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I am reading Pekar's "Research in Electron Theory of Crystals" and I came across a passage I find a bit unclear:

The theory developed below takes into account the dielectric polarization of a an ionic crystal by the electric field of the conduction electron. The local polarization that results from this is related with the displacement of the ions and consequently is inertial. It cannot follow the relatively rapidly moving electron and therefore forms a potential well for the electron. The depth of this potential well turns out to be sufficient for discrete energy levels of the electron to exist in it. The electron, being in a local state on one of these levels, can maintain with it sown field the aforementioned local polarization of the crystal. Because of their inertia, the ions are sensitive not to the instantaneous value of the electron field, but to the average field. The latter can be calculated as the static field of the $|\psi|^2$ cloud of the electron; it produces a static polarization potential well, which in turn maintains the electron stationarily in a local state. Such states of the crystal with the polarization potential well, which in turn maintains the electron stationarily in the local state. Such states of the crystal with a polarization potential well, in which the electron is localized, were called by the author polarons

Now, what exactly does he mean by local vs. conduction electrons? Are local electrons those that are not moving and are in the crystal? That doesn't seem right. What does it mean for an electron to be in a local state? (Also what does he mean by "inertial"?) IN fact, it would be nice if one explains this passage in understandable terms so that I can have some intuitive picture in mind.

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up vote 1 down vote accepted

Generally speaking, localized electrons are confined to one particular orbital, while conduction electrons are "free" to float from orbital to orbital due to the nearly degenerate nature of the states in the conduction band. So, when one discusses a localized electron it isn't that the electron is pinned to a precise location in space, but isolated to a particular orbital or small region of space when compared to the finite extent of the material.

The passage you cite above, the introduction of additional materials into the structure varies the potential felt by the electron, creating a local potential from which the electron may not have sufficient energy to escape, which is a way of introducing localization.

The wiki article on Anderson Localization is accessible, and may help with some additional insight.

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This is not quite right: the localized electrons here are not confined to one orbital, which is one atom, but to a region of space which can be the size of dozens or hundreds of atoms by an external perturbation. The definition you give is the one people use too often, though. – Ron Maimon Dec 29 '11 at 18:53 can easily answer this for you. Conduction electron is the non-local antithesis of "local electron" ... it's basically saying whether the electron is associated to a particular atom or whether it can move freely in the material (i.e. a conductor).

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Think of the local state as the electron having its own energy values for a localized Hamiltonian with the potential due to its presence in the crystal lattice site. – Chris Gerig Dec 28 '11 at 3:09
WHY was this downvoted, especially without remark... seems to go against the ethics of this forum. – Chris Gerig Dec 28 '11 at 7:39

Well, you probably know that a single atom consists of a nucleus and a bunch of electrons in orbit around that nucleus. In this sense, you can say the electrons are located around the nucleus.

Now, when you bring a bunch of atoms close together, something interesting happens: Because the electrons start to feel not only "their" atom but also that of other atoms, the electron orbitals are modified:

Some orbitals, typically those with low energies, retain their atomic-like character, maybe with a few deformations, so electrons living in those orbitals still stay close to the nucleus. We call these "localized" electrons. Some orbitals, on the other hand, are smeared out over the entire material. These are extended or delocalized orbitals.

What he means by "inertia" is that ions are at least a factor of ~2000 heavier than electrons, and therefore they are very slow to react to changes in the electron states. Therefore, they don't follow every little fluctuation of the electrons. Their slowness averages out the tiny fluctuations.

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