User:Brews ohare/Sandbox13

Discrete formulation
For an array of point charges {qj} the dipole moment of the array is defined as:
 * $$\mathbf{d} = \sum_j\ \mathbf{r}_jq_j \, $$

where the vectors {rj} locate the charge from some origin. Assuming only pairs of opposite charges are present, the overall charge is zero:
 * $$\sum_j\ q_j=0 \, $$

which has the interesting consequence that the polarization doesn't depend upon the choice of origin. To show this fact, suppose the origin is shifted so all the charges are now located at rj − R. The dipole moment is then:
 * $$\mathbf{d} = \sum_j\ \left(\mathbf{r}_j-\mathbf R \right) q_j \ = \sum_j\ \mathbf{r}_jq_j - \sum_j\ \mathbf{R}q_j \, $$

and because of overall neutrality the last sum is zero, so the dipole moment is the same as originally.

Some authors introduce the notion of an induced dipole moment, defined in terms of the equilibrium positions of charges by:


 * $$\mathbf d_{ind} = \sum_j\ \left(\mathbf{r}_j-\mathbf{r}_{j,eq}\right)q_j  = \mathbf{d}-\mathbf{d}_{eq}\, $$

with rj,eq the equilibrium location of charge qj, and rj its actual position, the difference caused by displacement of the charge by the force from an applied field.

Suppose we pair positive and negative charges that are close to each other, and express the dipole moment int terms of these pairs. For example,
 * $$\mathbf{d} = \sum_j \ \mathbf{r}_j q_j =\sum_k\ q_k \left( \mathbf{r}_{k+}  -  \mathbf{r}_{k-}\right) =\sum_k \ q_k \mathbf{s}_k =\sum_k \mathbf{d}_k   \, $$

where subscripts refer to the plus and minus charges of a particular dipole, and the last three sums are over pairs of opposite charges that are separated by vector displacements {sk}, so each pair k has dipole moment dk = qk sk.

To introduce the polarization density, we imagine the volume containing the charges to be divided into subregions, such that any given dipole is entirely in one subregion or another, and none straddle regions. Then the dipole moment of each subregion can be found using the above formula, and for a subregion of volume &Delta;V the polarization density is defined as:
 * $$P_{\Delta V} = \frac{1}{\Delta V} \sum_{j\in \Delta V}\ \mathbf{r}_jq_j \ . $$

If as above we confine ourselves to suregions composed of charge arrays consisting only of pairs of opposite charges with dipole moments {dk} we find:
 * $$P_{\Delta V} = \frac{1}{\Delta V} \sum_{k\in \Delta V}\ \mathbf{d}_k\, $$

where k now labels pairs of opposite charges, and not individual charges.

In the event that different subregions contain different arrays of dipoles; differing in orientation, strength or number; the polarization of each subregion will be different, and the polarization density at any given position will correspond to the particular polarization density at that location.

Integral formulation
The polarization density in a volume Ω often is described in terms of the charge density per unit volume defined at any location r in that region ρ(r) using:
 * $$\mathbf{P} = \frac{1}{\Omega} \int_\Omega \ d^3\mathbf{r} \ \mathbf{r}\rho (\mathbf{r}) \ . $$

The charge density in this integral can be taken to be the true microscopic charge density, or alternatively as a macroscopic charge density if less detail is wanted. Evaluation of the formula involves both the volume itself and the surface surrounding the volume. It should be cautioned that a clearly defined separation of the bulk and surface contributions can be problematic. In particular, a molecular dipole moment dm for a molecule m inside volume &Delta;V surrounding a point r composed of charges ρm can be defined as:
 * $$\mathbf{d}_m (\mathbf{r}) = \int_{\Delta V (\mathbf{r})} \ d^3\mathbf{r_o} \ \mathbf{r_o}\rho_m (\mathbf{r_o}) \, $$

and the dipole moment of a volume of molecules &Delta;V as:
 * $$\mathbf{d}_{\Delta V}(\mathbf{r}) = \sum_{m\in\Delta V (\mathbf{r}) \ } \ \mathbf{d}_m(\mathbf{r})$$

leading to a polarization density:
 * $$\mathbf {P}(\mathbf{r})\ =\ \frac{1}{\Delta V (\mathbf{r})} \mathbf{d}_{\Delta V}(\mathbf{r}) \ . $$

In a crystalline solid, this formulation often is taken using for Ω the volume of a unit cell. (The unit cell contains a representative group of atoms or ions that is repeated to form the entire solid. The cell can become distorted by strain, and this strain can vary with position inside a sample. For example, normally strain varies with position as the surface of the sample is approached.) Some ambiguity is introduced by the cell location unless the cell boundary lies in a region with zero charge.

Relation to bound charge
The concepts of bound and free charge densities are introduced when Maxwell's equations are applied to electrical media like dielectrics, semiconductors and so forth. The total charge density that includes all the microscopic charges is defined by the electric field. In SI units:
 * $$\mathbf{\nabla \cdot E} = \frac{\rho_t} {\varepsilon_0 } \, $$

where ε0 is the electric constant. In material media the displacement field D and polarization density P are introduced and their divergences are supplied by the so-called free charge density &rho;f and bound charge density &rho;b:
 * $$\mathbf{\nabla \cdot D} = \rho_f \  ,$$
 * $$\mathbf{\nabla \cdot P} = -\rho_b \ . $$

Using the definition of D,
 * $$\mathbf{ D} = \varepsilon_0\mathbf{ E}+\mathbf{ P} \  $$

the above divergence relations produce:
 * $$\rho_t = \rho_f +\rho_b \ ,$$

indicating the total charge is split into two populations, called free and bound charge. Because bound charge depends upon inhomogeneity in P (the divergence is zero unless P has a spatial dependence), bound charge cannot be interpreted simply as charges tied to a site, but instead is related to charges whose relative positions vary throughout the sample. For example, in a charge array of dipoles, the bound charge is not the individual, constituent charges forming each dipole, but the variation in orientation and/or strength of the dipoles from position to position. A particularly striking example is at the boundary separating a dielectric from a classical vacuum, where the polarization steps down from a finite value inside the dielectric where dipoles are located to a value of zero outside the dielectric, causing a step function variation of the polarization with position that results in a sheet of bound charge on the interface. (If the outside region is a quantum vacuum, the polarization in the vacuum is not exactly zero, but still it is very small.)

Some authors refer to bound charge as "charges of equal magnitude but of opposite signs that are held in close proximity and are free to move only atomic distances (roughly 1Å or less)." One may reasonably inquire whether this definition is equivalent to the div P definition. Looking at the definition of polarization above, a material composed only of a spatially independent array of dipoles will have a spatially independent polarization and hence a polarization with zero divergence and zero bound charge. In fact, even if the dipoles change strength or orientation under the influence of an applied electric field, but all shift by the same amount so the dipole moments still are distributed uniformly in space, there will be zero bound charge. That is the case, for example, when a dielectric sphere is introduced into a uniform electric field: the bound charge is everywhere zero except at the surface of the sphere where a surface charge appears because of the step in dipole moment upon leaving the dielectric. These results are incompatible with this paragraph's introductory word definition.

The original meaning of free and bound charges began with the observation of charges induced upon a metal electrode. When a charge was induced in the ends of a metal rod by bringing one end near an external charge, the charge on the far end left the rod when the far end was grounded. That was free charge. However, the charge at the near end next the inducing charge remained and was called bound charge.