipole moment per unit volume is mN ... 

P is the electric dipole moment per unit volume, P = P being the polarization charge per unit area normal to the vector P. D = 1 D 1 also expresses the charge density of a condenser required to maintain the field E at the interior of the dielectric (Fig. 4.11). 

In certain crystals we observe an electric polarization P (i.e. a dipole moment per unit volume) on the application of a stress o, due to the rearrangement of the charges in the unit cell (Fig. 4.15). This is known as the piezoelectric effect, characterized by a tensor of rank 3,... 

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