Editing Polarization density (section)

=== Crystalline materials ===
In a simple approach the polarization inside a solid is not, in general, uniquely defined. Because a bulk solid is periodic, one must choose a unit cell in which to compute the polarization (see figure).<ref name="Respa">{{cite journal |last=Resta |first=Raffaele |year=1994 |title=Macroscopic polarization in crystalline dielectrics: the geometric phase approach |journal=Rev. Mod. Phys. |volume=66 |issue=3 |pages=899–915 |doi=10.1103/RevModPhys.66.899 |url=http://inside.mines.edu/~zhiwu/research/papers/E04_berry2.pdf|bibcode = 1994RvMP...66..899R }} See also: [http://www.physics.rutgers.edu/~dhv/talks/rahman.pdf D Vanderbilt, ''Berry phases and Curvatures in Electronic Structure Theory''], an introductory-level powerpoint.</ref><ref name="Spaldin">{{cite journal | last=Spaldin |first=Nicola A. |author-link=Nicola Spaldin |year=2012 |title=A beginner's guide to the modern theory of polarization |journal=Journal of Solid State Chemistry |volume=195 |pages=2–10 |doi=10.1016/j.jssc.2012.05.010 |arxiv=1202.1831 |bibcode=2012JSSCh.195....2S |s2cid=55374298 |url=https://www.sciencedirect.com/science/article/abs/pii/S0022459612003234}}</ref> In other words, two people, Alice and Bob, looking at the same solid, may calculate different values of '''P''', and neither of them will be wrong. For example, if Alice chooses a unit cell with positive ions at the top and Bob chooses the unit cell with negative ions at the top, their computed '''P''' vectors will have opposite directions. Alice and Bob will agree on the microscopic electric field '''E''' in the solid, but disagree on the value of the displacement field <math>\mathbf{D} = \varepsilon_0 \mathbf{E} + \mathbf{P}</math>.

Even though the value of '''P''' is not uniquely defined in a bulk solid, ''variations'' in '''P''' ''are'' uniquely defined.<ref name=Respa/> If the crystal is gradually changed from one structure to another, there will be a current inside each unit cell, due to the motion of nuclei and electrons. This current results in a macroscopic transfer of charge from one side of the crystal to the other, and therefore it can be measured with an ammeter (like any other current) when wires are attached to the opposite sides of the crystal. The time-integral of the current is proportional to the change in '''P'''. The current can be calculated in computer simulations (such as [[density functional theory]]); the formula for the integrated current turns out to be a type of [[Berry's phase]].<ref name=Respa/>

The non-uniqueness of '''P''' is not problematic, because every measurable consequence of '''P''' is in fact a consequence of a continuous change in '''P'''.<ref name=Respa/> For example, when a material is put in an electric field '''E''', which ramps up from zero to a finite value, the material's electronic and ionic positions slightly shift. This changes '''P''', and the result is [[electric susceptibility]] (and hence [[permittivity]]). As another example, when some crystals are heated, their electronic and ionic positions slightly shift, changing '''P'''. The result is [[pyroelectricity]]. In all cases, the properties of interest are associated with a ''change'' in '''P'''.

In what is now called the ''modern theory of polarization'', the polarization is defined as a difference. Any structure which has inversion symmetry has zero polarization; there is an  identical distribution of positive and negative charges about an inversion center. If the material deforms there can be a polarization due to the charge in the charge distribution.<ref name="Spaldin" />